CT: We present a map-based method for estimating the yearly solar irradiation received by a PV modules
mounted on a sun-tracker with a single axis of movement. The method finds the annual in-plane irradiation and
calculates the optimum angle of inclination for a vertical-axis system with inclined modules or an inclined-axis
system with the plane of the modules parallel to the axis of rotation. The approach is applied to the PVGIS climate
database extending the European Subcontinent. Results indicate that optimised 1-axis tracking systems performs
almost as well as a full 2-axis sun-tracking system. We also demonstrate that the use of such tracking systems is
particularly advantageous at high latitudes (above 60� North), allowing tracking systems there to have the same yield
as systems placed in a Central Europe.
Keywords: sun-tracking, solar radiation, PV system


1 INTRODUCTION

Photovoltaics is at present experiencing very rapid
growth in several European regions with widely varying
climate from Southern Mediterranean, almost desert-
like conditions to more cloudy temperate conditions
found in Germany, the Netherlands and the Czech
Republic. This growth has sharpened interest in strategies
to maximize the energy output that can be obtained from
a given set of PV modules. One increasingly popular
strategy is to mount the PV modules on movable supports
that adjust the tilt and orientation of the modules during
the day to increase the amount of sunlight falling on the
modules. These are known as tracking mountings since
they let the PV modules follow the path of the sun
exactly or approximately.
Such tracking systems can increase the energy yield
for a PV system compared to a fixed mounting of the
same nominal power. However, tracking systems tend to
be more expensive than fixed systems. This is due both
to the increased cost of construction of mountings with
moving parts and motors, and to the increased
maintenance costs (fixed mountings rarely need any
maintenance at all). The choice of which mounting type
to use then depends both on the increase in energy yield
(and hence income) and on the price of tracking systems
relative to fixed ones.

In previous publications we have described a method
for calculating the energy yield from a PV system at any
location in large geographical areas [1,2,3], and applied
this system to make a comparison between fixed
mounted systems and 2-axis tracking systems [4].
However, two-axis tracking systems, which position the
modules so that the direct solar radiation is normal to the
plane of the modules, are by no means the only option
for sun-tracking systems. Other systems use movement
around a single axis to achieve an approximate tracking
of the position of the sun. Such systems are simpler to
construct and hence lower cost. The question then
becomes whether the two-axis tracking yields enough
extra energy to justify the increased cost. The present
paper will not deal with the cost issue, but will focus on a
calculation of the yields of a number of different 1-axis
tracking options and compare these to the fixed south-
facing mounting and to the 2-axis tracking option.
The paper is organized as follows: in section 2 we
present an overview of the method using for calculating
the in-plane irradiation for an arbitrary plane in any
location o Europe. Then follows an overview of a
number of different 1-axis tracking options and how the
sun-tracking of each option works. Section 3 presents the
results for a fixed system and compares this to the
various 1-axis tracking systems. An optimization
calculation of the 1-axis tracking options is also
presented. We then discuss the relative merits of the
various 1-axis options compared with full 2-axis
tracking. Finally section 4 presents the conclusions and
outlook for further work.

2 METHODS


2.1 Calculation of the solar irradiance through an
arbitrary plane at ground level
We have previously described the method for
calculating the solar irradiance impinging on a plane of a
given orientation (azimuth) and inclination at any
location in a geographical region at a given time and day
during the year [1,2]. Therefore only a brief description
will be given here.
The approach is based on a data set containing long-
term monthly averages of global and diffuse horizontal
irradiation, measured at ground stations, together with
information about the monthly average turbidity of the
atmosphere in the form of Linke turbidity. The ground
station irradiation data were obtained from the European
Solar Radiation Atlas [5], while the Linke turbidity
values have been described in [6].
Using the Linke turbidity values and the elevation,
we estimated the clear-sky global and diffuse irradiation
for a typical day of each month [5,1]. For ground stations
with irradiation measurements the clear-sky coefficients
for beam and diffuse radiation were derived. To extend
that result to other locations, a spatial interpolation was
performed on the point values of coefficients as well as
of Linke turbidity. The interpolation was made using 3-D
regularized spline with tension [7, 8], resulting in set of
24 maps of long-term monthly average of clear-sky
coefficients and 12 maps of Linke turbidity.
These spatial datasets were used as input for the solar
radiation model described in [1] to calculate the T. Huld, M. �ri, T. Cebecauer, and E.D. Dunlop, Optimal mounting strategy for single-axis tracking non-concentrating PV in Europe.
Proceedings of the 23
rd
European Photovoltaic Solar Energy Conference and Exhibition, 1-4 September 2008, Valencia, Spain (preprint).

2
irradiance at any time during the typical day in the
month. It should be noted that the irradiance at a given
time during the day represents the long term monthly
average at that time assuming the clear-sky index and
diffuse fraction stays constant during the day.
The total daily irradiation on a plane (fixed or
moving) can then be calculated from the irradiance using
numerical integration over the time from sunrise to
sunset. The time step chosen for the numerical
integration is 15 minutes.
All the methods used to calculate the irradiance or
irradiation have been assembled into the GRASS module
r.sun, which calculates a map of irradiance/irradiation for
a given time and/or day. r.sun also has the ability to read
information about terrain horizon height around each grid
point. Using this it is possible to calculate the effect of
shadowing from surrounding terrain features [9].

2.2 Axis rotation angle strategies
Depending on the type of tracking mounting, the
rotation of the axis will follow a certain mathematical
relation in order to let the PV modules capture the largest
amount of radiation coming directly from the sun or from
the circumsolar disk. This is not necessarily the optimum
strategy, as it depends on having a sufficient fraction of
the global irradiation coming from the beam or
circumsolar components. For example, if the modules are
placed in a location where nearby hills or mountains
block the direct sunlight for a part of the day, the
optimum orientation of the modules during these periods
would be fixed in the horizontal position. However, this
is outside the scope of the present investigation, and we
will in the following assume that the tracking system
tracks the sun in a way that always minimizes the angle
between the normal of the module plane and the direction
to the sun. In this paper we focus on three tracking
configurations:
-

2-axis tracking;
-

1-axis tracking with vertical axis and optimal
inclination of modules;
-

1-axis tracking with north-south oriented optimally-
inclined axis.
Let A
0
be the azimuth angle of the sun at a given
time, reckoned from south, and positive towards east. Let
h
0
be the elevation of the sun at the same point in time. In
a Cartesian (x,y,z) coordinate system with the x-axis
pointing south, the y-axis pointing east and the z-axis
pointing up, the unit vector pointing in the direction of
the sun will have coordinates
( ) ( )
( ) ( )
( )












0
0
0
0
0
0
0
0
0
sin
cos
sin
cos
cos
h
h
A
h
A
=
z
y
x
=
v

(1)

Similarly, for a PV module with orientation angle
and inclination the normal to the surface is given by:
( ) (
)
( ) (
)
(
)











=
z
y
x
=
n
2
/
sin
2
/
cos
sin
2
/
cos
cos

(2)

For a two-axis tracking system, the relation is trivial:
=A
0
, =/2-h
0
. For a system with a vertical rotating
axis, the rotation angle is likewise given as =A
0
. To
calculate the optimum rotation angle for a rotating
inclined axis aligned in the north-south direction and
with inclination angle (angle that the axis makes with
the horizontal plane), the unit normal of the plane is
given by:
( ) ( )
( )
( ) ( )





=
n
i
cos
cos
sin
cos
sin

(3)

The angle between the normal and the direction of
the sun is then given by:
( )
( ) ( ) ( ) ( )
( ) ( )
(
)
( ) ( ) ( )
0
0
0
0
0
0
cos
sin
sin
cos
sin
sin
cos
cos
cos
cos
h
A + h
+ h
A v
n
= i
= (4)

The minimum angle is found for maximum cos( ).
Thus we can find the optimum by differentiating with
respect to and setting equal to 0, which gives:

( )
( ) ( )
( ) ( ) ( )
( ) ( ) h
+ h
A
h
A
= opt
cos
sin
sin
cos
cos
cos
sin
tan
0
0
0
0
0

(5)

2.2 Calculation of optimum inclination angle
Just as