# Optimization of Active Distribution Network Operation with SOP Considering Reverse Power Flow
## 3 Actual Maximum Transfer Capacity of Reverse Power Flow in the Distribution Network
### Model Transformation and Solution Process  
In the model mentioned above, the variables involve both SOP real and reactive power outputs as continuous variables, and actions of the OLTC and CB components as discrete variables. Essentially, it is a mixed-integer nonlinear programming problem, difficult to solve directly for optimization. This paper employs second-order cone relaxation to transform it into a mixed-integer second-order cone programming model, thus allowing the direct use of the CPLEX commercial solver for resolution. Nonlinear elements such as \(I_{m,ij,t}^{2}\) and \(U_{m,i,t}^{2}\) in Equations (41)-(46) are replaced by \(l_{m,ij,t}\) and \(V_{m,j,t}\). However, Equation (42) remains nonlinear, so it is relaxed into the following second-order cone constraints:  
\[\left|\left|\left[2P_{m,ij,t}2Q_{m,ij,t}l_{m,ij,t}-V_{m,i,t}\right]^{T}\right| \right|_{2}<l_{m,ij,t}+V_{m,i,t} \tag{56}\]  
Constraints (27) containing quadratic nonlinearity can be transformed into the following rotated second-order cone constraints. \[\left(S_{i}^{AC}\right)^{2}\geq\left(P_{i,t}^{AC}\right)^{2}+\left(Q_{i,t}^{AC }\right)^{2} \tag{57}\]  
The OLTC-related constraint (53) will transform after introducing \(V_{i,t}\):  
\[V_{m,i,t}=k_{m,ij,t}^{2}V_{m,j,t} \tag{58}\]  
At this point, one needs to handle the quadratic terms, introducing the binary variable \(b_{m,ij,t,k}\) to represent them in the following form. \[K_{m,ij,t}=\sum_{k=0}^{2K_{m,ij,max}}\left(k-K_{m,ij,max}\right)b_{m,ij,t,k} \tag{59}\]  
\[\sum_{k=0}^{2K_{m,ij,max}}b_{m,ij,t,k}=1,\ b_{m,ij,t,k}\in\{0,1\} \tag{60}\]After substitution, the result is:  
\[V_{m,i,t}=\sum_{k=0}^{2K_{m,ij,max}}\left(k_{m,ij,0}+\left(k-K_{m,ij,max}\right) \Delta k_{ij}\right)^{2}V_{m,j,t}b_{m,ij,t,k} \tag{61}\]  
At this point, the product \(V_{j,t}b_{ij,t,k}\) remains nonlinear and can be expressed by the variable \(V_{ij,t,k}^{b}\). Accordingly, Equation (53) can be represented as:  
\[V_{m,i,t}=\sum_{k=0}^{2K_{ij,max}}\left(k_{m,ij,0}+\left(k-K_{m,ij,max}\right) \Delta k_{m,ij}\right)^{2}V_{m,i,j,t,k}^{b} \tag{62}\]  
Constraint (45) is expressed as:  
\[U_{i,min}^{2}b_{m,ij,t,k}<V_{m,ij,t,k}^{b}<U_{i,max}^{2}b_{m,ij,t,k} \tag{63}\]  
\[U_{i,min}^{2}\left(1-b_{m,ij,t,k}\right)<V_{m,i,t}-V_{m,ij,t,k}^{b}<U_{i,max}^ {2}\Big{(}1-b_{m,ij,t,k}\Big{)} \tag{64}\]  
For constraints (54) and (55), auxiliary variables \(K_{m,ij,t}^{+}\) and \(K_{m,ij,t}^{-}\) could be introduced, representing the number of tap positions for the OLTC adjusted upwards or downwards, respectively. Thus, constraints (54) and (55) can be expressed as:  
\[\sum_{t=1}^{24}\left|K_{m,ij,t}^{+}+K_{m,ij,t}^{-}\right|\leq\Delta K_{ij,max} \tag{65}\]  
\[\left(S_{i}^{AC}\right)^{2}\geq\left(P_{i,t}^{AC}\right)^{2}+\left(Q_{i,t}^{AC }\right)^{2} \tag{57}\]0 (6  
appearing during early morning and midday. In distribution network C, the DG output is low, with reverse power flow occurring only during the early morning. The OLTC parameters in the three distribution networks are identical, with a rated transformation ratio \(K_{N}\) of 110/11, a rated voltage \(U_{N}\) of 110 kV, a rated capacity \(S_{N}\) of 20 MVA, a rated power factor \(cos\varphi_{T}\) of 0.85, \(P_{k}\) of 112 kW, \(U_{k}\)% of 10.5%, high-voltage side taps of \(\pm\)1.25% \(\times\) 8, and a maximum number of actions within a scheduling cycle \(K_{m,j,max}=10\). In the reactive power compensation on the low-voltage side, the capacity of a single capacitor \(q_{m,CB,i}=1 MVA\), with a total installation group number \(N_{m,i,max}=5\), and a maximum number of operations per dispatch period \(\Delta N_{m,j,max}=10\). In the utilized SOP, each VSC has a capacity of 5.0 MVA with a loss coefficient of 0.02. For the distribution networks A, B, and C, nodes connected to the SOP are selected according to the equivalent source method: A.5, B.14, and C.6 respectively. The model was tested on a PC with a 2.2 GHz main frequency and 32 GB of memory, using the MATLAB R2020b simulation platform.