Chaotic ant swarm optimization for fuzzy-based tuning of power system stabilizer

Electrical Power and Energy Systems 33 (2011) 657–672

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Chaotic ant swarm optimization for fuzzy-based tuning of power system stabilizer A. Chatterjee a, S.P. Ghoshal b, V. Mukherjee c,⇑ a

Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal 713 305, India Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal 713 209, India c Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand 826 004, India b

a r t i c l e

i n f o

Article history: Received 30 July 2008 Received in revised form 21 December 2009 Accepted 9 December 2010 Available online 4 February 2011 Keywords: Ant colony optimization Chaotic ant swarm optimization Genetic algorithm Particle swarm optimization Power system stabilizer Sugeno fuzzy logic

a b s t r a c t In this paper, chaotic ant swarm optimization (CASO) is utilized to tune the parameters of both singleinput and dual-input power system stabilizers (PSSs). This algorithm explores the chaotic and selforganization behavior of ants in the foraging process. A novel concept, like craziness, is introduced in the CASO to achieve improved performance of the algorithm. While comparing CASO with either particle swarm optimization or genetic algorithm, it is revealed that CASO is more effective than the others in finding the optimal transient performance of a PSS and automatic voltage regulator equipped singlemachine-infinite-bus system. Conventional PSS (CPSS) and the three dual-input IEEE PSSs (PSS2B, PSS3B, and PSS4B) are optimally tuned to obtain the optimal transient performances. It is revealed that the transient performance of dual-input PSS is better than single-input PSS. It is, further, explored that among dual-input PSSs, PSS3B offers superior transient performance. Takagi Sugeno fuzzy logic (SFL) based approach is adopted for on-line, off-nominal operating conditions. On real time measurements of system operating conditions, SFL adaptively and very fast yields on-line, off-nominal optimal stabilizer variables. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Usage of fast acting, high gain automatic voltage regulator (AVR) in modern generator excitation system invites the problem of low frequency electromechanical oscillation. Transfer of bulk power across weak transmission lines; any disturbance such as sudden change in loads, change in transmission line parameters, fluctuation in the output of the turbine and faults, etc. also invites the problem of low frequency oscillations (typically in the range of 0.2–3.0 Hz) under various sorts of system operating conditions and configurations. The very common and widely accepted solution, prevailing in the utility houses to address this problem, is the usage of power system stabilizer (PSS). The PSS adds a stabilizing signal to AVR that modulates the generator excitation. Here, its main task is to create a damping electrical torque component (in phase with rotor speed deviation) in turbine shaft, which increases the generator damping. A practical PSS must be robust over a wide range of operating conditions and capable of damping the oscillation modes in power system. From this perspective, the conventional single-input PSS (machine shaft speed (Dxr) as single input to PSS) design approach based on a single-machine-

⇑ Corresponding author. Tel.: +91 0326 2235644; fax: +91 0326 2296563. E-mail addresses: [email protected] (A. Chatterjee), spghoshal nitdgp@ gmail.com (S.P. Ghoshal), [email protected] (V. Mukherjee). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2010.12.024

infinite-bus (SMIB) linearlized model in the normal operating condition has some deficiencies. The two inputs to dual-input PSS, unlike the conventional single-input (Dxr) PSS, are Dxr and DTe. The processed output of the PSS is DVpss that acts as an excitation modulation signal and the desired damping electrical torque component is produced. Modeling of IEEE type PSS2B, PSS3B, and PSS4B are reported in [1] and those models are taken in the present study. Pole-placement [2] or eigenvalue assignment for single-input single-output system has been reported in the literature. A robust PSS tuning approach [3] based upon lead compensator design has been carried out by drawing the root loci for finite number of extreme characteristic polynomials. In [3], such polynomials have been obtained by using Kharitonov theorem to reflect wide loading condition. An approach based on linear matrix inequalities (LMIs) for mixed H2/H1-design under pole region constraints has been reported by Werner et al. [4]. In [4], plant uncertainties are expressed in the form of a linear fractional transformation. Results obtained in [4] are compared to the results obtained in [5] based on quantitative feedback theory. Linear quadratic control [6] has been applied for coordinated control design. The problem has been formulated as a standard LQR and a full feedback control was obtained from the solution that retains the dominant modes of the closed loop system. Structural constraints, such as, simple and decentralized control, feedback of only measured variables, etc. have been in use in

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Nomenclature EB Et H Ka KD

infinite bus voltage generator’s LT side bus voltage inertia constant, MW-s/MVA thyristor exciter gain, damping torque coefficient, pu torque/pu speed deviation osh overshoot of change in rotor speed, pu OF1() first objective function OF2() second objective function P active power, pu Q reactive power, pu rand() random number in the interval [0, 1] tex execution time, s tst settling time of change in rotor speed, s system constant, s T0 Trr terminal voltage transducer time constant, s ush undershoot of change in rotor speed, pu v craziness a random number chosen uniformly in the interval i [v min ; v max ] for the ith ant i i v craziness maximum value of v craziness max i v craziness minimum value of v craziness min i v max ; v min maximum and minimum velocity of the ith ant, i i respectively

power systems for many years and cannot be addressed by a standard LQR. Such a structurally constrained optimal control problem has been solved using the generalized Riccati equation [7] and was applied to a power systems exploiting sparsity [8]. Fuzzy [9], GA-fuzzy [10], neuro-fuzzy [11] are just a few among the other numerous works reported in the literature to tune PSS. Most of these techniques are centered on angular speed deviation (Dxr) as single-input feedback to PSS. Some of these techniques suffer from complexity of computational algorithm, heavy computational burden, memory storage problem and non-adaptive tuning under various system operating conditions and configurations. Some suffer from robustness because of choice of limited number of control variables of PSS, limited number of optimization functions and on-line real time necessity for fast changing PSS variables. Recently, evolutionary programming and intelligent control techniques are being applied to solve many complex optimization problems in engineering applications. With high speed computing tools, these search methods are increasingly being applied in power system planning, design, operation and control problems. The advantage of these methods is that the objective function need not be explicit or differentiable and nonlinearity or non-convexity is not a problem and optimal damping in the closed loop can be obtained. Some algorithms like genetic algorithm (GA), simulated annealing suffer from settings of algorithm parameters and give rise to repeated revisiting of the same suboptimal solutions. Chaotic ant swarm optimization (CASO) is, essentially, a search algorithm that is based on the chaotic behavior of individual ant and the intelligent organization actions of ant colony. This algorithm is reported in the literature to obtain the economic dispatch of power systems by Cai et al. [12] for three different power systems. In the present work, this algorithm, (with some inherent modifications made by the authors of the present work to suit the present application area), is utilized for the purpose of optimal tuning of the PSS variables. The novelty of the present work is the study of the performance of the CASO in designing PSS. Bacteria foraging optimization (BFO), a bio-inspired technique, has been reported by Mishra et al. [13] to establish the potential

Xe equivalent transmission line reactance, pu ki ¼ ri þ jbi ith eigenvalue ri real part of ith eigenvalue bi imaginary part of ith eigenvalue xd damped frequency, rad/s xn undamped natural frequency, rad/s x0 rated speed = 2pf0, elect, rad/s n damping ratio DTe incremental change in electromagnetic torque, pu DTm incremental change in mechanical torque, pu Dv1 incremental change in terminal voltage, pu DVpss incremental change in power system stabilizer output voltage, pu DV max maximum value of incremental change in power system pss stabilizer output voltage, pu DV min minimum value of incremental change in power system pss stabilizer output voltage, pu DVref incremental change in reference voltage, pu Dd rotor angle deviation, pu Dxr speed deviation = xrx0x0 , pu

application of the BFO technique as a soft computing intelligence in power system optimization arena. The main focus of the article [14] was the tuning of single-input PSS installed in multi-machine power system. But what about the tuning of dual-input PSS? Thus, it is very much pertinent to explore a comparative study between these two configurations of PSSs with the assistance of the CASO. A few variants of particle swarm optimizations (PSOs), and GA may be taken as some standard algorithms for the sake of comparison. A fuzzy logic system-based PSS [14] can adjust its variables online according to the environment in which it works and can provide good damping over a wide range of operating conditions. The best PSS, ultimately, derived from this paper (PSS3B) proves to be the most robust model compared to either CPSS, or PSS2B, or PSS4B in damping all electromechanical modes of generator’s angular speed oscillations for all off-line and on-line conditions, step changes of mechanical torque inputs (DTm), reference voltage inputs (DVref) and during/after clearing of system faults. For the present work, off-line conditions are 34 (=81) sets of nominal system operating conditions which is given in SFL table (not shown in the paper). On the other hand, in real time environment these input conditions vary dynamically and become off-nominal. And this necessitates the use of very fast acting SFL to determine the offnominal PSS variables for off-nominal input conditions occurring in real time. Thus, the major and minor objectives of this paper may be documented as follow: Major objectives (pertaining to algorithm performance): (a) To study the performance of CASO in designing PSS. (b) To present the potential benefit of CASO over other PSOs and GA as optimizing technique. (c) To explore the suitability of fuzzy logic-based tuned PSS under various changes in system operating conditions, including occurrence of fault and its subsequent clearing. Minor objectives (pertaining to PSS performance): (a) To compare single-input PSS with dual-input PSS.

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(b) To contrast the generator’s angular speed oscillations for dual-input PSS (namely PSS2B, PSS3B, and PSS4B) equipped system model. (c) To critically examine the best type of PSS for practical implementation under any sort of system disturbances. The rest of the paper is organized as follows. In Section 2, SMIB system and various PSSs under investigation are presented. Mathematical problem for the present study is formulated in Section 3. Optimizing algorithms as implemented to optimal PSS tuning are described in Section 4. Sugeno fuzzy logic as applied to on-line tuning of PSS variables is narrated in Section 5. Input control parameters for the simulation are given in Section 6. Section 7 documents the simulation results. Finally, concluding remarks and scope of future work are outlined in Section 8.

2. SMIB system and various PSSs under investigation An SMIB [15] model, as considered in the present work, is shown in Fig. 1. As the purpose of PSS is to introduce damping torque component, speed deviation is used as logical signal to control generator excitation for conventional power system stabilizer (CPSS). On the other hand, speed deviation and torque deviation are taken as the best pair of inputs for dual-input PSS [16]. The

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block diagram of SMIB system with AVR, thyristor high gain exciter, synchronous generator and PSS is shown in Fig. 2. The generator including AVR, excitation system and transmission-circuit reactance is represented by a two-axis, fourth order model. IEEE type ST1A model of the static excitation system is considered. The block diagrams of different stabilizers under study are shown in Figs. 3–6. The generator with AVR and excitation system along with CPSS/PSS2B/PSS3B/PSS4B is represented by eighth/seventeenth/eighth/eleventh order state matrices, respectively.

3. Mathematical problem formulation The performance specifications of PSS, in terms of damping and speed of response, may be designed in terms of an admissible pole region for the linearized small-signal mode. Maintaining stability and performance over a range of uncertain parameters may be handled by imposing an upper bound on the H1-norm of the closed loop transfer function. In LMI-based approach, these may be handled by designing a single convex optimization problem. A pole region constraint and bound on the H1-norm may be expressed as LMIs. Minimization of a quadratic performance index under constraint, provided a solution exists, may be solved by applying an efficient algorithm. But in the context of the present work, two

Fig. 1. Single-machine-infinite-bus test system.

Fig. 2. Block diagram representation of SMIB system with AVR, thyristor high gain exciter, synchronous generator, and PSS.

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Fig. 3. Block diagram representation of conventional power system stabilizer.

Fig. 4. Block diagram representation of dual-input PSS2B.

Fig. 5. Block diagram representation of dual-input PSS3B.

optimization functions, as described below, are formulated to tune the variables of the PSS. The variables of the PSS (for CPSS (Fig. 3): Kpss, Td1, Td2, Td3, Td4, Td5, Td6; for PSS2B (Fig. 4): Ks1, T1, T2, T3, T4, T5; for PSS3B (Fig. 5): Ks1, Ks2, Td1, Td2, Td3, Td4; for PSS4B (Fig. 6): Ks1, Ks2, T1, T2, T3, T4) are to be so tuned that some degree of relative stability and damping of electromechanical modes of oscillations, minimized undershoot (ush), minimized overshoot (osh), and lesser settling time (tst) of transient oscillations of Dxr are achieved. So, to satisfy all these requirements, two multi-objective optimization functions, OF1() and OF2() which are to be minimized in succession are designed in the following ways. P 2 OF11 = i(r0  ri) if r0 > ri, ri is the real part of the ith eigenvalue. The relative stability is determined by r0. The value of r0 is taken as 6.0 for the best relative stability and optimal transient performance. P 2 OF12 = i(n0  ni) , if (bi, imaginary part of the ith eigenvalue) > 0.0, ni is the damping ratio of the ith eigenvalue and ni < n0. Minimum damping ratio considered, n0 = 0.3. Minimization of this objective function will minimize maximum overshoot.

P 2 OF13 = i(bi) , if ri P r0. High value of bi to the right of vertical line r0 is to be prevented. Zeroing of OF13 will increase the damping further. OF14 = an arbitrarily chosen very high fixed value (say, 106), which will indicate some ri values P0.0. This means unstable oscillation occurs for the particular variables of PSS. These particular PSS variables will be rejected during the optimization technique. So, first multi-objective optimization function is formulated as follows.

OF 1 ðÞ ¼ 10  OF 11 þ 10  OF 12 þ 0:01  OF 13 þ OF 14

ð1Þ

A sensitivity analysis is carried out and is presented in Table 1 to demonstrate the impact of OF11, OF12, OF13, and OF14 on OF1(). Results of interest are bold faced. This table helps us to formulate the various weighting coefficients of (1). The weighting factors ‘10’ and ‘0.01’ in (1) are chosen to impart more weights to OF11, OF12 and to reduce high value of OF13, to make them mutually competitive during optimization. By optimizing OF1(), closed loop system poles are consistently pushed further left of jx axis with simultaneous

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Fig. 6. Block diagram representation of dual-input PSS4B.

Table 1 Sensitivity analysis of OF1(). Operating conditions

Type of PSS

OF1() and its components OF1()

OF11

OF12

OF13

OF14

0.2, 0.2, 1.08, 1.1

CPSS PSS2B PSS3B PSS4B

1344.23 1295.8 415.63 671.71

134.42 118.6 41.56 55.15

0 0 0 0

0 10982.0 0 12021.37

0 0 0 0

0.5, 0.2, 0.4752, 1.0

CPSS PSS2B PSS3B PSS4B

1232.15 1112.30 467.82 597.06

123.22 101.0 46.78 47.25

0 0 0 0

0 10232.3 0 12456.71

0 0 0 0

1.2, 0.6, 0.93, 1.0

CPSS PSS2B PSS3B PSS4B

1228.66 1211.11 436.51 672.8183

122.86 110.4 43.65 54.6811

0 0 0 0

0 10714.1 0 12600.73

0 0 0 0

1.2, 0.6, 0.4752, 0.5

CPSS PSS2B PSS3B PSS4B

1213.17 1199.41 451.50 606.89

121.32 109.49 45.15 48.12

0 0 0 0

0 10451.8 0 12569.39

0 0 0 0

Fig. 7. D-shaped sector in the negative half of s plane.

reduction in imaginary parts also, thus, enhancing the relative stability and increasing the damping ratio above n0. Finally, all closed loop system poles should lie within a Dshaped sector (Fig. 7) in the negative half plane of jx axis for which ri   r0, ni  n0. Selection of such low negative value of r is purposefully chosen. The purpose is to push the closed loop system poles as much left as possible from the jx axis to enhance stability to a great extent. Thorough computation shows that optimization of OF1() is not sufficient for sharp tuning of PSS variables. So, it is essential to design second multi-objective optimization function for sharp tuning of PSS variables. Thus, the second multi-objective optimization function OF2() is formulated as follows:

OF 2 ðÞ ¼ ðosh  106 Þ2 þ ðush  106 Þ2 þ ðt st Þ2  2 d þ ðDxr Þ  106 dt

In (2); osh, ush, tst, dtd ðDxr Þ are all referred to the transient response of Dxr. A sensitivity analysis is carried out and is presented in Table 2 to demonstrate the impact of osh, ush, tst, and dtd ðDxr Þ on OF2(). This table helps us to formulate the various weighting coefficients of (2). Results of interest are bold faced. The constrained optimization problem for the tuning of PSSs is, thus, formulated as follows. Minimize OF1() and OF2() in succession with the help of any optimization technique to get optimal PSS variables, subject to the limits [1,15]:

ðaÞ For CPSS :



175:0  K pss  230:0 i ¼ 1 to 6 0:001  T di  1:0;

8 > < 10  K s1  30 ðbÞ For PSS2B : 0:01 6 T i 6 1:0; i ¼ 1 to 5 > : T 10 ¼ 0:0001 Other parameters are fixed.

 ðcÞ For PSS3B :

100:0  K s1  10:0; 10:0  K s2  100:0 0:005  T di  2:0;

i ¼ 1 to 4

No fixed parameters are required.

 ð2Þ

ðdÞ For PSS4B :

100:0  K s1  10:0; 10:0  K s2  100:0 0:005  T i  2:0;

i ¼ 1 to 4; T 0 ¼ 0:2

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Table 2 Sensitivity analysis of OF2(). Operating conditions

Type of PSS

OF2() and its components OF2()  107

osh  106

ush  106

tst

d ðD dt

xr Þð106 Þ

0.2, 0.2, 1.08, 1.1

CPSS PSS2B PSS3B PSS4B

3.79 3.57 1.46 1.88

896.91 830.30 179.10 860.31

1089.31 682.42 451.48 1505.31

5.999 5.881 3.797 3.976

6.71 4.87 4.86 7.71

0.5, 0.2, 0.4752, 0.9

CPSS PSS2B PSS3B PSS4B

3.76 2.82 1.21 1.59

727.31 657.81 660.30 174.76

1245.20 837.60 454.30 1211.12

5.965 5.224 3.378 3.795

7.02 5.01 4.31 7.87

1.2, 0.6, 0.93, 1.0

CPSS PSS2B PSS3B PSS4B

3.71 2.40 1.05 1.65

693.23 708.04 151.69 720.07

777.31 1938.60 816.07 1456.20

6.000 4.439 3.128 3.717

7.28 7.71 7.29 8.22

1.2, 0.6, 0.4752, 0.5

CPSS PSS2B PSS3B PSS4B

4.13 4.08 1.39 2.04

1280.9 1381.4 441.91 832.10

2854.60 3746.60 1538.20 1795.71

5.618 4.986 3.260 4.063

7.90 7.10 6.20 7.70

4. Optimization algorithms as applied to PSS tuning 4.1. Genetic algorithm Implementation steps of the GA algorithm are shown in Fig. 8. 4.2. Chaotic ant swarm optimization CASO [12] combines the chaotic and self-organization behavior of ants in the foraging process. It includes both the effects of chaotic dynamics and swarm-based search. In the present work, this algorithm is employed to tune the PSS (both single-input and dual-input) variables with some modifications to suit the present application. CASO is based on the chaotic behavior of individual ant and the intelligent organization actions of ant colony. Here, the search behavior of the single ant is ‘‘chaotic’’ at first and the organization variable, ri is introduced to achieve self-organization process of the ant colony. Initially, the influence of the organization variable on the behavior of individual ant is sufficiently small. With the

continual change of organization variable evolving in time and space, the chaotic behavior of the individual decreases gradually, via the influence of the organization variable and the communication of previously best positions with neighbors, the individual ant alters his position and moves to the best one they can find in the search space. The searching area of ants corresponds to the problem search space. In the search space Rl, which is the l-dimensional continuous space of real numbers, the algorithm searches for optima. A population of K ants is considered. These ants are located in a search space S and they try to minimize a function f: S ? R. Each point s in S is a valid solution to the considered problem. The position of an ant i is assigned the algebraic variable symbol Si = (zi1, . . . , zil), where i = 1, 2, . . . , K. Naturally, each variable can be of any finite dimension. During its motion, each individual ant is influenced by the organization processes of the swarm. In mathematical terms, the strategy of movement of a single ant is assumed to be a function of the current position, the best position found by itself. Any member of its neighbors and the organization variable are given by:

Step 1 Initialization a)

Input operating values of P, Q, Xe, Et. Input fixed SMIB parameters,

b) Setting of limits for PSS variables, c)

Setting of GA parameters like mutation probability, crossover ratio,

d) Maximum population number, maximum iteration cycles, e)

Binary value initialization of all the PSS variables strings of the population within limits,

f)

Decoding of the binary strings within parameters’ limits.

Step 2 Determination of SMIB parameters like K1, K2, K3, T3, K4, K5, K6 (Fig. 2) [12]. Step 3 Computation of misfitness function/objective function of each string of the total population. Step 4 Arraigning the values in increasing order from minimum, and selection of top 50% better strings. Step 5 Copying of 50% selected strings over the rest 50% inferior strings to form the total population. Step 6 Crossover. Step 7 Mutation. Step 8 Repeat from Step 3 till the end of the maximum iteration cycle. Step 9 Determine the optimal PSS variables string corresponding to the grand minimum misfitness.

Fig. 8. Implementation steps of GA algorithm for tuning of PSS variables.

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9 > > > =

yi ðnÞ ¼ yi ðn1Þ1þri   zid ðnÞ ¼ zid ðn1Þþ wc V i exp ðð1expðayi ðnÞÞÞð3wd ðzid ðn1Þ d >  > > þ wc V i V i þexp ð2ayi ðnÞþbÞðpid ðn1Þzid ðn1ÞÞ ;  7:5 w d

zid ðnÞ ¼ zid ðnÞ þ signðr4Þ  v craziness i The values of sign(r4) and v respectively.



d

ð3Þ where n is current iteration cycle, n1 previous iteration cycle, yi(n) current state of the organization variable (yi(0) = 0.999), a, b, c positive constants, ri e [0, 0.1] a positive constant and is termed as the organization factor of ant i, zid(n) current state of the dth dimension of the individual ant i, d = 1, 2, . . . , l, wd determines the selection of the search range of dth element of variable in search space, and Vi is determines the search region of ith ant and offers the advantage that ants could search diverse regions of the problem space (Vi = rand()) The neighbor selection can be defined in the following two ways. The first one is the nearest fixed number of neighbors. The nearest m ants are defined as the neighbors of single ith ant. The second way of the number of neighbor selection is to consider the situation in which the number of neighbors increases with iteration cycles. This is due to the influence of self-organization behaviors of ants. The impact of organization will become stronger than before and the neighbors of the ant will increase. That is to say, the number of nearest neighbors is dynamically changed as iteration progresses. The general CASO is a self-organizing system. When every individual trajectory is adjusted toward the successes of neighbors, the swarm converges or clusters in optimal regions of the search space. The search of some ants will fail if the individual cannot obtain information about the best food source from their neighbors. For the present work, the algorithm’s parameters like ri, wd, Vi, a, b, c are different from those reported in [12]. These are, respectively, 1 + ri is replaced by (1.02 + 0.04  rand()), wd = 2.0, Vi = rand(), a = 1, b = 0.1, c = 3. These values are pre-set after a lot of experiments to get the best convergence to optimal solution. In this work, finally, craziness is introduced by the authors of this work as given in (4), which is not present in [12] but taken from [17,18].

signðr4Þ ¼

1;

craziness i

ð4Þ are determined by (5) and (6),

ðr4  0:5Þ

1; ðr4 < 0:5Þ

v crazziness ¼ v crazziness þ ðv crazziness  v crazziness Þ  randðÞ i min max min

ð5Þ

ð6Þ

Introduction of craziness enhances CASO’s ability of searching and convergence to a global optimal solution. Variables’ upper and lower bound restrictions are always present. Ultimately, after a maximum iteration cycles the optimal solution of zid corresponds to global optimal value of fitness function under consideration. A suite of six benchmark test functions [19,20], broadly categorized into three groups viz. (a) unimodal functions, (b) multimodal functions with only a few local minima, and (c) multimodal functions with many local minima, have been tested using CASO by Mukherjee in [21]. In [21], unimodal functions tested are Sphere model and Generalized Rosenbrock’s function. In the same work, multimodal functions with only a few local minima are Mexican Hat function and Six-hump camel back function. Multimodal functions with many local minima, as tested in [21], are Generalized Rastrigin’s function and Generalized Griewank function. Promising performances have been obtained in [21] while applying novel CASO on these benchmark test functions. These results motivate the authors of the present work to apply the same algorithm for tuning the different PSSs equipped with AVR model. Implementation steps of CASO algorithm are shown in Fig. 9. A flow chart of the whole algorithm is depicted in Fig. 10.

5. Sugeno fuzzy logic as applied to on-line tuning of PSS variables The behavior of a synchronous generator connected to a network depends on, among other things, its position in this network, the operating conditions (in particular, the reactive power flow and the voltage map), the network topology and the generation schedule. Usage of the desired optimization technique yields a distinct set of controller variables for different operating conditions. Under drastic change in operating conditions (e.g. different circuit topologies) the nominal controller is not necessarily going to be tuned enough to yield satisfactory performance. For on-line tuning of the variables of PSS, very fast acting SFL is to be adopted.

Step 1 Initialization a) Input operating values of P, Q, X e, Et. Input fixed SMIB parameters, b) Setting of limits for PSS variables, c) Setting of CASO parameters, d) Maximum population number, maximum iteration cycles, e) Real value initialization of all the PSS variables strings of the population within limits. Step 2 Determination of SMIB parameters like K1, K2, K3, T3, K4, K5, K6 (Fig. 2) [12]. Step 3 Computation of misfitness function/objective function of each string of the total population. Step 4 Determination of the best string corresponding to minimum misfitness/objective function value. Step 5 Updating the strings of the population using CASO algorithm (described in section 4.2). Step 6 Repeat from Step 3 till the end of the maximum iteration cycle. Step 7 Determine the optimal PSS variables string corresponding to the grand minimum misfitness.

Fig. 9. Implementation steps of CASO algorithm for tuning of PSS variables.

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Start Initialize input operating parameters ( P , Q , X e and E t ), limits of PSS variables, CASO parameters, maximum population number, maximum iteration number Real value initialization of all the PSS variable strings of the population Compute SMIB parameters (K1, K2, K3, T3, K4, K5, K6)

Generation Index Repeat next generation

Population Index

Repeat next population

Compute misfitness function/ objective function of each string Set variables’ values equal to max/min values, as the case may be

Compute the best string corresponding to minimum misfitness

No

Are PSS variables within limits? Yes

Compute new values of PSS variables incorporating velocity vector using CASO algorithm and find pBest , gBest

Move each particle to new position using position updating equation No

Is pBest = gBest ? Yes

Output PSS variables

Stop Fig. 10. Flowchart of CASO algorithm as implemented for PSS tuning.

5.1. Sugeno fuzzy logic for on-line tuning of PSS variables The whole process of SFL [14] can be categorized into three steps viz. Fuzzification of input operating conditions, Sugeno fuzzy inference and Sugeno defuzzification. The whole process of SFL [14] involves three steps as discussed below. 5.1.1. Fuzzification The first step is fuzzification of input operating conditions as active power (P), reactive power (Q), equivalent transmission line reactance (Xe) and Generator’s LT side bus voltage (Et) in terms of fuzzy subsets (Low, Medium, High). These are associated with overlapping triangular membership functions. SFL rule base table is formed, each composed of four nominal inputs and corresponding nominal optimal PSS variables as output determined by any of the optimizing techniques dealt with. The respective nominal central values of the input subsets of P are (0.2, 0.7, 1.2), those of Q are (0.2, 0.6, 1.0), those of Xe are (0.4752, 0.77, 1.08) and those of Et are (0.5, 0.8, 1.1), respectively, at which membership values are unity (Fig. 11). These are nominal input conditions also. Sugeno fuzzy rule base table consists of 34 (=81) logical input conditions or sets (SFL tables calculated for different PSS structures investigated), each composed of four nominal inputs. Each logical input set corresponds to nominal optimal PSS variables as output.

5.1.2. Sugeno fuzzy inference For on-line imprecise values of input parameters, firstly their subsets in which the values lie are determined with the help of ‘‘IF’’, ‘‘THEN’’ logic and corresponding membership values are determined from the membership functions of the subsets. From Sugeno fuzzy rule base table, corresponding input sets and nominal PSS variables are determined. Now, for each input set being satisfied, four membership values like lP, lQ, lX e and lEt and their minimum lmin are computed. For the input logical sets, which are not satisfied because variables do not lie in the corresponding fuzzy subsets, lmin will be zero. For the non-zero lmin values only, nominal PSS variables corresponding to fuzzy sets being satisfied are taken from the Sugeno fuzzy rule base table. 5.1.3. Sugeno defuzzification Sugeno defuzzification yields the defuzzified, crisp output for each parameter of PSS. Final crisp PSS parameter output is given by:

P K crisp ¼

i

ðiÞ lmin  Ki ðiÞ i lmin

P

ð7Þ

where i corresponds to input logical sets being satisfied among 81 input logical sets, Ki is corresponding nominal PSS parameter. Kcrisp

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6.1. Input operating conditions In the present simulation work, P, Q and Et are varied individually and independently in steps of 0.1 over the respective intervals (0.2, 1.2), (0.2, 1.0) and (0.5, 1.1). The various values of Xe (refer Fig. 1) are (Tr + (Xe1//Xe2)), (Tr + Xe1) and (Tr + Xe2), respectively. These values are j0.4752, j0.65 and j1.08, respectively. Thus, 11 P values, 13 Q values, three Xe values and seven Et values are considered. Therefore, a family of 3003 (=11  13  3  7) input operating conditions is considered for the simulation. The main focus of this paper is to tune the PSS variables that maintain a damping ratio of at least 0.3 and the real part of all eigenvalues more than 6.0 simultaneously for all modes of the family.

Fig. 11. Fuzzification of input operating conditions (P, Q, Xe and Et). ðiÞ

is crisp PSS parameter. lmin is the minimum membership value corresponding to ith input logical set being satisfied. 5.2. Link between SFL and optimization algorithm SFL interpolates the variables accurately. Optimizing algorithm is not required to run in real time rather SFL is required to run in real time. The optimized variables corresponding to nominal model variables (P, Q, Xe and Et) as determined by the optimizing algorithm, are needed to be stored in memory (in the form of SFL table). SFL only utilizes this SFL table in real time. SFL simply acts as a piecewise linear interpolator within restricted region. SFL works on simple if-then-else logical loops and a very few additions and multiplications. Hence, the computational burden of SFL is very low. Its practical implementation is easy. It is to be noted here, in case of SFL application, that more the number of fuzzy subsets, more accurate result will be obtained at the cost of increased computational burden. For real time determination of optimal model variables fast, adaptive SFL is adopted.

6.2. For SMIB system [1, 15] Inertia constant, H = 5, M = 2H, nominal frequency, f0 = 50 Hz, 0.995 6 |Eb | 6 1.0, the angle of Eb = 0°, 0.2 6 P 6 1.2, 0.2 6 Q 6 1.0, 0.4752 6 Xe 6 1.08, 0.5 6 Et 6 1.1. In the block diagram representation of generator with exciter and AVR; Trr = 0.02s, Ka = 200.0. 6.3. For GA Number of parameters depends on problem variables (PSS configuration), number of bits = (number of parameters) 8 (for binary coded GA, as considered for the present work), population size = 50, maximum number of iteration cycles = 200, mutation probability = 0.001, crossover rate = 80%. 6.4. For CASO Number of problem variables depends upon the PSS structure under investigation. All the parameters of the algorithm are given in Section 4.

6. Input control parameters 7. Simulation and results For simulation, step perturbation of 0.01 pu is applied either in reference voltage or in mechanical torque. The simulation is implemented in MATLAB 7.1 software on a PC with PIV 3.0 G CPU and 512M RAM. The followings are the different input control parameters.

Input system operating conditions are P, Q, Xe, and Et. Each operating condition has three fuzzy logical sets as Low, Medium, and High. Thus, total number of nominal input operating condition sets are 34 (=81). Sugeno fuzzy rule base table (not shown) is obtained

Table 3 GA, HPSOIWA, HPSOCFA, and CASO-based comparison of OF2() values for CPSS, PSS2B, PSS3B and PSS4B-based systems. Value of OF2() (107)

Sl. no.

Operating conditions (P, Q, Xe, Et; all are in pu)

Algorithms

CPSS

PSS2B

PSS3B

PSS4B

1

0.2, 0.2, 0.4752, 1.1

GA-SFL HPSOIWA-SFL [22] HPSOCFA-SFL [22] CASO-SFL

7.45 5.91 5.16 3.24

6.23 5.54 4.42 2.97

1.42 1.35 1.05 0.99

2.90 2.64 1.71 1.47

2

0.5, 0.2, 0.4752, 1.0

GA-SFL HPSOIWA-SFL [22] HPSOCFA-SFL [22] CASO-SFL

7.85 7.56 5.46 4.19

7.33 7.13 4.39 4.04

2.47 1.51 1.35 1.12

3.17 2.86 1.60 1.15

3

0.75, 0.50, 0.4752, 0.50

GA-SFL HPSOIWA-SFL [22] HPSOCFA-SFL [22] CASO-SFL

7.16 5.46 4.23 3.14

5.18 4.38 2.38 2.47

2.71 1.62 1.09 0.87

3.88 2.58 1.74 1.19

4

0.95, 0.30, 0.4752, 0.5

GA-SFL HPSOIWA-SFL [22] HPSOCFA-SFL [22] CASO-SFL

7.29 6.45 3.45 3.01

5.85 5.17 2.84 2.06

2.02 1.61 1.51 1.15

4.58 3.54 2.01 1.45

5

1.2, 0.6, 1.08, 0.5

GA-SFL HPSOIWA-SFL [22] HPSOCFA-SFL [22] CASO-SFL

8.96 7.59 6.12 4.19

8.72 6.01 5.01 3.89

3.74 2.91 2.67 2.01

4.39 4.00 2.75 2.89

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for all nominal input operating condition sets by applying each optimization technique. Each table contains corresponding 81 sets of distinct nominal optimal PSS variables. Optimized PSS variables, as determined by any of the optimization technique, are substituted in MATLAB-SIMULINK model of the system to obtain the transient response profiles. Final values of OF1(), and OF2() are determined from the end of the optimization. Final eigenvalue, undamped frequencies, damped frequencies and damping ratio are also determined by the optimization technique at the end of the optimization. The major and minor observations of the present work are documented below. Results of interest are bold faced in the respective table. 7.1. Major observations (pertaining to algorithm performance) 7.1.1. Comparative optimization performance of the optimization techniques In [22], Mukherjee and Ghoshal have discussed two basic variants of PSOs like hybrid particle swarm optimization with inertia weight approach (HPSOIWA), and hybrid particle swarm optimiza-

tion with constriction factor approach (HPSOCFA) and have applied these PSOs for the tuning of dual-input PSSs. The same results are reproduced here for the sake of comparison to show the potential merits between the proposed approach in this paper using CASO and PSOs [22]. Table 3 depicts the comparative GA, HPSOIWA [22], HPSOCFA [22], and CASO-based optimal transient response characteristics (in terms of OF2() value) of different PSS equipped system model. From this table, it is observed that the CASO-based optimization technique offer lesser value of OF2() than either GA, or HPSOIWA [22], or HPSOCFA-based one [22]. Thus, CASO-based optimization technique offers better results than GA/HPSOIWA/ HPSOCFA-based results. With regard to optimization performances of the optimizing algorithms, as depicted in Table 3 and Table 4, it may be concluded that the CASO-based approach offers the lower values of OF1() and OF2() as compared to GA-based approach for the same input operating conditions. Comparative transient performances of Dxr and convergence profiles of OF1() for GA and CASO-based optimization for all the four PSS modules (CPSS, PSS2B, PSS3B, and PSS4B) are depicted in Fig. 12 and Fig. 13, respectively. From Fig. 12, it may be concluded that the transient

Table 4 GA, and CASO-based comparison of OF1() values under different operating conditions for CPSS, PSS2B, PSS3B, and PSS4B (operating condition: P = 1.0, Q = 0.6, Xe = 0.93, Et = 0.5; all are in pu). Type of PSS

Algorithm-SFL

PSS variables

OF1()

tex (s)

CPSS

GA-SFL CASO-SFL

181.15, 0.162, 0.080, 0.096, 0.053, 0.187, 0.198 175.00, 0.005, 0.005, 0.001, 0.001, 0.179, 0.001

1449.69 1428.61

13.45 5.20

PSS2B

GA-SFL CASO-SFL

10.00, 0.174, 0.015, 0.013, 0.191, 0.044 10.00, 0.195, 0.306, 0.010, 0.010, 0.0100

1252.42 1228.45

20.07 10.56

PSS3B

GA-SFL CASO-SFL

10.35, 99.65, 0.854, 0.098, 0.340, 0.192 10.00, 24.00, 2.000, 1.493, 0.005, 0.093

535.20 417.93

10.42 5.15

PSS4B

GA-SFL CASO-SFL

12.11, 33.20, 0.922, 0.005, 0.961, 0.005 10.00, 10.00, 0.005, 0.188, 0.206, 0.005

785.41 581.03

17.12 6.46

Fig. 12. Comparative GA and CASO-based transient response profiles of Dxr for CPSS, PSS2B, PSS3B, and PSS4B for 0.01 pu simultaneous change in DTm and DVref (P = 1.0, Q = 0.2, Xe = 1.08;, Et = 0.9, all are in pu).

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stabilization performance of CASO-based optimization is better than that of GA-based one. From Fig. 13, it is observed that CASO-based optimization technique offers lesser OF1() value as compared to GA-based one, for whatever may be the PSS model under consideration. Thus, CASO offers much better optimal performance and it may be accepted as a true optimizing algorithm for the power system-based application as considered in the present work. 7.1.2. Performance improvement of CASO with craziness In the present work, the concept of craziness (a novel concept proposed by Mukherjee and Ghoshal in the literature [14]) is blended with original CASO algorithm [12] with an attempt to have improved performance of CASO algorithm reported in [12]. Table 5 shows how improved performance is taking place with the inclusion of craziness concept. This table shows for a particular input operating condition, improved OF1() value is achieved for all the four PSS structures. The comparative convergence profiles of OF1() based on CASO without craziness and CASO with craziness

for CPSS, PSS2B, PSS3B, and PSS4B are depicted in Fig. 14 corresponding to a nominal operating condition of P = 0.2, Q = 0.2, Xe = 1.08, Et = 1.0 (all are in pu). From this figure, the objective function value, OF1() corresponding to CASO with craziness is found to converge to the lesser minimum value faster than the other. It is also noticed from this figure that the OF1() value of PSS3B corresponding to CASO with craziness is the lowest one. 7.1.3. Fuzzy logic-based tuning of PSS variables under changes in system operating conditions LT bus fault of duration 220 ms at the instant of 2.0 s is simulated for different PSS equipped system model and the corresponding comparative transient response profiles of Dxr for both GA-SFL and CASO-SFL-based approaches are plotted in Fig. 15 for all the PSSs. Fig. 15 reveals that CASO-SFL response exhibits better response than GA-SFL based one irrespective of the PSS model. A close look into this figure also reveals that after the creation of the fault, the CASO-SFL-based response for PSS3B equipped system model recovers from this abnormal situation with much lesser

Fig. 13. Comparative GA and CASO -based transient response profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B for 0.01 pu simultaneous change in DTm and DVref (P = 1.0, Q = 0.2, Xe = 1.08, Et = 0.9, all are in pu).

Table 5 Comparison of OF1() values based on CASO without craziness, and CASO with craziness for all the four PSSs, nominal input operating conditions being P = 0.2, Q = 0.2, Xe = 1.08, Et = 1.0; all are in pu. Type of PSS

Concept of craziness

PSS variables

OF1()

CPSS

CASO without craziness CASO with craziness

175.00, 0.734, 1.00, 0.001, 0.001, 0.001, 0.001 175.00, 0.082, 0.005, 0.001, 0.001, 0.001, 0.001

1375.53 1299.39

PSS2B

CASO without craziness CASO with craziness

10.00, 0.375, 0.010, 1.00, 0.010, 0.010 10.00, 1.00, 0.011, 0.253, 0.010, 0.010

1227.92 1200.61

PSS3B

CASO without craziness CASO with craziness

10.00, 10.00, 0.965, 1.510, 0.040, 0.005 10.00, 10.00, 0.934, 2.00, 0.012, 0.005

449.23 442.41

PSS4B

CASO without craziness CASO with craziness

10.00, 10.00, 0.229, 0.202, 0.191, 0.019 10.00, 10.00, 0.271, 0.103, 0.005, 0.017

560.42 518.35

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Fig. 14. Comparative CASO (with craziness and without craziness)-based convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B (P = 0.2, Q = 0.2, Xe = 1.08, Et = 1.0, all are in pu).

Fig. 15. Comparative GA-SFL, and CASO-SFL-based transient response profiles of Dxr for the generator equipped with different PSS under change in operating conditions.

fluctuation in angular speed as compared to that of GA-SFL-based counter part. Table 6 depicts the system model parameters as determined by SFL for all the PSS models. Thus, CASO-SFL-based

model exhibits better response having lesser amplitude of angular speed deviation under fault and subsequent clearing condition yielding better dynamic transient performance than GA-SFL-based

A. Chatterjee et al. / Electrical Power and Energy Systems 33 (2011) 657–672

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one for PSS3B equipped system model. Hence, PSS3B proves to be much less susceptible to faults because PSS3B settles all the state deviations to zero much faster than any other PSSs. 7.2. Minor observations (pertaining to PSS performance) 7.2.1. Analytical eigenvalue-based comparison of PSSs GA-SFL, and CASO-SFL based comparison of OF1() values of CPSS, PSS2B, PSS3B, and PSS4B are shown in Table 4 for different system operating conditions. From Table 4, it is observed that the value of OF1(), irrespective of the optimization technique adopted, is the least one for PSS3B. It establishes the performance of PSS3B to be the best one. For PSS3B equipped system model, majority of the eigenvalues are within D-shaped sector (Fig. 7) which yield lesser values of OF11(), OF12(), and OF13(). Hence, the value of OF1() is very less for PSS3B equipped system model. On the other hand, majority of the eigenvalues for PSS2B-based system are outside the Dshaped but very close to and right side of (r0, j0) point. This yields higher values of OF11(), OF12(), and OF13(). The value of OF1() is more for this system. Thus, from the eigenvalue analysis it may be concluded that a considerable improvement has occurred in the transient performance for the PSS3B-based system. 7.2.2. Comparative performances of PSSs in terms of analytical transient responses characteristics From Table 3 and Table 4, it may also be inferred that the transient stabilization performance of dual-input PSS equipped system model is better than single-input counter part. Comparing dual-input PSSs, it is also observed that the transient stabilization performance of PSS3B equipped system model is superior to that of others. PSS3B equipped system model offers lesser values of osh, ush, tst, dtd ðDxr Þ and, thereby, lesser values of OF2(). Fig. 16 depicts that the comparative optimal transient performance of the different PSS equipped power system model corresponding to an operating condition of P = 0.5, Q = 0.2, Xe = 0.93, Et = 1.0 (all are in pu) for 0.01 pu simultaneous change in DTm and DVref. From this figure, it is noticed that the transient stabilization performance of dual-input PSS is better than that of single input one. Among the dual-input PSSs, the performance of PSS3B is established to be the best one for this specific application of SMIB system model under study. From Fig. 16, it is apparently observed that CPSS outperforms the other three PSSs in terms of magnitudes of ush and osh. But the duration of undershoot and tst are very large for CPSS. The second objective function OF2(), given in (2), is designed in such a fashion

Fig. 16. Comparative CASO-based transient response profiles of Dxr for CPSS, PSS2B, PSS3B and PSS4B for 0.01 pu simultaneous change in DTm and DVref (P = 0.5, Q = 0.2, Xe = 0.93, Et = 1.0, all are in pu).

Fig. 17. Comparative CASO-based convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B (P = 1.0, Q = 0.2, Xe = 0.4752, Et = 1.0, all are in pu).

Table 6 PSS variables under change in off-nominal operating conditions. Off-nominal operating conditions (P, Q, Xe, Et; all are in pu)

Type of PSS

Algorithms-SFL

PSS variables

1.0, 0.5, 0.4752, 1.0 (pre-fault)

CPSS

GA-SFL CASO-SFL GA-SFL CASO-SFL GA-SFL CASO-SFL GA-SFL CASO-SFL

179.68, 0.030, 0.032, 0.245, 0.005, 0.144, 0.401 230.00, 0.177, 0.005, 0.001, 0.001, 0.001, 0.001 10.00, 0.143, 0.047, 0.089, 0.011, 0.138 10.00, 0.098, 0.010, 0.010, 0.039, 0.022 10.00, 83.47, 1.992, 0.995, 0.005, 0.036 10.00, 10.00, 2.00, 1.142, 0.005, 0.077 27.23, 55.70, 0.440, 0.005, 0.044, 0.005 10.00, 10.00, 0.579, 0.005, 0.202, 0.005

PSS2B PSS3B PSS4B LT bus fault of duration 220 ms, and subsequent clearing

CPSS/PSS2B/PSS3B/PSS4B

GA-SFL CASO-SFL

No change in variables

0.2, 0.2, 1.08, 1.1 (post-fault)

CPSS

GA-SFL CASO-SFL GA-SFL CASO-SFL GA-SFL CASO-SFL GA-SFL CASO-SFL

182.32, 0.109, 0.090, 0.071, 0.038, 0.088, 0.075 175.00, 0.065, 0.005, 0.001, 0.001, 0.001, 0.001 10.39, 0.022, 0.150, 0.177, 0.179, 0.077 10.00, 0.282, 0.010, 0.795, 0.010, 0.010 18.09, 10.00, 1.525, 0.994, 0.005, 0.005 10.00, 10.00, 0.861, 2.00, 0.005, 0.005 10.35, 69.06, 0.153, 0.005, 0.852, 0.005 10.00, 10.00, 0.139, 0.005, 0.076, 0.014

PSS2B PSS3B PSS4B

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that it will take care of ush, osh, tst dtd ðDxr Þ and as the value of tst is very large for CPSS, it offers the highest value of OF2() among all the PSSs taken into consideration for this study. Thus, CPSS is exhibits poor performance for the specific application under study.

7.2.3. Comparative performances of PSSs based on convergence profiles The comparative CASO-based convergence profiles of OF1() for CPSS, PSS2B, PSS3B, and PSS4B are depicted in Fig. 17 corresponding to an operating condition of P = 1.0, Q = 0.2, Xe = 0.4752, Et = 1.0 (all are in pu). From this figure, the objective function value, OF1() corresponding to PSS3B is found to be the least one and, hence,

PSS3B proves to be the best PSS among the PSSs considered for the specific application under study.

7.2.4. Transient performances of PSSs under different perturbations Fig. 18 displays MATLAB-SIMULINK-based transient response profiles of Dv1 and Dxr for PSS3B equipped generator. This figure helps to conclude that PSS3B damps the oscillations of Dv1 and Dxr very quickly under the system perturbations considered. Real parts of some eigenvalues for CPSS/PSS2B are always either equal to, or greater than r0 in the negative half plane of jx axis. A few eigenvalues are always outside D-shaped sector (Fig. 7) for any operating condition. So, objective function values (OF1()) are

Fig. 18. MATLAB-SIMULINK-based transient response profiles of Dv1 and Dxr under different perturbation conditions for generator equipped with PSS3B.

Table 7 GA, and CASO-based results of eigenvalue analysis corresponding to operating conditions P = 0.95, Q = 0.30, Xe = 1.08, Et = 0.5; all are in pu. Type of PSS

Algorithms-SFL

Damping ratio (n)

Undamped natural frequency (xn), rad/s

Corresponding damped frequency (xd), rad/s

Lowest

Highest

Lowest

Lowest

Highest

Highest

CPSS

GA-SFL CASO-SFL

0.16 0.41

0.57 0.67

0.38 0.61

2.45 1.49

0.42 0.52

0.59 4.89

PSS2B

GA-SFL CASO-SFL

0.26 0.69

0.97 1.19

0.48 0.78

3.33 13.59

0.32 0.63

0.79 7.99

PSS3B

GA-SFL CASO-SFL

0.36 0.79

0.98 0.99

0.17 0.51

1.93 0.99

0.15 0.53

1.8 0.58

PSS4B

GA-SFL CASO-SFL

0.2 0.72

0.95 1.05

0.55 0.45

1.22 3.15

0.54 0.38

0.36 3.25

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8.2. Minor conclusions

Fig. 19. Comparative terminal voltage, Dvt (pu) profiles for PSS equipped nonlinear system model and linear system model, for 0.01 pu change in DTm and no change in DVref (P = 1.0, Q = 0.2, Xe = 1.08, Et = 1.0;, all are in pu).

always higher (Table 4). Much lower negative real parts of eigenvalues of PSS3B and PSS4B (not shown) cause higher relative stability than CPSS/PSS2B. Larger reductions of xn and xd for some electromehanical oscillations are due to higher damping ratios ni  n0 for those particular modes, in case of PSS3B and PSS4B (Table 7). 7.2.5. Comparative performances of nonlinear and linear model for PSS3B Terminal voltage response profile of an SMIB power system based on the complete 7th order nonlinear model of the generator system [23,24] is compared to that of the same with twoaxis, fourth order linear model [15]. This comparison is carried out for PSS3B equipped power system model for 0.01 pu change in DTm and no change in DVref, operating conditions being P = 1.0, Q = 0.2, Xe = 1.08, Et = 1.0, all are in pu. The comparative response profiles are portrayed in Fig. 19. From this figure it is evident that the terminal voltage response under the simulated input operating conditions exhibits improved stabilization performance for nonlinear system model as compared to that for linear one. 7.2.6. Optimization time For the same iteration cycle, the optimization time taken by CASO is very much lesser as compared to GA (as shown in Table 4, last column). 8. Conclusion The present work concludes the followings. 8.1. Major conclusions i. Among the optimization techniques considered, chaotic ant swarm optimization with craziness yields the best optimization performance. ii. Under change in operating conditions, faulted or post faulted conditions, the variables of PSS as determined by CASO-SFL is found to be superior than those determined by GA-SFL counter part. iii. For on-line, off-nominal system operating conditions, fast acting Takagi Sugeno fuzzy logic is suitable for determination of off-nominal PSS variables. Moreover, the computational burden of the SFL is very low and its practical implementation is easy.

i. Dual-input PSS offers better transient performance than single-input counter part. ii. Transient performances of dual-input PSS3B among dualinput PSS family (namely PSS2B, PSS3B, and PSS4B) offer less undershoot, less overshoot and less settling time as compared to other dual-input PSSs. iii. Results obtained from the analysis of PSS3B equipped system configuration shows that the dynamic stabilization performance offered by PSS3B is better with change in operating conditions and configurations, faulted or postfault conditions. iv. PSS3B equipped system model damps out the system perturbations very quickly. v. The seventh order nonlinear model of the system exhibits better performance over linear fourth order model of the system. Thus, CASO with craziness may be appreciated as a powerful optimizing technique for power systems application and the practical applicability of dual-input PSS, especially PSS3B, may be established for SMIB system. More rigorous testing of PSS3B for multi-generator system with the help of any optimizing tool may be under consideration for some future works. Acknowledgement The comments of the reviewers were instrumental in improving this paper from its original version. References [1] IEEE Digital Excitation System Subcommittee Report. Computer models for representation of digital-based excitation systems. IEEE Trans EC 1996;11(3): 607–15. [2] El-Sherbiny MK, Hasan MM, El-Saady G, Yousef AM. Optimal pole shifting for power system stabilization. Electr Power Sys Res 2003;66(3):253–8. [3] Soliman H, Elshafei AL, Shaltout A, Morsi M. Robust power system stabilizer. IEE Proc Elect Power Appl 2000;147(5):285–91. [4] Werner H, Korba P, Chen Yang T. Robust tuning of power system stabilizers using LMI techniques. IEEE Trans CST 2003;11(1):147–52. [5] Shrikant Rao P, Sen I. Robust tuning of power system stabilizers using QFT. IEEE Trans CST 1999;7(4):478–86. [6] Hopkins WE, Medanic J, Perkins WR. Output feedback pole placement in the design of suboptimal linear quadratic regulators. Int J Control 1981;34:593–612. [7] Geromel JC. Methods and techniques for decentralized control systems. Italy: Clup; 1987. [8] Costa AS, Freitas FD, Silva AS. Design of decentralized controllers for large power systems considering sparsity. IEEE Trans PS 1997;12(1):144–52. [9] Hiyama T, Kugimiya M, Satoh H. Advanced PID type fuzzy logic power system stabilizer. IEEE Trans EC 1994;9(3):514–20. [10] Abido MA, Abdel-Magid YL. A genetic based fuzzy logic power system stabilizer for multimachine power systems. In: Proc IEEE conf; 1997. p. 329– 34. [11] Abido MA, Abdel-Magid YL. A hybrid neuro-fuzzy power system stabilizer for multimachine power systems. IEEE Trans PS 1998;13(4):1323–30. [12] Cai Jiejin, Ma Xiaoqian, Li Lixiang, Yang Yixian, Peng Haipeng, Wang Xiangdong. Chaotic ant swarm optimization to economic dispatch. Electr Power Sys Res 2007;77:1373–80. [13] Mishra S, Tripathy M, Nanda J. Multi-machine power system stabilizer design by rule based bacteria foraging. Electr Power Sys Res 2007;77(12):1595–607. [14] Mukherjee V, Ghoshal SP. Intelligent particle swarm optimized fuzzy PID controller for AVR system. Electr Power Sys Res 2007;77(12):1689–98. [15] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994. [16] Mukherjee V, Ghoshal SP. Towards input selection and fuzzy based optimal tuning of multi-input power system stabilizer. In: Proceedings of the fourteen national power sys conf (NPSC 2006), paper no. 51, IIT Roorkee, India; 2006. [17] Mukherjee V, Ghoshal SP. Comparison of intelligent fuzzy based AGC coordinated PID controlled and PSS controlled AVR system. Int J Electr Power Energy Syst 2007;29(9):679–89. [18] Roy Ranjit, Ghoshal SP. A novel crazy swarm optimized economic load dispatch for various types of cost functions. Int J Electr Power Energy Syst 2008;30(4):242–353.

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