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Comparative seeker and bio-inspired fuzzy logic controllers for power system stabilizers
Electrical Power and Energy Systems 33 (2011) 1728–1738
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Comparative seeker and bio-inspired fuzzy logic controllers for power system stabilizers Binod Shaw a, Abhik Banerjee a, S.P. Ghoshal b, V. Mukherjee c,⇑ a
Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal, India Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, India c Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand, India b
a r t i c l e
i n f o
Article history: Received 18 May 2010 Received in revised form 16 January 2011 Accepted 12 August 2011 Available online 20 October 2011 Keywords: Bacteria foraging optimization Genetic algorithm Power system stabilizer Seeker optimization algorithm Sugeno fuzzy logic
a b s t r a c t Seeker optimization algorithm (SOA) is a new heuristic population-based search algorithm. In this paper, SOA is utilized to tune the parameters of both single-input and dual-input power system stabilizers (PSSs). In SOA, the act of human searching capability and understandings are exploited for the purpose of optimization. In SOA-based optimization, the search direction is based on empirical gradient by evaluating the response to the position changes and the step length is based on uncertainty reasoning by using a simple fuzzy rule. Conventional PSS (CPSS) and the three dual-input IEEE PSSs (namely PSS2B, PSS3B and PSS4B) are optimally tuned to obtain the optimal transient performances. From simulation study it is revealed that the transient performance of the dual-input PSS is better than the single-input PSS. It is further explored that among the dual-input PSSs, PSS3B offers the best optimal transient performance. While comparing the SOA with recently reported optimization algorithms like bacteria foraging optimization (BFO) and genetic algorithm (GA), it is revealed that the SOA is more effective than either BFO or GA in finding the optimal transient performance. 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 parameters. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The problem of low frequency electromechanical oscillations arises from the usage of fast acting, high gain automatic voltage regulator (AVR) in modern generator excitation system [1]. Any form of disturbances such as sudden change in loads, change in transmission line parameters, fluctuation in the output of the turbine and faults, 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. Transfer of bulk power across weak transmission lines may also invite this problem of low frequency oscillation. The usage of power system stabilizer (PSS) is a very common and widely accepted solution, prevailing in the utility houses, to tackle this problem. The PSS adds a stabilizing signal to AVR which modulates the generator excitation. 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 conven⇑ Corresponding author. Tel.: +91 0326 2235644; fax: +91 0326 2296563. E-mail addresses: [email protected] (B. Shaw), abhik_banerjee@ rediffmail.com (A. Banerjee), [email protected] (S.P. Ghoshal), vivek_ [email protected] (V. Mukherjee). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.08.015
tional single-input PSS (machine shaft speed (Dxr) as single input to the PSS) design approach based on a single-machineinfinite-bus (SMIB) linearized model in the normal operating condition has some deficiencies. On the other hand, the two inputs to dual-input PSS are machine shaft speed (Dxr) and the change in electrical torque (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 [2] and those models are taken in the present study. Pole-placement or eigenvalue assignment for single-input single-output system has been reported in literature [3]. A robust PSS tuning approach [4] based upon lead compensator design has been carried out by drawing the root loci for finite number of extreme characteristic polynomials. In [4], 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. [5]. In [5], plant uncertainties are expressed in the form of a linear fractional transformation. Results obtained in [5] are compared to the results obtained in [6] based on quantitative feedback theory. Linear quadratic control [7] has been applied for coordinated control design. The problem has been formulated as a standard
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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, have been in use in 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 [8] and was applied to power systems exploiting sparsity [9]. Fuzzy, GA-fuzzy, neuro-fuzzy 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 online 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 GA, simulated annealing suffer from settings of algorithm parameters and give rise to repeated revisiting of the same suboptimal solutions. The application of particle swarm optimization (PSO) [10] for PSS tuning has attracted more focus of the researchers for this purpose. Seeker optimization algorithm (SOA) [11,12] is, essentially, a new population based heuristic search algorithm. It is based on human understanding and searching capability for finding an optimum solution. In SOA, optimum solution is regarded as one which is searched out by a seeker population. The underlying concept of SOA is very easy to model and relatively easier than other optimization techniques prevailing in the literature. The highlighting characteristic features of this algorithm are the following:
Thus, it is very much pertinent to explore a comparative study between the SOA-based PSS tuning and the BFO/GA-based PSS tuning. Based on the transient performance, it is also important to draw some comparative logical conclusion between single-input PSS and dual-input PSSs with the assistance of SOA [11,12,15– 18], BFO and GA in line with [14]. A fuzzy logic system-based [14] PSS can adjust its parameters on-line 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 proves to be the most robust model 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 Sugeno fuzzy logic (SFL) table. 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 off-nominal PSS parameters for off-nominal input operating conditions occurring in real time. Thus, the major and minor objectives of this paper may be documented as follow:
(a) search direction and step length are directly used in this algorithm to update the position, (b) proportional selection rule is applied for the calculation of search direction which can improve the population diversity so as to boost global search ability and decrease the number of control parameters making it simpler to implement, and (c) fuzzy reasoning is used to generate the step length because the uncertain reasoning of human searching could be best described by natural linguistic variables, and a simple if–else control rule.
(a) to compare the transient performance of single-input PSS with dual-input PSSs, (b) to contrast the generator’s angular speed oscillations for dual-input PSSs (namely PSS2B, PSS3B, and PSS4B) equipped system model, and (c) to critically examine the best type of PSS for practical implementation under any sort of system disturbances.
In the present work, this SOA algorithm is utilized for the purpose of optimal tuning of the PSS parameters. The novelty of the present work is the study of the performance of the SOA in designing the PSS parameters and to compare the optimizing performance of this algorithm with those reported in the stateof-the-art literature for the same purpose. Bacteria foraging optimization (BFO), a bio-inspired technique, has been reported by Mishra et al. [13] to establish the potential application of the BFO technique as a soft computing intelligence in power system optimization arena. The main focus of this article was the tuning of single-input PSS. Ghoshal et al. [14] have explored the possibilities of this algorithm for tuning both the single-input, as well as, the dual-input PSSs. Is it possible to obtain more optimal results by exploiting the optimization capability of the SOA for this specific engineering optimization application?
Major objectives (pertaining to algorithm performance) are: (a) to study the performance of the SOA for tuning the PSS parameters, (b) to present the potential benefit of the SOA over bioinspired technique like BFO [14] and evolutionary technique like GA [14] as optimizing techniques, (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, and (d) to contrast the convergence profile of the SOA with the BFO, and the GA. Minor objectives (pertaining to PSS performance) are:
The rest of the paper is organized as follows. In Section 2, the 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 tuning of PSS parameters, are described in Section 4. Sugeno fuzzy logic as applied to on-line tuning of PSS parameters 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 the scope for future work are outlined in Section 8.
2. SMIB system and various PSS under investigation An SMIB [1] 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 PSS (CPSS). On the other hand, speed deviation and torque deviation are taken as the best pair
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Fig. 1. Single-machine-infinite-bus test system.
Fig. 4. Block diagram representation of dual-input PSS2B.
Fig. 5. Block diagram representation of dual-input PSS3B.
Fig. 2. Block diagram representation of the SMIB system with AVR, high gain thyristor exciter, synchronous generator and PSS.
of inputs for dual-input PSSs [2]. The block diagram of SMIB system with AVR, high gain thyristor 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 in this work. 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 parameters 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 way.
Fig. 3. Block diagram representation of conventional PSS.
Fig. 6. Block diagram representation of dual-input PSS4B.
P OF11() = i(r0 ri)2 if r0 > rq, 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 OF12() = i(n0 ni)2, 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 OF13() = i(bi)2, 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 P 0.0. This means unstable oscillation occurs for the particular parameters of PSS. These particular PSS parameters will be rejected during the optimization technique. So, first multi-objective optimization function is formulated as in following equation:
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These subpopulations search over several different domains of the search space. All seekers in the same subpopulation constitute a neighborhood.
Fig. 7. D-shaped sector in the negative half of s plane.
OF 1 ðÞ ¼ 10 OF 11 ðÞ þ 10 OF 12 ðÞ þ 0:01 OF 13 ðÞ þ OF 14 ðÞ
ð1Þ
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 [19,20]. By optimizing OF1(), closed loop system poles are consistently pushed further left of jx axis with simultaneous 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 D-shaped 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 greater extent. Thorough computation shows that optimization of OF1() is not sufficient for sharp tuning of PSS parameters. So, it is essential to design a second multi-objective optimization function for sharp tuning of PSS parameters. Thus, the second multi-objective optimization function OF2() is formulated as in following equation:
OF 2 ðÞ ¼ ðosh 106 Þ2 þ ðush 106 Þ2 þ ðt st Þ2 þ
d ðDxr Þ 106 dt
2 ð2Þ
d ðD dt
In (2); osh, ush, tst, xr Þ are overshoot, undershoot, settling time and time derivative of change in rotor speed, respectively. All these are referred to the transient response of Dxr. 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 parameters, subject to the limits as given in [1,14]. 4. Optimization algorithms as applied to PSS tuning parameters 4.1. Genetic algorithm Implementation steps of the GA algorithm are given in [14]. 4.2. Bacteria foraging optimization algorithm Implementation steps of the BFO algorithm are shown in [14]. 4.3. Seeker optimization algorithm SOA [11,12] is a population-based, heuristic search algorithm. It regards optimization process as an optimal solution obtained by a seeker population. Each individual of this population is called seeker. A neighborhood is defined for each seeker. This neighborhood represents the social component for social sharing of information. The population is randomly categorized into three subpopulations.
4.3.1. Steps of seeker optimization algorithm In SOA, a search direction dij(t) and a step length aij(t) are computed separately for each seeker i on each dimension j for each time step t, where aij ðtÞ P 0 and dij(t) e {1, 0, 1}. Here, i represents the population number and j represents the number of variables to be optimized. If the ith seeker goes towards the positive direction of the coordinate axis on the dimension j, dij(t) is taken as +1. If the ith seeker goes towards the negative direction of the coordinate axis on the dimension j, dij(t) is assumed as 1. The value of dij(t) is assumed as 0 if the ith seeker stays at the current position. In a population of size S, for of each seeker i (1 6 i 6 S), the position update on each dimension j is given by the following equation:
xij ðt þ 1Þ ¼ xij ðtÞ þ aij ðtÞ dij ðtÞ
ð3Þ
Each subpopulation is searching for optimal solution using its own information. It hints that the subpopulation may trap into local optima yielding a premature convergence. Subpopulations must learn from each other about the optimum information so far they have acquired in their domain. Thus, the positions of the worst seekers of each subpopulation are combined with the best one in each of the other subpopulations using the following binomial crossover operator as expressed in following equation:
( xkn j;worst ¼
xlj;best
if randj 6 0:5
xknj ;worst
else
ð4Þ
In (4), randj is a uniformly random real number within [0, 1], xkn j;worst is denoted as the jth dimension of the nth worst position in the kth subpopulation, xlj,worst is the jth dimension of the best position in the lth subpopulation with and n, k, l = 1, 2, . . . , K 1 and k – l. In order to increase the diversity in the population, good information acquired by each subpopulation is shared among the subpopulations. 4.3.2. Calculation of search direction The natural tendency of the swarm is to reciprocate in a cooperative manner while executing needs and deeds. Normally, there are two extreme types of cooperative behavior prevailing in swarm dynamics. One, egotistic is entirely pro-self and another, altruistic is entirely pro-group [15]. Every seeker, as a single sophisticated agent, is uniformly egotistic [16]. He believes that he should go toward his historical best position according to his own judgment. This attitude of ith seeker may be modeled by an empirical direction vector ~ di;ego ðtÞ as in following equation:
~ di;ego ðtÞ ¼ signð~ pi;best ðtÞ ~ xi ðtÞÞ
ð5Þ
In (5), sign(.) is a signum function on each dimension of the input vector. On the other hand, in altruistic behavior each seeker wants to communicate with each other, cooperate explicitly and adjust their behaviors in response to other seeker in the same neighborhood region for achieving the desired goal. That means the seekers exhibit entirely pro-group behavior. The population then exhibits a self-organized aggregation behavior of which the positive feedback usually takes the form of attraction toward a given signal source. Two optional altruistic directions may be modeled as in following equations:
~ di;alt1 ðtÞ ¼ signð~ g best ðtÞ ~ xi ðtÞÞ
ð6Þ
~ di;alt2 ðtÞ ¼ signð~lbest ðtÞ ~ xi ðtÞÞ
ð7Þ
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In (6) and (7), ~ g best ðtÞ represents neighbors’ historical best position, ~ lbest ðtÞ means neighbors’ current best position. Moreover, seekers enjoy the properties of pro-activeness. Seekers do not simply act in response to their environment rather they are able to exhibit goal-directed behavior [16]. In addition, future behavior can be predicted and guided by past behavior [17]. As a result, the seeker may be pro-active to change his search direction and exhibit goal-directed behavior according to his past behavior. Hence, each seeker is associated with an empirical direction called as pro-activeness direction as in following equation:
~ di;pro ðtÞ ¼ signð~ xi ðt 1 Þ ~ xi ðt2 ÞÞ
Fig. 9. The action part of the fuzzy reasoning.
ð8Þ
In (8), t1, t2 e {t, t 1, t 2} and it is assumed that ~ xi ðt1 Þ is better than ~ xi ðt 2 Þ: Aforementioned four empirical directions as mentioned in (5)–(8) direct human being to take a rational decision in search direction. Every dimension j of ~ di ðtÞ is selected applying the following proportional selection rule (shown in Fig. 8) as stated in following equation:
8 ð0Þ 0; if r j 6 pj > > < ð0Þ ðþ1Þ dij ¼ þ1; if pð0Þ 6 r j 6 pj þ pj j > > : ð0Þ ðþ1Þ 1; if pj þ pj 6 rj 6 1
ð9Þ
ðmÞ
In (9), rj is a uniform random number in [0, 1], pj (m e {0, +1 1} is the percent of the number of m from the set {dij,ego, dij,alt1, dij,alt2, dij,pro} on each dimension j of all the four ðmÞ empirical directions, i.e. pj = (the number of m)/4. 4.3.3. Calculation of step length From the view point of human searching behavior, it is understood that one may find the near optimal solutions in a narrower neighborhood of the point with lower fitness value and, on the other hand, in a wider neighborhood of the point with higher fitness value. A fuzzy system may be an ideal choice to represent the understanding and linguistic behavioral pattern of human searching tendency. Different optimization problems often have different ranges of fitness values. To design a fuzzy system to be applicable to a wide range of optimization problems, the fitness values of all the seekers are sorted in descending manner and turned into the sequence numbers from 1 to S as the inputs of fuzzy reasoning. The linear membership function is used in the conditional part since the universe of discourse is a given set of numbers, i.e. 1, 2, . . . , S. The expression is presented as in the following equation:
li ¼ lmax
S Ii lmin Þ ðl S 1 max
ð10Þ
In (10), Ii is the sequence number of ~ xi ðtÞ after sorting the fitness values, lmax is the maximum membership degree value which is equal to or a little less than 1.0. Here, the value of lmax is taken as 0.95. A fuzzy system works on the principle of control rule as ‘‘If {the conditional part}, then {the action part}. Bell membership function 2 lðxxÞ ¼ ex2 =2d (shown in Fig. 9) is well utilized in the literature to represent the action part. For the convenience, one dimension
Fig. 8. The proportional selection rule of search directions.
Fig. 10. Flowchart of the seeker optimization algorithm.
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B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 Table 1 Implementation steps of the SOA for the tuning of PSS parameters. Step 1 (a) (b) (c) (d) (e) Step 2 Step 3 Step 4 Step 5
Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12 Step 13
Initialization Input operating values of P, Q, Xe, Et. Input fixed SMIB parameters Setting of limits of variable PSS parameters Maximum population number of PSS parameter strings, maximum iteration cycles Real value initialization of all the PSS parameter strings of the population within limits Read the SOA parameters Determine the SMIB parameters like K1, K2, K3, K4, K5, K6 (Fig. 2) [1] Initialize the positions of the seekers in the search space randomly and uniformly Set the time step t = 0 Compute the objective function of the initial positions. The initial historical best position among the population is achieved. Set the personal historical best position of each seeker to his current position Let t = t + 1 Select the neighbor of each seeker Determine the search direction and step length for each seeker and update his position Compute the objective function for the new positions Update the historical best position among the population and historical best position of each seeker Repeat from Step 6 till the end of the maximum iteration cycles/stopping criterion Determine the best string corresponding to minimum objective function value Determine the optimal PSS parameters string corresponding to the grand minimum objective function value
Fig. 11. SOA-based comparative transient response profiles of Dxr for CPSS, PSS2B, PSS3B and PSS4B equipped system model with 0.01 pu change in DVref and no change in DTm.
0.1 during a run. The ~ xbest and ~ xrand are the best seeker and a randomly selected seeker, respectively, from the same subpopulation to which the ith seeker belongs. It is to be noted here that ~ xrand is different from ~ xbest and ~ d is shared by all the seekers in the same subpopulation. In order to introduce the randomness in each dimension and to improve local search capability, the following equation is introduced to convert li into a vector ~ li .
is considered. Thus, the membership degree values of the input variables beyond [3d + 3d] are less than 0.0111 ðlð3dÞ ¼ 0:0111Þ, and the elements beyond ½3d þ 3d, in the universe of discourse, can be neglected for a linguistic atom [18]. Thus, the minimum value lmin = 0.0111 is set. Moreover, the parameter ~ d of the Bell membership function is determined by the following equation:
~ d ¼ x absð~ xbest ~ xrand Þ
ð11Þ
In (11), the absolute value of the input vector as the corresponding output vector is represented by the symbol abs(.). The parameter x is used to decrease the step length with time step increasing so as to gradually improve the search precision. In the present experiments, x is linearly decreased from 0.9 to
lij ¼ RANDðli ; 1Þ
ð12Þ
In (12), RAND(li, 1) returns a uniformly random real number within [li, 1]. Eq. (13) denotes the action part of the fuzzy reasoning and gives the step length (aij) for every dimension j.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aij ¼ dj lnðlij Þ
ð13Þ
The flow-chart of the SOA is presented in Fig. 10.
4.3.4. Implementation of SOA for PSS tuning The steps of the SOA, as implemented the tuning of PSS parameters of present work, is shown in Table 1.
Table 2 GA–SFL, BFO–SFL and SOA–SFL based comparison of OF2() values for CPSS, PSS2B, PSS3B and PSS4B equipped system model. 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 [14] BFO–SFL [14] SOA–SFL
7.45 4.91 3.61
6.23 4.02 3.93
1.42 1.01 0.97
2.90 1.62 1.42
2
0.5, 0.2, 0.4752, 1.0
GA–SFL [14] BFO–SFL [14] SOA–SFL
7.85 4.87 4.16
7.33 4.14 3.99
2.47 1.21 1.02
3.17 1.52 1.46
3
0.75, 0.50, 0.4752, 0.50
GA–SFL [14] BFO–SFL [14] SOA–SFL
7.16 3.97 3.41
5.18 3.17 2.94
2.71 1.01 0.95
3.88 1.45 1.26
4
0.95, 0.30, 0.4752, 0.5
GA–SFL [14] BFO–SFL [14] SOA–SFL
7.29 3.15 2.97
5.85 2.17 2.01
2.02 1.42 1.14
4.58 1.96 1.45
5
1.2, 0.6, 1.08, 0.5
GA–SFL [14] BFO–SFL [14] SOA–SFL
8.96 5.16 4.91
8.72 4.83 4.16
3.74 2.57 1.91
4.39 2.43 2.14
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Table 3 Comparative BFO- and SOA-based comparison of OF1() values under different operating conditions for CPSS, PSS2B, PSS3B and PSS4B equipped system model. Operating conditions (P, Q, Xe, Et; all are in pu)
Type of PSS
Algorithm
PSS parameters
OF1()
tex (s)
1.2, 0.6, 0.4752, 0.5
CPSS
BFO [14] SOA BFO [14] SOA BFO [14] SOA BFO [14] SOA
175.00, 0.005, 0.005, 0.001, 0.001, 0.352, 0.001 175.00, 0.088, 0.005, 0.084, 0.001, 0.096, 0.001 10.00, 0.097, 0.010, 0.113, 0.01, 0.25 20.59, 0.155, 0.444, 0.603, 0.010, 0.097 10.00, 10.00, 0.271, 0.005, 0.005, 0.334 29.10, 34.37, 0.5, 0.005, 0.005, 0.706 10.00, 10.00, 0.273, 0.005, 0.138, 0.005 10.00, 20.19, 1.963, 0.005, 0.345, 0.005
1455.62 629.40 1251.60 1070.91 291.30 212.56 311.52 271.20
330.65 3.90 639.21 6.97 344.78 3.94 420.73 4.81
BFO [14] SOA BFO [14] SOA BFO [14] SOA BFO [14] SOA
175.00, 0.005, 0.005, 0.001, 0.001, 0.165, 0.001 175.00, 0.070, 0.005, 0.067, 0.001, 0.069, 0.001 10.00, 0.01, 0.253, 0.158, 0.01, 0.01 29.98, 0.161, 0.35, 0.408, 0.112, 0.059 10.00, 10.00, 2.00, 2.00, 0.005, 0.099 10.00, 37.99, 0.533, 0.005, 0.005, 0.945 10.00, 10.00, 0.005, 0.153, 0.249, 0.005 10.00, 38.01, 0.005, 0.135, 0.728, 0.005
1448.61 630.59 1243.00 1201.12 290.88 212.91 378.87 276.89
350.09 3.96 632.93 7.08 334.89 3.89 411.90 5.07
BFO [14] SOA BFO [14] SOA BFO [14] SOA BFO [14] SOA
230.00, 0.005, 0.005, 0.029, 0.001, 0.001, 0.001 10.00, 100.00, 0.704, 0.079, 0.005, 0.005 10.00, 0.01, 0.126, 0.01, 0.393, 0.204 10.00, 0.182, 0.076, 0.172, 0.142, 0.01 10.00, 10.00, 0.171, 0.005, 0.139, 0.131 31.14, 34.79, 0.578, 0.005, 0.031, 0.866 10.00, 10.00, 0.005, 0.168, 0.237, 0.005 10.00, 100.00, 0.704, 0.079, 0.005, 0.005
1491.69 530.52 1444.20 1210.36 267.77 247.79 372.49
345.16 4.76 638.75 6.91 335.89 3.89 415.62 4.84
PSS2B PSS3B PSS4B 1.0, 0.6, 0.93, 0.5
CPSS PSS2B PSS3B PSS4B
1.0, 0.2, 1.08, 0.5
CPSS PSS2B PSS3B PSS4B
5. Sugeno fuzzy logic as applied to on-line tuning of PSS parameters The whole process of SFL [14,19] can be categorized into three steps viz. Fuzzification of input operating conditions, Sugeno fuzzy inference and Sugeno defuzzification. The detailed of the SFL steps are given in [14,19]. 6. Input control parameters For simulation, step perturbation of 0.01 pu is applied either in reference voltage (DVref) or in mechanical torque (DTm). The simulation is implemented in MATLAB 7.1 software on a PC with P-IV 3.0 G CPU and 512 M RAM. The following are the different input control parameters. (a) For SMIB system: 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 [1,14]; Trr = 0.02 s, Ka = 200.0. (b) For GA: Number of parameters depends on the number of 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%. (c) For BFO: Number of problem variables depends upon the PSS structure under investigation. All the parameters of the algorithm are as in [14]. (d) For SOA: Number of problem variables depends upon the PSS structure under investigation. All the parameters of the algorithm are given in Section 4.3.
the system to obtain the transient response profiles. Final values of OF1() and OF2() are already obtained from the end of optimization. Final eigenvalues, final undamped and damped frequencies and final damping ratio are determined by the optimization technique at the end of optimization. Sugeno fuzzy rule base tables (not shown) are obtained by applying each optimization technique for distinct 81 number nominal input operating conditions. The outputs are 81 distinct nominal optimal PSS parameters sets. For optimization, the SOA technique is adopted. BFO [14] and GA [14] are utilized for the sake of comparison. The major observations of the present work are documented below. The results of interest are bold faced in the respective tables. 7.1. Analytical transient response characteristics Table 2 depicts the comparative GA–SFL [14], BFO–SFL [14] and SOA–SFL based optimal transient response characteristics (in terms
7. Simulation results and discussions Optimized PSS parameters determined by any of the optimization techniques are substituted in MATLAB–SIMULINK model of
Fig. 12. SOA-based comparative convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B equipped system model with 0.01 pu change in DVref and no change in DTm.
B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738
of OF2() value) of different PSS equipped system model. From this table, it may be inferred that the transient stabilization performance of dual-input PSS equipped system model is better than single-input counterpart. 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, and d ðDxr Þ. And thus, lesser value of OF2() is obtained for this PSS dt equipped system model. It may also be observed from this table that the SOA-based optimization technique offers lesser value of OF2() than BFO- and GA-based ones already reported in [14]. Hence, SOA–SFL based optimization technique offers better results than either BFO–SFL and GA–SFL based ones. Fig. 11 depicts the
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comparative optimal transient performance of the different PSS equipped power system model corresponding to an operating condition of P = 1.2, Q = 0.6, Xe = 0.4752, Et = 0.5 (all are in pu) for 0.01 pu change in DVref and no change in DTm. From this figure, it is noticed that the transient stabilization performance of dual-input PSS is better than that of the single-input one. Again, among the dual-input PSSs, the performance of PSS3B is established to be the best one. 7.2. Analytical eigenvalue based system performance analysis SOA-based comparison of OF1() values of CPSS, PSS2B, PSS3B and PSS4B are shown in Table 3 for different system operating
Fig. 13. Comparative GA-, BFO- and SOA-based transient response profiles of Dxr with 0.01 pu simultaneous change in DVref and DTm (P = 0.2, Q = 0.2, Xe = 0.4752, Et = 1.0, all are in pu): for (a) CPSS, (b) PSS2B, (c) PSS3B, and (d) PSS4B equipped system model.
Fig. 14. Comparative GA-, BFO- and SOA-based convergence profiles of OF1() for 0.01 pu simultaneous change in DVref and DTm (P = 0.2, Q = 0.2, Xe = 0.4752, Et = 1.0, all are in pu): for (a) CPSS, (b) PSS2B, (c) PSS3B and (d) PSS4B equipped system model.
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conditions. From Table 3, it is observed that the value of OF1() is the least one for PSS3B, establishing 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(). Thus, for PSS2B equipped system model value of OF1() is very much less for PSS3B equipped system model. On the other hand, majority of the eigenvalues for PSS2Bbased system are outside the D-shaped but very close to and right side of (r0, j0) point. This yields higher values of OF11(), OF12() and OF13() for PSS2B equipped system model. Thus, the value of OF1() is more for this system. Hence, from the eigenvalue analysis it may be concluded that a considerable improvement has occurred in the transient performance for the PSS3B-based system model.
7.3. Convergence profiles The comparative SOA-based convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B equipped system model are depicted in Fig. 12 corresponding to an operating condition of P = 1.2, Q = 0.6, Xe = 0.4752, Et = 0.5 (all are in pu). From this figure, it is
notice the value of OF1() corresponding to PSS3B equipped system model is found to converge faster than the others. 7.4. Comparative optimization performance of the optimization techniques With regard to the optimization performances of the optimizing algorithms, as depicted in Table 2, it may be concluded that the SOA-based approach offers the lower values of OF2() for a particular PSS equipped system model for the same input operating conditions. Comparative transient performances of Dxr and convergence profiles of OF1() for SOA-, BFO- [14] and GA- [14] based optimization for all the four PSS modules (CPSS, PSS2B, PSS3B, and PSS4B) are depicted in Figs. 13 and 14, respectively. These figures assist to conclude that the transient stabilization performance and the convergence profile of the objective function for the SOA-based optimization are better than those of BFO-based [14] and GA-based [14] ones. The SOA is offering much better optimal transient performance than BFO and GA. Thus, the SOA may be accepted as a true optimizing algorithm for the power system based application as considered in the present work.
Table 4 Comparative GA–SFL, BFO–SFL, and SOA–SFL based results of eigenvalue analysis corresponding to an input operating condition P = 0.95, Q = 0.30, Xe = 1.08, Et = 0.5; all are in pu. Type of PSS
Algorithms-SFL
Damping ratio (n)
Un-damped natural frequency (xn), rad/s
Corresponding damped frequency (xd), rad/s
Lowest
Highest
Lowest
Lowest
Highest
CPSS
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.16 0.38 0.41
0.57 0.65 0.74
0.38 0.59 0.62
2.45 1.45 1.56
0.42 0.45 0.56
0.59 4.31 5.21
PSS2B
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.26 0.49 0.52
0.97 1.00 1.23
0.48 0.63 0.69
3.33 13.42 14.01
0.32 0.56 0.62
0.79 7.79 7.99
PSS3B
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.36 0.72 0.81
0.98 0.98 1.12
0.17 0.47 0.59
1.93 0.96 1.12
0.15 0.33 0.41
1.8 0.38 0.56
PSS4B
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.2 0.37 0.45
0.95 0.95 1.45
0.55 0.34 0.49
1.22 2.97 3.15
0.54 0.28 0.31
0.36 2.98 3.02
(a) Fault 1
Highest
(b) Fault 2
Fig. 15. Comparative GA–SFL, BFO–SFL, and SOA–SFL based transient response profiles of Dxr for the generator equipped with PSS3B under change in operating conditions: (a) Fault 1, and (b) Fault 2.
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B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 Table 5 Off-nominal operating conditions, simulation of faults, algorithms-SFL, and optimal PSS3B parameters. Fault no.
Operating conditions (P, Q, Xe, Et; all are in pu)
Algorithm-SFL
PSS3B parameters
Fault 1
0.95, 0.3, 0.4752, 1.0
GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL
59.57, 59.22, 0.558, 0.013, 0.317, 0.169 13.50, 83.26, 0.529, 0.051, 0.008, 1.566 14.89, 31.86, 0.38, 0.005, 0.008, 0.544 No change in parameters
GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL
10.00, 53.24, 1.618, 0.192, 0.410, 0.254 10.88, 26.40, 1.279, 0.369, 0.070, 0.140 42.56, 33.81, 0.911, 0.005, 0.021, 0.693 No change in parameters
LT bus fault of duration 220 ms and subsequent clearing
0.95, 0.3, 1.08, 1.0
Fault 2
1.0, 0.6, 0.4752, 1.1
LT bus fault of duration 220 ms and subsequent clearing
0.2, 0.2, 0.4752, 1.1
14.22, 66.25, 0.208, 0.013, 0.161, 0.099 12.34, 83.67, 1.608, 0.239, 0.326, 0.473 14.09, 31.46, 1.17, 0.005, 0.008, 0.896
15.27, 12.11, 1.805, 0.893, 0.137, 0.005 13.07, 39.00, 1.322, 1.816, 1.006, 0.088 14.05, 35.25, 0.431, 0.005, 0.005, 0.621
7.5. Comparative eigenvalue analysis
8. Conclusion
Real parts of some eigenvalues for CPSS/PSS2B equipped system model are always either equal to or greater than r0 in the negative half plane of jx axis. A few eigenvalues are always outside the Dshaped sector (Fig. 7) for any operating condition. So, the value of OF1() are always higher (Table 3) for these two variants of PSS model. Much lower negative real parts of eigenvalues for PSS3B and PSS4B (not shown) cause higher relative stability than CPSS/PSS2B. In case of PSS3B and PSS4B larger reductions in (xn) and (xd) for some electromechanical oscillations are noticed due to higher damping ratios (ni n0) for those particular modes (Table 4). It is also noted from Table 4 that as compared to the results published in [14], SOA–SFL simulation offers better damping ratio (n), un-damped natural frequency (xn), and corresponding damped frequency (xd). These features for PSS3B equipped system model are better than CPSS/PSS2B/PSS4B counterpart. It establishes that the PSS3B damps the oscillation of Dxr very quickly under any sort of system perturbations.
The act of human searching ability is simulated in the SOA. It is a newly entrant heuristic stochastic optimization algorithm. This algorithm is easy to understand, simple to implement, very fast and can be used for a wide variety of optimization tasks. In this paper, a SOA-based fuzzy logic PSS tuning is presented. The results are compared with that of either BFO or GA based ones, reported in the literature. While comparing the benefits of SOA for PSS tuning it is revealed that this algorithm offers true optimal result with faster convergence ability. Thus, it may be proposed that the SOA is capable of efficiently and effectively solving PSS tuning problem. It has the capability of solving other engineering optimization problems.
7.6. Simulation of fault It is revealed that PSS3B is offering the best transient performance in damping all electromechanical modes of generator’s angular speed oscillations for all nominal and off-nominal system conditions, step changes of mechanical torque inputs (DTm) and reference voltage inputs (DVref). LT bus fault of duration 220 ms at the instant of 2.0 s is simulated for PSS3B equipped system model and the corresponding comparative transient response profiles of Dxr for SOA–SFL, GA–SFL and BFO–SFL based responses are plotted in Fig. 15 (Fig. 15a) corresponds to Fault 1 and Fig. 15b corresponds to Fault 2). A close look into these figures show that after the creation of the fault, the SOA–SFL based response recovers from this abnormal situation with much lesser fluctuation in angular speed as compared to those of BFO–SFL [14] and GA–SFL [14] based ones. Table 5 depicts the system model parameters as determined by SFL. Thus, SOA–SFL based model exhibits better response having lesser amplitude of angular speed deviation under fault and subsequent clearing condition yielding better dynamic robust transient performance than BFO–SFL [14] and GA–SFL [14] based ones. PSS3B equipped system model proves to be much less susceptible to faults because PSS3B settles all the state deviations to zero much faster than any other PSS.
References [1] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994. [2] IEEE digital excitation system subcommittee report. Computer models for representation of digital-based excitation systems. IEEE Trans Energy Conver 1996;11(3):607–15. [3] 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. [4] Soliman H, Elshafei AL, Shaltout A, Morsi M. Robust power system stabilizer. IEE Proc Elect Power Appl 2000;147(5):285–91. [5] Werner H, Korba P, Chen Yang T. Robust tuning of power system stabilizers using LMI techniques. IEEE Trans CST 2003;11(1):147–52. [6] Shrikant Rao P, Sen I. Robust tuning of power system stabilizers using QFT. IEEE Trans CST 1999;7(4):478–86. [7] Hopkins WE, Medanic J, Perkins WR. Output feedback pole placement in the design of suboptimal linear quadratic regulators. Int J Contr 1981;34: 593–612. [8] Geromel JC. Methods and techniques for decentralized control systems. Italy: Clup; 1987. [9] Costa AS, Freitas FD, Silva AS. Design of decentralized controllers for large power systems considering sparsity. IEEE Trans Power Syst 1997;12(1): 144–52. [10] El-Zonkoly AM. Optimal tuning of power system stabilizers and AVR gains using particle swarm optimization. Expert Syst Appl 2006;31(3):551–7. [11] Dai Chaohua, Chen Weirong, Zhu Yunfang, Zhang Xuexia. Seeker optimization algorithm for optimal reactive power dispatch. IEEE Trans Power Syst 2009;24(3):1218–31. [12] Daia Chaohua, Chena Weirong, Zhub Yunfang, Zhanga Xuexia. Reactive power dispatch considering voltage stability with seeker optimization algorithm. Electr Power Syst Res 2009;79:1462–71. [13] Mishra S, Tripathy M, Nanda J. Multi-machine power system stabilizer design by rule based bacteria foraging. Electric Power Syst Res 2007;77(12): 1595–607. [14] Ghoshal SP, Chatterjee A, Mukherjee V. Bio-inspired fuzzy logic based tuning of power system stabilizer. Expert Syst Appl 2009;36:9281–92. [15] Eustace D, Barnes DP, Gray JO. Co-operant mobile robots for industrial applications. Proc Int Conf Ind Electron Control Instrum 1993;1:39–44.
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[16] Wooldridge M, Jennings NR. Intelligent agents: theory and practice. Knowl Eng Rev 1995;10(2):115–52. [17] Ajzen I. Residual effects of past on later behavior: habituation and reasoned action perspectives. Pers Soc Psychol Rev 2002;6(2):107–22. [18] Li D. Uncertainty reasoning based on cloud models in controllers. Comput Math Appl 1998;35(3):99–123.
[19] Chatterjee AC, Ghosal SP, Mukherjee V. Chaotic ant swarm optimization for fuzzy-based tuning of PSS. Int J Electr Power Energy Syst 2011;33(3):657–72. [20] Chatterjee AC, Mukherjee V, Ghosal SP. Velocity relaxed and craziness-based swarm and optimized intelligent PID PSS controlled AVR system. Int J Electr Power Energy Syst 2009;31(7–8):323–33.
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Comparative seeker and bio-inspired fuzzy logic controllers for power system stabilizers Binod Shaw a, Abhik Banerjee a, S.P. Ghoshal b, V. Mukherjee c,⇑ a
Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal, India Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, India c Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand, India b
a r t i c l e
i n f o
Article history: Received 18 May 2010 Received in revised form 16 January 2011 Accepted 12 August 2011 Available online 20 October 2011 Keywords: Bacteria foraging optimization Genetic algorithm Power system stabilizer Seeker optimization algorithm Sugeno fuzzy logic
a b s t r a c t Seeker optimization algorithm (SOA) is a new heuristic population-based search algorithm. In this paper, SOA is utilized to tune the parameters of both single-input and dual-input power system stabilizers (PSSs). In SOA, the act of human searching capability and understandings are exploited for the purpose of optimization. In SOA-based optimization, the search direction is based on empirical gradient by evaluating the response to the position changes and the step length is based on uncertainty reasoning by using a simple fuzzy rule. Conventional PSS (CPSS) and the three dual-input IEEE PSSs (namely PSS2B, PSS3B and PSS4B) are optimally tuned to obtain the optimal transient performances. From simulation study it is revealed that the transient performance of the dual-input PSS is better than the single-input PSS. It is further explored that among the dual-input PSSs, PSS3B offers the best optimal transient performance. While comparing the SOA with recently reported optimization algorithms like bacteria foraging optimization (BFO) and genetic algorithm (GA), it is revealed that the SOA is more effective than either BFO or GA in finding the optimal transient performance. 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 parameters. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The problem of low frequency electromechanical oscillations arises from the usage of fast acting, high gain automatic voltage regulator (AVR) in modern generator excitation system [1]. Any form of disturbances such as sudden change in loads, change in transmission line parameters, fluctuation in the output of the turbine and faults, 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. Transfer of bulk power across weak transmission lines may also invite this problem of low frequency oscillation. The usage of power system stabilizer (PSS) is a very common and widely accepted solution, prevailing in the utility houses, to tackle this problem. The PSS adds a stabilizing signal to AVR which modulates the generator excitation. 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 conven⇑ Corresponding author. Tel.: +91 0326 2235644; fax: +91 0326 2296563. E-mail addresses: [email protected] (B. Shaw), abhik_banerjee@ rediffmail.com (A. Banerjee), [email protected] (S.P. Ghoshal), vivek_ [email protected] (V. Mukherjee). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.08.015
tional single-input PSS (machine shaft speed (Dxr) as single input to the PSS) design approach based on a single-machineinfinite-bus (SMIB) linearized model in the normal operating condition has some deficiencies. On the other hand, the two inputs to dual-input PSS are machine shaft speed (Dxr) and the change in electrical torque (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 [2] and those models are taken in the present study. Pole-placement or eigenvalue assignment for single-input single-output system has been reported in literature [3]. A robust PSS tuning approach [4] based upon lead compensator design has been carried out by drawing the root loci for finite number of extreme characteristic polynomials. In [4], 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. [5]. In [5], plant uncertainties are expressed in the form of a linear fractional transformation. Results obtained in [5] are compared to the results obtained in [6] based on quantitative feedback theory. Linear quadratic control [7] has been applied for coordinated control design. The problem has been formulated as a standard
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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, have been in use in 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 [8] and was applied to power systems exploiting sparsity [9]. Fuzzy, GA-fuzzy, neuro-fuzzy 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 online 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 GA, simulated annealing suffer from settings of algorithm parameters and give rise to repeated revisiting of the same suboptimal solutions. The application of particle swarm optimization (PSO) [10] for PSS tuning has attracted more focus of the researchers for this purpose. Seeker optimization algorithm (SOA) [11,12] is, essentially, a new population based heuristic search algorithm. It is based on human understanding and searching capability for finding an optimum solution. In SOA, optimum solution is regarded as one which is searched out by a seeker population. The underlying concept of SOA is very easy to model and relatively easier than other optimization techniques prevailing in the literature. The highlighting characteristic features of this algorithm are the following:
Thus, it is very much pertinent to explore a comparative study between the SOA-based PSS tuning and the BFO/GA-based PSS tuning. Based on the transient performance, it is also important to draw some comparative logical conclusion between single-input PSS and dual-input PSSs with the assistance of SOA [11,12,15– 18], BFO and GA in line with [14]. A fuzzy logic system-based [14] PSS can adjust its parameters on-line 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 proves to be the most robust model 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 Sugeno fuzzy logic (SFL) table. 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 off-nominal PSS parameters for off-nominal input operating conditions occurring in real time. Thus, the major and minor objectives of this paper may be documented as follow:
(a) search direction and step length are directly used in this algorithm to update the position, (b) proportional selection rule is applied for the calculation of search direction which can improve the population diversity so as to boost global search ability and decrease the number of control parameters making it simpler to implement, and (c) fuzzy reasoning is used to generate the step length because the uncertain reasoning of human searching could be best described by natural linguistic variables, and a simple if–else control rule.
(a) to compare the transient performance of single-input PSS with dual-input PSSs, (b) to contrast the generator’s angular speed oscillations for dual-input PSSs (namely PSS2B, PSS3B, and PSS4B) equipped system model, and (c) to critically examine the best type of PSS for practical implementation under any sort of system disturbances.
In the present work, this SOA algorithm is utilized for the purpose of optimal tuning of the PSS parameters. The novelty of the present work is the study of the performance of the SOA in designing the PSS parameters and to compare the optimizing performance of this algorithm with those reported in the stateof-the-art literature for the same purpose. Bacteria foraging optimization (BFO), a bio-inspired technique, has been reported by Mishra et al. [13] to establish the potential application of the BFO technique as a soft computing intelligence in power system optimization arena. The main focus of this article was the tuning of single-input PSS. Ghoshal et al. [14] have explored the possibilities of this algorithm for tuning both the single-input, as well as, the dual-input PSSs. Is it possible to obtain more optimal results by exploiting the optimization capability of the SOA for this specific engineering optimization application?
Major objectives (pertaining to algorithm performance) are: (a) to study the performance of the SOA for tuning the PSS parameters, (b) to present the potential benefit of the SOA over bioinspired technique like BFO [14] and evolutionary technique like GA [14] as optimizing techniques, (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, and (d) to contrast the convergence profile of the SOA with the BFO, and the GA. Minor objectives (pertaining to PSS performance) are:
The rest of the paper is organized as follows. In Section 2, the 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 tuning of PSS parameters, are described in Section 4. Sugeno fuzzy logic as applied to on-line tuning of PSS parameters 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 the scope for future work are outlined in Section 8.
2. SMIB system and various PSS under investigation An SMIB [1] 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 PSS (CPSS). On the other hand, speed deviation and torque deviation are taken as the best pair
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Fig. 1. Single-machine-infinite-bus test system.
Fig. 4. Block diagram representation of dual-input PSS2B.
Fig. 5. Block diagram representation of dual-input PSS3B.
Fig. 2. Block diagram representation of the SMIB system with AVR, high gain thyristor exciter, synchronous generator and PSS.
of inputs for dual-input PSSs [2]. The block diagram of SMIB system with AVR, high gain thyristor 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 in this work. 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 parameters 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 way.
Fig. 3. Block diagram representation of conventional PSS.
Fig. 6. Block diagram representation of dual-input PSS4B.
P OF11() = i(r0 ri)2 if r0 > rq, 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 OF12() = i(n0 ni)2, 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 OF13() = i(bi)2, 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 P 0.0. This means unstable oscillation occurs for the particular parameters of PSS. These particular PSS parameters will be rejected during the optimization technique. So, first multi-objective optimization function is formulated as in following equation:
B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738
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These subpopulations search over several different domains of the search space. All seekers in the same subpopulation constitute a neighborhood.
Fig. 7. D-shaped sector in the negative half of s plane.
OF 1 ðÞ ¼ 10 OF 11 ðÞ þ 10 OF 12 ðÞ þ 0:01 OF 13 ðÞ þ OF 14 ðÞ
ð1Þ
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 [19,20]. By optimizing OF1(), closed loop system poles are consistently pushed further left of jx axis with simultaneous 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 D-shaped 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 greater extent. Thorough computation shows that optimization of OF1() is not sufficient for sharp tuning of PSS parameters. So, it is essential to design a second multi-objective optimization function for sharp tuning of PSS parameters. Thus, the second multi-objective optimization function OF2() is formulated as in following equation:
OF 2 ðÞ ¼ ðosh 106 Þ2 þ ðush 106 Þ2 þ ðt st Þ2 þ
d ðDxr Þ 106 dt
2 ð2Þ
d ðD dt
In (2); osh, ush, tst, xr Þ are overshoot, undershoot, settling time and time derivative of change in rotor speed, respectively. All these are referred to the transient response of Dxr. 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 parameters, subject to the limits as given in [1,14]. 4. Optimization algorithms as applied to PSS tuning parameters 4.1. Genetic algorithm Implementation steps of the GA algorithm are given in [14]. 4.2. Bacteria foraging optimization algorithm Implementation steps of the BFO algorithm are shown in [14]. 4.3. Seeker optimization algorithm SOA [11,12] is a population-based, heuristic search algorithm. It regards optimization process as an optimal solution obtained by a seeker population. Each individual of this population is called seeker. A neighborhood is defined for each seeker. This neighborhood represents the social component for social sharing of information. The population is randomly categorized into three subpopulations.
4.3.1. Steps of seeker optimization algorithm In SOA, a search direction dij(t) and a step length aij(t) are computed separately for each seeker i on each dimension j for each time step t, where aij ðtÞ P 0 and dij(t) e {1, 0, 1}. Here, i represents the population number and j represents the number of variables to be optimized. If the ith seeker goes towards the positive direction of the coordinate axis on the dimension j, dij(t) is taken as +1. If the ith seeker goes towards the negative direction of the coordinate axis on the dimension j, dij(t) is assumed as 1. The value of dij(t) is assumed as 0 if the ith seeker stays at the current position. In a population of size S, for of each seeker i (1 6 i 6 S), the position update on each dimension j is given by the following equation:
xij ðt þ 1Þ ¼ xij ðtÞ þ aij ðtÞ dij ðtÞ
ð3Þ
Each subpopulation is searching for optimal solution using its own information. It hints that the subpopulation may trap into local optima yielding a premature convergence. Subpopulations must learn from each other about the optimum information so far they have acquired in their domain. Thus, the positions of the worst seekers of each subpopulation are combined with the best one in each of the other subpopulations using the following binomial crossover operator as expressed in following equation:
( xkn j;worst ¼
xlj;best
if randj 6 0:5
xknj ;worst
else
ð4Þ
In (4), randj is a uniformly random real number within [0, 1], xkn j;worst is denoted as the jth dimension of the nth worst position in the kth subpopulation, xlj,worst is the jth dimension of the best position in the lth subpopulation with and n, k, l = 1, 2, . . . , K 1 and k – l. In order to increase the diversity in the population, good information acquired by each subpopulation is shared among the subpopulations. 4.3.2. Calculation of search direction The natural tendency of the swarm is to reciprocate in a cooperative manner while executing needs and deeds. Normally, there are two extreme types of cooperative behavior prevailing in swarm dynamics. One, egotistic is entirely pro-self and another, altruistic is entirely pro-group [15]. Every seeker, as a single sophisticated agent, is uniformly egotistic [16]. He believes that he should go toward his historical best position according to his own judgment. This attitude of ith seeker may be modeled by an empirical direction vector ~ di;ego ðtÞ as in following equation:
~ di;ego ðtÞ ¼ signð~ pi;best ðtÞ ~ xi ðtÞÞ
ð5Þ
In (5), sign(.) is a signum function on each dimension of the input vector. On the other hand, in altruistic behavior each seeker wants to communicate with each other, cooperate explicitly and adjust their behaviors in response to other seeker in the same neighborhood region for achieving the desired goal. That means the seekers exhibit entirely pro-group behavior. The population then exhibits a self-organized aggregation behavior of which the positive feedback usually takes the form of attraction toward a given signal source. Two optional altruistic directions may be modeled as in following equations:
~ di;alt1 ðtÞ ¼ signð~ g best ðtÞ ~ xi ðtÞÞ
ð6Þ
~ di;alt2 ðtÞ ¼ signð~lbest ðtÞ ~ xi ðtÞÞ
ð7Þ
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In (6) and (7), ~ g best ðtÞ represents neighbors’ historical best position, ~ lbest ðtÞ means neighbors’ current best position. Moreover, seekers enjoy the properties of pro-activeness. Seekers do not simply act in response to their environment rather they are able to exhibit goal-directed behavior [16]. In addition, future behavior can be predicted and guided by past behavior [17]. As a result, the seeker may be pro-active to change his search direction and exhibit goal-directed behavior according to his past behavior. Hence, each seeker is associated with an empirical direction called as pro-activeness direction as in following equation:
~ di;pro ðtÞ ¼ signð~ xi ðt 1 Þ ~ xi ðt2 ÞÞ
Fig. 9. The action part of the fuzzy reasoning.
ð8Þ
In (8), t1, t2 e {t, t 1, t 2} and it is assumed that ~ xi ðt1 Þ is better than ~ xi ðt 2 Þ: Aforementioned four empirical directions as mentioned in (5)–(8) direct human being to take a rational decision in search direction. Every dimension j of ~ di ðtÞ is selected applying the following proportional selection rule (shown in Fig. 8) as stated in following equation:
8 ð0Þ 0; if r j 6 pj > > < ð0Þ ðþ1Þ dij ¼ þ1; if pð0Þ 6 r j 6 pj þ pj j > > : ð0Þ ðþ1Þ 1; if pj þ pj 6 rj 6 1
ð9Þ
ðmÞ
In (9), rj is a uniform random number in [0, 1], pj (m e {0, +1 1} is the percent of the number of m from the set {dij,ego, dij,alt1, dij,alt2, dij,pro} on each dimension j of all the four ðmÞ empirical directions, i.e. pj = (the number of m)/4. 4.3.3. Calculation of step length From the view point of human searching behavior, it is understood that one may find the near optimal solutions in a narrower neighborhood of the point with lower fitness value and, on the other hand, in a wider neighborhood of the point with higher fitness value. A fuzzy system may be an ideal choice to represent the understanding and linguistic behavioral pattern of human searching tendency. Different optimization problems often have different ranges of fitness values. To design a fuzzy system to be applicable to a wide range of optimization problems, the fitness values of all the seekers are sorted in descending manner and turned into the sequence numbers from 1 to S as the inputs of fuzzy reasoning. The linear membership function is used in the conditional part since the universe of discourse is a given set of numbers, i.e. 1, 2, . . . , S. The expression is presented as in the following equation:
li ¼ lmax
S Ii lmin Þ ðl S 1 max
ð10Þ
In (10), Ii is the sequence number of ~ xi ðtÞ after sorting the fitness values, lmax is the maximum membership degree value which is equal to or a little less than 1.0. Here, the value of lmax is taken as 0.95. A fuzzy system works on the principle of control rule as ‘‘If {the conditional part}, then {the action part}. Bell membership function 2 lðxxÞ ¼ ex2 =2d (shown in Fig. 9) is well utilized in the literature to represent the action part. For the convenience, one dimension
Fig. 8. The proportional selection rule of search directions.
Fig. 10. Flowchart of the seeker optimization algorithm.
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B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 Table 1 Implementation steps of the SOA for the tuning of PSS parameters. Step 1 (a) (b) (c) (d) (e) Step 2 Step 3 Step 4 Step 5
Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12 Step 13
Initialization Input operating values of P, Q, Xe, Et. Input fixed SMIB parameters Setting of limits of variable PSS parameters Maximum population number of PSS parameter strings, maximum iteration cycles Real value initialization of all the PSS parameter strings of the population within limits Read the SOA parameters Determine the SMIB parameters like K1, K2, K3, K4, K5, K6 (Fig. 2) [1] Initialize the positions of the seekers in the search space randomly and uniformly Set the time step t = 0 Compute the objective function of the initial positions. The initial historical best position among the population is achieved. Set the personal historical best position of each seeker to his current position Let t = t + 1 Select the neighbor of each seeker Determine the search direction and step length for each seeker and update his position Compute the objective function for the new positions Update the historical best position among the population and historical best position of each seeker Repeat from Step 6 till the end of the maximum iteration cycles/stopping criterion Determine the best string corresponding to minimum objective function value Determine the optimal PSS parameters string corresponding to the grand minimum objective function value
Fig. 11. SOA-based comparative transient response profiles of Dxr for CPSS, PSS2B, PSS3B and PSS4B equipped system model with 0.01 pu change in DVref and no change in DTm.
0.1 during a run. The ~ xbest and ~ xrand are the best seeker and a randomly selected seeker, respectively, from the same subpopulation to which the ith seeker belongs. It is to be noted here that ~ xrand is different from ~ xbest and ~ d is shared by all the seekers in the same subpopulation. In order to introduce the randomness in each dimension and to improve local search capability, the following equation is introduced to convert li into a vector ~ li .
is considered. Thus, the membership degree values of the input variables beyond [3d + 3d] are less than 0.0111 ðlð3dÞ ¼ 0:0111Þ, and the elements beyond ½3d þ 3d, in the universe of discourse, can be neglected for a linguistic atom [18]. Thus, the minimum value lmin = 0.0111 is set. Moreover, the parameter ~ d of the Bell membership function is determined by the following equation:
~ d ¼ x absð~ xbest ~ xrand Þ
ð11Þ
In (11), the absolute value of the input vector as the corresponding output vector is represented by the symbol abs(.). The parameter x is used to decrease the step length with time step increasing so as to gradually improve the search precision. In the present experiments, x is linearly decreased from 0.9 to
lij ¼ RANDðli ; 1Þ
ð12Þ
In (12), RAND(li, 1) returns a uniformly random real number within [li, 1]. Eq. (13) denotes the action part of the fuzzy reasoning and gives the step length (aij) for every dimension j.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aij ¼ dj lnðlij Þ
ð13Þ
The flow-chart of the SOA is presented in Fig. 10.
4.3.4. Implementation of SOA for PSS tuning The steps of the SOA, as implemented the tuning of PSS parameters of present work, is shown in Table 1.
Table 2 GA–SFL, BFO–SFL and SOA–SFL based comparison of OF2() values for CPSS, PSS2B, PSS3B and PSS4B equipped system model. 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 [14] BFO–SFL [14] SOA–SFL
7.45 4.91 3.61
6.23 4.02 3.93
1.42 1.01 0.97
2.90 1.62 1.42
2
0.5, 0.2, 0.4752, 1.0
GA–SFL [14] BFO–SFL [14] SOA–SFL
7.85 4.87 4.16
7.33 4.14 3.99
2.47 1.21 1.02
3.17 1.52 1.46
3
0.75, 0.50, 0.4752, 0.50
GA–SFL [14] BFO–SFL [14] SOA–SFL
7.16 3.97 3.41
5.18 3.17 2.94
2.71 1.01 0.95
3.88 1.45 1.26
4
0.95, 0.30, 0.4752, 0.5
GA–SFL [14] BFO–SFL [14] SOA–SFL
7.29 3.15 2.97
5.85 2.17 2.01
2.02 1.42 1.14
4.58 1.96 1.45
5
1.2, 0.6, 1.08, 0.5
GA–SFL [14] BFO–SFL [14] SOA–SFL
8.96 5.16 4.91
8.72 4.83 4.16
3.74 2.57 1.91
4.39 2.43 2.14
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Table 3 Comparative BFO- and SOA-based comparison of OF1() values under different operating conditions for CPSS, PSS2B, PSS3B and PSS4B equipped system model. Operating conditions (P, Q, Xe, Et; all are in pu)
Type of PSS
Algorithm
PSS parameters
OF1()
tex (s)
1.2, 0.6, 0.4752, 0.5
CPSS
BFO [14] SOA BFO [14] SOA BFO [14] SOA BFO [14] SOA
175.00, 0.005, 0.005, 0.001, 0.001, 0.352, 0.001 175.00, 0.088, 0.005, 0.084, 0.001, 0.096, 0.001 10.00, 0.097, 0.010, 0.113, 0.01, 0.25 20.59, 0.155, 0.444, 0.603, 0.010, 0.097 10.00, 10.00, 0.271, 0.005, 0.005, 0.334 29.10, 34.37, 0.5, 0.005, 0.005, 0.706 10.00, 10.00, 0.273, 0.005, 0.138, 0.005 10.00, 20.19, 1.963, 0.005, 0.345, 0.005
1455.62 629.40 1251.60 1070.91 291.30 212.56 311.52 271.20
330.65 3.90 639.21 6.97 344.78 3.94 420.73 4.81
BFO [14] SOA BFO [14] SOA BFO [14] SOA BFO [14] SOA
175.00, 0.005, 0.005, 0.001, 0.001, 0.165, 0.001 175.00, 0.070, 0.005, 0.067, 0.001, 0.069, 0.001 10.00, 0.01, 0.253, 0.158, 0.01, 0.01 29.98, 0.161, 0.35, 0.408, 0.112, 0.059 10.00, 10.00, 2.00, 2.00, 0.005, 0.099 10.00, 37.99, 0.533, 0.005, 0.005, 0.945 10.00, 10.00, 0.005, 0.153, 0.249, 0.005 10.00, 38.01, 0.005, 0.135, 0.728, 0.005
1448.61 630.59 1243.00 1201.12 290.88 212.91 378.87 276.89
350.09 3.96 632.93 7.08 334.89 3.89 411.90 5.07
BFO [14] SOA BFO [14] SOA BFO [14] SOA BFO [14] SOA
230.00, 0.005, 0.005, 0.029, 0.001, 0.001, 0.001 10.00, 100.00, 0.704, 0.079, 0.005, 0.005 10.00, 0.01, 0.126, 0.01, 0.393, 0.204 10.00, 0.182, 0.076, 0.172, 0.142, 0.01 10.00, 10.00, 0.171, 0.005, 0.139, 0.131 31.14, 34.79, 0.578, 0.005, 0.031, 0.866 10.00, 10.00, 0.005, 0.168, 0.237, 0.005 10.00, 100.00, 0.704, 0.079, 0.005, 0.005
1491.69 530.52 1444.20 1210.36 267.77 247.79 372.49
345.16 4.76 638.75 6.91 335.89 3.89 415.62 4.84
PSS2B PSS3B PSS4B 1.0, 0.6, 0.93, 0.5
CPSS PSS2B PSS3B PSS4B
1.0, 0.2, 1.08, 0.5
CPSS PSS2B PSS3B PSS4B
5. Sugeno fuzzy logic as applied to on-line tuning of PSS parameters The whole process of SFL [14,19] can be categorized into three steps viz. Fuzzification of input operating conditions, Sugeno fuzzy inference and Sugeno defuzzification. The detailed of the SFL steps are given in [14,19]. 6. Input control parameters For simulation, step perturbation of 0.01 pu is applied either in reference voltage (DVref) or in mechanical torque (DTm). The simulation is implemented in MATLAB 7.1 software on a PC with P-IV 3.0 G CPU and 512 M RAM. The following are the different input control parameters. (a) For SMIB system: 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 [1,14]; Trr = 0.02 s, Ka = 200.0. (b) For GA: Number of parameters depends on the number of 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%. (c) For BFO: Number of problem variables depends upon the PSS structure under investigation. All the parameters of the algorithm are as in [14]. (d) For SOA: Number of problem variables depends upon the PSS structure under investigation. All the parameters of the algorithm are given in Section 4.3.
the system to obtain the transient response profiles. Final values of OF1() and OF2() are already obtained from the end of optimization. Final eigenvalues, final undamped and damped frequencies and final damping ratio are determined by the optimization technique at the end of optimization. Sugeno fuzzy rule base tables (not shown) are obtained by applying each optimization technique for distinct 81 number nominal input operating conditions. The outputs are 81 distinct nominal optimal PSS parameters sets. For optimization, the SOA technique is adopted. BFO [14] and GA [14] are utilized for the sake of comparison. The major observations of the present work are documented below. The results of interest are bold faced in the respective tables. 7.1. Analytical transient response characteristics Table 2 depicts the comparative GA–SFL [14], BFO–SFL [14] and SOA–SFL based optimal transient response characteristics (in terms
7. Simulation results and discussions Optimized PSS parameters determined by any of the optimization techniques are substituted in MATLAB–SIMULINK model of
Fig. 12. SOA-based comparative convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B equipped system model with 0.01 pu change in DVref and no change in DTm.
B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738
of OF2() value) of different PSS equipped system model. From this table, it may be inferred that the transient stabilization performance of dual-input PSS equipped system model is better than single-input counterpart. 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, and d ðDxr Þ. And thus, lesser value of OF2() is obtained for this PSS dt equipped system model. It may also be observed from this table that the SOA-based optimization technique offers lesser value of OF2() than BFO- and GA-based ones already reported in [14]. Hence, SOA–SFL based optimization technique offers better results than either BFO–SFL and GA–SFL based ones. Fig. 11 depicts the
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comparative optimal transient performance of the different PSS equipped power system model corresponding to an operating condition of P = 1.2, Q = 0.6, Xe = 0.4752, Et = 0.5 (all are in pu) for 0.01 pu change in DVref and no change in DTm. From this figure, it is noticed that the transient stabilization performance of dual-input PSS is better than that of the single-input one. Again, among the dual-input PSSs, the performance of PSS3B is established to be the best one. 7.2. Analytical eigenvalue based system performance analysis SOA-based comparison of OF1() values of CPSS, PSS2B, PSS3B and PSS4B are shown in Table 3 for different system operating
Fig. 13. Comparative GA-, BFO- and SOA-based transient response profiles of Dxr with 0.01 pu simultaneous change in DVref and DTm (P = 0.2, Q = 0.2, Xe = 0.4752, Et = 1.0, all are in pu): for (a) CPSS, (b) PSS2B, (c) PSS3B, and (d) PSS4B equipped system model.
Fig. 14. Comparative GA-, BFO- and SOA-based convergence profiles of OF1() for 0.01 pu simultaneous change in DVref and DTm (P = 0.2, Q = 0.2, Xe = 0.4752, Et = 1.0, all are in pu): for (a) CPSS, (b) PSS2B, (c) PSS3B and (d) PSS4B equipped system model.
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conditions. From Table 3, it is observed that the value of OF1() is the least one for PSS3B, establishing 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(). Thus, for PSS2B equipped system model value of OF1() is very much less for PSS3B equipped system model. On the other hand, majority of the eigenvalues for PSS2Bbased system are outside the D-shaped but very close to and right side of (r0, j0) point. This yields higher values of OF11(), OF12() and OF13() for PSS2B equipped system model. Thus, the value of OF1() is more for this system. Hence, from the eigenvalue analysis it may be concluded that a considerable improvement has occurred in the transient performance for the PSS3B-based system model.
7.3. Convergence profiles The comparative SOA-based convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B equipped system model are depicted in Fig. 12 corresponding to an operating condition of P = 1.2, Q = 0.6, Xe = 0.4752, Et = 0.5 (all are in pu). From this figure, it is
notice the value of OF1() corresponding to PSS3B equipped system model is found to converge faster than the others. 7.4. Comparative optimization performance of the optimization techniques With regard to the optimization performances of the optimizing algorithms, as depicted in Table 2, it may be concluded that the SOA-based approach offers the lower values of OF2() for a particular PSS equipped system model for the same input operating conditions. Comparative transient performances of Dxr and convergence profiles of OF1() for SOA-, BFO- [14] and GA- [14] based optimization for all the four PSS modules (CPSS, PSS2B, PSS3B, and PSS4B) are depicted in Figs. 13 and 14, respectively. These figures assist to conclude that the transient stabilization performance and the convergence profile of the objective function for the SOA-based optimization are better than those of BFO-based [14] and GA-based [14] ones. The SOA is offering much better optimal transient performance than BFO and GA. Thus, the SOA may be accepted as a true optimizing algorithm for the power system based application as considered in the present work.
Table 4 Comparative GA–SFL, BFO–SFL, and SOA–SFL based results of eigenvalue analysis corresponding to an input operating condition P = 0.95, Q = 0.30, Xe = 1.08, Et = 0.5; all are in pu. Type of PSS
Algorithms-SFL
Damping ratio (n)
Un-damped natural frequency (xn), rad/s
Corresponding damped frequency (xd), rad/s
Lowest
Highest
Lowest
Lowest
Highest
CPSS
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.16 0.38 0.41
0.57 0.65 0.74
0.38 0.59 0.62
2.45 1.45 1.56
0.42 0.45 0.56
0.59 4.31 5.21
PSS2B
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.26 0.49 0.52
0.97 1.00 1.23
0.48 0.63 0.69
3.33 13.42 14.01
0.32 0.56 0.62
0.79 7.79 7.99
PSS3B
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.36 0.72 0.81
0.98 0.98 1.12
0.17 0.47 0.59
1.93 0.96 1.12
0.15 0.33 0.41
1.8 0.38 0.56
PSS4B
GA–SFL [14] BFO–SFL [14] SOA–SFL
0.2 0.37 0.45
0.95 0.95 1.45
0.55 0.34 0.49
1.22 2.97 3.15
0.54 0.28 0.31
0.36 2.98 3.02
(a) Fault 1
Highest
(b) Fault 2
Fig. 15. Comparative GA–SFL, BFO–SFL, and SOA–SFL based transient response profiles of Dxr for the generator equipped with PSS3B under change in operating conditions: (a) Fault 1, and (b) Fault 2.
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B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 Table 5 Off-nominal operating conditions, simulation of faults, algorithms-SFL, and optimal PSS3B parameters. Fault no.
Operating conditions (P, Q, Xe, Et; all are in pu)
Algorithm-SFL
PSS3B parameters
Fault 1
0.95, 0.3, 0.4752, 1.0
GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL
59.57, 59.22, 0.558, 0.013, 0.317, 0.169 13.50, 83.26, 0.529, 0.051, 0.008, 1.566 14.89, 31.86, 0.38, 0.005, 0.008, 0.544 No change in parameters
GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL GA–SFL [14] BFO–SFL [14] SOA–SFL
10.00, 53.24, 1.618, 0.192, 0.410, 0.254 10.88, 26.40, 1.279, 0.369, 0.070, 0.140 42.56, 33.81, 0.911, 0.005, 0.021, 0.693 No change in parameters
LT bus fault of duration 220 ms and subsequent clearing
0.95, 0.3, 1.08, 1.0
Fault 2
1.0, 0.6, 0.4752, 1.1
LT bus fault of duration 220 ms and subsequent clearing
0.2, 0.2, 0.4752, 1.1
14.22, 66.25, 0.208, 0.013, 0.161, 0.099 12.34, 83.67, 1.608, 0.239, 0.326, 0.473 14.09, 31.46, 1.17, 0.005, 0.008, 0.896
15.27, 12.11, 1.805, 0.893, 0.137, 0.005 13.07, 39.00, 1.322, 1.816, 1.006, 0.088 14.05, 35.25, 0.431, 0.005, 0.005, 0.621
7.5. Comparative eigenvalue analysis
8. Conclusion
Real parts of some eigenvalues for CPSS/PSS2B equipped system model are always either equal to or greater than r0 in the negative half plane of jx axis. A few eigenvalues are always outside the Dshaped sector (Fig. 7) for any operating condition. So, the value of OF1() are always higher (Table 3) for these two variants of PSS model. Much lower negative real parts of eigenvalues for PSS3B and PSS4B (not shown) cause higher relative stability than CPSS/PSS2B. In case of PSS3B and PSS4B larger reductions in (xn) and (xd) for some electromechanical oscillations are noticed due to higher damping ratios (ni n0) for those particular modes (Table 4). It is also noted from Table 4 that as compared to the results published in [14], SOA–SFL simulation offers better damping ratio (n), un-damped natural frequency (xn), and corresponding damped frequency (xd). These features for PSS3B equipped system model are better than CPSS/PSS2B/PSS4B counterpart. It establishes that the PSS3B damps the oscillation of Dxr very quickly under any sort of system perturbations.
The act of human searching ability is simulated in the SOA. It is a newly entrant heuristic stochastic optimization algorithm. This algorithm is easy to understand, simple to implement, very fast and can be used for a wide variety of optimization tasks. In this paper, a SOA-based fuzzy logic PSS tuning is presented. The results are compared with that of either BFO or GA based ones, reported in the literature. While comparing the benefits of SOA for PSS tuning it is revealed that this algorithm offers true optimal result with faster convergence ability. Thus, it may be proposed that the SOA is capable of efficiently and effectively solving PSS tuning problem. It has the capability of solving other engineering optimization problems.
7.6. Simulation of fault It is revealed that PSS3B is offering the best transient performance in damping all electromechanical modes of generator’s angular speed oscillations for all nominal and off-nominal system conditions, step changes of mechanical torque inputs (DTm) and reference voltage inputs (DVref). LT bus fault of duration 220 ms at the instant of 2.0 s is simulated for PSS3B equipped system model and the corresponding comparative transient response profiles of Dxr for SOA–SFL, GA–SFL and BFO–SFL based responses are plotted in Fig. 15 (Fig. 15a) corresponds to Fault 1 and Fig. 15b corresponds to Fault 2). A close look into these figures show that after the creation of the fault, the SOA–SFL based response recovers from this abnormal situation with much lesser fluctuation in angular speed as compared to those of BFO–SFL [14] and GA–SFL [14] based ones. Table 5 depicts the system model parameters as determined by SFL. Thus, SOA–SFL based model exhibits better response having lesser amplitude of angular speed deviation under fault and subsequent clearing condition yielding better dynamic robust transient performance than BFO–SFL [14] and GA–SFL [14] based ones. PSS3B equipped system model proves to be much less susceptible to faults because PSS3B settles all the state deviations to zero much faster than any other PSS.
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