Photovoltaic Cell Parameter Estimation Using Hybrid Particle Swarm Optimization and Simulated Annealing

Accurate parameter estimation of solar cells is vital to assess and predict the performance of photovoltaic energy systems. For the estimation model to accurately track the experimentally measured current-voltage (I-V) data, the parameter estimation problem is converted into an optimization problem and a metaheuristic optimization algorithm is used to solve it. Metaheuristics present a fairly acceptable solution to the parameter estimation but the problem of premature convergence still endures. The paper puts forward a new optimization approach using hybrid particle swarm optimization and simulated annealing (HPSOSA) to estimate solar cell parameters in single and double diode models using experimentally measured I-V data. The HPSOSA was capable of achieving a global minimum in all test runs and was significant in alleviating the premature convergence problem. The performance of the algorithm was evaluated by comparing it with five different optimization algorithms and performing a statistical analysis. The analysis results clearly indicated that the method was capable of estimating all the model parameters with high precision indicated by low root mean square error (RMSE) and mean absolute error (MAE). The parameter estimation was accurately performed for a commercial (RTC France) solar cell.


Introduction
Increasing power demands due to continuous population growth and industrial needs, and depleting fossil fuel reserves and environmental concerns have led to the use of renewable energy sources, particularly to solar energy.Solar energy, being pollution free, renewable and freely available, has attracted great attention all around the world.
Photovoltaic (PV) solar cells are used to harvest energy from solar radiation and convert it into electric energy.These cells are made up of semiconductor materials, traditionally silicon.The low efficiency, high cost and physical barriers of silicon limit the use of traditional solar cells.Extensive research has been carried out to improve the conversion efficiency of a solar cell; special attention has been paid to the materials used in the manufacture of solar cells.A new generation of solar cells, known as the 3rd generation solar cells, has evolved, which make use of sustainable materials and flexible architectures like dye-sensitized solar cells (DSSCs).DSSCs operate as an artificial photosynthetic system to convert solar light into electricity, and are reported to provide efficiencies of up to 14% [1].More insights on the materials and coatings employed for efficient DSSCs can be found in [2][3][4].
These cells are connected in series and parallel combinations to construct a solar module.Accurate modeling of solar cells is necessary to evaluate and forecast the performance of the PV systems.Many circuit models have been proposed.Among them, single diode models (SDM) and double diode This paper attempts to mitigate the problem of premature convergence.Each global best solution from PSO undergoes SA to further improve the solution in terms of better objective values.This approach sufficiently eliminated the premature convergence problem and achieved a better solution in less iteration.This paper is organized as follows: Section 2 discusses the solar cell modeling and formulation of the parameter estimation problem.Section 3 provides details of the HPSOSA algorithm.Simulation results are discussed in Section 4 along with analysis on the results.Section 5 provides concluding remarks on the research work.

Photovoltaic Cell Modeling and Parameter Estimation Problem Formulation
This section discusses modeling and mathematical formulation of parameter estimation for SDM and DDM of PV cell.

PV Cell Modeling
A mathematical model that precisely represents the characteristics of the PV cell is indispensable.Among the various proposed PV models, two models are prominent, i.e., SDM and DDM.The PV cell is modeled as a current source with a diode, ideally.In practice, the model is also equipped with a shunt resistance and a series resistance to accumulate partial short circuit current path near the cell's edges due to the semiconductor impurities and non-idealities, solar cell metal contacts and the semiconductor bulk resistance, respectively, whereas DDM of PV cell employs another diode shunted across existing diode to accumulate space charge recombination current [21].
SDM is a most widely used PV model and has been shown in Figure 1.I-V characteristics of SDM are expressed as: From Equation (1), SDM is characterized by five parameters: I ph (photocurrent), I 0 (diode saturation current), R s (series resistance), R sh (shunt resistance), and n (diode ideality factor).Estimation of these five parameters is essential for modeling of PV cell.
Energies 2017, 10, 1213 3 of 13 approach sufficiently eliminated the premature convergence problem and achieved a better solution in less iteration.This paper is organized as follows: Section 2 discusses the solar cell modeling and formulation of the parameter estimation problem.Section 3 provides details of the HPSOSA algorithm.Simulation results are discussed in Section 4 along with analysis on the results.Section 5 provides concluding remarks on the research work.

Photovoltaic Cell Modeling and Parameter Estimation Problem Formulation
This section discusses modeling and mathematical formulation of parameter estimation for SDM and DDM of PV cell.

PV Cell Modeling
A mathematical model that precisely represents the characteristics of the PV cell is indispensable.Among the various proposed PV models, two models are prominent, i.e., SDM and DDM.The PV cell is modeled as a current source with a diode, ideally.In practice, the model is also equipped with a shunt resistance and a series resistance to accumulate partial short circuit current path near the cell's edges due to the semiconductor impurities and non-idealities, solar cell metal contacts and the semiconductor bulk resistance, respectively, whereas DDM of PV cell employs another diode shunted across existing diode to accumulate space charge recombination current [21].
SDM is a most widely used PV model and has been shown in Figure 1.I-V characteristics of SDM are expressed as: From Equation ( 1), SDM is characterized by five parameters: Iph (photocurrent), I0 (diode saturation current), Rs (series resistance), Rsh (shunt resistance), and n (diode ideality factor).Estimation of these five parameters is essential for modeling of PV cell.DDM of a PV cell has been shown in Figure 2. The I-V characteristics of DDM are expressed as: From Equation (2), DDM is characterized by seven parameters: Iph (photocurrent), I01 (diode saturation current for diode D1), I02 (diode saturation current for diode D2), Rs (series resistance), Rsh (shunt resistance), (diode ideality factor for diode D1), and (diode ideality factor for diode D2).Similarly, estimation of these seven parameters is essential for modeling of PV cell.DDM of a PV cell has been shown in Figure 2. The I-V characteristics of DDM are expressed as: From Equation (2), DDM is characterized by seven parameters: I ph (photocurrent), I 01 (diode saturation current for diode D 1 ), I 02 (diode saturation current for diode D 2 ), R s (series resistance), R sh (shunt resistance), n 1 (diode ideality factor for diode D 1 ), and n 2 (diode ideality factor for diode D 2 ).Similarly, estimation of these seven parameters is essential for modeling of PV cell.

Parameter Estimation Problem Formulation
The parameter estimation problem is transformed into an optimization problem to minimize the difference between measured current and calculated current.A performance criterion or objective function is defined for the minimization; RMSE is used as objective function and is given by: where, is the measured current, is the calculated current and N is the number of measured data points.θ is the parameter vector (to be estimated) which has five elements in the case of SDM i.e., = and seven parameters in the case of DDM i.e., = .is the function of Vm, and θ.
A programming model computed and thus RMSE using the , and ; thus the parameters are estimated.The estimated parameters should strictly follow the actual I-V characteristics, ideally.However, there is a difference between the experimental current and the estimated current due to the measurement noise errors.Parameter bounds [22] used in this study have been tabulated in Table 1 for SDM and DDM.

HPSOSA Algorithm for PV Parameter Estimation
This section presents a new algorithm of HPSOSA based on the analysis of PSO and SA.

Particle Swarm Optimization
Particle swarm optimization is a swarm-based metaheuristic optimization algorithm.A swarm of particles (potential solutions) is used in the entire search space to find the solution with optimized (minimized) objective value.The particles are randomly initialized in the search space.The initial position and associated objective values are stored as their personal best solutions.The particle or position with the minimum objective function value is stored as global best.The position and velocity of each particle is updated according to following relations:

Parameter Estimation Problem Formulation
The parameter estimation problem is transformed into an optimization problem to minimize the difference between measured current and calculated current.A performance criterion or objective function is defined for the minimization; RMSE is used as objective function and is given by: where, I m is the measured current, I c is the calculated current and N is the number of measured data points.θ is the parameter vector (to be estimated) which has five elements in the case of SDM i.e., θ = R s R sh I ph I 0 n and seven parameters in the case of DDM i.e., θ = R s R sh I ph I 01 n 1 I 02 n 2 .I c is the function of V m , and θ.
A programming model computed I c and thus RMSE using the V m , I m and I c ; thus the parameters are estimated.The estimated parameters should strictly follow the actual I-V characteristics, ideally.However, there is a difference between the experimental current and the estimated current due to the measurement noise errors.Parameter bounds [22] used in this study have been tabulated in Table 1 for SDM and DDM.

HPSOSA Algorithm for PV Parameter Estimation
This section presents a new algorithm of HPSOSA based on the analysis of PSO and SA.

Particle Swarm Optimization
Particle swarm optimization is a swarm-based metaheuristic optimization algorithm.A swarm of particles (potential solutions) is used in the entire search space to find the solution with optimized (minimized) objective value.The particles are randomly initialized in the search space.The initial position and associated objective values are stored as their personal best solutions.The particle or position with the minimum objective function value is stored as global best.The position and velocity of each particle is updated according to following relations: Energies 2017, 10, 1213 5 of 14 In Equations ( 4) and (5) V is the velocity of the i-th particle, P b is the personal best solution, G b is the global best solution, θ is the position of current solution, c 1 and c 2 are the personal acceleration coefficient and social acceleration coefficient, respectively, ω is the inertia weight, r 1 and r 2 are random numbers ∈ [0, 1].
After changing velocity and position of each particle, objective associated with new position is evaluated and their P b and G b are updated using simple relations shown in Equations ( 6) and ( 7): where J is the objective function to be minimized.In this way G b is computed iteratively until a stopping criterion is satisfied.The stopping criterion may be a certain number of iterations or a predefined error tolerance.

Simulated Annealing
Simulated annealing (SA) is another metaheuristic, proposed by Kirkpatrick et al. in 1983 [23].Since then SA, a point to point based algorithm, has found diverse applications.In SA, a new solution is generated in the vicinity of the previous solution.For a minimization problem, all new solutions are evaluated for an objective function.Solutions that contribute to a minimized objective are accepted; solutions are also accepted which do not minimize the objective but with a certain probability based on the following inequality: In Equation ( 8) r is a random number ∈ [0, 1].θ k is the new solution and θ k−1 is the previous solution, and T SA is the temperature.Before starting SA an initial temperature T 0 and minimum temperature T min is defined.Accepting solutions based on Equation (8) enables SA to escape local minimum in early iterations and reach the global minimum.The temperature is lowered using a cooling schedule; commonly used one is geometric cooling schedule written as: where α is the temperature control factor.

Hybrid Particle Swarm Optimization and Simulated Annealing
The purpose of hybridizing PSO with SA is to alleviate the premature convergence problem.The G b obtained by PSO, at each iteration, is further processed and evaluated by SA.The obtained result thus presents an improved solution by virtue of optimal objective.Following is a description of step-wise procedure for HPSOSA.
Step 1.1: Initialize particles using a random generation system within parameter bounds.
Step 1.2: Evaluate the objective value of all the particles using Equation (3), assign P b and G b of the particles.
Step 2: Start the iteration cycle.
Step 2.1: Update particle velocity according to Equation (4).It is evident from Table 2 that HPSOSA performs better than CPSO in terms of the average, best, worst, standard deviation, and median of the objective values in all 20 runs.The HPSOSA achieved the average, minimum, maximum and median of objective values as low as 7.7301 × 10 −4 .HPSOSA achieved a standard deviation of 4.0768 × 10 −17 ; obviously it is far better than the standard deviation calculated for CPSO.The best values of the estimated parameters of SDM have been tabulated in Table 3 along with RMSE and MAE values.It can be seen that HPSOSA stands out as superior among all the algorithms with RMSE and MAE values as low as 7.7301 × 10 −4 and 6.7818 × 10 −4 , respectively.A measure of robustness of the HPSOSA and CPSO for 20 runs is shown in Figure 4.It is apparent from Figure 4 that the HPSOSA is able to achieve a minimum RMSE value in all 20 iterations and presents a robust solution for the parameter estimation problem.Whereas CPSO reached only once in the proximity of the minimum RMSE, achieved by HPSOSA.Objective function convergence curve for the best run of HPSOSA has been shown in Figure 5.It is evident from the figure that the HPSOSA is able to attain a stable minimum objective value in less than 20 iterations.A further insight of how close the estimated values are with the experimental values has been given in Table 4, Figures 6 and 7. Experimentally measured voltage, current and the error between measured current and estimated current, calculated by = − , have been tabulated in Table 4. Objective function convergence curve for the best run of HPSOSA has been shown in Figure 5.It is evident from the figure that the HPSOSA is able to attain a stable minimum objective value in less than 20 iterations.Objective function convergence curve for the best run of HPSOSA has been shown in Figure 5.It is evident from the figure that the HPSOSA is able to attain a stable minimum objective value in less than 20 iterations.A further insight of how close the estimated values are with the experimental values has been given in Table 4, Figures 6 and 7. Experimentally measured voltage, current and the error between measured current and estimated current, calculated by = − , have been tabulated in Table 4.
Table 4 shows that very low error values portray high precision of the estimated parameters.A further insight of how close the estimated values are with the experimental values has been given in Table 4, Figures 6 and 7. Experimentally measured voltage, current and the error between measured current and estimated current, calculated by e = I m − I c , have been tabulated in Table 4. Table 4 shows that very low error values portray high precision of the estimated parameters. Figure 6 plots experimentally measured I-V data points and I-V data obtained by estimated parameters. Figure 7 plots experimentally measured P-V (power-voltage) data points and P-V data obtained by estimated parameters.Figures 6 and 7 clearly portray that the estimated data is in close agreement with the experimentally measured data.
The RACF result for SDM using the HPSOSA is shown in Figure 8.The estimated SDM qualifies the test as the values are in the range of -1 and +1.The result for SDM using the HPSOSA is shown in Figure 8.The estimated SDM qualifies the test as the values are in the range of -1 and +1.

Results for Double Diode PV Model
For the double diode model of the PV cell, statistics of 20 runs are tabulated in Table 5.It is obvious from Table 5 that the HPSOSA presented better statistics when compared with CPSO.The HPSOSA attains a best RMSE value of 7.4532 × 10 −4 , which is far better than the best RMSE value attained by CPSO.The HPSOSA outperforms CPSO in all means of average, best, maximum, standard deviation and median.The HPSOSA achieved a good standard deviation of 5.8569 × 10 −5 while CPSO achieved a standard deviation of 5.0461 × 10 −4 .

Results for Double Diode PV Model
For the double diode model of the PV cell, statistics of 20 runs are tabulated in Table 5.It is obvious from Table 5 that the HPSOSA presented better statistics when compared with CPSO.The HPSOSA attains a best RMSE value of 7.4532 × 10 −4 , which is far better than the best RMSE value attained by CPSO.The HPSOSA outperforms CPSO in all means of average, best, maximum, standard deviation and median.The HPSOSA achieved a good standard deviation of 5.8569 × 10 −5 while CPSO achieved a standard deviation of 5.0461 × 10 −4 .To further examine that how close the currents are calculated by the HPSOSA with the experimentally measured currents, Figures 9 and 10    Table 7 shows another measure of how close are the estimated model and the experimentally measured data by virtue of error between them.Experimentally measured voltage, current and the error between the experimental current and the calculated current have been listed in Table 7.The very low error is an indication of the accuracy of the HPSOSA.Table 7 shows another measure of how close are the estimated model and the experimentally measured data by virtue of error between them.Experimentally measured voltage, current and the error between the experimental current and the calculated current have been listed in Table 7.The very low error is an indication of the accuracy of the HPSOSA.Table 7 shows another measure of how close are the estimated model and the experimentally measured data by virtue of error between them.Experimentally measured voltage, current and the error between the experimental current and the calculated current have been listed in Table 7.The very low error is an indication of the accuracy of the HPSOSA.The RACF result for DDM using the HPSOSA has been shown in Figure 11.The estimated DDM qualifies the test as the values are in the range of -1 and +1.The result for DDM using the HPSOSA has been shown in Figure 11.The estimated DDM qualifies the test as the values are in the range of -1 and +1.

Conclusions
This paper has presented a hybrid optimization approach using particle swarm optimization and simulated annealing for parameter estimation of photovoltaic solar cell single diode and double diode models.Experimentally measured data of a silicone solar cell (RTC France), measured at an irradiance of 1000 W/m 2 and a temperature of 33 °C, were used to estimate the models.The approach significantly improves the problem of premature convergence.The applied approach is compared with different metaheuristic algorithms, namely CPSO, IABC, HS, SA and PS.The HPSOSA outperformed all the compared algorithms by all means of statistical analysis used in this paper, i.e. average, best, maximum, standard deviation and median.The HPSOSA achieved very low values of RMSE and MAE comparatively.The HPSOSA successfully passed the RACF test and the test values lie within the confidence interval.
Author Contributions: All authors have contributed equally to this paper.The manuscript is submitted with approval of all the authors.

Conflicts of Interest:
The authors declare no conflict of interest.

Conclusions
This paper has presented a hybrid optimization approach using particle swarm optimization and simulated annealing for parameter estimation of photovoltaic solar cell single diode and double diode models.Experimentally measured data of a silicone solar cell (RTC France), measured at an irradiance of 1000 W/m 2 and a temperature of 33 • C, were used to estimate the models.The approach significantly improves the problem of premature convergence.The applied approach is compared with different metaheuristic algorithms, namely CPSO, IABC, HS, SA and PS.The HPSOSA outperformed all the compared algorithms by all means of statistical analysis used in this paper, i.e., average, best, maximum, standard deviation and median.The HPSOSA achieved very low values of RMSE and MAE comparatively.The HPSOSA successfully passed the RACF test and the test values lie within the confidence interval.

Energies 2017, 10 , 1213 8 of 13 A
measure of robustness of the HPSOSA and CPSO for 20 runs is shown in Figure 4.It is apparent from Figure 4 that the HPSOSA is able to achieve a minimum RMSE value in all 20 iterations and presents a robust solution for the parameter estimation problem.Whereas CPSO reached only once in the proximity of the minimum RMSE, achieved by HPSOSA.

Figure 4 .
Figure 4. Comparison of the HPSOSA and the CPSO for 20 runs.

Figure 5 .
Figure 5. Convergence of HPSOSA for parameter estimation of single diode PV cell circuit.

Figure 4 .
Figure 4. Comparison of the HPSOSA and the CPSO for 20 runs.

Energies 2017, 10 , 1213 8 of 13 A
measure of robustness of the HPSOSA and CPSO for 20 runs is shown in Figure 4.It is apparent from Figure 4 that the HPSOSA is able to achieve a minimum RMSE value in all 20 iterations and presents a robust solution for the parameter estimation problem.Whereas CPSO reached only once in the proximity of the minimum RMSE, achieved by HPSOSA.

Figure 4 .
Figure 4. Comparison of the HPSOSA and the CPSO for 20 runs.

Figure 5 .
Figure 5. Convergence of HPSOSA for parameter estimation of single diode PV cell circuit.

Figure 5 .
Figure 5. Convergence of HPSOSA for parameter estimation of single diode PV cell circuit.

Figure 6 .
Figure 6.Comparison of estimated model and experimental data I-V characteristics of single diode model by the HPSOSA.

Figure 6 .
Figure 6.Comparison of estimated model and experimental data I-V characteristics of single diode model by the HPSOSA.

Figure 6 .
Figure 6.Comparison of estimated model and experimental data I-V characteristics of single diode model by the HPSOSA.

Figure 7 .
Figure 7.Comparison of estimated model and experimental data PV characteristics of single diode model by HPSOSA.
have been shown.Figure 9 plotted the I-V characteristics of the estimated model and the experimental data while Figure 10 plotted the P-V characteristics of the estimated model and the experimental data.It is clear from Figures 9 and 10 that the current and power estimated by the HPSOSA closely trace the experimentally measured data.Energies 2017, 10, 1213 11 of 13 To further examine that how close the currents are calculated by the HPSOSA with the experimentally measured currents, Figures 9 and 10 have been shown.Figure 9 plotted the I-V characteristics of the estimated model and the experimental data while Figure 10 plotted the P-V characteristics of the estimated model and the experimental data.It is clear from Figures 9 and 10 that the current and power estimated by the HPSOSA closely trace the experimentally measured data.

Figure 9 .
Figure 9.Comparison of estimated model and experimental data I-V characteristics of double diode model by HPSOSA.

Figure 10 .
Figure 10.Comparison of estimated model and experimental data P-V characteristics of double diode model by HPSOSA.

Figure 9 .
Figure 9.Comparison of estimated model and experimental data I-V characteristics of double diode model by HPSOSA.

Figure 9 .
Figure 9.Comparison of estimated model and experimental data I-V characteristics of double diode model by HPSOSA.

Figure 10 .
Figure 10.Comparison of estimated model and experimental data P-V characteristics of double diode model by HPSOSA.

Figure 10 .
Figure 10.Comparison of estimated model and experimental data P-V characteristics of double diode model by HPSOSA.

Table 1 .
Parameter bounds for single diode and double diode PV cell circuit.

Table 1 .
Parameter bounds for single diode and double diode PV cell circuit.

Table 2 .
Statistics of objective values (RMSE) for single diode PV model using HPSOSA.

Table 3 .
Estimated parameters of single diode PV model obtained by HPSOSA and other five algorithms with RMSE and MAE values.

Table 4 .
Error respective to each measurement for single diode PV model.

Table 4 .
Error respective to each measurement for single diode PV model.

Table 5 .
Statistics of objective values (RMSE) for double diode PV model using HPSOSA.

Table 6
lists the best values of parameters estimated by HPSOSA and other five optimization algorithms along with and .Table 6 is evident of the superiority of the HPSOSA compared with other algorithms.It is obvious that the HPSOSA achieved significantly low values of and , 7.453163 × 10 −4 and 6.5556 × 10 −4 respectively.

Table 6 .
Estimated parameters of single diode PV model obtained by HPSOSA and other five algorithms with RMSE and MAE values.

Table 5 .
Statistics of objective values (RMSE) for double diode PV model using HPSOSA.

Table 6
lists the best values of parameters estimated by HPSOSA and other five optimization algorithms along with RMSE and MAE.Table 6 is evident of the superiority of the HPSOSA compared with other algorithms.It is obvious that the HPSOSA achieved significantly low values of RMSE and MAE, 7.453163 × 10 −4 and 6.5556 × 10 −4 respectively.

Table 6 .
Estimated parameters of single diode PV model obtained by HPSOSA and other five algorithms with RMSE and MAE values.

Table 7 .
Relative error respective to each measurement for single diode PV model.

Table 7 .
Relative error respective to each measurement for single diode PV model.