2.2.1. Photovoltaic DGs

The small-scale PV DGs are the most common renewable sources in the MGs. PV modules are usually modeled using single, double, and triple diode-equivalent circuits [46,47]. Unfortunately, the produced power is intermittent and varied in each hour with high uncertainty levels due to their dependency on the solar irradiances. Therefore, they should be treated with an effective way for each hour where the uncertainty of solar irradiance is modeled by the Beta Probability Density Function (PDF). Consequently, in each hour, different states of the solar irradiance are considered to generate its Beta-PDF. For each state of solar irradiance s, the output power from the PV module, *Ppv*0(*s*), can be expressed as follows [48,49]:

$$FF = \frac{V\_{MPP} \times I\_{MPP}}{V\_{OC} \times I\_{SC}} \tag{2}$$

where

$$P\_{pv0}(s) = N \times FF \times V\_y \times I\_y \tag{3}$$

$$V\_y = V\_{OC} - K\_v \times T\_{cy} \tag{4}$$

$$I\_y = s \left[ I\_{\rm sc} + K\_{\rm i} \times (T\_{\rm cy} - 25) \right] \tag{5}$$

$$T\_{cy} = T\_A + s \left(\frac{N\_{OT} - 20}{0.8}\right) \tag{6}$$

$$f\_{\boldsymbol{\theta}}(\boldsymbol{s}) = \begin{cases} \frac{\Gamma(\boldsymbol{a} + \boldsymbol{\beta})}{\Gamma(\boldsymbol{a})} \boldsymbol{s}^{(\boldsymbol{a} - 1)} \left(1 - \boldsymbol{s}\right)^{(\boldsymbol{\beta} - 1)} & \boldsymbol{0} \le \boldsymbol{s} \le \boldsymbol{1}, \boldsymbol{a}, \boldsymbol{\beta} \ge \boldsymbol{0} \\\boldsymbol{0} & \text{otherwise} \end{cases} \tag{7}$$

$$\beta = (1 - \mu) \left( \frac{\mu \left( 1 + \mu \right)}{\sigma^2} - 1 \right) \tag{8}$$

$$\mathfrak{a} = \frac{\mu \times \beta}{1 - \mu} \tag{9}$$

$$\rho(s) = \int\_{s\_1}^{s\_2} f\_b(s) \, ds \tag{10}$$

$$P\_{pv}(t) = \int\_0^1 P\_{pv0}\left(s\right)\rho(s)ds\tag{11}$$

 

where *N* is the module number; *s* is the solar irradiance kW/m<sup>2</sup> ; *K<sup>i</sup>* and *K<sup>v</sup>* are current and voltage temperature coefficients (A/◦C and V/◦C), respectively; *Tcy* and *T<sup>A</sup>* are cell and ambient temperatures (◦C), respectively; *NOT* is the nominal operating temperature of the cell in ◦C; *FF* is the fill factor; *Voc* and *Isc* are the open circuit voltage (V) and short circuit current (A), respectively; *VMPP* and *IMPP* are the voltage and current at the maximum power point, respectively; *f<sup>b</sup>* (*s*) is the Beta-PDF of *s*; α and *β* are the parameters of the Beta-PDF; *µ* and *σ* are the mean and standard deviation of the random variable *s*, respectively; *s*<sup>1</sup> and *s*<sup>2</sup> are the solar irradiance limits of state (*s*); and *ρ*(*s*) is the probability of the solar irradiance state (*s*) during any specific hour. α *β σ ρ*

 

*α*

*ρ*

*β μ*

*σ μ μ*

*μ μ β*

Figure 1 describes in detail the calculation of the total average power at each hour including the uncertainties of solar irradiance. As shown, the related uncertainties are modeled using Beta-PDF where the day is split into 24-h periods, each of which is 1 hr. From the collected historical data, the mean and standard deviation of the hourly solar irradiance of the day is estimated. For each hour, different states of the solar irradiance among the Beta-PDF with equal steps are taken. In this study, each hour has 20 states for solar irradiance with a step of 0.05 kW/m<sup>2</sup> . Accordingly, the PV output power is obtained for each state using Equation (3). Besides, the probability of the solar irradiance state (*s*) is estimated using Equation (10). Thus, the average output power of the PV module at any specific hour can be obtained using Equation (11) [48]. This study considers that a PV unit is associated with the type of converter that can deliver active power only (i.e., unity power factor) as the standard IEEE 1547 [50].

**Figure 1.** Evaluation of PV output power considering the uncertainties in solar irradiance.
