## 1 Introduction
In addition, the effect of DG technology on the improvement of voltage stability in the distribution network is studied by [13; 14; 15; 16; 17; 18]. Similar research, such as that conducted in [5], notably examines the impact that a DG unit's capacity and location has on the improvement of voltage stability in distribution networks. In [19] the authors conducted a pragmatic examination of the effect that DG units has on systems stability. Voltage stability index (VSI) is usually used for assessing load bus stability in a power system, and it can be formulated as follows [20],  
\[L_{i+1}=\frac{4\Big{[}V_{i}^{2}(P_{i+1}\times r_{i}+Q_{i+1}\times x_{i})+(P_{ i+1}\times x_{i}-Q_{i+1}\times r_{i})^{2}\Big{]}}{V_{i}^{4}}, \tag{1}\]  
where \(L\) is a stability index, \(r_{i}+jx_{i}\) is a branch with impedance and it is connected between sending end bus with voltage of \(V_{i}\) and receiving node with loading of \(P_{i+1}+jQ_{i+1}\). A modified voltage stability index formula is also given by [21] as follows,  
\[L=4\Big{[}\Big{(}P_{leq}\times r_{eq}+Q_{leq}\times x_{eq}\Big{)}+\Big{(}P_{leq }\times x_{eq}-Q_{leq}\times r_{eq}\Big{)}^{2}\Big{]}, \tag{2}\]where \(L\) is a stability index, \(P_{leq}+jQ_{leq}\) is the total real and reactive power respectively and \(r_{eq}+jx_{eq}\) is the impedance of the connecting branches of network and they are given as:  
\[r_{eq}=\sum P_{loss}/\bigg{[}\Big{(}P_{leq}+\sum P_{loss}\Big{)}^{2}+\Big{(}Q_{leq }+\sum Q_{loss}\Big{)}^{2}\bigg{]}, \tag{3}\]  
\[x_{eq}=\sum Q_{loss}/\bigg{[}\Big{(}P_{leq}+\sum P_{loss}\Big{)}^{2}+\Big{(}Q_ {leq}+\sum Q_{loss}\Big{)}^{2}\bigg{]}. \tag{4}\]  
For a stable system, the value of stability index, \(L\) should be less than 1.0. Additional formula of VSI is proposed by [PERSON] et al. in [22] for a simple power system with two busses. Voltage stability index was applied in the context of an expansive system incorporating numerous busses. The VSI is formulated in order to evaluate voltage stability in the general radial distribution network as follows,  
\[\text{VSI}=0.5\times V_{1}-\frac{\Big{(}P_{leq}\times r_{eq}+Q_{leq}\times x_ {eq}\Big{)}}{V_{1}}, \tag{5}\]  
where \(V_{1}\) is the sending end voltage of the reduced single line network. Different studies have utilized different optimization approaches to determine the optimal location of DG units and size with regards to voltage stability and loss reduction [23; 24; 25; 26]. The range of the VSI, between unity at no load and zero with regard to the voltage collapse point, can be determined with Equation (15) for line \(l\) or for node \(m\) as displayed in Figure 1.  
\[\text{VSI}(m)=V_{k}^{4}-4\{P_{km}x_{km}-Q_{km}r_{km}\}^{2}-4\{P_{km}r_{km}-Q_{ km}x_{km}\}V_{k}^{2}, \tag{6}\]  
where, \(V_{k}\) denotes the voltage magnitude at node-\(k\), \(P_{km}\) represents the sum of real power loads of all the nodes beyond node-\(m\) plus the real power load on node-\(m\) itself plus the sum of the real power losses of all the branches beyond node-\(m\), \(Q_{km}\) denotes the sum of reactive power loads of all the nodes beyond node-\(m\) plus the reactive power load on node-\(m\) itself, plus the sum of the reactive power losses of all the branches beyond node-\(m\), \(r_{km}-jx_{km}\) represents the resistance and reactance of line-\(l\), while \(P_{m}+jQ_{m}\) represents the real and reactive power load at node-\(m\). On the other hand, the power losses at each line and total power losses can be determined as in Equations (7) and (8), respectively. \[Pineloss_{ij}=\big{|}I_{ij}\big{|}^{2}\times R_{ij}, \tag{7}\]  
\[PTotalloss_{ij}=\sum_{i=1}^{nbr}\lvert I_{i}\rvert^{2}\times R_{i}, \tag{8}\]  
Figure 1: Sample distribution line.