# 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  
Sections 2.1-2.3 analyzed the impacts of the OLTC transformation ratio \(K^{*}_{t}\), the reactive power compensation capacity on the low-voltage side of the OLTC \(Q^{*}_{C2,re,t}\), the electrical distance from the DG to the low-voltage side of the OLTC \(Z^{*}_{DG}\), and the OLTC's low-voltage side voltage parameter \(b_{t}\) on \(S^{*}_{2,re,t}\). Section 2.1 states that \(S^{*}_{2,re,t}\) increases with larger \(K^{*}_{t}\) and \(b_{t}\), reaching up to approximately 84% of the OLTC's rated capacity when both are at maximum values. Section 2.2 explains that increasing \(K^{*}_{t}\) and \(b_{t}\) to enhance \(S^{*}_{2,re,t}\) requires reactive support from \(Q^{*}_{C2,re,t}\) on the low-voltage side, necessitating additional reactive equipment, which raises cost. However, given sufficient reactive power, \(S^{*}_{2,re,t}\) achieves the maximum value from Section 2.1. Section 2.3 notes that \(S^{*}_{2,re,t}\) decreases as \(Z^{*}_{DG}\) rises and points out a contradiction with prior conclusions when \(Z^{*}_{DG}\) is fixed and a smaller \(b_{t}\) results in a larger \(S^{*}_{2,re,t}\). In the above scenario, analyzing from the perspective of the OLTC, an increase in \(b_{t}\) results in a larger \(S^{*}_{2,re,t}\). However, when analyzed from the perspective of the line to the low-voltage side of the DG to OLTC, an increase in \(b_{t}\) results in a smaller \(S^{*}_{2,re,t}\). Therefore, it is necessary to select an appropriate \(b_{t}\) to maximize \(S^{*}_{2,re,t}\) after comprehensively considering  
Figure 5: The relationship between electrical distance and the magnitude of reverse flow delivery. various influencing factors. The actual maximum reverse power transfer capacity of distribution networks, \(S^{*}_{2,re,t,max}\), can be expressed as:  
\[S^{*}_{2,re,t,max}=\min\{(7),(9),(22)\} \tag{23}\]  
\[(7)=(9)=(22) \tag{24}\]  
Equation (23) elucidates that the value of \(S^{*}_{2,re,t,max}\) is determined as the minimum under three distinct constraints. Equation (24) indicates that \(S^{*}_{2,re,t}\) should remain equal across the same time period under three different constraints, as they represent the same power flow. Figure 6 demonstrates the relationship among \(S^{*}_{2,re,t}\), \(Q^{*}_{C2,re,t}\), \(Z^{*}_{DG}\), and \(b_{t}\) when \(K^{*}_{t}\) is set to \(K^{*}_{max}\). The horizontal axis of the graph has its positive direction labeled as \(Z^{*}_{DG}\), and its negative direction is labeled as \(Q^{*}_{C2,re,t}\). The vertical axis is \(S^{*}_{2,re,t}\). On the right side of the vertical axis, inverse proportion curves in different colors represent the variation of \(S^{*}_{2,re,t}\) with \(Z^{*}_{DG}\) under various \(b_{t}\) conditions. On the left side, a black linear function indicates the relationship of \(S^{*}_{2,re,t}\) with \(Q^{*}_{C2,re,t}\) under different \(b_{t}\) conditions. The colored horizontal lines on both sides of the vertical axis signify the value of \(S^{*}_{2,re,t}\) when \(K^{*}_{t}=K^{*}_{max}\), across different \(b_{t}\) scenarios. When \(K^{*}_{t}=K^{*}_{max}\) and reactive power compensation \(Q^{*}_{C2,re,t}\) is sufficient, without considering \(Z^{*}_{DG}\) constraints, \(S^{*}_{2,re,t,max}\) can reach approximately 84% of the OLTC's rated capacity. However, in practical operation, with \(Z^{*}_{DG}\) limitations considered, \(S^{*}_{2,re,t,max}\) is limited to about 50% of the OLTC's rated capacity. Refer to Figure 1 for an equivalent circuit diagram, taking the high-voltage side node of the OLTC as the balance node. The specific calculation process of \(S^{*}_{2,re,t,max}\) is illustrated in Table 1.  
The specific flowchart is shown in Figure 7. The aforementioned steps calculate the maximum reverse power flow transfer capacity for this distribution network. When the required reverse power flow exceeds this value, the DG output needs reduction. Conversely, if the reverse power flow is less than this value, it can be transferred smoothly.