## 3 Formulation of HRESs
### Total Annual Cost Modeling and Constraints
#### 3.6.1 Objective Function Formulation  
The objective function evaluates the optimal number of the HRES's components to satisfy the user's load at minimum TAC. The total TAC is achieved by summing the capital cost and maintenance cost as given by the following formula:  
\[Minimize\quad\xi^{tot}=\xi^{cap}+\xi^{mtn}, \tag{10}\]  
where \(\xi^{tot}\), \(\xi^{cap}\), and \(\xi^{mtn}\) show the total, capital, and maintenance costs, respectively.  
To convert the initial capital cost into the annual capital cost, the capital recovery factor (CRF) approach is used, which is obtained using the formula given below [29].  
\[CRF=\frac{i^{rat}(1+i^{rat})^{n}}{(1+i^{rat})^{n}-1}, \tag{11}\]  
where \(i^{rat}\) and \(n\) show the interest rate and the system's life span in years. The values of these parameters and other HRESs' components are summarized in Table 1. Furthermore, the life cycle \(n\) is twenty years for this study.  
Several components used in HRESs need to be replaced during the project life span, i.e., the batteries' estimated life is five years. Similar to the approach used in [29], the present worth factor via the single payment of a battery is derived using the formula:  
\[\zeta^{bat}=\rho^{bat}\times\left(1+\frac{1}{(1+i^{rat})^{5}}+\frac{1}{(1+i^{rat })^{10}}+\frac{1}{(1+i^{rat})^{15}}\right), \tag{12}\]  
where \(\zeta^{bat}\) and \(\rho^{bat}\) represent the battery's present worth and price, respectively.  
The life time of the inverter/converter used for the HRES system is estimated as ten years, which can be obtained by the formula:  
\[\zeta^{ic}=\rho^{ic}\times\left(1+\frac{1}{(1+i^{rat})^{10}}\right), \tag{13}\]  
where \(\zeta^{ic}\) and \(\rho^{ic}\) depict the present worth of the inverter/converter and their price, respectively.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Hybrid System Components** & **Parameters** & **Value** \\ \hline \multirow{6}{*}{PV panel} & \(p_{rat}^{pv}\) & 120 W \\  & \(\zeta^{pv}\) & \$614 \\  & \(\zeta^{pv}_{mtn}\) & \$0 \\  & \(A^{pv}\) & 1.07 m\({}^{2}\) \\  & \(\eta^{pv}\) & 12\% \\  & \(T^{cel}_{nct}\) & 33 °C \\ \hline \multirow{6}{*}{WT} & \(p_{mm}^{vt}\) & 1 kW \\  & \(v^{ct}\) & 2.5 m/s \\  & \(v^{rt}\) & 11 m/s \\  & \(v^{cuo}\) & 13 m/s \\  & \(\zeta^{bat}_{v}\) & \$3200 \\  & \(\zeta^{bat}_{vmtn}\) & \$100 \\ \hline \multirow{6}{*}{Battery} & _Voltage_ & 12 V \\  & Battery nominal capacity & 1.3 kWh \\ \cline{1-1}  & Life span & 5 years \\ \cline{1-1}  & \(\eta^{bat}\) & 85\% \\ \cline{1-1}  & \(\rho^{bat}\) & \$130 \\ \cline{1-1}  & \(DoD\) & 0.8 \\ \cline{1-1}  & \(\epsilon^{dr}\) & 0.0002 \\ \hline \multirow{6}{*}{Power inv/conv} & \(p_{rat}^{ic}\) & 3 kW \\ \cline{1-1}  & Life span & 10 years \\ \cline{1-1}  & \(\eta^{i}\) & 95\% \\ \cline{1-1}  & \(\rho^{ic}\) & \$2000 \\ \hline \multirow{2}{*}{Other parameters} & \(i^{rat}\) & 5\% \\ \cline{1-1}  & n & 20 years \\ \hline \hline \end{tabular}
\end{table}
Table 1: PV-WT-battery hybrid system components and parameters [29].  
By breaking apart the HRESs' system into the annual costs of PVs, WTs, batteries, and the inverter/converter, Equation (14) is achieved.  
\[\zeta^{cap}=CRF\times\left[N^{wt}\times\zeta^{wt}+N^{pv}\times\zeta^{pv}+N^{b} \times\zeta^{bat}+N^{ic}\times\zeta^{ic}\right], \tag{14}\]  
where \(\zeta^{wt}\), \(\zeta^{pv}\), \(\zeta^{bat}\), and \(\zeta^{ic}\) represent the unit cost of the WT, PV, battery unit, and inverter/converter, respectively. \(N\) depicts the number of each component in Equation (14). For this study, the number of inverters/converters \(N^{ic}\) used is assumed as one.  
The annual maintenance cost of the system is calculated using the formula:  
\[\zeta^{mtn}=N^{pv}\times\zeta^{pv}_{mtn}+N^{wt}\times\zeta^{wt}_{mtn}, \tag{15}\]  
where \(\zeta^{pv}_{mtn}\) and \(\zeta^{wt}_{mtn}\) represent the annual maintenance costs of PVs and WTs, respectively. In this study, the maintenance costs of the battery and inverter/converter are also not considered.