# 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
### Impact of OLTC Parameters on Reverse Power Flow Transmission Capacity
The real distribution network data of \(tan\varphi_{1,t}\), \(tan\varphi_{T}\), \(tan\varphi_{2,t}\), \(tan\varphi_{1,re,t}\), and \(tan\varphi_{2,re,t}\) show the relationship between power flow limits and reactive compensation, as depicted in \[U^{*}_{2,t}=\frac{U_{2,t}}{U_{1N}/K_{t}}=\frac{a_{t}\cdot U_{2N}\cdot K^{*}_{ t}\cdot K_{N}}{U_{1N}}=\frac{b_{t}\cdot K^{*}_{t}}{1.1},b_{t}\in(1.00,\;1.05] \tag{4}\]1  
In the diagram, the vertical axis represents the forward and reverse power flow capacities of the OLTC, while the horizontal axis indicates reactive compensation capacities on both sides of the OLTC, with the red and blue curves representing high-voltage and low-voltage sides, respectively. When \(S_{1,t}^{*}\) equals 1, the x-coordinate values of the red and blue curves denote the initial reactive compensation installed capacities on the high- and low-voltage sides of the OLTC. Under these installed capacities, the maximum reverse power flow capacity that the low-voltage side can compensate is about 33% of the OLTC's rated capacity, whereas the high-voltage side can compensate up to approximately 95%. This indicates that the reverse power flow capacity is primarily constrained by the reactive compensation capacity on the low-voltage side of the OLTC. Therefore, Equation (18) can be expressed as:  
\[U^{*}_{2,t}=\frac{U_{2,t}}{U_{1N}/K_{t}}=\frac{a_{t}\cdot U_{2N}\cdot K^{*}_{ t}\cdot K_{N}}{U_{1N}}=\frac{b_{t}\cdot K^{*}_{t}}{1.1},b_{t}\in(1.00,\;1.05] \tag{4}\]2  
To enhance the reverse power flow capacity, increase the reactive power compensation installation capacity on the low-voltage side. After increasing the installation capacity, the relationship between \(S_{2,re,Q,t}^{*}\) and \(Q_{C2,re,t}^{*}\) can be seen in \[U^{*}_{2,t}=\frac{U_{2,t}}{U_{1N}/K_{t}}=\frac{a_{t}\cdot U_{2N}\cdot K^{*}_{ t}\cdot K_{N}}{U_{1N}}=\frac{b_{t}\cdot K^{*}_{t}}{1.1},b_{t}\in(1.00,\;1.05] \tag{4}\]3  
\[U^{*}_{2,t}=\frac{U_{2,t}}{U_{1N}/K_{t}}=\frac{a_{t}\cdot U_{2N}\cdot K^{*}_{ t}\cdot K_{N}}{U_{1N}}=\frac{b_{t}\cdot K^{*}_{t}}{1.1},b_{t}\in(1.00,\;1.05] \tag{4}\]4 The relationship between OLTC power flow capacity and reactive power compensation. The graph's horizontal axis represents the reactive power compensation capacity on the OLTC's low-voltage side, while the vertical axis indicates the reverse power flow transfer capacity. One can observe that \(S_{2,re,t}^{*}\) increases with the rise in \(Q_{\text{C2,re,t}}^{*}\). The horizontal lines from bottom to top signify the maximum \(S_{2,re,t}^{*}\) achievable with increasing \(b_{t}\), reflecting the constraint imposed by \(K_{t}^{*}\) in Section 2.1. At this juncture, further escalation of \(Q_{\text{C2,re,t}}^{*}\) does not enhance \(S_{2,re,t}^{*}\). When \(b_{t}\) attains its peak value of 1.05, the corresponding \(Q_{\text{C2,re,t}}^{*}\) denotes the maximum installable reactive power compensation capacity on the low-voltage side. At this point, both \(b_{t}\) and \(K_{t}^{*}\) reach their limits, and \(S_{2,re,t}^{*}\) approaches the OLTC's boundary, rendering additional reactive compensation futile. The Influence of Electrical Distance from DG to Low-Voltage Side of OLTC on Reverse Power Flow Delivery Capacity  
The electrical distance from the DG to the low-voltage side of the OLTC refers to the impedance relationship between these two points. A greater electrical distance \(Z_{DG}^{*}\) implies a larger voltage difference between these points when transmitting power flow, thus reducing the capacity for reverse power flow. The relationship between reverse power flow transmission capacity and the electrical distance from the DG to the low-voltage side of the OLTC can be derived as follows.