Acta Polytechnica Vol. 52 No. 5/2012

Control Algorithms for Large-scale Single-axis Photovoltaic Trackers

Dorian Schneider

Institute of Imaging & Computer Vision, RWTH Aachen University, 52056 Aachen

Corresponding author: schneider@lfb.rwth-aachen.de

Abstract
The electrical yield of large-scale photovoltaic power plants can be greatly improved by employing solar trackers. While
fixed-tilt superstructures are stationary and immobile, trackers move the PV-module plane in order to optimize its
alignment to the sun. This paper introduces control algorithms for single-axis trackers (SAT), including a discussion for
optimal alignment and backtracking. The results are used to simulate and compare the electrical yield of fixed-tilt and
SAT systems. The proposed algorithms have been field tested, and are in operation in solar parks worldwide.

Keywords: single-axis solar tracker, backtracking, photovoltaic, sun tracking.

1 Introduction

The degree of efficiency of photovoltaic (PV) power
plants can be maximized by optimizing the align-
ment of the photovoltaic module plane to the cur-
rent position of the sun. Unlike fixed-tilt superstruc-
tures, where modules are placed stationary in the
field, tracking superstructures mount the modules on
carriers able to rotate along one or two axes in or-
der to maximize the electrical yield of the system.
Two different schemes for the construction of a PV-
tracker are in existence: Dual-Axis Trackers (DAT)
are provided with two degrees of freedom, which the-
oretically allows optimal module-sun alignment and
hence the maximum possible yield at any time. The-
oretically, DATs enable a rise of 30–45 % [7] in yield
compared to an optimally-positioned fixed-tilt plant
at the same location. The details for tracker perfor-
mances and extra yield have high variance, since they
depend strongly on the location where the system is
installed [4]. The benefits of DATs are achieved at
the cost of a need for more space and at the cost of
higher mechanical complexity, leading to higher costs
for construction, planning and maintainance. The
second scheme, Single-Axis Trackers (SAT), provides
only one degree of freedom, limiting the tracking mo-
tion so that perfect module-sun alignment cannot al-
ways be provided.
Although the tracking angle is limited, the system

can provide incremented yields of 10–20 % compared
to a perfectly adjusted fixed-tilt plant at the same lo-
cation. The constructional complexity is significantly
lower than for DATs, which has a positive effect on
the costs for construction and maintainance.

In order to choose the best system for a power plant
of given size and location, it is necessary to optimize
the cost-yield ratio of the system. While the costs
for construction and maintainance can be estimated

easily, expected yields can only be analysed using so-
phisticated simulations.
While several algorithms and control schemes for

high precision DAT control have been published
[1, 2, 8, 9], very few publications have handled the
control of SATs. In fact, the results found in [5, 6]
are estimations, and give not an exact solution for
the backtracking problem. For this reason, our paper
introduces control algorithms for SATs which allow
the system to be regulated to the optimal position at
any time, without the need for any additional sensor
technology or hardware, but using the calculated sun
position instead.
The problem of self-shadowing is addressed with

a high-precision, field-tested backtracking algorithm.
The results are used for a basic energy yield analysis.
The structure of the paper is as follows. In Sec-

tion 2, the nomenclature and coordinate system for
later sections are defined. Section 3 handles the basic
mathematical equations for naive tracking, allowing
the system to find the rotation angle that maximizes
the yield for a given sun position. Since the system
should avoid self-shadowing by all means, a back-
tracking algorithm that finds the optimal shadow free
rotation angle is discussed in Section 4. The results
are affiliated in Section 5 to simulate SAT yield ex-
pectations for different latitudes and superstructure
settings in Matlab/Simulink. Section 6 discusses the
results, and Section 7 concludes the work.

2 Coordinate system
We define the nomenclature and coordinate system
that is used throughout the subsequent sections. Due
to the rotational movement of the sun, a polar coor-
dinate system is used, as illustrated in Fig. 1: Here,
the azimuth angle φ is given by the angle between
the orthogonal projection of the sun vector s to the

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Acta Polytechnica Vol. 52 No. 5/2012

s

axis 
of ro

tatio
n

axis 
of ro

tatio
n

north

south

y

x

z

P

east

west

θ
90−θ

elevation angle

azimuth northsouth

zenith

Sun

sun rotation

φ

z-
ax

is

west

x-axis

A: 21
,6 m

D: 8,4 m

H:
 3

,5
 m

Figure 1: Schemes of the SAT coordinate system. Left: illustration of the sun azimuth and elevation angles.
Right: Illustration of three adjacent SATs with corresponding distances and annotations.

xy-plane and the x-axis (north-south axis). The az-
imuth takes positive values [+180°, 0°[ when the sun
is eastward, and negative values ]0°,−180°] when it
is westward. It is zero when the sun is in the south.
The zenith angle θ gives an indication for the sun
altitude by measuring the angle between the orthog-
onal projection of the sun vector s to the xy-plane
and the z-axis. The sun elevation θe angle is then
given by 90° −θ.
An optimal SAT alignment would be lengthwise,

parallel to the x-axis of the coordinate system, i.e.
perpendicular to the east-west axis. If the environ-
mental setting of the power plant prohibits optimal
alignment of the SAT superstructure, a deviation an-
gle η occurs, as illustrated in Fig. 2. As nomenclature
we define η < 0 for clockwise rotation and η > 0 for
counter-clockwise rotation. For simplicity, we do not
rotate the SAT system, but we rotate the sun az-
imuth instead. The new sun azimuth becomes

φη = φ−η. (1)

One can easily convert between polar and Carte-
sian coordinates using:

xs = r cos φη sin θe, (2)
ys = r sin φη sin θe, (3)
zs = r cos θe. (4)

2.1 Sun path mirroring

The sun starts its course early in the day in the east,
reaches the south around noon, and ends its path
in the evening in the west. The course of the sun
is always symmetric with regard to the north-south
axis. This fact is exploited to further simplify the
subsequent calculations: A variable δ is introduced,

according to:

δ =
{

+1 sun is in the east, i.e. ψη ∈ ]0◦, 180◦],
−1 sun is in the west, i.e. ψη ∈ ]−180◦, 0◦].

Using δ, the system behaviour must only be calcu-
lated once during simulation for either west or east.
The results are then mirrored to get the final result.

2.2 Plane inclination
Solar power plants are sometimes installed on non-
plane surfaces — for example in a hilly environment.
The inclination of the surface must be taken into ac-
count for precise tracking. The scheme for a non-
plane setting is illustrated in Fig. 3. To keep all cal-
culations as simple as possible, the plane inclination
is embedded into the SAT rotation angle by following
the update rule:

α = α−δ ·β. (5)

3 Basic tracking
According to Lambert’s law, the yield of a solar mod-
ule is directly proportional to the angle of incidence
γ between the sun light and the module normal vec-
tor for a given insolation intensity. The lower the
angle, the higher the irradiation intensity, and the
higher the yield. Since the loss in irradiation in-
tensity follows the function cos γ, maximal yield can
only be achieved when the sun vector is orthogonal
to the module plane, i.e. γ = 0. A single-axis tracker
(unlike a DAT) cannot always achieve orthogonality,
due to the mechanical restrictions that apply, but the
yield can be optimized during the span of one day.
This section aims to express the system rotation an-
gle α as a function of the sun position in order to
achieve maximum yield.

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Acta Polytechnica Vol. 52 No. 5/2012

0°

45°

- 45°

β

β

D

α

Figure 3: Illustration of the plane inclination angle β for hilly environments.

y

optimal
alignment

non optimalalignment

eastwest
x

η

Figure 2: Explanatory scheme for the SAT misalign-
ment angle η. The system is not optimally aligned
on the east-west axis (y-axis) for η 6= 0.

Let the module plane of one SAT be denoted as
P , cf. Fig. 1b. P can be rotated around the x-axis
with the rotation angle α, but has no other degrees of
freedom. In this way, P can be expressed in matrix
notation:

P =




1 0
0 cos α
0 sin α


 . (6)

The angle α is negative when P is inclined eastward,
positive when P is inclined westward. The origin of
co-ordinates O = [0, 0, 0]T is placed as indicated in
Fig. 4. A normal vector v of P can be expressed by:

v = detP =

∣∣∣∣∣∣∣∣
1 0
0 cos α
0 sin α

∣∣∣∣∣∣∣∣ =



0
−sin α

cos α


 . (7)

As mentioned above, the maximal yield is equivalent
to the minimal angle γ between the normal vector

v and the sun vector s. This can be achieved by
minimizing the absolute value of the cross product n
between the two vectors:

n = v×s =




0
−sin α

cos α


×



x

y

z




=



y cos α + z sin α

x cos α
x sin α


 . (8)

Minimizing the absolute value of (8) corresponds to
minimizing the area of the parallelogram that the two
vectors span, and hence corresponds to minimizing
the angle between them:

|n|2 = (y cos α + z sin α)2 + (x cos α)2 + (x sin α)2

= (y cos α + z sin α)2 + x2. (9)

From (9) it can be seen that perfect SAT alignment
(i.e. |n|2 = 0) can only be achieved when the sun is
positioned on the east-west axis (x2 = 0). The best
SAT rotation angle for a given sun position can be
found when

0 = y cos α + z sin α (10)

applies. Solving for α and applying (5) gives

α = −arctan
y

z
− δ ·β (11)

as the optimal rotation angle. Care needs to be taken
when the sun is rising or setting, i.e. z = 0.

4 Backtracking

Equation (11) allows the system to find an optimal
rotation angle for a given sun position at any time.

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Acta Polytechnica Vol. 52 No. 5/2012

w
S

F1

F2

s

H

A

D

y

x

z

p
B

O

Figure 4: Illustration of planes, lines and points needed for deriving the backtracking algorithm: A shadow is
cast from surface F1 to surface F2. By rotating SAT so that line s and line w intersect, shading can be avoided
by guaranteeing a minimal deviationfrom the optimal rotation angle.

However, the case is not considered when, for spe-
cific sun positions, self-shading may occur between
adjacent SATs. The problem is illustrated for a two-
unit SAT in Fig. 4. With more units, the problem
becomes even more important. In order to maximize
the yield, shading should be avoided by all means,
even though a shadow-free position is not optimal in
terms of the module-sun alignment, as discussed in
Section 3. Smart control algorithms should be able
to detect when SAT self-shading occurs and hence
update the current rotation angle so that no shading
can occur, while still optimizing the module align-
ment for maximum yield. This procedure is known as
backtracking. The next section derives the equations
for a shadow-free SAT backtracking control mecha-
nism that maximizes the electrical yield.

First, it is necessary to check whether self-shading
occurs for a given sun vector s. As illustrated in
Fig. 4, the module surface of two adjacent SATs of
width A and height H will be denoted as F1 and
F2, respectively. The origin of the co-coordinates is
placed at the center of gravity of F1. In order to
calculate whether a shadow is cast from F1 to F2,
one may project the corner point p of surface F1 onto
surface F2 along the sun vector s. Point p can be
expressed in Cartesian coordinates as:

p =
[
A
2 yp zp

]T
,

with yp =
D

2
cos α

and zp =
D

2
sin α.

The line s that passes through p and is parallel to
the sun vector s can be expressed as:

s : p + λs. (12)

According to equation (7), the normal vector v of
surface F1 corresponds to:

v =




0
−sin α

cos α


 . (13)

The plane that surface F2 is part of can be
parametrized by

F2,plane : v ·r− b = 0, (14)

where r is an arbitrary point on the plane and b is
a constant that must be determined. We choose r =
[0 D 0]T and rearrange for b to find

b = −D sin α. (15)

Insertion of (12) for vector r into (14) and rearrang-
ing for λ gives:

v · (p + λs) − b = 0,

thus λ =
b−v ·p
v ·s

=
b

vs
= −

D sin α
zs cos α−ys sin α

.

(16)

We are now in a position to detect whether shading
from surface F1 onto F2 occurs: By inserting (16) into
(12), one finds the Cartesian coordinates of the plane-
line intersection point B (cf. Fig. 4). Since B can lie
anywhere on the plane and must not necessarily lie
on surface F2, an interval check must be performed
to test whether B is inside the bounding box of F2.
If the check is positive, shading occurs and a new,
shadow-free rotation angle should be found.
If shading occurs, the tracker should be rotated as

far as the lower bound of surface F2 (green line in

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Acta Polytechnica Vol. 52 No. 5/2012

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
-200

-175

-150

-125

-100

-75

-50

-25

0

25

50

75

100

125

150

175

200

time [h]

an
g

le
[°

]

 

 

backtracking backtracking

tracking + backtracking
optimal tracking
sun azimuth
sun elevation

Figure 5: Tracking and backtracking angles, sun azimuth and elevation plotted for January 1st, 2012 in Berlin.
The maximal rotation angle has been clipped to ±45°. As the sun elevation angle is low, backtracking occurs
from 8 am to 10 am, and also from 14.30 pm and 16.30 pm.

Fig. 4 denoted as w) intersects with line s. Line w
can be parametrized by

w :



−A2 + µA
−yp + D
−zp


 . (17)

To find the rotation angle that avoids shading, the
intersection point between s and w must be found
by equalizing the line expressions w = s, thus


−A2 + µA
−D2 cos α + D
−D2 sin α


 =



−A2 + λxs

D
2 cos α + λys
D
2 sin α + λzs


 . (18)

This gives us a nonlinear system with three unknowns
(α,λ,µ) and three equations that we solve numeri-
cally to find two solutions for the shadow-free rota-
tion angle (including (5)) αsf:

αsf,1 = arccos
a−
√
b

c
−δβ, (19)

αsf,1 = arccos
a +
√
b

c
−δβ, (20)

where a = HDz2s,
b = H4y4s + H

4y2sz
2
s −H

2D2y2sz
2
s,

c = H2y2s + H
2z2s.

The actual values for λ and µ are of no interest
for our problem. It is intuitive that the system an-
gle αsf should be minimal. Obviously, equation (19)
does not meet this requirement, which leaves us with

2 4 6 8 10 12
0

10

20

30

40

50

60

70

80

90

100

month

yi
el

d
 (r

el
at

iv
e 

to
 t

h
e 

m
ax

im
al

 a
ch

ie
ve

ab
le

 y
ie

ld
) [

%
]

 

 

Fixed-tilt
SAT + BT

Figure 6: Comparison of fixed-tilt yields and SAT
yields for each month of a year. The results are
normed to the maximum yield that could be achieved
with a perfectly aligned module plane.

the final solution (20) for the best, shadow-free back-
tracking angle.
The proposed backtracking algorithm has been

tested in the Arizona desert with a real SAT sys-
tem. The algorithmic framework proved to be ex-
act at centimetre precision. Meanwhile, more than
100 MW of SATs worldwide are controlled using the
proposed scheme.

5 Results

Equations (20) and (11) allow us to find an optimal,
shadow-free operation angle for SAT. However, the
exact position of the sun is needed for a given time
and location. For this purpose, our paper uses the

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10 20 30 40 50 60 70 80 90
0

5

10

15

20

25

30

35

40

45

latitude[˚]

ex
tr

a 
yi

el
d

 [%
]

Figure 7: Averaged, annual extra yield of SAT compared to an optimally aligned fixed-tilt superstructure in
dependence on latitude.

sun position algorithm proposed in [10], which allows
us to determine the sun azimuth and elevation with
accuracy of ±0.0003°, without the need for any addi-
tional sensors or hardware. In fact, the sun position
algorithm can run on the same controller as the track-
ing control, which is beneficial in terms of costs and
installation complexity. Fig. 5 shows a simulation of
the SAT behaviour during one day in January for the
location in Berlin, Germany. The shadow-free angle
of operation αsf and the optimal angle α are plotted
against time. Additionally, the sun azimuth and ele-
vation are shown. The angle αsf is limited to a range
of [−45°, 45°], due to mechanical limitations that ap-
ply to real-world structures. As a matter of fact, all
SATs are provided with a rotation limit, which may
vary between 45° and 60°, depending on the design.
The system switches to backtracking as soon as the
sun rises, and continues shadow-free tracking until
about 10 am. Afterwards there is no longer any shad-
ing, and the system’s angle of operation corresponds
to the optimal angle until about 2.30 pm, when the
sun elevation has declined enough to cause shading
again.

The results were further used to simulate the rel-
ative extra yield of SAT compared to an optimally
aligned fixed-tilt superstructure at the same location,
cf. Fig. 6. The results shown here are normed to the
maximal achievable yield. It can be seen that SAT
clearly outperforms the fixed-tilt installation in the
summer months, when the sun stands high in the
sky and the days are long. In the winter months, the
fixed-tilt installation performs better because the sun
stands low in the sky, shadows are more likely and
the SAT system often switches to backtracking, hence
missing the optimal alignment to the sun. However,
the results are relative to the maximal achievable
yield and have no information about the absolute an-

nual extra yield. Fig. 7 shows the simulation results
for the total, absolute extra yield of SAT compared
to an optimally aligned fixed-tilt installation. The
plot shows the extra yield plotted against the lati-
tude. The clear-sky insolation model proposed in [3]
has been used for this purpose. It is interesting to
see how minimal gain can be achieved for latitudes
of around 50° to 60°, which correspond to central
Europe. Here, the simulation predicts an extra yield
of about 8–10 %, which is consistent with the results
found by other researchers [11]. According to Fig. 7,
the usage of a SAT system becomes very lucrative
for southern countries, where the gain may rise up to
30 %.
Finally, the effect of the spacing D between two

adjacent SATs has been investigated in this work.
Fig. 8 summarizes the results. It can be seen that
in general the best distance in terms of extra gain
and space efficiency can be achieved for a distance of
about 12 meters, when the slope of the curve flattens
out. However the best distance depends strongly on
the park layout, and generally needs to be selected
individually for each park. When dealing with diffuse
insolation (which forms a major part of central Eu-
rope’s insolation), a totally flat tracker would be the
best operational position. Theoretically, the maxi-
mum amount of light could be collected in this posi-
tion. However, to detect the current type of insola-
tion, extra sensors are required. Alternative solutions
could employ local weather broadcasts to adapt the
tracking behaviour in order to avoid external hard-
ware.

6 Conclusions
We have proposed an algorithmic framework for con-
trolling photovoltaic single-axis trackers using back-

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5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0

1

2

3

4

5

6

7

8

9

10

11

12

distance between two adjacent SATs [m]

g
ai

n
 in

 y
ie

ld
 [%

]

Figure 8: Averaged, annual extra yield of a SAT compared to an optimally aligned fixed-tilt superstructure in
dependence on the distance D between two adjacent SAT units.

tracking. The control schemes are able to maximize
the electrical yield of a SAT power plant by find-
ing the best, i.e. shadow-free, rotation angle for the
tracker in dependence on its location and environ-
mental settings. In the simulation, the SAT yields
have been compared to fixed-tilt yields, giving an in-
sight into the strengths and weaknesses of the sys-
tem. The proposed control scheme has been field
tested, and is in use in power plants with more than
100 MW output worldwide.

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