Page 1

Optimal PID Controller Design for AVR System

Ching-Chang Wong*, Shih-An Li and Hou-Yi Wang

Department of Electrical Engineering, Tamkang University,

Tamsui, Taiwan 251, R.O.C.

Abstract

In this paper, a real-valued genetic algorithm (RGA) and a particle swarm optimization (PSO)

algorithm with a new fitness function method are proposed to design a PID controller for the

Automatic Voltage Regulator (AVR) system. The proposed fitness function can let the RGA and PSO

algorithm search a high-quality solution effectively and improve the transient response of the

controlled system. The proposed algorithms are applied in the PID controller design for the AVR

system. Some simulation and comparison results are presented. We can see that the proposed RGA and

PSO algorithm with this new fitness function can find a PID control parameter set effectively so that

the controlled AVR system has a better control performance.

Key Words: PID Controller, Genetic Algorithms, Particle Swarm Optimization, Automatic Voltage

Regulator (AVR)

1. Introduction

There are three coefficients: proportional coefficient,

differential coefficient, and integral coefficient in the

PID controller. By tuning these three parameters (coef-

ficients), the PID controller can provide individualized

control requirements. In recent years, many intelligence

algorithms are proposed to tuning the PID parameters.

Tuning PID parameters by the optimal algorithms such

as the Simulated Annealing (SA), Genetic Algorithm

(GA), and Particle Swarm Optimization (PSO) algo-

rithm. Chent et al. proposed a method to tune PID para-

meters by SA [1]. However, it is slow to search the best

solution. Kwok and Sheng considered GA and SA for the

optimal robot arm PID control [2]. Some simulation re-

sults illustrate that not only the speed of operation but

also the system response by GA is better than that by SA.

Mitsukura et al. [3] and Krohling et al. [4] also used GA

to search the optimal PID control parameters and they

have nice performance in the simulation results. Genetic

algorithms are methods to obtain an optimal solution by

applying a theory of biological evolution [5,6]. Genetic

algorithms can be found in many applications in bio-

genetics, computer science, engineering, economics, che-

mistry, manufacturing, mathematics, physics, and other

fields. For example, GAs can be applied to discuss the

fuzzy modeling, data classification, and omni-directional

robot design problems [7-9]. The PSO algorithm, pro-

posed by Kennedy and Eberhart [10] in 1995, is an an-

other popular optimal algorithm. It was developed th-

rough a simulation of a simplified social system and

some papers were proposed to improve the PSO algo-

rithm [11-13]. The PSO technique can generate a high-

quality solution within a shorter calculation time and

have a stable convergence characteristic than other sto-

chastic methods [14-16]. It has many applications in en-

gineering fields. In the PID controller design, the PSO

algorithm is applied to search a best PID control para-

meters [17,18].

Many research papers provided many improvement

methods to improve the search performance of the GA

and PSO algorithms [19,20]. In this paper, a real-valued

GA (RGA) and a PSO algorithm with a new fitness func-

tion is proposed to find a PID control parameter set for

Tamkang Journal of Science and Engineering, Vol. 12, No. 3, pp. 259г270 (2009)

259

*Corresponding author. E-mail: wong@ee.tku.edu.tw

AVR system so that the controlled AVR system has a

better control performance than other methods. The rest

of this paper is organized as follows: The proposed RGA

and PSO algorithm are described in Section II. The op-

timal PID controller design by the proposed RGA and

PSO algorithm for AVR system is described in Section

III. Some MATLAB simulation results and some com-

parison results are shown in Section IV. Finally, some

conclusions are made in Section V.

2. Real-Valued GA and PSO Algorithm

In this paper, a real-valued GA (RGA) and a PSO al-

gorithm are proposed to choose an appropriate control

parameter set K = (kp, kd, ki) of the PID controller. A de-

fined fitness function will guide these two algorithms to

find an appropriate control parameter set to meet the de-

sired objective. They are described as follows:

In the RGA, the control parameter set K = (kp, kd, ki)

is viewed as an individual and each parameter value is

coded by a real number [21,22]. If there are L individuals

in a generation, the procedure of the proposed RGA can

be described by the following steps:

Step 1: Initialize RGA by setting the number of individ-

uals (L), the number of generations (N), the

crossover probability (Pc), and the mutation pro-

bability (Pm). The i-th individual of the popula-

tion with L individuals in the g-th generation is

denoted by

(1)

where the number of parameters in the parame-

ter set is 3 and K g

j

i ( ), i Î {1, 2, …, L}, j Î {1, 2,

3}, g Î {1, 2, …, N} is the j-th parameter of the

i-th individual in the g-th generation. Note that

K g

j

i ( ) is a real number in the real-valued GA.

Step 2: Set g = 1 for the first generation and randomly

generate the initial population with L individu-

als pop(1) = {K1(1), K2(1), …, KL(1)} by

(2)

where the searching range of the parameter Kj is

[Kj

min , Kj

max ] (i.e., K

K

K

j

j

j

Î[

,

]

min

max ) and rand( )

is an uniformly distributed random number in

[0,1].

Step 3: Calculate the fitness value of each individual in

the g-th generation by

(3)

where fit(.) is the fitness function.

Step 4: Find an index q of the individual with the high-

est fitness value by

(4)

and determine f best and Kbest by

(5)

and

(6)

where f best is the highest fitness value in the cur-

rent generation and Kbest is the individual with

the highest fitness value in the current genera-

tion.

Step 5: If g > N, then go to Step 11. Otherwise, go to

Step 6.

Step 6: Reproduce each individual in the reproduction

process by

(7)

where ni is the reproduced number of the i-th in-

dividual, L is the number of individuals in a po-

pulation, and Pi is the reproduce rate of the i-th

individual and is determined by

(8)

260

Ching-Chang Wong et al.

where f i is the fitness value of i-th individual.

Step 7: Choose two individuals Km(g) and Kn(g) from

the current population (m, n Î {1, 2, …, L}) to

be the parents and generate two new individuals

in the crossover process (the crossover proba-

bility Pc) by

(9)

where s1 is an uniformly distributed random

number in [0,1].

Step 8: Generate a new individual in the mutation pro-

cess (the mutation probability Pm) for each indi-

vidual by

(10)

where s

K

K

j

j

j

=

-

(

)

max

min

is a range value for

the searching range K

K

K

j

j

j

Î[

,

]

min

max of the j-th

searching parameter Kj. s2 is an uniformly dis-

tributed random number in [0,1].

Step 9: Bound each updated parameter Kj

i in its search-

ing range by

(11)

Step 10: Let g = g + 1 and go to Step 3.

Step 11: Determine the selected controller by the pro-

posed method based on the obtained parameter

set Kbest with the best fitness f best.

In the PSO algorithm, the control parameter set K =

(kp, kd, ki) is viewed as a position p = (p1, p2, p3) of a par-

ticle in a 3-dimensional searching space. If there are L

particles in a generation, the procedure of the proposed

PSO algorithm can be described by the following steps:

Step 1: Initialize the PSO algorithm by setting the num-

ber of particles (L), the number of iterations (N),

the searching range (p

p

p

j

j

j

Î[

,

]

min

max ), the ve-

locity constraint (v

v

v

j

j

j

Î[

,

]

min

max ), f

f

1

2

=

=

… = f L = 0, and c1 = c2 = 2. The i-th particle of

the population with L particles in the g-th itera-

tion is denoted by

(12)

where the number of parameters is 3, and p g

j

i ( ),

i Î {1, 2, …, L}, j Î {1, 2, 3}, g Î {1, 2, …, N},

is the j-th parameter of the i-th particle in the

g-th iteration.

Step 2: Set g =1 for the first generation and randomly

generate L particles pop(1) = {p1(1), p2(1), …,

pL(1)} in the initial generation by

(13)

where the searching range of the parameter Pj is

[

,

]

min

max

p

p

j

j

(i.e., p

p

p

j

j

j

Î[

,

]

min

max ) and rand( )

is an uniformly distributed random number in

[0,1].

Step 3: Calculate the fitness value of each particle in

the g-th generation by

(14)

where fit(.) is the fitness function.

Step 4: Determine f i and pi for each particle by

(15)

and

(16)

Optimal PID Controller Design for AVR System

261

where pi is the position vector of the i-th par-

ticle with the personal best fitness value f i

from the beginning to the current generation.

Step 5: Find an index q of the particle with the highest

fitness by

(17)

and determine f best and p gbest by

(18)

and

(19)

where pgbest is the position vector of the particle

with the global best fitness value f best from the

beginning to the current generation.

Step 6: If g > N, then go to Step 12. Otherwise, go to

Step 7.

Step 7: Update the velocity vector of each particle by

(20)

where v g

v g v g v g

i

i

i

i

( ) ( ( ), ( ), ( ))

= 1

2

3

is the cur-

rent velocity vector of the i-th particle in the g-

th generation. v g

v g

i

i

(

) ( (

)

+ =

+

1

1

1

, v g

i

2 1( )

+ ,

v g

i

3 1( ))

+

is the next velocity vector of the i-th

particle in the g+1-th generation. r1 and r2 are

two uniformly distributed random numbers in

[0,1]. w is a weight value and defined by

(21)

where wmax and wmin are respectively a maxi-

mum value and a minimum value of w. wmax =

0.9 and wmin = 0.2 are used in this paper.

Step 8: Check the velocity constraint by

where vj

max and vj

min are the maximum velocity

and minimum velocity of the j-th parameter, re-

spectively.

Step 9: Update the position vector of each particle by

(23)

where p g

p g p g p g

i

i

i

i

( ) ( ( ), ( ), ( ))

=

1

2

3

is the cur-

rent position vector of the i-th particle in the g-th

generation. p g

p g

p g

i

i

i

(

) ( (

), (

),

+ =

+

+

1

1

1

1

2

p g

i

3 1( ))

+

is the next position vector of the i-th

particle in the (g + 1)-th generation.

Step 10: Bound the updated position vector of each par-

ticle in the searching range by

where pj

max and pj

min are the maximum value

and minimum value of the j-th parameter, re-

spectively.

Step 11: Let g = g + 1 and go to Step 3.

Step 12: Determine the selected controller based on the

obtained parameter set pbest with the best fitness

f best.

The RGA and PSO algorithm only require the infor-

mation of the fitness function value of each parameter

set. These two algorithms are applied to choose a good

PID control parameter set for AVR system. They are de-

scribed in the next section.

262

Ching-Chang Wong et al.

(22)

(24)

3. PID Controller Design for AVR System

It is an important matter for the stable electrical

power service to develop the automatic voltage regulator

(AVR) of the synchronous generator with a high effi-

ciency and a fast response. Until now, the analog PID

controller is generally used for the AVR because of its

simplicity and low cost. However, these parameters of

PID controller are not easy to tune. Gaing [17] proposed

a method to search these parameters by using a particle

swarm optimization (PSO) algorithm. The AVR system

model controlled by the PID controller can be expressed

by Figure 1. where ns is the output voltage of sensor

model, ne is the error voltage between the ns and refer-

ence input voltage nref (S), nR is an amplify voltage by

amplifier model, nF is a output voltage by exciter model,

and nt is a output voltage by generator. There are five

models: (a) PID Controller Model, (b) Amplifier Model,

(c) Exciter Model, (d) Generator Model, and (e) Sensor

Model. Their transfer functions are described as follows:

(a) PID Controller Model

The transfer function of PID controller is

(25)

where kp, kd, and ki are the proportion coefficient, dif-

ferential coefficient, and integral coefficient, respec-

tively.

(b) Amplifier Model

The transfer function of amplifier model is

(26)

where KA is a gain and tA is a time constant.

(c) Exciter Model

The transfer function of exciter model is

(27)

where KE is a gain and tE is a time constant.

(d) Generator Model

The transfer function of generator model is

(28)

where KG is a gain and tG is a time constant.

(e) Sensor Model

The transfer function of sensor model is

(29)

where KR is a gain and tR is a time constant.

In this paper, the GA and PSO algorithm are applied

to search a best PID parameters so that the controlled

system has a good control performance. In [17], a perfor-

Optimal PID Controller Design for AVR System

263

Figure 1. Closed-loop block diagram of AVR system.

mance criterion W(K) is defined by

(30)

where K = (kp, kd, ki) is a parameter set of PID control-

ler, b is a weighting factor, Mp, tr, ts, and ess are re-

spectively the overshoot, rise time, settling time, and

steady-state error of the performance criteria in the

time domain. Mp and ess are positive values. Moreover,

the fitness function is proposed by Gaing [17] and de-

scribed by

(31)

When this fitness function fG is used in an over-damp-

ing system, as shown in Figure 2, the rise time will be

too long and the settling time will approach the rise

time. It will cause the value of W(K) is too small and a

wrong parameter set may be selected by the optimal al-

gorithm. In order to overcome this defect, a modified

fitness function is defined by

(32)

where W(K) is the performance criterion described by

Equation (30) and ITAE is an integral of time multiplied

by absolute-error value and it is defined by

(33)

where i Î {0, 1, 2, …, end time} is an index, ti is the i-th

sampling time, and ei is the absolute-error value in the

i-th sampling time. Equation (32) can increase the ef-

ficiency and accuracy of the intelligent algorithm to

search a high-quality solution.

4. Simulation Results

To verify the efficiency of the proposed fitness func-

tion in the RGA and PSO algorithm, a practical high-

order AVR system [17] as shown in Figure 3 is tested.

The AVR system has the following parameters. The lower

and upper bounds of the three control parameters are

shown in Table 1.

The following parameters are used for the real-

valued GA (RGA): the population size L = 50, the maxi-

mum generation number N = 100, the maximum iteration

number is 50, crossover rate Pc = 0.9, mutation rate Pm =

0.01. The following parameters are used for the PSO al-

gorithm: the particle swarm population size L = 50, max-

imum generation number N = 100, the maximum itera-

tion number is 50, c1 = c2 = 2. Each parameter set (indi-

264

Ching-Chang Wong et al.

Figure 3. A practical high-order AVR system controlled by a PID controller.

Figure 2. Time response of an over-damping system.

vidual) of the PID controller selected by the optimal al-

gorithm is K = (kp, kd, ki). The RGA and PSO algorithm

are used to search an optimal parameter set of the PID

controller. The searching range of each parameter is de-

scribed in Table 1. The simulation time is 2 second and

the sampling time is 0.01 second. In the simulation step,

one iteration has 100 generations. We run each algorithm

with 50 iterations and find the best fitness value, best

control parameters that selected by these two algorithms

in each iteration. The PID control parameter set selected

by the RGA and PSO algorithm in 50 iterations for the

AVR system based on the fitness function defined by

Equation (31) and Equation (32) with b = 1 are respec-

tively described in Table 2 and Table 3. Comparison of

the control performance of the AVR system controlled

by different controllers described in Table 2 (b = 1 and

f

W K

G =

1

( )

) and Table 3 (b = 1 and f

W K

ITAE

=

1( )

)

are respectively described in Table 4 and Table 5. The

best evaluation value in each iteration for the RGA and

PSO algorithm (b = 1 and f

W K

G =

1

( )

) and the best out-

put response in 50 iteration are shown in Figure 4. In this

paper, the performance evaluation criteria of two con-

trollers PIDRGA and PIDPSO selected by RGA and PSO

Optimal PID Controller Design for AVR System

265

Table 2. Best control parameters selected by two optimal

algorithms in 50 iterations for the AVR system

with b = 1 and

1

( )

G

f

W K

=

Selected Control Parameters

Controller

kp

kd

ki

PIDGA[17]

0.7722

0.3196

0.7201

PIDPSO[17]

0.6751

0.2630

0.5980

PIDRGA

0.0222

0.2451

0.2913

PIDPSO

0.0001

0.4226

0.3965

Table 3. Best control parameters selected by two optimal

algorithms in 50 iterations for the AVR system

with b = 1 and

1

( )

f

W K

ITAE

=

´

Selected Control Parameters

Controller

kp

kd

ki

PIDRGA

0.6311

0.2125

0.4615

PIDPSO

0.6443

0.2423

0.4700

Table 4. Comparison of the control performance of the AVR system controlled by different controllers described in

Table 2 (b = 1 and

1

( )

G

f

W K

=

)

Controller

Number of Generation

tr

ts

MP (%)

ess

W(K)

fG

PIDGA[17]

100

0.2138

0.8645

4.54

0

0.9002

01.1109

PIDPSO[17]

100

0.2648

0.3795

1.71

0

0.6851

01.4596

PIDRGA

100

1.9200

2.0100

0

0

0.0331

30.2031

PIDPSO

100

1.9400

2.0100

0

0

0.0258

38.8326

Table 5. Comparison of the control performance of the AVR system controlled by different controllers described in

Table 3 (b = 1 and

ITAE

KW

f

´

=

)(

1

)

Controller

Number of Generation

tr

ts

MP (%)

ess

ITAE

W(K)

f

PIDRGA

100

0.3100

0.4300

1.41

0

5.0159

0.0530

3.7583

PIDPSO

100

0.2800

0.4000

0

0

4.2040

0.0441

5.3882

Table 1. Searching range of each parameter

Parameter

Minimal Value

Maximal Value

kp

0.0001

1.5000

kd

0.0001

1.0000

ki

0.0001

1.0000

P

K

v

-0.7500

0.7500

d

K

v

-0.5000

0.5000

i

K

v

-0.5000

0.5000

algorithm in the time domain are executed on a Pentium

computer. We can find that the fitness values of PIDRGA

and PIDPSO obtained by the proposed algorithms based

on the fitness function defined by Equation (31) are

better than that of PIDGA and PIDPSO obtained in [17], but

the control performance of PIDRGA and PIDPSO are not

266

Ching-Chang Wong et al.

Figure 4. The best evaluation value in each iteration for b = 1

and f

W K

G =

1

( )

. (a) RGA, (b) PSO algorithm, (c)

Best output response in 50 iterations of two algo-

rithms.

Figure 5. The best evaluation value in each iteration for b = 1

and f

W K

ITAE

=

1( )

. (a) RGA, (b) PSO algo-

rithm, (c) Best output response in 50 iterations of

two algorithms.

better than that of PIDGA and PIDPSO obtained in [17].

This is because the rise time is too large and closed to the

settling time. Therefore, the fitness function defined by

Equation (31) lets the RGA and PSO algorithm easy to

search a bad parameter set with b = 1. The best evalua-

tion value in each iteration for the RGA and PSO algo-

rithm (b = 1 and f

W K

ITAE

=

1( )

) and the best output

response in 50 iteration are shown in Figure 5. From the

simulation and comparison results, we can find that the

new fitness function defined by Equation (32) helps the

RGA and PSO algorithm to search a best solution more

accurate than the fitness function defined by Equation

(31). The PID control parameter set selected by the RGA

and PSO algorithm in 50 iterations for the AVR system

based on the fitness function defined by Equation (31)

and Equation (32) with b = 1.5 are respectively described

in Table 6 and Table 7. Comparison of the control perfor-

mance of the AVR system controlled by different con-

trollers described in Table 6 (b = 1.5 and f

W K

G =

1

( )

)

and Table 7 (b = 1.5 and f

W K

ITAE

=

1( )

) are respec-

tively described in Table 8 and Table 9. The best evalua-

tion value in each iteration for the RGA and PSO algo-

rithm (b = 1.5 and f

W K

G =

1

( )

) and the best output re-

sponse in 50 iteration are shown in Figure 6. The best

evaluation value in each iteration for the RGA and PSO

algorithm (b = 1.5 and f

W K

ITAE

=

1( )

) and the best

output response in 50 iteration are shown in Figure 7.

From the simulation and comparison results, we can see

Optimal PID Controller Design for AVR System

267

Table 9. Comparison of the control performance of the AVR system controlled by different controllers described in

Table 7 (b = 1.5 and

1

( )

f

W K

ITAE

=

´

)

Controller

Number of Generation

tr

ts

MP (%)

ess

ITAE

W(K)

f

PIDRGA

100

0.3000

0.4300

0

0

4.1343

0.0290

8.3386

PIDPSO

100

0.3000

0.4200

0

0

4.1812

0.0268

8.9322

Table 8. Comparison of the control performance of the AVR system controlled by different controllers described in

Table 6 (b = 1.5 and

1

( )

G

f

W K

=

)

Controller

Number of Generation

tr

ts

MP (%)

ess

W(K)

fG

PIDGA [17]

100

0.1859

0.9396

6.17

0

0.9930

01.0071

PIDPSO [17]

100

0.2860

0.4168

0.92

0

0.8132

01.2297

PIDRGA

100

0.2800

0.4000

0

0

0.0268

37.3474

PIDPSO

100

0.2800

0.4000

0

0

0.0268

37.3474

Table 6. Best control parameters selected by two optimal

algorithms in 50 iterations for the AVR system

with b = 1.5 and

1

( )

G

f

W K

=

Selected Control Parameters

Controller

kp

kd

ki

PIDGA[17]

0.8372

0.3927

0.6973

PIDPSO[17]

0.6477

0.2375

0.5128

PIDRGA

0.6324

0.2464

0.5047

PIDPSO

0.6455

0.2458

0.4513

Table 7. Best control parameters selected by two optimal

algorithms in 50 iterations for the AVR system

with b = 1.5 and

1

( )

f

W K

ITAE

=

´

Selected Control Parameters

Controller

kp

kd

ki

PIDRGA

0.6193

0.2228

0.4589

PIDPSO

0.6300

0.2276

0.4538

that the new fitness function can let the RGA and PSO al-

gorithm find a high-quality PID control parameter set ef-

fectively so that the controlled AVR system has a better

control performance than the other methods.

5. Conclusion

In this paper, a RGA and a PSO algorithm with a new

fitness function are proposed to find a better PID control

268

Ching-Chang Wong et al.

Figure 6. The best evaluation value in each iteration for b =

1.5 and f

W K

G =

1

( )

. (a) RGA, (b) PSO algorithm,

(c) Best output response in 50 iterations of two al-

gorithms.

Figure 7. The best evaluation value in each iteration for b =

1.5 and f

W K

ITAE

=

1( )

. (a) RGA, (b) PSO algo-

rithm, (c) Best output response in 50 iterations of

two algorithms.

parameter set for AVR system. From the simulation and

comparison results, we can see that the proposed fitness

function can let the RGA and PSO algorithm find a

high-quality PID control parameter set effectively so that

the controlled AVR system has a better control perfor-

mance than the other methods. Moreover, some results

illustrate that a good fitness function can help the opti-

mal algorithms to find a high-quality solution effec-

tively.

Acknowledgement

This research was supported in part by the National

Science Council of the Republic of China under contract

NSC 95-2221-E-032-057-MY3.

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Manuscript Received: Nov. 6, 2007

Accepted: Aug. 1, 2008

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