Fuzzy optimal reactive power control

ELSEVIER

Electric Power Systems Research 30 (1994) 47-55

ELEGTRIG POW|n 8WSTEm8 R|8|nRCH

Fuzzy optimal reactive power control Rasool Kenarangui, Alireza Seifi Electrical Engineering Department, University of Tabriz, P,O.B. 711, Tabriz, Iran Accepted 22 January 1994

Abstract

Optimal reactive power control has been an active research area during the last decade. Where precise data are available, a specific objective with clear-cut constraints was the primary requirement. The sensitivity of a feasible solution in the constraint domain has been reported many times. Fuzziness and imprecision often arise due to poorly defined data, objectives, constraint boundaries and imperfect modeling. In the past few years, fuzzy modeling of such an optimization problem in mathematical programming has been developed and at present is still one of the active research areas in fuzzy set theory applications. In this paper, a fuzzy single-objective function with multiple fuzzy constraints is modeled by fuzzy set theory. Trapezoidal membership functions are defined for the objective function and constraints. Then the fuzzy mathematical programming of the reactive power control is reformulated as a standard linear programming problem and the 2-formulation method is applied for its solution. Ward-Hale 6-bus and IEEE 14-bus test systems are used as illustrative numerical examples and the results of the presented methodology are compared with conventional methods. The results of the computation are optimistic both in terms of minimizing the objective function and of the smaller number of iterations with respect to the previous methods. Keywords: Reactive power control; Fuzzy sets; Fuzzy linear programming

I. Introduction In electric power systems, like any other engineering field, in many real-life situations where planning is being carried out for the future, the planners encounter objectives and constraints which are not known precisely. While fuzzy set theory has found many applications in other engineering and nonengineering fields, the number of papers regarding the application of fuzzy set theory in electric power systems is still rather limited. In Refs. [1-3] uncertainty related to loads and power generation was considered with the help of fuzzy set theory in power flow analysis. A fuzzy mathematical programming technique is applied to optimal power flow calculation in Ref. [4]. Fuzzy set theory has been applied to power system transient security assessment with a pattern recognition method which assigns an index of power system transient stability to the system operating states [5]. Qualitative judgements and criteria are integrated into the program by the use of fuzzy set theory and applied in a long-range power system expansion planning program in Ref. [6]. In Ref. [7], fuzzy sets are used to modify the linear programming approach to voltage control and incorporate some heuristic concepts of the expert system approach. 0378-7796/94/$07.00 ~: 1994 Elsevier Science S.A. All rights reserved SSDI 0378-7796(94) 00835-R

In this paper fuzzy linear programming is applied to optimal reactive power control in electric power systems. Here, the upper and lower bounds of the objective function and constraints such as the voltage magnitude of the buses, the transformer tap changing domain, and reactive power source limits are considered to be fuzzy variables. The 2-formulation method is applied to solve the proposed fuzzy optimization problem. The results of the computation are optimistic both in terms of minimizing the objective function (minimizing the active power losses) and of the smaller number of iterations with respect to previous papers [8, 9]. Finally, improvement is made in determining the membership function of the objective function, which has had a considerable effect on the convergence.

2. Fuzzy linear programming An ordinary deterministic linear programming (LP) problem can be written as follows: minimize f ( X ) = C T X subject to A X <.b,

X>~O

(1)

R. Kenarangui, A. Seifi /Electric Power Systems Research 30 (1994)47-55

48

where X v = [xz, x2 . . . . . xn] is a control variable; f ( X ) is the objective function which can be a scalar when we have a single-objective function or a vector when we have a multiobjective function for the optimization problem; C v = [ G , C2. . . . . C,,] is the vector of objective function parameters; b v = [b~, b2 . . . . . bn] is an n-dimensional vector; and A = [a,j] is the sensitivity matrix. In real-life situations, specially where planning engineers are involved with future system development, the objective function(s) and/or constraints can take a fuzzy form. The conventional linear programming can be represented by different models, some of which are considered in the following. (a) LP problem with a fuzzy goal. Here, the objective function can be optimized by achievement of the objective function up to some acceptable level, then the general form of the mathematical programming can be expressed as minimize f ( X ) = c T x

(2)

subject to

A X <~b,

X >>,O

where the tilde, ~ , is used to denote that the operators or variables contain fuzzy information, and has a linguistic interpretation. For example, in electric power systems, it is desired to keep the system losses lower than a certain level as far as possible. This problem is solved by fuzzy mathematical programming with a fuzzy objective function. (b) LP problem with indefinite constraints. This form of general LP is minimize

C TX ~fo

(3)

subject to

AX~b,

pD(J0 = min{/~G(X), pc(X) } where /~(X)

= m i n #G,(X)

i = 1, 2 , . . . ,

n

(5)

is the membership function of the fuzzy objectives and ~c(X) = min pcj(X)

j = 1, 2 , . . . , m

(6)

is the membership function of the fuzzy constraints. An optimal solution x * e X must satisfy both the objective function(s) and constraints which can be obtained by taking the maximum of Eq. (4) such that 2 = max pD(X*) = max[min{#~(X), #c(X)}]

(7)

where 2 is the degree of satisfaction for the fuzzy objective function(s) and fuzzy constraints. Two methodologies, namely, the c~-cut approach and the 2-formulation can be utilized in order to solve the fuzzy optimization problem. e-cut approach. When the objective function(s) is precisely defined in a fuzzy optimization problem and only the constraints are fuzzified the c~-cut, or level-cut, approach is utilized. When only the constraints contain fuzzy information, the model can be represented by minimize f ( X )

(8)

subject to g,(X) e %

j = 1, 2 . . . . . m

where gi (X) e G; shows that gi(X) is a member of the fuzzy set Gj such that #Gj > 0. Then, the fuzzy feasible decision is defined such that it takes into account all the constraints _D= (~ %

X)O

where ~ denotes 'fuzzy less than or equal to' and indicates that the variables or parameters take a fuzzy form. For example, in electric power systems, some situations are encountered where planning engineers can only describe them in fuzzy linguistic terms, like "roughly smaller or larger than", or they set them in such a way that should not exceed certain values by too much. In the present paper, the second version of fuzzy linear programming is applied, where fuzzy objective function(s) and constraints are characterized by membership functions. The optimal solution (fuzzy decision D) is the one which can satisfy (optimize) the fuzzy objective function(s) as well as the fuzzy constraints simultaneously. If G and C are defined to be the fuzzy subset of variables x in set X which satisfy the fuzzy objective(s) and fuzzy constraints, respectively, then the membership function of the fuzzy decision, /~D, is defined [ 10, 11] as

(4)

j

(9)

1

and /tD(X) = "

min ,,{pQj[g/(X)]} j=l,2

.....

(10)

"

where Up(X) is the membership function of the fuzzy feasible decision D. An c~-cut of a fuzzy subset A~ of A is a fuzzy set whose membership values are greater than some threshold c~E [0, 1], such that A~ = {x eX, PA(X) /> e}

(11)

By using the e-cut approach, the optimal solution of the fuzzy feasible decision in Eq. (9) is considered to have at least a certain degree of membership in _D. On the other hand Qj can be defined as

Q/= {gjlg,~R, /z% :g ~ [ 0 , l]}

(12)

The c~-cut definition is extended to define the subinterval with a membership function of ~t%(gj) >~ ~,

~ ~ [0, 1]

(13)

R. Kenarangui, A. Seifi /Electric Power Systems Research 30 (1994) 47-55 Hence, the real-number universe of discourse is defined for it such that

Gj(a) = {g, lg,~R, uo,(g~) >/a}

(14/

where it is a closed interval with sharp boundaries Gj (a) = [g}')(a), g lY)(a)]

(15)

where Gj (a) is called the a-level of Qs, and g~))(a) and g(u)(a) are the lower and upper sharp boundaries, respectively, with a certain a-level. The ~-level can be interpreted in terms of a 'design level' in an engineering sense. Here, the lower value of a implies the larger interval of G~(a) such that a, < a2 ~ G s ( a , ) = Gj(a2)

(16)

3. Solution

g~') ~<&(X) ~
minimize f(X)

(18)

subject to g}')(a)

<~gy(X) <~g)U)(a)

j = 1, 2 . . . . . m

which can be solved by conventional nonlinear parametric methods. Then the optimization results for different a-levels are compared by solving series of this problem, such that 0~ 0, and the multiobjective function f(X) define a fuzzy domain S in D such that

j = 1, 2 . . . . . m

f:(X*)

f,(X*)

f,(X*),

P=

f2(X*).

... fk(X*)1 ' fk(X*).

fl(X*) f=(X*) ...

(23)

fk(X*)J

It can be observed that the diagonal elements are the minima in their respective columns. (c) Determine the best (minimum) and worst (maximum) solutions possible for each of the objective functions: fmax = maxf/(X, )

s

(24)

f m i n = m ! n f (X*) J

(d) Using the solution results in step (c) as boundary values the membership functions of the fuzzy objective functions are determined:

Uli(X)

fOfi(X}_4_fm,f i > * f m a X~.--Ji max fmin ~ f,.(X) <~fm.x / max min f,

-f,

L

_< tcmin

f ii "~-J i

1

i=1,2 ..... k (25) (e) The fuzzy constraints are stated as

g(')-Ag}l)<<.gj(X)<~gj J --(u) S = {i~l ~/}(X) } r%~ l UG,[gi(X)]}

(22)

and let the solutions be X*, i = 1, 2 . . . . . k. (b) Construct matrix P such that

(17)

and when a = 1 , Gj(1-) covers the restricted interval [g~), g)Uq. Thus, by substituting all fuzzy allowable intervals of Gj by a-level cuts Gj(a), the fuzzy optimization problem can be transformed into a nonfuzzy optimization problem, namely,

procedure

Once the membership functions of the objective and constraint functions are known, the practical solutions of Eq. (21) are determined, where the solution procedure is as follows. (a) By using conventional deterministic optimization procedures, minimize the individul objective functions f (X) subject to the constraints

when a = 0 +, Gs(0 +) covers the entire allowable range [gJ.') -- Ag}'), gJ~) + Ag{u)]

49

+Ag} u)

j=l,2

..... m

(19)

(26)

By maximizing the membership function of Eq. (19), the optimum solution X* from the fuzzy domain can be achieved such that

where Ag~l) and Ag~.u) are the distances through which the boundaries of the j t h constraints are moved. Then the membership functions of the j t h constraint can be represented as

Us(X*) = max Us(X)

(20)

U(1)t gj tX) -

where

gj(x) > g)'

Us(X) = min{&, (X) . . . . ,/%(X), #G, [gl (X)] . . . . .

flG,,,[gm (X)] }

=

(21)

where Uj}(X) and Uc~[g/(X)] are the membership functions of the ith objective and j t h constraint functions.

_ (X)

_ g},, + ~,gj

Ag}0

g~l) _ Ag~,) ~
gj(X) <~g~O _

Ag~l) (27)

50

gg ! 1

R. Kenarangui, A. Seij / Electric Power Systems Research 30 (1994) 47-55

and

S is an (n + ng + ns,, + n,ap) x n, matrix

4,

s,, St

%I

sgt ,,

1 0 0

0 1 0

0 1 0”

S

-g,(X)

Z

+ g;“’ + Agj”’ Ag!“) I

i

gj”’ 6 g-j(x) < g;“’ + Ag;.“’

s=

by

(33)

Refs. [8, 91 are used in the derivation of the mathematical programming problem which is given by Eq. (30). The details of the mathematical programming are given in the Appendix.

(f) The fuzzy optimization using conventional nonlinear as follows: maximize subject

problem can be solved by programming techniques 4. Fuzzy mathematical

ii to i=l,2,....k

2
j=l,2,...,m

1 G P;‘(x)

j=l,2,...,m

minimize

of the problem

(29)

subject

AP, = KTAX(’ + ‘)

Reactive power control in a power system is required in order to minimize the system losses while keeping the variables of the system within specified limits. Hence, the economic and security objectives in a power system must be achieved simultaneously. In this paper, the linear reactive power dispatch is formulated in terms of a full set of controllable variables which are as follows: generator voltages V,, transformer taps t,, switchable shunt capacitances and inductances Ys. The load bus voltage magnitudes, generator reactive power generation, and the system losses will be controlled through controllable variables. The linear reactive power control problem can be written as

minimize

(30)

to

b”‘,
subject

in another

KTAX”’ + ‘)

where the superscripts m and M stand for lower and upper limits, respectively. The vector AX is an n, x 1 vector of changes in the control variables, such that (31)

CT is a 1 x n, vector whose elements are the loss sensitivity coefficients with respect to the control variables n, and is given by

1

(32)

form as (35)

to

GAX” + 1)< $‘I where (36) and

h^=

(37)

The modeling in the form of Eq. (34) can be solved with any conventional linear programming method. The result of the solution for AX at every iteration is used to calculate the value of X, such that XV+ 1)= XV’+ AX”+”

AX = [AV,, AQ,, AtJ

(34)

b”’ _ 6 SAX” + 1)< jj”’ This can be written

{f( AX) = CrAX}

the for

to

061<1

minimize

programming

From the previous section and the Appendix, linear programming of the reactive power control the rth iteration can be written as

3, G P,,(X)

subject

defined

(38)

The iteration is continued till the vector norm of AX is less than the acceptable error. The upper and lower bounds in the problem constraints cannot be determined precisely due to many engineering facts. So, the upper and lower bounds are not precisely known, as mentioned in Section 2. Two methods, namely, the cx-cut and the A-formulation can be applied. The cc-cut method is not recommended for the present application, because it gives a parametric solution instead of a unique solution, and its computation time is very long. In addition, it just gives an

R. Kenarangui, A. Seifi / Electric Power Systems Research 30 (1994) 47 55

acceptable design level for the control variables. Hence it is just suitable for planning. So, in order to apply the A-formulation to our problem, we use an exact form of our mathematical programming, that is, maximize

A

(39)

subject to 2 ..< ~+;(X),

The membership function #y;.(X) is arbitrary and is assumed to have linear form. Then the constraint f~ (X) ~< di will have a fuzzy membership function in the fuzzy environment, such that 1 ],~[;(X) ~-

(42)

subject to S,AX+ ~A
1

d i <~f~(X) <~d,. + 6i (40)

where

bi

0

f i (X) >/d i + 6 i

where 6i is the threshold of the constraint f~(X) ~< di. From Eqs. (39) and (40) for the constraints we can write

A ~<

2
X(r + I) : x(r) ~_ AX~r + 1)

Z(X) ~ d,

(di + ¢~i) --fi(X)

•

losses should be zero, hence a membership function of unity will be assigned. Since our goal is to reduce the system losses from our first run (base case) of the load flow, the membership function of the base case is assumed to have a membership function of zero. The membership function is assumed to be linear between these two points. Hence the modeling of Eq. (34) will have the following form: maximize

O<~A~
51

( d i + (~i) --ft(X)

6i

(41)

= ~_ bcrl j

(44)

where Pg) is the active power loss in iteration r, and 6 is the threshold of the objective function and constraints. Since S is constant in each iteration, we will encounter only ordinary linear programming.

f~(x) + ~,~ ~ d, + ~ The membership function of the mathematical programming is determined by the planner. To assign membership function(s) to the objective function(s) of the optimization problem, some authors [7, 12] believe that first of all the problem should be solved by conventional methods without considering the fuzziness, then the solution should be used to determine the membership function(s) of the objective function(s). Others [4, 7] believe that the membership function(s) of the objective function(s) is subjective and should be determined based on the planner's past experience and preferences. That is, the best and worst cases are used to obtain the membership function of the objective function in a subjective manner. However, by selecting a suitable membership function for the objective function(s), considerable savings in computation time can be made [7]. In this paper, the second method is modified and a methodology is introduced which has proved to reduce the computation needs and time considerably. It should be noted, however, that if the worst case is assumed to be very low, the problem may not have a feasible solution. On the other hand, if the worst case is assumed to be very large, the problem may not have a reasonable solution. In our problem, since the objective function is the minimization of the active power losses in the system, the ideal case is that the

5. Fuzzy linear programming solution algorithm The general steps of the algorithm involved in the solution process are given below and the flowchart is presented in Fig. 1. Step I. Run a power ftow program for the base case. Step 2. Verify violations of the allowable limits of the constraints within the fuzzy domain, and the need for minimization of the losses. If there is no violation and no improvement is needed, stop; otherwise go to step 3. Step 3. Construct the Jacobian matrix without the reference bus in order to determine the coefficients of the objective function. Step 4. Construct the Jacobian matrix with the reference bus in order to evaluate the matrix S'. Step 5. Solve the fuzzy linear programming with the A-formulation and update the control variables. Step 6. Run a power flow program with the new operating condition and go to step 2.

6. Illustration example In this section, the developed methodology is applied to the W a r d - H a l e 6-bus and modified IEEE 14-bus test

52

R. Kenarangui, A. Seifi / Eleetric Power Systems Research 30 (1994)47 55

pERFORM A lld£E C.45E LOAD FLOW PROGRAM

,t ~ c K rB snrBu p ~ o ~ c E [ FoJ waO BUs voLrAoES, !oEt,'rauroR e,eacrtvs eov, xzs ~ # . S r s r ~ wssEs. .

I I 1

cvtarB ora~-i-~, FUNCTION OEFFICIENTS.

+

WITH FUZZY OPTIMIZ~ON ,.I

The allowable maximum value of the objective function was set to be the transmission loss for the base case with a membership function of zero, and for the ideal case where the transmission loss is zero a membership function of unity was assigned. The transmission loss and variables for each case are presented in Table 2. The transmission loss for each case is different, the lower value of the transmission loss being obtained in case I and the higher transmission loss in case 2, depending on the way the fuzzy domain for the constraints is defined. These results are compared with the results of Ref. [8], where the optimization result for the transmission loss is reported to be 8.93MW after 11 iterations. With the proposed methodology the number of iterations for each case was 3, and the transmission losses were lower than that reported in Ref. [8] except for case 2. Experience gained from these test cases show that the fuzzy optimal reactive power control does not always give a better value for the objective function than the deterministic reactive power control. This is because the objective function of the original problem constitutes a constraint equation of the fuzzy programming problem. In order to confirm the computation advantages gained from fuzzy reactive power modeling, the modified IEEE 14bus system was also tested. Table 3 gives the fuzzy domain and Table 4 the solutions obtained using the modified IEEE 14-bus system; similar observations were made.

TEC+P,'I~ TO F~VD TR~ ~Q,- { u1n+F+O . vSm+F r r.E I

co

oL v, uBu s.

4, MOD~Y ~E VALUES OF COIqIIIOL

[

+

PERFORM THE LO,,ID FLOW CALCULATIONS.

Fig. 1. Flowchart for the proposed algorithm.

systems. The network data and operating conditions for the 6-bus and 14-bus systems are taken from Refs. [8] and [13], respectively. A N e w t o n - R a p h s o n load flow program was run for the base case and the resulting transmission loss came out to be 11.44 MW for the 6-bus test system. The computer program was written in Turbo Pascal on an AT 80486 personal computer and the methodology as well as the computer program were tested under different conditions for each of the test cases. The fuzzy domain for each case in the 6-bus test system is shown in Table 1. The membership function used for the objective function was a linear type.

7. Conclusions

A fuzzy optimization programming methodology has been presented and applied for optimal reactive power control. This methodology is useful for problems dealing with cases where fuzzy information on the objective function is available and the upper and lower bounds have to be determined on the basis of the planner's subjective judgement. The W a r d - H a l e 6-bus and IEEE 14-bus systems are used as illustrative examples and it is concluded that the calculation task involved is much simpler than that for the deterministic structural optimization. Moreover, the proposed method is superior to the classical structural fuzzy optimization. The advantages of the proposed methodology compared with the ordinary optimization method are as follows: (1) fuzzy sets represent a power system's operating conditions more realistically; (2) the proposed method transfers the fuzzy modeling to a specific numerical algorithm; (3) the best solution is that which maximizes the least satisfied objective or constraint as determined by the fuzzy set membership function.

53

R. Kenarangui, A. Seifi / Electric Power Systems Research 30 (1994) 47 55 Table 1 Trapezoidal possibility distribution of variables for the W a r d - H a l e 6-bus system (reactive power and voltages in p.u.) Variable

Qc~J Qc2 v_ VIV4 V5 V6 VI QG4 Q~6 t43 t65

Case 1

Case 2

Case 3

Case 4

Max.

6M,x

Min.

•Min

Max,

6Max

Min.

(~Min

Max.

6Ma x

Min.

t~Min

Max.

t~Max

Min.

6Mi,

1.0 1.0 1.00 1.00 1.00 1.00 1.10 1.15 0.05 0.055 1.10 1.10

0.1 0.1 0.05 0.05 0.05 0.05 0.00 0.00 0.10 0.100 0.05 0.05

-0.2 -0.2 0.90 0.90 0.90 0.90 1.00 1.10 0.00 0.000 0.90 0.90

0.1 0.1 0.05 0.05 0.05 0.05 0.00 0.00 0.00 0.000 0.05 0.05

1.0 1.0 1.00 1.00 1.00 1.00 1.10 1.15 0.05 0.055 1.10 1.10

0.0 0.0 0.00 0.00 0.00 0.00 0.05 0.05 0.05 0.050 0.05 0.05

-0.2 -0.2 0.90 0.90 0.90 0.90 1.00 1.10 0.00 0.000 0.90 0.90

0.0 0.0 0.00 0.00 0.00 0.00 0.05 0.05 0.00 0.000 0.05 0.05

1.0 1.0 1.00 1.00 1.00 1.00 1.10 1.15 0.05 0.055 1.10 1.10

0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.10 0.100 0.90 0.90

-0.2 -0.2 0.90 0.90 0.90 0.90 1.00 1.10 0.00 0.000 0.05 0.05

0.1 0.1 0,05 0.05 0,05 0.05 0.05 0.05 0.00 0.000 0.05 0.05

1.0 1.0 1.00 1.00 1.00 1.00 1.10 1.15 0.05 0.055 1.10 1.10

0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.00 0.000 0.05 0.05

-0.2 -0.2 0.90 0.90 0.90 0.90 1.00 1.10 0.00 0.000 0.90 0.90

0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.00 0.000 0.05 0.05

Table 2 Fuzzy optimal solution for variables with optimal system losses for the Ward Hale 6-bus system (reactive power in MVAr and voltages in p.u.) Variable

Base case

Case 1

Case 2

Case 3

Case 4

Qcl QG2 V~ V4VI V5 V6 VI

35.350 34.063 0.895 0.975 0.903 0.937 1.050 1.100 0.000 0.000 I. 1O0 1.025 11.440

23.432 11.174 1.034 1.033 1.029 1.015 1.100 1.150 12. 573 13.073 0.980 0.950 8.086

29.507 23.594 0.997 0,995 0.998 0~992 1.081 1.161 0.000 9.617 0.984 0.975 9.009

21.783 21.839 1.032 1.031 1.033 1.028 1.101 1.187 3.675 12.854 0.985 0.976 8.318

33.343 21.831 1.034 1.032 1.033 1.022 1.121 1.188 0.000 5. 500 0.984 0.970 8.444

QG4 QG6 /43 /65 System losses (MW)

Table 3 Trapezoidal possibility distribution of variables for the IEEE 14-bus system (reactive power in MVAr and voltages in p.u.) Variable

Case 1

Case 2

Max.

t~Max

Min.

6Mi.

Max.

6M. x

Min.

6Mi.

Q~I Q~2 QG3 QG4 Qc5

1.0 0.5 0.4 0.24 0.24

0.1 0.1 o. 1 0.10 0.10

-0.3 -0.4 0.0 --0.06 -0.06

0.1 0.1 o. I 0.10 0.10

1.0 0.5 0.4 0.24 0,24

0.1 0.1 0.1 0.10 0.10

-0.3 -0.4 0.0 --0.06 0.95

0.1 0.1 o. 1 0.10 0.05

V7 V8

1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.10 1.10 1.10 1.10 1.10 0.2 1.10 1.10 1.10

0.05 0.05 0.05 0.05 0.05 0.05 0,05 0.05 0.05 0.05 0.05 0.05 0.05 O. 1 0.05 0.05 0.05

0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 1.00 1.00 1.00 1.00 1.00 0.0 0.90 0.90 0.90

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.0 0.05 0.05 0.05

1,05 1,05 1,05 1.05 1.05 1.05 1.05 1.05 1.10 1.10 1.10 1.10 1.10 0.2 1.10 1.10 1.10

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.0 0.05 0.05 0.05

0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 1.00 1.00 1.00 1.00 1.00 0.0 0.90 0.90 0.90

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.0 0.05 0.05 0.05

ii!l

V1t [ Vj21 VI 3 V14 V/ V2I/5 V3 V4 Q~9 167 t69 /84

105

005

095

005

105

005

095

005

R. Kenarangui, A . Seifi / Electric Power Systems Research 30 (1994) 47-55

54

Table 4 Fuzzy optimal solution for variables with optimal system losses for the IEEE 14-bus system (reactive power in MVAr and voltages in

p.u.) Variable

Base case

Case l

Case 2

QGI QG2 QG3 QG4 Qc5

-- 14.934 48.930 27.417 22.937 25.293

--39.909 51.669 47.769 -- 10.185 30.560

--38.963 56.632 47.943 --9.826 30.388

Iv61

1,012

1.073

1.072

IVTI

1.049 1.016 1.032 1.031 1.047 1.053 1.047 1,019 1.060 1.045

1.078 1.066 1.073 1.069 1.076 1.076 1.071 1.053 1.102 1.099 1.086 1.091 20.000 1.039 0.998 0.911

12.722

Iv, I

1.090

QG9 /67 t69 t84 System losses (MW)

0.000 0.978 0.969 0.932

1.080 1.066 1.076 1.072 1.079 1.078 1.073 1.056 1.100 1.097 1.087 1.093 1.128 25.578 1.038 1.020 0.910

13.596

12.754

Iv=l ivy] IV~ol

[Vii i

IV,21

]v:3l ]VI4 [ IV~i IV2[

Iv, I

1.010

IV4[

1.070

1.125

OPL

~PL c~Qi

(~[ViJ

~Qi ~lVt •]

i = 2. . . . .

Any change in the reference reactive power injections at and hence the reactive power buses connected to the slack ~PL

~

OPL //

0Q= "~

E ~PL + --

ng

(A2)

bus voltage will effect the all other generator buses injection errors at all load bus, thus

~PL ~Q2

E oIv, + OQ,=

(A3)

where c~ E 1 refers for all load buses connected to the reference bus. OPL/Otijis the loss sensitivity with respect to the transformer tap between buses i and j. It is assumed that changing the transformer tap will not affect the injected power at buses i and j. Hence, the total loss of the system will be changed and this coefficient can be evaluated as

~t~

- #e, \

Otij / -f- ~ i i

otij /

OPL ( OPj~ gPL (_OQj~ + ~fi] \ OtfiJ + g ~ \ c~t,jJ OPL/?~Q,=+.,. is the loss sensitivity with

(A4)

respect to the reactive power of the VAR sources at bus n e + ~ and can be evaluated by the incremental loss model. The constraints related to the control and dependent variables are as follows:

Appendix

AQ~, ~< AQG, ~< AQ~,

i = 1, 2 . . . . .

In this paper the objective function for reactive power control in a power system is the minimization of the real power loss, PL which is a nonlinear function of control variables like the generator bus voltages, transformer taps, and reactive power sources. A linear approximation of the real power loss function can be shown to be [8]

alvjl= ~ Albl ~ Alv, IM

j:ng+l,...,n

= r_~lPL ~PL

APE Lolv, I olv~l

~PL

~PL

elvo=l OQ,,=+x

OPL

C3PL]

~?Q~+,,=.~'V_]

I

(A5)

where

: eo~ - Q o, AQ~., = Q~,- Q°,

AQo~

AI~I~'= I~l"-I~l

(A6)

AIV~I'°= Iv, y - I~1 At/~ ~< At~s ~< At~

At~ = ti~ - to

Alv, I

..alv~l

×

ng

(AI)

/ AI g.=l /aQ,~+.,. L At,x where ng is the number of generator buses including the

reference bus, nsh is the number of buses with VAR sources, nt~p is the number of transformers with tap changers, a n d OPL/~]Vil is the loss sensitivity with respect to the voltages of the generator buses. ~PL/~[V~I is obtained by a modification of the VAR injection at the buses, such that

From the N e w t o n - R a p h s o n power flow equation the following set of equations can be written in matrix form:

AQ

=

I I 01vl ~JLAtij _1

0Q ~Q

eQ

(A7)

In reactive power control it is assumed that Ap = 0, thus,

AQ = [J.,, Jq,][-Ai VI ~ LAt,j J

(A8)

R. Kenarangui, A. Seifi / Electric Power Systems Research 30 (1994) 47-55

~Q

Jqv- lvl

~-~\-~j ~IV[

(A9)

0Q Jq, = ~to.

-~-~-\ ~ "*/

(AlO)

rSgg s,, /

Slg S. S= |1 0

[o°

c~t-~J

where Jqv and dq, are n × n and n × F/tap matrices, respectively. Then Eq. (A8) is written as

1

o

0

l

(A21)

~A[ fg[ 1

AX = / A Q ,

[AQg] [Jl,:JglJgt~2Jv~g}1

Sgt~ Slt,~ 0

55

/

(A22)

Lato J F A Q gMax

Subscripts 1 and g represent load and generator buses, respectively. The elements of the above matrices are Jacobian submatrices with the following dimensions: Jgg: F/g × F/g

Jgl ~ F/g × F/I

Jgt: F/g × F/tap

Jig: nl × n g

Jn: F/1 X n I

Jit: nl × F/tap

~rAIVgll

LAIr,l/

Slg Sll Sltu3[Att J

/

]

(A12)

where Sgg = Jgg - Jgl(Ju) IJlg

(A13)

Sgl : Jgl (Jll) -1

(A14)

Sgl = Jgt -- Jgl(Jll) 'Jit

(A15)

Sig = -(Ju) -1Jig

(AI6)

Su = (Jn) -l

(A17)

Sit =

-- (Jn)

- IJIt

(A

18)

The submatrices Sgg, Sgl, Sgt, Slg, Sn, and Sit are of dimensions ng × %, F/g × n~h, ng × F/tap, nl × F/g, F/1 × Hsh, and n l _ F/tap, respectively. For simplicity, Eq. (A12) is written in compact form as b ~ S AX-<
(A19)

where F A Q g in

/a[v,] man b =

/AlVg[min

[Atu

|AIvgl Ma* [AQ,

(A23)

L Ate"* References

The dependent variables in Eq. (A11) can be written in terms of the control variables, the resulting matrix equation representing the sensitivity relation between the control and dependent variables in the power system: [ AQg ] = [Sgg Sg' Sgt~J]/AQ,

/AIv, I-.*

(A20)

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