263 lines
14 KiB
Matlab
263 lines
14 KiB
Matlab
function [HSB,dhx,dgx,dgxT,GIJ,BIJ,L1,U1,LZ,UW,transG,transB,dT_Zt,mut,mub]=snd_formHSB(nodeNum,balNode,opfGoal,lineNum,transNum,lineI,lineJ,transI,transJ,...
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T,transG,transB,pgNode,pvNode,G,B,e,f,pgNum,pvNum,pgvNum,xNum,a,m,r,l,u,z,w,y,kt0,mut,Zt,oneTrans,oneCap,Zb,mub,q0,capBi,capNum,capI,tij,Bci,N1,H1,diagE,...
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diagF,b0,balNum)
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%程序功能：连续化处理过程，形成修正方程系数矩阵
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%编写时间：2010年10月
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%% 修正陡度参数μt
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dicVar=[tij;Bci];
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tmp1=dicVar.*(1-dicVar);
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if sum(tmp1)==0
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tmp2=0;
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else
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tmp2=tmp1/sum(tmp1);
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end
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tmp3=ones(transNum+capNum,1)./(ones(transNum+capNum,1)+tmp2); %计算每一步迭代中陡度参数的修正量
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bk=tmp3*b0;
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mut_bk=bk(1:transNum);
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mub_bk=bk(transNum+1:end);
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smallest=0.0000000001; %当陡度参数已经非常接近0时，让其以0.9的速率减小，保证其还能继续下降，
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large_mut=find(mut>smallest); %不过只要选择好适当的陡度参数初值和修正参数，陡度参数达到这个精度时基本上都已经收敛
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small_mut=find(mut<=smallest);
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mut(large_mut)=mut_bk(large_mut).*mut(large_mut);
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mut(small_mut)=0.9*mut(small_mut);
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large_mub=find(mub>smallest);
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small_mub=find(mub<=smallest);
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mub(large_mub)=mub_bk(large_mub).*mub(large_mub);
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mub(small_mub)=0.9*mub(small_mub);
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dT_Zt=oneTrans;
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d2T_Zt2=oneTrans;
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n_zt= Zt<0; %为了避免指数运算时出现的计算机溢出问题，在计算Sigmoid函数及其一、二阶导数时，
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p_zt= Zt>=0; %区分Zt的正负取值情况，采用不同的表达形式（两种形式完全等价），这是算法得以实现的关键。
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n_Ezu=exp(Zt(n_zt)./mut(n_zt));
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dT_Zt(n_zt)=kt0(n_zt).*(n_Ezu./(mut(n_zt).*(oneTrans(n_zt)+n_Ezu).^2)); %zt为负时，T对zt求一次偏导
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d2T_Zt2(n_zt)=(dT_Zt(n_zt)./mut(n_zt)).*(oneTrans(n_zt)-2*n_Ezu./(oneTrans(n_zt)+n_Ezu));%zt为负时，T对zt求二次偏导
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p_Ezu=exp(-Zt(p_zt)./mut(p_zt));
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dT_Zt(p_zt)=kt0(p_zt).*(p_Ezu./(mut(p_zt).*(oneTrans(p_zt)+p_Ezu).^2)); %zt为正时，T对zt求一次偏导
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d2T_Zt2(p_zt)=(dT_Zt(p_zt)./mut(p_zt)).*(oneTrans(p_zt)-2./(oneTrans(p_zt)+p_Ezu)); %zt为正时，T对zt求二次偏导
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transGmut=transG.*dT_Zt;
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transBmut=transB.*dT_Zt;
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dcapK_Zb=oneCap;
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d2capK_Zb2=oneCap;
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n_zb= Zb<0; %对于Zb，同样要区分其正负取值进行计算
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p_zb= Zb>=0;
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n_Ezbu=exp(Zb(n_zb)./mub(n_zb));
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dcapK_Zb(n_zb)=q0.*(n_Ezbu./(mub(n_zb).*(oneCap(n_zb)+n_Ezbu).^2)); %zb为负时，capK对zb求一次偏导
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d2capK_Zb2(n_zb)=(dcapK_Zb(n_zb)./mub(n_zb)).*(oneCap(n_zb)-2*n_Ezbu./(oneCap(n_zb)+n_Ezbu));%zb为负时，capK对zb求二次偏导
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p_Ezbu=exp(-Zb(p_zb)./mub(p_zb));
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dcapK_Zb(p_zb)=q0.*(p_Ezbu./(mub(p_zb).*(oneCap(p_zb)+p_Ezbu).^2)); %zb为正时，capK对zb求一次偏导
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d2capK_Zb2(p_zb)=(dcapK_Zb(p_zb)./mub(p_zb)).*(oneCap(p_zb)-2./(oneCap(p_zb)+p_Ezbu));%zb为正时，capK对zb求二次偏导
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capBi_dBci=capBi.*dcapK_Zb;
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%% 形成等式约束雅可比矩阵
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ef1=e(transI).*e(transJ)+f(transI).*f(transJ);
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ef2=e(transI).*f(transJ)-e(transJ).*f(transI);
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ef3=e(transI).*e(transI)+f(transI).*f(transI);
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ef1G=ef1.*transGmut;
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ef1B=ef1.*transBmut;
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ef2G=ef2.*transGmut;
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ef2B=ef2.*transBmut;
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dhPTi=-ef1G+ef2B+2*ef3.*transGmut.*T;
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dhPTj=-ef1G-ef2B;
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dhPT=sparse(1:transNum,transI,dhPTi,transNum,nodeNum)+sparse(1:transNum,transJ,dhPTj,transNum,nodeNum);%有功平衡方程P对变压器变比求偏导
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dhQTi=ef1B+ef2G-2*ef3.*transBmut.*T;
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dhQTj=ef1B-ef2G;
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dhQT=sparse(1:transNum,transI,dhQTi,transNum,nodeNum)+sparse(1:transNum,transJ,dhQTj,transNum,nodeNum);%无功平衡方程Q对变压器变比求偏导
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dhP=sparse(1:pgNum,pgNode,ones(pgNum,1),pgNum,m);
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dhQ1=sparse(1:pvNum,pvNode,ones(pvNum,1),pvNum,nodeNum);
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dhQ=[sparse(pvNum,nodeNum) dhQ1];
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dhcapK1=sparse(1:capNum,capI,(e(capI).^2+f(capI).^2).*capBi_dBci,capNum,nodeNum);%功率平衡方程对电容电抗器组数求偏导
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dhcapK=[sparse(capNum,nodeNum) dhcapK1];
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%功率方程雅可比矩阵
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deG=diagE*G;
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dfB=diagF*B;
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deB=diagE*B;
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dfG=diagF*G;
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dH1=sparse(1:nodeNum,1:nodeNum,H1,nodeNum,nodeNum);
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dN1=sparse(1:nodeNum,1:nodeNum,N1,nodeNum,nodeNum);
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Hij=-dH1-deG-dfB;
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Nij=-dN1+deB-dfG;
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Jij=dN1-dfG+deB;
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Lij=-dH1+dfB+deG;
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dhJacb=[Hij Nij;Jij Lij]'; %合并形成功率方程雅可比矩阵
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dhx=[dhPT dhQT;dhP;dhQ;dhcapK;dhJacb]; %合并形成等式约束雅可比矩阵h(x)
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%% 形成不等式约束雅可比矩阵▽g(x)
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gtransNum=transNum; %在原来变比和电容电抗器上下限不等式约束的模块，替换为Zt、Zb的上下限不等式约束
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gcapNum=capNum;
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dgT1=sparse(1:transNum,1:transNum,ones(transNum,1),transNum,r-lineNum);
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dglT=(e(transI).^2+f(transI).^2-e(transI).*e(transJ)-f(transI).*f(transJ))...
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.*transGmut+(e(transI).*f(transJ)-e(transJ).*f(transI)).*transBmut;
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dgT2=sparse(1:transNum,lineNum-transNum+1:lineNum,dglT,transNum,lineNum);
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dgT=[dgT1 dgT2]; %所有不等式对Zt求导的模块
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dgP=sparse(1:pgNum,gtransNum+1:gtransNum+pgNum,ones(pgNum,1),pgNum,r); %所有不等式约束对P求导的模块
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dgQ=sparse(1:pvNum,gtransNum+pgNum+1:gtransNum+pgvNum,ones(pvNum,1),pvNum,r);%所有不等式约束对Q求导的模块
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dgcapK=sparse(1:capNum,transNum+pgvNum+1:transNum+pgvNum+capNum,ones(capNum,1),capNum,r);%所有不等式约束对Zb求导的模块
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dgve=2*diagE;
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dgvf=2*diagF;
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sizeG=size(G);
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GIJ=G(sub2ind(sizeG,lineI,lineJ));
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BIJ=B(sub2ind(sizeG,lineI,lineJ));
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dgleI=(2*e(lineI)-e(lineJ)).*GIJ+f(lineJ).*BIJ;
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dgleJ=-e(lineI).*GIJ-f(lineI).*BIJ;
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dglfI=(2*f(lineI)-f(lineJ)).*GIJ-e(lineJ).*BIJ;
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dglfJ=-f(lineI).*GIJ+e(lineI).*BIJ;
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dgle=sparse(lineI,(1:lineNum),dgleI,nodeNum,lineNum)+sparse(lineJ,(1:lineNum),dgleJ,nodeNum,lineNum);
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dglf=sparse(lineI,(1:lineNum),dglfI,nodeNum,lineNum)+sparse(lineJ,(1:lineNum),dglfJ,nodeNum,lineNum);
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dg0=sparse(m,gtransNum+pgvNum+gcapNum);
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dgvl=[dgve dgle;dgvf dglf];
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dgef=[dg0 dgvl]; %所有不等式约束对电压实部、虚部求偏导的模块
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dgx=[dgT;dgP;dgQ;dgcapK;dgef]; %不等式约束雅克比矩阵
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%% 形成对角阵
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Z=sparse(1:r,1:r,z,r,r);
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W=sparse(1:r,1:r,w,r,r);
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L1=sparse(1:r,1:r,1./l,r,r);
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U1=sparse(1:r,1:r,1./u,r,r);
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LZ=L1*Z;
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UW=U1*W;
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%% 形成海森伯矩阵
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%目标函数的海森伯矩阵▽2f(x)
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if(opfGoal==2) %目标为有功出力最小
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HSdfx=sparse(xNum,xNum);
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else if(opfGoal==1)
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HSdfx=sparse(transNum+1:transNum+pgNum,transNum+1:transNum+pgNum,2*a(pgNode),xNum,xNum);%仅目标函数对PG求偏导处的对角元为非零元素，其他都为0(目标为发电耗量最小)
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else if(opfGoal==3)
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d2ftei=-2*transGmut.*(e(transI)-e(transJ));
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d2ftej=-d2ftei;
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d2ftfi=-2*transGmut.*(f(transI)-f(transJ));
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d2ftfj=-d2ftfi;
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d2fte=sparse(1:transNum,transI,d2ftei,transNum,nodeNum)+sparse(1:transNum,transJ,d2ftej,transNum,nodeNum);
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d2ftf=sparse(1:transNum,transI,d2ftfi,transNum,nodeNum)+sparse(1:transNum,transJ,d2ftfj,transNum,nodeNum);
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d2fT2=-transG.*d2T_Zt2.*((f(transI)-f(transJ)).^2+(e(transI)-e(transJ)).^2);
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d2fT0=sparse(1:transNum,1:transNum,d2fT2,transNum,transNum+pgvNum+capNum);
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d2fT=[d2fT0 d2fte d2ftf];
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d2fPGQR=sparse(pgvNum,xNum);
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d2fcapK=sparse(capNum,xNum);
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d2fee=2*(G-sparse(1:nodeNum,1:nodeNum,sum(G,2),nodeNum,nodeNum));
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d2fff=d2fee;
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zero_nodeNum=sparse(nodeNum,nodeNum);
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d2f_ef_ef=[d2fee zero_nodeNum;zero_nodeNum d2fff];
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d2efT=[d2fte d2ftf]';
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d2f_ef=[d2efT sparse(2*nodeNum,pgvNum+capNum) d2f_ef_ef];
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HSdfx=[d2fT;d2fPGQR;d2fcapK;d2f_ef];
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else if(opfGoal==4)
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d2ftei=2*transBmut.*(e(transI)-e(transJ));
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d2ftej=-d2ftei;
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d2ftfi=2*transBmut.*(f(transI)-f(transJ));
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d2ftfj=-d2ftfi;
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d2fte=sparse(1:transNum,transI,d2ftei,transNum,nodeNum)+sparse(1:transNum,transJ,d2ftej,transNum,nodeNum);
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d2ftf=sparse(1:transNum,transI,d2ftfi,transNum,nodeNum)+sparse(1:transNum,transJ,d2ftfj,transNum,nodeNum);
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d2fT2=transB.*d2T_Zt2.*((f(transI)-f(transJ)).^2+(e(transI)-e(transJ)).^2);
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d2fT0=sparse(1:transNum,1:transNum,d2fT2,transNum,transNum+pgvNum+capNum);
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d2fT=[d2fT0 d2fte d2ftf];
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d2fPGQR=sparse(pgvNum,xNum);
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d2fcapK=sparse(capNum,xNum);
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d2fee=-2*(B-sparse(1:nodeNum,1:nodeNum,sum(G,2),nodeNum,nodeNum));
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d2fff=d2fee;
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zero_nodeNum=sparse(nodeNum,nodeNum);
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d2f_ef_ef=[d2fee zero_nodeNum;zero_nodeNum d2fff];
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d2efT=[d2fte d2ftf]';
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d2f_ef=[d2efT sparse(2*nodeNum,pgvNum+capNum) d2f_ef_ef];
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HSdfx=[d2fT;d2fPGQR;d2fcapK;d2f_ef];
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end
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end
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end
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end
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%等式约束海森伯矩阵与拉格朗日乘子y的乘积▽2h(x)*y
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y1=y(1:nodeNum,1);
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y2=y(nodeNum+1:2*nodeNum,1);
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ef1Gd2=ef1.*transG.*d2T_Zt2;
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ef1Bd2=ef1.*transB.*d2T_Zt2;
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ef2Gd2=ef2.*transG.*d2T_Zt2;
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ef2Bd2=ef2.*transB.*d2T_Zt2;
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d2P_T2=(-ef1Gd2+ef2Bd2+2*ef3.*transG.*(dT_Zt.^2+T.*d2T_Zt2)).*y1(transI)+(-ef1Gd2-ef2Bd2).*y1(transJ);
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d2Q_T2=(ef1Bd2+ef2Gd2-2*ef3.*transB.*(dT_Zt.^2+T.*d2T_Zt2)).*y2(transI)+(ef1Bd2-ef2Gd2).*y2(transJ);
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dhT2=d2P_T2+d2Q_T2; %功率方程对变压器Zt求二次偏导所得的列向量
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HSdhT=sparse(1:transNum,1:transNum,dhT2,transNum+pgvNum+capNum,transNum+pgvNum+capNum)+...
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sparse(transNum+pgvNum+1:transNum+pgvNum+capNum,transNum+pgvNum+1:transNum+pgvNum+capNum,...
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(e(capI).^2+f(capI).^2).*d2capK_Zb2.*y2(capI),transNum+pgvNum+capNum,transNum+pgvNum+capNum);
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eiG=e(transI).*transGmut;
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ejG=e(transJ).*transGmut;
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eiB=e(transI).*transBmut;
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ejB=e(transJ).*transBmut;
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fiG=f(transI).*transGmut;
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fjG=f(transJ).*transGmut;
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fiB=f(transI).*transBmut;
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fjB=f(transJ).*transBmut;
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ejG_fjB=ejG-fjB;
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ejGfjB=ejG+fjB;
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fjGejB=fjG+ejB;
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ejB_fjG=ejB-fjG;
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eiGfiB=eiG+fiB;
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eiG_fiB=eiG-fiB;
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eiB_fiG=eiB-fiG;
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fiGeiB=fiG+eiB;
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dP2_ei_t=(4*eiG.*T-ejG_fjB).*y1(transI)-ejGfjB.*y1(transJ);
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dP2_ej_t=-eiGfiB.*y1(transI)-eiG_fiB.*y1(transJ);
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dQ2_ei_t=(-4*eiB.*T+fjGejB).*y2(transI)+ejB_fjG.*y2(transJ);
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dQ2_ej_t=eiB_fiG.*y2(transI)+fiGeiB.*y2(transJ);
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HSdh_et=sparse(1:transNum,transI,dP2_ei_t+dQ2_ei_t,transNum,nodeNum)+sparse(1:transNum,transJ,dP2_ej_t+dQ2_ej_t,transNum,nodeNum);%功率方程先对Zt求偏导，再对电压实部求偏导模块
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dP2_fi_t=(4*fiG.*T-fjGejB).*y1(transI)+ejB_fjG.*y1(transJ);
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dP2_fj_t=eiB_fiG.*y1(transI)-fiGeiB.*y1(transJ);
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dQ2_fi_t=(-4*fiB.*T-ejG_fjB).*y2(transI)+ejGfjB.*y2(transJ);
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dQ2_fj_t=eiGfiB.*y2(transI)-eiG_fiB.*y2(transJ);
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HSdh_ft=sparse(1:transNum,transI,dP2_fi_t+dQ2_fi_t,transNum,nodeNum)+sparse(1:transNum,transJ,dP2_fj_t+dQ2_fj_t,transNum,nodeNum);%功率方程先对Zt求偏导，再对电压虚部求偏导模块
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HSdh_eft=[HSdh_et HSdh_ft];
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HSdh_tef=HSdh_eft';
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HSdhPQ=sparse(pgvNum,2*nodeNum); %功率方程先对PG、QR求偏导，再对其他变量求偏导的部分
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d2hcap_e=sparse(1:capNum,capI,2*e(capI).*capBi_dBci.*y2(capI),capNum,nodeNum); %功率方程先对Zb求偏导，再对e求偏导的部分
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d2hcap_f=sparse(1:capNum,capI,2*f(capI).*capBi_dBci.*y2(capI),capNum,nodeNum); %功率方程先对Zb求偏导，再对f求偏导的部分
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HSdhcap_ef=[d2hcap_e d2hcap_f];
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d2htPQ_ef=[HSdh_eft;HSdhPQ;HSdhcap_ef];
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HSdhtPQ=[HSdhT d2htPQ_ef];
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HSdhx1=sparse(m,pgvNum);
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%潮流方程海森矩阵
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diagy1=sparse(1:nodeNum,1:nodeNum,y1,nodeNum,nodeNum);
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diagy2=sparse(1:nodeNum,1:nodeNum,y2,nodeNum,nodeNum);
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HSdhey=-diagy1*G+diagy2*B+(-G*diagy1+B*diagy2);
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HSdhefy=diagy1*B+diagy2*G-(B*diagy1+G*diagy2);
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HSdhx2=[HSdhey HSdhefy;HSdhefy' HSdhey];
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HSdhx=[HSdhtPQ;HSdh_tef HSdhx1 HSdhcap_ef' HSdhx2];
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%形成不等式约束二阶偏导的矩阵▽2g(x)*(z+w)
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ZWcl=z(gtransNum+pgvNum+gcapNum+nodeNum+1:end)+w(gtransNum+pgvNum+gcapNum+nodeNum+1:end);
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ZWv=z(gtransNum+pgvNum+gcapNum+1:gtransNum+pgvNum+gcapNum+nodeNum)+w(gtransNum+pgvNum+gcapNum+1:gtransNum+pgvNum+gcapNum+nodeNum);
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dg2T=((e(transI).^2+f(transI).^2-e(transI).*e(transJ)-f(transI).*f(transJ)).*transG.*d2T_Zt2+...
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(e(transI).*f(transJ)-e(transJ).*f(transI)).*transB.*d2T_Zt2).*ZWcl(lineNum-transNum+1:end);
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HSdg2T=sparse(1:transNum,1:transNum,dg2T,transNum,transNum); %不等式对Zt求二次偏导模块
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dgeiT=((2*e(transI)-e(transJ)).*transGmut+f(transJ).*transBmut).*ZWcl(lineNum-transNum+1:end);
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dgejT=(-e(transI).*transGmut-f(transI).*transBmut).*ZWcl(lineNum-transNum+1:end);
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dgeT=sparse(1:transNum,transI,dgeiT,transNum,nodeNum)+sparse(1:transNum,transJ,dgejT,transNum,nodeNum);%不等式约束先对Zt求偏导，再对e求偏导的模块
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dgfiT=((2*f(transI)-f(transJ)).*transGmut-e(transJ).*transBmut).*ZWcl(lineNum-transNum+1:end);
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dgfjT=(-f(transI).*transGmut+e(transI).*transBmut).*ZWcl(lineNum-transNum+1:end);
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dgfT=sparse(1:transNum,transI,dgfiT,transNum,nodeNum)+sparse(1:transNum,transJ,dgfjT,transNum,nodeNum);%不等式约束先对Zt求偏导，再对f求偏导的模块
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HSdgefT=[dgeT dgfT];
|
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HSdgT=[HSdg2T sparse(transNum,pgvNum+capNum) HSdgefT];
|
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HSdgee=sparse(lineI,lineI,2*GIJ.*ZWcl,nodeNum,nodeNum)-sparse(lineI,lineJ,GIJ.*ZWcl,nodeNum,nodeNum)-...
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sparse(lineJ,lineI,GIJ.*ZWcl,nodeNum,nodeNum)+sparse(1:nodeNum,1:nodeNum,2*ZWv,nodeNum,nodeNum);
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HSdgff=HSdgee; %所有不等式约束对e、f求二次偏导模块
|
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HSdgef=sparse(lineI,lineJ,BIJ.*ZWcl,nodeNum,nodeNum)-sparse(lineJ,lineI,BIJ.*ZWcl,nodeNum,nodeNum);
|
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HSdgfe=HSdgef';
|
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HSdgx1=[HSdgee HSdgef;HSdgfe HSdgff];
|
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HSdgxtPQ_0=sparse(pgvNum+capNum,xNum);
|
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HSdgef_0=[HSdgefT' sparse(2*nodeNum,pgvNum+capNum)];
|
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HSdgx=[HSdgT;HSdgxtPQ_0;HSdgef_0 HSdgx1]; %合成不等式约束海森矩阵
|
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%求和形成H(.)矩阵.
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dgxT=dgx'; %求dgx的转置
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MULdgx=dgx*(UW-LZ)*dgxT;
|
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H=HSdhx+HSdgx-HSdfx+MULdgx; %求和形成H(.)矩阵.
|
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%% 根据以上所求的各个矩阵合成海森伯矩阵
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HSB0=sparse(m,m);
|
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HSB=[H dhx;dhx' HSB0];
|
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balN=transNum+pgvNum+capNum+nodeNum+balNode;
|
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HSB(balN,:)=0; %保持平衡节点电压虚部不变，始终为0，即将海森矩阵中平衡节点虚部所对应的行列元素置0，对角元素置1
|
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HSB(:,balN)=0;
|
||
HSB_balN=sub2ind(size(HSB),balN,balN);
|
||
HSB(HSB_balN)=ones(balNum,1);
|
||
balN1=transNum+pgvNum+capNum+balNode;
|
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HSB(balN1,:)=0; %保持平衡节点电压实部不变，始终为0，即将海森矩阵中平衡节点实部所对应的行列元素置0，对角元素置1
|
||
HSB(:,balN1)=0;
|
||
HSB_balN1=sub2ind(size(HSB),balN1,balN1);
|
||
HSB(HSB_balN1)=ones(balNum,1); |