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


MkT\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=-bpstbkp′′-1+bpstbks′′-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.

bij′′-1b_{ij}^{{\prime\prime}{\thinspace-1}} is an element from the inverse of B′′{B^{\prime\prime}} as defined by Stott and Alsac (2).

t is magnitude of the transformer’s tap ratio.

Chan and Brandwajn (1) suggest that sensitivities computed from Equation 47 are useful for coordinating adjustments to a bus that is controlled by several transformers.