Fabrice Chabi

Elegbede

Alexandra Papadopoulos

Jocelyne Just

Denise Anne Moneret-Vautrin

Antoine Deschildre

Amélie Crepet
email: amelie.crepet@anses.fr

Gender, prick test size and rAra h 2 sIgE level may predict the eliciting dose in patients with peanut allergy: evidence from the Mirabel survey

Background: Peanut allergy management is based on active avoidance and access to emergency treatment including self-injectable adrenaline. Knowing the dose at which a patient is likely to react is crucial for risk assessment and could significantly improve management by integrating a personalized approach.

Objective: To develop a threshold dose distribution curve model from routinely collected data.

Methods:

The MIRABEL survey is an observational study of 785 patients with peanut allergy/sensitization conducted in France, Belgium and Luxemburg. The current analysis included the 238 participants for whom medical and oral food challenge data were available.

Several statistical models (Kaplan-Meier, Cox model, Weibull and Lognormal with predictive factors, basic Weibull and Lognormal) were compared to select the best model and predictive factor combination associated with the threshold doses. Inferences were made with a Bayesian approach.

Results: Patients were mainly children (mean age: 9 years [IQR: [START_REF] Rimbaud | Quantitative Risk Assessment Relating to Adventitious Presence of Allergens in Food: A Probabilistic Model Applied to Peanut in Chocolate[END_REF][START_REF] Crevel | Translating reference doses into allergen management practice: Challenges for stakeholders[END_REF][START_REF] Taylor | Establishment of Reference Doses for residues of allergenic foods: Report of the VITAL Expert Panel[END_REF][START_REF] Blumchen | Modified oral food challenge used with sensitization biomarkers provides more real-life clinical thresholds for peanut allergy[END_REF][START_REF] Boulay | A EuroPrevall review of factors affecting incidence of peanut allergy: priorities for research and policy[END_REF][START_REF] Crevel | Development and evolution of risk assessment for food allergens[END_REF]; 87% < 16 years) and males (62%). Median Ara h2 s IgE was of 8kUA/L [IQR: 1-55] and median skin prick test size of 10 mm [IQR: [START_REF] Crevel | Translating reference doses into allergen management practice: Challenges for stakeholders[END_REF][START_REF] Taylor | Establishment of Reference Doses for residues of allergenic foods: Report of the VITAL Expert Panel[END_REF][START_REF] Blumchen | Modified oral food challenge used with sensitization biomarkers provides more real-life clinical thresholds for peanut allergy[END_REF][START_REF] Boulay | A EuroPrevall review of factors affecting incidence of peanut allergy: priorities for research and policy[END_REF][START_REF] Crevel | Development and evolution of risk assessment for food allergens[END_REF][START_REF] Hourihane | Does severity of low-dose, double-blind, placebo-controlled food challenges reflect severity of allergic reactions to peanut in the community?[END_REF][START_REF] Hourihane | Thresholds of allergenic proteins in foods[END_REF]. OFC was positive in 204 patients (86%). The median threshold dose was of 67 mg of peanut protein [IQR: . The dose at which 1% of the patients are likely to react with objective symptoms was 0.26 [0.03; 2.24] mg of peanut protein. Gender, size of the skin prick test (SPT) and Ara h 2 specific IgE level had a significant impact on the threshold dose distribution curve. The Cox model was the most effective to predict threshold doses with this combination of factors. Girls react to lower doses than boys with a beta coefficient associated to the risk and a 95% credible interval of 0.44 [0.04; 0.77]. The higher the size of the SPT and the Ara h 2 specific IgE level are, the higher the risk of reacting to a small amount of peanut, with beta coefficients associated to the risk and 95% credible intervals of 0.05 [0.02; 0.08] and 0.01 [0.01; 0.02] respectively. Conclusion and clinical relevance: according to the model, routinely collected data could be used to estimate the threshold dose. The consequences could be the identification of high-risk patients who are susceptible to react to small amounts of peanut and a personalized management of peanut allergy integrating the risk of allergic reaction. Limitations of this study are that assessors of OFC outcome were aware of SPT and Arah2 results, and a further validation study is required to confirm the predictive value of these parameters. A dose distribution model including significant predictive factors improves threshold dose assessment in peanut allergic patients. Routinely collected data (gender, level of Ara h 2 sIgE and size of skin prick test) can be used by clinicians to estimate the allergic risk and consequently to personalize management.

Keywords

Key messages (highlights box)

 The threshold of the allergic population is often described using only one threshold distribution curve whatever distinctive features which may be observed between the patients in the study population.

 Significant predictive factors impacting OFC threshold dose were identified and the Cox model (integrating these factors) produces the best fit of the OFC threshold distribution curve.

 Physicians could use routinely collected data to identify patients with a higher risk of reaction at low doses (high-risk patients). This approach could result in a more personalized management integrating the risk of allergic reaction.

Introduction

Patients with food allergy react to a wide range of amounts of food. The dose of food below which no reaction is observed is known as the "threshold" dose [START_REF] Crevel | Chapter five -Thresholds or 'How Much Is Too Much?[END_REF]. Individual thresholds can be estimated from oral food challenges (OFC), a time consuming and expensive procedure which can also cause severe reactions. Clinical threshold doses lie between the highest dose observed not to produce any adverse effect (NOAEL) and the lowest dose to produce an adverse effect (LOAEL). In practice, the threshold dose is often estimated as being the LOAEL.

Thresholds can also be defined at a population level. Thus, thresholds are defined as the minimum eliciting dose (ED) which designates the amount of allergenic food to produce a reaction in a determined proportion 𝑝 of the allergic population (EDp). Population thresholds are characterized by fitting threshold distribution curves to individual thresholds determined by OFC. Accurately estimating a threshold at individual and population levels is crucial as it plays an important role in food allergy risk assessment and management [START_REF] Crevel | Hazard characterisation in food allergen risk assessment: The application of statistical approaches and the use of clinical data[END_REF][START_REF] Taylor | Factors affecting the determination of threshold doses for allergenic foods: how much is too much?[END_REF]. Indeed, threshold doses combined with consumed quantities and allergen concentrations in food are used by risk assessors to estimate the risk of the allergic population. [START_REF] Spanjersberg | Risk assessment and food allergy: the probabilistic model applied to allergens[END_REF], [START_REF] Madsen | Approaches to risk assessment in food allergy: report from a workshop ''developing a framework for assessing the risk from allergenic foods[END_REF], [START_REF] Rimbaud | Quantitative Risk Assessment Relating to Adventitious Presence of Allergens in Food: A Probabilistic Model Applied to Peanut in Chocolate[END_REF] Threshold doses can also be used by risk managers to establish safe limits, i.e. a maximum amount of unintended allergen in food to protect the majority of the allergic population [START_REF] Crevel | Translating reference doses into allergen management practice: Challenges for stakeholders[END_REF][START_REF] Taylor | Establishment of Reference Doses for residues of allergenic foods: Report of the VITAL Expert Panel[END_REF]. Finally, threshold doses are essential for the allergist to make diet recommendations to their patients.

Many individual factors -such as age, gender, comorbidities, stress, effort-could impact the threshold doses and thus the associated population threshold dose distribution [START_REF] Blumchen | Modified oral food challenge used with sensitization biomarkers provides more real-life clinical thresholds for peanut allergy[END_REF][START_REF] Boulay | A EuroPrevall review of factors affecting incidence of peanut allergy: priorities for research and policy[END_REF][START_REF] Crevel | Development and evolution of risk assessment for food allergens[END_REF][START_REF] Hourihane | Does severity of low-dose, double-blind, placebo-controlled food challenges reflect severity of allergic reactions to peanut in the community?[END_REF][START_REF] Hourihane | Thresholds of allergenic proteins in foods[END_REF]. This explains why individual variability should be taken into account when modeling threshold dose distribution curves. There are only a few published studies which focus on how factors related to allergy can be combined to predict individual threshold doses and the threshold distribution curve [START_REF] Van Der Zee | The eliciting dose of peanut in double-blind, placebo-controlled food challenges decreases with increasing age and specific IgE level in children and young adults[END_REF]. Indeed, methods are generally based on considering the best fitting of individual thresholds by several parametric probability distributions without using predictive factors [START_REF] Crevel | Hazard characterisation in food allergen risk assessment: The application of statistical approaches and the use of clinical data[END_REF][START_REF] Taylor | Establishment of Reference Doses for residues of allergenic foods: Report of the VITAL Expert Panel[END_REF]. Thus, the threshold of the allergic population is often described using only one threshold distribution curve whatever distinctive features may be observed between the patients in the study population.

The MIRABEL project was conducted by the French Agency for Food, Environmental and Occupational Health & Safety (ANSES) in partnership with the French National Institute for Agricultural Research (INRA) and the Allergy Vigilance Network (www.allergyvigilance.org).

The objectives and the methods of the MIRABEL survey have been previously published [START_REF] Crépet | MIRABEL: an integrated project for risk and cost/benefit analysis of peanut allergy[END_REF][START_REF] Guenard-Bilbault | Allergie à l'arachide en France : premiers résultats de l'étude pilote du programme MIRABEL : « Approche intégrée pour l'évaluation du risque et des coûts/bénéfices liés aux allergènes alimentaires[END_REF], as well as the characteristics of the population, mainly children (< 16 years: 86%) and primary peanut allergic patients [START_REF] Deschildre | Peanut allergic patients in the MIRABEL survey: comorbidities and specificities of eliciting dose in real-life[END_REF][START_REF] Just | studygroup M, Two severe peanut-allergy phenotypes with gender difference: Evidence from the MIRABEL survey[END_REF]. One of the objectives was to accurately define the population threshold distribution for peanut in order to assess allergic risk and to improve food labeling. This study presents an original approach to estimate threshold doses using predictive factors. The model is based on routinely collected medical data. It could be a tool for allergists to identify high-risk patients who are susceptible to react to small amounts of peanut and to improve peanut allergy management with a personalized approach integrating the risk of allergic reaction.

Materials and methods

Study design and population

The MIRABEL study is an observational multicenter survey based on the voluntary participation of patients from Metropolitan France, Belgium and Luxembourg, recruited from April 2012 to December 2013 during visits to their allergists. The 70 allergists who participated in the recruitment were affiliated to the Allergy Vigilance Network and were representative of the medical practice of all the members of the network (76% office-based and 24% hospital-based).

Details have previously been described [START_REF] Crépet | MIRABEL: an integrated project for risk and cost/benefit analysis of peanut allergy[END_REF][START_REF] Guenard-Bilbault | Allergie à l'arachide en France : premiers résultats de l'étude pilote du programme MIRABEL : « Approche intégrée pour l'évaluation du risque et des coûts/bénéfices liés aux allergènes alimentaires[END_REF][START_REF] Deschildre | Peanut allergic patients in the MIRABEL survey: comorbidities and specificities of eliciting dose in real-life[END_REF][START_REF] Just | studygroup M, Two severe peanut-allergy phenotypes with gender difference: Evidence from the MIRABEL survey[END_REF]. All the recruited patients were sensitized to peanut (wheal diameter of skin prick test (SPT) ≥ 3 mm and/or peanut specific IgE ≥ 0.35 kUA/L). Allergic patients were those who reported an allergic reaction to peanut at their first visit to the allergist. Severe or potentially severe reactions were: anaphylactic shock, laryngeal angioedema, acute asthma, and serious systemic reaction (involving two or more organs). Nonsevere reactions were: rash/dermatitis, urticarial or subcutaneous angioedema, gastro-intestinal symptoms, others. Patients without any previous clinical reaction in real life at the time of the first visit and before the OFC (if done) were considered to be sensitized rather than allergic.

The study population for this analysis was the subgroup of patients for whom a peanut OFC was performed. The study was approved by the French Data Protection Authority (CNIL) (Authorization no. DE-2011-048). All patients or parents signed an informed consent.

Data collection

An anonymized standard medical questionnaire was completed by the allergists. All the collected data are described elsewhere [START_REF] Crépet | MIRABEL: an integrated project for risk and cost/benefit analysis of peanut allergy[END_REF][START_REF] Guenard-Bilbault | Allergie à l'arachide en France : premiers résultats de l'étude pilote du programme MIRABEL : « Approche intégrée pour l'évaluation du risque et des coûts/bénéfices liés aux allergènes alimentaires[END_REF][START_REF] Deschildre | Peanut allergic patients in the MIRABEL survey: comorbidities and specificities of eliciting dose in real-life[END_REF][START_REF] Just | studygroup M, Two severe peanut-allergy phenotypes with gender difference: Evidence from the MIRABEL survey[END_REF]. The variables selected for the statistical analysis were those which could impact the threshold dose, in line with previous studies [START_REF] Deschildre | Peanut allergic patients in the MIRABEL survey: comorbidities and specificities of eliciting dose in real-life[END_REF][START_REF] Just | studygroup M, Two severe peanut-allergy phenotypes with gender difference: Evidence from the MIRABEL survey[END_REF] gender, diagnosis by the allergist at the first visit (allergy, sensitization), the age at which the OFC was performed, allergic comorbidities (atopic dermatitis, asthma, allergic rhinitis, other food allergy), and the results of the following allergic tests: wheal diameter of the SPT for peanut expressed in mm and the level of Ara h 2 sIgE expressed in kUA/L (immunoCAP system, Thermofisher, Uppsala, Sweden). Composite variables were used to distinguish patients according to the number of allergic comorbidities.

Threshold dose from oral food challenge

In this observational survey, threshold doses of peanut were obtained by single-blinded, doubleblinded placebo-controlled or open OFCs, according to the French guidelines [START_REF] Santos | Oral food challenge in children: who, when, and how? Procedures Revue Française d[END_REF][START_REF] Fauquert | Oral food challenge in children: who, when, and how? Interpretation[END_REF]. The threshold dose was defined as the lowest cumulative dose of roasted crushed peanuts causing an objective reaction, converted into mg of peanut protein (LOAEL). The OFC was considered negative in the absence of an objective reaction for a cumulative dose ≥ 7 g of peanut (1.75 g of peanut protein). The threshold dose of patients diagnosed as allergic but who did not react during an incomplete OFC were considered above the highest cumulative dose in the challenge trial, and treated as right-censored [START_REF] Van Der Zee | The eliciting dose of peanut in double-blind, placebo-controlled food challenges decreases with increasing age and specific IgE level in children and young adults[END_REF] in the analysis. The allergic or sensitized patients who did not react during a complete OFC and sensitized patients who did not react during an OFC which was stopped before the last incremental dose were excluded from the study to focus only on patients with confirmed allergy.

Statistical methods for threshold dose distribution modeling

Parametric Weibull, Lognormal, and Loglogistic distributions were used to model the threshold dose distribution [START_REF] Crevel | Hazard characterisation in food allergen risk assessment: The application of statistical approaches and the use of clinical data[END_REF]. In order to study the association of the predictive variables with the threshold doses, an approach integrating the predictive factors as covariables in the Weibull and Lognormal models was developed (Online Repository). The Cox regression model generally used in epidemiology to estimate the association between a disease and predictive factors during time [START_REF] Hosmer | Applied Survival Analysis: Regression Modeling of Time to Event Data[END_REF][START_REF] Cox | Regression Models and Life Tables (with Discussion)[END_REF] was adapted to the allergy topic. The occurrence of the allergic reaction was considered as the disease and the threshold dose was used instead of the time(details are provided in the Online Repository). All the models were adapted to account for right censored data. First, each variable was introduced singly in the different models to test their influence on the threshold dose. Variables with a 95% confidence interval of their estimated coefficient that does not contain 0 were considered as significant. These variables were then introduced together in the models and their interactions were tested. Basic modeling of the threshold dose distribution by Weibull, Lognormal and Loglogistic distributions was also conducted in order to compare our results with literature. The models were compared using two criteria: the Bayesian Information Criterion (BIC, commonly used for model selection and which penalizes the complexity of the models compared with the AIC), and the Root Mean Square Error (RMSE, commonly used to measure of the difference between values predicted by a model and the values observed). Inferences were performed using the Bayesian approach and were implemented in the OpenBUGS software version [START_REF] Lunn | The BUGS project: Evolution, critique and future directions[END_REF] via the BRugs package of R version 3.0.1. Simulations of the ED values from parameter estimates were done using second order Monte-Carlo simulations to separate variability and uncertainty [START_REF] Rimbaud | Quantitative Risk Assessment Relating to Adventitious Presence of Allergens in Food: A Probabilistic Model Applied to Peanut in Chocolate[END_REF] (details are provided in the Online Repository Online Repository).

Results

Study population

Overall, 785 patients were included in the MIRABEL survey [START_REF] Crépet | MIRABEL: an integrated project for risk and cost/benefit analysis of peanut allergy[END_REF][START_REF] Deschildre | Peanut allergic patients in the MIRABEL survey: comorbidities and specificities of eliciting dose in real-life[END_REF][START_REF] Just | studygroup M, Two severe peanut-allergy phenotypes with gender difference: Evidence from the MIRABEL survey[END_REF]. Out of these, 280 patients underwent an OFC (open OFC: 55%) and were considered for inclusion in the present analysis. One patient who did not react during a complete OFC, 14 sensitized patients with incomplete OFC, and 27 patients with missing data were not retained. The study population thus comprised 238 patients. The allergy was confirmed by a positive OFC in 86% of cases.

From the 238 patients, 29 initially considered as sensitized patients had a positive OFC (12.2%) and 34 allergic patients did not react during incomplete OFC (14.3%). The patients were mainly children (mean age at the date of the OFC: 9 years (6-11) and 87% under 16 years), and males (62%) (Table 1). The median threshold dose was of 67 mg (16-244) of peanut protein. . The minimum and the maximum threshold doses were 0.03 and 2404 mg of peanut protein, respectively.

Predictive factors of threshold doses

In univariate analysis, the level of Ara h 2 sIgE, the size of the SPT and presence of atopic dermatitis were significantly associated with the ED, whatever the model (Table 1): the greater the SPT size and level of Ara h 2 and the absence of atopic dermatitis, the lower the ED. Gender was also significantly associated with the threshold dose for the Cox model. Table 2 presents the combination of significant predictive variables for each model. Using the Weibull and Cox models, the combination of the variables significantly influencing the threshold dose were: gender, SPT size, level of Ara h 2 sIgE and the interaction between gender and the level of Ara h 2 sIgE. The Lognormal and Loglogistic models were composed of the combination of the SPT size and the level of Ara h 2 sIgE. Figure 1 depicts that girls react to lower doses than boys for identical SPT size and level of Ara h 2 sIgE (Cox model). For a given amount of peanut protein, SPT size and level of Ara h 2 sIgE, the risk of reaction for girls is 1.22-fold higher than for boys. Figure 2(a,b) shows that the larger the SPT, the higher the risk of reacting to a small amount of peanut. Figure 2(c,d) also shows the same risk increase according to the level of Ara h 2 sIgE.

Comparison of models

Table 2 shows comparison results of the Bayesian Information Criterion (BIC) and the Root Mean Square Error (RMSE). Both clearly demonstrated that the Cox model was the best model.

The BIC identified the Loglogistic models with and without predictive factors as giving the second best results, followed by the Weibull with predictive factors. In contrast, the Weibull model with and without predictive factors scored second best with the RMSE criterion followed by the Loglogistic models with predictive factors.

Predicted eliciting doses (EDp)

The 3 also shows that whatever the model used, females had lower EDs than males.

Prediction of threshold dose for a given patient in clinical practice

Figure 4 shows how the threshold dose range related to a given allergic risk can be predicted for a patient from the SPT size and level of Ara h 2 sIgE using the Cox model. For example, with an SPT size of 10 mm and a level of Ara h 2 sIgE of 42 kU/L, the threshold doses triggering a reaction in 1% of the allergic females range between 0.04 and 0.89 with a median value of 0.17 mg of peanut protein. For similar values of SPT size and Ara h 2 sIgE level, the threshold doses triggering a reaction in 10% and 50% of allergic females range between [2.36, 16.1] with a median of 6.33 and [18.3, 136] with a median of 67.3 mg of peanut protein, respectively.

Discussion

Our analysis shows that the Cox model results in the best predictions of EDp, close to the known EDs reported in the literature. The modeling we describe identifies significant predictive factors of the threshold dose in a large population, mainly children (86%: < 16 years of age), with primary peanut allergy. It is based on routinely collected medical data. This modeling could be a tool for allergists to approach allergic risk from routinely collected medical data, and consequently to better manage high-risk patients (identification of the risk to react at low doses, advices on the consumption of products with PAL, indication and procedure of OFC).

Factors influencing the threshold dose

Univariate analysis identified four variables which significantly influence the threshold dose.

In fact, the female gender, a diagnosis of peanut allergy without the presence of atopic dermatitis, a high level of Ara h 2 sIgE and a large SPT size all lead to a lower threshold dose of reaction. Similarly, van der Zee et al. [START_REF] Van Der Zee | The eliciting dose of peanut in double-blind, placebo-controlled food challenges decreases with increasing age and specific IgE level in children and young adults[END_REF] also reported that the level of peanut sIgE and the absence of atopic dermatitis were closely associated to lower threshold doses from a study of 126 OFCs in peanut allergic children modeled by a Cox model. In contrast to our results, they also observed an age effect, with lower threshold doses for teenagers, but not a gender effect.

The reason why atopic dermatitis may affect the threshold dose is not clearly understood. One hypothesis, by van der Zee, Dubois [START_REF] Van Der Zee | The eliciting dose of peanut in double-blind, placebo-controlled food challenges decreases with increasing age and specific IgE level in children and young adults[END_REF], is that the presence of atopic dermatitis may mask early, mild cutaneous symptoms in the setting of an OFC. Blumchen, Beder [START_REF] Blumchen | Modified oral food challenge used with sensitization biomarkers provides more real-life clinical thresholds for peanut allergy[END_REF] used a lognormal distribution to model individual peanut thresholds from a population of 63 peanut allergic children explored with a modified OFC protocol (dose increment every 2 hours). They observed that the threshold dose was significantly and inversely correlated with peanut and Ara h 2 sIgE levels, SPT size, basophil activation, and TH2 cytokine production by peripheral blood mononuclear cells. However, symptom severity did not correlate with the threshold or any of these markers. Finally, Santos, Du Toit [START_REF] Santos | Distinct parameters of the basophil activation test reflect the severity and threshold of allergic reactions to peanut[END_REF] assessed the basophil activation test (BAT) as a means of predicting the severity and threshold of reactivity to peanut during OFCs in 124 children, including 52 with a positive OFC. They concluded that BAT can be used not only to estimate the threshold of allergic reactions during OFCs, but also the severity of the reaction. However, this result was not confirmed by Reier-Nilsen, Michelsen [START_REF] Reier-Nilsen | Predicting reactivity threshold in children with anaphylaxis to peanut[END_REF], who evaluated the accuracy of clinical and/or immunological characteristics to predict OFC reactivity threshold and the severity of the reaction in a population of 96 children (5 to 15 years old) with a history of severe allergic reactions to peanut and/or sensitization to peanut. They observed that BAT had the best accuracy to predict reactivity threshold as well as LOAEL, but not the severity of the reaction. ED was also associated to the gender, with a lowest dose in females. We have previously shown that severe peanut-allergic phenotypes were more frequent in girls [START_REF] Just | studygroup M, Two severe peanut-allergy phenotypes with gender difference: Evidence from the MIRABEL survey[END_REF].

McWilliam, Koplin [START_REF] Mcwilliam | Self-reported adverse food reactions and anaphylaxis in the SchoolNuts study: A population-based study of adolescents[END_REF] found that female adolescents of the Australian "SchoolNuts" cohort were more likely to report experiencing any adverse food reaction in the past 12 months but found no significant sex difference associated with anaphylaxis and did not show data on ED [START_REF] Mcwilliam | Self-reported adverse food reactions and anaphylaxis in the SchoolNuts study: A population-based study of adolescents[END_REF]. The impact of gender on ED and FA severity are lacking and our finding has to be confirmed in prospective studies.

In our study, the best model was the Cox model composed of gender, SPT size, the level of Ara h 2 sIgE and the interaction between gender and the Ara h 2 sIgE level. This interaction implies that the association between the Ara h 2 sIgE and the threshold dose is different between girls and boys. The presence of atopic dermatitis disappeared with adjustment for other variables.

The fact that the combination of SPT size and Ara h 2 sIgE level is the best predictor of the threshold dose is a new concept. Several studies have been conducted to define predictive factors to diagnose peanut allergy [START_REF] Codreanu | A Novel Immunoassay Using Recombinant Allergens Simplifies Peanut Allergy Diagnosis[END_REF][START_REF] Dunngalvin | Highly accurate prediction of food challenge outcome using routinely available clinical data[END_REF][START_REF] Klemans | Ara h 2 Is the Best Predictor for Peanut Allergy in Adults[END_REF][START_REF] Klemans | The diagnostic value of specific IgE to Ara h 2 to predict peanut allergy in children is comparable to a validated and updated diagnostic prediction model[END_REF][START_REF] Nicolaou | Quantification of specific IgE to whole peanut extract and peanut components in prediction of peanut allergy[END_REF]. Our analysis contributes to improving the prediction of the threshold dose using a combination of routinely available data of allergic diagnosis such as Ara h 2 sIgE and SPT. These results have to be confirmed with further data and analyses.

The ED01 value

The criteria used to test the goodness-of-fit concluded that the Cox model followed by the Weibull model with predictive factors is the most appropriate one to model threshold data.

Taylor, Baumert [START_REF] Taylor | Establishment of Reference Doses for residues of allergenic foods: Report of the VITAL Expert Panel[END_REF] consented to a reference dose of 0.2 mg of peanut protein which is a consensus of ED01 values estimated from 16 published studies using basic Loglogistic and Lognormal distributions. This value is close to the one obtained with the Cox model and the Weibull model and comprised in their credible intervals, which reinforce our results. The basic Weibull proposed lower ED values and the basic Lognormal and Loglogistic higher ED values.

The fact that Weibull distribution produced lower values than other models has already been observed in previous studies [START_REF] Crevel | Hazard characterisation in food allergen risk assessment: The application of statistical approaches and the use of clinical data[END_REF][START_REF] Taylor | Threshold dose for peanut: Risk characterization based upon published results from challenges of peanutallergic individuals[END_REF]. Blumchen, Beder [START_REF] Blumchen | Modified oral food challenge used with sensitization biomarkers provides more real-life clinical thresholds for peanut allergy[END_REF] found a ED05 value of 1.95 mg of peanut protein which is lower than the ones obtained with our models (around 3 mg of peanut protein except for the Weibull one at 1.4 mg peanut protein). Blom, Vlieg-Boerstra [START_REF] Blom | Threshold dose distributions for 5 major allergenic foods in children[END_REF] reported a ED05 value of 1.6 mg in a population of 135 peanut allergic children. These differences can largely be explained by the characteristics of the populations, and the differences in the OFC protocol including subjective versus objective stopping criteria, as discussed by Taylor, Houben [START_REF] Taylor | Understanding food allergen thresholds requires careful analysis of the available clinical data[END_REF].

Variability and uncertainty

Our study was multicenter and patients were recruited by both office-and hospital-based allergists. This made it possible to integrate inter-individual variability. Bayesian modeling coupled with second order Monte-Carlo simulations makes it possible to account for both interindividual variability and model uncertainty [START_REF] Rimbaud | Quantitative Risk Assessment Relating to Adventitious Presence of Allergens in Food: A Probabilistic Model Applied to Peanut in Chocolate[END_REF]. With the basic model, the credible interval of the threshold distribution curves reflects only uncertainty, whereas with the models integrating predictive factors, they also reflect inter-individual variability. Indeed, for fixed values of model parameters randomly selected in their distribution, a cumulative distribution function can be calculated reflecting the differences between individual threshold doses in the MIRABEL population. If this process is repeated for different values of the model parameters, a credible interval reflecting model parameter uncertainty can be estimated. When the predictive factors are taken into account, a cumulative distribution function is obtained not only for a fixed value of model parameters but also for fixed values of the factors. Therefore, the credible interval accounts both for model parameter uncertainty and inter-individual variability.

However, our study has some limitations. As repeated measures of threshold dose for each patient were not available, we could not account for intra-individual variability. Moreover, patients selected to be tested by OFC might have specific characteristics and thus did not reflect general peanut allergic patients. Regarding MIRABEL patients, we did not observe any discrepancies between patients with and without OFCs, except for asthma comorbidity and age at diagnosis [START_REF] Deschildre | Peanut allergic patients in the MIRABEL survey: comorbidities and specificities of eliciting dose in real-life[END_REF]. Due to the recruitment process, allergic individuals who did not consult allergists could not have been included in our survey and lead to selection bias. Moreover, allergists who performed the OFC were aware of the peanut sIgE or prick test results and the assessment of the reaction occurring during the OFC was not standardized. Data on individual NOAELs were not available. Therefore, it was not possible to use the method proposed by Taylor, Crevel [START_REF] Taylor | Threshold dose for peanut: Risk characterization based upon published results from challenges of peanutallergic individuals[END_REF] to account for the fact that the reacting dose ranges between the NOAEL and the LOAEL and not necessarily equals the LOAEL value. This could lead to an underestimation of the frequency of lower threshold doses. Finally, it is important to mention that threshold doses in this work are related to reaction by ingestion and are not appropriate to other exposure routes such as inhalation.

Consequences for medical practices and risk management

Our findings could have implications for risk assessment (threshold dose modeling), for patient management (identification of the most sensitive patients, "low threshold reactors") and diet advice (products with precautionary labeling), OFC indications and procedure (low-dose challenge protocol, single dose protocol) and for policy makers (food labeling, precautionary allergen labeling) [START_REF] Epa | Clinical implications of food allergen thresholds[END_REF].

Integrating predictive factors by routinely collected data make it possible to integrate interindividual variability. This is particularly significant when threshold dose modeling is used to perform risk assessment. To be conservative, it is important to consider the lower bound of the threshold dose confidence interval integrating differences between individuals instead of a median value. In the same way, this type of modeling produces a threshold dose distribution per group of allergic individuals presenting similar predictive factors as opposed to the basic modeling which only takes into account one distribution for a whole population. This is particularly relevant for clinicians giving dietary advice or to choose the OFC procedure or to modulate the dose escalation of the OFC [START_REF] Cochrane | Development of a standardized low-dose double-blind placebo-controlled challenge vehicle for the EuroPrevall project[END_REF][START_REF] Hourihane | Peanut Allergen Threshold Study (PATS): Novel single-dose oral food challenge study to validate eliciting doses in children with peanut allergy[END_REF]. Thus, while the prediction of a threshold level with the proposed model provides additional information, it should be interpreted in the light of the patient's clinical history and presence of other risk factors. Furthermore, it does not give information about the severity of the reaction but can be used to identify patients who are at risk of reacting to small amounts of the allergen. It is consequently a step forward to a more personalized approach, integrating allergic reaction risk [38]. Finally, our study is also important for manufacturers and public health agencies as they could use threshold doses for improving labeling practices in the food industry and developing standardized policies, and consequently increasing food safety and peanut allergic consumer confidence. (2) 95% CI (2) 95% CI (2) 95% CI (2) Oral food challenges -Positive, n (%) -Threshold dose mg of peanut protein Median (IQR) (1) [0; 5[, n (%) (2) 𝛷: the cumulative distribution function of the standard normal distribution.

(3) h0(d): the baseline hazard, as defined in the repository material. (4) BIC: Bayesian Information Criteria, the lower the BIC value, the best the model (5) NA = Not Available. Sometimes, the bound of confidence interval for the Kaplan Meier approach cannot be estimated as Greenwood's formula, used to construct the bound of the confidence interval, does not work when the cumulative probability of reaction is close to 1.

  Abbreviations and definitions

  ED01 calculated from the Weibull with predictive factors and Cox models were close to the Kaplan Meier estimates: respectively 0.20 [0.02; 0.99] and 0.26 [0.03; 2.24] versus 0.19 [0.03; 2.25] mg of peanut protein (Table 3), with a smaller 95% credible interval for the Weibull model. The Lognormal model with predictive factors gave a higher value for the ED01 and the highest credible interval: 1.33 [0.15; 3.93] mg of peanut protein. The Loglogistic model with predictive factors produced intermediate results: 0.89 [0.08; 3.46] mg of peanut protein. The basic Weibull, Lognormal and Loglogistic distributions produced lower values for the ED01 and lower credible intervals than the ones with predictive factors: respectively 0.08 [0.03; 0.17], 0.72 [0.42; 1.14] and 0.40 [0.20; 0.74] mg of peanut protein. Similar tendencies were observed for the other EDp values. Figure 3 shows that the threshold distribution curve estimated with the Cox model was the closest to the Kaplan-Meier estimates and that all values from Kaplan-Meier were comprised in the Cox model 95% credible interval. It can also be seen that the Weibull model with predictive factors, which produced the higher probabilities of reaction and the higher credible interval, is more conservative than the Lognormal and Loglogistic models with predictive factors. The Cox model with the highest credible interval integrates the highest part of variability. Table

Table 1 :

 1 OFC results, patients characteristics and allergy tests (sIgE 1 , skin prick test) : associations with the threshold dose OFC results, patients characteristics and allergy tests

	Values	Association with the threshold dose
	Weibull	Lognormal	Loglogistic	Cox
	95% CI

Table 3 :

 3 Eliciting doses (EDp in mg of peanut protein) triggering an allergic reaction for p% of the population. The EDp are described by their median and the 2.5 th and 97.5 th percentiles. These percentiles define a 95% credible interval around the median estimate. The interval integrates all values for predictive variables and for model parameters.

		Models	Estimator	1	5	10	EDp 50	90	95	99
		Kaplan Meier	Median	0.19	3.62	8.13	126		2218	2392
			2.5 th percentile	0.03	1.81	4.00	89.8		2143	2321
			97.5 th percentile	2.25	5.97	11.1	177		NA	NA
		Weibull with covariables	Median	0.20	2.58	7.94	149		1448	2830
			2.5 th percentile	0.02	0.32	1.03	20.7		204	393
			97.5 th percentile	0.99	11.0	32.3	567		5612	11056
		Lognormal with covariables	Median	1.33	5.17	10.7	135		3349	12562
			2.5 th percentile	0.15	0.58	1.20	15.3		382	1426
			97.5 th percentile	3.93	14.2	28.5	343		9352	38011
		Log-logistic with covariables	Median	0.89	5.37	12.1	132		3165	18770
			2.5 th percentile	0.08	0.51	1.15	12.7		302	1737
	Entire population	Cox model	97.5 th percentile Median	3.46 0.26	18.9 3.9	42.1 11.1	441 127		11731 2191	76115 2218
			2.5 th percentile	0.03	0.94	2.93	16.4		194	251
			97.5 th percentile	2.24	11.1	25.0	676		2400	2403
		Basic Weibull	Median	0.08	1.38	4.94	137		1817	3896
			2.5 th percentile	0.03	0.74	2.96	105		1377	2789
			97.5 th percentile	0.17	2.39	7.77	176		2479	5556
		Basic Lognormal	Median	0.72	3.08	6.74	104		3472	14841
			2.5 th percentile	0.42	1.99	4.53	78.8		2224	8457
			97.5 th percentile	1.14	4.46	9.23	135		5863	28724
		Basic Loglogistic	Median	0.40	2.98	7.47	108		3890	28998
			2.5 th percentile	0.20	1.84	4.96	83.1		2461	15442
			97.5 th percentile	0.74	4.68	10.88	145		6880	62987

  RMSE: Root Mean Square Error, the lower the RMSE value, the best the model

		Kaplan Meier	Median 97.5 th percentile	0.03 1.34	3.22 4.57	6.32 8.77	64.7 110.4	752 1834	1444 4209
		(Female) Basic Loglogistic	2.5 th percentile Median	0.01 0.81	0.24 3.60	3.65 7.93	49.1 130	457 2086	739 4575
		(Male)	97.5 th percentile 2.5 th percentile	3.74 0.41	10.68 2.15	14.9 5.06	125 90.6	NA 1281	NA 2665	NA
		Kaplan Meier	Median 97.5 th percentile	0.22 1.54	3.54 5.87	7.96 12.1	153 183	2198 3887	2276. 9396
		(Male) Basic Loglogistic (Female)	2.5 th percentile Median 97.5 th percentile 2.5 th percentile	0.07 0.43 2.99 0.14	1.61 2.72 7.78 1.25	3.81 6.18 12.3 3.24	113 72.7 218 48.0	1238 846 NA 478	1938 2204 NA 993	NA
		Weibull with covariables	Median 97.5 th percentile	0.14 1.06	1.75 5.20	5.44 11.0	104 111	674 1626	1015 4253
		(Female) Basic Lognormal	2.5 th percentile Median	0.03 0.40	0.39 3.24	1.26 8.38	25.6 137	169 2232	254 5712
		(Male)	97.5 th percentile 2.5 th percentile	0.50 0.15	5.35 1.58	15.4 4.70	264 93.4	1708 1335	2592 3067
		Weibull with covariables	Median 97.5 th percentile	0.29 0.91	3.90 5.73	12.2 13.6	237 195	1534 4169	2300 12061
		(Male)	2.5 th percentile	0.02	0.30	0.94	18.2	117	177
			97.5 th percentile	1.10	11.9	34.6	608	4011	6083
		Lognormal with covariables	Median	1.26	4.85	9.95	124	1491	3014
		(Female)	2.5 th percentile	0.15	0.59	1.22	15.4	184	369
			97.5 th percentile	3.63	13.2	26.2	315	4064	8600
		Lognormal with covariables	Median	1.44	5.61	11.6	145	1752	3534
		(Male)	2.5 th percentile	0.15	0.59	1.22	15.6	192	387
			97.5 th percentile	4.01	14.4	28.7	342	4463	9396
	According to gender	Loglogistic with covariables (Female)	Median 2.5 th percentile	0.59 0.07	3.60 0.44	8.09 1.00	88.2 11.0	944 116	2101 252
			97.5 th percentile	2.20	11.9	26.1	268	3011	6974
		Loglogistic with covariables	Median	1.16	7.17	16.3	179	1920	4316
		(Male)	2.5 th percentile	0.12	0.77	1.75	19	202	446
			97.5 th percentile	3.78	21.33	46.5	483	5378	12457
		Cox model	Median	0.19	3.63	8.29	98.5	828	1250
		(Female)	2.5 th percentile	0.03	1.26	3.36	16.4	127	217
			97.5 th percentile	0.96	6.33	15.8	230	2217	2318
		Cox model	Median	0.68	6.10	13.5	220	2210	2270
		(Male)	2.5 th percentile	0.03	0.50	2.41	16.4	111	149
			97.5 th percentile	2.40	12.6	25.5	755	2392	2401
		Basic Weibull	Median	0.07	1.11	3.82	95.3	735	1149
		(Female)	2.5 th percentile	0.02	0.41	1.65	62.1	508	760
			97.5 th percentile	0.24	2.70	7.83	142	1176	1956
		Basic Weibull	Median	0.10	1.82	6.36	173	1423	2257
		(Male)	2.5 th percentile	0.03	0.77	3.24	123	1059	1632
			97.5 th percentile	0.28	3.68	11.5	243	2127	3493
		Basic Lognormal	Median	0.63	2.49	5.23	71.1	951	1981
		(Female)	2.5 th percentile	0.25	1.22	2.83	47.3	569	1104
			29

Acknowledgments

This work was funded by the French Research Agency (ANR) under reference ANR-10-ALIA-012-01. The authors wish to thank Sélina TSCHEILLER (analyst at the Allergy Vigilance Network) for her contribution to the data collection and all the allergists who participated in the MIRABEL survey: Dr ALT Roger, Strasbourg; BANOUN Laurence, Le Raincy ; BEAUMONT Pascale, Saint Maur des Fossés ; BEGON Isabelle, Paris ; BLANCHARD Paul, Desertines ; Boix Françoise, Angers ; BONNEFOY GUIONNET Bénédicte, Saint-Lo ; CHABANE Habib, Saint-Denis ; CHATEAU-WAQUET Dominique, Paris ; CORDEBAR Vanina, Thionville ; COUSIN Marie-Odile, Lille ; DE BLAY Fréderic, Strasbourg ; De Hauteclocque Cécile, Creil ; Delaval Yvonne, Rennes; DELEBARRE-SAUVAGE Christine, Lille ; DEVOISINS Jean Marc, Cholet ; DOYEN Virginie, Bruxelles ; DRON-GONZALVEZ Mireille, Martigues; DROUET Martine, Angers ; DUMOND Pascale, Nancy ; DUMOULIN Anne, Brest ; DZVIGA Charles, Saint Etienne ; EPSTEIN Madeleine, Paris ; FLABBEE Jenny-Anne, Thionville ; FONTAINE Monique, Reims ; FRENTZ Pascale, Thionville ; GAYRAUD Jacques, Tarbes ; GIBOURY LAFARGE Sophie, Verneuil sur Sein ; GRANET Patricia, Digne les Bains ; Guenard-Bilbaud Lydie, Strasbourg ; HALLET Jean-Louis, Luneville ; HATAHET Riad, Forbach ; Langlet Catherine, Bayonne ; Lepeltier -Canavan Laurence, Caen ; LEPRINCE Françoise, Saint Quentin ; LETELLIER Edouard, Paris ; LIABEUF Valérie, Marseille ; MARCHANDIN Anne, Luisant ; MASSABIE-BOUCHAT Yann-Patrick, Marseille ; Mazé Marie-Héléne, Plaisir ; MENETREY Céline, Limoges ; MERCIER Valérie, Toulouse ; MONSIGNY Monique, Chauny ; MOREL-CODREANU Françoise, Luxembourg ; MOUTON FAIVRE Claudie, Nancy ; MULLIEZ-PETITPAS Julie, Poitiers ; NICOLAS Pascale, Poissy; OUTTAS Omar, Clermont-Ferrand ; PASQUET-NOUALHAGUET Christine, Bois d'Arcy ; PEROTIN-COLLARD Jeanne-Marie, Reims ; PETIT Nicolas, Verdun ; PIRSON Francoise, Bruxelles ; POUVREAU Hélène, Poitiers ; PUILLANDRE Erick, Arcachon ; SABOURAUD-LECLERC Dominique, Reims ; SAINT MARTIN François, Villebon sur Yvette ; SANTOS Adjoke-Clarisse, Lille ; SERINGULIAN Alice, Paris ; SULLEROT Isabelle, Sens ; TERRIER Patrick, Marmomme ; THIERRY Marie-Hélène, Alberville ; THIS-VAISSETTE Christine, Massy ; VIGAN Martine, Besançon; VODOFF M.V, Mulhouse.

This work was funded by the French Research Agency (ANR) under the reference ANR-10-ALIA-012-01.

Contributors

Amélie Crépet: leader in the development and design of the MIRABEL project, involvement in the formulation of hypotheses, and in the acquisition, statistical analysis and interpretation of the data and in writing the manuscript.

Min -Max 0.01 -30 (1) sIgE: specific IgE, IQR = Interquartile range. (2) 95% CI: the credible interval defined by the 2.5th and the 97.5 th percentiles of the parameter distribution associated with each predictive variable. Boldfaced text indicates statistical significance. The variable is significantly associated with threshold dose if the 95% CI does not contain 0. For the Weibull and Cox models, a positive confidence interval of the parameter indicates a negative association with the threshold dose value and thus a positive association with the risk. For the Lognormal and Loglogistic models, a negative confidence interval of the parameter indicates a negative association with the threshold dose value and thus a positive association with the risk. d: Threshold dose (1) 95% CI: the credible interval defined by the 2.5th and the 97.5 th percentiles of the parameter distribution associated with each predictive variable. The variable is significantly associated with the threshold dose if the 95% CI does not contain 0. For the Weibull and Cox models, a positive confidence interval of the parameter indicates a negative association with the threshold dose value and thus a positive association with the risk. For the Lognormal and Loglogistic models, a negative confidence interval of the parameter indicates a negative association with the threshold dose value and thus a positive association with the risk. Threshold doses (mg of peanut protein) Cumulative probability of reaction Male Female Relative probability of reaction (Female compared with Male) = 1.22 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (a) Threshold doses (mg of peanut protein)

Cumulative probability of reaction q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (b) (b


Jean-Marc Nwesaty
email: nwesaty@gmail.com

Senior Member, IEEE Antoneta Iuliana Bratcu

Member, IEEE Alexandre Ravey
email: alexan-dre.ravey@utbm.fr

David Bouquain
email: david.bouquain@utbm.fr.

J.-M Nwesaty

A I Bratcu

Robust energy management system for multi-source DC energy systems -real-time setup and validation

Keywords: H∞ control, LPV systems, power source coordination, reduced-order controller, frequency separation, electric vehicle

This paper aims at providing a proof of concept of a systematically designed LPV/H∞-based energy management system (EMS) for coordinated multi-variable control of multi-source electrical systems. A three-source electrical system representing the power supply system on board of an electric vehicle has been chosen as a representative example of irregular and generally not a priori known load variation. The power supply system is composed of fuel cell, battery and supercapacitor. Each power source is coupled to a DC-DC converter, all converters being connected in parallel to a common DC-bus in order to feed the load represented by the vehicle's electrical motor. The system is modelled as an LPV system -as its operating point depends on the load -and the control objectives are cast into the H∞ formalism as a disturbance-rejection problem.

A dedicated hardware-in-the-loop system was built for proof of concept purpose, with real-world battery and supercapacitor being used, while the fuel cell system is entirely emulated. A dSPACE MicroAutoBox ® II device embeds the designed EMS, due to its flexibility and ease of programming with MATLAB ® . A driving cycle from IFSTTAR (Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux) is chosen as a pertinent scenario of load variation due to its rich frequency content able to challenge all the three sources. Effectiveness of the EMS is assessed in relation to the imposed control objectives -DC-bus voltage regulation, dynamical separation of power sources' current variations depending on the specialization range of each source, and imposing desired steady-state behavior for each of the three power sources -with very promising results.

I. INTRODUCTION

M ULTI-SOURCE -also named hybrid -power supply sys- tems are nowadays becoming ubiquitous, especially in applications where load variation is strongly irregular. The concept of "microgrid" has thus emerged, where the basic functional entities may operate either in the so-called "grid-connected" mode, or to supply power for a variety of stand-alone applicationssuch as electrical mobility -or for islanded/isolated zones when they operate in "grid-forming" mode. Renewable energies, whose intermittency requires hybridization with storage units, may also be present in microgrids. In particular, DC microgrids have gained popularity vs. AC ones due to their more simplified operation (e.g., no need for synchronization). Stand-alone applications in avionic, automotive and marine industries, but also power supply of remote areas may benefit from this advantage, among others [START_REF] Guerrero | Advanced control architectures for intelligent microgrids -part i: Decentralized and hierarchical control[END_REF], [START_REF] Justo | Ac-microgrids versus dcmicrogrids with distributed energy resources: A review[END_REF].

Multi-source power supply systems use combinations of different storage technologies, such as fuel cells, batteries, supercapacitors, flying wheels, etc.. Ragone's plot [START_REF] Kuperman | Battery/ultracapacitor hybrids for pulsed current loads: A review[END_REF], features two main classes of sources: high-energy-density sources that can supply power for long duration of time (e.g., fuel cells), and high-power-density sources, which can provide relatively high power for short period of time (e.g., supercapacitors). Batteries are classified in between the two classes depending on the battery type.

In the electromobility three main hybridization topologies are available depending on the number of components, energy management complexity and performance reliability: series, parallel, and cascaded [START_REF] Azib | Online energy management strategy of a hybrid fuel cell/battery/ultracapacitor vehicular power system[END_REF]- [START_REF] Aharon | Topological overview of powertrains for batterypowered vehicles with range extenders[END_REF]. In this paper, a parallel structure of three sources in a DC system -fuel cell, battery and supercapacitor -is chosen due to its flexibility (sources' independent operation, facility to replace/add power sources, etc.).

1) Related works: In multi-source systems an Energy Management System (EMS) should coordinate the different devices and ensure a desired power sharing between sources to satisfy the load demand, while preserving reliability of all sources and extending their remaining useful life [START_REF] Iannuzzi | Use of Supercapacitors, Fuel cells and Electrochemical Batteries for Electric Road Vehicles: A Control Strategy[END_REF].

Efficient EMS can be designed for a variety of applications. Model Predictive Control (MPC) -with its nonlinear and/or stochastic versions -may be applied at coordinated multivariable control strategies for stand-alone microgrids and/or multi-source systems, with or without renewable sources, as it successfully deals with multiple constraints [START_REF] Schuler | l1-optimal control of large wind turbines[END_REF]- [START_REF] Xie | An energy management strategy based on stochastic model predictive control for plug-in hybrid electric buses[END_REF]. But adequate plant mathematical modelling and also a priori knowledge of load variation to predict the behavior are needed. For electric vehicles, this results in drivingcycle-dependent solutions, which may be a disadvantage. Nonlinear methods have also been proposed, such as Lyapunov-based [START_REF] Rajabzadeh | Dynamic modeling and nonlinear control of fuel cell vehicles with different hybrid power sources[END_REF] or sliding mode [START_REF] Alnejaili | Sliding mode control of a multi-source renewable power system[END_REF]. More recently, interest of multi-agent-based approach and games theory has been revealed [START_REF] Khan | Multi-agent based distributed control architecture for microgrid energy management and optimization[END_REF].

Complex techno-economical criteria must be considered in the multi-source EMS design. Thus, reliability-aware design rely on complex degradation and ageing models of storage units, along with optimization methods [START_REF] Martel | Optimal economy-based battery degradation management dynamics for fuel-cell plug-in hybrid electric vehicles[END_REF]. The multi-scale character of multisource systems justify hierarchical control structures [START_REF] Xu | Hierarchical energy management system for multi-source multi-product microgrids[END_REF], with dynamically separated layers for enabling operation of sources in their "specialization" range of variations -according to Ragone's taxonomy -to improve their reliability [START_REF] Castaings | Decomposed energy management of a multi-source fuel cell vehicle using energetic macroscopic representation[END_REF]. In this way, highpower-density sources (like ultracapacitors), specialized in providing high-amplitude fast-varying currents, are able to protect highenergy-density sources (like fuel cells or batteries), whose current variations must be smoother. Dynamic separation can be done by filtering, where variations of current of each source are confined in a predefined frequency interval [START_REF] Silva | Application of a decoupling method based on online filtering technique for multi-source electric vehicles[END_REF]- [START_REF] Florescu | Adaptive frequency-separation-based energy management system for electric vehicles[END_REF]. The LQG optimal control strategy also works to dynamically separate two sources [START_REF] Florescu | LQG Optimal Control Applied to On-Board Energy Management System of All-Electric Vehicles[END_REF].

2) Paper contribution: Here a three-source -fuel cell, battery and supercapacitor -power supply on board of an electric vehicle is considered. Each source is controlled by means of a DC-DC converter to operate in its frequency specialization range; the electrical motor with its converter is the load (Fig. 1.a). Fuel cell and battery are thus protected from strong variations of power demand. The EMS design is cast into the linear parameter varying (LPV)/H ∞ framework methodology developed by authors in [START_REF] Nwesaty | Power sources coordination through multivariable linear parameter-varying/ control with application to multi-source electric vehicles[END_REF], which is a generic solution of EMS for multi-source systems with potentially any number of sources, that guarantees closed-loop robust stability. In this paper, the validation is done on a dedicated test-bench designed in collaboration with FEMTO-ST Laboratory in Belfort, France. Real battery and supercapacitor are used in the test bench, whereas a complete real-time emulator developed at FEMTO-ST Laboratory [START_REF] Gao | PEM fuel cell stack modeling for real-time emulation in hardware-in-the-loop applications[END_REF] represents the fuel cell. A reducedorder version of the controller [START_REF] Nwesaty | Reduced-order LPV controller for coordination of power sources within multi-source energy systems[END_REF] is employed in real time.

The methodology of implementation is discussed and comparison with the numerical simulation is conducted for IFSTTAR driving cycle (Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux), a suburban driving cycle. Effectiveness of the proposed EMS, as well as adaptations and limitations imposed by the real-world implementation, are discussed.

This paper is organized as follows. Section II provides a detailed description of the real-time test bench. Modelling of the LPV system, as well as control objectives, are presented in Section III. Section IV explains how the proposed LPV/H ∞ control design procedure is adapted to the real-time test bench. Section V discusses the real-time validation results including a comparison with the numerical simulation ones. Section VI concludes the paper.

II. DESCRIPTION OF THE REAL-TIME TEST BENCH

The physical setup of the dedicated hardware-in-the-loop system corresponds to Fig. 1.b. The three power sources are paralleled on a DC-bus through DC-DC buck/boost converters; they supply an active load -whose variation is imposed to represent the desired driving cycle of the electric vehicle -by means of a rapidprototyping platform based on dSPACE MicroAutoBox ® II device. Fig. 1.b shows an expanded view of the interconnections between the different parts used in the assembled test bench.

1) Power sources: This part is numbered as 1 in Fig. 1.b. The fuel cell is the main power source in the system. A 2-kW NEXA fuel cell with 50 V open-circuit voltage is here emulated by means of a fuel cell emulator [START_REF] Gao | PEM fuel cell stack modeling for real-time emulation in hardware-in-the-loop applications[END_REF]. In this way, behavior of the considered fuel cell can be replicated without hydrogen consumption. Consisting mainly of a programmable power supply driven by a real-time fuel cell model, the emulator is equipped by a human-machine interface, which allows on-line investigation of different parameters. Further details about fuel cell emulator can be found in [START_REF] Gao | PEM fuel cell stack modeling for real-time emulation in hardware-in-the-loop applications[END_REF], [START_REF] Fei | Proton Exchange Membrane Fuel Cells Modeling[END_REF]. The supercapacitor is used to provide/absorb the fast variations of load current. Two MAXWELL supercapacitors in series are used to cover the voltage interval [0,32] V, each of which has the capacity of 56 F with maximum voltage value of 16 V. The supercapacitor voltage range is limited in between [START_REF] Xu | Hierarchical energy management system for multi-source multi-product microgrids[END_REF][START_REF] Zitzler | SPEA2: Improving the strength Pareto evolutionary algorithm[END_REF] V, considered equivalent linearly to [0,100] % of state of charge (SOC). The battery is employed to handle the mid-range variations of the load current. Three batteries of 12 V and 110 Ah are here connected in series.

2) DC-DC converters: This part is numbered as 2 in Fig. 1.b. The fuel cell is connected to 1-quadrant DC-DC boost converter and the rest of sources are connected to 2-quadrant DC-DC buck/boost converters to allow bi-directional power flows for charging/discharging. In this implementation, each of the three arms of a three-phase DC-AC inverter works as a different DC-DC converter, each of which is connected to a common DC-bus. All inverter inputs/outputs can handle 25 A DC current and up to 350 V DC voltage. Each DC-DC converter consists of two IGBT transistors operated by two complementary pulse-width modulated (PWM) signals. Averaged duty cycle of each PWM signal is noted by α, where α ∈ [0, 1]. Practically, α ∈ [0.10, 0.90] in order to avoid extreme operating conditions of the converters.

3) DC-bus: Numbered as 3 in Fig. 1.b, the DC-bus represents the physical connection between all paralleled power converters and the load. The DC-bus voltage should be regulated around a desired value regardless of load current variations. Here this voltage reference is chosen to be V dc ref = 60 V, which can be ensured by all converters with duty cycle values in between admissible limits.

4) Active load: This part is numbered as 4 in Fig. 1.b; it is a device directly connected to the DC-bus, used to program a desired load current profile, a driving cycle profile in our case. The load current profile is provided as one-second-sampled table of values.

5) Sensors: This part is numbered as 5 in Fig. 1.b. Current sensors measure currents of all three sources (I f c , I bat and I sc ) to be controlled at low level; at high-level control both DC-bus voltage V dc and supercapacitor voltage V sc are measured, where the last one is a direct image of supercapacitor SOC.

6) Real-time embedded control system: Numbered as 6 in Fig. 1.b, this part consists of the proposed EMS. To this end, MicroAutoBox ® II from dSPACE is used as a flexible, MATLAB ® /Simulink ® -programmable real-time system, effective for hardware-in-the-loop-based rapid prototyping. The sampling time, 50 µs (20 KHz), is chosen in relation to the fastest dynamic in the system, namely that of PWM signals.

7) Human machine interface (HMI):

This part is numbered as 7 in Fig. 1.b; it is designed using the ControlDesk 4.2 program and allows communication with the real-time control system. Its main tasks are to read and visualize the actual values of all interest variables. Some of parameters can also be tuned in real time. A global two-level control structure [START_REF] Nwesaty | Power sources coordination through multivariable linear parameter-varying/ control with application to multi-source electric vehicles[END_REF] is here used: all three sources are current-controlled within low-level proportional-integral (PI) controller-based loops whose references come from the upperlevel LPV/H ∞ -based controller (Fig. 2). Hence, low-level dynamics are much faster than the upper-level ones. Fuel cell and battery are modelled as ideal current sources, with negligible dynamics.

III. SYSTEM MODELLING AND CONTROL OBJECTIVES

A. System modeling

I f c C a R m C c I bat C 1 R 2 C 2 R 1 I sc α f c α bt α sc V f c V bat V sc I load E f c R 1 -Qn Q E bat C dc C 0 R 0 R tc R ta R 4 R 3 C 3 C 4 R dc L f c R f c L bat R bat L sc R sc V dc
Fuel cell system Battery system Supercapacitor system

Load

Conversion stage

Output stage

Input stage Supercapacitor state of charge (SOC) is controlled to supply load variations during acceleration/braking phases. The EMS "sees" the load as the current drawn from or injected into the DC link. Average models of all the converters are used [START_REF] Bacha | Power Electronics Converters Modelling and Control: With Case Studies[END_REF]. The global model is:

V 0 V 2 V 1
                             dI f c dt = 1 L f c [V f c -R f c I f c -V dc (1 -α f c )] dI sc dt = 1 L sc [V sc -R sc I sc -V dc α sc ] dI bat dt = 1 L bat [V bat -R bat I bat -V dc α bat ] dV dc dt = 1 C dc [ -1 R dc V dc -I load + I f c (1 -α f c ) + I bat α bat + I sc α sc ], (1)
where I f c , I bat , and I sc are the currents of fuel cell, battery, and supercapacitor sources, respectively. α f c , α bat , and α sc are the corresponding converter average duty cycle values. V f c , V bat and V sc are fuel cell, battery and supercapacitor voltages, respectively. L f c , L bat , L sc , R f c , R bat and R sc are the inductances and resistances of smoothing inductors of each power converter, respectively. I Load is the load current, V dc is the DC-bus voltage, C dc and R dc are the DC-bus capacitor and resistance, respectively. V sc is the output of the supercapacitor electrical model in Fig. 1.a [START_REF] Wu | On-line supercapacitor dynamic models for energy conversion and management[END_REF]:

                   dV 0 dt = -1 C 0 I sc dV 1 dt = -1 C 1 R 1 V 1 - 1 C 1 I sc dV 2 dt = -1 C 2 R 2 V 2 - 1 C 2 I sc V sc = -I sc R 0 + V 0 + V 1 + V 2 , (2)
where R 0 , C 0 , R 1 , C 1 , R 2 , and C 2 are constant parameters of supercapacitor model, V 0 , V 1 and V 2 are sub-voltages in Fig. 1.a. [START_REF] Nwesaty | Power sources coordination through multivariable linear parameter-varying/ control with application to multi-source electric vehicles[END_REF], [START_REF] Nwesaty | Optimal frequency separation of power sources by multivariable lpv/h∞; control: Application to on-board energy management systems of electric vehicles[END_REF].

W eV dc W eI f c W eI bat W eVsc Wu Ifc W uI bat W uIsc K(ρ)
The high-level control handles the dynamics in (2) and V dc dynamics from (1), while the three currents are controlled within the low-level control loops (PI current controllers -see Section IV-A). The system composed of (2) and V dc equation from ( 1) is rewritten in LPV form as follows:

ẋ = A • x + B 1 • w + B 2 (ρ) • u y = C • x + D • u, (3)
with state vector x = [V dc V 1 V 2 V 0 ] T , disturbance input w = I load , control input vector u = [I f c I bat I sc ]
T composed of fuel cell, battery and supercapacitor currents, respectively.

ρ = [ρ 1 ρ 2 ρ 3 ] T = [α f c α bat α sc ]
T is the varying parameter vector, where each ρ i is bounded within [0.1, 0.9]. Each varying parameter is supposed to be independent from the others.

B. Control objectives

The main goals of the proposed EMS are: 1) to keep the DC-bus voltage around 60 V within an error of ±10% regardless of the load current variations;

2) to apply dynamical (frequency) separation of power sources, i.e., each power source must supply power with respect to its characteristic frequency according to Ragone's classification [START_REF] Kuperman | Battery/ultracapacitor hybrids for pulsed current loads: A review[END_REF] -this helps to protect fuel cell and battery from harmful fast variations of load current. Frequency separation is achieved due to some suitable choice of weighting functions associated to H ∞ control design (see Section IV-B);

3) to slowly keep the supercapacitor SOC around 50%, allowing to absorb/provide power to fulfil instantaneous load power demand; 4) to impose a desired steady-state behavior for the fuel cell and battery, according to some desired power sharing between sources in steady state. Fuel cell operating point can thus be imposed, e.g., the maximum-efficiency one. Steady-state behavior is involved in battery charging, depending on its type. Battery charging could be achieved by using the main power source (fuel cell).

Note that battery SOC is not controlled; instead, its charging/discharging current regimes are imposed. Indeed, in practice a battery SOC is not controlled linearly around a desired value like supercapacitor's one, but it is used as an indicator to start its charging/discharging according to some desired regime (e.g., C/5 or C/10, where C is the battery capacity in Ah).

IV. CONTROL DESIGN: SELECTION OF PARAMETERS AND ADAPTATIONS TO THE TEST BENCH

A. Curent control loops design (low-level control)

Each source current must be controlled and prevented from exceeding admissible limits. Supposing that the DC-link voltage V dc is kept constant at its reference value V dc ref by the EMS, dynamics of currents (1) are linear, which justifies their control by classical PI controllers. These loops are designed to be fast enough to be neglected at the slower-dynamic upper level and to ensure tracking of current references generated by upper-level LPV/H ∞ controller (Fig. 2). Hence, the following equalities hold in the sequel: I f c * =I f c , I bat * =I bat and I sc * =I sc . PI controllers' parameters can be found in Table II in the Appendix.

The three tracking low-level loops have the converters' duty cycles, α f c , α bat , α sc , as control inputs. It is thus reasonable to choose the varying parameter vector ρ = [α f c α bat α sc ] T to describe globally the time-varying system.

B. LPV/H ∞ control loop design (upper-level control)

The LPV controller K(ρ) in Fig. 2 should satisfy the control objectives in Section III-B for the parameter-varying plant G(ρ); its design is tackled in the H ∞ framework applied to LPV systems [START_REF] Apkarian | Self-scheduled H∞ control of linear parameter-varying systems: a design example[END_REF]. In Fig. 2 usual forms of the weighting functions are considered to represent the control objectives. A first-order weighting function 1/W eV dc is in charge to ensure performance of load variations rejection, i.e., both desired time response and acceptable error range for the DC-link voltage (V dc ) regulation. A first-order weighting function 1/W eV sc is used to maintain supercapacitor SOC around 50%. A supercapacitor's SOC may be expressed as:

SOC = (V sc -V sc,min )/(V sc,max -V sc,min ) • 100%, (4)
where V sc,max is the maximum allowed supercapacitor's voltage (SOC=100%) and V sc,min is the minimum allowed voltage input for the supercapacitor converter (SOC=0%). Here V sc ∈ [START_REF] Xu | Hierarchical energy management system for multi-source multi-product microgrids[END_REF][START_REF] Zitzler | SPEA2: Improving the strength Pareto evolutionary algorithm[END_REF] V.

Weighting functions 1/W uIf c , 1/W uIbat and 1/W uIsc shape the dynamic behavior of current references of fuel cell, battery and supercapacitor, respectively, in order to dynamically separate the sources depending on their characteristic frequencies. Sharp separation is done with fourth-order functions 1/W uIbat and 1/W uIsc , while 1/W uIf c is a constant. Weighting function 1/W eI f c is defined as a first-order one and 1/W eI bat is a constant value to determine fuel cell and battery long-term behaviors, respectively. Thus, a desired steady-state power sharing can be imposed by using I f c steadystate and I batsteadystate reference inputs. Plant G(ρ) together with the weighting functions form the extended plant P (ρ).

Parameters of weighting functions result from a geneticalgorithm (GA)-based optimization procedure [START_REF] Goldberg | Genetic Algorithms and Machine Learning[END_REF], [START_REF] Zitzler | SPEA2: Improving the strength Pareto evolutionary algorithm[END_REF] with a twofold objective function: to guarantee an arbitrarily imposed degree of closed-loop stability, while sizing a desired frequency separation between power sources according to each application [START_REF] Nwesaty | Optimal frequency separation of power sources by multivariable lpv/h∞; control: Application to on-board energy management systems of electric vehicles[END_REF]. Guess values for this algorithm are selected as presented in Section IV-D3. The resulting weighting functions are given in the Appendix.

C. LPV/H ∞ controller design and order reduction

The control design problem is cast into a set of Linear Matrix Inequalities (LMI) and solved in the context of LPV/H ∞ control polytopic approach [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]. System in (3) has a polytopic form with 2 3 = 8 vertices (since the parameter vector ρ has three bounded elements in [0.1,0.9]). The generalized LPV MIMO system is:

   ẋ z y    =    A B 1 B 2 (ρ) C 1 D 11 D 12 C 2 D 21 D 22       x w u    . (5)
The LMIs are solved off-line by using Yalmip/Sedumi solver as a convex optimization with a single Lyapunov function (framework of quadratic stabilization [START_REF] Apkarian | Self-scheduled H∞ control of linear parameter-varying systems: a design example[END_REF], [START_REF] Do | LPV modelling and control of semiactive dampers in automotive systems[END_REF]) at each vertex of the polytope, leading to vertex controllers

K i = A i B i C i D i with 1 ≤ i ≤ 8.
The LPV controller K(ρ) is computed on-line as a convex combination of the vertex controllers K i , i.e., K(ρ) = 8 1 γ i (ρ)K i , with:

γ i (ρ) = 3 j=1 |ρ j -C(w i ) j | 3 j=1 ρ j -ρ j > 0, 8 1 γ i = 1,
where w i are the vertices corresponding to the extreme values of the parameter vector ρ. C(w i ) j is the j th component of the vector

C(w i ) defined as C(w i ) j = ρ j if w i = ρ j ρ j otherwise
, where in this case ρ j = max(ρ j ) = 0.9, ρ j = min(ρ j ) = 0.1.

The full-order LPV/H ∞ controller is a combination of eight eighteenth-order controllers. For practical purposes, MORE toolbox [START_REF] Poussot-Vassal | Introduction to MORE: A MOdel REduction toolbox[END_REF] is used to find a reduced-order model that fits the original controller for bounded frequency range. The Iterative SVD Tangential Krylov Algorithm (ISTIA) is applied on each vertex controller for the whole closed-loop bandwidth, resulting in tenth-order vertex controllers. Stability and performance have been checked a posteriori (see [START_REF] Nwesaty | Reduced-order LPV controller for coordination of power sources within multi-source energy systems[END_REF] for more details).

D. Adaptations to the test bench 1) Control design: The proposed methodology has been applied to the test bench model to obtain a reduced-order controller for real-time implementation. Sources' currents saturation have been taken into account in the design. Comparison between real-time and numerical simulation results has been performed under the same scenario. Thus, samples of the measured load current (Fig. 3.a) were used to re-perform numerical simulation.

2) Selection of control references: Four exogenous inputs represent the references for the upper-level control loop (Fig. 2). These references are selected for real-time implementation as follows: a) DC-bus voltage reference V dc ref is chosen to be 60 V; b) fuel cell steady-state behavior is imposed such that fuel cell be able to supply the total average load current, according to:

I f c,ref (t) = average(I load (t))/(1 -α f c (t)), (6)
where average(I load (t)) is the load current suitably low-pass filtered, e.g., with 10 rad/s as cut-off frequency to remove noise. α f c (t) is the real-time value of the fuel cell converter average duty cycle. However, this reference choice is not unique and the fuel cell can be managed in a different way, e.g., to provide quasi-constant current corresponding to its maximum power efficiency; c) as the fuel cell provides all the load power in steady state, then the battery is here managed to provide no current in long term, thus preserving its SOC: I bat ref = 0 A. However, the battery should be able to replace the fuel cell on line in case of unavailability -in this case, another proper choice of I bat ref must be made; d) supercapacitor SOC reference is reasonably chosen as 50 %, so the supercapacitor is always ready to either provide or absorb power in case of fast and important variations of load current.

3) Selection of weighting functions: This step is crucial for implementing a reliability-aware design of the proposed EMS. Selection of 1/W eV dc , 1/W eV sc , 1/W eI f c and 1/W eI bat is explained above in Section IV-B. As regards the weighting functions 1/W uI f c , 1/W uI bat and 1/W uI sc , they must be chosen to represent a suitable conditioning of the source currents' response to load variations. Standard forms are here chosen, whose parameters -mainly their folding frequencies -result from a GA-based optimization, as explained above. Initial guess values are selected as follows.

Basically, the folding frequencies of these filters are chosen in relation with the characteristic frequencies of sources, which quantitatively express the maximum speed of variations each source can provide and still respect its reliability specifications. Practically, for a given storage technology, its characteristic frequency may be deduced from the datasheet, i.e., from the maximum current gradient recommended for maximizing its proper operation time.

Therefore, the fuel cell current must only have slow variations, so 1/W uI f c will be a low-pass filter with a roll-off frequency near its characteristic frequency. Battery current variations will be confined within a bandwidth including its characteristic frequency, so 1/W uI bat will be a bandpass filter. Finally, the ultracapacitor must work at high-frequency variations, so 1/W uI sc will ideally be a high-pass filter with a folding frequency near its characteristic frequency. In practice, however, it is about a bandpass filter, given the limited spectrum of supercapacitor. 1/W uI f c , 1/W uI bat and 1/W uI sc may be locally superposed, more or less -in frequency regions where the load is supplied by two of the three sources -and together they should cover the whole spectrum of load variations.

4) Expected differences between numerical simulation and realtime results: The modelling uncertainty of sources is expected to be the main source of differences here, that is, differences between nominal and real values of parameters characterizing each of the three sources. Indeed, widely-known equivalent electrical models of literature have been used in simulation for the fuel cell, battery and supercapacitor, with datasheet values of parameters of these models. It is however known that such parameters -resistances, inductances, capacities -vary in time and may not be equal to the datasheet ones any longer. Resuming identification of sources' models was beyond the scope of this paper. Therefore, currents of sources are expected to vary in real time differently from in simulation. Meanwhile, the proposed control strategy is expected to perform robustly against these parameter variations.

V. REAL-TIME VALIDATION RESULTS

This section discusses the real-time results obtained on the test bench. Different driving cycles can be used for EMS assessment.

Here the active load implements the extra-urban IFSTTAR driving cycle, where acceleration, speed variation on high-way, deceleration and full brake are frequent (Fig. 3.a); its rich frequency content is illustrative for the frequency separation of sources (see Section V-2). Results are analyzed in both time and frequency domains. DC-link voltage control (first control objective): Fig. 3.b shows the V dc time evolution. Note that real-time results differ by just 2.3% from simulation, due to filtering for noise reduction, and show V dc regulation within the imposed boundaries (60 V ± 10%).

Dynamical separation of the power sources (second control ob- jective): evolutions of the three currents in Fig. 4 show that fuel cell, battery and supercapacitor operate in low-, mid-and high-frequency range, respectively. As expected, fuel cell current is smoother compared to battery and supercapacitor ones. Supercapacitor current has the fastest variation, which complies with its role of high-frequency (high-power-density) source. Frequency-domain analysis in Section V-2 illustrates better the frequency separation goal. Fig. 5.a shows the time evolution of fuel cell current. Real-time results differ from simulation by around 20%; note also that their initial states are different. This can also be noticed for the battery current evolution shown in Fig. 5.b. Supercapacitor provided/absorbed current is shown in Fig. 6.a. Here, the difference between real time and simulation, especially concerning variations' magnitude, are just under 20%. As predicted, the largest differences between real time and simulation can effectively be seen in the behavior of sources' currents. This is potentially due to imprecise models and/or to differences between datasheet and real parameter values, as well as to noise from different sources (sensors, converters, etc.).

Supercapacitor SOC regulation (third control objective): Fig. 6.b shows that, as expected, the supercapacitor SOC is maintained around 50%. Table I shows that differences between the SOC realtime behavior and simulation are less than 10%; they are due to use of (4) in SOC computation: supercapacitor voltage is smoother in numerical simulation as average converter models are used. Despite these differences, the SOC is well regulated in both numerical simulation and real time, varying at maximum ±6% around the desired reference (50%). Hence, the system is able to handle larger load current variations; otherwise, a smaller supercapacitor can be used instead. Varying parameter vector ρ = [α f c , α bat , α sc ] T exhibits significant variations, but without reaching saturation.

2) Frequency-domain analysis: To illustrate the frequency separation feature of the proposed EMS, the power spectral density (PSD) and its normalized version are calculated for all current signals, including the load current (I f c , I bat , I sc , I load ):

   P SD(x) = (1/N ) X(f ) X(f ) * N ormalizedP SD = (P SD/max(P SD)) • 100%, (7)
where X(f ) is the discrete Fourier transform of the discrete-time signal x(k) of length N . X(f ) * is the complex-conjugated of X(f ). The normalized PSD of a source allows to identify the frequency range where most of that source's power is placed. Fig. 7 shows the normalized PSD of all three currents (I f c , I bat , I sc ) for IFSTTAR driving cycle. Fuel cell maximum power is provided at very low frequencies within [0, ω 1 ] = [0, 0.1] rad/s, practically representing a DC value. The battery maximum power is delivered within the range next to the fuel cell's one, i.e., [ω 4 , ω 5 ] = [0.1, 0.25] rad/s. Finally, as expected, the supercapacitor maximum use is within [0.25, 0.6] rad/s, i.e., in high frequency compared to the other two sources. Supercapacitor maximum power must usually be provided depending on the maximum frequency in the load spectrum.

VI. CONCLUSION

In this paper, the real-time validation of a multi-variable energy management system (EMS) based on LPV/H ∞ control is presented. A reduced-order version of a systematically designed LPV/H ∞ controller has been implemented on a dedicated test bench representing a three-source -fuel cell, battery and supercapacitorpower supply system on board of an electric vehicle. Real battery and supercapacitor have been used, while the fuel cell has been represented by a complete emulator. Sources are paralleled on a DC link by means of DC-DC converters; the DC bus feeds the load represented by the vehicle's electrical motor. A dSPACE MicroAutoBox ® II device embeds the designed EMS, due to its flexibility and ease of programming with MATLAB ® .

The EMS performance has been assessed against the imposed control objectives -DC-bus voltage regulation, dynamical separation of sources' current variations depending on specialization of each source, and imposing desired steady-state behavior for each source -with very promising results. IFSTTAR (Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux) driving cycle has been here chosen for illustration, due to its rich frequency content able to challenge all the three sources and thus to allow an as meaningful as possible validation. The EMS design does not require any previous knowledge of the load current, so the proposed strategy is not driving-cycle dependent.

Further work may aim at integrating some preliminary Fourier analysis results about the load profile, e.g., to find the maximum frequency in its spectrum. Further studies may also exploit the quasidiagonal form of the reduced-order LPV/H ∞ controller transfer matrix, consistent with the almost independent behavior of sources when paralleled on the DC link. A structured H ∞ control design may be interesting in this case, where a certain desired controller structure can a priori be imposed and optimized.

Fig. 1 .

 1 Fig. 1.a exhibits three main stages: the input stage including the power sources, the conversion stage including the DC-DC converters, and the DC link connected to the load as output stage.A global two-level control structure[START_REF] Nwesaty | Power sources coordination through multivariable linear parameter-varying/ control with application to multi-source electric vehicles[END_REF] is here used: all three sources are current-controlled within low-level proportional-integral (PI) controller-based loops whose references come from the upperlevel LPV/H ∞ -based controller (Fig.2). Hence, low-level dynamics are much faster than the upper-level ones. Fuel cell and battery are modelled as ideal current sources, with negligible dynamics.

Figure 1

 1 Figure 1: a) Electrical schematic of the considered on-board power supply system including power sources' equivalent electrical models. b) Schematics of the rapidprototyping test bench used to validate the energy management system.

Figure 2 :

 2 Figure 2: LPV/H∞ robust control design block diagram [21], [27].

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: a) Load current scenarii representing the IFSTTAR driving cycle, as provided by the programmable power supply/load used in the test bench. b) Realtime versus numerical-simulation behavior of the regulated DC-bus voltage in response to IFSTTAR driving cycle.1) Time-domain analysis: As mentioned before, the measured load current I load (Fig.3.a) is low-pass filtered and used in numerical simulation. Real-time results are here compared against

Figure 6 :

 6 Figure 6: Real-time versus numerical-simulation behavior of a) supercapacitor current and b) supercapacitor SOC corresponding to IFSTTAR driving cycle.

Figure 7 :

 7 Figure 7: Normalized power spectral density (PSD) of the three sources' currents under the IFSTTAR driving cycle scenario.

Table I :

 I Quantification of differences between simulation and real-time results, based on normalized difference signal norms: dif fx =x realtime -x simulation 2 x simulation 2 .

		V dc	I f c	I bat	Isc	SOCsc
	dif fx [%]	2.3	22.37	22.34	19.19	6.7

Table II :

 II Electrical system parameters used for simulation and real-time validation (notations refer to Fig. 1.a). Supercapacitor model's resistor 4.5 • 10 -4 Ω PIs local control loops Kp f c , Ki f c Fuel cell current control loop 0.5 , 0.1 Kp bat , Ki bat

	Parameter	Description	Real-time
	DC-bus		validation value
	V dc C dc R dc	DC-bus voltage DC-bus capacitor DC-bus discharging resistor	60 V 6.6 • 10 -3 F 10.0 • 10 +3 Ω
	Fuel cell
	L f c R f c	Fuel cell converter's inductor Fuel cell converter's resistor	2.0 • 10 -3 H 1.0 • 10 -2 Ω
	E f c Rm	Fuel cell open-circuit voltage Fuel cell model's resistor	50 V 7.63 • 10 -2 Ω
	Rta	Fuel cell model's resistor	2.0 • 10 -3 Ω
	Rtc	Fuel cell model's resistor	4.72 • 10 -4 Ω
	Ca	Fuel cell model's capacitor	2.12 • 10 -3 F
	Cc	Fuel cell model's capacitor	2.12 • 10 -2 F
	Battery
	L bat R bat	Battery converter's inductor Battery converter's resistor	1.0 • 10 -3 H 2.0 • 10 -3 Ω
	Qn	Initial SOC of battery	0 %
	C 3	Battery model's capacitor	2.92 • 10 -4 F
	R 3	Battery model's resistor	13.4 • 10 -3 Ω
	C 4	Battery model's capacitor	2.92 • 10 -4 F
	R 4	Battery model's resistor	13.4 • 10 -3 Ω
	Supercapacitor
	Lsc	Supercapacitor converter's inductor	1.0 • 10 -3 H
	Rsc	Supercapacitor model's resistor	0.8 • 10 -3 Ω
	C 0	Supercapacitor model's capacitor	56.0 F
	C 1	Supercapacitor model's capacitor	1.0 F
	C 2	Supercapacitor model's capacitor	1.0 F
	R 1	Supercapacitor model's resistor	6.0 • 10 -4 Ω
	R 2
		Battery current control loop	0.01 , 0.8
	Kpsc , Kisc	Supercapacitor current control loop	0.01 , 0.4

ACKNOWLEDGMENT

The authors would like to thank Professor Fei GAO, Mr. Patrice LUBERDA, and Mr. Pierre DIEBOLT for their help to apply the proposed methodology to the considered real-time application.

s 4 + 57.14s 3 + 816.3s 2 + 0.005714s + 10 -8 0.34s 4 + 0.0172s 3 + 2.858 • 10 -4 s 2 + 1.721