def fdidv(isat1, isat2, rs, rsh, ic, vc, vt):
"""
Derivative of IV curve and its derivatives w.r.t. Isat1, Isat2, Rs, Rsh, Ic,
Vc and Vt.
:param isat1: diode 1 saturation current [A]
:param isat2: diode 2 saturation current [A]
:param rs: series resistance [ohms]
:param rsh: shunt resistance [ohms]
:param ic: cell current [A]
:param vc: cell voltage [V]
:param vt: thermal voltage (kB * Tc / qe = 26[mV] at Tc=298K) [V]
:return: derivative of IV curve and its derivatives
"""
vd, _ = diode.fvd(vc, ic, rs) # vd = vc + ic * rs
vstar = vd / vt
rstar = rsh / rs
exp_vstar, exp_vstar_2 = np.exp(vstar), np.exp(0.5 * vstar)
v_sat1_sh, v_sat2_sh = isat1 * rsh, isat2 * rsh
v_sat1_sh_exp_vstar = v_sat1_sh * exp_vstar
v_sat2_sh_exp_vstar_2 = 0.5 * v_sat2_sh * exp_vstar_2
vsum = v_sat1_sh_exp_vstar + v_sat2_sh_exp_vstar_2 + vt
vsum_rstar = vsum + vt * rstar
combiterm1 = v_sat1_sh_exp_vstar + 0.5*v_sat2_sh_exp_vstar_2
combiterm2 = isat1*exp_vstar + 0.5*isat2*exp_vstar_2
combiterm3 = vsum / vsum_rstar - 1.0
combiterm4 = vsum_rstar * rs
combiterm5 = rstar * combiterm3 / vsum_rstar
combiterm6 = combiterm1 * combiterm3 / vt
combiterm7 = 1.0 / combiterm4
# dI/dV = derivative of IV curve
didv = -vsum / combiterm4
# jacobian
didv_isat1 = exp_vstar * combiterm5
didv_isat2 = 0.5 * exp_vstar_2 * combiterm5
didv__r_s = combiterm7 * (combiterm6 * ic + vsum**2.0 / combiterm4)
didv_rsh = combiterm7 * (combiterm2 * combiterm3 + vt * vsum / combiterm4)
didv_ic = combiterm6 / vsum_rstar
didv_vc = (didv + 1.0 / rs) * didv_ic
jac = np.array([
didv_isat1, didv_isat2, didv__r_s, didv_rsh, didv_ic, didv_vc
])
return didv, jac
def test_fvd():
"""
Test diode voltage.
"""
# make sympy symbols
vd, vc, ic, rs = sympy.symbols(['vd', 'vc', 'ic', 'rs'])
# diode voltage
vd = vc + ic * rs
d_vc = sympy.diff(vd, vc)
d_ic = sympy.diff(vd, ic)
d_rs = sympy.diff(vd, rs)
# evaluate
test_data = {'vc': VC, 'ic': IC, 'rs': RS_1}
fvd_test, jvd_test = diode.fvd(**test_data)
fvd_expected = np.float(vd.evalf(subs=test_data))
jvd_expected = np.array([
d_vc.evalf(subs=test_data),
d_ic.evalf(subs=test_data),
d_rs.evalf(subs=test_data)
],
dtype=np.float)
LOGGER.debug('test: %g = expected: %g', fvd_test, fvd_expected)
assert np.isclose(fvd_test, fvd_expected)
assert np.allclose(jvd_test, jvd_expected.reshape(-1, 1))
def residual_two_diode(x, isc, voc, imp, vmp, tc):
"""
Objective function to solve 2-diode model.
:param x: parameters isat1, isat2, rs and rsh
:param isc: short circuit current [A] at tc [C]
:param voc: open circuit voltage [V] at tc [C]
:param imp: max power current [A] at tc [C]
:param vmp: max power voltage [V] at tc [C]
:param tc: cell temperature [C]
:return: norm of the residuals its sensitivity
"""
# Constants
q = diode.QE # [C/electron] elementary electric charge
# (n.b. 1 Coulomb = 1 A * s)
kb = diode.KB # [J/K/molecule] Boltzmann's constant
tck = tc + 273.15 # [K] reference temperature
# Governing Equation
vt = kb * tck / q # [V] thermal voltage
# Rescale Variables
isat1_t0 = np.exp(x[0])
isat2 = np.exp(x[1])
rs = x[2]**2.0
rsh = x[3]**2.0
# first diode saturation current
isat1 = diode.isat_t(tc, isat1_t0)
# Short Circuit
vd_isc, _ = diode.fvd(vc=0.0, ic=isc, rs=rs)
id1_isc, _ = diode.fid(isat=isat1, vd=vd_isc, m=1.0, vt=vt)
id2_isc, _ = diode.fid(isat=isat2, vd=vd_isc, m=2.0, vt=vt)
ish_isc, _ = diode.fish(vd=vd_isc, rsh=rsh)
# Photo-generated Current
iph = isc + id1_isc + id2_isc + ish_isc # [A]
# Open Circuit
vd_voc, jvd_voc = diode.fvd(vc=voc, ic=0.0, rs=rs)
id1_voc, jid1_voc = diode.fid(isat=isat1, vd=vd_voc, m=1.0, vt=vt)
id2_voc, jid2_voc = diode.fid(isat=isat2, vd=vd_voc, m=2.0, vt=vt)
ish_voc, jish_voc = diode.fish(vd=vd_voc, rsh=rsh)
# Max Power Point
vd_mpp, jvd_mpp = diode.fvd(vc=vmp, ic=imp, rs=rs)
id1_mpp, jid1_mpp = diode.fid(isat=isat1, vd=vd_mpp, m=1.0, vt=vt)
id2_mpp, jid2_mpp = diode.fid(isat=isat2, vd=vd_mpp, m=2.0, vt=vt)
ish_mpp, jish_mpp = diode.fish(vd=vd_mpp, rsh=rsh)
# Slope at Max Power Point
dpdv, jdpdv = two_diode.fdpdv(isat1=isat1,
isat2=isat2,
rs=rs,
rsh=rsh,
ic=imp,
vc=vmp,
vt=vt)
# Shunt Resistance
frsh, jrsh = two_diode.fjrsh(isat1=isat1,
isat2=isat2,
rs=rs,
rsh=rsh,
vt=vt,
isc=isc)
# Residual
# should be (M, ) array with M residual equations (constraints)
f2 = np.stack(
[
(iph - id1_voc - id2_voc - ish_voc).T, # Open Circuit
(iph - id1_mpp - id2_mpp - ish_mpp - imp).T, # Max Power Point
dpdv.T, # Slope at Max Power Point
frsh.T # Shunt Resistance
],
axis=0).flatten()
# Jacobian
# should be (M, N) array with M residuals and N variables
# [[df1/dx1, df1/dx2, ...], [df2/dx1, df2/dx2, ...]]
jvoc = np.stack(
(
-jid1_voc[0], # d/disat1
-jid2_voc[0], # d/disat2
-jvd_voc[2] * (jid1_voc[1] + jid2_voc[1] + jish_voc[0]), # d/drs
-jish_voc[1] # d/drsh
),
axis=0).T.reshape(-1, 4)
jmpp = np.stack(
(
-jid1_mpp[0], # d/disat1
-jid2_mpp[0], # d/disat2
-jvd_mpp[2] * (jid1_mpp[1] + jid2_mpp[1] + jish_mpp[0]), # d/drs
-jish_mpp[1] # d.drsh
),
axis=0).T.reshape(-1, 4)
# Scaling Factors
scale_fx = np.array([np.exp(x[0]), np.exp(x[1]), 2 * x[2], 2 * x[3]])
# scales each column by the corresponding element
j2 = np.concatenate(
(jvoc, jmpp, jdpdv[:4].T.reshape(-1, 4), jrsh[:4].T.reshape(-1, 4)),
axis=0) * scale_fx
return f2, j2
def fdpdv(isat1, isat2, rs, rsh, ic, vc, vt):
"""
Derivative of PV curve and its derivatives w.r.t. Isat1, Isat2, Rs, Rsh, Ic,
Vc and Vt.
:param isat1: diode 1 saturation current [A]
:param isat2: diode 2 saturation current [A]
:param rs: series resistance [ohms]
:param rsh: shunt resistance [ohms]
:param ic: cell current [A]
:param vc: cell voltage [V]
:param vt: thermal voltage (kB * Tc / qe = 26[mV] at Tc=298K) [V]
:return: derivative of PV curve and its derivatives
"""
didv, _ = fdidv(isat1, isat2, rs, rsh, ic, vc, vt)
vd, _ = diode.fvd(vc, ic, rs) # vd = vc + ic * rs
dpdv = didv * vc + ic
dpdv_isat1 = 2.0*rs*rsh*vc*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)*np.exp(vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2 - 2.0*rsh*vc*np.exp(vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)
dpdv_isat2 = rs*rsh*vc*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)*np.exp(0.5*vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2 - rsh*vc*np.exp(0.5*vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)
dpdv_rs = -vc*(
2.0*isat1*rsh*ic*np.exp(vd/vt)/vt
+ 0.5*isat2*rsh*ic*np.exp(0.5*vd/vt)/vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - vc*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)*(
-2.0*isat1*rs*rsh*ic*np.exp(vd/vt)/vt
- 2.0*isat1*rsh*np.exp(vd/vt)
- 0.5*isat2*rs*rsh*ic*np.exp(0.5*vd/vt)/vt
- isat2*rsh*np.exp(0.5*vd/vt) - 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2
dpdv_rsh = -vc*(
2.0*isat1*np.exp(vd/vt) + isat2*np.exp(0.5*vd/vt)
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - vc*(
-2.0*isat1*rs*np.exp(vd/vt) - isat2*rs*np.exp(0.5*vd/vt) - 2.0*vt
)*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2
dpdv_ic = -vc*(
2.0*isat1*rs*rsh*np.exp(vd/vt)/vt
+ 0.5*isat2*rs*rsh*np.exp(0.5*vd/vt)/vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - vc*(
-2.0*isat1*rs**2*rsh*np.exp(vd/vt)/vt
- 0.5*isat2*rs**2*rsh*np.exp(0.5*vd/vt)/vt
)*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2 + 1.0
dpdv_vc = -vc*(
2.0*isat1*rsh*(rs*didv + 1)*np.exp(vd/vt)/vt
+ 0.5*isat2*rsh*(rs*didv + 1)*np.exp(0.5*vd/vt)/vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - vc*(
-2.0*isat1*rs*rsh*(rs*didv + 1)*np.exp(vd/vt)/vt
- 0.5*isat2*rs*rsh*(rs*didv + 1)*np.exp(0.5*vd/vt)/vt
)*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2 - (
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt)
+ isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) + didv
jac = np.array([
dpdv_isat1, dpdv_isat2, dpdv_rs, dpdv_rsh, dpdv_ic, dpdv_vc
])
return dpdv, jac
def fjrsh(isat1, isat2, rs, rsh, vt, isc):
"""
Shunt resistance residual and its derivatives w.r.t. Isat1, Isat2, Rs and
Rsh.
:param isat1: diode 1 saturation current [A]
:param isat2: diode 2 saturation current [A]
:param rs: series resistance [ohms]
:param rsh: shunt resistance [ohms]
:param vt: thermal voltage (kB * Tc / qe = 26[mV] at Tc=298K) [V]
:param isc: short circuit current [A]
:return: Rsh residual and its derivatives
Shunt resistance is assumed to be equal to the inverse of the slope of the
IV curve at short circuit.
.. math::
Rsh = \\frac{ -1 }{ \\left. \\frac{dI}{dV} \\right|_{V=0} }
This assumption is valid when [put condition here].
"""
didv, _ = fdidv(isat1, isat2, rs, rsh, ic=isc, vc=0, vt=vt)
vd, _ = diode.fvd(0.0, isc, rs) # vd = vc + ic * rs = 0.0 + isc * rs
# frsh = rsh + 1/didv
frsh = vd * (1.0/rsh + didv)
dfrsh_isat1 = vd*(
2.0*rs*rsh*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)*np.exp(vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt)
+ 2.0*rs*vt + 2.0*rsh*vt
)**2 - 2.0*rsh*np.exp(vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt)
+ 2.0*rs*vt + 2.0*rsh*vt
)
)
dfrsh_isat2 = vd*(
rs*rsh*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)*np.exp(0.5*vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt)
+ 2.0*rs*vt + 2.0*rsh*vt
)**2 - rsh*np.exp(0.5*vd/vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt)
+ 2.0*rs*vt + 2.0*rsh*vt
)
)
dfrsh_rs = (
vd*(
-(
2.0*isat1*rsh*isc*np.exp(vd/vt)/vt + 0.5*isat2*rsh*isc*np.exp(0.5*vd/vt)/vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - (
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)*(
-2.0*isat1*rs*rsh*isc*np.exp(vd/vt)/vt
- 2.0*isat1*rsh*np.exp(vd/vt)
- 0.5*isat2*rs*rsh*isc*np.exp(0.5*vd/vt)/vt
- isat2*rsh*np.exp(0.5*vd/vt) - 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2
) + (
-(2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) + 1.0/rsh
)*isc
)
dfrsh_rsh = (
vd*(
-(2.0*isat1*np.exp(vd/vt) + isat2*np.exp(0.5*vd/vt))/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - (
-2.0*isat1*rs*np.exp(vd/vt) - isat2*rs*np.exp(0.5*vd/vt) - 2.0*vt
)*(2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2 - 1.0/rsh**2
)
)
dfrsh_ic = (
rs*(
-(2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) + 1.0/rsh
) + vd*(
-(
2.0*isat1*rs*rsh*np.exp(vd/vt)/vt + 0.5*isat2*rs*rsh*np.exp(0.5*vd/vt)/vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - (
-2.0*isat1*rs**2*rsh*np.exp(vd/vt)/vt - 0.5*isat2*rs**2*rsh*np.exp(0.5*vd/vt)/vt
)*(
2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2
)
)
dfrsh_vc = (
vd*(-(
2.0*isat1*rsh*(rs*didv + 1)*np.exp(vd/vt)/vt
+ 0.5*isat2*rsh*(rs*didv + 1)*np.exp(0.5*vd/vt)/vt
)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) - (
-2.0*isat1*rs*rsh*(rs*didv + 1)*np.exp(vd/vt)/vt
- 0.5*isat2*rs*rsh*(rs*didv + 1)*np.exp(0.5*vd/vt)/vt
)*(2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
)**2) + (rs*didv + 1)*(
-(2.0*isat1*rsh*np.exp(vd/vt) + isat2*rsh*np.exp(0.5*vd/vt) + 2.0*vt)/(
2.0*isat1*rs*rsh*np.exp(vd/vt) + isat2*rs*rsh*np.exp(0.5*vd/vt) + 2.0*rs*vt + 2.0*rsh*vt
) + 1.0/rsh
)
)
jac = np.array([
dfrsh_isat1, dfrsh_isat2, dfrsh_rs, dfrsh_rsh, dfrsh_ic, dfrsh_vc
])
return frsh, jac