## 2 Mathematical Formulations
### Formulation for the Iterative-Based Load-Flow Solution
This method is fast, robust and characterized by better convergence properties [39]. The most suitable algorithm for distribution systems is the backward/forward sweep-based power-flow solution, whose derivation is presented as follows:  
With reference to Figure 1, the current of each load can be formulated as  
\[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]2  
But, to calculate the current of each line starting from the end of the feeder, backward sweep is used:  
\[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]3  
In the same way, voltage is calculated using the forward sweep method, starting from the first bus. \[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]4  
The backward and forward sweep modified method uses Kirchhoff's laws to formulate equations. With the formulated equations, solutions for power flow are obtained without the need to solve the simultaneous equations. Consider the simple radial distribution line diagram shown in \[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]5 The branch current is denoted by \(I_{n,\ n+1}\) and the injected current is denoted by \(I_{n}\). For simplicity, let \(I_{n,\ n+1}\) be '\(B\)' and \(I_{n}\) be '\(I\)'. With this information,  
\[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]6  
\[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]7  
\[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]8  
\[V_{ind}=\frac{1}{N}\sum_{i=1}^{N}V_{c} \tag{11}\]9  
\[\hat{V_{S}}=\hat{V_{R}}+\hat{I}(R+jX) \tag{12}\]0  
Based on the backward sweep of the branch current equation above, a matrix is formed. \[ \begin{bmatrix}B_{1}\\ B_{2}\\ B_{3}\\ B_{4}\\ B_{5}\end{bmatrix} = \begin{bmatrix}1&1&1&1\\ 0&1&1&1\\ 0&0&1&1\\ 0&0&0&1&1\\ 0&0&0&0&1\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\\ I_{3}\\ I_{4}\\ I_{5}\end{bmatrix}  \tag{30}\]  
The matrix formulated in (30) can be written compactly as  
\[[B]=[BBC][I] \tag{31}\]  
Figure 1: A simple distribution network to demonstrate the backward and forward sweep method. Figure 2: Radial distribution network showing the voltage at each bus. Using the same approach as in [40], the branch current to bus voltage is expressed in Figure 1 using forward sweep as follows:  
\[V_{2}=V_{s}<0-I_{12}Z_{12} \tag{32}\]  
\[V_{3}=V_{2}-I_{12}Z_{23} \tag{33}\]  
\[V_{4}=V_{3}-I_{34}Z_{34} \tag{34}\]  
\[V_{5}=V_{4}-I_{45}Z_{45} \tag{35}\]  
\[V_{6}=V_{3}-I_{36}Z_{36} \tag{36}\]  
From the above equations, it is obvious that the bus voltage is a function of line parameters, source voltage and branch currents. In matrix form,  
\[ \begin{bmatrix}V_{s}\\ V_{s}\\ V_{s}\\ V_{s}\end{bmatrix} - \begin{bmatrix}V_{2}\\ V_{3}\\ V_{4}\\ V_{5}\\ V_{6}\end{bmatrix} = \begin{bmatrix}Z_{12}&0&0&0&0\\ Z_{12}Z_{23}&0&0&0\\ Z_{12}Z_{23}Z_{34}&0&0\\ Z_{12}Z_{23}&2_{34}&Z_{45}&0\\ Z_{12}Z_{23}&0&0&Z_{56}\end{bmatrix}\begin{bmatrix}B_{1}\\ B_{2}\\ B_{3}\\ B_{4}\\ B_{5}\end{bmatrix}  \tag{37}\]  
Compactly,  
\[[\Delta V]=[BCBV][B] \tag{38}\]  
Substituting (31) into (38), we get a DG incorporation relationship that involves the bus voltage and injection current as  
\[[\Delta V]=[BCBV][BIBC][I] \tag{39}\]  
In a network involving DG, the injected power is given as  
\[S_{i}=S_{Di}-S_{Gi} \tag{40}\]  
where  
\(S_{i}\) = the injected power at the \(i\)th bus  
\(S_{Di}\) = the load power at the \(i\)th bus  
\(S_{Gi}\) = the generator power at the \(i\)th bus  
Since incorporating DG into the network system takes the form of current, then DG at the \(i\)th bus is given as  
\[I_{i}=\frac{\left[\left(P_{Di}-P_{Gi}\right)-j\left(Q_{Di}-Q_{Gi}\right) \right]^{*}}{V_{i}} \tag{41}\]  
The variation in voltage in the DG-incorporated network is given as follows  
\[\left[\Delta V^{K+1}\right]=[BCBV][BIBC]\Big{[}I^{K}\Big{]} \tag{42}\]  
where  
\[\left[V^{K+1}\right]=\left[V^{0}\right]-\Delta V^{K+1} \tag{43}\]