Actual source code: ex31.c

slepc-3.16.2 2022-02-01
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

189:     theta = PETSC_PI/2.0 - delta;

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

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

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

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

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

208:     idx = idx + 4;

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

213:     idx = idx + 2;

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

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

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

227:     idx = idx + 3;
228:   }

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

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

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

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

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

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

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

266:     idx = idx + 9;
267:   }

269:   start = user->neqs_gen;

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

278:   MatSeqAIJSetPreallocation(J,0,d_nnz);

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

284: /*
285:    J = [-df_dx, -df_dy
286:         dg_dx, dg_dy]
287: */
288: PetscErrorCode ResidualJacobian(Vec X,Mat J,void *ctx)
289: {
291:   Userctx        *user=(Userctx*)ctx;
292:   Vec            Xgen,Xnet;
293:   PetscScalar    *xgen,*xnet;
294:   PetscInt       i,idx=0;
295:   PetscScalar    Vr,Vi,Vm,Vm2;
296:   PetscScalar    Eqp,Edp,delta; /* Generator variables */
297:   PetscScalar    Efd;
298:   PetscScalar    Id,Iq;  /* Generator dq axis currents */
299:   PetscScalar    Vd,Vq;
300:   PetscScalar    val[10];
301:   PetscInt       row[2],col[10];
302:   PetscInt       net_start=user->neqs_gen;
303:   PetscScalar    Zdq_inv[4],det;
304:   PetscScalar    dVd_dVr,dVd_dVi,dVq_dVr,dVq_dVi,dVd_ddelta,dVq_ddelta;
305:   PetscScalar    dIGr_ddelta,dIGi_ddelta,dIGr_dId,dIGr_dIq,dIGi_dId,dIGi_dIq;
306:   PetscScalar    dSE_dEfd;
307:   PetscScalar    dVm_dVd,dVm_dVq,dVm_dVr,dVm_dVi;
308:   PetscInt          ncols;
309:   const PetscInt    *cols;
310:   const PetscScalar *yvals;
311:   PetscInt          k;
312:   PetscScalar PD,QD,Vm0,*v0,Vm4;
313:   PetscScalar dPD_dVr,dPD_dVi,dQD_dVr,dQD_dVi;
314:   PetscScalar dIDr_dVr,dIDr_dVi,dIDi_dVr,dIDi_dVi;

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

321:   VecGetArray(Xgen,&xgen);
322:   VecGetArray(Xnet,&xnet);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

419:     /* Exciter differential equations */

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

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

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

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

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

462:   MatAssemblyBegin(J,MAT_FLUSH_ASSEMBLY);
463:   MatAssemblyEnd(J,MAT_FLUSH_ASSEMBLY);

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

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

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

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

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

506:   VecRestoreArray(Xgen,&xgen);
507:   VecRestoreArray(Xnet,&xnet);

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

511:   MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
512:   MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
513:   return(0);
514: }

516: int main(int argc,char **argv)
517: {
518:   EPS            eps;
519:   EPSType        type;
521:   PetscMPIInt    size;
522:   Userctx        user;
523:   PetscViewer    Xview,Ybusview;
524:   Vec            X,Xr,Xi;
525:   Mat            J,Jred=NULL;
526:   IS             is0,is1;
527:   PetscInt       i,*idx2,its,nev,nconv;
528:   PetscReal      error,re,im;
529:   PetscScalar    kr,ki;
530:   PetscBool      terse;

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

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

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

553:   /* Read initial voltage vector and Ybus */
554:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"X.bin",FILE_MODE_READ,&Xview);
555:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"Ybus.bin",FILE_MODE_READ,&Ybusview);

557:   VecCreate(PETSC_COMM_WORLD,&user.V0);
558:   VecSetSizes(user.V0,PETSC_DECIDE,user.neqs_net);
559:   VecLoad(user.V0,Xview);

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

567:   PetscViewerDestroy(&Xview);
568:   PetscViewerDestroy(&Ybusview);

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

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

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

588:   MatCreate(PETSC_COMM_WORLD,&J);
589:   MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,user.neqs_pgrid,user.neqs_pgrid);
590:   MatSetFromOptions(J);
591:   PreallocateJacobian(J,&user);

593:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
594:      Set initial conditions
595:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
596:   SetInitialGuess(X,&user);

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

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

606:   if (!terse) {
607:     MatView(Jred,NULL);
608:   }

610:   MatCreateVecs(Jred,NULL,&Xr);
611:   MatCreateVecs(Jred,NULL,&Xi);

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

619:   /* Solve the eigenvalue problem */
620:   EPSSolve(eps);

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

629:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
630:                     Display solution and clean up
631:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
632:   if (terse) {
633:     EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL);
634:   } else {
635:     /* Get number of converged approximate eigenpairs */
636:     EPSGetConverged(eps,&nconv);
637:     PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %D\n\n",nconv);

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

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

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

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

687: /*TEST

689:    build:
690:       requires: !complex

692:    test:
693:       suffix: 1
694:       args: -terse
695:       requires: double !complex !defined(PETSC_USE_64BIT_INDICES)
696:       localrunfiles: X.bin Ybus.bin

698: TEST*/