## 2 Feasibility Analysis and Modeling of Energy Storage in Transmission Network Expansion Planning
### Constraints  
(1) Energy storage investment constraintsThe nodes in the grid suitable for installing energy storage stations are limited:  
\[\sum x_{ess,j,Y}\leq x_{ness}^{\max} \tag{14}\]  
where \(x_{ness}^{\max}\) represents the maximum number of energy storage installations allowed by the system. (2) Transmission Expansion Constraints  
Throughout the entire planning period, the constraints on the number of transmission lines must be met:  
\[0\leq\sum_{Y\in N_{Y}}\sum_{ \begin{array}{c}(i,j)\in N_{p}\\ i\
eq j\end{array} }x_{\text{line},ij,Y}^{p}\leq x_{\text{line},ij}^{\max} \tag{15}\]  
where \(x_{line,ij}^{\max}\) represents the maximum number of new lines between nodes \(i\) and \(j\). (3) Node Power Balance Constraints  
\[ \begin{array}{c}A^{o}P_{b,t,s,Y}^{o}+\sum\limits_{y=1}^{Y-1}A^{y}P_{b,t,s,Y}^ {y}+A^{p}P_{b,t,s,Y}^{p}+P_{gen,t,s,Y}+\\ P_{wind,t,s,Y}+P_{solar,t,s,Y}=P_{I,t,s,Y}+\sum\limits_{y=1}^{Y-1}P_{ess,t,s,Y}^{y}+P_{css,t,s,Y}\end{array}  \tag{16}\]  
where: \(A^{y}\) is the node-branch incidence matrix for new lines in year \(y\). \(A^{o}\) and \(A^{p}\) are the node-branch incidence matrices for the initial and candidate lines, respectively. \(P_{b,t,s,Y}^{o}\) and \(P_{b,t,s,Y}^{p}\) are the active power vectors for initial and candidate lines in scenario \(s\) during period \(t\) of year \(Y\). \(P_{b,t,s,Y}^{y}\) is the active power vector for new lines in year \(y\) during scenario \(s\) of period \(t\). \(P_{gen,t,s,Y}\), \(P_{wind,t,s,Y}\) and \(P_{solar,t,s,Y}\) are the active power output vectors of thermal plants, wind farms, and solar power stations in scenario \(s\) during period \(t\) of year \(Y\). \(P_{css,t,s,Y}\) is the charging and discharging power vector for new energy storage in scenario \(s\) during period \(t\) of year \(Y\). \(P_{i,t,s,Y}\) is the load vector at each node in scenario \(s\) during period \(t\) of year \(Y\). (4) Branch Power Flow Constraints  
\[P_{ij,t,s,Y}^{o}-b_{ij}n_{\text{line},ij}^{o}\big{(}\theta_{i,t,s,Y}-\theta_{ j,t,s,Y}\big{)}=0,\forall(i,j)\in N_{p}^{0} \tag{17}\]  
\[P_{ij,t,s,Y}^{y}-b_{ij}n_{\text{line},ij}^{y}\big{(}\theta_{i,t,s,Y}-\theta_{ j,t,s,Y}\big{)}=0,\ \forall y\in(1,2,\ldots,Y-1),\forall(i,j)\in N_{p} \tag{18}\]  
\[P_{ij,t,s,Y}^{p}-x_{\text{line},ij,Y}^{p}b_{ij}\big{(}\theta_{i,t,s,Y}-\theta_ {j,t,s,Y}\big{)}=0,\]  
\[\forall(i,j)\in N_{p},\forall p\in\left(1,2,\ldots,n_{\text{line},ij}^{\max}- \sum\limits_{y=1}^{Y-1}n_{\text{line},ij}^{y}\right) \tag{19}\]  
\[0\leq n_{\text{line},ij}^{y}\leq n_{\text{line},ij}^{\max} \tag{20}\]  
\[\theta_{\text{ref},t,s,Y}=0 \tag{21}\]  
where: \(b_{ij}\) is the susceptance of a single line between nodes \(i\) and \(j\). \(n_{\text{line},ij}^{o}\) and \(n_{\text{line},ij}^{\max}\) are the number of initial and candidate lines, respectively. \(n_{\text{line},ij}^{y}\) is the number of new transmission lines in \(y\). \(P_{ij,t,s,Y}^{o}\) is the total active power flow through the initial branch \(ij\) in scenario \(s\) during period \(t\) of year \(Y\). \(P_{ij,t,s,Y}^{y}\) is the total active power flow through new branches in the scenario \(s\) during period \(t\) of year \(Y\). \(P_{ij,t,s,Y}^{p}\) is the active power flow through the new transmission line in the scenario (\(s\)) during a period (\(t\)) of year \((Y)\). \(\theta_{i,t,s,Y}\), \(\theta_{j,t,s,Y}\) and \(\theta_{ref,t,s,Y}\) are the voltage phase angles at nodes \(i\), \(j\), and the reference node \(ref\) in scenario (\(s\)) during period (\(t\)) of year \((Y)\). Equation (17) represents the power flow constraint for the initial branches of the system; Equation (18) represents the power flow constraint for the new branches constructed in \(y\) for \(Y\); Equation (19) represents the power flow constraint for the candidate branches to be constructed in \(Y\); Equation (20) limits the number of new branches constructed in \(y\) to be less than the maximum number of branches that can be expanded. (5) Branch Power Limit Constraints  
\[\left\{