Page 262 - Mathematical Models and Algorithms for Power System Optimization
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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.
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