#### Abstract

This paper proposes an improved generalized method for evaluation of parameters, modeling, and simulation of photovoltaic modules. A new concept “Level of Improvement” has been proposed for evaluating unknown parameters of the nonlinear *I-V* equation of the single-diode model of PV module at any environmental condition, taking the manufacturer-specified data at Standard Test Conditions as inputs. The main contribution of the new concept is the improvement in the accuracy of values of evaluated parameters up to various levels and is based on mathematical equations of PV modules. The proposed evaluating method is implemented by MATLAB programming and, for demonstration, by using the values of parameters of the *I-V* equation obtained from programming results, a PV module model is build with MATLAB. The parameters evaluated by the proposed technique are validated with the datasheet values of six different commercially available PV modules (thin film, monocrystalline, and polycrystalline) at Standard Test Conditions and Nominal Operating Cell Temperature Conditions. The module output characteristics generated by the proposed method are validated with experimental data of FS-270 PV module. The effects of variation of ideality factor and resistances on output characteristics are also studied. The superiority of the proposed technique is proved.

#### 1. Introduction

Photovoltaics (PV) is a method of converting sunlight directly into electricity using semiconducting materials that exhibit the photovoltaic effect. A PV cell is the fundamental PV device. A PV cell is a specialized semiconductor diode that converts sunlight into direct current (DC) electricity. A PV module is a collection of PV cells wired in series/parallel combinations as required to meet current and voltage requirements. A PV panel includes one or more PV modules assembled as a prewired, field-installable unit. A PV array is the complete power-generating unit, consisting of any number of PV panels to form large PV systems.

PV cells are expensive, and the characteristics of PV devices are highly dependent on environmental conditions [1]. Therefore, to ensure the maximum use of the available solar energy by a PV power system, it is important to study its behaviour through modeling, before implementing it in reality. The mathematical model of the PV device is very useful in studying various PV technologies and in designing several PV systems along with their components for application in practical systems.

The equivalent circuit of the ideal PV cell is represented basically by (i) single-diode model [1–7] and (ii) two-diode model [8–12]. In single-diode model, the effect of the recombination loss of carriers in the depletion region is not considered, whereas in two-diode model, an additional diode is included to consider this effect. Many more sophisticated models have been developed so far to include the effects that are not considered by the earlier models, thus claiming more accuracy. The single-diode model is simple and accurate and is perfect for designers who are looking for a model for the modeling of PV devices where the intended result is achieved without great effort [4, 13].

The parameter identification of the single-diode model with five parameters has been enormously researched [1–7]. The information provided in the datasheet cannot saturate the necessary restrictions to calculate the five unknown parameters. Consequently, some presumptions and approximations are generally needed to initiate the parameter extraction. For example, in [4], the evaluation of both photovoltaic current () and diode reverse saturation current () is based on approximation, due to which the deviation at open and short conditions becomes inevitable, and the diode ideality constant () has been assumed on the basis of PV cell technology. Since this constant affects the curvature of the *I-V* curves and its correct estimation improves the model accuracy, it must be evaluated accurately without assumption. A separate DC circuit is constructed to determine ideality constant in [1], thus increasing the cost of the evaluation procedure. Some other techniques adopted by the authors to estimate the ideality constant are curve fitting [6, 14], iteration [7], trial and error [14], and the concept of the minimum sum of squares [15]. But for evaluating ideality constant by any of the above methods, manufacturer-specified output characteristics are required. Thus, these processes can be cumbersome. Also, in some PV module datasheets, output characteristics are not given [16]. Therefore, evaluating ideality constant of such modules becomes very difficult.

Apart from the evaluation of , , and , extensive studies have been conducted to determine the series resistance () and parallel resistance (). Some authors neglect to simplify the model as the value of this resistance is generally high [14–19], and sometimes, the is neglected, as its value is very low [20, 21]. The neglect of and has significant impact on the model accuracy. Several algorithms have been proposed to determine both and through iterative techniques [4, 12]. If the initialisation of the variables and the convergence conditions are not proper, then these iterative techniques require many iterations and, sometimes, may not converge. Curve fitting method can be utilized in the current density-voltage curves to estimate both and [22]. In [23], and are evaluated by using additional parameters which can be extracted from the current versus voltage curve of a PV module. These methods are quite poor, inaccurate, and tedious mainly because and are adjusted separately, which is not a good practice, if an accurate model is required. Moreover, these methods are applicable only if the manufacturer-specified output characteristics are provided. Differential evolution (DE) can be used to extract the excess seven parameters of a double-diode PV module model utilising only the information provided in the datasheets [8, 10]. An explicit modeling method based on Lambert W-function for PV arrays that has been used in [24] to find the values of parameters is intricate and time-consuming. Artificial intelligence (AI) such as fuzzy logic [25] and artificial neural network (ANN) [26, 27] and genetic algorithms such as particle swarm optimization (PSO) have also been proposed to model the *I-V* curves [28]. However, they are not widely adopted due to high computation burden. In [2], a comprehensive parameter identification method is proposed to enhance model accuracy while keeping the parameterization procedure in a simple form. Despite the accurate results, the approach requires extensive computation. A circuit-based piecewise linear PV device model has been developed and demonstrated using PSCAD/EMTDC for parameter identification [29]. As this method is based on trial and error for different values of irradiance and temperature and is based on a large number of approximations and assumptions, this method becomes complicated and less accurate.

Hence, to overcome the above drawbacks, an improved generalized method for evaluation of parameters, modeling, and simulation of photovoltaic modules has been proposed in this paper. A new concept “Level of Improvement” has been proposed for evaluating unknown parameters of the nonlinear *I-V* equation of the single-diode model of PV module including series and parallel resistances at any environmental condition (STC and NOCTC in this paper), taking the manufacturer-specified data at Standard Test Conditions as inputs. The new concept helps in improving the accuracy of the values of evaluated parameters up to various levels. The proposed evaluating method is based on mathematical equations of PV modules, thus making the method fast, simple, and accurate. The method for evaluating is based on the property that at the maximum power point, . The proposed evaluating method is implemented by MATLAB programming and by using the values of parameters of the *I-V* equation obtained from programming results, a PV module model is build with MATLAB. The parameters evaluated by the proposed technique are validated with the datasheet values of six different commercially available PV modules for irradiance and temperature at Standard Test Conditions and Nominal Operating Cell Temperature Conditions. The module output characteristics generated by the proposed method are validated with experimental data of FS-270 PV module at different environmental conditions (varying irradiance and temperature). The effects of variation of ideality factor and resistances on the output characteristics are also studied. The superiority of the proposed technique over three popular existing techniques [4, 6, 12] is proved.

#### 2. Mathematical Equation and Modeling of Photovoltaic Modules

Practical modules are composed of various PV cells connected in series or parallel. Figure 1 shows the single-diode equivalent circuit of a practical PV module.

The mathematical equation that describes the *I-V* characteristic of a practical PV module is

The equation that mathematically describes the *P-V* characteristic of a practical PV module is

For the observation of the characteristics of the PV module, it is required to evaluate all the parameters of (1) [6]. Datasheets generally give information about the parameters, characteristics, and performances of PV modules with respect to the Standard Test Condition (STC), which is taken as 1000 W/m^{2} solar irradiance and 25°C module temperature [30–36]. Some datasheets also give information about PV modules at Nominal Operating Cell Temperature Condition (NOCTC), which is generally taken as 800 W/m^{2} irradiance and 20°C ambient temperature. The parameters basically present in all PV module datasheets at STC and sometimes at NOCTC are as follows: , , , , , , , , , and for any environmental condition. Table 1 shows the values of the parameters of seven different commercially available photovoltaic modules at STC provided by the manufacturers [30–36].

Some of the parameters of (4) can be found in the manufacturer’s datasheets. The remaining parameters such as , , , , and have to be evaluated. They are usually not specified by the manufacturers because they cannot be measured and are unique for every module [19].

#### 3. Basic Evaluation of the Parameters

The proposed work attempts to evaluate the five unknown parameters ( and ) of a PV module at different environmental conditions.

##### 3.1. Evaluation of Photovoltaic Current

The parallel resistance is generally very high, so the last term of (1) can be eliminated for the further work.

By applying short-circuit condition () to (5), (6) can be derived as follows:

Since (), (6) can be written as

The photovoltaic current of a PV module is approximately equal to the short-circuit current at any environmental condition.

##### 3.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition () to (5), (8) can be derived as follows:

By rearranging (8), (9) is obtained as follows:

In the proposed method, the maximum power point is considered for evaluating “*a*” using the property at the maximum power point.

By applying maximum power condition to (11), (12) is obtained as follows:

By differentiating (5) and applying maximum power condition, (13) can be obtained as follows:

By substituting (13) in (12), (14) can be obtained as follows:

For (14) to be valid, the numerator of (14) must be zero as the denominator is finite.

By applying maximum power point condition to (5) and rearranging terms, (16) and (17) can be obtained as follows:

By combining (16) and (17), (18) can be obtained as follows:

Substituting (7) and (9) in (18), (19) is obtained as follows:

When the data specified in the manufacturer’s datasheets are used, the left hand side of (19) becomes a function of “*a*” and it is denoted by .

Thus, “*a*” is found by solving with the help of MATLAB programming.

##### 3.3. Evaluation of Diode Reverse Saturation Current

By (9), the diode reverse saturation current at any environmental condition can be obtained.

##### 3.4. Evaluation of Series and Parallel Resistances

Equating the maximum power calculated by the *P*-*V* model of (4) () to the power at the MPP from the datasheet () at any environmental condition, a relation between and will be obtained, that is,

According to (23), for any value of there will be a value of that satisfies (21). It is required to find an only pair of and for the desired environmental condition that satisfies the model accurately.

###### 3.4.1. Proposed Algorithm

In this work, and in (23) are calculated through the following steps:
(a)Eliminating the last term of (4), as the value of the parallel resistance is high, the power calculated by the *P*-*V* model excluding , is obtained as follows:(b)Iterations are performed on (24) where is slowly incremented starting from , till the maximum of power calculated by the *P*-*V* model excluding () becomes approximately equal to the power at the MPP from the datasheet ().(c)Putting in (23), the value of obtained is negative. Iterations are performed again on (23), where is slowly decremented starting from , until a positive value for is obtained. Hence, the maximum value of parallel resistance is obtained and the corresponding value of series resistance is the maximum value of series resistance, .(d)Putting in (23), the minimum value of parallel resistance is obtained as follows: The value of obtained by the proposed method is more accurate as compared to other methods [4, 12].(e)Initialising and , iterations are performed again on (4) where is slowly decremented and the corresponding value of is obtained, till the maximum of power calculated by the *P*-*V* model () becomes approximately equal to the power at the MPP from the datasheet (), while . Thus, and are obtained.

#### 4. Improving the Model

The accuracy of the values of evaluated parameters ( and ) is improved up to various levels by using “Level of Improvement.”

##### 4.1. First Level of Improvement

###### 4.1.1. Evaluation of Photovoltaic Current

By applying short-circuit condition () to (5), (26) can be derived as follows:

Since (), (26) can be written as

By (27), the first level of improved value of photovoltaic current of a PV module () at any environmental condition can be obtained.

###### 4.1.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition () to (5), (28) can be derived as follows:

By rearranging (28), (29) is obtained as follows:

By applying the same procedure as is applied on (5) to find the diode ideality constant on (1), an improved and more accurate value of diode ideality constant can be obtained, as the values of both and are taken into account.

The function , where is the first level of improved value of diode ideality constant, becomes

Thus, “*a*1” is found by solving with the help of MATLAB programming.

###### 4.1.3. Evaluation of Diode Reverse Saturation Current

By (29), the first level of improved value of diode reverse saturation current () at any environmental condition can be obtained.

###### 4.1.4. Evaluation of Series and Parallel Resistances

By applying the same procedure as is applied to evaluate and , a first level of improved values of series resistance () and parallel resistance () for the desired environmental condition can be found.

##### 4.2. Second Level of Improvement

Again by putting and in (27) and (30), a second level of improved value of photovoltaic current () and diode ideality constant () can be obtained. As a consequence, a second level of improved value of diode reverse saturation current (), series resistance (), and parallel resistance () can be obtained. The improvement can go up to level ( and ), till the errors between the values of parameters evaluated by the proposed method and the values of parameters provided in manufacturer’s datasheet become minimum.

It should also be noted that the value of diode ideality constant should not vary much in any level of improvement. This is possible only when the value of parallel resistance is high. In the proposed method, initialising the series and parallel resistances by their corresponding maximum values in every level of improvement helps in getting higher value of parallel resistance and also reduces the required number of iterations. In this paper, only the first level of improvement has been applied.

#### 5. Dependence of Parameters of the Characteristic Equation of the PV Module on Irradiance and Temperature

The , , , , and of the PV module depend on both solar irradiance and temperature and can be calculated using the following equations:

#### 6. Evaluation of Parameters of the Characteristic Equation of the PV Module at STC

Applying STC to the above procedure and using the manufacturer-specified data at Standard Test Condition, five unknown parameters (, and ) of a PV module at STC can be evaluated.

The flowchart for evaluation of five unknown parameters (, and ) of a PV module at STC is presented in Figure 2.

Table 2 shows the first level of improved values of the parameters of seven different commercially available PV modules for the proposed model and evaluated parameters of seven different commercially available PV modules for the model [6], and model [4], and two-diode model [12] at STC.

#### 7. Evaluation of Parameters of the Characteristic Equation of the PV Module at NOCTC

Using the manufacturer-specified data at STC and NOCTC in (31), (33), and (35) irradiance coefficients of , and of the PV module can be calculated.

Five unknown parameters of a PV module at NOCTC can be evaluated using the following equations:

By using the above equations and by applying the same procedure as is applied to find and , and can be obtained.

Table 3 shows the evaluated irradiance coefficients of , , and of seven different commercially available PV modules and evaluated parameters of seven different commercially available PV modules at NOCTC for the proposed model.

It is evident from the above equations that all the parameters of the characteristic equation of the PV module are subject to vary with irradiance and temperature, which is a truism. This increases the accuracy of the proposed method of evaluating parameters manifold as compared to existing methods in this area where the parameters are assumed constant [4, 6, 12]. The proposed model output characteristics very accurately match the experimental output characteristics and manufacturer’s datasheets which is authenticated from the results shown in the later sections.

#### 8. Curves and Inferences

As shown in Figures 3(a) and 3(b), *I-V* and *P-V* curves of the FS-270 PV module are generated for eight values of and the corresponding values of and . It can be noted that as the value of calculated by the proposed method is utilised, the value of obtained from the proposed model becomes closest to the datasheet value. *I-V* and *P-V* curves of the FS-270 PV module are plotted again for six values of and the corresponding values of and (see Figures 4(a) and 4(b)). As the value of calculated by the proposed method is utilised, the value of obtained from the proposed model becomes closest to the datasheet value. Similar is the case with . The accuracy of the model is highest, when the values of , and calculated by the proposed technique are utilised for modeling. The proposed method of evaluating the unknown parameters is far more accurate as compared to other methods mentioned in literature.

**(a) I-V characteristics**

**(b) P-V characteristics**

**(a) I-V characteristics**

**(b) P-V characteristics**

#### 9. Validation of the Model

The current and power output obtained from the proposed model is validated against measured current and power output data, respectively, for the FS-270 PV module provided by the National Institute of Technology Patna (NITP), India.

Figures 5(a) and 5(b) show the mathematical *I-V* and *P-V* characteristics of the FS-270 PV module [31] plotted with the experimental data by varying irradiance from 200 W/m^{2} to 1200 W/m^{2} at 25°C. Figures 6(a) and 6(b) show the *I-V* and *P-V* curves by varying temperature from 0°C to 75°C at 1000 W/m^{2}. The experimental points are represented by circular markers in the curves. As the model is not perfect, some points are not exactly matched, although it is sufficiently accurate for majority points.

**(a) I-V characteristics**

**(b) P-V characteristics**

**(a) I-V characteristics**

**(b) P-V characteristics**

The relative errors of the proposed model with respect to the experimental data for the FS-270 PV module at 25°C and 1000 W/m^{2} are shown in Figure 7. The model proposed in this paper is compared with the models proposed in [4, 6, 12]. The relative errors obtained by all the models are plotted on the same graphs. The model proposed in this paper is superior, because the values of relative errors obtained by the proposed model are very small as compared to other models.

Tables 4–9 show the comparison of model, and model, two-diode model, and proposed model output with manufacturer’s datasheets for six different commercially available PV modules [30–36].

In the proposed method, it has been assumed that all the PV cells of a module are perfectly made but this is not the case in reality. Still the proposed method confirms to be very accurate as can be clearly seen from Figure 7 and Tables 4–9. This is because the properties of the PV cells having the same parameters, manufactured from the same producer, do not vary much and hence can be assumed identical.

#### 10. Simulation of the PV Module

The PV module model can be simulated in any circuit simulator by implementing (1) and (4) using basic math blocks. Figures 8 and 9 show a PV module model, where these two equations are implemented in MATLAB/Simulink. In the complete model, irradiance and temperature along with the evaluated and manufacturer-specified parameters of the PV module are the inputs while the outputs are current and power. The proposed model is most generalized as compared to the models proposed in previous works.

#### 11. Conclusion

In this paper, an improved generalized method for evaluation of parameters, modeling, and simulation of photovoltaic modules is proposed.

The proposed PV module modeling method surpasses the other methods already published, as it has the following novelties:
(i)A new concept “Level of Improvement” has been proposed for evaluating unknown parameters of the nonlinear *I-V* equation of the single-diode model of PV module at any environmental condition (STC and NOCTC in this paper), taking the manufacturer-specified data at Standard Test Conditions as inputs.(ii)The new method of evaluation of unknown parameters is based on mathematical equations of PV modules. By implementing simple set of equations using any software, the parameters can be determined numerically simply by feeding few manufacturer-specified data as input to the program.(iii)In this paper, for the first time, the effects of varying ideality factor and resistances on the output curves have been observed. It has been inferred that as the values of , and calculated by the proposed method are utilised, the value of obtained from the proposed model becomes closest to the datasheet value. The accuracy of the model is maximum, when the values of , and calculated by the proposed technique are utilised for modeling.(iv)A most generalized PV module model is build with MATLAB/Simulink by using the values of parameters of the *I-V* equation obtained from programming results in order to show the practical use of the proposed model.(v)The proposed method proves to be more accurate in modeling commercially available PV modules as compared to other methods available in literature.

#### 12. Future Work

Under partial shading condition (PSC) on a PV array, the irradiance and temperature of the PV modules undergoing shading change. The parameters of the shaded and unshaded PV modules of the array can be easily evaluated by employing the proposed method. By applying a suitable maximum power point tracking (MPPT) technique, maximum power point (MPP) of a PV array under PSC can be reached and hence MPPT can be ensured which will be the subject of our further investigations.

#### Nomenclature

: | Photovoltaic current of the ideal PV cell (A) |

: | Reverse saturation or leakage current of the ideal PV cell (A) |

: | Photovoltaic current of the PV module (A) |

: | Reverse saturation or leakage current of the PV module (A) |

: | Diode ideality constant of the PV module |

: | Equivalent series resistance of the PV module (Ω) |

: | Equivalent parallel resistance of the PV module (Ω) |

: | Thermal voltage of the PV module (V) |

: | Electron charge (C) |

: | Boltzmann constant ( J/K) |

: | Irradiance (W/m^{2}) |

: | Temperature of the PV module (K) |

: | Number of cells connected in series in the PV module |

: | Number of parallel connections of cells in the PV module |

: | Power of the PV module at the maximum power point (W) |

: | Voltage of the PV module at the maximum power point (V) |

: | Current of the PV module at the maximum power point (A) |

: | Open-circuit voltage of the PV module (V) |

: | Short-circuit current of the PV module (A) |

: | Irradiance coefficient of |

: | Temperature coefficient of (W/K) |

: | Irradiance coefficient of |

: | Temperature coefficient of (V/K) |

: | Temperature coefficient of (A/K) |

: | Irradiance coefficient of |

: | Temperature coefficient of (V/K) |

: | Temperature coefficient of (A/K). |

*Index*

STC: | Subscripts indicate the parameters at Standard Test Condition |

NOCTC: | Subscripts indicate the parameters at Nominal Operating Cell Temperature Condition. |

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Acknowledgments

The authors wish to thank the Electrical Engineering Department of the National Institute of Technology Patna (India) for providing all the experimental facilities to carry out this work.