Actual source code: ex9adj.c

      2: static char help[] = "Basic equation for generator stability analysis.\n";

\begin{eqnarray}
\frac{d \theta}{dt} = \omega_b (\omega - \omega_s)
\frac{2 H}{\omega_s}\frac{d \omega}{dt} & = & P_m - P_max \sin(\theta) -D(\omega - \omega_s)\\
\end{eqnarray}
Ensemble of initial conditions
./ex9 -ensemble -ts_monitor_draw_solution_phase -1,-3,3,3 -ts_adapt_dt_max .01 -ts_monitor -ts_type rk -pc_type lu -ksp_type preonly
Fault at .1 seconds
./ex9 -ts_monitor_draw_solution_phase .42,.95,.6,1.05 -ts_adapt_dt_max .01 -ts_monitor -ts_type rk -pc_type lu -ksp_type preonly
Initial conditions same as when fault is ended
./ex9 -u 0.496792,1.00932 -ts_monitor_draw_solution_phase .42,.95,.6,1.05 -ts_adapt_dt_max .01 -ts_monitor -ts_type rk -pc_type lu -ksp_type preonly

     25: /*
     26:    Include "petscts.h" so that we can use TS solvers.  Note that this
     27:    file automatically includes:
     28:      petscsys.h       - base PETSc routines   petscvec.h - vectors
     29:      petscmat.h - matrices
     30:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
     31:      petscviewer.h - viewers               petscpc.h  - preconditioners
     32:      petscksp.h   - linear solvers
     33: */

     35: #include <petscts.h>

     37: typedef struct {
     38:   PetscScalar H,D,omega_b,omega_s,Pmax,Pm,E,V,X,u_s,c;
     39:   PetscInt    beta;
     40:   PetscReal   tf,tcl;
     41: } AppCtx;

     43: PetscErrorCode PostStepFunction(TS ts)
     44: {
     45:   PetscErrorCode    ierr;
     46:   Vec               U;
     47:   PetscReal         t;
     48:   const PetscScalar *u;

     51:   TSGetTime(ts,&t);
     52:   TSGetSolution(ts,&U);
     53:   VecGetArrayRead(U,&u);
     54:   PetscPrintf(PETSC_COMM_SELF,"delta(%3.2f) = %8.7f\n",(double)t,(double)u[0]);
     55:   VecRestoreArrayRead(U,&u);
     56:   return(0);
     57: }

     59: /*
     60:      Defines the ODE passed to the ODE solver
     61: */
     62: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec U,Vec F,AppCtx *ctx)
     63: {
     64:   PetscErrorCode    ierr;
     65:   PetscScalar       *f,Pmax;
     66:   const PetscScalar *u;

     69:   /*  The next three lines allow us to access the entries of the vectors directly */
     70:   VecGetArrayRead(U,&u);
     71:   VecGetArray(F,&f);
     72:   if ((t > ctx->tf) && (t < ctx->tcl)) Pmax = 0.0; /* A short-circuit on the generator terminal that drives the electrical power output (Pmax*sin(delta)) to 0 */
     73:   else Pmax = ctx->Pmax;

     75:   f[0] = ctx->omega_b*(u[1] - ctx->omega_s);
     76:   f[1] = (-Pmax*PetscSinScalar(u[0]) - ctx->D*(u[1] - ctx->omega_s) + ctx->Pm)*ctx->omega_s/(2.0*ctx->H);

     78:   VecRestoreArrayRead(U,&u);
     79:   VecRestoreArray(F,&f);
     80:   return(0);
     81: }

     83: /*
     84:      Defines the Jacobian of the ODE passed to the ODE solver. See TSSetIJacobian() for the meaning of a and the Jacobian.
     85: */
     86: static PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec U,Mat A,Mat B,AppCtx *ctx)
     87: {
     88:   PetscErrorCode    ierr;
     89:   PetscInt          rowcol[] = {0,1};
     90:   PetscScalar       J[2][2],Pmax;
     91:   const PetscScalar *u;

     94:   VecGetArrayRead(U,&u);
     95:   if ((t > ctx->tf) && (t < ctx->tcl)) Pmax = 0.0; /* A short-circuit on the generator terminal that drives the electrical power output (Pmax*sin(delta)) to 0 */
     96:   else Pmax = ctx->Pmax;

     98:   J[0][0] = 0;                                  J[0][1] = ctx->omega_b;
     99:   J[1][1] = -ctx->D*ctx->omega_s/(2.0*ctx->H);  J[1][0] = -Pmax*PetscCosScalar(u[0])*ctx->omega_s/(2.0*ctx->H);

    101:   MatSetValues(A,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
    102:   VecRestoreArrayRead(U,&u);

    104:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    105:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    106:   if (A != B) {
    107:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    108:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    109:   }
    110:   return(0);
    111: }

    113: static PetscErrorCode RHSJacobianP(TS ts,PetscReal t,Vec X,Mat A,void *ctx0)
    114: {
    116:   PetscInt       row[] = {0,1},col[]={0};
    117:   PetscScalar    J[2][1];
    118:   AppCtx         *ctx=(AppCtx*)ctx0;

    121:   J[0][0] = 0;
    122:   J[1][0] = ctx->omega_s/(2.0*ctx->H);
    123:   MatSetValues(A,2,row,1,col,&J[0][0],INSERT_VALUES);
    124:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    125:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    126:   return(0);
    127: }

    129: static PetscErrorCode CostIntegrand(TS ts,PetscReal t,Vec U,Vec R,AppCtx *ctx)
    130: {
    131:   PetscErrorCode    ierr;
    132:   PetscScalar       *r;
    133:   const PetscScalar *u;

    136:   VecGetArrayRead(U,&u);
    137:   VecGetArray(R,&r);
    138:   r[0] = ctx->c*PetscPowScalarInt(PetscMax(0., u[0]-ctx->u_s),ctx->beta);
    139:   VecRestoreArray(R,&r);
    140:   VecRestoreArrayRead(U,&u);
    141:   return(0);
    142: }

    144: static PetscErrorCode DRDUJacobianTranspose(TS ts,PetscReal t,Vec U,Mat DRDU,Mat B,AppCtx *ctx)
    145: {
    146:   PetscErrorCode    ierr;
    147:   PetscScalar       ru[1];
    148:   const PetscScalar *u;
    149:   PetscInt          row[] = {0},col[] = {0};

    152:   VecGetArrayRead(U,&u);
    153:   ru[0] = ctx->c*ctx->beta*PetscPowScalarInt(PetscMax(0., u[0]-ctx->u_s),ctx->beta-1);
    154:   VecRestoreArrayRead(U,&u);
    155:   MatSetValues(DRDU,1,row,1,col,ru,INSERT_VALUES);
    156:   MatAssemblyBegin(DRDU,MAT_FINAL_ASSEMBLY);
    157:   MatAssemblyEnd(DRDU,MAT_FINAL_ASSEMBLY);
    158:   return(0);
    159: }

    161: static PetscErrorCode DRDPJacobianTranspose(TS ts,PetscReal t,Vec U,Mat DRDP,AppCtx *ctx)
    162: {
    163:   PetscErrorCode    ierr;

    166:   MatZeroEntries(DRDP);
    167:   MatAssemblyBegin(DRDP,MAT_FINAL_ASSEMBLY);
    168:   MatAssemblyEnd(DRDP,MAT_FINAL_ASSEMBLY);
    169:   return(0);
    170: }

    172: PetscErrorCode ComputeSensiP(Vec lambda,Vec mu,AppCtx *ctx)
    173: {
    174:   PetscErrorCode    ierr;
    175:   PetscScalar       sensip;
    176:   const PetscScalar *x,*y;

    179:   VecGetArrayRead(lambda,&x);
    180:   VecGetArrayRead(mu,&y);
    181:   sensip = 1./PetscSqrtScalar(1.-(ctx->Pm/ctx->Pmax)*(ctx->Pm/ctx->Pmax))/ctx->Pmax*x[0]+y[0];
    182:   PetscPrintf(PETSC_COMM_WORLD,"\n sensitivity wrt parameter pm: %.7f \n",(double)sensip);
    183:   VecRestoreArrayRead(lambda,&x);
    184:   VecRestoreArrayRead(mu,&y);
    185:   return(0);
    186: }

    188: int main(int argc,char **argv)
    189: {
    190:   TS             ts,quadts;     /* ODE integrator */
    191:   Vec            U;             /* solution will be stored here */
    192:   Mat            A;             /* Jacobian matrix */
    193:   Mat            Jacp;          /* Jacobian matrix */
    194:   Mat            DRDU,DRDP;
    196:   PetscMPIInt    size;
    197:   PetscInt       n = 2;
    198:   AppCtx         ctx;
    199:   PetscScalar    *u;
    200:   PetscReal      du[2] = {0.0,0.0};
    201:   PetscBool      ensemble = PETSC_FALSE,flg1,flg2;
    202:   PetscReal      ftime;
    203:   PetscInt       steps;
    204:   PetscScalar    *x_ptr,*y_ptr;
    205:   Vec            lambda[1],q,mu[1];

    207:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    208:      Initialize program
    209:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    210:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
    211:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    212:   if (size > 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");

    214:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    215:     Create necessary matrix and vectors
    216:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    217:   MatCreate(PETSC_COMM_WORLD,&A);
    218:   MatSetSizes(A,n,n,PETSC_DETERMINE,PETSC_DETERMINE);
    219:   MatSetType(A,MATDENSE);
    220:   MatSetFromOptions(A);
    221:   MatSetUp(A);

    223:   MatCreateVecs(A,&U,NULL);

    225:   MatCreate(PETSC_COMM_WORLD,&Jacp);
    226:   MatSetSizes(Jacp,PETSC_DECIDE,PETSC_DECIDE,2,1);
    227:   MatSetFromOptions(Jacp);
    228:   MatSetUp(Jacp);

    230:   MatCreateDense(PETSC_COMM_WORLD,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,&DRDP);
    231:   MatSetUp(DRDP);
    232:   MatCreateDense(PETSC_COMM_WORLD,PETSC_DECIDE,PETSC_DECIDE,1,2,NULL,&DRDU);
    233:   MatSetUp(DRDU);

    235:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    236:     Set runtime options
    237:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    238:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Swing equation options","");
    239:   {
    240:     ctx.beta    = 2;
    241:     ctx.c       = 10000.0;
    242:     ctx.u_s     = 1.0;
    243:     ctx.omega_s = 1.0;
    244:     ctx.omega_b = 120.0*PETSC_PI;
    245:     ctx.H       = 5.0;
    246:     PetscOptionsScalar("-Inertia","","",ctx.H,&ctx.H,NULL);
    247:     ctx.D       = 5.0;
    248:     PetscOptionsScalar("-D","","",ctx.D,&ctx.D,NULL);
    249:     ctx.E       = 1.1378;
    250:     ctx.V       = 1.0;
    251:     ctx.X       = 0.545;
    252:     ctx.Pmax    = ctx.E*ctx.V/ctx.X;
    253:     PetscOptionsScalar("-Pmax","","",ctx.Pmax,&ctx.Pmax,NULL);
    254:     ctx.Pm      = 1.1;
    255:     PetscOptionsScalar("-Pm","","",ctx.Pm,&ctx.Pm,NULL);
    256:     ctx.tf      = 0.1;
    257:     ctx.tcl     = 0.2;
    258:     PetscOptionsReal("-tf","Time to start fault","",ctx.tf,&ctx.tf,NULL);
    259:     PetscOptionsReal("-tcl","Time to end fault","",ctx.tcl,&ctx.tcl,NULL);
    260:     PetscOptionsBool("-ensemble","Run ensemble of different initial conditions","",ensemble,&ensemble,NULL);
    261:     if (ensemble) {
    262:       ctx.tf      = -1;
    263:       ctx.tcl     = -1;
    264:     }

    266:     VecGetArray(U,&u);
    267:     u[0] = PetscAsinScalar(ctx.Pm/ctx.Pmax);
    268:     u[1] = 1.0;
    269:     PetscOptionsRealArray("-u","Initial solution","",u,&n,&flg1);
    270:     n    = 2;
    271:     PetscOptionsRealArray("-du","Perturbation in initial solution","",du,&n,&flg2);
    272:     u[0] += du[0];
    273:     u[1] += du[1];
    274:     VecRestoreArray(U,&u);
    275:     if (flg1 || flg2) {
    276:       ctx.tf      = -1;
    277:       ctx.tcl     = -1;
    278:     }
    279:   }
    280:   PetscOptionsEnd();

    282:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    283:      Create timestepping solver context
    284:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    285:   TSCreate(PETSC_COMM_WORLD,&ts);
    286:   TSSetProblemType(ts,TS_NONLINEAR);
    287:   TSSetEquationType(ts,TS_EQ_ODE_EXPLICIT); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */
    288:   TSSetType(ts,TSRK);
    289:   TSSetRHSFunction(ts,NULL,(TSRHSFunction)RHSFunction,&ctx);
    290:   TSSetRHSJacobian(ts,A,A,(TSRHSJacobian)RHSJacobian,&ctx);
    291:   TSCreateQuadratureTS(ts,PETSC_TRUE,&quadts);
    292:   TSSetRHSFunction(quadts,NULL,(TSRHSFunction)CostIntegrand,&ctx);
    293:   TSSetRHSJacobian(quadts,DRDU,DRDU,(TSRHSJacobian)DRDUJacobianTranspose,&ctx);
    294:   TSSetRHSJacobianP(quadts,DRDP,(TSRHSJacobianP)DRDPJacobianTranspose,&ctx);

    296:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    297:      Set initial conditions
    298:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    299:   TSSetSolution(ts,U);

    301:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    302:     Save trajectory of solution so that TSAdjointSolve() may be used
    303:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    304:   TSSetSaveTrajectory(ts);

    306:   MatCreateVecs(A,&lambda[0],NULL);
    307:   /*   Set initial conditions for the adjoint integration */
    308:   VecGetArray(lambda[0],&y_ptr);
    309:   y_ptr[0] = 0.0; y_ptr[1] = 0.0;
    310:   VecRestoreArray(lambda[0],&y_ptr);

    312:   MatCreateVecs(Jacp,&mu[0],NULL);
    313:   VecGetArray(mu[0],&x_ptr);
    314:   x_ptr[0] = -1.0;
    315:   VecRestoreArray(mu[0],&x_ptr);
    316:   TSSetCostGradients(ts,1,lambda,mu);

    318:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    319:      Set solver options
    320:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    321:   TSSetMaxTime(ts,10.0);
    322:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);
    323:   TSSetTimeStep(ts,.01);
    324:   TSSetFromOptions(ts);

    326:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    327:      Solve nonlinear system
    328:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    329:   if (ensemble) {
    330:     for (du[1] = -2.5; du[1] <= .01; du[1] += .1) {
    331:       VecGetArray(U,&u);
    332:       u[0] = PetscAsinScalar(ctx.Pm/ctx.Pmax);
    333:       u[1] = ctx.omega_s;
    334:       u[0] += du[0];
    335:       u[1] += du[1];
    336:       VecRestoreArray(U,&u);
    337:       TSSetTimeStep(ts,.01);
    338:       TSSolve(ts,U);
    339:     }
    340:   } else {
    341:     TSSolve(ts,U);
    342:   }
    343:   VecView(U,PETSC_VIEWER_STDOUT_WORLD);
    344:   TSGetSolveTime(ts,&ftime);
    345:   TSGetStepNumber(ts,&steps);

    347:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    348:      Adjoint model starts here
    349:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    350:   /*   Set initial conditions for the adjoint integration */
    351:   VecGetArray(lambda[0],&y_ptr);
    352:   y_ptr[0] = 0.0; y_ptr[1] = 0.0;
    353:   VecRestoreArray(lambda[0],&y_ptr);

    355:   VecGetArray(mu[0],&x_ptr);
    356:   x_ptr[0] = -1.0;
    357:   VecRestoreArray(mu[0],&x_ptr);

    359:   /*   Set RHS JacobianP */
    360:   TSSetRHSJacobianP(ts,Jacp,RHSJacobianP,&ctx);

    362:   TSAdjointSolve(ts);

    364:   PetscPrintf(PETSC_COMM_WORLD,"\n sensitivity wrt initial conditions: d[Psi(tf)]/d[phi0]  d[Psi(tf)]/d[omega0]\n");
    365:   VecView(lambda[0],PETSC_VIEWER_STDOUT_WORLD);
    366:   VecView(mu[0],PETSC_VIEWER_STDOUT_WORLD);
    367:   TSGetCostIntegral(ts,&q);
    368:   VecView(q,PETSC_VIEWER_STDOUT_WORLD);
    369:   VecGetArray(q,&x_ptr);
    370:   PetscPrintf(PETSC_COMM_WORLD,"\n cost function=%g\n",(double)(x_ptr[0]-ctx.Pm));
    371:   VecRestoreArray(q,&x_ptr);

    373:   ComputeSensiP(lambda[0],mu[0],&ctx);

    375:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    376:      Free work space.  All PETSc objects should be destroyed when they are no longer needed.
    377:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    378:   MatDestroy(&A);
    379:   MatDestroy(&Jacp);
    380:   MatDestroy(&DRDU);
    381:   MatDestroy(&DRDP);
    382:   VecDestroy(&U);
    383:   VecDestroy(&lambda[0]);
    384:   VecDestroy(&mu[0]);
    385:   TSDestroy(&ts);
    386:   PetscFinalize();
    387:   return ierr;
    388: }

    391: /*TEST

    393:    build:
    394:       requires: !complex

    396:    test:
    397:       args: -viewer_binary_skip_info -ts_adapt_type none

    399: TEST*/