nonlinear security constrained optimal power flow … · abstract the goal of this master thesis is...

eeh power systemslaboratory

Vasileios Saplamidis

Nonlinear Security Constrained OptimalPower Flow for Combined AC and HVDC

Grids

Master ThesisPSL 1417

EEH – Power Systems LaboratoryETH Zurich

Supervisor: Roger WigetExpert: Prof. Dr. Goran Andersson

Zurich, November 19, 2014

Abstract

The goal of this master thesis is the development of an algorithm for solvingthe non-linear Optimal Power Flow (OPF) and Security Constrained Opti-mal Power Flow (SCOPF) problems for combined AC and Muti-TerminalDC (MTDC) grids. Both grids are modeled using the non-linear power flowequations. Grid-level models for AC/DC converters are developed based onthe Voltage Source Converter (VSC) design with non-linear power through-put constraints and current-dependent losses. A new model for DC/DCconverters is proposed based on modern high power - low ratio converterdesigns. Such devices are used to improve the OPF solution for combinedAC/MTDC grids by controlling the DC power flows. The formulation of theSCOPF problem is such that it works for preventive OPF where no correc-tive actions are allowed after the contingency, as well as for corrective OPFwith adjustable actions limits. A contingency filtering approach to reducethe SCOPF problem size, is also developed and its impact on the conver-gence speed and accuracy is investigated.

Several case studies are performed to test both the new SCOPF algorithmand the developed models. Both the OPF and SCOPF solutions are used fora benefit analysis of a proposed MTDC grid expansion. In another study,the impact of DC/DC converters as power flow regulators on MTDC gridsis found to correlate strongly with the DC line lengths. A sensitivity analy-sis of the corrective actions limits on the generation cost is also performed.Generator and converter limits impact is investigated separately and the re-sults are compared.

Keywords: Contingency filtering, High Voltage DC (HVDC) Transmis-sion, Multi-Terminal HVDC (MTDC), Optimal Power flow (OPF), SecurityConstrained OPF (SCOPF).

iv

Acknowledgment

I would like to thank the people that contributed to the fulfillment of thisthesis and especially:

• Roger Wiget, PhD student at PSL, for our excellent collaboration andfruitful discussions during the whole period of this thesis.

• Prof. Dr. Goran Andersson for his support throughout my Masterstudies and for allowing my to work on this interesting topic.

I would also like to thank all my friends in Switzerland and Greece for theirsupport throughout the duration of my studies. Special thanks to Alina andVassilis.Finally I would like to express my gratitude to my parents, George andCleopatra and my sister Sissy. Without their total support and encourage-ment nothing of all this would be possible.

Zurich, November 2014Vasileios Saplamidis

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Contents

1 Introduction 11.1 Motivation and Literature Review . . . . . . . . . . . . . . . 31.2 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Models 52.1 Buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Generators and Loads . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 AC/DC Converters . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Active Power Exchange - Losses . . . . . . . . . . . . 142.4.2 Voltage Limits . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Power Throughput Constraints . . . . . . . . . . . . . 15

2.5 DC/DC Converters . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Optimal Power Flow 193.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Optimization Vector . . . . . . . . . . . . . . . . . . . 193.1.2 Objective Function . . . . . . . . . . . . . . . . . . . . 203.1.3 Equality Constraints . . . . . . . . . . . . . . . . . . . 203.1.4 Inequality Constraints . . . . . . . . . . . . . . . . . . 21

3.2 OPF Problem Formulation . . . . . . . . . . . . . . . . . . . 233.3 SCOPF Problem Formulation . . . . . . . . . . . . . . . . . . 24

3.3.1 Types of SCOPF Problems . . . . . . . . . . . . . . . 27

4 Algorithm Improvements 294.1 Contingency Filtering . . . . . . . . . . . . . . . . . . . . . . 30

5 Test Cases - Results 335.1 Test Cases Description . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 9-bus AC with 5-bus MTDC . . . . . . . . . . . . . . 335.1.2 IEEE 14-bus AC with 5-bus MTDC . . . . . . . . . . 34

vi

CONTENTS vii

5.1.3 RTS-96 one area with 8-bus MTDC . . . . . . . . . . 345.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1 Improved SCOPF Algorithm . . . . . . . . . . . . . . 365.2.2 Economic Benefit of MTDC Grid Expansion . . . . . 375.2.3 DC-DC Converters as Power Flow Control Devices . . 405.2.4 Influence of the Corrective Actions Limits on the SCOPF

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.5 Sensitivity Analysis of the ∆wk Components . . . . . 46

6 Conclusion 496.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Motivation for Future Research . . . . . . . . . . . . . . . . . 50

A Test systems data 51A.1 9-bus AC with 5-bus MTDC - System Data . . . . . . . . . . 51A.2 IEEE 14-bus AC with 5-bus MTDC - System Data . . . . . . 54A.3 RTS-96 one area with 8-bus MTDC - System Data . . . . . . 57

Bibliography 61

List of Figures

2.1 Venn diagram of different bus types . . . . . . . . . . . . . . 52.2 Π equivalent model of a power line . . . . . . . . . . . . . . . 62.3 Venn diagram of different branch types . . . . . . . . . . . . . 82.4 Power flows sign convention . . . . . . . . . . . . . . . . . . . 102.5 PQ - capability curve [24] . . . . . . . . . . . . . . . . . . . . 112.6 HVDC converter station [5] . . . . . . . . . . . . . . . . . . . 122.7 HVDC VSC station model . . . . . . . . . . . . . . . . . . . . 122.8 VSC converter station simplified model . . . . . . . . . . . . 132.9 VSC converter losses concept . . . . . . . . . . . . . . . . . . 142.10 Example of a VSC converter PQ capability curve . . . . . . . 162.11 DC/DC converter model . . . . . . . . . . . . . . . . . . . . . 17

3.1 Equivalent circuit of the VSC station model AC side . . . . . 233.2 SCOPF Algorithm including constraint building . . . . . . . 26

4.1 Improved SCOPF algorithm - Contingency filtering method . 31

5.1 14-bus AC/MTDC system . . . . . . . . . . . . . . . . . . . . 335.2 19-bus AC/MTDC system . . . . . . . . . . . . . . . . . . . . 345.3 32-bus AC/MTDC system . . . . . . . . . . . . . . . . . . . . 355.4 Severity Indexes on all contingencies in the 14-bus system . . 365.5 Economic benefit evaluation algorithm . . . . . . . . . . . . . 385.6 Power flow on line 27-32 . . . . . . . . . . . . . . . . . . . . . 415.7 Power losses on line 27-32 . . . . . . . . . . . . . . . . . . . . 415.8 Total generation cost . . . . . . . . . . . . . . . . . . . . . . . 425.9 Active generation dispatch . . . . . . . . . . . . . . . . . . . . 435.10 Cost of generation . . . . . . . . . . . . . . . . . . . . . . . . 445.11 AC bus voltages . . . . . . . . . . . . . . . . . . . . . . . . . 455.12 DC bus voltages . . . . . . . . . . . . . . . . . . . . . . . . . 455.13 Average voltage deviation . . . . . . . . . . . . . . . . . . . . 455.14 SCOPF cost increase due to converters’ maximum deviation

limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.15 SCOPF cost increase due to generators’ maximum deviation

limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

viii

LIST OF FIGURES ix

5.16 Influence of both ∆Pg and ∆Pc on the % generation costincrease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

List of Tables

2.1 Typical VSC station model parameters [21] . . . . . . . . . . 13

5.1 Comparison of the improved algorithm with the full pr-SCOPF 375.2 Generator dispatch comparison . . . . . . . . . . . . . . . . . 375.3 Results of economic benefit analysis of MTDC grid expansion

on the IEEE 14-bus test system . . . . . . . . . . . . . . . . . 395.4 Test Cases description . . . . . . . . . . . . . . . . . . . . . . 43

A.1 9-bus AC with 5-bus MTDC: Bus Data . . . . . . . . . . . . 51A.2 9-bus AC with 5-bus MTDC: Generator Data . . . . . . . . . 52A.3 9-bus AC with 5-bus MTDC: SVC Data . . . . . . . . . . . . 52A.4 9-bus AC with 5-bus MTDC: VSC Data . . . . . . . . . . . . 52A.5 9-bus AC with 5-bus MTDC: VSC Data (cont.) . . . . . . . . 52A.6 9-bus AC with 5-bus MTDC: Branch Data . . . . . . . . . . 53A.7 IEEE 14-bus AC with 5-bus MTDC: Bus Data . . . . . . . . 54A.8 IEEE 14-bus AC with 5-bus MTDC: Generator Data . . . . . 54A.9 IEEE 14-bus AC with 5-bus MTDC: SVC Data . . . . . . . . 55A.10 IEEE 14-bus AC with 5-bus MTDC: VSC Data . . . . . . . . 55A.11 IEEE 14-bus AC with 5-bus MTDC: VSC Data (cont.) . . . . 55A.12 IEEE 14-bus AC with 5-bus MTDC: Branch Data . . . . . . 56A.13 RTS-96 one area with 8-bus MTDC: Bus Data . . . . . . . . 57A.14 RTS-96 one area with 8-bus MTDC: Generator Data . . . . . 58A.15 RTS-96 one area with 8-bus MTDC: SVC Data . . . . . . . . 58A.16 RTS-96 one area with 8-bus MTDC: VSC Data . . . . . . . . 58A.17 RTS-96 one area with 8-bus MTDC: VSC Data (cont.) . . . . 59A.18 RTS-96 one area with 8-bus MTDC: Branch Data . . . . . . 59A.19 RTS-96 one area with 8-bus MTDC: Branch Data (cont.) . . 60

x

List of Symbols

Unless indicated otherwise, vectors and matrices are written with bold let-ters while scalar values with normal ones.

Latin letters

Symbol Units DescriptionA − Set including all AC busesC − Set including all AC buses with AC/DC converters

attachedD − Set including all DC busesE − Set including all DC buses with AC/DC converters

attachedG − Set including all buses with generatorsKA − Set including all branches between AC busesKD − Set including all branches between DC busesS − Set including all buses with SVCsT − Set including all branches with tap changing or

phase shifting transformers

b Ω−1 or p.u. Shunt susceptance of linec0 $/hourMW2 Cost coefficients for generatorsc1 $/hourMW Cost coefficients for generatorsc2 $/hour Cost coefficients for generatorsC $/hour Generation costDF − Discount factorg Ω−1 or p.u. Conductance of linei − Interest rateI A or p.u. Currentkv − Voltage relationship factor across a VSCkQ − Reactive power factor for a VSCKk − Set of buses adjacent to k, including kL W or p.u. LossesN − Number of ...

xii

LIST OF SYMBOLS xiii

p $/MWh Electricity priceP W or p.u. Active power flowQ VAr or p.u. Reactive power flowr Ω or p.u. Resistance of lineS VA or p.u. Apparent power flowt − Complex turns ratio of transformerTEB $ Total Economic BenefitU Volts or p.u. Voltage magnitudex Ω or p.u. Reactance of liney Ω−1 or p.u. Complex admittance of linez Ω or p.u. Complex impedance of line

x − Optimization vector1

u − Control vector1

z − State vector1

g − Equality constraints1

h − Inequality constraints1

f − Objective functionw − Subset of u containing active power generation and

converter active power throughput1

Greek letters

Symbol Units Descriptionα − Magnitude of transformer ratioθ radians Voltage angleκ0 MW Loss coefficients for a VSCκ1 MW/A Loss coefficients for a VSCκ2 MW/A2 Loss coefficients for a VSCφ radians Angle of transformer ratioΩk − Set of buses adjacent to k, excluding k

Subscripts

Symbol Descriptionb Bus(es)br Branch(es)c Converter AC side(s)cont Contingenciesd Load(s)

1For these variables, numerical subscripts refer to normal operation (0) or the k-th postcontingency case (k 6= 0).

LIST OF SYMBOLS xiv

dc Converter DC side(s)exp With MTDC grid expansionfilter VSC station filterfrom From busg Generator(s)inj Power injected on busk At2 bus kkk Between bus k and groundkm Between2 buses k and m, k 6= mloss Lossesnoexp Without MTDC grid expansionnom Nominal valuephr VSC station phase reactorref Reference values Static VAr Compensator(s)tf VSC station transformerto To bus

Superscripts

Symbol Description∗ At the optimum (or complex conjugate depending

on context)AC AC quantitiesDC DC quantitiessh Shunt elementy Year

Special symbols

Symbol Description· Maximum· Minimum[ · ] Diagonal matrix with the elements of · in the di-

agonal· Vector

2For symbols that represent power flows, a single subscript with the bus number indi-cates power injection, while a double subscript indicates power flow between two buses.For symbols that represent angles, a single subscript indicates the voltage angle betweenthis bus and the reference bus, while a double subscript is defined as the difference of theangle between two buses.

List of Acronyms

AC Alternating Currentc-SCOPF Corrective Security Constrained OPFCF Contingency FilteringDC Direct CurrentHVDC High Voltage DCIEEE Institute of Electrical and Electronics EngineersIGBT Insulated-Gate Bipolar TransistorLCC Line Commutated ConverterMTDC Multi-Terminal DCOPF Optimal Power Flowpr-SCOPF Preventive Security Constrained OPFPCC Point of Common ConnectionPWM Pulse-Width ModulationRTS Reliability Test SystemSCOPF Security Constrained OPFSI Severity IndexSVC Static VAr CompensatorVSC Voltage Source ConverterWSCC Western Systems Coordinating Council

xvi

Chapter 1

Introduction

The global energy demand is growing and there is a strong desire to moveaway from conventional energy sources towards environmentally friendly re-newables. Due to uncertainties that arise from the fluctuating nature of therenewable production, one of the main limitations of renewable penetrationis the capacity of the existing grids. Further, solar and wind energy sourcesare dependent on weather conditions and can usually not be built close tothe load centers.

The use of High Voltage DC (HVDC) technology for grid expansion andintegration of renewable energy sources has long been suggested and is re-garded as the optimal solution in many cases. Such cases include offshorewind generation further than 100km away from the nearest connection pointand connection in weak Alternating Current (AC) grid nodes [1]. The ad-vantages of DC transmission lines over the AC ones have been summarizedin [2] and include lower losses due to the elimination of the skin effect andline-to-line inductance phenomena, lower charging currents and cheaper subsea cable connections. In addition, HVDC interconnections are inherentlycontrollable since the Voltage Source Converters (VSCs) offer a large degreeof control in their power output characteristics.

Voltage Source Converters are a relatively recent development in the fieldof power converters. Most sources agree that the VSCs will soon be themain converter type used in HVDC grids, because of their superior con-trol characteristics over the now mainstream Line Commutated Converters(LCCs) [2–4]. The main advantages of a VSC over an LCC are its ability forsimultaneous and independent AC voltage and power flow control as well asalmost instantaneous power flow reversal without interruption and withoutchanging the polarity of the interconnection. Different types of VSCs havebeen proposed in the literature that range from 2-level converters using cus-tom Insulated-Gate Bipolar Transistors (IGBTs) [5] to modular multilevel

1

CHAPTER 1. INTRODUCTION 2

converters [6] as well as hybrid models [7]. Each of these different VSC tech-nologies has advantages and disadvantages concerning harmonics injection,fault blocking capabilities and they are in different stages of development.In the scope of this thesis the actual VSC implementation is not importantas long as the power and voltage controllability are provided.

The Optimal Power Flow (OPF) problem is one of the main ways to an-alyze and study a power system. The problem is based upon an objectivefunction which is to be minimized, by changing the values of certain con-trol variables while respecting some relevant constraints. The minimizedquantity usually has to do with the cost of generation but can also includeoptimal allocation of reactive power sources [8] or factors that affect thestability of the system [9].

The Security Constrained OPF (SCOPF) problem is an extension of theOPF problem which calculates the cost of steady state security. Steady statesecurity is defined in [10] as “the ability of the system to operate steady-state-wise within the specified limits of safety and supply quality followinga contingency, in the time period after the fast-acting automatic control de-vices have restored the system balance but before the slow-acting controlshave responded”. The additional constraints introduced in this OPF schemelead to a different optimum because the base-case generation dispatch ischanged to satisfy all the contingency cases. Due to the lack of computa-tional power, when these methods were first applied, only the most likelyor severe outages were taken into account when formulating the additionalconstraints. In this thesis the SCOPF formulation of [11] is expanded andthen applied with all possible contingencies including generator and AC/DCconverter outages and line failures. Only single element failures are studiedsince most transmission systems are designed around the N-1 security con-cept [12].

The power flow/optimal power flow problems can be solved either in theirfull non-linear form or in a simplified (linearized) version. The former gener-ally yields more accurate results and is able to calculate reactive power flowsand line losses but is much more computationally intensive. The latter, alsoknown as “DC power flow”, is generally much faster but due to the approxi-mations taken in the linearization process, it is not able to calculate reactivepower flows, line losses or voltage levels on the system buses. Within thisthesis, all the models and methods discussed make use of only the non-linearpower flow equations.


1.1 Motivation and Literature Review

Several methods for solving power flow problems in combined AC and HVDCgrids have been proposed in literature. Most power flow calculation meth-ods for combined AC and HVDC grids are developed around a VSC stationmodel. The modeling details vary among different approaches but generallyinclude a transformer, some passive current filtering and the actual converterdevice which can be modeled to include the power electronics induced lossesas well as voltage and power throughput limits.

In [13] the converter model only includes the power throughput and voltageconstraints but neglects the converter losses. A more complete model is pre-sented in [14] which includes phase reactors and filters on the AC side as wellas the transformer that connects to the AC grid. In [15] the converter lossesare also taken into account by using the simple method proposed in [16]that relates the VSC converter losses to the phase reactor current. Such anapproach is also made in this thesis, as the aforementioned losses model issimple enough to be used in a computationally intensive optimization pro-cess and relies on quantities that are already calculated on a grid level. Theaddition of multiple AC/DC converters in the system increases the problemcomplexity since the number of lines and buses of the system is increased.The main VSC station model can be reduced to a Π equivalent throughsuitable transformations in order to reduce the final system complexity [17].

The use of DC/DC converters in large DC grids has been proposed in [18].They can be used to connect DC grids of different voltage levels, or to pro-vide power flow control within a single Multi-Terminal HVDC (MTDC) grid.There are various DC/DC converter models proposed in literature based incontrol oriented approaches [19]. However, such a detailed converter modelis not needed for modeling the transmission grid level influence of such de-vices. A simple non-linear model has been used in this thesis based on theVSC converter model of [20] modified according to the DC/DC convertercharacteristics proposed in [18].

An algorithm for solving the OPF problem in combined AC and HVDCsystems, based on Newton’s method has been developed in [13]. Feng etal. [21] have proposed a benefit evaluation method based on a combinedOPF using a model similar to the one used in this thesis. Their algorithmuses the OPF solution to extract the power losses and generation cost un-der different scenarios. The monetary benefits are projected in the futurefor the lifetime of the investment to calculate the total economic benefit(or loss) of a planned MTDC system. Different MTDC topologies are alsoinvestigated. In another case study in [14], the OPF problem formulationdivides buses into AC buses and DC buses which have generally different


equality constraints. AC buses are further divided into Points of CommonConnection (PCC) and non-PCC. One of the converter nodes assumes therole of a “slack” converter. The results of this study show that the com-bined AC/DC system has generally better voltage profiles in the expense ofa slightly increased generation cost. Finally Wiget et al. [20] have proposeda unified approach to solve the combined AC/DC OPF problem that takesconverter losses into account and uses the non-linear power flow equations.

The SCOPF approach on combined AC/MTDC grids is a novel conceptthat (to the best of our knowledge) hasn’t been investigated until now. TheSCOPF problem for conventional (AC) transmission grids has been exten-sively studied in [10]. According to [11] the main issues with the traditionalSCOPF problem solution is the enormously increased size of the problem.The non-linear approach for solving the SCOPF problem in large grids whiletaking all possible contingencies into account demands unrealistically largecomputational power. In the same paper, alternative formulations and so-lution techniques are discussed to solve such problems, including Bendersdecomposition, linearization of post contingency constraints and networkcompression.

1.2 Goal

In this thesis an algorithm will be developed for solving the OPF and SCOPFproblem in mixed AC/DC grids. All the calculations assume a steady stateboth before and after the contingencies. The use of non-linear equations formodeling power flows enables losses to be calculated for both AC and DCgrids. Devices to be modeled include tap changing transformers and phaseshifters on the AC grid, AC/DC converters based on VSC technology andDC/DC converters that are controlled as equivalent DC transformers in theDC grid. Losses of all these converters will also be taken into account. TheSCOPF problem will be solved by using an interior-point solver in Matlab.

1.3 Thesis Structure

The rest of this thesis is organized as follows. Chapter 2 describes the basicequations that model the power flows in AC/DC grids and also the non-linear models used for the AC/DC and DC/DC converters. In Chapter3 the optimization problem is defined and all relevant constraints are dis-cussed. The OPF problem is also extended to include security constraints.A contingency filtering method is proposed in Chapter 4 to reduce the run-ning time of the SCOPF calculations. Chapter 5 describes the test casesstudied in this thesis and the the obtained results. Finally, the results arediscussed in Chapter 6, and suggestions are made for future research goals.

Chapter 2

Models

For the power flow problem to be formulated and solved, the transmission gridmust be mathematically modeled. In this Chapter, the models used for thevarious grid elements are presented and analyzed. The resulting equationswill be used to form the Optimal Power Flow problem.

2.1 Buses

In a transmission grid, buses are the connection points for all other parts,like transmission lines, loads and generators. In a mixed AC - MTDC gridthere are AC (set A) and DC buses (set D). The different bus sets and theirrelationships are shown in the Venn diagram of Fig. 2.1. The connection ofthe two grids takes place through AC/DC converters placed between someAC and DC nodes (sets C and E respectively). Generators are allowed onany bus (set G) but SVCs can be located only on AC buses (set S) sincethey need to inject reactive power in the grid.

A

C

D

E

G

S

ac dc

AC/DC converters

Figure 2.1: Venn diagram of different bus types

5

CHAPTER 2. MODELS 6

2.2 Lines

An AC transmission line shorter than 300km can be described with thesimplified Π model (also known as ”lumped-circuit model”) with enoughaccuracy for most power flow calculations [22]. A homogeneous power linebetween nodes k and m has an equivalent Π model as depicted in Fig. 2.2.

km km kmz r jx

shkmy sh

kmy

mIkI

kjkU e mj

mU e

:1kmt

km kt I

kjk

km

U et

Figure 2.2: Π equivalent model of a power line

The series impedance zkm = rkm + jxkm models the resistive behavior ofthe line materials as well as any inductance phenomena on the conductors.The shunt elements yshkm model the total shunt admittance of the line andby convention are defined as half of the total line shunt admittance each.

yshkm = j

bshkm2

(2.1)

This general definition of a branch also allows modeling of branch deviceslike transformers and phase shifters. The ideal transformer (tkm : 1) of Fig.2.2 has a complex turns ratio.

tkm = αkmejφkm (2.2)

In this model a tap changing transformer between k and m will have φkm = 0and αkm would be a control variable, while a phase shifter will have a con-trollable φkm. Obviously an αkm = 1 and φkm = 0 would imply a power linewithout any kind of transformer. In real world applications the ratio andangles of such devices can only take discreet values. In this thesis all con-trol variables including φkm and αkm are allowed to take continuous values.In most real world tap changers of phase shifters their discrete operationpoints are so close together that the continuous variable approximation isvalid. Moreover, the non-linear nature of the problem in conjunction withthe large scale implied by the SCOPF formulation would result in unrealisticrunning times if a discreet variable scheme was assumed (integer program-ming).

CHAPTER 2. MODELS 7

When modeling power flows in a transmission grid, it is often useful towork with power-voltage relationships. The apparent power flow betweentwo buses k and m can be calculated as in (2.3).

Skm = y∗kmUkejθk(Uke

−jθk − Ume−jθm)− jbshkmU2k ⇒

Skm = (gkm − jbkm)(U2k − UkUme−j(θk−θm))− jbshkmU2

k (2.3)

Separating real and imaginary parts yields:

Pkm = gkmU2k − UkUm(gkm cos(θkm) + bkm sin(θkm)) (2.4)

Qkm = −(bkm + bshkm)U2k + UkUm(bkm cos(θkm)− gkm sin(θkm)) (2.5)

where θkm ≡ θk − θm.

For a DC line, the AC related quantities are zero, and this leads to thefollowing equation for the active power flow:

PDCkm = 2gkmUk(Uk − Um) (2.6)

The factor 2 in (2.6) comes from the assumption that the cables on the DCside have bipolar structure [20].

Most calculations in the Matlab environment are inherently faster whenmatrix equations are used instead of sets of individual non-linear equations.By defining the admittance matrix Ybus, such matrix equations can be con-structed for the power flows and nodal power injections. Assuming that thegrid contains Nb buses and Nbr branches, the current injections at all buses(I) can be calculated as:

I = YbusU (2.7)

where I is an Nb × 1 vector containing the current injection at each bus, Uis an Nb × 1 vector containing each bus (complex) voltage, and Ybus is aNb ×Nb matrix whose elements are calculated through (2.8).

Ykm = −t∗kmtmkykmYkk = ysh

k +∑m∈Ωk

α2km(ysh

km + ykm) (2.8)

Where ykm = gkm + jbkm = 1/zkm, yshk are shunt elements connected to busk and Ωk is the set of buses adjacent to bus k, excluding k. Ybus is knownas the nodal admittance matrix and is usually sparse and (if transformerswith complex ratio are used) not symmetric.

CHAPTER 2. MODELS 8

The elements of this matrix consist of the branch connections between busesand any shunt elements. The connections between the AC and DC gridsthrough the converters do not influence the admittance matrix structurebut are taken into account through the equality constraints (see Section3.1.3). Therefore, the nodal admittance matrix of a mixed AC/DC systemhas the form of (2.9):

Ybus =

1 ∅

∅ 2

(2.9)

Where:

1 = Connections between AC buses and

2 = Connections between DC buses.

The indexes pairs of the non-zero elements of the two sub-matrices of theadmittance matrix define the set KA ∪ KD of bus connections (branches)shown in Fig. 2.3. The subset T ⊆ KA consists of the branches that havetap changing or phase shifting transformers. DC branches with DC/DCtransformers are again modeled through the OPF problem’s constraints aredo not affect the Ybus.

KA KD

T

Figure 2.3: Venn diagram of different branch types

By comparing (2.4) and (2.6), and knowing that a DC line has only a resis-tive part and no reactive components, it is apparent that the same equationcan be used to describe a power flow in both AC and DC grids. The factor×2 that is introduced by the assumed bipolar structure of the DC lines istherefore included in the 2 part of the Ybus matrix whose elements aredefined as in (2.10)

CHAPTER 2. MODELS 9

Ykm = −2gkm

Ykk = 2∑m∈Ωk

gkm(2.10)

It can be shown that the apparent power injection on each bus can be cal-culated as:

Sinj = [U]Y∗busU∗ (2.11)

Where [U] is a Nb ×Nb diagonal matrix containing the elements of U (busvoltages). The non-linear nodal power injection equations implied through(2.11) are:

Pk = Uk∑m∈Kk

Um (gkm cos θkm + bkm sin θkm) (2.12)

Qk = Uk∑m∈Kk

Um (gkm sin θkm − bkm cos θkm) (2.13)

where Kk is the set of buses adjacent to bus k, including k itself.

Apart from the nodal admission matrix, one can use the line admittances toconstruct additional Nbr ×Nb system admittance matrices (Yfrom and Yto)such that:

Ifrom = YfromU (2.14)

Ito = YtoU (2.15)

Where Ifrom and Ito are the currents at the “from” and “to” ends of allbranches. The definition of Yfrom and Yto has been done in [23]. Thebranch power flows can now be calculated as:

Sfrom = [U]Y∗fromU∗ (2.16)

Sto = [U]Y∗toU∗ (2.17)

This calculation method is very useful in a programming environment likeMatlab due to its ability to make fast matrix calculations.

2.3 Generators and Loads

The general convention followed throughout this thesis is that power flowsto a bus are positive while flows from a bus are negative. In the example

CHAPTER 2. MODELS 10

in Fig. 2.4, bus k is connected to a generator, a load and a line to bus m.With the power flow directions that are depicted, Pg and Qg are positivewhile Pkm, Qkm, Pd and Qd are considered negative.

kto bus m

Pd+jQd

Pg+jQg Pkm+jQkm

Figure 2.4: Power flows sign convention

2.3.1 Generators

Due to losses in the generator’s armature winding that generate heat, thestator current cannot be higher than a certain value thus limiting the ap-parent power generation within a circle that forms the rightmost boundaryof the curve in Fig. 2.5. The reactive power generation is further limited bythe field current heating limit while the reactive power consumption for anunder-exited generator is also limited by heating phenomena in the statorlaminations [24]. All these limits are also dependent on the operating volt-age and can be different for each generator.

For the power flow calculation within this thesis this detailed approach is notneeded however. Thus all the limitations described so far have been approx-imated by constant-value limits on the active and reactive power generation.Each generator is characterized by 4 limiting constants: The maximum andminimum active generation (Pg and Pg) and the maximum and minimum

reactive generation (Qg and Qg respectively).

Each generator is also characterized by a cost function Ci(Pgi). The cost ofgeneration is generally dependent on the active power output and for mosttypes of generators increases as the output increases. This is usually referredto as increasing marginal cost. In reality, cost functions are not smooth butin the majority of power system studies they are approximated either bypiecewise linear curves or quadratic functions. In this thesis the quadraticcost approach is adopted as shown in (2.18).

Ci(Pgi) = c2P2gi + c1Pgi + c0 (2.18)


k

Iksh

Iksh

k

Ikload

Load

Ik Ikm

yshk

yshk

Q

P

Field currentheating limit

Stator currentheating limit

End regionheating limit

Under

exci

ted

Ove

rexc

ited

Figure 2.5: PQ - capability curve [24]

2.3.2 Loads

In the scope of this thesis consumption is considered inflexible. This impliesthat all loads have constant complex values of Sd = Pd + jQd. Since all ofour calculations (both pre- and post-contingency) refer to steady states ofthe grid, there is no load fluctuation.

2.4 AC/DC Converters

While HVDC technology has been in use for almost half a century, onlyrelatively recently advances in AC/DC converter design have made MTDCgrids a realistic goal. The oldest still used technology is this of the LCC.In this type of power electronic devices the power flow is controlled throughthe DC voltage and the current flow is controlled to be constant.

After the invention of the IGBT, Pulse Width Modulation (PWM) tech-niques have been applied on the control of AC/DC converters leading tothe development of the VSC. This type of converter offers straightforwardAC side voltage and angle control, as well as independent current flow con-trol [25]. This means that power flow reversal can be achieved in a shorttime without large changes in the DC voltage. Despite LCC being still themost used technology in DC interconnections worldwide, VSC convertersare very likely to take over in the near future [2]. Thus only VSC converters


are modeled and used throughout this thesis as AC and DC grid connectionpoints.

HVDC converters are usually built within a station (Fig. 2.6) that containsthe converter itself complete with phase reactors, switches and transformersfor the AC grid interface as well as additional filters (if needed) and the DCcables connection interface [5].

Figure 2.6: HVDC converter station [5]

A VSC station model that takes into account all of the above and has beenwidely used in literature [21,26] is shown in Fig. 2.7.

k dccf

Uk, θk Uf, θf Uc, θc

Ic Idc

Udc

Ploss

Pc Pdc

ztf zphr

zfilter

Figure 2.7: HVDC VSC station model

The bus of the AC grid where the AC/DC converter is connected is labeledas bus k. The transformer that connects the station with the main AC gridis modeled with its equivalent impedance (typically a reactance) ztf . Thistransformer can generally be a controllable tap changing device but, becausethe voltage on bus c is assumed controllable, it is assumed that ztf representsa constant ratio transformer. On the station side of the transformer there is


an additional bus f for the AC filter (zfilter) to be connected. Such filtersare used to absorb high current harmonics. In modern VSC convertershowever, the current waveform is so close to sinusoidal that such filtersare usually considered optional. Lastly, another impedance zphr is used tomodel the phase reactors of the converter itself. Typical p.u. values forthese parameters were found in literature and are presented in Table 2.1.Bus c is the last AC bus before the converter and represents the converterterminals.

Table 2.1: Typical VSC station model parameters [21]

Parameter name Value (p.u.)

xphr 0.02rphr 0.0001bfilter = Im(1/zfilter) 2xtf 0.01rtf 0.0001

The control variables related to the AC/DC converter are the active andreactive power injections at bus c (Pc and Qc respectively) as well as theAC voltage on the same bus (Uc ∠θc). A Y −∆ transform of the ztf , zfilter,zphr impedances can further simplify the model and eliminate the need forextra voltage and angle variables for bus F . The transformed model can beseen in Fig. 2.8 and the new impedances can be calculated by (2.19).

z1 =ztfzphr + zfilterzphr + ztfzfilter

zphr


zfilter


ztf

(2.19)

k dcc

Uk, θk Uc, θc

Ic

Idc

Udc

Ploss

Pc Pdc

z2

z1 z3

Figure 2.8: VSC converter station simplified model


Ploss

DCC

DC

ACPc Pdc

Figure 2.9: VSC converter losses concept

2.4.1 Active Power Exchange - Losses

From a transmission grid perspective, an ideal VSC converter transfers theactive power it receives on the AC side (Pc) to its DC side (Pdc). A non-ideal converter has losses that will make the transmitted power less than theone that is entering the converter. From the sign convention introduced inSection 2.3 follows that the power balance of a lossy converter (Fig. 2.9) iscalculated by (2.20).

Pc + Pdc − Ploss = 0 (2.20)

The losses of a VSC converter have often been modeled with a quadraticequation of the phase reactor current (Ic) [16]. This model introduces 3losses components (κ0, κ1 and κ2) that correspond to no load losses, lossesdepending on Ic and losses depending on I2

c respectively.

Ploss = κ0 + κ1Ic + κ2I2c (2.21)

where

Ic =

√P 2c +Q2

c√3Uc

(2.22)

and the loss coefficients κ0 = 11.0331 × 10−3, κ1 = 3.464 × 10−3 and κ2 =5.5335× 10−3 (all p.u.) are taken from [15].

2.4.2 Voltage Limits

As with every bus on a transmission grid, limits on the voltage magnitudeexist to ensure system stability and over-voltage protection of the equipment(insulation limits). This is also true for the buses adjacent to an AC/DCconverter with the addition of one more constraint. In converters that usePWM modulation methods to produce the AC voltage, it is important to


avoid over-modulation because it creates unwanted harmonics in the ACgrid. Thus, there is a maximum AC voltage obtainable from a given DCvoltage on the other side of the converter. According to [27], a three-phaseVSC has a maximum AC voltage given by (2.23).

Uc =

√3

2√

2Udc (2.23)

Where Uc is the maximum line-to-line RMS value on the AC side. For aVSC in bipolar operation, the factor 2 on the denominator will cancel outwith the double DC voltage and thus:

Uc =

√3√2Udc ' 1.225Udc (2.24)

Equations (2.23) and (2.24) imply that both voltages are measured in volts.Since in a power flow study the per unit system is used, the above equationsmust change to reflect that. The nominal voltage on both AC and DCgrids is set at 1 p.u. and the maximum allowed voltage on any bus is 1.1p.u. According to [21] if no over-modulation is required, the ratio of themaximum AC side voltage to the DC voltage can be set to 1.1 thus:

Uc = kvUdc = 1.1Udc (2.25)

2.4.3 Power Throughput Constraints

The main limiting factor on the maximum power through an AC/DC con-verter is the maximum allowed current through the VSC valves (Ic). Theproduct of this current with the DC voltage yields the maximum apparentpower through the VSC.

|Sc| ≤ |UcIc| ⇒

P 2c +Q2

c ≤(UcIc

)2(2.26)

Values for Ic can be found in literature [5] and can be used to scale the VSCin various case studies.

A VSC converter is able to generate or consume reactive power in a control-lable fashion since the θc angle can be independently controlled. The phasereactor zphr is the main constraint on the reactive power flow. By applying(2.5) on the F -C branch, the reactive power flow is:

QFC = −bphrU2c + UcUf (bphr cos(θcf )− gphr sin(θcf )) (2.27)

Given that zphr is a reactive element we can assume 1/zphr = yphr =gphr + jbphr ' jbphr. In this case, the maximum reactive power flow on


the c-f branch of the full VSC model in Fig. 2.7 would be:

Qc = −bphrUc2

+ UcUfbphr cos(θc − θf ) (2.28)

The minimum reactive power of the VSC is represented in this study as apercent of its nominal power (kQ) as proposed in [21]. Both the nominalpower of various AC/DC converters and the value of kQ can be obtainedfrom [5].

Qc = −kQSnom (2.29)

By applying all these power constraints to a sample VSC converter thePQ capability curve shown in Fig 2.10 is produced. A value of zphr =0.0001 + j0.3 was used in this example.

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Active Power (p.u.)

ReactivePow

er(p.u.)

Maximum apparent power

Minimum reactive power

Maximum reactive power

Figure 2.10: Example of a VSC converter PQ capability curve

It appears that for small angle differences the change in the maximum re-active power limit is not very significant. However, because of the strongdependency of Qc on the value of zphr, the non-linear expression of themaximum reactive power limit will be used throughout this thesis.

2.5 DC/DC Converters

These types of converters are used to transform one DC voltage level toanother similarly to an AC transformer. They can be used to connect two


DC systems of different operating voltages, or provide power flow controlwithin a MT-HVDC grid.

From (2.6) it follows that the (active) power transmission between two DCnodes depends on their voltage difference. While AC/DC converters arecapable of controlling their DC voltage, in a true multi-terminal DC grid,there may be DC buses that aren’t connected to converters. In a complexenough system this will lead to lower voltages at these buses, depending onthe actual system topology and power flows. In order to influence the powerflows in such a system, DC/DC converters could be used as branch devicesto change the voltage difference between buses, thus effectively changing thepower flow in a way a phase shifting transformer can be used to change thepower flow on an AC grid.

According to [18], the most promising technologies for such ”high-power,low-ratio” devices are the Alternate Arm Converter or the Modular Multi-Level Converter. Most of these devices convert DC-AC-DC and, in thescope of this thesis, can be modeled as two AC/DC converters connectedback to back as seen in Fig. 2.11. Most of the additional equipment used ina AC/DC station is not needed in this model since the AC transformationtakes place internally in the converter.

Udc,m

DCk

Ploss,k

ACk DCmACm

Udc,k

Pdc,k Pc,k

Ploss,m

Pdc,m Pc,m

Figure 2.11: DC/DC converter model

Sometimes an isolation transformer is put between the two AC/DC com-ponents that provides galvanic separation. This can be modeled with theequivalent impedance shown in Fig. 2.11.

The mathematical modeling of such a device consists of the same equa-tions as the AC/DC converter although some elements like the transformerimpedance (ztf ) and filter (zfilter) are not included. The voltage trans-formation itself doesn’t need to be explicitly formulated since the voltagerelationships described in Section 2.4.2 can account for low ratio transfor-mation.

Chapter 3

Optimal Power Flow

The Optimal Power Flow problem deals with minimizing an objective func-tion while satisfying the power flow equations and any relevant limit that isimposed by the problem. In this chapter, a generic formulation of the OPFproblem will be given and then modified to suit the problems solved withinthis thesis.

3.1 Definitions

3.1.1 Optimization Vector

The optimization vector x shown in (3.1) contains all the variables that canbe used to describe the state of the system (state variables - z) and variablesthat will be used for control (control variables - u).Vector z contains the voltages and angles of every bus of the system, whileu contains the active and reactive power generations from generators andStatic Var Compensators - SVCs (Pg, Qg and Qs), active and reactivepower injections from the AC/DC converters (Pc and Qc) and ratios andphase angles for tap-changing or phase-shifting transformers (α and φ re-spectively). Vector w ⊆ u that contains only Pg and Pc is also defined hereand it will be used in the SCOPF problem formulation.

x =

[z

u

]=

UAC

θAC

UDC

Pg

Qg

Qs

Pc

Qc

αφ

, w =

[Pg

Pc

](3.1)

19

CHAPTER 3. OPTIMAL POWER FLOW 20

3.1.2 Objective Function

The objective function f(x) is a function of some of the variables in x. Thesolver tries to find the optimum by finding a combination of values in x thatwill minimize this function. A selection of examples of objective functionsis given in the following equations.

f(x) =

Ng∑i=1

c2P2gi + c1Pgi + c0 (3.2)

f(x) =

Ng∑i=1

Pgi (3.3)

f(x) =

Ng∑i=1

Pgi +

Nb∑i=1

πi |Uref − Ui| (3.4)

The objective (3.2) is the sum of the production costs as defined in (2.18)for generators and is used for the economic dispatch problem where theminimization of the total production costs is needed. The next equationis simply the sum of all active generation. By minimizing (3.3) we get thesolution that will result in the minimum possible amount of generation, thusminimizing system losses regardless of the generation cost.Combinations of optimization goals are also possible as seen in (3.4) wherethe voltage deviations from some reference values are minimized in parallelto the active generation. This generally doesn’t lead to loss minimizationbut depending on the values of the penalty factors πi a desired voltageprofile can be achieved. The πi factors impose a penalty on these deviationsby generating a “cost” that is included in the objective. According to themagnitude of the penalty factors πi, the maximum allowed deviations fromthe reference voltage can be controlled.

3.1.3 Equality Constraints

Power Balance Equations

Both AC and DC nodes have to fulfill the nodal balance equations (i.e. thesum of power inflow and outflow must be zero). Generators, loads, shuntelements and AC/DC converters all generate power inflow (resp. outflow)to (resp. from) the buses they are connected to. The total apparent powermismatch can be calculated by (3.5). The signs in all following equationsfollow the convention of chapter 2.3.

Smismatch = (Sg − Sd) + (Sc − Sdc − Sloss)− Sinj (3.5)

where: Sg = Pg + j(Qg + Qs) is the active and reactive generation (fromgenerators or SVCs) and Sd = Pd + jQd is the active and reactive demand


by loads. Equation (2.20) describes the relationship between the AC and DCside power flows through the converter and the VSC losses are calculatedthrough (2.21). Finally, Sinj is calculated through (2.11). All the variablesin (3.5) are in vector form.

Reference Angle

One AC node is used as a reference bus and its angle is set to some referencevalue, typically zero

θslack = θref = 0 (3.6)

3.1.4 Inequality Constraints

The inequality constraints impose limits on the variables of the optimizationvector and other quantities that are calculated through them. In this chap-ter, the various types of inequalities that limit the variables of the problemwill be analyzed.

Limits on State Vector Elements

All bus voltages in x are up and down limited both for equipment protectionagainst over-voltage and system protection against voltage instability.

UAC ≤ UAC ≤ UAC (3.7)

UDC ≤ UDC ≤ UDC (3.8)

The limits are taken ±10% of the nominal value i.e. 0.9 and 1.1 p.u. respec-tively as defined in [28], but can generally be different for each bus.

Bus voltage angles can also be limited but for this study they are left totake values between −π and π, thus:

−π ≤ θAC ≤ π (3.9)

Generator output is also limited as discussed in Section 2.3.1 thus producingthe following inequalities:

Pg ≤ Pg ≤ Pg (3.10)

Qg ≤ Qg ≤ Qg (3.11)

Qs ≤ Qs ≤ Qs (3.12)

Where the maximum and minimum active generation and reactive genera-tion are uniquely defined for each generator or SVC.


Tap changing and phase shifting transformers also offer a limited amount ofcontrol of their complex ratio t = αejφ. The limits of their control variablesare taken into account with (3.13) and (3.14).

α ≤ α ≤ α (3.13)

φ ≤ φ ≤ φ (3.14)

The elements of the state vector that correspond to the power throughAC/DC converters (Pc and Qc) are not bounded by fixed constraints withthe exception of the lower boundary for reactive power due to (2.29).

−kQSnom = Qc ≤ Qc (3.15)

Limits on Calculated Quantities

The first limiting factor of power flow in a transmission grid is the line ca-pacity. Each power line km is assigned a limit Skm in MVA that is thehighest amount of apparent power allowed. The limit results from the lim-ited thermal stress tolerance of the line. The power flows on each line arecalculated through (2.16) and (2.17) to produce the constraints shown in(3.16) and (3.17).

|Sf | ≤ S (3.16)

|St| ≤ S (3.17)

As discussed in chapter 2.4.3, an AC/DC converter must also obey the powerthroughput constraint (2.26). All the quantities in this formula are also inthe state vector (3.1) since C ⊆ A , and Ic is a converter constant that isconsidered known.

The maximum reactive power generation is taken into account with (2.28).This formula results from the power flow equations applied between the con-verter terminal - filter bus. However, as mentioned in chapter 2.4 the filterbus may be absent depending on the converter technology and the VSCstation configuration. Moreover, due to the VSC model simplification pre-sented in Fig. 2.8, the complex voltage on bus F is not a part of the systemvariables so it must be calculated from the adjacent AC voltages. Analysisof the equivalent circuit in Fig. 3.1 yields:

Uf =zfilterzphrUk + zfilterztf Uc

ztfzphr + ztfzfilter + zfilterzphr(3.18)

and if the filter zfilter is missing:

Uf = Uk −(Uk − Uc)ztfztf + zphr

(3.19)


If any of the other impedances are missing from the model, the equivalentfilter voltage is identical to either Uc or Uk.

Uk

Uf

Uc

ztf zphr

zfilter

Ik Ic

If

Figure 3.1: Equivalent circuit of the VSC station model AC side

The maximum AC side VSC voltage (Uc) is the minimum between the re-spective Uk of (3.7) and the one calculated by (2.25).

3.2 OPF Problem Formulation

Combining all the previous equalities and inequalities, the non-linear op-timization problem for economic dispatch can be formulated as following.

minimizex

Ng∑i=1

c2P2gi + c1Pgi + c0 (3.20a)

subject to Pgi − Pdi + Pci − Pdci −

κ0 + κ1

√P 2ci +Q2

ci√3Uci

+ κ2P 2ci +Q2

ci

3U2ci

− Ui

∑j∈Ki

Uj (gij cos θij + bij sin θij) = 0,

∀i ∈ A ∪ D (3.20b)

Qgi +Qsi −Qdi +Qci

− Ui∑j∈Ki

Uj (gij sin θij − bij cos θij) = 0,∀i ∈ A (3.20c)

θslack = θref (3.20d)∣∣∣y∗kmUkejθk (Uke−jθk − Ume−jθm)− jbshkmU2k

∣∣∣ ≤ Skm,∀(k,m) ∈ KA ∪KD (3.20e)


|Ui| ≤ kv|Uj |, ∀(i, j), i ∈ C, j ∈ E (3.20f)√P 2ci +Q2

ci ≤ |UciIci|,∀i ∈ C (3.20g)

− kQSnom i ≤ Qci ≤ −bphrUi2

+ Ui|Ufi|bphr cos(θc − θf ),

∀i ∈ C (3.20h)

Ui ≤ Ui ≤ minUi, kvUdc i, ∀i ∈ A ∪ D (3.20i)

Pgi ≤ Pgi ≤ Pgi,∀i ∈ G (3.20j)

Qgi ≤ Qgi ≤ Qgi, ∀i ∈ G (3.20k)

Qsi ≤ Qsi ≤ Qsi, ∀i ∈ S (3.20l)

αi ≤ αi ≤ αi,∀i ∈ T (3.20m)

φi ≤ φi ≤ φi, ∀i ∈ T (3.20n)

− π ≤ θi ≤ π,∀i ∈ A (3.20o)

3.3 SCOPF Problem Formulation

The concept of SCOPF is an extension of the OPF problem to include ad-ditional constraints for possible contingencies [29]. The SCOPF solutionprovides the optimal operating point in the base case so that if contingen-cies occur they would not create security violations. Thus all definitions inChapter 3.1 are valid, but some additional constraints need to be included.

In a generic form, the OPF problem is defined as:

minimizex0

f(z0,u0) (3.21a)

subject to g0(z0,u0) = 0 (3.21b)

h0(z0,u0) ≤ 0 (3.21c)

x0 ≤ x0 ≤ x0 (3.21d)

Where g0 = 0 summarizes the equality constraints (3.20b)-(3.20d), h0 ≤ 0summarizes the inequality constraints (3.20e)-(3.20h) and the upper andlower boundaries on x0 summarize equations (3.20i)-(3.20o). The subscript0 refers to the normal operation of the system i.e. all lines, generators andconverters are functioning normally.

A contingency is defined as a system configuration that differs from thenormal one because one system element has malfunctioned and is not oper-ating. Such elements can include generators or SVCs, AC/DC converters,


DC/DC converters or lines and it is assumed that there are Ncont such pos-sible contingencies. After each contingency the admittance matrix may bedifferent or the operation limits on various devices (generators or convert-ers) need to be changed. Thus the modified system equations need to beconstructed and new equality and inequality constraints to be formulated.All these constraints will be identical to the ones of the normal operationthe only difference being the different Ybus matrix which has changed toreflect the post-contingency system topology. If the broken elements haveno influence on the admittance matrix, their respective cases differ from thebase one in the sense that the upper and lower boundaries of all quantitiesrelated to the broken element are set to zero. Constraints that refer to thek-th post contingency configuration are shown with the subscript k in the(3.22) set of equations.

minimizex0

f(z0,u0) (3.22a)


h0(z0,u0) ≤ 0 (3.22c)

x0 ≤ x0 ≤ x0 (3.22d)

gk(zk,uk) = 0, k = 1, . . . , Ncont (3.22e)

hk(zk,uk) ≤ 0, k = 1, . . . , Ncont (3.22f)

xk ≤ xk ≤ xk, k = 1, . . . , Ncont (3.22g)

|w0 −wk| ≤ ∆wk, k = 1, . . . , Ncont (3.22h)

The vector w has been defined in chapter 3.1.1 as a subset of u that containsall active power generations and active power flows through AC/DC con-verters. Equation (4.1f) couples these variables pre- and post-contingencyto simulate realistic corrective actions by generators and converters. Thevalues in ∆wk can be adjusted to the maximum allowed variation betweenthe normal operation and the k-th post contingency configuration. Equation(4.1f) can therefore be written as (3.23)

∣∣∣∣∣∣P kgP kc

−P 0

g

P 0c

∣∣∣∣∣∣ ≤∆Pg

∆Pc

, k = 1, . . . , Ncont (3.23)

The developed algorithm constructs these additional constraints automati-cally. It is worth noting, that for some (not N-1 secure) systems, some faultyelements can lead to isolation of loads or generators thus resulting to a nonfeasible OPF sub-problem. Due to this fact, the algorithm (shown in Fig.3.2) recognizes such non-feasible sub-systems and doesn’t include them inthe final formulation.


If A is the adjacency matrix that corresponds to the studied transmissiongrid and n is a natural number, then the matrix An has the following prop-erty: the value in row i and column j is equal to the number of paths oflength n from node i to node j. Therefore, by choosing a large enough n, anyisolation of grid areas can be observed in the patterns of zero and non-zeroelements of An. In such cases the algorithm can recognize isolated loads orgenerators, and thus removing any non-feasible cases from the final set ofsecurity contingencies. The decision on the feasibility of a sub-problem istaken by comparing the loads and generators maximum or minimum valuesin the isolated grid areas. If there is not enough generation present, or if theload demand is lower than the minimum generation limit of the local gener-ators, the sub-problem is deemed non-feasible and the relevant constraintsare not constructed. A high level flowchart of the algorithm is presented inFig. 3.2.

Some isolated buses

Options

Case file

Base case

Break element

Isolation check

Feasibility check

K‐th contingency case

Finished?

Create final case file

Store case info

Store case info

Non feasible case

No isolation

Yes

No

Feasible solution

RUN OPF

Print results

Figure 3.2: SCOPF Algorithm including constraint building


The case file contains information about the system structure. It includesbranch and converter data, generation costs and all relevant limits and con-stants. The options structure contains options for the optimization solverand other output functions. With these inputs, the algorithm first cal-culates the admittance matrix and all other relevant system matrices de-scribed in Chapter 2 and stores them as base case information. In order forthe additional security constraints to be constructed, each system elementneeds to be disabled in sequence. For each contingency case, the admittancematrix of the resulting system is calculated and the algorithm checks forisolated busesas described above. If this is the case, a feasibility check isbeing performed and if the case is feasible, it’s respective case informationis stored, otherwise discarded. When all possible contingencies have beenexamined, the stored information about all feasible cases is collected and thefull SCOPF problem constraints are constructed, creating the final case fileto be used as an input to the OPF solver of Matlab. If the solver finishessuccessfully, the results are passed into a printing function to be displayedand evaluated.

3.3.1 Types of SCOPF Problems

If variation of the elements in w is not allowed post-contingency (i.e. ∆wk =0), the problem is called preventive SCOPF (pr-SCOPF) and if ∆wk 6= 0it’s called corrective SCOPF (c-SCOPF). The difference between these twotypes of SCOPF problem formulation is the range of corrective actions al-lowed to generators and converters. Most generators have inherent rampingconstraints i.e. they cannot change their output by and arbitrary amountbefore and after a contingency. These constraints depend upon the generatortechnology (nuclear plants are generally slower than hydro units etc). Theselimits may be set even lower than the technical maximum allowed valuesdue to safety considerations or other agreements. The VSC converters havepractically unlimited ramping as discussed in Chapter 1. It is however usefulto include the relevant constraints in ∆Pc as very large and rapid changesin their active power injection can lead to system stability issues.

As stated in the definition of the steady state security concept in Chap-ter 1, under the preventive SCOPF, no corrective actions are allowed inthe post-contingency state other than automatic actions like tap-changersor phase shifters. In this thesis, no distinction is being made between theprimary control and manual corrective actions taken by the system opera-tor. Thus if ∆wk = 0, the pr-SCOPF problem assumes no primary controlavailable. A small value of allowed corrective actions to one or some of thesystem generators can be used to simulate available primary control reserves.


According to [29], the value of the objective function in the optimum point(x∗) of the c-SCOPF problem is lower bounded by the one of the OPF prob-lem, while for pr-SCOPF the objective value is higher than any c-SCOPFsolution for the same system.

f(x∗OPF) ≤ f(x∗c-SCOPF) ≤ f(x∗pr-SCOPF) (3.24)

Chapter 4

Algorithm Improvements

The SCOPF problem is in general a large-scale, non-linear and non-convexoptimization problem. The introduction of the added security constraints inChapter 3.3 increases the size of the problem dramatically. If a system of Nb

buses, Nbr branches, Ng generators and Nc AC/DC converters is studied,the problem size will grow from Nb to Nb × (Nbr + Ng + Nc) if all of theseelements are taken into account when forming the security constraints. Fora 32-bus system like the one in Chapter 5.1.3 this implies an ×76 increase ofthe problem size if all possible contingencies are taken into account. Such bignon-linear, non-convex optimization problems are very hard to solve with atypical desktop computer within realistic timeframes.

After careful analysis of the effects of various contingencies in test system, itbecomes obvious that most contingencies don’t affect the system optimum.Only a relatively small subset of the (4.1f) set of inequality constraints be-come active in any given system and affect on the optimum.

One of the two primary approaches to reduce the complexity of the SCOPFproblem (the other being using simplified models for the post-contingencycases) is to use some sort of “Contingency Filtering” (CF) and include onlythe most severe contingencies in the SCOPF problem [11]. There are vari-ous ways to sort the different contingencies based on their severity by using“Severity Indexes”(SI) that are typically computed for all contingencies. Theapproach investigated in this chapter is based on one of the SIs presentedin [30]. While such an approach can be shown to improve the algorithm effi-ciency, some drawbacks still exist the most severe of them being the arbitrarydefinition of the SI threshold above which a contingency case is included inthe SCOPF problem. This implies that some fine-tuning may be needed foreach system.

29

CHAPTER 4. ALGORITHM IMPROVEMENTS 30

4.1 Contingency Filtering

The SCOPF problem can be compactly stated as in (4.1).

minimizex0

f(z0,u0) (4.1a)


h0(z0,u0) ≤ 0 (4.1c)

gk(zk,uk) = 0, k = 1, . . . , Ncont (4.1d)

hk(zk,uk) ≤ 0, k = 1, . . . , Ncont (4.1e)

|w0 −wk| ≤ ∆wk, k = 1, . . . , Ncont (4.1f)

The Severity Index for every contingency k can be defined as

SIk = 100

(f(x∗k)

f(x∗0)− 1

)(4.2)

Where the f(x∗k) is the objective function value at the optimum if only thek-th contingency is active and f(x∗0) is the objective at the optimum if nocontingency is active (base case). This index shows the normalized impactof each contingency on the objective function. The contingencies are filteredbased on this index, and only the ones that have the highest SI, are takeninto account for making the final SCOPF problem.

First the OPF problem is solved for the whole system. This part is rel-atively fast, because the size of the problem is limited to the actual systemsize. The base case optimum f(x∗0) is then stored in memory. Consequentlya series of small scale SCOPF problems are solved for each of the contingen-cies separately. The resulting optima f(x∗k) are also stored and the SIs arecalculated. This is done until all the contingencies are taken into account.Due to the basic inequality (3.24), the SIs are always positive or zero.

The SCOPF problem is constructed by taking into account only the mostsevere cases by comparing their respective SIs. The number of significantcontingencies used, is fine tuned for each system and depends on the load-ing of the lines and other system devices. This medium-scale optimizationproblem is then solved in less time that it would be required if all of thecontingencies were taken into account. The resulting total optimum may belower than that of the full SCOPF problem due to the possible exclusion ofconstraints that failed to produce a high enough SI but are still significantlyaffecting the SCOPF solution.

CHAPTER 4. ALGORITHM IMPROVEMENTS 31

The algorithm flowchart is presented in Fig. 4.1.

Start

Break element k

Run SCOPF between the base case and k‐

th contingencyStore f(x*k)

Run OPF on original system Store f(x*0)

k>Ncont ? Calculate SI

No

Select the most severe contingenciesYes

Create final case file

Run SCOPF

Figure 4.1: Improved SCOPF algorithm - Contingency filtering method

This method was found to work better for cases where the system in thebase case is not heavily loaded, and therefore only some of the contingenciesare affecting the optimum.

Chapter 5

Test Cases - Results

5.1 Test Cases Description

5.1.1 9-bus AC with 5-bus MTDC

The 14-bus test system is a modified version of the Western System Coordi-nating Council (WSCC) 3-Machines 9-bus system [31] and is shown in Fig.5.1. Buses 1-9 are AC and 10-14 DC. The bus and line parameters as well asdetailed cost functions of the generators are given in the appendix (SectionA.1).

14

2 8

7

VSC

10

6

VSC

13

3

9

VSC 11VSC 5VSCVSC

1

12

4

Figure 5.1: 14-bus AC/MTDC system

33

CHAPTER 5. TEST CASES - RESULTS 34

5.1.2 IEEE 14-bus AC with 5-bus MTDC

The 19-bus test system shown in Fig. 5.2, is a modified version of the IEEE14-bus test system [32] with an additional MTDC grid as in [20]. The busand line parameters as well as detailed cost functions of the generators aregiven in the appendix (Section A.2). The system consists of 14 AC busesand 5 additional DC ones, one of which is not connected to any VSC. Agenerator is also present on the DC grid and simulates renewable generation(e.g. a photovoltaic plant) directly connected to the DC grid.

19

9VSCVSC

12

11

0

1314

0

10

0

7

84

3VSCVSC

1

VSC

VSC

2

6

5

VSC

VSC 15

1617

18

=


5.1.3 RTS-96 one area with 8-bus MTDC

This 32-bus test system is a modified version of the RTS-96 one area test sys-tem [33] and is shown in Fig. 5.3. The system consists of 24 AC buses and 8additional DC buses and includes a DC generator and DC loads. There is aDC/DC converter on the line between buses 27 and 32 modeled as discussedin chapter 2.5. The characteristics of the DC lines were obtained from [5] tomatch the original system geographic scale. The bus and line parameters aswell as detailed cost functions of the generators are given in the appendix(Section A.3).The placement of the DC buses and lines is done in such a way that thecheap generators in the bottom have more connections to the loads. TheDC/DC converter was installed between buses 27 and 32 after analysis of theDC buses voltage profiles and the resulting power flows. A converter wouldhave the maximum impact placed between buses with large voltage differ-


ences. This does not imply a solution of the optimal converter placement inAC/MTDC grids, which is outside the scope of this thesis.

VSC

VSC

VSC

VSC

VSC

VSC

1 2 7

8

5

4

3 9 10

6

24 11 12

13

23

201916

15 14

17

18

2122

25

26

27

28

29

30

31

32

=

VSCVSC

3334

DC/DC converter

3536



5.2 Results

5.2.1 Improved SCOPF Algorithm

As discussed in Chapter 4.1, the Contingency filtering technique describedcan reduce the running time of the SCOPF algorithm significantly withoutsacrificing much on the accuracy of the optimum solution. In this chapter,this claim is investigated by using the“9-bus AC with 5-bus MTDC”system.

The algorithm of Fig. 4.1 was applied to this system and the Severity In-dexes for each contingency case are presented in Fig. 5.4. For this examplethe pr-SCOPF problem was assumed.

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

Seve

rity

Inde

x (S

I)

Contingency Type

Figure 5.4: Severity Indexes on all contingencies in the 14-bus system

The most severe cases can be easily identified as the outages of lines 8-2, 3-6and 14-12 as well as the failure of the AC/DC converter 5-12. The pr-SCOPFproblem is solved for the test system of Fig. 5.1 for a number of significantcontingencies and the results were compared to the ones obtained by solvingthe full pr-SCOPF problem in Table 5.1. The time needed to iterate throughall the possible contingencies and calculate the SIs is shown separately in thethird column. The outages of the generators were not included in this casestudy because the pr-SCOPF problem was infeasible for generator failures.


Table 5.1: Comparison of the improved algorithm with the full pr-SCOPF

f(x∗0)($/hour)

Elapsedtime (s)

Time im-provement

Distancefrom opti-mum

Full pr-SCOPF 9,870.417 2130 − −6 most severe cases 9,780.747 92 + 532 70.70% -0.91%

5 most severe cases 9,780.720 88 + 532 70.89% -0.91%




1 most severe case 8,656.643 9 + 532 74.60% -12.30%

It is apparent that there is a very significant time improvement as theSCOPF problem becomes smaller. The SCOPF with the filtered contin-gencies became more accurate as more of them were included, but aftersome point including more of the severe cases did not improve the optimum.At any case, the optimum provided by the filtered problem is lower thanthe full one as expected. As seen in Table 5.2, the generator dispatch at theoptimum with the filtered pr-SCOPF, is within 1% of the full pr-SCOPFproblem solution.

Table 5.2: Generator dispatch comparison

Pg1 (MW) Pg2 (MW) Pg3 (MW)

full pr-SCOPF 79.25 150.67 150.51

6 most severe cases 78.79 151.00 148.96



1 most severe case 43.39 151.00 180.92

The generator dispatch comes closer to the one of the full pr-SCOPF solutionas more cases are included but the improvement is again limited as moresevere cases are added.

5.2.2 Economic Benefit of MTDC Grid Expansion

An mentioned in Chapter 1, the transmission grid expansion is inevitableas the energy demand increases. In this chapter, a benefit evaluation ofa grid expansion using MTDC with VSC technology will be conducted on


the “IEEE 14-bus AC with 5-bus MTDC” test system. In order to haveaccurate indication of the benefit of only the construction of the MTDCgrid, the generator on DC bus 19 was not included in this case study. Thetotal economic benefit is calculated as in [21] and takes into account thereduction of the total generation cost as well as the (potential) reductionof the total system losses. The benefit evaluation approach is shown inFig. 5.5 where it is assumed that the system will operate under the securityconstrained optimum for the duration of the investment.

Case data(original 19‐bus grid)

Case data (with MTDC)

Load increase (+0.5%/year)

Unconstrained OPF (‘pure’ economic dispatch)

Corrective SCOPF (minimization of

gen. cost)

End of investment period?

Energy prices

Generation cost

No

Load increase (+0.5%/year)

Unconstrained OPF (‘pure’ economic dispatch)

Corrective SCOPF (minimization of

gen. cost)

End of investment period?

Energy pricesGeneration cost

No

Yes Yes

Total yearly energy cost

(present value)

Total yearly energy cost

(present value)

Total energy cost(original 19‐bus

grid)

Total energy cost(with MTDC)

Total benefit of MTDC grid expansion

Active power losses Active power losses

+‐

Figure 5.5: Economic benefit evaluation algorithm

For each system a c-SCOPF problem is solved to determine the generationcost under certain security constraints. The results also include the activepower losses for each case. It is assumed that these losses must be purchasedat the market price which is obtained by running an unconstrained OPF onthe same system as proposed in [34]. It is necessary to include the powerlosses in the calculation of the total energy cost, as they are not included


in the active load consumption, and cause additional cost to the system op-erator. The benefit analysis has been carried out with an discount rate of8% which can be considered typical for such a project [21]. The investmentperiod was chosen to be 15 years and a load increase of 0.5% per year hasbeen assumed. The total economic benefit (TEB) is calculated as in (5.1).

TEB =15∑y=1

(DFy

((Cy

exp + pyexpLyexp − Cynoexp − pynoexpLynoexp

)))(5.1)

Where

TEB : the total economic benefit (in present value)Cy : the generation cost for year yLy : the active power losses for year ypy : the electricity price for year y (calculated through the unconstrainedOPF)DFy : the discount factor for year y calculated in (5.2) from the discountrate r.

DFy =1

(1 + i)y(5.2)

This analysis has been performed under 2 different cSCOPF scenarios. Thefirst assumes a uniform ∆wk of 80 MW for all generators and converterswhile in the second this limit is halved for all devices. The results are pre-sented in Table 5.3.

Table 5.3: Results of economic benefit analysis of MTDC grid expansion onthe IEEE 14-bus test system

∆wk (MW) - same for all devices 80 40

Reduction of generation cost (million $) 19.369 14.843

Reduction of active power loss cost (million $) -0.421 -2.857

Total economic benefit (million $) 18.948 11.986

It is evident, that there are long term economic benefits in installing anMTDC grid in parallel to the existing AC infrastructure. Interestingly, theactive power losses generate higher costs in the combined AC/MTDC sys-tem due to the addition of more lines, but this loss is canceled out by theincrease in transmission capacity that allows cheaper generators to cover alarger part of the demand. This result is in line with the findings in [21]. Thetotal economic benefit is also decreasing as expected with tighter deviation


limits in the security constraints. This happens due to the more expensivegeneration dispatch produced by the stricter limits.

By comparing the calculated benefits with the construction cost of sucha grid expansion one can have an idea of the economic feasibility of suchan investment. It is rather difficult to get exact investment costs data forMTDC, but an estimation is done in [35].

5.2.3 DC-DC Converters as Power Flow Control Devices

In a conventional (AC) transmission system, phase shifters are used forpower flow control by changing the angles in congested ac lines. A DC/DCconverter can work in a similar way for an MTDC grid. To demonstrate thiseffect, the combined AC/MTDC OPF problem was solved for the modifiedRTS-96 one area system that includes a DC/DC converter. The physical linelengths (and therefore resistances) in the RTS-96 system are smaller thanthose which would allow a DC/DC converter to produce any benefit. Thisis because due to the assumed low resistance of such short HVDC lines,the voltages on the DC grid are normally very close even if large enoughpower flows are present. In order to better demonstrate the effect that aDC/DC converter would have on the DC voltages and thus on the powerflows, several test cases have been run with progressive up-scaling of the DClines resistances. The so called DC line resistance factor simulates longerDC lines or higher DC resistances.

The DC/DC converter has an effect on the voltage difference of the busesbetween which it is connected, and thus the 27-32 line flow is controlled.Figure 5.6 shows the 27-32 line loading in the presence and absence of theDC/DC converter. It is shown that given long enough lines, the presenceof a DC/DC converter keeps the line flow high despite the increased lineresistance. This may increase the line losses (see Fig. 5.7) but leads to amore economical generation dispatch for the whole system as shown in Fig.5.8. The line losses on the 27-32 line increase, as the equivalent length of theline increases when the DC/DC converter is present. When such a deviceis not used however, the losses begin to decrease after some point. Thisis due to the limitation of the power flow through this line induced by theoptimization, in order for the voltage constraints to be satisfied.


0 5 10 15 20 251

1.5

2

2.5

DC line resistance factor

Pow

erflow

(p.u.)

without DC/DC converter with DC/DC converter

Figure 5.6: Power flow on line 27-32

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Lineloss

(p.u.)

without DC/DC converter with DC/DC converter

Figure 5.7: Power losses on line 27-32

The DC line resistance factor of 10, after which the effects of the DC/DCconverter become apparent, in this system corresponds to a line length of700km given the same per km resistance.


0 5 10 15 20 251

1.1

1.2

1.3

1.4

1.5

1.6x 10

6


Generationcost

($/hour)

without DC/DC converter

with DC/DC converter

Figure 5.8: Total generation cost

5.2.4 Influence of the Corrective Actions Limits on the SCOPFSolution

As mentioned in Section 3.3.1, there are 2 ways to formulate the SCOPFproblem: preventive and corrective. In this chapter along with the OPFproblem, several versions of the SCOPF problem were solved for the “IEEE14-bus AC with 5-bus MTDC” test grid with the corrective actions limitspresented in Table 5.4.

In Case 1 cheap generators at buses 1 and 19 were not allowed to deviate fromtheir original settings in the event of a contingency while the more expensivegenerators at buses 2, 3, 6 and 8 where given more relaxed constraints. InCase 2 the opposite happens: cheap generators are free to deviate in orderto match the post-contingency optimal power flows and the more expensiveones have to stay in their pre-contingency values. Case 3 is similar but themaximum deviation of generator 1 has been lowered and that of generator 2increased. Finally, Case 4 is the solution of the preventive SCOPF problem,i.e. ∆wk = 0.

The distribution of active generation can be seen in Fig. 5.9 for all cases.The security constraints include all possible generator and line outages andconverter failures. Due to the mathematical formulation of the problem,zero values in the ∆wk vector were not permitted, thus small enough valueswere used instead.


Table 5.4: Test Cases description

Generators buses Converters

1 2 3 6 8 19 1-16 13-15 9-17 3-18

Pg 332.4 140 100 100 100 100see (2.26), Ic = 627A

Pg 0 0 0 0 0 0

∆Pg Case 1 ≈ 0 50 50 50 50 ≈ 0 100 100 100 100

∆Pg Case 2 100 ≈ 0 ≈ 0 ≈ 0 ≈ 0 100 100 100 100 100

∆Pg Case 3 50 20 ≈ 0 ≈ 0 ≈ 0 100 100 100 100 100

∆Pg Case 4 ≈ 0 ≈ 0 ≈ 0 ≈ 0 ≈ 0 ≈ 0 100 100 100 100

As the range of corrective actions of cheap generators is increased, whileexpensive ones are not allowed to perform any action, the production dis-tribution shifts from the cheap generators 1 and 2 to the more expensiveones. In the normal OPF solution generators on buses 3, 6 and 8 are notproducing at all and their contribution increases as the range of correctiveactions allowed on them is decreased. The total generation cost increase dueto the security constraints is shown in Fig. 5.10 for each case.

1 2 3 6 8 190

50

100

150

Generator bus number

Activegeneration(M

W)

OPFSCOPF Case 1SCOPF Case 2SCOPF Case 3SCOPF Case 4

Figure 5.9: Active generation dispatch


OPF SCOPF 1 SCOPF 2 SCOPF 3 SCOPF 40

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Case

Totalgenerationcost

($/hour)

Figure 5.10: Cost of generation

The considerable increase in cost (especially for the pr-SCOPF problem)results from the big cost difference between cheap and expensive generatorsthat was assumed for this example. The SCOPF problems’ objective valueobeys the boundaries shown by (3.24) as expected. The cost increase evidentin Fig. 5.10 results from the way the cheap and expensive generators cor-rective actions limits are treated. The cost generally increases if expensivegenerators are not allowed to deviate in the event of a contingency. This con-trol strategy causes the base case dispatch to include expensive generators(buses 3, 6 and 8) in the base case dispatch. A security strategy like Case1, where the cheap generators after the contingency are fixed in their pre-contingency values, is much more economical. The pr-SCOPF case (Case4) is the most expensive as it forces the base case optimum to include theconstraints of all contingency cases.

The change in the range of allowable corrective actions also has and in-fluence on the bus voltages at the optimum. This effect is shown in Figures5.11 and 5.12 for all cases.The bus voltage on average decreases from a valueof 1.09 p.u. to 0.98 p.u. as the restriction of the range of corrective actionsleads to a more expensive optimum (see Fig. 5.13). This is a side-effect ofrestricting the range of possible states for the system when strict securityconstraints are imposed.


1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.9

0.95

1

1.05

1.1

Bus number

Voltage(p.u.)

OPF SCOPF Case 1 SCOPF Case 2 SCOPF Case 3 SCOPF Case 4

Figure 5.11: AC bus voltages

15 16 17 18 19

0.9

0.95

1

1.05

1.1

Bus number

Voltage(p.u.)

OPF SCOPF Case 1 SCOPF Case 2 SCOPF Case 3 SCOPF Case 4

Figure 5.12: DC bus voltages

0.9

0.95

1

1.05

1.1

OPF SCOPF 1 SCOPF 2 SCOPF 3 SCOPF 4

Test case

Voltage(p.u.)

Figure 5.13: Average voltage deviation


5.2.5 Sensitivity Analysis of the ∆wk Components

A sensitivity analysis has been performed on the “9-bus AC with 5-busMTDC” test grid, to illustrate the influence of the two components of the∆wk vector in the optimization results. The SCOPF problem has beensolved for this system for different combinations of the values in ∆wk. Allgenerators have been given the same relative deviation limits, and the sameis true for the AC/DC converters. According to (3.23) the two differentcomponents of ∆wk (∆Pg and ∆Pc) affect the constraints of different con-trol variables (Pg and Pc respectively). The amount of influence of theseconstraints on the SCOPF solution depends on the operation point of thesedevices. In this test case, the generators were operating at ∼50% of theirmaximum capacity while the greatest loading of the AC/DC converters was∼25%. This explains the greater influence of the ∆Pg components on thevalue of the objective while the effects of ∆Pc are not so pronounced.

Figure 5.14 shows the hourly generation cost as the converter maximumdeviation limits are changed. Each line represents a different set limits asshown in the legend. Until a low enough ∆Pc, the generation cost is notaffected by the converter deviation constraints. This means that the samesecure base case optimum can be feasible for all contingencies if the converterpower throughput is allowed to vary more than ∼20%. If ∆Pc is lower thanthis value, the old base case optimum cannot satisfy some contingency casesand is therefore set to a more expensive one.

0 10 20 30 40 50 60 70 80 90 1008000

8200

8400

8600

8800

9000

9200

9400

9600

9800

10000

Converter deviation limits - ∆Pc (% Capacity)

Generationcost

($/hour)

∆Pg =100%

∆Pg =75%

∆Pg =50%

∆Pg =25%

∆Pg =20%

∆Pg =15%

∆Pg =10%

∆Pg =5%

∆Pg =0%

Figure 5.14: SCOPF cost increase due to converters’ maximum deviationlimits


A similar behavior can be observed as the generators’ maximum deviationlimits are changed while the converters’ stay fixed as shown in the legend ofFig. 5.15. Here, the increase in the generation cost is more prominent as Pgis directly affecting the objective’s value.

0 10 20 30 40 50 60 70 80 90 1008000

8200

8400

8600

8800

9000

9200

9400

9600

9800

10000

Generator deviation limits - ∆Pg (% Capacity)

Generationcost

($/hour)

∆Pc =100%

∆Pc =75%

∆Pc =50%

∆Pc =25%

∆Pc =20%

∆Pc =15%

∆Pc =10%

∆Pc =5%

∆Pc =0%

Figure 5.15: SCOPF cost increase due to generators’ maximum deviationlimits

Finally, in Fig. 5.16, the % cost increase is plotted against both ∆Pc and∆Pg. As expected, a combination of strict generator and converter max-imum deviation limits leads to a bigger increase of the generation cost atthe secure base case optimum. The cost at the (0,0) point in this plot is infact the pr-SCOPF problem solution. Corrective SCOPF solutions can beseen to vary from equal to the base case OPF if the ∆wk constraints arerelaxed enough, to close to the pr-SCOPF solution especially for strict ∆Pgconstraints.


10075

5025

2015

105

0

10075

5025

2015

105

00

5

10

15

20

25

∆Pg (% Capacity)

∆Pc (% Capacity)

Generationcost

increase

(%)

Figure 5.16: Influence of both ∆Pg and ∆Pc on the % generation cost in-crease

Chapter 6

Conclusion

6.1 Conclusion

In this thesis an algorithm was developed for the solution of the SecurityConstrained Optimal Power Flow problem in combined AC/MTDC grids.The non-linear equations were used for calculating the power flow both inthe AC and DC grid and non-linear models have been used for the simulationof AC/DC and DC/DC converters. The problem formulation is such thatthe combined AC/MTDC grid is simulated in the same set of equations forboth grids. The non-linearity and large scale of the problem typically resultsto long running times. One method of contingency filtering was investigatedthat had positive results regarding the improvement in running time of thealgorithm. The impact of this method on the accuracy of the results is notsevere although a slightly lower optimum is reached.

Several case studies have been investigated to show the impact of expandingexisting AC transmission grids with MTDC grids. The combined AC/MTDCsystem was found to yield a total economic benefit as an investment althoughdue to the addition of an additional meshed grid, the cost of the losses isincreased.

DC/DC converters were found to function as power rerouting devices inDC grids, reducing the generation cost. This effect however becomes moreapparent as the DC voltage differences between the buses of the DC systembecome bigger. (e.g. long DC lines or heavily loaded DC system).

As far as the SCOPF problem is concerned, the maximum deviation limitsbetween the base and the contingency cases, have a big impact on the opti-mum generation dispatch. The impact is greater if the cheapest generatorsare allowed to deviate after contingencies while the expensive ones are heldto their pre-contingency optimum. This implies that a lower base case oper-

49

CHAPTER 6. CONCLUSION 50

ation cost can be achieved if expensive generators are free to deviate duringcontingencies and cheaper ones are more or less bound within a narrow op-eration window.

Both generators’ and AC/DC converters’ maximum deviation limits influ-ence the base case optimum in a SCOPF problem. The influence of eachdevice type is different and depends on the % loading of the respective de-vice in the base case. The most expensive SCOPF solution was found tooccur for the pr-SCOPF problem as expected.

6.2 Motivation for Future Research

The main problem arising from the formulation of the full non-linear SCOPFproblem is its large size and non-linear, non-convex form. This problem ismagnified by the increased system size induced by the non-linear modelsfor the AC/DC converters that also introduce additional control and statevariables.

The method of contingency filtering presented in Chapter 4 is one of manysolutions that have been investigated in the relevant literature over time andit has the disadvantage that the contingency cases are pre-filtered before themain SCOPF optimization solver starts the iterations. The investigation ofinter-iteration filtering approaches like the ones presented in [36] would bea potential new study target as well as different ways to reduce the systemcomplexity (simplified models for the system in the post-contingency casesor network compression).

Appendix A

Test systems data

The data presented for the various test systems in this appendix are derivedfrom the relevant literature for the AC grid. For the DC grid, the convertercurrent and voltage limits as well as nominal power are taken from [5]. TheDC lines resistance characteristics are based on DC cables manufacturer datafrom the same document also taking into account the geographic scale of thesystem where such information was available.

A.1 9-bus AC with 5-bus MTDC - System Data

Table A.1: 9-bus AC with 5-bus MTDC: Bus Data

Busno.

Pd(MW)

Qd(MVAr)

Gs(MW)

Bs(MVA)

BaseVoltage

(kV)

V (p.u.) V (p.u.)

1 0 0 0 0 345 1.1 0.92 0 0 0 0 345 1.1 0.93 0 0 0 0 345 1.1 0.94 0 0 0 0 345 1.1 0.95 190 80 0 0 345 1.1 0.96 0 0 0 0 345 1.1 0.97 50 25 0 0 345 1.1 0.98 0 0 0 0 345 1.1 0.99 125 50 0 0 345 1.1 0.910 0 0 0 0 320 1.1 0.911 0 0 0 0 320 1.1 0.912 0 0 0 0 320 1.1 0.913 0 0 0 0 320 1.1 0.914 0 0 0 0 320 1.1 0.9

51

APPENDIX A. TEST SYSTEMS DATA 52

Table A.2: 9-bus AC with 5-bus MTDC: Generator Data

Busno.

Qg(MVAr)

Qg(MVAr)

Pg(MW)

Pg(MW)

∆Pg c2 c1 c0

1 400 -400 400 10 ... 0.51 5 1502 300 -300 400 10 ... 0.085 1.2 6003 300 -300 400 10 ... 0.1225 2 335

Table A.3: 9-bus AC with 5-bus MTDC: SVC Data

Busno.

Qs(MVAr)

Qs(MVAr)

9 50 0

Table A.4: 9-bus AC with 5-bus MTDC: VSC Data

ACbusno.

DCbusno.

rtf xtf bfilter rphr xphr κ0 κ1 κ2

5 12 0.0001 0.01 0.2 0.0001 0.02 0.011033 0.003464 0.0053356 13 0.0001 0.01 0.2 0.0001 0.02 0.011033 0.003464 0.0053357 10 0.0001 0.01 0.2 0.0001 0.02 0.011033 0.003464 0.0053359 11 0.0001 0.01 0.2 0.0001 0.02 0.011033 0.003464 0.005335

Table A.5: 9-bus AC with 5-bus MTDC: VSC Data (cont.)

Ic (A) Uc (kV) Snom(MVA)

∆Pc(MW)

580 320 427 ...580 320 427 ...580 320 427 ...580 320 427 ...


Table A.6: 9-bus AC with 5-bus MTDC: Branch Data

Frombus

Tobus

r(p.u.)

x(p.u.)

bsh

(p.u.)Branch device α α Capacity

(MVA)

1 4 0 0.0576 0 tap changer 0.9 1.1 2004 5 0.017 0.092 0.0158 − − − 2005 6 0.039 0.17 0.0358 − − − 2003 6 0 0.0586 0 tap changer 0.9 1.1 2006 7 0.0119 0.1008 0.0209 − − − 2007 8 0.0085 0.072 0.0149 − − − 2008 2 0 0.0625 0 tap changer 0.9 1.1 2008 9 0.032 0.161 0.0306 − − − 2009 4 0.01 0.085 0.0176 − − − 20010 11 0.00732 0 0 − − − 27611 14 0.01099 0 0 − − − 27614 12 0.00916 0 0 − − − 27614 13 0.00659 0 0 − − − 2763 7 0.039 0.17 0 − − − 1502 9 0.01 0.085 0 − − − 1505 1 0.032 0.161 0 − − − 150

Base Power for the whole system: 100 MVA


A.2 IEEE 14-bus AC with 5-bus MTDC - SystemData

Table A.7: IEEE 14-bus AC with 5-bus MTDC: Bus Data

Busno.

Pd(MW)

Qd(MVAr)

Gs(MW)

Bs(MVA)

BaseVoltage

(kV)

V (p.u.) V (p.u.)

1 0 0 0 0 150 1.1 0.92 21.7 12.7 0 0 150 1.1 0.93 94.2 19 0 0 150 1.1 0.94 47.8 -3.9 0 0 150 1.1 0.95 7.6 1.6 0 0 150 1.1 0.96 11.2 7.5 0 0 150 1.1 0.97 0 0 0 0 150 1.1 0.98 0 0 0 0 150 1.1 0.99 29.5 16.6 0 0 150 1.1 0.910 9 5.8 0 0 150 1.1 0.911 3.5 1.8 0 0 150 1.1 0.912 6.1 1.6 0 0 150 1.1 0.913 13.5 5.8 0 0 150 1.1 0.914 14.9 5 0 0 150 1.1 0.915 0 0 0 0 320 1.1 0.916 0 0 0 0 320 1.1 0.917 0 0 0 0 320 1.1 0.918 0 0 0 0 320 1.1 0.919 0 0 0 0 320 1.1 0.9

Table A.8: IEEE 14-bus AC with 5-bus MTDC: Generator Data

Busno.

Qg(MVAr)

Qg(MVAr)

Pg(MW)

Pg(MW)

∆Pg c2 c1 c0

1 10 0 332.4 0 ... 0.043029 20 02 50 -40 140 0 ... 0.25 20 03 40 0 100 0 ... 0.01 40 06 24 -6 100 0 ... 0.01 40 08 24 -6 100 0 ... 0.01 40 019 0 0 100 0 ... 0 0 1000


Table A.9: IEEE 14-bus AC with 5-bus MTDC: SVC Data

Busno.

Qs(MVAr)

Qs(MVAr)

14 50 0

Table A.10: IEEE 14-bus AC with 5-bus MTDC: VSC Data

ACbusno.

DCbusno.


1 16 0.0001 0.01 0 0.0001 0.02 0.011033 0.003464 0.0053353 18 0.0001 0.01 0 0.0001 0.02 0.011033 0.003464 0.0053359 17 0.0001 0.01 0 0.0001 0.02 0.011033 0.003464 0.00533513 15 0.0001 0.01 0 0.0001 0.02 0.011033 0.003464 0.005335

Table A.11: IEEE 14-bus AC with 5-bus MTDC: VSC Data (cont.)


∆Pc(MW)

3333 300 1000 ...3333 300 1000 ..3333 300 1000 ...3333 300 1000 ...


Table A.12: IEEE 14-bus AC with 5-bus MTDC: Branch Data

Frombus

Tobus

r(p.u.)

x(p.u.)

bsh


(MVA)

1 2 0.01938 0.05917 0.0528 − 0.9 1.1 501 5 0.05403 0.22304 0.0492 − 0.9 1.1 502 3 0.04699 0.19797 0.0438 − 0.9 1.1 502 4 0.05811 0.17632 0.034 − 0.9 1.1 502 5 0.05695 0.17388 0.0346 − 0.9 1.1 503 4 0.06701 0.17103 0.0128 − 0.9 1.1 504 5 0.01335 0.04211 0 − 0.9 1.1 504 7 0 0.20912 0 tap changer 0.9 1.1 504 9 0 0.55618 0 tap changer 0.9 1.1 505 6 0 0.25202 0 tap changer 0.9 1.1 506 11 0.09498 0.1989 0 − 0.9 1.1 506 12 0.12291 0.25581 0 − 0.9 1.1 506 13 0.06615 0.13027 0 − 0.9 1.1 507 8 0 0.17615 0 − 0.9 1.1 507 9 0 0.11001 0 − 0.9 1.1 509 10 0.03181 0.0845 0 − 0.9 1.1 509 14 0.12711 0.27038 0 − 0.9 1.1 5010 11 0.08205 0.19207 0 − 0.9 1.1 5012 13 0.22092 0.19988 0 − 0.9 1.1 5013 14 0.17093 0.34802 0 − 0.9 1.1 5015 16 0.0036 0 0 − 0.9 1.1 5016 19 0.0036 0 0 − 0.9 1.1 5015 19 0.0036 0 0 − 0.9 1.1 5015 17 0.0036 0 0 − 0.9 1.1 5017 19 0.0036 0 0 − 0.9 1.1 5017 18 0.0036 0 0 − 0.9 1.1 5018 19 0.0036 0 0 − 0.9 1.1 50



A.3 RTS-96 one area with 8-bus MTDC - SystemData

Table A.13: RTS-96 one area with 8-bus MTDC: Bus Data

Busno.

Pd(MW)

Qd(MVAr)

Gs(MW)

Bs(MVA)

BaseVoltage

(kV)

V (p.u.) V (p.u.)

1 208 22 0 0 138 1.1 0.92 197 20 0 0 138 1.1 0.93 180 37 0 0 138 1.1 0.94 74 15 0 0 138 1.1 0.95 71 14 0 0 138 1.1 0.96 136 28 0 1 138 1.1 0.97 125 25 0 0 138 1.1 0.98 171 35 0 0 138 1.1 0.99 175 36 0 0 138 1.1 0.910 195 40 0 0 138 1.1 0.911 0 0 0 0 230 1.1 0.912 0 0 0 0 230 1.1 0.913 265 54 0 0 230 1.1 0.914 194 39 0 0 230 1.1 0.915 317 64 0 0 230 1.1 0.916 100 20 0 0 230 1.1 0.917 0 0 0 0 230 1.1 0.918 133 68 0 0 230 1.1 0.919 181 37 0 0 230 1.1 0.920 128 26 0 0 230 1.1 0.921 0 0 0 0 230 1.1 0.922 0 0 0 0 230 1.1 0.923 0 0 0 0 230 1.1 0.924 0 0 0 0 230 1.1 0.925 0 0 0 0 320 1.1 0.926 50 0 0 0 320 1.1 0.927 0 0 0 0 320 1.1 0.928 0 0 0 0 320 1.1 0.929 0 0 0 0 320 1.1 0.930 0 0 0 0 320 1.1 0.931 0 0 0 0 320 1.1 0.932 0 0 0 0 320 1.1 0.933 0 0 0 0 320 1.1 0.934 0 0 0 0 320 1.1 0.9


Table A.14: RTS-96 one area with 8-bus MTDC: Generator Data

Busno.

Qg(MVAr)

Qg(MVAr)

Pg(MW)

Pg(MW)

∆Pg c2 c1 c0

1 80 -50 192 62 ... 13.02071 -89.2364

2497.285

2 80 -50 192 15.8 ... 13.02071 -89.2364

2497.285

7 180 0 300 75 ... 0.407455 284.3264 -92.0355

13 240 0 591 206.85 ... 0.16845 286.1954 190.233315 110 -50 215 14.4 ... 12.85612 599.1948 396.546716 80 -50 155 54.25 ... 0.075408 91.07841 220.159218 200 -50 400 100 ... 0.022498 172.529 218.689121 200 -50 400 100 ... 0.022498 172.529 218.689122 96 -60 300 0 ... 0.01 6 123 310 -125 660 248.5 ... 0.157363 197.4855 519.43226 0 0 200 0 ... 0.01 6 1

Table A.15: RTS-96 one area with 8-bus MTDC: SVC Data

Busno.

Qs(MVAr)

Qs(MVAr)

14 200 0

Table A.16: RTS-96 one area with 8-bus MTDC: VSC Data

ACbusno.

DCbusno.


17 25 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.0053351 27 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.00533523 30 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.00533521 29 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.0053357 32 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.00533513 31 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.00533535 33 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.00533536 34 0.0001 0.01 0.1 0.0001 0.02 0.011033 0.003464 0.005335


Table A.17: RTS-96 one area with 8-bus MTDC: VSC Data (cont.)


∆Pc(MW)

1881 320 1215 ...1881 320 1215 ...1881 320 1215 ...1881 320 1215 ...1881 320 1215 ...1881 320 1215 ...1881 320 1215 ...1881 320 1215 ...

Table A.18: RTS-96 one area with 8-bus MTDC: Branch Data

Frombus

Tobus

r(p.u.)

x(p.u.)

bsh


(MVA)

1 2 0.003 0.014 0.461 − 0.9 1.1 1751 3 0.055 0.211 0.057 − 0.9 1.1 1751 5 0.022 0.085 0.023 − 0.9 1.1 1752 4 0.033 0.127 0.034 − 0.9 1.1 1752 6 0.05 0.192 0.052 − 0.9 1.1 1753 9 0.031 0.119 0.032 − 0.9 1.1 1753 24 0.002 0.084 0 tap changer 0.9 1.1 4004 9 0.027 0.104 0.028 − 0.9 1.1 1755 10 0.023 0.088 0.024 − 0.9 1.1 1756 10 0.014 0.061 2.459 − 0.9 1.1 1757 8 0.016 0.061 0.017 − 0.9 1.1 1758 9 0.043 0.165 0.045 − 0.9 1.1 1758 10 0.043 0.165 0.045 − 0.9 1.1 1759 11 0.002 0.084 0 tap changer 0.9 1.1 4009 12 0.002 0.084 0 tap changer 0.9 1.1 40010 11 0.002 0.084 0 tap changer 0.9 1.1 40010 12 0.002 0.084 0 tap changer 0.9 1.1 40011 13 0.006 0.048 0.1 − 0.9 1.1 50011 14 0.005 0.042 0.088 − 0.9 1.1 50012 13 0.006 0.048 0.1 − 0.9 1.1 50012 23 0.012 0.097 0.203 − 0.9 1.1 50013 23 0.011 0.087 0.182 − 0.9 1.1 50014 16 0.005 0.059 0.082 − 0.9 1.1 500


Table A.19: RTS-96 one area with 8-bus MTDC: Branch Data (cont.)

Frombus

Tobus

r(p.u.)

x(p.u.)

bsh


(MVA)

15 16 0.002 0.017 0.036 − 0.9 1.1 50015 21 0.006 0.049 0.103 − 0.9 1.1 50015 21 0.006 0.049 0.103 − 0.9 1.1 50015 24 0.007 0.052 0.109 − 0.9 1.1 50016 17 0.003 0.026 0.055 − 0.9 1.1 50016 19 0.003 0.023 0.049 − 0.9 1.1 50017 18 0.002 0.014 0.03 − 0.9 1.1 50017 22 0.014 0.105 0.221 − 0.9 1.1 50018 21 0.003 0.026 0.055 − 0.9 1.1 50018 21 0.003 0.026 0.055 − 0.9 1.1 50019 20 0.005 0.04 0.083 − 0.9 1.1 50019 20 0.005 0.04 0.083 − 0.9 1.1 50020 23 0.003 0.022 0.046 − 0.9 1.1 50020 23 0.003 0.022 0.046 − 0.9 1.1 50021 22 0.009 0.068 0.142 − 0.9 1.1 50025 26 0.009 0 0 − 0.9 1.1 40025 29 0.009 0 0 − 0.9 1.1 40028 30 0.0158 0 0 − 0.9 1.1 40026 28 0.0128 0 0 − 0.9 1.1 40029 30 0.0117 0 0 − 0.9 1.1 40028 31 0.0109 0 0 − 0.9 1.1 40030 32 0.0223 0 0 − 0.9 1.1 40027 34 0.006 0 0 − 0.9 1.1 40032 33 0.006 0 0 − 0.9 1.1 40035 36 0 0.001 0 − 0.9 1.1 400


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Attachments

In the attached CD the following files are included:

• Source code of the implementation of the algorithm described in thisthesis (Matlab files).

• LATEX files of this report.

• Presentation files (PowerPoint).

65

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