A new formulation for facts allocation for security ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003

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A New Formulation for FACTS Allocation for Security Enhancement Against Voltage Collapse Naoto Yorino, E. E. El-Araby, Hiroshi Sasaki, and Shigemi Harada

control vector of the FACTS devices; control parameter vector excluding FACTS devices; number of contingencies; power-flow equations; eqality constraints “nominal load”; bounds of all the state and control variables “nominal load”; a set of conditions for the point of collapse; bounds of all the state and control variables “collapse point”; fixed cost and variable cost of FACTS device ; maximum VAR compensation of the FACTS device .

Abstract—This paper proposes a new formulation for reactive power (VAR) planning problem including the allocation of flexible ac transmission systems (FACTS) devices. A new feature of the formulation lies in the treatment of security issues. Different from existing formulations, we directly take into account the expected cost for voltage collapse and corrective controls, where the control effects by the devices to be installed are evaluated together with the other controls such as load shedding in contingencies to compute an optimal VAR planning. The inclusion of load shedding into the formulation guarantees the feasibility of the problem. The optimal allocation by the proposed method implies that the investment is optimized, taking into account its effects on security in terms of the cost for power system operation under possible events occurring probabilistically. The problem is formulated as a mixed integer nonlinear programming problem of a large dimension. The Benders decomposition technique is tested where the original problem is decomposed into multiple subproblems. The numerical examinations are carried out using AEP-14 bus system to demonstrate the effectiveness of the proposed method. Index Terms—Benders decomposition, FACTS devices, genetic algorithms, VAR planning, voltage stability.

NOMENCLATURE refer to base case, just after contingency and corrective states, respectively; occurrence probability of contingency ;; breakdown probabilities through states , , and ; investment cost; base-case operating cost; just after contingency expected cost; expected corrective state cost; breakdown cost; production or power losses costs; control cost; load margin cost; load shedding cost; load parameter values, ( at base case); load parameter value for states , and , respectively (collapse points); base case load; load direction vector; state variables vector (voltage); load shedding vector; Manuscript received December 15, 2000; revised January 11, 2002. N. Yorino, E. E. El-Araby, and H. Sasaki are with the Department of Electrical Engineering, Hiroshima University, Hiroshima, Japan. S. Harada is with the Department of Electrical and Electronic Engineering, University of the Ryukyus, Okinawa, Japan. Digital Object Identifier 10.1109/TPWRS.2002.804921

I. INTRODUCTION

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ECENTLY, several network blackouts have been related to voltage collapse. This phenomenon tends to occur from lack of reactive-power supports in heavily stressed conditions, which are usually triggered by system faults. Therefore, the voltage collapse problem is closely related to a reactive-power planning problem including contingency analyses, where suitable conditions of reactiv-power reserves are analyzed for secure operations of power systems. In the conventional reactivepower planning problem, two kinds of constraints have been used. They are voltage feasibility constraints which guarantee bus voltages within permissible limits, and voltage stability constraints which guard the system against voltage collapse. Traditionally, the objective of the reactive-power (VAR) planning problem is to provide a minimum number of new reactivepower supplies to satisfy only the voltage feasibility constraints in normal and post-contingency states. Various researches have been carried out for this subject [1] and [2]. Recently, due to a necessity to include the voltage-stability constraint, a few researches have been reported concerning new formulations considering the voltage-stability problem [3] and [4], which provides more realistic solutions for the VAR planning problem. However, the obtained solutions are sometimes too expensive since they satisfy all of the specified feasibility and stability constraints. This implies that the existing formulation cannot adjust the security level so as to meet the amount of investments. Another issue is that the existing formulations do not evaluate the cost reduction effect in operation by the devices to be invested. In general, such a device can reduce the amount of load shedding in contingency states, which may considerably contribute to the cost reduction. Therefore, it is preferable to count on the emergency control costs in the VAR planning problem.

0885-8950/03$17.00 © 2003 IEEE

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Fig. 1. System transition states. Fig. 2.

In this paper, a new formulation and solution method are presented for the VAR planning problem including FACTS devices, taking into account the issues just mentioned. TCSC and SVC are used to keep bus voltages and to ensure the voltage-stability margin. The voltage stability is treated in terms of cost functions rather than treated as constraints. By doing so, the coordination is performed between the amount of investment and the level of security. The formulation takes into consideration the transition states such as the contingency states as well as the normal state. The load-power margins are explicitly treated in the formulation in order to calculate the probability of voltage collapse as well as the expected cost for collapse as a function of the loading margin. The objective function is to minimize the sum of the installation costs and the operating costs under the normal and contingency states, which include the costs of load shedding, the cost of other emergency controls, and the expected costs for the voltage collapse. Thus, the problem is formulated as a mixed integer nonlinear programming problem. Since a main purpose of this paper is to show the feasibility and solvability for the new formulation, typical solution techniques for this kind of problem are applied. The Benders decomposition technique [5] is used to decompose the original problem into a master subproblem and multiple operation subproblems. Then, a simple genetic algorithm (SGA) is applied to the master subproblem combined with the linear programming. On the other hand, the successive linear programming (SLP) is used to solve the operation subproblems. A case study with AEP-14 bus system is presented and discussed. II. PRELIMINARY OF THE PROPOSAL A. System Transition States As is mentioned in the introduction, our proposal is to evaluate the cost effects of newly installed equipments in the possible transitions of power system states. The transitions may be simplified as shown in Fig. 1, where three states , , and are defined. We assume that, in the base-case operating state , there have been a number of reactive-power resources already installed in the system and a contingency will happen with probability . When it happens, the proceeding state is assumed to be state , which will result in either voltage collapse with

Nose curve.

probability or corrective state with . In the corrective state, controls are carried out to meet all operational constraints using all available control devices including the devices under investigation. Since there are possibilities for the system to proceed to voltage collapse through the base case and through the corrective state , they are denoted as probabilities, and . Although the exact computation of the breakdown probabilities , , and would include complicated procedure taking into account various probabilistic factors as well as dynamic factors, we avoid this discussion in this paper to concentrate on more essential problems. That is, we simply assume that each breakdown probability is a function of the loading margin only. The loading margin is defined in each transition state as shown in using loading paramFig. 2, where the margin is given by eter for each state. B. Objective Function An objective is to minimize the sum of the expected costs in some period of power system operations, which include investment costs and operating costs associated with every state as shown in Fig. 1. The overall objective function may be represented by the following equations: (1) ,

where ,

and

A detailed description for every objective function and its associated constraints will be discussed Section III. C. Operating Constraints Fig. 2 describes the bifurcation diagrams (nose curves or P–V base case, , curves) for the power system states for after corrective control. Muljust after contingency, and tiple operation subproblems are defined for individual states , , and . For each state, we define two target points, the operfor nominal load and the point of collapse ating point as shown in Fig. 2. Note that the superscript refers

YORINO et al.: FORMULATION FOR FACTS ALLOCATION FOR SECURITY ENHANCEMENT

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to contingency and notation “ ” represents voltage collapse point for each state. For the nominal load in states and , we assume one for each state, the following power-flow problem including equality and inequality constraints

(2) It is noted that the power-flow problem (2) is always feasible in our formulation even for state since the controls including load shedding are performed to guarantee the existence of the solution. At the points of collapse for each state, a set of the equality and inequality constraints is written as

(3) The equality constraints of (3) represent the conditions for the saddle-node bifurcations, consisting of the power-flow equations and the singularity condition of the power flow Jacobian [6]. It is noted that conditions (3) is useful to identify . Although there are other bifurcations than saddl-node bifurcations, we shall treat only saddle-node bifurcations in this paper.

Fig. 3. (a) Probability of voltage collapse. (b) Load margin cost.

from to very quickly on the condition that the other emergency controls also respond fast enough. Therefore, in the formulation of this paper, we will neglect state to simplify the problem. According to this assumption, the probability is setis eliminated from the ting to be zero and, consequently, overall objective function. A detailed description of the previous objective functions, and its associated constraints are presented. A. Investment Cost and Constraints The total investment cost is expressed as

D. Treatment of Breakdown Probability As is well known, a simple yet effective method in analyzing voltage collapse is to carry out a power-flow analysis, where a collapse point is detected as a tip of nose curve, which in general corresponds to the solution of (3). It is, however, a fact that various dynamic factors as well as human factors are concerned with this phenomenon. Furthermore, voltage characteristics of loads, which are usually unknown, can also make the bifurcation point move. In this paper, to simplify the too complex problem, it is assumed that the breakdown probability is as regarded as a function of loading margin shown in Fig. 3(a). This assumption implies that voltage collapse will occur at the nose point with probability 0.5; the larger the margin is, the smaller the probability will be and vice versa. Thus, probabilities , , and are treated as the function of , , and , respectively. According to the probability curve, the expected voltage collapse cost from each state is represented in the same way as described in Fig. 3(b). This cost will be referred to as load margin cost hereafter in this paper, which is defined for each state as follows:

(5) if the site is selected for FACTS device expanwhere ; is a set of all candidate sites, and sion; otherwise, is the size of the additional VAR of the device . The constraints are represented by the folof the additional VAR amount lowing equation: for each

(6)

B. Base Case Operation Subproblem The objective is to minimize the sum of the production cost or power losses and the loading margin cost, while (2) and (3) are the constraints. This problem may be formulated as (7)

minimize subject to

(8)

(4) (9)

III. PROBLEM FORMULATION Based on the basic concept proposed in Section II, a detailed formulation for the FACTS allocation problem is presented in this section. An exceptional feature of FACTS devices that is different from the conventional equipments lies in the control ability to react quickly against disturbances [7]. Therefore, when focusing on the FACTS devices, the transitions of power system states of Fig. 1 may be simplified. Namely, as soon as a contingency occurs, the system state directly changes

where the base case.

is the load margin cost of

C. Post-Contingency Corrective Control Subproblems It is assumed that as soon as a contingency occurs, the system proceeds to the corrective state directly, where all of the violations of the constraints will be removed by using all available controls including load shedding to satisfy these constraints.

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Therefore, the objective is to minimize the expected cost in the corrective state as shown in (1), whose detailed expression is written as (10). The objective takes into account the probability of the contingency , the total amount of the load shedding, the load margin cost, and the control costs for reactive powers. The reactive-power controls selected in this work are those by generators, LTC transformer taps, shunt capacitor/reactor reactances, and SVC/TCSC susceptances. A detailed formulation of this problem can be written as minimize

; ; ;

IV. BENDERS DECOMPOSITION The optimization problem represented by (13) is a large-scale mixed integer nonlinear programming problem. A well-known technique for solving this kind of problems is the Generalized Benders Decomposition method [5]. In this technique, the problem is separated into master subproblem and operation subproblems. In this paper, the operation subproblems imply the base-case operation subproblem (7)–(9) and post-contingency subproblems (10)–(12). They are nonlinear programming problems. On the other hand, the master subproblem deals with the investments for new FACTS devices, and therefore, it becomes a mixed-integer linear programming problem. The master problem may be formulated as

(10) (14)

minimize subject to 1) nominal load operating point constaints for contingency

subject to for (15) (11)

2) voltage collapse point constraints for contingency

(12) is load margin cost

where of the contingency . D. Overall Problem Formulation

V. FACTS DEVICES CANDIDATE SITES SELECTION

The overall problem may be stated as minimize subject to (6) rewritten as for each (8) and (9) rewritten as

(11) and (12) rewritten as (13) where

and are the Lagrange multipliers associated with where the base case and contingency cases, respectively. All of the linear constraints in (15) are known as Benders cuts [5], where and are the lower bounds of the operating costs and , respectively; subscript is the iteration number in the iterative procedure between the master subproblem and operation subproblems to find global solution of the problem (13). It is important to note that the convergence of the Benders decomposition algorithm can only be guaranteed under some convexity assumptions of the objective functions of the operation subproblems and, therefore, great care should be necessary when applying this technique [1], [2], [5].

A critical step in the VAR planning problem is the preselection of the candidate locations for installing new devices. In this paper, candidate sites for installing TCSC and SVC have been pre-examined for the most severe contingencies. The severity of contingency is evaluated in terms of the load margin. TCSC Set: Since the TCSC is a series device; it should be effective to adjust the load margin. Therefore, the candidate lines are determined for the severe contingencies by computing the sensitivity of the load margin with respect to the line susceptance. In this study, during the steady state, the TCSC is represented as a variable series capacitance, which is able to change the net susceptance of the connected line. SVC Set: Since the SVC is a voltage-control device, the participation factor [8] is used as an index for controllability of the voltages at the nose point. The participation factors are computed using the right and the left eigenvectors of the Jacobian


corresponding to the zero eigenvalue at the nose point. A candidate set is decided, related to the highest participation factors. VI. SOLUTION ALGORTHIM It is assumed that the candidate sites for the SCV and TCSC have already been decided using the previous strategies. The algorithm is summarized as follows: for arbi1) Set initial conditions of trary FACTS device combinations. 2) For individual states including the contingency cases, solve base case and independently the operation subproblems , (7)–(12) with available controls: , etc. , , and 3) Use the obtained values of the corresponding Lagrangian multipliers of all the subproblems to set the master subproblem, (14) and (15). 4) Solve the master subproblem (14) and , which implies a (15) to determine new new FACTS devices expansion. 5) Check the difference between the optimal objective functions of the operation subproblem (13) and master subproblem (14). If the difference is within a prespecified tolerance, stop. Otherwise, go step 2. A. Solution Algorithm for Operation Subproblems The operation subproblems (7)–(12) are nonlinear optimization problems. The successive linear programming is used to solve these problems, where the equations are linearized around the power-flow solution successively until the convergence is obtained. The linearized formulation for the corrective state subproblem is given by

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1) Solve load flow (2) with the available control variables ( , , and ). 2) Compute the maximum loading point for the post-fault condition to obtain the and , etc. of sensitivity matrices (16) and (17). 3) Solve optimization problem (16) and (17) using LP method. 4) If converged, stop. Otherwise, update all control variables and go to step 1. B. Master Problem Solution Algorithm Using GA The master problem is a mixed-integer linear programming problem, where the integer variables are restricted to 1 or 0, and the number of candidate sites for installing FACTS devices is relatively small even for large-scale networks. The procedure of the suggested SGA combined with the linear programming method is briefly explained as follows 1) String Representation: The binary bits are used, where each chromosome contains a number of bits, each of which represents to the candidate site of FACTS. The population of chromosomes is randomly selected in the binary-coded domain. 2) Solution of Continuous Variables Using LP: When the locations of the FACTS devices are fixed, the master subproblem becomes a continuous linear problem, for which the existing methods for solving this class of the problem are very efficient. Therefore, for each chromosome, an LP is used to solve the problem. 3) Evaluation Function: The evaluation of each individual is based on the concept of penalty function representing the sum of the violated constraints. Individuals are evaluated for each generation by the following formula: (19)

EVAL and

minimize otherwise (20) (16)

(17) and are the sensitivity matrices associated with where (11) and (12), respectively. The similar linearization is used for the base-case operation subproblem. It is noted that in the acis eliminated from (16) and (17) by using tual computation, and control the sensitivity relation among the load margin , , and as follows: variables, (18) where the sensitivities are computed at the nose point as proposed in [9]. The procedure used for solving each subproblem is summarized as follows:

where , , and are constants; is the generation number; is the objective function of (14); and is equivalent to the inequality constraints of (15). 4) Reproduction: According to their fitness values, the strings will be copied in the mating pool. The higher the fitness value is, the more likely is its selection. The biased roulette wheel is used in this paper to select mates. 5) Crossover: Crossover is performed on each randomly selected pair in the mating pool. Each pair of strings would swap their bits according to the crossover probability. The one-point crossover type [10] is used in this study. 6) Mutation: A mutation is a random alternation of bit of string position. At each bit position of chromosome, the bit will be changed from 1 to 0 or vice versa if a uniformly distributed is less than a specified probability of random number in mutation.

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TABLE I MASTER PROBLEM AND OPERATION SUBPROBLEMS TEST RESULTS WITH SVC CANDIDATE SET

TABLE II MASTER PROBLEM AND OPERATION SUBPROBLEMS TEST RESULTS WITH TCSC-SVC CANDIDATE SET Fig. 4. AEP-14 bus system.

VII. NUMERICAL RESULTS The proposed method has been applied to the AEP-14 bus system shown in Fig. 4. In order to demonstrate the performance of the proposed method, two cases are set as below. In case 1, only SVC will be chosen as candidate devices, while in case 2, a combination of SVC and TCSC will be examined. Parameters pu, used for the examinations are given as follows: pu, , , , , , , voltage collapse cost 100 000, and load shedding cost 1000 per 100 MW. As a preliminary computation, the contingency analysis mentioned in Section V was carried out first. According to these results, the most severe contingencies were the outages of lines [(1–2), (1–5), and (2–3)]. Then, the modal analysis was performed to rank buses according to their participation factors. Then, the best five candidate buses to site new SVC were [9, 10, 11, 13, and 14]. On the other hand, the sensitivity analysis of the load-power margin was carried out to determine the best few candidate lines to site new TCSC, which were lines [(7–8), (7–9), (9–10), and (9–14)]. For these candidate buses and lines, the algorithm presented in Section V was carried out. Table I shows the results for case 1, where the optimal SVC allocations are 0.19, 0.25, and 0.25 pu at buses 10, 13, and 14, respectively, and these reactive power are fully used for the outage of line (1–2). On the other hand, the optimal FACTS allocations for case 2 are shown in Table II, indicating TCSC of 0.091 pu at line (7–8) and SVC of 0.3 pu at buses 13 and 14. Table III lists the minimum bus voltage magnitude and load margin for each contingency, showing the effect of the FACTS installation for both cases 1 and 2 compared with no installation case. It is observed that voltages are violated and load margin is too small for the case without FACTS, where the voltage collapse is likely to occur with high probability. The contingency costs to maintain bus voltage magnitudes and to keep the load margin within the specified value are given in Table IV for the case without VAR expansion. It is observed

TABLE III MINIMUM VOLTAGE MAGNITUDE AND LOAD MARGIN VALUES IN ALL CASES

that the load shedding is required in this case, where the existing reactive-power sources are not adequate to keep the voltage stability of the system, resulting in a large amount of contingency cost. The contingency costs for the cases with SVC and TCSC-SVC optimal VAR allocations are given in Tables V and VI, respectively. In these cases, the reactive-power controls are successfully performed without load shedding, resulting in a small amount of total cost. Finally, some comments should be added concerning the convergence characteristics of the proposed method. Although the validity of the formulation and the feasibility of the problem have been confirmed in this paper, we have experienced bad convergence on the iterations between the master and operation subproblems for some situations. This problem might appear


TABLE IV CONTINGENCY COSTS WITHOUT VAR EXPANSION

TABLE V CONTINGENCY COSTS WITH SVC VAR EXPANSION SET

TABLE VI CONTINGENCY COSTS WITH TCSC-SVC COMBINED VAR EXPANSION SET

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the operation subproblems are solved using successive linear programming. Numerical examinations using the AEP-14 bus system for SVC and TCSC allocations show the feasibility as well as the validity of the formulation. The optimal allocation of the FACTS devices suppresses the operation costs in contingencies, maintaining the system security for contingencies. Although the validity of the formulation and the feasibility of the problem have been confirmed in this paper, some numerical problems are left to be solved in the future. A critical issue is that the problem size is considerably increased for the proposed formulation compared with the existing formulations due to the inclusion of the contingency analysis into the problem. Therefore, the Benders decomposition method has been applied in order for the efficient solution. However, the computational experience indicated that nonconvexity rarely appears depending on system parameters. In such a case, an improper selection of the initial condition of the algorithm may cause a convergence problem. Therefore, great care is necessary when applying the Benders decomposition method to our new formulation. This also implies that for a solution method, the robustness is more important than the efficiency in some case. A direct application of modern heuristics methods to this formulation [11] is currently being investigated to be applied to nonconvex cases. REFERENCES

due to the nonconvexity of the problem. This tends to be the case for a larger system. For example, in IEEE-57 bus system, we have met this problem for some settings of the weighting factors of the objective functions associated with the cost functions for load shedding, load margin, and control. This means that great care must be taken when applying the Benders decomposition method to our formulation. To avoid this difficulty, the authors are currently investigating another solution method [11] based on modern heuristics techniques. Although time consuming, that method has been confirmed to be useful for nonconvex cases including IEEE-57 bus system. VIII. CONCLUSION AND DISCUSSION A new formulation and solution method for VAR planning problem including FACTS devices are presented. The formulation takes into consideration the transitions of power system sates in contingencies. The objective function is to minimize the sum of the installation costs and the operating costs, which include the expected costs for voltage collapse, the emergency control costs for load shedding, and so on. Some technical features of the formulation are as follows. The expected voltage collapse cost is calculated as a function of the load margin, which works like a penalty function. This treatment, together with the inclusion of load shedding cost, guarantees the feasibility of the power-flow solutions and the feasibility of the problem. The problem is formulated as a mixed integer nonlinear programming problem, which is decomposed into master subproblem and multiple operation subproblems using the Benders decomposition method. The master subproblem is solved by using a simple GA combined with linear programming, while

[1] T. Gomez, J. Lumbreas, and V. Parra, “A security-constrained decomposition approach to optimal reactive power planning,” IEEE Trans. Power Syst., vol. 6, pp. 1069–1076, Aug. 1991. [2] S. Granville and M. Lima, “Application of decomposition techniques to VAR planning: Methodological & computational aspects,” IEEE Trans. Power Syst., vol. 9, pp. 1780–1787, Nov. 1994. [3] Y. Chen, “Weak bus-oriented optimal multi-objective VAR planning,” IEEE Trans. Power Syst., vol. 11, pp. 1885–1890, Nov. 1996. [4] E. Vaahedi et al., “Large scale voltage stability constrained optimal VAR planning and voltage stability application using existing OPF/ optimal VAR planning tools,” IEEE Trans. Power Syst., vol. 14, pp. 65–74, Feb. 1999. [5] A. Geoffrion, “Generalized benders decomposition,” J. Optim. Theory Appl., vol. 10, no. 4, 1972. [6] C. A. Canizares and F. L. Alvarado, “Point of collapse and continuation methods for large AC/DC systems,” IEEE Trans. Power Syst., vol. 8, pp. 1–8, Feb. 1993. [7] “Static VAR compensator models for power flow and dynamic performance simulation,” IEEE Trans. Power Syst., vol. 9, pp. 229–240, Feb. 1994. [8] B. Gao, G. K. Morison, and P. Kunder, “Voltage stability evaluation using modal analysis,” IEEE Trans. Power Syst., vol. 7, pp. 1529–1539, Nov. 1992. [9] S. Greene, I. Dobson, and F. L. Alvarado, “Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters,” IEEE Trans. Power Syst., vol. 12, pp. 262–272, Feb. 1997. [10] Z. Michalewicz, Genetic Algorithms Data Structure Evolution Programs. New York: Springer–Verlag, 1996. [11] E. E. El-Araby, N. Yorino, H. Sasaki, and H. Sugihara, “A hybrid genetic algorithm/SLP for voltage stability constrained VAR planning problem,” in Proc. of the Bulk Power Syst. Dyn. Contr.—V , Onomichi, Japan, Aug. 26–31, 2001.

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Naoto Yorino received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Waseda University, Tokyo, Japan, in 1981, 1983, and 1987, respectively. Currently, he is an Associate Professor in the Department of Electrical Engineering of Hiroshima University, Hiroshima, Japan. He was a Visiting Professor at McGill University, Montreal, QC, Canada, from 1991 to 1992. He joined Fuji Electric Co. Ltd., Tokyo, Japan, from 1983 to 1984. His research interests are power system planning, stability, and control problems. Dr. Yorino is a member of IEE of Japan and SICS.

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E. E. El-Araby received the B.Sc. and M.Sc. degrees in electrical engineering from Suez Canal University, Port Said, Egypt, in 1990 and 1996, respectively. Currently, he is pursuing the Ph.D. degree in the area of the application of modern heuristics to power system operation and planning at the Department of Electrical Engineering of Hiroshima University, Hiroshima, Japan.

Hiroshi Sasaki received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Waseda University, Tokyo, Japan, in 1963, 1965, and 1979, respectively. Currently, he is a Professor in the Department of Electrical Engineering, Hiroshima University, Hiroshima, Japan. He was a Visiting Professor at the University of Texas, Arlington, from 1984 to 1985. He was also given a Visiting Lecturership from the University of Salford, Salford, U.K., from 1971 to 1972. He has been studying various problems in the power engineering field, especially expert systems and neural network applications to power systems, etc. Dr. Sasaki is a member of CIGRE and several academic societies.


Shigemi Harada received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Waseda University, Tokyo, Japan in 1994, 1996, and 1999, respectively. Currently, he is a Research Associate of Department of Electrical and Electronic Engineering, University of the Ryukyus, Okinawa, Japan. His research interest is mainly power system voltage stability. Dr. Harada is a member of IEE Japan.