A New Strategy for Accurately Predicting 𝐼-𝑉 Electrical Characteristics of PV Modules Using a Nonlinear Five-Point Model

Abstract

This paper presents the modelling of electrical 𝐼-𝑉 response of illuminated photovoltaic crystalline modules. As an alternative method to the linear five-parameter model, our strategy uses advantages of a nonlinear analytical five-point model to take into account the effects of nonlinear variations of current with respect to solar irradiance and of voltage with respect to cells temperature. We succeeded in this work to predict with great accuracy the 𝐼-𝑉 characteristics of monocrystalline shell SP75 and polycrystalline GESOLAR GE-P70 photovoltaic modules. The good comparison of our calculated results to experimental data provided by the modules manufacturers makes it possible to appreciate the contribution of taking into account the nonlinear effect of operating conditions data on 𝐼-𝑉 characteristics of photovoltaic modules.

1. Introduction

The performance of photovoltaic modules is highly dependent on both the availability of solar radiation in the locality and the temperature of photovoltaic cells [1]. A real knowledge and mastery of photovoltaic modules are important as are the design and technology choice for a project operating photovoltaic electricity.

Important research has been carried out throughout the world in view of mastering electrical performance of photovoltaic modules [2, 3]. Numerical models are developed for this purpose [4, 5], mainly to reduce the costs of experimental studies. Unfortunately, these studies are very useful, particularly to obtain accurate results [6, 7]. It is obvious that the application of the best model will combine in a better way simplicity and accuracy [8, 9].

This work consists in doing comparative study between a five-parameter three-point model hereafter referred to as five-parameter model characterized by both the linear variation of the current with respect to the irradiance and the linear variation of the voltage with respect to the temperature of the cells and an adjusted five-parameter five-point model hereafter referred to as five-point model which takes into account the nonlinear effects of both the variation of the current with respect to the irradiance and the variation of the voltage with temperature.

2. Single-Diode Equivalent Circuit of a PV Module

A photovoltaic module under illumination can be represented either by two-diode [10] or by one-diode equivalent circuit. In normal operating conditions, the one-diode model is the most common [11], and the photovoltaic module is equivalent to the following electrical circuit (see Figure 1).

Figure 1 

Equivalent electrical circuit for the one-diode model.

The current-voltage (𝐼-𝑉) characteristic under a solar radiation and temperature set is expressed in 𝐼=𝐼_ph−𝐼_𝑑−𝐼_sh, (1) where 𝐼_𝑑 is the current through the diode and 𝐼_sh is the current through the shunt resistor both given by (2) as follows: 𝐼_𝑑=𝐼₀exp𝑉+𝐼𝑅_𝑆𝑎,𝐼−1_sh=𝑉+𝐼𝑅_𝑆𝑅_sh. (2) Therefore, 𝐼=𝐼_ph−𝐼₀exp𝑉+𝐼𝑅_𝑆𝑎−1−𝑉+𝐼𝑅_𝑆𝑅_sh, (3) where 𝐼_ph, 𝐼₀, 𝑅_𝑠, and 𝑅_sh are the photocurrent, the reverse saturation current of the diode, the series resistance and shunt resistance, respectively. The modified ideality factor of the junction is given as 𝑁𝑎=^𝑠𝐴₀𝐾𝑇_𝑐𝑞, (4) where 𝐴₀ is the normal ideality factor; 𝑁_𝑠, 𝑘, 𝑇_𝑐, and 𝑞 are the number of cells in series, the Boltzmann constant, the temperature of the cells, and elementary charge, respectively.

The determination of the electrical behaviour of a photovoltaic module in given operating conditions is based on the current-voltage (𝐼-𝑉) curve. However, (1) that gives this characteristic is implicit and cannot be easily solved. Many numerical algorithms to model the 𝐼-𝑉 curve from one-diode representation of photovoltaic module have been proposed. Some of these models are known as “resistive companion” [12], the nonlinear least squares optimization, and other iterative solutions described in [8, 9, 13]. Although many improvements have been done, these approaches still require a great effort of numerical calculations. Recently, some authors have used artificial intelligence methods (AI) such as fuzzy logic [14] and artificial neural network (ANN) [15, 16] for modelling (𝐼-𝑉) curves.

Another interesting approach is based on the evaluation of unknown parameters of (1) based on information related to the field operation of the module in order to obtain an explicit 𝐼-𝑉 relationship. To this end, an iterative process may be necessary to evaluate the different parameters [17, 18]. One may also perform a direct analytical calculation of each parameter [19, 20]. The latter approach requires information often available in the data sheets provided by photovoltaic modules manufacturers or experimental data collected on site. Both five-parameter and five-point models described in this study use only data provided by module manufacturers, together with semiempirical correlation equations and weather data, to predict 𝐼-𝑉 characteristics of photovoltaic crystalline modules.

3. Five-Parameter Model

The five parameters that give name to the model are the photocurrent, the reverse saturation current, series resistance, shunt resistance, and modified ideality factor. These parameters are firstly evaluated at so-called reference operating conditions or standard test conditions (STC) for which the irradiance on the modules is 1000 W/m² and the cells temperature is 25°C with an air mass number AM = 1.5. This environmental situation is used by manufacturers of the modules to measure the data laid out on the data sheet.

The five parameters defined above corresponding to STC are designated by 𝐼_phref, 𝐼_0ref, 𝑅_𝑠ref, 𝑅_shref, and 𝑎_ref (the index is relative to reference conditions). We obtain the (𝐼-𝑉) characteristic equation at STC as 𝐼=𝐼_phref−𝐼_0refexp𝑉+𝐼𝑅_𝑠ref𝑎_ref−1−𝑉+𝐼𝑅_𝑠ref𝑅_shref. (5)

3.1. Calculation of Reference Parameters

Data provided by module manufacturers for STC are used when writing the following five equations [21].

At short-circuit point: 𝑉=0, 𝐼=𝐼_sc, 𝐼_sc0=𝐼_phref−𝐼_0ref𝐼exp^sc0𝑅_𝑠ref𝑎_ref−𝐼−1^sc0𝑅_𝑠ref𝑅_shref. (6) At the open-circuit point: 𝐼=0, 𝑉=𝑉_oc, 0=𝐼_phref−𝐼_0ref𝑉exp^ocref𝑎_ref−𝑉−1^ocref𝑅_shref. (7) At maximum power point (MPP): 𝐼=𝐼_mp, 𝑉=𝑉_mp, 𝐼_mpref=𝐼_phref−𝐼_0ref𝑉exp^mpref+𝐼_mpref𝑅_sref𝑅_shref−𝑉−1_mpref+𝐼_mpref𝑅_sref𝑅_shref. (8) At MPP, the derivative of the power with respect to the voltage is equal to zero, that is, 𝑑(𝐼𝑉)𝑑𝑉_mpp,ref=0. (9) At the short-circuit point, the slope of the 𝐼-𝑉 curve at short circuit is given by 𝑑𝐼𝑑𝑉_𝐼sc1=−𝑅_shref. (10) A simultaneous resolution of (6)–(10) yields the five reference parameters, and then the 𝐼-𝑉 curve for STC is built using (5). As one can notice, in this strategy we made use of only three critical points of the 𝐼-𝑉 curve, namely, the short-circuit point, the maximum power point, and the open-circuit point.

3.2. Dependence of Operating Conditions

The determination of the 𝐼-𝑉 characteristic in all other operating conditions can be made possible by evaluating the various parameters of the 𝐼-𝑉 equation in the new weather environment using the semiempirical relationships that follow as shown in [21]. One may consider the following: 𝑎𝑎_ref=𝑇_𝑐𝑇_𝑐ref,𝐼₀𝐼_0ref=𝑇_𝑐𝑇_𝑐ref³exp𝐸_𝐺𝑁_𝑠𝑎_ref𝑇1−^𝑐𝑇_𝑐ref,𝐼_ph=𝐺𝐺_ref𝐼_ph,ref+𝛼_𝑐𝑐𝑇_𝑐−𝑇_𝑐ref,𝐺𝐺_ref=−𝐼₀/𝑎exp𝑉_mp+𝐼_mp𝑅_𝑠/𝑎−1/𝑅_sh1+𝐼₀𝑅_𝑠/𝑎exp𝑉_mp+𝐼_mp𝑅_𝑠/𝑎+𝑅_𝑠/𝑅_sh×−𝐼_0ref𝑎_ref𝑉exp^mpref+𝐼_mpref𝑅_𝑠ref𝑎_ref−1𝑅_shref×𝐼1+^0ref𝑅_𝑠ref𝑎_ref𝑉exp^mpref+𝐼_mpref𝑅_𝑠ref𝑎_ref+𝑅_𝑠ref𝑅_shref⁻¹⁻¹,𝑅_sh=𝑅_sh,ref. (11) Finally at the MPP, we write the following two equations: 𝐼_mp=𝐼_mpref𝐺𝐺_ref,𝑉_mp=𝑉_mpref+𝛽_𝑉oc𝑇_𝑐−𝑇_𝑐ref. (12)

4. Five-Point Model

The five-point model is an improved method to extract the five unknown parameters 𝑎, 𝐼_ph,  𝐼₀,  𝑅_𝑠,  and  𝑅_sh of (3) using five key physical quantities of the 𝐼-𝑉 characteristic: the short-circuit current, the open-circuit voltage, current and voltage at the maximum power point, and the nominal series and shunt resistances 𝑅_𝑠0 and 𝑅_sh0, respectively. This method was first introduced in [22] using experimental values of key physical quantities. Data issued by manufacturers of modules do not contain the values of the nominal series and shunt resistances. Indeed, the exact five-point model and the approximate five-point model that use approximate values of the nominal series and shunt resistances yield comparable results [22], provided that 𝑅_𝑠0 and 𝑅_sh0 are well defined.

4.1. Calculation of Key Quantities

The nominal series and shunt resistances are determined using the slope method. The quantities 𝑅_𝑠0 and 𝑅_sh0 are defined at STC as the slope of 𝐼-𝑉 curves at 𝐼=0 and at 𝑉=0, respectively, as it can be seen in following equations [22]: 𝑅_𝑠0=−𝑑𝑉𝑑𝐼_𝑉=𝑉oc,𝑅_sh0=−𝑑𝑉𝑑𝐼_𝐼=𝐼sc. (13)(14) In this paper, the concept of nonlinearity is introduced for a better prediction of the behaviour of the tested photovoltaic modules. Thus, the dependency of the others key quantities on temperature and irradiance is given by the following equations: 𝐼_sc=𝐼_scref+𝜇_𝐼sc𝑇_𝑐−𝑇_𝑐ref𝐺𝐺_ref^𝛼,𝑉_oc=𝑉_ocref1+𝛽ln𝐺_ref𝑇/𝐺𝑇^𝑐ref_𝑐^𝛾,𝐼_mp=𝐼_mpref+𝜇_𝐼mp𝑇_𝑐−𝑇_𝑐ref𝐺𝐺_ref^𝛼,𝑉_mp=𝑉_mpref1+𝛽ln𝐺_ref𝑇/𝐺𝑇^𝑐ref_𝑐^𝛾. (15) Exponents 𝛼 and 𝛾 are constants describing all effects related to the nonlinear dependence of current on the intensity of solar radiation and considering all the nonlinear effects of voltage variation with the cells temperature, respectively [23]. 𝛽 is a dimensionless coefficient related to the specific technology of photovoltaic module [24, 25].

4.2. Calculation of Parameters

After expressing the above essential quantities as functions of operating conditions, the five parameters required are evaluated using the following equations [26, 27]: 𝐼_ph=𝐼_sc𝑅1+^𝑠𝑅_sh+𝐼₀𝐼exp^sc𝑅_𝑠𝑎,𝐼−1₀=𝐼_sc+𝑉_oc𝑅_sh−𝑉exp^oc𝑎,𝑎=𝑉_mp+𝐼_mp𝑅_𝑠0−𝑉_oc×ln𝐼_sc−𝑉_mp𝑅_sh−𝐼_mp−ln𝐼_sc−𝑉_oc𝑅_sh+𝐼_mp𝐼_sc−𝑉_oc/𝑅_sh⁻¹,𝑅_𝑠=𝑅_𝑠0−𝑎𝐼₀−𝑉exp^oc𝑎,𝑅_sh=𝑅_sh0𝐺_ref𝐺. (16)

4.3. Calculation of the Constants

The constants 𝛼, 𝛽, and 𝛾 are related to important effects in the photovoltaic module. They are empirical, and the precision of model depends on the acuity of their calculated results. Accurate determination of these constants requires significant experimentation to find the values of current and voltage at different points for different operating conditions in view to calculate them using (17)–(19): 𝐼𝛼=ln_sc1−𝜇_𝐼sc𝑇₁−𝑇₀/𝐼_sc0ln𝐺₁/𝐺₀, (17) where 𝐼_sc, 𝐺, and 𝑇 are values of short-circuit current, irradiance, and temperature, respectively; and the indices 0 and 1 are related to the fact that these quantities have to be measured for two different operating conditions.

The variation of the open-circuit voltage is related to the variations of the solar radiation intensity and cells temperature. To calculate 𝛽, temperature is kept constant while irradiance varies from 𝐺₀ to 𝐺₁. 𝛽 is obtained as 𝑉𝛽=^oc0/𝑉_oc1−1ln𝐺₀/𝐺₁. (18) For the same value of irradiance, the temperature of the cells varies from 𝑇₀ to 𝑇₁, and 𝛾 is obtained as follows: 𝑉𝛾=ln_oc0/𝑉_oc1ln𝑇₁/𝑇₀. (19)

5. Results and Discussion

The calculations were made for the Shell SP75 module from Shell Solar and the GES-P70 module from GESOLAR. Shell SP75 is a commercial module with an output of 75 𝑊_𝑝, consisting of 36 single crystal silicon cells connected in series. GES-P70 is also a commercial module of 36 cells connected in series with an output of 70 𝑊_𝑝 but in polycrystalline silicon material. Both panels have similar high fill factor values (≈0.75) and at first glance, their curves seem to be quite similar, but if more accurately examined, they show some differences, probably due to their different processing technology. The data of the modules are presented in Table 1.

Table 1 

Data of the photovoltaic modules used.

Manufacturers usually provide only limited operational data for PV modules. These data are available at STC and sometimes under nominal operating cell temperature (NOCT) conditions, that is, irradiance of 800 W/m², temperature of 45 or 47°C (depending on the manufacturer), ambient temperature of 20°C, wind speed of 1 m/s, and AM 1.5 spectrum.

The constant 𝛼, 𝛽, and 𝛾 can be calculated following the constant estimation procedures given above, and the results are shown in Table 2.

Table 2 

Constants estimation for PV modules.

Utilization of available data aforementioned in the equations above presented for the five-parameter and the five-point models yields computed parameters of the 𝐼-𝑉 characteristic at given operating conditions for both modules used. Following Figures 2 and 3 shows that 𝐼-𝑉 curves fitted from computed parameters of five-parameter and five-point models and experimental data issue by the manufacturers of both the photovoltaic modules used.

(a)

(b)

Figure 2 

Calculated 𝐼-𝑉 curves and experimental data of Shell SP75 (a) and GES-P70 (b) at different irradiance, 25°C.

(a)

(b)

Figure 3 

Calculated 𝐼-𝑉 curves and experimental data of Shell SP75 (a) and GES-P70 (b) at different temperatures, 1000 W/m2.

Figure 2 shows the 𝐼-𝑉 curves for the modules Shell SP75 Figure 2(a) and GES-P70 Figure 2(b) for different levels of irradiance. It can be seen that the five-point model strongly agrees with experimental data than the five-parameter model for both types of modules. Another significant observation is the very accurate prediction in low irradiance of the five-point model.

Besides verifying the accuracy of the models to fit with the solar irradiance, it is also very important to analyze their capabilities to reproduce the way photovoltaic panel performance is affected by the silicon temperature. The 𝐼-𝑉 curves of both modules have been realized by both mathematical models used in this work when subjected to the variation of temperature as it is shown in Figure 3. The accuracy of both models was tested for different levels of temperature. All measurements were performed at 1000 W/m².

As it has been noticed previously with the variation of irradiance, Figure 3 shows that five-point model better fits 𝐼-𝑉 curves of both modules than five-parameter model when subjected to the temperature variation. It is well known that irradiance and temperature of the cells strongly affect the performance of operating photovoltaic devices.

However, latest observations made on modelled 𝐼-𝑉 characteristics when subjected to irradiance and temperature of the cells variations are insufficient to make a true judgement about the capability of both the five-point and the five-parameter models to reproduce the behaviour of operating photovoltaic modules. To further investigate the performance of both the models in various operating conditions of both analyzed photovoltaic modules, inaccuracies on 𝐼-𝑉 curves prediction may be quantified.

Root mean square error (RMSE) is used as the statistical tool for assessing the performance of the models to predict the current. The %RMSE compares the predicted (𝐼-𝑉) characteristic at different operating conditions to the measured one, thus providing a clear picture of the precision of the model. Figures 4 and 5 show %RMSE for five-point and five-parameter models for each operating condition analyzed.

(a)

(b)

Figure 4 

%RMS for Shell SP75 (a) and GES-P70 (b) at different irradiations, 25°C.

(a)

(b)

Figure 5 

%RMS for Shell SP75 (a) and GES-P70 (b) at different temperatures, 1000 W/m².

Figures 4 and 5 show the measuring deviation of computed 𝐼-𝑉 curves from experimental 𝐼-𝑉 data issue by manufacturers of modules for the whole of operating conditions analyzed. In general, five-point model exhibits lower %RMSEs than five-parameter model for both the studied photovoltaic modules for all environmental conditions. These observations are as expected because it has been above observed that computed 𝐼-𝑉 curves using five-parameter model are more unfasten from experimental 𝐼-𝑉 data than those computed using five-point model.

In this work, it has been noticed that the five-parameter model for which nonlinearities are neglected either over-estimates or underestimates the values of current while the five-point model that takes into account the effects of nonlinear shows a remarkable fitting around the key points and a sufficiently accurate prediction at the other points of the 𝐼-𝑉 characteristic for the whole set of the weather conditions analyzed. This is to be foreseeable because the formulation of the equations of the five-parameter model presents an obsolete dependence on the environmental variations. The five-parameter model like many photovoltaic models used up to now, in addition to considering that the variations of the current and voltage with irradiance and temperature of cells, respectively, at the key operating points are made in linear ways or deviate slightly from their linearities, does not well define the dependence on irradiance and temperature of the cells of certain parameters. The shunt resistance and the ideality factor are considered independent from irradiance and temperature of the cells; the variations of the parameter 𝑎 (modified ideality factor) with respect to temperature are not related to ideality factor but to its definition such as the above mentioned in (4). It is obvious that the use of constant parameters determined under STC conditions must bring about inaccuracies in other operating conditions [8, 28, 29].

In conventional one-diode representation of photovoltaic module, physical phenomena are considered globally, so that parameters (which are related to these phenomena) lose their meaning [30]. In the five-parameter model, variations of parameters with environmental data (irradiance and temperature) are from their physical conception. Contrary to this technique, five-point model uses only experimental data and information of key operating points at STC and at any other operating conditions. Furthermore, the effects of nonlinearities considered in key operating points are transported in any other point of the characteristic through the calculation of parameters using key physical quantities. Methods that use a minimum simplification assumption from (3) such as five-point model are expected to give more realistic values for parameters under investigation [22, 30].

6. Conclusion

In this work five-parameter and five-point models were used to predict electrical 𝐼-𝑉 characteristics of illuminated commercial photovoltaic modules Shell SP75 and GES-P70 of monocrystalline and polycrystalline silicon material, respectively. Validation of these models was carried out with comparison to the data provided by module manufacturers. The following conclusions were drawn from our analysis.(i)Most parameters depend on both the cell temperature and the solar irradiance. Thus, the mastery of their behavior is crucial to correctly predict the performance of the photovoltaic cells and arrays.(ii)For both models used in this work, the precision varies on one hand from one module to another and on the other hand from one operating condition to another. Furthermore, the five-point model is more precise than the five-parameter model as well for the two photovoltaic module types studied for the whole set of weather conditions analyzed.(iii)Taking into account the effects of nonlinearities has made it possible to achieve better fitting of the 𝐼-𝑉 characteristics. This study therefore shows that the five-point model is an excellent tool for the prediction of the electrical response of the illuminated crystalline photovoltaic modules.

Nomenclature

Symbols

𝐴0:	Ideality factor
𝑎:	Modified ideality factor
𝐸𝐺:	Band gap of the material
𝐺:	Irradiance
𝐼:	Current
MPP:	Maximum power point
𝑅:	Resistance
STC:	Standard test conditions
𝑉:	Voltage
𝐼0:	Saturation current
𝐼ph:	Photocurrent
𝑁𝑠:	Number of cells in series
𝑉oc:	Open-circuit voltage
𝐼sc:	Short-circuit current.

Subscript

ref:	Reference conditions
mpp:	Maximum power point
oc:	Open circuit
𝑠:	Series
sc:	Short  circuit
sh:	Shunt.

Greek Letters

𝜇:	Temperature coefficient
𝛼:	Constant
𝛽:	Constant
𝛾:	Constant.

Conflict of Interests

There is no competing interest in the validity of this paper due to the choice of the Shell SP75 and GES-P70 modules. However, the choice of the Shell SP75 and GES-P70 module was made with reference to other research carried on these modules.

Acknowledgments

The authors are grateful to Dr. Obounou and Dr. Akana for their efforts in the realization of this work.