Modeling and seeker optimization based simulation for intelligent reactive power control of an isolated hybrid power system

Swarm and Evolutionary Computation 13 (2013) 85–100

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Regular Paper

Modeling and seeker optimization based simulation for intelligent reactive power control of an isolated hybrid power system Abhik Banerjee a, V. Mukherjee b,n, S.P. Ghoshal c a

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

art ic l e i nf o

a b s t r a c t

Article history: Received 23 May 2012 Received in revised form 8 April 2013 Accepted 5 May 2013 Available online 28 May 2013

Seeker optimization algorithm (SOA) is a novel heuristic population-based search algorithm based on the concept of simulating the act of human searching. In SOA, the acts of human searching capability and understanding are exploited for the purpose of optimization. In this algorithm, 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. In this paper, effectiveness of the SOA has been tested for optimized reactive power control of an isolated wind–diesel hybrid power system model. In the studied power system model, a diesel engine based synchronous generator (SG) and a wind turbine based induction generator (IG) are used for the purpose of power generation. IG offers many advantages over the SG but it requires reactive power support for its operation. So, there is a gap between reactive power demand and its supply. To minimize this gap between reactive power generation and its demand, a variable source of reactive power such as static VAR compensator (SVC) is used. The SG is equipped with IEEE type-I excitation system and dual input power system stabilizer (PSS) like IEEE-PSS3B. The performance analysis of a Takagi–Sugeno fuzzy logic (TSFL)-based controller for the studied isolated hybrid power system model is also carried out which tracks the degree of reactive power compensation for any sort of input perturbation in real-time. In time-domain simulation of the investigated power system model, the proposed SOA–TSFL yields on-line, off-nominal coordinated optimal SVC and PSS parameters resulting in on-line optimal reactive power control and terminal voltage response. The performance of the proposed controller, with the influence of signal transmission delay, has also been investigated. & 2013 Elsevier B.V. All rights reserved.

Keywords: Induction generator Power system stabilizer Seeker optimization algorithm Static VAR compensator Synchronous generator Wind–diesel hybrid power system

1. Introduction Many energy companies are considering installation of distributed generation (DG) as the energy sector is being restructured with a significantly more rapid pace inviting healthy competition in the energy market. DG is small-scale power generation that is, usually, connected to the distribution system [1]. These could be reciprocating engine driven generators, microturbines, cogeneration or combined heat and power, fuel cells, wind turbines, solar panels and other non-dispatchable forms of energy sources. DGs may be installed within the distribution system or at a customer’s site to improve reliability [2] by (a) adding system generation capacity, (b) freeing up additional system generation, transmission and distribution capacity, thereby, relieving transmission and n

Corresponding author. Tel.: +91 326 2235644; fax: +91 326 2296563. E-mail addresses: [email protected] (A. Banerjee), [email protected] (V. Mukherjee), [email protected] (S.P. Ghoshal). 2210-6502/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.swevo.2013.05.003

distribution bottlenecks, (c) adding generation capacity at the customer site for continuous power supply backup and (d) supporting power system maintenance and restoration operations with generation of temporary backup power. But DG in the power system will change the structure of the grid network and has a great impact on real-time operation and planning for traditional power system. It increases the complexity of controlling, protecting and maintaining distribution systems. Connection of the DG to the weak parts of power network will [3] (a) increase fault levels, (b) induce voltage variations, (c) degrade network transient stability, (d) reverse the power flow and (f) increase losses, depending on the relative size of the plant and the local loads. The steady-state, slow and transient variations of voltage levels related to the connection of a DG may lead to undesired operation of voltage control equipment on transformers at the primary substations of the network. A net reversal or increase of power flow in the line due to the presence or the loss of a DG, respectively, may cause intolerable voltage fluctuations [3–4].

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List of symbols abs ð⋅Þ returns the absolute value of the input vector Bc ¼ ωC susceptance of the fixed capacitor Bl ¼ 1=ωL susceptance of the fixed reactor Bsvc equivalent susceptance of the SVC dij ðtÞ search direction for the ith seeker on the jth variable at time t Ess steady-state error, p.u. FOD Figure of demerit (a time domain performance index) Ka exciter gain Kv gain of energy balance loop Mp overshoot, p.u. r j ; randj random number in [0, 1] sign ð⋅Þ signum function on each variable of the input vector S population size tr rising time, s ts settling time, s Ta exciter time constant, s 0 T do direct-axis open circuit transient time constant, s Tr voltage transducer time constant, s Tv time constant of energy balance loop, s Xd direct-axis reactance of synchronous generator under steady state condition, p.u.

In any hybrid energy system, there may be more than one type of electrical generators [5]. In such circumstances, it is normal although not essential for diesel engine based generator(s), usually, to be synchronous and wind-turbine based generator (s) to be asynchronous (induction generator (IG)). An IG offers many advantages over the conventional synchronous generator (SG) as a source of isolated power supply. Reduction in unit cost, ruggedness, absence of brushes (in squirrel cage construction), absence of separate DC source for excitation, easy maintenance, self-protection against severe overloads and short circuits etc. are the main advantages of an IG [6] but it requires reactive power support for its operation. Due to the mismatch between generation and consumption of reactive powers, more voltage fluctuations occur at generator terminal in an isolated system which reduces the stability and quality of the supply. The problem becomes more complicated in hybrid system having both IGs and SGs. In the present investigated isolated hybrid power system model, SGs and IGs are chosen with diesel generators and wind turbines, respectively. The maximum efficiency may be achieved by using the full reactive power capability of the wind system for decreasing the system losses and improving the post-fault voltage profile. Various flexible AC transmission systems (FACTs) devices are available which may supply fast and continuous reactive power support [7]. For stand-alone applications, effective capacitive VAR controller has become central to the success of the IG system. Switched capacitors, static VAR compensator (SVC) and static synchronous compensator may provide the requisite amount of reactive power support. Prior to the development of SVC, the adjustment of voltage in transmission system, other than generator and synchronous compensator, was made possible only by mechanically switched shunt reactors and capacitors. The switching of shunt reactors and capacitors is, normally, crude causing abrupt voltage changes along with voltage and current transient. Incidentally, a synchronous condenser may raise the fault level while providing controllable reactive power. The SVC, on the other hand, provides rapid and fine adjustment of voltage, which is desirable in power system control and operation [7]. In a stand-

X d0

direct-axis reactance of synchronous generator under transient state condition, p.u. ΔEf d incremental change in exciter voltage, p.u. ΔEq incremental change in internal armature emf under steady state, p.u. ΔP ig ; ΔQ ig incremental changes in active and reactive powers, respectively, of induction generator, p.u. ΔP load ; ΔQ load incremental changes in active and reactive powers, respectively, of load, p.u. ΔP sg ; ΔQ sg incremental changes in active and reactive powers, respectively, of synchronous generator, p.u. ΔQ svc incremental change in reactive power of static VAR compensator, p.u. ΔT e incremental change in electromagnetic torque, p.u. ΔT m incremental change in mechanical torque, p.u. ΔV incremental change in load voltage, p.u. ΔV pss incremental change in PSS output, p.u. ΔV ref incremental change in reference voltage, p.u. ΔV t incremental change in terminal voltage, p.u. Δωr incremental change in rotor speed, p.u. αij ðtÞ step length Δδ incremental change in rotor angle, p.u.

alone hybrid power system, the reactive power device has to fulfill the variable reactive-power requirements for the operation of the IG and that of the load. In the absence of proper reactive power support and its proper control, the system may be subjected to large voltage fluctuations, which is not desirable. Various types of SVC controllers have been proposed in the literature like lead-lag controllers [8,9], proportional-controllers [10,11], proportional-integral controllers [12,13] and proportionalintegral-derivative (PID) controllers [14]. Generally, the parameters of the SVC controller are selected on the basis of a typical load but these values may not be optimum for different voltage characteristics. Therefore, the parameters of the SVC controllers require proper tuning to have always optimum settings for variations in the load voltage characteristics with load. The constant impedance model is not accurate and is not the proper approximation in view of the strong influence of load voltage sensitivity on the dynamic performance of the power system [15]. The SVC damping controller, designed under constant impedance model, may become unstable under other values of loads. Power system stabilizer (PSS) is linked with a generator whereas SVC is used with transmission service providers. There are many PSSs and limited number of SVCs in a power system. The objectives of their operations are very different. PSS tackles local mode problems whereas SVC, mainly, tackles inter-area problems. Simultaneous tuning of these devices is a challenging problem for the power industry. But with the advent of fast acting and low price communication technology, it may be possible to provide the control center with the real time signals from remote areas. However, the use of centralized controller entails inputs that may arrive after a certain time delay. Time delays may make the control system to have less damping features. In order to satisfy specifications for wide-area control systems, the design of a controller should take into account this time delay in order to provide a controller that is robust, not only for the range of operating conditions desired, but also for the uncertainty in delay. In view of the above, this paper will focus on the design of the SVC-based damping controller considering the potential time delays [16–18].

A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

Seeker optimization algorithm (SOA) [19] is, essentially, a novel 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 the 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 present work focuses on the performance of the SOA as an optimizing tool in tuning the different tunable parameters of different test cases for the studied isolated hybrid power system model. A Takagi–Sugeno fuzzy logic (TSFL) based controller may adjust its parameters on-line according to the environment in which it works and may provide a good damping over a wide range of operating conditions. Takagi–Sugeno fuzzy model for on-line tuning of the PID controller has been adopted by Mukherjee and Ghoshal in [20]. Off-line conditions are the sets of nominal system operating conditions which are given in the TSFL table. On the other hand, these input operating conditions vary dynamically in real time environment and become off-nominal. This necessitates the use of very fast acting TSFL to determine the off-nominal controller parameters for off-nominal input operating conditions occurring in real-time. Thus, the main motivation of the present work is (i) to develop a strategy, based on SOA, for reactive power control of an isolated wind–diesel hybrid power system model comprising of a diesel engine based SG and a wind turbine based IG, (ii) to minimize the gap between reactive power generation and its demand through a variable reactive power source like SVC, (iii) to analyze the performance of TSFL-based controller that tracks the degree of reactive power compensation for any sort of input perturbation in real-time and (iv) to propose a strategy that yields on-line, off-nominal coordinated optimal SVC and PSS parameters resulting in on-line optimal reactive power control and terminal voltage response.

87

The rest of this paper is organized as follows. The modeling of the studied isolated hybrid power system is carried out in Section 2. The mathematical problem of the present work is formulated in Section 3. SOA and its application for the purpose of reactive power management are detailed in Section 4. A review of TSFL for on-line tuning of the different parameters of the studied model is presented in Section 5. In Section 6, simulation results are presented and discussed. Finally, the conclusions of the present work are drawn in Section 7.

2. Studied hybrid power system model The studied hybrid wind–diesel power system comprises of SG coupled with a diesel engine, the IG coupled with a wind turbine, electrical loads and reactive power compensating device such as SVC and its proper control mechanism. Fig. 1 depicts the single-line diagram of the studied isolated hybrid power system model. It is to be noted here both the SG and the IG fulfill the active power demanded by the load while the reactive power requirement for the operation of the IG and that of the load is provided by the SG and the SVC. The real and reactive power demand equations for the studied power system model, in the s-domain, may be modeled as ΔP ig ðsÞ þ ΔP sg ðsÞ−ΔP load ðsÞ ¼ 0

ð1Þ

ΔQ sg ðsÞ þ ΔQ svc ðsÞ−ΔQ load ðsÞ−ΔQ ig ðsÞ ¼ 0

ð2Þ

In view of the above, the objectives of the present work are two folds and may be documented as follows.

Any sort of disturbance in the reactive power demanded by the load (ΔQ load ) may lead to the system voltage change which, in turn, results in incremental change in reactive power demand of the other components. The left hand side of (2) represents the net incremental reactive power surplus and this surplus in reactive power demand will have its immediate effect on the change in system voltage profile. But as per the recommendation of the grid, the voltage change should be within its permissible limit [4]. This necessitates the proper monitoring of the terminal voltage profile by reactive power control of the whole stand-alone system. Fig. 2 depicts the transfer-function block diagram for the reactive power control of the studied wind–diesel hybrid power

(a) Objectives pertaining to the reactive power management of the studied isolated hybrid power system model are to (i) plot, compare and analyze the terminal voltage response profiles of the different studied test cases, (ii) maintain the real-time terminal voltage profiles for the different test cases by proper management of the reactive power support using SVC, (iii) explore the suitability of intelligent fuzzy logic-based tuned controller parameters under various changes in system operating conditions occurring in real-time and (iv) critically examine the performance of the studied isolated hybrid power system model under any sort of input disturbances. (b) Objectives pertaining to the performance of the optimization algorithms are to (i) tune the different tunable parameters of the different studied test cases with the help of SOA and genetic algorithm (GA) [20] individually, (ii) compare the figure of demerit (FOD) values and various performance indices yielded by SOA and GA, (iii) compare the convergence profile of the objective function value yielded by SOA to that yielded by other optimization metaheuristic recently surfaced in the state-of-the-art literature and (iv) establish the efficiency of SOA, as an optimization tool, with the help of statistical tool.

Fig. 1. Single-line diagram of the studied isolated wind–diesel hybrid power system.

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In (4), the values of K 1 and K 2 may be found out, respectively, by using K1 ¼

V cos δ X d0

ð5Þ

K2 ¼

ðE′ cos δ−2VÞ X d0

ð6Þ

It is to be noted here that ΔEq ðsÞ is proportional to the change in direct-axis field flux under steady state condition and is given in (7) for the small perturbation by solving the equation of flux linkages:
   1 ΔEq ðsÞ ¼ ð7Þ K 3 ΔEf d ðsÞ þ K 4 ΔVðsÞ 1 þ sT g where K3 ¼

X d0 Xd

ð8Þ

K4 ¼

½ðX d −X d0 Þ cos δ X d0

ð9Þ

and 0 T g ¼ T d0

X d0 Xd

ð10Þ

2.2. Modeling of the IG (Block B, Fig. 2) The reactive power requirement of an IG under constant slip condition (slip) in terms of generator terminal voltage and its parameters is given by ΔQ ig ðsÞ ¼ K 5 ΔVðsÞ

ð11Þ

where 2VX eq

K5 ¼

ð12Þ

ðR2Y þ X 2eq Þ

RY ¼ Rp þ Req Rp ¼

ð13Þ

r 20 ð1−SlipÞ Slip

ð14Þ

Req ¼ r 1 þ r ′2

ð15Þ

X eq ¼ x1 þ x′2 Fig. 2. Transfer-function block diagram for reactive power control of the studied isolated wind–diesel hybrid power system.

Here,

r 1 , x1 , r ′2

ð16Þ and

x′2

are the parameters of the IG.

2.3. Modeling of the SVC (Block A, Fig. 2) system model. From this figure, the governing transfer function equation for the incremental change in load voltage (ΔVðsÞ), in the s-domain, may be written as
   Kv ΔVðsÞ ¼ ð3Þ ΔQ sg ðsÞ þ ΔQ svc ðsÞ−ΔQ load ðsÞ−ΔQ ig ðsÞ 1 þ sT v

2.1. Modeling of the SG (Block C, Fig. 2) The incremental change in reactive power of the SG may be given by ΔQ sg ðsÞ ¼ K 1 ΔEq ðsÞ þ K 2 ΔVðsÞ

ð4Þ

In the studied wind–diesel isolated hybrid power system model, IG draws reactive power apart from the fact that most of the loads are inductive in nature and the inductive loads draw reactive power from the line. So, there is always a chance of deficit of reactive power in the studied model. In the present work, SVC is used to control the generator terminal voltage of the wind–diesel hybrid power system model by compensating the mismatch between reactive power generation and its demand [7,21]. In the present work, SVC (Block A of Fig. 2) [21] is used in the studied isolated hybrid power system model. The mathematical model of the SVC, in s-domain, is presented in (17)–(18) while the limits of the SVC output may be given by (19)
   Ks ΔBs1 ðsÞ ¼ ΔV ref ðsÞ−ΔVðsÞ ð17Þ 1 þ sT s

A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

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Table 1 Range of controller parameters. Name of the controller

Controller parameters

Minimum value

Maximum value

PSS3B [22]

Ks1 Ks2 Td1 Td2 Td3 Td4

−10.0 10.0 0.001 0.001 0.001 0.001

100.0 100.0 0.005 0.005 0.005 0.005

SVC [21]

Ks Ts Ts1 Ts2

10.0 0.005 0.005 0.005

 1 þ sT s1 ΔBs1 ðsÞ ΔBs2 ðsÞ ¼ 1 þ sT s2

100.0 0.05 0.05 0.05

Fig. 3. D-shaped sector in the negative half of the s-plane.




If

ðBc −Bl Þ o Bs2 o Bc ;

If

Bs2 ≥Bc ;

If

Bs2 ≤ðBc −Bl Þ;

Bsvc ¼ Bs2

ð18Þ 9 > =

Bsvc ¼ Bc > Bsvc ¼ ðBc −Bl Þ ;

ð19Þ

The upper limit of the SVC corresponds to the point at which the thyristor is completely shut off while the lower limit of the SVC corresponds to the point at which the thyristor is a lossless conductor. The position between the limits corresponds to a point at which the thyristor is partially closed. The different tunable parameters of the SVC are Ks, Ts, Ts1 and Ts2. The minimum and the maximum limits of these parameters, as used in the present work, are given in Table 1 [21]. 2.4. Modeling of the dual-input PSS (Block F, Fig. 2) The two inputs to the dual-input PSS are Δωr and ΔT e , unlike the same for the conventional single-input PSS is Δωr . In [22], the performance of dual-input PSS has been found to be better than single-input counterpart and among the dual-input PSSs (PSS2B, PSS3B and PSS4B), IEEE type PSS3B has been established to be the best one within the periphery of the studied power system model. This dual-input PSS3B configuration of [22] is equipped with the SG of the present work and its block diagram representation is shown in Block F of Fig. 2. The different tunable parameters of this dual-input PSS are Ks1, Ks2, Td1, Td2, Td3 and Td4. The minimum and the maximum limits of these parameters are also featured in Table 1 [22]. 2.5. State space modeling approach The state differential equations in standard from may be written as ΔX ¼ AΔX þ BΔU þ ΓΔP

ð20Þ

where ΔX , ΔU and ΔP are the state, control and disturbance vectors, respectively, while A, Band Γ are the system, control and disturbance matrices, respectively. The different components of these ΔX , ΔU and ΔP for the studied power system model are presented in Section 6.

3. Mathematical problem formulation 3.1. Eigenvalue analysis The tunable parameters of the studied isolated hybrid power system model are to be so tuned that some degree of relative stability and damping of electromechanical modes of oscillations are to be obtained [22,23]. So, eigenvalue analysis based approach

is carried out to satisfy these requirements. Based on the eigenvalue analysis, an objective function is formulated in [22] Min

J ¼ 10  J 1 þ 10  J 2 þ 0:01  J 3 þ J 4

ð21Þ

The weighting factors involved in (21) are suitably chosen to impart proper weights to J 1 , J 2 and J 3 , thereby, making them mutually competitive during the process of optimization.1 The different components involved in (21) are stated in [22]. It is to be noted here that by optimizing J, closed loop system poles are, consistently, pushed further left of jω axis with simultaneous reduction in imaginary parts also, thus, enhancing the relative stability and increasing the damping ratio above ξ0 . Finally, all the closed loop system poles should lie within a D-shaped sector (Fig. 3) [22] in the negative half plane of jω axis for which si ⪡− s0 and ξi ⪢ ξ0 . Selection of such low negative value of s is, purposefully, chosen. The purpose is to push the closed loop system poles as much left as possible from the jω axis to enhance the stability to a greater extent. 3.2. Design of FOD The prime requirements of the minimization of (21) are to obtain higher relative stability and to achieve better damping of the electromechanical modes of oscillations. It ensures minimal incremental change in terminal voltage (ΔV t (p.u.)) response profile. This may be achieved when minimized overshoot (M p ), minimized settling time (t s ), lesser rising time (t r ) and lesser steady state error (Ess ) of the terminal voltage response profile are achieved. So, to assess the performance of the eigenvalue analysis based minimization approach, modal analysis [22,23] is adopted. Thus, based on the results obtained from the modal analysis, a time domain performance index, called as FOD, is designed as in [24]. FOD ¼ ð1−e−β ÞðM p þ Ess Þ þ e−β ðt s −t r Þ

ð22Þ

It may be noted here that in (22), the value of β is set to be larger than 0.7 to reduce the overshoot and the steady-state error. On the other hand, it may be recommended that the value of β is to be smaller than 0.7 to reduce the rise time and the settling time. In the present work, β is set to 1.0. In (22), M p , Ess , t s and t r are referred to the transient response of ΔV t determined by modal analysis subsequent to Δδ ¼ 5∘ ¼ 0:0857 rad state perturbation [22,23]. 3.3. Measure of performance Apart from the eigenvalue analysis and FOD calculation, three more performance indices like integral absolute error (IAE), 1 An explanation for the selection of the weighting factors involved in (21) is given in Table A.1 of Appendix Section.

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integral square error (ISE) and integral square time error (ITSE) [24] are also considered in the present work and the definitions of these indices are given as Z ∝    IAE ¼ ð23Þ V t ðtÞdt

Start

Real coded initialization of S seekers

0

Z ISE ¼

∝ 0

Z ISTE ¼

V t 2 ðtÞdt ∝

0

t  V t 2 ðtÞdt

Set t = 0

ð24Þ ð25Þ

Divide the population into K subpopulations randomly

Calculate the objective function value for each seeker

3.4. Constraints of the problem The constrained optimization problem for the tuning of the parameters of the studied isolated hybrid power system model is subject to the limits of the different tunable parameters as given in Table 1. In the literature, several constraint handling techniques have been proposed to be used with evolutionary algorithms [25–27]. 3.5. Mathematical optimization problem The optimal values of the tunable parameters of the studied isolated hybrid power system model are obtained by minimizing the value of J (i.e. eigenvalue analysis approach) as given in (21) with the help of any of the optimizing technique with due regard to the constraints of the model as laid down in Table 1. And, subsequently, by adopting the modal analysis [22,23], the value of the FOD is obtained with the help of (22) by utilizing the optimal controller parameters yielded by any optimization technique and modal analysis. Thus, it is clear that modal analysis is not a part of the optimization task/loop rather it is used to calculate the value of the FOD only. In addition, the values of the performance indices, as given in (23)–(25), are also calculated.

Calculate the personal best position, neighborhood best position and population best position

Compute search direction for each seeker

Compute step length for each seeker

Update the position of each seeker

Calculate the objective function value for each seeker

Update the personal best position, neighborhood best position and population best position

4. SOA and its application for reactive power management Subpopulations learn from each other

4.1. SOA SOA [19,22,28] is a population-based heuristic search algorithm. It regards the optimization process as an optimal solution obtained by a seeker population. Each individual of this population is called a seeker. The total population is randomly categorized into three subpopulations. These subpopulations search over several different domains of the search space. All the seekers in the same subpopulation constitute a neighborhood. This neighborhood represents the social component for the social sharing of information. The flowchart of the algorithm [19,22,28] is depicted in Fig. 4. 4.2. Review of SOA-based works As a new population-based heuristic search algorithm, SOA has been, successfully, applied to function optimization problem [29,30], proton exchange membrane fuel cell model optimization [31], optimal reactive power dispatch [19,32], digital IIR filter design [33], tuning the structure and parameters of neural networks [34], power system stabilizer tuning [22], economic load dispatch [35,36], optimal assembly tolerance design [37], loadtracking performance of an autonomous power system [28] etc. A hybrid technique combining SOA and sequential quadratic programming method for solving dynamic economic dispatch problem with valve-point effects has been proposed in [38]. It is to be noted here that unlike particle swarm optimization (PSO) and differential evolution, SOA deals with search direction and step length independently.

Increment t = t + 1

Meet stopping criterion?

No

Yes Display the optimal fitness value and the optimal solution

Stop Fig. 4. Flowchart of SOA [19,22,28].

Ref. [28] deals with a standard simplified power system model comprising of a typical diesel engine and SG. The model used in [28] consists of a speed governor and a very simplified model of AVR with a PID controller, whose gains are only optimized by SOA. The present work is an extension of the same reactive power compensation task but for a more complicated practical type wind–diesel isolated hybrid power system model equipped with PSS and SVC, whose parameters are simultaneously optimized by SOA. So, both the power system models as well as the parameter

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91

Table 2 Implementation steps of the SOA for reactive power management.

Step 1 Initialization (a) Input operating conditions like V and X eq . (b) Input fixed parameters of the model (refer Fig. 2). (c) Input limits of the variables (refer Table 1). (d) Real value initialization of all the variable strings of the population within limits. (e) Read the SOA parameters like S ¼ 60, K¼ 3, μmax ¼ 0.95, μmin ¼ 0.0111, ωmax ¼ 0.8, ωmin ¼ 0. 0.2 [19,22,28]. (f) Determine the model parameters (refer Fig. 2). Step 2 Initialize the positions of the seekers in the search space randomly and uniformly. Step 3 Set the time step t ¼ 0 Step 4 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. Step 5 Let t ¼ t þ 1. Step 6 Select the neighbor of each seeker. Step 7 Determine the search direction and step length for each seeker and update his position. Step 8 Compute the objective function for the new positions. Step 9 Update the historical best position among the population and historical best position of each seeker. Step 10 Repeat from Step 5 till the end of the maximum iteration cycles/stopping criterion. Step 11 Determine the best string corresponding to minimum objective function value. Step 12 Determine the optimal variable string corresponding to the grand minimum objective function value. Step 13 Calculate FOD. Step 14 Calculate IAE, ISE and ISTE. Step 15 Plot terminal voltage response profile.

optimization task of the present work are different from those presented in [28]. 4.3. Implementation of SOA for reactive power management The steps of the SOA, as implemented for the reactive power management of the studied hybrid power system model of the present work, are shown in Table 2. 5. Review of TSFL for on-line tuning of controller parameters The whole process can be categorized into three steps viz. fuzzification, Takagi–Sugeno fuzzy inference and defuzzification. The details of these steps may be found in [20].

There are no tunable parameters for this test case. Case II: Model+PSS (i.e. the SVC block is absent in Fig. 2) The different vector components of the standard state differential equations, as expressed in (20), for this test case are given by h iT ΔX ¼ ΔX 1 ΔX 2 ΔX 3 ΔX 4 Δωr Δδ ΔEf d ΔEq ΔV t ΔV ð29Þ h ΔU ¼ ΔV ref

ΔT m

iT

ð30Þ

ΔP ¼ ½ΔQ ref 

ð31Þ

Simulations are carried out on an isolated hybrid wind–diesel power system model and the observations of the present work are presented in this section.

The tunable parameters for this test case are K s1 , K s2 , T d1 , T d2 , T d3 and T d4 . Case III: Model+SVC (i.e. the PSS block is absent in Fig. 2) The different vector components of the standard state differential equations, as expressed in (20), for this test case are given by h iT ΔX ¼ Δωr Δδ ΔEf d ΔEq ΔV t ΔV ΔBs1 ΔBs2 ð32Þ

6.1. Test cases considered

h ΔU ¼ ΔV ref

Based on Fig. 2, the following four test cases are considered in the present work.

ΔP ¼ ½ΔQ ref 

6. Simulation results and discussions

Case I: Only model (i.e. both the PSS and the SVC blocks are absent in Fig. 2) The different vector components of the standard state differential equations, as expressed in (20), for this test case are given by h iT ΔX ¼ Δωr Δδ ΔEf d ΔEq ΔV t ΔV ð26Þ h

ΔU ¼ ΔV ref ΔP ¼ ½ΔQ ref 

ΔT m

iT

ΔT m

iT

ð34Þ

The tunable parameters for this test case are K s , T s , T s1 , T s2 and T s2 . Case IV: Model+PSS+SVC (i.e. both the PSS and the SVC blocks are present in Fig. 2) The different vector components of the standard state differential equations, as expressed in (20), for this test case are given by h ΔX ¼ ΔX 1

ΔX 2

ΔX 3

ΔX 4

ð27Þ ð28Þ

ð33Þ

Δωr

Δδ

ΔEf d

ΔEq

ΔV t

ΔV

ΔBs1

ΔBs2

iT

ð35Þ h ΔU ¼ ΔV ref

ΔT m

iT

ð36Þ

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A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

ΔP ¼ ½ΔQ ref 

ð37Þ

The tunable parameters for this test case are K s1 , K s2 , T d1 , T d2 , T d3 , T d4 , K s , T s , T s1 and T s2 . 6.2. Parameter setting The SOA has been applied to control the reactive power of the studied isolated hybrid power system for investigating its optimization capability. GA is taken for the sake of comparison. The software has been written in MATLAB-7.3 language and executed on a 3.0-GHz Pentium IV personal computer with 512-MB RAM. The different input parameters of the present work are presented below. (a) For GA: The number of parameters depends on the different cases under study (viz. Case II–Case IV) .2 In this work number of bits ¼(number of parameters)  8 (for binary coded GA, as considered in the present work), population size ¼60, number of the fitness function evaluations (NFFEs) ¼500, runtime ¼30, crossover rate¼80% and mutation probability¼0.001 are chosen. (b) For SOA: The main parameters involved in SOA are the population size S, the number of subpopulations and the parameters of membership function of Fuzzy reasoning (including the limits of membership degree value, i.e., μmax and μmin in [19,22,28] and the limits of ω, i.e., ωmax and ωmin in [19,22,28]). In this paper, S ¼60, K¼ 3, NFFEs¼ 500, runtime¼ 30, μmax ¼ 0.95, μmin ¼0.0111, ωmax ¼0.8 and ωmin ¼0. 0.2 are chosen. The number of problem variables depends upon the model under investigation. (c) For the studied power system model: The values for the different constants of the studied hybrid power system model are presented in Appendix Section (Table B.1)

(b)

(c)

6.3. Discussion on results The simulation is carried out based on the varying load voltage (V, in p.u.) and the equivalent reactance (Xeq, in p.u.). For timedomain plots of ΔV t (in p.u.) in MATLAB–SIMULINK, input step perturbation of 0.01 p.u. is applied either in incremental change in reference voltage (ΔV ref ), or in incremental change in mechanical torque (ΔT m ) or simultaneously in both of them. All the simulations of the present work are carried out based on the same number NFFEs for all the algorithms. The results of interest including modal analysis based transient response characteristics (SOA–TSFL-based) are bold faced in the respective tables. The major observations of the present work are presented below. (a) Eigenvalue-based system performance analysis: Table 3 includes ten different sets of input operating conditions on sample basis (for test Cases II–IV) of the investigated isolated hybrid power system model. This table presents the optimal parameter values for these ten sets of input conditions (for test Cases II–IV). All the contents of this table form the Takagi– Sugeno rule base table on sample basis. From this table, it may be noted that the proposed SOA-based optimization technique offers lesser value of J as compared to the GA-based technique for all the test cases considered. Also, it is observed that the value of J is the least one for the proposed SOA-based approach for each test case, establishing the optimization performance 2

There is no as such tunable parameter in Case I.

(d)

(e)

of the SOA-based approach to be the better than the GA-based one. For the SOA-based approach, majority of the eigenvalues are within the D-shaped sector (Fig. 3), which yield lesser values of J1, J2, and J3. On the other hand, majority of the eigenvalues for the GA-based system are outside the D-sector but very close to and right side of ð−s0 ; j0Þ point. This yields higher values of J1, J2, and J3 for the GA-based approach. Thus, the value of J is more for the GA-based approach. Hence, from the eigenvalue analysis it may be observed that a considerable improvement has occurred in the transient performance for the proposed SOA-based approach and, thus, this technique yields optimal parameter values for the different cases considered. FOD-based performance analysis: The modal analysis based transient response characteristics (in terms of M p , Ess , t s , t r etc.) of the incremental changes of terminal voltages of the studied test model for the adopted approaches are also calculated and presented in Table 4 for V¼1.0 p.u., Xeq ¼1.08 p.u. on sample basis. From this table it may be noted that the value of the FOD is the minimum for the Case IV (model+PSS+SVC) for any particular approach which indicates that the best optimal transient response profile is achieved for this test case of the hybrid power system model. Moreover, the proposed SOA-based optimization technique offers lesser value of the FOD for a particular test case. Performance analysis based on performance indices: As a measure of performances of the comparative algorithms and the adopted test cases, the values of IAE, ISE and ISTE, as defined in (23)–(25), are calculated for all the input operating conditions at the end of the developed programs and the results are analyzed. The values of these performance indices for the input operating condition of V ¼1.0 p.u., Xeq ¼1.08 p.u. are also featured in Table 4 on sample basis. A look into the values of these performance indices reveals that the optimal transient response is achieved for Case IV as compared to either Case II or Case III for any particular optimization algorithm. Thus, both FOD discussed in Section 6.3(b) and performance indices discussed in Section 6.3(c) help to conclude that the SOAbased optimization technique shows better optimization performance and, hence, better optimal transient response as compared to the GA-based approach. Analysis of the time-domain responses: Fig. 5 is pertaining to the comparative SIMULINK-based time-domain response profiles of ΔV t (p.u.) for the studied test cases with 1% simultaneous step change in reference voltage and in load demand. From this figure, it is prominent that among the different studied test cases, test Case IV yields the optimal voltage response profile for the given nominal input operating condition and any algorithm (GA or SOA). This has happened with the help of true reactive power support received from the SVC and a true stabilizing impact offered by the PSS loop. This figure also assists to remark that the proposed SOA-based optimal controller parameters settle the ΔV t (p.u.) response more quickly. Thus, the SOA-based controller parameters perform better than the GA-based controller parameters in achieving the optimal voltage response profile. TSFL-based response: For on-line, off-nominal input sets of parameters, the TSFL model is utilized to get the on-line, optimal controller parameters and these controller parameters also yield the on-line incremental change in terminal voltage response profile as presented in Fig. 6. Table 5 illustrates the TSFL-based off-nominal, on-line optimal output parameters, J values and the corresponding modal analysis based performance of the incremental change in terminal voltage (using SOA-based optimal parameters of TSFL Table 3) for offnominal input sets of parameters. Thus, the suitability of the

A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

93

Table 3 TSFL-based table, optimal model parameters and J values yielded by GA and SOA. Input operating condition V, Type of model Xeq (both are in p.u.) 1.0, 1.08

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

1.0, 0.752

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

1.0, 0.93

J

tex (s)

−10.00, 55.00, 0.0050, 0.0038, 0.0029, 0.0050 −9.45, 98.45, 0.0010, 0.0023, 0.0050, 0.0050 78.55, 0.0491, 0.0050, 0.0500 89.45, 0.0049, 0.0500, 0.0500 −10.00, 99.64, 0.0050, 0.0050, 0.0050, 0.0500, 94.37, 0.0059, 0.0079, 0.0500 −9.45, 96.42, 0.0011, 0.0015, 0.0049, 0.0050, 96.45, 0.0500, 0.0050, 0.0500

866.6424 798.6554 563.2229 515.5836 922.8867

15.1726 7.1497 16.7894 7.6148 15.3192

887.5077

7.3700

−10.00, 70.48, 0.0048, 0.0050, 0.0050, 0.0050 −8.45, 94.15, 0.0011, 0.0017, 0.0050, 0.0048 77.50, 0.0058, 0.0067, 0.0058 96.45, 0.0050, 0.0487, 0.0467 −6.32, 99.84, 0.0050, 0.0049, 0.0050, 0.0050, 88.39, 0.0490, 0.050, 0. 0380 −7.00, 93.19, 0.0011, 0.0046, 0.0013, 0.0051, 94.16, 0.0512, 0.0051, 0.0500

861.6070 803.9820 565.8729 488.5856 917.7534

15.4645 8.1448 14.9413 7.6349 15.1790

888.9734

8.4512

−10.00, 88.75,0.0027, 0.0044, 0.0049, 0.0046 −9.45, 98.46, 0.0012, 0.0022, 0.0050, 0.0050 66.25, 0.0049, 0.0362, 0.0500 97.12, 0.0051, 0.0500, 0.0500 −10.00, 99.64, 0.0046, 0.0050, 0.0050, 0.0040, 78.57,0.0460, 0.0500, 0.0128 −9.45, 96.16, 0.0023, 0.0014, 0.0016, 0.0050, 94.45, 0.0500, 0.0055, 0.0500

885.2372 806.5307 576.2175 508.9289 924.8265

14.0055 8.1568 15.9042 7.6177 15.1050

875.5765

7.5453

Algorithm Optimal model parameters

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

0.97, 1.08

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

−10.00, 40.58,0.0044, 0.0050, 0.0049, 0.0046 −9.45, 96.12, 0.0015, 0.0024, 0.0050, 0.0050 55.00, 0.0480, 0.0470, 0.0490 99.55, 0.0055, 0.0500, 0.0500 −9.7109, 99.64, 0.0049, 0.0050, 0.0050, 0.0035, 68.71, 0.0480,0.0500, 0.0490 −9.45, 98.19, 0.0018, 0.0014, 0.0015, 0.0045, 97.46, 0.0500, 0.0055, 0.0500

865.1424 797.0054 562.1129 514.0014 921.1456 886.1456

14.0392 7.1513 15.9328 7.6460 15.1221 8.3262

0.97, 0.752

Case II: (Model+PSS)

−9.70, 55.00, 0.0049, 0.0036, 0.0050, 0.0050 −9.45, 97.49, 0.0015, 0.0018, 0.0050, 0.0050 67.30, 0.0490, 0.0470, 0.0500 97.55, 0.0054, 0.0500, 0.0500 −10.00, 99.64, 0.0049, 0.0050, 0.0050, 0.0500, 55.00, 0.0496, 0.0476, 0.0489 −9.78, 99.45, 0.0015, 0.0014, 0.0011, 0.0055, 97.44, 0.0500, 0.0059, 0.0500

860.1598 802.1459 564.1489 487.1981 916.1489

14.3091 8.1223 14.9774 7.5734 15.1287

887.9810

7.3975

−10.00, 40.58,0.0044, 0.0050, 0.0049, 0.0046 −9.45, 96.12, 0.0015, 0.0024, 0.0050, 0.0050 55.00, 0.0480, 0.0470, 0.0490 99.55, 0.0055, 0.0500, 0.0500 −9.7109, 99.64, 0.0049, 0.0050, 0.0050, 0.0035, 68.71, 0.0480,0.0500, 0.0490 −9.45, 98.19, 0.0018, 0.0014, 0.0015, 0.0045, 97.46, 0.0500, 0.0055, 0.0500

865.8963 14.0392 799.1460 7.1513 564.5897 15.9328 514.15896 7.6460 921.48967 15.1221

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

0.97, 1.08

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

886.48963

8.3262

0.97, 0.752

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

−9.70, 55.00, 0.0049, 0.0036, 0.0050, 0.0050 −9.45, 97.49, 0.0015, 0.0018, 0.0050, 0.0050 67.30, 0.0490, 0.0470, 0.0500 97.55, 0.0054, 0.0500, 0.0500 −10.00, 99.64, 0.0049, 0.0050, 0.0050, 0.0500, 55.00, 0.0496, 0.0476, 0.0489 −9.78, 99.45, 0.0015, 0.0014, 0.0011, 0.0055, 97.44, 0.0500, 0.0059, 0.0500

860.5897 14.3091 802.7986 8.1223 564.5890 14.9774 487.4598 7.5734 916.8933 15.1287 887.89634 7.3975

0.97, 0.93

Case II: (Model+PSS)

−8.71, 54.64, 0.0050,0.0050, 0.0038, 0.0046 −9.78, 98.88, 0.0011, 0.0023, 0.0050, 0.0050 55.00, 0.0470, 0.0490,0.0467 98.99, 0.0054, 0.0500, 0.0500 −10.00, 99.64, 0.0047, 0.0050, 0.0050, 0.0500, 88.75, 0.0478,0.0466, 0.0478 −9.99, 79.99, 0.0019, 0.0014, 0.0018, 0.0050, 99.45, 0.0500, 0.0050, 0.0500

884.8001 14.0232 805.8963 7.1471 575.48965 15.9305 507.0012 7.6186 923.4865 15.1168 874.8596

7.3828

−5.00, 79.60, 0.0050,0.0012, 0.0050, 0.0050 −9.78, 98.45, 0.0011, 0.0023, 0.0050, 0.0050 55.00, 0.0049, 0.0410, 0.0490 98.46, 0.0050, 0.0500, 0.0145 −10.00, 99.64, 0.0011, 0.0050, 0.0050, 0.0050, 99.64, 0.0490, 0.0430, 0.0500 −8.96, 96.42, 0.0011, 0.0018, 0.0017, 0.0051, 98.45, 0.0500, 0.0050, 0.0500

864.0015 796.2233 561.7896 513.0444 920.8891

14.1821 8.1638 15.9407 8.5863 15.1442

885.1000

7.2674

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

1.01, 1.08

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

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A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

Table 3 (continued ) Input operating condition V, Type of model Xeq (both are in p.u.)

1.01, 0.752

Algorithm Optimal model parameters

Case II: (Model+PSS)

GA SOA Case III: (Model+SVC ) GA SOA Case IV: (Model+PSS+SVC) GA SOA

−10.00, 76.09,0.0025, 0.0020, 0.0032,0.0047 −9.87, 86.45, 0.0014, 0.0017, 0.0050, 0.0050 45.50, 0.0500, 0.0050, 0.0490 98.45, 0.0051, 0.0500, 0.0489 −10.00, 99.64, 0.0041, 0.0050, 0.0050, 0.0050, 98.94,0.0498, 0.0417, 0.0456 −9.85, 97.46, 0.0014, 0.0014, 0.0013, 0.0051 98.45, 0.0500, 0.0050, 0.0500

J

tex (s)

861.8844 803.4596 563.5877 486.8855 915.9111 886.7894

14.1585 7.1574 15.8242 7.6041 15.1345 7.2044

Table 4 Comparative performance analysis of GA and SOA for the different test cases corresponding to the input operating condition of V ¼ 1.0 p.u., Xeq ¼ 1.08 p.u. Type of model

Algorithm

Overshoot Mp (%)

Steady state error Ess (s)

Rise time tr (s)

FOD

IAE

ISE

ISTE

Case II: (Only model)

–

0.0362

0.0068

0.0932

1.3514

964.3370

23.3409

16.9156

Case II: (Model+PSS)

GA SOA

0.0181 0.0124

0.0057 0.0045

0.0365 0.6752

1.3491 1.2013

961.9573 832.2174

21.2605 11.4737

16.6919 15.2887

Case III: (Model+SVC )

GA SOA

0.0177 0.0091

0.0068 0.0068

0.3594 0.8543

1.2776 1.1605

835.9681 603.5889

20.5427 10.5951

13.9596 9.1991

Case IV: (Model+PSS+SVC)

GA SOA

0.006 0.0047

0.0043 0.0047

5.0956 5.9990

0.2098 0.0075

745.5003 456.5935

12.8955 2.9761

11.2292 7.4150

Fig. 5. Comparative time-domain simulation responses of the incremental change in terminal voltage (p.u.) with 1% simultaneous step changes in reference voltage and in load demand for nominal input parameters based on (a) GA-based response and (b) SOA-based response.

proposed TSFL controller during real-time operation of the studied hybrid power system model is demonstrated. (f) Convergence profile: Based on the same NFFEs, Fig. 7 portrays the comparative convergence profiles of the minimum J values yielded by the GA and the SOA for the studied test cases. From this figure it may be noted that the proposed SOA-based approach offers faster convergence profile and also lesser final value of J as compared to the GA-based approach for any particular test case. GA yields suboptimal higher values of J. (g) Comparison of accuracy: Two sample t-test is a hypothesis testing method for determining the statistical significance of

the difference between two independent samples of an equal sample size [39]. Since its inception in 1995, PSO [40] has received increasing interest by the researchers' pool and is being, successfully, applied to many power system optimization problems [41]. So, the proposed SOA is compared, mainly, to some recently modified variants of PSO techniques. Since the original PSO is prone to suffer from the so-called “explosion” phenomena [42], two improved versions of PSO, i.e., PSO with adaptive inertia weight (PSOIWA) and PSO with a constriction factor approach (PSOCFA), were proposed by Shi and Eberhart [43] and Clerc and Kennedy [42], respectively

A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

95

Moreover, a real coded genetic algorithm (RGA) [44], asexual reproduction optimization (ARO) [45,46], artificial bee colony optimization (ABC) [47], bacteria foraging optimization (BFO) [48,49], gravitational search optimization (GSA) [50] and firefly algorithm (FFA) [51,52] are also considered for the sake Table 6 Results of the response of incremental change in terminal voltage for various algorithms and SOA (bolded) corresponding to the input operating condition of V ¼1.0 p.u., Xeq ¼1.08 p.u. over 30 runs for Case II: (Model+PSS).

Fig. 6. Comparative GA–TSFL and SOA–TSFL-based time-domain simulation response profiles of the incremental change in terminal voltage (p.u.) with 1% simultaneous step changes in reference voltage and in load demand for off-nominal input parameters.

Algorithms

Best

Worst

Mean

Std.

t-value

h

GA [20] RGA [44] ARO [45,46] ABC [47] BFO [48,49] GSA [50] PSOCFA [42] PSOIWA [43] FFA [51,52] SOA [Proposed]

0.128 0.0885 0.1316 0.0849 0.0843 0.0882 0.0877 0.0886 0.0853 0.0836

0.0113 0.0164 0.0136 0.0207 0.0083 0.0177 0.0171 0.0177 0.0084 0.0177

0.0308 0.0083 0.0085 0.0079 0.0081 0.00769 0.0079 0.0077 0.0080 0.0049

0.01662 0.00125 0.00183 0.00083 0.00098 0.001142 0.00114 0.000115 0.00098 0.0115

9.0615 2.0783 2.1860 1.8398 1.8999 1.7071 1.8356 1.7216 1.8992 0

1 1 1 1 1 1 1 1 1 0

Table 5 TSFL-based off-nominal, on-line optimal controller gains, J values and transient response characteristics of incremental change in output terminal voltage (using GA/SOAbased optimal parameters of Table 3) corresponding to the input operating condition of V ¼ 0.99 p.u., Xeq ¼1.072 p.u. Type of model

Algorithm Optimal model parameters

Case II: (Model+PSS)

GA-TSFL SOA-TSFL

−10.00, 99.64, 1.9922, 1.9932, 0.5038, 0.1609 −9.73, 98.45, 0.0014, 0.0024, 0.0053, 0.0051

862.12876 1.4567 963.1673 21.1587 16.7019 799.4563 1.2113 831.4589 12.4597 14.4589

Case III: (Model+SVC )

GA-TSFL SOA-TSFL

35.31, 0.0596, 0.0050, 0.9713 99.45, 0.0051, 0.0544, 0.0511

566.0053 517.4895

1.3776 836.1678 20.1876 13.7606 1.1705 602.4515 11.7889 9.1130

−10.00, 99.64, 0.0596, 0.0050, 0.0051, 0.0518, 99.64, 0.4414, 0.3089, 0.9947 918.1089 −7.89, 98.43, 0.0014, 0.0012, 0.0045, 0.0305, 98.43, 0.0519, 0.0057, 0.0514 888.4569

0.2025 746.5111 12.6755 11.3867 0.0073 455.4621 3.4526 6.4382

Case IV: (Model+PSS+SVC) GA-TSFL SOA-TSFL

J

FOD

IAE

ISE

ISTE

Fig. 7. Comparative GA- and SOA-based convergence profiles of minimum J values for the models like (a) Case II: (Model+PSS), (b) Case III: (Model+SVC) and (c) Case IV: (Model+PSS+SVC).

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A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

Table 7 Results of the response of incremental change in terminal voltage for various other algorithms and SOA (bolded) corresponding to the input operating condition of V¼1.0 p.u., Xeq ¼ 1.08 p.u. over 30 runs for Case III: (Model+SVC). Algorithms

Best

Worst

Mean

Std.

t-value

h

GA [20] PSOCFA [42] GSA [50] RGA [44] PSOIWA [43] BFO [48,49] ABC [47] ARO [45,46] FFA [51,52] SOA [Proposed]

6.584  10−4 −1.2456  10−4 0.02157 −0.0011 −0.0011 −0.0011 −0.0233 −0.0504 −0.1225 −0.0011

0.0151 0.0062 0.0044 0.0057 0.0057 0.0056 0 0.0058 0.0217 0.0057

0.005688 0.0056 0.00981 0.0056 0.0056 0.0057 0.0061 0.0056 0.0056 0.0054

3.64  10−4 3.1027  10−4 128.9  10−4 4.3549  10−4 4.3549  10−4 4.3550  10−4 22.0  10−4 6.1  10−4 7.3271  10−4 4.3549  10−4

3.5880 2.6448 2.4178 2.2968 2.2963 2.2961 2.2071 1.8869 1.6592 0

1 1 1 1 1 1 1 1 1 0

Table 8 Results of the response of incremental change in terminal voltage for various other algorithms and SOA (bolded) corresponding to the input operating condition of V¼ 1.0 p.u., Xeq ¼ 1.08 p.u. over 30 runs for Case IV: (Model+PSS+SVC). Algorithms

Best

Worst

Mean

Std.

t-value

h

GA [20] RGA [44] ARO [45,46] BFO [48,49] ABC [47] GSA [50] PSOIWA [43] PSOCFA [42] FFA [51,52] SOA [Proposed]

−0.05504 −0.0329 −0.0594 −0.0591 −0.0088 0.00925 −0.0329 −0.0329 −0.0019 −0.0332

0.0131 0.0147 0 0 0 0.0147 0.0147 0.0147 0.0015 0.0148

0.0065 0.0058 0.0059 0.0060 0.0059 0.005886 0.0058 0.0059 0.0058 0.0056

0.003375 0.00021 0.00047 0.00048 3.8642  10−4 0.0002808 0.00021 0.00021 2.2208  10−4 6.8141  10−4

1.8483 1.9834 2.5627 2.5689 2.7080 2.7440 1.9834 2.9751 1.9733 0

1 1 1 1 1 1 1 1 1 0

Fig. 8. Comparative convergence profiles of minimum J values yielded by different algorithms for the models like (a) Case II: (Model+PSS), (b) Case III: (Model+SVC) and (c) Case IV: (Model+PSS+SVC).

A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

of comparison with the SOA. The different input parameters of the comparative algorithms are presented below. (i) For PSOIWA [43]: population size ¼ 60, NFFEs ¼500, runtime ¼30, learning rate c1 ¼c2 ¼2.05; inertia weight linearly decreases from 0.9 to 0.4 with increasing runtime; the maximum velocity vmax is set at 20% of the dynamic range of the variable on each dimension. (ii) For PSOCFA [42]: population size ¼60, NFFEs ¼500, runtime ¼30, learning rate learning rate c1 ¼c2 ¼2.01 and constriction factor¼0.729844. (iii) For RGA [44]: population size¼60, NFFEs¼500, runtime¼ 30, crossover rate¼80%., mutation probability¼ 0.001, crossover¼single point crossover, mutation¼ Gaussian mutation, selection¼Roulette wheel and selection probability¼1/3. (iv) For ARO [45,46]: population size ¼60, NFFEs ¼500, runtime ¼30, number of bits ¼(number of parameters)  8. (v) For ABC [47]: number of bees ¼60, NFFEs ¼500, runtime ¼30 and limit ¼40 (a control parameter in order to abandon the food source). (vi) For BFO [48,49]: total number of bacteria (numBact)¼60; NFFEs¼ 500; runtime¼30, maximum reproduction cycle (maxreprod )¼ 10; maximum chemo tactic cycle (maxchemo )¼ 20; maximum dispersal cycle (maxdispersal )¼ 1; some positive constants dattract ¼2.0, wattract ¼ 0.2, drepelent ¼2.0, wrepelent ¼0.1; selection ratio (sr )¼0.5; probability of elimination (P ed )¼0.3; cmax ¼0.1; cmin ¼0.0001; d1 ¼ 0.00001; d2 ¼0.00001 and maximum swim length (integer value) (maxswim )¼4. (vii) For GSA [50]: population size ¼60, NFFEs ¼500, runtime ¼30, τ ¼ 20, G0 ¼100, rNorm ¼ 2, rPower ¼1 and ε ¼ 0.0001. (viii) For FFA [51,52]: swarm size¼ 60, NFFEs ¼500, runtime ¼30, α ¼ 1, β0 ¼10 and γ ¼1. To compare the proposed method with other algorithms, the concerned performance indices including the best, worst, mean and standard deviation (Std.) of the Case II–Case IV are summarized in Tables 6–8, respectively. In order to determine whether the results obtained by the SOA are statistically different from the results generated by other algorithms, the t-tests [53] are conducted. An h value of one indicates that the performances of the two algorithms are statistically different with 95% certainty, whereas h value of zero implies that the performances are not statistically different. The corresponding h values are also presented in Tables 6–8, respectively, for test Cases II–IV. Thus, from statistical analysis, it is clear that the SOA-based optimization technique offers robust and promising results. Comparative convergence profiles for minimum J values yielded by all the comparative algorithms for the models like (a) Case II: (Model+PSS), (b) Case III: (Model+SVC) and (c) Case IV: (Model+PSS+SVC) are depicted in Fig. 8, corresponding to an operating condition of V ¼1.0 p.u., Xeq ¼ 0.93 p.u. From this figure it may be noted that the proposed SOA-based approach offers faster convergence profile and also lesser final value of J as compared to all the compared algorithms for a particular test case of the studied isolated hybrid power system model. (h) Performance evaluation with sinusoidal load pattern: The performance evaluation of the studied hybrid power system model with sinusoidal load pattern and SOA-based optimization technique for on-line, off-nominal input sets of parameters is also carried out. The expression for the sinusoidal load change containing low sub-harmonic terms [54] is assumed as in (38) and its variation with time (for 100 s) is

97

plotted in Fig. 9(a). ΔP d ¼ 0:03 sin ð4:36tÞ þ 0:05 sin ð5:3tÞ−0:1 sin ð6tÞ

ð38Þ

Under the on-line, off-nominal input sets of parameters with sinusoidal load pattern, the TSFL extrapolates the nominal optimal parameters intelligently and linearly in order to

Fig. 9. Dynamic performance evaluation with 1% simultaneous step changes in reference voltage and in load demand (a) applied sinusoidal load pattern, (b) SOA– TSFL-based time domain simulation response of incremental change in terminal voltage (p.u.) for off-nominal input parameters like V ¼ 1:01 p:u andX eq ¼ 0:93 p:u and (c) zoomed view of (b).

Fig. 10. Comparative SOA-based time-domain simulation responses of the incremental change in terminal voltage (p.u.) with 1% simultaneous step changes in reference voltage and in load demand for nominal input parameters.

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determine the off-nominal optimal controller parameters. These controller parameters yield the optimal incremental change in terminal voltage response profile as presented in Fig. 9(b). In this figure, it may be noted that a continuous fluctuation is always persisting in Case II. For Case III, though the situation is improved but the fluctuation is further improved with Case IV. Fig. 9(c) shows the incremental change in voltage response in zooming mode for small time duration (0–0.1 s) for better pictorial representation and easy understanding. (i) Performance evaluation considering time delay: Fig. 10 depicts the time domain comparative response of incremental change in terminal voltage of the studied power system model between the cases without and with a time delay (of duration 100 ms). This figure, basically, investigates the impact of remote signal on the local one. Lesser deviation in incremental change in terminal voltage is noticed when remote signal is considered with a proper time delay. Remote signal, thus, seems to be a better choice compared to the local one as the power system oscillations are quickly damped out with the remote one.

meters of the power system model. Takagi–Sugeno fuzzy logic is applied to obtain the on-line transient response of the incremental change in terminal voltage for off-nominal input parameters. The function of this fuzzy logic is to extrapolate the controller nominal optimal output parameters intelligently and linearly in order to determine off-nominal optimal output parameters with noticeably low on-line computational burden. The proposed seeker optimization algorithm may become a very promising algorithm for solving more complex engineering optimization problems in future research.

Acknowledgment The authors would like to thank the editor-in-chief and the anonymous reviewers for their insightful comments and suggestions.

Appendix 7. Conclusion

A. Sensitivity analysis

In this paper, the optimal reactive power control of an isolated hybrid power system model is investigated. The induction generator has been considered for electric power generation from wind turbine and the SVC for providing variable reactive power required by the system. A complete dynamic model of the system has been derived to study the effect of load disturbances. The different tunable output parameters of the investigated model are, individually, optimized by GA and the proposed novel seeker optimization algorithm. The potential benefits yielded by the proposed algorithm are compared to GA and the other state-ofthe-art algorithms surfaced in the recent literature. A time-domain simulation of the studied model is also carried out under different forms of input perturbations. From the simulation study, it is revealed that the coordinated proper tuning of the SVC and the PSS parameters yields the true optimal transient response of the incremental change in terminal voltage for nominal input para-

A sensitivity analysis in support of choosing the appropriate weights of the different components involved in (21) is presented in Table A.1 which justifies the fact that J1 and J2 should be given more weights while J3 should be given less weights to make them mutually competitive during the process of optimization. Normally, the value of J4 is zero.

B. Power system data The values of the constants used for the simulation (Fig. 2) are K a5 ¼ 2V  Bsvc , K v ¼ 6:667, T v ¼ 7:855  10−4 s, T r ¼ 0:02 s, K a ¼ 200, T a ¼ 0:01 s, H ¼ 1:0, D ¼ 0:8, ω0 ¼ 314. The other data of the studied hybrid isolated power system model are presented in Table B.1.

Table A.1 Sensitivity analysis of J. Input operating condition V, Xeq (both are in p.u.)

Type of modeln

Algorithm

J1

J2

J3

J4

J

1.0, 1.08

Case II: (Model+PSS)

GA SOA GA SOA GA SOA

86.1835 79.4258 55.8553 51.1110 91.8041 88.2707

0.0872 0.0872 0.0886 0.0789 0.0831 0.0864

393.5378 352.5378 378.3905 368.4556 401.4710 393.6715

0 0 0 0 0 0

866.6424 798.6554 563.2229 515.5836 922.8867 887.5077

GA SOA GA SOA GA SOA

85.7000 79.9692 56.1003 48.4512 91.2946 88.4255

0.0899 0.0851 0.0986 0.0489 0.0872 0.0843

370.8000 343.9036 388.3905 358.4556 393.5378 387.5378

0 0 0 0 0 0

861.6070 803.9820 565.8729 488.5856 917.7534 888.9734

GA SOA GA SOA GA SOA

88.0462 80.2404 57.1456 50.4780 92.0001 87.1000

0.0833 0.0800 0.0886 0.0549 0.0814 0.0645

394.2167 332.6718 387.5467 359.9876 401.1478 393.1456

0 0 0 0 0 0

885.2372 806.5307 576.2175 508.9289 924.8265 875.5765

Case III: (Model+SVC ) Case IV: (Model+PSS+SVC)

1.0, 0.752

Case II: (Model+PSS) Case III: (Model+SVC ) Case IV: (Model+PSS+SVC)

1.0, 0.93

Case II: (Model+PSS) Case III: (Model+SVC ) Case IV: (Model+PSS+SVC)

n

The details of different models are presented in Section 6.

A. Banerjee et al. / Swarm and Evolutionary Computation 13 (2013) 85–100

99

Table B.1 Data for the proposed hybrid power system model. Synchronous generator

Induction generator

Load

SVC

P sg ¼0.4 p.u kW Q sg ¼ 0.2 p.u kVAR Eq ¼ 1.12418 p.u. δ ¼17.24831 Eq0 ¼0.9804 p.u. V ¼1.0 p.u. X d ¼ 1.0 p.u. X d0 ¼0.15 p.u. 0 T do ¼ 5.0 s

P ig ¼ 0.6 p.u kW Q ig ¼0.291 p.u kVAR P in ¼0.667 p.u kW η ¼90% Power factor¼0.9 r 1 ¼r 20 ¼ 0.19 p.u. x1 ¼ x20 ¼0.56 p.u. S ¼−3.5% –

P load ¼1.0 p.u kW Q load ¼ 0.75 p.u. kVAR Power factor¼ 0.8

Q ¼0.841 p.u. kVAR α ¼138.81

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