# Optimization of Active Distribution Network Operation with SOP Considering Reverse Power Flow
## 2 Analysis of Factors Affecting Reverse Power Flow Transmission Capacity in Distribution Networks  
Reverse power flow in a distribution network occurs when the output from the DG exceeds the load demand of the distribution network, causing power to flow backward to the upstream grid. The reverse power flow capacity in distribution networks is influenced by the following three factors: The first factor is the parameters of the OLTC connected between the distribution network and the upstream grid, including the rated capacity of the OLTC, the difference in the per unit values of voltage on both sides of the OLTC, as well as the allowable maximum and minimum tap changer ratios. The second factor is the capacity of reactive power compensation that can be deployed on the low-voltage side of the OLTC. The third factor is the electrical distance from the DG to the low-voltage side of the OLTC.  
First, the nameplate-rated capacity of the OLTC directly determines the maximum reverse power flow that can be transmitted. Additionally, the magnitude of reverse power flow during transmission is proportional to the difference in the per unit values of voltage on both sides of the OLTC. Therefore, the greater the difference, the larger the reverse power flow capacity. Since the per unit value of the high-voltage side remains constant and is lower than that of the low-voltage side during operation, with increasing transmission capacity, the per unit value of the low-voltage side voltage increases. Consequently, the OLTC's transformation ratio also increases. When this ratio reaches its upper limit, further increase in reverse power flow capacity is not possible.  
Regarding the reactive power compensation capacity on the low-voltage side of the OLTC, to maintain the power factor of the OLTC primary side (low-voltage side during reverse flow) at the required value, as the reverse power flow capacity increases, more reactive compensation capacity is required on the low-voltage side. Once the full compensation capacity is utilized, further increase in reverse power flow capacity is not possible.  
Concerning the electrical distance from the DG to the OLTC's low-voltage side, as the reverse power flow capacity increases, the voltage drop along the line from the DG to OLTC also increases [25]. When this voltage drop reaches its allowable maximum, further increase in transmission capacity necessarily requires reduced allowable electrical distance between the DG and the OLTC to prevent voltage limit violations [26; 27]. When the electrical distance is reduced to zero, meaning the DG node coincides with the OLTC's low-voltage side node, any further capacity increase is not possible.  
The circuit diagram in Figure 1 shows the equivalent representation of all distributed generation (DG) and loads in the distribution network. Here, \(R_{T}^{*}+jX_{T}^{*}\) denotes the internal impedance of the OLTC, \(Z_{DG}^{*}\) refers to the per unit value of the impedance from the equivalent DG source to the OLTC's low-voltage side, and \(Z_{L}^{*}\) represents the impedance from the equivalent DG source to the equivalent load. \(Q_{C1}^{*}\) and \(Q_{C2}^{*}\) are the reactive power compensation on the high-voltage and low-voltage sides of the OLTC, respectively. \(U_{1}^{*}\), \(U_{2}^{*}\), and \(U_{3}^{*}\) denote the voltage values at the OLTC's high-voltage side, low-voltage side, and equivalent DG nodes, respectively. \(S_{1}^{*}\) and \(S_{2}^{*}\) are the magnitudes of the power flow injected into and flowing out of the OLTC under forward power flow conditions. \(S_{2,re}^{*}\) and \(S_{1,re}^{*}\) represent the apparent power injected into and flowing out of the OLTC during reverse power flow conditions. \(K^{*}\) signifies the transformer turns ratio.