电力系统短路电流计算例题与程序

1、电力系统短路电流计算例题与程序编写余名寰本文用导纳矩阵求逆计算节点阻抗矩阵，运用复合序网络图计算各节点对称故障和不对 称故障时短路电流、节点电压和各支路故障电流。2.1用阻抗矩阵计算短路的基本公式：节点三相对称短路，注入节点的短路电流ld=-V d(°)/Zdd(2-1)式中Vd(0)故障点在短路发生前的电压，简化计算时Vd(0)=1Zdd 故障点d的自阻抗负号表示电流从节点流出故障点短路电流在各节点所产生的电压分量V=ZI(2-2)式中Z节点阻抗矩阵I节点注入电流的列矩阵当只有一点故障时上述电压分量为V i(d) =Z di I d (i=1,2,3, n)(2-3)式中Zdi 故

2、障点d与节点i的互阻抗短路故障后的节点电压Vi=Vi(°)+Vi(d)(2-4)式中Vi(0)节点i故障发生前的电压短路故障时通过各支路的电流lij=(V i-Vj)/z ij(2-5 )式中Zij联系节点i和节点j的支路阻抗 单相接地短路故障点的电流和电压：A相单相接地故障Ia0 = I al =l a2= -V al (°(Z dd0 +Z dd1 +Z dd2 )(2-6)Zddo , Zddi , Zdd2 零序、正序、负序网络故障节点的自阻抗Va0= Z dd0 Ia0(2-7)Va1 =V a1(0)+Zdd1 Ia1(2-8)Va2= Z dd2 I a2(2

3、-9)Ia=3I a1(2-10)两相接地短路：B C相短路接地故障增广正序网的综合等值阻抗 Z刀Z X=Z ddO Zdd2 / (Z ddO +Z dd2 )(2-11)Ia1=-V a1 (0)/(Z dd1 + Z E)(2-12)Ia0= -I a1 Zdd2 / (Zdd0 +Z dd2 )(2-13)Ia2= -I a1 Zdd0 / (Zdd0 +Z dd2 )(2-14)Ib=I a0 + a 2Ia1 +aI a2(2-15)a=(-1/2+jV3/2)a2=(-1/2-j v3/2)两相短路：B.C两相短路故障|ai=|a2= -V a1(°)/(Z dd1 +

4、Z dd2)(2-18)Ib=j v3la1(2-19) 支路ij间的某一点d发生故障时，视 d点为新的节点d点与节点k的互阻抗ZdkZdk=(1-L)Z ik+LZ jk(2-20)d点的自阻抗ZddZdd=(1-L) 2Zii+L2Zjj+2L(1-L)Z u+L(1-L)z ij(2-21)式中L为端点i到故障点d的距离所占线路全长的百分数Zik，Zjk分别为节点i和节点j与节点k的互阻抗Zii，,Zjj为节点i和节点j的自阻抗Zij为节点i与节点j的互阻抗zij是节点i和节点j间的线路阻抗2.2短路电流计算时用导纳矩阵求逆计算节点阻抗矩阵参考文献介绍了从网络的原始阻抗矩阵求节点导纳矩阵

5、的方法和相关程序。与潮流计算时不同的是网络图参考节点为发电机中性点，计及发电机次暂态电抗；零序网络图参考节点为变压器中性点，计及变压器零序电抗，双回线有零序互感阻抗。包括中性点的节点导纳矩阵是不定导纳矩阵，在求逆矩阵前要去掉参考节点(中性点)转化为定导纳矩阵后再求阻抗矩阵。以下举例说明从导纳矩阵计算节点阻抗矩阵的方法和程序。【例2.1】 图2-1所示为一个3节点网络，两台发电机，四条线路。发电机一台中性点接地，另一台不接地。发电机次暂态电抗和线路阻抗原始数据在表2-1、表2-2中给出，线路3、4间有零序互感电抗。试计算该网络图的正序和零序节点阻抗矩阵。图2-1例2.1网络图表2-1正序网络数据

6、首端末端回路编号自感标么阻自感标么阻节点编号节点编号抗 R (pu.)抗 X (pu.)4160.00.20004310.00.16001220.00.08002330.00.06002340.00.06001350.00.1300表2-2零序网络数据首端末端回路编号自感标么自感标么互感标么互感标么节点编号节点编号阻抗R阻抗X阻抗Rm阻抗Xm4310.00.02001220.00.14002330.00.10000.00.05002340.00.12000.00.05001350.00.1700例2.1节点正序负序和零序阻抗矩阵计算源程序clearglobal Np NbNp=4;Nb=6;

7、Np1=3;n start=1,2,2,1,4,4; nen d=2,3,3,3,1,3;mm=ffm( nstart, nen d);% zb1 zbO,the series impedances of transmission line zb1i=0.08,0.06,0.06,0.13,0.2,0.16;zb1=j*zb1i;yb1=zb1.*1);ys=diag(c onj( yb1');y=mm*ys*(mm)'Nb=5;nstart0=1,2,2,1,4; nen d0=2,3,3,3,3;mmO=ffm(n startO ,nen d0);zb0i=0.14,0.1

8、0,0.12,0.17,0.02; zbO=j*zbOi;zsO=diag(zbO.');zs0(2,3)=0.05*i;zs0(3,2)=0.05*i;ys0=in v(zsO);yO=mmO*ysO*(mmO)'for i=1:Np1for j=1:Np1丫1(i,j)=y(i,j);YO(i,j)=yO(i,j);endendY2=Y1;z1=i nv(Y1);z2=i nv(Y2);z0=i nv(YO);程序说明：正序网络有4个节点6条支路，所以 Np=4;Nb=6;独立节点数Np1=3;发电机中性点编号设为最大值编号4。零序网络只有5条支路，因为只有一侧发电机中性点

9、接地。因为有零序互感电抗，采用在对角矩阵zs0增加互感元素zs0(2,3);zs0(3,2)的方法先导出零序原始阻抗矩阵zso,再求逆矩阵得到零序原始导纳矩阵ysOZ1,z2,z0分别为节点正序负序Y1，Y0为去掉中性节点（参考点）后的正序和零序定导纳矩阵和零序阻抗矩阵，假定负序阻抗和正序阻抗相同 程序运算结果:Z0 =>> Z1Z1 =>> Z2Z2 =0 + 0.1157i0 + 0.0546i0 + 0.0200i0 + 0.0546i0 + 0.0831i0 + 0.0200i0 + 0.0200i0 + 0.0200i0 + 0.0200i0 + 0.1047

10、i0 + 0.0840i0 + 0.0763i0 + 0.0840i0 + 0.1122i0 + 0.0928i0 + 0.0763i0 + 0.0928i0 + 0.0990i0 + 0.1047i0 + 0.0840i0 + 0.0763i0 + 0.0840i0 + 0.1122i0 + 0.0928i0 + 0.0763i0 + 0.0928i0 + 0.0990i2.3短路电流计算例题例 2.4.2例题为如图2-6所示的6节点网络。节点2、1间和节点4、3间通过变压器连接，变压器和发 电机接线如图所示，支路1和支路6之间有零序互感，其正序网络和零序网络数据见表2.5、2.6试计算各节

11、点单相和三相短路时节点短路电流、节点电压和相邻支路故障电流。表2-5正序网络数据（中性点编号为7）首端末端回路编号自感标么阻自感标么阻节点编号节点编号抗 R (pu.)抗 X (pu.)1410.16000.74001620.24001.03602331.44602.10002540.56401.28004350.00.26604660.19400.81406570.00.60007180.02000.24007290.03000.4800表2-6零序网络数据首端末端回路自感标自感标互感支互感支互感标互感标节点节点编号么阻抗么阻抗路首端路末端么阻抗么阻抗编号编号RX节点节点RmXm7290.0

12、0.03201620.9842.084014101.8500460.50000.95002540.8003.840023305.26004662.8202.0600140.50000.950076700.60007453.7800.266000.90000.00.02.4.2.1例2.4.2短路电流计算程序shortE2e.m% shortE2e.mclearglobal Np Nb %正序网络包括参考节点数和支路数a=-0.5+j*sqrt(3)/2;T=111aA2a1aaA21;%T为对称分量法的合成矩阵Np=7;Nb=9; %最大节点编号为参考点n start=1,1,2,2,4,4,

13、6,7,7; nen d=4,6,3,5,3,6,5,1,2; mm=ffm( nstart, nen d);% zb1 zb0,the series impedances of transmission line zb1r=0.16,0.246,1.446,0.564,0.0,0.194,0.0,0.020,0.030; zb1i=0.74,1.036,2.10,1.28,0.266,0.814,0.60,0.24,0.48;zb1=zb1r+j*zb1i;yb仁zb"(-1);ys仁 diag(conj(yb1');y1=mm*ys1*(mm)：Nb=8;%零序网络支路数

14、nstart0=7,1,1,2,2,4,7,7;nen d0=2,6,4,5,3,6,6,4;mmO=ffm(n start O,nen d0);zb0r=0.0,0.984,0.80,2.82,3.78,0.90,0.0,0.0;zb0i=0.032,2.084,1.850,3.840,5.260,2.060,0.60,0.266; zb0=zb0r+j*zb0i;ybO=zbO4(-1);ys01=diag(conj(yb0');y01= mmO*ysO1*(mmO)：zsO=diag(zbO.');zs0(3,6)=0.5+0.95*i;zs0(6,3)=0.5+0.95

15、*i;%互感支路ys0=in v(zs0);yO=mmO*ysO*(mmO)'N1=6;%输入网络独立节点数for i=1:N1for j=1:N1Y1(i,j)=y1(i,j);YO(i,j)=yO(i,j);YO1(i,j)=yO1(i,j);endendY=(abs(Y1)>0);%获得Y1非零元素ZM= 0.8000 + 1.8500i,0.5000 + 0.9500i;0.5000 + 0.9500i,0.9000 + 2.0600i ;YM=i nv(ZM); %输入互感导纳矩阵Y2=Y1;%输入负序网络节点导纳矩阵z1=i nv(Y1);z2=i nv(Y2);z0

16、=in v(Y0);'正序和零序网络节点阻抗矩阵z1,z0'z1z0'短路类型说明fault(f(1)=1 ，f(3)=2,f(2)=3,f(1,1)=4)'for k=1:N1fprintf（'短路点的节点编号k= %.0gn',k）%输入短路点的节点编号for fault=1:2 ;%l输入短路类型fprintf（'短路类型fault= %.0gn',fault）%第一部份，计算短路时所有节点的a,b,c三相电压Z1(:,k)=z1(:,k);Zk 1=Z1(:,k);%正序网络中节点m的自阻抗和互阻抗Z2(:,k)=z2(:

17、,k);Zk2=Z2(:,k);%负序网络中节点m的自阻抗和互阻抗Z0(:,k)=z0(:,k);Zk0=Z0(:,k);%零序网络中节点m的自阻抗和互阻抗if fault=1 %根据故障类型选择不同的计算公式lk1=1/(Z1(k,k)+Z2(k,k)+Z0(k,k);lk2=lk1;lk0=lk1;elseif fault=2lk1=1/Z1(k,k);lk2=0;lk0=0;elseif fault=3lk1=1/(Z1(k,k)+Z2(k,k);lk2=-lk1;lk0=0;elseif fault=4 lk1=1/(Z1(k,k)+Z2(k,k)*Z0(k,k)/(Z2(k,k)+Z

18、0(k,k); lk2=-lk1*Z0(k,k)/(Z2(k,k)+Z0(k,k); lk0=-lk1*Z2(k,k)/(Z2(k,k)+Z0(k,k);endendendendlk1;%所计算的短路点正序电流lk=lk1 lk2 lk0;'短路点正序负序和零序电流幅值和角度abs(lk)an gle(lk)*180/pilabc=T*lk.'labc;%所计算的短路点三相电流'短路点A相短路电流幅值和角度abs(labc(1,1)an gle(labc(1,1)*180/pifor p=1:N1if p=kI1(p)=-lk1;I2(p)=-lk2;I0(p)=-lk

19、0;elseI1(p)=0;I2(p)=0;I0(p)=0;endenduu1(:,k)=Y1l1.'uu2(:,k)=Y22：uuO(:,k)=YOIO.'u1=1+uu1(:,k);% 计算所有节点k点故障时的正序电压u2=uu2(:,k);%计算所有节点k点故障时的负序电压uO=uuO(:,k);%计算所有节点k点故障时的零序电压'节点编号，节点正序负序和零序电压幅值和角度p Up Dp'for p=1:N1U=u1(p) u2(p) uO(p);Up=abs(U);Dp=a ngle(U)*180/pi;Uabc=T*(U.');UUabc=ab

20、s(Uabc);%UUabc 表示p节点的三相电压有效值 if Y(p,k)=1p, Up,Dpend end%第二部分：计算支路电流 '支路节点编号，正序、负序和零序电流幅值和角度for n=1:N1if (Y( n,k)&n =k)=1mn (1)=n;m n( 2)=k;mn正序支路电流的实用计算负序支路电流的实用计算零序支路电流的实用计算IB1( n,k)=(u1( n)-u1(k)*(-Y1( n,k);%IB2( n,k)=(u2( n)-u2(k)*(-Y2( n,k);%IB0( n,k)=(u0( n)-u0(k)*(-Y01( n, k);%if(k=1 &

21、amp;n=4)=1IBO( n,k)=YM(1,1)*(uO( n)-uO(k)+YM(1,2)*(uO(6)-uO( n);%IB04 仁YM11*u41+YM12*u64elseif (k=4&n=1)=1IB0( n,k)=YM(1,1)*(u0( n)-u0(k)+YM(1,2)*(u0(k)-u0(6);%IB014=YM11*u14+YM12*u46elseif (k=4&n=6)=1IB0( n,k)=YM(2,1)*(uO(k)-uO(1)+YM(2,2)*(uO( n)-u0(k);%IB064=YM21*u41+YM22*u64elseif (k=6&am

22、p;n=4)=1IB0( n,k)=YM(2,1)*(uO(1)-uO( n)+YM(2,2)*(u0( n)-uO(k); %IB046=YM21*u14+YM22*u46endendendendIB=IB1(n,k) IB2(n,k) IB0(n,k).'abs(IB)an gle(IB)*180/pilabc=T*IB1( n,k) IB2( n,k) IB0( n, k).'Iabc;%labc 表示支路(m,n)的a,b,c三相电流abs(Iabc);an gle(Iabc)*180/pi;endendendend242.2程序说明由原始矩阵和节点关联矩阵求节点导纳矩

23、阵，去掉参考节点转为定导纳矩阵。节点阻抗矩阵z1，z0由定导纳矩阵求逆得到。ZM，YM为互感支路阻抗和导纳矩阵，用于计算互感支路电流。Y01为不计及互感的零序导纳矩阵，用于计算非互感支路电流。矩阵丫是Y1绝对值不为零关系运算结果，用于去除与故障点无关的节点电压和支路电流计算；按2.1节的基本公式计算各种短路故障正序负序和零序电流；由节点注入电流列矢量与阻抗矩阵的乘积计算节点电压，注意节点注入电流与节点故障电流方向相反，取负值。 对互感支路按以下公式计算支路电流：若支路1与支路2互耦，其导纳矩阵为丫11丫12，则"=丫11丫12?U121丫221221丫22 U2242.3程序运行结果

24、(为省篇幅只摘录部分数据)ans =正序和零序网络节点阻抗矩阵z1,zO z1 =0.0225 + 0.2150i-0.0061 + 0.0497i0.0263 + 0.1612i0.0225 + 0.1727i0.0212 + 0.1283i0.0183 +0.1638i-0.0061 + 0.0497i0.0442 + 0.3809i-0.0159 + 0.1571i-0.0079 + 0.1343i-0.0070 + 0.2231i0.0002 +0.1522i0.0263 + 0.1612i-0.0159 + 0.1571i0.1624 + 0.7391i0.1433 + 0.5337

25、i0.0619 + 0.2701i0.0729 +0.3279i0.0225 + 0.1727i-0.0079 + 0.1343i0.1433 + 0.5337i0.1327 + 0.5769i0.0651 + 0.2782i0.0688 +0.3473i0.0212 + 0.1283i-0.0070 + 0.2231i0.0619 + 0.2701i0.0651 + 0.2782i0.1657 + 0.8065i0.1326 +0.4654i0.0183 + 0.1638i0.0002 + 0.1522i0.0729 + 0.3279i0.0688 + 0.3473i0.1326 + 0.4

26、654i0.1303 +0.6112iz0 =0.3602 + 1.0440i000.0122 + 0.2111i0-0.0276 +0.1237i0-0.0000 + 0.0320i0.0000 + 0.0320i00 + 0.0320i000.0000 + 0.0320i3.7800 + 5.2920i00 + 0.0320i00.0122 + 0.2111i000.0079 + 0.2278i0-0.0178 +0.0861i00.0000 + 0.0320i0.0000 + 0.0320i02.8200 + 3.8720i0-0.0276 + 0.1237i00-0.0178 + 0.

27、0861i00.0402 +0.4059ians =短路类型说明 fault(f(1)=1 , f(3)=2,f(2)=3,f(1,1)=4)短路点的节点编号k= 1短路类型fault= 1ans =短路点正序负序和零序电流幅值和角度ans =0.65410.65410.6541ans =-74.6288-74.6288-74.6288ans =短路点A相短路电流幅值和角度ans =1.9624ans =-74.6288ans =节点编号，节点正序负序和零序电压幅值和角度 p Up Dpp =1Up =0.86080.14140.7224Dp = -1.5359 -170.6112176.33

28、78p =4Up =0.88730.11390.1383Dp = -1.0151 -172.0674 -167.9478p =6Up =0.89370.10780.0829Dp = -1.0805 -171.0064 -152.0449ans =支路节点编号，正序、负序和零序电流幅值和角度mn =41ans =0.03660.03660.3711ans =-62.4159-62.4159-76.4814mn =61ans =0.03160.03160.2834ans =-65.9867-65.9867-72.2029短路类型fault= 2ans =短路点正序负序和零序电流幅值和角度ans =

29、4.625100ans = -84.017600ans =短路点A相短路电流幅值和角度ans =4.6251ans = -84.0176ans =节点编号，节点正序负序和零序电压幅值和角度 p Up Dpp =1Up =1.0e-015 *0.444100Dp = 18000p =4Up =0.196000Dp =5.994800p =6Up =0.237800Dp =1.266900ans =支路节点编号，正序、负序和零序电流幅值和角度mn =41ans =0.258800ans =-71.804700mn =6 1ans =0.223300ans =-75.375500短路点的节点编号k=

30、 2短路类型fault= 1ans =短路点正序负序和零序电流幅值和角度ans =1.25191.25191.2519ans =-83.6430-83.6430-83.6430ans =短路点A相短路电流幅值和角度ans =3.7557ans =-83.6430ans =节点编号，节点正序负序和零序电压幅值和角度 p Up DpP =2Up =0.51990.48010.0401Dp =0.2444179.7353 -173.6430p =3Up =0.80780.19770.0401Dp =-2.9526 -167.8515 -173.6430p =5Up =0.72450.27940.04

31、01Dp =-3.1336 -171.8502 -173.6430ans =支路节点编号，正序、负序和零序电流幅值和角度mn =32ans =0.11380.11380.0000ans =-64.1380-64.138035.7022mn =52ans =0.14860.14860.0000ans =-77.8323-77.8323119.1676短路类型fault= 2ans =短路点正序负序和零序电流幅值和角度ans =2.607600ans =-83.378300ans =短路点A相短路电流幅值和角度ans =2.6076ans =-83.3783ans =节点编号，节点正序负序和零序电

32、压幅值和角度 p Up DpP =2Up =1.0e-015 *0.223200Dp =-5.869300p =3Up =0.604300Dp =-8.423500p =5Up =0.432800Dp =-11.347200ans =支路节点编号，正序、负序和零序电流幅值和角度mn =32ans =0.23700ans =-63.873300mn =52ans =0.309400ans =-77.567700短路点的节点编号k= 3短路类型fault= 1ans =短路点正序负序和零序电流幅值和角度ans =0.12630.12630.1263ans =-58.7711-58.7711-58.

33、7711ans =短路点A相短路电流幅值和角度ans =0.3789ans =-58.7711ans =p Up Dp节点编号，节点正序负序和零序电压幅值和角度P =2Up =0.98410.01990.0040Dp =-0.6992 -142.9796 -148.7711P =3Up =0.91010.09560.8214Dp =-1.9430 -161.1659175.6912p =4Up =0.93320.06980Dp =-1.1953 -163.80400ans =支路节点编号，正序、负序和零序电流幅值和角度mn =23ans =0.03020.03020.1263ans =-41.2605-41.2605-58.7711mn =43ans =0.09800.09800ans =-64.0848-64.08480短路类型fault= 2ans =短路点正序负序和零序电流幅值和角度ans =1.321400ans =-77.605200ans =短路点A相短路电流幅值和角度ans =1.3214ans =-77.6052ans =节点编号，节点正序负序和零序电压幅值和角度 p Up DpP =2Up =0.80440Dp =-4.64480P =3Up =1.0e-015 *0.44430Dp =-178.26240P =4Up =0.27260Dp =7