5.3 Controlled Bus Sensitivity

The proportionality constant α \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 α \alpha is one (the controlled voltage is perfectly sensitive to tap changes). An alternate assumption is that

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


M k T \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

α = - b p s t b k p ′′ - 1 + b p s t b k s ′′ - 1 \alpha=-b_{ps}t{b}_{kp}^{{\prime\prime}{\thinspace-1}}+b_{ps}t{b}_{ks}^{{% \prime\prime}{\thinspace-1}} (47)


bij is an element from the nodal susceptance matrix Bbus.

b i j ′′ - 1 b_{ij}^{{\prime\prime}{\thinspace-1}} is an element from the inverse of B ′′ {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.