Page 244 - Mathematical Models and Algorithms for Power System Optimization

236 Chapter 7
^
(2) Truncation criteria of ^ ρ and ϕ : Theoretically, the truncation of autocorrelation function
k
kk
ρ k and partial correlation function ϕ kk mean the tails are absolutely zero. However, ^ ρ k
^
and ϕ are only approximate estimates to ρ k and ϕ kk , including some errors. Therefore,
kk
even the ^ ρ of MA model will not be all zero after step q but fluctuate around zero. It can
k
be assumed that such fluctuation is of normal distribution. Judging from the normal
h i
2
2
distribution, as P j ^ ρ j < ffiffi ¼ 95:5%, for k > q, if the number of |^ ρ | > ffiffi is not greater
p
p
k
k
n
n
^
than 4.5%, the ^ ρ can be truncated. ϕ can be judged following the same method.
kk
k
^
(3) Identification of order d of the nonstationarity stochastic model: If the ^ ρ and ϕ calculated
k
kk
from the sample sequence are neither truncated nor tailing, then its first-order sequence
^
2
rZ t , second order sequence r Z t ,…, are tested until getting desired ^ ρ and ϕ . Generally,
kk
k
satisfactory results can be obtained when the difference d is no more than three times.
(4) Identification of p and q for ARMA: p and q have to be identified by tests from low to high
orders, for example, sequential tests (1,1), (1,2), (2,1),…, until appropriate p and q are
selected.
The load disturbance model obtained from identification is generally expressed with the
following difference equation:
ð
ΔP L kðÞ + a 1 ΔP L k 1ð Þ + ⋯ + a n ΔP L k nÞ ¼ ξ + b 1 ξ k 1 + b 2 ξ k 2 + ⋯ + b n ξ k n (7.30)
k
where ξ k is the white noise sequence. Eq. (7.30) can be transformed into the following discrete
state equation:
∗
ð
yk +1Þ ¼ Φ ykðÞ + B ∗ ξ kðÞ
T (7.31)
ΔP L kðÞ ¼ H ∗ ykðÞ
where
Eq. (7.31) can be further transformed into the following equivalent continuous state equation:

_ ytðÞ ¼ A ∗ ytðÞ + B ∗ ξ tðÞ
T (7.32)
ΔP L tðÞ ¼ H ∗ ytðÞ
Section 7.7 gives the method to calculate A from the given ϕ∗.

7.3.4 Parameter Estimation of the Model ΔP L

After identifying the order of models, the parameter estimation can be made. This section only
discusses the parameter estimation of AR models.
Substituting the covariance function ^ ρ of sample for ρ k into Eq. (7.23), the following equation,
k
that is, the Yule-Walker equation is obtained: