Optimal Power Flow Using Particle Swarm Intelligence Algorithm and Non-Stationary Multi-Stage Assignment Penalty Function

Lijing Zhang1

Information & Network Management Center North China Electric Power University

Baoding, Hebei Province, China, zhanglijing@ncepubd.edu.cn

YuLing Guo2

North China Electric Power University Baoding, Hebei Province, China,

guoyuling18@126.com

Abstract—With the rapid development of national economy, modern industry develops towards a nonlinear, large-scale and totalization direction, resulting in an increasing need for fast, efficient and robust optimization algorithms. The appearance of the intelligent computing methods brings hope to the solutions of these complex issues. This paper introduces the applications of improved artificial fish and improved particle swarm algorithms in flow calculation, as well as the comparison of the two methods' advantages and disadvantages.

Keywords- Swarm intelligence algorithm; Optimal Power Flow; Particle Swarm Optimization; Artificial Fish Swarm Algorithm

I. INTRODUCTION Along with the scale of power system expanding and

some rather large accidents, more and more attention is paid to the operation safety problem of the power system. People will consider the requirement of economic and security issues in a united way urgently. The optimal powerflowbased on a mathematical programming problem as a basic model hasstrong ability on the processing ofconstraints,and it can bring inall kinds of inequality constraints which can be expressed as state variables and control variables .So that it can unify therequirements ofpower system to the economy, safety and power qualityperfectly. This will givestrong powerresearch of the best the trend, at the same time it also put forward higher request to the algorithm tothe optimal power flow.

II. FLOW CALCULATION

A. Optimal Power Flow The Optimal Power Flow is refers to when the structure

of the system parameters and load is given,by optimization control variables found to meet all the specified constraints, and lead to one or more performance indicators of the system, the objective function to achieve optimal flow distribution.

B. The mathematical model of Optimal Power Flow calculation The mathematical model of optimal power flow is as Follows

inf( , ) (1)( , ) 0 (2)( , ) 0

M x ust g x u

h x u��

where x——The vector of dependent variables u——The vector of independent variables

The OPF equality constraints represent the physical and technical conditions of the power system.

Optimal power flow is optimized trend of distribution, must meet the basic flow equations constitute the optimal power flow equality constraints.

( cos sin ) ( )

( cos sin ) ( )

B

B

Ns

p i j i j j i j j ij i

Ns

p i j i j j i j j ij i

p VV G B p i N

Q VV G B Q i N

� �

� �

�

�

�� � � �

� � � ��

�

�

p Vp V�

VVpQp �

Inequality constraints is mainly controlled by the. adjustable control variables within the allowable adjustment range to meet the security of the system is running.

� � �

min max

min max

min max

1, (4)

1, (5)

1, (6)

Gi Gi Gi

Gi Gi Gi

i i i

P P P i

Q Q Q i

V V V l

� � �

� � �

� � �

maxl iS S� (7)

GiP GiQ are each generator node injected active and reactive power. minGiP are active lower and upper limit of the generator, the reactive lower and upper limit. miniV are node i voltage magnitude lower limit and upper limit, maxlS transmission line capacity limits,(3) is the flow equations. (4-6)are generating capacity constraints and node voltage constraints and(7) is for system security constraints.

III. NON-STATIONARY MULTI-STAGE ASSIGNMENT PENALTY FUNCTION

Generally, the OPF problem includes a great dealof equality constraints and inequality constraints. The most common approach for solving the constrainedoptimization problems is use of a penalty function.The constrained problem is transformed to anunconstrained one, by penalizing the constraints and building a single objective function, which in turn isminimized by using an unconstrained optimization algorithm.

2012 Fourth International Conference on Computational and Information Sciences

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DOI 10.1109/ICCIS.2012.197

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In this paper, a non-stationary multi-stage assignment penalty function is adopted. The penalty values are dynamically modified according to equality constraints

g x u and inequality constraints h x u . In OPF problem, a penalty function is defined as F x u =f x u +h k H x u 8

where f( x u is the original objective function of the OPF problem in Eq(1) h k is a dynamically modified value, where k is the algorithm’s current iteration number. In general, h k is set to h k =k, k ,H x u is penalty factor, defined as

( ( , ))

1

( , ) ( ( , )) ( , ) i

mq x u

i ii

H x u q x u q x u ���

�� (9)

� �( , ) max 0, ( , ) 1,2i iq x u g x u i� � (10)

�� �( , ) max 0, , 1,2i iq x u h x u i� � (11) where m n are the number of equality constraints and

inequality constraints respectively. The function ( , )iq x u is a relative violated function of the constraints; ( ( , ))iq x u� is a multi-stage assignment function; ( ( , ))iq x u� is the power of the penalty function. The function h(.),� ()and r (.) are problem dependent.

IV. OPF USING PARTICLE SWARM OPTIMIZATION Particle swarm optimization (PSO) is a

noveloptimization method developed by Eberhart, et al. It isa member of the wide category of Swarm Intelligence methods that traces its evolution to the emergentmotion of a flock of birds searching for food. It uses a number of particles that constitute a swarm. Each particle traverses the search space looking for the global minimum (or maximum). In a PSOsystem,particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and the experience of neighboring particles, making use of the best position encountered by itself and its neighbors. The swarm direction of a particle is defined by the set of particles neighboring the particle and its history experience.

Let x and v denote a particle coordinates positionand its corresponding flight speed velocity in a search space, respectively. The best previous position of a particle is recorded and represented as pBest. The index of the best particle among all the particles in the group is represented as gBest. To ensure convergence of PSO, Clerc indicates that use of a constriction function may be necessary At last, the modified velocity and position of each particle can be calculated as shown in the following formulas:

�1

12

() ( )()

d dd

d

V rand pBest xV k

rand gBest x� ���

� � � �� �� �� �

� � � �� �

(12)

1 1d d dx x v� �� � (13) where d—Pointer of iterations

d x —Current position of particle at iteration d

dv —Velocity of particle at iteration d �—Inertia weight factor

1 2� � —Acceleration constant rand()—Uniform random value in the range [0,1] k——Constriction factor which is a function of

1 2� � as reflected in(14)

2

2 142 4

k� � �

�� � �

Where 1 2, 4� � � �� � �

Suitable selection of inertia weight provides a balance between global and local explorations. In general, the inertia weight is set according to the following equation

max minmax

max

iteriter

� �� � �� � �

15 where

maxiter —The maximum number of iterations,and iter is the current number of iterations

In the above procedures, the particle velocity is limited by some maximum value maxv . This limit enhances the local exploration of the problem space and it realistically simulates the incremental changes of human learning. To ensure uniform velocity through all dimensions, the maximum velocity is as

max max min( ) /v v v N� � (16) where N—A chosen number of iterations

V. SOLVING THE ARTIFICIAL FISH SWARM ALGORITHM MAFSF-BASED OPTIMAL POWER FLOW

The individual's state of artificial fish can be expressed as a vector 1 2 iX (x ,x , x ),� �� where i x is the desire of optimizing control variables; the current location of the center of artificial fish food concentration is expressed as Y=f(x),artificial fish between the individual distance can be expressed as

ij i jd X X� � ; said artificial fish perception of distance for Visual; said artificial fish moving step of the step, said the crowding factor is of δ

A. Behavioral description 1 Foraging behavior Current state of artificial fish xi in its field of view within

the arbitrary choice of a state xj, the current state ix in the optimization process to find the next random state xj comparison, if the state xj better than the statexi to state xj, Maifurther, otherwise re-randomly selected state xj; so repeated attempts until it reaches the set number of times try_number still not satisfy the forward condition, then any move step. The mathematical expressions are:

When Yj>Yi ()( ) /inextk ik jk ik i jx x rand x x X X� � � � (17)

When Yj<Yi ()inextk ikx x rand� � (18)

Where: k=1 2… N jkx jkx and inextkx respectively, the state vector and artificial fish next state vector of the k-th vector of rand() random number between 0.and Step. The following kinds of symbols defined with the same.

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2)Behavior of clusters Suppose the current state of artificial fish behavior for the

set, within sight of ijd Visual� looking for partners’s number

Xi, at fn and Intermediate positioncX ,if /j f iY n Y�� , that

shows the partner center is not crowded and have more food to meet the forward condition, and further toward the location of the partner center before; otherwise perform foraging behavior.

3)Rear-end behavior Suppose the current artificial fish status for the X, Yj

partners as the largest partner in the jdi Visual� field of view,If /j f iY n Y�� , indicating that the partners around the less crowded and has a high food concentration, and then further towards the direction of partners set; otherwise jump implementation of the foraging behavior.

4)Random behavior The behavior is easier to select a state, that is in any

artificial fish’s vision, and then move in that direction, in essence, the foraging behavior of a default.

5)bulletin board In setting a bulletin board of algorithm, an artificial fish

food concentration and near position artificialfish individual's own state records. Each artificial fish in action after a will have their own state and thecorresponding bulletin board, compared with the current is better than bulletin board if state on bulletinboards recorded state update, otherwise don't record, bulletin board keep original state.

B. Algorithm to improve Artificial fish swarm algorithm in the field of view, the

decision to artificial fish access to the surrounding conditions; artificial fish foraging behavior there is a certain blindness because of its foraging behavior randomness. These two causes difficult to improve the accuracy of the optimization. In addition, the crowding factor is not properly handled are likely to cause the plight of falling into local optimum. To this end, the right step and crowding factor to make the appropriate adjustments to improve the accuracy of convergence.

Whether the parameters change as measured using the adjustment factor k and the change in variance σ. Which is defined as follows:

( ) ( )( ) t t n

f t t n X Xf t n �

� �� �

� t t nX Xt tX 19

f(t) mean the second-best when the t-generation f(t-n) mean the second-best when the (t-n) generation.

( ( ), ( ))D f t f t n � � 20 The equation represents the variance of these sub-

optimal value. Based on the above two criteria of step and crowding

factor adjusted, the expression as follows

( ), ( ), , (21), , , (22)

step f step g Kstep step K

� � � � �� � � � �

� � � �� � � �

Where ,� � represents the evaluation coefficient.,given different values according to the specific questions used to control the process of step speed and iteration.

VI. ADVANTAGE ANALYSIS AND COMPARISON OF TWO METHODS

Flow calculation based on PSO algorithm

Flow calculation based on MAFSF

Advantage

Strong robustness, parallel computing features and adaptability, and rapid convergence

Speed optimization, Search performance, not easy to fall into local optimal solution

Disadvantage

Search space is small, easy to fall into local optimal solution

Slow convergence, convergence and low accuracy, slow speed

VII. .CONCLUSION In the introduction of particle swarm optimization

algorithm-based optimal power flow calculation and the calculation of Optimal Power Flow Based on Improved Artificial Fish School, we can find both of them have their own advantages and disadvantages, so we can assume the combination of the two algorithms: the population is divided into two sub-groups,PSO algorithm in one iteration, and ASFA algorithm evolution in the other sub-groups , then calculate the optimal solution of the entire group which they have searched. Not only this algorithm use the PSO algorithm to chase the current global best to ensure the convergence of the algorithm, but also take advantage of the randomness of the ASFA algorithm search, increase the search range , overcome the speed of PSO into a local extreme points and AFSA algorithm slow and so on, thus taking the optimization of the algorithm accuracy and efficiency into account, and improve the algorithm to optimize performance.

[1] Poa-Sepulveda C A, Pavez-Lazo B J. A solution to theoptimal power

flow using simulated annealing.Electrical Power & Energy System2003 25 1 47 47

[2] Momoh J A, Zhu J Z. Improved interior point methodfor OPF problems. IEEE Transactions on Power System 1999 14 31114 1120

[3] Alsac O, Stott B. Optimal load flow with steady state security. IEEE Trans. on Power Apparat.Syst. 1974 5 170 195.

[4] Zhao Dongmei,ZhuoJunfeng . Survey of algorithms for optimal power flow problem)[J] Modern Electric Power , 2002, 19(3) : 28- 34.

[5] .Dommel H W,Tinney W F.Optimal power flow solution[J].IEEE Trans on Power Apparatus and Systems1968,87(10):1866-1876.

[6] Price J E Sheffin A Adapting Califomia senergy markets togrowth in renewable resources Power and Energy Society GeneralMeeting 2010 IEEE 2010[C] 25—29 July 2010IEEE

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