Date: 2022-03-04T17:39:43+00:00

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

     22: /*
     23:    Include "petscts.h" so that we can use TS solvers.  Note that this
     24:    file automatically includes:
     25:      petscsys.h       - base PETSc routines   petscvec.h - vectors
     26:      petscmat.h - matrices
     27:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
     28:      petscviewer.h - viewers               petscpc.h  - preconditioners
     29:      petscksp.h   - linear solvers
     30: */

     32: #include <petscts.h>

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

     40: PetscErrorCode PostStepFunction(TS ts)
     41: {
     42:   PetscErrorCode    ierr;
     43:   Vec               U;
     44:   PetscReal         t;
     45:   const PetscScalar *u;

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

     56: /*
     57:      Defines the ODE passed to the ODE solver
     58: */
     59: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec U,Vec F,AppCtx *ctx)
     60: {
     61:   PetscErrorCode    ierr;
     62:   PetscScalar       *f,Pmax;
     63:   const PetscScalar *u;

     66:   /*  The next three lines allow us to access the entries of the vectors directly */
     67:   VecGetArrayRead(U,&u);
     68:   VecGetArray(F,&f);
     69:   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 */
     70:   else Pmax = ctx->Pmax;

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

     75:   VecRestoreArrayRead(U,&u);
     76:   VecRestoreArray(F,&f);
     77:   return(0);
     78: }

     80: /*
     81:      Defines the Jacobian of the ODE passed to the ODE solver. See TSSetIJacobian() for the meaning of a and the Jacobian.
     82: */
     83: static PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec U,Mat A,Mat B,AppCtx *ctx)
     84: {
     85:   PetscErrorCode    ierr;
     86:   PetscInt          rowcol[] = {0,1};
     87:   PetscScalar       J[2][2],Pmax;
     88:   const PetscScalar *u;

     91:   VecGetArrayRead(U,&u);
     92:   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 */
     93:   else Pmax = ctx->Pmax;

     95:   J[0][0] = 0;                                  J[0][1] = ctx->omega_b;
     96:   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);

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

    101:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    102:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    103:   if (A != B) {
    104:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    105:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    106:   }
    107:   return(0);
    108: }

    110: static PetscErrorCode RHSJacobianP(TS ts,PetscReal t,Vec X,Mat A,void *ctx0)
    111: {
    113:   PetscInt       row[] = {0,1},col[]={0};
    114:   PetscScalar    J[2][1];
    115:   AppCtx         *ctx=(AppCtx*)ctx0;

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

    126: static PetscErrorCode CostIntegrand(TS ts,PetscReal t,Vec U,Vec R,AppCtx *ctx)
    127: {
    128:   PetscErrorCode    ierr;
    129:   PetscScalar       *r;
    130:   const PetscScalar *u;

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

    141: static PetscErrorCode DRDUJacobianTranspose(TS ts,PetscReal t,Vec U,Mat DRDU,Mat B,AppCtx *ctx)
    142: {
    143:   PetscErrorCode    ierr;
    144:   PetscScalar       ru[1];
    145:   const PetscScalar *u;
    146:   PetscInt          row[] = {0},col[] = {0};

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

    158: static PetscErrorCode DRDPJacobianTranspose(TS ts,PetscReal t,Vec U,Mat DRDP,AppCtx *ctx)
    159: {
    160:   PetscErrorCode    ierr;

    163:   MatZeroEntries(DRDP);
    164:   MatAssemblyBegin(DRDP,MAT_FINAL_ASSEMBLY);
    165:   MatAssemblyEnd(DRDP,MAT_FINAL_ASSEMBLY);
    166:   return(0);
    167: }

    169: PetscErrorCode ComputeSensiP(Vec lambda,Vec mu,AppCtx *ctx)
    170: {
    171:   PetscErrorCode    ierr;
    172:   PetscScalar       sensip;
    173:   const PetscScalar *x,*y;

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

    185: int main(int argc,char **argv)
    186: {
    187:   TS             ts,quadts;     /* ODE integrator */
    188:   Vec            U;             /* solution will be stored here */
    189:   Mat            A;             /* Jacobian matrix */
    190:   Mat            Jacp;          /* Jacobian matrix */
    191:   Mat            DRDU,DRDP;
    193:   PetscMPIInt    size;
    194:   PetscInt       n = 2;
    195:   AppCtx         ctx;
    196:   PetscScalar    *u;
    197:   PetscReal      du[2] = {0.0,0.0};
    198:   PetscBool      ensemble = PETSC_FALSE,flg1,flg2;
    199:   PetscReal      ftime;
    200:   PetscInt       steps;
    201:   PetscScalar    *x_ptr,*y_ptr;
    202:   Vec            lambda[1],q,mu[1];

    204:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    205:      Initialize program
    206:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    207:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
    208:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    209:   if (size > 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");

    211:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    212:     Create necessary matrix and vectors
    213:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    214:   MatCreate(PETSC_COMM_WORLD,&A);
    215:   MatSetSizes(A,n,n,PETSC_DETERMINE,PETSC_DETERMINE);
    216:   MatSetType(A,MATDENSE);
    217:   MatSetFromOptions(A);
    218:   MatSetUp(A);

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

    222:   MatCreate(PETSC_COMM_WORLD,&Jacp);
    223:   MatSetSizes(Jacp,PETSC_DECIDE,PETSC_DECIDE,2,1);
    224:   MatSetFromOptions(Jacp);
    225:   MatSetUp(Jacp);

    227:   MatCreateDense(PETSC_COMM_WORLD,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,&DRDP);
    228:   MatSetUp(DRDP);
    229:   MatCreateDense(PETSC_COMM_WORLD,PETSC_DECIDE,PETSC_DECIDE,1,2,NULL,&DRDU);
    230:   MatSetUp(DRDU);

    232:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    233:     Set runtime options
    234:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    235:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Swing equation options","");
    236:   {
    237:     ctx.beta    = 2;
    238:     ctx.c       = 10000.0;
    239:     ctx.u_s     = 1.0;
    240:     ctx.omega_s = 1.0;
    241:     ctx.omega_b = 120.0*PETSC_PI;
    242:     ctx.H       = 5.0;
    243:     PetscOptionsScalar("-Inertia","","",ctx.H,&ctx.H,NULL);
    244:     ctx.D       = 5.0;
    245:     PetscOptionsScalar("-D","","",ctx.D,&ctx.D,NULL);
    246:     ctx.E       = 1.1378;
    247:     ctx.V       = 1.0;
    248:     ctx.X       = 0.545;
    249:     ctx.Pmax    = ctx.E*ctx.V/ctx.X;
    250:     PetscOptionsScalar("-Pmax","","",ctx.Pmax,&ctx.Pmax,NULL);
    251:     ctx.Pm      = 1.1;
    252:     PetscOptionsScalar("-Pm","","",ctx.Pm,&ctx.Pm,NULL);
    253:     ctx.tf      = 0.1;
    254:     ctx.tcl     = 0.2;
    255:     PetscOptionsReal("-tf","Time to start fault","",ctx.tf,&ctx.tf,NULL);
    256:     PetscOptionsReal("-tcl","Time to end fault","",ctx.tcl,&ctx.tcl,NULL);
    257:     PetscOptionsBool("-ensemble","Run ensemble of different initial conditions","",ensemble,&ensemble,NULL);
    258:     if (ensemble) {
    259:       ctx.tf      = -1;
    260:       ctx.tcl     = -1;
    261:     }

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

    279:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    280:      Create timestepping solver context
    281:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    282:   TSCreate(PETSC_COMM_WORLD,&ts);
    283:   TSSetProblemType(ts,TS_NONLINEAR);
    284:   TSSetEquationType(ts,TS_EQ_ODE_EXPLICIT); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */
    285:   TSSetType(ts,TSRK);
    286:   TSSetRHSFunction(ts,NULL,(TSRHSFunction)RHSFunction,&ctx);
    287:   TSSetRHSJacobian(ts,A,A,(TSRHSJacobian)RHSJacobian,&ctx);
    288:   TSCreateQuadratureTS(ts,PETSC_TRUE,&quadts);
    289:   TSSetRHSFunction(quadts,NULL,(TSRHSFunction)CostIntegrand,&ctx);
    290:   TSSetRHSJacobian(quadts,DRDU,DRDU,(TSRHSJacobian)DRDUJacobianTranspose,&ctx);
    291:   TSSetRHSJacobianP(quadts,DRDP,(TSRHSJacobianP)DRDPJacobianTranspose,&ctx);

    293:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    294:      Set initial conditions
    295:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    296:   TSSetSolution(ts,U);

    298:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    299:     Save trajectory of solution so that TSAdjointSolve() may be used
    300:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    301:   TSSetSaveTrajectory(ts);

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

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

    315:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    316:      Set solver options
    317:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    318:   TSSetMaxTime(ts,10.0);
    319:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);
    320:   TSSetTimeStep(ts,.01);
    321:   TSSetFromOptions(ts);

    323:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    324:      Solve nonlinear system
    325:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    326:   if (ensemble) {
    327:     for (du[1] = -2.5; du[1] <= .01; du[1] += .1) {
    328:       VecGetArray(U,&u);
    329:       u[0] = PetscAsinScalar(ctx.Pm/ctx.Pmax);
    330:       u[1] = ctx.omega_s;
    331:       u[0] += du[0];
    332:       u[1] += du[1];
    333:       VecRestoreArray(U,&u);
    334:       TSSetTimeStep(ts,.01);
    335:       TSSolve(ts,U);
    336:     }
    337:   } else {
    338:     TSSolve(ts,U);
    339:   }
    340:   VecView(U,PETSC_VIEWER_STDOUT_WORLD);
    341:   TSGetSolveTime(ts,&ftime);
    342:   TSGetStepNumber(ts,&steps);

    344:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    345:      Adjoint model starts here
    346:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    347:   /*   Set initial conditions for the adjoint integration */
    348:   VecGetArray(lambda[0],&y_ptr);
    349:   y_ptr[0] = 0.0; y_ptr[1] = 0.0;
    350:   VecRestoreArray(lambda[0],&y_ptr);

    352:   VecGetArray(mu[0],&x_ptr);
    353:   x_ptr[0] = -1.0;
    354:   VecRestoreArray(mu[0],&x_ptr);

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

    359:   TSAdjointSolve(ts);

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

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

    372:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    373:      Free work space.  All PETSc objects should be destroyed when they are no longer needed.
    374:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    375:   MatDestroy(&A);
    376:   MatDestroy(&Jacp);
    377:   MatDestroy(&DRDU);
    378:   MatDestroy(&DRDP);
    379:   VecDestroy(&U);
    380:   VecDestroy(&lambda[0]);
    381:   VecDestroy(&mu[0]);
    382:   TSDestroy(&ts);
    383:   PetscFinalize();
    384:   return ierr;
    385: }

    387: /*TEST

    389:    build:
    390:       requires: !complex

    392:    test:
    393:       args: -viewer_binary_skip_info -ts_adapt_type none

    395: TEST*/