Validation
Contents
Validation#
This document shows how the shading calculations using the twoaxistracking package compare to existing methods found in the literature. Specifically, the annual shading loss is calculated for nine different field layouts and compared with the values presented in Cumpston and Pye (2014).
Import necessary packages:
import pandas as pd
import numpy as np
from shapely import geometry
import twoaxistracking
Definition of collector geometry#
The study by Cumpston and Pye (2014) simulated shading for a circular aperture. The study made no distinction between gross and aperture area, i.e., the two geometries were assumed to be the same.
radius = np.sqrt(1/np.pi) # radius for a circle with an area of 1
collector_geometry = geometry.Point(0,0).buffer(radius)
collector_geometry
Reference dataset#
The simulations carried out in the reference study were based on the 1976 irradiance dataset from Barstow.
A stringent quality-control of this dataset was carried out as described in the Reference irradiance dataset section. This step was necessary as the dataset contained several periods of erroneous data, e.g., irradiance at night and periods with unfeasible irradiance levels.
Load the quality-controlled dataset:
filename = '../../../twoaxistracking/data/barstow_1976_irradiance_data_quality_controlled.csv'
df = pd.read_csv(filename, index_col=0, parse_dates=[0])
df.head() # print the first five rows of the dataframe
| ghi | dni | dhi_calc | temp_air | apparent_zenith | zenith | apparent_elevation | elevation | azimuth | dni_extra | |
|---|---|---|---|---|---|---|---|---|---|---|
| time | ||||||||||
| 1976-01-01 00:15:00-08:00 | 0.88 | 0.94 | 0.88 | -1.59 | 167.092519 | 167.092519 | -77.092519 | -77.092519 | 25.278534 | 1413.981805 |
| 1976-01-01 00:30:00-08:00 | 0.00 | 0.94 | 0.00 | -1.94 | 165.467857 | 165.467857 | -75.467857 | -75.467857 | 38.153710 | 1413.981805 |
| 1976-01-01 00:45:00-08:00 | 0.88 | 1.88 | 0.88 | -2.36 | 163.355080 | 163.355080 | -73.355080 | -73.355080 | 48.333521 | 1413.981805 |
| 1976-01-01 01:00:00-08:00 | 0.00 | 0.94 | 0.00 | -2.50 | 160.917391 | 160.917391 | -70.917391 | -70.917391 | 56.311453 | 1413.981805 |
| 1976-01-01 01:15:00-08:00 | 0.88 | 0.00 | 0.88 | -2.78 | 158.265917 | 158.265917 | -68.265917 | -68.265917 | 62.652770 | 1413.981805 |
Furthermore, it is important to note that the reference study only performed simulations “for solar elevation angles greater than 10°”.
Therefore, a new column of the direct normal irradiance (DNI) is derived where the irradiance below 10° elevation is set to zero.
df['dni>10deg'] = df.loc[df['elevation']>10, 'dni']
df['dni>10deg'] = df['dni>10deg'].fillna(0)
Field layouts#
Cumpston and Pye (2010) provided the annual shading loss (ASL) for nine different field layouts (GCR ranging from 0.1 to 0.9). The field layout parameters and their annual shading loss were:
dfc = pd.DataFrame(
columns=['gcr', 'aspect_ratio', 'rotation', 'offset', 'asl_cumpston'],
data=[
[0.1, 2.3, 0.0, 0.0, 0.04],
[0.2, 1.61, 0.0, 0.0, 1.53],
[0.3, 1.11, 1.0, -0.5, 4.50],
[0.4, 1.0, 0.0, -0.5, 8.14],
[0.5, np.sqrt(3)/2, 0.0, -0.5, 12.4],
[0.6, np.sqrt(3)/2, 0.0, -0.5, 17.2],
[0.7, np.sqrt(3)/2, 0.0, -0.5, 22.3],
[0.8, np.sqrt(3)/2, 89.0, -0.5, 27.6],
[0.9, np.sqrt(3)/2, 167.0, -0.5, 33.1]])
dfc.round(2)
| gcr | aspect_ratio | rotation | offset | asl_cumpston | |
|---|---|---|---|---|---|
| 0 | 0.1 | 2.30 | 0.0 | 0.0 | 0.04 |
| 1 | 0.2 | 1.61 | 0.0 | 0.0 | 1.53 |
| 2 | 0.3 | 1.11 | 1.0 | -0.5 | 4.50 |
| 3 | 0.4 | 1.00 | 0.0 | -0.5 | 8.14 |
| 4 | 0.5 | 0.87 | 0.0 | -0.5 | 12.40 |
| 5 | 0.6 | 0.87 | 0.0 | -0.5 | 17.20 |
| 6 | 0.7 | 0.87 | 0.0 | -0.5 | 22.30 |
| 7 | 0.8 | 0.87 | 89.0 | -0.5 | 27.60 |
| 8 | 0.9 | 0.87 | 167.0 | -0.5 | 33.10 |
For comparison the annual shading loss for the nine field layouts were therefore calculated using the twoaxistracking package:
field_parameters = ['gcr', 'aspect_ratio', 'rotation', 'offset']
for index, (gcr, aspect_ratio, rotation, offset) in dfc[field_parameters].iterrows():
# Define the tracker field for each configuration
tracker_field = twoaxistracking.TrackerField(
total_collector_geometry=collector_geometry,
active_collector_geometry=collector_geometry,
neighbor_order=2,
gcr=gcr,
aspect_ratio=aspect_ratio,
rotation=rotation,
offset=offset)
# Calculate shaded fraction for each timestamp
df['shaded_fraction'] = tracker_field.get_shaded_fraction(df['elevation'], df['azimuth'])
# Calculate annual shading loss
dfc.loc[index, 'asl_twoaxistracking'] = (df['shaded_fraction'].multiply(df['dni>10deg'], axis='rows').sum() /
df['dni>10deg'].sum())*100
/home/docs/checkouts/readthedocs.org/user_builds/twoaxistracking/envs/stable/lib/python3.7/site-packages/twoaxistracking/layout.py:176: RuntimeWarning: invalid value encountered in arcsin
(tracker_distance * np.cos(delta_gamma_rad)))) + relative_slope
Comparison of annual shading loss#
The annual shading loss (ASL) presented in Cumpston and Pye (2014) and those calculated using the twoaxistracking package are shown below. The deviations between the shading fractions are presented both in absolute and relative terms.
# Limit asl_twoaxistracking to three significant figures and two decimal places
dfc['asl_twoaxistracking'] = dfc['asl_twoaxistracking'].\
apply(lambda s: f"{s:.3g}").astype(float).round(2)
# Calculate absolute and relatieve deviation
dfc['absolute_deviation'] = (dfc['asl_twoaxistracking'] - dfc['asl_cumpston'])
dfc['relative_deviation'] = (dfc['asl_twoaxistracking'] - dfc['asl_cumpston']).\
divide(dfc['asl_cumpston']).multiply(100)
result_columns = ['asl_cumpston', 'asl_twoaxistracking', 'absolute_deviation', 'relative_deviation']
dfc.set_index('gcr')[result_columns].round(2).applymap(lambda s: f"{s:.2f} %")
| asl_cumpston | asl_twoaxistracking | absolute_deviation | relative_deviation | |
|---|---|---|---|---|
| gcr | ||||
| 0.1 | 0.04 % | 0.04 % | 0.00 % | 0.00 % |
| 0.2 | 1.53 % | 1.51 % | -0.02 % | -1.31 % |
| 0.3 | 4.50 % | 4.49 % | -0.01 % | -0.22 % |
| 0.4 | 8.14 % | 8.09 % | -0.05 % | -0.61 % |
| 0.5 | 12.40 % | 12.30 % | -0.10 % | -0.81 % |
| 0.6 | 17.20 % | 17.10 % | -0.10 % | -0.58 % |
| 0.7 | 22.30 % | 22.20 % | -0.10 % | -0.45 % |
| 0.8 | 27.60 % | 27.40 % | -0.20 % | -0.72 % |
| 0.9 | 33.10 % | 32.90 % | -0.20 % | -0.60 % |
The comparison above shows that the differences between the annual shading losses are very low (always lower than 1.3%). The discrepancies are believed to be due to minor differences in the quality control procedure applied to reference data.
The reference study (Cumpston and Pye, 2014) utilized the shading calculation algorithm developed by Meller (2010). Meller’s algorithm has previously been compared to results from Pons & Dugan (1984).