## 2 Materials and Methods
### Case Study of the Selected Area: The Skikda Region
#### 2.2.3 Problem Formulation Based on a Multi-Objective Optimization Approach  
Multi-objective optimization is defined as the task of identifying a vector of decision variables that satisfies a set of constraints while optimizing a vector-valued objective function ([PERSON] [26]). Each component of this function represents a distinct performance criterion, which is typically in conflict with the others. In this context, \"optimization\" refers to finding a solution that yields values for all objectives that are acceptable to the designer. In mathematical terms, a multi-objective optimization problem can be represented as follows:  
**Find**: a vector \(X^{*}\) that:
**Minimize**: \(F(X)=[f_{1}(P),f_{2}(X),\cdots,f_{k}(X)]^{T}\) subject to \(g_{m}(X)\leq 0\) (\(m\) inequality constraints). and \(h_{l}(X)=0\) (\(l\) equality constraints). \(X^{*}\in\mathbf{R}^{n}\) : Vector of the decision variables. \(F(X^{*})\in\mathbf{R}^{k}\) : Vector of the objectives function. In this study, the decision variables are represented by the vector \(X\), which is defined as follows:  
\[X=(x_{1},x_{2}),\]where: \(x_{1}\): Number of PV panels. \(x_{2}\): Number of wind turbines. The primary focus in designing the hybrid power system is to determine the optimal sizing of each component to ensure the load is met both economically and reliably. The renewable energy sources considered include PV panels (\(E_{pv}\)) and wind turbines (\(E_{e}\)), along with the unit costs of each component, \(C_{pv}\) and \(C_{e}\). The optimization model intrinsically incorporates site-specific meteorological inputs--hourly wind-speed and solar irradiation time series from previous studies (see references [3; 27])--which feed directly into the annual energy-yield simulations and, in turn, inform our equipment cost estimates. Therefore, two objective functions can be defined: * The aim to maximize the electrical energy produced by the hybrid system, which is represented by \(F_{1}\) and is defined as follows: \[F_{1}(x)=E_{pv}x_{1}+E_{e}x_{2}.\] (3) * The purpose to minimize the total cost represented by \(F_{2}\), which is defined as follows: \[F_{2}(x)=C_{pv}x_{1}+C_{e}x_{2}.\] (4) The constraints that can affect the system are defined as follows: Number of PV: \(x_{1_{min}}\leq x_{1}\leq x_{1_{max}}\). Number of wind turbines: \(x_{2_{min}}\leq x_{2}\leq x_{2_{max}}\). Unit cost of PV: \(C_{pv}=161.19\) $. Unit cost of wind turbine: \(C_{e}=1492.54\) $. Maximum energy produced by a PV: \(E_{pv}=0.240\) kWh. Maximum energy produced by a wind turbine: \(E_{e}=0.900\) kWh. Minimum load to be satisfied \(E_{D}=24\) kWh. _Determination of_ \(\textbf{x}_{1_{min}},\textbf{x}_{1_{max}},\textbf{x}_{2_{min}},\textbf{x}_{2_ {max}}\). These extreme values are determined using the complementarity study, and, according to Table 2 (see Appendix B.2), it can be seen that the following apply: * The temperature is important during the period from March to September. Hence, the electrical energy produced by the PVs is at the maximum (maximum efficiency), whereas the electrical energy produced by the wind turbines is negligible. In the interest of satisfying the load of the system in this period, the energy produced will be based solely on the PVs, which allows us to determine the maximum number of PVs: \[x_{1_{max}}=\frac{system\ load}{E_{pv}}=\frac{24}{0.24}=100.\] (5) * The wind speed is very important between the months of January to April; therefore, the electrical energy produced by the wind turbines is at the maximum (maximum efficiency). On the other hand, the electrical energy produced by the PVs is negligible. In order to satisfy the load of the system in this period, the energy produced will be based solely on the wind turbines. This allows us to determine the maximum number of wind turbines: \[x_{2_{max}}=\frac{Charge}{E_{e}}=\frac{24}{0.9}=27.\] (6)Finally, the problem can be formulated as follows:  
**Find the vector \(X=[x_{1},x_{2}]\)** that  
\[\left\{ \begin{array}{l}max\ E_{p}=E_{pv}\ x_{1}+E_{e}\ x_{2}\\ min\ C_{tot}=C_{pv}\ x_{1}+C_{e}\ x_{2}\end{array}