Actual source code: ex5.c


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

\begin{eqnarray}
T_w\frac{dv_w}{dt} & = & v_w - v_we \\
2(H_t+H_m)\frac{ds}{dt} & = & P_w - P_e
\end{eqnarray}
 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:   TSGetApplicationContext(ts,&user);
 80:   TSGetTime(ts,&t);
 81:   TSGetSolution(ts,&X);
 82:   idx      =  3*user->stepnum;
 83:   MatDenseGetArray(user->Sol,&mat);
 84:   VecGetArrayRead(X,&x);
 85:   mat[idx] = t;
 86:   PetscArraycpy(mat+idx+1,x,2);
 87:   MatDenseRestoreArray(user->Sol,&mat);
 88:   VecRestoreArrayRead(X,&x);
 89:   user->stepnum++;
 90:   return 0;
 91: }

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

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

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

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

136: /* Sets the parameters for wind turbine */
137: PetscErrorCode SetWindTurbineParams(AppCtx *user)
138: {
139:   user->Rt  = 35;
140:   user->Ar  = PETSC_PI*user->Rt*user->Rt;
141:   user->nGB = 1.0/89.0;
142:   user->rho = 1.225;
143:   user->Ht  = 1.5;
144:   return 0;
145: }

147: /* Sets the parameters for induction generator */
148: PetscErrorCode SetInductionGeneratorParams(AppCtx *user)
149: {
150:   user->np = 4;
151:   user->Xm = 3.0;
152:   user->Xs = 0.1;
153:   user->Xr = 0.08;
154:   user->Rs = 0.01;
155:   user->Rr = 0.01;
156:   user->Xp = user->Xs + user->Xm*user->Xr/(user->Xm + user->Xr);
157:   user->Hm = 1.0;
158:   user->Te = 0.011063063063251968;
159:   return 0;
160: }

162: /* Computes the power extracted from wind */
163: PetscErrorCode GetWindPower(PetscScalar wm,PetscScalar vw,PetscScalar *Pw,AppCtx *user)
164: {
165:   PetscScalar temp,lambda,lambda_i,cp;

167:   temp     = user->nGB*2*user->Rt*ws/user->np;
168:   lambda   = temp*wm/vw;
169:   lambda_i = 1/(1/lambda + 0.002);
170:   cp       = 0.44*(125/lambda_i - 6.94)*PetscExpScalar(-16.5/lambda_i);
171:   *Pw      = 0.5*user->rho*cp*user->Ar*vw*vw*vw/(MVAbase*1e6);
172:   return 0;
173: }

175: /*
176:      Defines the ODE passed to the ODE solver
177: */
178: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,AppCtx *user)
179: {
180:   PetscScalar       *f,wm,Pw,*wd;
181:   const PetscScalar *u,*udot;
182:   PetscInt          stepnum;

184:   TSGetStepNumber(ts,&stepnum);
185:   /*  The next three lines allow us to access the entries of the vectors directly */
186:   VecGetArrayRead(U,&u);
187:   VecGetArrayRead(Udot,&udot);
188:   VecGetArray(F,&f);
189:   VecGetArray(user->wind_data,&wd);

191:   f[0] = user->Tw*udot[0] - wd[stepnum] + u[0];
192:   wm   = 1-u[1];
193:   GetWindPower(wm,u[0],&Pw,user);
194:   f[1] = 2.0*(user->Ht+user->Hm)*udot[1] - Pw/wm + user->Te;

196:   VecRestoreArray(user->wind_data,&wd);
197:   VecRestoreArrayRead(U,&u);
198:   VecRestoreArrayRead(Udot,&udot);
199:   VecRestoreArray(F,&f);
200:   return 0;
201: }

203: int main(int argc,char **argv)
204: {
205:   TS                ts;            /* ODE integrator */
206:   Vec               U;             /* solution will be stored here */
207:   Mat               A;             /* Jacobian matrix */
208:   PetscMPIInt       size;
209:   PetscInt          n = 2,idx;
210:   AppCtx            user;
211:   PetscScalar       *u;
212:   SNES              snes;
213:   PetscScalar       *mat;
214:   const PetscScalar *x,*rmat;
215:   Mat               B;
216:   PetscScalar       *amat;
217:   PetscViewer       viewer;

219:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
220:      Initialize program
221:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
222:   PetscInitialize(&argc,&argv,(char*)0,help);
223:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

226:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227:     Create necessary matrix and vectors
228:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229:   MatCreate(PETSC_COMM_WORLD,&A);
230:   MatSetSizes(A,n,n,PETSC_DETERMINE,PETSC_DETERMINE);
231:   MatSetFromOptions(A);
232:   MatSetUp(A);

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

236:   /* Create wind speed data using Weibull distribution */
237:   WindSpeeds(&user);
238:   /* Set parameters for wind turbine and induction generator */
239:   SetWindTurbineParams(&user);
240:   SetInductionGeneratorParams(&user);

242:   VecGetArray(U,&u);
243:   u[0] = vwa;
244:   u[1] = s;
245:   VecRestoreArray(U,&u);

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

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

252:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
253:      Create timestepping solver context
254:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
255:   TSCreate(PETSC_COMM_WORLD,&ts);
256:   TSSetProblemType(ts,TS_NONLINEAR);
257:   TSSetType(ts,TSBEULER);
258:   TSSetIFunction(ts,NULL,(TSIFunction) IFunction,&user);

260:   TSGetSNES(ts,&snes);
261:   SNESSetJacobian(snes,A,A,SNESComputeJacobianDefault,NULL);
262:   /*  TSSetIJacobian(ts,A,A,(TSIJacobian)IJacobian,&user); */
263:   TSSetApplicationContext(ts,&user);

265:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
266:      Set initial conditions
267:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
268:   TSSetSolution(ts,U);

270:   /* Save initial solution */
271:   idx=3*user.stepnum;

273:   MatDenseGetArray(user.Sol,&mat);
274:   VecGetArrayRead(U,&x);

276:   mat[idx] = 0.0;

278:   PetscArraycpy(mat+idx+1,x,2);
279:   MatDenseRestoreArray(user.Sol,&mat);
280:   VecRestoreArrayRead(U,&x);
281:   user.stepnum++;

283:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
284:      Set solver options
285:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
286:   TSSetMaxTime(ts,20.0);
287:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);
288:   TSSetTimeStep(ts,.01);
289:   TSSetFromOptions(ts);
290:   TSSetPostStep(ts,SaveSolution);
291:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
292:      Solve nonlinear system
293:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
294:   TSSolve(ts,U);

296:   MatCreateSeqDense(PETSC_COMM_SELF,3,user.stepnum,NULL,&B);
297:   MatDenseGetArrayRead(user.Sol,&rmat);
298:   MatDenseGetArray(B,&amat);
299:   PetscArraycpy(amat,rmat,user.stepnum*3);
300:   MatDenseRestoreArray(B,&amat);
301:   MatDenseRestoreArrayRead(user.Sol,&rmat);

303:   PetscViewerBinaryOpen(PETSC_COMM_SELF,"out.bin",FILE_MODE_WRITE,&viewer);
304:   MatView(B,viewer);
305:   PetscViewerDestroy(&viewer);
306:   MatDestroy(&user.Sol);
307:   MatDestroy(&B);
308:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
309:      Free work space.  All PETSc objects should be destroyed when they are no longer needed.
310:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
311:   VecDestroy(&user.wind_data);
312:   VecDestroy(&user.t_wind);
313:   MatDestroy(&A);
314:   VecDestroy(&U);
315:   TSDestroy(&ts);

317:   PetscFinalize();
318:   return 0;
319: }

321: /*TEST

323:    build:
324:       requires: !complex

326:    test:

328: TEST*/