a ne arithmetic based methods for power systems analysis

Affine Arithmetic Based Methods forPower Systems Analysis Considering

Intermittent Sources of Power

by

Juan Carlos Munoz Guerrero

A thesispresented to the University of Waterloo

in fulfillment of thethesis requirement for the degree of

Doctor of Philosophyin

Electrical and Computer Engineering

Waterloo Ontario Canada 2013

ccopy Juan Carlos Munoz Guerrero 2013

I hereby declare that I am the sole author of this thesis This is a true copy of the thesis includingany required final revisions as accepted by my examiners

I understand that my thesis may be made electronically available to the public

ii

Abstract

Intermittent power sources such as wind and solar are increasingly penetrating electrical gridsmainly motivated by global warming concerns and government policies These intermittent andnon-dispatchable sources of power affect the operation and control of the power system becauseof the uncertainties associated with their output power Depending on the penetration level ofintermittent sources of power the electric grid may experience considerable changes in powerflows and synchronizing torques associated with system stability because of the variability ofthe power injections among several other factors Thus adequate and efficient techniques arerequired to properly analyze the system stability under such uncertainties

A variety of methods are available in the literature to perform power flow transient and volt-age stability analyses considering uncertainties associated with electrical parameters Some ofthese methods are computationally inefficient and require assumptions regarding the probabilitydensity functions (pdfs) of the uncertain variables that may be unrealistic in some cases Thusthis thesis proposes computationally efficient Affine Arithmetic (AA)-based approaches for volt-age and transient stability assessment of power systems considering uncertainties associatedwith power injections due to intermittent sources of power In the proposed AA-based methodsthe estimation of the output power of the intermittent sources and their associated uncertainty aremodeled as intervals without any need for assumptions regarding pdfs This is a more desirablecharacteristic when dealing with intermittent sources of power since the pdfs of the output powerdepends on the planning horizon and prediction method among several other factors The pro-posed AA-based approaches take into account the correlations among variables thus avoidingerror explosions attributed to other self-validated techniques such as Interval Arithmetic (IA)

The AA-based voltage stability method proposed in the thesis computes the hull of PVcurves associated with the assumed uncertainties and is tested using two study cases first a5-bus test system is used to illustrate the proposed technique in detail and thereafter a 2383-bustest system to demonstrate its practical application The results are compared with those obtainedusing conventional Monte Carlo Simulations (MCS) to verify the accuracy and computationalburden of the proposed AA-based method and also with respect to a previously proposed tech-nique to estimate parameter sensitivities in voltage stability assessment On the other hand theproposed AA-based transient stability assessment method solves the set of Differential-Algebraic

iii

Equations (DAEs) in affine form using a trapezoidal integration approach which leads to the hullof the dynamic response of the system for large disturbances on the system This approach istested using a Single Machine Infinite Bus (SMIB) test system with simplified models and atwo-area test system with variable input powers from synchronous generators and wind tur-bines based on Doubly Fed Induction Generators (DFIGs) In all study cases MCS is used forcomparison purposes

The results obtained using the proposed AA-based methods for voltage and transient stabilityassessment depict a reasonably good accuracy at significantly lower computational costs whencompared to those obtained using simulation based techniques These AA-based methods can beused by system operators to efficiently estimate the system dynamic response and PV curves as-sociated with the input-power uncertainties and thus devise countermeasures in case of insecureoperation

iv

Acknowledgments

First I would like to dedicate my deepest appreciation and admiration to my supervisor ProfessorClaudio Canizares for his continuous support encouragement trust and guidance during theresearch and writing of this thesis I would also like to acknowledge with much appreciationto my co-supervisor Professor Kankar Bhattacharya for all his support advice kindness anddedication It has been a huge honor and enriching experience to conduct my research undertheir supervision

I want to express my sincere gratitude to Professor Alfredo Vaccaro from the University ofSannio for all his kind help and advice I also want to acknowledge the following members ofmy comprehensive or examination committees for their valuable comments and input Profes-sor Srinivasan Keshav from the Computer Science Department of the University of WaterlooProfessor Shesha Jayaram and Professor Sheryas Sundaram from the Electrical and ComputerEngineering Department of the University of Waterloo Also thanks Professor Stephen Smithfrom the Electrical and Computer Engineering Department of the University of Waterloo andProfessor Venkataramana Ajjarapu from the Department of Electrical and Computer Engineer-ing of the Iowa State University for kindly agreeing to be part of the examination committee

I would like to acknowledge the Natural Sciences and Engineering Research Council (NSERC)Canada Hydro One IBM Canada ABB Corporate Research USA and the Universidad de LosAndes in Venezuela for providing the funding necessary to carry out this research

I want to gratefully and sincerely thank the following professors of the Engineering Facultyof the Universidad de Los Andes for all their support and guidance during my studies ErnestoMora Ricardo Stephen Marisol Davila Jose Contreras Jaime Gonzalez Pedro Mora JesusVelazco Luz Moreno Nelson Ballester and Jean Carlos Hernandez I also want to thank MarielaMonsalve for her friendship and support

I extend my gratitude to all my colleagues and friends in the Electricity Market Simula-tion and Optimization Laboratory (EMSOL) for their support and the pleasant work environ-ment provided special mention deserve Mauricio Restrepo Daniel Olivares Jose Daniel LaraBehnam Tamimi and Nafeesa Mehboob for their friendship and willingness to help

I also want to express my deeply gratitude to my lovely mother Eciodila and my wife Yeliana

v

for all the patience and love they offer to me and the sacrifice they made to support me duringmy studies

Special thanks and admiration go to my daughter Sharon Valentina for her love and enthusi-asm that made me extremely happy in the ups and downs of this journey

Finally and most importantly I would like to thank God who has guided me to make thisthesis possible

vi

Dedication

This thesis is dedicated to the memory of my grandmother Julia and my grandfather EzioAlso this thesis is dedicated to my lovely family my wife Yeliana my mother Eciodila mybrother Jesus and my sons Sharon Valentina and Juan Marcos

vii

Table of Contents

Authorrsquos Declaration ii

Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii

List of Figures xiii

List of Abbreviations xvi

Nomenclature xvii

1 Introduction 1

11 Research Motivation 1

12 Literature Review 4

121 Stochastic Power Flow Analysis 4

122 Stochastic Voltage Stability Assessment 7

viii

123 Stochastic Transient Stability Assessment 11

13 Research Objectives 16

14 Thesis Outline 16

2 Background Review 18

21 Introduction 18

22 Power System Analysis 18

221 Power Flow 19

222 Voltage Stability 22

223 Transient Stability 29

23 Self-validated Numerical Methods for Uncertainty Analysis 32

231 Interval Arithmetic 33

232 Affine Arithmetic 35

24 Monte Carlo Simulations 40

25 Summary 45

3 Affine Arithmetic Based Power Flow Assessment 49

31 Introduction 49

32 Problem Formulation 49

33 Improved Reactive Power Limit Representation 54

34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix

4 Affine Arithmetic Based Voltage Stability Assessment 67

41 Introduction 67


43 Simulation Results 72

431 5-Bus Test System 72


44 Sources of Error 78

45 Summary 79

5 Affine Arithmetic Based Transient Stability Assessment 80

51 Introduction 80



531 Single Machine Infinite Bus Test System 87

532 Two-area Test System with Synchronous Generators 89

533 Two-area Test System with Wind Turbines 92

54 Summary 95

6 Conclusions Contributions and Future Work 96

61 Summary and Main Conclusions 96

62 Main Contributions 97

63 Future Work 98

Appendices 98

A AA Model of Single-Machine-Infinite-Bus Test System 99

x

B AA Model of Two-Area Test System with Synchronous Generators 101

C AA Model of Two-Area Test System with Wind Turbines 104

Bibliography 107

xi

List of Tables

41 Comparison of load changes using different approaches for the 5-bus test system 74

42 Comparison of load changes using different approaches for the 2383-bus testsystem 77

51 SMIB test system data 87

52 Sub-transient model data for synchronous generators 91

53 Synchronous generators data 93

54 DFIG-based wind turbines data 93

xii

List of Figures

11 Long-term energy supply for Ontario in 2030 [9] 2

12 Wind power forecast example [10] 3

21 PV curves for 2-bus system 23

22 Two-bus system example 23

23 Perpendicular intersection technique for the correction step 26

24 Parametrization technique for the correction step 27

25 Conventional power flow technique 28

26 Trapezoidal method 33

27 Convex polygon defining the bounds of KA and KB bounds for mA = 4 38

28 IEEE 14 bus system in PSAT 42

29 DFIG collector system in PSAT 43

210 (a) Wind speed pdf and corresponding (b) power output characteristics of a windturbine 43

211 Pdfs and cdfs for the static load margins when the wind farm is operating involtage control mode (a) pdf and (b) cdf in normal operating conditions (c) pdfand (d) cdf for a contingency 47

212 Cdfs of dominant-mode damping ratio for the DFIG wind turbine operating involtage control mode (a) normal operating conditions and (b) contingency con-ditions 48

xiii

213 Eigenvalues for synchronous generator at Bus 1 for different wind power outputswhen the DFIG is operated in voltage control mode (a) AVR relevant modes and(b) rotor speed relevant modes 48


32 Case A with rectangular coordinates (a) bus voltage magnitudes and (b) busvoltage angles 62

33 Case A with polar coordinates (a) bus voltage magnitudes and (b) bus voltageangles 62

34 Case B with rectangular coordinates (a) bus voltage magnitudes and (b) busvoltage angles 63

35 Case B with polar coordinates (a) bus voltage magnitudes and (b) bus voltageangles 64

36 Case C with polar coordinates (a) bus voltage magnitudes and (b) bus voltageangles 65

37 Generator reactive power intervals in Case C 65

41 PV curves (a) using the load power as the parameter (b) using a PQ-bus voltagemagnitude as the parameter 69

42 Algorithm for the AA-based PV curve computation method 73

43 PV curves for the IEEE 5-bus test system 75

44 PV curves for the 2383-bus test system 76

51 AA-based time domain simulation algorithm 86

52 SMIB test system [94] 87

53 SMIB test system 3-cycle fault with plusmn5 variation in generation 88



xiv



58 Two-area test system with synchronous generators 91

59 Two-area test system with synchronous generators results 71-cycle fault withplusmn10 variation in generation 92

510 Two-area test system results with synchronous generators results 1-cycle faultwith plusmn10 variation in generation 93

511 Two-area test system with DFIG-based wind turbines 93

512 Two-area test system with DFIG-based wind turbines results 3-cycle fault withplusmn10 variation in generation 94


xv

List of Abbreviations

AA Affine ArithmeticAESO Alberta Electric System OperatorARMA Auto Regressive Moving AverageATC Available Transfer CapabilityCBM Capacity Benefit Margincdf Cumulative Density FunctionDAE Differential-Algebraic EquationsDFIG Doubly Fed Induction GeneratorETC Existing Transmission CommitmentsIA Interval ArithmeticMCR Maximum Capacity RatingMCS Monte Carlo SimulationsNERC North America Reliability CorporationODE Ordinary Differential Equationpdf Probability Density FunctionPIS Population-based Intelligent SearchSF Sensitivity FormulaSMIB Single Machine Infinite BusTRM Transmission Reliability MarginTTC Total Transfer CapabilityWECS Wind Energy Conversion Systems

xvi

Nomenclature

Indices

D Index for dynamic variables parameters and functions

h Index for approximation noise symbols

IC Index indicating initial conditions

i j Bus indices

k s Index associated with differential-algebraic equations

m Index for buses with intermittent sources of power

max Index indicating maximum values

min Index indicating minimum values

n Index associated with trapezoidal integration method

nc Index for buses with voltage control settings

nl Index for load buses

o Index indicating central values and reference values

PF Index for power flow variables parameters and functions

r Index indicating iteration number

xvii

ra Index for affine form uA

rb Index for affine form uB

rh Index for Chebyshev approximations

sa Index for saddle node bifurcation

sl Index for noise symbols in power flow analysis

Parameters

Ar Rotor blades swept area (m2)

B Susceptance matrix (pu)

Da Damping coefficient (Nmsrad)

∆PA Variation of injected active power (pu)

∆PG Direction of generator dispatch (pu)

∆PGA Variation of generated power (pu)

∆PL Direction of the active power variations of load (pu)

∆QA Variation of injected reactive power (pu)

∆QL Direction of reactive power variations of load (pu)

∆t Time step (s)

∆VA Variation of voltage settings (pu)

∆Vp Step size used in the parametrization technique (pu)

εA Noise symbols introduced by the approximation of non-affine operations

G Conductance matrix (pu)

xviii

H Synchronous generator inertia constant (pu-s)

Hm DFIG Rotor inertia (kWskVA)

Ka Excitation system controller gain

KG Power share in the distributed slack bus approach

Kp Pitch angle controller gain

Kv DFIG Voltage controller gain

mA Number of shared noise symbols of two affine forms

N Number of buses

na Number of noise symbols for the affine form uA

nb Number of noise symbols for the affine form uB

NDA Number of differential-algebraic equations

NIG Set of buses with intermittent sources of power

np Number of polynomials of the Chebyshev approximations

NP Set of buses where the injected active power is specified (PV and PQ buses)

NPV Set of buses with voltage control settings

NPL Set of buses considering active power load uncertainties

NQ Set of PV buses where the injected reactive power is specified (PQ buses)

NQL Set of buses considering reactive power load uncertainties

nu Number of noise symbols for the affine form u

ωr DFIG Reference rotor speed (pu)

xix

ωs Synchronous speed (pu)

pD Controllable parameters such as AVR set points (pu)

pD Affine form of pD (pu)

pPF Specified active and reactive powers as well as terminal generator voltage set points (pu)

pPF Affine forms of pPF (pu)

PX f Probability density function of the variable X f

rr Rotor resistance (pu)

rs Stator reactance (pu)

tc Fault clearing time (s)

tprimedo Open-circuit transient time constant in direct axis (s)

tprimeprimedo Open-circuit sub-transient time constant in direct axis (s)

Tec Time constant of static excitation controls (s)

t f Final simulation time for time domain simulations (s)

t f ault Instant of fault (s)

Tp Pitch control time constant (s)

tprimeqo Open-circuit transient time constant in quadrature axis (s)

tprimeprimeqo Open-circuit sub-transient time constant in quadrature axis (s)

Tr Power control time constant (s)

Vp Magnitude of the voltage used as a parameter (pu)

V2 Magnitude of the infinite-bus voltage (pu)

xx

Vre f Reference Voltage(pu)

Xd Direct axis reactance of synchronous generators (pu)

Xprimed Direc axis transient reactance of synchronous generators (pu)

Xprimeprimed Direct axis sub-transient reactance of synchronous generators (pu)

Xq Quadrature axis reactance of synchronous generators (pu)

Xprimeq Quadrature axis transient reactance of synchronous generators (pu)

Xprimeprimeq Quadrature axis sub-transient reactance of synchronous generators (pu)

xm Magnetizing reactance (pu)

xr Rotor reactance (pu)

xs Stator reactance (pu)

XT Transformer reactance (pu)

X12 Transmission line reactance (pu)

Y Admittance matrix in affine form (pu)

Y Admittance matrix in phasorial form (pu)

Variables

A Matrix of coefficients for affine forms used in AA-based power flow analysis (pu)

AL Matrix of coefficients for affine forms used to account for generatorrsquos limits (pu)

Avs Matrix of coefficients for affine forms used in AA-based voltage stability assessment (pu)

α β Coefficients of the Chebyshev approximations

brh Floating-point series coefficients of Chebyshev polynomials

xxi

C Auxiliary vector for linear programming formulation used in AA-based power flow anal-ysis (pu)

CL Auxiliary vector for linear programming formulation used to account for generatorrsquos lim-its (pu)

cp Performance coefficient

Cvs Auxiliary vector for the linear programming formulation used in AA-based voltage sta-bility assessment (pu)

δ Generator rotor angular position in affine form (electrical rad)

∆FA Errors of the differential-algebraic equations in affine form

∆FH Equivalent approximation errors of non-affine operations for ∆FA

∆λ Variation of bifurcation parameter (pu)

Dλ fPF Jacobian of fPF with respect to the bifurcation parameter (pu)

∆P Mismatch of active power (pu)

∆Pp Mismatch of active power in polar form (pu)

∆Pr Mismatch of active power in rectangular form (pu)

∆Q Mismatch of reactive power (pu)

∆Qp Mismatch of reactive power in polar form (pu)

∆Qr Mismatch of reactive power in rectangular form (pu)

∆θ Variation of bus voltage angle (pu)

∆V Variation of bus voltage magnitude (pu)

∆xPF Corrections of power flow variables (pu)

DxPF fPF Jacobian of fPF with respect to power flow variables (pu)

xxii

∆xD Corrections of state variables in affine form (pu)

∆y Corrections of algebraic variables in affine form (pu)

∆z Corrections of the state and algebraic variables in affine form (pu)

∆zH Equivalent approximation errors introduced by non-affine operations of state and alge-braic variables (pu)

Eprimeprime

dq Synchronous generator sub-transient internal voltages associated with damping windingsin affine form (pu)

Eprime

dq Synchronous generator transient internal voltages associated with field and rotor-core in-duced windings in affine form (pu)

E f Voltage associated with field and rotor-core induced currents in affine form (pu)

e Error measured with respect to MC simulations ()

Eprime

Generator voltage behind the generator transient reactance in affine form (pu)

ε Noise symbols associated with AA-based power flow analysis

εA Vector of noise symbols associated with approximation errors

εA Noise symbols associated with affine form uA

εB Noise symbols associated with affine form uB

ε f Individual noise symbols associated with approximation errors of non-affine operationsof ∆FA

ε fH Equivalent noise symbols associated with approximation errors of non-affine operationsof ∆FA

εH Equivalent noise symbols associated with approximation errors of non-affine operationsof ∆z

εIG Noise symbols associated with intermittent sources of power

xxiii

ε j Individual noise symbols associated with approximation errors of non-affine operationsof J

εL Vector of noise symbols to account for generatorsrsquo limits associated with AA-based powerflow analysis

εP Noise symbols associated with active power injections

εPL Noise symbols associated with load active power

εPVab Noise symbols associated with voltage control settings

εQ Noise symbols associated with reactive power injections

εQL Noise symbols associated with load reactive power

εu Noise symbols associated with affine form u

εVS Vector of noise symbols associated with AA-based voltage stability assessment

εξ Independent noise symbol of the multiplication affine approximation

Fa Individual approximation error of non-affine operations for ∆F

Hc Chebyshev polynomial approximation of sinusoidal functions

I Synchronous generator current injections in affine form (pu)

I Synchronous generator current injections in phasorial form (pu)

Id Synchronous generator direct axis current in affine form (pu)

Iq Synchronous generator quadrature axis current in affine form (pu)

idr DFIG rotor current in direct axis (pu)

idr DFIG affine form of rotor current in direct axis (pu)

iqr DFIG rotor current in quadrature axis (pu)

xxiv

iqr DFIG affine form of rotor current in quadrature axis (pu)

ids DFIG stator current in direct axis (pu)

ids DFIG affine form of stator current in direct axis (pu)

iqs DFIG stator current in quadrature axis (pu)

iqs DFIG affine form of stator current in quadrature axis (pu)

Jc Jacobian matrix in affine form

Jaug Augmented Jacobian matrix

KAB Centered affine forms

L Vector of active and reactive power injections for AA-based power flow analysis (pu)

LL Vector of active and reactive power injections used to account generatorrsquos limit (pu)

Lvs Vector of active and reactive power injections for AA-based voltage stability assessment(pu)

λ Bifurcation parameter (pu)

λ Bifurcation parameter in affine form (pu)

λp tip speed ratio

micro Maximum absolute error for univariate Chebyshev approximations

Nh Set of noise symbols introduced by approximations of non-affine operations

ωm DFIG rotor speed (pu)

ω Synchronous generator rotor speed (pu)

ωm DFIG rotor speed in affine form (pu)

ΨCh Chebyshev approximation for quadratic terms

xxv

P Injected active power generated (pu)

P Injected active power generated in affine form(pu)

PA Computed partial deviation of injected active power related to approximation errors (pu)

Pe Generator electrical output power in affine form (pu)

PG Active power generated (pu)

PG Active power generated in affine form (pu)

PNIG Computed partial deviation of active power injections related to intermittent sources ofpower (pu)

PNPL Computed partial deviation of active power injections related to active power loads (pu)

PNQL Computed partial deviation of active power injections related to reactive power loads(pu)

Poptω Optimal active power extracted from the wind (pu)

PL Active power loads (pu)

Pm Generator mechanical input power in affine form (pu)

PNPV Computed partial deviation of active power injections related to PV buses (pu)

Q Injected reactive power (pu)

Q Injected reactive power in affine form (pu)

QA Computed partial deviation of reactive power injections related to approximation errors(pu)

QG Reactive power generated (pu)

QL Reactive power load (pu)

QNPV Computed partial deviation of reactive power injections related to PV buses (pu)

xxvi

QNPL Computed partial deviation of reactive power injections related to active power loads(pu)

QNQL Computed partial deviation of reactive power injections related to reactive power loads(pu)

q Matrix of errors introduced by approximations of non-affine operations in power flowAA-based analysis

qe Lumped approximation error associated with the affine forms of generator reactive power(pu)

qp Components of generator reactive power associated with uncertain active power injections(pu)

qq Components of generator reactive power associated with uncertain reactive power injec-tions (pu)

qL Matrix of errors introduced by approximations of non-affine operations to account forgeneratorrsquos limits (pu)

QL Reactive power loads (pu)

qvs Matrix of errors introduced by approximations of non-affine operations in AA-based volt-age stability assessment (pu)

R Resistance (pu)

RIC Vector of central values for active and reactive power in AA-based power flow analysis(pu)

RLIC Vector of central values for active and reactive power used to account generatorrsquos limits(pu)

RvsIC Vector of central values for active and reactive power in AA-based voltage stability as-sessment (pu)

xxvii

S Complex power in phasorial form (pu)

σ Auxiliary variable for univariate Chebyshev approximations

Ψds Stator flux in direct axis (pu)

Ψqs Stator flux in quadrature axis (pu)

t Time (s)

θ Bus voltage angles (rad)

θ Bus voltage angles in affine form (rad)

θIa Current angles in affine form (rad)

θp Pitch angle (rad)

u Real variable

U Variable used to represent the bounds of the product of two zero centered affine forms inthe Chebyshev approximation approach

u Interval associated with the variable u

uA Interval associated with the variable uA

uA Variable uA in affine form

uB Interval associated with the variable uB

uB Variable uB in affine form

v Wind speed (ms)

V Bus voltage magnitudes (pu)

V Bus voltage magnitudes in affine form (pu)

V Bus voltages in phasorial form (pu)

xxviii

Vab Auxiliary variables used to modify voltage control settings (pu)

Vd Synchronous generator terminal voltage in direct axis (pu)

vdr DFIG direct axis rotor-voltage (pu)

vdr DFIG direct axis rotor-voltage in affine form (pu)

vds DFIG direct axis stator-voltage (pu)

vds DFIG direct axis stator-voltage in affine form (pu)

VI Imaginary components of bus voltages (pu)

VI Bus voltage imaginary components in affine form (pu)

vm Mean wind speed (ms)

VPQ Variable associated with the voltage of the PQ-bus used as parameter (pu)

Vq Synchronous generator quadrature axis terminal-voltage (pu)

vqr DFIG quadrature axis rotor-voltage (pu)

vqr DFIG quadrature axis rotor-voltage in affine form (pu)

vqs DFIG quadrature axis stator-voltage (pu)

vqs DFIG quadrature axis stator-voltage in affine form (pu)

VR Real component of bus voltages (pu)

VR Real component of bus voltages in affine form (pu)

Vre f Reference of voltage for voltage controllers in affine form (pu)

Vre f Reference of voltage for voltage controllers (pu)

ω Generator rotor angular speed in affine form (rads)

xxix

ωL Left eigenvector associated with the zero eigenvalue of DxPF fPF

xD State variables (pu)

xD State variables in affine form (pu)

X f Random variable

X fab Bounds of the random variable X f

x f Integration variable

xPF Bus voltage magnitudes angles and other unknown variables such as generator reactivepowers (pu)

xPF Affine forms for xp f (pu)

y Algebraic variables such as static voltages and angles (pu)

y Algebraic variables in affine form (pu)

YL Load admittance (pu)

YL Load admittance in affine form (pu)

z Vector of state and algebraic variables (pu)

z Vector of state and algebraic variables in affine form (pu)

zch Auxiliary variable for the Chebyshev polynomial expansion

Functions

F Bounded function twice differentiable and whose second derivative does not change signwithin its domain

FA Set of algebraized DAEs in affine form

Fap Chebyshev approximation of F

xxx

fD Set of differential equations

fPF Set of power flow equations

g Set of algebraic equations

Hc Chebyshev approximation function

hode Invertible function

PX f Probability density function of the random variable X f

rL function of the line used in the univariate Chebyshev approximation of non-affine func-tions

Trh Chebyshev polynomials

xxxi

Chapter 1

Introduction

11 Research Motivation

Renewable energy sources are increasingly penetrating electrical grids because of global warm-ing concerns volatility of fuel prices and their effect on conventional generation cost and gov-ernmental financial incentives such as the Green Energy Act in Ontario Canada [1] Large-scalerenewable energy projects are being developed in several countries aimed at harnessing the po-tential of renewable energy sources and reducing CO2 emissions according to established goalsFor instance the Ontario Power Authority (OPA) is planning the long-term supply mix shown inFigure 11 which eliminates coal as a generation source and increases wind solar and bioenergygeneration sources

Globally wind power is one of the fastest growing renewable energy sources this is mainlymotivated by improved wind turbine technologies and the short capital investment recovery pe-riod [2] However wind as well as solar generation exhibits intermittency of their output powerwhich imposes a huge challenge to power system planning operation and control due to the ad-ditional uncertainties involved [2ndash5] In an attempt to predict the output power associated withthese intermittent sources solar and wind power forecast tools are used by system operators andplanners [6ndash8] However the accuracy of these tools particularly for wind power depends onfactors such as the quality of the prediction model the level of the predicted power the predic-

1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation

Figure 11 Long-term energy supply for Ontario in 2030 [9]

tion horizon and spatial distribution of wind farms Consequently different prediction tools canlead to different expected productions and different probability density functions (pdfs) to modelthe uncertainty of the prediction error [6ndash8] Similarly the accuracy of solar power forecastmethods are affected by factors such as the prediction horizon and the environmental conditionsamong others For these reasons uncertainties associated with the power prediction are betterexpressed in terms of intervals instead of point estimates [6] For instance Figure 12 illustratesa 12 hours ahead aggregated wind power forecast corresponding to Alberta Electric System Op-erator (AESO) these forecasted values are based on near real-time meteorological data and aregiven in terms of maximum minimum and most likely values

In order to maintain the power balance at any time instant conventional power plants arenormally used to cope with renewable power production uncertainties As a result conventionalunit commitment and dispatch of generators are reformulated to account for these uncertaintiesby means of enough power reserve provisions [2] [4] In this case conventional generatoroutputs are adjusted up and down to compensate for the variability of intermittent energy sourcesAn alternative solution is the use of energy storage technologies which can smooth the output ofintermittent energy sources [11] however technical and economical factors are a limitation forenergy storage deployment specially at transmission system levels

2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0

Alberta 12 Hour Wind Power forecast updated as of

7172013 94400AM MT

MCR MOST LIKELY MAX MIN

Figure 12 Wind power forecast example [10]

Depending on the penetration level of intermittent sources of power the electricity grid mayexperience considerable and uncertain changes in power flows These changes may lead to limitviolations that affect system security [2] Moreover the location of renewable resources whichmay be far from the main load centers also affects the system limits since these generation mayhave to be transported over long distances leading to system congestion that have a direct impacton system stability margins [2 3 12 13]

Besides the changing synchronizing forces that the system experiences due to the uncertainpower flow variations associated with intermittent sources of power the effective system inertiachanges when a large amount of variable speed wind turbines and solar power is connectedwhich may affect the system transient stability This change in the effective inertia will affectthe angular acceleration of the synchronous generators when a disturbance occurs [13] Alsothe rate of frequency decay can be accelerated for generator trip events if proper controls arenot implemented Adequate analysis tools are thus required to properly analyze power system

3

stability when subjected to penetration of intermittent sources of power

A variety of methods are available in the literature to perform power flow transient andvoltage stability analyses considering uncertainties associated with electrical parameters [14ndash83]Most of these methodologies are better suited for reliability and planning studies where detailedprobabilistic methods are used moreover some of them require a high volume of computationsas in sampling based methods A different approach based on Affine Arithmetic (AA) whichdoes not rely on detailed probabilistic or sampling methods is proposed in [14] for power flowassessment considering uncertainties associated with the operating conditions This approach isable to compute bounds of bus voltage magnitudes angles and line flows when the generatoroutput powers and load power levels are assumed to vary within established intervals Thereforethe application of this technique to stability analysis is studied here in detail

12 Literature Review

In this section previously proposed techniques and tools aimed at assessing power flow voltagestability and transient stability considering uncertainties are briefly discussed highlighting theirmain advantages and disadvantages

121 Stochastic Power Flow Analysis

Stochastic power flow assessment based on Monte Carlo Simulations (MCS) are reported in[15ndash17] Although these methods allow simulating stochastic events with any level of complex-ity they impose a large computational burden Some analytical techniques are reported in theliterature seeking to reduce the computational time associated with MCS [18ndash22] These tech-niques use properties and theorems of probability to compute the pdfs of the desired variablesbased on the pdfs of the input data Some simplifications such as linearization of the power flowequations and assuming a normal distribution for the stochastic variables are reported in [18]Linearization of the power flow equations introduces errors in the pdf calculations thus consid-erable effort has been put into improving the accuracy of analytical approaches For instancein [19] a multilinearization technique is formulated to reduce the inaccuracies associated with

4

the linear approximation of the non-linear power flow equations Furthermore in [20] a secondorder approximation for the power flow equations together with a technique of moments is em-ployed to improve the accuracy associated with linear approximation based analytical methodsAlternatively in [21] and [22] the line flow pdfs are approximated by means of Gram-Charlierexpansions together with cumulants for nodal powers In general these analytical methods arebased on probabilistic theorems and elaborated computations that add some mathematical com-plexity when compared to MCS Furthermore most of them are only accurate for particular inputsvariable pdfs such as normal distributions

As an alternative to analytical and MCS in [23] the point estimate method is used to ac-count for the uncertainties in the transmission line parameters operating conditions and topolo-gies This method exhibits a good accuracy and computational efficiency when compared withanalytical and sampling approaches Likewise in [24] the point estimate method is appliedconsidering unbalanced three-phase power systems and correlations between the input variablesin this paper it is concluded that the proposed method performance in terms of computationaltime is better than the performance offered by MCS Furthermore in [25] a probabilistic optimalpower flow proposed for a market environment and based on the two-point estimate method isfound to be better suited for a relatively low number of stochastic variables or alternatively forstochastic variables with small standard deviations

Methods that combine analytical and MCS are also reported in the literature For instance in[26] a sampling method based on latin hypercube and Cholesky decomposition is used togetherwith MCS in order to reduce the computational cost attributed to conventional MCS Similarlyin [27] a multilinearization technique is combined with MCS for the purpose of reducing theinaccuracies of linear approximation based analytical approaches

As an alternative to the aforementioned methods fuzzy power-flow methods are widely ap-plied in stochastic power flow calculations [28ndash31] In these methods stochastic variables aremodeled by means of fuzzy sets which are associated with a membership function The finaloutput is a possibility distribution rather than a pdf

Stochastic power flow based on self-validated methods such as IA and AA has also receivedattention in the literature In [32] IA is applied to compute the power flow bounds of a five-bustest system when loads and generator power uncertainties are modeled using intervals a Newton

5

operator is used to deal with the nonlinear equations The authors observe that although the IAoffers a better performance than MCS in terms of simulation time it may produce excessivelyconservative results when the input interval sizes increase Similarly in [33] IA is used to solvethe power flow problem by means of the Krawczyk method the authors conclude that IA is anefficient technique to solve the power flow problem considering uncertainties In [34] a radialpower flow solution by means of IA is proposed for balanced systems In this methodology theload uncertainties are modeled as intervals and a complex approach for the IA operations is usedthe IA results are observed to be very close to MCS results in this case Also in [35] a radialpower flow is solved by using IA combined with an interval constraint propagation technique inorder to obtain narrower intervals this technique is based on finding a restriction function thatproduces thinner intervals by a successive contraction process Since IA-based methods are notable to account for correlations among variables they may lead to error explosion resulting inimpractical bounds Aimed at avoiding the error explosion attributed to IA-based methods anAA-based approach for power flow assessment is proposed in [14] Herein analysis carried outconsidering a 50-bus test system assuming uncertainties in the injected powers demonstrate theaccuracy of the method as compared to MCS and IA reporting considerable improvements incomputational efficiency when compared to MCS

In [36] a fast approximation of the power flow solution bounds is assessed by sampling areduced set of deterministic power flow solutions based on a worst case analysis of the gen-eration and load scenarios These scenarios correspond to maximum generation and minimumload maximum generation and maximum load minimum generation and minimum load andminimum generation and maximum load The deterministic power flow solutions correspondingto these generationload scenarios are then combined to determine an approximation of the hullboundary of the power flow solutions

From the above described methods sampling based approaches such as MCS have beenwidely used for power flow appraisal they can manage complex and accurate probabilistic mod-els at the cost of a considerable computational burden Analytical methods are also an optionaimed to be more computationally efficient than MCS however their accuracy is in generalaffected due to the simplifications needed to enable the use of probabilistic theorems Thesesimplifications are in some cases based on unrealistic representations of the stochastic nature ofthe input variables For instance for the particular case of wind power the probability distri-

6

bution for the forecast errors depends on the considered time scale among other factors thusassuming normal distributions for all time scales as in most of the analytical methods maylead to inaccurate results [6] The AA-based method previously mentioned for power flow as-sessment considering uncertainties associated with loads and power injections does not requireassumptions regarding the pdfs of the uncertain variables furthermore it is able to improve thecomputational efficiency when compared to MCS while providing quite reasonable accuracybecause correlations among variables are considered [14] Therefore this thesis proposes animproved AA-based power flow methodology with a more efficient representation of generatorreactive power limits

122 Stochastic Voltage Stability Assessment

MCS have been used to model a variety of system complexities for voltage stability assess-ment [37] [38] However the high computational cost of MCS renders it unsuitable for certainapplications Thus most of the effort in the literature has concentrated on reducing the compu-tational cost of MCS while achieving acceptable accuracy For instance the two-point estimatemethod used in [39] computes the transfer capability associated with uncertainties of transmis-sion line parameters and bus injections This method computes approximations for the momentsof the output variable using only two probability concentrations for each uncertain input vari-able However in certain cases for a relatively large number of uncertain input variables andorlarge statistical dispersion of these variables the two-point estimate method is not sufficientlyaccurate as compared to MCS [25] Moreover a higher order point estimate method is requiredfor the cases where the pdfs of the input variables are not normal which adversely affects thecomputational efficiency of the method In this sense one of the advantages of the proposedAA-based technique is its independence with respect to the pdfs associated with the uncertainvariables which are modeled as intervals with no assumptions regarding their probabilities

Enumeration techniques are alternate approaches to MCS for online applications For in-stance in [40] an enumeration technique is used to compute the voltage stability indicatorsassuming uncertainties in the loads the stability indicators are derived from the power flow equa-tions based on various simplifications such as constant generator voltage magnitudes and phasesIn [41] online risk indices are proposed to evaluate the system security from the perspective of

7

low voltage violation overload cascading overload and long-term voltage stability

Truncated Taylor series expansion methods are used in [42] to efficiently account for vari-ations in the transfer capability due to variations in system parameters A formula to computereliability margins is derived assuming conditions associated with the central limit theoremAccording to this theorem the uncertain parameters must be independent and their number besufficiently large in order to approximate the output variable as normally distributed In theproposed AA-based method for voltage stability assessment the uncertain input variables arenot necessarily assumed to be statistically independent in fact correlations among uncertainvariables are explicitly represented in the model

In [43] and [44] the effect of system control parameters and system data on voltage stabil-ity margins is accounted for by means of linearization techniques and direct formulas In [45]these formulas are used to efficiently estimate changes in saddle node bifurcations and limit in-duced bifurcations due to changes in power system transactions in a market environment Thesemethods based on sensitivity analysis and formulas are able to efficiently estimate voltage sta-bility indices due to system parameter variations but within certain limits in which the assumedlinearizations are valid Moreover they are not intended to compute full PV curves under uncer-tainties which is the main contribution of the AA-based method proposed here

Some research work is reported on the computation of loadability margins considering un-certainties associated with intermittent sources of generation For instance in [46] a stochasticresponse surface is used to estimate the pdf of the loadability margins considering stochastic gen-eration variations attributed to renewable energy sources and assuming non-conforming loadsThis method is able to reduce the computational burden of MCS by using a polynomial chaosexpansion of a relatively low order As the number of uncertain variables increase the order ofthis polynomial chaos expansion may increase and hence improve the accuracy of the methodAlternatively various papers are focused on finding maximum loadabilities in terms of load un-certainties For example in [47] the stochastic nature of loads is modeled using a hyper-conemodel whose thickness represents the uncertainty of future loading and the vertex is the cur-rent operating point This approach is focused on finding the worst case scenario for maximumloadabilities in terms of load uncertainties while the AA-based method proposed in this thesiscomputes the bounds of PV curves and associated maximum loadabilities considering uncer-tainties due to power injections In [48] a cumulant method is used for assessment of saddle

8

node bifurcations considering uncertainties in load forecast The method is used to solve theresultant stochastic non-linear programming formulation that leads to the estimation of the pdffor maximum loadability The cumulant method requires information on the cumulants of theuncertain variables which may not be accurately available for uncertainties associated with inter-mittent sources of generation where there is no general agreement on the pdf to best representthe forecasting errors [6 7]

In order to overcome some of the limitations of sampling- and statistical-based methods theuse of soft-computing-based methodologies for uncertainty representation in voltage stabilityassessment has been proposed in the literature In particular the application of fuzzy set theoryto represent uncertainties has been proposed in [49 50] In this paradigm the input data aremodeled by fuzzy numbers which are special types of fuzzy sets Defining the connectionbetween interval analysis and fuzzy set theory is not a trivial task [51] In [52] the ideas of fuzzysets and interval analysis are both connected to a general topological theory Similarly in [53]it is argued that the theory of fuzzy information granulation the rough set theory and intervalanalysis can all be considered as subsets of a conceptual and computing paradigm of informationprocessing called Granular Computing An essential aspect of this paradigm is that its constituentmethodologies are complementary and symbiotic rather than competitive and exclusive Basedon these principles the main contributions of the present thesis lie on the application of advancedinterval based solution approaches to voltage stability analysis with data uncertainties

More recently computational intelligence based techniques have been proposed in both stateselection (as an alternative to MCS) and in state evaluation (as an aid to MCS) The applica-tion of these techniques allows deploying directed intelligent search paradigms which are analternative to the proportionate sampling approach characterizing standard MCS Amongst thesetechniques the most promising for the problem under study is the PIS-based algorithm [54]This algorithm tries to generate the dominant failure states and minimize the generation of suc-cess states Thanks to this feature fewer states need to be evaluated compared to standard MCS(where the majority of the states sampled are success states) The proposed AA-based approachis intrinsically different from this computing paradigm since it does not integrate either samplingbased methods or evolutionary computation processes By using the methodology presented inthis thesis a reliable estimation of the problem solution hull can be directly computed takinginto account the parameter uncertainty interdependencies as well as the diversity of uncertainty

9

sources The main advantage of this solution strategy is that it neither requires derivative compu-tations nor interval systems and hence are suitable in principle for large scale problems whererobust and computationally efficient solution algorithms are required

Another general class of statistical methods aimed at empirically assessing the parametricconfidence intervals of a sampling distribution are based on resampling theory The most com-mon technique in this class is the so called bootstrap method a technique for using the datacollected from a single experiment to simulate what the results might be if the experiment wasrepeated by sampling (with replacement) data from the original dataset [55] This paradigm hasbeen recognized as a powerful tool for testing or avoiding parametric assumptions when com-puting confidence intervals in many engineering problems [56 57] Although the adoption ofthe bootstrap method is expected to be more effective compared to standard MCS especially interms of computational efforts it still requires repetitive simulations

The detailed chronological nature of intermittent power sources can be considered using se-quential MCS and time series models instead of analytical approaches or simple state-samplingMCS For instance in [58] an Auto Regressive Moving Average (ARMA) based technique isused to estimate the wind speed and the corresponding output power of a Wind Energy Conver-sion System (WECS) on an hourly basis These hourly point estimates are used together withthe random failure of generating units and load profiles to compute the hourly available systemreserve using sequential MCS The goal of this approach is the computation of reliability indicesand generating capacity adequacy assessment over a long-term time frame Since this methodrequires repetitive MCS for each time instant considered it imposes a high computational bur-den The proposed AA-based approach for voltage stability assessment can indeed considerthe chronological issues associated with intermittent sources of power since it uses intervalsto model the uncertain output power of these intermittent and non-dispatchable sources withno assumptions regarding their probabilities nor the associated time windows For a differenttime window a new estimate of the interval that properly reflects the uncertainty can be easilyobtained The larger the time window the larger the interval of uncertainty and vice versa Itis also important to point out that the proposed method is intended to efficiently compute thehull of PV curves associated with intermittent sources of power considering short operationaltime frames and it is not intended to compute reliability indices or evaluate generating adequacycapacity as in [58] and other similar applications of sequential MCS

10

Various probabilistic voltage stability studies are reported in the literature that use machinelearning techniques For instance in [59] a trained radial basis function network is used to per-form online appraisal of the probabilistic risk of voltage collapse given the loading condition andthe settings for the reference voltages at PV buses a radial basis function is trained by consid-ering uncertainties in transmission line parameters and reference voltages for PV buses In [60]probabilistic insecurity indices considering voltage stability restrictions are computed for severalload levels considering reactive power redispatch and single- and double-line contingencies Thismethodology uses artificial neural networks to evaluate the system voltage security margins inaddition corrective actions aimed to improve the voltage security are introduced by evaluatingthe sensitivity of the probabilistic insecurity indices with respect to the reactive power controlactions The general shortcoming of machine learning based techniques is the relatively elevatednumber of training cases that may be needed in order to produce accurate results which can addsome level of complexity to the problem

In this thesis a new AA-based computing paradigm is proposed that is an alternative to sam-pling based approaches to represent the uncertainties of the power system state variables Thisallows expressing the system equations in a more convenient formalism compared to the tradi-tional and widely used linearization frequently adopted in interval Newton methods By usingthe proposed methodology a reliable estimation of the problem solution hull can be computedtaking into account the parameter uncertainty interdependencies as well as the diversity of un-certainty sources without requiring repetitive simulations and assumptions regarding pdfs

123 Stochastic Transient Stability Assessment

Transient stability analysis for operation and planning purposes evaluates the capability of thepower system to remain in synchronism when subjected to large and sudden system disturbancesTo account for the uncertainties associated with power system variables probabilistic methodsfor transient stability analysis have been widely proposed in the literature Most of these meth-ods account for the probabilities of different factors such as the fault type and location clearingtime operating conditions fault duration and reclosing times Thus various papers addressthe probabilistic transient stability problem by means of MCS In [61] MCS are used togetherwith transient energy functions for transient stability assessment considering uncertainties in

11

fault characteristics and the reliability of circuit breakers In [62] MCS are employed to deter-mine the probability of instability for a large-scale power system probabilistic models for theload level fault type fault location and successful reclosing time are developed by means ofavailable system historical data The accuracy of the probabilistic models is achieved becauseof the availability of the historical data for this particular system In [63] a MCS-based ap-proach is proposed where the equal area criterion is used to calculate the critical clearing timesIn [64] a deterministic approach is combined with MCS in order to analyze the system securitywhile reducing the computational time Most of these MCS-based approaches are mainly fo-cus on probabilistic transient stability assessment for medium and long term planning purposesFurthermore they require enough historical data to assess the pdfs of the uncertain variablesConsidering that MCS involve a long computational burden the proposed AA-based methodol-ogy for transient stability assessment significantly improves the computational efficiency whichmay render it suitable for online applications

Special attention has also been paid in the literature to conditional probability based meth-ods In [65] the probability indices for an 8-bus 3-machine system are determined using theconditional probability theorem considering the probabilities of faults and assuming the loadvariations in three discrete levels the critical clearing times are computed using Lyapunov meth-ods The main drawback of this approach is the difficulty in properly computing the transientenergy functions and the stability regions for large scale systems In [66] the conditional proba-bility method is used for assessment of transient stability of a multimachine system consideringuncertainties in the fault types locations and clearing times which are modeled using availablehistorical data however the effect of the stochastic prefault conditions on transient stability in-dices is not addressed In [67] conditional probability approach and the Bayesrsquo theorem are usedto compute the probability of instability based on the historical data for the occurrence of eachconsidered event In [68] the pdfs for the reclosing time and the fault duration are convolvedto obtain the pdf of successful reclosing uncertainties in fault type fault location and loadinglevels are considered In this paper the conditional probability theorem is applied to calculatethe stability probability indices with successful and unsuccessful reclosing In an attempt toreduce the volume of computation associated with the calculation of critical clearing times afixed clearing time which is equal to the maximum clearing time given by its normal probabil-ity distribution is assumed in [69] accordingly the critical clearing times are only computed if

12

the system is unstable for a particular combination of stochastic events These critical clearingtimes and the transient stability assessment are carried out by means of a corrected transient en-ergy function strategy For the evaluation of the instability indices the uncertainties associatedwith fault characteristics are considered but those linked to pre-fault conditions are not takeninto account In [70] the conditional probability approach is used to estimate the probabilityof stability indices in long term planning for a 10-bus system considering wind power penetra-tion besides uncertainties in the fault characteristics and protection operation uncertainties inoperating conditions are accounted for in this case

Similarly to MCS conditional probability approaches are mainly applied to stability stud-ies for medium and long term planning purposes Even though the conditional probability ap-proaches reduce the simulation times as compared to MCS their accuracy relies on the availablehistorical data in order to provide accurate statistics of the transient stability indices In generalconditional probability based approaches exhibit some difficulties in properly modeling the pdfsof the uncertainties associated with the operating conditions This is not an issue for the pro-posed AA-based methodology where the uncertainties associated with the operating conditionsare modeled as intervals

Some research is reported on the application of machine learning techniques for probabilistictransient stability assessment aimed at improving the computational efficiency when comparedto conventional MCS For instance in [71] a self organizing map neural network is used to re-duce the size of the samples required in MCS this map is built considering transient and voltagestability constraints It is noted that the self-organizing map neural network is able to reducethe computational times compared to the conventional MCS Alternatively to the use of artifi-cial neural networks in [72] a Bayes decision function is used to classify the level of securityof the system states The method of calculation of the Bayes decision function is based on theestimation of the mean and variances of chosen security groups which are calculated from thecharacterization of feature vectors finally the Bayes decision function is used to classify newfeature vectors according to the established security groups and compute transient stability se-curity indices In [73] the polynomial representation of the transient stability boundary regionobtained in [74] by means of a machine learning technique is used together with a linearizationapproach to compute a risk-based index for dynamic security considering uncertainties in the op-erating conditions this approach also reduces the computational time when compared to MCS

13

These machine learning based techniques are designed to reduce the computational burden asso-ciated with MCS however their accuracy relies on the quality and number of cases used in thetraining stage which may present a computational challenge This training stage is not requiredin the proposed AA-based methodology for transient stability assessment which represents animportant advantage with respect to machine learning based techniques

In [75] pdfs for the maximum relative rotor angles considering stochastic loads stochasticclearing times and a particular fault type and location are computed by means of the two-pointestimate method In this paper the cumulative probabilities of individual relative rotor angles arecomputed in order to determine the probabilities of stability according to a predefined thresholdThe basic idea is to reduce the number of samples of the stochastic variables to only two rep-resentative samples for each stochastic variable which are then used to compute the individualrelative maximum rotor angles for each generator by means of a transient simulation packageFor pdfs other than normal pdfs a higher order point estimate method is required adversely af-fecting the computational efficiency of the two point estimate method as discussed before TheAA-based methodology proposed in this work is intended to be independent of the pdfs assumedto model the uncertain input variables resulting in a more flexible approach

Analytical approaches to probabilistic transient stability assessment are also reported in theliterature In [76] a method to compute transient stability indices in terms of relevant systemfixed parameters and statistical parameters is presented to determine an analytical expressionfor the critical clearing time The nonlinear relationship between the critical clearing time andsystem load is approximated using a log-linear model whose parameters are computed by linearregression Similarly in [77] an analytical approach to probabilistic transient stability assess-ment considering uncertainties associated with fault clearing times and system loads is proposedthis approach uses the potential energy boundary surface method to assess the transient stabilityLikewise in [78] the authors propose a methodology to calculate the pdf of the critical clearingtime using a linear function with respect to the system load the linear function is computed us-ing the equal area criterion together with linearization techniques around an equilibrium pointThese analytical approaches for transient stability assessment are mainly theoretical approachesin practical power systems these methods are mathematically difficult to implement Thus theaccuracy of these analytical methods relies on the validity of the assumed simplifications fur-thermore the mathematical complexities of some of these probabilistic methods render them

14

unsuitable for applications to real size power systems In this context the proposed AA-basedmethod for transient stability assessment is intended to avoid the need of elaborated probabilisticmethods while providing reasonable accurate results for the transient stability bounds regardlessof the system size

As an alternative to sampling-and statistical-based methods fuzzy set theory is applied in [79]to determine transient stability indices for a defined number of system conditions Similarlyreachability analysis is used in [80 81] to determine a bounded set of possible trajectories ofthe system state variables when the system is perturbed by uncertainties which are modeled asbounded sets Likewise in [82] reachability analysis is used to generate the possible trajectoriesof multi area frequency dynamics considering uncertainties associated with intermittent renew-able sources One of the advantages of reachability analysis based approaches is the modelingof the uncertain variables as bounded sets without the need of assumptions regarding pdfs how-ever the large errors attributed to interval operations and the difficulties associated with modelingof system discontinuities in reachability analysis may be an issue in some applications

An IA-based method is applied in [83] to transient stability assessment considering uncertain-ties in the system parameters the resultant interval differential equations are solved by means ofan interval Taylor series method It is noted that the proposed interval methodology may lead toexcessively conservative results with relatively long simulation times In fields other than powersystems a methodology to evaluate the impact of parameter variations in the transient and DCresponse of analog circuits using AA is proposed in [84] solving a set of Differential AlgebraicEquations (DAEs) that models the analog circuit using an iterative approach

Based on the aforementioned detailed literature review the present thesis proposes the useof AA as a novel method for transient stability appraisal of power system with intermittent re-newable sources This method is more accurate than the IA-based methodology previously men-tioned since correlations among variables are considered It is conceived to enclose the outputvariables according to intervals that yield reasonably fast and adequate output variable boundswhen the input variables are modeled as intervals without any assumptions regarding their prob-abilities

15

13 Research Objectives

This thesis proposes novel methods for adequate voltage and transient stability assessment basedon AA aimed at reducing the computational burden associated with MCS-based methods andavoiding the need of elaborated probabilistic methods and pdf assumptions The proposed meth-ods are conceived to be reliable and computational efficient tools for transient and voltage stabil-ity assessment of power systems with intermittent and renewable sources of power representinga new computing paradigm which is an alternative to sampling-based approaches Thereforethe objectives of this thesis are the following

bull Improve the existing AA-based power flow analysis method particularly with respect tothe representation of generator reactive power limits

bull Develop a computationally efficient method based on AA for voltage stability assessmentof power systems with intermittent renewable sources This method is based on the com-putation of the bounds of PV curves given the uncertainties associated with the powerproduction of intermittent sources of power which are modeled as intervals without anyassumptions regarding their probabilities

bull Develop a computationally efficient method based on AA for transient stability assessmentof power systems with intermittent renewable sources This method is aimed at comput-ing the hull of the transient stability time domain dynamic responses associated with theintervals that models the uncertainties of the power production

bull Validate the results of the proposed methods by comparison with respect to conventionalprobabilistic methods such as MCS-based approaches

bull Study the computational effort of the proposed methods using realistic test systems

14 Thesis Outline

The rest of this thesis is organized as follows Chapter 2 presents a review of relevant back-ground specially of power flow voltage stability and transient stability analyses In addition

16

self-validated numerical techniques are explained with special attention to some of the availableapproaches to approximate non-affine operations Also MCS are explained in some detail pre-senting and example of a stability analysis of a power grid with intermittent sources of power todemonstrate its relevant application in power systems

Chapter 3 discusses the AA-based power flow analysis and the results of the applicationof this method to three study cases In Chapter 4 the proposed AA-based method for voltagestability assessment of power systems considering uncertainties associated with power injectionsis presented discussing the results from applying it to small- and a large-size test systems

Chapter 5 presents and discusses the proposed AA-based method for transient stability as-sessment of power systems considering uncertainties associated with power injections resultsare discussed for two study cases using various generator models including Doubly Fed Induc-tion Generator (DFIG) based wind turbines Finally Chapter 6 presents the summary of thethesis highlighting its contributions and possible future research work

17

Chapter 2

Background Review

21 Introduction

In this chapter fundamental concepts and solution techniques concerning power flow voltagestability and transient stability are briefly reviewed In addition a section is devoted to thediscussion of self-validated IA and AA numerical techniques with an special emphasis on theapproaches used to approximate non-affine operations Finally MCS are briefly discussed andan example of the application of MCS to system stability analysis considering uncertain powerinjections due to wind generation is presented The notation of all equations presented in thischapter and the rest of the thesis are defined at the beginning of this document in the nomencla-ture section

22 Power System Analysis

Power systems contain a variety of components aimed at generating transmitting and consumingthe electrical power in an efficient an reliable manner In order to properly design plan and op-erate a power system analysis tools which commonly rely on numerical methods are employedThis section concentrates on the discussion of power flow voltage stability and transient stabilityanalysis tools which are the main focus of this thesis

18

221 Power Flow

Power flow studies are aimed at computing the steady-state voltage magnitudes and angles aswell as the complex powers across the elements that comprise a power grid Power flows arethe basic tools in power system analysis and form the basis for stability analysis The generalformulation of the power flow problem is as follows [85]

fPF(xPF pPF) = 0 (21)

This vector equation represents a system of nonlinear equations referred to as the power flowequations where xPF represents the vector of bus voltage magnitudes angles and other relevantunknown variables such as generator reactive powers and p stands for the vector of specifiedactive and reactive powers injected at each node as well as terminal generator voltage set points

A detailed representation of (21) in polar form is given by

∆PPi = PGi minus PLi minus

Nsumj=1

ViV j[Gi j cos(θi minus θ j) + Bi j sin(θi minus θ j)] foralli = 1 N (22)

∆QPi = QGi minus QLi minus

Nsumj=1

ViV j[Gi j sin(θi minus θ j) minus Bi j cos(θi minus θ j)] foralli = 1 N (23)

Equations (22) and (23) can also be written in rectangular form as follows

∆Pri = PGi minus PLi minus

Nsumj=1

VRi(VR jGi j minus VI j Bi j) + VIi(VI jGi j + VR j Bi j) foralli = 1 N (24)

∆Qri = QGi minus QLi minus

Nsumj=1

VIi(VR jGi j minus VI j Bi j) minus VRi(VI jGi j + VR j Bi j) foralli = 1 N (25)

In PV buses the injected active power and bus voltage magnitude are specified with the busvoltage angle and reactive power injection being the unknown variables to be calculated In PQbuses both the injected active and reactive powers are specified and the voltage magnitude and

19

angle are unknown Additionally a slack bus has to be defined in order to provide a referenceangle and balance the total power injections active and reactive power injections are typicallythe unknown variables in this bus

Because of the nonlinear characteristics of the power flow equations these are usually solvedusing numerical techniques involving iterative methods of which the Newton-Raphson methodis the most commonly used This method can be formulated with respect to equation (21) asfollows

bull The solution method starts with an initial guess for the unknown variables At each iter-ation starting with the initial guess then this initial guess the elements of the Jacobianmatrix and the power mismatches are computed and the following linear equation is usedto compute the corrections of the unknown variables at an iteration r

part fPF(xrminus1PF prminus1

PF )partxPF

∆xrPF = minus fPF(xrminus1

PF prminus1PF ) (26)

bull This process is repeated using the updated values of the unknown variables until the con-vergence criteria is met or the maximum number of iterations is reached

In the solution process different strategies can be employed to account for the reactive powerlimits associated with generator buses The most commonly used strategy is based on the switch-ing of the bus type According to this strategy the generator reactive power limit violations arechecked at each iteration If a violation occurs the corresponding generator cannot longer con-trol the voltage at the corresponding bus and thus the PV bus is switched to PQ Similarly ifthe voltage recovers in the subsequent iterations the PQ bus type is switched back to PV and soon until a feasible solution is obtained without any limit violations

As an alternative to the bus type switching strategy the terminal voltage at generator busescan be treated as a variable so that PV buses where reactive limits have been violated can bemodified until no limit violations are observed while still treating the bus as PV This approachcan be implemented either by modifying the terminal voltage whenever necessary in the iterativeprocess or by using an optimization approach which is based on the representation of PV busesdiscussed in [86 87] where the voltage at these buses is defined as

20

Vnc = Vonc + Vanc minus Vbnc forallnc isin NPV (27)

where

Vanc gt 0Vbnc = 0 for QGnc lt QGminnc

Vanc = 0Vbnc gt 0 for QGnc gt QGmaxnc

Vanc = Vbnc = 0 for QGmaxncle QGnc le QGminnc

(28)

the first condition in (28) is a strict complementarity condition and guarantees that the minimumreactive power constraint of the generator is not violated this is achieved by increasing themagnitude of the terminal voltage by means of the auxiliary variable Vanc according to (27)Similarly the second condition avoids violation of the upper reactive power limit by adjustingthe magnitude of the auxiliary variable Vbnc which reduces the terminal voltage of the generatorWhen no reactive power limits are violated the auxiliary variables in (27) are zero as in thethird condition of (28) thus keeping the terminal voltage at its reference value VoIC This modelyields the following Mixed Complementarity Problem (MCP) [87]

min 12∆P22 + 1

2∆Q22

st (QGnc minus QGminnc)Vanc = 0

(QGnc minus QGmaxnc)Vbnc = 0

QGmaxncle QGnc le QGminnc

Vnc = Vonc + Vanc minus Vbnc

Vanc Vbnc ge 0

forallnc isin NPV

(29)

21

222 Voltage Stability

Voltage stability deals with the ability of the power system to maintain acceptable voltage lev-els under normal operating conditions and after the system is perturbed by small or large dis-turbances [88] Voltage stability can be associated with short-term or long-term phenomenaShort-term voltage stability is studied considering system dynamics whereas long-term voltagestability is commonly studied by means of steady-state solution techniques For instance volt-age collapse which is usually associated with long-term phenomena is commonly analyzed bypower flow techniques and linearization of the system equations Even though voltage stabil-ity is a dynamic process steady-state analysis techniques are used to identify the absence of along-term equilibrium in the post-contingency state which leads to voltage instability [89] Thisthesis is focused on this particular voltage stability phenomenon

The maximum load that the system can withstand without experiencing a voltage collapseis commonly used to compute voltage-stability indices This maximum loadability is associatedwith a saddle-node bifurcation or a limit-induced bifurcation point [90] Saddle-node bifurca-tions refer to the operating point where the system state matrix becomes singular This singularityof the system state matrix usually coincides with the singularity of the power flow Jacobian andthus no power flow solution can be found using conventional power flow techniques On theother hand limit-induced bifurcations arise when generators reach their reactive power genera-tion limits thus losing voltage control capability

Figure 21 illustrates saddle-node and limit-induced bifurcations for the 2-bus test systemshown in Figure 22 Observe that limit-induced bifurcations tend to decrease the maximumloadability when compared to saddle-node bifurcations In this example limit-induced bifurca-tion are computed for two different values of the maximum generator reactive power associatedwith the generator connected at Bus 1 with QGmax1

gt QGmax2 yielding as expected a lower static

load margin when the maximum generator reactive power limit is lower

The most representative methods devised to compute the maximum loadability are the con-tinuation power flow and direct methods which are briefly discussed next

22

Figure 21 PV curves for 2-bus system

Ѳ

Figure 22 Two-bus system example

23

Continuation Power Flow Method [90ndash93]

Continuation power flow methods are designed to overcome the difficulties associated with thecomputation of power flow solutions in the vicinity of the maximum loadability point Thesedifficulties are linked with the singularity of the power flow Jacobian matrix or the reactive powerlimits at this point The general idea behind continuation power flow methods is the solution ofthe power flow equations given by

fPF(xPF pPF λ) = 0 (210)

where the parameter λ is used to generate different scenarios for loads and generator outputsaccording to the following equations

bull For generator powers excluding the slack bus

PGi(λ) = PGoi + λ∆PGi foralli = 1 N minus 1 (211)

bull For generator powers including the slack bus (distributed slack bus)

PGi(λ) = PGoi + (λ + KG)∆PGi foralli = 1 N (212)

bull For load powers

PLi(λ) = PLoi + λ∆PLi foralli = 1 N (213)

QLi(λ) = QLoi + λ∆QLi foralli = 1 N (214)

The solution of (210) for different values of λ is obtained using two basic steps predictorand corrector In the predictor step an approximate power flow solution for a small increase inthe loading level is obtained from an initial operation point Two methods are typically used toperform the predictor step the tangent vector method and the secant method The tangent vec-tor method is based on the computation of an approximate solution for x and λ from a previousknown solution in a direction that is tangent to the solution path On the other hand the secantmethod requires two previous solutions in order to compute the next approximate solution In

24

the corrector step the approximate solution obtained from the predictor step is used as an initialguess to solve the power flow equations plus an additional equation that allows to determinethe value of λ and facilitates convergence around the maximum loadability point Two tech-niques that may be used in the corrector step are the perpendicular intersection technique and theparametrization technique The perpendicular intersection technique which is illustrated in Fig-ure 23 uses a perpendicular vector at the approximated solution in order to define the requiredadditional equation to prevent convergence problems that may arise when a large predictor stepis used this method can be accompanied by a step cutting technique In the parametrizationtechnique the convergence issues close to the maximum loading point are avoided by fixing asystem variable such as the voltage magnitude at a specific load bus Figure 24 illustrates theparametrization technique when a load voltage is fixed

Conventional power flow methods are also used to compute the upper side of the PV curveThis technique depicted in Figure 25 used by many commercial packages is based on a seriesof conventional power flows as the load is increased together with a load step-cutting techniqueto obtain solutions in the vicinity of the maximum loading points [94]

Direct Methods

Direct methods rely on the direct computation of the maximum loadability point by means ofan optimization approach The general formulation for this optimization approach is as follows[86 95]

max λ

st fPF(xPF pPF λ) = 0

xPFmin le xPF le xPFmax

pPFmin le pPF le pPFmax

λ ge 0

(215)

The system controllable parameters pPF such as PV bus voltage set points are considered asvariables in this formulation in order to improve the maximum loadability

If the controllable parameters pPF are fixed (215) can be reformulated to compute the max-

25

Figure 23 Perpendicular intersection technique for the correction step

imum loadability due to saddle-node or limit-induced bifurcations [86 95 96]

The maximum loadability obtained using continuation power flow methods and direct meth-ods may not be the same because of the particular treatment of voltage controlled buses used oneach approach In continuation power flow methods the reference voltage at voltage-controlledbuses are kept constant unless a reactive power limit is violated On the contrary in typicaldirect methods the voltage reference at voltage-controlled buses are allowed to change withinlimits [86 96]

26

Figure 24 Parametrization technique for the correction step

These and other tools are used by system operators and planners to guarantee the powersystem security which is defined as the ability of the power system to withstand a set of realisticcontingencies without violating acceptable operating limits [97] These contingencies may leadto changes in power flows and bus voltages that affect the system maximum loadabilities thussingle-element and multiple-element contingencies are used by system planners and operatorsto guarantee the system well-being Typically an N-1 contingency criterion is used to analyzethe system and prevent insecure operation this criterion states that for any single componentfailure of a system with n components the system can still reliably operate However multiple

27

Figure 25 Conventional power flow technique

contingencies that are likely to occur are also considered One important concept used to securelyoperate the system is the Available Transfer Capability (ATC) which is described next

Available Transfer Capability

Transfer capability refers to the ability of the power system to reliably transfer power betweentwo areas [98] These areas may be those formed by individual electric systems power pools

28

control areas or subregions The Available Transfer Capability is computed as follows [98]

ATC=TTC-TRM-ETC-CBM (216)

where

bull TTC or Total Transfer Capability is the maximum loadability of the system defined as theminimum of the thermal voltage or stability limits for the worst N-1 contingency

bull TRM or Transmission Reliability Margin takes into account uncertainties associated withother contingencies and is usually assigned a fixed value as a percentage of the TTC orcomputed using probabilistic methods

bull ETC or Existing Transmission Commitments denotes the current power transferred be-tween the concerned areas

bull CBM or Capacity Benefit Margin is the reserve required by load-serving entities to meettheir generation reliability requirements

223 Transient Stability

Transient stability deals with ability of the power system to remain in synchronism when sub-jected to large and sudden disturbances [88] The severity of this disturbance makes it impossibleto linearize the system dynamic model equations thus in practical systems numerical analysistechniques are commonly used to solve the corresponding nonlinear equations

The general form for the nonlinear Differential Algebraic Equations (DAEs) that describe thesystem behavior is as follows [99ndash106]

xD(t) = fD(xD y pD λ) for xD(0) = xDIC (217)

0 = g(xD y pD λ) for y(0) = yIC (218)

where xD represents the state variables such as generator internal angles and speeds y refers toalgebraic variables such as static load voltages and angles pD stands for controllable parameters

29

such as AVR set points and λ represents uncontrollable parameters such as load levels A differ-ent set of (217) and (218) is needed when a discrete variable change occurs which correspondsto discontinuities in the algebraic variables associated with limits and external disturbances Forinstance when a transmission line is tripped at the instant of line trip there is a discontinuitywith the algebraic variables showing instantaneous step changes while the state variables remainunchanged

The network model can be written in terms of algebraic equations for power balance or interms of algebraic equations for current balance The complex power balance model has thefollowing form

S (xD y pD λ) minus V Ylowast(xD y pD)V

lowast= 0 (219)

On the other hand the current balance form is obtained by dividing both sides of the previousequation by V and taking the complex conjugate which results in

I(xD y pD λ) minus Ylowast(xD y pD)V

lowast= 0 (220)

Depending on the study the loads could be treated as constant power or constant impedanceloads The former is suitable for simulation times less than the under load tap changer time con-stants and the latter is used for long-term stability studies where the effect of load tap changersis important

There are different approaches for the numerical solution of these DAEs These approachesare generally classified into two main groups namely partitioned and simultaneous approachesas described next

Partitioned Approach [100ndash102]

The general procedure of the partitioned approach requires finding the exact solution of the al-gebraic equations and then using this solution to numerically integrate the differential equationsThe solution of the differential equations is then used to solve the algebraic equations again andthe cycle is repeated until the stopping criterion is met In this procedure there is a delay between

30

the algebraic equations and the differential equation solutions In order to overcome this delayan iterative partitioned approach is used which is based on the reintegration of the differentialequations at each time step until convergence of all variables is achieved With the aim of re-ducing the computational cost that involves the exact solution of the algebraic equations at eachreintegration step extrapolated values for the solution of the algebraic equations are generallyused Moreover further simplifications are achieved if a single iteration for the differential andalgebraic equation solutions is taken per cycle

The most evident advantage of the partitioned approach is the possibility of choosing differentmethods for solving the differential and the algebraic equations Thus the approach can beapplied either to implicit or explicit integration methods In the implicit integration methods thedifferential equations are algebraized and then solved as a whole while in the explicit integrationmethods each differential equation is individually solved

Simultaneous Approach [100ndash102]

In the simultaneous approach all the equations namely differential and algebraic equations aresimultaneously solved Therefore a single numerical method is used together with an implicitintegration method The main advantage of the simultaneous approach is the elimination of thecomputation delay between the algebraic and the differential equations at the cost of a lowercomputational speed when compared to the partitioned approach However superior numericalstability is achieved

Numerical Methods for Integration

The available numerical methods for solving the set of DAEs offer differing advantages anddisadvantages in terms of complexity and numerical stability Numerical stability is a measureof the ability of the method to avoid error accumulation over the whole process These errors canoriginate from truncation roundoff finite time increments or specific errors introduced by themethod itself The impact of these errors on the accuracy of the final results is highly dependenton the system stiffness which refers to the ratio of the maximum and minimum time constantsof the model [100ndash102] For systems with high stiffness numerical integration methods with

31

low numerical stability produce inaccurate results unless the integration time step is sufficientlysmall If an integration method with a better numerical stability is used the error propagation isnot as much as for a method with lower numerical stability and thus the integration time stepcan be increased

Some of the most common explicit integration methods used in transient stability analysisare the Euler methods while one of the most common implicit integration method is the trape-zoidal method Since the trapezoidal method offers good numerical stability it is used in thisthesis to formulate the proposed AA-based transient stability method This method is based onan approximated solution of the integrals by means of trapezoids whose areas can be readily cal-culated [99 101] A graphical representation of this method is depicted in Figure 26 where thedefinite integral between to and t1 is approximated by the area of the corresponding trapezoid thesame applies for the definite integral between t1 and t2 and so on Thus the trapezoidal methodyields the following numerical discrete solution for (217) and (218)

xDn minus xDnminus1 minus (∆t2)( fD(xDnminus1 ynminus1 pD λ tnminus1) + fD(xDn yn pD λ tn)) = 0 (221)

g(xDn yn pD λ) = 0 (222)

In this case the solution for xDn and yn not only depends on the solution at the previous timeinstant xDnminus1 and ynminus1 but also on xDn and yn themselves Thus the Newton method is used toiteratively solve the resultant system of equations This approach facilitates the handling of DAEmodels since all nonlinear equations can be solved simultaneously

23 Self-validated Numerical Methods for Uncertainty Anal-ysis

Self-validated numerical methods are numerical methods that automatically keep track of thesources of error in such a way that the computed variables are bounded [107108] These sources

32

Figure 26 Trapezoidal method

of error can be internal or external The external sources of error typically correspond to mea-surement errors or uncertainties while the internal sources of error are introduced within thecomputation itself such as errors caused by truncated series rounded arithmetic finite stepsetc The term self-validated refers to the fact that all sources of error are taken into account inthe process of computing the variables yielding guaranteed enclosures for these variables Theseenclosures are obtained after computations are performed thus they are initially unknown Twoself-validated numerical methods are described next in this section namely Interval Arithmeticand Affine Arithmetic

231 Interval Arithmetic

Interval Arithmetic defines the real variables as intervals of floating point numbers [107] Themain mathematical operations and functions can be applied to these intervals resulting in new

33

intervals that enclose the output variables The mathematical formulation for the intervals is asfollows

u = [umin umax] u isin R umin le u le umax (223)

The bounds umin and umax can be infinite

The basic operations of addition subtraction multiplication and division are straightforwardin IA these are based on finding the maximum and minimum values for the output intervals asfollows

uA + uB = [uAmin uAmax] + [uBmin uBmax] = [uAmin + uBmin uAmax + uBmax] (224)

uA minus uB = [uAmin uAmax] + [uBmin uBmax] = [uAmin minus uBmax uAmax minus uBmin] (225)

uAuB = [uAmin uAmax] times [uBmin uBmax] = [upmin upmax] (226)

where

upmin = minuAminuBmin uAminuBmax uAmaxuBmin uAmaxuBmax

upmax = maxuAminuBmin uAminuBmax uAmaxuBmin uAmaxuBmax

AnduA

uB=

[uAmin uAmax][uBmin uBmax]

= [udmin udmax] (227)

where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34

These operations can be coded for use in various applications considering particular caseswhere the bounds for uA or uB are defined by infinite or zero values The volume of computationsfor multiplication and division can be reduced by coding a routine that checks the signs of thebounds

Some of the monotonic mathematic functions such as logarithms exponential and squareroots are also simple to implement in IA The only consideration that is usually required is thefiltering of the input intervals according to the domains associated with each function Particularattention should be paid to sinusoidal functions which can be non-monotonic depending on theinterval selected For intervals where the function is non-monotonic the local minimum andmaximum have to be found whereas for monotonic intervals the minimum and maximum arelocated at the end-points

The main drawback of IA is the representation of the variables as independent intervals Thisissue is commonly referred to as the dependency problem and is the main factor leading toexcessively conservative results which can be impractical in many applications This problemis accentuated when there is a long chain of operations involved In this case IA is prone toproduce error explosion at the final calculation of the chain Some measures can be useful insome cases to avoid error explosion for instance long operation chains can be rearranged inthe most efficient way in order to produce thinner intervals and interval splitting can also beapplied However these measures become more difficult to implement when the operation chainlength increases

232 Affine Arithmetic

Affine Arithmetic is a numerical analysis technique where the variables of interest are representedas affine combinations of data uncertainties andor approximation errors AA was proposedin [107 108] and a similar approach was previously proposed in [109] It is intended to reducethe excessively conservative width of the output intervals that arise in some cases by IA Thisinterval width reduction is achieved in AA by tracking the correlation between the computedquantities which is not possible in IA However AA requires more complex operations andlarger computational cost than IA [107 108]

35

Affine forms are characterized by a linear equation which has the following general expres-sion [107 108]

u = uo + u1εu1 + u2εu2 + + unuεunu (228)

The noise symbol coefficients u1 u2 unu are calculated as the partial derivatives of the vari-able u with respect to each of the independent components associated with the noise symbolsεu1 εu2 εunu By definition the noise symbols can vary within the interval [-1 1]

In general affine forms for the different variables under study can fully or partially sharethe noise symbols This characteristic allows reduction of the width of the AA output intervalswhen compared to IA-based methods However AA-based techniques deal with the issue ofapproximating nonlinear functions by adding an independent noise symbol representing the errorof the approximation Since most of the applications in power system analysis deal with longchains of nonlinear operations such as multiplications and sinusoidal functions the next sectionsfocus on the approximation methods of the affine representation of these non-affine operationspresenting some basic affine operations

Sum of Affine Forms

The summation or subtraction of two affine forms given by uA = uAo +uA1εA1 +uA2εA2 ++uAnaεAna

and uB = uBo + uB1εB1 + uB2εB2 + + uBnbεBnb is defined as

u = uA plusmn uB = uAo plusmn uBo plusmn

nasumra=1

(uAraεAra) plusmnnbsum

rb=1

(uBrbεBrb) (229)

If the considered affine forms uA and uB share all the noise symbols (229) can be simplified asfollows

u = uA plusmn uB = uAo plusmn uBo +

nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36

Multiplication of Affine Forms

The multiplication of two affine forms uA = uAo + uA1εA1 + uA2εA2 + + uAnaεAna and uB =

uBo + uB1εB1 + uB2εB2 + + uBnbεBnb is given by

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1

uArauBrbεAraεBrb (231)

This multiplication operation produces quadratic terms for the noise symbols which make ua non-affine form This issue is solved by means of two commonly adopted approaches theChebyshev approximation and the trivial approximation [107 108 110 111] The Chebyshevapproximation produces the highest accuracy with a high computational cost while the trivialapproximation is computationally efficient but exhibits a higher error

Chebyshev Approximation [110]

Chebyshev approximations for the multiplication of uA and uB are based on finding the best affineapproximation for the quadratic terms

sumnara=1

sumnbrb=1 uArauBrbεraεrb This approximation denoted by

ΨCh can be written as

ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)

where Umax and Umin are the bounds of the product of the zero centered affine forms KA = uAminusuAo

and KB = uB minus uBo The computation of Umax and Umin is not straightforward as it requires thetracking of the extreme values across the edges and intersection points of a convex polygonrepresenting the bounded domain of the affine forms KA and KB Thus Figure 27 shows anexample of a convex polygon that bounds the joint combination of affine forms uA and uB Thenumber of edges E is at most 2mA + 4 hence the greater the number of shared noise symbolsbetween uA and uB the longer is the computation time required to numerically obtain Umin andUmax

37

Figure 27 Convex polygon defining the bounds of KA and KB bounds for mA = 4

Trivial Affine Approximation

The trivial affine approximation is the most computationally efficient method to approximate themultiplication of two affine forms however it is not as accurate as the Chebyshev approximationAccording to this method the quadratic terms are conservatively approximated as follows [107108]

nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)

By substituting (233) into (231) one gets

38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)

As in the previously studied approximations the new noise symbol εξ is independent of theoriginal noise symbols and can exhibit a maximum error four times greater than the error givenby the Chebyshev approximation [107 108]

Various approaches seeking to improve the accuracy of the trivial approximation are reportedin [111] For instance the so called quadratic form is based on the inclusion of new quadraticnoise symbols into the trivial affine approximation These new quadratic noise symbols improvethe accuracy of the approximations however the computational efficiency is reduced comparedto the trivial affine approximation Since the trivial affine approximation offers a good trade off

between accuracy and computational cost this approach is used in this thesis to approximate themultiplication of affine forms

Chebyshev Orthogonal Polynomial-Based Approximations

Sine and cosine functions as well as other non-affine functions such as exponential functions areapproximated in this thesis using a series of Chebyshev orthogonal polynomials as follows

Hc(zch) =

npsumrh=0

brhTrh(zch) (235)

where the Chebyshev polynomials Trh(zch) are computed using the following recurrence relation[112]

T0(zch) = 1

T1(zch) = zch

Trh+1(zch) = 2zchTrh(zch) minus Trhminus1(zch)

(236)

The number of polynomials np in (235) are truncated depending on the number of digit

39

accuracy Thus for 5-digit accuracy as in this thesis the error of the Chebyshev approxima-tion is 10minus5 This error is attributed to the truncation of the Chebyshev series only Becausethe polynomials of this approximation involves non-affine operations additional errors are in-troduced [108] therefore the final affine forms of the sine and cosine operations exhibit errorsgreater than those attributed to the Chebyshev approximations only

24 Monte Carlo Simulations

MCS refers to the repetitive evaluation of deterministic problems such as mathematical expres-sions or physical models by sampling random variables [113114] This method has been appliedin many fields such as optimization evaluation of definite integrals estimation of statistical pa-rameters and in general in problems where an analytical solution is intractable One of theadvantages of MCS is their ability to model system complexities subjected to many uncertain-ties which are modeled using pdfs The pdfs are sampled to evaluate deterministic modelsand the obtained results are lumped in order to compute parameters such as mean and varianceof the concerned variables andor obtain their corresponding pdfs which provide informationregarding the likeliness of a random variable to take on different values

If a random variable is discrete the pdf is referred to as discrete probability density functionIn this case the pdf of the different values of the discrete random variable are associated witha probability On the other hand if a random variable is continuous the pdf is referred to ascontinuous probability density function and the information regarding the probability of therandom variable to lie within an interval is given by the integral of the associated pdf over thisinterval as follows

Prob[X f a le X f le X f b] =

int X f b

X f a

PX f (x f ) dx f (237)

Besides the mean variance and other statistical parameters a qualitative analysis of the pdfscan be achieved computing their skewness and excess kurtosis The skewness of a pdf is ameasure of its asymmetry with respect to its mean value and the excess kurtosis represents ameasure of its peakiness

40

Conventional MCS are used in this thesis as benchmark for comparison purposes since bydefinition it does not produce any spurious trajectories and for a large enough number of trialsthe union of the uncertainty region described by MCS is a very close approximation of the correctproblem solution To demonstrate its application in this thesis an example of MCS for systemstability analysis is presented next based on the Power System Analysis Toolbox (PSAT) [115]which is used to perform stability studies with variable wind power generator in the IEEE 14-bussystem shown in Figure 28 This benchmark system described in detail in [116] comprises twosynchronous generators connected at Buses 1 and 2 and three synchronous condensers connectedat Buses 3 6 and 8 The synchronous generator at Bus 2 is replaced by an aggregated DFIG-based wind turbine as shown in Figure 29 the transformers from 480 V to 25 kV and from 25kV to 69 kV as well as the impedance of the collector system in a wind farm for better modellingaccuracy A brief description of the DFIG model and associated controls as well as the relevantsystem data can be found in [117] For all simulations the loads are modeled using exponentialrecovery models

The wind speed distribution is modeled using the typical Rayleigh pdf used for these pur-poses which is expressed in terms of the wind speed v and mean wind speed vm as follows [118]

PX f (v) =2v

1128vmev

1128vm2 (238)

Figure 210 (a) shows the windrsquos pdf for a mean wind speed of 10 ms This function is usedto approximate the stochastic nature of the considered wind speed Figure 210 (b) shows theturbine output power as a function of the wind speed The rated wind speed is 15 ms while thecut-in and cut-out wind speeds are 5 ms and 25 ms respectively When the wind speed is abovethe rated speed the pitch angle controller comes into effect in order to maintain the output powerconstant at its rated value Moreover for wind speeds above the cut-out wind speed (which is notvery likely according to Figure 210 (a)) the wind turbine is prevented from generating power inorder to avoid any damage and for wind speeds under 5 ms the wind power output is null

Three study cases are presented and discussed here

bull In Case A the maximum loadability variations of the above described system for the windfarm with terminal voltage control are assessed by using MCS

41

Figure 28 IEEE 14 bus system in PSAT

bull In Case B MCS are used to compute the maximum system loadability variations for thewind farm with constant power factor mode

bull In Case C the variations in the damping ratio of the dominant mode are assessed usingMCS for the wind farm with terminal voltage control

Normal and contingency operating conditions are considered in the above described cases The

42

Figure 29 DFIG collector system in PSAT

Figure 210 (a) Wind speed pdf and corresponding (b) power output characteristics of a windturbine

transmission line connected between Buses 2 and 4 is disconnected to simulate a contingency

Case A

The static load margins obtained using PV curves for the wind turbine in terminal voltage controlmode are bounded by a minimum value of 42492 MW and a maximum value of 43146 MWwhen the wind speed varies according to the specified pdf The probabilities associated with

43

loading margins within these bounds are depicted in Figures 211 (a) and (b) The occurrence ofthese static load margins are associated with the probabilities of having a wind speed below thecut-in speed and above the cut out speed and the probability of having a wind speed within therated and cut-out speeds respectively When the line 2-4 is tripped the static load margins lie inthe interval of 38788 MW and 38332 MW for wind power output variations within 45 MW and0 MW The probabilities associated with loading margins within these bounds are illustrated inFigures 211 (c) and (d)

The observed pdfrsquos peaks are due to the relatively high probabilities associated with havingeither zero output power for the wind farm (left peak) or nominal output power (right peak) Thisis mainly due to the Rayleigh pdf assumed to model the wind speed stochastic behavior sinceaccording to this pdf there exists a relatively high probability of having a wind speed below thecut-in wind speed or above the cut-out wind speed and also a high probability associated withhaving a wind speed within the nominal speed and the cut-out wind speed These heavy taileddistributions lead to a high kurtosis which is 741e5 and 229e6 for normal and contingencyconditions respectively

The mean and standard deviations for the maximum loadability in normal operating conditionare 42766 MW and 263 MW respectively with a skewness of 0384 This positive skewnessindicates that the distribution is more stretched on the side above the mean For contingencyconditions the mean and standard deviations are 38531 MW and 182 MW respectively theskewness in this case is 023 and thus the distribution is more stretched on the side above themean

Case B

When the DFIG is operated at constant power factor mode the static load margins in normaloperating condition are within the interval given by [41930 42411] MW which is smaller thanthe system with voltage control wind turbine as expected Moreover the static load margininterval is reduced even further under contingency conditions to the interval [3787 3824 MW]

The pdfs and cdfs in this case are very similar to those shown in 211

44

Case C

Figure 212 (a) depicts the cdf of the dominant mode damping ratio under normal operatingconditions for a loading level of 3315 MW Participation factors associated with this dominantmode correspond to the AVR of the synchronous generator connected at Bus 1 When the windpower output varies within 0 MW and 45 MW the dominant-mode damping ratio is boundedby the interval given by [112 33] in this case the probability of occurrence of a dampingratio lower or equal than 3 is approximately 082 On the other hand the damping ratiosunder contingency conditions vary within the interval [059 26] from Figure 212 (b) theprobability of occurrence for a damping ratio lower than 2 is approximately 077

Figure 213 (a) shows the trajectories for the dominant-mode damping ratios associated withthe AVR of the synchronous generator connected at Bus 1 As the wind power output levelincreases the damping ratio improves not only for normal operating conditions but also forcontingency conditions Figure 213 (b) shows the eigenvalues associated with the rotor speedof the synchronous generator connected at Bus 1 In this case the damping ratio decreases whenthe wind power output level rises however the minimum damping ratio is sufficiently large toassure system stability

25 Summary

The basic tools for power flow long-term voltage stability and transient stability analysis havebeen briefly discussed in this chapter Particularly for voltage stability assessment differentstrategies commonly used for the computation of the maximum loadability were presentednamely direct methods and continuation power flow techniques Also the definitions associ-ated with ATC computation were briefly discussed

Relevant aspects concerning the formulation and solution of the DAEs that describe the sys-tem dynamics were also addressed in this chapter specifically the trapezoidal method was brieflydescribed this method offers good numerical stability and thus is considered in this thesis forthe formulation of the AA-based transient stability assessment

45

Self-validated numerical methods for uncertainty analysis were presented as well Specif-ically the main characteristics of IA and AA were pointed out It was explained that IA ischaracterized by its simplicity and low computational cost however its inability to properly rep-resent the dependency among variables may lead to excessively conservative results making itimpractical On the other hand AA was shown to represent the correlations among variableswhich may lead to better results than IA however non-linear operations and functions in AAhave to be approximated by means of affine forms which affect its accuracy In this regard someof the available techniques intended to approximate non-affine operations and functions werediscussed

Finally MCS were briefly discussed and their application to voltage and transient stabilityassessment of a test system considering uncertainties associated with wind power generation wasdemonstrated

46

Figure 211 Pdfs and cdfs for the static load margins when the wind farm is operating in voltagecontrol mode (a) pdf and (b) cdf in normal operating conditions (c) pdf and (d) cdf for acontingency

47

1 15 2 25 30

02

04

06

08

1

DominatminusMode Damping Ratio ()(a)

Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1

DominantminusMode Damping Ratio ()(b)

Cum

ulat

ive

Pro

babi

lity

Figure 212 Cdfs of dominant-mode damping ratio for the DFIG wind turbine operating involtage control mode (a) normal operating conditions and (b) contingency conditions

Figure 213 Eigenvalues for synchronous generator at Bus 1 for different wind power outputswhen the DFIG is operated in voltage control mode (a) AVR relevant modes and (b) rotor speedrelevant modes

48

Chapter 3

Affine Arithmetic Based Power FlowAssessment

31 Introduction

This chapter discusses the formulation of the AA-based power flow analysis method basedon [14] proposing a novel modeling of reactive power limits The power flow equations areformulated in both rectangular and polar form for comparison purposes and simulations are car-ried out using the IEEE-30 Bus test system and assuming uncertainties associated with loadsand generation which are modeled as intervals

32 Problem Formulation

AA is used in [14] to solve the power flow problem considering uncertainties in the load andgeneration levels According to this approach the initial affine approximation for bus voltagemagnitudes and angles can be written as follows

bull In polar form [14]

49

Vi = Voi +sumjisinNP

partVi

partP j

∣∣∣∣∣o∆PA jεP j +

sumjisinNQ

partVi

partQ j

∣∣∣∣∣o∆QA jεQ j foralli isin NP (31)

θi = θoi +sumjisinNP

partθi

partP j


sumjisinNQ

partθi

partQ j

∣∣∣∣∣o∆QA jεQ j foralli isin NP NQ (32)

bull In rectangular form (new)

VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j


The number of noise symbols in (31) to (34) depends on the number of uncertainties to beanalyzed For instance if only uncertainties of generation outputs are considered the numberof noise symbols will be restricted to the number of generators It is important to point out thatcorrelations amongst generators as in the case of close-by wind farms can be easily representedin (31) to (34) by means of a single noise symbol associated with the correlated variations ofpower of these wind farms The coefficients of the noise symbols given by the derivatives ofvoltage magnitudes and angles with respect to the injected active and reactive power can becalculated by using the Jacobian matrix or numerical sensitivity studies These derivatives areevaluated at the central values of the active and reactive powers injected at PQ buses and theactive powers injected at PV buses which are defined as follows

Poi =PGmaxi

minus PLmaxi+ PGmini

minus PLmini

2foralli isin NP (35)

Qoi =QGmaxi

minus QLmaxi+ QGmini

minus QLmini

2foralli isin NQ (36)

The prediction intervals used to model the uncertainty associated with the predicted valuescan be obtained by using non-parametric approaches which do not need any assumptions re-garding the pdf of forecast errors and thus are very appropriate to compute the intervals of the

50

uncertainties associated with intermittent renewable energy sources [119] These intervals areassociated with a confidence level thus depending on this confidence level parts of the possibleoutcomes of the uncertain variables are necessarily neglected Parametric approaches can alsobe used to compute the bounds associated with the forecasting uncertainty [120] however thesemethods rely on assumptions regarding the pdfs of the forecasting errors that may be unrealisticin some cases Since in this thesis the uncertain variables are modeled as prediction intervalsthe expected predictions are not relevant

Numerical sensitivity studies are used in this thesis to obtain an approximation of the deriva-tives in (31) to (34) using a small perturbation of the system around the central values of theuncertain parameters and then solving the standard power flow problem to compute the cor-responding changes in the respective variables This requires as many power flow solutionsas uncertain variables which in the case of large wind and solar power plants are not manyComputing the changes to approximate derivatives leads to more accurate results than using theJacobian matrix where the derivatives are evaluated at the central values

The numerical sensitivity approach yields initial approximations for the affine forms of thepower flow variables In order to compute the final affine forms and their associated bounds theinitial affine approximations obtained using the sensitivity approach are corrected based on thesolution of the Linear Programming (LP) problems described later in this Chapter These LPproblems account for the physical restrictions imposed by the power flow equations generatorlimits as well as the bounds of the noise symbols Also aimed at accounting for the nonlinear-ities of the power flow equations additional noise symbols are introduced by the approximationof non-affine operations [107] Thus the resultant final affine forms are different to the initialapproximations obtained using the sensitivity approach

The active and reactive power affine forms Pi and Qi are obtained by replacing the voltageand angle affine forms given in (31) and (32) in the following equations

Pi =

Nsumj=1

ViV j

[Gi j cos(θi minus θ j) + Bi j sin(θi minus θ j)

]foralli isin NP (37)

Qi =

Nsumj=1

ViV j

[Gi j sin(θi minus θ j) minus Bi j cos(θi minus θ j)

]foralli isin NQ (38)

51

Alternatively the power flow equations in rectangular coordinates can be written as follows

Pri =

Nsumj=1

(VRi

(VR jGi j minus VI j Bi j

)+ VIi

(VI jGi j + VR j Bi j

))foralli isin NP (39)

Qri =

Nsumj=1

(VIi


)minus VRi


))foralli isin NQ (310)

The affine forms for the active (37) or (39) and reactive powers (38) or (310) exhibit thefollowing compact form after all affine functions operations and approximations

Aε = C (311)

C = L minus (Ro + q) (312)

The maximum and minimum values of C are given by the corresponding maximum and minimumvalues of active and reactive power injections as follows

Cmax = Lmax minus Ro + |q| (313)

Cmin = Lmin minus Ro minus |q| (314)

where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)

Pmax = PGmax minus PLmin (317)

52

Pmin = PGmin minus PLmax (318)

Qmax = QGmax minus QLmin (319)

Qmin = QGmin minus QLmax (320)

Equation (311) is a linear system which can be solved for ε by means of the two followingLP problems

minsumεslmin

st minus1 le εslmin le 1

Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax

st minus1 le εslmax le 1

Cmin le Aε le Cmax

forallsl isin NP NQ

(322)

The values obtained for εslmin and εslmax from these solutions are used in (31) and (32) or al-ternatively in (33) and (34) to calculate the minimum and maximum voltage magnitudes andangles respectively A contraction domain technique is proposed in [14] to produce narrowerintervals for the voltage and angle bounds According to this formulation the two optimizationapproaches are consecutively repeated and the coefficients for the voltage and angle affine formsare contracted at each iteration until convergence is achieved

After the voltage and angle bounds are computed the interval of maximum and minimumreactive powers generated at the PV buses and slack bus can be calculated as follows in order toverify any limit violations

53

QGmaxi= qpoi +

sumslisinNp

qpislεslmax +sumslisinNq

qqislεslmax + |qei|εhi + QLmaxiforalli = 1 NPV + 1 (323)

QGmini= qpoi +

sumslisinNp

qpislεslmin +sumslisinNq

qqislεslmin minus |qei|εhi + QLminiforalli = 1 NPV + 1 (324)

In case of a limit violation the associated bus or buses are treated as PQ buses as in standardpower flow methods and the methodology explained before is repeated until no more limit vi-olations are observed A similar procedure can be used to compute the generator active powerinterval at the slack bus

33 Improved Reactive Power Limit Representation

According to the AA-based power flow approach proposed in [14] and previously describedthe power flow equations are initially solved without limits and in case of a limit violation theassociated bus or buses are treated as PQ buses as in standard power flow methods applyingagain the procedure to compute the affine forms for voltages and angles this is repeated until nomore limit violations are observed A different approach is proposed in this thesis to account forgenerator reactive power limits based on the following polar formulation of voltages and anglesin affine form [121 122]

Vi = Voi +sum

misinNIG

partVi

partPGm

∣∣∣∣∣o∆PGA mεIGm +

sumncisinNPV

partVi

partVnc

∣∣∣∣∣o∆VAnc

(εPVanc minus εPVbnc

)+

sumnlisinNPL

partVi

partPLnl

∣∣∣∣∣o∆PLA nlεPLnl

+sum

nlisinNQL

partVi

partQLnl

∣∣∣∣∣o∆QLA nlεQLnl

foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)

where the noise symbols εPVa and εPVb are introduced to account for the reactive power limitsof generators and thus avoid the need for PV-PQ bus switching This approach is based on therepresentation of PV buses discussed in Chapter 2 which yields the following formulation

Vpvnc = Vopvnc+ ∆VAnc


)(327)

where0 lt εPVanc lt 1 εPVbnc = 0 for QG lt QGmin

εPVanc = 0 0 lt εPVbnc lt 1 for QG gt QGmax

εPVanc = 0 εPVbnc = 0 for QGmax le QG le QGmin

forallnc isin NPV (328)

the noise symbols εPVanc and εPVbnc in (327) act as the auxiliary variables Vanc and Vbnc in (27) tokeep the reactive power within limits The effect of the variation of the voltage specified at PVnodes due to these noise symbols is accounted for in (325) and (326) by means of the partialderivatives of the bus voltages and angles with respect to the voltage reference set points at PVbuses This approach is implemented using the Linear Programming (LP) formulations describedin detail at the end of this section

The coefficients of the noise symbols in (325) and (326) given by the partial derivatives ofvoltage magnitudes and angles with respect to the injected active power and reference voltages atvoltage controlled buses can be calculated from a ldquobaserdquo power flow solution and the numericalsensitivity analysis described in Section 32 Since the derivatives of the initial affine forms areapproximated by computing the changes in the respective variables using small perturbationsill-defined derivatives that arise when reactive power limits are reached at a power flow solutioncorresponding to the central values can be avoided The central values are defined as follows

55

PGo m =PGmaxm

+ PGminm

2forallm isin NIG (329)

PLo nl =PLmaxnl

+ PLminnl

2forallnl isin NL (330)

QLo nl =QLmaxnl

+ QLminnl


The active and reactive power affine forms Pi and Qi are obtained by replacing the voltageand angle affine forms given in (325) and (326) in (37) and (38) resulting in the followingequations after after all affine function operations and approximations given in [107 108]

Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh

PAihεAh foralli = 1 N

(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh

QAihεAh foralli = 1 N

(333)

These equations may be written in compact form as follows

ALεL = CL (334)

CL = LL minus (RLo + qL) (335)

where

56

AL =

PNIG11 middot middot middot PNIG

1NIGP

NPL11 middot middot middot P

NPL1NPL

PNQL11 middot middot middot P

NQL1NQL

PNPV11 middot middot middot PNPV

1NPVminusPNPV

11 middot middot middot minusPNPV1NPV

PNIGNminus11 middot middot middot P

NIGNminus1NIG

PNPLNminus11 middot middot middot P

NPLNminus1NPL

PNQLNminus11 middot middot middot P

NQLNminus1NQL


1NPVminusPNPV

Nminus11 middot middot middot minusPNPVNminus1NPV

QNIG11 middot middot middot QNIG

1NIGQ

NPL11 middot middot middot Q

NPL1NPL

QNQL11 middot middot middot Q

NQL1NQL

QNPV11 middot middot middot QNPV

1NPVminusQNPV

11 middot middot middot minusQNPV1NPV

QNIGNminus11 middot middot middotQ

NIGNminus1NIG

QNPLNminus11 middot middot middotQ

NPLNminus1NPL

QNQLNminus11 middot middot middotQ

NQLNminus1NQL

QNPVNminus11 middot middot middotQ

NPVNminus1NPV

minusQNPVNminus11 middot middot middotminusQNPV

Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][

PGminNminus1minus PLmaxNminus1

][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][

QGminNminus1minus QLmaxNminus1

][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =

PA11 middot middot middot PA

1Nh

PANminus11 middot middot middot PA

Nminus1Nh

QA11 middot middot middot QA

1Nh

QANminus11 middot middot middot QA

Nminus1Nh

εA1

εANh

To obtain the maximum and minimum values of the noise vector ε in (334) the following LPproblems need to be solved

minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc

st minus1 le εIGminm εPLminnl

εQLminnlle 1

0 le εPVaminnc εPVbminnc

le 1

CLmin le ALεLmin le CLmax

forallm isin NIGforallnc isin NPV

(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc

st minus1 le εIGmaxm εPLmaxnl

εQLmaxnlle 1

minus1 le εPVamaxnc εPVbmaxnc

le 0

CLmin le ALεLmax le CLmax


(337)

The values of εIGmin εPLmin εQLmin

εPVamin εPVbmin and εIGmax εPLmax εQLmax

εPVamax εPVbmax obtainedby solving these simple LP problems are used in (325) and (326) to compute the maximumand minimum bus voltages and angles Notice that the objective functions in (336) and (337)are mathematically formulated to contract the vector ε according to the physical restrictionsimposed by the power flow equations and the bounds of the noise symbols thus there is noparticular physical meaning attributed to the objective functions Since (336) corresponds to aminimization problem and the bounds of εPVaminnc

and εPVbminncare restricted to the interval [01]

these variables will tend to zero unless there is any reactive power limit violation Similarly

58

because in (337) the objective function is formulated as a maximization problem and the boundsof εPVamaxnc

and εPVamaxncare restricted to the interval [-10] these noise symbols tend to zero

unless a reactive power limit is violated On the other hand the noise symbols εIGminm εIGmaxm

εPLminnl

εPLmaxnl εQLminnl

εQLmaxnlwhich are related to the intermittent power injections are allowed

to vary within the interval [-11] which is the hull assigned to affine arithmetic variables

34 Example

The IEEE-30 bus benchmark system shown in Figure 31 is utilized for testing the solution ofthe AA-based power flow equations There are two synchronous generators connected at Buses1 and 2 and four synchronous condensers connected at Buses 5 811 13 respectively and thesystem base load is 2834 MW and 1262 MVAR The following three study cases are considered

bull Case A The generators output power and all the loads are allowed to vary within a rangeof plusmn35 with respect to their central values in order to stress the system The power flowequations are solved using both rectangular and polar representation of the power flowequations Generator reactive power limits are not taken into account in this case

bull Case B The generator output power and all the loads for Case A are allowed to vary withina range of plusmn10 with respect to their central values Generator reactive power limits arenot taken into account in this case

bull Case C The generator reactive power limits are considered and the generator output pow-ers and all the loads are allowed to vary within a range of plusmn15 with respect to their centralvalues The maximum reactive power of the generator at Bus 2 is set to 50 MVA and themaximum reactive powers of the generators at Buses 5 and 8 are set to 40 MVA

In the three cases described above and the rest of this thesis MCS are used for comparisonpurposes because of their flexibility and simplicity Moreover for a large enough number oftrials the union of the uncertainty region described by MCS is a very close approximation of thecorrect problem solution In these MCS a uniform pdf is used to represent the uncertainties inpower injections since it is the most appropriate pdf used to yield solution intervals [14]

59


The Trivial Affine Approximation is used to deal with the multiplications of affine forms andthe sine and cosine functions are approximated by using the Chebyshev Orthogonal Polynomialapproach

60

341 Case A

Figures 32 (a) and 32 (b) depict the upper and lower bounds of bus voltages and angles re-spectively when the power flow equations are formulated in rectangular form The maximumdiscrepancies of the AA solution with respect to MCS results are 194 for the voltage mag-nitudes and 162 for the voltage angles The computational time is 638 s for the AA-basedmethod and 1549 s for MCS (5000 iterations to guarantee convergence)

Aimed at illustrating the differences between the initial affine forms obtained using the sen-sitivity approach and the final affine forms obtained by solving the LP maximization problemthe initial and final affine forms of the real part of the bus voltage magnitude at bus 30 truncatedat the fifth noise symbol are presented as example

bull Initial affine form

VR30 = 09498 + 00007εL1 minus 00004εL2 minus 00001εL3 minus 00003 lowast εL4 minus 00033 lowast εL5

bull Final affine form

VR30 = 09498 + 00007εL1 minus 00004εL2 minus 00068e minus 2εL3 minus 00002 lowast εL4 minus 00022 lowast εL5

Notice that the initial affine form has been corrected resulting in a different final affine form

The bounds for the bus voltages and angles when the polar form representation is used toformulate the power flow equations are depicted in Figures 33 (a) and 33 (b) respectivelyIn this case the maximum errors obtained from the AA-based method with respect to MCS are227 and 3712 for bus voltage magnitudes and angles respectively Note that the rectangularcoordinates representation exhibits a better accuracy than the polar form representation this canbe attributed to the relatively high volume of multiplications associated with the expansion ofsinusoidal functions in the Chebyshev Orthogonal Polynomial approach The computationaltime is 1790 s in this case

61

Figure 32 Case A with rectangular coordinates (a) bus voltage magnitudes and (b) bus voltageangles

Figure 33 Case A with polar coordinates (a) bus voltage magnitudes and (b) bus voltageangles

62

Figure 34 Case B with rectangular coordinates (a) bus voltage magnitudes and (b) bus voltageangles

342 Case B

Observe in Figures 34 (a) and 34 (b) that the bus voltages and angles obtained with the AA-based method are almost superimposed on those obtained with MCS Indeed the maximumerrors for the voltage magnitudes and angles are 023 and 206 respectively which is asignificant improvement compared to Case A The computational time is 251 s for the AA-based method and 1664 s for MCS Figures 35 (a) and 35 (b) depicts the results obtained usingthe polar form representation of power flow equations In this case the maximum error for thevoltage magnitudes and angles are 184 and 180 respectively with a computational time of375 s These reduced errors are mainly due to the fact that the intervals of uncertainty have beenreduced by over 13 with respect to Case A

343 Case C

Figures 36 (a) and 36 (b) depict the bus voltage magnitude and angle bounds respectivelywhen generator reactive power limits are considered and the improved reactive power limit

63

Figure 35 Case B with polar coordinates (a) bus voltage magnitudes and (b) bus voltageangles

representation is used The maximum discrepancy of the AA-based method results with respectto MCS is 52 which shows good accuracy The Computational time in this case is 455s which is approximately half the computational time obtained when the bus type switchingstrategy is used to account for generator reactive power limits (898 s) The computational timeof MCS is 12616 s

Figure 37 shows bounds for reactive power in synchronous generators Observe that thesynchronous generator connected at Bus 2 and the synchronous condensers connected at Buses5 and 8 reach their upper reactive power limits

The accuracy of the proposed AA based methods is affected by the approximation of non-affine operations the size of the intervals that model the uncertain variables the number of un-certain variables and the size of the system being analyzed As demonstrated in the simulationsfor relatively small sizes of the intervals that model the uncertain variables the bounds obtainedusing MCS and AA are very close however when the size of these intervals increases the erroralso increases

64

Figure 36 Case C with polar coordinates (a) bus voltage magnitudes and (b) bus voltageangles

Figure 37 Generator reactive power intervals in Case C

65

35 Summary

In this chapter the AA-based power flow analysis was discussed in both polar and rectangularform and various study cases were presented to compare the performance of different solutionapproaches

A novel and more efficient way to deal with reactive power limits was also introduced Theimpact of the input affine intervals width on the AA-based power flow method accuracy wasinvestigated concluding that the AA-based power flow method exhibits a reasonable accuracyand is more computationally efficient than MCS Furthermore this method automatically boundsthe output variables according to the intervals that model the uncertain input variables withoutthe need to make assumptions regarding the pdf of the uncertainties The use of rectangular-coordinate equations in the AA-based power flow problem was also studied demonstrating to besignificantly more accurate than the polar-coordinate equations approach The basic principlesof this AA-based power flow approach is used in this thesis to formulate the AA-based voltagestability methods which is discussed next

66

Chapter 4

Affine Arithmetic Based Voltage StabilityAssessment

41 Introduction

In this chapter the proposed AA-based method for voltage stability assessment of power sys-tems considering uncertainties associated with operating conditions is discussedThe power flowequations in affine form based on the more efficient representation of generator reactive powerlimits are used to formulate the AA-based method for voltage stability assessment which yieldsthe hull of the PV curves associated with the assumed uncertainties A parametrization approachwhich allows obtaining feasible solutions for the whole hull of PV curves is presented and dis-cussed Finally the proposed method is tested using two study cases first a 5-bus test systemis used to illustrate the proposed technique in detail and thereafter a 2383-bus test system isstudied to demonstrate its practical application The results are compared with those obtainedusing MCS to verify the accuracy and computational burden of the proposed AA-based methodand also with respect to a previously proposed technique to estimate the impact of parametervariations in voltage stability assessment

67


The bus voltage magnitude and angle equations given by (325) and (326) are reformulated inthis section to consider only uncertainties due to intermittent sources of power as follows

Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)

In the proposed AA-based voltage stability method (211) can be written for dispatchablegenerators excluding the slack bus as follows


Similarly the load equations can be written as



whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)

Notice that the loading parameter λ is now in affine form and thus is written as a function of theuncertainty of the output power of intermittent generators This uncertainty is modeled as inter-vals that represent the maximum and minimum power injections associated with the simulationtime period as follows

PGm =[PGo m minus ∆PGA m PGo m + ∆PGA m

]forallm isin NIG (47)

68

Figure 41 PV curves (a) using the load power as the parameter (b) using a PQ-bus voltagemagnitude as the parameter

or equivalently in AAPGm = PGo m + ∆PGA mεIGm forallm isin NIG (48)

For a different period of time a different interval for the uncertain power injections may beneeded for instance ∆PGA may be chosen according to the forecasting error associated withthe concerned planning horizon On the other hand ∆VAnc may be decided so that there is asufficient margin for the variation of the reference voltage setting of PV buses in order to keepgeneratorrsquos reactive power within limits In this thesis ∆VAnc has been assumed to be a 10 ofthe pre-established reference voltage control settings for all study cases

For the computation of the central values and the upper and lower bounds of the PV curves aparametrization technique is used in this thesis Figure 41 (a) illustrates the problems when theload power is used as a parameter to compute the lower and upper parts of the PV curves It isnoted that varying the load by varying the power from PL1 to PL7 and computing the respectiveupper voltages Vu1 to Vu7 and lower voltages Vl1 to Vl7 based on AA or MCS can only be doneuntil the lower bound of the PV curve reaches the maximum loading point given by PL7 Beyondthis point the dotted parts of the PV curves cannot be readily computed To overcome thisproblem the voltage magnitude at a PQ bus is chosen as the parameter as in Figure 41(b) based

69

on a parametrization approach [90ndash93] Notice that the voltage magnitude at the chosen PQ busvaries from V1 to V9 and for each of these voltages the associated lower power loads PLl1 toPLl9 and upper power loads PLu1 to PLu9 can be readily computed The PQ-bus voltage of thebus that exhibits the largest variation in voltage magnitude with respect to load variations is usedas the parameter The affine form for the voltage of this bus can be written as

VPQ = Vp (49)

where Vp is updated for the computation of the PV curves as follows

Vp = Vp minus ∆Vp (410)

The step size ∆Vp varies depending on the proximity to the maximum loadability thus thecloser to this maximum the smaller the step size The proximity to maximum loadability isdetected by computing the difference between the former and the current loading factor intervalλ the lower this difference the closer the system is to a saddle node bifurcation Notice that forcomputations beyond the maximum loadability the loading factor starts decreasing and thus itis straightforward to detect when the maximum loadability has been reached

In standard PV computations the parametrization approach is implemented by augmentingthe power flow equations by one equation which refers to the value assigned to the voltagemagnitude chosen as the parameter Thus the set of equations to be solved is

fPF(θV λ)VPQ minus Vp

= 0 (411)

which is solved using a Newton approach as follows

fPF(θrVr λr)VPQ minus Vp

= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)

where r refers to the iteration number These equations are iteratively solved until convergenceis attained

The proposed PV-curve computation approach is based on the following AA form of powerflow equations

PGi minus PLoi minus λ∆PLi minus Pi = 0 foralli = 1 N (414)

QGi minus QLoi minus λ∆QLi minus Qi = 0 foralli = 1 N (415)

plus the additional equation corresponding to the parametrization approach

VPQ minus Vp = 0 (416)

where Pi and Qi are written according to (37) and (38) and PGi is given by (43) for dispatchablegenerators excluding the slack bus In case of non-dispatchable generators PGi is modeled as aninterval which represents the uncertainties associated with the output power as in (48)

Equations (414) and (415) exhibit the following compact form after all affine functionoperations and approximations

Avsεvs = Cvs (417)

Cvs = Lvs minus (Rvso + qvs) (418)

where

Rvso =

P1o + λo(∆PL1 minus ∆PG1)

PNo + λo(∆PLn minus ∆PGn)Q1o + λo(∆QL1 minus ∆QG1)

QNo + λo(∆QLn minus ∆QGn)

and Avs εvs Cvs Lvs and qvs have the form of the matrices used in (334) and (335) neglecting

71

the load uncertainties

Using a similar approach to the one previously described to account for generator reactivepower limits the LP formulation for the proposed AA-based voltage stability assessment is ob-tained by substituting CL by Cvs and AL by Avs in (336) and (337) and ignoring the noise symbolsfor load variations εPLmin

εPLmaxand εQLmin

εQLmax The values for εIGmin εPVamin εPVbmin and εIGmax

εPVamax εPVbmax obtained by solving the resultant LP formulations are used in (46) to compute theloadability interval λ Figure 42 depicts the algorithm used to compute the PV-curve intervalsThe stopping criteria for this algorithm is decided based on the number of solution points of thehull of the PV curves beyond the maximum loadabilities

43 Simulation Results

To test the proposed AA-based methodology for voltage stability assessment two test systemsare considered The first is a 5-bus test system [85] which allows a thorough test and demon-stration of the proposed methodology without the complexities associated with system size Thesecond corresponds to a portion of the Polish system and comprises 2383 buses [123] allowingto test and demonstrate the proposed approach in a realistic system To validate and comparethe results obtained with the proposed AA-based technique PV curves are computed using anMCS approach assuming uniform distribution for the non-dispatchable generators intervals andare thus treated as the benchmark for comparison purposes The convergence of MCS was de-termined by assuming a tolerance of 00001 in the change of the expected value The sensitivityformula proposed in [44] is also used to compute the maximum loadability intervals for compar-ison purposes

431 5-Bus Test System

This system comprises two generators which supply a base load of 440 MW One of these gener-ators is assumed to be an intermittent power source while the second is assumed to be dispatch-able and is the system slack bus For the computation of PV curves the load directions in (414)and (415) are ∆PLi = PLoi

and ∆QLi = QLoi

72

Start

Read system data

Write the power flow equations in

affine form (414)-(415)

Solve the linear programming problems

to compute and

Compute affine forms using (41)-(4-13)

Yes Print results End

Choose the PQ-bus voltage magnitude as varying parameter

Compute and using (44)

No Stop

Rearrange the power flow equations

according to (416) and (417)

Figure 42 Algorithm for the AA-based PV curve computation method

Table 41 shows the effect of increasing the size of the interval that models the uncertainvariable on the computed maximum and minimum load changes For comparison purposesMCS and the sensitivity formula (SF) results are also depicted (MCS required 50 samples toconverge) Notice that as the margin of variation associated with the uncertain power injectionincreases the error for the upper bound of the maximum loadabilities using AA and SF alsoincreases with the error corresponding to the AA-based method being slightly lower than theerror computed using the SF approach Also notice that the lower bound for the maximum

73

Table 41 Comparison of load changes using different approaches for the 5-bus test systemMargin ()

5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456

loadability is pessimistic for the AA-based approach and optimistic for the SF approach whichis an advantage of the proposed approach

Figure 43 shows the PV curves obtained using MCS and the proposed AA-based methodfor a 30 margin variation of the uncertain variable Notice that the AA-based approach allowsto efficiently compute the upper and lower bounds of the PV curves providing information re-garding the hull of voltage profiles No other technique proposed in the literature is capable ofquickly generating such curves which could be used by power system operators as standard PVcurves are used now [90] (eg to determine voltage profiles as the maximum system loadabilityis approached)


The system comprises 323 generators and 4 synchronous condensers supplying a base load of24558 MW A total of 50 generators with total capacity of 76772 MW (31 of the total systemgeneration capacity) are assumed to be intermittent power sources with 15 margin variationsThe size and number of uncertainties of this system allows evaluating the performance of theproposed AA-based voltage stability assessment for practical systems As in the previous ex-ample the following load and dispatch directions are assumed ∆PLi = PLoi

∆QLi = QLoiand

∆PGi = PGoiforalli

74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)

MC Lower BoundMC Upper BoundAA Lower BoundAA Upper Bound

Figure 43 PV curves for the IEEE 5-bus test system

The PV curves depicted in Figure 44 are computed using the proposed AA-based methodand MCS for normal operating conditions The probability distribution of the uncertain variablesin MCS are assumed to be uniform distributions requiring 2000 samples to attain convergenceObserve again that the proposed AA-based approach yields slightly pessimistic results

Table 42 shows the bounds for load changes obtained using MCS the proposed AA-basedmethod and the SF approach for the worst single line trip and single generator trip Becauseof the difficulties encountered in the computation of the Jacobian matrices the SF formula wasapproximated using the following equation

dλdp

∣∣∣∣∣oasymp

∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)


Figure 44 PV curves for the 2383-bus test system

with ∆p small enough to have a good approximation It is observed that the worst contingencyoccurs when Line 67-138 is tripped Observe that the errors in the values obtained using theproposed AA-based technique are lower than those obtained with the SF approach furthermorethere are no underestimation of the margins by the AA-based method as opposed to those ob-tained using the SF technique In this case the ATC for transmission line 31-32 which is themost loaded line in the system computed using (216) and assuming CBM=0 varies within theintervals [3183 7872] MW [2917 8256] MW and [2618 8531] MW for MCS the AA-based method and the SF approach respectively this results in a maximum error for the AA-

76

based approach of 837 while the maximum error for the SF approach is 1777 Notice thatthe difference between the loading changes computed using MCS and the AA-based approachtends to increase as the size of the system and number of uncertain variables increases Thisresult is to be expected because as the number of uncertain variables and operations increasethe approximation errors of the non-affine operations become larger

Table 42 Comparison of load changes using different approaches for the 2383-bus test systemNormal Line Gen

67-138 trip Gen 131 tripMCS ∆Lmax(MW) 9585 4160 6532

AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109

These simulations based on MATLAB were carried out on a computer with 8 GB of RAMand a processor of 340 GHz The proposed AA-based approach was computationally moreefficient than MCS being 196 times faster (0889 s vs 1744 s) and 11 times faster (2707 s vs29757 s) for the 5-bus and the 2383-bus test system respectively These simulation times arereferred to the system in normal operating conditions without contingencies

It is important to point out that the proposed AA-based methodology computes ATCs con-sidering uncertainties attributed to intermittent sources of power for a system topology Hencebased on the N-1 contingency criterion the AA-based method was repeatedly applied for eachcontingency to compute the ATC Note that the uncertainty associated with equipment outages(eg lines and transformers) can also be modeled using the proposed AA-based computingparadigm According to a mathematical approach widely adopted in circuit analysis litera-ture the equipment outages can be described by proper variations of the admittance matrixelements defining the parameters of the equipment equivalent circuits using proper intervals(eg Ri = [RminRmax] = [R 1e6] where Ri = 1e6 corresponds to the i-th equipment out ofservice)

77

44 Sources of Error

The main sources of error of the proposed AA-based method for voltage stability assessment areas follows

bull Errors introduced by sinusoidal function approximations In this thesis sinusoidal func-tions are written in terms of Chebyshev polynomial approximations for 5-digit accuracyThis approximation given by (235) comprises a finite number of polynomials whosecoefficients are floating-points The number of polynomials N p are truncated dependingon the number of digits accuracy For 5-digit accuracy as in this thesis the Chebyshevapproximation error is 10minus5 because of the truncation of the Chebyshev series approxi-mation only Since the polynomials of this approximation involves non-affine operationsadditional truncation approximation errors are introduced [107] Therefore the final affineforms of the sine and cosine operations exhibit approximation errors greater than those at-tributed to the Chebyshev approximations only Therefore increasing the number of digitsof the Chebyshev approximation beyond 5 digits does not improve the overall accuracy ofthe sine and cosine affine approximations

bull Errors due to the size of the intervals that model the uncertain variables The greater thesize of the interval that models the uncertain variables the greater the error This is ex-pected since the initial coefficients of the affine forms are obtained using a sensitivityanalysis approach Thus the greater the deviation of the central value of the affine formswith respect to their minimum and maximum values the greater the error of the initialcoefficients of the affine forms

bull Errors due to the number of uncertain variables and size of the system The greater thenumber of uncertain variables and size of the system the greater the error This is also ex-pected since with the number of equations and coefficients of the affine forms increasingthe truncation errors associated with non-affine operations increase

78

45 Summary

A novel method for voltage stability assessment of power systems with intermittent sources ofgeneration such as wind and solar power has been presented in this chapter Specifically theproposed methodology computes the bounds of the PV curves and associated static load marginswhen the system presents uncertainties due to operating conditions without any assumption ontheir pdfs The results depict a reasonable good accuracy at significantly lower computationalcosts when compared to those obtained using simulation based techniques Comparisons alsoshow that the lower bound of the maximum loadability obtained using the AA-based approach ispessimistic while it is optimistic in some cases for the SF approach thus making the AA-basedapproach more appropriate for practical applications

The proposed AA-based method takes into account the correlations among variables whichis not possible with other self-validated methods such as IA-based methods The accuracy of theproposed method is mainly attributed to this characteristic However as the number of uncertainvariables size of the system and size of the uncertainty intervals increase the accuracy of themethod decreases This should not be a significant problem for the practical application of theproposed technique since the number and sizes of uncertain generation sources in real systemsis not expected to be more than 30 of the total system generation

79

Chapter 5

Affine Arithmetic Based Transient StabilityAssessment

51 Introduction

In this chapter the proposed AA-based method for transient stability assessment of power sys-tems considering uncertainties associated with operating conditions is presented and discussedThis method is based on the solution of the DAEs that describe the system dynamic behaviorusing a trapezoidal integration approach which is described in detail Two test cases are dis-cussed first a Single Machine Infinite Bus (SMIB) test system with simplified models is usedto illustrate the proposed technique in detail thereafter a two-area test system using detailedsynchronous generator models and wind turbines based on Doubly Fed Induction Generators(DFIGs) is used to demonstrate its more practical application The results are compared withthose obtained using simple MCS to verify the accuracy of the proposed AA-based method Thecomputational burden of the AA-based technique and its possible application to large systemsare also discussed

80


The set of DAEs (217) and (218) describing a power system can be formulated in affine formas follows [122]

˙xD(t) = fD

(xD y pD λ

)for xD(0) = xDIC (51)

0 = g(xD y pD λ) for y(0) = yIC (52)

The purpose of this formulation is to compute the hull of the system response due to systemuncertainties such as intermittent sources of power to examine the effect of large disturbanceson the system To achieve this goal the set of AA DAEs (51) and (52) are solved using thetrapezoidal integration approach resulting in the following formulation for the DAE system

xDn minus xDnminus1 minus (∆t2)(

fD

(xDnminus1 ynminus1 pD λ tnminus1

)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)

Based on a similar representation of the affine forms for voltages and angles used in [14]and [121] the initial affine forms of the state and algebraic variables are computed using thefollowing affine relations

xD = xDo +sum

misinNIG

partxD

partPGAm

∣∣∣∣∣o∆PGAm

εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)

which represent the impact of the various uncertain sources ∆PG in the corresponding variablesThe load admittances are then modeled as

Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81

The central values in (55)-(57) are computed by solving the following set of steady-state equa-tions

fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)

The affine forms of the intermittent sources of power are modeled as in (48) and the centralvalues of the intermittent sources of power are computed using (331) For a different period oftime a different affine interval for the uncertain power injections may be needed for instance∆PGA may be chosen according to the forecasting error associated with the operating horizon ofinterest The partial derivatives in (55)-(57) are computed using sensitivity analysis

The resultant set of algebraic equations (53) and (54) can be solved using a Newton approachas follows

J(zr) (zr+1 minus zr

)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)

This linear system of equations is solved iteratively updating the state and algebraic variablesuntil convergence is attained The set of DAEs are simultaneously solved achieving better nu-merical stability than partitioned techniques Unlike the Krawczykrsquos or the Interval Gauss SeidelIteration approaches used to solve the system of equations in IA [124] the Newton-based ap-proach used in this thesis does not require certain special classes of matrices such as M-matricesand H-matrices to solve the linearized set of equations

The components of the vector ∆FA in (511) exhibit the following form after all affine opera-tions and non-affine approximations are carried out

∆FAs = ∆FAos+ ∆FA1s

εIG1 + ∆FA2sεIG2 + + ∆FAms

εIGm

+sumhisinNh

∣∣∣(Fahs)∣∣∣ ε f hs

= 0 forallm isin NIG s = 1NDA(512)

82

The last term of this equation which corresponds to the independent noise symbols resultingfrom the non-affine operations can be reduced to a single noise symbol as follows

∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)

Similarly the components of the Jacobian matrix can be written as

Jsk = Josk + J1skεIG1 + J2skεIG2 + + JmskεIGm

+sumhisinNh

∣∣∣(Jahsk)∣∣∣ ε jhsk forallm isin NIG s k = 1NDA

(514)

JHskε jHsk =sumhisinNh

|(Jahsk)|ε jhsk s k = 1NDA (515)

And the affine forms of the vector ∆z are

∆zs = ∆zos + ∆z1sεIG1 + + ∆zmsεIGm + ∆zHsεHs

s = 1NDAforallm isin NIG

(516)

This vector is comprised of the state and algebraic variables and thus has the following form

∆z =

∆xD

∆y

(517)

Applying equation (234) to (511) yields the following components of the affine form ∆F

bull Central values

∆FAos=

NDAsumk=1

Josk∆zok s = 1NDA (518)

83

bull The m-th components

∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)

bull Approximation errors

∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)

Equations (518)-(520) stated in matrix form are used to compute the components of the affineform ∆z (517) as follows

∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr

m11minus minus ∆zr

oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1

minus minus ∆zroNDA

JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA

forallm isin NIG (522)

84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr

H21minus minus ∆zr

oNDAJr

H2NDA

∆FrHNDAminus ∆zr

o1Jr

HNDA 1minus minus ∆zr

oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)

The state and algebraic variables are iteratively updated as follows until convergence is attained

xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)

where ∆yr and ∆xrD are computed using (521)-(524) Figure 51 illustrates the proposed al-

gorithm for the AA-based transient stability assessment This algorithm is very similar to thecommonly used approach to solve the DAEs for time domain simulations but in affine form


To test the proposed AA-based methodology for transient stability assessment three test casesare considered The first is a Single Machine Infinite Bus (SMIB) test system [94] which al-lows a thorough test and demonstration of the proposed methodology using simplified modelsthe second corresponds to a two-area system [99] aimed at testing the proposed method usingdetailed models for the synchronous generators In the third study the two are system is mod-ified to consider wind turbines based on Doubly Fed Induction Generators (DFIGs) allowing

85

Start

Read system data

Update the state variables using

(5-25)(5-26)

Compute the initial affine forms of state and algebraic variables using (55)-(5-9)

Compute corrections of the state and algebraic variables using (5-21)-(5-24)

Yes

The set of DAEs are algebraized using the trapezoidal rule (5-3)

No convergence

Store Variables

End

Contingeny or Topology change

Yes

No

Limit Violation Solve (54) for algebraic

variables

No

No Yes

Yes

Figure 51 AA-based time domain simulation algorithm

to more realistically test and demonstrate the proposed approach using intermittent sources ofpower To validate and compare the results obtained using the proposed AA-based techniqueMCS are used as the benchmark assuming uniform probability distributions of the intermittentsources of power 3000 samples for the first study case and 5000 samples for the second andthird study cases were required for MCS to converge

86

Figure 52 SMIB test system [94]

531 Single Machine Infinite Bus Test System

The system shown in Figure 52 comprises a synchronous generator which is assumed to beintermittent and connected through a transformer and two parallel lines to an infinite bus Thedata for this system is given in Table 51 A three phase fault is assumed in the middle of oneof the lines connecting Buses 1 and 2 which is subsequently cleared by tripping the faultedline The initial central values for the affine forms are computed assuming that the infinite bus isdrawing 55 pu of real power at unity power factor

Table 51 SMIB test system dataH D Xprimed XT X12 V2 pf Po

(pu middot s) (pu) (pu) (pu) (pu) (pu) (pu)

30 01 0025 0025 01 10 1 55

In this example the generator was represented using a classical synchronous machine modelHowever it is important to point out that the proposed AA-based approach for transient stabilityassessment is not model-dependent as demonstrated with the second test system The set ofalgebraized DAEs in affine form for this study case is given in Appendix A

Figure 53 shows the upper and lower bounds of the system dynamic response when theintermittent source varies within the interval [5225 5775] MW ie a plusmn5 variation from thepower drawn by the infinite bus Notice that the bounds obtained using the AA-based approachclosely follows the bounds obtained using MCS with a maximum error of 098

87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)

AA Upper BoundAA Lower BoundAA Lower BoundMC Lower Bound

Figure 53 SMIB test system 3-cycle fault with plusmn5 variation in generation

Figures 54 and 55 depict the effect of the size of the interval that models the intermittentsource of power on the accuracy of the proposed AA-based technique In these cases the in-tervals considered are [495 605]MW ie plusmn10 variation and [440 660] MW ie plusmn20respectively resulting in a maximum error of 388 for plusmn10 variation and 869 for plusmn20variation when compared to MCS Thus it can be readily inferred that as the size of the intervalthat models the uncertain variables increases the error increases as well as expected Also noticethat the AA-based approach is slightly conservative as compared to MCS which is a desirablefeature of the proposed method

Figure 56 illustrates a case where the system becomes unstable for the plusmn5 range of powervariation [5225 5775] MW of the intermittent source Notice that the AA-based approach isalso able to accurately represent unstable cases with a maximum error of 498 for the simula-tions considered

Finally Figure 57 illustrates the bounds obtained when stable and unstable cases are con-tained within the assumed interval of power variation of the intermittent source Thus observethat the upper bound is unstable while the lower bound is stable It is important to point outthat the accuracy of the method for simulation times beyond 045 s decreases because of the

88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)

AA Upper BoundAA Lower BoundMC Upper BoundMC Lower Bound


significant difference between the curve that bounds unstable cases and the curve that boundsstable cases as time progresses Nevertheless for the simulation time considered the proposedAA-based approach is able to identify stable and unstable bounds

The proposed AA approach was computationally more efficient than MCS for all cases yield-ing 23 times faster simulations (0798 s vs 1836 s) for the results depicted in Figure 57

532 Two-area Test System with Synchronous Generators

The system shown in Figure 511 is comprised of 11 buses and four generators that supply a totalbase load of 2734 MW [99] Two of these generators (G2 and G4) are assumed to be intermittentsources of power and are represented using detailed synchronous generator sub-transient modelsas well as fast static excitation systems The set of algebraized DAEs in affine form for this studycase is given in Appendix B The data for the generators is shown in Table 52

Figure 59 illustrates the dynamic response of the system for a three-phase fault in the middleof one of the parallel lines connecting buses 8 and 9 the fault is cleared after 71 cycles by

89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)

AA Upper BoundAA Lower BoundMC Upper BoundMc Lower Bound


Table 52 Sub-transient model data for synchronous generatorsXd(pu) Xq(pu) Xl(pu) Xprimed(pu) Xprimeq(pu) Xprimeprimed (pu) Xprimeprimeq (pu) Ra(pu) T primed0(s) T primeq0(s)

18 17 02 03 055 025 025 00025 8 04

T primeprimed0(s) T primeprimeq0(s) HG1(pu middot s) HG2(pu middot s) HG3(pu middot s) HG4(pu middot s) Ka Tec(s) Da S n(MVA)

003 005 65 65 6175 6175 200 001 001 900

Figure 58 Two-area test system with synchronous generators

tripping the faulted line Notice that the system is unstable for plusmn10 variation of power at G2

91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)


Figure 59 Two-area test system with synchronous generators results 71-cycle fault with plusmn10variation in generation

and G4 The AA-based approach closely follows the hull of dynamic curves with a maximumerror of 602 and being 292 times faster (1348 s vs 393633 s) with respect to MCS

Figure 510 depicts the dynamic response of the system for a three-phase fault close to Bus 9The fault is cleared after 1 cycle by tripping the faulted line The system is stable for the plusmn10range of variation of the assumed intermittent power sources The AA-based approach closelyfollows the hull of dynamic curves with a maximum error of 83

533 Two-area Test System with Wind Turbines

In the two-area test system system in Figure (511) two of these generators (G2 and G4) arereplaced by wind turbines based on DFIG which are modeled using the dq equations presentedin Appendix C The set of algebraized DAEs in affine form for this study case is also givenin Appendix C Generators G1 and G2 are modeled using a classical representation and thecorresponding data is given in Table 53 The wind turbine data is shown in Table 54

Figure 512 illustrates the dynamic response of the system for a three-phase fault in the middle

92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)


Figure 510 Two-area test system results with synchronous generators results 1-cycle fault withplusmn10 variation in generation

Table 53 Synchronous generators dataxprimed(pu) HG2(pu middot s) HG4(pu middot s) Da

03 65 6175 01

Figure 511 Two-area test system with DFIG-based wind turbines

Table 54 DFIG-based wind turbines dataxs(pu) xr(pu) xm(pu) rr(pu) rs(pu) Kp Kv Tp(s) Tr(s) Hm(kWskVa)

010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)


Figure 512 Two-area test system with DFIG-based wind turbines results 3-cycle fault withplusmn10 variation in generation

of one of the parallel lines connecting Buses 8 and 9 the fault is cleared after 3 cycles by trippingthe faulted line Notice that the system is stable for plusmn10 variation of power at G2 and G4 TheAA-based approach exhibits a maximum and mean error of 2067 and 767 respectivelybeing 238 times faster than MCS Figure 513 depicts the dynamic response of the system fora three-phase fault close to Bus 9 which is cleared after 8 cycles by tripping the faulted lineIn this case the system is unstable for the plusmn10 range of variation of the assumed intermittentpower sources Notice that the AA-based approach closely follows the hull of dynamic curveswith a maximum error of 565

The sources of error of the proposed AA-based method are associated with the truncatedapproximation of non-affine functions such as sinusoidal functions the size of the intervals of theuncertain variables and errors due to the number of uncertain variables and size of the systemThe greater the number of uncertain variables and size of the system the greater the error asdemonstrated for the two-area test system This is expected since with the number of equationsand coefficients of the affine forms increasing the truncation errors associated with non-affineoperations also increase However given the limited number of uncertain generators and limited

94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)



range of variation in real systems one would expect reasonable results in real-sized systems

54 Summary

This chapter presented a novel AA-based method for transient stability assessment of powersystems with intermittent sources of generation such as wind and solar power Similar to the AA-based methods for power flow analysis and voltage stability assessment discussed in Chapters 3and 4 respectively uncertainties were modeled as intervals without any assumptions regardingtheir probabilities The proposed AA-based method was able to compute the hull of the systemdynamic response when the system presents uncertainties due to operating conditions based ona trapezoidal integration technique It was noticed that stable and unstable bounds of the systemdynamic response are adequately represented by the AA-based approach and the results depicteda reasonable good accuracy at significantly lower computational costs when compared to thoseobtained using simulation based techniques

95

Chapter 6

Conclusions Contributions and FutureWork

61 Summary and Main Conclusions

Most of the available techniques for probabilistic stability assessment of power systems are basedon analytical and sampling based approaches which may be computationally demanding andorrely on assumptions that may be unrealistic in some cases To overcome these difficulties novelAA-based methods for voltage and transient stability assessment of power systems with inter-mittent sources of generation such as wind and solar power were proposed in this thesis

A new representation of generator reactive power limits was proposed in this thesis which isa more efficient alternative to the bus type switching strategy adopted in the existing AA-basedpower flow analysis This new representation is based on the introduction of additional noisesymbols associated with voltage control settings which allows keeping the generator reactivepowers within limits by modifying their terminal voltages

The proposed AA-based method for voltage stability assessment computes the bounds of thePV curves and associated static load margins when the system presents uncertainties due to oper-ating conditions without any assumption on their pdfs This method is based on a parametriza-tion technique which allows obtaining feasible solutions for the whole hull of PV curves The

96

results depicted a reasonable good accuracy at significantly lower computational costs whencompared to those obtained using simulation based techniques Comparisons also showed thatthe lower bound of the maximum loadability obtained using the AA-based approach is pes-simistic while it was optimistic in some cases for the SF approach thus making the AA-basedapproach more appropriate for practical applications

The proposed AA-based method for transient stability assessment is able to compute the hullof the system dynamic response when the system presents uncertainties in operating conditionsThis method is based on a trapezoidal integration approach which yields a good numerical stabil-ity and is not model-dependent thus any model for synchronous generators wind turbines andPV arrays can be used as demonstrated in the study cases Similar to the AA-based method forvoltage stability assessment results depicted a reasonable good accuracy at significantly lowercomputational costs when compared to those obtained using simulation based techniques Sta-ble and unstable bounds of the system dynamic response were adequately represented by theAA-based approach

The proposed AA-based methods represent computationally efficient alternative to the well-known sampling and analytical approaches and unlike most of these approaches uncertaintiesare modeled as intervals without any need for assumptions regarding their probabilities Theseintervals reflects the uncertainty of the estimated output power and avoids the difficulties inproperly representing the uncertainties of wind and solar power using pdfs

62 Main Contributions

This thesis conceptualized and proposed new AA-based computing paradigms for PV curve com-putation and transient stability assessment of power systems considering uncertainties associatedwith intermittent sources of power The main contributions are as follows

bull A more efficient representation of generator reactive power limits which avoids the busswitching strategy used in the existing AA-based power flow analysis method has beenproposed in this thesis The use of rectangular-coordinate equations in the AA-based powerflow problem is also studied demonstrating to be more efficient and accurate than thepolar-coordinate equations approach

97

bull A novel computationally efficient and reasonably accurate approach based on AA for volt-age stability assessment of power systems with intermittent power sources has been pro-posed in this thesis as a better alternative to popular simulation-based tools This methodis also more adequate as compared with a previously proposed technique to estimate theimpact of parameter variations in voltage stability assessment

bull Finally a new computationally efficient and reasonably accurate AA-based method methodfor transient stability assessment of power systems with power sources has also been pro-posed in this thesis This method is not model-dependent and as the proposed AA-basedvoltage stability technique uncertainties are modeled as intervals without any need forassumptions regarding their probabilities

The proposed AA-based methods as well as part of the results of this thesis have been pub-lished in [117] [121] and [122] and the journal paper [125] has been submitted for publication

63 Future Work

The following research topics could be considered to improve the proposed AA-based methods

bull In order to efficiently account for system contingencies the uncertainty associated withequipment outages (ie lines and transformers) could be modeled using AA by describ-ing the equipment outages with proper variations of the admittance matrix elements Forinstance the parameters of the equipment equivalent circuits could be defined using theintervals [X 1e6] where X is the value of the impedance and 1e6 corresponds to theequipment out of service This approach needs further research to show its feasibility

bull Code development to facilitate the general handling of model equations in affine form isneeded This code will allow testing the proposed AA-based methods with larger testsystems

98

Appendix A

AA Model of Single-Machine-Infinite-BusTest System

Algebraized DAEs in affine form

δn minus δnminus1 minus∆t2

(ωn minus ωs + ωnminus1 minus ωs

)= 0 (A1)

ωn minus ωnminus1 minus∆t2

(f2a + f2b

)= 0 (A2)

where

f2a =[Pm minus Pesin(δn minus Da(ωn minus ωs)

] ω2s

2Hωn(A3)

f2b =[Pm minus Pesin(δnminus1 minus Da(ωnminus1 minus ωs)

] ω2s

2Hωnminus1(A4)

where

bull Power delivered before the fault

Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99

bull Power delivered during the fault

Pe =EprimeV2X13(X13 + X12)

Xprimed + XT + X12X13(X12 + X13)(A6)

bull Power delivered after the fault

Pe =EprimeV2

Xprimed + XT + X12(A7)

100

Appendix B

AA Model of Two-Area Test System withSynchronous Generators


Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)

wheref5a = minusIdnVnsin

(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)

f5b = minusIdnminus1Vnminus1sin(θnminus1 minus δnminus1

)minus Iqnminus1Vnminus1cos

(θnminus1 minus δnminus1

)minus Da

(ωnminus1 minus ωs

)(B15)


(ωn minus ωnminus1

)= 0 (B16)

E fn minus E fnminus1 minus∆t2

(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec

(Ka(Vre f minus Vn) minus E fn

)(B18)

f7b =1

Tec

(Ka(Vre f minus Vnminus1) minus E fnminus1

)(B19)

Eprimeprime

q minus(rs Iq minus X

primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime

d minus(rs Id minus X

primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations

bull For generator buses

V j I jcos(θ j minus θIa j

)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)

V j I jsin(θ j minus θIa j

)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C

AA Model of Two-Area Test System withWind Turbines

The DFIG generator DAEs are

ωm =

(Pω

ωmminus (Ψdsiqs minus Ψqsids)

)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where

Ψds = minus((xs + xm)ids + xmidr) (C5)

104

Ψqs = minus((xs + xm)iqs + xmiqr) (C6)

Pω =ρ

2cp

(λp θp

)Arv3 (C7)

cp = 022(116λiminus 04 lowast θp minus 5

)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)

and the following algebraic equations

vds = minusrsids + (xs + xm)iqs + xmiqr (C10)

vqs = minusrsiqs minus (xs + xm)ids minus xmidr (C11)

vdr = minusrridr + (1 minus ωm)((xr + xm)iqr + xmiqs) (C12)

vqr = minusrriqr minus (1 minus ωm)((xr + xm)idr + xmids) (C13)

The algebraized DAEs in affine form are as follows

bull For wind turbines based on DFIGs

ωn minus ωnminus1 minus∆t2lowast

Pω

ωmn

minus (Ψdsniqsnminus Ψqsn

idsn)1

2Hm

+

∆t2

Pω

ωmnminus1

minus (Ψdsnminus1iqsnminus1minus Ψqsnminus1

idsnminus1)1

2Hm

= 0

(C14)

105

Ψdsn = minus((xs + xm)idsn + xmidrn) (C15)

Ψqsn= minus((xs + xm)iqsn

+ xmiqrn) (C16)

iqrnminus iqrnminus1

minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)

idrn minus idrnminus1 minus∆t2

Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv

(Vnminus1 minus Vre f

)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)

θpn minus θpnminus1 minus∆t2

(Kp

Tp

(ωmn minus ωre f n

)+

Kp

Tp

(ωmnminus1 minus ωre f

))= 0 (C19)

vdsn minus(minusrsidsn + (xs + xm)iqsn

+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn

minus (xs + xm)idsn minus xmidrn

)= 0 (C21)

vdrn minus(minusrridrn + (1 minus ωmn)((xr + xm)iqrn

+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn

minus (1 minus ωmn)((xr + xm)idrn + xmidsn))

= 0 (C23)

bull For Synchronous Generators

106


(Pm minus E lowast Vn lowast sin(δn minus θn) minus Da lowast (ωrn minus ωs)+

Pm minus E lowast Vnminus1 lowast sin(δnminus1 minus θnminus1) minus Da lowast (ωrnminus1 minus ωs)) = 0(C24)


(ωn minus ωs + ωnminus1 minus ωs) = 0 (C25)

107

Bibliography

[1] ldquoGreen Energy Actrdquo 2009 [Online] Available httpwwwe-lawsgovoncahtmlsourcestatutesenglish2009elaws src s09012 ehtm

[2] ldquoGrid Integration of Wind Generationrdquo Working Group C 608 CIGRE 2011

[3] D Gautam V Vittal and T Harbour ldquoImpact of Increased Penetration of DFIG-BasedWind Turbine Generators on Transient and Small Signal Stability of Power SystemsrdquoIEEE Trans Power Syst vol 24 no 3 pp 1426ndash1434 Aug 2009

[4] K Methaprayoon C Yingvivatanapong L Wei-Jen and J Liao ldquoAn Integration of ANNWind Power Estimation Into Unit Commitment Considering the Forecasting UncertaintyrdquoIEEE Trans on Industry Applications vol 43 no 6 pp 1441ndash1448 Nov 2007

[5] J Matevosyan and L Soder ldquoMinimization of Imbalance Cost Trading Wind Power on theShort-Term Power Marketrdquo IEEE Trans on Power Syst vol 21 no 3 pp 1396ndash1404Aug 2006

[6] H Bri-Mathias and M Milligan ldquoWind Power Forecasting Error Distributions Over Mul-tiple Timescalesrdquo in Proc Power and Energy Society General Detroit Michigan 2011pp 1ndash7

[7] H Bludszuweit J Dominguez-Navarro and A Llombart ldquoStatistical Analysis of WindPower Forecast Errorsrdquo IEEE Trans on Power Syst vol 23 no 3 pp 983ndash991 Aug2008

108

[8] S Tewari C Geyer and N Mohan ldquoA Statistical Model for Wind Power Forecast Errorand its Application to the Estimation of Penalties in Liberalized Marketsrdquo IEEE Trans onPower Syst vol 26 no 4 pp 2031ndash2039 Nov 2011

[9] [Online] Available httpwwwenergygovoncadocsenMEI LTEP enpdf

[10] [Online] Available httpwwwaesocagridoperations18286html

[11] L Ming-Shun C Chung-Liang L Wei-Jen and L Wang ldquoCombining the Wind PowerGeneration System With Energy Storage Equipmentrdquo IEEE Trans on Industry Applica-tions vol 45 no 6 pp 2109ndash2115 Nov 2009

[12] V Akhmatov ldquoAnalysis of Dynamic Behaviour of Electric Power Systems With LargeAmount of Wind Powerrdquo PhD Thesis Technical University of Denmark Denmark2003

[13] D Gautam C Goel and D Villacci ldquoControl Strategy to Mitigate the Impact of ReducedInertia Due to Doubly Fed Induction Generators on Large Power Systemsrdquo IEEE Transon Power Syst vol 26 no 1 pp 214ndash224 Feb 2011

[14] A Vaccaro C Canizares and D Villacci ldquoAn Affine Arithmetic-Based Method for Re-liable Power Flow Analysis in the Presence of Data Uncertaintyrdquo IEEE Trans on PowerSyst vol 25 no 2 pp 624ndash632 May 2010

[15] P Jorgensen J S Christensen and J O Tande ldquoProbabilistic Load Flow CalculationUsing Monte Carlo Techniques for Distribution Network With Wind Turbinesrdquo in Proc8th International Conference on Harmonics and Quality of Power 1998 pp 1146ndash1151

[16] R N Allan C H Grigg and M R G Al-Shakarchi ldquoNumerical Techniques in Proba-bilistic Load Flow Problemsrdquo International Journal for Numerical Methods in Engineer-ing vol 10 no 4 pp 853ndash860 Mar 1976

[17] S Conti and S Raiti ldquoProbabilistic Load Flow Using Monte Carlo Techniques for Distri-bution Networks With Photovoltaic Generatorsrdquo Solar Energy vol 81 no 12 pp 1473ndash1481 Dec 2007

109

[18] P Chen Z Chen and B Bak-Jensen ldquoProbabilistic Load Flow A Reviewrdquo in Proc 3rdInt Conf Electric Utility Deregulation and Restructuring and Power Technologies 2008pp 1586ndash1591

[19] R Allan and A Leite da Silva ldquoProbabilistic Load Flow Using Multilinearisationsrdquo IEEGeneration Transmission and Distribution vol 128 no 5 pp 280ndash287 Sep 1981

[20] M Brucoli F Torelli and R Napoli ldquoQuadratic Probabilistic Load Flow With Lin-early Modelled Dispatchsrdquo International Journal of Electrical Power and Energy Systemsvol 81 pp 138ndash146 Jul 1985

[21] D Dondera R Popa and C Velicescu ldquoThe Multi-Area Systems Reliability EstimationUsing Probabilistic Load Flow by Gramm-Charlier Expansionrdquo in Proc EUROCON theInternational Conference on Computer as a Tool 2007 pp 1470ndash1474

[22] P Zhang and S T Lee ldquoProbabilistic Load Flow Computation Using the Method of Com-bined Cumulants and Gram-Charlier Expansionrdquo IEEE Trans on Power Syst vol 19no 1 pp 676ndash682 Feb 2004

[23] S Chun-Lien ldquoProbabilistic Load-Flow Computation Using Point Estimate MethodrdquoIEEE Trans on Power Syst vol 20 no 4 pp 1843ndash1851 Nov 2005

[24] P Caramia G Carpinelli and P Varilone ldquoPoint Estimate Schemes for ProbabilisticThree-Phase Load Flowrdquo Electr Power Syst Res vol 80 pp 168ndash175 Feb 2010

[25] G Verbic and C Canizares ldquoProbabilistic Optimal Power Flow in Electricity MarketsBased on a Two-Point Estimate Methodrdquo IEEE Trans on Power Syst vol 21 no 4 pp1883ndash1893 Nov 2006

[26] H Yu C Y Chung K P Wong H W Lee and J H Zhang ldquoProbabilistic Load FlowEvaluation With Hybrid Latin Hypercube Sampling and Cholesky Decompositionrdquo IEEETrans on Power Syst vol 24 no 2 pp 661ndash667 May 2009

[27] A M L da Silva and V L Arienti ldquoProbabilistic Load Flow by a Multilinear SimulationAlgorithmrdquo vol 137 no 4 pp 276ndash282 Jul 1990

110

[28] P R Bijwe and G K V Raju ldquoFuzzy Distribution Power Flow for Weakly Meshed Sys-temsrdquo IEEE Trans on Power Syst vol 21 no 4 pp 1645ndash1652 Nov 2006

[29] A Dimitrovski and K Tomsovic ldquoBoundary Load Flow Solutionsrdquo IEEE Trans onPower Syst vol 19 no 1 pp 348ndash355 Feb 2004

[30] M Cortes-Carmona R Palma-Behnke and G Jimenez-Estevez ldquoFuzzy Arithmetic forthe DC Load Flowrdquo IEEE Trans on Power Syst vol 25 no 1 pp 206ndash214 Feb 2010

[31] L Hong L Shi L Yao Y Ni and M Bazargan ldquoStudy on Fuzzy Load Flow WithConsideration of Wind Generation Uncertaintiesrdquo in Proc Transmission and DistributionConference and Exposition Asia and Pacific 2009 pp 1ndash4

[32] Z Wang and F Alvarado ldquoInterval Arithmetic in Power Flow Analysisrdquo IEEE Trans onPower Syst vol 7 no 3 pp 1341ndash1349 Aug 1992

[33] L Barboza G P Dimuro and R Reiser ldquoTowards Interval Analysis of the Load Uncer-tainty in Power Electric Systemsrdquo in Proc 8th International Conference on ProbabilisticMethods Applied to Power Sytems 2004 pp 538ndash544

[34] B Das ldquoRadial Distribution System Power Flow Using Interval Arithmeticrdquo Elect PowerSyst vol 24 no 10 pp 827ndash836 Dec 2002

[35] A Vaccaro and D Villaci ldquoRadial Power Flow Tolerance Analysis by Interval ConstraintPropagationrdquo IEEE Trans on Power Syst vol 24 no 1 pp 28ndash31 Feb 2009

[36] A Vaccaro C Canizares and D Villaci ldquoA Simple and Reliable Algorithm for Comput-ing Boundaries of Power Flow Solutions due to System Uncertaintiesrdquo in Proc Powertech- Innovative ideas toward the Electrical Grid of the Future 2009 pp 1ndash6

[37] A Rodrigues R Prada and M Da Guia da Silva ldquoVoltage Stability Probabilistic As-sessment in Composite Systems Modeling Unsolvability and Controllability Lossrdquo IEEETrans on Power Syst vol 25 no 3 pp 1575ndash1588 Aug 2010

[38] A Leite da Silva I Coutinho A C Zambroni de Souza R B Prada and A Rei ldquoVoltageCollapse Risk Assessmentsrdquo Electr Power Syst Res vol 54 no 3 pp 221ndash227 Jun2000

111

[39] S Chun-Lien and L Chan-Nan ldquoTwo-Point Estimate Method for Quantifying TransferCapability Uncertaintyrdquo IEEE Trans on Power Syst vol 20 no 2 pp 573ndash579 May2005

[40] R Billinton and S Aboreshaid ldquoVoltage Stability Considerations in Composite PowerSystem Reliability Evaluationrdquo IEEE Trans on Power Syst vol 13 no 2 pp 655ndash660May 1998

[41] M Ni J D McCalley V Vittal and T Tayyib ldquoOnline Risk-Based Security AssessmentrdquoIEEE Trans Power Syst vol 18 no 1 pp 258ndash265 Feb 2003

[42] J Zhang I Dobson and F L Alvarado ldquoQuantifying Transmission Reliability MarginrdquoInt J Elect Power Energy Syst vol 26 no 9 pp 697ndash702 Nov 2004

[43] S Greene S G Dobson and F Alvarado ldquoSensitivity of Transfer Capability MarginsWith a Fast Formulardquo IEEE Trans on Power Syst vol 17 no 1 pp 34ndash40 Feb 2002

[44] mdashmdash ldquoSensitivity of the Loading Margin to Voltage Collapse With Respect to ArbritaryParametersrdquo IEEE Trans on Power Syst vol 12 no 1 pp 262ndash272 Feb 1997

[45] H Chen ldquoSecurity Cost Analysis in Electricity Markets Based on Voltage Security Cri-teria and Web-Based Implementationrdquo PhD Thesis University of Waterloo WaterlooCanada 2002

[46] E Haesen C Bastiaensen J Driesen and R Belmans ldquoA Probabilistic Formulation ofLoad Margins in Power Systems With Stochastic Generationrdquo IEEE Trans on PowerSyst vol 24 no 2 pp 951ndash958 May 2009

[47] Y Kataoka ldquoA Probabilistic Nodal Loading Model and Worst Case Solutions for ElectricPower System Voltage Stability Assessmentrdquo IEEE Trans on Power Syst vol 18 no 4pp 1507ndash1514 Nov 2003

[48] A Schellenberg W Rosehart and J Aguado ldquoCumulant-Based Stochastic NonlinearProgramming for Variance Constrained Voltage Stability Analysis of Power SystemsrdquoIEEE Trans on Power Syst vol 21 no 2 pp 579ndash585 May 2006

112

[49] S Senthil Kumar and R PAjay-D-Vimal ldquoFuzzy Logic Based Stability Index Power Sys-tem Voltage Stability Enhancementrdquo International Journal of Computer and ElectricalEngineering vol 2 no 1 pp 24ndash31 Feb 2010

[50] Z Jing G Yun-Feng and M-H Y ldquoAssessment of Voltage Stability for Real-TimeOperationrdquo in Proc IEEE Power India Conference 2006 p 5

[51] R Moore and W Lodwick ldquoInterval Analysis and Fuzzy Set Theoryrdquo Fuzzy Sets andSyst vol 135 no 1 pp 5ndash9 Apr 2003

[52] R Albrecht ldquoTopological Theory of Fuzzinessrdquo in Proc Int Conf Computational Intel-ligence Springer Heidelberg 1999 pp 1ndash11

[53] L Zadeh ldquoSome Reflections on Soft Computing Granular Computing and Their Roles inthe Conception Design and Utilization of InformationIntelligent Systemsrdquo Soft Comput-ing vol 2 no 1 pp 23ndash25 Apr 1998

[54] L Wang and C Singh ldquoPopulation-Based Intelligent Search in Reliability Evaluation ofGeneration Systems With Wind Power Penetrationrdquo IEEE Trans on Power Syst vol 23no 3 pp 1336ndash1345 Aug 2008

[55] Z Yang M Zwolinski and C Chalk ldquoBootstrap an Alternative to Monte Carlo Simula-tionrdquo Electronics Letters vol 34 no 12 pp 1174ndash1175 Jun 1998

[56] M M Othman S Kasim N Salim and I Musirin ldquoRisk Based Uncertainty (RibUt)Assessment of a Power System Using Bootstrap Techniquerdquo in Proc IEEE InternationalPower Enineering and Optimization Conference (PEDCO) Melaka Malaysia 2012 pp460ndash464

[57] N Zhou J Pierre and D Trudnowski ldquoA Bootstrap Method for Statistical Power SystemMode Estimation and Probing Signal Selectionrdquo in Proc IEEE PES Power Syst Confer-ence and Exposition PSCE 2006 pp 172ndash178

[58] R Billinton H Chen and R Ghajar ldquoA Sequential Simulation Technique for AdequacyEvaluation of Generating Systems Including Wind Energyrdquo IEEE Tran on Power Systvol 11 no 4 pp 728ndash734 Dec 1996

113

[59] L Arya L Titare and D P Kothari ldquoDetermination of Probabilistic Risk of VoltageCollapse Using Radial Basis Function (RBF) Networkrdquo Electric Power System Researchvol 76 pp 426ndash434 Apr 2006

[60] mdashmdash ldquoProbabilistic Assessment and Preventive Control of Voltage Security Margins Us-ing Artificial Neural Networkrdquo International Journal of Electrical Power and Energy Sys-tems vol 29 no 2 pp 99ndash105 Feb 2007

[61] K Timko A Bose and P Anderson ldquoMonte Carlo Simulation of Power System Sta-bilityrdquo IEEE Trans Power Apparatus and Syst vol 102 no 10 pp 3453ndash3459 Oct1983

[62] E Vaahedi W Li T Chia and H Dommel ldquoLarge Scale Probabilistic Transient StabilityAssessment Using BC Hydrorsquos On-Line Toolrdquo IEEE Trans Power Syst vol 15 no 2pp 661ndash667 May 2000

[63] C Ferreira J A Dias Pinto and F P M Barbosa ldquoDynamic Security Analysis of an Elec-tric Power System Using a Combined Monte Carlo-Hybrid Transient Stability Approachrdquoin Proc IEEE Power Tech Proceedings Porto 2001

[64] W Wangdee and R Billinton ldquoBulk Electric System Well-Being Analysis Using Sequen-tial Monte Carlo Simulationrdquo IEEE Trans Power Syst vol 21 no 1 pp 188ndash193 Feb2006

[65] R Billinton P R S Kuruganty and M F Carvalho ldquoAn Approximate Method for Prob-abilistic Assessment of Transient Stabilityrdquo IEEE Trans Reliability vol 28 no 3 pp255ndash258 Aug 1979

[66] R Billinton and P R S Kuruganty ldquoProbabilistic Assessment of Transient Stability in aPractical Multimachine Systemrdquo IEEE Trans Power Apparatus and Syst vol 100 no 7pp 3634ndash3641 Jul 1981

[67] H Yuang-Yih and C Chang ldquoProbabilistic Transient Stability Studies Using the Condi-tional Probability Approachrdquo IEEE Trans on Power Syst vol 3 no 4 pp 1565ndash1572Nov 1988

114

[68] R Billinton and S Aboreshaid ldquoStochastic Modelling of High-Speed Reclosing in Proba-bilistic Transient Stability Studiesrdquo IEE Proc Generation Transmission and Distributionvol 142 no 4 pp 350ndash354 Jul 1995

[69] D Fang L Jing and T Chung ldquoCorrected Transient Energy Function-Based Strategy forStability Probability Assessment of Power Systemsrdquo IET Proc Generation Transmissionand Distribution vol 2 no 3 pp 424ndash432 May 2008

[70] S Faried R Billinton and S Aboreshaid ldquoProbabilistic Evaluation of Transient Stabilityof a Power System Incorporating Wind Farmsrdquo IET Proc Renewable Power Generationvol 4 no 4 pp 299ndash307 Jul 2010

[71] H Kim and C Singh ldquoProbabilistic Security Analysis Using SOM and Monte Carlo Sim-ulationrdquo in Proc Power Engineering Society Winter Meeting 2002 pp 755ndash760

[72] mdashmdash ldquoPower System Probabilistic Security Assessment Using Bayes Classifierrdquo ElectrPower Syst Res vol 74 no 1 pp 157ndash165 Apr 2005

[73] A Dissanayaka U Annakkage B Jayasekara and B Bagen ldquoRisk-Based Dynamic Se-curity Assessmentrdquo IEEE Trans Power Syst vol 26 no 3 pp 1302ndash1308 Dec 2011

[74] B Jayasekara and U Annakkage ldquoDerivation of an Accurate Polynomial Representationof the Transient Stability Boundaryrdquo IEEE Trans Power Syst vol 21 no 4 pp 1856ndash1863 Aug 2006

[75] A Karimishad and T Nguyen ldquoProbabilistic Transient Stability Assessment Using Two-Point Estimate Methodrdquo in Proc 8th International Conference on Advances in PowerSystem Control Operation and Management (APSCOM) 2009 pp 1ndash6

[76] E Chiodo F Gagliardi and D Lauria ldquoProbabilistic Approach to Transient StabilityEvaluationrdquo vol 141 no 5 pp 537ndash544 Sep 1994

[77] F Allella E Chiodo and D Lauria ldquoTransient Stability Probability Assessment and Sta-tistical Estimationrdquo Electr Power Syst Res vol 67 no 1 pp 21ndash33 Oct 2003

115

[78] S Ayasun Y Liang and C O Nwankpa ldquoA Sensitivity Approach for Computation ofthe Probability Density Function of Critical Clearing Time and Probability of Stability inPower System Transient Stability Analysisrdquo Applied Mathematics and Computation vol176 no 2 pp 563ndash576 May 2006

[79] J L Soufis A V Machias and B Papadias ldquoAn Application of Fuzzy Concepts to Tran-sient Stability Evaluation of Power Systemsrdquo IEEE Trans on Power Syst vol 4 no 3pp 1003ndash1009 Aug 1989

[80] Y Chen and A Dominguez-Garcia ldquoA Method to Study the Effect of Renewable ResourceVariability on Power Systems Dynamicsrdquo IEEE Trans on Power Syst vol 27 no 4 pp1978ndash1989 Nov 2012

[81] M Althoff M Cvetkovic and M Ilic ldquoTransient Stability Analysis by Reachable SetComputationrdquo in Proc 3rd IEEE PES International Conference and Exhibition on Inno-vative Smart Grids Technologies ISGT Europe 2012 pp 14ndash17

[82] H N Villegas D C Aliprantis and E C Hoff ldquoReachability Analysis of Power SystemFrequency Dynamics with New High-Capacity HVAC and HVDC Transmission Linesrdquo inProc IREP Symposium-Bulk Power System Dynamics and Control-IX Rethymnon Greece2013 pp 1ndash9

[83] Z Wang Sh Zheng and C Wang ldquoPower System Transient Stability Simulation Un-der Uncertainty Based on Interval Methodrdquo in Proc International Conference on PowerSystem Technology PowerCon 2006 pp 1ndash6

[84] D Grabowski M Olbrich and E Barke ldquoAnalog Circuit Simulation Using Range Arith-meticrdquo in Proc Asia and South Pacific Design Automation Seoul Korea 2008 pp 762ndash767

[85] J Duncan-Glover S S Sarma and T Overbye Power System Analysis and DesignUSA CENGAGE Learning 2008

[86] J Avalos-Munoz ldquoAnalysis and Applications of Optmization Techniques to Power Sys-tem Electricity Marketsrdquo PhD Thesis University of Waterloo Waterloo Canada 2008

116

[87] M Pirnia C Canizares and K Bhattacharya ldquoRevisiting the Power Flow Problem Basedon a Mixed Complementarity Formulation Approachrdquo IET Proc Generation Transmis-sion and Distribution vol 7 no 11 pp 1194ndash1201 Nov 2013

[88] P Kundur J Paserba V Ajjarapu A Bose C Canizares N Hatziargyriou D HillA Stankovic C Taylor T Van Cutsem and V Vittal ldquoDefinition and Classification ofPower System Stability IEEECIGRE Joint Task Force on Stability Terms and Defini-tionssrdquo IEEE Trans Power Syst vol 19 no 3 pp 1387ndash1401 Aug 2004

[89] T Van Cutsem and C Vournas Voltage Stability of Electric Power Systems New YorkUSA Springer 1998

[90] ldquoVoltage Stability Assessment Concepts Practices and Toolsrdquo IEEEPES Power SystemStability Subcomitte Tech Rep Aug 2002

[91] V Ajjarapu and C Christy ldquoThe Continuation Power Flow A Tool for Steady StateVoltage Stabilityrdquo IEEE Trans on Power Syst vol 7 no 1 pp 406ndash423 Feb 1992

[92] C Canizares and F L Alvarado ldquoPoint of Collapse and Continuation Methods for LargeACDC Systemsrdquo IEEE Trans on Power Syst vol 8 no 1 pp 1ndash8 Feb 1993

[93] H-D Chiang A J Flueck K Shah and N Balu ldquoCPFLOW a Practical Tool for TracingPower System Steady-State Stationary Behavior Due to Load and Generation VariationsrdquoIEEE Trans on Power Syst vol 10 no 2 pp 623ndash634 May 1995

[94] A Gomez-Exposito A J Conejo and C Canizares Electric Energy Systems Analysisand Operation Vol I Boca Raton CRC press 2009

[95] C Vournas M Karystianos and N Maratos ldquoBifurcations Points and Loadability Limitsas Solutions of Constrained Optimization Problemsrdquo in Proc Power Engineering SocietySummer Meeting 2000 IEEE vol 3 2000 pp 1883ndash1888

[96] R Avalos C Canizares and M Anjos ldquoA Practical Voltage-Stability-Constrained Op-timal Power Flowrdquo in Proc Power and Engineering Society Meeting Conversion andDelivery of Electrical Energy in the 21st Century vol 3 Jul 2008 pp 1ndash6

117

[97] NERC ldquoReliability conceptsrdquo 2007 [Online] Available httpwwwnerccomfilesconcepts v102pdf

[98] mdashmdash ldquoAvailable Transfer Capability Definitions and Determinationrdquo 1996 [Online]Available httpwwwwestgovorgwiebwind06-96NERCatepdf

[99] P Kundur Power System Stability and Control McGraw-Hill 1994

[100] B Stott ldquoPower System Dynamic Response Calculationsrdquo Proc of the IEEE vol 67no 2 pp 219ndash 241 Feb 1979

[101] F Milano Power System Modelling and Scripting London Springer 2010

[102] P Sauer and M Pai Power System Dynamics and Stability Prentice Hall 1998

[103] T Noda K Takenaka and T Inoue ldquoNumerical Integration by the 2-Stage DiagonallyImplicit Runge-Kutta Method for Electromagnetic Transient Simulationsrdquo IEEE Transon Power Delivery vol 24 no 1 pp 390ndash399 Jan 2009

[104] Y Cho J Park and G Jang ldquoA Novel Tool for Transient Stability Analysis of Large-ScalePower Systems Its Application to the KEPCO Systemrdquo Simulation Modelling Practiceand Theory vol 15 pp 786ndash800 Aug 2007

[105] A A Nimje and C K Panigrahi ldquoTransient Stability Analysis Using Modified EulerrsquosIterative Techniquerdquo in Proc International Conference on Power Syst ICPS 2009 pp1ndash4

[106] H Yi S Cheng Z Du L B Shi and Y Ni ldquoModeling and Simulation on Long-TermDynamics of Interconnected Power System Using Area COI Conceptrdquo Electr Power SystRes vol 78 no 8 pp 1369ndash1377 Aug 2008

[107] L de Figueiredo and J Stolfi ldquoSelf-Validated Numerical Methods and Applicationsrdquo inProc Brazilian Mathematics Colloq Monograph IMPA Rio de Janeiro Brazil 1997 pp1ndash103

118

[108] J Comba and J Stolfi ldquoAffine Arithmetic and its Applications to Computer Graphicsrdquo inProc Brazilian Symposium on Computer Graphics and Image Processing VI SIBGRAPI1993 pp 9ndash18

[109] E Hansen ldquoA generalized interval arithmeticrdquo Lecture Notes in Computer ScienceSpringer vol 29 pp 7ndash18 1975

[110] L Zhang Y Zhang and W Zhou ldquoTradeoff Between Approximation Accuracyand Complexity for Range Analysis Using Affine Arithmeticrdquo Journal of SignalProcessing Systems vol 61 no 3 pp 279ndash291 Dec 2010 [Online] Availablehttpwwwspringerlinkcomcontentt4445742873p0273

[111] F Messine ldquoExtensions of Affine Arithmetic Application to Unconstrained GlobalOptimizationrdquo Journal of Universal Computer Science vol 8 no 11 pp 992ndash10152002 [Online] Available httphttpwwwciteseerxistpsuedu

[112] J Mason and D Handscomb Chebyshev Polynomials Florida USA Chapman ampHallCRC 2002

[113] J Gentle Random Number Generation and Monte Carlo Methods New York USASpringer 2003

[114] J A Bucklew Introduction to Rare Event Simulation New York USA Springer 2004

[115] F Milano L Vanfretti and J Morataya ldquoAn Open Source Power System VirtualLaboratory The PSAT Case and Experiencerdquo IEEE Trans on Education vol 51 no 1pp 17ndash23 Feb 2008 [Online] Available httpwwwpoweruwaterloocasimfmilano

[116] S Kodsi and C Canizares ldquoModeling and Simulation of IEEE 14 bus System with FACTSControllersrdquo Tech Rep Univ Waterloo ON Canada 2003

[117] J Munoz and C Canizares ldquoComparative Stability Analysis of DFIG-Based Wind Farmsand Conventional Synchronous Generatorsrdquo in Proc Power Syst Conference and Expo-sition 2011 pp 1ndash7

119

[118] M X Prabhakar and R Jiang Weibull Model New Jersey USA Willey-Interscience2010

[119] A Khosravi and S Nahavandi ldquoCombined Nonparametric Prediction Intervals for WindPower Generationrdquo IEEE Trans on Sustainable Energy vol 4 no 4 pp 849ndash856 Oct2013

[120] A Khosravi S Nahavandi D Creighton and D Srinivasan ldquoOptimizing the Qualityof Bootstrap-Based Prediction Intervalsrdquo in Proc Joint Conf Neural Networks (IJCNN)2011 pp 3072ndash3078

[121] J Munoz C Canizares K Bhattacharya and A Vaccaro ldquoAn Affine Arithmetic BasedMethod for Voltage Stability Assessment of Power Systems with Intermittent GenerationSourcesrdquo IEEE Trans on Power Syst vol 28 no 4 pp 4475ndash4487 Nov 2013

[122] mdashmdash ldquoAffine Arithmetic Based Methods for Voltage and Transient Stability Assessmentof Power Systems with Intermittent Generation Sourcesrdquo in Proc IREP Symposium-BulkPower System Dynamics and Control- IX Rethymnon Greece 2013

[123] R Zimmerman C Murillo-Sanchez and R Thomas ldquoMatpower Steady-State Opera-tions Planning and Analysis Tools for Power Systems Research and Educationrdquo IEEETrans on Power Syst vol 26 no 1 pp 12ndash19 Feb 2011

[124] F Alvarado and Z Wang ldquoDirect Sparse Interval Hull Computations for Thin Non-M-Matricesrdquo Interval Computations vol 2 pp 5ndash28 1993

[125] J Munoz C Canizares K Bhattacharya and A Vaccaro ldquoTransient Stability Assessmentof Power Systems with Intermittent Generation Sources Using Affine Arithmeticrdquo IEEETrans on Power Syst 8 pages Manuscript submitted for review on October 17 2013

120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review

Stochastic Power Flow Analysis

Stochastic Voltage Stability Assessment

Stochastic Transient Stability Assessment

Research Objectives

Thesis Outline

Background Review

Introduction

Power System Analysis

Power Flow

Voltage Stability

Transient Stability

Self-validated Numerical Methods for Uncertainty Analysis

Interval Arithmetic

Affine Arithmetic

Monte Carlo Simulations

Summary

Affine Arithmetic Based Power Flow Assessment

Introduction

Problem Formulation

Improved Reactive Power Limit Representation

Example

Case A

Case B

Case C

Summary

Affine Arithmetic Based Voltage Stability Assessment

Introduction

Problem Formulation

Simulation Results

5-Bus Test System

2383-Bus Test System

Sources of Error

Summary

Affine Arithmetic Based Transient Stability Assessment

Introduction

Problem Formulation

Simulation Results

Single Machine Infinite Bus Test System

Two-area Test System with Synchronous Generators

Two-area Test System with Wind Turbines

Summary

Conclusions Contributions and Future Work

Summary and Main Conclusions

Main Contributions

Future Work

Appendices

AA Model of Single-Machine-Infinite-Bus Test System

AA Model of Two-Area Test System with Synchronous Generators

AA Model of Two-Area Test System with Wind Turbines

Bibliography

I hereby declare that I am the sole author of this thesis This is a true copy of the thesis includingany required final revisions as accepted by my examiners

I understand that my thesis may be made electronically available to the public

ii

Abstract




iii



iv

Acknowledgments







v




vi

Dedication


vii

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography

Abstract




iii



iv

Acknowledgments







v




vi

Dedication


vii

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography



iv

Acknowledgments







v




vi

Dedication


vii

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography

Acknowledgments







v




vi

Dedication


vii

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography




vi

Dedication


vii

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography

Dedication


vii

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography

Table of Contents


Abstract iv

Acknowledgments v

Dedication vii

List of Tables xii



Nomenclature xvii

1 Introduction 1





viii





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography





21 Introduction 18


221 Power Flow 19







25 Summary 45


31 Introduction 49



34 Example 59

341 Case A 61

342 Case B 63

343 Case C 63

35 Summary 66

ix


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography


41 Introduction 67






45 Summary 79


51 Introduction 80






54 Summary 95




63 Future Work 98

Appendices 98


x



Bibliography 107

xi

List of Tables







xii

List of Figures















xiii


















xiv









xv



xvi

Nomenclature

Indices




i j Bus indices











xvii






Parameters










∆t Time step (s)





xviii








N Number of buses













xix





















xx















Variables






xxi




















xxii





Eprimeprime


Eprime




Eprime










xxiii




















xxiv














λp tip speed ratio







xxv




















xxvi










R Resistance (pu)




xxvii





t Time (s)





u Real variable







v Wind speed (ms)




xxviii





















xxix




X f Random variable












Functions




xxx









xxxi

Chapter 1

Introduction




1

46

7 10

15

13

20

14 Nuclear

Gas

Wind

Solar PV

Bionergy

Water

Conservation




2

MW

1100

1000

900

800

700

600

500

400

300

200

100

0


7172013 94400AM MT





3







4






5




6





7





8




9




10





11



12




13




14





15








14 Thesis Outline


16




17

Chapter 2

Background Review

21 Introduction




18

221 Power Flow






Nsumj=1



Nsumj=1




Nsumj=1



Nsumj=1



19





PF )partxPF






20


where




(28)


min 12∆P22 + 1

2∆Q22





Vanc Vbnc ge 0

forallnc isin NPV

(29)

21








22


Ѳ


23













24



Direct Methods


max λ




λ ge 0

(215)



25




26



27





28



where









0 = g(xD y pD λ) for y(0) = yIC (218)


29




lowast= 0 (219)



lowast= 0 (220)





30







31




g(xDn yn pD λ) = 0 (222)




32





33








where



AnduA

uB=



where

udmin = min

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

udmax = max

uAmin

uBmin

uAmin

uBmax

uAmax

uBmin

uAmax

uBmax

34






35





Sum of Affine Forms




nasumra=1


rb=1

(uBrbεBrb) (229)



nasumra=1

[(uAra plusmn uBra

)εAra

](230)

36




u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

nbsumrb=1





sumnara=1



ΨCh =Umax + Umin

2+

Umax minus Umin

2εξ (232)



37




nasumra=1

nbsumrb=1

uArauBrbεAraεBrb

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (233)


38

u = uAuB = uAouBo +

nbsumrb=1

(uAouBrb)εBrb +

nasumra=1

(uBouAra)εAra +

nasumra=1

uAra

nbsumrb=1

uBrb

εξ (234)






Hc(zch) =

npsumrh=0

brhTrh(zch) (235)


T0(zch) = 1

T1(zch) = zch


(236)


39






int X f b

X f a

PX f (x f ) dx f (237)


40



PX f (v) =2v

1128vmev

1128vm2 (238)




41





42




Case A


43




Case B



44

Case C



25 Summary



45



46


47

1 15 2 25 30

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity

05 1 15 2 250

02

04

06

08

1


Cum

ulat

ive

Pro

babi

lity



48

Chapter 3


31 Introduction





49


partVi

partP j


sumjisinNQ

partVi

partQ j



partθi

partP j


sumjisinNQ

partθi

partQ j



VRi = VRoi+

sumjisinNP

partVRi

partP j


sumjisinNQ

partVRi

partQ j


VI i = VIoi+

sumjisinNP

partVIi

partP j


sumjisinNQ

partVIi

partQ j



Poi =PGmaxi


minus PLmini


Qoi =QGmaxi


minus QLmini



50





Pi =

Nsumj=1

ViV j



Qi =

Nsumj=1

ViV j



51


Pri =

Nsumj=1

(VRi


)+ VIi



Qri =

Nsumj=1

(VIi


)minus VRi




Aε = C (311)

C = L minus (Ro + q) (312)




where

Lmax =

Pmax

Qmax

(315)

Lmin =

Pmin

Qmin

(316)


52





minsumεslmin


Cmin le Aε le Cmax

forallsl isin NP NQ

(321)

maxsumεslmax


Cmin le Aε le Cmax

forallsl isin NP NQ

(322)



53

QGmaxi= qpoi +

sumslisinNp



QGmini= qpoi +

sumslisinNp






Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)+

sumnlisinNPL

partVi

partPLnl


+sum

nlisinNQL

partVi

partQLnl


foralli = 1 N

(325)

54

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)+

sumnlisinNPL

partθi

partPLnl


+sum

nlisinNQL

partθi

partQLnl


foralli = 1 N

(326)




)(327)







55

PGo m =PGmaxm

+ PGminm


PLo nl =PLmaxnl

+ PLminnl


QLo nl =QLmaxnl

+ QLminnl



Pi = Pio +sum

misinNIG

PNIGim εIGm +

sumnlisinNPL

PNPLinl εPLnl

+sum

nlisinNQL

PNQLinl εQLnl

+sum

ncisinNPV

PNPVinc εPVanc

minussum

ncisinNPV

PNPVinc εPVbnc

+sumhisinNh


(332)

Qi = Qio +sum

misinNIG

QNIGim εIGm +

sumnlisinNPL

QNPLinl εPLnl

+sum

nlisinNQL

QNQLinl εQLnl

+sum

ncisinNPV

QNPVinc εPVanc

minussum

ncisinNPV

QNPVinc εPVbnc

+sumhisinNh


(333)


ALεL = CL (334)


where

56

AL =


1NIGP


NPL1NPL


NQL1NQL


1NPVminusPNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


1NPVminusPNPV



1NIGQ


NPL1NPL


NQL1NQL


1NPVminusQNPV



NIGNminus1NIG


NPLNminus1NPL


NQLNminus1NQL


NPVNminus1NPV


Nminus1NPV

εL =

εIG1

εIGNIG

εPL1

εPLNPL

εQL1

εQLNQL

εPVa1

εPVaNPV

εPVb1

εPVbNPV

LL =

[PGmin1

minus PLmax1

][PGmax1

minus PLmin1

][


][PGmaxNminus1

minus PLminNminus1

][QGmin1

minus QLmax1

][QGmax1

minus QLmin1

][


][QGmaxNminus1

minus QLminNminus1

]

57

RLo =

P1o

PNo

Q1o

QNo

qL =


1Nh


Nminus1Nh


1Nh


Nminus1Nh

εA1

εANh


minsumεIGminm

+sumεPLminnl

+sumεQLminnl

+sumεPVaminnc

+sumεPVbminnc


εQLminnlle 1


le 1



(336)

maxsumεIGmaxm

+sumεPLmaxnl

+sumεQLmaxnl

+sumεPVamaxnc

+sumεPVbmaxnc


εQLmaxnlle 1


le 0



(337)






58




εPLminnl

εPLmaxnl εQLminnl



34 Example






59



60

341 Case A









61



62


342 Case B


343 Case C


63





64



65

35 Summary



66

Chapter 4


41 Introduction


67



Vi = Voi +sum

misinNIG

partVi

partPGm


sumncisinNPV

partVi

partVnc



)foralli = 1 N (41)

θi = θoi +sum

misinNIG

partθi

partPGm


sumncisinNPV

partθi

partVnc



)foralli = 1 N (42)






whereλ = λo +

summisinNIG

partλ

partPGm


sumncisinNPV

partλ

partVnc



)(46)




68





69


VPQ = Vp (49)






= 0 (411)



= minusJaugr

∆θr

∆Vr

∆λr

(412)

70

θr+1

Vr+1

λr+1

=

θr

Vr

λr

+

∆θr

∆Vr

∆λr

(413)









Avsεvs = Cvs (417)


where

Rvso =





71



εPLmaxand εQLmin







and ∆QLi = QLoi

72

Start

Read system data




to compute and





No Stop





73


5 15 25 35MCS ∆Lmax(MW) 33546 33667 33752 33805

AA∆Lmax(MW) 33547 33697 33847 33997

e() 0003 0090 0280 0566

SF∆Lmax(MW) 33555 33716 33877 34038

e() 0026 0146 0369 0688MCS ∆Lmin(MW) 33393 33215 33004 32761

AA∆Lmin(MW) 33367 33157 32951 32745

e() -0082 -0172 -0162 -0049

SF∆Lmin(MW) 33394 33233 33072 32911

e() 0003 0055 0204 0456





∆QLi = QLoiand


74

160 180 200 220 240 260 280 300 320 340

065

07

075

08

085

09

LOADING CHANGE (MW)

BU

S 2

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)





dλdp


∆λ

∆p

∣∣∣∣∣o

(419)

75

4000 5000 6000 7000 8000 900009

091

092

093

094

095

096

097

098

099

LOADING CHANGE (MW)

BU

S 2

383

VO

LT

AG

E M

AG

NIT

UD

E (

pu

)




76




AA∆Lmax(MW) 9622 4300 6577

e() 039 337 069

SF∆Lmax(MW) 9702 4400 6410

e() 122 577 -187MCS ∆Lmin(MW) 8072 2452 5120

AA∆Lmin(MW) 8038 2355 5111

e() -042 -396 -018

SF∆Lmin(MW) 7958 2246 5176

e() -141 -84 109



77

44 Sources of Error





78

45 Summary



79

Chapter 5


51 Introduction


80



˙xD(t) = fD

(xD y pD λ


0 = g(xD y pD λ) for y(0) = yIC (52)



fD


)+ fD

(xDn yn pD λ tn

))= 0 (53)

g(xDn yn pD λ) = 0 (54)


xD = xDo +sum

misinNIG

partxD

partPGAm


εIGm (55)

y = yo +sum

misinNIG

partypartPGAm


εIGm (56)


Y = Yo +sum

misinNIG

partYpartPGAm


εIGm (57)

81


fD(xDo yo pDo δo

)= 0 (58)

g(xDo yo pDo δo

)= 0 (59)




)= 0 minus FA

(zr) (510)

or equivalently

∆FA = J∆z (511)





εIGm

+sumhisinNh



82


∆FHsε f Hs=

sumhisinNh


s = 1NDA (513)



+sumhisinNh


(514)






(516)


∆z =

∆xD

∆y

(517)


bull Central values

∆FAos=

NDAsumk=1


83


∆FAms=

NDAsumk=1

Josk∆zmk +

NDAsumk=1

Jmsk∆zok


(519)


∆FHs =

NDAsumk=1

Josk∆zHk +

NDAsumk=1

JHsk∆zok


(520)


∆Fr

Ao1

∆FrAo2

∆FrAoNDA

=

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

∆zr

o1

∆zro2

∆zroNDA

(521)

∆Fr

Am1minus ∆zr

o1Jr


oNDAJr

m1NDA

∆FrAm2minus ∆zr

o1Jr


oNDAJr

m2NDA

∆FrAmNDA

minus ∆zro1

JrmNDA 1


JrmNDA NDA

= Aro

∆zr

m1

∆zrm2

∆zrmNDA


84

∆Fr

H1minus ∆zr

o1Jr

H11 minus ∆zr

oNDAJr

H1NDA

∆FrH2minus ∆zr

o1Jr


oNDAJr

H2NDA


o1Jr


oNDAJr

HNDA NDA

= Aro

∆zr

H1

∆zrH2

∆zrHNDA

(523)

where

Aro =

Jr

o11Jr

o12 Jr

o1NDA

Jro21

Jro22

Jro2NDA

JroNDA 1

JroNDA 2

JroNDA NDA

(524)


xr+1D = xr

D + ∆xrD (525)

yr+1 = yr + ∆yr (526)





85

Start

Read system data


(5-25)(5-26)



Yes


No convergence

Store Variables

End


Yes

No


variables

No

No Yes

Yes



86






30 01 0025 0025 01 10 1 55



87

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)






88

0 05 1 15 2 2504

05

06

07

08

09

1

11

12

13

Time (s)

δ (r

ad)








89

0 05 1 15 2 25

05

1

15

Time (s)

δ (r

ad)



0 01 02 03 04 050

05

1

15

2

25

3

35

4

Time (s)

δ (r

ad)



90

0 01 02 03 04 050

1

2

3

4

5

6

Time (s)

δ (r

ad)




18 17 02 03 055 025 025 00025 8 04


003 005 65 65 6175 6175 200 001 001 900



91

0 01 02 03 04minus05

0

05

1

15

2

25

3

Time (s)

δ 1 minus δ

2 (ra

d)








92

0 1 2 301

015

02

025

03

035

Time (s)

δ 3 minus δ

4 (ra

d)




03 65 6175 01



010 008 3 001 001 50 50 001 001 3

93

0 2 4 6 8 10

minus15

minus1

minus05

0

05

Time (s)

δ 3minus δ 1 (

rad)





94

006 008 01 012 014 016 018minus05

0

05

1

15

2

25

3

Time (s)

δ 3minusδ1 (

rad)




54 Summary


95

Chapter 6






96







97




63 Future Work




98

Appendix A





)= 0 (A1)


(f2a + f2b

)= 0 (A2)

where


] ω2s

2Hωn(A3)


] ω2s

2Hωnminus1(A4)

where


Pe =EprimeV2

Xprimed + XT + X12X13 (X12 + X13)(A5)

99



Xprimed + XT + X12X13(X12 + X13)(A6)


Pe =EprimeV2


100

Appendix B



Eprimeprime

qnminus E

primeprime

qnminus1minus

∆t2

(f1a + f1b

)= 0 (B1)

wheref1a =

1T primeprime

do

(Eprime

qnminus

(Xprime

d minus Xprimeprime

d

)Idn minus E

primeprime

qn

)(B2)

f1b =1

T primeprime

do

(Eprime

qnminus1minus

(Xprime

d minus Xprimeprime

d

)Idnminus1 minus E

primeprime

qnminus1

)(B3)

Eprimeprime

dnminus E

primeprime

dnminus1minus

∆t2

(f2a + f2b

)= 0 (B4)

wheref2a =

1T primeprime

qo

(Eprime

dnminus

(Xprime

q minus Xprimeprime

q

)Iqn minus E

primeprime

dn

)(B5)

f2b =1

T primeprime

qo

(Eprime

dnminus1minus

(Xprime

q minus Xprimeprime

q

)Iqnminus1 minus E

primeprime

dnminus1

)(B6)

Eprime

qnminus E

prime

qnminus1minus

∆t2

(f3a + f3b

)= 0 (B7)

101

wheref3a =

1T prime

do

(E fn +

(Xd minus X

prime

d

)Idn minus E

prime

qn

)(B8)

f3b =1

T prime

do

(E fnminus1 +

(Xd minus X

prime

d

)Idnminus1 minus E

prime

qnminus1

)(B9)

Eprime

dnminus E

prime

dnminus1minus

∆t2

(f4a + f4b

)= 0 (B10)

wheref4a =

1T prime

qo

(minus

(Xq minus X

prime

q

)Iqn minus E

prime

dn

)(B11)

f4b =1

T prime

qo

(minus

(Xq minus X

prime

q

)Iqnminus1 minus E

prime

dnminus1

)(B12)

ωn minus ωprime

n minus∆t2

1M

(2Pm + f5a + f5b

)= 0 (B13)


(θn minus δn

)minus IqnVncos

(θn minus δn

)minus Da

(ωn minus ωs

)(B14)




)minus Da


)(B15)



)= 0 (B16)


(f7a + f7b

)= 0 (B17)

wheref7a =

1Tec


)(B18)

f7b =1

Tec


)(B19)

Eprimeprime


primeprime

d Id + Vq

)= 0 (B20)

Eprimeprime


primeprime

d Iq + Vd

)= 0 (B21)

102

Vq minus(Vcos

(θIa minus δ

))= 0 (B22)

Vd minus(V sin

(θIa minus δ

))= 0 (B23)

Iq minus(Icos

(θIa minus δ

))= 0 (B24)

Id minus(I sin

(θIa minus δ

))= 0 (B25)

Network equations



)minus PL j minus

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))= 0 (B26)


)minus QL j minus

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))= 0 (B27)

bull For load buses

Nsumi=1

(V jVi

(G jicos

(θ j minus θi

)+ Bi jsin

(θ j minus θi

)))+ PLi = 0 (B28)

Nsumi=1

(V jVi

(G jisin

(θ j minus θi

)minus Bi jcos

(θ j minus θi

)))+ QLi = 0 (B29)

103

Appendix C



ωm =

(Pω


)1

2Hm(C1)

iqr =

(minus

xs + xm

xmVPoptω (ωm)ωm

minus iqr

)1Tr

(C2)

idr = Kv

(V minus Vre f

)minus

Vxmminus idr (C3)

θp =Kp

Tp

(ωm minus ωre f

)(C4)

where


104


Pω =ρ

2cp

(λp θp

)Arv3 (C7)


)e(minus125λi

)(C8)

1λi

=1

λp + 008θpminus

0035λ3

p + 1(C9)









Pω

ωmn


idsn)1

2Hm

+

∆t2

Pω

ωmnminus1


idsnminus1)1

2Hm

= 0

(C14)

105



+ xmiqrn) (C16)


minus∆t2lowast

minus xs + xm

xmV

Poptωn

(ωmn)

ωmn

minus iqrn

1Tr

+

minus xs + xm

xmV

Poptωnminus1

(ωmnminus1)

ωmnminus1

minus iqrnminus1

1Tr

= 0

(C17)


Kv

(Vn minus Vre f

)minus

Vn

xmminus idrn + Kv


)minus

Vnminus1

xmminus idrnminus1

= 0 (C18)


(Kp

Tp


)+

Kp

Tp


))= 0 (C19)


+ xmiqrn

)= 0 (C20)

vqsnminus

(minusrsiqsn


)= 0 (C21)


+ xmiqsn))

= 0 (C22)

vqrnminus

(minusrriqrn


= 0 (C23)


106






107

Bibliography








108











109











110












111











112











113










114











115










116











117












118











119









120

Authors Declaration

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures


Nomenclature

Introduction

Research Motivation

Literature Review




Research Objectives

Thesis Outline

Background Review

Introduction


Power Flow

Voltage Stability

Transient Stability


Interval Arithmetic

Affine Arithmetic


Summary


Introduction

Problem Formulation


Example

Case A

Case B

Case C

Summary


Introduction

Problem Formulation

Simulation Results

5-Bus Test System


Sources of Error

Summary


Introduction

Problem Formulation

Simulation Results




Summary



Main Contributions

Future Work

Appendices




Bibliography

a ne arithmetic based methods for power systems analysis

Documents