Page 263 - Mathematical Models and Algorithms for Power System Optimization
P. 263
Optimization Method for Load Frequency Feed Forward Control 255
Theorem 1 If matrix ϕ(τ) has n different eigenvalues ϕ i (i¼1, 2, …, n) with modulo less
than 1 and n eigenvectors e i (i¼1, 2, …, n) corresponding to ϕ i and mutually independent, then
a constant matrix A can be found, which satisfies the equivalent condition to make
the continuous model Eq. (7.69) equal to the discrete model Eq. (7.68):
_
XtðÞ ¼ AXtðÞ (7.69)
Proof
Let
ð
T ¼ e 1 , e 2 , ⋯, e n Þ (7.70)
because (e 1 , e 2 , …, e n ) are independent of each other, so the inverse of matrix T exists. From
linear algebra, we have:
φ 1
2 3
φ
6 7
ϕ ϕ ¼ T 6 2 7 1 (7.71)
T
ðÞ
t
4 ⋱ 5
φ
n
Let
φ ¼ e λ i τ ¼ x i +jy i (7.72)
i
where
1 1 i
y
λ i ¼ lnφ ,lnφ ¼ ln φ jj + jargφ , argφ ¼ tan (7.73)
i
i
i
i
i
τ x i
Hence
2 λ 1 τ 3
e
6 e λ 2 τ 7
T
φ τðÞ ¼ T 6 7 1 (7.74)
4 ⋱ 5
e λ n τ
Expand the above expression to get
2 2 2 1 K K 1
ϕ τðÞ ¼ T I + Λτ + Λ τ + ⋯ + Λ τ + ⋯ T (7.75)
2! K!
where
2 3
λ 1
λ 2
6 7
Λ ¼ 6 7 (7.76)
4 ⋱ 5
λ n
Let
A ¼ TΛT 1 (7.77)