1、电力网节点导纳矩阵计算例题与程序电力网节点导纳矩阵计算例题与程序例图1-1对于具有n个节点b条支路的有向图，它的关联矩阵为一个NB的矩阵Aa：Aa=aij若支路j与节点i相关，且箭头背离节点i，则aij=1，若箭头指向节点则aij=-1，若支路j与节点i无关，则aij=0，图1-1所示的有向图的关联矩阵为1 支路编号Aij=行编号从上到下为1 2 3 4 5节点编号（5为参考节点）去掉第5行即为独立节点的关联矩阵。以下介绍生成网络关联矩阵的M函数文件 ffm.m：% M FUNCTION ffm.m% Np is number of node point,Nb is number of bra
2、ches% nstart-the start point of branches ,nend - the end point,% A - network incidence matrix functionA=ffm(nstart,nend) global Np Nb n=length(nstart); A=zeros(Np,Nb); for i=1:n A(nstart(i),i)=1; A(nend(i),i)=-1; end 以例图1-1网络为例调用ffm.m文件求其关联矩阵运算以上程序可得关联矩阵 mmij如下：mm = -1 0 1 0 0 1 1 0 -1 0 0 -1 -1 0 -
3、1 0 0 0 1 0 0 0 0 0 1Mmij明显与Aij是相同的。1.2生成节点导纳矩阵程序： 由网络原始矩阵计算节点导纳矩阵公式 Y=AYs0At (1-1) Y -节点导纳矩阵 A-网络关联矩阵 At-A的转置矩阵 YS0-网络原始导纳矩阵 若网络各支路阻抗为 Zb=zb1,zb2,zbn 则ZS0= Ys0=Zs0-1 (1-2) Y=A Zs0-1At (1-3) 节点导纳计算程序以例1-1网络为例，在不计对地电容和变压器变比假定为1条件下，节点导纳矩阵计算程序如下： clearglobal Np Nb % Np is number of node point,Nb is num
4、ber of braches, Np=5;Nb=5;% nstart-the start point of branches ,nend - the end point,% mm - network incidence matrix nstart=2,2,1,4,5; nend=1,3,3,2,3; mm=ffm(nstart,nend);% zb1,the series impedances of transmission line % yb1,the series admittances of transmission line zb1r=0.04,0.08,0.1,0.0,0.0; zb
5、1i=0.25,0.30,0.35,0.015,0.03; zb1=zb1r+zb1i*j; yb1=zb1.(-1); yb=diag(conj(yb1); y=mm*yb*(mm);运算程序后可得节点导纳矩阵y= 1.3787 - 6.5417i -0.6240 + 3.9002i -0.7547 + 2.6415i 0 0 -0.6240 + 3.9002i 1.4539 -73.6789i -0.8299 + 3.1120i 0 +66.6667i 0 -0.7547 +2.6415i -0.8299+ 3.1120i 1.5846 -39.0869i 0 0 +33.3333i 0
6、0 +66.6667i 0 0 -66.6667i 0 0 0 0 +33.3333i 0 0 -33.3333i线路对地并联电容的处理例图1-1网络中，线路，两端对地电容导纳均为j0.25,节点1,2,3自导纳相应增加j0.25,j0.5,j0.25。程序处理如下：% yb0 ,the shunt admittances of transmission line yb0i=0.25,0.25,0.0,0.0,0.0; yb0=0+yb0i*j; y0=diag(conj(yb0); yg=mm*y0*(mm); yn=diag(diag(yg); yy=yn-yg; y=y+yn; 程序中y
7、b0为各线路对地电容导纳，y0为原始导纳矩阵，yn对角矩阵对应各节点自导纳增加值，yy相应各支路两端对地并联导纳，用于计算支路电流。y为计及对地电容后的节点导纳矩阵。程序运算结果yn = 0 + 0.2500i 0 0 0 0 0 0 + 0.5000i 0 0 0 0 0 0 + 0.2500i 0 0 0 0 0 0 0 0 0 0 0 0 yy = 0 0 + 0.2500i 0 0 0 0 + 0.2500i 0 0 + 0.2500i 0 0 0 0 + 0.2500i 0 0 0 0 0 0 0 0 0 0 0 0 0 y =1.3787 - 6.2917i -0.6240 + 3
8、.9002i -0.7547 + 2.6415i 0 0 -0.6240 + 3.9002i 1.4539 -73.1789i -0.8299 + 3.1120i 0 +66.6667i 0 -0.7547 + 2.6415i -0.8299 + 3.1120i 1.5846 -38.8369i 0 0 +33.3333i0 0 +66.6667i 0 0 -66.6667i 0 0 0 0 +33.3333i 0 0 -33.3333i 非标准变压器支路的处理 当支路有变压器并且标么变比不为1时，可将变压器化作形等值电路，如图1-2所示。图中理想变压器变比为k:1,接入节点i,高压侧经漏抗Z
9、T与节点j相连。 图1-2变压器等值电路等值电路图中YT=1/ZT。i侧自导变为yii=kyt+k(k-1)yt=k2yt，改变量为yii=(k2-1)yt; I j间互导变为yij=-kyt,yij=-kyt+yt=-(k-1)yt。j侧自导变为yjj=kyt+(1-k)yt=yt,与K=1时相同;变压器变比不为1时修正导纳矩阵的M函数文件 fdt1.m% M FUNCTION fdt1.m% the transformer turns ratio,with off-nominal tap-setting,fig 1-2 % bt two-dimensional array ,line on
10、e is point i,line two is point j% p is the transformer turns ratio,k is number of transformers functionA,B=fdt1(a,b) global bt p m k for i=1:m for j=1:m A(i,j)=a(i,j); B(i,j)=b(i,j); end end for i1=1:k k1=bt(1,i1); k2=bt(2,i1); t=-a(k1,k2); p1=p(i1); for i=1:m for j=1:m if i=k1 if j=k1 A(i,j)=A(i,j)
11、+(p1*p1-1)*t; elseif j=k2 A(i,j)=A(i,j)-(p1-1)*t; B(i,j)=p1*(p1-1)*t; end elseif i=k2 if j=k1 A(i,j)=A(i,j)-(p1-1)*t; B(i,j)=(1-p1)*t; end end end end end 程序中bt是2行K列数组，第一行为变压器接入点i的节点编号，第二行为经漏抗后所接j节点编号,k为变压器支路数。P为一维k列数组，输入各个变压器的变比；m 为矩阵 a,b,A,B的阶数a，A是进行变比修正前后的节点导纳矩阵， b，B是变比修正前后支路对地并联导纳矩阵。1.3 例图1-1节点导
12、纳和支路对地并联导纳计算程序例图1-1共有5个节点，5条支路，所以Np=5，Nb=5支路，为变压器支路，其 i 节点编号为2，j节点编号为4，变比为1：1/1.05；支路也为变压器支路，其 i 节点编号为3，j节点编号为5，变比为1：1/1.05；所以数组bt=2,3;4,5，p=1/1.05,1/1.05，k=2;5条支路按箭头方向首末端节点分别为2,2,1,4,5和1,3,3,2,3，所以nstart=2,2,1,4,5;nend=1,3,3,2,3;输入以上数据可得以下程序：%* NU11.m , example 1-1, fig 1-1 * % The following Progra
13、m for node impedances calculation is based on MATLAB2007clear% bus 1,2,3 is PQ bus,bus 4 is PV bus,bus 5 is slack busglobal Np Nb bt p m k% Np is number of node point,Nb is number of braches,% p the transformer turns ratio,with off-nominal tap-setting,fig 1-1 % bt is two-dimensional array ,line one
14、is point i,line two is point j,% m is number of node point, k is number of transformers Np=5;Nb=5; bt=2,3;4,5;p=1/1.05,1/1.05;m=5;k=2;% nstart-the start point of branches ,nend - the end point,% mm - network incidence matrix nstart=2,2,1,4,5; nend=1,3,3,2,3; mm=ffm(nstart,nend);% zb1,the series impe
15、dances of transmission line % yb1,the series admittances of transmission line zb1r=0.04,0.08,0.1,0.0,0.0; zb1i=0.25,0.30,0.35,0.015,0.03; zb1=zb1r+zb1i*j; yb1=zb1.(-1); yb=diag(conj(yb1); y=mm*yb*(mm); % yb0 ,the shunt admittances of transmission line yb0i=0.25,0.25,0.0,0.0,0.0; yb0=0+yb0i*j; y0=dia
16、g(conj(yb0); yg=mm*y0*(mm); yn=diag(diag(yg); yy=yn-yg; y=y+yn; Y,YY=fdt1(y,yy); % YY is the shund admittances of the line and transformers at each end,% Y is node-admittance matrix for network运行结果为：Y = 1.3787 - 6.2917i -0.6240 + 3.9002i -0.7547 + 2.6415i 0 0 -0.6240 + 3.9002i 1.4539 -66.9808i -0.82
17、99 + 3.1120i 0 +63.4921i 0 -0.7547 +2.6415i -0.8299 + 3.1120i 1.5846 -35.7379i 0 0 +31.7460i 0 0 +63.4921i 0 0 -66.6667i 0 0 0 0 +31.7460i 0 0 -33.3333i YYYY = 0 0 + 0.2500i 0 0 0 0 + 0.2500i 0 0 + 0.2500i 0 + 3.0234i 0 0 0 + 0.2500i 0 0 0 + 1.5117i 0 0 - 3.1746i 0 0 0 0 0 0 - 1.5873i 0 0 参考文献：1MATLAB基础与应用简明教程 张平等编著 北京航空航天大学出版社2001年2燃气工程电算方法，冯良 张同 全惠君 编著，同济大学出版社，1998年3电力系统计算，西安交通大学 清华大学 等合编，水利电力出版社，1978年
