Rule extraction for voltage security margin estimation

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Rule Extraction for Voltage Security Margin Estimation Mu-Chun Su, Member, IEEE, Chih-Wen Liu, Member, IEEE, and Chen-Sung Chang

Abstract—Lately, research efforts have been devoted to estimating voltage security margins which show how close the current operating point of a power system is to a voltage collapse point as assessment of voltage security. One main disadvantage of these techniques is that they require large computations, therefore, they are not efficient for on-line use in power control centers. In this paper, we propose a technique based on hyperrectangular composite neural networks (HRCNN’s) and fuzzy hyperrectangular composite neural networks (FHRCNN’s) for voltage security margin estimation. The technique provides us with much faster assessments of voltage security than conventional techniques. The two classes of HRCNN’s and FHRCNN’s integrate the paradigm of neural networks with the rule-based approach, rendering them more useful than either. The values of the network parameters, after sufficient training, can be utilized to generate crisp or fuzzy rules on the basis of preselected meaningful features. Extracted rules are helpful to explain the whole assessment procedure so the assessments are more capable of being trusted. In addition, the power system operators or corresponding experts can delete unimportant features or add some additional features to improve the performance and computational efficiency based on the evaluation of the extracted rules. The proposed technique was tested on 3000 simulated data randomly generated from operating conditions on the IEEE 30-bus system to indicate its high efficiency. Index Terms— Fuzzy systems, neural networks, phasor measurement unit, power systems, voltage security.

I. INTRODUCTION

I

N RECENT years, voltage instabilities which are responsible for several major network collapses have been reported in many countries [1], [2]. The phenomenon was in response to an unexpected raise in the load level, sometimes in combination with an inadequate reactive power support at critical network buses. With the advent of systems capable of making synchronized phasor measurements, the ultrafast assessments of voltage stability become feasible [3]. Commercial systems based on a global positioning system (GPS) can provide synchronization to 1 ms accuracy. By communicating timetagged phasor measurements to a central location, the state of the system can be tracked in real time. Many significant research efforts have been devoted to voltage security margins which show how close the current operating point of a power Manuscript received March 24, 1997; revised March 14, 1998. Abstract published on the Internet August 20, 1999. M.-C. Su is with the Department of Electrical Engineering, Tamkang University, Tamsui 25137, Taiwan, R.O.C. (e-mail: [email protected]). C.-W. Liu and C.-S. Chang are with the Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. Publisher Item Identifier S 0278-0046(99)08476-2.

system is to a voltage collapse point as assessment of voltage security. Tiranuchit proposed the minimum singular value of the Jacobian of the load flow equation as a voltage security margin [4]. The concept of multiple load flow solutions was proposed to deal with voltage stability margin problems [5], [6]. Van Cutsem used the solution of a reactive power optimization problem as the voltage security margin [7]. One main disadvantage of the above-mentioned techniques is that they require large computations, therefore, they are not appropriate tools to be used on-line in central centers, since advance warning signals are required for operators to steer a system away from a developing voltage collapse whenever possible. Neural networks provide a feasible alternative, since they can be constructed off-line from a given training set and then be used on-line in control centers [8], [9]. Basically, a neural network is a massively parallel distributed processor, which can improve its performance by adjusting its synaptic weights. A feedforward multilayer neural network with sufficient hidden nodes has been proven to be a universal approximator [10]–[13]. Although neural networks have many appealing properties, there are three main disadvantages in neural networks. The first one is that there is no systematic way to set up the topology of a neural network. The second is that it usually takes a long time to train a neural network. The third and the most apparent disadvantage is that a trained neural network is unable to explain its response. In other words, the knowledge is encoded in the values of synaptic weights, therefore, a neural network cannot justify its response on the basis of explicit rules or logical reasoning process. There have been several attempts to overcome the problem. One approach is to interpret or extract rules from a trained backpropagation network [14]. The algorithm proposed by Bochereau and Bourgine can only extract some sets of the Boolean logic rules [15]. Goodman et al. considered problems of extracting rules that relate to a set of discrete feature variables [16]. Gallant developed a neural network expert system matrix controlled inference engine (MACIE), which possesses features that are usually associated with conventional expert systems [17], [18]. Basically, the above-mentioned methods extract crisp rules. Another approach to voltage security margin estimation (VSME) is to formulate VSME as a classification problem. Classification problems can be solved either exclusively or nonexclusively. An exclusive or crisp classification is a partition of the set of objects. Each object belongs to exactly one class. A nonexclusive or overlapping classification can assign

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SU et al.: RULE EXTRACTION FOR VSME

an object to several classes. Fuzzy classification is a type of nonexclusive classification in which a pattern is assigned a degree of belongingness to each class in a partition. That is, the decision boundaries of the classes are vague. The incorporation of fuzziness introduced by Zadeh [19] into class assignments broadens their flexibility and robustness. The major question of fuzzy classification is how to design a fuzzy classifier. A rule-based approach is to generate fuzzy if–then rules to construct a fuzzy classifier. The most straightforward method of generating fuzzy classification rules involves two phases: 1) fuzzy partition of a pattern space and 2) identification of a fuzzy rule for each fuzzy subspace. The number of partitions of each feature greatly affect the classification power of the fuzzy classifier consisting of the fuzzy rules. Another problem associated with this method is how we identify the consequent part of each rule. Generally, the policies and heuristic strategies of the corresponding decision-making experts or common sense dictate the assignments of the consequents. In most cases, these rules are too crude for engineering purposes, therefore, they have to be fine tuned. A more complicated but appealing method is to generate fuzzy rules directly from numerical data. In [20] and [21], fuzzy rules with variable fuzzy regions are extracted for classification problems. In [20], each class is represented by a set of hyperrectangles (hyperboxes), in which overlaps among hyperrectangles for the same class are allowed, but no overlaps are allowed between different classes. However, this approach may not easily handle patterns where complicated separative boundaries exist. To overcome this problem, two types of hyperrectangles, activation hyperrectangles and inhibition hyperrectangles, were proposed in [21]. These hyperrectangles are defined recursively. It allows inhibition hyperrectangles to be stored inside activation hyperrectangles, and inhibition to inhibition can be nested any number of levels deep. However, during the training procedure, if the activation hyperrectangle at level is identical to the inhibition hyperrectangle at level , they need to define a set of activation hyperrectangles which include only one datum. Therefore, it will require a large memory for storing parameters defining these hyperrectangles and large computational resources to classify patterns. In [22] and [23], we have developed a method for extracting crisp classification rules directly from numerical data. The method is based on applying the supervised decision-directed learning (SDDL) algorithm to train a class of hyperrectangular composite neural networks (HRCNN’s). Each generated hyperrectangle which shows the existence region of data corresponds to a crisp rule. In this paper, we propose a reasonable membership to fuzzify crisp rules extracted from a trained HRCNN so as to implement a fuzzy classifier for VSME. A two-layer fuzzy hyperrectangular composite neural network (FHRCNN) is proposed to implement a fuzzy system utilizing the fuzzy rules. This paper is organized as follows. The architecture of HRCNN’s and the SDDL algorithm are briefly introduced in Section II. The concept of incorporating fuzzy logic into HRCNN’s is presented in Section III. In Section IV, the proposed scheme for VSME is simulated on the IEEE 30-bus system to illustrate its performance. Finally, a summary and conclusion are given in Section V.

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Fig. 1. A symbolic representation of a two-layer HRCNN.

II. HRCNN’S The construction of a rule-based system consisting of either crisp or fuzzy rules unavoidably involves the process of acquiring if–then rules. The most straightforward and simplest way is to ask human experts, however, the difficulties of obtaining an adequate set of rules from human experts are well known. First, experts may not know or may be unable to articulate what knowledge they actually used in solving their problems. Second, the performance of rule-based systems at best cannot be better than the expert on which it was modeled (although it might be more consistent because computer programs do not have “bad days”). Third, the average development time for a rule-based system is lengthy. This leads to the need of machine learning systems which make possible the automatic generation of rules from available examples. Successful machine learning systems will solve the knowledge bottleneck which is seriously handicapping the development of truly capable rule-based systems. Neural networks have long been considered a suitable framework for machine learning. Among the many appealing properties of a neural network, the property that is of prime significance is the ability of the neural network to inductively learn concepts from given numerical data. The problem is that conventional feedforward neural networks (e.g., backpropagation networks) do not arrive at an inference structure for this kind of highlevel knowledge representation. This motivated us to propose a class of HRCNN’s which integrate the paradigm of neural networks with the rule-based approach. This kind of hybrid system may neutralize the disadvantages of each alternative. A symbolic representation of a two-layer HRCNN is illustrated in Fig. 1. The mathematical description of a two-layer HRCNN is given as follows:

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(1) (2) (3)

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Fig. 2. An initial rectangular concept for a two-dimensional case. Fig. 3. The process of preventing overgeneralization formed by an initial rectangular concept.

where if otherwise

(4)

and are adjustable synaptic weights of th hidden node with the property is an input pattern, is a positive constant is the output function of the th hidden less than 1, is the output signal of a node, and two-layer HRCNN with hidden nodes. According to (2) and is 1 if and only if the following (3), we know that condition is satisfied: the

Fig. 4. The process of generalization for a rectangular concept.

(5)

.. .

Then, the classification knowledge can be described in the form of a production rule If Then

(6)

The domain defined by the antecedent of (6) is a hyperrect. It is easy to angle defined by is 1 if and only if is in at least one of the find that hyperrectangles. Understandably, the if–then classification rules extracted from a trained HRCNN with hidden nodes can be represented as Then

If .. .

Then

If Else

(7) Note that the antecedent conditions in each if–then rule (i.e., each hyperrectangle) are combined by the minimum operator if–then rules (i.e., the hyperrectangles) in (3) and the are combined by the maximum operator in (1). Besides, the representativeness of each rule can be measured by calculating the number of patterns contained in the corresponding hyperrectangle. The larger the number is, the more representative the rule is. The supervised decision-directed learning (SDDL) algorithm generates a two-layer feedforward HRCNN in a sequential manner by adding hidden nodes as needed. As long as there are no identical data over different classes, we can obtain a

100% recognition rate for training data. First of all, the training patterns are split into two sets: 1) a positive class from which we want to extract the concept and 2) a negative class which provides the counterexamples with respect to the concept. A seed pattern is used as the base of the initial concept (the seed pattern can be arbitrarily chosen from the positive class). When a pattern is selected as the seed pattern, a kind of generalization occurs automatically, since the seed pattern naturally partitions the space into two regions: 1) a relevant region surrounding the seed pattern contains all points closer to that seed pattern than from 2) a nonrelevant region. Fig. 2 illustrates this kind of generalization represented in terms of a rectangle for the two-dimensional case. After a seed pattern is chosen, the weights are initialized in the following way: for for where , a small positive real number, is the control parameter which decides the degree or generalization contributed by a positive pattern. After that, we use all counterexamples from the negative class to prevent overgeneralization (the induced concept should not be so general as to include any counterexample) formed by the initialization of weights. Fig. 3 illustrates the process of preventing the overgeneralization formed by an initial concept. The following step is to fetch the second pattern from the positive class and to generalize the initial concept to include the new positive pattern. This process involves expanding the original hyperrectangle to make it larger to include the new positive pattern. Fig. 4 illustrates the process of generalization. Here, we assume every positive pattern contributes some degree of generalization to the concept so that the new pattern lies inside the expanded hyperrectangle. After the process of generalization, we again use patterns from

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Fig. 5. The process of preventing overgeneralization for an expanded rectangular concept.

the negative class to prevent overgeneralization. Fig. 5 shows the method of preventing overgeneralization; this process is repeated for all the remaining patterns in the positive class. Here, it should be pointed out that during the process of preventing overgeneralization, the most important thing is that whenever we shrink the expanded hyperrectangle (i.e., at time ) in order to exclude any pattern from the negative ) class, the new shrunk hyperrectangle (i.e., at time should include the original hyperrectangle (i.e., at time ). This criterion guarantees that at least one positive pattern (seed pattern) be recognized after the training procedure. In other words, it prevents the new learning from washing away the memories of prior learning. Thus, after one cycle of presenting positive patterns, if there is any unrecognized positive pattern, another hidden node is self-generated and the process of learning is repeated again and again until all positive patterns are recognized. In the worst case, the number of hidden nodes is equal to the number of positive patterns. In fact, the number of the extracted intermediate rules is decided by the inherent properties of the patterns. If we hope that hyperrectangles do

not overlap each other too much, we can make the recognized positive patterns negative and then continue the process of the learning process. The flow chart of the algorithm is shown in Fig. 6. Here, pseudocode descriptions of two important procedures are given at the bottom of the page. can be equal to or greater than zero. The value of the equal to The most straightforward method is to make . As for the specification of the value of the parameter , one of the simplest methods is to make equal if or to if . There are similarities between the fuzzy min–max neural network classifiers [20] and HRCNN’s. Both approaches utilize hyperrectangles as rule representation elements. However, there are still many major differences between these two classes of neural networks [22]. First, HRCNN’s were originally developed to generate crisp if–then rules. After crisp rules have been generated, a reasonable membership function is utilized to fuzzify these crisp rules. On the contrary, the fuzzy min-max neural networks focus on finding fuzzy rules.

Procedure of generalization begin ( is a positive pattern) for from 1 to dimensions-of-input begin if then else if then end; end; end. Procedure of prevention-of-overgeneralization begin ( is a counterexample) for from 1 to dimensions-of-input begin if then ( should be chosen to ensure else if then ( should be chosen to ensure end; end; end. Authorized licensed use limited to: National Taiwan University. Downloaded on March 9, 2009 at 04:09 from IEEE Xplore. Restrictions apply.

); );

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form of a set of hyperrectangles. In order to fuzzify these crisp rules, a membership function is required to measure the degree to which an input pattern is close to a hyperrectangle. As the membership function approaches 1, an input pattern is closer to the th hyperrectangle defined by , with the value 1 representing complete hyperrectangle containment. Basically, any function producing a value between 0–1 can be a membership function. Membership functions can have different shapes (trapezoidal, triangular, or bell-shaped) depending on the designer’s preference or experience. According to our simulations on many pattern recognition problems such as speech recognition, character recognition, etc., we found the following function is a good choice in terms of computation burden and recognition rates: (8) where (9) (10)

Fig. 6. The flow chart of the SDDL algorithm.

Second, the fuzzy min-max neural network classifiers find a set of hyperrectangles under a special expansion criterion so that the sizes of hyperrectangles are limited. On the contrary, HRCNN’s find hyperrectangles whose sizes are as large as possible. Third, the contraction procedures are different in these two classes of neural networks.

is a sensitivity parameter which regulates how fast the and similarity value decreases as the distance between and the th hyperrectangle. Note that the values of and are proportional to the perimeters of the th hyperrectangle and the hyperrectangle which is expanded to include the pattern , respectively. If the pattern falls inside the th hyperrectangle, is 1 and, if not, is always larger the value of , therefore, the larger the difference between than and , the smaller the value of . Fig. 7 illustrates an for the two-dimensional case. example of fuzzy rules for class Now, suppose we have extracted , for . To numerically combine the fuzzy rules in order to compute the final membership value, one of the following two kinds of defuzzifiers can be used. Finally, a pattern is assigned to the class which has the largest membership value. Type 1—Maximum method:

III. FHRCNN’S A crisp classification attempts to determine the exact boundary inside which patterns of the same class lie. However, much of the information we have to deal with in engineering applications or in real life is in the form of imprecise, disturbed, or uncertain patterns. Obviously, it is not appropriate to use crisp classification rules to manipulate such kinds of data because vagueness may exist in the decision space, therefore, fuzzy logic is an appropriate tool to cope with these kinds of data. The incorporation of fuzzy logic into classifications provides flexibility and robustness. Fuzzy classification rules regard the decision surface between two different classes as a gray region. A pattern falling in the gray region has partial membership value representing the degree to which class it belongs. As discussed in the preceding section, the values of the synaptic weights of a trained HRCNN can be utilized to extract crisp classification rules of which antecedents are represented in the

(11) indicates the degree that an input where the value belongs to class . It is the simplest and most pattern straightforward method. Type 2—Modified center average method: (12) and are adjustable parameters and the value where is an integer which is less than or equal to the of . The reason why we use instead of is value of that some hyperrectangles have negligible representativeness, so we can discard these rules so as to omit unnecessary computational resources and memory. The defuzzifier defined in (12) is computationally equivalent to the most popular

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(a)

Fig. 7. (a) A step-like

(b)

mj (x). (b) A Gaussian-like mj (x).

Fig. 8. A symbolic representation of a two-layer FHRCNN.

Fig. 9. The large network for combining

“center average defuzzifier” except the bias term . The reason for including the bias in the output is to add one more free parameter to the network so as to increase the modeling flexibility of the network. A two-layer FHRCNN as shown in Fig. 8, a fuzzified version of an HRCNN, was proposed to implement a fuzzy system utilizing such a kind of defuzzifier [24]. two-layer Originally, we have to separately train classes, however, we may FHRCNN’s since there are networks into a large network, as shown in combine these Fig. 9. The output of the large network is given as

K subnetworks for K classes.

for each sample. The updating rules for the weights and the bias are given as follows:

(15) and

(16) (13) We can use the least-mean-squares (LMS) algorithm to update and . The goal of the LMS algorithm is to find the value of ’s and the bias that minimize the values of the weights the criterion function

and are the learning rates for the weights and where the bias, respectively. According to our experience, we suggest that it is better to first use the Type 1 defuzzifier since it involves the least amount of computational resources. If the resulting performance is not good enough, then it is better to use the Type 2 defuzzifier instead via training an FHRCNN.

(14) IV. EXPERIMENTAL RESULTS is the number of training patterns and is the where . The desired output desired output for the input pattern is assigned to be if the input pattern belongs to class . The recursive LMS algorithm uses the steepest descent minimization procedure for adapting the weights and the bias

The IEEE 30-bus system with phasor measurement units (PMU’s) (shown in Fig. 10) was used to test the effectiveness of HRCNN’s for VSME. A detailed description of PMU’s utilizing time-synchronized sampling over an entire power system to simultaneously obtain the phasor measurements can

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NUMBER

OF

TABLE I EXTRACTED CRISP RULES FOR EACH SECURITY LEVEL.

is of the following form:

Fig. 10.

IEEE 30-bus system with 10 PMU’s.

be found in [3], [25], and [26]. There are 2500 patterns and 500 patterns in the training set and the testing set, respectively. These patterns were generated from an extensive power flow by considering random load changes (light load to critical load), effects of on-load tap changers (OLTC’s), generator power limits (reactive power limits on generators), reactive power support compensators, and various contingencies. We installed ten PMU’s 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 condition. The identification of weak buses is based on the right singular vector from base case load flow. The weakest bus would imply the largest change in voltage magnitudes, therefore, the system buses are arranged in order of weakness. We choose the set of voltages and phases of monitored buses as the input vector. Thus, the input vector

where and are the voltage and angle of the th PMU, respectively. We quantized the voltage stability margin into five levels based on the magnitude of the minimum singular value of the power flow Jacobian matrix. Fast calculations of the minimum singular value and corresponding singular vector based on [27] were adopted to prepare the 3000 patterns. The five voltage security levels are as follows: ; class 1 (very dangeous level) ; class 2 (dangeous level) ; class 3 (alert level) ; class 4 (secure level) ; class 5 (very secure level) is the minimum singular value. Of course, the where determination of the ranges of the minimum singular value depends heavily on the specific power system under operation. It should be emphasized that the range of the magnitudes for each security level selected above is just for the of ease of illustration of the numerical test. One possible way for each security to determine the appropriate ranges of level is to extensively conduct simulations off-line for the study system under various operating conditions to statistically determine appropriate ranges. Each training pattern is then labeled to be one of the five levels. The simulations were conducted on a SUN SPAC II. After sufficient training, a total of 70 crisp rules were extracted from the 2500 training patterns. The number of extracted crisp rules and the most representative rule for each security level are tabulated in Tables I and II, respectively. The successful classification rate for the testing set of the 70 crisp rules was 87.2%. In order to improve the performance, the membership function in (8) was utilized to fuzzify these crisp rules. For comparison purposes, the proposed two types of defuzzifiers were then employed to compute the final membership value. Finally, we assign patterns to the security level which has the largest membership value. The performance was greatly improved to be 90.1%, and 98.15%, respectively. It should be emphasized that the 21 most representative rules of the original 70 crisp rules were employed in the second type of defuzzifier. After 10 000 iterations, we stopped the training and , were all set procedure. The learning rates, such as up to be 0.0004. The classification rates for the five security levels obtained by the trained HRCNN’s and the FHRCNN are tabulated in Table III. We also used the backpropagation algorithm to train a two-layer neural network with 20 hidden nodes using the same training set and the testing set. After 100 000 iterations, the average recognition rates are 95.15% and 87.32% for the training set and the testing set, respectively. The detailed classification rate for each security level is given

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TABLE II MOST REPRESENTATIVE RULE EXTRACTED

COMPARISONS

OF

FOR

TABLE III CLASSIFICATION PERFORMANCE OF

CLASSIFICATION PERFORMANCE

THE

EACH SECURITY LEVEL.

HRCNN’S

AND THE

FHRCNN.

TABLE IV TRAINED BACKPROPAGATION NETWORK.

OF THE

in Table IV. From the numerical simulations, one makes the following observations. • The fuzzy rules are more effective than the crisp rules. • The Type 2 defuzzifier is better than Type 1 in terms of the recognition performance. • The trained FHRCNN has a fairly high classification rate for VSME.

• The trained FHRCNN using the phasor measurements as inputs has the potential to be an effective tool for VSME. V. CONCLUSION In this paper, we have demonstrated the success of extracting rules for voltage security monitoring based on synchronized phasor measurements. Extensive testing was performed

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on the IEEE 30-bus system under various operating conditions. The class of HRCNN’s studied in the paper integrate the paradigm of neural networks with the rule-based approach. After sufficient training, the values of the synaptic weights of an HRCNN can be utilized to extract crisp rules. In order to incorporate fuzzy logic into the crisp rules so as to increase flexibility and robustness, a special membership function was proposed. We may discard unimportant rules and then use the LMS algorithm to train a two-layer FHRCNN to make a satisfactory compromise between the classification performance and the number of rules. According to the simulation results, the rule-based approach to VSME opens up new possibilities for power system protection and control. ACKNOWLEDGMENT The authors wish to thank H.-C. Yu and H.-T. Chang for their programming assistance.

[17] S. I. Gallant, “Connectionist expert systems,” Commun. ACM, vol. 31, pp. 152–169, Feb. 1988. [18] S. I. Gallant, Neural Network Learning and Expert Systems. Cambridge, MA: MIT Press, 1993. [19] L. A. Zadeh, “Fuzzy sets,” Inform. Contr., vol. 8, pp. 338–353, 1965. [20] P. K. Simpson, “Fuzzy min-max neural networks-part 1: Classification,” IEEE Trans. Neural Networks, vol. 3, pp. 776–786, Sept. 1992. [21] S. Abe and M. S. Lan, “A classifier using fuzzy rules extracted directly from numerical data,” in Proc. 2nd IEEE Int. Conf. Fuzzy Systems, 1993, pp. 1191–1198. [22] M. C. Su, “A novel neural network approach to knowledge acquisition,” Ph.D. dissertation, Dep. Elect. Eng., Univ. Maryland, College Park, Aug. 1993. [23] M. C. Su, “Use of neural networks as medical diagnosis expert systems,” Comput. Bio. Med., vol. 24, no. 6, pp. 419–429, 1994. [24] M. C. Su and C. J. Kao, “Time series prediction based on a novel neurofuzzy system,” in Proc. 4th Golden West Conf. Intelligent Systems, 1995, pp. 228–233. [25] A. G. Phadke, “Synchronized phasor measurements in power systems,” IEEE Comput. Applicat. Power, vol. 6, pp. 10–15, May 1993. [26] A. G. Phadke and J. S. Thorp, Computer Relay for Power Systems. New York: Wiley, 1988. [27] P.-A. L¨of, T. Semed, G. Andersson, and D. J. Hill, “Fast calculation of a voltage stability index,” IEEE Trans. Power Syst., vol. 7, pp. 54–64, Feb. 1992.

REFERENCES [1] Voltage Stability of Power Systems: Concepts, Analytical Tools, and Industry Experience, IEEE, Piscataway, NJ, IEEE Pub. 90TH0358-2PWR, 1990. [2] C. W. Taylor, Power System Voltage Stability. New York: McGrawHill, 1994. [3] A. G. Phadke and J. S. Thorp, “Improved control and protection of power system through synchronized phasor measurements,” in Control and Dynamic Systems, vol. 43. New York: Academic, 1991, pp. 335–376. [4] A. Tiranuchit and R. J. Thomas, “A posturing strategy against voltage instabilities in electric power systems,” IEEE Trans. Power Syst., vol. 3, pp. 87–93, Feb. 1988. [5] Y. Tamura, H. Mori, and S. Iwamoto, “Relationship between voltage instability and multiple load flow solutions in electric power systems,” IEEE Trans. Power App. Syst., vol. PAS-102, pp. 1115–1125, May 1983. [6] C. L. DeMarco and T. J. Overbye, “An energy based stability measure for assessing vulnerability to voltage collapse,” IEEE Trans. Power Syst., vol. 5, pp. 419–452, May 1990. [7] T. Van Cutsem, “A method to compute reactive power margins with respect to voltage collapse,” IEEE Trans. Power Syst., vol. 6, pp. 145–156, Feb. 1991. [8] A. A. El-Keib and X. Ma, “Application of artificial neural networks in voltage stability assessment,” IEEE Trans. Power Syst., vol. 10, pp. 1890–1896, Nov. 1995. [9] B. Jeyasurya, “Artificial neural networks for power system steady-state voltage instability evaluation,” Elect. Power Syst. Res., vol. 29, no. 2, pp. 85–90, Mar. 1994. [10] K. Hornik, M. Stinchcommbe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, pp. 359–366, 1989. [11] K. Funahashi, “On the approximation realization of continuous mappings by neural networks,” Neural Networks, vol. 2, pp. 183–192, 1989. [12] G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math., Contr., Signals, Syst., vol. 2, pp. 315–341, 1989. [13] I. Park and I. W. Sandberg, “Universal approximation using radialbasis-function networks,” Neural Computation, vol. 3, pp. 246–255, 1991. [14] L. M. Fu, “Rule learning by searching on adapted nets,” in Proc. AAAI’91, 1991, pp. 590–595. [15] L. Bochereau and P. Bourgine, “Extraction of semantic features and logical rules from a multilayer neural network,” in Proc. IJCNN’90, Washington, DC, 1990, pp. 579–582. [16] R. M. Goodman, C. M. Higgins, and J. W. Miller, “Rule-based neural networks for classification and probability estimation,” Neural Computation, vol. 4, pp. 781–804, 1992.

Mu-Chun Su (S’89–M’89) received the B.S. degree in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., and the M.S. and Ph.D. degrees in electrical engineering from the University of Maryland, College Park, in 1986, 1990, and 1993, respectively. He is an Associate Professor of Electrical Engineering, Tamkang University, Tamsui, Taiwan, R.O.C. His current interests include neural networks, fuzzy systems, computer-aided medical systems, pattern recognition, and man–machine interfaces. Prof. Su was a co-recipient of the 1992 IEEE Franklin V. Taylor Award for the most outstanding paper presented at the 1991 IEEE SMC Conference.

Chih-Wen Liu (S’93–M’96) received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., and the Ph.D. degree from Cornell University, Ithaca, NY, in 1987 and 1994, respectively, both in electrical engineering. Since 1994, he has been with National Taiwan University, where he is an Associate Professor of Electrical Engineering. His research interests include power system computer applications.

Chen-Sung Chang received the diploma in electrical engineering from National Taipei Institute of Technology, Taipei, Taiwan, R.O.C. He is currently working toward the Ph.D. degree at National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include power system voltage stability and applications of artificial intelligence in power systems.

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