The impact of spectral variation on the thermodynamic limits to photovoltaic energy conversion

Highlights

• The Shockley-Queisser model is extended to account for all of the spectral energy.

• Thermodynamic losses are described and cell temperature is accounted for.

• Real spectra can be analyzed, accounting for geographical and climatic variation.

Abstract

Research into the fundamental limitations to photovoltaic power conversion has historically used a single predetermined set of conditions to define device performance limitations. This fails to account for the many variables involved in real-world situations. Previous work describing thermodynamic losses in solar energy conversion has typically used an analytical approach, precluding the use of real-world spectra. This paper describes a model which marries the advantages of the analytical approach with a numerical detailed balance calculation, enabling analysis of maximum attainable power conversion efficiency and associated loss mechanisms in photovoltaics under more representative conditions.

Input spectra in the model are treated as separate beam and diffuse components, both in terms of power and subtended angle. Differences in conversion show that diffuse light is effectively under maximum concentration. This does not result in an efficiency gain since the equivalent energy is instead accounted for in the Carnot loss. The Carnot limit for the diffuse portion of the spectrum is therefore lower than that for direct light.

Simulated hourly “clear sky” spectra across a year were analysed for five geographically disparate locations. Results showed that at higher latitudes narrower band gap devices have a similar maximum efficiency to those with wider band gaps, whilst at lower latitudes wider band gap devices have a slightly higher maximum efficiency. This is compounded by increased irradiance at lower latitudes. Irrespective of band gap, annual energy conversion shows little variation at lower latitudes, with greater conversion in summer being offset by reductions in winter at higher latitudes.

Introduction

The first serious examination of the physical constraints on solar cell power conversion efficiency (PCE) was carried out by Shockley and Queisser in 1961 (Shockley and Queisser, 1961), and was referred to as the detailed balance limit. This identified the upper boundary for photovoltaic device efficiency at around 33% for a band gap of approximately 1.3 eV. This approach contained a set of theoretically reasonable yet practically improbable assumptions, in particular: that the cell is both optically and electrically perfect; and that the solar spectrum is equivalent to that emitted by a Planck blackbody at 5800 K. Whilst this latter could plausibly be considered an acceptable approximation of the extraterrestrial spectrum, it fails to take into account the various atmospheric attenuations that determine the terrestrially received spectrum (Fig. 1). Early improvements to this model included the use of the Air Mass 1.5 global (AM1.5G) spectrum (Gueymard, 1995), which produces a double peak with maximum efficiencies of 33% at 1.15 eV and 33.7% at 1.45 eV. However this still fails to account for the very significant daily, seasonal and geographical variations in both spectrum and ambient temperature, and therefore in device PCE. It also fails to account for additional spectral variations caused by local and regional atmospheric fluctuations such as weather, particularly clouds, and pollution. Many of these effects have a significant impact not only on the total irradiance but also on the balance between the direct and diffuse spectral components. As demonstrated in section 3, this can have a disproportionate impact on the PCE of concentrator devices in particular.

More recent modifications have attempted to redefine the power conversion efficiency in terms of fundamental thermodynamic theory using an analytical rather than numerical approach (Dupré et al., 2016, Hirst and Ekins-Daukes, 2011, Markvart, 2016, Markvart, 2008, Markvart, 2007). This enables identification of the energy loss mechanisms, in addition to calculation of the PCE. However, these analytical methods require the use of a Planck blackbody spectrum, rendering them unsuitable for considerations of terrestrially bound energy generation limitations.

The model presented in this paper combines the key advances from the analytical approach outlined by Dupré et al; and Hirst and Ekins-Daukes (Dupré et al., 2016, Hirst and Ekins-Daukes, 2011); with the numerical method introduced by Shockley and Quiesser (Shockley and Queisser, 1961). The advantage of this model is that it enables numerical calculation of both the PCE and the intrinsic losses for any given band gap range, input spectrum and initial device temperature.

Standard test conditions (STC) for solar cell device testing specify that the cell should be held at 25 °C and illuminated normally and uniformly using the AM1.5G spectrum (BSI Standards Publication, 2019). Whilst this has the advantage of enabling direct comparison between devices both within a technology and between different technologies, its utility as a simulation (and hence predictor) of real-world behaviour is open to debate. There are two significant flaws to this approach, namely: problems surrounding the validity of the conditions themselves along with the associated solar simulator setup; and the questionable value of measuring devices solely under a single spectrum and temperature regarding the real world application of such metrics.

Typical simulator setups fail to accurately simulate real usage in three key ways. Firstly, cells under outdoor illumination are rarely actively cooled. The AM1.5G spectrum has a total power value of approximately 1 kW/m². Even for a cell operating at the nominal maximum 33% efficiency, this will cause a dramatic rise in temperature (e.g., Dupré et al., 2016). Solar cells are temperature-sensitive (with typical absolute efficiency losses around 0.5% per 1 °C increase), hence the device temperature has a significant impact on the resulting device efficiency (Dupré et al., 2016, Dupré et al., 2015). Cells under AM1.5G illumination with no active cooling have experimentally been demonstrated to reach temperatures of 25–45 °C above ambient (e.g., (Faiman, 2008, Kurnik et al., 2011)). Secondly, devices are measured under direct illumination. As the solar zenith angle for an air mass of 1.5 is 48.2° this in effect simulates a module angle of 41.8° from the horizontal. Solar modules are normally installed at a range of angles, with the optimum angle being a function of latitude, local topography, climate and weather patterns (Calabrò, 2009, Gharakhani Siraki and Pillay, 2012), with the practically achievable angle often being different from this (e.g. rooftop installation). In the UK for instance the ideal angle is usually nearer 30–35°, meaning that light collection for a real-world AM1.5G spectrum would be less effective and the performance of devices optimised for this spectrum effectively reduced. Finally, the simulated AM1.5G spectrum does not accurately represent the mixture of direct and diffuse light present under real outdoor conditions. As we will demonstrate, this can be important in some circumstances because of the thermodynamic differences in the interaction of a real device with diffuse and direct beam light.

For locations with minimal atmospheric disturbance, the daily and yearly variation in spectrum for a given location can be calculated to a reasonable degree of accuracy relatively easily using basic orbital information and standard atmospheric compositional data. Analysis of cell performance and losses for such “clear sky” spectral data is given below for five different global locations. Obtaining a complete picture of the spectrum at any given time however requires details of a variety of atmospheric variables such as haze, degree and type of cloud cover, pollution, gas concentrations etc. Whilst measurement of devices under all possible conditions is neither practical nor particularly helpful, use of a range of location-typical spectra would help to identify how a given device would perform under different conditions and in different locations. An important consideration, and something that few groups have examined in any detail, is the effect of an increased diffuse component. This is particularly important in locations such as the UK, where upwards of 50% of the total available light can be diffuse (Armstrong and Hurley, 2010, Gibson et al., 2017). The model presented here enables facile analysis of the impact of these types of spectral variation where suitable spectral data can be obtained.