## 3 Problem Formulation
### Constraints
#### 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].  
\(\bullet\) Limits on the Control Variables  
The upper limit (\(V_{g_{i}}^{max}\)) and lower limit (\(V_{g_{i}}^{min}\)) of a generator voltage magnitude for the \(s^{th}\) scenario can be applied, as follows:\[V_{g_{i}}^{min}\leq V_{g_{i,s}}\leq V_{g_{i}}^{max},\qquad\forall i\in\Omega_{g}, \forall s\in\Omega_{s} \tag{31}\]  
For all tap-changing transformers in each scenario, the following constraint should be satisfied.  
\[t_{k_{i}}^{min}\leq t_{k_{i}}\leq t_{k_{i}}^{max},\qquad\forall i\in\Omega_{Tq \mathcal{C}_{h}},\forall s\in\Omega_{s} \tag{32}\]  
where \(t_{k_{i}}^{min}\) and \(t_{k_{i}}^{max}\) show the minimum and maximum settings of the \(i^{th}\) tap-changing transformer, respectively.  
The output reactive power of the VAR sources in each scenario is as follows:  
\[Q_{C_{i}}^{min}\leq Q_{C_{i,s}}\leq Q_{C_{i}}^{max},\qquad\forall i\in\Omega_{ Comp},\forall s\in\Omega_{s} \tag{33}\]  
where \(Q_{C_{i}}^{min}\) and \(Q_{C_{i}}^{max}\) show the minimum and maximum output reactive power of the \(i^{th}\) VAR compensator device, respectively.  
\(\bullet\) 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:  
\[P_{G_{Slack}}^{min}\leq P_{G_{Slack}}\leq P_{G_{Slack}}^{max},\qquad\forall s\in \Omega_{s} \tag{34}\]  
\[Q_{C_{i}}^{min}\leq Q_{G_{i,s}}\leq Q_{G_{i}}^{max},\qquad\forall i\in\Omega_{ g},\forall s\in\Omega_{s} \tag{35}\]  
where \(P_{G_{Slack}}^{min}\) and \(P_{G_{Slack}}^{max}\) indicate the maximum and minimum generated active power of the slack bus for the \(s^{th}\) scenario, respectively. In addition, \(Q_{C_{i}}^{min}\) and \(Q_{G_{i}}^{max}\) show the maximum and minimum generated reactive power of the \(i^{th}\) generator for the \(s^{th}\) 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:  
\[V_{L_{i}}^{min}\leq V_{L_{i,s}}\leq V_{L_{i}}^{max},\qquad\forall i\in\Omega_{ TQ},\forall s\in\Omega_{s} \tag{36}\]  
where \(V_{L_{i}}^{min}\) and \(V_{L_{i}}^{max}\) are considered as the lower and upper limits of the voltage magnitude at the \(i^{th}\) load bus for the \(s^{th}\) 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:  
\[S_{l,s}^{From}\leq S_{l}^{max},\qquad\forall l\in\Omega_{Lines},\forall s\in \Omega_{s} \tag{37}\]  
\[S_{l,s}^{To}\leq S_{l}^{max},\qquad\forall l\in\Omega_{Lines},\forall s\in \Omega_{s} \tag{38}\]  
where \(S_{l}^{max}\) indicates the maximum apparent flow of the \(l^{th}\) transmission line.