Abstract
Hyperloop system design is a uniquely coupled problem because it involves the simultaneous design of a complex, high-performance vehicle and its accompanying infrastructure. In the clean-sheet design of this new mode of high-speed mass transportation there is an excellent opportunity for the application of rigorous system optimization techniques. This work presents a system optimization tool, HOPS, that has been adopted as a central component of the Virgin Hyperloop design process. We discuss the choice of objective function, the use of a convex optimization technique called geometric programming, and the level of modeling fidelity that has allowed us to capture the system’s many intertwined, and often recursive, design relationships. We also highlight the ways in which the tool has been used. Because organizational confidence in a model is as vital as its technical merit, we close with a discussion of the measures taken to build stakeholder trust in HOPS.
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Notes
How small (i.e. how many passengers) is obviously one of the most important variables to optimize, but intuitively we can say that they should carry more people than a car and fewer people than a regional jet.
Ultra-high throughput is not only important for designing a system that is equipped to handle the demands of future population growth, but it is also a key part of reducing the total cost per passenger by enabling much higher utilization than a conventional rail or maglev system.
Virgin Hyperloop is the latest name for a company that has previously also been known as Hyperloop Technologies, Hyperloop One, and Virgin Hyperloop One.
We call the sum of CapEx per passenger-km and OpEx per passenger-km the total hard cost per passenger-km or the Levelized Cost of Transportation (LCOT). This is analogous to a concept in power systems engineering called the Levelized Cost of Energy (LCOE)(Ashuri et al. 2014).
These inputs are referred to as “substitutions” in GPkit verbiage to reflect the notion that any free variable can be substituted with a fixed value.
The dimensionality also scales with the number of routes. For example, when optimizing over three routes, each Pod Performance variable has six elements.
The hyperstructure name derives not only from hyperloop, but also from the generalization of the substructure (i.e. columns) and superstructure (i.e. tube).
References
Agrawal A, Diamond S, Boyd S (2019) Disciplined geometric programming. Optim Lett 13(5):961–976
Andersen MS, Dahl J, Vandenberghe L (2013) Cvxopt: a python package for convex optimization. https://cvxopt.org
Ashuri T, Zaaijer MB, Martins JR, Van Bussel GJ, Van Kuik GA (2014) Multidisciplinary design optimization of offshore wind turbines for minimum levelized cost of energy. Renew Energy 68:893–905
Bierlaire M, Axhausen K, Abay G (2001) The acceptance of modal innovation: the case of swissmetro. In: Swiss transport research conference
Boyd S, Kim SJ, Vandenberghe L, Hassibi A (2007) A tutorial on geometric programming. Optim Eng 8(1):67–127
Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, New York
Boyd SP, Kim SJ, Patil DD, Horowitz MA (2005) Digital circuit optimization via geometric programming. Operat Res 53(6):899–932
Brooks L, Liscow ZD (2019) Infrastructure costs. Available at SSRN 3428675
Burnell E (2020) A worker-centered approach to convex optimization in engineering design. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge
Burnell E, Damen NB, Hoburg W (2020) GPkit: a human-centered approach to convex optimization in engineering design. Conference on human factors in computing systems (CHI). Association for Computing Machinery, Honolulu. https://doi.org/10.1145/3313831.3376412
Burnell E, Pillai PP, Yang MC (2020) Maps, mirrors, and participants: design lenses for sociomateriality in engineering organizations. arXiv:2008.06616, 9 . arxiv:2008.06616
Chiang M (2005) Geometric programming for communication systems. Now Publishers Inc, Delft
Goddard EC (1950) Vacuum tube transportation system. US Patent 2,511,979
Henderson RP, Martins JR, Perez RE (2012) Aircraft conceptual design for optimal environmental performance. Aeronaut J 116(1175):1–22
Hoburg W, Abbeel P (2014) Geometric programming for aircraft design optimization. AIAA Journal 52(11):2414–2426
Hoburg W, Kirschen P, Abbeel P (2016) Data fitting with geometric-programming-compatible softmax functions. Optim Eng 17(4):897–918
Jedinger A (2020) Homopolar linear synchronous machine. U.S. Patent 20200091807A1
Kirschen PG, York MA, Ozturk B, Hoburg WW (2018) Application of signomial programming to aircraft design. J Aircr 55(3):965–987
Kroo I, Altus S, Braun R, Gage P, Sobieski I (1994) Multidisciplinary optimization methods for aircraft preliminary design. In: 5th symposium on multidisciplinary analysis and optimization
Lee D Jr (2000) Methods for evaluation of transportation projects in the USA. Transp Policy 7(1):41–50
Leeham (2018) Don’t look for commercial bwb airplane any time soon, says boeing’s future airplanes head. https://leehamnews.com/2018/04/03/dont-look-for-commercial-bwb-airplane-any-time-soon-says-boeings-future-airplanes-head/
Levy A (2020) Low- and medium-cost countries. https://pedestrianobservations.com/2020/09/25/low-and-medium-cost-countries
Martins JR, Alonso JJ, Reuther JJ (2004) High-fidelity aerostructural design optimization of a supersonic business jet. J Aircr 41(3):523–530
Martins JR, Lambe AB (2013) Multidisciplinary design optimization: a survey of architectures. AIAA Journal 51(9):2049–2075
Meluso J, Austin-Breneman J, Uribe J (2020) Estimate uncertainty: miscommunication about definitions of engineering terminology. J Mech Design 142(7):13. https://doi.org/10.1115/1.4045671
MOSEK: Mosek optimizer api for c, version 8.1 (2019). https://docs.mosek.com/8.1/capi/index.html
Oster D, Kumada M, Zhang Y (2011) Evacuated tube transport technologies (et3) tm: a maximum value global transportation network for passengers and cargo. J Modern Transp 19(1):42–50
Pillai PP, Burnell E, Wang X, Yang MC (2020) Early-stage uncertainty: effects of robust convex optimization on design exploration. J Mech Design. https://doi.org/10.1115/DETC2020-22626
Small KA (2012) Valuation of travel time. Econ Transp 1(1–2):2–14
Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Struct Optim 14(1):1–23
SpaceX: Hyperloop alpha (2013). https://www.tesla.com/sites/default/files/blog_images/hyperloop-alpha.pdf
Wardman M (2004) Public transport values of time. Transp Policy 11(4):363–377
White V (2016) Revised departmental guidance on valuation of travel time in economic analysis. Office of the Secretary of Transportation, US Department of Transportation, Available at: https://www.transportation.gov/sites/dot.gov/files/docs/2016%20Revised%20Value%20of%20Travel%20Time%20Guidance.pdf
Acknowledgements
This work would not have been possible without the dozens of colleagues who provided models and inputs to be used in HOPS. We thank them for their support and their trust. We also thank Alex Esseveld and Naveen D’souza Lazar for the renderings used in this paper. Finally, we thank Jim Coutre for his leadership, his enthusiastic use of HOPS, and the insightfulness with which he approaches hyperloop system design.
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Kirschen, P., Burnell, E. Hyperloop system optimization. Optim Eng 24, 939–971 (2023). https://doi.org/10.1007/s11081-022-09714-7
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DOI: https://doi.org/10.1007/s11081-022-09714-7