An Improved Empirical Model for Estimation of Temperature Effect on Performance of Photovoltaic Modules

It is prerequisite to predict the behaviour of photovoltaic (PV) modules in a particular geographical area where the system is to be installed for their better performance and increasing lifetime. For that, models are the easiest and acceptable tools to characterise the behaviour of PV modules in any location. The purpose of this study was to develop an empirical model to predict the influence of temperature on the performance of four different PV module technologies, namely, polycrystalline, monocrystalline, amorphous, and thin film in an outdoor environment. The model has been developed by fitting of one year experimental data using the least squares method. The estimated results of the developed model were validated with real-time data (winter and summer season) and a comparison of other existing model estimates using error analysis with 95% confidence interval. The proposed model estimations confirm that the monocrystalline module performs better in winter and polycrystalline in summer as compared to amorphous and thin film in the study area. During analysis, it is revealed that developed model results are more precise and appropriate among other existing model estimations. It is concluded that the proposed model estimations could be used for the prediction of PV module temperature in similar environmental conditions as that of the study area with more accuracy and confidence. It ultimately helps to develop cost-effective and efficient PV systems.


Introduction
The intensities of solar radiation, ambient temperature, wind speed, relative humidity, configuration, and method of mounting are considered to be responsible for variations in the power output of photovoltaic (PV) modules [1][2][3][4][5][6]. PV module temperature is one of the key parameters which affect the performance of photovoltaic (PV) modules after solar radiations [2,4,7,8]. Photovoltaic power output is proportional to the PV module operating temperature [9,10]. Since the change of PV module temperature depends on the variation of ambient temperature, as ambient temperature increases, the module temperature increases and vice versa [4,11]. It is because, the increase of temperature reduces the band gap of a PV module and increases the energy of the electrons in the material, which ultimately increases the recombination rate of internal carriers caused by the increasing amount of carrier concentrations [9,[12][13][14]. Consequently, it slightly increases the short-circuit current and considerably decreases open-circuit voltage [2,9,15]. Weather conditions affect the PV module temperature; therefore, its influence is necessary to be quantified. This can be done with the help of modeling, which eventually helps to design better systems for proper functioning. Several attempts have been made by different authors from different countries to exemplify the behaviour of PV modules. Some models are intuitive, and others are analytical, numerical, or empirical. Nevertheless, the majority of models are validated in indoor environments of developed countries with the exception of a few in outdoor conditions. It is a very challenging task to develop a model which represents the behaviour of various module technologies simultaneously in outdoor environments. An exact module temperature estimation model is indispensable to achieve reliable data of PV module power output [5,11,[16][17][18][19][20][21][22]. The models used for the prediction of module temperature can be categorized in different ways: steady-state or dynamic, explicit or implicit, etc. [2,9,11]. In steady-state modeling, all parameters are assumed to be independent of time (with small time interval, i.e., an hour). However, such models are useful for specific locations and module technologies, while in the dynamic models, some parameters are considered to be varied with respect to time. Dynamic models are preferable for high-resolution input data. Explicit models predict the value of photovoltaic module temperature directly, whereas the implicit correlations involve variables that themselves depend on module temperature. In implicit models, an iteration procedure is compulsory to get the outputs [2,5,9,[23][24][25][26][27][28][29][30]. Nevertheless, the selection of an appropriate model is crucial for the design and sizing of photovoltaic systems. The use of an inappropriate model gives faulty predictions thus making the systems over-or undersized. The oversized system becomes costly alternative, whereas undersizing causes malfunctioning of the system. This problem can be controlled through proper sizing and designing of system components with the help of precise modeling and using of long-term reliable data [9,20,[31][32][33][34]. Unfortunately, long-term data are not available in developing countries [31] including Pakistan [35], and the reliability of data is also questionable. Actually, photovoltaic module temperature models are submodels of power output models, as these models predict the effect of temperature on the performance of photovoltaic modules. Most of such models estimate the temperature of photovoltaic modules in indoor conditions but not in outdoor environments [36][37][38]. The main objective of this study was to develop a simple empirical model for the estimation of the temperature effect on four different PV module technologies, namely, polycrystalline, monocrystalline, amorphous, and thin film in an outdoor environment.

Existing Photovoltaic Module Temperature Models
In [39], the researchers consider only one basic climatic variable such as the ambient temperature (T a ) in their study. It is clear that one input variable does not reflect the whole behave of the environment. The developed model is given in equation (1) and also used by [40].
T m = 1 411 × T a − 6 414 1 Muzathik [38] suggested three variable models with ambient temperature T a (°C), global solar radiation G sr (W/m 2 ), and wind speed W v (m/s). The model and coefficients of each variable are provided as given in Equation 2.
In addition, [2] proposed a simple and semiempirical model for the calculation of module temperature as given in equation (3). The author considered T a in (°C), G sr in (W/m 2 ), and W v in (m/s). The same model is reported by [41].
Duffie and Beckman [42] proposed a novel mathematical approach for the calculation of photovoltaic module temperature in controlled nominal operating cell temperature (NOCT) conditions: 0.8 kW/m 2 solar radiation, 20°C ambient temperature, and 1 m/s wind speed. The model depends on the input of T a (°C), G sr (W/m 2 ), W v (m/s), and NOCT conditions as given in equation (4). Furthermore, the model is adopted by [9].
Risser and Fuentes [43] also proposed three variable models with the same variables as that of Muzathik [38] as given in equation 5. The author considered T a in (°C), G sr in (W/m 2 ), and W v in (m/s). The same model is tested by [19].
The authors [2,38,43] proposed new temperature models which were based on three basic input variables (solar radiations "G sr ," ambient temperature "T a ," and wind speed "W v "). The researchers proposed linear models in their studies, but the behaviour of climatic data is parabolic with respect to time. In the morning hours, the intensities of G sr and T a are directly proportional, but in the evening, these are less related due to the slight decreasing trend of temperature as compared to the sharp decrease of solar radiations. The authors [9,42] proposed a mathematical approach for the calculation of photovoltaic module temperature based on NOCT conditions. Such conditions could not be familiarized with a real outdoor condition.

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International Journal of Photoenergy Almaktar et al. [40] proposed a temperature model which depends on four climatic variables, namely, solar radiations "G sr ," ambient temperature "T a ," wind speed "W v ," and relative humidity "R h " as given in Equation 6.
It is already mentioned that the behaviour of climatic data is parabolic in nature with respect to the time of the day. Therefore, in this study, an empirical, nonlinear, multivariate, and least squares model was developed and proposed to calculate the PV module temperature in an outdoor envi-ronment. Table 1 shows the well-known PV module temperature models.

Materials and Methods
3.1. Study Area. The study was conducted in Nawabshah city, Shaheed Benazirabad District, Sindh, Pakistan, as shown in Figure 1. It is one of the hottest places and located at 26.14°N, 68.23°E [44] and mean 37 m above sea level [45].

Experimental
Setup. An experimental setup was installed at the Energy and Environment Engineering Department, QUEST, Nawabshah. Four generic photovoltaic modules (polycrystalline, monocrystalline, amorphous, and thin film) were used in this study, and their specifications are given in Table 1: Existing PV module models.

Proposed Empirical Model
In this section, we develop a model for the estimation of the temperature effect on different photovoltaic (PV) module technologies, namely, polycrystalline, monocrystalline, amorphous, and thin film in the outdoor environment. In model development, one dependent variable (module temperature) and four basic independent climatic variables (global solar radiation, ambient temperature, wind speed, and relative humidity) were adopted. Furthermore, the correlation of the dependent variable with each independent variable was analyzed. The correlation of module temperature (T m ) with the global solar radiation (G sr ) was found to be 0.89217, ambient temperature (T a ) 0.73765, wind speed (W v ) 0.075766, and relative humidity (R h ) -0.55918. The relationship between climatic parameters and module temperature was found to be nonlinear because of the parabolic curve. Thus, it was deduced from the curve fitting that polynomial models might be suitable models, as these cover the maximum number of measured data points. Further scrutiny of models was made by fitting the data with different degrees of polynomials (1-9 degrees). It was found that the 2 nd degree polynomial model covers the maximum number of data points of the measured data. Thus, an empirical second degree multivariate nonlinear model was proposed with fitting of data with the least squares method. It was assumed that photovoltaic module temperature (T m ) is the function of four variables, namely, G sr , T a , W v , and R h . Thus, the basic function of PV module temperature (T m ) is given in equation (7).
The general form of the model would be given in Equation 8.
By expanding equation (8) with the combination of all four independent variables, equation (9) is developed, which demonstrates the output and input parameters and all involved coefficients.
Let T m meas be the measured module temperature and T m est be the estimated module temperature. The least squares method assumes that the sum of the squares of the residuals (error) is less. Therefore, it can be estimated using Equation 10.
where i = 1, 2, ⋯, n, as n = 4392 and β is the set of the coefficients of the model. The minimum value of E occurs when the gradient is zero. The model contains m = 25 parameters; therefore, the gradient equation is 25. Furthermore, the minimum values of E and r i are calculated through equations (11) and (12).
The model equation (12) is complex and timeconsuming. Thus, it requires to be simplified for easy computation and application. For that, symbolic derivatives of equation 12 were put in MAPLE software, by producing a system of equations with the coefficients β j . Then, the obtained system of equations from MAPLE software was solved iteratively in MATLAB software. The coefficients of the developed model were approximated with an error tolerance of 0.0001. The general form of the developed model for the estimation of all four types of photovoltaic (PV) module temperatures is shown in equation (13), and model coefficients are given in Table 4.

Statistical Analysis
Statistical analysis was conducted to see the variation between models' estimated and measured results. The coefficient of determination (R 2 ) [48,49], root mean square error (RMSE), and mean absolute error (MAE) [40,[48][49][50] were used as statistical indicators as given in equation (14), respectively. The root mean square error (RMSE) and mean absolute error (MAE) are considered in°C. The statistical analysis was done at 95% confidence level.
where T m_est is the average estimated module temperature and T m_meas is the average measured module temperature.  Figure 5. Similarly, Figure 6 displays the wind speed. The maximum yearly average W v was recorded as 2.60 m/s at 16 hours and the minimum as 1.30 m/s at 07 hours with a yearly average of 2.14 m/s. Likewise, the maximum yearly average R h was noted as 76.90% at 07 hours and the minimum as 26.25% with a yearly average of 42.66%. The R h is given in Figure 7. The yearly average values of climatic conditions like G sr , T a , W v , and R h are given in Table 5. Relative humidity was found inversely proportional to the intensity of global solar radiation and ambient temperature.

Error Analysis
The error analysis of the proposed model estimations was checked with measured data of winter season and summer season and with estimations of other existing models. The coefficient of determination (R 2 ), root mean square error (RMSE) (°C), and mean absolute error (MAE) (°C) of each PV module of winter season (months of December and January) are summarized in Tables 6-8 and of the season of summer (months of May and June) in Tables 9-11, respectively.

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International Journal of Photoenergy     [39] model shows the least behavior of module temperature than the measured, proposed model estimations and other existing model estimation values in both seasons. The proposed model gave around 0.998 coefficient of determination for monocrystalline and low root mean square error and mean absolute error in both seasons. It is concluded that the proposed model is more appropriate for the estimation of photovoltaic module temperature in outdoor conditions because the proposed model gave a maximum coefficient of determination and minimum root mean square error and mean absolute error in both seasons. It is recommended that the time interval of data recording may be reduced from 1 hour to minutes and PV module technologies with the same ratings may be used for a comparison purpose. The performance and effect of temperature on both free standing and building integrated systems may be checked and verified in outdoor environments.   Quaid-e-Awam University of Engineering, Science and Technology

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.