**Design
of solar thermophotovoltaic power generators using a
genetic algorithm**

*John
DeSutter, Michael P. Bernardi
and Mathieu Francoeur*

*Radiative** Energy Transfer Lab, Department of
Mechanical Engineering, University of Utah, Salt Lake City, Utah 84112*

**I. Introduction
**

Solar thermophotovoltaic (STPV) power generators are conceptually similar to photovoltaic (PV) solar cells in which solar radiation is converted into electricity. However, STPVs have an intermediate layer between the sun and the STPV cell. This intermediate layer absorbs solar radiation and reradiates this energy towards the STPV cell. The intermediate layer, also called the radiator, allows for the radiation to be spectrally emitted and/or filtered in a way that better matches the cell's absorption characteristics. It has been shown that the theoretical maximum efficiency for a STPV power generator that radiates monochromatically at a frequency matching the absorption bandgap of the cell is 85.4 % [1]. A record STPV efficiency of 3.2% has recently been experimentally demonstrated [2]. Despite this important milestone, solar PV cells still largely outperform STPV power generators.

Many papers have been devoted to the design of selectively emitting radiators for potential use in TPV power generators (e.g., see Ref. [3]). However, these radiators are usually designed without considering radiative, electrical and thermal losses in the cell. Radiation absorbed by the cell with frequency smaller than the bandgap that does not contribute to photocurrent generation is a radiative loss. Electrical losses are caused by electron-hole pairs (EHPs) recombining before reaching the depletion region. Thermal losses are due to absorption of radiation with frequency smaller than the cell's bandgap by the free carriers and the lattice, non-radiative recombination of EHPs and thermalization of radiation with frequency larger than the bandgap. Thermal losses negatively affect STPV performance by increasing the dark current, due to an increase of the cell temperature, opposing the generated photocurrent [4].

In this work, optimum radiator
emission spectra maximizing STPV power output and conversion efficiency are
designed by accounting for radiative, electrical and
thermal losses. This is achieved by coupling a multi-physics model combining
thermal radiation, electrical and thermal transport [5] with the publicly
available PIKAIA genetic algorithm [6]. The STPV power generator analyzed in this
work consists of a cell made of gallium antimonide (GaSb) having an absorption bandgap
of 0.723 eV at 293 K, a radiator that has a temperature
of 2000 K, and a cell thermal management system characterized by a heat
transfer coefficient of 600 Wm^{-2}K^{-1} and a fluid
temperature of 293 K. It is shown that the spectra maximizing conversion efficiency
and power output greatly differ from each another.

**II. Theoretical
Framework**

When radiative, electrical, and thermal losses are taken into account, determining an emission spectrum maximizing STPV performance is a difficult task, since these three loss mechanisms are coupled with each other [4]. This makes an analytical solution for the optimum spectrum very difficult, if not impossible. To find this optimum spectrum, a multi-physics model combining thermal radiation, electrical and thermal transport in TPV devices, described in Ref. [5], is coupled with the genetic algorithm PIKAIA [6].

The genetic algorithm PIKAIA is
an optimization tool utilizing the concept of evolution. PIKAIA determines a
set of input parameters maximizing a user defined objective function. In this
work, the thermal spectrum emitted by the radiator is discretized into *N* bands, where the emissivity is assumed
to be uniform in a given band. The input parameters in PIKAIA are therefore the
*N* values of emissivity. A given set
of *N *emissivities
is a single individual and each emissivity corresponds to a trait of the
individual. The total number of sets is the population size. Two objective
functions are investigated, namely STPV power output and conversion efficiency.
The objective function for each set
of emissivities is computed via the TPV model, and
PIKAIA ranks each set based on objective function. Parent sets are selected
based on rank to create offspring sets that consist of traits from both parent
sets. If an offspring set produces a higher objective function than a set in
the parent generation, the set in the parent generation is replaced by the
offspring set. The population is evolved in this manner over a user specified
number of generations in order to determine a set of spectral emissivities that maximize the objective function.

**III.
Results**

The spectrum obtained from
PIKAIA maximizing STPV power output for *N
*= 104 spectral bands is shown in Fig. 1. The power output, cell temperature
and conversion efficiency for this case are 41673 Wm^{-2}, 393 K and
24.4%, respectively. The spectrum takes the form of a step function in which
all radiation below 0.685 eV and above 1.082 eV has an emissivity of zero, while all radiation between these
limits has the maximum emissivity of 1. Note that at a temperature of 393 K, the
bandgap of the GaSb cell
reduces to 0.685 eV, thus explaining why the
low-energy cutoff is at 0.685 eV instead of 0.723 eV.

Figure
1. Thermal emission spectrum maximizing STPV power output for a radiator
temperature of 2000 K, and a thermal management system having a fluid
temperature and a heat transfer coefficient of 293 K and 600 Wm^{-2}K^{-1},
respectively.

The high-energy cutoff of 1.082 eV for the optimal spectrum can be explained by analyzing Fig. 2, where the power output, the cell temperature, the current density at maximum power, the bias voltage at maximum power, and the dark current density are plotted as a function of the high-energy cutoff. Note that each curve has been normalized by its own maximum. As the high-energy cutoff increases, the current density increases due to an increasing amount of radiation absorption with energy larger than the cell's bandgap. The increased amount of radiation absorption, however, causes the temperature of the cell to increase due to non-radiative recombination of EHPs and thermalization. This results in an increase of the dark current density which, in turn, causes the bias voltage at maximum power to decrease. Since power output is the product of the current density and the bias voltage, there is a spectral limit beyond which the decrease in bias voltage outweighs the increase in current density. Beyond that limit, the power output starts to decrease with increasing high-energy cutoff.

The spectrum obtained by PIKAIA
when maximizing STPV conversion efficiency results in power output, cell
temperature and conversion efficiency of 9591 Wm^{-2}, 302 K and 38.9%,
respectively. Interestingly, this spectrum is not simply monochromatic at the
cell's bandgap. Instead, the spectrum takes the form
of a narrow step function, where radiation below 0.723 eV** **and above 0.761 eV
has an emissivity of zero, while radiation between these limits has the maximum
emissivity of 1.

Figure 2. Normalized power output, cell temperature, current density at maximum power, bias voltage at maximum power, and dark current density as a function of the high-energy cutoff of the radiator spectrum.

Figure 3 shows STPV power output obtained from different radiators: blackbody, tungsten, optimum spectrum A obtained by maximizing power output and optimum spectrum B obtained by maximizing conversion efficiency. It can be seen that the power output produced by the optimum spectrum A is nearly twice that of the blackbody and tungsten radiators, and exceeds the optimum spectrum B by a factor of four. In addition, the cell temperature of 505 K with a blackbody radiator greatly exceeds that of the tungsten, optimum spectrum A and optimum spectrum B cell temperatures of 380 K, 393 K, and 302 K, respectively. The high thermal losses for the blackbody radiator result in a very low conversion efficiency of 8.44% compared to that of tungsten, optimum spectrum A and optimum spectrum B values of 19.3%, 24.4% and 38.9%, respectively.

Figure 3. Comparison of power output obtained with various STPV radiators: blackbody, tungsten, optimum spectrum A and optimum spectrum B.

**IV.
Conclusions**

This work shows the importance of taking radiative, electrical and thermal losses into account when designing optimal STPV power generators. The next step is to examine how other system level variables such as the temperature of the fluid and the heat transfer coefficient in the thermal management system affect the radiator emission spectrum when maximizing power output and conversion efficiency.

**V.
References**

[1] N.-P.
Harder, P. Wurfel, *Semicond**. Sci. Technol. ***18**,** **S151, 2003.

[2] A. Lenert,
D. M. Bierman, Y. Nam, W. R. Chan, I. Celanovic, M. Soljacic, E. N.
Wang, *Nat. Nanotechnol.
***9**, 126, 2014.

[3] B. Zhao, L. Wang, Y. Shuai, Z. M. Zhang, *Int.
J. Heat Mass Tran.* **67**,
637, 2013.

[4] M.P. Bernardi,
O. Dupr, E. Blandre,
P.-O. Chapuis, R. Vaillon, M.
Francoeur, *Sci. Rep.*, under review,
2015.

[5] M. Francoeur, R. Vaillon, M.P. Meng, *IEEE T. Energy Conver.
***26**, 686-698,
2011.

[6] PIKAIA: http://www.hao.ucar.edu/modeling/pikaia/pikaia.php. May 10 2015.

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