Question 7.AE.16: Using the results of Application Example 7.15, evaluate the ......
Using the results of Application Example 7.15, evaluate the mismatch vector for the system of Fig. E7.14.1.
Step #3 of Harmonic Load Flow. Evaluation of the mismatch vector
\begin{gathered} \Delta \bar{M}^0=\left(\Delta \bar{W}, \Delta \bar{I}^{(5)}, \Delta \bar{I}^{(1)}\right)^t=\left(P_2^{(1)}+F_{r, 2}^{(1)}, Q_2^{(1)}+F_{i, 2}^{(1)},\right. \\ P_3^{(1)}+F_{r, 3}^{(1)}, Q_3^{(1)}+F_{i, 3}^{(1)}, P_4^t+F_{r, 4}^{(1)}+F_{r, 4}^{(5)}, Q_4^t+F_{i, 4}^{(1)}+F_{i, 4}^{(5)}, \\ I_{r, 1}^{(5)}, I_{i, 1}^{(5)}, I_{r, 2}^{(5)}, I_{i, 2}^{(5)}, I_{r, 3}^{(5)}, I_{i, 3}^{(5)}, I_{r, 4}^{(5)}+G_{r, 4}^{(5)}, I_{i, 4}^{(5)}+G_{i, 4}^{(5)}, \\ \left.I_{r, 4}^{(1)}+G_{r, 4}^{(1)}, I_{i, 4}^{(1)}+G_{i, 4}^{(1)}\right) . \end{gathered}To compute these mismatches use equations Eqs. 7-80 and 7-81
\Delta P_i=P_i+F_{r, i}=P_i+\sum_{j=1}^n y_{i j} V_j V_i \cos \left(-\theta_{i j}-\delta_j+\delta_i\right) (7-80)
\Delta Q_i=Q_i+F_{i, i}=Q_i+\sum_{j=1}^n y_{i j} V_j V_i \sin \left(-\theta_{i j}-\delta_j+\delta_i\right) (7-81)
and Eqs. 7-119a,b
\Delta I_{r, i}^{(h)}=I_{r, i}^{(h)}+G_{r, i}^{(h)}=\sum_{j=1}^n Y_{i j}^{(h)}\left|\widetilde{V}_j^{(h)}\right| \cos \left(\theta_{i j}^{(h)}+\delta_j^{(h)}\right)+G_{r, i}^{(h)}, (7-119a)
\Delta I_{i, i}^{(h)}=I_{i, i}^{(h)}+G_{i, i}^{(h)}=\sum_{j=1}^n y_{i j}^{(h)}\left|\widetilde{V}_j^{(h)}\right| \sin \left(\theta_{i j}^{(h)}+\delta_j^{(h)}\right)+G_{i, i}^{(h)}, (7-119b)
Hence,
\begin{aligned} & \Delta P_2^{(1)}=P_2^{(1)}+F_{r .2}^{(1)}=P_2+y_{21} V_1 V_2 \cos \left(-\theta_{21}-\delta_1+\delta_2\right)+ \\ & y_{22} V_2 V_2 \cos \left(-\theta_{22}-\delta_2+\delta_2\right)+y_{23} V_3 V_2 \cos \left(-\theta_{23}-\delta_3+\delta_2\right) \\ & +y_{24} V_4 V_2 \cos \left(-\theta_{24}-\delta_4+\delta_2\right)=-0.0170 \mathrm{pu} \\ & \Delta Q_2^{(1)}=Q_2^{(1)}+F_{i, 2}^{(1)}=Q_2+y_{21} V_1 V_2 \sin \left(-\theta_{21}-\delta_1+\delta_2\right)+ \\ & y_{22} V_2 V_2 \sin \left(-\theta_{22}-\delta_2+\delta_2\right)+y_{23} V_3 V_2 \sin \left(-\theta_{23}-\delta_3+\delta_2\right) \\ & +y_{24} V_4 V_2 \sin \left(-\theta_{24}-\delta_4+\delta_2\right)=0.083 \mathrm{pu} \\ & \Delta P_3^{(1)}=P_3^{(1)}+F_{r, 3}^{(1)}=P_3+y_{31} V_1 V_3 \cos \left(-\theta_{31}-\delta_1+\delta_3\right)+ \\ & y_{32} V_2 V_3 \cos \left(-\theta_{32}-\delta_2+\delta_3\right)+y_{33} V_3 V_3 \cos \left(-\theta_{33}-\delta_3\right. \\ & \left.+\delta_3\right)+y_{34} V_4 V_3 \cos \left(-\theta_{34}-\delta_4+\delta_3\right)=-0.00325 \mathrm{pu} \\ & \end{aligned}\\\begin{aligned} & \Delta Q_3^{(1)}=Q_3^{(1)}+F_{i, 3}^{(1)}=Q_3+y_{31} V_1 V_3 \sin \left(-\theta_{31}-\delta_1+\delta_3\right)+ \\ & y_{32} V_2 V_3 \sin \left(-\theta_{32}-\delta_2+\delta_3\right)+y_{33} V_3 V_3 \sin \left(-\theta_{33}-\delta_3+\delta_3\right) \\ & +y_{34} V_4 V_3 \sin \left(-\theta_{34}-\delta_4+\delta_3\right)=0.00192 \mathrm{pu} \\ & \Delta P_4^t=P_4^t+F_{r, 4}^{(1)}+F_{r, 4}^{(5)} \\ & =P_4^t+\sum_{j=1}^4 y_{4 j} V_j^{(1)} V_4^{(1)} \cos \left(-\theta_{4 j}-\delta_j^{(1)}+\delta_4^{(1)}\right) \\ & +\sum_{j=1}^4 y_{4 j}^{(5)} V_j^{(5)} V_4^{(5)} \cos \left(-\theta_{4 j}^{(5)}-\delta_j^{(5)}+\delta_4^{(5)}\right) \\ & =P_4^t+y_{41} V_1^{(1)} V_4^{(1)} \cos \left(-\theta_{41}-\delta_1^{(1)}+\delta_4^{(1)}\right) \\ & +y_{43} V_3^{(1)} V_4^{(1)} \cos \left(-\theta_{43}-\delta_3^{(1)}+\delta_4^{(1)}\right) \\ & +y_{44} V_4^{(1)} V_4^{(1)} \cos \left(-\theta_{44}-\delta_4^{(1)}+\delta_4^{(1)}\right) \\ & +y_{41}^{(5)} V_1^{(5)} V_4^{(5)} \cos \left(-\theta_{41}^{(5)}-\delta_1^{(5)}+\delta_4^{(5)}\right) \\ & +y_{43}^{(5)} V_3^{(5)} V_4^{(5)} \cos \left(-\theta_{43}^{(5)}-\delta_3^{(5)}+\delta_4^{(5)}\right) \\ & +y_{44}^{(5)} V_4^{(5)} V_4^{(5)} \cos \left(-\theta_{44}^{(5)}-\delta_4^{(5)}+\delta_4^{(5)}\right)=0.037 \mathrm{pu} \\ & \Delta Q_4^{t}=Q_4^{t}+F_{i, 4}^{(1)}+F_{i, 4}^{(5)} \\ & =Q_4^t+\sum_{j=1}^4 y_{4 j} V_j^{(1)} V_4^{(1)} \sin \left(-\theta_{4 j}-\delta_j^{(1)}+\delta_4^{(1)}\right) \\ & +\sum_{j=1}^4 y_{4 j}^{(5)} V_j^{(5)} V_4^{(5)} \sin \left(-\theta_{4 j}^{(5)}-\delta_j^{(5)}+\delta_4^{(1)}\right) \\ & =Q_4^{t}+y_{41} V_1^{(1)} V_4^{(1)} \sin \left(-\theta_{41}-\delta_1^{(1)}+\delta_4^{(1)}\right) \\ & +y_{43} V_3^{(1)} V_4^{(1)} \sin \left(-\theta_{43}-\delta_3^{(1)}+\delta_4^{(1)}\right) \\ & +y_{44} V_4^{(1)} V_4^{(1)} \sin \left(-\theta_{44}-\delta_4^{(1)}+\delta_4^{(1)}\right) \\ & +y_{41}^{(5)} V_1^{(5)} V_4^{(5)} \sin \left(-\theta_{41}^{(5)}-\delta_1^{(5)}+\delta_4^{(5)}\right) \\ & +y_{43}^{(5)} V_3^{(5)} V_4^{(5)} \sin \left(-\theta_{43}^{(5)}-\delta_3^{(5)}+\delta_4^{(5)}\right) \\ & +y_{44}^{(5)} V_4^{(5)} V_4^{(5)} \sin \left(-\theta_{44}^{(5)}-\delta_4^{(5)}+\delta_4^{(5)}\right)=-0.188 \mathrm{pu} \\ & \end{aligned}\begin{aligned} & I_{r, 1}^{(5)}-\sum_{j=1}^4 y_{1_j}^{(5)}\left|\nabla_j^{(5)}\right| \cos \left(\theta_{1 j}^{(5)}+\delta_j^{(5)}\right) \\ & =y_{11}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \cos \left(\theta_{11}^{(5)}+\delta_1^{(5)}\right) \\ & +y_{12}^{(5)}\left|\widetilde{V}_2^{(5)}\right| \cos \left(\theta_{12}^{(5)}+\delta_2^{(5)}\right) \\ & +y_{13}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \cos \left(\theta_{13}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{14}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \cos \left(\theta_{14}^{(5)}+\delta_4^{(5)}\right)=0.201 \mathrm{pu} \\ & I_{i, 1}^{(5)}=\sum_{j=1}^4 y_{1 j}^{(5)}\left|\widetilde{V}_j^{(5)}\right| \sin \left(\theta_{1 j}^{(5)}+\delta_j^{(5)}\right) \\ & =y_{11}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \sin \left(\theta_{11}^{(5)}+\delta_1^{(5)}\right) \\ & +y_{12}^{(5)}\left|\widetilde{V}_2^{(5)}\right| \sin \left(\theta_{12}^{(5)}+\delta_2^{(5)}\right) \\ & +y_{13}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \sin \left(\theta_{13}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{14}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \sin \left(\theta_{14}^{(5)}+\delta_4^{(5)}\right)=-201.98 \mathrm{pu} \\ & I_{r, 2}^{(5)}=\sum_{j=1}^4 y_{2 j}^{(5)}\left|\widetilde{V}_j^{(5)}\right| \cos \left(\theta_{2 j}^{(5)}+\delta_j^{(5)}\right) \\ & =y_{21}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \cos \left(\theta_{21}^{(5)}+\delta_1^{(5)}\right) \\ & +y_{22}^{(5)}\left|\widetilde{V}_2^{(5)}\right| \cos \left(\theta_{22}^{(5)}+\delta_2^{(5)}\right) \\ & +y_{23}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \cos \left(\theta_{23}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{24}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \cos \left(\theta_{24}^{(5)}+\delta_4^{(5)}\right)=0.000255 \mathrm{pu} \\ & \end{aligned}.
\begin{aligned} I_{i, 2}^{(5)}= & \sum_{j=1}^4 y_{2 j}^{(5)}\left|\widetilde{V}_j^{(5)}\right| \sin \left(\theta_{2 j}^{(5)}+\delta_j^{(5)}\right) \\ = & y_{21}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \sin \left(\theta_{21}^{(5)}+\delta_1^{(5)}\right) \\ & +y_{22}^{(5)}\left|\widetilde{V}_2^{(5)}\right| \sin \left(\theta_{22}^{(5)}+\delta_2^{(5)}\right)+y_{23}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \sin \left(\theta_{23}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{24}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \sin \left(\theta_{24}^{(5)}+\delta_4^{(5)}\right)=-0.000620 \mathrm{pu} \end{aligned}.
\begin{aligned} & I_{r, 3}^{(5)}=\sum_{j=1}^4 y_{3 j}^{(5)}\left|\widetilde{V}_j^{(5)}\right| \cos \left(\theta_{3 j}^{(5)}+\delta_j^{(5)}\right) \\ & =y_{31}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \cos \left(\theta_{31}^{(5)}+\delta_1^{(5)}\right) \\ & +y_{32}^{(5)}\left|\widetilde{V}_2^{(5)}\right| \cos \left(\theta_{32}^{(5)}+\delta_2^{(5)}\right) \\ & +y_{33}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \cos \left(\theta_{33}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{34}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \cos \left(\theta_{34}^{(5)}+\delta_4^{(5)}\right)=0.000104 \mathrm{pu} \\ & I_{i_{i, 3}}^{(5)}=\sum_{j=1}^4 y_{3 j}^{(5)}\left|\widetilde{V}_j^{(5)}\right| \sin \left(\theta_{3 j}^{(5)}+\delta_j^{(5)}\right) \\ & =y_{31}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \sin \left(\theta_{31}^{(5)}+\delta_1^{(5)}\right) \\ & +y_{32}^{(\mathbf{s})}\left|\widetilde{V}_2^{(5)}\right| \sin \left(\theta_{32}^{(\mathbf{5})}+\delta_2^{(5)}\right) \\ & +y_{33}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \sin \left(\theta_{33}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{34}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \sin \left(\theta_{34}^{(5)}+\delta_4^{(5)}\right)=0.0000082 \mathrm{pu} \\ & I_{r, 4}^{(5)}+G_{r, 4}^{(5)}=\sum_{j=1}^4 y_{4 j}^{(5)}\left|\widetilde{V}_j^{(5)}\right| \cos \left(\theta_{4 j}^{(5)}+\delta_j^{(5)}\right)+G_{r, 4}^{(5)} \\ & =y_{41}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \cos \left(\theta_{41}^{(5)}+\delta_1^{(5)}\right)+y_{43}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \cos \left(\theta_{43}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{44}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \cos \left(\theta_{44}^{(5)}+\delta_4^{(5)}\right)+G_{r, 4}^{(5)}=0.2996 \mathrm{pu} \\ & I_{i, 4}^{(5)}+G_{i, 4}^{(5)}=\sum_{j=1}^4 y^{(5)}\left|\widetilde{V}_j^{(5)}\right| \sin \left(\theta_{4 j}^{(5)}+\delta_j^{(5)}\right)+G_{i, 4}^{(5)} \\ & =y_{41}^{(5)}\left|\widetilde{V}_1^{(5)}\right| \sin \left(\theta_{41}^{(5)}+\delta_1^{(5)}\right)+y_{43}^{(5)}\left|\widetilde{V}_3^{(5)}\right| \sin \left(\theta_{43}^{(5)}+\delta_3^{(5)}\right) \\ & +y_{44}^{(5)}\left|\widetilde{V}_4^{(5)}\right| \sin \left(\theta_{44}^{(5)}+\delta_4^{(5)}\right)+G_{i, 4}^{(5)}=-0.00325 \mathrm{pu} \\ & \end{aligned}.
\begin{aligned} I_{r, 4}^{(1)}+G_{r, 4}^{(1)}= & \sum_{j=1}^4 y_{4 j}^{(1)}\left|\widetilde{V}_j^{(1)}\right| \cos \left(\theta_{4 j}^{(1)}+\delta_j^{(1)}\right)+G_{r, 4}^{(1)} \\ = & y_{41}\left|\widetilde{V}_1^{(1)}\right| \cos \left(\theta_{41}^{(1)}+\delta_1^{(1)}\right)+y_{43}\left|\widetilde{V}_3^{(1)}\right| \cos \left(\theta_{43}^{(1)}+\delta_3^{(1)}\right) \\ & +y_{44}\left|\widetilde{V}_4^{(1)}\right| \cos \left(\theta_{44}^{(1)}+\delta_4^{(1)}\right)+G_{r, 4}^{(1)}=0.0027 \mathrm{pu} \\ I_{i, 4}^{(1)}+G_{i, 4}^{(1)}= & \sum_{j=1}^4 y_{4 j}^{(1)}\left|\widetilde{V}_j^{(1)}\right| \sin \left(\theta_{4 j}^{(1)}+\delta_j^{(1)}\right)+G_{i, 4}^{(1)} \\ = & y_{41}\left|\widetilde{V}_1^{(1)}\right| \sin \left(\theta_{41}^{(1)}+\delta_1^{(1)}\right)+y_{43}\left|\widetilde{V}_3^{(1)}\right| \sin \left(\theta_{43}^{(1)}+\delta_3^{(1)}\right) \\ & +y_{44}\left|V_4^{(1)}\right| \sin \left(\theta_{44}^{(1)}+\delta_4^{(1)}\right)+G_{i, 4}^{(1)}=-0.0054 \mathrm{pu} . \end{aligned}The mismatch vector \Delta \bar{M}^0 is now
\begin{gathered} \Delta \bar{M}^0=\left(\Delta \bar{W}, \Delta \bar{I}^{(5)}, \Delta \bar{I}^{(1)}\right)^t=\left(P_2^{(1)}+F_{r, 2}^{(1)}, Q_2^{(1)}+F_{i, 2}^{(1)},\right. \\ P_3^{(1)}+F_{r, 3}^{(1)}, Q_3^{(1)}+F_{i, 3}^{(1)}, P_4^t+F_{r, 4}^{(1)}+F_{r, 4}^{(5)}, Q_4^t+F_{i, 4}^{(1)}+ \\ F_{i, 4}^{(5)}, I_{r, 1}^{(5)}, I_{i, 1}^{(5)}, I_{r, 2}^{(5)}, I_{i, 2}^{(5)}, I_{r, 3}^{(5)}, I_{i, 3}^{(5)}, I_{r, 4}^{(5)}+G_{r, 4}^{(5)}, I_{i, 4}^{(5)}+G_{i, 4}^{(5)}, \\ \left.I_{r, 4}^{(1)}+G_{r, 4}^{(1)}, I_{i, 4}^{(1)}+G_{i, 4}^{(1)}\right)^t \end{gathered}or
\begin{gathered} \Delta \bar{M}^0=(-0.0170,0.083,-0.00325,0.00192,0.037,-0.188,0.201,- \\ 201.98,0.000255,-0.000620,0.000104,0.0000082,0.2996 \\ -0.00325,0.0027,-0.0054)^t . \end{gathered}Note that for a mismatch residual of 0.0001 pu the solution has not converged yet.