## Abstract

The objective of this paper is to assess the techno-economic performance of different cycle configurations for pumped thermal energy storage (PTES), including the effects of charging electricity costs. Reversible turbomachinery was employed to reduce the capital cost of the system. Brayton cycles with different working fluids and a subcritical Rankine cycle operating with ammonia were compared. Both liquid and packed bed thermal storages were investigated. A new cost correlation for turbomachines, initially established for the turbines of organic Rankine cycles, was developed for compressors and reversible machines. This correlation is based on the number of stages and physical size of the machine, which were estimated considering thermodynamic as well as mechanical limitations. The results indicate that for a plant size of 50 MW and a discharge duration of 8 h, the Brayton system with liquid storage and helium as a working fluid has the lowest levelized cost of storage at 0.138 $/kWh, mainly due to the high thermal conductivity of the fluid. Packed bed thermal energy storage systems were found to be more expensive than liquid storage systems due to the large cost of the pressure vessels, with cost parity reached at a discharge duration of 4 h. However, at this duration, lithium-ion batteries are likely to be cheaper. The results suggest that the levelized cost of storage for the Rankine cycle-based system is slightly higher at 0.151 $/kWh.

## 1 Introduction

Current scenarios for the decarbonization of the energy sector predict that non-dispatchable energy sources such as wind and solar energy will provide most of the electricity generation, reaching a share of 70% by 2050 in the International Energy Agency net-zero scenario [1]. Grid-scale energy storage will then be required to provide flexibility in matching production and demand [1]. There is no single technology that can comprehensively address all the energy storage needs. Different groups of technologies have emerged based on storage duration; short-duration energy storage is exemplified by lithium-ion batteries, while long-duration storage, typically exceeding 6–10 h, can be achieved through thermo-mechanical energy storage systems (TMESs) [2,3].

Pumped hydro, a form of mechanical energy storage, as of 2023 represents around 90% of the available installed storage capacity on an energy basis [4], however, further adoption is limited by geographical constraints. A-CAES, which is currently considered by various authors [3,5,6] as the cheapest long-duration energy storage, is also limited by geographical constraints.

Pumped thermal electricity storage (PTES) holds promise as a long-duration energy storage technology due to its independence from both geographical limitations and the need for critical materials. Compared to other TMES technologies, PTES has low maturity but is rapidly evolving, moving from a technology readiness level [7] of 2–5 in 2021 [3] to technology readiness level of 6–7 in 2023, when data were published regarding two independently developed demonstrators [8,9].

PTES systems are comprised of two thermodynamic cycles: the charging cycle is like a heat pump, using low-cost electricity in periods of high electricity generation to transfer heat from a low-temperature storage to a high-temperature storage, while the discharging cycle recovers part of the electricity using a heat engine. While in principle resistive heating [10] or a different cycle could be used for the charging process, most studies consider a common cycle operated in opposite directions for the two modes of operation. Due to the large flowrates involved, turbomachines are considered the cheapest and most efficient alternative for the charging and discharging cycles. Since the turbomachinery represents a significant cost in the system, and the two modes of operation do not happen concurrently, using the same machine for both modes of operation can lead to significant cost savings [3]. This principle, while employed at commercial scale for pumped hydro plants [11], has been studied only recently for compressible flow applications. Previous studies [12–14] have confirmed that it is possible to obtain efficiency levels similar to those of dedicated machines, with the resulting designs being similar to those of dedicated compressors. The main requirement for efficient reversible operation is that the work coefficients and the density ratio of the inlet to the outlet are similar in both modes of operation.

A wide array of cycles has been proposed for PTES, including Brayton and both subcritical and transcritical Rankine cycles [15]. While Rankine cycles can achieve greater efficiencies for a given temperature level, Brayton cycles have the advantage of allowing part load operation at close to nominal efficiency. High efficiency at part load operation was obtained with inventory control [16], i.e., changing the pressure levels in the system while maintaining similar temperature levels as the design values. As for Brayton cycles, it has also been proposed to operate the cycle close to the critical point of the working fluid. McTigue et al. [17] observed an increase in the round-trip (electricity to electricity) efficiency from 61% with argon operating close to ideal gas, to 78% with supercritical CO_{2}, while making the system more compact and less sensitive to the efficiency of the compression and expansion machines. At the same time, the capability for inventory control is lost, and the maximum pressure reaches 25 MPa, making the development of such system more challenging than one operating at lower pressure.

Both liquid and solid thermal energy storage materials have been considered, with solid materials typically using a packed bed configuration with direct contact heat exchange. Direct contact packed bed thermal energy storage has the advantage of not requiring separate heat exchangers, however, the entire storage volume needs to be pressurized, leading to rapidly rising costs as the operating pressure or storage duration increases. Using an intermediate fluid, as proposed by Klasing et al. [18], eliminates the need for pressurizing the packed bed, but necessitates the use of heat exchangers between the working and heat transfer fluid.

Previous studies on the thermo-economic assessment of PTES, such as Zhao et al. [15], suggest that the turbomachinery represents the largest cost, accounting for 55–82% of the total capital cost in Brayton cycle PTES systems, both with liquid and packed bed storage, and 53% for a transcritical Rankine system using ammonia.

McTigue et al. [19] performed a techno-economic analysis of a PTES focusing on a Brayton cycle with nitrogen as a working fluid and liquid storage. They used numerous equations to estimate the components' cost, obtaining a distribution of values from Monte Carlo simulations with different combinations. In this study, the turbomachines were found to be the components with the highest cost, but also those with the highest cost variability between different simulations, spanning over an order of magnitude.

Frate et al. [20] compared Brayton cycle PTES with packed bed and liquid storage at different plant sizes. They concluded that air is the optimal working fluid, and packed bed storage was found to be competitive only up to power ratings in the order of 10 MW, while liquid storage was found to be more cost-effective at larger sizes.

Zhang et al. [21] analyzed a packed bed-based PTES with helium as a working fluid, considering representative charging and discharging cycles, and accounting for off-design performance of the turbomachinery. They concluded that the system capacity significantly affects the levelized cost of storage (LCOS), with a 63% reduction from the 10 MW to the 1000 MW system.

The previously mentioned studies show large variations in the techno-economic performance, mainly due to the turbomachinery, with different cost correlations proposed for the same component leading to very different cost estimates. Nevertheless, even if the absolute values are uncertain, comparisons among different systems can still be meaningful when they are done within the same model, as long as the models can capture the correct trends. The turbomachinery cost models [19,22,23] employed in previous studies (see Sec. 2.2.1) were developed for specific fluids and different operating conditions than those of PTES systems, therefore can present unphysical trends when applied to PTES systems. For this reason, a novel cost correlation was developed in this work.

The main objective of this paper is to identify the optimal PTES cycle from a thermo-economic perspective, comparing different working fluids and storage media, by minimizing the levelized cost of storage. In the present study, three different combinations were analyzed: Brayton cycle with packed bed storage, Brayton cycle with liquid storage, and subcritical Rankine cycle with liquid storage and ammonia as a working fluid. The working fluids considered for the Brayton cycles are air, helium, and carbon dioxide; all fluids operate in the gas region far from the critical point. This leads to a total of seven different configurations being analyzed. The systems were modeled at the design point with focus on turbomachinery and heat exchanger performance. The reversible turbomachinery was modeled with a lumped parameter approach, using the number of stages to estimate both the efficiency and cost. The heat exchangers were discretized with a 1D model using correlations for the pressure loss and heat transfer coefficient within a circular pipe.

Our work introduces two significant novel contributions by: (i) carrying out the first techno-economic analysis of an energy system employing reversible compressible-flow turbomachinery; (ii) developing a novel cost correlation for turbomachinery, based on physically meaningful parameters, suitable for the operating conditions of PTES systems.

In Sec. 2, the cycle layouts are described, followed by the models for the turbomachinery, the models for all the other components, and the optimization routine. Section 3 presents the results of the system with the baseline assumptions, followed by a comparison of the effects of different assumptions. Section 4 presents the conclusions.

## 2 Methods

### 2.1 Cycle Layouts.

In a Brayton cycle PTES plant, such as the one shown in Fig. 1, the charging cycle is composed of the following steps: compression at high temperature (1 to 2), heat transfer to the high-temperature storage (2 to 3), cooling of the working fluid in the regenerator (3 to 4), expansion of the cold fluid in a turbine (5 to 6), heat transfer from the cold storage (6 to 7), and heating in the regenerator (7 to 1). The discharging cycle is realized by reversing the direction of all the flows: in this case, the hot machine (2 to 1) operates as a turbine and the cold one (6 to 5) as a compressor. If packed bed storage is used, regeneration is not employed. In addition to the steps mentioned above, heat rejection to ambient is necessary. Multiple configurations are available regarding the heat rejection. In this work, the heat rejection is done before the turbine inlet (4 to 5) in charging mode only. This configuration was chosen as it has a lower optimal pressure ratio, avoiding very low temperatures, and achieves a high round-trip efficiency [20]. For the liquid-based system, regeneration is used to limit the temperature range of the storage, as there is no single material that can cover the entire range of temperatures considered. Regeneration is not used in the packed bed system; however, the temperature ranges of the hot and cold storage are allowed to overlap. A simplified loss model, based on information obtained from industry partners, was considered for the packed bed, assuming a 2% pressure loss and a 5 K temperature difference between the rocks and the working fluid.

The Rankine cycle plant, depicted in Fig. 2, uses a multi-tank heat storage with five heat exchangers in order to follow the temperature profile of the working fluid on the high-pressure side. In charging mode the fluid is compressed (1 to 2), then cooled as a gas (2 to 3), condensed (3 to 4), cooled as a liquid (4 to 5), further cooled in a chiller that rejects heat to ambient (5 to 6), then expanded in the liquid state (6 to 7), evaporated (7 to 8), and preheated before the compressor inlet (8 to 1). In discharge mode, the cycle is reversed, with additional heat rejection (between 1 and 8) and bypassing the chiller. Since the tanks that store the latent heat have a limited temperature difference and very large volume, a single thermally stratified tank was considered for these cases. Water at ambient pressure was chosen as storage material for the liquid and two-phase region, while thermal oil was used for the gas phase. The chiller is used in charging mode to allow saturated liquid conditions at the expander outlet.

For both cycles, the heat rejection temperature was fixed at 298.15 K, while the compressor outlet during charging was fixed at 840 K for the Brayton cycle and 648 K for the Rankine cycle, based on the storage material maximum temperature. All the other temperatures and pressure levels are either optimized or computed with energy balances and applying the equations of state.

### 2.2 Turbomachinery Models

#### 2.2.1 Previous Turbomachinery Cost Models.

In this section, some of the cost correlations for turbomachines used in previous works are presented, indicating the need for an improved cost correlation. In Sec. 3.3, the results obtained with the different correlations are compared.

*n*is a scaling exponent taken equal to 1. The currency is USD refers to the year 1995. These equations were modified from the ones originally developed by Agazzani and Massardo [23], based on open-cycle gas turbine data; the original equations do not consider the density dependence (scaling exponent equal to 0). Also in this case, however, the parameters used are insufficient to relate to the physical size of the machine. For example, if two machines with the same parameters are compared, one using air and one using helium, the machine with helium will have a larger number of stages since, due to the higher speed of sound, the pressure ratio that can be obtained per stage is much lower.

#### 2.2.2 New Turbomachinery Model.

The model considered in this paper aims to capture the trends in performance and cost among different fluids, for this reason, the cost correlation is based on physical parameters, and the performance is measured using polytropic efficiency and accounting for real-gas effects.

Due to the large flowrates involved, multistage axial machines were considered for this study. In order to estimate the cost and performance of the machine, it is important to estimate the number of stages required. The minimum number of stages is governed by mechanical as well as fluid dynamic considerations. Since the reversible machine is similar to a compressor, the following methodology to determine the number of stages considers only the compressor mode of operation.

*U*is considered for all the stages.

A typical value for the loss coefficient in the diffuser is $\zeta =0.35$, with $Nst$ being the number of stages in the machine and $\eta p,st$ being the combined efficiency of the stages without diffuser. The overall specific enthalpy change $\Delta h$ is the sum of the specific enthalpy differences in all the stages. When substituting the numerical values, it can be noticed that the efficiency penalty for a single stage machine is over 15%, but less than 3% for six stages machine. The latter result corresponds to the experimental results reported in Ref. [27]. In this study, the number of stages was optimized allowing for a non-integer number, since the selected optimization algorithm requires a function with a continuous domain space. The stage efficiency considered in the present work is $\eta p,st=92%$. This value corresponds to the machine efficiency only when the diffuser loss is zero, which happens at the limit of an infinite number of stages leading to the same cost asymptote of Eqs. (3) and (4). The value considered for the stage efficiency is slightly higher than the 91% obtained for reversible turbomachinery in a computational fluid dynamics study by Harris et al. [12]. However, the authors of the study concluded that further improvements are likely.

The value computed for the compressor efficiency is also used for the turbine mode of operation, following the results obtained in Ref. [12].

$Q\u02d9$ is the volumetric flowrate based on the total density at the low-pressure end of the machine, i.e., the outlet for a turbine but the inlet for a compressor. With this correlation, the physical parameters of the machine are correctly taken into consideration when determining the cost. The main weakness in this case is the lack of a parameter to correct for the pressure, since a machine operating at very high pressure will be more expensive due to the need of more advanced and expensive casing, sealing, and bearings. For the Brayton cycle PTES, the cost of the hot machine is twice as that of the cold machine, since more expensive materials are required to withstand the high temperature, with a threshold of 625 K corresponding to a typical maximum operating temperature of carbon steel [20].

The reference values of Eq. (12) were modified to match the results of Eq. (3) when applied to the low-pressure compressor of a mid-sized gas turbine, the LM6000, and are presented in Table 1. The reference cost is in 1995 USD. When corrected for inflation, the estimated cost is within 15% of the one estimated in Ref. [29] for the same component. It can be noticed that the cost computed with the current coefficients is one order of magnitude lower than the one that would be obtained using the coefficients presented in Ref. [28]. This significant difference is expected, since the original correlation was developed for ORC turbines which are often customized machines, in contrast to the large series ones modeled in Eq. (3). Moreover, for the same size parameter a dedicated turbine will be larger than a compressor or reversible machine.

The cost obtained with the previous correlation considers only the component and not the installation, piping, and other auxiliaries. These costs are considered by multiplying the component cost by an appropriate coefficient, in this case 4, to obtain the installed or bare module cost, as suggested by Turton et al. [30].

The model presented above was used for both machines in the Brayton cycles and the hot machine in the Rankine cycles. For the cold machine in the Rankine cycles, a different model was used since the fluid is in liquid state. For the machines operating with a liquid an isentropic efficiency of 70% was considered, while the cost model is defined in the next section.

### 2.3 Models for Other Components.

*A*an appropriate measure for the size of the equipment.

The heat exchangers operating without phase change are modeled using the procedure presented by McTigue et al. [19], i.e., the heat exchanger is discretized using different segments with equal heat duty, imposing the same frontal area for both fluids. The hydraulic diameter is fixed to 5 mm. Thermophysical properties are computed for each section, and correlations for flow inside a pipe are used to compute the local convective heat transfer coefficients on both sides. The overall heat transfer coefficient is then computed, and from that the length, heat transfer area, and pressure loss in each segment. The frontal area is used as iteration variable. A larger frontal area leads to a lower fluid velocity and therefore losses, but at the same time lower heat transfer coefficients due to the reduced Reynolds number. For the two-phase heat exchanger, the global heat transfer coefficient is fixed to 1200 W/(m^{2} · K), which is within the typical range for ammonia condensers [31]. A pinch point greater than 0.5 K is imposed for every heat exchanger in this study. When modeling the cost, the correlation for a fixed plate shell and tube heat exchanger was used, with a pressure correction coefficient that considers only the high-pressure fluid inside the tubes.

The power $W\u02d9$ is expressed in kilowatts and is the highest of the two modes of operation.

Air, helium, carbon dioxide, and ammonia were investigated as working fluids. The cost of the working fluid is neglected, while the thermodynamic model uses the Helmholtz equation of state implemented in coolprop [32].

The packed bed storage is modeled as a pressure vessel with internal insulation, with the basalt rocks occupying 40% of the pressurized volume.

The liquid storage is modeled with a fixed roof multi-tank system at ambient pressure. For the Brayton cycle hexane was chosen as the cold storage fluid, and the HITEC molten salt as the hot storage fluid, while for the Rankine cycle, water was used for the cold storage and Therminol 66 oil for the hot storage. For water and hexane, the equations of state available from coolprop [32] were used, while for thermal oil and solar salt average values of the properties over the temperature range of interest were considered [15]; the values are reported in Table 2. The cost of water is considered negligible, while for hexane a value of 1 $/kg was used [15].

Fluid | $\rho (kg/m3)$ | c (J/kg)_{p} | k (W/(m · K)) | μ (mPa · s) | Cost ($/kg) |
---|---|---|---|---|---|

HITEC | 1637 | 1562 | 0.382 | 2.60 | 0.93 |

Therminol 66 | 909 | 2072 | 0.109 | 1.20 | 1.0 |

Basalt | 2640 | 1231 | 1.50 | — | 0.12 |

Fluid | $\rho (kg/m3)$ | c (J/kg)_{p} | k (W/(m · K)) | μ (mPa · s) | Cost ($/kg) |
---|---|---|---|---|---|

HITEC | 1637 | 1562 | 0.382 | 2.60 | 0.93 |

Therminol 66 | 909 | 2072 | 0.109 | 1.20 | 1.0 |

Basalt | 2640 | 1231 | 1.50 | — | 0.12 |

The cost of the recirculation pumps for the storage fluids and the insulation are not included, since in previous studies [19] they were found to be negligible.

It is assumed that the rejected heat is used for district heating. The cost of the heat rejection equipment is considered allocated to the waste heat, and therefore it is not included in the cost estimation for the stored electricity.

### 2.4 Performance Metrics and Optimization Routine.

^{®}[33]. The optimization variable is the LCOS, defined as

*i*represents the year, $Ccap,i$ the capital cost and $OMi$ the operation and maintenance costs, $Cel,i$ the price of electricity during charging, $Ein,i$ and $Eout,i$ the quantity of electricity input and output, and

*r*the discount rate. Assuming that the whole capital cost is concentrated in the first year and operation and maintenance costs are constant, the previous equation can be simplified to

All the parameters for the cost correlations have been converted using the chemical engineering plant cost index [30] to the value of the year 2020. Table 3 presents the baseline parameters used in the study.

Parameter | Unit | Value |
---|---|---|

Yearly discharge cycles | 1/year | 300 |

Discharge duration | h | 8 |

Interest rate | % | 6 |

Plant life | years | 25 |

Discharge power rating | MW | 50 |

Maximum working fluid pressure | MPa | 10 |

Parameter | Unit | Value |
---|---|---|

Yearly discharge cycles | 1/year | 300 |

Discharge duration | h | 8 |

Interest rate | % | 6 |

Plant life | years | 25 |

Discharge power rating | MW | 50 |

Maximum working fluid pressure | MPa | 10 |

## 3 Results and Discussion

### 3.1 Model Verification.

Considering the lack of confident manufacturer data, in this section, the results of the present model are compared with results from previous studies.

In McTigue et al. [19], the minimum LCOS computed for an air Brayton cycle operating with nitrate molten salts is (0.13 ± 0.03) $/kWh, with a round-trip efficiency of 59%. The results of the present work, presented in Table 4, indicate a higher LCOS and lower efficiency obtained, both due to the higher bare module factor considered, which increases the heat exchanger installed cost. In both cases, the technical potential for round-trip efficiency is over 65%, at an LCOS exceeding 0.20 $/kWh.

Type | Brayton liquid storage | Brayton liquid storage | Brayton packed bed | Rankine |
---|---|---|---|---|

Fluid | He | Air | Air | NH_{3} |

LCOS ($/kWh) | 0.138 | 0.148 | 0.185 | 0.151 |

$\eta rt(%)$ | 51.3 | 51.2 | 58.0 | 53.2 |

Pressure ratio | 2.23 | 3.81 | 6.98 | 21.8 |

Base pressure (bar) | 44.5 | 26.2 | 1.01 | 4.25 |

Capital cost (M$) | 72.9 | 86.1 | 141.6 | 92.5 |

Machinery cost (%) | 17.8 | 13.1 | 27.6 | 6.8 |

Heat exchanger cost (%) | 43.3 | 57.0 | 0 | 46.2 |

Storage cost (%) | 22.4 | 15.9 | 64.9 | 30.8 |

Other costs (%) | 16.5 | 14.0 | 7.5 | 16.2 |

Type | Brayton liquid storage | Brayton liquid storage | Brayton packed bed | Rankine |
---|---|---|---|---|

Fluid | He | Air | Air | NH_{3} |

LCOS ($/kWh) | 0.138 | 0.148 | 0.185 | 0.151 |

$\eta rt(%)$ | 51.3 | 51.2 | 58.0 | 53.2 |

Pressure ratio | 2.23 | 3.81 | 6.98 | 21.8 |

Base pressure (bar) | 44.5 | 26.2 | 1.01 | 4.25 |

Capital cost (M$) | 72.9 | 86.1 | 141.6 | 92.5 |

Machinery cost (%) | 17.8 | 13.1 | 27.6 | 6.8 |

Heat exchanger cost (%) | 43.3 | 57.0 | 0 | 46.2 |

Storage cost (%) | 22.4 | 15.9 | 64.9 | 30.8 |

Other costs (%) | 16.5 | 14.0 | 7.5 | 16.2 |

For a 100 MW Brayton cycle system with 6 h discharge capacity, Zhao et al. [15] found that both the heat exchanger and turbomachinery capital costs are lower for a helium-based system than for a nitrogen (which has almost identical properties to air) based system when liquid storage is considered. The total capital cost for the helium-based system was 2620 $/kW, which is significantly higher than the 1458 $/kW obtained in the present paper. The higher capital expenditure can be explained by different cost correlations used for the turbomachinery, as Ref. [15] used a power-based correlation, which was found to give significantly higher cost estimates as detailed in Sec. 3.3.

These comparisons with the results of previous works indicate that the models presented in this paper give reasonable results.

### 3.2 Results With Baseline Parameters.

Table 4 contains a summary of the optimized system using the baseline parameters. The liquid Brayton system operating with helium, according to the results, has the lowest LCOS among all the fluids and cycle configurations. The percentage cost shares are defined as the ratio of the bare module cost to the total capital cost. The storage cost includes the tanks and vessels as well as the storage material. The pressure ratio indicated in the table is defined across the compressor in the charging mode, while the base pressure is the pressure at the compressor inlet.

When the values obtained are compared with the existing literature [15–21], the main difference that can be noticed is the very low share of the turbomachinery cost in the present work. Only part of this difference can be attributed to the usage of reversible turbomachinery, which is expected to approximately cut the cost by half. The remaining difference is caused by the cost correlation used, and this aspect is analyzed in detail in Sec. 3.3. The results suggest that air is the optimal working fluid for the packed bed system, however for the liquid storage, helium is predicted to have the lowest LCOS. The working fluid choice is analyzed in Sec. 3.4. Figure 3 depicts a temperature–entropy diagram for the Brayton cycle with liquid storage and helium as a working fluid.

The temperature–entropy diagram of the Rankine system is depicted in Fig. 4. This cycle has a competitive LCOS but has two significant drawbacks: (i) the temperature and pressure cannot be varied independently, therefore the turbomachinery will operate far from the design point at part load, reducing the efficiency, and (ii) to operate efficiently, the temperature difference in the storage fluid should be very small, which requires a very large storage tank to store the latent heat at the cold temperature. For the Rankine cycle system presented in Table 4, the computed storage volume for the cold tank is 3.3 × 10^{5} m^{3}, which is more than ten times bigger than the sum of the storage volumes for the Brayton cycle. Since the cold storage temperature is limited by the freezing point of water, an alternative solution that would limit the storage volume is to employ a water–ice slurry, which would also result in a higher efficiency due to better matching of the temperature profile.

Since previous studies [12–14] focused on Brayton cycles with smaller pressure ratios, further research would also be required to confirm that reversible machinery can be used in the conditions found in Rankine cycles.

At the selected storage duration, the results indicate that the packed bed system is not competitive due to the very large cost of the pressurized storage, but if the duration is decreased to 4 h the LCOS becomes equal to 0.22 $/kWh for both the packed bed and liquid storage. At this discharge duration, however, both solutions are unlikely to be cost competitive with lithium-ion batteries, that according to Lazard [35] have an estimated LCOS in the range of 0.132–0.245 $/kWh. The packed bed system is also characterized by a higher pressure ratio with respect to the liquid system, which has the effect of reducing the mass flowrate for the same output power: this in turn decreases the required amount of energy storage.

The thermodynamic performance levels are quite low when compared with the thermodynamic optimum for each system. The round-trip efficiency can be increased significantly with a small increase in LCOS, as shown in Fig. 5, which agrees with the results obtained by McTigue et al. [19]. Improving the round-trip efficiency past the economic optimum, requires an increase in capital cost, mainly due to an increased heat exchange area that outweighs the reduction in charging electricity cost. The round-trip efficiency is of secondary importance when compared to the cost, especially when considering that the energy stored might otherwise be curtailed.

However, it needs to be stressed that both the present and previous works [15,16,19] do not consider that the price of electricity changes with time. If a more realistic assumption was made, a higher round-trip efficiency would likely be favored, since more efficient systems could operate with a smaller price difference between charging and discharging, therefore operating for more hours in a year.

The results obtained are highly sensitive to several assumptions, but the most important one is the cost of charging electricity. Figure 6 shows the optimum values found by imposing different electricity prices between 0 $/kWh and 0.05 $/kWh for the Brayton system with air and liquid thermal storage. If the input electricity is free, for example, in a situation where grid congestion is common, the LCOS decreases to 0.058 $/kWh and the optimum efficiency decreases to 36.6%. In this extreme situation, the minimization of the LCOS coincides with the minimization of the capital cost.

The more relevant competing technology to PTES for location independent thermo-mechanical energy storage is the liquid air energy storage (LAES) technology. The results of the present study are in line with the range of economic performances proposed for LAES systems in comparable studies [3,5]. The more suitable storage technology is governed by the operating conditions, with PTES being more competitive at higher cost of charging electricity, due to the higher potential round-trip efficiency [34].

### 3.3 Comparison of Turbomachinery Cost Models.

In order to compare the turbomachinery cost models, the air cycle with liquid storage is considered. Table 5 shows the results with four different cost models: the new model developed for this paper, the model of Eq. (3) with density correction, the original model from Agazzani and Massardo [23] without density correction, and the model based on fluid power of Eq. (1).

Correlation | Present work | Density correction | Agazzani | Power based |
---|---|---|---|---|

LCOS ($/kWh) | 0.148 | 0.1455 | 0.1897 | 0.2363 |

$\eta rt(%)$ | 51.2 | 51.6 | 48.7 | 55.1 |

Pressure ratio | 3.81 | 3.11 | 2.47 | 3.53 |

Base pressure | 26.2 | 31.7 | 37.5 | 27.9 |

Machinery cost (M$) | 11.2 | 6.8 | 42 | 121 |

Correlation | Present work | Density correction | Agazzani | Power based |
---|---|---|---|---|

LCOS ($/kWh) | 0.148 | 0.1455 | 0.1897 | 0.2363 |

$\eta rt(%)$ | 51.2 | 51.6 | 48.7 | 55.1 |

Pressure ratio | 3.81 | 3.11 | 2.47 | 3.53 |

Base pressure | 26.2 | 31.7 | 37.5 | 27.9 |

Machinery cost (M$) | 11.2 | 6.8 | 42 | 121 |

The different parameters considered in the various correlations are clear from the results obtained. With the new correlation, a higher pressure ratio is favored with respect to Eq. (3), because a high number of stages in turbomachinery increases its efficiency. The correlation of Eq. (3) instead favors low-pressure ratios and efficiencies. When density is considered in Eq. (3), the cost share of the turbomachinery is low, therefore the optimal point will be determined mostly by the requirements of the cycle.

Very high values for the base pressure are present also for the correlations that do not account for the physical size of the machines. An analysis of the heat transfer coefficients computed confirms that this is due to the improved heat transfer characteristics of the gases at high density, due to the high Reynolds numbers in the channels.

The cost obtained from the power-based correlation is an order of magnitude higher than the one obtained from the new correlation, which may still be reasonable for machines produced in very small quantities, but is almost certainly overestimating the cost when series production is considered.

In the present work, only the correlations for compressors were implemented since the design of the reversible machine is based on a compressor. A standalone turbine with the same diameter and rotational speed as those of a compressor would attain the same efficiency and enthalpy difference using approximately one-third of the number of stages as the compressor. This is because for the same flow coefficient the turbine can have a work coefficient three times higher than the one of the compressor [26]. Therefore, it is expected that a turbine would be less expensive than a compressor operating under the same conditions. However, the cost correlations of Eqs. (2) and (4) give a higher cost for the turbine than the corresponding compressor correlations for the conditions in the PTES cycles. This is because these correlations were originally developed for gas turbines, where the turbine is made of more expensive material and employs cooling as it operates at much higher temperatures than the compressor.

In the absence of more accurate data from manufacturers, we consider the correlations that include physically meaningful terms to account for the actual machine size more reliable. Therefore, either the equation of the present work or Eq. (3) with density correction is preferred. However, we recommend using the latter only for air-based turbomachinery since it does not include the effect of the working fluid which can have a significant influence (see Sec. 3.4).

### 3.4 Comparison of Working Fluids.

For the Brayton cycle, helium and carbon dioxide operating in the gas region were considered as alternatives to air. Helium was selected because it has the highest thermal conductivity among gases. Carbon dioxide instead was selected because of the lower specific heat ratio, which increases the heat capacity, leading to lower flowrates. The results suggest that the LCOS with carbon dioxide is 0.206 $/kWh, which is significantly higher than for the other working fluids. The high LCOS is due to the high cost of the low-temperature heat exchangers, which is due to the low heat transfer coefficient of carbon dioxide at low temperatures.

When comparing the results for air and helium with liquid storage presented in Table 4, helium has a lower LCOS. This is explained by considering the higher heat transfer coefficients observed, leading to smaller and cheaper heat exchangers. Despite having different volumetric flowrates, the helium and air turbomachinery have a similar size parameter, and therefore diameter. This is because in the case of helium, the peripheral speed is higher, leading to higher work per stage according to Eq. (8), which in Eq. (13) compensates for the higher volumetric flowrate. The helium machinery is still more expensive due to the higher number of stages: 28 compared to 15 for air in the hot machine. The increased number of stages requires a more complex bearing arrangement, which will increase the cost further. Storage costs are also higher for the helium cycle than for the air cycle due to the lower pressure ratio, as the ratio of the heat stored to the work increases at lower pressure ratios.

Despite the higher predicted capital cost, air might still be preferred as a working fluid as it is cheaper and easier to operate: no storage is required for the working fluid, sealing requirements are less strict, and the technology for air-based turbomachinery is already mature.

## 4 Conclusions

Different configurations of pumped thermal energy storage were analyzed, using the levelized cost of storage as a performance metric. Reversible turbomachinery was considered as a way to reduce costs, and a new correlation based on physical parameters was developed to estimate the cost of turbomachinery.

The new turbomachinery cost correlation developed results in lower turbomachinery costs than those correlations that do not include scaling parameters related to the physical size of the machine. The results indicate that the correlations used in previous studies, especially the ones developed for turbines, are unsuitable for cost estimates for pumped thermal energy storage systems, as they were developed mostly from gas turbines operating in significantly different conditions. This is confirmed by comparing the trend obtained between different working fluids of previous works. Further work is needed to obtain more accurate correlations for the cost and performance of turbomachinery for storage applications—particularly correlations that can capture the effects on the efficiency when a reversible machine is operated far from the best efficiency point are needed.

The results of the cycle optimization suggest that for a 50 MW discharge power and an 8 h discharge duration, Brayton cycles with liquid energy storage have the lowest levelized cost of storage, while packed bed energy storage systems are not competitive due to an excessive cost of the pressure vessel. At 4 h discharge duration, the performance gap between the liquid energy storage and packed bed energy storage systems reduces, however, lithium-ion batteries are likely to become the cheapest option for this case. Helium was found to be the working fluid with the lowest levelized cost of storage, mainly due to the lower cost of the heat exchangers than those of the other configurations.

Moreover, the results suggest that a subcritical Rankine cycle with ammonia and thermal oil storage achieves a levelized cost of storage close to that of the Brayton cycle, however, this configuration is not considered promising, mainly due to its poor part load efficiency.

## Acknowledgment

The research presented in this paper was developed as part of the project “Grid-scale—A Cost-effective Large-scale Power to Power Storage” (Grant No. 64020-2120) funded by the Energy Technology Development and Demonstration Program (EUDP) under the Danish Energy Agency. The financial support is gratefully acknowledged.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- $h$ =
enthalpy per unit mass, J/kg

- $s$ =
entropy per unit mass, J/(kg · K)

- $C$ =
cost, $

- $E$ =
energy, kWh

- $T$ =
temperature, K

- $U$ =
mean line peripheral speed, m/s

- $m\u02d9$ =
mass flowrate, kg/s

- $Q\u02d9$ =
volumetric flowrate, m3/s

- $W\u02d9$ =
power, kW

- $cp$ =
specific heat capacity, J/kg

- $td$ =
discharge duration for a single cycle, h

- CRF =
capital recovery factor, y

^{−1}- LCOS =
levelized cost of storage, $/kWh