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: */
     25: /*T

     27: T*/

     29: #include <petscts.h>

     31: #define freq 50
     32: #define ws (2*PETSC_PI*freq)
     33: #define MVAbase 100

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

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

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

     62:   Mat      Sol;   /* Solution matrix */
     63:   PetscInt stepnum;   /* Column number of solution matrix */
     64: } AppCtx;

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

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

     84:   TSGetApplicationContext(ts,&user);
     85:   TSGetTime(ts,&t);
     86:   TSGetSolution(ts,&X);
     87:   idx      =  3*user->stepnum;
     88:   MatDenseGetArray(user->Sol,&mat);
     89:   VecGetArrayRead(X,&x);
     90:   mat[idx] = t;
     91:   PetscArraycpy(mat+idx+1,x,2);
     92:   MatDenseRestoreArray(user->Sol,&mat);
     93:   VecRestoreArrayRead(X,&x);
     94:   user->stepnum++;
     95:   return(0);
     96: }

     99: /* Computes the wind speed using Weibull distribution */
    100: PetscErrorCode WindSpeeds(AppCtx *user)
    101: {
    103:   PetscScalar    *x,*t,avg_dev,sum;
    104:   PetscInt       i;

    107:   user->cw       = 5;
    108:   user->kw       = 2; /* Rayleigh distribution */
    109:   user->nsamples = 2000;
    110:   user->Tw       = 0.2;
    111:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Wind Speed Options","");
    112:   {
    113:     PetscOptionsReal("-cw","","",user->cw,&user->cw,NULL);
    114:     PetscOptionsReal("-kw","","",user->kw,&user->kw,NULL);
    115:     PetscOptionsInt("-nsamples","","",user->nsamples,&user->nsamples,NULL);
    116:     PetscOptionsReal("-Tw","","",user->Tw,&user->Tw,NULL);
    117:   }
    118:   PetscOptionsEnd();
    119:   VecCreate(PETSC_COMM_WORLD,&user->wind_data);
    120:   VecSetSizes(user->wind_data,PETSC_DECIDE,user->nsamples);
    121:   VecSetFromOptions(user->wind_data);
    122:   VecDuplicate(user->wind_data,&user->t_wind);

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

    128:   /* Wind speed deviation = (-log(rand)/cw)^(1/kw) */
    129:   VecSetRandom(user->wind_data,NULL);
    130:   VecLog(user->wind_data);
    131:   VecScale(user->wind_data,-1/user->cw);
    132:   VecGetArray(user->wind_data,&x);
    133:   for (i=0;i < user->nsamples;i++) x[i] = PetscPowScalar(x[i],(1/user->kw));
    134:   VecRestoreArray(user->wind_data,&x);
    135:   VecSum(user->wind_data,&sum);
    136:   avg_dev = sum/user->nsamples;
    137:   /* Wind speed (t) = (1 + wind speed deviation(t) - avg_dev)*average wind speed */
    138:   VecShift(user->wind_data,(1-avg_dev));
    139:   VecScale(user->wind_data,vwa);
    140:   return(0);
    141: }

    143: /* Sets the parameters for wind turbine */
    144: PetscErrorCode SetWindTurbineParams(AppCtx *user)
    145: {
    147:   user->Rt  = 35;
    148:   user->Ar  = PETSC_PI*user->Rt*user->Rt;
    149:   user->nGB = 1.0/89.0;
    150:   user->rho = 1.225;
    151:   user->Ht  = 1.5;
    152:   return(0);
    153: }

    155: /* Sets the parameters for induction generator */
    156: PetscErrorCode SetInductionGeneratorParams(AppCtx *user)
    157: {
    159:   user->np = 4;
    160:   user->Xm = 3.0;
    161:   user->Xs = 0.1;
    162:   user->Xr = 0.08;
    163:   user->Rs = 0.01;
    164:   user->Rr = 0.01;
    165:   user->Xp = user->Xs + user->Xm*user->Xr/(user->Xm + user->Xr);
    166:   user->Hm = 1.0;
    167:   user->Te = 0.011063063063251968;
    168:   return(0);
    169: }

    171: /* Computes the power extracted from wind */
    172: PetscErrorCode GetWindPower(PetscScalar wm,PetscScalar vw,PetscScalar *Pw,AppCtx *user)
    173: {
    174:   PetscScalar temp,lambda,lambda_i,cp;

    177:   temp     = user->nGB*2*user->Rt*ws/user->np;
    178:   lambda   = temp*wm/vw;
    179:   lambda_i = 1/(1/lambda + 0.002);
    180:   cp       = 0.44*(125/lambda_i - 6.94)*PetscExpScalar(-16.5/lambda_i);
    181:   *Pw      = 0.5*user->rho*cp*user->Ar*vw*vw*vw/(MVAbase*1e6);
    182:   return(0);
    183: }

    185: /*
    186:      Defines the ODE passed to the ODE solver
    187: */
    188: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,AppCtx *user)
    189: {
    190:   PetscErrorCode    ierr;
    191:   PetscScalar       *f,wm,Pw,*wd;
    192:   const PetscScalar *u,*udot;
    193:   PetscInt          stepnum;

    196:   TSGetStepNumber(ts,&stepnum);
    197:   /*  The next three lines allow us to access the entries of the vectors directly */
    198:   VecGetArrayRead(U,&u);
    199:   VecGetArrayRead(Udot,&udot);
    200:   VecGetArray(F,&f);
    201:   VecGetArray(user->wind_data,&wd);

    203:   f[0] = user->Tw*udot[0] - wd[stepnum] + u[0];
    204:   wm   = 1-u[1];
    205:   GetWindPower(wm,u[0],&Pw,user);
    206:   f[1] = 2.0*(user->Ht+user->Hm)*udot[1] - Pw/wm + user->Te;

    208:   VecRestoreArray(user->wind_data,&wd);
    209:   VecRestoreArrayRead(U,&u);
    210:   VecRestoreArrayRead(Udot,&udot);
    211:   VecRestoreArray(F,&f);
    212:   return(0);
    213: }

    215: int main(int argc,char **argv)
    216: {
    217:   TS                ts;            /* ODE integrator */
    218:   Vec               U;             /* solution will be stored here */
    219:   Mat               A;             /* Jacobian matrix */
    220:   PetscErrorCode    ierr;
    221:   PetscMPIInt       size;
    222:   PetscInt          n = 2,idx;
    223:   AppCtx            user;
    224:   PetscScalar       *u;
    225:   SNES              snes;
    226:   PetscScalar       *mat;
    227:   const PetscScalar *x,*rmat;
    228:   Mat               B;
    229:   PetscScalar       *amat;
    230:   PetscViewer       viewer;

    234:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    235:      Initialize program
    236:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    237:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
    238:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    239:   if (size > 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"Only for sequential runs");

    241:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    242:     Create necessary matrix and vectors
    243:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    244:   MatCreate(PETSC_COMM_WORLD,&A);
    245:   MatSetSizes(A,n,n,PETSC_DETERMINE,PETSC_DETERMINE);
    246:   MatSetFromOptions(A);
    247:   MatSetUp(A);

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

    251:   /* Create wind speed data using Weibull distribution */
    252:   WindSpeeds(&user);
    253:   /* Set parameters for wind turbine and induction generator */
    254:   SetWindTurbineParams(&user);
    255:   SetInductionGeneratorParams(&user);

    257:   VecGetArray(U,&u);
    258:   u[0] = vwa;
    259:   u[1] = s;
    260:   VecRestoreArray(U,&u);

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

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

    267:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    268:      Create timestepping solver context
    269:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    270:   TSCreate(PETSC_COMM_WORLD,&ts);
    271:   TSSetProblemType(ts,TS_NONLINEAR);
    272:   TSSetType(ts,TSBEULER);
    273:   TSSetIFunction(ts,NULL,(TSIFunction) IFunction,&user);

    275:   TSGetSNES(ts,&snes);
    276:   SNESSetJacobian(snes,A,A,SNESComputeJacobianDefault,NULL);
    277:   /*  TSSetIJacobian(ts,A,A,(TSIJacobian)IJacobian,&user); */
    278:   TSSetApplicationContext(ts,&user);

    280:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    281:      Set initial conditions
    282:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    283:   TSSetSolution(ts,U);

    285:   /* Save initial solution */
    286:   idx=3*user.stepnum;

    288:   MatDenseGetArray(user.Sol,&mat);
    289:   VecGetArrayRead(U,&x);

    291:   mat[idx] = 0.0;

    293:   PetscArraycpy(mat+idx+1,x,2);
    294:   MatDenseRestoreArray(user.Sol,&mat);
    295:   VecRestoreArrayRead(U,&x);
    296:   user.stepnum++;

    299:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    300:      Set solver options
    301:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    302:   TSSetMaxTime(ts,20.0);
    303:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);
    304:   TSSetTimeStep(ts,.01);
    305:   TSSetFromOptions(ts);
    306:   TSSetPostStep(ts,SaveSolution);
    307:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    308:      Solve nonlinear system
    309:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    310:   TSSolve(ts,U);

    312:   MatCreateSeqDense(PETSC_COMM_SELF,3,user.stepnum,NULL,&B);
    313:   MatDenseGetArrayRead(user.Sol,&rmat);
    314:   MatDenseGetArray(B,&amat);
    315:   PetscArraycpy(amat,rmat,user.stepnum*3);
    316:   MatDenseRestoreArray(B,&amat);
    317:   MatDenseRestoreArrayRead(user.Sol,&rmat);

    319:   PetscViewerBinaryOpen(PETSC_COMM_SELF,"out.bin",FILE_MODE_WRITE,&viewer);
    320:   MatView(B,viewer);
    321:   PetscViewerDestroy(&viewer);
    322:   MatDestroy(&user.Sol);
    323:   MatDestroy(&B);
    324:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    325:      Free work space.  All PETSc objects should be destroyed when they are no longer needed.
    326:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    327:   VecDestroy(&user.wind_data);
    328:   VecDestroy(&user.t_wind);
    329:   MatDestroy(&A);
    330:   VecDestroy(&U);
    331:   TSDestroy(&ts);

    333:   PetscFinalize();
    334:   return ierr;
    335: }

    338: /*TEST

    340:    build:
    341:       requires: !complex

    343:    test:

    346: TEST*/