推导公式错了

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

run.asv

@@ -37,11 +37,13 @@ cmpBranchI=(cmpV(lineI)-cmpV(lineJ))./(lineR+1j*lineX);%
 rBranchI=abs(cmpBranchI);% 支路电流幅值
 mBranchI=rBranchI.*(normrnd(0,sigma,length(rBranchI),1)+1);%支路电流量测量
 %% 支路功率
-rBranchP=real((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI));
+rBranchP=real((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI))+real(cmpV(lineI) .*conj(1j*lineB2.*cmpV(lineI) ) );
-rBranchP=rBranchP(abs(rBranchP)>1e-5);
+rV=abs(cmpV);
+rA=imag(
+% rBranchP=rBranchP(abs(rBranchP)>1e-5);
 mBranchP=rBranchP.*(normrnd(0,sigma,length(rBranchP),1)+1);%支路功率量测量
-rBranchQ=imag((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI));
+rBranchQ=imag((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI))+imag(cmpV(lineI) .*conj(1j*lineB2.*cmpV(lineI) ) );
-rBranchQ=rBranchQ(abs(rBranchQ)>1e-5);
+% rBranchQ=rBranchQ(abs(rBranchQ)>1e-5);
 mBranchQ=rBranchQ.*(normrnd(0,sigma,length(rBranchQ),1)+1);%支路功率量测量
 %% 注入功率
 rPD=PD;
@@ -97,56 +99,92 @@ Busnum=length(Volt);
 % 以下都是Jacobi矩阵
 SEVolt=sparse(ones(length(mVolt),1));
 SEVAngel=sparse(zeros(length(mVolt),1));
-
+maxD=1000;
+Iteration=0;
+while maxD>1e-3;
 dV_dV=sparse(1:length(mVolt),1:length(mVolt),1,length(mVolt),length(mVolt));%电压量测量的微分
 dV_dTyta=sparse(length(mVolt),length(mVolt));
 dLPij_dVi=sparse(1:length(lineI),lineJ, ...
     -SEVolt(lineJ).*( ...
-    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineX.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
     )... 
     +2*(lineG).*SEVolt(lineI) ...
     ,length(lineI),length(mVolt));%线路的
 dLPij_dVj=sparse(1:length(lineI),lineJ, ...
     -SEVolt(lineI).*( ...
-    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineX.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    ) ...
+    ,length(lineI),length(mVolt));%线路的
+dLPij_dThetai=sparse(1:length(lineI),lineJ, ...
+    SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
+    )... 
+    ,length(lineI),length(mVolt));%线路的
+dLPij_dThetaj=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
+    )... 
+    ,length(lineI),length(mVolt));%线路的
+
+dLQij_dVi=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineJ).*( ...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    )... 
+    -2*(lineB+lineB2).*SEVolt(lineI) ...
+    ,length(lineI),length(mVolt));%线路的
+dLQij_dVj=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineI).*( ...
+    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
+    ) ...
+    ,length(lineI),length(mVolt));%线路的
+dLQij_dThetai=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    ) ...
+    ,length(lineI),length(mVolt));%线路的
+dLQij_dThetaj=sparse(1:length(lineI),lineJ, ...
+    SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
     ) ...
     ,length(lineI),length(mVolt));%线路的
 % 处理线路电阻或电抗为0的情况，即消除NaN
 zerosRXInd=union(find(abs(lineR)<1e-5),find(abs(lineX)<1e-5));
 dLPij_dVi=dLPij_dVi(setdiff( 1:size(dLPij_dVi,1),zerosRXInd),:);
 dLPij_dVj=dLPij_dVj(setdiff( 1:size(dLPij_dVj,1),zerosRXInd),:);
-dLQij_dVi=sparse(1:length(lineI),lineJ, ...
+dLPij_dThetai=dLPij_dThetai(setdiff( 1:size(dLPij_dThetai,1),zerosRXInd),:);
-    -SEVolt(lineJ).*( ...
+dLPij_dThetaj=dLPij_dThetaj(setdiff( 1:size(dLPij_dThetaj,1),zerosRXInd),:);
-    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) -lineX.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
-    )... 
-    -2*(lineB+lineB2).*SEVolt(lineI) ...
-    ,length(lineI),length(mVolt));%Ïß·µÄ
-dLQij_dVj=sparse(1:length(lineI),lineJ, ...
-    -SEVolt(lineI).*( ...
-    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineX.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
-    ) ...
-    ,length(lineI),length(mVolt));%Ïß·µÄ
 dLQij_dVi=dLQij_dVi(setdiff( 1:size(dLQij_dVi,1),zerosRXInd),:);
 dLQij_dVj=dLQij_dVj(setdiff( 1:size(dLQij_dVj,1),zerosRXInd),:);
+dLQij_dThetai=dLQij_dThetai(setdiff( 1:size(dLQij_dThetai,1),zerosRXInd),:);
+dLQij_dThetaj=dLQij_dThetaj(setdiff( 1:size(dLQij_dThetaj,1),zerosRXInd),:);
 %% 进入迭代
-H=[dV_dV ;dV_dTyta]';
+H=[dV_dV ,dV_dTyta;
+  dLPij_dVi+dLPij_dVj, dLPij_dThetai+dLPij_dThetaj;
+  dLQij_dVi+dLQij_dVj, dLQij_dThetai+dLQij_dThetaj];%jacobi
+% H=[dV_dV];
 h=SEVolt;
 z=mVolt;
 W=sparse(1:length(mVolt),1:length(mVolt),1/sigma.^2,length(mVolt),length(mVolt));
 G=H'*W*H;
 g=-H'*W*(z-h);
 % 平衡节点相角恒定;
-G(length(mVolt)+Balance,:)=0;
+% G(length(mVolt)+Balance,:)=0;
-G(:,length(mVolt)+Balance)=0;
+% G(:,length(mVolt)+Balance)=0;
-G=G+sparse(length(mVolt)+Balance,length(mVolt)+Balance,1,length(mVolt)*2,length(mVolt)*2);
+% G=G+sparse(length(mVolt)+Balance,length(mVolt)+Balance,1,length(mVolt)*2,length(mVolt)*2);
-g(length(mVolt)+Balance)=0;
+% g(length(mVolt)+Balance)=0;
-dX=-g\G;
+dX=G\-g;
+maxD=max(abs(dX))
 % 更新变量
-SEVolt=SEVolt+
+SEVolt=SEVolt+dX(1:length(mVolt));
+Iteration=Iteration+1;
+SEVAngel=SEVAngel+dX(length(mVolt)+1:end);
+end
 %% 输出结果
+fprintf('迭代%d次\n',Iteration);
+h=SEVolt;
+fval=full((z-h)'*W*(z-h));
 fprintf('目标函数为:%f\n',fval);
 fprintf('相对误差\n');
 (abs(rVolt-double(SEVolt)))./(rVolt)
 MaxDeviation(rVolt,SEVolt,rVAngel,SEVAngel)
-plotAndComparison( rVolt,rVAngel,SEVolt,SEVAngel )
+plotAndComparison( rVolt,rVAngel,SEVolt,SEVAngel )
-seOpti.fun(x);

run.m

@@ -31,17 +31,21 @@ lineJ=newwordParameter.line.lineJ;
 lineR=newwordParameter.line.lineR;
 lineX=newwordParameter.line.lineX;
 lineB2=newwordParameter.line.lineB2;
-lineG=1./lineR;
+lineG=real(1./(lineR+1j*lineX));
-lineB=1./lineX;
+lineB=imag(1./(lineR+1j*lineX));
 cmpBranchI=(cmpV(lineI)-cmpV(lineJ))./(lineR+1j*lineX);%复数支路电流
 rBranchI=abs(cmpBranchI);% 支路电流幅值
 mBranchI=rBranchI.*(normrnd(0,sigma,length(rBranchI),1)+1);%支路电流量测量
 %% 支路功率
-rBranchP=real((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI));
+rBranchP=real((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI))+real(cmpV(lineI) .*conj(1j*lineB2.*cmpV(lineI) ) );
-rBranchP=rBranchP(abs(rBranchP)>1e-5);
+rV=abs(cmpV);
+rA=angle(cmpV);
+a=rV(lineI).^2.*(lineG)-rV(lineI).*rV(lineJ).*(lineG.*cos(rA(lineI)-rA(lineJ) )+lineB.*sin(rA(lineI)-rA(lineJ) ) );
+b=-rV(lineI).^2.*(lineB+lineB2)-rV(lineI).*rV(lineJ).*(lineG.*sin(rA(lineI)-rA(lineJ) )-lineB.*cos(rA(lineI)-rA(lineJ) ) );
+% rBranchP=rBranchP(abs(rBranchP)>1e-5);
 mBranchP=rBranchP.*(normrnd(0,sigma,length(rBranchP),1)+1);%支路功率量测量
-rBranchQ=imag((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI));
+rBranchQ=imag((cmpV(lineI)-cmpV(lineJ)).*conj(cmpBranchI))+imag(cmpV(lineI) .*conj(1j*lineB2.*cmpV(lineI) ) );
-rBranchQ=rBranchQ(abs(rBranchQ)>1e-5);
+% rBranchQ=rBranchQ(abs(rBranchQ)>1e-5);
 mBranchQ=rBranchQ.*(normrnd(0,sigma,length(rBranchQ),1)+1);%支路功率量测量
 %% 注入功率
 rPD=PD;
@@ -104,34 +108,61 @@ dV_dV=sparse(1:length(mVolt),1:length(mVolt),1,length(mVolt),length(mVolt));%
 dV_dTyta=sparse(length(mVolt),length(mVolt));
 dLPij_dVi=sparse(1:length(lineI),lineJ, ...
     -SEVolt(lineJ).*( ...
-    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineX.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
     )... 
     +2*(lineG).*SEVolt(lineI) ...
     ,length(lineI),length(mVolt));%线路的
 dLPij_dVj=sparse(1:length(lineI),lineJ, ...
     -SEVolt(lineI).*( ...
-    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineX.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    ) ...
+    ,length(lineI),length(mVolt));%线路的
+dLPij_dThetai=sparse(1:length(lineI),lineJ, ...
+    SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
+    )... 
+    ,length(lineI),length(mVolt));%线路的
+dLPij_dThetaj=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
+    )... 
+    ,length(lineI),length(mVolt));%线路的
+
+dLQij_dVi=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineJ).*( ...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    )... 
+    -2*(lineB+lineB2).*SEVolt(lineI) ...
+    ,length(lineI),length(mVolt));%线路的
+dLQij_dVj=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineI).*( ...
+    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineB.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
+    ) ...
+    ,length(lineI),length(mVolt));%线路的
+dLQij_dThetai=sparse(1:length(lineI),lineJ, ...
+    -SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
+    ) ...
+    ,length(lineI),length(mVolt));%线路的
+dLQij_dThetaj=sparse(1:length(lineI),lineJ, ...
+    SEVolt(lineI).*SEVolt(lineJ).*( ...
+    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) +lineB.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
     ) ...
     ,length(lineI),length(mVolt));%线路的
 % 处理线路电阻或电抗为0的情况，即消除NaN
 zerosRXInd=union(find(abs(lineR)<1e-5),find(abs(lineX)<1e-5));
 dLPij_dVi=dLPij_dVi(setdiff( 1:size(dLPij_dVi,1),zerosRXInd),:);
 dLPij_dVj=dLPij_dVj(setdiff( 1:size(dLPij_dVj,1),zerosRXInd),:);
-dLQij_dVi=sparse(1:length(lineI),lineJ, ...
+dLPij_dThetai=dLPij_dThetai(setdiff( 1:size(dLPij_dThetai,1),zerosRXInd),:);
-    -SEVolt(lineJ).*( ...
+dLPij_dThetaj=dLPij_dThetaj(setdiff( 1:size(dLPij_dThetaj,1),zerosRXInd),:);
-    lineG.*cos(SEVAngel(lineI)-SEVAngel(lineJ)) -lineX.*sin(SEVAngel(lineI)-SEVAngel(lineJ))...
-    )... 
-    -2*(lineB+lineB2).*SEVolt(lineI) ...
-    ,length(lineI),length(mVolt));%Ïß·µÄ
-dLQij_dVj=sparse(1:length(lineI),lineJ, ...
-    -SEVolt(lineI).*( ...
-    lineG.*sin(SEVAngel(lineI)-SEVAngel(lineJ)) -lineX.*cos(SEVAngel(lineI)-SEVAngel(lineJ))...
-    ) ...
-    ,length(lineI),length(mVolt));%Ïß·µÄ
 dLQij_dVi=dLQij_dVi(setdiff( 1:size(dLQij_dVi,1),zerosRXInd),:);
 dLQij_dVj=dLQij_dVj(setdiff( 1:size(dLQij_dVj,1),zerosRXInd),:);
+dLQij_dThetai=dLQij_dThetai(setdiff( 1:size(dLQij_dThetai,1),zerosRXInd),:);
+dLQij_dThetaj=dLQij_dThetaj(setdiff( 1:size(dLQij_dThetaj,1),zerosRXInd),:);
 %% 进入迭代
-H=[dV_dV ,dV_dTyta];
+H=[dV_dV ,dV_dTyta;
+  dLPij_dVi+dLPij_dVj, dLPij_dThetai+dLPij_dThetaj;
+  dLQij_dVi+dLQij_dVj, dLQij_dThetai+dLQij_dThetaj];%jacobi
 % H=[dV_dV];
 h=SEVolt;
 z=mVolt;

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-. 91.
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-. 
-. Defining document command \newfontfeature with arg. spec. 'mm' on line 118.
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-. 
-. Defining document command \newICUfeature with arg. spec. 'mmm' on line 132.
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-. 
-. Defining document command \aliasfontfeature with arg. spec. 'mm' on line
-. 159.
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-. 
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-. line 164.
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-. 
-. Defining document command \newfontscript with arg. spec. 'mm' on line 169.
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-. 
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-. 
-. Defining document command \DeclareFontsExtensions with arg. spec. 'm' on
-. line 211.
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- -> C:/Windows/Fonts/simsun.ttc
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- -> C:/Windows/Fonts/simsun.ttc
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- -> C:/Windows/Fonts/simsun.ttc
-.................................................
-. fontspec info: "no-font-shape"
-. 
-. Could not resolve font 宋体/I (it probably doesn't exist).
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-Requested font "é»^^91ä½^^93/ICU" at 10.0pt
- -> C:/Windows/Fonts/simhei.ttf
-Requested font "é»^^91ä½^^93/I/ICU" at 10.0pt
- -> C:/Windows/Fonts/simhei.ttf
-.................................................
-. fontspec info: "no-font-shape"
-. 
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-.................................................
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-. 
-. Font family '宋体(0)' created for font '宋体' with options [BoldFont=黑ä
-½“].
-. 
-. This font family consists of the following shapes:
-. 
-. * 'normal' with NFSS spec.:
-. <->"宋体/ICU:script=latn;language=DFLT;"
-. 
-. * 'bold' with NFSS spec.:
-. <->"黑体/ICU:script=latn;language=DFLT;"
-.................................................
-Requested font "å®^^8bä½^^93/ICU:script=latn;language=DFLT;" at 10.0pt
- -> C:/Windows/Fonts/simsun.ttc
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-File: t3cmr.fd 2001/12/31 TIPA font definitions
-)
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-
-*** you should *not* be loading the inputenc package
-*** XeTeX expects the source to be in UTF8 encoding
-*** some features of other encodings may conflict, resulting in poor output.
-.................................................
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-. 
-. Adjusting the maths setup (use [no-math] to avoid this).
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-11.
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-(D:\Matlab\StateEstimation-self-derivatives\å…¬å¼<C3A5>\å…¬å¼<C3A5>.aux) ) 
-Here is how much of TeX's memory you used:
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- 297453 string characters out of 3191503
- 292163 words of memory out of 3000000
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- 44i,8n,31p,604b,132s stack positions out of 5000i,500n,10000p,200000b,50000s
-
-Output written on å…¬å¼<C3A5>.pdf (1 page).

公式/公式.tex

@@ -9,13 +9,146 @@
 \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}
+%\begin{equation}
+%\begin{aligned}
+%\frac{\partial P_{ij}}{\partial V_i}
+%&=
+%-V_{ij}(g_{ij}cos\theta_{ij}+b_{ij}sin\theta_{ij})+
+%2(g_{ij}+g_{si})V_i
+%\end{aligned}
+%\end{equation}
+线路功率（不考虑接地导纳）
+\newline
 \begin{equation}
 \begin{aligned}
-\frac{\partial P_{ij}}{\partial V_i}
+\dot{I}_{12}&=(V_1 e^{j\theta_1} -  V_2 e^{j\theta_2} )
-&=
+(G+jB) \\
--V_{ij}(g_{ij}cos\theta_{ij}+b_{ij}sin\theta_{ij})+
+&=[V_1 (cos\theta_1+jsin\theta_1) - V_2 (cos\theta_2+jsin\theta_2)](G+jB) \\
-2(g_{ij}+g_{si})V_i
+&=(V_1 cos\theta_1+ jV_1 sin\theta_1- V_2 cos \theta_2 -j V_2 sin \theta_2)(G+jB) \\
+&=(V_1 cos \theta_1 -V_2 cos \theta_2)G + j(V_1 sin \theta_1- V_2 sin \theta_2)G
+\\&+j( V_1 cos \theta_1 -V_2 cos \theta_2)B + j(jV_1 sin \theta_1-jV_2 sin \theta_2)B \\
+&=G(V_1 cos \theta_1- V_2 cos \theta_2) - B(V_1 sin \theta_1 -V_2 sin \theta_2) \\
+&+jB(V_1 cos \theta_1 - V_2 cos \theta_2) + jG(V_1 sin \theta_1 - V_2 sin \theta_2) \\
+&= V_1( G cos \theta_1-B sin \theta_1) -V_2 (G cos \theta_2 - B sin \theta_2) \\
+&+jV_1(Bcos \theta_1+Gsin\theta_1)-jV_2(Bcos\theta_2+Gsin\theta_2)
 \end{aligned}
 \end{equation}
-
+电流的共轭复数
-\end{document}
+\begin{equation}
+\begin{aligned}
+\dot{I}^*_{12}&=V_1( G cos \theta_1-B sin \theta_1) -V_2 (G cos \theta_2 - B sin \theta_2) \\
+&-jV_1(Bcos \theta_1+Gsin\theta_1)+jV_2(Bcos\theta_2+Gsin\theta_2)
+\end{aligned}
+\end{equation}
+线路复功率
+\begin{equation}
+\begin{aligned}
+\dot{S}_{12}&=(V_1 e^{j\theta_1} -  V_2 e^{j\theta_2} ) 
+\dot{I}^*_{12} \\
+&=[V_1 (cos\theta_1+jsin\theta_1) - V_2 (cos\theta_2+jsin\theta_2)]\dot{I}^*_{12} \\
+&=[V_1 (cos\theta_1+jsin\theta_1) - V_2 (cos\theta_2+jsin\theta_2)] \\
+&[V_1( G cos \theta_1-B sin \theta_1) -V_2 (G cos \theta_2 - B sin \theta_2) 
+-jV_1(Bcos \theta_1+Gsin\theta_1)+jV_2(Bcos\theta_2+Gsin\theta_2)]\\
+&=V_1( G cos \theta_1-B sin \theta_1) 
+[V_1( G cos \theta_1-B sin \theta_1) -V_2 (G cos \theta_2 - B sin \theta_2)]\\
+&-V_1( G cos \theta_1-B sin \theta_1)
+[jV_1(Bcos \theta_1+Gsin\theta_1)-jV_2(Bcos\theta_2+Gsin\theta_2)] \\
+&- V_2 (cos\theta_2+jsin\theta_2)
+[V_1( G cos \theta_1-B sin \theta_1) -V_2 (G cos \theta_2 - B sin \theta_2)]\\
+&V_2 (cos\theta_2+jsin\theta_2)
+[jV_1(Bcos \theta_1+Gsin\theta_1)-jV_2(Bcos\theta_2+Gsin\theta_2)]
+\end{aligned}
+\end{equation}
+另一种推导方式
+\begin{equation}
+\begin{aligned}
+\dot{I}_{12}=(V_1 e^{j\theta_1} -  V_2 e^{j\theta_2} )
+(G+jB)
+\end{aligned}
+\end{equation}
+电流的共轭复数
+\begin{equation}
+\begin{aligned}
+\dot{I}^*_{12}=(V_1 e^{-j\theta_1} -  V_2 e^{-j\theta_2} )
+(G-jB)
+\end{aligned}
+\end{equation}
+线路复功率
+\begin{equation}
+\begin{aligned}
+\dot{S}_{12}&=(V_1 e^{j\theta_1} -  V_2 e^{j\theta_2} )\dot{I}^*_{12} \\
+&=(V_1 e^{j\theta_1} -  V_2 e^{j\theta_2} )
+(V_1 e^{-j\theta_1} -  V_2 e^{-j\theta_2} )
+(G-jB) \\
+&=(V_1 e^{j\theta_1}V_1 e^{-j\theta_1} - V_1 e^{j\theta_1}V_2 e^{-j\theta_2} 
+-  V_2 e^{j\theta_2}V_1 e^{-j\theta_1}+  V_2 e^{j\theta_2}V_2 e^{-j\theta_2}
+)
+(G-jB) \\
+&=(V_1^2+V_2^2-V_1V_2e^{j(\theta_1-\theta_2)} -V_1V_2e^{j(\theta_2-\theta_1)} )
+(G-jB) \\
+&=[V_1^2+V_2^2
+-V_1V_2(e^{j(\theta_1-\theta_2)}+e^{j(\theta_2-\theta_1)})
+](G-jB) \\
+&=\{V_1^2+V_2^2-V_1V_2[cos(\theta_1-\theta_2)+jsin(\theta_1-\theta_2)
++cos(\theta_2-\theta_1)+jsin(\theta_2-\theta_1)]\}(G-jB) \\
+&=\{V_1^2+V_2^2-V_1V_2[cos(\theta_1-\theta_2)
++cos(\theta_1-\theta_2)]\}(G-jB) \\
+&=[V_1^2+V_2^2-2V_1V_2cos(\theta_1-\theta_2)](G-jB) \\
+&=(V_1^2+V_2^2)G-j(V_1^2+V_2^2)B-j2V_1V_2cos(\theta_1-\theta_2)G-j2V_1V_2cos(\theta_1-\theta_2)(-jB) \\
+&=(V_1^2+V_2^2)G-2V_1V_2cos(\theta_1-\theta_2)B
+-j(V_1^2+V_2^2)B-j2V_1V_2cos(\theta_1-\theta_2)G
+\end{aligned}
+\end{equation}
+\begin{equation}
+\begin{aligned}
+P_{12}=(V_1^2+V_2^2)G+2V_1V_2sin(\theta_1-\theta_2)B
+\end{aligned}
+\end{equation}
+\begin{equation}
+\begin{aligned}
+Q_{12}=-(V_1^2+V_2^2)B+2V_1V_2sin(\theta_1-\theta_2)G
+\end{aligned}
+\end{equation}
+考虑接地支路,接地电流为
+\begin{equation}
+\label{接地电流}
+\dot{I}_{1}=V_1e^{j\theta_1}(G_d+jB_d)
+\end{equation}
+(\ref{接地电流})共轭为
+\begin{equation}
+\dot{I}_{1}^{*}=V_1e^{-j\theta_1}(G_d-jB_d)
+\end{equation}
+%Sd
+\begin{equation}
+\begin{aligned}
+\dot{S}_{1}&=V_1e^{j\theta_1}\dot{I}_{1}^{*} \\
+&=V_1e^{j\theta_1} 
+V_1e^{-j\theta_1}(G_d-jB_d) \\
+&=V_1^2(G_d-jB_d)
+\end{aligned}
+\end{equation}
+因而
+\begin{equation}
+\begin{aligned}
+P_{1}&=V_1^2 G_d \\
+\end{aligned}
+\end{equation}
+\begin{equation}
+\begin{aligned}
+Q_{1}&=-V_1^2 B_d \\
+\end{aligned}
+\end{equation}
+考虑节点支路后的注入电流
+\begin{equation}
+\begin{aligned}
+\tilde{P}_{12}&=V_1^2 G_d 
++(V_1^2+V_2^2)G+2V_1V_2sin(\theta_1-\theta_2)B
+\end{aligned}
+\end{equation}
+\begin{equation}
+\begin{aligned}
+\tilde{Q}_{12}&=-V_1^2 B_d 
+-(V_1^2+V_2^2)B+2V_1V_2sin(\theta_1-\theta_2)G
+\end{aligned}
+\end{equation}
+\end{document}

公式/公式.tex.bak

@@ -1,21 +0,0 @@
-\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}
-\begin{equation}
-\begin{aligned}
-\frac{\partial P_{ij}}{\partial V_i}
-&=
--V_{ij}(g_{ij}cos\theta_{ij}+b_{ij}sin\theta_{ij})+
-2(g_{ij}+g_{si})V_i
-\end{aligned}
-\end{equation}
-
-\end{document}