2014 IEEE International Conference Power & Energy (PECON)
Grey Wolf Optimizer for Solving Economic Dispatch Problems 1
L.I. Wong, 2M.H. Sulaiman, 3M.R.Mohamed, 4M.S. Hong Faculty of Electrical & Electronics Engineering (FKEE) Universiti Malaysia Pahang, 26600, Pekan, Pahang, Malaysia 1
[email protected],
[email protected],
[email protected],
[email protected] may cause to suboptimal operation and huge revenue loss over time[9].
Abstract—This work proposes a new meta-heuristic called Grey Wolf Optimizer (GWO) which inspired by grey wolves (Canis lupus). The GWO algorithm mimics the leadership hierarchy and hunting mechanism of grey wolves in nature. Four types of grey wolves such as alpha, beta, delta, and omega are employed for simulating the leadership hierarchy. In addition, the three main steps of hunting, searching for prey, encircling prey, and attacking prey, are implemented. The algorithm is then benchmarked on 20 generating units in economic dispatch, and the results are verified by a comparative study with Biogeography-based optimization (BBO), Lambda Iteration method (LI), Hopfield model based approach (HM), Cuckoo Search (CS), Firefly, Artificial Bee Colony (ABC), Neural Networks training by Artificial Bee Colony (ABCNN), Quadratic Programming (QP) and General Algebraic Modeling System (GAMS). The results show that the GWO algorithm is able to provide very competitive results compared to these well-known meta-heuristics.
Hence, to solve the ED problem by using meta-heuristic optimization techniques have become very popular over the last two decades especially Genetic Algorithm (GA), Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) which have been applied in various fields of study. There are four reasons meta-heuristic have become remarkably common. There are simplicity, flexibility, derivation-free mechanism, and local optima avoidance [10]. First, they have been inspired by simple concepts with respect to physical phenomena, animals’ behaviors, or evolutionary concepts. Second, flexibility refers to the applicability of meta-heuristics to different problems without any special changes in the structure of the algorithm. Third, the majority of meta-heuristics have derivation-free mechanisms. In contrast to gradient-based optimization approaches, metaheuristics optimize problems stochastically. Finally, metaheuristics have superior abilities to avoid local optima compared to conventional optimization techniques. This is due to the stochastic nature of meta-heuristics which allow them to avoid stagnation in local solutions and search the entire search space extensively. Thus, the new meta-heuristic, GWO proposed by S. Mirjalili [10] is implemented in solving ED problems.
Keywords— Economic Dispatch; Grey Wolf Optimizer; Loss minimization; Meta-heuristic technique;
I. INTRODUCTION Optimization problems are widely encountered in various fields in science and technology. Sometimes such problems can be very complex because of the actual and practical nature of the objective function or the model constraint. ED (Economic dispatch) is one of the most important optimization problems in power system operation and planning by scheduling of generators to minimize the total operating cost and to meet load demand of the power system over some appropriate period while satisfying various equality and inequality constraint. The ED basically considers the load balance constraint beside the generating capacity limits. However, in practical ED, ramp rate limits as well as prohibited operating zones (POZ), valve point effects, and multi-fuel option must be taken into the account to provide the completeness for the ED problem formulation [1].
II. ECONOMIC DISPATCH PROBLEMS The Objective of Economic Dispatch is to minimize the fuel cost while satisfying several equality and inequality constraints. Hence, the problem is formulated as below. A. Economic Load Dispatch Formulation The primary concern of ED problem is to minimize of its objective function. The objective function is formulated as below, where Ft is total fuel cost, N is number of generating unit and Fi (PGi) is operating fuel cost of generating unit i. N
Over the past few years, a number of approaches have been developed for solving the ED using classical mathemathical programming methods [2-8]. However, conventional method failed to solve the problem because they are highly sensitive to starting points and frequently converge to local optimum solution or diverge altogether. Besides, conventional method usually have simple mathematical model and high search speed. But, it will use approximation to search for the algorithms that have the required characteristics. This
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min( FT ) = min ∑ Fi ( PGi )
(1)
i =1
B. Minimization of Fuel Cost The generator cost curve is represented by quadratic functions and the total fuel cost F(PG) in (RM/h) can be expressed as:
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2014 IEEE International Conference Power & Energy (PECON)
N
F ( PGi ) = ∑ ai + bi PGi + ci PGi2
(2)
i =1
where N is the number of generators; ai, bi, ci are the cost coefficients of the i-th generator and PG is the vector of real power outputs of generators and defined as:
PG = [PG1 , PG 2 ,..., PGN ]
3) Fig. 1. Hierarchy of grey wolf (dominance decreses from top)
C. •
Constraints
The leader can be a male or a female, called alpha. The alpha’s decisions are dictated to the pack. However, some kind of democratic behavior has also been observed which means the alpha is not necessarily the strongest but the best in term of managing the pack. Hence, it shows that the discipline of the pack is much more important than its strength.
Power Balance/Equality Constraint
The total generated power must cover the total power demand PD and the real power of transmission loss, Ploss which can be defined as: N
∑P
Gi
− PD − Ploss = 0
The second level in the hierarchy of grey wolf is beta. He or she probably is the best candidate to be the alpha in case one of the alpha wolves passes away or becomes very old. The beta wolf plays the role of an advisor to the alpha and discipliner for the pack. The beta strengthens the alpha’s commands throughout the pack and gives feedback to the alpha.
(4)
i =1
To achieve accurate economic dispatch, the transmission loss can be formulated by B-matrix method. N
N
N
Ploss = ∑∑ Pi Bij Pj + ∑ Bi 0 Pi + B00 i =1 j =1
(5)
The third level is delta. Delta wolves have to submit to alphas and betas, but they dominate the omega. Scouts, sentinels, elders, hunters, and caretakers belong to this category. Scouts are responsible for watching the boundaries of the territory and warning the pack in case of any danger. Sentinels protect and guarantee the safety of the pack. Elders are the experienced wolves who used to be alpha or beta. Hunters help the alphas and betas when hunting prey and providing food for the pack. Lastly, the caretakers are responsible for caring for the weak, ill, and wounded wolves in the pack.
i =1
where, Pj = the output generation of unit j (MW). Bij = the ij-th element of the loss coefficient square matrix. Bi0 = the i-th element of the loss coefficient. B00 = the loss coefficient constant. •
Generation Capacity/Inequality Constraint
The lowest ranking grey wolf is omega. The omega plays the role of scapegoat. They are the last wolves that are allowed to eat. It may seem the omega is not an important individual in the pack, but it help to maintain the dominance structure of the entire pack. In some cases the omega is also the babysitters in the pack.
For stable operation, the real power output of each generator is restricted by lower and upper limits as follows:
PGimin ≤ PGi ≤ PGimax
i = 1,2,..., N
(6)
Group hunting is another interesting social behavior of grey wolves. According to Muro et al. [11] the main phases of grey wolf hunting are as follows:
III. GREY WOLF OPTIMIZER (GWO) In this section the inspiration of the proposed method is first discussed. Then, the mathematical model is provided.
• Tracking, chasing, and approaching the prey. • Pursuing, encircling, and harassing the prey until it stops moving. • Attack towards the prey
A. Inspiration Grey wolves (Canis lupus) are considered as predators, meaning that they are at the top of the food chain. They live in group approximately 5-12 on average. The particular interest is they have a very strict social dominant hierarchy as shown in Fig. 1.
This hunting techniques and the social hierarchy of grey wolves are mathematically modeled in order to design GWO. IV. MATHEMATICAL MODEL AND ALGORITHM In this section, the mathematical models of social hierarchy, tracking, encircling, and attacking prey are provided.
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2014 IEEE International Conference Power & Energy (PECON)
V. METHODOLOGY A. Social Hierarchy We consider the fittest solution as the alpha (α). Consequently, the second and third best solutions are named beta (β) and delta (δ) respectively. The rest of the candidate solutions are assumed to be omega (ω). In the GWO algorithm the hunting (optimization) is guided by α, β, and δ. The ω wolves follow these three wolves.
Firstly, a set of candidate for solution, Xsa,ng is initialized. This comprises of the number of generations of the system that will be optimized which resulted a minimum cost by fulfilling all the constraints. The variables of the optimal ED are expressed as follows:
B. Encircling Prey When the wolves do hunting, they tend to encircle their prey. The following equations depicted the encircling behavior [5]: →
→
→
→
→
→ →
(8)
X (t + 1) = X p (t ) − A⋅ D
where D is position of each hunter from ω or any other →
hunters, t is the current iteration, X is the position vector of →
→
grey wolf, Xp is the position of the prey and A and C are coefficient vectors calculated as below: →
→ →
→
A = 2 a ⋅ r1 − a →
n
(9)
F = ( F ) + PF * abs[( ∑ PGi ) − PD − PLoss ]
→
(10) Where r1 and r2 are random vectors [0, 1] and is linearly decreased from 2 to 0 over the course of iterations. The three best solutions (X1, X2, and X3) are saved including the latest positions of omegas according to the current best position. X1 is the best position of α, X2 is the best position of β, and X3 is the best position of δ. The final position X(t+1), is defined by the positions of alpha, beta, and delta in the search space. These situations are expressed in the following expressions: →
→
→
→
→
→
→
→
→
→
→
→
→ → → ⎛→⎞ → → → ⎛→⎞ → → → ⎛→⎞ X1 = Xα − A1⋅ ⎜ Xα ⎟, X2 = X β − A2 ⋅ ⎜ Xβ ⎟, X3 = Xδ − A3 ⋅ ⎜ Xδ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ →
→
3
VI. SIMULATION RESULTS & DISCUSSION A system with 20 generators is used to show the effectiveness of GWO. The system data are tabulated in Table 1 [12, 13]. The valve point loading effect is not considered for this system but transmission loss is considered. For this test system load demand is 2500 MW. The results reported in the literature viz. BBO [14], Li [12], HM [12], QP and GAMS [15], ABCNN [16], ABC [17], CS [18] and Firefly [19] are compared with the GWO-based results and the potential benefit of the GWO as an optimizing algorithm for this specific application is established. The simulation results for GWO, CS, Firefly, ABCNN, ABC, BBO, LI, HM, QP and GAMS are tabulated in the Table 2 where the real power generation by each generator unit for the given demand and the total cost are described. It can be observed from Table 2 that the minimum costs achieved by the GWO based method for test system is 60413.0014 $/h. Again, power mismatches are the third least ones in the GWO as compared to others algorithm. It also can be noted that the power generated by GWO are within the range of the minimum and maximum bounds at each generator. Hence, it can be concluded that for all the mentioned test system the performance of the GWO is found to be the best one.
(11) (12) (13)
To sum up, the optimization approach for GWO is starting with creating a random population of grey wolves which can be called as candidates of solution. During the simulation, alpha, beta and delta wolves estimate the possible position of the prey. Exploration and exploitation are guaranteed by the adaptive values of and A. Candidate solutions are diverged →
(15)
The algorithm will continue until the maximum iteration is met and the optimum result is obtained.
→
X 1+ X 2 + X X ( t + 1) = 3 →
(14)
i =1
C = 2 ⋅ r2
Dα = C1⋅ Xα − X , Dβ = C2 ⋅ Xβ − X , Dδ = C3⋅ Xδ − X
." x1,ng ⎤ ⎥ % # ⎥ " xng ,sa ⎥⎦
where sa is the number of search agent and ng is the number of generator plant in the system which is generated randomly for initialization. Eq. (2) was applied in the performance evaluation of the ED problem until the optimum cost is achieved. For inequality constraints, similar to any other techniques, when the solutions obtained for any iteration are out of boundaries, GWO chooses the boundaries values, while for equality constraint, when it is violated, the penalty factor, PF is implemented and embedded in the cost function, as follows:
(7)
D = C ⋅ X p (t ) − X (t ) →
X i, j
⎡ x1,1 ⎢ =⎢ # ⎢x ⎣ 1,sa
→
from the prey if A > 1 and converged towards the prey if A < 1 . Finally GWO algorithm is terminated by the criterion that has been set initially.
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2014 IEEE International Conference Power & Energy (PECON)
TABLE I.
SYSTEM DATA FOR 20 GENERATORS SYSTEM min
max
Unit
PGi (MW)
PGi (MW)
a ($/MW)
b ($/MW)
c ($/MW)
1
150
600
0.00068
18.19
1000
2
50
200
0.00071
19.26
970
3
50
200
0.0065
19.8
600
4
50
200
0.005
19.1
700
5
50
160
0.00738
18.1
420
6
20
100
0.00612
19.26
360
7
25
125
0.0079
17.14
490
8
50
150
0.00813
18.92
660
9
50
200
0.00522
18.27
765
10
30
150
0.00573
18.92
770
11
100
300
0.0048
16.69
800
12
150
500
0.0031
16.76
970
13
40
160
0.0085
17.36
900
14
20
130
0.00511
18.7
700
15
25
185
0.00398
18.7
450
16
20
80
0.0712
14.26
370
17
30
85
0.0089
19.14
480
18
30
120
0.00713
18.92
680
19
40
120
0.00622
18.47
700
20
30
100
0.00773
19.79
850
ACKNOWLEDGMENT This work was supported by Ministry of Education (MOE) Malaysia under Fundamental Research Grant (FRGS) #RDU 130104. REFERENCES [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12]
VII. CONCLUSION The various optimization techniques have been applied to economic problem in this paper. The results obtained show that GWO have been successfully implemented to solve different ED problems besides GWO is able to provide very competitive results in terms of minimizing total fuel cost and lower transmission loss. It has been observed that the GWO has the ability to converge to a better quality near-optimal solution and possesses better convergence characteristics than other prevailing techniques reported in the recent literatures. It is also clear from the results obtained by different trials that the GWO shows a good balance between exploration and exploitation that result in high local optima avoidance. This superior capability is due to the adaptive value of A. It is because half of the iterations are devoted to exploration, →
[13]
[14] [15]
[16] [17]
→
A > 1 and the rest to exploitation A < 1 . Thus, this algorithm may become very promising for solving some more complex engineering optimization problems for future researches.
[18] [19]
153
Sharma, J., & Mahor, A. (2013). Particle Swarm Optimization Approach for Economic Load Dispatch: A Review. 3 (1). Chen CL, Wang CL. Branch-and-bound scheduling for thermal generating units.IEEE Trans Energy Convers 1993;8(2):184–9. Lin CE, Viviani GL. Hierarchical economic dispatch for piecewise quadratic cost functions. IEEE Trans Power Apparatus Syst 1984;103(6):1170–5. Granville S. Optimal reactive dispatch through interior point methods. In: IEEE Summer Meeting, Paper no. 92 SM 416-8 PWRS, Seattle, WA,USA; 1992. Yang HT, Chen SL. Incorporating a multi-criteria decision procedure into the combined dynamic programming/production simulation algorithm for generation expansion planning. IEEE Trans Power Syst 1989;4(1):165–75. Liang ZX, Glover JD. A zoom feature for a programming solution to economic dispatch including transmission losses. IEEE Trans Power Syst 1992;7(3):544–50. Yan X, Quintana VH. An efficient predictor-corrector interior point algorithm for security-constrained economic dispatch. IEEE Trans Power Syst 1997;12(2):803–10. Lin CE, Chen ST, Huang CL. A direct Newton–Raphson economic dispatch. IEEE Trans Power Syst 1992;7(3):1149–54. G. Zwe-Lee, "Particle swarm optimization to solving the economic dispatch considering the generator constraints," IEEE Transactions on Power Systems, vol. 18, pp. 1187-1195, 2003. Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey Wolf Optimizer. Advances in Engineering Software, vol. 69, pp. 46-61, March 2014. Muro C, Escobedo R, Spector L, Coppinger R. Wolf-pack (Canis lupus) hunting strategies emerge from simple rules in computational simulations. BehavProcess 2011;88:192–7. Su, C. T., & Lin, C. T. (2000). New approach with a Hopfield modeling framework to economic dispatch.IEEE Transactions on Power Systems, 15(2), 541–545. Coelho, L. D. S., & Lee, C.-S. (2008). Solving economic load dispatch problems in power systems using chaotic and Gaussian particle swarm optimization approaches. International Journal of Electrical Power & Energy System, 30(5), 297–307. Bhattacharya, A., & Chattopadhyay, P. K. (2010a). Biogeography-based optimization for different economic load dispatch problems. IEEE Transactions on Power Systems, 25(2), 1064–1077. Bisen, Devendra, et al. "Solution of Large Scale Economic Load Dispatch Problem using Quadratic Programming and GAMS: A Comparative Analysis. “Journal of Information and Computing Science 7.3 (2012): 200-211. D. Karaboga, C. Ozturk, Neural Networks Training by Artificial Bee Colony Algorithm on Pattern Classification, Neural Network World, 19(3), 279-292, 2009. D. Karaboga, B. Akay, A Comparative Study of Artificial Bee Colony Algorithm, Applied Mathematics and Computation, 214, 108-132, 2009. X.-S. Yang, S. Deb, Engineering optimization by cuckoo search, Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 1, No. 4, 330-343, 2010. Yang, Xin-She. Engineering optimization: an introduction with metaheuristic applications. John Wiley & Sons, 2010.
2014 IEEE International Conference Power & Energy (PECON)
TABLE II.
TOTAL POWER GENERATON FOR EACH UNIT AND TOTAL COST FOR VARIOUS META-HEURISTIC TECHNIQUES
Unit
GWO
CS
Firefly
ABCNN
ABC
BBO
LI
HM
QP
GAMS
P1
599.9991
599.8830128
599.998
599.9972
599.882
513.0892
512.7805
512.7804
600
512.782
P2
160.1612
148.2233017
147.925
172.4309
172.866
173.3533
169.1033
169.1035
200
169.102
P3
50
50.21865734
50.0011
50
106.993
126.9231
126.8898
126.8897
50
126.891
P4
50.0197
51.00308676
51.1681
50
63.1275
103.3292
102.8657
102.8656
56.92
102.891
P5
93.2573
94.63829957
92.491
115.8288
70.9701
113.7741
113.6386
113.6836
94.28
113.683
P6
26.7292
28.94186485
25.8761
39.5502
52.1022
73.06694
73.571
73.5709
33.72
73.572
P7
125
124.1803253
125
120.0216
119.142
114.9843
115.2878
115.2876
125
115.29
P8
50.0177
50.54153747
50.0286
71.7034
50
116.4238
116.3994
116.3994
60.24
116.4
P9
108.6717
110.8490887
110.066
129.4385
76.3559
100.6948
100.4062
100.4063
103.28
100.405
P10
54.5728
53.14430866
56.4774
30
102.403
99.99979
106.0267
106.0267
79.49
106.027
P11
266.1245
267.8709763
268.714
230.4784
263.905
148.977
150.2394
150.2395
221.14
150.239
P12
414.621
413.2721485
414.467
469.0286
362.23
294.0207
292.7648
292.7647
347.05
292.766
P13
124.3059
126.4160821
122.542
104.1452
123.52
119.5754
119.1154
119.1155
127.38
119.114
P14
70.6027
71.18035605
69.4262
80.0902
47.7657
30.54786
30.834
30.8342
60.29
30.832
P15
98.069
95.1474043
93.9302
59.3637
56.4597
116.4546
115.8057
115.8056
116.7
115.805
P16
36.6912
36.70498828
37.338
34.0204
34.0936
36.22787
36.2545
36.2545
36.25
36.254
P17
30.018
30.73434539
36.9973
41.623
31.4734
66.85943
66.859
66.859
30
66.859
P18
42.6504
47.47232556
48.8782
30
30
88.54701
87.972
87.972
58.21
87.971
P19
81.598
82.57521179
81.7705
55.3963
118.464
100.9802
100.8033
100.8033
85.52
100.803
P20 Total power output (MW) Total transmission loss(MW) Total generation cost ($/h)
30.0049
30.10386566
30.0038
30
30
54.2725
54.305
54.305
30
54.305
2513.10
2513.10
2513.1
2513.1164
2511.8
2592.1011
2591.967
2591.967
2515.48
2591.967
13.1145
13.10118718
13.0987
13.1163
11.7527
92.1011
91.967
91.9669
15.48
91.967
60413.0014
60414.10387
60415
60446.37744
60540
62456.779
62456.6391
62456.6341
62456.63
62456.63
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