## 3 Energy Storage Siting Based on Sensitivity Analysis  
The selection of an energy storage site is based on sensitivity analysis, which expands the power balance equation of system nodes in accordance with [PERSON]'s first-order expansion. This is achieved by setting the reactive power change \(\Delta Q=0\) and the active power change \(\Delta P=0\) in order to obtain the relationship between voltage change \(\Delta V\), \(\Delta P\), and \(\Delta Q\). \[\Delta V=\left(\frac{\partial\Delta P}{\partial V}-\frac{\partial\Delta P}{ \partial\theta}\left(\frac{\partial\Delta Q}{\partial\theta}\right)^{-1} \frac{\partial\Delta Q}{\partial V}\right)^{-1}\Delta P \tag{1}\]  
\[\Delta V=\left(\frac{\partial\Delta Q}{\partial V}-\frac{\partial\Delta Q}{ \partial\theta}\left(\frac{\partial\Delta P}{\partial\theta}\right)^{-1} \frac{\partial\Delta P}{\partial V}\right)^{-1}\Delta Q \tag{2}\]  
In (1) and (2), V and \(\theta\) are the voltage amplitude and phase angle, respectively. Further expansion of (1) and (2) leads to the following equation:  
\[\left[\Delta V_{1}\;\cdots\;\Delta V_{j}\;\cdots\;\Delta V_{(n-1)}\right]^{T} = \begin{bmatrix}\lambda_{11}&\cdots&\lambda_{1i}&\cdots&\lambda_{1(n-1)}\\ \vdots&&\vdots&&\vdots\\ \lambda_{j1}&\cdots&\lambda_{ji}&\cdots&\lambda_{j(n-1)}\\ \vdots&&\vdots&&\vdots\\ \lambda_{(n-1)1}&\cdots&\lambda_{(n-1)i}&\cdots&\lambda_{(n-1)(n-1)}\end{bmatrix} \begin{bmatrix}\Delta P_{1}\\ \vdots\\ \Delta P_{i}\\ \vdots\\ \Delta P_{(n-1)}\end{bmatrix}  \tag{3}\]  
\[\left[\Delta V_{1}\;\cdots\;\Delta V_{j}\;\cdots\;\Delta V_{(n-1)}\right]^{T} = \begin{bmatrix}\tau_{11}&\cdots&\tau_{1i}&\cdots&\tau_{1(n-1)}\\ \vdots&&\vdots&&\vdots\\ \gamma_{j1}&\cdots&\gamma_{ji}&\cdots&\gamma_{j(n-1)}\\ \vdots&&\vdots&&\vdots\\ \tau_{(n-1)1}&\cdots&\gamma_{(n-1)i}&\cdots&\gamma_{(n-1)(n-1)}\end{bmatrix} \begin{bmatrix}\Delta Q_{1}\\ \vdots\\ \Delta Q_{i}\\ \vdots\\ \Delta Q_{(n-1)}\end{bmatrix}  \tag{4}\]  
In (3) and (4), \(n\) is the number of nodes. \(\Delta V_{j}\) is the voltage variation of node \(j\). \(\Delta P_{i}\) and \(\Delta Q_{i}\) represent the active power change and reactive power change of node \(i\), respectively. \(\lambda_{ij}\) and \(\gamma_{ij}\) are the sensitivity of the active power change and the reactive power change of node \(i\) to the voltage of node \(j\), respectively. The matrix composed of \(\lambda_{ij}\) and \(\gamma_{ij}\) is the sensitivity matrix of active and reactive power, respectively. Considering the impact of power changes at a certain node on the voltage of each node in the distribution system, the sensitivity of power changes at node \(i\) to the voltage at node \(j\) in the system is as follows. \[\delta_{ji}=\omega_{1}\lambda_{ji}+\omega_{2}\gamma_{ji} \tag{5}\]  
In (5), \(\omega_{1}\) and \(\omega_{2}\) are the sensitivity weight coefficients for active and reactive voltage, respectively, and \(\omega_{1}+\omega_{2}=1\). The calculation method for the weight coefficients A and B will be explained below. The ideal point method [15] represents a comprehensive evaluation approach that is capable of objectively, fairly, and reasonably assessing the evaluated object. In the ideal point method, assuming there are m evaluation indicators, the optimal value of each indicator is defined as the ideal point \(X^{*}=(x_{1}^{*},x_{2}^{*},\cdots,x_{m}^{*})\), and the actual value of the \(i\)th evaluated object for each indicator is \(\mathbf{X}_{i}=(x_{i1},x_{i2},\cdots,x_{im})\). The optimality of the \(i\)th evaluatedobject among all evaluated objects is evaluated using the weighted distance between \(X_{i}\) and the ideal point \(X^{*}\). In the majority of cases, the Euclidean distance is employed, which is defined as follows. \[y_{i}=\sum_{j=1}^{m}\omega_{j}(x_{ij}-x_{j}^{*})^{2} \tag{6}\]  
In (6), \(y_{i}\) is the Euclidean distance of the \(i\)th evaluated object. \(\omega_{j}\) is the weight coefficient of the \(j\)th evaluation indicator. \(x_{ij}\) is the calculated value of the \(j\)th evaluation indicator for the \(i\)th evaluated object.