Misalignment Error

How much transposition error is caused by small sensor misalignments?

This example simulates how alignment error translates to transposition error.

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


lat, lon = 40, -80
times = pd.date_range('2019-01-01 06:00', '2019-01-01 21:00',
                      freq='5min', tz='Etc/GMT+5')

dni_extra = pvlib.irradiance.get_extra_radiation(times)
location = pvlib.location.Location(lat, lon)

solpos = location.get_solarposition(times)
clearsky = location.get_clearsky(times, model='ineichen',
                                 solar_position=solpos)


def transpose(surface_tilt, surface_azimuth):
    transposed = pvlib.irradiance.get_total_irradiance(
        surface_tilt=surface_tilt,
        surface_azimuth=surface_azimuth,
        solar_zenith=solpos['apparent_zenith'],
        solar_azimuth=solpos['azimuth'],
        dni=clearsky['dni'],
        ghi=clearsky['ghi'],
        dhi=clearsky['dhi'],
        dni_extra=dni_extra,
        albedo=0.15,
        model='haydavies',
    )
    return transposed['poa_global']

true_poa = transpose(20, 180)
true_total = true_poa.resample('h').mean().sum()

df_tilt = pd.DataFrame()
df_azi = pd.DataFrame()
heatmap_results = []

for azimuth_delta in np.linspace(-5, 5, 11)[::-1]:
    for tilt_delta in np.linspace(-5, 5, 11)[::-1]:

        surface_tilt = 20 + tilt_delta
        surface_azimuth = 180 + azimuth_delta

        misaligned_poa = transpose(surface_tilt, surface_azimuth)

        if tilt_delta == 0:
            label = f'{surface_azimuth:0.01f}'
            df_azi[label] = misaligned_poa

        if azimuth_delta == 0:
            label = f'{surface_tilt:0.01f}'
            df_tilt[label] = misaligned_poa

        misaligned_total = misaligned_poa.resample('h').mean().sum()
        heatmap_results.append({
            'surface_tilt': surface_tilt,
            'surface_azimuth': surface_azimuth,
            'max_abs_error': (misaligned_poa - true_poa).abs().max(),
            'total_error': (misaligned_total / true_total - 1) * 100
        })

def plot_ts(df, case, base):
    fig, axes = plt.subplots(2, 1, sharex=True, figsize=(6, 6))

    df.plot(ax=axes[0])
    axes[0].set_ylabel('Transposed Irradiance [W m^-2]')
    axes[0].legend(loc='upper right')

    error = (df.divide(df[base], axis=0) - 1) * 100
    error.plot(ax=axes[1])
    axes[1].set_ylabel('Error [%]')
    axes[1].legend(loc='upper right')

    fig.tight_layout()
    fig.show()

plot_ts(df_tilt, 'Tilt', '20.0')

plot_ts(df_azi, 'Azimuth', '180.0')

df = pd.DataFrame(heatmap_results)


def plot(df, column):
    df2 = df.pivot_table(index='surface_azimuth',
                         columns='surface_tilt',
                         values=column)
    fig = plt.figure()
    plt.pcolormesh(df2.columns, df2.index, df2, shading='auto')
    plt.colorbar()
    plt.ylabel(r'Surface Azimuth [$\degree$]')
    plt.xlabel(r'Surface Tilt [$\degree$]')
    return fig

fig = plot(df, 'max_abs_error')
fig.suptitle('Max Absolute Error [W m^-2]')
fig.show()

fig = plot(df, 'total_error')
fig.suptitle('Total Error [%]')
fig.show()

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Diffuse self-shading