## 2 Modeling for OMEF
### Power System
#### 2.1.2 Power Balance Equation Convex Relaxation  
The classic power system model is non-convex, and its non-convexity comes from Equations (3) and (4). In order to ensure the efficiency and quality of model solving, it is necessary to carry out convex processing for that equations. First, equivalent substitutions are made to the variables in Equations (3) and (4), as follows:  
\[\left\{ \begin{array}{l}U_{i,t}=V_{i,t}^{2}\\ W_{ij,t}=V_{i,t}V_{j,t}\cos\theta_{ij,t}\\ M_{ij,t}=V_{i,t}V_{j,t}\sin\theta_{ij,t}\end{array} \right. \tag{9}\]  
Then, Equations (3) and (4) can be equivalent to the following equations:  
\[P_{ij,t}=g_{ij}U_{i,t}-g_{ij}W_{ij,t}-b_{ij}M_{ij,t} \tag{10}\]  
\[Q_{ij,t}=-b_{ij}U_{i,t}-g_{ij}M_{ij,t}+b_{ij}W_{ij,t} \tag{11}\]  
\[4M_{ij,t}^{2}+4W_{ij,t}^{2}+\left(U_{i,t}-U_{j,t}\right)^{2}=\left(U_{i,t}+U_{ j,t}\right)^{2} \tag{12}\]  
\[\sin\theta_{ij,t}/\cos\theta_{ij,t}=M_{ij,t}/W_{ij,t} \tag{13}\]  
For general mesh networks, the cycle conditions (for each loop in the network, the values of the voltage phase angle difference accumulated cyclically in a certain direction is 0 (modulo 2\(\pi\))) [26] need to be considered. That is, it is necessary to consider Equation (13). In the equivalent equations, Equations (12) and (13) are non-convex. Aiming at Equations (12) and (13), paper [25] proposed a convex relaxation method for single-period economic dispatch of power system. Based on this, this paper studies the convex processing method of Equations (12) and (13) in the multi-period OMEF model. For Equation (13), the equivalent transformation is performed as follows:  
\[S_{ij,t}=\sin\theta_{ij,t} \tag{14}\]  
\[C_{ij,t}=\cos\theta_{ij,t} \tag{15}\]  
\[S_{ij,t}^{2}+C_{ij,t}^{2}=1 \tag{16}\]  
\[(W_{ij,t}+S_{ij,t})^{2}+(M_{ij,t}-C_{ij,t})^{2}=(W_{ij,t}-S_{ij,t})^{2}+(M_{ ij,t}+C_{ij,t})^{2} \tag{17}\]  
According to the voltage phase angle difference between the two ends of the branch in the actual power system is generally not more than 10\({}^{\circ}\)[27], by using the Taylor expansion formula of the trigonometric function and ignoring the higher-order terms, Equations (14) and (15) can be transformed into the following equations:  
\[S_{ij,t}\approx\theta_{ij,t} \tag{18}\]  
\[C_{ij,t}\approx 1-\theta_{ij,t}^{2}/2 \tag{19}\]  
Through the above transformations, Equations (12), (16), (17), and (19) are still non-convex, and further transformations are needed. Among them, Equation (12) can be equivalent to the following two inequalities:  
\[Q_{ij,t}=-b_{ij}U_{i,t}-g_{ij}M_{ij,t}+b_{ij}W_{ij,t} \tag{11}\]0  
\[Q_{ij,t}=-b_{ij}U_{i,t}-g_{ij}M_{ij,t}+b_{ij}W_{ij,t} \tag{11}\]1  
Equation (20) is a standard second-order cone constraint and has convexity. Equation (21) is a difference-of-convex constraint, which is non-convex and can be accurately convex relaxed by usingthe PCCP method [25]. The idea is to deal with the difference-of-convex constraint through the Taylor expansion formula, and add relaxation variables to make the constraint hold, as follows:  
\[ \begin{array}{c}(U_{i,t}+U_{j,t})^{2}-8M_{ij,t}M^{k}_{ij,t}+4(M^{k}_{ij,t})^{2 }-8W_{ij,t}W^{k}_{ij,t}+4(W^{k}_{ij,t})^{2}-\\ 2(U_{i,t}-U_{j,t})(U^{k}_{i,t}-U^{k}_{j,t})+(U^{k}_{i,t}-U^{k}_{j,t})^{2}\leq \varepsilon_{1,ij,t}\end{array}  \tag{22}\]  
where \(M^{k}_{ij,t}\), \(W^{k}_{ij,t}\), and \(U^{k}_{i,t}\) are the value obtained by the \(k\)th iteration, and are constant; \(\varepsilon_{1,ij,t}\) is a relaxation variable and is a positive number. In the same way, Equations (16), (17), and (19) can be transformed into two inequalities, and then transformed as above. In order to ensure the accuracy of relaxation, a penalty term of relaxation variables is added to the objective function, and the relaxation field is continuously tightened by continuously punishing the relaxation variables until the accuracy is satisfied.