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A novel technique to design cuckoo search based FOPID controller for AVR in power systems
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A novel technique to design cuckoo search based FOPID controller for AVR in power systemsR A. Sikander a, P. Thakur a,∗, R.C. Bansal b, S. Rajasekar c a
Department of Electrical Engineering, Graphic Era University, Dehradun-248002, India Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria 0002, South Africa c NEC Laboratories, Singapore b
a r t i c l e
i n f o
Article history: Received 12 February 2017 Revised 5 July 2017 Accepted 5 July 2017 Available online xxx Keywords: Fractional order proportional integral derivative controller Automatic voltage regulator Optimization Cuckoo search algorithm
a b s t r a c t Firstly, this paper develops a robust fractional order proportional integral derivative (FOPID) controller for automatic voltage regulator (AVR) in power systems. In the proposed controller, optimum design parameters, such as, proportional, integral, derivative gains, integral and derivative constants, have been achieved by using cuckoo search (CS) algorithm. Then, the time response characteristics of AVR with and without proposed controller have been investigated to reveal the effectiveness of the proposed controller. Also, the performance of the proposed controller has been tested under the condition of the parameter uncertainty and improved performance has been revealed. Finally, the accuracy and robustness of the proposed controller has been investigated through the comparative analysis of the results, achieved from proposed and other existing controllers using the same sets of data. It is shown that the proposed controller provides the better time response characteristics than the other existing techniques. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The fractional order proportional integral derivative controller, being similar to conventional PID controllers, gives more design flexibility, large stability region, selectivity etc., due to the acceptance of fraction orders of derivatives and integrals [1–4]. Also, FOPID is found more proficient than conventional PID under the conditions of parameter uncertainties and external disturbances. Firstly, the capability of FOPID controller, in precise assessment of linear systems with constant coefficient, has been investigated in [1]. Afterward, myriad of research, such as, frequency domain approach [2,3], pole distribution [4], state space design approach [5], internal model control [6], sensitivity constraint [7], etc., were contributed in the design and development of FOPID. In addition, various MATLAB toolboxes [8] were also suggested for the identification and control of FOPID. Also, the performance of FOPID controller in various systems, such as, motor control [9], robotics [10], time delay system [11] etc., has been investigated and it is found that it gives better time response characteristics in comparison with systems comprising of conventional PID controller. Furthermore, numerous contributions have been made by the researchers, in the literature, to develop design approach of FOPID controller based on meta-heuristics search algorithms, such as, particle swarm optimization (PSO) [12], ant colony/
R ∗
Reviews processed and recommended for publication to the Editor-in-Chief by Guest Editor Dr. J. Yogapriya. Corresponding author. E-mail address: [email protected] (P. Thakur).
http://dx.doi.org/10.1016/j.compeleceng.2017.07.005 0045-7906/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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bee colony and genetic algorithm (GA) [13,14]. In these approaches the authors utilise the concept of optimization by minimizing a defined cost function to obtain the parameters of FOPID controllers. The design method of FOPID controller for a practical AVR is suggested in [15] which is based on PSO and the results obtained by this method are comparable with conventional PID controller. The conventional PID controller is also synthesised by PSO for AVR to overcome the shortcomings of premature convergence of GA [16,17]. Recently mixed H2 /H∞ and chaotic multi-objective optimization based fractional order PID controllers are also designed for AVR system [18,19]. Therefore, despite of having so many FOPID controller tuning techniques, the determination of the best FOPID controller parameters is still a challenging task. Therefore, this paper contributes a novel optimum design technique of FOPID controller for AVR. The novel contributions of this work can be summarized in the following points: (i) Firstly, cuckoo search algorithm has been used for the optimization of FOPID parameters for the AVR system. The numbers of parameters of cuckoo search are lesser than the parameters used in other meta-heuristic techniques, such as, GA, PSO, ant colony. Therefore, the use of lesser parameters in cuckoo search results in fast convergence of FOPID parameters. (ii) Then, the time response of characteristics of the proposed cuckoo search based FOPID, namely, CS-FOPID, are evaluated and compared with PSO-FOPID, PSO-PID, GA-PID, and conventional PID for AVR system for the same data sets. The comparative analysis results in better time response characteristics than the other existing controllers used with AVR. (iii) Also, uncertainties in AVR parameters have been taken into consideration to show robustness of the proposed CS-FOPID controller for AVR. It has been shown that the proposed CS-FOPID is least affected than the other FOPID controller if uncertainty in AVR model is considered. (iv) Finally, the performance of AVR systems is assessed for three different values of generation and weighting factor. It has been observed that the proposed CS-FOPID gives better performance results than the other existing controlled such as PSO-PID, GA-PID. The optimal design of FOPID controller is achieved by minimizing the time response specifications using cuckoo search algorithm. The results obtained in this paper are comparable with recently published and other existing design methods. Furthermore, the performance of the CS-FOPID controller with respect to exciter system uncertainty is also evaluated. The suggested CS-FOPID controller exhibits superior response as compared to conventional PSO-PID and PSO FOPID controllers not only in normal conditions but with exciter system uncertainty also. The superiority and robustness of the suggested controller is observed by time response specifications. This paper is organized in seven sections. Section 1 comprises introduction and the detailed literature review. Section 2 covers the description of FOPID controllers and in Section 3, the overview of AVR system are explained in detail. The performance measures of the controllers are given in Section 4. The proposed design methodology of CS-FOPID controller followed by the description of cuckoo search algorithm which is to be used to obtain the optimized parameters of the controller is discussed in Section 5. The superiority and powerfulness of the proposed CS-FOPID controller with practical example is depicted in Section 6 followed by the application to AVR system. Section 7 comprises the discussion and conclusion along with the future aspects of the present work. 2. FOPID controller 2.1. Mathematical modelling Fractional order PID controllers consist of integral and derivative parts with fractional order which provide more flexibility to obtain the additional design specifications. Generally, the FOPID controller is given as-
GF OPID (s ) =
TD sλ+μ + KP sλ + TI sλ
(1)
where KP , TI & TD are proportional, integral and derivative gains respectively, λ is the fractional orders of integral part whereas μ is the fractional orders of derivative part of the FOPID controller. The graphical representation of FOPID controller is shown in Fig. 1 where the x-axis shows the order of the integrator and the y-axis shows the order of the differentiator of the controller. Depending upon the values of λ and μ, it is observed that the conventional P, PI, PD, and PID controllers can be obtained easily from FOPID controller. 2.2. Hardware implementation Most of the controllers are represented by their mathematical model for analysis however the hardware implementation of a system is necessary for practical point of view. Therefore, investigation of new methods for practical circuit realisation of the controllers is in demand. A simple technique for the circuit realisation of fractional order controllers is suggested by Charef [20] in which a rational function is used for approximation of fractional order operators. The fractional order integrator (FOI) is represented as follows-
GF OI (s ) =
1 sn
(2)
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 1. Graphical representation of FOPID controller.
Fig. 2. Circuit realisation of FOI.
where n is a positive real number. This transfer function is first approximated using fractional power pole (FPP) method Charef [20] in the predefined frequency band (ωL , ωH ) . The transfer function of FPP is given as follows-
GF PP (s ) =
KI
(3)
(1 + (s/ωc ))n
where ωc is the −3n decibel corner frequency of FPP and KI is the integral gain. Let us consider that ω ∈ (ωL , ωH ), ωc . So Eq. (3) can be written as-
GF PP (s ) =
KI
(s/ωc )
n
=
KI ωcn sn
& ω
(4)
If we put KI ωcn = 1 i.e. KI = ω1n then c
GF PP (s ) =
1 = GF OI (s ) sn
(5)
Hence the transfer function representation of fractional order integrator is as followsm −1
1 KI i=0 GI ( s ) = n = KI m s (1 + (s/ωc ))n i=0
(1 + (s/z0 (cd )i )) (6)
(1 + (s/ p0 (cd )i ))
The parameters of Eq. (6) are obtained by the graphical method suggested by Charef [21] and given as follows-
c = 10(δ /(10(1−n ))) ,
d = 10(δ /10n) ,
z0 = c p0 , and
m = Integer
p0 = ωc 10(δ /20n) ,
log(ωmax / p0 ) log(cd )
+1
ωc = ωL (10(δ/10n) − 1 )
where ωmax = 100ωH and δ is the specified error between FOI and FPP in the predefined frequency band. Now, the circuit realisation of FOI using resistor and capacitor is depicted in Fig. 2 [20] where V(s) is the terminal voltage and I(s) is the current. The values of resistors and capacitors are calculated using the following relationsm −1
Ri = KI Ci =
(1−( (cd )(i− j ) /c ))
j=0 m j=0,i= j
1 (cd )i p0 Ri
(1−(cd )(i− j ) )
,
(7)
f or i = 0, 1, . . . , . . . m
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 3. Circuit realisation of FOD.
Similarly, the fractional order differentiator (FOD) is represented as follows-
GF OD (s ) = sn
(8)
This transfer function is first approximated using fractional power zero (FPZ) method Charef [20] in the predefined frequency range (ωL , ωH ) . The transfer function representation of fractional power zero (FPZ) is given as-
n
s
GF PZ (s ) = KD 1 +
(9)
ωc
where ωc is the 3n decibel corner frequency of fractional power zero and KD is the derivative gain. Let us consider that ω ∈ (ωL , ωH ), & ω ωc . So Eq. (9) can be written as-
GF PZ (s ) = KD
s n K D = n sn ωc ωc
(10)
If we put ωDn = 1 i.e. KD = ωcn then c K
GF PZ (s ) = sn = GF OD (s )
(11)
Hence the fractional order differentiator is represented as followsm
(1 + (s/z0 (cd ) )) s n i=0 GD (s ) = sn = KD 1 + KD m ωc (1 + (s/ p0 (cd )i ))
i
(12)
i=0
The parameters of Eq. (12) are obtained by the graphical method suggested by Charef [21] and given as follows-
c = 10(δ /(10(1−n ))) ,
d = 10(δ /10n) ,
p0 = c z0 , and
m = Integer
z0 = ωc 10(δ /20n) ,
log(ωmax /z0 ) log(cd )
+1
ωc = ωL (10(δ/10n) − 1 )
where ωmax = 100ωH and δ is the specified error between FOD and FPZ in the predefined frequency band. Now the circuit realisation of FOD using resistor and capacitor is depicted in Fig. 3 [20] where V(s) is the terminal voltage and I(s) is the current and R p = KD . The values of resistors and capacitors are calculated using the following relationsm
Ci = Ri =
KD i p0 (cd )
1− c ( c d )
j=0 m
j=0,i= j
1 (cd )i p0Ci
( i− j )
1−(cd )
( i− j )
,
(13)
f or i = 0, 1, . . . , . . . m
Furthermore, in order to achieve the circuit realisation of fractional order PID controller the FOPID controller represented by Eq. (1) is modified in the following form-
GF OPID (s ) = KP +
where
TI s
, (TD s),
1 snI
T 1 I
s
snI
+ (TD s )(snD )
(14)
& (snD ) are first order and fractional order integrator and differentiator respectively with 0 < nI ,
nD < 1 . Hence the transfer function of FOPID controller is approximated using fractional power pole (FPP) and fractional power zero (FPZ) method and is represented as follows-
⎛
⎞
⎛ ⎞ m (1 + (s/z0 (cd ) )) ⎟ (1 + (s/z0 (cd )i )) T ⎜ ⎟ I ⎜ i=0 i=0 ⎟ + (TD s )⎜ GF OPID (s ) = KP + KI m ⎝KD ⎠ m ⎠ s ⎝ i i (1 + (s/ p0 (cd ) )) (1 + (s/ p0 (cd ) )) m −1
i=0
i
(15)
i=0
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 4. Design structure of FOPID.
Fig. 5. Analogue circuit design of proportional element.
Fig. 6. Analogue circuit design of integral element.
Fig. 7. Analogue circuit design of derivative element.
All the three elements of FOPID controller are realised individually and then their combined effect is analysed using the structure specified in the Fig. 4. The individual circuit design of proportional, integral and derivative element is specified in the Figs. 5, 6 and 7, respectively. Therefore, the analogue realisation of FOPID controller is achieved by using Fig. 8 and hence
TI R2 Z2 Z3 GF OPID (s ) = KP + λ + TD sμ = + + R1 Z1 Z4 s
(16)
3. Overview of automatic voltage regulator The maintenance of output voltage of synchronous generator at a predefined level is very essential. This is achieved by automatic voltage regulator (AVR) which acts as a controller to maintain the level of the terminal voltage within the excitation system. Therefore, the design of stable automatic voltage regulator is of great interest. The block diagram representation Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 8. Analogue circuit design of FOPID controller.
Fig. 9. Block diagram representation of AVR with CS-FOPID controller.
of automatic voltage regulator with CS-FOPID controller is shown in Fig. 9 which consists of four components of automatic voltage regulator namely amplifier, exciter, generator and sensor/feedback. The role of sensor, as shown in Fig. 9, is to sense terminal voltage variations, existing either due to change in load demand or power system faults, and feedback the same to the comparator after being rectified. This feedback signal is first compared with reference signal at the comparator to generate error signal and then amplified and given to the exciter. The major role in complete loop is played by the exciter which control the field current in such a way that the error is reduced to zero. Finally, the output of the exciter is given to alternator to achieve constant terminal voltages. For analysis and design of FOPID controller for AVR system, the linearized models of the components of AVR system are chosen as follows [16]. The transfer function representation of amplifier is as follows-
Ga ( s ) =
Ka 1 + τa s
(17)
where amplifier gain (Ka ) and time constant (τ a ) are such that 10 < Ka < 400 & 0.02s < τ a < 0.1s. The transfer representation of the exciter model is as follows-
Ge ( s ) =
Ke 1 + τe s
(18)
where exciter gain (Ke ) and time constant (τ e ) are such that 10 < Ke < 400 & 0.5s < τ e < 1s. The transfer function representation of linearised model of generator is given as follows-
Gg ( s ) =
Kg 1 + τg s
(19)
where Kg & τ g are gain and time constant of generator which depends upon load. The range of gain is 0.7 to 1 and time constant is 1 to 2s. Similarly, the transfer function representation of sensor/feedback model is as follows-
H f (s ) =
Kf 1 + τf s
(20)
where feedback gain (Kf ) and time constant (τ f ) are such that Kf ࣃ 1 & 0.001s < τ f < 0.06s. Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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4. Performance measure of CS-FOPID controller There are various performance indices available in the literature to analyse and design of controllers. Some of the most popular performance indices are integral of absolute error (IAE), integral of squared error (ISE) and integral of time multiplied by squared error (ITSE). But these criterions have their own benefits and drawbacks such as small overshoot can be achieved using IAE & ISE but sacrifices the settling time whereas ITSE may resolve this problem but it never guarantee a required stability boundary. The IAE, ISE and ITSE are express as -
IAE =
∞ 0
ISE =
∞ 0
IT SE =
Vre f (t ) − Vt (t )dt
(21)
(Vre f (t ) − Vt (t ))2 dt
(22)
∞ 0
t (Vre f (t ) − Vt (t )) dt 2
(23)
where Vref (t), Vt (t) are reference and terminal voltage of synchronous generator respectively. Therefore, in this paper a time domain performance criterion is chosen which was previously considered by Gaing et. al. [16] in which time response specifications are included. The performance criterion Z(K) is expressed as-
Min Z (K ) = (1 − e−β ).(M p + ess ) + e−β (ts − tr )
(24)
where K is [KP , TI , TD ] and β is the weighting factor such that 0.8 < β < 1.5. Mp , ess, ts and tr denotes maximum overshoot, steady state error, rise time and settling time respectively. 5. Proposed design methodology of CS-FOPID controller Cuckoo Search (CS) is an evolutionary algorithm which is inspired by common behaviour of Cuckoo bird of laying their eggs in the host nest birds and recently developed by Yang and Deb [22]. This algorithms is developed based on the concept that all the cuckoo birds lay down their eggs into to the other bird’s nest for fertilization. It is possible that the host birds might recognize that it is not their eggs. Therefore, either they throw the alien eggs from their nests or form a new nest at new place. [23]. A set of host nest show the cuckoo breading analogy. Each nest carries an egg which is considered as a solution. A new nest is formed using Lévy flight [24] i.e. random walk. Success of random searches of resources can be optimize using the Lévy flight movements [25]. In order to utilize this natural behaviour of Cuckoo species including the Lévy flight, Yang and Deb [22] suggested the following three rules: • For laying down the egg, the nest should be selected at random and dump by every cuckoo. • The nest must be transferred to the next generation if good quality eggs are found in it. • The probability of an alien egg which can be observed by the fixed number of host nest is pa ∈ [0, 1]. In such cases the host nest either throw the alien egg or form a new nest at any other place. For easiness, the fraction of pa of n nests, which are being replaced by new nests, is considered the approximation of previous assumption. The Lévy flight can be represented by the following relation for the generation of new solution Y (t + 1 ) of cuckoo i [23]-
Yi (t+1) = Yi (t ) + a Lévy(λ )
(25)
where a (a > 0) is a step size which is related to the level of the problem optimized by the technique. The random step size of the Lévy flights are calculated as follows-
Lévy u˜ = t −λ
; (1 < λ ≤ 3 )
(26)
CS algorithm has been utilised widely in different areas which includes steel structure design [26], structural optimization problem [27], applications to business optimization [28], model order reduction [29] etc., due to its computational efficiency and fast convergence. The flow chart of cuckoo search algorithm is shown in Fig. 10 and the design of the FOPID controller using cuckoo search algorithm consists of the following stepsStep 1 Specify objective function and the parameters for cuckoo search such as decision variables and their lower and upper boundaries. Set the probability of the worst nests and step size also. Initialization of a population of p host nests then the problem is summarised asMinimize Z(K), subject to KPL < KP < KPU TIL < TI < TIU Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 10. Flow chart of cuckoo search algorithm.
TDL < TD < TDU where KPL , TIL , TDL and KPU , TIU , TDU are the lowest and highest values of the chosen variables of the controller respectively. Step 2 Obtain the value of Zα for a randomly selected cuckoo (α ) and select a nest (β ) randomly among p. Step 3 if (Zα > Zβ ) then interchange β by the current obtained solution else go to Step 4. Step 4 Check whether the predefined stopping criterion is arrived or the maximum generation occurred or not if yes, then the solution obtained in the current generation would be the best solution. Step 5 -Abandon a fraction of worse nests with optimal value of probability pa and step size a. Step 6 - Using Eq. (25) the obtained solution must be updated by calculating Yi (t+1 ) and repeat this algorithm, until the predefined condition is arrived or the maximum generation occurred. 6. Practical example and results A practical high order automatic voltage regulator is considered in this paper, to evaluate the performance and efficacy of the proposed CS-FOPID controller, with the following specifications [16]- Ka = 10, τa = 0.1, Ke = 1, τe = 0.4, Kg = 1, τg = 1, K f = 1 and τ f = 0.01. The basic block diagram representation of automatic voltage regulator with CS-FOPID controller is presented in Fig. 9 and the range of its parameters are depicted in Table 1. The output voltage step response of a practical automatic voltage regulator in the absence of controller is depicted in Fig. 11 in which it is found that M p = 65.72%, ess = 0.0909, tr = 0.2607 s and ts = 6.9865 s. The time responses of automatic voltage regulator with different controllers are depicted in Figs. 12–14. The values of performance criterion for different controllers are tabulated in Table 2. Total three cases are considered depending upon Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Table 1 Parameters of CS-FOPID controller. Controller parameter
Min. value
Max. value
KP TI TD
0 0 0 0 0
3 1 1 2 2
λ μ
Fig. 11. Output voltage step response of AVR in the absence of controller. Table 2 Comparison of different controllers for AVR. Case
β
Number of generation
Type of controller
KP
TI
TD
Mp (%)
ts
tr
ess
1
1.5
50
2
1
100
3
0.8
150
CS-FOPID PSO-PID [16] GA-PID [16] CS-FOPID PSO-PID [16] GA-PID [16] CS-FOPID PSO-PID [16] GA-PID [16]
2.5490 0.6568 0.8861 2.515 0.6751 0.7722 2.4676 0.6570 0.8663
0.1759 0.5393 0.7984 0.1629 0.5980 0.7201 0.3020 0.5390 0.7531
0.3904 0.2458 0.3158 0.3888 0.2630 0.3196 0.4230 0.2458 0.3365
0.0014 1.17 8.66 0 1.71 4.54 0 1.16 7.34
0.4415 0.4027 0.5980 0.4507 0.3795 0.8645 0.5002 0.4025 0.8519
0.1035 0.2768 0.2019 0.1042 0.2648 0.2138 0.0960 0.2767 0.1959
0 0 0 0 0 0 0 0 0
the value of weighting factor β and the number of generations. For each case, the efficiency and superiority of the proposed method for controller design is observed as the AVR system exhibits better performance with CS-FOPID controller (λ = 0.97, μ = 1.38 ) as compared to other well-known PSO-PID and GA-PID [16] controllers available in the literature. Furthermore, the proposed CS-FOPID controller is also compared with classical PID and PSO-FOPID controller [15] for AVR and found better. The comparative analysis is depicted in Table 3. The robustness and powerfulness of the CS-FOPID controller with exciter uncertainty is examined by considering the exciter transfer model as follows [15]-
Ge ( s ) =
1 0.5 + 0.5s
(27)
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 12. Comparison of unit step response of AVR system with proposed and previously designed controllers (CASE-1).
Fig. 13. Comparison of unit step response of AVR system with proposed and previously designed controllers (CASE-2). Table 3 Comparison of CS-FOPID controller with classical PID and PSO-FOPID controller for AVR. Type of controller
KP
TI
TD
λ
μ
Mp (%)
ts
tr
es s
CS-FOPID PSO-FOPID [15] PID [15]
2.515 0.17 0.0046
0.1629 0.03 0.0268
0.3888 0.014 0.18
0.97 0.97 1
1.38 1.38 1
0 0 2
0.4507 52 27
0.1042 27 17.6
0 0 0
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 14. Comparison of unit step response of AVR system with proposed and previously designed controllers (CASE-3). Table 4 Comparison of controllers with parameter uncertainty in exciter model of AVR. Type of controller
KP
TI
TD
λ
μ
Mp (%)
ts
tr
es s
CS-FOPID PSO-FOPID [15] PSO-PID [16]
0.8912 0.17 0.6570
0.2485 0.03 0.5389
0.5105 0.014 0.2458
0.97 0.97 1
1.38 1.38 1
0.1779 2.0279 18.03
1.4591 17.6056 1.6581
0.0986 1.1185 0.2578
0 0 0
Further, for parameter variation in exciter model, the terminal voltage step responses of proposed CS-FOPID, PSO-FOPID [15] and PSO-PID [16] are shown in Fig. 15. It is observed that the PSO-FOPID and PSO-PID exhibits an overshoot of 2.0279% and 18.03% respectively whereas the proposed CS-FOPID controller exhibits an overshoot of 0.1779% only which shows the superiority of the proposed controller over others with parameter variation in exciter model. Also, a comparative analysis of different controllers in terms of time domain performance criterion with parameter uncertainty in exciter model of AVR is depicted in Table 4.
7. Conclusion With the widespread use of power-electronic devices and different types of nonlinear loads, voltage flickering has now become one of the most pressing concern to the industries which are operating in a highly competitive business environment. Therefore, the quality of voltage for these equipment is highly desirable to keep themselves in tune with the market demand. Nowadays, AVR has received utmost attention in the industries to maintain constant voltage at the equipment terminals under all the conditions. However, to achieve improved transient characteristics, a precise controller, associated with AVR, is highly desirable. In addition, precise controller design plays vital role for proper monitoring of reactive power sharing between the parallel connected generators. Considering, all of these issues and to obtain improved dynamic performance, robustness, and stability, this paper considers a novel design technique for FOPID controller of AVR, using cuckoo search algorithm. The time response characteristic parameters, such as, maximum overshoot, rise time, settling time, and steady state error, of AVR have been evaluated with and without proposed FOPID controller to check its effectiveness. It is revealed that, the proposed controller improves time response characteristic parameters. Further, the time response characteristic parameters are evaluated with the other existing controllers, such as, PSO-FOPID, PSO-PID, GA-PID and conventional PID controller, to present comparative analysis. It is found that the proposed controller provides more improved dynamic performance than the existing controllers, PSO-FOPID, PSO-PID, GA-PID and conventional PID. Also, the performance of the proposed as well as other controllers has been tested Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 15. Comparison of step response of AVR system with proposed and previously designed controllers under exciter uncertainty.
under the condition of parameter variation in exciter model of AVR. It is found that, the proposed controller exhibits more robust and improved dynamic performance than the other existing controller under the condition of parameter variation. References [1] Podlubny I. Fractional-order systems and PIλ Dμ controllers. IEEE Trans Automat Contr 1999;44(1):208–14. [2] Vinagre BM, Podlubny I, Dorcak L, Feliu V. On fractional pid controllers: a frequency domain approach. IFAC workshop; 20 0 0. [3] Pan I, Das S. Frequency domain design of fractional order PID controller for AVR system using chaotic multi-objective optimization. Int J Electr Power Energy Syst 2013;51:106–8. [4] Petras I. The fractional order controllers: methods for their synthesis and application. J Electr Eng 20 0 0;50(9-10):284–8. [5] Dorcak L, Petras I, Kostial I, Terpak J. State space controller design for the fractional order regulated system. In: proceedings of ICCC; 2001. p. 15–20. [6] Vinopraba T, Sivakumaran N, Narayanan S, Radhakrishnan TK. Design of internal model control based fractional order PID controller. J Control Theory Appl 2012;10(3):297–302. [7] Wang DJ, Li W, Guo ML. Tuning of PIλ Dμ controllers based on sensitivity constraint. J Process Control 2013;23(6):861–7. [8] Tepljakov A, Petlenkov E, Belikov J. FOMCON: a MATLAB toolbox for fractional-order system identification and control. Int J Microelectron Comput Sci 2011;2(2):51–62. [9] Rajasekhar A, Kumar JR, Abraham A. Design of intelligent PID/PIλ Dμ speed controller for chopper fed DC motor drive using opposition based artificial bee colony algorithm. Eng Appl Artif Intell 2014;29:13–32. [10] Dumlu A, Erenturk K. Trajectory tracking control for a 3-DOF parallel manipulator using fractional-Order PIλ Dμ control. IEEE Trans Ind Electron 2014;61(7):3417–26. [11] Luo Y, Chen Y. Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica 2012;48(9):2159–67. [12] Cao JY, Cao BG. Design of fractional order controller based on particle swarm optimization. Int J Control Autom Syst 2006;4(6):775–81. [13] Padhee YSS, Gautam A, Kaur G. A novel evolutionary tuning method for fractional order pid controller. Int J Soft Comput Eng 2011;1(1):1–9. [14] Das S, Pan I, Das S, Gupta A. Improved model reduction and tuning of fractional-order PIλ Dμ controllers for analytical rule extraction with genetic programming. ISA Trans 2012;51(2):237–61. [15] Zamani M, Karimi-Ghartemani M, Sadati N, Parniani M. Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Eng Pract 2009;17(12):1380–7. [16] Gaing Z. A particle swarm optimization approach for optimum design of PID controller in AVR system. IEEE Trans Energy Convers 2004;19(2):384–91. [17] Eberhart RC, Shi Y. Comparison between genetic algorithms and particle swarm optimization. In: Proc. IEEE Int. Conf. Evol. Comput., Anchorage, AK; 1998. p. 611–16. [18] Pan I, Das S. On the mixed H2 /H∞ l loop-shaping tradeoffs in fractional-order control of the AVR system. IEEE Trans Ind Inf 2014;10(4):1982–91. [19] Pan I, Das S. Chaotic multi-objective optimization based design of fractional order PIλ Dμ controller in AVR system. Int J Electr Power Energy Syst 2012;43(1):393–407. [20] Charef A. Analogue realization of fractional order integrator, differentiator and fractional PIλ Dμ controller. In: IEE Proc. Control Theory Appl.; 2006. p. 714–20. [21] Charef A, Sun HH, Tsao YY, Onaral B. Fractal system as represented by singularity function. IEEE Trans Automat Contr 1992;37(9):1465–70. [22] Yang XS, Deb S. Nature-Inspired Metaheuristic Algorithms. United Kingdom: Luniver Press; 2008. [23] Yang XS, Deb S. Engineering optimisation by cuckoo search. Int J Math Modell Numer Optim 2010;1:330–43. [24] Brown CT, Liebovitch LS, Glendon R. Lévy flights in dobe ju/hoansi foraging patterns. Hum Ecol 2007;35(1):129–38. [25] Humphries NE, Weimerskirch H, Queiroz N, Southall EJ, Sims DW. Foraging success of biological lévy flights recorded in situ. Proc Nat Acad Sci 2012;109(19):7169–74.
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[26] Kaveh A, Bakhshpoori T, Ashoory M. An efficient optimization procedure based on cuckoo search algorithm for practical design of steel structures. Int J Optim Civil Eng 2012;2(1):1–14. [27] Gandomi AH, Yang XS, Alavi AH. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 2013;29(1):17–35. [28] Yang XS, Deb S, Karamanoglu M, He X. Cuckoo search for business optimization applications. In: Proceeding of National Conference on Computing and Communication Systems (NCCCS 2012); 2012. p. 29–33. ISBN 9781467319539. [29] Sikander A, Prasad R. A novel order reduction method using cuckoo search algorithm. IETE J Res 2015;61(2):83–90.
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Afzal Sikander has over 10 years of teaching and research experience and currently he is working as Associate Professor with the Department of Electrical Engineering, Graphic Era University, Dehradun, India. He is acting as Editorial board member of various journals. His area of interest includes circuit design, control system, model order reduction, controller design, system Engineering and optimization etc. Padmanabh Thakur has over 17 years of experience and currently he is a Professor and head in the Department of Electrical, Graphic Era University, Dehradun. Also, Prof. Thakur is holding responsibility of Associate Editor of IEEE-Access and JIPS, South Korea. Ramesh Bansal has over 25 years of experience and currently he is a Professor and group head (Power) in the Department of EEC Engineering at University of Pretoria. He has published over 250 papers. Prof. Bansal is an Editor of IET-RPG & Electric Power Components and Systems. He is a Fellow and CEngg IET-UK, Fellow Engineers Australia and Senior Member-IEEE. Selvamuthukumaran Rajasekar was the recipient of the IEEE Industrial Electronics Society Travel grant award for the year 2012. He also received POSOCO research award in year 2015 for his research work. During 2015–2017, he was a postdoctoral researcher in NEC Laboratories Singapore and conducted research project in collaborating with Nanyang Technological University, Singapore.
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Contents lists available at ScienceDirect
Computers and Electrical Engineering journal homepage: www.elsevier.com/locate/compeleceng
A novel technique to design cuckoo search based FOPID controller for AVR in power systemsR A. Sikander a, P. Thakur a,∗, R.C. Bansal b, S. Rajasekar c a
Department of Electrical Engineering, Graphic Era University, Dehradun-248002, India Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria 0002, South Africa c NEC Laboratories, Singapore b
a r t i c l e
i n f o
Article history: Received 12 February 2017 Revised 5 July 2017 Accepted 5 July 2017 Available online xxx Keywords: Fractional order proportional integral derivative controller Automatic voltage regulator Optimization Cuckoo search algorithm
a b s t r a c t Firstly, this paper develops a robust fractional order proportional integral derivative (FOPID) controller for automatic voltage regulator (AVR) in power systems. In the proposed controller, optimum design parameters, such as, proportional, integral, derivative gains, integral and derivative constants, have been achieved by using cuckoo search (CS) algorithm. Then, the time response characteristics of AVR with and without proposed controller have been investigated to reveal the effectiveness of the proposed controller. Also, the performance of the proposed controller has been tested under the condition of the parameter uncertainty and improved performance has been revealed. Finally, the accuracy and robustness of the proposed controller has been investigated through the comparative analysis of the results, achieved from proposed and other existing controllers using the same sets of data. It is shown that the proposed controller provides the better time response characteristics than the other existing techniques. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The fractional order proportional integral derivative controller, being similar to conventional PID controllers, gives more design flexibility, large stability region, selectivity etc., due to the acceptance of fraction orders of derivatives and integrals [1–4]. Also, FOPID is found more proficient than conventional PID under the conditions of parameter uncertainties and external disturbances. Firstly, the capability of FOPID controller, in precise assessment of linear systems with constant coefficient, has been investigated in [1]. Afterward, myriad of research, such as, frequency domain approach [2,3], pole distribution [4], state space design approach [5], internal model control [6], sensitivity constraint [7], etc., were contributed in the design and development of FOPID. In addition, various MATLAB toolboxes [8] were also suggested for the identification and control of FOPID. Also, the performance of FOPID controller in various systems, such as, motor control [9], robotics [10], time delay system [11] etc., has been investigated and it is found that it gives better time response characteristics in comparison with systems comprising of conventional PID controller. Furthermore, numerous contributions have been made by the researchers, in the literature, to develop design approach of FOPID controller based on meta-heuristics search algorithms, such as, particle swarm optimization (PSO) [12], ant colony/
R ∗
Reviews processed and recommended for publication to the Editor-in-Chief by Guest Editor Dr. J. Yogapriya. Corresponding author. E-mail address: [email protected] (P. Thakur).
http://dx.doi.org/10.1016/j.compeleceng.2017.07.005 0045-7906/© 2017 Elsevier Ltd. All rights reserved.
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bee colony and genetic algorithm (GA) [13,14]. In these approaches the authors utilise the concept of optimization by minimizing a defined cost function to obtain the parameters of FOPID controllers. The design method of FOPID controller for a practical AVR is suggested in [15] which is based on PSO and the results obtained by this method are comparable with conventional PID controller. The conventional PID controller is also synthesised by PSO for AVR to overcome the shortcomings of premature convergence of GA [16,17]. Recently mixed H2 /H∞ and chaotic multi-objective optimization based fractional order PID controllers are also designed for AVR system [18,19]. Therefore, despite of having so many FOPID controller tuning techniques, the determination of the best FOPID controller parameters is still a challenging task. Therefore, this paper contributes a novel optimum design technique of FOPID controller for AVR. The novel contributions of this work can be summarized in the following points: (i) Firstly, cuckoo search algorithm has been used for the optimization of FOPID parameters for the AVR system. The numbers of parameters of cuckoo search are lesser than the parameters used in other meta-heuristic techniques, such as, GA, PSO, ant colony. Therefore, the use of lesser parameters in cuckoo search results in fast convergence of FOPID parameters. (ii) Then, the time response of characteristics of the proposed cuckoo search based FOPID, namely, CS-FOPID, are evaluated and compared with PSO-FOPID, PSO-PID, GA-PID, and conventional PID for AVR system for the same data sets. The comparative analysis results in better time response characteristics than the other existing controllers used with AVR. (iii) Also, uncertainties in AVR parameters have been taken into consideration to show robustness of the proposed CS-FOPID controller for AVR. It has been shown that the proposed CS-FOPID is least affected than the other FOPID controller if uncertainty in AVR model is considered. (iv) Finally, the performance of AVR systems is assessed for three different values of generation and weighting factor. It has been observed that the proposed CS-FOPID gives better performance results than the other existing controlled such as PSO-PID, GA-PID. The optimal design of FOPID controller is achieved by minimizing the time response specifications using cuckoo search algorithm. The results obtained in this paper are comparable with recently published and other existing design methods. Furthermore, the performance of the CS-FOPID controller with respect to exciter system uncertainty is also evaluated. The suggested CS-FOPID controller exhibits superior response as compared to conventional PSO-PID and PSO FOPID controllers not only in normal conditions but with exciter system uncertainty also. The superiority and robustness of the suggested controller is observed by time response specifications. This paper is organized in seven sections. Section 1 comprises introduction and the detailed literature review. Section 2 covers the description of FOPID controllers and in Section 3, the overview of AVR system are explained in detail. The performance measures of the controllers are given in Section 4. The proposed design methodology of CS-FOPID controller followed by the description of cuckoo search algorithm which is to be used to obtain the optimized parameters of the controller is discussed in Section 5. The superiority and powerfulness of the proposed CS-FOPID controller with practical example is depicted in Section 6 followed by the application to AVR system. Section 7 comprises the discussion and conclusion along with the future aspects of the present work. 2. FOPID controller 2.1. Mathematical modelling Fractional order PID controllers consist of integral and derivative parts with fractional order which provide more flexibility to obtain the additional design specifications. Generally, the FOPID controller is given as-
GF OPID (s ) =
TD sλ+μ + KP sλ + TI sλ
(1)
where KP , TI & TD are proportional, integral and derivative gains respectively, λ is the fractional orders of integral part whereas μ is the fractional orders of derivative part of the FOPID controller. The graphical representation of FOPID controller is shown in Fig. 1 where the x-axis shows the order of the integrator and the y-axis shows the order of the differentiator of the controller. Depending upon the values of λ and μ, it is observed that the conventional P, PI, PD, and PID controllers can be obtained easily from FOPID controller. 2.2. Hardware implementation Most of the controllers are represented by their mathematical model for analysis however the hardware implementation of a system is necessary for practical point of view. Therefore, investigation of new methods for practical circuit realisation of the controllers is in demand. A simple technique for the circuit realisation of fractional order controllers is suggested by Charef [20] in which a rational function is used for approximation of fractional order operators. The fractional order integrator (FOI) is represented as follows-
GF OI (s ) =
1 sn
(2)
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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3
Fig. 1. Graphical representation of FOPID controller.
Fig. 2. Circuit realisation of FOI.
where n is a positive real number. This transfer function is first approximated using fractional power pole (FPP) method Charef [20] in the predefined frequency band (ωL , ωH ) . The transfer function of FPP is given as follows-
GF PP (s ) =
KI
(3)
(1 + (s/ωc ))n
where ωc is the −3n decibel corner frequency of FPP and KI is the integral gain. Let us consider that ω ∈ (ωL , ωH ), ωc . So Eq. (3) can be written as-
GF PP (s ) =
KI
(s/ωc )
n
=
KI ωcn sn
& ω
(4)
If we put KI ωcn = 1 i.e. KI = ω1n then c
GF PP (s ) =
1 = GF OI (s ) sn
(5)
Hence the transfer function representation of fractional order integrator is as followsm −1
1 KI i=0 GI ( s ) = n = KI m s (1 + (s/ωc ))n i=0
(1 + (s/z0 (cd )i )) (6)
(1 + (s/ p0 (cd )i ))
The parameters of Eq. (6) are obtained by the graphical method suggested by Charef [21] and given as follows-
c = 10(δ /(10(1−n ))) ,
d = 10(δ /10n) ,
z0 = c p0 , and
m = Integer
p0 = ωc 10(δ /20n) ,
log(ωmax / p0 ) log(cd )
+1
ωc = ωL (10(δ/10n) − 1 )
where ωmax = 100ωH and δ is the specified error between FOI and FPP in the predefined frequency band. Now, the circuit realisation of FOI using resistor and capacitor is depicted in Fig. 2 [20] where V(s) is the terminal voltage and I(s) is the current. The values of resistors and capacitors are calculated using the following relationsm −1
Ri = KI Ci =
(1−( (cd )(i− j ) /c ))
j=0 m j=0,i= j
1 (cd )i p0 Ri
(1−(cd )(i− j ) )
,
(7)
f or i = 0, 1, . . . , . . . m
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Fig. 3. Circuit realisation of FOD.
Similarly, the fractional order differentiator (FOD) is represented as follows-
GF OD (s ) = sn
(8)
This transfer function is first approximated using fractional power zero (FPZ) method Charef [20] in the predefined frequency range (ωL , ωH ) . The transfer function representation of fractional power zero (FPZ) is given as-
n
s
GF PZ (s ) = KD 1 +
(9)
ωc
where ωc is the 3n decibel corner frequency of fractional power zero and KD is the derivative gain. Let us consider that ω ∈ (ωL , ωH ), & ω ωc . So Eq. (9) can be written as-
GF PZ (s ) = KD
s n K D = n sn ωc ωc
(10)
If we put ωDn = 1 i.e. KD = ωcn then c K
GF PZ (s ) = sn = GF OD (s )
(11)
Hence the fractional order differentiator is represented as followsm
(1 + (s/z0 (cd ) )) s n i=0 GD (s ) = sn = KD 1 + KD m ωc (1 + (s/ p0 (cd )i ))
i
(12)
i=0
The parameters of Eq. (12) are obtained by the graphical method suggested by Charef [21] and given as follows-
c = 10(δ /(10(1−n ))) ,
d = 10(δ /10n) ,
p0 = c z0 , and
m = Integer
z0 = ωc 10(δ /20n) ,
log(ωmax /z0 ) log(cd )
+1
ωc = ωL (10(δ/10n) − 1 )
where ωmax = 100ωH and δ is the specified error between FOD and FPZ in the predefined frequency band. Now the circuit realisation of FOD using resistor and capacitor is depicted in Fig. 3 [20] where V(s) is the terminal voltage and I(s) is the current and R p = KD . The values of resistors and capacitors are calculated using the following relationsm
Ci = Ri =
KD i p0 (cd )
1− c ( c d )
j=0 m
j=0,i= j
1 (cd )i p0Ci
( i− j )
1−(cd )
( i− j )
,
(13)
f or i = 0, 1, . . . , . . . m
Furthermore, in order to achieve the circuit realisation of fractional order PID controller the FOPID controller represented by Eq. (1) is modified in the following form-
GF OPID (s ) = KP +
where
TI s
, (TD s),
1 snI
T 1 I
s
snI
+ (TD s )(snD )
(14)
& (snD ) are first order and fractional order integrator and differentiator respectively with 0 < nI ,
nD < 1 . Hence the transfer function of FOPID controller is approximated using fractional power pole (FPP) and fractional power zero (FPZ) method and is represented as follows-
⎛
⎞
⎛ ⎞ m (1 + (s/z0 (cd ) )) ⎟ (1 + (s/z0 (cd )i )) T ⎜ ⎟ I ⎜ i=0 i=0 ⎟ + (TD s )⎜ GF OPID (s ) = KP + KI m ⎝KD ⎠ m ⎠ s ⎝ i i (1 + (s/ p0 (cd ) )) (1 + (s/ p0 (cd ) )) m −1
i=0
i
(15)
i=0
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Fig. 4. Design structure of FOPID.
Fig. 5. Analogue circuit design of proportional element.
Fig. 6. Analogue circuit design of integral element.
Fig. 7. Analogue circuit design of derivative element.
All the three elements of FOPID controller are realised individually and then their combined effect is analysed using the structure specified in the Fig. 4. The individual circuit design of proportional, integral and derivative element is specified in the Figs. 5, 6 and 7, respectively. Therefore, the analogue realisation of FOPID controller is achieved by using Fig. 8 and hence
TI R2 Z2 Z3 GF OPID (s ) = KP + λ + TD sμ = + + R1 Z1 Z4 s
(16)
3. Overview of automatic voltage regulator The maintenance of output voltage of synchronous generator at a predefined level is very essential. This is achieved by automatic voltage regulator (AVR) which acts as a controller to maintain the level of the terminal voltage within the excitation system. Therefore, the design of stable automatic voltage regulator is of great interest. The block diagram representation Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 8. Analogue circuit design of FOPID controller.
Fig. 9. Block diagram representation of AVR with CS-FOPID controller.
of automatic voltage regulator with CS-FOPID controller is shown in Fig. 9 which consists of four components of automatic voltage regulator namely amplifier, exciter, generator and sensor/feedback. The role of sensor, as shown in Fig. 9, is to sense terminal voltage variations, existing either due to change in load demand or power system faults, and feedback the same to the comparator after being rectified. This feedback signal is first compared with reference signal at the comparator to generate error signal and then amplified and given to the exciter. The major role in complete loop is played by the exciter which control the field current in such a way that the error is reduced to zero. Finally, the output of the exciter is given to alternator to achieve constant terminal voltages. For analysis and design of FOPID controller for AVR system, the linearized models of the components of AVR system are chosen as follows [16]. The transfer function representation of amplifier is as follows-
Ga ( s ) =
Ka 1 + τa s
(17)
where amplifier gain (Ka ) and time constant (τ a ) are such that 10 < Ka < 400 & 0.02s < τ a < 0.1s. The transfer representation of the exciter model is as follows-
Ge ( s ) =
Ke 1 + τe s
(18)
where exciter gain (Ke ) and time constant (τ e ) are such that 10 < Ke < 400 & 0.5s < τ e < 1s. The transfer function representation of linearised model of generator is given as follows-
Gg ( s ) =
Kg 1 + τg s
(19)
where Kg & τ g are gain and time constant of generator which depends upon load. The range of gain is 0.7 to 1 and time constant is 1 to 2s. Similarly, the transfer function representation of sensor/feedback model is as follows-
H f (s ) =
Kf 1 + τf s
(20)
where feedback gain (Kf ) and time constant (τ f ) are such that Kf ࣃ 1 & 0.001s < τ f < 0.06s. Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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4. Performance measure of CS-FOPID controller There are various performance indices available in the literature to analyse and design of controllers. Some of the most popular performance indices are integral of absolute error (IAE), integral of squared error (ISE) and integral of time multiplied by squared error (ITSE). But these criterions have their own benefits and drawbacks such as small overshoot can be achieved using IAE & ISE but sacrifices the settling time whereas ITSE may resolve this problem but it never guarantee a required stability boundary. The IAE, ISE and ITSE are express as -
IAE =
∞ 0
ISE =
∞ 0
IT SE =
Vre f (t ) − Vt (t )dt
(21)
(Vre f (t ) − Vt (t ))2 dt
(22)
∞ 0
t (Vre f (t ) − Vt (t )) dt 2
(23)
where Vref (t), Vt (t) are reference and terminal voltage of synchronous generator respectively. Therefore, in this paper a time domain performance criterion is chosen which was previously considered by Gaing et. al. [16] in which time response specifications are included. The performance criterion Z(K) is expressed as-
Min Z (K ) = (1 − e−β ).(M p + ess ) + e−β (ts − tr )
(24)
where K is [KP , TI , TD ] and β is the weighting factor such that 0.8 < β < 1.5. Mp , ess, ts and tr denotes maximum overshoot, steady state error, rise time and settling time respectively. 5. Proposed design methodology of CS-FOPID controller Cuckoo Search (CS) is an evolutionary algorithm which is inspired by common behaviour of Cuckoo bird of laying their eggs in the host nest birds and recently developed by Yang and Deb [22]. This algorithms is developed based on the concept that all the cuckoo birds lay down their eggs into to the other bird’s nest for fertilization. It is possible that the host birds might recognize that it is not their eggs. Therefore, either they throw the alien eggs from their nests or form a new nest at new place. [23]. A set of host nest show the cuckoo breading analogy. Each nest carries an egg which is considered as a solution. A new nest is formed using Lévy flight [24] i.e. random walk. Success of random searches of resources can be optimize using the Lévy flight movements [25]. In order to utilize this natural behaviour of Cuckoo species including the Lévy flight, Yang and Deb [22] suggested the following three rules: • For laying down the egg, the nest should be selected at random and dump by every cuckoo. • The nest must be transferred to the next generation if good quality eggs are found in it. • The probability of an alien egg which can be observed by the fixed number of host nest is pa ∈ [0, 1]. In such cases the host nest either throw the alien egg or form a new nest at any other place. For easiness, the fraction of pa of n nests, which are being replaced by new nests, is considered the approximation of previous assumption. The Lévy flight can be represented by the following relation for the generation of new solution Y (t + 1 ) of cuckoo i [23]-
Yi (t+1) = Yi (t ) + a Lévy(λ )
(25)
where a (a > 0) is a step size which is related to the level of the problem optimized by the technique. The random step size of the Lévy flights are calculated as follows-
Lévy u˜ = t −λ
; (1 < λ ≤ 3 )
(26)
CS algorithm has been utilised widely in different areas which includes steel structure design [26], structural optimization problem [27], applications to business optimization [28], model order reduction [29] etc., due to its computational efficiency and fast convergence. The flow chart of cuckoo search algorithm is shown in Fig. 10 and the design of the FOPID controller using cuckoo search algorithm consists of the following stepsStep 1 Specify objective function and the parameters for cuckoo search such as decision variables and their lower and upper boundaries. Set the probability of the worst nests and step size also. Initialization of a population of p host nests then the problem is summarised asMinimize Z(K), subject to KPL < KP < KPU TIL < TI < TIU Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 10. Flow chart of cuckoo search algorithm.
TDL < TD < TDU where KPL , TIL , TDL and KPU , TIU , TDU are the lowest and highest values of the chosen variables of the controller respectively. Step 2 Obtain the value of Zα for a randomly selected cuckoo (α ) and select a nest (β ) randomly among p. Step 3 if (Zα > Zβ ) then interchange β by the current obtained solution else go to Step 4. Step 4 Check whether the predefined stopping criterion is arrived or the maximum generation occurred or not if yes, then the solution obtained in the current generation would be the best solution. Step 5 -Abandon a fraction of worse nests with optimal value of probability pa and step size a. Step 6 - Using Eq. (25) the obtained solution must be updated by calculating Yi (t+1 ) and repeat this algorithm, until the predefined condition is arrived or the maximum generation occurred. 6. Practical example and results A practical high order automatic voltage regulator is considered in this paper, to evaluate the performance and efficacy of the proposed CS-FOPID controller, with the following specifications [16]- Ka = 10, τa = 0.1, Ke = 1, τe = 0.4, Kg = 1, τg = 1, K f = 1 and τ f = 0.01. The basic block diagram representation of automatic voltage regulator with CS-FOPID controller is presented in Fig. 9 and the range of its parameters are depicted in Table 1. The output voltage step response of a practical automatic voltage regulator in the absence of controller is depicted in Fig. 11 in which it is found that M p = 65.72%, ess = 0.0909, tr = 0.2607 s and ts = 6.9865 s. The time responses of automatic voltage regulator with different controllers are depicted in Figs. 12–14. The values of performance criterion for different controllers are tabulated in Table 2. Total three cases are considered depending upon Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Table 1 Parameters of CS-FOPID controller. Controller parameter
Min. value
Max. value
KP TI TD
0 0 0 0 0
3 1 1 2 2
λ μ
Fig. 11. Output voltage step response of AVR in the absence of controller. Table 2 Comparison of different controllers for AVR. Case
β
Number of generation
Type of controller
KP
TI
TD
Mp (%)
ts
tr
ess
1
1.5
50
2
1
100
3
0.8
150
CS-FOPID PSO-PID [16] GA-PID [16] CS-FOPID PSO-PID [16] GA-PID [16] CS-FOPID PSO-PID [16] GA-PID [16]
2.5490 0.6568 0.8861 2.515 0.6751 0.7722 2.4676 0.6570 0.8663
0.1759 0.5393 0.7984 0.1629 0.5980 0.7201 0.3020 0.5390 0.7531
0.3904 0.2458 0.3158 0.3888 0.2630 0.3196 0.4230 0.2458 0.3365
0.0014 1.17 8.66 0 1.71 4.54 0 1.16 7.34
0.4415 0.4027 0.5980 0.4507 0.3795 0.8645 0.5002 0.4025 0.8519
0.1035 0.2768 0.2019 0.1042 0.2648 0.2138 0.0960 0.2767 0.1959
0 0 0 0 0 0 0 0 0
the value of weighting factor β and the number of generations. For each case, the efficiency and superiority of the proposed method for controller design is observed as the AVR system exhibits better performance with CS-FOPID controller (λ = 0.97, μ = 1.38 ) as compared to other well-known PSO-PID and GA-PID [16] controllers available in the literature. Furthermore, the proposed CS-FOPID controller is also compared with classical PID and PSO-FOPID controller [15] for AVR and found better. The comparative analysis is depicted in Table 3. The robustness and powerfulness of the CS-FOPID controller with exciter uncertainty is examined by considering the exciter transfer model as follows [15]-
Ge ( s ) =
1 0.5 + 0.5s
(27)
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 12. Comparison of unit step response of AVR system with proposed and previously designed controllers (CASE-1).
Fig. 13. Comparison of unit step response of AVR system with proposed and previously designed controllers (CASE-2). Table 3 Comparison of CS-FOPID controller with classical PID and PSO-FOPID controller for AVR. Type of controller
KP
TI
TD
λ
μ
Mp (%)
ts
tr
es s
CS-FOPID PSO-FOPID [15] PID [15]
2.515 0.17 0.0046
0.1629 0.03 0.0268
0.3888 0.014 0.18
0.97 0.97 1
1.38 1.38 1
0 0 2
0.4507 52 27
0.1042 27 17.6
0 0 0
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 14. Comparison of unit step response of AVR system with proposed and previously designed controllers (CASE-3). Table 4 Comparison of controllers with parameter uncertainty in exciter model of AVR. Type of controller
KP
TI
TD
λ
μ
Mp (%)
ts
tr
es s
CS-FOPID PSO-FOPID [15] PSO-PID [16]
0.8912 0.17 0.6570
0.2485 0.03 0.5389
0.5105 0.014 0.2458
0.97 0.97 1
1.38 1.38 1
0.1779 2.0279 18.03
1.4591 17.6056 1.6581
0.0986 1.1185 0.2578
0 0 0
Further, for parameter variation in exciter model, the terminal voltage step responses of proposed CS-FOPID, PSO-FOPID [15] and PSO-PID [16] are shown in Fig. 15. It is observed that the PSO-FOPID and PSO-PID exhibits an overshoot of 2.0279% and 18.03% respectively whereas the proposed CS-FOPID controller exhibits an overshoot of 0.1779% only which shows the superiority of the proposed controller over others with parameter variation in exciter model. Also, a comparative analysis of different controllers in terms of time domain performance criterion with parameter uncertainty in exciter model of AVR is depicted in Table 4.
7. Conclusion With the widespread use of power-electronic devices and different types of nonlinear loads, voltage flickering has now become one of the most pressing concern to the industries which are operating in a highly competitive business environment. Therefore, the quality of voltage for these equipment is highly desirable to keep themselves in tune with the market demand. Nowadays, AVR has received utmost attention in the industries to maintain constant voltage at the equipment terminals under all the conditions. However, to achieve improved transient characteristics, a precise controller, associated with AVR, is highly desirable. In addition, precise controller design plays vital role for proper monitoring of reactive power sharing between the parallel connected generators. Considering, all of these issues and to obtain improved dynamic performance, robustness, and stability, this paper considers a novel design technique for FOPID controller of AVR, using cuckoo search algorithm. The time response characteristic parameters, such as, maximum overshoot, rise time, settling time, and steady state error, of AVR have been evaluated with and without proposed FOPID controller to check its effectiveness. It is revealed that, the proposed controller improves time response characteristic parameters. Further, the time response characteristic parameters are evaluated with the other existing controllers, such as, PSO-FOPID, PSO-PID, GA-PID and conventional PID controller, to present comparative analysis. It is found that the proposed controller provides more improved dynamic performance than the existing controllers, PSO-FOPID, PSO-PID, GA-PID and conventional PID. Also, the performance of the proposed as well as other controllers has been tested Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005
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Fig. 15. Comparison of step response of AVR system with proposed and previously designed controllers under exciter uncertainty.
under the condition of parameter variation in exciter model of AVR. It is found that, the proposed controller exhibits more robust and improved dynamic performance than the other existing controller under the condition of parameter variation. References [1] Podlubny I. Fractional-order systems and PIλ Dμ controllers. IEEE Trans Automat Contr 1999;44(1):208–14. [2] Vinagre BM, Podlubny I, Dorcak L, Feliu V. On fractional pid controllers: a frequency domain approach. IFAC workshop; 20 0 0. [3] Pan I, Das S. Frequency domain design of fractional order PID controller for AVR system using chaotic multi-objective optimization. Int J Electr Power Energy Syst 2013;51:106–8. [4] Petras I. The fractional order controllers: methods for their synthesis and application. J Electr Eng 20 0 0;50(9-10):284–8. [5] Dorcak L, Petras I, Kostial I, Terpak J. State space controller design for the fractional order regulated system. In: proceedings of ICCC; 2001. p. 15–20. [6] Vinopraba T, Sivakumaran N, Narayanan S, Radhakrishnan TK. Design of internal model control based fractional order PID controller. J Control Theory Appl 2012;10(3):297–302. [7] Wang DJ, Li W, Guo ML. Tuning of PIλ Dμ controllers based on sensitivity constraint. J Process Control 2013;23(6):861–7. [8] Tepljakov A, Petlenkov E, Belikov J. FOMCON: a MATLAB toolbox for fractional-order system identification and control. Int J Microelectron Comput Sci 2011;2(2):51–62. [9] Rajasekhar A, Kumar JR, Abraham A. Design of intelligent PID/PIλ Dμ speed controller for chopper fed DC motor drive using opposition based artificial bee colony algorithm. Eng Appl Artif Intell 2014;29:13–32. [10] Dumlu A, Erenturk K. Trajectory tracking control for a 3-DOF parallel manipulator using fractional-Order PIλ Dμ control. IEEE Trans Ind Electron 2014;61(7):3417–26. [11] Luo Y, Chen Y. Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica 2012;48(9):2159–67. [12] Cao JY, Cao BG. Design of fractional order controller based on particle swarm optimization. Int J Control Autom Syst 2006;4(6):775–81. [13] Padhee YSS, Gautam A, Kaur G. A novel evolutionary tuning method for fractional order pid controller. Int J Soft Comput Eng 2011;1(1):1–9. [14] Das S, Pan I, Das S, Gupta A. Improved model reduction and tuning of fractional-order PIλ Dμ controllers for analytical rule extraction with genetic programming. ISA Trans 2012;51(2):237–61. [15] Zamani M, Karimi-Ghartemani M, Sadati N, Parniani M. Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Eng Pract 2009;17(12):1380–7. [16] Gaing Z. A particle swarm optimization approach for optimum design of PID controller in AVR system. IEEE Trans Energy Convers 2004;19(2):384–91. [17] Eberhart RC, Shi Y. Comparison between genetic algorithms and particle swarm optimization. In: Proc. IEEE Int. Conf. Evol. Comput., Anchorage, AK; 1998. p. 611–16. [18] Pan I, Das S. On the mixed H2 /H∞ l loop-shaping tradeoffs in fractional-order control of the AVR system. IEEE Trans Ind Inf 2014;10(4):1982–91. [19] Pan I, Das S. Chaotic multi-objective optimization based design of fractional order PIλ Dμ controller in AVR system. Int J Electr Power Energy Syst 2012;43(1):393–407. [20] Charef A. Analogue realization of fractional order integrator, differentiator and fractional PIλ Dμ controller. In: IEE Proc. Control Theory Appl.; 2006. p. 714–20. [21] Charef A, Sun HH, Tsao YY, Onaral B. Fractal system as represented by singularity function. IEEE Trans Automat Contr 1992;37(9):1465–70. [22] Yang XS, Deb S. Nature-Inspired Metaheuristic Algorithms. United Kingdom: Luniver Press; 2008. [23] Yang XS, Deb S. Engineering optimisation by cuckoo search. Int J Math Modell Numer Optim 2010;1:330–43. [24] Brown CT, Liebovitch LS, Glendon R. Lévy flights in dobe ju/hoansi foraging patterns. Hum Ecol 2007;35(1):129–38. [25] Humphries NE, Weimerskirch H, Queiroz N, Southall EJ, Sims DW. Foraging success of biological lévy flights recorded in situ. Proc Nat Acad Sci 2012;109(19):7169–74.
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[26] Kaveh A, Bakhshpoori T, Ashoory M. An efficient optimization procedure based on cuckoo search algorithm for practical design of steel structures. Int J Optim Civil Eng 2012;2(1):1–14. [27] Gandomi AH, Yang XS, Alavi AH. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 2013;29(1):17–35. [28] Yang XS, Deb S, Karamanoglu M, He X. Cuckoo search for business optimization applications. In: Proceeding of National Conference on Computing and Communication Systems (NCCCS 2012); 2012. p. 29–33. ISBN 9781467319539. [29] Sikander A, Prasad R. A novel order reduction method using cuckoo search algorithm. IETE J Res 2015;61(2):83–90.
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Afzal Sikander has over 10 years of teaching and research experience and currently he is working as Associate Professor with the Department of Electrical Engineering, Graphic Era University, Dehradun, India. He is acting as Editorial board member of various journals. His area of interest includes circuit design, control system, model order reduction, controller design, system Engineering and optimization etc. Padmanabh Thakur has over 17 years of experience and currently he is a Professor and head in the Department of Electrical, Graphic Era University, Dehradun. Also, Prof. Thakur is holding responsibility of Associate Editor of IEEE-Access and JIPS, South Korea. Ramesh Bansal has over 25 years of experience and currently he is a Professor and group head (Power) in the Department of EEC Engineering at University of Pretoria. He has published over 250 papers. Prof. Bansal is an Editor of IET-RPG & Electric Power Components and Systems. He is a Fellow and CEngg IET-UK, Fellow Engineers Australia and Senior Member-IEEE. Selvamuthukumaran Rajasekar was the recipient of the IEEE Industrial Electronics Society Travel grant award for the year 2012. He also received POSOCO research award in year 2015 for his research work. During 2015–2017, he was a postdoctoral researcher in NEC Laboratories Singapore and conducted research project in collaborating with Nanyang Technological University, Singapore.
Please cite this article as: A. Sikander et al., A novel technique to design cuckoo search based FOPID controller for AVR in power systems, Computers and Electrical Engineering (2017), http://dx.doi.org/10.1016/j.compeleceng.2017.07.005