*3.1. Variables*

The same as the other optimization problems in power systems, such as Optimal Power Flow (OPF), two types of variables, named control variables and state variables, are defined for the RPP.

Normally, for a typical RPP, control variables are defined as generators' voltage magnitudes, transformers tap settings, and output reactive power of VAR sources. Considering a scenario-based approach to model the uncertainty of the problem, the control variables set (*U*) for a probabilistic RPP are expressed as Equation (8) [14–17].

$$\mathcal{U} = \begin{cases} V\_{\mathcal{G};s^\*} & i \in \Omega\_{\mathcal{G}'} \text{ s} \in \Omega\_{\mathfrak{s}} \\ t\_{k\_{i,s^\*}} & i \in \Omega\_{\text{TapCh}\_{\mathcal{H}}} \text{ s} \in \Omega\_{\mathfrak{s}} \\ Q\_{\mathbb{C}i\_{s^\*}} & i \in \Omega\_{\text{Cowp}\_{\mathcal{H}}} \text{ s} \in \Omega\_{\mathfrak{s}} \end{cases} \tag{8}$$

where *Vgi*,*<sup>s</sup>* shows the voltage magnitude of the *ith* generator for the *sth* scenario, *tki*,*<sup>s</sup>* is used to assign the settings of the *ith* tap-changing transformer for the *sth* scenario, and *QCi*,*<sup>s</sup>* shows the output reactive power of the *ith* VAR compensator device for the *sth* scenario. Likewise, <sup>Ω</sup>*g*, Ω*<sup>s</sup>*, <sup>Ω</sup>*TapCh*, and <sup>Ω</sup>*Comp* symbolize the set of generators, set of scenarios, set of tap-changing transformers, and set of VAR compensator devices, respectively.

The state variables in a typical RPP consist of the generated active power by the slack bus, the generated reactive power by each of the existing generators, the voltage magnitude of the load buses, and the flow of the transmission lines.

Using a scenario-based approach to model the uncertainty of the problem, the state variables set (*X*) for a probabilistic RPP are expressed as Equation (9) [14–17].

$$X = \begin{cases} P\_{G\_{\text{Slink},s}} & \text{s} \in \Omega\_{\text{s}} \\ Q\_{G\_{\text{i},s'}} & \text{i} \in \Omega\_{\text{s}'} \text{ s} \in \Omega\_{\text{s}} \\\ V\_{L\_{i,s'}} & \text{i} \in \Omega\_{PQ\_{\text{s}'}} \text{ s} \in \Omega\_{\text{s}} \\\ S\_{I\_{\text{s}}}^{From} & \text{l} \in \Omega\_{L \text{lines}}, \text{ s} \in \Omega\_{\text{s}} \\\ S\_{I\_{\text{i},s'}}^{To} & \text{l} \in \Omega\_{L \text{lines}}, \text{ s} \in \Omega\_{\text{s}} \end{cases} \tag{9}$$

where *PGSlack*,*<sup>s</sup>* indicates the generated active power by the slack generator (bus) for the *sth* scenario, *QGi*,*<sup>s</sup>* is used to denote the generated reactive power by the *ith* generator for the *sth* scenario, *VLi*,*<sup>s</sup>* shows the voltage magnitude of the *ith* load bus for the *sth* scenario, and *SFrom l*,*s* and *STo l*,*s* show the apparent power flow of the sending and receiving ends of the *lth* line for the *sth* scenario, respectively. Additionally, <sup>Ω</sup>*PQ* and Ω*Lines* specify the set of the load buses and the set of transmission lines, respectively.
