# 5.3 Controlled Bus Sensitivity

The proportionality constant $\alpha$ (referenced in Equation 45 and Equation 42) represents the sensitivity of the controlled voltage to tap changes at the regulating transformer. It commonly assumed that $\alpha$ is one (the controlled voltage is perfectly sensitive to tap changes). An alternate assumption is that

 $\alpha=\mathrm{M}_{k}^{T}{\left(\mathrm{B^{\prime\prime}}\right)}^{-1}\mathrm{N}$ (46)

where

$\mathrm{M}_{k}^{T}$ is a row vector with 1 in the kth position.

N is a column vector with -bpst in position p and bpst in position s.

Carrying out these matrix operations yields

 $\alpha=-b_{ps}t{b}_{kp}^{{\prime\prime}{\thinspace-1}}+b_{ps}t{b}_{ks}^{{% \prime\prime}{\thinspace-1}}$ (47)

where

bij is an element from the nodal susceptance matrix Bbus.

$b_{ij}^{{\prime\prime}{\thinspace-1}}$ is an element from the inverse of ${B^{\prime\prime}}$ as defined by Stott and Alsac (1).

t is magnitude of the transformer’s tap ratio.

Chan and Brandwajn (2) suggest that sensitivities computed from Equation 47 are useful for coordinating adjustments to a bus that is controlled by several transformers.