### 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{↓}{\underbrace{{\delta }_{2}^{\left(1\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{2}^{\left(1\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{{\delta }_{3}^{\left(1\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{3}^{\left(1\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{{\delta }_{4}^{\left(1\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{4}^{\left(1\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{{\delta }_{1}^{\left(5\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{1}^{\left(5\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{{\delta }_{2}^{\left(5\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{2}^{\left(5\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{{\delta }_{3}^{\left(5\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{3}^{\left(5\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{{\delta }_{4}^{\left(5\right)}}}\hfill & \hfill \underset{↓}{\underbrace{\left|{\stackrel{~}{V}}_{4}^{\left(5\right)}\right|}}\hfill & \hfill \underset{↓}{\underbrace{\alpha }}\hfill & \hfill \underset{↓}{\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}$

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