Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012

PID Design for AVR System by PSO and Imperialist Competitive Algorithms Saeed Jalilzadeh1 , Saeed Behzadpoor1 , Mohammad Hashemi1 1

Electrical Engineering Department, Zanjan University, Iran Email: jalilzadeh@znu.ac.ir , S.behzadpoor@znu.ac.ir , hashemi.mhz@gmail.com Abstract—In thermal power plants, biomedical instrumentation the popular use of proportional-integralderivative (PID) controllers can be noted. Proper tuning of such controllers is obviously a prime priority as any other alternative situation will require a high degree of industrial expertise. So in order to get the best results of PID controllers the optimal tuning of PID gains is required. This paper, thus, deals with the determination of off-line, nominal, optimal PID gains of a PID controller of an automatic voltage regulator (AVR) for nominal system parameters and step reference voltage input. Particle swarm optimization (PSO) and Imperialist competitive algorithm (ICA) are the two props used to get the optimal PID gains. ICA proves to be more robust than PSO in performing optimal transient performance even under various nominal operating conditions. Computational time required by PSO is more than that of ICA.

II. PID CONTROLLER AND AVR MODEL PID controllers are being extensively used by industries today owing to their simplicity. Its main focus here is reduction/elimination of steady state error as well as an improvement in the dynamic response. Reduction/elimination of steady state error is achieved by adding a pole at the origin with the help of integral controller, by increasing the system type by one. Transient response improvement may be achieved from the action of derivative controller which adds a finite zero to the open loop transfer function. As modeled in this paper, the transfer function of PID controller [8] is: (1) Table.1 depicts parameters of PID controller and AVR model as considered in this paper, transfer function of each item including limits of parameters [8]. In [8], Gaing has taken the

I. INTRODUCTION Even though several control theories have been developed significantly, we can see the widely popular use of proportional-integral-derivative (PID) controllers in process control, motor drives, flight control, and instrumentation. The reason of this acceptability is for its simple structure which can be easily understood and implemented. Industries too can boast of the extensive use of PID controllers because of its robustness and simplicity. The past decades witnessed many advancing improvements keeping in mind the requirement of the end users. Easy implementation of hardware and software has helped to gain its popularity. Several approaches have been documented in literatures for determining the PID parameters of such controllers. Genetic Algorithm [4], neural network [1], fuzzy based approach [3], neuro-fuzzy approach [7], evolutionary computational techniques [8,9] are just a few among these numerous works. This paper focuses on optimal tuning of PID controller for the AVR using particle swarm optimization (PSO) and imperialist competitive algorithm (ICA) and we’ll compare the results of these two algorithms. Particle swarm optimization (PSO) [3,5,6] is a population based evolutionary algorithm. Instead of the survival of the fittest, it is the simulation of the social behavior that motivates PSO. ICA [10] is an algorithm based on the competitions between imperialists to own the countries. In this paper, at first we obtain the results of each algorithm then we’ll have a comparison between them.

© 2012 ACEEE DOI: 02.ACE.2012.03.17

generator transfer function as Kg / (1 +

) where Kg

depends on load (0.7–1.0) and 1.0 s 2.0 s. The same model has been taken in the present work. III. AVR WITH PID CONTROLLER Incorporating the above models “Table.I” a composite AVR system along with PID controller is obtained. The block diagram representation is shown in “Fig. 1”. Equation of the incremental change in terminal voltage (ΔVt(s)) with an incremental change in reference voltage input (_ΔVref(s)) is as follows: (2)

(3)

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Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012 TABLE I . PARAMETERS

OF

PID CONTROLLER AND AVR MODEL WITH TRANSFER FUNCTION AND PARAMETER LIMITS

Figure 1. Block diagram of AVR system along with PID controller

IV. DESIGN OF MISFITNESS FUNCTION

these colonies start moving toward their relevant imperialist. The way of movement is discribed in section B. The total power of an empire depends on both imperialist countray and its colonies. We will model this fact by defining the total power of an empire by the power of imperialist countray plus a percentage of mean power of its colonies.Then the imperialistic competitions begin among all the empires. Any empire that is not be able to succeed in this competition and can’t increase its power will be eliminated from the competition. Weak empires will lose their power and ultimately they will collapse. The movement of colonies toward their relevant imperialists along with competition among empires and also the collapse mechanism will hopefully cause all the countries to converge to a state in which there exist just one empire in the world.In this ideal world colonies have the same position and power as the imperialist.

The performance criterion is to be judged based on a misfitness function (MF). Optimization of PID gains by applying any of the optimization techniques corresponds to minimum misfitness function value. The MF is being formulated as follows [11]: (4) Minimization of MF with the help of any optimization technique corresponds to minimum overshoot (osh), minimum settling time (tst), and maximum max dv. Repetitive trial run of the optimizing algorithms reveals that osh is having the minimum and maximum value of 0.0000 and 0.0002, respectively. Thus, the first term in the right hand side of (4) is in the order of 0–4. The numerical value of tst lies from 1.6400 to 5.4194. Thus, the second term in the right hand side of (4) is in the range of 2.6896–29.3699. The value of max dv lies from 0.0116 to 0.0400, yielding the third term of (4) in the range of 7.4316–0.6250. Thus, incorporation of appropriate weighting factors to the right hand individual terms facilitates to make each term competitive during the optimization process. Any other choice of the weighting factors lead to incompatible numerical values of each term involved in the definition of MF which gives misleading result. V. IMPERIALIST COMPETITIVE ALGORITHM FOR PARAMETER IDENTIFICATION

“Figure.2” shows the flowchart of the proposed algorithm [11]. Like other evolutionary ones, the proposed algorithm starts with an initial population (countries in the world). Some of the best countries are selected to be imperialists and the rest are colonies of these imperialists. All the colonies of the initial population are divided among imperialists based on their power. After dividing all colonies among imperialists, © 2012 ACEEE DOI: 02.ACE.2012.03.17

Fig 2. flowchart of the proposed algorithm

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Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012 A. Generating Initial Empire The goal of optimization is to find and optimal solution in terms of variable values to be optimized. In an N-dimensional optimization problem, a countray is a 1×N array. To form the initial empires, we divide the colonies amonge the imperialists based on their power.:

empires and an increase in the power of more powerful ones.To start the competition, first we find the possession probability of each empire based on its total power. The normalized total cost is simply obtained by: NTCn= TCn –max{TCi} (10) Where TC and NTC are respectively total cost and normalized total cost of nth empire. Having the normalized total cost , the possession probability of each empire is given by:

(5) Where is the cost of n-th imperialist and is its normalized cost. the normalized power of each imperialist is defined by:

= So we have this vector

(6)

P= (12) Then we create a vector with the same size as P whose elements are random numbers.

the initial number of an imperialist is defined by N.C = round {Pn.Ncol} (7) Where N.C is the initial number of colonies of nth empire and Ncol is the number of all colonies.

R= (13) Then we form the vector D by simply sub tracting R fromP D=P-R

B. Moving the Colonies of an Empire toward the Imperialist In proposed algorithm each imperialist try to improve its colonies. This fact is modeled by moving all colonies toward their relevant imperialist. This movement is shown in “Figure.3” in which the colony moves toward the imperialist by x units. In this figure x is a random variable with uniform distribution. for x we have : x~U (0, β×d) (8) where β is a number greater than 1 and d is the distance between colony and imperialist. β>1 causes the colonies to get closer to the imperialist state from both sides.

VI. PARTICLE SWARM OPTIMIZATION The PSO was first introduced by Kennedy and Eberhart [2]. It is an evolutionary computational model, a stochastic search technique based on swarm intelligence. Social behavioral pattern of organisms such as bird flocking and fish schooling inspired them to look into the effect of collaboration of species when achieving their goals as a group. Dynamics of bird flocking resulted in the possibilities of utilizing this behavior as an optimization tool. These have been used to solve a range of optimization problems. Review of PSO algorithm The PSO [2,5,6] is a population-based optimization technique, where the population is called ‘swarm’. In a PSO system, multiple candidate solutions coexist and collaborate simultaneously. Each solution candidate, called a ‘particle’, flies in the problem space (similar to the search process for food of a bird swarm) looking for the optimal position. A ‘particle’ with time adjusts its position to its own ‘experience’, while adjusting the ‘experience’ of neighboring particles. If a particle discovers a promising new solution, all the other particles will move closer to it, exploring the region more thoroughly in the process. Based on PSO concept, mathematical equations for the searching process are: (Velocity updating equation(15) ,Position updating equation (16) )

Fig 3. Moving colonies toward their relevant imperialist in a randomly devided direction

C. Total Power of an Empire Total power of an empire is mainly affected by the power of imperialist country. But the power of colonies of an empire has an effect. We have modeled this fact by defining the total cost by T.C = Cost(imperialist) + ξ mean{Cost(colonies of empire) } (9) Where T.C is the total cost of nth empire and ξ is a positive number which is considered to be less than 1.A little value for ξ causes the total power of the empire to be determined by just the imperialist and increasing it will increase the role of the colonies in determining the total power of an empire.

(15) (16) VII. SIMULATION RESULTS

D. Imperialistic competition All empires try to take possession of colonies of other empires and control them. This imperialistic competition gradually bring about a decrease in the power of weaker © 2012 ACEEE DOI: 02.ACE.2012.03.17

(11)

Step perturbation of 0.01 p.u. of reference voltage has been applied to get the transient response of incremental change in terminal voltage in the present work.”Fig. 4" shows the step response of incremental change in terminal voltage 26

Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012 of the system without inclusion of PID-controller. Oscillatory transient response with non-zero settling error is observed. By usage of PID we can improve maximum overshoot, rise time and settling time.

Figure 7.Convergence profile of ICA and PSO (Kg=0.7 and τg=1)

Figure 4. Step response of incremental change in terminal voltage of the system without PID

We have assume the parameters of block diagram as follow TABLE II. VALUES OF

PARAMETERS TO OBTAIN RESULTS

Figure 8. Step response of incremental change in terminal voltage of PID controller by ICA (Kg=0.8 and τg=1.2)

For different values of and the results are obtained with both algorithms. Simulation results are shown in table 3 for different values of and . Step response of incremental change in terminal voltage of PID controller based AVR system and Convergence profile of PSO and ICA algorithms are shown in “figures 5- 13”. Figure 9. Step response of incremental change in terminal voltage of PID controller by PSO (Kg=0.8 and τg=1.2)

Figure 5 . Step response of incremental change in terminal voltageof PID controller by ICA (K_g=0.7 and τ_g=1) Figure 10.Convergence profile of ICA and PSO (Kg=0.8 and τg=1.2)

Figure 6. Step response of incremental change in terminal voltage of PID controller by PSO (Kg=0.7 and τg=1) Figure 11. Step response of incremental change in terminal voltage of PID controller by ICA (Kg=1 and τg=1.2)

© 2012 ACEEE DOI: 02.ACE.2012.03. 17

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Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012 CONCLUSION For tuning of PID controller gains with off-line, as well as, nominal input conditions, this paper represents a novel evolutionary search technique, it is imperialist competitive algorithm (ICA) . As “ TABLE. III” and figures are shown PID controller for AVR system is design in a excellent way with both algorithms. The system has a good step response in both of them. If we want to compare these algorithms in this paper we can say ICA is so faster than PSO. Also from 9 cases in table 3, ICA could reach better answer in 6 cases.

Figure 12. Step response of incremental change in terminal voltage of PID controller by PSO (Kg=1 and τg=1.2)

REFERENCES [1] Q.H. Wu, B.W. Hogg, G.W. Irwin, A neural network regulator for turbogenerators, IEEE Trans. Neural Networks 3 (1992) 95–100. [2] J. Kennedy, R.C. Eberhart, Particle swarm optimization, in: Proceedings of IEEE International Conference on Neural Networks, Perth, Australia, 1995, pp. 1942–1948. [3] G.K.I. Mann, B.-G. Hu, R.G. Gosine, Analysis of direction fuzzy PID controller structures, IEEE Trans. Syst., Man, Cybern. Part B 29 (3) (1999) 371–388. Figure 13 .Convergence profile of ICA and PSO (Kg=1 and τg=1.2) TABLE III. SIMULATION RESULTS

FOR DIFFERENT VALUES OF

[4] R.A. Krohling, J.P. Rey, Design of optimal disturbance rejection PID controllers using genetic algorithm, IEEE Trans. Evol. Comput. 5 (February) (2001) 78–82. [5] R.C. Eberhart, Y. Shi, Particle swarm optimization: developments, applications and resources, in: Evolutionary Computation Proceedings of the 2001 Congress, vol. 1, May 2001, pp. 81–86. [6] H. Yoshida, K. Kawata, Y. Fukuyama, A particle swarm optimization for reactive power and voltage control considering voltage security assessment, in: Proceedings of IEEE Power Engineering Society Winter Meeting, vol.2, 2001, pp. 815–820. [7] G.K. Venayagamoorthy, R.G. Harley, D.C. Wunsch, Implementation of adaptive critic-based neurocontrollers for turbogenerators in a multimachine power system, IEEE Trans. Neural Networks 14 (September (5)) (2003) 1047–1064.

© 2012 ACEEE DOI: 02.ACE.2012.03.17

K_G AND Τ_G

[8] Z.L. Gaing, A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Trans. Energy Convers. 19 (June (2)) (2004) 384–391. [9] S.P. Ghoshal, Optimization of PID gains by particle swarm optimization in fuzzy based automatic generation control, Electr. Power Syst. Res. 72 (2004) 203–212. [10] E.Atashpaz-Gargari ,Caro Lucas Imperialist Competitive Algorithm: An algorithm for optimization inspired by imperialistic competition 1-4244-1340-0/07$25.00_c 2007 IEEE. [11] V. Mukherjee, S.P. Ghoshal , Intelligent particle swarm optimized fuzzy PID controller for AVR system, 0378-7796/ $ – see front matter © 2006 Elsevier.

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