Design optimisation of PID controller in automatic voltage regulator ...

Int. J. Power and Energy Conversion, Vol. 7, No. 4, 2016

Design optimisation of PID controller in automatic voltage regulator system using teaching-learning based optimisation algorithm Mohammad Jafar Hadidian Moghaddam Young Researchers and Elite Club, Zanjan Branch, Islamic Azad University, Zanjan, Iran Email: [email protected]

Mehdi Bigdeli* Department of Electrical Engineering, Zanjan Branch, Islamic Azad University, Zanjan, Iran Fax: +98-24-33460463 Email: [email protected] *Corresponding author

Saber Arabi Nowdeh Golestan Technical and Vocational Training Center, Gorgan, Iran Email: [email protected]

Mehdi Golshani Monfared Department of Electrical Engineering, Zanjan Branch, Islamic Azad University, Zanjan, Iran Email: [email protected] Abstract: It is becoming increasingly difficult to ignore the role of the optimal design of the proportional-integral-derivative (PID) controller in achieving a satisfactory response in the automatic voltage regulator (AVR) system. The main aim of this paper is to determine the optimal gains of a PID controller in the AVR system using teaching-learning based optimisation (TLBO) algorithm. Linearly decreasing inertia particle swarm optimisation (LDIPSO) is also employed for comparison purposes. The simulation results are provided to compare the effectiveness of these two different algorithms. MATLAB toolboxes are employed in this paper. Simulation results show that the TLBO algorithm has a considerable potential when compared to one of the best-known heuristic algorithms for optimisation problems. It proves to be more robust than PSO in performing optimal transient performance even under various nominal operating conditions. With the TLBO method, the step response of the AVR system can be improved.

Copyright © 2016 Inderscience Enterprises Ltd.

377

378

M.J.H. Moghaddam et al. Keywords: proportional-integral-derivative; PID controller; automatic voltage regulator; AVR; particle swarm optimisation; PSO; teaching-learning based optimisation; TLBO; optimal control. Reference to this paper should be made as follows: Moghaddam, M.J.H., Bigdeli, M., Nowdeh, S.A. and Monfared, M.G. (2016) ‘Design optimisation of PID controller in automatic voltage regulator system using teaching-learning based optimisation algorithm’, Int. J. Power and Energy Conversion, Vol. 7, No. 4, pp.377–390. Biographical notes: Mohammad Jafar Hadidian Moghaddam received his MSc degree from Islamic Azad University, Science and Research Branch, in 2014. His research interests are in renewable energy, design optimisation of power systems and microgrids. Mehdi Bigdeli received his BSc in Electrical Engineering from Iran University of Science and Technology (IUST), in 2004, an MSc degree from Faculty of Engineering of Zanjan University, in 2006 and a PhD degree from Islamic Azad University, Sciences and Research Branch, in 2012. His research interests are in fault detection, transient modelling and application of nature inspired algorithms in power systems analysis. He is currently an Assistant Professor at the Zanjan Branch, Islamic Azad University, Zanjan, Iran. Saber Arabi Nowdeh received his BSc degree and MSc degree in Electrical Power Engineering from Urmia University, Iran in 2008 and 2010, respectively. He received his PhD degree in Electrical Power Engineering from Tehran University, Iran, in 2014. He works in Golestan Technical and Vocational Training Center, Gorgan, Iran. His research interests are in renewable energy, distribution system planning and reliability assessment of power system. Mehdi Golshani Monfared received his MSc degree from Islamic Azad University, Science and Research Branch, in 2014. His research interests are in power transformer and renewable energy.

1

Introduction

The automatic voltage regulator (AVR) is an important component in the synchronous generator, and plays a key role in maintaining constant terminal voltage of the generator at a desired level and under different loading conditions (Ula et al., 1992). Basically, it operates based on the regulation of the exciter voltage in synchronous generators. Recent developments in the power system have heightened the need for proposing a suitable control strategy for the AVR system. A considerable amount of literature has been published on control of the AVR system in recent years. The proportional-integral-derivative (PID) controller is one of the most widely used regulators among them, which provides numerous advantages for control applications (Moghaddam et al., 2015; Miveh et al., 2015; Li et al., 2006). There is today a trend toward using this controller in the AVR system owing to the salient features of this regulator such as robustness and accurate stability margin.

Design optimisation of PID controller

379

The main challenge concerning the PID controller is to obtain proper gains, especially in the systems faces nonlinearities, high order and delay time. Hence, numerous methods have already been proposed in the literature to fine-tune the parameters of the PID controller. In Li et al. (2006), a novel tuning approach on the basis of the Ziegler-Nichols is suggested. Basically, it is impossible to achieve a suitable tuning response through the Ziegler-Nichols’ approach. It is due to the fact that obtaining a satisfactory response in this method greatly depends on the designer experience. In order to fine-tune the PID parameters, some approaches based on the artificial intelligence (AI) techniques such as the neural network and the fuzzy logic are proposed in Qi et al. (2004), Rubaai et al. (2008) and Rubaai and Young (2011). However, a serious weakness with the artificial neural network is that it suffers from the length of the training process and the convergence time. Moreover, the capability of the designer in realising the fuzzy logic approach plays an important role in fine-tuning of the membership functions. The Lambda method is also another approach, which is used for tuning in the PID controllers (Lennartson and Kristiansson, 2009). Nonetheless, it has proper performance only for second-order non-minimum phase systems. Recently, a variety of methods have been employed to fulfil the optimal tuning of the PID parameters using the evolutionary computation methods (Li et al., 2014; Krohling and Rey, 2001; Devaraj and Selvabala, 2014; Zhang et al., 2009; Gaing, 2004; Miveh et al., 2014; Chiou and Liu, 2009). Genetic algorithm (GA) and particle swarm optimisation (PSO) are widely used in published papers to deal with optimisation issues (Mukherjee and Ghoshal, 2007). However, they suffer from memory capability and computational burden. The teaching-learning based optimisation (TLBO) algorithm is proposed in this study for design optimisation of the PID controller in the AVR system. The main advantage of this algorithm is its simplicity. Moreover, the TLBO algorithm is effective and robust and has a great potential for solving multi-objective problems. The PSO is also employed to get the optimal PID gains. The simulation results are provided to compare the effectiveness of these two different algorithms. This paper is organised as follows. The AVR system model and the PID controller are reviewed in Section 2. The AVR system with the PID controller is presented in Section 3. Design of misfitness function (MF) is proposed in Section 4. In Section 5, the optimal design of the PID controller using the TLBO and the PSO is described. Input data as well as simulation results are given in Sections 6 and 7, respectively. Some conclusions are drawn in Section 8.

2

System modelling

Generally, the AVR system can be used to keep constant the terminal voltage of a synchronous generator at a preferable level. It contains four different parts, including amplifier, exciter, generator and sensor. It should be noted that the model of these four components is considered linear in this study.

380

M.J.H. Moghaddam et al.

The PID controller is applied to enhance the dynamic response of the system and to eliminate the steady-state error. The transfer function of a PID controller can be written as (Gaing, 2004): G(S ) = K p +

Ki + Kd s S

(1)

where Kp, Ki and Kd are the integral gain, the proportional gain and the derivative gain, respectively. The AVR model and the PID parameters comprising the transfer function of each item and its limitation are shown in Table 1. Table 1

Parameters of PID controller and AVR model with transfer function and parameter limits

Item PID controller

Transfer function Ki + Kd s S

G(S ) = K p +

10≤Ka ≤40; 0.02s ≤ τa ≤ 0.1 s

Ka 1 + τa s

1 ≤ Ke ≤10; 0.4 s ≤ τe ≤ 1.0 s

Kg 1 + τg s

Kg depends on load (0.7–1.0); 1.0 s ≤ τg ≤ 2.0 s

TFamplifier =

Exciter

TFexciter =

Sensor

3

TFgenerator = TFsensor =

0.2 ≤ Kp, Ki, Kd ≤ 2.0

Ka 1 + τa s

Amplifier

Generator

Parameter limits

Ks 1 + τss

0.001 s ≤ τs ≤ 0.06 s

The AVR system with PID controller

The block diagram of the AVR system with the PID controller is depicted in Figure 1. The main reason behind using the PID controller is to mitigate the steady-state error and to improve the dynamic response of the AVR system. By adding a pole at the origin with the help of the integral controller the steady-state error can be eliminated. Furthermore, improving the transient response can be achieved by adding a finite zero to the open-loop transfer function and the help of the derivative controller. Figure 2 shows how to change eigenvalues when a PID controller is employed for the AVR system. Table 2 presents the relative information of the system before and after using the PID controller.

Design optimisation of PID controller

381

Figure 1

Block diagram of the AVR system along with PID controller

Figure 2

Changes of eigenvalues when a PID controller is employed for the AVR system (see online version for colours)

Table 2

Parameters of the AVR system before and after using the PID controller

Parameters Damping ratio Angular frequency Eigenvalues

Before using PID controller

After using PID controller

22.67024

74.26928

0.5373848

0.6839602

–0.7859202 ± J 3.376488

–4.766323 ± J 4.297449

382

4

M.J.H. Moghaddam et al.

Design of MF

Parameter optimisation of the PID controller can be done using meta-heuristic methods. These methods based on their capabilities can be minimised the relevant MF value. In this regards, a MF is defined as the performance criterion. In this paper, the MF can be formulated as follows (Mukherjee and Ghoshal, 2007): OF = ( osh × 10, 000 ) + tst ′ 2 +

0.001

( max ′ dν )2

(2)

Optimisation of the MF corresponds to the minimum overshoot (osh), minimum settling time (tst), and maximum derivative of change in terminal voltage (max-dv). Repetitive trial run of the optimising algorithms reveals that osh is having the minimum and maximum value of 0.0000 and 0.0012, respectively. Consequently, the first term in the right-hand-side of (2) is in the order of 0–4. The numerical value of tst lies from 0.4955 to 1.5424. Therefore, incorporation of appropriate weighting factors in the right hand individual terms facilitates making each term competitive during the optimisation process.

5

Optimal design

The target of this paper is to optimise the parameters of the PID controller for the AVR system. Moreover, it can be ensured the voltage stability of the system. The TLBO and linearly decreasing inertia particle swarm optimisation (LDIPSO) methods are used to determine the optimal PID gains of the PID controller. Furthermore, the performances of these two optimisation methods are critically compared.

5.1 Optimisation by the TLBO It is becoming increasingly difficult to ignore the role of metaheuristic algorithms in optimising the parameters of the PID controller. TLBO algorithm is one of the most widely used evolutionary computation techniques and has been extensively utilised in the literature as a novel population oriented metaheuristic algorithm (Rao et al., 2011, 2012; Rao and Kalyankar, 2011; Rao and Patel, 2011). The TLBO is a smart optimisation algorithm based on an inspiration from the teaching-learning process. This method has recently been suggested by Rao et al. (2011) on the basis of the effect of teacher to students to enhance the scientific level of class. The flowchart of the TLBO algorithm is depicted in Figure 3. Basically, it imitates the teaching capability of students and their teacher in a class. Indeed, in this populationbased algorithm, the teacher attempts to improve the level of class to himself and the students. This algorithm comprises two important components, containing teacher and learners. In this regard, two different modes of learning can be introduced, including teacher learning based mode (known as teacher phase) and interacting with the other learners (known as learner phase). Note that, the grades of the students are considered as the output in TLBO algorithm. Clearly, such output depends on the quality of the teacher. Moreover, the interaction among students also plays an important role in enhancing the output results. Students comprise the population of this algorithm. Different design

Design optimisation of PID controller

383

variables are considered as different subjects offered to the learners, and learners’ result is analogous to the ‘fitness’ value of the optimisation problem. In the entire population, the best solution is considered as the teacher. The functioning of TLBO is divided into two phases, including teacher phase and student phase. Figure 3

Flowchart of the TLBO algorithm

Teacher phase: in this phase, the teacher attempts to enhance the class average close to himself. Since the increase of the class average from Mi to Mnew is so difficult, each set of problem variables are updated on the basis of the difference of these two values. Difference of these two values can be saved by the parameter Diff_Mean as follows. Diff _ Meani = ri ( M _ new − T f M i )

(3)

where Tf is the teacher parameter that is selected randomly between 1 and 2. The ri is a random number between 0 and 1. Applying the following equation each set of variables is updated. X new,i = X old ,i + Diff _ Meani

(4)

384

M.J.H. Moghaddam et al.

Student phase: students in addition to teacher’s knowledge, benefits from each other’s knowledge. The mathematical expression of this approach is that in each step and in each repetition, each variable (student) selects one of the students randomly. For instance, student i selects student j and this i is opposite of j. If the student j has more knowledge respects to student i then the student i updates his status based on the following equation: X new,i = X old ,i + ri ( X i − X j )

(5)

Else the student status is varied as follows. X new,i = X old ,i + ri ( X j − X i )

(6)

After changing the status, their level is evaluated by the objective function. Under these conditions, the best student is compared to the teacher of the previous step and if a better result has, is replaced with the previous iteration teacher. This process is continued to obtain convergence conditions.

5.2 Optimisation by the LDIPSO The PSO is presented by Kennedy and Eberhart (1995). The PSO can be defined as an effective metaheuristic algorithm to cope with highly complicated optimisation and search problems using a population of solutions from the search space which is called swarm. This algorithm can be initialised using a population of random solutions. These candidates called particles. Particles move in the n-dimensional search space (n is the number of optimisation variables). Each particle can be a possible solution for the optimisation problem. Since each particle can be adjusted based on its historical behaviours, the mathematical model of the algorithm can be expressed as: Si (t ) = ( Si ,1 (t ), Si ,2 (t ),… , Si , n (t ) )

(7)

where S is an n-dimensional vector, which present the location of ith particle and n is the number of optimisation variables. The speed of each particle at t (time) is expressed by the following equation. Vi (t ) = (Vi ,1 (t ), Vi ,2 (t ),… , Vi , n (t ) )

(8)

Indeed, equation (8) represents the rate of change of the position of each particle. Initially, the PSO is initialised with a group of random solutions with different location and velocity, afterwards each particle is updated by following two best values: Pi (t ) = ( Pi ,1 (t ), Pi ,2 (t ),… , Pi , n (t ) ) Pgb (t ) = ( S gb,1 (t ), S gb ,2 (t ),… , S gb , n (t ) )

(9) (10)

In the above-mentioned equations, P is the best previous location, which is obtained up to now. The Pgb(t) is also an n-dimensional vector, which shows the best previous location among particles so far. It should be noted that both Pi(t) and Pgb(t) are updated at each iteration. After finding the values of the Pi(t) and Pgb(t), each particle updates its new velocity and position based on the following equations:

Vi (t + 1) = k ⎡⎣ w(t )Vi (t ) + C1rand1 ( Pi (t ) − Si (t ) ) + C2 rand 2 ( Pgb (t ) − Si (t ) ) ⎤⎦

(11)

Design optimisation of PID controller Si (t + 1) = Si (t ) + Vi (t + 1)

385 (12)

where C1 and C2 are two learning factors, which control the influence of the social and cognitive components. w(t) is the inertia factor. rand1 and rand2 are also two random numbers independently generated within the range of [0, 1].

6

Input data and parameters

Maximum population size = 50, maximum allowed iteration cycles = 100, these two parameters are common for both the TLBO and the PSO algorithms. For the TLBO, number of parameters = 6; number of bits = (number of parameters) × 8; mutation probability = 0.04; crossover rate = 100%; selection ratio, Sr = 0.3. For the PSO, craziness probability = 0.2, c1 = c2 = 2.05. The value of Vi lies between 0.1 and 0.4. The parameters of the block diagram are chosen as Ka = 10, Ke = Kg = Ks = 1.0, τa = 0.1, τe = 0.4 s, τs = 0.01 s, τg = 1.0 s. Only Kg is the load dependent. Simulation results are obtained by MATLAB software on a 3.0 GHz P4 computer. The suggested optimisation method is presented to fine-tune the PID controller parameters with an AVR system. The four main components of the AVR system are amplifier, exciter, generator, and sensor. A step reference voltage signal of amplitude 1 pu is applied to the system.

7

Simulation results and discussion

The MATLAB-Simulink model of the AVR system with the PID controller is utilised in this study to confirm the effectiveness of the proposed optimisation algorithm. In order to show the comparative performance characteristics of TLBO-PID controller and PSO-PID controller, Table 3 is provided. It should be noted that MATLAB-Simulink-based simulation results illustrate the instability of the performance of the AVR system for higher value of Ka. Kg has been varied from 0.7 to 1.0 in steps of 0.1. τg has been varied from 1.0 to 2.0 in steps of 0.2. Consequently, Table 3 contains 24 different sets of input conditions of the AVR system. From Table 3, it may be noted that the TLBO-based optimisation technique offers lesser overshoot of change in terminal voltage and lesser settling time of change in terminal voltage over the PSO. Moreover, Figures 4 to 6 show the values of MF of terminal response profiles are less for TLBO-based optimisation than those of PSO-based optimisation technique. Figures 4 to 6 depict TLBO-based PID controller exhibits better optimal transient response characteristics in respect of step response of incremental change in terminal voltage in comparison with the PSO. Figure 7 also shows the step response of incremental change in terminal voltage of the system without inclusion of PID-controller. Oscillator transient response with non-zero settling error is observed. The minimum MF against the number of iteration cycles of the TLBO is recorded in each iteration to get the convergence profile of the algorithm. Figure 8 portrays the convergence profiles of minimum MF of TLBO and PSO. From Figure 8, it is clear that TLBO converges faster than PSO. PSO yields suboptimal higher values of MF.

386

M.J.H. Moghaddam et al.

Table 3

Performance judgement of PID controller with proposed TLBO and PSO

Kg

Tg

0.7

1.0

0.7

1.2

0.7

1.4

0.7

1.6

0.7

1.8

0.7

2

0.8

1

0.8

1.2

0.8

1.4

0.8

1.6

0.8

1.8

0.8

2

0.9

1

0.9

1.2

0.9

1.4

0.9

1.6

0.9

1.8

0.9

2

Type of controller

Kp

Ki

Kd

Osh

tst(s)

ITAE

MF

PSO

0.6752

0.4822

0.2052

0.0000

0.6699

6.5011e-004

1.4102

TLBO

0.7040

0.4898

0.2130

0.0000

0.6294

6.0976e-004

1.3169

PSO

0.8903

0.5522

0.3089

0.0000

0.5242

6.0073e-004

1.1265

TLBO

0.8888

0.5382

0.3035

0.0000

0.5252

5.7496e-004

1.1156

PSO

0.9657

0.5392

0.3266

0.0000

0.5591

5.6411e-004

1.6693

TLBO

0.7527

0.4208

0.2338

0.0000

0.5261

7.6943e-004

1.5075

PSO

1.0989

0.4480

0.3067

0.0003

0.9847

0.0016

16.6861

TLBO

1.1310

0.5871

0.4308

0.0000

1.0401

7.0364e-004

2.0457

PSO

1.8811

0.8766

0.8057

0.0001

0.9856

5.7278e-004

104.2414

TLBO

1.3243

0.6240

0.6126

0.0000

0.8548

8.8184e-004

2.4300

PSO

1.9950

0.8687

0.8465

0.0009

1.0134

5.8357e-004

88.3965

TLBO

1.2991

0.5281

0.4960

0.0000

0.5415

5.9629e-004

1.1446

PSO

0.5884

0.4005

0.2261

0.0000

1.5424

0.0011

3.2680

TLBO

0.5682

0.3507

0.1516

0.0000

0.7142

0.0015

1.5179

PSO

0.8343

0.5240

0.2802

0.0000

0.4955

5.2627e-004

4.4278

TLBO

0.7364

0.4428

0.2501

0.0001

0.5913

6.5422e-004

1.2155

PSO

0.7964

0.4395

0.2801

0.0000

0.6554

6.8444e-004

1.3465

TLBO

0.7876

0.4241

0.2594

0.0000

0.6195

6.6788e-004

1.3246

PSO

1.0913

0.5483

0.4183

0.0002

1.1376

5.7101e-004

6.4909

TLBO

0.9259

0.4479

0.3385

0.0000

1.1133

6.4477e-004

2.1264

PSO

1.6184

0.7572

0.6788

0.0009

0.9960

5.6457e-004

94.5376

TLBO

0.8096

0.3285

0.2480

0.0000

0.8291

0.0015

1.8582

PSO

1.6242

0.7005

0.6845

0.0007

1.0567

5.8382e-004

56.5779

TLBO

1.0658

0.4337

0.3974

0.0000

0.6041

6.5455e-004

1.3210

PSO

0.5264

0.3785

0.1546

0.0000

0.6443

6.0282e-004

1.5254

TLBO

0.6158

0.4294

0.2018

0.0000

1.0730

5.6483e-004

1.2488

PSO

0.7341

0.4597

0.2378

0.0002

0.7258

4.9906e-004

6.4057

TLBO

0.6419

0.3900

0.2103

0.0000

0.5992

6.1153e-004

1.2461

PSO

0.6755

0.3683

0.2050

0.0001

0.6681

7.0877e-004

3.1650

TLBO

0.7451

0.3991

0.2606

0.0000

0.5670

6.2898e-004

1.1859

PSO

0.6357

0.3148

0.1963

0.0000

0.8374

8.8237e-004

1.9538

TLBO

0.8389

0.3898

0.2972

0.0000

0.5628

8.1913e-004

1.1911

PSO

1.5402

0.7248

0.7109

0.0012

0.9593

6.3542e-004

144.0205

TLBO

0.8959

0.4337

0.3974

0.0000

1.3090

0.0011

2.8371

PSO

0.8838

0.3768

0.2992

0.0000

0.6890

6.9043e-004

2.0242

TLBO

1.0313

0.4380

0.4046

0.0000

0.5165

6.5846e-004

1.1505

Design optimisation of PID controller Table 3

Performance judgement of PID controller with proposed TLBO and PSO (continued)

Kg

Tg

1

1

1 1 1 1 1

Figure 4

387

1.2 1.4 1.6 1.8 2

Type of controller

Kp

Ki

Kd

Osh

tst(s)

ITAE

MF

PSO

0.3741

0.2685

0.1000

0.0000

0.8989

9.8627e-004

2.2348

TLBO

0.4008

0.2556

0.1017

0.0000

0.8296

7.5244e-004

1.7563

PSO

0.5672

0.3482

0.2041

0.0000

1.2629

7.9552e-004

2.4597

TLBO

0.6363

0.4012

0.2239

0.0000

1.0152

6.2154e-004

1.8801

PSO

0.7032

0.3978

0.2452

0.0000

0.5364

5.9535e-004

1.8467

TLBO

0.7216

0.4209

0.2693

0.0000

0.4964

7.2898e-004

1.2286

PSO

0.6796

0.3408

0.2183

0.0001

0.6652

5.9014e-004

2.4697

TLBO

0.7683

0.3723

0.2814

0.0000

0.5277

5.9285e-004

1.1194

PSO

1.2730

0.5952

0.5671

0.0009

1.0335

6.2287e-004

84.5070

TLBO

0.8210

0.3569

0.3037

0.0000

0.5622

6.9750e-004

1.1925

PSO

0.8367

0.3540

0.2886

0.0000

0.6226

6.1224e-004

2.0280

TLBO

0.8434

0.3388

0.2997

0.0000

0.6191

7.1364e-004

1.3360

Step response of incremental change in terminal voltage of PID controller-based AVR system for Kg = 0.9 and Tg = 1.4 (see online version for colours) 0. 012

0.01

0. 008

0. 006

0. 004

0. 002

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Note: Solid line: TLBO, dotted line: PSO. Figure 5

Step response of incremental change in terminal voltage of PID controller-based AVR system for Kg = 0.7 and Tg = 1 (see online version for colours)

Note: Solid line: TLBO, dotted line: PSO.

388 Figure 6

M.J.H. Moghaddam et al. Step response of incremental change in terminal voltage of PID controller based AVR system for Kg = 1 and Tg = 1 (see online version for colours) 0. 012

0.01

0. 008

0. 006

0. 004

0. 002

0 0

0.5

1

1.5

2

2. 5

3

3.5

4

4.5

5

Note: Solid line: TLBO, dotted line: PSO. Figure 7

Step response of incremental change in terminal voltage of an AVR system without PID controller (see online version for colours) 0.014

0.012

0. 01

0.008

0.006

0.004

0.002

0 0

Figure 8

5

10

15

Convergence profile of TLBO and PSO (see online version for colours) 60

50

40

30

20

10

0

0

20

Note: Blue line: TLBO, red line: PSO.

40

60

80

100

Design optimisation of PID controller

8

389

Conclusions

TLBO algorithm was proposed in this paper to optimally design the PID controller in the AVR system for enhancing the step response of terminal voltage. The proportional gain, the integral gain and the derivative gain were chosen to define the search space for the optimisation problem. Besides, the PSO method is also used to compare the performance of these two evolutionary computation techniques. The optimised step response of terminal voltage using the TLBO method is improved, compared to the PSO model. As a result of the proposed approach, fast design and an accurate performance prediction were achieved. Therefore, when this proposed approach is applied, it is more efficient in raising the precision of optimisation.

References Chiou, J.S. and Liu, M-T. (2009) ‘Numerical simulation for fuzzy-PID controllers and helping EP reproduction with PSO hybrid algorithm’, Simulation Modeling Practice Theory, November, Vol. 17, No. 10, pp.1555–1565. Devaraj, D. and Selvabala, B. (2014) ‘Real-coded genetic algorithm and fuzzy logic approach for real-time tuning of proportional-integral-derivative controller in automatic voltage regulator system’, IET Generation Transmission Distribution, Vol. 3, No. 7, pp.641–649. Gaing, Z-L. (2004) ‘A particle swarm optimization approach for optimum design of PID controller in AVR system’, IEEE Trans. Energy Convers., June, Vol. 19, No. 2, pp.384–391. Kennedy, J. and Eberhart, R.C. (1995) ‘Particle swarm optimization’, Proceedings of IEEE International Conference on Neural Networks, Perth, Australia, pp.1942–1948. Krohling, R.A. and Rey, J.P. (2001) ‘Design of optimal disturbance rejection PID controllers using genetic algorithms’, IEEE Trans. Evol. Comput., February, Vol. 5, No. 1, pp.78–82. Lennartson, B. and Kristiansson, B. (2009) ‘Evaluation and tuning of robust PID controllers’, IET Control Theory Appl., Vol. 3, No. 3, pp.294–302. Li, X.H., Yu, H.B., Yuan, M.Z. and Wang, J. (2014) ‘Design of robust optimal proportional-integral-derivative controller based on new interval polynomial stability criterion and Lyapunov theorem in the multiple parameters’ perturbations circumstance’, IET Control Theory Appl., Vol. 4, No. 11, pp.2427–2440. Li, Y., Ang, K.H. and Chong, G.C.Y. (2006) ‘PID control system analysis and design, problems, remedies, and future directions’, IEEE Control Syst. Mag., Vol. 26, No. 1, pp.32–41. Miveh, M.R., Rahmat, M.F. and Mustafa, M.W. (2014) ‘A new per-phase control scheme for threephase four-leg grid-connected inverters’, Electronics World, Vol. 120, No. 1939, pp.30–36. Miveh, M.R., Rahmat, M.F., Ghadimi, M.A.A. and Mustafa, M.W. (2015) ‘Power Quality improvement in autonomous microgrids using multi-functional voltage source inverters: a comprehensive review’, Journal of Power Electronics, Vol. 15, No. 4, pp.1054–1065. Moghaddam, M.J.H., Bigdeli, M. and Miveh, M.R. (2015) ‘A review of the primary-control techniques for the islanded microgrids’, Elektrotehniški vestnik, Vol. 82, No. 4, pp.169–175. Mukherjee, V. and Ghoshal, S.P. (2007) ‘Intelligent particle swarm optimized fuzzy PID controller for AVR system’, Electric Power Systems Research, Vol. 77, No. 12, pp.1689–1698. Qi, S-R., Wang, D-F., Han, P. and Li, Y-H. (2004) ‘Grey prediction based RBF neural network self-tuning PID control for turning process’, Proc. Int. Conf. Mach. Learning Cybern., pp.802–805. Rao, R.V. and Kalyankar, V.D. (2011) ‘Parameters optimization of advanced machining processes using TLBO algorithm’, International Conference on Engineering, Project, and Production Management, Singapore, pp.21–32.

390

M.J.H. Moghaddam et al.

Rao, R.V. and Patel, V. (2011) ‘Thermodynamic optimization of plate-fin heat exchanger using teaching-learning-based optimization (TLBO) algorithm’, Int. J. Adv. Therm. Sci. Eng., Vol. 2, No. 2, pp.91–96. Rao, R.V., Savsani, V.J. and Vakharia, D.P. (2012) ‘Teaching-learning-based optimization: an optimization method for continuous non-linear large scale problems’, Inform. Sci., Vol. 183, No. 1, pp.1–15. Rao, R.V., Vakharia, D.P. and Savsani, V.J. (2011) ‘Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems’, Comput. Aided Des., Vol. 43, No. 3, pp.303–315. Rubaai, A. and Young, P. (2011) ‘EKF-based PI-/PD-like fuzzy-neural-network controller for brushless drives’, IEEE Trans. Ind. Appl., November-December, Vol. 47, No. 6, pp.2391–2401. Rubaai, A., Castro-Sitiriche, M.J. and Ofoli, A.R. (2008) ‘Design and implementation of parallel fuzzy PID controller for high-performance brushless motor drives: an integrated environment for rapid control prototyping’, IEEE Trans. Ind. Appl., July–August, Vol. 44, No. 4, pp.1090–1098. Ula, A.H.M.S. and Hasan, A.R. (1992) ‘Design and implementation of a personal computer based automatic voltage regulator for a synchronous generator’, IEEE Trans. Energy Convers., March, Vol. 7, No. 1, pp.125–131. Zhang, J., Zhuang, J., Du, H. and Wang, S. (2009) ‘Self-organizing genetic algorithm based tuning of PID controllers’, Inform. Sci., March, Vol. 179, No. 7, pp.1007–1018.