*2.1. Bus Sensitivity Factor*

The bus sensitivity factor (BSF) is expressed as the ratio of incremental real power change flowing in bus '*i*' connected between buses '*j*' and '*k*' to the incremental change in *m*th power of the bus as given below. BSF provides the optimal location for the placement of PHSU based on highest negative sensitive indexes.

$$BSF^i\_m = \frac{\Delta P\_{jk}}{\Delta P\_m} \tag{11}$$

where, *BSF<sup>i</sup> m* indicates the quantum of real power change in real power flows in a transmission line in accordance to real power injection at bus *m*.

BSF can be derived from Equation (2) as illustrated below.

$$
\Delta P\_{\vec{j}k} = \frac{\partial P\_{\vec{j}k}}{\partial \delta\_{\vec{j}}} \Delta \delta\_{\vec{j}} + \frac{\partial P\_{\vec{j}k}}{\partial \delta\_{\vec{k}}} \Delta \delta\_{\vec{k}} + \frac{\partial P\_{\vec{j}k}}{\partial V\_{\vec{j}}} \Delta V\_{\vec{j}} + \frac{\partial P\_{\vec{j}k}}{\partial V\_{\vec{k}}} \Delta V\_{K} \tag{12}
$$

$$
\Delta P\_{\text{jk}} = a\_{\text{jk}} \Delta \delta\_{\text{j}} + b\_{\text{jk}} \Delta \delta\_{\text{k}} + c\_{\text{jk}} \Delta V\_{\text{j}} + d\_{\text{jk}} \Delta V\_{\text{k}} \tag{13}
$$

$$
\Delta P\_{\text{jk}} = a\_{\text{jk}} \Delta \delta\_{\text{j}} + b\_{\text{jk}} \Delta \delta\_{\text{k}} + c\_{\text{jk}} \Delta V\_{\text{j}} + d\_{\text{jk}} \Delta V\_{\text{k}} \tag{14}
$$

where,

$$a\_{\vec{\boldsymbol{k}}} = V\_{\vec{\boldsymbol{\beta}}} V\_{\vec{\boldsymbol{k}}} \boldsymbol{Y}\_{\vec{\boldsymbol{\beta}}\vec{\boldsymbol{k}}} \sin(\theta\_{\vec{\boldsymbol{\beta}}\vec{\boldsymbol{k}}} + \delta\_{\vec{\boldsymbol{k}}} - \delta\_{\vec{\boldsymbol{\beta}}}) \tag{15}$$

$$b\_{\vec{\beta}k} = -V\_{\vec{\beta}}V\_{\vec{k}}Y\_{\vec{\beta}k}\sin(\theta\_{\vec{\beta}} + \delta\_{\vec{k}} - \delta\_{\vec{\beta}})\tag{16}$$

$$c\_{jk} = -V\_k Y\_{jk} \cos(\theta\_{jk} + \delta\_k - \delta\_j) - 2V\_k Y\_{jk} \cos \theta\_{jk} \tag{17}$$

$$d\_{\vec{\boldsymbol{jk}}} = V\_{\vec{\boldsymbol{j}}} Y\_{\vec{\boldsymbol{jk}}} \cos(\theta\_{\vec{\boldsymbol{jk}}} + \delta\_{\vec{\boldsymbol{k}}} - \delta\_{\vec{\boldsymbol{j}}}) \tag{18}$$

The Jacobian Matrix using Newton–Raphson (NR) method is given in Equation (19).

$$
\begin{pmatrix} \Delta P\\ \Delta Q \end{pmatrix} = [J] \begin{pmatrix} \Delta \delta\\ \Delta V \end{pmatrix} = \begin{pmatrix} J\_{11} & J\_{12} \\ J\_{21} & J\_{22} \end{pmatrix} \begin{pmatrix} \Delta \delta\\ \Delta V \end{pmatrix} \tag{19}
$$

where,

$$
\Delta \delta = \left[ I\_{11} \right]^{-1} \left[ \Delta P \right] \\
= \left[ M \right] \left[ \Delta P \right] \tag{20}
$$

<sup>Δ</sup>δ*j* = *n l*=1 *mjl*Δ*Pl j* = 1, 2, ......... ., *n*, *j* - *s* (21)

Thus

$$BSF^i\_m = a\_{jk}m\_{jl} + b\_{jk}m\_{jl} \tag{22}$$
