Using the results of Application Examples 7.15 and 7.16, evaluate the Jacobian matrix for the system of Fig. E7.14.1.

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

[latex] J^0=\left|\begin{array} {lllllllllllllllll|l}& \underbrace{\delta_2^{(1)}}_1 &\underbrace{\left| ilde{V}_2^{(1)} ight|}_I& \underbrace{\delta_3^{(1)}}_1& \underbrace{\left| ilde{V}_3^{(1)} ight|}_I& \underbrace{\delta_4^{(1)}}_1& \underbrace{\left| ilde{V}_4^{(1)} ight|}_1 &\underbrace{\delta_1^{(5)}}_1& \underbrace{\left| ilde{V}_1^{(5)} ight|}_I& \underbrace{\delta_2^{(5)}}_1 &\underbrace{\left| ilde{V}_2^{(5)} ight|}_1& \underbrace{\delta_3^{(5)}}_1& \underbrace{\left| ilde{V}_3^{(5)} ight|}_1& \underbrace{\delta_4^{(5)}}_1& \underbrace{\left| ilde{V}_4^{(5)} ight|}_I& \underbrace{\alpha}_1& \underbrace{\beta}_1 \ & J_{1,1}^0 & J_{1,2}^0 & J_{1,3}^0 & J_{1,4}^0 & J_{1,5}^0 & J_{1,6}^0 & J_{1,7}^0 & J_{1,8}^0 & J_{1,9}^0 & J_{1,10}^0 & J_{1,11}^0 & J_{1,12}^0 & J_{1,13}^0 & J_{1,14}^0 & J_{1,15}^0 & J_{1,16}^0 & \leftarrow \Delta P_2^{(1)} \ &J_{2,1}^0 & J_{2,2}^0 & J_{2,3}^0 & J_{2,4}^0 & J_{2,5}^0 & J_{2,6}^0 & J_{2,7}^0 & J_{2,8}^0 & J_{2,9}^0 & J_{2,10}^0 & J_{2,11}^0 & J_{2,12}^0 & J_{2,13}^0 & J_{2,14}^0 & J_{2,15}^0 & J_{2,16}^0 & \leftarrow \Delta Q_2^{(1)} \ &J_{3,1}^0 & J_{3,2}^0 & J_{3,3}^0 & J_{3,4}^0 & J_{3,5}^0 & J_{3,6}^0 & J_{3,7}^0 & J_{3,8}^0 & J_{3,9}^0 & J_{3,10}^0 & J_{3,11}^0 & J_{3,12}^0 & J_{3,13}^0 & J_{3,14}^0 & J_{3,15}^0 & J_{3,16}^0 & \leftarrow \Delta P_3^{(1)} \ & J_{4,1}^0 & J_{4,2}^0 & J_{4,3}^0 & J_{4,4}^0 & J_{4,5}^0 & J_{4,6}^0 & J_{4,7}^0 & J_{4,8}^0 & J_{4,9}^0 & J_{4,10}^0 & J_{4,11}^0 & J_{4,12}^0 & J_{4,13}^0 & J_{4,14}^0 & J_{4,15}^0 & J_{4,16}^0 & \leftarrow \Delta Q_3^{(1)} \ & J_{5,1}^0 & J_{5,2}^0 & J_{5,3}^0 & J_{5,4}^0 & J_{5,5}^0 & J_{5,6}^0 & J_{5,7}^0 & J_{5,8}^0 & J_{5,9}^0 & J_{5,10}^0 & J_{5,11}^0 & J_{5,12}^0 & J_{5,13}^0 & J_{5,14}^0 & J_{5,15}^0 & J_{5,16}^0 & \leftarrow \Delta P_4^t \ & J_{6,1}^0 & J_{6,2}^0 & J_{6,3}^0 & J_{6,4}^0 & J_{6,5}^0 & J_{6,6}^0 & J_{6,7}^0 & J_{6,8}^0 & J_{6,9}^0 & J_{6,10}^0 & J_{6,11}^0 & J_{6,12}^0 & J_{6,13}^0 & J_{6,14}^0 & J_{6,15}^0 & J_{6,16}^0 & \leftarrow \Delta Q_4^t \& J_{7,1}^0 & J_{7,2}^0 & J_{7,3}^0 & J_{7,4}^0 & J_{7,5}^0 & J_{, 6}^0 & J_{7,7}^0 & J_{7,8}^0 & J_{7,9}^0 & J_{7,10}^0 & J_{7,11}^0 & J_{7,12}^0 & J_{7,13}^0 & J_{7,14}^0 & J_{7,15}^0 & J_{7,16}^0 & \leftarrow \Delta I_{r, 1}^{(5)} \ &J_{8,1}^0 & J_{8,2}^0 & J_{8,3}^0 & J_{8,4}^0 & J_{8,5}^0 & J_{8,6}^0 & J_{8,7}^0 & J_{8,8}^0 & J_{8,9}^0 & J_{8,10}^0 & J_{8,11}^0 & J_{8,12}^0 & J_{8,13}^0 & J_{8,14}^0 & J_{8,15}^0 & J_{8,16}^0 & \leftarrow \Delta I_{i, 1}^{(5)} \& J_{9,1}^0 & J_{9,2}^0 & J_{9,3}^0 & J_{9,4}^0 & J_{9,5}^0 & J_{9,6}^0 & J_{9,7}^0 & J_{9,8}^0 & J_{9,9}^0 & J_{9,10}^0 & J_{9,11}^0 & J_{9,12}^0 & J_{9,13}^0 & J_{9,14}^0 & J_{9,15}^0 & J_{9,16}^0 & \leftarrow \Delta I_{r, 2}^{(5)} \& J_{10,1}^0 & J_{10,2}^0 & J_{10,3}^0 & J_{10,4}^0 & J_{10,5}^0 & J_{10,6}^0 & J_{10,7}^0 & J_{10,8}^0 & J_{10,9}^0 & J_{10,10}^0 & J_{10,11}^0 & J_{10,12}^0 & J_{10,13}^0 & J_{10,14}^0 & J_{10,15}^0 & J_{10,16}^0 & \leftarrow \Delta I_{i, 2}^{(5)}\& J_{11,1}^0 & J_{11,2}^0 & J_{11,3}^0 & J_{11,4}^0 & J_{11,5}^0 & J_{11,6}^0 & J_{11,7}^0 & J_{9,8}^0 & J_{11,9}^0 & J_{11,10}^0 & J_{11,11}^0 & J_{11,12}^0 & J_{11,13}^0 & J_{11,14}^0 & J_{11,15}^0 & J_{11,16}^0 & \leftarrow \Delta I_{r, 3}^{(5)}\ & J_{12,1}^0 & J_{12,2}^0 & J_{12,3}^0 & J_{12,4}^0 & J_{12,5}^0 & J_{12,6}^0 & J_{12,7}^0 & J_{12,8}^0 & J_{12,9}^0 & J_{12,10}^0 & J_{12,11}^0 & J_{12,12}^0 & J_{12,13}^0 & J_{12,14}^0 & J_{12,15}^0 & J_{12,16}^0 & \leftarrow \Delta I_{i, 3}^{(5)} \& J_{13,1}^0 & J_{13,2}^0 & J_{13,3}^0 & J_{13,4}^0 & J_{13,5}^0 & J_{13,6}^0 & J_{13,7}^0 & J_{13,8}^0 & J_{13,9}^0 & J_{13,10}^0 & J_{13,11}^0 & J_{13,12}^0 & J_{13,13}^0 & J_{13,14}^0 & J_{13,15}^0 & J_{13,16}^0& \leftarrow \Delta I_{r, 4}^{(5)}\ & J_{14,1}^0 & J_{14,2}^0 & J_{14,3}^0 & J_{14,4}^0 & J_{14,5}^0 & J_{14,6}^0 & J_{14,7}^0 & J_{14,8}^0 & J_{14,9}^0 & J_{14,10}^0 & J_{14,11}^0 & J_{14,12}^0 & J_{14,13}^0 & J_{14,14}^0 & J_{14,15}^0 & J_{14,16}^0 & \leftarrow \Delta I_{i, 4}^{(5)} \& J_{15,1}^0 & J_{15,2}^0 & J_{15,3}^0 & J_{15,4}^0 & J_{15,5}^0 & J_{15,6}^0 & J_{15,7}^0 & J_{15,8}^0 & J_{15,9}^0 & J_{15,10}^0 & J_{15,11}^0 & J_{15,12}^0 & J_{15,13}^0 & J_{15,14}^0 & J_{15,15}^0 & J_{15,16}^0 & \leftarrow \Delta I_{r, 4}^{(1)} \& J_{16,1}^0 & J_{16,2}^0 & J_{16,3}^0 & J_{16,4}^0 & J_{16,5}^0 & J_{16,6}^0 & J_{16,7}^0 & J_{16,8}^0 & J_{16,9}^0 & J_{16,10}^0 & J_{16,11}^0 & J_{16,12}^0 & J_{16,13}^0, & J_{16,14}^0 & J_{16,15}^0 &J_{16,16}^0 & \leftarrow \Delta I_{i, 4}^{(1)} \end{array} ight.[/latex]

The top row, which is not part of the Jacobian, lists the variables:

[latex]\delta_2^{(1)},\left|\widetilde{V}_2^{(1)} ight|, \delta_3^{(1)},\left|\widetilde{V}_3^{(1)} ight|, \delta_4^{(1)},\left|\widetilde{V}_4^{(1)} ight|, \delta_1^{(5)},\left|\widetilde{V}_1^{(5)} ight|, \delta_2^{(5)},\left|\widetilde{V}_2^{(5)} ight|, \delta_3^{(5)},\left|\widetilde{V}_3^{(5)} ight|, \delta_4^{(5)},\left|\widetilde{V}_4^{(5)} ight|, \alpha, \beta[/latex]

and the right-hand side column, which is not part of the Jacobian, lists the mismatch quantities that must be satisfied:

[latex]\left[\Delta P_2^{(1)}, \Delta Q_2^{(1)}, \Delta P_3^{(1)}, \Delta Q_3^{(1)}, \Delta P_4^t, \Delta Q_4^t, \Delta I_{r, 1}^{(5)}, \Delta I_{i, 1}^{(5)}, \Delta I_{r, 2}^{(5)}, \Delta I_{i, 2}^{(5)}, \Delta I_{r, 3}^{(5)}, \Delta I_{i, 3}^{(5)}, \Delta I_{r, 4}^{(5)}, \Delta I_{i, 4}^{(5)}, \Delta I_{r, 4}^{(1)}, \Delta I_{i, 4}^{(1)} ight]^t .[/latex]

These two identifications help in defining the partial derivatives of the Jacobian. For example,

[latex]\begin{gathered} J_{3,5}^0=\frac{\partial\left(\Delta P_3^{(1)} ight)}{\partial\left(\delta_4^{(1)} ight)}, J_{3,10}^0=\frac{\partial\left(\Delta P_3^{(1)} ight)}{\partial\left(\Delta V_2^{(5)} ight)} \ J_{8,5}^0=\frac{\partial\left(\Delta I_{i, 1}^{(5)} ight)}{\partial\left(\delta_4^{(1)} ight)}, J_{8,10}^0=\frac{\partial\left(\Delta I_{i, 1}^{(5)} ight)}{\partial\left(\Delta V_2^{(5)} ight)} \ J_{9,11}^0=\frac{\partial\left(\Delta I_{r, 2}^{(5)} ight)}{\partial\left(\delta_3^{(5)} ight)}, ext { and } J_{12,16}^0=\frac{\partial\left(\Delta I_{i, 3}^{(5)} ight)}{\partial(\beta)} \end{gathered}[/latex]

In the solution below submatrices are defined, which is an alternative way of defining the Jacobian entries.

Step #4 of Harmonic Load Flow (continued). Evaluate the Jacobian matrix [latex]\bar{J}[/latex] and compute [latex]\Delta \bar{U}^0[/latex] . The Jacobian matrix has the following form ([latex]\left(\overline{0}_{n imes m} ight.[/latex] denotes an n by m zero submatrix with zero entries):

[latex]\bar{J}=\left|\begin{array}{ccc} \bar{J}^{(1)} & \bar{J}^{(5)} & \overline{0}_{6,2} \ & & \overline{0}_{6,2} \ \bar{G}^{(5,1)} & \bar{Y}^{(5,5)}+\bar{G}^{(5,5)} & \bar{H}^{(5)} \ \bar{Y}^{(1,1)}+\bar{G}^{(1,1)} & \bar{G}^{(1,5)} & \bar{H}^{(1)} \end{array} ight| .[/latex]

The Jacobian submatrix for the fundamental is

[latex]\vec{J}^{(1)}=\left|\begin{array}{cccccc} 61.68 & 52.932 & -11.764 & -2.944 & 0 & 0 \ -52.94 & 61.945 & 2.944 & -11.764 & 0 & 0 \ -11.764 & -2.944 & 51.772 & 22.917 & -40.0076 & -19.98 \ 2.944 & -11.764 & -22.926 & 51.773 & 19.982 & -40.0076 \ 0 & 0 & -40.0076 & -19.982 & 80.015 & 39.961 \ 0 & 0 & 19.982 & -40.0076 & -39.963 & 80 \end{array} ight| .[/latex]

The Jacobian submatrix for the 5th harmonic is

[latex]\bar{J}^{(5)}=\left[\begin{array}{ccccccccc}&&&{\overline{0}_{4 imes 8}} \ -0.099 & -0.099 & 0 & 0 & -0.099 & -0.09 & 0.198 & 0.198 \ 0.010 & -0.099 & 0 & 0 & 0.010 & -0.99 & -0.020 & 1.98 \end{array} ight][/latex]

Also

[latex]\begin{gathered} \bar{Y}^{(1,1)}=\left|\begin{array}{ccccccc} 0 & 0 & -39.889 & -19.89 & 79.81 & 39.73 \ 0 & 0 & -19.81 & 40.06 & 39.56 & -80.14 \end{array} ight| \ \bar{Y}^{(5,5)}=\left|\begin{array}{cccccccc} 202.91 & 4.84 & -1.92 & -3.85 & 0 & 0 & -0.99 & -0.99 \ 0.48 & -2029.1 & -0.39 & 19.23 & 0 & 0 & -0.10 & 9.90 \ -1.92 & -3.846 & 2.17 & 3.97 & -0.25 & -0.13 & 0 & 0 \ -0.39 & 19.23 & 0.40 & -21.73 & -0.01 & 2.45 & 0 & 0 \ 0 & 0 & -0.25 & -0.13 & 1.24 & 1.12 & -0.99 & -0.99 \ 0 & 0 & -0.01 & 2.49 & 0.11 & -12.40 & -0.10 & 9.90 \ -0.99 & -0.99 & 0 & 0 & -0.99 & -0.99 & 1.98 & 1.98 \ -0.01 & 9.90 & 0 & 0 & -0.01 & 9.90 & 0.20 & -19.80 \end{array} ight| . \end{gathered}[/latex]

In this example, [latex]\bar{H}^{(1)} ext { and } \bar{H}^{(5)}[/latex] are zero submatrices since there are no nonlinear device variables defined for the system of Fig. E7.14.1. However, setting [latex]\bar{H}^{(1)} ext { and } \bar{H}^{(5)}[/latex] to zero introduces two columns of zeros in the Jacobian matrix, which makes it singular. In order to avoid this situation, [latex]\bar{H}^{(1)} ext { and } \bar{H}^{(5)}[/latex] are set to random matrices. Note that the solution does not depend on the values of these submatrices.
In order to compute [latex]\bar{G}^{(1,1)} ext { and } \bar{G}^{(1,5)} ext {, we need to differentiate } G_{r, 4}^{(1)} ext { and } G_{i, 4}^{(1)}[/latex] as follows:

[latex]\begin{aligned} & \frac{\partial G_{r, 4}^{(1)}}{\partial \delta_4^{(1)}}=\frac{P_4^{(1)} \cos \gamma_4^{(1)} an \left(\delta_4^{(1)}-\gamma_4^{(1)} ight)}{V_4^{(1)} \cos \left(\delta_4^{(1)}-\gamma_4^{(1)} ight)}=0.00167 \ & \frac{\partial G_{r, 4}^{(1)}}{\partial V_4^{(1)}}=-\frac{P_4^{(1)} \cos \gamma_4^{(1)}}{\left(V_4^{(1)} ight)^2 \cos \left(\delta_4^{(1)}-\gamma_4^{(1)} ight)}=-0.252 \ & \frac{\partial G_{i, 4}^{(1)}}{\partial \delta_4^{(1)}}=\frac{P_4^{(1)} \sin \gamma_4^{(1)} an \left(\delta_4^{(1)}-\gamma_4^{(1)} ight)}{V_4^{(1)} \cos \left(\delta_4^{(1)}-\gamma_4^{(1)} ight)}=-0.0000112 \ & \frac{\partial G_{i, 4}^{(1)}}{\partial V_4^{(1)}}=-\frac{P_4^{(1)} \sin \gamma_4^{(1)}}{\left(V_4^{(1)} ight)^2 \cos \left(\delta_4^{(1)}-\gamma_4^{(1)} ight)}=0.00169 \end{aligned}[/latex]

The partial derivatives of [latex]G_{r, 4}^{(1)} ext { and } G_{i, 4}^{(1)} ext { with respect to } V_4^{(5)} ext { and } \delta_4^{(5)}[/latex] are zero. Therefore, [latex]\bar{G}^{(1,5)} ext { is a } 2 imes 8[/latex] submatrix of zeros and

[latex]\bar{G}^{(1,1)}=-\left|\begin{array}{ccc} & 0.10026 & -0.2517 \ \overline{0}_{2 imes 4} &&\ & -0.0405 & 0.25415 \end{array} ight| .[/latex]

In order to compute [latex]\bar{G}^{(5,1)} ext { and } \bar{G}^{(5,5)} ext {, we need to differentiate } \bar{G}_{r, 4}^{(5)} ext { and } \bar{G}_{i, 4}^{(5)}[/latex] as follows:

[latex]\begin{aligned} & \frac{\partial G_{r, 4}^{(5)}}{\partial \delta_4^{(1)}}=-0.9\left(V_4^{(1)} ight)^3 \sin \left(3 \delta_4^{(1)} ight)=0.00896 \ & \frac{\partial G_{r, 4}^{(5)}}{\partial V_4^{(1)}}=0.9\left(V_4^{(1)} ight)^2 \cos \left(3 \delta_4^{(1)} ight)=0.893 \ & \frac{\partial G_{i, 4}^{(5)}}{\partial \delta_4^{(1)}}=0.9\left(V_4^{(1)} ight)^3 \cos \left(3 \delta_4^{(1)} ight)=0.8894 \ & \frac{\partial G_{i, 4}^{(5)}}{\partial V_4^{(1)}}=0.9\left(V_4^{(1)} ight)^2 \sin \left(3 \delta_4^{(1)} ight)=-0.009 . \end{aligned}[/latex]

Also

[latex]\begin{aligned} & \frac{\partial G_{r, 4}^{(5)}}{\partial \delta_4^{(5)}}=-0.9\left(V_4^{(5)} ight)^2 \sin \left(3 \delta_4^{(5)} ight)=0.00 \ & \frac{\partial G_{r, 4}^{(5)}}{\partial V_4^{(5)}}=0.6 V_4^{(5)} \cos \left(3 \delta_4^{(5)} ight)=0.06 \ & \frac{\partial G_{i, 4}^{(5)}}{\partial \delta_4^{(5)}}=0.9\left(V_4^{(5)} ight)^2 \cos \left(3 \delta_4^{(5)} ight)=0.009 \ & \frac{\partial G_{i, 4}^{(5)}}{\partial V_4^{(5)}}=0.6 V_4^{(5)} \sin \left(3 \delta_4^{(5)} ight)=0.00 \end{aligned}[/latex]

Therefore,

[latex]\bar{G}^{(5.1)}=\left|\begin{aligned} & \overline{0}_{6 imes 8} && \ & &0.00896 \quad 0.893 \ \overline{0}_{2 imes 6} &&&\ && 0.8894-0.009 \end{aligned} ight| [/latex]

[latex] \bar{G}^{(5.5)}=\left|\begin{aligned} &\overline{0}_{6 imes 8} &&\ && 0 \quad 0.06 \ \overline{0}_{2 imes 6}&&& \& & 0.009 \quad 0 \ \end{aligned} ight|[/latex]

The Jacobian matrix is

[latex]\left|\begin{array}{ccccccccccccccccc} -61.57 & 52.71 & -11.70 & -2.90 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ -52.79 & 61.52 & 2.89 & -11.74 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ -11.69 & -2.69 & 51.37 & 22.86 & -39.71 & -19.90 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 2.95 & -11.72 & -22.77 & 51.58 & 19.82 & -39.87 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & -39.66 & -19.94 & 79.45 & 39.59 & -0.099 & -0.099 & 0 & 0 & -0.099 & -0.099 & 0.198 & 0.198 & 0 & 0 \ 0 & 0 & 19.86 & -39.83 & -39.93 & 79.58 & 0.010 & -0.99 & 0 & 0 & 0.010 & -0.99 & -0.020 & 1.98 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 202.9 & 4.84 & -1.92 & -3.85 & 0 & 0 & -0.99 & -0.99 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0.48 & -2029 & -0.39 & 19.23 & 0 & 0 & -0.10 & 9.90 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & -1.92 & -3.846 & 2.17 & 3.97 & -0.25 & -0.13 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & -0.39 & 19.23 & 0.40 & -21.73 & -0.01 & 2.45 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.25 & -0.13 & 1.24 & 1.12 & -0.99 & -0.99 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.01 & 2.49 & 0.11 & -12.40 & -0.10 & 9.90 & 0 & 0 \ 0 & 0 & 0 & 0 & 0.009 & 0.893 & -0.99 & -0.99 & 0 & 0 & -0.99 & -0.99 & 1.98 & 2.04 & 0 & 0 \ 0 & 0 & 0 & 0 & 0.889 & -0.009 & -0.01 & 9.90 & 0 & 0 & -0.01 & 9.90 & 0.201 & -19.80 & 0 & 0 \ 0 & 0 & -39.89 & -19.89 & 79.91 & 39.48 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \ 0 & 0 & -19.81 & 40.06 & 39.81 & -80.04 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 \end{array} ight|[/latex]

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