def mass_hpa_propeller(diameter,
max_power,
include_variable_pitch_mechanism=False):
"""
Returns the estimated mass of a propeller assembly for low-disc-loading applications (human powered airplane, paramotor, etc.)
:param diameter: diameter of the propeller [m]
:param max_power: maximum power of the propeller [W]
:param include_variable_pitch_mechanism: boolean, does this propeller have a variable pitch mechanism?
:return: estimated weight [kg]
"""
smoothmax = lambda value1, value2, hardness: cas.log(
cas.exp(hardness * value1) + cas.exp(hardness * value2)
) / hardness # soft maximum
mass_propeller = (
0.495 * (diameter / 1.25)**1.6 *
smoothmax(0.6, max_power / 14914, hardness=5)**2
) # Baselining to a 125cm E-Props Top 80 Propeller for paramotor, with some sketchy scaling assumptions
# Parameters on diameter exponent and min power were chosen such that Daedalus propeller is roughly on the curve.
mass_variable_pitch_mech = 216.8 / 800 * mass_propeller
# correlation to Daedalus data: http://journals.sfu.ca/ts/index.php/ts/article/viewFile/760/718
if include_variable_pitch_mechanism:
mass_propeller += mass_variable_pitch_mech
return mass_propeller
def __cost_saturation_l(self, x, x_ref, covar_x, u, covar_u, delta_u, Q, R,
S):
""" Stage Cost function: Expected Value of Saturating Cost
"""
Nx = ca.MX.size1(Q)
Nu = ca.MX.size1(R)
# Create symbols
Q_s = ca.SX.sym('Q', Nx, Nx)
R_s = ca.SX.sym('Q', Nu, Nu)
x_s = ca.SX.sym('x', Nx)
u_s = ca.SX.sym('x', Nu)
covar_x_s = ca.SX.sym('covar_z', Nx, Nx)
covar_u_s = ca.SX.sym('covar_u', ca.MX.size(R))
Z_x = ca.SX.eye(Nx) + 2 * covar_x_s @ Q_s
Z_u = ca.SX.eye(Nu) + 2 * covar_u_s @ R_s
cost_x = ca.Function('cost_x', [x_s, Q_s, covar_x_s], [
1 - ca.exp(-(x_s.T @ ca.solve(Z_x.T, Q_s.T).T @ x_s)) /
ca.sqrt(ca.det(Z_x))
])
cost_u = ca.Function('cost_u', [u_s, R_s, covar_u_s], [
1 - ca.exp(-(u_s.T @ ca.solve(Z_u.T, R_s.T).T @ u_s)) /
ca.sqrt(ca.det(Z_u))
])
return cost_x(x - x_ref, Q, covar_x) + cost_u(u, R, covar_u)
def RBF(n):
if n == 1:
sX = 1
sY = n
n_features = self.n_features
X = cs.SX.sym('X', sX, n_features)
Y = cs.SX.sym('Y', sY, n_features)
length_scale = cs.SX.sym('l', 1, n_features)
X_ = X / cs.repmat(length_scale, sX, 1)
Y_ = Y / cs.repmat(length_scale, sY, 1)
dist = cs.SX.zeros((sX, sY))
for i in xrange(0, sX):
for j in xrange(0, sY):
dist[i, j] = cs.sum2((X_[i, :] - Y_[j, :])**2)
K = cs.exp(-.5 * dist)
self.RBF1 = cs.Function('RBF1', [X, Y, length_scale], [K])
else:
sX = 1
sY = n
n_features = self.n_features
X = cs.SX.sym('X', sX, n_features)
Y = self.model.X_train_
length_scale = cs.SX.sym('l', 1, n_features)
X_ = X / cs.repmat(length_scale, sX, 1)
Y_ = Y / cs.repmat(length_scale, sY, 1)
dist = cs.SX.zeros((sX, sY))
for i in xrange(0, sX):
for j in xrange(0, sY):
dist[i, j] = cs.sum2((X_[i, :] - Y_[j, :])**2)
K = cs.exp(-.5 * dist)
self.RBFn = cs.Function('RBFn', [X, length_scale], [K])
def obstacle_cost(U_var, T):
U_tot = casadi.reshape(U_var, 2, T)
X_tot = find_X_tot(U_var, T)
obs = env_setup()
obs = casadi.DM(obs)
cost = 0
for i in range(T):
x = X_tot[0, i]
y = X_tot[1, i]
x0 = obs[0, 0]
y0 = obs[0, 1]
r0 = obs[0, 2]
x1 = obs[1, 0]
y1 = obs[1, 1]
r1 = obs[1, 2]
x2 = obs[2, 0]
y2 = obs[2, 1]
r2 = obs[2, 2]
cost = cost + r0 * 5000 * casadi.exp(-((x - x0)**2 +
(y - y0)**2 - r0**2))
cost = cost + r1 * 5000 * casadi.exp(-((x - x1)**2 +
(y - y1)**2 - r1**2))
cost = cost + r2 * 5000 * casadi.exp(-((x - x2)**2 +
(y - y2)**2 - r2**2))
return cost
def cost_func_SSP(self, robot, U, X0):
cost = 0
U = c.reshape(U, robot.nu, self.T)
X = c.MX(robot.nx, self.T + 1)
X[:, 0] = X0
for i in range(self.T):
cost = (cost + c.mtimes(c.mtimes(U[:, i].T, self.R), U[:, i]) +
c.mtimes(c.mtimes((self.Xg - X[:, i]).T, self.Q),
(self.Xg - X[:, i])))
X_temp = X[0:2, i]
if params.OBSTACLES:
obstacle_cost = params.M*(c.exp(-(c.mtimes(c.mtimes((params.c_obs_1 - X_temp).T, params.E_obs_1),(params.c_obs_1 - X_temp))))+ \
c.exp(-(c.mtimes(c.mtimes((params.c_obs_2 - X_temp).T, params.E_obs_2),(params.c_obs_2 - X_temp)))) + \
c.exp(-(c.mtimes(c.mtimes((params.c_obs_3 - X_temp).T, params.E_obs_3),(params.c_obs_3 - X_temp)))) + \
c.exp(-(c.mtimes(c.mtimes((params.c_obs_4 - X_temp).T, params.E_obs_4),(params.c_obs_4 - X_temp)))))
cost = cost + obstacle_cost
X[:, i + 1] = robot.kinematics(X[:, i], U[:, i])
cost = cost + c.mtimes(c.mtimes((self.Xg - X[:, self.T]).T, self.Qf),
(self.Xg - X[:, self.T]))
return cost
def Cl_rae2822(alpha, Re_c):
# A curve fit I did to a RAE2822 airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9857
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < 10^6.
# See: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\rae2822
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
a1l = 5.5686866813855172e-02
a1t = 9.7472055628494134e-02
a4l = -7.2145733312046152e-09
a4t = -3.6886704372829236e-06
atr = 8.3723547264375520e-01
atr2 = -8.3128119739031697e-02
c0l = -4.9103908291438701e-02
c0t = 2.3903424824298553e-01
ctr = 1.3082854754897108e+01
rtr = 2.6963082864300731e+00
a = alpha
r = log10_Re
Cl = (c0t + a1t * a + a4t * a**4) * 1 / (
1 + cas.exp(ctr - rtr * r - atr * a - atr2 * a**2)) + (
c0l + a1l * a + a4l * a**4) * (
1 - 1 / (1 + cas.exp(ctr - rtr * r - atr * a - atr2 * a**2)))
return Cl
def Cl_e216(alpha, Re_c):
# A curve fit I did to a Eppler 216 (e216) airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9994
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < Inf.
# See: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\e216
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
a1l = 3.0904412662858878e-02
a1t = 9.6452654383488254e-02
a4t = -2.5633334023068302e-05
asl = 6.4175433185427011e-01
atr = 3.6775107602844948e-01
c0l = -2.5909363461176749e-01
c0t = 8.3824440586718862e-01
ctr = 1.1431810545735890e+02
ksl = 5.3416670116733611e-01
rtr = 3.9713338634462829e+01
rtr2 = -3.3634858542657771e+00
xsl = -1.2220899840236835e-01
a = alpha
r = log10_Re
Cl = (c0t + a1t * a + a4t * a**4) * 1 / (
1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)) + (
c0l + a1l * a + asl / (1 + cas.exp(-ksl * (a - xsl)))) * (
1 - 1 / (1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)))
return Cl
def Cd_profile_e216(alpha, Re_c):
# A curve fit I did to a Eppler 216 (e216) airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9995
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < 10^6.
# see: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\e216
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
a1l = 4.7167470806940448e-02
a1t = 7.5663005080888857e-02
a2l = 8.7552076545610764e-04
a4t = 1.1220763679805319e-05
atr = 4.2456038382581129e-01
c0l = -1.4099657419753771e+00
c0t = -2.3855286371940609e+00
ctr = 9.1474872611212135e+01
rtr = 3.0218483612170434e+01
rtr2 = -2.4515094313899279e+00
a = alpha
r = log10_Re
log10_Cd = (c0t + a1t * a + a4t * a**4) * 1 / (
1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)) + (
c0l + a1l * a + a2l * a**2) * (
1 - 1 / (1 + cas.exp(ctr - rtr * r - atr * a - rtr2 * r**2)))
Cd = 10**log10_Cd
return Cd
def Cd_cylinder(Re_D, subcritical_only=False):
"""
Returns the drag coefficient of a cylinder in crossflow as a function of its Reynolds number.
:param Re_D: Reynolds number, referenced to diameter
:param subcritical_only: Determines whether the model models purely subcritical (Re < 300k) cylinder flows. Useful, since
this model is now convex and can be more well-behaved.
:return: Drag coefficient
"""
csigc = 5.5766722118597247
csigh = 23.7460859935990563
csub0 = -0.6989492360435040
csub1 = 1.0465189382830078
csub2 = 0.7044228755898569
csub3 = 0.0846501115443938
csup0 = -0.0823564417206403
csupc = 6.8020230357616764
csuph = 9.9999999999999787
csupscl = -0.4570690347113859
x = cas.log10(Re_D)
if subcritical_only:
Cd = 10**(csub0 * x + csub1) + csub2 + csub3 * x
return Cd
else:
log10_Cd = (
(cas.log10(10**(csub0 * x + csub1) + csub2 + csub3 * x)) *
(1 - 1 / (1 + cas.exp(-csigh * (x - csigc)))) +
(csup0 +
csupscl / csuph * cas.log(cas.exp(csuph * (csupc - x)) + 1)) *
(1 / (1 + cas.exp(-csigh * (x - csigc)))))
Cd = 10**log10_Cd
return Cd
def Cd_profile_rae2822(alpha, Re_c):
# A curve fit I did to a RAE2822 airfoil, 2D XFoil data. Incompressible flow.
# Within -2 < alpha < 12 and 10^4 < Re_c < 10^6, has R^2 = 0.9995
# Likely valid from -6 < alpha < 12 and 10^4 < Re_c < Inf.
# see: C:\Projects\GitHub\firefly_aerodynamics\Gists and Ideas\XFoil Drag Fitting\e216
Re_c = cas.fmax(Re_c, 1)
log10_Re = cas.log10(Re_c)
# Coeffs
at = 8.1034027621509015e+00
c0l = -8.4296746456429639e-01
c0t = -1.3700609138855402e+00
kart = -4.1609994062600880e-01
kat = 5.9510959342452441e-01
krt = -7.1938030052506197e-01
r1l = 1.1548628822014631e-01
r1t = -4.9133662875044504e-01
rt = 5.0070459892411696e+00
a = alpha
r = log10_Re
log10_Cd = (c0t + r1t * (r - 4)) * (
1 / (1 + cas.exp(kat * (a - at) + krt * (r - rt) + kart * (a - at) * (r - rt)))) + (
c0l + r1l * (r - 4)) * (
1 - 1 / (1 + cas.exp(kat * (a - at) + krt * (r - rt) + kart * (a - at) * (r - rt))))
Cd = 10 ** log10_Cd
return Cd
def plotLogProbVsDistance():
spchar = " "
rspace = np.linspace(-0.1, 0.75, num=1000)
for theta_val in (theta_low, theta_high):
uspace_type = [
-log(1 + exp(-beta[0] - beta[1] * theta_val - beta[2] * rval))
for rval in rspace
]
ax3.plot(rspace,
uspace_type,
color=color_type_map[theta_val],
label=f"Utility{spchar}for Type {theta_val}",
zorder=-1)
low_index = theta.index(theta_low)
high_index = theta.index(theta_high)
# yvals_inverted = [0 if v==1 else 1 for v in yvals]
# gp_inverted = [i for i,v in enumerate(yvals_inverted) if v==1]
# xlist_inverted = [ gp_inverted[np.argmin([dist[i][j] for j in gp_inverted])] for i in range(nIndiv)]
# r_inverted = [dist[i][xlist_inverted[i]] for i in range(nIndiv)]
r_inverted = list(reversed(rvals))
r_inverted_and_not = {1: rvals, 0: r_inverted}
utility_inverted_and_not = {
inversion: sum([
-log(1 + exp(-beta[0] - beta[1] * theta[ind] -
beta[2] * rvals_possibly_inverted[ind]))
for ind in range(nIndiv)
])
for inversion, rvals_possibly_inverted in
r_inverted_and_not.items()
}
for inversion, rvals_possibly_inverted in r_inverted_and_not.items():
for ind in reversed(range(nIndiv)):
ax3.scatter(
rvals_possibly_inverted[ind], [
-log(1 + exp(-beta[0] - beta[1] * theta[ind] -
beta[2] * rvals_possibly_inverted[ind]))
],
marker="+",
c=[facility_color_map[inversion]],
s=[facility_size_options[inversion]],
label=
f"Utility from{spchar}{facility_states[inversion].lower()} facility (Sum: {utility_inverted_and_not[inversion]:.02})",
zorder=1)
# for ind in range(len(theta)):
# ax3.scatter(r_inverted[ind],[1/(1+exp(-beta[0] - beta[1]*theta[ind] - beta[2]*r_inverted[ind]))],marker = "+",c=[facility_color_map[0]],s=[facility_size_options[0]],label=f"Utility from {facility_states[0]} facility")
legend_dict = {
artist.properties().get('label'): artist
for artist in ax3.collections.copy() + ax3.lines.copy()
if "no_legend" not in artist.properties().get('label')
}
# ax3.legend(legend_dict.values(),legend_dict.keys(),loc='upper center', bbox_to_anchor=(0.5, -0.2), ncol=len(legend_dict),fontsize='small')
ax3.legend(legend_dict.values(),
legend_dict.keys(),
loc='best',
fontsize='small')
ax3.set_xlabel("distance from nearest selected facility")
ax3.set_ylabel("$log[P(Success)]$")
ax3.set_title("$log[P(success)]$ vs Distance Differs by Type")
def obstacle_cost_func(self, X_temp):
cost = params.M*(c.exp(-(c.mtimes(c.mtimes((params.c_obs_1 - X_temp).T, params.E_obs_1),(params.c_obs_1 - X_temp))))+ \
c.exp(-(c.mtimes(c.mtimes((params.c_obs_2 - X_temp).T, params.E_obs_2),(params.c_obs_2 - X_temp)))) + \
c.exp(-(c.mtimes(c.mtimes((params.c_obs_3 - X_temp).T, params.E_obs_3),(params.c_obs_3 - X_temp)))) + \
c.exp(-(c.mtimes(c.mtimes((params.c_obs_4 - X_temp).T, params.E_obs_4),(params.c_obs_4 - X_temp)))))
return cost
def trig_prop(m, v, idx, a=1.0):
""" Exact moment-matching for trig function with Gaussian input
Compute E(a*sin(x)), E(a*cos(x)), V(a*sin(x)), V(a*cos(x)) and cross-covariances
for Gaussian inputs x \sim N(m_idx,v_idx) as well as input-output covariances
using exact moment-matching
Parameters
----------
m : dx1 ndarray[float | casadi.Sym]
The mean of the input Gaussian
v : dxd ndarray[float | casadi.Sym]
idx: int
The index to be trigonometrically augmented
a: float [optional]
A scalar coefficient
Returns
-------
m_out: 2x1 ndarray[float | casadi.Sym]
The mean of the trigonometric output
v_out: 2x2 ndarray[float | casadi.Sym]
The variance of the trigonometric output
c_out: inv(v) times input-output-covariance
"""
m_out = MX(2, 1)
v_out = MX(2, 2)
v_exp_0 = exp(-v[idx, idx] / 2)
m_sin = v_exp_0 * sin(m[idx, 0])
m_cos = v_exp_0 * cos(m[idx, 0])
m_out[0] = a * m_sin
m_out[1] = a * m_cos
v_exp_1 = exp(-2 * v[idx, idx])
e_s_sq = (1 - v_exp_1 * cos(2 * m[idx, 0])) / 2
e_c_sq = (1 + v_exp_1 * cos(2 * m[idx, 0])) / 2
e_s_times_c = v_exp_1 * sin(2 * m[idx, 0]) / 2
v_out[0, 0] = e_s_sq - m_sin**2
v_out[1, 1] = e_c_sq - m_cos**2
v_out[1, 0] = e_s_times_c - m_sin * m_cos
v_out[0, 1] = v_out[1, 0]
v_out = a**2 * v_out
d = np.shape(m)[0]
c_out = MX(d, 2)
c_out[idx, 0] = m_out[1]
c_out[idx, 1] = -m_out[0]
return m_out, v_out, c_out
def generate_bird(n=2, func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("bird is only defined for n=2")
x = data_type.sym("x", n)
f = cs.sin(x[0]) * cs.exp((1 - cs.cos(x[1]))**2)
f += cs.cos(x[1]) * cs.exp((1 - cs.sin(x[0]))**2)
f += (x[0] - x[1])**2
func = cs.Function("bird", [x], [f], ["x"], ["f"], func_opts)
glob_min = [[4.70104, 3.15294], [-1.58214, -3.13024]]
return func, [[-2 * cs.np.pi, 2 * cs.np.pi]] * n, glob_min
def generate_ackley(n=2,
a=20,
b=0.2,
c=2 * cs.np.pi,
func_opts={},
data_type=cs.SX):
x = data_type.sym("x", n)
sum1 = cs.sumsqrt(x)
sum2 = cs.sum1(cs.cos(x))
f = -a * cs.exp(-b * cs.sqrt((1. / n) * sum1)) - cs.exp(
(1. / n) * sum2) + a + cs.exp(1)
func = cs.Function("ackley", [x], [f], ["x"], ["f"], func_opts)
return func, [[-32., 32.]] * n, [[0.] * n]
def generate_ackleyn3(n=2,
a=200,
b=0.02,
c=5.,
d=3.,
func_opts={},
data_type=cs.SX):
if n != 2:
raise ValueError("ackleyn3 is only defined for n=2")
x = data_type.sym("x", n)
f = -a * cs.exp(-b * cs.sqrt(x[0]**2 + x[1]**2))
f += c * cs.exp(cs.cos(d * x[0]) + cs.sin(d * x[1]))
func = cs.Function("ackleyn3", [x], [f], ["x"], ["f"], func_opts)
glob_min = [[0.682584587365898, -0.36075325513719],
[-0.682584587365898, -0.36075325513719]]
return func, [[-32., 32.]] * n, glob_min
def generate_ackleyn2(n=2, a=200, b=0.02, func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("ackleyn2 is only defined for n=2")
x = data_type.sym("x", n)
f = -a * cs.exp(-b * cs.sqrt(x[0]**2 + x[1]**2))
func = cs.Function("ackleyn2", [x], [f], ["x"], ["f"], func_opts)
return func, [[-32., 32.]] * n, [[0.] * n]
def solve_enumerative(obj_fn):
bbax = bax()
yvals = np.zeros((nFac), dtype=int)
rvals = np.zeros((nIndiv))
uvals = np.full((nIndiv), -np.inf)
for ind, gp in enumerate(bbax.bax_gen(nFac, nSelectedFac)):
gp = gp.toList()
xlist = [
gp[np.argmin([dist[i][j] for j in gp])] for i in range(nIndiv)
]
r = [dist[i][xlist[i]] for i in range(nIndiv)]
u = obj_fn(r)
print(f"Obj is {sum(u)}")
if sum(u) > sum(uvals):
uvals = u
rvals = r
yvals = np.zeros((nFac), dtype=int)
yvals[gp] = 1
print(f"Best obj is {sum(uvals)}")
fstar = sum(uvals)
prob_success = [
1 / (1 + exp(-beta[0] - beta[1] * theta[i] - beta[2] * rvals[i]))
for i in range(nIndiv)
]
print(f"Solved enumeratively")
return yvals, rvals, uvals, prob_success, fstar, obj_fn.extra_info
def Model(param, x, args):
T = x[:, 0]
A, B = param[0], param[1]
return exp(
A / 8.31446 + B / (8.31446 * T) -
(68.2 / 8.31446) * log(T / 298.15)) # Pvp calculation - vectorized
def _k_mat52(x,
y=None,
variance=1.,
lengthscale=None,
diag_only=False,
ARD=False):
""" Evaluate the Matern52 kernel function symbolically using Casadi"""
n_x, dim_x = x.shape
if diag_only:
ret = SX(n_x, )
ret[:] = variance
return ret
if y is None:
y = x
n_y, _ = np.shape(y)
if lengthscale is None:
if ARD:
lengthscale = np.ones((dim_x, ))
else:
lengthscale = 1.
if ARD is False:
lengthscale = lengthscale * np.ones((dim_x, ))
lens_x = repmat(lengthscale.reshape(1, -1), n_x)
lens_y = repmat(lengthscale.reshape(1, -1), n_y)
r = _unscaled_dist(x / lens_x, y / lens_y)
# GPY: self.variance*(1+np.sqrt(5.)*r+5./3*r**2)*np.exp(-np.sqrt(5.)*r)
return variance * (1. + sqrt(5.) * r + 5. / 3 * r**2) * exp(-sqrt(5.) * r)
def Cd_profile_2412(alpha, Re_c):
# A curve fit I did to a NACA 2412 airfoil in incompressible flow.
# Within -2 < alpha < 12 and 10^5 < Re_c < 10^7, has R^2 = 0.9713
print(
"Warning: Cd_profile_e216() recommended over Cd_profile_2412(); those are MUCH more accurate fits."
)
Re_c = cas.fmax(Re_c, 1)
log_Re = cas.log(Re_c)
CD0 = -5.249
Re0 = 15.61
Re1 = 15.31
alpha0 = 1.049
alpha1 = -4.715
cx = 0.009528
cxy = -0.00588
cy = 0.04838
log_CD = CD0 + cx * (alpha - alpha0)**2 + cy * (log_Re - Re0)**2 + cxy * (
alpha - alpha1) * (log_Re - Re1
) # basically, a rotated paraboloid in logspace
CD = cas.exp(log_CD)
return CD
def cal(self, x, style="casadi"):
# return self.f(x)
if (style == "casadi"):
exp_x = csd.exp(x)
elif (style == "numpy"):
exp_x = np.exp(x)
return exp_x / (1 + exp_x)
def generate_brent(n=2, a=10., b=10., func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("brent is only defined for n=2")
x = data_type.sym("x", n)
f = (x[0] + a)**2 + (x[1] + b)**2 + cs.exp(-cs.sumsqr(x))
func = cs.Function("brent", [x], [f], ["x"], ["f"], func_opts)
return func, [[-20., 0.]] * n, [[-10.] * n]
def generate_easom(n=2, func_opts={}, data_type=cs.SX):
if n != 2:
raise ValueError("easom is only defined for n=2")
x = data_type.sym("x", n)
exponential = cs.exp(-(x[0] - cs.pi)**2 - (x[1] - cs.pi)**2)
f = -cs.cos(x[0]) * cs.cos(x[1]) * exponential
func = cs.Function("easom", [x], [f], ["x"], ["f"], func_opts)
return func, [[-100., 100.]] * n, [[cs.pi, cs.pi]]
def solve_casadi():
opti = Opti()
x = [[opti.variable() for j in range(nFac)]
for i in range(nIndiv)] # nIndiv X nFac
y = [opti.variable() for j in range(nFac)]
r = [opti.variable() for i in range(nIndiv)]
u = [opti.variable() for i in range(nIndiv)]
discrete = []
discrete += [False for j in range(nFac) for i in range(nIndiv)
] #x variables - will be binary without integer constraint
discrete += [True for j in range(nFac)] #y variables
discrete += [False for i in range(nIndiv)] #r variables
discrete += [False for i in range(nIndiv)] #u variables
opti.minimize(-sum(u)) #maximize sum u
opti.subject_to([
u[i] == -log(1 + exp(-beta[0] - beta[1] * theta[i] - beta[2] * r[i]))
for i in range(nIndiv)
]) #log prob(success)
# opti.subject_to([u[i] == 1/(1+exp(-beta[0] - beta[1]*theta[i] - beta[2]*r[i])) for i in range(nIndiv)]) #prob(success)
opti.subject_to([
r[i] >= sum([dist[i, j] * x[i][j] for j in range(nFac)])
for i in range(nIndiv)
])
opti.subject_to([sum(x[i]) == 1 for i in range(nIndiv)])
opti.subject_to(
[x[i][j] <= y[j] for i in range(nIndiv) for j in range(nFac)])
opti.subject_to(sum(y) <= nSelectedFac)
opti.subject_to([opti.bounded(0, y[j], 1) for j in range(nFac)])
opti.subject_to([
opti.bounded(0, x[i][j], 1) for i in range(nIndiv) for j in range(nFac)
])
p_options = {"discrete": discrete, "expand": True}
s_options = {"max_iter": 100, 'tol': 100}
opti.solver('bonmin', p_options, s_options)
sol = opti.solve()
# possibilities for tolerance in s_options (https://web.casadi.org/python-api/):
#only 'tol' works...
# for key in ['tol','boundTolerance','epsIterRef','terminationTolerance','abstol','opttol']:
# try:
# s_options = {"max_iter": 100,key: .1}
# opti.solver('bonmin',p_options,s_options)
# sol = opti.solve()
# print(f"NOTE: '{key}' works in s_options!!")
# except Exception as e:
# print(e)
# print(f"NOTE: '{key}' is not a valid s_option!")
xvals = np.round(
np.array([[sol.value(x[i][j]) for j in range(nFac)]
for i in range(nIndiv)])).astype(int)
yvals = np.array([sol.value(y[j]) for j in range(nFac)]).astype(int)
rvals = np.array([sol.value(r[i]) for i in range(nIndiv)])
uvals = np.array([sol.value(u[i]) for i in range(nIndiv)])
# prob_success = np.array([1/(1+exp(-beta[0] - beta[1]*theta[i] - beta[2]*rvals[i])) for i in range(nIndiv)])
prob_success = np.exp(uvals)
fstar = sol.value(opti.f)
print(f"Solved using CasADI")
return xvals, yvals, rvals, uvals, prob_success, fstar
def mvg_mgf(t, mu, sigma):
"""Multivariate Gaussian Moment Generating Function.
Args:
t (1D array):
mu (1D array):
sigma (2D array):
"""
return casadi.exp(mu.T @ t + 0.5 * t.T @ sigma @ t)
def incidence_angle_function(latitude, day_of_year, time, scattering=False):
"""
What is the fraction of insolation that a horizontal surface will receive as a function of sun position in the sky?
:param latitude: Latitude [degrees]
:param day_of_year: Julian day (1 == Jan. 1, 365 == Dec. 31)
:param time: Time since (local) solar noon [seconds]
:param scattering: Boolean: include scattering effects at very low angles?
"""
# Old description:
# To first-order, this is true. In class, Kevin Uleck claimed that you have higher-than-cosine losses at extreme angles,
# since you get reflection losses. However, an experiment by Sharma appears to not reproduce this finding, showing only a
# 0.4-percentage-point drop in cell efficiency from 0 to 60 degrees. So, for now, we'll just say it's a cosine loss.
# Sharma: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6611928/
elevation_angle = solar_elevation_angle(latitude, day_of_year, time)
theta = 90 - elevation_angle # Angle between panel normal and the sun, in degrees
cosine_factor = cosd(theta)
if not scattering:
return cosine_factor
else:
# Calculate scattering knockdown (See C:\Users\User\Google Drive\School\Grad School\2020 Spring\16-885\Solar Panel Scattering Rough Fit)
theta_rad = theta * cas.pi / 180
# Model 1
c = (0.27891510500505767300438719757949,
-0.015994330894744987481281839336589,
-19.707332432605799255043166340329,
-0.66260979582573353852126274432521)
scattering_factor = c[0] + c[3] * theta_rad + cas.exp(
c[1] * (cas.tan(theta_rad - 1e-8) + c[2] * theta_rad))
# Model 2
# c = (
# -0.04636,
# -0.3171
# )
# scattering_factor = cas.exp(
# c[0] * (
# cas.tan(theta_rad-1e-8) + c[1] * theta_rad
# )
# )
# Model 3
# p1 = -21.74
# p2 = 282.6
# p3 = -1538
# p4 = 1786
# q1 = -923.2
# q2 = 1456
# x = theta_rad
# scattering_factor = ((p1*x**3 + p2*x**2 + p3*x + p4) /
# (x**2 + q1*x + q2))
# scattering_factor = cas.fmin(cas.fmax(scattering_factor, 0), 1)
return cosine_factor * scattering_factor
def get_h_i_t():
w_i, alpha_i, beta_i, b_i, c_i, mu_i, v_i, h_tm1_agg_i, delta_t_agg_i = casadi.SX.sym('w_i'), casadi.SX.sym('alpha_i'), casadi.SX.sym('beta_i'), casadi.SX.sym('b_i'), casadi.SX.sym('c_i'), casadi.SX.sym('mu_i'), casadi.SX.sym('v_i'), casadi.SX.sym('h_tm1_agg_i'), casadi.SX.sym('delta_t_agg_i'),
boolM = mu_i > 0
sqrthtpowmu = casadi.sqrt(h_tm1_agg_i) ** mu_i
zTrue = (w_i + alpha_i * sqrthtpowmu * f_i(delta_t_agg_i, b_i, c_i) ** v_i + beta_i * sqrthtpowmu) ** (2./mu_i)
sqrtht = casadi.sqrt(h_tm1_agg_i)
zFalse= (casadi.exp(w_i + alpha_i * f_i(delta_t_agg_i, b_i, c_i) ** v_i + beta_i * casadi.log(sqrtht)) ** (2.))
z = casadi.if_else(boolM, zTrue, zFalse)
return casadi.Function('h_i_t',
[w_i, alpha_i, beta_i, b_i, c_i, mu_i, v_i, h_tm1_agg_i, delta_t_agg_i],
[z])
def prod(x, axis: int = None):
"""
Return the product of array elements over a given axis.
See syntax here: https://numpy.org/doc/stable/reference/generated/numpy.prod.html
"""
if not is_casadi_type(x):
return _onp.prod(x, axis=axis)
else:
return _cas.exp(sum(_cas.log(x), axis=axis))
def fermination_ode(t, x, u, p, z):
"""
Fermination model
"""
k_dc = p[0]
k_dm = p[1]
s_i = p[2]
T = u[0]
x_lag = x[0]
x_active = x[1]
x_bottom = x[2]
s = x[3]
e = x[4]
acet = x[5]
diac = x[6]
mu_x0 = ca.exp(108.31 - 31934.09 / (T+273.15))
mu_eas = ca.exp(89.92 - 26589 / (T+273.15))
mu_s0 = ca.exp( -41.92 + 11654.64/ (T+273.15))
mu_lag = ca.exp(30.72 - 9501.54 / (T+273.15))
k_dc = 0.000127672
k_m = ca.exp( 130.16 - 38313 / (T+273.15))
mu_D0 = ca.exp( 33.82 - 10033.28/ (T+273.15))
mu_a0 = ca.exp(3.72 - 1267.24 / (T+273.15))
k_s = ca.exp(-119.63 + 34203.95 / (T + 273.15))
k_dm = 0.00113864
mu_x = mu_x0 * s / (0.5 * s_i + e)
mu_D = 0.5 * s_i * mu_D0 / (0.5 * s_i + e)
mu_s = mu_s0 * s / (k_s + s)
mu_a = mu_a0 * s / (k_s + s)
f = 1 - e / (0.5 * s_i)
dx_lag = - mu_lag * x_lag
dx_active = mu_x * x_active - k_m * x_active + mu_lag * x_lag
dx_bottom = k_m * x_active - mu_D * x_bottom
ds = - mu_s * x_active
de = mu_a * f * x_active
dacet = - mu_eas * mu_s * x_active
ddiac = k_dc * s * x_active - k_dm * diac * e
rhs = [dx_lag,
dx_active,
dx_bottom,
ds,
de,
dacet,
ddiac
]
return ca.vertcat(*rhs)
def CasadiRBF(X, Y, model):
""" RBF kernel in CasADi
"""
sX = X.shape[0]
sY = Y.shape[0]
length_scale = model.kernel_.get_params()['k1__k2__length_scale'].reshape(1,-1)
constant = model.kernel_.get_params()['k1__k1__constant_value']
X = X / cs.repmat(length_scale, sX , 1)
Y = Y / cs.repmat(length_scale, sY , 1)
dist = cs.repmat(cs.sum1(X.T**2).T,1,sY) + cs.repmat(cs.sum1(Y.T**2),sX,1) - 2*cs.mtimes(X,Y.T)
K = constant*cs.exp(-.5 * dist)
return K
def get_f():
r_t, h_i_tm1, lambda_i, gamma_i = casadi.SX.sym('r_t'), casadi.SX.sym('h_i_tm1'), casadi.SX.sym('lambda_i'), casadi.SX.sym('gamma_i')
z = 1/casadi.sqrt(2*numpy.pi * h_i_tm1) * casadi.exp(-((r_t - (lambda_i + gamma_i * casadi.sqrt(h_i_tm1))) ** 2.) / (2 * h_i_tm1))
return casadi.Function('f', [r_t, h_i_tm1, lambda_i, gamma_i], [z])