"""
Backtracking on sloped terrain
==============================
Modeling backtracking for single-axis tracker arrays on sloped terrain.
"""
# %%
# Tracker systems use backtracking to avoid row-to-row shading when the
# sun is low in the sky. The backtracking strategy orients the modules exactly
# on the boundary between shaded and unshaded so that the modules are oriented
# as much towards the sun as possible while still remaining unshaded.
# Unlike the true-tracking calculation (which only depends on solar position),
# calculating the backtracking angle requires knowledge of the relative spacing
# of adjacent tracker rows. This example shows how the backtracking angle
# changes based on a vertical offset between rows caused by sloped terrain.
# It uses :py:func:`pvlib.tracking.calc_axis_tilt` and
# :py:func:`pvlib.tracking.calc_cross_axis_tilt` to calculate the necessary
# array geometry parameters and :py:func:`pvlib.tracking.singleaxis` to
# calculate the backtracking angles.
#
# Angle conventions
# -----------------
#
# First let's go over the sign conventions used for angles. In contrast to
# fixed-tilt arrays where the azimuth is that of the normal to the panels, the
# convention for the azimuth of a single-axis tracker is along the tracker
# axis. Note that the axis azimuth is a property of the array and is distinct
# from the azimuth of the panel orientation, which changes based on tracker
# rotation angle. Because the tracker axis points in two directions, there are
# two choices for the axis azimuth angle, and by convention (at least in the
# northern hemisphere), the more southward angle is chosen:
#
# .. image:: ../../_images/tracker_azimuth_angle_convention.png
# :alt: Image showing the azimuth convention for single-axis tracker arrays.
# :width: 500
# :align: center
#
# Note that, as with fixed-tilt arrays, the axis azimuth is determined as the
# angle clockwise from north. The azimuth of the terrain's slope is also
# determined as an angle clockwise from north, pointing in the direction
# of falling slope. So for example, a hillside that slopes down to the east
# has an azimuth of 90 degrees.
#
# Using the axis azimuth convention above, the sign convention for tracker
# rotations is given by the
# `right-hand rule `_.
# Point the right hand thumb along the axis in the direction of the axis
# azimuth and the fingers curl in the direction of positive rotation angle:
#
# .. image:: ../../_images/tracker_rotation_angle_convention.png
# :alt: Image showing the rotation sign convention for single-axis trackers.
# :width: 500
# :align: center
#
# So for an array with ``axis_azimuth=180`` (tracker axis aligned perfectly
# north-south), pointing the right-hand thumb along the axis azimuth has the
# fingers curling towards the west, meaning rotations towards the west are
# positive and rotations towards the east are negative.
#
# The ground slope itself is always positive, but the component of the slope
# perpendicular to the tracker axes can be positive or negative. The convention
# for the cross-axis slope angle follows the right-hand rule: align
# the right-hand thumb along the tracker axis in the direction of the axis
# azimuth and the fingers curl towards positive angles. So in this example,
# with the axis azimuth coming out of the page, an east-facing, downward slope
# is a negative rotation from horizontal:
#
# .. image:: ../../_images/ground_slope_angle_convention.png
# :alt: Image showing the ground slope sign convention.
# :width: 500
# :align: center
#
# %%
# Rotation curves
# ---------------
#
# Now, let's plot the simple case where the tracker axes are at right angles
# to the direction of the slope. In this case, the cross-axis tilt angle
# is the same as the slope of the terrain and the tracker axis itself is
# horizontal.
from pvlib import solarposition, tracking
import pandas as pd
import matplotlib.pyplot as plt
# PV system parameters
tz = 'US/Eastern'
lat, lon = 40, -80
gcr = 0.4
# calculate the solar position
times = pd.date_range('2019-01-01 06:00', '2019-01-01 18:00', freq='1min',
tz=tz)
solpos = solarposition.get_solarposition(times, lat, lon)
# compare the backtracking angle at various terrain slopes
fig, ax = plt.subplots()
for cross_axis_tilt in [0, 5, 10]:
tracker_data = tracking.singleaxis(
apparent_zenith=solpos['apparent_zenith'],
apparent_azimuth=solpos['azimuth'],
axis_tilt=0, # flat because the axis is perpendicular to the slope
axis_azimuth=180, # N-S axis, azimuth facing south
max_angle=90,
backtrack=True,
gcr=gcr,
cross_axis_tilt=cross_axis_tilt)
# tracker rotation is undefined at night
backtracking_position = tracker_data['tracker_theta'].fillna(0)
label = 'cross-axis tilt: {}°'.format(cross_axis_tilt)
backtracking_position.plot(label=label, ax=ax)
plt.legend()
plt.title('Backtracking Curves')
plt.show()
# %%
# This plot shows how backtracking changes based on the slope between rows.
# For example, unlike the flat-terrain backtracking curve, the sloped-terrain
# curves do not approach zero at the end of the day. Because of the vertical
# offset between rows introduced by the sloped terrain, the trackers can be
# slightly tilted without shading each other.
#
# Now let's examine the general case where the terrain slope makes an
# inconvenient angle to the tracker axes. For example, consider an array
# with north-south axes on terrain that slopes down to the south-south-east.
# Assuming the axes are installed parallel to the ground, the northern ends
# of the axes will be higher than the southern ends. But because the slope
# isn't purely parallel or perpendicular to the axes, the axis tilt and
# cross-axis tilt angles are not immediately obvious. We can use pvlib
# to calculate them for us:
# terrain slopes 10 degrees downward to the south-south-east. note: because
# slope_azimuth is defined in the direction of falling slope, slope_tilt is
# always positive.
slope_azimuth = 155
slope_tilt = 10
axis_azimuth = 180 # tracker axis is still N-S
# calculate the tracker axis tilt, assuming that the axis follows the terrain:
axis_tilt = tracking.calc_axis_tilt(slope_azimuth, slope_tilt, axis_azimuth)
# calculate the cross-axis tilt:
cross_axis_tilt = tracking.calc_cross_axis_tilt(slope_azimuth, slope_tilt,
axis_azimuth, axis_tilt)
print('Axis tilt:', '{:0.01f}°'.format(axis_tilt))
print('Cross-axis tilt:', '{:0.01f}°'.format(cross_axis_tilt))
# %%
# And now we can pass use these values to generate the tracker curve as
# before:
tracker_data = tracking.singleaxis(
apparent_zenith=solpos['apparent_zenith'],
apparent_azimuth=solpos['azimuth'],
axis_tilt=axis_tilt, # no longer flat because the terrain imparts a tilt
axis_azimuth=axis_azimuth,
max_angle=90,
backtrack=True,
gcr=gcr,
cross_axis_tilt=cross_axis_tilt)
backtracking_position = tracker_data['tracker_theta'].fillna(0)
backtracking_position.plot()
title_template = 'Axis tilt: {:0.01f}° Cross-axis tilt: {:0.01f}°'
plt.title(title_template.format(axis_tilt, cross_axis_tilt))
plt.show()
# %%
# Note that the backtracking curve is roughly mirrored compared with the
# earlier example -- it is because the terrain is now sloped somewhat to the
# east instead of west.