Page 262 - Mathematical Models and Algorithms for Power System Optimization
P. 262
254 Chapter 7
two solution methods: one is to first transform the discrete model into an equivalent
continuous one, which is solved to derive the required discrete model; the other is to
determine the desired discrete model by directly using the mutually transformation.
Furthermore, this chapter has brought us to the computer solution of discrete transfer
function when the model is of higher orders.
For this purpose, three models for discrete-continuous and discrete-discrete model
transformations are proposed in this section, including the eigenvalue method, the logarithmic
matrix expansion method and the successive approximation method. The condition of
equivalence between models is theoretically proved and the mutual conversion of linear models
is solved practically. Three methods have different scopes of application. The eigenvalue
method can obtain the accurate results, independent of the time interval, but not applicable for
the higher order equations. The logarithmic matrix method is affected by the time interval. The
successive approximation method, whose calculation results are more accurate for the
examples with a smaller sampling time, surely depends on the accuracy required for the
practical problems. In addition, this section also proposes a solution method to transform a
differential transfer function into a difference transfer function, which is suitable for various
complicated situations even for higher-order equations, and only requires inverse, and is
superior to the conventional method. These theoretical innovations and improvements to the
existing linear model transformations have shown that they are applicable to engineering
practice needs by the practical results in section 7.8.
7.7.1 Transformation of Difference Equation into Differential Equation
Formulation of the known difference equation, a discrete linear time-invariant model is of the
form:
ð
Xk +1Þ ¼ ϕτðÞXkðÞ
Formulate the corresponding equivalent steady continuous equation:
_
X ¼ AX
where A is an unknown constant coefficient matrix. The problem is to determine the constant
matrix A by the transition matrix ϕ(τ). Two methods, that is, eigenvalue method and
logarithmic matrix method are proposed.
7.7.1.1 Eigenvalue method
Consider the constant linear discrete model:
Xk +1Þ ¼ ϕτðÞXkðÞ (7.68)
ð
where ϕ(τ) is known, and τ is the sampling time interval.