Can Rocks Replace Batteries? An Analysis of Lifted Weight Energy Storage

Motivation

Fig. 1: Illustration of residential energy usage for a full day. The first valley represents Sleep Time, the period in which a majority of the population is asleep. The second valley occurs during typical working hours and is indicative of residents working at an office away from home. Finally, the two peaks are indicative of residents using power to prepare for their mornings and relax in the evening (humorously termed "TV Hours"). (Source: S. Barnes)

Solar energy is a promising technology that can be used to generate electricity without greenhouse gas emissions. The technology is already gaining traction in many countries, with worldwide and U.S. solar capacity growing by 19.0% and 27.3%, respectively, from 2020 to 2021. [1] However, solar energy faces a major challenge: it only generates electricity during the day. Specifically, solar energy follows a daily cycle (assuming a sunny day) where the amount of energy generated increases starting at dawn, reaches a plateau around midday, and decreases in the afternoon until reaching zero at sunset. [2] Moreover, if it's a cloudy day, the amount of energy generation will be significantly reduced.

In contrast, humans use energy in nearly the opposite cycle. A study based on energy data from 114 apartments in Western Massachusetts showed that residential energy consumption exhibited two peaks and two valleys throughout a given day. [3] (A qualitative reproduction of the data is shown in Fig. 1.) While Fig. 1. is only representative of residential energy demand, it highlights that humans use power according to their own schedules, not the cycle of the sun. Thus, energy storage is necessary if solar energy is to become a dominant energy source.

To solve this problem, several startup companies, including Gravitricity and Energy Vault, are pursuing lifted weight energy storage (LWES). As the name suggests, this technique stores energy by lifting large masses into the air and releases that energy by lowering the masses back down. Gravitricity plans to implement this technology in abandoned mineshafts. [4] On the other hand, Energy Vault is building a modular system to hoist a series of masses around 100 m into the air. [5] Using fundamental physics principles, the subsequent analysis aims to evaluate the feasibility of this type of energy storage, with a specific focus on the design proposed by Gravitricity.

Analysis and Discussion

Fig. 2: Simplified model of lifted weight energy storage (LWES). (Source: S. Barnes)

In a financial analysis, the Imperial College of London concluded that Gravitricity's LWES system design was the most cost-effective energy storage method (in terms of cost per unit of energy) for facilities with a capacity greater than 1 GWh. [6] This begs the question, is a system like this possible to build? To answer this, a LWES system will be modeled as a motor/generator, support structure, and dense mass (Fig. 2.). Additionally, to simplify analysis, the present discussion will neglect all nonidealities in the system, including friction, air resistance, and motor/generator losses. In light of these simplifying assumptions, the energy stored by LWES can be derived from conservation of energy and is equal to

where M is the mass in kg, g = 9.8 m sec^-2 is the acceleration due to gravity, and h₁ and h₂ are the initial and final heights of the mass in meters. The second key equation relates the weight of the dense mass to the diameter of the supporting cables. Assuming the mass is not accelerating, we have for the cable diameter D

where σ is the yield strength of the material. The yield strength represents the value at which a material undergoes irreversible deformation and can be used as a quantitative point of system failure. Combining the two equations, we find that

The height difference, (h₂-h₁), will be assumed to be 1 km, which is the mine shaft depth in Gravitricity's design. [4] From the financial analysis, E = 1 GWh (minimum) to be economically competitive. [6] Lastly, assuming a steel cable, σ = 250 MPa. [7] Substituting numbers, we obtain

The mass of such a cable would be 7,800 kg m^-3 × π × (4.3 m / 2)² × 1,000 m = 1.13 × 10⁸ kg. Thus, this is a massive cable that is clearly infeasible to construct (especially considering it must be at least a kilometer long).

Suppose instead that the dense mass is supported by 100 cables. In this case, each cable would need a diameter of D = (4/π×(1 GWh × 3.6×10¹² J/GWh)/((1,000 m) × (250×10⁶ Pa)×100))^0.5 = 0.43 m = 43 cm. In comparison, submarine power cables, which are some of the thickest in the world, only have a diameter of 30 cm. [8] Similar to the previous calculation, this result shows that the thickness of cable required makes a 1 GWh LWES system utterly infeasible. Thus, Gravitricity's LWES system is not the most cost effective energy storage solution and will likely be supplanted by cheaper technologies if it is pursued on a smaller scale.

Conclusion

Based on this simple model, it has been shown that the LWES system proposed by Gravitricity is not economically feasible if it is operating under the conditions assumed in this analysis. To improve the performance of the LWES system, the height change could be increased by building an above ground structure. However, this leads to increased structural concerns and associated costs. Additionally, it is prudent to mention that the above analysis was heavily idealized, with a real system storing less electricity than calculated. While this rudimentary analysis demonstrated an issue with LWES technology, it is recommended that a parametric study concerning height differences, exotic materials, alternative designs, and inefficiencies be conducted before completely ruling out this unique energy storage technique.

© Spencer Barnes. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] "BP Statistical Review 2022," British Petroleum, June 2022.

[2] Kh. B. Ashurov et al., "Solving the Problem of Energy Storage for Solar Photovoltaic Plants (Review)," Appl. Sol. Energy 55, 119 (2019).

[3] P. Amin et al., "Analysis and Demand Forecasting of Residential Energy Consumption at Multiple Time Scales," IEEE 8717811, 2019 IFIP/IEEE Symp. on Integrated Network and Service Management, 8 Apr 19.

[4] S. K. Moore, "The Ups and Downs of Gravity Energy Storage," IEEE Spectrum 58, 38 (2021).

[5] S. O'Neill, "Weights-Based Gravity Energy Storage Looks to Scale Up," Engineering 14, 3 (2022).

[6] "Levelized Cost of Storage Gravity Storage," Imperial College London, October 2018.

[7] R. C. Hibbeler, Mechanics of Materials (Pearson, 2015).

[8] B. Taormina et al., "A Review of Potential Impacts of Submarine Power Cables on the Marine Environment: Knowledge Gaps, Recommendations and Future Directions," Renew. Sustain. Energy Rev. 96, 380 (2018).