# Progress in Improving Photovoltaics Longevity
## 3 Theory: PV Electricity Generation and Degradation Rate
### Degradation Rate Model for PV Modules  
In order to evaluate the effect of stress variables on the PV module's performance degradation, a model for these degradation rates needs to be established. Over the past few decades, especially for the most common degradation mechanisms, several models that study the electrical behavior of the module components under climatic stress conditions have been developed [28]. [PERSON] et al. [29], using the model proposed by [PERSON] et al. [30], focused on determining outdoor degradation rate. In outdoor conditions, PV modules are mostly susceptible to the impact of temperature, ultraviolet (UV) irradiance, and relative humidity. These three factors were thoroughly examined by [PERSON] et al. [30], who proposed a model for a quantitative description of the physical degradation rate.  
According to this model [29; 30], three key degradation processes related to climatic conditions and stresses occur on a PV module during its real-world operation: hydrolysis, photo-degradation, and thermo-mechanical degradation. The export of an aggregate model is based on the proper combination of individual models for each degradation factor. Thus, the total physical degradation rate is given as follows:  
\[K_{T}=A_{N}\cdot(1+k_{H})\cdot(1+k_{P})\cdot(1+k_{Tm})-1, \tag{5}\]  
where \(K_{T}\left[\text{year}^{-1}\right]\) and \(A_{N}\left[\text{year}^{-2}\right]\) being the total degradation rate and the normalization constant, respectively. By \(k_{Tm}\), \(k_{H}\) and \(k_{P}\), respectively, the thermo-mechanical, hydrolysis, and photo-degradation rates are denoted. The expressions for the aforementioned degradation processes are given below:  
\[k_{Tm}=A_{t}\cdot C_{N}\cdot(273.15+\Delta T)^{\theta}\cdot\exp\left(\frac{- E_{at}}{k_{B}T_{max}}\right),\;\Delta T=T_{max}-T_{min}, \tag{6}\]  
\[k_{H}=A_{h}\cdot RH^{n}\cdot\exp\left(\frac{-E_{ah}}{k_{B}T_{mod}}\right), \tag{7}\]  
\[k_{P}=A_{p}\cdot UV^{X}\cdot(1+RH^{n})\cdot\exp\left(\frac{-E_{ap}}{k_{B}T_{ mod}}\right), \tag{8}\]  
where \(A_{t}\left[{}^{\circ}\text{C}^{-1}\text{cycle}^{-1}\right]\), \(A_{h}\left[\text{year}^{-1}\right]\) and \(A_{p}\left[\text{m}^{2}/\text{kWh}\right]\) are the thermo-mechanical, hydrolysis, and photo-degradation pre-exponential constants, respectively. By \(E_{at}\) [eV], \(E_{ah}\) [eV], \(E_{ap}\) [eV] the activation energies of power degradation induced by the thermo-mechanical, hydrolysis and photo-degradation processes, respectively, are denoted, while \(k_{B}\) is the Boltzmann constant. \(T_{mod}\) [K], \(T_{max}\) [K], and \(T_{min}\) [K] are the PV module temperature, the maximum temperature of the module, and the minimum temperature of the module, respectively. \(UV\) and \(RH\) are the integral UV dose \(\left[\text{kW}/\text{m}^{2}\right]\) and the relative humidity, respectively, and \(X\), \(n\), and \(C_{N}\) [cycles/year] are model parameters related to UV irradiation dose and relative humidity on the power degradation and cycling rate, respectively.  
Having calculated the total physical degradation rate, \(K_{T}\left[\text{year}^{-1}\right]\), the model also provides an estimation of the lifespan of a PV module. To achieve this, the failure time \(\text{f}_{f}\) [years] is calculated by considering a loss of the order of 20% in the initial maximum output power according to the manufacturer and is given as follows [31]:  
\[t_{f}=\frac{\Gamma}{K_{T}\cdot\left[\left|\log(0.2)\right|\right]^{\frac{1}{ \mu}}}, \tag{9}\]  
where \(\Gamma\) and \(\mu\) are model parameters for the construction material and the shape of the module, respectively.