Thermoeconomic Analysis of Concentrated Solar Power Plants Based on Supercritical Power Cycles

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Introduction
The Levelised Cost of Electricity (LCOE) generated by Solar Thermal Power Plants (STPPs) has been reduced by over 50% in the last decade, primarily driven by economies of scale and improvements in the operation and maintenance (O&M) of the plants [1]. However, further research efforts are needed to make the energy provided by STPPs competitive, dispatchable, and safe. In this regard, the Gen3 program [2] proposes the use of supercritical CO2 (sCO2) power cycles coupled to solar central receiver systems as a way to increase the overall efficiency; furthermore, four different pathways are identified, depending on the working fluid in the solar receiver. This study focuses on one of these research lines, proposing the use of pressurised CO2 as Heat Transfer Fluid (HTF) in the solar receiver coupled to a sCO2 power cycle, as shown in Figure 1. As shown in Figure 1, the STPP under study exhibits an indirect coupling. This is due to the significant differences in operating conditions between the HTF in the solar field (pressurised CO2) and the power cycle (supercritical CO2). The primary heat exchanger should be a Printed Circuit Heat Exchanger [3], as this type of heat exchanger performs well under high pressure difference conditions (250 bar in the supercritical cycle compared to 55 bar in the solar receiver).
In STPP schemes with indirect coupling to sCO2 cycles, the working fluid in the solar receiver can be molten salts, with special attention to the new generation of advanced ternary salts [4], a fluidised bed of particles [5], pressurised gas [6], and, finally, liquid metals [7].
Direct coupling STPP schemes are less frequently mentioned in technical literature; in these layouts, the working fluid in the solar receiver is directly supercritical CO2. Although there are initial studies based on tubular external receivers [8], the large thickness required in the tubes is a challenge to the thermal performance of the receiver. Therefore, an option currently under investigation is the use of receivers based on compact structures [9], and this will be further analysed in this section.
Supercritical power cycles have been extensively studied, mainly for nuclear applications [10] and more recently for solar applications [11]. Within solar applications, notable research includes the study presented by Wang [12], in which different recompression sCO2 cycles are selected and compared based on various criteria, such as cycle efficiency, complexity in terms of additional elements compared to the simplest cycle, and, finally, the sCO2 temperature increase in the primary heat exchanger as it determines the volume of the working fluid to be heated in the solar receiver. A later work [13] specifically compares the conventional recompression cycle with the partial-cooling cycle, with the latter presenting very competitive advantages: higher efficiency, less complexity, and, finally, a significant increase in the sCO2 temperature in the primary heat exchanger, which leads to a smaller solar field with a reduced investment. For this reason, as shown in Figure 1, the chosen power cycle is the partial cooling configuration.
Regarding the solar receiver, there are also different designs to work with pressurised gases or supercritical fluids. This study focuses on microchannel receivers, whose compact structures have been extensively studied for working with these fluids [6,9], highlighting two prototypes. The first prototype has been developed in the Clean Energy Research Center [14] and consists of a 3 MWth cavity receiver. The second has been developed by the National Renewable Energy Laboratory [15] and proposes a 10 MWe external receiver.
In both cases, one of the major challenges is the thermal gradient across the thickness of the compact structure. The materials used for these panels, such as stainless steel and nickel or titanium alloys, can withstand high temperatures, but they have the disadvantage of low thermal conductivity [16]. Additionally, in the case of the external receiver, it is necessary to include a light-trapping geometry to reduce radiation heat loss, which is significant due to the high working temperatures. For these reasons, a radial receiver is proposed [17], where the absorber panels, consisting of compact structures, converge on the tower axis, as shown in Figure 2. This radial configuration allows the absorber panels to be irradiated on both sides, thereby reducing the thermal gradient. In addition, the prism-shaped cavity formed between two consecutive and converging absorber panels acts as a light-trapping geometry, reducing radiation heat loss to the outside. This effect is favoured by the fact that the panel region with the highest surface temperature is the inner part, closer to the tower axis, which has a lower view factor with the cavity aperture, as can be seen in Figure 3a. Lastly, this design uses the concept of increasing compactness, which means that the hydraulic diameter of the channels decreases from one flow pass to the next one as the sCO2 is heated, as shown in Figure 3b. This approach improves the cooling of the absorber panel in the region subjected to harsher thermal conditions (higher concentrated solar flux and poorer fluid thermal characteristics due to higher temperature) without excessively penalising pressure drop.  The work is structured as follows: in Section 2, the models used to characterise the supercritical power cycle, the heliostat field, and the solar receiver are explained. These models provide a comprehensive understanding of the previously mentioned systems, allowing for a detailed analysis of their performance and optimisation. The last point of Section 2 proposes the objective functions to be analysed as well as the methodology used. Section 3 presents and discusses the results, providing a scientific explanation for them. Finally, there is a section dedicated to the conclusions.

The sCO2 Partial-Cooling Cycle
The sCO2 cycle model includes the thermodynamic analysis of the cycle components, such as the supercritical compressor, the heat exchangers, and the turbine. Various equations and parameters are considered to calculate the efficiency and performance of the cycle, which are implemented in the software Engineering Equation Solver (EES) [18]. The partial cooling layout is depicted in Figure 4.  Figure 4, the sCO2 partial-cooling cycle is based on the recompression configuration, but it differs from this layout because the Main Compressor (MC) is split in two, introducing an intermediate cooling between the two compressors. Only a portion of the sCO2 is cooled, while another percentage is introduced into the Auxiliary Compressor (AC). The main compressor provides the more pressurised sCO2 stream to the Low Temperature Recuperator (LTR), while the auxiliary compressor connects directly to the High Temperature Recuperator (HTR). Heat is supplied through the Primary Heat Exchanger (PHE), located upstream the Turbine (T). The cycle is refrigerated by an air-cooled Pre-Cooler (PC). The equations governing the behaviour of all the elements of the cycle can be found in the technical literature [10], so they are not summarised in this article. The thermodynamic properties at each of the points marked in Figure 4 are listed in Table 1. Table 1. Thermodynamic properties of the state points of simulated partial-cooling cycle (Source: [19]). As seen in Table 1, to generate 50 MWe, the power cycle requires 103.42 MWth from the solar subsystem at the design point. For this analysis, thermal storage has not been considered, although it would be necessary to ensure the dispatchability of the plant. Since the study conducted is under design conditions, considering or not considering thermal storage would involve considering or not considering the additional cost of such storage, which would be the same for all the STPP configurations analysed.

The Solar Receiver
The solar receiver model describes the thermal behavior of the receiver when exposed to concentrated solar radiation [20]. This model has been implemented in Matlab [21], and the thermofluidodynamic properties of CO2 are provided by the database NIST REFPROP [22]. The model is based on an energy balance in the direction of fluid heating and a thermal resistance model that characterises the temperature gradient across the panel thickness. The common iteration variable of both models is the temperature at the external surface of the absorber panel, as shown in Figure 5. To characterise the fluid heating along the flow direction, it is necessary to divide the channel into Heat Control Elements (HCEs), in each of which the fluid properties are assumed to be constant. For each of these HCEs, the energy balance is given by Equations As observed in Equation (3), the total heat losses are the sum of the radiation, reflection, and convection heat loss. Radiation heat loss is calculated using the simplified formula of Stefan-Boltzmann law [23], accounting for the view factor between the panel external surface and the aperture of the prism-shaped cavity. The reflection loss represents a percentage of the incident heat on the panel, depending on the solar reflectivity, and also includes the view factor. Convection losses are calculated using Siebers and Kraabel correlation [24], which considers both forced convection and free convection.
The thermo-fluid dynamic behavior of the CO2 is characterised by the convection heat gain and pressure drop in the flow direction. Both effects depend on the fluid regime: laminar, turbulent, or transitional. The equations used for each case are summarised in Table 2. Table 2. Correlations for the convection heat transfer coefficient and friction factor of the fluid in each heat control element (Source: [3,10]).

Convection Heat Transfer
In Table 2, Nu is the Nusselt number; ReDh is the Reynolds number based on the hydraulic diameter of the duct, Dh; Pr is the Prandtl number at the fluid temperature; Prsi is the Prandtl number based on the duct inner surface temperature; L is the HCE length; Dh is the hydraulic diameter of the duct; ρ is the average fluid density; u is the average fluid velocity; and fD is the Darcy friction factor, which is four times the Fanning friction factor, fF.
The thermal resistance model across the panel thickness characterises the temperature gradient between parallel channels rows in the direction of concentrated solar flux. The equivalent overall thermal resistance for a panel with a number of parallel channels rows equal to Nc,rows is given by Equation (4). The specific value of each of these resistances is given in Table 3.

Rp,cond
Thermal resistance due to conduction through the wall thickness of the intermediate (frontal) plate , = · · Rc,conv Thermal resistance due to convection between the channel base and top surface Rf,cond Thermal resistance due to conduction through the fin half length Rf,conv Thermal resistance due to convection from the fin surface to the fluid , = 1 2 · · ℎ · RHTF Thermal resistance due to the fluid heat gain = 1 · · · · In Table 3, tp is the intermediate plate thickness (tp0 for frontal plate); tf and lf are the fin thickness and length, respectively; and LHCE and WHCE are the HCE length and width, respectively. These geometric parameters are also shown in Figure 5. Regarding the thermal parameters: ρ, cp, and u are the average fluid density, specific heat, and velocity in each HCE; krec is the absorber thermal conductivity; and hconv is the convection heat transfer coefficient to the fluid.
The solar receiver is sized to provide the thermal power required by the power cycle. The optimal CO2 pressure at the inlet of the receiver is approximately 55 bar [25]. Assuming that the primary heat exchanger is balanced with a Terminal Temperature Difference (TTD) of 12 °C, and that the interconnection pipes are adiabatic, the temperature rise of the CO2 in the solar receiver ranges from 494.8 °C to 700 °C. This solar receiver model has been validated [20] with data from a Thermal Resistance Model (TRM) and a Computational Fluid Dynamics (CFD) model and some limited experimentation [26].

The Heliostat Field
The heliostat field is circular, as in the case of the external receivers. The heliostat field was modelled using two open-source software tools developed by NREL: SolarPilot [27] and SolTrace [28]. SolarPilot is used for heliostat layout optimization and economic analysis, while SolTrace is employed for detailed ray tracing and optical analysis of the heliostat field.
SolarPilot has been used to obtain the optimal heliostats layout based on the design point, the thermal power, the tower height, and the receiver configuration, which is dif- The heliostats chosen for this calculation are the default ones in SolarPilot, and their main characteristics are shown in Table 4. For each STPP analysed, the heliostat field obtained with SolarPilot has been exported to SolTrace in order to perform a detailed ray tracing calculation of the solar flux map on each absorber panel and the optical efficiency. As will be seen in Section 3, the number of heliostats will vary depending on the receiver thermal performance and the solar field optical efficiency, which directly impacts the cost.

The Comparative Analysis as Function of the Receiver Configuration
The comparative analysis is based on different receiver configurations, keeping the receiver thermal power constant. The CO2 temperature at the receiver inlet and outlet as well as the CO2 pressure at the receiver inlet are also constant, as well as the geometric characteristics of the compact structure inside the panel, as shown in Table 5. Table 5. Geometric and thermo-fluid dynamic parameters of the solar receiver that remain constant in the comparative analysis. By keeping the previous characteristics constant, a comparative analysis can be conducted to evaluate the performance of the solar receiver under different number and size of the absorber panel by means of the following parameters: the energy and exergy efficiencies referred to the receiver, the solar subsystem, and the solar thermal power plant (Equations (6)- (10)). This type of analysis, including both energy and exergy efficiencies, is very useful for analysing STPPs, particularly when the working fluid in the solar receiver is a gas, and pressure drop is therefore an important factor to consider [30,31].

Global Receiver Parameters
, In previous equations, ̇, 2, is the total heat gain by convection in the receiver; ̇, is the concentrated solar radiation incident on the receiver; ∆ 2, is the total exergy gain in the receiver; ∆ , is the exergy associated to the solar radiation incident on the receiver, given by Parrot equation [32]; is the optical efficiency of the heliostat field; and is the thermal efficiency of the sCO2 cycle, equal to 48.41% in all the cases, as seen in Table 1.
The comparative analysis will be conducted from a thermal and optical perspective in Section 3.1, while an economic analysis will be performed in Section 3.2.

Thermal and Optical Analysis of the Solar Thermal Power Plant as a Function of the Receiver Configuration
The results presented in this section only refer to the optical and thermal performance of STPPs that differ in the receiver configuration (number of panels and their dimensions) but that have the same thermal power, as well as the other parameters enumerated in Section 2.4. Table 6 shows the panel dimensions, heat losses, and pressure drop as a function of the number of panels in the receiver. As observed in Table 6, as the number of panels increases, their dimensions decrease. This ultimately leads to higher optical loss due to spillage, which decreases the optical efficiency, as observed in Figure 6a,b. On the other hand, heat losses decrease as the number of panels increases, as the view factor from the panel to the outside decreases due to a reduction in the aperture of the prism-shaped cavity. The pressure drop also decreases because the parallel fluid circuits increase as the number of panels increases, while their length, specifically the panel height, decreases. As a result, both the receiver energy and exergy efficiency improve with an increasing number of absorber panels. The overall solar subsystem performance, which is the convolution of the solar field optical efficiency and the receiver energy/exergy efficiency, exhibits maximum values for configurations between 8 and 10 panels, approximately, as observed in Figure 6a,b. Finally, Figure 7 displays the overall energy efficiency of each STPP considered, which follows the same trend as the energy efficiency of the solar subsystem, since the power cycle efficiency is the same for all the configurations analysed: 48.41%, as seen in Table 1. As expected, there is also a maximum efficiency between 8 and 10 panels, which is the result of two opposing effects: on the one hand, energy and exergy efficiency, which increase as the number of panels increases, as heat loss and fluid pressure drop decrease due to shorter parallel circuits; on the other hand, there is a reduction in optical efficiency as the number of panels increases as the spillage loss increases.

Economic Analysis of the Solar Thermal Power Plant as a Function of the Receiver Configuration
A comparative economic analysis of the entire STPP has been conducted for each receiver configuration considered in the previous section. For this purpose, the Total Capital Cost (TCC) [33], the Payback period (PB) [34,35], and the Levelised Cost of Electricity (LCOE) [33] of each STPP have been estimated.
The PB is the length of time that it takes to recover the initial capital investment in a project, and it is calculated by means of Equation (11).
In Equation (11), AEP is the Annual Electricity Production, which is estimated assuming that the plant has a Capacity Factor (CF) of 25%, as it is a stand-alone CSP plant [36]. Based on this, the annual production for all plants is equal to 109.5·10 3 MWhe/year. CE is the electricity cost, 124.9 $/MWhe, which is based on the average market price at USA [37]. Finally, OMC is the O&M costs per MWhe, taken as 23 $/MWhe [1].
The LCOE is calculated using Equation (12): In Equation (12), CRF (%) is the Capital Recovery Factor, calculated as Equation (12): In Equation (13), WACC refers to the Weighted Average Capital Cost, which is assumed to be 8%; and n represents the economic lifetime, which is assumed to be 30 years. With the aforementioned values, the CRF is calculated to be 0.088 years −1 .
The TCC of the sCO2 partial-cooling cycle has been calculated using the methodology described in [38,39]. Table 7 summarises the TCC for the primary heat exchanger, recuperators, precooler, and turbomachinery. This TCC is constant for all the STPPs analysed in the previous section, as the power cycle is the same for all of them. The TCC of the solar field has been calculated according to the equations and reference values provided in the SolarPilot program. The most significant costs are associated with the heliostat field, the receiver, and the tower, as shown in Equations (14) is the receiver cost scaling exponent, equal to 0.7 in SolarPilot; is the receiver area, which depends on the number of absorber panels and their size; , is the fixed tower cost, with a default value in SolarPilot of 3 Mio. $; k is the tower cost scaling exponent, equal to 0.0113 in SolarPilot; and ℎ is the tower height, equal to 109 m for all the cases analysed.
As seen in Table 8, as the number of absorber panels increases, the receiver area decreases, and, consequently, its cost decreases, since its energy and exergy efficiencies increase, as discussed in the previous section. The tower height and its cost are constant, while the total reflective area increases as the receiver size decreases because of the larger optical losses due to spillage. Although initially reducing the receiver area improves the overall STPP performance, beyond a certain number of absorber panels, the optical efficiency decreases significantly, and the additional investment cost in the solar field does not compensate for the savings in the receiver cost. In the specific example being studied, this occurs from 14 panels onwards, although it will depend on the reference cost and the values adopted for the different parameters. Finally, Table 9 shows the PB and the LCOE for all the STPPs considered. As expected, the PB is minimal for the plant with the lowest TCC, as AEP and OMC are the same for all plants. Although this PB is high for standard projects, it is in line with other STPPs, and it should be noted that the economic lifetime is 30 years. The LCOE follows an opposite trend compared to the solar subsystem TCC, which is logical, since the power cycle TCC is the same for all plants as well as the annual estimate of electricity produced. Table 9. Summary of the total capital cost and levelised cost of energy of each solar thermal power plant, based on a partial-cooling sCO2 cycle.

Conclusions
This work focuses on the study of a STPP configuration that is included in the new generation of STPPs, aiming to reduce costs through improved performance by employing supercritical power cycles [2].
Specifically, this STPP is based on a central solar receiver refrigerated by a pressurised gas and coupled to a supercritical power cycle. Compared to more conventional configurations of the aforementioned scheme, this STPP presents two innovative elements. Firstly, the power cycle is a SCO2 partial-cooling layout, which presents significant advantages compared to conventional recompression, such as lower investment due to the smaller recuperator volume, higher thermal efficiency, and higher temperature increment of the sCO2 in the primary heat exchanger. This leads to a smaller volume of gas to be heated in the receiver, which directly reduces the receiver sizing and the solar field cost. Secondly, the STPP features a novel receiver design based on compact structures in a radial configuration. This design is particularly suitable for working with pressurised gas.
The comparative analysis conducted in this study involves keeping a constant thermal power supplied by the solar receiver while varying its design through the number and dimensions of the absorber panels. The analysis demonstrates that there is an optimal configuration that balances two opposing effects: the increase in receiver thermal efficiency as the number of absorber panels increases, and the decrease in optical efficiency due to increased spillage as the receiver size is reduced. Although this optimum will depend on the specific working conditions chosen, it provides an idea of the range in which the solar thermal plant operates optimally, resulting in lower electricity generation costs.
In the specific case under study, the optimal energy and exergy efficiency is achieved with around 10-12 panels, while the configuration with the minimum LCOE corresponds to 14 panels. This result depends on the reference values and other parameters assumed for the economic study. In any case, the obtained LCOE values are of the same order as the currently estimated values [1], and it is expected that these values will further decrease as economies of scale are developed for these STPPs.
Future lines of work include a more detailed study of the annual analysis, considering transients, startups, and shutdowns. This will involve examining the behavior of the plant throughout the year and optimising its performance under different operating conditions. Additionally, the study aims to explore the integration of thermal energy storage systems within the STPP. This will investigate the feasibility and the benefits of incorporating storage technologies in order to enhance the dispatchability and flexibility of the solar thermal power plant. Furthermore, a more detailed optimisation study of specific components is planned, such as the solar receiver and the primary heat exchanger.