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Procedia Engineering 23 (2011) 828 834

1877-7058 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2011.11.2589

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ProcediaEngineering

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2011 International Conference on Power Electronics and Engineering Application

An Efficient Cultural Particle Swarm Optimization for Economic Load Dispatch with Valve-point Effect

Qun Niu, Xiaohai Wang, Zhuo Zhoua

Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation, Shanghai University,Shanghai, 200072, China

Abstract

This paper presents a hybrid method by combining cultural algorithm (CA) and particle swarm optimization (PSO) to solve the economic load dispatch (ELD) problem with valve-points effect. The proposed method is called SCAPSO. In SCAPSO, a new velocity update strategy is employed to replace the standard one in PSO, which can enlarge the search space and speed up the convergence. SCAPSO not only has the advantages of PSO, but also utilizes the CA to increase the diversity of the population so as to improve the global search ability. To verify the performance of the proposed method, the power system with 40 generating units is tested. Simulation results show the efficiency, convergence characteristic and robustness of proposed SCAPSO.

2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords: Cultural algorithm, Particle swarm optimization, Economic load dispatch, Valve-point effect.

1. Introduction

Economic load dispatch (ELD) is an important optimization task in power system. The objective ofELD is to reduce the total power generation cost, while satisfying all unit equality and inequality constraints [1]. However, the practical ELD with valve-point effect is described as a non-smooth optimization problem which has complex features with heavy equality and inequality constraints. Therefore, many conventional methods are not efficient to solve ELD problems since they are sensitive to initial estimates and easy to converge into the local optimal solution [2].

* Corresponding author. Tel.: +86-21-56334241; fax: +86-21-56334241. E-mail address: [email protected].

Open access under CC BY-NC-ND license.

Open access under CC BY-NC-ND license.

http://creativecommons.org/licenses/by-nc-nd/3.0/http://creativecommons.org/licenses/by-nc-nd/3.0/

829Qun Niu et al. / Procedia Engineering 23 (2011) 828 8342 Qun Niu et al. / Procedia Engineering 00 (2011) 000000

In the recent years, ELD problem with valve-point effect has been solved using various meta-heuristic search algorithms such as evolutionary programming (EP) [3], genetic algorithm (GA) [4], tabu search (TS) [5], deferential evolution (DE) [6] and Particle swarm optimization (PSO) [7-9].

Cultural algorithm (CA) was first introduced by Robert G. Reynlods in 1994 [10]. CA has been applied to many optimization problems, including function optimization, scheduling and data mining etc. Reynolds and Al-Shehri employed cultural algorithm with evolutionary programming to guide decision tree induction in large database. Knowledge-based cultural algorithms were used to solve nonlinear constraints optimization problems [11].

Even though many versions of PSO have ability to find a good quality solution, their global search ability is comparatively weak when compared with other evolutionary optimization methods. Hence to increase the global search ability, the cultural algorithm based on particle swarm optimization has been presented (CAPSO) to solve the continuous optimization problems. A cultural cooperative particle swarm optimization (CCPSO) was proposed for the functional link neural fuzzy network in several predictive applications [12]. A new cultural particle swarm optimization algorithm with feedback mechanism was designed to solve economic load dispatch (ELD) problems in power system [13]. In this paper, we propose a simple cultural algorithm based on particle swarm optimization, namely SCAPSO, to solve the ELD problem with valve-point effect. SCAPSO can enhance the diversity of swarm as well as convergence performance.

The reminding sections of this paper are organized as follows. In section two, the ELD modal is described. In section three, an introduction to CA and SCAPSO is presented. Numerical results are given in section four. Section 5 includes this paper.

2. Problem Description

The economic load dispatch (ELD) is a complex non-linear sub-problem of the unit commitment problem. The ELD problem is to find the optimal combination of power generation that minimizing the total fuel cost while each power unit has to satisfy the system load demand and operation constraints. The ELD problem can be formulated by the mathematical terms as follows:

Minimize ( )1

n

i ii

f F P=

= (1)Where f is the total generation cost ($/h) and iF is the total fuel cost of the i th generator ($/h). Generally, the fuel cost of the generating units is represented in polynomial function.

2( )i i i i i i iF P a P b P c= + + (2)The equality and inequality constraints are given as follows:

Real power balance constraint: 1

N

ui D Lossi

P P P=

= + (3) Where DP is the total active power demand, LossP is total active loss of network. LossP can be calculated

by using B-matrix loss coefficients. It is not considered in this work. Thus, the 0LossP = .Real power generation limit: min maxi i iP P P (4)

Where miniP and maxiP are the minimum and maximum operating limits of the i th generator. To obtain an accurate and practical ELD solution, the valve-point effect is to be considered. The fuel

cost functions of the generation units with valve-point loading contains higher order nonlinearity and

830 Qun Niu et al. / Procedia Engineering 23 (2011) 828 834 Qun Niu et al./ Procedia Engineering 00 (2011) 000000 3

discontinuity. A recurring rectifying sinusoid is added to the basic quadratic cost as given by: ( )2 min( ) sin( )i i i i i i i i i i iF P a P b P c e f P P= + + + (5)

Where ia , ib , ic , ie and if are the fuel cost coefficients of the i th unit with valve-point effect.

3. Proposed Simple Cultural Algorithm with Particle Swarm Optimization

3.1 Cultural Algorithm

Cultural algorithm was first introduced by Robert G. in 1994. The CA is made up with three principal components: a population space, a belief space and the communication protocol. First, there is a population component that contains a set of individuals to be evolved. In this paper, the species are generated by a version of PSO.

Then, the belief space is made up with the knowledge and experience of the individuals. There are five basic knowledge types: normative knowledge, situational knowledge, domin knowledge, history knowledge, and topographical knowledge. In this paper, only normative knowledge is used. The normative knowledge contains the intervals for each domain variable in order to move the new individuals towards to the good solutions, which can be given as:

1 2, , , DNK N N N= i i i i iN l u L U= (6)

Where D is the dimension of decision variables, and il and iu are the lower and upper bounds of i th

variable. iU and iL are the values of the fitness function corresponding to the bound il and iu . Then the normative knowledge is renewed according to the Eq. (7)-(8) under the assumption that the lower bound of i th particle is affected by j th particle and upper bound is affected by k th particle.

, , , ,1 1, ,( ) ( )t t t t t t t tt tj i i j i i k i i k t it tj i k i

i it ti i

if x l or f x L if x u or f x Ux xl u

l lotherwise otherwise+ +

<


3.2 Modified Cultural Particle Swarm Optimization

In this paper, the previous velocity term in the classical PSO is removed and the new velocity update strategy in [14] is employed to generate a new swarm. Using the new velocity, particles track the global history optimal solution and personal history optimal solution. Since the velocity is simpler than the standard PSO, a new hybrid method of CA and PSO is proposed, namely SCAPSO. In the SCAPSO, the belief space guides the population space and increases the diversity of population, which can avoid the local convergence. In return, the population space supplies their elites with the belief space. Though the dual evolution mechanism and mutual influence of the belief space and population space, the SCAPSO not only have advantages of PSO, but also utilizes the CA to improve the ability of global search.

In this paper, SCAPSO is extended to solve ELD by modifying velocity vector and making it simple. The proposed SCAPSO is presented as follows:

Step1: Initialize the population space and belief space according to the limit of each unit. Step2: Handling the inequality and equality constraint. The details of handling procedure are as

follows: Step2.1. The amount of power balance is calculated by Eq. (11), where df is the difference between

the demand and the total output. 1

N

D uii

df P P=

= (11) Step2.2. If 0df = , go to step3. Step2.3. If 0df , the value of df is adjusted by assigning it to a randomly chosen units load so

that the generating constraint is met. If power of units exceeds the boundary constraints, it is modified by the following express:

,min ,min

,max ,max

i i ii

i i i

P if P PP

P if P P

(12)

Step3: Evaluate the fitness of each particle in population space. Step4: Update the particle swarm by means of influence function. The position of each particle is

given by Eq. (10) and the velocity vector is updated as follows:

( )1 1 1 2 2* *( ) * *t t t t ti i i iv c r pbest x c r gbest x+ = + (13) Step5: Adjust position of each particle by step 2. Step6: Evaluate the fitness of updated particle value. Step7: Update pbest and gbest. Step8: Update belief space which gain messages by means of Acceptance function. Step9: Go back to step 3 until the stop criterion is satisfied.

4. Numerical Results

In order to show the effectiveness of the proposed method, 40 generating units with valve-point effect is described in this section. The results have been compared with several previously published approaches, such as EP [3], EP-SQP [9], PSO [9], PSO-SQP [9], MPSO [8], NPSO-LRS [15], GA [16], DE [6], IGA-MU [16] and GA-PS-SQP [17]. According to the experiences of a lot of experiments and literature, the main parameter settings of the methods are given as follows:


SCAPSO: 1 22, 1.6c c= = , 0w = ; PSO: 1 2 2c c= = , 0.8w = ;

PSO-w: 1 2 2c c= = , max 0.9w = , min 0.4w = ; The generation is set to 2000 and the population size is 50 for all the three methods. The software is

written in Matlab language and executed on 2.6 GHz, Pentium(R) Dual-core processor, 2 GHz RAM PC. The case study consists of 40 generating units with quadratic cost functions. The data for the 40

generators are given in [3] and the total demand of the system is 10500MW. The SCAPSO has been executed 50 times. Table 1 summarizes the comparison results including

minimum costs, maximum costs and mean costs obtained by other methods reported in the recent literature. The minimum cost achieved by SCAPSO is lower than many other methods except for DE and GA-PS-SQP. In terms of the mean cost, the proposed method outperforms all the other method except for DE. Fig. 1 shows the best solutions for 50 trials. For this power system, the SCAPSO performs best. Among the three PSO strategies, the SCAPSO achieves the minimum and mean generation cost. Fig. 2 illustrates the convergence characteristic of SCAPSO.

Table1. Comparison of results for the 40 generating units

Methods Minimum cost ($/h)

Mean cost ($/h)

Maximum cost ($/h)

CEP [3] 123488.29 124793.48 126902.89 FEP [3] 122679.71 124119.37 127245.59 MFEP [3] 122647.57 123489.74 124356.47 IFEP [3] 122624.35 123382.00 125740.63 EP-SQP [9] 122323.97 122379.63 PSO [9] 123930.45 124154.49 PSO-SQP [9] 122094.67 122245.25 MPSO [8] 122252.27 NPSO-LRS [15] 121664.43 122209.31 122981.59 IGA-MU [16] 121819.25 DE [6] 121416.29 121422.72 121431.47 GA-PS-SQP [17] 121458.14 122039.48 PSO 122302.32 122766.58 122936.15 PSO-w 121870.67 122224.78 122609.94 SCAPSO 121473.76 121755.02 122201.32

0 10 20 30 40 501.21

1.215

1.22

1.225

1.23

1.235

1.24x 10

5

Iteration

SCAPSOPSOPSO-w

0 500 1000 1500 2000

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34x 10

5

Number of generation

Cos

t($)

SCAPSOPSOPSO-w

Fig. 1. Objective function value for the 40 generating units. Fig. 2. Convergence characteristics for 40 generating units.


5. Conclusion

This paper proposes a novel method based on cultural algorithm (CA) and particle swarm algorithm (PSO) to solve the economic dispatch problems with valve-point effect. Simulations have been performed to verify the performance of the proposed SCAPSO over test system with 40 generating units. The comparison results with other existing relevant approaches show the robustness and proficiency of the developed method. From the comparative study, it can be concluded that the proposed SCAPSO can be effectively used to solve the non-smooth constrained ELD problems.

In the future, efforts will be made to incorporate more realistic constraints to the ELD problem structure and the other optimization problems would be also attempted by the proposed SCAPSO.

Acknowledges

This work is supported by the National Natural Science Foundation of China (60804052), Chen Guang project from Shanghai Municipal Education Commission and Shanghai Education Development Foundation (09CG39), the project of Shanghai Municipal Education commission (12YZ020) and the projects of Shanghai Science &Technology Community (08160512100 & 08DZ2272400).

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