An Improved Particle Swarm Optimization Algorithm and Application in Available Transfer Capability

Qing-ran Wang Research Center for Electricity Market North China Electric Power University

Beijing, China wangqingran@163.com

Li-zi Zhang Research Center for Electricity Market North China Electric Power University

Beijing, China lizizhang2001@sina.com

Abstract—With electricity market reformation more deepening, Available Transfer Capability is important public information for transaction participants in the deregulated power industry. Considering the calculation difficulties, the paper analyses the essence of ATC and the defects of existing solution methods, and a new improved particle swarm optimization model is established, the particle swarm optimization algorithm is introduced to reduce the computing time of settling the model which elevates the applicability of the particle swarm optimization algorithm and accelerates its convergence. At last, an improved dissipative operation model is constructed with the theory of dissipation which effectively overcomes the shortcoming of earliness.

Keywords-Available Transfer Capability; particle swarm optimization; economic dispatch; regulation

I. INTRODUCTION With the restructuring of the electric power industry,

many optimization methods have been proposed to evaluate ATC successfully, which include several classic optimal algorithms, such as Interior Point method [1], Newton method [2] and Quadratic programming approach [3]. Some modern intelligence algorithms are also involved, such as Genetic Algorithm (GA) [4] and Particle Swarm Optimization (PSO) [5]. However, dealing with the problem of large-scale nonlinear system, almost all classic optimal algorithms have common defects such as slowly computing speed and unsatisfactory robustness for their single-searching strategy [6]. The Available Transfer Capability (ATC) of an interconnected power system is becoming an important concern of both system planners and operators. Due to the advent of open transmission access, the ATC calculations should produce commercially viable results with a reasonable and dependable indication of the transfer capabilities available to the electric power market.

Comparing with optimization algorithms, PSO is easy to be trapped by local optimum. Besides, the initial value of the particles [7], in some degree, decides whether it could find the global optimal solution. Moreover, in the basic PSO, particle adjusts its velocity only according to its best experiences and that of the best one in the population, without considering other particle’s information [8]. Based on the analysis above, this paper proposes a particle swarm

optimization (PSO) algorithm. The method and process to solve ATC problem are presented.

II. PARTICLE SWARM OPTIMIZATION ALGORITHM

A. Particle Swarm Optimization Algorithm The PSO was first introduced by Kennedy and Eberhart

[9]. It was developed through simulation of social behaviors such as fish schooling and bird flocking. The PSO provides a population-base search procedure in which the individuals, called particles, change their positions with time. Each particle adjusts its position according to its own best experience and the best experience of neighboring particles.

Assuming that the search space is D-dimensional, the i th particle of the swarm is represented by the D-dimensional vector 1 2( , , ... , ..., )i i i id inX x x x x= and the best particle in the swarm, i.e., the particle with the smallest function value, which is denoted by the index bestg . The best previous position of the i th particle is recorded and represented as

1 2( , , ..., )i i i inP p p p= , while the position change (velocity)

of the i th particle is represented as 1 2( , , ..., )i i i inV v v v= which is clamped to a maximum velocity

max max1 max 2 max( , , ..., )nV v v v= specified by the user. Following this notation, the particles are manipulated according to the following equations:

11 2( ) ( )t t

id id id id best idV wv c rand p x c rand g x+ = + − + − (1)

1 1 1,2,... ;t t tid id idx x v d n+ += + = (2)

Where w can be expressed by the inertia weights approach, 1c and 2c are the acceleration constants which influence the convergence speed of each particle, and rand is a random number in the range of [0,1] . The first part represents the inertia of the previous velocity, the second part is the “cognition” part which represents the private thinking by itself, and the third part is the “social” part which

2009 Second International Conference on Intelligent Computation Technology and Automation

978-0-7695-3804-4/09 $26.00 © 2009 IEEE

DOI 10.1109/ICICTA.2009.65

237

2009 Second International Conference on Intelligent Computation Technology and Automation

978-0-7695-3804-4/09 $26.00 © 2009 IEEE

DOI 10.1109/ICICTA.2009.65

237

2009 Second International Conference on Intelligent Computation Technology and Automation

978-0-7695-3804-4/09 $26.00 © 2009 IEEE

DOI 10.1109/ICICTA.2009.65

237

2009 Second International Conference on Intelligent Computation Technology and Automation

978-0-7695-3804-4/09 $26.00 © 2009 IEEE

DOI 10.1109/ICICTA.2009.65

237

2009 Second International Conference on Intelligent Computation Technology and Automation

978-0-7695-3804-4/09 $26.00 © 2009 IEEE

DOI 10.1109/ICICTA.2009.65

237

2009 Second International Conference on Intelligent Computation Technology and Automation

978-0-7695-3804-4/09 $26.00 © 2009 IEEE

DOI 10.1109/ICICTA.2009.65

237

represents the cooperation among the particles. If the summation causes the velocity idv , on that dimension, to

exceed max,dv , then idv will be limited to max,dv . maxV determines the resolution with which regions between the present position and the target position are searched. If maxV is too large, the particles might fly over the past good solutions. If maxV is too small, the particles may not explore sufficiently beyond local solutions. In many experiences with PSO, maxV is often set to maximum dynamic range of the

variables on each dimension. The constants 1c and 2c represent the weighting of the stochastic acceleration terms that pull each particle toward ip and gp positions. Low values allow particles to roam far from the target regions before being tugged back. On the other hand, high values result in abrupt movement toward or passed the target regions. Hence, the acceleration constants 1c and 2c are often set to be 2.0 according to the past experiences. 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.

B. Algorithm Process The steps of standard PSO algorithm are as follows: Step 1: To initialize a group of particles including the

speed and random location; Step 2: To evaluate the fitness of each particle; Step 3: To compare their fitness with the best location of

their experience for each particle, and then update the current location bestg if it is better.

Step 4: To compare its fitness with the best position

bestg of their overall experience for each particle, and then

set bestg newly if it is better. Step 5: Change the speed and location of particles; Step 6: If there is no interruptive condition to be satisfied

(the maximum value or the number of iteration), then return to the step 2.

C. Parameters Setting One advantage of PSO is using real-coded and without

too much adjusted parameters [10]. The basic parameters in formula are: inertia factor, the weight factor 1c 2c , the

maximum number of iteration maxk and population size m . The most important one is inertia factor that could meet the following equation:

max minmax

max.

w ww w k

k−

= − (3)

Where: maxw minw : the largest and the smallest

allowed values respectively; k , maxk : the present and the largest number of iteration.

From the linear relationship between the number of iteration and w we know w could affect the convergence and its rate greatly. After several rounds of tests, we get

max 1w = min 0.2w = max 200k = . When w decline from 1 to 0.3 linearly, PSO search the optimal solution in the large region at the beginning and fast take the target. With gradually decreasing of w , the speed of particles slow down, the local search start. The method accelerates the rate of convergence and improves the performance of the PSO algorithm [11]. Acceleration constants 1c and 2c represent the weight of statistics speed which push particles to

bestg and bestg . According to tests and experience, 1c is

equivalent to 2c and general in scope between 0 and 4. The

paper adopts 1 2 1.5c c= = . It should be noted that in PSO, the “mutation” operator is guided by the particle’s own “flying” experience and also benefits from the swarm’s “flying” experience. In other words, PSO is considered as performing mutation with a “conscience”, as pointed out by Eberhart and Shi [12].

III. ATC MATHEMATIC MODEL For a given operating mode, ATC between two areas can

be described as the maximization the incremental sum of the active power of the loads in receiving area, if various system constrains are satisfied, when the active power of the generators in receiving area and the loads in sending area is constant, meanwhile, the active power of the generators in sending area and the loads in receiving area are increased at the same time [13].

The optimal model of ATC evaluation including objective function, equality constraints and inequality constraints are described as follows:

A. Objective Function

( ) max( )i

BL

if x P

∈Γ= Δ (4)

B. Equality Constraints

1( cos sin ) 0

i i

nj ij ij ij ijG D i

jP P V V G Bθ θ

=− = + = (5)

1( cos sin ) 0

i i

nj ij ij ij ijG D i

jQ Q V V G Bθ θ

=− = − = (6)

238238238238238238

WhereiDPΔ is active power increment of load-bus i ;

iGP and iGQ are respectively active power and reactive

power of generator-bus i ; iDP and

iDQ are respectively active power and reactive power of load-bus i ; n is the number of nodes in system; iV and iθ are respectively

voltage magnitude and phase angle of bus i ; ij i jθ θ θ= −

and ij ijG jB+ are respectively elements in bus admittance matrix of network.

C. Inequality Constraints

min maxi i iV V V≤ ≤ (7)

min maxGi Gi GiP P P≤ ≤ (8)

min maxGi Gi GiQ Q Q≤ ≤ (9)

min maxDi Di DiP P P≤ ≤ (10)

min maxDi Di DiQ Q Q≤ ≤ (11)

Where nS is set of all nodes; GS is set of generator nodes in sending area; DS is set of load nodes in receiving area; superscripts, min and max are respectively power flow value of basic state, lower limit and upper limit of corresponding variables.

IV. APPLICATION EXAMPLE The performance of the constructed model is tested on

IEEE-30 bus test system. The system is partitioned into 3 areas and Fig.1 shows the test system. The results for ATC evaluation between areas are given in table I. Upper-limit and lower-limit of all node voltage are uniformly 1.05pu and 0.9pu.

Figure 1. IEEE-30 bus test system

A. Comparison of Results with IPSO, and TA The value of ATC can be computed respectively by

using PSO and TA, and table I shows the results of comparison.

TABLE I. INPUTS AND OUTPUTS OF DMUS

Partnership of the areas

ATC/MW TA PSO

1-2 103.72 111.78 2-1 40.22 60.12 1-3 110.54 112.32 3-1 55.78 92.89 2-3 29.98 39.45 3-2 49.99 64.61

B. Comparison of Results with PSO and TA Limited to the length of paper, the results for ATC

evaluation from area 2 to area 3 were given merely calculated by PSO and TA.

V. CONCLUSION In this paper, we propose a modified particle swarm

optimization algorithm considering economic dispatch. Penalty function is assigned dynamically according to the amount overrunning the upper-limit or lower limit of inequality restriction during the calculating process, which accelerates greatly the speed of convergence. Moreover, precision and stability of the algorithm are both obviously.

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