Date: 2021-02-01T14:42:04+00:00

Actual source code: ex31.c
slepc-3.14.2 2021-02-01
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2020, 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*/