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增加线路二阶导数的公式 

Signed-off-by: facat <facat@facat.cn>
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facat 2013-08-22 15:14:22 +08:00
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公式/公式.tex
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@ -204,5 +204,56 @@ Q_{ij}&=-[V_1^2-V_1V_2cos(\theta_1 - \theta_2)]B_{ij}-V_1V_2sin (\theta_1 - \the
\end{equation}
ps.已检验过线路的公式。
\par
以上公式已经可以完成状态估计，若要实现更好的收敛性，需要利用二阶导数。
\par
线路支路功率二阶导数
\begin{equation}
\begin{aligned}
\frac{\partial^2 P_{12}}{\partial V_1^2}&=
\frac{-2}{k^2}B_{12}\\
&=\frac{-2B_{12}}{k^2}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\frac{\partial^2 P_{12}}{\partial V_1 \partial V_2 }&=
\frac{-1}{k}[sin(\theta_1 - \theta_2)G_{ij}-cos (\theta_1 - \theta_2)B_{ij}] \\
&=
\frac{-[sin(\theta_1 - \theta_2)G_{ij}-cos (\theta_1 - \theta_2)B_{ij}]}{k}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\frac{\partial^2 P_{12}}{\partial V_2^2}&=0
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\frac{\partial P_{12}}{\partial \theta_1^2}&=
V_1V_2[cos(\theta_1 - \theta_2)G_{ij}+sin (\theta_1 - \theta_2)B_{ij}]
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\frac{\partial P_{12}}{\partial \theta_1 \partial \theta_2}&=
V_1V_2[-cos(\theta_1 - \theta_2)G_{ij}-sin (\theta_1 - \theta_2)B_{ij}] \\
&=
-V_1V_2[cos(\theta_1 - \theta_2)G_{ij}+sin (\theta_1 - \theta_2)B_{ij}]
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\frac{\partial P_{12}}{\partial \theta_2^2}&=
-V_1V_2[-cos(\theta_1 - \theta_2)G_{ij}-sin (\theta_1 - \theta_2)B_{ij}] \\
&=
V_1V_2[cos(\theta_1 - \theta_2)G_{ij}+sin (\theta_1 - \theta_2)B_{ij}]
\end{aligned}
\end{equation}
\end{document}