为了改善收敛性，利用内点法进行求解。

Signed-off-by: facat <facat@facat.cn>

公式/内点法公式.tex

@@ -0,0 +1,104 @@
+\documentclass[10pt,a4paper,final]{report}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{fontspec}%使用xetex
+\setmainfont[BoldFont=黑体]{宋体}               % 使用系统默认字体
+\XeTeXlinebreaklocale "zh"                      % 针对中文进行断行
+\XeTeXlinebreakskip = 0pt plus 1pt minus 0.1pt  % 给予TeX断行一定自由度
+\linespread{1.5}                                % 1.5倍行距
+\begin{document}
+由于之前的Gauss-Newton对1047节点收敛性不好（dX为0的时候，最优条件不为0），准备改用内点法求解。
+\begin{equation}
+f(x)=[z-h(x)]^T W [z-h(x)]
+\end{equation}
+最优条件等价为
+\begin{equation}
+\bigtriangledown f(x)=J^T W [z-h(x)]=0
+\end{equation}
+其中$J$是$h(x)$的$Jacobi$矩阵
+\newline
+对于线路功率
+线路有功功率Jacobi
+\begin{equation}
+\begin{aligned}
+\frac{\partial P_{ij}}{\partial V_1}=
+2V_1G_{ij}-V_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 V_2}=
+-V_1[cos(\theta_1 - \theta_2)G_{ij}+sin (\theta_1 - \theta_2)B_{ij}]
+\end{aligned}
+\end{equation}
+
+
+利用Newton法对上式进行展开，写成元素形式。以两个变量为例
+\begin{equation}
+\bigtriangledown f= [ \bigtriangledown f_1\,, ... \,,\bigtriangledown f_n]^T
+\end{equation}
+\begin{equation}
+\begin{aligned}
+\bigtriangledown^2 f_1=(\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_1}
++\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_1} ) \Delta x_1 \\
++ (\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_2}
++\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_2} ) \Delta x_2 \\
++( \omega_1 f_1 \frac{ \partial ^2 f_1 }{ \partial x_1 \partial x_1 } + \omega_2 f_2 \frac{ \partial ^2 f_2 }{ \partial x_1 \partial x_1 } ) \Delta x_1 \\
++ ( \omega_1 f_1 \frac{ \partial ^2 f_1 }{ \partial x_1 \partial x_2 } + \omega_2 f_2 \frac{ \partial ^2 f_2 }{ \partial x_1 \partial x_2 } ) \Delta x_2
+\end{aligned}
+\end{equation}
+利用matlab的语句可以表示为
+\begin{equation}
+J ^T W J + \tilde{Q}
+\end{equation}
+其中
+\begin{equation}
+J ^T W J=
+(\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_1}
++\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_1} ) \Delta x_1 \\
++ (\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_2}
++\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_2} ) \Delta x_2
+\end{equation}
+
+\begin{equation}
+\label{非矢量化二阶导数}
+\tilde{Q}=
+( \omega_1 f_1 \frac{ \partial ^2 f_1 }{ \partial x_1 \partial x_1 } + \omega_2 f_2 \frac{ \partial ^2 f_2 }{ \partial x_1 \partial x_1 } ) \Delta x_1 \\
++ ( \omega_1 f_1 \frac{ \partial ^2 f_1 }{ \partial x_1 \partial x_2 } + \omega_2 f_2 \frac{ \partial ^2 f_2 }{ \partial x_1 \partial x_2 } ) \Delta x_2
+\end{equation}
+式(\ref{非矢量化二阶导数})可以用Matlab语言在不需要循环的情况下处理好。因为
+% 线路功率二阶导数
+\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}
+利用
+sparse([1 1],[1 1],[$ \omega_1 f_1 \frac{ \partial ^2 f_1 }{ \partial x_1 \partial x_1 } \quad \omega_2 f_2 \frac{ \partial ^2 f_2 }{ \partial x_1 \partial x_1 }  $] ) 可以表示
+$
+\omega_1 f_1 \frac{ \partial ^2 f_1 }{ \partial x_1 \partial x_1 } + \omega_2 f_2 \frac{ \partial ^2 f_2 }{ \partial x_1 \partial x_1 }
+$
+在海森矩阵中与 $\Delta x_1$ 对应的元素，也就是第一行第一列的元素，其他的类似。用这种办法可以不利用矢量化公式任然依靠Matlab语句实现矢量化运算。
+
+\end{document}
+