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Mathematical modelization of an azimuthal -elevation
tracking system of small scale heliostat
M.Debbache1, A.Takilalte 2, O. Mahfoud 3,H. Karoua4, S. Bouaichaoui 5, M. Laissaoui 6, A. Hamidat 7
Centre de Développement des Energies Renouvelables, CDER,
BP 62 Route de l’observatoire Bouzaréah, 16340, Algiers, Algeria
m.debbache@cder.dz
a.takilalte@cder.dz
o.mahfoud@cder.dz
h.karoua@cder.dz
s.bouaichaoui@cder.dz
m.laissaoui@cder.dz
a.hamidat@cder.dz
Abstract—The heliostat is an assembly of mirrors or a single
mirror above a pedestal. It is oriented mechanically toward the
displacement of the sun. It have a relative position depended to
the sun and the tower position. For the high ratio of reflection
solar rays to the top of a high tower, where the incident
solar energy is converted to thermal energy, which is used to drive
steam turbines and produce electricity. Heliostat has two
motions, an azimuthal and an elevation motion. In this paper we
have established a mathematical modelization method to define
the important control parameters which is the rotational speed
and torque engine must be provided by an azimuthal-elevation
tracking system to guide a small scale Heliostat.
Keywords—Solar; Sun; Heliostat; tracking system; Azimuth;
Elevation;
I. INTRODUCTION
The huge world demand of the electricity, and the
catastrophic effect of pollution on environment due by the uses
of the fossil energy sources and the limitation of this sources,
push us to look for a friendly a renewable energy sources to
product the electricity. The sun one of this sources, many
technologies used to exploit this source of energy such us the
central solar tower [1].
The central solar tower is an infrastructure used to produce
the electricity by the concentration the solar reflected rays at
the top of the tower, where an absorption system used to
convert this rays to the heat energy to turn drive stream turbine.
The reflection of sun rays is due by the heliostats. The heliostat
is a machine has a reflection area, it has a studied position in
the field of the central solar, used to follow the sun motion by
a tracking system [2] [3].
A many types of solar tracking system with different
accuracies exist , the tracking system can be implemented by
using one-axis with higher accuracy, two-axis sun-tracking
systems, in this work a mathematical modelization of a
heliostat with biaxial tracking system has been presented,
reserved to guide a heliostat by two independent stepper
engines. An applied example has presented in this work to
define the torque provided by the stepper engines of a prototype
of a heliostat designed by SolidWorks at Renewable Energy
Development Center (CDER -Algiers).
II. DESIGN MODEL
A proposed design is a prototype of a heliostat have been
designed by SolidWorks at Renewable Energy Development
Center (CDER-Algiers).it have 1m² reflection area ,1 m of
high , 7.5m of tower’s distance and 0° of facing angle, used in
a central solar of tower , have 10 m of tower’s high guided by
an azimuthal- elevation tracking system. The elevation
mechanism is a screw-nut system controlled by a stepper
engine where the screw is 0.025 m in diameter, 0.005m in
thread step and 0.7m in length. The azimuthal mechanism is a
screw- gear system controlled by a second stepper engine. The
gear is 0.15 m and screw is 0.03m in diameter and 0.008m in
thread step “Fig. 1”.
Fig. 1. Heliostat design model.
III. MATHEMATICAL MODELISATION OF AN AZIMUTHAL-
ELEVATION HELIOSTAT TRACKING SYSTEM
A. Sun angle
All configuration of heliostat tracking system bases on the
Sun position, where defined by two principal’s angles, altitude
angle (α) and an azimuthal angle (A), the “Fig. 2” presents the
coordinate system attached to the center of earth and her
surface.
Fig. 2. Locating the sun-related coordinate system center of the earth and
surface of the earth [3].
Fig. 3. Angles of AE tracking system [1].
: The sun vector is defined by the time angle and
decline angle“Fig. 3”.
Where the decline angle equal [4]:
(1)
N: the rank number of days in the year. For example: N=1
correspond the first January and N=42 correspond the
eleventh of February.
The time angle equal: (2)
TSV: The solar time in 24 hours, it can written by [5] [6]:
(3)
Where is the time given by the clock, is the deference
in time to the Greenwich line, is the site’s longitude and
is the correction of time equation.
(4)
Where: ) (5)
In the position situated in the earth surface at the
altitude angle , the position of the vector on the
coordinate system attached to surface of the earth defined by
the altitude angle and the azimuthal angle of the sun
“Fig. 3” [3].
Where:
(6)
(7)
If:
B. Heliostats angle
The heliostat position to solar tower is defined by the front
angle and focal angle which have a relationship with
the distance between a heliostat and tower “Fig. 4” [3] [7].
(8)
Whereis the tower’s high and is the heliostat’s high.
To define the configuration of the azimuthal-elevation tracking
system in any position of a heliostat in solar central field, a
coordinate system attached to the underground and heliostat
surface plans. Which the normal vector of the heliostat
is defined by the angle azimuthal angle and the
elevation angle , this angles can be derived exclusively
by a reflection law, concerning the position of the sun vector,
the vector of target position at the tower and the normal vector
of the heliostat [8].
(9)
(10)
Fig. 4. Incident and reflected rays [1].
C. The rotation velocities
The heliostat states have been determinate by the
azimuthally and the elevation angle, where the variation of the
two angles have been discussed in the term of the time
necessary to refreshing the value of this parameters for
deferent position in solar central field and deferent annual
states. “Fig. 12,13” present an example the distributions of the
azimuthally and the elevation rotation velocity for the heliostat
have 7.5 m of distance from the solar tower and 0° of facing
angle. TRCPE is the time necessary of the variation of the
elevation angle and TRCPA for the azimuthal angle, we define
this tow factor like a step time of de variation of the position
to define the elevation an azimuthal velocity. Where maximum
of TRCPA is almost 90 seconds at two day instance between
8H00 to 9H00, and 16H00 to 18H00, and his minimum is at
between 12H to 13H00 with the value of 50 seconds. For
TRCPE increases from 25 seconds in the morning and get the
maximum between 12H00 to 13H00, and it starts decrease in
afternoon to return to same value of the starting at the morning.
“Fig. 5”.
(11)
(12)
Fig. 5. The variation of TRCPE/TRCPA.
Then the speed provided by the elevation and azimuthal
stepper engine is defined by:
(13)
(14)
Where is the length of heliostat reflection area and is a
screw step “Fig. 11”.
D. Torque engine
The start moving of the heliostat requires a value of a
torque engine greater than the loads value applied by the
heliostat structure in all instance of changing the mirror
position obtained by the synchronizing of the elevation and
azimuthal mechanism in motion. The elevation mechanism
moves the reflection area by screw- nut system controlled by a
stepper engine where the torque in up and down moving is
defined by the equations [9] [10]:
Down:
(15)
Up:
(16)
Fig. 6. The elevation mechanism.
Fig. 7. The model of elevation mechanism.
Fig. 8. The azimuthal mechanism.
Fig. 9. The model of azimuthal mechanism.
is the elevation force, is the screw step of the elevation
mechanism, is the diameter of screw, is the friction
coefficient and is the security factor mostly choose it equal
(1.2~1.3) “Fig. 6,7” .
For the azimuthal mechanism the torque engine is defined
by the same equation of the moving up in elevation mechanism
but here the active force is the azimuthal force acting in screw-
gear system used), the torque applied in the opposite way
to return to the start point is the maximum torque can be
provided by the azimuthal engine with maximum velocity
“Fig. 8, 9” [10].
(17)
The “Fig. 10, 11” presents a simple model was estimated to
define the forces, which are the elevation force and the
azimuthal force, can be applied to move a heliostat. The
application of the first Newton low gives:
(18)
Where:
(19)
Fig. 10. Kinematic model of the elevation heliostat tracking mechanism.
Fig. 11. Kinematic model of the azimuthal heliostat tracking mechanism.
With applying the same method to define the maximum value
of the azimuthal force, the results were:
(20)
That gives: (21)
(22)
The maximum elevation force equal the potential load value
where the heliostat reflation area take the horizontal position
which called the security position where the elevation angle
equal π/2, and for azimuthal force the maximum being at an
elevation angle equal π/4 .we define the effective azimuthal
force () acting in screw-gear gives from the equivalent
torque applied in screw-gear system where:
= (23)
(24)
E. The power engines
When the variation of the torque engines the rotation
velocities are defined then the total power engines is:
(25)
(26)
IV. RESULTS AND ANALYSE
We have established the Matlab programming codes to
define the variation of the azimuthal and elevation angles,
velocities and torques and power engines for the proposed
heliostat design in the solstice summer day in Ghardaïa where
the latitude angle is 32.4deg and the longitudinal angle is
3.8deg [10] [11] , the results give the range of the variation of
the tow angles, [-55.78: 41.53deg] for the azimuthal angle,
[29.3:55.36deg] for the elevation angle in down moving and
[55.36:39.27deg] in up moving “Fig. 12, 13”.
The “Fig.14,15” shows the variation of the engines velocity
in the range of [0.021:0.032rd/s] for the azimuthal motion with
a mean value 0.026rd/s, and [0.005:0.027rd/s] for the elevation
motion with a mean value 0.0188rd/s, the mean value
equivalent in degrees to 1.5deg/s for azimuthal velocity and
1.08deg/s for the elevation velocity, this results proves with the
commercial engines properties that the value 0.9deg/s is the
best velocity choice for the high accuracy.
The “Fig. 16,17” presents the variation of the azimuthal and
elevation torque engine, where the azimuthal engine starts
moving the heliostat reflection area with maximum value
(7N.m) for reason of the passing by the critical position at
elevation angle value 45deg in moving down motion where
the potential effect is the maximum. A low value in the
instances between [12H00/13H00] where the reflection area
position far to the critical position and after starts up to the final
value (4N.m) at the instance 18H00 passing by the critical
position in moving down motion.
For the elevation torque, the engine works by two phases,
the first phase when the engine moves down the reflection area
with a start torque value (3N.m) to the low value (1.75N.m) at
the instance 13H00 where the elevation angle get the
maximum (55.36deg) the engine starts enter in the second
phase of moving up of the reflection area with a value of torque
greater than the values of the first phases (3.5N.m) and
continues increase to the value (4.5N.m) at the final instance
18H00.in this phase the engine resists the maximum loads
applied by the heliostat reflection area.
Fig. 12. The variation of the heliostat azimuthal angle.
Fig. 13. The variation of the heliostat elevation angle.
Fig. 14. The variation of the azimuthal velocity.
Fig. 15. The variation of the elevation velocity.
Fig. 16. The variation of the azimuthal torque engine.
Fig. 17. The variation of the elevation torque engine.
V. CONCLUSION
The define of the control parameters of the heliostat tracking
system based on the technological and design solutions
implanted in this system and the mathematical modelization
method. This work is a particular example to define the control
parameters, which are the rotational speed and the torque
engine. Where we established a mathematical modelization
method established for a small scale heliostat controlled by an
azimuthal-elevation tracking system.
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