Page 268 - Mathematical Models and Algorithms for Power System Optimization
P. 268
260 Chapter 7
where α is the step factor. Generally, take α¼τ 1 /τ, or adjust based on test results. The iterative
process shall continue until the norm of residual matrix Δϕ(τ):
k Δϕ τ ðÞk ¼ ϕ τ ðÞ ϕ τ ðÞk (7.103)
k
k
k
is less than a given error.
Apparently, based upon the finalized ϕ(τ 1 ), it is easy to solve the transition matrix of any time
m
interval t ¼ mτ 1 ¼ τ:
n
m
ðÞ
ϕ tðÞ ¼ ϕ τ 1 (7.104)
If n is large enough, based on the transition matrix equation:
_
ϕ tðÞ ¼ Aϕ tðÞ (7.105)
the equation of A can be approximately determined as
1
A ½ ϕ τ 1 I (7.106)
ðÞ
τ 1
7.7.3 Transformation Method From Differential Into Difference Transfer Function
Subject to the control by digital computer, the computer can conveniently access the sampling
signals in the past moment, and also easily perform the arithmetic operations of addition,
subtraction, multiplication, and division. Thus, in the sampling adjustment, differential
equations can be replaced by difference ones to describe the objects. For an engineering system,
what we often encounter is the Laplace transform of transfer function, by which a method is
proposed to obtain the Z transform of transfer function, that is, difference transfer type is
proposed.
7.7.3.1 Method of direct Z transform
Traditionally, the differential transfer function G(S) of any system is directly transformed into
G(Z) by direct Z transform, that is:
f
GZðÞ ¼ ZG SðÞg (7.107)
In fact, obtaining of the difference equation has taken into account the retainer, without
which the system output is in a pulse form, and the coefficients of difference transfer equations
are somewhat different; therefore, the relationship between the continuous differential and
difference transfer functions is:
Differential transfer function Sampling switch Retainer ¼ Difference transfer function
