## 2 Modeling the Problem under Study
### Assessing Reliability and Proposed Indicators  
Based on the purpose of this work, which is to improve the reactive power of the network in the planning stage, first, the effect of reactive power on a reliability parameter is investigated. Then, four reliability indicators, including unsecured energy and unsecured clock, were combined to form a single index due to a lack of active and reactive power. Providing reactive power locally in a power system causes the reactive power component of the current passing through the lines to be relatively compensated. Consequently, the current size is reduced. Reducing the current size will reduce the active and reactive power losses proportional to the square of the current and the feeder temperature. According to [4], the expected life of insulation materials decreases exponentially with an increasing temperature. As a result, the feeder failure rate will decrease after the optimal placement of reactive power sources, and the distribution network reliability level will increase. Here, the branch has a component (\(I\!b^{\prime}=I\!b^{0\prime}\)) and \(\lambda b^{0}\) is an error rate. After compensation, the reactive current component of this branch will be zero, and the error rate \(\lambda b^{f}\) will be changed. The mathematical modeling is as follows [4]:  
\[\left\{\begin{array}{l}I_{b}^{r}=I_{b}^{0,r}\Rightarrow\lambda_{b}=\lambda_ {b}^{0}\\ I_{b}^{r}=0\Rightarrow\lambda_{b}=\lambda_{b}^{f}\end{array}\right.\forall b \in Sb\quad Sb=\{1,\ldots,Number\ of\ branches\} \tag{16}\]  
To evaluate the effect of compensating the reactive power component of the current on the failure rate, four mathematical functions, including linear, power, exponential, and logarithmic functions, can be considered. Fixed values of these functions \((A,B)\) are obtained by using test results. In this study, it is hypothesized that complete compensation of the reactive power component of the current reduces the feeder failure rate by a maximum of 28% [29]. The normalized feeder failure rate curves versus the normalized value of the reactive power component of the current are demonstrated in Figure 6.  
Here, the linear function is used to model the error rate, while other functions are readily applicable to the problem. After modeling the effect of reactive power on the initial index (failure rate) and reliability for active and reactive power planning, we need to  
Figure 6: Branch failure rate (pu) versus reactive current component (pu).  
examine some additional indicators. For this purpose, the energy supply indicators and the hour supply due to a lack of active power and reactive are used as follows [30]:  
\[ENS_{P} =\sum_{i=1}^{nc}LC_{P(i)}\cdot\lambda_{i} \tag{17}\] \[ENS_{Q} =\sum_{i=1}^{nc}LC_{Q(i)}\cdot\lambda_{i}\] (18) \[VNS_{P} =\sum_{i=1}^{nc}QC_{P(i)}\cdot\lambda_{i}\] (19) \[VNS_{Q} =\sum_{i=1}^{nc}QC_{Q(i)}\cdot\lambda_{i} \tag{20}\]  
In the above relations, \(\lambda_{i}\) is the annual unavailability of the \(i\) load point, \(nc\) is events, \(QC_{P(i)},LC_{P(i)}\) is the amount of active and reactive load not applied, \(QC_{Q(i)},LC_{Q(i)}\) is the amount of active and reactive load not applied, \(ENS_{p}\) and \(ENS_{Q}\) are the energy applied due to a lack of active power, \(VNS_{p}\) and \(VNS_{Q}\) are reactive, and VAR represents unsecured clocks caused by a lack of active and reactive power.