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40) and assuming sampling at At second interval A(t2) = A(tx) + M [ v(t2) + v( tl )] - L [ i(t 2 ) -i( t l )] (108) With subscript k denoting the # h sample A k+i = Ak + H i v k+i + vki - L i \+i - y (109) The voltage in Eq. (109) is the winding voltage while the current in equation is the trip current for each phase of the three-winding transformer. Given the initial flux linkage, Eq. (109) can be used to track the flux-current plot of the transformer at each sample time. An 51 COMPUTER RELAYING IN POWER SYSTEMS open circuit magnetizing curve of a transformer is shown in Figure 17 along with the flux-current curve that would be associated with an internal fault.

The validity of these impressions can be checked by considering the sources of error in the calculations. Currents and voltages which do not satisfy the differential equation (Eq. (37)) will cause errors in the estimated R and L. Included in the sources of such signals are the effects produced by the shunt capacitances which would be included in the π section model of the transmission lines along with transducer errors, A/D errors, and errors in the anti-aliasing filters. The shunt capacitors are more important in models of longer, higher voltage lines.

THORP AND ARUN G. PHADKE Observing that F N = S S (compare Eqs. (11)-(13) with Eqs. (70) and (75)) and that the bracketed quantity in Eq. (75) is Γ Σ cos(k0)zk 1 [ Σ sin(k0)z k J it can be seen that the two-state Kaiman filter with no initial information and a measurement error with a constant covariance is identical to the equivalent least-squares solution. In fact, in this situation, if No is a multiple of a half-cycle, then the Kaiman filter is the same as the appropriate Fourier estimates. The same conclusions apply to Kaiman filter solution with more terms  since S in Eqs.