同步电机三相短路电流和转矩计算

同步电机三相短路电流和电磁转矩计算编写佘名寰本文是按照陈珩教授所著的‘同步电机运行基本理论与计算机算法’一书介绍的算法和例题计算同步电机的三相短路电流。计算程序用MATLAB语言编写，计算结果与书中结果基本一致。本文可供电力系统电气技术人员和大专院校电力专业学生参考。1．计算方法1.1初始数据计算由短路前的机端电压u[0],定子绕组电流i[o],和功率因数角φ[0]求得短路前的功率角δ0=𝐭𝐚𝐧−𝟏𝐮[𝟎]𝐬𝐢𝐧𝛗[𝟎]+𝐱𝐪𝐢[𝟎]𝐮[𝟎]𝐜𝐨𝐬𝛗[𝟎]+𝐫𝐢[𝟎]−𝛗[𝟎]从而得u[0],i[0]的正、交轴分量ud[0]=u[0]sinδ0uq[0]=u[0]cosδ0id[0]=i[0]sin(δ0+φ[0])iq[0]=i[0]cos(δ0+φ[0])短路前的空载电势是Eq[0]=uq[0]+riq[0]+xdid[0]励磁电流为if[0]=Eq[0]/xaf式中xd为同步电机正轴同步电抗xq同步电机交轴同步电抗xaf定子绕组与劢磁绕组间的互感电抗r定子绕组电阻1.2电流变化量的状态空间方程式同步电机突然短路时各绕组电流的变化量∆id∆iq∆if∆iD∆iQ的计算可运用以派克分量表示的状态空间方程式[∆𝐮𝐝∆𝐮𝐪∆𝐮𝐟𝟎𝟎]=[−𝐱𝐝𝐱𝐚𝐟𝐱𝐚𝐃−𝐱𝐪𝐱𝐚𝐐−𝐱𝐚𝐟𝐱𝐟𝐱𝐟𝐃−𝐱𝐚𝐃𝐱𝐟𝐃𝐱𝐃−𝐱𝐚𝐐𝐱𝐐][∆𝐢̇𝐝∆𝐢̇𝐪∆𝐢̇𝐟∆𝐢̇𝐃∆𝐢̇𝐐]+[−𝐫𝐱𝐪−𝐱𝐚𝐐−𝐱𝐝−𝐫𝐱𝐚𝐟𝐱𝐚𝐃𝐫𝐟𝐫𝐃𝐫𝐐][∆𝐢𝐝∆𝐢𝐪∆𝐢𝐟∆𝐢𝐃∆𝐢𝐐]方程中各下标变量的含义为d---纵轴，q---横轴，f----励磁绕组，D---纵轴阻尼绕组，Q---横轴阻尼绕组，a---定子绕组上式可简化为∆udq0=Xdq0(3)∆İdq0+Zdq0(3)∆Idq0化作电流变化量的常系数一阶微分方程组形式∆İdq0=−Xdq0（3）−1Zdq0(3)∆Idq0+Xdq0（3）−1∆udq0在三相短路时若励磁电压不可调，则∆udq0=[−ud[0]−uq[0]000]t由于电流不能突变，t=0瞬间电流变化量的初值∆idq00=[00000]t将电压变化量和电流变化量的初值代入微分方程，用数值计算的龙格---库塔法即可求出t=0+Δh时刻的各电流变化量，反复计算则可求得各个时刻的∆idq0，叠加短路前绕组电流idq0[0]=[id[0]iq[0]if[0]00]t可得短路时电流全量idq0=[idiqifiDiQ]t用派克逆变换可得定子三相电流，以a相为例ia=idcos(t+θ0)−iqsin(t+θ0)θ0短路t=0时转子位置角2．.同步电机三相短路电流计算例题与程序电机参数r=0.005,rf=0.000656，rD=0.00151,rQ=0.00159xd=1,0,xq=0.60,xf=1.03,xD=0.95,xQ=0.70xaf=0.85,xaD=0.85,xfD=0.85,xaQ=0.45原始运行条件为额定负载U[0]=1,i[0]=1,φ[0]=0.5548(单位为弧度，相对于cosφ=0.8)短路时的转子位置角θ0=3.1416三相短路计算程序：CMSHORT3.M%part1ra=0.005;rf=.000656;rzd=.00151;rzq=.00159;xd=1.0;xq=.60;xf=1.03;xzd=.95;xzq=.70;xaf=.85;xazd=.85;xfzd=.85;xazq=.450;u0=1.0;i0=1.0;phas=.5548;cita0=3.1416;p=31.4160;h=.5236;x1=[-1.0,0.0,0.85,0.85,0.0;0.0,-0.60,0.0,0.0,0.45;-0.85,0.0,1.03,0.85,0.0;-0.85,0.0,0.85,0.95,0.0;0.0,-0.45,0.0,0.0,0.70];z1=[-0.005,0.6,0.0,0.0,-0.45;-1.0,-0.005,0.85,0.85,0.0;0.0,0.0,0.000656,0.0,0.0;0.0,0.0,0.0,0.00151,0.0;0.0,0.0,0.0,0.0,0.00159];g0=(u0*sin(phas)+xq*i0)/(u0*cos(phas)+ra*i0);g0=atan(g0)-phas;ud0=u0*sin(g0);uq0=u0*cos(g0);di0=i0*sin(g0+phas);qi0=i0*cos(g0+phas);eq0=uq0+ra*qi0+xd*di0;fi0=eq0/xaf;du=[-ud0,-uq0,0.0,0.0,0.0];x2=inv(x1);z2=-x2*z1;i1=x2*du';y=[0.0,0.0,0.0,0.0,0.0];%part2t=0.0;fori=1:5b(i)=y(i);enddy=z2*(y).'+i1;tt(1)=t;di(1)=y(1)+di0;qi(1)=y(2)+qi0;fi(1)=y(3)+fi0;zdi(1)=y(4);zqi(1)=y(5);ai(1)=di(1)*cos(t+cita0)-qi(1)*sin(t+cita0);tm(1)=qi(1)*(-xd*di(1)+xaf*fi(1)+xazd*zdi(1))-di(1)*(-xq*qi(1)+xazq*zqi(1));%while(tt(n)p)%k=0;forn=2:10a(1)=h/2;a(2)=a(1);a(3)=h;a(4)=h;fork=1:3fori=1:5c(i)=b(i)+a(k)*dy(i);y(i)=y(i)+a(k+1)*dy(i)/3.0;enddy=z2*(c).'+i1;endfori=1:5y(i)=y(i)+h*dy(i)/6.0;endt=t+h;fori=1:5b(i)=y(i);enddy=z2*(y).'+i1;%k=k+1;%part3tt(n)=t;di(n)=y(1)+di0;qi(n)=y(2)+qi0;fi(n)=y(3)+fi0;zdi(n)=y(4);zqi(n)=y(5);ai(n)=di(n)*cos(t+cita0)-qi(n)*sin(t+cita0);tm(n)=qi(n)*(-xd*di(n)+xaf*fi(n)+xazd*zdi(n))-di(n)*(-xq*qi(n)+xazq*zqi(n));endplot(tt,ai);xlabel('t(rad)');ylabel('ia');title('3-phasefaultofsynchronousmachine');grid3．程序说明：X1为状态空间方程的Xdq0矩阵Z1为状态空间方程的Zdq0矩阵X2=Xdq0（3）−1I1=Xdq0（3）−1∆udq0程序%part1第一部分输入初始数据，建立状态空间方程；程序%part2第二部分用定步长四阶龙格---库塔法解一阶微分方程组，计算tn+1=tn+h时刻各绕组电流的变化量。时间步长h用标么值，单位为弧度,h取1/6π定步长四阶龙格---库塔法解一阶微分方程组的基本公式：y1’=f1(t,y1,y2,…,ym),y1(t0)=y10y2’=f2(t,y1,y2,…,ym),y2(t0)=y20…….Ym’=fm(t,y1,y2,…,ym),ym(t0)=ym0微分方程组由tj积分一步到tj+1=tj+hYi,j+1=yij+(k1i+2k2i+2k3i+k4i)h/6,i=1,2,…,mk1i=fi(tj,y1j,y2j,…,ymj),i=1,2,…,mk2i=fi(tj+h/2,y1j+k11h/2,…,ymj+k1mh/2),i=1,2,…,mk3i=fi(tj+h/2,y1j+k21h/2,…,ymj+k2mh/2),i=1,2,…,mk4i=fi(tj+h,y1j+k31h,…,ymj+k3mh),i=1,2,…,m程序%part3第三部分输出计算结果，tm为转矩Tm=iqψd+idψqψd=-xdid+xafif+xaDiDψq=-xqiq+xaQiQ本程序计算时间周期为t=0—10X1/6π,绘画取20π4.程序计算结果：T=0--10X1/6π,取10个点纵轴短路电流didi=Columns1through70.7966190047267142.2367130993432224.4478626375923196.8263978366408118.7369322382886849.6813436639196989.427860674567469Columns8through108.0668926315905455.9803449299180053.734356462313833A相短路电流aiai=Columns1through7-0.796614563934470-0.9644241603207350.0404061250097802.4496686213438025.6777204819841818.4172515171420659.427882627330178Columns8through108.3622653501447395.9184325579004333.227399551484802横轴短路电流qiqi=Columns1through70.6044817295073681.9452043621400782.6145733115566602.4495933972144211.5117004293321450.065795066843252-1.494184215750709Columns8through10-2.752315328026902-3.381332555555734-3.227468137315201励磁绕组电流fifi=Columns1through72.0389317955928842.5178174245469513.2548509777309844.0501318468609244.6925015330335965.0157162661537904.941547215637447Columns8through104.4986640266012453.8128055242439333.071441629933557纵轴阻尼绕组电流zdizdi=Columns1through700.8597117550616702.1774129105476173.5916009770288544.7227836673887345.2743848475096735.109576170838786Columns8through104.2841844368405483.0278754729489831.679680254626384横轴阻力尼绕组电流zqizqi=Columns1through700.8613477700684861.2903273689965281.1827458178930910.578725790607257-0.350939846737101-1.352763638980671Columns8through10-2.159442618662395-2.560957386046311-2.459010179356190电磁转矩tmtm=Columns1through70.8550058119311612.9771766601552444.8382496751246415.5892558297652774.5397477838432851.849594513339010-1.391560982877246Columns8through10-3.827143999454925-4.680476918568964-4.08070