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

/*F

\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
   ./ex2 -ensemble -ts_monitor_draw_solution_phase -1,-3,3,3      -ts_adapt_dt_max .01  -ts_monitor -ts_type rosw -pc_type lu -ksp_type preonly

  Fault at .1 seconds
   ./ex2           -ts_monitor_draw_solution_phase .42,.95,.6,1.05 -ts_adapt_dt_max .01  -ts_monitor -ts_type rosw -pc_type lu -ksp_type preonly

  Initial conditions same as when fault is ended
   ./ex2 -u 0.496792,1.00932 -ts_monitor_draw_solution_phase .42,.95,.6,1.05  -ts_adapt_dt_max .01  -ts_monitor -ts_type rosw -pc_type lu -ksp_type preonly

F*/

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

#include <petsctao.h>
#include <petscts.h>

typedef struct {
  TS          ts;
  PetscScalar H, D, omega_b, omega_s, Pmax, Pm, E, V, X, u_s, c;
  PetscInt    beta;
  PetscReal   tf, tcl, dt;
} AppCtx;

PetscErrorCode FormFunction(Tao, Vec, PetscReal *, void *);
PetscErrorCode FormGradient(Tao, Vec, Vec, void *);

/*
     Defines the ODE passed to the ODE solver
*/
static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec U, Vec F, AppCtx *ctx)
{
  PetscScalar       *f, Pmax;
  const PetscScalar *u;

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

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

  PetscCall(VecRestoreArrayRead(U, &u));
  PetscCall(VecRestoreArray(F, &f));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(U, &u));
  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 */
  else Pmax = ctx->Pmax;

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

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

  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  if (A != B) {
    PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

  PetscFunctionBeginUser;
  J[0][0] = 0;
  J[1][0] = ctx->omega_s / (2.0 * ctx->H);
  PetscCall(MatSetValues(A, 2, row, 1, col, &J[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CostIntegrand(TS ts, PetscReal t, Vec U, Vec R, AppCtx *ctx)
{
  PetscScalar       *r;
  const PetscScalar *u;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(U, &u));
  PetscCall(VecGetArray(R, &r));
  r[0] = ctx->c * PetscPowScalarInt(PetscMax(0., u[0] - ctx->u_s), ctx->beta);
  PetscCall(VecRestoreArray(R, &r));
  PetscCall(VecRestoreArrayRead(U, &u));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(U, &u));
  ru[0] = ctx->c * ctx->beta * PetscPowScalarInt(PetscMax(0., u[0] - ctx->u_s), ctx->beta - 1);
  PetscCall(VecRestoreArrayRead(U, &u));
  PetscCall(MatSetValues(DRDU, 1, row, 1, col, ru, INSERT_VALUES));
  PetscCall(MatAssemblyBegin(DRDU, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(DRDU, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode DRDPJacobianTranspose(TS ts, PetscReal t, Vec U, Mat DRDP, AppCtx *ctx)
{
  PetscFunctionBegin;
  PetscCall(MatZeroEntries(DRDP));
  PetscCall(MatAssemblyBegin(DRDP, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(DRDP, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode ComputeSensiP(Vec lambda, Vec mu, AppCtx *ctx)
{
  PetscScalar       *y, sensip;
  const PetscScalar *x;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(lambda, &x));
  PetscCall(VecGetArray(mu, &y));
  sensip = 1. / PetscSqrtScalar(1. - (ctx->Pm / ctx->Pmax) * (ctx->Pm / ctx->Pmax)) / ctx->Pmax * x[0] + y[0];
  y[0]   = sensip;
  PetscCall(VecRestoreArray(mu, &y));
  PetscCall(VecRestoreArrayRead(lambda, &x));
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  Vec          p;
  PetscScalar *x_ptr;
  PetscMPIInt  size;
  AppCtx       ctx;
  Vec          lowerb, upperb;
  Tao          tao;
  KSP          ksp;
  PC           pc;
  Vec          U, lambda[1], mu[1];
  Mat          A;    /* Jacobian matrix */
  Mat          Jacp; /* Jacobian matrix */
  Mat          DRDU, DRDP;
  PetscInt     n = 2;
  TS           quadts;

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Initialize program
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
  PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Set runtime options
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Swing equation options", "");
  {
    ctx.beta    = 2;
    ctx.c       = PetscRealConstant(10000.0);
    ctx.u_s     = PetscRealConstant(1.0);
    ctx.omega_s = PetscRealConstant(1.0);
    ctx.omega_b = PetscRealConstant(120.0) * PETSC_PI;
    ctx.H       = PetscRealConstant(5.0);
    PetscCall(PetscOptionsScalar("-Inertia", "", "", ctx.H, &ctx.H, NULL));
    ctx.D = PetscRealConstant(5.0);
    PetscCall(PetscOptionsScalar("-D", "", "", ctx.D, &ctx.D, NULL));
    ctx.E    = PetscRealConstant(1.1378);
    ctx.V    = PetscRealConstant(1.0);
    ctx.X    = PetscRealConstant(0.545);
    ctx.Pmax = ctx.E * ctx.V / ctx.X;
    PetscCall(PetscOptionsScalar("-Pmax", "", "", ctx.Pmax, &ctx.Pmax, NULL));
    ctx.Pm = PetscRealConstant(1.0194);
    PetscCall(PetscOptionsScalar("-Pm", "", "", ctx.Pm, &ctx.Pm, NULL));
    ctx.tf  = PetscRealConstant(0.1);
    ctx.tcl = PetscRealConstant(0.2);
    PetscCall(PetscOptionsReal("-tf", "Time to start fault", "", ctx.tf, &ctx.tf, NULL));
    PetscCall(PetscOptionsReal("-tcl", "Time to end fault", "", ctx.tcl, &ctx.tcl, NULL));
  }
  PetscOptionsEnd();

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    Create necessary matrix and vectors
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
  PetscCall(MatSetSizes(A, n, n, PETSC_DETERMINE, PETSC_DETERMINE));
  PetscCall(MatSetType(A, MATDENSE));
  PetscCall(MatSetFromOptions(A));
  PetscCall(MatSetUp(A));

  PetscCall(MatCreateVecs(A, &U, NULL));

  PetscCall(MatCreate(PETSC_COMM_WORLD, &Jacp));
  PetscCall(MatSetSizes(Jacp, PETSC_DECIDE, PETSC_DECIDE, 2, 1));
  PetscCall(MatSetFromOptions(Jacp));
  PetscCall(MatSetUp(Jacp));
  PetscCall(MatCreateDense(PETSC_COMM_WORLD, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, &DRDP));
  PetscCall(MatSetUp(DRDP));
  PetscCall(MatCreateDense(PETSC_COMM_WORLD, PETSC_DECIDE, PETSC_DECIDE, 1, 2, NULL, &DRDU));
  PetscCall(MatSetUp(DRDU));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Create timestepping solver context
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSCreate(PETSC_COMM_WORLD, &ctx.ts));
  PetscCall(TSSetProblemType(ctx.ts, TS_NONLINEAR));
  PetscCall(TSSetEquationType(ctx.ts, TS_EQ_ODE_EXPLICIT)); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */
  PetscCall(TSSetType(ctx.ts, TSRK));
  PetscCall(TSSetRHSFunction(ctx.ts, NULL, (TSRHSFunctionFn *)RHSFunction, &ctx));
  PetscCall(TSSetRHSJacobian(ctx.ts, A, A, (TSRHSJacobianFn *)RHSJacobian, &ctx));
  PetscCall(TSSetExactFinalTime(ctx.ts, TS_EXACTFINALTIME_MATCHSTEP));

  PetscCall(MatCreateVecs(A, &lambda[0], NULL));
  PetscCall(MatCreateVecs(Jacp, &mu[0], NULL));
  PetscCall(TSSetCostGradients(ctx.ts, 1, lambda, mu));
  PetscCall(TSSetRHSJacobianP(ctx.ts, Jacp, RHSJacobianP, &ctx));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set solver options
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSetMaxTime(ctx.ts, PetscRealConstant(1.0)));
  PetscCall(TSSetTimeStep(ctx.ts, PetscRealConstant(0.01)));
  PetscCall(TSSetFromOptions(ctx.ts));

  PetscCall(TSGetTimeStep(ctx.ts, &ctx.dt)); /* save the stepsize */

  PetscCall(TSCreateQuadratureTS(ctx.ts, PETSC_TRUE, &quadts));
  PetscCall(TSSetRHSFunction(quadts, NULL, (TSRHSFunctionFn *)CostIntegrand, &ctx));
  PetscCall(TSSetRHSJacobian(quadts, DRDU, DRDU, (TSRHSJacobianFn *)DRDUJacobianTranspose, &ctx));
  PetscCall(TSSetRHSJacobianP(quadts, DRDP, (TSRHSJacobianPFn *)DRDPJacobianTranspose, &ctx));
  PetscCall(TSSetSolution(ctx.ts, U));

  /* Create TAO solver and set desired solution method */
  PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
  PetscCall(TaoSetType(tao, TAOBLMVM));

  /*
     Optimization starts
  */
  /* Set initial solution guess */
  PetscCall(VecCreateSeq(PETSC_COMM_WORLD, 1, &p));
  PetscCall(VecGetArray(p, &x_ptr));
  x_ptr[0] = ctx.Pm;
  PetscCall(VecRestoreArray(p, &x_ptr));

  PetscCall(TaoSetSolution(tao, p));
  /* Set routine for function and gradient evaluation */
  PetscCall(TaoSetObjective(tao, FormFunction, (void *)&ctx));
  PetscCall(TaoSetGradient(tao, NULL, FormGradient, (void *)&ctx));

  /* Set bounds for the optimization */
  PetscCall(VecDuplicate(p, &lowerb));
  PetscCall(VecDuplicate(p, &upperb));
  PetscCall(VecGetArray(lowerb, &x_ptr));
  x_ptr[0] = 0.;
  PetscCall(VecRestoreArray(lowerb, &x_ptr));
  PetscCall(VecGetArray(upperb, &x_ptr));
  x_ptr[0] = PetscRealConstant(1.1);
  PetscCall(VecRestoreArray(upperb, &x_ptr));
  PetscCall(TaoSetVariableBounds(tao, lowerb, upperb));

  /* Check for any TAO command line options */
  PetscCall(TaoSetFromOptions(tao));
  PetscCall(TaoGetKSP(tao, &ksp));
  if (ksp) {
    PetscCall(KSPGetPC(ksp, &pc));
    PetscCall(PCSetType(pc, PCNONE));
  }

  /* SOLVE THE APPLICATION */
  PetscCall(TaoSolve(tao));

  PetscCall(VecView(p, PETSC_VIEWER_STDOUT_WORLD));
  /* Free TAO data structures */
  PetscCall(TaoDestroy(&tao));
  PetscCall(VecDestroy(&p));
  PetscCall(VecDestroy(&lowerb));
  PetscCall(VecDestroy(&upperb));

  PetscCall(TSDestroy(&ctx.ts));
  PetscCall(VecDestroy(&U));
  PetscCall(MatDestroy(&A));
  PetscCall(MatDestroy(&Jacp));
  PetscCall(MatDestroy(&DRDU));
  PetscCall(MatDestroy(&DRDP));
  PetscCall(VecDestroy(&lambda[0]));
  PetscCall(VecDestroy(&mu[0]));
  PetscCall(PetscFinalize());
  return 0;
}

/* ------------------------------------------------------------------ */
/*
   FormFunction - Evaluates the function

   Input Parameters:
   tao - the Tao context
   X   - the input vector
   ptr - optional user-defined context, as set by TaoSetObjectiveAndGradient()

   Output Parameters:
   f   - the newly evaluated function
*/
PetscErrorCode FormFunction(Tao tao, Vec P, PetscReal *f, PetscCtx ctx0)
{
  AppCtx      *ctx = (AppCtx *)ctx0;
  TS           ts  = ctx->ts;
  Vec          U; /* solution will be stored here */
  PetscScalar *u;
  PetscScalar *x_ptr;
  Vec          q;

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(P, (const PetscScalar **)&x_ptr));
  ctx->Pm = x_ptr[0];
  PetscCall(VecRestoreArrayRead(P, (const PetscScalar **)&x_ptr));

  /* reset time */
  PetscCall(TSSetTime(ts, 0.0));
  /* reset step counter, this is critical for adjoint solver */
  PetscCall(TSSetStepNumber(ts, 0));
  /* reset step size, the step size becomes negative after TSAdjointSolve */
  PetscCall(TSSetTimeStep(ts, ctx->dt));
  /* reinitialize the integral value */
  PetscCall(TSGetCostIntegral(ts, &q));
  PetscCall(VecSet(q, 0.0));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSGetSolution(ts, &U));
  PetscCall(VecGetArray(U, &u));
  u[0] = PetscAsinScalar(ctx->Pm / ctx->Pmax);
  u[1] = PetscRealConstant(1.0);
  PetscCall(VecRestoreArray(U, &u));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSolve(ts, U));
  PetscCall(TSGetCostIntegral(ts, &q));
  PetscCall(VecGetArray(q, &x_ptr));
  *f = -ctx->Pm + x_ptr[0];
  PetscCall(VecRestoreArray(q, &x_ptr));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FormGradient(Tao tao, Vec P, Vec G, PetscCtx ctx0)
{
  AppCtx      *ctx = (AppCtx *)ctx0;
  TS           ts  = ctx->ts;
  Vec          U; /* solution will be stored here */
  PetscReal    ftime;
  PetscInt     steps;
  PetscScalar *u;
  PetscScalar *x_ptr, *y_ptr;
  Vec         *lambda, q, *mu;

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(P, (const PetscScalar **)&x_ptr));
  ctx->Pm = x_ptr[0];
  PetscCall(VecRestoreArrayRead(P, (const PetscScalar **)&x_ptr));

  /* reset time */
  PetscCall(TSSetTime(ts, 0.0));
  /* reset step counter, this is critical for adjoint solver */
  PetscCall(TSSetStepNumber(ts, 0));
  /* reset step size, the step size becomes negative after TSAdjointSolve */
  PetscCall(TSSetTimeStep(ts, ctx->dt));
  /* reinitialize the integral value */
  PetscCall(TSGetCostIntegral(ts, &q));
  PetscCall(VecSet(q, 0.0));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Set initial conditions
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSGetSolution(ts, &U));
  PetscCall(VecGetArray(U, &u));
  u[0] = PetscAsinScalar(ctx->Pm / ctx->Pmax);
  u[1] = PetscRealConstant(1.0);
  PetscCall(VecRestoreArray(U, &u));

  /* Set up to save trajectory before TSSetFromOptions() so that TSTrajectory options can be captured */
  PetscCall(TSSetSaveTrajectory(ts));
  PetscCall(TSSetFromOptions(ts));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Solve nonlinear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSSolve(ts, U));

  PetscCall(TSGetSolveTime(ts, &ftime));
  PetscCall(TSGetStepNumber(ts, &steps));

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     Adjoint model starts here
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
  PetscCall(TSGetCostGradients(ts, NULL, &lambda, &mu));
  /*   Set initial conditions for the adjoint integration */
  PetscCall(VecGetArray(lambda[0], &y_ptr));
  y_ptr[0] = 0.0;
  y_ptr[1] = 0.0;
  PetscCall(VecRestoreArray(lambda[0], &y_ptr));
  PetscCall(VecGetArray(mu[0], &x_ptr));
  x_ptr[0] = PetscRealConstant(-1.0);
  PetscCall(VecRestoreArray(mu[0], &x_ptr));

  PetscCall(TSAdjointSolve(ts));
  PetscCall(TSGetCostIntegral(ts, &q));
  PetscCall(ComputeSensiP(lambda[0], mu[0], ctx));
  PetscCall(VecCopy(mu[0], G));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

   build:
      requires: !complex !single

   test:
      args: -viewer_binary_skip_info -ts_adapt_type none -tao_monitor -tao_gatol 0.0 -tao_grtol 1.e-3 -tao_converged_reason

   test:
      suffix: 2
      args: -viewer_binary_skip_info -ts_adapt_type none -tao_monitor -tao_gatol 0.0 -tao_grtol 1.e-3 -tao_converged_reason -tao_test_gradient

TEST*/