Aiding Prosumers by Solar Cell Parameter Optimization Using a Hybrid Technique for Achieving Near Realistic P-V Characteristics

The correct optimization of the solar cell electrical-model parameters is the key to produce better and more realistic P-V characteristics. This helps prosumers to select solar panels having better comparative efficiency, which in turn increases electricity production. Evolutionary Algorithms have shown comparatively good results in the estimation of these parameters. The Photon-current, Diode dark saturation current, Series resistance, Shunt resistance and Diode ideality factor, constitute the single diode’s unknown electrical model parameters. The mathematical-model of the P-V cell is derived in terms of Series-resistance and Diode-ideality factor. These two parameters are then used in a 2-variable single objective function. Using this derived model, Genetic Algorithm and Numerical method, a new parameter estimation technique has been proposed. Making use of machine learning and combination of two algorithms highlights the usefulness of the intended hybrid technique. P-V characteristic and relative maximum power point error of different solar cells, have been compared. The relative analysis disclosed that the proposed method offers more pragmatic P-V characteristics, as compared to the existing methods.


I. INTRODUCTION
To cope with the expanding requirements of electrical energy, solar photo-voltaic cells have gained a noteworthy research attention to amplify its application. Photovoltaic, being a part of Distributed Energy Resources (DERs) has made it possible for prosumers to take an active part in fulfilling energy demands. For the past decade, a great amount of research has been done in order to estimate unknown solar cell parameters accurately. To produce accurate P-V electrical model parameters, Analytical, Numerical and Meta-heuristic methods are usually opted. Analytical method [1], [2] is reliant on the right positioning of the data for the accurate estimation of solar cell parameters. So before the installation of the solar panel it is important to simulate the P-V characteristics to better optimize and control the PV systems. This data is acquired from the manufacturer's data-sheet and Solar cell characteristic curve. Having number of attributes such as accuracy of results and less computational time makes these methods a good option but on the other hand, in case of large number of unknown parameters erroneous results should be expected [3]. Also higher computation is required in case of more variables. On contrary to the Analytical method, each sample points from the characteristic curve of the solar cell are considered, when using numerical methods. This provides more accurate results in comparison to the Analytical methods [3], [4]. So Numerical-methods such as Gauss Seidel [1], [7] and Newton-Raphson [5], [6] have often been used in researches for the estimation of solar cell parameters. Even though the degree of accuracy is high but the dependence of this method on accurate initial guess in case of more unknowns has been found difficult to predict [8], [9]. Because of this reason the solution can converge to a local-minima rather than a global-minima, in case of a wrong initial guess. The use of machine learning algorithms is necessary to solve current day challenges related to optimization or other problems [26], [27], [28].In order to sort out these certain problems in Numerical and Analytical techniques, several Meta-heuristic techniques were recommended which included Simulated-Annealing (SA) [13], Genetic-Algorithm (GA) [10], Particle-Swarm-Algorithm [11], [12], Teaching-Learning-Algorithm [15], Differential-Algorithm [14] etc. These methods proved to be more proficient in estimating the unknown parameters, as compared to the Numerical and Analytical techniques. But correspondingly these methods were slow in convergence and sometimes incapable of tracking actual characteristics [3], [4]. The contributions of this research are as follows: • Computational efforts have been reduced in some of the research works by neglecting Shunt resistance "Rsh" [19], [ 20], Series Resistance "Rs" [17], [18] or by assuming the ideality factor "n" of the diode [3]. All these approaches lead to a less accurate estimation of MPPE [9], [21].To increase the accuracy of results, this paper considers all five electrical model parameters. • Furthermore, only two Parameters are estimated using Genetic Algorithm and the remaining 3 are evaluated using the least square method, which helps to reduce the computational efforts and increases precision of results. • Resistance connected in series "rs" and the ideality -factor (Modified) "Vdi" are two variables which are undetermined, so the derivation of the solar model is done accordingly. The derived model (objective function) has been applied to estimate the unknown "rs" and "Vdi " using the Genetic Algorithm method . • Using the least square method the parameters yet to be estimated namely Photon-Current (Iphn), Shunt-resistance (rsh) and Saturation-current (Id) are calculated. • Finally, five different PV cells are considered to highlight the performance of the provided method. Performance Indices such as MPPE and P-V characteristic curve are compared with the existing and proposed method to reveal the accuracy of results. The organization of the paper is as follows: Section II is about the solar cell optimization related work that has been published. Section III includes the derivation of the single diode mathematical model and the calculation of the objective function. Also discussion related to the estimation of parameters using Genetic and Least square algorithm will be presented in this section. In section IV, the proposed technique is applied on different solar panels to highlight its effectiveness. And comparison on the basis of Maximum Power Point Error (MPPE) has been devised. Section V provides with the concluding remarks.

II. Related Work
A good number of research papers have been published to determine the solar cell unknown parameters using a number of methods. These methods proved to be beneficial but each method had some set-backs associated with it.

A. Numerical Methods
These methods still have great demand for optimizing solar cell parameters. But still, for good results they are dependent on the accurate initial guess. In case of a wrong initial guess, the solution converges to local minima rather than global minima, which is considered as a setback. Gauss Seidel [1], [7] and Newton-Raphson [5], [6] methods are some of the examples of numerical methods.

B. Meta-Heuristic Methods
These methods are inspired by nature. Using these methods for optimization problems, help in more accurate estimations even with a slightly wrong initial guess these algorithms prove to be more accurate. The chances of the solution to converge at global optima are brighter. But still the number of iterations or the time consumed by the algorithm is comparatively more. Genetic-Algorithm (GA) [10], Particle-Swarm-Algorithm [11], [12], Simulated-Annealing (SA) [13], Teaching-Learning-Algorithm [15], Differential-Algorithm [14] are examples of meta-heuristic algorithms.
So it can be observed that each method has a certain liability associated with it. The proposed research offers a method, which uses both the numerical and Meta-heuristic algorithms in a hybrid manner to achieve better results.

III. Least Squares and GA based parameter determination and Mathematical Modeling
Because of simplicity and accuracy, single diode model is considered for estimation of parameters [22], [24].Single diode model is depicted in Fig.1.

FIGURE 1. Single Diode Equivalent Model
The resistance between the bulk material and metal contact is labeled 'rs', as shown in the Fig.1 and recombination of electron hole pairs is represented by 'rsh'. The relationship between IO (Output Current) and VL (Voltage) in case of a single diode model is given in (1): Where, Vdi= n.k.t.Scs/qe. n=Ideality factor of the diode. t =temperature (kelvins). k=1.3806x10-23J/K (Boltzmann constant). qe =1.6021x10-19C (charge on an electron). Id = Dark saturation current (Ampere). Iphn = Photon current (Ampere). Scs= Number of series connected cells.
The remaining 'Iphn', 'Id', 'rs' , 'rsh' and 'Vdi' are yet to be determined as they have not been stated in the manufacturers datasheet. From (1) it is evident that the characteristic-curve is reliant on these undetermined parameters mentioned above. So in order to achieve results within close range to the real-characteristics, precise and accurate estimation of the above mentioned five unknown parameters is essential.

I.
By substituting VL=0 in (1) and short circuiting the load terminal, Is (Short-Circuit Current) is obtained.
By substituting Io=0 in (1) and placing an open circuit at the load terminal of the solar PV, following equation is obtained.

III.
By substituting 'vmp' and 'imp' in (1)  The tangent drawn parallel to the voltage axis provides PV curve at mpp. dp dv mpp = 0 After solving (5) the equation becomes as follows The final Equation can be formulated by using (6) and (1) At short circuit, current is differentiated w.r.t to voltage which provides the slope.

A. Derived Mathematical Model and Calculation of Objective Function
The parameters 'rs' and 'Vdi' are the two unknown parameters according to which the mathematical model or the characteristic equation of solar P-V will be derived. After the derivation of the characteristic equation, GA has been applied to estimate the two unknown values. The convergence is faster and the results are accurate because the proposed characteristic equation has only two unknowns, rather than five. The proposed characteristics equation derivation steps are as follows: To obtain the expression for 'Is' and 'imp', the value of 'Iphn' from (3) is substituted in (2) and (4). In order to find 'Is' and 'imp' in terms of 'rs', 'rsh' and 'Vdi', the value of 'Io' from (9) is substituted in (7), (10) and ( Furthermore, by equating (13) and (14) 'rsh' can also be found by (12) and (13) By using (15) and (16), the proposed equation can be derived as: So the proposed equation (17) can only be used to find 'rs' and 'Vdi'. Instead of five unknowns, using this proposed model to find two unknowns helps the GA method to converge faster and provide more accurate results. (17) is considered a Non-linear single objective optimization function, which is minimized using the genetic algorithm. Also this equation helps to reduce computational burden because now only four equations (rather than five) are required to determine 5 parameters.

B. Extraction of Parameters Using Genetic Algorithm
From Charles Darwin's theory of natural evolution a search heuristic named as Genetic Algorithm was inspired. In order to produce offspring of the next generation the fittest individuals are selected for reproduction, this process of natural selection is reflected in this algorithm.
In contrast with analytical methods, the genetic algorithm carefully evades the capture of the unknown parameters in the local-minima, which helps to achieve the global minima. To find the electrical model parameters of solar PV, the steps mentioned in the Fig.2 are followed [25]. In this research paper the Genetic algorithm has been used on 3 different mono and 2 poly-crystalline solar panels to demonstrate and compare the accuracy of this method. The combination of Genetic and Analytical algorithm helps to achieve desirable accurate results.
The results are far better than the parameters estimated by a single analytical technique for all five parameters. This reveals the competency of this hybrid algorithm. In this paper, Population Size, Number of Generations, Tour Size and Cross-over Probability are considered as 75, 95, 2 and 0.9 respectively for the GA based method.

C. Extraction of Parameters Using Least square Algorithm
In this method, the sum of the squares of the residuals, made in the results of every single equation is minimized to approximate the solution of over-determined systems .This method is a special approach of regression analysis. So the behavior of dependent variables is predicted using this technique. In order to implement this method, Matlab's built in command "Lsqnonlin" is used providing upper and lower bounds to further enhance the efficiency of the algorithm. The results for Least Square Method on all five equations and on only three equations have been depicted in Table II. It is clear from the results that using Least Square Method simultaneously with genetic algorithm helps to achieve greater accuracy in results and computational inefficiencies are reduced.

A. Attainment of P-V Characteristic-Curve using Proposed, Genetic and Least Square Method
Five different solar cells have been considered from [16] out of which three solar panels are mono-crystalline and the remaining two are polycrystalline. These solar cells are summarized in Table I. Using the given parameters in the datasheet from Table I, estimation of parameters is carried out using all the above mentioned methods and a comparison is devised on the basis of accuracy with respect to the maximum power point in Table II (A, B, C, D, and E).
Using these results from Table II (A, B, C, D, E), the Power-Voltage characteristic curves have been obtained in Fig.3, Fig.4, Fig.5, Fig.6 and Fig.7. It is evident from the PV curve in Fig.3, Fig.4, Fig.5, Fig.6 and Fig.7 that the suggested approach is in close vicinity to the MPP (Maximum Power Point) as compared to the Genetic algorithm and Least Square Method. So considering the evaluation of five unknown parameters the proposed method highlights more realistic PV characteristics as compared to Genetic and Least Square method. So it can be said with certainty that more accurate results have been provided using the proposed approach.

B. Approximation of MPE of the PV Cell using Proposed, Genetic and Least Square Method.
The difference between the actual and the calculated power is known as relative error or maximum power error. This MPPE has been listed in Table III, which provides the values of actual power, calculated power and the error between them with respect to least, genetic and proposed algorithm. It is evident from Table III that the proposed-method produces smaller values of MPPE in comparison to other methods with regard to the five solar panels mentioned in this study.

V. Conclusions
At first the Characteristic equation of the Solar PV in terms of two unknowns, namely Series Resistance and Diode Ideality Factor has been derived in this paper. Furthermore the proposed characteristic equation is used in the Genetic algorithm to find the two unknowns 'rs' and 'Vdi'. The remaining three parameters have been estimated using the Least Square Method. Five PV solar cells have been considered and compared to reveal the effectiveness of this approach. The PV curve and the MPPE have been estimated with the Least-square-method, Genetic algorithm and the proposed method. In comparison with the above mentioned methods, the proposed hybrid method highlights and offers PV characteristic curves closer to the real Characteristics. Also "rsh" and "rs" have not been neglected and ideality factor is also not assumed which has been done in previous works to reduce complexity but leads to less accurate estimations. Additionally, four equations instead of five equations for estimating five unknown parameters make the technique to be computationally less exhaustive. This will facilitate the prosumers in the acquirement of efficient solar panels thus increasing electricity generation.