# From ideal to realistic compact hybrid PV-CSP systems: A technoeconomic evaluation

Alexis Vossier, Stella Verne, Audrey Soum-Glaude, A. Dollet

► To cite this version:

Alexis Vossier, Stella Verne, Audrey Soum-Glaude, A. Dollet. From ideal to realistic compact hybrid PV-CSP systems: A technoeconomic evaluation. Renewable Energy, 2024, 236, pp.121380. 10.1016/j.renene.2024.121380. hal-04716789

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Submitted on 1 Oct 2024

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# Highlights

## From ideal to realistic compact hybrid PV-CSP systems: a technoeconomic evaluation

Alexis Vossier, Stella Verne, Audrey Soum-Glaude, Alain Dollet

* A techno-economic evaluation of compact hybrid PV/CSP systems is realized

* The behavior of idealized and realistic devices is scrutinized

* The power output, the power balance and the cost of these systems is discussed
# From ideal to realistic compact hybrid PV-CSP systems: a technoeconomic evaluation

Alexis Vossier<sup>a,*</sup>, Stella Verne<sup>a</sup>, Audrey Soum-Glaude<sup>a</sup> and Alain Dollet<sup>a</sup>

<sup>a</sup>PROMES-CNRS, Processes, Materials and Solar Energy laboratory, 7 rue du Four solaire, Odeillo, 66120, France

## ARTICLE INFO

**Keywords:**

Hybrid system

PV

CSP

Techno-economic evaluation

## ABSTRACT

Compact PV/CSP hybrid systems, based on the integration of both PV and CSP technologies within a single power plant, have been attracting interest in recent years, because of their potential to lower the cost of solar electricity and minimize the land footprint required. However, the objective evaluation of these technologies is complicated by the fact that they involve two separate converters, each delivering a different form of energy (dispatchable for CSP, non-dispatchable for PV). We propose here an original evaluation and optimization approach based on 3 techno-economic indicators quantifying 1) the overall conversion efficiency of these systems compared to their standalone CSP counterparts, 2) the energy production balance between PV and CSP, and 3) the extra manufacturing cost compared to standalone CSP systems. These different indicators are evaluated for two families of hybrid systems (PV Topping and PV Mirror) and for 3 geometries of concentrating optics (Linear Fresnel Lenses, Parabolic Troughs and Solar Towers). One originality of our approach lies in its ability to determine the optoelectronic properties of PV cells likely to lead to the best hybrid performance, taking into account whether or not they are able to reach their own theoretical limits. Another original feature is that the hybrid systems considered in our analysis are compared with CSP equivalents which have been independently optimized. It is shown that the integration of realistic solar cell technologies whose performances deviate significantly from their theoretical limits notably restricts the range of operating conditions for which hybrid systems could outperform their standalone CSP counterparts. We conclude this work by discussing a number of key challenges that need to be overcome in order to develop more competitive and efficient hybrid systems.

## Nomenclature

2

AOI Angle of Incidence, (°)

C Cost, ($)

CPP Cost Per Power, ($/W)

CSP Concentrated Solar Power

DHI Diffuse Horizontal Irradiance, (W/m²)

DNI Direct Normal Irradiance, (W/m²)

E Photon energy, (eV)

E<sub>g</sub> Electronic gap, (eV)

f(λ) Spectral distribution, (W/m²/nm)

h<sub>conv</sub> convective heat transfer coefficient, (W/m²/K)

L Length, (m)

LFR Linear Fresnel Reflector

P Power, (W)

<figure><img src="image_2.png" /></figure>

<footnote>*Corresponding author</footnote>
PT Parabolic Trough

PVM Photovoltaïc Mirror

PVT Photovoltaïc Topping

S Surface, (m²)

ST Solar Tower

V Voltage, (V)

X Concentration ratio of sunlight, (suns)

Z Sun zenith angle, (°)

f fraction of radiation collected

I Solar resource (W/m²) or Electrical current (A)

## Greek

α sun elevation, (°)

βs Surface tilt angle, (°)

η₀ nominal optical efficiency

γ Sun azimuth angle, (°)

γs Surface azimuth angle, (°)

λ angle between the line passing through the top of the tower and the center of the heliostat, (°)

θ incident angle of the direct beam on the reflector, (°)

ε Surface emissivity

εⱼ incident angle of the direct beam on the jth reflector, (°)

ξ Shockley-Queisser index

## Physical Constants

σ Stefan-Boltzmann constant, $5.67 \times 10^{-8}$ (W/m²K⁴)

c Speed of light in vacuum, $3 \times 10^8$ (m/s)

h Planck constant, $6.62 \times 10^{-34}$ (J s)

k Boltzmann constant, $1.38 \times 10^{-23}$ (J/K)

q electronic charge, $1.6 \times 10^{-19}$ (C)

T Temperature, (K)

## Subscripts

al arc-length

amb ambiant

beam beam

c cell

Conc concentration

conv convective

dif diffuse

dir direct

e electrical

g gap

hyb hybrid
loss loss

opt optimal

pos position

PV Photovoltaic

rad radiative

rec receiver

ref reflector

S Surface

sc short-circuit

SF Solar Field

sky sky

th thermal

tot total

# 1. Introduction

Compact hybrid systems PV-CSP have been proposed as a promising way to better exploit solar radiation [1, 2, 3] and typically involve two distinct types of solar converters, namely a) PV converters and b) CSP converters, merged into a single system. These technologies take advantage of the joint presence of both types of converter, either by redirecting incident photons to the appropriate converter, depending on their wavelength, or by making better use of residual heat and minimizing thermalization losses in the photovoltaic cells. At first glance, these hybrid systems may offer several advantages over corresponding stand-alone systems, including: a) smaller footprint; b) greater dispatch capacity; c) higher efficiency; and d) lower cost. The evaluation of compact hybrid systems has been the subject of various research projects over the last few decades: the superiority of compact hybrid systems over conventional photovoltaic and solar thermal systems was first demonstrated on theoretical grounds by Luque *et al* [1].

The performance of a PV mirror hybrid power plant (where PV cells behave both as converters and solar reflectors), based on the retrofitting of a CSP plant has been evaluated by Liew *et al* [4]. The authors take into account the spectral characteristics of the glass used, as well as the angular dependence of the sun beams, by calculating the fraction of power sent to each of the two converters over the course of a full year. On this basis, they calculate an energy output 9% higher than that of an equivalent CSP plant, and estimate that the LCOE of the retrofitted hybrid plant could be 9% lower. The authors consider mono-cristalline silicon technology, and indicate that PV conversion efficiency is an influential key parameter affecting the attractiveness of hybrid systems compared with conventional PV systems (higher PV cell efficiency leading to a lower benefit of hybridization). Yu *et al* [5] evaluate the optimal optical characteristics of the optical filters used on mirror PV hybrid systems, guaranteeing the highest possible electrical output while ensuring balanced electrical output between the two converters. The authors show that the constraint of balance between PV and CSP significantly lowers the maximum electrical power that can be generated by the system (which is comparable to, or even lower than, that delivered by a conventional PV system), but point out that the value placed on dispatchable electricity compared with non-dispatchable PV electricity (and therefore injected directly on the electrical grid with any storage possibility) could motivate the development of hybrid systems. Branz *et al* [6] discuss the various ways in which PV and CSP can be hybridized, distinguishing between systems based on spectral light splitting, and systems involving integrated PV-CSP receivers. The authors also suggest that the superiority of hybrid systems compared to their constitutive standalone PV and CSP systems will necessarily be correlated to the value attributed to the storable form of energy (i.e. heat) relative to electricity. The evaluations carried out by these authors are based on the use of ideal PV cells operating at their theoretical efficiency within the limits of the detailed balance model [7]. Hybrid performances are compared with the performances of PV and CSP systems under similar operating conditions (in terms of illumination level and temperature), and predictions do not take into account non-idealities likely to affect system performance in reality.

Many of the previous evaluations carried out were based on a specific solar cell technology, considered reliable and proven under standard operating conditions, but not necessarily optimal under the conditions characteristic of
hybrid systems (i.e., at temperature levels significantly higher than ambient (PV topping) or under significantly altered spectral conditions (PV Mirror)). Most studies have focused on retrofitting existing technologies, or developing hybrid technologies based on cell technologies adapted for conventional applications [4, 5, 2]. Fewer studies have examined the optoelectronic characteristics of semiconductor materials likely to lead to optimal performance of compact hybrid systems [6, 8, 9], and these studies generally overlook the existence of limiting mechanisms that can affect the performance of hybrid systems under real-life conditions, and therefore struggle to quantify the benefits of realistic hybrid systems compared with PV or CSP systems that are themselves penalized by operational losses. An inherent characteristic of hybrid systems, discussed by several authors, lies in the difficulty of maximizing power output while guaranteeing a satisfactory balance between PV and CSP [5, 8, 9]. The existence of limiting mechanisms, in addition to adversely affecting the performance of hybrid systems, may significantly alter the optimal characteristics of these systems, as well as the optimal operating conditions likely to lead to the best hybrid system performance. In this work, we set out to fill these gaps by identifying the optimal characteristics (in particular the bandgap $E_g$ of the PV cells) and operating parameters of realistic hybrid systems, i.e. operating at only a fraction of their theoretical performance. To do this, we first derive the performances of hybrid systems integrating ideal PV cells, before quantifying how a gradual decrease in the performance of these cells will affect the various metrics 1) the electrical power generated by the hybrid system 2) the PV/CSP production balance 3) the installation cost of these systems. We select two distinct hybrid technologies that have demonstrated theoretical interest, namely PV-topping (PVT) hybrid systems (involving an integrated dual PV-CSP receiver which consists in a high-temperature PV module also serving as thermal absorber) and PV-Mirror (PVM) technology (involving the use of photovoltaic mirrors acting as both solar cells and sunlight reflectors). Each of these hybrid technologies is broken down into 3 sub-families associated with the 3 main solar concentration technologies (LFR, PT and ST), in order to identify which operating conditions are the most suitable and what is the added value of hybridization compared with the corresponding standalone CSP technologies. For a fair comparison between the hybrid technologies considered and their standalone CSP counterparts, the latter have been optimized independently in order to derive their best theoretical performances. In a second step, we evaluate how taking into account limiting mechanisms leading to a decrease in PV cell performance results in a modification of the optimal operating conditions, as well as a possible degradation of the added value of hybrid systems compared to the standalone systems on which they are based. In the light of these results, we discuss the ranges of operating conditions likely to lead to good enough performance indicators to justify the deployment of hybrid systems. We finally provide a number of recommendations concerning the choice of cell technologies most likely to lead to sufficiently high performance under realistic operating conditions, and we raise a number of technological issues to which we have to pay close attention in order to guarantee the added value of hybrid systems.

## 2. Modelling

The models developed in this work are based upon the hybridization of each of three conventional CSP technologies (namely Linear Fresnel Receivers (LFR), Parabolic Troughs (PT) and Solar Power Towers (ST)) to include both a PV converter and a thermal converter. Two hybrid approaches are considered here:

* The *PV Mirror* approach (PVM), where PV cells are integrated onto the outer surface of the CSP solar concentrator, and act both as reflectors, redirecting below bandgap photons on the thermal receiver, and PV converters. The operation conditions of the PV cells are basically independent of the concentrating technology, since they are assumed to be operated at ambient temperature, and exposed to direct and diffuse sunlight, as in conventional PV systems.

* The *PV topping* approach (PVT), where PV cells are integrated into the thermal receiver and behave both as thermal receiver and PV converter. The operation conditions to which the cells are exposed (illumination, temperature) are dictated by the geometry of the solar concentrator to which the PV cells are integrated.

The PVM and PVT approaches are schematically depicted in Fig. 1.

### 2.1. Optical modelling

The nature and the amplitude of the optical losses may alter both the performance of the hybrid systems as well as their added-value relative to conventional PV and CSP systems. In PVT, optical losses affect the PV and the CSP converters in a similar fashion, both of them being integrated into a compact receiver located at the focus of the
<figure><img src="image_3.png" /><figcaption>**Figure 1:** Schematic drawing of the PVM (left) and the PVT (right) approaches. The PVM approach involves semi-reflecting mirrors absorbing high-energy photons, and reflecting the sub-bandgap photons onto the thermal receiver located at the focus of the concentrator. The fraction of the incident spectrum directed onto the thermal receiver is primarily dictated by the electronic gap of the PV cells used as PV mirrors. In the PVT approach, the whole solar spectrum is reflected onto an integrated PV/thermal receiver encompassing a high-temperature PV module acting as a thermal receiver.</figcaption></figure>

concentrator. Conversely, the PVM approach involves PV and CSP converters located at different positions along the optical path, thus undergoing different optical losses. Consequently, a rigorous evaluation of the electrical power likely to be delivered by each strategy requires, on the one hand, a precise knowledge of the available solar resource at the reflector and the receiver levels, and on the other hand, a detailed knowledge of the optical losses along the optical path followed by the light rays. To do so, we evaluate the typical solar resource effectively collected at the reflector and at the receiver levels for the three concentrating technologies under investigation, considering realistic solar plants run over a typical year of operation (a detailed explanation of the method used to evaluate the solar resources and the optical losses for the various technologies is given in the supplementary material). The site of Targasonne (France) is selected here as a case study. This site is characterized by a moderately high solar resource (~ 2000 kWh/m²/year) making it potentially suitable for such systems (the added value of hybrid systems compared with CSP systems being slightly higher on moderately sunny sites [10]).

### 2.1.1. Estimation of the solar resource

Three parameters encompassing the ability of the different technologies investigated to harness the solar radiation at the concentrator (reflector) level and at the receiver level will be used:

* $f_{dif}$ is defined as the ratio between the diffuse radiation impinging on the reflectors and the Direct Normal Irradiance (DNI), and can simply be written as:

$$f_{dif} = \frac{I_{dif,ref}}{DNI} \qquad (1)$$

where $I_{dif,ref}$ is the diffuse solar resource effectively collected at the reflector level.

* $f_{dir,ref}$ is defined as the ratio between the fraction of the direct radiation effectively collected at the reflector level, and the DNI available. This indicator thus quantifies the amplitude of the cosine losses, stemming from non-normal incidence of the solar rays onto the reflector surface, and is given by:

$$f_{dir,ref} = \frac{I_{dir,ref}}{DNI} \qquad (2)$$

* $f_{dir,rec}$ is defined as the ratio between the fraction of the direct radiation effectively collected at the receiver level, and the DNI available. This indicator thus accounts for both the cosine losses and the optical losses between the
reflector and the receiver. $f_{dir,rec}$ is given by:

$$f_{dir,rec} = \frac{I_{dir,rec}}{DN I} = \frac{I_{dir,ref} \times \eta_0}{DN I} = f_{dir,ref} \times \eta_0 \quad (3)$$

where $\eta_0$ is the nominal optical efficiency, which translates the fraction of the direct radiation impinging on the reflectors which is effectively absorbed at the receiver level. The nominal optical efficiency encompasses spillage losses, imperfect reflection of the incident rays on the reflector, shading and blocking losses, reflection losses on the receiver, as well as end losses (i.e., when the extremities of the receiver are not illuminated by solar radiation) for both PT and LFR technologies. The optical efficiency values for each concentrator family investigated is retrieved from [11].

The values of the main optical parameters considered for each concentrator technology under investigation are summarized in Table 1.

<table><caption>**Table 1**
Main optical parameters for each concentrator technology</caption><thead><tr><th></th><th>Linear Fresnel Reflector</th><th>Parabolic Trough</th><th>Solar Tower</th></tr></thead><tbody><tr><td>Collector surface (m²)</td><td>1,290,000</td><td>1,290,000</td><td>1,290,000</td></tr><tr><td>Concentration ratio X</td><td>37 [11]</td><td>26 [11]</td><td>1089 [11]</td></tr><tr><td>η₀</td><td>0.74 [11]</td><td>0.78 [11]</td><td>0.79*</td></tr><tr><td>f<sub>dif</sub></td><td>0.30*</td><td>0.28*</td><td>0.27*</td></tr><tr><td>f<sub>dir,ref</sub></td><td>0.88*</td><td>0.83*</td><td>0.68*</td></tr><tr><td>f<sub>dir,rec</sub></td><td>0.65*</td><td>0.65*</td><td>0.52*</td></tr></tbody></table>

## 2.2. Thermal modelling

A realistic description of the thermal losses affecting the different systems is required to evaluate the thermal power effectively available, and to determine the maximum receiver temperature likely to be reached under real operating conditions. We adopt different physical descriptions of the thermal losses depending on the concentrating technology considered: the thermal losses for both LFR and PT technologies are estimated using empirical correlations relating the power lost by convection or radiation, and the mean temperature of the heat transfer fluid. The thermal power loss for LFR technology is derived from [12]:

$$P_{loss,LFR} = S \times (u_0 \Delta T + u_1 \Delta T^2) \quad (4)$$

where $u_0$ and $u_1$ are empirical coefficients respectively equal to 0.056 W/(m²K) and 0.000213 W/(m²K²), while $\Delta T$ refers to the temperature difference between the heat transfer fluid mean temperature and the ambient.

The heat loss in the case of the PT technology is derived from [13]

$$P_{loss,PT} = \frac{S}{L_{al}} \times (0.141T + 6.48 \times 10^{-9}T^4) \quad (5)$$

where $L_{al}$ is the arc-length of the parabolic trough and $T$ refers to the absorber temperature in °C.

Heat losses for ST technology are modelled assuming convective and radiative heat losses between the receiver surface $S$ and the environment. The total thermal losses can be written as [14]:

$$P_{loss,ST} = P_{rad,ST} + P_{conv,ST} = S \times \frac{\underbrace{P_{rad}}_{\varepsilon \sigma (T_{rec}^4 - T_{sky}^4)} + \underbrace{P_{conv}}_{h_{conv} (T_{rec} - T_{amb})}}{X} \quad (6)$$

where $\varepsilon$ and $\sigma$ are respectively the receiver surface emissivity and the Stefan-Boltzmann constant, while $h_{conv}$ is the convective heat transfer coefficient computed using the physical properties of air, $T_{sky}$ is the sky temperature (calculated as $= 0.0552 (T_{amb})^{1.5}$ [15]), and $X$ is the optical concentration ratio of the ST technology.

<footnote>* the calculations are detailed in the supplementary material.</footnote>
## 2.3. PV Electrical model

The electricity generated by the PV cells is first calculated in the detailed balance limit [7], assuming ideal PV cells (ie, without any non-radiative recombination, nor series resistance losses, and assuming perfect absorption of the solar photons with energy above the bandgap of the PV cells). In this limit, the maximum electrical current extractable from the PV cells as a function of the voltage V can simply be written:

$$I(V) = I_{sc} - I_{rad}(V) \qquad (7)$$

where $I_{sc}$ is the short-circuit current, while $I_{rad}(V)$ is the diode current, arising from radiative recombination in the cell. $I_{sc}$ can be written as [16]:

$$I_{sc} = qfS \int_{0}^{\frac{hc}{E_g}} f(\lambda) \frac{\lambda}{hc} d\lambda \qquad (8)$$

where $q$ is the elementary charge and $f$ accounts for the fraction of the incoming sunlight effectively collected by the PV cells, the value of which being dependent upon the hybrid strategy considered (see next subsection for further details), and $f(\lambda)$ refers to the spectral distribution of sunlight (taken equal to AM1.5D reference solar spectrum in this work).

Finally, the maximum electrical power extractable from the PV cell, i.e. the maximal value of the product between the voltage V applied to the cell and the current $I(V)$ can simply be written as:

$$P_{PV} = \max(V \times I(V)) \qquad (9)$$

In a second step, a correction factor will be introduced to derive the electricity generated from non-ideal solar cells, as explained in section 4.

## 2.4. CSP electrical production

Realistic turbines are commonly assumed to operate at a fraction of their Carnot limit (a value of 2/3 of the Carnot limit is selected here, which is representative of large-scale solar thermal power plants operation [6]). The thermal input available is calculated as the difference between the optical power collected at the receiver level (taking into account the absorption of the PV cells in the PVM approach), the optical and thermal losses, and the absorption of the high-temperature PV cells in the PVT approach. The hybrid power is then simply calculated as the sum of the PV and CSP power outputs.

## 2.5. Combined PV/CSP electrical production

### 2.5.1. PV Mirror technology

The short-circuit current delivered by the PV cells can simply be written:

$$I_{sc,PVM} = qS(f_{dir,ref} + f_{dif}) \int_{0}^{\frac{hc}{E_g}} f(\lambda) \frac{\lambda}{hc} d\lambda \qquad (10)$$

where $f_{dir,ref} + f_{dif}$ is the solar resource collected at the reflector level, normalized relative to the DNI. The operating temperature of the PV cells in the PVM approach is taken equal to ambient temperature.

The radiative recombination current is calculated in the detailed balance limit as [17]:

$$I_{rad}(V) = \frac{2\pi}{h^3 c^2} qS \int_{E_g}^{\infty} \frac{E^2 dE}{\exp\left(\frac{E-qV}{kT_c}\right) - 1} \qquad (11)$$

where $h$ and $k$ are respectively the Planck and Boltzmann constants, $c$ the speed of light and $E_g$ the bandgap of the PV cells used. $E$ refers to the energy of photons, while $T_c$ is the temperature of the PV cells.
The power output of the CSP converter can then be calculated from:

$$P_{CSP,PVM}(E_g, T) = \frac{2\eta_{Carnot}}{3} \times \left( S f_{dir,rec} \int_{\frac{hc}{E_g}}^{\infty} f(\lambda) d\lambda - P_{loss}(T) \right) \quad (12)$$

where $\eta_{Carnot}$ refers to the Carnot efficiency of the thermal converter.

### 2.5.2. PV Topping technology

The short-circuit current delivered by the PV cells in the PV topping is given by:

$$I_{sc,PVT} = qS f_{dir,rec} \int_{0}^{\frac{hc}{E_g}} f(\lambda) \frac{\lambda}{hc} d\lambda \quad (13)$$

The temperature of the PV cells in the PVT approach is taken as a variable (together with the bandgap value of the PV cell). The radiative recombination current is calculated as:

$$I_{rad}(V) = \frac{2\pi}{h^3 c^2} q \frac{S}{X} \int_{E_g}^{\infty} \frac{E^2 dE}{\exp\left(\frac{E-qV}{kT_c}\right) - 1} \quad (14)$$

where $\frac{S}{X}$ is the effective area of the PV cells integrated onto the high-temperature receiver, and exposed to an optical concentration of X.

The power output of the CSP converter can be written:

$$P_{CSP,PVT}(E_g, T) = \frac{2\eta_{Carnot}}{3} \times \left( S f_{dir,rec} \int_{0}^{\infty} f(\lambda) d\lambda - P_{loss}(T) \right) \quad (15)$$

In the following, conventional CSP systems are also optimised by identifying the optimum fluid temperature leading to the maximum electrical power assuming optical and thermal losses similar to those affecting hybrid systems based on identical concentrator geometries.

## 3. Ideal PV cells: Results

In this section, all the results presented assume hybrid systems encompassing solar cells operated in the radiative limit.

### 3.1. Optimizing the power output

A first simple optimization strategy for hybrid systems is based upon the maximization of their power output. This is practically achieved by researching the $(T, E_g)$ combination leading to a maximum electrical power of the hybrid system relative to the optimized standalone CSP power output (the optimal PV electronic gap is searched in the range 0.3-3 eV, with a calculation step taken equal to 0.01 eV). Fig. 2 shows the optimal PV bandgaps of hybrid PV-CSP systems tailored to provide the largest electrical power output relative to conventional CSP technologies (left: LFR technology, center: PT technology, right: ST technology). The solid lines represent the optimal gap value for the PVM approach (in red) and for the PVT approach (in blue). The symbols represent the electronic gaps of specific semiconductor materials (round: GaAs, cross: Si, asterisk: InGaP). The symbols in blue (PVT approach) take into account the temperature dependence of the electronic gap of these materials.

Overall, the PVM approach shows weak variations in the optimal PV bandgap throughout the range of temperature investigated, independent of the technology of interest. The optimal PV bandgap is typically comprised between 1.37
<figure><img src="image_4.png" /><figcaption>**Figure 2:** Optimal PV bandgaps of hybrid PV/CSP systems designed to provide the highest electrical power output relative to their conventional CSP counterparts. The PV cells are supposed to operate in the radiative limit. The solid lines represent the optimal gap value for the PVM approach (in red) and for the PVT approach (in blue). The symbols represent the electronic gaps of specific semiconductor materials (round: GaAs, cross: Si, asterisk: InGaP). The symbols in blue (PVT approach) take into account the temperature dependence of the electronic gap of these materials.</figcaption></figure>

and 1.4 eV, thus coinciding with the bandgap of GaAs. The PVT approach is characterized by stronger variations in the optimal PV bandgap across the range of temperatures investigated, with optimal values ranging between ~ 1.12-1.15 eV in the low temperature range of each technology, to ~ 1.4 eV for the highest temperature values. The abrupt variation in the optimum bandgap observed at T ~ 200°C (LFR and PT technologies) and T ~ 500°C (ST technology) can be explained by the existence of two local maxima in the value of the bandgap leading to the maximum PV conversion efficiency (around 1.1 and 1.3 eV at ambient temperature). The threshold temperature at which this transition occurs is also a function of the concentration ratio of the system, which explains why the transition takes place at a significantly higher temperature in the case of ST technology (which is characterised by a concentration factor exceeding 1000 suns). Finally, the temperature range across which a specific semiconductor material is likely to correspond to the optimal bandgap is drastically narrowed in the PVT approach, due to the strong bandgap dependence on the operating temperature. This observation suggests that the choice of a specific semiconductor material for the PVT approach should be guided by the operating temperature at which the system is expected to work (such a requirement being far less restrictive in the PVM approach). Fig. 3 (left) illustrates the improvement in the maximum electrical power delivered by the hybrid PV/CSP systems relative to their (optimized) conventional CSP counterparts. The relative improvement in the power output of hybrid PV-CSP system can thus simply be calculated as:

$$ \Delta P_e = \frac{P_{hyb}(T)}{P_{CSP}(T_{opt})} \qquad (16) $$

where T and $T_{opt}$ are the Hybrid temperature and the CSP optimal temperature (for which the electrical power output of the CSP systems are maximized). For sake of clarity, the region inside which the power delivered by the hybrid systems exceed the one of conventional CSP technologies has been highlighted in purple (curves or points falling outside of this region translate electrical power of the associated Hybrid PV-CSP systems lower than their pure CSP counterparts). The right hand-side figures show the PV fraction $f_{PV}$, which is simply defined as the ratio between the electrical power output of the PV subsystem, and the total power output of the hybrid system, and can be written
$$f_{PV} = \frac{P_{PV,Hyb}}{P_{tot,Hyb}} = \frac{P_{PV,Hyb}}{P_{PV,Hyb} + P_{CSP,Hyb}} \quad (17)$$

where $P_{PV,Hyb}$, $P_{PV,Hyb}$ and $P_{PV,Hyb}$ respectively refer to the PV power output, the CSP power output and the total power output of the hybrid PV/CSP system under investigation. We arbitrarily set a target PV fraction value comprised between 0.4 and 0.6, a region highlighted in purple in the corresponding graphs. Hybrid system showing $f_{PV}$ values deviating noticeably from this target region are likely to demonstrate weak dispatchabilities (for high $f_{PV}$ values) as well as high costs (a point left for the final section of this study). Overall, hybrid systems encompassing PV cells operating in the radiative limit significantly outperform their pure CSP counterparts in terms of power output, regardless of the operating temperature or the technology considered. PVM Hybrid systems show the highest performance, with power outputs exceeding those of their CSP counterparts by 50-130%, owing to the extra solar resource these systems are likely to harness, together with their reduced optical losses. On the other hand, the balance between PV and CSP seems to be largely dependent on the technology considered and the temperature range targeted. We observe a strong imbalance in favor of PV for PVM technologies, whatever the type of concentrator (LFR, PT or ST) or the temperature range considered. In the case of PVT technologies, a more favorable balance between PV and CSP is observed for PT and ST technologies over a wide temperature range. Thanks to their characteristic bandgap, Si and GaAs solar cells appear as good PV materials candidates for ensuring balanced electricity conversion between PV and CSP. However, we must keep in mind that only ideal solar cells have been considered at this stage; furthermore, Si being fundamentally limited by Auger recombination [18] and resistive losses [19, 20], the use of silicon solar cells will practically be restricted to low and intermediate concentration technologies, such as PT).

## 3.2. Optimizing the power balance

In the radiative limit, the power optimization strategy described above leads to a noticeable improvement in the power output of hybrid PV/CSP systems relative to the conventional CSP systems from which the former are derived. However, this gain in the electrical power output may be achieved at the expense of the balance between the two subsystems (a point which is particularly clear with PVM systems, showing $f_{PV}$ values systematically exceeding 0.8). An unbalanced hybrid PV/CSP system is likely to entail large cost penalties, because of the inefficient exploitation of one of the converter involved, or to suffer from poor dispatchability (in the case of high $f_{PV}$ values). A second optimization strategy can thus be orientated toward ensuring the best power balance possible between each converter. This is practically achieved by researching the operating conditions ($E_g$, $T$) leading to PV fraction values as close as possible to 0.5.

Fig. 4 illustrates the optimal PV electronic gaps leading to the best power balance between each subsystem, for LFR (right), PT (center) and ST (right) technologies. These curves are plotted both for the PVM (red solid line) and the PVT (blue solid line) approaches. This figure also includes two candidate semiconductor materials (InGaP and CdS) selected here because of their electronic gap values consistent with the optimal bandgap curves. The temperature range across which the curves are plotted denote hybrid systems showing $f_{PV}$ values comprised between 0.4 and 0.6. The absence of data points thus underlines the inability of the specific approach (solid line) or material (symbols) to provide balanced electricity generation over the corresponding temperature range.

The optimization of hybrid systems aimed at delivering balanced power between each of its subsystem leads to noticeably different conclusions. Optimal designs of power-balanced PVM systems should integrate high-bandgap PV cells (~ 2.2-2.7 eV), as a result of the large solar resource to which these cells are exposed (owing to their ability to harness diffuse radiation). The integration of wide band gap PV cells allows a larger fraction of the incoming sunlight to be sent onto the thermal receiver, thus leading to a satisfying balance between PV and CSP electrical generation. The analysis of fig. 4 also reveals the existence of two different $E_g$ values leading to the same optimal design of power-balanced PVT systems. This phenomenon finds its roots in the fact that two distinct $E_g$ values are associated with a given PV output (an illustration of which being the famous PV maximum efficiency curve published by Shockley and Queisser in 1961 [7]). In the PVT approach, the incident solar energy which is not converted by the PV cell is assumed to be fully dissipated as heat and used by the CSP converter. The two optimal bandgap values therefore lead to identical values of both the PV and the thermal power output. Conversely, the PVM approach is based on the assumption that the unconverted fraction of the incident power is reflected back to the thermal absorber. The reflected power increasing monotonically with the band gap of the PV cell, there is only one $E_g$ value leading to a power-balanced PVM system. Several additional comments are in order:
<figure><img src="image_5.png" /><figcaption>**Figure 3:** Left: electrical power improvement of power-optimized PV/CSP systems relative to their optimized CSP counterpart for LFR (upper graph), PT (middle graph) and ST (lower graph) technologies. Hybrid PV/CSP system outperform conventional CSP systems for Δ*P*<sub>e</sub> values exceeding 1 (region highlighted in purple). Right: PV fraction *f*<sub>PV</sub> of the PVT and PVM hybrid approaches considering LFR (upper graph), PT (middle graph) and ST (lower graph) technologies. A satisfying power balance between PV and CSP is achieved for *f*<sub>PV</sub> values typically comprised between 0.4 and 0.6 (region highlighted in purple). For both set of curves, red and blue solid lines are respectively associated with optimized PVM and PVT systems, while asterisk and cross symbols depict the behaviour of GaAs and Si solar cells. The PV cells are supposed to operate in the radiative limit.</figcaption></figure>

* The power-balance optimization leads to optimal PV bandgaps which are not compatible with the most largely deployed PV technologies today (such as Si, CIGS or CdTe). Indeed, very high bandgap PV materials are still in their infancy, and are currently considered for niche markets only (such as space applications[21, 22]).
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4:** Optimal PV bandgaps of hybrid PV/CSP systems designed to achieve the best power balance between the PV and CSP subsystems. The PV cells are supposed to operate in the radiative limit. The solid lines represent the optimal gap value for the PVM approach (in red) and for the PVT approach (in blue). The symbols represent the electronic gaps of specific semiconductor materials (round: CdS, triangle: InGaP), and the temperature range across which the symbols and the curves are plotted denote $f_{PV}$ values comprised between 0.4 and 0.6. The symbols in blue (PVT approach) take into account the temperature dependence of the electronic gap of these materials</figcaption></figure>

* The practical integration of low bandgap semiconductor materials, such as GaSb, Ge or InAs, in the PVT approach, appears as highly speculative due to the limited ability of these materials to withstand extreme temperature or solar concentration levels [23, 24].

* The temperature range across which the different approaches and materials investigated are likely to provide balanced power production is highly dependent upon the concentrator technology considered. LFR and PT technologies may show balanced power production across the whole temperature range investigated. Conversely, the PVT approach based on ST technology, which is characterized by operating temperatures significantly higher than those of LFR and PT technologies, is not able to provide balanced power production above 800°C. In the case of the PVM approach integrated onto ST technology, none of the materials investigated here appear to deliver balanced power production, regardless of the target temperature. These results suggest that the quest for very high bandgap materials constitute a mandatory condition to be fulfilled to ensure balanced electricity generation using the ST technology.

* The optimal $E_g$ value in the PVM approach is likely to vary noticeably depending on the diffuse resource available: for a given bandgap value, a high diffuse solar resource will necessarily favor PV production at the expense of CSP (more diffuse photons being available for PV generation). As a result, sites characterized by a low DHI will logically require lower bandgap materials to achieve a satisfying balance between PV and thermal power output.

Fig. 5 (left) illustrates the improvement in the maximum electrical power delivered by hybrid PV/CSP systems tailored to deliver balanced electrical power production between each subsystem, relative to their conventional CSP counterparts. The PV cells are assumed to operate in the radiative limit, and the behaviour of hybrid PV/CSP systems integrating PV cells technologies consistent with the optimal PV bandgaps shown in Fig. 4 are plotted on the same figure. The right hand-side figures show the PV fraction $f_{PV}$ for ideal power-balanced hybrid systems (red and blue solid lines for PVM and PVT systems), as well as representative $f_{PV}$ values for hybrid PV/CSP systems integrating CdS (rounds) and InGaP (triangles) PV technologies.
<figure><img src="image_8.png" /><figcaption>**Figure 5:** Left: electrical power improvement of power-balanced PV/CSP systems relative to their optimized CSP counterpart for LFR (upper graph), PT (middle graph) and ST (lower graph) technologies. Hybrid PV/CSP system outperform conventional CSP systems for $\Delta P_e$ values exceeding 1 (region highlighted in purple). Right: PV fraction $f_{PV}$ of the PVT and PVM hybrid approaches considering LFR (upper graph), PT (middle graph) and ST (lower graph) technologies. A satisfying power balance between PV and CSP is achieved for $f_{PV}$ values typically comprised between 0.4 and 0.6 (region highlighted in purple). For both set of curves, red and blue solid lines are respectively associated with optimized PVM and PVT systems, while round and triangle symbols respectively refer to CdS and InGaP PV technology. The PV cells are supposed to operate in the radiative limit.</figcaption></figure>

In comparison with power-optimized hybrid systems discussed in the first section, power-balanced hybrid systems appear to demonstrate weaker improvement in the power output relative to conventional CSP systems. This translates into $\Delta P_e$ showing peak values of ~1.5 for PVM systems, depending on the concentrator technology considered, and ~1.4 for PVT systems. The drop in the power output relative to the power-optimization scheme is particularly noticeable
in the low-end temperature range. This decrease in the power is logically achieved at the benefit of the PV fraction $f_{PV}$, which is much more balanced in the design approach followed. It should be underlined that:

* In the case of the PVM approach, the InGaP technology shows unbalanced power in favor of the PV subsystem. In contrast, the power balance for the CdS technology is noticeably better, or slightly in favor of the CSP in the case of the ST concentrator technology. This observation militates for the requirement of high bandgap PV materials (~ 2.0-2.5 eV) toward achieving balanced hybrid PVM systems.

* For both the PT and the ST concentrator technologies, the power-balance optimization in the PVT approach does not allow a perfectly balanced power production across the whole temperature range. In particular, we observe a progressive drop in the $f_{PV}$ value for temperatures exceeding 360°C (PT), and 500°C (ST). Provided that the bandgaps are carefully selected, these two technologies however demonstrate $f_{PV}$ between 0.4 and 0.6 up to 500°C (PT) and 800°C (ST). The use of high bandgap solar cells, such as CdS, doesn't allow a good power balance in PVT systems, but InGaP offers a more balanced power production, and could consequently represent a promising candidate material for the PVT approach.

## 4. From ideal to realistic PV cells

The results discussed above rely upon the assumption of PV cells operated in the radiative limits. Realistic PV modules demonstrate performance which are noticeably lower, because of series resistance, non-radiative recombination, or imperfect absorption. To better account for the non-idealities affecting realistic PV cells we introduce an index, denoted $\xi_{PV}$, which basically translates the ability of a given PV cell technology to operate close to its theoretical limit [24], and that can simply be calculated as:

$$ \xi_{PV} = \frac{\eta_{PV,real}}{\eta_{PV,ideal}} \qquad (18) $$

where $\eta_{PV,real}$ refers to the practical efficiency of a realistic PV cell technology, while $\eta_{PV,ideal}$ is the corresponding ideal efficiency, calculated in the radiative limit described above. A $\xi_{PV}$ value approaching 1 indicates close-to-ideal operation of the PV cells considered. $\xi_{PV}$ values are likely to vary noticeably depending on the PV technology, but are typically comprised between 0.5 and 0.8 at the module level for the most widely deployed PV technologies today (representative values of $\eta_{PV}$ and $\xi_{PV}$ for different PV technologies are given in table 2. Note that the efficiencies reported in this table correspond to standard values, not to record efficiencies).

Fig. 6 illustrates how the integration of realistic PV cells deviating from the ideal performance lead to a shift in the optimal bandgap associated with balanced-power generation of each converter. In the PVM approach, we observe a decrease in the optimal bandgap of ~ 0.5 eV throughout the $\xi_{PV}$ range investigated, stemming from the lower ability of realistic PV cells to convert sunlight into electricity. Achieving a given PV output to fulfill a satisfying balance between PV and CSP thus require a larger fraction of the solar spectrum to be exploited and in turn, lower bandgap PV cells to be operated. In the case of the PVT approach, taking into account non-ideal PV cells brings the two branches of the curve representing the optimum bandgap closer from each other, the faster $\xi_{PV}$ moves away from unity. Because of their temperature dependence, which affects their ability to convert solar radiation efficiently at increasing temperatures, PV cells are the limiting means of conversion. Consequently, the quest for a balanced power between PV and CSP leads to electronic gaps ensuring maximum PV power for increasing temperatures.

Fig. 7 and 8 depict the decrease in the maximum hybrid power attainable considering PV cells operating between 50 and 100% of their theoretical limit (clear colors indicate PV cells operating far from the theoretical limit, while darker curves illustrate the behaviour of PV cells progressively approaching their own theoretical limit), and assuming cells designed following the power-balance optimization. Similarly to Fig. 4, the temperature extent across which the curves are plotted are characteristic of balanced PV/CSP systems showing $f_{PV}$ values between 0.4 and 0.6.

The drop in the cell' ability to approach its own theoretical limit logically translates into hybrid powers progressively decreasing from the maximum values discussed above. Overall, the temperature range across which hybrid systems surpass conventional CSP technologies narrows with decreasing $\xi_{PV}$ values. The resilience of PVT systems to decreasing PV cell efficiency also appears to be higher than that of PVM systems, owing to the ability of the former to benefit from the residual heat stemming from the imperfect conversion of sunlight by the PV cells. However,
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 6:** Optimal PV bandgaps of hybrid PV/CSP systems designed to achieve the best power balance between the PV and CSP subsystems. PV cells are supposed to operate at a fraction $\xi_{PV}$ of their efficiency in the radiative limit. The solid lines represent the optimal gap value for the PVM approach (upper subfigure, in red) and for the PVT approach (lower subfigure, in blue).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 7:** Relative improvement in the power output of PVM hybrid systems integrating PV cells operating at a fraction $\xi_{PV}$ of their efficiency in the radiative limit (left: LFR, center: PT, right: ST). The hybrid PV/CSP systems considered here are designed to achieve the best power balance between the PV and CSP subsystems.</figcaption></figure>

despite their improved tolerance to non-ideal losses in terms of power-output, the temperature range over which PVT systems show balanced power output appears to shrink with decreasing $\xi_{PV}$ (which translates into a significant drop in the maximum operating temperature allowing balanced PV/CSP production).
<figure><img src="image_12.png" /><figcaption>Figure 8: Relative improvement in the power output of PVT hybrid systems integrating PV cells operating at a fraction $\xi_{PV}$ of their efficiency in the radiative limit (left: LFR, center: PT, right: ST). The hybrid PV/CSP systems considered here are designed to achieve the best power balance between the PV and CSP subsystems.</figcaption></figure>

## 4.1. Economical model

The economical interest of the different hybrid approaches evaluated in this work is constrained by their ability to provide electrical energy at a lower cost than the individual PV or CSP devices on which they are based. A rigorous economic evaluation would require the Levelized Cost Of Electricity (LCOE) to be estimated and compared over the entire lifetime of the different hybrid plants investigated. However, some key cost components, related for instance to the integration costs, the operation and the maintenance of these systems are still largely unknown, precluding any precise estimation of this cost indicator. For this reason, we base our economical analysis on the evaluation of the cost per power (in $ per Watt) which basically reflects the total installation cost of the different systems per Watt of electrical power delivered to the grid. The evaluation is based on several assumptions:

* The hybridization costs, associated for instance with the integration of a spectral-splitting component in the PVM cells, or with the integration of high-temperature concentrator solar cells onto the thermal receivers, are neglected. Because these costs are largely unknown, the estimates provided here thus constitute an optimistic estimation in the unlikely situation where the integration costs are null.

* The cost-per-power (CPP) for the different CSP technologies is defined as:

$$CPP = \frac{P_e \sum C_e + P_{th} \sum C_{th} + S \sum C_S}{P_e} \quad (19)$$

where $P_e$ and $P_{th}$ are the electrical and the thermal power associated with a specific plant configuration, $S$ is the collecting area, and $C_e$, $C_{th}$ and $C_S$ are the specific electrical, thermal and areal cost, expressed in $/kW_e$, $/kW_{th}$ and $/m^2$ of reflector, respectively. Typical values of these cost components for the different technologies investigated are reported in table 3.

* For the PVM approach, the CPP is calculated by assuming that the mirrors, which typically represent ~ 15-30% of the total cost of the solar field in a CSP plant[25, 26], are removed and replaced by semi-reflecting PV modules. Here, the cost of these semi-reflecting PV modules is taken equal to those of conventional c-Si module technologies, which are reported to be 199 $/m^2$ at the system level in 2020[27]. Assuming that the cost of the mirrors in conventional CSP technologies amounts to ~ 20% of the total cost of the solar field [26, 28, 29, 30], the CPP of the PVM approach is calculated by modifying the solar field cost to account for the presence of PV Mirrors:

$$C_{SF,PVM} = 0.8 \times C_{SF,CSP} + C_{PV} \quad (20)$$

where $C_{SF,PVM}$ and $C_{SF,CSP}$ are the solar field costs for the PVM and CSP technologies, and $C_{PV}$ is the PV system cost (both of them being expressed in $/m^2$). The 0.8 factor accounts for the 20% drop in the cost of the solar field relative to standalone CSP technology.
* In the PVT approach, the thermal receiver incorporates PV cells acting both as PV converters and thermal receivers. Because of the extreme temperature and illumination conditions to which PV cells are exposed, we assume that the High-temperature PV modules use III-V cells, owing to their improved ability to withstand such extreme operating conditions [31] (note that low and medium concentration technologies including Si cells have been developed in the past [32], but because of the elevated temperature to which PVT cells are likely to be exposed, this material does not appear as a relevant choice due to its weak temperature behaviour [33]). The CPP of the PVT approach must account for the integration of concentrator solar cells onto a conventional thermal receiver. To do so, the cost associated with the PV system is subdivided into two components, namely

1. *the PV module cost*, which typically represents 30% of the total system cost [34], and which is assumed to vary with the typical concentration ratio to which the cells are exposed as:

$$C_{PVT,PVmodule} = C_{PV,Conc}/X \quad (21)$$

where $C_{PV,Conc}$ is the concentrator cell cost (in $/m^2$) and X is the typical concentration factor of the hybrid system considered.

2. *the remaining system cost*, which includes the labor and equipment, the inverter, the electrical and structural Balance of System, and which represent ~70% of the total system cost. This cost component can simply be written:

$$C_{PVT,PVother} = 0.7 \times C_{PV,Si} \quad (22)$$

where $C_{PV,Si}$ is the typical cost of a conventional Si PV system [27]. The total cost of the high temperature PV module in the PVT approach, expressed in $/m^2$ of reflector, can thus simply be written:

$$C_{PVT,PVsystem} = C_{PVT,PVmodule} + C_{PVT,PVother} = C_{PV,Conc}/X + 0.7 \times C_{PV,Si} \quad (23)$$

<table><caption>**Table 2**

Performance and economical data of the different PV cell technologies [27]</caption><thead><tr><th></th><th>Bandgap (eV)</th><th>η (@ 298K)</th><th>ξ</th><th>Cost ($/m²)</th></tr></thead><tbody><tr><td>cSi (PERC)</td><td>1.12</td><td>20.5 %</td><td>0.62</td><td>198.9</td></tr><tr><td>CdTe</td><td>1.42</td><td>18%</td><td>0.62</td><td>189</td></tr><tr><td>CIGS</td><td>1.08</td><td>16 %</td><td>0.5</td><td>211.2</td></tr><tr><td>Perovskites</td><td>2.22</td><td>10.4%</td><td></td><td></td></tr><tr><td>III-V</td><td>1.4-2</td><td>f(X)</td><td>~ 0.35@673K</td><td>33000</td></tr></tbody></table>

The economic value of any compact hybrid approach can be gauged through its cost-per-power, which should be lower than the cost per power of comparable *non-compact* hybrid plants. To do so, we calculate the cost ratio (CR) between the cost-per-power of compact hybrid systems and the cost-per-power of standalone CSP systems based on similar geometry designs. This cost ratio can simply be written:

$$CR = \frac{CPPhyb}{CPCSP} \quad (24)$$

where $CPC_{hyb}$ and $CPC_{CSP}$ respectively refer to the cost-per-power of the compact hybrid and the conventional CSP plants (the values of which are derived from equations 19 to 23). A CR lower than 1 is required to guarantee installation costs of the hybrid systems lower than those of their optimized CSP counterparts, from which they are derived.

Fig.9 shows the cost ratio of a PVM hybrid system based on the PT technology (the evaluations corresponding to the LFR and ST technologies are shown in figs S11 and S13 of the supplementary material), as a function of the operating temperature of the turbine and the electronic gap of the PV cells, and for $\xi_{PV}$ values taken equal to 1 (left subfigure), 0.8 (central subfigure), and 0.6 (right subfigure). Blueish colors indicate ($E_g$, T) regions associated with
<table><caption>**Table 3**
Economical data of the different CSP technologies.</caption><thead><tr><th></th><th>Value</th><th>Unit</th><th>Cost component</th></tr></thead><tbody><tr><td colspan="4">Linear Fresnel Reflector (LFR) [25]</td></tr><tr><td>Site improvement</td><td>20</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Solar Field</td><td>170</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Heat Transfer fluid</td><td>47</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Power block</td><td>1190</td><td>$/kW<sub>e</sub></td><td>C<sub>e</sub></td></tr><tr><td>Balance of plant</td><td>340</td><td>$/kW<sub>e</sub></td><td>C<sub>e</sub></td></tr><tr><td colspan="4">Parabolic Trough (PT) [35]</td></tr><tr><td>Site improvement</td><td>25</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Solar Field</td><td>150</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Heat Transfer fluid</td><td>60</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Power block</td><td>910</td><td>$/kW<sub>e</sub></td><td>C<sub>e</sub></td></tr><tr><td>Balance of plant</td><td>90</td><td>$/kW<sub>e</sub></td><td>C<sub>e</sub></td></tr><tr><td colspan="4">Solar Tower (ST) [35, 36]</td></tr><tr><td>Site improvement</td><td>16</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Heliostat field</td><td>140</td><td>$/m²</td><td>C<sub>S</sub></td></tr><tr><td>Tower+Receiver</td><td>180</td><td>$/kW<sub>th</sub></td><td>C<sub>th</sub></td></tr><tr><td>Power block</td><td>1040</td><td>$/kW<sub>e</sub></td><td>C<sub>e</sub></td></tr><tr><td>Balance of plant</td><td>290</td><td>$/kW<sub>e</sub></td><td>C<sub>e</sub></td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Figure 9:** Cost ratio of a Hybrid PVM system based on the PT technology as a function of the operating temperature of the turbine and the electronic gap of the PV cells, and for 3 different values of the η<sub>PV</sub> index (left: 1, center: 0.8, right: 0.6). The bandgaps derived in the power-optimization and in the power balance strategy are respectively plotted in grey and in red. Finally, the orange, grey and red circles indicate the (E<sub>g</sub>, T) combinations respectively leading to the minimum CR, the maximum power in the power-optimization strategy, and the maximum power in the power-balance strategy.</figcaption></figure>

CR lower than 1, where hybrid systems are likely to be more cost-effective than their CSP counterpart (the darker colors being associated with lower costs). Clear colors correspond to (E<sub>g</sub>, T) regions where the cost of both systems are comparable, while reddish colors are associated with CR exceeding 1.

Several important conclusions can be drawn from the analysis of Fig 9. First, the (E<sub>g</sub>, T) region associated with CR lower than 1 is significantly narrowed with decreasing ξ<sub>PV</sub> values. This stresses the fact that cost-effective hybrid PVM systems are constrained by the ability of the PV mirror modules to operate close to their theoretical limit. Overall, PVM systems designed following the power-optimization strategy show CR values close to the minimum cost values, and
thus the highest cost-reduction potential in comparison with conventional CSP systems. Conversely, PVM systems designed following the power-balance approach demonstrate CR values close to 1 even for a $\xi_{PV}$ value of 1. This suggests that the design of hybrid PVM systems demonstrating a balanced production between PV and CSP is likely to entail large cost penalties limiting the economical added-value of these systems.

Fig.10 shows the cost ratio of a PVT hybrid system based on the PT technology (the evaluations corresponding to the LFR and ST technologies are shown in figs S12 and S14 of the supplementary material), as a function of the operating temperature of the turbine and the electronic gap of the PV cells, and for $\xi_{PV}$ values taken equal to 1 (left subfigure), 0.8 (central subfigure), and 0.6 (right subfigure). Similar to the PVM system shown above, color gradients illustrate the cost gap with conventional PT technology, darker colors being associated with lower costs. The evaluations are realized assuming a PV module cost taken equal to the one of conventional Si technology, which means that the values of CR reported here are optimistic, i.e. lower than the values that would be obtained by considering more realistic prices for III-V cells. The bandgaps derived in the power-optimization and in the power balance strategy are respectively plotted in grey and in red (these curves being only plotted for (E<sub>g</sub>, T) combinations leading to $f_{PV}$ values comprised between 0.4 and 0.6).

<figure><img src="image_14.png" /><figcaption>**Figure 10:** Cost ratio of a Hybrid PVT system based on the PT technology as a function of the operating temperature of the turbine and the electronic gap of the PV cell, and for 3 different values of the $\eta_{PV}$ index (left: 1, center: 0.8, right: 0.6). The bandgaps derived in the power-optimization and in the power balance strategy are respectively plotted in grey and in red. Finally, the orange, grey and red circles indicate the (E<sub>g</sub>, T) combinations respectively leading to the minimum CR, the maximum power in the power-optimization strategy, and the maximum power in the power-balance strategy.</figcaption></figure>

As in the case of PVM systems, we observe a drop in the cost-ratio for increasing $\xi_{PV}$ values, translating the lower ability of practical PVT systems encompassing realistic PV technologies to offer an affordable alternative to conventional PT systems. PVT systems integrating PV cells operating at their theoretical limit ($\xi_{PV}=1$) show CR values as low as 0.6 for cost-optimized systems, while power-optimized (grey curves) and power-balanced systems (red curves) demonstrate minimum CR values close to 0.7. If the optimal bandgaps show similar values independent of the optimization strategy followed, one can notice large discrepancies in the optimal operating temperatures: while cost-minimized systems require operating temperatures as low as 100 °C (a temperature which is unrealistically low regarding the turbine operation), systems designed toward achieving balance power or maximum power generation are likely to demonstrate the lowest CR for operating temperatures in the range 280 – 400°C. For decreasing $\xi_{PV}$ values, the CR appears to increase until a minimum value of ~0.8 at $\xi_{PV}=0.6$. This observation highlights how the use of PV technologies that deviate from their theoretical limits are likely to affect the economic competitiveness of PVT approaches compared with the CSP technology on which they are based. Finally, the decrease in $\xi_{PV}$ leads to a shift in the optimal operating temperature toward higher temperatures, for both the cost-minimization and the power-maximization strategies. Conversely, the operating temperature leading to the highest power generation while ensuring a balanced power generation between PV and CSP is shown to decrease from 330°C for $\xi_{PV}=1$, to 180°C for $\xi_{PV}=0.6$. Achieving cost-effective and power-balanced PVT systems considering PV cells deviating noticeably from
their theoretical limit thus require the operating temperature of the receiver to be decreased significantly below 300°C, a temperature level practically not suitable with an efficient turbine operation.

**5. Discussion**

Judging the performance of hybrid PV-CSP systems on the basis of one single performance indicator (such as the conversion efficiency) is misleading, and could ultimately lead to poor system design preventing its efficient operation. Overall, hybrid systems designed toward ensuring the highest power output possible usually favor one converter at the expense of the other (especially for the PVM approach, which is mainly driven by PV conversion): this unbalance power production raises concerns regarding the profitability of the other converter involved, and the ability of the system to provide dispatchable power (in the case where the latter is dominated by PV production). One should keep in mind that PV + short duration battery storage has recently reached prices below 0.1€/kWh [37, 38, 34]. Hybrid PV/CSP systems showing unbalanced power generation in favor of PV are less likely to store thermal energy over extended periods of time, and constitute in this sense direct competitors of the PV + short duration battery storage combination. Conversely, ensuring balanced power generation and promoting long storage duration (for which electrochemical batteries are less competitive than thermal storage due to their high cost per energy) is theoretically possible with both PVM and PVT approaches (provided that the bandgap of the PV cell technology selected is high enough, in the case of the PVM approach). As mentioned before, high bandgap PV materials typically exceeding 2 eV remains largely unexplored, and the path toward balanced hybrid PV/CSP systems will thus require this specific family of materials to be investigated. The economical interest of PVM systems designed to provide a satisfying balance between PV and CSP is however questionable, since the CR of those systems does not drop noticeably below 1, even for systems encompassing the most efficient PV technologies (see red curves in Fig. 9). If PVT systems do not necessarily require high bandgap PV materials toward achieving balance power production, those systems should fulfill an additional constraint related to the ability of the PV cells used to sufficiently approach their own theoretical limit: inefficient PV cells are unlikely to guarantee balanced electricity production over a broad range of temperatures, and the operating temperature range across which PVT systems are balanced decreases noticeably with decreasing cell efficiency. The development of efficient HT PV cells is thus a key hurdle to be overcome, which appears as particularly challenging today because of the large efficiency gap currently separating state-of-the-art HT PV cells from their own theoretical limit ($\xi_{PV}$ values close to 0.35 being reported for III-V cells operating at 400°C [31, 8]). The cost-effectiveness of PVT systems is also called into question: the estimates provided in this study assume *Si* PV cells as HT PV converters, which is practically unrealistic due to the poor temperature dependence of this material. *III-V* cells are currently the most mature cell technology for high-temperature applications, but suffer from their high cost (typically two orders of magnitude higher than *Si* cost, which would entail CR close to 3). The successful development of PVT systems will either require to drastically reduce manufacturing costs of III-V technologies, or to develop alternative PV technologies demonstrating both reduced fabrication costs and strong resilience to high temperatures. Finally, the economical evaluation of the different systems investigated in this work did not include any *hybridization* costs, associated for instance with 1) the integration of mirrors onto the PVM modules (either through highly reflective mirrors at the rear side of the cell, or using selective coatings at the front-side) 2) the integration of high-temperature PV cells onto the thermal absorber (which include the electrical and thermal management of multiple cells connected to each other). These costs are largely unknown, and the evaluation realized here suggests that they should be kept extremely low to allow the financial viability of these systems, provided that the other technical and financial issues addressed above are fulfilled. A detailed sensitivity analysis centred around these hybridisation costs will necessarily have to be carried out in order to better understand the financial flexibility needed to guarantee the economic viability of these systems.

**6. Conclusions**

The rigorous evaluation of hybrid PV-CSP systems is inherently difficult, because the association of two different solar energy converters harnessing different types of solar radiation (diffuse and/or direct), and delivering different forms of energy (electricity and heat) does not allow any simple comparison. In this work, we've suggested, implemented and discussed an original evaluation approach aiming to provide key technical and economical informations based on 3 complementary indicators 1) the improvement in the power output of a given hybrid system relative to its optimized CSP counterpart 2) the power balance between each converter and 3) the *Cost-Ratio* of the hybrid system, translating the ratio between the installation cost of a specific hybrid system (expressed in $/W), and the installation
cost of an optimized CSP system. Based on these indicators, we then derived the range of operating conditions likely to satisfy each performance indicator for two families of hybrid PV-CSP systems (PVM and PVT). The incorporation of ideal solar cells allows a noticeable improvement in the power output of hybrid PV-CSP systems relative to their conventional CSP counterparts. This improvement appears to be more modest in the case of hybrid systems designed to achieve balanced power output between each converter, keeping in mind that the interest of combining a PV and a CSP system in a single compact plant arise, for a large part, from the possibility of reaching a good balance between both production modes. The integration of PV cells showing conversion efficiency deviating from the theoretical limit, and more representative of realistic solar cells that are commercially available, suggest that the added-value of compact hybrid system may become marginal:

* the improvement in the power output of compact hybrid PV/CSP systems relative to conventional CSP systems appears to decrease noticeably with decreasing cell performance. Overall, the PVM approach integrated into PT technology shows the highest power improvement relative to its CSP counterpart, considering non-ideal solar cells.

* the satisfaction of the power balance is achievable with the PVM approach encompassing non-ideal solar cells, provided that the electronic gaps are tailored to harness a larger fraction of the solar spectrum. In the PVT approach, the integration of non-ideal solar cells drastically narrows the range of operating temperatures across which the power generation between PV and CSP may be balanced.

* the economical benefit of compact hybridization, expressed in this work as a ratio between the installation cost of a specific hybrid system and the installation cost of its pure CSP counterpart, appears to strongly vary depending on the PV technology and on the optimization strategy followed. Taking into account realistic solar cells operating at a fraction of their theoretical limit, the economic advantage of hybrid systems over conventional CSP progressively vanishes as the efficiency of PV cells decreases. The cost ratio of PVM systems designed to ensure a satisfying power balance between PV and CSP was also shown to approach 1 even for the most efficient PV cells, stressing the limited economic advantage of this specific approach over conventional CSP.

This study does not take into account the technical constraints and the costs specific to the combination of two PV and CSP converters in a single system (hybridization costs). For instance, the thermal and electrical management of PVT modules involving hundred or thousands of PV cells will necessarily be challenging, and will have to be addressed. The performances of the spectral-splitting component in the PVM approach must also be scrutinized, and should simultaneously ensure 1) efficient spectrum-splitting 2) balanced-power 3) minimum optical losses, while keeping the cost of the optical component as low as possible. These evaluations are not in the scope of this work, but these additional constraints could narrow even more the operating conditions range across which hybrid PV/CSP may outperform conventional CSP systems. Finally, a limitation of this study stems from the fact that the instantaneous power delivered by the system, which is the metric of comparison here, fails to capture the ability of the different systems studied to better meet electricity demand at a competitive cost. Assessing the energy output of hybrid systems incorporating thermal energy storage will logically be the next step towards a more precise understanding of the added value of these systems compared with standalone PV and CSP systems.

# A. Supplementary information

## S1.1. Estimation of the solar resource

A precise estimation of the power output of the different systems under investigation requires the solar resource to be precisely assessed. A rigorous estimation of the different solar resource components harnessed by the different systems would require a site-specific estimation of the energy over an extended period of time (a 1 year period, for instance). Since we decide to focus on the power output rather than on the energy yield, we must account for the ability of each hybrid system and each PV or CSP subsystem to efficiently collect DNI or DHI, through appropriate indicators. Here, the reference AM1.5D solar spectrum is considered as a reference solar spectrum in our calculations. Each system is characterized by its ability to collect more or less efficiently the Direct Normal Irradiation (DNI) (depending for instance on the integration of 1D or 2D solar trackers), as well as the DHI. Two distinct indicators $f_{dir}$ and $f_{dif}$ are introduced to quantify the ability of each system and subsystem to efficiently collect DNI and DHI.

* $f_{dir}$ is defined as the typical fraction of the DNI effectively collected by each absorber. $f_{dir}$ is estimated on an annual basis for each technology and two distinct $f_{dir}$ values are calculated for hybrid systems, namely 1)
$f_{dir,ref}$, which translates the fraction of the direct power effectively collected at the reflector level and, 2) $f_{dir,rec}$, which quantifies the fraction of the DNI effectively reaching the thermal absorber receiver.

* $f_{dif}$ is defined as the typical fraction of diffuse irradiance effectively collected at the reflector level, relative to the DNI reference value. Its accurate estimation is necessary for technologies exploiting diffuse radiation (PVM hybrid systems and conventional PV modules).

$f_{dir}$ and $f_{dif}$ are estimated on an annual basis for each technology. The DNI and DHI data over a full year are retrieved at Targasonne (France) from PVSyst [39] and the $f_{dir}$ and $f_{dif}$ factors are derived for each technology following the methodology described below.

## S1.2. Flat plate PV

The total irradiance on an inclined PV module can simply be written [40]:

$$I_{tot} = I_{beam} + I_{dif} + I_{ref} \qquad (25)$$

where $I_{beam}$, $I_{dif}$ and $I_{ref}$ respectively refer to the beam, the diffuse, and the reflected irradiance impinging on the PV module. For the sake of simplicity, the reflected radiation, which is highly dependent upon the albedo of the site selected, is neglected here and $I_{ref}$ is thus taken equal to 0. $I_{beam}$ can be calculated as:

$$I_{beam} = DNI \times \cos(AOI) \qquad (26)$$

where AOI is the angle of incidence defined as the angle between beam irradiance and the normal to the PV module[41]:

$$AOI = \begin{cases} \pi & \text{if } a < -1 \\ 0 & \text{if } a > 1 \\ \arccos a & \text{if } -1 \le a \le 1. \end{cases} \qquad (27)$$

where a is given by:

$$a = \sin Z \cos(\gamma - \gamma_s) \sin \beta_s + \cos Z \cos \beta_s \qquad (28)$$

where Z, γ, γs and βs respectively refer to the sun zenith angle, the sun and surface azimuth angles, and the surface tilt angle.

$f_{dir}$ is then calculated as:

$$f_{dir} = \frac{\sum_{i=1}^{8760} I_{beam}(i)}{\sum_{i=1}^{8760} DNI(i)} \qquad (29)$$

where i refers to the hour of the year (the $f_{dir}$ value is representative of a typical year of operation, consequently it is calculated over 8760 hours). In the isotropic diffuse sky irradiance model, the diffuse irradiation on a tilted PV plane can simply be written [42]:

$$I_d = DHI \frac{1 + \cos \beta_s}{2} \qquad (30)$$

$f_{dif}$ is then calculated as:
$$f_{dif} = \frac{\sum_{i=1}^{8760} I_d(i)}{\sum_{i=1}^{8760} DNI(i)} \qquad (31)$$

$f_{dif}$ thus represents the fraction of the diffuse irradiance effectively absorbed by the PV module, relative to the DNI, over a year of operation.

### S1.3. Linear Fresnel Reflectors

The solar radiation reflected by a given reflector in a LFR system is given by [43]:

$$I_{beam}(j) = DNI \times \cos(\epsilon_j) \qquad (32)$$

where $\epsilon_j$ is the incident angle of the direct beam striking the jth reflector (whose value is determined following [43]). The total direct power impinging on the reflectors at any time can simply be written:

$$I_{beam,ref} = \sum_{j=1}^{N} I_{beam}(j) \qquad (33)$$

where N is the number of reflectors, taken equal to 16 here [12].

The total direct power absorbed as the receiver level is given by:

$$I_{beam,rec} = \eta_0 \times I_{beam,ref} \qquad (34)$$

where $\eta_0$ is the optical efficiency taking into account all the optical losses along the optical path between the reflector and the receiver [11].

Two distinct $f_{dir}$ values can thus be deduced, at the reflector and at the receiver level, respectively:

$$f_{dir,ref} = \frac{\sum_{i=1}^{8760} I_{beam,ref}(i)}{\sum_{i=1}^{8760} DNI(i)} \qquad (35)$$

$$f_{dir,rec} = \frac{\sum_{i=1}^{8760} I_{beam,rec}(i)}{\sum_{i=1}^{8760} DNI(i)} \qquad (36)$$

The diffuse irradiation collected at the reflector level is given by:

$$I_d(j) = DNI \frac{1 + \cos \beta_{ref}(j)}{2} \qquad (37)$$

where $\beta_{ref}(j)$ is the inclination angle of the jth reflective element (whose value is determined at any time following [43]).
The total diffuse power impinging on the reflectors at any time can simply be written:

$$I_{dif,ref} = \sum_{j=1}^{N} I_{d}(j) \qquad (38)$$

and the $f_{dif}$ value for the LFR system is then calculated as:

$$f_{dif} = \frac{\sum_{i=1}^{8760} I_{dif,ref}(i)}{\sum_{i=1}^{8760} DNI(i)} \qquad (39)$$

### S1.4. Parabolic Troughs

The solar radiation collected by a parabolic trough is given by:

$$I_{beam,ref} = DNI \times \cos(\theta) \qquad (40)$$

where $\theta$ is the incident angle of the direct beam striking the parabolic trough. Assuming a north-south axis system, this angle can simply be written as [44]:

$$\cos(\theta) = \sqrt{1 - \cos^2(\alpha) \cos^2(\gamma)} \qquad (41)$$

where $\alpha$ is the sun elevation and $\gamma$ is the sun azimuth.

The diffuse irradiation collected at the reflector level is given by:

$$I_{dif,ref} = DHI \frac{1 + \cos \beta_s}{2} \qquad (42)$$

where $\beta_s$ is the inclination angle of the trough, and is simply derived from [45]:

$$\tan \beta_s = \tan \theta_z |\cos(\gamma_s - \gamma)| \qquad (43)$$

with $\gamma_s$ the trough azimuth angle which is taken equal to 90° or -90° depending on the sign of the solar azimuth angle $\gamma$:

$$\gamma_s = \begin{cases} 90^\circ & \text{if } \gamma > 0 \\ -90^\circ & \text{if } \gamma \le 0 \end{cases} \qquad (44)$$

The $f_{dif}$ value for the PT system is then derived from eq. 52 (vide infra), considering the $I_{dif,ref}$ value provided in eq. 42.

### S1.5. Solar Tower

The incidence angle of the solar rays on the heliostat $\theta_{i,hel}$ (i.e. the angle formed between the solar rays and the normal to the heliostat), is deduced from equation 45 [46, 47]:

$$\cos(\theta_{i,hel}) = \frac{\sqrt{2}}{2} \cdot \sqrt{((\sin(\alpha) \cos(\lambda) - \cos(\gamma_h - \gamma) \cos(\alpha) \sin(\lambda) + 1)} \qquad (45)$$
where $\lambda$ is the angle between 1) the line passing through the top of the tower (where the receiver is located) and the center of the heliostat and, 2) the vertical line passing through the tower, see equation 46:

$$ \lambda = \arctan\left(\frac{d_{h-t}}{z_{tower}}\right) = \arctan\left(\frac{\sqrt{x_{pos}^2 + y_{pos}^2}}{z_{tower}}\right) \qquad (46) $$

with $d_{h-t}$, $z_{tower}$, $x_{pos}$, and $y_{pos}$, respectively referring to the distance between the heliostat and the tower base, the tower height, and the x and y coordinates of the heliostat relative to the tower. $\gamma_h$ is the azimuth angle of the heliostat relative to the tower base (anticlockwise from the South), see equation 47:

$$ \gamma_h = \begin{cases} -90 - \arctan\left(\frac{y_{pos}}{|x_{pos}|}\right) & \text{if } x_{pos} \le 0 \\ 90 - \arctan\left(\frac{y_{pos}}{|x_{pos}|}\right) & \text{if } x_{pos} > 0 \end{cases} \qquad (47) $$

The total direct power impinging on the heliostat field at any time can simply be written:

$$ I_{beam,ref} = \sum_{hel=1}^{N} DNI \times \cos(\theta_{i,hel}) \qquad (48) $$

where N is the number of heliostats in the ST power plants, taken equal to 8739 here (a heliostat field dimension compatible with a peak power of 100 MW in Targassonne [10]).

The $f_{dir,ref}$ and $f_{dir,rec}$ values are then derived from equations 35 and 36, considering the $I_{beam,ref}$ value calculated with equation 48. The diffuse irradiation collected at the individual heliostat level is given by:

$$ I_d(hel) = DNI \frac{1 + \cos \beta_{hel}}{2} \qquad (49) $$

where $\beta_{hel}$ is the inclination angle of the heliostat, which is given by [48]:

$$ \beta_{hel} = \arcsin\left(\frac{\cos(\lambda) + \sin(\alpha)}{2 \cos(\gamma_h)}\right) \qquad (50) $$

The total diffuse power impinging on the whole heliostat field at any time can simply be written:

$$ I_{dif,ref} = \sum_{hel=1}^{N} I_d(hel) \qquad (51) $$

and the $f_{dif}$ value for the ST system is finally calculated as:

$$ f_{dif} = \frac{\sum_{i=1}^{8760} I_{dif,ref}(i)}{\sum_{i=1}^{8760} DNI(i)} \qquad (52) $$
<figure><img src="image_15.png" /><figcaption>**Figure S11:** Cost ratio of a Hybrid PVM system based on the LFR technology as a function of the operating temperature of the turbine and the electronic gap of the PV cell, and for 3 different values of the η<sub>PV</sub> index (left: 1, center: 0.8, right: 0.6). The bandgaps derived in the power-optimization and in the power balance strategy are respectively plotted in grey and in red. Finally, the orange, grey and red circles indicate the (E<sub>g</sub>, T) combinations respectively leading to the minimum CR, the maximum power in the power-optimization strategy, and the maximum power in the power-balance strategy.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure S12:** Cost ratio of a Hybrid PVT system based on the LFR technology as a function of the operating temperature of the turbine and the electronic gap of the PV cells, and for 3 different values of the η<sub>PV</sub> index (left: 1, center: 0.8, right: 0.6). The bandgaps derived in the power-optimization and in the power balance strategy are respectively plotted in grey and in red. Finally, the orange, grey and red circles indicate the (E<sub>g</sub>, T) combinations respectively leading to the minimum CR, the maximum power in the power-optimization strategy, and the maximum power in the power-balance strategy.</figcaption></figure>

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