Genetic Algorithms as a Reactive Power Source Dispatching Aid for Voltage Security Enhancement

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 25, No. 1, pp Genetic Algorithms as a Reactive Power Source Dispatching Aid for Voltage Security Enhancement CHIH-WEN LIU *, CHEN-SUNG CHANG *, AND JOE-AIR JIANG ** * Department of Electrical Engineering National Taiwan University Taipei, Taiwan, R.O.C. ** Department of Electrical Engineering Kuang-Wu Institute of Technology and Commerce Taipei, Taiwan, R.O.C. (Received March 13, 2000; Accepted May 12, 2000) ABSTRACT This paper presents an efficient computing algorithm for enhancing voltage security. The algorithm uses the genetic algorithm (GA) to dispatch reactive power sources under various system conditions. GA aids the dispatching of reactive power sources so as to maintain the specified security level. The reactive power sources used in the proposed genetic algorithm are transformer tap changers, static capacitors, static Var compensators (SVCs) and generator terminal voltages. The proposed method has been tested on an IEEE 30-bus power system, and successful results have been obtained. Key Words: voltage security indices, voltage security enhancement, genetic algorithms, phasor measurement unit I. Introduction In recent years, an instability problem, usually termed voltage instability, has been observed and has been found to be responsible for several major network collapses in different countries (IEEE Publication, 1990; Taylor, 1994). This phenomenon did not always occur in response to a contingency, such as the loss of an important transmission line or a generator, but rather in response to an unexpected rise in the load level, sometimes in combination with inadequate reactive power support at critical network buses. A large amount of research has focused on voltage instability or voltage collapse. Indeed, many researchers have proposed voltage security margins which show how close the current operating point of a power system is to the voltage collapse point (Schlueter, 1998; Canizares and Alvarado, 1993; Gao et al., 1992; Chebbo et al., 1992a; Löf et al., 1992; Van Cutsem, 1991), which serves as an assessment of voltage security. In addition to the identification of voltage security, system operators are also interested in knowing how much and where external reactive support is needed and located with regard to both security and economics. Reactive power can be dispatched effectively to maintain acceptable voltage levels throughout the system and to reduce the overall real power loss of the system. Therefore, many researchers have pointed out that the security margin of power systems can be enlarged to reduce the possibility of voltage collapse by providing a sufficient amount of reactive power (Van Cutsem, 1991; Ajjarapu et al., 1994; Chang and Huang, 1998; Begovic and Phadke, 1992; Nanba et al., 1998; Chebbo et al., 1992b). Specifically, Van Cutsem (1991) used the solution of a reactive power optimization problem as the voltage security margins. Ajjarapu et al. (1994) introduced a method for determining the minimum amount of shunt reactive power support needed, which indirectly imizes real power transfer before voltage collapse is encountered. Chang and Huang (1998) used optimal multi-objective static Var compensator planning to enhance voltage stability. Begovic and Phadke (1992) used sensitivity analysis to control voltage security. Nanba et al. (1998) proposed a control method for improving voltage security based on the concept of the voltage instability proximity index. In addition, Chebbo et al. (1992b) described a linear reactive power dispatch algorithm to reduce the risk of voltage collapse. The system operators usually apply proper operations to controllers to enhance voltage security when the power network has voltage instability or shows vulnerability to voltage collapse. Due to mismatch problems between the searched optimal or near optimal control decisions and practicable dispatches under these limitations, such as placement, expenditure, supplementary instruments etc., it is necessary that multi-solutions be selected by operators to enhance voltage security. If we use the traditional optimal approaches, which are the linear (Hegdt and Grady, 1983), nonlinear (Billiington and Sachdev, 1983), mixed integer programming (Aoki et al., 1988), and decomposition (Mangooli et al., 1993) methods, 53

2 C.W. Liu et al. to obtain optimal solutions, then the only optimal solution is found, and dispatch selection is limited. Furthermore, some of these conventional optimal methods, such as the Newton method, Broyden s method, and the gradient descent method, are classified as greedy search techniques and often get stuck at the local optimum rather than at the global optimum. Recently, the stochastic optimization technique, known as simulated annealing (SA), has been applied to many power system constrained optimization problems (Chang and Huang, 1998; Hsiao et al., 1993). However, SA uses much CPU time to find the global optimum. Therefore, SA is not suitable for on-line use. Genetic Algorithms (GAs) are stochastic searching algorithms that employ models of deoxyribonucleic acid (DNA) selection observed in nature. GAs have been applied to many search and optimization applications (Goldberg, 1989; Lee et al., 1995; Chen and Chang, 1995; Syswerda, 1989) due to their flexibility and efficiency. The advantage of GAs is their use of stochastic operators instead of deterministic rules to search for fitness solutions. The searching process jumps randomly from point to point, thus allowing escape from the local optimum, in which other conventional optimization algorithms might land, and it searches for many sub-optimum points in parallel. Since GAs can provide many sub-optimal dispatch solutions, they enable the operator to enhance voltage security in a flexible and practical manner. In this paper, we propose a voltage security enhancement (VSE) scheme that consists of a security monitoring level and a dispatching level, which are able to raise the voltage security level by means of reactive power source dispatch. The genetic algorithm is proposed in this paper to aid dispatching of reactive power sources. Voltage security monitoring by means of fuzzy hyper-rectangular composite neural networks (FHRCNNs) has been developed by Liu et al. (1998). Therefore, in this paper, we focus on reactive power dispatch using a genetic algorithm to enhance voltage security. The VSE scheme is shown in Fig. 1. This paper is organized as follows. Section II explains the overall VSE scheme. The control decisions generated by the genetic algorithm are described in Section III. In Section IV, we present simulation of the proposed scheme on the IEEE 30-bus system. Finally, summary discussion and a conclusion are given in Section V. II. Voltage Security Enhancement Scheme Voltage security indices are alternative techniques for the detection of collapse and insecurity points. These indices are scalar variables that are continuously monitored to determine how close a system is to voltage collapse. Simple examples of voltage security indices are the minimum singular value and real eigenvalue of the power flow Jacobian matrix: the closer these values are to zero, the closer the system is to collapse. In this paper, we use five fuzzy linguistic variables (i.e., Very Small (VS), Small (S), Medium (M), Large (L) and Fig. 1. The VSE scheme. Very Large (VL)) to specify the degree of the voltage security margin of a power system. To quantify these linguistic variables, an existing voltage security index is used to describe the relative severity of each variable. Also, a numerical secure value (NSV) is assigned to each of the five possibility levels: the secure value for VL is 5, the secure value for L is 4, the secure value for M is 3, the secure value for S is 2, and the secure value for VS is 1. The operating conditions of a power system are characterized by a number, n, of variables; therefore, each operating condition can be represented by an input vector of patterns in the n-dimensional pattern space R n. In this paper, we let x = [x 1, x 2,..., x n ], which is acquired based on synchronized phasor measurement units (PMUs), be an input vector of the FHRCNNs. The magnitude of an output variable is employed to indicate the level of voltage security, and power system voltage security is divided into 5 levels according to the range of the values of the voltage security indices. Take the minimum singular value, s n, as an example: we have a very secure level (s n VL), secure level (s n L), alert level (s n M), dangerous level (s n S), and very dangerous level (s n VS). After sufficient training, the synaptic of a trained FHRCNN with hidden nodes can be utilized to extract the classification knowledge, which is then represented by a set of IF-THEN rules for voltage security monitoring, such as the following fuzzy rules: IF (x HR 5 VL), THEN voltage security is at the very secure level. IF (x HR 4 L), THEN voltage security is at the secure level. IF (x HR 3 M), THEN voltage security is at the alert level. IF (x HR 2 S), THEN voltage security is at the dangerous level. 54

3 Power System Voltage Security IF (x HR 1 VS), THEN voltage security is at the very dangerous level. Here, HR j represents an n-dimensional hyperrectangle defined by m j1, M j1... m jn, M jn in R n, j = 1, 2,..., 5, where m ij, M ij are parameters determined after sufficient training. A detailed description of voltage security monitoring by means of FHRCNN was given by Liu et al. (1998). If voltage security is neither very secure nor secure, then the voltage security enhancement algorithm is triggered. The voltage security enhancement strategy is formulated as the following search problem: {NSV} 4 (1) x,u Γ subject to L(x, u) = 0, (2) G(x, u) 0, (3) where NSV is the numerical secure value and is chosen as the objective function, x denotes the vector of state variables, and u denotes the vector of control variables, which are the voltage magnitudes of all the generators, V g, the ratios of the transformer tap changers, T, and the switched capacitor/reactor settings, λ. Γ is the feasible set, which is determined by means of power flow constraints L( ) and inequality constraints G( ). The inequality constraints belong to the following categories: 1. Control Variable Constraints The control variables constraints include V g min V g V g T t min T t T t 0 q ck q ck 0 q lk q lk, g N g, (4) t N T, (5) k Ω L, (6) where N g is the set of generator buses and N T is the set of on-line tap changer (OLTC) transformers; q ck and q lk are the additional capacitive and inductive compensations at bus k; and Ω L is the set of all candidate buses for VAR support. 2. Operating Constraints The system operating constraints consist of the available range of active and reactive generated power, bounds on voltage magnitudes, phase angle difference limits, line flow limits etc., which are shown as Eqs. (7) (10): S g min S g S g g N g, (7) V i min V i V i θ i θ j θ i,j P 2 ij + Q 2 2 ij S ij, i N, (8) i, j N, (9) i, j N, (10) where N is the set of the numbers of total buses. 3. Control Decisions Generated by Genetic Algorithms The control strategy for VSE is formulated using the GA to maintain the specified security level. The controllers in the proposed genetic algorithm are transformer tap changers, static capacitors, SVCs and generator terminal voltages. The reason why we use genetic algorithms to make control decisions is that genetic algorithms are different from conventional optimization and search procedures in four ways: (1) GAs work with a coding of the parameter set, not the parameters themselves. (2) GAs search from a population of points, not a single point. The computation for each individual in the population is independent of those for others. Therefore, GAs have inherent parallel computation ability. (3) GAs only use fitness or objective function information directly as a basis for the searching direction, not differentiation or other auxiliary knowledge. Gas, therefore, can deal with non-continuous, non-smooth and non-differentiable functions. Gas offer an alternative to other traditional methods for solving realworld search and optimization problems. (4) GAs use probabilistic transition rules, not deterministic rules, so they can search a complicated and uncertain area to find global optimal or near-optimal solutions. Therefore, GAs are more flexible and robust than conventional optimization and search procedures. In this section, Eqs. (1) (3) are solved to obtain the control vector u using a genetic algorithm. In practical applications, the variables of problems are encoded into a finite string corresponding to chromosomes of biological systems. Also, each string (chromosome) represents a possible solution to the problem being optimized, and these solutions are classified based on a fitness function, better fitness, corresponding to better solutions. Generation of the optimal control vector by a GA is explained in the following: A. The Representation of Strings Each string (chromosome) represents a possible voltage security enhancement strategy, and each element (gene) represents a value of control variables. Therefore, the numerical string in this paper is that shown in Fig. 2. The length of the 55

4 C.W. Liu et al. V1 V2... Vg T1 T2... Tt λ 1 λ 2... λ k string is equal to the total number of control variables. Usually, the value of an element in the string is represented by a binary bit. However, this paper proposes the use of a real number string. B. Fitness Function In a genetic algorithm, a fitness function is a mapping which determines the fitness of each string in the population. The GA proceeds to evolve better-fitting strings, and the fitness value is the only information available to the GA. The strings with large fitness values offer better solutions to the problem and have a higher probability of being selected. Therefore, the adopted fitness function of the GA methodology for VSE is a measure of how close a specific operating point is to the point of voltage instability. We will use a voltage security index as a fitness function as an example. We will consider the power flow equation and use the strong coupling relations between the reactive power and voltage magnitude in a stressed power system; then the relation between the changes in reactive injection and voltage magnitude can be described as Q = J s V, where J s = J 4 J 3 J 1 1 J 2. In addition, the singular value decomposition (SVD) is applied to the matrix J s. That is, J s = LSR T = n Σ L i S i R i T i =1 numerical string Fig. 2. Representation of a numerical string of a GA., where L and R are n n orthonormal matrices, S is a diagonal matrix, and S i is a singular value of matrix J s with the order s 1 s 2 s 3... s n 0. We know from Löf et al. (1992) that the minimum singular value, S n, of J s can be used as a voltage security margin measure; that is, the smaller the singular value is, the closer to voltage collapse the operating point is. Thus, the minimum singular value, S n, is an indicator of proximity to the voltage security limit. Thus, we can take S n as the fitness function of GA. Since the GA is basically an unconstrained search procedure in the original problem domain, to incorporate the constraints into the objective function, any string that violates the constraints is penalized by setting its fitness value to zero. In this manner, the GA implicitly becomes a constrained search procedure. to the rest of the population. Stings with higher fitness values (S n ) have a higher probability of contributing offspring and are simply copied into the next generation. In this paper, we employ an elitist policy ; that is, the sting with the highest fitness value in the population is directly copied into the next generation. (2) The crossover operator recombines the extremely important features of two strings to make offspring (child) strings. Not only do they inherit some important characteristics from their parent strings, but they also have a chance of getting closer to the optimal decisions. Crossover is performed on two strings that are selected from the population randomly at one time, its frequency being controlled by a crossover probability, P c. Crossover can occur at a single position (single crossover) or at a number of different positions (multiple crossover). In this paper, we adopt a uniform crossover technique, which exchanges elements between the two selected parent strings to create new offspring strings by means of a random mask. In this manner, the elements are swapped when the value of the random mask is 1. On the contrary, the elements remain unchanged when the value of the random mask is 0. Figure 3 illustrates such a uniform crossover technique. Uniform crossover was first presented by Syswerda (1989), and its main advantage is that the convergence speed is faster than that in one-point crossover or two-point crossover. (3) The mutation operator is a mean used to avoid losing important information at a particular position in the decisions. The mutation operator with small mutation probability, P m, is applied to all the elements (genes) that are in each offspring (child). The mutation operator of a GA for VSE is the element added or subtracted by one step (scale) of regulating devices, such as the load ratio of OLTCs, the excitation control of generators or VAR source installation whenever a mutation occurs, as shown in Fig. 4. Therefore, we can escape from the local optimal solution and search for the C. The Three Genetic Operators of a GA (1) The reproduction operator is a probabilistic selection process in which strings are selected so as to produce offspring (child) based on their fitness value (S n ). A conventional method for this is called roulette wheel selection and was described in detail by Goldberg (1989). This ensures that the expected number of a string will be in proportion to the string s fitness relative Fig. 3. The uniform crossover operator. 56

5 Power System Voltage Security optimal or near-optimal control decisions of VSE using mutation skills. After mutation, the new generation is complete, and the procedure begins again with fitness evaluation of the population. IV. Numerical Results The IEEE 30-bus system with PMUs (shown in Fig. 5) was used to test the effectiveness of the proposed algorithm for VSE. The test system used in this study has six generation buses (buses 1, 2, 5, 8, 11 and 13), 21 load buses, 4 OLTC transformers (branches T1(6, 9), T2(6, 10), T3(4, 12) and T4 (28, 27)) and 37 transmission lines. The test system, system parameters and initial buses data were discussed in detail by Freris and Sasson (1968). Fast calculations of the minimum singular value and corresponding singular vector based on Löf et al. (1992) were adopted to prepare the training patterns and test patterns. One Fig. 4. The mutation operator. Fig. 5. IEEE 30-bus system with 10 PMUs. Fig. 6. The voltage security index for the IEEE 30-bus system. result of fast calculations of the minimum singular value (voltage security index) from light load to critical load is shown in Fig. 6. In Fig. 6, the jumps in the numerical value of the voltage security index occur because PV-buses change into PQ-buses when limits for reactive power generation are hit, reflecting the importance of reactive power resources, which if inadequate can contribute to voltage security problems in power systems. The simulation program was developed on a SUN workstation using C ++. Ten installed PMUs were simulated on buses 30, 26, 29, 25, 27, 24, 23, 19, 18 and 20 based on weak bus ranking of the test system under heavy load conditions (shown in Table 1). In this paper, the voltage security levels are classified into 5 levels according to the magnitude of the minimum singular value, S n : very secure level (S n ), secure level ( S n ), alert level ( S n ), dangerous level ( S n ), and very dangerous level ( S n 0.0). The ranges of the voltage security index values that are dependent on the studied power system under various operating conditions are statistically determined. It should be emphasized that the range of the magnitude of S n for each of the security levels selected above is set so as to ease presentation of the numerical tests. Four tested cases were considered in VSE under heavy load and credible contingency conditions. In these tests, the bus voltages were selected as the state variables, and the transformer tap setting T, generator bus voltage and 5 possible VAR source installations of the weakest buses were selected as the control variables. Therefore, the control variables strings of the GA are p i = [V gi, T i, λ i ], i = 1, 2,..., m, where m is the population size. The fitness function is the voltage security index, S n. The mutation vector is M = V g, T, λ. The initial settings and the regulating scales of different controllers are shown in Table 2. The setting limits of the controllers and the operating limits of the tested system are given in Table 3. In Table 3, we list the limits of the controllers and bus voltage magnitudes for the four cases. The simulation parameters in the GA for different cases are given in Table 4. In practical applications, a small mutation probability can only result in premature convergence while a search using a large mutation probability will not converge. 57

6 C.W. Liu et al. Table 1. Weak Bus Ranking under Heavy Load Conditions for the IEEE 30-Bus System Rank a r i index load demand voltage angle Bus r i MW MVAR PMU Note: +, Bus in which a PMU is to be installed. a The voltage phasors of buses ranked 1 0, which are obtained from PMUs, are the input patterns of FHRCNNs. Table 2. The Initial Settings of Controllers Capacitor / Reactor settings ( λ = 0.04 a (p.u.)) VAR λ1 λ2 λ3 λ4 λ5 initial Transformer taps settings ( T = 0.015) OLTC T1 T2 T3 T4 initial Generator bus voltage settings ( V g = 0.01) Bus Vg (case 1 & 2) Vg (case 3) $ b 0.95 Vg (case 4) $ b $ b a VAR source settings in p.u. on a 100 MVA base. b outage contingency condition. An adaptive mutation probability was given for different voltage security levels. However, the different results were V g Table 3. The Limits of Controllers and Bus Voltages V g min Voltage and Tap setting limits Table 4. Simulation Parameters in GA parameter case Maximum generation number Population size, m Crossover probability, p c Mutation probability, p m obtained in different runs using the genetic algorithm. Thus, these results of the GA were the multi-selection and nearoptimum solutions, shown in Table 5. In Table 5, we only list three near-optimum solutions as control decision selections (CDS) for VSE in every case. The best fitness in each generation was calculated. For example, the searching process for case 3 is shown in Fig. 7. In the figure, it is seen that the best fitness of these generations clearly increased, and that the oscillatory phenomenon occurred because the voltage security index had a large jump (refer to Fig. 6) at the alert level. Case 1. The test system was investigated under heavy load conditions, and the system was operating at the alert level. The load demand and power supply were: P load + Q load = MW MVAR, P g + Q g = MW MVAR. T T min Capacitor / Reactor limits λ c min λ c (p.u.) One bank capability (p.u.) Generator bus imum MVAR capability limits Bus No MVAR Low voltage limits Bus No V (case 1 & 2) V (case 3) V (case 4) Bus No V(case 1 & 2) V (case 3) $ a $ a $ a 0.79 V (case 4) Bus No V(case 1 & 2) V (case 3) V (case 4) a outage contingency condition 58

7 Power System Voltage Security Table 5. The Results of VSE Using GA on the IEEE 30-Bus System Objective function Fitness function min. voltage Controller Numerical secure value Min. singular value Load bus voltage Generator bus voltage OLTC setting Capacitor/Reactor setting Cass 1 NSV S n Bus #30 Bus # T1 T2 T3 T4 Bus # Before CDS CDS CDS Cass 2 NSV S n Bus #30 Bus # T1 T2 T3 T4 Bus # Before CDS CDS CDS Case 3 NSV S n Bus #30 Bus # T1 T2 T3 T4 Bus # Before CDS CDS CDS Case 4 NSV S n Bus #30 Bus # T1 T2 T3 T4 Bus # Before CDS CDS CDS

8 C.W. Liu et al. Table 6. Comparison between the Stochastic Approach and Deterministic Approach Stochastic approach Deterministic approach performance GA SA Newton method Broyden s method accuracy ** ** * * flexibility *** ** * * efficiency ** * *** ** stability *** *** * ** simplicity *** * *** *** Notes: ***distinguished, **mediocre, *disappointing The controller settings and network states are shown in Table 5. After the GA process finished, the objective function (NSV) improved from 3 (alert level) to 4 (secure level); the fitness function (S n (J s )) increased from to (CDS1), (CDS2) or (CDS3); and the lowest voltage of the load bus also increased from to (CDS1), (CDS2) or (CDS3). Case 2. The test system was investigated under heavy load conditions and was operating at the alert level. The load demand was the same as in Case 1, but we did not use weak buses as candidate buses for switching to new capacitors/reactors. After the GA process finished, the objective function (NSV) improved from 3 (alert level) to 4 (secure level); the fitness function (S n (J s )) increased from to (CDS1), (CDS2) or (CDS3); and the lowest voltage of the load bus increased from to (CDS1), (CDS2) or (CDS3). Table 5 compares the results of Cases 1 and 2. Case 3. The test system was investigated under heavy load conditions, and generator #11 and line #18-21 were studied under outage contingency conditions. The system was operating at the dangerous level. The load demand and power supply were: P load + Q load = MW MVAR, P g + Q g = MW MVAR. After the GA process finished, the objective function (NSV) improved from 2 (dangerous level) to 4 (secure level); the fitness function (S n (J s )) increased from to (CDS1), (CDS2) or (CDS3); and the lowest voltage of the load bus increased from to (CDS1), (CDS2) or (CDS3). Therefore, in this case, an improvement in the voltage security level from the dangerous level to the secure level was obtained. Case 4. The test system was investigated under peak load, and generators #5 and #13 were studied under outage contingency conditions. The system was operating at the very dangerous level. The system was on the verge of collapse. The load demand and power supply were: P g + Q g = MW MVAR. After the GA process finished, the objective function (NSV) improved from 1 (very dangerous level) to 3 (alert level); the fitness function (S n (J s )) increased from to (CDS1), (CDS2) or (CDS3); and the lowest voltage of the load bus increased from to (CDS1), (CDS2) or (CDS3). Thus, in this case, an improvement in the voltage security could be obtained to prevent voltage collapse. In the above test cases, we could obtain results very close to many results of numerical tests. Therefore, these results are nearly global optimal solutions. V. Discussion and Conclusion With the increased loads on existing power transmission systems, VSE problems have become a major concern in power network operations. Most of the early works on VSE were based on determinism. The deterministic approach often cannot find optimal or even near optimal solutions, and the searching process tends to lead to divergence or instability. It is difficult to deal with non-continuous, non-differentiable and complex real-world optimization problems, and it is not easy to deal with inequality constraints or obtain algorithms for VSE. Also, only a feasible solution can usually be obtained, which may lack sufficient robustness to cope with the contingencies encountered. On the other hand, stochastic methods offer more than one optimal solution, can deal with non-differentiable constrained search and optimization procedures, and can search a complicated and uncertain area to find global optimal or near optimal solutions. In addition, the stochastic methods have been shown to be more flexible, accurate and stabile than deterministic methods (Chang and Huang, 1998; Hsiao et al., 1993; Goldberg, 1989; Lee et al., 1995; Chen and Chang, 1995; Syswerda, 1989). Therefore, the stochastic approach has become the prime candidate for constrained optimization and search problems in recent years. In particular, both the SA and GAs have been used in VAR planning applications (Chang and Huang, 1998; Hsiao et al., 1993; Lee et al., 1995), and P load + Q load = MW MVAR, Fig. 7. Searching process of a GA for Case 3. 60

9 Power System Voltage Security the SA has been shown to be able to obtain optimal VSE solutions (Chang and Huang, 1998). However, the SA has drawbacks in that a tremendous amount of execution time is needed to obtain a near global solution and its control parameters are complex and difficult to handle. GAs, on the other hand, have inherent parallel computation ability, so they can search for many optimum solutions in parallel and use fitness or objective function information directly according to the searching directions. GAs are more efficient, flexible and simple than the SA. Therefore, we find that GAs provide a powerful search technique, unequalled by any other approach in terms of performance based on accuracy, flexibility, efficiency, stability and simplicity. A comparison between stochastic and deterministic methods is shown in Table 6. To sum up, based on the numerical examples and test results given in Section IV, we can make the following observations: (1) Searching for control decisions for VSE using GAs is efficient and effective. (2) These searched control decisions for VSE are near optimal, flexible solutions. (3) Control decisions searched using GAs are accurate and stable. (4) Reactive power sources can be dispatched using GAs to enhance voltage security. (5) Dispatching of reactive power sources using GAs is easy to implement. (6) GAs have potential as efficient on-line tools for VSE. Acknowledgment We thank the National Science Council of the Republic of China for financial support under grant NSC E References Ajjarapu,V., P. L. Lau, and S. Battula (1994) An optimal reactive power planning strategy against voltage collapse. IEEE Trans. on Power Systems, 9(2), Aoki, K., M. Fan, and A. Nishikori (1988) Optimal Var Planning by approximation method for recursive mixed integer linear planning. IEEE Trans. on Power Systems, 3(4), Begovic, M. M. and A. G. Phadke (1992) Control of voltage stability using sensitivity analysis. IEEE Trans. on Power Systems, 7(4), Billiington, R. and S. S. Sachdev (1983) Optimum network Var planning by nonlinear programming. IEEE Trans. on Power Apparatus and Systems, PAS-102(5), Canizares, C. A. and F. L. Alvarado (1993) Point of collapse and continuation methods for large AC/DC systems. IEEE Trans. on Power Systems, 8(1), 1-8. Chang, C. S. and J. S. Huang (1998) Optimal multi-objective SVC planning for voltage stability enhancement. IEE Proceeding-C, 145(2), Chebbo, A. M., M. R. Irving, and M. J. H. Sterling (1992a) Voltage collapse proximity indicator: behavior and implications. IEE Proceeding-C, 139 (3), Chebbo, A. M., M. R. Irving, and M. J. H. Sterling (1992b) Reactive power dispatch incorporating voltage stability. IEE Proceeding-C, 139(3), Chen, P. K. and H. C. Chang (1995) Large-scale economic dispatch by genetic algorithm. IEEE Trans. on Power Systems, 10(4), Freris, L. L. and A. M. Sasson (1968) Investigation of the load-flow problem. IEE Proceeding-C, 115(10), Gao, B., G. K. Morison, and P. Kundur (1992) Voltage stability evaluation using Modal analysis. IEEE Trans. on Power Systems, 7(4), Goldberg, D. E. (1989) Genetic algorithms in search. In: Optimization and Machine Learning. Addison-Wesley, Reading, MA, U.S.A. Hegdt, G. T. and W. M. Grady (1983) Optimal Var siting using linear load flow formulation. IEEE Trans. on Power Apparatus and Systems, PAS- 102(5), Hsiao,Y. T., C. C. Liu, H. D. Chiang, and Y. L. Chen (1993) A new approach for optimal Var sources planning in large scale electric power systems. IEEE Trans. on Power Systems, 8(3), IEEE Publication (1990) Voltage Stability of Power Systems: Concepts, Analytical Tools, and Industry Experience. 90TH PWR, IEEE Service Center, Piscataway, NJ, U.S.A. Lee, K. Y., X. Bai, and Y. M. Park (1995) Optimization method for reactive power planning by using a modified simple genetic algorithm. IEEE Trans. on Power Systems, 10(4), Liu, C. W., C. S. Chang, and M. C. Su (1998) Neuro-fuzzy networks for voltage security monitoring based on synchronized phasor measurements. IEEE Trans. on Power Systems, 13(2), Löf, P. A., T. Semed, G. Andersson, and D. J. Hill (1992) Fast calculation of a voltage stability index. IEEE Trans. on Power Systems, 7(1), Mangooli, M. K., K. Y. Lee, and Y. M. Park (1993) Optimal long-term reactive power planning using decomposition network techniques. Electric Power Systems Research, 26, Nanba, M., Y. Huang, T. Kai, and S. Iwamoto (1998) Studies on VIPI based control methods for improving voltage stability. Electrical Power & Energy System, 20(2), Schlueter, R. A. (1998) A voltage security assessment method. IEEE Trans. on Power Systems, 13(4), Syswerda, G. (1989) Uniform crossover in genetic algorithms. Proceeding of the 3rd International Conference on Genetic Algorithms, pp. 2-9, George Mason University, Fairfax, VA, U.S.A. Taylor, C. W. (1994) Power System Voltage Stability. McGraw-Hill, Inc., New York, NY, U.S.A. Van Cutsem, T. (1991) A method to compute reactive power margins with respect to voltage collapse. IEEE Trans. on Power Systems, 6(1),

10 C.W. Liu et al. GA SVCs IEEE 30 62