Page 241 - Mathematical Models and Algorithms for Power System Optimization

Optimization Method for Load Frequency Feed Forward Control 233

This is a difference equation with the coefficient of ϕ k . Assume that the solution of Eq. (7.18) is:

k k k
ρ ¼ A 1 G + A 2 G + ⋯ + A p G ð k > pÞ (7.19)
k
p
2
1
1
1
1
where G 1 , G 2 ,…, G p are different single roots under the stationary condition of ϕ(B)¼0,
1
and their modules |G i | >1. A 1 , A 2 , …, A p in Eq. (7.19) are simultaneous solutions of the
following equations:
9
A 1 + A 2 + ⋯ + A p ¼ 1
>
=

k
k
k
A 1 G G k + A 2 G G k + ⋯ + A p G G k ¼ 0 (7.20)
p
1
2
2
p
1
>
k ¼ 1,…, p +1 ;
1
Because |G j | >1, namely the roots of φ(B) are outside the unit circle, from which jG k j
declines along the exponential curve, approach zero, thus, the absolute value of autocorrelation
function determined by Eq. (7.20) follows the same rules, approaching zero. Different from the
MA process, ρ k is not identically equal to zero after a certain k. Such autocorrelation function is
called tailing.
Further, based on the first P equations of Eq. (7.17), allowing for ρ k ¼ ρ k , we have:
ρ ¼ ϕ ρ + ϕ ρ + ⋯ + ϕ ρ 9
1 1 0 2 1 p p 1 >
ρ ¼ ϕ ρ + ϕ ρ + ⋯ + ϕ ρ >
=
2
p p 2
1 1
2 0
⋮ (7.21)
>
ρ ¼ ϕ ρ + ϕ ρ + ⋯ + ϕ ρ >
;
2 p 2
p
p 0
1 p 1
It can be seen clearly that ϕ 1 , ϕ 2 , ϕ 3 ,…, ϕ p is the solution of linear equation Set (7.21), the
coefficients of which are made up of ρ 0 , ρ 1 ,…, ρ p . Eq. (7.21) is called the Yule-Walker
equation.
The truncation of autocorrelation function ρ k is the basic characteristic of MA process. But the
tail of ρ k is not the unique characteristic of AR process, because the autocorrelation function in
ARMA process is also tailing. The partial autocorrelation function of AR process will be
discussed in (3).
(2) MA process. Rewrite the expansion of Eq. (7.14):
Z t ¼ a t θ 1 a t 1 θ 2 a t 2 ⋯ θ q a t q
Z t k ¼ a t k θ 1 a t k 1 θ 2 a t k 2 ⋯ θ q a t k q
Multiply these two equations, then take the mathematical expectation to get the variance
function:
8
2
2
σ 2 1+ θ + θ + ⋯ + θ 2
> ð k ¼ 0Þ
< 1 2 q
γ ¼ 2 ð 0 < k qÞ (7.22)
k
> σ θ k + θ 1 θ k +1 + ⋯ + θ q k θ q
:
0 ð k > qÞ