# On the Determination of Meshed Distribution Networks Operational Points after Reinforcement
## 2 System under Examination and Problem Formulation  
Tables 1-3 present the technical characteristics and the configuration of the distribution network under study [34]. An abstract graphical representation of the network is provided at Figure 1. The voltage level of the grid is 20 kV, the total length is 55 km and the installed capacity is 12 MVA. The network includes 45 20/0.4 kV distribution transformers and 24 renewable energy sources plants (photovoltaics). The resistance and the reactance of the conductors is R = 1.268 \(\Omega\)/km and X = 0.422 \(\Omega\)/km for ACSR 16 mm\({}^{2}\), R = 1.071 \(\Omega\)/km and X = 0.393 \(\Omega\)/km for ACSR 35 mm\({}^{2}\) and R = 0.215 \(\Omega\)/km and X = 0.334 \(\Omega\)/km for ACSR 95 mm\({}^{2}\). In order to improve the performance of this line [34] and to demonstrate the capability of the proposed method suitable for meshed networks, it is being reinforced in a meshed manner, connecting the nodes 51 and 84 using ACSR 95 mm\({}^{2}\) conductors. The distance between these nodes is 8 km, creating a resistance of 1.72 \(\Omega\) and impedance of 2.672 \(\Omega\), or pu resistance and impedance 0.43 pu and 0.668 pu, respectively. Figure 1: Abstract graphical representation of the network. AC power flow is used to perform the calculations for this analysis [35], by using an appropriate computer tool [25; 26]. All nodes are PQ, being able to offer active and reactive power, except from the slack node. In this case, there is no generator connected. The distributed generators are simulated as negative loads given that based on research questions, their production is known. The following equations used to simulate the branches [36]:  
\[\left[\frac{\mathbf{i}_{\mathrm{f}}}{\mathbf{i}_{\mathrm{t}}}\right]=Y_{ \mathrm{br}}\left[\frac{\mathbf{v}_{\mathrm{f}}}{\mathbf{v}_{\mathrm{t}}}\right] \tag{1}\]  
where: \(\mathbf{v}_{\mathrm{f}}\), \(\mathbf{v}_{\mathrm{t}}\), \(\mathbf{i}_{\mathrm{f}}\) and \(\mathbf{i}_{\mathrm{t}}\) are the terminal voltages and currents. Equation (1) connects voltages at all nodes and currents. The impedance matrix (\(Y_{\mathrm{br}}\)) expresses the impedances across all branches forming a table that is unique for each network. The admittance matrix \(Y_{\mathrm{br}}\) is as follows:  
\[Y_{\mathrm{br}}=\left[ \begin{array}{cc}y_{\mathrm{ff}}^{\mathrm{i}}&y_{ \mathrm{ft}}^{\mathrm{i}}\\ y_{\mathrm{ff}}^{\mathrm{i}}&y_{\mathrm{tt}}^{\mathrm{i}}\end{array} \right] \tag{2}\]  
Figure 2: Flow chart of the developed calculation procedure. Eventually, the balance equation based on [PERSON]'s laws can be written as follows [35], f(V, Sg) needs to equal to zero:  
\[\text{f}\left(\text{V},\text{S}_{\text{g}}\right)=\text{S}_{\text{bus}}(\text{V })+\text{S}_{\text{d}}-\text{C}_{\text{g}}\text{-S}_{\text{g}} \tag{3}\]  
where Sg, and Sd are the generators' and loads' apparent power, then (3) becomes [35] fp(\(\Theta\), Vm, Pg) and fq(\(\Theta\), Vm, Pg) that also need to equal to zero:  
\[\text{f}_{\text{P}}\left(\Theta,\text{ V}_{\text{m}},\text{ P}_{\text{g}}\right)=\text{P}_{\text{bus}}(\Theta,\text{ V}_{\text{m}})+\text{P}_{\text{d}}-\text{C}_{\text{g}}\text{-S}_{\text{g}} \tag{4}\]  
\[\text{f}_{\text{P}}\left(\Theta,\text{ V}_{\text{m}},\text{ P}_{\text{g}}\right)=\text{Q}_{\text{bus}}(\Theta,\text{ V}_{\text{m}})+\text{Q}_{\text{d}}-\text{C}_{\text{g}}\text{-Q}_{\text{g}} \tag{5}\]  
and:  
\[\text{g}(\text{x})=\left[ \begin{array}{c}\text{f}_{\text{P}}^{[i]}\left( \Theta,\text{ V}_{\text{m}},\text{P}_{\text{g}}\right)\\ \text{f}_{\text{Q}}^{[i]}\left(\Theta,\text{ V}_{\text{m}},\text{P}_{\text{g}}\right)\end{array} \right] \tag{6}\]  
\[\text{\forall i}\in\text{I}_{\text{PV}}\cup\text{I}_{\text{PV}},\text{ \forall j}\in\text{I}_{\text{PQ}}\]  
where vector x equals to:  
\[\text{x}=\left[ \begin{array}{c}\Theta_{\{\{i\}}}\\ \text{U}_{\text{m}}^{[i]}\end{array}