Page 266 - Mathematical Models and Algorithms for Power System Optimization

258 Chapter 7

or

0
λ +1
1
ð j
I ϕ τðÞ 1 λ Þj ϕ τðÞ + I½ Š ¼ 0 (7.89)
0

0
ð 1 λ Þ

Consequently, we have:
j λI ϕ τ ðÞj ¼ 0 (7.90)
where
λ +1
0
λ ¼ (7.91)
1 λ 0
Apparently, λ is the eigenvalue of matrix ϕ(τ). From Eq. (7.89), we know
λ 1
0
λ ¼ (7.92)
λ +1
From Eq. (7.91), we obtain:
j
j λ +1j > λ 1j (7.93)
Let
λ ¼ α +jβ (7.94)

Substituting it into Eq. (7.93), we have:
α > 0 (7.95)

That is, the real parts of eigenvalues λ of matrix ϕ(τ) are positive. Hence the theorem is
proven. □

The difference between the eigenvalue method and logarithmic matrix method is that the
former is of computational complexity, whereas the latter requires real parts of all eigenvalues
are positive. Because the eigenvalues of most systems are dominated by real parts, and real
roots are normally meeting the condition, the simple method of logarithmic matrix can
normally be used to solve. In case of failure to satisfy the previous condition, the following
successive approximation method can be used to solve.

7.7.2 Mutual Transformation Method Between Difference Equations
(Successive Approximation Method)

Formulation of the known difference equation is:

Xk +1Þ ¼ ϕ τðÞXkðÞ
ð