Hailong Wu
email: hailong.wu@utbm.fr

Daniel Depernet
email: daniel.depernet@utbm.fr

Vincent Lanfranchi
email: vincent.lanfranchi@utc.fr

Analysis of torque ripple reduction effect on acoustic noise of a synchronous reluctance machine

Keywords: torque ripple, vibration, acoustic noise, synchronous reluctance machine, radial magnetic pressure

this paper aims to present the relation between acoustic noise and torque ripple. An equation evaluating the variation of acoustic noise caused by reducing torque ripple in synchronous reluctance machine (SynRM) is proposed. First, qualitative relation between the torque and the radial magnetic pressure (RMP) is presented. A quantitative relation between the RMP and torque is defined. Thirdly, based on the proposed equation, the influence of torque ripple reduction on electromagnetic noise is studied. The evaluation method is also computed thanks to the vibro-acoustic software Manatee. It can be concluded that the proposed approach can analyze effect of reducing torque ripple on acoustic noise.

INTRODUCTION

Synchronous reluctance machine (SynRM) has been studied widely because of its high efficiency, robust structure and less cost [START_REF] Wu | A Survey of Synchronous Reluctance Machine used in Electric Vehicle[END_REF]. However, this machine also has several disadvantages such as torque ripple and lower power factor [START_REF] Wu | A Survey of Synchronous Reluctance Machine used in Electric Vehicle[END_REF].To improve its performances, researches are mainly performed from two aspects: design and control. Authors [START_REF] Vagati | Synchronous reluctance electrical motor having a low torqueripple design[END_REF] [START_REF] Rakgati | Torque performance of optimally designed three and five phase reluctance synchronous machines with two rotor structures[END_REF] pay more attention to the design of its rotor structure such as rib, flux barriers and insulation to get higher saliency ratio, higher power factor and lower torque ripple. Other authors [START_REF] Foo | Robust Direct Torque Control of Synchronous Reluctance Motor Drives in the Field-Weakening Region[END_REF] [START_REF] Senjyu | High efficiency control of synchronous reluctance motors using extended Kalman filter[END_REF] focus on gaining higher performance with different control methods such as direct torque control and Kalman filter.

Besides, when SynRM is applied for electrical vehicle, another important requirement is its vibration and acoustic noise which can greatly affects the comfortableness. The electromagnetic acoustic noise existing in electrical machines has been studied extensively. Authors [START_REF] Fakam | Electromagnetic noise comparaison between 'SPM'and 'IPM'concentrated winding synchronous machine[END_REF] [START_REF] Islam | Analytical model for predicting noise and vibration in permanent magnet synchronous motors[END_REF] study the noise theoretically while some authors [START_REF] Islam | Analytical model for predicting noise and vibration in permanent magnet synchronous motors[END_REF][10] [START_REF] Dupont | Noise radiated by a permanent magnet synchronous motor: Simulation methodology and influence of motor defects[END_REF] [START_REF] Besnerais | Multiobjective optimization of induction machines including mixed variables and noise minimization[END_REF] analyze and reduce noise by optimizing the design with finite element software such as Flux 2D, Nastran and MANATEE. In addition, other authors [START_REF] Pellerey | Active reduction of electrical machines magnetic noise by the control of low frequency current harmonics[END_REF] [START_REF] Takiguchi | Acoustic noise and vibration reduction of SRM by elimination of third harmonic component in sum of radial forces[END_REF] apply optimal currents to reduce the noise. A widely validated and accepted conclusion [START_REF] Besnerais | Prediction of audible magnetic noise radiated by adjustable-speed drive induction machines[END_REF] is that the magnetic acoustic noise is mainly caused by radial magnetic pressure which acts on the stator teeth. Therefore, we can study acoustic noise by analyzing the radial magnetic pressure.

However, when it comes to SynRM, there are few researches about its vibration and noise. As stated in [START_REF] Cameron | The origin and reduction of acoustic noise in doubly salient variable-reluctance motors[END_REF], shaping currents can be an important method to diminish the acoustic noise in switched reluctance machine. But how to determine the optimal shaped current which is restricted by other performances such as efficiency and torque is not given. Authors [START_REF] Rabinovici | Torque ripple, vibrations, and acoustic noise in switched reluctance motors[END_REF] present that reduction of noise by different methods can also lead to lower torque. What's more, it has been pointed out that torque ripple can cause acoustic noise [START_REF] Gieras | Noise of polyphase electric motors[END_REF]. However, noise and torque ripple are analyzed independently and how the torque ripple affect the emission of noise is not analyzed. Authors [START_REF] Inanc | Torque ripple minimization of a switched reluctance motor including mutual inductances via sliding mode control technique[END_REF] decrease the torque ripple of switched reluctance machine and point out that the acoustic noise is also reduced. But the authors do not give any proofs to present the reduction of noise. Consequently, the relation between diminishing torque ripple and variation of acoustic noise is still not clear. As the electromagnetic noise is mainly produced by the radial magnetic pressure, it's interesting to know the relation between reduction of torque ripple and variation of radial magnetic pressure. In this paper, we propose a method to evaluate the variation of electromagnetic acoustic noise when the torque ripple in SynRM is reduced.

In addition, the reduction of torque ripple of the studied SynRM is the first goal. During the reduction of torque ripple, SynRM can be controlled by different strategies such as maximum torque per ampere control (MTPAC), maximum efficiency control (MEC) and maximum power factor control (MPFC). The MTPAC method is usually applied because MEC can be obtained at the same current angle (β=45°) with MTPAC. So the MTPA approach is used during the analysis of the relation between torque ripple reduction and acoustic noise. This paper is organized as four parts. The first part presents the qualitative relation and quantitative relation between RMP and torque; the second part gives the acoustic equation which is calculated by RMP and also expresses the acoustic variation by the electromagnetic torque; the third part presents the acoustic noise and vibration obtained by vibro-acoustic software Manatee; at last, the fourth part concludes the paper.

RELATION BETWEEN RMP AND TORQUE

Qualitative relation

In order to define the relation between RMP and torque, the presentation of the origins of electromagnetic noise and electromagnetic torque is necessary. The electromagnetic force (F(α,t) or magnetic pressure) shown in Fig. 1 can be expressed by two components: radial magnetic force (Fn(α,t) or radial magnetic pressure Pn(α,t)) and tangential magnetic force (Ft(α,t) or tangential magnetic pressure (TMP) Pt(α,t)). The magnetic pressure can be calculated by the flux density

, = ² , t - , t , = , t , t (1)
Where α is the spatial position of the electromagnetic force in the airgap and t is the time that rotor rotates.

, t and , t are the radial and tangential flux density.

As presented in the introduction, the vibration and electromagnetic acoustic noise which come from electromagnetic source are mainly caused by radial magnetic force, namely radial magnetic pressure. Besides, TMP can also produce vibration and noise; however, the amplitude caused by tangential pressure is too small compared with that resulted by radial pressure. Thereby, in this paper we neglect the effect of tangential pressure on vibration and electromagnetic acoustic noise and use the variation of radial magnetic pressure to represent the change of vibration and acoustic noise.

The angle ε , between RMP and TMP is defined in Fig. 1.

, , = , (2)
On the other hand, many researches have pointed out that the sum of tangential magnetic pressures is the origin of electromagnetic torque of electrical machine [START_REF] Camus | Prédiction des vibrations du stator d'une machine à réluctance variable en fonction du courant absorbé[END_REF] [START_REF] Zhu | Investigation of force generation in a permanent magnet synchronous machine[END_REF] as shown in [START_REF] Vagati | Synchronous reluctance electrical motor having a low torqueripple design[END_REF]. For example, the authors [START_REF] Camus | Prédiction des vibrations du stator d'une machine à réluctance variable en fonction du courant absorbé[END_REF] obtain nearly the same torque by co-energy method and by TMP for a switched reluctance machine.

= , ! !" #$ ( 3
)
Where r is the radius of the integrating contour, Ls is the stack length of the machine (along the axial direction), ω is the rotor speed.

Hence, considering that torque is produced by TMP, a parameter k' is added to represent the link between TMP and torque as shown in [START_REF] Rakgati | Torque performance of optimally designed three and five phase reluctance synchronous machines with two rotor structures[END_REF]. According to the origins of acoustic noise and torque, the qualitative relation between RMP and torque can be defined by [START_REF] Foo | Robust Direct Torque Control of Synchronous Reluctance Motor Drives in the Field-Weakening Region[END_REF].

, + , = -. , (5)

Quantitative relation

Based on the analysis of the origins of acoustic noise and torque, a qualitative relation has been defined and in this part, the quantitative relation is proposed.

To define the quantitative relation, at first we have to introduce the equations to calculate torque and RMP. The torque expression for synchronous reluctance machine can be expressed by ( 6) for the MTPA control strategy. It has been validated and the detailed deduction can be found in [START_REF] Wu | Etude de l'influence de saturation sur la réduction des ondulations de couple d'une machine synchrone à reluctance variable[END_REF].

= / 0 1 (6)
Where Kt(t) is the torque function which is a function of rotor position or time; id is the current of d-axis.

The RMP can be calculated by the Maxwell stress tensor method with neglecting the tangential flux density [START_REF] Islam | Analytical model for predicting noise and vibration in permanent magnet synchronous motors[END_REF][24]

, = 2 ² ,3 (7)
In order to find the quantitative relation between torque and RMP, the RMP should be represented by the current id which is similar to [START_REF] Senjyu | High efficiency control of synchronous reluctance motors using extended Kalman filter[END_REF]. For the synchronous reluctance machine, the windings exist only on stator and there are no windings on rotor. The flux density can be expressed by surfacic permeance Λ and the magnetomotive force fmm [START_REF] Fakam | Improved method to compute air-gap magnetic pressure of the Interior Permanent Magnet Synchronous Machine[END_REF]. General speaking, the surfacic permeance Λ is a function of the structure of the machine and the magnetomotive force is a function of the distribution of the spth stator winding Nsp and stator currents 0 45 . The analytical equation of flux density is [START_REF] Tan-Kim | Influence of the manufacturing process of a claw-pole alternator on its stator shape and acoustic noise[END_REF] , = Λ 7 88 , t

The radial and tangential surfacic permeances are

9 Λ = Λ & + ∑ Λ ) < )= >?@ -A 4 Λ = ∑ Λ ) < )= >?@ -A 4 (9)
Where Λ & , Λ ) and Λ ) are the amplitudes and A 4 is the number of stator teeth.

7 88 , t = ∑ B 45 C 45= 0 45 ( 10
)
Where sp is the stator phase.

Substituting ( 9) and ( 10) into (8) in order to represent the radial flux density by stator currents,

, = Λ & + ∑ Λ ) < )= >?@ -A 4 ∑ B 45 C 45= 0 45 (11)
Then, replacing the flux density in ( 7) by [START_REF] Dupont | Noise radiated by a permanent magnet synchronous motor: Simulation methodology and influence of motor defects[END_REF], the RMP can be presented as

, = DEF G∑ F H I HJK LM4 )N ( O ∑ P (Q R (QJK S (Q T # (12)
In the machine, stator currents can be represented by the currents id and iq in the rotating dq-frame which can be obtained by the Park transformation. What's more, the SynRM is usually controlled to get the maximum torque per ampere which means id equals iq.

U 0 0 V 0 > W = X C Y >?@Z >?@ EZ - [ C O >?@ EZ + [ C O -@0 Z -@0 EZ - [ C O -@0 EZ + [ C O \ ] 0 ! 0 ^_ ( 13
)
Where θ is the rotor electrical position. As a result,

Where / c dd , represents the calculation of the square value of the magnetomotive force fmm by current id. fsp(t) represents the change from id to stator currents. For example when sp=1, fsp(t) is

7 45 = X 2 3 g >?@Z -@0 Z (15)
Therefore, substituting ( 14) into ( 12), the RMP can be calculated Where Kpn , represents the relation between id and RMP and it's a function of spatial position and time.

According to ( 6) and ( 16), both the torque and RMP are expressed as function of id. So the quantitative relation between torque and RMP is introduced

, + , = h Q , h = / 5 , (18)
Where Kpt , is the pressure torque function which is 2D and presents the relation between RMP and torque. The pressure torque function of the studied SynRM, is obtained and shown in Fig. 2 .Besides, if the torque is known, the RMP can be expressed by In order to verify that for different currents (without saturation) Kpt( , ) is the same, we calculate this parameter for different sinusoidal stator currents whose maximal values (Im) are 20A, 30A and 40A. To compare its values for different currents simply, we just compare the values when α equals 78.3° as shown in Fig. 3 (at this point the radial pressure is maximal). It can be concluded that the three Kpt of 20A, 30A and 40A are independent of current when the current angle is fixed (in this paper β=45°). On the other hand, for different current angles, pressure torque function is different. But SynRM is usually controlled by MTPA. Therefore, β=45° is the most important case and other cases are not considered in this paper.

, = / 5 , (19)
Therefore, the relation between electromagnetic torque and RMP is defined in [START_REF] Inanc | Torque ripple minimization of a switched reluctance motor including mutual inductances via sliding mode control technique[END_REF]. Besides, all the RMP mainly produce the vibro-acoustic phenomena. As a result, the noise could be evaluated based on the electromagnetic torque which will be presented in the following section.

ACOUSTIC NOISE ANALYSIS

The electromagnetic noise has been analyzed theoretically by [START_REF] Besnerais | Acoustic noise of electromagnetic origin in a fractional-slot induction machine[END_REF]. The analytical equation of sound power is

o p = 10s?t & D ( (u,i T ( 20
)
Where Ps is the sound power caused by radial force; Psref is the reference sound power (10 -12 W).

Based on the expression [START_REF] Inanc | Torque ripple minimization of a switched reluctance motor including mutual inductances via sliding mode control technique[END_REF], the RMP can be calculated thanks to the torque. Then, the effect of torque ripple reduction on noise can be evaluated by torque and the derivation is presented in the Appendix.

∆o w = 20s?t & x yy+ zEh Q , + ,({ O yy+ zEh Q , + ,|Q { O } (21)
Where Tesin(t) is the uncompensated torque and Teopti(t) is the compensated torque.

Therefore, the variation of acoustic noise resulted by reducing torque ripple with supplying optimal currents can be evaluated by torque. In the next part, we will compare the variation of acoustic noise when the torque ripple of synchronous reluctance machine is reduced.

SIMULATION

Simulation and result

To test the proposed method, the maximal value of sinusoidal stator current is 30A and the speed is 1500rpm. The sinusoidal stator currents and optimal stator currents are presented in. Fig. 4. The uncompensated torque and compensated torque are introduced in Fig. 5. It can be seen that the optimal currents are effective for the purpose of reducing torque ripple which is reduced from 109.08% to 5.08%. Besides, the RMP corresponding with these two torques are calculated by [START_REF] Inanc | Torque ripple minimization of a switched reluctance motor including mutual inductances via sliding mode control technique[END_REF] and presented in Fig. 6 and Fig. 7. It's obvious that the RMP are changed by reducing torque ripple. What's more, the stator natural frequencies are analyzed by the finite element method. They are presented in Fig. 8. The natural frequencies and the vibrational modes of the stator cannot change when the stator has been designed. Hence, in the proposed method, the natural frequencies of stator are viewed as constant for uncompensated torque and compensated torque. It has been presented out that the significant vibration and acoustic noise are usually caused by the RMP whose frequency is close to the natural frequency of the stator and the force order is the same as the circumferential vibrational mode of stator [START_REF] Gieras | Noise of polyphase electric motors[END_REF] [START_REF] Rasid | Contribution to multi-physical studies of small synchronous-reluctance machine for automotive equipment[END_REF].The spatial order m and temporal order n of radial force can be calculated [START_REF] Rasid | Contribution to multi-physical studies of small synchronous-reluctance machine for automotive equipment[END_REF] ~, = sA 4 ± 2-€, 2-€

Where l and k are natural number, p is the number of pole pairs. The temporal frequency 7 8 is calculated by the temporal order and the rotor's mechanical frequency or rotor speed nr

7 8 = 7 8 = u •& (23)
Therefore, the resonance will occur when m equals the stator's circumferential vibrational mode and 7 8 are close to the stator's natural frequencies at the same time. At these resonance frequencies, if ∆Lw is greater than zero, it means that the noise has been reduced; on the contrary, a negative ∆Lw indicates that the noise has been increased. At other frequencies which cannot cause resonance the value of ∆Lw is not important for the noise.

Based on (21) the variation of noise is calculated and is presented in Fig. 9. Besides, the natural frequencies of stator are also introduced in Fig. 9. The x-axis is the spatial order of ∆o ' and y-axis is the temporal frequencies of ∆o ' . It can be seen from Fig. 9 that ∆o ' is increased for some frequencies while it is decreased for other frequencies. But the frequencies do not correspond with the stator natural frequencies. It means that the noise will not be influenced for most frequencies. When the frequency is about 21 kHz, ∆o ' has the identical frequency as the stator natural frequency whose spatial order is 4. For the frequency 21 kHz, the spatial order of ∆o ' should be ±4 in order to produce resonance. The corresponding temporal order of 21 kHz is 72 [START_REF] Rasid | Contribution to multi-physical studies of small synchronous-reluctance machine for automotive equipment[END_REF], hence according to [START_REF] Zhu | Investigation of force generation in a permanent magnet synchronous machine[END_REF], the number l can be calculated and the results are l =±6.33 or ±5.67 which is not an integer. As a result, the radial pressure at 21 kHz will not cause resonance and the noise of the studied SynRM will not be influenced by supplying current harmonics to minimize torque ripple.

Fig. 8 Natural frequencies of stator [START_REF] Wu | A Survey of Synchronous Reluctance Machine used in Electric Vehicle[END_REF] On the other hand, if it is supposed that l is an integer at 21 kHz which means the resonance occurs, the rotor speed can be calculated by [START_REF] Wu | Etude de l'influence de saturation sur la réduction des ondulations de couple d'une machine synchrone à reluctance variable[END_REF] and is 17500rpm. This speed is too high for the studied machine. Therefore, the resonance will also not appear for the lower speed even if l is an integer.

Computation with Manatee

To verify the result, the vibro-acoustic software, Manatee, is applied to simulate the vibration and noise of the studied machine without compensation of torque ripple and with compensation of torque ripple. This software has been proved as an effective tool to test and evaluate the vibration and noise [START_REF] Besnerais | Fast prediction of variable-speed acoustic noise due to magnetic forces in electrical machines[END_REF]. The stator model of the studied SynRM in Manatee is presented in Fig. 10. In order to save the effort, the flux density without torque ripple reduction and with torque ripple reduction obtained in Flux 2D are imported into Manatee directly.

Then the noise level can be calculated by Manatee. The noise without torque ripple reduction and with torque ripple reduction are presented in Fig. 11 and Fig. 12. It can be concluded by comparing these two figures that the noise is 29.5 dB and is not affected by reducing torque ripple. Therefore, the conclusion introduced by the proposed method is validated.

Fig. 1

 1 Fig.1The magnetic forces or magnetic pressures

Fig. 2 Fig. 3

 23 Fig. 2 The pressure torque function Kpt ,

Fig. 4

 4 Fig. 4 Sinusoidal currents and non-sinusoidal currents

Fig. 5 Fig. 6

 56 Fig. 5 Torque without compensation and torque with compensation

Fig. 9

 9 Fig. 9 Variation of noise by reducing torque ripple

Fig. 10

 10 Fig. 10 Stator model in Manatee

& ' ( &

Besides, the vibrations of the two cases are also calculated by Manatee which are shown in Fig. 13 and Fig. 14. Compared with vibration without torque ripple reduction, the vibration in Fig. 14 is increased at low frequencies while it is not changed for the high frequencies. Because the amplitudes of RMP at low frequencies are increased by decreasing torque ripple. So the vibration can be increased by supplying current harmonics in the studied SynRM to minimize torque ripple. According to Manatee, the total vibration level without torque ripple reduction is 88dB while the value with torque ripple reduction is increased to 93dB.

CONCLUSION

This paper analyzed the relation between RMP and torque. Based on the relation, a method is proposed to study the effect of torque ripple reduction on acoustic noise. This method could evaluate the variation of acoustic noise of synchronous reluctance machine by using the uncompensated torque and compensated torque. A segmented SynRM with the usually used MTPA strategy (current angle is 45°) is studied by the proposed method in order to analyze the influence of torque ripple reduction on acoustic noise. It showed that torque ripple compensation do not affect the noise of the studied machine. Besides, the conclusion obtained by the proposed method is verified by the software Manatee. It means that the proposed method is effective when saturation can be negligible. But, the result of Manatee showed that the vibration was increased by minimizing torque ripple.

This method can also be applied for other current angles, but the pressure torque function will be different and should be recalculated. The author would state that for other machines the effect of decreasing torque ripple may be different.

When saturation is noticeable, the procedures of the proposed method is identical. However, the pressure torque function could be different for different saturation which means that the evaluation of acoustic noise variation should be performed several times for different saturation.

In the future, the method should be validated by experiment. It should also be tested for other machines.

ACKNOWLEDGEMENT

The authors would like to thank the Eomys for providing the software Manatee and their help during the use of Manatee.

APPENDIX

The theoretical acoustic equation is rewritten

Where Ps is the sound power caused by radial force; Psref is the reference sound power (10 -12 W).

The sound power Ps can be calculated

Where Rout is the outer radius of the stator yoke; ρa is the density of air; c is the speed of sound in air and fexc is the frequency of each radial magnetic pressure harmonic.

The frequency of radial magnetic pressure is a multiple of the rotor rotational frequency

The static displacements for different order m are presented [START_REF] Besnerais | Prediction of audible magnetic noise radiated by adjustable-speed drive induction machines[END_REF]

Where Pmω is the harmonics of radial magnetic pressure.

The dynamic displacement is

Where the empirical equation of the modal damping ratio is [START_REF] Gieras | Noise of polyphase electric motors[END_REF] Ÿ 8 = [ 2.76 × 10 -¡ 7 8 + 0.062 (A8)

For our machine which has been manufactured, the parameters of geometry are fixed. And the mode frequency is also constant. What's more, the speed of rotor, namely the frequency of radial magnetic pressure, for two situations: compensated torque and uncompensated torque, is also unchanged. When the torque ripple is decreased, the currents of stator are changed and therefore the flux density of air gap is changed. It means that the only variable parameter in the above expressions is the radial magnetic pressure. Therefore, the static displacements (A5) (A6) and dynamic displacements (A7) can be expressed by different radial magnetic pressure Where ksp represents the relation between static displacements and radial magnetic pressure; kds represents the relation between dynamic displacements and static displacements.

Besides, the acoustic power (A2) can be rewritten as

Where kd is the function between acoustic density and dynamic displacement.

To evaluate the variation of acoustic noise resulted by minimizing torque ripple, the sound power (A14) can be rewritten as

Where kp is the parameter which defines the relation between sound power and radial magnetic pressure. At each harmonic of radial magnetic pressure, kp is constant for the same machine at the same speed.

As a result, the variation of acoustic noise can be introduced Where FFT2D represents the two dimension fast Fourier transform; Pn_sin and Pn_opti are the radial magnetic pressures with uncompensated torque Tesin(t) and compensated torque Teopti(t) respectively.


Fabien Lacressonniere

Andy Varais

Xavier Roboam

Scaling electro-thermal model of a lithium-ion battery for time-accelerated experiments in a HIL system

Keywords: Li-ion battery, electro-thermal modeling, similarity, scaling, time acceleration

This paper describes a scaled-size electrothermal model of a Lithium-ion (Li-ion) battery. This model is used in order to develop a physical emulation of the scaled battery in a Hardware In the Loop (HIL) process. Physical emulation allows scaling the actual process but also accelerating HIL experiments through an innovative concept: the "compacted virtual time". According to the management strategies implemented in the HIL process and to environmental conditions (temperature) of the battery, the electrical performance (energy/power) of the actual device and its physical emulator will be different. The electrical model is coupled with a 1D thermal model of a cell. The model parameters are determined from experimental data and by using a nonlinear least-squares solver. A robustness analysis, with voltage, current and time scaling is presented in order to determinate the accuracy of the scale model.

INTRODUCTION

The use of a Battery Energy Storage System (BESS) is becoming a technical solution in the actual and future electrical applications in order to improve the fuel efficiency and reductions in CO2 emissions.

In embedded systems, the BESS are widely integrated. Indeed, in electric vehicle applications, for several years, the market growth rate increases: the global electrical vehicle stock surpassed 2 million units in 2016 and the battery fed electrical vehicles still account for the majority of the electric car stock [START_REF]Global EV Outlook[END_REF]. In the More-Electrical Aircraft (MEA) context, the BESS can be used in an aircraft hybrid propulsion system to improve the overall propulsion system efficiency and to reduce the fuel consumption [START_REF] Rashed | Conceptual Design of Battery Energy Storage for Aircraft Hybrid Propulsion System[END_REF]. For the emergency electrical architecture in the aircraft, the hybridization of the Ram Air Turbine (RAT) with a BESS, associated with an optimized energy management strategy (power sharing), would reduce the mass of the overall system [START_REF] Mariani | Design of a medium voltage power converter-storage devices embedded in a hybrid emergency network for more electrical aircraft[END_REF].

The BESS presents also other advantages for stationary systems and more particularly in large power grids and microgrids. Energy storage devices can harvest energy excess during periods of low demand and inject the stored energy in the grid during power peak periods [START_REF] Reihani | Energy management at the distribution grid using a Battery Energy Storage System (BESS)[END_REF][START_REF] Leadbetter | Battery storage system for residential electricity peak demand shaving[END_REF][START_REF] Koller | Review of grid applications with the Zurich 1 MW battery energy storage system[END_REF]. The BESS can also be used to manage intermittent renewable energy devices (wind and solar power smoothing) in microgrids and isolated systems [START_REF] Branco | Battery energy storage systems as a way to integrate renewable energy in small isolated power systems[END_REF][START_REF] Alharbi | Optimal Sizing of Battery Energy Storage Systems for microgrids[END_REF]. This storage device can also provide a frequency regulation in power grids [START_REF] Oudalov | Optimizing a Battery Energy Storage System for Primary Frequency Control[END_REF].

Preparing the actual implementation of the BESS in electrical systems as mentioned above, offline simulations then HIL process (also known as semi-physical real-time simulation) can be applied in order to study, optimize and verify the effectiveness of the energy management strategies which will be implemented in the real system. In complex systems, there are several loads and storage devices and the HIL real time simulation presents more advantages than offline simulations: it is a low cost technique rather than testing on real (scale 1) system, the simulation can perform multiple tests for the same test bench. Moreover, it provides methods to accomplish non-destructive testing under extreme and fault conditions and also has strong experimental reproducibility [START_REF] Khazaei | Real-time simulationand hardware-in-the-loop tests of a battery system[END_REF].

The HIL tests have been extensively used in the automotive industry for component development [START_REF] Yi1 | Hardware-in-Loop Simulation for the energy management system development of a Plug-in hybrid electric bus[END_REF][START_REF] Taksale | Low cost hardware-in-loop for automotive application[END_REF]. For instance, to avoid electric and thermal abuse on the BESS used in the electrical vehicle (or in another application), a Battery Management System (BMS) has to be associated with the storage device. In an advanced BMS, in order to maximize battery capacity and to limit the aging effects, there are several functions implemented like as active or passive balancing, thermal management, charging process and diagnosis analysis (State Of Charge -SOC and State Of Health -SOH). To develop a sophisticated BMS, manufacturers may require extensive testing (software and hardware). To test the diagnostic functions of the BMS, some faults (for instance over-charge and over-discharge) has to be provoked. In comparison with tests conducted on real battery, the HIL tests appear more cost and time effective and it is easier to reproduce the same electrical and environment (i.e. thermal) conditions for BMS tests [START_REF] Dai | Cell-BMS validation with a hardware-in-the-loop simulation,of lithium-ion battery cells for electric vehicles[END_REF][START_REF] Barreras | An Advanced HIL Simulation Battery Model for Battery Management System Testing[END_REF][START_REF] Dai | A Hardware-in-the-Loop System for Development of Automotive Battery Management System[END_REF][START_REF] Haupt | Hardware-in-the-Loop Test of Battery Management Systems[END_REF]. The HIL simulations are also built to test power electronics devices [START_REF] Wang | Hardware-in-the-loop real-time plateform[END_REF] for MEA power supply system and to test the automatic flight control system of the aircraft [START_REF] Kaden | Hardware-in-the-Loop Flight Simulator -An Essential Part in the Development Process for the Automatic Flight Control System of a Utility Aircraft[END_REF]. Likewise, for microgrids applications, the HIL platforms become essential for testing and validating controls and energy management strategies [START_REF] Vijay | Real-Time Testing Approaches for Microgrids[END_REF]. In [START_REF] Kumar | Alternative Hardware-In-the-Loop (HIL) Setups for Real-Time Simulation and Testing of Microgrids[END_REF], some alternative HIL setups are proposed for real-time simulation and testing of microgrids. For the development and testing microgrid control and protection functions, a microgrid model (including diesel generators, BESS, PV plant and wind turbine) in a HIL environment has been developed [START_REF] Baccino | Real-time Hardware-in-the-Loop Modeling for Microgrid Applications[END_REF].

The real time emulation is one method to provide the realistic environments [START_REF] Vijay | Real-Time Testing Approaches for Microgrids[END_REF]. The emulation involves mimicking the behavior of the subsystems to be represented such as sources (PV system, wind turbine), storage (BESS, etc) and loads (consuming profiles). Hence, to emulate a complete power system (microgrid, electrical vehicle) a number of emulators, each replacing a physical subsystem, are connected together with some physical components. Sometimes, a real subsystem, instead of an emulator, is coupled in the HIL simulator test bench. For instance, to evaluate the performance of the battery in a virtual vehicle for different ambient temperatures, a real hardware battery is used in a HIL simulator [START_REF] Shidore | Battery in the Loop: Battery Evaluation in a systems context[END_REF]. In this case, the method is called Battery In the Loop (BIL) or Battery HIL. This method can be used if the electrical characteristics of the battery are adapted to those of the HIL test bench. Otherwise, a battery emulator (or virtual battery), with scaling factors, has to be developed [START_REF] Razavian | A battery Hardware-in-the-Loop setup for concurrent design and evaluation of real-time optimal HEV power management controllers[END_REF][START_REF] Seitl | Power hardware-in-theloop implementation and verification of a real time capable battery model[END_REF]. To exchange bidirectional power flow between virtual battery and the real electric test bench, the battery emulator is composed of bidirectional power supply operating in voltage source. The control signal of the voltage source can be created by the voltage response of a battery model [START_REF] Barreras | An Advanced HIL Simulation Battery Model for Battery Management System Testing[END_REF] or the voltage measured from a real battery [START_REF] Khazaei | Real-time simulation and hardware-in-the-loop tests of a battery system[END_REF].

The paper deals with a scalable electro-thermal model of a Li-ion battery. This latter can be used for replacing at real time a physical battery to simulate its thermal environment and especially to accelerate HIL simulations. Indeed, thanks to the similitude process and based on the original idea proposed in [START_REF] Cabello | Scaling Electrochemical Battery Models for Time-Accelerated and Size-Scaled Experiments on Test-Benches[END_REF][START_REF] Varais | Scaling of wind energy conversion system for time accelerated and size scaled experiments[END_REF], a "time acceleration" by considering a "virtual compressed time" is possible together with the reduction of physical scales (power, voltage, current, etc). The paper is organized as follows. In section 2, the electro-thermal battery model is presented and the scaling factors (voltage, current, time) are applied to the parameters of the model. In section 3, the parameterization of the model is detailed, based on experimental data from 18650 format 1.6Ah Li-ion cells with lithium iron phosphate (LFP) cathode. The experimental test bench is also described in this section. In section 4, a robustness analysis of the scaling electro-thermal model is conducted by using a typical current profile extracted from microgrid application. First, without time accelerating, a comparison of the measured and simulated battery voltage is presented. Then, in addition of both voltage and current scaling factors, the time scaling factor is applied to the model in order to reduce the total time of experimental test. Finally, section 5 gives the conclusion.

SCALING ELECTRO-THERMAL MODEL OF LI-ION BATTERY

The complexity of the battery model can affects the simulation time, especially for large and complex electrical systems. Hence, there is a tradeoff between accuracy and simulation time. Thus, for complex HIL simulation, simple models are used.The most common approach for battery modeling methods is electrical models bases on Equivalent Circuit Model (ECM) [START_REF] Barreras | An Advanced HIL Simulation Battery Model for Battery Management System Testing[END_REF]. They are preferred to the mathematical or physic-chemical models.

The electro-thermal model presented here is used in order to develop a battery emulator in a HIL system especially for testing microgrid management strategies. The model has to provide various information about the macroscopic quantities as SOC, the battery voltage (Vbat) and the surface temperature of the battery (TS). Fig. 1 presents the concept of the battery emulator connected to the power HIL.

Electrical model

The electrical model presented in this study is based on an extended modified shepherd model [START_REF] Tremblay | A Generic Battery Model for the Dynamic Simulation of Hybrid Electric Vehicles[END_REF] in which several modifications have been made [START_REF] Cabello | Battery dynamic model improvement with parameters estimation and experimental validation[END_REF]. This model is commonly used in the modeling of multi-physical energy systems incorporating a battery and in particular for studies of energy management in hybrid systems [START_REF] Camara | Design and New Control of DC/DC Converters to Share Energy Between Supercapacitors and Batteries in Hybrid Vehicles[END_REF][START_REF] Boulon | Multi physics modelling and representation of power and energy sources for Hybrid Electric Vehicles[END_REF][START_REF] Motapon | A Comparative Study of Energy Management Schemes for a Fuel-Cell Hybrid Emergency Power System of More-Electric Aircraft[END_REF]. Indeed, by its simplicity and its accuracy, this model is easy to implement in a multi-physics and multi-scale system. Nevertheless, this model cannot completely describe all phenomena in the battery like the dynamic behavior such as the diffusion phenomena, which, in an ECM, can be taken into account with several RC parallel networks [START_REF] Kuhn | Modelling Ni-mH battery using Cauer and Foster structures[END_REF] (for instance, Cauer or Foster structures). Furthermore, this model is based on specific assumptions noted below:  the Open Circuit Voltage (OCV) hysteresis phenomenon [START_REF] Roscher | OCV Hysteresis in Li-ion Batteries inclunding Two-Phase Transition Materials[END_REF] is not taken into account,  the OCV is not parameterized as a function of the temperature (entropy changes are neglected [START_REF] Thomas | Measurement of the Entropy of Reaction as a Function of State of Charge in Doped and Undoped Lithium Manganese Oxide[END_REF]),  the aging phenomena are not taken into account,  the internal resistances are supposed constant during the charge and discharge cycles and are not varied with the amplitude of the current,  the capacity of the battery is constant (No Peukert effect),  the self-discharge of the battery is neglected.

In spite of these assumptions, the internal resistances will be dependent on the temperature.

The battery model is summarized as follows:

(

(

where the variables are the battery current i (A), the filtered battery current i * (A), the battery capacity Q (Ah), the battery constant voltage E0 (V), the exponential zone amplitude A (V), the exponential zone inverse capacity B (Ah -1 ), the polarization constant K (V/Ah), the internal resistance R1 (Ω), the polarization resistance R2 (Ω), the current filter time constant Tf = R2.C2 (s), the discharge logic variable isdch and the charge logic variable isch. The logic variable isdch is equal to 1 in discharging mode while in charging mode, it is equal to 0. The logic variable isch is equal to 1 in charging mode while in discharging mode, it is equal to 0. From the equation ( 1), the OCV relationship can be obtained:

(4)

𝑉 𝑏𝑎𝑡 = 𝐸 0 + 𝐴𝑒 -𝐵𝑄 1-𝑆𝑜𝐶 -𝐾𝑄 1 𝑆𝑜𝐶 -1 -𝑅1𝑖 -𝑅2𝑖 * 𝑖𝑠 𝑑𝑐 ℎ 𝑆𝑜𝐶 + 𝑖𝑠 𝑐ℎ 1.1 -𝑆𝑜𝐶 𝑂𝐶𝑉 𝑆𝑜𝐶 = 𝐸 0 + 𝐴𝑒 -𝐵𝑄 1-𝑆𝑜𝐶 -𝐾𝑄 1 𝑆𝑜𝐶 -1 𝑖 = 𝑖 * + 𝑅2𝐶2 𝑑 𝑑𝑡 𝑖 * 𝑑 𝑑𝑡 𝑆𝑂𝐶 = - 1 𝑄 𝑖
Likewise, from equation ( 1), the total battery voltage drop is: [START_REF] Leadbetter | Battery storage system for residential electricity peak demand shaving[END_REF] It is possible to translate these equations by an ECM as shown in Fig. 2. The R2C2 parallel network, in the Fig. 2, allows taking into account the dynamic behavior on the battery voltage response. However, it is not possible to dissociate the activation dynamics (charge transfer process) and the diffusion phenomena.

In the equation ( 2), the SOC reflects the residual capacity and it is calculated by using the common Coulomb counting method. Hence, the SOC can be expressed as the following equation (with a coulombic efficiency equal to 1. Indeed, for LFP battery, this value is near to 1 [START_REF] Chang | A New State of Charge Estimation Method for LiFePO4 Battery Packs Used in Robots[END_REF]): [START_REF] Koller | Review of grid applications with the Zurich 1 MW battery energy storage system[END_REF] where SOCinit is the initial capacity of the battery. For a correct use of the battery, the minimum value of the SOC (SOCmin) cannot reach zero. In our study, the values of the SOCmin and SOCmax (maximum SOC) are limited in order to limit the aging effect. Generally, the energy management strategy of the microgrid limits the battery depth of discharge (DOD) at 80% with a SOCmax close to 90% [START_REF] Zhang | An improved charging/discharging strategy of lithium batteries considering depreciation cost in day-ahead microgrid scheduling[END_REF].

The OCV curve, obtained by equation ( 4), is shown in Fig. 3. For the maximum SOCs, there is an exponential potential drop on the OCV. In our case study (battery emulator for HIL microgrid application), the SOCmax has been fixed at 90%. Therefore, in order to limit the parameters of the model, it is possible to remove the exponential expression in the equation ( 4) thus to simplify both model and parameter identification. For the rest of the study, the OCV relationship is: [START_REF] Branco | Battery energy storage systems as a way to integrate renewable energy in small isolated power systems[END_REF] The overall parameters of the electrical model are summarized in the Table 1. Electrical parameters are determined from experimental data by using a nonlinear least-squares solver (Matlab's optimization toolbox). On the one hand, E0, K and Q are obtained by the measurement of the OCV characteristic, and on the other hand, R1, R2 and Tf are determined from the voltage measurement of the battery when this latter is excited with a current profile of the Hybrid Pulse Power Characteristic (HPPC) test. The HPPC profile was designed in order to measure the dynamic power capability over a device's usable charge and voltage range [START_REF] Hunt | PNGV battery test manual[END_REF][START_REF] Shim | Characterization of high-power lithium-ion cells during constant current cycling: Part I. Cycle performance and electrochemical diagnostics[END_REF].

Thermal model

The goal of the thermal model is to get an estimate of the surface temperature of the battery from a simple measurement of the surface temperature. The battery used for this study is composed of several 18650 format 1.6Ah Li-ion cells. As shown in Fig. 4, these cells are assembled in several modules (connected in series and parallel) in order to obtain the characteristics of the battery (voltage and capacity). The surface temperature of the battery is measured by using a thermocouple attached on the cell (placed at half height). Hence, the thermal model of the battery is based on the thermal cylindrical cell. The heat source and Rthv can be expressed as:

(8) (9)
where hv (W/m²/K) is the thermal convection coefficient and Surf (m²) is the cell surface area (surfaces of negative and positive terminal of the cell are neglected).

The surface temperature can be calculated by: [START_REF] Khazaei | Real-time simulationand hardware-in-the-loop tests of a battery system[END_REF] where Tamb (K) is the ambient temperature of the cell (being also the ambient temperature of the battery).

The overall parameters of the thermal model are Cp, Rthc and hv. The parameters are obtained from experimental surface temperature measurement and by using a nonlinear leastsquares solver.

Scaling of the electro-thermal model

In order to develop the physical emulator of a scaled battery and to reduce the testing time in the HIL test bench, a previous study [START_REF] Cabello | Scaling Electrochemical Battery Models for Time-Accelerated and Size-Scaled Experiments on Test-Benches[END_REF] was conducted. The scaling model is based on a dimensional analysis (well established method in fluid and thermal systems) and on the Vaschy-Buckingham's Pi theorem. The readers are invited to consult specific literature for more details [START_REF] Zohuri | Dimensional Analysis Beyond the Pi Theorem[END_REF]. Also, the same scaling methodology based on a dimensional analysis has been used to scale a wind energy conversion system [START_REF] Varais | Scaling of wind energy conversion system for time accelerated and size scaled experiments[END_REF]. About the thermal model, the format of the cell is always the same regardless of the size of the battery pack. Hence, these scaling factors do not affect Cp, Rthc and Rthv (the thermal convection coefficient is supposed to be the same for the overall cell). Only the time constant of the thermal model will be scaled by kt (as for the current filter time constant Tf).

By applying the dimensional analysis [START_REF] Cabello | Scaling Electrochemical Battery Models for Time-Accelerated and Size-Scaled Experiments on Test-Benches[END_REF] and the scaling factors at the electro-thermal model, the new parameters of the scaling electro-thermal model are presented in the Table 2.

Table 2. Parameters of the scaling electro-thermal model.

Original electrothermal model

Scaled electro-thermal model

E0 [Kv]E0 K [Kv/KiKt]K Q [KiKt]Q R1 [Kv/Ki]R1 R2 [Kv/Ki]R2 R2C2 [kt]R2C2 (Rthc + Rthv)Cp [kt](Rthc + Rthv)Cp
The method to obtain the parameters of the original model is explained in the next section.

PARAMETER EXTRACTION

The nonlinear least-squares solver of the Matlab's optimization toolbox is used in order to identify parameters of the electro-thermal model. In this study several objective functions to be minimized are proposed: [START_REF] Yi1 | Hardware-in-Loop Simulation for the energy management system development of a Plug-in hybrid electric bus[END_REF] (12) [START_REF] Dai | Cell-BMS validation with a hardware-in-the-loop simulation,of lithium-ion battery cells for electric vehicles[END_REF] Parameters in table 1 are obtained with the minimization of equations ( 11) and ( 12) whereas the minimization of equation ( 13) allows determining Cp, Rthc and hv.

Experimental setup for parameter extraction

The experimental setup is shown in Fig. 6, which includes a remote controllable bidirectional source of voltage/current (configured by a Power Source PSI 9200-210 and a Controlled Load ELR 9250-210) to adjust charge and discharge profiles, a computer (with DSPACE supervisor) to control the bidirectional source and to secure the experimental test bench, a multi-channel recorder (SEFRAM -DAS 240) for data storage, a thermal chamber (CLIMATS) for environment (thermal) control and finally several LFP batteries under study (Table 3). The used batteries are not equipped neither with a battery management system nor a balancing circuit. Hence, for safety consideration in all tests, both cell voltages and surface temperatures are measured. The upper and lower cut-off voltages are 3.65V and 2.5V to fulfill the safe operating area.

OCV parameter extraction

To extract the relationship between the OCV and the SOC in order to obtain parameters E0, K and Q, a pulsed current profile has been used (Fig. 7). Each pulse current allows reducing the SOC by 10% with a constant rate (C/5). A rest time (toff) of one hour has been fixed for obtaining an OCV in equilibrium [START_REF] Baronti | Open Circuit Voltage of Lithium-ion batteries for energy storage in DC microgrids[END_REF].

First of all, the LFP cell was fully charged (SOC=1) with a constant current constant voltage profile (CCCV) with a constant current charge at 0.2C until the voltage reaches 3.65V; then, a constant voltage of 3.65V charge until the current is below 0.02C. Then, the battery was discharged with constant current (0.2C) until reaching the desired initial SOC at 0.9. During this test, the temperature in the thermal chamber was regulated at 20°C. The test result is reported in Fig. 8 with the OCV hysteresis (the blue curve represents the OCV measured in charging mode while the orange curve represents the OCV measured in discharging mode). To simplify the model, the OCV hysteresis is not taken into account despite its importance for the SOC estimation accuracy [START_REF] Marongiu | On-board capacity estimation of lithium iron phosphate batteries by means of half-cell curves[END_REF]. Then, the OCV characteristic used to extract parameters is based on an average between the two measured OCV curves (reported in dashed line in Fig. 8). The results of the optimization problem with the objective function of equation ( 11) are presented in table 4. For qualifying the accuracy of the OCV model a weighted Root Mean Square Error (RMSE) is calculated (Eq. 14). A comparison of OCV curves, between measurements (dashed line) and model (solid line) with parameters in table 4, is represented by solid line in Fig. 8.

Voltage drop parameter extraction

This second parameter identification method is based on the HPPC profile which is widely used for parameter identification [START_REF] Hariharan | A nonlinear equivalent circuit model for lithium ion cells[END_REF]. The test procedure is conducted on the LFP battery (25.6V -13Ah) at 10% of SOC intervals (by constant current 1C discharge segments) starting from 90% to 10% SOC, each interval being followed by 30min rest time before applying the next sequence. A sequence (Fig. 9) is composed with symmetrical charge and discharge pulses which magnitudes are inside the range from 0.1C to 2C. For each HPPC profile, the battery was placed in the climatic chamber where the room temperature was regulated. In order to study the influence of the temperature on the parameters (R1, R2, Tf), the parameters have been identified for several ambient temperatures (0°C, 10°C, 20°C, 30°C).

The battery is composed with 8 LFP modules in series each module being equipped with 8 LFP cells in parallel (module characteristics: 3.2V -13Ah). To solve the optimization problem (minimization of the equation 12) in order to obtain overvoltage parameters, an experimental measurement is necessary. This measure is based on the voltage across one module (Vmodule) of the battery. OCV parameters determined in section 3.2 have been implemented in the electrical model with a current scaling factor equal to 8 (1.6Ah  13Ah). A comparison of Vmodule curves, between measurement and model, for the ambient temperature at 0°C is represented in Fig. 10. 5. The RMSE values show that the simulated results are well fitted with the experimental results. During the HPPC tests, the average surface temperature (TS) of the cell is close to the ambient temperature. Hence, from the internal resistances values, noted in Table 5, the influence of the temperature on the internal resistances can be given by the following equations: (15) [START_REF] Haupt | Hardware-in-the-Loop Test of Battery Management Systems[END_REF] where R is the gas constant (8.314J/mol). The temperature dependence of resistance R1 can be given by an Arrhenius law while, for the resistance R2, the temperature dependence is given by an exponential curve fitting.The fitting parameters (K11, K12, K21, K22) of equations ( 15) and ( 16) are determined with a nonlinear least-squares solver and presented in Table 6. Table 6. Parameters of internal resistances.

K11

K12 K21 K22 0.26mΩ 8600J/mol 4033Ω 0.05K -1 In the electrical model, Tf is considered as constant and is calculated from the average of the 4 values in table 5 (i.e. 45s).

Thermal parameter extraction

The parameters (Cp, Rthc and hv) are obtained with the same method as the previous parameter extraction. Data measurements used in the optimization problem presented in [START_REF] Dai | Cell-BMS validation with a hardware-in-the-loop simulation,of lithium-ion battery cells for electric vehicles[END_REF] are the surface temperature measurements (on a cell in a LFP module). The surface temperature has been measured during the HPPC test detailed in the section 3.3. The simulated surface temperature is based on the equation [START_REF] Khazaei | Real-time simulationand hardware-in-the-loop tests of a battery system[END_REF]. Thermal parameters are shown in Table 7 for different ambient temperature. 7. In [START_REF] Saw | Electro-thermal analysis of Lithium Iron Phosphate battery for electric vehicles[END_REF], for 18650 format LFP cell, the specific heat capacity value is around 36J/K and the thermal convection coefficient (for natural convection) is around 4W/m²/K. The ratio Rthc/Rthv can be calculated with parameters in the last row of Table 8. The value of Rthc/Rthv is equal to 0.33. This value is close to that calculated in [START_REF] Forgez | Thermal modeling of a cylindrical LiFePO4/graphite lithium-ion battery[END_REF] for cylindrical LFP cell.

ROBUSNESS ANALYSIS AND TIME SCALING

Robustness analysis of the electo-thermal model

In order to assess the robustness of the identification process, a validation profile was used to ensure that the electrothermal battery model works well in realistic settings. The current profile based on microgrid application with a specific energy management strategy [START_REF] Hernandez-Torres | Optimal techno-economical storage sizing for wind power producers in dayahead markets for island networks[END_REF] is shown in Fig. 11. The battery current magnitude is adapted for the physical battery configuration. The battery used for this analysis has the following characteristics: 25.6V -13Ah. A comparison between the voltage battery measurement and voltage response of the battery model is made in this section. This comparison is also conducted for the surface temperature. Initially, the battery SOC is set to 80% and the battery was placed in the thermal chamber with a regulated temperature. The analysis was conducted with different ambient temperatures from 10°C to 30°C.

Parameters R1, R2 and Tf were obtained from a LFP module (3.2V -13Ah) while parameters E0, K and Q were obtained from LFP cell (3.2V -1.6Ah). Hence, to obtain parameters of scaled electro-thermal battery model (25.6V -13Ah), voltage and current scaling factors have to be applied as mentioned in Table 8. Neither voltage nor current scaling factors are applied on the thermal model because the surface temperature of the battery is the same as the surface temperature of the cell.

; Ki=1 [Kv]E0 Cp [Kv/Ki]R1 [Kv/KiKt]K Rthc [Kv/Ki]R2 [KiKt]Q hv [kt]R2C2
A comparison between the simulation and the measurements at 10°C is shown in Fig. 12: there is only a small difference between the measured and simulated values for both battery voltage and surface temperature of the cell. During the test, the SOC variation of the battery is depicted in Fig. 13 The performance of the scaled model was tested for several ambient temperatures (10°, 20°C and 30°C). The Table 9 presents the weighted RMSE of the battery voltage and the surface temperature (based on equation 14). The weighted RMSE of the surface temperature was calculated by the ratio between the RMSE value and the ambient temperature. These results validate the performance of the scaling electro-thermal model for large SOC variations and different ambient temperatures. The scaled model presented in this section can be implemented in a real time simulator in order to develop a physical emulator for HIL testing (Fig. 1). Nevertheless, in order to accelerate the physical emulation of the battery and therefore to reduce the development cost and testing efforts, the time scaling is an original concept for reduce the testing time.

To accelerate the testing time with a real battery, the magnitude of the current profile has to be increased. However, the internal voltage drop in the battery generated by the magnitude of the current will cause a capacity reduction and a possible thermal abuse in the battery. So, if a physical emulator is based on a real battery [START_REF] Khazaei | Real-time simulation and hardware-in-the-loop tests of a battery system[END_REF], the time scaling is not possible. So, the time scaling can only be done with a scalable battery model.

Time accelerated with the scaled electro-thermal model

The total time of the current profile, presented in Fig. 11, is 2 hours. For instance, in order to reduce the test time by 60 (i.e, 2 hours at real time equal to 2 minutes in "compacted virtual time"), a time axis compression (by 60) is applied on the current profile. The results of the scaled electro-thermal battery model (at 10°C) with kt=1/60 (and with the same voltage and current scaling factors as in the robustness analysis) are shown in Fig. 14. The simulated response of the scaled battery model is very close to the experimental measurements for which the total time is 2h. Hence, with the scaled electro-thermal model presented in this paper, it is possible to reduce testing times for HIL simulation. However, for the development of the physical emulator and for the HIL experiments, some limits can appear on kt values. Indeed, the bandwidth of the power supply or the bandwidth of the control loops of power electronic converters can limit the performance of the accelerated tests.

CONCLUSION

A scaled electro-thermal model of a cylindrical LFP/graphite Li-ion battery was presented. This model is used in order to develop a physical emulator in power HIL and to predict the thermal performance of the LFP cell under dynamic current profile and for different ambient temperatures. Hence, the surface temperature information can be implemented in the energy management systems of the microgrid. Good agreement was obtained, despite of simplifying assumptions, between simulated and experimental results. Moreover, the accuracy of the scaled model has been tested for several ambient temperature and for a time scaling allowing to shorten testing times.

Fig. 1 .

 1 Fig. 1. Battery emulator : power supply interfaced to a real time simulator.

Fig. 2 .

 2 Fig. 2. Equivalent circuit for electrical modeling.

Fig. 3 .

 3 Fig. 3. OCV discharge characteristic of the battery model.

Fig. 4 .Fig. 5 .

 45 Fig.4. Module of 8 Li-ion LFP cells developped by TYVA ENERGY.By constructing this thermal model, a series of assumptions were made: the temperature inside the cylindrical cell is uniform[START_REF] Jeon | Thermal modeling of cylindrical lithium ion battery during discharge cycle[END_REF],  there is no cell-to-cell temperature variance in each battery modules,  the predominant head mode transport in the cell is the conduction (the convection and radiation modes are neglected),  the heat source in the cell is governed by the irreversible heat source (the reversible entropic heat source which depends on the entropy change in the electrode is neglected),  each module have the same inlet air temperature,  parameters of the thermal model are independent of the temperature.The thermal model of the cell[START_REF] Forgez | Thermal modeling of a cylindrical LiFePO4/graphite lithium-ion battery[END_REF] (as shown in Fig.5) is composed by a heat source p (W), a specific heat capacity Cp (J/K), an internal heat transfer resistance Rthc (K/W) and an external heat transfer resistance Rthv (K/W).

  To obtain a new set of parameters for the electro-thermal model, several scaling factors are introduced:  voltage scaling factor (kv): vscaled/voriginal,  current scaling factor (ki) : iscaled/ioriginal,  time scaling factor (kt) : tscaled/toriginal.

Fig. 6 .

 6 Fig. 6. Exprimental test bench for parameter extraction.Table 3. Characteristics of LFP Batteries and cells. LFP Battery LFP Cell

Fig. 7 .

 7 Fig.7. Pulse current profile for OCV measurement.

Fig. 8 .

 8 Fig.8. OCV curves for LFP cell (dashed line : OCVexperimental, solid line : OCVmodel).

Fig. 9 .

 9 Fig.9. Pulse current profile sequence.

Fig. 10 .

 10 Fig.10. Comparison of measured (green line) and simulated (blue line) of module voltage at 0°C. The parameter values, for different ambient temperatures, and the weighted RMSE (based on equation 14) are reported in Table5. The RMSE values show that the simulated results are well fitted with the experimental results.

  𝑗 ) -𝑂𝐶𝑉 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 (𝑗 ) 2 𝑁 𝑗 =1 𝑉𝑐𝑒𝑙𝑙 𝑛𝑜𝑚 (= 3.2𝑉)

Fig. 11 .

 11 Fig.11. Current battery profil for robustness analyis.

Fig. 12 .

 12 Fig.12. (a) voltage battery, (b) surface temperature of a cell during the cycle profile (green line : experimental measurements, blue line : responses of the scaled battery model).

Fig. 13 .

 13 Fig.13. SOC varaitaion for the validation profile.

Fig. 14 .

 14 Fig.14. (a) voltage battery, (b) surface temperature of a cell (green line : experimental measurements presented in Fig.12; blue line : responses of scaled battery model).

Table 1 .

 1 Parameters of the electrical model.

	Electrical parametres
	OCV relationship voltage drop relationship
	E0, K, Q	R1, R2, Tf

Table 4 .

 4 Results of OCV parameter extraction.

	E0	K	Q	RMSE
	3.323V 0.01V/Ah 1.5Ah 0.35%
				(14)

Table 5 .

 5 Results of voltage drop parameter extration.

	Ambient temperature	R1	R2	Tf	RMSE
	0°C	13.4mΩ	4.8mΩ	46s 0.84%
	10°C	10.5mΩ	2.9mΩ	47s 0.89%
	20°C	9.1mΩ	1.3mΩ	51s 0.82%
	30°C	7.3mΩ	0.63mΩ 36s 0.66%

Table 7 .

 7 Parameters of thermal model.

	Ambient temperature	Cp	Rthc	hv
	0°C	27J/K 22.6K/W 4.7W/m²/K
	10°C	26J/K 23.3K/W	5W/m²/K
	20°C	27J/K 21.3K/W 4.8W/m²/K
	30°C	28J/K 23.5K/W	4W/m²/K
	Parameters used in thermal model 27J/K 22.6K/W 4.6W/m²/K
	Parameter values used in the model is based on the average
	of the four values in Table

Table 8 .

 8 Parameters of scaled electro-thermal battery model.

	OCV Kv =8; Ki=8	Thermal	Voltage drop Kv=8

  .

				15
				10
	𝑅1 = 𝐾 11 𝑒 𝑅2 = 𝐾 21 𝑒 -𝐾 𝐾 12 𝑅𝑇 𝑆 22 𝑇 𝑆	Current (A)	-10 -5 0 5
			-15
			0 -20	0.2	0.4	0.6	0.8	1 Time (h)	1.2	1.4	1.6	1.8	2
				28
				27
		Vbat (V)	25 26
				24
				0 23	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2
									Time (h)
				17
				16
				15
		Ts (°C)	13 14
				12
				11
				0 10	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2
									Time (h)

Table 9 .

 9 RMSE of the scaled electro-thermal model.

	Ambient temperature RMSE (Vbat) RMSE (Ts)
	10°C	0.9%	3.2%
	20°C	0.6%	1.75%
	30°C	0.5%	3.3%