## 2 Optimal Power Flow Problem
Objective functions:**  
The mono-objective optimization problem is considered for eight cases of objective function \(f(x,u)\) in Equation (1):  
Case 1: minimization of total fuel cost  
The total cost of electrical power generation is expressed as an objective function (quadratic function) [9] by:  
\[f\ =\ F_{\rm C}\ =\ \sum_{i\ =\ 1}^{Ng}F_{i}(P_{Gi})\ =\ \sum_{i\ =\ 1}^{Ng}(a_{i}+b_{i}P_{Gi}+c_{i}P_{Gi}^{2}) \tag{12}\]  
where \(a_{i}\), \(bi\), and \(c_{i}\) are the cost coefficients of the \(i\)-\(th\) generating unit. Case 2: minimizing total gas emissions  
The gasses emitted by each generating unit can be expressed as a combination of the quadratic and exponential functions of the generated active power [10], and the total amount of gas emissions produced by all generating units:  
\[f\ =\ F_{Em}\ =\ \sum_{i\ =\ 1}^{Ng}(\alpha_{i}+\beta_{i}P_{Gi}+y_{i}P_{Gi}^{2 }+\delta_{i}exp(\epsilon_{i}P_{Gi}))\ \ (\text{ton/h}) \tag{13}\]  
where \(F_{Em}\) is the total gas emissions (ton/h) and \(\alpha_{i}\), \(\beta_{i}\), \(\gamma_{i}\), \(\delta_{i}\), and \(\epsilon_{i}\) are the emission coefficients of the \(i\)-th unit. Case 3: minimization of total fuel cost considering the valve point effect  
A sine component is added to the expression of the objective function in (12) to consider the effect of the valve point to evaluate the total fuel cost as follows [11]:  
\[f\ =\ F_{\rm C}\ =\ \sum_{i\ =\ 1}^{Ng}F_{i}(P_{Gi})\ =\ \sum_{i\ =\ 1}^{Ng}(a_{i}+b_{i}P_{Gi}+C_{i}P_{Gi}^{2}+f_{epi}) \tag{14}\]  
where \(f_{epi}\ =\ 1_{i}\cdot\sin\left[\epsilon_{i}(P_{Gi,min}-P_{Gi})\right]1\), while \(e_{i}\) and \(d_{i}\) are the fuel cost coefficients of the since component. Case 4: minimization of total active losses  
The total active transmission losses for the power system can be expressed as an objective function:  
\[f\ =\ F_{Plosses}\ =\ \sum_{k\ =\ 1}^{N_{l}}g_{k}\left[V_{i}^{2}+V_{j}^{2} -2V_{i}V_{j}\cos(\delta_{i}-\delta_{j})\right] \tag{15}\]  
where \(g_{k}\) is the conductance of the \(k\)-th transmission line between buses \(i\) and \(j\). Case 5: minimization of the total load bus voltage deviation  
This optimization case is devoted to the minimization of total voltage magnitude deviation related to load buses with the goal of enhancing power system security; the total voltage deviation function for all load buses is given by:  
\[T_{i,min}\leq T_{i}\leq T_{i,max}\ \ i\ =\ 1,\ldots N_{t} \tag{7}\]0  
Case 6: improvement of voltage stability index  
The stability of a power system is its capacity to maintain all bus voltages within normal limits. The power system reaches a voltage instability state due to severe perturbations, rising electric load, or network configuration changes, which occasions a progressive and unexpected voltage drop. The improvement of the system voltage stability for each bus is a fundamental measure to ensure a safe and stable power system using voltage stability (L-index). Its value varies between 0 and 1 (0 corresponds to a zero-load situation, while 1 means a voltage drop). If the power system has \(N_{PQ}\) load buses (PQ buses) and \(N_{PV}\) generator buses (PV buses), the L-index is evaluated by [32; 33]:  
\[T_{i,min}\leq T_{i}\leq T_{i,max}\ \ i\ =\ 1,\ldots N_{t} \tag{7}\]1  
where \(V_{Gi}\) is the complex voltage of the generation bus \(i\) and \(V_{Lj}\) is the complex voltage of the load bus \(j\). \[T_{i,min}\leq T_{i}\leq T_{i,max}\ \ i\ =\ 1,\ldots N_{t} \tag{7}\]2  
where the sub-matrices \(Y_{LL}\) and \(Y_{LG}\) have been obtained from the YBUS matrix after rearranging nodal current injections with respect to nodal voltages, as defined as:  
\[ \begin{bmatrix}I_{L}\\ I_{G}\end{bmatrix} \ =\ \begin{bmatrix}Y_{LL}Y_{LG}\\ Y_{GL}Y_{GG}\end{bmatrix}\begin{bmatrix}V_{L}\\ V_{G}\end{bmatrix}