# 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
To facilitate the analysis of changes in active and reactive power, based on Figure 1, this paper employs the tangent value of the power factor angle to describe the aforementioned compensation process. For forward power flow, the tangent values of the power factor angle on both sides of the OLTC can be expressed as follows:  
\[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]0  
\[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]1 Relationship diagram of the OLTC reverse power transmission capability and the OLTC tap ratio. \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]2 \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]3 (11) \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]4  
In forward power flow conditions, \(P_{1,t}^{*}\) and \(Q_{1,t}^{*}\) are the per unit values of active and reactive power injected from the higher-level grid into the high-voltage side. \(\varphi_{1,t}\) represents the power factor angle, \(\varphi_{T}\) is the required power factor angle on the primary side of the OLTC, and \(\varphi_{2,t}\) is the required power factor angle on the low-voltage side of the OLTC. \(Q_{C1,t}^{*}\) and \(Q_{C2,t}^{*}\) are the per unit values for reactive compensation capacity on the high- and low-voltage sides. \(\Delta P_{T,t}^{*}\) and \(\Delta Q_{T,t}^{*}\) denote the per unit active and reactive power losses on the OLTC at this time. Below reverse flow conditions, the tangent of the power factor angle for each node is represented as:  
\[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]5 \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]6 (14) \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]7 (15) \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]8  
\(P_{2,ret}^{*}\) and \(Q_{2,re,t}^{*}\) represent the per unit active and reactive power injected into the distribution network under reverse power flow at the low-voltage side of the OLTC. \(\varphi_{2,re}\) denotes the power factor angle at this moment, while \(\varphi_{1,re,t}\) indicates the required power factor angle at the high-voltage side at this time. \(Q_{C1,re,t}^{*}\) and \(Q_{C2,re,t}^{*}\) represent the per unit reactive compensation capacity at the high- and low-voltage sides at this time. \(\Delta P_{T,re,t}^{*}\) and \(\Delta Q_{T,re,t}^{*}\) represent the active and reactive power losses on the OLTC at this moment. The relationship between the per unit values of power flow transfer capacity and reactive power compensation at both ends in positive and negative sequences is derived after arranging Equation (9) through (16) according to the aforementioned relationships. \[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]9  
\[U^{*}_{2,t}=\frac{U_{2,t}}{U_{1N}/K_{t}}=\frac{a_{t}\cdot U_{2N}\cdot K^{*}_{ t}\cdot K_{N}}{U_{1N}}=\frac{b_{t}\cdot K^{*}_{t}}{1.1},b_{t}\in(1.00,\;1.05] \tag{4}\]0  
The OLTC reverse power flow limits, \(S_{1,t}^{*}\) and \(S_{2,ret,t}^{*}\) represent forward and backward power flow scenarios. Both values increase as the reactive power compensation on the high and low sides increases. When the forward power flow \(S_{1,t}^{*}\) reaches the OLTC's rated capacity (equal to 1), \(Q_{L1,t}^{*}\) and \(Q_{C2,t}^{*}\) denote the initial installation capacity on high and low sides. If the reactive power compensation needed for reverse flow exceeds this, the reverse power limit peaks. Equation (18) seeks the maximum \(S_{2,re,t}^{*}\), and hence uses \(K_{max}^{*}\).