From 6de744f51ca3b8f02df3a6ae75717da162d55c73 Mon Sep 17 00:00:00 2001
From: facat <facat@facat.cn>
Date: Thu, 22 Aug 2013 15:14:22 +0800
Subject: [PATCH] =?UTF-8?q?=E5=A2=9E=E5=8A=A0=E7=BA=BF=E8=B7=AF=E4=BA=8C?=
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Signed-off-by: facat <facat@facat.cn>
---
 公式/公式.tex | 51 +++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 51 insertions(+)

diff --git a/公式/公式.tex b/公式/公式.tex
index 33f19dd..73dc7ff 100644
--- a/公式/公式.tex
+++ b/公式/公式.tex
@@ -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}