Page 242 - Mathematical Models and Algorithms for Power System Optimization

234 Chapter 7

The corresponding autocorrelation function is:

8
1 ð k ¼ 0Þ
>
>
θ k + θ 1 θ k +1 + ⋯ + θ q k θ q
<
ρ ¼ 1+ θ + ⋯ + θ 2 ð 0 < k qÞ (7.23)
k
2
> 1 q
>
0 ð k > qÞ
:
0
Eq. (7.23) shows, when the steps between Z t and Z t are greater than q, they are no longer
correlated to each other. Seen from ρ 0 , ρ 1 , …, ρ q , …, the origin part of such sequence is zero as
from q, and this autocorrelation function is called truncation.
(3) Partial autocorrelation function. The approximation of Z k is represented by a linear
combination of Z t 1 , …, Z t k and the coefficient ϕ kz is chosen so that:
! 2
k
X
minS ¼ EZ t ϕ Z t j (7.24)
kj
j¼1
To obtain ϕ kj , take partial derivative to be zero:
∂S
¼ 0 j ¼ 1, 2, …, kÞ (7.25)
ð
∂ϕ kj
Expand S to get:
k k k
X 2 X X
S ¼ γ + ϕ γ 2 ϕ γ +2 ϕ ϕ γ (7.26)
0 kj 0 kj i kj ki j i
j¼1 j¼1 j>i
Substitute it into Eq. (7.25) and then divide by γ 0 to get:
2 32 3 2 3
1 ρ 1 ⋯ ρ k 1 ϕ k1 ρ 1
ρ ⋮ ρ ⋮ ⋮
6 1 76 7 6 7
⋮ ⋮ ⋮ ⋮ ⋮
6 k 2 76 7 ¼ 6 7 (7.27)
4 54 5 4 5
ρ ρ ⋮ ϕ ρ
k k 2 kk k
ϕ 11 , ϕ 22 ,…, ϕ kk is called partial autocorrelation function of Z k . The following shows how to
solve ϕ kj , proceeding directly from the definition of S, substitute Eq. (7.13) into S to get:
" # 2
k
X
S ¼ E ϕ Z t 1 + ϕ Z t 2 + ⋯ + ϕ Z t p + a t
1 2 p ϕ Z t j
kj
j¼1
" # 2
p k
X

X
¼ Ea t + ϕ ϕ kj Z t j ϕ Z t j
j
kj
j¼1 j¼p +1 (7.28)
" # 2
p k
X
2

X
¼ σ + E ϕ ϕ Z t j ϕ Z t j
kj
j
a
kj
j¼1 j¼p +1
σ 2
a