1.1 Tethered aerostat

For an aerostat to support PV devices, this has to be able to produce enough lift via its buoyancy to overcome its weight, the weight of the solar cells plus any control system and that of the tether, still leaving enough margin to produce an appropriate tension in the tether to avoid excessive sag.

Neglecting any aerodynamic lift which could be generated by the shape of the Aerostat, the lifting force due to the aerostat buoyancy is:

B = (pMrpeJgVol (3)

where Vol is the volume of the aerostat and pair and pgas the densities of air and gas (helium or hydrogen can be used) filling the aerostat envelop at the specific conditions of operations (e. g. pressure, altitude), and g is the gravity acceleration (9.81m/s2). Here it is assumed that there is a negligible pressure differential between inside and outside the aerostat envelope, and for simplicity is also taken that the whole volume of the envelope is occupied by the gas (i. e. ballonets for altitude control completely empty).

Fig. 5. Schematic configuration of an Aerostat for Electrical Power Generation, as a gimballed tethered balloon – the grey area represents PV cells cladding. |

Typically aerostats have streamlined bodies to reduce the aerodynamic drag, however when such shaped aerostats are moored they then tend to rotate to the oncoming flow direction like a weathervane. Here the aerostat is required to maintain its orientation towards the Sun, therefore a spherical shape pointed through a system of gimbals seems more appropriate (see Fig. 5). A spherical aerostat generates more aerodynamic drag and clearly would require a more substantial structure and tether, but these issues can be tackled by its structural design (Miller & Nahon, 2007). A tethered sphere also suffers substantial vortex induced vibrations (Williamson & Govardhan, 1997). However a previous study (Aglietti, 2009) has shown that due to the non-linearity of the structural problem (mainly the sag of the tether) and the very slow frequency response characterized by a high value of damping, the force oscillations in the tether line (produced by relatively rapid force transients, e. g. gusts) are relatively small. The resultant rotations of the aerostat are only a few degrees which in turn produces a drop in the energy production of less than 1%.

Given its spherical shape, from the volume it is possible to calculate the surface area, and from this, taking an appropriate material area density, it is possible to estimate the weight of the envelope. The area density of the material for the skin can then be increased by 33% as suggested in (Khoury & Gillett, 2004), to account for the weight of various reinforcements, support for the payload etc.

With a similar approach the weight of the PV cells can be estimated by the surface covered, and assuming that a fraction у of the whole aerostat envelope is covered by the cells, knowing the area density of the cells (also here including wiring etc), it is possible to estimate the weight of the cells. Therefore the weight of aerostat and PV devices can be written as:

WAero = (l-33daero+d ceIISy) g 4яК2 (4)

where Saero and Scells are the area density of the envelope material and PV cells respectively, у is the fraction of the envelope surface that is covered by the PV cells, g is gravity acceleration and R is the radius of the balloon.

To assess the weight of the tether it is necessary to estimate the weight of the electrical conductors (taken as aluminum for this high conductivity over mass ratio) plus that of the strengthening fibers (e. g. some type of Kevlar). The size of the required conductor can be estimated from the electrical current (that is the ratio between the power generated by the PV devices on the aerostat and the transmission voltage) and setting the electrical losses permitted in the cable to a specific value. Therefore the cross section of the conductor will be

4^ = rAI (5)

П trans V

where rAl is the resistivity of the aluminum, S is the overall length of the conductor, qtrans is the ratio between the power lost in the cable and that generated by the PV devices (that is Pgen), and V is the voltage.

The power generated by the PV system can be estimated from the area covered by the cells (that is a fraction у of the whole aerostat surface), their efficiency (ncells), an efficiency parameter (qarea) that considers that the cells are on a curved surface and therefore the angle of incidence of the sun beam varies according to the position of the cells and finally the solar flux Ф at the aerostat operational altitude that is the irradiance discussed in the previous sections:

Pgen = 4П-гїП cellaretФ (6)

Finally the weight of the conductor will be its cross sectional area multiplied by length and by its specific weight (density times g), so substituting equation (6) in (5) the weight of the conductor can be written as:

The weight of the reinforcing fibres can be calculated from the strength necessary to keep the aerostat safely moored.

The maximum tensile force on the tether can be calculated as:

T = 7 (B – WAem )2 + D2 (8)

where D is the aerostat drag force, equal to:

D = ра, УСпР2 (9)

In the above expression v is the maximum wind velocity and Cd is the drag coefficient. From the maximum expected tension in the tether, knowing the fibres strength (7u) it is possible to calculate the required cross section and from that the weight of the reinforcing fibres.

Wflb = SmST T

where 5fib is the density of the fibers and St the length of the tether. So that the overall weight of the tether will be:

W £ gr S 2 П’ cellsП’ areaAcells^ , £ S T

VVTether ~ °cond3icond ° T/2 i" ° fib^T

11 V (7

I trans gen u

Aerodynamic forces will also act on the tether line, and they will produce further sagging (see Aglietti, 2009). However this effect does not modify significantly the maximum tension in the tether that will still be at the attachment between the balloon and the tether (equation 8).

5.2 Engineering parameters

In the previous sections, the equations that govern the preliminary sizing of the aerostatic platform have been derived and these equations can be combined, for example, to design a facility with a specified power output. Overall, as the lift and the weight are proportional to the aerostat volume and surface respectively, it will always be possible to design an aerostat large enough to "fly". Here the volume as been set at 179,000 m3 (that is a 35 m radius sphere) which gives a suitable ratio between lift and drag (using helium as a gas filler would give a buoyancy of 1 MN).

In order to reduce the interference with the aviation industry and international air traffic the maximum altitude will be set to 6 km.

The values of the specific engineering parameters which appear in the equations (like for example the area density of the skin) have a crucial role in defining the size of the aerostat. In this section realistic and sometimes conservative values for these parameters will be discussed and utilized in the equations to size a viable platform.

Starting with the solar cells, there are various types available on the market. These range from light weight amorphous silicon triple junction cells (with an efficiency of up to 7%) that could be directly integrated on the skin (see for example Amrani et al., 2007), with a mass penalty that could be as low as 25 g/m2, to heavier but more efficient cells (e. g. TripleJunction with Monolithic Diode High Efficiency Cells (www. emcore. com) efficiency 28%), which require some rigid backing and could be used with a mass penalty that can be in the region of 850 g/m2. These types of cells could be mounted on light weight carbon fibre reinforced plastic tiles that would be used to clad part of the aerostat envelope. Although amorphous silicon cells seem more appropriate, judging by the efficiency over area density ratio, there are issues concerning the ease of installations, repairs, amount of surface available and finally costs that have to be considered. In this study, an efficiency of 15% and an overall area density of the PV cells Scells (including connectors etc.) of 1 kg/m2 will be considered.

Taking a maximum peak solar irradiation of 1.2 kW/m2, from the equations in the previous section it is possible to calculate WAero (18.9×103 kg) and the peak power generated (~0.5 MW). However to size the conductor in the tether it is necessary to set a transmission voltage V, and this should be high enough in order to reduce the losses in the cable. One option is to connect the solar arrays to obtain a voltage in the region of 500V DC and use a converter to bring it up to a few kV. However the converter will introduce some electrical losses and its weight might be an issue as it has to be supported by the aerostat (although the weight of the converter might be compensated by a lighter cable). The other option is to "simply" connect identical groups of solar arrays in series, to maintain the same current and obtain a DC voltage in the region of 1.5-3kV. The solar panels would be provided with bypass and blocking diodes and other circuitry that might be necessary to protect the elements of the system. Setting the transmission voltage at 3kV and allowing for 5% electrical losses in the cable (i. e. ntmns = 0.05) enables the cross section of the aluminium conductor and its weight to be calculated as 388 mm2 and 13.0×103 kg respectively.

Using the results in the previous section and taking a maximum wind speed of 55 m/s (3 sigma value) and using equation 9 it is possible to calculate the weight of the fibres as 2.7×103 kg, so that the overall weight of the tether will be 15.7×103 kg. It should be stressed that the 55 m/ s value is quite conservative, in fact this corresponds to the peak wind speed during a gust and due to the highly non linear behaviour of the tether system (see Aglietti, 2009) the force in the tether will be considerably smaller. On the other hand this level of conservatism is more than justified by the catastrophic effect that the tether rupture would have.

6. Conclusion

This chapter has investigated the possibility of using a high altitude aerostatic platform to support PV modules to increase substantially their output by virtue of the significantly enhanced solar radiation at the operating altitude of the aerostat.

Although the figures presented for the analysis of the radiation have been obtained for a specific set of data relative to a well defined location in the UK (and the calculations presented involve some approximations, justified by the preliminary character of the analysis). The results obtained illustrate the advantages, in terms of irradiation, of collecting solar energy between 6km and 12 km altitude, rather than on the ground. The general conclusions can be extended, with a certain degree of approximation, to other countries at the same latitude and with similar climates.

Based on realistic values for the relevant engineering parameters that describe the technical properties of the materials and subsystems, a static analysis of the aerostat in its deployed configuration has been carried out. The results of the computations, although of a preliminary nature, demonstrate that the concept is technically feasible.

As the AEPG requires minimum ground support and could be relatively easily deployed, there are several applications where these facilities could be advantageous respect to other renewables.

It is acknowledged that the concept mathematical model and its concept design are of a preliminary nature. However they do indicate that there is the potential for a new facility to enter the renewable energy market, and further work should be carried out to investigate this possibility more in depth.