For the system of Fig. E7.8.1, compute the Jacobian matrix.

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[latex]\begin{gathered} J_{11}=\frac{\partial \Delta P_2}{\partial \delta_2}=-\sum_{j=1}^4 \gamma_{2 j} V_2 V_j \sin \left(\delta_2-\delta_j- heta_{2 j} ight) \ eq 2 \ =-y_{21} V_2 V_1 \sin \left(\delta_2-\delta_1- heta_{21} ight) \ -y_{23} V_2 V_3 \sin \left(\delta_2-\delta_3- heta_{23} ight) \ -y_{24} V_2 V_4 \sin \left(\delta_2-\delta_4- heta_{24} ight) \end{gathered}[/latex].

[latex]\begin{aligned} J_{11} & =\frac{\partial \Delta P_2}{\partial \delta_2}=-70.72 \cdot 1 \cdot 1 \cdot \sin (0-0-2.356)-12.127 \cdot 1 \cdot 1 \cdot \sin (0-0-1.816)-0 \ & =61.68 \end{aligned}[/latex].

[latex]\begin{aligned} J_{12}= & \frac{\partial \Delta P_2}{\partial V_2}=\sum_{j=1}^4 y_{2 j} V_j \cos \left(\delta_2-\delta_j- heta_{2 j} ight)+2 y_{22} V_2 \cos \left(- heta_{22} ight) \ & eq 2 \ = & y_{21} V_1 \cos \left(\delta_2-\delta_1- heta_{21} ight)+y_{23} V_3 \cos \left(\delta_2-\delta_3- heta_{23} ight) \ & +y_{24} V_4 \cos \left(\delta_2-\delta_4- heta_{24} ight)+2 V_2 y_{22} \cos \left(- heta_{22} ight) \end{aligned}[/latex].

[latex]\begin{aligned} J_{12}= & \frac{\partial \Delta P_2}{\partial V_2}=70.72 \cdot 1 \cdot \cos (0-0-2.356) \ & +12.127 \cdot 1 \cdot \cos (0-0-1.816)+0+2 \cdot 1 \cdot 81.42 \cos (0.863) \ = & -49.996-2.944+0+105.872=52.932 \ J_{13}= & \frac{\partial \Delta P_2}{\partial \delta_3}=y_{23} V_1 V_3 \sin \left(\delta_2-\delta_3- heta_{23} ight) \ = & 12.127 \cdot 1 \cdot 1 \cdot \sin (0-0-1.816)=-11.764 \end{aligned}[/latex].

[latex]\begin{aligned} & J_{14}= \frac{\partial \Delta P_2}{\partial V_3}=y_{23} V_2 \cos \left(\delta_2-\delta_3- heta_{23} ight) \ &= 12.127 \cdot 1 \cdot \cos (0-0-1.816)=-2.944 \ & J_{15}= \frac{\partial \Delta P_2}{\partial \delta_4}=y_{24} V_1 V_5 \sin \left(\delta_2-\delta_5- heta_{25} ight)=0 \ & J_{16}=\frac{\partial \Delta P_2}{\partial V_4}=y_{24} V_2 \cos \left(\delta_2-\delta_4- heta_{24} ight)=0 \ & J_{21}= \frac{\partial \Delta Q_2}{\partial \delta_2}=\sum_{j=1}^4 y_{2 j} V_2 V_j \cos \left(\delta_2-\delta_j- heta_{2 j} ight) \ & \quad eq 2 \ &= y_{21} V_2 V_1 \cos \left(\delta_2-\delta_1- heta_{21} ight) \ &+y_{23} V_2 V_3 \cos \left(\delta_2-\delta_3- heta_{23} ight) \ &+y_{24} V_2 V_4 \cos \left(\delta_2-\delta_4- heta_{24} ight) \end{aligned}[/latex].

[latex]\begin{aligned} & J_{21}= \frac{\partial \Delta Q_2}{\partial \delta_2}=70.72 \cdot 1 \cdot 1 \cdot \cos (0-0-2.3560) \ &+12.127 \cdot 1 \cdot 1 \cdot \cos (0-0-1.816)+0 \ &=-49.996-2.944+0=-52.94 \ & J_{22}=\frac{\partial \Delta Q_2}{\partial V_2}= \sum_{j=1}^4 y_{2 j} V_j \sin \left(\delta_2-\delta_j- heta_{2 j} ight)+2 V_2 y_{22} \sin \left(- heta_{22} ight) \ & eq 2 \ &=y_{21} V_1 \sin \left(\delta_2-\delta_1- heta_{21} ight)+y_{23} V_3 \sin \left(\delta_2-\delta_3- heta_{23} ight) \ &+y_{24} V_4 \sin \left(\delta_2-\delta_4- heta_{24} ight)+2 V_2 y_{22} \sin \left(- heta_{22} ight) \ & J_{22}=\frac{\partial \Delta Q_2}{\partial V_2}= 70.72 \cdot 1 \cdot \sin (0-0-2.356) \ &+12.127 \cdot 1 \cdot \sin (0-0-1.816)+0+2 \cdot 1 \cdot 81.42 \sin (0.863) \ &=-50.016-11.764+0+123.725=61.945 \ & J_{23}=\frac{\partial \Delta Q_2}{\partial \delta_3}=-y_{23} V_2 V_3 \cos \left(\delta_2-\delta_3- heta_{23} ight) \ &=-12.127 \cdot 1 \cdot 1 \cdot \cos (0-0-1.816) \ &=2.944 \ & J_{24}= \frac{\partial \Delta Q_2}{\partial V_3}=y_{23} V_2 \sin \left(\delta_2-\delta_3- heta_{23} ight) \ &= 12.127 \cdot 1 \cdot \sin (0-0-1.816)=-11.764 \end{aligned}[/latex].

[latex]\begin{aligned} J_{25} & =\frac{\partial \Delta Q_2}{\partial \delta_4}=-y_{24} V_2 V_4 \cos \left(\delta_2-\delta_4- heta_{24} ight)=0 \ J_{26} & =\frac{\partial \Delta Q_2}{\partial V_4}=y_{24} V_2 \sin \left(\delta_2-\delta_4- heta_{24} ight)=0 \ J_{31} & =\frac{\partial \Delta P_3}{\partial \delta_2}=y_{32} V_3 V_2 \sin \left(\delta_3-\delta_2- heta_{32} ight) \ & =12.127 \cdot 1 \cdot 1 \cdot \sin (0-0-1.816)=-11.764 \ J_{32} & =\frac{\partial \Delta P_3}{\partial V_2}=y_{32} V_3 \cos \left(\delta_3-\delta_2- heta_{32} ight) \ & =12.127 \cdot 1 \cdot \cos (0-0-1.816)=-2.944 \ J_{33} & =\frac{\partial \Delta P_3}{\partial \delta_3}=-\sum_{j=1}^4 y_{3 j} V_3 V_j \sin \left(\delta_3-\delta_j- heta_{3 j} ight) \end{aligned}[/latex].

≠ 3

[latex]\begin{aligned} & J_{33}=\frac{\partial \Delta P_3}{\partial \delta_3}=-y_{31} V_3 V_1 \sin \left(\delta_3-\delta_1- heta_{31} ight) \ & -y_{32} V_3 V_2 \sin \left(\delta_3-\delta_2- heta_{32} ight) \ & -y_{34} V_3 V_4 \sin \left(\delta_3-\delta_4- heta_{34} ight) \ & =0-12.127 \cdot 1 \cdot 1 \cdot \sin (0-0-1.816) \ & -44.72 \cdot 1 \cdot 1 \cdot \sin (0-0-2.034)=51.772 \ & \begin{aligned} J_{34}=\frac{\partial \Delta P_3}{\partial V_3}=\sum_{\substack{j=1 \ eq 3}}^4 y_{3 j} V_j \cos \left(\delta_3-\delta_j- heta_{3 j} ight)+2 V_3 y_{33} \cos \left(- heta_{33} ight) \end{aligned} \ & =y_{31} V_1 \cos \left(\delta_3-\delta_1- heta_{31} ight)+y_{32} V_2 \cos \left(\delta_3-\delta_2- heta_{32} ight) \ & +y_{34} V_4 \cos \left(\delta_3-\delta_4- heta_{34} ight)+2 V_3 y_{33} \cos \left(- heta_{33} ight) \ & =0+12.127 \cdot 1 \cdot \cos (0-0-1.186)+44.72 \cdot 1 ext {. } \ & \cos (0-0-2.034)+2 \cdot 1 \cdot 56.62 \cos (1.154) \ & =0-2.944-19.982+45.843=22.917 \ & J_{35}=\frac{\partial \Delta P_3}{\partial \Delta_4}=y_{34} V_3 V_4 \sin \left(\delta_3-\delta_4- heta_{34} ight) \ & =44.72 \cdot 1 \cdot 1 \cdot \sin (0-0-2.034)=-40.0076 \ & \end{aligned}[/latex].

[latex]\begin{aligned} & J_{36}=\frac{\partial \Delta P_3}{\partial V_4}=y_{34} V_3 \cos \left(\delta_3-\delta_4- heta_{34} ight) \ & =44.72 \cdot 1 \cdot \cos (0-0-2.034)=-19.98 \ & J_{41}=\frac{\partial \Delta Q_3}{\partial \delta_2}=-y_{32} V_3 V_2 \cos \left(\delta_3-\delta_2- heta_{32} ight) \ & =-12.127 \cdot 1 \cdot 1 \cdot \cos (0-0-1.816)=2.944 \ & J_{42}=\frac{\partial \Delta Q_3}{\partial V_2}=y_{32} V_3 \sin \left(\delta_3-\delta_2- heta_{32} ight) \ & =12.127 \cdot 1 \cdot \sin (0-0-1.816)=-11.764 \ & J_{43}=\frac{\partial \Delta Q_3}{\partial \delta_3}=\sum_{j=1}^4 y_{3 j} V_3 V_j \cos \left(\delta_3-\delta_j- heta_{3 j} ight) \ & eq 3 \ & \end{aligned}[/latex].

[latex]\begin{aligned} & J_{43}=\frac{\partial \Delta Q_3}{\partial \delta_3}=y_{31} V_3 V_1 \cos \left(\delta_3-\delta_1- heta_{31} ight) \ & +y_{32} V_3 V_2 \cos \left(\delta_3-\delta_2- heta_{32} ight) \ & +y_{34} V_3 V_4 \cos \left(\delta_3-\delta_4- heta_{34} ight) \ & J_{43}=\frac{\partial \Delta Q_3}{\partial \delta_3}=0+12.127 \cdot 1 \cdot 1 \cdot \cos (0-0-1.816) \ & +44.72 \cdot 1 \cdot 1 \cdot \cos (0-0-2.034) \ & =-2.944-19.982=-22.926 \ & \begin{aligned} & J_{44}=\frac{\partial \Delta Q_3}{\partial V_3}= \sum_{j=1}^4 y_{3 j} V_j \sin \left(\delta_3-\delta_j- heta_{3 j} ight)+2 V_3 y_{33} \sin \left(- heta_{33} ight) \ & eq 3 \end{aligned} \ & =y_{31} V_1 \sin \left(\delta_3-\delta_1- heta_{31} ight)+y_{32} V_2 \sin \left(\delta_3-\delta_2- heta_{32} ight) \ & +y_{34} V_4 \sin \left(\delta_3-\delta_4- heta_{34} ight)+2 V_3 y_{33} \sin -\left( heta_{33} ight) \ & J_{44}=\frac{\partial \Delta Q_3}{\partial V_3}=0+12.127 \cdot 1 \cdot \sin (0-0-1.816) \ & +44.72 \cdot 1 \cdot \sin (0-0-2.034) \ & +2 \cdot 1 \cdot 56.62 \sin (1.154) \ & =-11.764-40.0076+103.546=51.773 \ & \end{aligned}[/latex].

[latex]\begin{aligned} & J_{45}= \frac{\partial \Delta Q_3}{\partial \delta_4}=-y_{34} V_3 V_4 \cos \left(\delta_3-\delta_4- heta_{34} ight) \ &=-44.72 \cdot 1 \cdot 1 \cdot \cos (0-0-2.034)=19.982 \ & J_{46}=\frac{\partial \Delta Q_3}{\partial V_4}=y_{34} V_3 \sin \left(\delta_3-\delta_4- heta_{34} ight) \ &=44.72 \cdot 1 \cdot \sin (0-0-2.034)=-40.0076 \ & J_{51}=\frac{\partial \Delta P_4}{\partial \delta_2}=y_{42} V_4 V_2 \sin \left(\delta_4-\delta_2- heta_{42} ight)=0 \ & J_{52}=\frac{\partial \Delta P_4}{\partial V_2}=y_{42} V_4 \cos \left(\delta_4-\delta_2- heta_{42} ight)=0 \ & J_{53}=\frac{\partial \Delta P_4}{\partial \delta_3}=y_{43} V_4 V_3 \sin \left(\delta_4-\delta_3- heta_{43} ight) \ &=44.72 \cdot 1 \cdot 1 \cdot \sin (0-0-2.034)=-40.0076 \ & J_{54}=\frac{\partial \Delta P_4}{\partial V_3}=y_{43} V_4 \cos \left(\delta_4-\delta_3- heta_{43} ight) \ &=44.72 \cdot 1 \cdot \cos (0-0-2.034)=-19.982 \ & J_{55}=\frac{\partial \Delta P_4}{\partial \delta_4}=-\sum_{j=1}^4 \gamma_{4 j} V_4 V_j \sin \left(\delta_4-\delta_j- heta_{4 j} ight) \ & eq 4 \end{aligned}[/latex].

[latex]\begin{gathered} =-y_{41} V_4 V_1 \sin \left(\delta_4-\delta_1- heta_{41} ight) \ \quad-y_{42} V_4 V_2 \sin \left(\delta_4-\delta_2- heta_{42} ight) \ -y_{43} V_4 V_3 \sin \left(\delta_4-\delta_3- heta_{43} ight) \ J_{55}=\frac{\partial \Delta P_4}{\partial \delta_4}=-44.72 \cdot 1 \cdot 1 \cdot \sin (0-0-2.034)-0 \ -44.72 \cdot 1 \cdot 1 \cdot \sin (0-0-2.034) \ =40.0076-0+40.0076=80.015 \ J_{56}=\frac{\partial \Delta P_4}{\partial V_4}=\sum_{j=1}^4 y_{4 j} V_j \cos \left(\delta_4-\delta_j- heta_{4 j} ight) \ eq 4 \ \quad+2 V_4 y_{44} \cos \left(- heta_{44} ight) \end{gathered}[/latex].

[latex]\begin{aligned} & J_{56}=\frac{\partial \Delta P_4}{\partial V_4}=y_{41} V_1 \cos \left(\delta_4-\delta_1- heta_{41} ight) \ & +y_{42} V_2 \cos \left(\delta_4-\delta_2- heta_{42} ight) \ & +y_{43} V_3 \cos \left(\delta_4-\delta_3- heta_{43} ight) \ & +2 V_4 y_{44} \cos \left(- heta_{44} ight) \ & J_{56}=\frac{\partial \Delta P_4}{\partial V_4}=44.72 \cdot 1 \cdot \cos (0-0-2.034)+0 \ & +44.72 \cdot 1 \cdot \cos (0-0-2.034) \ & +2 \cdot 1 \cdot 89.44 \cos (1.1076) \ & J_{56}=\frac{\partial \Delta P_4}{\partial V_4}=-19.982-19.982+79.925=39.961 \ & J_{61}=\frac{\partial \Delta Q_4}{\partial \delta_2}=-y_{42} V_4 V_2 \cos \left(\delta_4-\delta_2- heta_{42} ight)=0 \ & J_{62}=\frac{\partial \Delta Q_4}{\partial V_2}=y_{42} V_4 \sin \left(\delta_4-\delta_2- heta_{42} ight)=0 \ & J_{63}=\frac{\partial \Delta Q_4}{\partial \delta_3}=-y_{43} V_4 V_3 \cos \left(\delta_4-\delta_3- heta_{43} ight) \ & =-44.72 \cdot 1 \cdot 1 \cdot \cos (0-0-2.034)=19.982 \ & J_{64}=\frac{\partial \Delta Q_4}{\partial V_3}=y_{43} V_4 \sin \left(\delta_4-\delta_3- heta_{43} ight) \ & =44.72 \cdot 1 \cdot \sin (0-0-2.034)=-40.0076 \ & J_{65}=\frac{\partial \Delta Q_4}{\partial \delta_4}=\sum_{j=1}^4 y_{4 j} V_4 V_j \cos \left(\delta_4-\delta_j- heta_{4 j} ight) \ & eq 4 \ & \end{aligned}[/latex].

[latex]\begin{aligned} & J_{65}=\frac{\partial \Delta Q_4}{\partial \delta_4}=y_{41} V_4 V_1 \cos \left(\delta_4-\delta_1- heta_{41} ight) \ & +y_{42} V_4 V_2 \cos \left(\delta_4-\delta_2- heta_{42} ight) \ & +y_{43} V_4 V_3 \cos \left(\delta_4-\delta_3- heta_{43} ight) \ & J_{65}=\frac{\partial \Delta Q_4}{\partial \delta_4}=44.72 \cdot 1 \cdot 1 \cdot \cos (0-0-2.034)+0 \ & +44.72 \cdot 1 \cdot 1 \cdot \cos (0-0-2.034) \ & =-19.982-19.982=-39.963 \ & \end{aligned}[/latex].

[latex]\begin{aligned} & J_{66}=\frac{\partial \Delta Q_4}{\partial V_4}=\sum_{j=1}^4 y_{4 j} V_j \sin \left(\delta_4-\delta_j- heta_{4 j} ight) \ & eq 4 \ & +2 V_4 y_{44} \sin \left(- heta_{44} ight) \ & J_{66}=\frac{\partial \Delta Q_4}{\partial V_4}=y_{41} V_1 \sin \left(\delta_4-\delta_1- heta_{41} ight) \ & +y_{42} V_2 \sin \left(\delta_4-\delta_2- heta_{42} ight)+y_{43} V_3 \sin \left(\delta_4-\delta_3- heta_{43} ight) \ & +V_4 y_{44} \sin \left(- heta_{44} ight) \ & J_{66}=\frac{\partial \Delta Q_4}{\partial V_4}=44.72 \cdot 1 \cdot \sin (0-0-2.034) \ & +0+44.72 \cdot 1 \cdot \sin (0-0-2.034) \ & +2 \cdot 1 \cdot 89.44 \sin (1.1076) \ & J_{66}=\frac{\partial \Delta Q_4}{\partial V_4}=-40.0076-40.0076+160.031=80 . \ & \end{aligned}[/latex].

The Jacobian matrix is

[latex]\bar{J}=\left[\begin{array}{llllll} J_{11} & J_{12} & J_{13} & J_{14} & J_{15} & J_{16} \ J_{21} & J_{22} & J_{23} & J_{24} & J_{25} & J_{26} \ J_{31} & J_{32} & J_{33} & J_{34} & J_{35} & J_{36} \ J_{41} & J_{42} & J_{43} & J_{44} & J_{45} & J_{46} \ J_{51} & J_{52} & J_{53} & J_{54} & J_{55} & J_{56} \ J_{61} & J_{62} & J_{63} & J_{64} & J_{65} & J_{66} \end{array} ight][/latex].

[latex]\bar{J}^0=\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]

Question 7.AE.10: For the system of Fig. E7.8.1, compute the Jacobian matrix....

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