Actual source code: ex5.c


  2: static char help[] = "Basic equation for an induction generator driven by a wind turbine.\n";

  4: /*F
  5: \begin{eqnarray}
  6:           T_w\frac{dv_w}{dt} & = & v_w - v_we \\
  7:           2(H_t+H_m)\frac{ds}{dt} & = & P_w - P_e
  8: \end{eqnarray}
  9: F*/
 10: /*
 11:  - Pw is the power extracted from the wind turbine given by
 12:            Pw = 0.5*\rho*cp*Ar*vw^3

 14:  - The wind speed time series is modeled using a Weibull distribution and then
 15:    passed through a low pass filter (with time constant T_w).
 16:  - v_we is the wind speed data calculated using Weibull distribution while v_w is
 17:    the output of the filter.
 18:  - P_e is assumed as constant electrical torque

 20:  - This example does not work with adaptive time stepping!

 22: Reference:
 23: Power System Modeling and Scripting - F. Milano
 24: */

 26: #include <petscts.h>

 28: #define freq    50
 29: #define ws      (2 * PETSC_PI * freq)
 30: #define MVAbase 100

 32: typedef struct {
 33:   /* Parameters for wind speed model */
 34:   PetscInt  nsamples;  /* Number of wind samples */
 35:   PetscReal cw;        /* Scale factor for Weibull distribution */
 36:   PetscReal kw;        /* Shape factor for Weibull distribution */
 37:   Vec       wind_data; /* Vector to hold wind speeds */
 38:   Vec       t_wind;    /* Vector to hold wind speed times */
 39:   PetscReal Tw;        /* Filter time constant */

 41:   /* Wind turbine parameters */
 42:   PetscScalar Rt;  /* Rotor radius */
 43:   PetscScalar Ar;  /* Area swept by rotor (pi*R*R) */
 44:   PetscReal   nGB; /* Gear box ratio */
 45:   PetscReal   Ht;  /* Turbine inertia constant */
 46:   PetscReal   rho; /* Atmospheric pressure */

 48:   /* Induction generator parameters */
 49:   PetscInt    np; /* Number of poles */
 50:   PetscReal   Xm; /* Magnetizing reactance */
 51:   PetscReal   Xs; /* Stator Reactance */
 52:   PetscReal   Xr; /* Rotor reactance */
 53:   PetscReal   Rs; /* Stator resistance */
 54:   PetscReal   Rr; /* Rotor resistance */
 55:   PetscReal   Hm; /* Motor inertia constant */
 56:   PetscReal   Xp; /* Xs + Xm*Xr/(Xm + Xr) */
 57:   PetscScalar Te; /* Electrical Torque */

 59:   Mat      Sol;     /* Solution matrix */
 60:   PetscInt stepnum; /* Column number of solution matrix */
 61: } AppCtx;

 63: /* Initial values computed by Power flow and initialization */
 64: PetscScalar s = -0.00011577790353;
 65: /*Pw = 0.011064344110238; %Te*wm */
 66: PetscScalar vwa  = 22.317142184449754;
 67: PetscReal   tmax = 20.0;

 69: /* Saves the solution at each time to a matrix */
 70: PetscErrorCode SaveSolution(TS ts)
 71: {
 72:   AppCtx            *user;
 73:   Vec                X;
 74:   PetscScalar       *mat;
 75:   const PetscScalar *x;
 76:   PetscInt           idx;
 77:   PetscReal          t;

 79:   PetscFunctionBegin;
 80:   PetscCall(TSGetApplicationContext(ts, &user));
 81:   PetscCall(TSGetTime(ts, &t));
 82:   PetscCall(TSGetSolution(ts, &X));
 83:   idx = 3 * user->stepnum;
 84:   PetscCall(MatDenseGetArray(user->Sol, &mat));
 85:   PetscCall(VecGetArrayRead(X, &x));
 86:   mat[idx] = t;
 87:   PetscCall(PetscArraycpy(mat + idx + 1, x, 2));
 88:   PetscCall(MatDenseRestoreArray(user->Sol, &mat));
 89:   PetscCall(VecRestoreArrayRead(X, &x));
 90:   user->stepnum++;
 91:   PetscFunctionReturn(PETSC_SUCCESS);
 92: }

 94: /* Computes the wind speed using Weibull distribution */
 95: PetscErrorCode WindSpeeds(AppCtx *user)
 96: {
 97:   PetscScalar *x, *t, avg_dev, sum;
 98:   PetscInt     i;

100:   PetscFunctionBegin;
101:   user->cw       = 5;
102:   user->kw       = 2; /* Rayleigh distribution */
103:   user->nsamples = 2000;
104:   user->Tw       = 0.2;
105:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Wind Speed Options", "");
106:   {
107:     PetscCall(PetscOptionsReal("-cw", "", "", user->cw, &user->cw, NULL));
108:     PetscCall(PetscOptionsReal("-kw", "", "", user->kw, &user->kw, NULL));
109:     PetscCall(PetscOptionsInt("-nsamples", "", "", user->nsamples, &user->nsamples, NULL));
110:     PetscCall(PetscOptionsReal("-Tw", "", "", user->Tw, &user->Tw, NULL));
111:   }
112:   PetscOptionsEnd();
113:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user->wind_data));
114:   PetscCall(VecSetSizes(user->wind_data, PETSC_DECIDE, user->nsamples));
115:   PetscCall(VecSetFromOptions(user->wind_data));
116:   PetscCall(VecDuplicate(user->wind_data, &user->t_wind));

118:   PetscCall(VecGetArray(user->t_wind, &t));
119:   for (i = 0; i < user->nsamples; i++) t[i] = (i + 1) * tmax / user->nsamples;
120:   PetscCall(VecRestoreArray(user->t_wind, &t));

122:   /* Wind speed deviation = (-log(rand)/cw)^(1/kw) */
123:   PetscCall(VecSetRandom(user->wind_data, NULL));
124:   PetscCall(VecLog(user->wind_data));
125:   PetscCall(VecScale(user->wind_data, -1 / user->cw));
126:   PetscCall(VecGetArray(user->wind_data, &x));
127:   for (i = 0; i < user->nsamples; i++) x[i] = PetscPowScalar(x[i], (1 / user->kw));
128:   PetscCall(VecRestoreArray(user->wind_data, &x));
129:   PetscCall(VecSum(user->wind_data, &sum));
130:   avg_dev = sum / user->nsamples;
131:   /* Wind speed (t) = (1 + wind speed deviation(t) - avg_dev)*average wind speed */
132:   PetscCall(VecShift(user->wind_data, (1 - avg_dev)));
133:   PetscCall(VecScale(user->wind_data, vwa));
134:   PetscFunctionReturn(PETSC_SUCCESS);
135: }

137: /* Sets the parameters for wind turbine */
138: PetscErrorCode SetWindTurbineParams(AppCtx *user)
139: {
140:   PetscFunctionBegin;
141:   user->Rt  = 35;
142:   user->Ar  = PETSC_PI * user->Rt * user->Rt;
143:   user->nGB = 1.0 / 89.0;
144:   user->rho = 1.225;
145:   user->Ht  = 1.5;
146:   PetscFunctionReturn(PETSC_SUCCESS);
147: }

149: /* Sets the parameters for induction generator */
150: PetscErrorCode SetInductionGeneratorParams(AppCtx *user)
151: {
152:   PetscFunctionBegin;
153:   user->np = 4;
154:   user->Xm = 3.0;
155:   user->Xs = 0.1;
156:   user->Xr = 0.08;
157:   user->Rs = 0.01;
158:   user->Rr = 0.01;
159:   user->Xp = user->Xs + user->Xm * user->Xr / (user->Xm + user->Xr);
160:   user->Hm = 1.0;
161:   user->Te = 0.011063063063251968;
162:   PetscFunctionReturn(PETSC_SUCCESS);
163: }

165: /* Computes the power extracted from wind */
166: PetscErrorCode GetWindPower(PetscScalar wm, PetscScalar vw, PetscScalar *Pw, AppCtx *user)
167: {
168:   PetscScalar temp, lambda, lambda_i, cp;

170:   PetscFunctionBegin;
171:   temp     = user->nGB * 2 * user->Rt * ws / user->np;
172:   lambda   = temp * wm / vw;
173:   lambda_i = 1 / (1 / lambda + 0.002);
174:   cp       = 0.44 * (125 / lambda_i - 6.94) * PetscExpScalar(-16.5 / lambda_i);
175:   *Pw      = 0.5 * user->rho * cp * user->Ar * vw * vw * vw / (MVAbase * 1e6);
176:   PetscFunctionReturn(PETSC_SUCCESS);
177: }

179: /*
180:      Defines the ODE passed to the ODE solver
181: */
182: static PetscErrorCode IFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, AppCtx *user)
183: {
184:   PetscScalar       *f, wm, Pw, *wd;
185:   const PetscScalar *u, *udot;
186:   PetscInt           stepnum;

188:   PetscFunctionBegin;
189:   PetscCall(TSGetStepNumber(ts, &stepnum));
190:   /*  The next three lines allow us to access the entries of the vectors directly */
191:   PetscCall(VecGetArrayRead(U, &u));
192:   PetscCall(VecGetArrayRead(Udot, &udot));
193:   PetscCall(VecGetArray(F, &f));
194:   PetscCall(VecGetArray(user->wind_data, &wd));

196:   f[0] = user->Tw * udot[0] - wd[stepnum] + u[0];
197:   wm   = 1 - u[1];
198:   PetscCall(GetWindPower(wm, u[0], &Pw, user));
199:   f[1] = 2.0 * (user->Ht + user->Hm) * udot[1] - Pw / wm + user->Te;

201:   PetscCall(VecRestoreArray(user->wind_data, &wd));
202:   PetscCall(VecRestoreArrayRead(U, &u));
203:   PetscCall(VecRestoreArrayRead(Udot, &udot));
204:   PetscCall(VecRestoreArray(F, &f));
205:   PetscFunctionReturn(PETSC_SUCCESS);
206: }

208: int main(int argc, char **argv)
209: {
210:   TS                 ts; /* ODE integrator */
211:   Vec                U;  /* solution will be stored here */
212:   Mat                A;  /* Jacobian matrix */
213:   PetscMPIInt        size;
214:   PetscInt           n = 2, idx;
215:   AppCtx             user;
216:   PetscScalar       *u;
217:   SNES               snes;
218:   PetscScalar       *mat;
219:   const PetscScalar *x, *rmat;
220:   Mat                B;
221:   PetscScalar       *amat;
222:   PetscViewer        viewer;

224:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
225:      Initialize program
226:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
227:   PetscFunctionBeginUser;
228:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
229:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
230:   PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Only for sequential runs");

232:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
233:     Create necessary matrix and vectors
234:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
235:   PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
236:   PetscCall(MatSetSizes(A, n, n, PETSC_DETERMINE, PETSC_DETERMINE));
237:   PetscCall(MatSetFromOptions(A));
238:   PetscCall(MatSetUp(A));

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

242:   /* Create wind speed data using Weibull distribution */
243:   PetscCall(WindSpeeds(&user));
244:   /* Set parameters for wind turbine and induction generator */
245:   PetscCall(SetWindTurbineParams(&user));
246:   PetscCall(SetInductionGeneratorParams(&user));

248:   PetscCall(VecGetArray(U, &u));
249:   u[0] = vwa;
250:   u[1] = s;
251:   PetscCall(VecRestoreArray(U, &u));

253:   /* Create matrix to save solutions at each time step */
254:   user.stepnum = 0;

256:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, 3, 2010, NULL, &user.Sol));

258:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
259:      Create timestepping solver context
260:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
261:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
262:   PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
263:   PetscCall(TSSetType(ts, TSBEULER));
264:   PetscCall(TSSetIFunction(ts, NULL, (TSIFunction)IFunction, &user));

266:   PetscCall(TSGetSNES(ts, &snes));
267:   PetscCall(SNESSetJacobian(snes, A, A, SNESComputeJacobianDefault, NULL));
268:   /*  PetscCall(TSSetIJacobian(ts,A,A,(TSIJacobian)IJacobian,&user)); */
269:   PetscCall(TSSetApplicationContext(ts, &user));

271:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
272:      Set initial conditions
273:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
274:   PetscCall(TSSetSolution(ts, U));

276:   /* Save initial solution */
277:   idx = 3 * user.stepnum;

279:   PetscCall(MatDenseGetArray(user.Sol, &mat));
280:   PetscCall(VecGetArrayRead(U, &x));

282:   mat[idx] = 0.0;

284:   PetscCall(PetscArraycpy(mat + idx + 1, x, 2));
285:   PetscCall(MatDenseRestoreArray(user.Sol, &mat));
286:   PetscCall(VecRestoreArrayRead(U, &x));
287:   user.stepnum++;

289:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
290:      Set solver options
291:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
292:   PetscCall(TSSetMaxTime(ts, 20.0));
293:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));
294:   PetscCall(TSSetTimeStep(ts, .01));
295:   PetscCall(TSSetFromOptions(ts));
296:   PetscCall(TSSetPostStep(ts, SaveSolution));
297:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
298:      Solve nonlinear system
299:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
300:   PetscCall(TSSolve(ts, U));

302:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, 3, user.stepnum, NULL, &B));
303:   PetscCall(MatDenseGetArrayRead(user.Sol, &rmat));
304:   PetscCall(MatDenseGetArray(B, &amat));
305:   PetscCall(PetscArraycpy(amat, rmat, user.stepnum * 3));
306:   PetscCall(MatDenseRestoreArray(B, &amat));
307:   PetscCall(MatDenseRestoreArrayRead(user.Sol, &rmat));

309:   PetscCall(PetscViewerBinaryOpen(PETSC_COMM_SELF, "out.bin", FILE_MODE_WRITE, &viewer));
310:   PetscCall(MatView(B, viewer));
311:   PetscCall(PetscViewerDestroy(&viewer));
312:   PetscCall(MatDestroy(&user.Sol));
313:   PetscCall(MatDestroy(&B));
314:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
315:      Free work space.  All PETSc objects should be destroyed when they are no longer needed.
316:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
317:   PetscCall(VecDestroy(&user.wind_data));
318:   PetscCall(VecDestroy(&user.t_wind));
319:   PetscCall(MatDestroy(&A));
320:   PetscCall(VecDestroy(&U));
321:   PetscCall(TSDestroy(&ts));

323:   PetscCall(PetscFinalize());
324:   return 0;
325: }

327: /*TEST

329:    build:
330:       requires: !complex

332:    test:

334: TEST*/