Daniel Jost
email: daniel.jost@pompiersparis.fr

Vivien Hong

Tuan Ha

Julie Trichereau

Benoit Frattini

Clément Derkenne

Sabine Lemoine

Frédéric Lemoine

Romain Jouffroy

Frédéric Lemoine

Romain Kedzierewicz

Frédérique Briche

Pascal Diegelmann

René Bihannic

Olivier Stibbe

Bertrand Prunet

Frederic Lemoine

Vincent Lanoë

Frederique Briche

Xavier Lesaffre

Ludovic Delhaye

Laurent Prieux

Stéphane Travers

Contributing factors to early recurrence of ventricular fibrillation during out-of-hospital cardiac arrest: An observational retrospective study

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

ABSTRACT Introduction

In out-of-hospital cardiac arrest (OHCA), external electric shock (EES) is recommended for treating ventricular fibrillation (VF). Refibrillation commonly occurs within one minute postshock. We aimed to investigate refibrillation times and identify clinical and electrical factors associated with them.

Materials and Methods

This retrospective observational study, based on the Paris Fire Brigade database, included non-traumatic OHCA over 18 years of age who received at least one shock with an AED from Basic Life Support (BLS) rescuers and from which we randomly selected a sample to measure the refibrillation-times. Without prior reference to it in the literature, we classified the refibrillation-time into two modalities according to whether it was above or below the mediantime. We performed multiple regression analysis to assess associations between refibrillation-time and potential explanatory factors.

Results

Among 13,181 patients who experienced OHCA from January 2010 to January 2014, we analysed AED data from 215 patients (590 shocks), 82.1% males, median age 61[IQR: 52-75] years. Most of them occurred at home (57%), were witnessed (87%), and were shockable (88.8%). A median of 5 [START_REF] Valenzuela | Outcomes of rapid defibrillation by security officers after cardiac arrest in casinos[END_REF][START_REF] Caffrey | Public use of automated external defibrillators[END_REF][START_REF] Van Alem | VF recurrence: characteristics and patient outcome in out-of-hospital cardiac arrest[END_REF][START_REF] Shanmugasundaram | Analysis of amplitude spectral area and slope to predict defibrillation in out of hospital cardiac arrest due to ventricular fibrillation (VF) according to VF type: recurrent versus shock-resistant[END_REF] EES/patients were delivered. The median-time from shock to refibrillation was 25 sec. Multivariate analysis showed that a shorter postshock hands-off time favoured earlier refibrillation (p=0.034), as well as older age (p=0.002) (Figure 2, supplementary table).

Conclusion

In non-traumatic OHCA, most refibrillations occurred within 45-sec post-shock. Age and postshock hands-off time were the two contributing factors to time to refibrillation.

MANUSCRIPT

Survival after OHCA varies depending on many factors. In a recent report, survival to hospital discharge ranged from 3.1% to 20.4% across 16 OHCA registries across the world [START_REF] Kiguchi | Out-ofhospital Cardiac Arrest across the World: First Report from the International Liaison Committee on Resuscitation (ILCOR)[END_REF]. The prognosis differs markedly for patients with initially shockable rhythms, such as ventricular fibrillation (VF), or pulseless ventricular tachycardia (VT) [START_REF] Hasselqvist-Ax | Early Cardiopulmonary Resuscitation in Out-of-Hospital Cardiac Arrest[END_REF][START_REF] Sasson | Predictors of survival from out-ofhospital cardiac arrest: a systematic review and meta-analysis[END_REF]. External electric shock (EES) is recommended for treating VF/VTs. The sooner defibrillation, the better the chance of survival [START_REF] Valenzuela | Outcomes of rapid defibrillation by security officers after cardiac arrest in casinos[END_REF][START_REF] Caffrey | Public use of automated external defibrillators[END_REF]. However, many subjects experience rapid VF/VTs recurrence [START_REF] Van Alem | VF recurrence: characteristics and patient outcome in out-of-hospital cardiac arrest[END_REF][START_REF] Shanmugasundaram | Analysis of amplitude spectral area and slope to predict defibrillation in out of hospital cardiac arrest due to ventricular fibrillation (VF) according to VF type: recurrent versus shock-resistant[END_REF] defined as a new VF episode that occurs after an initial VF termination [START_REF] Callaway | Part 4: Advanced Life Support: 2015 International Consensus on Cardiopulmonary Resuscitation and Emergency Cardiovascular Care Science With Treatment Recommendations[END_REF]. A VF recurs, in part, due to a persisting underlying cause. The only recourse for Basic Life Support (BLS) teams is to repeat EES as many times as necessary, without any recommended limitation.

Refibrillation mostly occurs within one minute post-shock [START_REF] Telesz | Recurrent ventricular fibrillation: Experience with first responders prior to advanced life support interventions[END_REF][START_REF] Conover | Resumption of Chest Compressions After Successful Defibrillation and Risk for Recurrence of Ventricular Fibrillation in Out-of-Hospital Cardiac Arrest[END_REF]. The cardiac rhythm is analysed with an automated external defibrillator (AED) every 2 min. The patient must wait until the cardiopulmonary resuscitation CPR-cycle ends before receiving the next shock [START_REF] Berdowski | Chest Compressions Cause Recurrence of Ventricular Fibrillation After the First Successful Conversion by Defibrillation in Out-of-Hospital Cardiac Arrest[END_REF][START_REF] Wilkins | Part 4: Adult Basic Life Support[END_REF]. However, CPR provides only partial blood flow; thus, waiting is harmful [START_REF] Telesz | Recurrent ventricular fibrillation: Experience with first responders prior to advanced life support interventions[END_REF][START_REF] Berdowski | Chest Compressions Cause Recurrence of Ventricular Fibrillation After the First Successful Conversion by Defibrillation in Out-of-Hospital Cardiac Arrest[END_REF][START_REF] Berdowski | Time in Recurrent Ventricular Fibrillation and Survival After Out-of-Hospital Cardiac Arrest[END_REF].

This study aimed to describe the time to refibrillation in OHCAs and identify clinical and electrical factors associated with them.

Materials and methods

Study design

This retrospective observational study retrieved data from the Paris Fire Brigade database on OHCAs that occurred in Paris and suburbs. The French Committee for Ethics in Anaesthesia-Resuscitation-Research approved the protocol (IRB 00010254-2020-033).

Setting

The Paris Fire Brigade covers 62 km 2 (7 million inhabitants) with a two-tiered physicianmanned emergency system. The first tier is provided by BLS crews, which are staffed with three to five firefighter-rescuers who can apply an AED (not switchable to manual mode).

They respect the international guidelines despite providing CPR with an active compressiondecompression device.

Participants

We used a two-step inclusion procedure. The first step was focused on the patient, and the second step was focused on the CPR-cycle patterns. First, we included patients over 18 years that experienced non-traumatic OHCAs and received at least one EES with an AED.

Then, we randomly selected 40 patients in each of the strata of the variable "number of shocks administered" (1,2…7, and >7). This 40 number per-strata was intended to provide robustness to the normality of the distribution within each stratum. In the end, we included all patients in the last stratum, due to their small number. Second, since we were given access to the traces only after randomisation, we secondary excluded patients with non-interpretable electrocardiographs (ECGs), non-readable thoracic-impedance variation lines, and CPRcycle patterns that showed refibrillation-times without the observation window between 5 and 120-seconds post-shock (Figure 1, supplementary file)

Variables

We collected the Utstein variables (age, gender, location, bystander status, response time) and AED data, including the ECG trace, thoracic impedance variation curve, initial heart rhythm, VF amplitude, pre-and post-shock hands-off time, CPR-ratio, chest compression (CC) frequency, post-shock rhythm, number of analysis, and time to refibrillation.

Defibrillation was considered successful when VF terminated for over 5 sec, after shock delivery. We defined refibrillation as recurrence of VF within 5 to 120-sec post-shock [START_REF] White | Refibrillation, resuscitation and survival in out-of-hospital sudden cardiac arrest victims treated with biphasic automated external defibrillators[END_REF]. In the absence of any prior reference in the literature, we classified the refibrillation-time into two modalities according to whether it was above or below the median time.

The post-shock hands-off times were categorized into four groups: 1-5, 6-10, 11-30,>30sec [START_REF] Conover | Resumption of Chest Compressions After Successful Defibrillation and Risk for Recurrence of Ventricular Fibrillation in Out-of-Hospital Cardiac Arrest[END_REF]. The CPR-ratio was defined as the time to achieve CPR / total-CPR-cycletime before shock delivery (Supplementary Figure ).

Data Sources

For each OHCA, firefighters transmitted a case-report form to the Sudden Death Expertise Centre (SDEC) [START_REF] Marijon | Sudden death expertise centre: A multi disciplinary approach for sudden death[END_REF][START_REF] Bougouin | Characteristics and prognosis of sudden cardiac death in Greater Paris[END_REF]. We retrieved ECGs and impedance records from AED internal storage files. We used specific software to measure the hands-off times; physicians assessed the times to VF recurrence manually, with twofold-checking for difficult cases.

Statistical Analysis

Continuous variables were expressed as mean (standard deviation) or median [interquartilerange 25th-75 th ] depending on their distribution, categorical data as percentage.

Comparisons used the chi-square test for categorical variables, and the Student or Mann-Whitney-Wilcoxon or Kruskal-Wallis test, when appropriate, for continuous variables. The linearity of quantitative variables was checked with fractional polynomial regression. For nonlinear data, continuous variables were evaluated as categorical variables.

We used generalized linear latent and mixed models (GLAMM) to analyse repeated data, with the link function logit and the binomial family. Associations between time to VF recurrence and potential explanatory factors were evaluated with multiple regression analysis. We constructed a multivariate logistic regression model to identify variables with p<0.2 and applied a stepwise-descending method to select variables with p<0.05 for the final multivariate model. All tests were bilateral, with a 5% significance level. Randomisation was a computer-based process. Analyses were performed with STATA 14.0 (StataCorp,College Station,TX).

Results

We identified 13,181 OHCAs that occurred from January 2010 to January 2014 (49/100,000 inhabitants/year). Of the 1,590 patients that received at least one EES, we randomly selected data (Figure 1). Their epidemiological characteristics were comparable to those of the initial 1,590 patients. They were mostly men (82.1%), with a median age of 61(IQR: 52-75) years.

Most OHCAs occurred at home (57%), were witnessed (87%), and the initial cardiac rhythm was VF/VT(88.8%). The median number of EES per patient was 5[IQR:4-7]. Among 590 EES patterns, 260 showed pre-shock coarse VF. The median pre-and post-shock hands-off-times were, respectively, 23.3 [19.3-27.3] and 7.4[5.9-11.2] sec. The median-time from shock to refibrillation was 25 sec (Table 1). In univariate analysis, female gender (p=0.02), older age (p=0.03), and shorter post-shock hands-off times (p=0.02), favoured earlier refibrillation. No other variables were associated with refibrillation precocity, including the number of EESs administered (p=0.68). Multivariate analysis showed that a shorter postshock hands-off times favoured earlier refibrillation (p=0.034), as well as an older age (p=0.002)(Figure 2

and supplementary table).

Discussion

To date, this retrospective study was the largest (590 refibrillation patterns) to investigate factors associated with refibrillation precocity. The median time to refibrillation (25 sec,IQR:13-44)was early in the currently recommended 2-min interval between CPR-cycles.

Age and post-shock hands-off time were associated with refibrillation precocity. The postshock hands-off time did not depend on the initial AED rhythm (e.g., asystole, pulseless electrical activity (PEA), VF), the post-shock rhythm (asystole versus PEA), or the number of shocks administered.

-Methodology

To measure refibrillation time, we needed to collect as many refibrillation episodes as possible, based on technical limitations for ECG pattern extraction. Patients with fewer shocks were the least informative on this point. For our cohort, the previously decided number of patients randomised per-stratum (n=40), led to over-represent multi-shocked patients, but allowed recovery of more refibrillation episodes. Regarding the model's post-estimation, the residuals appear graphically close to a Gaussian distribution; additional research may identify further variables to improve the model.

-Time to refibrillation

Our observed refibrillation precocity was consistent with recent studies. Conover reported a time to refibrillation of 39 sec, Telesz a 55% recurrence rate within a 30-sec postshock period [START_REF] Telesz | Recurrent ventricular fibrillation: Experience with first responders prior to advanced life support interventions[END_REF][START_REF] Conover | Resumption of Chest Compressions After Successful Defibrillation and Risk for Recurrence of Ventricular Fibrillation in Out-of-Hospital Cardiac Arrest[END_REF]. We observed that recurrence precocity led to VFs that lasted until the end of the CPR-cycle imposed by the AED.

-

Post-shock CC resumption and refibrillation time

Refibrillation was all the more delayed as the BLS team was late in resuming post-shock CC.

It is well-documented that the force generated by CC can cause ventricular capture followed by a VF recurrence [START_REF] Berdowski | Chest Compressions Cause Recurrence of Ventricular Fibrillation After the First Successful Conversion by Defibrillation in Out-of-Hospital Cardiac Arrest[END_REF][START_REF] Osorio | Effect of chest compressions on ventricular activation[END_REF]; that delaying CPR after the shock can result in later VF recurrences [START_REF] Berdowski | Time in Recurrent Ventricular Fibrillation and Survival After Out-of-Hospital Cardiac Arrest[END_REF]; and that most VF recurrences occur within 10 sec of initiating CC [START_REF] Telesz | Recurrent ventricular fibrillation: Experience with first responders prior to advanced life support interventions[END_REF]. In general, it may be plausible that CPR could be linked to VF recurrence [START_REF] Hill | The environment and disease: association or causation?[END_REF]. Prolonged cumulative post-shock CC times lead to coronary perfusion pressures and a critical mass of excited myocardium, which condition the heart muscle to be conducive to VF recurrences.

-Association between age and refibrillation precocity

Myocardial aging create a substrate for re-entrant ventricular arrhythmias. Ischemia can reduce the resting threshold of myocardial cells and facilitate ventricular arrhythmias [START_REF] Sangaralingham | The aging heart, myocardial fibrosis, and its relationship to circulating C-type natriuretic Peptide[END_REF]. This phenomenon might explain previous evidence showing older patients needing fewer stimuli to refibrillate.

-

Perspectives

The refibrillation precocity may suggest AED rhythm analysis should be anticipated to limit the time in VF. The consequences of such a change would need to be balanced with the capacity for lay responders to adhere to.

-

Limitations

This study had limitations. First, it was retrospective designed. Second, refibrillation-time was expressed as a dichotomous variable, the cut-off corresponding to the median-time value.

Since this might not be clinically relevant, it ensured a balanced distribution between groups.

As the median-time may vary between populations, we cannot suggest systems use the same cut-off value, but rather consider analysing their population-specific refibrillation-times.

Third, the sample was not representative of the population in terms of number of patients per-stratum of number of shocks. Fourth, the study may have been undersized due in part to secondary exclusions, increasing the challenge of replicating our results.

Conclusion

For non-traumatic adult OHCAs in VF/VT, age and post-shock hands-off time were significant contributing factors to earlier VF recurrence. Although these findings need confirmation, they may contribute to further research on the causes and consequences of time spent in VF during OHCA.

Figure 1 .

 1 Figure 1. Flow chart Patients from the Paris OHCA cohort, from January 2010 to January 2014. Abbreviations: OHCA, Out-of-hospital cardiac arrest ; ALS, advanced life support; BLS, basic life support; AED, automated external defibrillator.

Figure 2 ..

 2 Figure 2. Factors associated with the refibrillation status (early vs. late) in multivariate analysis, after stepwise descending procedure. A longer post-shock hands-off time favoured late refibrillation, while older age favoured early refibrillation..

Table 1 .

 1 Baseline characteristics and outcomes (a), AED and Resuscitation characteristics (b). Abbreviations: AED, automated external defibrillator; CC, chest compression; CPR, cardiopulmonary resuscitation; VF, ventricular fibrillation; VT ventricular tachycardia. *median [25th-75th percentile]; **mean (standard deviation); ***7 missing data (N=208); †10 missing data (N=205) ; † †8 lost to follow-up ;

		61 [52-75]
	**	62.8 ± 15
	Response time: Call to AED switch on (min)*	9.2 [4.7-13]
			Time from shock to
		n (%)	VF recurrence* (sec)
	Gender
	Men	170 (82.1)	25 [13-46]
	Women	37 (17.9)	21 [12-38]
	Bystander***
	No Bystander	27 (13.0)	25 [14-47]
	Bystander without CC performed	100 (48.1)	19.5 [12-38.5]
	Bystander with CC performed	81 (38.9)	27 [13-42]
	Location †
	Home	118 (57.6)	23 [12-42]
	Public	69 (33.7)	28 [14-44]
	Other	18 (8.7)	34 [15-58]
	Rhytm at AED Switch on
	Not shockable	24 (11.2)	17.5 [10-33.5]
	Shockable	191 (88.8)	25 [13-46]
	No. Of shocks delivered per patient*	5 [4-7]
	Outcomes
	Death on site	113 (52.6)	23 [13-42]
	Survival to hospital admission	102 (47.4)	25 [13-47]
	Survival to hospital discharge † †	28 (13.0)	35 [16-62]
	(b)
	Number of analyses with shockable rhythm	590
	Shock outcomes -n (%)
	VT	8 (1.4)
	Asystole	359 (60.8)
	Organized rhythm	223 (37.8)
	Time from shock to refibrillation (sec)*	25 [13-44]
	Chest compression rate per min*	101 [99-106]
	Ratio (%) *
	Pre-shock CPR Ratio	76.0 [72.7-80]
	Pre-shock CC Ratio	61.8 [55-67.4]
	Times (sec)*
	Preshock CC hands off time	23.3 [19.3-27.3]
	Postshock CC hands off time	7.4 [5.9-11.2]
	AED-Analysis duration	10.3 [9.1-10.3]
	Time from Analysis to Shock	20.4 [17.0-23.2]
	VF Amplitude -n (%)
	Fine	324 (55.5)
	Coarse	260 (44.5)

Acknowledgments:

the Paris Fire Brigade and their Firefighters who struggle every day to save lives.

Conflict of interest statement

No conflicts of interest to declare.

Funding

No source of funding


Leïla Bensalah

Stéphanie Caillies
email: stephanie.caillies@univ-reims.fr

Leila Bensalah
email: leila.bensalah@univ-reims.fr

High and low theory-of-mind scores of child-teachers: Which teaching strategies are efficient?

Keywords: Teaching, Tutoring, Theory of mind, False belief, Preschoolers

While some authors investigating teaching in children suggest that teaching strategies are more efficient in terms of learners' performances when child-teachers have achieved falsebelief understanding, others argue that even before they reach this point, children already use efficient teaching strategies. To test these two hypotheses, we investigated the efficiency of teaching strategies adopted by child-teachers who had either succeeded or failed on falsebelief (FB) tasks. A total of 26 5-year-old child-teachers were divided into two groups according to their FB score (FB+ vs. FB-) and paired with 3-year-old learners who had not yet acquired FB understanding. The learners were asked to perform a jigsaw task. Teaching strategies displayed by the child-teachers included conveying information, giving verbal explanations coupled with demonstrations, and engaging in nonverbal demonstration/modelling. FB understanding was measured with classic FB tasks, and a composite verbal score was calculated as a control measure. Results showed that childteachers who successfully performed the FB tasks above all used conveying information and verbal explanation coupled with demonstrations strategies. However, correlations revealed that conveying information, which was more frequently used in the FB+ group, was significantly and positively correlated with learners' performances in both groups. This finding lessens the impact of child-teachers' FB understanding in the tutoring situation.

Introduction

Teaching is consensually viewed as a critical condition for promoting learning (e.g., [START_REF] Bidell | Vygotsky, Piaget and the dialectic of development[END_REF][START_REF] Bruner | Toward a theory of instruction[END_REF][START_REF] Csibra | Natural pedagogy[END_REF][START_REF] Leblanc | Teaching and learning as a bi-directional activity: Investigating dyadic interactions between child teachers and child learners[END_REF]Strauss, one's own or others' false-belief understanding and, for some of them, ignorance understanding [START_REF] Bartsch | Young children's attribution of action to beliefs and desires[END_REF]. Accordingly, when children successfully perform a battery of FB tasks, we can assume that they understand both ignorance and FB, and would thus be able to identify the mental states of learners in a tutoring situation and adjust their teaching strategies accordingly, thereby improving the learners' performances. However, studies in young children (e.g., [START_REF] Scott | How fresh a look? A reply to Heyes[END_REF][START_REF] Warneken | The emergence of contingent reciprocity in young children[END_REF] have suggested that even before they are able to successfully complete these tasks, childteachers already use efficient teaching strategies, in terms of learners' performances, thus generating controversy regarding the impact of ToM on teaching by children.

Our goal was to investigate how FB understanding influences child-teachers' use of teaching strategies and child-learners' performances. To the best of our knowledge, no study had previously probed the efficiency of the teaching strategies exhibited by child-teachers who can successfully perform FB tasks, and none had examined the efficiency of these strategies when child-teachers are unable to perform such tasks. This research is important for several reasons. First, it could shed light on the impact of FB understanding, defined as a high level of understanding of mental states, including ignorance, on children's teaching abilities, and more generally it could contribute to our understanding of teaching strategy development in children. Second, it could be informative for teachers. Teachers can use tutoring between pupils to efficiently complement their teaching in the classroom, and train pupils to adopt more efficient teaching strategies to help learners succeed. If teachers do not realize that their pupils are using inefficient teaching strategies, they may eventually abandon tutoring practice in the classroom. Finally, knowing which teaching strategies are efficient could inform adultteachers' practices.

In tutoring between children, the issue of teaching strategy efficiency raises the question of how these strategies vary according to whether child-teachers can successfully perform FB tasks. Currently, what little findings available are mixed. We therefore set out to document the teaching strategies employed by child-teachers and their impact on learners' performances during the acquisition of FB understanding.

How do child-teachers' teaching strategies change with FB understanding?

Despite substantial literature on tutoring between children (e.g., [START_REF] Bensalah | Les bénéfices du tutorat entre enfants [The benefits of tutoring between children[END_REF][START_REF] Cohen | Educational outcomes of tutoring: A meta-analysis of findings[END_REF][START_REF] Hartup | Peer relations[END_REF] and ToM in preschoolers (e.g., [START_REF] Astington | Sometimes necessary, never sufficient: False belief understanding and social competence[END_REF][START_REF] Miller | Children's understanding of preexisting differences in knowledge and belief[END_REF][START_REF] Wellman | Sequential progressions in a theory-ofmind scale: Longitudinal perspectives[END_REF], and a growing interest in the impact of ToM on how children approach teaching (e.g., [START_REF] Bensalah | Acquisition of the concept of teaching and its relationship with theory of mind in French 3-to 6-year olds[END_REF][START_REF] Ziv | Children's understanding of teaching: The role of knowledge and belief[END_REF][START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF], few studies have probed how child-teachers' strategy use changes as a result of acquiring FB. Moreover, while a qualitative analysis by [START_REF] Bensalah | The emergence of the teaching/learning process in preschoolers: Theory of mind and age effect[END_REF] showed that child-teachers gradually become more interested in learners' needs, other studies [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF][START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF][START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF] have reported mixed findings concerning the three main types of teaching strategies probed, namely conveying information (e.g., verbal explanations, repetitions, rule explanations, instructions), giving verbal explanations coupled with demonstrations (e.g., showing learners how to perform the task based on verbal explanations), and engaging in nonverbal demonstration/modelling (e.g., simply showing learners how to perform the task). Several studies have demonstrated that once children have understood FB at around 4-5 years and are enrolled in a teacher role, they display more conveying information strategies than 3-year-olds who cannot yet successfully perform FB tasks, providing more verbal explanations and repetitions (i.e., reminding learners of tools and instructions; [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF], and more rule explanations [START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF]. Conversely, [START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF]'s study showed that 3-to 4-year-olds provide instructions more often than 5-year-olds do, and another study found no significant difference between these two age groups on verbal instruction use (e.g., "Get a bigger one"; [START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF]. Mixed results regarding verbal explanations coupled with demonstrations (i.e., explicitly showing learners how to perform the task, and doing it instead of them) have also been reported. [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF] found that 3-year-olds who failed on FB tasks displayed less verbal explanations coupled with demonstrations strategies than 5-year-olds who succeeded on FB tasks, whereas [START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF] reported that 5-year-olds made less use of demonstrations with verbalizations than 3-year-olds. Here again, two other studies showed that this strategy use remained stable during this period [START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF][START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF]. Regarding nonverbal demonstration, [START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF] and [START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF] also found that 5-year-olds with FB understanding made less use of nonverbal demonstration/modelling (i.e., when a teacher actively shows a learner what to do, carrying out the task without any verbalizations) than 3-year-olds. Once again, however, Wood et al. found no such significant difference between these two age groups [START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF]. The mixed findings of these scant studies in this area are probably due to the fact that in every case, the child-teachers were each associated with a same-age peer (3-vs. 5-year-old dyads), hence a high variability in the teaching strategies collected. At least two other methodological flaws limited these studies' scope: either the effects of the child-teachers' FB understanding and age were probed concurrently [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF][START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF], or only the age effect was examined, but it was interpreted in terms of ToM ability [START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF]. It was therefore impossible to disentangle the specific effect of FB/ignorance understanding on the development of teaching strategies.

To shed light on these mixed results concerning possible changes in teaching strategies as a result of improvements in FB understanding, we investigated the three above-mentioned strategies: conveying information, giving verbal explanations coupled with demonstrations, and engaging in nonverbal demonstration/modelling. To address the three methodological flaws, we studied child-teachers at 5 years, as this is the age when children start to successfully perform FB tasks, but it is still possible to find some with low FB task scores. To reduce dyadic variability, we paired these 5-year-old teachers with 3-year-old learners who failed on FB tasks.

Efficiency of teaching strategies adopted by child-teachers with or without FB understanding

While authors have assessed gains resulting from tutoring among children (e.g., [START_REF] Bensalah | Les bénéfices du tutorat entre enfants [The benefits of tutoring between children[END_REF], they have so far neglected the issue of teaching strategy efficiency in terms of learners' performances, depending on whether the child-teacher has understood FB. Several findings suggest that more efficient teaching strategies are adopted after FB acquisition, but the question of strategy efficiency in child-teachers who cannot yet successfully perform FB tasks has so far been tackled separately. Studies investigating teaching strategies after FB acquisition have highlighted correlations between the FB scores of 5-year-old child-teachers and both verbal explanation strategies [START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF] and the number of different teaching strategies used [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF]. Given the methodological limitations mentioned earlier, these sparse findings need to be further documented, focusing on learners' performance.

Researchers have yet to explore the question of teaching strategy efficiency before children have acquired mental states related to teaching, as measured with the usual FB tasks, including ignorance questions. Although the findings described above suggest that children without FB understanding do not adopt efficient teaching strategies, there is growing evidence that this is not the case. Studies have shown that, from an early age, children experiment with strategies, especially conveying information to unknowledgeable persons, and are also able to attribute false belief to others in nonverbal implicit FB tasks, suggesting that their strategies can be efficient. In line with authors arguing in favor of the early emergence of teaching [START_REF] Koster | Why do infants help? A simple action reveals a complex phenomenon[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF], studies examining the teaching strategies of young children who are able to attribute FB to others in implicit FB tasks have shown that children as young as 12 months are able to convey information to an unknowledgeable person, pointing to the location of an object in order to inform an unknowing adult [START_REF] Liszkowski | 12-and 18-month-olds point to provide information for others[END_REF]. At 2 years, toddlers more often supply information to a parent who does not know where a toy is than to a parent who does know [START_REF] O'neill | Two-year-old children's sensitivity to a parent's knowledge state when making requests[END_REF], and can spontaneously hand over an out-of-reach object to an adult [START_REF] Warneken | The emergence of contingent reciprocity in young children[END_REF]. These findings are consistent with those yielded by an ecological design [START_REF] Verba | Beginnings of collaboration in peer interaction[END_REF], which showed that toddlers aged 18-24 months who have mastered a specific activity spontaneously facilitate the activity of a child who has not yet done so (e.g., handing the correct object to the partner, taking away an object that is hindering the partner's activity). This early ability to convey information to an unknowledgeable person continues to develop in preschoolers, who also start to display demonstration strategies.

Children aged 3; 6 years have been found to convey more explicit information about what should or should not be done in a task than their younger counterparts aged 2;6 or 3 years do [START_REF] Ashley | Cooperative problem-solving and teaching in preschoolers[END_REF]. They also more actively show learners how to perform the task.

These findings are supported in the recent literature on infants. For although the issue is still subject to some debate (e.g., [START_REF] Allen | How to help: Can more active behavioral measures help transcend the infant false-belief debate?[END_REF], there is growing evidence that infants are able to implicitly attribute FBs to others, as assessed with nonverbal FB tasks, before they can successfully perform the usual explicit FB tasks [START_REF] Baillargeon | Invited commentary: Interpreting failed replications of early false-belief findings: Methodological and theoretical considerations[END_REF][START_REF] Scott | How fresh a look? A reply to Heyes[END_REF]. As early as 15 months, infants can predict the action of an actor on the basis of the latter's FB [START_REF] Onishi | Do 15-month-old infants understand false beliefs?[END_REF], and at 18 months, they understand that an actor's FB can serve as the basis for their own helping behavior [START_REF] Buttelmann | Eighteen-month-old infants show false belief understanding in an active helping paradigm[END_REF] or can be corrected by appropriate communication involving words and gestures [START_REF] Song | Can an agent's false belief be corrected by an appropriate communication? Psychological reasoning in 18-monthold infants[END_REF].

Toddlers aged 18-24 months can spontaneously inform an adult who falsely believes that a toy is in a particular place [START_REF] Knudsen | 18-month-olds predict specific action mistakes through attribution of false belief, not ignorance, and intervene accordingly[END_REF]. During their third year of life, children seem able to understand others' FBs, as measured both with implicit FB tasks [START_REF] He | False-belief understanding in 2.5-year-olds: Evidence from violation-of-expectation change-of-location and unexpected-contents tasks[END_REF] and with an adapted version of a classic verbal FB task [START_REF] Rubio-Fernandez | How to pass the false-belief task before your fourth birthday[END_REF]. Taken together, these findings show that before 3 years, children can already convey tuned information to an unknowledgeable person and exhibit an implicit FB understanding. This suggests that after 3 years, both abilities continue to improve, becoming more explicit, and therefore making the children's teaching strategies more efficient.

To further document previous findings [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF], we examined the effect of child-teachers' FB task success (including on ignorance questions) on learners' performances. More specifically, to address the question of the efficiency of teaching strategies implemented by child-teachers with or without successful FB task performances, we explored whether teaching strategies can be identified as efficient in terms of learners' performances. More specifically, we used learners' performances during the tutoring session to gauge the immediate effect of teaching strategies, and scrutinized their performances on a posttest in which they were asked to perform the task on their own. First, to further extend the scant literature on the implications of FB and ignorance understanding in a tutoring situation, we set out to shed light on previous mixed findings by examining two groups of dyads in which 5-year-old child-teachers with either a low (FB-) or a high (FB+) score on FB/ignorance tasks were each paired with a 3-year-old learner-child without FB/ignorance understanding. We then looked at differences in teaching strategies (conveying information, nonverbal demonstration/modelling, and verbal explanations coupled with demonstrations) between the FB-and FB+ dyad groups. Second, we examined whether learners performed better when they were associated with a child-teacher who had acquired FB/ignorance understanding, rather than a child-teacher who had not. Third, we looked at whether some teaching strategies were more closely linked to learners' performances in the FB+ group that in the FB-group. Fourth and last, we controlled for the effect of the childteachers' verbal abilities.

Method

Participants

We recruited 58 French preschoolers from a middle-class district of Reims (France).

The majority of the children came from middle-class families. All their parents provided their written informed consent, in accordance with the Declaration of Helsinki. Six children were excluded, as they did not meet the matching criteria. The final sample therefore comprised 26 3-year-olds (12 girls, 14 boys; Mage = 39 months, range = 36-42 months) and 26 5-year-olds (12 girls, 14 boys; Mage = 60 months, range = 57-63 months).

Materials

Jigsaw task. We chose a jigsaw that would elicit both information-telling and demonstrations, in order to elicit a range of instructional strategies among the preschoolers [START_REF] Bensalah | The emergence of the teaching/learning process in preschoolers: Theory of mind and age effect[END_REF]. The jigsaw, representing a clown, was made up of 12 blocks that fitted together and were painted on one side. FB tasks. We administered three FB tasks to participants: two unexpected-content tasks (Band-Aid box task including an ignorance test question; [START_REF] Bartsch | Young children's attribution of action to beliefs and desires[END_REF]and Candies task;[START_REF] Perner | Three-year-olds' difficulty with false belief: The case for a conceptual deficit[END_REF] and a change-of-location task [START_REF] Wimmer | Beliefs about beliefs: Representations and constraining function of wrong beliefs in young children's understanding of deception[END_REF].

French versions of all three had previously been used by [START_REF] Caillies | Children's understanding of idioms and theory of mind development[END_REF]. The maximum overall score was 9.

Verbal tasks. We administered three subtests drawn from the Wechsler Preschool and Primary Scale of Intelligence (WPPSI-III; Wechsler, 2004) to establish a composite verbal score (CVS). The Receptive Vocabulary subtest assessed participants' knowledge of the names of objects that might be involved in the task, such as the parts of the jigsaw or its colors. The Verbal Reasoning subtest assessed their ability to infer words from verbal cues and thence their ability to reason from verbal information, such as that provided by learners about performing a task. The Information subtest appraised academic and cultural acquisition, and included questions about everyday life, such as the bodily characteristics featured in our clown jigsaw task (e.g., "How many ears do you have?").

Procedure

First, all the participants were individually pretested by an experimenter to see whether they could complete the jigsaw. Second, we randomly administered the three FB tasks to all the children, in order to form two groups of teachers and make sure that none of the 3-year-olds understood FB. Third, we administered the verbal tasks to the 5-year-olds.

Fourth, we randomly paired the 5-year-old child-teachers with same-sex 3-year-old childlearners to obtain 12 dyads (i.e., seven girl dyads and five boy dyads) for the FBU+ group, and 14 dyads (i.e., five girl dyads and nine boy dyads) for the FB-group, in order to videotape a dyadic tutoring session.

Pretesting the jigsaw task. All the children were pretested with the jigsaw task. They were shown the completed clown jigsaw by the experimenter, who turned it over so that they could see it was only painted on one side. The experimenter then took the blocks apart and jumbled them up. The children were given up to 5 minutes to put the jigsaw back together. This period could be curtailed if the older children completed the jigsaw particularly quickly, or if the younger children displayed a lack of interest in the task or showed signs of distress.

All the 5-year-old children successfully completed the jigsaw, correctly positioning all 12 blocks before the 5 minutes were up. In order to verify the stability of their performances (see [START_REF] Verba | Tutoring interactions between young children: How symmetry can modify asymmetrical interactions[END_REF], we asked them to do the jigsaw three more times. All the 5-year-olds successfully performed these three trials and were therefore considered sufficiently expert to take on the teacher's role, while the 3-year-old children, none of whom correctly positioned more than six blocks, were deemed to be learners. FB tasks. The three FB tasks were administered to all the children in random order. In the change-of-location story, the lion figurine of the original test [START_REF] Caillies | Children's understanding of idioms and theory of mind development[END_REF] was replaced with a dog figurine, but the rest of the task was the same. The experimenter explained that Pierre has a plastic dog that he puts away in a box with a blue top before going out. While he is out, his sister enters the room, takes the dog out and places it in a box with a green top. Pierre comes back into the room and wants to play with his plastic dog. We asked the children two test questions, followed by two control questions. The test questions were: "Where will Pierre look for his plastic dog?" (Q1) and "Why will Pierre look for his dog in (child's answer)?" (Q2). The control questions were: "Where did Pierre put his dog before going out?" and "Where is the dog really?" Each test question was scored 1 point, with a maximum score of 2, providing the control questions were also answered correctly.

In the Band-Aid box task, the experimenter showed the children two boxes, one plain, the other with a picture of Band-Aids on it. She asked the children to indicate the box containing Band-Aids, and most of the children pointed to the labeled box. She then offered to open both boxes, and they established that the labeled box was empty, whereas the plain box contained Band-Aids. She closed the two boxes. The experimenter then produced two dolls, named Camille and Hugo, that both needed Band-Aids because they had cut themselves, but did not know what was in the boxes.

Next, she asked four test questions. A prediction question (Q1: "Where will Camille look for the Band-Aids first?") was followed by a control question ("Will she find the Band-Aids?"). The prediction question was scored 1 point, providing the control question was answered correctly. Then came an explanatory test question (Q2: "Why should Hugo look in this box [the box containing Band-Aids]?"), an ignorance test question (Q3: "Does Hugo know where the Band-Aids are?"), and an explanatory test question (Q4: "What does Hugo believe?"). These three questions were followed by a control question about the objects' location: "Where are the Band-Aids?" Each correct answer to a test question was scored 1, providing the children had correctly answered the control questions, with a maximum score of 4.

In the Candies task, the children were shown the contents of two boxes. One had a picture of sweets on it but contained a pencil, while the other one was plain but contained sweets. We asked the children a prediction test question about the time when they did not yet know what the boxes contained (Q1: "What did you think the box contained before you saw inside it?"). We then asked a prediction test question about a classmate who did not know about the boxes' contents (Q2: "What would a classmate expect to find in the boxes?"). This was followed by an explanation test question (Q3: "Why would s/he think that there was (previous response to Q2) in the box?"). The final question was a control question ("What is in the box?"). Correct answers to the three test questions were each scored 1, providing the control question was correctly answered, with a maximum score of 3.

Twenty-six 5-year-olds were allocated to one or other of the groups according to their FB scores. Those who scored 6-9 points on the three FB tests were included in the FB+ group (12 dyads), and those who scored 0-3 joined the FB-group (14 dyads). None of the 3-year-olds scored more than 2 and were all randomly assigned to one or other of the dyad groups.

Six 5-year-olds scored 4 or 5 points and were excluded from the sample.

Verbal tasks. We administered the three different subtests, always in the same order, to the 5-year-old children. The Information standard subscore ranged from 0 to 15, the Receptive Vocabulary standard subscore from 0 to 15, and the Verbal Reasoning standard subscore from 0 to 15. The maximum CVS ranged from 0 to 45.

Tutoring session. For each dyad, the 5-year-old child-teacher and 3-year-old learner were placed side by side in front of the completed clown jigsaw so that the learner could see the goal to be attained. The experimenter then said that one of them (name of the older child) knew how to put the clown together and could teach the other one (name of the younger child). The adult took the blocks apart and jumbled them up. The session was videotaped and lasted as long as it took to complete the jigsaw. The duration of the sessions ranged from 80 to 364 seconds. The learner was then asked to do the jigsaw unaided in an immediate posttest.

Measures

Analysis of teaching strategies

The teaching strategies displayed by the child-teachers were divided into the three categories identified in the Introduction: conveying information, engaging in nonverbal demonstration/modelling, and giving verbal explanations coupled with demonstrations. These three categories were intended to correspond to three forms of teaching displayed by childteachers. The conveying information strategy was identified when the learner states were consistently monitored from a distance, and included both verbal and nonverbal modes. The nonverbal demonstration/modelling strategy involved showing the learner how to do the jigsaw by actually putting the blocks together. The intermediary category (verbal explanations coupled with demonstrations) was a combination of the two preceding ones. These strategies, which took the form of actions and verbalizations, were only taken into account if the learner was watching the child-teacher at the time. Conveying information strategies were defined as teaching behaviors that allowed the learner's activity to be monitored or followed from a distance by the child-teacher (i.e., the latter did not actually do the jigsaw). This strategy included monitoring the learner by enrolment, where the teacher attempted to direct the learner's attention to the task (e.g., "Can we begin?"), responses to the learner's questions about performing the task (e.g., "Where do I put this block?" "It's the hand"), evaluation of the learner's performance (e.g., "This block isn't in the right place"), offers of help to the learner (e.g., "Do you want me to help you?"), verbal guidance (e.g., "You have to turn the block round first"), nonverbal guidance of the learner's activity (e.g., pointing to the place where a block picked up by the learner should go), and following the learner by observing his or her activity (e.g., observing the learner's handling of the blocks). Nonverbal demonstration/modelling strategies exclusively comprised the child-teacher's actions (e.g., picking up and positioning the blocks to construct the jigsaw). Verbal explanations coupled with demonstrations also included the child-teacher's actions, but this time combined with verbalizations (e.g., "Look at this block, it belongs here", or "Not this block, I'll show you").

As task duration and the number of behaviors varied across the dyads, we calculated the amount of time taken up by each behavior in relation to the total duration of each dyad's tutoring session. As in [START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF], in order to check that the behaviors had been accurately identified in terms of teaching strategies, two judges coded five dyads' behaviors (20% of videotapes): one Master's student who was trained to code behaviors according to the categories, and one of the authors, who had not participated in the data collection. The coding was based on transcripts of the videotapes. In the cases where a doubt persisted, the coders consulted the video, and when two coders disagreed, a discussion generally led to agreement.

Learners' performances

were correctly positioned during the tutoring session, in which learners were assisted by their child-teacher, and then in a posttest administered after the tutoring session, in which learners were given 5 minutes to do the clown jigsaw unaided, unless they displayed a lack of interest in the task or showed signs of distress before this time had elapsed.

Results

Group differences and reliability

Preliminary analyses did not reveal any significant difference between the two FB groups on any of the following variables: teacher age (MFB-= 58.71 months, MFB+ = 60 months), t(24) = 1.19, ns, learner age (MFB-= 38.93 months, MFB+ = 38.83 months), t(24) = -.12, ns, jigsaw pretest performances of the learners (all FB-), who were paired with either FBor FB+ teachers (MFB-= 1.93, MFB+ = 2.17), t(24) = .33, ns, jigsaw completion times (in seconds) (MFB-= 200.29, MFB+ = 185. 17), t(24) = -0.71, ns, or length (in seconds) of the three jigsaw completion trials (MFB-= 381.71, MFB+ = 349.92), t(24) = -.93, ns. Furthermore, there was no significant difference between the dyads on either mean videotape duration (172.79 s for the FB-group vs. 215.83 s for the FB+ group), t(24) = 1.53, ns or teacher CVS (MFB-= 31.57, MFB+ = 34.17), t(24) = 1.34, p = .19, 95% CI [-2.12, -9.95]. To check that the behaviors had been accurately identified in terms of teaching strategies, we used Cohen's kappa to measure interrater reliability. For teaching strategies, overall agreement was 98.09% (range: 94.59-100%), yielding a kappa of .97.

Effects of FB success on teaching strategies

To explore how teaching strategies differed between the FB-and FB+ dyad groups, we conducted three analyses of covariance (ANCOVAs) on the proportion of time taken up by each type of teaching strategy used by the child-teachers (i.e., conveying information, nonverbal demonstration/modelling, and verbal explanations coupled with demonstrations) as a function of FB group, with CVS as the covariate. Results (see Fig. 1) revealed significantly greater use of conveying information, MFB-= 33.89, MFB+ = 62.96, F(1, 23) = 6.80, p < .02, and verbal strategy coupled with demonstrations and MFB-= 1.17, MFB+ = 13.59, F(1, 23) = 11.99, p < .001, with FB task success.

Insert Figure 1 about here

Effects of child-teachers' FB success on learners' performances

To further document previous findings [START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF] and examine the effect of child-teachers' FB on learners' performances, we conducted an ANOVA on the mean numbers of blocks correctly positioned by the learners during the tutoring session and at the posttest, and an ANCOVA with CVS as the covariate. As illustrated in Table 1, results revealed significant main effects of both FB group, F(1, 24) = 8.31, p = .01, and session (i.e., tutoring session or posttest), F(1, 24) = 7.60, p < .01, but no significant interaction, F(1, 24) = 0.91, ns. However, when CVS was introduced as a covariate, only the FB group effect remained significant, F(1, 23) = 5.97, p < .02. These results supported previous findings.

Insert Table 1 about here

Relationship between teaching strategies and learners' performances as a function of FB group

To test the prediction that teaching strategies would be more efficient in the FB+ dyad group than in the FB-dyad group, we calculated Pearson's correlation coefficients between each type of teaching strategy and learners' performances as a function of FB dyad group. Results (set out in Table 2) showed that when teachers scored highly on the FB tasks, their use of conveying information strategies was positively and significantly correlated with learners' tutoring session performances, while their use of verbal explanations coupled with demonstrations strategies was negatively and significantly correlated with learners' tutoring session performances. These results remained significant when language (CVS) was kept constant (Table 3). When teachers scored poorly on the FB tasks, their use of conveying information strategies was positively and significantly correlated with learners' performances, not only during the tutoring session like the FB+ teachers, but also at the posttest. These results remained significant when language (CVS) was kept constant. Moreover, results revealed that for FB-teachers, verbal explanations coupled with demonstrations strategies were not significantly linked to learners' performances, but when CVS was kept constant, this link was positively and significantly correlated with learners' tutoring session performances.

These results suggest that whatever the child-teachers' FB level, their teaching strategies were efficient enough for learners to perform well.

Insert Table 2 about here

Insert Table 3 about here

Discussion

We conducted a novel investigation into the impact of FB/ignorance understanding, as measured by the usual tasks, on child-teachers in a tutoring situation, and specifically on their use of efficient teaching strategies. Before tackling this issue, we documented two points to clarify previous sparse but mixed literature findings and address methodological limitations of previous studies.

First, in order to probe how teaching strategies differed according to FB task success, we compared FB-and FB+ dyad groups on the use of three types of strategies. In line with Davis-Unger and Carlson (2008)'s, [START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF][START_REF] Ziv | Relations between the development of teaching and theory of mind in early childhood[END_REF]'s findings, our results showed that FB+ child-teachers made more use of conveying information and verbal explanations coupled with demonstrations strategies than FB-child-teachers. As in [START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF]'s study, however, the two FB groups did not differ significantly on demonstration/modelling strategy use. Our results showed that child-teachers with FB or ignorance understanding spend more time applying teaching strategies than FB-childteachers did, and each strategy was adapted to specific teaching sequences [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF]. For instance, conveying information allowed the FB+ child-teachers in our study to guide the learners from a distance and facilitate their handling of the blocks to construct the jigsaw. Giving verbal explanations coupled with demonstrations provided an alternative to the other types of strategies when, for instance, conveying information proved to be inadequate. Once the learners had begun to grasp how to construct the jigsaw, the childteachers could go back to using the conveying information strategies, as they were now better adapted to the learners' needs. Moreover, while we cannot exclude the possibility that demonstration/modelling was adapted to learners' needs, the fact that the two groups did not differ significantly on this strategy, probably because of the low number of dyads in each group, leads us to suggest that further studies would yield similar findings.

Second, to further document sparse findings [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF], we examined the effect of child-teachers' FB success on learners' performances during the tutoring session and at the posttest. We found that the learners' performances during the tutoring session and at posttest were better when their child-teachers had FB or ignorance understanding, even when we controlled for CVS. This means that the impact of child-teachers' FB or ignorance understanding can be both immediate and deferred, thus doubly benefiting learners. The effect of child-teachers' FB task success on learners' performances appeared to be substantial and instantaneous. Despite the small size of our sample, we can cautiously suggest that these results generally confirm the impact of childteachers' FB or ignorance understanding both on their strategy use and on learners' performances.

However, our main goal was to test alternative hypotheses in terms of teaching strategies and learners' performances. First, we hypothesized that teaching strategies are more efficient in terms of learners' performances when child-teachers have achieved FB understanding. However, we also assumed that before they reach this point, children's teaching strategies are already efficient in terms of learners' performances, as a result of their prior experience of teaching strategies and their early implicit FB understanding. Furthermore, our study was designed to establish which teaching strategies (conveying information, verbal explanations coupled with demonstrations, or demonstration/modelling) are most efficient, depending on the child-teachers' FB or ignorance understanding. Results

showed that FB+ child-teachers' use of conveying information was positively linked to learners' performances during the tutoring session, while their use of verbal explanations coupled with demonstrations strategies was negatively linked to learners' performances during the tutoring session when CVS was kept constant. This result suggests that the use of conveying information strategies by child-teachers with FB or ignorance understanding allows learners both to benefit from the teachers' information or guidance, and to gain direct experience of performing the task. The fact that the child-teachers' use of verbal explanations coupled with demonstrations strategies was negatively linked to learners' performances during the tutoring session is not surprising, given that when they used these strategies, they placed the blocks instead of the learner, thereby reducing the learners' performance during the tutoring session. Although this may be an interesting alternative strategy for child-teachers, particularly when learners have obvious difficulty placing the blocks, overuse of this strategy may be counterproductive. Bearing in mind our small sample size, we cautiously suggest that conveying information strategies allow learners to quickly gain in efficiency, even during the tutoring session. These results also complement a previous study [START_REF] Bensalah | The emergence of the teaching/learning process in preschoolers: Theory of mind and age effect[END_REF] showing that child-teachers with FB understanding are more interested in learners' needs, by suggesting that the use of these distant teaching strategies [START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF] indicates that child-teachers who have acquired FB and ignorance understanding are able to continually detect them in learners. It would be interesting for future studies to dissociate the FB and ignorance measures, in order to confirm this interpretation. Nevertheless, our results showed that when the child-teachers had low FB scores, their conveying information strategies were also positively correlated with learners' performances, both during the tutoring session and at posttest, and that verbal explanations coupled with demonstrations strategies were positively linked to learners' performances, but only during the tutoring session, even when CVS was kept constant. This suggests that even when the child-teachers failed on the FB tasks, the learners benefited from both types of strategies during the tutoring session, and continued to benefit from conveying information strategies at posttest. These results are consistent with the hypothesis whereby children acquire experience of both conveying information [START_REF] Ashley | Cooperative problem-solving and teaching in preschoolers[END_REF][START_REF] Liszkowski | 12-and 18-month-olds point to provide information for others[END_REF][START_REF] O'neill | Two-year-old children's sensitivity to a parent's knowledge state when making requests[END_REF][START_REF] Verba | Beginnings of collaboration in peer interaction[END_REF][START_REF] Warneken | The emergence of contingent reciprocity in young children[END_REF] and reading other people's minds before the age of 4-5 years [START_REF] Baillargeon | Invited commentary: Interpreting failed replications of early false-belief findings: Methodological and theoretical considerations[END_REF][START_REF] Buttelmann | Eighteen-month-old infants show false belief understanding in an active helping paradigm[END_REF][START_REF] He | False-belief understanding in 2.5-year-olds: Evidence from violation-of-expectation change-of-location and unexpected-contents tasks[END_REF][START_REF] Knudsen | 18-month-olds predict specific action mistakes through attribution of false belief, not ignorance, and intervene accordingly[END_REF][START_REF] Onishi | Do 15-month-old infants understand false beliefs?[END_REF][START_REF] Rubio-Fernandez | How to pass the false-belief task before your fourth birthday[END_REF][START_REF] Scott | How fresh a look? A reply to Heyes[END_REF][START_REF] Song | Can an agent's false belief be corrected by an appropriate communication? Psychological reasoning in 18-monthold infants[END_REF]. Nevertheless, the fact that verbal explanations coupled with demonstrations strategies were positively linked to learners' performances in the tutoring session when child-teachers had low FB scores, but negatively linked when childteachers had high FB scores, suggests that beyond a certain threshold, this type of strategy is counterproductive. Further studies may support this view. These results therefore indicated that whatever the child-teachers' FB scores, all the learners seemed to make maximum use of the conveying information strategies to perform the task in hand, whether or not they were accompanied by verbal explanations coupled with demonstrations strategies. This suggests that this type of strategy is efficient at a particularly early age, presumably as a result of children's observation and/or experience of conveying information strategies displayed by parents, siblings, and relatives in everyday life. Above all, however, the fact that more teaching strategies were positively linked to learners' performances, and that one type influenced performances both during and after the tutoring session, even when the childteachers had a low FB score, lessens the importance of child-teachers' FB or ignorance understanding. Here again, however, these explanations will have to be further documented, owing to the small size of our sample.

To conclude, in addition to supporting and extending previous findings [START_REF] Bensalah | The emergence of the teaching/learning process in preschoolers: Theory of mind and age effect[END_REF][START_REF] Davis-Unger | Development of teaching skills and relations to theory of mind in preschoolers[END_REF][START_REF] Strauss | Teaching as a natural cognition and its relations to preschoolers' developing theory of mind[END_REF][START_REF] Wood | On becoming a tutor: Toward an ontogenetic model[END_REF], the main contribution of our study is to highlight specific links between teaching strategies and learners' performances in a tutoring situation, depending on whether the child-teachers have acquired FB and ignorance understanding. Consistent with a growing number of findings in young children (e.g., [START_REF] Warneken | The emergence of contingent reciprocity in young children[END_REF] and infants (e.g., [START_REF] Baillargeon | Invited commentary: Interpreting failed replications of early false-belief findings: Methodological and theoretical considerations[END_REF], our study revealed an early, pre-FB/ignorance understanding ability to teach efficiently, through the use of conveying information strategies and, to a less extent, verbal explanations coupled with demonstrations. We also assessed the actual impact of acquiring this ToM ability, highlighting a rapid but also limited gain in efficiency among learners as a result of child-teachers using distant teaching strategies. These findings now need to be confirmed by further studies with larger samples. Moreover, several other control measures will have to be added in future studies, such as executive function assessments. In the education field, our findings indicate that adult teachers should encourage pupils to engage in tutoring situations, as these could be a good way of enabling learners to continue learning, at least in this type of task, whatever the child-teachers' level of FB understanding. They probably also tell us something about how child-teachers understand adult teachers in an

FB-FB+

Demonstration/ Modelling

Verbal explanations and demonstrations

Conveying information

Figure 1

 1 Figure 1 Time taken up by each type of teaching strategy in each type of dyad as a mean

Table 1

 1 Mean numbers (standard deviation) of blocks* correctly placed by the learners according to type of dyad and session

		FB-	FB+
	Learners' performance	3.36 (3.54)	6.92 (3.20)
	during tutoring session
	Learners' performance at	5.07 (3.20)	7.75 (1.36)
	posttest

* The jigsaw comprised 12 blocks.

Table 2

 2 Pearson's correlation coefficients between each type of teaching strategy (mean proportion of total time) and the two measures of learner performance, according to the type

	of dyad group
			FB-group		FB+ group
		Learners'	Learners'	Learners'	Learners'
		performance	performance at	performance	performance at
		during	posttest	during	posttest
		tutoring		tutoring
		session		session
	Conveying Information	.79***	.73**	.75**	.34
	Verbal Explanations and
	Demonstrations	.49	.35	-.64*	.06
	Demonstration/Modelling	-.43	-.36	-.45	-.58

*p < .05. ** p < .01. *** p < .001.

Table 3

 3 Partial Pearson's correlation coefficients between each type of teaching strategy (mean proportion of total time) and the two measures of learner performance, according to the type of dyad group, keeping language (CVS) constant

			FB-group		FB+ group
		Learners'	Learners'	Learners'	Learners'
		performance	performance at	performance	performance at
		during	posttest	during	posttest
		tutoring		tutoring
		session		session
	Conveying Information	.75**	.67*	.76**	.29
	Verbal Explanations and
	Demonstrations	.69**	.55	-.72*	-.02
	Demonstration/Modelling	-.35	-.27	-.45	-.54

*p < .05. ** p < .01. *** p < .001.


K B K Cao

V Boitier
email: vboitier@laas.fr

A novel simple GMPPT method based on probability distribution of global maximum power point under partial shading conditions

Keywords: Global maximum power point tracking algorithm (GMPPT), partial shading, Lambert W function, probability distribution, MATLAB/Simulink

A photovoltaic (PV) array having multiple cells in series with bypass diodes may exhibit multiple power peaks under uneven irradiation, therefore an algorithm is required to reach the global maximum power point (GMPP). While a lot of methods have been proposed in the literature, they are usually quite complex and does not fully utilize the characteristics of the PV array. This paper first highlights a rapid and optimized mathematical simulation of the PV array using MATLAB to find the probability distribution of voltage at GMPP under multiple irradiation conditions and temperatures. The resulting GMPP distribution in 22E6 irradiation and temperature conditions checked for an array of 4 PV modules in series with bypass diodes is presented, as well as the optimizations in MATLAB necessary to keep simulation time in check. From the obtained result, we propose a simple and efficient probability-based algorithm capable of reaching GMPP up to 93.64% of the time.

Introduction

Harvesting energy through PV is usually the simplest and most viable source for mobile embedded systems away from the grid. Some examples are remote sensors to monitor a jungle's ecosystem and a mountaineer up in the Alps trying to charge his/her phone. This mobility and unstable operating conditions mean that such PV systems must be efficient, resilient against rapidly changing irradiation and partial shading, yet simple enough to be put on a low power microcontroller (µC) (e.g., PIC18 8-bit µC family) Since PV cells are connected in series to increase harvesting potential, uneven insolation across the array may cause hotspots that accelerate degradation as well as significantly reduce the array's power output [START_REF] Kim | Photovoltaic hot spot analysis for cells with various reverse-bias characteristics through electrical and thermal simulation[END_REF]. Therefore, bypass diodes are usually used to provide a current path around the shaded modules, but this leads to multiple power peaks at the detriment of control efficiency (Figure 1) [START_REF] Mohammed | Bypass diode effect on temperature distribution in crystalline silicon photovoltaic module under partial shading[END_REF], [START_REF] Teo | Impact of Partial Shading on the P-V Characteristics and the Maximum Power of a Photovoltaic String[END_REF]. . In general, these algorithms are simple to implement and are efficient under good and stable weather conditions, but they cannot guarantee optimal power acquisition in less-than-ideal conditions. The multiple peaks capable methods on the other hand are capable of identify the true MPP when the array is partially shaded, and they can be divided into several sub-classes of algorithms. The first are improved versions of single peak capable methods like improved P&O or incremental conductance with zoning to multiples of 0.8Voc [10]- [START_REF] Ahmad | A Variable Step Size Perturb and Observe Method Based MPPT for Partially Shaded Photovoltaic Arrays[END_REF], which take advantage of the fact that the MPP of a single module can be found in this region. These approaches are also simple and decently good under stable weather conditions, but they fail to consider the voltage drop caused by activated bypass diodes and their convergence times are not great. The second type of GMPP methods are optimization algorithm based like genetic algorithm . These approaches can identify GMPP, but they suffer from several drawbacks, most notably their implementation complexity (e.g., floating points operations, exponentials, logarithms) and wild power swings during the search phase. Finally, there are fuzzy logic controllers methods [29]- [START_REF] Yilmaz | PV system fuzzy logic MPPT method and PI control as a charge controller[END_REF] and neural network MPPT [START_REF] Md | Artificial Neural Network Based Maximum Power Point Tracking of a Photovoltaic System[END_REF], which are all too heavy on a low-power µC. Another problem missing in the current literature was the distribution of GMPP for a particular PV installation. Currently, our models for PV arrays are quite precise with the simple single diode model [33]- [START_REF] Ding | A simplified model for photovoltaic modules based on improved translation equations[END_REF] and Simulink has proven to be an effective simulation tool for PV applications [START_REF] Nguyen | Mathematical modeling of photovoltaic cell/module/arrays with tags in Matlab/Simulink[END_REF]. Moreover, there exists a computational method based on the Lambert W function that would vastly reduce the time needed to solve the PV equation [37]- [START_REF] Upadhyay | Parameter extraction of solar photovoltaic system using Lambert-W function for different environmental condition[END_REF]. However, most of the efforts to solve the PV equation mainly focus on obtaining the current output of a single PV module without any bypass diode and there was not yet any proposition on how to efficiently build the current output profile of an entire array with multiple bypass diodes numerically besides using Simulink. This method is acceptable if we are to perform a few simulations, but to observe the array's performance under a wider range of weather conditions, it quickly becomes inefficient (i.e, 375000 conditions simulation take 8h45min using MATLAB/Simulink 2020b on 8 parallel threads using a Ryzen 7 3700X 8-cores processor, 32GB of DDR4 3200MT/s). In this paper, we propose an optimized method to simulate the power output of a PV array and help to visualize the distribution of GMPP under a very wide range of irradiation and weather conditions. From this result, we will also be proposing a probabilistic approach to improve the traditional P&O method that should have a fast convergence time as well as simple enough to be easily implemented on a low power µC. The preliminary proofof-concept will be done in simulation via Simulink and compared to deterministic PSO, GWO and ABC algorithms. All results presented were tested using 4 PV monocrystalline modules in series with bypass diodes whose characteristics will be detailed in the later sections (Figure 2).

Modelling PV arrays with bypass diodes under partial shading

Since each one of our PV modules consist of 6 smaller cells in series, its model can be obtained via the modified single diode model as shown in Figure 3 and from the work of [34], we obtain the mathematical equations modelling our PV modules (1).

I 0 = I scn + k i (T -T ref ) e ( q AkT (V ocn +k v (T-T ref ))) -1
Solving the methods proposed in [39], we obtain the explicit expression of current I for a given V, G and T (2).

I = X - Lambert(KYR s e KXR s ) KR s (2) with X = I L +I 0 - V R p (1+ R s R p ) , Y = I 0 e KV (1+ R s R p ) and K = q AkT
Next, we have the bypass diode which will be modeled using the simplified version of the Shockley diode equation [START_REF] Teo | Impact of Partial Shading on the P-V Characteristics and the Maximum Power of a Photovoltaic String[END_REF].

I b = I r e -qV NkT (3)
The parameters and their description as well as values used for the later simulations are detailed in Table 1.

Combining the PV module current with the bypass diode current gives us total current output of each one of these PV + bypass diode block [START_REF] De Brito | Evaluation of the Main MPPT Techniques for Photovoltaic Applications[END_REF].

𝐼 𝑡𝑜𝑡𝑎𝑙 = 𝑋 - 𝐿𝑎𝑚𝑏𝑒𝑟𝑡(𝐾𝑌𝑅 𝑠 𝑒 𝐾𝑋𝑅 𝑠 ) 𝐾𝑅 𝑠 + 𝐼 𝑟 𝑒 -𝑞𝑉 𝑁𝑘𝑇 (4)
Since these blocks of PV + bypass are connected in series, it means that their combined voltage output equals the sum of their respective voltage when operating under the same current. For this reason, we need to interpolate the above obtained I = f(V,G,T) into V = g(I,G,T), so that we can add the voltage of each block for a given current. An illustration of the method is given in Figure 4. This work can be done using MATLAB via the function 'interp1()'.

GMPP distribution of the PV array

Using the previously described method, we can vary the irradiation for each module from 0 to 1000Wm -2 in steps of 10Wm -2 , find the GMPP and register the voltage where GMPP occurred. To further improve its predicting capabilities, we also vary the temperature from 0°C to 80°C in steps of 20°C, supposing homogenous temperatures for all modules, which should cover a wider operating range. The result obtained is presented in Figure 6. It confirms that for a given temperature, the GMPP can be found in clear clusters on the voltage range and consistent with the approximation that it should falls in the vicinity of 0.8Vocn plus some multiples of Vocn, sans some voltage drop caused by activated bypass diodes. We can also observe that for increasing temperature, the peak distribution shifts toward the left (lower voltage) which is consistent with the negative voltage coefficient of monocrystalline silicon. Another distribution result based on real world irradiation measurement can be found in Figure 7, acquired using 4 SP Lite2 pyranometers strapped to the back of a bicycle during a ride of around 1 hour around Toulouse, France (10h34 to 11h22, 21/06/2021). The sampling frequency is 5kHz per channel with subsequent filtering making the effective sampling frequency 100Hz. The resulting graph was also filtered of power peaks below 1W because they are not realistically extractable.

However, we should also note about several optimization taken during this step to avoid excessive runtimes. The first problem is with 'interp1()' function which is computationally intensive comparing to other vector operations and should be done only once. Therefore, the program calculates the I/V profile of the PV module + bypass for all irradiation and temperatures only once and store these I/V into memory. To construct the overall, I/V of the entire array, a simple addition is necessary which drastically reduces runtime (about a factor of 1000) (Table 2). The second optimization is the removal of redundant conditions. For an array of PV modules in series, whether they receive 1000-800-400-200Wm -2 respectively or 1000-400-800-200Wm -2 respectively should give the same output. The time gained from this can be observed in Figure 5, plotting the amount of time needed to sweep an array of N modules in series with a precision of 20 levels of irradiation per module. By choosing the arbitrary limit of simulation time to 1 day, we see that instead being capable of checking only 8 modules in series without optimization, we can check up to 20 modules in series. This means that the presented method can be scaled up to more powerful systems beyond 100W.

Proposed improved P&O fast GMPPT

With the distribution graph clearly denominating zones where the GMPP can be found, we can conclude that randomly searching the voltage range, or calculating the power at each iteration is not necessary. Instead, we propose a fast GMPPT method where we only measure the power at 3 specific voltage values and continue with P&O where the power obtained is the highest. To evaluate the feasibility of such algorithm, we devise the following criteria (Figure 8):

• If the power gradient from the chosen voltage point and the GMPP is constantly increasing, it means that a P&O from that point can reach GMPP.

• Otherwise, P&O will be stuck in a local peak and fail to reach GMPP. By choosing 3 voltage values 5.2V, 8.2V and 12.2V on the sweep performed in Figure 6, and by applying the aforementioned criteria to evaluate the probability of finding GMPP, we have a success rate of 93.64% over 22106375 distinct irradiation/temperature conditions checked, 3.46% failure rate and 2.89% are power at GMPP too low to be extractable (i.e. below 1W) or inaccessible peaks . This is because we will be testing this method on a buck converter that output to a Li-ion battery operating at around 3.2-4V. Moving on to the implementation on a buck converter, reaching only 3 voltage points is not a trivial task because there is no linear relationship between the duty cycle and input voltage, but we can regulate the duty cycle so that the input voltage approaches the desired values. Given that during this regulation only a comparison needs to be made (whether voltage measured equals what we want), the algorithm still retains its simplicity. Furthermore, by registering the duty cycles that correspond to each one of the desired voltage points during the last search, subsequent scans should take considerably less time to finish than the initial initialization. For the converter, we use a DC/DC buck converter whose averaged model can be found in Figure 9. For the preliminary evaluation of our proposed method, we are going to compare it against P&O [4], deterministic PSO (DPSO) [START_REF] Ishaque | A Deterministic Particle Swarm Optimization Maximum Power Point Tracker for Photovoltaic System Under Partial Shading Condition[END_REF], GWO [20] and ABC [START_REF] Pilakkat | An improved P&O algorithm integrated with artificial bee colony for photovoltaic systems under partial shading conditions[END_REF]. All algorithms are written in our application context using the description from their respective articles. Since the converter's response time to any duty cycle step is at most 3ms, the commands will be sent at 5ms intervals. The test conditions would be for 3 irradiation conditions in Figure 1, initially starting at condition (1), changing to (2) and then to (3). Temperature is fixed for all modules at 60°C. For condition [START_REF] Kim | Photovoltaic hot spot analysis for cells with various reverse-bias characteristics through electrical and thermal simulation[END_REF] where the array is evenly irradiated (Figure 10), P&O converged rapidly and without oscillation thanks to its optimal starting position, then we have GWO and DPSO converges relatively quickly around 80ms, our method converges a bit slower at around 120ms and finally ABC converges late at around 170ms. Concerning the amount of power obtained, there is no clear edge for any of the methods. When the condition changes from (1) to (2) (Figure 11), the GMPP position changes rapidly from around 12V down to 5V with a local peak around 9V, and it is here that P&O shows its weakness, with GWO also failing to converge toward global peak, however this seems to vary from run to run. Our proposed method converges around 20ms later than DPSO at around 90ms with ABC the last to converge toward global peak. Finally, when the condition changes from (2) to (3) (Figure 12), the GMPP changes to around 9V and here we have P&O, our method, DPSO and GWO all converges at around the same speed, with ABC again trailing behind. We also observe the effects of memorization of the previous duty cycles on the voltage search phase: 8 steps, then 7 steps then 5 steps. With this result, our method is not strictly superior in terms of convergence time, nor does it guarantee that the system will arrive at GMPP all the time. However, in terms of implementation complexity, the most complex operation is integer multiplication to obtain the power values, which is on par with P&O. This is also true for DPSO, however it has more comparisons and multiplications at each control cycle. Finally, we have GWO and ABC integrating a lot of randomized numbers, floating point multiplications and divisions.

Conclusions

In this paper, an efficient method to quickly simulate the power output of an array under a large amount of weather conditions was presented together with optimizations that made it possible. Using this capability, we plotted the distribution of the GMPP on the voltage range proving clear zone delineation for this application of 4 PV modules with 4 bypass diodes and estimated that our proposed fasttracking method can reach GMPP in 93.64% of the time.

The Simulink simulation result compared versus other lightweight algorithms like DPSO, P&O, GWO, ABC

shows promise and warrants further research with testing on real hardware as well as real world weather conditions.

Figure 1

 1 Figure 1 Power output of 4 PV modules in series with bypass diodes under different irradiation conditions Considering that under perfect irradiation the PV array still has a maximum power peak, there are great interest in optimizing PV systems in the literature which can be classified into 2 categories of algorithms: single peak capable and multiple peaks capable. The most well-known amongst the former are Perturb and Observe (P&O) [4], Incremental Conductance (IC) [5], β-parameter method [6], fractional open circuit voltage (Voc) [7], MPP locus characterization [8] and temperature based approximation of MPP [9]. In general, these algorithms are simple to implement and are efficient under good and stable weather conditions, but they cannot guarantee optimal power acquisition in less-than-ideal conditions. The multiple peaks capable methods on the other hand are capable of identify the true MPP when the array is partially shaded, and they can be divided into several sub-classes of algorithms. The first are improved versions of single peak capable methods like improved P&O or incremental conductance with zoning to multiples of 0.8Voc [10]-[START_REF] Ahmad | A Variable Step Size Perturb and Observe Method Based MPPT for Partially Shaded Photovoltaic Arrays[END_REF], which take advantage of the fact that the MPP of a single module can be found in this region. These approaches are also simple and decently good under stable weather conditions, but they fail to consider the voltage drop caused by activated bypass diodes and their convergence times are not great. The second type of GMPP methods are optimization algorithm based like genetic algorithm [14], differential evolution [15], particle swarm optimization (PSO) [16], [17], artificial bee colony (ABC) [18],

  [14], differential evolution [15], particle swarm optimization (PSO) [16], [17], artificial bee colony (ABC) [18], grasshopper optimization [19], grey-wolf optimization (GWO) [20]-[22], flower pollination algorithm [23], student psychology based optimization [24], dragonfly algorithm [25], ant colony optimization [26], Henry gas solubility optimization [27] and cuckoo search algorithm [28]

Figure 2 PVFigure 3

 23 Figure 2 PV modules whose parameters were used for the simulations

Figure 4 Figure 5 Figure 6 Figure 7

 4567 Figure 4 Illustration of how to obtain the current output of the array of 4 PV modules + bypass diodes

Figure 8 Figure 9 Figure 10 2 Figure 11

 8910211 Figure 8 Illustration of success and failure criteria for propose fast GMPPT method

Table 1 Parameter description and numerical values of the PV module and bypass diodes used

 1

	Name Description	Value	Unit
	G	PV module irradiation	Variable	Wm -2
	Gref	Standard irradiation	1000	Wm -2
	T	PV module temperature Variable	K
	Tref	Reference temperature	298.15	K
	Vocn	Open circuit voltage	3.8	V
	Iscn	Short circuit current	1	A
	Rs	Equivalent serial	0.2	Ω
		resistance of PV module
	Rp	Equivalent parallel	1200	Ω
		resistance of PV module
	kv	Voltage temperature	-0.0026	VK -1
		coefficient
	ki	Current temperature	0.0023	AK -1
		coefficient
	A	PV module equivalent	9.5
		diode ideality factor
	q	Electron charge	1.6x10 -19	C
	k	Boltzmann constant	1.38x10 -23 JK -1
	Ir	Bypass diode reverse	0.017	A
		saturation current
	N	Bypass diode ideality	4.23
		factor

Table 2 Average execution time of each iteration in the loop for 3 different simulation methods

 2

	Method	Time (µs)
	Simulink	672000
	'interp1() at each iteration'	8000
	Precalculated table	9