Page 265 - Mathematical Models and Algorithms for Power System Optimization
P. 265
Optimization Method for Load Frequency Feed Forward Control 257
Definition 1 Let
ϕ τðÞ ¼ e Aτ
(7.80)
By contrast with the logarithmic definition of real field, the logarithm matrix is defined as:
lnϕ τðÞ ¼ Aτ (7.81)
Definition 2 Define the series expansion of logarithmic matrix as:
h i 3
1 1 1
lnϕ τðÞ ¼ 2 ½ ϕ τðÞ I ϕ τðÞ + I + ð ϕ τðÞ IÞ ϕ τðÞ + IÞ + ⋯
½
ð
3
(7.82)
1 h 1 i 2k 1
+ ð ϕ τðÞ IÞ ϕ τðÞ + IÞ + ⋯ ð k ¼ 1, 2, ⋯Þ
ð
2k 1
where the real part of eigenvalue of matrix ϕ(τ) is greater than 0.
Theorem 3 If the real parts of all eigenvalues of matrix ϕ(τ) are positive, according to
Eqs. (7.81) and (7.82), then matrix A can be expressed as:
2 1 1 h 1 i 3
ð
½
A ¼ ½ ϕ τðÞ I ϕ τðÞ + I + ð ϕ τðÞ IÞ ϕ τðÞ + IÞ + ⋯
τ 3
(7.83)
1 h 1 i 2k 1
+ ð ϕ τðÞ IÞ ϕ τðÞ + IÞ + ⋯ ð k ¼ 1, 2, ⋯Þ
ð
2k 1
Proof Using the proof by contradiction, that is, assuming Eq. (7.83) is satisfied, then the real
parts of all eigenvalues of ϕ(τ) are positive.
Let
1
½
½
B ¼ ϕ τðÞ I ϕ τðÞ + I (7.84)
Judging from Eq. (7.81), the necessary and sufficient condition for matrix A to exist is:
k
lim B ¼ 0 (7.85)
k!∞
0
It means that the modulo of eigenvalue of matrix B must be less than 1. Let λ represent the
eigenvalues of matrix B, then:
0
λ jj < 1 (7.86)
Based on Eq. (7.84), the characteristic equation of matrix B is
1
½
½
λ I ϕ τ ðÞ I ϕ τ ðÞ + I ¼ 0 (7.87)
0
or
0 0 1
½
ð j λ +1ÞI 1 λð Þϕ τðÞj ϕ τðÞ + I ¼ 0 (7.88)