Alexandre Richard
email: alexandre.richard@safrangroup.com-florent.rougier@safrangroup.com

Florent Rougier

Xavier Roboam

Nicolas Roux

Hubert Piquet

Hubert Piquet Ac Electrical

AC electrical architecture propelling systems for aeronautics

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Introduction

With the evolution of aircraft propulsion towards greener solutions, fully electrical configurations are more and more studied. Most architectures are proposed around a DC bus with or without hybridization with auxiliary electric sources (batteries or fuel cells) [1]. They need AC/DC and DC/AC converters to interface AC electric generators and motors. This paper proposes a "power electronic-less" architecture that directly links through an AC bus two PMSM or more, one in generator mode driven by a gas turbine and the other one(s) in motor mode driving one (or several) propeller(s). This would enable suppressing power electronics thus making potential savings in mass and costs. However, this AC architecture may be too "rigid" to be implemented; indeed removing power electronics blocks creates a constraint between the speeds of the two PMSM, which raises questions on the stability and the dynamic behaviour of this architecture.

Development of the analytical model

In order to be as close as possible to the simple (single channel) application case considered here, the studied system is composed of the electrical architecture previously described to which is added a representation of a speed controlled gas turbine. This addition is done to take in to account the gas turbine that is linked to the system and the influence it could have on it. An IP controller thus represents the gas turbine, as this structure enable to depict the behaviour of a gas turbine from the point of view of the electrical architecture. As presented in Figure 2 this system has therefore two inputs: -The speed set point as a gas turbine input (𝜔 The document will now focus on the electrical part of the system that can be seen in Figure 3. This approach has as starting point the Behn-Eschenburg model of the electric architecture, putting end-to-end the Behn-Eschenburg models of the two PMSM. In order to establish the model, the two PMSM are firstly considered separately in their particular reference frame and at first, their ABC reference frame (generator equations (2) and motor equations (3)):

𝑈 𝑎𝑏𝑐 = -𝑅 𝑔 𝐼 𝑎𝑏𝑐 + 𝑑 𝑑𝑡 Φ g abc ; Φ g 𝑎𝑏𝑐 = -𝐿 𝑔 𝐼 𝑎𝑏𝑐 + 𝜙 𝑎 𝑔 (2) 𝑈 𝑎𝑏𝑐 = 𝑅 𝑚 𝐼 𝑎𝑏𝑐 + 𝑑 𝑑𝑡 Φ m abc ; Φ m 𝑎𝑏𝑐 = 𝐿 𝑚 𝐼 𝑎𝑏𝑐 + 𝜙 𝑎 𝑚 (3) With 𝜙 𝑎 𝑖 = 𝜙 𝑎 𝑖 ( cos (𝜃 𝑖 ) cos (𝜃 𝑖 - 2𝜋 3 ) cos (𝜃 𝑖 - 4𝜋 3
) )

the permanent magnet flux, 𝑅 𝑖 = (

𝑅 𝑖 0 0 0 𝑅 𝑖 0 0 0 𝑅 𝑖 ), 𝐿 𝑖 = ( 𝐿 𝑖 𝑎𝑎 𝑀 𝑖 𝑎𝑏 𝑀 𝑖 𝑎𝑐 𝑀 𝑖 𝑏𝑎 𝐿 𝑖 𝑏𝑏 𝑀 𝑖 𝑏𝑐 𝑀 𝑖 𝑐𝑎 𝑀 𝑖 𝑐𝑏 𝐿 𝑖 𝑐𝑐
) and 𝜃 𝑖 = 𝜔 𝑖 𝑡 with 𝜔 𝑖 the pulsation. The system is then described thanks to alternative values. It is therefore necessary to carry changes of reference frame with a Park's transformation in order to describe the system with continuous quantities in the reference frame DQ0. After using the Park's transformation the equations describing the generator (( 4) & (5)) and the motor (( 6) & (7)) are expressed in the particular reference frame DQ0 of each PMSM (' g ' for generator and ' m ' for the motor) :

𝑈 𝑑𝑞0 𝑔 = -𝑅 𝑔 𝐼 𝑑𝑞0 𝑔 + 𝑑 𝑑𝑡 (Φ g 𝑑𝑞0 𝑔 ) + 𝜔 𝑔 ( 0 -1 0 1 0 0 0 0 0 ) Φ g 𝑑𝑞0 𝑔 (4) Φ g 𝑑𝑞0 𝑔 = -𝐿 𝑔 𝐼 𝑑𝑞0 𝑔 + 𝜙 𝑎 𝑔 ( 1 0 0 ) (5) 𝑈 𝑑𝑞0 𝑚 = 𝑅 𝑚 𝐼 𝑑𝑞0 𝑚 + 𝑑 𝑑𝑡 (Φ m 𝑑𝑞0 𝑚 ) + 𝜔 𝑚 ( 0 -1 0 1 0 0 0 0 0 ) Φ m 𝑑𝑞0 𝑚 (6) Φ m 𝑑𝑞0 𝑚 = 𝐿 𝑚 𝐼 𝑑𝑞0 𝑚 + 𝜙 𝑎 𝑚 ( 1 0 0 ) (7)
Until this point, the two PMSM are described separately in their particular DQ0 reference frames.

Consequently, the two equations can not be linked at this point and in order to describe the whole system, the same DQ0 reference frame must be used to couple both machine electrical equations. The DQ0 reference frame of the generator is chosen to this common DQ0 reference frame as it can be used in a similar way in the case where there would be several motors, it also enables to describe easily the position of motor compared to the generator with only the variable 𝛿. This means that the equations of the motor have to undergo a transformation from the DQ0 reference frame of the motor to the DQ0 reference frame of the generator. Finally, after simplifications, the equations of the generator and the motor in the DQ0 reference frame of the generator can be expressed as follows: The two systems of equations being in the same DQ0 reference frame, it is possible to couple them finally obtaining: This result in having: In the same way, the equations from the controller standing for the gas turbine have to be added (the case of an IP controller is presented here) according to the following scheme :

𝑈 𝑑𝑞0 𝑔 = -𝑅 𝑔
(𝐿 𝑔 + 𝐿 𝑚 ) 𝑑 𝑑𝑡 (𝐼 𝑑𝑞0 𝑔 ) = -(𝑅 𝑔 + 𝑅 𝑚 )𝐼 𝑑𝑞0 𝑔 -(𝐿 𝑔 + 𝐿 𝑚 )𝜔 𝑔 ( 0 -1 0 1 0 0 0 0 0 ) 𝐼 𝑑𝑞0 𝑔 -
𝑇 𝑒𝑚 𝑚 =
+ - Ωg Ωgref TTAG 1 TIPp kIP
𝑉 𝑑𝑞0 𝑔 = -𝐿 𝑔 𝑑 𝑑𝑡 (𝐼 𝑑𝑞0 𝑔 ) -𝐿 𝑔 𝜔 𝑔 ( 0 -1 0 1 0 0 0 0 0 ) 𝐼 𝑑𝑞0 𝑔 + 𝜔 𝑔 𝜙 𝑎 𝑔 ( 0 1 0 ) (17)
This result in having: Thanks to all these equations, the state model ( 1) is complete and can describe the performances of the AC architecture.

𝑇 𝑒𝑚 𝑔 = 3𝑝 𝑔 2 𝛷 𝑎 𝑔 (

Stability Study: linearized approach with root locus

After establishing the analytical model, the main idea is to study the stability of the AC architecture thanks to the analytical model using the root-locus analysis to characterize possible oscillations or stall situations. The objective of this approach is to find a stability criterion or at least to clarify the conditions of use that would enable a system operation without any instability.

As it can be seen in the development above, the state model is non-linear and needs first to be "small signal linearized" before using the root locus approach.

Once the system is linearized around a steady state point, the root-locus can be displayed as on the After sensibility studies done by varying main parameters (electrical, mechanical and input parameters) it is possible to identify the origins of each pole expressed from the state model.

In the example of Figure 6, a variation of inductances values is done; one pair of poles varies with a high intensity compared to the others. This indicates that this pair of poles is linked to the electrical equations describing the evolution of currents, seen previously. Due to its position and its evolution according to parametric variations, the pole pairs that can create oscillations and potentially instabilities are the ones associated to the mechanical domain (in green on the previous figure). For instance, the previous figures were established given a particular steady state point.

The idea is now to study the evolution of the poles with respect to the system operation at steady state and to compare this evolution to a transient time simulation of the system crossing the corresponding steady states.

The evolution of the pole pair related to the mechanical

Figure 1 :

 1 Figure 1 : DC (left) -AC (right) architectures for a single propeller channel

Figure 3 :

 3 Figure 2 : Considered system

Figure 5 .

 5 On the Figure 5, six poles are visible as the state model is a six order model.

Figure 5 :Figure 6 :

 56 Figure 5 : Root locus display

Figure 7 :

 7 Figure 7 : Association of the poles with the physical field (red : electrical, green : mechanical, black : control of the gas turbine)From parameter sensitivity analysis, it is possible to associate the different pole pairs to the different domainselectrical, mechanical, control of the gas turbine. The results are presented in Figure7.Due to its position and its evolution according to parametric variations, the pole pairs that can create oscillations and potentially instabilities are the ones associated to the mechanical domain (in green on the previous figure). For instance, the previous figures were established given a particular steady state point.The idea is now to study the evolution of the poles with respect to the system operation at steady state and to compare this evolution to a transient time simulation of the system crossing the corresponding steady states.The evolution of the pole pair related to the mechanical

of the currents of the AC bus (expressed in a Park's reference frame: 𝐼 𝑑 , 𝐼 𝑞 ), both common for generator and motor - The speeds of the generator and of the motor (𝜔 𝑔 , 𝜔 𝑚

		𝑔 𝑟𝑒𝑓 )
	-	The load torque set point as a propeller input
		(𝛤 𝑐ℎ )
	And six state variables:
	-	The two components )
	-	The load torque of the gas turbine (GT) (𝑇 𝑇𝐴𝐺 )
	-	The difference between rotor positions of
		generator and motor named the rotor angle
		(𝛿)
	The objective is to obtain a state model representation
	in the following form:
			𝐼 𝑑		𝐼 𝑑
		𝑑 𝑑𝑡	𝐼 𝑞 𝛿 𝜔 𝑚 𝜔 𝑔	= 𝐴	𝐼 𝑞 𝛿 𝜔 𝑚 𝜔 𝑔	+ 𝐵 ( 𝜔 𝑔 𝑟𝑒𝑓 𝛤 𝑐ℎ	) (1)
		(	𝑇 𝑇𝐴𝐺 )	(	𝑇 𝑇𝐴𝐺 )

  𝑇 𝑇𝐴𝐺 -𝑇 𝑒𝑚 𝑔 ) avec 𝐽 𝑒𝑞 𝑔 = 𝐽 𝑔 + 𝐽 𝑇𝐴𝐺(15)

	𝑑𝜔 𝑔 𝑑𝑡 (𝑑𝑇 𝑇𝐴𝐺 = 𝑝 𝑔 𝐽 𝑒𝑞 𝑔 𝑑𝑡 = 𝑘 𝐼𝑃 𝑇 𝐼𝑃	*	𝜔 𝑔 𝑟𝑒𝑓 𝑝 𝑔	-	𝑘 𝐼𝑃 𝑇 𝐼𝑃	*	𝜔 𝑔 𝑝 𝑔	-	𝑘 𝐼𝑃 𝑝 𝑔	*	𝑑𝜔 𝑔 𝑑𝑡	(16)
	With 𝑇 𝑒𝑚 𝑔 =	3𝑝 𝑔 2𝜔 𝑔	𝑅𝑒(𝑉 𝑑𝑞0 𝑔 𝐼 𝑑𝑞0 𝑔 * ) and

+ -Figure 4 Speed controller representing the GT

mode is displayed below for a range of steady states: After the study of this approach, it appears that the root locus approach coincides under approximations of few radians per second with any transient simulations.

For example, on these figures, the oscillations coincide with the passing in the right-hand plane with an approximation of a few radians per second (comparison here on the motor mechanical speed), approximation mostly due to the "small signal linearization" that gives the root locus approach : From this model, it has been analysed that oscillations and sometimes system stall may appear with this architecture, certainly because of the stiffness of this AC architecture. These situations are depending on the parameters of the AC architecture (both machines but also especially the controller parameters driving the gas turbine).

In order to describe these oscillations a stability analysis is derived thanks to a small signal linearization of the previous state model allowing to display the system root-locus. The complete study of this linearized approach allows to conclude that it is appropriate for a first characterization of the stability of this AC architecture.

Beyond that methodological aspect, one can conclude so far that stable operation range can be reached for this simple AC architecture, the stable operation being depending to the parametrization of the system. The future issue is then to prove that a safe and stable operation can be obtained by setting an appropriate sizing of both machines and also by adjusting the speed controller.


Fabien Halter

Chaimae Bariki

Christian Chauveau

Heinz Pitsch

Joachim Beeckman

Joachim Beeckman Towards

Towards a universal method for flame speeds determination

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Energy production from most renewable energy sources, such as wind, solar, and water (wave, tidal), is typically intermittent, thus energy storage is necessary. Chemical storage as one option is more economical in comparison to batteries. Energy carriers such as ammonia and methanol could be considered. The combustion of such energy vectors, which is the most effective way to use them as an energy source, is a key technology. Challenges, e.g. low flammability, heat release rate and emission, can be overcome by knowing the dynamics and chemistry of ammonia combustion. Flame speed is a sizing metric of a fuel to be considered in designing energetic industrial systems such as IC engines and gas turbines. Indeed, this physicochemical property of a combustible mixture notably influences the heat release rate and flame stabilization. Moreover, its significance makes it a major target for the validation of kinetic mechanisms. For example, the flame speed of ammonia or ethanol is around one fourth compared to traditional hydrocarbon fuels. Flame stabilization difficulties, possibly encountered in industrial systems, can be solved by technical adjustments (i.e. air preheating, compression ratio increase, oxygen-enriched air, co-firing with methane or syngas). The accuracy of experimental flame speed determination (target data for numerical schemes) for slow flames is the main issue to be addressed here. The ultimate goal is to characterize the reactivity of slow flames over a wide range of unexplored pressure and temperature conditions (p > 1 MPa & T > 500 K) with the following objectives:

-Ob1 Evaluate radiation effects for weakly reactive fuels using two complementary setups.

-Ob2 Evaluate buoyancy effects on the determination of flame speed for low reactivity mixtures. -Ob3 Propose semi-physical models to subsequently correct all data available in the literature for slow flames and evaluate potential applicability for fuels with comparable combustion physics (e.g. refrigerants). The research group composed of ITV (RWTH Aachen University) and ICARE (CNRS University of Orléans) have developed experimental facilities to tackle these scientific objectives. Flame speed measurements under microgravity conditions will be performed both in the Bremen drop tower facility and in the airplane. The drop tower is operated by ZARM, the Center of Applied Space Technology and Microgravity of the University Bremen. The main objective of this study is to investigate the reactivity of flames strongly influenced by buoyancy under earth gravitational conditions (1 g), which can be obtained by varying the oxygen content in the oxidizer mixture and approaching the fuel-rich flammability limit. The gravitational force can lead to a deformation of the initially spherical flame shape, making the determination of laminar flame speed, a fundamental metric of a combustible mixture, quite challenging. The upward motion of buoyant flames induces a complex flow resulting in flames with mushroom-like shapes. Therefore, the validity of the existing flame speed extraction techniques for slowly propagating flames is unclear. Consequently, the accuracy of the experimental data obtained under normal gravity should be addressed. By suppressing the effect of buoyant flame motion under microgravity (µg), laminar flame speeds can be extracted accurately. The comparison of flame speed data obtained with and without gravitational forces will lead to the development of novel post-processing routines for buoyant flames. To achieve this goal, experiments were conducted during a collaborative campaign between RWTH Aachen University, Germany, and ICARE, France, using two fuels: (i) ethanol as a representative of a typical high-temperature branch reacting fuel, (ii) and diethoxymethane being a bio-hybrid fuel and for its multistage ignition behavior.

Colloque Annuel du GDR MFA 2799 -02 au 04 Novembre 2021 -Carry-Le-Rouet

Preliminary Results

Experiments were conducted in a pressure-release-type dual chamber, which is adequate for aircraftrelated safety requirements. A conventional shadowgraphy system is used to visualize the flame morphology, as can be seen in Fig. 1(left). After spark ignition, the flame propagates spherically under microgravity conditions, whereas it is highly deformed under terrestrial gravity (cf. Fig. 1(right)). Due to the density difference between the burned and unburned gas, these slowly propagating flames develop an upward motion, and the initially spherical flame deforms and attains a mushroom-like shape. The upward motion of buoyant flames induces a complex flow. In contrast to ideally spherical flames, different points of the flame front propagate with a different absolute speed, making the extraction of laminar flame speed extremely difficult. The methods found in the literature to assess buoyant flames differ quite significantly between studies and seem somehow arbitrarily chosen, as recently demonstrated by Berger et al. [1]. To determine the flame radius R from the raw images recorded by the high-speed camera, the surface of the spherical flame is projected, and the instantaneous flame front radius can be deduced. The evolution of flame propagation speed as a function of flame stretch for ethanol flames at various equivalence ratios is shown in Fig. 2. By increasing the equivalence ratio, the burned-gas Markstein length ℒ 𝑏 , which represents the sensitivity of flame speed to stretch, decreases significantly. Considering these low flame speeds, radiation effects are of major concern as previously highlighted by ITV and should be considered, as it was shown in the previous study of Hesse et al. [2]. Further work should be done to optimize the analytical model.

Figure 1 :

 1 Figure 1: Exemplary images of fuel-rich ethanol flame under µg (left) and highly buoyant at the same conditions under terrestrial gravity. (𝜙 = 1.4, T 0 = 373 K, P 0 = 3 bar, oxidizer: 12% O 2 -88% N 2 in volume).

Figure 2 :

 2 Figure 2: Evolution of burned-gas flame speed as a function of flame stretch for ethanol flames at various equivalence ratios. (T 0 = 373 K, P 0 = 5 bar, oxidizer: 12% O 2 -88% N 2 in volume).


S Madad

PhD) b C Guiguet-Auclair

PhD) a , D Gallot

M-P Celse

MD, PhD F Vendittelli

MD) a . Anne Debost-Legrand

Validation of the 'EPICES' social deprivation score in a population of women who have just given birth: a French cross-sectional study

Keywords: social deprivation, EPICES score, pregnancy, threshold

Objectives: To assess the diagnostic performance of the EPICES score for identifying social deprivation during pregnancy in a population of women in the immediate postpartum period.

Study design: This cross-sectional survey took place between 05 June and 05 August 2017, among women who had just given birth in either of the maternity units in Clermont-Ferrand, France.

Methods: A self-administered questionnaire was completed by women. The questionnaire came in two parts: the EPICES index and the criteria for social deprivation defined by French law. These criteria were chosen to define the reference standard. The women were classified in two groups, living in precarious circumstances or not, according to the criteria defined by the French law (reference standard). To determine the most relevant threshold of the EPICES score, the precision associated with the threshold (the fraction of those predicted positive who are true positives: positive predictive value) was balanced with its sensitivity. EPICES scores above the threshold were classified as deprived, those below as non-deprived.

Results: Of the 947 women who gave birth during the study period, 700 (73.9%) completed the self-administered questionnaire. The best trade-off between precision and sensitivity was obtained with a threshold of 22. For this threshold value, the positive predictive value was 42.3% and the sensitivity 70.3%.

Conclusions:

The EPICES score with a threshold validated in the population of pregnant women is a useful, rapid and easy to use tool that makes it possible to identify maternal deprivation at an individual level.

Introduction

Several studies have established an association between social deprivation and adverse perinatal outcomes. [1][2][START_REF] Charreire | Poor prenatal care in an urban area: a geographic analysis[END_REF][START_REF] Cosson | Psychosocial deprivation in women with gestational diabetes mellitus is associated with poor fetomaternal prognoses: an observational study[END_REF][5][START_REF] Hesselman | Neighborhood deprivation and adverse perinatal outcomes in Sweden: A populationbased register study[END_REF][START_REF] Kapaya | Deprivation and poor psychosocial support are key determinants of late antenatal presentation and poor fetal outcomes--a combined retrospective and prospective study[END_REF][START_REF] Vos | Deprived neighborhoods and adverse perinatal outcome: a systematic review and meta-analysis[END_REF] Social deprivation has been measured with indices constructed by small geographic areas. 2,[START_REF] Charreire | Poor prenatal care in an urban area: a geographic analysis[END_REF][START_REF] Deguen | Using a Clustering Approach to Investigate Socio-Environmental Inequality in Preterm Birth-A Study Conducted at Fine Spatial Scale in Paris (France)[END_REF][START_REF] Kihal-Talantikite | Adverse Birth Outcomes as Indicators of Poor Fetal Growth Conditions in a French Newborn Population-A Stratified Analysis by Neighborhood Deprivation Level[END_REF] They describe the associations between the characteristics of the residential area, its match to the geographic availability of care and health-care facilities, while trying to control for the role of socioeconomic factors in the analysis of the local environment's effects on its inhabitants' health with consistent results about the prevalence of deprivation. [START_REF] Vos | Deprived neighborhoods and adverse perinatal outcome: a systematic review and meta-analysis[END_REF] These indices, generally global, are on the whole poorly adapted to mother-child pairs. Other indices have been created from essentially economic data that do not take into account the social dimension of deprivation, as defined by Wresinski as the lack of one or more securities enabling people to assume their elementary responsibilities and enjoy their fundamental rights. [START_REF] Opatowski | New index of social deprivation during pregnancy: results from a national study in France[END_REF] Thus the concept of deprivation appears to be multidimensional and may be difficult to be defined. In France, an individual score of social deprivation was developed in 2002 and called the EPICES index (Evaluation de la Précarité et des Inégalités de santé dans les Centres d'Examen de santé, Evaluation of Deprivation and Inequalities in Health Examination Centers) . This reliable index, validated on cohort of 197 389 persons, is composed of 11 questions regarding material, psychosocial and social problems. [START_REF] Labbe | A new reliable index to measure individual deprivation: the EPICES score[END_REF][START_REF] Sass | Le score Epices : un score individuel de précarité. Construction du score et mesure des relations avec des données de santé, dans une population de 197 389 personnes[END_REF][START_REF] Sass | Comparaison du score individuel de précarité des Centres d'examens de santé, EPICES, à la définition socio-administrative de la précarité[END_REF] It has been used in several studies but very little among pregnant women. [START_REF] Bihan | Factors associated with screening for glucose abnormalities after gestational diabetes mellitus: baseline cohort of the interventional IMPACT study[END_REF][START_REF] Lelong | Is individual social deprivation associated with adverse perinatal outcomes? Results of a French multicentre cross-sectional survey[END_REF] In one study of women with gestational diabetes, with EPICES score greater than 30.17 (validated threshold), had more complications. [START_REF] Bihan | Factors associated with screening for glucose abnormalities after gestational diabetes mellitus: baseline cohort of the interventional IMPACT study[END_REF] Another study found that prenatal care was poorer among women with higher EPICES scores and that the three principal causes of unfavourable pregnancy outcomes (preterm delivery, gestational diabetes and pregnancy-related hypertension) were more frequent among women in precarious situations. [START_REF] Lelong | Is individual social deprivation associated with adverse perinatal outcomes? Results of a French multicentre cross-sectional survey[END_REF] Nonetheless these studies have mainly targeted subpopulations already at risk among pregnant women. Moreover, the threshold of deprivation used in these studies was that established at the initial validation study of the EPICES score, conducted in a population consulting at a health insurance care centre. [START_REF] Sass | Le score Epices : un score individuel de précarité. Construction du score et mesure des relations avec des données de santé, dans une population de 197 389 personnes[END_REF][START_REF] Sass | Comparaison du score individuel de précarité des Centres d'examens de santé, EPICES, à la définition socio-administrative de la précarité[END_REF] Consequently, the prevalence of deprivation among pregnant women in the general population is not well estimated, mainly because most estimates use only economic indicators. The hypothesis of our study is that the threshold for identifying women in social vulnerability is different from that determined in a population consulting in a national health insurance health care centre.

Our principal objective was to assess the diagnostic performance of the EPICES score for identifying social deprivation during pregnancy in a population of women in the immediate postpartum period.

Materials and methods

Study design

This cross-sectional survey was carried out between 05 June and 05 August 2017, among women who had just given birth in either of the maternity units (one public level III and one private level II) in Clermont-Ferrand, France.

A medical practitioner (obstetrician or midwife) informed women about the study, distributed the questionnaire and collected it each day during the study period in the postpartum units, before women were discharged.

Study Population

Women were eligible if they were older than 18 years old, spoke and read French fluently, and had provided written informed consent to participate. Women with terminations of pregnancy for medical indications were not included.

Questionnaire

A questionnaire was self-administered to participants and came in two parts: the EPICES index and the criteria for socioeconomic deprivation defined by the French law. The French law in article 2 of the decree of 1992 defined deprived populations as young people aged 16 to 25 years old excluded from the school system in a process of professional integration, unemployed, beneficiaries of the active solidarity income, people on solidarity employment contracts or consolidated employment contracts, homeless (1998), and beneficiaries of universal health coverage (for very low-income individuals) (2000). [START_REF]Ministère des Affaires sociales[END_REF] These criteria were chosen to define the reference standard in our study because they create eligibility for social rights and a specific pathway for care for the population recognised as deprived in France.

The EPICES index is composed of 11 binary items on marital status (1 item), health insurance status (1 item), economic status (3 items), family support (3 items) and leisure activities during the last 12 months (3 items) (supplementary file 1). The EPICES score is calculated by adding the coefficient for question to the intercept whenever the answer is "yes". It ranged from 0 to 100, from the lowest to the highest level of deprivation.

Statistical analysis

Categorical data are presented as frequencies and percentages and continuous variables as means and standard deviations (SD). The distribution of the EPICES score was also evaluated using quintiles, the first quintile (Q1), corresponding to the least deprived women, to the fifth quintile (Q5) for those most deprived. The quintiles used were those already used in the literature as the distribution of the score is the same since scoring was established following the initial quotation. [START_REF] Sass | Le score Epices : un score individuel de précarité. Construction du score et mesure des relations avec des données de santé, dans une population de 197 389 personnes[END_REF] The women were classified in two groups: living in precarious circumstances or not, according to the criteria defined by the French law (reference standard). These groups were compared using Chi-square or Fisher's exact tests for categorical variables, Student's t tests for normally distributed continuous data and Mann-Whitney non-parametric tests otherwise.

To determine the most relevant threshold of EPICES score to discriminate women according deprivation, precision and recall curve was used as the groups were imbalanced in order to display the trade-off between precision and sensitivity (also called recall) over all possible test threshold values. [START_REF] Saito | The precision-recall plot is more informative than the ROC plot when evaluating binary classifiers on imbalanced datasets[END_REF] The precision associated with the threshold (the fraction of those predicted positive who are true positives, that is, the positive predictive value, PPV) was balanced with its sensitivity (= recall: fraction of true positives who are predicted to be positive). If no threshold simultaneously achieved good sensitivity and good precision, then we assessed a threshold that identified a subset of situations with a higher-than average rate of being at risk for deprivation. Figures were plotted showing both enrichment (ratio of the precision associated with the threshold to the mean positive rate) and sensitivity as functions of the threshold. [START_REF] Sing | ROCR: visualizing classifier performance in R[END_REF] Then we evaluated the results of the threshold by studying the confusion matrix. Women with EPICES scores above the threshold were classified as deprived, those below as non-deprived.

All analyses were performed using R software (R 4.0.2, ggplot2, ROCR, and grid packages) and conducted at a two-sided alpha=0.05 significance level.

Results

Of the 947 women who gave birth during the study period, 700 (73.9%) completed the selfadministered questionnaire. According to the reference standard (social deprivation defined by the French law) 145 of these 700 women, that is, 20.7%, were classified as deprived.

The socio-demographic characteristics of participants are presented in Table 1. The mean age of the participants was 32.5 years (±5.0), 52.3% had at least post-secondary education, 95.3% were living with a partner, and 79.3% were born in metropolitan France. The mean preconception BMI was 23.7 (± 5.4) and 15.0% were nulliparous. The characteristics of pregnancy and delivery are described in Table 2. We note that 35.4% had a pregnancy-related disorder, that the mean term at delivery was 39.4 (± 2.2) weeks, and that 16.4% had a caesarean delivery. The new-born characteristics are presented in Table 3. Mean birth weight was 3256.4 g (± 508.4) and 48.4% of the infants were being exclusively breast-fed at discharge (Table 3).

Deprived women according to the reference standard were compared to non-deprived ones.

Deprived women were younger (P=0.0003), had a lower educational level (P<0.0001), and were more often unemployed (P<0.0001), more often lived alone (P<0.0001), , and more often born abroad (P<0.0001). The proportion of obesity (BMI>30 kg/m 2 ) was higher among deprived women (P=0.003). They had smoked more frequently at the beginning of their pregnancy (P=0.04). Finally, the proportion of multiparous was higher among them (P<0.0001) (Table 1). The women socially deprived had more frequent pregnancy-related disorders (P=0.009) and more preterm births (P=0.005) (Table 2). Non-deprived mothers had small-for-gestational-age infants (P=0.01) more frequently compared to deprived ones, and breastfed more often (P=0.01) (Table 3).

The EPICES index was compared between deprived and non-deprived groups (Table 4). The mean EPICES score was significantly higher in the deprived group (P<0.0001). The distribution between the two groups in quintiles varied significantly: in the non-deprived group, 65.7% of the women had a score in the first two quintiles (i.e., the lowest EPICES scores), while in the deprived group, 61.4% had a score in the last two quintiles (i.e., the highest EPICES scores) (P<0.0001). Distributions of the items of the EPICES index were significantly different between deprived and non-deprived groups (P<0.0001). More specifically, more than 50% of deprived women did not have leisure activity including physical activity and more than 80% of them were not homeowner. Finally, nearly half of the deprived women had financial difficulties during the month. For non-deprived women, less than 30% of them were in those situations (no leisure activity, not owner and financial difficulties).

The best trade-off between precision and sensitivity was obtained with a threshold of 22 (Figures 1 and2). For this threshold value, the sensitivity was 70.3% and the positive predictive value 42.3% (Table 5).

Discussion

Our study enabled us to validate a threshold score for the EPICES index to use during pregnancy. A sensitivity of 70% was obtained for a threshold value of 22. This sensitivity is lower than that calculated during the initial validation of the EPICES score, which was 80%. This difference is probably due to the score's initial validation within a specific population. [START_REF] Sass | Le score Epices : un score individuel de précarité. Construction du score et mesure des relations avec des données de santé, dans une population de 197 389 personnes[END_REF] more than 20% of the deprived women according to the reference standard had a score lower than 30.17; this may be associated with their greater social and family isolation. That is, the women defined as deprived more often lived alone; [START_REF] Kapaya | Deprivation and poor psychosocial support are key determinants of late antenatal presentation and poor fetal outcomes--a combined retrospective and prospective study[END_REF][START_REF] Opatowski | New index of social deprivation during pregnancy: results from a national study in France[END_REF] questions about the family environment on the EPICES score therefore had a greater weight in the calculation of the score in this population of mothers.

To our knowledge, the criteria of internal and external validity of other deprivation indices have never been compared with an administrative reference standard. The other social deprivation indices explore the marital situation but not the family structure or the social fabric of precarious populations. 2,[START_REF] Charreire | Poor prenatal care in an urban area: a geographic analysis[END_REF] That is, geographic indices are based on ecological data about the neighbourhood, which makes it difficult to take their individual variability into account. [START_REF] Hesselman | Neighborhood deprivation and adverse perinatal outcomes in Sweden: A populationbased register study[END_REF][START_REF] Deguen | Using a Clustering Approach to Investigate Socio-Environmental Inequality in Preterm Birth-A Study Conducted at Fine Spatial Scale in Paris (France)[END_REF][START_REF] Kihal-Talantikite | Adverse Birth Outcomes as Indicators of Poor Fetal Growth Conditions in a French Newborn Population-A Stratified Analysis by Neighborhood Deprivation Level[END_REF] Most economic indicators consider household income but not the number of household members nor their family relationships. [START_REF] Opatowski | New index of social deprivation during pregnancy: results from a national study in France[END_REF] The deprived population in our sample presents the same social and demographic characteristics already described among this population: they are younger, less educated, and more often born abroad. [START_REF] Lindquist | Experiences, utilisation and outcomes of maternity care in England among women from different socio-economic groups: findings from the 2010 National Maternity Survey[END_REF][START_REF] Yang | Neighborhood deprivation and maternal psychological distress during pregnancy: a multilevel analysis[END_REF] They are also more often obese and active smokers. [START_REF] Clausen | Pregnancy complications by overweight and residential area. A prospective study of an urban Norwegian cohort[END_REF][START_REF] Räisänen | Smoking during pregnancy was up to 70% more common in the most deprived municipalities -a multilevel analysis of all singleton births during 2005-2010 in Finland[END_REF] Their pregnancies more often had related diseases, especially diabetes, and they tended to give birth at a lower gestational age. Few differences were observed between the two groups for neonatal outcome. The higher rate of preterm deliveries among deprived women is an unfavourable outcome already described in the literature, 2,[START_REF] Vos | Deprived neighborhoods and adverse perinatal outcome: a systematic review and meta-analysis[END_REF][START_REF] Deguen | Using a Clustering Approach to Investigate Socio-Environmental Inequality in Preterm Birth-A Study Conducted at Fine Spatial Scale in Paris (France)[END_REF] as is the clear association between pregnancy-related diseases and social deprivation. [START_REF] Bihan | Factors associated with screening for glucose abnormalities after gestational diabetes mellitus: baseline cohort of the interventional IMPACT study[END_REF][START_REF] Vinikoor-Imler | The effects of exposure to particulate matter and neighbourhood deprivation on gestational hypertension[END_REF] Among these diseases, gestational diabetes nonetheless appears more frequent among these women in other studies than in ours. [START_REF] Cosson | Psychosocial deprivation in women with gestational diabetes mellitus is associated with poor fetomaternal prognoses: an observational study[END_REF][START_REF] Sampson | The association between socio-demographic marginalization and plasma glucose levels at diagnosis of gestational diabetes[END_REF] A difference between our results and the literature 6,8,10 is that we did not observe a higher frequency of fetal growth restriction in the deprived population. These results must be interpreted with caution and are one of our study's limitations: because these events are rare, the number of individuals affected is very low. Our study probably lacks the power to be able to explain the associations between maternal and neonatal outcomes and social deprivation.

Similarly, because this was a written questionnaire, women who did not speak French or were unable to read could not participate, and this group probably included women in the most precarious situations, for whom particular attention is necessary. It is not possible to know how many women were eligible for our study and how many refused to participate. Our participation rate is nonetheless high.

The EPICES questionnaire has already been used in telephone or face-to-face interviews; its acceptability among the respondents was good and the results about the prevalence of social deprivation consistent with ours. [START_REF] Cosson | Psychosocial deprivation in women with gestational diabetes mellitus is associated with poor fetomaternal prognoses: an observational study[END_REF][START_REF] Henrotin | Deprivation, occupational hazards and perinatal outcomes in pregnant workers[END_REF] Calculation of the EPICES score as part of a prenatal interview can therefore be envisioned for early screening of deprivation and specific management for it. In France, the early prenatal interview is the first session of preparation for childbirth and parenting. It is recommended that these interviews be planned with a midwife or physician, from the beginning of pregnancy, starting at the fourth month, either alone or with the partner. This interview is fully covered (100%) by the French national health insurance fund. Among other things, it makes it possible to identify situations of social deprivation among both mothers and fathers and has been mandatory since May 2020, [START_REF]Ministères des Affaires sociales[END_REF] just like the 7 visits (for a full-term pregnancy) of medical antenatal care.

In conclusion, the EPICES score with a threshold validated in the population of pregnant women is a useful, rapid and easy to use tool that makes it possible to identify maternal social deprivation at an individual level. The routine use of this questionnaire during pregnancy would make it possible to study the consequences of deprivation on maternal and fetal health status.

Ethical approval

The relevant CPP ethics committee (Patient Protection Committee, Southeast 3, no. 2017-012B) approved this study. TP=true positive; FP=False Positive; FN=False negative; TN=True Negative

1 Table 2

 12 Social and demographic characteristics of participants in the immediate postpartum, in the overall study population and in deprived and non-deprived groups as defined by the Characteristics of pregnancy and delivery of women in the overall study population and in deprived and non-deprived groups as defined by the French law

Figure 1 :

 1 Figure 1: Ratio of the precision associated with the threshold to the mean positive rate: enrichment.

Figure 2 :

 2 Figure 2: Sensitivity analysis as functions of the threshold of the EPICES score

Term at delivery (weeks of gestation)

		[39.4 ± 2.2]	[39.1 ± 1.5]	[39.5 ± 2.3]	0.007
	< 37	4.6	5.5	4.3	0.005
	≥37 -≤40	73.9	82.8	71.5
	>40	21.6	11.7	24.1
	Mode of delivery				0.55
	Caesarean	16.4	19.3	15.7
	Operative vaginal	14.4	13.1	15.7
	delivery
	Vaginal delivery,	69.1	67.6	69.6
	spontaneous

Table 3

 3 Medical characteristics of newborns in the overall study population and in deprived and non-deprived groups as defined by the French law

		Total	Deprived	Non-deprived	P
				N=564	value
		N=710	N=146	n %
		n %	n %	[m±SD]
		[m±SD]	[m±SD]
	Preterm birth (<37 weeks of	5.1	5.5	5.0	0.80
	gestation)
	Sex
	Female	51.1	49.3	51.6	0.62
	Birth weight
		[3256.4 ± 508.4]	[3286.7 ± 492.9]	[3248.5 ± 512.4]	0.42
	< 2500 g	6.3	6.2	6.4	0.45
	≥2500-<3500 g	62.5	58.2	63.7
	≥3500 g	25.6	30.8	24.3
	≥4000 g	5.5	4.8	5.7
	Small for gestational-age *	6.3	2.1	7.5	0.02
	One-minute Apgar score	n= 699	n= 144	n= 555	0.63
		[8.6 ± 1.3]	[8.6 ± 1.4]	[8.7 ±1.3]
	Five-minute Apgar score	n= 708	n= 146	n=562	0.82
		[9.6 ± 1.2]	[9.6 ± 1.1]	[9.6 ± 1.3]
	<7	2.0	1.4	2.1	0.55
	≥7	98.0	98.6	97.9
	Immediate resuscitation	n= 702	n= 143	n= 559	0.14
		6.4	9.1	5.7
	Milk source at discharge	n= 680	n= 139	n= 541	0.005
	Breast milk exclusively	48.4	44.6	49.4
	Mixed	11.0	18.7	9.1
	Artificial	40.6	36.7	41.6

* Birthweight below the 5th percentile according to the gestational age and sex

Table 4 :

 4 Distribution of the EPICES items and score in the overall study population and in deprived and non-deprived groups as defined by the French law

	Total	Deprived	Non-deprived	P
	N= 700	N=145	N=555
	n (%)	n (%)	n (%)
	[m±SD]	[m±SD]	[m±SD]

Table 5

 5 Ability of the EPICES score to discriminate between deprived women and nondeprived with social deprivation defined by the French law

	Social deprivation	Social deprivation defined by the French law
	defined by the
	EPICES index *
		Deprived	Non-deprived	Total
	Deprived	TP	FP
		102	139	241
	Non-deprived	FN	TN
		43	416	459
	Total	145	555	700

* Deprived for EPICES score ≥ 22; non-deprived for EPICES score < 22.


El Hassan

Youssfi Polyanalytic Reproducing

E H Youssfi

M R C N R S 7373

POLYANALYTIC REPRODUCING KERNELS IN C n

Keywords: 2000 Mathematics Subject Classification. Primary 47B35, 32A36, 32A37 Reproducing kernel, Polyanalytic, Bergman space, Fock space I2M, U

come

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INTRODUCTION AND STATEMENT OF THE MAIN RESULT

Polyanalytic function theory in higher dimensions is still not understood. This is essentially due to the lack of explicit formula for related integral kernels. On the other hand, despite this extensive study, there were only two examples of explicit polyanalytic reproducing kernels; the first one was established by Koshelev [START_REF] Koshelev | The kernel function of a Hilbert space of functions that are polyanalytic in the disc[END_REF] in the disc, and the second one was given in [START_REF] Balk | Polyanalytic functions and their generalizations, Complex analysis I[END_REF] and [START_REF] Abreu | Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions[END_REF], and proved in [START_REF] Haimi | The polyanaytic Ginibre ensembles[END_REF], [START_REF] Haimi | Asymptotic expansion of polyanalytic Bergman kernels[END_REF]. It was only in the very recent work by Hachadi and the author [START_REF] Hachadi | The polyanalytic reproducing kernels[END_REF] that a wide source of examples was constructed. There it is proved that to a given a positive integer q and sequence of orthogonal polynomials associated to a non-negative measure in the half real line [0, +∞[ whose support contains at least q strictly positive points, is naturally associated a Hilbert space of q-analytic functions and established a general formula for its reproducing kernel. It is also shown that the formula generalizes to the higher dimensional setting of (q 1 , . . . , q n )analytic functions , where q 1 , • • • , q n are positive integers. These functions were considered by Avanissian and Traor [START_REF] Avanissian | Sur les fonctions polyanalytiques de plusieurs variables[END_REF] and [START_REF] Avanissian | Extension des théorèmes de Hartogs et de Lindelöf aux fonctions polyanalytiques de plusieurs variables[END_REF]. We recall that a function f is said to be polyanalytic of order (q 1 , . . . , q n ) or just ((q 1 , . . . , q n )-analytic) in the domain Ω ⊂ C n if in this domain it satisfies the generalized Cauchy-Riemann equation

∂ q 1 +•••+qn f ∂ zq 1 1 • • • ∂ zqn n = 0. (1.1)
They can be uniquely expressed as

f (z) =
(q 1 -1,...,qn-1)

j=(0,••• ,0) z j φ j (z) (1.2)
where the φ j (z) are holomorphic in Ω where for j = (j 1 , . . . ,

j n ), k = (k 1 , . . . , k n ) ∈ N n 0 and z = (z 1 , . . . , z n ) ∈ C n , the inequality j ≤ k means that j l ≤ k l for all l = 1, • • • , n and z j := z j 1 1 • • • z jn n .
Another notion of polyanalyticity was considered recently by Daghighi [START_REF] Daghighi | Polyanalytic functions on Banach spaces[END_REF]. Namley, for a positive integer q, a function f defined on an open set Ω in C n is called analytic of absolute order q (or just q-analytic) if it is of class C q that satisfies the generalized Cauchy-Riemann equation

∂ α f ∂z α (z) = 0, on Ω for all multi-indices α = (α 1 , . . . , α n ) ∈ N n 0 with length |α| := α 1 + • • • + α n = q.
He proved that these functions f are precisely the slice q-analytic functions in the sense that for all (a, ξ) ∈ Ω × C n , the one variable functions λ → f (a + λξ) are q-analytic in some open neighborhood of 0 in C. It is clear that an absolute q-analytic function f on B can be expressed as a sum

f (z) = |α|<q zα f α (z), (1.3)
where for all α ∈ N n with |α| ≤ q -1 the functions f α are holomorphic on Ω called the holomorphic components of the polyanalytic function f . To the best of our knowledge, no integral representations are available for q-analytic functions in C n . The main goal of this work is to consider this question when Ω is C n or the unit ball in C n .

In this note, we let B := {z ∈ C n : |z| < 1} be the unit ball in C n and denote by dv(z) the Lebesgue measure on C n normalized so that v(B) = 1.

A first result of this paper establishes an explicit expression of the weighted polyanalytic (in the absolute sense) Bergman kernel of the unit ball B. More precisely, for s > -1, we consider the space A 2 s,q (B) of all square integrable of q-analytic functions with respect to the measure dν s (z) := (1 -|z| 2 ) s dv(z). More precisely, we will prove the following Theorem A. The space A 2 s,q (B) is a Hilbert space which coincides with the closure of the qanalytic polynomials in L 2 (ν s ) and its reproducing kernel is given by

K s,q,n (z, w) = Γ(q+s+n) n!Γ(s+q) (1-w,z ) q-1 (1-z,w ) s+q+n P (n,s) q-1 (1 -2|ϕ z (w)| 2 ),
for all z, w ∈ B, where P (n,s) q-1 is the classical Jacobi polynomial with parameters n, s and degree q -1, and ϕ w is the biholomorphic automorphism of B exchanging 0 and w.

We point out that when n = 1, this result was established recently by Hachadi and the author [START_REF] Hachadi | The polyanalytic reproducing kernels[END_REF]. In this case, the one dimensional weighted polyanalytic Bergman kernel reduces

K s,q,1 (z, w) = (q + s) (1-w,z ) q-1 (1-z,w ) s+q+1 P (1,s) q-1 (1 -2|ϕ z (w)| 2 ), (1.4)
where ϕ z (w) := z -w 1 -z, w for all elements z, w of the unit disc D in C.

To state our second result, we consider the Gaussian measure µ defined on C n by dµ (z) := e -|z| 2 dv (z) (1.5) and denote by F 2 q (C n ) the weighted q-polyanalytic Fock space on C n where q is a positive integer. This is the space of all q-analytic functions f on C n which are square integrable with respect to dµ(z).

Theorem B. The space F 2 q (C n ) is a Hilbert space which coincides with the closure of the qanalytic polynomials in L 2 (µ) and its reproducing kernel is given by

K F,q (z, w) = e z,w (n -1)! L n q-1 |z -w| 2 (1.6)
for all z, w ∈ C n , where L n q-1 is the classical weighted Laguerre polynomial with weight n and degree q -1.

After submitting this paper, I found a work by Lea-Pacheco, Maximenko and Ramos-Vazquez [START_REF] Leal-Pacheco1 | Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain[END_REF] refering to an earlier version of my work [START_REF] Youssfi | Polyanalytic reproducing kernels in C n[END_REF]. The authors have found another way to get some of the results presented in this work. Namely, they also compute the reproducing kernel the unit ball.

PRELIMINARIES

Let D n be the unit polydisc in C n . This is the open unit ball in C n with respect to the norm

z ∞ := max 1≤j≤n |z j |, z = (z 1 , . . . , z n ) ∈ C n .
For a positive integer q, we let A 2 q (D n ) denote the space of those (q, . . . , q)-analytic functions on D n which are square integrable with respect to the Lebesgue measure dv(z) on D n . As a consequence of Theorem D of [START_REF] Hachadi | The polyanalytic reproducing kernels[END_REF], we have the following Lemma 2.1. The space A 2 q (D n ) is the closure of the (q, . . . , q)-analytic polynomials in the space L 2 (D n ) of all square integrable functions on D n with respect to the measure dv(z). Furthermore, its reproducing kernel is given by

K D n ,q (z, w) = n j=1 K 0,q,1 (z j , w j ) for all z = (z 1 , . . . , z n ), w = (w 1 , . . . , w n ) ∈ D n
, where K 0,q,1 is the one dimensional polyanalytic weighted Bergman kernel of the unit disc given by (1.4) with s = 0.

Lemma 2.2. Let Ω be an open set in C n and let : Ω →]0, +∞[ be a continuous weight function.

For each compact subset C of Ω, and each α ∈ N n 0 , there is a positive constant c α > 0 such that

max C ∂ α f ∂ zα ≤ c α f L 2 (Ω, dv) ,
for all q-analytic functions in L 2 (Ω, dv). In particular, the space A 2 ,q (Ω) of all such functions, equipped the L 2 (Ω, dv) inner product is Hilbert space and for all z ∈ Ω, the evaluation functional f → f (z) is continuous on A 2 ,q (Ω).

Proof. It is sufficient to consider compact polydiscs C = a + rD n , for a ∈ Ω and r > 0 such that a + 2rD n ⊂ Ω. Considering this, we let f ∈ A 2 ,q (Ω). Then the function g(z) := f (a + 2rz) is well-defined (q, • • • , q)-analytic function on a neighborhood of D n . By Lemma 2.1

g(z) = D n g(w)K D n ,q (z, w)dv(w), from which it follows that ∂ α g(z) ∂ zα = D n g(w) ∂ α ∂ zα (K D n ,q (z, w)) dv(w),
so that by Cauchy-Schwarz inequality we see that

∂ α f ∂ zα (a + rz) ≤ f L 2 (Ω, dv) r |α| D n ∂ α ∂ zα (K D n ,q (z/2, w)) 2 (a + rz) dv(w) 1/2 . Hence max C ∂ α f ∂ zα ≤ c α f L 2 (Ω, dv) where c α := sup z∈D n D n ∂ α ∂ zα (K D n ,q (z/2, w)) 2 r |α| (a + rz) dv(w) 1/2 .
This completes the proof of the lemma.

Lemma 2.3. Let Ω be a domain in C n and D a domain in C p . Suppose that ψ : Ω → D is a holomorphic map of the form

ψ(z) = g(z)
h(z) where g : Ω → C p and h : Ω → C are polynomials of respective degrees d g and d h such that d g ≤ d h and h(z) = 0 for all z ∈ Ω. Suppose that q and l are positive integers such that l ≥ q. If f is a q-analytic function on D, then the function hl (f

• ψ) is (ld h +1)-analytic on Ω. Proof. Writing f in the form f (w) = |α|<q wα f α (w), it follows for all z ∈ D, we have hl (z)(f • ψ)(z) = |α|<q h(z) l-|α| (g(z)) α f α (ψ(z))
which is q-analytic since for each α, the function h(z) l-|α| (g(z))

α is a polynomial of degree at most ld h + |α|(d g -d h ) ≤ ld h . This completes the proof.

Let Aut(B) be the group of all biholomorphic automorphisms of the unit ball B. Let ϕ ∈ Aut(B) and let a := ϕ -1 (0). Its is well-known (see [START_REF] Rudin | Function Theory on the open unit ball in C n[END_REF], p28) that ϕ can be written in the form

ϕ = U ϕ a (2.1)
where U : C n → C n is a unitary linear transformation and ϕ 0 := -I and for a = 0,

ϕ a (z) = a -(1 -|a| 2 )P a z -1 -|a| 2 (Q a (z) 1 -z, a , for all z ∈ B. (2.2)
where P a z : z, a a/|a| 2 and Q a := I -P a .

Lemma 2.4. Suppose that ϕ is an automorphism of the unit ball and let J ϕ be its complex jacobian. If f is a q-analytic function on B, then the function J ϕ (1-q)/(n+1) (f • ϕ) is also q-analytic on B.

Proof. Suppose that ϕ ∈ Aut(B) and let a := ϕ -1 (0). Then it follows from Theorem 2.2 in [START_REF] Rudin | Function Theory on the open unit ball in C n[END_REF] that

J ϕ (z) = 1 -|a| 2 1 -z, a n+1
, for all z ∈ B.

(2.3)

In addition, in view of (2.2) we see that there are a C-linear mapping L : C n → C n and a vector b ∈ C n such that for all z ∈ B,

ϕ(z) = L(z) + b 1 -z, a , for all z ∈ B.
Applying Lemma 2.3, with g(z) = L(z) + b and h(z) = 1 -z, a shows that if f is q-analytic function on B, then

J ϕ (z) (1-q)/(n+1) (f • ϕ)(z) = 1 (1 -|a| 2 ) q-1 ( h(z) q-1 (f • ϕ)(z)
which also q-analytic. This completes the proof.

PROOF OF THEOREM A

In this section we will prove the following transformation rule stated for the unit ball. Lemma 3.1. Suppose that ϕ is an automorphism of the unit ball and let J ϕ be its complex jacobian. Then the Bergman kernel K q,s of A s q (B) follows the transformation rule

K q,s (z, ξ) = J ϕ (z)J ϕ (ξ) s+q+1 n+1 J ϕ (z)J ϕ (ξ) q-1 n+1 K q,s (ϕ(z), ϕ(ξ)) (3.1)
for all z, ξ ∈ B.

Proof. It is sufficient to assume that ϕ • ϕ(z) = z, for all z ∈ D. We recall that the measure dv(z)

(1-|z| 2 ) n+1 is invariant under the action of the automorphism group of the unit ball. We also observe that for any fixed ξ ∈ B, the function

z → (J ϕ (z)) s+q+1 n+1 J ϕ (z) q-1 n+1 K q,s (ϕ(z), ξ)
is an element of A s q (B). By the reproducing property and change of variables formula we see that

(J ϕ (z)) s+q+1 n+1 J ϕ (z) q-1 n+1 K q,s (ϕ(z), ξ) = B (J ϕ (w)) s+q+1 n+1 J ϕ (w) q-1 n+1 K q,s (ϕ(w), ξ)K q,s (z, w)dν s (w) = B J ϕ (w) (s+q+1)/(n+1 ) (J ϕ (w)) (q-1)/2 K q,s (w, ξ)K q,s (z, ϕ(w))dν s (w) = J ϕ (ξ) (s+q+1)/(n+1) (J ϕ (ξ)) (q-1)/(n+1) K q,s (z, ϕ(ξ))
Replacing ξ by ϕ(ξ) the latter equalities yield

J ϕ (z)J ϕ (ξ) (s+q+1)/(n+1) J ϕ (z)J ϕ (ξ) (q-1)/(n+1) K q,s (ϕ(z), ϕ(ξ)) = K q,s (z, ξ).
This completes the proof.

Lemma 3.2. The Bergman kernel K q,s of A s q (B) satisfies the equality

K q,s (z, w) = (1 -w, z ) q-1 (1 -z, w ) s+n+q K q,s (ϕ z (w), 0) (3.2)
for all z, ξ ∈ B.

Proof. Let z, w ∈ B. Applying Lemma 3.1 with ϕ = ϕ z and ξ = ϕ z (w) and using (2.3) yields

K q,s (z, w) = J ϕ (z)J ϕ (ξ) s+q+1 n+1 J ϕ (z)J ϕ (ξ) q-1 n+1 K q,s (0, ϕ z (w)) = (1 -w, z ) q-1 (1 -z, w ) s+n+q K q,s (0, ϕ z (w)).
Since K q,s (z, w) = K q,s (w, z), for all z, w ∈ B, the lemma follows.

We shall make use of the classical Jacobi polynomials P (s,d) m with parameters (s, d) and degree m. An explicit formula for these polynomials is given by

P (s,d) m (x) = 1 2 m m k=0 s + m k d + m m -k (x -1) m-k (x + 1) k . (3.3)
This can be found in [START_REF] Ismail | Classical and Quantum Orthogonal Polynomials in One Variable[END_REF]. It is well-known by formula (3.96) in ( [START_REF] Shen | Spectral methods: Algorithms, Analysis and Applications[END_REF], p. 71) that these polynomials verify the equality

P (s,d) m (1 -2x) = Γ(m + s + 1) m! m j=0
(-1) j m j Γ(m + j + s + d + 1) Γ(m + s + d + 1)Γ(j + s + 1)

x j .

(3.4)

The Jacobi polynomials satisfy the orthogonality condition

1 0 P (s,d) m (2x -1)P (s,d) m (2x -1)x d (1 -x) s dx = δ m,m h s,d m (3.5)
where

h s,d m := Γ (s + m + 1)) Γ (d + m + 1) m!Γ (s + d + m + 1) (s + d + 2m + 1) . (3.6) Lemma 3.3.
For each z ∈ B, we have

K q,s (z, 0) = Γ(q+s+n) n!Γ(q+s) P n,s q-1 (1 -2|z| 2 ). (3.7)
Proof. Due to Lemma 3.1 we see that the q-analytic function z → K q,s (z, 0) invariant under unitary linear transformations and hence it is of the form K q,s (z, 0) = q-1 j=0 c j |z| 2j for some complex coefficients c 0 ,

• • • , c q-1 . Set Q s,n q-1 (x) := q-1 j=0 c j x j , x ∈]0, 1[. (3.8)
We equip the space P q-1 of polynomials of degree at most q -1 with the L 2 -inner product associated to the measure x n (1 -x) s dx in the interval ]0, 1[. Integrating in polar coordinates shows that for all j = 0, • • • , q -2, one has that

1 0 x j Q s,n q-1 (x)x n (1 -x) s dx = 2 1 0 x 2j Q s,n q-1 (x 2 )x 2n (1 -x 2 ) s dx = 1 n B |z| 2j Q s,n q-1 (|z| 2 )|z| 2 (1 -|z| 2 ) s dv(z) = 1 n B |z| 2j+2 K s,n q-1 (z, 0)(1 -|z| 2 ) s dv(z) = 0,
where the latter equality holds due to the reproducing property at 0 of the q analytic polynomial |z| 2j+2 . Hence Q s,n q-1 is orthogonal in L 2 (0, 1[, x n (1 -x) s dx) to all polynomials with strictly smaller than q -1 so that by (3.5) we see that Q s,n q-1 (x) is proportional to the polynomial P (s,n) q-1 (2x -1) and thus for some complex constant C(s, n, q -1) we have

Q s,n q-1 (x) = C(s, n, q)(-1) q-1 P (s,n) q-1 (2x -1) = C(s, n, q)P (n,s) q-1 (1 -2x).
To compute the constant, we apply the reproducing property to the constant function 1 and get

1 0 Q s,n q-1 (x)x n-1 (1 -x) s dx = 2 1 0 Q s,n q-1 (x 2 )x 2n-1 (1 -x 2 ) s dx = 1 n B Q s,n q-1 (|z| 2 )|z| 2 (1 -|z| 2 ) s dv(z) = 1 n B K s,n q-1 (z, 0)(1 -|z| 2 ) s dv(z) = 1 n ,
and thus

1 n = C(s, n, q)(-1) q-1 1 0 P (s,n) q-1 (2x -1)x n-1 (1 -x) s dx.
To compute the constant C(s, n, q), let

I(s, n -1, q -1) := 1 0 P (s,n) q-1 (2x -1)x n-1 (1 -x) s dx.
The the orthogonality property of Jacobi polynomials, combined with integration by parts, yields

I(s, n -1, q -1) = - q + s + n n I(s + 1, n, q -2)
so that by induction we get I(s, n -1, q -1) = (-1) q-1 (n -1)!Γ(2q + s + n -1) Γ(q + s + n)Γ(n + q -1) I(s + q -1, n + q -2, 0).

Since

I(s + q -1, n + q -2, 0) = Γ(q + s)Γ(n + q -1) Γ(2q + s + n -1) it follows that

I(s, n -1, q -1) = (-1) q-1 (n -1)!Γ(q + s) Γ(q + s + n) showing that C(s, n, q) = Γ(q + s + n) n!Γ(q + s) and hence Q s,n q-1 (x) = Γ(q + s + n) n!Γ(q + s) P (n,s)
q-1 (1 -2x). This completes the proof of the lemma.

Proof of Theorem A. Combining Lemmas 3.2 and 3.3 gives the expression of the reproducing kernel K q,s . To show that the q-analytic polynomials are dense in A s q (B) it suffices to observe that for all α, β ∈ N n 0 , and any q-analytic function f ∈ A s q (B) we have ∂ α+β f ∂z α ∂z β (0) = f, P where P (w) := ∂ α+β ∂z α ∂z β K q,s (•, w) z=0 .

On the other hand, a little computing shows that P is a q-analytic polynomial provided that |β| < q. Therefore, if f is orthogonal to q-analytic polynomials in A s q (B), then f must vanish everywhere, showing that q-analytic polynomials are dense in A s q (B)

THE POLYANALYTIC FOCK SPACE

We consider the Gaussian measure µ given by (1.5) and the corresponding Fock space F 2 q (C n ) of all q-analytic functions f on C n which are square integrable with respect to dµ(z). From Lemma 2.2, we see that F 2 q (C n ) furnished with the L 2 (µ) scalar product turns out to be a Hilbert space for which the evaluation functional f → f (z) is continuous for all z ∈ C n . Hence F 2 q (C n ) admits a reproducing kernel K F, (•, •).

On the other hand, by the change of variable formula, it is clear that if a ∈ C n and f ∈ F 2 q (C n ), then the function z → e -z,a f (z + a) remains in F 2 q (C n ). Lemma 4.1. The reproducing kernel K F,q of F q (C n ) follows the transformation rule e -z,a K F,q (U z + a, ξ) = e a,ξ K F,q (z, U ξ -U a) (

for all z, ξ ∈ C n , and all unitary transformations U : C n → C n . In particular, K F,q (z, w) = e z,w K F,q (z -w, 0) (4.2) and K F,q (U z, U w) = K F,q (z, w)

for all z, ξ ∈ C n and all unitary transformations U :

C n → C n .
Proof. If a, z, ξ ∈ C n , then applying twice the reproducing formula and a change of variable yield e -z,a K F,q (U z + a, ξ) =

C n e -U w,a K F,q (U w + a, ξ)K F,q (z, w)dµ(w)

= C n e -a,w K F,q (z, U w -U a)K F,q (ξ, w)dµ(w) = e -a,w K F,q (z, U ξ -U a).
This completes the proof.

In view of the latter lemma, in order compute the Fock reproducing kernel it is sufficient to calculate K F,q (z, 0) for all z ∈ C n . To do so, we shall make use of the classical weighted Laguerre polynomials L m d of degree d and weight m. These polynomials satisfy,

+∞ 0 L m d (x) L m d (x) x m e -x dx = Γ (d + m + 1) d! δ d,d . (4.4)
They have the following explicit representation

L m d (x) = d l=0 (n + m)! l!Γ (d + m + 1 -l) (-x) n-l (n -l)! . (4.5)
Lemma 4.2. For all z ∈ C n , we have

K F,q (z, 0) = 1 n! L n q-1 |z| 2 . (4.6)
Proof. In virtue of (4.3) we see that z → K F,q (z, 0) is invariant under unitary linear transformations and hence it is of the form

K F,q (z, 0) = q-1 j=0 c j |z| 2j
for some complex coefficients c 0 , • • • , c q-1 . To compute the coefficients c j , we consider the space P q-1 of polynomials with degree at most q -1 and equip it with the L 2 -inner product associated to the measure e -x dx in the interval [0, +∞[. Set

R q (x) := q-1 j=0 c j x j , x ≥ 0.
Integrating in polar coordinates shows that for all j = 0, • • • , q -2, one has that +∞ 0

x j R q (x)x n e -x dx = 1 n C n |z| 2j+2 R q-1 (|z| 2 )dµ(z) = 1 n C n |z| 2j+2 K F,q (z, 0)dµ(z) = 0, where the latter equality holds due to the reproducing property at 0 of the q analytic polynomial |z| 2j+2 . Hence R q-1 (x) is orthogonal in L 2 (µ) to all polynomials with strictly smaller than q -1 so that by (4.4 ) we see that R q-1 is proportional to the polynomial L n q-1 and thus for some complex constant c(n, q) we have R q-1 (x) = c(n, q)L n q-1 (x). To compute the constant, we apply the reproducing property to the constant function 1 and get +∞ 0 R q-1 (x)x n-1 e -x dx = 1 n C n K F,q (z, 0)dµ(z) = 1 n , and thus 1 n = c(n, q) +∞ 0

L n q-1 (x)x n-1 e -x dx.

To compute the constant c(n, q), let I(n, q) := +∞ 0 L n q-1 (x)x n-1 e -x dx.

Using the observation that d dx e -x L n-1 q (x) = -e -x L n q-1 (x)

integrations by parts shows that I(n, q) = (n -1)!I(0, q + n -1) = (n -1)!.

The lemma follows.

Proof of Theorem B. Combining Lemmas 4.1 and 4.2 gives the expression of the reproducing kernel K F,q . To show that the q-analytic polynomials are dense in F q (C n ) it suffices to observe that for all α, β ∈ N n 0 , and any q-analytic function f ∈ F 2 q (C n ) we have ∂ α+β f ∂z α ∂z β (0) = f, P where P (w) := ∂ α+β ∂z α ∂z β K F,q (•, w) z=0 .

On the other hand, a little computing shows that P is a q-analytic polynomial provided that |β| < q. Therefore, if f is orthogonal to q-analytic polynomials in F 2 q (C n ), then f must vanish everywhere, showing that q-analytic polynomials are dense in F 2 q (C n ).


Robert Griñó

Romeo Ortega

Emilia Fridman

Jin Zhang

Frédéric Mazenc

A Behavioural Dynamic Model for Constant Power Loads in Single-Phase AC Systems

Keywords: AC electrical networks, constant power load, behavioural model, stability, delay-differential equations

Constant power loads (CPLs) in power systems have a destabilizing effect that gives rise to significant oscillations or to network collapse, motivating the development of new methods to analyse their effect in AC and DC power systems. A sine qua non condition for this analysis is the availability of a suitable mathematical model for the CPL. In the case of DC systems power is simply the product of voltage and current, hence a CPL corresponds to a first-third quadrant hyperbola in the loads voltagecurrent plane. The same approach is applicable for balanced three-phase systems that, after a rotation of the coordinates to the synchronous frame, can be treated as a DC system. Modelling CPLs for single-phase (or unbalanced poly-phase) AC systems, on the other hand, is a largely unexplored area because in AC systems (active and reactive) power involves the integration in a finite window of the product of the voltage and current signals. In this paper we propose a simple dynamic model of a CPL that is suitable for the analysis of single-phase AC systems. We give conditions on the tuning gains of the model that guarantee the CPL behaviour is effectively captured.

Introduction

In many electric power distribution systems and particularly in microgrids, stability problems may occur when a major proportion of the loads are electronic equipment. This kind of equipment is usually powered by cascade

The work of Robert Griñó was supported in part by the Government of Spain through the Agencia Estatal de Investigación Project DPI2017-85404-P and in part by the Generalitat de Catalunya under Project 2017 SGR 872. The work of Emilia Fridman and Jin Zhang was supported by Israel Science Foundation (grant no. 673/19) and by the Planning and Budgeting Committee Fellowship from the Council for Higher Education, Israel. The material in this paper was not presented at any conference. Corresponding author R. Griñó.

Email addresses: roberto.grino@upc.edu (Robert Griñó), romeo.ortega@itam.mx (Romeo Ortega), emilia@tauex.tau.ac.il (Emilia Fridman), zhangjin1116@126.com (Jin Zhang), frederic.mazenc@l2s.centralesupelec.fr (Frédéric Mazenc).

distributed architectures which are characterized by the presence of different voltage levels and power electronic converters. These converters act as interfaces between sections of different voltages in which, at last stage, loads are a combination of power electronic converters tightly regulating their output voltage, behaving as Constant Power Loads (CPLs). These architectures are common in information and communication technology facilities where the many telecom switches, wireless communication base stations, and data center servers act as CPLs. It is well-known that CPLs introduce a destabilizing effect that gives rise to significant oscillations or even voltage collapse, and hence they are considered to be the most challenging component of the standard load modelreferred to as the ZIP model in power system stability analysis. See [START_REF] Molinas | Constant power loads in ac distribution systems: An investigation of stability[END_REF][START_REF] Karimipour | Stability analysis of ac microgrids with constant power loads based on popov's absolute stability criterion[END_REF][START_REF] Machado | Some Problems on the Analysis and Control of Electrical Networks with Constant Power Loads[END_REF][START_REF] Singh | Constant power loads and their effects in dc distributed power systems: A review[END_REF] for a recent review of the literature and [START_REF] Matveev | A tool for analysis of existence of equilibria and voltage stability in power systems with constant power loads[END_REF] for a detailed analysis of the effect of CPLs on the power systems behavior.

The growing presence of CPLs in modern installations significantly aggravates this issue, hence motivating the development of new methods to analyze their effect in AC and DC power systems. To carry out this analysis it is necessary to dispose of a mathematical model that suitably describes the behaviour of the CPL. In the case of a DC system a suitable model for the CPL is simply a firstthird quadrant hyperbola in the loads voltage-current plane, i(t) = P v(t) , and, consequently, the load incrementally behaves as a negative resistance contributing to reduce the relative stability of the electrical network and even to destabilize it [START_REF] Marx | Large signal stability analysis tools in dc power systems with constant power loads and variable power loads-a review[END_REF][START_REF] Riccobono | Comprehensive review of stability criteria for dc power distribution systems[END_REF]. The same model is applicable for balanced three-phase systems that, after a rotation of the coordinates to the synchronous frame, can be treated as a DC system. Modeling CPLs for single-phase (or unbalanced multi-phase) AC systems, on the other hand, is a largely unexplored area because of the definition of active power (P (t)) in AC systems (IEEE 2010[START_REF] Garcia-Canseco | Power factor compensation of electrical circuits: The nonlinear non-sinusoidal case[END_REF], which involves the integration in a finite moving window of the product of the voltage and current signals, more precisely, for a singlephase AC system 1 we have

P (t) = 1 T t t-T v(s)i(s)ds, (1)
where v(t) is the voltage and i(t) the current. Throughout the paper we make the following assumption.

Assumption 1 The voltage v(t) is T -periodic, i.e., v(t) = v(t + T ), for all t ≥ 0.
In this paper we propose a simple dynamic model of a CPL that is suitable for the analysis of single-phase AC systems 2 . Because of space limitations we have concentrated in the emulation of active power. However, the extension to include reactive power is immediate and is explained in Section 6. To derive the model we adopt a control theory perspective and reformulate the problem as the design of a negative feedback controller wrapped around the output of a dynamical system that computes P (t). The voltage, required in the computation of P (t), is viewed as an external periodic signal. The control objectives are to drive P (t) to some constant desired value and to ensure that the current is in phase with the voltage. In spite of the apparent simplicity of the aforementioned control problem, it is a far from trivial task because the dynamics of the power computation model

1 It is worth to remark that this definition computes the active power (W) for sinusoidal and nonsinusoidal (general T -periodic) AC single-phase systems and it is the causal version of the expression that appears in (IEEE 2010).

2 For AC poly-phase systems definition (1) changes to

P (t) = 1 T t t-T v (s)i(s)
ds where v(t) and i(t) are, for example, the line-to-neutral voltages vector and the line currents vector, respectively. is a (T -periodic) linear time-varying delay-differential equation (LTV-DDE), complicating the design of the required controller.

We consider in the paper two controllers: a simple proportional one with a constant bias input and a classical proportional-integral (PI) controller. A detailed analysis of the stability of the closed-loop is carried out using advanced techniques of delay-differential systems. Thus, the stability analysis under PI controller employs a recent time-delay approach to averaging (Fridman & Zhang 2020). The outcome of this study is the definition of regions in the space of the controller parameters for which asymptotic stability is ensured. To check how conservative the bounds derived from the theoretical analysis are, we carry out extensive numerical simulations and the computation of the more relevant characteristic multipliers of the linear periodic DDEs that arise.

To the best of the authors' knowledge this is the first attempt to develop a mathematically well-founded model for AC CPL's. Most of the results been restricted to numerical derivations (Y. Dong, W. Liu, Z. Gao & X. Zhang 2008) or invoking various kinds of approximations to the actual phenomenon [START_REF] Molinas | Constant power loads in ac distribution systems: An investigation of stability[END_REF].

Problem Formulation

In this section we describe the dynamical system that computes the active power P (t) defined in (1) as an LTV operator Σ : (v, i) → P . From (1) we get the DDE satisfied by P (t) as

Ṗ (t) = 1 T [p(t) -p(t -T )]. (2)
where, for ease of reference, we have defined the instantaneous active power

p(t) = v(t)i(t). (3)
A block diagram description of the system Σ is given in the green box of Fig. 1, where v(t) is a fixed T -periodic, external signal and i(t) as the "control" to be defined.

As explained in the introduction, to derive our dynamic model of the AC CPL we adopt a control theory perspective. To generate the input signal i(t) of the system Σ we propose a classical output feedback configuration with Σ c : P → u the controller to be defined, and P (t) := P -P (t), with P the desired value for P (t). To capture our objective of ensuring that, in steady-state, the voltage and the current are in phase, we define the latter as

i(t) = u(t)v(t). (4)
In this case, the "in-phase" requirement translates to the condition that u(t) converges to a constant value. A block diagram description of the overall system is given in Fig. 1.

To complete the description of the mathematical model of the system Σ, we note that from ( 3) and ( 4) we have p(t) = v 2 (t)u(t). Then (1) can be rewritten as

P (t) = 1 T t t-T v 2 ( )u( )d , (5)
revealing that P (t) is just an average of v 2 (t)u(t). Differentiating the latter equation, we arrive at the final description of our system

Ṗ (t) = v 2 (t) T [u(t) -u(t -T )]. (6)
We are now in position to formulate our controller design problem.

Control problem formulation Consider the LTV, delay-differential, periodic, scalar system (6). Given P > 0, design a controller Σ c : P → u such that lim t→∞

P (t) = P lim t→∞ u(t) = u (7)
where u > 0 is an arbitrary constant.

In the following two sections we will present two controllers Σ c : a simple proportional one with a constant bias input and a classical proportional-integral (PI) controller. For both schemes, we give conditions on their tuning gains that ensure the control objective ( 7) is satisfied with all signals remaining bounded.

Remark 1 A particular case of practical interest is

v(t) = √ 2 V sin 2π T t (8)
where V > 0 is the RMS value of the voltage v(t) defined as

V 2 := 1 T T 0 v 2 ( )d . ( 9
)

A Simple Proportional plus Constant Bias Controller

The main result of this section is given in the proposition below whose proof, to enhance readability, is given in Appendix A.

Proposition 1 Consider the simple proportional plus constant bias control law

u(t) = k p P (t) + u (10)
with u = 1 V 2 P and V the RMS value of the voltage v(t) given by (9), which satisfies Assumption 1. For all values of k p in the interval 0, 1

V the conditions (7) are satisfied-with the convergence being exponential. 222

Remark 2 It is shown in the proof that, when P is fixed, the u given in Proposition 1 is the unique value that can be obtained for the limit of u(t) if both P (t) and u(t) converge to constant values. From the proof it also follows that ln kpV T is an upperbound on the exponential rate of convergence.

Remark 3 As it is well-known, to enhance the robustness of the system, it is necessary to implement feedback-as opposed to open-loop-control laws. In any case, it is of mathematical interest to observe from (A.1) that setting k p = 0, that is, considering the feedforward controller u = u , convergence of the error to zero is achieved in finite time. Actually, it is also possible to show that convergence is preserved even for negative values of k p in the interval -1

V , 1

V and that the trajectories of the closed-loop system remain bounded for all k p . Since the analysis is quite involved, and the boundedness property not very informative, it is omitted for brevity.

A Proportional-Integral Controller

In this section we analyze the behavior of the closed-loop system when Σ c is a PI controller. That is, ẋc (t) = P (t)

u(t) = k p P (t) + k i x c (t) (11)
with k p ≥ 0 and k i > 0. As in the previous controller our objective is to find conditions on (k p , k i )-given in terms of feasibility of linear matrix inequalities (LMIs)-such that the control objective ( 7) is satisfied. We consider two possible scenarios, when v 2 (t) is T -periodic, or T 2periodic. The motivation to consider the two cases is two-fold: on one hand, the LMIs are less conservative when considering smaller periods. On the other hand, the case of practical interest ( 8) is3 , indeed, T 2 -periodic.

The proposition below, whose proof is given in Appendix B, pertains to the T -periodic case.

Proposition 2 Let v 2 (t) be T -periodic.

(i) Consider the PI controller given by (11) with k p > 0 and k i > 0. Given k M p > 0 and k M i > 0, if there exist scalars r > 0, r 1 > 0, s > 0 and q > 0 such that the following LMIs are satisfied

Ξ 0 = -s + 2v 4 M [(k M p ) 2 s + (k M i ) 2 (r + q)] + 1 T 2 r 1 < 0, (12) and Ξ(k M p , k M i ) =       Ξ 11 (k M p , k M i ) V 2 1 1 * -4 T r -1 -1 * * -1 v 4 M T q 0 * * * -1 v 4 M T r 1       < 0,
(13) where we defined the function

Ξ 11 (k M p , k M i ) := -2V 2 + 2v 4 M T [(k M p ) 2 s + (k M i ) 2 (r + q)],
and

v M := max t∈[0,T ] v(t), (14)
then (7) is satisfied for all k p ∈ [-k M p , k M p ] and k i ∈ (0, k M i ].
(ii) Consider the I controller given by (11) with k p = 0 and k i > 0. Given k M i > 0, if there exist scalars r > 0 and q > 0 such that the following LMI

   -2V 2 + v 4 M T (k M i ) 2 (r + q) V 2 1 * -4 T r -1 * * -1 v 4 M T q    < 0 (15) holds, then (7) is satisfied for all k i ∈ (0, k M i ]. 222
As indicated above, if v 2 (t) has a smaller fundamental period T 2 , a less conservative LMI condition may be derived. This result is contained in the corollary below, whose proof is given in Appendix C.

Corollary 1 Let v 2 (t) be T 2 -periodic.
(i) Consider the PI controller given by (11) with k p > 0 and k i > 0. Given k M p > 0 and k M i > 0, if there exist scalars r > 0, r 1 > 0, s > 0 and q > 0 such that the following LMIs are satisfied

Ξ0 = -s + 2v 4 M [(k M p ) 2 s + (k M i ) 2 ( r 2 + q)] + 1 T 2 r 1 < 0, ( 16
) and

Ξ(k M p , k M i ) =       Ξ11 (k M p , k M i ) V 2 1 1 * -8 T r -1 -1 * * -16 9v 4 M T q 0 * * * -1 v 4 M T r 1       < 0, ( 17
)
where we defined the function

Ξ11 (k M p , k M i ) := -2V 2 + 2v 4 M T [(k M p ) 2 s + (k M i ) 2 ( r 2 + q)],
and v M is given by (14), then (7) is satisfied for all

k p ∈ [-k M p , k M p ] and k i ∈ (0, k M i ].
(ii) Consider the I controller given by (11) with k p = 0 and k i > 0. Given k M i > 0, if there exist scalars r > 0 and q > 0 such that the following LMI

   -2V 2 + v 4 M T (k M i ) 2 ( r 2 + q) V 2 1 * -8 T r -1 * * -16 9v 4 M T q    < 0 (18) holds, then (7) is satisfied for all k i ∈ (0, k M i ]. 222
Remark 4 As it is seen from LMIs of Proposition 2 and Corollary 1, given T > 0, these LMIs hold with k M p = k M i = 0. Thus, these LMIs hold for sufficiently small PI gains, i.e. ( 7) is always guaranteed for small enough k p ≥ 0 and k i > 0.

Remark 5 The claim that LMIs ( 16) and ( 17) are less conservative than the LMIs ( 12) and ( 13) can be established noting that Ξ0 -Ξ 0 ≤ 0 and Ξ(•) -Ξ(•) ≤ 0 hold. Note also that LMI (13) with k p = 0 is more conservative than LMI (15) in Proposition 2 because its 11term 15). The same holds for LMIs ( 17) and ( 18) in Corollary 1.

-2V 2 + 2v 4 M T (k M i ) 2 (r + q) is larger than 11-term -2V 2 + v 4 M T (k M i ) 2 (r + q) in (

Numerical Simulation Results

This section shows some simulation results and numerical stability assessment, using the characteristic multipliers of the monodromy matrix of the closed-loop delaydifferential system, for the AC sinusoidal voltage (8) with V = 230.0 V, T = 0.02 s. The computation of the characteristic multipliers has been done with the algorithm eigTMN [START_REF] Breda | Stability of Linear Delay-differential Equations -A Numerical Approach with MATLAB[END_REF].

Remark 6 The proposed behavioural model has been extensively tested numerically against step variations of the RMS value of the grid voltage, V , and there are no stability problems for reasonable variations of that value around the nominal one. Besides, the main theoretical result in Proposition 2 can be extended to a variable in time V ≥ V 0 > 0 that takes values in the given interval. This extension is not in the scope of the current paper.

P controller

Using the controller in eq. ( 10), the closed-loop dynamics is

Ṗ (t) = -kp T v 2 (t) P (t) + kp T v 2 (t) P (t -T ) ( 19
)
and the sufficient condition in Proposition 1 states that it would be stable for k p ∈ 0, 1 V = 0, 4.348 • 10 -3 .

The numerical study of the stability of eq. ( 19) is shown in Fig. 2 where the maximum absolute value characteristic multiplier is plotted against k p . As it can be observed, the closed-loop system is stable for all positive values of k p -revealing that, at least for this particular numerical scenario, the theoretical analysis is very conservative. As it can be seen from the figure, the closedloop system tends to be marginally stable as k p → ∞.

Fig. 3 shows the power, P (t), and the control signal, u(t), time responses for a step reference in power, P = 1000 W, for three different values of k p . As it can be observed, the steady-state error is zero and the overshoot in the transient increases as k p gets bigger. In Fig. 4 we plot the voltage applied to the CPL and the computed current for the same three different values of k p . After the transient, with bigger overshoot as k p grows, the current is sinusoidal and perfectly in-phase with the voltage.

PI controller

In this subsection we investigate numerically the PI controller (11) with k p > 0, k i > 0, which results in the closed-loop dynamics

Ṗ (t) ẋc (t) = -kp T v 2 (t) -ki T v 2 (t) 1 0 P (t)
x c (t)

+ kp T v 2 (t) ki T v 2 (t) 0 0 P (t -T ) x c (t -T ) . ( 20
)
Fig. 5 (left) shows the stability chart in the parameter plane (k p , k i ), where the shaded area in solid grey corresponds to unstable behaviour. This information has been obtained looking for the maximum absolute value characteristic multiplier of a rectangular sampling of the parameter space k p -k i . In Fig. 5 (right) we show the contour lines for the maximum absolute value characteristic multiplier in the plane (k p , k i ) for k p and k i close to zero. It is worth to mention that in the other three quadrants of the plane k p -k i the system is unstable. Looking for the k p and k i values that give the minimum maximum absolute value characteristic multiplier (min kp,ki max l |µ l (k p , k i )|) results 4 in k p = 2.95767 • 10 -6 , k i = 0.92594•10 -3 with a maximum absolute value characteristic multiplier, max l |µ l | = 0.048368.

Fig. 6 shows responses of the power, P (t), and the control signal, u(t), to a step reference in power, P = 1000

4 These optimum has been found using several tries of a Nelder-Mead optimization algorithm. So, it can not be assured that it is the global minimum but it is the best minimum found.

W, for different values of k p and k i . As it can be observed, the steady-state error is zero due to the integrator included in the controller. The red curve in the left plot and the blue curve in the right plot correspond to the k p and k i values of the minimum maximum absolute value characteristic multiplier commented in the previous paragraph. As seen from these plots the resulting transient performance is excellent with no overshoot. In Fig. 7 appears the (scaled) voltage (v/10) applied to the CPL and the computed current for the same three different values of k p and k i . After the transient, the currents are sinusoidal and perfectly in-phase with the voltage. Table 1 below summarizes the numerical LMI results with v M = √ 2 V running the solver with different k M p .

Concluding Remarks and Future Research

We have presented in the paper a possible scenario for the emulation of CPLs in single-phase AC systems. The inputs of the system are the voltage signal, which is only assumed to be periodic, and the desired value of the constant power load. The in-phase current that will generate this active power is calculated with a dynamic controller.

We have tried three different versions of the latter: pro-portional+bias, integral and proportional+integral, for which we dispose of a rigorous theoretical analysis to determine bounds on their tuning gains, hence these results are compared with numerical validation.

Because of space limitations, we have concentrated in the emulation of active power. However, the extension to include reactive power is immediate. Indeed, reactive power can be calculated5 as Q(t) = -1 2π t t-T v(s)i(s)ds, see [START_REF] Garcia-Canseco | Power factor compensation of electrical circuits: The nonlinear non-sinusoidal case[END_REF]) for a control-oriented discussion on reactive power. Taking into account this definition, a single-phase constant reactive power load can be emulated changing, in Fig. 1, the voltage carrier, v(t), by its derivative, v(t). Obviously, it is also possible to emulate a constant active and reactive power load using two closed-loop systems: one to compute the current for the constant active power load and the other to compute the current for the constant reactive power load. Then, the total current is the sum of the currents from each system as they are in quadrature.

M p 0 1 • 10 -6 2 • 10 -6 4 • 10 -6 6 • 10 -6
Proposition 2 2.2923 • 10 -4 1.6027 • 10 -4 1.5466 • 10 -4 1.2985 • 10 -4 0.7140 • 10 -4

Corollary 1 3.1077 • 10 -4 2.1727 • 10 -4 2.0967 • 10 -4 1.7604 • 10 -4 0.9680 • 10 -4

Fridman, E. & Shaikhet, L. ( 2016), 'Delay-induced stability of vector second-order systems via simple Lyapunov functionals', Automatica 74, 288-296.

Fridman, E. & Zhang, J. ( 2020), 'Averaging of linear systems with almost periodic coefficients: A time-delay approach', Automatica 122, 109287.

A Proof of Proposition 1

Substituting u(t) = k p [P -P (t)] + 1 V 2 P into (5) we obtain

P (t) = kp T t t-T v 2 ( )[P -P ( )]d + P , (A.1)
or, equivalently,

P (t) = - kp T t t-T v 2 ( ) P ( )d . (A.2)
Consequently, taking into account that the function v(t) is T -periodic, we obtain

| P (t)| ≤ kp T t t-T v 2 ( )| P ( )|d ≤ kp T T 0 v 2 ( )d sup ∈[t-T,t] | P ( )| = k p V sup ∈[t-T,t] | P ( )|, t ≥ 0. By using Lemma 1 in (Mazenc, Malisoff & Niculescu 2017) with the condition k p V < 1, we arrive at | P (t)| ≤ sup ∈[-T,0] | P ( )|e ln(kp V ) T
t . Thus, (7) holds. 222

B Proof of Proposition 2

We will employ Lemma 1 given in Appendix D.

(i) To complete the proof of (i) we make the following simple, but important, observation. For the PI control law (11), we have the relation 6) and taking into account Ṗ (t) = -Ṗ (t), we arrive at

u(t) -u(t -T ) = t t-T u(ξ)dξ = k p t t-T Ṗ (ξ)dξ + k i t t-T P (ξ)dξ. (B.1) Substituting (B.1) into (
Ṗ (t) = - kp T v 2 (t) t t-T Ṗ (ξ)dξ -ki T v 2 (t) t t-T P (ξ)dξ. (B.2) Then, conditions (7) are satisfied if (B.2) is asymptoti- cally stable.
For the stability analysis of the time-varying system (B.2) with the T -periodic coefficient v 2 (t), we suggest to use the averaging method. We will employ a constructive time-delay approach to averaging introduced recently in (Fridman & Zhang 2020). Following this approach, we will integrate (B.2) on t ∈ [t -T, t] for t ≥ T . Note that similar to (Fridman & Shaikhet 2016), we get

1 T t t-T Ṗ (ξ)dξ = 1 T [ P (t) -P (t -T )] = d dt [ P (t) -G], (B.3) where G = 1 T t t-T (ξ -t + T ) Ṗ (ξ)dξ. (B.
ż(t) = - kp T 2 t t-T v 2 (ξ) ξ ξ-T Ṗ (θ)dθdξ -ki T 2 t t-T v 2 (ξ) ξ ξ-T P (θ)dθdξ = - kp T 2 t t-T v 2 (ξ) ξ ξ-T Ṗ (θ)dθdξ -ki T 2 t t-T v 2 (ξ) ξ ξ-T dθdξ P (t) + ki T 2 t t-T v 2 (ξ) ξ ξ-T [ P (t) -P (θ)]dθdξ, t ≥ T.
The latter equation can be written as

ż(t) = -k i V 2 P (t) + k i Y + k i X, t ≥ T, (B.5)
where

Y = 1 T 2 t t-T ξ ξ-T t θ v 2 (ξ) Ṗ (τ )dτ dθdξ, X = - kp kiT 2 t t-T ξ ξ-T v 2 (ξ) Ṗ (θ)dθdξ. (B.6)
Summarizing, if P (t) is a solution to (B.2), then it satisfies the time-delay system (B.5). Therefore, the stability of the time-delay system guarantees the stability of the original system.

We will derive the stability conditions for the time-delay system (B.5) via Lyapunov-Krasovskii's method. Towards this end, we choose the function

V z = z 2 (t). (B.7) Differentiating V z along (B.5) we have Vz = 2[ P (t) -G][-k i V 2 P (t) + k i Y + k i X]. (B.8)
By applying Jensen's inequality (D.1), we obtain

2G 2 ≤ t t-T (ξ -t + T ) Ṗ 2 (ξ)dξ, (B.9)
whereas via Jensen's inequalities (D.2) and (D.3) we find

X 2 = k 2 p k 2 i T 4 ( t t-T ξ ξ-T v 2 (ξ) Ṗ (θ)dθdξ) 2 ≤ k 2 p v 4 M k 2 i T 4 ( t t-T ξ ξ-T Ṗ (θ)dθdξ) 2 ≤ k 2 p v 4 M k 2 i T 2 t t-T ξ ξ-T Ṗ 2 (θ)dθdξ, (B.10) Y 2 = 1 T 4 ( t t-T ξ ξ-T t θ v 2 (ξ) Ṗ (τ )dτ dθdξ) 2 ≤ v 4 M T 4 ( t t-T ξ ξ-T t θ Ṗ (τ )dτ dθdξ) 2 ≤ v 4 M T t t-T ξ ξ-T t θ Ṗ 2 (τ )dτ dθdξ (B.11)
with v M given by ( 14). To compensate the G-term, similarly to (Fridman & Shaikhet 2016), we will use

V G = ki T r t t-T (ξ -t + T ) 2 Ṗ 2 (ξ)dξ, r > 0. (B.12)
We have

VG = k i rT Ṗ 2 (t) -2ki T r t t-T (ξ -t + T ) Ṗ 2 (ξ)dξ. (B.13) Then, due to (B.9) VG ≤ k i rT Ṗ 2 (t) -4ki T rG 2 . (B.14)
For the X-term, we consider

V X = k 2 p kiT 3 r 1 t t-T ξ ξ-T (ξ -t + T ) Ṗ 2 (θ)dθdξ, r 1 > 0. (B.15) Then, VX = k 2 p kiT 3 r 1 (T t t-T Ṗ 2 (ξ)dξ - t t-T ξ ξ-T Ṗ 2 (θ)dθdξ). (B.16) Via (B.10), we have VX ≤ k 2 p kiT 2 r 1 t t-T Ṗ 2 (ξ)dξ -ki v 4 M T r 1 X 2 . (B.17)
To compensate the positive term in the right-hand side of (B.17), we employ

ṼX = k 2 p ki s t t-T (ξ -t + T ) Ṗ 2 (ξ)dξ, s > 0. (B.18) We have VX = k 2 p ki s(T Ṗ 2 (t) - t t-T Ṗ 2 (ξ)dξ). (B.19)
From (B.17) and (B.19), it follows that

VX + VX ≤ k 2 p T ki s Ṗ 2 (t) -ki v 4 M T r 1 X 2 - k 2 p ki (s -1 T 2 r 1 ) t t-T Ṗ 2 (ξ)dξ. (B.20)
To compensate Y in (B.8), we consider

V Y = ki T 2 q t t-T ξ ξ-T t θ (ξ -t + T ) Ṗ 2 (τ )dτ dθdξ, q > 0. (B.21) Then, VY = ki T q t t-T ξ ξ-T dθdξ • Ṗ 2 (t) + ki T q t t-T t θ Ṗ 2 (τ )dτ dθ -ki T 2 q t t-T ξ ξ-T t θ Ṗ 2 (τ )dτ dθdξ = kiT 2 q Ṗ 2 (t) + ki T q t t-T t θ Ṗ 2 (τ )dτ dθ -ki T 2 q t t-T ξ ξ-T t θ Ṗ 2 (τ )dτ dθdξ.
Further, by using (B.11) we have

VY ≤ kiT 2 q Ṗ 2 (t) + ki T q t t-T t θ Ṗ 2 (τ )dτ dθ -ki v 4 M T qY 2 . (B.22)
To cancel the double integral term in the right-hand side of (B.22), we additionally employ

ṼY = ki T q t t-T t θ (θ -t + T ) Ṗ 2 (τ )dτ dθ. (B.23) Then, VY = ki T q t t-T (θ -t + T )dθ Ṗ 2 (t) -ki T q t t-T t θ Ṗ 2 (τ )dτ dθ = kiT 2 q Ṗ 2 (t) -ki T q t t-T t θ
Ṗ 2 (τ )dτ dθ.

≤ k i T (r + q+ k 2 p k 2 i s) Ṗ 2 (t) -4ki T rG 2 -ki v 4 M T r 1 X 2 -ki v 4 M T qY 2 - k 2 p ki (s -1 T 2 r 1 ) t t-T Ṗ 2 (ξ)dξ. (B.25)
Employing (B.2) and further using Young's inequality and Jensen's inequality (3.87) in [START_REF] Fridman | Introduction to Time-delay Systems: Analysis and Control[END_REF] we have

Ṗ 2 (t) ≤ v 4 M T 2 [k p t t-T Ṗ (ξ)dξ + k i t t-T P (ξ)dξ] 2 ≤ 2v 4 M T 2 [k 2 p ( t t-T Ṗ (ξ)dξ) 2 + k 2 i ( t t-T P (ξ)dξ) 2 ] ≤ 2v 4 M T [k 2 p t t-T Ṗ 2 (ξ)dξ + k 2 i t t-T P 2 (ξ)dξ] (B.26)
with v M given by ( 14). Substitution of (B.26) into (B.25) leads to the term 2k 3

i v 4 M (r + q + k 2 p k 2 i s) t t-T P 2 (ξ)dξ.
To cancel the latter term, we additionally employ

V v M = 2k 3 i v 4 M (r + q + k 2 p k 2 i s) t t-T (ξ -t + T ) P 2 (ξ)dξ (B.27) that leads to Vv M = 2k 3 i v 4 M (r + q + k 2 p k 2 i s)(T P 2 (t) - t t-T P 2 (ξ)dξ).
(B.28) Define a Lyapunov functional as

V := V z + V G + V X + ṼX + V Y + ṼY + V v M , (B.29) where V z , V G , V X , ṼX , V Y ,
t t-T φ 2 (ξ)dξ ≥ 1 T [ t t-T φ(ξ)dξ] 2 with φ(s) = (ξ-t+T ) Ṗ (ξ) leads to V G ≥ k i rG 2 , whereas Jensen's inequality (D.1) leads to ṼX ≥ 2k 2 p ki sG 2 . Hence V ≥ V z + V G + ṼX ≥ P (t) G   1 -1 * 1 + k i r + 2k 2 p ki s   P (t) G ≥ c P 2 (t)
with T -independent c > 0. Thus, V is positive definite. Taking into account (B.8), (B.25) and (B.28), via (12), we arrive at

V ≤ k i η Ξ(k p , k i )η + k 2 p ki [-s + 2v 4 M (k 2 i (r + q) +k 2 p s) + 1 T 2 r 1 )] t t-T Ṗ 2 (ξ)dξ ≤ k i η Ξ(k p , k i )η, η := col{ P (t), G, Y, X},
where Ξ(•, •) is defined by (13). Therefore, conditions (7) are satisfied.

Since k p and k i appear only in the positive terms of ( 12) and Ξ 11 (•, •), the feasibility of LMIs ( 12) and ( 13) with the maximum value of k M p > 0 and k M i > 0 implies their feasibility for all -k M p ≤ k p ≤ k M p and 0 < k i ≤ k M i .

(ii) For k M p = 0, (B.5) holds with X = 0, whereas (B.26) is changed by

Ṗ 2 (t) ≤ k 2 i v 4 M T 2 [ t t-T P (ξ)dξ] 2 ≤ k 2 i v 4 M T t t-T P 2 (ξ)dξ.
Then choosing V as in (B.29) with V X = ṼX = 0 and with V v M changed by 1 2 V v M , and using arguments of (i) we arrive at the result.

222 t t-T (ξ -t + T ) P 2 (ξ)dξ (C.1) with r > 0, r 1 > 0, s > 0 and q > 0, we arrive at V ≤ k i η Ξ(k p , k i )η, η := col{ P (t), G, Y, X}, where Ξ(•, •) is given by ( 17). PROOF. Inequality (D.1) was proved in [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF]). We will prove (D.2) and (D.3). By Schur complement, the following holds

C Proof
x (τ )Rx(τ ) x (τ ) * R -1 ≥ 0. (D.4) Integration of (D.4), respectively, from ξ -T to ξ in θ, from t -T to t in ξ, and from θ to t in τ , from ξ -T to ξ in θ, from t -T to t in ξ, where

Fig. 1 .

 1 Fig. 1. Block diagram representation of the system (6) in closed-loop with feedback controller Σc. The light blue box encloses all the CPL device with port voltage and desired active power inputs and current output.

Fig. 2 .

 2 Fig. 2. Maximum absolute value characteristic multiplier vs. kp.

Fig. 5 .

 5 Fig. 5. (kp, ki) stability charts: (left) stability regions and (right) contour lines for the maximum absolute value characteristic multiplier near kp = 0 and ki = 0.

Fig. 6 .

 6 Fig. 6. P and P (t) (left) and u(t) (right) for the PI controller.

Fig. 7 .

 7 Fig. 7. Voltage, v(t)/10 and CPL current i(t) for the PI controller.

  4) Denote z(t) := P (t) -G. Then, integrating (B.2) and taking into account (B.3) we arrive at

  14), (B.20), (B.22) and (B.24), we obtain VG + VX + VX + VY + VY

(x

  ii) The proof of (ii) is similar to (ii) of Proposition 2τ )dτ dθdξ, where a ≤ b, f : [a, b] → R, x(τ ) ∈ R n and the integration concerned is well defined. Then, for any n × n matrix R > 0 the following Jensen's inequalities hold: (τ )Rx(τ )dτ dθdξ.(D.3)

Table 1

 1 Maximum value of k M i for different k M p via LMIs of Proposition 2 and Corollary 1. k

  of Corollary 1 (i) Averaging of (B.2) over [t -T 2 , t] for t ≥ T 2 leads to (B.5) with

	G = 2 T Y = 2 T 2 X = -kiT 2 t t-T 2 t t-T (ξ -t + T 2 ) Ṗ (ξ)dξ, ξ ξ-T t θ v 2 (ξ) Ṗ (τ )dτ dθdξ, 2 2kp t t-T 2 ξ ξ-T v 2 (ξ) Ṗ (θ)dθdξ.
	Then, choosing Lyapunov functional
	V = z 2 (t) + 2ki T r + 2k 2 p kiT 3 r 1 t t-T 2 + k 2 p ki s t t-T (ξ -t + T ) Ṗ 2 (ξ)dξ t t-T (ξ -t + T 2 ) 2 Ṗ 2 (ξ)dξ 2 ξ ξ-T (ξ -t + T 2 ) Ṗ 2 (θ)dθdξ + 8ki 3T 2 q t t-T ξ ξ-T t θ (ξ -t + T 2 ) Ṗ 2 (τ )dτ dθdξ 2 + 4ki 3T q t t-T t θ (θ -t + T ) Ṗ 2 (τ )dτ dθ +2k 3 i v 4 M ( r 2 + q + k 2 p i k 2 s)

Another real case of interest is when v(t) is a T -periodic signal with only odd harmonics (with zero average). In this situation v 2 (t) is also a T 2 -periodic signal.

This definition characterizes the reactive power of a load that can be fully compensated using a shunt capacitor or a shunt inductor at its port variables and, for AC sinusoidal systems, single-phase and poly-phase, coincides with the usual equations to compute the reactive power.


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