Reverse transposition using one year of hourly data

With a brief look at accuracy and speed.

Author: Anton Driesse

Introduction

When irradiance is measured on a tilted plane, it is useful to be able to estimate the GHI that produces the POA irradiance. The estimation requires inverting a GHI-to-POA irradiance model, which involves two parts: a decomposition of GHI into direct and diffuse components, and a transposition model that calculates the direct and diffuse irradiance on the tilted plane. Recovering GHI from POA irradiance is termed “reverse transposition.”

In this example we start with a TMY file and calculate POA global irradiance. Then we use pvlib.irradiance.ghi_from_poa_driesse_2023() to estimate the original GHI from POA global. Details of the method found in 1.

Another method for reverse tranposition called GTI-DIRINT is also available in pvlib python (pvlib.irradiance.gti_dirint()). More information is available in 2.

References

1

Driesse, A., Jensen, A., Perez, R., 2024. A Continuous form of the Perez diffuse sky model for forward and reverse transposition. Solar Energy vol. 267. DOI: 10.1016/j.solener.2023.112093

2

B. Marion, A model for deriving the direct normal and diffuse horizontal irradiance from the global tilted irradiance, Solar Energy 122, 1037-1046. DOI: 10.1016/j.solener.2015.10.024

import os
import time
import pandas as pd

import matplotlib.pyplot as plt

import pvlib
from pvlib import iotools, location
from pvlib.irradiance import (get_extra_radiation,
                              get_total_irradiance,
                              ghi_from_poa_driesse_2023,
                              aoi,
                              )

Read a TMY3 file containing weather data and select needed columns.

PVLIB_DIR = pvlib.__path__[0]
DATA_FILE = os.path.join(PVLIB_DIR, 'data', '723170TYA.CSV')

tmy, metadata = iotools.read_tmy3(DATA_FILE, coerce_year=1990,
                                  map_variables=True)

df = pd.DataFrame({'ghi': tmy['ghi'], 'dhi': tmy['dhi'], 'dni': tmy['dni'],
                   'temp_air': tmy['temp_air'],
                   'wind_speed': tmy['wind_speed'],
                   })

Shift the timestamps to the middle of the hour and calculate sun positions.

df.index = df.index - pd.Timedelta(minutes=30)

loc = location.Location.from_tmy(metadata)
solpos = loc.get_solarposition(df.index)

Estimate global irradiance on a fixed-tilt array (forward transposition). The array is tilted 30 degrees and oriented 30 degrees east of south.

TILT = 30
ORIENT = 150

df['dni_extra'] = get_extra_radiation(df.index)

total_irrad = get_total_irradiance(TILT, ORIENT,
                                   solpos.apparent_zenith,
                                   solpos.azimuth,
                                   df.dni, df.ghi, df.dhi,
                                   dni_extra=df.dni_extra,
                                   model='perez-driesse')

df['poa_global'] = total_irrad.poa_global
df['aoi'] = aoi(TILT, ORIENT, solpos.apparent_zenith, solpos.azimuth)

Now estimate ghi from poa_global using reverse transposition. The algorithm uses a simple bisection search, which is quite slow because scipy doesn’t offer a vectorized version (yet). For this reason we’ll process a random sample of 1000 timestamps rather than the whole year.

df = df[df.ghi > 0].sample(n=1000)
solpos = solpos.reindex(df.index)

start = time.process_time()

df['ghi_rev'] = ghi_from_poa_driesse_2023(TILT, ORIENT,
                                          solpos.apparent_zenith,
                                          solpos.azimuth,
                                          df.poa_global,
                                          dni_extra=df.dni_extra)
finish = time.process_time()

print('Elapsed time for reverse transposition: %.1f s' % (finish - start))

Elapsed time for reverse transposition: 8.1 s

This graph shows the reverse transposed values vs. the original values. The markers are color-coded by angle-of-incidence to show that errors occur primarily with incidence angle approaching 90° and beyond.

Note that the results look particularly good because the POA values were calculated using the same models as used in reverse transposition. This isn’t cheating though. It’s a way of ensuring that the errors we see are really due to the reverse transposition algorithm. Expect to see larger errors with real-word POA measurements because errors from forward and reverse transposition will both be present.

df = df.sort_values('aoi')

plt.figure()
plt.gca().grid(True, alpha=.5)
pc = plt.scatter(df['ghi'], df['ghi_rev'], c=df['aoi'], s=15,
                 cmap='jet', vmin=60, vmax=120)
plt.colorbar(label='AOI [°]')
pc.set_alpha(0.5)

plt.xlabel('GHI original [W/m²]')
plt.ylabel('GHI from POA [W/m²]')

plt.show()

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