522
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
A Novel Objective Function and Algorithm for Optimal PSS Parameter Design in a Multi-Machine Power System Sheng-Kuan Wang Abstract—This paper proposes a novel objective function and algorithm to obtain a set of optimal power system stabilizer (PSS) parameters that include a feedback signal of a remote machine and local and remote input signal ratios for each machine in a multi-machine power system under various operating conditions. A novel function called the damping scale is proposed and formulated as an objective function to increase system damping after the system undergoes a disturbance. Three existing objective functions of the damping factor, damping ratio, and a combination of the damping factor and damping ratio were analyzed and compared with the proposed objective function. A novel algorithm called gradual self-tuning hybrid differential evolution (GSTHDE) was developed for rapid and efficient searching of an optimal set of PSS parameters. GSTHDE uses the gradual search concept on STHDE to enhance the probability of searching for an optimal solution. Eigenvalue analysis and nonlinear time domain simulation results demonstrated the effectiveness of the proposed objective function and algorithm. Index Terms—Differential evolution, dynamic response, eigenvalue analysis, optimization method, power system stability.
I. INTRODUCTION
I
N a power system, the power system stabilizer (PSS) of a speed deviation type was widely used to reduce the effects of electromechanical oscillation modes [1]–[3]. Various parameter tuning methods of PSSs have been developed to solve the problem of system dynamic response. The simultaneous coordinated tuning method was used to obtain the set of optimal PSS parameters under various operating conditions with an optimization algorithm [4], [5]. It can simultaneously displace and coordinate all eigenvalues under various operating conditions according to the limit of the objective function. The objective function plays a crucial role in the solving process. Currently, both damping factors and damping ratios are often used as an objective function on the PSS parameter tuning problem [4]–[11]. Such objective functions may produce eigenvalues near an imaginary axis or higher/high-frequency oscillations. A novel function called the damping scale is proposed to manage these problems. The damping scale was formulated as an objective function for PSS parameter optimization. Manuscript received December 30, 2011; revised March 29, 2012 and April 29, 2012; accepted April 29, 2012. Date of publication June 14, 2012; date of current version January 17, 2013. This work was supported by the National Science Council of the R.O.C. under contract number NSC-99-2221-E-131-041. Paper no. TPWRS-01253-2011. The author is with the Ming Chi University of Technology, New Taipei 24301, Taiwan (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRS.2012.2198080
Khaleghi et al. [4] proposed a modified artificial immune network (MAINet) algorithm and multiobjective immune algorithm (MOIA) to optimize the parameters of PSSs in a 16-machine 68-bus system. Abido and Abdel-Magid [5], [7]–[10] used the tabu search (TS), particle swarm optimization (PSO), evolutionary programming (EP), genetic algorithm (GA), and Combinatorial Discrete and Continuous Action Reinforcement Learning Automata (CDCARLA) to optimize the parameters of the PSSs in a 10-machine 39-bus or two-area four-machine test system. Wang [11] proposed a mixed-integer ant direction hybrid differential evolution (MIADHDE) algorithm to optimize PSS parameters in the 10-machine 39-bus system. Local and remote input signals were considered. Bomfim [12] used Gas to simultaneously tune multiple power system damping controllers for various operating conditions. Zhang [14] also used gas to coordinate syntheses of PSS parameters by using a set of inequalities as an objective function. These studies all used objective functions of the rectangle, D-shaped (or trapezoid), or fan-shaped, with the tip at the origin to optimize and/or damping ratio of the desired damping factor the lightly damped and undamped modes. The remote input signal was demonstrated to improve the dynamic stability in a power system [11], [15], [16]. To increment system stability, both local and remote measurement signals were regarded as input signals of the damping controllers in this study. The remote feedback signals were provided by synchronized phasor measurement units in the test system. Wang et al. [13] developed the self-tuning hybrid differential evolution (STHDE) method to solve economic dispatch problems on a 40-machine system, which used the concept of the 1/5 success rule of evolution strategies (Ess) in the original hybrid differential evolution to accelerate the search for the global optimum. STHDE was proven to obtain excellent computation results for real decision variables [13]. Therefore, STHDE is also used in this study. Because the machine numbers of remote feedback signals are considered the integer variable, STHDE must be modified to enable simultaneous searching of a set of optimal solutions of integer and real variables. All decision variables in the 16-machine 68-bus system are 128-real number and 16-integer. This type of optimization problem is difficult to solve. The disadvantage of the simultaneous coordinated tuning method is that it increases the computational burden and may be unable to determine the global optimum solution for a large power system. The gradual search concept is proposed to overcome this disadvantage in this study, and it was applied to STHDE algorithm, and renamed GSTHDE. GSTHDE uses gradual increments of the dimension of decision variables to increase the probability
0885-8950/$31.00 © 2012 IEEE
WANG: NOVEL OBJECTIVE FUNCTION AND ALGORITHM FOR OPTIMAL PSS PARAMETER DESIGN
523
of searching for the global optimum solution and to decrease the computational burden. The GSTHDE algorithm was used to determine the optimal set of PSS parameters with the objective function of the damping scale for the test system under various system operating conditions. II. PROBLEM FORMULATION A. Power System Model In the test system, each generator was modeled as a two-axis model, which is 6 state variables in the state-space model, the exciter used IEEE type 1 model of 3 state variables, and PSS used traditional lead-lag model of 3 state variables [17]. Therefore, each machine had 12 state variables. These device models of a power system can be formulated as follows: (1) where is the vector of all state variables, and is the vector of input variables. In the PSS design, the power system is usually linearized to perform small signal analysis. Therefore, (1) can be represented as follows: (2) where is the vector of the network bus voltages, and is the current vector of injection into the network from the device [17]. The PSS structure of the two lead-lag compensators with speed deviation input signal was considered. The transfer function of the th PSS was derived by (3)[11]:
Fig. 1. Convergence regions of four objective functions. (a) Rectangular region of damping factor. (b) Fan-shaped region with tip at the origin of damping ratio. (c) D-shaped region of damping factor. (d) Fan-shaped region with tip at and of damping scale. damping ratio the
operating conditions of the test system are , the difference values of all operating conditions are summed as (6). Equation (6) is further represented as (7). When the function value of (7) is equal to or less than zero, the response is that the maximal damping factors in each operating conditions are exactly or less than the excepted value , respectively: (4)
(3) where , , , , , and are the time constants and gain of the th PSS, is the machine numbers, is the number of machines, are the speed deviations of the th local and remote machine, and and are the weights of the th local and remote machine, respectively. B. Existing Objective Function When a disturbance occurs in a power system, the duration of an oscillation in the time domain is determined by several eigenvalues at the most right sides on the -plane. Objective functions are currently formulated often by the damping factor and/or the damping ratio on the problem of PSS parameter tuning [4]–[11]. Three objective functions have often been used in several studies, as follows: 1) The first objective function with the damping factor may be derived as follows: In a power system, if the number of eigenvalues is under the th operating condition, then the maximal damping factor in all eigenvalues is represented as (4), where is the damping factor of the th eigenvalue. Equation (4) then subtracts the expected damping factor constant (negative value) to obtain a difference value as (5). If the total
(5)
(6) (7) All damping factors are limited in a rectangular region, which is no more than , as shown in Fig. 1(a). This may produce higher frequency modes close to the line , which may cause numerous up/down oscillations and reduce the lifetime of system devices. 2) The second objective function with the damping ratio can be derived as follows: The minimal damping ratio in all eigenvalues is represented as (8), where is the damping ratio of the th eigenvalue. The expected damping ratio constant (positive value) then subtracts (8) to obtain a difference value as (9). The difference values of all operating conditions are summed as (10). Equation (10) is further represented as (11). When the function value of (11) is equal to or less
524
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
than zero, the response is that the minimal damping ratios in each operating conditions are exactly or more than the excepted value , respectively:
TABLE I FEATURES OF FOUR OBJECTIVE FUNCTIONS
(8) (9)
(10) (11) A fan-shaped region with the tip at the origin was structured by (11), in which all damping ratios are no less than , as shown in Fig. 1(b). This damping ratio is defined by (12), and the slope of the straight line can be derived by (13)–(15), which has an inverse ratio with . The function may produce low-frequency inferior damped modes that cause continued oscillation over a long time:
, is proposed to overcome the disadvantage. The damping scale function was derived as shown in (17). It was obtained by subtracting from the eigenvalue, and was then derived as the damping ratio function (12). The damping scale may be formulated as an objective function (18): (17) (18)
(12) (13) (14) (15) 3) The third objective function with the damping ratio and damping factor may be formulated as (16), which is same with the third objective function of [11]. The derivation can be obtained by the derivation of F1 and F2:
where is the damping scale of the th eigenvalue, and is a constant value of the expected damping scale. A fan-shaped region with the tip at is structured by (18), as shown in Fig. 1(d), in which all damping scales and damping factors are no less than and no more than , respectively, and the slope of the straight line is exhibited as (19). Fig. 1(d) shows that F4 does not have these problems of higher/high-frequency modes or low-frequency inferior damped modes. The relation between damping ratio and damping scale can be observed from (12) and the last two terms of (19). If is 0, is equal to ; otherwise, is a negative value and the magnitude of is smaller than that of . When the restriction conditions were and , the objective function implied F3, rather than F4:
(16) where is the weight for combining both damping factor and damping ratio. Objective function (16) forms a D-shaped region, as shown in Fig. 1(c), in which all damping factors and damping ratios are no more than and no less than , respectively. The objective function might produce high-frequency modes close to the line , which might reduce the lifetime of some devices in a power system.
(19)
In summary, the features of four objective functions are listed in Table I. The four objective functions not only relocate unstable or lightly damped oscillation modes, but also shift other oscillation modes to the left side of the -plane. Therefore, the optimal values of the four objective functions are less than zero.
C. Proposed Objective Function These existing objective functions of (7), (11), and (16) comprise the damping factor and/or damping ratio, which may obtain higher/high-frequency modes or low-frequency inferior damped modes. A novel function called the damping scale,
D. Constraints of All Objective Function The constraint set is composed of bounds of each PSS parameter, which can be formulated as (20). The gain implies that machine does not need to install the PSS. The
WANG: NOVEL OBJECTIVE FUNCTION AND ALGORITHM FOR OPTIMAL PSS PARAMETER DESIGN
or implies that machine local or remote signals, respectively:
does not need
(20) III. PROPOSED OPTIMIZATION ALGORITHM GSTHDE The gradual search method is used to mitigate the computational burden and to increase the probability of a successful search for an optimal solution. The prime concept of the method was used to increase the dimension of decision variables gradually, to improve the probability and search speed. All decision variables were categorized to three stages according to their properties. These parameters of , , and were searched with , , and in the first stage. In the second stage, the RN were also added to search with and . In the third stage, and were also added to search for a set of optimal solutions. The STHDE algorithm is discussed in [13]. The main concept of STHDE is to use the variable scaling factor based on the 1/5 success rule [18], [19] to overcome the disadvantage of the fixed and random scaling factor used in the HDE. In contrast to the HDE, every individual during the solving process has the corresponding scaling factor in STHDE. The rule of updating the scaling factor based on the 1/5 success rule of ESs was used to adjust the scaling factor. The 1/5 success rule emerged as a conclusion of the process of optimizing the convergence rate of two functions (the so-called corridor mode and sphere model [18]–[20]). The original STHDE must be modified for all PSS parameter tuning problems to enable searching for the set of optimal solutions of integer and real variables by simultaneously using a gradual search method. The revised STHDE algorithm is called gradual STHDE (GSTHDE), which is briefly described as follows: Step 1: Initialization of various stages The initial populations are created. A number of initial populations are randomly chosen to cover the complete parameter space uniformly. The symbol stg represents the present stage, the value of which is 1 to 3 in this paper. The elements of every set integer and real random variables were assumed as (21). If the stg is more than one as (22), that first population is replaced with the optimal individual of the previous stage:
525
and are real and integer variables, respectively. The initial process can create individuals of randomly. In the test system, 96 real variables of , , and were searched in the first stage. In the second stage, the 16 integer variables of RN were added to the search. In the third stage, the 32 real variables of and were added to the search. Step 2: Mutation operation Five strategies of mutation operator have been introduced by [21]. The essential ingredient in the mutation operation is the difference vector. Each population pair at the th generation defines a difference vector as follows: (23) The mutation process at the th generation begins by randomly selecting either two or four populations , , , and for any , , , and , according to five mutation strategies. These four populations are subsequently combined to form a difference vector , as follows:
(24) A perturbed population is subsequently generated based on in the mutation process by the present population
(25) where the scaling factor, , is a constant, and , , , and are randomly selected. The perturbed population in (25) is essentially a noisy replica of . The parent population depends on the circumstance in which type of mutation operation is used. Step 3: Crossover operation To extend the diversity of further individuals at the next generation, the perturbed population of and the present population of are selected by a binomial distribution to progress the crossover operation to generate the offspring. Each individual of the th population is reproduced from the perturbed population and the present population . That is, otherwise
(21)
(22) is a random number, and INT operator is where represented to acquire the nearest integer. The two variables
(26)
otherwise ; ; , and where the crossover factor is assigned by the user. Step 4: Estimation and selection The evaluation function of a child is one-to-one competed to that of its parent. This competition indicates that the parent is replaced by its child if the fitness of the child is superior to that
526
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
of its parent. Conversely, the parent is retained in the next generation if the fitness of the child is inferior to that of its parent; that is,
(27) (28) where arg min indicates the argument of the minimum. Step 5: Migrating operation if necessary To effectively enhance the investigation of the search space and reduce the choice pressure of a small population, a migration phase is introduced to regenerate a new diverse population. The optimal population is selected to create the new population. The th or th individual of the th population is shown in (29), where and are randomly generated numbers uniformly distributed in the range of ; see (29) at the bottom of the page. The migrating operation is executed only if a measure fails to match the desired tolerance of population diversity. The measure is defined as follows:
(25) and (26). Therefore, the migration operation in GSTHDE must be performed to redevelop a new diversified population if a measure of population diversity fails to satisfy the desired tolerance of (30) and (31). Step 6: Accelerated operation if necessary When the optimal population at the present generation is no longer improved by the mutation and crossover operations, a descent method is used subsequently to push the present optimal population toward attaining a superior point. Thus, the accelerated phase is expressed as follows:
otherwise
denotes the optimal population, as obwhere tained using (28). The gradient of the objective function can be approximately calculated by finite difference. The step size in (32) is determined by the descent property. Initially, was set to 1 to obtain the new population. Step 7: Updating the scaling factor if necessary The rule of updating the scaling factor is as follows: if if if
(30)
where if otherwise if
(31)
otherwise. express the desired tolerance for Parameters the population diversity and individual diversity with respect to the optimal population, and and are the scale indices. Equation (30) and (31) indicate that the value is in the range of . If is less than , the migrating operation is executed to generate a new population to escape the local point; otherwise, the migrating operation is turned off. The convergence rate of GSTHDE can be improved by using the 1/5 success rule. This faster descent results in a local minimum or premature convergence. The candidate populations gradually cluster around the optimal population to decrease population diversity quickly. These closely clustered populations cannot reproduce to the next superior populations through mutation and crossover operations, as obtained using
(32)
(33)
where is the frequency of successful mutations measured of the th population. The successful mutation of the th population indicates that the fitness value of the th population is superior to that of its parent. The initial value of the scaling factor was set to 1.2. The factors of and [18], [19] were used for adjustment, which must be updated as (33) in every iteration. The suggested by [19] is equal to , where is a constant. When the performed migrating operation or the scaling factor is too small to obtain the optimal solution, the scaling factor is reset as (34): (34) are the numbers of the current iterwhere iter and ation in this stage and the maximal iteration of each stage, respectively. Step 8: Promotion operation if the maximal iteration number arrives in each stage If the number of the current iteration does not reach the maximal iteration number of this stage , repeat steps 2 to 7. If the number of current iterations is equal to that of
if otherwise if otherwise.
(29)
WANG: NOVEL OBJECTIVE FUNCTION AND ALGORITHM FOR OPTIMAL PSS PARAMETER DESIGN
527
Fig. 3. 5-area, 16-machine, 68-bus system.
The solution procedures begin by conducting calculations of power flows for various operating conditions. Calculations of the eigenvalues, oscillation frequencies, damping ratios, and damping scales of small signal analysis for various operating conditions are subsequently executed. This computational process of GSTHDE for determining the optimal small signal stability solution is shown in a flowchart in Fig. 2. IV. STUDIED SYSTEM AND PARAMETERS SETTING
Fig. 2. Main calculation procedures of the proposed method.
, increase stg, set , add necessary decision variables to be searched together in the following stage, and return to step 1 until the total iteration quantity is obtained. The total iteration quantity is a summation of all in the study system.
The 5-area, 16-machine, 68-bus system [23] is used to illustrate the performance of the proposed method. Fig. 3 shows the one-line diagram of the test system. Experimental results demonstrate that, when the load model of the system is set to 10% constant power, 10% constant current, 80% constant impedance for active power, and 100% constant power for reactive power, the system is at its weakest and it collapses after the system undergoes a disturbance. This status of operation is selected as Case0. The worst operation conditions of N-1 and N-2 in Case0 were regarded as Case1 and Case2, respectively. The three operating conditions are listed as follows: • Case0: Base case • Case1: The tie line 1-2 is out-of-service. • Case2: Tie lines 16-17 and 2-25 are out-of-service. All generators were equipped with PSS for performance estimation of the proposed method in the test system. In the PSS tuning process, the constraints on PSS time constants , , , and were set within 0.001-2 s, the time constants were set within 10–20 s, and the gain ranged from 0 to 50. In this study, 128 real variables and 16 integer variables were optimized. Parameters , , and are not indispensable, and can be set to zero in four objective functions. In this study, they were set to , 5%, and 2% to observe the values of objective function during the solving process. The weight parameter was set to 0.05 after several experiments for satisfaction conditions of more than and less than on F3, simultaneously. In the GA, PSO, MIADHDE, STHDE, and GSTHDE, several parameters must be established before performing the optimization process. These parameters include population size , crossover factor , the desired tolerance of
528
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
TABLE II OPTIMAL VALUES OF OBJECTIVE FUNCTIONS FOR FIVE ALGORITHMS
TABLE III CONVERGENCE RATES OF OBJECTIVE FUNCTIONS FOR FIVE ALGORITHMS Fig. 4. Convergence characteristics for five algorithms in F4.
the population diversity , the desired tolerance of chromosome diversity for the integer variable, and for real variables, mutation rate for the GA, number of stage , maximal iteration numbers for GSTHDE, and 15 000 for GA, MIADHDE, and STHDE. The initial-setting scaling factor , and the iteration number of the scaling factor were updated . The and for STHDE and GSTHDE are derived from Schwefel [19] and Wang [11]. In addition, the PSO parameters are the same as those of [8], expect that population size and maximal iteration numbers are set to 5 and 15 000, respectively, for comparative purposes.
TABLE IV AVERAGE CPU TIME OF FIVE ALGORITHMS WITH
V. SIMULATION RESULTS AND DISCUSSIONS
TABLE V OPTIMAL PSS PARAMETERS BY F4 WITH GSTHDE
The study system using four objective functions was performed 50 times by the GA, PSO, MIADHDE, STHDE, and GSTHDE algorithms. These algorithms were implemented using Matlab, and they were executed on a Core 2 Duo 3.0-GHz PC. The optimal values of each objective function and the convergence rates are shown in Tables II and III, respectively. Table II shows that GSTHDE can obtain an optimal solution comparison with four other algorithms for four objective functions. In Table II, F4 with STHDE is 2.6645 more than zero, which indicates that it did not reach the objective of by more than 2%. Fig. 4 shows the convergence characteristics of the five algorithms with F4; the convergence characteristics of GSTHDE are evidently the best, compared to the other four algorithms. Table III shows the convergence rates of four objective functions with five algorithms, in which the dash symbol (-) is used to substitute zero for easy identification. The convergence rate is defined as the percentage of the number of objective function values less than or equal to zero divided by the number of total runs. The objective function F2 is easier to converge, whereas F4 is most difficult. The cause of weaker PSO convergence is the population size of only 5 in Table III. The convergence rates of GSTHDE for F1-F4 are all 100%. Thus, the convergence ratio of GSTHDE is superior to that of the other four algorithms. Table IV shows the average CPU time for five algorithms with four objective functions when population size is set to 5. The calculation time of GSTHDE is slightly longer than that of PSO, MIADHDE, and STHDE,
but its convergence rates with F1-F4 are the best in Table III, compared to those of the other four algorithms. When the PSO population size is set to 10, its average CPU time and the best value of objective function are 17295.51 s and 9.6496 for F4, respectively. The calculation time and function value are more than that of GSTHDE. An increase in population size or iteration numbers in MIADHDE and STHDE might enhance their convergence rate, but this is more time-consuming. Therefore, GSTHDE can obtain a better solution with less time, which can reduce computational burden.
WANG: NOVEL OBJECTIVE FUNCTION AND ALGORITHM FOR OPTIMAL PSS PARAMETER DESIGN
529
TABLE VI DAMPING FACTORS, OSCILLATION FREQUENCIES, DAMPING RATIOS, AND DAMPING SCALES FOR FOUR OBJECTIVE FUNCTIONS
Table V shows PSS parameters searched by GSTHDE for F4, in which the dash symbol is used to substitute zero for easy identification. Table V indicates the position and amount of PMUs in
F4 and the input signal ratios of the local and remote machine. If is 0, PSS gain can be directly set to . Most magnitudes of are near 1.0; however, the fourth machine
530
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
Fig. 5. Eigenvalues of used F1 search result by three cases.
Fig. 7. Eigenvalues of used F3 search results by three cases.
Fig. 6. Eigenvalues of used F2 search results by three cases.
Fig. 8. Eigenvalues of used F4 search results by three cases.
of F4 is 0.727, as shown in Table V. Table VI shows the oscillation frequencies, damping factors, damping ratios, and damping scales under four objective functions and three operating conditions, which demonstrate the worst three values of damping factors, damping ratios, and damping scales. Table VI shows that all damping factors of F1, F3, and F4 are smaller than or equal to the value of , and all damping ratios of F2 and F3 are more than or equal to . The damping scales of F4 are more than . These results are also shown in Figs. 5–8. The results show that F1, F3, and F4 not only relocated unstable or lightly damped modes, but also shifted other modes to the left side in the -plane. The test system with and without PSSs is also included in these figures for comparison. The eigenvalues are enclosed in the rectangular frame for F1 in Fig. 5. A number of high damped and higher-frequency eigenvalues were observed near the line , the modes of which may cause cancellation, growth, or higher-frequency oscillation in the time domain. The eigenvalues are enclosed in the fan-shaped region with the tip at the origin frame for F2 in Fig. 6. A number of low-frequency eigenvalues were observed near the origin, which cause lower-frequency oscillation of inferior damping. The eigenvalues are enclosed in the D-shaped frame for F3 in Fig. 7. A number of high damped eigenvalues were observed near the line , the modes of which may cause cancellation, growth, or high-frequency oscillation in the time domain. The eigenvalues are enclosed in the fan-shaped region with the tip at the frame for F4 in Fig. 8; i.e., only eigenvalues of high damped and low frequency are near the point , which can mitigate and reduce oscillation in the time domain. A number of time-domain simulations were performed to demonstrate the effectiveness of PSS parameters designed by
Fig. 9. Line power flows from buses 1 to 27 for Case0.
using the proposed GSTHDE method. In the test system, assume that a three-phase fault occurs at bus 1 to pass through the four-cycle. Fig. 9–11 show the line powers from buses 1 to 27 for three different operation conditions. Fig. 9 shows that F1-F4 are similar in Case0 for a normal operation condition. F2 has a number of eigenvalues near the imaginary axis, as shown in Fig. 6; thus, oscillations are considerable in Case1 and Case2. Because a number of eigenvalues are close at the high-frequency area, which causes amplitude growth in Fig. 7, F3 has slight oscillation in Case2. The oscillation shapes of F1 and F4 are approximate in Figs. 9–11. However, the F4 amplitude is smaller than that of F1. Therefore, F4 can obtain the optimal result in the four objective functions. These time-domain simulations were consistent with the results of eigenvalue analysis. VI. CONCLUSION Currently, a number of power utility units use restrictions of the minimum damping ratio as a planning criterion on dynamic
WANG: NOVEL OBJECTIVE FUNCTION AND ALGORITHM FOR OPTIMAL PSS PARAMETER DESIGN
Fig. 10. The line power flows from bus 1 to 27 for Case1.
Fig. 11. Line power flows from bus 1 to 27 for Case2.
stability for a power system. Oscillation can be damped after a long time; however, these association devices of the power system undergo an effect of power oscillation, which influences their lifetime. A novel function and algorithm, called the damping scale and GSTHDE, respectively, are proposed to determine the optimal parameters of power system stabilizers in a power system. Four objective functions were applied to the problem of PSS parameter tuning for analysis and comparative purposes on performance. The set of optimum PSS parameters must be able to satisfy three operation conditions simultaneously. The feedback signal of the remote machine and local and remote signal ratios were also optimized, the values of which cannot be obtained using any current formula. The results show that GSTHDE is superior to STHDE, and the objective function of the damping scale is excellent in these objective functions. Eigenvalue analysis and nonlinear time domain simulation results demonstrate the effectiveness of the proposed algorithm. For coordinated control of PSS and Flexible AC Transmission System, the objective function of the damping scale can be used further to improve the system’s dynamic response. REFERENCES [1] A. R. Messina, J. M. Ramirez, and J. M. C. Canedo, “An investigation on the use of power system stabilizers for damping inter-area oscillations in longitudinal power systems,” IEEE Trans. Power Syst., vol. 13, no. 2, pp. 552–559, May 1998. [2] M. Klein, G. J. Rogers, and P. Kundur, “A fundamental study of interarea oscillations in power systems,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 914–921, Aug. 1991.
531
[3] P. Kundur, “Effective use of power system stabilizers for enhancement of power system reliability,” in Proc. IEEE Power Eng. Soc. Summer Meeting, Jul. 1999, vol. 1, pp. 96–103. [4] M. Khaleghi, M. M. Farsangi, H. Nezamabadi-pour, and K. Y. Lee, “Pareto-optimal design of damping controllers using modified artificial immune algorithm,” IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., vol. 41, no. 2, pp. 240–250, 2011. [5] Y. L. Abdel-Magid and M. A. Abido, “Optimal multiobjective design of robust power system stabilizers using genetic algorithms,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1125–1132, Aug. 2003. [6] K. Hongesombut, Y. Mitani, S. Dechanupaprittha, I. Ngamroo, K. Pasupa, and J. Tippayachai, “Power system stabilizer tuning based on multiobjective design using hierarchical and parallel micro genetic algorithm,” in Proc. IEEE Int. Conf. Power System Technology, PowerCon 2004, Nov. 21–24, 2004, vol. 1, pp. 402–407. [7] M. A. Abido and Y. L. Abdel-Magid, “Robust design of multimachine power system stabilisers using tabu search algorithm,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 147, no. 6, pp. 387–394, 2000. [8] M. A. Abido, “Optimal design of power-system stabilizers using particle swarm optimization,” IEEE Trans. Energy Convers., vol. 17, no. 3, pp. 406–413, Sep. 2002. [9] M. A. Abido and Y. L. Abdel-Magid, “Optimal design of power system stabilizers using evolutionary programming,” IEEE Trans. Energy Convers., vol. 17, no. 4, pp. 429–436, Dec. 2002. [10] M. Kashki, Y. L. Abdel-Magid, and M. A. Abido, “Parameter optimization of multimachine power system conventional stabilizers using CDCARLA method,” Int. J. Elect. Power Energy Syst., vol. 32, no. 5, pp. 498–506, Jun. 2010. [11] S.-K. Wang, J.-P. Chiou, and C.-W. Liu, “Parameters tuning of power system stabilizers using improved ant direction hybrid differential evolution,” Int. J. Elect. Power Energy Syst., vol. 31, no. 1, pp. 34–42, Jan. 2009. [12] A. L. B. Do Bomfim, G. N. Taranto, and D. M. Falcao, “Simultaneous tuning of power system damping controllers using genetic algorithms,” IEEE Trans. Power Syst., vol. 15, no. 1, pp. 163–169, Feb. 2000. [13] S. K. Wang, J. P. Chiou, and C. W. Liu, “Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm,” IET Gen., Transm., Distrib., vol. 1, no. 5, pp. 793–803, 2007. [14] P. Zhang and A. H. Coonick, “Coordinated synthesis of PSS parameters in multi-machine power systems using the method of inequalities applied to genetic algorithms,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 811–816, May 2000. [15] D. P. Ke, C. Y. Chung, and Y. Xue, “An eigenstructure-based performance index and its application to control design for damping interarea oscillations in power systems,” IEEE Trans. Power Syst., to be published. [16] D. Dotta, A. S. eSilva, and I. C. Decker, “Wide-area measurementsbased two-level control design considering signal transmission delay,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 208–216, Feb. 2009. [17] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. [18] T. Back, F. Hoffmeister, and H.-P. Schwefel, “A survey of evolution strategies,” in Proc. Int. Conf. Genetic Algorithms, San Diego, CA, Jul. 1991, pp. 2–9. [19] T. Back and H. P. Schwefel, “An overview of evolutionary algorithms for parameter optimization,” Evol. Computat., vol. 1, no. 1, pp. 1–23, 1993. [20] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 3rd ed. New York: Springer, 1999. [21] R. Storn and K. Price, “Minimizing the real functions of the ICEC ’96 contest by differential evolution,” in Proc. IEEE Conf. Evolutionary Computation, Nagoya, Japan, 1996, pp. 842–844. [22] R. Storn, “On the Usage of Differential Evolution for Function Optimization,” NAFIPS 1996, Berkeley, CA pp. 519–523. [23] R. Graham, Power System Oscillations. Boston, MA: Kluwer, 2000, pp. 314–317. Sheng-Kuan Wang was born in Chiayi County, Taiwan, in 1969. He received the M.S. and Ph.D. degrees from the Electrical Engineering Department of National Taiwan University, Taipei, Taiwan, in 1998 and 2007, respectively. Currently, he is an Assistant Professor of electrical engineering at Ming Chi University of Technology, New Taipei, Taiwan, where he has been since 1998. His research interests include power system dynamic stability, application of FACTS devices, economic dispatch, development and application of optimization algorithm, and wind power generation system.
