An Efficient Technique for Reactive Power Dispatch Using a Revised Linear Programming Approach

Electric Power Systems Research, 15 (1988) 121 - 134

121

An Efficient Technique for Reactive Power Dispatch Using a Revised Linear Programming Approach N. I. DEEB and S. M. SHAHIDEHPOUR

Department of Electrical and Computer Engineering, Illinois Institute of Technology, HT Center, Chicago, IL 60616 (U.S.A.) (Received March 15, 1988)

ABSTRACT

This paper presents a mathematical formulation o f the reactive p o w e r operation/ planning problem. The outstanding features o f the proposed m e t h o d are represented by the fact that it requires no matrix inversion, will save computation time and m e m o r y space, and hence can be implemented on very large-scale p o w e r systems. The m e t h o d employs a linearized objective function and constraints, and its approach is based on adjusting control variables which are tap positions o f transformers and reactive p o w e r injection. Linear programming is used to calculate voltage increments which minimize transmission losses, and adjustments o f control variables are obtained by a modified Jacobian matrix. This approach would greatly simplify the application o f decomposition methods in p o w e r systems planning and operation. The proposed algorithm is applied to a six-bus system and the IEEE 30-bus system and numerical results are presented. These results verify the superiority o f the proposed m e t h o d over the existing ones.

I. INTRODUCTION

The purpose of an optimal reactive power dispatch is mainly to improve the voltage profile in the system and to minimize system losses. This goal is accomplished by reactive p o w e r sources which are installed at appropriate locations in a large power network to minimize power losses and improve the quality and reliability of the system operation. In the past, extensive studies using nonlinear programming (NLP) approaches have been proposed to solve this complex problem. 0378-7796/88/$3.50

However, the traditional NLP-based procedures have many drawbacks, such as insecure convergence properties and difficulties in the evaluation of system losses. Shoults and Sun [1] developed a nonlinear optimizing strategy based upon the gradient method employing the sequential unconstrained minimization technique. To make the m e t h o d suitable for a nonfeasible starting point, an outside-in penalty was chosen to force the dependent functions to be feasible. Mansour and AbdelRahman [2] divided the original problem into several subproblems. Lebow et al. [3] presented a hierarchical approach which used a projected augmented Lagrangian technique. Dommel and Tinney [4] developed a nonlinear optimization technique to determine the optimal power flow solution. They minimized a nonlinear objective function of production costs using K u h n - T u c k e r conditions. On the other hand, linear programming has also been employed as a proper tool for finding a solution to this problem. This numerical method has proven to present many features which include its convergence properties. Shoults and Chen [5] developed an algorithm using least-square minimization to find suitable changes of dependent variables such as voltage magnitudes and currents flowing in different branches. H o b s o n [6] presented a m e t h o d of finding the network reactive power distribution. He used the incremental transmission line and transformer models and linearized network equations. Then the problem was solved b y a special LP technique b y giving priorities to the generatots in the system. This method seems to maintain only soft limits on transformer taps, generator voltages, generator reactive power, etc. Mamandur and Chenoweth [7] presented a mathematical formulation suitable for LP © Elsevier Sequoia/Printed in The Netherlands

122

and developed a systematic formulation to minimize system losses and improve the voltage profile. This method uses a dual linear programming technique to determine the optimal adjustments of the control variables, and simultaneously satisfy the constraints. Ramalyer e t al. [8] presented an algorithm to minimize system losses and improve the voltage profile without incorporating power flow calculations in each iteration. The algorithm incorporates a method which avoids zigzagging of the solution around the optimal point. Iba e t al. [9] presented a new method for reactive power planning. It is suitable for both loss and investment cost minimizations. The formulation of the method is based on the N e w t o n - R a p h s o n load flow technique. However, all these methods require specific numerical algorithms which present various difficulties for large-scale systems. These problems include the inversion of the Jacobian matrix which is regarded as a time-consuming process and requires a large m e m o r y space. This paper presents a revised LP algorithm suitable for both loss minimization and voltage profile improvement, which will satisfy network performance requirements. Control variables are defined as voltage magnitudes at reactive generating buses and buses connected to transformer terminals, and dependent variables are represented by voltage magnitudes at load and junction buses not connected to transformer terminals. The power loss equation of the system represents the objective function of the proposed method. The objective function is linearized, and power loss sensitivities with respect to all bus voltages in the system are calculated. In previous studies, loss sensitivities were calculated with respect to the voltages of the reactive power generating buses, and other bus voltages (load and junction buses) were represented as functions of reactive power generating bus voltages. The proposed criterion will relax the necessity for inversion of the Jacobian matrix. The mathematical features of this algorithm are as follows: (1) buses of the system are categorized into three different types: reactive power generating buses, buses connected to tap changing transformer terminals, and junction and load buses not connected to transformer terminals;

(2) linearized sensitivity relations are calculated by utilizing elements of the Jacobian matrix; (3) load effects and tap changing transformer factors are added to the Jacobian matrix to formulate a modified Jacobian matrix; (4) dependent variables are incorporated in the formulation of equality constraints; (5) the state of the system with an optimal real power generation schedule is the initial condition for the solution of this system.

2. APPROACH TO THE PROBLEM

The formulation of the proposed method is based on adding the load and tap changing transformer factors to the Jacobian matrix. The linearized network performance constraints are derived from all elements of the Jacobian matrix to form a modified Jacobian matrix. The linearized objective function is utilized for real power loss minimization in the system by controlling the VAR generator voltages and transformer taps. Power loss is compensated by the power injection from the slack bus. This consideration, however, is valid only if the power injected to other buses is k e p t constant during the optimization process. The real power is distributed optimally before the proposed method is applied. Bus voltage angles are assumed to remain constant in each iteration of the process. This assumption is required so that for changes in the reactive power or voltage magnitudes, the couplings between phase angles and the reactive variables, voltage magnitudes and reactive power generating changes are regarded as small. The proposed method models the load and the tap changing transformer as follows. 2.1. T h e load m o d e l

The effect of voltage variation on the power consumed by system loads depends to a great extent on t h e type of load that is being supplied b y the power system. Reactive loads can be represented as a function of bus voltage magnitudes as follows: Qd = Qds

(1)

123

Initial or base case reactive p o w e r and voltage are denoted b y Qo~ and V~ respectively. The characteristic of the load can be specified by the value of q. The typical values for q are: {i

(45)

Yil (4c)

constant current load constant impedance load

So, the variation of the load with respect to changes of the bus voltage magnitude can be represented by AQd-

Y'ii = g'ii + Jb~i = TilYi~ + (Ti~ 2 - - Tiz)Yiz = T~z2Yil

Y'n = g'u + jbli = T~zYiz + (1 -- T~)~ u =

cOnstant pOwer lOad

q =

(4a)

Y~l = glz + jb'iz = T~fc~z

~g

V~

(2)

AV

Equation (4) should be added to other self admittance elements to form the overall admittance matrix (for bus l there is no effect). This will be illustrated in the examples. From Fig. 2, the complex power injection to bus i is Si = Pi + JQi = ~ *

where * indicates the complex conjugate of the variable. So,

2.2. T h e tap c h a n g i n g t r a n s f o r m e r m o d e l

The modeling of voltage transformer taps must be incorporated in any discussions related to reactive p o w e r dispatch because they represent a vital part of the automatic control process. Transformer tap changing is more difficult to model since two buses are directly involved in the tap changing process. Let us consider a transformer connecting buses i and l with tap Ti~, as shown in Fig. 1. This branch can be represented b y an equivalent ~r circuit as shown in Fig. 2. An alternative technique for modeling the tap changing transformer is discussed in Appendix A. Since (3)

Yil = gil + j b i l

= ~ [ ~ ( T i z 2 - - T,z)Yil]*

Si = V~:(T~z: - - Tii)gii -- jV/:(T~i ~ -- T~i)biz

(5)

Similarly, the complex power injection to bus l is represented as S~ = V~:(1 -- Tu)gi~ -- jV~:(1 -- Tiz)biz

(6)

From (5) and (6), the equations for Q~ and Ql are

(7)

Qi = --bi[ Vi2(T~l 2 - - Ti~ )

and Qz = - - b u Vz2(1 -- Tu)

(8)

If AQ~ is the increment of Qi with respect to voltage and tap position changes, then ~Qi AVi +

'the new branch admittances affected by the transformer modeling are

AT u

H o w e v e r , for the power flow in Fig. 2, we have i

I

1:Tit

Y

AQti = - - A Q i

il

So, differentiating (7) with respect to Vi and Tiz, we have AQti = 2buVi(Ti~ ~ - - T~) A V i

Fig. 1. Model of tap changing transformer.

+ b ~ I V ~ : ( 2 T ~ - 1) ATiz

I

Til 1

(Til-

Til) Yil

Yil

:

J (1-Til)Yil

Fig. 2. Equivalent ~r circuit for the branch of Fig. 1.

(9)

Similarly, differentiating (8) with respect to Vz and Tiz, AQtl

=

2bizVl(1 -- T~;) AVz

- -

biIVl 2 ATi;

(10)

These equations for the load and tap changing transformer will be incorporated in the formulation of the Jacobian matrix to form the constraint equations. This will be discussed in the following section.

124 3. PROBLEM FORMULATION

The problem formulation consists of the objective function formulation and the formulation of the constraint equations.

3.1. Objective function The objective is to minimize real power losses during the operation and control of a network. The real power loss PL is represented by

sion of the Jacobian matrix. In our approach the objective function is linearized with respect to all bus voltages of the system. Equality constraints are presented for all non-generating buses. The power loss increment AP L is related to changes in bus voltages as follows:

] ~kPL

[~Vl

~V2

"'"

~Vnb j

"

nr

PL = ~ Vk[Vi 2+ Vz2-- 2ViVl cos(Si--Sz)] k=s (11) where G~ is the conductance of line k which is connected between buses i and I. In (11) the losses are represented by a nonlinear function of the bus voltages and phase angles which is indirectly a function of the controllable VAR sources. Losses may also be expressed as a function of the values in the bus impedance matrix of the network. This representation results in an expression of losses in terms of bus voltages, phase angles, real power injections, and reactive power injections. Other methods for modeling the loss equation include a Taylor series expansion of the objective function in terms of the variables that are to be minimized. These alternatives were not considered in this study because of their computer storage and computation time requirements. In order to use the LP, the objective function is linearized as follows: OPL

- Gk [2Vi -- 2V~ cos(5~ -- ~it)]

(12)

Gk [2Vl -- 2V~ cos(~ i -- 81) ]

(13)

3PL

aVz

-

For every transmission line, the partial derivatives of PL with respect to the voltages at buses i and l are calculated. Partial derivatives pertaining to a certain bus are summed to form the power loss sensitivities with respect to all bus voltages in the system. In ref. 7 the loss sensitivities were calculated with respect to the voltages of the reactive generator buses (control variables). Then the voltages of other buses were represented as functions of these variables in the form of inequality constraints. This type of modeling would introduce the requirement for inver-

(14a) or AP L = M" AV

(14b)

An alternative approach to the problem of reducing the power loss is to reduce the power generation AP s of the slack bus. The results would be the same using either form of objective function.

3.2. Ne tworh performance constraints The following are the inequality and equality constraints imposed on different buses of a power network: Qmin < Q~ ~< Qmax (15a) (15b)

Qi = Qsi

Vrain ~< V~ ~< VF ax

i = 1, 2 . . . . , nb

(15C)

In eqn. (15), the first set of inequality constraints are for reactive power sources and tap changing transformer terminals. The equality constraints are for load and junction buses not connected to transformer terminals. These constraints can be rewritten in the form of increments: A Q rain • AQ~ = J" AVi ~< A(J max.~,

(16a)

AQ~ = J" AV~ = 0

(16b)

A V rain < h Y / < A V max

i = 1, 2, . . . , n b

(16c) The formulation of the constraints follows two steps: formation of the modified Jacobian matrix J", and determination of the limits.

3.2.1. Formulation of the modified Jacobian matrix J" Formulation of the modified Jacobian matrix follows three steps: formation of the

125 Jacobian matrix, addition of load effects, and addition of the tap changing transformer factor to the Jacobian matrix.

3.2.1.1. Formation o f the Jacobian matrix J. The reactive power injection into bus i in the system is nb

Q~ = ~ V~VIYn sin(Si -- 6, -- 0~,)

(17)

,=1

The linearized form of eqn. (17) is OQ~ Ji, -ViYi, sin(6i - - 6, - - On) -

-

(18)

Furthermore,

Ji, -

-

2 Vi Yii sin Oi,

nb

+ ~

VzYn sin(51 - - 5, - - 0i, )

,=1 l--,e i

Equation (2) indicates t h a t 3Qa/3V is a diagonal matrix. Therefore, only the diagonal elements in the J matrix will be modified by the load effect.

3.2.1.3. Addition o f the tap changing transformer effect (forming J"). Changing the tap ratio of the transformer is equivalent to the injection of two reactive power increments into buses which are connected to the transformer terminals. There are three cases which must be considered in this problem. First, consider the case in which neither of the two buses is a VAR generating bus; second, the case when at most one of the two buses is a VAR generating bus; and third, when both buses are VAR generating buses. In all cases, assume that the transformer is connecting buses i and l, and if one of them is a reactive generating bus, bus i would have the VAR source. In general, Qgti = Qgi + Qtl

or

Jii

= -- Vi Yii sin 0 H nb

+ ~

VzYtz sin(Si - - 6, - - On)

(19)

i=l

So, in general, the variations of the reactive power injections may be given as a function of the variation of the voltage magnitudes: AQ = J AV

(20)

3.2.1.2. Addition o f the load effect (forming J'). The reactive load is represented by eqn. (1) as a function of the different bus voltages. This function is modeled as a negative injected reactive power. Therefore, Q = Qg -- Qd and AQg

= AQ

+

(23)

where Qg, represents the total reactive power generated at bus i which includes the effect of transformer modeling. However, Qg~ = 0 if bus i is not connected to a VAR source. For bus l,

Qgtl = Qgz + Qtz

(24)

and Qgz = 0 if bus I is not connected to a VAR source. Equations (23) and (24) can be rewritten to reflect the incremental changes of the reactive powers as follows:

AQgti = AQg i + AQti

(25)

and

AQgtz = AQg I + AQtl

(26)

Derivations of AQti and AQ n are given as follows: From (10) we have 1 ATn - bnVs2 [2b~tVl(1 -- Tn) AVz -- AQtt]

AQd

Substituting for AQ from (20) and for AQd from (2) we have

AQg = J' AV

(21)

and, if we substitute ATi~ into (9), then AQti will be given by AQti = 2blzVi(Tiz 2 - Tit) AV~

where

J;, = Ji,

i, l = 1, 2, ..., rib;

i =/=l

(22a)

+ 2bi, ~

(2Ti, -- 1)(1 -- T~,) AVl

q,Qds, (Vi l q i - ' J~i -- J i i -t-

--

Vz: (2Til -- 1) AQtt

(27)

127 A QYilax tl

--bilyl2[(ATil) min]

(34a)

AQ mi". = - - b . Vz2[(AT.)max ]

(345)

AQ maxti = biz V~2(2Tn -- 1) [(A Tiz )max ]

(34c)

AQ minti = b i z V t 2 ( 2 T ~ z - 1)[(ATn) min]

(34d)

=

nb

where (AT) max

This will increase the number of variables b y one. The coefficient of the new variable in the objective function is the negative of the sum of all the coefficients of the other variables, Mnb+l

= T max

=-

-- Ts

( A T ) rain _- T r a i n __

~_lMi i=l

(38)

The effect of adding a new variable in a certain constraint equation is given b y

Ts

and Ts is the initial value of the tap ratio.

nb

A ( i , n b + 1) = - - ~ A ( i , l)

(39)

I=1 4. M A T H E M A T I C A L S T A T E M E N T O F T H E PROBLEM

Based on the formulation of the objective function, constraint equations, and the modified Jacobian matrix, the optimization problem is represented as follows: minimize

AP L = M. A V

(35)

subject to A Q min ~< AQz

t = J" AV ~< A Q max

(36a)

J" AV = 0

(36b)

A V rain ~< A V ~< A V max

(36c)

The inequality constraints are for reactive power source buses, and for buses connected to transformer terminals. The equality constraints are for junction and load buses not connected to transformer terminals. Furthermore, J" in the equality constraint represents a part of the modified Jacobian concerning junction and load buses n o t connected to VAR source or transformer terminals. In LP formulation, the variables should be positive. However, this is not the case for the physical behavior of corresponding variables. In order to readjust the constraint equations and ensure the existence of positive solutions for the state of the system, t w o methods have been implemented and b o t h have given the same results. One of these methods will be described in this section, the other one is discussed in Appendix B. A new variable AVntb+l is introduced in the problem such that the voltage increment at bus i (AVe) is given as A V i = A V ; -- AV'%+ 1

i = 1, 2, . . . , n b

(37)

where AVe' is the o u t c o m e of the optimization problem with the new variable introduced.

where A ( i , l) is the ilth element in the coefficient matrix that represents all the constraints. 5. S T E P S IN T H E D I G I T A L S O L U T I O N

The discrete steps in the digital solution process of the optimal reactive power dispatch are summarized in Fig. 3. These steps are described below. START )

I

i

Perform an initial loadI flow solution

J

I

Formulate the modified Jacobian matrix

equalityand inequality constraints

Set up the

Formulate LP problem by ~ equatons (35 and (36

I Solvethe LP problem I no ~

yes

-M.~('-"Stop )

Fig. 3. F l o w c h a r t o f t h e digital s o l u t i o n p r o c e s s o f t h e o p t i m a l reactive p o w e r dispatch.

128

Step 1. Perform an initial load flow solution. Step 2. (a) Formulate the primitive Jacobian matrix J; (b) add load and tap changing transformer effects to form the modified Jacobian matrix J". Step 3. The system losses vary nonlinearly with generator voltages. So, t h e y can be linearized in a small region around the operating point. Values for variations of control variables should be monitored at every iteration (AQstep, AYstep , and ATstep ). If any of the step sizes is beyond the limits, the value of the limit is taken into account. Step 4. Formulate the LP problem for the given objective function and the set of constraints. Step 5. Solve the LP problem and use the results as base values for the next iteration. Step 6. Check whether the real power loss in the system is significantly different from that of the previous iteration. If so, repeat the process, otherwise calculate all the variables in the system and stop.

6. T E S T R E S U L T S

A computer program implementing the proposed algorithm was prepared and used on a VAX digital computer to test the method on the Ward and Hale six-bus system [7] and a modified IEEE 30-bus system [10]. Many

(•)

different studies were performed on the six-bus system to show the flexibility and convenience of the proposed method, and how it compares with other conventional methods. Results of the test cases performed on the six-bus system are presented in this section. Results of the modified 30-bus system are presented in Appendix C. The basic system state of the six-bus system is shown in Fig. 4. The line and bus data are given in Tables 1 and 2 respectively. Table 3 gives the power loss convergence of the system for different values of AYstep. For all cases presented we have considered ATstep = 5~Tit , AQstep = 0.1, and the load was assumed to be of constant power load type. As shown in Table 3, when we decreased AVstev, the number of iterations TABLE 1 Line data o f t h e six-bus s y s t e m Line No.

From bus

To bus

R (p.u.)

X (p.u.)

Tap ratio T a

1 2 3 4 5 6 7

1 1 4 3 5 2 4

3 4 3 5 2 6 6

0.123 0.080 0.097 0.000 0.282 0.723 0.000

0.518 0.370 0.407 0.300 0.640 1.050 0.133

1.025

a F o r b o t h t r a n s f o r m e r s in t h e system, 0.9 ~< Til ~ 1.1.

95.5+j37.1

(1)

1.o51o I

55.0+113.0 0.955 I -9.86

(4'

I

'

I

(6)

o.8581 -!,3.87

I

1 : 1.1O0

(3)

IX

50.0+j5.0

Fig. 4. Six-bus system.

;

J

0.902 ! -13.25

30.0÷il 8.0

1.100

(2)

t•.101 -5.91

50.1 +j34.3

129 TABLE 2 Limits on the variables Bus

TABLE 3 Power loss convergence of the six-bus system

V min

vmax

Qmin

Qmax

1.000 1.100 0.900 0.900 0.900 0.900

1.100 1.150 1.000 1.000 1.000 1.000

--0.200 --0.200 0.000 0.000

1.000 1.000 0.055 0.050

No.

1 2 3 4 5 6

increased, a n d zigzagging w a s e l i m i n a t e d . T h e c o n v e r g e n c e b e h a v i o r s are s h o w n in Fig. 5, in w h i c h v i o l a t i o n o f PL is t h e d i f f e r e n c e b e t w e e n t h e result o f e a c h i t e r a t i o n a n d t h e final result o f t h e p r o b l e m , a n d t h e final states o f t h e v o l t a g e m a g n i t u d e s , t a p positions, and V A R sources are p r e s e n t e d in T a b l e 4. I t is i m p o r t a n t t o n o t e t h a t t h e o p t i m i z e d t r a n s m i s s i o n loss is 8 . 9 3 M W , w h i c h is similar t o t h e result o b t a i n e d in ref. 7, and t h e n u m b e r o f i t e r a t i o n s r e q u i r e d to o b t a i n this result is 7, w h i c h is smaller t h a n 11, given in ref. 7. S t u d i e s o f half a n d q u a r t e r load levels are p r e s e n t e d in T a b l e 5. T h e s e results are f o r AVstep -- 0 . 2 5 w i t h t h e s a m e values f o r ATstep a n d AQstep as b e f o r e . F o r these studies t h e final t r a n s m i s s i o n loss o f e v e r y case is t h e s a m e as t h e values o b t a i n e d in ref. 7. T h e n u m b e r o f i t e r a t i o n s f o r t h e half load level is 4 c o m p a r e d w i t h 7, a n d f o r

2.8 I 2.4

O :

AVstep

=

0.15

r"1:

AVstep

.

0.1

2.0 D_--I U-

0 Z

1.6

~ >

1.2

_o

.8

Full load level, AYstep = 0.25 Iteration No.

System losses (MW)

Initial state 1 2 3 4 5 6 7

11.45 10.50 9.20 8.90 9.28 8.89 8.93 8.93

Full load level, AVstep = 0.15 Iteration No.

System losses (MW)

Initial state 1 2 3 4 5 6 7 8 9 10

11.45 10.90 10.55 10.20 9.91 9.72 9.50 9.21 8.92 8.91 8.91

Full load level, AVstep = 0.1 Iteration No.

System losses (MW)

Initial state 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11.45 11.16 10.90 10.76 10.45 10.20 10.02 9.82 9.66 9.51 9.37 9.16 9.01 8.92 8.89 8.89

.4

5

6

7

8

9

10

11

12

13

14

ITERATION #

Fig. 5. Convergence behavior for the six-bus system for different values o f Agstep.

t h e q u a r t e r load level is 3 c o m p a r e d w i t h 7 in ref. 7. O t h e r i m p o r t a n t a d v a n t a g e s are r e p r e s e n t e d b y t h e c o m p u t a t i o n a l a s p e c t a n d t h e eliminat i o n o f zigzagging. In t h e p r o p o s e d a l g o r i t h m , t h e t i m e p e r single i t e r a t i o n is a b o u t 0.32 s.

130 TABLE 4

TABLE 6

Final results

Decomposition for a full load a n d AYstep = 0.15

Variables

Full load level

Half load level

Quarter load level

V1 V2 V3 V4 Vs V6 QI Q2 Q3 Q4 T35 T4~

1.09 1.15 1.00 1.00 1.00 0.98 36.30 19.32 5.00 5.50 0.96 0.98

1.03 1.07 1.00 1.00 1.00 0.99 7.00 7.75 5.00 5.50 0.98 0.99

1.00 1.02 1.00 1.00 1.00 0.99 --0.50 --0.20 5.00 5.50 0.98 0.98

TABLE 5 Power loss convergence of the six-bus system Half load level, AYstep = 0.25 Iteration No.

System losses (MW)

Initial state 1 2 3 4

3.06 2.72 2.44 2.23 2.23

Iteration No.

System losses (MW)

Initial state 1 2 3 4 5 6 7 8 9

11.45 10.88 10.52 10.15 9.85 9.65 9.34 9.01 8,90 8.90

time per single iteration is about 0.33 s. The number of iterations required to obtain the optimal value is 9 compared with 11 before using the decomposition. Decomposition advantages are not obvious in the six-bus system because the problem deals with a small number of variables. For large-scale systems decomposition will have a greater effect on the time and convergence aspects. More detailed studies on applying decomposition to the proposed method are planned.

7. CONCLUSIONS Quarter load level, AVstep = 0.25 Iteration No.

System losses (MW)

Initial state 1 2 3

1.31 0.86 0.55 0.55

As noticed in Table 3, zigzagging is found when AVs~p=0.25; however, for smaller values of AVsup no zigzagging is obtained. The small computation time per iteration and fast convergence rates are due to the particular mathematical formulation adopted in this study and do not depend on a specific optimization technique. From the mathematical formulation of eqns. (35) and (36), the problem can be decomposed into a main problem and one subproblem. Bender's decomposition has been applied to the problem based on the dependent and independent types of variables in the objective function. The results are represented in Table 6 for AVstep = 0.15. The

A new method is presented to find the optimal reactive power dispatch to minimize transmission losses and improve the voltage profile by adjusting VAR sources and transformer tap positions. It is concluded in this study that the proposed methodology has distinct advantages which are summarized below. (1) Fast computation and small m e m o r y space: the sensitivity matrix is not required in this study, because the effect of all variables has been introduced in the Jacobian matrix. (2) Convergence: this method provides faster convergence in the optimal power dispatch problem than do other conventional methods. (3)Decomposition: as a result of the problem formulation, Bender's decomposition can be applied to the problem. The problem is decomposed into dependent and independent variables. This procedure will save a lot of computation time and provides faster convergence.

131

(4) Applications: the m e t h o d has been tested on the six-bus and 30-bus systems. It can be implemented on large-scale systems in which the advantages of decomposition will be clear. (5) Possibility for on-line application: the fast and reliable characteristics of the computations mentioned above present the possibility for on-line applications for reactive power-voltage control.

REFERENCES 1 R. R. Shoults and D. T. Sun, Optimal power flow based upon P - Q decomposition, IEEE Trans., PAS-101 (1982) 397 - 405. 2 M. O. Mansour and T. M. Abdel-Rahman, Nonlinear VAR optimization using decomposition and coordination, IEEE Trans., PAS-103 (1984) 246 - 255. 3 W. M. Lebow et al., Optimization of VAR sources in system planning, EPRI Rep. EL-3729, Nov. 1984. 4 H. W. Dommel and W. F. Tinney, Optimal power flow solution, IEEE Trans., PAS-87 (1968) 1866 - 1876. 5 R. R. Shoults and M. S. Chen, Reactive power control by least squares minimization, IEEE Trans., PAS-95 (1976) 325 - 334. 6 E. Hobson, Network constrained reactive power control using linear programming, IEEE Trans., PAS-99 (1980) 868 - 877. 7 K. R. C. Mamandur and R. D. Chenoweth, Optimal control of reactive power flow for improvements in voltage profiles and for real power losses minimization, IEEE Trans., PAS-IO0 (1981) 3185 - 3193. 8 S. Ramalyer, R. Ramachandran and S. Haribaron, New technique for optimal reactive-power allocation for loss minimization in power system, Proc. Inst. Electr. Eng., Part C, 130 (1983) 178 - 182. 9 K. Iba, H. Suzuki, Ke. Suzuki and Ka. Suzuki, Practical reactive power allocation/operation planning using successive linear programming, IEEE PES Winter Meeting, New Orleans, LA, 1987, Paper No. 87 WM 055-7. 10 K. Y. Lee, Y. M. Park and J. L. Ortiz, A united approach to optimal real and reactive power dispatch, IEEE Trans., PAS-104 (1985) 1 1 4 7 1153. 11 S. M. Shahidehpour and J. Qiu, A new approach for minimizing power losses and improving voltage profile, IEEE Trans., PAS-106 (1987) 287 - 295.

NOMENCLATURE

J J' J"

Jacobian matrix Jacobian matrix with added load effect modified Jacobian matrix (Jacobian matrix with added load effect and tap changing transformer factor) nb number of buses in system nr number of lines in system P~ real power injection at bus i Pil real p o w e r flow from bus i to bus l PL real power transmission loss in system Qi reactive power injection at bus i Qiz reactive power flow from bus i to bus l Qdi reactive load at bus i Qdsi initial value of reactive load at bus i Qgi injected reactive power, from generator or capacitor, to bus i Qti injected reactive power equivalent to tap position adjustment to bus i St complex power injection at bus i ~l complex p o w e r flow from bus i to bus l T~ tap ratio for transformer b e t w e e n buses i and 1 Vi voltage magnitude at bus i V~ initial value of voltage at bus i Yii magnitude of self admittance of bus i YI~ modified magnitude of self admittance of bus i :Yiz admittance of line between buses i and 1 ~}~ modified admittance of line between buses i and l 8i 0iz

voltage angle at bus i line admittance angle that buses i and l

APPENDIX A

The tap changing transformer can be modeled as shown in Fig. A-1. The change of the p o w e r flow through the line due to the change of tap position can be modeled b y two incremental reactive p o w e r injections at buses i and I.

i

l:Til

connects

Boldface italic letters denote vectors, boldface roman letters denote matrices, and letters with bars represent phasors.

Zil I

l

" Qt i

I

I t~Qtl

Fig. A-1. Model of tap changing transformer.

132

Assume that the transformer is adjacent to bus i. This assumption will create a new bus of voltage TnVi instead of V~. The nonlinear reactive power flow from bus i to bus 1 can be expressed as

Qil = TiiViVz sin(~t -- 51 -- On) +

Tiy2Vi2Yil sin Oil

Qli = TnViVz sin(~i -- 61 -- Oil) + Vl2yil sin Oil (A-2) According to Fig. A-l, we can write

aT~

ATil

(A-3)

From eqn. (A-l), the sensitivity of Q~l with respect to Tn can be written as OQil

~)Tit

- ViVzYn s i n ( S i - ~ z - 0il) + 2TnVi2Yn sin Oil

0 < AU ~< AU m~x Therefore, the expressed as

(B-4) objective function will be

APL = M" A V + M" A V rain (A-l)

Similarly, Q . can be expressed as

AQti =--AQil = - - - -

and the constraints for the voltages will be changed to

Note that the objective function can be represented as before w i t h o u t any changes because the term M . A V rain is a constant. This constant will be added again to get the exact value of the objective function after solving the problem with the new variables.

APPENDIX C

The proposed m e t h o d has been implemented on a modified IEEE 30-bus system (Fig. C-I). Line and bus data are shown in Tables C-1 and C-2 respectively. The results are shown in Table C-3.

(A-4)

Similarly, AQtz can be written as

AQtl =--AQH = - ~Qli ATil ~Tn = V~V~Yn s i n ( S i - 5 z - Oil)

(A-5)

ATil can be expressed as a function of either AQti or AQtl using the above equations. So, if we know the value of AQt i or AQtt, the tap position (see Appendix D) can be calculated for the optimal voltage profile.

APPENDIX B

In the linear programming formulation, the variables have to be non-negative. To overcome this difficulty we defined new variables as follows: AU = AV--

A V rain

(B-l)

Referring to eqns. (35) and (36), the reactive power increments in the inequalities will be expressed as AQ = J"(AU + • V rain)

(B-2)

So, the equality constraints axe modified as follows: J" A V = --J" A V rain

(B-3)

(B-5)

Fig. C-1. Modified IEEE 30-bus system.

133 TABLE C-2 (continued)

TABLE C-1 Line data for the 30-bus system

Bus

Line No.

From bus

To bus

R (p.u.)

X (p.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

1 1 2 2 2 3 4 4 5 6 6 6 6 6 8 9 9 10 10 10

2 3 4 5 6 4 6 12 7 7 8 9 10 28 28 10 11 17 20 21 22 13 14 15 16 15 18 23 17 19 20 22 24 24 25 26 27 27 29 30 30

0.0192 0.0452 0.0570 0.0472 0.0581 0.0132 0.0119 0.0000 0.0460 0.0267 0.0120 0.0000 0.0000 0.0169 0.0636 0.0000 0.0000 0.0324 0.0936 0.0348 0.0727 0.0000 0.1231 0.0662 0.0945 0.2210 0.1070 0.1000 0.0824 0.0639 0.0340 0.0116 0.1150 0.1320 0.1885 0.2544 0.1093 0.0000 0.2198 0.3202 0.2399

0.0575 0.1852 0.1737 0.1983 0.1763 0.0379 0.0414 0.2560 0.1160 0.0820 0.0420 0.2080 0.5560 0.0599 0.2000 0.1100 0.2080 0.0845 0.2090 0.0749 0.1499 0.1400 0.2559 0.1304 0.1987 0.1997 0.2185 0.2020 0.1932 0.1292 0.0680 0.0236 0.1790 0.2700 0.3292 0.3800 0.2087 0.3960 0.4153 0.6027 0.4533

I0

12 12 12 12 14 15 15 16 18 19 21 22 23 24 25 25 28 27 27 29

Tap ratio T a

0.9610 0.9560

0.9650

0.9530 0.9700 0.9850

0.9635

--5.60 --7.00 --5.80 --7.00 --9.30 --4.50 --7.80 --5.00 --9.00 --9.30 --8.80 --9.50 --10.30 --10.60 --10.30 --10.00 --10.00 --10.10 --10.60 --11.00 --11.70 --10.80 --6.20 --12.30 --13.50

Qmax

6 7 G8 9 10 Gll 12 G13 14 15 16 C17 C18 19 20 21 22 C23 24 25 26 C27 28 29 30

--0.150

0.150

--0.100

0.100

--0.150

0.150

--0.050 0.000

0.050 0.055

--0.050

0.055

--0.055

0.055

0.9655 TABLE C-3 Power loss convergence of the 30-bus system Full load level, A Y s t e p = 0.35 0.9810

Bus data for the 30-bus system ~

0.960 0.948 0.959 0.955 0.914 0.975 0.949 0.936 0.935 0.933 0.921 0.902 0.930 0.889 0.894 0.900 0.900 0.929 0.888 0.901 0.881 0.919 0.945 0.884 0.875

Qmin

0.9590

TABLE C-2

Vb

~

aG ffi generator bus; C = capacitor bus. b0.00 < V1 < 1.06, and for all other buses 0.90 < V~ < 1.10.

aFor all transformers in the system, 0.9 < Tiz < 1.1.

BUS

Vb

No. a

Qmin

Qmax

No. a G1

1.060

0.00

--0.200

0.000

G2 3 4 G5

1.030 0.976 0.967 0.988

--2.20 --4.00 --4.80 --7.50

--0.200

0.200

--0.150

0.150

(continued)

Iteration No.

System losses (MW)

Initial state 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10.24 10.01 9.66 9.38 9.08 8.69 8.24 8.01 7.72 7,43 7.09 6.63 6.85 6.64 6.51 6.51

134 APPENDIX D In this appendix we introduce a lemma to calculate the tap position of a transformer from the information acquired from the buses connected to the terminals of the transformer. Lemma. The value of ATiz calculated by eqn. (9) is the same as t h a t calculated by eqn. (10). Proof. Eqn. (9) can be rewritten as follows:

A Q n = 2bi~VtTn(Ti~ -- 1) AV~ + bilV~2(Ti! -- 1) ATn + b~iVi2Til ATiz (D-l) Since the value of T~l is close to unity, and AV/and AT H are small, AT n can be written as

ATn-

AQn biiV.2Tl z

(D-2)

Similarly, from eqn. (10) ATiz can be written as

AT~I-

--AQtl biiVi2

(D-3)

If we assume that these two values of A T n are equal, then from eqns. (D-2) and (D-3) we will have

AQti _ Vi 2 AQtz Vz2 (--Tiz)

(D-4)

However, from Fig. 2, AQti and AQtt can be written as

AQti = V/2(Til 2 -- Tn)biz

(D-5)

and

AQ u = Y/2(1 -- Til)bil

(D-6)

Dividing (D-5) by (D-6), we will get the same equation presented by (D-4). So, equality (D-4) holds, and this proves that both the ATil from eqns. (D-2) and {D-3) are the same. This lemma relaxes the necessity to present a subprogram to solve the problem when we have two VAR generating buses connected to the terminals of a transformer [ 11 ].