Actual source code: ex31.c

slepc-3.11.2 2019-07-30
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2019, Universitat Politecnica de Valencia, Spain

  6:    This file is part of SLEPc.
  7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
  8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  9: */

 11: static char help[] = "Power grid small signal stability analysis of WECC 9 bus system.\n\
 12: This example is based on the 9-bus (node) example given in the book Power\n\
 13: Systems Dynamics and Stability (Chapter 8) by P. Sauer and M. A. Pai.\n\
 14: The power grid in this example consists of 9 buses (nodes), 3 generators,\n\
 15: 3 loads, and 9 transmission lines. The network equations are written\n\
 16: in current balance form using rectangular coordinates. It uses the SLEPc\n\
 17: package to calculate the eigenvalues for small signal stability analysis\n\n";

 19: /*
 20:    This example is based on PETSc's ex9bus example (under TS).

 22:    The equations for the stability analysis are described by the DAE

 24:    \dot{x} = f(x,y,t)
 25:      0     = g(x,y,t)

 27:    where the generators are described by differential equations, while the algebraic
 28:    constraints define the network equations.

 30:    The generators are modeled with a 4th order differential equation describing the electrical
 31:    and mechanical dynamics. Each generator also has an exciter system modeled by 3rd order
 32:    diff. eqns. describing the exciter, voltage regulator, and the feedback stabilizer
 33:    mechanism.

 35:    The network equations are described by nodal current balance equations.
 36:     I(x,y) - Y*V = 0

 38:    where:
 39:     I(x,y) is the current injected from generators and loads.
 40:       Y    is the admittance matrix, and
 41:       V    is the voltage vector

 43:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

 45:    The linearized equations for the eigenvalue analysis are

 47:      \dot{\delta{x}} = f_x*\delta{x} + f_y*\delta{y}
 48:              0       = g_x*\delta{x} + g_y*\delta{y}

 50:    This gives the linearized sensitivity matrix
 51:      A = | f_x  f_y |
 52:          | g_x  g_y |

 54:    We are interested in the eigenvalues of the Schur complement of A
 55:      \hat{A} = f_x - g_x*inv(g_y)*f_y


 58:    Example contributed by: Shrirang Abhyankar
 59: */

 61: #include <petscdm.h>
 62: #include <petscdmda.h>
 63: #include <petscdmcomposite.h>
 64: #include <slepceps.h>

 66: #define freq 60
 67: #define w_s (2*PETSC_PI*freq)

 69: /* Sizes and indices */
 70: const PetscInt nbus    = 9; /* Number of network buses */
 71: const PetscInt ngen    = 3; /* Number of generators */
 72: const PetscInt nload   = 3; /* Number of loads */
 73: const PetscInt gbus[3] = {0,1,2}; /* Buses at which generators are incident */
 74: const PetscInt lbus[3] = {4,5,7}; /* Buses at which loads are incident */

 76: /* Generator real and reactive powers (found via loadflow) */
 77: const PetscScalar PG[3] = {0.716786142395021,1.630000000000000,0.850000000000000};
 78: const PetscScalar QG[3] = {0.270702180178785,0.066120127797275,-0.108402221791588};
 79: /* Generator constants */
 80: const PetscScalar H[3]    = {23.64,6.4,3.01};   /* Inertia constant */
 81: const PetscScalar Rs[3]   = {0.0,0.0,0.0}; /* Stator Resistance */
 82: const PetscScalar Xd[3]   = {0.146,0.8958,1.3125};  /* d-axis reactance */
 83: const PetscScalar Xdp[3]  = {0.0608,0.1198,0.1813}; /* d-axis transient reactance */
 84: const PetscScalar Xq[3]   = {0.0969,0.8645,1.2578}; /* q-axis reactance Xq(1) set to 0.4360, value given in text 0.0969 */
 85: const PetscScalar Xqp[3]  = {0.0969,0.1969,0.25}; /* q-axis transient reactance */
 86: const PetscScalar Td0p[3] = {8.96,6.0,5.89}; /* d-axis open circuit time constant */
 87: const PetscScalar Tq0p[3] = {0.31,0.535,0.6}; /* q-axis open circuit time constant */
 88: PetscScalar M[3]; /* M = 2*H/w_s */
 89: PetscScalar D[3]; /* D = 0.1*M */

 91: PetscScalar TM[3]; /* Mechanical Torque */
 92: /* Exciter system constants */
 93: const PetscScalar KA[3] = {20.0,20.0,20.0};  /* Voltage regulartor gain constant */
 94: const PetscScalar TA[3] = {0.2,0.2,0.2};     /* Voltage regulator time constant */
 95: const PetscScalar KE[3] = {1.0,1.0,1.0};     /* Exciter gain constant */
 96: const PetscScalar TE[3] = {0.314,0.314,0.314}; /* Exciter time constant */
 97: const PetscScalar KF[3] = {0.063,0.063,0.063};  /* Feedback stabilizer gain constant */
 98: const PetscScalar TF[3] = {0.35,0.35,0.35};    /* Feedback stabilizer time constant */
 99: const PetscScalar k1[3] = {0.0039,0.0039,0.0039};
100: const PetscScalar k2[3] = {1.555,1.555,1.555};  /* k1 and k2 for calculating the saturation function SE = k1*exp(k2*Efd) */

102: PetscScalar Vref[3];
103: /* Load constants
104:   We use a composite load model that describes the load and reactive powers at each time instant as follows
105:   P(t) = \sum\limits_{i=0}^ld_nsegsp \ld_alphap_i*P_D0(\frac{V_m(t)}{V_m0})^\ld_betap_i
106:   Q(t) = \sum\limits_{i=0}^ld_nsegsq \ld_alphaq_i*Q_D0(\frac{V_m(t)}{V_m0})^\ld_betaq_i
107:   where
108:     ld_nsegsp,ld_nsegsq - Number of individual load models for real and reactive power loads
109:     ld_alphap,ld_alphap - Percentage contribution (weights) or loads
110:     P_D0                - Real power load
111:     Q_D0                - Reactive power load
112:     V_m(t)              - Voltage magnitude at time t
113:     V_m0                - Voltage magnitude at t = 0
114:     ld_betap, ld_betaq  - exponents describing the load model for real and reactive part

116:     Note: All loads have the same characteristic currently.
117: */
118: const PetscScalar PD0[3] = {1.25,0.9,1.0};
119: const PetscScalar QD0[3] = {0.5,0.3,0.35};
120: const PetscInt    ld_nsegsp[3] = {3,3,3};
121: const PetscScalar ld_alphap[3] = {0.0,0.0,1.0};
122: const PetscScalar ld_betap[3]  = {2.0,1.0,0.0};
123: const PetscInt    ld_nsegsq[3] = {3,3,3};
124: const PetscScalar ld_alphaq[3] = {0.0,0.0,1.0};
125: const PetscScalar ld_betaq[3]  = {2.0,1.0,0.0};

127: typedef struct {
128:   DM       dmgen, dmnet; /* DMs to manage generator and network subsystem */
129:   DM       dmpgrid;      /* Composite DM to manage the entire power grid */
130:   Mat      Ybus;         /* Network admittance matrix */
131:   Vec      V0;           /* Initial voltage vector (Power flow solution) */
132:   PetscInt neqs_gen,neqs_net,neqs_pgrid;
133:   IS       is_diff;      /* indices for differential equations */
134:   IS       is_alg;       /* indices for algebraic equations */
135: } Userctx;

137: /* Converts from machine frame (dq) to network (phase a real,imag) reference frame */
138: PetscErrorCode dq2ri(PetscScalar Fd,PetscScalar Fq,PetscScalar delta,PetscScalar *Fr,PetscScalar *Fi)
139: {
141:   *Fr =  Fd*PetscSinScalar(delta) + Fq*PetscCosScalar(delta);
142:   *Fi = -Fd*PetscCosScalar(delta) + Fq*PetscSinScalar(delta);
143:   return(0);
144: }

146: /* Converts from network frame ([phase a real,imag) to machine (dq) reference frame */
147: PetscErrorCode ri2dq(PetscScalar Fr,PetscScalar Fi,PetscScalar delta,PetscScalar *Fd,PetscScalar *Fq)
148: {
150:   *Fd =  Fr*PetscSinScalar(delta) - Fi*PetscCosScalar(delta);
151:   *Fq =  Fr*PetscCosScalar(delta) + Fi*PetscSinScalar(delta);
152:   return(0);
153: }

155: PetscErrorCode SetInitialGuess(Vec X,Userctx *user)
156: {
158:   Vec            Xgen,Xnet;
159:   PetscScalar    *xgen,*xnet;
160:   PetscInt       i,idx=0;
161:   PetscScalar    Vr,Vi,IGr,IGi,Vm,Vm2;
162:   PetscScalar    Eqp,Edp,delta;
163:   PetscScalar    Efd,RF,VR; /* Exciter variables */
164:   PetscScalar    Id,Iq;  /* Generator dq axis currents */
165:   PetscScalar    theta,Vd,Vq,SE;

168:   M[0] = 2*H[0]/w_s; M[1] = 2*H[1]/w_s; M[2] = 2*H[2]/w_s;
169:       /*      D[0] = 0.1*M[0]; D[1] = 0.1*M[1]; D[2] = 0.1*M[2];
170:        */
171:   D[0] = D[1] = D[2] = 0.0;
172:   DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet);

174:   /* Network subsystem initialization */
175:   VecCopy(user->V0,Xnet);

177:   /* Generator subsystem initialization */
178:   VecGetArray(Xgen,&xgen);
179:   VecGetArray(Xnet,&xnet);

181:   for (i=0; i < ngen; i++) {
182:     Vr  = xnet[2*gbus[i]]; /* Real part of generator terminal voltage */
183:     Vi  = xnet[2*gbus[i]+1]; /* Imaginary part of the generator terminal voltage */
184:     Vm  = PetscSqrtScalar(Vr*Vr + Vi*Vi); Vm2 = Vm*Vm;
185:     IGr = (Vr*PG[i] + Vi*QG[i])/Vm2;
186:     IGi = (Vi*PG[i] - Vr*QG[i])/Vm2;

188:     delta = atan2(Vi+Xq[i]*IGr,Vr-Xq[i]*IGi); /* Machine angle */

190:     theta = PETSC_PI/2.0 - delta;

192:     Id = IGr*PetscCosScalar(theta) - IGi*PetscSinScalar(theta); /* d-axis stator current */
193:     Iq = IGr*PetscSinScalar(theta) + IGi*PetscCosScalar(theta); /* q-axis stator current */

195:     Vd = Vr*PetscCosScalar(theta) - Vi*PetscSinScalar(theta);
196:     Vq = Vr*PetscSinScalar(theta) + Vi*PetscCosScalar(theta);

198:     Edp = Vd + Rs[i]*Id - Xqp[i]*Iq; /* d-axis transient EMF */
199:     Eqp = Vq + Rs[i]*Iq + Xdp[i]*Id; /* q-axis transient EMF */

201:     TM[i] = PG[i];

203:     /* The generator variables are ordered as [Eqp,Edp,delta,w,Id,Iq] */
204:     xgen[idx]   = Eqp;
205:     xgen[idx+1] = Edp;
206:     xgen[idx+2] = delta;
207:     xgen[idx+3] = w_s;

209:     idx = idx + 4;

211:     xgen[idx]   = Id;
212:     xgen[idx+1] = Iq;

214:     idx = idx + 2;

216:     /* Exciter */
217:     Efd = Eqp + (Xd[i] - Xdp[i])*Id;
218:     SE  = k1[i]*PetscExpScalar(k2[i]*Efd);
219:     VR  =  KE[i]*Efd + SE;
220:     RF  =  KF[i]*Efd/TF[i];

222:     xgen[idx]   = Efd;
223:     xgen[idx+1] = RF;
224:     xgen[idx+2] = VR;

226:     Vref[i] = Vm + (VR/KA[i]);

228:     idx = idx + 3;
229:   }

231:   VecRestoreArray(Xgen,&xgen);
232:   VecRestoreArray(Xnet,&xnet);

234:   /* VecView(Xgen,0); */
235:   DMCompositeGather(user->dmpgrid,INSERT_VALUES,X,Xgen,Xnet);
236:   DMCompositeRestoreLocalVectors(user->dmpgrid,&Xgen,&Xnet);
237:   return(0);
238: }

240: PetscErrorCode PreallocateJacobian(Mat J,Userctx *user)
241: {
243:   PetscInt       *d_nnz;
244:   PetscInt       i,idx=0,start=0;
245:   PetscInt       ncols;

248:   PetscMalloc1(user->neqs_pgrid,&d_nnz);
249:   for (i=0; i<user->neqs_pgrid; i++) d_nnz[i] = 0;
250:   /* Generator subsystem */
251:   for (i=0; i < ngen; i++) {

253:     d_nnz[idx]   += 3;
254:     d_nnz[idx+1] += 2;
255:     d_nnz[idx+2] += 2;
256:     d_nnz[idx+3] += 5;
257:     d_nnz[idx+4] += 6;
258:     d_nnz[idx+5] += 6;

260:     d_nnz[user->neqs_gen+2*gbus[i]]   += 3;
261:     d_nnz[user->neqs_gen+2*gbus[i]+1] += 3;

263:     d_nnz[idx+6] += 2;
264:     d_nnz[idx+7] += 2;
265:     d_nnz[idx+8] += 5;

267:     idx = idx + 9;
268:   }

270:   start = user->neqs_gen;

272:   for (i=0; i < nbus; i++) {
273:     MatGetRow(user->Ybus,2*i,&ncols,NULL,NULL);
274:     d_nnz[start+2*i]   += ncols;
275:     d_nnz[start+2*i+1] += ncols;
276:     MatRestoreRow(user->Ybus,2*i,&ncols,NULL,NULL);
277:   }

279:   MatSeqAIJSetPreallocation(J,0,d_nnz);

281:   PetscFree(d_nnz);
282:   return(0);
283: }

285: /*
286:    J = [-df_dx, -df_dy
287:         dg_dx, dg_dy]
288: */
289: PetscErrorCode ResidualJacobian(Vec X,Mat J,void *ctx)
290: {
292:   Userctx        *user=(Userctx*)ctx;
293:   Vec            Xgen,Xnet;
294:   PetscScalar    *xgen,*xnet;
295:   PetscInt       i,idx=0;
296:   PetscScalar    Vr,Vi,Vm,Vm2;
297:   PetscScalar    Eqp,Edp,delta; /* Generator variables */
298:   PetscScalar    Efd;
299:   PetscScalar    Id,Iq;  /* Generator dq axis currents */
300:   PetscScalar    Vd,Vq;
301:   PetscScalar    val[10];
302:   PetscInt       row[2],col[10];
303:   PetscInt       net_start=user->neqs_gen;
304:   PetscScalar    Zdq_inv[4],det;
305:   PetscScalar    dVd_dVr,dVd_dVi,dVq_dVr,dVq_dVi,dVd_ddelta,dVq_ddelta;
306:   PetscScalar    dIGr_ddelta,dIGi_ddelta,dIGr_dId,dIGr_dIq,dIGi_dId,dIGi_dIq;
307:   PetscScalar    dSE_dEfd;
308:   PetscScalar    dVm_dVd,dVm_dVq,dVm_dVr,dVm_dVi;
309:   PetscInt          ncols;
310:   const PetscInt    *cols;
311:   const PetscScalar *yvals;
312:   PetscInt          k;
313:   PetscScalar PD,QD,Vm0,*v0,Vm4;
314:   PetscScalar dPD_dVr,dPD_dVi,dQD_dVr,dQD_dVi;
315:   PetscScalar dIDr_dVr,dIDr_dVi,dIDi_dVr,dIDi_dVi;


319:   MatZeroEntries(J);
320:   DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet);
321:   DMCompositeScatter(user->dmpgrid,X,Xgen,Xnet);

323:   VecGetArray(Xgen,&xgen);
324:   VecGetArray(Xnet,&xnet);

326:   /* Generator subsystem */
327:   for (i=0; i < ngen; i++) {
328:     Eqp   = xgen[idx];
329:     Edp   = xgen[idx+1];
330:     delta = xgen[idx+2];
331:     Id    = xgen[idx+4];
332:     Iq    = xgen[idx+5];
333:     Efd   = xgen[idx+6];

335:     /*    fgen[idx]   = (Eqp + (Xd[i] - Xdp[i])*Id - Efd)/Td0p[i]; */
336:     row[0] = idx;
337:     col[0] = idx;           col[1] = idx+4;          col[2] = idx+6;
338:     val[0] = 1/ Td0p[i]; val[1] = (Xd[i] - Xdp[i])/ Td0p[i]; val[2] = -1/Td0p[i];

340:     MatSetValues(J,1,row,3,col,val,INSERT_VALUES);

342:     /*    fgen[idx+1] = (Edp - (Xq[i] - Xqp[i])*Iq)/Tq0p[i]; */
343:     row[0] = idx + 1;
344:     col[0] = idx + 1;       col[1] = idx+5;
345:     val[0] = 1/Tq0p[i]; val[1] = -(Xq[i] - Xqp[i])/Tq0p[i];
346:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

348:     /*    fgen[idx+2] = - w + w_s; */
349:     row[0] = idx + 2;
350:     col[0] = idx + 2; col[1] = idx + 3;
351:     val[0] = 0;       val[1] = -1;
352:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

354:     /*    fgen[idx+3] = (-TM[i] + Edp*Id + Eqp*Iq + (Xqp[i] - Xdp[i])*Id*Iq + D[i]*(w - w_s))/M[i]; */
355:     row[0] = idx + 3;
356:     col[0] = idx; col[1] = idx + 1; col[2] = idx + 3;       col[3] = idx + 4;                  col[4] = idx + 5;
357:     val[0] = Iq/M[i];  val[1] = Id/M[i];      val[2] = D[i]/M[i]; val[3] = (Edp + (Xqp[i]-Xdp[i])*Iq)/M[i]; val[4] = (Eqp + (Xqp[i] - Xdp[i])*Id)/M[i];
358:     MatSetValues(J,1,row,5,col,val,INSERT_VALUES);

360:     Vr   = xnet[2*gbus[i]]; /* Real part of generator terminal voltage */
361:     Vi   = xnet[2*gbus[i]+1]; /* Imaginary part of the generator terminal voltage */
362:     ri2dq(Vr,Vi,delta,&Vd,&Vq);

364:     det = Rs[i]*Rs[i] + Xdp[i]*Xqp[i];

366:     Zdq_inv[0] = Rs[i]/det;
367:     Zdq_inv[1] = Xqp[i]/det;
368:     Zdq_inv[2] = -Xdp[i]/det;
369:     Zdq_inv[3] = Rs[i]/det;

371:     dVd_dVr    = PetscSinScalar(delta); dVd_dVi = -PetscCosScalar(delta);
372:     dVq_dVr    = PetscCosScalar(delta); dVq_dVi = PetscSinScalar(delta);
373:     dVd_ddelta = Vr*PetscCosScalar(delta) + Vi*PetscSinScalar(delta);
374:     dVq_ddelta = -Vr*PetscSinScalar(delta) + Vi*PetscCosScalar(delta);

376:     /*    fgen[idx+4] = Zdq_inv[0]*(-Edp + Vd) + Zdq_inv[1]*(-Eqp + Vq) + Id; */
377:     row[0] = idx+4;
378:     col[0] = idx;         col[1] = idx+1;        col[2] = idx + 2;
379:     val[0] = -Zdq_inv[1]; val[1] = -Zdq_inv[0];  val[2] = Zdq_inv[0]*dVd_ddelta + Zdq_inv[1]*dVq_ddelta;
380:     col[3] = idx + 4; col[4] = net_start+2*gbus[i];                     col[5] = net_start + 2*gbus[i]+1;
381:     val[3] = 1;       val[4] = Zdq_inv[0]*dVd_dVr + Zdq_inv[1]*dVq_dVr; val[5] = Zdq_inv[0]*dVd_dVi + Zdq_inv[1]*dVq_dVi;
382:     MatSetValues(J,1,row,6,col,val,INSERT_VALUES);

384:     /*  fgen[idx+5] = Zdq_inv[2]*(-Edp + Vd) + Zdq_inv[3]*(-Eqp + Vq) + Iq; */
385:     row[0] = idx+5;
386:     col[0] = idx;         col[1] = idx+1;        col[2] = idx + 2;
387:     val[0] = -Zdq_inv[3]; val[1] = -Zdq_inv[2];  val[2] = Zdq_inv[2]*dVd_ddelta + Zdq_inv[3]*dVq_ddelta;
388:     col[3] = idx + 5; col[4] = net_start+2*gbus[i];                     col[5] = net_start + 2*gbus[i]+1;
389:     val[3] = 1;       val[4] = Zdq_inv[2]*dVd_dVr + Zdq_inv[3]*dVq_dVr; val[5] = Zdq_inv[2]*dVd_dVi + Zdq_inv[3]*dVq_dVi;
390:     MatSetValues(J,1,row,6,col,val,INSERT_VALUES);

392:     dIGr_ddelta = Id*PetscCosScalar(delta) - Iq*PetscSinScalar(delta);
393:     dIGi_ddelta = Id*PetscSinScalar(delta) + Iq*PetscCosScalar(delta);
394:     dIGr_dId    = PetscSinScalar(delta);  dIGr_dIq = PetscCosScalar(delta);
395:     dIGi_dId    = -PetscCosScalar(delta); dIGi_dIq = PetscSinScalar(delta);

397:     /* fnet[2*gbus[i]]   -= IGi; */
398:     row[0] = net_start + 2*gbus[i];
399:     col[0] = idx+2;        col[1] = idx + 4;   col[2] = idx + 5;
400:     val[0] = -dIGi_ddelta; val[1] = -dIGi_dId; val[2] = -dIGi_dIq;
401:     MatSetValues(J,1,row,3,col,val,INSERT_VALUES);

403:     /* fnet[2*gbus[i]+1]   -= IGr; */
404:     row[0] = net_start + 2*gbus[i]+1;
405:     col[0] = idx+2;        col[1] = idx + 4;   col[2] = idx + 5;
406:     val[0] = -dIGr_ddelta; val[1] = -dIGr_dId; val[2] = -dIGr_dIq;
407:     MatSetValues(J,1,row,3,col,val,INSERT_VALUES);

409:     Vm = PetscSqrtScalar(Vd*Vd + Vq*Vq); Vm2 = Vm*Vm;

411:     /*    fgen[idx+6] = (KE[i]*Efd + SE - VR)/TE[i]; */
412:     /*    SE  = k1[i]*PetscExpScalar(k2[i]*Efd); */

414:     dSE_dEfd = k1[i]*k2[i]*PetscExpScalar(k2[i]*Efd);

416:     row[0] = idx + 6;
417:     col[0] = idx + 6;                     col[1] = idx + 8;
418:     val[0] = (KE[i] + dSE_dEfd)/TE[i];  val[1] = -1/TE[i];
419:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

421:     /* Exciter differential equations */

423:     /*    fgen[idx+7] = (RF - KF[i]*Efd/TF[i])/TF[i]; */
424:     row[0] = idx + 7;
425:     col[0] = idx + 6;       col[1] = idx + 7;
426:     val[0] = (-KF[i]/TF[i])/TF[i];  val[1] = 1/TF[i];
427:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

429:     /*    fgen[idx+8] = (VR - KA[i]*RF + KA[i]*KF[i]*Efd/TF[i] - KA[i]*(Vref[i] - Vm))/TA[i]; */
430:     /* Vm = (Vd^2 + Vq^2)^0.5; */

432:     dVm_dVd = Vd/Vm; dVm_dVq = Vq/Vm;
433:     dVm_dVr = dVm_dVd*dVd_dVr + dVm_dVq*dVq_dVr;
434:     dVm_dVi = dVm_dVd*dVd_dVi + dVm_dVq*dVq_dVi;
435:     row[0]  = idx + 8;
436:     col[0]  = idx + 6;           col[1] = idx + 7; col[2] = idx + 8;
437:     val[0]  = (KA[i]*KF[i]/TF[i])/TA[i]; val[1] = -KA[i]/TA[i];  val[2] = 1/TA[i];
438:     col[3]  = net_start + 2*gbus[i]; col[4] = net_start + 2*gbus[i]+1;
439:     val[3]  = KA[i]*dVm_dVr/TA[i];         val[4] = KA[i]*dVm_dVi/TA[i];
440:     MatSetValues(J,1,row,5,col,val,INSERT_VALUES);
441:     idx     = idx + 9;
442:   }

444:   for (i=0; i<nbus; i++) {
445:     MatGetRow(user->Ybus,2*i,&ncols,&cols,&yvals);
446:     row[0] = net_start + 2*i;
447:     for (k=0; k<ncols; k++) {
448:       col[k] = net_start + cols[k];
449:       val[k] = yvals[k];
450:     }
451:     MatSetValues(J,1,row,ncols,col,val,INSERT_VALUES);
452:     MatRestoreRow(user->Ybus,2*i,&ncols,&cols,&yvals);

454:     MatGetRow(user->Ybus,2*i+1,&ncols,&cols,&yvals);
455:     row[0] = net_start + 2*i+1;
456:     for (k=0; k<ncols; k++) {
457:       col[k] = net_start + cols[k];
458:       val[k] = yvals[k];
459:     }
460:     MatSetValues(J,1,row,ncols,col,val,INSERT_VALUES);
461:     MatRestoreRow(user->Ybus,2*i+1,&ncols,&cols,&yvals);
462:   }

464:   MatAssemblyBegin(J,MAT_FLUSH_ASSEMBLY);
465:   MatAssemblyEnd(J,MAT_FLUSH_ASSEMBLY);

467:   VecGetArray(user->V0,&v0);
468:   for (i=0; i < nload; i++) {
469:     Vr      = xnet[2*lbus[i]]; /* Real part of load bus voltage */
470:     Vi      = xnet[2*lbus[i]+1]; /* Imaginary part of the load bus voltage */
471:     Vm      = PetscSqrtScalar(Vr*Vr + Vi*Vi); Vm2 = Vm*Vm; Vm4 = Vm2*Vm2;
472:     Vm0     = PetscSqrtScalar(v0[2*lbus[i]]*v0[2*lbus[i]] + v0[2*lbus[i]+1]*v0[2*lbus[i]+1]);
473:     PD      = QD = 0.0;
474:     dPD_dVr = dPD_dVi = dQD_dVr = dQD_dVi = 0.0;
475:     for (k=0; k < ld_nsegsp[i]; k++) {
476:       PD      += ld_alphap[k]*PD0[i]*PetscPowScalar((Vm/Vm0),ld_betap[k]);
477:       dPD_dVr += ld_alphap[k]*ld_betap[k]*PD0[i]*PetscPowScalar((1/Vm0),ld_betap[k])*Vr*PetscPowScalar(Vm,(ld_betap[k]-2));
478:       dPD_dVi += ld_alphap[k]*ld_betap[k]*PD0[i]*PetscPowScalar((1/Vm0),ld_betap[k])*Vi*PetscPowScalar(Vm,(ld_betap[k]-2));
479:     }
480:     for (k=0; k < ld_nsegsq[i]; k++) {
481:       QD      += ld_alphaq[k]*QD0[i]*PetscPowScalar((Vm/Vm0),ld_betaq[k]);
482:       dQD_dVr += ld_alphaq[k]*ld_betaq[k]*QD0[i]*PetscPowScalar((1/Vm0),ld_betaq[k])*Vr*PetscPowScalar(Vm,(ld_betaq[k]-2));
483:       dQD_dVi += ld_alphaq[k]*ld_betaq[k]*QD0[i]*PetscPowScalar((1/Vm0),ld_betaq[k])*Vi*PetscPowScalar(Vm,(ld_betaq[k]-2));
484:     }

486:     /*    IDr = (PD*Vr + QD*Vi)/Vm2; */
487:     /*    IDi = (-QD*Vr + PD*Vi)/Vm2; */

489:     dIDr_dVr = (dPD_dVr*Vr + dQD_dVr*Vi + PD)/Vm2 - ((PD*Vr + QD*Vi)*2*Vr)/Vm4;
490:     dIDr_dVi = (dPD_dVi*Vr + dQD_dVi*Vi + QD)/Vm2 - ((PD*Vr + QD*Vi)*2*Vi)/Vm4;

492:     dIDi_dVr = (-dQD_dVr*Vr + dPD_dVr*Vi - QD)/Vm2 - ((-QD*Vr + PD*Vi)*2*Vr)/Vm4;
493:     dIDi_dVi = (-dQD_dVi*Vr + dPD_dVi*Vi + PD)/Vm2 - ((-QD*Vr + PD*Vi)*2*Vi)/Vm4;


496:     /*    fnet[2*lbus[i]]   += IDi; */
497:     row[0] = net_start + 2*lbus[i];
498:     col[0] = net_start + 2*lbus[i];  col[1] = net_start + 2*lbus[i]+1;
499:     val[0] = dIDi_dVr;               val[1] = dIDi_dVi;
500:     MatSetValues(J,1,row,2,col,val,ADD_VALUES);
501:     /*    fnet[2*lbus[i]+1] += IDr; */
502:     row[0] = net_start + 2*lbus[i]+1;
503:     col[0] = net_start + 2*lbus[i];  col[1] = net_start + 2*lbus[i]+1;
504:     val[0] = dIDr_dVr;               val[1] = dIDr_dVi;
505:     MatSetValues(J,1,row,2,col,val,ADD_VALUES);
506:   }
507:   VecRestoreArray(user->V0,&v0);

509:   VecRestoreArray(Xgen,&xgen);
510:   VecRestoreArray(Xnet,&xnet);

512:   DMCompositeRestoreLocalVectors(user->dmpgrid,&Xgen,&Xnet);

514:   MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
515:   MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
516:   return(0);
517: }

519: int main(int argc,char **argv)
520: {
521:   EPS            eps;
522:   EPSType        type;
524:   PetscMPIInt    size;
525:   Userctx        user;
526:   PetscViewer    Xview,Ybusview;
527:   Vec            X,Xr,Xi;
528:   Mat            J,Jred=NULL;
529:   IS             is0,is1;
530:   PetscInt       i,*idx2,its,nev,nconv;
531:   PetscReal      error,re,im;
532:   PetscScalar    kr,ki;
533:   PetscBool      terse;

535:   SlepcInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
536:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
537:   if (size > 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");
538:   /* show detailed info unless -terse option is given by user */
539:   PetscOptionsHasName(NULL,NULL,"-terse",&terse);

541:   user.neqs_gen   = 9*ngen; /* # eqs. for generator subsystem */
542:   user.neqs_net   = 2*nbus; /* # eqs. for network subsystem   */
543:   user.neqs_pgrid = user.neqs_gen + user.neqs_net;
544:   PetscPrintf(PETSC_COMM_WORLD,"\nStability analysis in a network with %D buses and %D generators\n\n",nbus,ngen);

546:   /* Create indices for differential and algebraic equations */
547:   PetscMalloc1(7*ngen,&idx2);
548:   for (i=0; i<ngen; i++) {
549:     idx2[7*i]   = 9*i;   idx2[7*i+1] = 9*i+1; idx2[7*i+2] = 9*i+2; idx2[7*i+3] = 9*i+3;
550:     idx2[7*i+4] = 9*i+6; idx2[7*i+5] = 9*i+7; idx2[7*i+6] = 9*i+8;
551:   }
552:   ISCreateGeneral(PETSC_COMM_WORLD,7*ngen,idx2,PETSC_COPY_VALUES,&user.is_diff);
553:   ISComplement(user.is_diff,0,user.neqs_pgrid,&user.is_alg);
554:   PetscFree(idx2);

556:   /* Read initial voltage vector and Ybus */
557:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"X.bin",FILE_MODE_READ,&Xview);
558:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"Ybus.bin",FILE_MODE_READ,&Ybusview);

560:   VecCreate(PETSC_COMM_WORLD,&user.V0);
561:   VecSetSizes(user.V0,PETSC_DECIDE,user.neqs_net);
562:   VecLoad(user.V0,Xview);

564:   MatCreate(PETSC_COMM_WORLD,&user.Ybus);
565:   MatSetSizes(user.Ybus,PETSC_DECIDE,PETSC_DECIDE,user.neqs_net,user.neqs_net);
566:   MatSetType(user.Ybus,MATBAIJ);
567:   /*  MatSetBlockSize(user.Ybus,2); */
568:   MatLoad(user.Ybus,Ybusview);

570:   PetscViewerDestroy(&Xview);
571:   PetscViewerDestroy(&Ybusview);

573:   /* Create DMs for generator and network subsystems */
574:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.neqs_gen,1,1,NULL,&user.dmgen);
575:   DMSetOptionsPrefix(user.dmgen,"dmgen_");
576:   DMSetFromOptions(user.dmgen);
577:   DMSetUp(user.dmgen);
578:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.neqs_net,1,1,NULL,&user.dmnet);
579:   DMSetOptionsPrefix(user.dmnet,"dmnet_");
580:   DMSetFromOptions(user.dmnet);
581:   DMSetUp(user.dmnet);

583:   /* Create a composite DM packer and add the two DMs */
584:   DMCompositeCreate(PETSC_COMM_WORLD,&user.dmpgrid);
585:   DMSetOptionsPrefix(user.dmpgrid,"pgrid_");
586:   DMCompositeAddDM(user.dmpgrid,user.dmgen);
587:   DMCompositeAddDM(user.dmpgrid,user.dmnet);

589:   DMCreateGlobalVector(user.dmpgrid,&X);

591:   MatCreate(PETSC_COMM_WORLD,&J);
592:   MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,user.neqs_pgrid,user.neqs_pgrid);
593:   MatSetFromOptions(J);
594:   PreallocateJacobian(J,&user);

596:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
597:      Set initial conditions
598:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
599:   SetInitialGuess(X,&user);

601:   /* Form Jacobian */
602:   ResidualJacobian(X,J,(void*)&user);
603:   MatScale(J,-1);
604:   is0 = user.is_diff;
605:   is1 = user.is_alg;

607:   MatGetSchurComplement(J,is1,is1,is0,is0,MAT_IGNORE_MATRIX,NULL,MAT_SCHUR_COMPLEMENT_AINV_DIAG,MAT_INITIAL_MATRIX,&Jred);

609:   if (!terse) {
610:     MatView(Jred,NULL);
611:   }

613:   MatCreateVecs(Jred,NULL,&Xr);
614:   MatCreateVecs(Jred,NULL,&Xi);

616:   /* Create the eigensolver and set the various options */
617:   EPSCreate(PETSC_COMM_WORLD,&eps);
618:   EPSSetOperators(eps,Jred,NULL);
619:   EPSSetProblemType(eps,EPS_NHEP);
620:   EPSSetFromOptions(eps);

622:   /* Solve the eigenvalue problem */
623:   EPSSolve(eps);

625:   EPSGetIterationNumber(eps,&its);
626:   PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the eigensolver: %D\n",its);
627:   EPSGetType(eps,&type);
628:   PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n", type);
629:   EPSGetDimensions(eps,&nev,NULL,NULL);
630:   PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %D\n",nev);

632:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
633:                     Display solution and clean up
634:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
635:   if (terse) {
636:     EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL);
637:   } else {
638:     /* Get number of converged approximate eigenpairs */
639:     EPSGetConverged(eps,&nconv);
640:     PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %D\n\n",nconv);

642:     if (nconv>0) {
643:       /* Display eigenvalues and relative errors */
644:       PetscPrintf(PETSC_COMM_WORLD,
645:            "           k          ||Ax-kx||/||kx||\n"
646:            "   ----------------- ------------------\n");

648:       for (i=0;i<nconv;i++) {
649:         /* Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
650:           ki (imaginary part) */
651:         EPSGetEigenpair(eps,i,&kr,&ki,Xr,Xi);
652:         /* Compute the relative error associated to each eigenpair */
653:         EPSComputeError(eps,i,EPS_ERROR_RELATIVE,&error);

655: #if defined(PETSC_USE_COMPLEX)
656:         re = PetscRealPart(kr);
657:         im = PetscImaginaryPart(kr);
658: #else
659:         re = kr;
660:         im = ki;
661: #endif
662:         if (im!=0.0) {
663:           PetscPrintf(PETSC_COMM_WORLD," %9f%+9fi %12g\n",(double)re,(double)im,(double)error);
664:         } else {
665:           PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g\n",(double)re,(double)error);
666:         }
667:       }
668:       PetscPrintf(PETSC_COMM_WORLD,"\n");
669:     }
670:   }

672:   /* Free work space */
673:   EPSDestroy(&eps);
674:   MatDestroy(&J);
675:   MatDestroy(&Jred);
676:   MatDestroy(&user.Ybus);
677:   VecDestroy(&X);
678:   VecDestroy(&Xr);
679:   VecDestroy(&Xi);
680:   VecDestroy(&user.V0);
681:   DMDestroy(&user.dmgen);
682:   DMDestroy(&user.dmnet);
683:   DMDestroy(&user.dmpgrid);
684:   ISDestroy(&user.is_diff);
685:   ISDestroy(&user.is_alg);
686:   SlepcFinalize();
687:   return ierr;
688: }

690: /*TEST

692:    build:
693:       requires: !complex

695:    test:
696:       suffix: 1
697:       args: -terse
698:       requires: double !complex !define(PETSC_USE_64BIT_INDICES)
699:       localrunfiles: X.bin Ybus.bin

701: TEST*/