Problem 2.15

Fundamentals of Solar Cells and Photovoltaic Systems Engineering

Solutions Manual - Chapter 2

Problem 2.15

Estimate the global irradiance G(α,β) reaching the plane of array (POA) of a north-facing, 20° tilted generator over grass (reflectivity \(ρ_{alb}\)=0.2) located at a PV plant in Antananarivo, Madagascar (18° 56′ 0″ S, 47° 31′ 0″ E) at midday (true solar time) on May 12, 2022.

Solve the problem using pvlib-python and the default clear-sky model.

We will use the packages pvlib, pandas and matplotlib.pyplot to plot the results.

import pvlib
import pandas as pd
import matplotlib.pyplot as plt

We start by defining the location, date and time. We will implement the calculation for every hour on May 12, 2022.

# Antananarivo, Madagascar
lat, lon = -18.93, 47.52
tz = 'Indian/Antananarivo' #existing timezones can be checked using pytz.all_timezones[::20]

date = '2022-05-12'

# surface angles beta, alpha
tilt, orientation = 20, 0 # pvlib sets orientation origin at North -> North=0

# location
location = pvlib.location.Location(lat, lon, tz=tz)

# albedo
albedo = 0.20 # grass reflectivity

# datetimes
times = pd.date_range(start=date, freq='h', periods=24, tz=tz)

We calculate the clear-sky irradiance using the default options in pvlib.

# generates clear-sky ghi, dni, dhi irradiances (decomposition using Ineichen model and turbidity index; pvlib's default)
clearsky = location.get_clearsky(times, model='ineichen')

We calculate the Sun’s coordinates and calculate the irradiance on the plane of array (POA)

# calculates Sun's coordinates
solar_position = location.get_solarposition(times=times)

# calculates POA (transposition using isotropic model)
poa_fixed_irradiance = pvlib.irradiance.get_total_irradiance(
                        surface_tilt=tilt,
                        surface_azimuth=orientation,
                        dni=clearsky['dni'],
                        ghi=clearsky['ghi'],
                        dhi=clearsky['dhi'],
                        albedo=albedo,
                        solar_zenith=solar_position['apparent_zenith'],
                        solar_azimuth=solar_position['azimuth'],
                        model='isotropic')

We can plot the daily evolution of direct, diffuse, and global irradiance

poa_fixed_irradiance.plot()
plt.ylabel('Irradiance [$W/m^2$]')

Text(0, 0.5, 'Irradiance [$W/m^2$]')

And obtain the value at midday.

poa_fixed_irradiance.loc['2022-05-12 12:00']

poa_global            1015.612935
poa_direct             917.173423
poa_diffuse             98.439512
poa_sky_diffuse         93.244835
poa_ground_diffuse       5.194677
Name: 2022-05-12 12:00:00+03:00, dtype: float64

We can also obtain the values at any other time step.

poa_fixed_irradiance

	poa_global	poa_direct	poa_diffuse	poa_sky_diffuse	poa_ground_diffuse
2022-05-12 00:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 01:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 02:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 03:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 04:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 05:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 06:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 07:00:00+03:00	193.627419	168.924254	24.703165	23.942678	0.760487
2022-05-12 08:00:00+03:00	479.117336	427.054165	52.063171	49.833871	2.229300
2022-05-12 09:00:00+03:00	716.583932	643.657557	72.926375	69.398793	3.527582
2022-05-12 10:00:00+03:00	891.691469	803.759942	87.931526	83.430202	4.501324
2022-05-12 11:00:00+03:00	993.550532	896.973748	96.576784	91.505547	5.071237
2022-05-12 12:00:00+03:00	1015.612935	917.173423	98.439512	93.244835	5.194677
2022-05-12 13:00:00+03:00	956.450393	863.029044	93.421350	88.558725	4.862625
2022-05-12 14:00:00+03:00	819.902753	738.119574	81.783180	77.683603	4.099576
2022-05-12 15:00:00+03:00	614.653185	550.614295	64.038890	61.074724	2.964165
2022-05-12 16:00:00+03:00	353.090633	312.627069	40.463564	38.904784	1.558780
2022-05-12 17:00:00+03:00	58.729342	50.030709	8.698633	8.522135	0.176498
2022-05-12 18:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 19:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 20:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 21:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 22:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000
2022-05-12 23:00:00+03:00	0.000000	0.000000	0.000000	0.000000	0.000000