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International Review of Electrical Engineering (I.R.E.E.), Vol. 5, N. 6 November-December 2010

Gravitational Search Algorithm for Economic Dispatch with Valve-Point Effects S. Duman1, U. Güvenç1, N. Yörükeren3 Abstract – In recent years, various heuristic optimization methods have been proposed to solve Economic Dispatch (ED) problem in power systems. This paper presents the well-known power system ED problem solution considering valve-point effect by a new optimization algorithm called as Gravitational Search Algorithm (GSA). The proposed approach has been applied to various test systems with incremental fuel cost function taking into account the valve-point effects. The results shows that performance of the proposed approach reveal the efficiently and robustness when compared results of other optimization algorithms reported in literature. Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Gravitational

Search

Algorithm,

Economic

Dispatch,

Valve-Point

Effect,

Optimization

Nomenclature ai, bi, ci ei, fi Fi(Pi) N PD PL Pimin, Pimax

Cost coefficients of the ith unit Constants of the valve-point effect of ith unit Cost of producing real power output ith unit Number of generating units Load demand Transmission loss Lower and upper generation limits of the ith unit

I.

Introduction

In general, economic dispatch (ED) problem is one of the most important in the operation of power systems. The objective of ED problem of electric power generation is to schedule outputs of all generating units so as to meet load demand at minimum operating total fuel cost, subject to equality constraints on power balance and inequality constraints on power outputs. This makes the ED problem a large-scale highly nonlinear constrained optimization problem. Improvements in scheduling of the generator power outputs can lead to very important fuel cost savings [1]-[4]. Previous efforts on solving ED problems have been applied classical mathematical programming techniques such as interior point algorithm, linear programming and dual quadratic programming [5]-[7]. In these mathematical techniques, the main assumption is that the fuel cost curve is considered as monotonically increasing one. However, when the problem is highly nonlinear or has non-smooth cost functions, some of these techniques may not be able to produce good solutions. Manuscript received and revised November 2010, accepted December 2010

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Approximately for the past twenty years, many of researchers have been used heuristic optimization techniques unlike conventional mathematical techniques in solving ED problems in power systems [8-9]. Dakuo et al. a hybrid genetic algorithm approach based on differential evolution algorithm have proposed to solve economic dispatch with valve-point effect [10]. Coelho et al. utilized particle swarm approach based on quantum mechanics and harmonic oscillator potential to solve economic load dispatch with valve-point effects [11]. AlSumait et al. used patern search method to solve this problem [12]. Su and Lin investigated to solve economic dispatch using Hopfield model [13]. Zhisheng proposed quantum behaved particle swarm optimization algorithm for economic load dispatch in power system [14]. Bhattacharya and Chattopadhyay utilized biogeographybased optimization algorithm to solve complex economic load dispatch problems [15]. Hosseini et al. used a novel mathematical-heuristic method for non-convex dynamic economic dispatch [16]. Subramanian and Anandhakumar used to solve dynamic economic dispatch solution using composite cost function [17]. Hooshmand et al. investigated emission and economic dispatch [18]. Recently, a new heuristic search algorithm, namely gravitational search algorithm (GSA), motivated by the gravitational law and laws of motion has been proposed by Rashedi et al. [20]-[21]. They have been applied successfully in solving various nonlinear functions. The obtained results confirm the high performance and efficiently of proposed method in these problems. GSA has a flexible and well-balanced mechanism to enhance exploration and exploitation abilities [21]. The main objective of this study is to present the use of GSA optimization technique to the subject of economic dispatch in power systems. In this paper, the Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved

S. Duman, U. Güvenç, N. Yörükeren

GSA method has been proposed to solve economic dispatch problem with valve-point effect for 3 and 13 units test systems. In general terms, the contribution of this paper is the new efficient GSA approach for ED problem with valve-point effect. The results obtained with the proposed GSA approach were analyzed and compared other optimization results reported in literature. The remainder of this paper is structured as follows: In Section II, the mathematical formulation of ED problem with valve-point is presented. The concept GSA and application of the proposed GSA is explained in Section III. The parameter settings for the test system to evaluate the performance of GSA and the results are discussed in Section IV. The conclusion is drawn in Section V.

II.

Formulation of the Economic Dispatch Problem

( (

Fi ( Pi ) = Fi ( Pi ) + ei sin fi Pi min − Pi

))

( (

Fi ( Pi ) = ai Pi 2 + bi Pi + ci + ei sin fi Pi min − Pi

(5)

))

(6)

where ei and fi are constants of the valve-point effect of generators. Therefore, total fuel cost that must be minimized according to Eq. (7), is modified to [11]: min F =

N

∑ Fi ( Pi )

(7)

i =1

where Fi is the cost function of generator ith ($/h) defined by Eq. (7). The system losses are ignored for all test systems considered in this study. The fuel cost function curves considering without valve effects and with valve effects are shown in Fig. 1.

The classical formulation of the standard economic dispatch problem is an optimization problem of determining the schedule of the fuel costs of real power outputs of generating units subject to the real power balanced with the total load demand as well as the limits on generators outputs. In mathematical terms the problem can be defined as following: min F =

N

∑ Fi ( Pi )

(1)

i =1

where Fi is the total fuel cost for the generator units, which is defined by: Fi ( Pi ) = ai × Pi 2 + bi × Pi + ci

(2)

Fig. 1. The valve-point effect

III. Gravitational Search Algorithm

where PD is the system load demand and PL is the transmission loss, and of each generator generating capacity constraints must be between its minimum and maximum values:

Rashedi et al. proposed one of the newest heuristic algorithms, namely Gravitational Search Algorithm (GSA) in 2009. GSA is based on the physical law of gravity and the law of motion [20, 22]. The gravitational force between two particles is directly proportional to the product of their masses and inversely proportional to the square of the distance between them [19]. GSA a set of agents called masses has been proposed to find the optimum solution by simulation of Newtonian laws of gravity and motion [20]. In the GSA, consider a system with m masses in which position of the ith mass is defined as follows:

Pi min ≤ Pi ≤ Pi max for i=1,2,3, …, N

X i = x1i ,...,xid ,...,xin , i = 1, 2,...,m

where ai, bi, and ci, are cost coefficients of generator i. Pi subject to power balance constraints: D=

N

∑ Pi − PD − PL = 0

(3)

i =1

(4)

The inclusion of valve-point loading effects makes the modeling of the incremental fuel cost function of the generators more practical. This increases the nonlinearity as well as number of local optima in the solution space. The incremental fuel cost function of the generating units with valve-point loadings are represented as follows [4],[10],[11],[22]:

Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved

(

)

(8)

where xid is position of the ith mass in the dth dimension and n is dimension of the search space. At the specific time ‘t’ a gravitational force from mass ‘j’ acts on mass ‘i’, and is defined as follows [19],[22]: Fijd ( t ) = G ( t )

M pi ( t ) xM aj ( t ) Rij ( t ) + ε

(x

d j

( t ) − xid ( t ) )

(9)

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where Mi is the mass of the object i, Mj is the mass of the object j, G(t) is the gravitational constant at time t, Rij (t) is the Euclidian distance between the two objects i and j, and ε is a small constant. The total force acting on agent i in the dimension d is calculated as follows: Fi d ( t ) =

m

∑

rand j Fijd ( t )

where fiti(t) represents the fitness value of the agent i at time t, and the best(t) and worst(t) in the population respectively indicate the strongest and the weakest agent according to their fitness route. For a minimization problem: (18) best ( t ) = min fit j ( t ) j∈{1,...,m}

(10)

worst ( t ) = max

where randj is a random number in the interval [0,1]. According to the law of motion, the acceleration of the agent i, at time t, in the dth dimension, aid(t) is given as follows: F d (t ) (11) aid ( t ) = i M ii ( t )

For a maximization problem:

j =i j ≠ i

Furthermore, the next velocity of an agent is a function of its current velocity added to its current acceleration. Therefore, the next position and the next velocity of an agent can be calculated as follows [22]: vid ( t + 1) = randi x vid ( t ) + aid ( t )

(12)

xid ( t + 1) = xid ( t ) + vid ( t + 1)

(13)

where randi is a uniform random variable in the interval [0, 1]. The gravitational constant, G, is initialized at the beginning and will be decreased with time to control the search accuracy. In other words, G is a function of the initial value (G0) and time (t): G ( t ) = G ( G0 ,t )

G ( t ) = G0 e

−α

t T

(14)

j∈{1,...,m}

best ( t ) = max

j∈{1,...,m}

worst ( t ) = min

j∈{1,...,m}

fit j ( t )

fit j ( t ) fit j ( t )

(19)

(20) (21)

III.1. Simulation Methodology The proposed GSA approach for economic dispatch problem with valve-point effects can be summarized as follows: Step 1. Search space identification. Step 2. Generate initial population between minimum and maximum values. Step 3. Fitness evaluation of agents. Step 4. Update G(t), best(t), worst(t) and Mi(t) for i = 1,2,. . .,m. Step 5. Calculation of the total force in different directions. Step 6. Calculation of acceleration and velocity. Step 7. Updating agents’ position. Step 8. Repeat step 3 to step 7 until the stop criteria is reached. Step 9. Stop. The general steps of the gravitational search algorithm are given in Fig. 2 [19].

(15)

The masses of the agents are calculated using fitness evaluation. A heavier mass means a more efficient agent. This means that better agents have higher attractions and moves more slowly. Supposing the equality of the gravitational and inertia mass, the values of masses is calculated using the map of fitness. The gravitational and inertial masses are updating by the following equations [19], [22]: fit ( t ) − worst ( t ) (16) mi ( t ) = i best ( t ) − worst ( t ) M i (t ) =

mi ( t ) m

∑ m j (t )

(17)

j =1

Fig. 2. Flow chart of the GSA



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IV.

Numerical Results

In order to verify the feasibility and efficiency of the proposed algorithm was tested two test cases for solving ED problem with valve-point effects. These are 3 and 13 units systems ignored transmission loss, including valvepoint loading. The setup for the proposed algorithm is executed with following parameters: m=100 (masses), G is set using in Eq.(15), where G0 is set to 100 and α is set to 10, and T is the total number of iterations. Maximum iteration numbers are 2000 for case studies. In the following the results are presented. IV.1. Test Case I This test case study considered of three thermal units of generation with effects of valve-point as given Table I [24]. In this case, the load demand expected to be determined was PD = 850 MW.

Units 1 2 3

TABLE I GENERATOR DATAS OF THE TEST CASE I Pimax a b c e Pimin 100 600 0.001562 7.92 561 300 50 200 0.004820 7.97 78 150 100 400 0.001940 7.85 310 200

f 0.0315 0.063 0.042

The results obtained for this case study are given Table II, which shows that the GSA has approximately good solution for power demand of 850 MW. The best fuel cost result obtained from proposed GSA and other optimization algorithms are compared in Table III. From the Table III it is clear that GA and PS approaches did not meet the load demand. IV.2. Test Case II This test case study considered of thirteen thermal units of generation with effects of valve-point as given Table IV [11],[24]. The complexity and nonlinearity to solution procedure is increased. The required load demands to be met by all the thirteen generating units are 1800 and 2520 MW. The results obtained for this case study are given Table V and Table VII, which show that the simulation results obtained by GSA for the best solution for power demand of 1800 and 2520 MW respectively. TABLE II RESULTS OBTAINED BY THE PROPOSED METHOD FOR TEST CASE I Units (MW) Proposed GSA 1 300.2102 2 149.7953 3 399.9958 Total Power Output(MW) 850.0013 Total Cost ($/h) 8234.1

TABLE III COMPARISON OF PROPOSED METHOD FOR TEST CASE I Method P1 (MW) P2 (MW) P3 (MW) PD (MW) Cost ($/h) GA [21] 398.700 50.100 399.600 848.400 8222.07 EP [21] 300.264 149.736 400.000 850.000 8234.07 EP-SQP [21] 300.267 149.733 400.000 850.000 8234.07 PSO [21] 300.268 149.732 400.000 850.000 8234.07 PSO-SQP [21] 300.267 149.733 400.000 850.000 8234.07 GAB [24] 8234.08 GAF [24] 8234.07 CEP [24] 8234.07 FEP [24] 8234.07 MFEP [24] 8234.08 IFEP [24] 8234.07 PS [12] 300.2663 149.7331 399.9996 849.9990 8234.05 Proposed 300.2102 149.7953 399.9958 850.0013 8234.1 GSA

Units 1 2 3 4 5 6 7 8 9 10 11 12 13

TABLE IV GENERATOR DATAS OF THE TEST CASE II Pimax a b c e Pimin 0 680 0.00028 8.10 550 300 0 360 0.00056 8.10 309 200 0 360 0.00056 8.10 307 150 60 180 0.00324 7.74 240 150 60 180 0.00324 7.74 240 150 60 180 0.00324 7.74 240 150 60 180 0.00324 7.74 240 150 60 180 0.00324 7.74 240 150 60 180 0.00324 7.74 240 150 40 120 0.00284 8.60 126 100 40 120 0.00284 8.60 126 100 55 120 0.00284 8.60 126 100 55 120 0.00284 8.60 126 100

f 0.035 0.042 0.042 0.063 0.063 0.063 0.063 0.063 0.063 0.084 0.084 0.084 0.084

The best fuel cost result obtained from proposed GSA and other optimization algorithms are compared in Table VI and Table VIII for load demand 1800 and 2520 MW respectively. It appears that the proposed algorithm performs better as the problem becomes larger and more complex. In Figure 3 and Figure 4 show that convergence characteristic curve of the best case with valve point effect for load demand 1800 and 2520 MW respectively. TABLE V RESULTS OBTAINED BY THE PROPOSED METHOD FOR TEST CASE II (1800 MW) Units (MW) Proposed GSA 1 628.3185 2 149.5996 3 222.7492 4 109.8666 5 109.8665 6 109.8665 7 109.8665 8 60.0000 9 109.8666 10 40.0000 11 40.0000 12 55.0000 13 55.0000 Total Power Output(MW) 1800 Total Cost ($/h) 17960.3684

V.

Conclusion

GSA is one of the recently stochastic methods are developed by Rashedi et al. for solving optimization



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problems. In this paper a new stochastic search technique named GSA is proposed to solve the ED problem with valve point effect. Two test cases have been considered, the simulation results demonstrate the effectiveness and robustness of the proposed algorithm to solve ED problem in power systems. Moreover, the results of the proposed GSA algorithm have been compared to those published methods in the literature.

Fig. 4. Convergence of fitness value with valve-point effects for load demand 2520 MW TABLE VIII COMPARISON OF PROPOSED METHOD FOR TEST CASE II (2520 MW) Method Total Cost ($/h) SA[21] 24970.91 GA [21] 24398.23 GA-SA[21] 24275.71 EP-SQP [21] 24266.44 PSO-SQP[21] 24261.05 UHGA [4] 24172.25 GA-MU [27] 24170.755 IGAMU [27] 24169.979 ACO [3] 24169.93 HGA [10] 24169.92 EDSA[28] 24169.92 DE [26] 24169.9177 Proposed GSA 24164.251357

Fig. 3. Convergence of fitness value with valve-point effects for load demand 1800 MW TABLE VI COMPARISON OF PROPOSED METHOD FOR TEST CASE II (1800 MW) Method Total Cost ($/h) CEP [24] 18048.21 PSO [21] 18030.72 MFEP [24] 18028.09 FEP [24] 18018.00 IFEP [24] 17994.07 EP-SQP [21] 17991.03 HDE [23] 17975.73 CGA-MU [25] 17975.34 PSO-SQP [21] 17969.93 PS [12] 17969.17 UHGA [4] 17964.81 QPSO [14] 17964 IGA_MU [25] 17963.98 ST-HDE [23] 17963.89 HGA [10] 17963.83 HQPSO(5) [11] 17963.9571 DE [26] 17963.83 Proposed GSA 17960.3684

The comparison confirms the effectiveness highquality solution, stable convergence characteristic, good computation efficiency and the superiority of the proposed GSA approach over the other techniques in terms of solution quality.

References [1] [2]

[3]

TABLE VII RESULTS OBTAINED BY THE PROPOSED METHOD FOR TEST CASE II (2520 MW) Units (MW) Proposed GSA 1 628.3185 2 299.1993 3 294.5730 4 159.7331 5 159.7331 6 159.7331 7 159.7331 8 159.7331 9 159.7331 10 77.3999 11 77.3999 12 92.3999 13 92.3999 Total Power Output(MW) 2520.08899 Total Cost ($/h) 24164.251357

[4] [5] [6]

[7] [8]


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Authors’ information 1

Department of Electrical Education, Duzce University, Duzce, Turkey. E-mails: [email protected] [email protected] 2 Department of Electrical Engineering, Kocaeli University, Kocaeli, Turkey. E-mail: [email protected]

S. Duman was born in Bandırma, Turkey, in 1981. He received the B.Sc. degree in electrical education from Abant Izzet Baysal University, Bolu, Turkey, in 2008 and M.Sc. degree from the Department of Electrical Education, Duzce University, Duzce, Turkey in 2010. He is currently student of Ph.D. in the Department of Electrical Engineering, Kocaeli University, Turkey. His areas of research include power system transient stability, power system dynamic stability, FACTS, optimization techniques, voltage stability, optimization problems in power systems and artificial intelligent. U. Guvenc was born in Zile, Turkey, in 1980. He received the B.Sc. degree in electrical education from Abant İzzet Baysal University, Bolu, Turkey in 2002, M.Sc. degree from Gazi University, Turkey in 2005 and the Ph.D. degree from Gazi University, Turkey in 2008. He is currently an Assistance Professor in the Department of Electrical Education, Faculty of Technical Education, Duzce University, Turkey. His main interests are in artificial intelligent, power system and image processing. N. Yörükeren received the B.S. degree in electrical engineering in 1985 from the Technical University of Yildiz. She received the M.S. and Ph. D. degrees in 1989 and 1994 from the University of Kocaeli respectively. Since 1994, she has been working as Assistance Professor at the University of Kocaeli.


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