Once good parameter values for the PV cell model are obtained using the above-mentioned various algorithms, the values can be further polished using gradient-based techniques. In this study, we adopted generalized reduced gradient GRG method as the gradient-based local search technique. The GRG algorithm was first developed by Abadie and Carpentier [ 23 ] as an extension of the reduced gradient method to solve a general constrained nonlinear programming problem, which can be stated as follows [ 24 ]: The and denote the lower and upper bounds of the variable , respectively.
It is assumed that all problem functions , , and are twice continuously differentiable. GRG transforms inequality constraints into equality constraints by introducing slack variables. Hence all the constraints are of equality form and can be represented as follows: Variables are divided into dependent variables and independent ones or basic and nonbasic variables, resp.
The names of basic and nonbasic variables are from linear programming. Similarly, the gradient of the objective function, bounds, and the Jacobian matrix may be partitioned as follows: Let be an initial feasible solution, which satisfies equality constraints and bound constraints.
Note that basic variables must be selected so that is nonsingular. The reduced gradient vector is determined as follows: The search directions for the independent and the dependent variables are given by A line search is performed to find the step length as the solution to the following problem: The optimal solution to the problem gives the next solution: A more detailed description of the GRG method can be found in [ 25 ].
For this PV parameter optimization problem, various metaheuristic and mathematical algorithms have so far proposed their solutions, as summarized in Table 2. Table 1 shows corresponding currents calculated by various algorithms, and Figure 3 shows the errors between calculated and measured currents from various algorithms.
As observed in the figure, the magnitude of the errors abruptly increases after around 0. Presumably this is because representing rapid change is more difficult than representing monotonic change. The measured current abruptly drops after around 0. However, is it possible to further improve the solution quality? The memetic algorithm, which combines metaheuristic and gradient-based algorithms, may answer to this question because the memetic algorithm can complement the weakness of metaheuristic algorithm by using gradient-based algorithm.
Thus, we can obtain even better solutions if we introduce gradient-based algorithm to the metaheuristic algorithm. The gradient-based algorithm GRG in this study by itself cannot find good solution unless it starts with good and feasible initial vector. Otherwise, it will get stuck in a local optimum or will diverge instead of converging. For this PV parameter optimization problem also, finding good solutions using gradient-based algorithm requires proper feasible initial vector, which is not very easy and tedious task.
Thus, the result vectors from metaheuristic or other mathematical algorithms can be used as the proper initial vectors for the gradient-based algorithm. The results from the memetic approach were tabulated in Table 3. As seen in the table, some initial vectors could be further improved while others could not because of getting stuck in local optimal LO. Here, the computation time of GRG part is less than one second, so it does not become a burden. Figure 4 shows the errors between calculated and measured currents from various memetic approaches.
As observed in the figure, the magnitude of the errors is less than single algorithm approach. As seen in the table, and are fluctuating, is increasing, is decreasing, and is almost constant.
In this study, a memetic approach has proposed for optimally determining the parameter values of single-diode-equivalent solar cell model. The memetic approach considered various metaheuristic and mathematical algorithms and combined them with GRG technique.
The results validated the performance of the memetic approach, which further improved the quality of solutions originally obtained by various algorithms. The proposed memetic approach in this work can be also further applied to more complicated and realistic PV models with various module types [ 32 ] in the future. And theoretically the balance between global search and local search can be a good future research topic because too much exploited solutions do not appear to be good initial vectors for local search algorithms.
In addition, parameter-setting-free technique [ 33 , 34 ] can be also incorporated in metaheuristic approach in order to eliminate tedious algorithm parameter setting task. The authors declare that there is no conflict of interests regarding the publication of this paper. International Journal of Photoenergy. Indexed in Science Citation Index Expanded. Subscribe to Table of Contents Alerts. Table of Contents Alerts. Abstract This study proposes a memetic approach for optimally determining the parameter values of single-diode-equivalent solar cell model.
Introduction Determining the parameter values of photovoltaic PV cell models is very important when designing solar cells and estimating their performance. Optimization Formulation of Photovoltaic Model Solar cells are made of various semiconductor materials. Measured - data and computation results. Experimental curve of a PV module. Errors between calculated and measured currents single algorithm. Proper care is to be taken of the feedback loop to get quicker convergence.
In this paper simplification of equation is done by excluding. All the above four blocks are interconnected to get Simulink model of for the PV module. This model takes insolation, temperature, and as inputs and calculates.
Simulink model of is simulated with the setup shown in Figure 7. Detailed discussion of simulation steps of model for obtaining I-V and P-V characteristics under varying irradiation with constant temperature and constant irradiation with varying temperature is available in [ 18 ].
The hardware for validating the results obtained in developed Simulink model is given in Figure 9. The description of experimental circuit given in Figure 8 is as follows. The maximum power point for all these temperatures lies between these voltages.
In the equivalent circuit of a PV cell, as shown in Figure 1 , the voltage available across the PV cell is nothing but the PN junction forward bias voltage of 0. The open-circuit voltage of the PV module is Simulink model of , developed in Section 2 , provides the module current. This PV current is calculated from irradiation and temperature and is the input to be used directly in the circuit model. The voltage at the output terminal of the model is fed back as the voltage input for Simulink model of [ 10 ].
A small resistance of 0. The detailed circuit model of PV module is shown in Figure The circuit model block of PV module is given in Figure Here, a voltage value is chosen initially and the iteration of power equation is carried out as done in normal functional PV model as it involves the algebraic loop problem.
With the variation of irradiation and temperature, the power output of PV module varies continuously. The maximum power point tracking MPPT algorithm is used for extracting the maximum power from the solar PV module and transferring that power to the load [ 15 ]. By changing the duty cycle of the PWM control signal, the load impedance as seen by the source varies and matches the point of the peak power of the source so as to transfer the maximum power.
The types of converters used are buck, boost, and buck-boost. For battery charging applications buck-boost configuration is preferred where as boost converters are used for grid-connected applications. DC-DC boost converters are used often in PV systems to step up the low module voltage to higher load voltages. The boost converter configuration, as shown in Figure 15 , consists of a DC input voltage source , boost inductor , controlled switch , diode , filter capacitor , and load resistance.
The boost converter operates in the continuous conduction mode for value of inductance where, where is the minimum value of inductance for continuous conduction.
The current supplied to the output RC circuit is discontinuous. Thus, a larger filter capacitor is required to limit the output voltage ripple. The minimum value of filter capacitor that provides the output DC current to the load when the diode is off is given by.
The minimum value of the filter capacitance, that results in the ripple voltage , is given by. The DC-DC converter with configuration given in Figure 15 and component values in Table 5 is simulated with battery supply as shown in Figure The battery supply in circuit shown in Figure 16 is replaced by the developed circuit model in Section 3 and simulated as shown in Figure The detailed experimental verification with circuit response of this developed circuit model is available in [ 19 ].
For the design of MPPT, the data is collected through simulation with the developed circuit model and results are tabulated in Table 6. From Table 6 , it can be seen that for lower values of irradiation and constant load, the duty cycle has to be reduced from 0. This variation coincides with the graph shown in Figure 18 , as found in [ 20 ], where the duty cycle variation with respect to irradiation is reported. So it is used in this paper.
This perturbation causes the power of the solar module to change. If the power increases due to the perturbation, then the perturbation is continued in that direction. After the peak power is reached, the power at the next instant decreases and after that the perturbation reverses. The above MPPT unit is placed as closed-loop control in the simulation circuit, as shown in Figure 17 and the detailed Simulink model for closed-loop control of developed circuit model of PV module with MPPT control unit is shown in Figure The schematic diagram of the proposed hardware system is shown in Figure The pulse generated given is to the gate of the power semiconductor device MOSFET , thereby changing the duty cycle of the converter.
The hardware setup of the proposed system is shown in Figure The microcontroller programming should be fed with the required range of duty cycle as given in Table 6 for quicker response. The experimental values of PV module power and current are lower by about 2 to 5 percent compared to the simulation values, as shown in Figure Thus the performance of the developed circuit model, in closed-loop control, follows the simulation values with reasonable accuracy.
In Section 2 , in 1 and Figure 1 , it can be seen that the PV current is a function of the solar irradiation and is the only energy conversion process in which light energy is converted to electrical energy. The next two equations, 2 and 3 , indicate that the PV voltage is a function of the junction voltage of diode, which is the material property of the semiconductors, susceptible to failure at higher temperatures. The physical equations governing the PV module also applicable to PV cell is elaborately presented with numerical values of module saturation current at various temperatures.
Hence, this circuit model presents the relationship between module parameters and circuit performance. However, for functional PV models used in other papers, the voltage level for the iterative process is chosen as per the convenience of end-circuit requirements and affect the circuit performance considerably.
This has an effect on the temperature performance of the circuit. So the selection of this voltage level which is very important has to be selected appropriately. Circuit model of photovoltaic PV module is presented in this paper, which can be used as a common platform by material scientists as well as power electronic circuit designers to develop the better PV power plant.
International Journal of Photoenergy. Indexed in Science Citation Index Expanded. Subscribe to Table of Contents Alerts. Table of Contents Alerts. Abstract Circuit model of photovoltaic PV module is presented in this paper that can be used as a common platform by material scientists and power electronic circuit designers to develop better PV power plant. Introduction The field of photovoltaics PV has experienced a remarkable growth for past two decades in its widespread use from standalone to utility interactive PV systems.
Modeling of PV Module 2.