Power Quality in Power Systems and Electrical Machines, 2nd Edition

Solution to Application Example 7.17

Step #4 of Harmonic Load Flow. The matrix equation shown below makes it easy to identify the entries of the Jacobian matrix:

${\overline{J}}^0=\begin{array}{c}\hfill \left|\begin{array}{cccccccccccccccc}\hfill \underset{\downarrow }{\underbrace{{\delta }_2^{\left(1\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_2^{\left(1\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{{\delta }_3^{\left(1\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_3^{\left(1\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{{\delta }_4^{\left(1\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_4^{\left(1\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{{\delta }_1^{\left(5\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_1^{\left(5\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{{\delta }_2^{\left(5\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_2^{\left(5\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{{\delta }_3^{\left(5\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_3^{\left(5\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{{\delta }_4^{\left(5\right)}}}\hfill & \hfill \underset{\downarrow }{\underbrace{\left|{\tilde{V}}_4^{\left(5\right)}\right|}}\hfill & \hfill \underset{\downarrow }{\underbrace{\alpha }}\hfill & \hfill \underset{\downarrow }{\underbrace{\beta }}\hfill \\ \hfill J_{1,1}^0\hfill & \hfill J_{1,2}^0\hfill & \hfill J_{1,3}^0\hfill & \hfill J_{1,4}^0\hfill & \hfill J_{1,5}^0\hfill & \hfill J_{1,6}^0\hfill & \hfill J_{1,7}^0\hfill & \hfill J_{1,8}^0\hfill & \hfill J_{1,9}^0\hfill & \hfill J_{1,10}^0\hfill & \hfill J_{1,11}^0\hfill & \hfill J_{1,12}^0\hfill & \hfill J_{1,13}^0\hfill & \hfill J_{1,14}^0\hfill & \hfill J_{1,15}^0\hfill & \hfill J_{1,16}^0\hfill \\ \hfill J_{2,1}^0\hfill & \hfill J_{2,2}^0\hfill & \hfill J_{2,3}^0\hfill & \hfill J_{2,4}^0\hfill & \hfill J_{2,5}^0\hfill & \hfill J_{2,6}^0\hfill & \hfill J_{2,7}^0\hfill & \hfill J_{2,8}^0\hfill & \hfill J_{2,9}^0\hfill & \hfill J_{2,10}^0\hfill & \hfill J_{2,11}^0\hfill & \hfill J_{2,12}^0\hfill & \hfill J_{2,13}^0\hfill & \hfill J_{2,14}^0\hfill & \hfill J_{2,15}^0\hfill & \hfill J_{2,16}^0\hfill \\ \hfill J_{3,1}^0\hfill & \hfill J_{3,2}^0\hfill & \hfill J_{3,3}^0\hfill & \hfill J_{3,4}^0\hfill & \hfill J_{3,5}^0\hfill & \hfill J_{3,6}^0\hfill & \hfill J_{3,7}^0\hfill & \hfill J_{3,8}^0\hfill & \hfill J_{3,9}^0\hfill & \hfill J_{3,10}^0\hfill & \hfill J_{3,11}^0\hfill & \hfill J_{3,12}^0\hfill & \hfill J_{3,13}^0\hfill & \hfill J_{3,14}^0\hfill & \hfill J_{3,15}^0\hfill & \hfill J_{3,16}^0\hfill \\ \hfill J_{4,1}^0\hfill & \hfill J_{4,2}^0\hfill & \hfill J_{4,3}^0\hfill & \hfill J_{4,4}^0\hfill & \hfill J_{4,5}^0\hfill & \hfill J_{4,6}^0\hfill & \hfill J_{4,7}^0\hfill & \hfill J_{4,8}^0\hfill & \hfill J_{4,9}^0\hfill & \hfill J_{4,10}^0\hfill & \hfill J_{4,11}^0\hfill & \hfill J_{4,12}^0\hfill & \hfill J_{4,13}^0\hfill & \hfill J_{4,14}^0\hfill & \hfill J_{4,15}^0\hfill & \hfill J_{4,16}^0\hfill \\ \hfill J_{5,1}^0\hfill & \hfill J_{5,2}^0\hfill & \hfill J_{5,3}^0\hfill & \hfill J_{5,4}^0\hfill & \hfill J_{5,5}^0\hfill & \hfill J_{5,6}^0\hfill & \hfill J_{5,7}^0\hfill & \hfill J_{5,8}^0\hfill & \hfill J_{5,9}^0\hfill & \hfill J_{5,10}^0\hfill & \hfill J_{5,11}^0\hfill & \hfill J_{5,12}^0\hfill & \hfill J_{5,13}^0\hfill & \hfill J_{5,14}^0\hfill & \hfill J_{5,15}^0\hfill & \hfill J_{5,16}^0\hfill \\ \hfill J_{6,1}^0\hfill & \hfill J_{6,2}^0\hfill & \hfill J_{6,3}^0\hfill & \hfill J_{6,4}^0\hfill & \hfill J_{6,5}^0\hfill & \hfill J_{6,6}^0\hfill & \hfill J_{6,7}^0\hfill & \hfill J_{6,8}^0\hfill & \hfill J_{6,9}^0\hfill & \hfill J_{6,10}^0\hfill & \hfill J_{6,11}^0\hfill & \hfill J_{6,12}^0\hfill \end{array}\right|\hfill \end{array}$