Page 240 - Mathematical Models and Algorithms for Power System Optimization

232 Chapter 7

(1) Multiorder linear AR model:

(7.13)
ϕ BðÞZ t ¼ a t
P
P k
where a t is white noise, Z t is output, ϕ BðÞ ¼ 1 ϕ B , B means the delay of one-step
k
operation, P refers to the order. k¼1
Condition: When the module of each root in ϕ(B)¼0 is greater than 1, the stationary solution Z t
of such model is called stationary autoregressive process, denoted as AR model.

(2) MA model:

(7.14)
Z t ¼ θ BðÞa t
g
P k
where θ BðÞ ¼ 1 θ k B , k means the order.
k¼1
Condition: When the module of each root in θ(B)¼0 is greater than 1, the stationary solution of
such model is called reversible moving average process, denoted as MA model.
(3) ARMA model:

(7.15)
ϕ BðÞZ t ¼ θ BðÞa t
If satisfying the two previous conditions, the stationary solution Z t of such model is called an
ARMA process. (p,q) means the orders of AR and MA models.

7.3.2.2 Autocorrelation functions and partial autocorrelation functions of AR, MA,
ARMA process
(1) AR process. Rewrite the expansion of Eq. (7.13)

Z t ¼ ϕ Z t 1 + ϕ Z t 2 + … + ϕ Z t p + a t
2
p
1
Multiply Z t k with two sides of the equation to take the mean and allow for E(Z t k a t )¼0,
k>0 to get the covariance function:
γ ¼ ϕ γ + ϕ γ + ⋯ + ϕ γ (7.16)
1 k 1
2 k 2
p k p
k
Denoted as:
Bρ ¼ ρ
k k 1
And autocorrelation function:
ρ ¼ ϕ ρ + ϕ ρ + ⋯ + ϕ ρ ð k > 0Þ (7.17)
p k p
1 k 1
k
2 k 2
Eq. (7.17) can be transformed into:
ϕ BðÞρ ¼ 0 (7.18)
k