3.3.2. Inequality Constraints

To keep both the control and state variables within their specific limits, another set of constraints is added to the problem, named as inequality constraints. Those constraints involve Equations (31)–(38) [24,25].

### • Limits on the Control Variables

The upper limit (*Vmax gi* ) and lower limit (*Vmin gi* ) of a generator voltage magnitude for the *sth* scenario can be applied, as follows:

*Appl. Sci.* **2020**, *10*, 2859

$$V\_{\mathcal{G}i}^{\min} \le V\_{\mathcal{G}i,s} \le V\_{\mathcal{G}i}^{\max}, \qquad \forall i \in \Omega\_{\mathcal{G}'} \,\forall s \in \Omega\_s \tag{31}$$

For all tap-changing transformers in each scenario, the following constraint should be satisfied.

$$t\_{k\_i}^{\min} \le t\_{k\_{i^\*}} \le t\_{k\_i}^{\max}, \qquad \forall i \in \Omega\_{\text{TapCh}^\*} \,\forall s \in \Omega\_s \tag{32}$$

where *tmin ki* and *tmax ki* show the minimum and maximum settings of the *ith* tap-changing transformer, respectively.

The output reactive power of the VAR sources in each scenario is as follows:

$$\mathcal{Q}\_{\mathbb{C}\_i}^{\min} \le \mathcal{Q}\_{\mathbb{C}\_{i^\*}} \le \mathcal{Q}\_{\mathbb{C}\_i}^{\max}, \qquad \forall i \in \Omega\_{\mathbb{C}\_{\text{supp}}}, \forall s \in \Omega\_s \tag{33}$$

where *Qmin Ci* and *Qmax Ci* show the minimum and maximum output reactive power of the *ith* VAR compensator device, respectively.

### • Limits on the State Variables

In terms of the generation units, for each scenario, two important constraints should be satisfied; (1) the limitation on the generated active power of the slack bus and (2) the limitation on the generated reactive power of each generation unit. Those constraints are given as follows:

$$P\_{G\_{\rm Slack}}^{\rm min} \le P\_{G\_{\rm Slack},s} \le P\_{G\_{\rm Slack}'}^{\rm max} \qquad \forall s \in \Omega\_{\rm \\$} \tag{34}$$

$$\Omega \bigotimes\_{\mathcal{G}\_i}^{\text{min}} \le \mathcal{Q}\_{\mathcal{G}\_{i\boldsymbol{\mu}}} \le \mathcal{Q}\_{\mathcal{G}\_i}^{\text{max}}, \qquad \forall i \in \Omega\_{\mathcal{G}'} \,\forall \mathbf{s} \in \Omega\_{\mathbf{s}} \tag{35}$$

where *Pmin GSlack* and *Pmax GSlack* indicate the maximum and minimum generated active power of the slack bus for the *sth* scenario, respectively. In addition, *Qmin Gi* and *Qmax Gi* show the maximum and minimum generated reactive power of the *ith* generator for the *sth* scenario, respectively.

In order to prevent the voltage collapse or insulating problems, it is required to limit the voltage magnitude of loads for each scenario, as follows:

$$V\_{L\_i}^{\min} \le V\_{L\_{i\rho}} \le V\_{L\_i}^{\max}, \qquad \forall i \in \Omega\_{PQ\_{\prime}}\\ \forall s \in \Omega\_s \tag{36}$$

where *Vmin Li* and *Vmax Li* are considered as the lower and upper limits of the voltage magnitude at the *ith* load bus for the *sth* scenario, respectively.

To reduce the risk of overload in transmission lines, the apparent flow of the transmission lines should be lower than a specified value. Equations (37) and (38) enforce the apparent flow of transmission lines to be at the secure level, as follows:

$$S\_{l,s}^{\text{Fround}} \le S\_l^{\text{max}}, \qquad \forall l \in \Omega\_{l,\text{lines}},\\\forall s \in \Omega\_s \tag{37}$$

$$
\Delta S\_{l,s}^{To} \le S\_l^{\max}, \qquad \forall l \in \Omega\_{l.\text{inv}s}, \forall s \in \Omega\_{\text{\\$}} \tag{38}
$$

where *Smax l*indicates the maximum apparent flow of the *lth* transmission line.

### *3.4. Other Considerations in the Problem Formulation*

There are other considerations in the problem formulation, which are listed as follows:

• The transformers tap settings and output reactive power of the VAR sources are treated as continuous variables. Therefore, the whole problem is stated as a probabilistic multi-objective nonlinear problem.

