115 lines
4.5 KiB
TeX
115 lines
4.5 KiB
TeX
\documentclass[10pt,a4paper,final]{report}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{fontspec}%使用xetex
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\setmainfont[BoldFont=黑体]{宋体} % 使用系统默认字体
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\XeTeXlinebreaklocale "zh" % 针对中文进行断行
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\XeTeXlinebreakskip = 0pt plus 1pt minus 0.1pt % 给予TeX断行一定自由度
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\linespread{1.5} % 1.5倍行距
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\begin{document}
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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。
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\begin{equation}
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f(x)=[z-h(x)]^T W [z-h(x)]
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\end{equation}
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最优条件等价为
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\begin{equation}
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\bigtriangledown f(x)=J^T W [h(x)-z]=0
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\end{equation}
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其中$J$是$h(x)$的$Jacobi$矩阵
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\newline
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对于线路功率
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线路有功功率Jacobi
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\begin{equation}
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\begin{aligned}
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\frac{\partial P_{ij}}{\partial V_1}=
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2V_1G_{ij}-V_2[cos(\theta_1 - \theta_2)G_{ij}+sin (\theta_1 - \theta_2)B_{ij}]
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\frac{\partial P_{12}}{\partial V_2}=
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-V_1[cos(\theta_1 - \theta_2)G_{ij}+sin (\theta_1 - \theta_2)B_{ij}]
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\end{aligned}
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\end{equation}
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利用Newton法对上式进行展开，写成元素形式。以两个变量为例
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\begin{equation}
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\bigtriangledown f= [ \bigtriangledown f_1\,, ... \,,\bigtriangledown f_n]^T
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\bigtriangledown^2 f_1=(\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_1}
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+\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_1} ) \Delta x_1 \\
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+ (\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_2}
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+\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_2} ) \Delta x_2 \\
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+( \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 \\
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+ ( \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
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\end{aligned}
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\end{equation}
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利用matlab的语句可以表示为
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\begin{equation}
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J ^T W J + \tilde{Q}
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\end{equation}
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其中
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$J$具有以下形式
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\begin{equation}
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J=\left[ %左括号
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\begin{array}{cc} %该矩阵一共3列，每一列都居中放置
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\frac{\partial f_1}{ \partial x_1} & \frac{\partial f_1}{ \partial x_2}\\ %第一行元素
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\frac{\partial f_2}{ \partial x_1} & \frac{\partial f_2}{ \partial x_2}\\ %第二行元素
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\end{array}
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\right] %右括号
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\end{equation}
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\begin{equation}
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J ^T W J=
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(\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_1}
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+\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_1} ) \Delta x_1 \\
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+ (\omega_1 \frac{ \partial f_1 }{\partial x_1} \frac{ \partial f_1 }{\partial x_2}
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+\omega_2 \frac{ \partial f_2 }{\partial x_1} \frac{ \partial f_2 }{\partial x_2} ) \Delta x_2
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\end{equation}
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\begin{equation}
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\label{非矢量化二阶导数}
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\tilde{Q}=
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( \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 \\
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+ ( \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
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\end{equation}
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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
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% 线路功率二阶导数
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\begin{equation}
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\begin{aligned}
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\frac{\partial^2 P_{12}}{\partial V_1^2}&=
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\frac{-2}{k^2}B_{12}\\
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&=\frac{-2B_{12}}{k^2}
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\frac{\partial^2 P_{12}}{\partial V_1 \partial V_2 }&=
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\frac{-1}{k}[sin(\theta_1 - \theta_2)G_{ij}-cos (\theta_1 - \theta_2)B_{ij}] \\
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&=
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\frac{-[sin(\theta_1 - \theta_2)G_{ij}-cos (\theta_1 - \theta_2)B_{ij}]}{k}
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\frac{\partial^2 P_{12}}{\partial V_2^2}&=0
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\end{aligned}
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\end{equation}
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利用
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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 } $] ) 可以表示
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$
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\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 }
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$
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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。
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\end{document}
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