Several reliability models, including the Weibull model, can be used in the context of these studies. According to the following relationship (2) [5] , the reliability of equipment is expressed as the probability that it is still operational at a given time t:

$R\left(t\right)=\exp \left(-{\left(\frac{t}{\eta }\right)}^{\beta }\right)$(2)

where:

beta (β) is the shape parameter (dimensionless) and eta (η) is the scale parameter (time unit).

The analysis of a PV module’s reliability focuses on the failures of this PV module. Its primary goal is to determine the expected operating lifespan of PV modules. However, the Weibull distribution is the best method for representing the distribution of lifetimes. Indeed, it is applicable in both electronics and mechanics, and it allows you to describe the behavior of a PV module in three phases: the initial phase, the productive operational period, and the deterioration or aging phase [12] . In the context of our study, we assume that the lifetime distribution of solar panels follows a Weibull distribution.

The average lifetime of a PV module represents the expected time before a PV module is likely to cease operating reliably. It is essential for determining reliability and estimating the expected lifetime of a PV module. This average lifetime noted μ, can be estimated using the parameters and of the Weibull distribution, as well as the gamma function using Equation (3) [13] :

$\mu =\eta \cdot \Gamma \left(1+\frac{1}{\beta }\right)$(3)

where:

μ is the average lifetime (time unit), beta (β) is the shape parameter (dimensionless), eta (η) the scale parameter (time unit) and Γ is the gamma function defined by:

$\Gamma \left(\alpha \right)={\displaystyle \underset{0}{\overset{+\infty }{\int }}{x}^{\alpha -1}{\text{e}}^{-x}\text{d}x}$(4)

According to the study by Laronde, Charki and Bigaud [5] , the majority of photovoltaic module manufacturers offer an electrical performance guarantee covering more than 80% (or even 90%) of the value specified on the label for a period of 25 years (or 20 years) [5] . By expressing the average lifetime in years, it becomes easier to compare differents PV modules and make informed decisions on the choice of technologies and manufacturers. In fact, a PV module with a higher average lifetime can be interpreted as more reliable and durable.

In this study, we estimate the average lifetime of the PV module when its reliability reaches a tolerable level of 80%, which corresponds to a 20% degradation of its rated power, as mentioned in the work of Laronde, Charki, Bigaud, Ndiaye et al. [5] .

Our research is based on the two (2) parameters Weibull model, with the goal of developing a mathematical model that takes into account the effect of temperature and relative humidity on the reliability of solar panels in real-world usage scenarios. We begin with the assumption that photovoltaic module lifespan follows a Weibull distribution. The mathematical model that we propose for this purpose is represented by Equation (4):

$R\left(t\right)=\exp \left(-C{\left(\frac{t}{\eta }\right)}^{\beta }\right)$(5)

with:

$C=\frac{T_{\mathrm{mod}}+293\left(\frac{t}{\eta }\right)}{T_{\text{amb}}}\times \left(hr\right)$(6)

and

$T_{\mathrm{mod}}=T_{\text{amb}}+\frac{G}{800}\left(T_{\text{NOCT}}-20\right)$(7)

where:

C is the correction factor related to the temperature of the PV module, ambient temperature and relative humidity, T_mod is the PV module temperature over time in Kelvin (K), T_amb is the ambient temperature in Kelvin (K), G is the solar irradiation (W/m²), T_NOCT is the nominal operating temperature of a photovoltaic cell, hr is the relative humidity in percentage (%), β is the shape parameter of the Weibull distribution, η is the scale parameter of the Weibull distribution in hours (h), τ is the lifetime of the PV module (time unit), which corresponds to the scale parameter η of the Weibull distribution in hours (h), ${\gamma }_0$, ${\gamma }_1$ and ${\gamma }_2$ are the parameters of the Peck model.

The expression $293\left(\frac{t}{\eta }\right)$ in Equation (5) corresponds to a linear increase in the temperature of the PV module over time. Here, we multiply the time t by $\frac{293}{\eta }$ achieve a 293 Kelvin increase over the nominal lifetime of the PV module.

The use of Matlab Simulink in modeling and estimating the temporal reliability of a PV module is a well-established approach in the fields of solar energy engineering and research. Figure 2 , which represents the model proposed for this study, is the usual diagram of the Simulink model developed to model the behavior of the PV module.

We experimented with accelerated aging during this study. The simulations are running using Matlab software, based on the following criteria:

By examining the influence of temperature and relative humidity on the survival function of photovoltaic modules, it is observed that the scale parameter η of the Weibull distribution ideally adjusts to the lifespan τ of these PV modules.

Our research will look into how temperature and relative humidity affect the reliability of a photovoltaic module. As a result, during our simulations, we will only change the temperature of the PV module while maintaining the relative humidity constant, and vice versa. In the simulation, the meteorological data (temperature and relative humidity) used come from the national meteorological agency (ANAM). These annual average weather data for fifteen years from 2007 to 2021 for Ouagadougou station are recorded in Table 1 .

During our investigation of the effect of temperature on the PV module in this publication, we kept relative humidity at 55 percent. Furthermore, we investigated how relative humidity affects the reliability and lifespan of the PV module while maintaining the module’s temperature at 32˚C. Indeed, Burkina Faso enjoys high temperatures all year round. Dry air blows in from the northeast in winter, with maximum temperatures of around 30/32˚C in the north and 32/33˚C in the south [15] .

We note that we deliberately selected the pair of values ( $T_{\text{amb}}=32˚\text{C}$; $hr=55\%$)

with the aim of analyzing the behavior of the reliability function over time for different values of temperature or relative humidity.

This section describes the simulation results for the analysis of the effect of ambient temperature and relative humidity on reliability.

In this section, we investigate the effect of photovoltaic module temperature on module reliability, as determined by Equation (3). To demonstrate this, Figure 3 depicts the module’s reliability as a function of time for various temperature values while keeping the relative humidity constant at 55 percent. The reliability curves in Figure 3 reflect the results of our simulations regarding the analysis of the influence of ambient temperature on reliability.

It is worth noting that the patterns of the curves shown in Figure 3 and those from Ndiaye’s study are strikingly similar. The reliability of a photovoltaic module remains constant at R = 1 for the first two years, then begins to decline. The rate of decrease in reliability accelerates with increasing temperature, eventually leading to failure at 28, 30, and 34 years. Elevated temperatures have a negative impact on the performance of PV modules in general. As ambient temperature rises, so does the temperature of the PV module, and so does its output voltage, leading to a reduction in the efficiency of the conversion of solar energy into electricity. Furthermore, high temperatures can accelerate the aging and degradation of PV module materials, such as solar cells and encapsulation materials. This deterioration can shorten the life of the photovoltaic module and compromise its long-term reliability.

After determining the PV module’s dependability, we use equation (3) to calculate the average lifetime. Table 2 shows the reliability and average lifetime values for the PV module.

The examination of the various average lifetime data in Table 2 reveals that an increase in ambient temperature has a significant impact on its lifespan and, as a result, its reliability. Indeed, a lower average lifetime value indicates a shorter time between failure incidents, which leads to lower reliability. In summary, the analysis of the various average lifetime values in Table 2 reveals a significant relationship between ambient temperature and lifespan, resulting in a decrease in average lifetime value as ambient temperature rises.

Figure 4 depicts the time-dependent reliability curves of a PV module for various relative humidity values while keeping the ambient temperature constant at 32˚C.

The graphs presented in Figure 3 show the evolution of the reliability of a PV module for different relative humidity values. These graphs are typical for reliability functions, with a change in shape after five (5) years. In this representation, the curves exhibit linear behavior for the first two years before showing a slight decrease until the fifth year. After this, they begin a gradual decline as a result of the increase in relative humidity, and tend to fail between 25 and 30 years. These curve models reflect the effects of relative humidity on PV module reliability. In reality, relative humidity can have an impact on the reliability of PV modules, particularly the integrated electronic components in the system. High humidity levels can corrode metal contacts, resulting in electrical connection losses and lowering PV module performance. Furthermore, humidity can enter the PV module through microcracks or encapsulation flaws, damaging the solar cells and other constituent elements. Prolonged exposure to high humidity levels can hasten the deterioration of the PV module, resulting in decreased reliability. Table 3 shows the reliability values derived from the reliability curves over time, as well as the average lifetime values calculated using Equation (3).

The analysis of the various average lifetime values listed in the Table 3 reveals a significant relationship between relative humidity increase and PV module lifespan. Indeed, we see a decrease in average lifetime as relative humidity rises. From the above, we conclude that the reliability of solar panels depends on both ambient temperature and relative humidity.

The combination of temperature and humidity can have a significant impact on photovoltaic module reliability. High temperatures combined with high humidity levels, on the other hand, have the potential to accelerate degradation processes within PV modules. This can include material degradation, corrosion of metal contacts, layer delamination, and other issues. Humidity can also promote the growth of mold or microorganisms, which can harm the PV module’s components. Ambient temperature and humidity can also have an impact on the electrical performance of PV modules. High temperatures can cause internal resistance to increase, open-circuit voltage to decrease, power output to decrease, and overall efficiency to decrease. Humidity can affect material electrical conductivity, amplify ohmic losses, and cause short circuits. Furthermore, the combined effect of temperature and humidity can shorten the useful lifespan of PV modules. Harsh environmental conditions can hasten the aging and deterioration processes, resulting in an early decline in PV module performance and reliability, as illustrated in Figure 5 .

Figure 5 depicts the gradual decline in the reliability of a photovoltaic module over time. These graphs clearly show that the combination of high temperatures and high humidity has a negative effect on the reliability of photovoltaic technology. This decrease in reliability can be explained by the fact that this combination of environmental conditions hastens the deterioration of the PV module’s performance, resulting in a shorter useful lifespan. It should be noted that when the module’s reliability deteriorates, the photovoltaic system’s electricity production suffers immediately. Table 4 contains information such as reliability results extracted from reliability curves over time, lifespan estimation obtained from Equation (1), and average lifetime calculation performed using Equation (3).

Table 4 shows that for high levels of temperature and relative humidity, with a specific reliability threshold, the estimated lifespan and average lifetime are reduced. It is worth noting that the estimated lifespan values and various average lifetime values listed in the table clearly show the significant impact of increasing temperature and relative humidity on the PV module’s lifespan.