Abstract—This paper presents a comparative performance study of different variants of particle swarm optimization techniques for economic load dispatch problem. Economic load dispatch is one of the fundamental issues in power system operation. Many evolutionary techniques such as particle swarm optimization (PSO), differential evolution have been applied to solve this problem and found to perform in a better way in comparison with conventional optimization methods.
But often these methods converge to a sub-optimal solution prematurely. Several variants of particle swarm optimization have been developed in the recent years to address the issue of premature convergence. This paper also introduces a new variant of PSO called New Improved Self Adaptive PSO (NISAPSO) to avoid premature convergence for economic load dispatch problem. A five-unit test system is considered to demonstrate the effectiveness of the proposed method and a comparison study is made with other variants of PSO.
Index Terms— Economic load dispatch (ELD), premature convergence, Time varying acceleration coefficient (TVAC), New improved self adaptive particle swarm optimization (NISAPSO).
I. INTRODUCTION
Economic load dispatch is one of important tasks in the power system operation and planning. The main objective of the economic load dispatch (ELD) is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints. Thus, it is a large-scale highly nonlinear constrained optimization problem. The cost of power generation from fossil fuel plants is very high and economic dispatch can be used as tool in saving a significant amount of revenue. Traditionally, a Lagrangian augmented function is first formulated [1] – [2], and the optimal conditions are obtained by partial derivation of this function. Different mathematical as well as classical optimization techniques methods have been applied to solve the ELD problem. These include dynamic programming (DP), quadratic programming (QP), linear programming, homogenous linear programming, nonlinear programming techniques and interior point method [3]–[6].
In order to make numerical methods more efficient artificial
intelligent techniques, like the Hopfield neural networks, have been successfully employed to solve ELD problems [7].
In the recent year several heuristic techniques such as genetic algorithms (GA) [8]-[9], evolutionary programming (EP) [10], artificial neural network (ANN) [11] have been successfully applied to solve economic load dispatch problems. Wong et. al. [12] presented simulated annealing (SA) based optimization technique for ELD problem. An ant colony optimization (ACO) based method was proposed by Song et al. [13] and presented promising results. Coelho et al. [14] applied differential evolution [DE] based algorithm for solving large scale economic dispatch problem with generator constraints. More recently, Bhattacharya et al. [15] presented a bio-geography based optimization technique to solve complex ELD problem. A chaotic differential evolution based method was presented by Lu et al. [16] for solving dynamic ELD problem and presented promising results. Recently, Cuckoo search algorithm was successfully applied to solve economic dispatch problem by several researchers [17]-[18].
Particle swarm optimization (PSO) is one of the modern heuristic algorithms and has gained lots of attention in various power system optimization problems. Kennedy and Eberhart [19] originally developed the PSO concept based on the behavior of individuals (i.e. particles or agents) of a swarm or group. The PSO technique has been applied to various fields of power system optimization. Yoshida et al.
applied PSO to control reactive power and voltage considering voltage security arrangement [20] and have presented promising results. Abido proposed a revised PSO technique for optimal design of voltage stabilizer [21].
Gaing used PSO to solve economic dispatch problem considering generator constraints with line loss [22]. Park et al. presented a method for solving economic dispatch with non-smooth cost functions without considering line loss [23].
Since the introduction of Particle swarm optimization (PSO), several variants to PSO have been proposed to improve the convergence rate and to avoid the premature convergence [24]. This paper presents a comparative performance study of several variants PSO for economic load dispatch problem. A new improved self adaptive
Comparative Performance Study of Different Variants of Particle Swarm Optimization Techniques for Economic Load Dispatch
S. Mandal1, K.K. Mandal2#
1Department of Electrical Engineering, Jadavpur University, Kolkata- 32,
2 Department of Power Engineering, Jadavpur University, Kolkata-98,
International Journal of Advanced Engineering Science and Technological Research (IJAESTR) particle swarm optimization technique (NISAPSO) with
time-varying acceleration coefficients (TVAC) is also proposed in this paper to address the problem of premature convergence in ELD problems. In this case, the inertia weight is made self adaptive in terms of fitness function and number of particles [25]. A relatively high value of the cognitive component results in excessive wandering of particles while a higher value of the social component causes premature convergence of particles. Hence, time- varying acceleration coefficients (TVAC) [24] are employed to strike a proper balance between the cognitive and social component during the search. The proposed approach was applied on a simple test system to determine its effectiveness. The results have been compared with other PSO variants. It is found that it is capable to avoid premature convergence and can produce better results.
II. PROBLEM FORMULATION
In this section, we describe the problem formulation for economic load dispatch (ELD) problem. The primary objective of ELD problem is to minimize the total fuel cost of the generating units and to meet the system demand under several operating constraints. The fuel cost curve for any unit is assumed to be approximated by segments of quadratic functions of the active power output of the generator for simplicity. Thus, the problem may be described as the minimization of the total fuel cost as defined by (1)
1)
( 2
1
i gi i gi n
i i
g aP bP c
P
FC
where, FC(Pg)is total fuel cost of generation in the system ($/hr), ai,bi,ci are the fuel cost coefficients of the i th generating unit, Pgi is the power generated by the ith unit and n is the number of thermal units.
The cost is minimized with the following constraints.
(i) Generator upper and lower limit
The power output of each generating unit must be greater than or equal to the minimum power permitted and also be less than or equal to maximum power permitted on that specified unit. This can be defined as follows.
2max min
gi gi
gi P P
P
where, Pgimin is the minimum power generation by ith unit, Pgimaxis the maximum power generation by the ith unit.
(ii)
Power balanceTotal power generated by all the units must be equal to power demand plus losses and the same can be expressed as
31
L D n
i
gi P P
P
where, PDis the total power demand and PLis the total transmission loss. The transmission lossPL can be calculated by using B matrix and is defined by (4)
41 1 j ij n
i n
j i
L PB P
P
where, Bijs are the elements of loss coefficient matrix.
III. OVERVIEW OF SOME PSO STRATEGIES
In this section, we briefly describe some PSO techniques that are commonly employed to solve optimization problems.
A. Classical PSO
The Particle Swarm Optimization (PSO) is one of the recent developments in the category of heuristic optimization technique. The method is based on the backgrounds of artificial life and is inspired by the natural phenomenon of fish schooling or bird flocking. Kennedy and Eberhart [22] originally developed the PSO concept based on the behaviour of individuals (i.e. particles or agents) of a swarm or group. PSO, as an optimization tool, provides a population-based search procedure in which individuals called agents or particles change their position with time. In a PSO algorithm, the particles fly around the multidimensional search space in order to find the optimum solution. Each particle adjusts its position according to its own experience and the experience of neighbouring particle.
PSO is basically based on the fact that in quest of reaching the optimum solution in a multidimensional space, a population of particles is created whose present coordinate (position) determines the objective function to be minimized. After each iteration the new velocity and hence the new position of each particle is updated on the basis of a summated influence of each particle’s present velocity, distance of the particle from its own best performance achieved so far during the search process and the distance of the particle from the leading particle, i.e. the particle which at present is globally the best particle producing till now the best performance i.e. minimum of the objective function achieved so far.
Let in a physical d-dimensional search space, the position and velocity of the ith particle (i.e. i th individual in the population of particles) be represented as the vectors
i i id
i x x x
X 1, 2,..., and Vi
vi1,vi2,...,vid
respectively. The previous best position of the i-th particle isrecorded and represented as
i i id
i pbest pbest pbest
pbest 1, 2,..., . The index of the best particle among all the particles in the group is represented by the gbestd. The modified velocity and position of each particle can be calculated using the current velocity and the distance from pbestid to gbestd as shown below:
5 ,
, 2 , 1 , , , 2 , 1
2 1
1
g p
k id d k
id id k
id k id
N d
N i
X gbest rand c X pbest rand c V w V
where Np
is the number of particles in a swarm or group, Ng
is the number of members or elements in a particle,
k
Vid is the velocity of individual i at iteration k, w is the weight parameter or swarm inertia, c1 , c2 are the acceleration constant and rand
is uniform random number in the range [ 0 1]. The updated velocity can be used to change the position of each particle in the swarm as depicted in (6) as:
61
1
idk idk
k
id X V
X
Suitable selection of inertia weight w provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution.
In general, the inertia weight w the inertia weight w is set according to the following equation:
7max min
max max iter
iter w w w
w
where itermaxis the maximum iteration number and iteris the current iteration number.
B. Concept of Self Adaptive Inertia Weight (SAIW) The inertia weight w is utilized to adjust the influence of the previous velocity on the current velocity, and also to balance between global and local exploration capabilities.
The diversity in the population decreases during the end of the optimization procedure. Consequently, the velocities of the particles gradually decrease. To prevent this situation, the utilization of different inertia weights for different particles can play an important role. In Classical PSO, particle with better fitness value is closer to global optimum.
For the particle with a better fitness, it is required to impose stronger local exploration ability in order to search its local surrounding region in a better way. On the other side, the particle with inferior fitness, it is required to impose stronger global exploration ability in order to search the entire solution space in a better way.
The swarm size also plays an important role in finding the optimal value. When the swarm size is very small, the PSO algorithm may not be able to search the entire solution space effectively due lack of searching agents. On the other side, very large swarm size may slow down the convergence rate.
In order to address the above issues, inertia weight can be adjusted according to the swarm size and fitness function.
This can be implemented as follows [25].
3exp /200 /1002
1
8 p r
w
where p is the population size and r is the fitness rank of the particle. In this paper, we refer this method as self adaptive inertia weight particle swarm optimization
(SAIWPSO).
C. Concept of Time-Varying Acceleration Coefficients (TVAC)
It is observed from (5) that the search toward the optimum solution is heavily dependent on the two stochastic acceleration components (i.e. the cognitive component and the social component). Thus, it is very important to control these two components properly in order to get optimum solution efficiently and accurately. It has been reported [25]
that in PSO, problem-based tuning of parameters is a key factor to find the optimum solution accurately and efficiently. Kennedy and Eberhart [19] reported that a relatively higher value of the cognitive component, compared with the social component, results in excessive roaming of individuals through a larger search space. On the other hand, a relatively high value of the social component may lead particles to rush toward a local optimum prematurely. They also reported each of the acceleration coefficients be set at 2, in order to make the mean of both acceleration constants in (5) equal to one, so that particles would over fly only half the time of search.
In general, for any population-based optimization method like PSO, it is always desired to encourage the individuals to wander through the entire search space, during the initial part of the search, without clustering around local optima. In contrast, during the latter stages, it is desirable to enhance convergence towards the global optima so that optimum solution can be achieved efficiently. For this, the concept of parameter automation strategy called time varying acceleration coefficients (TVAC) had been introduced [24].
The main purpose of this concept is to enhance the global search capability during the early part of the optimization process and to promote the particles to converge toward the global optimum at the end of the search. For this, the cognitive component should be reduced while the social component should be increased during search procedure. In TVAC, this can be achieved by changing the acceleration coefficients with time. With a large cognitive component and small social component at the beginning, the particles are encouraged to move around the search space, instead of moving towards the population best prematurely. On the other hand, during the latter stage of optimization, a small cognitive component and a large social component encourage the particles to converge towards the global optimum. The concept of time varying acceleration coefficients (TVAC) can be introduced mathematically as follows [24].
1
9max 1 1
1 f i Ci
iter C iter C
C
2
10max 2 2
2 f i Ci
iter C iter C
C
where C1i,C1f ,C2i,C2f are constants representing initial and final values of cognitive and social acceleration factors respectively. Hence onward, we call this as time varying acceleration coefficient particle swarm optimization
International Journal of Advanced Engineering Science and Technological Research (IJAESTR) (TVACPSO).
D. New Improved Self Adaptive Particle Swarm Optimization (NISAPSO) with TVAC)
It is seen that the classical PSO is either based on a constant inertia weight factor or a variable inertia factor. It is reported that the particles in classical PSO may converge to a local minimum prematurely due to lack of diversity in the population, particularly for complex problems. The above situation may be avoided and can be implemented as follows:
2 2
11max 2 2
1 1 max 1 1 1
k id d i
i f
k id id i
i f k id k
id
X gbest rand iter C
C iter C
X pbest rand iter C
C iter C V w V
Here, w is varied according to (8) depending on the current population size and fitness of the particle.
IV. RESULTS AND DISCUSSIONS
The proposed algorithm based on NISAPSO was implemented using in house Matlab code on 3.0 MHz, 2.0 GB RAM PC. To demonstrate the effectiveness and feasibility of the proposed algorithm, it was applied on a test system consisting of five generating units [26]. The system data including minimum and maximum generation limits and cost coefficients are shown in Table I. The B- coefficients are as shown below.
TABLEI.UNIT DATA FOR FIVEUNIT SYSTEM
Unit min
Pgi Pgimax ai bi ci
1 10 75 0.0080 2.00 25
2 20 125 0.0030 1.80 60
3 30 175 0.0012 2.10 100
4 40 250 0.0010 2.00 120
5 50 300 0.0015 1.80 140
B = [0.000049 0.000014 0.000015 0.000015 0.000020 0.000014 0.000045 0.000016 0.000020 0.000018 0.000015 0.000016 0.000039 0.000010 0.000012 0.000015 0.000020 0.000010 0.000040 0.000014 0.000020 0.000018 0.000012 0.000014 0.000035]
The performance of PSO based algorithm is quite sensitive to the various parameter settings. Tuning of parameters is essential in all PSO based methods. Based on empirical studies on a number of mathematical benchmark functions [22], it has been reported the best range of variation as 2.5–0.5 forC1 and 0.5–2.5 forC2
. The idea is to use a high initial value of the cognitive coefficient to make use of full range of the search space and to avoid premature convergence with a low social coefficient. We experimented with the same range and the best results were obtained for 2.5 – 1.1 forC1
and 1.1 – 2.5 forC2
out of 50 trial runs. The optimization is done with a randomly initialized population of 40 swarms. The maximum iteration was set at 500. The number of iteration was increased in
step of 50 and no significant improvement in result was noticed beyond 500. Hence maximum number of iteration was set at this value. Large number of population is used to allow the algorithm to search the solution space thoroughly.
Table II shows the comparative results in terms of fuel cost and losses for NISAPSO, TVACPSO, SAIWPSO and Classical PSO for the demand of 250 MW, 550 MW and 850 MW. It is seen from the Table II that NISAPSO produces better results in terms of fuel cost and losses.
Table III shows the results for optimized fuel cost, generation schedule, losses and CPU time for a demand of 250 MW, 550MW and 850 MW obtained by NISAPSO. The randomness of the proposed method has been verified by testing with same demand for several times.
TABLEIICOMPARISON OF DIFFERENT VARIANTS OF PSO
Methods 250 MW 550 MW 850 MW
NISAPSO Fuel Cost ($./hr)
848.02 1524.10 2283.10 Losses
(MW)
0.8678 3.7054 14.2421 TVACPSO Fuel Cost
($./hr)
848.60 1524.50 2284.20 Losses
(MW)
0.8780 3.8023 14.3643 SAIWPSO Fuel Cost
($./hr)
849.57 1524.92 2286.34 Losses
(MW)
0.8933 3.9045 15.2361 Classical
PSO
Fuel Cost ($./hr)
850.12 1525.20 2288.52 Losses
(MW)
0.9014 4.0365 15.8026
TABLE III.SOLUTION FOR ECONOMIC LOAD DISPATCHBYNISAPSO
Unit (MW) Demand (MW)
250 550 850
P1 (MW) 10.0000 21.9157 44.3410
P2 (MW) 55.1649 94.1465 125.0000
P3 (MW) 30.0000 97.6160 175.0000
P4 (MW) 62.7293 181.1937 250.0000
P5 (MW) 92.9736 158.8335 269.9011
Total Generation (MW) 250.8678 553.7054 864.2421
Losses (MW) 0.8678 3.7054 14.2421
CPU Time (Sec). 4.25 3.24 4.10
Iterations 500 500 500
Fuel Cost ($./hr) 848.02 1524.10 2283.10
. Fig. 1 Convergence characteristics for optimal fuel cost
The convergence characteristic for the proposed algorithm based on NISAPSO is shown in Figure 1. Figure 1 also compares the convergence characteristics for other variants of PSO like classical PSO, self adaptive inertia weight PSO (SAIWPSO), time variant acceleration coefficient PSO (TVACPSO). It is clearly seen that proposed method is capable to avoid premature convergence.
V. CONCLUSIONS
Economic load dispatch (ELD) is one of the important issues in modern day power system operation. The basic objective is to reduce fuel cost while satisfying the several constraints. The economic load dispatch is a complex optimization problem and many optimization techniques and algorithm have been applied to solve it. In this paper, a comparative study of several variants of PSO is presented for economic load dispatch problem. Also a new improved self adaptive particle swarm optimization technique with time-varying acceleration coefficients have been proposed for solving economic load dispatch problem to avoid premature convergence. To evaluate the performance of the proposed algorithm, it has been applied on a test system.
The results obtained by the proposed method are compared with other variants of PSO. It is found that the proposed method is capable to avoid premature convergence and can produce improved results.
ACKNOWLEDGMENT
We would like to thank Jadavpur University, Kolkata, India for providing all the necessary help to carry out this work. This work is supported by the program DRS and University with Potential for Excellence (UPE), Phase II, UGC, Govt. of India.
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