A precise mathematical model plays a pivotal role in the simulation, evaluation, and optimization of photovoltaic (PV) power systems. Different from the traditional linear model, the model of PV module has the features of nonlinearity and multiparameters. Since conventional methods are incapable of identifying the parameters of PV module, an excellent optimization algorithm is required. Artificial fish swarm algorithm (AFSA), originally inspired by the simulation of collective behavior of real fish swarms, is proposed to fast and accurately extract the parameters of PV module. In addition to the regular operation, a mutation operator (MO) is designed to enhance the searching performance of the algorithm. The feasibility of the proposed method is demonstrated by various parameters of PV module under different environmental conditions, and the testing results are compared with other studied methods in terms of final solutions and computational time. The simulation results show that the proposed method is capable of obtaining higher parameters identification precision.
Due to the increasing price of fossil fuels and their possible depletion, the growing trend to use renewable energy sources has gained attention in recent years [
It is well known that numerous solar cells are connected in series and parallel to form a PV module. Accurate solar cell modeling has aroused sufficient attention over the past years. The output curves of a solar cell exhibit nonlinear and multivariable characteristics determined by the solar cell parameters that describe its model under different operating conditions [
During the last two decades, many methods for parameters identification of solar cells have been proposed. There are two main methods used in the literatures to solve the parameters identification for solar cell models: the traditional and intelligent optimization methods. Newton’s method is commonly used in identifying the parameters by mathematical equations [
Motivated by the behaviors of fish swarm, a novel heuristics called artificial fish swarm algorithm (AFSA) is introduced into this topic for the first time [
The rest of this paper is arranged as follows. Section
An accurate mathematical model describing the electrical characteristics of solar cell is needed in advance. So far, there are several equivalent circuit models which are proposed to simulate voltagecurrent (
Generally speaking, an ideal solar cell model under illumination is a photogenerated current source connected in parallel with a rectifying diode. Nevertheless, the fact that the current source is also shunted by another diode modeled the space charge recombination current and a shunt leakage resistor to take into account the partial short circuit path near the cell’s edges because of the semiconductor impurities and nonidealities. Moreover, a resistor is connected in series with the cell shunt elements due to the solar cell metal contacts and the semiconductor material bulk resistance. The equivalent circuit of this double diode model is shown in Figure
Equivalent circuit of a double diode model.
In this double diode model, the terminal current of solar cell
Taking the Shockley equation into consideration, the two related diode currents are illustrated in (
Substituting (
As can be seen, there are seven unknown parameters to be estimated for such a solar cell model, namely,
Due to the simplicity and accuracy, the single diode model is also used widely to represent the solar cell behavior. The concept of this model is inspired by combining together both diode currents, under the introduction of a nonphysical diode ideality factor
Equivalent circuit of a single diode model.
The representation of this model can be formulated as follows:
In this model, there are five unknown parameters to be identified, namely,
In conclusion, the double diode model significantly improves the accuracy but at the expense of additional parameter calculation. The single diode model is known to have a reasonable tradeoff between simplicity and accuracy under normal weather conditions. Therefore, this paper employs the single diode model to identify the parameters of PV module.
The PV module is formed in the way that parallel connection arrays consisting of solar cells are first connected in series and then connected in parallel. A blocking diode is connected in series with each PV string to prevent excess current produced by other strings from flowing back in a failed string. In series strings, a bypass diode is connected across each PV module or number modules, providing energy releasing route to prevent power mismatching losses even when partially shaded or soiled. Furthermore, the mathematical expression of the terminal equation related to the currents and voltages of a PV module with
However, for the PV module, it should be noted that the identified parameters show a good association with the experimental data; they cannot be exactly related to the physical phenomena due to the differences between the solar cells connected to form the modules.
It is obviously noted that (
In order to extract the parameters of PV module by the collected
Moreover, the new objective function that sums RMSE for any given set of measurements is defined as
In this paper, the objective function expressed in (
During the process of AFSA optimization, the objective function is introduced to be minimized with respect to the parameter ranges. The upper and lower boundaries of each parameter, provided by the literature survey, are tabulated in Table
Upper and lower ranges of the PV module parameters.
Parameter  Lower bound  Upper bound 


0  0.01 

0  10 

0  10 

0  1 

1  2 
In theory, the accurate values of unknown parameters are obtained, along with the objective function which will be close to zero. That is to say, in order to make the simulation results better fit the actual values, the objective function in (
According to the above analysis, the parameters identification for solar cell models belongs to a class of NPhard problems and also is very difficult to solve. Artificial fish swarm algorithm (AFSA), a novel intelligent algorithm, was first proposed in 2003 [
In a water area, fish are most likely distributed around the region where foods are the most abundant. A fish swarm completes its food foraging process by taking several simple social behaviors. It is found that there are three most common fish behaviors: (1) searching behavior, that is, fish tend to head towards food; (2) swarming behavior, gregarious fish tend to concentrate towards each other while avoiding overcrowding; (3) following behavior, the behavior of chasing the nearest buddy. Inspired by swarm intelligence, AFSA is an artificial intelligent algorithm based on the simulation of collective behavior of real fish swarms. It simulates the behavior of a single artificial fish (AF) and then constructs a swarm of AF. Each AF will search its own local optimum, pass on information in its selforganized system, and finally achieve the global optimum.
Suppose that the searching space is Ddimensional and there are
In the initial state of the algorithm, the variable of trial number should be defined as the trial times of AF searching for food. Then, the following steps are described the fish swarm behaviors.
Searching behavior of AFSA.
Swarming behavior of AFSA.
Following behavior of AFSA.
In the process of AFSA, searching behavior lays the foundation for the AF; swarming behavior enhances the convergence of stability; following behavior ensures the convergence of quickness; behavior selection guarantees the high efficiency and stability of the algorithm. Through the behavior selection, the AFSA can form an optimization strategy with high efficiency. The procedure of the AFSA is shown in Algorithm
Procedure Artificial_Fish_Swarm_Algorithm
AF_initialization()
while the result is satisfied do
switch (AF_foodconsistence())
case value 1
AF_Following();
case value 2
AF_Swarming();
default
AF_Searching();
end switch
AF_move()
get_result();
end while
end Artificial_Fish_Swarm_Algorithm
The AFSA has the ability to grasp the searching direction and avoid falling into the local optimum. But when some fish move in aimless randomly or gather around the local optima, the convergence speed will be slowed down greatly and the searching accuracy is greatly reduced. To avoid premature convergence, an intelligent MO similar to the genetic algorithm is employed to enhance the ability escaping from the local optima in this paper [
Randomly select one of the variables in the position to plus one and choose the nonnull to minus one.
If the value of state is better than that of the current state, update the state of position; otherwise, go back to step (1) until the initial number of the mutating operation is satisfied.
By adding the mutation mechanism into the standard AFSA, it achieves the aim of altering the state of each AF. Through adjusting the swarms, the rate of convergence and the global searching ability of AFSA are both improved. The selection of mutating probability will have a great influence on the performance of the proposed algorithm, which has a positive correlation with the elapsed time. According to the experimental experience, the probability of mutation operator (PMO) selected as
The steps of the proposed algorithm used in this work to obtain the optimal parameters of solar cell models can be summarized as follows.
Initialize the state of fish swarm, such as
The values of the objective function for all individuals are evaluated, and the best individual is assigned to the bulletin board.
The state of each fish is updated by the behaviors of searching, swarming, and following. A new fish swarm is generated.
All the individuals are reevaluated. If one individual is superior to the bulletin board, it would replace the individual on the bulletin board.
Add the mutation operator to the standard AFSA.
Evaluate objective function and refresh the bulletin board.
Steps
The best individual of the fish swarm is selected as the optimal solution to the solar cell parameters identification.
In conclusion, the flowchart of identifying the PV module parameters is shown in Figure
Flowchart of identifying PV module parameters with the improved AFSA.
The improved AFSA technique is proposed to identify the parameters of PV module in this section. The efficiency of the improved AFSAbased parameters identification method is verified by identifying the experimental data of PV module under different irradiance and temperature conditions. Comparisons with other optimization algorithms for identification are also presented for the experimental data, which is generated using the single diode PV module model.
Since we cannot practically guarantee optimal parameters for the experimental PV module, we consider the results with the minimum RMSE, defined in (
In order to study the performance of the proposed AFSA, four nonlinear functions have been used to investigate its optimization capacity, as shown in (
Comparisons of convergence speed and precision among the five algorithms in four benchmark test functions.
Comparisons of convergence speed and precision among the five algorithms in (
Comparisons of convergence speed and precision among the five algorithms in (
Comparisons of convergence speed and precision among the five algorithms in (
Comparisons of convergence speed and precision among the five algorithms in (
The iteration times
The simulation results demonstrate that four improved AFSAs can effectively enhance the convergence speed and optimizing precision by comparison with the standard AFSA. Meanwhile, the four improved AFSAs have the ability to avoid the problem of premature convergence. However, because the basic ideas of the four improved AFSAs are slightly different, the optimization effect may appear to have significant difference, especially in solving different optimization problems.
For LAFSA, the AFs gather around the extreme point in the later period of convergence, while the leaping step at this moment changes in a relative large range, so it does not avoid the problems of erratic fluctuation in the nearby of the optimal value. Therefore, it is difficult to obtain more accurate results of approximation by LAFSA.
For CAFSA, as evolution continues, the fish swarm tends to simplification. The effect of new individual AF generated by crossover operation might gradually disappear; the iteration process of AFSA will terminate soon. When the group size
For GAFSA, when there exists an extraordinary local extremum (the fitness value of this local extremum is significantly bigger than others), this individual AF will be selected repeatedly with the action of adding global information in the position increment vector. Fish swarm in the following generation would be soon brought under control by the extraordinary individual AF, which results in low competitiveness of fish swarm. And the AFs may stay stuck in local extreme point easily.
Meanwhile, for MAFSA, the history of best AF is retained, and the other AFs mutate in a given probability. In this way, the best individual AF will not be lost in convergence process, and also the searching scope is enlarged. As a consequence, the MAFSA ensures population stability and diversity and increases the accuracy and convergence speed.
In comparison with the other improved AFSAs, it can be seen that the new AFSA with MO has no obvious deficiency and increases the possibility of searching the global optimum. Hence, it is feasible to identify the unknown parameters for PV module based on the AFSA with MO.
In order to verify the effectiveness of proposed algorithm, several experiments were implemented on the established outdoor test platform on the roof of the Experiment Building in Hohai University Changzhou Campus (latitude 31.82 N, longitude 119.98 E) as shown in Figure
Outdoor test platform of PV modules on the roof.
The actual PV module temperature is relatively higher than the temperature of ambient temperature. So the following equation is used to transform PV module temperatures from measured ambient temperatures:
Meanwhile, it is assumed that the environmental conditions of all solar cells connected in one module are identical. The NewtonRaphson (NR) method in MATLAB software programmed by Mfile is utilized for the parameters identification. For all the evolutionary algorithms, the population size
During the parameter identification process for the PV module, the values of the objective function in different optimization algorithms are shown in Table
Comparison of different methods for parameter identification with experimental data (TSM250PC05A,
Parameter  Method  

NR  GA  PSO  ABSO  AFSA  MAFSA  

0.0047  0.0025  0.0021  0.0029  0.0028  0.0028 

4.8973  4.9996  2.6191  4.9104  5.0017  5.0094 

8.7984  8.7746  8.8808  8.7948  8.7950  8.7949 


0.1044  0.8180  0.0922  0.1020  0.1021 

0.9256  1.5123  1.5226  1.3416  1.3488  1.3488 
RMSE  0.1517 





Moreover, in order to further evaluate the accuracy and effectiveness of the MAFSAbased parameters identification, the parameters identified by the standard AFSA and AFSA with MO are shown in Figure
Convergence process of standard AFSA and AFSA with MO for parameters identification.
For further effectiveness of the proposed algorithm, the MAFSA is evaluated by using this type of PV module operating under different irradiance and temperature conditions. In this paper, four different irradiance conditions (
Parameters identification for the TSM250PC05A PV module under different irradiance and temperature conditions (
Parameter  TSM250PC05A  












0.0028  0.0028  0.0028  0.0028 

4.9989  5.0020  6.6044  6.6075 

8.7950  8.8889  3.5180  3.5556 

0.1010  0.9470  0.1010  0.9470 

1.3483  1.3417  1.3484  1.3417 
RMSE 




A comparison of
As can be seen, the RMSE values of the testing PV modules are much lower, which indicates that the best objective function value can be obtained at each iteration process.
In addition, the
A comparison of
A comparison of
Here, we report not only the number of function evaluations but also the computational time. The reason is that the computational time can reflect the efficiency of algorithms more comprehensively than the number of function evaluations in direct search methods.
However, how to fairly compare the GA, PSO, SA, and AFSA is a problem, as their mechanisms and parameters are different. Firstly, for all the evolutionary algorithms, the population size is set to 30 numbers and the maximum iteration number is set to 100 in Section
Comparison of the computational time for parameters identification with other methods.
Parameter  Method  

GA  PSO  ABSO  AFSA  MAFSA  
Time/s  42  59  104  89  67 
RMSE 





From Table
In addition, to ensure the impartiality of comparisons, the stop criteria of all the evolutionary algorithms are equal to the iterative precision with
For GA,
For PSO,
For ABSO,
For AFSA,
For MAFSA,
Obviously, we can see that the individual code computing for different algorithms is 50 × 200 times, 30 × 100 times, 50 × 500 times, 50 × 100 times, and 30 × 100 times, respectively. The computational time of the MAFSA is in direct proportion to the computational complexity. The corresponding computational times of the algorithms are 87 s, 64 s, 205 s, 77 s, and 53 s, respectively.
In the iterative processes, the PSO and ABSO for parameters identification have a large distribution of results, which cannot guarantee consistency in the identified solutions. Meanwhile, the MAFSA generated variation in a relatively small range and the standard deviation are small and tolerable. It is demonstrated that the proposed algorithm has a better quality of solution and robustness for parameters identification of PV module. In other words, the proposed method is able to effectively obtain the parameters of PV module and thus can always estimate the parameters with good accuracy and consistency.
This paper proposes parameters identification for PV module based on an improved AFSA. The feasibility of the proposed algorithm has been verified by identifying the parameters of one commercial PV module with single diode model under different operating conditions. The simulated current values are in good agreement with the experimental data, which mean that the proposed algorithm has high precision and fast convergence speed. Comparing with the other methods from the literatures, the results obtained by the MAFSA are quite superior and promising.
As a result, the improved AFSA algorithm is a useful way and can be efficiently applied to parameters identification for various types of PV modules. In future work, the fault diagnosis for PV modules will highlight the importance of this accurate parameters identification.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to Associate Professor Kun Ding and Changzhou Key Laboratory of Photovoltaic System Integration and Production Equipment Technology. This work is supported by Graduate Education Innovation Project in Jiangsu Province (no. CXZZ12_0228).