def execute(self):
ctrl = self.control
n = self.npts
# finer power curve
self.V, _, _ = linspace_with_deriv(ctrl.Vin, ctrl.Vout, n)
spline = Akima(self.Vcoarse, self.Pcoarse)
self.P, dP_dV, dP_dVcoarse, dP_dPcoarse = spline.interp(self.V)
self.J = hstack([dP_dVcoarse, dP_dPcoarse])
def solve_nonlinear(self, params, unknowns, resids):
n = params['npts']
# finer power curve
V, _, _ = linspace_with_deriv(params['control:Vin'],
params['control:Vout'], n)
unknowns['V'] = V
spline = Akima(params['Vcoarse'], params['Pcoarse'])
P, dP_dV, dP_dVcoarse, dP_dPcoarse = spline.interp(unknowns['V'])
unknowns['P'] = P
J = {}
J['P', 'Vcoarse'] = dP_dVcoarse
J['P', 'Pcoarse'] = dP_dPcoarse
self.J = J
def execute(self):
nc = len(self.chord_sub)
nt = len(self.theta_sub)
Rhub = self.Rhub
Rtip = self.Rtip
idxc = self.idx_cylinder
r_max_chord = Rhub + (Rtip-Rhub)*self.r_max_chord
r_cylinder = Rhub + (Rtip-Rhub)*self.r_af[idxc]
# chord parameterization
rc_outer, drc_drcmax, drc_drtip = linspace_with_deriv(r_max_chord, Rtip, nc-1)
r_chord = np.concatenate([[Rhub], rc_outer])
drc_drcmax = np.concatenate([[0.0], drc_drcmax])
drc_drtip = np.concatenate([[0.0], drc_drtip])
drc_drhub = np.concatenate([[1.0], np.zeros(nc-1)])
# theta parameterization
r_theta, drt_drcyl, drt_drtip = linspace_with_deriv(r_cylinder, Rtip, nt)
# spline
chord_spline = Akima(r_chord, self.chord_sub)
theta_spline = Akima(r_theta, self.theta_sub)
self.r = Rhub + (Rtip-Rhub)*self.r_af
self.chord, dchord_dr, dchord_drchord, dchord_dchordsub = chord_spline.interp(self.r)
theta_outer, dthetaouter_dr, dthetaouter_drtheta, dthetaouter_dthetasub = theta_spline.interp(self.r[idxc:])
theta_inner = theta_outer[0] * np.ones(idxc)
self.theta = np.concatenate([theta_inner, theta_outer])
self.r_af_spacing = np.diff(self.r_af)
self.precurve = np.zeros_like(self.chord) # TODO: for now I'm forcing this to zero, just for backwards compatibility
# gradients (TODO: rethink these a bit or use Tapenade.)
n = len(self.r_af)
dr_draf = (Rtip-Rhub)*np.ones(n)
dr_dRhub = 1.0 - self.r_af
dr_dRtip = self.r_af
dr = hstack([np.diag(dr_draf), np.zeros((n, 1)), dr_dRhub, dr_dRtip, np.zeros((n, nc+nt))])
dchord_draf = dchord_dr * dr_draf
dchord_drmaxchord0 = np.dot(dchord_drchord, drc_drcmax)
dchord_drmaxchord = dchord_drmaxchord0 * (Rtip-Rhub)
dchord_drhub = np.dot(dchord_drchord, drc_drhub) + dchord_drmaxchord0*(1.0 - self.r_max_chord) + dchord_dr*dr_dRhub
dchord_drtip = np.dot(dchord_drchord, drc_drtip) + dchord_drmaxchord0*(self.r_max_chord) + dchord_dr*dr_dRtip
dchord = hstack([np.diag(dchord_draf), dchord_drmaxchord, dchord_drhub, dchord_drtip, dchord_dchordsub, np.zeros((n, nt))])
dthetaouter_dcyl = np.dot(dthetaouter_drtheta, drt_drcyl)
dthetaouter_draf = dthetaouter_dr*dr_draf[idxc:]
dthetaouter_drhub = dthetaouter_dr*dr_dRhub[idxc:]
dthetaouter_drtip = dthetaouter_dr*dr_dRtip[idxc:] + np.dot(dthetaouter_drtheta, drt_drtip)
dtheta_draf = np.concatenate([np.zeros(idxc), dthetaouter_draf])
dtheta_drhub = np.concatenate([dthetaouter_drhub[0]*np.ones(idxc), dthetaouter_drhub])
dtheta_drtip = np.concatenate([dthetaouter_drtip[0]*np.ones(idxc), dthetaouter_drtip])
sub = dthetaouter_dthetasub[0, :]
dtheta_dthetasub = vstack([np.dot(np.ones((idxc, 1)), sub[np.newaxis, :]), dthetaouter_dthetasub])
dtheta_draf = np.diag(dtheta_draf)
dtheta_dcyl = np.concatenate([dthetaouter_dcyl[0]*np.ones(idxc), dthetaouter_dcyl])
dtheta_draf[idxc:, idxc] += dthetaouter_dcyl*(Rtip-Rhub)
dtheta_drhub += dtheta_dcyl*(1.0 - self.r_af[idxc])
dtheta_drtip += dtheta_dcyl*self.r_af[idxc]
dtheta = hstack([dtheta_draf, np.zeros((n, 1)), dtheta_drhub, dtheta_drtip, np.zeros((n, nc)), dtheta_dthetasub])
drafs_dr = np.zeros((n-1, n))
for i in range(n-1):
drafs_dr[i, i] = -1.0
drafs_dr[i, i+1] = 1.0
drafs = hstack([drafs_dr, np.zeros((n-1, 3+nc+nt))])
dprecurve = np.zeros((len(self.precurve), n+3+nc+nt))
self.J = vstack([dr, dchord, dtheta, drafs, dprecurve])
def apply_nonlinear(self, params, unknowns, resids):
n = params['npts']
Vrated = unknowns['Vrated']
# residual
spline = Akima(params['Vcoarse'], params['Pcoarse'])
P, dres_dVrated, dres_dVcoarse, dres_dPcoarse = spline.interp(Vrated)
resids['Vrated'] = P - params['control:ratedPower']
if True:
P1, _, _, _ = spline.interp(params['control:Vin'])
P2, _, _, _ = spline.interp(params['control:Vout'])
resids1 = P1 - params['control:ratedPower']
resids2 = P2 - params['control:ratedPower']
if ((resids1 < 0) == (resids2 < 0)):
if Vrated == params['control:Vout']:
resids['Vrated'] = 10000
elif Vrated != params['control:Vin']:
resids['Vrated'] = 0
## Test on
# region 2
V2, _, dV2_dVrated = linspace_with_deriv(params['control:Vin'], Vrated,
n / 2)
P2, dP2_dV2, dP2_dVcoarse, dP2_dPcoarse = spline.interp(V2)
# region 3
V3, dV3_dVrated, _ = linspace_with_deriv(Vrated,
params['control:Vout'],
n / 2 + 1)
V3 = V3[1:] # remove duplicate point
dV3_dVrated = dV3_dVrated[1:]
P3 = params['control:ratedPower'] * np.ones_like(V3)
# concatenate
unknowns['V'] = np.concatenate([V2, V3])
unknowns['P'] = np.concatenate([P2, P3])
R = params['R']
# rated speed conditions
Omega_d = params['control:tsr'] * Vrated / R * RS2RPM
OmegaRated, dOmegaRated_dOmegad, dOmegaRated_dmaxOmega \
= smooth_min(Omega_d, params['control:maxOmega'], pct_offset=0.01)
splineT = Akima(params['Vcoarse'], params['Tcoarse'])
Trated, dT_dVrated, dT_dVcoarse, dT_dTcoarse = splineT.interp(Vrated)
unknowns['ratedConditions:V'] = Vrated
unknowns['ratedConditions:Omega'] = OmegaRated
unknowns['ratedConditions:pitch'] = params['control:pitch']
unknowns['ratedConditions:T'] = Trated
unknowns['ratedConditions:Q'] = params['control:ratedPower'] / (
OmegaRated * RPM2RS)
unknowns['azimuth'] = 180.0
# gradients
ncoarse = len(params['Vcoarse'])
dV_dVrated = np.concatenate([dV2_dVrated, dV3_dVrated])
dP_dVcoarse = vstack([dP2_dVcoarse, np.zeros((n / 2, ncoarse))])
dP_dPcoarse = vstack([dP2_dPcoarse, np.zeros((n / 2, ncoarse))])
dP_dVrated = np.concatenate([dP2_dV2 * dV2_dVrated, np.zeros(n / 2)])
drOmega = np.concatenate(
[[dOmegaRated_dOmegad * Vrated / R * RS2RPM],
np.zeros(3 * ncoarse),
[
dOmegaRated_dOmegad * params['control:tsr'] / R * RS2RPM,
-dOmegaRated_dOmegad * params['control:tsr'] * Vrated / R**2 *
RS2RPM, dOmegaRated_dmaxOmega
]])
drQ = -params['control:ratedPower'] / (OmegaRated**2 *
RPM2RS) * drOmega
J = {}
J['Vrated', 'Vcoarse'] = np.reshape(dres_dVcoarse,
(1, len(dres_dVcoarse)))
J['Vrated', 'Pcoarse'] = np.reshape(dres_dPcoarse,
(1, len(dres_dPcoarse)))
J['Vrated', 'Vrated'] = dres_dVrated
J['V', 'Vrated'] = dV_dVrated
J['P', 'Vrated'] = dP_dVrated
J['P', 'Vcoarse'] = dP_dVcoarse
J['P', 'Pcoarse'] = dP_dPcoarse
J['ratedConditions:V', 'Vrated'] = 1.0
J['ratedConditions:Omega',
'control:tsr'] = dOmegaRated_dOmegad * Vrated / R * RS2RPM
J['ratedConditions:Omega',
'Vrated'] = dOmegaRated_dOmegad * params['control:tsr'] / R * RS2RPM
J['ratedConditions:Omega', 'R'] = -dOmegaRated_dOmegad * params[
'control:tsr'] * Vrated / R**2 * RS2RPM
J['ratedConditions:Omega', 'control:maxOmega'] = dOmegaRated_dmaxOmega
J['ratedConditions:T', 'Vcoarse'] = np.reshape(dT_dVcoarse,
(1, len(dT_dVcoarse)))
J['ratedConditions:T', 'Tcoarse'] = np.reshape(dT_dTcoarse,
(1, len(dT_dTcoarse)))
J['ratedConditions:T', 'Vrated'] = dT_dVrated
J['ratedConditions:Q', 'control:tsr'] = drQ[0]
J['ratedConditions:Q', 'Vrated'] = drQ[-3]
J['ratedConditions:Q', 'R'] = drQ[-2]
J['ratedConditions:Q', 'control:maxOmega'] = drQ[-1]
self.J = J
def evaluate(self):
ctrl = self.control
n = self.npts
Vrated = self.Vrated
# residual
spline = Akima(self.Vcoarse, self.Pcoarse)
P, dres_dVrated, dres_dVcoarse, dres_dPcoarse = spline.interp(Vrated)
self.residual = P - ctrl.ratedPower
# functional
# place half of points in region 2, half in region 3
# even though region 3 is constant we still need lots of points there
# because we will be integrating against a discretized wind
# speed distribution
# region 2
V2, _, dV2_dVrated = linspace_with_deriv(ctrl.Vin, Vrated, n/2)
P2, dP2_dV2, dP2_dVcoarse, dP2_dPcoarse = spline.interp(V2)
# region 3
V3, dV3_dVrated, _ = linspace_with_deriv(Vrated, ctrl.Vout, n/2+1)
V3 = V3[1:] # remove duplicate point
dV3_dVrated = dV3_dVrated[1:]
P3 = ctrl.ratedPower*np.ones_like(V3)
# concatenate
self.V = np.concatenate([V2, V3])
self.P = np.concatenate([P2, P3])
# rated speed conditions
Omega_d = ctrl.tsr*Vrated/self.R*RS2RPM
OmegaRated, dOmegaRated_dOmegad, dOmegaRated_dmaxOmega \
= smooth_min(Omega_d, ctrl.maxOmega, pct_offset=0.01)
splineT = Akima(self.Vcoarse, self.Tcoarse)
Trated, dT_dVrated, dT_dVcoarse, dT_dTcoarse = splineT.interp(Vrated)
self.ratedConditions.V = Vrated
self.ratedConditions.Omega = OmegaRated
self.ratedConditions.pitch = ctrl.pitch
self.ratedConditions.T = Trated
self.ratedConditions.Q = ctrl.ratedPower / (self.ratedConditions.Omega * RPM2RS)
# gradients
ncoarse = len(self.Vcoarse)
dres = np.concatenate([[0.0], dres_dVcoarse, dres_dPcoarse, np.zeros(ncoarse), np.array([dres_dVrated]), [0.0, 0.0]])
dV_dVrated = np.concatenate([dV2_dVrated, dV3_dVrated])
dV = hstack([np.zeros((n, 1)), np.zeros((n, 3*ncoarse)), dV_dVrated, np.zeros((n, 2))])
dP_dVcoarse = vstack([dP2_dVcoarse, np.zeros((n/2, ncoarse))])
dP_dPcoarse = vstack([dP2_dPcoarse, np.zeros((n/2, ncoarse))])
dP_dVrated = np.concatenate([dP2_dV2*dV2_dVrated, np.zeros(n/2)])
dP = hstack([np.zeros((n, 1)), dP_dVcoarse, dP_dPcoarse, np.zeros((n, ncoarse)), dP_dVrated, np.zeros((n, 2))])
drV = np.concatenate([[0.0], np.zeros(3*ncoarse), [1.0, 0.0, 0.0]])
drOmega = np.concatenate([[dOmegaRated_dOmegad*Vrated/self.R*RS2RPM], np.zeros(3*ncoarse),
[dOmegaRated_dOmegad*ctrl.tsr/self.R*RS2RPM, -dOmegaRated_dOmegad*ctrl.tsr*Vrated/self.R**2*RS2RPM,
dOmegaRated_dmaxOmega]])
drpitch = np.zeros(3*ncoarse+4)
drT = np.concatenate([[0.0], dT_dVcoarse, np.zeros(ncoarse), dT_dTcoarse, [dT_dVrated, 0.0, 0.0]])
drQ = -ctrl.ratedPower / (self.ratedConditions.Omega**2 * RPM2RS) * drOmega
self.J = vstack([dres, dV, dP, drV, drOmega, drpitch, drT, drQ])
def execute(self):
nc = len(self.chord_sub)
nt = len(self.theta_sub)
Rhub = self.Rhub
Rtip = self.Rtip
idxc = self.idx_cylinder
r_max_chord = Rhub + (Rtip - Rhub) * self.r_max_chord
r_cylinder = Rhub + (Rtip - Rhub) * self.r_af[idxc]
# chord parameterization
rc_outer, drc_drcmax, drc_drtip = linspace_with_deriv(
r_max_chord, Rtip, nc - 1)
r_chord = np.concatenate([[Rhub], rc_outer])
drc_drcmax = np.concatenate([[0.0], drc_drcmax])
drc_drtip = np.concatenate([[0.0], drc_drtip])
drc_drhub = np.concatenate([[1.0], np.zeros(nc - 1)])
# theta parameterization
r_theta, drt_drcyl, drt_drtip = linspace_with_deriv(
r_cylinder, Rtip, nt)
# spline
chord_spline = Akima(r_chord, self.chord_sub)
theta_spline = Akima(r_theta, self.theta_sub)
self.r = Rhub + (Rtip - Rhub) * self.r_af
self.chord, dchord_dr, dchord_drchord, dchord_dchordsub = chord_spline.interp(
self.r)
theta_outer, dthetaouter_dr, dthetaouter_drtheta, dthetaouter_dthetasub = theta_spline.interp(
self.r[idxc:])
theta_inner = theta_outer[0] * np.ones(idxc)
self.theta = np.concatenate([theta_inner, theta_outer])
self.r_af_spacing = np.diff(self.r_af)
self.precurve = np.zeros_like(
self.chord
) # TODO: for now I'm forcing this to zero, just for backwards compatibility
# gradients (TODO: rethink these a bit or use Tapenade.)
n = len(self.r_af)
dr_draf = (Rtip - Rhub) * np.ones(n)
dr_dRhub = 1.0 - self.r_af
dr_dRtip = self.r_af
dr = hstack([
np.diag(dr_draf),
np.zeros((n, 1)), dr_dRhub, dr_dRtip,
np.zeros((n, nc + nt))
])
dchord_draf = dchord_dr * dr_draf
dchord_drmaxchord0 = np.dot(dchord_drchord, drc_drcmax)
dchord_drmaxchord = dchord_drmaxchord0 * (Rtip - Rhub)
dchord_drhub = np.dot(
dchord_drchord, drc_drhub) + dchord_drmaxchord0 * (
1.0 - self.r_max_chord) + dchord_dr * dr_dRhub
dchord_drtip = np.dot(dchord_drchord,
drc_drtip) + dchord_drmaxchord0 * (
self.r_max_chord) + dchord_dr * dr_dRtip
dchord = hstack([
np.diag(dchord_draf), dchord_drmaxchord, dchord_drhub,
dchord_drtip, dchord_dchordsub,
np.zeros((n, nt))
])
dthetaouter_dcyl = np.dot(dthetaouter_drtheta, drt_drcyl)
dthetaouter_draf = dthetaouter_dr * dr_draf[idxc:]
dthetaouter_drhub = dthetaouter_dr * dr_dRhub[idxc:]
dthetaouter_drtip = dthetaouter_dr * dr_dRtip[idxc:] + np.dot(
dthetaouter_drtheta, drt_drtip)
dtheta_draf = np.concatenate([np.zeros(idxc), dthetaouter_draf])
dtheta_drhub = np.concatenate(
[dthetaouter_drhub[0] * np.ones(idxc), dthetaouter_drhub])
dtheta_drtip = np.concatenate(
[dthetaouter_drtip[0] * np.ones(idxc), dthetaouter_drtip])
sub = dthetaouter_dthetasub[0, :]
dtheta_dthetasub = vstack([
np.dot(np.ones((idxc, 1)), sub[np.newaxis, :]),
dthetaouter_dthetasub
])
dtheta_draf = np.diag(dtheta_draf)
dtheta_dcyl = np.concatenate(
[dthetaouter_dcyl[0] * np.ones(idxc), dthetaouter_dcyl])
dtheta_draf[idxc:, idxc] += dthetaouter_dcyl * (Rtip - Rhub)
dtheta_drhub += dtheta_dcyl * (1.0 - self.r_af[idxc])
dtheta_drtip += dtheta_dcyl * self.r_af[idxc]
dtheta = hstack([
dtheta_draf,
np.zeros((n, 1)), dtheta_drhub, dtheta_drtip,
np.zeros((n, nc)), dtheta_dthetasub
])
drafs_dr = np.zeros((n - 1, n))
for i in range(n - 1):
drafs_dr[i, i] = -1.0
drafs_dr[i, i + 1] = 1.0
drafs = hstack([drafs_dr, np.zeros((n - 1, 3 + nc + nt))])
dprecurve = np.zeros((len(self.precurve), n + 3 + nc + nt))
self.J = vstack([dr, dchord, dtheta, drafs, dprecurve])