# 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 OLTC's rated capacity determines its transmission capacity limit during power flow, regardless of the flow direction. The transmission capacity is proportional to the per unit voltage difference between the OLTC's high- and low-voltage sides, again irrespective of flow direction. The relationship between the OLTC's transmission capacity under forward and reverse power flows and the per unit voltage difference across both sides is:  
\[ \begin{split} dU_{t}^{*}&=U_{1,t}^{*}-U_{2,t}^{*}=U_ {1,t}^{*}-\sqrt{\left(U_{1,t}^{*}-\frac{P_{1,t}^{*}R_{T}^{*}+Q_{1,t}^{*}X_{T} ^{*}}{U_{1,t}^{*}}\right)^{2}+\left(\frac{P_{1,t}^{*}X_{T}^{*}-Q_{1,t}^{*}R_{T }^{*}}{U_{1,t}^{*}}\right)^{2}}\\ &\quad\quad\approx\frac{P_{1,t}^{*}R_{T}^{*}+Q_{1,t}^{*}X_{T}^{*} }{U_{1,t}^{*}}=\frac{S_{1,t}^{*}Z_{T}^{*}(cos\varphi_{T}sin\varphi_{Z_{T}^{*} +sin\varphi_{T}cos\varphi_{Z_{T}}})}{U_{1,t}^{*}}=\frac{S_{1,t}^{*}Z_{T}^{*} \circ T}{U_{1,t}^{*}}\end{split}  \tag{1}\]  
\[ \begin{split} dU_{re,t}^{*}&=U_{2,t}^{*}-U_{1,t}^{*} =U_{2,t}^{*}-\sqrt{\left(U_{2,t}^{*}+\frac{P_{1,t}^{*}R_{T}^{*}+Q_{1,t}^{*}X_{T }^{*}}{U_{2,t}^{*}}\right)^{2}+\left(\frac{P_{1,t}^{*}X_{T}^{*}-Q_{1,t}^{*}R_{ T}^{*}}{U_{2,t}^{*}}\right)^{2}}\\ &\quad\quad\approx\frac{P_{1,t}^{*}R_{T}^{*}+Q_{1,t}^{*}X_{T}^{*} }{U_{2,t}^{*}}=\frac{S_{2,re,t}^{*}Z_{T}^{*}\left(cos\varphi_{T}sin\varphi_{Z_ {T}^{*}}+sin\varphi_{T}cos\varphi_{Z_{T}}\right)}{U_{2,t}^{*}}=\frac{S_{2,re, t}^{*}Z_{T}^{*}\circ T}{U_{2,t}^{*}}\end{split}  \tag{2}\]  
Figure 1: Equivalent circuit diagram of a distribution network. In this context, \(S^{*}_{1,t}\) and \(S^{*}_{2,r,t}\) represent the per unit values of forward and reverse flow capacities of the OLTC during period \(t\). Their values are less than the rated per unit capacity of the OLTC, i.e., less than 1. \(U^{*}_{1,t}\) and \(U^{*}_{2,t}\) are the peer unit values of the high- and low-voltage sides of the OLTC during period \(t\). \(dU^{*}_{t}\) and \(dU^{*}_{re,t}\) are the differences in the per unit values of voltages on the high and low sides for forward and reverse flows, respectively. \(Z^{*}_{T}\) is the per unit internal impedance of the OLTC. The power factor \(cos\varphi_{T}\) is required on the primary side of the OLTC, while \(cos\varphi_{Z_{T}}\) is the power factor angle of the OLTC internal impedance. It can be observed that regardless of the flow direction, the flow capacity of the OLTC increases as the difference in the per unit voltage values between the high- and low-voltage sides increases. The high- and low-voltage sides can be represented as per unit values:  
\[U^{*}_{1,t}=\frac{U_{1,t}}{U_{1N}}=1 \tag{3}\]  
\[U^{*}_{2,t}=\frac{U_{2,t}}{U_{1N}/K_{t}}=\frac{a_{t}\cdot U_{2N}\cdot K^{*}_{ t}\cdot K_{N}}{U_{1N}}=\frac{b_{t}\cdot K^{*}_{t}}{1.1},b_{t}\in(1.00,\;1.05] \tag{4}\]  
\[K^{*}_{min}\;\leq K^{*}_{t}\leq K^{*}_{max} \tag{5}\]  
In this context, \(U_{1N}\) and \(U_{2N}\) are the nominal voltage values at the high- and low-voltage sides, respectively. \(K^{*}_{t}\), \(K^{*}_{max}\), and \(K^{*}_{min}\) denote the per unit tap ratios, maximum tap ratio, and minimum tap ratio for the on-load tap changer (OLTC) at time period t, while \(K_{N}\) refers to the rated tap ratio of the OLTC. Since the OLTC connecting the distribution network and the upper grid is designed to function as a step-down transformer, it follows that \(K_{N}=U_{1N}/1.1U_{2N}\). The coefficient \(b_{t}\) represents the ratio between \(U_{2,t}\) and \(U_{2N}\). Under forward power flows, to ensure the voltage at the end of the distribution feeder does not drop below the lower limit, \(U_{2,t}\) is generally set to 1.05 times \(U_{2N}\), resulting in \(a_{t}=1.05\). During reverse power flow, to guarantee \(U^{*}_{1,t}<U^{*}_{2,t}\), \(U_{2,t}\) is chosen between 1.00 to 1.05 times \(U_{2N}\), making \(b_{t}\in(1.00,1.05]\).