A Complex-Method-based PSO algorithm for the maximum power point tracking in photovoltaic system

Qiang Fu

College of Science and Technology, Ningbo University Ningbo, China

E-mail goodwill99@163.com

Nan Tong College of Science and Technology, Ningbo University

Ningbo, China E-mail tongnan@nbu.edu.cn

Abstract—-In order to deal with problems of premature convergence and slow convergence rate, this paper proposes a Complex-method-based Particle Swarm Optimization algorithm, namely CPSO. At the beginning of the evolution, the PSO is applied to implement global search, and when the group of particles has entered into the local optimum region, the Complex method is used to quickly get the local optimal point, which can effectively improve the ability of local search. The mutation inertia weight is also used in the CPSO to jump out of the local optimum, which can increase the diversity of the population and solve the premature convergence problem. This new algorithm is applied for the maximum power point tracking (MPPT) in the photovoltaic system, and the effectiveness of this algorithm is demonstrated in the experimental findings.

Keywords-Particle Swarm optimization; Complex method; Mutation; Photovoltaic system; MPPT

I. INTRODUCTION

The Particle Swarm Optimization Algorithm is a swarm intelligence-based algorithm in an attempt to locate the global optimal solutions. One of the reasons that PSO is attractive is that there are only a very few parameters that need to be adjusted. Although PSO is still in its infancy, it has been used across a wide range of applications. Generally speaking, the PSO like other evolutionary optimization algorithms is applicable to most optimization problems and circumstances that can be cast into optimization problems.

However, in the PSO there are two major weaknesses. First of all, it is easy to be premature convergence and fall into local minimum points when the PSO handle with high-dimensional complex optimization problems. Second, the local convergence accuracy of the PSO is not high. The algorithm can organize efficient search in the early stages, but later on the convergence rate will slow down or even stagnate and poor of the ability to do local optimization.

To overcome all these problems, the paper presents a novel PSO algorithm based on the Complex method and mutation inertia weight mechanism, which has been applied to tack the Maximum Power Point (MPP) of the photovoltaic (PV) system. The numerical results show the effectiveness and efficiency of the proposed algorithm.

II. MODELING THE PHOTOVOLTAIC CELL SIMULATION

A. simulation model Worldwide the conventional energy sources are rapidly

depleting, while population growth and scientific and technological developments have dramatically increased the global demand for renewable energy resources. In particular, solar photovoltaic energy is attracting a lot of attention, since it is clean, pollution-free, and inexhaustible. Photovoltaic cell is the core component of the PV system, which has the ability to convert sunlight radiation into electrical energy directly.

As shown in Fig. 1, the equivalent circuit of a photovoltaic cell [1] is a current source in parallel with a diode, which includes a series resistance giving a more accurate shape between MPP and the open circuit voltage. According to the characteristics of the circuit and the internal structure of photovoltaic cell, the I-V characteristics of the cell can be described as follows (1) ~ (8):

SR

)1( /)(0 ��� � nkTIRVq

LSeIII (1)

))(1( 10)1( TTKII TLL ��� (2)

)(),1()1( /* nomnomISCTL GIGI � (3)

)/()( 12)1()2(0 TTIIK TSCTSC ��� (4) )/1/1(//3

1)1(001*)/(* TTnkqVn

TgeTTII ��� (5)

)1/( 1)1(

1

/)()1(0 �� nkTqV

TSCTTOCeII (6)

VVS XdIdVROC

/1/ ��� (7)

1)1(

1

/1)(0 */* nkTqV

TVTOCenkTqIX � (8)

Where, : photocurrent of the PV panel; : saturation current of the diode; : series resistance;

LI 0IVSR SIR� : the

voltage across the diode; : the voltage equivalent of temperature; : irradiance; : temperature of panel(

qKT /TG K ); :

Boltzmann constant ( ); : elementary charge ( ); : diode factor (usually between 1 and 2);

: short circuit current; : open circuit voltage, normally provided by PV panel manufacturers.

kK.1 J /10 23�

OC

38�

V

qC19106.1 ��

SCIn

LI I

SR

DIV

2010 Second International Conference on Information Technology and Computer Science

978-0-7695-4074-0/10 $26.00 © 2010 IEEE

DOI 10.1109/ITCS.2010.39

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Figure1. The circuit diagram of the PV model

B. Analyzing the output characteristics of the photovoltaic cell

The photovoltaic cell is by nature a non-linear power source. As it is well known, the MPP of a PV power generation system depends on temperature, irradiance and other external conditions. By using (1) ~ (8), the simulation model of the photovoltaic cell is built up and used to analyze its output characteristics. Fig. 2 shows P-V curves at the same irradiance value (1 ), but at various temperatures (0 75 ), and Fig. 3 shows at constant temperature (25 ), but under increasing irradiance (0.25 1 ).

2/ mkw C�C�

2/ mkwAs seen in Fig. 2 and Fig. 3, under different conditions, there

is a unique point on the curve, called the maximum power point (MPP), at which the photovoltaic cell operates with maximum efficiency and produces maximum output power. At the same irradiance value, the voltage of the MPP decreases as the temperature increases, the same as the maximum output power. Under the conditions of constant temperature and increasing irradiance, needless to say there is any change of the voltage at which the MPP occurs, however, the value of the maximum output power increases obviously.

C. Maximum Power Point (MPP) Analysis In order to obtain mathematic characteristics of the MPP for

the photovoltaic cell, this paper provides a theoretical analysis with (1) ~ (8) as follows:

Form (1), the expression of the output voltage can be obtained:

SL IRIIIIqnkTV ���� )/)ln((*/)( 00 (9) From (1) and (9), the following expression for P-I

characteristic can be obtained: SL RIIIIIIqnkTVIP 2

00 *)/)ln((*/)( ����� (10)

From (10), the first order derivative of I can be obtained as:

)/)(ln((*/)(/ 00 IIIIqnkTdIdP L ���

02))/( 0 ����� SL IRIIII (11)

Obviously, I that fits (11) is the current of the MPP, marked as . maxI

Substituting I = into (9), the voltage of the MPP can be obtained, marked as . Then, the maximum output power

can be calculated as follows:

maxIVmax

maxP

maxmaxmax * IVP � (12) However, (11) is a complex transcendental equation, cannot

be easily solved through conventional methods or the numerical solution.

III. THE COMPLEX-METHOD-BASED PSO ALGORITHM(CPSO)

A. Particle swarm optimization(PSO) In the PSO algorithm [2], the system is initialized as a set of

random solutions. The optimal particle in the group transmits information to other particles, then, the entire search process

follows the current updating optimal solution to gain the goal.

Figure2. PV curves at various temperatures. (Temperature is 0

75 ; Irradiance is 1 2 ) C� / mkw

Figure3. PV curves at various irradiation levels. (Temperature is

25 ; Irradiance is 0.25 1 ) C� 2/ mkwThe position and velocity of each particle are initialized

randomly, and then each particle in the population moves towards the global optimum and individual optimum. Equations are given as follows:

)(*()*)(*()** 211 t

iti

ti

tg

ti

ti xprandcxprandcvwv ������

(13) 11 �� �� t

it

it

i vxx (14)

Where indicates iteration number, is the best

position in the population at time t (global best), represents the best ever position of each particle in the population at time (best individual position), is a random value inside (0, 1), which provides randomness to the movement of the swarm.

t gp

ip

t ()rand

The acceleration coefficients and adjust the velocity

to move towards gp d ip . inertia weight w contro the impact of the previous particle velocity on the current particle velocity. Linear decrease of w an slow the movement of particles, and convenient for convergence when particles move close to .

1cThe

2c an ls

c

gpAfter doing further analysis on (1) and (2), there are two

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major weaknesses of the PSO algorithm. Firstly, when the particles enter into the local optimum areas,

the global optimal value is similar with the individual optimal value, and the inertia weight has been set smaller value to catch the local optimum, so the velocity update for the global best particle through (1) and (2) generates merely a tiny jump for further improvement such that those particles are very unlikely to pull themselves out of the local optimum.

w

Secondly, when the particles get into the local optimum area, according to (1) and (2), particles randomly adjust their velocity and position without certainty direction of search. Therefore, particles will do a slight swing around the optimal point, and difficult to converge on the optimal point.

To overcome these problems, many researchers take further analyses on the PSO algorithm, and propose corresponding improvements: Y. Shi[3] pointed out that the adjustment of inertia weight could help particles to jump out of local optimum area, and then enhanced the capability to reach the global optimum; [4] integrated PSO with hybrid method employed in the genetic algorithms, which greatly increased the diversity of population and improved the ability to locate the global optimal solution; [5] proposed the population of particles was divided into several independent sub-groups with different evolutionary mode, and then cooperated to search for the global optimum. This approach could effectively avoid being trapped into one of the local optima; [6] verified that the simplex algorithm could effectively improve the accuracy of local convergence. All these researches put forward many improvements to enhance the ability of the PSO algorithm, but they are difficult to balance the relationship between jumping out of premature convergence and improving convergence rate.

w

B. Complex Method The Complex method is essentially a deterministic local

search algorithm with high efficiency search ability [7]. It does not need to calculate the gradient of the objective function, so the computation cost is small. It is mainly through reflection, contraction, expansion and other operations to adjust positions of the vertices. The new complex keeps moving and contracting to the optimum point.

When using the Complex method to solve an optimization problem of variables, an initial Complex with

vertices, must first be constructed. All vertices will be evaluated for their objective function values and accordingly sorted to locate the direction of the next search. The vertex with good performance will be located to replace the worst one. The iterative process goes over and over until all vertices occupy almost the same location, and the Complex shrinks to one point. This point represents the optimum vertex that can be found.

n)2n~1( nkk ��

Assume constraint optimization problem:

mnxgtsRxxf

n

n

,,2,10)(..),(min

����

In order to solve the problem above, an initial Complex

configuration with vertices, marked as , should be generated randomly.

)2~1( nnkk ��

),,2,1()( kjX j ��

)( )( jXf ),,2,1( kj �� are the objective function value of

each vertex, in which the best point )(LX and the worst point

)(HX can be obtained according to (15)~(16): )(LX : (15) )},)(( kjXf j � ,2,1 �min{)( )(Xf L �)(HX : (16) )},)(( kjXf j �,2,1�max{)( )(Xf H �

Suppose the central point )(cX of vertices except for k)(HX ,

)(1

11

)( HjXk

k

j

j �

� �

)(X c (17)

The worst point )(HX should be removed from the Complex, and the substituted vertex )R(X could be found according to (18):

)( )()()( Hcc XXX ��� �(R)X (18) (� is the mapping coefficient from the range [1,2]) If < , ))(RX(f )( )(HXf )(RX is the new vertex of the

Complex in the next iteration. Otherwise, an appropriate )(RX should be searched by modifying � through expansion, contraction and other operations to make superior to

. The procedure is repeated to remove the worst point )( )(RXf

)( )(HXf)(HX from each new Complex, and find the new vertex )(RX with better objective function value to construct a new

Complex. The algorithm is terminated when the convergence criterion has been met.

C. The procedure of Complex-method-based PSO(CPSO) After the group of particles enters into the optimal region,

each particle is still in the state of random search without clear direction, and takes a lot of time to reach the optimum. So it is not enough to fundamentally solve the problem of slow convergence rate only by reducing the inertia weight . Therefore, when the particles have entered into the optimal region, the Complex method is adopted in this paper to replace random search process, which could efficiently improve the local search ability of the PSO algorithm, and balance the relationship between local performance and global search ability, namely CPSO. The solution process of the CPSO algorithm is as follows:

w

Step 1: Generate a population of size 6N. Randomly initialize the position and velocity of each particle, and set the convergence criterion.

Step 2: Evaluate the fitness value of each particle, and

record and . if

ip gpStep 3: Determine whether the convergence criterion has

been met. If satisfied, go to step 9; otherwise execute step 4. Step 4: To all of the particles in the population, carry out

following operations: Apply PSO operator for updating position and velocity

of particles.

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Evaluate the fitness of each particle, and update

and at the same time. if ip

gp

Step5: Calculate fitness variance according to (19): 2�2

1

2 �

��

l

i

averi

fff� (19)

Where, is the number of the particles,l �

�l

iiaver f

lf

1

1,

is the normalized scaling factor, which is to limit the size

of . The value of is as follows:

f2� f

�� ���

�others

fffff averiaveri

,11}max{},max{

(20)

Step 6: Estimate the size of . If the algorithm is trapped into the local optimum region, then execute Step 7. Otherwise, go back to step 3.

2�

Step 7: Rank all of the particles based on fitness . Apply

Complex algorithm to the top N particles, and then update with the result.

if

gp

Step 8: According to (21), adjust the inertia weight of remained particles. Go back to step 3.

w

)*6.01(* ��� ww (21) ( � is the mutation factor from the range [1,2])

Step 9: Output , end of running. gp

IV. EXPERIMENTAL SIMULATION AND RESULT ANALYSIS In order to train and test the CPSO algorithm for the MPP

prediction of PV systems, the actual data included irradiation, temperature taken at half an hour intervals from 8:00 to 17:00 on a certain day at China. The CPSO algorithm was used to solve (11) to evaluate the solution of the MPP at each collected time. Each CPSO maintained 30 populations and 100 independent runs and the average statistical data are collected as the final results. At the same time, the ideal values were obtained from the previous simulation model. Comparison between the solutions of the MPP by the CPSO and the ideal values by the model are displayed in Fig. 4.

Fig. 4 illustrates that the CPSO algorithm can effectively track the MPP of the photovoltaic cell under different conditions. The convergence rate is rapid, and the convergence precision is high.

V. CONCLUSION This paper presents a Complex-method-based Particle

Swarm Optimization algorithm, namely CPSO, which integrates the global search ability of the PSO and the local search ability of the Complex method. Considering both requirements of the search speed and population diversity, the novel algorithm effectively solves problems of local search efficiency and premature convergence in the PSO.

Figure 4. the CPSO prediction comparing with the ideal values from the simulation model

The CPSO algorithm is applied to predict the MPP of the

photovoltaic system. The experimental results verify that the proposed algorithm, which can accurately track the movement of the MPP, is a practical and efficient search algorithm.

ACKNOWLEDGMENT The work was supported by Foundation of Ningbo

University(XYL09007, XYL09009); Ningbo Natural Science Foundation (2009A610061); Pre-Research Foundation of college of Science and Technology of Ningbo University (XYY09005), China.

REFERENCES [1] Walker,Geoff, Evaluating MPPT converter topologies using a matlab PV model. Journal of Electrical & Electronics Engineering, Australia, Vol. 21, 2001, pp. 49-55. [2] Kennedy J, Eberhart RC, Particle swarm optimization. Proc. IEEE International Conf. on Neural Networks (Perth, Australia), IEEE Service Center, Piscataway, NJ, 1995, pp. 1942~1948. [3] Shi Y, Eberhart RC, A modified particle swarm optimizer. Proceedings of IEEE International Conference on Evolutionary Computation, Anchorage, AK, 1998, pp. 69–73. [4] Amgeline PJ, Evolutionary Optimization Versus Paticle Swarm Optimization: Philosophy and Performance Differences. Evolutionary Programming VII, UK, Vol. 1447, 1998, pp. 601-610. [5] Lovbjerg M,Rasmussen T K, Krink T. Hybird Particle Swarm Optinmiser With Breeding and Subpopulations. Proc of the third Genetic and Evolutionary Computation Conference,2001. [6] Begambre O,Laier JE, A hybrid Particle Swarm Optimization-Simplex algorithm(PSOS) for structural damage identification. Adcances in Engineering Software, UK, 2009, pp. 883-891. [7] Ssu-yuan Hu and Jung-Ho Cheng, Development of the unlocking mechanisms for the complex method. Computers & Structures, UK, Vol. 83, 2005, pp. 1991-2002

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