# 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
By substituting Equations (3) and (4) into (1) and (2), the relationship between the per unit values of forward and reverse power transmission capacity and \(K^{*}_{t}\) is obtained as follows:  
\[S^{*}_{1,t} =\frac{\left(\;1-\frac{b_{t}\;K^{*}_{t}}{1.1}\;\right)}{Z^{*}_{T} \epsilon\;\tau} \tag{6}\] \[S^{*}_{2,re,K,t} =\frac{\left(\;\frac{b_{t}K^{*}_{t}}{1.1}-1\right)\frac{b_{t}K^{*} _{t}}{1.1}}{Z^{*}_{T}\epsilon\;\tau} \tag{7}\]  
In expressions (6) and (7), when the forward flow occurs, a smaller \(K^{*}_{t}\) results in a larger \(S^{*}_{1,t}\); conversely, during reverse flow, a larger \(K^{*}_{t}\) results in a larger \(S^{*}_{2,re,t}\). This phenomenon arises because the greater the flow transmitted through the OLTC, the larger the difference between the per unit values of the high-voltage and low-voltage side voltages becomes. Assuming \(U^{*}_{1,t}\) always equals 1, as transmission capacity increases, \(U^{*}_{2,t}\) varies more significantly. Given \(U^{*}_{2,t}=b_{t}K^{*}_{t}/1.1\), where \(b_{t}\) is primarily determined by the operational state of the distribution network, the transmission capacity is thus determined solely by \(K^{*}_{t}\). Additionally, the maximum difference in the per unit values of high-voltage and low-voltage side voltages during forward flow exceeds that during reverse flow, resulting in OLTC transmission capacity being smaller in reverse flow compared to forward flow, as demonstrated below:  
\[dU^{*}_{max}\;-\;dU^{*}_{re,max} =\left(\;1-\frac{b_{t}K^{*}_{min}}{1.1}\;\right)-\left(\;\frac{b_ {t}K^{*}_{max}}{1.1}\;-1\right) \tag{8}\] \[=2-\frac{b_{t}\left(K^{*}_{max}\;+\;K^{*}_{min}\;\right)}{1.1}\] \[=2-\frac{2b_{t}}{1.1}>0,b_{t}\in(1.00,\;1.05]\]The maximum differences in the per unit voltage values for the OLTC's high- and low-voltage sides during forward and reverse power flow are denoted as \(dU_{max}^{*}\) and \(dU_{re,max}^{*}\), respectively. When \(K_{max}^{*}\) and \(K_{min}^{*}\) are set to 1.1 and 0.9, the relationship between forward and reverse power flow transmission capacity and \(K_{t}^{*}\) can be observed in Figure 2. The horizontal axis represents the per unit value of the OLTC tap ratio, while the vertical axis represents the per unit value of the OLTC power flow transmission capacity. The blue curve illustrates the variation of \(S_{1,t}^{*}\) with \(K_{t}^{*}\) when \(b_{t}=1.05\) (as it is generally assumed that the OLTC's low-voltage side voltage is 1.05 times its rated voltage during operation). The cluster of red curves illustrates the changes in \(S_{2,re,t}^{*}\) with \(K_{t}^{*}\) as \(b_{t}\) gradually increases from bottom to top. Both \(S_{1,t}^{*}\) and \(S_{2,re,t}^{*}\) must remain greater than zero. In the forward power flow scenario, as \(K_{t}^{*}\) decreases, \(S_{1,t}^{*}\) reaches the OLTC's rated capacity, achieving a value of 1. However, in the reverse power flow scenario, when both \(b_{t}\) and \(K_{t}^{*}\) are at their maximum, the maximum value of \(S_{2,re,t}^{*}\) is approximately 84% of the OLTC's rated capacity. Reactive Power Compensation on the Low-Voltage Side of Oil-Inmersed Tap Changers Influences Reverse Power Flow Capacity  
During OLTC operation, reactive power compensation must be applied on both sides to maintain power factor. In the presence of forward power flow, the power factor of the upstream grid might not meet the OLTC's primary requirements. Hence, reactive power is added on the high-voltage (primary) side to adjust the power factor. Also, considering the mismatch between the OLTC's low-voltage (secondary) side power factor and the distribution network's power factor, reactive compensation is introduced on the low-voltage side as well. In reverse power flow scenarios, due to the high-power factor derived from DG units, capacitive reactive power is introduced on the low-voltage (primary) side to lower the power factor. Similarly, as the OLTC's internal reactive losses surpass active losses, capacitive reactive power is added to the high-voltage (secondary) side to maintain the power factor.