Cuckoo Search algorithm based load frequency controller design for nonlinear interconnected power system

Electrical Power and Energy Systems 73 (2015) 632–643

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Cuckoo Search algorithm based load frequency controller design for nonlinear interconnected power system A.Y. Abdelaziz a, E.S. Ali b,c,⇑ a

Electric Power and Machine Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt Electric Power and Machine Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt c Currently, Electrical Department, Faculty of Engineering, Jazan University, KSA b

a r t i c l e

i n f o

Article history: Received 1 October 2014 Received in revised form 19 May 2015 Accepted 26 May 2015

Keywords: Cuckoo Search algorithm Load Frequency Control PI controller Time delay Generation Rate Constraint Three area system

a b s t r a c t A new optimization technique called Cuckoo Search (CS) algorithm for optimum tuning of PI controllers for Load Frequency Control (LFC) is suggested in this paper. A time domain based-objective function is established to robustly tune the parameters of PI-based LFC which is solved by the CS algorithm to attain the most optimistic results. A three-area interconnected system is investigated as a test system under various loading conditions where system nonlinearities are taken into account to confirm the effectiveness of the suggested algorithm. Simulation results are introduced to show the enhanced performance of the developed CS based controllers in comparison with Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and conventional integral controller. These results denote that the proposed controllers offer better performance over others in terms of settling times and various indices. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction In the large scale power systems, LFC plays a serious role. The LFC is aimed to ensure the frequency of each area and the inter area tie line power within tolerable limits to deal with the fluctuation of load demands and system disturbances [1,2]. These important functions are delegated to LFC due to the fact that a well-designed power system should keep voltage and frequency in scheduled range while supplying an acceptable level of power quality [3,4]. Several researches and techniques had been applied to the field of LFC during the last decades. Robust control [5–9], decentralized control [10,11], linear quadratic problem [12], pole placement approach [13,14], variable structure control [15], and state feedback [16], are used to deal with LFC problem design. However, these strategies have many problems which limit their applicability. In an effort to overcome these problems, many researches have used artificial intelligence as Fuzzy Logic Controller (FLC) [17–21] and Artificial Neural Network (ANN) [21–23]. Although these methods are effective in dealing with the nonlinear characteristics of the power system, they have their own problems. For example, ⇑ Corresponding author at: Electrical Department, Faculty of Engineering, Jazan University, KSA. E-mail addresses: [email protected] (A.Y. Abdelaziz), [email protected] (E.S. Ali). http://dx.doi.org/10.1016/j.ijepes.2015.05.050 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

ANN suffers from the long training time, the selecting number of layers and the number of neurons in each layer. Also, FLC requests a hard work to catch the efficient signals and it needs fine tuning and simulation before operation. Another approach is to use Evolutionary Algorithm (EA) techniques. EA is visualized to be very effective to deal with LFC problem due to its ability to treat nonlinear objective functions. Among the EA techniques, GA [24–29], PSO [30–33], Bacteria Foraging [34–38], Artificial Bee Colony [39], and Ant Colony Optimization [40] have attracted the attention in LFC controller design. Although these algorithms appear to be efficient for the design problem, they suffer from slow convergence problem in refined search stage, weak local search ability, which may lead them to get trapped in local minimum solution. A new evolutionary computation algorithm, called CS algorithm has been presented by [41] and further formed newly by [42–44]. In addition, it is simple and population based stochastic optimization algorithm. Moreover, it requires less control parameters to be tuned. Also, it is a compatible optimization tool for power system controller design [45,46]. This paper introduces a modern optimization algorithm called CS for the optimum tuning of PI controller parameters in LFC problem. The motivation behind this research is to ensure and prove the robustness of CS based PI, and to enhance the performance of frequency deviation and tie line power under various loading conditions in presence of system nonlinearities.

A.Y. Abdelaziz, E.S. Ali / Electrical Power and Energy Systems 73 (2015) 632–643

633

Nomenclature f i Ri T gi T ti T ri K ri

the system frequency in Hz, subscript referring to area (i = 1, 2, 3) the regulation constant (Hz/p.u MW) for area i, the speed governor time constant in second for area i the turbine time constant in second for area i, the reheat time constant in second for area i, the p.u megawatt rating of high pressure stage for area i Tw the hydro turbine time constant, K d , K p , K i the electric governor derivative, proportional and integral gains, respectively T Pi , K Pi the time constant and gain of power system respectively for area i DPtiei the difference between the actual tie-line power and scheduled one B the biasing factor in pu MW/Hz K PPi , K IIi the gains of PI controller of area i N the number of area in power systems tsim the simulation time in second t time in second T ij synchronizing coefficient J objective function Ui the control signal of area i Ki the controller of area i Three area power system The system under study consists of 3 areas of equal sizes. Areas 1 and 2 are reheat thermal systems and area 3 is a hydro system

max K min PPi ; K PPi

the lower and the upper limit of proportional gain of area i max K min the lower and the upper limit of Integral gain of area i IIi ; K IIi Pa the probability to abandon a nest n number of nests xti the current solution xtþ1 a new solution i List of abbreviations LFC Load Frequency Control GA Genetic Algorithm PSO Particle Swarm Optimization CS Cuckoo Search FLC Fuzzy Logic Controller ANN Artificial Neural Network ANFIS adaptive neuro-fuzzy inference system PI proportional plus integral GRC Generation Rate Constraint ACE Area Control Error IAE the integral of absolute value of the error ITAE the integral of the time multiplied absolute value of the error ISE the integral of square error ITSE the integral of time multiply square error

[17]. The detailed designed model of 3 area hydro-thermal power system for load frequency control is shown in Fig. 1. The thermal plant has a single stage reheat steam turbine and the hydro plant is equipped with an electric governor. In this model, nonlinearities

Generation Rate Constraint

Generation Rate Constraint

Generation Rate Constraint

Fig. 1. A three area model.

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case [47]. Sometimes, it can reach to several seconds in conventional systems [4]. The system parameters are given in appendix. The transfer functions of different blocks used in power system model are given below [17,21]: The transfer function of the hydraulic turbine is

Start Initialization of CS parameters

T w S þ 1 : 0:5T w S þ 1

Initial generation of host nest population

ð1Þ

Transfer function of the hydraulic governor is

Evaluation of fitness function for the generated host nest population

K d S2 þ K p S þ K i 2

K d S þ ðK p þ f =RÞS þ K i

Move all cuckoos towards better environment

Modification of nest position using Lévy flight equation

ð2Þ

:

Transfer function of the governor is

1 : TgS þ 1

Evaluate the new solution

ð3Þ

Transfer function of the steam turbine is

K rTrS þ 1 : TrS þ 1

No

Check the condition is satisfied?

ð4Þ

Transfer function of the Re-heater is

1 ; TtS þ 1

Yes Print the result

ð5Þ

and transfer function of the Generator is

KP : TPS þ 1

Stop

ð6Þ

For the ith area, the Area Control Error (ACE) signal made by frequency and tie-line power variations is stated by:

Fig. 2. Flowchart of Cuckoos Search optimization algorithm.

are represented in Generation Rate Constraint (GRC) and communication time delay. GRC illustrates the limitation on the generation rate of change in the output generated power due to the limitation of thermal and mechanical movements [4], for thermal stations it is taken to be 0.1 Pu Mw per minute [40]. On the other hand, communication time delay is introduced by many signal processing and data exchanging operations. In this paper, a time delay between the control center and each unit is stimulated. Time delay is in the order of 100 ms and can extend to 250 ms at the worst

ACEi ¼ B  Df i þ DPtiei :

ð7Þ

Optimization model of load frequency control system For the three area considered, the conventional integral controller was exchanged by a PI one with the given equation:

K i ðSÞ ¼ K PPi þ

K IIi : S

ð8Þ

0.02 CS GA

0.01

PSO Integral

Change in f1

0

-0.01

-0.02

-0.03

-0.04

-0.05

0

10

20

30

40

50

60

Time in second Fig. 3. Change in f 1 for case 1.

70

80

90

100

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The control signal is specified by equation:

U i ðSÞ ¼ K i ðSÞACEi ðSÞ:

ð9Þ

The sum of time multiple absolute errors in ACE is considered as performance index, hence J can be outlined as:

J¼

N Z X i¼1

tjACEi jdt;

ð10Þ

0

ð11Þ

The breeding behavior of cuckoo species and the basic items of this algorithm are discussed below. Cuckoo breeding behavior CS is a metaheuristic search technique that has been introduced by Yang and Deb in 2010 [42]. The algorithm is inspired by obligate brood parasitism of cuckoo species by laying their eggs in the nests of other host birds. At the proper minute, the hen cuckoo flies down to the host’s nest, ejects one egg out of the nest, lays an

0.02 CS GA

0.01

PSO Integral

0

Change in f3

6

K PPi 6 K max PPi ; max K IIi 6 K IIi :

-0.01

-0.02

-0.03

-0.04

-0.05

0

10

20

30

40

50

60

70

80

90

100

Time in second Fig. 4. Change in f 3 for case 1.

0.005

0

-0.005

Change in ACE1

6

Cuckoo Search algorithm

tsim

To enhance the system response in terms of the settling time and overshoots, it is necessary to minify equation (10). The design process can be formed as the following constrained optimization problem. Lessen J subject to:

K min PPi K min IIi

The regular ranges of the optimized gains are [2 to 2] as obtained in [35–37] and the objective function are calculated under disturbance of 1% in all areas.

-0.01

-0.015

-0.02 CS

-0.025

GA PSO Integral

-0.03

0

10

20

30

40

50

60

Time in second Fig. 5. Change in ACE1 for case 1.

70

80

90

100

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egg and flies off. This procedure takes about 10 s. A female may visit up to 50 nests during a breeding season. The host birds may detect that the eggs are not their own and either throw them away or abandon the nest. This has resulted in the evolution of cuckoo eggs which mimic the eggs of local host birds [43]. Moreover, the timing of egg laying of some species is amazing. Parasitic cuckoos select a nest where the host bird just laid its own eggs. In general, the cuckoo eggs hatch slightly earlier than their host eggs. It dislodges all host progeny from host nests. It is a larger bird than its hosts, and needs to monopolize the food supplied by the parents. The chick will roll the other eggs out of the nest by pushing them with its back. If the host’s eggs hatch before the cuckoos, the cuckoo chick will push the other chicks out of the nest in an identical manner.

from a probability density function which has a power law tail. This procedure demonstrates the optimal random search pattern and is found in nature [45]. When generating a new egg, a Lévy flight is done starting at the location of a randomly chosen egg, if the objective function value at these new coordinates is better than another chosen egg then that egg is shifted to this new location. One of the advantages of CS over other optimization algorithms are that only one parameter, the fraction of nests to abandon P a ; requires to be inspected. The use of Lévy flights as the search method means that the CS can detect each optimum in a design space and it has been proved in comparison with other algorithms [46].

Lévy flights

To apply this search as an optimization algorithm, three presented approximation rules are adopted as below [45,46]:

The use of Lévy flights for local and global searching is a vital portion of the CS [44]. The Lévy flight process is a random walk that is characterized by a series of instantaneous jumps elected

1. Cuckoos select random nest for laying their eggs. Artificial cuckoo can lay only one egg at the time.

Cuckoo Search implementation

0.015 CS GA PSO

0.01

Integral

Change in ACE3

0.005

0

-0.005

-0.01

-0.015

-0.02

0

10

20

30

40

50

60

70

80

90

100

Time in second Fig. 6. Change in ACE3 for case 1.

8

x 10-3 CS GA

7

PSO Integral

6

Change in P2

5 4 3 2 1 0 -1 -2

0

10

20

30

40

50

60

Time in second Fig. 7. Change in P2 for case 1.

70

80

90

100

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0.1 CS GA PSO

0.05

Integral

Change in f1

0

-0.05

-0.1

-0.15

-0.2

0

10

20

30

40

50

60

70

80

90

100

Time in second Fig. 8. Change in f 1 for case 2.

0.1 CS GA PSO

0.05

Integral

Change in f3

0

-0.05

-0.1

-0.15

-0.2

0

10

20

30

40

50

60

70

80

90

100

Time in second Fig. 9. Change in f 3 for case 2.

2. Elitist selection process is applied, so only the eggs with highest quality are passed to the next generation. 3. The number of available hosts nests is specified and a host can reveal a foreigner egg with a probability Pa 2 ½0; 1. If cuckoo egg is detected by the host, it may be thrown away, or the host may leave its own nest and commit it to the cuckoo intruder. The last assumption can be approximated by a factor Pa of the n nests being exchanged by new nests. The quality of a solution is relative to the objective function. Based on the previous rules, the flowchart of CS is displayed in Fig. 2. The parameters of CS are shown in appendix. Other forms of fitness can be determined in GA [48,49] and PSO [50,51]. When generating a new solution xtþ1 for the ith cuckoo, a Lévy i flights is carried out

vyðkÞ; xitþ1 ¼ xti þ a  Le

ð12Þ

where a > 0 is the step size which should be proportional to the scales of the optimization problem. The product  intends entrywise walk during multiplications. Lévy flights supply a random walk while their random steps are drawn from a Lévy distribution for large steps.

vy  u ¼ tk ; ð1 < k < 3Þ; Le

ð13Þ

which has an infinite variance with an infinite mean. Here the steps form a random walk process with a power law step-length distribution with a heavy tail. Some of the new solutions should be generated by Lévy walk round the best solution; this will accelerate the local search. However, a substantial fraction of the new solutions should be generated by far field randomization and whose places should be far enough from the current best solution, this will ensure the system will not be trapped in a local optimum.

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Results and simulations Various comparative cases are tested in this section to prove the efficiency of the developed CS algorithm for optimizing controller parameters. The proposed CS methodology, PSO and GA are programmed in MATLAB 7.1 with I5, CPU 2.53 GHz and 4.0 GB RAM. The obtained results are the best for all algorithms depending on value of J. The convergence times for CS, PSO and GA are 23.3, 28.7 and 40.2 s respectively. Case 1: step increase in DPD1 A 1% step increment in demand of the first area (DPD1 ) is employed as the first operating point. The system responses are shown in Figs. 3–7. In these Figures, the responses with

conventional integral controller are suffered from high settling time and unwanted oscillations. Moreover, the developed controllers are more successful in enhancing the damping feature of power system compared with PSO and GA. Thus, power system oscillations are efficiently attenuated with the implementation of the developed controllers. Hence, CS provides more satisfactory results than other controllers. Case 2: step increase in all areas In this case, a 1% step increment in demand of all areas simultaneously is applied as the second operating point. The signals of the closed loop system are introduced in Figs. 8–12. It is clear from these Figures, that the system oscillations are poorly damped for conventional controller. Also compared with PSO and GA, the

0.03 CS GA

0.02

PSO Integral

0.01

Change in ACE1

0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07

0

10

20

30

40

50

60

70

80

90

100

Time in second Fig. 10. Change in ACE1 for case 2.

0.04 CS GA PSO

0.02

Integral

Change in ACE3

0

-0.02

-0.04

-0.06

-0.08

0

10

20

30

40

50

60

Time in second Fig. 11. Change in ACE3 for case 2.

70

80

90

100

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developed controllers have a lower settling time and system response is driven back to zero speedily. Moreover, the seniority of the suggested algorithm in tuning controllers is demonstrated. Case 3: parameter variation A parameter variation test is utilized to affirm the effectiveness of the suggested CS algorithm in tuning controllers. Figs. 13–16 show the responses of frequency and ACE deviation of the second and third area respectively with variation in all time constants by

10%. It is clear that the system is stable with the developed controllers. In addition, the designed controllers have the ability of adding adequate damping to the system oscillatory modes under various conditions and the robustness of these controllers is believed. Case 4: step response with different times delay The communication time delay is added here, as a second source of nonlinearities, to investigate its effect on the closed loop

x 10-3

3

CS

2.5

GA PSO

2

Integral

1.5

Change in P3

1 0.5 0 -0.5 -1 -1.5 -2 -2.5

0

10

20

30

40

50

60

70

80

90

100

Time in second Fig. 12. Change in P3 for case 2.

0.02 0 -0.02

Change in f1

-0.04 -0.06 -0.08 -0.1 -0.12 -0.14 Normal

-0.16

Increase by 10% Decrease by 10%

-0.18

0

5

10

15

20

25

30

Time in second Fig. 13. Change in f 1 for case 3.

35

40

45

50

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0.02 0 -0.02

Change in f3

-0.04 -0.06 -0.08 -0.1 -0.12 -0.14 Normal

-0.16

Increase by 10% Decrease by 10%

-0.18

0

5

10

15

20

25

30

35

40

45

50

Time in second Fig. 14. Change in f 3 for case 3.

0.01 0

Change in ACE2

-0.01

-0.02

-0.03 -0.04

-0.05 Normal

-0.06

Increased by 10% Decrease by 10%

-0.07

0

10

20

30

40

50

60

Time in second Fig. 15. Change in ACE2 for case 3.

system of LFC. The performance of the system with different times delay is offered in Figs. 17 and 18. The simulation results detect that, the developed controllers provide adequate performance and they compensate the time delay which donates a more enhancement in system performance.

ITAE ¼

3 Z X

3 Z X i¼1

Performance indices and robustness To prove the superiority of the developed controllers, some indices like IAE, ITAE, ISE and ITSE are being used as:

IAE ¼

3 Z X i¼1

0

100

ðjDACEi jÞdt;

ð14Þ

ITSE ¼

100

ð15Þ

ðDACEi Þ2 dt;

ð16Þ

0

3 Z X i¼1

tðjDACEi jÞdt;

0

i¼1

ISE ¼

100

100

tðDACEi Þ2 dt

ð17Þ

0

It is remarkable that the lower the value of these indices is, the better the system response in terms of time domain characteristics. Parameters of controllers are recorded in Table 1 while the values of various indices are illustrated in Table 2. It can be

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0.02 Normal

0.01

Increase by 10% Decrease by 10%

0

Change in ACE3

-0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08

0

10

20

30

40

50

60

Time in second Fig. 16. Change in ACE3 for case 3.

0 -0.02

Change in f1

-0.04 -0.06 -0.08 -0.1 -0.12 -0.14

Without time delay Time delay=0.1 sec Time delay=0.3 sec Time delay= 0.6 sec

-0.16 0

5

10

15

20

25

30

35

40

Time in second Fig. 17. Change in f 1 for case 4.

seen that the values of these system performance with the proposed controllers are smaller compared with those of PSO, GA and conventional controller. This demonstrates that the overshoot and settling time are largely minimized by applying the proposed CS algorithm. Merits of CS over others The settling times of disperse states for each area under different operating conditions are shown in Table 3. It is clear that the settling time associate with CS is the smallest one compared with PSO, GA and conventional integral control. Moreover, the values of these times are smaller than those obtained in [17,21]. Consequently, CS based controllers provide better performance than other controllers. Therefore, the proposed controllers approach using CS are more accurate and quicker than the other schemes for complex dynamical system.

Conclusions CS algorithm is suggested in this paper to tune the parameters of PI controllers for LFC problem. An integral time absolute error of the ACE for all areas is chosen as the objective function to enhance the system response in terms of the settling time and overshoots. The major contributions of this paper are: 1. Establishment of the dynamic model for 3 area power system considering GRC and communication time delay as nonlinearities sources with LFC based CS to assure the superiority of the developed controllers over PSO, GA and conventional integral controller throughout different disturbances for various signals. 2. The robustness of the developed controllers are confirmed through parameter variations.

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0.01 0 -0.01

Change in ACE1

-0.02 -0.03 -0.04 -0.05 -0.06 Without time delay Time delay=0.1 sec Time delay=0.3 sec Time delay=0.6 sec

-0.07 -0.08

0

5

10

15

20

25

30

35

40

45

50

Time in second Fig. 18. Change in ACE1 for case 4.

Table 1 Gains for different algorithms.

CS PSO GA

K PP1

K II1

K PP2

K II2

K PP3

K II3

0.2674 0.299 0.2198

0.2219 0.232 0.1997

0.2194 0.234 0.1986

0.1841 0.168 0.1587

0.1818 0.192 0.1876

0.0681 0.088 0.1976

computational time and gets trapped in local minimum solution. Also, PSO suffers from weak local search ability and algorithm may lead to possible entrapment in local minimum solutions. 4. The capability of the developed controllers to compensate the communication time delay and preserve its satisfactory performance is demonstrated. 5. The effectiveness of the developed controllers in terms of various indices and settling time is proved.

Table 2 Values of performance indices.

CS PSO GA Integral

IAE

ITAE

ISE

ITSE

4.1453 4.2616 5.0199 6.462

28.9463 31.2114 44.7268 91.7536

0.4882 0.4892 0.5297 0.5726

2.7914 2.8225 3.5032 4.276

Appendix (a) The parameters of system under study are given below [17]: T P1 = T P2 = T P3 = 20 s; T t1 = T t2 = 0.3 s; T r1 = T r2 = 10 s; T 12 = T 31 = T 23 = 0.545 p.u; T g1 = T g2 = 0.08 s; K P1 =K P2 =K P3 = 120 Hz/p.u MW; B1 = B2 = B3 = 0.425 p.u MW/Hz; a12 =a31 =a23 =-1; R1 =R2 =R3 =2.4 Hz/p.u MW; T W = 1 s; K r1 =K r2 =0.5 P.u MW; f =60 Hz; K d =4.0; K i = 5.0; K p =1.0. (b) CS parameters: Max generation = 100; Number of nests = 50; P a =0.25. (c) GA parameters: Max generation = 100; Population size = 50; Crossover probabilities = 0.75; Mutation probabilities = 0.1. (d) PSO parameters: Max generation = 100; Population in swarm = 50; C1 = C2 = 2; x = 0.9.

3. CS outperforms PSO, GA in solving LFC problem due to only one parameter required to fine-tune. Thus, it has the priority to reach the optimal solution rapidly. On the other hand, GA deals with a population of solutions, thus leading to the disadvantage of requiring a great number of function evaluations, large

Table 3 Settling time of each variable for different controllers and operating conditions. Area 1 demand change

CS PSO GA Integral ANFIS [17] ANN [21] Fuzzy [21]

All areas change

Df 1

Df 3

DACE2

17.94 18.41 21.49 48.97 36.9 – –

18.14 18.87 31.88 50.84 62.7 – –

17.26 17.68 20.71 69.88 62.7 – –

DP 2

Df 1

Df 3

DACE2

DP 3

20.79 21.46 46.75 59.8 55.2 – –

25.89 26.11 27.71 56.15 31.6 35 50

26.29 26.74 27.49 63.29 32.4 45 50

17.26 18.36 20.71 69.94 62.7 – –

38.4 42.18 47.88 68.74 66.3 45 50

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