Page 264 - Mathematical Models and Algorithms for Power System Optimization
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256 Chapter 7

            Eq. (7.69) has a steady-state solution because λ i and φ i corresponds one by one, that is, λ i (i ¼1,
            2, …, n) are different from each other, and their real parts are located at the left half plane.
            Substituting Eq. (7.77) into Eq. (7.75) results in:

                                                1     2       1    K
                                  ϕτðÞ ¼ I + Aτ +  ð AτÞ + ⋯ +  ð AτÞ + ⋯                (7.78)
                                                2!           K!
                                                                                             □
            This equation shows that the continuous model Eq. (7.69) determined by matrix A is equivalent
            to the previous discrete model Eq. (7.68), so the Theorem is proven. The above discussion
            is the case where the eigenvectors of ϕ(τ) have no multiple eigenvectors.

            As for the case of multi-eigenvalues of ϕ(τ), the following two conditions shall be considered
            respectively.

            (1) ϕ(τ) has a complete set of eigenvectors.
            Theorem 2 For the necessary and sufficient condition of the n   n matrix φ to be similar to a
            diagonal matrix is that φ must have a complete set of eigenvectors. Thus, A can be solved
            using the foregoing method.
            (2) ϕ(τ) has no complete set of eigenvectors.
                 In this case, ϕ(τ) cannot be transformed into a diagonal form but a jordan form. Then
                 lnϕ(τ) can be solved directly following the definition of matrix function, that is:

                                   2                                             3
                                                    1               1
                                              1         2 ðÞ ⋯            n 1
                                     ln λ i    ðÞ     ln λ i            ln   ðÞ
                                                                              λ i
                                       ðÞ ln λ i
                                                    2!            n 1Þ!
                                   6                             ð               7
                                   6                                             7
                    1           1 6    0     ln λ i  ln λ i  ⋯          ⋮        7
                                                       1
                                               ðÞ
                                                        ðÞ
                                                                                  T
                A ¼ T lnJT   1  ¼ T 6                                            7  1    (7.79)
                    τ           τ  6   ⋮      ⋮              ⋱          ⋮        7
                                       ⋮      ⋮              ⋯          1
                                   6                                             7
                                                                         ðÞ
                                   4                                  ln λ i     5
                                       0      0              ⋯        ln λ i
                                                                        ðÞ
            If eigenvalues have complex roots, the conjugate complex roots shall be treated as a single root,
            and multiple conjugate complex roots are taken as multiple roots. All arguments are taken
            according to the principal value region to avoid the multivalued solution of logarithmic
            function lnϕ(τ), such as ln1 ¼ 0 instead of ln1 ¼ 0+2πi.
            The above problem can also be solved by the minimum polynomial method, since it not only
            needs to do the above main work, but also needs to solve a group of linear equations, the details
            will not be elaborated here.
            7.7.1.2 Logarithmic matrix expansion method
            To find a numerical solution of ln ϕ(τ), avoiding the multivalued solution of eigenvectors
            and logarithms, thereby use the derivation of the logarithm matrix method as follows.
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