## 2 Methodology
### Model Formulation
#### 2.1.2 The Objective Function  
The objective is formulated to balance profitability with risk management. It is described in (1) as:  
\[Maximize\sum_{t\in T}\sum_{icP}\sum_{seqS}\pi_{s}\cdot R_{i,t,s}\cdot I_{i,t}- \lambda\cdot\left(\theta+\frac{1}{1-\alpha}\sum_{seq}\pi_{s}\cdot\eta_{s}\right) \tag{1}\]  
where the key symbols and parameters are summarized in Table 1.  
Equation (1) is adopted based on the optimization framework introduced by [PERSON], [PERSON] [54], which integrates risk-adjusted returns into portfolio optimization. The first term, weighted for its relative probabilities, reflects the predicted returns across all projects, time phases, and scenarios. The second term includes CVaR, penalizing the target depending on the risk of severe losses and matching the optimization with risk-averse investing methods. The objective function described in (1) is subject to the following constraints defined in the preceding equations.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(T\) & set of time stages \\ \hline \(P\) & set of renewable projects \\ \hline \(S\) & set of scenarios \\ \hline \(\pi_{s}\) & probability of the scenario \(S\) \\ \hline \(R_{i,t,s}\) & expected return on investment for the project \\ \(i\) at time stage \(t\) \\ \hline \(\lambda\) & risk aversion parameter \\ \hline \(\alpha\) & confidence level for \(CVaR\) \\ \hline \(\theta\) & Value at Risk (VaR) at the confidence level \(\alpha\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key Symbols and Parameters.  
The budget constraints described in (2) make sure the investment in each time stage does not exceed the allocated budget, where \(B_{t}\) represents the budget allocated for the time stage \(t\).  
\[\sum_{i\in P}I_{i,t}\leq B_{t}\qquad\forall t\in T \tag{2}\]  
The investment in each project is restricted to its utmost additional capacity due to the capacity constraints outlined in (3), where \(C_{i}^{max}\) and \(C_{i}^{existing}\) are the maximum and existing capacities of the project \(i\), respectively.  
\[C_{i,t}\leq C_{i}^{max}-C_{i}^{existing}\quad\forall i\in P,\ \forall t\in T \tag{3}\]  
The diversification constraints, as shown in (4), mitigate excessive investment in any singular technology to foster portfolio diversification, where \(P_{tech}\) denotes the collection of projects associated with a particular technology.  
\[\sum_{i\in P_{tech}}I_{i,t}\leq 0.5\cdot B_{t}\quad\forall tech\ \in\ \{solar,\ wind,\ hydro\},\ \forall t\in T \tag{4}\]  
The policy constraints, as shown in (5), make sure operational capacities under all scenarios comply with regulatory requirements, where \(\delta_{i}\) denotes the minimum operational capacity fraction for the project \(i\).  
\[O_{i,t,s}\geq\delta_{i}\cdot\left(C_{i}^{existing}+C_{i,t}\right)\ \forall i\in P,\ \forall t\in T,\ \forall s\in S \tag{5}\]  
The constraints described in (6) ensure that operational capacity does not exceed available capacities.  
\[O_{i,t,s}\leq C_{i}^{existing}+C_{i,t}\ \ \forall i\in P,\ \forall t\in T,\ \forall s\in S \tag{6}\]  
The constraints described in (7) and (8) show the relationship between \(\theta\), \(\ \eta_{s}\), and portfolio returns to accurately model the CVaR. It captures how much a portfolio in adverse scenarios and is key to measuring average loss beyond cetain threshold.  
\[\eta_{s}\leq\theta-\sum_{t\in T}R_{i,t,s}\cdot I_{i,t}\ \ \forall s\in S \tag{7}\]  
\[\eta_{s}\geq 0\ \ \ \forall s\in S \tag{8}\]  
The minimum CVaR restriction depicted in Equation (8) guarantees that it is nonzero, allowing for insightful risk evaluations.  
\[\theta+\frac{1}{1-\alpha}\sum_{s\in S}\pi_{s}\cdot\eta_{s}\geq Minimum\ \ \text{CVaR} \tag{9}\]