Page 263 - Mathematical Models and Algorithms for Power System Optimization
P. 263

Optimization Method for Load Frequency Feed Forward Control 255

               Theorem 1 If matrix ϕ(τ) has n different eigenvalues ϕ i (i¼1, 2, …, n) with modulo less
               than 1 and n eigenvectors e i (i¼1, 2, …, n) corresponding to ϕ i and mutually independent, then
               a constant matrix A can be found, which satisfies the equivalent condition to make
               the continuous model Eq. (7.69) equal to the discrete model Eq. (7.68):
                                                    _
                                                   XtðÞ ¼ AXtðÞ                              (7.69)

               Proof
               Let
                                                    ð
                                                 T ¼ e 1 , e 2 , ⋯, e n Þ                    (7.70)
               because (e 1 , e 2 , …, e n ) are independent of each other, so the inverse of matrix T exists. From
               linear algebra, we have:
                                                     φ 1
                                                   2              3
                                                         φ
                                                   6              7
                                          ϕ ϕ ¼ T  6      2       7  1                       (7.71)
                                                                   T
                                            ðÞ
                                              t
                                                   4        ⋱     5
                                                               φ
                                                                 n
               Let
                                                 φ ¼ e λ i τ  ¼ x i +jy i                    (7.72)
                                                  i
               where
                                    1                                         1 i
                                                                               y
                                λ i ¼  lnφ ,lnφ ¼ ln φ jj + jargφ , argφ ¼ tan               (7.73)
                                                                       i
                                                        i
                                                i
                                          i
                                                                i
                                    τ                                          x i
               Hence
                                                  2  λ 1 τ         3
                                                   e
                                                  6     e λ 2 τ    7
                                                                    T
                                          φ τðÞ ¼ T 6              7  1                      (7.74)
                                                  4         ⋱      5
                                                               e λ n τ
               Expand the above expression to get

                                                   2  2 2      1   K K        1
                                  ϕ τðÞ ¼ T I + Λτ +  Λ τ + ⋯ +  Λ τ + ⋯ T                   (7.75)
                                                   2!          K!
               where
                                                   2            3
                                                     λ 1
                                                        λ 2
                                                   6            7
                                               Λ ¼  6           7                            (7.76)
                                                   4       ⋱    5
                                                              λ n
               Let

                                                   A ¼ TΛT   1                               (7.77)
   258   259   260   261   262   263   264   265   266   267   268