**2. Uncertainty Modeling**

In this section, the uncertainties in load demand and wind power generation in the RPP problem are modeled to cope with the stochastic nature of the load demand and wind power generation. In the following subsections, modeling of both the load demand and wind power uncertainties are described. Finally, modeling of the system uncertainty via scenario generation is presented.

### *2.1. Modeling the Load Demand Uncertainty*

The uncertainty of the load is usually modeled by the normal distribution with mean (μ) and standard deviation (σ) [18]. In this paper, it is assumed that all the loads have constant PF, the same mean, and standard deviation. Therefore, for simplicity, a normal distribution is applied at the load level (λ) instead of applying in each load independently. The probability of each load level is shown by (<sup>π</sup>*l*), and is calculated using Equation (1). The associated value of each load level is denoted by (λ*l*), and can be obtained using Equation (2) [19]. It is worth mentioning that λ*Min*,*<sup>l</sup>* and λ*Max*,*<sup>l</sup>* are known as the minimum and maximum levels of the system loading at the *lth* load level, respectively.

$$
\pi\_l = \int\_{\lambda\_{\rm Min}}^{\lambda\_{\rm Max,l}} \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{\left(\lambda - \mu\right)^2}{2\sigma^2}\right) d\lambda \tag{1}
$$

$$
\lambda\_l = \frac{1}{\pi\_l} \int\_{\lambda\_{\rm Min}}^{\lambda\_{\rm Max}} \lambda \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{\left(\lambda - \mu\right)^2}{2\sigma^2}\right) d\lambda \tag{2}
$$

### *2.2. Modeling the Wind Power Generation Uncertainty*

Considering the intermittent nature of the wind speed, the Weibull distribution is often considered as the probability density function that can approximate the behavior of the wind with a reasonable error. Therefore, by defining the Weibull distribution for wind speed, the probability of wind speed at different intervals (scenarios) can easily be calculated. Equation (3) is a general expression for Weibull distribution [20]. Equation (4) can be used to calculate the probability of a wind speed interval (scenario). The corresponding value of each wind speed interval can be achieved using Equation (5).

$$PDF(v) = \frac{\beta}{a} \left(\frac{v}{a}\right)^{\beta - 1} \exp\left(-\left(\frac{v}{a}\right)^{\beta}\right) \tag{3}$$

$$
\pi\_W = \int\_{v\_{l,w}}^{v\_{f,w}} \frac{\beta}{\alpha} \left(\frac{v}{\alpha}\right)^{\beta - 1} \exp\left(-\left(\frac{v}{\alpha}\right)^{\beta}\right) dv \tag{4}
$$

$$w\_w = \frac{1}{\pi\_w} \int\_{v\_{i,w}}^{v\_{f,w}} v \frac{\beta}{\alpha} \left(\frac{v}{\alpha}\right)^{\beta - 1} \exp\left(-\left(\frac{v}{\alpha}\right)^{\beta}\right) dv \tag{5}$$

where *v* denotes the wind speed, and α and β are the wind speed parameters that vary depending on the region in which the wind blows. Considering *vi*,*<sup>w</sup>* and *vf*,*<sup>w</sup>* as the initial speed and final speed of the hypothetical scenarios for wind speed, the probability (<sup>π</sup>*w*) of occurrence of any wind speed scenario can simply be obtained. Thereafter, the wind speed (*vw*) associated with each scenario is gained using the calculated probabilities.

The output power of a wind turbine is highly dependent on wind speed. Therefore, any wind turbine has a characteristic named power curve that exactly shows the capability of a wind turbine in power generation versus existing wind speed. Knowing a specific wind speed (*vw*), one can estimate the output power of a wind turbine (*Pes<sup>t</sup> w* ) through its power curve. The power curve is generally defined by a set of equations as it is stated in Equation (6) [21], which in terms, (*vcin*), (*vrated*), and (*vcout*) denote the cut-in wind speed, rated wind speed, and cut-out wind speed for a wind turbine, respectively. The rated power (*Prw*) and the estimated output power of the wind turbine are also evident from Equation (6).

$$P\_w^{\rm ext} = \begin{cases} 0, & \upsilon\_w \le \upsilon\_{\rm in}^c \\ \frac{\upsilon\_{\rm w} - \upsilon\_{\rm in}^c}{\upsilon\_{\rm rated} - \upsilon\_{\rm in}^c}, & \upsilon\_{\rm in}^c < \upsilon\_w < \upsilon\_{\rm rated} \\ P\_{\rm w}^r, & \upsilon\_{\rm rated} < \upsilon\_w < \upsilon\_{\rm out}^c \\ 0, & \upsilon\_{\rm w} \ge \upsilon\_{\rm out}^c \end{cases} \tag{6}$$

In most research studies, the concept of the power curve is extended to a wind farm. Hence, instead of studying a single wind turbine, it is preferable to focus on a group of wind turbines that are in a special area and usually known as wind farms.

Considering several scenarios for a probabilistic problem is generally not an easy procedure. Depending on the problem type, various methods exist for scenario generation [22,23]. However, in this paper, a technique based on [19,24,25] is applied to generate a desirable number of scenarios with reasonable accuracy. In order to have a combination of load and wind scenarios, the following steps are taken:


$$
\pi\_{\mathfrak{k}} = \pi\_{l} \ltimes \pi\_{w} \tag{7}
$$

### **3. Problem Formulation**

As mentioned earlier, a wide range of objective functions for the RPP in power systems can be represented. This matter enormously affects the control variables, state variables, and all constraints of the RPP problem. Thus, by a proper formulation, all objectives can be achieved, all constraints can be satisfied, and the feasibility of the problem can be ensured. Due to the fact that the probabilistic nature of the problem has a major impact on its formulation, it is very important to use the probabilistic variables accurately in the problem formulation.
