Masoud Ardestani
email: ardestani.masoud90@gmail.com

Hamid Reza Izadfar
email: hrizadfar@semnan.ac.ir

Reduction of Air-Gap Flux Density Distortion for a 20 kW HTS Induction Motor

Keywords: High-temperature superconductor, Air-gap flux density, Harmonic component, Total harmonic distortion, Finite element method

Nowadays, High-Temperature Superconducting (HTS) electric machines are widely used in industry due to the unique properties of HTS materials. Besides, the effect of space harmonics is a very serious challenge that must be considered in the design of HTS electric machines. In this paper, a high-temperature superconductor-induction/synchronous machine (HTS-ISM) with 20 kW power is provided to reduce air-gap flux density distortion. For a precise comparison, the main parameters of the machine are studied under the same conditions such as frequency, core material, pole number, critical current density, and voltage by the Finite Element Method. In addition to winding arrangement, the geometrical parameters of the machine also affect the space harmonics, so by modifying each of these parameters, the amount of Total Harmonic Distortion (THD) is obtained in each case. In the proposed model, the amount of THD is significantly reduced, which makes the airgap flux density more sinusoidal. Torque ripple has also been improved in the proposed machine.

Introduction

Today, the development of High-Temperature Superconductor (HTS) machines has increasingly drawn the attention of researchers [START_REF] Umemoto | Development of 1 MW-class HTS motor for podded ship propulsion system[END_REF][START_REF] Gamble | Full power test of a 36.5 MW HTS propulsion motor[END_REF][START_REF] Ueno | Race-track coils for a 3 MW HTS ship motor[END_REF][START_REF] Huang | Control and operation of a high temperature superconducting synchronous motor[END_REF]. These types of HTS machines can be used in different applications such as ship propulsion [START_REF] Snitchler | The performance of a 5 MW high temperature superconductor ship propulsion motor[END_REF], electric vehicles [START_REF] Karashima | Multidisciplinary analysis of the transient performance of a 20 kW class HTS induction / synchronous motor cooled with a cryocooler and gaseous air-gap coolant[END_REF], wave energy converters [START_REF] Ardestani | A new HTS dual stator linear permanent magnet Vernier machine with Halbach array for wave energy conversion[END_REF], wind energies [START_REF] Go | Structural design of a module coil for a 12-MW class HTS generator for wind turbine[END_REF], and so on. However, although superconducting electric machines have been increasingly used in various industries and have unique properties, they are not immune to harmonics, Nevertheless, superconducting machines perform better than conventional machines against harmonic distortion [START_REF] Singh | Applications of High Temperature Superconductors to Electric Power Equipment[END_REF]. The problem of harmonics is the main issue in electric machines that must be considered in the design since it affects the performance of electric machines, both superconducting, and conventional machines.

Generally, harmonics are divided into two categories, time, and space harmonics. Time harmonics are caused by nonlinear loads and power electronics devices, but space harmonics are created inside the electric machines. The space harmonics are related to the design, the type of winding (concentrated or distributed), the full or fractional pitch, and the geometry of the machine such as air-gap length, slot opening, eccentricity, and lamination stack, magnetic saturation, etc. [START_REF] Lukasik | Finite element assisted method to reduce harmonic content in the airgap flux density of a high temperature superconducting coreless rotor generator[END_REF][START_REF] On | Guidance Note to Control of Harmonics in Electrical Power Systems[END_REF][START_REF] Wakileh | Harmonics in rotating machines[END_REF]. Apart from that, another issue that must be considered in HTS machines is the design and implementation of HTS coils, which is also effective in reducing the distortion of the air-gap flux density, so that the perpendicular flux to their surface is reduced. Since the anisotropy of HTS coils is completely different from copper winding and their characteristics as critical current density are strongly affected by perpendicular fluxes which leads to a decrease in critical current density, an increase in AC losses, and distortion in the air-gap flux density. Therefore, the proper design of an HTS coil in the superconducting machine is very important to reduce the harmonic component of the air-gap flux density [START_REF] Ikeda | DC and AC Current Transport Characteristics of the HTS Stator Coils in an HTS Induction/Synchronous Motor[END_REF][START_REF] Yazdani-Asrami | Influence of fielddependent critical current on harmonic AC loss analysis in HTS coils for superconducting transformers supplying non-linear loads[END_REF][START_REF] Wei | Comparison of Different Stator Winding Configurations of Fully High-Temperature Superconducting Induction/Synchronous Motor[END_REF][START_REF] Lukasik | Finite element assisted method to reduce harmonic content in the airgap flux density of a high temperature superconducting coreless rotor generator[END_REF][START_REF] Kulkarni | Design improvisation for reduced harmonic distortion in a flux pump-integrated HTS generator[END_REF]. One of these machines discussed in this paper is High-Temperature Superconductor-Induction/Synchronous Machine (HTS-ISM), which is an ideal candidate in the electrical transportation industry. Concerning this, a 20 kW HTS induction motor which was designed and built by Japanese researchers [START_REF] Sekiguchi | Trial test of fully HTS induction/synchronous machine for next generation electric vehicle[END_REF] examined from the harmonic perspective. In fact, we analyzed a partial HTS induction motor in which superconducting materials only used in the rotor and stator winding are made of copper exactly like a conventional electric motor [START_REF] Sekiguchi | Trial test of fully HTS induction/synchronous machine for next generation electric vehicle[END_REF][START_REF] Ardestani | Electromagnetic Analysis of Partial and Fully HTS Induction Motor Using Finite Element Method[END_REF]. On the other hand, since the stator winding was pre-designed as distributed, we have only focused on the motor geometry to reduce the harmonic components in the air-gap. Furthermore, space harmonics in the HTS-ISM cause noise due to the pulsating torque, and increased copper and core losses, which leads to heat and decrease efficiency. For this reason, if the harmonic components of the air-gap can be reduced, the vibration, noise, copper, and core losses of the motor can be decreased [START_REF] Wakileh | Harmonics in rotating machines[END_REF][START_REF] Cordier | Modelling space harmonics in induction machines for real-time applications[END_REF]. Regarding this, the HTS-ISM air-gap magnetic flux distortion is investigated, and a new structure is proposed to reduce the harmonic component of air-gap flux density. For this purpose, the geometric parameters that were effective on the harmonic component of air-gap flux density were investigated and the most appropriate ones were selected from the THD point of view. Also, with decreasing the harmonic components of the air-gap, the torque ripple has decreased. The rest of the paper is as follows. In section 2, the relation of the conference theme with the paper is given. In section 3 the structure of HTS-ISM will be described. In section 4, the HTS-ISM operation is explained. In section 5, the geometry optimization of HTS-ISM is given. In section 6, electromagnetic analysis of the initial and proposed HTS-ISM is investigated. Finally, the conclusion and future work will be explained in sections 7 and 8, respectively.

Relation with Technological Innovation for Life Improvement

The technological innovation proposed in this work is related to life improvement by contributing to the development of high-temperature superconducting machines.

Superconducting materials allow the development of more compact and light devices with the same power when compared to their conventional counterparts. This has impacts on several sectors, such as industry or transportation. The development of superconductingbased technologies has also direct impact on life improvement, as related to all the links of the energy chain, besides end-use, like generation, transmission, distribution, and storage of electrical energy, fostering the Energy Transition.

Machine Structure

The initial machine structure is shown in Fig. 1 [START_REF] Sekiguchi | Trial test of fully HTS induction/synchronous machine for next generation electric vehicle[END_REF]. The stator and rotor cores are siliconsteel and laminated. The stator coil is made of copper and the distributed winding is adopted to reduce the harmonic component of the air-gap magnetic flux density [START_REF] Karashima | Experimental and analytical studies on highly efficient regenerative characteristics of a 20-kW class HTS induction/synchronous motor[END_REF]. Also, rotor bars with HTS wires (Bi-2223) are located in the rotor slots. Since the core of the motor passes almost all of the main magnetic flux from the slots, it has two major advantages over the air-core motor: 1-There are only self-magnetic fields in the slots i.e., magnetic field leakage due to the current passing through the HTS rotor bars, in other words, there is no external magnetic field in the rotor bars. Therefore, the destructive effects of magnetic fields on the current transport property can be ignored. 2-Reducing the electromagnetic force applied on rotor HTS bars, therefore does not need a complex structure to protect the conductors [START_REF] Laksar | Numerical Calculation of the Effect of the Induction Machine Load on the Air Gap Magnetic Flux Density Distribution[END_REF]. The HTS rotor bars are mounted on the top of the rotor near the air-gap to reduce leakage flux [START_REF] Ainslie | Transport AC loss in high-temperature superconducting coils[END_REF]. Rotor bars are made of eleven HTS tapes, isolated copper bars are also used in the rotor slots for thermal stability, mechanical support, and current pass in the non-superconducting area of the HTS bars [START_REF] Shimura | Calculated characteristics of HTS induction/synchronous machine below and above its critical temperature[END_REF]. At the time of starting, the motor has to be cooled to 77 k. The length of the cryostat is the length of the shaft with a diameter of 428 mm. The cryostat structure is such that the HTS-ISM can rotate slowly even below the Cryogenic temperature and the cryostat is equipped with a nitrogen reservoir [START_REF] Sekiguchi | Trial test of fully HTS induction/synchronous machine for next generation electric vehicle[END_REF]. Since the stator and rotor cores are laminated, the effects of eddy losses are neglected. All the analyses of the proposed and initial machines were carried out under the identical conditions such as the coil number, speed, core materials, and poles number. The specification of HTS wire and the main parameter of the HTS-ISM are presented in Table 1 andTable 2, respectively.

Machine Operation

When AC voltage is applied to an HTS induction motor, the rotor bars are exposed to timevarying magnetic flux due to the MMF caused by the stator winding. On the other hand, since the rotor bars are made of high-temperature superconducting material and type Bi-2223, they show losses against AC fields. Also, another type of loss due to thermal activation occurs in currents below Jc, called Flux creep. In addition, superconducting materials have inherent properties and impurities, so their interaction with time-varying magnetic fields creates a force called the pinning force.

This force keeps in place the vortices that cause losses in superconducting materials. At the opposite point, there is another force called the Lorenz force that wants to move the vortices. When these two forces are equal, that point is the critical current density. Thus, in the initial moments, since the motor has not yet started to rotate and the magnetic field applied to the rotor bars is less than Hc1, the flux applied to the rotor bars is shielded because of pinning centers and they are not allowed to enter the bars, which this is called initial mode (shielding). At this moment, the magnetic flux inside the rotor bars is zero (B = 0). After a while, as the motor starts to rotate due to high slip (s = 1), high currents are created in the rotor bars. In this situation, the Lorenz force is greater than the pinning force, which causes the vortices to flow in the rotor bars, and the bars show resistance losses, which is called Flux-Flow or Asynchronous mode. At this moment, rotor bars current also greater than the critical current density of the bars, which is 2090 amps, so the bars are out of superconductivity for a short time. In this case, the motor starts with a slip, which is obtained as follows [START_REF] Umemoto | Development of 1 MW-class HTS motor for podded ship propulsion system[END_REF].

N -N S S= N S (1)
Where, Ns is the synchronous speed and Nr is the speed of rotor. Moments later, due to the reduction of slip, the current of the rotor bars becomes less than the critical current density of the bars and the resistance of the rotor bars becomes almost zero. At this time, the Lorentz force is less than the pinning force, and all magnetic flux is trapped by the pinning centers in the rotor bars, which is called persistent current mode or Synchronous mode. At this time, the rotor is locked with a magnetic field and reaches synchronous speed. In this case, the rotor bars act like PM. In fact, this motor has 3 modes of shielding, asynchronous and synchronous due to the unique properties of superconducting materials that have nonlinear resistance. This nonlinear resistance leads the motor to have high starting torque (due to high resistance) and synchronous torque (due to almost zero resistance) [START_REF] Karashima | Experimental and analytical studies on highly efficient regenerative characteristics of a 20-kW class HTS induction/synchronous motor[END_REF][START_REF] Morita | Theoretical analysis of a YBCO squirrel-cage type induction motor based on an equivalent circuit[END_REF][START_REF]Magnetic properties of high temperature superconductor[END_REF][START_REF] Ainslie | Transport AC loss in high-temperature superconducting coils[END_REF][START_REF] Arish | Optimum structure of rotor slot for a 20 kW HTS Induction motor[END_REF].

The Relationship between Voltage-Current of Superconductor and Torque-Speed of HTS-ISM

To use a superconductor, its electromagnetic behavior must be described. Quantitative calculation of the electromagnetic behavior of HTS materials is complex because their voltage-current relationship is non-linear. The E-j power-law method uses the nonlinear voltage-current superconductor relation to model it as a nonlinear conductor. The voltagecurrent curve of the superconductor and torque-speed of HTS-ISM is shown in Fig 2 . the superconductor voltage-current curve can be fit by the power-law given in Equation 2.

I N E(I) = E ( ) c I c (2)
Where E(I) is the longitudinal voltage drop through the superconductor, EC is the electric field criterion at 1µv/cm, I is the current in the conductor, IC is the critical current, and N is the exponent. A higher N-value produces a sharp transition in the V-I curve and demonstrates the quality of the superconductor. High-quality superconductors will have numerous N-value. A superconductor is categorized by a V-I curve or the voltage drop across the length of it as a function of its current. The critical current of the superconductor is defined as the current that produces a voltage drop of 1 µv/cm as depicted in Fig. 2. According to Fig. 2., this motor not only has a slip torque but also synchronous torque. In simple terms, the rotational synchronous mode is achieved in zero resistivity state and rotational slip mode is achieved in the flux-flow state. These exclusive features are due to the nonlinear voltage current of HTS materials [START_REF] Singh | Applications of High Temperature Superconductors to Electric Power Equipment[END_REF][START_REF] Ikeda | Hysteretic rotating characteristics of an HTS induction/synchronous motor[END_REF].

Motor Geometry Optimization

Generally, in an electric machine, the geometrical parameters of the machine and the structure of the winding affect the space harmonics. So, by choosing these parameters properly, the effects of the space harmonics, which result in vibration, parasitic torque, noise, and losses, can be reduced. In this section we only focused on the geometrical parameters of the motor because the winding arrangement of the initial motor has been designed as distributed with a short pitch configuration. Therefore, to investigate the airgap magnetic flux density in terms of harmonic, first, air-gap magnetic flux density was acquired based on FEM with the ANSYS-Maxwell software. After that, the Fourier transform was obtained from the air-gap magnetic flux by MATLAB software, and the THD value of air-gap flux was attained for each variable which is presented in Fig. 4. The parameters examined are air-gap (g), tooth gap width (Tgw), slot width (Sw), tooth tang angle (α), and tooth tang depth (Ttd) as shown in Fig. 3. Fig. 4 (a) shows the rate of THD changes of parameters which their unit is length (air-gap, tooth gap width, slot width, tooth tang depth). Fig . 4 (b) also shows the THD value of tooth tang angle parameter for different angles in degree. In addition, an explanation of all the examined parameters is given below. Finally, the THD values for different parameters are given in Table 3.

Air-gap

Air-gap is the most important factor in reducing air-gap magnetic flux density distortion.

The length of the air-gap of the initial HTS-ISM is 0.3 mm. By increasing this parameter from 0.3 to 1 mm, the odd harmonics components of the air-gap magnetic flux decreased, the reason behind this, is the variation of permeance which affects the amplitude of the harmonic component [START_REF] Cordier | Modelling space harmonics in induction machines for real-time applications[END_REF]. Also, the amplitude of the fundamental component is reduced due to the increase in the air-gap magnetic reluctance. The lowest value of THD belongs to the air-gap of 1 mm.

Tooth Gap Width

After the air-gap, the gap between the teeth which is called tooth gap width is the most important parameter in reducing THD of the air-gap magnetic flux. By changing this parameter to 5.5 mm, the THD reaches its minimum value which is 0.5064. Reduction of Air-Gap Flux Density Distortion for a 20 kW HTS Induction Motor 173

Slot Width

The slot width is designed according to machine limitations such as specific magnetic loading and specific electric loading. As the slot width increases, the specific electric loading increases, and by decreasing it, the specific magnetic loading increases. Therefore, the slot width should be selected according to the limitations of the machine such as the standard current density in the stator coil and the maximum allowable magnetic flux density; regarding the need and the limitations of the machine, a compromise should be made between the specific magnetic loading and specific electric loading. To optimally select the dimensions of the slot, equation 3 must be satisfied.

Where ZSS is the number of conductors, aS is the conductor cross-section, 0.4 is a constant coefficient and ASS is the slot area. By varying the slot width from 10 to 13 mm, the THD value of the air-gap flux belongs to the slot width of 11 mm with a value of 0.5927.

Tooth Tang Angle

The tooth tang angle is the angle between the tang and the bottom of the slot. The variation of the THD of tooth tang angle changes in terms of degree is shown in Fig. 4 (b), the lowest of which is at 20 °.

Tooth Tang Depth

Tooth tang depth is the length of the tang. The lowest THD value of the air-gap belongs to the tooth tang depth of 1.5 mm. As shown in Table 3, the minimum THD values for the examined parameters (air-gap, tooth gap width, slot width, tooth tang angle, and tooth tang depth) are 1mm, 5.5 mm, 11 mm, 20˚, and 1.5 mm respectively. Therefore, the proposed machine is designed with these values.

Za ss s A= ss 0.4

(3)

Table 3. THD values for the examined machine parameters in different range.

2D FEM Analysis

Analytical equations cannot be used when the geometry is complicated, and the field condition is not uniform. Numerous numerical methods have been proposed to solve this problem. These numerical methods usually use the FEM and FDM methods to solve the Maxwell equations. FEM is the most popular method for solving partial differential equations. These methods can be categorized by solving the Maxwell equations that are the same: A-V (based on the magnetic vector potential A), T-Ω (based on the current vector potential T), E (based on the electric field E), and H (based on directly solving the magnetic field components) formulations. The final answer to each of these equations is the same.; the only difference is their solutions [START_REF] Ainslie | Transport AC loss in high-temperature superconducting coils[END_REF][START_REF] Asghar | Calculation of AC Magnetizing Loss of ReBCO Superconducting Tapes Subjected to Applied Distorted Magnetic Fields[END_REF][START_REF] Wang | Numerical Simulation of AC Loss in 2G High-Temperature Superconducting Coils with 2D-Axisymmetric Finite Element Model by Magnetic Field Formulation Module[END_REF]. The numerical model used in this paper for modeling of HTS-ISM is based on solving the set of Maxwell equations in 2D implementing the A-V formulation using the software package ANSYS-Maxwell. Therefore, by solving the partial differential equation (PDE) in 2D, the amount of electric and magnetic fields in each node can be accurately obtained from the equation ( 4).

dA × υ × A = J -σ -σ v + ×H sc dt     ( 4
)
Where A is the magnetic vector potential, Js is the source current density (one component in z-direction for 2D), HC is the coercive magnetic field (A/m) and v is the velocity of the moving part. For a simple definition of the HTS materials, a material called perfect conductor which also exist in the ANSYS-Maxwell Library, is included with a magnetic relative permeability of almost zero. The proposed machine is designed based on the initial machine as shown in Fig. 5.

According to the results of the previous section, the lowest THD values were obtained by changing air-gap, tooth gap width, slot width, tooth tang angle, and tooth tang depth values by 1mm, 5.5mm, 11mm, 20˚ degree, and 1.5mm, respectively. In this section, the electromagnetic parameters of the initial and proposed machine are presented and investigated, such as the distribution of the air-gap magnetic flux and its harmonic components, magnetic flux lines and magnetic flux density distribution, and torque ripple.

Air-gap Flux Density Distribution and Harmonic Component

The interaction between the stator and the rotor MMF leads to magnetic flux density in the air-gap of the HTS induction motor. So that the voltage applied to the stator winding, which causes induced current in the cage rotor, creates MMF in the stator and rotor winding, respectively. Space harmonics make the air-gap magnetic flux distorted, so it is not purely sinusoidal. To detect the decrease and increase of air-gap magnetic flux density harmonics, Fourier transforms which are known as the total harmonic distortion (THD) criterion, could be utilized. Generally, the THD relation in terms of air-gap magnetic radial flux (Br) can be defined as below. The tangential flux is ignored. Where k is an integer and refers to harmonics (k = 1,2,3,4, ...), Brk is the Kth amplitude of the Kth order harmonic and Br1 is the fundamental component [START_REF] Laksar | Numerical Calculation of the Effect of the Induction Machine Load on the Air Gap Magnetic Flux Density Distribution[END_REF][START_REF] Kulkarni | Design improvisation for reduced harmonic distortion in a flux pump-integrated HTS generator[END_REF]. pursuant to equation ( 5), the lower the amplitude of the harmonic components, the lower the THD, and the waveform gets closer to the main component, which is a pure sine The distribution of air-gap flux density is shown in Fig. 6 (a). As can be seen, the RMS value of air-gap flux density in the proposed machine is lower than the initial one, and that is the main disadvantage of air-gap increases which leads to reducing the air-gap magnetic flux magnitude. On the other hand, by reducing the air-gap distance the harmonic ratio also decreases significantly. Fig. 6 (b) shows the odd harmonic components of the initial and the proposed machines. As shown in Fig. 6 (b), the amplitude of components 3, 5, and 7th decreased, which made the magnetic flux of the air-gap more sinusoidal. Also, the amplitude of the first component, which is the fundamental component is reduced due to the elimination of space disturbing harmonic components. The obtained amount of THD in the proposed machine was 0.4126 and in the initial machine was 0.6267 under 112 Nm load, which was reduced by 34.17% in the proposed machine.

2 Brk k=2 THD = Br1  (5)

Magnetic Flux Density Distribution and Magnetic Flux Lines

Fig. 7 shows the distribution and flux line density of magnetic flux in the stator and rotor of the initial and proposed machine under a load of 112 Nm. As can be seen from the figure, the magnetic flux density distribution is higher in the initial machine due to the smaller airgap of the initial machine compared to the proposed machine. On the other hand, since the analysis of both machines was performed under the same conditions, the applied voltage to both machines was equal, which caused a small saturation in the stator teeth of the initial machine. In general, increasing the air-gap increases the magnetizing current of the motor, which causes the motor to draw more current from the grid, so it needs to apply more voltage. The maximum magnetic flux density is set at 2.5 tesla, which is less than 2.5 tesla in both machines. Besides, the flux line intensities in the proposed machine are weaker than the initial machine due to the increase in air-gap [START_REF] Chen | Design and Analysis of a New Fully Stator-HTS Motor[END_REF].

Torque Ripple

Torque ripple is one of the effects that can be observed in the majority of motors and the reason is the periodical increment and reduction of the output torque in the motor shaft. The amount of torque is often expressed as a percentage, which can be calculated by the difference between the maximum and minimum electromagnetic torques to their average over a period, using the following relation:

Where, Tmax is the maximum torque, Tmin is the minimum torque and Tave is the average torque [36]. Based on Fig. 8, when the motor is in synchronous speed and in steady state, a load equal to maximum synchronous torque (112 Nm) which was measured in [START_REF] Sekiguchi | Trial test of fully HTS induction/synchronous machine for next generation electric vehicle[END_REF] is applied to the motor shaft at the time of 0.6 seconds. The motor power, in this case, is about 8.7 kW. In Fig. 8 (a), at 1.3 to 1.4 seconds (s), the maximum torque and the minimum torque, and average torque are 137.95, 110.79, and 124.37 respectively. Fig. 8 (b) shows the maximum torque of 131.77 and minimum torque of 120.11 and a mean torque of 125.94 at 1.3 and 1.4 seconds, respectively. So that the torque ripple is 9.2% in the proposed machine and 21.8% in the initial machine. As can be seen from Fig. 8, the torque ripple in the proposed machine is 12.6% lower than in the initial machine.

Conclusions

In this paper, a 20 kW HTS induction motor with the new physical model is presented to reduce the air-gap flux density distortion. The cause of distortion in the air-gap flux density of electric machines is related to space harmonics. These harmonics can be reduced or eliminated by a modification of the winding configuration and motor geometry. Concerning this, we only focused on motor geometry parameters, since the stator winding was predesigned as distributed with a short pitch factor. The geometrical parameters which we investigated were air-gap, tooth gap width, slot width, tooth tang depth, and tooth tang angle, respectively. So that, by changing each of these parameters and selecting the best ones, the optimal model of the machine is obtained from the harmonic point of view. In addition, the most important parameter that reduced the harmonic component of the air-gap flux density was the air-gap changes. Also, all analyses were performed under the same conditions such as voltage, current, pole number, current density, etc. by ANSYS-Maxwell software. Eventually, In the proposed model, THD decreases from 0.6267 to 0.4126 with a decrease of 34.17%. Torque ripple has also decreased from 21.8% to 9.2% by a 12.6% decrease, which results in improving the performance of the machine.

Future Work

This HTS motor has been improved by Japanese researchers for many years. Besides, this motor was made in two types of partial and fully HTS induction motors. In the partial structure, HTS materials are used in the rotor, and copper is used only in the stator windings, but in the fully structure, HTS materials are used in both the stator and the rotor. This approach, which is used in this paper to reduce the harmonic component of air-gap flux density, has the potential to be implemented in other structures of HTS induction motors such as fully HTS induction motors or higher power HTS induction motors with both structures (partial and fully) which also built-in recent years. Although in general all parameters must be considered for the design of electric motors and compromise between the parameters is necessary, nevertheless, this approach, regardless of its disadvantages, is a good candidate to reduce the harmonic component of air-gap flux density.

Fig. 1 .

 1 Fig.1. The structure of the initial machine

Fig. 2 .

 2 Fig.2. The voltage-current of superconductor relationship with torque-speed of HTS-ISM [25].

Fig. 3 .

 3 Fig.3. Examined parameters of HTS-ISM.

Fig. 4 .

 4 Fig.4. The THD values for different parameters a) parameters with different unit lengths b)Tooth tang angle parameter with unit of degree.

Fig. 5 .

 5 Fig.5. The structure of the proposed machine.

Fig. 6 .

 6 Fig.6. Air-gap flux density a) air-gap flux density distribution b) air-gap flux density harmonic component.

Fig. 7 .

 7 Fig.7. Magnetic flux lines and magnetic flux density distribution a) initial machine b) proposed machine.

Fig. 8 .

 8 Fig.8. Transient curve of torque. a) initial machine. b) proposed machine. b)

Table 1 .

 1 Specification of HTS wire

	HTS wire

Table 2 .

 2 Main parameter of HTS-ISM

	Item


Olivier Dudas
email: olivier.dudas@univ-amu.fr

Emily Norton
email: e.norton@kent.ac.uk

DECOMPOSITION NUMBERS FOR UNIPOTENT BLOCKS WITH SMALL sl 2 -WEIGHT IN FINITE CLASSICAL GROUPS

We show that parabolic Kazhdan-Lusztig polynomials of type A compute the decomposition numbers in certain Harish-Chandra series of unipotent characters of finite groups of Lie types B, C and D over a field of non-defining characteristic ℓ. Here, ℓ is a "unitary prime" -the case that remains open in general. The bipartitions labeling the characters in these series are small with respect to d, the order of q mod ℓ, although they occur in blocks of arbitrarily high defect. Our main technical tool is the categorical action of an affine Lie algebra on the category of unipotent representations, which identifies the branching graph for Harish-Chandra induction with the sl d -crystal on a sum of level 2 Fock spaces. Further key combinatorics has been adapted from Brundan and Stroppel's work on Khovanov arc algebras to obtain the closed formula for the decomposition numbers in a d-small Harish-Chandra series.

Introduction

For a finite group G, decomposition numbers encode how ordinary irreducible representations behave after reduction modulo a prime number ℓ. In this paper, we study the case where G is one the finite classical groups SO 2n+1 (q), Sp 2n (q), O ± 2n (q) and ℓ is prime to q (non-defining characteristic).

Computing the whole decomposition matrix can be done blockwise, and by the recent work of Eteve one can even restrict to the unipotent blocks, which are the ones containing unipotent characters (see also [START_REF] Bonnafé | Catégories dérivées et variétés de Deligne-Lusztig[END_REF]). However, even for those blocks, computing the decomposition numbers explicitly seems out of reach. Still, for finite general linear groups, they can be related to Kazhdan-Lusztig polynomials, at least when ℓ is large enough with respect to n. Our main result gives a similar interpretation for classical groups, see Corollary 4.2, but in a much more restrictive case.

Let d be the multiplicative order of q in F × ℓ . When d is odd (the linear prime case), the representation theory of unipotent blocks of classical groups is governed by that of finite general linear groups. When d is even (the unitary prime case), much less is known. For example, the decomposition numbers for unipotent blocks of Sp 2n (q) in the unitary case have been determined up to n = 4 only, and quite recently: in 1998 for n = 2 [START_REF] Okuyama | Decomposition numbers of Sp(4, q)[END_REF], in 2014 for n = 3 [START_REF] Himstedt | Decomposition numbers of SO 7 (q) and Sp 6 (q)[END_REF] and in 2022 for n = 4 [START_REF] Dudas | Decomposition matrices for groups of Lie type in non-defining characteristic[END_REF]. The standard strategy for computing these numbers is the following:

(1) Compute the projective cover of cuspidal representations ;

(2) Decompose the module obtained by Harish-Chandra induction of that projective cover.

We will work under two assumptions which will allow to solve these two problems: first, we will restrict our ordinary irreducible characters to a given Harish-Chandra series. Second, we will impose a smallness condition with respect to d on the Lusztig symbols in the series. The exact condition is given in Section 3.1. In this setting, cuspidal representations, when they occur, are always the "smallest" within the series with respect to the partial order on symbols. Thus problem [START_REF] Bonnafé | Catégories dérivées et variétés de Deligne-Lusztig[END_REF] is no problem thanks to the unitriangular shape of the decomposition matrix.

Everything then hinges on problem (2) -and it turns out to be tractable for these series. Note that we end up working with blocks of any defect, even though in our situation the decomposition numbers turn out to be either 0 or 1.

Let us be more precise on the behaviour of [START_REF] Brunat | Unitriangular shape of decomposition matrices of unipotent blocks[END_REF]. Using the sl d -action constructed in [START_REF] Dudas | Categorical actions on unipotent representations of finite classical groups[END_REF], one can break the Harish-Chandra induction functor into pieces, and obtain an i-induction functor F i for each i ∈ Z/dZ. If P S is the projective cover of a simple module S of highest weight (for the action of sl 2 corresponding to some i ∈ Z/dZ), then F n i (P S ) is very close to being the projective cover of F n i (S). The difference involves only characters which are in the image of F n+1 i , see Proposition 1.3 for an exact statement. With our assumption on the Harish-Chandra series, all these extra characters lie in another series, hence do not contribute to the decomposition numbers we are interested in.

The explicit computation of the decomposition numbers now boils down to knowing exactly how to compute the i-induction on unipotent characters and simple modules in characteristic ℓ.

Under our assumptions, one can attach to each irreducible character/module an up-down diagram and a cup diagram as in [START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra I: cellularity[END_REF][START_REF] Brundan | An orthogonal form for level two Hecke algebras with applications[END_REF] and obtain an elementary description of the action of F i on these combinatorial data as in [START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra III: category O. Represent[END_REF]. See section 3, which takes its cue from the analogous situation for Hecke algebras at d = ∞ [START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra III: category O. Represent[END_REF][START_REF] Brundan | An orthogonal form for level two Hecke algebras with applications[END_REF]. This combinatorial set-up puts us in position to prove our main theorem, Theorem 4.1, in an inductive way using Proposition 1.3. The latter exploits the small sl 2 -weight of the sl 2 -strings in which such characters are found with respect to the categorical action on the category of representations. We observe that we get the exact same formula as in [START_REF] Brundan | An orthogonal form for level two Hecke algebras with applications[END_REF][START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra I: cellularity[END_REF], thus relating the decomposition numbers within the Harish-Chandra series to the parabolic Kazhdan-Lusztig polynomials for a maximal parabolic in type A.

Categorical sl 2 -action

In this section we prove a formula relating the character of the projective cover of a simple module S and the character of the projective cover of its highest weight support. Only part of the character can be controlled, but this will be sufficient to show in Section 4 how to compute decomposition numbers from the case of cuspidal simple modules.

1.1. Recollection on categorical actions. We recall here some of the features of sl 2 -actions on categories as defined in [START_REF] Chuang | Derived equivalences for symmetric groups and sl 2 -categorification[END_REF][START_REF] Rouquier | Quiver Hecke algebras and 2-Lie algebras[END_REF].

Let Λ be a ring with unit and V be a Λ-linear abelian category. A categorical datum on V is given by a pair of biadjoint exact endofunctors E and F of V, together with two natural transformations X ∈ End(E) and T ∈ End(E 2 ) satisfying the relations given in [23, 3.3.3] in the case of sl 2 . Equivalently, we require that X and T induce an action of the affine nil-Hecke algebra of S n on the functor E n for all n ≥ 0. Using that structure one can define the divided power functors E (n) and F (n) which are still exact and biadjoint. They satisfy

E n ≃ E (n) ⊕n! and F n ≃ F (n) ⊕n! .
Assume now that Λ is a field and that V has finite length. An sl 2 -categorical action on V is given by a categorical datum (E, F, X, T ) and a decomposition

V = ω∈Z V ω
of V into abelian categories (which we will call weight categories). Furthermore, the functors E and F should shift the weights by 2 and -2 respectively

V ω E , , k k F V ω+2 such that in the Grothendieck group K 0 (V ω ), the commutator [E][F ] -[F ][E]
acts by multiplication by ω. In particular the class e = [E] and f = [F ] of the functors in the complexified Grothendieck group V = C ⊗ Z K 0 (V) induce an action of sl 2 for which the weight space of weight ω is exactly V ω = C ⊗ Z K 0 (V ω ). Note that we will always assume that this action is integrable, so that e and f are locally nilpotent.

For such a notion of sl 2 -action on a category, the divided power functors satisfy the following identity on weight spaces of weight ω ≥ 0, see [START_REF] Rouquier | Quiver Hecke algebras and 2-Lie algebras[END_REF]Lem. 4.8] (1.1)

E (n) F (n) Vω ≃ 1 ⊕( ω n ) Vω ⊕ F E Vω ⊕( ω n-1 ) ⊕ F (2) E (2) Vω ⊕( ω n-2 ) ⊕ • • • ⊕ F (n) E (n)

Vω

.

We now assume that End(S) ≃ Λ for all simple objects S ∈ Irr V. In that case it is proven in [START_REF] Chuang | Derived equivalences for symmetric groups and sl 2 -categorification[END_REF]Prop. 5.20] that E(S), if non-zero, has simple socle and head and that they are isomorphic. Successive applications of E give a highest weight semi-simple object. We will use the further following properties, which are also proved in [START_REF] Chuang | Derived equivalences for symmetric groups and sl 2 -categorification[END_REF]Prop. 5.20].

Lemma 1.2. Let S ∈ Irr V ω and n ≥ 0 be such that E n+1 (S) = 0 and E n (S) = 0.

(1)

E (n) (S) is simple. (2)
The socle and head of F (n) E (n) (S) are isomorphic to S.

(3) The simple module S occurs in F (n) E (n) (S) with multiplicity ω+2n n as a composition factor.

Decomposition numbers.

Let O be a complete discrete valuation ring, with residue field k of positive characteristic and fraction field K of characteristic zero. Let {G r } r∈N be a family of finite groups. We consider the category ΛG = r≥0 ΛG r -mod which is the sum of the categories of finitely generated representations of G r over Λ, where Λ is any ring among K, O, k. If k and K are large enough for all the finite groups encountered, the following conditions will be satisfied:

• For Λ = K, k, the category ΛG has finite length and End(S) = Λ for all S ∈ Irr k G.

• Every S ∈ Irr k G has a projective cover P S in kG, unique up to isomorphism.

• Every projective module P in kG lifts uniquely to a projective module P in OG.

• KG is semisimple. If S ∈ Irr k G and ∆ ∈ Irr K G, the decomposition number of S in ∆ is the multiplicity of ∆ as a direct summand (equivalently, a composition factor) of K ⊗ O P S . We denote it by [P S : ∆] in a way which will look familiar to the reader interested in highest weight categories. Now, let V be a direct summand of OG. We assume that (E, F, X, T ) is a categorical datum on V inducing an sl 2 -categorical action on kV. This is to ensure that the divided powers E (n) and F (n) defined in the previous section can be lifted to exact and biadjoint endofunctors of V, even though V itself does not have an action of sl 2 since its Grothendieck group might be too big. The image of these functors by extension of scalars will be still denoted by E (n) and F (n) . Proposition 1.3. In the previous setting, let S ∈ Irr k V and n ≥ 0 be such that E n+1 (S) = 0 and E n (S) = 0. Then

[P S : ∆] = [P E (n) S : E (n) ∆]
for all irreducible characters ∆ ∈ Irr K V such that E n+1 ∆ = 0.

Proof. Let ω be the weight of the simple module T = E (n) S. Note that ω ≥ 0 since the class of T is a highest weight vector. Since S is the head of F (n) E (n) S by Lemma 1.2(2) we have that F (n) P T contains P S as a direct summand. We write (1.4)

F (n) P E (n) S = P S ⊕ Q
for some projective module Q. We want to have some control on the character of Q, so for that we compute the image of (1.4) by E (n) . Using (1.1) we have

E (n) P S ⊕ E (n) Q = E (n) F (n) P T = P ⊕( ω n ) T ⊕ F EP ⊕( ω n-1 ) T ⊕ • • • ⊕ F (n) E (n) P T .
Now we claim that E (n) P S contains ω n copies of P T . Indeed, the weight of S equals ω -2n and we have

Hom(E (n) P S , T ) = Hom(E (n) P S , E (n) S) ≃ Hom(P S , F (n) E (n) S)
which has dimension ω n by Lemma 1.2.(3). This proves that E (n) Q is a direct summand of a sum of modules of the form F (k) E (k) (P T ) where k ≥ 1. By the lifting property of projective modules, the same holds over O. Now, let ∆ be an irreducible character of KV such that [Q : ∆] = 0. In other words, ∆ is isomorphic to a submodule of K ⊗ O Q, and since

E (n) is exact, E (n) ∆ is isomorphic to a submodule of K ⊗ O E (n) Q.
Furthermore, it must be non-zero since by definition Q is a direct summand of F (n) P T , putting ∆ in the image of F (n) . Now by the previous paragraph, there is k ≥ 1 such that [F (k) E (k) (P T ) : E (n) ∆] = 0, which forces E (k) E (n) ∆, and therefore E n+k ∆ to be non-zero. We showed that E n+1 ∆ = 0 implies [Q : ∆] = 0, which gives the result by (1.4).

Remark 1.5. The same result (with analogous proof) would hold for highest weight categories and standard objects ∆.

Unipotent representations of finite classical groups

By a finite classical group G n (q) we will always mean one of the following groups:

Group SO 2n+1 (q) Sp 2n (q) O + 2n (q) O - 2n (q) Type B n C n D n 2 D n
where q > 1 is a power of some odd prime. The convention for small values of n is that SO

1 (q) = Sp 0 (q) = O ± 0 (q) = 1 is the trivial group, O + 2 (q) = GL 1 (q) ⋊ Z/2 and O - 2 (q) = GU 1 (q)
⋊ Z/2 are 1-dimensional tori (of respective orders q -1 and q + 1) extended by Z/2.

We will work with modular representations, therefore with different coefficient rings for our representations. As in Section 1.2 we fix a complete discrete valuation ring O with residue field k and fraction field K. We assume that K has characteristic zero, k has characteristic ℓ > 0 and that they are both large enough for all the finite groups encountered. In particular, every irreducible representation over K or k will be absolutely irreducible. We will always work under the assumption that ℓ is odd and does not divide q. The multiplicative order d of q in k × will be assumed to be even, so that we work in the unitary prime case.

Note that some of the results contained in this section for the groups O ± 2n (q) have not been published yet. Still, we have decided to include these groups since the combinatorics is not much different from the groups of type B/C, and since there are work in progress by Li-Shan-Zhang [START_REF] Li | Categorical action for finite classical groups and its applications: characteristic 0[END_REF] and Cia-Luvecce [START_REF] Cia-Luvecce | Categorical action on unipotent representations of finite orthogonal groups[END_REF] which will contain all the results we need here.

2.1. Combinatorics. We recall here the combinatorics that is used to classify the unipotent characters and unipotent blocks of finite classical groups in the unitary prime case.

Partitions and symbols.

Let m be a non-negative integer. A partition λ of m is a nonincreasing sequence of non-negative integers λ

= (λ 1 ≥ λ 2 ≥ • • • ≥ 0) which add up to m, the size |λ| of λ. A bipartition λ = λ 1 .λ 2 of m is a pair of partitions (λ 1 , λ 2 ) such that |λ 1 | + |λ 2 | = m. A charged bipartition |λ,
s is the data of a bipartition and a pair of integers s = (s 1 , s 2 ) ∈ Z 2 , called the charge.

To a charged bipartition |λ, s one can attach a charged symbol Θ(λ, s) corresponding to the pair of charged β-sets coming from the two partitions. More precisely, we have Θ(λ, s) = (X 1 , X 2 ) where

X k = {s k + λ j -j + 1 | j ≥ 1}
. A symbol will be represented by the corresponding 2-abacus, with the first row X 1 on the bottom. The defect of a charged symbol Θ

= Θ(λ, s) is def(Θ) = s 2 -s 1 .
Example 2.1. The charged symbol of charge (-4, 3) and bipartition 1 3 .2 2 1 will be represented as follows row 2 row 1

6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7
2.1.2. Adding and removing boxes. The unipotent characters will be parametrized by such symbols. In order to explain how the induction and restriction of unipotent representations behave on symbols, we will need the notion of addable/removable boxes. Recall that d is a fixed even integer. Let i ∈ Z/d and Θ = (X 1 , X 2 ) be a charged symbol. An addable i-box of Θ in the row k ∈ {1, 2} is an integer x such that

• x ≡ i + (k -1) d 2 mod d; • x ∈ X k and x + 1 / ∈ X k . Adding the i-box x in the symbol Θ consists in replacing x by x + 1 in X k . A removable i-box of Θ in the row k ∈ {1, 2} is an integer x such that • x ≡ i + (k -1) d 2 mod d; • x / ∈ X k and x + 1 ∈ X k . Removing the i-box x in the symbol Θ consists in replacing x + 1 by x in X k .
There is a notion of good addable/removable i-box, see for example [19, Section 3], [14, Theorem 2.8]. Given i, there is at most one good addable/removable i-box in a charged symbol, and it can be used to describe the Kashiwara operators f i and e i on charged symbols (or charged bipartitions in other contexts).

Remark 2.2. The reader might be surprised by the occurrence of d/2 in the definition of addable/removable boxes. This comes from the fact that the charges of our symbols will come from parameters (q s1 , -q s2 ) in the Hecke algebra. Using the fact that -1 = q d/2 , one should rather work with symbols of charge (s 1 , s 2 + d/2), but that will remove the symmetry from the combinatorics of d/2-co-hooks and d/2-co-cores. We have chosen to work with the original symbol parametrizing a characteristic 0 unipotent character, while shifting the notion of i-boxes to account for its behavior in quantum characteristic d. This way we are consistent with the usual description of unipotent ℓ-blocks. The discrepancy also appears in Section 2.6 where the charge on bipartitions and symbols differs by (0, d/2).

Adding and removing boxes does not change the charge. Removing all the possible boxes yields a symbol containing in each row only consecutive integers. Such a symbol corresponds to the empty bipartition, and the charge equals (max(X 1 ), max(X 2 )) where (X 1 , X 2 ) is the β-set of the empty bipartition.

2 row 1 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7
2.1.3. Co-hooks and co-cores. Let e ≥ 1 be a positive integer. Given a charged symbol Θ = (X 1 , X 2 ), a e-co-hook in row k is a pair (x, xe) where x ∈ X k and xe / ∈ X k+1 (here k + 1 must be understood modulo 2). Removing the e-co-hook to Θ amounts to removing x from X k and adding xe to X k+1 and swapping X 1 and X 2 . If Θ has charge (s 1 , s 2 ) then removing a e-co-hook yields a symbol of charge (s 2 ± 1, s 1 ∓ 1). A charged symbol with no e-co-hook is called a e-co-core. 2.2. Harish-Chandra series of unipotent characters. We follow [START_REF] Fong | The blocks of finite classical groups[END_REF] for the groups of type B/C and [START_REF] Waldspurger | Une conjecture de Lusztig pour les groupes classiques[END_REF] for the groups of type D. We start by describing the cuspidal irreducible unipotent characters of G n (q).

• SO 2n+1 (q) and Sp 2n (q) have a cuspidal unipotent character if and only if n = t 2 + t for some t ∈ N. In that case it is unique, and we denote it by ∆ t . • O + 2n (q) has a cuspidal unipotent character if and only if n = t 2 for some t ∈ 2N. If t = 0, there is a unique one, denoted by ∆ 0 . If t = 0, there are exactly two, which we denote by ∆ t and ∆ -t .

• O - 2n (q) has a cuspidal unipotent character if and only if n = t 2 for some t ∈ 2N + 1. In that case there are exactly two, which we denote again by ∆ t and ∆ -t . Note that t 2 + t is unchanged by the transformation t → -t -1, so writing ∆ t with t ∈ Z makes sense in all cases, with the convention that ∆ t = ∆ -1-t for groups of type B/C.

Let n, m, t ≥ 0 be such that G n (q) has a cuspidal unipotent character ∆ t . Then the unipotent characters of G n+m (q) above ∆ t are parametrized by bipartitions of m.

Classification of unipotent characters.

In order to have a global treatment of all the Harish-Chandra series, we define, for each t ∈ Z, the following charge

σ t =        (t, -1 -t) if t is even and G n is of type B/C, (-1 -t, t) if t is odd and G n is of type B/C, (t, -t) if t is even and G n is of type D, (-t, t) if t is odd and G n is of type 2 D.
Then unipotent characters are classified by charged symbols of charged σ t , see Section 2.1 for the definition or properties of symbols. In a series above ∆ t , the symbols will have defect 2t + 1, -2t -1, 2t, -2t depending on the type of groups and the parity of t.

Given a charged symbol Θ with charge σ t (t ∈ Z), we will denote by ∆ Θ the corresponding unipotent character. Note that there are no unique such parametrization, but the one we will choose comes from a categorical action and it will satisfy the properties given in the following sections.

2.4. Unipotent blocks. Recall that the multiplicative order of q in k × , denoted by d, is assumed to be even. A unipotent ℓ-block is an ℓ-block containing at least one unipotent character. The partition of unipotent characters into ℓ-blocks can be read off from the labelling of unipotent characters by symbols. By [START_REF] Fong | The blocks of finite classical groups[END_REF][START_REF] Srinivasan | Isolated blocks in finite classical groups[END_REF], one can chose the parametrization such that two unipotent characters are in the same ℓ-block if and only if the corresponding symbols have the same d/2-cocore.

2.5. Classification of unipotent simple modules over k. Under the assumptions on q and ℓ, the decomposition matrix of the unipotent blocks of classical groups have a unitriangular shape. This is proven for finite classical groups coming from connected reductive algebraic groups in [START_REF] Brunat | Unitriangular shape of decomposition matrices of unipotent blocks[END_REF], and the case of O ± 2n (q) follows for example from [START_REF] Feng | Späeth, Unitriangular basic sets, Brauer characters and coprime actions[END_REF]Thm. 3.1]. In particular, the parametrization of unipotent characters by charged symbols yields a parametrization of the irreducible unipotent representations over k as well. Given a charged symbol Θ with charge σ t (t ∈ Z), we will denote by S Θ the corresponding simple representation.

2.6. Categorical action. Let G n (q) be a finite classical group. We denote by OG n (q)-umod the category of finitely generated unipotent representations of G n (q) over O. It is the direct summand of OG n (q)-mod corresponding to the sum of all unipotent blocks. The type of finite classical group being fixed, we denote by V the category

V := n≥0 OG n (q)-umod.
Recall that we work in the unitary prime case, where d, the multiplicative order of q in k × , is even. In that case, there is, for every i ∈ Z/d, a categorical datum on V inducing an sl 2 -categorical action on kV. We will denote by E i and F i the corresponding functors (over any ring of coefficient between K, O and k). These functors are defined using Harish-Chandra induction and restriction functors, see [START_REF] Dudas | Categorical actions on unipotent representations of finite unitary groups[END_REF][START_REF] Dudas | Categorical actions on unipotent representations of finite classical groups[END_REF] for more details for their construction. These various sl 2 -categorical actions come from an sl d -categorical action. We shall not use that fact since we will be working with each action separately. However, it is important to know that one can chose the parametrisation of unipotent characters in such a way that one can actually compute the action of each E i , F i on irreducible unipotent characters and unipotent Brauer characters. More precisely, if we define (2.5)

s t = σ t + (0, d/2)
then there exists an isomorphism of sl d -modules

(2.6) C ⊗ Z K 0 (kV) ∼ -→ t F(s t ) ∆ Θ(λ,σt) -→ |λ, s t
where F(s t ) is the level 2 Fock space of charge s t , and ∆ denotes the class of any ℓ-reduction of the character ∆. Note that depending on the type of the classical group G n , the integer t will run over the non-negative integers only (for type B/C), over all the even integers (for type D) or over all the odd integers (for type 2 D).

Note that the action of sl d on the Fock space F(s t ) depends only on the charge up to shifts by (0, d) and (d, 0). However, s t ∈ Z 2 is chosen so that the map S Θ(λ,σt) -→ |λ, s t induces an isomorphism between the crystals (which are sensitive to the charge, not only the residue class of the components of the charge). This is proven for type B/C in [10, Thm. 1.7], and the same proof should work for type D and 2 D once we have a good parametrization. 2.7. Explicit formulas. The categorification results contained in the previous section were proven to get an explicit description of the Harish-Chandra induction and restriction functors on the unipotent representations in both characteristic zero and ℓ. Given a charged symbol Θ with charge σ t , we have (2.7)

E i ∆ Θ = j≡i mod d Θ Ψ=j ∆ Ψ
so that E i removes all the possible removable i-boxes from the charged symbol Θ.

(2.8)

soc E i S Θ = hd E i S Θ = 0 Θ has no good removable i-box, S Ψ if Ψ is obtained from Θ by removing the good i-box.
Similar formulas are obtained for the action of F i by adjunction.

The diagrammatics of d-small symbols

In this section, we adapt the combinatorics in [START_REF] Brundan | An orthogonal form for level two Hecke algebras with applications[END_REF][START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra I: cellularity[END_REF][START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra III: category O. Represent[END_REF] to an appropriate class of symbols labeling unipotent characters of types B, C, D and 2 D. This will position us to prove the formula for decomposition numbers in Theorem 4.1 using Proposition 1.3.

3.1. d-small symbols. Given a symbol Θ, we work with its graphical representation as two rows of beads and spaces, and so we will often refer colloquially to the elements β ∈ Θ as beads and to the elements of (Z, Z) \ Θ as spaces. Definition 3.1. Let Θ = (X 1 , X 2 ) be a symbol. For each i = 1, 2, define the interval

J i = [c i , d i ] ⊂ Z by d i = max{β ∈ Z | β ∈ X i } and c i = max{z ∈ Z | β ∈ X i for all β < z}. Fix d ∈ 2N. We say Θ is d-small if there exist intervals I 1 := [a 1 , b 1 ], I 2 := [a 2 , b 2 ] ⊂ Z satisfying the following conditions: • b i -a i = d 2 -1 for i = 1, 2, • J i ⊆ I i for i = 1, 2, • b 2 ≡ b 1 + d 2 mod d.
Given a d-small symbol Θ, we define the right region to be the region of Θ containing beads in the larger interval of integers I j , and the left region to be the region of Θ containing beads in the smaller subset of integers I j+1 (taking the subscript mod 2). We define the middle region to region of the symbol between the left and right regions. We note that the length of the middle region is always a multiple of d. The middle region contains only beads in the row containing the beads of the right region, and only spaces in the row containing the spaces of the left region.

The left region always contains only beads in one row (the same row as the beads of the middle region) while the right region always contains only spaces in the opposite row. By abuse of language, we will often refer to the intervals I j and I j+1 of Z as the right and left regions, and the interval of integers between them as the middle region.

Example 3.2. The following symbol is d-small for d = 12:

row 2 row 1 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7
The left region must be chosen to be [-6, -1] and the right region to be [0, 5], and we have marked these regions with the vertical lines in the abacus diagram. The middle region is empty in this example.

Example 3.3. We take d = 10 and consider the following symbol. We see that it is d-small, with right region [START_REF] Brundan | Highest weight categories arising from Khovanovs diagram algebra III: category O. Represent[END_REF][START_REF] Dudas | Decomposition matrices for groups of Lie type in non-defining characteristic[END_REF] • if β i ∈ Θ and β ikd -d 2 occurs in both rows of Θ, then It is uniquely determined from w ∧∨ (Θ) and consists of non-crossing counterclockwise arcs and rays attached to the ∨'s and ∧'s of w ∧∨ (Θ) by the following recursive procedure. Start with a pair ∨ • • • ∧ which are either adjacent (i.e. w i w i+1 = ∨∧ for some i), or have only × and • symbols between them. Connect the ∨ to the ∧ with a curved arc (a "cup"). Considering the ∨ and ∧ symbols as directional, the cup is counterclockwise-oriented as its left endpoint is ∨ and its right endpoint is ∧. Next, continue to connect with cups any ∨ • • • ∧ pairs which are adjacent, or have only × and • symbols and previously constructed cups between them. When no more counterclockwise cups can be constructed by this rule, attach vertical rays below the remaining ∧ and ∨ symbols.

w i = ×, • if β i ∈ Θ and β i -kd -d 2 occurs only once in Θ, then w i = ∧, • if β i / ∈ Θ and β i -kd -d 2 occurs in both rows of Θ, then w i = ∨, • if β i / ∈ Θ
Example 3.9. We continue with Example 3.2. We have w ∧∨ (Θ) = ∧ ∨ × ∨ ∧ ∧ and the associated cup diagram c ∧∨ (Θ) is:

c ∧∨ (Θ) = ∧ ∨ × ∨ ∧ ∧
w ∧∨ (Θ) = ∧ ∧ ∨ ∨ • ∧ ∨ ∧ ∧ × ∨ ∨ ∧ • and c ∧∨ (Θ) = ∧ ∧ ∨ ∨ • ∧ ∨ ∧ ∧ × ∨ ∨ ∧ •
We remark that the rays will not play any role in computing decomposition numbers; what is important for our purposes are the cups in c ∧∨ (Θ). Likewise the × and • symbols play a placeholder role. In the situation of Example 3.4 where there is more than one choice for the left and right regions, different choices are reflected in w ∧∨ (Θ) by replacing a prefix of ×'s with a suffix of the same number of •'s or vice versa.

3.3.

The cocore of a d-small symbol. Lemma 3.11. Let Θ be a d-small symbol and let Θ • be its cocore. Then Θ • is a d-small symbol, its middle region has length 0, and w ∧∨ (Θ • ) is obtained from w ∧∨ (Θ) by replacing all ∧ by ∨.

Proof. We will induct on k where kd is the length of the middle region of the d-small symbol Θ. Suppose k = 0, so the middle region has length 0. Given an occurrence of ∧ in w ∧∨ (Θ), there is a bead in the corresponding position of the right region and a space d 2 to its left and in the opposite row. Thus each ∧ yields a removable cohook, and conversely, any removable cohooks are given in this way by the ∧'s in w ∧∨ (Θ). Removing such a cohook moves the bead in from the right region to the left region. Do this for each ∧ in w ∧∨ (Θ), then flip the resulting symbol upside-down if the number of ∧'s in w ∧∨ (Θ) is odd. No more cohooks can be removed from the resulting symbol, so we have found the cocore Θ • of Θ. It is clear that Θ • has the desired description.

Next, suppose by induction that if Ψ is a d-small symbol whose middle region has length kd then Ψ • has the desired description. Let Θ be a d-small whose middle region has length (k + 1)d. Partition the middle region into equal segments of length d 2 . For the leftmost such segment of length d 2 (adjacent to the left region), move the i'th bead d 2 to the left and to the opposite row if there's a space in the i'th position there, thus, if ∧ or • is the i'th letter of w ∧∨ (Θ). For all other beads in the middle and right regions, they all belong to the same row. The opposite row has only empty spaces in the middle and right regions. We move these beads d 2 to the left and to the opposite row. Finally, flip the symbol upside down if the total number of beads moved was odd. Call this symbol Ψ. We have removed some number of cohooks from Θ to obtain Ψ, so Ψ has the same cocore as Θ. We observe that Ψ has the same pattern of beads and spaces in its left and right regions as Θ. Thus w ∧∨ (Ψ) = w ∧∨ (Θ). However, the middle region of Ψ has length kd. Applying induction, Θ • has the desired description.

A d-small symbol Θ is determined by the length of its middle region together with w ∧∨ (Θ), up to flipping the symbol upside-down. In type B, that is when the defect of the symbol is odd, by convention the flip of the symbol upside-down labels the same unipotent character. Thus when Θ is the symbol of a type B unipotent character, the preceding lemma completely characterizes the cocore of Θ. In the case of type D or 2 D unipotent characters, that is when the defect of the symbol is even, the lemma characterizes the cocore up to flipping it upside-down (these label two different unipotent characters in that case). Proof. We use the description of blocks in terms of weight subspaces for the sl d -action given by [START_REF] Dudas | Categorical actions on unipotent representations of finite classical groups[END_REF]Lem. 6.10]. By [12, (2.4)] the weight of a charged bipartition |λ, s for this action is given by

Λ s1 + Λ s2 - i∈Z/d n i (λ, s)α i -∆(s, d)δ
where {Λ i } i∈Z/d (resp. {α i } i∈Z/d ) are the fundamental weights (resp. the simple roots), δ is the imaginary root, n i (λ, s) is the number of boxes of content i of the charged bipartition |λ, s and ∆(s, d) is an integer defined in [12, (2.3)]. In particular, if |λ, s and |µ, s have the same then n i (λ, s) = n i (µ, s) for all i ∈ Z/d. In particular, if Θ(λ, s) is d-small then so is Θ(µ, s) by Remark 3.5.

Let Θ ′ be a symbol with the same charge as Θ and belonging to the same block. By the above argument, Θ ′ is d-small. By Lemma 3.11, the up-down diagrams w ∧∨ (Θ) and w ∧∨ (Θ ′ ) agree up to changing some ∧'s into ∨'s and vice versa. Since they have the same charge, the number of ∧'s (hence the number of ∨'s) in their diagrams must be equal. Consequently w ∧∨ (Θ) and w ∧∨ (Θ ′ ) differ only by a permutation of ∧'s and ∨'s.

Conversely, swapping one ∧ with one ∨ can be obtained as a succession of removing 2k + 1 co-hooks starting from the right region followed by a succession of adding 2k + 1 co-hooks starting from the left region, where k is such that kd is the size of the middle region. We have w ∧∨ (Θ) = ∧ × ∧ ∨ ∧. There are four permutations of the set of ∧'s and ∨'s in w ∧∨ (Θ):

{∨ × ∧ ∧ ∧, ∧ × ∨ ∧ ∧, ∧ × ∧ ∨ ∧, ∧ × ∧ ∧ ∨}.
By permuting the beads in the left and right regions of the symbol Θ to match these permutations of its up-down diagram, we obtain all the symbols in the same series and block. For instance, ∨ × ∧ ∧ ∧ gives the symbol row 2 row 1 4 -1 -6 3.4. Action of the crystal operators on d-small symbols. In this section, we explain how the action of the crystal operators ẽi on symbols translates to w ∧∨ (Θ) for a d-small symbol Θ.

To describe the action of crystal operators using w ∧∨ (Θ), we consider i ∈ Z/dZ where i labels the residue mod d of a position in the right region if the right region lies in row 1 of Θ. Otherwise, the left region lies in row 1 of Θ, and then we let i label the residue mod d of a position in the left region. Match up the letters w 1 , . . . , w d 2 in w ∧∨ (Θ) from left to right with the residues of the appropriate region. In Example 3.12, we'd have:

1 2 3 4 5 6 • × ∧ • ∨ ∧
We then consider the two adjacent letters w j , w j+1 ∈ {∧, ∨, •, ×} in w ∧∨ (Θ) corresponding to the residues i, i + 1: (3.15) i i + 1 w j w j+1

Continuing our example and taking i = 5, we'd consider:

5 6 ∨ ∧
Then the effect of ẽi on w ∧∨ (λ) is given by acting locally on these two letters as described in the next lemma.

Lemma 3.16. Let Θ be a d-small symbol and let w ∧∨ (Θ) be its up-down diagram. Label the letters of w ∧∨ (Θ) with the residues of the appropriate region as described above. The action of the crystal operators ẽi and fi on Θ is computed locally in w ∧∨ (Θ) on w j w j+1 as follows:

(3.17)

⋆× ei * * j j fi ×⋆ •⋆ ei ) ) i i fi ⋆• •× ei * * j j fi ∨∧ ei * * j j fi ו
for ⋆ ∈ {∧, ∨}. The action on ∧∨ is zero. Proof. Suppose the Kashiwara i-word of Θ is non-empty for some i. Then it consists of at most two letters from the alphabet {+, -}, with at most one contributed by an addable or removable box in left region and at most one contributed by an addable or removable box in right region of Θ. We need to check that if the i-word has two letters, then the letter contributed by the right region is larger than the letter contributed by the left region.

Suppose that the right region occurs in row 2. Recall that |λ, s t = |λ, σ t + (0, d 2 ) . Then any addable or removable i-box in λ 2 has charged content at least d greater than any addable or removable i-box of λ 1 . when comparing addable and removable i-boxes of λ, the one in λ 2 is larger than the one in λ 1 . Next, suppose that the right region occurs in row 1. If the middle region has length at least d, then as in the previous case, an addable or removable i-box of λ 1 will have charged content at least d greater than an addable or removable i-box of λ 2 , so will be larger. Otherwise, the right region is d 2 to the right of the left region. Then in |λ, s t , the addable/removable i-boxes of λ 1 and λ 2 have the same charged content. In this situation, the box in λ 1 is consider larger. So again, the letter contributed to the i-word from the right region is larger.

The lemma implies that the class in the Grothendieck group of a simple module corresponding to a d-small symbol lies in a representation of sl 2 of dimension at most 3. The action on the standard modules is the same as for simple modules unless we are in the 3-dimensional representation of sl 2 , where we have Assume by induction on the rank of the group that the theorem holds for P Θ ′ . The multiplicity [P Θ : ∆ Ψ ] is non-zero if and only if there exists a charged symbol Ψ ′ such that ∆ Ψ ′ occurs in E (n) i ∆ Ψ and such that w ∧∨ (Ψ ′ ) is obtained from w ∧∨ (Θ ′ ) by reversing the orientation on a subset of cups of c ∧∨ (Θ ′ ). The constraints on w ∧∨ (Θ ′ ) given in the previous paragraph force (3.18), so [P Θ : ∆ Ψ ] = 1. In the configuration ו for Ψ ′ (and hence Θ ′ ), suppose we are in the case n = 1. We must have the configuration ∨∧ for Θ, as Θ = fi ẽi Θ = fi Θ ′ . That is, in w ∧∨ (Θ) we have w j w j+1 = ∨∧, where w j corresponds to residue i as in (3.15). Hence in c ∧∨ (Θ) there is a cup connecting w j and w j+1 . However, in w ∧∨ (Θ ′ ) we have w ′ j w ′ j+1 = ו so there is no cup connecting w ′ j and w ′ j+1 in c ∧∨ (Θ ′ ). Since all other letters in w ∧∨ (Θ) and w ∧∨ (Θ ′ ) are the same, the cup diagram c ∧∨ (Θ) is thus obtained from the cup diagram c ∧∨ (Θ ′ ) by inserting a cup at positions i and i + 1. On the other hand, by (3.18) there are two possibilities for Ψ in positions i and i + 1, namely ∨∧ and ∧∨. It follows that w ∧∨ (Ψ) is obtained from w ∧∨ (Θ) by reversing the orientation on a subset of cups of c ∧∨ (Θ).

E (n) i ∆ Ψ = ∆ Ψ ′ by
For all the other configurations, Ψ is uniquely determined from Ψ ′ . We conclude that in all the cases, we have that w ∧∨ (Ψ) is obtained from w ∧∨ (Θ) by reversing the orientation on a subset of cups of c ∧∨ (Θ).

We deduce from [START_REF] Stroppel | Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology[END_REF] that the decomposition numbers can be computed from Kazhdan-Lusztig polynomials. For d-small symbols belonging to the same block in the setting of Theorem 4.1, let h be the number of ∧'s and w be the number of ∨'s in their up-down diagrams.

Corollary 4.2. The entries of the square submatrix of the decomposition matrix given by the formula of Theorem 4.1 are given by parabolic Kazhdan-Lusztig polynomials of type (W, P ) = (S h+w , S h × S w ) evaluated at 1.

Example 2 . 3 .

 23 Let us consider the symbol drawn in Example 2.1. Assume d = 8. Then there are two addable 1-boxes, the one in the top row being a good addable 1-box.

  row

Example 2 . 4 .

 24 Let us again consider the symbol drawn in Example 2.1. There are four 4-cohooks, all being in the top row.

Remark 3 . 5 . 2 consisting of a word of length d 2

 3522 and left region [-10, -6]. The middle region has length d in this case. d = 16. Then this symbol is d-small with four different possible choices for the left and right regions. They are: (1) left region = [-9, -2], right region = [-1, 6], (2) left region = [-8, -1], right region = [0, 7], (3) left region = [-7, 0], right region = [1, 8], (4) left region = [-6, 1], right region = [2, 9]. Recall that if Θ = Θ(λ, σ) is the symbol attached to the charged bipartition |λ, σ , then |λ, (σ 1 , σ 2 + d 2 ) =: |λ, s is the charged bipartition for the Fock space as in (2.6). Then Θ is d-small if and only if the set of residues of the charged contents of boxes in |λ, s is contained in a closed interval of length d 2 -1 modulo d. 3.2. Diagrams. Definition 3.6. Fix d ∈ 2N. Let Θ be a d-small symbol. Write the right region as I j = m + [1, d 2 ] ⊂ Z for some m ∈ Z. Let kd be the length of the middle region, k ∈ N. To Θ we associate an up-down diagram w ∧∨ (Θ) = w 1 w 2 • • • w d in the alphabet {∧, ∨, •, ×}. Each β i = m + i ∈ I j determines the i'th letter w i of w ∧∨ (Θ) according to the rule:

Definition 3 . 8 .

 38 and βkd -d 2 occurs only once in Θ, then w i = •. Thus, the up-down diagram of Θ records what happens in the left and right regions of Θ, in the opposite rows where a mix of beads and spaces is possible. Putting the right region on top of the left region, the rules corresponds to the following possibilities: Given a d-small symbol Θ, we further associate to it a cup diagram c ∧∨ (Θ).

Example 3 . 10 .

 310 Let d = 28 and consider the following d-small symbol: region on top of the left region we have hence according to (3.7) we get the following up-down diagram:

Example 3 . 12 .row 1 Lemma 3 . 13 .

 3121313 Take d = 12 and the following d-small symbol Θ: havew ∧∨ (Θ) = • × ∧ • ∨ ∧ . Then w ∧∨ (Θ • ) = • × ∨ • ∨∨ and Θ • is:row 2 Let Θ be a d-small symbol. The set of symbols {Θ ′ | Θ and Θ ′ belong to the same block and have the same charge } consists of those d-small symbols with the same charge whose up-down diagrams w ∧∨ (Θ ′ ) are given by all possible permutations of the ∧'s and ∨'s in w ∧∨ (Θ).

Example 3 . 14 .

 314 Take d = 10 and consider the following d-small symbol Θ:

(3. 18 )Example 3 . 19 .

 18319 e i ∆ •× = ∆ ∨∧ + ∆ ∧∨ = f i ∆ וin the Grothendieck group. Here the subscripts correspond to the changes in the ∧∨-diagram of the corresponding symbol. Continuing with Example 3.12, the lemma tells us thatẽ5 (• × ∧ • ∨∧) = • × ∧ • וand thus ẽ5 acts on Θ by moving the rightmost bead in the symbol one position to the left. Now assume that S Θ is not annihilated by every E i . Then there exists i ∈ Z/d and n ∈ {1, 2} such that E n i S Θ = 0 and E n+1 i S Θ = 0. By Proposition 1.3 and the remark following Lemma 3.16 we have[P Θ : ∆ Ψ ] = [P e n i Θ : E (n) i ∆ Ψ ]. Write Θ ′ = e n i Θ. By Lemma 3.16, Θ ′ is again d-small. By the rules in (3.17), the only possible configurations in positions i, i + 1 in w ∧∨ (Θ ′ ) are ⋆• with ⋆ ∈ {×, ∧, ∨} and ×⋆ with ⋆ ∈ {•, ∧, ∨}. In particular the configurations ∧∨ and ∨∧ cannot occur.

3.5. Highest weights for the sl d -action. For every charge σ = (σ 1 , σ 2 ) ∈ Z 2 , the symbol |∅.∅, σ labels a cuspidal unipotent character of the appropriate group (depending on the charge of the symbol) in characteristic 0, see §2.3. When reduced module ℓ, it remains cuspidal [START_REF] Dudas | Modular irreducibility of cuspidal unipotent characters[END_REF].

We now explain the classification of modular cuspidals labeled by non-empty d-small symbols. The bipartition of such a symbol is a rectangle concentrated in one component, with a condition on the charged content of the removable box of the rectangle. In the case that Θ is a d-small symbol, it follows from [START_REF] Dudas | Categorical actions on unipotent representations of finite classical groups[END_REF][START_REF] Norton | On modular Harish-Chandra series of finite unitary groups[END_REF] and the rule for computing the Heisenberg or sl ∞ crystal [START_REF] Gerber | Heisenberg algebra, wedges and crystals[END_REF][START_REF] Gerber | Norton The sl∞-crystal combinatorics of higher level Fock spaces[END_REF] that S Θ is cuspidal if and only if [S Θ ] is a highest weight vector for sl d , that is, if and only if E i S Θ = 0 for all i ∈ Z/d. However, we will not need the condition that S Θ is cuspidal, but only the condition of its being annihilated by every E i . The description of such Θ follows immediately from Lemma 3.16.

Corollary 3.20. Let Θ be a d-small symbol. We have ẽi Θ = 0 for all i ∈ Z/d if and only if in w ∧∨ (λ), all ×'s precede all ∧'s which precede all ∨'s which precede all •'s.

Remark 3.21. Let Θ be a d-small symbol on which ẽi acts by 0 for all i ∈ Z/d. Set w = #{∧'s in w ∧∨ (λ)} and h = #{∨'s in w ∧∨ (λ)}. Set λ = (w h ). Then Θ = Θ(λ, σ) where λ = λ.∅ if the left region is in row 1 and λ = ∅.λ if the left region is in row 2. The charge σ = (σ 1 , σ 2 ) is given by:

(1)

Decomposition numbers in a d-small Harish-Chandra series of a block

Our main theorem is a closed combinatorial formula for the entries of the square submatrix of the decomposition matrix cut out by a d-small Harish-Chandra series within a block. Theorem 4.1. Suppose Θ and Ψ are symbols of the same charge, and that Θ is d-small. Then

by reversing the orientation on a subset of the cups of c ∧∨ (Θ), 0 otherwise. Proof. We can assume without loss of generality that Θ and Ψ are in the same block, otherwise the decomposition number is zero. Consequently, Ψ is also d-small by Lemma 3.13. If S Θ is annihilated by E i for every i ∈ Z/d, then w ∧∨ (Θ) has no cup by Corollary 3.20 ; on the other hand, we claim that any charged symbol Ψ in the same block and with the same charge as Θ will satisfy Ψ Θ for the order defined in [10, §1.

is a non-trivial permutation of the ∧'s and ∨'s of w ∧∨ (Θ), some ∧ in w ∧∨ (Ψ) is to the right of every ∧ in w ∧∨ (Θ). Writing Θ = (X 1 , X 2 ) and Ψ = (Y 1 , Y 2 ), it follows that max(Y 1 , Y 2 ) > max(X 1 , X 2 ) , so Ψ Θ. Therefore by unitriangularity [P Θ : ∆ Ψ ] = 0 unless Ψ = Θ.


Carolina Lagartinho-Oliveira

Filipe Moutinho

Luís Gomes

Towards Digital Twin in the Context of Power Wheelchairs Provision and Support

Keywords: Digital Twin, Monitoring, Power Wheelchair, Remote Control

Recent advances in digital technologies have been triggered by Industry 4.0, deeply focused on connectivity, real-time data and digitization, with Digital Twin (DT) at the forefront of this. A DT is a virtual counterpart seamlessly linked to a physical entity, both sharing the same properties, characteristics and behavior, using data and information. Applications of DT include real-time monitoring, design/planning, optimization, maintenance, prediction, remote access, etc. This paper gives an overview of the Digital Twin concept and seeks to synthesize the technologies frequently used to enable it. It is also proposed the creation and use of a Digital Twin of a Power Wheelchair to promote the health and wellness of wheelchair users and facilitate services by healthcare professionals in the provision and support of wheelchairs.

Introduction

The World Health Organization estimates that about 15% of the world's population has a disability [1]. In Portugal, it corresponds to almost 1 million people with the highest prevalence related to mobility [START_REF]Censos 2011[END_REF]. For these people, operating a wheelchair can be a way to be able to live with freedom and independence. To accommodate those who are unable to use a Manual Wheelchair, one can opt for a Power Wheelchair (PWC), which generally offers a wider range of features to meet different needs and user capabilities. The main obstacles are that the industry is currently limited to high-income markets, and those with lower economic resources end up relying on charity services, which in most scenarios only provide low-quality or used products, unsuitable for the user in the context, with lack of repair and follow-up mechanisms. In addition, in many low-and middle-income countries, access to trained healthcare professionals (HCPs) is limited, but they are essential for the proper provision and support of PWCs [START_REF]Assistive technology[END_REF]. Even now, with the COVID-19 pandemic situation, the scenario got worse, with a significant impact on budgets that concern access to PWCs; and by restricting in-person clinic or home visits. Given this, the idea of performing some parts of the provision of wheelchairs remotely, via telehealth and videoconferencing, has been encouraged [START_REF] Watanabe | Includes Telehealth Options for CRT Clinicians[END_REF]. Nowadays, the concept of Digital Twin [START_REF] Grieves | Digital Twin: Mitigating Unpredictable, Undesirable Emergent Behavior in Complex Systems[END_REF] is emerging applied to different areas as a promising approach to gather insightful data shared between the physical and virtual worlds, and facilitate the means to design, monitor, analyze, optimize, predict, control, etc. physical entities [START_REF] Singh | Digital Twin: Origin to Future[END_REF]. In this way, the following research questions aroused: "How to use Digital Twin in the context of Power Wheelchairs provision and support?" and "To what extent does the Digital Twin represent a real Power Wheelchair?". In view of this, the objective of this paper is to propose the creation and use of a Digital Twin of a PWC to promote the health and wellness of people who use wheelchairs, as well as to facilitate and improve the comprehensive delivery of remote services by HCPs.

Relation and Contribution to Digitalization and Virtualization

The use of Digital Twin (DT) in the current days had its boom around 2016, a few years after the proposal of Industry 4.0 in 2013 [START_REF] Kagermann | Recommendations for implementing the strategic initiative INDUSTRIE 4.0[END_REF]. Since then, the concept has expanded to different areas, in which new DT applications have emerged to support digitalization as a way of innovation, detecting problems, anticipating challenges and increasing efficiency [START_REF] Hinduja | Industry 4.0: Digital Twin and its Industrial Applications[END_REF]. Digital Twin allows the virtualization of anything in the real world to be represented in the virtual world, with physical and virtual entities connected sharing the same properties, characteristics and behavior by means of data and information; changes that occur at the physical entity (PE) or at the virtual entity (VE) lead directly to changes in the other. Some of the DT advantages are [START_REF] Liu | Review of digital twin about concepts, technologies, and industrial applications[END_REF]: (1) it considers large amounts of data derived from different sources, providing a reliable way to model VE and mirror the PE for the duration of its lifecycle; (2) VE synchronizes with the counterpart, which allows DT to monitor and analyze PE in real-time, supporting design, maintenance, control, etc.; (3) it can help optimize the PE, using simulations and predictions, assessing the impact of different scenarios, that can be verified and validated before they occur; (4) it is able of performing diagnostics and intelligent control over the PE, with VE sending instruction when necessary; (5) it allows remote access for human interaction with the PE; just to mention the most relevant advantages. Indeed, due to its advantages, DT can be used in different contexts. This paper proposes the use of DT to promote digitalization and virtualization in the Power Wheelchair sector, in the context of provision and support.

Digital Twin Overview

Three common elements to represent a Digital Twin are [START_REF] Grieves | Product lifecycle management: the new paradigm for enterprises[END_REF]: a real space, corresponding to the physical world where PEs exist; a virtual space, where VE mirrors PE; and a bidirectional connection for data synchronization between spaces. In [START_REF] Tao | Five-Dimension Digital Twin Modeling and Its Key Technologies[END_REF], a particular proposal highlights the presence of PE and VE interacting with services (Ss) and data (DD) using connections (CN) that link elements. In short, DT supports the representation of PE through VE if: (1) it guarantees the entanglement of entities; (2) PE values are timely mapped to VE; and (3) VE reproduces the relevant PE attributes in the context. These are the properties of entanglement, reflection and representativeness. Optional properties (e.g. replication, composability, memorization or augmentation) [START_REF] Minerva | Digital Twin in the IoT Context: A Survey on Technical Features, Scenarios, and Architectural Models[END_REF] will depend on the application and DT type. A DT can be classified according to its position in the lifecycle [START_REF] Jones | Characterising the Digital Twin: A systematic literature review[END_REF]: Digital Twin Prototype (DTP) or Digital Twin Instance (DTI); or based on individual values [START_REF]High Value Manufacturing Catapult Visualization[END_REF]: Supervisory Digital Twin (SDT), Interactive Digital Twin (IDT) or Predictive Digital Twin (PDT); etc. In the first case, DTP is the DT of a physical prototype to be used early in the lifecycle to support tests that are difficult to perform during traditional prototyping. In the realization of PE, DTP becomes a DTI, corresponding to the DT already defined; it lasts beyond PE life. In the second group, SDT monitors PE and collects data to support the perception of PE attributes and their relationship to its performance. IDT leads to a more reliable PE by diagnosing and controlling processes. PDT can help drive PE by simulating or predicting the impact of planned or potential scenarios.

Mapping a PE in the digital world starts with cognition of the physical world and data acquisition: measurement technologies capture PE characteristics; sensory technologies and Internet of Things (IoT) technologies collect PE data in real-time; other data is collected through application programming interfaces or web crawlers; etc. [START_REF] Qi | Enabling technologies and tools for digital twin[END_REF]. Thus, different data can be used to build and maintain VE, for it to operate faithfully and control PE via power technologies, drive systems, control technologies, among others. To support the data flow and connectivity of all DT elements, transmission technologies are used together with appropriate network architectures, communication and security protocols, middleware platforms, etc. [START_REF] Lim | A state-of-the-Art Survey of Digital Twin: Techniques, Engineering Product Lifecycle Management and Business Innovation Perspectives[END_REF]. In addition, all data are made available for further processing, analysis and visualization using storage technologies [START_REF] Qi | Enabling technologies and tools for digital twin[END_REF] and ontologies for knowledge representation and management [START_REF] Lim | A state-of-the-Art Survey of Digital Twin: Techniques, Engineering Product Lifecycle Management and Business Innovation Perspectives[END_REF]. Data processing helps to eliminate redundant and inconsistent data using data cleaning and transformation technologies [START_REF] Tao | Background and Concept of Digital Twin[END_REF], to later extract useful information using statistical methods, artificial intelligence (AI) methods [START_REF] Qi | Enabling technologies and tools for digital twin[END_REF] and Big Data technologies, such as cloud, fog and edge computing [START_REF] Pires | Digital Twin in Industry 4.0: Technologies, Applications and Challenges[END_REF]. Data fusion is also suggested to help filter and correlate the processed data [START_REF] Qi | Enabling technologies and tools for digital twin[END_REF], which can then be used in functions provided by the DT (e.g. prediction, optimization and validation), encapsulated in services to facilitate their use [START_REF] Tao | Background and Concept of Digital Twin[END_REF]. Services are based on: virtualization, verification, validation, and simulation technologies [START_REF] Lim | A state-of-the-Art Survey of Digital Twin: Techniques, Engineering Product Lifecycle Management and Business Innovation Perspectives[END_REF]; platformrelated technologies used to manage resource services, knowledge services, or application services; and application interface technologies to enable user interaction with the services. Even so, mirroring a PE is a long process where it is also up to the VE to reproduce its characteristics, properties and behaviors through different models. Section 5 builds on the aforementioned aspects, focusing on how to enable VE.

Digital Twin for Power Wheelchairs Provision and Support

The provision of wheelchairs is a process for users to obtain and maintain their wheelchairs. Since the outbreak of COVID-19, the option to remotely carry out some parts of wheelchair provision is being valued; users can stay at home, reducing the risk of exposure and helping professionals determine day-to-day challenges; and it may be possible for users who do not live near a clinic to have access to HCPs. Still, telehealth do not cover important phases of wheelchair provision, such as when additional modifications and adjustments are required, or HCPs are asked to provide maintenance and repair of technical issues. To deal with this, the use of Digital Twin is proposed in the context of Power Wheelchairs provision and support. Fig. 1 presents each step of the provision process, where a DT of a PWC can support the digitalization and virtualization of the product preparation, fitting, training, and follow-up, maintenance and repair steps; 3 groups of participants are also highlighted. First, a DT of a PWC can be used during product preparation and fitting to help understand the functioning of the wheelchair and improve the planning and testing of various scenarios for its use. Concretely, assistive technology professionals (ATPs) can use DT to justify, design, and validate new or existing features and parts of the PWC; and simulate PWC's health conditions, to determine and eliminate the cause of problems and undesired situations. ATPs can also use DT to predict the performance of PWC, including the occurrence of unforeseen scenarios. Its predictive capability can also be used during training to anticipate future risks, with PWC users being advised on how to avoid accidents based on PWCs usage data and the computation of patterns of usage and misuse situations. The DT can also guide PWC users with personalized training services, based on the users' health data, therapists' assessments, etc. In the last step, the DT can acquire real-time operational data from PWC and increase user engagement by showing PWC users and caregivers what is happening inside their PWCs, and giving insight on possible consequences of their use. It can also help therapists monitor users' rehabilitation activities, tracing usage patterns and reporting which PWC configurations and functionalities are used. This can influence the planning of the rehabilitation process, and help PWC users to follow it. Also during follow-up, maintenance and repair, a DT of a PWC can trigger alerts to help diagnose problems, as well as identify actions to improve PWC performance. In case of a wheelchair repair, DT facilitates PWC analysis and remote intervention with real-time control over the PWC. Here are some ways to use a DT of a PWC, as a possible answer to the question "How to use Digital Twin in the context of Power Wheelchairs provision and support?".

Digital Twin of a Power Wheelchair

In Fig. 2 a conceptual model for a Digital Twin of a Power Wheelchair is shown; the model is based on [START_REF] Tao | Five-Dimension Digital Twin Modeling and Its Key Technologies[END_REF] and has been extended to consider the user element and its interactions. While DT abstracts the complexity of a PWC, offering specific services, users continue to interact with the PWC and contribute with different knowledge. To implement the model, PDT is the type of DT considered; as well as a DTI to adapt PWCs that come with predefined functionalities, or a DTP to add them to those that don't [START_REF] Singh | Digital Twin: Origin to Future[END_REF]. Properties that characterize the proposed model are: augmentation and servitization for the PWC to be modified, updated and improved by VE augmentation as new services are offered; persistency so if the PWC malfunctions, VE continues reporting to services, and restores it to a known acceptable state; and composability for when a VE represents only a subpart of a PWC, and an aggregation of VEs represents the PWC itself [START_REF] Minerva | Digital Twin in the IoT Context: A Survey on Technical Features, Scenarios, and Architectural Models[END_REF]. In fact, the PWC can be seen as an aggregation of the parts that make it up (e.g. joystick module, power module or seating module) and each one can have its VE; the VE in Fig. 2 corresponds to a series of sub-models integrated to mimic specific parts of PWC. To deal with this modularity, the proposal considers a Model-Driven Development (MDD) approach that supports all stages of systems development using models as the primary artifact [START_REF] Meyers | A Model-Driven Engineering Framework to Support the Functional Safety Process[END_REF].

MDD ensures a high level of specification abstraction while attaining the level of detail required for a DT; it has the ability to [START_REF] Kleidermacher | Secure Embedded Software Development[END_REF]: (1) enable hardware and software design; (2) eliminate errors early in development; (3) facilitate testing with automated verification and validation techniques; and (4) support automatic code generation for target platforms. Besides, MDD supports various modeling formalisms, enabling the use of technologies and tools that are essential for VE. The VE has to represent the PWC (or part of it) in a way that users can analyze and manage it through the services, based on the data; so, the DT model in Fig. 2 represents a real PWC through VE, considering three types of models: geometric, behavioral and rule model.

The geometric model will represent the global solid appearance of the PWC based on data structures of topological and geometric information, used to feed software for three-dimensional modeling, computer-aided design, or virtual reality. The behavioral model will describe the PWC's behaviors according to its state model, dynamic model and kinematic model. The state model will be specified using the Petri net formalism, suitable for the development of complex and concurrent systems. A new class of nonautonomous high-level Petri nets will be created based on IOPT nets [START_REF] Gomes | Refining IOPT Petri Nets class for embedded system controller modeling[END_REF], colored Petri nets [START_REF] Jensen | Colored Petri nets: A Graphical Language for Formal Modeling and Validation of Concurrent Systems[END_REF], and DS-Pnets [START_REF] Pereira | The IOPT-Flow modeling framework applied to Power Electronics controllers[END_REF], which have the potential to reduce the complexity of data modeling and data processing; also, the use of Petri nets will provide evidence on the representation of requirements for the VE state models through their verification and validation, using automation tools supported by MDD. Regarding to the dynamic model and kinematic model, they will be described through mathematical equations; initially, it will be considered a unicycle model to deal with the motion of the PWC. Additionally, the rule model will provide VE means to describe patterns and make predictions. This will be based on rule-based learning algorithms under the unsupervised branch of machine learning [START_REF] Fürnkranz | A Brief Overview of Rule Learning[END_REF]. The rule model will allow associate PWC users actions with virtual model of PWC. In this way, it is believed that a DT can represent a real Power Wheelchair and may answer the question "To what extent does the Digital Twin represent a real Power Wheelchair?".

Related Work and Discussion

Many users find it difficult to use a wheelchair, even for daily activities. They need to have continue access to HCPs for proper provision, training and follow-up; and that's sometimes limited. Additionally, many users suffer injuries from tips or falls near stairs, ramps or curb [START_REF] Sung | Falls among full-time wheelchair users with spinal cord injury and multiple sclerosis: a comparison of characteristics of fallers and circumstances of falls[END_REF]; other injuries occur in the scenario of wheelchair breakdown. So, although a PWC is designed to promote the well-being of many users, it often ends up being abandoned, or it does not bring any benefit. In this sense, innovations and proposals for wheelchair revolve around technological advances; examples are: alternative control devices, assessment and prescribing tools, simulators and training applications, or mobility aids. Some examples of recent technology include: LUCI, which is a smart technology accessory that is mounted on a PWC to allow the user to navigate with greater safety and independence [START_REF] Luci | LUCI | Mobility[END_REF]; blind spot sensor systems used to easily provide obstacle detection and avoidance [START_REF] Mobility | Braze Mobility -Blind Spot Sensors for Wheelchairs[END_REF]; and a fleet management system to provide access to valuable data and analysis of customers' wheelchairs and improve diagnostic [START_REF] Permobil | Permobil Connect Fleet Management[END_REF]. None of these proposals use Digital Twin which can offer multiple benefits: (1) promote the well-being and safety of PWC users; (2) increase their engagement; (3) support PWCs adaptation and repair, increasing their safety and reliability, while reducing maintenance costs; and (4) offer new mechanisms for HCPs to remotely assist more people.

Conclusions and Future Work

This paper presents a proposal for the creation of a Digital Twin of a Power Wheelchair, enabling new services for wheelchairs users, as well as their HCPs. So far, there is no evidence of its application in this sector. Several ways of using DT in the context of provision and support of PWCs were pointed out. The proposal of a conceptual model identifies the Model-Driven Development approach to combine different sub-models and allow specification, verification, simulation and implementation of the virtual entity in the DT. In particular, a new extended Petri net formalism will aim to model VE state model. As future work, it is intended to analyze different descriptions of VE dynamic and kinematic models while associating them to a geometric model; in parallel, the new formalism of Petri net will begin to be defined; and it is intended to call participation of users, therapists, and ATPs to assess their opinions, comments and recommendations on the values of the proposed DT.

Fig. 1 .

 1 Fig. 1. Steps and participants in the wheelchair provision process.

Fig. 2 .

 2 Fig. 2. Proposed conceptual model for a Power Wheelchair DT.

Acknowledgements. This work was partially financed by Portuguese Agency "Fundação para a Ciência e a Tecnologia" (FCT), in the framework of project UIDB/00066/2020, and under the PhD scholarship with the reference 2020.08462.BD, through national funds from the MCTES ("Ministério da Ciência, Tecnologia e Ensino Superior").