Date: 2017-05-17T08:26:25+00:00

Actual source code: ex31.c
slepc-3.7.4 2017-05-17
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2016, Universitat Politecnica de Valencia, Spain

      6:    This file is part of SLEPc.

      8:    SLEPc is free software: you can redistribute it and/or modify it under  the
      9:    terms of version 3 of the GNU Lesser General Public License as published by
     10:    the Free Software Foundation.

     12:    SLEPc  is  distributed in the hope that it will be useful, but WITHOUT  ANY
     13:    WARRANTY;  without even the implied warranty of MERCHANTABILITY or  FITNESS
     14:    FOR  A  PARTICULAR PURPOSE. See the GNU Lesser General Public  License  for
     15:    more details.

     17:    You  should have received a copy of the GNU Lesser General  Public  License
     18:    along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
     19:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     20: */

     22: static char help[] = "Power grid small signal stability analysis of WECC 9 bus system.\n\
     23: This example is based on the 9-bus (node) example given in the book Power\n\
     24: Systems Dynamics and Stability (Chapter 8) by P. Sauer and M. A. Pai.\n\
     25: The power grid in this example consists of 9 buses (nodes), 3 generators,\n\
     26: 3 loads, and 9 transmission lines. The network equations are written\n\
     27: in current balance form using rectangular coordinates. It uses the SLEPc\n\
     28: package to calculate the eigenvalues for small signal stability analysis\n\n";

     30: /*
     31:    This example is based on PETSc's ex9bus example (under TS).

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

     35:    \dot{x} = f(x,y,t)
     36:      0     = g(x,y,t)

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

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

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

     49:    where:
     50:     I(x,y) is the current injected from generators and loads.
     51:       Y    is the admittance matrix, and
     52:       V    is the voltage vector

     54:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     55:
     56:    The linearized equations for the eigenvalue analysis are

     58:      \dot{\delta{x}} = f_x*\delta{x} + f_y*\delta{y}
     59:              0       = g_x*\delta{x} + g_y*\delta{y}

     61:    This gives the linearized sensitivity matrix
     62:      A = | f_x  f_y |
     63:          | g_x  g_y |

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

     69:    Example contributed by: Shrirang Abhyankar
     70: */

     72: #include <petscdm.h>
     73: #include <petscdmda.h>
     74: #include <petscdmcomposite.h>
     75: #include <slepceps.h>

     77: #define freq 60
     78: #define w_s (2*PETSC_PI*freq)

     80: /* Sizes and indices */
     81: const PetscInt nbus    = 9; /* Number of network buses */
     82: const PetscInt ngen    = 3; /* Number of generators */
     83: const PetscInt nload   = 3; /* Number of loads */
     84: const PetscInt gbus[3] = {0,1,2}; /* Buses at which generators are incident */
     85: const PetscInt lbus[3] = {4,5,7}; /* Buses at which loads are incident */

     87: /* Generator real and reactive powers (found via loadflow) */
     88: const PetscScalar PG[3] = {0.716786142395021,1.630000000000000,0.850000000000000};
     89: const PetscScalar QG[3] = {0.270702180178785,0.066120127797275,-0.108402221791588};
     90: /* Generator constants */
     91: const PetscScalar H[3]    = {23.64,6.4,3.01};   /* Inertia constant */
     92: const PetscScalar Rs[3]   = {0.0,0.0,0.0}; /* Stator Resistance */
     93: const PetscScalar Xd[3]   = {0.146,0.8958,1.3125};  /* d-axis reactance */
     94: const PetscScalar Xdp[3]  = {0.0608,0.1198,0.1813}; /* d-axis transient reactance */
     95: 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 */
     96: const PetscScalar Xqp[3]  = {0.0969,0.1969,0.25}; /* q-axis transient reactance */
     97: const PetscScalar Td0p[3] = {8.96,6.0,5.89}; /* d-axis open circuit time constant */
     98: const PetscScalar Tq0p[3] = {0.31,0.535,0.6}; /* q-axis open circuit time constant */
     99: PetscScalar M[3]; /* M = 2*H/w_s */
    100: PetscScalar D[3]; /* D = 0.1*M */

    102: PetscScalar TM[3]; /* Mechanical Torque */
    103: /* Exciter system constants */
    104: const PetscScalar KA[3] = {20.0,20.0,20.0};  /* Voltage regulartor gain constant */
    105: const PetscScalar TA[3] = {0.2,0.2,0.2};     /* Voltage regulator time constant */
    106: const PetscScalar KE[3] = {1.0,1.0,1.0};     /* Exciter gain constant */
    107: const PetscScalar TE[3] = {0.314,0.314,0.314}; /* Exciter time constant */
    108: const PetscScalar KF[3] = {0.063,0.063,0.063};  /* Feedback stabilizer gain constant */
    109: const PetscScalar TF[3] = {0.35,0.35,0.35};    /* Feedback stabilizer time constant */
    110: const PetscScalar k1[3] = {0.0039,0.0039,0.0039};
    111: const PetscScalar k2[3] = {1.555,1.555,1.555};  /* k1 and k2 for calculating the saturation function SE = k1*exp(k2*Efd) */

    113: PetscScalar Vref[3];
    114: /* Load constants
    115:   We use a composite load model that describes the load and reactive powers at each time instant as follows
    116:   P(t) = \sum\limits_{i=0}^ld_nsegsp \ld_alphap_i*P_D0(\frac{V_m(t)}{V_m0})^\ld_betap_i
    117:   Q(t) = \sum\limits_{i=0}^ld_nsegsq \ld_alphaq_i*Q_D0(\frac{V_m(t)}{V_m0})^\ld_betaq_i
    118:   where
    119:     ld_nsegsp,ld_nsegsq - Number of individual load models for real and reactive power loads
    120:     ld_alphap,ld_alphap - Percentage contribution (weights) or loads
    121:     P_D0                - Real power load
    122:     Q_D0                - Reactive power load
    123:     V_m(t)              - Voltage magnitude at time t
    124:     V_m0                - Voltage magnitude at t = 0
    125:     ld_betap, ld_betaq  - exponents describing the load model for real and reactive part

    127:     Note: All loads have the same characteristic currently.
    128: */
    129: const PetscScalar PD0[3] = {1.25,0.9,1.0};
    130: const PetscScalar QD0[3] = {0.5,0.3,0.35};
    131: const PetscInt    ld_nsegsp[3] = {3,3,3};
    132: const PetscScalar ld_alphap[3] = {0.0,0.0,1.0};
    133: const PetscScalar ld_betap[3]  = {2.0,1.0,0.0};
    134: const PetscInt    ld_nsegsq[3] = {3,3,3};
    135: const PetscScalar ld_alphaq[3] = {0.0,0.0,1.0};
    136: const PetscScalar ld_betaq[3]  = {2.0,1.0,0.0};

    138: typedef struct {
    139:   DM       dmgen, dmnet; /* DMs to manage generator and network subsystem */
    140:   DM       dmpgrid;      /* Composite DM to manage the entire power grid */
    141:   Mat      Ybus;         /* Network admittance matrix */
    142:   Vec      V0;           /* Initial voltage vector (Power flow solution) */
    143:   PetscInt neqs_gen,neqs_net,neqs_pgrid;
    144:   IS       is_diff;      /* indices for differential equations */
    145:   IS       is_alg;       /* indices for algebraic equations */
    146: } Userctx;

    148: /* Converts from machine frame (dq) to network (phase a real,imag) reference frame */
    151: PetscErrorCode dq2ri(PetscScalar Fd,PetscScalar Fq,PetscScalar delta,PetscScalar *Fr,PetscScalar *Fi)
    152: {
    154:   *Fr =  Fd*PetscSinScalar(delta) + Fq*PetscCosScalar(delta);
    155:   *Fi = -Fd*PetscCosScalar(delta) + Fq*PetscSinScalar(delta);
    156:   return(0);
    157: }

    159: /* Converts from network frame ([phase a real,imag) to machine (dq) reference frame */
    162: PetscErrorCode ri2dq(PetscScalar Fr,PetscScalar Fi,PetscScalar delta,PetscScalar *Fd,PetscScalar *Fq)
    163: {
    165:   *Fd =  Fr*PetscSinScalar(delta) - Fi*PetscCosScalar(delta);
    166:   *Fq =  Fr*PetscCosScalar(delta) + Fi*PetscSinScalar(delta);
    167:   return(0);
    168: }

    172: PetscErrorCode SetInitialGuess(Vec X,Userctx *user)
    173: {
    175:   Vec            Xgen,Xnet;
    176:   PetscScalar    *xgen,*xnet;
    177:   PetscInt       i,idx=0;
    178:   PetscScalar    Vr,Vi,IGr,IGi,Vm,Vm2;
    179:   PetscScalar    Eqp,Edp,delta;
    180:   PetscScalar    Efd,RF,VR; /* Exciter variables */
    181:   PetscScalar    Id,Iq;  /* Generator dq axis currents */
    182:   PetscScalar    theta,Vd,Vq,SE;

    185:   M[0] = 2*H[0]/w_s; M[1] = 2*H[1]/w_s; M[2] = 2*H[2]/w_s;
    186:       /*      D[0] = 0.1*M[0]; D[1] = 0.1*M[1]; D[2] = 0.1*M[2];
    187:        */
    188:   D[0] = D[1] = D[2] = 0.0;
    189:   DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet);

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

    194:   /* Generator subsystem initialization */
    195:   VecGetArray(Xgen,&xgen);
    196:   VecGetArray(Xnet,&xnet);

    198:   for (i=0; i < ngen; i++) {
    199:     Vr  = xnet[2*gbus[i]]; /* Real part of generator terminal voltage */
    200:     Vi  = xnet[2*gbus[i]+1]; /* Imaginary part of the generator terminal voltage */
    201:     Vm  = PetscSqrtScalar(Vr*Vr + Vi*Vi); Vm2 = Vm*Vm;
    202:     IGr = (Vr*PG[i] + Vi*QG[i])/Vm2;
    203:     IGi = (Vi*PG[i] - Vr*QG[i])/Vm2;

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

    207:     theta = PETSC_PI/2.0 - delta;

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

    212:     Vd = Vr*PetscCosScalar(theta) - Vi*PetscSinScalar(theta);
    213:     Vq = Vr*PetscSinScalar(theta) + Vi*PetscCosScalar(theta);

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

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

    220:     /* The generator variables are ordered as [Eqp,Edp,delta,w,Id,Iq] */
    221:     xgen[idx]   = Eqp;
    222:     xgen[idx+1] = Edp;
    223:     xgen[idx+2] = delta;
    224:     xgen[idx+3] = w_s;

    226:     idx = idx + 4;

    228:     xgen[idx]   = Id;
    229:     xgen[idx+1] = Iq;

    231:     idx = idx + 2;

    233:     /* Exciter */
    234:     Efd = Eqp + (Xd[i] - Xdp[i])*Id;
    235:     SE  = k1[i]*PetscExpScalar(k2[i]*Efd);
    236:     VR  =  KE[i]*Efd + SE;
    237:     RF  =  KF[i]*Efd/TF[i];

    239:     xgen[idx]   = Efd;
    240:     xgen[idx+1] = RF;
    241:     xgen[idx+2] = VR;

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

    245:     idx = idx + 3;
    246:   }

    248:   VecRestoreArray(Xgen,&xgen);
    249:   VecRestoreArray(Xnet,&xnet);

    251:   /* VecView(Xgen,0); */
    252:   DMCompositeGather(user->dmpgrid,X,INSERT_VALUES,Xgen,Xnet);
    253:   DMCompositeRestoreLocalVectors(user->dmpgrid,&Xgen,&Xnet);
    254:   return(0);
    255: }

    259: PetscErrorCode PreallocateJacobian(Mat J,Userctx *user)
    260: {
    262:   PetscInt       *d_nnz;
    263:   PetscInt       i,idx=0,start=0;
    264:   PetscInt       ncols;

    267:   PetscMalloc1(user->neqs_pgrid,&d_nnz);
    268:   for (i=0; i<user->neqs_pgrid; i++) d_nnz[i] = 0;
    269:   /* Generator subsystem */
    270:   for (i=0; i < ngen; i++) {

    272:     d_nnz[idx]   += 3;
    273:     d_nnz[idx+1] += 2;
    274:     d_nnz[idx+2] += 2;
    275:     d_nnz[idx+3] += 5;
    276:     d_nnz[idx+4] += 6;
    277:     d_nnz[idx+5] += 6;

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

    282:     d_nnz[idx+6] += 2;
    283:     d_nnz[idx+7] += 2;
    284:     d_nnz[idx+8] += 5;

    286:     idx = idx + 9;
    287:   }

    289:   start = user->neqs_gen;

    291:   for (i=0; i < nbus; i++) {
    292:     MatGetRow(user->Ybus,2*i,&ncols,NULL,NULL);
    293:     d_nnz[start+2*i]   += ncols;
    294:     d_nnz[start+2*i+1] += ncols;
    295:     MatRestoreRow(user->Ybus,2*i,&ncols,NULL,NULL);
    296:   }

    298:   MatSeqAIJSetPreallocation(J,0,d_nnz);

    300:   PetscFree(d_nnz);
    301:   return(0);
    302: }

    304: /*
    305:    J = [-df_dx, -df_dy
    306:         dg_dx, dg_dy]
    307: */
    310: PetscErrorCode ResidualJacobian(Vec X,Mat J,void *ctx)
    311: {
    313:   Userctx        *user=(Userctx*)ctx;
    314:   Vec            Xgen,Xnet;
    315:   PetscScalar    *xgen,*xnet;
    316:   PetscInt       i,idx=0;
    317:   PetscScalar    Vr,Vi,Vm,Vm2;
    318:   PetscScalar    Eqp,Edp,delta; /* Generator variables */
    319:   PetscScalar    Efd;
    320:   PetscScalar    Id,Iq;  /* Generator dq axis currents */
    321:   PetscScalar    Vd,Vq;
    322:   PetscScalar    val[10];
    323:   PetscInt       row[2],col[10];
    324:   PetscInt       net_start=user->neqs_gen;
    325:   PetscScalar    Zdq_inv[4],det;
    326:   PetscScalar    dVd_dVr,dVd_dVi,dVq_dVr,dVq_dVi,dVd_ddelta,dVq_ddelta;
    327:   PetscScalar    dIGr_ddelta,dIGi_ddelta,dIGr_dId,dIGr_dIq,dIGi_dId,dIGi_dIq;
    328:   PetscScalar    dSE_dEfd;
    329:   PetscScalar    dVm_dVd,dVm_dVq,dVm_dVr,dVm_dVi;
    330:   PetscInt          ncols;
    331:   const PetscInt    *cols;
    332:   const PetscScalar *yvals;
    333:   PetscInt          k;
    334:   PetscScalar PD,QD,Vm0,*v0,Vm4;
    335:   PetscScalar dPD_dVr,dPD_dVi,dQD_dVr,dQD_dVi;
    336:   PetscScalar dIDr_dVr,dIDr_dVi,dIDi_dVr,dIDi_dVi;

    340:   MatZeroEntries(J);
    341:   DMCompositeGetLocalVectors(user->dmpgrid,&Xgen,&Xnet);
    342:   DMCompositeScatter(user->dmpgrid,X,Xgen,Xnet);

    344:   VecGetArray(Xgen,&xgen);
    345:   VecGetArray(Xnet,&xnet);

    347:   /* Generator subsystem */
    348:   for (i=0; i < ngen; i++) {
    349:     Eqp   = xgen[idx];
    350:     Edp   = xgen[idx+1];
    351:     delta = xgen[idx+2];
    352:     Id    = xgen[idx+4];
    353:     Iq    = xgen[idx+5];
    354:     Efd   = xgen[idx+6];

    356:     /*    fgen[idx]   = (Eqp + (Xd[i] - Xdp[i])*Id - Efd)/Td0p[i]; */
    357:     row[0] = idx;
    358:     col[0] = idx;           col[1] = idx+4;          col[2] = idx+6;
    359:     val[0] = 1/ Td0p[i]; val[1] = (Xd[i] - Xdp[i])/ Td0p[i]; val[2] = -1/Td0p[i];

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

    363:     /*    fgen[idx+1] = (Edp - (Xq[i] - Xqp[i])*Iq)/Tq0p[i]; */
    364:     row[0] = idx + 1;
    365:     col[0] = idx + 1;       col[1] = idx+5;
    366:     val[0] = 1/Tq0p[i]; val[1] = -(Xq[i] - Xqp[i])/Tq0p[i];
    367:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

    369:     /*    fgen[idx+2] = - w + w_s; */
    370:     row[0] = idx + 2;
    371:     col[0] = idx + 2; col[1] = idx + 3;
    372:     val[0] = 0;       val[1] = -1;
    373:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

    375:     /*    fgen[idx+3] = (-TM[i] + Edp*Id + Eqp*Iq + (Xqp[i] - Xdp[i])*Id*Iq + D[i]*(w - w_s))/M[i]; */
    376:     row[0] = idx + 3;
    377:     col[0] = idx; col[1] = idx + 1; col[2] = idx + 3;       col[3] = idx + 4;                  col[4] = idx + 5;
    378:     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];
    379:     MatSetValues(J,1,row,5,col,val,INSERT_VALUES);

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

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

    387:     Zdq_inv[0] = Rs[i]/det;
    388:     Zdq_inv[1] = Xqp[i]/det;
    389:     Zdq_inv[2] = -Xdp[i]/det;
    390:     Zdq_inv[3] = Rs[i]/det;

    392:     dVd_dVr    = PetscSinScalar(delta); dVd_dVi = -PetscCosScalar(delta);
    393:     dVq_dVr    = PetscCosScalar(delta); dVq_dVi = PetscSinScalar(delta);
    394:     dVd_ddelta = Vr*PetscCosScalar(delta) + Vi*PetscSinScalar(delta);
    395:     dVq_ddelta = -Vr*PetscSinScalar(delta) + Vi*PetscCosScalar(delta);

    397:     /*    fgen[idx+4] = Zdq_inv[0]*(-Edp + Vd) + Zdq_inv[1]*(-Eqp + Vq) + Id; */
    398:     row[0] = idx+4;
    399:     col[0] = idx;         col[1] = idx+1;        col[2] = idx + 2;
    400:     val[0] = -Zdq_inv[1]; val[1] = -Zdq_inv[0];  val[2] = Zdq_inv[0]*dVd_ddelta + Zdq_inv[1]*dVq_ddelta;
    401:     col[3] = idx + 4; col[4] = net_start+2*gbus[i];                     col[5] = net_start + 2*gbus[i]+1;
    402:     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;
    403:     MatSetValues(J,1,row,6,col,val,INSERT_VALUES);

    405:     /*  fgen[idx+5] = Zdq_inv[2]*(-Edp + Vd) + Zdq_inv[3]*(-Eqp + Vq) + Iq; */
    406:     row[0] = idx+5;
    407:     col[0] = idx;         col[1] = idx+1;        col[2] = idx + 2;
    408:     val[0] = -Zdq_inv[3]; val[1] = -Zdq_inv[2];  val[2] = Zdq_inv[2]*dVd_ddelta + Zdq_inv[3]*dVq_ddelta;
    409:     col[3] = idx + 5; col[4] = net_start+2*gbus[i];                     col[5] = net_start + 2*gbus[i]+1;
    410:     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;
    411:     MatSetValues(J,1,row,6,col,val,INSERT_VALUES);

    413:     dIGr_ddelta = Id*PetscCosScalar(delta) - Iq*PetscSinScalar(delta);
    414:     dIGi_ddelta = Id*PetscSinScalar(delta) + Iq*PetscCosScalar(delta);
    415:     dIGr_dId    = PetscSinScalar(delta);  dIGr_dIq = PetscCosScalar(delta);
    416:     dIGi_dId    = -PetscCosScalar(delta); dIGi_dIq = PetscSinScalar(delta);

    418:     /* fnet[2*gbus[i]]   -= IGi; */
    419:     row[0] = net_start + 2*gbus[i];
    420:     col[0] = idx+2;        col[1] = idx + 4;   col[2] = idx + 5;
    421:     val[0] = -dIGi_ddelta; val[1] = -dIGi_dId; val[2] = -dIGi_dIq;
    422:     MatSetValues(J,1,row,3,col,val,INSERT_VALUES);

    424:     /* fnet[2*gbus[i]+1]   -= IGr; */
    425:     row[0] = net_start + 2*gbus[i]+1;
    426:     col[0] = idx+2;        col[1] = idx + 4;   col[2] = idx + 5;
    427:     val[0] = -dIGr_ddelta; val[1] = -dIGr_dId; val[2] = -dIGr_dIq;
    428:     MatSetValues(J,1,row,3,col,val,INSERT_VALUES);

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

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

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

    437:     row[0] = idx + 6;
    438:     col[0] = idx + 6;                     col[1] = idx + 8;
    439:     val[0] = (KE[i] + dSE_dEfd)/TE[i];  val[1] = -1/TE[i];
    440:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

    442:     /* Exciter differential equations */

    444:     /*    fgen[idx+7] = (RF - KF[i]*Efd/TF[i])/TF[i]; */
    445:     row[0] = idx + 7;
    446:     col[0] = idx + 6;       col[1] = idx + 7;
    447:     val[0] = (-KF[i]/TF[i])/TF[i];  val[1] = 1/TF[i];
    448:     MatSetValues(J,1,row,2,col,val,INSERT_VALUES);

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

    453:     dVm_dVd = Vd/Vm; dVm_dVq = Vq/Vm;
    454:     dVm_dVr = dVm_dVd*dVd_dVr + dVm_dVq*dVq_dVr;
    455:     dVm_dVi = dVm_dVd*dVd_dVi + dVm_dVq*dVq_dVi;
    456:     row[0]  = idx + 8;
    457:     col[0]  = idx + 6;           col[1] = idx + 7; col[2] = idx + 8;
    458:     val[0]  = (KA[i]*KF[i]/TF[i])/TA[i]; val[1] = -KA[i]/TA[i];  val[2] = 1/TA[i];
    459:     col[3]  = net_start + 2*gbus[i]; col[4] = net_start + 2*gbus[i]+1;
    460:     val[3]  = KA[i]*dVm_dVr/TA[i];         val[4] = KA[i]*dVm_dVi/TA[i];
    461:     MatSetValues(J,1,row,5,col,val,INSERT_VALUES);
    462:     idx     = idx + 9;
    463:   }

    465:   for (i=0; i<nbus; i++) {
    466:     MatGetRow(user->Ybus,2*i,&ncols,&cols,&yvals);
    467:     row[0] = net_start + 2*i;
    468:     for (k=0; k<ncols; k++) {
    469:       col[k] = net_start + cols[k];
    470:       val[k] = yvals[k];
    471:     }
    472:     MatSetValues(J,1,row,ncols,col,val,INSERT_VALUES);
    473:     MatRestoreRow(user->Ybus,2*i,&ncols,&cols,&yvals);

    475:     MatGetRow(user->Ybus,2*i+1,&ncols,&cols,&yvals);
    476:     row[0] = net_start + 2*i+1;
    477:     for (k=0; k<ncols; k++) {
    478:       col[k] = net_start + cols[k];
    479:       val[k] = yvals[k];
    480:     }
    481:     MatSetValues(J,1,row,ncols,col,val,INSERT_VALUES);
    482:     MatRestoreRow(user->Ybus,2*i+1,&ncols,&cols,&yvals);
    483:   }

    485:   MatAssemblyBegin(J,MAT_FLUSH_ASSEMBLY);
    486:   MatAssemblyEnd(J,MAT_FLUSH_ASSEMBLY);

    488:   VecGetArray(user->V0,&v0);
    489:   for (i=0; i < nload; i++) {
    490:     Vr      = xnet[2*lbus[i]]; /* Real part of load bus voltage */
    491:     Vi      = xnet[2*lbus[i]+1]; /* Imaginary part of the load bus voltage */
    492:     Vm      = PetscSqrtScalar(Vr*Vr + Vi*Vi); Vm2 = Vm*Vm; Vm4 = Vm2*Vm2;
    493:     Vm0     = PetscSqrtScalar(v0[2*lbus[i]]*v0[2*lbus[i]] + v0[2*lbus[i]+1]*v0[2*lbus[i]+1]);
    494:     PD      = QD = 0.0;
    495:     dPD_dVr = dPD_dVi = dQD_dVr = dQD_dVi = 0.0;
    496:     for (k=0; k < ld_nsegsp[i]; k++) {
    497:       PD      += ld_alphap[k]*PD0[i]*PetscPowScalar((Vm/Vm0),ld_betap[k]);
    498:       dPD_dVr += ld_alphap[k]*ld_betap[k]*PD0[i]*PetscPowScalar((1/Vm0),ld_betap[k])*Vr*PetscPowScalar(Vm,(ld_betap[k]-2));
    499:       dPD_dVi += ld_alphap[k]*ld_betap[k]*PD0[i]*PetscPowScalar((1/Vm0),ld_betap[k])*Vi*PetscPowScalar(Vm,(ld_betap[k]-2));
    500:     }
    501:     for (k=0; k < ld_nsegsq[i]; k++) {
    502:       QD      += ld_alphaq[k]*QD0[i]*PetscPowScalar((Vm/Vm0),ld_betaq[k]);
    503:       dQD_dVr += ld_alphaq[k]*ld_betaq[k]*QD0[i]*PetscPowScalar((1/Vm0),ld_betaq[k])*Vr*PetscPowScalar(Vm,(ld_betaq[k]-2));
    504:       dQD_dVi += ld_alphaq[k]*ld_betaq[k]*QD0[i]*PetscPowScalar((1/Vm0),ld_betaq[k])*Vi*PetscPowScalar(Vm,(ld_betaq[k]-2));
    505:     }

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

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

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

    517:     /*    fnet[2*lbus[i]]   += IDi; */
    518:     row[0] = net_start + 2*lbus[i];
    519:     col[0] = net_start + 2*lbus[i];  col[1] = net_start + 2*lbus[i]+1;
    520:     val[0] = dIDi_dVr;               val[1] = dIDi_dVi;
    521:     MatSetValues(J,1,row,2,col,val,ADD_VALUES);
    522:     /*    fnet[2*lbus[i]+1] += IDr; */
    523:     row[0] = net_start + 2*lbus[i]+1;
    524:     col[0] = net_start + 2*lbus[i];  col[1] = net_start + 2*lbus[i]+1;
    525:     val[0] = dIDr_dVr;               val[1] = dIDr_dVi;
    526:     MatSetValues(J,1,row,2,col,val,ADD_VALUES);
    527:   }
    528:   VecRestoreArray(user->V0,&v0);

    530:   VecRestoreArray(Xgen,&xgen);
    531:   VecRestoreArray(Xnet,&xnet);

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

    535:   MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
    536:   MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
    537:   return(0);
    538: }

    542: int main(int argc,char **argv)
    543: {
    544:   EPS            eps;
    545:   EPSType        type;
    547:   PetscMPIInt    size;
    548:   Userctx        user;
    549:   PetscViewer    Xview,Ybusview;
    550:   Vec            X,Xr,Xi;
    551:   Mat            J,Jred=NULL;
    552:   IS             is0,is1;
    553:   PetscInt       i,*idx2,its,nev,nconv;
    554:   PetscReal      error,re,im;
    555:   PetscScalar    kr,ki;
    556:   PetscBool      terse;

    558:   SlepcInitialize(&argc,&argv,NULL,help);
    559:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    560:   if (size > 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");
    561:   /* show detailed info unless -terse option is given by user */
    562:   PetscOptionsHasName(NULL,NULL,"-terse",&terse);

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

    569:   /* Create indices for differential and algebraic equations */
    570:   PetscMalloc1(7*ngen,&idx2);
    571:   for (i=0; i<ngen; i++) {
    572:     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;
    573:     idx2[7*i+4] = 9*i+6; idx2[7*i+5] = 9*i+7; idx2[7*i+6] = 9*i+8;
    574:   }
    575:   ISCreateGeneral(PETSC_COMM_WORLD,7*ngen,idx2,PETSC_COPY_VALUES,&user.is_diff);
    576:   ISComplement(user.is_diff,0,user.neqs_pgrid,&user.is_alg);
    577:   PetscFree(idx2);

    579:   /* Read initial voltage vector and Ybus */
    580:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"X.bin",FILE_MODE_READ,&Xview);
    581:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,"Ybus.bin",FILE_MODE_READ,&Ybusview);

    583:   VecCreate(PETSC_COMM_WORLD,&user.V0);
    584:   VecSetSizes(user.V0,PETSC_DECIDE,user.neqs_net);
    585:   VecLoad(user.V0,Xview);

    587:   MatCreate(PETSC_COMM_WORLD,&user.Ybus);
    588:   MatSetSizes(user.Ybus,PETSC_DECIDE,PETSC_DECIDE,user.neqs_net,user.neqs_net);
    589:   MatSetType(user.Ybus,MATBAIJ);
    590:   /*  MatSetBlockSize(user.Ybus,2); */
    591:   MatLoad(user.Ybus,Ybusview);

    593:   PetscViewerDestroy(&Xview);
    594:   PetscViewerDestroy(&Ybusview);

    596:   /* Create DMs for generator and network subsystems */
    597:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.neqs_gen,1,1,NULL,&user.dmgen);
    598:   DMSetOptionsPrefix(user.dmgen,"dmgen_");
    599:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.neqs_net,1,1,NULL,&user.dmnet);
    600:   DMSetOptionsPrefix(user.dmnet,"dmnet_");
    601:   /* Create a composite DM packer and add the two DMs */
    602:   DMCompositeCreate(PETSC_COMM_WORLD,&user.dmpgrid);
    603:   DMSetOptionsPrefix(user.dmpgrid,"pgrid_");
    604:   DMCompositeAddDM(user.dmpgrid,user.dmgen);
    605:   DMCompositeAddDM(user.dmpgrid,user.dmnet);

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

    609:   MatCreate(PETSC_COMM_WORLD,&J);
    610:   MatSetSizes(J,PETSC_DECIDE,PETSC_DECIDE,user.neqs_pgrid,user.neqs_pgrid);
    611:   MatSetFromOptions(J);
    612:   PreallocateJacobian(J,&user);

    614:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    615:      Set initial conditions
    616:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    617:   SetInitialGuess(X,&user);

    619:   /* Form Jacobian */
    620:   ResidualJacobian(X,J,(void*)&user);
    621:   MatScale(J,-1);
    622:   is0 = user.is_diff;
    623:   is1 = user.is_alg;

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

    627:   if (!terse) {
    628:     MatView(Jred,NULL);
    629:   }

    631:   MatCreateVecs(Jred,NULL,&Xr);
    632:   MatCreateVecs(Jred,NULL,&Xi);

    634:   /* Create the eigensolver and set the various options */
    635:   EPSCreate(PETSC_COMM_WORLD,&eps);
    636:   EPSSetOperators(eps,Jred,NULL);
    637:   EPSSetProblemType(eps,EPS_NHEP);
    638:   EPSSetFromOptions(eps);
    639:
    640:   /* Solve the eigenvalue problem */
    641:   EPSSolve(eps);

    643:   EPSGetIterationNumber(eps,&its);
    644:   PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the eigensolver: %D\n",its);
    645:   EPSGetType(eps,&type);
    646:   PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n", type);
    647:   EPSGetDimensions(eps,&nev,NULL,NULL);
    648:   PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %D\n",nev);

    650:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    651:                     Display solution and clean up
    652:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    653:   if (terse) {
    654:     EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL);
    655:   } else {
    656:     /* Get number of converged approximate eigenpairs */
    657:     EPSGetConverged(eps,&nconv);
    658:     PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %D\n\n",nconv);
    659:
    660:     if (nconv>0) {
    661:       /* Display eigenvalues and relative errors */
    662:       PetscPrintf(PETSC_COMM_WORLD,
    663:            "           k          ||Ax-kx||/||kx||\n"
    664:            "   ----------------- ------------------\n");
    665:
    666:       for (i=0;i<nconv;i++) {
    667:         /* Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
    668:           ki (imaginary part) */
    669:         EPSGetEigenpair(eps,i,&kr,&ki,Xr,Xi);
    670:         /* Compute the relative error associated to each eigenpair */
    671:         EPSComputeError(eps,i,EPS_ERROR_RELATIVE,&error);

    673: #if defined(PETSC_USE_COMPLEX)
    674:         re = PetscRealPart(kr);
    675:         im = PetscImaginaryPart(kr);
    676: #else
    677:         re = kr;
    678:         im = ki;
    679: #endif
    680:         if (im!=0.0) {
    681:           PetscPrintf(PETSC_COMM_WORLD," %9f%+9fi %12g\n",(double)re,(double)im,(double)error);
    682:         } else {
    683:           PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g\n",(double)re,(double)error);
    684:         }
    685:       }
    686:       PetscPrintf(PETSC_COMM_WORLD,"\n");
    687:     }
    688:   }

    690:   /* Free work space */
    691:   EPSDestroy(&eps);
    692:   MatDestroy(&J);
    693:   MatDestroy(&Jred);
    694:   MatDestroy(&user.Ybus);
    695:   VecDestroy(&X);
    696:   VecDestroy(&Xr);
    697:   VecDestroy(&Xi);
    698:   VecDestroy(&user.V0);
    699:   DMDestroy(&user.dmgen);
    700:   DMDestroy(&user.dmnet);
    701:   DMDestroy(&user.dmpgrid);
    702:   ISDestroy(&user.is_diff);
    703:   ISDestroy(&user.is_alg);
    704:   SlepcFinalize();
    705:   return ierr;
    706: }