Parameters identification of a photovoltaic module in a thermal system using metaheuristic optimization methods
Abstract
Experimental studies confirm that the obtained electrical power by a conventional photovoltaic PV system is progressively degraded when the temperature of its cells is increased. The watercooled photovoltaic thermal PVT system is therefore proposed to avoid the voltage drop at high temperature. The use of single diode PV/PVT models in simulation software becomes indispensable to analyze its performances where several climatic conditions such as environmental temperature and solar radiation variations should be considered. An optimal set of PV/PVT model parameters are determined through experimental data using two evolutionary computation algorithms; genetic algorithm and particle swarm optimization algorithm. Furthermore, the robustness of the given PV/PVT model should be analyzed. The predicted electrical properties by the proposed PVT model are compared with those given by the conventional PV model at its operating cell conditions and also at several rigid atmospheric conditions.
Keywords
Photovoltaic system Photovoltaic thermal system Modelization Identification Genetic algorithm Particle swarm optimization algorithmIntroduction
The main applications of solar energy can be classified into two categories: thermal and photovoltaic systems. In the nature, only 20% of solar radiations incident on a PV module can increase the operating cell temperature, in which its performances are deteriorated [1].
Consequently, the obtained energy conversion is reduced with order of 0.4–0.5% when environmental temperatures are progressively increased [1]. To avoid this drawback, the overheating problem of the conventional PV cells is solved using the proposed cooling system which drops its cell temperatures to those neighboring the nominal temperature range.
The proposed solar system uses the water in the closed circuit in which its cells are cooled down in high temperatures. The advantages of this system are better heat absorption and lower production cost [2, 3]. Therefore, our study focuses on the comparison between the obtained electrical powers by both conventional PV and proposed PVT systems in different atmospheric conditions.
In the modeling step of actual PV/PVT systems, a good choice of the efficient model ensuring more accuracy of the actual system behavior is a key success factor for several analysis studies [4], such as diagnosis, synthesis and robustness of PV/PVT control law step against sensor noises, model parameter uncertainties and PV output power forecast [5]. Therefore, various electrical circuits’ oriented PV models have been proposed in the literature providing some optimal models where different intrinsic physical phenomena occurred in the electricity generation process. Among them, the equivalent circuit based upon a single diode is the most commonly adopted model for PV cells, accounting for the photongenerated current and the physics of the P–N junction of the PV cell.
In the design phase of single diode PV models, some unknown parameters should be well optimized such as the photogenerated current, the diode quality factor, the series and parallel resistors and others. An optimal set of these parameters is determined through solving an optimization problem which is previously formulated by the designer. Its fitness metric function (to be minimized) presents the mean square error given through discrepancy value between model prediction and actual measurement for each sampling time.
In the recent years, many researchers have been interested in designing efficient single diode PV models using some evolutionary computation algorithms such as GA or PSO algorithm or others [6, 7]. Among them, Askarzadeh et al. identified the PV model parameters using the Bird Mating Optimizer BMO algorithm [8]. Fialho et al. determined these parameters through some analytical approaches where the PV system was linked to the electric grid [9]. Ogliari et al. estimated the model parameters by adopting the particle filter in the conventional PV power output forecast [10]. Soon and Low identified the single diode KC65T PV model given by three unknown electrical components, which were optimized by the PSO algorithm based upon log barrier constraint [11]. These unknown electrical components have been identified by Qin and Kimball from field test data using PSO algorithm in which both total solar irradiance and environmental temperature variations are taken into account [12]. These parameters have been identified from combining the GA by the InteriorPoint Method IPM by Dizqah et al. [13]. Unfortunately, all proposed models are imprecisely described; the actual solar system behaviors when atmospheric conditions are changed in a wide range, particularly at high environmental temperature as well as the robustness of the developed models have not been considered.
This paper investigates the analysis of the above mentioned problem in which two following main contributions are proposed. The first one is to enhance the obtained electrical properties of the conventional PV system, regardless the effect of various atmospheric conditions. The second one is to decrease the obtained sensitivity of model parameters against environmental temperature variations. Therefore, the obtained electrical properties become depending only on the total solar irradiance variations. As a result, the validity of the proposed model will be extended in wide time range for different weathers such as hot and hazy weathers. The latter presents an important capital, especially, in synthesis control laws ensuring a good tracking of maximum power point MPPT.
The current paper starts in “Tools used for optimization” by introducing the mechanism of both GA and PSO algorithm. In “Circuit model of PV/PVT cells”, the design problem of single diode PV/PVT models is formulated. Its model parameters are then determined through experimental data recorded at different operating points in the “Experimental tests and study cases”. Robustness analysis of obtained PV/PVT models is established where other experimental data recorded at high temperatures and different total solar irradiances are taken into account. Finally, the current paper is ended by a conclusion given in “Conclusion”.
Tools used for optimization
GA optimization
The GA is a heuristic method that simulates the biological evolution, browsing the parameter space. The design of set model parameters are changed according to an evolutionary process based upon genetic rules where some chromosomes may be modified (crossover, mutation, selection…, etc.). In the optimization problem, each variable defines a gene in chromosome. However, the set of chromosomes evolves by different operations modeled on genetic laws to an optimal chromosome [6]. The GA algorithm procedure consists of the following steps:
Step 2: Calculate the fitness function for each chromosome.
 a.
Perform reproduction, i.e., select the best chromosomes with probabilities based upon its fitness function values.
 b.
Perform crossover on chromosomes selected in the above step by crossover probability.
 c.
Perform mutation on chromosomes generated in the above step by mutation probability.
Step 4: If the stopping condition is reached or the optimum solution is obtained, the process can be stopped. Otherwise, repeat Steps 2–4 until the stop condition is achieved.
Step 5: Get the optimal solution \( X^{*} \) corresponding to the best fitness function value \( X^{*} = \mathop {\hbox{min} }\nolimits_{{X_{i}^{j} }} (J(X_{i}^{j} ), \forall i, j). \)
PSO optimization
The PSO algorithm consists of the following stepprocedures [16]:
Step 1: Initialize the \( n_{\text{p}} \) particles with randomly chosen position, which should be previously constrained by \( \varOmega \in (X_{ \hbox{min} } , X_{ \hbox{max} } ) \) where \( X_{ \hbox{min} } \le X_{i} \le X_{ \hbox{max} } \). Afterward, evaluate the corresponding objective function at each position. Finally, set the iteration number \( \ell = 0 \) and determine the initial solutions \( X_{i}^{{{\text{best}}, 0}} \) and \( X_{\text{swarm}}^{{{\text{best}}, 0}} \) using Eq. (2). Go to the next step.
Step 2: Check termination criterion. If it is satisfied, the algorithm terminates with the solution. Otherwise, go to the next step.
Step 3: Apply updates (1) and (2) to all particles and evaluate the corresponding objective function at each position again. Afterward, set the iteration number \( \text{ }\ell \text{ } \) \( \text{ }\ell + 1 \) and determine \( X_{i}^{{{\text{best}},\text{ }\ell }} \) and \( X_{\text{swarm}}^{{{\text{best}},\text{ }\ell }} \). Go back to step 2. For simplicity, the termination criterion in step 2 is set as a maximum number of iteration \( \ell_{\hbox{max} } \)
Circuit model of PV/PVT cells
Mathematical model of PV/PVT modules
The following equivalent electrical circuit based on a single diode is commonly used in modeling step of PV/PVT cells:
Values used in equivalent electrical circuit
Parameter  Quantity identification (unity)  Value 

I _{sc}  Short resistor current (A)  2.99 
q  Elementary charge (c)  1.60 × 10^{−19} 
K  Boltzmann’s constant (J/K)  1.38 × 10^{−23} 
V _{oc}  Opencircuit voltage (V)  20.80 
V _{g}  Energy gap (eV)  1.20 
Formulation of the optimization problem
Experimental tests and study cases
The comparative study has been presented here for two solar systems based upon ISOFOTON I50 PV modules. The first one is the conventional PV cell operating without cooling. However, the second one is the proposed PVT cell that previously reinforced against high temperatures by means of the closed water circuit. These solar systems are positioned on the building roof of the applied research unit in renewable energy located in the south of Algeria.
In this study, both PV and PVT panels are inclined by an angle equals to the latitude of the area and each one has two sensors. The first sensor is a Ktype thermocouple which measures the absolute temperature using the Campbell CS215 instrument. The second one is installed to measure the total solar irradiance using the Kipp and Zonen CMP21 pyranometer.
Typical electrical characteristics of PV/PVT modules
Characteristic  Value 

Maximum power \( P_{ \hbox{max} } \)  39.10 W 
Maximum voltage \( V_{ \hbox{max} } \)  14.90 V 
Maximum current \( I_{ \hbox{max} } \)  2.620 A 
Number of cells  36 
PSO parameters
Parameter  Value 

Number of executions of PSO algorithm  20 
Swarm size \( n_{\text{p}} \)  100 
Maximum iteration number \( l_{ \hbox{max} } \)  200 
Inertia factor \( c_{0} \)  0.90 
Cognitive learning rate \( c_{1} \)  0.25 
Social learning rate \( c_{2} \)  1.25 
GA parameters
Parameter  Value  

Number of executions of GA  20  
Population size  100  
Generation number  200  
Reproduction  
Elite count  2  
Crossover  0.8  
Mutation function  Constrain dependent  
Crossover function  Scattered  
Migration  
Direction  forward  
Fraction  0.2 
Note that the GA and PSO algorithm are executed 20 times. After that, the best obtained fitness value is considered to design the single diode PV and PVT models.
Design of PV and PVT models
Design of first PV and PVT models
Identification results of first PV and PVT models
Model parameters  J _{min}  

I _{ph}  n  R _{s}  R _{p}  
PVT  GA  2.0993  1.0000  0.1817  2.8967  5.95 × 10^{−4} 
PSO  2.1626  1.0018  0.1936  2.1599  5.67 × 10^{−4}  
PV  GA  1.9728  1.5734  0.0343  4.7437  5.08 × 10^{−4} 
PSO  2.0180  1.0000  0.1167  3.7548  4.94 × 10^{−4} 
According to Fig. 6, it is easy to observe that the obtained actual power–voltage characteristics are closely matching those determined through the corresponding models. This figure confirms also that the obtained power energy is enhanced by the actual PVT system with a maximal value of \( P_{\text{PVT}} = 26.19 {\text{Watts}} \) given at the voltage \( V = 14.66 {\text{Volts}} \). This maximal power is better than the one provided by the conventional PV system in which its maximal power reaches \( P_{\text{PV}} = 25.9 {\text{Watts}} \) at the voltage \( V = 14.50 {\text{Volts}} \). Note that, this comparison does not reduce the GA efficiency, as it will be shown in the next section.
Design of second PV and PVT models
Identification results of second PV/PVT models
Parameters  J _{min}  

I _{ph}  n  R _{s}  R _{p}  
PVT  GA  2.9822  1.4644  0.3237  2.1342  7.40 × 10^{−4} 
PSO  2.9900  1.0000  1.2007  7.1147  31.70 × 10^{−4}  
PV  GA  2.8508  1.3110  0.4191  3.2142  8.54 × 10^{−4} 
PSO  2.9900  1.0000  1.0825  5.5967  22.30 × 10^{−4} 
According to Table 6, it is easy to observe that the best minimization of the MSE criterion is performed by using the GA.
According to Figs. 8 and 9, the obtained current–voltage characteristics by the second PV and PVT models are matched as close as possible with those given through the actual PV and PVT systems where the GA gives the best models.
For this reason, only the second PV and PVT models based upon the GA are used to compare its power–voltage characteristics with those determined through the actual PV and PVT systems.
Validation of the obtained PV and PVT models
In this section, both first PV and PVT models based upon PSO algorithm and both second PV and PVT models based upon GA are validated in severe atmospheric conditions, which are recorded in July 2015.
Absolute temperatures and total solar irradiances used for PV and PVT models validation
Time  08H30  09H00  11H00  13H30  15H30 

\( G_{\text{a}} ({\text{W/m}}^{2} ) \)  672.6580  686.1710  883.4676  1047.9080  936.4730 
\( T_{\text{a}} (^\circ {\text{C}}) \)  34.09  36.01  37.88  40.92  39.09 
According to Figs. 11 and 12 the maximal powers provided by the actual PV and PVT systems can be arranged as the following histogrammes:
The obtained maximal powers in different weather conditions compared with those given by the proposed models where the best results are mentioned in bold
\( P_{ \hbox{max} } \) (PVT model1) = 26.19 \( {\text{Watt}} \)  \( P_{ \hbox{max} } \) (PV model1) = 25.90 \( {\text{Watt}} \)  

\( G_{\text{a}} \) \( ({\text{W/m}}^{2} ) \)  \( T_{\text{a}} \) \( (^\circ {\text{C)}} \)  Actual \( P_{{{ \hbox{max} }_{\text{PVT}} }} \)  Match ratio \( P_{{{ \hbox{max} }_{{{\text{PVT\% }}}} }} \)  Match ratio error \( \xi_{{{\text{PVT}}\% }} \)  Actual \( P_{{{ \hbox{max} }_{\text{PV}} }} \)  Match ratio \( P_{{{ \hbox{max} }_{{{\text{PV\% }}}} }} \)  Match ratio error \( \xi_{{{\text{PV}}\% }} \) 
672.658  34.09  24.54  93.700  6.3  17.58  67.87  32.124 
686.171  36.10  25.16  96.067  3.93  19.43  75.019  24.981 
\( P_{ \hbox{max} } \) (PVT model2) = 30.48 \( {\text{Watt}} \)  \( P_{ \hbox{max} } \) (PV model2) = 29.75 \( {\text{Watt}} \)  

883.4678  37.88  29.21  95.830  4.166  22.50  75.63  21.008 
936.473  39.09  29.90  98.097  1.706  25.47  85.613  14.387 
1047.908  40.92  29.68  97.375  2.6247  24.02  80.739  19.261 

In high temperatures, the proposed PVT models ensure better robustness properties than those provided by the conventional PV models.

The proposed PVT models have the ability to well model the actual PVT measurement regardless the severe atmospheric conditions.

The proposed cooling system ensures the best electrical powers which become stationary in two different irradiation ranges and independently of temperature variations.
Conclusion
In this paper, the watercooled PVT system is well modeled by two single diode PVT models according to the two total solar irradiance ranges and the absorbed temperature system. The optimal set of the PVT model parameters are identified through experimental data using both evolutionary optimization algorithms such as GA and PSO. The given current–voltage and power–voltage curves by the actual PV and PVT systems are compared to those given by the proposed PV and PVT models in nominal atmospheric conditions. The robustness of the best PV and PVT models are verified in severe atmospheric conditions in which the PVT model becomes more advantageous than the conventional PV one from an energetic point of view. So, the proposed PVT model becomes interesting for practical uses.
Notes
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable suggestions that enhance the technical and scientific quality of this paper.
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