Actual source code: ex31.c

      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-, 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\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: {
    139:   PetscFunctionBegin;
    140:   *Fr =  Fd*PetscSinScalar(delta) + Fq*PetscCosScalar(delta);
    141:   *Fi = -Fd*PetscCosScalar(delta) + Fq*PetscSinScalar(delta);
    142:   PetscFunctionReturn(PETSC_SUCCESS);
    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: {
    148:   PetscFunctionBegin;
    149:   *Fd =  Fr*PetscSinScalar(delta) - Fi*PetscCosScalar(delta);
    150:   *Fq =  Fr*PetscCosScalar(delta) + Fi*PetscSinScalar(delta);
    151:   PetscFunctionReturn(PETSC_SUCCESS);
    152: }

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

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

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

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

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

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

    188:     theta = PETSC_PI/2.0 - delta;

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

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

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

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

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

    207:     idx = idx + 4;

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

    212:     idx = idx + 2;

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

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

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

    226:     idx = idx + 3;
    227:   }

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

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

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

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

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

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

    260:     d_nnz[idx+6] += 2;
    261:     d_nnz[idx+7] += 2;
    262:     d_nnz[idx+8] += 5;

    264:     idx = idx + 9;
    265:   }

    267:   start = user->neqs_gen;

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

    276:   PetscCall(MatSeqAIJSetPreallocation(J,0,d_nnz));

    278:   PetscCall(PetscFree(d_nnz));
    279:   PetscFunctionReturn(PETSC_SUCCESS);
    280: }

    282: /*
    283:    J = [-df_dx, -df_dy
    284:         dg_dx, dg_dy]
    285: */
    286: PetscErrorCode ResidualJacobian(Vec X,Mat J,void *ctx)
    287: {
    288:   Userctx        *user=(Userctx*)ctx;
    289:   Vec            Xgen,Xnet;
    290:   PetscScalar    *xgen,*xnet;
    291:   PetscInt       i,idx=0;
    292:   PetscScalar    Vr,Vi,Vm,Vm2;
    293:   PetscScalar    Eqp,Edp,delta; /* Generator variables */
    294:   PetscScalar    Efd;
    295:   PetscScalar    Id,Iq;  /* Generator dq axis currents */
    296:   PetscScalar    Vd,Vq;
    297:   PetscScalar    val[10];
    298:   PetscInt       row[2],col[10];
    299:   PetscInt       net_start=user->neqs_gen;
    300:   PetscScalar    Zdq_inv[4],det;
    301:   PetscScalar    dVd_dVr,dVd_dVi,dVq_dVr,dVq_dVi,dVd_ddelta,dVq_ddelta;
    302:   PetscScalar    dIGr_ddelta,dIGi_ddelta,dIGr_dId,dIGr_dIq,dIGi_dId,dIGi_dIq;
    303:   PetscScalar    dSE_dEfd;
    304:   PetscScalar    dVm_dVd,dVm_dVq,dVm_dVr,dVm_dVi;
    305:   PetscInt          ncols;
    306:   const PetscInt    *cols;
    307:   const PetscScalar *yvals;
    308:   PetscInt          k;
    309:   PetscScalar PD,QD,Vm0,*v0,Vm4;
    310:   PetscScalar dPD_dVr,dPD_dVi,dQD_dVr,dQD_dVi;
    311:   PetscScalar dIDr_dVr,dIDr_dVi,dIDi_dVr,dIDi_dVi;

    313:   PetscFunctionBegin;
    314:   PetscCall(MatZeroEntries(J));
    315:   PetscCall(DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet));
    316:   PetscCall(DMCompositeScatter(user->dmpgrid,X,Xgen,Xnet));

    318:   PetscCall(VecGetArray(Xgen,&xgen));
    319:   PetscCall(VecGetArray(Xnet,&xnet));

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

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

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

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

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

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

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

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

    361:     Zdq_inv[0] = Rs[i]/det;
    362:     Zdq_inv[1] = Xqp[i]/det;
    363:     Zdq_inv[2] = -Xdp[i]/det;
    364:     Zdq_inv[3] = Rs[i]/det;

    366:     dVd_dVr    = PetscSinScalar(delta); dVd_dVi = -PetscCosScalar(delta);
    367:     dVq_dVr    = PetscCosScalar(delta); dVq_dVi = PetscSinScalar(delta);
    368:     dVd_ddelta = Vr*PetscCosScalar(delta) + Vi*PetscSinScalar(delta);
    369:     dVq_ddelta = -Vr*PetscSinScalar(delta) + Vi*PetscCosScalar(delta);

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

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

    387:     dIGr_ddelta = Id*PetscCosScalar(delta) - Iq*PetscSinScalar(delta);
    388:     dIGi_ddelta = Id*PetscSinScalar(delta) + Iq*PetscCosScalar(delta);
    389:     dIGr_dId    = PetscSinScalar(delta);  dIGr_dIq = PetscCosScalar(delta);
    390:     dIGi_dId    = -PetscCosScalar(delta); dIGi_dIq = PetscSinScalar(delta);

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

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

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

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

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

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

    416:     /* Exciter differential equations */

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

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

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

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

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

    459:   PetscCall(MatAssemblyBegin(J,MAT_FLUSH_ASSEMBLY));
    460:   PetscCall(MatAssemblyEnd(J,MAT_FLUSH_ASSEMBLY));

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

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

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

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

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

    503:   PetscCall(VecRestoreArray(Xgen,&xgen));
    504:   PetscCall(VecRestoreArray(Xnet,&xnet));

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

    508:   PetscCall(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY));
    509:   PetscCall(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY));
    510:   PetscFunctionReturn(PETSC_SUCCESS);
    511: }

    513: int main(int argc,char **argv)
    514: {
    515:   EPS            eps;
    516:   EPSType        type;
    517:   PetscMPIInt    size;
    518:   Userctx        user;
    519:   PetscViewer    Xview,Ybusview;
    520:   Vec            X,Xr,Xi;
    521:   Mat            J,Jred=NULL;
    522:   IS             is0,is1;
    523:   PetscInt       i,*idx2,its,nev,nconv;
    524:   PetscReal      error,re,im;
    525:   PetscScalar    kr,ki;
    526:   PetscBool      terse;

    528:   PetscFunctionBeginUser;
    529:   PetscCall(SlepcInitialize(&argc,&argv,NULL,help));
    530:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD,&size));
    531:   PetscCheck(size==1,PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");
    532:   /* show detailed info unless -terse option is given by user */
    533:   PetscCall(PetscOptionsHasName(NULL,NULL,"-terse",&terse));

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

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

    550:   /* Read initial voltage vector and Ybus */
    551:   PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,"X.bin",FILE_MODE_READ,&Xview));
    552:   PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD,"Ybus.bin",FILE_MODE_READ,&Ybusview));

    554:   PetscCall(VecCreate(PETSC_COMM_WORLD,&user.V0));
    555:   PetscCall(VecSetSizes(user.V0,PETSC_DECIDE,user.neqs_net));
    556:   PetscCall(VecLoad(user.V0,Xview));

    558:   PetscCall(MatCreate(PETSC_COMM_WORLD,&user.Ybus));
    559:   PetscCall(MatSetSizes(user.Ybus,PETSC_DECIDE,PETSC_DECIDE,user.neqs_net,user.neqs_net));
    560:   PetscCall(MatSetType(user.Ybus,MATBAIJ));
    561:   /*  PetscCall(MatSetBlockSize(user.Ybus,2)); */
    562:   PetscCall(MatLoad(user.Ybus,Ybusview));

    564:   PetscCall(PetscViewerDestroy(&Xview));
    565:   PetscCall(PetscViewerDestroy(&Ybusview));

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

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

    583:   PetscCall(DMCreateGlobalVector(user.dmpgrid,&X));

    585:   PetscCall(MatCreate(PETSC_COMM_WORLD,&J));
    586:   PetscCall(MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,user.neqs_pgrid,user.neqs_pgrid));
    587:   PetscCall(MatSetFromOptions(J));
    588:   PetscCall(PreallocateJacobian(J,&user));

    590:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    591:      Set initial conditions
    592:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    593:   PetscCall(SetInitialGuess(X,&user));

    595:   /* Form Jacobian */
    596:   PetscCall(ResidualJacobian(X,J,(void*)&user));
    597:   PetscCall(MatScale(J,-1));
    598:   is0 = user.is_diff;
    599:   is1 = user.is_alg;

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

    603:   if (!terse) PetscCall(MatView(Jred,NULL));

    605:   PetscCall(MatCreateVecs(Jred,NULL,&Xr));
    606:   PetscCall(MatCreateVecs(Jred,NULL,&Xi));

    608:   /* Create the eigensolver and set the various options */
    609:   PetscCall(EPSCreate(PETSC_COMM_WORLD,&eps));
    610:   PetscCall(EPSSetOperators(eps,Jred,NULL));
    611:   PetscCall(EPSSetProblemType(eps,EPS_NHEP));
    612:   PetscCall(EPSSetFromOptions(eps));

    614:   /* Solve the eigenvalue problem */
    615:   PetscCall(EPSSolve(eps));

    617:   PetscCall(EPSGetIterationNumber(eps,&its));
    618:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the eigensolver: %" PetscInt_FMT "\n",its));
    619:   PetscCall(EPSGetType(eps,&type));
    620:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n", type));
    621:   PetscCall(EPSGetDimensions(eps,&nev,NULL,NULL));
    622:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %" PetscInt_FMT "\n",nev));

    624:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    625:                     Display solution and clean up
    626:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    627:   if (terse) PetscCall(EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL));
    628:   else {
    629:     /* Get number of converged approximate eigenpairs */
    630:     PetscCall(EPSGetConverged(eps,&nconv));
    631:     PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %" PetscInt_FMT "\n\n",nconv));

    633:     if (nconv>0) {
    634:       /* Display eigenvalues and relative errors */
    635:       PetscCall(PetscPrintf(PETSC_COMM_WORLD,
    636:            "           k          ||Ax-kx||/||kx||\n"
    637:            "   ----------------- ------------------\n"));

    639:       for (i=0;i<nconv;i++) {
    640:         /* Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
    641:           ki (imaginary part) */
    642:         PetscCall(EPSGetEigenpair(eps,i,&kr,&ki,Xr,Xi));
    643:         /* Compute the relative error associated to each eigenpair */
    644:         PetscCall(EPSComputeError(eps,i,EPS_ERROR_RELATIVE,&error));

    646: #if defined(PETSC_USE_COMPLEX)
    647:         re = PetscRealPart(kr);
    648:         im = PetscImaginaryPart(kr);
    649: #else
    650:         re = kr;
    651:         im = ki;
    652: #endif
    653:         if (im!=0.0) PetscCall(PetscPrintf(PETSC_COMM_WORLD," %9f%+9fi %12g\n",(double)re,(double)im,(double)error));
    654:         else PetscCall(PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g\n",(double)re,(double)error));
    655:       }
    656:       PetscCall(PetscPrintf(PETSC_COMM_WORLD,"\n"));
    657:     }
    658:   }

    660:   /* Free work space */
    661:   PetscCall(EPSDestroy(&eps));
    662:   PetscCall(MatDestroy(&J));
    663:   PetscCall(MatDestroy(&Jred));
    664:   PetscCall(MatDestroy(&user.Ybus));
    665:   PetscCall(VecDestroy(&X));
    666:   PetscCall(VecDestroy(&Xr));
    667:   PetscCall(VecDestroy(&Xi));
    668:   PetscCall(VecDestroy(&user.V0));
    669:   PetscCall(DMDestroy(&user.dmgen));
    670:   PetscCall(DMDestroy(&user.dmnet));
    671:   PetscCall(DMDestroy(&user.dmpgrid));
    672:   PetscCall(ISDestroy(&user.is_diff));
    673:   PetscCall(ISDestroy(&user.is_alg));
    674:   PetscCall(SlepcFinalize());
    675:   return 0;
    676: }

    678: /*TEST

    680:    build:
    681:       requires: !complex

    683:    test:
    684:       suffix: 1
    685:       args: -terse
    686:       requires: double !complex !defined(PETSC_USE_64BIT_INDICES)
    687:       localrunfiles: X.bin Ybus.bin

    689: TEST*/