Anh Tuan Le

Quentin Govignon
email: quentin.govignon@mines-albi.fr

Samuel Rivallant
email: samuel.rivallant@isae-supaero.fr

Thierry Cutard
email: thierry.cutard@mines-albi.fr

Nano-engineered prepreg manufacturing: control of capillary rise of resin into VACNTs' forests

The incorporation of vertically aligned carbon nanotubes (VACNTs) between composites plies has been said to enhance the through-thickness strength, and it can also decrease the risk of interply delamination and reduce crack initiation. Thanks to these high mechanical performances, nano-engineered hybrid composites are seen as promising for highly demanding structural reinforcement applications. This paper is part of a study that focuses specifically on the methodology for trans-ferring VACNTs onto a prepreg surface while maintaining their initial vertically aligned morphology. The chosen method involved bonding the VACNTs' forest through capillary impregnation of the forest by the prepreg's resin. Key parameters for an effective transfer and to achieve a partial capillary rise of the resin into the VACNTs will be discussed here.

Introduction

Composite materials play an increasingly important role in a wide range of applications, including aerospace, due to their high mechanical properties as well as their lightness [START_REF] Rubino | Marine application of fiber reinforced composites: a review[END_REF][START_REF] Pendhari | Application of polymer composites in civil construction: a general review[END_REF][START_REF] Liu | Shape memory polymers and their composites in aerospace applications: a review[END_REF][START_REF] Kausar | Review of Applications of polymer/carbon nanotubes and Epoxy/CNT composites[END_REF]. However, delamination during fatigue or impact loading is one of the major drawbacks of such materials [5][START_REF] Sørensen | 11 -Delamination fractures in composite materials[END_REF][START_REF] Dransfield | Improving the delamination resistance of CFRP by stitching-a review[END_REF].

Since the turn of the twenty-first century, the discovery and scientific investigation of nanomaterials have been increasingly prevalent across a wide range of applications, including chemistry [START_REF] Nouruzi | In situ organized pd and au nanoparticles in a naphthalene-based imine-linked covalent triazine framework for catalytic suzuki reactions and H2 generation from formic acid[END_REF][9][START_REF] Doustkhah | Organosiloxane tunability in mesoporous organosilica and punctuated Pd nanoparticles growth; theory and experiment[END_REF], biology [START_REF] Karimi-Maleh | Guanine-based DNA biosensor amplified with Pt/SWCNTs nanocomposite as analytical tool for nanomolar determination of daunorubicin as an anticancer drug: a docking/experimental investigation[END_REF], and environment [START_REF] Sabaghnia | Euphorbia leaf extract-assisted sustainable synthesis of Au NPs supported on exfoliated GO for superior activity on water purification: reduction of 4-NP and MB[END_REF].

In this context, the introduction of nanoconstituents has been proposed to reinforce composite structures throughout their thickness [START_REF] Domun | Improving the fracture toughness and the strength Fig. 5 VACNTs' forests transferred onto prepreg at: a 20 kPa pressure and b 200 kPa pressure of epoxy using nanomaterials-a review of the current status[END_REF]. For example, several publications demonstrate that the presence of carbon nanotubes (CNTs) in matrix improves both the mechanical [START_REF] Nam | Improved mechanical properties of aligned multiwalled carbon nanotube/thermoplastic polyimide composites by hot stretching[END_REF][START_REF] Tong | Surface modification of single-walled carbon nanotubes with polyethylene via in situ Ziegler-Natta polymerization[END_REF][START_REF] Jia | Study on poly(methyl methacrylate)/carbon nanotube composites[END_REF], electrical [START_REF] Mezdour | Electrical properties of polyamide/polyaniline composite films[END_REF][START_REF] Sandler | Development of a dispersion process for carbon nanotubes in an epoxy matrix and the resulting electrical properties[END_REF], and thermal [START_REF] Dong | A review of aligned carbon nanotube arrays and carbon/carbon composites: fabrication, thermal conduction properties and applications in thermal management[END_REF][START_REF] Kaur | Enhanced thermal transport at covalently functionalized carbon nanotube array interfaces[END_REF][START_REF] Peacock | Characterization of transferred vertically aligned carbon nanotubes arrays as thermal interface materials[END_REF][START_REF] Tong | Dense vertically aligned multiwalled carbon nanotube arrays as thermal interface materials[END_REF] properties of composites. Improvements of in-plane and interlaminar properties of the composites with adjunction of VACNTs between composite plies are also observed [START_REF] Zhu | Improving the dispersion and integration of single-walled carbon nanotubes in epoxy composites through functionalization[END_REF][START_REF] Zhang | Toughening of cycloaliphatic epoxy resin by multiwalled carbon nanotubes[END_REF][START_REF] Zhu | Reinforcing epoxy polymer composites through covalent integration of functionalized nanotubes[END_REF].

As far as we are concerned, literature on hybrid composite VACNTs-prepreg manufacturing details are limited so far. In the previous works, Garcia, Wardle et al. [START_REF] Garcia | Joining prepreg composite interfaces with aligned carbon nanotubes[END_REF][START_REF] Wardle | Nano-engineered material architectures: Ultra-tough hybrid nanocomposite system[END_REF] described the VACNTs' transfer process from their initial rigid Si substrate onto prepreg bonding on a cylinder. The VACNTs' forests separate from Si wafer and then bond on prepreg surface by capillary force. In consequence, the nanotransferred prepreg surface is limited by cylinder's lateral surface.

Furthermore, Garcia et al. [START_REF] García | Fabrication of composite microstructures by capillarity-driven wetting of aligned carbon nanotubes with polymers[END_REF] suggest nano-engineered composites based on the capillarity principle, where the liquid impregnated to VACNTs ends by capillary rise. The authors have not fully given the relationship between resin capillary rise height and process parameters (temperature, pressure, etc.). Beard et al. [START_REF] Beard | Control and modelling of capillary flow of epoxy resin in aligned carbon nanotube forests[END_REF] estimated capillary rise height versus liquid viscosity. The authors used VACNTs with heights ranging from 400 to 700 µm, an electrically conductive chromium-complex dye additive was blended with low viscosity resin to simplify the resin capillary rise height. According to Sojoudi et al. [START_REF] Sojoudi | Stable wettability control of nanoporous microstructures by iCVD coating of carbon nanotubes[END_REF], one of the major risks of this process is VACNTs' coalescence due to high resin-VACNTs' surface tension during capillary rise, which degrades the VACNTs' initial morphology.

The aforementioned works give an idea to impregnate resin into VACNTs' forest by resin capillary rise. However, the determination of capillary rise height over time and the consideration of substrate roughness have not fully been reported. The main objective in this work consists of observing the resin impregnation into VACNTs' forest by capillary rise while taking into consideration material properties, such as dense VACNTs' forest, limited VACNTs height, highly viscous resin without electrically conductive additive, and prepreg roughness, which could make the impregnation process and the resin capillary rise height determination difficult. These criteria allow us to manufacture a semi-product which helps to study nano-engineered composites manufacturing afterwards. Hence, this paper deals with manufacturing process which transfers VACNTs' forests from Al substrate to prepreg surface. Process parameter intervals are suggested to partially impregnate resin into VACNTs' forests. SEM images allow to observe VACNTs' morphology after impregnation process as well as capillary rise height.

To achieve this objective, capillary rise of resin from prepreg into VACNTs' forests could help maintain nanotubes' morphology and partial capillary height is targeted to reach the optimal transfer: half rise from inferior prepreg ply and another half rise from superior prepreg ply fill completely resin into VACNTs' forests. In fact, the resin impregnated at VACNTs ends forms a bond to keep VACNTs' forests on prepreg surface. The partial capillary rise also represents impregnation control, which is useful to avoid coalescence phenomenon in VACNTs' forests due to surface tension. Moreover, VACNTs ends not yet impregnated with resin could be lightly pressed to create reinforced contact between VACNTs and prepreg surface.

Experimental

Materials

The prepreg reference used for the experiment is Hex-ply® M18/32%/UD116/M55J/300 mm, which is made of unidirectional M55J carbon fibre and M18 thermosetting epoxy resin. The average filament diameter of M55J is 5 µm (SEM images in Fig. 1), and the viscosity of M18 at 60 °C and 80 °C is 280 Pa.s and 60 Pa.s, respectively.

Multi-walled carbon nanotubes' forests are synthesised by Continuous Chemical Vapour Deposition (CCVD) method [31,[START_REF] Mayne-L'hermite | Nanotubes de carbone alignés synthétisés par CVD d'aérosol : mécanismes de croissance et applications[END_REF], by our industrial partner NAWA Technologies, on a 50 µm-thick aluminium (Al) substrate. The density of VACNTs ( D VACNTs ) is about 2.10 11 tubes/ cm 2 , with an average height of 30 µm (a higher value of height increases spacing between two prepreg plies and it is likely to make delamination), and VACNTs have an average diameter ( d VACNTs ) of 8 nm. The volume fraction of VACNTs can therefore be estimated at approximately 10% following Eq. ( 1):

with VACNTs volume fraction of VACNTs, d VACNTs their aver- age diameter, and D CNTs their density.

Assuming that the interior of the VACNTs cannot be filled with resin, the amount of resin necessary to impregnate the VACNTs' forests can be estimated to ~ 27 ml/m 2 .

(1)

VACNTs = d VACNTs 2 2 D VACNTs ,

Impregnation and transfer process

To transfer the VACNTs from Al substrate to the prepreg surface, the controlled impregnation of resin in the VAC-NTs' forests by capillary rise is carried out. Partially impregnating the resin at only one end of the VACNTs is privileged. Indeed, prepreg's resin impregnated in VACNTs' forests gives high bonding and the VACNTs can remain on prepreg surface after substrate removal. Controlling resin impregnation can avoid VACNTs-resin coalescence and make it possible to preserve VACNTs' morphology after the impregnation process. Figure 2 illustrates the impregnation process by resin capillary rise, where the viscosity of the resin is adjusted by controlling the temperature and the capillary rise height is controlled by the contact time. Pressure is also applied on the transfer zone to ensure good contact between the VAC-NTs and the prepreg surface. After a given impregnation time, the platens are separated, and the sample is removed and cooled to freeze the resin. The growth substrate can then be peeled off. To enable precise control of the temperature and pressure applied on a small sample, a Nanotest™ Thermal Interface Material Analyser (TIMA) [START_REF] Tima | Nanotest Homepage[END_REF], which is designed for thermo-mechanical characterisation of thermal interface materials, was used.

Table 1 summarises the ranges of selected parameters during the impregnation process and provides a priori justification for the choices.

Results and discussion

Capillary rise

The capillary impregnation of the VACNTs was determined by SEM observations of the samples, as illustrated in Fig. 3. While the presence of resin cannot be directly observed and no chemical contrast appears, several clues can help determine the resin level in the VACNTs' forests. A slight coalescence of the carbon nanotubes caused by the capillary impregnation can be observed in the impregnated areas. While the dry nanotubes appear slightly frizzy, they appear much straighter in the impregnated zone due to VACNTsresin surface tension. At a larger scale, a variation of the grey levels on the SEM images can also give hints of the resin front position. The level of capillary rise displays a significant variability within a sample. This can be partly explained by local variation in the VACNTs' density caused by whether growth condition or impregnation induced coalescence, as well as the variation in the contact between the VACNTs' forests and the prepreg, and the local resin availability at the surface of the prepreg.

Figure 3a shows VACNTs' forests transferred onto prepreg surface and resin partial capillary rise at the bottom of VACNTs, with the resin level highlighted by a yellow dashed line. Figure 3b illustrates total resin impregnation into VACNTs' forests, where the resin is visible all the way to the aluminium substrate, and the VACNTs' forests remained on Al growth substrate. This demonstrates that total resin impregnation is not favourable to the transfer process as the resin might adhere to the growth substrate. For each impregnated sample, the capillary rise of the resin in the VACNTs' forests is measured at approximately ten locations to determine an average capillary impregnation level.

Capillary impregnation is most commonly described using Lucas-Washburn (LW) model [START_REF] Washburn | The dynamics of capillary flow[END_REF]. For non-circular capillaries, with varying pore sizes and introducing a level of tortuosity in the capillaries, the LW equation can be expressed as [START_REF] Ahn | Simultaneous measurements of permeability and capillary pressure of thermosetting matrices in woven fabric reinforcements[END_REF][START_REF] Lebel | Experimental characterization by fluorescence of capillary flows in the fiber tows of engineering fabrics[END_REF] with h resin capillary rise height, F geo geometric factor, D h equivalent hydraulic diameter, surface tension of the fluid, contact angle between resin and fibre, fluid dynamic viscosity, tortuosity of the VACNTs. and t VACNTs-resin contact time.

While the equivalent diameter can be calculated as The surface tension of the prepreg resin, the contact angle, the tortuosity, and the geometric factor can be challenging to determine due to the nanoscale size of individual nanotubes. However, these parameters could be assumed as a constant C to study the effects of time and resin capillary rise height. LW equation [Eq. ( 2)] can then be formulated as with C empirical coefficient that can be found by experi- mental results, t VACNTs-resin contact time, and fluid dynamic viscosity.

(2) h 2 = F geo .D h . .cos 16 2 .t, (3)
D h = d VACNTs 1 -VACNTs VACNTs ,
Figure 4 summarises the measurements of capillary rise height over time in the VACNTs based on the SEM observations. Two values of temperature (60 °C and 80 °C) correspond to 280 Pa.s and 60 Pa.s of resin viscosity, respectively.

Based on the measured capillary rise at 60 °C, the constant C can be identified to draw a regression curve at both 60 °C and 80 °C with known values of resin viscosity. It can be observed that the capillary rise height progresses gradually in the targeted interval of time, from 0 to 20 min. Therefore, even in case of dense VACNTs' forests ( 2 × 10 11 nanotubes/cm 2 ), resin partial capillary rise height can be controlled when resin viscosity value is in the vicinity of 280 Pa.s. Constituent material properties, such as prepreg surface roughness, VACNTs' tortuosity, and VACNTs' limited height, could be the main factors which exacerbate the deviation in resin capillary height determination.

When heating the resin to 80 °C, due to low resin viscosity, uncontrolled capillary rise has occurred and resin from prepreg has completely impregnated VACNTs' forests upon 5 min, as shown in Fig. 3b. This observation is consistent with theoretical curve in Fig. 4, which shows that resin would reach 30 µm of height in less than 6 min. Therefore,

(4) h = C √ t ,
the experimental results in capillary rise height cannot be reported in 80 °C heating in our study.

Buckling of VACNTs

Both the LW model and the experimental measurements show no effect of the contact pressure. However, while carbon nanotubes demonstrate tremendous tensile properties, long and slender nanotubes can be prone to buckling when subjected to longitudinal compression. At microscopic scale, the unidirectional prepreg surface can also demonstrate a certain level of roughness. While increasing the applied pressure can improve the contact between the prepreg and the VACNTs' forest, care must be taken not to damage the vertical alignment of the carbon nanotubes. Figure 5 presents observations of VACNTs transferred on the prepreg after capillary impregnation at two levels of compaction.

Both were impregnated at 60 °C with a contact duration of 5 min. On Fig. 5a, the applied pressure was 20 kPa, and the VACNTs remain straight and vertically aligned. On Fig. 5b, after compaction at 200 kPa, some buckling can be observed on the VACNTs' forest. While the buckling remains minimal, this demonstrates the limit to ensure proper vertical alignment of the nanotubes.

Conclusion

The feasibility and control of capillary impregnation of VACNTs' forest using the resin of the prepreg have been demonstrated, and a few preconisations can be made. The characterisation of the height and density of VAC-NTs' forests is necessary to calculate the quantity of resin needed for impregnation. The chosen prepreg needs to have sufficient resin flow to ensure good impregnation.

The capillary impregnation of the resin into the VAC-NTs can be controlled by careful selection of temperature and contact time. A partial impregnation is more favourable to ensure efficient transfer of VACNTs from their growth substrate to the prepreg. The capillary impregnation in the VACNTs follows the Lucas-Washburn model with impregnation height increasing as a function of t 1/2 . Although a temperature of 80 °C was not favourable in this study, due to the chosen minimal time and the experimental apparatus used, faster heating and cooling rates could allow efficient impregnation and subsequent transfer at higher temperature.

The contact pressure applied between the VACNTs and prepreg does not significantly impact the speed of capillary impregnation. While sufficient pressure is needed to ensure a uniform and intimate contact between the prepreg and the VACNTs' forests, the applied pressure should not exceed the buckling stress of the VACNTs' forest. The feasibility of nano-structured hybrid composite impregnation aims to determine the range of process parameters. These factors are key to study the scalability of VAC-NTs-prepreg transfer. Mechanical characterisations of this novel material will be investigated afterwards.

Fig. 1

 1 Fig. 1 SEM images showing prepreg roughness due to 5 µm elementary filament and resin distribution before the process

Fig. 2

 2 Fig. 2 Four-step impregnation process. a Placement of materials and pre-heating of the top head; b contact between heated top head, Al substrate, VACNTs, prepreg, and ambient bottom head; c capillary rise of resin in the VACNTs; d end of process by moving the two

Fig. 3

 3 Fig. 3 SEM images illustrating a vertically aligned morphology of carbon nanotubes on prepreg surface and resin partial capillary rise at the bottom of these nanotubes; b total resin impregnation in VACNTs, and sample process parameters: 80 °C-5 min-200 kPa

Fig. 4

 4 Fig. 4 Resin capillary rise curve (h-t relation)

Table 1

 1 Three process parameters and their ranges

	Impregnation parameters	Below inferior threshold	Inferior threshold	Superior threshold Above superior threshold
	Temperature	High resin viscosity, difficult	60 °C (Viscos-	80 °C (Viscos-	Low resin viscosity leading to fast
		impregnation of the VACNTs'	ity of the	ity of the	capillary rise, resin impregnation
		forests	resin ~ 280 Pa.s)	resin ~ 60 Pa.s)	uncontrolled
	Contact time (step of 5 min) Lack of repeatability due to han-	5 min	20 min	Strong possibility of full capillary
		dling variability and transient			rise
		phase of heating, loading, and
		unloading
	Pressure	Higher risk of uneven contact	20 kPa	200 kPa	High risk of VACNTs' buckling
		between VACNTs' forests and
		prepreg

Acknowledgements This work was carried out within the framework of the FUI ATIHS project (improving satellite structures strength against hypervelocity impacts from space debris) funded by Bpifrance and Région Occitanie. The authors also thank the partners of the ATIHS project (DynaS+, Thiot Ingenierie, CEA Saclay, CEA Cesta, NAWATechnologies, Armines, and Airbus Defence&Space) for the fruitful exchanges carried out in this context. Special thanks are due to Alexandre Sangar (NAWATechnologies) for helpful discussions and instructions on the TIMA apparatus, and to Karine Vieillevigne for performing the SEM observations.

Data availability

The datasets analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.


Yacine Amara

Sami Hlioui

Hamid Ben Ahmed

Mohamed Gabsi

Pre-optimization of hybridization ratio in hybrid excitation synchronous machines using electrical circuits modelling

Keywords: hybrid excitation, synchronous machines, permanent magnet, hybridization ratio, optimization, electric vehicles, efficiency

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Hybrid excited synchronous machines (HESM) are synchronous machines where two magnetic excitation sources co-exist: permanent magnets (PM) and wound-field (WF) excitation. Thanks to this particularity, these machines possess an additional degree of freedom, which is the hybridization ratio α. This parameter reflects the contribution of PM to the total excitation flux. It can be advantageously exploited for the optimized design of electrical machines in very demanding applications [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF].

The electric traction is chosen as the case study, in this contribution. For this application, the traction motor is often operating in partial load regions, which requires optimizing the power efficiency in these regions for achieving high energy efficiency. If classical synchronous machines are used, this additional requirement will be difficult to achieve without compromising other ones. Thanks to its additional degree of freedom HESM allow answering this new requirement without compromising others.

The value of α allowing optimizing efficiency around a given (Torque, Speed) operating point is determined thanks to an electrical circuits model based on first harmonics of the different quantities. Since this investigation only concerns the predesign of HESM, the parameters of adopted model are considered constant [START_REF] Morimoto | Loss minimization control of permanent magnet synchronous motor drives[END_REF][START_REF] Fernandez-Bernal | Determination of parameters in interior permanent-magnet synchronous motors with iron losses without torque measurement[END_REF]. More precise models could be adopted [START_REF] Mohammadi | A computational study of efficiency map calculation for synchronous AC motor drives including cross-coupling and saturation effects[END_REF], but these require more complicated and time consuming approaches, which may not be necessary at the predesign stages. Nevertheless, the adopted model, even if it can be considered as too simple, is classical, widely spread and has been validated by many works [START_REF] Morimoto | Loss minimization control of permanent magnet synchronous motor drives[END_REF][START_REF] Fernandez-Bernal | Determination of parameters in interior permanent-magnet synchronous motors with iron losses without torque measurement[END_REF]. Furthermore, its exploitation in a parametric study helps answer this relative drawback.

This model is first introduced. Per-unit quantities are used in order to generalise the findings of this study. The algorithm allowing the determination of the optimal value of α is then detailed. Although already presented and used in a previous contribution [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF], the software tool based on this algorithm has never been detailed. The first goal of this contribution is to present this tool, and allow the reader to replicate it. This tool, coded under MATLAB environment, is also made available through the link given as reference [5]. Finally, a parametric study intended to investigate the effect of some design specifications and parameters on this optimal value is presented. Variations of design parameters, mainly resistances and inductances, could be due to, for example, temperature variations for resistances or magnetic saturation for inductances. This study will help analyse the effect of physical phenomena inherent to electrical machines operation on the hybridization ratio optimal value.

Pre-design model

The model used in this study is the classical, widely known first harmonic model for synchronous machines based on a synchronous d-q reference frame, including iron loss [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF][START_REF] Morimoto | Loss minimization control of permanent magnet synchronous motor drives[END_REF][START_REF] Fernandez-Bernal | Determination of parameters in interior permanent-magnet synchronous motors with iron losses without torque measurement[END_REF]. Figures 1(a) and 1(b) show equivalent circuits for armature windings, and Figure 1(c) shows an equivalent circuit for the wound field excitation. Steady state electric equations of this model could be easily derived from these figures [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF][START_REF] Fernandez-Bernal | Determination of parameters in interior permanent-magnet synchronous motors with iron losses without torque measurement[END_REF]. Symbols in these figures are defined as: id, iq d and q axes components of armature current, Ie excitation current, ifd, ifq d and q axes components of iron loss current, vd, vq d and q axes components of terminal voltage, Ve excitation coils terminal voltage, Ra armature winding resistance per phase, Rf iron loss resistance, Re excitation coils resistance, Фa permanent magnet flux linkage, Фexc total excitation flux linkage, ke "Armature/Excitation windings" mutual inductance, Ld, Lq d and q axes components of synchronous inductance.

Thanks to the presence of the wound-field excitation, the total excitation flux varies with the excitation current; it can be expressed as

Φ Φ Φ exc a e e f e max k I k = + ⋅ = ⋅ , (1)
where, Фe max and kf are the maximum total excitation flux and the excitation coefficient, respectively. The excitation coefficient varies ideally between 1, as an upper limit, and 0. Nevertheless, depending on how the HESM is designed, it may not be possible to completely cancel the excitation flux (kf min > 0). Magnetic saturation, thermal limits or the demagnetization limit may set bound to the lower value of the excitation coefficient.

Per-unit system

Per unit system model allows a better understanding of parameters effect on machines performance. It is also a powerful tool for electric machines drives classification [START_REF] Schiferl | Power capability of salient pole permanent magnet synchronous motors in variable speed drive applications[END_REF][START_REF] Soong | Field-weakening performance of brushless synchronous AC motor drives[END_REF][START_REF] Soulard | Maximum power limits of small single-phase permanent magnet drives[END_REF]. Base values of EMF and current are chosen as the rated values for the motor at rated speed (base speed Ωb).

Per-unit values of the model parameters (resistances and inductances) are defined as

Φ Ω Φ Ω f m a m e em an fn en e max b e max b em R I R I R I R , R , R p p V ⋅ ⋅ ⋅ = = = ⋅ ⋅ ⋅ ⋅ , (2)
Φ Φ d m e em dn qn dn en e max e max L I k I L , L L , k ⋅ ⋅ = = ρ ⋅ = , (3) where
+ + = = = ⋅ ⋅ . ( 4
)
Per-unit values of output power P, copper loss PCu, and iron loss PFe are given by

V I V I V I = = = ⋅ ⋅ ⋅ , (5)
where, Vm is the maximum armature terminal voltage (in d-q referential). Finally, per-unit values of speed and torque are defined as

Ω Ω Ω Ω n n n b n P , T = = . ( 6
)
Another dimensionless parameter specific to synchronous machines with a wound field excitation, i.e., pure wound field excited synchronous machines and HESM, has also to be defined

m m em em V I V I ⋅ β = ⋅ . ( 7
)
β is the power ratings ratio between converters supplying the armature and excitation windings, respectively. Hybridization ratio α, which is specific to HESM, is an additional degree of freedom from a design point of view; it is defined as

Φ Φ a e max α = . ( 8
)

Hybridization ratio α

To illustrate this central parameter (hybridization ratio α) and its link with the design of HESM, examples of HESM structures combining PM and wound field excitations are presented in this subsection.

Different criteria can be used for classification of HESM [START_REF] Amara | A new topology of hybrid synchronous machine[END_REF][START_REF] Amara | Comparison of open circuit flux control capability of a series double excitation machine and a parallel double excitation machine[END_REF]. Two are specific to HESM: the first concerns the way the two magnetic excitation flux sources are combined, i.e., series and parallel hybrid excitation, and the second concerns the localization of excitation flux sources in the machine, i.e., both sources in the stator, both sources in the rotor and mixed localization. The first criterion is used here in order to illustrate the hybridization ratio α and how it can be adjusted.

Figure 2 illustrates the series hybrid excitation principle and two examples of structures belonging to this class. Figure 3 shows similar illustrations for the parallel hybrid excitation principle. In [START_REF] Amara | Comparison of open circuit flux control capability of a series double excitation machine and a parallel double excitation machine[END_REF], a comparison between two machines belonging to each of the two classes has shown the superiority of the principle of parallel hybrid excitation. In fact, in series hybrid excitation, the flux generated by the wound-field excitation should cross the PM region which has a relatively high reluctance [START_REF] Amara | Comparison of open circuit flux control capability of a series double excitation machine and a parallel double excitation machine[END_REF]. Flux control capability is then better for parallel HESM as compared to the series HESM.

The hybridization ratio α, in these machines, could be adjusted by choosing adequate values of the parameters related to the PM and the wound-field excitation circuits, e.g. PM grade, PM thickness, number of turns of the wound-field excitation, and the excitation current limitations.

Optimization of hybridization ratio

HESM offers an additional degree of freedom in the control and design of synchronous machines as compared to other synchronous machines.

Electromagnetic loss in hybrid excited machines depends on two groups of parameters (9): one group corresponds to control parameters, the other one to design parameters. Design Control ( )

Cu Fe f d P P f I , , k , , L , , + = ψ α ρ β 14243 1 424 3 (9)
From the control point of view, hybrid excitation machines are similar to wound field synchronous machines [START_REF] Amara | Power capability of hybrid excited synchronous motors in variable speed drives applications[END_REF]. From the design point of view, hybrid excitation machines have an additional degree of freedom, which is the hybridization ratio α [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF].

Figure 4 shows an algorithm used to optimize hybridization ratio. This algorithm includes design and control parameters. It allows defining the value of hybridization ratio α which maximizes efficiency for desired "Speed -Torque" point (Ωn0, Tn0). ψ is the phase shift between the armature current and EMF, and Vnmax is the normalized value of armature windings terminals maximum voltage.

This algorithm allows choosing, for α varying between 0 and 1, combinations of excitation coefficient kf, normalized armature current In, and phase shift ψ, which maximizes efficiency η while respecting the voltage limit. Optimal value of hybridization ratio αopt is the one which maximizes efficiency.

The entry data of this algorithm are the design specifications quantities (Ωn0, Tn0). The design parameters or variables are embedded in the inner loop of this algorithm, i.e., the one allowing computing the efficiency. It is the first harmonic model which is used for that purpose. The design variables and their variations ranges are described in the next section "2.4".

The efficiency computation is at the heart of this optimization algorithm. It will be detailed in this section, and the files allowing the computation of efficiency maps are also provided along with this contribution [5]. Before starting describing the process allowing the computation of efficiency maps, it should be stressed that this algorithm has only been developed for non-salient machines, i.e., ρ = 1. Indeed, having ρ = 1 helps reducing the number of numerical computations, since some quantities required during the efficiency computation algorithm can be derived analytically, while having ρ ≠ 1 requires these quantities to be determined numerically, which implies a longer computation duration. Nevertheless, explanations to extend it to salient poles machines are also provided.

The first aspect which should be given attention to is the fact that some parameters are interdependent regarding the way the per-unit system has been defined. Indeed, given the equation of excitation flux and its per-unit version [START_REF] Amara | Comparison of open circuit flux control capability of a series double excitation machine and a parallel double excitation machine[END_REF], parameters ken, normalized values of ke, and β will be dependent on the value of hybridization ratio α. Equation [START_REF] Amara | Power capability of hybrid excited synchronous motors in variable speed drives applications[END_REF] gives relations between these parameters and α. A detailed explanation of these relations is provided in Appendix A.

k I = α + ⋅ (10) [ ] ( ) [ ] [ ] ( ) [ ] 2 1 2 1 if 0 5 1 1 if 0 0.5 if 0 5 1 1 if 0 5 1 en . , k , . , . ,  α α ∈   =   -α α ∈       β α α ∈   β =   β -α α ∈    (11)
In equation [START_REF] Amara | Power capability of hybrid excited synchronous motors in variable speed drives applications[END_REF], β1 is the value of β when wound field excitation is only used for weakening the total excitation flux, i.e., α = 1. The second step towards the computation of efficiency maps is the determination of the maximum normalized value of armature voltage Vnmax. The base speed should be first determined. The base speed is the maximum speed which could be reached while adopting the maximum torque control strategy. For non-salient poles machines (ρ = 1), the maximum torque is obtained when maximum excitation flux and maximum value of i0q (Figure 1) [START_REF] Amara | Power capability of hybrid excited synchronous motors in variable speed drives applications[END_REF], while respecting current limitation, are imposed. Figure 5 shows the algorithm used to compute Vnmax. A detailed explanation of this algorithm is provided in Appendix B. Vnmax computation should be done prior to any efficiency computation algorithm implementation.

Figure 6 shows the efficiency maps computation algorithm. This general algorithm could be applied for non-salient and salient poles machines. It allows to determine the maximum efficiency for a given (Ωn, Tn) combination. Normalized torque can vary between 0 and 1 as a maximum value. Per-unit value of speed varies between 0 and Ωn max > 1. For (Ωn, Tn) combinations for which there is no means of respecting the voltage limit, the efficiency is set to 0 [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF]. Mechanical losses are neglected in efficiency calculation, but they could be easily incorporated. The main problematic in this algorithm is the selection of (kf , In, ψ) combinations.

This algorithm is different from the one shown in Figure 4, even if the efficiency computation is exactly similar in both, and it's at the heart of the two. The difference between them is their function. While the algorithm of Figure 4 is used for getting the optimal hybridization ratio for a given (Ωn0, Tn0) combination (used for design purposes), the one shown in Figure 6 is used to generate the efficiency maps for the entire (Torque, Speed) plane, for a given machine, i.e., a given parameters combination.

The efficiency computation being at the heart of these two algorithms, the aim in what follows is to clearly explain the efficiency estimation methodology. Furthermore, in order to help the reader to replicate it, the algorithms, shown in Figures 4 and6, are both made available through the link given as reference [5].

For non-salient poles machines, this problematic is simplified and many operations could be done analytically. For these machines, the selection of (kf , In, ψ) combinations is done through four simple steps. These steps are explained here and further detailed in Appendix C. In fact, for each (Ωn, Tn) combination, the selection process only requires the implementation of a single loop on kf. The two other components (In, ψ) are determined analytically. The four steps are [START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF]:

1. analytical determination of i0dn allowing maximizing efficiency for a given (Ωn, Tn, kf) combination; 2. analytical determination of i0dn range allowing respecting current limitation (In ≤ 1); 3. analytical determination of i0dn range allowing respecting voltage limitation (Vn ≤ Vnmax); 4. selection of (kf , In, ψ) combination which maximizes efficiency.

The three first steps are independent, and can be processed in any order. In the fourth step, selection step, results of the three previous steps are exploited in order to select (kf , In, ψ) combinations allowing maximizing efficiency, or set η(Ωn, Tn) = 0, if none of the (kf , In, ψ) combinations are respecting current and/or voltage limitations. For the first step, it is possible to express the total normalized electromagnetic losses (PCun + PFen) as a function of i0dn. The value of i0dn allowing maximizing efficiency, for a given (Ωn, Tn, kf) combination, is obtained by finding the optimum of the losses function. This optimum is obtained for the value of i0dn0 given by [START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF] [START_REF] Morimoto | Loss minimization control of permanent magnet synchronous motor drives[END_REF][START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF]. More details are provided in Appendix C.

( ) ( ) 2 0 0 2 2 2 Ω Ω n dn an fn f dn an fn n dn an fn L R R k i R R L R R -⋅ ⋅ + ⋅ =   ⋅ + ⋅ ⋅ +   (12)
For the second step, intervals for which the current limit is respected, if ever existing, are determined. This is done by solving equation In ≤ 1, which corresponds to equation [START_REF] Amara | Measured performances of a new hybrid excitation synchronous machine[END_REF]. Obtaining of this equation is explained in Appendix C [START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF].

2 0 0 0 I dn I dn I A i B i C   ⋅ + ⋅ + ≤   (13) with, ( ) ( ) ( ) ( ) (
)

2 2 2 2 2 2 2 1 Ω 2 Ω Ω Ω 1 I n dn fn I n f dn fn I n dn n n max n n max fn n f f fn A L R B k L R C L T V T V R k k R     = + ⋅       = ⋅ ⋅ ⋅          = ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ -                  
In case the discriminant of equation ( 13) is positive, let consider that it is verified for i0dn ∈ [i0dn1, i0dn2]. For the third step, intervals for which the voltage limit is respected, if ever existing, are determined by solving equation Vn ≤ Vnmax, which corresponds to equation [START_REF] Williamson | Comprehensive drive train efficiency analysis of hybrid electric and fuel cell vehicles based on motorcontroller efficiency modeling[END_REF]. Obtaining of this equation is explained in Appendix C [START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF].

2 0 0 0 V dn V dn V A i B i C   ⋅ + ⋅ + ≤   (14) with, ( ) ( ) ( ) ( ) ( ) ( )
( )

2 2 2 2 2 2 2 2 2 Ω 1 2 Ω 1 Ω 1 Ω 1 V dn n an fn an V n f dn an fn f n an fn an n n max f V dn n n n max an fn f n max A L R R R B k L R R k R R R T V k C L T V R R k V        = ⋅ ⋅ + +            = ⋅ ⋅ ⋅ ⋅ +        ⋅ ⋅ + + ⋅ ⋅    =    + ⋅ ⋅ ⋅ ⋅ + -      
In case the discriminant of equation ( 14) is positive, let consider that it is verified for i0dn ∈ [i0dn3, i0dn4]. The final step consists of finding, for a given (Ωn, Tn) combination, the higher efficiency. In addition to outer loops allowing incrementing the speed and torque values, there is only one loop on kf (inner loop). For a given set of kf values, this loop will help identify values of (i0dn, i0qn), and subsequently (In, ψ), allowing to maximize efficiency. This loop will help to generate a vector containing efficiencies η(kf) corresponding to each value of kf. Within this set of kf values, the one which will be kept, at the end, is the one maximizing the efficiency. The following table (Table 1) describes what is principally done inside the inner loop (loop on kf), in order to estimate η(kf). More details are given in Appendix C, and files allowing the computation of efficiency maps could be downloaded from [5].

Table 1 Algorithm inside inner loop for a given (Ωn, Tn, kf) combination

Start

If discriminant of equation ( 13) is negative ⇒ η(kf) = 0 Or if discriminant of equation ( 14)

is negative ⇒ η(kf) = 0 Else If [i0dn1, i0dn2] ∩ [i0dn3, i0dn4] = ∅ ⇒ η(kf) = 0 Else If i0dn0 ∈ [i0dn1, i0dn2] ∩ [i0dn3, i0dn4] ⇒ η(kf) = η(Ωn, Tn, kf, i0dn0)
Else η(kf) = η(Ωn, Tn, kf, i0dn5), i0dn5 is equal to the intersection interval terminal which is closer to the value of i0dn0.

End

Parameters variations

Table 2 gives normalized parameters variations intervals. These intervals are bounded by values between which the different parameters could reasonably vary. The two first lines, in this table, correspond to the design specifications quantities, and the rest to design variables.

The parametric study, presented in this contribution, is mainly conducted in order to establish the separate effects of each parameter on the optimal value of hybridization ratio. In addition to the effect of design specifications quantities (Ωn0, Tn0), only losses parameters, within the design variables, are considered, i.e., Ran and Rfn. Several references [START_REF] Fernandez-Bernal | Determination of parameters in interior permanent-magnet synchronous motors with iron losses without torque measurement[END_REF][START_REF] Schiferl | Power capability of salient pole permanent magnet synchronous motors in variable speed drive applications[END_REF][START_REF] Soong | Field-weakening performance of brushless synchronous AC motor drives[END_REF][START_REF] Soulard | Maximum power limits of small single-phase permanent magnet drives[END_REF] have been analysed in order to establish reasonable variations intervals of the different parameters. Variations intervals, presented in Table 2, are chosen so as to include parameters found in these references. Authors of the reference [START_REF] Soulard | Maximum power limits of small single-phase permanent magnet drives[END_REF] have identified some designs, 2 over 11 analysed designs, where the normalized values of armature windings resistance Ran are higher than 0.5. They pointed out that these designs correspond to small-power machines. It should be recalled that the machines concerned are these used for electric traction, which are classified within horsepower drives.

The initial values of the different design variables are given as follows: Ldn = 0.5; ρ = 1; Ran = 0.1; and Rfn = 20, ken = 1; Ren = 1; β = 27, and α = 1. These parameters have been derived from an existing prototype [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF][START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF][START_REF] Amara | Measured performances of a new hybrid excitation synchronous machine[END_REF].

Table 2 Normalized parameters variations intervals

Parameter

Variations interval

Ωn0 [0, 4] Tn0 [0, 1] Ldn [0, 5] ρ [0, 5] Ran [0, 0.5] Rfn [5, +∞[ Ren 1 ken [0, 1] β [5, +∞[

Design specifications and variables effects

The effects are evaluated using drawings of isovalues of some quantities, efficiency among others, on the (Torque, Speed) plane.

As explained earlier, the chosen hybridization ratio αopt allows efficiency maximization in a given (Torque, Speed) region. This efficiency is evaluated by neglecting mechanical losses, and it does not include the static converter losses.

Figure 7 shows the efficiency map of the HESM with previously defined parameters. Efficiency maps constitute a convenient way to assess motor designs and their control strategies [START_REF] Amara | Hybrid excitation synchronous machines: Energy-efficient solution for vehicles propulsion[END_REF][START_REF] Mohammadi | A computational study of efficiency map calculation for synchronous AC motor drives including cross-coupling and saturation effects[END_REF][START_REF] Amara | Power capability of hybrid excited synchronous motors in variable speed drives applications[END_REF][START_REF] Amara | Contribution à la conception et à la commande des machines synchrones à double excitation[END_REF][START_REF] Williamson | Comprehensive drive train efficiency analysis of hybrid electric and fuel cell vehicles based on motorcontroller efficiency modeling[END_REF][START_REF] Sugii | Applicability of various motors to electric vehicles[END_REF][START_REF] Fonseca | New modelling methodology for different PM motors for electric and hybrid vehicles[END_REF][START_REF] Menne | Energy-efficiency evaluation of traction drives for electric vehicles[END_REF][START_REF] Vido | Recherche des paramètres optimaux d'une machine synchrone à aimants permanents[END_REF][START_REF] Chu | Investigation on operational envelops and efficiency maps of electrically excited machines for electrical vehicle applications[END_REF][START_REF] Zhu | Quantitative comparison of electromagnetic performance of electrical machines for HEVs/EVs[END_REF][START_REF] Jung | Optimal design and validation of IPMSM for maximum efficiency distribution compatible to energy consumption areas of HD-EV[END_REF][START_REF] Mohammadi | Hybridisation ratio for hybrid excitation synchronous motors in electric vehicles with enhanced performance[END_REF][START_REF] Krasopoulos | Multicriteria PM motor design based on ANFIS evaluation of EV driving cycle efficiency[END_REF]. Effects of the different parameters and variables are studied in the following sections.

Design specifications

In order to investigate the design specifications effects on the value of optimal hybridization ratio αopt, a first optimization study is conducted. The optimization algorithm has been used in order to determine the value of α allowing maximizing the efficiency for (Ωn0 = 2, Tn0 = 0.2). It has been found that αopt should be equal to 0.5.

Figure 8 shows efficiency maps for previous machine with a modified value of hybridization ratio, i.e., α = 0.5. As can be seen, the adoption of this new hybridization ratio allowed to shift the high efficiency zone to the desired operating (Ωn0, Tn0) point [1] [22].

In order to investigate the separate effects of speed and torque on the value of optimal hybridization ratio, a first study is conducted by keeping the speed Ωn0 constant (Ωn0 = 2) and varying the torque value Tn0 (Figure 9), and then keeping the torque constant (Tn0 = 0.2) and varying the speed value Ωn0 (Figure 10). The value of optimal hybridization ratio is decreasing as the torque Tn0 is decreasing for a constant speed, or as the speed Ωn0 increases for a constant value of Tn0. Figure 11 shows isovalues of αopt in the (Torque, Speed) plane.

For Figure 9, the optimal hybridization ratio variations stop for a value of Tn0 ≈ 0.433, because beyond this value the machine is not able to deliver a torque while respecting current or/and voltage limits.

Obtained results are logical and conform to the electrical machines general behaviour. Indeed, for high values of torque it would be logically better to produce excitation flux from permanent magnets and therefore having minimum copper losses. For high speed operation where both the iron and copper losses may be significant, the flux weakening is made easier as the PM flux is reduced.

Design variables

As stated earlier, in section "2.4", only losses parameters, within the several design variables, are considered, i.e., Ran and Rfn, in this sensitivity study. Effects of these parameters are studied separately. For that purpose, curves shown in Figures 9 and10 Figure 12(a) shows variations of optimal hybridization ratio, when Ωn0 = 2, for three values of normalized armature resistance Ran, while Rfn is kept constant (Rfn = 20). It could be noticed that, for a large torque range when this resistance is higher the optimization algorithm tends to give a higher value of αopt, which is quite coherent since the efficiency is higher when torque production is mainly insured by PM excitation flux. The curves stop at a given normalized torque value, which vary with Ran, because beyond this value the torque production could not be insured while respecting the current and/or the voltage limits. The machine could produce higher torque as the armature joule losses are lower (lower values of Ran). The value of αopt is higher when Ran = 0, for Tn0 < 0.1, is mainly due to the fact that under a certain value of Ran, the armature joule losses are so low, that the torque could be insured with the minimum iron loss, by imposing a large armature current in d axis while opposing PM excitation flux (flux weakening operation).

In order to further investigate this phenomenon, variations of optimal values of different design and control quantities, i.e., hybridization ratio α (Figure 13), and armature current amplitude In [Figure 14(a)] and phase-shift ψ [Figure 14(b)], with Ran, when (Ωn0 = 2, Tn0 = 0.05), have been computed. Figures 13 and14 show these variations and it could be seen that for low values of Ran, the optimization algorithm imposes a flux weakening operation [Figures 14(a) and 14(b)]. It should also be noticed that this type of operation is limited to a narrow variation range of low values of Ran (Figure 13).

When Tn0 = 0.2 [Figure 12(b)], the value of αopt is higher as armature joule losses are higher. The torque Tn0 = 0.2 could not be maintained beyond a certain speed (Ωn0 ≈ 3.2) for the machine with Ran = 0.5. For the machine with Ran = 0, the optimal value of the hybridization ratio is relatively constant (αopt ≈ 0.55). The torque production for this machine should be insured by the PM to reduce excitation joule losses.

Figures 15(a) and 15(b) show similar variations as Figures 12(a) and 12(b) respectively, but it is Rfn which varies this time, while Ran is kept constant (Ran = 0.1). It can be noticed from Figure 15(a), that when the iron loss could be neglected (higher values of Rfn), the algorithm tends to give a higher value of αopt.

Indeed, when iron loss could be neglected, it is more convenient to produce the torque by increasing the excitation flux and reducing the electrical load (armature current). The torque production is improved as the iron losses are getting lower. For machines with Rfn = 5 and Rfn = 20, the optimization algorithm gives more or less same values of αopt.

Figure 15(b) shows variations of optimal hybridization ratio, when Tn0 = 0.2. As previously, the algorithm tends to give a higher value of αopt when iron losses could be neglected. This is again quite coherent. When the speed increases, the optimal value of hybridization ratio is decreasing, to lessen iron loss and allow an easier flux weakening operation. The torque Tn0 = 0.2 could not be maintained beyond a certain speed (Ωn0 ≈ 3.6) for the machine with Rfn = 5.

Conclusions

The study presented in this contribution helped assessing the validity and efficacy of the algorithm allowing the computation of optimal hybridization ratio. The efficiency computation, which is at the heart of this algorithm, has been detailed, to allow the reader to replicate it. This algorithm has proven to be reliable, and has been used to analyse the effect of design specifications and variables on αopt. The parametric study undertaken has clearly shown the usefulness of the developed tool. This tool could be used for both design and analyses purposes, as has been seen in sections "3.1" and "3.2". This quantitative study, related to the pre-optimization of the hybridization ratio, has shown that the results are consistent with the physics of electric machines. Furthermore, no contradictions with previous literature were found, which comfort the trust on the developed algorithms. This tool will be further improved in order to reduce simplifying assumptions on which the used model of HESM is based. The next steps will be to consider salient poles machines, and include a consideration of the magnetic saturation. Nevertheless, the presented tool has clearly showed how the additional degree of freedom, offered by the HESM, could be advantageously used to optimize the energy efficiency of electric drives used in electric vehicles.

V I V I V I k V I R R I  ⋅ ⋅ = = =        ⋅ ⋅ ⋅  β = = = ⋅ = β      ⋅ ⋅      (A.1) A.2. α ∈ ]1, 0.5]
If the machine is designed with same maximum excitation flux Φe max and a hybridization ratio α ∈ ]1, 0.5], it could be easily demonstrated that the maximum excitation current is still obtained for kf = 0 (A.2).

(

) ( )

Φ 0 Φ 0 Φ Φ Φ 1 Φ Φ Φ Φ 1 e max f a
  = ⇒ + ⋅ = ⇒ = - = -α ⋅     ⇒ = α ⋅     = ⇒ + ⋅ = ⇒ = - = -α ⋅     (A.2)
In this case:

( )

V I V I V I k V I R R I  ⋅ α ⋅ ⋅ = = = α        ⋅ ⋅ ⋅ β  β = = = ⋅ =      ⋅ α ⋅ ⋅ α      (A.3) A.3. α ∈ ]0.5, 0]
If the machine is designed now with same maximum excitation flux Φe max, but with a hybridization ratio α ∈ ]0.5, 0], it could be easily demonstrated that the maximum excitation current is obtained for kf = 1 (A.4).

( ) ( ) ( ) Φ 0 Φ Φ 1 Φ 1 Φ Φ 1 e max f e
I k k I k k   = ⇒ = - = -α ⋅     ⇒ = -α ⋅     = ⇒ = - = -α ⋅     (A.4)
In this case:

( ) ( ) ( ) ( ) ( ) 2 1 2 2 1 Φ 1 Φ Φ 1 Φ 1
V I V I V I k V I R R I  ⋅ -α ⋅ ⋅ = = = -α        ⋅ ⋅ ⋅ β  β = = = ⋅ =      ⋅ -α ⋅ ⋅   -α    (A.5)

Appendix B

The definition of base speed should be first recalled here; base speed is the maximum speed which could be reached while adopting the maximum torque control strategy, so its determination is equivalent to the determination of conditions allowing imposing the maximum torque. Torque expression, for nonsalient poles synchronous machines, is given by: 0

Φ f e max q T p k i = ⋅ ⋅ ⋅ (A.6)
From a first sight, it is easily understandable that the torque is maximized when kf = 1 (maximum excitation flux), and when maximum value of i0q is imposed. But rigorously, as regards to the adopted electrical circuits model, which includes iron losses, i0q is dependent on kf, and therefore the maximum (optimum) of the function T(kf) should be determined in order to maximize the torque. Nevertheless, most electrical machines, for not saying all, are designed in order to minimize the iron loss, which are then modelled with a quite high value resistance Rf. The dependence of i0q could then be neglected, in particular at relatively low speeds, and the torque is indeed maximized when kf = 1 (maximum excitation flux), and when maximum value of i0q is imposed.

In order to determine the maximum value of iq0 while respecting the armature current limit, relations between (id0, iq0) and (id, iq) should be first determined. These relations are given by equation (A.7).

(

) ( ) ( ) ( ) ( ) ( ) 2 2 2 0 2 0 Φ 1 Φ 1 d d d q f d f e max f d f q q d d f f e max f d f i i L i R L k R L R i i L i R k R L R      = + ⋅ ω⋅ - ⋅ ⋅ ⋅ ω + ⋅ ω                 = - ⋅ ω⋅ - ⋅ ⋅ ω + ⋅ ω         (A.7)
It could be noticed from equation (A.7), that if Rf is high, the dependence of i0q on kf could be effectively neglected. At the base speed and by imposing kf = 1, the normalized value of i0q is given by:

( ) ( ) ( ) 2
i i L i R R L R     = - ⋅ - +       (A.8)
Once the maximum value of i0qn is determined, the computation of Vnmax is straight forward. The Matlab scripts used in order to determine the maximum value of i0qn and subsequently the value of Vnmax are given in following tables (Table A

.1 and Table A.2).

C.1. Minimizing total losses

For a given (Torque, Speed) operating point, and a given value of excitation current, the losses are given by: ( )

( ) ( )
L i L i k R  = ⋅ + + ⋅     = ρ ⋅ ⋅ ω⋅ + ⋅ ω⋅ + ⋅ ⋅ ω       (A.9)
For non-salient poles synchronous machines at a given excitation flux, the value of i0q is given by: ( )

0 Φ q f e max i T p k = ⋅ ⋅ (A.10)
Using equations (A.9), (A.10) and relations between (id, iq) and (id0, iq0) (A.14), the total losses could be expressed as a function of id0 as shown by equation (A.11).

(

L i R L T p k L i k R       + ⋅ ω⋅ ⋅ ⋅ ⋅       ⋅ + ⋅           + = + ⋅ ⋅ - ⋅ + ⋅ ⋅ ω           + ⋅ ω⋅ ⋅ ⋅ + ⋅ ω⋅ + ⋅ ⋅ ω         (A.11)
It could be then shown that:

( ) ( ) ( ) ( ) ( ) ( ) 2 0 2 2 0 Φ 0 a f d f e max Cu Fe d d a f a f d R R L k P P i i R R R R L - + ⋅ ⋅ ⋅ ⋅ ω ∂ + = ⇒ = ∂ ⋅ + + ⋅ ⋅ ω (A.12)
The normalized value of i0d, given by equation (A.12), is i0dn0 shown in equation (4).

C.2. i0dn for which In ≤ 1

To find values of i0dn allowing respecting the current limit constraint, the normalized armature current amplitude In should be expressed as a function of i0dn. Knowing that:

( )

2
0 0 0 0 Φ d d d q f q q f e max d d f i i L i R i i k L i R ⋅    = -ω⋅ ρ ⋅ ⋅       = + ω⋅ ⋅ + ω⋅ ⋅     , (A.14)
by combining these two equations (equation (A.14) should be used in its normalized version) and replacing i0qn by its expression deduced from equation (A.10), a quadratic equation of i0dn is obtained (5). It should be recalled that these developments are only true for non-salient poles synchronous machines.

It should be noticed that AI > 0, which implies:

[ ] 0 0 1 0 2

Fig. 1 .

 1 Fig. 1. Synchronous machines equivalent circuits model under motor mode operation.

  (a) Principle of series hybrid excitation (b) Equivalent magnetic equivalent circuit (c) 2D series HESM (d) 3D series HESM Fig. 2. Series HESM. (a) Principle of parallel hybrid excitation (b) Equivalent magnetic equivalent circuit (c) 2D parallel HESM (d) 3D parallel HESM Fig. 3. Parallel HESM.

Fig. 4 .

 4 Fig. 4. Optimization algorithm.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Vnmax computation algorithm.

Fig. 7 .

 7 Fig. 7. Efficiency map of the HESM (α = 1). Fig. 8. Efficiency map of the HESM (α = 0.5).

Fig. 9 .Fig. 11 .

 911 Fig. 9. Optimal hybridization ratio for Ωn0 = 2. Fig. 10. Optimal hybridization ratio for Tn0 = 0.2.

  are plotted for different values of Ran and Rfn. Figures 12 and 15 show variations of the optimal hybridization ratio for different values of Ran and Rfn, respectively.

2 Fig. 12 .

 212 Fig. 12. Optimal hybridization ratio variations for different values of Ran.

Fig. 13 .Fig. 14 . 2 Fig. 15 .

 1314215 Fig. 13. Optimal hybridization ratio αopt variations with Ran for (Ωn0 = 2, Tn0 = 0.05).

  Note the use of the subscript n to indicate per unit value. Per-unit values of excitation and armature currents and the armature terminal voltage are given by

	,
	p	number of poles pairs,
	Im	maximum armature current (in d-q referential),
	Iem	maximum excitation current,
	Vem	maximum excitation coils terminal voltage,
	ρ	saliency ratio.
		I	n	2 2 d q m i i I	, I	en	e em I I	n , V	Φ	2 d e max v	v p	2 Ω q	b

Appendix A

Within the hybridization ratio variation range α ∈ [0, 1], three domains could be identified: α = 1, α ∈ ]1, 0.5], and α ∈ ]0.5, 0]

In this case, wound field excitation is only used for weakening the PM excitation flux. The higher value of excitation current is obtained for kf = 0, i.e., Iem = | Ie | = Фa / ke. In this case: For i = 1 : length(In)

C.3. i0dn for which Vn ≤ Vnmax

To find values of i0dn allowing respecting the voltage limit constraint, the normalized armature voltage amplitude Vn should be expressed as a function of i0dn. Knowing that:

( )

and, ( )

by combining these two equations (equation (A.17) should be used in its normalized version) and replacing i0qn by its expression deduced from equation (A.10), a quadratic equation of i0dn is obtained [START_REF] Schiferl | Power capability of salient pole permanent magnet synchronous motors in variable speed drive applications[END_REF]. It should be recalled that these developments are only true for non-salient poles synchronous machines.

It should be noticed that AV > 0, which implies:

[ ] .18)


Charlotte Nachtergaele

MD Eric Vicaut

MD Sara Pinto

MD Sopio Tatulashvili

MD PhD Hélène Bihan

MD Meriem Sal

Narimane Berkane

MD Lucie Allard

MD Camille Baudry

Narimane Berkanemd

MD PhD Lionel Carbillon

Professor Emmanuel Cosson
email: emmanuel.cosson@aphp.fr

COVID-19 pandemic: Can fasting plasma glucose and HbA1c replace the oral glucose tolerance test to screen for hyperglycaemia in pregnancy?

Keywords: Colle ge National des Gyne ćologues Obste triciens Français (CNGOF), COVID-19, HbA1c, oral glucose tolerance test, Société Francophone du Diabète

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

To slow the spread of Coronarovirus Disease 19 (COVID- [START_REF] Agarwal | Gestational diabetes screening: the low-cost algorithm[END_REF], it is critical to practice social distancing and to reduce contacts, including in phlebotomy centres. This is crucial regarding pregnant women and screening for hyperglycaemia in pregnancy (HIP). The oral glucose tolerance test (OGTT) -reference standard test-requires measurement of fasting (FPG), 1hour (1h-PG), 2-hour (2h-PG) and sometimes 3-hour plasma glucose [START_REF] Panel | International association of diabetes and pregnancy study groups recommendations on the diagnosis and classification of hyperglycemia in pregnancy[END_REF][START_REF]Diagnostic criteria and classification of hyperglycaemia first detected in pregnancy: a World Health Organization Guideline[END_REF][START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF][START_REF] Benhalima | Survey by the European Board and College of Obstetrics and Gynaecology on screening for gestational diabetes in Europe[END_REF][START_REF] Hod | The International Federation of Gynecology and Obstetrics (FIGO) Initiative on gestational diabetes mellitus: A pragmatic guide for diagnosis, management, and care[END_REF][START_REF]American Diabetes Association. 2. Classification and Diagnosis of Diabetes: Standards of Medical Care in Diabetes-2020[END_REF], time that the patient may spent waiting in the crowded phlebotomy centres. In this context, UK [START_REF]-guidance-for-maternal-medicine-in-the-evolving-coronavirus-covid-19pandemic[END_REF], France [START_REF] Vambergue | Three alternative ways to screen for hyperglycaemia in pregnancy during the COVID-19 pandemic[END_REF] and Japan [START_REF] Kasuga | A new diagnostic strategy for gestational diabetes during the COVID-19 pandemic for the Japanese population[END_REF] proposed to temporarily replace OGTT by FPG and HbA1c measurement.

The rationale behind this proposal is to combine tests and to reduce the time spent in phlebotomy centres.

Such a proposal should be balanced by the need to provide appropriate care to ensure the best possible pregnancy outcomes for women and their infants. Therefore, screening procedures based on FPG and HbA1c measurement should be sensitive enough to diagnose most of the women with HIP. As a matter of fact, not looking for HIP might lead to a doubling of the rate of events during pregnancy [START_REF] Landon | A multicenter, randomized trial of treatment for mild gestational diabetes[END_REF][START_REF] Crowther | Effect of treatment of gestational diabetes mellitus on pregnancy outcomes[END_REF]. However, missing a few HIP diagnoses could be less deleterious than expected if the false negative cases were at lower risk of HIP-related adverse events than the true positive ones.

The aim of the study was to retrospectively evaluate in a large cohort of women with HIP [START_REF] Cosson | Early screening for gestational diabetes mellitus is not associated with improved pregnancy outcomes: an observational study including 9795 women[END_REF][START_REF] Cosson | Performance of a selective screening strategy for diagnosis of hyperglycaemia in pregnancy as defined by IADPSG/WHO criteria[END_REF] (i) the sensitivity of the French-speaking Society of Diabetes (SFD: Société Francophone du Diabète) / French National College of Obstetricians and Gynecologists (CNGOF: Collège National des Gynécologues et Obstétriciens Français) temporary COVID-19 proposal for HIP diagnosis and (ii) the occurrence of HIP-related events in false negative and true positive cases of HIP when applying this proposal.

Material and methods

-Data collection

We have conducted this observational study in our University hospital in a suburban area of Paris, Bondy, France, where medical electronic records of maternal and neonatal events at birth have been routinely collected between January 2012 and October 2016 [START_REF] Cosson | Early screening for gestational diabetes mellitus is not associated with improved pregnancy outcomes: an observational study including 9795 women[END_REF][START_REF] Cosson | Performance of a selective screening strategy for diagnosis of hyperglycaemia in pregnancy as defined by IADPSG/WHO criteria[END_REF]. In addition, data on HIP screening were available for all women. Women were informed that their medical records could be used for research, unless they opposed [START_REF] Cosson | Early screening for gestational diabetes mellitus is not associated with improved pregnancy outcomes: an observational study including 9795 women[END_REF][START_REF] Cosson | Performance of a selective screening strategy for diagnosis of hyperglycaemia in pregnancy as defined by IADPSG/WHO criteria[END_REF]. We analyzed the data anonymously. Our database was declared to the French Committee for computerized data (CNIL: Commission Nationale de l'Informatique et des Libertés, number 1704392v0).

-Screening for and management of hyperglycaemia in pregnancy

In our centre, we have been following the French recommendations for HIP screening, except that our policy is to universally screen every woman, both at the beginning of pregnancy and after 24 weeks of gestation (WG) if prior screening was normal or not done. Early screening during pregnancy is based on FPG measurement. Women with FPG level ≥ 5.1 mmol/L are diagnosed with HIP and immediately managed appropriately [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF]. Those without earlydiagnosed HIP are planned to undergo a 75g OGTT between 24 and 28 WG, with measurement of FPG, 1h-PG and 2h-PG [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF]. International Association of Diabetes Pregnancy Study Group (IADPSG) [START_REF] Panel | International association of diabetes and pregnancy study groups recommendations on the diagnosis and classification of hyperglycemia in pregnancy[END_REF] / World Health Organization (WHO) [START_REF]Diagnostic criteria and classification of hyperglycaemia first detected in pregnancy: a World Health Organization Guideline[END_REF] criteria are used for HIP diagnosis, as they have been endorsed in France [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF]. Accordingly, gestational diabetes mellitus (GDM) is defined by FPG 5.1-6.9 mmol/L and/or 1h-PG ≥ 10.0 mmol/L and/or 2h-PG 8.5-11.0 mmol/L during OGTT, whereas diabetes in pregnancy (DIP) is defined by FPG ≥ 7.0 and/or 2h-PG value ≥ 11.1 mmol/L [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF].

After HIP diagnosis, all women are referred to our multidisciplinary team including a diabetologist, an obstetrician, a midwife, a dietician and a nurse educator and are managed according to French recommendations. They receive individualized dietary advice, education for performing self-monitoring of blood glucose levels six times per day and visit the diabetologist every 2-4 weeks. At the beginning of this educational program, HbA1c level is centrally measured in our hospital (turbidimetric inhibition immunoassay for the in vitro determination of hemoglobin A1c and total hemoglobin in whole blood; Cobas 6000; Roche).

Insulin treatment is initiated when pre-prandial or 2-hour post-prandial glucose levels are respectively above 5.0 or 6.7 mmol/L during follow up, according to the French guidelines [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF]. Obstetrical care also is also managed according to the French recommendations [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF].

-Reference standards and selection criteria

Inclusion criteria were age 18 to 50 years, singleton pregnancies, no personal history of either diabetes or bariatric surgery, no early HIP during this current pregnancy, OGTT performed between 22 and 30 WG.

We then selected among the women who had HIP according to IADPSG/WHO criteria (as described earlier in the article; reference standard) those who had an HbA1c level measurement (additional Figure 1).

-Description of tested algorithm

According to SFD/CNGOF COVID-19 proposal [START_REF] Vambergue | Three alternative ways to screen for hyperglycaemia in pregnancy during the COVID-19 pandemic[END_REF], no OGTT is performed and women with either HbA1c ≥ 5.7% (39 mmol/mol) or FPG ≥ 5.1 mmol/l are diagnosed with HIP.

We then explored whether the results would be similar if selective screening would be applied, which is recommended by SFD/CNGOF proposal. For this sensitivity analysis, we only selected the women who had any of the following risk factors (reference standard in case of selective screening): body mass index ≥ 25 kg/m²; age ≥ 35 years; first-degree relative with history of diabetes; previous pregnancy with HIP or with macrosomic infant [START_REF]Expert consensus on gestational diabetes mellitus. Summary of expert consensus[END_REF].

-HIP-related events

The main predefined endpoint was the occurrence of a HIP-related event. This composite criterion included at least one of the following events: (i) preeclampsia (blood pressure ≥ 140/90 mmHg on two recordings four hours apart and proteinuria at or above 300 mg/24 hours or 3+ on dipstick testing in a random urine sample), (ii) large-for-gestational-age infant (birth weight greater than the 90 th percentile for a standard French population [START_REF] Cosson | Early screening for gestational diabetes mellitus is not associated with improved pregnancy outcomes: an observational study including 9795 women[END_REF][START_REF] Cosson | Performance of a selective screening strategy for diagnosis of hyperglycaemia in pregnancy as defined by IADPSG/WHO criteria[END_REF]), (iii) shoulder dystocia defined as the use of obstetrical manoeuvers (McRoberts manoeuver, episiotomy after delivery of the foetal head, suprapubic pressure, posterior arm rotation to an oblique angle, rotation of the infant by 180 degrees, or delivery of the posterior arm) and neonatal hypoglycaemia, defined as at least one blood glucose value below 2.2 mmol/L during the first two days of life [START_REF] Cosson | Early screening for gestational diabetes mellitus is not associated with improved pregnancy outcomes: an observational study including 9795 women[END_REF][START_REF] Cosson | Performance of a selective screening strategy for diagnosis of hyperglycaemia in pregnancy as defined by IADPSG/WHO criteria[END_REF]. Each one of the previous events was also considered separately, the need for insulin during pregnancy, a preterm delivery (delivery before 37 completed weeks) and admission to a neonatal intensive care unit.

-Statistics

Baseline continuous variables were expressed as mean ± standard deviation. Categorical variables were expressed as frequencies (percentages). To explore the presence of any selection bias, the baseline characteristics of the women who were included were compared to those who were not. To compare continuous variables ANOVA and Chi-squared (X 2 ) test or Fisher-exact test for categorical variables were used. The reference standard was the results of OGTT according IADPSG/WHO criteria. The sensitivity of the COVID-19 proposal for HIP diagnosis was evaluated.

A sensitivity analysis by restricting inclusion to women at high-risk for HIP, as SFD/CNGOF recommend selective screening [START_REF]-guidance-for-maternal-medicine-in-the-evolving-coronavirus-covid-19pandemic[END_REF][START_REF] Vambergue | Three alternative ways to screen for hyperglycaemia in pregnancy during the COVID-19 pandemic[END_REF] was also made. Finally, another sensitivity analysis considering the same statistical analyses only in women with HbA1c measured within four (and not six) weeks after OGTT was also made.

The sensitivities by using different thresholds of FPG or HbA1c to diagnose HIP were also evaluated.

Finally, characteristics and event rates of true positive and false negative HIP diagnoses applying SFD/CNGOF proposal were compared. Student t test or the Mann Whitney test for Gaussian or non-Gaussian continuous variables respectively were used, and chi-squared (X 2 ) test or the Fisher-exact tests for categorical variables.

All tests were two-sided and used a significance level of p value at 0.05. Analyses were conducted using and R 3.6.3 software (www.r-project.org).

Results

-Population characteristics

As shown in the flow chart (Additional Figure 1), 467 women were included, and their characteristics are described in Table 1.

The baseline characteristics of these included women and the non-included women were compared with the ones of the 88 women who had HbA1c measured > 6 weeks after the OGTT and the 441 who had no HbA1c measured (additional Table 1). Globally, the highest 1h-PG and 2h-PG levels during diagnostic OGTT was observed in the study population.

HbA1c level was higher in the women who had HbA1c measured > 6 weeks after OGTT (non-included women) than in those for whom HbA1c was measured within 6 weeks. The included women were also slightly older and were more prone to have had hyperglycaemia in previous pregnancy.

For sensitivity analyses, 397 women with risk factors (selective screening) were included.

Table 2 shows the characteristics of these women.

-Sensitivity of SFD-CNGOF COVID-19 proposal to diagnose HIP cases

Using universal screening, SFD-CNGOF COVID-19 proposal would have identified 266/467 women with HIP (sensitivity 57% [95% confidence interval 52-62]). Out of the 32 women having DIP according to OGTT (reference standard), 9 women would have been classified as not having HIP, 18 women with GDM and 5 women with DIP.

Using selective screening (sensitivity analysis), SFD-CNGOF COVID-19 proposal would have identified 232/397 women with HIP (sensitivity 58% [95% confidence interval 53-64]).

Out of the 30 women having DIP according to OGTT, 8 women would have been classified

as not having HIP, 17 women with GDM and 5 women with DIP. Additional Table 2 shows (i) the sensitivity of HbA1c ≥ 5.7% (39 mmol/mol) (15% [95% confidence interval 12-19]) or FPG ≥ 5.1 mmol/l alone (54% [95% confidence interval 50-59]) for HIP diagnosis and (ii) that the results were globally similar when only women for whom HbA1c was measured within 4 weeks after OGTT were considered (sensitivity analysis).

-Sensitivities applying different thresholds of FPG or HbA1c to diagnose HIP cases

Tables 3 and4 show to what extent applying lower thresholds of FPG (Table 3) or HbA1c (Table 4) would increase the sensitivities of SFD/CNGOF COVID-19 proposal to diagnose HIP cases.

-Characteristics of true positive and false negative cases of HIP applying SFD-CNGOF COVID-19 proposal

Table 1 shows the comparison of HIP true positive and false negative case subgroups with universal screening, while Table 2 shows the results with selective screening (sensitivity analysis). When universal or selective screening were used, lower FPG and HbA1c levels, as well as a lower mean body mass index; and higher 1h-PG and 2h-PG were found in the false negative case subgroup compared to the true positive case subgroup. The percentage of women who needed insulin therapy during pregnancy was also lower.

-Prognosis of true positive and false negative HIP case subgroups applying SFD-CNGOF COVID-19 proposal

The percentage of HIP-related events was similar in true positive and false negative case subgroups of HIP considering universal (Table 1) or selective screening (Table 2). The percentage of each outcome was also similar in both groups in case of universal screening (Table 1). The results were similar when selective screening was used (Table 2, sensitivity analysis).

Discussion

OGTT is the cornerstone of the diagnosis of HIP. Besides its inconvenience and a high variability of 2h-PG [START_REF] Selvin | Short-term variability in measures of glycemia and implications for the classification of diabetes[END_REF], it is time-consuming and not appropriate for social distancing. The results show that the SFD/CNGOF proposal to substitute OGTT for HbA1c and FPG measurement in the context of COVID-19 pandemic has a poor sensitivity to detect HIP. Furthermore, screening based on HbA1c and FPG does not appear to select women with the highest rate of adverse events during pregnancy.

Several studies have explored the accuracy of FPG [START_REF] Agarwal | Gestational diabetes screening: the low-cost algorithm[END_REF][START_REF] Agarwal | Gestational diabetes mellitus: simplifying the international association of diabetes and pregnancy diagnostic algorithm using fasting plasma glucose[END_REF][START_REF] Rüetschi | Fasting glycaemia to simplify screening for gestational diabetes[END_REF] and HbA1c measured between 24 and 28 GW for HIP diagnosis defined according to IADPSG/WHO criteria [START_REF] Ye | The utility of HbA1c for screening gestational diabetes mellitus and its relationship with adverse pregnancy outcomes[END_REF][START_REF] Soumya | HbA1c: A Useful Screening Test for Gestational Diabetes Mellitus[END_REF][START_REF] Rajput | Utility of HbA1c for diagnosis of gestational diabetes mellitus[END_REF][START_REF] Pastakia | Prevalence of gestational diabetes mellitus based on various screening strategies in western Kenya: a prospective comparison of point of care diagnostic methods[END_REF]. Like in our study, FPG measurement alone, with a threshold of 5.1mmol/L, was shown to be not highly sensitive [START_REF] Agarwal | Gestational diabetes screening: the low-cost algorithm[END_REF][START_REF] Agarwal | Gestational diabetes mellitus: simplifying the international association of diabetes and pregnancy diagnostic algorithm using fasting plasma glucose[END_REF][START_REF] Rüetschi | Fasting glycaemia to simplify screening for gestational diabetes[END_REF][START_REF] Pastakia | Prevalence of gestational diabetes mellitus based on various screening strategies in western Kenya: a prospective comparison of point of care diagnostic methods[END_REF][START_REF] Ikomi | Diagnosis of gestational diabetes during the pandemic: what is the risk of falling through the net?[END_REF][START_REF] Gemert Te Van | Gestational diabetes mellitus testing in the COVID-19 pandemic: The problems with simplifying the diagnostic process[END_REF]. For example, a recent study from UK reported that sensitivity of FPG 5.1 mmol/L or more was 63.8% [START_REF] Ikomi | Diagnosis of gestational diabetes during the pandemic: what is the risk of falling through the net?[END_REF]. Regarding the use of HbA1c, when HIP was defined with National Institute for Health and Care Excellence (NICE) criteria, HbA1c ≥ 6.0% (42 mmol/mol) was reported to have a sensitivity of 22% but lower thresholds of HbA1c were not tested [START_REF] Hanna | Diagnosing gestational diabetes mellitus: implications of recent changes in diagnostic criteria and role of glycated haemoglobin (HbA1c)[END_REF]. HbA1c ≥ 5.7% (39 mmol/mol) was reported to have a sensitivity of 73.3% in India [START_REF] Soumya | HbA1c: A Useful Screening Test for Gestational Diabetes Mellitus[END_REF] but only 9% in China [START_REF] Ye | The utility of HbA1c for screening gestational diabetes mellitus and its relationship with adverse pregnancy outcomes[END_REF] to identify women with HIP according to IADPSG/WHO criteria. Overall, the sensitivity of an isolated HbA1c measurement with this threshold is therefore considered to be poor. This may be partially due to physiological changes that occur during pregnancy, including erythrocyte turnover, erythropoietin production and anaemia [START_REF] Mendes | Beyond self-monitored plasma glucose and HbA1c: the role of non-traditional glycaemic markers in gestational diabetes mellitus[END_REF][START_REF] Sinha | Effect of iron deficiency anemia on hemoglobin A1c levels[END_REF]. It has also been reported that HbA1c levels show some variations according to ethnic origin [START_REF] Bleyer | Ethnic variation in the correlation between random serum glucose concentration and glycated haemoglobin[END_REF][START_REF] Herman | Differences in A1C by race and ethnicity among patients with impaired glucose tolerance in the Diabetes Prevention Program[END_REF].

To compensate the lack of sensitivity when using FPG or HbA1c alone at current thresholds, some national societies proposed to associate both measurements to diagnose HIP [START_REF]-guidance-for-maternal-medicine-in-the-evolving-coronavirus-covid-19pandemic[END_REF][START_REF] Vambergue | Three alternative ways to screen for hyperglycaemia in pregnancy during the COVID-19 pandemic[END_REF][START_REF] Kasuga | A new diagnostic strategy for gestational diabetes during the COVID-19 pandemic for the Japanese population[END_REF]. We show here that this strategy has not a good sensitivity either (59%), as recently reported in Japan (39%) [START_REF] Kasuga | A new diagnostic strategy for gestational diabetes during the COVID-19 pandemic for the Japanese population[END_REF]. We found only one published study exploring whether associating both parameters was more sensitive than considering each parameter alone: the area under the curve to identify HIP (defined by Carpenter and Coutsan criteria after 100g OGTT) of FPG, HbA1c and both of them were 0.833, 0.784 and 0.863, respectively [START_REF] Kwon | HbA1c for diagnosis and prognosis of gestational diabetes mellitus[END_REF].

Missing HIP diagnoses would be less deleterious than expected if women with false negative HIP diagnosis were at lower risk for HIP-related adverse events than women with a true positive diagnosis. This could have been expected as both high FPG and HbA1c [START_REF] Ye | The utility of HbA1c for screening gestational diabetes mellitus and its relationship with adverse pregnancy outcomes[END_REF][START_REF] Soumya | HbA1c: A Useful Screening Test for Gestational Diabetes Mellitus[END_REF][START_REF] Dubey | Mid-trimester glycosylated hemoglobin levels (HbA1c) and its correlation with oral glucose tolerance test (World Health Organization 1999)[END_REF] levels after 24 WG are associated with a poor prognosis. Interestingly, our results have shown that the prognosis of false negative and true positive cases was in fact similar.

Considering that (i) all the women included in this study had HIP according to IADPSG/WHO criteria and were therefore managed for this condition and that (ii) not caring for HIP in low-risk women might lead to twice as much events during pregnancy [START_REF]-guidance-for-maternal-medicine-in-the-evolving-coronavirus-covid-19pandemic[END_REF][START_REF] Vambergue | Three alternative ways to screen for hyperglycaemia in pregnancy during the COVID-19 pandemic[END_REF], missing diagnosis for those women might be deleterious. As a matter of fact, in our study, more than 40% of women with false negative diagnoses received insulin therapy in addition to diet and physical activity counselling. The incidence of insulin requirement was also similar in true positive and false negative cases of HIP in the Japanese series [START_REF] Kasuga | A new diagnostic strategy for gestational diabetes during the COVID-19 pandemic for the Japanese population[END_REF]. A recently published study based on Hyperglycaemia and Adverse Pregnancy Outcome dataset [START_REF] Mcintyre | Testing for gestational diabetes during the COVID-19 pandemic. An evaluation of proposed protocols for the United Kingdom, Canada and Australia[END_REF] have evaluated the accuracy of FPG ≥ 5.6 mmol/L and/or HbA1c ≥ 5.7 % (39 mmol/mol) to diagnose HIP, using the NICE guidance as reference Standard (FPG ≥ 5.6mmol/L and/or 2-h plasma glucose ≥ 7.8 mmol/L) [35]. Women whose HIP would remain undetected post COVID-19 (missed HIPs) displayed similar rates of large-for-gestational-age infant, neonatal hypoglycaemia and preterm delivery, but a lower rate of pregnancy-related hypertension, to those with post COVID-19 HIP. To note, women in this study were untreated [START_REF] Mcintyre | Testing for gestational diabetes during the COVID-19 pandemic. An evaluation of proposed protocols for the United Kingdom, Canada and Australia[END_REF]. However, randomized studies are required to draw definite conclusions.

The strengths of our studies include the large number of subjects with HIP and a pragmatic guidance-based approach. The prospectively collected standardized data provide a robust investigational data set and they could apply selective (sensitivity analysis) or universal screening for reference Standard. Our evaluation was limited to women who underwent OGTT in the late second and early third trimester [START_REF] Ye | The utility of HbA1c for screening gestational diabetes mellitus and its relationship with adverse pregnancy outcomes[END_REF][START_REF] Soumya | HbA1c: A Useful Screening Test for Gestational Diabetes Mellitus[END_REF][START_REF] Rajput | Utility of HbA1c for diagnosis of gestational diabetes mellitus[END_REF][START_REF] Pastakia | Prevalence of gestational diabetes mellitus based on various screening strategies in western Kenya: a prospective comparison of point of care diagnostic methods[END_REF][START_REF] Ikomi | Diagnosis of gestational diabetes during the pandemic: what is the risk of falling through the net?[END_REF][START_REF] Gemert Te Van | Gestational diabetes mellitus testing in the COVID-19 pandemic: The problems with simplifying the diagnostic process[END_REF][START_REF] Hanna | Diagnosing gestational diabetes mellitus: implications of recent changes in diagnostic criteria and role of glycated haemoglobin (HbA1c)[END_REF][START_REF] Sinha | Effect of iron deficiency anemia on hemoglobin A1c levels[END_REF][START_REF] Bleyer | Ethnic variation in the correlation between random serum glucose concentration and glycated haemoglobin[END_REF]. Finally, our study could compare not only characteristics but also prognosis of true positive and false negative cases of HIP. We however have to consider while interpreting the results that all included women were cared for HIP in our observational series.

Our study has limitations. Actually, HbA1c level was measured in our centre only in women with HIP according to IADPSG/WHO criteria. Therefore, sensitivity but not specificity, nor positive and negative predictive values of COVID-19 proposals could be investigated.

Another issue could be that HbA1c was not measured at the same time as FPG. In pregnant women, the time course of HbA1c is actually biphasic with a decrease during the second trimester, a nadir at 24 WG and an increase during the last trimester [START_REF] Mendes | Beyond self-monitored plasma glucose and HbA1c: the role of non-traditional glycaemic markers in gestational diabetes mellitus[END_REF][START_REF] Phelps | Biphasic changes in hemoglobin A1c concentrations during normal human pregnancy[END_REF][START_REF] Worth | Glycosylated haemoglobin in normal pregnancy: a longitudinal study with two independent methods[END_REF][START_REF] Law | Translating HbA1c measurements into estimated average glucose values in pregnant women with diabetes[END_REF]. This is the reason why women for whom HbA1c was measured more than 6 weeks after OGTT were not included. Indeed, these non-included women had higher HbA1c level than women who had their HbA1c measured within 6 weeks after the OGTT. Our sensitivity analysis showed that sensitivity was similar when women had their HbA1c measured either within 4 weeks or 6 weeks after OGTT. Finally, around one half of women with HIP had no HbA1c measured but their characteristics were globally similar as those of women who were included.

To conclude, due to their low sensitivities, even during current and future pandemics, we do not recommend the routine application of SFD/CNGOF current proposals, i.e. to use FPG and HbA1c measurement with proposed thresholds as a substitute for OGTT. This proposal should all the more be temporary and other options might be considered [START_REF]ADIPSADSCOVID-19GDMDiagnosisUpdated250420[END_REF][START_REF] Ardilouze | Self-Monitoring of Blood Glucose: A Complementary Method Beyond the Oral Glucose Tolerance Test to Identify Hyperglycemia During Pregnancy[END_REF]. As shown in our study, considering lower FPG [START_REF] Rüetschi | Fasting glycaemia to simplify screening for gestational diabetes[END_REF][START_REF] Pastakia | Prevalence of gestational diabetes mellitus based on various screening strategies in western Kenya: a prospective comparison of point of care diagnostic methods[END_REF][START_REF] Kwon | HbA1c for diagnosis and prognosis of gestational diabetes mellitus[END_REF] and HbA1c thresholds [START_REF] Ye | The utility of HbA1c for screening gestational diabetes mellitus and its relationship with adverse pregnancy outcomes[END_REF][START_REF] Soumya | HbA1c: A Useful Screening Test for Gestational Diabetes Mellitus[END_REF][START_REF] Rajput | Utility of HbA1c for diagnosis of gestational diabetes mellitus[END_REF] would increase sensitivity. However, it would also decrease specificity, which is an issue during pandemics. Indeed, it is critical to practice social distancing not only at OGTT testing centres but also within the health care setting -where false positive cases of HIP would be managed. Limiting the proportion of women addressed for OGTT according to FPG [START_REF] Gemert Te Van | Gestational diabetes mellitus testing in the COVID-19 pandemic: The problems with simplifying the diagnostic process[END_REF][START_REF]ADIPSADSCOVID-19GDMDiagnosisUpdated250420[END_REF] and/or HbA1c level [START_REF] Ye | The utility of HbA1c for screening gestational diabetes mellitus and its relationship with adverse pregnancy outcomes[END_REF][START_REF] Soumya | HbA1c: A Useful Screening Test for Gestational Diabetes Mellitus[END_REF][START_REF] Rajput | Utility of HbA1c for diagnosis of gestational diabetes mellitus[END_REF] might be better options. For example, FPG thresholds of ≤ 4.4 mmol/L have been reported to rule out HIP in 50 to 65% of women with a sensitivity of 80 to 95% [START_REF] Agarwal | Gestational diabetes screening: the low-cost algorithm[END_REF][START_REF] Agarwal | Gestational diabetes mellitus: simplifying the international association of diabetes and pregnancy diagnostic algorithm using fasting plasma glucose[END_REF][START_REF] Rüetschi | Fasting glycaemia to simplify screening for gestational diabetes[END_REF]. Also, as suggested in the case of a personal history of bariatric surgery [START_REF] Cosson | Diagnosis and management of patients with significantly abnormal glycaemic profiles during pregnancy after bariatric surgery: PRESAGE (Pregnancy with significantly abnormal glycaemic exposure -bariatric patients)[END_REF], women could self-monitor their blood glucose at home but this implies education to do so [START_REF] Ardilouze | Self-Monitoring of Blood Glucose: A Complementary Method Beyond the Oral Glucose Tolerance Test to Identify Hyperglycemia During Pregnancy[END_REF]. Thus, there is an urgent need to validate new methods to diagnose and manage HIP in order to be ready to face future pandemics/lockdowns.

Preconception body mass index (kg/m 2 ) 26. The threshold of HbA1c to diagnose hyperglycaemia in pregnancy is constant (≥ 5.7%, 39 mmol/mol), while the threshold of fasting plasma glucose value varies. The threshold of fasting plasma glucose to diagnose hyperglycaemia in pregnancy is constant (≥ 5.1 mmol/L), while the threshold of HbA1c varies.

Table 1 :

 1 Characteristics of the women by true positive and false negative HIP diagnoses applying universal screening.

	Total	True positive	False negative	p
		diagnoses	diagnoses
	n=467	n=266	n=201

Table 2 :

 2 Characteristics of the women by true positive and false negative HIP diagnoses applying selective screening (sensitivity analysis). DIP: diabetes in pregnancy; GDM: gestational diabetes mellitus; HIP: hyperglycaemia in pregnancy; LGA: large for gestational age; OGTT: oral glucose tolerance test; WG: weeks of gestation HIP-related event is a composite endpoint: preeclampsia or LGA infant or shoulder dystocia or neonatal hypoglycaemia Selective screening according to SFD/CNGOF guidelines: body mass index ≥ 25 kg/m²; age ≥ 35 years; firstdegree relative with history of diabetes; previous pregnancy with HIP or with macrosomic infant.

	Total	True positive	False negative	p
		diagnoses	diagnoses
	n=397	n=232	n=165

*Yes vs No

Table 3 :

 3 Sensitivity applying different thresholds of fasting plasma glucose, with constant HbA1c threshold, to diagnose cases of hyperglycaemia in pregnancy.

	Fasting plasma glucose threshold, mmol/L	Sensitivity [95% confidence interval]
	4.6	0.78 [0.74 -0.82]
	4.7	0.76 [0.72-0.80]
	4.8	0.70 [0.66-0.74]
	4.9	0.66 [0.61-0.70]
	5.0	0.60 [0.56 -0.65]
	5.1	0.57 [0.52 -0.62]

Table 4 :

 4 Sensitivity applying different thresholds of HbA1c, with constant fasting plasma glucose threshold, to diagnose cases of hyperglycaemia in pregnancy

		Sensitivity [95% confidence
	HbA1c threshold, % (mmol/mol)	interval]
	5.0 (31)	0.80 [0.76 -0.83]
	5.1 (32)	0.76 [0.72 -0.80]
	5.2 (33)	0.70 [0.66 -0.74]]
	5.3 (34)	0.67 [ 0.63-0.71]
	5.4 (36)	0.64 [0.60-0.68]
	5.5 (37)	0.62 [0.57 -0.66]
	5.6 (38)	0.57 [0.54-0.63]
	5.7 (39)	0.57 [0.52-0.62]

Acknowledgements

Emmanuel Cosson and Lionel Carbillon are the guarantors of this work. The authors thank Lilly France for funding. The authors do not report any conflict of interest. Especially, apart funding, Lilly France did not participate in any part of this study. We thank Sylvie Picard, MD, PhD, Dijon, France, for her help in the preparation of the English manuscript and Didier André, AP-HP, Unité de Recherche Clinique GHU-SSPD, for data management. Finally, we thank Phuc Thu Trang Nguyen, AP-HP, Unité de Recherche Clinique St-Louis-Lariboisière, Université Denis Diderot, Paris, France for help in collecting HbA1c results.