IAM Model Fitting

Illustrates how to fit an IAM model to data using fit().

An incidence angle modifier (IAM) model quantifies the fraction of direct irradiance is that is reflected away from a module’s surface. Three popular IAM models are Martin-Ruiz martin_ruiz(), physical physical(), and ASHRAE ashrae(). Each model requires one or more parameters.

Here, we show how to use fit() to estimate a model’s parameters from data.

Model fitting require a weight function that can assign more influence to some AOI values than others. We illustrate how to provide a custom weight function to fit().

import numpy as np
from random import uniform
import matplotlib.pyplot as plt

from pvlib.tools import cosd
from pvlib.iam import (martin_ruiz, physical, fit)

Fitting an IAM model to data

Here, we’ll show how to fit an IAM model to data. We’ll generate some data by perturbing output from the Martin-Ruiz model to mimic measured data and then we’ll fit the physical model to the perturbed data.

# Create some IAM data.
aoi = np.linspace(0, 85, 10)
params = {'a_r': 0.16}
iam = martin_ruiz(aoi, **params)
data = iam * np.array([uniform(0.98, 1.02) for _ in range(len(iam))])

# Get parameters for the physical model by fitting to the perturbed data.
physical_params = fit(aoi, data, 'physical')

# Compute IAM with the fitted physical model parameters.
physical_iam = physical(aoi, **physical_params)

# Plot IAM vs. AOI
plt.scatter(aoi, data, c='darkorange', label='Data')
plt.plot(aoi, physical_iam, label='physical')
plt.xlabel('AOI (degrees)')
plt.ylabel('IAM')
plt.title('Fitting the physical model to data')
plt.legend()
plt.show()

The weight function

pvlib.iam.fit() uses a weight function when computing residuals between the model and data. The default weight function is \(1 - \sin(aoi)\). We can instead pass a custom weight function to pvlib.iam.fit().

# Define a custom weight function.  The weight function must take ``aoi``
# as its argument and return a vector of the same length as ``aoi``.
def weight_function(aoi):
    return cosd(aoi)


physical_params_custom = fit(aoi, data, 'physical', weight=weight_function)

physical_iam_custom = physical(aoi, **physical_params_custom)

# Plot IAM vs AOI.
fig, ax = plt.subplots(2, 1, figsize=(5, 8))
ax[0].plot(aoi, data, '.', label='Data (from Martin-Ruiz model)')
ax[0].plot(aoi, physical_iam, label='With default weight function')
ax[0].plot(aoi, physical_iam_custom, label='With custom weight function')
ax[0].set_xlabel('AOI (degrees)')
ax[0].set_ylabel('IAM')
ax[0].legend()

ax[1].plot(aoi, physical_iam_custom - physical_iam, label='Custom - default')
ax[1].set_xlabel('AOI (degrees)')
ax[1].set_ylabel('Diff. in IAM')
ax[1].legend()
plt.tight_layout()
plt.show()

print("Parameters with default weights: " + str(physical_params))
print("Parameters with custom weights: " + str(physical_params_custom))

Parameters with default weights: {'n': 1.2031780765297881, 'K': 4, 'L': 0.015386312188580494}
Parameters with custom weights: {'n': 1.202753515577044, 'K': 4, 'L': 0.01545043069076259}

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Diffuse IAM Calculation

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IAM Model Conversion