Page 267 - Mathematical Models and Algorithms for Power System Optimization

Optimization Method for Load Frequency Feed Forward Control 259

its corresponding difference equation is:

ð
ðÞXk ðÞ
Xk +1Þ ¼ ϕ τ 1
where τ6¼τ 1 , that is, with different sampling intervals.
The following two cases are to be considered, respectively:

(1) τ 1 >τ, τ 1 is the integral multiple of τ, that is, τ 1 ¼nτ
From the definition of transition matrix, we know:
n
ðÞ
ϕ τ 1 ¼ ϕ τðÞ
ϕ(τ 1 ) can be solved using the above equation.

(2) τ 1 <τ, τ¼nτ 1
Aτ
Take the first two terms of polynomial expansion of e
2 2
e Aτ ¼ I + Aτ + 1 A τ + ⋯ (7.96)
2!
to get the approximate equation of matrix ϕ(τ)
ϕτðÞ I + Aτ (7.97)

Hence,
1
A ½ ϕτðÞ IŠ (7.98)
τ
Because τ 1 ¼τ/n, correspond to Eq. (7.98), the approximate expression of A and the transition
matrix ϕ τ 1 of τ 1 is:
ðÞ
1
ðÞ
A ½ ϕ τ 1 IŠ (7.99)
τ 1
Combining Eqs. (7.98) and (7.99), we know:
τ 1
ðÞ
ϕ τ 1 ¼ ½ ϕ τðÞ IŠ + I (7.100)
τ
Apparently, ϕ(τ 1 ) obtained from Eq. (7.100) is very rough and needs to be further refined
generally by iteration. Let the k-th iterative values ϕ(τ 1 ) and ϕ(τ)be ϕ k (τ 1 ) and ϕ k (τ). Let ϕ(τ 1 )
solved from Eq. (7.100) be ϕ 1 (τ 1 ). Judging from the properties of transition matrix, we have
n
ðÞ
ϕ τ 1 ¼ ϕ τ ðÞ (7.101)
k k
In the k-th iteration, if the error lies between ϕ k (τ) and ϕ(τ), where the former is solved by
Eq. (7.101) and the latter is the given one, is not satisfactory.
ϕ k+1 (τ 1 ) can be rectified using the following:
ϕ (7.102)
τ 1 ¼ ϕ τ 1 + α ϕ τðÞ ϕ τðފ
k +1 ðÞ k ðÞ ½ k