Reactive power optimization using fuzzy load ... - Semantic Scholar

IEEE Transactions on Power Systems, Vol. 9, No. 2, May 1994

898

REACTIVE POWER OPTIMIZATION USING FUZZY LOAD REPRESENTATION K. H. Abdul-Rahman S . M. Shahidehpour Department of Electrical and Computer Engineering Illinois Institute of Technology Chicago, Illinois 60616

Abstract- This paper presents a mathematical formulation for the optimal voltage/reactive power control problem taking into account linguistic declaration of system load values. The fuzzy set theory which is based on the feasibility rather than the frequency of occurrence of an outcome is considered, and possibility distributions are assigned to load values and bus voltages. The objective is to minimize power losses considering various load conditions. The problem is decomposed into four subproblems via the Dantzig-Wolfe decomposition for reducing the dimensions of the problem. A second Dantzig-Wolfe decomposition divides each subproblem into several areas leading to a considerable reduction in the dimensions of subproblems. An illustrative example demonstrates the applicability of the approach. The fuzzy approach provides a global solution for the system behavior under various load conditions. Keywords- Reactive Power Optimization, Fuzzy Sets, DantzigWolfe Decomposition 1. INTRODUCTION

Due to limited transmission capabilities for accommodating additional loads, reactive power allocation has received an ever increasing attention from the electric utility industry in recent years. Any changes in the system configuration or system demand may result in higher or lower voltage profiles. In order to maintain desired levels of voltage and reactive flow under' various operating conditions and system configurations, power system operators may utilize a number of control tools such as switching var sources, changing generator voltages, and/or adjusting transformer tap settings. By an optimal adjustment of these controls, the redistribution of the reactive power would minimize transmission losses. Several methods had emerged in the literature for finding the optimal settings of control variables that will eliminate voltage violations and minimize real power losses in a power system. In the past, several approaches proposed the non-linear formulation as a solution to this problem. Others suggested the possibility of linearizing the constraints while maintaining a non-linear objective function (minimize real power losses). Recently, the Linear Programming (LP) approach has become dominant in the field for problems with separable and non-separable objective functions, with satisfactory solutions within a reasonable computation time [2-81. The LP formulation is found to be much more reliable with a faster rate of convergence than any alternative approaches. Reference [l]discusses the advantages and the drawbacks of most of the existing techniques to the reactive power optimization for the operation and planning of power systems.

A common drawback in previous approaches was based on the assumption of fixed load values in the LP formulation. Unfortunately, this is not the case in real-life situations where uncertainty in data are often encountered. If, for the data under consideration, some values are known to occur more often than 93 SM 502-5 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1993 Summer Meeting, Vancouver, B.C., Canada, July 18-22, 1993. Manuscript submitted Sept. 1, 1993; made available for printing April 14, 1993 PRINTED IN USA

others, then the nature of this uncertainty is obviously probabilistic. Thus, a random variable can be assigned to each datum and a probabilistic model would be used [9,10]. However, such a knowledge is not always available, and the pertinent information may be limited to some linguistic declarations about the data (e.g., load at bus i is approximately 15 Mvar, load at bus j is mainly an industrial type). This type of data is clearly neither deterministic nor probabilistic. The situation is encountered most often in forecasting problems where the reflection of data into the future is not a stationary phenomenon, and human decisions are involved in an environment that is always fluctuating. This type of data is said to be fuzzy and the nature of the uncertainty is described as possibilistic. The possibility theory that is based on fuzzy sets was first introduced in [ll],where the feasibility rather than the frequency of assessment of a given datum was considered. The first attempt to apply fuzzy concepts to power systems decision analysis was in [12]. Since that time, fuzzy sets have been applied to different fields of power systems [22]. Recently, fuzzy sets were used to model the soft voltage constraints and the multiple objective functions of the voltage/reactive control problem, and a standard LP was used to solve the new fuzzy based formulation [13]. An expert system approach was proposed in [14] where the approximate reasoning of voltage/reactive power control based on fuzzy sets was introduced. The heuristic controls were introduced by a set of rules, and the adaptability of rules was measured by membership functions. Reference [15] used fuzzy sets as a tool to control reactive power flow via a heuristic membership function for bus voltages. The objective was to minimize real power losses, and the problem was modified into a max-min format. Although the above formulations used fuzzy sets to reflect more realistic circumstances, they did not treat the load uncertainty and assumed fixed values for loads. Fuzzy load flow analysis incorporating load uncertainties was introduced in [16]. The outcome of the fuzzy load flow was a set of fuzzy voltages, angles, active and reactive flows and losses as well as active and reactive power generations. Reference [17] managed to use the fuzzy DC load flow model, that was developed in [16],to model real power demand uncertainties in optimizing the cost of real power generation subject to generation and line flow limits. A new approach for solving the ower flow problem with uncertain The method was based on inload values was described in terval arithmetic which was viewed as a special case of fuzzy sets. The values inside the interval were assigned a membership equal to 1, and zero membership was considered for values outside the interval.

181.

This paper presents a rigorous solution to the optimal voltage/reactive power control problem taking into account the uncertainty associated with the reactive power demand. The objective is to minimize real power losses under various loading conditions. There are two ways of solving this problem. The first one would be to try several load values within a specific range, optimize each case and prepare a set of guidelines for the optimal control. This option would be expensive and time consuming in which some essential features of the data would not be taken into consideration. A more reasonable method of analysis, which is considered in this paper, is the one that depends on fuzzy set theory for analyzing different load scenarios. It is concluded that the arbitrary reduction of fuzzy values to ordinary closed intervals may result in misleading forecasts or unclear risky decisions.

0885-8950/94/$04.00 0 1993 IEEE

899 straints which are the inequality constraints (i.e., limits on the variables), and the equality constraints (i.e., reactive power demands). These constraints are written as,

List of Symbols: n = number of system buses I = number of load buses m = number of generator buses nl = number of transmission lines in the system N A = number of areas conductance of line k between buses i and j gk M = row vector relating real power loss increments to bus voltage increments

J" = Jacobian matrix with added load effect and tap changing transformer factor U = row vector containing the simplex multipliers (dual solution) PL = real power losses Q L i = reactive load at bus i Q . - initial value of reactive load at bus i Lt Qi = injected reactive power at bus i v;= voltage magnitude at bus i v;= initial value of voltage magnitude at bus i 6i = angle of the voltage at bus i G.. - real part of the ijth element of the Y-bus matrix I1 Bij = imaginary part of the ijth element of the Y-bus matrix P ( X ) = membership function of the x variable max = maximum value of the variable min = minimum value of the variable

2. DETERMINISTIC REACTIVE POWER

The purpose of the optimal reactive power control is to improve the voltage profile and minimize system losses. This goal is achieved by proper adjustments of reactive power variables in a large power network. In the past, non-linear programming approaches were proposed, which encountered the convergence problem in evaluating system losses. In this paper, the hypersurface of the non-linear power loss function is approximated by its tangent hyper-plane at the current operating point and the LP approach to the reactive power control problem is adopted. This linear approximation is found to be valid over a small defined region which is formulated by imposing limits on the deviations of the control variables from their current values. Most of the proposed L P approaches formulated the problem by a sensitivity matrix resulting from the inversion of the Jacobian matrix, which is essentially a time consuming process and requires a large memory space. The approach implemented in this study for the fuzzy set application is based on [6] which incorporates all the variables in formulating a Jacobian matrix, and hence does not require any matrix inv&ions. The objective function in this case is represented by, nl

yz- 2VT/'jCOS(6;- 6j)]

k=l

Thus, the linearized objective function will be,

AP, =

min

APL = M

or,

. AV

+ +

There are rn n 1 constraints. The first m constraints are for reactive power sources and tap changing transformer terminals. We will refer to the matrix of reactive power injections at these buses as Q1. The next n constraints are the bus voltage constraints. The last 1 equality constraints are for loads and junction buses that are not connected to transformer terminals. We will refer to the matrix of reactive power injections at these buses as Q2. So, the linearized form of the constraints is found to be,

AQF'" 5 AQ1= JYAV 5 AQF"

Avmin

(5)

5 A V 5 AVmau

(7)

where JY and Ji are submatrices of J" which is a modified Jacobian matrix. The load and the tap changing effects are formulated in terms of voltage increments, thus their effects are implicitly considered in J". Further information regarding the derivation of this model is given in Appendix A.

UNCERTAIN LOAD CONDITIONS

The fuzzy set theory was introduced to various system engineering problems in which uncertainties were represented as intrinsic ambiguities. In Appendix B, some basic definitions of fuzzy sets, which are of direct relevance to the paper, are reviewed. 3.1 Fuzzy Load Representation

In this study, linguistic declarations of variables are translated into possibility distributions by assigning a degree of membership to each possible value of the variable. Possibility distribution refers to the mapping of a fuzzy variable on the [0,1] interval. In the power system analysis, some loads and generations are determined precisely and others are described in terms of "more or less" expressions. To model such fuzzy quantities, we use trapezoidal or triangular possibility distribution where the latter is a special case of the former. So, the fuzzy reactive load at a certain bus that would never exceed is always is higher than Q i ) .and typically falls between Q!) and represented by Fig. 1. The possibility distribution will have a value of 1 for the load values that are highly possible, and will drop as possibility diminishes. A zero possibility is assigned to the values that are rather impossible to occur, which are located beyond the two extremes.

QP,

't

[%

min

=-QL~

(4)

k = 1,...,1

3. REACTIVE POWER OPTIMIZATION WITH

OPTIMIZATION

g k [ K 2f

Qk

i = l , ...,m

j = 1,...,n

The following section will discuss the formulation of the reactive power optimization problem under uncertain load conditions described by fuzzy sets.

Vectors and matrices are denoted by boldface letters.

PI,= c

Q;'" L Qt L QTa" 5 v, 5 l y a "

I y n

QE),

-

(3)

The minimization problem is subjected to operating con-

Fig. 1 possibility distribution of reactive power load

900 subject to :

3.2 Mathematical Model 3.2.1 O p e r a t i n g constraints Based on the discussion in Appendix C concerning the relation between incremental changes in loads and bus voltages, the uncertainty in system voltages is depicted in Fig. 2 corresponding to reactive loads in Fig 1. The voltage possibility distribution signifies four break-points, i.e., V(l), V(’) ,V(3),V(4) and their possibilities. This representation of voltages concurs with our experience in power systems indicating that as the system demand decreases, the load bus voltages will increase and smaller voltages would be required at generators to maintain specific load voltages. However, if the system demand increases, the load bus voltages will decrease and higher generator voltages will be needed to raise load bus voltages to the specified values. The above discussion can be verified by a fuzzy load flow [16]. The fuzzy load flow is an alternative to our previous argument regarding the correlation between bus voltages and reactive load increments. The relations between voltage magnitudes at four break points in Fig. 2 are subjected to their minimum and maximum limits. Therefore, voltage constraints are,

v(k)

where is the vector of bus voltages for break-point IC at the current operating state. The successive solution to this problem will determine break-points of the voltage possibility distribution. A follow up power flow will determine the break-points for the possibility distribution of transmission losses, which corresponds to the possibility distribution of loads. It is conceivable that the number of the problem variables increase due to the fuzzy modeling. However, our specific problem formulation utilizes the Dantzig-Wolfe decomposition as a mathematical tool to overcome this dimensionality problem. 4. D A N T Z I G - W O L F E D E C O M P O S I T I O N

In addition, the minimization problem is subjected to the operating conditions given by (5-6) for each break-point. In this capacity, the Jacobian matrix will be evaluated for each break point.

4.1 Decomposition of t h e Original P r o b l e m It is seen in (13)-(15) that there are four different sets of variables, each describing one of the break-points. Thus, we decompose the equations into 4 subproblems which are linked together through the linking constraints (13). After adding the necessary slack variables, the fuzzy optimization problem is formulated as, M(k)AV(k) min

2

k= 1

subject to :

Fig. 2 Possibility distribution of voltage

3.2.2 Objective function The objective function for the deterministic case was given in (3), where AV was a crisp variable. In the fuzzy case, APL is a function of fuzzy voltages and is given as,

AL ?

=

M-AV

(9)

The minimization of A L ? is imposed by taking the the removal of the fuzzy objective function as discussed in Appendix B. Therefore, the objective function is,

which can be written as,

3.2.3 P r o b l e m formulation In this study, the overall formulation of the loss minimization is given as follows,

This angular structure of four diagonal blocks is solved by the Dantzig-Wolfe (DW) decomposition [21 , which is a powerful technique for reducing the dimension o the original problem. DW decomposition coordinates the alternate solutions of subproblems via a master problem that is obtained by a linear transformation. The dimension of the master problem is equal to the number of linking constraints plus the number of subproblems, while each subproblem has a dimension equal to the number of constraints in the corresponding block. The solution of the master problem generates new simplex multipliers (dual solutions) that will adjust the cost function of the subproblems. The solution of the subproblems with the adjusted objective function will provide the master problem with new columns primal solutions to enter the master basis matrix. The revise simplex metho is used to solve the master problem and either the simplex or the revised simplex methods is used to solve the subproblems. The interaction between the subproblems and the master problem is shown in Fig. 3. Each subproblem will have its objective function and the respective constraints. For instance, the formulation of the kfh subproblem is as follows,

f’

d

subject to

d

901

Master problem

Master Problem

I

Subproblem 1 Subproblem 2

1

linking Constraints for the subproblems

- Objective function

- linkingo f constraints areas

>

(

t

Fig. 3 Interaction between the master problem and the subproblems The last set of constraints are added for accuracy as we are dealing with a linearized form of non-linear equations. So, we restrict the changes in the control variables to a small range in each iteration, otherwise the LP solution will greatly deviate from that of load flow. The load flow solution after each iteration finds another operating point according to the revised settings of control variables obtained from the LP. To minimize the total number of iterations for the final solution, more than one candidate columns are introduced into the basis of the master problem in each iteration. This technique is referred to as the multiple column strategy.

Suboroblem 4

linking constraints

4-97

I

Area

4.2 Decomposition of Subproblems

.NA .. .

The objective of each subproblem is to minimize the corresponding power losses. The problem formulation of the kth subproblem was described in (17). However, each subproblem in our analysis may consist of multi areas that are linked together through transmission lines. In this situation, the reactive power control devices may be located locally as each area tries its own localized control. We can decompose multi-area systems in each subproblem by applying a second order DW decomposition. The model consists of a master problem for linking constraints associated with buses that link these areas, and N A independent blocks representing areas. The objective function for each area is to minimize area power losses. This decomposition of a large problem into subproblems, and the further decomposition of these subproblem into several small areas reduces the dimension of the problem to the level where personal computers can be used. Fig. 4 shows the first and second order decompositions, and the overall structure of the problem. 4.2.1 Enhancing the system security

The basic function of an electric power system is to provide an adequate supply of electrical energy to all its customers as economically as possible with a reasonable level of quality and continuity. In our study, we have added another feature as the final solution of bus voltages are to be within security limits. Let us assume that a certain operating state of a power system is found to be most secured according to the worst case scenario analysis, and denote the voltage at bus i corresponding to this secured operating state as vi.. In order to direct the voltage profile, we specify the degree of satisfaction with different variables for being closer to the secured operating point within their operating ranges. A fuzzy LP implemented in [15 is used to modify the objective function of each area into severa objective functions and assign a membership function to each objective. The limits on variables are fuzzified as well in favor of a more reasonable solution with minimum losses. The feasible

i

I

Fig. 4 Overall structure of the problem region is the one defined by the intersection of the constraints and the objectives. This intersection is simulated as the minimum membership for the constraints and the objectives. The problem is solved by the standard LP to maximize the intersection, which will direct the final solution towards the values of variables with higher memberships. In this study, the proposed method is used to enhance the security of the system by offering a tighter control on variables within their operating ranges.

5. ILLUSTRATIVE EXAMPLE In this section, the application of the proposed approach to a 30-bus system is presented. The system is shown in Fig. 5, and line and bus data are given in [6,15]. The permissible voltage range is 0.9 to 1.1 pu. The possibility distributions of the reactive loads are given in Table 1 which are used to find the possibility distributions of bus voltages and system losses. The solution starts by computing the elements of J" and identifying the 30-bus voltage constraints which represent the linking constraints of the subproblems. Each bus voltage is represented by inequality constraints (13). For this example, we have 20 inequalities representing (14) and 10 equalities for (15) at each value of k . The DW decomposition is applied, and the problem is decomposed into 4 subproblems and one master problem. The master problem provides the subproblems with the dual solutions, and the subproblems will feed the primal solutions to the master problem. The second DW decomposition is applied and each subproblem is divided into 3 areas. Area 1 contains buses 1 to 10, area 2 contains buses 11 to 20 and area 3 contains buses 21 to 30. Buses 4,6,8,9 and 10 link area 1 to other areas, buses 11,12,15,17 and 20 link area 2 to other areas and buses 21,22,23 and 28 link area 3 to other areas. So, each subproblem will consist of linking constraints corresponding to

For the sake of comparison, we have also included the optimized values, using the conventional methods, for minimum and maximum loads (i.e., &(I) and Qf' in Table 1). It is seen in Tables 2 and 3 that the fuzzy solution provides a smaller range of voltages and generations than that of the fixed interval corresponding to the two-extreme values, indicating that the fixed load interval leads to an overestimate of the system behavior in an uncertain environment. Table 2 Final possibility distributions for the 30 bus-system voltages (pu) corresponding to Fig. 2

Bus

v

G1 G2 3 4 G5 6 7 G8 9 10 G11 12 G13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.097 1.093 1.062 1.061 1.092 1.068 1.066 1.088 1.083 0.987 1.097 0.974 1.075 0.985 0.986 0.979 0.994 0.982 0.981 0.982 0.991 0.968 0.974 0.972 0.982 0.976 0.991 1.011 0.981 0.971

(1)

v

(2)

v

(3)

v (4)

-

loads (pu) corresponding to Fig. 1

3

4 7 10 12 14 15 16 17 18 19 20 21 23 24 26 29 30

0.000 0.010 0.010 0.020 0.050 0.100 0.010 0.020 0.050 0.070 0.010 0.015 0.010 0.025 0.005 0.010 0.030 0.050 0.005 0.010 0.020 0.030 0.000 0.005

0.050 0.010 0.040 0.005 0.000 0.005

0.100 0.015 0.050 0.010 0.005

0.015

0.015 0.025 0.110 0.027 0.080

0.017 0.030 0.020 0.070 0.015 0.050 0.010 0.110 0.020 0.070 0.020 0.010 0.025

0.020 0.030 0.150 0.030 0.089 0.020 0.035 0.025 0.080 0.020 0.060 0.015 0.150 0.025 0.090 0.025 0.020 0.035

the reactive power constraints for linking buses, and three sets of independent equations for areas 1,2 and 3. The first set con-. tains the reactive power constraints for buses 1,2,3,5 and 7, the second set of constraints represents buses 13,14,16,18 and 19, and the third set contains constraints for buses 24,25,26,27,29 and 30. In this example, we use the restricted step sizes as AVatep= .02pu ,ATstep= .05pu ,AQc = .05pu. After each iteration of the first DW decomposition, new settings for the control variables are computed. To account for the non-linearity of the loss equation, a new operating point is computed based on the updated variables. The final solution is reached after 11 iterations, and the final possibility distributions for bus voltages and reactive power generation are given in Tables 2 and 3, respectively.

1.097 1.093 1.083 1.070 1.095 1.074 1.078 1.090 1.085 0.991 1.098 0.979 1.080 0.991 0.995 0.985 1.001 0.989 0.987 0.987 1.011 0.974 0.988 0.985 0.991 0.983 0.999 1.027 0.989 0.985

1.098 1.094 1.091 1.083 1.097 1.087 1.081 1.092 1.089 1.012 1.098 0.991 1.099 1.010 1.021 0.999 1.015 0.995 0.994 0.994 1.031 0.985 1.001 0.998 1.010 0.997 1.021 1.050 0.997 0.992

Results of minimum and maximum loads

1.100 1.094 1.100 1.100 1.100 1.098 1.092 1.093 1.100 1.026 1.098 1.040 1.100 1.050 1.060 1.050 1.038 1.041 1.021 1.022 1.078 1.020 1.036 1.023 1.041 1.024 1.061 1.082 1.033 1.025

1.095 1.093 1.100 1.100 1.090 1.100 1.094 1.087 1.100 1.028 1.096 1.043 1.073 1.052 1.063 1.051 1.041 1.045 1.024 1.026 1.081 1.023 1.039 1.027 1.045 1.027 1.061 1.085 1.037 1.029

1.100 1.094 1.058 1.057 1.100 1.065 1.062 1.094 1.082 0.984 1.098 0.971 1.100 0.984 0.981 0.977 0.992 0.979 0.978 0.977 0.986 0.965 0.970 0.968 0.981 0.974 0.990 1.009 0.976 0.967

Table 3 Final possibility distributions of reactive power generation for the 30 bus-wstem Bus 1

2 5 8

11

Q:')

Q,'"

Q,(3)

Results O f minimum and maximum loads

Q,'"

-0.1943 -0.1548 -0.1314 -0.1211 0.3994 0.4312 0.4987 0.5434 0.2801 0.2998 0.3114 0.3402 0.2654 0.2898 0.3102 0.3334 0.1542 0.1699 0.1823 0.2112

-0.2000 0.3991 0.2799 0.2652 0.1540

-0.1208 0.5434 0.3402 0.3335

0.2114

The fuzzy real power loss, which corresponds to fuzzy loads, and voltages obtained in (12)-(15), has a possibility distribution as shown in Fig. 6. The optimal values obtained from the minima and maxima of the load values are included in Fig. 6 to provide a comparison between the results. In this case the fived load interval provides a wider range of system losses than that of the fuzzy model, indicating that fixed load intervals result in overestimated forecasts and may lead to a higher system operation cost.

TPP,

4.86 4.89

5.1 1

6.61

7.32 7.39

t w o e x t r e m e cases Fig. 6 Possibility distribution of real power losses for the 30 bus-system

903 6. CONCLUSION

Uncertainty in data is often encountered as we deal with real-life situations. In majority of cases, as the available data are insufficient for the solution of the problem, the linguistic declaration may be used to describe the validity of the data. The uncertainty in this kind of modeling is based on intrinsic ambiguity rather than the frequency of assessment of the data under consideration. Such uncertainty is suitably modelled via fuzzy sets. Forecasting future demands is one area where this kind of uncertainty can be encountered. The proposed formulation of the optimal voltage/reactive power control problem indicated that the fuzzy modeling of loads would enable power system operators to operate the system more economically in an uncertain environment. It should be emphasized that the fuzzy set modeling is not a replacement for the probabilistic approach. Each model is used to describe a different type of uncertainty, and it may even be possible to incorporate both models in one approach. The ability of fuzzy sets to discriminate between different values of variables inside a given range via the variable membership is an important feature of the proposed approach that cannot be obtained in the standard LP. This feature can be best utilized for controlling the behavior of power system variables in the optimization process. Initial steps towards this direction were reported in [13-151. Finally, fuzzy set theory has merits for incorporating heuristics and powerful numerical methods under one framework for large scale applications.

REFERENCES

L.A Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, pp. 338-353, 1965. S. B. Dhar, ‘LPowerSystem Long-Range Decision Analysis Under Fuzzy Environment,” IEEE Trans. on Power Apparatus & Systems, Vol. PAS-98, No. 2, pp. 585-596, Mar. 1979.

1131 K. Tomsovic, “A Fuzzy Linear Programming Approach to the Reactive Power/Voltage Control Problem,” IEEE Trans. on Power Systems, Vol. 7, No. 1, pp. 287-293, Feb. 1992. R. Yokoyama, T. Niimura and Y. Nakanishi, “ A Coordinated Control of Voltage and Reactive Power by Heuristic Modeling and Approximate Reasoning,’’ Paper # 92 WM 114-9 PWRS, Presented at the IEEE/PES 1992 Winter Meeting, New York, New York, Jan. 1992.

K. H. Abdul-Rahman and S. M. Shahidehpour, “A Fuzzy -

Based Optimal Reactive Power Control,” Paper # 92 SM 402-8 PWRS, Presented at the IEEE/PES 1992 Summer Meeting, Seattle, WA, July 1992. V. Miranda, M. Matos and J. T. Saraiva, L‘Fuzzy Load Flow - New Algorithms Incorporating Uncertain Generation and Load Representation”, Proceedings of the 10th PSCC, Graz, Austria, pp. 621-627, Aug. 1990. 1171 V. Miranda and J.T. Saraiva, “Fuzzy Modeling of Power System Optimal Load Flow,” IEEE Trans. on Power Systems, Vol. 7, No. 2, pp. 843-849, May 1992.

S.M. Shahidehpour and N. Deeb, L‘AnOverview of the Reactive Power Allocation in Electric Power Systems,” Electric Machines and Power Systems, Vol. 18, pp. 495-518, 1990.

Z. Wang and F. L. Alvarado, “Interval Arithmetic in Power Flow Analysis,” IEEE Trans. on Power Systems, Vol. 7, NO. 3, pp. 1341-1349, Aug. 1992.

K. Mamandur and R. Chenoweth, “Optimal Control of Reactive Power Flow for Improvements in Voltage Profiles and for Real power Loss Minimization,” IEEE Trans. on Power Apparatus & Systems, Vol. PAS-100, No. 7, pp. 31853194, July 1981.

H. J. Zimmermann, “Fuzzy Sets, Decision Making, and Expert Systems,” Kluwer Academic Publishers, Boston, 1987.

J. Qiu and S.M. Shahidehpour, “A New Approach for Minimizing Power Losses and Improving Voltage Profile,” IEEE Trans. on Power Systems, Vol. 2, No. 2, pp. 287-295, May 1987.

0. Alsac, J. Bright, M. Prais and B. Stott, Vurther Developments in LP-Based Optimal Power Flow,” IEEE Trans. on Power Systems, Vol. 5, No. 3, pp. 697-711, Aug. 1990. Walter L. Snyder, Jr., ‘LLinearProgramming Adapted for Optimal Power Flow,” IEEE Tutorial # 90EH0328-5-PWR, pp. 20-36, 1990.

N. Deeb and S.M.Shahidehpour, “An Efficient Technique for Reactive Power Dispatch Using a Revised Linear Programming Approach,” Electric Power Systems Research Journal, Vol. 15, pp. 121-134,1988.

N. Deeb and S.M. Shahidehpour, L‘LinearReactive Power Optimization in a Large Power Network Using The Decomposition Approach,” IEEE Trans. on Power Systems, Vol. 5 , No. 2, pp. 428438, May 1990.

A. Kaufmann, M. M. Gupta, “Fuzzy Mathematical Models in Engineering and Management Science,” North-Holland Publishing Company, Amsterdam, 1988. G. B. Dantzig and P. Wolfe, L‘TheDecomposition Algorithm for Linear Programs,” Econornetrica, Vol. 29, No. 4, pp. 767-778, Oct. 1961. [22] S.M. Shahidehpour, “A Fuzzy Set Approach to Heuristic Power Generation Scheduling with Uncertain Data,” Proceedings of 1991 NSF/EEI Workshop, Norman, OK., July 1991.

APPENDIX A For buses i and j and a tap changing transformer in between, the Jacobian (J) elements are given by (A.1). These elements are derived assuming that the bus phase angles are fixed in each optimization iteration, which disregard the coupling between variables. The elements of J are modified to take into account the effect of voltage increments on reactive loads. Assuming that the load is at bus i, the modification is given in (A.2). The effect of tap changing transformers is added in (A.3) and AA), to form the modified Jacobian (J“) elements. The detai s of formulation are given in [6].

I

N. Deeb and S.M. Shahidehpour, “A Decomposition Approach for Minimizing Real Power Losses in Power SYStems,” IEE Proceedings, Part C, Vol. 138, No. 1, pp. 27-38, Jan. 1991.

J. F. Dopazo, 0. A. Klitin and A. M. Sasson, “Stochastic Load Flow,” IEEE Trans. on Power Apparatus & Systems, Vol. PAS-94, No. 2, pp. 299-309, Mar./April 1975. [lo] A. P. Meliopoulos, G. J. Cokkinides and X. Y. Chao, ‘LA New Probabilistic Power Flow Analysis Method,” IEEE Trans. on Power Systems, Vol. 5, No. 1, pp. 182-190, Feb. 1990.

J,!j=J,j

i , j = 1 , ..., n, i # j )

904

where qi is 0 for constant power load, 1 for constant current load and 2 for constant impedance load. If the injected power at bus i is changed by AQ,, due to a capacitor switching or load changes, the corresponding load bus voltage increments will be found by linearizing (C.l) and (C.2) with respect to load voltages, as follows,

1'

i, j = rn + 1,..., n or in the matrix form,

where,

D=-+ aQ-

~ Q L

avL avL

APPENDIX B The idea of fuzzifying a variable is to replace the concept that a variable has a precise value by the fuzzy concept indicating that a variable has a degree of membership assigned to each possible value of the variable. In this paper the term possibility distribution, [19], refers to the mapping of a set X to [0,1]. We will represent the possibility distribution by its break-point in Fig. B1. values (i.e., Z ~ , Z ~ , Q . , Zas~ shown ), The minimization of a fuzzy variable X , given in Fig. B1, is translated into the minimization of its removal from the px axis. The concept of removal in fuzzy sets, can be interpreted as the distance of the fuzzy variable from the pz axis. It is proved in [20] that the distance is given as,

where A1 and A2 are given in Fig. B1. For the fuzzy variable

X in Fig. B1, the distance given by (B.l) is computed as, 21

+ 322 + + z3

24

4

39.

2 = -2XBii

av;

n

+

[Gij sin(& - Sj) j=1

and

A quick procedure to check the signs of equations (C.7)-(C.9) is to assume unity voltage magnitudes and a small difference between their angles. If we assume all Gij are equal to zero, we can write the following,

aQi = -Bij

av,

50

(C.10)

(B.2)

where (B.2) is the same as (9) for the minimization of losses. It should be emphasized that no matter how many points we used to describe the possibility distribution in Fig. B1, the distance from the pz axis will always be given as (B.2).

T px

Fig. B1 Trapezoidal possibility distribution of a fuzzy variable

Thus, D has positive diagonal elements that are equal to the negative s u m of its off-diagonal elements. So, matrix D is a Mmatrix and by definition D-' is a non-negative matrix, which suggests that if the incremental reactive power injected into a load bus (AQ,,) is positive due to aload reduction, then voltages at load buses will increase and AVL will be positive. On the other hand, if the injected reactive power decreases due to a load increase, then the load bus voltages will decrease. For generator buses, we can easily verify that an increase in the injected power will cause the generator voltages to decrease and vice versa.

APPENDIX C

BIOGRAPHIES

Consider a power system with m generation buses and 1 load buses summing up to a total of 12 buses. The reactive power injection at bus i is given as,

K.H. Abdul-Rahman was born on June 2nd, 1964 in Jor-

n

Qi =

c y [G;jsin(Si - Si) - Bijcos(6i - Sj)]

(C.l)

j=1

The reactive power demands are, in general, voltage dependent. Thus, the reactive load at bus i is given as,

dan. He received his BS and MS in Electrical Engineering from Kuwait University in 1986 and 1990, respectively. He is currently a Ph.D. student in the ECE Dept. at IIT. His research interests include optimization and control of power system.

S.M. Shahidehpour received his Ph.D. in electrical engineering from the University of Missouri-Columbiain 1981. He has been with the IIT since 1983, where he is currently a professor in the ECE Department. He has published over 120 papers on power systems planning and operation. He was the vicechairman for the 1992 IEEE International Conference on Systems, Man and Cybernetics, and serves as the associate director of the American Power Conference.

905

Discussion T. Niimura (University of British Columbia, Vancouver, Canada): The authors are to be commended for an interesting paper. I would like to supplement the work by raising two possible improvements for hrther development of firzzy set’s applications: (1) One of the advantages to apply f i z z y sets for voltage-reactive power control modeling is the explanation of state by defining linguistic terms such as “high voltage” corresponding to numerical values [Dl]. For example, in Fig. 1 the authors define the possible reactive power by “always higher than QL(’)’’ and so on. However, the possibility distribution of voltage (Fig. 2) is just given by four “break-points”. To facilitate the comprehension they could be correspondingly referred to such that voltage V will “never be lower than V(1) but always lower than V(4) “ and “most possibly between V(z)and V(3).” (2) Another advantage of fizzy sets’ application is $he flexibility to conform with changing power system states. However, it is a sort of perpetual concern how the re-definition of the fizzy sets ~ Q L and pv or “the break-points” affects the overall solution. Due to the authors’ modeling, based on linear membership finctions and LP frame work, it seems relatively straight-forward to introduce sensitivity or marginal cost of the adjustment in parameters for the change of solution. I would appreciate the authors’ comments or insight for the above discussions. [Dl] R. Yokoyama, T. Niimura, and Y. Nakanishi, “A Coordinated Control of Voltage and Reactive Power by Heuristic Modeling and Approximate Reasoning,” IEEE Trans. PWRS, Vol. 8, “0.2, pp. 636-645, May 1993. Manuscript received August 10, 1993.

K.H.Abdul-Rahman a n d S.M.Shahidehpour- We would like to express our appreciation to Dr. Niimura for his interest in our paper and his valuable comments. In the following, we will provide our response to his comments. (1) The voltage possibility distribution as seen in Fig. 2 indicates that voltage is always higher than V(’)and lower than V(4) and most likely to be between V ( 2 )and V ( 3 ) .This interpretation of voltage possibility distribution is similar to that of the load given in Fig. 1. However, the break points of the voltage possibility distributions are determined according to the given load possibility distribution. Hence the range of voltage values or the break points in Fig. 2 may vary between V”’” and V“””. The importance of the fuzzy set approach is that it brings forth the notion of imprecision that is overlooked in conventional approaches. (2) We agree with the discusser that artificial costs of changes in the solution for variations in parameters may be introduced to find the flexibility of the approach. Alternatively, the optimization problem can be formulated as a parametric LP to study the effect of varying the break points. As more complex and additional constraints are imposed on the system operation, there will be a need for enhancing the existing technology to reduce the computation time, meet the increasing complexity and provide proper options for the operation of power systems. In [C2], advances in AI technology are utilized for the solution of the var control problem. The approach is based on modeling load uncertainty by fuzzy sets, which is given as input to artificial neural network to identify the closest control solution corresponding to uncertain loads. Expert system is used to assure the feasibility of the solution. AbduI-Rahman, S.M.Shahidehpour, “Application of artificial intelligence to optimal var control in electric power systems,” in Proceedings of Expert System Applications for the Electric Power Industry: International Conference and Exhibition, Phoenix, AZ, Dec. 1993

[CZ]K.H.

Manuscript received October 11, 1993.