# 5.2 Error Feedback Formulation

Stott and Alsac (2) and Chan and Brandwajn (1) describe transformer tap adjustments as an error feedback formula

 $a^{new}-a^{old}=\alpha\left(\frac{V_{k}^{sp}}{V_{k}}\right)$ (44)

where all quantities are expressed in per unit. The per unit voltage transformation ratio apu is computed as follows

 $a^{pu}=\frac{V_{p}^{pu}}{V_{s}^{pu}}=\frac{\frac{V_{p}}{V_{p}^{nom}}}{\frac{V_% {s}}{V_{s}^{nom}}}$

which simplifies to

 $a^{pu}=\frac{V_{s}^{nom}\frac{V_{p}}{V_{p}^{nom}}}{V_{s}}=\left(\frac{V_{p}}{V% _{s}}\right)\left(\frac{V_{s}^{nom}}{V_{p}^{nom}}\right)=\frac{a}{a^{nom}}$

where anom is the nominal voltage transformation ratio. Therefore, Equation 44 is equivalent to

 $\frac{a^{new}}{a^{nom}}-\frac{a^{old}}{a^{nom}}=\alpha\frac{V_{k}-V_{k}^{sp}}{% V_{k}^{base}}$

or

 $a^{new}-a^{old}=a^{nom}\alpha\frac{V_{k}-V_{k}^{sp}}{V_{k}^{base}}$

in physical quantities. Solving for the updated transformation ratio yields

 $a^{new}=a^{old}+a^{nom}\alpha\frac{V_{k}-V_{k}^{sp}}{V_{k}^{base}}$ (45)

Computation of $\alpha$, the sensitivity of the regulated voltage to changes in transformation ratio, is discussed in Section 5.3.