Solution to Application Example 7.11

Calculation of the inverse of ${\overline{J}}^0$, that is. ${\left({\overline{J}}^0\right)}^{-1}$.

${\left({\overline{J}}^0\right)}^{-1}=\frac{\mathrm{A}\mathrm{d}\mathrm{j}\left({\overline{J}}^0\right)}{det\left({\overline{J}}^0\right)}\text{,}$

where Adj $\left({\overline{J}}^0\right)$ is the adjoint of matrix ${\overline{J}}^0$ and det$\left({\overline{J}}^0\right)$ = |${\overline{J}}^0$| is the determinant of ${\overline{J}}^0$.

The adjoint of ${\overline{J}}^0$ is obtained by first taking the transpose of ${\overline{J}}^0$, that is, ${\left({\overline{J}}^0\right)}^t$, and then replacing each element in the transpose with its cofactor:

${\left[{\left({\overline{J}}^0\right)}^t\right]}_{ij}={\left[{\overline{J}}^0\right]}_{ji}\text{.}$

The transpose of ${\overline{J}}^0$ is

${\left({\overline{J}}^0\right)}^t=\left[\begin{array}{cccccc}\hfill J_{11}\hfill & \hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{12}\hfill & \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right]$

${\left({\overline{J}}^0\right)}^t=\left[\begin{array}{cccccc}\hfill 61.68\hfill & \hfill -52.94\hfill & \hfill -11.764\hfill & \hfill 2.944\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 52.932\hfill & \hfill 61.945\hfill & \hfill -2.944\hfill & \hfill -11.764\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -11.764\hfill & \hfill 2.944\hfill & \hfill 51.772\hfill & \hfill -22.926\hfill & \hfill -40.0076\hfill & \hfill -19.982\hfill \\ \hfill -2.944\hfill & \hfill -11.764\hfill & \hfill 22.917\hfill & \hfill 51.773\hfill & \hfill -19.982\hfill & \hfill -40.0076\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -40.0076\hfill & \hfill 19.982\hfill & \hfill 80.015\hfill & \hfill -39.963\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -19.98\hfill & \hfill -40.0076\hfill & \hfill 39.961\hfill & \hfill 80\hfill \end{array}\right]$

$Adj\left({\overline{J}}^0\right)=\left[\begin{array}{cccccc}\hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{11}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{12}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{13}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{14}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{15}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{16}\hfill \\ \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{21}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{22}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{23}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{24}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{25}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{26}\hfill \\ \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{31}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{32}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{33}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{34}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{35}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{36}\hfill \\ \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{41}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{42}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{43}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{44}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{45}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{46}\hfill \\ \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{51}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{52}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{53}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{54}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{55}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{56}\hfill \\ \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{61}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{62}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{63}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{64}\hfill & \hfill -\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{65}\hfill & \hfill +\mathrm{A}\mathrm{d}\mathrm{j}{\left({\overline{J}}^0\right)}_{66}\hfill \end{array}\right]$

where

$Adj{\left({\overline{J}}^0\right)}_{11}=\left|\begin{array}{ccccc}\hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{12}=\left|\begin{array}{ccccc}\hfill J_{12}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{13}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{13}=\left|\begin{array}{ccccc}\hfill J_{12}\hfill & \hfill J_{22}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{14}=\left|\begin{array}{ccccc}\hfill J_{12}\hfill & \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{15}=\left|\begin{array}{ccccc}\hfill J_{12}\hfill & \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{62}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{16}=\left|\begin{array}{ccccc}\hfill J_{12}\hfill & \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{21}=\left|\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{22}=\left|\begin{array}{ccccc}\hfill J_{11}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{13}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{23}=\left|\begin{array}{ccccc}\hfill J_{11}\hfill & \hfill J_{21}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{24}=\left|\begin{array}{ccccc}\hfill J_{11}\hfill & \hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{25}=\left|\begin{array}{ccccc}\hfill J_{11}\hfill & \hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{61}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{63}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{64}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{65}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{26}=\left|\begin{array}{ccccc}\hfill J_{11}\hfill & \hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill \\ \hfill J_{13}\hfill & \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill \\ \hfill J_{14}\hfill & \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill \\ \hfill J_{15}\hfill & \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill \\ \hfill J_{16}\hfill & \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{31}=\left|\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{41}=\left|\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}$

$Adj{\left({\overline{J}}^0\right)}_{51}=\left|\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right|\text{,}Adj{\left({\overline{J}}^0\right)}_{61}=\left|\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{31}\hfill & \hfill J_{41}\hfill & \hfill J_{51}\hfill & \hfill J_{61}\hfill \\ \hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{56}\hfill & \hfill J_{65}\hfill \end{array}\right|\text{.}$

As an example, the calculation of the first entry of the adjonint Adj${\left({\overline{J}}^0\right)}_{11}$ will be outlined:

$\begin{array}{c}Adj{\left({\overline{J}}^0\right)}_{11}=\left[\begin{array}{ccccc}\hfill J_{22}\hfill & \hfill J_{32}\hfill & \hfill J_{42}\hfill & \hfill J_{52}\hfill & \hfill J_{62}\hfill \\ \hfill J_{23}\hfill & \hfill J_{33}\hfill & \hfill J_{43}\hfill & \hfill J_{53}\hfill & \hfill J_{63}\hfill \\ \hfill J_{24}\hfill & \hfill J_{34}\hfill & \hfill J_{44}\hfill & \hfill J_{54}\hfill & \hfill J_{64}\hfill \\ \hfill J_{25}\hfill & \hfill J_{35}\hfill & \hfill J_{45}\hfill & \hfill J_{55}\hfill & \hfill J_{65}\hfill \\ \hfill J_{26}\hfill & \hfill J_{36}\hfill & \hfill J_{46}\hfill & \hfill J_{56}\hfill & \hfill J_{66}\hfill \end{array}\right]\\ =\left[\begin{array}{ccccc}\hfill 61.945\hfill & \hfill -2.944\hfill & \hfill -11.764\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2.944\hfill & \hfill 51.772\hfill & \hfill -22.926\hfill & \hfill -40.0076\hfill & \hfill 19.982\hfill \\ \hfill -11.764\hfill & \hfill 22.917\hfill & \hfill 51.773\hfill & \hfill -19.982\hfill & \hfill -40.0076\hfill \\ \hfill 0\hfill & \hfill -40.0076\hfill & \hfill 19.982\hfill & \hfill 80.015\hfill & \hfill -39.963\hfill \\ \hfill 0\hfill & \hfill -19.98\hfill & \hfill -40.0076\hfill & \hfill 39.961\hfill & \hfill 80.0\hfill \end{array}\right]\end{array}$

$\begin{array}{cc}Adj{\left({\overline{J}}^0\right)}_{11}& =61.945\left[\begin{array}{cccc}\hfill 51.772\hfill & \hfill -22.926\hfill & \hfill -40.0076\hfill & \hfill 19.982\hfill \\ \hfill 22.917\hfill & \hfill 51.772\hfill & \hfill -19.982\hfill & \hfill -40.0076\hfill \\ \hfill -40.0076\hfill & \hfill 19.982\hfill & \hfill 80.015\hfill & \hfill -39.963\hfill \\ \hfill -19.98\hfill & \hfill -40.0076\hfill & \hfill 39.961\hfill & \hfill 80\hfill \end{array}\right]\hfill \\ & +2.944\left[\begin{array}{cccc}\hfill 2.944\hfill & \hfill -22.917\hfill & \hfill -40.0076\hfill & \hfill 19.98\hfill \\ \hfill -11.764\hfill & \hfill 51.773\hfill & \hfill -19.982\hfill & \hfill -40.0076\hfill \\ \hfill 0\hfill & \hfill 19.982\hfill & \hfill 80.015\hfill & \hfill -39.963\hfill \\ \hfill 0\hfill & \hfill -40.0076\hfill & \hfill 39.961\hfill & \hfill 80\hfill \end{array}\right]\hfill \\ & -11.764\left[\begin{array}{cccc}\hfill 2.944\hfill & \hfill 51.772\hfill & \hfill -40.0076\hfill & \hfill 19.98\hfill \\ \hfill -11.764\hfill & \hfill 22.917\hfill & \hfill -19.982\hfill & \hfill -40.0076\hfill \\ \hfill 0\hfill & \hfill -40.0076\hfill & \hfill 80.015\hfill & \hfill -39.963\hfill \\ \hfill 0\hfill & \hfill -19.982\hfill & \hfill 39.961\hfill & \hfill 80\hfill \end{array}\right]\text{.}\hfill \end{array}$

The determinant is

$\begin{array}{c}det\left(\overline{J}\right)=J_{11}\left[\begin{array}{ccccc}\hfill J_{22}\hfill & \hfill J_{23}\hfill & \hfill J_{24}\hfill & \hfill J_{25}\hfill & \hfill J_{26}\hfill \\ \hfill J_{32}\hfill & \hfill J_{33}\hfill & \hfill J_{34}\hfill & \hfill J_{35}\hfill & \hfill J_{36}\hfill \\ \hfill J_{42}\hfill & \hfill J_{43}\hfill & \hfill J_{44}\hfill & \hfill J_{45}\hfill & \hfill J_{46}\hfill \\ \hfill J_{52}\hfill & \hfill J_{53}\hfill & \hfill J_{54}\hfill & \hfill J_{55}\hfill & \hfill J_{56}\hfill \\ \hfill J_{62}\hfill & \hfill J_{63}\hfill & \hfill J_{64}\hfill & \hfill J_{65}\hfill & \hfill J_{66}\hfill \end{array}\right]-J_{12}\left[\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{23}\hfill & \hfill J_{24}\hfill & \hfill J_{25}\hfill & \hfill J_{26}\hfill \\ \hfill J_{31}\hfill & \hfill J_{33}\hfill & \hfill J_{34}\hfill & \hfill J_{35}\hfill & \hfill J_{36}\hfill \\ \hfill J_{41}\hfill & \hfill J_{43}\hfill & \hfill J_{44}\hfill & \hfill J_{45}\hfill & \hfill J_{46}\hfill \\ \hfill J_{51}\hfill & \hfill J_{53}\hfill & \hfill J_{54}\hfill & \hfill J_{55}\hfill & \hfill J_{56}\hfill \\ \hfill J_{61}\hfill & \hfill J_{63}\hfill & \hfill J_{64}\hfill & \hfill J_{65}\hfill & \hfill J_{66}\hfill \end{array}\right]\\ +J_{13}\left[\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{22}\hfill & \hfill J_{24}\hfill & \hfill J_{25}\hfill & \hfill J_{26}\hfill \\ \hfill J_{31}\hfill & \hfill J_{32}\hfill & \hfill J_{34}\hfill & \hfill J_{35}\hfill & \hfill J_{36}\hfill \\ \hfill J_{41}\hfill & \hfill J_{42}\hfill & \hfill J_{44}\hfill & \hfill J_{45}\hfill & \hfill J_{46}\hfill \\ \hfill J_{51}\hfill & \hfill J_{52}\hfill & \hfill J_{54}\hfill & \hfill J_{55}\hfill & \hfill J_{56}\hfill \\ \hfill J_{61}\hfill & \hfill J_{62}\hfill & \hfill J_{64}\hfill & \hfill J_{65}\hfill & \hfill J_{66}\hfill \end{array}\right]-J_{14}\left[\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{22}\hfill & \hfill J_{23}\hfill & \hfill J_{25}\hfill & \hfill J_{26}\hfill \\ \hfill J_{31}\hfill & \hfill J_{32}\hfill & \hfill J_{33}\hfill & \hfill J_{35}\hfill & \hfill J_{36}\hfill \\ \hfill J_{41}\hfill & \hfill J_{42}\hfill & \hfill J_{43}\hfill & \hfill J_{45}\hfill & \hfill J_{46}\hfill \\ \hfill J_{51}\hfill & \hfill J_{52}\hfill & \hfill J_{53}\hfill & \hfill J_{55}\hfill & \hfill J_{56}\hfill \\ \hfill J_{61}\hfill & \hfill J_{62}\hfill & \hfill J_{63}\hfill & \hfill J_{65}\hfill & \hfill J_{66}\hfill \end{array}\right]\\ +J_{15}\left[\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{22}\hfill & \hfill J_{23}\hfill & \hfill J_{24}\hfill & \hfill J_{26}\hfill \\ \hfill J_{31}\hfill & \hfill J_{32}\hfill & \hfill J_{33}\hfill & \hfill J_{34}\hfill & \hfill J_{36}\hfill \\ \hfill J_{41}\hfill & \hfill J_{42}\hfill & \hfill J_{43}\hfill & \hfill J_{44}\hfill & \hfill J_{46}\hfill \\ \hfill J_{51}\hfill & \hfill J_{52}\hfill & \hfill J_{53}\hfill & \hfill J_{54}\hfill & \hfill J_{56}\hfill \\ \hfill J_{61}\hfill & \hfill J_{62}\hfill & \hfill J_{63}\hfill & \hfill J_{64}\hfill & \hfill J_{66}\hfill \end{array}\right]-J_{16}\left[\begin{array}{ccccc}\hfill J_{21}\hfill & \hfill J_{22}\hfill & \hfill J_{23}\hfill & \hfill J_{24}\hfill & \hfill J_{25}\hfill \\ \hfill J_{31}\hfill & \hfill J_{32}\hfill & \hfill J_{33}\hfill & \hfill J_{34}\hfill & \hfill J_{35}\hfill \\ \hfill J_{41}\hfill & \hfill J_{42}\hfill & \hfill J_{43}\hfill & \hfill J_{44}\hfill & \hfill J_{45}\hfill \\ \hfill J_{51}\hfill & \hfill J_{52}\hfill & \hfill J_{53}\hfill & \hfill J_{54}\hfill & \hfill J_{55}\hfill \\ \hfill J_{61}\hfill & \hfill J_{62}\hfill & \hfill J_{63}\hfill & \hfill J_{64}\hfill & \hfill J_{65}\hfill \end{array}\right]\text{.}\end{array}$

The ${\left(J_{11}^0\right)}^{-1}$ entry is now${\left(J_{11}^0\right)}^{-1}=\frac{Adj\left({\overline{J}}^0\right)}{det\left({\overline{J}}^0\right)}$.

7.3.9 Application Example 7.12: Inversion of a 3 × 3 Matrix

In order to demonstrate the entire procedure for the computation of the inverse of a matrix the following 3 × 3 matrix will be used

$\overline{A}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill -4\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]\text{.}$

Solution to Application Example 7.12

Transpose of Ā is

${\overline{A}}^t=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill -4\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\text{.}$

The cofactors of Ā^t are

$A_{11}^t={\left(-1\right)}^{\left(1+1\right)}\left|\begin{array}{cc}\hfill -4\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right|=\left(-4-2\right)=-6\text{,}$

$A_{12}^t={\left(-1\right)}^{\left(1+2\right)}\left|\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right|=-\left(3+1\right)=-4\text{,}$

$A_{13}^t={\left(-1\right)}^{\left(1+3\right)}\left|\begin{array}{cc}\hfill 3\hfill & \hfill -4\hfill \\ \hfill -1\hfill & \hfill 2\hfill \end{array}\right|=\left(6-4\right)=2\text{,}$

$A_{21}^t={\left(-1\right)}^{\left(2+1\right)}\left|\begin{array}{cc}\hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right|=\left(-2+6\right)=4\text{,}$

$A_{22}^t={\left(-1\right)}^{\left(2+2\right)}\left|\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right|=\left(1+3\right)=4\text{,}$

$A_{23}^t={\left(-1\right)}^{\left(2+3\right)}\left|\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 2\hfill \end{array}\right|=-\left(2+2\right)=-4\text{,}$

$A_{31}^t={\left(-1\right)}^{\left(3+1\right)}\left|\begin{array}{cc}\hfill 2\hfill & \hfill 3\hfill \\ \hfill -4\hfill & \hfill 1\hfill \end{array}\right|=\left(2+24\right)=14\text{,}$

$A_{32}^t={\left(-1\right)}^{\left(3+2\right)}\left|\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 1\hfill \end{array}\right|=-\left(1-9\right)=8\text{,}$

$A_{33}^t={\left(-1\right)}^{\left(3+3\right)}\left|\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill -4\hfill \end{array}\right|=\left(-4-6\right)=-10\text{.}$

The determinant det$\left(\overline{A}\right)=\left|\overline{A}\right|$ is

$\begin{array}{c}\left|\overline{A}\right|=1\cdot \left|\begin{array}{cc}\hfill -4\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right|-3\cdot \left|\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 1\hfill \end{array}\right|-1\cdot \left|\begin{array}{cc}\hfill 2\hfill & \hfill -4\hfill \\ \hfill 3\hfill & \hfill 1\hfill \end{array}\right|\\ =-4-2-6+18-2-12=-26+18=-8\text{.}\end{array}$

The inverse of Ā is now

${\left(\overline{A}\right)}^{-1}=\frac{\mathrm{A}\mathrm{d}\mathrm{j}\left(\overline{A}\right)}{\left|\overline{A}\right|}=-\frac{1}{8}\left[\begin{array}{ccc}\hfill -6\hfill & \hfill -4\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 4\hfill & \hfill -4\hfill \\ \hfill 14\hfill & \hfill 8\hfill & \hfill -10\hfill \end{array}\right]\text{.}$

7.3.10 Application Example 7.13: Computation of the Correction Voltage Vector

For the system of Fig. E7.8.1, compute the correction voltage vector$\mathrm{\Delta }{\overline{x}}^0$, updating the mismatch power vector, and check the convergence of the load flow algorithm (using a convergence tolerance of 0.0001).

Solution to Application Example 7.13

Based on the results of Application Examples 7.9 to 7.11, the correction voltage vector $\mathrm{\Delta }{\overline{x}}^0$ can be computed from

$\begin{array}{l}\mathrm{\Delta }{\overline{x}}^0={\left({\overline{J}}^0\right)}^{-1}\mathrm{\Delta }\overline{W}\left({\overline{x}}^0\right)=(2.68\mathrm{E}-4,2.3\mathrm{E}-3,\\ 2.79\mathrm{E}-3,3.57\mathrm{E}-3,3.39\mathrm{E}-3,4.03\mathrm{E}-3){}^t\text{.}\end{array}$

The updated mismatch power vector is

$\begin{array}{l}\mathrm{\Delta }{\overline{W}}^1=(1.41\mathrm{E}-4,3.01\mathrm{E}-4,-9.63\mathrm{E}-6,7.62\mathrm{E}-5,\\ 1.20\mathrm{E}-3,5.96\mathrm{E}-4){}^t\text{.}\end{array}$

Since the components of $\mathrm{\Delta }{\overline{W}}^1$ are larger than the specified convergence tolerance (e.g., 0.0001), the Newton–Raphson solution has not converged. However, it will converge in two iterations:

Fundamental power flow iteration summary (Fig. E7.8.1):

Fundamental power flow output solution (Fig. E7.8.1), where δ angles are rounded (approximate) values:

7.4.1 Harmonic Bus Admittance Matrix and Power Definitions

For nonsinusoidal operating conditions power system matrices will be defined at harmonic frequencies. These matrices include the harmonic bus admittance matrix ${\overline{Y}}_{\mathrm{bus}}^{\left(h\right)}\text{,}$ the harmonic bus impedance matrix ${\overline{Z}}_{\mathrm{bus}}^{\left(h\right)}$ = $1/{\overline{Y}}_{\mathrm{bus}}^{\left(h\right)}$, and the harmonic Jacobian ${\overline{J}}^{\left(h\right)}$ matrix.

The harmonic bus admittance matrix ${\overline{Y}}_{\mathrm{bus}}^{\left(h\right)}$ is identical to the fundamental bus admittance matrix ${\overline{Y}}_{\mathrm{bus}}^{\left(h\right)}={\overline{Y}}_{\mathrm{bus}}$ with the difference that line admittances are evaluated at the hth harmonic frequency f_h = h · f₁:

${\overline{Y}}_{\mathit{bus}}^{\left(h\right)}=\left[\begin{array}{cccccc}\hfill y_{11}^{\left(h\right)}\hfill & \hfill y_{12}^{\left(h\right)}\hfill & \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill y_{1n}^{\left(h\right)}\hfill \\ \hfill y_{21}^{\left(h\right)}\hfill & \hfill y_{22}^{\left(h\right)}\hfill & \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill y_{2n}^{\left(h\right)}\hfill \\ \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill .\hfill \\ \hfill y_{n1}^{\left(h\right)}\hfill & \hfill y_{n2}^{\left(h\right)}\hfill & \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill y_{nn}^{\left(h\right)}\hfill \end{array}\right]\text{,}$

where ω_h = 2πhf₁ = h · 377 rad/s.

Power Definitions at Harmonic Frequencies

Consider a voltage v(t) and a current i(t) expressed in terms of their rms harmonic components

$v\left(t\right)=V_0+{\displaystyle \sum _{h=1}^{\infty }\sqrt{2}{V}_{h}sin}\left(h{\omega }_{1}t+{\theta }_h^v\right)\text{,}$

  (7-84)

$i\left(t\right)=I_0+{\displaystyle \sum _{h=1}^{\infty }\sqrt{2}{I}_{h}sin}\left(h{\omega }_{1}t+{\theta }_h^i\right)\text{.}$

  (7-85)

Then

• the active (real, average) power is given by

$P=V_0{I}_0+{\displaystyle \sum _{h=1}^{\infty }{V}_h{I}_{h}cos\left({\theta }_h^v-{\theta }_h^i\right)\text{,}}$

  (7-86)

• the reactive power is given by

$Q={\displaystyle \sum _{h=1}^{\infty }{V}_h{I}_{h}sin\left({\theta }_h^v-{\theta }_h^i\right)\text{,}}$

  (7-87)

• and the apparent (voltampere) power is

$S=\sqrt{\left({\displaystyle \sum _{h=0}^{\infty }{I}_h^2}\right)\left({\displaystyle \sum _{h=0}^{\infty }{V}_h^2}\right)\text{.}}$

  (7-88)

In case of sinusoidal v(t) and i(t) (h = 1 only), we have

$S^2=P^2+Q^2\text{.}$

  (7-89)

For nonsinusoidal cases that relation does not hold and must be replaced by

$D=\sqrt{S^2-P^2-Q^2}\text{,}$

  (7-90)

where D is called the distortion power. Other definitions have also been proposed for harmonic powers, which are not discussed here.

Classification of Time Harmonics

Time harmonics can be classified into three types (Table 7.2):

Table 7.2

Classification of Time Harmonics

Positive (+) sequence	Negative (–) sequence	Zero (0) sequence
1	2	3
4	5	6
7	8	9
10	11	12
13	14	15
16	17	18
19	20	21
$⋮$	$⋮$	$⋮$

• positively rotating harmonics (+),

• negatively rotating harmonics (–), and

• zero-sequence harmonics (0).

Triplen (multiples of three) harmonics (3h, for h = 1, 2, 3, …) in a balanced three-phase system are predominantly of the zero-sequence type. With a few exceptions (discussed in preceding chapters), zero-sequence currents cannot flow from the network into a delta or an ungrounded wye-connected device (e.g., transformer), and therefore triplen harmonics are excluded from this analysis.

7.4.2 Modeling of Nonlinear and Linear Loads at Harmonic Frequencies

Before proceeding with the reformulation of the Newton–Raphson load flow to include harmonic frequencies, it is necessary to properly model linear and nonlinear loads at fundamental and harmonic frequencies.

Harmonic Modeling of Nonlinear Loads

The (v–i) relationship of nonlinear loads (such as line-commutated converters) will be modeled as coupled harmonic current sources. The injected harmonic currents of a nonlinear load (I_injected^(h)) at bus m will be a function of its fundamental and harmonic voltages:

$\left\{\begin{array}{c}\hfill \mathrm{Real}\left(I_{\mathrm{injected}}^{\left(h\right)}\right)=g_{r,m}^{\left(h\right)}\left({\tilde{V}}_m^{\left(1\right)},{\tilde{V}}_m^{\left(5\right)},{\tilde{V}}_m^{\left(7\right)},\dots ..,{\tilde{V}}_m^{\left(L\right)},{\alpha }_m,{\beta }_m\right)\hfill \\ \hfill \mathrm{Imag}\left(I_{\mathrm{injected}}^{\left(h\right)}\right)=g_{i,m}^{\left(h\right)}\left({\tilde{V}}_m^{\left(1\right)},{\tilde{V}}_m^{\left(5\right)},{\tilde{V}}_m^{\left(7\right)},\dots ..,{\tilde{V}}_m^{\left(L\right)},{\alpha }_m,{\beta }_m\right)\hfill \end{array}\right.$

  (7-91)

where α_m and β_m are the nonlinear load control parameters and L is the maximum harmonic order considered L = h_max. The currents of Eq. 7-91 are referred to the nonlinear bus m to which the nonlinear load is connected. The nonlinearity of ${\tilde{g}}_{\mathrm{m}}^{\left(\mathrm{h}\right)}=g_{\mathrm{r},\mathrm{m}}^{\left(\mathrm{h}\right)}+j{g}_{i,m}^{\left(h\right)}$ is due to strong couplings between harmonic currents and voltages dictated by the nonlinear load. In some cases (e.g., decoupled harmonic power flow formulation), it is convenient to ignore harmonic couplings ${\tilde{g}}_m^{\left(h\right)}\left({\tilde{V}}_m^{\left(h\right)},{\alpha }_m,{\beta }_m\right)$ and simplify the problem.

As an example, for a full-wave bridge rectifier connected to bus m (shown in Fig. 7.13), α_m is the firing angle of semiconductor-controlled rectifiers (SCRs), and β_m is either the commutating impedance (inductance) or the DC voltage (E). Note that the equivalent circuit models a three-phase full-wave rectifier/inverter with a general load, where L_com is the commutating inductance, that is, the leakage inductance of the transformer, and

• for a passive load, only resistance R and filter inductance F are considered,

• for a DC motor drive, only R and E are considered (E > 0) when motoring, and

• for inverter operation with a DC generator, only R and E are considered (E < 0).

Modeling Linear Loads at Harmonic Frequencies

For the formulation of harmonic power flow, equivalent circuits of linear loads are required. At the fundamental frequency, linear loads are modeled as conventional PQ and PV buses. However, shunt admittances are used to model them at harmonic frequencies [7,8]. The admittance of a linear load connected to bus k at the hth harmonic is

${\tilde{y}}_k^{\left(h\right)}=\frac{\left(P_k^{\left(1\right)}-j{Q}_k^{\left(1\right)}\right)/h}{\left|{\tilde{V}}_k^{\left(1\right)}\right|{}^2}\text{.}$

  (7-92)

where j represents the operator $\sqrt{-1}$.

Capacitor banks are modeled as a fixed shunt reactance. Transformers are approximated by linear leakage inductances, and their nonlinearities and losses due to eddy currents, hysteresis, and saturation are neglected.

7.4.3 The Harmonic Power Flow Algorithm (Assembly of Equations)

Power system components, loads, and generators that produce time harmonics in an otherwise harmonic-free network are termed nonlinear. Other buses such as the standard PQ and PV buses with no converter or other nonlinear devices connected are termed linear. Consider

• Bus 1 to be the conventional swing (slack) bus with specified values for voltage magnitude |${\tilde{V}}_1$| and phase angle δ₁.

• Buses 2 through (m – 1) to be the conventional linear PQ (or PV) buses. As with the conventional power flow problem, active P_i⁽¹⁾ and reactive Q_i⁽¹⁾ powers are assumed to be known at these linear buses.

• Buses m through n to be nonlinear buses. At these buses, the fundamental real power P_i⁽¹⁾ and the total apparent power S_i,total (or the total reactive power Q_i,total) are specified.

Network voltages and currents are represented by Fourier series components, and the bus voltage vector $\overline{x}$ of the fundamental power flow (Eq. 7-42) is redefined as

$\begin{array}{l}\overline{U}=[{\delta }_2^{\left(1\right)},|{\tilde{V}}_2^{\left(1\right)}|,\dots ,{\delta }_n^{\left(1\right)},|{\tilde{V}}_n^{\left(1\right)}|,{\delta }_1^{\left(5\right)},|{\tilde{V}}_1^{\left(5\right)}|,\dots ,\hfill \\ {\delta }_n^{\left(5\right)},|{\tilde{V}}_n^{\left(5\right)}|,\dots ,{\delta }_1^{\left(L\right)},|{\tilde{V}}_1^{\left(L\right)}|,\dots ,{\delta }_n^{\left(L\right)},|{\tilde{V}}_n^{\left(L\right)}|,\hfill \\ {\alpha }_m,{\beta }_m,\dots ,{\alpha }_n,{\beta }_n]{}^t={\left[{\overline{V}}^{\left(1\right)},{\overline{V}}^{\left(5\right)},\dots ,{\overline{V}}^{\left(L\right)},\overline{\mathrm{\Phi }}\right]}^t\hfill \end{array}$

where (superscripts and subscripts denote harmonic orders and bus numbers, respectively)

$\begin{array}{l}\mathrm{Fundamental}\mathrm{voltages}\mathrm{are}\\ {\overline{V}}^{\left(1\right)}={\delta }_2^{\left(1\right)},\left|{\tilde{V}}_2^{\left(1\right)}\right|,\dots ,{\delta }_n^{\left(1\right)},\left|{\tilde{V}}_n^{\left(1\right)}\right|\text{,}\end{array}$

$\begin{array}{l}\mathrm{Fifth}\mathrm{harmonic}\mathrm{voltages}\mathrm{are}\\ {\overline{V}}^{\left(5\right)}={\delta }_1^{\left(5\right)},\left|{\tilde{V}}_1^{\left(5\right)}\right|,\dots ,{\delta }_n^{\left(5\right)},\left|{\tilde{V}}_n^{\left(5\right)}\right|\text{,}\end{array}$

$\begin{array}{l}\mathrm{L}\mathrm{t}\mathrm{h}\mathrm{harmonic}\mathrm{voltages}\mathrm{are}\\ {\overline{V}}^{\left(L\right)}={\delta }_1^{\left(L\right)},\left|{\tilde{V}}_1^{\left(L\right)}\right|,\dots ,{\delta }_n^{\left(L\right)},\left|{\tilde{V}}_n^{\left(L\right)}\right|\text{,}\end{array}$

$\begin{array}{c}\mathrm{Nonlinear}\mathrm{device}\mathrm{variables}\mathrm{vector}\mathrm{is}\\ \overline{\mathit{\Phi }}={\alpha }_m,{\beta }_m,\dots ,{\alpha }_n,{\beta }_n\text{.}\end{array}$

Harmonic bus voltage magnitudes, phase angles, and the nonlinear device parameters (α_m, β_m) are unknowns; therefore the conventional formulation (forcing the fundamental mismatch powers to zero) is not sufficient to solve the harmonic power flow problem. Three additional relations are used:

• Kirchhoff’s current law for the fundamental frequency (fundamental current balance) at nonlinear buses.

• Kirchhoff’s current law for each harmonic (harmonic current balance) at all buses.

• Conservation of apparent voltamperes (apparent power or voltampere balance) at nonlinear buses.

Note that Kirchhoff’s current law for the fundamental frequency is not applied to linear buses since active and reactive power (mismatch power) balance has already been applied at these buses.

Current Balance for Fundamental Frequency

The current balance is written for fundamental frequency (h = 1) at the nonlinear buses as indicated in Fig. 7.14, where the nonlinear currents g_r,m⁽¹⁾, g_i,m⁽¹⁾ are referred to bus m. Therefore one obtains for the nonlinear buses (from bus m to bus n) for the fundamental current balance:

$\left[\begin{array}{c}\hfill I_{r,m}^{\left(1\right)}\hfill \\ \hfill I_{i,m}^{\left(1\right)}\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill I_{r,n}^{\left(1\right)}\hfill \\ \hfill I_{i,n}^{\left(1\right)}\hfill \end{array}\right]=-\left[\begin{array}{c}\hfill g_{r,m}^{\left(1\right)}\left({\tilde{V}}_m^{\left(1\right)},{\tilde{V}}_m^{\left(5\right)},\dots ,{\tilde{V}}_m^{\left(L\right)},{\alpha }_m,{\beta }_m\right)\hfill \\ \hfill g_{i,m}^{\left(1\right)}\left({\tilde{V}}_m^{\left(1\right)},{\tilde{V}}_m^{\left(5\right)},\dots ,{\tilde{V}}_m^{\left(L\right)},{\alpha }_m,{\beta }_m\right)\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill g_{r,n}^{\left(1\right)}\left({\tilde{V}}_n^{\left(1\right)},{\tilde{V}}_n^{\left(5\right)},\dots ,{\tilde{V}}_n^{\left(L\right)},{\alpha }_n,{\beta }_n\right)\hfill \\ \hfill g_{i,n}^{\left(1\right)}\left({\tilde{V}}_n^{\left(1\right)},{\tilde{V}}_n^{\left(5\right)},\dots ,{\tilde{V}}_n^{\left(L\right)},{\alpha }_n,{\beta }_n\right)\hfill \end{array}\right]\text{,}$

  (7-96)

where I_r,m⁽¹⁾ and I_i,m⁽¹⁾ are the fundamental real and reactive line currents at the mth nonlinear bus, respectively, and g_r,m⁽¹⁾ and g_i,m⁽¹⁾ are the fundamental real and reactive injected load currents at the same bus. Note that the line and injected load currents are positive if they leave the bus.

Current Balance at Harmonic Frequencies

For all (linear and nonlinear) buses one can write

$\left[\begin{array}{c}\hfill I_{r,1}^{\left(h\right)}\hfill \\ \hfill I_{i,1}^{\left(h\right)}\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill I_{r,m-1}^{\left(h\right)}\hfill \\ \hfill I_{i,m-1}^{\left(h\right)}\hfill \\ \hfill ---\hfill \\ \hfill I_{r,m}^{\left(h\right)}\hfill \\ \hfill I_{i,m}^{\left(h\right)}\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill I_{r,n}^{\left(h\right)}\hfill \\ \hfill I_{i,n}^{\left(h\right)}\hfill \end{array}\right]=-\left[\begin{array}{c}\hfill 0\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill \left(\mathrm{swing}\mathrm{bus}\mathrm{and}\mathrm{linear}\mathrm{buses}\right)\hfill \\ \hfill .\hfill \\ \hfill .\hfill \\ \hfill 0\hfill \\ \hfill ----------------\hfill \\ \hfill g_{r,n}^{\left(h\right)}\left({\tilde{V}}_m^{\left(1\right)},{\tilde{V}}_m^{\left(5\right)},\dots ,{\tilde{V}}_m^{\left(L\right)},{\alpha }_m,{\beta }_m\right)\hfill \\ \hfill g_{i,m}^{\left(h\right)}\left({\tilde{V}}_m^{\left(1\right)},{\tilde{V}}_m^{\left(5\right)},\dots ,{\tilde{V}}_m^{\left(L\right)},{\alpha }_m,{\beta }_m\right)\hfill \\ \hfill .\hfill \\ \hfill \left(\mathrm{nonlinear}\mathrm{buses}\right)\hfill \\ \hfill .\hfill \\ \hfill g_{r,n}^{\left(h\right)}\left({\tilde{V}}_n^{\left(1\right)},{\tilde{V}}_n^{\left(5\right)},\dots ,{\tilde{V}}_n^{\left(L\right)},{\alpha }_n,{\beta }_n\right)\hfill \\ \hfill g_{i,n}^{\left(h\right)}\left({\tilde{V}}_n^{\left(1\right)},{\tilde{V}}_n^{\left(5\right)},\dots ,{\tilde{V}}_n^{\left(L\right)},{\alpha }_n,{\beta }_n\right)\hfill \end{array}\right]$

  (7-97)

The line currents I_r,m^(h) and I_i,m^(h) will be computed in the next section.

Apparent Power or Voltampere Balance

The third type of relation is the apparent power (voltampere) balance at each nonlinear bus j:

$S_j^2={\left(P_j^t\right)}^2+{\left(Q_j^t\right)}^2+D_j^2\text{,}$

  (7-98)

where j = m, …, n and

$\begin{array}{c}{\left(P_j^t\right)}^2={\left({\displaystyle \sum _h{P}_j^{\left(h\right)}}\right)}^2=\left(V_j^{\left(1\right)}{I}_j^{\left(1\right)}cos{\theta }_j^{\left(1\right)}\right)+V_j^{\left(5\right)}{I}_j^{\left(5\right)}cos{\theta }_j^{\left(5\right)}+V_j^{\left(7\right)}{I}_j^{\left(7\right)}cos{\theta }_j^{\left(7\right)}+\dots ){}^2\hfill \end{array}$

  (7-99a)

$\begin{array}{l}{\left(Q_j^t\right)}^2={\left({\displaystyle \sum _h{Q}_j^{\left(h\right)}}\right)}^2=\left(V_j^{\left(1\right)}{I}_j^{\left(1\right)}cos{\theta }_j^{\left(1\right)}\right)+V_j^{\left(5\right)}{I}_j^{\left(5\right)}sin{\theta }_j^{\left(5\right)}+V_j^{\left(7\right)}{I}_j^{\left(7\right)}sin{\theta }_j^{\left(7\right)}+\dots ){}^2\end{array}$

  (7-99b)

are the total (fundamental plus harmonics) real and reactive load powers at nonlinear buses. D_m to D_n can be computed from g_r,m, g_i,m, …, g_r,n, g_i,n; therefore, distortion powers are not independent variables. Equation 7-98 is used to update (between iterations) the nonlinear-device injected powers P and Q at the nonlinear buses.

The number of unknowns for the n buses are as follows (K = number of harmonics considered not including fundamental, e.g., for h = 1, 5, 7, 11 we get K = 3):

• Fundamental bus voltage magnitudes and phases for all buses but the swing bus	2(n – 1)
• Fundamental real and reactive powers at the swing bus (losses P1, Q1)	2
• Harmonic (not including fundamental) voltage magnitudes and phases at all (total) n buses	2nK
• Total reactive powers Qjt at each nonlinear bus, where n is the number of all buses and (m – 1) the number of linear buses	n – (m – 1) = (n – m + 1)
• Two variables (αj, βj) associated with each nonlinear bus	2(n – m + 1)
• Total number of unknowns	2n(K + 1) + 3(n – m + 1)

The number of available equations are as follows:

• Fundamental real and reactive power balance at all (m – 1) linear buses but the swing bus	2(m – 2)
• Fundamental voltage magnitude and phase at the swing bus (|${\tilde{V}}_1$| = 1 pu, δ1 = 0 radians)	2
• Fundamental real and imaginary current balance at n – (m – 1) nonlinear buses (at (m – 1) linear buses the Pi(1) and Qi(1)are specified)	2(n – (m – 1)) = 2(n – m + 1)
• Harmonic real and reactive current (except the fundamental) balance at all n buses	2nK
• Apparent voltampere balance at n – (m – 1) nonlinear buses to compute reactive power (if reactive power is specified at nonlinear buses, however, this apparent voltampere balance is not required at nonlinear buses)	(n – m + 1)
• Total number of equations	2n(K + 1) + (n – m + 1)

There remain 2(n – m + 1) equations to be defined. These can be the total real and reactive mismatch powers at the n – (m – 1) nonlinear buses:

$\mathrm{\Delta }{P}_j^{\mathrm{nonlinear}}=P_j^t+{\displaystyle \sum _h{F}_{r,j}^{\left(h\right)}}\text{,}$

  (7-100a)

$\mathrm{\Delta }{Q}_j^{\mathrm{nonlinear}}=Q_j^t+{\displaystyle \sum _h{F}_{i,j}^{\left(h\right)}}\text{,}$

  (7-100b)

where j = m, …, n and P_j^t and Q_j^t are the total (fundamental and harmonics) nonlinear load real and reactive powers at bus j, respectively. ${\displaystyle \sum _h{F}_{r,j}^{\left(h\right)}}$ and ${\displaystyle \sum _h{F}_{i,j}^{\left(h\right)}}$ (for h = 1, 5, 7, …, L) are the total real and reactive line powers, respectively. Note that the formulas for F_r,j^(h) and F_i,j^(h) are identical to those for F_r,j and F_i,j of the fundamental power flow problem with the difference that ${\overline{Y}}_{\mathrm{bus}}^{\left(h\right)}$ and the hth harmonic voltages must be used. The application of the newly defined mismatch powers at the nonlinear buses results in 2(n – m + 1) additional equations, and the number of equations is identical to the number of unknowns.