Appendix – Shaded fraction and backtracking in single-axis trackers on rolling terrain

Author: Jensen, Adam R.

Publisher: Zenodo

DOI: 10.5281/zenodo.17536794

Source: https://zenodo.org/records/17536794/files/appendix_shaded_fraction_and_backtracking_in_single_axis_trackers_on_rolling_terrain.pdf

APPENDIX
Shaded ac ion and back acking in single-axis acke s
on olling e ain
Adam R. Jensen
Ma ch 2024
Appendix – Shaded ac ion and back acking in single-axis acke s on olling e ain
2024
By
Adam R. Jensen
Copy igh : Rep oduc ion o his publica ion in whole o in pa mus include he cus oma y
bibliog aphic ci a ion, including au ho a ibu ion, epo i le, e c.
Published by: DTU Cons uc , Anke Engelunds Vej 101, Building 101, 2800 Kongens Lyngby
Denma k
DOI: h ps://doi.o g/10.5281/zenodo.10787366
0 Upda e - 2025-11-05
This appendix was upda ed on 2025-11-05 by adding an ex a decimal o he ZR alues in Table 3.
Speci ically, he ZR alues which we e p e iously lis ed as 0.1 should in ac be 0.12 o he es
cases o ma ch. The same mis ake occu s in he jou nal a icle (doi: 10.1063/5.0202220).
1 O e iew
This documen is an appendix o he jou nal pape ”Shaded ac ion and back acking in single-
axis acke s on olling e ain” au ho ed by Ke in S. Ande son and Adam R. Jensen. The pape
was submi ed o he Jou nal o Renewable and Sus ainable Ene gy in Janua y 2024.
In addi ion o his documen , he appendix consis s o se e al supplemen a y iles which a e
made a ailable a he open esea ch da a eposi o y Zenodo (doi: 10.5281/zenodo.10513988). An
o e iew o he supplemen a y iles is p o ided in Table 1 and desc ibed in he ollowing sec ions.
Desc ip ion Filename
bi acial_ adiance ou pu bi acial_ adiance_shaded_ ac ion_ ixed_ il .cs
bi acial_ adiance_shaded_ ac ion_single_axis.cs
Py hon unc ions a iable_ e ain_shaded_ ac ion.py
a iable_ e ain_back acking.py
Tes cases cases_shaded_ ac ion.cs
cases_bac acking.cs
Py hon es s es _ a iable_ e ain_shaded_ ac ion.py
es _ a iable_ e ain_back acking.py
Table 1: O e iew o supplemen a y iles.
2 bi acial_ adiance ou pu
The shaded ac ions calcula ed using bi acial_ adiance o he wo alida ion cases in he pape
(Sec ion III B) a e p o ided as CSV iles. These iles allow u u e s udies o compa e al e na i e
so wa e o me hods o ou simula ion esul s. The necessa y de ails o eplica e he simula ions
a e p o ided in he pape .
The bi acial_ adiance shaded ac ion CSV iles each con ains h ee columns. The i s column
is he simula ion imes amp. The las wo columns co espond o he lowe (min) and uppe (max)
bounds o he es ima ed shaded ac ion.
3 Py hon unc ions
To make adop ion o he a iable e ain shaded ac ion me hod as easy as possible, we p o ide
a simple implemen a ion in he Py hon p og amming language. The unc ion shaded_ ac ion is
con ained wi hin he a iable_ e ain_shaded_ ac ion.py ile and calcula es he shaded ac ion
gi en inpu s o acke angles and geome y as well as he p ojec ed sola zeni h angle. Simila ly,
he a iable_ e ain_back acking.py ile con ains he unc ion back acking_ he a_1 o calcula -
ing he back acking angle o a on acke (θ1) gi en he maximum accep able shading ac ion
and geome ic pa ame e s o he ea acke . All unc ions a e documen ed using in-code docu-
men a ion ollowing he numpydoc s yle and ely only on s anda d Py hon lib a ies.
1
4 Tes cases
To acili a e implemen a ion o he shading calcula ion and back acking me hod, we p o ide a se
o es cases wi h expec ed calcula ion esul s. Tes cases o he shaded ac ion calcula ion a e
lis ed in Table 2 and isualized in Figu e 1. Tes cases o he back acking calcula ion a e lis ed in
Table 3 and isualized in Figu e 2. The isualiza ions o he es cases can be ound in Sec ion 6.
No e, o he back acking es cases, i is assumed ha he collec o s a e posi ioned pa allel o
he sun line i shading canno be a oided. Fo a desc ip ion o he a iables wi hin he ile, he
eade is e e ed o he jou nal a icle o he Py hon code documen a ion.
Case xLzLθLxRzRθRz0ℓ θs ∗
s
1 1 0.2 50 0 0.0 25 0.00 0.5 80 1.000000
2 1 0.1 50 0 0.0 25 0.05 0.5 80 0.937191
3 1 0.0 50 0 0.1 25 0.00 0.5 80 0.306050
4 1 0.0 50 0 0.2 25 0.00 0.5 80 0.000000
5 1 0.2 -25 0 0.0 -50 0.00 0.5 -80 0.000000
6 1 0.1 -25 0 0.0 -50 0.00 0.5 -80 0.306050
7 1 0.0 -25 0 0.1 -50 0.10 0.5 -80 0.881549
8 1 0.0 -25 0 0.2 -50 0.00 0.5 -80 1.000000
9 1 0.2 5 0 0.0 25 0.05 0.5 80 0.832499
10 1 0.2 -25 0 0.0 25 0.05 0.5 80 0.832499
11 1 0.2 5 0 0.0 -45 0.05 0.5 80 0.832499
12 1 0.2 -25 0 0.0 -45 0.05 0.5 80 0.832499
13 1 0.0 -25 0 0.2 25 0.05 0.5 -80 0.832499
14 1 0.0 -25 0 0.2 -5 0.05 0.5 -80 0.832499
15 1 0.0 45 0 0.2 25 0.05 0.5 -80 0.832499
16 1 0.0 45 0 0.2 -5 0.05 0.5 -80 0.832499
Table 2: Tes cases o he shaded ac ion calcula ion.
2
Case xLzLxRzRz0ℓ θs ∗
sθ2θ1
1 1 0.1 0 0.0 0.025 0.5 80 0.00 30 -10.000000
2 1 0.0 0 0.0 0.025 0.5 80 0.00 30 -8.369714
3 1 0.0 0 0.12 0.025 0.5 80 0.00 30 21.025781
4 1 0.0 0 0.2 0.025 0.5 80 0.00 30 50.031945
5 1 0.1 0 0.0 0.025 0.5 80 0.25 30 -10.000000
6 1 0.0 0 0.0 0.025 0.5 80 0.25 30 10.877359
7 1 0.0 0 0.12 0.025 0.5 80 0.25 30 50.915129
8 1 0.0 0 0.2 0.025 0.5 80 0.25 30 80.000000
9 1 0.1 0 0.0 0.025 0.5 80 0.50 30 6.338550
10 1 0.0 0 0.0 0.025 0.5 80 0.50 30 34.407694
11 1 0.0 0 0.12 0.025 0.5 80 0.50 30 80.000000
12 1 0.0 0 0.2 0.025 0.5 80 0.50 30 80.000000
13 1 0.1 0 0.0 0.025 0.5 -80 0.00 -30 -15.604247
14 1 0.0 0 0.0 0.025 0.5 -80 0.00 -30 8.369714
15 1 0.0 0 0.12 0.025 0.5 -80 0.00 -30 10.000000
16 1 0.0 0 0.2 0.025 0.5 -80 0.00 -30 10.000000
17 1 0.1 0 0.0 0.025 0.5 -80 0.25 -30 -41.380899
18 1 0.0 0 0.0 0.025 0.5 -80 0.25 -30 -10.877359
19 1 0.0 0 0.12 0.025 0.5 -80 0.25 -30 10.000000
20 1 0.0 0 0.2 0.025 0.5 -80 0.25 -30 10.000000
21 1 0.1 0 0.0 0.025 0.5 -80 0.50 -30 -80.000000
22 1 0.0 0 0.0 0.025 0.5 -80 0.50 -30 -34.407694
23 1 0.0 0 0.12 0.025 0.5 -80 0.50 -30 -1.567397
24 1 0.0 0 0.2 0.025 0.5 -80 0.50 -30 10.000000
Table 3: Tes cases o he back acking calcula ion.
5 Py hon es s
Compa ison o he ou pu s om he p o ided Py hon unc ions desc ibed in Sec ion 3 a e p o ided
in he wo es iles ( es _ a iable_ e ain_shaded_ ac ion.py and es _ a iable_ e ain_back acking.py).
Execu ing he wo es iles equi es ha all o he supplemen al iles be loca ed in he wo king di-
ec o y.
3

6 Visualiza ion o es cases
This sec ion con ains he illus a ion o he es cases in Sec ion 4.
50.0°
25.0°
Case 1
50.0° 25.0°
Case 2
50.0°
25.0°
Case 3
50.0°
25.0°
Case 4
-25.0°
-50.0°
Case 5
-25.0°
-50.0°
Case 6
-25.0° -50.0°
Case 7
-25.0°
-50.0°
Case 8
5.0°
25.0°
Case 9
-25.0°
25.0°
Case 10
5.0°
-45.0°
Case 11
-25.0°
-45.0°
Case 12
-25.0°
25.0°
Case 13
-25.0°
-5.0°
Case 14
45.0°
25.0°
Case 15
45.0°
-5.0°
Case 16
Figu e 1: Visualiza ion o he shaded ac ion es cases.
4
-10.0°
30.0°
Case 1
-8.4° 30.0°
Case 2
21.0°
30.0°
Case 3
50.0°
30.0°
Case 4
-10.0°
30.0°
Case 5
10.9° 30.0°
Case 6
50.9°
30.0°
Case 7
80.0°
30.0°
Case 8
6.3°
30.0°
Case 9
34.4° 30.0°
Case 10
80.0°
30.0°
Case 11
80.0°
30.0°
Case 12
-30.0° -15.6°
Case 13
-30.0° 8.4°
Case 14
-30.0°
10.0°
Case 15
-30.0°
10.0°
Case 16
-30.0° -41.4°
Case 17
-30.0° -10.9°
Case 18
-30.0°
10.0°
Case 19
-30.0°
10.0°
Case 20
-30.0°
-80.0°
Case 21
-30.0° -34.4°
Case 22
-30.0°
-1.6°
Case 23
-30.0°
10.0°
Case 24
Figu e 2: Visualiza ion o he back acking es cases.
5

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