Operational-driven optimal-design of a hyperloop system ☆

We present an operational-driven optimal-design framework of a Hyperloop system. The novelty of the proposed framework is in the problem formulation that links the operation of a network of Hyperloop capsules, the model of the Hyperloop infrastructure, and the model of the capsule’s propulsion and kinematics. The objective of the optimisation is to minimize the energy consumption of the whole Hyperloop system for diﬀerent operational strategies. By considering a network of energy-autonomous capsules and various depressurization control strategies of the Hyperloop infrastructure, the constraints of the optimisation problem represent the capsule’s battery energy storage system response, the capsule’s propulsion system and its kinematic model linked with the model of the depressurization system of the Hyperloop infrastructure. Depending on the operational scheme and lengths of the trajectories, the proposed framework determines optimal operating pressures of the Hyperloop infrastructure between 1 . 5 − 80 𝑚𝑏𝑎𝑟 along with the maximum capsules cruising speeds. Furthermore, the proposed framework determines maximum operational power of the capsule’s propulsion system in the range between 1 . 7 − 5 𝑀𝑊 with a minimum energy need of 25 𝑊 ℎ ∕ 𝑝𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 ∕ 𝑘𝑚 .


Introduction
The Hyperloop transportation system comprises a set of capsules traveling at (almost) sonic speed in a constrained space characterized by a low-pressure environment (i.e., a tube) housing a dedicated set of rails that enable the capsules' guidance, levitation and/or suspension.The Hyperloop is characterised by higher speeds, compared to existing ground transportation systems and, due to the large reduction of the drag aerodynamic losses, can require lower energy needs with respect to electric trains and intra-continental aircrafts.Since 2015, when the research activity on the Hyperloop system was relaunched, very few papers have addressed the problem of the optimal design of this very peculiar transportation system.The very first technical question was whether the Hyperloop capsules can be designed to be energy-autonomous in order to avoid the electrification of the rail with the obvious consequences on the simplification of the tube design and cost.In [1] , the actual authors reply to this question by proposing an optimisation framework capable of designing the propulsion system (PS) of a Hyperloop capsule that is supplied by a battery energy storage system (BESS).The approach proposed in [1] did not take into consideration, however, the energy needs of the tube (i.e., associated with its depressurization) or the impact of the Hyperloop tube operation on the total energy consumption and the associated capsule PS design.Indeed, in order to minimise the overall energy needs of both the capsules' PS and tube depressurization, the operation of the Hyperloop infrastructure (i.e., the tube operational pressure, the interval between subsequent depressurization processes, as well as the number of capsules occupying the tube per day) could be coupled with the capsule's PS optimal design.Furthermore, the operation of the infrastructure and the capsules' PS design could change as a function of the length of the tube, and determining whether this is the case is a technical question that deserves to be addressed.
Therefore, there are four fundamental questions that need to be addressed.(i) What is the optimal operating pressure inside a Hyperloop tube in order to minimize its global energy consumption ?(ii) What is the achievable minimal energy consumption of a Hyperloop system ?(iii) Is there a strong dependency between the infrastructure operation and capsule's PS design ?(iv) Which is the impact of the magnetic levitation on the energy consumption of the capsule?
Although the existing literature has not yet produced specific contributions to address these questions, it has produced a number of publications that addressed similar problems.Indeed, it is worth observing that the design of a Hyperloop system presents similarities to an inverted Maglev train (the rail of the Maglev is the main source of power generating the capsule thrust).In [2 , 3] and [4] , other researchers have shown how the main characteristic of Maglev levitation and guidance systems can be determined by solving a suitable optimization problem.In [4] and in [5] , Cassat and Jufer studied how the propulsion and energy transfer to an in-motion vehicle can be jointly modeled in order to be suitably op- timised.The operational performance and safety standards for Maglev system have been further studied in [6] by Cassat and Jufer.Publications addressing the problem of the design of wireless energytransfer systems into in-motion vehicles are also worth mentioning: in [7] by other researchers, this type of systems is studied for vehicles that host a BESS and travelling at atmospheric pressure.An economic viability and environmental study about wireless power transfer is presented in [8] by Limb.In [9] , He proposed a multiobjective co-optimization problem for a vehicle with a hybrid power supply with the main purpose of improving the energy efficiency of the vehicle propulsion system along with it ride comfort.As a result, a Pareto front is obtained to analyse the best compromise solutions between the power consumption and ride comfort.
A hybrid power supply system of an electric train is discussed in [10] by other researchers, where the BESS has the role of a dedicated energy buffer for transfering power between the kinematic energy stored in the train and its regenerative braking system.The model also contains a dissipative braking system due to the limited energy capacity of the onboard BESS.The proposed design determines the best trade-off between the optimal BESS capacity, energy-saving rate, volume of the system, capital cost, maximum power and mass of the train.
Note that none of the aforementioned manuscripts presents a complete framework that can be used to address the Hyperloop-specific questions listed above.In this respect, our original contribution in this paper is to fill this gap by proposing a non-convex and non-linear optimization framework.The multi-objective function of the proposed optimization targets the minimization of the total energy consumption of the Hyperloop system (composed by the capsules' and tube's depressurization energy needs), which is subject to a comprehensive set of constraints modeling in detail the Hyperloop capsule's kinematic, and its PS, the tube depressurization process, leaks compensation, and the operation of the Hyperloop system.
The structure of the paper is the following: In Section II, we illustrate the model of the Hyperloop infrastructure where the energy consumption of the vacuum pumps (both for the initial depressurization process and air leaks compensation) is taken into account with the model of the energy consumption of a network of capsules travelling on a trajectory of generic length.We also represent the operation of the whole infrastructure.In the Section III, we propose the optimization framework targeting the minimization of the global energy need of the Hyperloop system subject to the capsule's infrastructure and operational constraints illustrated in Section II.In Section IV, we present a numerical example used to obtain the results discussed in Section V. We complete this last sections with a comprehensive analysis related to the average energy consumption (Wh/passenger/km).In Section VI, we conclude the paper with a summary of the findings.

Model of the hyperloop infrastructure
In order to model the operation of the Hyperloop system, we represent two processes: (i) the depressurization that refers to the decrease of the pressure inside the Hyperloop tube from the atmospheric pressure,  0 , to the desired one,   ( <  0 ) , and (ii) the compensation of the air leaks in order to maintain the pressure   as the material used to manufacture the tube is assumed to not be perfectly airtight.The depressurization and leak compensation processes are considered independent (see Section II.C for further details).
The system is supposed to have available a total number of vacuum pumps, say  ′ , between Stations A and B of the Hyperloop tube.All the  ′ pumps are used for the depressurization process and a subset of them, say  ′′ , is used to compensate the air leaks. in view of the different air volumes that these pumps have to process, the need for  ′′ ≤  ′ is obvious.Fig. 1 provides a schematic view associated with the assumptions reported above.
As shown in Fig. 2 , the parameters of the tube geometry are as follows:  is the length of the Hyperloop tube,  represents the thickness of the tube,   corresponds to the inner diameter of the tube, and   represents the outer diameter of the tube, with   = 2 ⋅  +   .The inner cross section of the tube is simply The  ′ pumps are expected to operate periodically when the Hyperloop tube needs to be completely depressurized, starting from the atmospheric pressure  0 .This period of time is named   .Assuming the depressurization process to be an adiabatic thermodynamic transformation, we can easily compute the time necessary,   , to bring the Hyperloop tube pressure from its initial value,  0 , to the final one,   shown by other researchers in [11][12][13] .This computation is expressed by (1) where   represents the pumped-air volume flow expressed in ℎ ] , usually a known quantity from the pump's manufacturer expressed as a function of the pressure.
Equation ( 2) quantifies the energy needed for the depressurization of the Hyperloop tube between subsequent maintenance periods, where   represents the power of a single vacuum pump (during the operations, the pump's power is constant).
By referring to a daily operation horizon (i.e., 24h) of the Hyperloop tube, we assume the capsules to be launched within an operation period named   .Therefore, it is reasonable to impose the following inequality   ≤ 24 ℎ −   as we might want the initial depressurization process to be finalised in a relatively short time and, then, to start the scheduled operation.
The next step is to determine a model of the Hyperloop air leaks.The material used to build the Hyperloop tube should be characterized by a known air permeability.In view of the evident impact that the tube material has on the Hyperloop infrastructure cost, we consider the case of concrete as strongly advocated in [14] by Heller and we assume its permeability to be isotropic.The use of concrete corresponds to a worstcase scenario regarding the influence of tube's air leaks.On the contrary, the use of steel tubes defines the situation where the corresponding airpermeability is close to zero.However, steel tubes presents the main disadvantage represented by the associated cost of the infrastructure.In view of the above, here below we mainly refer to the case of concrete tubes and the reader may refer to the sensitivity analysis regarding the air permeability of the tube's material shown in Section V.
The material and fluid parameters taken into account to estimate the leaks are   , which represents the concrete's air permeability, and , which corresponds to the dynamic air viscosity.
In order to compute the energy needed to compensate for the air leaks, we need to express the air-leak volumetric flow rate,   .This quantity can be estimated using the Darcy's law, assuming the compressible characteristic and the radial bidirectional air flow.According to [11][12][13] , the air leaks volumetric flow rate is given by (3) .
To compensate for the air leaks, we need to activate  ′′ pumps in order to satisfy the following inequality   ≤  ′′ ⋅   .The daily energy needs to supply the  ′′ pumps,   , is given by ( 4) , where   represents the time the air leakage is occurring.As we are modeling the operation of the Hyperloop tube over a 24h horizon,   = 24 ℎ because leaks are always present.
The energy required to operate the Hyperloop tube is expressed by   (needed between two subsequent complete depressurization periods) and   (daily need).These two quantities are used next within a suitably defined optimization problem that enables us to asses the optimal parameters associated with the Hyperloop propulsion and the optimal parameters associated with the operation of the Hyperloop tube along a given time horizon.

Model of the hyperloop capsule
The operation of a Hyperloop system involves the launching of several capsules travelling in series in the same tube.This operation is required as the number of passengers per capsule is limited to a few dozen.Hence, a set of travelling Hyperloop capsules in the above-mentioned tube is considered, where   represents the number of launched capsules per unit of time.A detailed model of the capsule propulsion can be found in [1] and the main equations can be found in this section, as they are used in the optimization problem proposed here.

Trajectory
As shown in Fig. 3 , the total length of the trajectory,  , is split into  different zones: {  1 ,  2 , … ,   } each one corresponding to a given state of the capsule propulsion system: acceleration, constant speed, and deceleration.In Fig. 3 ,  represents the generic position of a capsule along the trajectory and  the elapsed time relative to the generic discrete position  with respect to the trajectory origin.The trajectory's space interval [0 ,  ] is sampled at regular intervals Δ.Correspondingly, the time intervals Δ for the capsule to travel each discrete space interval Δ, can be simply computed from the following equation: Δ =  (  − 1)Δ + 1 2  ( )Δ 2 (being  and  the capsule's speed and acceleration) such that the discrete positions are  = 0 , 1 , … ,  Δ .As the capsules can move only forward, for each , we can associate a corresponding unique discrete time index  = 0 , … ,    , … ,   where    = ∑   Δ  .

The model of the capsule propulsion system
the Hyperloop PS is assumed to be composed by three main components: (i) an energy reservoir represented by a BESS, (ii) a DC/AC power electronic converter (usually a voltage source inverter (VSI)) and (iii) an electrical machine consisting of a linear induction motor (LIM) as studied by other researchers in [15][16][17][18] .a) Model of the BESS: the capsule's source of power is supposed to be a BESS that is modeled at the cell level.As discussed in [19] by the actual Authors, due to the numerical complexity and large number of equations and corresponding state variables, we choose a simple equivalent circuit of a cell where the charge diffusion dynamics are not taken into account as the obtained results are not affected by more sophisticated cell models [20] as shown by Einhorn et al.
Assuming the BESS to be composed by identical cells where   and   represent the number of cells in series and in parallel, respectively, Fig. 3.The generic trajectory of Hyperloop capsules.Adapted from [1] . the equations of the BESS are given by (5) .(5) where: •     represents the open-circuit voltage of the cell, and it varies with the state-of-charge ( ) (e.g.Dees in [21] ).
•   represents the drawn current through a single cell.
•   represents the equivalent series resistance of the cell.It embeds the equivalent resistance of the cell's terminals' connections with the next cell too.  is assumed to be known and constant (e.g., Zhao in [22] ).
•   corresponds to the voltage accessible in the correspondence of the cell's terminals; it is affected by the voltage drop produced by the   .

• 𝑉 𝑏𝑎𝑡𝑡
is the open-circuit voltage, which is a function of the cells .•   embeds the all the cells' and connectors' resistances.
•   represents the capacitance of the BESS.
•   is the total current provided (or absorbed) by the BESS.
•   and   are the accessible voltage and power at the BESS terminals.
The function    ( ) is available from the cell's manufacturer.b) Propulsion: acceleration and speed profiles are dependent on the traction force, where the most important parameters are: , respectively.Thus, the final expression of total mass is given by (6) , where  0 is constant and considered a passive mass,   is the associated cell's mass embedding the unitary mass plus cells' wiring,   represents the supplementary power transfer efficiency from BESS to LIM (i.e., the efficiency of the power electronics converter),  ( ) is the power factor of the VSI and   the maximum power provided by a cell.Naturally, we have that  ∼     who are two of the main control variables for the design of the capsule's PS.In order to keep the proposed optimisation problem tractable, these efficiencies and VSI power factor are assumed constant.
The kinematic model of the capsule is represented by the acceleration,  , and the speed,  , both sampled at every Δ (or Δ since these two indexes have a unique 1:1 correspondence).The trajectory length is  divided into three main zones: acceleration, constant speed, and braking zones; with a total number of discrete points [  Δ ], considering Δ ≪  .
The capsule is subjected to two main forces: the traction provided by the propulsion, and the drag force.The latter is given by (7) , where  is the fluid density,   represents the capsule's drag coefficient, and  is the cross section capsule's surface.  is function of  , as discussed in [23] by Kang et al.
The traction force and mechanical traction power,   and   , respectively, are given in (8) by using a mono-dimensional Newtonian kinematic model of the capsule.
The levitation drag of the capsule may be considered to be null due to a potential usage of a suspended capsule solution as mentioned in Fig. 4. Operational scheme for the Hyperloop infrastructure.[24] .In practice, a drag-less magnetic levitation can be realized by using in the capsule's propulsion system a Single Sided Linear Induction Motor (SSLIM).Indeed, a SSLIM may provide the necessary thrust and levitation force with no magnetic drag.Nevertheless, the presence of a magnetic levitation system is analyzed in the section V in order to quantify the effects of the magnetic levitation drag on the power/energy requirements of the capsule propulsion system (see the sub-section V.F).
Through the reduction of the pressure inside of depressurized tubes, the Hyperloop system reduces the density of the tube's fluid.The expression of  given by ( 9) assumes (i) the air to behave as an ideal gas, where  0 represents the fluid density at standard atmospheric conditions (  0 = 1 .225   3 for  0 = 1 .013  , and  = 288 .15 ) and (ii) the operating temperature of the depressurized Hyperloop tube to be equal to The electrical power provided by the BESS,   , is directly related with   through the efficiency of the LIM,   , as in (10) .
The total energy consumption for one capsule,   , is calculated in (11) .
Assuming a set of capsules traveling in the same tube and launched within a given period of time, we can define   as the number of launched capsules in one day during the   .The total energy consumption per day of these capsules is simply   =   ⋅   .

Accounting for the kantrowitz limit to determine the upper-bound of capsules' speed
The study of a high speed capsule in a confined environment (i.e., tunnel or tube) implies compressibility effects.As capsules travel at high speed through a tube, they can choke the flow of the fluid in the area between the capsules' cross section, , and the tube's cross section,   .The assessment of the choke flow regime plays a major role in the de-termination of the maximum speed of the capsules,  max , which is a fundamental constraint of the capsule kinematic model.
By making reference to the speed of sound,   , in the air for a given temperature,  , we make use of the standard definition of the Mach number . Furthermore, we introduce the quantity We assume that the flow of the fluid around the capsule obeys the conventional isoentropic gas equations.With this assumption, the limitation of the maximum capsule's speed results from the Mach number   = 1 of the fluid flow around the capsule, which represents the maximum value of the Mach number before the choke flow.Equation (12) allows to compute   as a function of  ∞ ,  =     is the isoentropic expansion factor of the gas in the tube environment and   and   represent the specific heats of the gas at constant pressure and volume.Therefore, (12) allows to link the cross-sectional dimensions of the tube/capsule with the maximum speed of the capsules to avoid the choke-flow regime to take place.
It is worth observing that the detrimental effects of pressure waves generated by a supersonic fluid between the capsules and the tube can be neglected since the maximum speed of the capsule enforced by (12) avoids the choke-flow regime to take place.This analysis has been shown by Kang and Ham in Fig. 7 and Fig. 8 of reference [23] .

Hyperloop infrastructure operation
The energy needs of the whole Hyperloop system is given by adding (i) the energy associated to the initial depressurization of the tube from  0 to   , (ii) the energy needed to compensate the air leaks (i.e., to maintain   inside the tube), and (iii) the energy used by the capsules' PSs.
As we are interested in defining the operational parameters of the Hyperloop infrastructure (essentially   ) and the characteristics of its PS, which that minimize the whole Hyperloop energy consumption, we need to define the ways to operate the infrastructure with respect to the vacuum pumps.We can identify two main ways to operate the infrastructure: (i) At the end of the daily operations, the  ′ pumps are shut down; hence, a new stage of tube depressurization to achieve   is needed.(ii) The  ′′ pumps continue to maintain the   inside the tube independently of the operations.This option represents a relatively more advantageous solution with respect to the objective of minimizing the total energy consumption.Therefore, the nominal operation process involves an initial depressurization stage from  0 to   including  ′ vacuum pumps, maintaining of the nominal pressure (parasitic air leakage in the tubes) at the level of   and   traveling per one direction per day during the   .  is independent of   as the air leakage always occurs when   ≠  0 .The entire operations process of the Hyperloop system is periodical with the period   , as shown by the operations' diagram of Fig. 4 .

Formulation of the optimization problem
Given the model of the Hyperloop infrastructure, capsules and operations, we formulate operational-driven optimal-design problem of the Hyperloop system, as in (13) .The objective function is to minimize total energy requirement of the whole Hyperloop system.This is expressed by the objective   +   (   +   ) , where: •   is the energy required by the  ′ pumps to depressurize the Hyperloop tube between subsequent maintenance periods   (expressed in days); •   is the daily energy need of the  ′′ Hyperloop tube pumps compensating for the air leaks; •   is daily energy need of the Hyperloop capsules.

The decision variables of the problem are:
•   : number of cells in series in the capsule's BESS; •   : number of cells in parallel in the capsule's BESS; •  : capsule acceleration along the trajectory; •   : pressure inside the Hyperloop tube; •  ′ : number of pumps of know rated power to depressurize the hyperloop tube; It is worth observing that the single elements of the objective function have conflicting behaviours as   decreases with the decrease of   , whereas   and   increase.Therefore, the problem (13) determines the best the trade-off between the energy required by the depressurization process and the losses due to the drag force of the energyautonomous capsules.As in [1] , in (13) we consider the constraints associated with the model of the capsule's PS, in addition to the constraints of the infrastructure model and its operation.
The kinematic variables of the capsule are constrained as follows: The maximum speed is limited by  max , namely by the establishment of the choke-flow regime in the air surrounding the travelling capsules and the tube, the acceleration is limited to the value presented in [1] derived from civil air crafts, and the traveling time at the end of the constant speed zone,  =    (  =   Δ ), is limited to    .Regarding the constraints of the capsule's PS, the discharge rate of the cells that compose the BESS have to be lower than the maximum admissible discharge rate of the selected cell's type.   is bounded with respect to the railway electrification system standard through the control variable   multiplied by the maximum    .As this latter parameter is known once the cell technology is selected, we simply require that   min ≤   ≤   max .Finally, the BESS  should be in the range between  min and  max .The minimum value of the  can be found at position  =   Δ , which represents the end of the constant speed zone.After this point, the capsules enter the deceleration zone where a part of the braking is ensured by a regenerative one [25] - [26] , but limited by the maximum charging rate of the considered cell,     ℎ .The regenerative braking zone is not taken into account in (13) , as it is a consequence of acceleration and constant speed zones.The  at the end of the trajectory,    when  =  Δ , it is in any case computed.For the infrastructure, the volume flow of the  ′′ pumps should compensate at least the   , and the time to depressurize the tube from  0 to   ,   , is constrained to be less than 24 ℎ −   .
As discussed in [1] , the optimisation problem is non-convex and it has been solved using a gradient-based method [27] - [28] .The presence of mixed integer decision variables   ,   ,  ′ and  ′′ was solved by treating these as continuous variables that, once determined, are rounded to the nearest integer.The problem ( 13) can be solved by different numerical solvers.We opted to use Yalmip coupled with the   solver in Matlab.The initialization of the solver is performed by fixing the initial values of the decision variables (see section IV.A).Then, for all solutions of ( 13) obtained in correspondence of each of the initialisation of the decision variables, we retain only the one with the least objective value.

Numerical assumptions
In this section, we provide the numerical assumptions used to solve the problem (13) .

Infrastructure assumptions
The diameter of the tube is selected based on the values the values reported in [29] , [30] .Therefore, we assume for the simulations   = 4  .
For the thickness of the tube, the lower bound is given by reinforced concrete adopted by the tunnelling industry:  = 25  (e.g., as used for the Lötschberg tunnel in Switzerland and mentioned in [31] ).The average value for the permeability of the reinforced concrete is   = 5 ⋅ 10 −18  2 as reported in [32][33][34] .
Regarding the other physical quantities associated with the infrastructure,  0 = 1  is the standard atmospheric pressure and, for the dynamic air viscosity, we assumed  = 1 .85 ⋅ 10 −5   ⋅  (at 298.15 K).The rated power of the vacuum pumps and the associated characteristics are taken from real data: We refer to the Dessin Cobra NC 2500 B for which   = 55  and the characteristic   =  (   ) are both documented in [35] .
Finally,  represents the length of the tube (or trajectory) for which we have considered the following three values:  = 226  ,  = 500  and  = 1000  as they are associated to typical distances of intracontinental flights.

Capsule assumptions
Most of the numerical assumptions for capsules are those made in [1] .

Assumptions on the capsule trajectory
as we mentioned in the above section, we consider  = 226  (14) for the first selected length of the Hyperloop trajectory as it represents the distance between the two largest economical poles in Switzerland: Geneva and Zürich.The actual timing for this trip with the Swiss federal railways is in the order of 2h30min, whereas time travel by plane is around 45 minutes (not including the boarding time).In order to extensively validate the optimization process, we also consider  = 500  and  = 1000  .The discrete space sampling of the trajectory is Δ = 100  , resulting into 2260 points for  = 226  , 5000 points for  = 500  and 10,000 points for  = 1000  .The value of Δ can be determined as the capsule acceleration is upper-bounded to 0 .3204  (being this value derived from civil air crafts [1] ) and because we would like to have an upper bound on the variations in the capsule speed we would like to observe in correspondence of the point in the trajectory with the maximum acceleration.Since we would like to observe a maximum difference of speed of 50 km/h between two equidistant discretization points along the trajectory, for a maximum acceleration of 0 .3204 , the corresponding Δ = 100  .Such a computation has been done in correspondence of the first and the second node of the discretized trajectory where we have the capsule's maximum acceleration.
In ( 14) -( 16)  1 and  2 represent the acceleration zones,  3 the constant speed zone, and  4 the deceleration one.The chosen values for the extension of these zones is to have them large enough to allow the optimization finding the optimal values of the capsule's speed.To be specific, for the maximum acceleration of 0 .3204  and and the extensions of the acceleration zones  1 and  2 of total 12km, the maximum potential speed is of 988 km/h which is larger than the upper bound we have chosen for the maximum speed to avoid choked-flow conditions (see Section IV.C).Regarding the deceleration zone, its extension is of 20 km, i.e. a value larger than the extension of the acceleration zones allowing for the optimisation problem to have ample margin to determine the optimal speed profile while satisfying the constraints on the maximum speed and acceleration.It is worth noting that the extension of the acceleration/constant speed/deceleration zones can be also imposed by the modeler according to safety requirements of the Hyperloop infrastructure.

Assumptions on the capsule and PS
each of the capsules launched per day (   ) carry a payload mass equivalent to 50 persons [37] (the corresponding mass can be also a cargo).The average mass payload attributed for a single person is 80kg, which means a total payload mass of 4000kg; the considered mass of mechanics is 6000kg.These two assumptions translate into  0 = 10000 .
According to [38] , the frontal cross-section surface of capsules is assumed to be  = 3 .14  2 .Regarding the dependency of the drag coefficient with the speed of the capsule,   (  ) , we adopted the values shown in  .3 ( "3  _ ") of [23] for a blockage ratio   = 0 .25 (as it is in our case).For the reader's convenience, such dependency   (  ) is also shown in Fig. 5 .Considering the lack of available literature regarding Hyperloop capsules dependency of aerodynamic drag coefficient with speed, a dedicated sensitivity analysis on the obtained optimal solutions is reported in Section V.
Regarding the PS, the efficiency of the power transfer is assumed to be   = 0 .95 , whereas the efficiency of a high-speed LIM is assumed to be   = 0 .65 (this value has been inferred by preliminary tests at the Authors' laboratory).The parameter  1 for LIM is selected according to a Hyperloop capsule prototype realised by our laboratory and assumed to be  1 = 0 .091   .The values for  2 is chosen with respect to industry-grade VSI used by the automotive sector:  2 = 0 .075   , as well as  ( ) = 0 .6 .Furthermore, in Section V it is shown how the dependency of both   and  ( ) with the capsule's speed influences the solution of (13) .Fig. 5.   dependency with the Mach number,  , adapted from [23] .
In (17) , we indicate the upper bounds for the accelerations 1 in the Sections  1 ,  2 and  3 (the values of these upper bounds are the same used in [1] ).In (17) , we also indicate the maximum speed,  max , and maximum travel time at the end of the constant speed zone,   1 ,   2 and   3 for the three selected trajectory lengths  = 226  ,  = 500  and  = 1000  .The maximum travel times at the end of constant speed zone are bounded.
Regarding the BESS, the numerical assumptions shown in ( 18) are based on a real cell: the Kokam SLPB 11543140H5.This cell is graded for a continuous discharge rate up to 30C and exhibits remarkable performance in terms of ageing (more than 1000 cycles at 90% depth-ofdischarge).The constraints on the    are based on the IEC 60850 standard [39] and the EN 50163 standard [40] and are translated into a direct constraints on the decision variable   given the maximum open-circuit voltage of the selected cell.The bounds on the  are chosen with respect to the safety operations of the cell.The other cell's parameters have been characterized at our laboratory.

Assumptions for the computation of the kantrowitz limit and associated capsules' maximum speed
By considering  = 288 .15  and  = 1 .4032 , Fig. 6 illustrates the dependency of  ∞ with   =  been obtained by numerically inputting the above-mentioned values into (12) .

Hyperloop infrastructure operational assumptions
For the infrastructure operation, the most relevant parameter to fix is   , as it represents the total number of hours-per-day where capsules are launched into the Hyperloop infrastructure.In order to define a value for this parameter, we made reference to the actual daily time for operations adopted by the Swiss Federal railways for which   = 16 ℎ .The other parameter to fix is the number of capsules launched per day (   ).In [37] , it is reported a rate of 1  2  which translates to a   = 480   , equivalent to a maximum number of passengers per day of   = 24000   .Furthermore, the value adopted for   can be coupled with the capsules distance at cruising speed vs the distance needed to decelerate them in case of an emergency braking.Indeed, it is reasonable to suppose the network of capsules being controlled by an automatic system capable to estimate the position of each capsule along their trajectories (i.e., a classical state estimator).Such an automatic system governs the entire network of capsules and is capable to handle any error appearing from any capsule.In this case, all the capsules enter in an "Error State " where a safe braking is applied to all capsules to stop them.Assuming the capsules traveling at a maximum speed of 616  ℎ ( 171   ), namely the maximum cruising speed we have determined in Section IV.C, a safety braking deceleration should not exceed 0 .5  (i.e., 4.905   2 ) (such a value for an emergency deceleration has been proposed to prevent passengers' injuries in the document by C. Grover, I. Knight in [36] ).Therefore, the corresponding safety braking distance would be:   = 2457 .8  .For the assumed   = 1  2  , the time difference between two capsules is of 120  and, at the maximum speed of 171   , the distance between two subsequent capsules is   = 20520  .It is evident that   <<   guaranteeing a safe emergency braking with ample margin.Regardless of the above reasoning, a sensitivity analysis on the influence of   on the solutions provided by the optimization problem is contained in Section V.

Results
This section illustrates the results obtained by solving (13) with respect to different values of the main parameters of the proposed optimisation problem.The results are show with respect to the quantities shown in Table 1 as they represent the main operational characteristics of both the Hyperloop infrastructure and capsule's propulsion system.

Main numerical assumptions
As the total energy consumption of the Hyperloop system is largely influenced by   , the analyses discussed in this section are carried out by increasing this parameter up to a certain value until the total energy consumption of the infrastructure tends to an asymptotic value.Hence,   is varied in the following set:   = { 1 , 7 , 14 , 21 , 28 , 35 , 42 , 70 , 84 , 168 }  for each of the three considered trajectory lengths.As the optimisation problem is non-convex, yet numerically tractable, it is solved by using a gradient-descent method where the initial conditions were varied within intervals that have a technical feasible meaning.Then, the obtained solutions were ranked according to their objective value in order to determine the one with the least value.For every   and  , the initialization of the control variables were made accordingly to the intervals show in Table 2 .The   ,   and  initialization values were chosen accordingly to [1] , and for the initialization for   , we assumed the range indicated in [1] and [29] .

General observations
Fig. 7 shows the optimal values of   as a function of   for each trajectory length.The optimal pressure inside the tube,   varies from 2 .82  to 76 .92  for  = 226  , from 1 .17  to 54 .5  for  = 500  and from 1 .14  to 17 .25  for  = 1000  .As a first general conclusion we can see that for lower values of   , the largest fraction of used energy is associated to the Hyperloop infrastructure, namely   +   .For values of   in the range between 42  to 168  , the energy needed by the infrastructure is of the same order of magnitude of the energy used by the capsules, whereas, for higher values of   (and of   ), the energy used by the capsules is dominant (this dependency is expected as   are lower for higher values of   ).It is also worth observing that, for the same value of   , the optimisation problem determines optimal tube's pressures that decrease with the increase of the trajectory length as, for longer trajectories, capsules' aerodynamic energy losses become more important compared to the energy used by the Hyperloop infrastructure (i.e.,   +   ) that tends to a constant value for   → ∞ (see Fig. 8, Fig. 9 ).Furthermore, as the value of   Fig. 7. Dependency of the optimal operational internal pressure of the tube,   , with   .Fig. 8. Dependency of the tu be depressurization energy,   , with   .increases, the value of   increases as well.This trend is due to the nonlinear behaviour of   and   as a function of the tube's operating pressure.In particular, for increasing values of   ,   and   both decrease and tend to have comparable magnitudes.Indeed, in (13) the best trade-off between the energy used for by the capsules,   , and the energy used by the Hyperloop infrastructure,   +   , determines the value of   that tends to a constant value for   → ∞.These trends are shown in Fig. 8, Fig. 9 and Fig. 10 that quantify, respectively,  the dependency of   with   , the dependency   with   and the dependency   with   .Fig. 11 shows the optimal maximum power required by the PS of a Hyperloop capsule as a function of   : it can be seen that it increases with the increase of   since larger values of   results in larger tube operational pressures,   , and consequent larger aerodynamic drag.It is also worth observing that, for the various considered trajectories and   , the optimal maximum power required by the PS of a Hyperloop capsule is in the range between 1 .8 − 5 . 1  .These values of maximum power appear to be compatible with technologies nowadays available for both BESS and power electronics.Fig. 12 shows the optimal maximum cruising speeds of the capsules for the various considered trajectories and   .The results shown in this figure allows to draw an important conclusion: in order to optimise the energy needs of the whole Hyperloop system, the maximum speed of Hyperloop capsules has to be subsonic.Such a conclusion appears to hold also for relatively long trajectories.

Pressure vs. masses
Fig. 13 illustrates the dependency of the active masses of the capsule:  ,   and    as a function of the infrastructure's operational pressure,   .Observe a linear increase of the masses as a function of   with steeper trends for longer trajectories.It is important to remember that the proposed optimisation problem considers both the capsule's mass increase with   (as the capsules are energy-autonomous) along with the increase of the infrastructure's energy with the decrease of   .As a matter of fact, the identified optimal solutions for the capsule's masses and   represent the best trade-off that makes the solution of the proposed problem non-trivial and less intuitive.

Profiles of speed, travel time, BESS SoC and power
In this subsection, we show the profiles of the most important internal variables of the optimisation problem (13) as a function of the capsule's position along its trajectory for the specific case of  = 226  (as for the other graphs, the results are also shown for the various   ).Fig. 14 , Fig. 15 , Fig. 16 , Fig. 17 show the profile of speed, travel time, BESS SoC and BESS power as a function of the capsule position for every   .
As already observed, due to the nonlinear increase of the aerodynamic losses with the speed associated with   and   , the optimal cruising speed of the capsules is in the order of 612  ℎ .Such an optimal cruising speed is linked with   , due to the various optimal values of   .Fig. 16 shows the trend of the capsule's BESS .For very short maintenance periods, i.e.,   = 1  and partially   = 7  , the  does not reach the minimum binding value of 10%, because the BESS is constrained by the maximum discharge rate of the cell.Indeed, the optimal solutions identified for these short maintenance periods have a peculiar BESS design for which the binding constraints in (13) are those associated with the power, rather than with the energy of the BESS.This occurs because the optimal   reaches its lowest values and the energy consumption for one capsule,   , is the lowest too.Therefore, for low values of   (and corresponding   ), the Hyperloop system can be associated with a power-intensive application, as the constraints associated with the cells discharge are binding.Whereas, for higher values of   (and corresponding   ), the Hyperloop system can be associated to an energy-intensive application as the constraints on the BESS  are binding.
Fig. 17 shows the different optimal profiles of   , depending on   .The maximum power vary in the interval between 1 .8 − 3 .2  .Smaller maximum values of   correspond to lower   and lower maximum speed (see Fig. 14 ).

Energy needs and infrastructure operation
For a given   and trajectory length, the total energy need per number of passengers and per km is given in (19) and the results are shown in Fig. 18 .
The operational strategy of the Hyperloop infrastructure plays an important role on the energy consumption of the entire system.For short   ,   can reach high values, especially for long trajectory lengths, as for  = 1000  the best values of   are in the range of 100 − 225  ℎ  ⋅ .Therefore, independently of the infrastructure length, it is suggested to have   ≥ 21  as, depending on the length of the trajectory,   ranges between [20 , 30]  ℎ  ⋅ .

Impact of the levitation drag
The optimisation model (13) considers suspended capsules where the levitation drag is null.However, it is worth analysing the impact of the magnetic levitation drag force,   on the optimal solution of Fig. 19.  dependency with the speed of the capsule,  , adapted from [41] .
(13) since the BESS power profile might be influenced as well as the energy consumption of the capsule.
In this sub-section, we analyse such an impact for a trajectory length of  = 226  .
is given by (20) where   represents the levitation drag coefficient that has a dependency with the speed of the capsule,  , as discussed in [41] .The representation of   (  ) is shown in Fig. 19 adapted from [41] .
Therefore, compared with (8) , the new capsule's traction force is represented by (21) and the traction power by (22) .
With the same conditions imposed in (13) , except for the definition of   and   that, in this case, include the magnetic levitation drag, the problem (13) has been solved for all the   for the trajectory length  = 226  .
The profiles of   ,   and  are presented in Fig. 20 , Fig. 21 , respectively in Fig. 22 .In Fig. 23 , it is worth observing the  ,   ,    dependencies with   for  = 226  .
By comparing the results of Fig. 20 with those in Fig. 7 , the tube operational pressure   shows a slight decrease of up to 10% for each   .Regarding the capsule masses, by comparing the results of Fig. 13 with those in Fig. 23 , we can observe that, for each operational pressure,   , the total mass increase of up to 4 times.The peak BESS power required for the case with magnetic levitation is about 14 .5  whereas, for the case without the magnetic levitation, is 3  .For the constant speed zone, the BESS power required for the case with magnetic levitation is about 5 .5  while, for the case without the magnetic levitation, is about 0 .5  .Regarding the speed profiles, the comparison between Fig. 14 vs Fig. 22 shows that they are quite similar.
In view of the obtained results, it is worth observing that the magnetic levitation is responsible of a dramatic increase the capsule's masses, energy needs as well as peak power requirements.It is quite clear from these results that Hyperloop capsules have to rely on dragless magnetic levitation solutions especially if the energy reservoir is embedded in the capsule.

Sensitivity analysis
This section contains a comprehensive sensitivity analysis with respect to parameters that have an influence on the solutions of the proposed optimisation problem.In particular, these parameters are: (i) the number of capsules per unite of time   , (ii) the aerodynamic drag coefficient   , (iii) the permeability of tube's material   , (iv) the passive mass of the capsule  0 and (v) the LIM efficiency and power factor dependency on capsule's speed   (  ) ,  ( )(  ) .

Variable 𝑟 𝑐𝑎𝑝𝑠 (number of capsules per unite of time)
In this sub-section we analyse the influence on the solution of ( 13) of different values of the variable   .For this purpose, with respect to the original value of   = 1  2  chosen for this parameter, we have considered two other values, namely: (i)   = 1  the tube pressure (less than 15% ) is obtained.The differences for   ,  and  are presented in Fig. 25 , Fig. 26 and Fig. 27 , respectively.Lower flows of capsules produce a decrease of the energy consumption of the capsules.As a consequence, in order to minimize the total energy need, the optimization identifies slightly larger tube pressure (see Fig. 24 ) that limit the increase of the energy needed to operate the infrastructure (i.e.,   +   ).The values obtained for the energy perpassenger-per-km shown in Fig. 28 , do not exceed 70  ℎ ∕  ∕  for   = 1  10  and   > 21  and are slightly higher compared to the case of   = 1  2  .

Variable 𝐶 𝑑 (aerodynamic drag coefficient)
A sensitivity analysis regarding this parameter is here carried out by adding an offset to the original aerodynamic drag coefficient as shown in Fig. 29 where (i)   =  , + 0 . 1 and (ii)  , =   + 0 . 2 .
A variation in the drag coefficient directly impacts the energy required by the capsules.In order to minimise such an impact, the optimisation identifies optimal tube pressures that, in correspondence of     large values of   , have a slight change (less than 15% -see Fig. 30 ) with respect to the optimal solutions obtained for the original aerodynamic drag coefficient.The obtained values for   shown in Fig. 31 exhibit changes in the order of 10% with respect to the original value.It is also interesting to observe that the variations on the tube are small enough not influence the energy consumption related to the infrastructure (i.e.,   in Fig. 32 and   in Fig. 33 ).The energy per-passenger-per-km shown in Fig. 34 is very similar with respect to values obtained with the original aerodynamic drag coefficient.

Variable 𝑘 𝑝𝑒𝑟𝑚 (permeability of tube's material)
it is worth observing that the construction of an Hyperloop tube may require to impose the value of   in order to guarantee a given performance of the tube regarding its depressurization and air leaks.Although the value adopted for   is inferred from the existing literature, a sensitivity analysis on this parameter is shown in this sub-section.The sensitivity analysis is considering variations of ±25% with respect to the original value assumed for this parameter, namely: (i)   = 3 .75 ⋅ 10 −18  2 and (ii)   = 6 .25 ⋅ 10 −18  2 .As expected, higher values of the tube permeability involve higher amount of energy needed for the operation of the infrastructure (i.e.,   +   ).As a consequence, the proposed optimization manages to adjust the tube pressure,   (see Fig. 35 ) such that, for a higher value of   , the tube pressure is increased with respect to the values obtained for the original value assumed for   in order to minimize the increase of   (see Fig. 36 ) and   (see Fig. 37 ).On the contrary, for a lower value of   , the optimization identifies lower tube pressure levels with respect to those obtained in correspondence of the original value adopted for   .As a result,   and   present an average difference of 7% with respect to the values obtained with the original value adopted for   .It is also worth observing that different tube's operating pressures have an impact on the capsules' energy consumption   (see Fig. 38 ) that exhibits changes of 10% − 11% with respect to the values obtained to the original value adopted for   .In Fig. 39 , the average energy consumption does not exhibit a significant change.

Variable 𝑚 0 (passive mass of the capsule)
As for the other parameters, the sensitivity analysis is carried out by varying the original value assumed for  0 (i.e., 6000  for the passive mechanics + 4000  for the passengers = 10000kg) by adding a weight of 20% and 40% more with respect to the original value assumed for this parameter, namely: (i)  0 = 12000  and (ii)  0 = 14000 .
Fig. 40 shows that the optimisation problem identifies solutions with a difference in the tube pressure of about 20% with respect to the results obtained with the original value of  0 .The energy consumption of the capsules,   , is correspondingly decreased by 6% with respect to the results obtained with the original value of  0 as shown in Fig. 41 .The reduction of the energy needed by the capsules is shifted in a corresponding increase of the energy needed by the infrastructure (   and   shown in Fig. 42 and Fig. 43 ) respectively.The overall result is, however, unchanged as the optimization identifies operating conditions for which the energy per-passenger-per-km remains practically the same compared to the one obtained in correspondence of the original mass  0 as shown in Fig. 44 .

Variables 𝜂 𝐿𝐼𝑀 ( 𝑣 ) , 𝑐𝑜𝑠 ( 𝜙)( 𝑣 ) (LIM efficnecy and power factory expressed as a function of the capsule's speed)
The efficiency (   ) and power factor (  ( ) ) of a high-speed LIM designed at the Authors' laboratory for Hyperloop applications are shown here below as a function of the capsule speed (see Fig. 45 and Fig. 46 ).In the optimisation problem (13) , instead of using constant values, these two functions we have been imposed for the LIM's efficiency and power factor as a function of the speed of the capsule.As shown in Fig. 47 , the proposed optimization identifies a lower tube pressure (in the range of −15% ) with respect to the results obtained for   = 0 .65 and  ( ) = 0 .6 .The lower tube operating pressure is identified to compensate for the larger losses in the capsule propulsion system associated to   (  ) ,  ( )(  ) .Indeed, the energy needed for the capsules,   , exhibit an increase of 20% as shown in Fig.48

Conclusions
In this paper, we have proposed an operational-driven optimaldesign framework of a Hyperloop system.The framework is capable of addressing fundamental questions related to the design of this new transportation mode, namely: (i) assess the pressure inside a Hyperloop tube to minimize its global energy consumption; (ii) determine the minimal energy need passenger and per km, (iii) understand whether there is a strong link between the infrastructure operation and the capsule's design and (iv) understand whether, or not, a magnetic levitation system affects the energy consumption of the capsule.
In this respect, we have proposed a comprehensive optimisation framework capable of linking the operation of a network of Hyperloop capsules, the model of the Hyperloop infrastructure and the model of the capsule's propulsion and kinematics in view of the inherent coupling between the tube environmental conditions (i.e., its operational pressure) and the motion of the capsules along the trajectory.
We have quantified the (strong) impact of the operational strategy of the Hyperloop infrastructure on the energy consumption of the entire system.More specifically, for depressurization periods in the range of few days, the energy-per-passenger-per-km can reach high values (especially for long trajectories) with best values in the range of 100 − 225  ℎ  ⋅ .Therefore, the first conclusion is to enforce depressurization periods ≥ 21  as, depending on the length of the trajectory, the energy-per-passenger-per-km can fall in the range be- tween [20 , 30]  ℎ  ⋅ , thus making this transportation mode energycompetitive with respect to electric trains.Indeed, we have shown that for values of depressurization periods in the order of few days, the dominant use of energy is given by both process energy and depressurization energy, and for values of depressurization periods in the range between 42   to 168   , the energy need of the infrastructure is of the same order of magnitude of the energy used by the capsules.For higher values of depressurization periods, the energy used by the capsules becomes dominant Regarding the capsule's optimal cruising speed, the obtained results have shown that, in order to minimise the total energy required by the whole Hyperloop system, the capsules have to travel at a sub-sonic speed.
The proposed framework has also allowed to determine the optimal pressures inside the tube to be in the range from 2 .82  to 76 .92  for a length of trajectory of 226  , from 1 .17  to 54 .5  for a length of trajectory of 500  and from 1 .14  to 17 .25  for a length of trajectory of 1000  .Note that, for the same value of depressurization period, the optimisation problem determines the increasing tube operating pressures for larger trajectory length.
Furthermore, the proposed optimisation has shown that the magnetic levitation is responsible of a dramatic increase the capsule's masses, energy needs as well as peak power requirements.Therefore, it is quite clear from these results that Hyperloop capsules have to rely on dragless magnetic levitation solutions especially if the energy reservoir is embedded in the capsule.
Compared to other electrical transportation modes, such as electrical vehicles (EVs) and electrical trains (ETs), it is worth mentioning that Hyperloop represents an energy-efficient and high-speed solution.Such a conclusion is supported by the Gabrielli-Kármán diagram shown in Fig. 52 where the best energy need per-passenger-per-km obtained in this paper for the Hyperloop system are compared with the corresponding values for ETs and EVs taken from Ruangjirakit [42] and Andersson [43] .Indeed, for a similar or lower values of energy need per-passenger-per-km, the Hyperloop system can offer higher maximum speeds compared to the other two main electrical transportation modes making it a viable solution for intra-continental travels.

Disclosure of conflicts of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

2 Fig. 6 .
Fig. 6.Assessment of the chocked flow regime of the fluid around the hyperloop capsule.

Fig. 18 .
Fig. 18.Total energy need per number of passengers and per km,   , as a function of   for each trajectory length.

Fig. 20 .
Fig. 20.Infrastructure   for each   (profiles refers to  = 226  ) including the losses of the passive levitation.

Fig. 21 .
Fig. 21.Capsule   along its position for each   (profiles refers to  = 226  ) including the losses of the passive levitation.

Fig. 22 .
Fig. 22. Speed  along its position for each   (profiles refers to  = 226  ) including the losses of the passive levitation.

Fig. 23 .
Fig. 23.The dependency of masses (  ,   ,    ) with   for  = 226  including the losses of the passive levitation.

Fig. 24 .
Fig.24.Dependency of the optimal operational internal pressure of the tube,   , with   (   sensitivity analysis).

Fig. 28 .
Fig. 28.Total energy need per number of passengers and per km,   , as a function of   for each trajectory length (   sensitivity analysis).

Fig. 30 .
Fig.30.Dependency of the optimal operational internal pressure of the tube,   , with   (   sensitivity analysis).

Fig. 34 .
Fig. 34.Total energy need per number of passengers and per km,   , as a function of   for each trajectory length (   sensitivity analysis).

Fig. 35 .
Fig.35.Dependency of the optimal operational internal pressure of the tube,   , with   (   sensitivity analysis).

Fig. 39 .
Fig.39.Total energy need per number of passengers and per km,   , as a function of   for each trajectory length (   sensitivity analysis).

Fig. 40 .
Fig. 40.Dependency of the optimal operational internal pressure of the tube,   , with   (  0 sensitivity analysis).

Fig. 44 .
Fig. 44.Total energy need per number of passengers and per km,   , as a function of   for each trajectory length (  0 sensitivity analysis).
while the energy needed by the infrastructure,   +   , remains practically unvaried as shown by Fig.49and Fig.50.As shown in Fig.51, the energy perpassenger-per-km exhibit a small increase with respect to the results obtained for   = 0 .65 and  ( ) = 0 .6 .

Fig. 47 .
Fig. 47.Dependency of the optimal operational internal pressure of the tube,   , with   (propulsion sensitivity analysis).

Fig. 51 .
Fig. 51.Total energy need per number of passengers and per km,   , as a function of   for each trajectory length (propulsion sensitivity analysis).

Fig. 52 .
Fig. 52.Gabrielli-Kármán diagram showing the comparison of Hyperloop with electrical vehicles and electrical trains.
0 , BESS,   , and PS represented by the VSI and LIM,    , compose the total mass of one capsule,  =  0 +   +    .

Table 2
Initialization of the control variables.
Fig. 9. Dependency of the air leaks compensation energy,   , with   .Fig. 10.Dependency of capsules network energy,   , with   .