## 3 Solution Method
### Brief Introduction of Second-Order Cone Programming and Second-Order Cone Programming Relaxation  
The standard form of the SOCP is  
\[\left\{\begin{array}{l}\min\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!  
Firstly, two new variables, \(V_{j}\) and \(l_{ij}\), are introduced. Let \(V_{j}=U_{j}^{2}\) and \(l_{ij}=(P_{ij}^{2}+Q_{ij}^{2})/V_{i}\). In this way, constraints (4), (6), (8), and (26) can be restated as follows:  
\[C_{\text{loss}}=c_{\text{loss}}\sum\limits_{t=1}^{T}\sum\limits_{(i,j)\in \psi_{j}}r_{ij}l_{ij,t}, \tag{28}\]  
\[C_{\Delta U}=c_{\Delta U}\sum\limits_{t=1}^{T}\Delta U_{t}=c_{\Delta U}\sum \limits_{t=1}^{T}\sum\limits_{i\in\psi_{b}}\left|1-(V_{i,t}/U_{N}^{2})\right|, \tag{29}\]  
\[\left\{\begin{array}{l}\sum\limits_{i\in u(j)}P_{ij}-r_{ij}l_{ij}=\sum \limits_{k\in v(j)}P_{jk}+P_{j}\\ \sum\limits_{i\in u(j)}Q_{ij}-x_{ij}l_{ij}=\sum\limits_{k\in v(j)}Q_{jk}+Q_{j }\\ V_{j}=V_{i}-2(r_{ij}P_{ij}+x_{ij}Q_{ij})+(r_{ij}^{2}+x_{ij}^{2})l_{ij}\end{array} \right., \tag{30}\]  
\[\left\{\begin{array}{l}U_{i,\text{min}}^{2}\leq V_{i}\leq U_{i,\text{max}}^{ 2},i\in\psi_{b}\\ 0<l_{ij}\leq\left(I_{l}^{\text{max}}\right)^{2}\end{array}\right.. \tag{31}\]  
After the above deformation, the original nonconvex power flow Equation (8) has been transformed to a linear Equation (30). However, a new equality constraint is produced as Equation (32), with which the proposed model is still nonconvex.  
\[l_{ij}=\left(P_{ij}^{2}+Q_{ij}^{2}\right)/V_{i} \tag{32}\]  
Then the SOCP relaxation is applied to Equation (32) as  
\[(P_{ij}^{2}+Q_{ij}^{2})/V_{i}\leq l_{ij}. \tag{33}\]  
Note that, Equation (33) is equal to Equation (34)  
\[(2P_{ij})^{2}+(2Q_{ij})^{2}+(l_{ij}-V_{i})^{2}\leq(l_{ij}+V_{i})^{2}. \tag{34}\]  
Therefore, the standard second-order cone format of Equation (33) is  
\[\left\|\begin{array}{c}2P_{ij}\\ 2Q_{ij}\\ l_{ij}-V_{i}\end{array}\right\|_{2}\leq l_{ij}+V_{i}. \tag{35}\]  
After the conic relaxation (Equation (35)) and the transformation (Equations (28)-(31)), the original dynamic coordinated active-reactive power optimization model (Equations (3)-(26)) is transformed into a MISOCP problem as follows:  
\[\begin{array}{ll}\min f=&\sum\limits_{s=1}^{N_{s}}\pi_{s}(\omega_{1}C_{loss, s}+\omega_{2}C_{\text{cos},s}+\omega_{3}C_{\Delta U,s})\\ =&\sum\limits_{s=1}^{N_{s}}\pi_{s}(\omega_{1}c_{\text{loss}}\sum\limits_{t=1} ^{T}\sum\limits_{(i,j)\in\psi_{j}}r_{ij}l_{ij,s,t}+\omega_{2}(\sum\limits_{t=1 }^{T}c_{Grid,t}P_{Grid,s,t}+c_{\text{DG}}\sum\limits_{t=1}^{T}P_{DG,s,t}+c_{ \text{ESS}}\sum\limits_{t=1}^{T}P_{ESS,s,t})\\ &+\omega_{3}c_{\Delta U}\sum\limits_{t=1}^{T}\sum\limits_{i\in\psi_{b}}\left| 1-(V_{i,s,t}/U_{N}^{2})\right|)\\ &\text{s.t. Equations (\ref{eq:1}),(\ref{eq:2}),(\ref{eq:30}),(\ref{eq:31}),(\ref{eq:35})} \end{array} \tag{36}\]  
It is no doubt that the solutions obtained in the relaxed convex domain after the SOCP relaxation are the lower bound solutions of original problem since the original feasible region is enlarged. If the solutions obtained after the SOCP relaxation are within the feasible domain of the original problem, they are the optimum solutions of the original problem as well.