This tutorial explores some of the functions available in the pvlib module irradiance.py.

This tutorial requires pvlib >= 0.6.0.

Authors:

• Will Holmgren (@wholmgren), University of Arizona. July 2014, April 2015, July 2015, March 2016, July 2016, February 2017, August 2018.
In [1]:
%matplotlib inline
import matplotlib.pyplot as plt

# built in python modules
import datetime

import numpy as np
import pandas as pd

import pvlib


Many solar power algorithms start with the irradiance incident on the top of the Earth's atmosphere, often known as the extraterrestrial radiation. pvlib has four different algorithms to calculate the yearly cycle of the extraterrestrial radiation given the solar constant. As of pvlib 0.4, each method can accept many different input types (day of year, arrays of day of year, datetimes, DatetimeIndex, etc.) and will consistently return the appropriate output type.

In [2]:
# DatetimeIndex in yields a TimeSeries out
times = pd.date_range('2014-01-01', '2015-01-01', freq='1h')


In [3]:
spencer.plot(label='spencer')
asce.plot(label='asce')
ephem.plot(label='pyephem')
nrel.plot(label='nrel')
plt.legend()

Out[3]:
Text(0,0.5,'Extraterrestrial radiation (W/m^2)')

The pyephem and nrel methods are the most accurate. However, as shown in the plot below, the difference between them and the spencer method is only +/-2 W/m^2 over the entire year.

In [4]:
et_diff = spencer - ephem
et_diff.plot()
plt.ylabel('spencer-ephem (W/m**2)')

Out[4]:
Text(0,0.5,'spencer-ephem (W/m**2)')

The intraday squiggles are due to the fact that the asce and spencer methods will cast a DatetimeIndex into integer days of year, while the pyephem and nrel methods also use the time of day.

The difference between the nrel and pyephem methods is negligible.

In [5]:
et_diff = nrel - ephem
et_diff.plot()
plt.ylabel('nrel-ephem (W/m**2)')

Out[5]:
Text(0,0.5,'nrel-ephem (W/m**2)')

You can also control the solar constant. Recent literature suggests that the solar constant is 1361 $W/m^2$ rather than the commonly accepted 1367 $W/m^2$.

In [6]:
spencer_1361 = pvlib.irradiance.get_extra_radiation(times, method='spencer', solar_constant=1361)

spencer.plot(label='default 1366.7')
spencer_1361.plot(label='1361')
plt.legend()
plt.title('Impact of solar constant')

Out[6]:
Text(0,0.5,'ET Irradiance (W/m^2)')

Compare the time it takes to do the calculations.

In [7]:
times = pd.DatetimeIndex(start='2015', end='2016', freq='1min')

In [8]:
%timeit spencer = pvlib.irradiance.get_extra_radiation(times, method='spencer')

55.9 ms Â± 406 Âµs per loop (mean Â± std. dev. of 7 runs, 10 loops each)
32.6 ms Â± 444 Âµs per loop (mean Â± std. dev. of 7 runs, 10 loops each)
7.98 s Â± 94.5 ms per loop (mean Â± std. dev. of 7 runs, 1 loop each)
1.04 s Â± 15.9 ms per loop (mean Â± std. dev. of 7 runs, 1 loop each)
263 ms Â± 14.2 ms per loop (mean Â± std. dev. of 7 runs, 1 loop each)


In addition to DatetimeIndex input, the methods also work for various scalar datetime-like formats as well as scalar and array day of year input.

In [9]:
methods = ['spencer', 'asce', 'pyephem', 'nrel']

# pandas timestamp input
times = pd.Timestamp('20161026')
for method in methods:
assert isinstance(dni_extra, float)
print(times, method, dni_extra)

# date input
times = datetime.date(2016, 10, 26)
for method in methods:
assert isinstance(dni_extra, float)
print(times, method, dni_extra)

# integer doy input
times = 300
for method in methods:
assert isinstance(dni_extra, float)
print(times, method, dni_extra)

# array doy input
times = np.arange(1, 366)
for method in methods:
assert isinstance(dni_extra, np.ndarray)
plt.plot(times, dni_extra, label=method)

plt.legend()

2016-10-26 00:00:00 spencer 1383.6362029
2016-10-26 00:00:00 asce 1385.08377469
2016-10-26 00:00:00 pyephem 1382.3920240844304
2016-10-26 00:00:00 nrel 1382.3912017480877
2016-10-26 spencer 1383.6362028955045
2016-10-26 asce 1385.0837746866198
2016-10-26 pyephem 1382.3920240844304
2016-10-26 nrel 1382.3912017480877
300 spencer 1383.6362029
300 asce 1385.08377469
300 pyephem 1382.618664623793
300 nrel 1382.6174658748046

Out[9]:
Text(0,0.5,'Extraterrestrial radiation (W/m^2)')

## Clear sky models¶

See the online documentation for clear sky modeling examples.

Here we only generate data for the functions below.

In [10]:
tus = pvlib.location.Location(32.2, -111, 'US/Arizona', 700, 'Tucson')
times = pd.DatetimeIndex(start='2016-01-01', end='2016-01-02', freq='1min', tz=tus.tz)
ephem_data = tus.get_solarposition(times)
plt.ylabel('Irradiance $W/m^2$')
plt.title('Ineichen, climatological turbidity')

Out[10]:
Text(0.5,1,'Ineichen, climatological turbidity')

## Diffuse ground¶

The grounddiffuse function has a few different ways to obtain the diffuse light reflected from the ground given an surface tilt and the GHI.

First, you can specify the albedo of ground.

In [11]:
ground_irrad = pvlib.irradiance.get_ground_diffuse(40, irrad_data['ghi'], albedo=.25)

Out[11]:
Text(0,0.5,'Diffuse ground irradiance (W/m^2)')

Alternatively, you can specify the surface type with a string such as 'concrete' or 'snow'. All of the available surface_type options are show in the plot below.

In [12]:
for surface, albedo in sorted(pvlib.irradiance.SURFACE_ALBEDOS.items(), key=lambda x: x[1], reverse=True):

plt.legend()
plt.title('Surface types')

Out[12]:
Text(0.5,1,'Surface types')

Next, vary the tilt angle. We expect to see maximum ground diffuse irradiance at a 90 deg tilt, and no ground diffuse irradiance at 0 tilt.

In [13]:
for surf_tilt in np.linspace(0, 90, 5):

plt.legend()
plt.title('Ground diffuse as a function of tilt')

Out[13]:
Text(0.5,1,'Ground diffuse as a function of tilt')

### Diffuse sky¶

pvlib has many different ways to calculate the diffuse sky component of GHI.

The API for some of these functions needs some work.

### Isotropic model¶

The isotropic model is the simplest model.

In [14]:
sky_diffuse = pvlib.irradiance.isotropic(40, irrad_data['dhi'])
sky_diffuse.plot(label='isotropic diffuse')
plt.legend()

Out[14]:
Text(0,0.5,'Irradiance (W/m^2)')

Compare just the POA diffuse to the input DHI.

In [15]:
sky_diffuse = pvlib.irradiance.isotropic(40, irrad_data['dhi'])
sky_diffuse.plot(label='isotropic diffuse')
plt.legend()

Out[15]:
Text(0,0.5,'Irradiance (W/m^2)')

### Klucher model¶

In [16]:
surf_tilt = 40
surf_az = 180

ephem_data['apparent_zenith'], ephem_data['azimuth'])
sky_diffuse.plot(label='klucher diffuse')
plt.legend()

Out[16]:
Text(0,0.5,'Irradiance (W/m^2)')
In [17]:
surf_tilt = 40
surf_az = 180 # south facing

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

plt.legend()

Out[17]:
Text(0,0.5,'Irradiance (W/m^2)')

Klucher as a function of surface azimuth.

In [18]:
surf_tilt = 40

iso_diffuse.plot(label='isotropic')

for surf_az in np.linspace(0, 270, 4):
ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher: {}'.format(surf_az))

plt.legend()

Out[18]:
<matplotlib.legend.Legend at 0x11801c7b8>

Surface azimuth should not matter if tilt is 0.

In [19]:
surf_tilt = 0

iso_diffuse.plot(label='isotropic')

for surf_az in np.linspace(0, 270, 4):
ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher: {}'.format(surf_az))

plt.legend()

Out[19]:
<matplotlib.legend.Legend at 0x117d0ee10>

### Reindl model¶

South facing at latitude.

In [20]:
surf_tilt = 32
surf_az = 180 # south facing

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
reindl_diffuse.plot(label='reindl diffuse')

plt.legend()

Out[20]:
<matplotlib.legend.Legend at 0x117c15470>

East facing

In [21]:
surf_tilt = 32
surf_az = 90

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
reindl_diffuse.plot(label='reindl diffuse')

plt.legend()

Out[21]:
<matplotlib.legend.Legend at 0x117b2b390>

### Hay-Davies model¶

Hay-Davies facing south.

In [22]:
surf_tilt = 32
surf_az = 180

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
haydavies_diffuse.plot(label='haydavies diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
reindl_diffuse.plot(label='reindl diffuse')

plt.legend()

Out[22]:
<matplotlib.legend.Legend at 0x115346ef0>

Facing east.

In [23]:
surf_tilt = 32
surf_az = 90

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
haydavies_diffuse.plot(label='haydavies diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
reindl_diffuse.plot(label='reindl diffuse')

plt.legend()

Out[23]:
<matplotlib.legend.Legend at 0x115206630>

Hay-Davies appears to be very similar to Reindl. Too similar?

### King model¶

In [24]:
surf_tilt = 32
surf_az = 90

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
haydavies_diffuse.plot(label='haydavies diffuse')

king_diffuse.plot(label='king diffuse')

plt.legend()

Out[24]:
<matplotlib.legend.Legend at 0x112ed3a20>

### Perez model¶

This section walks through the Perez algorithm.

In [25]:
sun_zen = ephem_data['apparent_zenith']
sun_az = ephem_data['azimuth']
AM = pvlib.atmosphere.get_relative_airmass(sun_zen)

surf_tilt = 32
surf_az = 180

kappa = 1.041 #for sun_zen in radians

#Dhfilter = DHI > 0

# epsilon is the sky's clearness
eps = ( (DHI + DNI)/DHI + kappa*(z**3) ) / ( 1 + kappa*(z**3) )

In [26]:
eps.plot()

Out[26]:
<matplotlib.axes._subplots.AxesSubplot at 0x117c12b38>
In [27]:
ebin = eps.copy()
ebin[(eps<1.065)] = 1
ebin[(eps>=1.065) & (eps<1.23)] = 2
ebin[(eps>=1.23) & (eps<1.5)] = 3
ebin[(eps>=1.5) & (eps<1.95)] = 4
ebin[(eps>=1.95) & (eps<2.8)] = 5
ebin[(eps>=2.8) & (eps<4.5)] = 6
ebin[(eps>=4.5) & (eps<6.2)] = 7
ebin[eps>=6.2] = 8

ebin.plot()
plt.ylim(0,9)

Out[27]:
(0, 9)
In [28]:
ebin = ebin - 1
ebin = ebin.dropna().astype(int)
ebin.plot()

Out[28]:
<matplotlib.axes._subplots.AxesSubplot at 0x114e8e0f0>
In [29]:
delta = DHI * AM / DNI_ET
delta.plot()

Out[29]:
<matplotlib.axes._subplots.AxesSubplot at 0x114daf390>
In [30]:
modelt = 'allsitescomposite1990'

F1 = F1c[ebin,0] + F1c[ebin,1]*delta[ebin.index] + F1c[ebin,2]*z[ebin.index]
F1[F1<0]=0;
F1=F1.astype(float)

#F2= F2c[ebin,0] + F2c[ebin,1]*delta[ebinfilter] + F2c[ebin,2]*z[ebinfilter]
F2= F2c[ebin,0] + F2c[ebin,1]*delta[ebin.index] + F2c[ebin,2]*z[ebin.index]
F2[F2<0]=0
F2=F2.astype(float)

F1.plot(label='F1')
F2.plot(label='F2')
plt.legend()

Out[30]:
<matplotlib.legend.Legend at 0x114d0e7f0>
In [31]:
from pvlib import tools

In [32]:
A = tools.cosd(surf_tilt)*tools.cosd(sun_zen) + tools.sind(surf_tilt)*tools.sind(sun_zen)*tools.cosd(sun_az-surf_az) #removed +180 from azimuth modifier: Rob Andrews October 19th 2012
#A[A < 0] = 0

B = tools.cosd(sun_zen);
#B[B < pvl_tools.cosd(85)] = pvl_tools.cosd(85)

A.plot(label='A')
B.plot(label='B')
plt.legend()

Out[32]:
<matplotlib.legend.Legend at 0x114de3ac8>
In [33]:
sky_diffuse = DHI*( 0.5* (1-F1)*(1+tools.cosd(surf_tilt))+F1 * A[ebin.index]/ B[ebin.index] + F2*tools.sind(surf_tilt))
sky_diffuse[sky_diffuse < 0] = 0
sky_diffuse[AM.isnull()] = 0

sky_diffuse.plot()

Out[33]:
<matplotlib.axes._subplots.AxesSubplot at 0x113354908>

Compare the Perez model to others.

In [34]:
sun_zen = ephem_data['apparent_zenith']
sun_az = ephem_data['azimuth']
AM = pvlib.atmosphere.get_relative_airmass(sun_zen)

surf_tilt = 32
surf_az = 180

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
haydavies_diffuse.plot(label='haydavies diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'],
AM)
perez_diffuse.plot(label='perez diffuse')

plt.legend()

Out[34]:
<matplotlib.legend.Legend at 0x112feb8d0>
In [35]:
sun_zen = ephem_data['apparent_zenith']
sun_az = ephem_data['azimuth']
AM = pvlib.atmosphere.get_relative_airmass(sun_zen)

surf_tilt = 32
surf_az = 90

iso_diffuse.plot(label='isotropic diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
klucher_diffuse.plot(label='klucher diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'])
haydavies_diffuse.plot(label='haydavies diffuse')

ephem_data['apparent_zenith'], ephem_data['azimuth'],
AM)
perez_diffuse.plot(label='perez diffuse')

plt.legend()

Out[35]:
<matplotlib.legend.Legend at 0x112f7ef28>

Examine the impact of the coeffecient selection.

In [36]:
perez_diffuse = pvlib.irradiance.perez(surf_tilt, surf_az,
ephem_data['apparent_zenith'], ephem_data['azimuth'],
AM, model='allsitescomposite1990')
perez_diffuse.plot(label='allsitescomposite1990')

ephem_data['apparent_zenith'], ephem_data['azimuth'],
AM, model='phoenix1988')
perez_diffuse.plot(label='phoenix1988')

plt.legend()

Out[36]:
<matplotlib.legend.Legend at 0x113182208>

## Angle of incidence functions¶

The irradiance module has some convenience functions to help calculate the angle of incidence.

First, the angle of incidence.

In [37]:
proj = pvlib.irradiance.aoi(32, 180, ephem_data['apparent_zenith'], ephem_data['azimuth'])
proj.plot()

#plt.ylim(-1.1,1.1)
plt.legend()

Out[37]:
<matplotlib.legend.Legend at 0x113182128>

AOI projection: the dot production of the surface normal and the vector to the sun.

In [38]:
proj = pvlib.irradiance.aoi_projection(32, 180, ephem_data['apparent_zenith'], ephem_data['azimuth'])
proj.plot()

plt.ylim(-1.1,1.1)
plt.legend()

Out[38]:
<matplotlib.legend.Legend at 0x1120b5ef0>

The ratio between POA projection and the horizontal projection.

In [39]:
ratio = pvlib.irradiance.poa_horizontal_ratio(32, 180, ephem_data['apparent_zenith'], ephem_data['azimuth'])
ratio.plot()
plt.ylim(-4,4)

Out[39]:
(-4, 4)

This plot shows that an explicit dot product calculation gives the same result as aoi_projection.

In [40]:
surf_tilt = 90
surf_az = 90

dotprod.plot(label='dotprod')

proj = pvlib.irradiance.aoi_projection(surf_tilt, surf_az, ephem_data['apparent_zenith'], ephem_data['azimuth'])
proj.plot()

plt.ylim(-1.1,1.1)
plt.legend()

Out[40]:
<matplotlib.legend.Legend at 0x112b81208>

There is a convenience function get_total_irradiance that aims to make it easier to play with different models. For now, we use it to make summary plots of the models explored above.

South facing with latitude tilt.

In [41]:
def get_total_irradiance_per_model(surface_tilt, surface_azimuth):
models = ['isotropic', 'klucher', 'haydavies', 'reindl', 'king', 'perez']
totals = {}

for model in models:
surface_tilt, surface_azimuth,
ephem_data['apparent_zenith'], ephem_data['azimuth'],
dni_extra=dni_et, airmass=AM,
model=model,
surface_type='urban')
totals[model] = total
total.plot()
plt.title(model)
plt.ylim(-50, 1100)

get_total_irradiance_per_model(32, 180)