電力網(wǎng)節(jié)點導納矩陣計算例題與程序

1、7電力網(wǎng)節(jié)點導納矩陣計算例題與程序佘名寰 編寫用計算機解算電力網(wǎng)潮流電壓和短路電流問題首先需確定電力網(wǎng)的節(jié)點導納矩陣或節(jié)點阻抗矩陣。本文通過例題介紹用網(wǎng)絡(luò)拓撲法計算節(jié)點導納矩陣的方法和程序，程序考慮了線路并聯(lián)電容和變壓器支路標么變比不為1時的影響。程序用MATLAB語言編寫，線路參數(shù)均采用標么值。本文稿用office word 2007 版編寫，可供電氣專業(yè)人員計算相關(guān)問題時參考。1.用網(wǎng)絡(luò)拓撲計算節(jié)點導納矩陣 1.1網(wǎng)絡(luò)拓撲矩陣：【例1.1】 例圖1-1是有5 個節(jié)點和5條支路的網(wǎng)絡(luò)，節(jié)點5作為基準參考點，1 ,2, 3, 4為獨立節(jié)點，支路編號和方向圖中已標識。Z21=0.04+J0.25

2、Z23=0.08+J0.30Z13=0.1+J0.35Z42=J0.015Z53=J0.03YC1=J0.25YC2=J0.50YC3=J0.254Z23Z13Z21YC2YC3YC1Z42Z531:1.051.05:142315例圖1-1對于具有n個節(jié)點b條支路的有向圖，它的關(guān)聯(lián)矩陣為一個N×B的矩陣Aa：Aa=aij若支路j與節(jié)點i相關(guān)，且箭頭背離節(jié)點i，則aij=1，若箭頭指向節(jié)點則aij=-1，若支路j與節(jié)點i無關(guān)，則aij=0，圖1-1所示的有向圖的關(guān)聯(lián)矩陣為 支路編號Aij=-10100110-100-1-10-100.01000001行編號從上到下為1 2 3 4 5節(jié)

3、點編號（5為參考節(jié)點）去掉第5行即為獨立節(jié)點的關(guān)聯(lián)矩陣。以下介紹生成網(wǎng)絡(luò)關(guān)聯(lián)矩陣的M函數(shù)文件 ffm.m：% M FUNCTION ffm.m% Np is number of node point,Nb is number of braches% 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(n

4、start(i),i)=1; A(nend(i),i)=-1; end 以例圖1-1網(wǎng)絡(luò)為例調(diào)用ffm.m文件求其關(guān)聯(lián)矩陣運算以上程序可得關(guān)聯(lián)矩陣 mmij如下：mm = -1 0 1 0 0 1 1 0 -1 0 0 -1 -1 0 -1 0 0 0 1 0 0 0 0 0 1Mmij明顯與Aij是相同的。1.2生成節(jié)點導納矩陣程序： 由網(wǎng)絡(luò)原始矩陣計算節(jié)點導納矩陣公式 Y=AYs0At (1-1) Y -節(jié)點導納矩陣 A-網(wǎng)絡(luò)關(guān)聯(lián)矩陣 At-A的轉(zhuǎn)置矩陣 YS0-網(wǎng)絡(luò)原始導納矩陣 若網(wǎng)絡(luò)各支路阻抗為 Zb=zb1,zb2,zbn 則ZS0=zb100000zb200000:0000.0:0

5、0000zbn Ys0=Zs0-1 (1-2) Y=A Zs0-1At (1-3) 節(jié)點導納計算程序以例1-1網(wǎng)絡(luò)為例，在不計對地電容和變壓器變比假定為1條件下，節(jié)點導納矩陣計算程序如下： clearglobal Np Nb % Np is number of node point,Nb is number 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

6、; 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; 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)'運算程序后可得節(jié)點導納矩陣y= 1.3787 - 6.5417i -0.6240 + 3.90

7、02i -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 0 +66.6667i 0 0 -66.6667i 0 0 0 0 +33.3333i 0 0 -33.3333i 線路對地并聯(lián)電容的處理例圖1-1網(wǎng)絡(luò)中，線路，兩端對地電容導納均為j0.25,節(jié)點1,2,3自導納相應(yīng)增加j0.25,j0.5,j0.25。程序處理如下：% yb0

8、 ,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yb0為各線路對地電容導納，y0為原始導納矩陣，yn對角矩陣對應(yīng)各節(jié)點自導納增加值，yy相應(yīng)各支路兩端對地并聯(lián)導納，用于計算支路電流。y為計及對地電容后的節(jié)點導納矩陣。程序運算結(jié)果yn = 0 + 0.2500i 0 0 0 0 0 0 + 0.5000i 0

9、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 + 3.9002i -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 + 3.1120

10、i 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 非標準變壓器支路的處理 當支路有變壓器并且標么變比不為1時，可將變壓器化作形等值電路，如圖1-2所示。圖中理想變壓器變比為k:1,接入節(jié)點i,高壓側(cè)經(jīng)漏抗ZT與節(jié)點j相連。 JK:1ZTi(1-K)YTK(K-1)YTKYTi J 圖1-2變壓器等值電路等值電路圖中YT=1/ZT。i側(cè)自導變?yōu)閥ii=kyt+k(k-1)yt=k2yt，改變量為yii=(k2-1)yt; I j間互導變?yōu)閥ij=-kyt,yij=

11、-kyt+yt=-(k-1)yt。j側(cè)自導變?yōu)閥jj=kyt+(1-k)yt=yt,與K=1時相同;變壓器變比不為1時修正導納矩陣的M函數(shù)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 one is point i,line two is point j% p is the transformer turns ratio,k is number of transformers functi

12、onA,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)+(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)=

13、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列數(shù)組，第一行為變壓器接入點i的節(jié)點編號，第二行為經(jīng)漏抗后所接j節(jié)點編號,k為變壓器支路數(shù)。P為一維k列數(shù)組，輸入各個變壓器的變比；m 為矩陣 a,b,A,B的階數(shù)a，A是進行變比修正前后的節(jié)點導納矩陣， b，B是變比修正前后支路對地并聯(lián)導納矩陣。1.3 例圖1-1節(jié)點導納和支路對地并聯(lián)導納計算程序例圖1-1共有5個節(jié)點，5條支路，所以Np=5，Nb=5支路，為變壓器支路，其 i 節(jié)點編號為2，j節(jié)點編號為4，變比為1：1/1.05；支路也為變壓器支路，其 i 節(jié)點編號

14、為3，j節(jié)點編號為5，變比為1：1/1.05；所以數(shù)組bt=2,3;4,5，p=1/1.05,1/1.05，k=2;5條支路按箭頭方向首末端節(jié)點分別為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;輸入以上數(shù)據(jù)可得以下程序：%* NU11.m , example 1-1, fig 1-1 * % The following Program for node impedances calculation is based on MATLAB2007clear% bus 1,2,3 is PQ bus,bus 4 is PV bus,bu

15、s 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 is point i,line two is point j,% m is number of node point, k is number of transformers Np=5;Nb=5; bt

16、=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 impedances of transmission line % yb1,the series admittances of transmission line zb1r=0.04,0.08,0.1,0.0,

17、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=diag(conj(yb0'); yg=mm*y0*(mm)' yn=diag(diag(yg); yy=yn-yg; y=y+yn; Y,YY=fdt1(y,yy); %

18、YY is the shund admittances of the line and transformers at each end,% Y is node-admittance matrix for network運行結(jié)果為：Y = 1.3787 - 6.2917i -0.6240 + 3.9002i -0.7547 + 2.6415i 0 0 -0.6240 + 3.9002i 1.4539 -66.9808i -0.8299 + 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