Original Article
Sine-cosine algorithm-based optimization for automatic voltage regulator system
Transactions of the Institute of Measurement and Control 1–11 Ó The Author(s) 2018 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0142331218811453 journals.sagepub.com/home/tim
Baran Hekimog˘lu
Abstract A novel design method, sine-cosine algorithm (SCA) is presented in this paper to determine optimum proportional-integral-derivative (PID) controller parameters of an automatic voltage regulator (AVR) system. The proposed approach is a simple yet effective algorithm that has balanced exploration and exploitation capabilities to search the solutions space effectively to find the best result. The simplicity of the algorithm provides fast and high-quality tuning of optimum PID controller parameters. The proposed SCA-PID controller is validated by using a time domain performance index. The proposed method was found efficient and robust in improving the transient response of AVR system compared with the PID controllers based on ZieglerNichols (ZN), differential evolution (DE), artificial bee colony (ABC) and bio-geography-based optimization (BBO) tuning methods.
Keywords Sine-cosine algorithm, optimization, automatic voltage regulator
Introduction Research background Synchronous generators have been designed for nominal or rated voltage in electric power systems. In practice, disturbances including changes in loads and fluctuations in the output of turbine will influence the status of synchronous generators, thereby leading the synchronous generators to oscillate about the equilibrium state. The terminal voltage of generators cannot be sustained to a constant level as the immediate consequence of this oscillation. It is damaging the stability of power system and reduces the power quality. The generators are frequently equipped with an automatic voltage regulator (AVR) so as to deal this problem (Saadat, 1999). Nevertheless, the AVR system contains problems with inefficient oscillated transient response, a maximum overshoot and steady-state errors. These problems can be overcome through making a closed-loop system with the AVR system and the controller.
Literature survey Since it is difficult to find optimal or near optimal controller parameters with classical tuning methods, such as gain-phase margin, Cohen Coon and Ziegler-Nichols (ZN) (Ogata, 2002; Yuksel, 2012), to achieve better dynamic response, various heuristic optimization methods have been proposed for tuning controller parameters during the past two decades. The controller types that have been studied for improving the dynamic response of AVR system are proportional-integralderivative (PID), fractional order PID (FOPID), gray PID
(GPID), and fuzzy logic PID (FLPID). In literature, heuristic optimization-based tuning methods that have been applied to improve performance of the fore mentioned controller types are particle swarm optimization (PSO) (Gaing, 2004), artificial bee colony (ABC) (Gozde and Taplamacioglu, 2011), teaching learning-based optimization (TLBO) (Chatterjee and Mukherjee, 2016; Priyambada et al., 2014), gravitational search algorithm (GSA) (Duman et al., 2016; Kumar and Shankar, 2015), chaotic ant swarm (CAS) optimization (Zhu et al., 2009), chaotic optimization based on Lozi map (COLM) (Coelho, 2009), pattern search algorithm (PSA) (Sahu et al., 2012), anarchic society optimization (ASO) (Shayeghi and Dadashpour, 2012), many optimising liaisons (MOL) (Panda et al., 2012), Taguchi combined genetic algorithm (TCGA) (Hasanien, 2013), local unimodal sampling (LUS) optimization (Mohanty et al., 2014), firefly algorithm (FA) (Bendjeghaba, 2014), bio-geography-based optimization (BBO) (Guvenc et al., 2016), Nelder-Mead algorithm (NMA) (Verma et al., 2015), ant colony optimization (ACO) (Suri babu and Chiranjeevi, 2016), cuckoo search (CS) algorithm (Sikander et al., 2018), grasshopper optimization algorithm (GOA) (Hekimoglu and Ekinci, 2018) and genetic algorithm (GA) tuned neural networks (NN) (Al Gizi et al., Department of Electrical & Electronics Engineering, Faculty of Engineering and Architecture, Batman University, Turkey Corresponding author: Baran Hekimog˘lu, Department of Electrical & Electronics Engineering, Faculty of Engineering and Architecture, Batman University, Batman, 72060, Turkey. Email:
[email protected]
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2015). It is worth mentioning that, in literature, the most studied heuristic optimization methods that have either been proposed or used for comparison with other existing methods for AVR system are PSO, GA, ABC and DE. In addition, the ZN tuning method is the only classical method used for comparison purposes. Table 1 gives a concise literature survey of tuning methods used for AVR system in the last decade. On the contrary, there is no definite algorithm to find the fitting solution for the AVR system. Thus, studying a new heuristic optimization algorithm to find the optimal parameters of a PID controller in an AVR system is an observable problem. Sine-cosine algorithm (SCA) is a population-based heuristic optimization technique proposed by Mirjalili (2016) for solving optimization problems. It is not only simple but is also an effective algorithm to optimize real problems with unknown spaces. As its name suggests, SCA is based on mathematical functions of sine-cosine, and uses them to explore and exploit the space among two solutions effectively to find better solutions in the search space. Since its first proposal, SCA has attracted many researchers. Chen et al. (2018), Nenavath and Jatoth (2017), Sindhu et al. (2017), Elaziz et al. (2017) and Bureerat and Pholdee (2017) hybridized or improved versions of SCA; in Reddy et al. (2018) a binary variant of SCA, and in Tawhid and Savsani (2017) multi-objective version of SCA have been proposed. Other works found in literature are the implementation of SCA to real problems, such as training of a feedforward neural network system for improved prediction of liver enzyme concentrations of fish fed by nano-selenite (Sahlol et al., 2016), re-entry trajectory optimization for space shuttle vehicles (Banerjee and Nabi, 2017), peak power detection of partially shaded solar photovoltaic (PV) system (Kumar et al., 2017), parameter tuning of support vector regression (SVR) (Li et al., 2018), loading margin stability improvement for power system stability (Mahdad and Srairi, 2017), short-term hydrothermal scheduling problem of power generation units for power system economics (Das et al., 2018), feature selection (Sindhu et al., 2017; Hafez et al., 2016), object tracking (Nenavath and Jatoth, 2017), structural damage detection (Bureerat and Pholdee, 2017), and solving profit-based unit commit problem (Reddy et al., 2018).
advantages of the SCA technique, a SCA-based PID (SCAPID) controller is proposed for a high-order AVR system to minimize the maximum percentage overshoot, the settling time, the rise time, and the steady-state error of the terminal voltage of synchronous generator in this study. To the best knowledge of the author, there is no such study that has been proposed in the literature before. Therefore, in order to reveal the proposed method’s effectiveness and robustness, some comparison results between the proposed SCA-PID controller and DE-PSS, ABC-PID, BBO-PID and ZN-PID controllers are presented, as well. The comparative results show that the step voltage response of AVR system can be improved with proposed SCA technique. The rest of this paper is organized as follows. Section 2 describes the AVR system and its transfer function model with PID controller. Section 3 describes SCA-based optimization. Section 4 explains the implementation of SCA-PID controller in AVR system and presents numerical results and comparisons of simulations. Finally, Section 5 concludes the paper with a brief summary of key findings of the work.
Contribution of the present work The SCA technique is not mostly affected by the magnitude and nonlinear nature of the problem, and even in most cases where other global optimization techniques show early con-
AVR system In a power system, a change in real power mainly impacts the frequency and a change in reactive power mainly impacts the voltage magnitude. Since the mutual influence between voltage control and frequency control of a power generation system is usually weak enough, their analysis can be done separately (Saadat, 1999). Generators are one of the main sources of reactive power and the principal means of reactive power control of a generator is to control its excitation via an AVR. Thus, the function of an AVR is to keep the terminal voltage magnitude of a generator at a specified level. Figure 1 illustrates an AVR system. As seen from the figure, the AVR system mainly consists of four components; amplifier, exciter, sensor and generator. Hence, an AVR system with PID controller can be modeled by the transfer function of all its components, as shown in Figure 2. Here, DVref (s), DVs (s), DVe (s) and DVt (s) are reference input voltage, sensor output voltage, error voltage and generator terminal voltage, respectively. The transfer functions of AVR system without and with PID controller are given as Ka Ke Kg (1 + sTs ) DVt (s) = DVref (s) (1 + sTa )(1 + sTe )(1 + sTg )(1 + sTs ) + Ka Ke Kg Ks ð1Þ and
(s2 Kd + sKp + Ki )(Ka Ke Kg )(1 + sTs ) DVt (s) = DVref (s) s(1 + sTa )(1 + sTe )(1 + sTg )(1 + sTs ) + (Ka Ke Kg Ks )(s2 Kd + sKp + Ki )
vergence, the SCA finds the best solution more efficiently with faster convergence. The balance between the exploration and the exploitation is the basic benefit of this optimization technique (Mirjalili, 2016). Considering the aforementioned
ð2Þ
respectively. The boundary values of AVR system components, PID controller parameters and the used values in AVR system are all given in Table 2. It is worth mentioning that the used values in this paper are chosen in accordance with Gozde and Taplamacioglu (2011) and Guvenc et al. (2016). When these values are used, although it is stable, the step
Sikander et al. (2018)
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o
o
o
o
o
o
o
Guvenc et al. (2016)
Duman et al. (2016)
Suri babu and Chiranjeevi (2016)
o
o
o
o
o
Hekimoglu and Ekinci (2018)
o
o o
Kumar and Shankar (2015)
o
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Al Gizi et al. (2015)
Refers to the proposed method. o Refers to the method used for comparison with the proposed method.
FOPID PID Non-linear FLPID GPID NN Tuning Classical ZN methods Heuristic ABC optimization ACO ASO BBO CAS COLM CS DE FA GA GOA GSA GWO ICA LUS MOL NMA PSA PSO TCGA TLBO Performance IAE indices ISE ITAE ITSE ZG Other Analysis Transient methods Root locus Bode Robustness
Controller Linear
Reference Chatterjee and Mukherjee (2016)
Table 1. A concise literature survey of tuning methods used for AVR system. Verma et al. (2015)
o
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o
Tang et al. (2016)
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o
Bendjeghaba (2014)
o
Mohanty et al. (2014)
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o
o
Priyambada et al. (2014)
Hasanien (2013)
o
o
Panda et al. (2012)
o
o
o
Shayeghi and Dadashpour (2012)
o
Sahu et al. (2012)
o
Gozde and Taplamacioglu (2011)
o
o
Coelho (2009)
Zhu et al. (2009)
o
Gaing (2004)
o
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Figure 1. AVR system.
Figure 2. Transfer function model of AVR system with PID controller.
Table 2. Boundary values of AVR system with PID controller. Model
Parameter ranges
Used values in AVR system
Controller
0:2 < Kp , Ki , Kd < 2:0
Amplifier Exciter Generator Sensor
10 < Ka < 40, 0:02 < Ta < 0:1 1:0 < Ke < 10, 0:4 < Te < 1:0 0:7 < Kg < 1:0, 1:0 < Tg < 2:0 1:0 < Ks < 2:0, 0:001 < Ts < 0:06
Optimal values of Kp , Ki , Kd Ka = 10, Ta = 0:1 s Ke = 1:0, Te = 0:4 s Kg = 1:0, Tg = 1:0 s Ks = 1:0, Ts = 0:01 s
voltage response of AVR system without controller is highly oscillatory, as shown in Figure 3. The system has two real poles at s = 99:971 and 212.489 and two complex poles at s = 0:519864:6642i with a damping ratio of 0.111. The system has a rise time of 0.261 s, a peak time of 0.752 s, a
Figure 3. Step response of AVR system without controller.
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Figure 4. Effects of sine-cosine functions in equation (3) on the next position (Mirjalili, 2016).
settling time of 6.99 s, an overshoot of 65.7% and a steady state value of 0.909 pu, which corresponds to a steady state error of 0.091 pu. From these figures, it is clear that the dynamic response of AVR system needs to be improved and the steady state error must be cancelled by using a PID controller.
SCA-based optimization SCA is a recently proposed population-based heuristic algorithm (Mirjalili, 2016). It initially creates multiple random solutions and then makes them oscillate towards or outwards the best solution. In addition, in order to emphasize exploration and exploitation of search space, various random and adaptive variables are integrated to the algorithm. Exploration and exploitation are two commonly used phases of stochastic population-based optimization process and for SCA; both phases are included in the following position updating equations Xit + 1 =
Xit + r1 sin (r2 ) r3 Pti Xit , Xit + r1 cos (r2 ) r3 Pti Xit ,
r4 \0:5 r4 ø 0:5
ð3Þ
Here, Xit is the current solution’s position in i-th dimension at t-th iteration, Pti is the destination point’s position in i-th dimension at t-th iteration, and |.| is the absolute value. In the above equations, r1 defines the region of next position and has a range of [22, 2], in this paper. r2 defines how distant the movement should be towards or outwards the destination and has a range of [0, 2p]. r3 defines a random weight for destination with a stochastic influence emphasizing (r3 . 1) or reducing (r3 \ 1) the distance. Lastly, r4 defines equal switching from sine to cosine or vice versa and has a range of [0, 1]. The effects of sine and cosine functions in equation (3) on the next position are shown in Figure 4. As illustrated in the figure, equation (3) defines a space among two solutions in the search space. The solutions are able to search outside the space among their corresponding
destinations by changing the amplitude of sine-cosine functions, which guarantees exploration of the search space. But then, the periodic pattern of sine-cosine functions guarantees exploitation of the space among two solutions, by letting a solution to be shifted close to another solution. The effects of sine-cosine functions within a range of [22, 2] are shown in Figure 5. In order to achieve balanced exploration and exploitation phases in algorithm, the amplitude of sine-cosine functions in equation (3) is changed adaptively by the following equation r1 = a t
a T
ð4Þ
Here, t is the current iteration, T is the total number of iterations, which is equal to 100, and a is a constant, which is equal to 2 in this paper. In equation (4) r1 decreases linearly from a to 0.
Implementation of the proposed SCA-PID controller in AVR system The proposed SCA tuned PID controller, which is called SCA-PID controller is presented to make the transient response of an AVR system better, in this section. Figure 6 illustrates the block diagram of AVR system with SCA-PID controller. To analyze and design controllers there are different performance indices available in literature. The most popular ones are integral of absolute error (IAE), integral of time multiplied absolute error (ITAE), integral of squared error (ISE), and integral of time multiplied squared error (ITSE). All of these performance indices have their own benefits and drawbacks such as small overshoot can be achieved by using IAE and ISE but sacrifices the settling time whereas time weighted indices ITSE and ITAE may resolve this problem but calculation process is more complex with longer time and they never guarantee a required stability boundary (Gaing, 2004; Sikander et al., 2018). In this paper, a time domain
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Figure 5. Sine-cosine functions within a range of [-2, 2] letting a solution go beyond (outside the space among them) or around (inside the space among them) the destination (Mirjalili, 2016).
Figure 6. AVR system with SCA-PID controller.
performance index that includes time response specifications, which was proposed by Gaing (2004) is considered and given in equation (5) Min of J (Kp , Ki , Kd ) = (1 eb )(Mp + Ess ) + eb (ts tr ) ð5Þ Here, Ess, Mp, tr, and ts represent steady-state error, overshoot, rise time and settling time, respectively. b is a weighting factor within a range of [0.5, 1.5], which is set as 1.0 in this paper (Chatterjee and Mukherjee, 2016). The detailed computational flowchart of SCA implementation applied for optimizing the AVR performance is shown in Figure 7. The simulation has been done on MATLAB/Simulink 7.11.0 (R2010b) environment running on a personal computer with 2.50 GHz, Intel CoreTM Processor and 16.00 GB of RAM. The population size and total number of iterations are chosen as 40 and 100, respectively. After the simulation process, the optimized PID controller parameters obtained by using SCA method is given in Table 3. Figure 8 shows the typical convergence profile of the proposed SCA method. From this figure, it is observed that the SCA method converges at a J(Kp, Ki, Kd) value of 0.2844 after only 19 iterations. The transfer functions of AVR system with its PID controller tuned by the proposed SCA method is given in equation (6)
DVt (s) 0:04982s3 + 5:08s2 + 9:909s + 8:337 = DVref (s) 0:0004s5 + 0:0454s4 + 0:555s3 + 6:492s2 + 10:83s + 8:337
ð6Þ The comparative simulation results obtained for step response of AVR system with different controllers are shown in Figure 9.
Transient response analysis Maximum overshoot, settling time, rise time and peak time are all performance features that define the transient response of a unit step input. Therefore, the obtained results of these features are given in Table 4 to emphasize the efficiency of the proposed SCA-PID controller compared with the other controllers. For maximum overshoot, SCA has better results by 36.00% compared with ZN, 19.39% compared with DE, 12.21% compared with ABC, and 4.13% compared with BBO. For settling time, SCA has better results by 221% compared with ZN, 31.49% compared with DE, 27.07% compared to ABC, and 5.80% compared with BBO. For rise time, SCA has better results by 60.14% compared with ZN, 2.70% compared with DE, 5.41% compared with ABC, and 0.68% compared with BBO. For peak time, SCA has better
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Figure 7. SCA implementation block diagram for optimizing the AVR performance.
Table 3. Optimized PID parameters. Controller type
Kp
Ki
Kd
ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID
1.0210 1.9499
1.8743 0.4430
0.1390 0.3427
1.6524
0.4083
0.3654
1.2464 0.9826
0.5893 0.8337
0.4596 0.4982
Figure 9. Step response of AVR system with different controllers.
results by 111.84% compared with ZN, 18.42% compared with DE, 18.42% compared with ABC, and 4.28% compared with BBO. These results confirm that the proposed controller tuned by SCA has better performance than the other controllers tuned by ZN, DE, ABC and BBO.
Pole/zero map analysis Figure 8. Convergence curve in SCA method.
The pole/zero map analysis gives the location of the closed loop poles and their corresponding damping ratios and
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Table 4. Transient response analysis results of AVR system. Controller type
Maximum overshoot
Settling time (5% band)
Rise time
Peak Time
ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID
1.515 1.330
2.324 0.952
0.237 0.152
0.644 0.360
1.250
0.920
0.156
0.360
1.160
0.766
0.149
0.317
1.114
0.724
0.148
0.304
Figure 10. Pole/zero map plot of AVR system with SCA-PID controller. Table 5. Closed loop poles and damping ratios of AVR system. Controller type
Closed loop poles
Damping ratio
ZN-PID
2100.37 21.23 + j4.35 21.23 2 j4.35 27.72 22.96 2100.91 23.02 + j8.19 23.02 2 j8.19 26.29 20.22 2100.98 23.75 + j8.40 23.75 2 j8.40 24.74 20.25 2100.00 24.80 + j10.2 24.80 2 j10.2 22.1 20.585 2101.37 25.16 + j10.52 25.16 2 j10.52 20.91 + j0.82 20.91 + j0.82
1.00 0.27 0.27 1.00 1.00 1.00 0.34 0.34 1.00 1.00 1.00 0.40 0.40 1.00 1.00 1.00 0.427 0.427 1.00 1.00 1.00 0.44 0.44 0.74 0.74
DE-PID (Gozde and Taplamacioglu, 2011)
ABC-PID (Gozde and Taplamacioglu, 2011)
BBO-PID (Guvenc et al., 2016)
Proposed SCA-PID
Figure 11. Bode plot of AVR system with SCA-PID controller.
closed loop poles for all systems are on the left-hand plane. For the dominant complex conjugate poles, SCA acquired the left most poles on the s-plane with the highest damping ratio. It is greater than those acquired by ZN, DE, ABC and BBO by 62.96%, 29.41%, 10% and 3.04%, respectively.
Bode analysis defines the stability characteristics of a system. The pole/zero map plot for the proposed SCA method is shown in Figure 10. The comparative simulation results obtained for AVR system are given in Table 5. From this table, it is obvious that all methods provide stability for the AVR system, since the
Bode analysis is used to analyze stability of a system in frequency domain by observing its gain margin, phase margin, delay margin and bandwidth. The Bode plot for the proposed method is shown in Figure 11. The comparative simulation results obtained for AVR system are given in Table 6. As seen from the table, SCA has minimum peak gain, maximum phase
Table 6. Peak gain, phase margin, delay margin and bandwidth of AVR system. Controller type
Peak gain
Phase margin (deg)
Delay margin
Bandwidth
ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID
6.96 dB (4.18 rad/s) 4.20 dB (7.61 rad/s) 2.87 dB (7.52 rad/s) 1.56 dB (8.65 rad/s) 1.09 dB (9.17 rad/s)
51.9 58.4 69.4 81.6 87.3
0.14 s (6.48 rad/s) 0.092 s (11.10 rad/s) 0.111 s (10.90 rad/s) 0.122 s (11.7 rad/s) 0.128 s (11.9 rad/s)
7.429 12.800 12.880 14.284 14.821
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Table 7. Transient response analysis of AVR system under parameter changes. Parameter
Rate of change (%)
Controller type
Maximum overshoot
Settling time (5% band)
Rise time
Peak time
Tg
250
ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID
1.435 1.407 1.341 1.281 1.255 1.479 1.361 1.287 1.206 1.172 1.544 1.304 1.223 1.118 1.072 1.569 1.284 1.203 1.090 1.041 1.497 1.273 1.194 1.093 1.048 1.506 1.302 1.223 1.125 1.082 1.523 1.352 1.275 1.183 1.143 1.531 1.375 1.298 1.209 1.171
1.093 1.096 1.052 0.926 0.875 1.885 0.805 0.781 0.681 0.653 3.276 1.083 1.036 0.778 0.736 3.747 1.201 1.129 0.539 0.288 2.647 1.099 1.078 0.867 0.812 2.473 1.017 0.988 0.811 0.764 2.196 0.897 0.865 0.729 0.691 2.084 0.851 0.820 0.696 0.662
0.164 0.099 0.099 0.092 0.090 0.203 0.126 0.129 0.120 0.118 0.266 0.175 0.182 0.177 0.177 0.294 0.197 0.207 0.205 0.209 0.273 0.178 0.186 0.181 0.183 0.253 0.164 0.169 0.163 0.163 0.223 0.142 0.145 0.137 0.136 0.211 0.133 0.136 0.128 0.126
0.427 0.247 0.245 0.216 0.204 0.545 0.318 0.310 0.271 0.254 0.729 0.419 0.417 0.372 0.353 0.810 0.468 0.472 0.426 0.402 0.734 0.422 0.415 0.373 0.347 0.684 0.388 0.391 0.341 0.317 0.613 0.356 0.345 0.301 0.290 0.579 0.330 0.323 0.286 0.276
225
+25
+50
Kg
220
210
+10
+20
margins, maximum delay margin and maximum bandwidth compared with ZN, DE, ABC and BBO.
Robustness analysis To illustrate the robustness of the proposed method, the time constant and gain constant of generator have been changed separately in the range of 650 and 620, respectively. The comparative simulation results obtained for AVR system are given in Table 7. From the table, it can be seen that SCA has less change in terminal voltage with better performance compared with ZN, DE, ABC and BBO in terms of maximum overshoot, settling time, rise time and peak time when a change in a system parameter occurs. These results validate the robustness of the proposed SCA-based controller for AVR system.
Table 8. Performance comparison of different controllers. Controller type
J(Kp, Ki, Kd) value
ITSE value
ZN-PID DE-PID (Gozde and Taplamacioglu, 2011) ABC-PID (Gozde and Taplamacioglu, 2011) BBO-PID (Guvenc et al., 2016) Proposed SCA-PID
1.0936 0.5029
0.1070 0.0223
0.4391
0.0180
0.3281
0.0077
0.2844
0.0064
Performance index comparison Two different performance index values for different controllers are given in Table 8. As seen from this table, the proposed
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Transactions of the Institute of Measurement and Control 00(0)
SCA-based controller gives the minimum J(Kp, Ki, Kd) and ITSE values compared with other controllers. These results confirm that the proposed controller tuned by SCA has better performance than the other controllers tuned by ZN, DE, ABC and BBO.
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Conclusion A novel parameter tuning method based on SCA algorithm is presented for the first time to determine the optimal or near optimal PID controller parameters of an AVR system. In the parameter tuning process, the SCA algorithm is repeatedly run to give the optimal parameters of the PID controller based on a time domain performance index. From simulations, it is obvious that the proposed SCA-PID controller can perform a fast and efficient search for the optimal controller parameters. Also, the SCA-PID controller is compared with DE-PID, ABC-PID, BBO-PID and ZN-PID controllers with their results taken from transient analysis, pole/zero map analysis, Bode analysis and robustness analysis. The results showed that the proposed SCA-PID controller can obtain a better solution than the compared methods and have an excellent robustness in case of parameter uncertainty in the AVR system.
Declaration of conflicting interest The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The authors received no financial support for the research, authorship, and/or publication of this article.
ORCID iD Baran Hekimog˘lu
https://orcid.org/0000-0002-1839-025X
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