AN ABSTRACT OF THE THESIS OF

Young-Hyun Moon for the degree of Doctor of Philosophy

in The Department of Electrical and Computer Engineering

presented on October 28, 1982

Title: OPTIMAL LOAD SHEDDING AND GENERATION RESCHEDULING

FOR OVERLOAD SUPPRESSION IN LARGE POWER SYSTEMS

Abstract approved:

Redacted for Privacy

C Hi an K. Lauw

Ever-increasing size, complexity and operation costs in modern

power systems have stimulated the intensive study of an optimal

Load Shedding and Generator Rescheduling (LSGR) strategy in the

sense of a secure and economic system operation.

The conventional approach to LSGR has been based on the appli-

cation of LP (Linear Programming) with the use of an appropriately

linearized model, and the LP algorithm is currently considered to be

the most powerful tool for solving the LSGR problem. However, all

of the LP algorithms presented in the literature essentially lead to

the following disadvantages: (i) piecewise linearization involved

in the LP algorithms requires the introduction of a number of new

inequalities and slack variables, which creates significant burden

to the computing facilities, and (ii) objective functions are not

formulated in terms of the state variables of the adopted models,

resulting in considerable numerical inefficiency in the process of

computing the optimal solution.

A new approach is presented, based on the development of a new

linearized model and on the application of QP (Quadratic Program-

ming).

The changes in line flows as a result of changes to bus injec-

tion power are taken into account in the proposed model by the

introduction of sensitivity coefficients, which avoids the mentioned

second disadvantages. A precise method to calculate these sensi-

tivity coefficients is given.

A comprehensive review of the theory of optimization is

included, in which results of the development of QP algorithms for

LSGR as based on Wolfe's method and Kuhn-Tucker theory are evaluated

in detail.

The validity of the proposed model and QP algorithms has been

verified and tested on practical power systems, showing the signifi-

cant reduction of both computation time and memory requirements

as well as the expected lower generation costs of the optimal solu-

tion as compared with those obtained from computing the optimal

solution with LP.

Finally, it is noted that an efficient reactive power compensa-

tion algorithm is developed to suppress voltage disturbances due to

load sheddings, and that a new method for multiple contingency simu-

lation is presented.

Optimal Load Shedding and Generation Reschedulingfor Overload Suppression in Large Power Systems

by

Young-Hyun Moon

A THESIS

submitted to

Oregon State University

in partial fulfillment ofthe requirements for the

degree ofDoctor of Philosophy

Commencement June 1983

APPROVED:

Redacted for PrivacyProfessor of Electrical and Compu r Engineering

in charge of major

Redacted for Privacy

Head of DeartMent of Elec rical and Computer Engineering

Redacted for Privacy

Dean of Gradu School

Date thesis is presented October 28, 1982

Typed by Frances Gallivan for Young-Hyun Moon

ACKNOWLEDGEMENT

I wish to express my deepest appreciation to my advisor,

Professor Hian K. Lauw, for his constant guidance and encourage-

ment throughout the course of this study. Special thanks are ex-

tended to the members of my graduate committee for their valuable

advise and interest in my study.

I am indebt to the scholarship foundation of Korea Electric

Association for their support. Partial support of Bonneville

Power Administration(BPA) under Project No. DE-AP79-82BP34709, is

gratefully acknowledged.

I would like to thank Professor Young Moon Park in Seoul

National University,Korea, for his initial guidance of this study,

and Mr. John W. Walker in BPA for his special kindness of helping

me use the BPA computer system.

I am sincerely grateful to my parents for their encouragement

and wishes for my academic fulfilment. I would like to dedicate

this thesis to my father.

TABLE OF CONTENTS

Introduction 1

Chapter 1 General Consideration of Static Optimization

Problems 7

1.1 Introduction 8

1.2 Theories of Optimality 11

1.2.1 Necessary and SufficientConditions for Optimality 11

1.2.2 The Lagrangean Function andDuality of the OptimizationProblem 21

1.3 Linear Programming (LP) 29

1.4 Quadratic Programming (QP) 43

1.5 Nonlinear Programming (NLP) 50

Chapter 2 Load Flow Analysis 52

2.1 Introduction 53

2.2 Formulation of Load Flow Problems 55

2.2.1 Modeling of System Components 55

2.2.2 Mathematical Modeling 57

2.3 Load Flow Calculation 63

2.4 Optional Load Flow 69

Chapter 3 Contingency Analysis 73

3.1 Introduction 74

3.2 Bus Outage 77

3.3 Line Outage80

3.3.1 Single Line Outage 80

3.3.2 Multiple Line Outage 86

3.4 Line Flow Monitoring 90

3.5 Conclusion 91

Chapter 4 Load Shedding and Generator Rescheduling (LSGR) 93

4.1 Introduction94

4.2 General Approach and Basic Physical

Assumptions98

4.3 Sensitivity Coefficients of Line Flows

to Bus Injection Power102

4.4 Linearized Model by Using the Sensitivity

Coefficients114

Chapter 5 Application of Optimization Techniques to

the LSGR Problem120

5.1 Introduction122

5.2 General Approach125

5.3 Optimal Load Shedding Algorithm 128

5.4 Optimal Reactive Power CompensationAlgorithm 133

5.5 Optimal Generation Rescheduling Algorithm 142

5.6 Comparison with Conventional LP Approach 151

Chapter 6 Numerical Results 157

Chapter 7 Conclusions 164

Bibliography 166

Appendices 171

Appendix I 171

A. Proof of Theorem 1.1 171

B. Pivot Operation of Simplex Method 175

Appendix II 184

A. Test of Sensitivity Coefficients 186

B. Numerical Results for the 17-Bus System 186

C. Numerical Results for the 50-Bus System 198

LIST OF FIGURES

Figure Page

Fig. 2.1 Generator model 56

Fig. 2.2 Transformer model 56

Fig. 2.3 Transmission line model 57

Fig. 2.4 System of configuration 58

Fig. 2.5 Flow Chart for Load Flow Algorithm 72

Fig. 3.1 H- equivalent model of a transmission line 81

Fig. 4.1 11-equivalent model of transmission line k 105

Fig. 5.1 Flow chart of LSGR program 127

Fig. 5.2 Optimal reactive power compensation algorithm 142

Fig. 5.3 Optimal generation rescheduling algorithm 150

Fig. 5.4 Comparison of QP and LP 154

Fig. 6.1 5-bus system configuration 161

Fig. 6.2 17-bus system configuration 162

Fig. 6.3 50-bus system configuration 163

LIST OF TABLES

Table Page

Table 5.1 Essential Memory Requirements 15&

Table 6.1 Summary of the Numerical Results 160

Table A.1 Input Data 184

Table A.2 Results of Load Flow Calculation 184

Table A.3 Base Case Line Flows 184

Table A.4 Sensitivity Coefficients 185

Table A.5 Input Data for Operating Point 185

Table A.6 Comparison of Load Flows by Sensitivitywith Accurate Load Flows 185

OPTIMAL LOAD SHEDDING AND GENERATION RESCHEDULING FOR

OVERLOAD SUPPRESSION IN LARGE POWER SYSTEMS

Introduction

With the increase in size, complexity and operating costs of

modern power systems, the problem of establishing a secure and low-

cost operation has become more significant than ever before.

Several serious blackouts in wide areas of the U.S., England

and Japan in the late 1960's showed the disastrous consequences of

the aggravation of initially small disturbances. This led to the

recognition that the increasing size and complexity of power systems

through interconnections between adjacent areas required a careful

system design which would ensure an adequate safety margin. The

sudden increase of oil prices in the 1970's, followed by the so-

called oil shock, has given great impetus to the study of economic

operation of power systems, especially to the reduction of genera-

tion costs.

In this respect, the importance of developing optimal and

secure control strategies for load shedding and generation resched-

uling has been highly appreciated [53-60].

The challenge to the development of a proper LSGR (Load Shed-

ding and Generation Rescheduling) strategy is to trigger, in an

emergency or alert state, some optimal control actions which

2

(i) suppress all overloads and abnormal voltages, and

(ii) minimize the sum of eventual load curtailments and total

generation cost,

subject to the following constraints:

(a) power balance condition,

(b) line current (or line flow) constraints,

(c) real and reactive power generation limits,

(d) load constraints, and

(e) bus voltage constraints.

Due to the primary requirements of system integrity, the min-

imization of the sum of load curtailments is to be secured regard-

less of generation cost [57-59]. Consequently, the LSGR problem

needs to be formulated in terms of an hierarchical optimization

problem, which generally involves a high degree of nonlinearity.

The LSGR problem is well formulated by many contributors [53-

60], and the most effective approach has been considered to be a

so-called LP (Linear Programming) algorithm based on linearized

models relevant to the LSGR problem. Most of the LP algorithms

found in the literature [53-60] adopt piecewise-linearized genera-

tion cost functions, for which it is necessary to introduce a

number of slack variables and new inequalities for all line seg-

ments of the piecewise-linearized curves. This creates a signifi-

cant burden on computing facilities.

A great number of linearized models have been presented in

3

earlier papers [53-60]. A trend which can be observed in the re-

cent literature [55-59] is that of using the Jaccobian matrix of

load flow equations in establishing a model relevant to the LSGR

problem, making it possible to apply powerful numerical techniques

such as optimal triangular factorization. However, most of these

models employ some constraints which are expressed in terms of

currents and/or phase angles, while objective functions are formu-

lated in terms of power generation and loads at system buses. This

also leads to numerical inefficiency in finding optimal solutions.

In this thesis, a new approach for solving the LSGR problem is

pursued, based on the development of a new linearized model and on

the application of QP (Quadratic Programming).

The linearized model presented herein is developed with the

use of sensitivity coefficients of line flows to injection power at

system buses, obtained by linearizing the line flow equations. By

means of the sensitivity coefficients, all the system constraints

are converted to equivalent linearized constraints which are

expressed in terms of the injection power to buses, that is, in

terms of power generation and load at each system bus. This sig-

nificantly simplifies the algorithm of the optimization procedures.

The application of QP to the solution of the LSGR problem sig-

nificantly reduces the computation requirements as set forth by LP,

due to the lack of necessity for piecewise linearization of genera-

tion cost functions. Moreover, by the use of quadratic cost func-

tions, QP produces an optimal solution which yields lower genera-

tion costs as compared with those obtained from an optimal solution

4

by LP. Under these considerations, two QP algorithms are developed

on the basis of the proposed model:

(i) Wolfe's method [1], and

(ii) the Lagrangean multiplier method with the

Kuhn-Tucker necessary conditions [3, 6, 18],

respectively.

With the first algorithm, the feasibility of applying QP to

the LSGR problem is investigated from a theoretical point of view.

The second algorithm is successfully developed and tested for sev-

eral systems to show its applicability to practical power systems.

In this study, a new technique to suppress bus voltage distur-

bances due to load curtailment is presented.

In the case where load curtailments are initiated as reactive

load curtailments as well for the suppression of overloads on

the system, it is necessary to readjust reactive injection power

at system buses in order to maintain all bus voltages within a cer-

tain tolerance limit of bus voltage deviation. In conventional

algorithms for solving LSGR problems, this reactive power control

strategy is determined by conducting a load flow program, which

leads to a considerable increase in computation time. In this

thesis, an efficient algorithm for optimal reactive power compensa-

tion is developed by employing the least square deviation method

for the purpose of minimizing bus voltage deviations from the

specified bus voltage. The significant reduction in required

computation time will be demonstrated.

5

In Chapter 1, the theories of optimization will be introduced

and various optimization techniques will be discussed in terms of

their merits for the LSGR problem. Chapter 1 will focus on dis-

cussions of the Kuhn-Tucker necessary conditions, the simplex

method for LP, and Wolfe's method for QP, due to their important

roles in the solution algorithm to be proposed.

Load flow analysis is dealt with in detail in Chapter 2.

This will provide a theoretical background to the analysis of the

LSGR problem, as well as to the presentation of a new linearized

model.

Contingency analysis is covered in Chapter 3, including the

presentation of a new simulation method for multiple contingency

cases which improves the conventional simulation method consider-

ably in terms of computation time and simulation accuracy. Con-

tingency analysis is necessary to obtain information of overload

conditions and system state during a particular emergency situa-

tion.

The LSGR problem is treated in Chapters 4 and 5 on the basis

of the theory presented in the previous chapters. Chapter 4 is

devoted to the development of a new linearized model, formulated

in terms of sensitivity coefficients. A precise method to calcu-

late these coefficients is also included. In Chapter 5, QP

algorithms for solving LSGR problems are developed by using the

proposed linearized model, and a scheme for implementation of an

optimal reactive power compensation algorithm is presented.

6

In Chapter 6, the validity of the proposed model and algo-

rithms is numerically verified for illustrative and practical power

system configurations. The results of the numerical tests are com-

pared with those obtained from conventional methods. Conclusions

reached as a result of this study are presented in Chapter 7.

7

CHAPTER 1

General Consideration of

Static Optimization Problems

asterisked letter : optimal value

underlined letter : a vector

Rn

: n-dimensional real Euclidian space

E : belongs to

[ ,b] : closed interval

v : gradient

v : gradient with respect to x

: scalar product

V : for an arbitrary

3 : there exist

: such that

superscript T : transpose

1.1 Introduction

The problem of optimizing (either maximizing or minimizing) a

numerical function subject to certain constraints on the variables,

is called the optimization problem or the mathematical programming.

The general static optimization problem is stated as follows:

where

Maximize (or minimize) f(x) (1.1)

subject to

0 i

gi(x) 0 , i =t+1,...,m

x = [xl,x2,...,xnjT

(1.2)

8

In the mathematical context, f(x) is a given function f: RnR1

and g.(x) also a given function g:Rn-,R , where n denotes the

dimensional real Euclidian space and x is a vector representing a

point in x.

The goal of the optimization problem is to find a point x which

maximizes (or minimizes) f(x) over the region restricted by the given

constraints (1.2).

The optimization problems are encountered in various fields such

as engineering, science, economics, etc., and are applicable in

9

numerous areas such as (i) the optimal planning of an electric power

system to maximize the profit margin subject to certain service

requirements and technological limitations, (ii) the optimal design

of structures to minimize the total cost subject to given load condi-

tions and specified strength limitations of supplied materials.

Optimization problems for real systems can be classified into

various categories in the following terms: (i) deterministic or

probabilistic, (ii) static or dynamic, (iii) continuous or discrete,

(iv) linear or nonlinear, and (v) smooth or nonsmooth. Since this

study is concerned with a static optimization in the LSGR problem

introduced in Chapter 4, only deterministic and static optimization

problems with smooth functions of continuous variables will be dealt

with in this chapter.

Due to the wide applicability and great complexity of optimiza-

tion problems, a number of techniques have been developed for solving

various static optimization problems. The Largrangean multiplier

method is the most popular technique for equality-constrained optimi-

zation problems and the steepest descent method for unconstrained

optimization problems. The Kuhn-Tucker theorem, the gradient method

with feasible directions and the duality theory provide a theoretical

background for the general optimization problems.

Linear programming is a well known technique for linear optimi-

zation problems. Since the introduction of the simplex method by G.

B. Dantzig, the linear programming technique has been widely applied

10

to various fields, especially to economics. Another contribution of

the simplex method is stimulating the development of Wolfe's method

for quadratic programming problems.

In this chapter, the theories related to the optimization will

be discussed. Linear programming and quadratic programming techni-

ques will be introduced in detail since linear and quadratic optimi-

zation problems are included in the LSGR problem. Finally, various

nonlinear optimization techniques will be briefly discussed with

their features.

11

1.2 Theories of Optimality

1.2.1 Necessary and Sufficient Conditions for Optimality

In this subsection, the necessary and sufficient condition for

optimality will be discussed with important theorems after the intro-

duction of definitions relating to this topic.

Definition 1.1 Direction vector

A vector d with unit length is called a direction vector

since x = x +T1 (T 0) determines half a straight line

emerging from point x1.

Definition 1.2 Convex set

Suppose x1 and x2 are arbitrary elements of a set C. Then,

the set C is said to be convex if, and only if

exi + (1_0 E2 C for all 0 E 0,1] (1.3)

Definition 1.3 Concave and convex function

Suppose a set C is convex and a function f(x) is well

defined in C. Then, function f(x) is said to be concave in C

if and only if

Aexi + (1.4)x2) > e Axi) + (1-0) gx2)(1.4)

for al 1 321 , 322 F C and 0 E- [0,1]

A function f(x) is convex in C if - f(x) is concave in C.

Here it is remarked that the convexity of sets is closely

12

related to the concavity of functions.

An important property of concave functions is that any non-

negative linear combination of concave functions is again concave.

Since the concavity of function plays an important role in

obtaining sufficient conditions for optimality, several important

properties of concave functions will now be presented.

(a) Let f(x) be a concave function. Then, for any c, the

set S(c) =fx1f(x) > c } is convex.=

(b) A function f(x) is concave, if and only if the set

S = f(x,E) I f(x) > 0 is convex where is any arbi-

trary real number or a real variable.

(c) LetasetS=fxlAx > b,x> 0 } whereAis an (mon)

matrix and b E Rn. The set S is then a convex set.

(d) If all the functions 4:(x), are concave

functions, then the function f(x) = minffi(x),,477(x)}

is also concave.

(e) Let x is,i=1,...,m be elememts of a convex set C, and

let all a.'s be positive constants.

If E a. = 1 holds, then a point x_ = E ot.x.s is also

s=1 s i =l=1s-

an element of C for all x.'s E C.

(f) Suppose f(x) is a concave function and let x = E

i=1

where all a.'s are positive constants.

13

If E ai = 1 holds, then it follows thati=1

mf(x) > E ot,f(x.)- i=1

(The proof of the above properties can be found inZangwill [ 6] Chapter 2.)

Definition 1.4 Feasible set

For an optimization problem, a set of all points which

satisfies all given constraints, is called the feasible set of

the optimization problem and is denoted by F.

The feasible set of the optimization problem (1.1)-(1.2),

can be represented as follows:

F = fx1 gi(x).= 0, i=1,2,-..4 and g. (r) > 0, i= k+1,... 1(1.5)

Definition 1.5 Feasible direction

A direction d is said to be a feasible direction at x E F

if there exists a 5 such that x + T dE F for all (5>T> 0. The

set of all feasible directions at x1 is denoted by D (x1) and

the closure of D (x1) is denoted by D (xi).

The optimality conditions will now be discussed on the basis of

previously introduced definitions.

Proposition 1.1

Let f(x) be the objective function for a given maximization

problem and D(x) a set of feasible directions at x.

If x is optimal, then it follows that

Vf(x) d < 0 for all d E D(x*) (1.6)

where . represents the inner product of two vectors.

Proof) Given in Zangwill [6] Chapter 2.

Kuhn-Tucker Necessary Conditions

In order to set up the Kuhn-Tucker necessary conditions, the

following optimization problem will be considered.

Maximize f(x)

subject to

gi 0, i= 1,2,...,

gi (x_) 1 0,

where x E Rn

(1.7)

14

At some feasible point xi, the given constraints can be divided

into three sets: a set of equality constraints; a set of active

inequality constraints; and a set of inactive inequality constraints.

For a simple treatment of the given constraints, the following

sets will be defined as follows:

15

Ieq

= f I gi(x) 0 xe F 1

(1.8)

Ia(xl)

Ia(D21)

IE )

=filgi(xl)

=fil

= Ieq

= 0 and

gi(xl) > 0 and

U Ia(x-1

)

gi(x) ?0,

gi(x) > 0 ,

xeFl

x e F }

where x1

is an arbitrary feasible point.

Since gi(x) > 0 for all ieIa.i(x) where x E F, it is concluded from

the continuity of all gi(x)'s that the inactive inequality con-

straints do not influence D(x). Also, from the definition of

I

a(x) it can be proved that if d E p(x) then Vgl.,(3.7).d 0 for all

i E I a (x) .

For an arbitrary feasible point x, a set (x) is defined as

1)(x) = d I Vgi(x)d = 0, i e Ieq

and Va .(x)d > 0, la}

Then v(x) satisfies the following proposition.

Proposition 1.2

D(x) c: D(x) for all xe F

(1.9)

(1.10)

16

Proof)

Assume deD(x).

If 7g.s (x).d < 0 for some i E Ia, then

+ 'td) < g.(x) = 0 for all sufficiently small T.

Hence g7:(x + id) < 0, which contradicts the assumption of

d E D(x).

Thus, Vg (x) d 0 for all i e Ia and all feasible

directions.

For equality constraints all feasible directions must be

tangential to al 1 equality constraint hyperplanes. Thus

Vgi(x) d = 0 for al 1 i e Ieg

Therefore D(x)c 12(x)

It can be remarked that D(x) = 12(x), which might be conjectured

from the above proposition, does not hold in general. However, it

seems to be justified that the deviating cases can be disregarded

from the practical point of view. Hence, the following assumption,

the so-called "constraint qualification" [2, 6, 10, 17], will be

adopted in this study.

Assumption: D(x) = 17(x) (1.11)

As a result, D(x) can be expressed in terms of constraints

gi(x), i = 1, ,m from Eqs. (3.9) and (1.11).

D(x) 0(x)

= d I Vgi(x)d= 0, iE Ie9

and Vg.(x).d > 0, jel }

(1.12)

17

The following theorem with respect to necessary conditions for

optimality, which was originated by Kuhn and Tucker, can now be

presented.

Theorem 1.1

Let all gi(x)'s and f(x) be continuously differenciable.

If x is an optimal solution to a general maximization

problem defined by Eq. (1.7), then x* satisfies the following

conditions, which is referred to as the Kuhn-Tucker necessary

conditions.

(i) gi(x*) = 0 for all i E IE(x*)

(ii) gi(x*) > 0 for all i E Izi(x*)

(iii) There exist multipliers Xi's such that

a) A. > 0 for all ie Ia (x*)=

(1.13)

b) 7f(x*) + E XiVgi(x*)= 0 (1.14)ie I

E

Proof) The proof of this theorem can be found in many standard

books [1-8] [17], but a proof using the differential geome-

try is inserted in the appendix since it illustrates the

precise economical interpretation.

In the above theorem, some important economical aspects of the

optimization problem can be observed by examining the increment of

the total profit corresponding to an arbitrary differential change dx

at a feasible point x. From the viewpoint of differential geometry,

18

the incremental profit can be decomposed as follows (see Appendix I).

Vf(x)dx = E Xi Vai(x).dx + an(x)dxIE

(1.15)

where IE denotes 'E(x) for brevity

n(x) is a unit vector perpendicular to Vgi(x)

for all je IF

In the above equation, Xi Vgi(x)dth represents the differential

increment of the profit which comes from changing the investment of

the i-th resource. Since Vgi(x),dx corresponds to the differential

increment of the i-th resource, Xi

can be interpreted as the differ-

ential unit price of the i-th resource at an investment state x. The

last term an(x).dx corresponds to the differential profit increment

for all other resources which are not included in the first term.

It should be noted that, unless x is optimal, n(x)dx can be

set positive because of the freedom of dx while V4c).dx > 0 (i F IE)

for all dx E D(x). From this observation, it can be found that

* * *E X. g.(x ).dx = 0 and an(x )-dx = 0 for all drE=D(x ),where x*

_ _ _

denotes the optimal solution. (Otherwise, the total profit can be in-

creased.)

Consequently, it is concluded that each term in the right side

of Eq. (1.15) represents the incremental cost for the change of the

corresponding resource, and that all the multipliers X's in the

Kuhn-Tucker conditions are just the unit prices for the corresponding

19

resources.

The multiplier X.is usually termed as the marginal cost,

shadow price or imputed cost in economics, and as the Lagrangian

multiplier or dual variable in mathematics.

Consideration of the Sufficient Condition for Optimality

From the definition of optimality, the sufficient condition of

optimality for a maximization problem can be generally stated as

follows:

f(x*) ? f(x) for all x e F (1.16)

where x is an optimal solution.

The above inequality is obviously a sufficient.condition, but there

is no general way to check the above condition for all x in F.

Subsequently, there is a need to establish some other sufficient

conditions which can be easily checked.

There are two approaches to developing sufficient conditions for

optimality: one is an approach utilizing the properties of concave

functions and convex sets for the concave optimization problem with a

convex feasible set, and the other is an approach based on the duali-

ty theorem. The former method will be discussed now, while the

latter will be presented after the duality theory for the optimiza-

tion problem has been introduced.

20

Theorem 1.2

Suppose the functions f(x) and gj(x), are

differentiable and concave functions. Let f(x) be the objective

function and gi (x) be the constraint functions in the general

optimization problem (1.7).

If X* satisfies the Kuhn-Tucker conditions, then x* is the

optimal solution for the given problem.

Theorem 1.3

For the given optimization problem (1.7), let .f(x) be a

differentiable pseudo-concave functiont and let every g2(x) be

a differentiable quasi-concave function:

If x* satisfies the Kuhn-Tucker conditions, then x is the

optimal solution for the given maximization problem.

Proofs of Theorem 1.2 and 1.3 can be found in many standard

books [1-7].

fi Pseudo-concave Function

A function f%1R71--iR1 is said to be pseudo-concave if, once adirectional derivative indicates a decrease of the function, the func-tion continues to decrease in that direction.

ft Quasi Concave Function

A function f: Rn-PR1 is said to be quasi concave in domain S iffor two arbitrary points xi, x2 e S it holds that

f( ex + (1-e)x2) > min [ f(x1), f(x2)] for all e <0,2].

21

It should be mentioned that the quasi-concavity [6] of the

constraint function and the pseudo-concavity [6] of the objective

function implies that there is neither any local minimum nor any

saddle point in the feasible convex set. Accordingly, only one

solution can be obtained by solving the Kuhn-Tucker conditions, and

this solution is optimal for the given problem.

1.2.2 The Lagrangean Function and Duality of the Optimization

Problem.

The Lagrangean function and the definition of a saddle point

will be introduced here since it is highly convenient to treat the

duality of the optimization problem.

Consider a following optimization problem with only inequality

constraints:

Maximize f(r)

subject to

gi(x) 0, = 1,2,...,m

(1.17)

Note that this formulation is a generalized optimization problem

since every equality constraint can be replaced by two inequality

constraints.

Definition 1.6 Lagrangean Function

The Lagrangean function L ( 0/3 X) corresponding to the

22

optimization problem (2.17), is defined as

L(x, X) = f(x) + E Xigi(x)i=1

f(x) + X.g(x)

where X = [X1,X2,..,Xm]T

> 0

g(x) [ (x)'g2(x)''gm(x):IT

(1.18)

Definition 1.7 Saddle Points

A point (x°,X°) is said to be a saddle point of L(x,X)

if L(x,X) satisfies that

t OX - f_0,, ON < f_0,1L(x ,A °) LkT ) LkT

for all xeX , eA

where X =fen and Ate

(1.19)

Definition 1.8 Primal Function and Dual Function

Given a Lagrangean function Eq. (1.18), the primal

and the dual functions are defined as follows:

Primal function:

L*(x) = min L(x,X)

A

whereA = f(XI,X2,,Xm)T I Xi > 0, '1=1,2,...,70

(From now on, A will denote the set as defined above.)

(1.20)

Dual function:

L (x) = max L(x,X) where Xc Rri (1.21)

x E X

From the definition of the primal function, it is observed that

f(x) if x is feasibleL*(x)--

- ... otherwise(1.22)

23

This implies that the original optimization problem is equivalent to

max L*(x). Moreover the definitions of the primal and dual functionsxE-x

give the following inequality:

L*(x) < L (X) for any XEX, XE A (1.23)

Definition 1.9 Primal Problem and Dual Problem

Given an optimization problem in (1.17), the primal and

dual problems corresponding to the optimization problem are

defined as follows:

Primal problem:

Dual problem:

max L*(x)x X

min L *( X)

X E A

(1.24)

(1.25)

24

It should be noted that the primal problem is actually equi-

valent to the original optimization problem while the dual problem

induces the same solution as that of the original problem.

Proposition 1.3

If x° is the solution of the primal problem, then x° is the

optimal solution of the original optimization problem.

Proof) The proposition follows directly from the defini-

tions.

Theorem 1.4

If the Lagrangean function corresponding to an optimization

problem has a saddle point at (x° ,X° ,) then x° is the optimal

solution of the optimization problem and the evaluated value of

its objective function is given by L(x°, 0).

Proof) Given in Zangwill[6] Chapter 2.

Sufficient conditions for optimality by means of the Lagrangean

function.

From theorem 1.4, it is concluded that the existence of a saddle

point for the Lagrangean function gives the sufficient condition of

optimality. Furthermore, many other sufficient conditions can be

established by the duality, some of which will be presented in the

next theorem.

25

Theorem 1.5

If x0

Rn

satisfies any of the following conditions, then

xo

is an optimal solution of the corresponding optimization

problem

o.=

o.*,.(a) L*(x k, for some X EA

(b) x° is feasible for problem (1.17), and it holds that

I(T ) f L,

(_.E

1,.A 1),

for any xle X, and Xle A,

(c) 1) x° is feasible for problem (2.17), and

2) E X.Vg.(2) =

i=1 "where X

o

i> 0, i=1,,m, and

3) L (x°, ft°'A ) = max L (x, A °)

x EX

(d) 1) xosatisfies the Kuhn-Tucker conditions for problem

(1.17), and

2) V L (x°, X) = 0 implies L(x°, X) = max L(x, X)x eX

where X is the Lagrangean multiplier vector in the

Kuhn-Tucker condition.

Proof) The proofs are given in Zangwill [6] Chapter 2.

For some kinds of optimization problems, it is easier to solve

the corresponding dual problem than the primal problem. The dual

problem, simply defined as min L* (k), is approachable in theX EA

following manner.

26

From the definition of the dual problem, the following relations

are obtained:

min L (X) = min max L(x,X)Xe A XEA xEX

. min max [ f(x) + E X.g.(x)XEA x(iX i=1 "

The above formulation can be restated as

min ff(x) + E X4.g.(x)}

XA /7=1

subject to

m mf(x) + E X.g.(x) = max ff(x) + E Xi g-(x)}

s ii=1 x E X i=i

X. > 0, i= , ,m=

Dual Concave Problem

(1.26)

(1.27)

Suppose all functions f(x) and pi(x) are concave and differ-

entiable. Then it is evident that

V L(x°_,A) = 0 if and only if L(x°,X) = max L(xdO and X e Ax xeR

(1,28)

The above equation gives the following dual problem for the con-

cave optimization.

27

mmin f f(x) + E X.g.(x)}X E A 2=1

subject to

(1.29)

7f(x) +i1

X.Vg.(x) X.0 ( X > 0, i=1,...,m)=

LP (Linear Programming) Dual Problem

Consider the following linear optimization problem:

MaximizeTx-

subject to (1.30)

A x = b

Theorem 1.5 can be applied to the above LP problem. Hence, the

Lagrangean and the dual functions corresponding to (1.30) are given

by

L(x,X) = qTx

+ XT( b - Ax), x > 0

L* (x) = maxTx

+ XT( b - Ax )

x e X

(1.31)

(1.32)

= qT x + XT( b Ax) subject to qT

- xTA = 0

From the above result, the LP dual problem is given by

min L*(X) = min f qTx-XT ( b Ax ))

AEA (1.33)

subject to

qTx - XTA = 0 and X > 0

28

Furthermore, the objective function in Eq. (1.23) can be reduced by

using the constraints as follows:

qTx - xT(b - A x) = XTb + (q

T- X

TA) = XTb (1.34)

Finally, the LP dual problem is given by

minimize XTb

subject to (1.35)

TAx = q, X > 0

29

1.3 Linear Programming (LP)

Linear programming is the study of optimization problems with a

linear objective function and linear constraints, and is a product of

the computer-age. Although the LP problem was posed a long time ago,

because of its computational complexity it had been relegated to a

pure mathematical topic with little applications until the computer-

era began in the 1950's.

The practical application of computers to business and industry

motivated the vigorous study of linear programming. As a result, an

elegant method for solving LP problems, the so-called simplex method,

was presented in 1951 by George Dantzig, which made linear program-

ming suddenly spring to life. In this section the simplex method

and its theoretical background will be briefly discussed.

In general, a linear programming problem consists of an objec-

tive function, equality and/or inequality constraints, and sign con-

straints on some of the given variables. However, any linear pro-

gramming problem can be changed to the canonical form defined as

follows.

Definition 1.10 Canonical form of LP problems (Primal LP problem)

The following form of LP problem is called the canonical form:

30

MinimizeT

c x

subject to (1.36)

Ax = b

x > 0

where A is an (mxn) matrix with column vectors a--s . E Rn,

i=1,2,...,n; b E Rm and c E Rn.

(It is assumed that all dependent equations are excluded in

this problem.)

The above LP problem will also be referred to as a primal LP

problem in contrast to the corresponding dual LP problem, which will

be introduced later.

The procedures of finding the canonical form for a given LP

problem will be illustrated in the following example.

Example) Consider the following LP problem:

Minimize 3x1 - 2x2 + 5x3

Subject to xl - 2x2 < 3

x1

+ x2

5

2x1 + x2 + x3 > 10

xi > 0 , x3 > 0

(1.37)

Since x2 has no sign constraint, define new variablesand x"

2> 0 for x2.

x+2

> 0=

31

Let

x2

= x2

x2

with x+

2> 0 , x

2> 0 (1.38)

= =

By introducing a slack variable wl 0, the first equation can berewritten as

+x

1

- 2x2

2x2

+ w1

= 3 (1.39)

The second equation is directly changed to+ -

xi + x2 - x2 = 5(1.40)

For the last inequality, multiplying both sides by (-1) and substi-tuting x, by Eq. (1.38) gives

-2x1 2

- x+

+ x2

- x3< 10

By introducing another slack variable w2 > 0, it follows that

+ --2x

1

- x2 + x2 - x3 + w2 = 10 (1.41)

Consequently, the canonical form for the given LP problem isobtained from Eqs. (1.38)-(1.41) as follows:

+ -Minimize 3x1 - 2x

2+ 2x

2+ 5x

3+ Ow

1

+ Ow2

+ -subject to x

1- 2x

2+ 2x

2+ w

1

= 3

xl x2+

x2= 5 (1.42)

+-2xl - x2 + x2 + x3 + w3 = 10

x1

> 0 , x2 >0,x2> 0 ,x

3>0, wl > 0 , w2 0

= =

As discussed in the previous section, the duality of the LP

problem can provide the sufficient conditions for optimality. The LP

dual problem will now be discussed for the canonical LP problem.

Definition 1.11 LP Dual Problem

The LP dual problem for the canonical LP problem is defined

as follows:

32

Maximize

2! b

subject to LTA < c

T

(1.43)

Proposition 1.3 (Farka's Lemma)

Let A be an (mxn) matrix and c E Rn.

Then the following statements are equivalent.

(i) For alix E Rn, Ax > 0 implies cTx > 0

(ii) There exists some > 0 such that aTA = cT

(Proof is given in many references [1], [10-15].)

For the discussion of duality, two sets X and Y will be defined

as follows:

X = { x Ax = b , x > 0 }(1.44)

YyTA cT}

(These definitions will be adopted throughout this section.)

Theorem 1.6

Suppose X # ci) for a given canonical LP problem.

If cTx has finite minimum in X, then

cTx > yTb for all x(EX and a(EY, and

(ii) There exist optimal solutions x* and y*

such that

cTx*

= (y*

)

Tb (1.45)

33

(Proof is given in Franklin [1] Chapter 1, pp. 62-66)

It is remarked that Theorem 1.6, (ii) gives the sufficient con-

dition for the given canonical LP problem, and thus this theorem

plays an important role in developing an algorithm to solve the LP

problem.

Definition 1.12 Basic Solution and Basis of the LP Problem

Suppose that a nonzero vector x is a feasible solution

(xE=X) for a given cononical LP problem, and its nonzero compo-

nents are xal

, xa2

,... Then Ax is represented by a linear com-

bination of the corresponding column vectors of A:

Ax = x aal + x aa2 + + x (Pk = b (1.46)

ak

If the column vectors aal, aa2, ...aam are linearly

independent in the case of k = m, then x is called a basic

solution.

The set of the vectors aal,aa2 ...aam is called a basis

and is denoted as

B = {aal , aa2

... aam}

Theorem 1.7

(1.47)

Consider a canonical LP problem as given in Eq. (1.36).

i) If there is any feasible solution (X (I)), then

there exists at least a basic feasible solution.

34

ii) If there is any optimal solution, then there

exists a basic optimal solution.

Proof: Given in Franklin [1], Chapter 1.4 (pp. 29-31).

Definition 1.3 Degeneracy and Non-degeneracy

Let A be an (mxn) matrix for the given canonical LP problem

and suppose m < n.

If rank of A is m , and b cannot be represented by any

linear combination of fewer than m column vectors of A, then

the canonical LP problem is said to be non-degenerated. Other-

wise, the LP problem is said to be degenerated.

When a given LP problem is degenerated in mathematical struc-

ture, great difficulties will be encountered in developing an algo-

rithm, and the powerful algorithm of the simplex method may fail to

yield an optimal solution. Though the degenerated case may occur in

the mathematical point of view, such cases seldom arise in practical

problems, and so the non-degeneracy assumption is generally adopted

in the study of LP problems as also will be admitted here.

Admitting the non-degeneracy assumption, it follows that any

feasible solution with m positive components is a basic feasible

solution, and vice versa.

Theorem 1.8

A feasible solution x is optimal for the canonical LP

problem if and only if there exists y such that

mE yiaii < ej

i=1j= 1, 2, ..., m (1.48)

35

and the equality holds for every j which corresponds to x.>0.

The equality condition in Eq. (1.48) is referred to as

the equilibrium condition.

Proof) Given in Franklin [1], Chapter 1.

For the further discussions, define two index sets as follows:

IB= the set of all basic column vectors' indices

IB- = the set of all nonbasic column vectors' indices

This notation will be used throughout this section.

Theorem 1.9

Let x* be a basic optimal solution for the canonical LP

problem given in Eq. (1.36).

Suppose xj > 0 for all jEIB and xs = 0 for all 8E1E.

If an optimal dual vector y * satisfies that

(y*)Tai

(y*

)

Tas

< es

for all j E IB and (1.49)

for all s E I§ (1.50)

then x is the unique basic optimal solution of the primal

problem and u. is the unique basic optimal solution of the

dual problem. (Proof: Franklin [1], Chapter 1.9, pp. 73-74.)

36

From the above theorem, it can be observed that all non-basic

variables can be set to zero without destroying the optimality of the

solution. Furthermore, not all solutions will be examined for

optimality but only all possible basic solutions. By setting all

nonbasic variables to zero, the equality constraints of the canonical

LP problem can be changed to

E x.ai = biI

B

(1.51)

From Theorem 1.8, the equilibrium conditions corresponding to the

basic solution given by Eq. (1.51), can be represented as

iE 1

yiaii = cij=

for all j IB

(1.52)

This condition can be rewritten by use of the so-called basis matrix

M as follows:

yTM = cT

where

al a2 aM = [a , a a m] , a

iE IB

(1.53)

It should be noted here that vector is determined from Eq. (1.53) is

referred to as the shadow price vector corresponding to basis B.

37

Theorem 1.10

Suppose that a basis B is chosen in some manner for a given

canonical LP problem. Let x be the basic solution and y the

shadow price corresponding to the basis B.

If yTas

> as for some s E le , then

(i) the cost given by cTx can be reduced by replacing aP in

the basis by as and the reduced cost is given by

,

new cost =cx-x*kz

s-cs ) (1.54)

(ii) the basic solution for the new basis x is modified as fol-

lows:

*x' = A , x! = x. - A t. for jEI

B,j#p (1.55)

where x* t. and z are determined as:

t = M-las

z = E t .c.s jIB t7

- if t. < 0 for all j E IB

A =

min {x ./t ; t > 0} , otherwisejEIB

a j j

(1.56)

Proof) Given in Franklin [1], Chapter 1, pp. 37-39.

Here it is noteworthy to observe that the most effective change

of bases can be achieved by choosing aP which gives the largest

38

The above theorem gives the guidelines for developing a linear

programming algorithm. The simplex and the revised simplex methods

will now be briefly outlined.

Simplex Algorithm

Suppose a basic solution has been found in some manner with the

basis B = {21 aa2, aam}. (How to find a basic solution will

be discussed later.)

Then the simplex algorithm takes the following steps.

(i) Establish a tableau corresponding to the basis

associated with the basic solution

TE

t11tin tio g11..-g1

tml

...tmn tmo mi

...qmm

where T and Q are determined by

with

Q = P-1

T = Q[P1/4 7;,]

= [I;Q](1.57)

P = [aal, aa2 aam]

(ii) Calculate shadow prices with

p-lc

(1.58)

(1.59)

If y. < 0 for all i E IR , then x = E t. aaii=1

is an optimal solution. Thus the computation stops.

39

Otherwise, go to step (iii).

(iii) Bring some nonbasic vector as such that

yTa s

> 0 into the basis, and modify the_

the tableau through the pivot operation, which is

based on Theorem 2.11. (Pivot operation is des-

cribed in detail in Appendix I and can be found in

many books [1, 5, 7, 12].) Go to step (ii).

Through the above procedures, the optimal solution can be

attained when any optimal solution with a finite optimal value

exists. This subalgorithm is referred to as Phase II of the simplex

method.

At this stage, the simplex algorithm requires another sub-

algorithm, the so-called Phase I in order to get a starting basic

solution. This subalgorithm can be established through the transfor-

mation of a given canonical LP problem to the so-called preliminary

problem.

Definition 1.14 Preliminary Problem

For canonical LP problem as given in Eq. (1.36), the

corresponding preliminary problem is defined as

minimize W.i

subject to (1.60)

Ax + w = b

40

where b > 0 will be assumed because it is possible to

make all b.'s non-negative.

This preliminary problem is also a canonical LP problem with a

known feasible solution, which is

xo = 0 , wo = b (1.61)

Hence, Phase II can be applied to the above preliminary problem

and generates two possible cases.

Case 1): When min E w2 = 0;i

The optimal solution xp for the preliminary problem

satisfies Ax = b , x >0. Thus the original LP problem can be

solved by starting with the known basic solution 4 .

Case 2): When min E w. > 0;i

In this case the original problem has no feasible solution,

so the computation stops.

Therefore, any LP problem satisfying non-degeneracy assumptions

can be solved by the simplex method.

Revised Simplex Algorithm

Dantzig presented a more efficient algorithm by modifying his

simplex tableau algorithm.

41

For a given canonical LP problem, the revised simplex algorithm

employs an abbreviated simplex tableau in the following form.

t10

tnn

q11.

.

qM 1

. .

chm

.

. amm

7 Y1

. . . ym

(1.62)

This algorithm adopts the same procedures as the simplex method

except the modification of tableaus, and is described in detail in

many books [1, 5, 7, 12, 14, 18]. Its merits can be summarized as

follows compared with the simplex algorithm.

(i) The abbreviated simplex tableau reduces the memory re-

quirements considerably.

(ii) Computation time can be saved since the algorithm requires

the modification of only (m+1) x (m+1) elements of the

employed tableau.

In the LSGR problem, this algorithm will be adopted in order to

find an optimal load shedding policy due to the above advantages.

Degeneracy Problem

If a given LP problem does not satisfy the non-degeneracy

assumptions, serious complications may arise in conducting the sim-

plex algorithm as the result of indefinite computational cyclings.

Such cases are explained in detail in standard books [1, 12].

42

The lexicographic algorithm [1] has been developed in order to

remove cyclings in solving degenerated LP problems, but is not widely

applied because of the complexity of the algorithm. The pertubation

method [1, 12] is another feasible technique for solving degenerated

LP problems.

Dimensionality Problem

The simplex method requires the introduction of slack variables,

which increases the dimensionality of matrix A in the canonical form

of LP problems. For large-scale LP problems, the increase of dimen-

sionality causes remarkable increments of both computation time and

memory requirements. Hence, great attention has recently been di-

rected toward the application of the triangular factorization tech-

nique [38-29] to the large-scale LP problems. The use of sparsity-

oriented solution methods reduces significantly computation time and

memory requirements. A well-known method is the Bartels-Golub decom-

position technique [21], and a number of variants of this method can

be found in the literature [20-22].

43

1.4 Quadratic Programming (QP)

A general quadratic optimization problem is described by the

following form:

maximize - (pT

+ 1/2xTQx)

(namely min pTx +1/2.x Qx)

subject to Ax > b , x > 0

(1.63)

where P: (nxl) vector

Q: (nxm) symmetric matrix with element qii

A: (mxn) matrix, b : (mxl) vector

A method to solve the QP problems was presented by Philip Wolfe

in 1959. Since Wolfe's method [1] is a variant of the simplex meth-

od, his algorithm is easy to program and is considered an elegant

and practical one.

Some other methods such as the conjugate direction method [6]

and the convex simplex method [6] are applicable to the QP problem,

but both of them have difficulty in treating given constraints or a

great disadvantage resulting from repeated optimization by the sim-

plex method.

In this study, Wolfe's method will be adopted for the quadratic

optimization of the generation costs.

Wolfe's algorithm can be derived by considering the Kuhn-Tucker

condition with the Lagrangean function of the given QP problem.

From Definition 1.6, the Lagrangean function is given by

L(x,X) = -pTx - 1/2x

IQx +

T(Ax - b) + p

Tx

a >0 , a >0

(1.64)

44

From Theorem 1.1, the Kuhn-Tucker condition can be represented

as follows:

DL _p _ Qx AX p = 0ax

p x = 0

AT(Ax - b) = 0

(1.65)

(1.66)

(1.67)

Inequality constraints in Problem (1.63) will be changed to the

following equality constraints by introducing new variables w.

Ax - w = b , w> 0 (1.68)

The substitution of Eq. (1.68) into Eq. (1.67) gives

Tw = 0 (1.69)

Since all variables are non-negative, Eqs. (1.66) and (1.69) implies

p.x. = 0 , i=1, n (1.70)

X .w = 0 , j=1, m

Just as in the LP problem, Eq. (1.68) can be changed so that all

numbers in the right hand side become positive.

Ax Gw =

where b. = lb.Is s

(1.71)

As a result, a set of the following LP-like equations is

obtained.

-0x + AX +p = P

Ax - Gw =

subject to

(1.72)

p.x. = 0 , i=1, n (1.73)

X 20. = 0 , j=1, M

x> 0 , w> 0 , a> 0 , p>0

45

Any solution of Eq. (1.72) can be a candidate for the optimal

solution of QP problem (1.63) since it certainly satisfies the Kuhn-

Tucker condition.

On the other hand, it is observed from the properties of convex

sets that the set X ={xlAx > b , x > 0) is convex, and that if Q is

semi-positive definite the objective function - ( PTx 1/2x

TQx ) is

pseudo-concave. (See Subsection 1.2.1.) Consequently, from Theorem

1.3 it is concluded that if Q is semi-positive definite the feasible

solution of Eq. (1.72) is optimal for the QP problem 0.63).

Equality constraints (1.70) are interpreted as exclusion rules

for the involved variables.

The Wolfe's algorithm is composed of two phases similar to the

46

simplex method: Phase I which determines the existence of a feasible

solution for a given QP problem, and Phase II which attains an opti-

mal solution by using the simplex algorithm with the exclusion rulesV

as given in Eq. (1.70).

Phase I

The second constraint in Eq. (1.72) will be considered by

introducing slack variable z since the existence of a feasible

solution is determined by only the constraints of Problem

(1.63). The simplex method will be applied to the following LP

problem.

. Minimize E z,.`i (1.74)

zwAxA - G + = b+. subject to

x> 0 , w> 0 , z> 0.

with a starting feasible solution, x = 0,

w = 0, and z = b+

.

Then Phase I gives rise to two possible cases.

Case 1) min ( E z,. ) = 0:i

In this case, the QP problem has at least one feasible

solution. Thus Phase II will succeed Phase I to obtain the

optimal solution.

47

Case 2) min ( E

`

) > 0:

This means that the QP problem has no feasible solution,

so the computation stops here.

Phase II

A diagonal matrix Ep will be defined as follows:

E = [e..] where e.2. '2= - sign (o)6.. (1.75)2j

From the above definition, it follows that

EP!PI = (1.76)

where IP! denotes [11111, IP21, 1Pril]T

The introduction of slack variables y and z to Eq.

(1.72) gives the following LP problem with the exclusion rules.

n mMinimize E y. + E z.

i=1 I j=1

subject toQx + AX +p + Epy = -p

Ax - Gw + z = b

(1.77)

x>0 ,y>0 ,z> 0 ,w>0 , X> 0 ,p >0

with exclusion rulesp.x. = 0 , i=1, n

'7. 2

.Wa

= 0 j =1, , m

From Eq. (1.75), a feasible solution of Problem (1.77) can be

intuitively found.

o= 0 , xo = 0 , po = 0 , w°x = 0 ,

y = 1pf , zo

=+

(1.78)

48

Since the above optimization problem has the same structure

as the LP problem except for exclusive rules, the simplex method

can be applied to this problem with the following consideration:

WI If p. # 0, then the column vector corresponding to x.

must not be brought into the basis.

(ii) If xi # 0, then the column vector corresponding to P.

must not be brought into the basis.

(iii) The same rule must be applied to all xi and wi .

The above rules directly follow from the fact that both

xi and Pi can not be simultaneously positive and neither

can xi and wi .

A pivot column will be searched for by the simplex algo-

rithm and next the validity of the selected pivot column will be

examined by the exclusion rules. If the pivot column violates

the rule, then it will be discarded and another available pivot

column will be sought in the same way. If the selected pivot

column does not violate the exclusion rules, it will be brought

into the basis and the pivot operation will be executed by the

simplex method. Thus an optimal solution, of Problem (1.74) can

be attained.

49

By virtue of Theorem 1.1, it is expected that min ( qz. ),7- j

is equal to zero since X = fxlAx > b , x > 01 is not empty due to

Phase I.

Therefore, the final value of

the given QP problem.

x for conducting Phase II solves

50

1.5 Nonlinear Programming (NLP)

The various aspects of NLP problems due to the geometrical

complicacy have necessitated the development of a number of NLP

algorithms. General NLP techniques will be discussed here briefly.

For unconstrained NLP problems, there are several well-known

solution methods such as the steepest descent method, the Newton

method, and the Davidon-Fletcher-Powell method [8, 17, 19]. The

steepest descent method is reliable but converges slowly. Although

the Newton method yields convergence rapidly, its algorithm is un-

reliable and also requires second-order derivatives and matrix inver-

sion procedures.

The Davidon-Fletcher-Powell method incorporates the merits of

both methods so that the algorithm starts like the steepest descent

method and terminates like the Newton method. Moreover, it requires

neither the second-order derivatives nor matrix inversions.

Many variants of these methods for special purposes can be found

in the literature [8, 17, 19].

The conjugate direction method [6] is considered to be a

powerful tool for unconstrained quadratic optimization problems.

The penalty and the barrier methods are well-known techniques

for inequality-constrained NLP problems [4, 6, 17]. Both of them are

based on the steepest descent method. The latter has an advantage in

computation time but can only be applied to those problems in which

the starting feasible solutions can be easily found.

51

Fiacco and McCormick presented the SUMT (sequential uncon-

strained minimization technique) algorithm [2, 16] which is under

wide application. The outstanding feature of this method is its

applicability to any kind of static optimization problem.

The feasible direction method [9] also provides good computation

algorithms for only inequality-constrained problems. Its greatest

merit is the ability to produce stable covergency.

The convex simplex method is another useful optimization techni-

que but has limited applications since it can accept only linear

constraints [6].

From the above discussion, it can be observed that no NLP algo-

rithms are suitable for the LSGR problem since they require long

computation time resulting from indispensable iterative calculation.

52

CHAPTER 2

Load Flow Analysis

Notations

Rn

: n-dimensional real vector space

P,Q: real and reactive bus injection powers

F,G: real and reactive line flows

S: line flow

Vsp specified bus voltage

c: bus voltage tolerance limit

ei: phase angle of bus voltage Vi

eij =I _I

phase difference between bus voltages Viand V.

H,N,J,L: Jacobian matrices

[B]: loss coefficient matrix

N: number of system buses

n=N-1: dimensionality of bus admittance matrix orJacobian matrix

A: incremental value

Y,G,B: bus admittance, conductance and succeptance

y,g,b: line admittance, conductance and succeptance

subscript G,L,C: control units of generator, load and condensor

subscript u,k: upper and lower limits

subscript s: specified value

superscript o: evaluation at the ambient operating point

superscript T: matrix transpose

underlined letter: a vector

upperdotted letter: a complex number

53

2.1 Introduction

Load flow analysis deals with the calculation of load flows and

bus voltages in a power system subject to the regulating capability

constraints of generators and capacitors, and a given bus voltage

tolerance limits under certain loading conditions. The purpose of

load flow calculations is to quantify the evaluation of the current

system performance as well as to set forth criteria important to the

planning of power system expansions.

The formulation of the load flow problem results in a set of

nonlinear simultaneous equations. Several methods are known for

solving them.

The Gauss and Gauss-Seidel methods [23-31], both of which are

based on repeated substitution scheme for the calculations of bus

voltages and phase angles, have advantages with respect to program-

ming simplicity and in the little requirements of computer memory.

However, both methods converge rather slowly in achieving the solu-

tion.

The Ward-Hale method [61], which can be considered a modified

Gauss method, just marginally improves the convergence speed without

any other worthwhile improvements.

Nonlinear programming techniques of load flow calculations can

be found in the literature but are still in embryonic stage because

of the complexity of their algorithms.

54

The Newton-Raphson method [23 -31] based on the iterative modifi-

cation scheme by the Jacobian matrix, is superior to the above

mentioned methods in convergence speed, and is considered extremely

powerful for large systems with a great number of buses. However,

its algorithm requires a large computer memory capacity to store the

Jacobian matrix, and also involves some difficulties in the inverse

operation of the Jacobian matrix.

A recent trend shown in the literature [32-34, 40] is the modi-

fication of the Newton method as an effort to reduce both computation

time and required memory capacity. The optimally ordered triangular

technique [38 -40] was presented by Tinney and his coworkers and is

shown to be able to reduce significantly the computational burden in

load flow analysis. The decoupled load flow method [32-33] has been

proposed by Stott and is to be considered as another powerful contri-

bution to the struggle of bringing computation time and core memory

for the solution of large systems to within acceptable limits.

In this study the decoupled load flow method as well as the

optimally ordered triangular factorization technique will be applied

due to the demand of fast computation speed in solving the LSGR

problem.

For simplicity, the change of tap ratios of transformers [23,

29, 31] will be disregarded. The [p.u.](per unit) unit system will

be adopted throughout this study.

55

2.2 Formulation of Load Flow Problems

For the formulation of the load flow problem, several kinds of

models can be established depending on whether the reference frame is

based on a loop or a bus frame representation with either an impe-

dence or an admittance matrix. However, the bus admittance descrip-

tion is in widespread application because of the convenience of data

preparation and admittance matrix modification in the event of net-

work change [23, 29].

2.2.1 Modeling of System Components

For the purpose of load flow analysis, all system components are

represented by simplified models that are known to be practically

justified. Simplification is important in order to keep the number

of system equations within reasonable limits as governed by the

computing facilities.

A. The synchronous machine: A synchronous machine will beconsid-

ered as an ideal power supplier which controls the magnitude and

phase angle of the bus voltage arbitrarily

56

SG = PG + jQG

Fig 2.1 Generator model

B. Power transformer: A power transformer circuit shown in Fig.

2.2(a) can be changed to the following II -equivalent circuit [23,

29, 31].

(a) Transformer circuit (b) Tr-equivalent circuit

Fig. 2 Transformer model

In Fig. 2.2, equivalent admittances are determined by

Y1 = 1)

V2 = aZ( a - 1) = -aY

1L

1Y3 aZ

L

where a is the off-nominal tap ratio of the transformer.

57

In this study, the unit tap ratio a is always considered as 1

due to the disregard of tap changes.

C. Transmission line: The following ir equivalent model will be

adopted for a transmission line:

VP I

jbck

/2

line k

q + jb-k k

11111.

9PV9

jbck

/2

Fig 2.3 Transmission line model

2.2.2 Mathematical Modeling

0

In the formulation of the power flow problem, the real and

reactive bus power injections are usually considered as the control

variables, whereas the magnitudes and phase angles of bus voltages

are designated as the state variables.

For a given N-bus system (Fig. 2.4), each bus injection power is

given by

.*P. + JQ. = I. V. ( ) (2.1)

where is used to denote complex variables

v.:Nodal voltage of bus

I. : Bus injection current of bus i

PG1+DI

G1

Fig. 2.4 System Configuration

58

Bus injection currents are obtained from the bus-admittance node

equation.

where

= [Y] v (2.2)

= [Il'

I2" N]T

'

[f

2' ' N]l

Y : (NxN) indefinite bus admittance matrix with elements:

Y.. = G.. + jB..1j 1j 13

Y.. = G.. + jB.. (» Y..)11 11 11 lj

(2.3)

(Note the exceptional use of Y which represents a complexnumber even without a dot).

From Eq. (2.2), every bus injection current can be calculated as

follows:

N

I. = Y..V. + E (G. +i

jB1 11 1 k=1 lk k

)Vk

kOi

(2.4)

59

By letting vi = Vie-ie

1 for every bus i the real and reactive

bus injection powers are given by

N

Pi = Gii Vf + Vi kEl Vk (Gik cos eik+ Bik sin Oik)

kOiPGi PLi

N

Qi = -Bii Vi2 + Vi kEl Vk (Gik sin eik- Bik COS en)

kg= QGi

where Oik = Oi Ok ,

(2.5)

The real bus injection power is determined from the generation

schedule and load condition, and the bus voltages are mainly

control led by the reactive bus injection power. In the ideal case

that every system bus has enough phase compensating capacitance, the

power flow problem can be solved by letting Pi = Psi and Vi = Vsp,

where Psi and Vsp represent the specified real injection power to

bus i and bus voltage, respectively. However, it is usually impos-

sible to adjust all bus voltages to the specified voltage in

practical systems. Moreover, all practical systems have certain

restrictions not only on the reactive bus injection power but on some

other variables and line flows as well.

The system constraints can be described as follows:

PG9A sPGi PGui for all generator buses i

(ii) ()Giza AGi < QGui for all generator buses i

Qcti Vcjggcui for all capacitor buses j

(iii) Phi and Qu are specified, and not con-

trollable for all load buses i

(iv) 1Vi - Vspl< E for all buses i

where Vsp : specified bus voltage for all buses

: the limit of bus voltage tolerance

(2.6)

60

(v) Sk = Sk,max for all transmission lines k+

where Sk is the power flow of line k

The power balance condition is another constraint that needs to

be satisfied by any power system. That is

N N

E

k 1P

k=1P - P

Gi Li loss==0

where the power loss is determined by

N

Ploss = Pi(2.7)

2= E [ G..V.

1+ V. Z Vk (Gikcos ea Biksin ea)]

k=111 1

k=1kOi

Since Ploss is determined after completion of the load flow

calculation, it is necessary to select one slack bus the generation

of which will be changed to compensate the power imbalance. The bus

which has the largest generation capacity, is usually chosen as the

slack bus. Moreover, the slack bus is taken as the reference bus for

The line flow constraints will be discussed in detail in Section4.3.

61

the phase angles (i.e. eN=0) and will be assigned to the highest

numbered bus N.

By assuming that the slack bus N can supply unlimited real and

reactive power, bus power PN, QN will be left undetermined until

completion of all load flow calculations. Let us further set VN=1

[pu]. This results in a reduction of two control variables and two

state variables.

Consequently, the above discussions lead to the following

formulation of the load flow problem.

Solve the following simultaneous equations:

2

k

P.1

= G.. V. + V.1

E

1

Vk(G

ikcose

ik+ B. sine. )

11 1 1k 1k=

k#i

NQi

11 1+ V.

1 kZiVk ik( G sine

ik- B. cose

ik)

=

(i=1,2,...,N)

subject to

N N 0

(i)E P

GiE P

Li- P = 0

lossk=1 k=1

with the rated generation on the slack bus.

(ii) P - P < P. < P - PGti Li 1 Gui Li

Qti Qi Qui

Ivi - vspl < c

(v) S < Sk = k,max

for all lines k

(2.9)

(2.8)

62

where

0

loss: rough estimation of the power loss

Qti QGti

qui QGui QC11,

Subscripts u and i denote the upper and lower limits, respec-

tively.

The high degree of nonlinearity in the above formulation re-

quires the introduction of an algorithm which should solve Eq. (2.9)

subject to constraints (2.9). This will be discussed in the next

section.

63

2.3 Load Flow Calculation

The Newton-Raphson method will first be introduced to represent

the theoretical background of the decoupled load flow method. Next,

the decoupled load flow method will be discussed in detail with the

optimally ordered triangular factorization technique for the use in

the LSGR problem.

Newton-Raphson Method

Consider the load flow problem formulated as in Eq. (2.8) sub-

ject to constraints (2.9).

All the real injection power to every bus will now be assumed to

have already been specified by virtue of the optimal load flow calcu-

lation (See Sec. 2.4). The specified real injection power to bus

will be denoted by P

For some estimated solution [VO], the mismatches between the

estimated and specified injection powers to bus i are given by:

AP. = P . - P.1 Si x

2

Si ik 1

k( G

ikcos9. + B. sin

11=

ik ik

k#i

AQi Qsi Qi

N

=

1k=Si

4.

1B.

1 1 1-- 'V. E Vk( GiksinOik - Bikcoseik)

ki

(2.10)

(2,11)

where Qsi is determined by

ki if bus i is voltage-uncontrollable

Qi if (hi Qi Qui

Qsi(44 if Qi < Q94

f ' QuiUzi i.

(2.12)

64

The Newton-Raphson

tation scheme.

AO{

method

H , N

J , L

takes the following iterative compu-

Ae

(2.13)Ae

where

AP = [ AP1, AP2,..., API1] T

AQ = [ AQ1, AQ2,..., AQA T

Ae = [ Ael' 2'

Ae'

Ae n] T

A V = [ AV1, A V2,..., A VII] T

H, N, J, L: (nxn) partitioned Jacobian matrices

The elements of the Jacobian matrix are given as follows:

h..I

= -B.I

V. - Q1j I I

h.. = V.V.(G..sine.. B..cose..) jj j j j j j

2 = -B..V. + Q./V.

Q.I. = h../V. , i JIJ J

n

t

n.I.t

= 2G..V. +k

E

1

Vk(Gikcoseik + Biksiheik)

kOi

nil j= V.(G..cose. + B..sine..) , j

j

k

.. = V. E

1

Vk(Gikcose. + Bi sine ) = -G..V.2

+ Pik k ik It

P1

=kOi

j..tjij= -n/V. , i j

From Eq. (2.13), AO and AV are obtained by:

IAGH

-1

AV J L AQ

The estimated solution can be modified as follows:

Ae

V new dold AV

(2.14)

(2.15)

(2.16)

65

If the iterative modification of the variables by Eqs. (2.10) -

(2.16) converges, the final values of e and V in the iterations

solves the given load flow problem.

66

Decoupled Load Flow Method

In the Newton-Raphson method, the submatrices N and J are

negligible since nii and are very small compared with elements

in matrices H and L. (Note Gik >> Bik , Bik >> Bii and sineik>> 1

for all i, k.) The neglect of N and J in the Newton Raphson method

leads to the decoupled load flow method, which adopts the following

iterative computation scheme.

de = [H] -1 AP

AV = [L]-1 A

(2.17)

where AP and AQ are determined by Eqs. (2.10) and (2.11).

The validity of this method has been verified through wide

applications [29-33], and the following advantages can be noted.

(i) Reducing computer memory requirements.

(ii) Saving computation time considerably.

For the convenient treatment of the Jacobian matrix, Eq. (2.17)

will be changed to the following form:

A; = IN] -1 A13

LS/ = 1 A 'a

(2.18)

where

= [V1 A01, V2 A02,..-., Vn AOn]T

Ai = [AP01, AP2/V2,, APn/Vn]l-

AQ [AQ1/V1, AQ2/V2,, AQn /Vn]T

AV = [AV1,02,...,AVX

and the elements of H and L are given by

2hii = -Bjj - Qj/Vj

hid = -(Gib sin eij Bid cos eij)

Qii = -Bii + Qi/14A

Zij = hij

(2.19)

(2.20)

67

It is to be noted that in the new Jacobian matrices, all off-

diagonal elements are independent of the bus voltages.

In order to execute the inverse matrix operation effectively,

the optimally ordered triangular factorization technique is usually

adopted in studies of large-scale linear systems.

Consider the following simultaneous equation:

Ax =bb

where A: (nxn) matrix, and x, b E Rn

(2.21)

The matrix A can be factorized into the product of a lower

triangular matrix [L] and an upper triangular matrix [U], and Eq.

(2.21) is rewritten as

where

[L] [U] x = b

68

0

0 0

.

6

1

U =

(2.22)

u11 11 12

0 u22

0

0 0

uIn

.2n

. .

0 unn/

Once the matrices [..] and [U] are established, the solution x can be

rapidly computed by solving equations [L] /. = b and [U] x = / in

order [38]. The so-called triangular factor table is established

from combining nonzero parts of [L] and [U].

In a load flow program, the factor table is usually stored in

place of the Jacobian matrix.

Study of the optimal ordering of bus numbers to keep sparsity in

the Jacobian matrix, has been an important topic in the load flow

study, and stimulated the presentation of a number of papers. Some

important results can be found in the references [38 -42].

69

2.4 Optimal Load Flow

The problem of optimal generation scheduling, which was reserved

Min Section 2.3, will now be considered.

From the economical point of view, it is desirable to reduce the

total generation cost in addition to satisfying all system con-

straints. In a practical power system, the total generation cost

is given by

N

C =i1

F.(PGi.) (2.23)

1

where Fi (PGi) is the generation cost function for a generator

at bus i.

From the system constraints given in Eq. (2.9), the optimal load

flow problem can be formulated as follows:

N

Minimize E Fi(PGi)

i=1

subject to

(1) power balance condition

N N

PGii1

PLi Ploss ==

(ii) the generation limits

PGki 5 PGi PGui for all buses i.

(iii) line flow limits

Sk Sk max for all lines k

(2.24).

(2.25)

(2.26.)

(2.27)

70

However great difficulty is encountered in the calculation of

gloss because gloss can be calculated exactly only after completion

of the load flow calculation. Accordingly, it is required to develop

some method to approximately evaluate gloss in terms of PGi and PIA.

The loss coefficient matrix B is usually adopted for this purpose and

gives the following power loss formula [43-45].

where

gloss * PT [B] p

P = [Pl, P

Pi = PGi - PIA for i=1,....,N

B: NxN matrix with elements Bij

(2.28)

The generation function Fi (PGi) is highly nonlinear but usually

approximate to a quadratic function from the practical point of view.

F.(p .) ao bo p ,o ,2G1 i i Gi rGi (2.29)

Consequently, the optimal load flow problem is changed to a

quadratic minimization problem.

N

Minimize JG =i=E 1

(a1 + bi PGi + Ci PGi2)

subject to

-(i) 7 PGi E PLi

(PG-EL)

T[B](12G PL) 0

i=1 i=1

(ii) PGki `< PG; 5.PGui for all buses i

(2.30)

71

where PG [PG1, PG2,"DT

GNI

i

!L = [PL1, PL2''''''PLN1T

In the above formulation, the line flow constraints are reserved

as a way of checking the validity of solutions. It is not possible

to develop a general systematic procedure for estimating the line

flows before conducting the corresponding load flow calculation.

It should be remarked that normally the optimal load flow for

the normal system seldom causes any line flow to exceed its power

flow capacity because most transmission lines are designed with

enough capacity in order to be capable of overcoming the most

important contingency cases.

The optimal load flow problem can be solved effectively by the

application of the quadratic optimization technique as introduced in

Chapter I.

In the cases that a certain line flow exceeds its capacity, a

method for the modification of the generation schedule is to be

developed in order to relieve such line overloads. One possible

method is the trial and error method, which may require a multi-

plication of the computation requirements for the normal case. It

should be noted that the algorithm for the LSGR problem presented in

Chapter 5 can be an effective method for such a situation in the

optimal load flow problem.

State Variables:ei> vi

Control Variables:

PGi' (1Gi'

MODIFY:

OCi

READ:-

Input Data4

CALCULATE:

Bus Admittance Matrixand Line Admittances

1

CALCULATE:Jaccobian Matrix

(Construct: Factor Table

ISet State Variables toStarting values

f

Ml-ODIFY:

Jaccobian !

Matrix and '

Factor Table!if necessary I

Invalid

MODIFY: Phase Angles

MODIFY: Bus Voltages

4

CALCULATE:

Bus Injection Power

Checkvalidity of

solution

Valid

CALCULATE:Line Flows andPower loss

PRINT:Calculated Results

72

Information of1.System configuration,2.System parameters,3.Generators and Loads.

Starting values:

8ij=0 for all buses i,j, and

V.1 =

if bus i is voltagesp

controllable

0 otherwise

IVi-V5DkEv or not(?)

IPsi.-P.1< s

Dor not(?)

1 =

Bus injection Power:Pi,OiBus voltagesLine slowsPower loss.

Fig. 2.5 Flow Chart for Load Flow Algorithm

73

CHAPTER 3

Contingency Analysis

Notations

Rn: n-dimensional real vector space

P,Q: real and reactive bus injection powers

F,G: real and reactive line flows

S: line flow

V sp : specified bus voltage

E: bus voltage tolerance limit

ei: phase angle of bus voltage Vi

eij = 0.-e.j : phase difference between bus voltages Viand V.

H,N,J,L: Jacobian matrices

[B]: loss coefficient matrix

N: number of system buses

n=N-1: dimensionality of bus admittance matrix orJacobian matrix

A: incremental value

Y,G,B: bus admittance, conductance and succeptance

y,g,b: line admittance, conductance and succeptance

subscript G,L,C: control units of generator, load and condensor

subscript u,2: upper and lower limits

subscript 1: specified value

superscript 0: evaluation at the ambient operating point

superscript T: matrix transpose

underlined letter: a vector

upperdotted letter: a complex number

74

3.1 Introduction

The purpose of contingency analysis is to predict the ultimate

operation state of a power system subjected to various disturbances.

The transient behavior of the system will be disregarded in this

study, not because it is considered to be a matter of no importance.

Rather, the scope of the study is to determine if the new steady-

state operation, as a result of the disturbance, would not jeopardize

the integrity of the system. It certainly would be meaningless to

conduct a transient analysis if this new operation state would not be

acceptable to the system. On the other hand, if an acceptable opera-

tion state ensues, a transient analysis would be a required second

step to the study. This second step is beyond the scope of this

thesis.

In contingency analysis, the computation time is crucial to

insure appropriate control action should be taken before the system

disturbance causes a breakdown anywhere in the system.

Known simulation methods can be summarized as follows:

(i) Z-matrix method based on the DC flow technique[48,52]

(ii) Bus real injection power substitution method with Z

matrix [47]

(iii) Line outage simulation method by modification of the

Jacobian matrix with the Newton-Raphson load flow

[50]

75

(iv) Linearization method with the use of optimally ordered

triangular factorization [46]

The first two methods are mathematically attractive due to the

preciseness of the employed models, but have limited application

because of the inaccuracy of solutions as a result of the neglect of

reactive power flows. Method (iii) can provide exact solutions, but

has the serious disadvantage of excessive computation time require-

ments due to the necessity to perform repeated calculations of the

inverse Jacobian matrix.

Matrix (iv) employs an approximated model with the Jacobian

matrix, and yields fast contingency load flow calculation with suffi-

cient accuracy. The lack of need for modification of the factor

table is the most outstanding advantage of this method. Moreover,

single-line-outage contingencies are dealt with under the assumption

of symmetry of the incremental Jacobian matrices [Ali] and [AL] ,

which may lead into additional solution errors. It can be remarked

that the method is not valid for multiple line outage contingencies.

In this study, a new approach to single-line-outage cases will

be pursued which facilitates the network changes to be exactly ref-

lected in the power flow calculation. The resulting simulation

technique will be generalized to multiple contingency cases. Final-

ly, the validity of the proposed simulation technique will be veri-

fied.

76

Single contingency cases can be classified into two main cate-

gories: bus outages and line outages. With regard to multiple

contingency cases, multiple bus outage and multiple line outage cases

can arise and a mixed type of multiple contingency which includes

both line outages and bus outages might occur. The simulation of the

two categories of single contingency cases will be separately dealt

with in the next sections. The generalization of the single line

outage simulation to multiple line outage cases will be given in

subsection 3.3.2. The treatment of multiple bus outage and mixed

type contingencies will be discussed in the last section.

77

3.2 Bus-Outage

Bus outages can be considered to be the result of either a

generator outage or a load shedding.

(a) Generator outage

Suppose a generator at bus f is subjected to a fault

causing the changes in bus generation power APGf and AQGf.

Then, the changes of bus injection power are given by:

where

APf = (PGf APGf) PLf Pf APGf

AQf = min [Xf QLf-- = AQcf + AQGf

X fe[X9,f' Xuf ](3.2)

X2f = QG)2.1 QGJ

Xuf = QGuf QGuf QCuf

P, Q, AP, AQ: real and reactive power and theirincrements.

Subscripts G, C: Generator and capacitor, respec-tively.

Nonprimed variables: pre-fault state.

Primed variables: faulted state.

For the power flow calculation of the contingency, the

specified bus real injection power (indicated by the index s) and

the upper and the lower limits of bus reactive injection power

are modified as follows:

78

PI = P AP fPsf sf - G

Quf Quf Quf

QQf

(3.2)

(3.3)

where Psf denotes the specified real bus injection power at bus f.

(b) Load outage

Consider a load outage case with load changes APLf and

AQLf at bus f.

Then the changes of bus injection power at bus f are determined

by

APf = PGf (PLf APLf) Pf APLf

where

min 4-(QLf- AQLf) -Qf

fE [Xmf'

Mf]

if bus f is voltage controllable.

AQLf if bus f is voltage-uncontrollable.

XMf = QGtf QC9,f

XMf QGuf QCuf

(3.4)

The specified real bus injection power and the upper and lower

limits of reactive bus injection power are changed to

Psf Psf APLf

Quf = Quf AQLf

QQf = Qtf AQkf

(3,5)

(3.6)

79

(c) Contingency load flow for bus outages

From the above simulation of bus outages, the load flow

problem for the contingency case is well defined. It should be noted

that for both generator and load outage cases, the Jacobian matrix

is left unchanged since the topological configuration of the network

is not affected.

Therefore, the contingency load flow can be rapidly calculated

by conducting the load flow program with the aid of the already

established factor table. Moreover, the calculation can be termi-

nated just after one or two iterations, yielding the sufficient

accuracy of the solution.

80

3.3 Line-Outage

For line outage cases, it is indispensable to modify the Jaco-

bian matrix in order to reflect the corresponding network changes.

Contingency load flow calculation by conventional techniques

requires long computation time since computation of the inverse of

the new Jacobian matrix is involved in the solution procedures. A

new method will be introduced which makes it possible to calculate

the contingency load flows by using the original factor table without

modification. The generalization of the method for a single line

outage to the multiple contingencies will be presented.

3.3.1 Single Line Outage

Suppose a transmission line f between buses i and j, is

disconnected by a fault.

Four elements of the bus admittance matrix should then be modi-

fied as follows:

Vii = Gii "I" Rii =Vii Yf Ycf

= (Gii-gf) + j(Bii - bf - 0.5bcf)

Y.ij = Gij + jBij = Yii + Yf

= (Gij + gf) + j (Bij + bf)

= Yji

Y-ij = Gii + jen = Yjj - Yf - Ycf

= (Gjj - gf) +j (Bii - bf - 0.5 bcf)

(3.7)

81

where the primed characters denote the modified quantities

with respect to the unprimed characters and

Yf gf + jbf :

Ycf = jbcf/2

Ik

line admittance of the faultedline f

capacitive admittance of the 1T-modelof the faulted line f,

line k

gk ibk I

jbck

/2

Iji

Cj

jbck

/2

Fig. 3.1 it -equivalent model of line k.

Let H' and L denote the modified Jaccobian matrices. Then, the

elements of H' are determined from Eqs. (2.3) and (2.20) as follows:

hii = -Bu t - QI/Vi2

= hii - Vj fgf sin eij - (bf 0.5 bcf)cos eij} /Vi

hij = - (Glij sin eij - Bij cos eij)

= hij + fgf sin eij - (bf + 0.5bcf)coseii),hji = - (Glji sin eji - Bpi cos Op)

A (3.8)= hji + fgf sineji (bf +0,5bcf) cos eji}

hij = - 13:jj - (6/Vj2

= hjj - vi fgj sin 0.0- (bf +0.5 bcf)cos eji }/VjA, A

hkQ = hk9, if k e fi,j} or ZE fi,j }

Similarly, the elements of L' can be calculated.

/N.

82

The incremental Jaccobian matrices AN and AL will now be de-

fined as:

N'AH'

ALA Ls L(3.9)

For the single line outage under consideration, AH and AL

have the following form:

AN=

IAhi ---Ahi

j

j --Ah.Jr JJ

At..Ii

At ..I

AL =

(3.10)At ... A

Zij..

Ji

The power flow calculation as involved in the contingency

analysis requires the inverse of the modified Jaccobian matrices HI

and L . An efficient method to evaluate these matrices will now be

presented.

The following relation is easily proved to hole:

[A+BCBT]- 1 = A-1 - A-1 B IC-1+BTA-1B]-1 BT A-1 (3.11)

In order to apply this relation to the calculation of the inverse

+It can be easily proven by checking the result of the following

multiplication:

[A+BCBT]{A-1-A-1B

[C-1+BTA-1B] -1 BTA-11(2)I

83

Jaccobian matrix, it is desirable to decompose the matrices AH and

AL such that the dimension of the term [C-1+BTA-1B]-1 is minimal.

Let Mij .. be an (nx2) matrix defined as:

Then, AH and AL can be decomposed as follows:

where

Mij =0 0 1 0. .0 0 0 0

AH = Mij [Ab] Mid

AL = M.- [A ] M1J t lj

0 0 0 0 .0 1 0 0

[

Ahii

Ahij

[AO A A [At]

Ah.. Ah.

Atii Atij

At. At.J1 Jj

(3.12)

(3.13)

(3.14)

From Eqs. (3.11) and (3.13), it follows that

[H1]1 = +

T -1= + Mij [A ] M..}

h ij

=- H-1 mijf[A0-104Tj 11-1 mii1-1

-1 ^ ^ -1= EL + AL]

- 1 -1 -14ITS1m..1-1 RAT ;-1= L - L M..-E1Aij i` "ij) "ijL

-In both equations above, it is noted that the inverse of [Ah

1

+M..T

H-1

rM. and LAk

1+M.T .L

-1 m..] can be easily calculated, since the dimen-

sion of both matrices is just (2x2). Moreover, the factor tables for

H and L have already been established for the base case appropriate

84

to the considered single line outage.

The contingency load flow calculation will now be considered for

a single line outage case.

r 0 OnLet LV , e J represent the system state corresponding to the

base case appropriate to the outage. This state will be considered

the initial guess for the solution of the faulted system.

From Eqs. (2.10) and (2.11), the mismatches of bus injection

power for an arbitrary bus k as caused by the initial guess DP, 60],

are given by:

N

APk = Psk Vk E Vj (Gkj cos Okj + Bkj sin Okj)j=1

N

AQk = Qsk Vk

jE 1

Vj (Gkj cos ekj + Bkj sin eki)=

The substitution of [11°,2°] and CAP°, Ag) into Eq. (2.18) gives

the following equations.

1-'111 -0= LH J AP

11-1 AoAV = J AQ

=where Ae [V91A01,112A02,,eriAen]T

AV = [A1/1,AV2,...,AVX

= [AP1 /V7,AP°2/V2,--,APn/V0]T

Ai = [A(11/1/7,AQ2/V°2,-...,AQn/eX

(3.18)

(3.19)

85

From Eqs. (3.15) and (3.16), the above equations can be rewrit-

ten as:

r T A- 1Ae = H AP - H-1

ij h ijM.LA 1

+ M..H M..] -1 m.T .H-1AP

(3.20)

1 A1j

1 -1 -1 T ^-1AV = L AQ - L M..[A + M..L M..] M..H AQ

r,The term [H]-1 AP of the first equation above can be immediately

computed with the aid of the factor table. The term lq-1M1 j can be

changed to the following form:

-1 -1H-1

= [H e1. e.] (3.21)Mij .9 H

where < is a unit vector with entries equal to zero, except the

k-th entry equal to 1.

Hence-1Mi- can also be efficiently calculated.

The inverse matrices of Ah and [Ah1

+MiTj H-1

Mij] can be

directly computed as mentioned before. The same logic can be applied

to similar terms in the second equation of Eq. (3.20). Hence, it is

shown that [AV, A8] can be computed with rapidity. From Eq.

(3.18), it follows that

Aei = 00i/Vi

AVi = AVi

(3.22)

This shows that the entire contingency load flow calculation can

be efficiently executed through the iterative scheme as given by Eqs.

(3.17), (3.18) and (3.22).

86

3.3.2 Multiple Line Outage

Consider an outage case on m lines: f1, f2,...., fm. Let line

fk be connected to bus ik and bus jk.

In the following discussion a bus pair is meant to be 2 buses,

in which the connecting line is subject to a line outage. Hence, the

following set of bus pairs can be introduced.

f(ii.j1)'(i2.j2)' ,(im,41)1 (3.23)

Suppose the faulted lines and the above bus pair set are re-

arranged so that

ik < jk for all k and(3.24)

1 5. i2 im

The procedure as indicated earlier to solve the single line

outage case can be generalized by systematically decomposing the

incremental Jaccobian matrices AH and AL to a similar form as given

in Eq. (3.13). For this purpose, an ordered set K and a matrix Ek is

defined as follows:

K = k1, k2, , kp : an ordered set consisting of all

possible k.'s such that

(i) ki < k2<...< kp < N and(3.25)

(ii) every ki { il, i2,...,im "'jm }

where N is the slack bus number.

(Note the highest bus number is assigned to the slack bus.)

Ek =It(1, 11(2,-41] (3.26)

where eki is defined similarly to ek in Eq. (3.21)

A

Then, AH and AL are to be decomposed as follows:

AH = Ek

[Ah] Ek

AL = Ek

[At] E

k

(3.27)

where [Ah]and pyare (pxp) matrices with the following

elements.

AI

il , 1kif k. and . belong to the

I-

Ahij

= /same bus pair or ki = ki.

0 otherwise.

AZij

if k. and kJ belong to the

same bus pair or ki = kJ.

0 otherwise.

87

EXAMPLE

Consider a triple contingency case for a 10-bus system.

Suppose faulted lines fl, f2, f3 are connected between bus pairs

(2,4), (2,8) and (7,10), respectively.

Then, the ordered set K is found as:

K = f2, 4, 7, 8 } (3.29)

From the ordered set K, matrices Ek,[Ah] and [At]are easily

established

EK =

0 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

(Note EK is an (9x4) matrix since n=10-1).

A A

Ah22

Ah24

Ah27

0

Ai-i42

A1144

0 0

Ai;72

0Ah77

0

0 0 0 Ai;88

A

Ak22

Ak24

At27

0

A42

AZ44

0 0

A72

0 A77

0

0 0 0 Ak88

(3.30)

(3.31)

(3.32)

88

On the other hand, it is found from the definition given by Eq.

(3.27) that AH has the following scheme of nonzero elements

89

AN =

0

0

0

0

0

0

0

0

0

0

Ah22

0

A;42

0

0

A;72

0

0

0

0

0

0

0

0

0

0

0

0

Aii24

0

LSI44

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Ah27

0

0

0

0

Ah77

0

0

0

0

0

0

0

0

0

Ah88

0

0

0

0

0

0

0

0

0

0

(3.33)

By substituting the results given in Eqs. (3.30) - (3.33) into

Eq. (3.27), one can check the validity of the tecknique pre-

sented in this section.

3.1.

90

3.4 Line Flow Monitoring

Consider a line k which connects a pair of buses i and j in Fig.

The line flow Sij

from bus i to bus j is then given by

SkJ Iij Vi (ik ici)* Vi Fikj jqj

where (3.34)

Fij = (Vf)2 Vi Vf fa cos(ef - ef) + b sin(ef ef)1k -ki-ij-kijkijGk = -b

k(Vi f)2 +

ij fbkcos(Oif - ej!) - g

k l

sin(ef. - ejf)1- 1/2bck

(Vf)2

(The superscript f indicates the final value for each variable

obtained from the contingency load flow.)

Since it is possible to determine the lines that are expected to

be overloaded for the considered contingency cases, line flow Sk is

examined for such lines only. This selective calculation reduces the

computation requirements.

The monitor observes whether the line flow of every transmission

line exceeds its capacity, in which case an alarm signal including

the number of the overload line is sent to the system operator for

the initiation of some remedical action.

91

3.5 Conclusion

An exact simulation for single line contingencies has been

developed in this chapter. This model only results in a negligible

increase of computation requirements in comparison with the conven-

tional methods as introduced in Section 3.1. The common consensus is

that Method (iv) by Peterson et al. yields good accuracy of solutions

with the fastest computation speed. The proposed simulation method

is superior to Method (iv) in solution accuracy due to the full

consideration of unsymmetry of the incremental Jaccobian matrix.

The resulting additional computation requirements are negligible.

Another feature of the proposed method is that this method can be

generalized systematically to multiple line outage cases, which is

not possible with Method (iv).

The presented technique is a powerful tool to simulate all types

of practical line outage contingencies. Although the simulation

technique may demand a rather high computation time for an outage

case in a large number of lines, the probability for occurrence of

such contingencies is extremely small. Common contingency analyses

deal with multiple line outage contingencies for only up to three

lines which apparently is considered to be sufficient to the design

of a practical power system.

Multiple bus outages or mixed type contingencies have a rela-

tively high probability of occurrence. These outages can be simu-

lated as well by applying the technique as developed in this chapter

92

through the decouplingt of the outage. This does not result in

additional complexity to the simulation procedure. It should be

noted here that any multiple bus outage can be treated as a collec-

tion of single bus outages.

For a multiple or mixed type of contingency case, each bus outagecan be considered a decoupled outage since it can be simulated with-out regard of other outages. However, any multiple line outageshould be treated as one decoupled case.

93

CHAPTER 4

Load Shedding and Generator Rescheduling

NOTATIONS

ai, bi : generation cost coefficients for incrementalpower generation of generator i

D : diagonal matrix

Da

: diagonal matrix with elements a.

P, Q : real and reactive bus injection powers

F, G : real and reactive line flows

S : line flow

Vsp

: specified bus voltages

: bus voltage tolerance limit

6i : phase angle of bus voltage Vi

Oij = Oi - Oj : phase difference between busvoltages V. and V.

H, N, J, L : Jaccobian matrices

B : loss coefficient matrix

N : number of system buses

n = N - 1 : dimensionality of bus admittance matrixor Jaccobian matrix

A : incremental value

Y, G, B : bus admittance, conductance and succeptance

y, g, b : line admittance, conductance and succeptance

subscript G, L, C : control units of generator, load and condensor

subscript sp : specified

subscript u, : upper and lower limits

superscript o : evaluation at the ambient operating point

underlined letter : a vector

upperdotted letter : a complex number

94

4.1 Introduction

With ever-increasing size and complexity of power systems due to

interconnections between adjacent areas, the importance of developing

adequate control strategies for load shedding and generator resched-

uling has been highly appreciated.

The challenge to the load shedding and generator rescheduling

(LSGR) problem is finding such an optimal load shedding and generator

rescheduling that will

(i) suppress all overloads and abnormal bus voltages and

(ii) minimize subsequently the sum of load curtailments and

the total generation cost, subject to some given

constraints.

Due to the fact that system integrity is the primary requirement

for any electric power system, the minimization of the sum of load

curtailments is to be secured regardless of the generation cost [6,

7]. In a second instance, the total generation cost should be min-

imized under the given load curtailments. In this analysis, the

increment of the total generation cost is minimized rather than the

total generation cost.

(i) Primary objective function:

Minimize JL . Li

where APLi

is load curtailment at bus i

95

(ii) Secondary objective function:

Minimize Jn = I [Fi(PGi + LPG.) Fi ((PG ?)]

' i

where F.'is the generation cost function for genera-

tor i, and superscript o denotes the ambient

operating point.

The system constraints have already been discussed in Chapter 2

and include the following items:

(i) Power balance condition,

(ii) Line current constraints,

(iii) Real and reactive power generation limits,

(iv) Load conditions, and

(v) Bus voltage constraints.

As the system model defined by the constraints (i) through (v)

contains a high degree of nonlinearity, quite a few approaches have

been investigated.

Shen and Laughton [60] presented a linearized model by using the

line succeptance matrix B. However, their model does not yield accu-

rate solutions because the reactive power flows are disregarded in

their approach.

Subramanian [54] presented a sensitivity model and a linear

programming model. Kaltenbach and Hadju [53] presented a Q-matrix

model with line current constraints. Both models are mainly based on

the description of system constraints in terms of bus injection power

96

and line currents. However, none of these models seem to have found

wide application because of excessive computer requirements.

In their two consecutive papers, Stott and Hobson [55, 56]

proposed an approach based on the use of a load flow calculation

model. In the second paper, various aspects of the LSGR problem were

discussed, including the indication of a number of qualified objec-

tive functions.

Most of the models presented in the recent literature [55-59]

are established on the basis of the adoption of the Jaccobian matrix

in the load flow equations, making it possible to employ powerful

numerical methods such as the optimally ordered triangular factoriza-

tion. These contributions to finding the optimal solution of the

LSGR problem, however, are based upon the formulation of equality and

inequality constraints, some of which are described in terms of

currents or phase angles while the objective functions are given in

terms of the injection power to the system buses. Such constraints

can be utilized only to check the validity of the optimal solution

that is established without regard to these constraints.

A trend that can be found in the literature [55-59] is the

conversion of any possible constraint expressed in terms of currents

and/or phase angles into the corresponding constraints in terms of

the bus injection power. This creates the convenience of analyzing

overloads in a power system by means of a load flow calculation.

With regard to the application of optimization techniques, most

of the algorithms proposed in the literature [53-60] adopt the LP

97

(Linear Programming) technique by piecewise linearization of the

generation cost functions. The necessity of introducing many slack

variables and new inequality for every line segment of the piecewise

linearized curves creates a significant burden to the computing

facilities.

The application of QP (Quadratic Programming) algorithms, which

will be discussed in detail in Chapter 5, enables us to avoid these

disadvantages of conventional LP algorithms due to the lack of need

for piecewise linearization of the generation cost functions. More-

over, QP algorithms yield lower generation costs as compared with

those obtained from the use of conventional algorithms since the

actual generation costs can be more accurately reflected to quadratic

functions.

In this chapter, a new linearized approach to the solution of

the LSGR problem will be presented. This approach will produce a

linearized model appropriate for the QP applications.

The general approach and the basic physical assumptions are

discussed in Section 4.2. In Section 4.3, sensitivity coefficients

of line flows to bus injection power are introduced through the

linearization approach; and in Section 4.4 a new linearized model

relevant to the LSGR problem is established with the use of those

sensitivity coefficients. It is to be noted that the influence of

reactive as well as active bus power changes to the power flows can

be effectively taken into account through sensitivity coefficients

in the proposed model.

98

4.2 General Approach and

Basic Physical Assumptions

The LSGR problem can be generally formulated to an optimization

problem as follows:

(i) Primary objective function:

MinimizeL

= I AtiPLi

.

.

(ii) Secondary objective function:

Minimize JG

= E [F. (P. (PG?

Constraints:

(i) Power balance condition:

p. -P. -PGl . Ll loss

(ii) Line current constraints:

forIk

Ik,max

(iii) Real and reactive power generation

Pati < PGi < PGui

Clad < QGi < QGui

< QCi < QCui

G? + APGi.) - F. (P

Gi?)]

0

all lines k

limits:

for all buses i

( 4.1 )

(4.2)

(4.3)

(4.4)

(4.5)

99

(iv) Load condition:

Lei PLi PLui

for all buses i (4.6)

QL.ei QLi QLui

(v) Bus voltage constraints:

1 V. - Vsp

e for all buses i (4.7)

where the subscripts above follow the

notations below:

G, L, C : Generator, Load and Capacitor,respectively,

u, : upper and lower limits, and

Vsp

: specified bus voltage.

It is remarked that, in the above formulation, the increment

of the total generation cost rather than the total generation cost

is supposed to be minimized.

Throughout this study the equations of Constraints (4.3) to

(4.7) will be considered to constitute the model relevant to the

LSGR problem.

The computation speed of finding the solution to the LSGR

problem is crucial. In the event of an overload, the solution is

required to be established before this overload could lead into a

system breakdown such as line or bus outages. Because of the high

nonlinearity of the objective functions and the equations of con-

straints, linearly-approximated models are usually adopted in the

100

approach to solving the LSGR problem for the purpose of reducing com-

putation time and yet guaranteeing sufficient accuracy of solutions.

The following assumptions for the linearization approach seem

to be practically justified and lead to further reduction of the

required computation time:

(i) Although the system is under some overload situation,

the power generation is considered to be optimally

allocated due to the effectiveness of the ELD (Economic

Load Dispatch) program;

(ii) Coupling effects between the Q-V and P-f controls are

negligible, so that the decoupled load flow analysis

method [32, 33] guarantees sufficiently accurate results;

(iii) System faults or overload conditions under consideration

'are not so severe that a first-order approximation to

the solution of the LSGR problem would be invalid for

the problem analysis; and

Even under an overload situation, the reactive power

(Q - V) control is assumed to still have the capabil-

ity of maintaining all bus voltages within the

specified limits.

Under the above assumptions, a new linearization approach to

the LSGR problem will be pursued on the basis of the following

considerations:

(i) The objective function reflecting the generation costs

101

will be approximated as a second-order polynomial, and

full benefit will be taken from the application of the

quadratic programming (QP) technique. This leads to

the reduction of computation requirements and genera-

tion costs, as mentioned earlier;

(ii) All equations of system constraints will be linearized

in terms of bus power changes; and

(iii) When a generator outage occurs or certain load curtail-

ments are activated, the reactive bus injection power

will be controlled to minimize the mean square deviation

of bus voltages.

Otherwise the bus voltage constraints will only be

utilized for checking the validity of the optimal solu-

tion which is established without taking them into

consideration. This approach is quite reasonable,

considering the insensitivity of bus voltages if no

changes occur to the reactive bus injection power.

102

4.3 Sensitivity Coefficients of Line

Flows to Bus Injection Power

In this section, the line flow changes due to the changes in

bus injection power will be analyzed. The effects will be captured

by the introduction of sensitivity coefficients through the approxi-

mated relations of power flow changes in terms of bus power changes.

Suppose line k is connecting buses p and q. Then the power

flow from bus p to bus q is given by

13ci = I* V = F" + G"k k p k

where superscripts pq denote the considered

direction of line flow,

and the magnitude of the line flow is

(4.8)

Skq = IIkIIVpI = [(Fr)2 + (Gr)2]1/2 (4.9)

where Fk'

" G" real and reactive power flow ofk

line k with the direction from

bus p to bus q.

(Note that Skk

" * ScIP because of the line loss.)

By multiplying both sides of Constraints (4.4) by IVpl,

the following inequalities are obtained:

SM I 111:11(4.10)

Since IV I is controlled to be constant regardless of the line

current, the maximum allowable line flow can be defined as

SPq = Ik,max k,max

103

for all lines k (4.11)

As a result, the line current constraints can be replaced by

the following equivalent line flow constraints:

SPq Sk k,max

for all lines k (4.12)

Since the line flows SPq is a function of injection

power to the system buses, it follows that

Sk k kM [(FIn2 (0q)2]1/2 = Skq (P1,...,PN,

where P., Q. : active and reactive bus injection

powers at bus i, and

N : number of all system buses. (4.13)

The bus power changes APi, 401i which result from the bus

injection power control for the suppression of the imposed over-

load, are assumed to be small. Therefore, line flow change due

to bus injection power changes can be approximated to

N 3S13,(1 aSPq4sPq /

k AD. k An.l (4.14)k j=1 `aPi "4-1 aQ. '411

The subscripts pq were introduced to denote the direction

of power flow. They will be dropped from now on, with the under-

standing that the power flow is considered to flow from bus p

104

to bus q with p < q.

Since Equation (4.14) can be rewritten as:

the sensitivity of the k-th line flow to real power change

at bus i can be defined as:

aSk aFk aSk aGk

flow to reactive power

0

are to be

point.

be rewritten as

i]

(4.16)

(4.17)

(4.18)

KPk,i (aFk

aPi

aGk aPido

and the sensitivity of the k-th line

change at bus i as:

aSk aGk aSk aFk

=K (

36K a71i BFK iaQ)Qk,iQko

where 0 denotes that all differentiations

evaluated at the ambient operating

As a result, Equation (4.15) can

N

Ask

=

1I1

[KPk,i

APi+ K

Qk,i6Q

=

For the calculation of the sensitivity coefficients in

Equations (4.15) and (4.16), consider the power flow of line k

which is connecting buses p and q (p < q) as shown in Fig. 4.1.

105

I I I

P9 k 9P© >

I

Igk + j bid______,

icp

Icq

j bck/2 j bck/2

Fig. 4.1 II model of transmission line k.

The power flow of line k is given by:

- * - * -

Sk

= Ipq

Vp

= (Ik

+ Icp

) Vp

2

= gk Vp - VpVcifgk cos(Op - Oq) + bk sin(Op - Oc)}

2

+j [-bk Vp + VpVq

2

fbk

cos(0p

0q) - gk sin(0P

- 09

) - libck

Vp}] (4.19)

The above equation gives the following real and reactive power

flows of line k:

2

Fk = gkVp - VpVq fgk

cos(0p- 0q) + bk sin(0

P- 8q)}

2Gk

= -6 kVp + VpVq fbk

cos(0p- 0q) - gk sip(0

P- 9q)}

(4.20)

2

1/26ck

Vp

(4.21)

In the derivation of the sensitivity coefficients, the follow-

ing step-by-step procedure will be followed:

(a) Derive the linearized equations through the partial

differentiation of line flows with respect to the bus

voltages and phase angles.

(b) By using the relations between the increments of the

state variables [le, Av] and the increments of bus

injection powers APi,AQi in the decoupled load flow

equations, eliminate the variables Api, AVi from

the equations obtained from Step (a). Determine all

the partial derivatives of:

aFk ,

aQk ,

aFk and

aGk

aP. aPaQi

30.for all buses i.

(c) Derive compact sensitivity formulas from Equations

(4.16) and (4.17) by using the results of Step (b).

Step (a): Equations (4.20) and (4.21) are linearized at the am-

bient operating point, which will be indicated by variables

provided by superscript o.

From the total differentials of Equations (4.20) and

(4.21), the following equations are obtained:

AVAF

k= V°V° {Ai (AO ) + A2 + A3 Tq}

V

106

(4.22)

4u 4mP

AGk

= V°V°q

{131

(410p

- 6,0q

) + B2 o

+ B3 o

9---} (4.23)

V VP 9

with:

Al = gk

sin(0° - 0°q) -k

cos(0° - 0°)p P

A2 = 2gk fgk cos(0°,3 - 000 + bk sin(ep - 0°01

(4.24)

A3 = -{gk cos(0°,3 - 00q) + bk sin(ep - 0°1)1

B1 = -gk sin(0°,3 - 00(1) + bk cos(0°13 - 0°)

2 k k ksin(0° - 0°)}

P qB = -26 + {6 cos (9p - -

B = bk

cos(0° - 0°) - gk

sin(0° - 0°)3 p q P q

Step (b): The decoupled load flow equations are discussed in

107

(4.24)

(cont'd.)

Section 3.3 and can be formulated as:

A-po Ho AO()

OQo

where AP° = [AP1/V.°1, APn/eri]T ,

4[(3 [41/V7' "" 40n] '

El° = [V.°1 A01, AO n] , and

AVo

= [AV1, AVn] .

The incremental bus angles and voltages are easily

found as follows:

1 nA0i

=a

cik APk/V°

V. k=1k

n

V. = E d AQ /V°1.

k=1

dik k

-1where c

ik= element of C = H

-1dik

= element'of D = L

(4.25)

for all buses i (4.26)

(4.27)

Substituting Equation (4.26) into Equations (4.22) and

(4.23) gives:

or

n

AFk

= Al(V9 pl

. - V°P

cqi 1

) AP./V°=zg 1

n+ A

2Vo I d

pi4. /e + A V' d 4)./V°q

i=1 1 3 p

1=1qi

n

AFk

= Al (V° c . - V°i=1 q pl

c .) AP./v?

n

+ (A V d . + A V° d ) AQ./V?Iq p1 3 p qki=1 1

108

(4.28)

and, in a similar way, the following equation is obtained:

n

AG = B1 (V c -Voc .)AP./v?1=1

Gk1 q pi p

n

+ Voq

dpk

+ B3

Vop

dqk

) AQ./v?

i=1(4.29)

All the partial derivatives of Fk and Gk with respect

to the injection power of all system buses can now be found

by:

aFk

0

0

0

= Al(V°0 cpi ep co)/V7

= (A2 eci dpi + A3 VP dpi)/V7

= B1(V° c

pi- V° c

qi)/V°

(4.30)

aPi

aFk

aQi

aGk

aP.1

aGk

aQi= (B

2V°q

dpi

+ B3 pV° d

qi)/V°

109

Step (c): Substituting this result into Equations (4.16) and

(4.17) provides us with the desired sensitivity coeffi-

cients, i.e.:

0 2 0Kpk,i = E(Fk) (Gk)

2]-%

2. 7 fF0 Al

cpiV0

cqi)

V.

o+ G

kB

1(V

qcpi

Vp

cqi

)1

= (Sc0 (V?) (A./F(1)( + B1 G(1)()(eq cpi - VP cqi) (4.31)

for all buses i

KQk,i = (Sk)-1

(V0 )-1

FA2

G°B2)

V°dpik 2 k 2 q

+ (F°kA3 + G° B ) V° d .

3 k 3 p (of1

for all buses i (4.32)

Both equations above can be rewritten in a matrix

form as follows:

KPk,1

KPk,N

-KQk,1

KQk,N

Al Fk + B1

S°

1

Cok

0

0

1

V°n

V1.

0

[D]r

0

1

Vn

0

..p-th

(4.33)

..q7th

..p-th

(4.34)

. .qth

At this stage, a diagonal matrix qt will be defined as:

D V0

V°

0

0

1

V0Vn

With the use of Equations (4.27) and (4.35), Equations

(4.33) and (4.34) can be rewritten as:

AlFk + BlGko

So

[DV] x

H

with

1 1 0

(4.35)

k (4.36)

= [D(0 iLSk

[11]T xH = 644 and [1_]T it = (4.37)

where b and 1'4_

are determined by

b . =

Vq

for i = p

-VP

for i = q

0 otherwise

(4.38)

(q A2 + G(1)( B2) eci for i = p

bLi

= (F°k

A3 k

G° B3 p

) V° for i = q (4.39)

0 otherwise

111

In the above senstivity formulation given by Equations

(4.36) - (4.39), the calculation of x and x can be carried

out by means of the powerful optimal triangular factorization

technique originated by Tinney and his co-workers [38-40].

The treatment of the transposed Jacobian matrix and the

detailed calculation procedures will now be discussed.

Consider the following simultaneous equations with the Jaco-

bian matrix H.

[H]T (4.40)

By the use of the triangular factor table, [H] can be decom-

posed as:

[H] [L] [U] =

where

0

, .

u11

. .0ln

0 unn

[L]: lower triangular matrix

[U]: upper triangular matrix

From the relation of [H]T

= [U]T

[L]T

, Equation (4.40) is re-

written as:

u11

1 t . .

.2n 1211

0

(4.41)

x = b (4.42)

By letting = MT x, / is given by

yi

1

i-1

=U

(b. E uki

yk

)'

i=2, n

ii1

k=1

Then, x can be calculated from y =T

x as follows:

Xn

= un

n

xn-1

yn-1

. tk n-i

xk '

i=1, n-1

k=n-i+1 '

112

(4.43)

(4.44)

As a result, 4 can be found by replacing b by in equa-

tion (4.40), and the similar procedures can be applied for the

calculation of x{.

Consequently, it is concluded that the sensitivity coeffi-

cients given by Equations (4.36) (4.39) can be efficiently

calculated with the use of the factor tables stored for load

flow analysis.

For the resultant sensitivity coefficients, it is to be

noted that

Kpk,N = 0

KQ,kN

= 0

for all lines k (4.45)

where N is the number assigned to the slack bus.

113

This is a result of the fact that the power injected to the slack

bus cannot be independently changed. The power imbalance caused

by the changes of bus power to all other buses of the system is to

be compensated by the injection power at the slack bus.

4.4 Linearized Model by Using the

Sensitivity Coefficients.

As indicated in the discussion regarding the general approach

in Section 4.2, all the system constraints will be linearized and

the objective functions will be approximated to second-order poly-

nomials for the application of the QP technique.

The primary objective function as given in Equation (4.1) is

linear and will, therefore, not be subject to any approximation.

In most practical cases, the nonlinear generation cost function

Fi(PGi) can be reasonably approximated to a quadratic function:

114

2F.(P

Gi.) = ao. P

Gi+ b. P

Gl .+ c. for all generators i . (4.46)

1 1

Substitution of this function into the secondary objective func-

tion (4.2) gives

jG I [Z ai 6P2Gi bi 6PGi]

where

a. 2aI

b. = 2a° ?1 1 Gi

? + b1

In the power balance condition, the power loss can be approx-

imately estimated by the loss coefficient matrix B as follows [43-

45]:

Ploss

= PT

[B] P

where

Using this relation, Equation (4.3) can be rewritten in the

following matrix form:

1T PG

1T P - (P -t

) [B] (PG

-t) =0

115

(4.48)

where 1 is an (n)-dimensional vector with all entries

equal to identity.

The linearization of this equation at the ambient operating

point gives

1T AP - 1T AP - 2(P° -po)T pp°

-G -t -G(4.49)

-G -t

The line flow constraints (4.12) will now be linearized

by using the sensitivity coefficients as introduced earlier.

Suppose a certain line k is overloaded with the power

flow Skk k,max

After some appropriate control action to suppress the line

overload has been initiated, the power flow on line k is re-

quired to satisfy the constraint:

or

Sk = Sk + ASk< S

k,max

k< -S

k,ovt

where Sk,ovt = Sk Sk,max: overloaded amount of

power flow on line k.

(4.50)

Therefore, by substitution of the result of Equation (4.18)

into Equation (4.50), and taking into consideration Equation

(4.45), the following linearized line flow constraints are

obtained:

n

.ASk

i

I1

An1 ] -Sk,ov[Kpk,i APi KQk,i

=

for all overloaded or vulnerable lines k

where

APi APGi APLi

AQi AQGi AQCi

116

(4.51)

It should be noted here that Inequality (4.51) must be

considered in the optimization process for all vulnerable lines.

It can, however, be disregarded for lines considered to be well

within the so-called secure limits of operation.

All the power generation constraints (4.5) can be directly

rewritten as follows:

PGiPGi PGui PG?

QUA QG7 QGui QG7for all buses i (4.52)

Qati QC7 11Ci QCui QC7

The load constraints (4.6) need not be linearized, but

the relationship between real and reactive load curtailments

should be considered. When some real load curtailment APLi is

is indispensable at bus i, the reactive load must be cut off in

proportion to the APLi since it is not possible to control the

real and reactive loads respectively. That is,

AQLi ai APLifor all buses i

where the proportional constant ai can be determined

by [58]

QLi

117

(4.53)

(4.54)

Consequently, the load constraints (4.6) can be replaced by

P P° AP P - P°Le L L Lu L

with 4Li ai APLi

(4.55)

Finally, linearization of the bus voltage constraints given

in Inequality (4.7) can be achieved with the aid of Equations

(4.25) and (4.35), which leads to:

-6 < V° + [L] 1 [q] Vsp <

where

= [c, c, E]TE Rn

Vsp

= [Vsp'

.., Vsp

]T Rn

(4.56)

These linearized voltage constraints are related to the

inverse of Jacobian matrix. An efficient algorithm to deal

with this requirement to determine the optimal reactive compen-

sation in order to suppress bus voltage changes due to load

118

curtailments will be discussed in Chapter 5.

It can be remarked as a conclusion that, by the results of

Equations (4.1), (4.47), (4.49), (4.51), (4.52), (4.55) and (4.56),

a linearized model with quadratic objective functions has been

established, i.e., the objective functions are approximated to

second-order polynomials and all the constraints are linearized

in terms of the control variables AEG, Apt, AgG, Agt and Aiac.

This considerably simplifies the optimization process of the load

shedding and generator rescheduling.

The linearized formulation will be summarized here with the

quadratic objective functions for the discussions of the solution

algorithms for the LSGR problem in Chapter 5.

(i) Objective functions to be minimized:

Primary objective function

JL

= I AP. Lii

Secondary objective function

J = APT

D APG

+ bTAP

G 2

1G a G

where Da

=

Ia.

10

0 an

(4.57)

119

(ii) Constraints:

Power balance condition

1T 41E 1T 64 2[4 2.10)T B [AEG Al2.] = 0 (4.58)

Line flow constraints

I [Kpk,i 6,13i 1<c)k,i gi] <k, ovt

i=1

for all overloaded or vulnerable lines k.

Power generation limits

po <40 D DO

-G -G 21.0

e(3 < '42G < gGu

gf.f. <6gC < 2Cu RC

Load constraints

PL 2AP <

PLu - -L

with AQL., = ai APLi for all buses i

Bus voltage constraints

-E < e [E]-' [q] AQ Vsp <

(4.59)

(4.60)

(4.61)

(4.62)

120

CHAPTER 5

Application of Optimization Techniques to

the LSGR Problem

NOTATIONS

ai, bi : generation cost coefficients for incrementalpower generation of generator i

D : diagonal matrix

Da

: diagonal matrix with elements a.

a. : proportional ratio between active and reactiveloads

P, Q : real and reactive bus injection powers

F, G : real and reactive line flow

S : line flow

Vsp

: specified bus voltage

E : bus voltage tolerance limit

O. : phase angle of bus voltage V.

Ou = Oi - 0i : phase difference between bus voltages Vi and Vj

H, N, J, L : Jacobian matrix

B : loss coefficient matrix

N : number of system buses

n = N - 1 : dimensionality of bus admittance matrix orJacobian matrix

O : incremental value

Y, G, B : bus admittance, conductance and succeptance

y, g, b : line admittance, conductance and succeptance

Kpk,i , KQk,i : sensitivities of power flow of line k to realand reactive injection powers at bus i

K : sensitivity coefficient matrix for all over-loaded and vulnerable lines

121

IGc : Indicator of the existence of reactive powergeneration unit

QGC1 QGi QCi: sum of reactive power generation at bus i

0 2a = [

1 i=1a.

Ux

: diagonal matrix with elements equal to 0 or 1(See page 145)

U : Identity matrix

1 : a vector with all elements equal to unity

subscript G, L, C : control units of generator, load and condensor

subscript sp : specified

subscript u, 2 : upper and lower limits

subscript ovt : overload

subscript D : demand

superscript o : evaluation at the ambient operating point

superscript M : modified

underlined letter : a vector

upperdotted letter : a complex number

asterisked letter : optimal value

$ = 2[B]T

LpG :loss coefficient vector for incremental bus

power changes

122

5.1 Introduction

The most effective approach to finding optimal solutions to the

LSGR problem is considered to be the so-called LP (Linear Program-

ming) [53, 55-60]. However, the LP algorithm requires the piecewise

linearization of generation cost functions, which creates the same

number of additional inequality constraints as the number of piece-

wise line segments for each generator cost function. Since the

standardization of LP problems needs the introduction of slack vari-

ables for each inequality constraint, the piecewise linearization

approach is confronted with a large-dimensionality problem which

significantly increases the computation time and memory requirements.

This dimensionality problem has been avoided by Chan and Yip [59] by

the use of a sparse linear programming algorithm on the basis of the

Bartels-Golub decomposition [20-22].

Subramanian [54] attempted to apply the Lagrangean multiplier

method to the LSGR problem. His algorithm, however, does not seem

to have a wide application, as mentioned in Chapter 4.

In this study, two QP algorithms are developed:

1. on the basis of Wolfe's method, and

2. on the basis of the Kuhn-Tucker theory.

These applications do not require piecewise linearization of genera-

tion cost functions. Despite the fact that the application of QP

allows for the use of quadratic cost function, the optimal solution

123

is obtained by a set of linear equations. Therefore, the proposed

QP algorithms are expected to have computation time and memory re-

quirements comparable to those of LP algorithms. Moreover, optimal

solutions with less generation cost will be established than the

cost as obtained from the use of LP algorithms.

Rather than establishing the optimal solution by the process of

minimizing one cost function as subject to a number of system con-

straints, a hierarchical approach will be pursued. Several cost

functions will be scrutinized, reflecting the urgency level as set

forth by practical operational management. This hierarchy is set up

based on the following three subalgorithms:

(i) Optimal load shedding algorithm, using the

simplex method,

(ii) Optimal reactive power compensation algorithm

for bus voltage changes due to load shedding, and

(iii) Optimal generation rescheduling algorithm, using

QP techniques.

Both of the algorithms that will be developed are based on the

same approach as far as load shedding and reactive power compensa-

tion are concerned. Generation rescheduling will, however, be

established differently, i.e., by the use of either Wolfe's method

or Kuhn-Tucker theory.

In Section 5.2, the function of these subalgcrithms will be

outlined, and interactions between each subalgorithm will be de-

scribed by a flow chart. Each of the subalgcrithms will be developed

124

in subsequent sections. In the last section, the results of applying

the proposed algorithms are compared with those obtained from the

conventional LP algorithm.

125

5.2 General Approach

The linearized formulation of the LSGR problem as given in

Section 4.4 includes two objective functions: the primary objective

function associated with the total sum of load curtailments, and the

secondary objective function reflecting the total generation cost.

As mentioned in Chapter 4, the primary objective function should be

minimized regardless of the secondary objective function due to the

prerequisite of system integrity. The optimal load shedding algo-

rithm will be developed in Section 5.3 on the basis of the simplex

method.

Applying the load shedding algorithm yields an optimal load

shedding policy. Moreover, the existence of a feasible solution

for the LSGR problem can be searched for by conducting Phase I of

the LP algorithm. (See Section 1.3)

When certain load curtailments are indispensable, the corres-

ponding reactive power compensation should be aimed at readjusting

all bus voltage deviations within the specified tolerance. The

reactive load curtailments which are accompanying real curtailments

may cause considerable changes to bus voltages of the system. In

order to satisfy this requirement, the optimal reactive power compen-

sation algorithm is developed to minimize the sum of squared bus

voltage deviations.

In case that no load curtailments need to be initiated, the

reactive power compensation algorithm can be bypassed. It was

126

assumed in Section 4.2 that, in the absence of changes to the reac-

tive bus injection, any changes to the bus voltages can be considered

negligibly small.

On completion of the optimal load shedding and optimal reactive

power algorithm, the next stage in the solution to the LSGR problem

is finding an optimal generation rescheduling which will minimize

the total generation cost under the predetermined load shedding and

reactive power control policies. This will be accomplished by apply-

ing QP techniques. In Section 5.5, two different QP algorithms will

be presented for this optimal generation rescheduling.

The optimal solution to the LSGR problem is usually established

with the aid of a load flow program and a contingency analysis pro-

gram. The interactions between various algorithms in the LSGR pro-

gram are indicated in the following flow chart.

At the top of the flow chart, the abnormal voltage suppression

is outlined with the use of the load flow program.

It can be remarked that Loop 2 is rarely initiated in practice,

i.e., an invalid solution is expected to be obtained rather rarely

for relatively small system disturbances. (See assumptions in Sec-

tion 4.2)

127

Go to thenext case

START

Conduct the contin-gency analysis orsystem monitoring

Loop II

Are

there any Yesabnormal volt

ages?

No

Is

any overloadfound?

Loop

Update the oldoperating pointby the new stateresulting fromthe last loadflow calculation

invalid

es

Suppress all abnormalvoltages by conductingthe load flow program

Calculate sensitivitycoefficients by using

the factor table

Optimal Load Sheddingalgorithm

No

Yes

Optimal Reactive PowerCompensation algorithm

Optimal GenerationRescheduling

Check validity of thesolution by the load

flow calculation

valid

Print the solution

Print: No solution

STOP

Fig. 5.1 LSGR Program Flow Chart.

128

5.3 Optimal Load Shedding Algorithm

This algorithm concerns the minimization of the primary objec-

tive function only. Under the system constraints, as formulated by

the linearized model in Sec. 4.4, the relevant optimization problem

can be described as follows:

Minimize

JL= ZAP

. Li

subject to

(i) Power balance condition

(5.1)

iT-

1T _2[4-

fct).]T APL] 0 (5.2)

(ii) Line flow constraints

E PPGi - APLi)PK

KOk,i (41QGi+AQ*

-41Q L03 -Sk,OV.e.

for all overloaded or vulnerable lines k (5.3)

(iii) Power generation limits

Pad PGi "PGi 4 PGui PGi

(iv) Load constraints

Lei PLi "PLi PLui PLi

(5.4)

(5.5)

AQL; ai APLi

nLi o

'where a. =

1 PoLi

129

(5.6)

where superscript o denotes the ambient operating

point,

AQGi , : optimal reactive power compensationCi

at bus i, and

41(ki : reactive load curtailment at bus i.

The bus voltage constraints are disregarded since the adjust-

ment of bus voltages will be dealt with by the reactive bus powers

compensation algorithm. It is, however, to be noted that the con-

straints involve gGi's and41Qci's. This calls for an iterative

calculation scheme between the load shedding and reactive power

compensation algorithms. Since such an iterative procedure results

in a considerable increase of computation time, the load shedding

algorithm will be implemented with the practical consideration that

the reactive power changes due to load curtailments can be fully

compensated by the reactive power generation facilities at the

corresponding bus. At the buses where no reactive power generation

is produced, AOIGi and are both zero.

Full benefit of this observation can obviously be taken with

respect to constraint (ii) as formulated in Equation (5.3):

Kpk

'

(APGi - APLi ) - KQk [1 - IGc( )] AQLi } < Sk,ovt.

i

where

(5.7)

1 if bus i has any reactive power-generating

I

GC(i) =

facility (since AQGi .

+AQCi

-AQLi

= 0).

0 otherwise (since AQ0i = 0 , AQ0 0).

By introducing a loss coefficient vector s, the power balance

condition (5.2) is rewritten as:

[i e]r 4 aoir

130

(5.8)

e = 2BT 2.00 -where

From Equations (5.6) and (5.7), it follows that

T[KPk,iPGi "Pk,i ai(1 IGC(i))KOik,i1APLi] < -S .0 (5.9)

As a summary, the optimization problem (5.1)-(5.6) reduces

to:

Minimize JL = 1T Aft

subject tosof aoyi (5.10)

f[Kpk,i4AP0i -"Pk,i ai(1 IGC(i))1(0k,i1APLi] < -Sk,ovt

for all overloaded or vulnerable lines k

P - Po

P - Po

Gti Gi Gi Gui Gi

for all buses i

PLiAD n

r Li rn

LUi r Li

It is to be observed that this formulation of the load shedd-

ding optimization problem is completely linear and can, therefore,

131

be solved by the LP technique as introduced in Chapter 1.

By introducing slack variables and changing variables appro-

priately, the above optimization problem can be reduced to the

following canonical form:

Minimize c x

subject to (5.11)

Ax = b

x >0

In this thesis, the revised simplex method is adopted to

solve this LP problem since it requires less computation than

the simplex method.

Both the simplex and the revised simplex methods consist

of two phases: Phase I determines a beginning feasible solution,

and Phase II searches for an optimal solution. If Phase I cannot

provide any feasible solution, it means that the given optimization

problem has no feasible solution. In the LSGR problem, however,

such cases are seldom expected to occur, due to the following

observations:

(i) In the case where Pui = 0 for every load bus, cutting

all loads of the system and generating no active and

reactive power at all systems buses obviously satis-

fies all the system constraints and can thus be a

feasible solution to Problem (5.11).

(ii) In practical power systems, each Pai is very small

132

compared with Phi, and every transmission line is designed

to have enough line flow capacity for provision against

various contingency outages, as mentioned in Chapter 3.

Thus, it is almost certain that there exists a feasible

solution with the minimum load at each load bus, except

for a severely faulted system.

As a conclusion, conducting Phase I and Phase II successively

yields an optimal solution to Problem (5.11) for most practical

cases. This provides an optimal load shedding policy.

133

5.4 Optimal Reactive Power

Compensation Algorithm

In the case where load curtailments are initiated, readjustment

of reactive bus injection power is required for the purpose of keep-

ing all bus voltages within the specified tolerance limit of bus

voltage deviation due to the fact that the load curtailments also

involve a curtailment of reactive power.

In conventional algorithms, this phenomenon is taken into

account by conducting a load flow program, after determination of

all load curtailments and generator rescheduling. However, the use

of the load flow program solely for this purpose leads to a consid-

erable increase in computation time. Moreover, the neglect of reac-

tive power changes may cause errors in the solution such that the

final solution would fail to suppress the given line overloads. In

such a case, there is no way to avoid rerunning the LSGR program

which requires starting over again from a different set of reactive

bus injection power values. This usually doubles or triples the

computation time.

In order to overcome this problem, as well as that of using

conventional algorithms, the development of an optimal reactive

power compensating method has been undertaken in this study. The

proposed reactive power compensation algorithm is based on the

application of the least square deviation method, which is applied

to all bus voltages of the system.

The approximate bus voltage changes can be obtained from the

decoupled load flow method given in Section 2.3

[L]

where

o`,6,1/

1 1/V1

AVn 4n/q1,

, 6Qci : reactive power changes of genera-

tor and capacitor at bus i, respectively.

134

(5.12)

From the above equation, it is observed that if all AQi's are

adjusted to zero by some appropriate control of 41C)Gi's and

AQci's, all bus voltages remain unchanged. This control can be

realized easily from the consideration of 4/4)Gi + ACki =

and can be considered to be the best control with respect to

simple bus voltage management and minimum computation time.

The control AQGi + 6tki --41QLi will be referred to as complete

reactive power compensation. However, it is not always possible

to achieve this control, since the control of each igGi or AQci

is limited by the reactive power generation capacity of the

corresponding bus. Hence, it is desirable to find a central-

ized optimal reactive power control which minimizes the devia-

tion of each bus voltage from the desired operating point. This

leads to the development of the following optimal reactive bus

power compensation algorithm.

Since the load curtailments have already been determined from

the optimal load shedding algorithm, the optimal reactive power

compensation problem can be formulated as follows:

Minimize

subject to

where

JV

= E AV..

[L]

4/1*

(AQGC1 AQL1)/111

*AN/ (AQGCn 41A)/Vn

; i=1, 2, ..., nACIGCi AQGCUi

QGCi AQGi "QCi

AQGCti AQGti AQUI

4GCui =AQGui AQCui

no An =n o

AQG.ei QGti 'Gi ' 'Gui 'Gul "(G1

AQUI QC 2i QCi ' AQ Q Q (3Cui Cul Ci and,

t and u denote the lower and upper bounds of the

corresponding variables.

135

(5.13)

(5.14)

This formulation results in a quadratic optimization problem

and, thus, one may try to apply QP techniques such as Wolfe's

method. However, QP problems require, in general, the calculation

of a matrix inversion. In this case, the calculation of the inverse

of [L] requires considerable computation time.

136

In this study, an approximated algorithm is developed on the

basis of the application of the least square deviation method, which

will be shown to be a straightforward and efficient method to use

in dealing with this problem.

Consider a standard least square deviation problem.

1

Minimize J =2

xTx

subject to

where

(5.15)

Ax = b (5.16)

X E Rn , b E Rm

A : (m x n) matrix

(It is assumed that all dependent equations are removed.)

The corresponding Lagrangean function is established as

L (x , X) = xT x + X (Ax - b)

and the necessary conditions for optimality are given by

or

T *

ax (

*

+AT

°

x = -A A

From the substitution of Equation (5.17) into Equation (5.16),

it follows that

-AT

A x* = b

(5.17)

(5.18)

137

Since Equation (5.16) includes independent equations only, [AAT]

is non-singular. Hence X can be calculated as

*= -[AA

T] b (5.19)

From the substitution of Equation (5.19) into Equation (5.17),

the optimal solution is obtained as follows:

x*

= AT[AA

T] b (5.20)

In order to use the above solution method, the optimization

problem (5.14-5.15) will be reduced to the standard least square

deviation problem through the following considerations.

Suppose reactive load curtailments [IQLi'

..., AQLn

] have

been initiated.

The complete reactive power compensation (AQGi

+AQCi

= AQLi

for all i's) will first be attempted in order to obtain the

reactive power changes. Then either of the two following cases

arises:

(i) Complete reactive power compensation is achieved,

i.e., the following relation is satisfied:

An-Q Li'Li for all buses i , or (5.21)

(ii) The complete reactive power compensation cannot be

achieved.

In the second case, the following procedure will be followed

to attain an optimal solution which minimizes bus voltage deviations:

138

Collect all equations for those buses kl, ..., km at which

the reactive power changes are not completely compensated:

.A A

.

tkil , . tkin

.

44/1

fki

(5.22)

k 1.

kmn4u

n

Af,

"Wkm, .

where

n f fAn-

4-44 vm"`IGCuk. /QLk.)Ali(5.23)

On the other hand, the rest of the equality constraints in

Equation (5.14) will be disregarded in finding optimal values of

6V.'s. Instead, those equations will be appended to the calcula-

tions of optimal AQi's after optimal AVi's have been calculated.

For simplicity of calculation, the inequality constraints will

be reserved only for checking the validity of the solution.

As a result, Problem (5.13-5.14) reduces to the following:

Minimize

subject to

where

J = AV.

A OV = 64f

A

0-

kil. .

kin

A A

tk*1

tk'n

(5.24)

f f

Ag. Egklf ' AQk 3T

m

From Equation (5.20), the optimal solution of (5.24) is given by

-T I 1 fAN = A [AA ] Ag.

The optimal reactive bus injection power to attain these

least voltage deviations is determined by Equation (5.13).

-* A *&Q =LAN

* * -* Twhere Ak = [441, AQ2, - - , AQn]

oAg*. =AQ* ./V.

1 1 i

*and gi's are calcualted by

* g-*AQ. = v. .

1 1 1

139

(5.25)

(5.26)

for all i (5.27)

FiTm Equation (5.12), the optimal reactive power compensation

is given by

* *

Gei + AQ Lifor all i (5.28)

*As mentioned before, the reactive control AQGo's must be

checked to see whether all of them satisfy the inequality

constraints in problem (5.13-5.14).

*Since some of AQGci may not meet these inequality con-

straints, the following algorithm will be adopted:

140

(i) Calculate all AQGo's through Equations (5.25)-(5.28).

If all AQGci's satisfy the inequality constraints in

Equation (5.14), then the calculation is complete.

Otherwise, go to (ii).

(ii) Select all buses having AQ:ci's which do not satisfy

the inequality constraints in Equation (5.14). Add

all equality constraints corresponding to these buses

to the simultaneous equation set (5.22) by setting

fAQi

(AQGCui

('6QGCti

°QLOM

)/v°i

' if AQGCi >AQGCui

An' 'GCi

< A n(5.29)

Then, go to step (i).

The optimization procedure outlined can be summarized accord-

ing to the flow chart on the following page.

141

Start with the previously

determined aLi's

the

complete reactive Yes

ower compensatiopossible

No

Set up equality constraints (5.22) and

determine A and Ag.

Modify A and Lgf

in the equality

constraints (5.22)

Calculate AV

Calculate Ag. and Ag.

1

Set AQui=

for all buses i

Is

* anyAQGci out of

'ts boun

Go to the next step.

Fig. 5.2 Optimal reactive power compensation

algorithm

142

5.5 Optimal Generation Rescheduling Algorithm

The conduction of the load shedding and reactive power compen-

sation algorithms determines the optimal load shedding and optimal

reactive power compensation policies.

Under both predetermined policies, the optimal generation re-

scheduling problem for reduction of the total generation cost will

now be discussed.

Suppose the previously-determined load curtailments and reac-

tive power compensation are given as follows:

* * *AL* = [4311 , APL2, ..., APL]

T

0 R*1. = [alAP*Li, ..., arIAP*Ln]T= [AeLl, ..., AeuX

* * T

A-g:C [AQGC1' "'' AQGCn](5.30)

From Equations (4.57-4.62), the optimal generation rescheduling

problem can then be described as follows:

Minimize

J = 1 APT D AP + bT APG 2 G a G G '

subject to constraints (5.2)-(5.4),

and constraints (5.2)-(5.4) can be rewritten as follows by

using the results of Equation (5.30).

(5.31)

(i) Power balance condition: From Equation (5.9),

eir aolT 612-11.,

(ii) Line flow constraints: From Equation (5.10),

(KPk,i 4IPGi) (Sk,ov2 ASLQ,k)

for all overloaded and vulnerable lines.

where

ASLQ,k (KPk,i KQk,i) 'APLi 66Q,k

(iii) Power generation limits

P . - P° 4 AP . P - P°Gi Gui Gi

143

(5.32)

(5.33)

for all buses i (5.34)

Conseqently, the optimal generation rescheduling problem

reduces to a quadratic optimization problem which can be solved

by Wolfe's method, as introduced in Chapter 1. Although Wolfe's

method works well for solving most QP problems, the degeneracy

problem (which is described in Section 1.3) may be encountered in

special cases. Wolfe's method is also involved in a large-dimen-

sionality problem.

An alternative algorithm has been investigated as a possible

solution for the above QP problem, based on the Kuhn-Tucker condi-

tions.

144

The Lagrangean function corresponding to the optimization prob-

lem (5.31)-(5.34) is given by:

where

L (A'

x x) = AP TDa GAP + b

TAP

o ' 2

1G G

(5.35)

xo

([1 - o ] APG

- APD

) - xT

(K OPT, +km ,ovt

)

K = coefficient matrix for all inequalities in (5.33)

pp = ri _ noJ TAn*

ASk,ovt = SASLQ,k

The constraint (5.34) will not be included in the above

Lagrangean function since it requires the introduction of 2n

multipliers which would result in excessive computer require-

ments. However, it will be considered later for the verifica-

tion of the validity of solutions.

that

or

From the Kuhn-Tucker necessary conditions, it follows

aL

;OPT

(5.36)

DaOPT + b - ho [1 - e] KT X = 0 (5.37)

From Equation (5.37), AEG is calculated as t

,AP

*

G= D

a

1

{X [1 -13o.1 + K x-L (5.38)

tFor all buses i with no generator, dii

1will be considered zero.

In order to determine and A from Equations (5.37), (5.32)

and (5.33), a matrix Ux will be defined as follows:

UX

: diagonal matrix with elements given by

U.. =11

0 if AP satisfies the strict inequality of

the i-th constraint in (5.33)

1 if AP satisfies the equality of the

i-th constraint in (5.33)

Then, the constraints (5.33) can be rewritten as

Ux

{K AP,*

+ S ovmx.o} = 0

* m[U - Ux] {K ApG

5-ovt} <

where equality holds only for zero rows in [U - Ux].

145

(5.39)

(5.40)

Moreover, X can be replaced by Ux A by setting every xi corres-

ponding to uii = 0 (i = 1, 2, ..., n) to zero. (This does not

change the optimal solution. See Appendix I, proof of Theorem

1.1).

The substitution of Equations (5.36) and (5.38) into Equa-

tion (5.32) gives

[1 - e]T Dal {7,0[1 e] + KTx - = APD

By replacing X with Ux X, the following equation is obtained:

-e]Tcalp

-en

-e]Tcal[ijuxx

-

Since xT A x = trace [A x xT], it follows that

Let

0 2n (1-.)

EicIT13

i=1a.

1

N (1-q)2a = 1/ E

1 i=1a.

Then can be calculated from

x = a {_a - (31:1T D-1 [KT " -Ll OP D}

1 LUx A L.'

DI

It is remarked here that X can be interpreted as the

incremental unit generation cost.

The substitution of Equations (5.38) and (5.42) into Equation

(5.39) leads to

-Ux K Dal [ [1 - p_

,0] r OoNTDal (KT " -

I

+ APD1 + K

T UX-61+U = 0X

Sovt

This equation can be rearranged as:

K [Cal -cal

-.ao) 130)T cal] KT LIT),

= UX a

(K D-1 {b - (1 -OON

.1 Lk p- noNT 1

p

+ AP ]) - S v)D v.2)

146

(5.41)

(5.42)

(5.43)

Since the above equation includes zero multipliers and depen-

dent equations, it can be reduced by eliminating such variables

147

and equations as follows:

Let a = [xi, x, at], which includes all undeter-

mined Xk's.

Ux : matrix consisting of all nonzero rows of Ux. (5.44)

Then, Equation (5.43) reduces to

MX = UX a

(K D-1

{b - (1 -P

0)1 I

- eNT

where

M = Ux

[DalKa `

1 fi OoN (.1

- 2-) '

111(ToT;(5.45)

As a result, X can be calculated as

= M-1 ux (K D-c-L1 {6 - (1 -r3o) - e)T b

+APD]}-S

-ovt ) (5.46)

(Note: M is non-singular unless K has any dependent

row.)

By substituting Equation (5.46) into Equation (5.42), X0 can be

determined and, finally, LEG is obtained from Equation (5.38).

However, it is necessary to check whether the final AP0

satisfies constraints (5.34) and (5.40) since these constraints

*were not taken into consideration in the calculation of APB

From the Kuhn-Tucker condition (Theorem 2.1), it is ob-

vious that there exists some UXyielding OPT which satisfies

constraints (5.40). Such a UXwill be searched for by trial and

error. The prediction of some appropriate Ux is also possible

since the system overload usually arises from one or two simple

factors.

With regard to constraints (5.34), AP*Gi will be set to an

appropriate limit value in case it falls out of its range, i.e.,

PGi=

ApGui

if AP*. > ApGui

APG-ei if APGi APG.ei

148

(5.47)

If any APGi

is set to its limit value, all of the unset variables

should be recalculated by restarting the entire procedure as de-

scribed. Moreover, setting variables successively in this manner,

the algorithm may fail to produce any feasible solution. Hence,

the following algorithm is included:

(i) Calculate Al4 by Equation (5.38).

If all APGi's satisfy Constraint (5.34), the computation

stops and an optimal solution is successfully established.

Otherwise, go to step (ii).

(ii) Set each APGi that exceeds its available ranges to its

appropriate limit value (upper or lower limits). When-

ever a certain APGi is set to its limit value, the exis-

tence of a feasible solution should be checked. In the

stage of setting APGi'

APGi

is allowed to be free if no

feasible solution can be found with the APGi set to a

limit value, and APGi

is regarded as being determined so

149

as never to be changed if any feasible solution exists.

(iii) Modify system parameters and constraints appropriately,

and go to step (i).

The flow chart in Fig. 4.3 is a summary of the optimal genera-

tion rescheduling algorithm as developed in this section.

150

i Start with optimal

Lt., and 200

Choose a suitable UPI

CONSTRUCT: Vector 4atrices M

COMPUTE: Xo, X

COMPUTE: 5F4

Choose anotherproper Ux,

ILIMIT = 0

1

N

MODIFY: ovl

Sk,:vvI/s7essN\

\than zero

o I

LPGi

PGA.( PGi

ILIMIT

P* P P3Gi Gui Gi

I Check tne existence offeasible solutions forremainino variables

YES thereany feasible

solution?

Let eP reef,NO

NO

YES

I STOP

Fig, 5.3 Optimal Generation Rescheduling

Algorithm

151

5.6 Comparison with Conventional LP Approach

In the previous sections, two QP algorithms for the LSGR prob-

lem have been developed: one is an application of Wolfe's method,

and the other is based on the Lagrangean multiplier method with the

Kuhn-Tucker necessary conditions. In this section, these algo-

rithms will be compared with the conventional LP algorithms which,

according to many contributors, have been considered to be most

effective for solving the LSGR problem.

The LP algorithm requires piecewise linearization of every gen-

eration cost function which leads to numerical inefficiency, as

mentioned earlier, while QP algorithms directly incorporate quad-

ratic cost functions. The lack of necessity for piecewise linear-

ization with QP algorithms saves computation time and memory

requirements, in spite of additional complications associated with

their use. The dimensionality of the LP algorithm increases pro-

portionally with the number of line segments resulting from piece-

wise linearization. Each line segment creates an additional set

of inequality constraints which are boundary conditions for the

certain number of used line segments[6][8]. Introduction of slack

variables is required for every inequality, and this causes a rapid

increase in the dimension of system variables. In order to solve

their large-dimensionality problem, Chan and Yip [7] proposed a

sparse linear programming algorithm based on the Bartels-Golub

decomposition method. The Bartels-Golub decomposition utilizes

152

the sparsity of the matrix by employing triangular factorization

and one-dimensional alignment of non-zero elements, resulting in

a remarkable reduction of computation time as well as computer

memory requirements. However, this technique can only reduce the

required computation time and memory requirements by converting

an exponentially-increasing function of the number of variables

into a proportionally-increasing function. Thus, the computation-

al burden is considered to be aggravated in rough proportion to

the dimensionality.

The QP algorithm based on Wolfe's method also needs the intro-

duction of slack variables which, however, does not result in an

increase in dimensionality to the same extent as with conventional

LP algorithms since there is no need for piecewise linearization.

Wolfe's method produces a stable solution, except for degeneracy

cases, and consequently can be considered to be as reliable as the

conventional LP algorithm method.

The QP algorithm, being based on the Kuhn-Tucker necessary

conditions, needs neither the introduction of slack variables nor

piece-wise linearization. It can, therefore, be expected to be

superior as far as computer memory savings is concerned. Improper

prediction of UXmight, however, lead to an increase in the number

of iterations.

The figures on the following page illustrate comparisons of the

QP algorithms with conventional LP algorithms in terms of computa-

tion time, memory requirements and resulting generation cost. In

(a) Computation Time

C

(b) Computer MemoryRequirements

153

1

.

- - LP-

N : Number of line segmentsfor each generation costfunction due to piece-wise linearization.

QP

N

(c) Generation Cost andPenalty Cost, for

Load Shedding

Fig. 5.4 Comparison of QP and LP

154

these figures, the computation time and memory requirements for the

QP algorithms are chosen as references and are normalized to unity.

Generation cost is also compared in the same manner. NTO and NMO

represent the number of line segments with which the LP algorithm

meets, respectively, the same computation time and memory require-

ments as when QP algorithms are employed. The comparative study has

shown that NTO

is less than 4 and NMO

less than 3 for a system with

more than 50 buses.

Both QP algorithms produce solutions with less generation cost

than LP algorithms since the quadratic generation cost functions

adopted by QP algorithms reflect actual generation cost more accu-

rately than any piecewise linearized cost function. It is impor-

tant to realize that, although LP may result in small computation

requirements by piecewise-linearizing each generation cost

function to two or three line segments, the optimum solution that

is obtained will be associated with higher generation cost. More-

over, since the computation requirements of QP algorithms are

highly acceptable, obtaining lower generation costs by using QP

algorithms is considered to be more significant than the reduction

of computation time in LP with its intolerably small number of

segments.

Comparisons of essential memory requirements for 10- and 50-bus

systems are given in Table 5.1. For each system, the number of over-

loaded or vulnerable lines is assumed to be 3 and 10 respectively.

Since LSGR programs are executed in conjunction with a load

155

flow program, the memory portion occupied by the load flow program

is not taken into account. Thus, the figures in the tables denote

pure additional memory requirements when running LSGR algorithms.

QP(I) and QP(II) indicate the QP algorithms employing Wolfe's

method and the Kuhn-Tucker theory, respectively.

It should be remarked here that the sparsity factor for memory

requirements is assumed to be 0.1. For instance, a (50 x 50) real

matrix normally occupies (50 x 50) x 2 words. If the matrix is

sparse, the required memory is calculated as (50 x 50) x 2 x 0.1

words due to the sparsity factor.

156

Table 5.1 Essential Memory Requirements

Unit : byte words

System Adoptedalgorithm

No. of linesegments for

genera-each gfor costfunction

Memory

Requirements

10-bus system with20 lines5 generators8 loads3 capacitors

LP

2 2392

3 2456

4 3068

6 3974

QP(I) 2364

QP(II) 1008

50-bus system with100 lines20 generators35 loads15 capacitors

LP

2 16,696

3 20,836

4 25,016

6 37,496

QP(I) 21,760

QP(II) 5,600

157

CHAPTER 6

Numerical Results

The linearized model and the proposed QP algorithm have been

tested for several systems. Some results are essential to verifi-

cation of the validity of the linearized model and applicability

of the QP algorithm to the LSGR problem. The test methods employed

and their numerical results are summarized in this chapter.

First of all, the sensitivity coefficients have been tested,

since they are crucial to the formulation of the linearized model.

In this test, a base case load flow is assumed for a typical load

flow pattern with a specific bus injection power, and an operating

point is appropriately selected to deviate from the base case load

flow by about 10 percent. For both the base case and the operating

state, power flows are calculated by conducting the load flow pro-

gram, which yields accurate power flow calculations. On the other

hand, power flows for the operating state are computed by using the

sensitivity coefficients and the results of the base case load flow.

The results are compared with accurate power flows obtained from

execution of the load flow program.

This test has been carried out for 5-bus, 17-bus and 50-bus

systems. The average error of power flows obtained from the model

with sensitivity coefficients turns out to be about 2 percent.

158

This error can indeed be considered sufficiently small from a prac-

tical point of view.

The computation time required in power flow calculation with

the use of sensitivity coefficients is almost negligible when com-

pared with that required in optimization procedures or the load

flow program.

Numerical results for the 5-bus system are listed in Appendix

II, which demonstrates precisely the effectiveness of using sensi-

tivity coefficients.

For purposes of testing the proposed QP algorithms, the follow-

ing procedures are employed:

(i) conduct a contingency analysis for the power system

under a certain situation,

(ii) calculate sensitivity coefficients for all overloaded

and vulnerable lines, then

(iii) set up the linearized model with quadratic generation

cost functions and apply both of the proposed QP algo-

rithms.

Algorithm QP(I), based on Wolfe's method, has excessive compu-

ter core memory requirements when sparsity of matrices is not taken

into account in development of the coding. This results in an in-

surmountable dimensionality problem for even a 17-bus system. It

is, therefore, vitally important to include a sparsity-oriented ma-

trix treatment when applying QP(I) to LSGR for large power systems.

159

Algorithm QP(II) has been successfully applied to large power

systems. This algorithm reduces computer memory requirements re-

markably and saves considerable computation time as compared with

conventional LP algorithms. Numerical results obtained from test-

ing this algorithm demonstrate that QP(II) works fine for those

cases in which only several overloaded or vulnerable lines occur on

a given system. This algorithm is particularly powerful for the

case of single-line overloads. Computation time for the QP(II) al-

gorithm varies significantly, depending on the overload situation.

For a 50-bus system, average execution time for the case of a

single-line overload is about 3 seconds. The following table sets

forth the computation time required for various overload cases.

It should be noted that the computation times given do not include

execution time for the load flow program or contingency analysis.

All programs in the tests have been executed using the Oregon

State University Cyber System (CDC Cyber 170-230) and the Bonneville

Power Administration CDC. These computers have similar computation

speeds.

Table 6.1 Summary of Numerical Results

SYSTEMSFAULT

CASES

OVERLOAD

SITUATION

OVERLOAD SUPPRESSION

MEASURES

COMPUTATION TIME (Seconds)

LOAD

FLOW

CONTIN-GENCY

ANALYSYS

0.4

LSGR

0.7

TOTALTIME

1.6

17-BUS

SYSTEM

Line outageon Line 22

Line 6: 30.0%Line 19: 0.9% Generator Rescheduling 0.5

Line outageon Line 11

Line 9: 6.1%Generator Rescheduling,

Load Shedding at Bus 110.5"

0.8 2.1

Line outageon Line 6

Line 7: 18.0% Generator Rescheduling 0.5 0.3 0.4 1.3

Generatoroutage atBus 13

No overload 0.5 - 0.5---

Line outageon Line 23

Line 24: 0.15% Generator Rescheduling 0.5 0.3 0.4 1.2

50-BUS

SYSTEM

Generatoroutage atBus 10

Line 36: 2.8%Generator Rescheduling,Load Shedding atBus 7 and Bus 10

3.4 0.9 2.7 6.0

Load outageat Bus 13 Line 36: 20.0%

Gerator Rescheduling,Load Shedding at Bus 7

3.4 1.3 2,4 7.1

Line outageon Line 37

Line 5: 15.7% Generator Rescheduling,Load Scheding at Bus 14 3.4 1.1 2,3 6.7

Line outageon Line 29 Line 81: 31.3% Generator Rescheduling 3.4 1.4 1.6 6.4

Fig. 6.1 5-Bus System Configuration

162

Fig. 6.2 17-Bus System Configuration

163

F A q . 6.3 50-Bus System Configurationonf i gurati on

164

CHAPTER 7

Conclusions

The following conclusions summarize the results of analysis and

verifications as presented throughout the previous chapter.

(i) A new linearized model has been formulated on the basis

of sensitivity coefficients. In this model, all linear-

ized constraints are expressed in terms of system control

variables which form a highly convenient description for

establishing an optimal solution to the LSGR problem.

(ii) An efficient method for calculating sensitivity coeffi-

cients has been devised and successfully tested with

practical systems.

(iii) The feasibility of applying QP algorithms to the LSGR

problem has been justified both theoretically and in

practice. Two QP algorithms, based on Wolfe's method

and Kuhn-Tuckery theory, were developed. Results ob-

tained in numerical testing indicate that the second

QP algorithm is applicable to practical systems.

(iv) It has been demonstrated that, by the application of QP,

both computation time and memory requirements can be

reduced and a better optimal solution, in terms of lower

generations costs, can be achieved in comparison to

165

results obtained from the application of conventional

LP algorithms.

(v) A reactive power compensation algorithm has been success-

fully devised to compensate for bus voltage changes due

to load curtailments.

(vi) Power loss was effectively taken into account in the LSGR

problem, with negligible increase in computation time.

(vii) As a by-product of the analysis required in Chapter 5, a

powerful simulation method for multiple-contingency anal-

ysis has been developed.

166

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APPENDIX I

APPENDIX I (Chapter 1)

A. Proof of Theorem 1.1 (Kuhn-Tucker necessary conditions).

Theorem 1.1

Let all gi(x)IS and f(x) be continuously differentiable.

If x is an optimal solution to a general maximization

problem defined by Equation (1.7), then x* satisfies the fol-

lowing conditions:

gi(x*) = 0 for all i E IE(x*)

(ii) gj(x*) > 0 for all j E la(x*)

(iii) There exist multipliers Xi's such that

(a) Ai > 0 for all i E Ia(x*)

(b) vf(x*) + E AVgi(x ) = 0iEIE

171

Proof)

The conditions (i) and (ii) directly follow from the definitions

of IE

and I- in Equation (1.8).

In order to prove condition (iii), the necessary condition of

optimality will be examined in the neighborhood of x*. Since the

necessary conditions are related to the feasible directions only,

and since constraints gi(x)>0 for all i E I5(x ) do not influence

a set of feasible directions at a point xl in the neighborhood of

x*, any of the constraints gi(x) for all i E la(X ) need not be

172

considered.

Hence, the optimization problem (1.7) is changed to a new equiv-

alent optimization with inequalities eliminated in the vicinity of

x*

as follows:

Maximize f(x)

subject to

gi(x) = 0 for all i E IE(x*)

(A.3)

*Since x is optimal, it follows from the well-known Lagrangean

multiplier method that there exist multipliers Xi

such that

* *vf(x ) + E X. vg.(x ) = 0

iEIE

where IE

denotes IE(02 ).

(A.4)

From Equation (A.4), Ai can be determined uniquely unless a

certain Vgi(x ) is linearly dependent on some other vgi(x )'s.

IfallX.'s corresponding to linearly dependent Vg.(x )'s are

settozero,thehasetofx.'s is determined from Equation

(A.4). This provesX.

exists without sign constraint (A.1).

The sign constraints on Xi

will now be proved by considering

*

the set of feasible directions at x , which is equivalent to

*1)(x ).

Since x*

is an optimal solution, Proposition 1.1 gives the

following necessary condition for optimality:

vf(x )-dx < 0 for all dx E D(x ) = D(x*

) (A.5)

173

From Equation (1.9), the foregoing inequality can be rewritten as

Vgx )-dx < 0 for all dTE{dxlvgi(x )-dx = 0eq

and vg.(x*)-dx > 0 ,vjEIa(x )1 (A.6)a

*From the viewpoint of differential geometry, vj(x ) can be rep-

resentedValihearcombinationofv.(x )1s and a unit vectorgs

n(x)whichisverticaltoeverVv.gs

(x ) for all i E IE(x ).

(Note that IE(x ) = I

eq+ I

a(x ).)

Vf(03 ) = E ai Vgi(x ) + an(x )iEIE

where

n(x ) L vgi(x ) for all i E IE

IE represents IE(x_ ) for brevity

: orthogonal relation

(A.7)

Let us define a new set Da(x ) as a set of all feasible di-

rections which are tangential to all hyperplanes gi(x)=0 for all

i E Ieq

, and perpendicular to n(x ). That is

Da(th ) = {dthldth E Nth ) and vgi(x)dX = 0, V i E leg

and n(x*)-dx = 01 (A.8)

It is noted here that Da(x ) is not empty unless problem

(1.7) has no inequality constraint, which deviates from present

arguments. (Recall that condition (iii) b) is being proved now.)

174

From the definition of '0a(x ), it is obvious that

Da(x ) c D(x ) (A.9)

By multiplying both sides of Equation (A.7), the following

equation is obtained:

* a, * a *.

dxa (A.10)Vflx ).dx = E X Vg kx )-dx + ankx pi

iEIE

In this equation, the last term vanishes since n(x ) dxa.

Since Ieq

and Ia

are exclusive and IE = Ieq

+ Ia

, the above

equation can be rewritten as

Vf(x*)dxa = E X Vg.(x*)-dxa + E X Vg.(x*).dtha

iEIeq

jEIa

'1

The first sum of this equation also vanishes since dxa E Da

As a result, it follows that

) dola = Z A.Vg (x* ) Ix('

jIa

cl

(A.11)

*

(A.12)

Suppose that xk

< 0 for some k E Ia. Then, it is evident

from the previous discussions that Vgk(x ) can be neither null

nordependentonanyotherv.gs

(x ) for all i Ea

. Since the

number of elements ofa

is less than the dimensionality of

Da (x *) ) , it is possible to choose some non-zero differential vec-

torcisP"a g(x)sucilth c72 perpendicular to all-s(x )

for all i E 1a but not to vgk(oi ). (See Finkbeiner [13], Chap-

ter 8.) By replacing dxa with dxP in Equation (A.12), it

175

follows that

Vgx .dx = Aj vg(x*).dx

p+ Ak vgkdx (A.13)

jEI

From the orthogonality of dxP.and vgj(x *) , the above equation

is reduced to

vgx*)dxP = xk vgk(x*)dxP (A.14)

Since vgk(x)dx > 0 for all x E F and all dx E D(X), the fol-

lowing result is obtained:

vgx*)-dx

p< 0 for some dxP which is not perpendic-

ular to vgk(x ).

*The above inequality contradicts the assumption that x is optimal,

which completes the proof of the theorem.

B. Pivot operation of the simplex method.

Suppose the current basic vector aar is replaced by a non-

basic vector as

such that yTas > es.s

Since as

= Mtn, , trs

is required to be non-zero in order to,

be a basis vector. (If trs

= 0, as is dependent on other basis

vectors than aar.) Hence, aar can be represented as follows:

aar = trs

1[a

s- E t. aai]

i=1ss

(A.16)

176

In terms of the old basis, all the the column vectors of A can be

represented as:

aj = E tij

aai = trj aar + t.. aai j=1, 2, ...,n (A.17)

i=1 i=12,0

i#r

All the column vectors of A can now be expressed in terms of new

basic vectors in behalf of the old tableau and Equation (A.16):

a3 = tr .trs

1(as - E t. aai) +

sd, j=1, n (A.18)ss

fr i#r

From this equation, the elements of the new tableau can be easily

calculated as follows:

t' = trj rj rs

=ti

t.. (t.rs

)-trd

. , i=1, m, i#rj sj ss

(A.19)

In order to get the extended tableau, consider the natural unit

vectors as columns of an extended matrix [A,I], i.e.,

an+1

= e1

, an+2

= e2

, ..., an+m

= em

(A.20)

Now the relationship (A.18) can be directly applied to finding

elements of the extended tableau, yielding:

q;j griArs

q: = q. - (t. )qrjsj ss rs

(A.21)

As a result, the above procedure reduces the cost by Theorem 1.10

updating the extended tableau only if an appropriate as is chosen.

177

This is the so-called pivot operation. Element trs is called a

pivot element and vector as a pivot column.

At this stage, finding the most effective pivot element, in

the case where as

is chosen as a pivot column, will be considered.

Since t0

is the current basic solution, all t. in Equationso

(A.19) are positive. Since to is to be a new basic solution,

sofor all i is required to be positive. Thus, a pivot ele-

menttrs

must satisfy the following condition:

trs

> 0

t.o

= (t.rs

) tro

> 0 for all i, i#rs ss

(A.22)

From this condition, only tis can be non-positive. Hence, the

following two cases are possible:

(i) If t. <=

0, then any of the positive trs's is accept-

able

ss

for a pivot element.

(ii) If tis > 0, troArs < tio/tis (A.23)

Therefore, a t must be chosen so that it satisfies either ofrs

the two cases. Moreover, the most effective pivot element can be

uniquely determined by Theorem 1.10 if the pivot element is chosen

to reduce the cost as much as possible. Theorem 1.10 gives the

following rule for the best choice of a pivot element:

Choose r E IB which satisfies(A.24)

tro

/trs

= min {t. A. : t. > 0, i=1,so ss ss

This can be proved as follows:

Suppose the minimum in Equation (A.24) occurs for both i=r

and i=q.

The following relation holds:

tro

trs

= tqo

tqs

with tqs

> 0

178

(A.25)

This contradicts the strict inequality in Equation (A.23).

In summary, the cost cTx can be reduced by the pivot opera-

tion through straightforward procedures if as is selected such

that yTas > Cs.

Now, a method to select an appropriate pivot column with the

use of the simplex tableau will be established.

Let us define vectors c and x as follows:

c = [cal

, ca2

, ca m

X = [x , x , xam

]I

al a2

where

variables.

al, a2, am

The old cost is then given by:

are the indices of basic

(A.26)

"TcTx = E cx. =cTx=ct (A.27)

EIB

(Note that to

is a basic solution, i.e., x = to .)

From Equation (1.56), any pivot column satisfies:

ma3 - E t. aai = 0 and

i=1

E t.

"aai = Ax = b

i=1

By adding X times of the first equation to the second, it

follows that:

179

(A.28)

Aas

+ E

i=1

A vector x(X) will

xi(X) =

With the use of x(X),

A x (X) =

-

(tio

- t. ) aa1- = b

ss

now be defined by its elements as:

t.0

- at. for i E IB1.

X for i = s

I-B

i$s0 for i E'

Equation (A.29) can be rewritten as

b

(A.29)

(A.30)

(A.31)

This implies that x(X) is a new feasible solution if and

only if x(X) > 0. The new cost associated with x(A) is given

by

Tc x(x) = Ac + E (t. - At. ) c.

8 2=1SO ZS 2

= old cost - x(ys - cs)

with

7 tiss i=1

(A.32)

180

From this equation, it can now be concluded that it pays to insert

as

in the basis only if 7s

>s

.

As shown in Theorem 1.10, if there is no positive tis in the

pivot column, A can be set to infinity and the cost can be driven

to negative infinity when 73 cs > 0. Thus the computation stops

if this situation is met with.

If there exists any tis > 0 for some as which satisfies

that 7s

> c3, such as can be selected as a pivot column. This

selection reduces the cost by A (7s

- c ) through the pivot

operation, where A is given by Equation (1.56). Moreover, A

is the coefficient of as in the new basic solution, and thus A

is equal to troArs. Therefore, the following cost relation can

be obtained:

70 = 70

- (tro/t

rs) (7

s- C

s)

where70

: old cost

7"0

: new cost

(A.33)

Intheabovediscussions,7.-cs

plays the role of criterion for

determining the pivot row. Hence, the following supplementary row

will be added to the bottom of the tableau to keep this information:

71 c1, "" 7n cn, 70 : Y1' Y(A.34)

This is the so-called criterion row. Here the first n components

are determined by the use of the tableau as:

7 - c . = E t.. - c.a a i=1

(Note that 7. - c. = 0 for all j E IB).a

181

(A.35)

The next element indicates the current cost which can be calculated

from the tableau by

T T m7 =c x=c x= E c.0 1.o 1,

i=1

(A.36)

The last m components in the supplementary row represent the

shadow costs, which can be given by

y =P-1 c=Qc or

(A.37)

y. = q.. C.a i=1

From Theorem 1.8, it is obvious that y is the solution of equi-

librium conditions. From this y, the dual cost for the dual

problem can be calculated by

n=yTb= E y b. (A.38)

From Theorem 1.6, it is concluded that the optimization is com-

pleted if and only if 70

= n is achieved. Moreover, the optimal

solution can be attained through a finite number of pivot opera-

tions, providing the non-degeneracy assumptions [1, 5, 12].

It is now necessary to consider the modification of the criter-

ion row corresponding to a base change.

In the following discussions, the primed variable denotes

the modified value of the non-prime variable, as usual.

From Equations (A.19) and (A.32), 7:i is given by

7 = E c:i=1 1-a a

= E (t.. - t. t ./t c. + (tj rs s

) cza 2s raj rs r

182

(A.39)

Since (t. - t. tr3./t

rs) c. = 0 for i=r, this equation can

ls

be rewritten as

m m ..

7". = E t.. c. - (t ./t ) ( E t. c. c

2t7=1

1-J a ra rs 2s 2 s

= 7a

. - (t ./trs

) (7s

- cs

)

As a result, 7: can be modified by

7:a

= 7a. - e

0try, j=1, m

where eo= (7s - c

s)/t

rs

(A.40)

(A.41)

Equation (A.33) gives the modification rule of 70 as follows:

7' = 7 e t0 0 o tro

Similarly, the following modification of marginal costs yi's

can be derived:

(A.42)

(A.43)

183

As a conclusion, the modification rules (A.41), (A.42) and

(A.43) can be summarized as follows:

"The modification of the criterion row can be carried

out just like all other non-pivot rows," [1 ], i.e.,

the new criterion row = old criterion row - eo

(pivot row)

with eo

= (7 - cs)/t

rs

APPENDIX II

184

* *

APPENDIX II

Numerical Results

All units adopted : [p.u.] (Base ; 50 KVA)

Numbers in parentheses denotes the numbers assigned

to system buses.

A. Test of Sensitivity Coefficients

Table A.1 Input Data

LINE ADAIITANCI AND LIME CIANGS ADKITIIMICZ

LZMZ 9k Ak Dy, LINZ N 5, bc1 (1. 2) 1.9608 .7.8431 .0054 2 (2. 3) 1.9446 -6.6444 .00503 (3. 41 1.9115 -9.1233 .0048 4 (2, 41 1.8690 -7.5365 .006)5 (2. 1.4345 -7.2266 .0055 6 (1, 51 1.3426 -9.0530 .00597 (4. 5) 1.8182 -7.3529 .0064

SC13221232'D r CZNIRATION

111 0.0000 (2) 3.5000 (31 0.0000 (41 1.7000 (51 3.500024412,ATI7IG 241A8I2.IT0

(11 0.0000 (2) 3.9000 (31 0.0000 (41 3.0000 (5) 4.0000MIR UNIT Or 0 GTO AND C

(1) 2.4000 (2) 3.5814 (31 0.0000 (41 3.2730 (S1 4.3750LOOM LIMIT OF Q CMS AND C

-Acrrn(1) -0.7000 (21 -1.0000 :3) 0.0000 (4) -0.8000 '51 -1.5000

1013(11 -2.4000 2) -2.0000 (3) -1.7000 (41 -2.1000 (51 -1.5000

RIACTITZ LOAD

(11 0.8040 (21 0.4500 (3) 0.6000 (4) 0.5500 (51 0.3000

Table A.2 Results of Load Flow Calculation

POwZROS rACTIVE 1124 INXICTICM

(1) -2.4000 (21 1.5000 (3) -1.7000 (41 0.6000 (51 2.1575

AZACTIVZ SOS rCNIZA

(11 0.5704 (21 -0.5164 (3) 1.5999 ;41 -0.2154 ,51 -0.7963

XIS 701.04G2

(11 1.0000 (2) 1.0229 (31 1.0842 (4) 1.03311 (3) 1.0000

Table A.3 Base Case Line Flows

LINZ ACTIVE ROM LEACTIVE moo LIEF FLON LINK CAPACITY MERCENT LOAD

, -1.0118 0.1231 1.0193 1.6000 63.7042 0.8336 -0.7124 1.0965 '.5000 71.1003 -0.8951 0.7671 1.1766 1.5000 79.5854 -0.0220 -0.0790 0.0820 1.0000 8.1955 -0.3546 0.2790 0.4512 1.0000 45.1186 -1.3982 0.4469 1.4584 1.4000 '04.1717 -0.3433 0.3593 0.4969 1.0000 49.696

?Ian 575 ;,... 0.6421

185

Table A.4 Sensitivity Coefficients

3101821.291T7 OF LINE FLOWS TO

ACTIVZ 805 POwnt CHANGES (8,,.)

SZOSITTPITI OF LINT ROWS TO1WACTIV2 In POWER CHANGZ15 (Kg,....)

2 3 4 S '...-::,.,....US 1 2 3 4 5

, -0.374 0.236 0.200 0.150 0.000 1 0.048 -0.027 -0.019 -0.016 0.000

2 I 0.047 0.067 -0.412 -0.132 0.000 2 .0.054 -0.107 0.268 0.071 0.000

3 I -0.044 -0.083 -0.366 0.126 0.000 3 0.023 0.046 0.259 -0.139 0.000

4 -0.026 .0.048 0.015 0.074 0.000 4 -0.092 .0.164 0.040 0.360 0.000

.4.190 .0.355 -0.301 -0.326 0.000 5 0.136 0.272 0.192 0.161 0.000

6 -0.594 -0.226 -0.192 .0.144 0.000 6 0.194 0.072 0.051 0.042 3.000

i -0.109 -0.204 -0.309 -0.382 0.000 0.096 0.193 0.260 0.382 0.000

Table A.5 Data Input for Operating Point

sammumtcomumm(1) 0.0000 (2) 2.2000 (31 0.0000 (41 2.4000 (5) 3.2000

P Loam(11 -2.0000 (21 - 1.9000 (31 -1.5000 (4) -2.0000 (51 -1.5000

; LOU(11 0.6500 (21 0.4500 (31 0.7000 (41 0.3500 (5) 0.3000

Table A.6 Comparison of Load Flows by Sensitivity

with Accurate Load Flows

=2 10. CASZL.MIZ FLOW

23101,21a3r153. 1:1318 !LAX ST

5101.92227/227

AC=ILATZ LabFLAX

LENZ:74/PACICTf

3

4

5

O

7

1.01937.09651.1788

0.08200.45121.4384

0.4970

-0.1747-0.00e9-0.08700.0051

-0.0616-0.25780.0005

2.84461.38761.0918

0.08710.30961.20060.4975

0.94891.09621.1027

0.10540.41501.2035

0.5210

'.60001.3000

1.50001.0000

1.00001.4000

1.0000

B. Numerical Results for 17-Bus System

BASE CASE LOAD FLOW

INPUT DATALINE ADMITTANCE, LINE CHARGE ADMITTANCE AND T4P OFF - NOMINAL RATIOS

11 1. 11 29.4118*J-117.6471 .0040 1.0040 1.0000 21 1.151 27.52291) -91.7431 .0093 1.0030

31 2, 31 23.5294J-105.8824 .0040 1.0000 1.0000 41 3. 91 14.7159.J -54.8235 .0130 1.1laL

51 3.111 19.607e*J -78.4314 .0040 1.0000 1.0000 61 4. 51 21.E216*J -70.2703 .0090 1.0000

71 4.041 23.5244.J-135.8824 .0060 1.0000 1.0040 81 5.161 29.41.8*J-117.6471 .0670 1.0060

91 6. 71 29.26830J-130.5454 .0646 1.0000 1.0040 101 6,151 25.6291*J-104.8462 .0070 1.0000

111 7,131 34.4564,J-131.9969 .0070 1.0030 1.0000 121 8,131 38.4615*J-192.3077 .0030 1.0000

131 8.151 19.2300*) -96.1518 .0080 i:0000 . 1.0000 141 9.141 -11.1/02+J -44.0689 .0040 1.0600

151 9.151 16.8539*J - 73.0337 .0070 1.0000 1.0000 16110,111 16.1451*J -87.7123 .0040 1.0033

17(10.16) 27.5229*J -91.7431 .0100 1.0000 1.0300 181.1.121 24.0385*J-120,924 .0060 1.6000

19112.141 17.8465*J -79.3179 .0120 1.0000 1.0000 20112.161 32.0513*J-160.25E4 .0050 1.0006

21113.171 16.6389*J -71.8659 .0050 1.0000 1.0000 22114.161 19.6674+J -70...31. .0040 1.4000

23114,171 34.49190J-159.1414 .0080 1.0000 1.0300 24(15.171 14.6341.J -68.2927 .0120 1.0000

1.44401.10341.0400.4 074

1.00001.0(Ju1.00;,0

1.00001.0(011.060u1.0(731.3J43

IUS ADMITTANCE1. 11. 56.93471) - 209.3837 1. 21= -29.41181J 117.6471 1 1.151= -27.5229*3 91.7431 1 2, 11. -19.4110*J 117..471

2. 21. 52.94120.1 -223.5254 2, 31. -23.53940J 105.8024 I 3. 21. -23.52949J 145.8824 1 3, 31. 57.0431*J -243.12E3

3, 91= -14.7C59*J 58.8235 3.111. -19.607893 78.4314 1 4. 41. 45.1510*3 -17E4451 1 4. 51= -21.6216*3 70.2703

4,141= -23.52941J 165.8024 5, 41. -21.6216*3 70.2703 1 5, 51. 51.0334*3 -187.9093 1 5.161. -29.4118*3 117.6471

6, 41. 54.89741.3 -241.4261 6. 71. -29.268393 136.5854 1 6,151. -25.6291*3 104.8462 1 7, 61. -29.268303 134.53.

7, 71. 63.72464.3 -270.5768 7.131. -34:4564+J 133;9969 t 8-.-1110--57;092317 L283:4500 1-8;131. ;33.461507 192.3077

0.151. -19.23081J 96.1538 9, 31. -14.7059*J 58.8235 1 9, 91. 42.6700.J -172.7147 1 9,141. -11.110293 44.8660

9.151. -16.0539*J 73.0337 10,101= 44.2680.3 -179.4464 110,121. -16.7451,3 67.7123 (10,161= -27.5229*J 91.7431

11. 31= -19.60784J 78.4314 11.101. -16.7451*3 87.1123 111.111. 60.3914.3 -286.3290 111,121. -24.0385*3 12C.1123

12.111. - 24.03851.3 120.1923 12,121. 73.936343 -355.7551 112.141. -17.6465,,, 79.3179 112,161. -32.0513.3 160.2564

03. 71= -34.45549J 133.9969 13. 81. -18.461593 192.3077 113.131= 89.556813 -406.1640 (13.171. -16.638013 74.8664

14, 41. -23.5294*J 105.8824 14.-91. =11.1102+J---44.16tio----ii6a211-=17:541s+J-114.141. 106.5850.J -467.6740

1'4,161. -19.6078*J 78.4314 14,171. -34.4919.3 159.1934 114. 11. -27.5229*J 91.7431 115. 61. -25.4291*3 104.2462

15. 81. -19.2308*J 96.1533 15, 91. -16.8539+J 73.0337 115.151= 103.8709*3 -434.0481 115.171= -14.034143 64.24,7

16. 51. -29.411893 117.6471 16.101. -27.522943 91.7431 116.121. -32.0513*3 160.2564 116,141. -19.6078*3 78.4314

16.161. 108.5938*3 -448.0630 17.131. -16.63891.1 79.8669 117,141. - 34.4919J 159.1934 117.151= -14.6341*3 64.2927

17.171. 25.7650.3 - 307.3405

9JS TYPE

1131 2121 3121 4131 5121 6121 7121 8131 9(31 10(31 11121 12131 13131 14131 15(31 16131 17411

SCHEDULED P 06,40681109

11 11.600000 1 21 0.000000 1 31 0.000000 I 41 11.800000 1 51 0.000000 4 61 0.000000 I 71 0.000u00 I 81 4.800000

I 91 9.970000 (101 3.000010 1111 0.000000 (121 0.000000 1131 15.900000 1141 11.400000 101 0.000040 1161 23.000040

1171 31.490000 1

GENERATING CAPABILITY

I 11 16.000000 1 21 0.000000 1 31 0.000000 1 41 15.040000 I 51 0.000003 1 61 0.000440 1 71 3.003000 1 81 8.004003

I 91 12.000000 (101 0.000000 1111 0.000000 1121 0.000000 1131 20.060000 1141 14.000440 1151 0.000000 1161 28.004000

1171 35.000000 1

GENERATOR COI,STANTS

41 11 = .06E0 Al 41 = .0010 Al 81 = .1810 Al 91 .1000 41131

-T5510---BI -11 -=--1;710I--atim. .0530 4(141 = .0430 41151 = .0306 41171 = .0,50

01 11 1.7500 B1 41 1:7500.-17:-Asur-73t14r-7--1:75oo- )1151-t--3:1036--0117) . 2.1000

CI 11 = 2.9000 CI 41 2.8000 CI 81 = 2.4000 CI 91 . 2.5(00 C1131 = 3.2006 C1141 = 2.7000 CI161 = 3.4000 cu11 4.6400

UPPED 111110 OF 0 6E0 rmn C

1 11 14.560000 1 41 10.930000 1 61 4.930001 1 41 6.474590 1101 0.009030 11,1 9.909040 (13) 11.515060 (141 9.6400001t51 0.000106 1161 14.550000 (171 25.530760 1

109))) 11011 OF 0 GEN AND C

1 1) 0.090900 1 41 0.404030 4 St 0.0176080 1 4) G.000093 1.0) -4.00000J 1121 -8.003000 1131 9.4991900 1141 6.9..490011,'.1-10.000000 1

L 1)4,1

1 10 -6.300000 1 20 -5.050000 1 3) -7.800000 I 41 0.800000 1 51 -1.318000 1 61 5.030040 4 11-10.400000 1 81 0.0600001 9) 0.000000 1101-17.560000 1111 -6.280000 1121-13.200800 113) 6.000000 1141 0.080018 1151-17.660000 1161-22.5006401171 0.000000 1

O LuAD

1 .1k.0000 1 21 .280808 4 31 .180000 1 41 0.800818 1 51 1.300000 1 61 0.000E40 I ro 1.450100 1 61 6.0094001 91 0.0(0060 4101 1.006000 III, .500300 1121 2.000000 (131 0.000000 1141 0.000000 4151 6.100400 1161 .900000II?) 0.000000 1

BUS VOLTAGE SPECIFIED

1 11 1.000000 1 41 1.0818110 1 81 1.000008 I 91 1.000000 1101 1.000000 1121 1.000000 (131 _1.000000 1141 1.000380.1151 1.000000 I

CALCULATED RESULTS

Bus VOLTAGE

1 II 1.00000 I 2)

I 91 1.00000 1101(171 1.0000u 1

BUS P

.97384 1 31 .96884 1 41 1.01723 1 51

.92874 1111 .94382 (121 .95688 11311.00521 1 6) 1.04003 1 71 .99903 1 01 1.2111761.01266 (141 1.00000 1151 1.00000 1161 1.00040

I 11 5.30000 1 21 - 5.05000 1 31 -7.09000 I 41 11.80018 1 51 -1.31004 t 611 91 9.97008 1101 -1/.56000 1111 -6.28000 1121 -13.19919 1131 15.90000 11411171 23.37303 I

BUS ri

1 11 2.51392 4 214 91 .78763 11011171 -4.09126 1

LINE

2

SS

e

40(11 11

.00000 1 71 -18.41000 1 81 5.8000011.40000 (151 -17.66000 1161 .5L001

.19936 1 31 .11929 1 41

-2.11136 1111 .50699 1121.076E5 1 5)

-4.54011 11311.30419 I 61 .00000 I 71 1.45082 1 81 .04314.03956 1141 1.67000 1151 3.49249 rat 15.30961

N007 N PRP OMN LOAD 10.11106 PERCENT1 2 8.991509 1.328980 9.089193 15.030002 60.5951 15 - 3.691509 1.184931 3, 07 len 17.080080 22.0062 3 2.976271 .0/0995 1.101100 16.000300 19.3023 9 -4.236256 .645205 -9.256764 1..000600 ----77.1563 li 4.392239 .317522 4.4.03701 15.404000 29.3584 5 6.857424 -1.244916 6.944403 10.000000 69.4454 14 2.142576 1.321172 3.225562 10.004000 32.2565 06 7.238231 -.926503 7.298534 14.000000 52.132

9 6 7 -1.796304 .41.1556 1.845564 11..054..34.

10 6 15 1.798304 -.42052 1.445563 11.40403020.001444

14.4E6

11 7 13 -12.201415 1.854523 12.34154712 8 13 -4.013449 4.037549

5.82161115.006000

16.77::

13 8 15 P.81.3449.4404(15

24.006330 24..N:6.995

-.4003471S.00000C14 9 14 .261790

15 9 15-1..1157441.377390

1.64893712.0301/00

16 10 11 -4.339003 ::;1337179g

1.4113414.351673 11.000000 43:75617

17 10 16 -13.22095-6.335995

-1.785450 13.341009 20.000000 56.705

10 11 12 .005731 6.335999 16.300110014./49281 20.00003019 12 14 -14.1i6272

- 5...918!0

:965923-5.855413 16.000300

39.60070.74o

20 12 16 8.027824 51..741.19826921 13 17 -.605107 1.342388 18.000000

6;:1161:22 14 16 12.246472 12.422167 21.00000023 14 17 -13.653246

-2.0819813.580035 14.114807 18.000400 78.411

18.005006 49.70524 15 17 -8.611407 2.425661 8.946901_ _ _ .

P LOSS = 3.983851 0 LOSS . 16.101309

IP= 6 10 = 5 8 5

LINE OUTAGE NO. = 23

CALCULATED PiSOLTS

9US VOLTAGE

1 11 1.00000 1 21 .96528 1 3) .94980 1 41 1.01727 1 51 1.00623 1 61 .99072 1 71 .99275 1 41 1.00.931 91 1.00000 1101 .92458 (111 .93553 1121 .95332 (131 1.01065 1141 1.04006 1151 .98823 1161 1.714361171 1.00000 t

BUS P

1 11 5.30190 1 21 -5.85057 1 31 -7.80059 1 41 11.79994 1 51 -1.30985 1 61 .00005 1 7) -13.40006 1 81 5.40040

1 91 9.97189 1101 -17.55999 1111 -6.28051 1121 -13.19933 1131 15.90072 1141 11.40027 (151 -17.65919 (161 .560591171 25.22453 1

nos

1 11 5.88025 4 21 .19526 1 31 .17751 I 41 .00213 I 51 1.29976 161 .00671 1 71 1.45591 I 8) 1.650081 91 3.35050 1101 -2.42534 1111 .50532 4121 -5.21101 1131 1.12169 1141 1.36689 1151 1.99928 1161 15.447211171

LINE

-2:660107

NODE M NODE N PM OflO LOAC RATING PERCENT

V 1 2 13.9507114 1.613092 14.043663 15.000000 93.62.

2 1 15 -8.648818 4.267155 9.644204 17.000000 56.731

3 2 s 7 10567-8- .234773 --7-:7r9/41 16.000607 ---48.1834 3 9 -10.322125 .523636 10.335399 12.00000 86.1205 3 11 10.099638 -.682155 10.122643 15.410006 67.484

6 4 5 7.938152 -1.145851 8.020426 10.000000 80.2047 4 14 3.961792 1.147979 4.028803 10.000300 40.288

a 5 16 6.379699 -.614850 6.412206 14.000400 45.8019 F -7 =G:17F890 .E60152 1-717?3,7- 10-.000000 -41-.794

10 6 15 4.126944 -.653446 4.178356 11.000000 37.98511V 7 13 -14.553644 1.995411 14.689800 26.000000 73.449 CO12 8 13 -7.752258 1.359999 7.870648 15.000000 52.471 CO13 a 15 13.552757 .290686 13.555874 23.000000 67.779

14 9 14 4.893417 -1..64334 6.191057 15.300003 46.667

15 9 15 -7.717320 3.153299 8.33E645 1i.000336 69.472

16 10 11 -5.969918 .450710 5.980907 10.000000 59.869

17 16 16 -11.590075 -2.076052 11.941587 26.000000 59.708

18 11 12 =2.579593 -1.543361 -3;006C34 16:400000 16.784

19 12 14 -11..98755 -.29982. 11.502664 20.000300 57.513

20 12 16 -4.297271 -6.531713 7.51855o 16.320000 48.066

211 13 17 -6.460177 2.650115 7.354260 18.006000 40.657

22 14 16 9.976133 -1.842271 10.1467713 20.300006 56.734

23 14 17 0.306008 0.060000 0.000601 18.000006. .

0.600

ovLRLoAn ALARM 24105,171 -17.233519.J 5.284021 10,0442630

24 15 17 -17.233519 5.208021 16.026573 16.0000013 130.148

P LOSS . 5.839714 0 LOSS = 24.212579

IP. 4 10 . 2

SENSITIVITY

PKPK124. 11= -.694 PK124. 21= -.734 PK124, 31= -.758 PK124. 41= -.732 PK124. 51= -.776

PK124. 61= -.539 pN(24;-71. -.451- Pk124;-111. PK(24; 9ii -.611---PN(24;Idi= -.844

PK124.111. -.813 P1(124.121= -.801 P1(124.131. -.352 P10124.141. -.745 P10124.151. -.655

PK124.061. -.794 PK124.171= 0.000 PK(

OKOK124. 11= .150 OK124, 21= .150 B1I24, 31= .147 00124. 41. .116 01124. 51= .118

OK124. 61= .137 OK124. 71= .1i3 Okr26; Al= OK(24-; 91= --aid 01124,131; .1.40

01124.111= .140 01124.121= .133 01124.131. .087 01(124441= .122 01124.151= .169

OK124.1.61 .120 06024.171= 0.000 0100

POWER BALANCE . -.000001 OVERLOAD LINE = 24 OVERLOAD .00000

OVERLOAD PREVENTIVE MEASURES

CHANGES OF ACTIVE POWER GLMERATION

1 11 -2.12096 1 zi 0.00000 1 31 0.00400 1 41 3.20006 1 51 0.00000 1 61 0.013000 1 71 0.064.0 1 81 1.35680

1 91 2.02011 (101 6.00300 1111 0.00000 1121 0.00000 (131 3.59019 (141 2.59973 1151 0.00000 (161 -5.52868

1171 -5.52523 1

CHANGES OF nus REACIVE POWER

I 11 0.00000 I 2) 0.00000 1 31 0.00000 1 41 0.06400 1 51 0.00000 1 61 0.00000 1 71 0.00000 1 81 0.66030

1 91 0.06000 1101 0.00400 1111 0.00300 (121 0.0(000 1131 0.63000 1141 6.00660 (131 0.00000 (161 0.06666

(171 0.00000 (

LOAD SNEOCING

ACTIVE LOAD CUPTAILM,AITS

1 11 6.00003 1 21 6.00000 1 11 0.00000 1 41 0.00000 1 51 0.00000 1 61 0.00060 1 71 0.00000 1 81 0.00000

91 0.00600 (101 0.00000 1111 0.00000 1121 0.00060 1131 0.00000 1141 0.30000 (151 0.00660 (161 0.66000 CO

1171 6.60006 1

LID

REE:TIVE LOAD CURIAILHLuTS

1 11 0.00000 1 21 0.03000 1 31 0.00003 1 41 1.00300 51 0.00000 1 61 0.00040 1 71 0.00010 1 81 0.01400

1 91 0.00000 1101 0.00000 1111 0.00001 1121 3.00000 1131 0.00000 1141 0.30310 1151 0.60000 1161 0.300001171 0.00000 1

CHECK OF SOLUTIONS nr LOAD FLOW PROGRAM

CALCULATED FESULTS

nus WOLTAGI

1 11 1.00000 1 21 .96488 I 31 .94760 1 41 1.019.14 1 51 11.00500 1 61 .97723 1 71 .98210 1 51 1.0E0301 91 1.00000 1101 .91996 1111 .95154 1121 .94952 1151- 1.00596 1141 1.00006 1151 .97142 11,51 1.0E3031171 1.00000 1

BUS P

1 11 3.18309 1 21 -5.05048 1 31 -7.66071 1 41 15.00042 51 -1:5i016-1 61 .00036 1 71 -10.55582 1 61 7.157301 91 12.00124 1101 -17.56051 1111 -6.25061 1121 -13.20059 1131 19.69032 1141 14.00008 1151 -17.65954 1161 -5.027131171 20.23994 1

9.1S

1 11 8.35136 1 21 .19947 1 31 .17990 1 41 .00029 1 51 1.30020 1 61 .00038 1 71 1.45795 1 81 1.325:,41 91 4.44683 1101 -2.91517 1111 .50035 (121 -5.87190 1131 .50077 1141 1.38798 1151 -3.24738 1161 15.452861071 - .53719 1

LINE NODE H NODE N PHN OHN L060 RATING PL.-CENT

1. 1 2 13.187164 1.754474 13.303364 15.006000 85.6592 1 15 -14.004073 6.596086 11.983338 17.000000 70.4903 2 3 6.982711 .541922 7.003709 16.400000 43.7734 3 9 -11.236763 .837004 11.26E426 12.000000 93.8475 3 11 10.313404 -.565734 10.324909 15.000000 68.4596 4 5 9;56504Z -1;205604 -9.91E802- 10:000006- 59.3110-7 4 14 5.135378 1.2E8596 5.275750 10.300000 52.7588 5 16 8.175034 -1.133694 8.253269 14.300036 56.9529 li 7 -4.765080 .524980 4,783994 10.000000 47.841

10 6 15 4.765139 -.4E4599 4.744019 11.000000 43.49111 7 13 - 15.20C955 1.718996 15.297743 20.000000 76.48912 8 IT ---=783727e .44-64313 7.1146113 1331113111 52-;627-

15 a 15 14.994478 .988505 155.027..31 20.000104 75.13514 9 14 6.906943 -1.165571 7.004600 15.000000 46.69715 9 15 -6.707935 4.126999 7.801066 12.000030 65.67616 10 11 -6.245456 .479923 6.263865 10.000000 62.63917 10 16 -11.314802 - 3.395098 :1.813237 20.000000 59.066

-6516 II 12 -7.66E4411 -17514176 -3:1TIE3-711 16.000003 19:119 12 14 - 12.921600 -.046257 1e.921762 20.000000 64.61920 12 16 -2.962651 -7.421121 7.994652 16.000000 41.942It 13 17 -3.646778 1.174067 3.831112 18.000004 21.24422 14 16 12.311826 - 2.067937 12.447615 20.000000 62.43023 14 17 0.000000 0.000000 0.000000 15.006003 0.01024 15 17 -17.733692 3.362175 1-6:1I-57191T --1u000000 89:353-

P LOSS = 6.343111 9 LOSS = 26.149E64

IP= 6 In = 5 0 7

LINE OUTAGE NO. . 22

CALCULATED RESULTS

941S VOLTAGE

1 11 1.00000 1 21 .97012 1 1) .95343.1 41- -1.013091 51 -.99783 0 61' .99974 1 71 .99933 1 81 1.0:4251 91 1.00000 1101 .92217 1111 .93408 1121 .94551 1131 1.41224 1141 1.600uu 1151 1.00000 1161 1.040301171 11.00000 1

NUS P

1 11 5.30007 1 21 -5.85001 1 31 - 7.80000 t 41 11.80015 1 51 -1.34469 1 61 .00004 1 71 -10.40000 1 81 5.84.,J41 91 9.97005 1141 -17.55944 1111 -6.28001 1121 -13.19974 1131 15.90000 1141 11.41002 1151 -17.65495 1161 .541901171 24.36090

NUS

1 11 3.16210 1 21 .199114 1 31 .18026 1 41 .11132 1 51 1.30816 1 61 .00001 1 71 1.45082 1 81 .041631 91 1.46036 1101 -2.72600 1111 .50330 1121 -5.47485 1131 .04273 1141 3.73840 1151 3.69515 1161 17.533751170 -4.97245 1

LINE NODE M NODE N PHN OMN LOAD R4TING PtFCtNT1 1 2 10.054131 1.603898 10.181259 15.000000 67.8752 1 15 -4.754066 1.558205 5.002913 17.300000 29.4293 2 3 3.996791 .978303 4.114700 16.000000 25.7171/4 3 9 -10.564325 .844264 10.598007 12.000000 88.3175 3 11 5;775130 .156646 -6;72E940 15.J06000 44.846

OVERLOAD ALARM 61 4. 51 12.981020,3 -1.632034 130.8321116 4 5 12.981020 -1.632034 13.083411 10.000100 134.832f 4 14 -1.180871 1.743353 2.106644 10.000000 21.6568 5 16 11.004276 -2.482681 11.200860 14.000000 84.5709 6 7 -2.032594 .5[0102 2.095134 10.00000, 24.95110 6 15 2.032596 -:568088 2.095137 11.000006 19.84711 7 13 -12.439184 1.932157 12.58(31.9 20.000000 62.94212 8 13 -4.396073 .524002 4.427193 15.000000 29.51513 a 15 10.196075 -.482377 10.207474 20.000u00 51.03714 9 14 -2.049039 .555634 2.123038 15.006000 14.15415 9 15 .960501 -.218522 .985445 12.000000 0.20916 10 11 -6:055890 .396342- 6.066846 --io.a01100; 66.60817 10 16 -11.544050 -3.122338 11.920241 20.000000 59.60118 11 12 -5.851067 -.011096 5.851077 16.300000 36.569

OVERLOAD ALARM 19112.141 -19.9717384'3 2.944927 100.92846219 12 14 -19.971738 2.914927 20.107692 20.000000 100.93020 12 16 .858197 -8.739473 8.181509 16.000000 54.88421 13 17 -1:240106 1.302161 1.798189 18.068606 9.99022 14 16 0.000000 0.010000 0.030400 20.000006 0.00023 14 17 -13.064702 3.359159 13.49°656 18.000000 74.99824 15 17 -9.516709 2.752075 3.906648 18.000000 55.037

P LOSS = 4.173680 I) LOSS = 20.254530

IP= 5 10= 4 0 4

SENSITIVITY

PKPK1 6, 11= -.062 PK( 6. 21= -.088 PK( 0. 31= -.117 PK1 6, 41= .237 PKI 6. 51. -.513

PK1 6, 51= -.C26 PK( 6, 71' -.022 P*1 6, 81. -.C22 PK1 6, 9) . -.042 PK( 1,101. -.294

PK1 6,111= -.215 PK1 6,121= -.219 PK( 6,13)0 PKi 6,141. .022 PK1 6.191. -.032

PK1 6.151. -.337 PK1 6,171. 0.000 PK1PK119. 11= -.136 PK119. 21= -.134 PK119, 31* -.257 P0(119, 41. -.093 PK119. 51. -.J3E

PK119. 61= -.058 PK119, 71. -.049 PK119, 81= -.048 PK119, 91. -.093 PK119,101. -.502

PK119,11/= -.471 P1(111,121. -.584 P1(119,131. -.016 PK119,141. .049 PK119,151. -.064

PK119,11(1. -.480 P1119,171. 0.400 PK1

OKOK( 6, 11= .051 OKI 6, 21= .069 OKI 6, 31= .089 13(1 6, 41= .260 OKI b, 51. .200

OKI 6. 61= .027 OK1 6. 71. .023 OK1 6, 81. .023 OKI 6, 91= .067 OK( 6.101. .154

01(1 6,14). .131 OKI 5,121. .141 OK( 6,131. .018 OK( 6,141= .108 OKI 6,151= .033

01(1 6,161= .150 OK1 6,171. 0.000 OKI01(119. 11= .086 01(119. 21. -.117 01(119, 31= .152 OK110. 41= .175 OK119-. 51. .230

(min, 61= .046 01(114. 71= .018 OK119, 81. .038 01(119, 91= .111 OK119.10. .243

OK119,111. .229 01(119,121. .266 01(119,131. .030 OK119.141. .17: OK119,151. .056

OK1119.1f1= .217 nx119,671= 0.400 (30(1

POWER BALANCE -.000001 OVERLOAD LINE = 6 OVERLOAD = -.000002 OVERLOAD LINE = 19 OVERLOAD = .54695

POWER BALANCE . -.000001 OVERLOAD LINE . 6 OVERL340 = -.000002 OVERLOAD LINE . 19 OVERLOAD . 0.00000

OVERLOAD PREVENTIVE MEASURES

CIANGES OF ACTIVE POWER GENERATION

1 11 .99694 1 21 0.00000 1 31 0.00000 1 41 -8.95978 1 51 0.00000 1 61 C.00620 1 71 0.00006 1 81 .86869

1 91 2.02995 1101 0.00000 1111 0.00000 1121 0.00000 1131 2.24E93 1141 -3.37675 1151 0.00060 1161 4.99820

1171 -9.93432 1

CdANGES OF BUS REACIVE POWER

1 1/ 0.00000 1 21 6.40600 1 31 0.30004 1 41 0.00000 1 51 0.00000 1 61 C.00000 ( 71 0.00000 1 11 0.00020

1 91 0.00000 (101--0ro0000--itrr--1-14o0on--TT71---uTutru0---mr5.ouu u 1141 17.011000 IrST--0.6.00W---11117---U60000

1171 0.00000 1

LOAD SHEDDING

ACTIVE LOAD CUFTAILMENTS

1 11 0.00004 6 21 0.03000 t 31 0.00000 1 41 0.03000 1 51 0.00030 1 61 0.7ac02 1 71 0.00100 1 61 0.006116

1 91 0.00000 1101 0.00630 1111 0.00000 1121 0.00000 1131 2.00034 1141 0.43466 1151 0.00000 1161 6.06100

117) 0.00000 1

REACTIVE LOAD CURTAILMENTS

1 14 0.00000 1 21 6.00000 1 31 0.00000 1 41 0.00000 1 51 0.00000 1 61 6.006J0 1 71 2.00610 1 61 u.00.1.16

1 91 0.00000 1101 0.00600 1111 0.00000 1121 0.00000 1131 0.00000 1141 0.00000 1151 0.33000 1.61 0.00000

1171 3.00003 1k.f.)

N.)

CHECK Of SOLUTIONS BY LOAD FLON PR3GA114

CALCULATED RESULTS

4US VOLTAGE

1 11 1.00000 1 21 .96937 I 31 .95251 1 40 1.00560 1 51 1:060501 6) 1.6.055 1 71 1.60077 I 81 1.01202

I 91 1.60000 1101 .92077 1111 .93331 1121 .94567 1131 1.01455 1141 1.00062 1151 1.00000 (IL) 1.0(300

1171 1.00000 I

'WS P

1 11 6.29699 1 21 -5.85001 1 II -7.79997 1 41 2.04003 I 51 -1.30992 I 61 -.00006 I 71 -10.40000 1 d1 6.66769

1 91 11.99598 1101 - 17.56031 1111 -6.27999 112/ -13.20006 1131 16.14894 1141 8.62264 1151 -17.66000 (16) 5.45326

11/1 25.05471 1

BUS n

1 1) 2.99460 1 21 .20020 4 31 .10066 I 41 .39612 1 51 1.30771 1 61 -.00001 1 71 1.45165 I 61 .00638

1 91 .98078 1161 -2.87803 1111 .50051 1121 -5.99639 1131 .00161 1141 5.32357 1151 3.32432 1161 15.71542

1111 -5.11761 I

LINE NODE M 400E N P44 OMN LOAT RATING PERCENT

1 1 2 10.545306 1.615693 10.668392 15.000040 71.123

2 1 15 -4.248310 1.378767 4.466459 17.000000 26.2733 2 3 4.467656 .909407 4.559276 16.000000 26.4954 3 9 -10.664435 .636176 10.697166 12.000000 89.143

5 3 11- 7.2670E13 ;058462 7;788097 35.000000 48.687s 4 5 8.948937 -1.778740 9.124001 10.000000 91.240

7 4 14 -6.108911 2.174664 6.484507 10.800004 64.845

8 5 16 7.309785 -1.531968 7.466593 14.000000 53.347

9 F 7 -2.296953 .477980 2.346661 10.000200 23.461

10 6 15 2.296852 -.477985 2.346041 11.000000 21.32611 7 13 -12:70510i 1.695135 ic.445666 c0.34000c 64.226

12 8 13 -4.227737 .376635 4.244456 15.000000 28.29713 6 15 10.096427 -.370754 10.902732 20.000000 54.514

14 9 14 -.022906 .013678 .023200 15.000000 .155

L5 9 15 .653918 -.195314 .675970 12.000000 7.300

15 10 11 -5.948616 .312864 5.956036 10.000000 59.566

17 10 16 -11:611389 =3;200851 17.044503 70.000000 62.22318 11 12 -5.204281 -.264146 5 16.000000 32.575

19 12 14 -17.977686 1.940173 18.062078 20.000000 90.410

20 12 16 -.476545 -8.464691 6.478294 16.000000 52.969

21 13 17 .901921 1.042427 1.376447 16.000000 7.656

22 14 16 0.000000 0.000000 0.000600 ci.000JAJ 0.006

23 14 17 =17;157274 4:707710 17:791419 18.000000 96.64124 15 17 -8.167516 2.270527 8.477240 18.002000 47.0%

P LOST = 4.471259 0 LOSS = 18.487261

10. 3 10 = 2 0 6

LINE OUTAGE NO. = 11

CALCULATE° RESULTS

gin VOLTAGE

1 11 1'.00000 1 21 .91346 1 31 .95846 1 41 1.01664 1 51 1.00475 1 61 .93850 ( 7) .92)23 1 dl 1.0100.11 91 1.00000 1101 .92080 (111 .93861 1121 .94979 (131 1.0u983 1141 1.00004 (151 .96334 1:61 1.000641171 1.00000 1

nus P

1 1) 5.33090 1 21 -5.84946 1 31 -7.79997 1 41 11.80000 1 5) -1.31000 1 61 .00089 1 71 -10.39996 I 81 5.8040491 9.97038 1101 -17.56638 1111 -6.27997 (121 -13.20010 1131 15.89729 1141 :1.40010 1151 -17.65435 1161 .50016

1171 24.66395 1

RIPS 0

1 11 6.11975 1 21 .19445 1 31 .18008 1 41 .00157 1 51 1.30026 1 61 .00425 1 71 1.4b689 1 81 Z.352361 91 3.77385 1101 -2.96467 (111 .49992 1121 -5.94838 1131 .03629 114) 2.61765 1151 -3.33177 (16) 17.371341171 -1.46049 1

LIME NODE H 4000 II PMN ONN LOAO RATING PEI-CENTI' 1 2 6.132163 1.516797 8.272428 15.000000 55.1502 1 15 - 2.931266 4.602949 5.401010 17.000000 31.7883 2 3 2.145347 1.172631 2.444909 16.000000 15.2614 3 9 -9.137155 .512106 1;151495 12.000000 76.2625 3 11 3.469911 .767537 3.558159 .5.000100 23.7216 4 5 9.031159 -1.286445 9.122323 10.000300 91.2237 4 14 2.766937 1.288016 3.053759 10.300000 30.5388 5 16 7.399140 -1.023548 7.469600 14.000000 53.354OVERLOAD ALARM 91 6. 71 10.591449/3 -.566320 106.0657629 6 7 10;591449 -.566320 10.606570 16.006000 136.06618 E 15 -13.590556 .570573 10.605915 11.000000 96.41711 7 13 0.000000 8.000202 0.000000 20.300000 4.00012 6 13 -11.412833 .644346 11.431006 15.000300 76.20713 6 15 17.212677 1.706017 17.297412 21.000000 86.48714 9 14 -2.241260 .613174 2.323624 15.000000 15.49115 9 15 2;709794 2:223605 3.505341----- 12.000000 29.21116 10 11 -4.029122 -.6.5504 4.077387 10.000000 40.77417 10 16 -13.530954 -2.339166 13.731657 20.000300 68.65816 il 12 - 6.921706 .267931 6.927694 16.000000 43.29619 12 14 -14.532057 .664347 14.547234 20.001006 72.73621 12 16 -5.676920 -6.755274 8.823896 16.000000 55.14521 13 IT 4.353(40 .040313 4.3531{95 10.001400 24.16820 14 16 12.685576 -2.120965 12.862056 20.000003 64.31023 14 17 -15.969921 4.315793 16.542807 18.000000 91.90424 15 17 -12.162569 1.208890 12.222500 10.300000 67.903

P LOSS = 5.274417 0 LOSS . 22.215262

IP= 9 10 = 8 0

ST POSIT1911V

PKPKI 9, 11= .000 Pr( 9, 21= .000 PK1 9, 31= .000 PK1 9, 41= .000 PK1 9, 51= .000

PK1 9. 61= .000 PK1 g, 71=-1.043 PKI 9, 81= .000 PKI 9, 91= .000 PK1 9,101= .000PK1 9,111= .000 PK1 3.121= .000 PKI 9,131= .020 PKI 9,141= .000 PK1 9.151= .030PKI 9.111= .000 PKI 3,171= 0.000 PKI

OKOKI 9. 1)= .113

OK( 9. 61= .!70OKI 9,111= .057OK) 9,1161. .035

OKInk(nkl901

9, 21= .002 OKI 9. 31. .046 OK1 9. 4. .J28 nkt 0, 51= .03[

I, 71. .301 OKI 9, 81. .082 OKI 9, 91= .088 OK( 9,141. .050

1421i ;042 OKI 9.111= ;356 OKI 9,141.. ;$27 Mei 9.151.9,171= 0.000 OK(

POWER 9ALANEE = -.040701 OVERLOAD LINE

POWER 9ALANCE . -.000001 OVERLOAO LINE

. 9 OVERLOAD = 1.60658

= 9 OVERLOAD = -.77134

OVERLOAD PREVENTIVE MEASURES

CHANGES OF ACTIVE POWER GENERATION

1 11 -4.13504 0 21 0.00003 31

1 91 2.02962 (101 0.00000 1111

0171 1.40949 1

CHANGES OF MIS REACIVE POWER

1 11 5.25289 1 21 0.00000 1 31

1 91 4.41765 (101 .96467 Ili)

1171 1.46049 1

LOAD SHEOCIIIG

ACTIVE LOAD CURTAILMENTS

1 IA 0.00000 1 21 0.00000 1 31

1 91 0.00000 (101 0.00000 (111

(171 0.00000 1

REACTIVE LOAD CURTAILMENTS

1 11 0.000001 91 0.00000(171 0.00000

1 21 0.00000 1 II

1101 0.00000 1111

1

LINE OUTAGE NO. = 6

CALCULATED PESULTS

9U5 VOLTAGE

1 11 1.000001 91 1.000001111 1.00000

.97205

.92471

1 .00000 --1 -2:67515 51 0.00004--r-61 -o.locoo I 71 0.10000 1 II 1.230320.00000 (121 0.00000 1131 4.10271 1141 2.5)990 1151 0.00000 1161-11.41709

0.00000 1 41 .12608 1 51 0.00000 1 61 0.00000 1 71 0.00000 1 81 5.544000.00003 1121 1.94838 1131 18.E2376 (141 16.30324 1151 5.69177 1161 8.15627

0.00000 1 41 0.00000 1 51 0.00000 ( 61 0.00660 1 71 1.41720 1 11 0.000000.00000 1121 0.00000 1131 0.00E00 1141 0.00060 1151 0.00000 1161 0.00000

0.00001 1 41 0.00000 ( 51 0.00000 1 61 4.00000 1 71 .42516 1 81 0.000100.00000 1121 0.00000 (131 0.00000 1141 0.00000 1151 0.00000 1161 0.E0000

1 31 .95683 ( 41 1.01777 151 1 61 .99996 1 71 .99461 ( 81 1.014501.007661111 .91889 1171 .95087 1131 1.01250 1141 1.00000 1151 1.40000 1161 1.04000

9J5 P

I 11 5.30003 1 21 -5.65001 1 31 -7.84002 I 4/ 11.60001 1 51 -1.30982 1 61 .08000 1 71 -10.48800 ( 81 5.66800I 91 9.97002 (101 -17.55917 101 -6.26002 (121 ...13.19981 (131 15.91000 1141 11.44011 (151 -17.65998 1101171 24.05613 1

9tIS

11 2.82425 I 21 .20001 I 31 .16006 1 41 .00102 51 1.29417 61 .40001 1 71 1.45097 1 61 .04675I 91 0.10223 1101 -2.55268 1111 .50444 1121 -5.14151 1141 .04550 1141 4.02548 1151 3.56495 110 16.7E04511171 -4.96169 I

LINE NO01 H 1100E N PHN 0H9 RATING PTTGENT1 1 2 9.511400 1.408426 9.622642 15.0300100 64.1512 i 15 -4.211445 1.365630 4.427366 17.000000 26.0443 2 3 3.476269 .921515 3.596337 16.000000 26.4774 3 9 ..9.875630 .734947 9.902939 12.000030 F2.5i45 3 11 5.524496 .247125 5.530021 15.0001100 36.6075 4 5 6:000000 0.000000 0.003020 10.000000 0.006OVERLOAD ALARN 71 4.141 11.8000110 .001022 116.0001127 4 14 11.800011 .001022 11.900011 10.000000 116.0000 5 16 -1.309617 1.299165 1.844844 14.000000 13.1779 6 7 .1.914664 .414235 1.170163 10.000000 19.701

10 6 15 1.914685 -.464226 1.970158 11.000000 17.91111 7 13 -12.320510 1.892010 12.464938 20.400004 62.32512 8 13 ..4.204890 .481262 4.232352 15.800000 28.21613 8 15 10.004841 -.426509 10.014446 20.300000 5C.07216 14 -1.689339 .393248 1.548361 15.0000110 10.26915 9 15 1.155231 -.260503 1.184239 12.000000 9.86916 10 11 -5.172232 -.037264 5.172366 10.000003 51.7c417 16 16 -12:387734 =2.515615 12;640581- 26.000000 63.20310 11 12 -6.093670 ..0i3637 6.093715 16.000000 38.06619 12 14 -16.940039 1.836744 17.035344 20.300000 85.19720 12 16 -2.420621 -7.301526 7.692376 16.000000 48.07721 13 17 - .922800 1.252641 1.555860 16.000000 8.64422 14 16 17.163755 -2.380678 17.328100 20.000000 96.6.123 14 17 -13.54E305 3.545464 135997754- 18.000300 77.76524 15 17 - 0.064626 2.567240 9.426625 18.000000 52.370P LOSS . 4.667791 n LOSS = 19.397140

/1". 4 10= 2 8 2

SENSITIVITY

PKPK( 1. 11= -.000 PK( 7, 21= -.000 PK1 7. 31. -.000 PK( 7. 41. 1.000 PK( 1. 51= -.000

PK( 7. 61= .:000 51-1 -.olio PRI .. gry-=-0140 PK( 1,131-B-7-7000-PK( 7.1111= -.000 PK( r.iri= -.000 PK( 7.131= -.000 PK( 7,141= -.000 PK( 7.151= -.000PK( 7,161= -.030 PK( 7.171= 0.000 PK1

no(

910 7, 11. .040 OK1 7, 21= .052 OK( 7, 31. .066 OK( 7, ..1= .423 OK( 7, 51= .089OKI 7. 61. .023 nr( 7, 71= .019 nKt-7.--8)= .019-- OKI-7i 91= .058---08T-7.f01= ;098

7.111. .090 061 7.121= .I96 011 7.131= .015 CIK1 7,141= .105 OK( 7.151= .028OKI 7,161= .790 061 7.171= 0.000 061

PLWER BALANCE = -.000001 OVORLOAD LINE = 7 OVERLOAD = -.00800

OVERLOAD PREVENTIVE MEASURES

CHANGES OF ACTIVi POWER GENERATION

1 11 - 3.00967 1 21 0.00000 1 31 0.00014-1-41--2.00004 i 54.- 0:00cila -1 61-o.3occo i 71 0.00060 1 81 1.640.71 91 2.02990 1101 0.00000 1111 0.00000 1121 0.00000 1131 4.10000 1141 2.5)98( 1151 0.00000 1:61 -8.941,81121 4.38046 1

CHANGES OF OW REAEIVE POWER

1 11 0.00000 I 21 0.00600 1 31 0.00000 1 41 0.00000 0 51 0.00000 1 61 0.03000 1 21 0.u0000 1 81 0.000001 41 0.00000 1101 0.000001171 0.00000 1

1111 0.00000 1121 0.00000 1131 0.00000 (141 0.30400 1151 3.00000 1161 0.00000

LOAD SHEDDING

ACTIVE LOAD CURTAILMENTS

1 11 0.00000 ( 21 0.03000 1 31 0.00000 1 41 0.00000 1 51 0.(0000 1 61 0.00000 1 21 0.00000 1 81 0.040061 91 0.00000 1101 0.00000 1111 0.00000 1121 0.00000 1131 0.00E03 1141 0.00000 1151 0.00060 1161 0.04.0004171 0.00000 1

REACTIVE 1OAD CURTAILTIENTT,

1 11 0.00000 1 21 0.00000 1 31 0.00000 1 41 0.00000 1 51 0.00000 1 61 0.001.1.0 1 71 0.00000 1 01 0.660061 41 0.00000 (101 0.03000 1111 0.00000 1121 0.00006 1131 0.01010 1141 0.00000 1151 0.00000 1161 0.000001171 G.00000 1

C, Numerical Results for 50-Bus System

!Alm CAS( LOAD FLOW

1wInT1 DATA

IcloomOID r (wfA110ft

--1-0. 1.foioo0 4 0) 0.000000 4 Si 0.0 19/__2.11 .1.1 ____1101_1.0000011.1111__$.11 1111

(1) 1.266(loo III) 0.400000 410 3Am:woo 120)

- no_ 1.1____Iiii_iii______Ain_a.F.__IIII__(34) 0.'00000 00 0. 04)__10.1__0.120920_1111_1,1110000_4(21__1.11 _lift

0.000000 1-3I--iTiT g 0 0.0__4111111_11

...woo 00 o.o 011 o.soofoo (121 0.0iisammuiti

00 0.30.020112_00_1,11 410_11.00012.22_0.11__A.O___ J10 CI

ihip-41140 c4,40(ifi

--(-21---7110000 1 1) 3.1161-61 171 0.0 4 41

11:1 -1:1111111111-2:giewcfil,1111 1:1------11:1--i!9soia-1111--81----11114111_ i.300fl0 L2) 0.00100_143) 0.0 1441

1-1.1i6o5i-1111 11412MP-1111 --1:1!"92----------

11 0.0 1 21 27111120201.00000o 1 11 0.0 ( a) 0.0

1.8"Cpa-031-ti--1111-1:104u6-91F-1.:--1111-110.200011 ( 0 0t0 itil_11,111911 0.) 29289

POPO L1011 Of r aft AN0 C

11 V3110011 b-1.1155iITIF-17350600 141 -0.1iobaa -i-o-c-oulour-r-u-17365ocio-11/1- 1:1311111 1gg-tilg22S1111-911111-1i11-1:1P,81g-111F-1:111181E-111-1:11111_ min oAlgrarna_roal_pli_l,1"0.0_1") pAinfnLt"4"!_i.4__g."909iiq pin13.1

TLamm L1H11 OE a lits_IbikC

i-ii _9.11 Ai_ MIL t_lt_2.1 i_a_v.a______1_,I__!.O______1_1)--04-_----4 11---141------1 2) 2.2--110) 0.0 ill) 0.2 (1/1 C.0 *141 0.0 1114 9.0 114/ CO 1111 o.o 4131 0.011!) 2.2______4101 _Mt__ 11/1 1).1_ (1)1__2.1------1141--0112

1111 !!! 1111_1.0_ __.1;VI 5.11007 /1.0 (30) 0.0 ()At Q.0 U)) 0.0 ISA) 0.0 (AO) 0.0 fAI) 0.0____illi__1,0 (441 C.11_a*L_11.0 I'll t

t LOAD

i 1) 0.0 1 2) -0.11000 i-I-31-.71.241000 4) -0.004000 4 Tr -4151000 I -im°0s0 t 1) 4.110000 ( f) 0.0

--l_11-:1.200110_41.01_=1.41110000_1111=1.0100011_110_241.01000_1111_=1.011100_101_4.11011111_1411_:11.$1(0011_110_m0.1110011MI -0.090000 1111 -0.00/000 (11) -0.202022 112/ -0.315000 al/ 0.0 1111 -1.442000 031 -11.120000 (12/ -0022220_1111_m0.1110011_1241_z1.011000_142_=0,102000_1111_,A.il0000_1111_.1.11900o_1301 22.114000_1111.r.0.111000_1/0_20.)0)000.___0)) -0.01000 (14) -(000000 00 -1.04000 04) -0.111000 (0 0.0 (701 0.0 (11) -0.44)000 44 -1.0 luuo441_11000_1421,0.2apoaa_ilt-a.a__1411_ma.0000a_1111,1302000_001_sl.1230001111 a a_

,1!

! LOAD

( 1) 05 21 -0.092000 1 2/ -0.101000 1 -0.001000 g 11 -0.11000 4 0) -0.6)4000 1) .6.14,601) ( 44 - 0.017000__1_21-.0.2311100 Ito) ..0.).4)000 00 .1.0210oatin.a.112000_110_.1.01)ooa_t01 .1..20040_110_.0.01.00_0111_..o.140000

4111 .41,054000 (III -0.001000 Ili) -0.11/O00 (30) -0.2)1000 121) 0.0 1211 -0.03000 021 -0.110000 4141 -040020-11.1.1.11zona_1241-lauwanatuLmo-Olona_tall_ALototuuLlal.,LItt000..1.1a_clAlaostoo 4311 .D 440100 1212 0A212006

(33) -0.159000 (34) .0.192000I1 I

(25) -0.240000 (I.) .0,471000 (37) 0.0 (30) 0.0 09) -0,0000 (40) .0,644000

!Us 90L7AGE SPECIFIED

( 1) 0.930000 ( 2) 1.0c3000 ( 3) 0.9u1000 ( 4) 0.945000 ( 4) 0.939000 7) 0.116000 ( I) 0,920(100 ( 9) 0.9620000 944000 1111 0.2310110 (1)1 0.1,91ouu (241 0.111000 (1)1 0444000 410 0 343000 a71 1.010000 (111 0.921000_1101

(19) 1.010000 (20) 1.040000 (22) 0.934000 (22) 14912000 (24) 0.171000 (25) 0.17)000 (26) 0,152000 (21) 0.952000(201__10,s72000 (321 0.0740010 04/ 0.4.10000 (351 0.112000 (361 0.653000 (32) 1.004000 (401 n 792000 141) 0.102111111._(42) 0.916000 (44) 0.197000 (46) 0.453000 (44) 1.040000 II

LA' C uLAItatilESuLIS

(Nis 90LIAck

( 1) 093001211_12/1.110.11001 1LSa06433 ( 4) 0.20.001) ( s) 0.9012/4 4) 0.921644 ( /) 0.926000 ( .120000( 9) 0,962499 (10) 0.443927 (11) 0.932201 (12) 0.915627 (13) 0.697450 (14) 0.163778 (15) 0.946499 ( 6) 0,190000

___11.12_1,001241_1411_0,1220an_11911,0109(20 110)_1104000.1i1L_Ii01123.1.(P) 0,9.4000_12P -11.2ilat027 0.647442 (26) 0.922000 (17) 0.94720 (76) 0.952000 (29) 1.017947 (30) 0.879000 (31) 0.444641 (12) 0.902480

___12314.196120311 0-122721_11:10.221.4.11A.361_0_,D1212V1711_0L2411))_012_1,031402_092_0,091!! 4401_9,02)14(41) 0.093503 (42) 0.972527 (45) 0.979461 (44) 0.197000 OW 0.94444/ (96) 0,114424 (47) 0,94344) (46) 4.040000

1_11_1,107998 ( 21 1.921401 ( 31:4-L42!MO 1_!1_11,171109.42_1_) 21P.309000 (_4)1404225 111-2.2359'9_Lilka00001( 9)-1,5)3994 (10) 2,939490 (11)-2,099978 (12)-0.469001 (13)-2.890006 (14)-1.647999 (15)-0.123993 (110-0.7529961112_1.410001 (1t) 0.394000 1191 3.5150 (20) 4.624992 (L11 0.0 (12)-2,941991 (13)-0.2811000 (211:9.432499(25)-0.277999 ( 6)-2.01097 (27)-0.664005 (26) 1.579999 (291-2.ITTEW5( 0)-2.194000 (31)-0.97 014 (32)-0.442996

___11.11:0,2_?1,211_0110,221001. (351-).2761011361=07719,90_177,1_0.000002 (3) _14!1297_1117- 4429p1 (401-1..0709!9(41)- 0.933991 (42)-0,470001 (43) 0.1,00004 144)-0.168000 (45)-(.301000 (44)-0.92299s (47) 0.000014 (48) 3.31-4181

PSIS

_(_11_1/9444 (.2) 4.7:4111_i_51:0.434941 i 41 0.493370 1_52-0.191911 6)-0,476004 ( 7)-0.517354 ( it) 2.360453( 9)-0.92011 (10) 2.573415 (11)-1.276946 lib-U.414861 (13):T.412404( 14)-2.010897 (15)-0.467314 (14)-0.344310

_1171 0.794202 (111 .4.04)226 114) 2..44115 4p)2 1.014120 612) 0.0 1122-0.117litin1-0.972244 1111_:1.153541(23)-0.143441 ( 43-0.9.4740 (27)-0.415000 (20) 0.699251 (197-1;200416 t 02-1475044 022-0.270327 c32)-0.12051'Op-0.240494 L242-0414067 001-0.430941 041-0.644942 072 0.000113 067 1.906116 /397-0.116202 1404-0.614079(41)-0.404611 (41)-1.244242 (62) 0.001041 146)-0.760724 (40)-0.611490 047-0.671102 (47) 0.000173 140) 1.71044

LINE N014. m miut N PM61

1.169)33-1.516905

anti0.416656

:91.(i4691

LOAD1,2413.51.7290340.627517

_J2.42161/0.36531112.2.0421

ShI1h61.1700002.11000n5.0000001.0000041.170000

PER(.2101____64.212§1.07716.55014.1/2__19.53510,442

1

11

6213411

3

4I6j9

40411413

-0,3041950.2131790.324765-O./I1371

0.769970.103110.1672720-200412

5

____IL_7

____IL.

133

1113

1316

0.24.0460,191364

0.197703-0.0114156

0.3156360,216494

__3.1/04001,1100001.400000

16,07911.511

9

1011

1213

__14

3

11114_____

-0.5949270.11iQ.

-0.4141520.071611

0.727179 1.920000 37.87410. 16133.?19

___ 23.104 __4.94110.15112.046lb./NI__

_15

16

0.4,17460.149234

0.245166_0.1621140.5342290.401)11

_Lenonno1.600000

1.4_000001.6000001.4100AP1,600000

142265=0.01210944

444

0.070423_17.;06712-0.u10)030.111449

0.071173n.112112

15-lc

24

3L

_

...1.1.0 %%%%%

v.25291A1.109999

41.114116____

-0.176644-0.0211112.0.794203-Q 016612

0,209007____(62212.711

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LAICulA71.11 RESULTS

1104 VOLTAGE

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112) 0.909283 (13) 0,191000 (14) 0.211781(20) 1.04111100 (212 1.012110 (22) 0.3)0000_11/1_0411112A111_211111,17Os) 0.9)2000 (29) 2.017110 (30) 0.179000(21.1.3.42.6.510_137) 0.941)21_1211 1.034000(44) n.867000 (45) 0.431.264 (46) 0,164266

(29) 0.00635(31) 0.123111

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