# 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
### Fundamental Principles and Mathematical Models of SOP  
This paper employs a back-to-back voltage source converter (B2B VSC) to implement the SOP. The number of ports is determined by the count of AC/DC bidirectional converters, which are internally connected through a DC/DC converter. The AC side of the AC/DC converter links to the distribution network feeders, while the DC side connects to the DC/DC converter, enabling interconnection between different feeders. The basic topology is illustrated in the Figure 9.  
Figure 8: The SOP interconnects two distribution networks. Figure 9: The SOP’s basic topological structure. Under normal operating conditions, the two-end SOP selects a PQ-Vdc control mode, while the three-end SOP employs a PQ-PQ-Vdc Q control mode. This means that one VSC regulates the DC-side voltage, and the other VSCs control the power being transmitted. The operational variables of the SOP include the active and reactive power transmitted by the VSCs, with specific operational constraints as follows [28; 29]:  
\[\sum_{i\in\Omega_{SOP}}\left(P_{i,t}^{AC}-P_{i,t,loss}^{AC}\right)=0 \tag{25}\]  
\[S_{i}^{AC}\geq\sqrt{\left(P_{i,t}^{AC}\right)^{2}+\left(Q_{i,t}^{AC}\right)^{2}} \tag{26}\]  
\[S_{i}^{AC}\geq P_{i,t}^{AC}\geq-S_{i}^{AC} \tag{27}\]  
\[S_{i}^{AC}\geq Q_{i,t}^{AC}\geq-S_{i}^{AC} \tag{28}\]  
The set \(\Omega_{SOP}\) contains the port nodes of the SOP. The terms \(P_{i,t}^{AC}\) and \(Q_{i,t}^{AC}\) represent the active and reactive power injected by port \(i\) of the SOP at time period t. The terms \(P_{t}^{DC}\) and \(P_{t,loss}^{SOP}\) are the active power injected by the SOP's DC/DC converter and the total active power loss at time period t, respectively. The variable \(S_{i}^{AC}\) denotes the capacity of the AC/DC converter at port \(i\) of the SOP, and \(S^{DC}\) represents the capacity of the SOP's DC/DC converter. When the SOP transmits a significant amount of power, its internal power loss becomes non-negligible. Given that multiple VSCs are identical in model and symmetrically positioned, the analysis focuses on one \(VSC_{i}\), and its equivalent model is illustrated in Figure 10.  
In the Figure 10, \(S_{i1}\) represents the SOP bus's injected SOP power, while \(S_{i2}\) denotes the apparent power injected by the VSC at the converter bus. \(U_{i1}\angle\delta_{1}\) and \(U_{i2}\angle\delta_{2}\) correspond to the voltage and phase angle at the SOP bus and VSC bus, respectively. The phase angle difference \(\Delta\delta\) indicates the phase angle of VSC pulse modulation, determining the magnitude and direction of SOP injected power. \(U_{dc}\) and \(I_{dc}\) stand for the DC-side current and voltage of the VSC, and \(g_{DC}\) represents the equivalent conductance on the VSC's DC side. \(R_{VSC}\) and \(X_{VSC}\) are the equivalent resistance and reactance of the VSC. Specific constraints include:  
* DC-side KCL balance constraint: \[I_{dc,t}=g_{DC}\cdot U_{dc,t}\] (29)
* AC/DC voltage pulse modulation constraints: \[U_{i2,t}=\frac{\mu M_{i}}{\sqrt{2}}U_{dc,t}\] (30)  
In this context, \(\mu\) represents the direct current voltage utilization rate, while \(M_{i}\) denotes the modulation index of the pulse signal. Figure 8:0 Diagram of VSC AC/DC model, considering impedance. * Alternating current measurement involves balancing constraints of active and reactive power: Figure 8:1 (31) Figure 8:2 (32) where \(G_{i}\) and \(B_{i}\) represent the AC filter admittance of the SOP, denoted as \(Y_{SOP}\). * Power loss constraint of VSC: Figure 8:3 (33)  
The active power loss of VSC equates to the power injected on the AC side minus the power injected on the DC side. In the context of SOP involvement in distribution network optimization, there are five unknowns in the five constraints stated above. When employing PQ control, the unknowns are: \(U_{dc}\), \(I_{dc}\), \(P_{loss,i}\), \(\Delta\delta\), and \(M_{i}\). Under QU control, the unknowns are: \(P_{i1}\), \(I_{dc}\), \(P_{loss,i}\), \(\Delta\delta\), and \(M_{i}\), which can be solved based on optimization results.