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**Additional info for Analysis and Control System Techniques for Electric Power Systems, Part 4 of 4**

**Example text**

Sigma z Note that the sampling rate does not appear in Eq. (100). The explanation lies with the assumption about the noise and anti-aliasing filter. The assumption that the noise before filtering is white ( S(u) = 46 JAMES S. THORP AND ARUN G. PHADKE S v for ail ώ) coupled with the ideal anti-aliasing filter produces a situation where a higher sampling rate yields more samples each of which has a larger variance. Since the noise is ultimately band-limited, for example, by the transducers, there is a sampling frequency at which there would be a gain in faster sampling.

Is invertible. the specific form of the two-state system in Eq. (54): φ± = I Hk = [ cos(k^) sin(k^) ] with Rk = I, the recursion for information matrix becomes F k+1 = F cos^(k+l)0 cos(k+l)0 s i n ( k + l ) 0 k + the For and the cos(k+l)0 s i n ( k + l ) 0 sin2(k+l)0 (69) or F cos k0 coskô sinkfl Σ N = coskö sink# sin k0 (70) k=l and Eq. (68) simplifies to K k+1 - F H k+1 (71) k+1 and using Eq. (71) in Eq. (57) K+i = \ + F kii H k+it «k+i - H k+i «kl (72) or x k+i - F k+i L [F k + r H k+l H k+li x k + Hk + 1 "k+lj (73) With φ = I and R = I Eq.

THORP AND ARUN G. PHADKE ¿(M) . ¿(M-i) + (XM . XM_K , ¡κ ή -m (29) Equation (29) is valid for any Θ or any K. If ΚΘ is not a multiple of a half-cycle then the real and imaginary parts of Eq. (29) must be T —1 multiplied by the matrix (S S) to form the estimates. With ΚΘ = 2π (a full -cycle window), the recursive form of the algorithm becomes ¿(M) = ¿(M-l) + [XM _ XM_K] c o s { m (30) ¿(M) = ¿(M-l) + [XM _ XM_K] M (31) ¿(M) = ¿(M-l) + [XM + XM_K] c o s { m (32) χ(Μ) = ¿(M-l) + [XM + XM_K] M (33) m while with Κθ = τ m The bracketed quantity in Eqs.