## 3 Problem Formulation
### Objective Functions
#### 3.2.2 Minimization of Voltage Stability Index  
In this paper, the \(L\)-index is proposed as the voltage stability index that is a well-known static voltage index [31]. In order to estimate the static voltage stability of the power system, the \(L\)-index should be calculated for all load buses (\(PQ\) buses). All the load buses that have higher values of the \(L\)-index than others are considered as the weak buses. Weak buses mostly suffer from a lack of reactive power and are prone to the voltage collapse. Equation (14) can be used to calculate the \(L\)-index (\(L_{j}\)) for the \(j^{th}\) load bus, as follows [31]:  
\[L_{j}=\left|1-\sum_{i=1}^{\Omega_{s}}F_{ji}\frac{\overline{V_{i}}}{\overline{ V_{j}}}\right|,\qquad\forall j\in\Omega_{PQ} \tag{14}\]  
where \(\overline{V_{i}}\) shows the voltage of the \(i^{th}\) generator, and \(\overline{V_{j}}\) represents the voltage of the \(j^{th}\) load bus. \(F_{ji}\) can be derived from \(Y_{bus}\) matrix of the system. Thus, by rearranging the current and voltage equations in power systems, as shown in Equation (15), the consecutive \(Y_{bus}\) matrix is achieved. Thereafter, using the arrays of the consecutive \(Y_{bus}\) matrix, the \(F_{ji}\) matrix can be calculated as Equation (16). \[\left[ \begin{array}{c}I_{g}\\ I_{l}\end{array} \right]=\left[ \begin{array}{cc}Y_{gg}&Y_{gl}\\ Y_{lg}&Y_{ll}\end{array} \right]\left[ \begin{array}{c}V_{g}\\ V_{l}\end{array} \right] \tag{15}\]  
\[F_{ji}=-[Y_{ll}]^{-1}Y_{lg},\qquad\forall j\in\Omega_{PQ},\forall i\in\Omega_{g} \tag{16}\]  
where \(I_{g}\) and \(I_{l}\) show the current of generators and loads, respectively, and \(V_{g}\) and \(V_{l}\) are the voltage of generators and loads, respectively. In addition, \(Y_{gg}\), \(Y_{gl}\), \(Y_{lg}\), and \(Y_{ll}\) are the submatrices of the consecutive \(Y_{bus}\) matrix. It should be noted that only \(Y_{ll}\) arrays of the consecutive \(Y_{bus}\) matrix are related to the PQ nodes. Also, the consecutive \(Y_{bus}\) matrix is a symmetric matrix. Therefore, \(Y_{lggl}=Y_{lg}\). By minimizing the values of the \(L\)-index at the weak buses, there is a possibility to increase the level of static voltage stability in power systems. The voltage stability of power systems can be determined by the \(L\)-index when the maximum value of the \(L\)-index (\(L_{max}\)) is assigned to the static voltage stability level in power systems, as follows:  
\[L_{max}=\max\Bigl{(}L_{j}\Bigr{)},\qquad\forall j\in\Omega_{PQ} \tag{17}\]  
To improve the static voltage stability of power systems, it is necessary to minimize \(L_{max}\). It should be noted that the equations proposed for the \(L\)-index are related to the deterministic problem. In the case of a probabilistic problem, considering all necessary modifications on the \(Y_{bus}\) matrix in each scenario, after re-formulating Equations (14)-(17), new equations can be rewritten as follows:  
\[L_{js}=\left|1-\sum_{i=1}^{\Omega_{S}}F_{ji,s}\frac{\overline{V_{i,s}}}{ \overline{V_{j,s}}}\right|,\qquad\forall j\in\Omega_{PQ},\ \forall s\in\Omega_{s} \tag{18}\]  
where \(L_{j,s}\) indicates L-index value for the \(j^{th}\) load bus and \(s^{th}\) scenario, \(\overline{V_{i,s}}\) and \(\overline{V_{j,s}}\) are the voltage of the \(i^{th}\) generator and \(j^{th}\) load bus for the \(s^{th}\) scenario, respectively. For each scenario, the \(F_{ji,s}\) matrix can be calculated as Equation (20). \[\left[ \begin{array}{c}I_{g,s}\\ I_{l,s}\end{array} \right]=\left[ \begin{array}{cc}Y_{gg,s}&Y_{gl,s}\\ Y_{lg,s}&Y_{ll,s}\end{array} \right],\qquad\forall s\in\Omega_{s} \tag{19}\]  
\[F_{ji,s}=-\left[Y_{ll,s}\right]^{-1}Y_{lg,s},\qquad\forall j\in\Omega_{PQ}, \forall i\in\Omega_{g},\forall s\in\Omega_{s} \tag{20}\]  
where \(I_{g,s}\) and \(I_{l,s}\) denote the current of generators and loads for the \(s^{th}\) scenario, respectively, \(V_{g,s}\) and \(V_{l,s}\) are the voltage of generators and loads for the \(s^{th}\) scenario, respectively.