Amanda Edmonds

Aarnes Gudmestad

Paul-Valéry Montpellier

Operationalizing variables

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

Introduction

In the field of second language acquisition (SLA), we are often faced with the challenge of how to go about coding the data that are at the heart of any empirical approach to linguistics. It is crucial that data-coding decisions allow researchers to address new hypotheses and research questions, while at the same time allowing them to compare and contrast with past research findings, thereby building the knowledge base within the field [START_REF] Révész | Coding second language data validly and reliably[END_REF]. This tension between new and sometimes innovative operationalization decisions and the need for generalization (and, thus, points of comparison with previous research) tends to go unaddressed in much research published in SLA. By neglecting to thoroughly attend to this crucial aspect within SLA research method head on, we inevitably head towards the curious situation of using the same (or similar) terms to talk about variables that have been operationalized in quite different manners (for examples in the field of phraseology, see [START_REF] Myles | Formulaic sequence (FS) cannot be an umbrella term in SLA[END_REF]; in the field of corrective feedback, see [START_REF] Ranta | A cognitive approach to improving immersion students' oral language abilities: The Awareness-Practice-Feedback sequence[END_REF]. This state of affairs leads to claims of knowledge building that may be unfounded or overstated.

The current chapter reflects on the issue of data operationalization within SLA, using the specific example of future-time expression in second-language (L2) French as an illustration. We begin with a general reflection on the challenges associated with data coding, the potential for lack of consistency, and the dangers posed to the ability to generalize and draw valid conclusions. We support these observations with examples of data-coding decisions in variationist studies on future-time expression in native-speaker (NS) French [START_REF] Blondeau | La trajectoire de l'emploi du futur chez une cohorte de Montréalais francophones entre 1971 et 1995[END_REF][START_REF] Blondeau | La référence temporelle au futur dans les bulletins météo en France et au Québec : regard variationniste sur l'oral préparé[END_REF][START_REF] Comeau | A window on the past, a move toward the future: sociolinguistic and formal perspectives on variation in Acadian French[END_REF][START_REF] Grimm | A real-time study of future temporal reference in spoken Ontarian French[END_REF][START_REF] Grimm | Grammatical variation and change in spoken Ontario French: The subjunctive mood and the expression of future temporal reference[END_REF][START_REF] Grimm | Le présent à valeur de futur en français parlé[END_REF]Grimm & Nadasdi, 2011;Gudmestad, Edmonds, Donaldson, & Carmichael, in press;[START_REF] King | Back to the future in Acadian French[END_REF][START_REF] Poplack | Prescription vs. praxis: The evolution of future temporal reference in French[END_REF][START_REF] Poplack | Does the futur have a future in (Canadian) French?[END_REF][START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF][START_REF] Roberts | The future of Martinique French: The role of random effects on the variable expression of futurity[END_REF]Villeneuve & Comeau, 2016) and in L2 French [START_REF] Blondeau | Future temporal reference in the bilingual repertoire of Anglo-Montrealers: A twin variable[END_REF][START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF][START_REF] Gudmestad | Variable future-time reference in French: A comparison of learners in a study-abroad and a foreign-language context[END_REF][START_REF] Nadasdi | Emploi du 'futur' dans le français parlé des élèves d'immersion française[END_REF]. In the second portion of the chapter, we illustrate the significance of data-coding decisions by proposing two reanalyses of data from published studies. We conclude the chapter with a call for more explicitness in reporting and for clear justifications when linguists depart from previously established data-coding practices, arguing that both are necessary in order to move forward SLA research in a meaningful way.

Challenges associated with data coding: The example of future-time expression in

French

In a discussion of frequent shortcomings in SLA method sections, Mackey (2012, p. 26) remarked that "coding systems vary widely and are not always represented in sufficient detail." Concerning first the wide variety of data-coding practices in the field of SLA, [START_REF] Mackey | Second language research: Methodology and design[END_REF] suggest that this can lead to lack of comparability across different studies.

For this reason, they advise researchers to adopt existing data-coding schemas, unless changes in those schemas are clearly motivated. The second observation in the previously cited quote highlights the fact that descriptions of data coding are not always sufficiently precise. While this is sometimes the result of constraints over which researchers have no control (e.g., length limits placed on publications), the results of this lack of precision can be serious and weaken the ability of the scientific community to judge the importance of the study in question and to make relevant comparisons across studies. Lack of clarity may exist with respect to what is being studied (i.e., the dependent variable), as well as how the phenomenon under study was analyzed (e.g., independent variables). Thus, the variety in practices and the fact that sufficient details are not always provided together have the potential to undermine the ability to interpret the findings presented, as well as the potential for replication [START_REF] Mackey | Why (or why not), when, and how to replicate research[END_REF][START_REF] Marsden | The IRIS Repository: Advancing research practice and methodology[END_REF][START_REF] Polio | Replilcation and reporting: A commentary[END_REF][START_REF] Révész | Coding second language data validly and reliably[END_REF]. In other words, the fuzziness that exists with respect to data coding can impede attempts at building knowledge that is generalizable and verifiable within the field. Moreover, while any investigation into linguistic phenomena may suffer from data-coding problems, the coding of interlanguage poses particular challenges, given its "unstable" and "transitional" nature (Corder, 1981, p. 18).

Although we expect that the preceding comments will not be particularly controversial, the actual impact of data-coding decisions is rarely given center stage in the literature (for a recent exception, see [START_REF] Myles | Formulaic sequence (FS) cannot be an umbrella term in SLA[END_REF]. In the current chapter, we provide two concrete illustrations of how coding decisions may influence results and, ultimately, constrain conclusions drawn with respect to SLA. To do so, we have chosen the domain of future-time expression in L2 French, an area that we have researched within variationist [START_REF] Geeslin | Sociolinguistics and second language acquisition: Learning to use language in context[END_REF] and concept-oriented approaches [START_REF] Von Stutterheim | A concept-oriented approach to second language studies[END_REF] to SLA (see the sections Future-time expression and Reanalyzing adverbial specification, respectively, for definitions of these two approaches). When we began examining future-time expression, we were struck by the high level of diversity with respect to data-coding practices and reporting, diversity that characterized both the identification of the dependent variable (i.e., what counts as a future-time context) and the operationalization of numerous independent variables. This variety is present in studies of both first-language (L1) and L2 French and complicates our ability to engage in dialogue with previously published research. Faced with multiple ways of operationalizing certain variables, for which coding practices were not always sufficiently explicit, we found ourselves struggling with these decisions. We have published reflections on the coding of the dependent variable in a previous article (see [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF]. In the current chapter, we use the example of two of the most frequently studied independent variables -namely, temporal distance (also sometimes referred to as temporal reference) and the presence or absence of a lexical temporal indicator (LTI, often referred to as adverbial specification) -in order to highlight these challenges and, then, to examine how different decisions lead to different findings. In the remainder of this section, we offer a concise presentation of variable future-time reference in French, followed by an in-depth discussion of how temporal distance and LTIs are thought to influence this case of variation. In particular, we highlight differences in how these two factors have been operationalized in previous variationist research. Finally, we present the rationale for the current chapter.

Future-time expression in French

Future-time reference may be expressed in French using a variety of verb forms.

Whereas [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF], using a concept-oriented approach, identify no fewer than 13 such forms in the NS portion of a corpus of informal conversations, most previous studies have focused on two or three of the most frequent verb forms found in future-time contexts.

These include the periphrastic future (PF: 1a), the inflectional future (IF: 1b), and the futurate present (1c). In each of these three examples, the same NS is making reference to future time with the verb faire 'to make/do', although three different verb forms are used. This particular case of alternation has received much attention, particularly from researchers interested in examining morphosyntactic variation within a variationist perspective. A variationist approach to futuretime expression recognizes that certain instances in which reference to future time is made can be expressed using more than one form (called variants). Those instances in which variation is possible make up the envelope of variation, and variationist studies generally 1 Examples come from the corpus used in [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF]. 'NS5 refers to the fifth native speaker in the corpus and the token number 249 in the dataset.

employ quantitative methods (especially regression models) with the aim of identifying the internal and external factors that influence frequency of use of a given variant within that envelope [START_REF] Tagliamonte | Variationist sociolinguistics: Change, observation, interpretation[END_REF].

While variationist linguistics began in the 20 th century [START_REF] Labov | The social stratification of English in New York City[END_REF], the interest in variation in the future-time sector in French can be traced back hundreds of years. As shown by [START_REF] Poplack | Prescription vs. praxis: The evolution of future temporal reference in French[END_REF], grammarians writing between 1530 and 1999 addressed the variation in future-time expression, most often attempting to explain it away by evoking factors that constrain which variant will be used.

Future-time reference and temporal distance

The distance between time of speaking and the expected moment at which the future event (be it an action or a state) will be realized has long been thought to influence which verb form is used to make reference to future time. This is particularly clear with the PF, which is also called the futur proche or futur prochain 'close future', revealing the presumed connection between this form and events that are set to occur in the near future. In their review of 163 grammars, [START_REF] Poplack | Prescription vs. praxis: The evolution of future temporal reference in French[END_REF] found that "the proximity reading for PF enjoys the highest rate of agreement (59%) among grammarians" (p. 574). However, when the presumed connection between PF use and temporal distance has been examined in actual language use, findings diverge along geographic lines. Comeau and Villeneuve (2016, p. 234) go so far as to speak of a "temporal distance-polarity divide" when it comes to linguistic constraints on future-time reference. As they point out, it has generally been found that futuretime reference in varieties of French spoken in Québec, Ontario, and Western Canada are most strongly conditioned by sentential polarity (e.g., Blondeau, 2006; Grimm, 2015 2 ; Grimm & Nadasdi, 2011; Poplack & Turpin, 1999; Poplack & Dion, 2009), whereas the expression of future time in Acadian varieties of French has been found to be constrained by temporal distance [START_REF] Comeau | A window on the past, a move toward the future: sociolinguistic and formal perspectives on variation in Acadian French[END_REF][START_REF] King | Back to the future in Acadian French[END_REF]. As for French spoken in France, the results for these two variables are currently mixed: Roberts (2012) reports that sentential polaritybut not temporal distance -is operative, [START_REF] Villeneuve | Breaking down temporal distance in a Continental French variety: Future temporal reference in Vimeu[END_REF] find that only temporal distance significantly influences future-time expression, and Gudmestad et al. (in press) find that both variables are significant. As for the role played by temporal distance in future-time expression L2 French, we know of two variationist studies that have looked at oral data. Whereas [START_REF] Blondeau | Future temporal reference in the bilingual repertoire of Anglo-Montrealers: A twin variable[END_REF] found that the PF was significantly favored in the expression of proximal future events in the speech of Anglo-Montrealers, this factor had no significant effect in Nadasdi et al.'s (2003) analysis of speech produced by students living in Ontario and attending a French immersion program.

It bears noting that this body of evidence regarding temporal distance is based on at least seven different ways of dividing up the temporal distance spectrum, from the binary proximal versus distal opposition seen in ( 2a), to the six temporal categories coded by Roberts (2013 and2016, see [2g]). Clearly, King and Nadasdi (2003, p. 333) were correct when they wrote that "the exact definition of what constitutes 'near' is difficult to pin down."

(2) a. Proximal (< 24 hours) versus distal (> 24 hours) (Blondeau, 2006, p. 86;Blondeau et al., 2014, p. 679;Blondeau & Labeau, 2016, p. 252;Grimm, 2015, p. 236;Grimm, 2016, p. 7;Nadasdi et al., 2003, p. 207;Poplack & Dion, 2009, p. 571;Poplack & Turpin, 1999, p. 150;Roberts, 2012, p. 99;Villeneuve & Comeau, 2016, p. 325, fn. 9) b. < 1 hour, > week, indeterminate, continuous (Grimm & Nadasdi, 2011, pp. 178-179) c. < 1 hour, < 24 hours, < week, > week, continuous (King & Nadasdi, 2003, p. 328) d. < 1 hour, < 24 hours, < week, > week, unspecified (Grimm, 2010, p. 86) e. < 24 hours, < week, < month, > month, ambiguous (Gudmestad et al., in press) f. < 1 hour, < 24 hours, < week, < year, > year, indeterminate (Comeau, 2011, pp. 218-219) g. < 24 hours, < week, < year, > year, continuous, indeterminate (Roberts, 2013, p. 142;Roberts, 2016, p. 293) There is a clear tendency in this literature to opt for a two-way distinction between proximal and distal. However, this trend must be nuanced by at least three observations concerning the studies mentioned in (2a). First, there is some ambiguity around the coding of future events for which temporal distance cannot be determined. Whereas many linguists opt for an explicit "indeterminate" or "ambiguous" category, others appear to have grouped indeterminate examples with distal ones (Blondeau & Labeau, 2016, p. 253; see also [START_REF] King | Back to the future in Acadian French[END_REF] for a similar coding decision). Other researchers simply have not addressed the question of indeterminate future-time reference in their publications. Second, certain linguists (such as [START_REF] Nadasdi | Emploi du 'futur' dans le français parlé des élèves d'immersion française[END_REF] use the terms proximal and distal but do not define what is meant by these terms. Although we presume, given the previous studies they cite, that Nadasdi et al.

have adopted the within a day / beyond a day distinction, the fact that this is not made explicit is problematic, especially when we consider that other authors working on future-time reference in French may use these terms with different definitions. This was the case, for example, in Moses' ( 2002) concept-oriented analysis in which proximal referred to events occurring within the week following speech time. Third, [START_REF] Poplack | Does the futur have a future in (Canadian) French?[END_REF], Roberts (2012), [START_REF] Grimm | Grammatical variation and change in spoken Ontario French: The subjunctive mood and the expression of future temporal reference[END_REF][START_REF] Grimm | Le présent à valeur de futur en français parlé[END_REF], and [START_REF] Villeneuve | Breaking down temporal distance in a Continental French variety: Future temporal reference in Vimeu[END_REF] originally opted for a finergrained coding of temporal distance, only later collapsing to a binary distinction either because there was no evidence that the finer coding provided additional insight (Poplack & Turpin) or because the distribution of the data did not allow the researchers to retain the more detailed coding (Roberts; Villeneuve & Comeau).3 Thus, if a majority of studies has ultimately opted for the binary opposition, this distinction is not always the first coding choice preferred by scholars.

The tendency to start with a multi-categorial variable before reducing to a binary or ternary opposition is relatively common in this literature, where collapsing categories appears most often to be motivated by the need to avoid small cells in the regression analyses that characterize variationist linguistics. In such cases, and space permitting, it would be useful for authors to also report the tendencies found with the original coding, as was done by [START_REF] Villeneuve | Breaking down temporal distance in a Continental French variety: Future temporal reference in Vimeu[END_REF] with respect to temporal distance. After having presented their multivariate analysis with a binary temporal distance variable, these authors returned to their original coding for a follow-up analysis. In their own words, they recognize that the original finer breakdown reveals a finding that is unfortunately masked by the broader binary coding necessary for the multivariate analysis: the high frequency of the periphrastic future in the most immediate contexts.

[…] These results show that the PF is used at an extremely high rate (93.3%) with events anticipated to occur within a minute, which suggests that the PF is marking imminence, and appears to confirm early grammarians' descriptions: the periphrastic future does mark le futur proche. (p.

329)

In discussing this finding, the researchers highlight the difficulty in comparing results across studies: "In the other varieties [of French] which display a strong temporal distance effect, it is impossible to establish whether or not the PF is as highly favoured in imminent contexts (i.e., within the minute) due to the fact that the authors did not adopt as fine a temporal breakdown" (p. 330). In other words, comparison of results concerning temporal distance is compromised by the wide variety of coding schemas adopted in this literature. Moreover, the justification for the different ways of coding temporal distance presented in (2a-g) has, to the best of our knowledge, not been addressed.

Future-time reference and LTIs

Temporal adverbials (also referred to as LTIs in this chapter) that have the possibility of being future-looking, such as tout de suite 'right away', demain 'tomorrow', and un jour 'one day', are one means of establishing reference to future time. The importance of such LTIs in modern-day French has been extensively discussed with respect to future-time reference, and especially as concerns the futurate present (e.g., [START_REF] Grimm | Le présent à valeur de futur en français parlé[END_REF][START_REF] Goffic | Le présent « pro futuro[END_REF]. The futurate present does not carry explicit future morphology, and it has been suggested that future-time reference must be established by other means, such as temporal adverbials (see [START_REF] Blondeau | La trajectoire de l'emploi du futur chez une cohorte de Montréalais francophones entre 1971 et 1995[END_REF][START_REF] Moses | The development of future expression in English-speaking learners of French[END_REF]. In recent variationist accounts of future-time reference, datasets have been coded for the presence versus absence of temporal adverbials, with most linguists making a three-way distinction between absence of an adverbial, presence of a specific adverbial (e.g., demain 'tomorrow'), and presence of a non-specific adverbial (e.g., bientôt 'soon'). Research that has conducted multivariate analyses for only the PF and the IF in future-time contexts has most often found no significant effect for adverbial specification [START_REF] Blondeau | La trajectoire de l'emploi du futur chez une cohorte de Montréalais francophones entre 1971 et 1995[END_REF]Grimm & Nadasdi, 2011;[START_REF] King | Back to the future in Acadian French[END_REF][START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF][START_REF] Roberts | The future of Martinique French: The role of random effects on the variable expression of futurity[END_REF]Villeneuve & Comeau, 2016). [START_REF] Poplack | Prescription vs. praxis: The evolution of future temporal reference in French[END_REF] Results from this study look different as a multinomial regression was run. This type of analysis makes a three-way dependent variable possible, allowing us to examine the full dataset in a single model. For the other four studies, researchers ran three different models (one for each form of the dependent variable).

On its surface, the operationalization of this factor appears quite straightforward: A futuretime context either is or is not accompanied by a temporal adverbial making reference to future time. However, the difficulty -and the ambiguity -lie precisely in what is meant by accompanied. As we began coding our own data, this ambiguity became apparent, and we questioned how close a temporal adverbial had to occur to a future-time context in order to count as present. Did the temporal adverbial specifying when a future event was to occur have to appear within the same clause, or could it occur within the same utterance, the same turn, or even within the same discourse context as the verb expressing the event in question?

Unfortunately, previous studies that have provided details about the coding of this factor are rare: [START_REF] Poplack | Does the futur have a future in (Canadian) French?[END_REF] and [START_REF] Poplack | Prescription vs. praxis: The evolution of future temporal reference in French[END_REF] state that adverbial specification was coded at the level of syntax in their studies, whereas Grimm (2015, pp. 229-230) Opting for a clause-level coding of LTI could be justified on practical grounds, but perhaps less so on theoretical ones. Generally speaking, data-coding schemas should be both reliable (i.e., allow for consistent categorization of data) and theoretically valid [START_REF] Révész | Coding second language data validly and reliably[END_REF]. A clause-level coding of LTI certainly allows for consistent categorization of the variable, constituting an excellent example of what Révész (2012, p. 213) refers to as "low-inference categories," that is coding categories "which require little judgment" on the part of the coder. However, the justification of clause-level coding on a more theoretical level is less clear. Indeed, restricting this factor to the level of the clause effectively implies that the presence of temporal adverbials within the larger discourse beyond the clause does not influence verb forms used to express future time. Although this may very well be the case in interview data, in looking at our own conversation data, we saw numerous instances of futuretime reference being established via a temporal adverbial and being maintained across several turns, without repetition of LTIs. A short extract from an example that will be analyzed later in the paper illustrates this phenomenon (3):

(

Rationale

As demonstrated throughout this section, two important variables thought to influence future-time reference in French -namely, temporal distance and adverbial specificationhave been subject to a variety of data-coding decisions. Temporal distance has been sliced into anywhere from two to six categories of various temporal lengths, clearly complicating our ability to build knowledge across studies. The issue at hand with adverbial specification is somewhat different, as what is meant by presence and absence of an LTI has largely gone unaddressed. In other words, in these two variables, we find both an example of "coding systems [that] vary widely" and one of coding systems that "are not always represented in sufficient detail" (Mackey, 2012, p. 26). In what remains of this chapter, we offer two reanalyses of previously published data. First, we explore how the operationalization of temporal distance may change outcomes in terms of findings reported for an elicitation task [START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF]. We then return to conversation data that we analyzed in [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] in order to offer an alternative approach to operationalizing adverbial specification. In both cases, we discuss the implications that these critical assessments of data coding have for knowledge of L2 acquisition.

Reanalyzing temporal distance

As shown previously, the factor of temporal distance has been defined in numerous ways in studies looking at future-time reference in French, with the binary opposition between proximal (< 24 hours) and distal (> 24 hours) being the most frequently adopted. In this section, we explore the impact of two different operationalizations of this variable on a single dataset [START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF]. We will begin by presenting the study, in which temporal distance was originally divided into five categories. For the current chapter, we recoded the variable of temporal distance, collapsing it into a two-way distinction, before reanalyzing the dataset. We present these new findings and then end by comparing the results.

Presentation of original study

For our first publications on future-time expression, we opted to use a data-elicitation task in order to examine the role that three linguistic factors may play in future-time expression: temporal distance, LTIs, and (un)certainty markers (see IRIS repository for access to the task). The written contextualized task that we created contained a series of 30 short contexts that together told a story (for the original, Spanish version of the task, see [START_REF] Gudmestad | Second-language development of variable futuretime expression in Spanish[END_REF]. Each context ended with an expression that introduced direct dialogue (e.g., "he said", "she asked"), followed by three possible formulations for the dialogue. The formulations differed only in the form of the verb (PF, IF, or present), and the order in which the forms was presented was counter-balanced across items. As an example, item 24 is provided in (4):

(4) For each item, participants were asked to choose the formulation that they preferred. Across the 30 contexts, we manipulated temporal distance, LTIs, and (un)certainty markers. With respect to temporal distance, five different categories were examined: immediate, < 24 hours, < week, < month, and > year. The temporal distance of the future event was made clear in each context, and the five categories were evenly distributed across the 30 items, resulting in six items per temporal distance category. This task was completed by 30 NSs of French and a total of 116 NNSs with 24 different L1s, all of whom were studying in France at time of testing (see [START_REF] Gudmestad | Variable future-time reference in French: A comparison of learners in a study-abroad and a foreign-language context[END_REF], for additional analyses of this task). The NNSs were assigned to four levels of proficiency on the basis of a c-test (see [START_REF] Tremblay | Proficiency assessment standards in second language acquisition research: "Clozing" the gap[END_REF], for a discussion of c-tests as proficiency measures). This type of task presents learners with a passage in the target language from which a certain number of words has been deleted (50, in the case of the c-test used in the project in question). Learners were provided with half of the letters of the missing word and asked to complete all blanks.

In order to analyze the responses, a multinomial regression was conducted. The advantage of this type of analysis, when compared to binary regressions used in most variationist studies of future-time reference, is that a single model of the dataset is generated, thus taking into account the relationships between the three categories of the dependent variable (IF, PF, present). In Table 2, we present the temporal-distance results from [START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF]; for the full models, see the original study. In this type of model, both the dependent and independent variables have a base category against which the other categories are compared. In the model presented here, the base category for the dependent variable was the PF, with the results for the IF versus PF comparison presented on the left and the results for the present versus PF comparison presented on the right. As concerns the independent variable of temporal distance, < 24 hours was the reference point. For each comparison with this base, the odds of choosing the comparison category of the dependent variable can be higher, lower, equal, or not significant. 6 As an example, we see in the first column that for Levels 1 and 3, the odds of choosing the IF over the PF are the same for an action set to occur in the immediate future (vs. one set to occur within 24 hours). However, for Levels 2 and 4 and for the NSs, the odds of choosing the IF over the PF are lower when the future event is expected to occur immediately, when compared to events that should occur within 24 hours.

Table 2. Results for the temporal distance variable [START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF] IF vs. PF Present vs. PF Group Immed < 24h < week < month > year Immed < 24h < week < month > year

Level 1 = Base = = > = Base < = = Level 2 < Base = > > = Base < = = Level 3 = Base = > > = Base < < < Level 4 < Base = > > = Base = < < NSs < Base = > > = Base < = <
These results led us to two conclusions. First, when examined within the context of the full model, these findings suggested that patterns of variation similar to those seen for NSs developed more slowly for these learners with respect to the present-for-future than for the IF, compared to the PF. More specifically, NNSs at the higher proficiency levels showed close approximation to NS patterns of variation in the IF versus PF comparison, whereas patterns identified when selection of the present was compared with that of the PF showed more variability (for all three independent variables examined). The second conclusion concerned the factor of temporal distance more specifically. In examining the IF versus PF comparison, we noted that the NSs dispreferred the IF in proximal contexts (i.e., immediate contexts), whereas this form was preferred in more distal ones (< month and > year). This pattern is also category of that variable is not shown to be significantly different from another, the odds are found to be "equal."

In contrast, when an independent variable is not included in the regression model, this means that the factor does not condition the dependent variable and the variable is found to be "not significant." instantiated in the learner data, where we see non-linear development across levels. For example, the learners at the lowest level of proficiency (Level 1) prefer the IF to the PF in the most distal contexts (> year), with no differences seen for the other categories. Level 3 participants prefer the IF to the PF in the two most distal contexts (< month, > year) but show no significant preference between the two forms identified for the other categories. Finally, Levels 2 and 4 show the NS pattern, in which the IF is disfavored in the immediate contexts and favored in the two most distal contexts. These findings may mean that, with regard to the IF-PF comparison, learners first show sensitivity in the expression of the most distal events before developing sensitivity to IF versus PF use in proximal ones.

Reanalysis

For our reanalysis, we examined the same dataset using the most common operationalization of temporal distance: proximal (< 24 hours) versus distal (> 24 hours). This means that each participant responded to 12 proximal contexts (6 immediate, 6 < 24 hours) and 18 distal ones (ranging from < week to > year). We then ran a second multinomial regression. The results for temporal distance analyzed as a binary variable are presented in Table 3 (the results for the two other variables analyzed in this project did not change from those found in [START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF]. When reanalyzed as a binary variable, we find that all four groups of learners and the NSs preferred the IF (over the PF) in distal contexts, whereas in those same contexts, the present was dispreferred (compared with the PF). In other words, the NSs and all four learner groups show preference for the IF and dispreference for the present in distal contexts. Whereas this general tendency is also visible in the results provided in Table 2, the two analyses differ crucially with respect to the patterns of development. Indeed, in the reanalysis using a binary proximal versus distal opposition (Table 3), all groups of participants showed similar selection patterns. Thus, even the learners with the lowest levels of proficiency match the NS patterns, meaning that no conclusions concerning development can be drawn. The analysis presented in Table 2, on the other hand, shows that a finer division of the temporal distance spectrum reveals developmental patterns, particularly in the IF versus PF comparison: the IF is preferred first in the most distal contexts (see Level 1) before becoming dispreferred in the most proximal ones (see Levels 2 and 4). Both findings are of interest to the field of SLA: (a) learners appear to be able to approximate the macro-distinction between proximal and distal contexts in an elicitation task even at low levels of proficiency (Table 3) and (b) at a finer level of distinction, developmental patterns are evident (Table 2). These findings obviously derive directly from data-coding decisions, and those decisions reflect what a researcher judges to be most important: in this case, a high level of detail versus comparability with previous studies. Whereas our original study gave preference to the use of "as finely grained a measurement as possible" (Mackey & Gass, 2005, p. 230), thus revealing information about development, the reanalysis sought to facilitate comparisons with previous studies by adopting a similar coding scheme. In so doing, the reanalysis allowed us to more easily note that patterns from our dataset are more in line with Blondeau et al.'s (2014) results from oral interviews conducted with Anglo-Montrealers, in which they found that the PF was favored in proximal contexts, than those reported by [START_REF] Nadasdi | Emploi du 'futur' dans le français parlé des élèves d'immersion française[END_REF], who found no significant effect for temporal distance. Thus, the issue at stake is not which one of the two coding decisions is "better", but rather to demonstrate empirically that these two examples lead to different conclusions about L2 acquisition.

Reanalyzing adverbial specification

Most

Presentation of original study

In [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF], we examined how NSs and NNSs of French expressed future-time reference in informal oral conversations. Ten English NSs with high L2 (nearnative) proficiency were recruited. All had been living in France for at least 4 years. Each participant selected a NS of French with whom they were comfortable speaking in French (e.g., spouse, friend, co-worker). Every dyad was instructed that they would be left alone for 45-60 minutes with a microphone and that they could talk about any subject that they wished.

The resulting corpus contains over 8 hours of recordings (more than 77,000 words), with the NSs and NNSs contributing similar numbers of words (see [START_REF] Donaldson | Syntax and discourse in near-native French: Clefts and focus[END_REF], for additional details on this corpus). A concept-oriented approach was adopted for the coding of each transcript [START_REF] Bardovi-Harlig | One functional approach to second language acquisition: The concept-oriented approach[END_REF][START_REF] Von Stutterheim | A concept-oriented approach to second language studies[END_REF]. Such approaches "are interested in the range of linguistic devices that speakers use to express a particular concept" (Bardovi-Harlig, 2007, p. 58). In the case of [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF], the analysis aimed to identify how NSs and NNSs expressed the concept of future time. In order to do so, we began by identifying all future-time contexts, which was defined as any finite predicate set to occur after the moment of speaking. Each transcript was coded minimally by two researchers and any disagreements were resolved with the third researcher. A total of 947 future-time contexts were identified in the ten transcripts.

All future-time contexts were coded for a variety of variables, including the presence of a specific or non-specific LTI. In [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF], the temporal adverbial had to occur within the same clause as the future-time context in order to be coded as present.

Following this coding schema, we found that 24.2 percent of future-time contexts in the NS portion of the corpus were accompanied by a LTI. The percentage for the NNSs was similar (23.2%). We then looked more specifically at the co-occurrence of futurate present forms with temporal adverbials. We found that 40.8 percent of futurate present forms produced by NSs and 32.2 percent of such forms produced by NNSs were used in the presence of a clauseimmediate temporal adverbial. Thus, we found that use of LTIs was higher with futurate present forms than with the full corpus for both groups of speakers. However, the rates of cooccurrence between futurate present forms and temporal adverbials were strikingly lower than those reported by researchers such as [START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF], who found near categorical cooccurrence of LTIs with futurate present forms in a corpus of native Hexagonal French.

Reanalysis

We sought to compare the coding of LTI at the clause level adopted in [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] with a new coding of this variable, which takes discourse context as its scope. In examining the transcripts of the ten conversations, it was clear that future temporal reference was often established by one speaker and maintained by both as they continued to speak about the same topic. One such example is provided in ( 5), with all verbs occurring in future-time contexts indicated in bold. In this example, the NS speaker has introduced a new topic, namely a play that will be performed on the 28 th by middle school students. The LTI -le vingt-huit 'the 28 th ' -appears in the NSs' first turn, but the talk that occurs in the seven turns that follow maintains this temporal reference.

( For the reanalysis of this independent variable, we redefined what we understood by "presence" of a LTI, extending its scope from the clause level to the discourse context. For this reanalysis, a future-time context is considered to be accompanied by a LTI when, in the same discourse context, a temporal adverbial is used that establishes when that future event is expected to occur. Discourse context was defined as a theme that extends more than one clause and provides continuity to a part of the conversation (cf. [START_REF] Givón | Topic continuity in discourse: A quantitative cross language study[END_REF][START_REF] Lambrecht | Information structure and sentence form: Topic, focus, and the mental representations of discourse referents[END_REF][START_REF] Reinhart | Pragmatics and linguistics: An analysis of sentence topics[END_REF]. We began by identifying discourse contexts across the ten transcripts.

We then reexamined each future-time context identified in our initial analysis, searching for LTIs within the discourse context. For this coding, we followed previous research and distinguished between specific and non-specific adverbials. Logically, all LTIs identified using our original clause-immediate coding were also included in this new operationalization of the variable. If we return to the example provided in (5), all eight future-time contexts were coded as occurring in the presence of a LTI for the reanalysis, as le vingt-huit is when each is expected to occur. Table 4 presents the results from the clause-level coding of LTI [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] and the discourse level coding (reanalysis). A comparison of the results from the two approaches to operationalizing the presence of LTI brings to light two observations. First, as concerns questions of SLA, we note that the NNSs who participated in these conversations show similar results to those found for the NSs, regardless of how adverbial specification is operationalized. In other words, in terms of both clause-level and discourse-level patterns, this group of NNSs closely approximates the distributional patterns seen in the NS corpus. This is already remarkable, as one of the challenges inherent in the acquisition of variable structures in a L2 is the acquisition of the rates of use of each variant [START_REF] Gudmestad | Variationist approaches to second language Spanish[END_REF]. For the second observation, we consider our results within the larger literature on future-time reference in French. We note that although the new definition of the LTI variable increases the number of present-for-future tokens that occur in the company of such adverbials, the rates reported remain far from an "almost categorical cooccurrence with future adverbials", which is how Roberts (2012, p. 97) described the use of the futurate present in his corpus of Hexagonal French. Thus, whether defined as occurring within the clause or within the discourse context, the results from our analysis are clearly different from those reported by [START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF] for Hexagonal French.

We presume that this difference either reflects actual differences between the two datasets (in data type or in participant characteristics) or is an artifact of diversity in data coding of the dependent variable in the two studies. As concerns the first possibility, it is plausible that the interview data analyzed by Roberts (and in most other variationist analysis mentioned in this chapter) may show distinct ways of establishing and maintaining temporal reference, likely dependent on the interviewer, when compared to informal conversation between peers who know each other well. This hypothesis suggests that genre or task type may be a factor that should be taken into consideration in future research [START_REF] Geeslin | Beyond 'naturalistic': On the role of task characteristics and the importance of multiple elicitation methods[END_REF][START_REF] Schilling | Investigating stylistic variation[END_REF]. It may, however, also be the case that the differences in rate of adverbial specification derive at least in part from the coding of the dependent variable (i.e., what counts as a future-time context).

Indeed, little detail is provided in [START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF] that would allow us to determine how present-for-future forms were identified. In other words, it is impossible for us to know if the two studies went about identifying future-time contexts in the same way (see Polio & Gass, 1997, p. 502, for a similar point concerning replication studies). If the identification criteria for the dependent variable adopted in [START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF] and in [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] differ, this may help explain the divergence in our conclusions.

Reflections on data coding

Decisions made when researchers code their data form a prism through which interlanguage is analyzed and interpreted. One reflection of the importance of these decisions is the space dedicated to data coding in how-to guides to conducting SLA research [START_REF] Dörnyei | Research methods in applied linguistics[END_REF][START_REF] Mackey | Second language research: Methodology and design[END_REF]. However, publications adopting a more critical posture concerning data-coding practices are less common (cf. [START_REF] Mackey | Why (or why not), when, and how to replicate research[END_REF][START_REF] Myles | Formulaic sequence (FS) cannot be an umbrella term in SLA[END_REF][START_REF] Révész | Coding second language data validly and reliably[END_REF], despite the fact that all researchers are confronted with these difficult questions. In the current chapter, we have attempted to provide two compelling examples from the literature on future-time expression in L2 French demonstrating the room for improvement that exists with respect to consistency and transparency in data coding. In the first example, we showed that the coding of temporal distance has been subject to many different approaches and that this variety in coding practices has two important results. First, it is difficult to build upon previous research, as the categories of this variable are not always the same. Second, we showed, using data published in [START_REF] Edmonds | What the present can tell us about the future: A variationist analysis of future-time expression in native and nonnative French[END_REF], that the way in which temporal distance is coded leads to different -and in this case complementary -conclusions about the development of future-time expression in L2 French.

We argued that conclusions drawn on the basis of the two approaches to coding temporal distance were valid and of interest. In the second example, we returned to conversation data that we had analyzed for [START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF]. The coding of spontaneous production data comes with many challenges, and in the current chapter, we revisited decisions made concerning the LTI variable. A discourse-level definition of LTI allowed us to better account for how future-time reference was established and maintained across discourse in our dataset.

Moreover, results from both the clause-level and the discourse-level analyses of LTI showed that NNSs used temporal adverbials at rates similar to those found for the NSs. However, when we compared the percentage of futurate present forms accompanied by a LTI for NSs and NNSs with what has been reported elsewhere for Hexagonal French [START_REF] Roberts | Future temporal reference in Hexagonal French[END_REF], we saw that even with the new discourse-level operationalization of the LTI variable, our results are very different. At this point, it is impossible to know whether the differences reported reflect actual differences in the two datasets and, thus, warrant additional research, or whether these differences are artifacts of coding decisions that were made (with respect to the dependent variable, in this case). This observation is crucial to our argument. Unless coding practices are clear and decisions justified, readers and researchers find themselves in the position of having to take the authors at their word. Without sufficient detail, which may entail including fully coded contextualized examples, sample items, and/or full tasks, we cannot verify coding decisions, nor can we be sure to be able to replicate what has been done in previous research (as was the case with respect to the LTI variable). And although departures from previous coding decisions may be justified, it is too often the case that such decisions are left unexplained (as was the case for temporal distance, including in some of our own previous research). To this end, we hope that this chapter has succeeded in illustrating the importance of data-coding decisions and practices within SLA.

  pense qu'il va le faire (NS5, t249) 1 'I think that he is going to do it' b. mais tu feras ça samedi (NS5, t225) 'but you will do it Saturday' c. qu'est-ce que tu fais demain après-midi (NS5, t269) 'what are you doing tomorrow afternoon'

  considered that a temporal adverbial was present if used in the same turn as the futuretime context or in the immediately preceding question by the interviewer. In the remaining studies, data-coding practices have not been made explicit, although the examples cited show temporal adverbials occurring in the same clause as the future-time context. It is possible that coding in these studies actually went gone beyond the clause level, and that space constraints led authors to choose short examples. However, given that descriptions of coding did not clarify this possibility, we deduced that most previous studies had likely adopted a clauselevel coding for LTI, as suggested by the examples provided. This is thus how we ultimately operationalized this factor for[START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] andGudmestad et al. (in press).

  3) NS1: et le vingt-huit il y a une avant-première y a une pièce de théâtre du collège Jeanne d'Albret 'and the 28 th there is premiere there is a play by the Jeanne d'Albret middle school' NNS1: ah ouais j'y vais et Chloé elle joue? 'ah yeah I'm going and is Chloé performing?' NS1: ouais Chloé y joue ouais 'yeah Chloé's performing yeah'Here, the NS establishes future-time reference with the LTI le vingt-huit (the 28 th ), and this temporal frame in maintained over the next several clauses and turns, by both the NS and the NNS. In light of such examples, we question whether a clause-level definition of LTI, which appears to correspond to current practice in this literature, actually reflects how temporal adverbials are used in conversation data in order to establish and maintain temporal reference.These reflections have led us to explore the possibility of offering a new operationalization of this variable (see[START_REF] Grimm | Le présent à valeur de futur en français parlé[END_REF], for a discussion of a similar issue in interview data andGudmestad et al., in press, for a first attempt at addressing this issue with conversation data).

  variationist analyses into future-time expression in French have coded for the presence or absence of a LTI in future-time contexts. For this particular variable, coding practices tend to be underspecified, and although most examples given show the temporal adverbial in the same clause as the future-time context, it is not clear if all examples of adverbial specification are clause immediate. As discussed previously, as well as by Grimm (2016), there is good reason to expand this definition to include LTIs beyond the immediate clause: Temporal reference established by a LTI can be maintained across long stretches of discourse. Thus, in our second reanalysis, we will apply two different operationalizations of what is meant by the presence versus the absence of a LTI in future-time contexts to a dataset of casual conversations. The first coding schema (a clause-level definition) was adopted in Edmonds et al. (2017), whereas the second (a discourse-level definition) is presented here for the first time.

  ) NS1: et le vingt-huit il y a une avant-première y a une pièce de théâtre du collège Jeanne d'Albret NNS1: ah ouais j'y vais et Chloé elle joue? NS1: ouais Chloé y joue ouais NNS1: et c'est avant quel film alors? NS1: avant Kelly euh j'sais pas NNS1: ah oui le secret de Kelly ou je sais pas NS1: oui je sais pas ce que c'est NNS1: oui une petite fi oui oui d'accord et Chloé va jouer? [oui] tu vas y aller alors NS1: and the 28 th there is premiere there is a play by the Jeanne d'Albret middle school NNS1: ah yeah I'm going and is Chloé performing? NS1: yeah Chloé's performing yeah NNS1: and so it's before which movie? NS1: before Opal uh I dunno NNS1: ah yes Opal Dream or I don't know NS1: yes I don't know what it is NNS1: yes a little gir yes yes ok and Chloé is going to perform? [yes] so you're going to go In applying our original coding schema for LTI, only the very first future-time context (il y a une avant-première) was coded as being accompanied by the temporal adverbial le vingt-huit.

Table 1 .

 1 Results from variationist studies having examined the PF, IF, and present in oral production

	Study	Population	PF	IF	Present
	Poplack & Turpin (1999)	NSs (Ottawa)	Favored when LTI is	Favored when a non-	Favored when a specific
			absent	specific LTI is present	or non-specific adverbial
					is present
	Grimm (2016)	NSs (Ontario)	Favored when LTI is	Non-significant	Favored when a specific
			absent		adverbial is present
	Gudmestad et al. (in press) 5 NSs (France)	Presence of specific LTI: IF < Present indicative (non-significant for non-specific
			LTI)
			Presence of specific or non-specific LTI: PF < Present indicative
	Nadasdi et al. (2003)	NNSs (Ontario)	Favored when LTI is	Non-significant	Favored when a specific
			absent		or non-specific adverbial
					is present
	Blondeau et al. (2014)	NNSs (Québec)	Favored when LTI is	Favored when a non-	Favored when a non-
			absent	specific LTI is present	specific LTI is present
	5

,

[START_REF] Comeau | A window on the past, a move toward the future: sociolinguistic and formal perspectives on variation in Acadian French[END_REF]

, and

[START_REF] Blondeau | La référence temporelle au futur dans les bulletins météo en France et au Québec : regard variationniste sur l'oral préparé[END_REF]

constitute exceptions, and in each of these studies, the IF was found to be favored in contexts containing an LTI.

At least five studies have used regression models to examine the futurate present, in addition to the PF and IF, in the expression of future time by NSs or non-native speakers (NNSs) of French: Poplack and Turpin (1999) using the Ottawa-Hull corpus,

Grimm (2016) 4

for the community of Hawkesbury (Ontario),

Gudmestad et al. (in press)

for Hexagonal French,

[START_REF] Nadasdi | Emploi du 'futur' dans le français parlé des élèves d'immersion française[END_REF]

for French spoken by Canadian school children in an immersion program, and

[START_REF] Blondeau | Future temporal reference in the bilingual repertoire of Anglo-Montrealers: A twin variable[END_REF]

for the French of Anglo-Montrealers. The results from these five studies are summarized in Table

1

, which illustrates that the present has consistently been favored in the presence of a temporal adverbial, for both NSs and NNSs.

Table 3 .

 3 Results for the temporal distance variable reanalyzed

		IF vs. PF		present vs. PF
	Group	Proximal	Distal	Proximal	Distal
	Level 1	Base	>	Base	<
	Level 2	Base	>	Base	<
	Level 3	Base	>	Base	<
	Level 4	Base	>	Base	<
	NSs	Base	>	Base	<

Table 4 .

 4 Distribution of LTI in[START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] and in the reanalysis.The change in scope for the variable of LTI unsurprisingly led to the identification of more instances of the presence of LTI. Whereas we reported that 24.2 percent of future-time contexts in the NS corpus and 23.2 percent in the NNS corpus were accompanied, within the same clause, by a LTI, these percentages increase to 36.7 for NSs and 39.8 for the NNSs in the reanalysis. For NSs, 56 predicates were reanalyzed as occurring within the discourse frame established by the time adverbial, and 83 in the case of NNSs. When we narrow in on uses of the futurate present only, we also see a clear increase in the co-occurrence of this form with a LTI. Whereas[START_REF] Edmonds | A concept-oriented analysis of futuretime reference in native and near-native Hexagonal French[END_REF] reported that futurate present forms produced by NSs were accompanied 40.8 percent of the time by a clause-immediate LTI and that the figure for NNSs was 32.2 percent, a discourse-level definition of LTI brings these percentages to over 50 percent: 52.5 percent for NSs (74 out of 141 instances of futurate present forms) and 52.7 percent for NNSs (77 of 146 total instances of present-for-future).

		Edmonds et al. (2017): Clause level		Reanalysis: Discourse level
		Specific	Nonspecific	Absent	Specific	Nonspecific	Absent
	Group	n	%	n	%	n	%	n	%	n	%	n	%
	NSs	81	18.1	27	6	339 75.8 132 29.5	32	7.2	283 63.3

In his dissertation, Grimm looked at a corpus from 1978 and one from 2005, as well as data from several different communities in Ontario. Most analyses showed sentential polarity (and not temporal distance) to be operative. However, certain types of speakers (i.e., restricted speakers in the 1978 corpus) and certain communities (i.e., Cornwall, North Bay, and Pembroke) were found to be sensitive to both sentential negation and temporal distance.

Grimm simply states that "it was not necessary to scrutinize the data to this level of detail" (2015, p. 236).

[START_REF] Grimm | Grammatical variation and change in spoken Ontario French: The subjunctive mood and the expression of future temporal reference[END_REF] also analyzed the present-for-future in several different datasets. Because of space restrictions, we choose here to present only the results reported in his 2016 article.

When an independent variable is included in the regression model, this usually means that there is at least one significant effect for which the odds of choosing the comparison category of the dependent variable are lower or higher than the base case. In some cases, though, some categories of the variables are not shown to be significantly different from each other. Thus, when the independent variable is included in the model but a given


Yijing Chen
email: yijing.chen@lss.supelec.fr

Gilney Damm
email: gilney.damm@ibisc.fr

Abdelkrim Benchaib
email: ab-delkrim.benchaib@alstom.com

Franc ¸oise Lamnabhi-Lagarrigue
email: francoise.lamnabhi-lagarrigue@lss.supelec.fr

F Lamnabhi-Lagarrigue

Multi-Time-Scale Stability Analysis and Design Conditions of a VSC Terminal with DC Voltage Droop Control for HVDC Networks

This paper addresses the problem of analyzing the dynamic limitations of a VSC terminal with DC voltage droop control. Thanks to its fast response, DC voltage droop control is usually used to adjust the DC voltage in case of power imbalance. To ensure the stability of the VSC terminal, the dynamics between the dq currents (i l,dq ) and the DC voltage (uc) must be well distinguished. Therefore, the choice of the droop control gain Ku and the dq current control gains K dq is still a challenging problem. First, we assume that there exist Ku and K dq such that a slow and a fast dynamics can be imposed on uc and i l,dq , respectively. Then, we propose a novel methodology based on separation of dynamics to prove the existence of these control gains. Subsequently, a detailed stability analysis is carried out via Lyapunov theory, from which a necessary condition and a sufficient condition are derived for the control gains. To verify this methodology, numerical simulations are carried out in MATLAB/SIMULINK.

I. INTRODUCTION

Voltage-source converter high voltage direct current (VSC-HVDC) systems have been greatly promoted in today's electrical networks due to their flexibility, controllability and efficiency. They are specially suitable for long-distance power transmission, interconnection between asynchronous AC networks, underground/undersea cable transmission systems, etc. In order to enlarge power transmission capacity and realize power exchange between multi-suppliers and multi-consumers, the use of multi-terminal HVDC (MT-HVDC) systems has been increasingly widely used [START_REF] Long | Consideration for implementing multiterminal DC systems[END_REF], [START_REF] Chen | A detailed study on a DC-voltage-based control scheme using a multi-terminal HVDC system for frequency control[END_REF]. However, the control design for MT-HVDC systems is still a major problem. The control scheme should be strong enough to maintain the system in normal operation in case of small-signal perturbations [START_REF] Kalcon | Smallsignal stability analysis of multi-terminal VSC-based DC transmission systems[END_REF] or loss of some terminals [START_REF] Liang | Control of multi-terminal VSC-HVDC transmission for offshore wind power[END_REF]. In general, a master controller and local controllers compose the control system where the master one provides information such as the reference values for DC voltage, the active/reactive power to coordinate the operations of the local ones. In an MT-HVDC system, at least one terminal is in charge of regulating the DC-bus voltage level, and hence various control configurations are proposed such as masterslave control scheme, voltage margin method [START_REF] Nakajima | A control system for HVDC transmission by voltage sourced converters[END_REF].

In this paper, we consider a very popular control technique called DC voltage droop control. It is characterized by the change of power flow in proportion to the change of the DC voltage. The main advantage of DC voltage droop control is that it involves more than one terminal in regulating the DCbus voltage based on their predetermined DC voltage droop characteristics such that more terminals share the duty.

As a matter of fact, the use of DC voltage droop control has been widely studied in the literature [START_REF] Haileselassie | Impact of DC line voltage drops on power flow of mtdc using droop control[END_REF], [START_REF] Prieto-Araujo | Methodology for droop control dynamic analysis of multiterminal VSC-HVDC grids for offshore wind farms[END_REF]. Actually, these control structures belong to the family of vector current control since two loops are usually involved, i.e. an inner current (i) loop and an outer loop. DC voltage droop control considered as an outer loop is used to regulate the DC voltage (u c ) and provide a current reference i * to the current loop. Then, different current loop controllers [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF], [START_REF] Chen | Nonlinear control design for a multi-terminal VSC-HVDC system[END_REF], [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF] are proposed to make i converge to i * . It is very important to notice that vector current control is derived under the assumption that the current loop has much faster dynamics than the outer one. Thus, the dynamics of i * is neglected when designing the current control scheme. Unfortunately, there is no theoretical demonstration of this assumption. Moreover, it is still unknown which dynamic limitations should be imposed on the current and the DC voltage. This paper is devoted to the stability and dynamic limitation analysis of a VSC station equipped with DC voltage droop control. A sufficient condition for selecting the droop and the current control gains is derived based on singular perturbation theory and Lyapunov theory.

The paper is organized as follows. An averaged statespace model is introduced in Section II. Vector current control scheme with DC voltage droop control is presented in Section III. Section IV gives a detailed stability analysis from which the dynamic limitations on the current and the DC voltage are clarified. We also derive a necessary and a sufficient conditions for choosing the control gains and present the steady-state analysis of the system. The theoretical analysis in this paper is verified by numerical simulations in Section V. Conclusions are drawn in Section VI.

II. MODELING

Figure 1 depicts a simplified configuration of a VSC terminal connected to an AC system where R l + jL l represents the phase reactor and C is the DC capacitor used to reduce the ripple of the DC voltage.

An averaged state-space model established in a synchronous dq reference frame for the VSC terminal is given by [START_REF] Blasko | A new mathematical model and control of a three-phase AC-DC voltage source converter[END_REF]

di ld dt = - R l L l i ld + ωi lq + u c 2L l M d - v ld L l ( 1
)
di lq dt = - R l L l i lq -ωi ld + u c 2L l M q - v lq L l ( 2
)
du c dt = - 1 C [i c + 3 2
v ld i ld + v lq i lq u c ] (3)
where v l,dq , v c,dq and i l,dq are the AC system voltages, the converter voltages and the currents through the phase reactor, respectively, M dq are the modulation indices considered as the control inputs and ω is the frequency of AC system. By convention, i c is positive if the power flows from left to right as shown in Fig. 1.

Remark 1: Due to the physical limitations of the converter itself, the modulation indices should always satisfy M 2 d + M 2 q ≤ 1 and the domain of operation for u c is restricted to a narrow area, i.e. u c ∈ [u min , u max ].

Remark 2: In the present paper, we want to investigate the local behavior of the converter and hence, for convenience, we consider that the discussed terminal is connected to a controlled current source represented by i c . It can be either negative (as a supplier) or positive (as a consumer).

Remark 3: For the sake of simplicity, the dq reference frame is chosen such that v lq = 0 and v ld = V l,rms . As a consequence, the instantaneous active and reactive powers on the AC side of the converter are given by

P l = 3 2 v ld i ld Q l = - 3 2 v ld i lq (4)
III. CONTROL STRUCTURE An inner current loop and an outer loop compose a vector current control system. Various outer loops exist for different control objectives [START_REF] Xu | Multi-terminal DC transmission systems for connecting large offshore wind farms[END_REF], [START_REF] Haileselassie | Multi-terminal VSC-HVDC system for integration of offshore wind farms and green electrification of platforms in the North Sea[END_REF]. In this paper, the control goals are to regulate u c in its normal operation region and at the same time, to make Q l follow its reference Q * l . Thus, the considered VSC station operates in DC voltage-reactive power mode. By solving (4), the q current reference is obtained as

i * lq = - 2Q * l 3v ld (5)
As to the d current reference, it is directly given by DC voltage droop control

i * ld = K u (u c -u * c ) (6)
where K u is the droop gain and u * c is the DC voltage reference.

Remark 4: The reference values Q * l and u * c are considered constant in this paper and actually, they are provided by a higher level control called secondary control.

The objective of the current loop is to make i l,dq quickly converge to their varying references i * l,dq . Different current controllers have been proposed [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF], [START_REF] Chen | Nonlinear control design for a multi-terminal VSC-HVDC system[END_REF], [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF]. In this paper, the conventional current control in [START_REF] Thomas | Analysis of a robust DCbus voltage control system for a VSC transmission scheme[END_REF] is selected. The control signals applied to the converter contain the traditional PI controllers and some compensation terms as

M d = 2L l u c (v d + R l L l i ld -ωi lq + v ld L l ) (7)
M q = 2L l u c (v q + R l L l i lq + ωi ld + v lq L l ) (8)
where v dq are

v dq = -K p,dq (i l,dq -i * l,dq ) -K i,dq (i l,dq -i * l,dq ) (9)
Finally, the vector current control scheme is built based on ( 5)-( 9). Remark 5: Note that the dynamics of i * ld is neglected in the current control design since we assume that the currents have much faster dynamics that the DC voltage. Substituting ( 7)-( 9) into the current subsystem leads to

di l,dq dt = v dq (10)
where v dq act as additional control inputs to regulate the behaviors of i l,dq . It is obvious that the autonomous system (10) is locally exponentially stabilized at i * l,dq if i * l,dq are constant or varying quite slowly. However, as seen in ( 6), i * ld is given by DC voltage droop control, which means that its dynamics is regulated by K u . If K u is quite large, i * ld would have fast dynamics. In this case, the system (10) may become unstable. Therefore, it is necessary to consider the dynamic effect of i * ld on the system stability. In the following section, we will investigate what dynamics i l,d and i * ld should have so that the VSC terminal can operate properly.

IV. STABILITY ANALYSIS

As seen in ( 6)- [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF], the dynamics of i * ld and i l,dq mainly depend on K u and K dq . In this section, we investigate the feasible region for these control gains. Before going further, let us study the steady-state condition first.

A. Steady-state condition

The steady-state values of the system (1)-( 3) can be obtained by solving the following algebraic equation

0 = - 1 C [i c + 3 2 v ld K u (u c -u * c ) + v lq i * lq u c ] (11)
which leads to

ūc = 3v ld K u u * c 3v ld K u + 2i c = u * c - 2u * c i c 3v ld K u + 2i c (12) īld = - 2u * c i c K u 3v ld K u + 2i c (13)
The equilibrium value of i lq , denoted by īlq , is equal to i * lq . Taking the practical feasibility into account, i.e. ūc ∈ [u min , u max ], K u must satisfy the following condition Condition 1:

K u > Max(- 2i c 3v ld , 2i c u max 3v ld (u * c -u max ) , 2i c u min 3v ld (u * c -u min ) ) (14
) Remark 6: Notice that Condition 1 is a necessary condition for K u . It is obvious that the equilibrium point heavily depends on the choice of the droop gain K u and the DC voltage setpoint u * c . As mentioned in the previous section, u * c is provided by the secondary control. In the present paper, u * c is considered as a known constant parameter, which is not used to regulate ūc . Therefore, if a smaller deviation between ūc and u * c is desired, a larger value for K u should be chosen.

B. Analysis of dynamic limitations and system stability

In this section, in order to state the main results, two steps are carried out. We first prove that there exist control gains such that the original system (1)-( 3) can present a multitime-scale behavior. Subsequently, a sufficient condition for choosing K u and K dq is given to guarantee that the system can be locally stabilized around the equilibrium point in Section IV-A.

Theorem 1: Consider the nonlinear system (1)-(3). For any K u satisfying Condition 1, there exists K min such that for any K dq > K min , the proposed control strategy ( 5)-(9) stabilizes the system around the equilibrium point ( īld , īlq , ūc ). Moreover, a multi-time-scale behavior can be found, i.e. a slow and a fast transients.

Proof: To demonstrate this claim, let us substitute the developed control laws (5)-( 9) into the actual system (1)-(3). We then get the following closed-loop system

di ld dt = -K pd [i ld -K u (u c -u * c )] + γ d (•) (15)
di lq dt = -K pq (i lq -i * lq ) + γ q (•) (16)
du c dt = - 1 C [i c + 3 2 v ld i ld + v lq i lq u c ] (17)
where γ dq (•) represent the integral parts and for convenience, we set

K pd = K pq = K d . Let us introduce a new variable • K d = 1 (18)
Then, the system ( 15)-( 17) becomes

di ld dt = -[i ld -K u (u c -u * c )] + γ d (•) (19) di lq dt = -(i lq -i * lq ) + γ q (•) (20) du c dt = - 1 C [i c + 3 2
v ld i ld + v lq i lq u c ] (21)
When is small enough, the derivatives of i l,dq are quite large. By setting = 0 in ( 19) and (20), i l,dq quickly enter their manifolds i * l,dq and then a reduced model can be deduced from the DC-voltage subsystem as

du r c dt = - 1 C [i c + 3 2 v ld K u (u r c -u * c ) + v lq i * lq u r c ] (22)
Remark 7: The reduced model ( 22) is locally exponentially stabilized at its equilibrium point u r c = ūc . Since the reduced model is deduced when i l,dq converge to i * l,dq , it is also called a quasi-steady-state model. i * l,dq are the quasi-steady states of i l,dq . It is expected that the behavior of the original DC voltage system can be approximated by its reduced model due to its simpler form. Therefore, this requires that i l,dq converge to i * l,dq during the initial interval. In order to analyze the problem more conveniently, let us introduce some new variables:

ĩl,dq = i l,dq -i * l,dq (23) ũc = u c -u * c (24)
that shift the quasi-steady states of i l,dq to the origin. In these new variables, the full system problem becomes:

d ĩld dt = -ĩld + K u [ 3K u 2C u * c ūc v ld ūc + ũc ũc + 3v ld 2C ĩld ūc + ũc ] + γ d f d ( 25
)
d ĩlq dt = -ĩlq + γ q f q (26) dũ c dt = - 3K u 2C u * c ūc v ld ūc + ũc ũc - 3v ld 2C ĩld ūc + ũc f u (27) Now we set d ĩl,dq dt = d ĩl,dq dτ (28)
where

dτ dt = 1 (29)
τ is the introduced new time variable which can be approximated by τ = t . In the τ time scale, the current subsystem is expressed as

d ĩld dτ = -ĩld + γ d + K u C [ 3v ld 2u c K u (u c -u * c ) + i c + 3v ld 2u c ĩld ] (30) d ĩlq dτ = -ĩlq + γ q (31)
The variables t and ũc are considered to be varying slowly in the τ time scale. Setting = 0 restricts t and u c to their initial values, and then the current subsystem is reduced to the following autonomous system

d ĩl,dq dτ = -ĩl,dq (32)
which is called the boundary-layer model. It is expected that the solutions of (32) can be close to the origin during the boundary-layer interval (t ∈ [0, t b ]), and that they can still remain around the origin for the future time (t > t b ) when t and u c slowly vary. This can be achieved since we have Remark 8: The boundary-layer model (32) is exponentially stable at the origin, uniformly in (t, ũc ).

By combining Remarks 7-8 and referring to Theorem 11.4

in [START_REF] Khalil | Nonlinear Systems[END_REF], there exists min such that for any

1 K d = <
min , the translated system (25)-( 27) is locally asymptotically stable around the origin. It means that the equilibrium of the system (19)-(21), i.e. ( īld , īlq , ūc ), is also asymptotically stable. Thus, there exists K min = 1 min . This concludes the proof. Remark 9: From the above proof, i ld exhibits a two-timescale behavior. It starts with a fast transient of i ld moving from the initial value towards i * ld . Then, i ld remains close to i * ld . Finally, i ld and i * ld converge to their equilibrium īld together. In the simulation section, this two-time-scale behavior will be clearly illustrated.

In the following section, we carry out a further study on the relationship between K u and K d . Investigation of feasible regions of K u and K d is made by applying Lyapunov theory. A Lyapunov function is selected as

V = d 2 V 1 + (1 -d) 2 V 2 (33)
with d ∈ (0, 1) where V 1 and V 2 are derived from the boundary-layer model and the reduced model, respectively.

V 1 = ĩ2 ld + ĩ2 lq + K id φ 2 d + K iq φ 2 q (34) V 2 = ũ2 c (35)
where φ dq = ĩl,dq . Then, the derivative of V can be deduced from ( 25)-( 27)

V = -a 1 ĩ2 ld -d 1 ĩ2 lq -a 2 ũ2 c + a 3 ũc ĩld (36)
with

a 1 = d( 1 - 3K u 2C v ld ūc + ũc ) (37)
a 2 = (1 -d) 3K u 2C u * c ūc v ld ūc + ũc (38)
a 3 = d 3K 2 u 2C u * c ūc v ld ūc + ũc -(1 -d) 3 2C v ld ūc + ũc (39) Rewrite V as V = -ĩld ũc P ĩld ũc -d 1 ĩ2 lq ( 40
)
where

P =   a 1 - a 3 2 - a 3 2 a 2   (41)
Lemma 1: If is selected such that the matrix P is positive definite, the translated system (25)-( 27) asymptotically converges to the origin.

Remark 10: The above lemma is obvious since a positive definite matrix P would result in a negative V except at the origin. By applying Barbalat's lemma or LaSalle theorem, it can be proved that the system is asymptotically stable.

In fact, the sufficient condition to be given later is mainly based on Lemma 1. It is known that a symmetric n × n real matrix P is positive definite if and only if the determinants of all leading principal minors of P are positive. Thus, in order to get a positive definite P , we need

a 1 > 0 (42) a 1 a 2 -( a 3 2 ) 2 > 0 (43)
These two inequalities indicate that if K u and K d satisfy the following condition Condition 2:

K d = 1 > 3K u 2C v ld ūc + ũc (44) K d = 1 > a 2 3 4da 2 + 3K u 2C v ld ūc + ũc (45)
the system ( 19)-( 21) is asymptotically stable at the origin. Remark 11: It is clear that heavily depends on ũc . If the system is expected to be stable in a larger region, a smaller value should be assigned to , i.e. a larger value for K d . It is also found that a small (a large K d ) makes the system have fast responses. However, due to some physical consideration, we cannot improve the responses of the system unlimitedly.

Remark 12: It is worth noticing that the Lyapunov function V is deduced from the reduced model and the boundarylayer model. When K u is fixed, a large value of K d will make V quickly converge to the origin. It also implies that i l,dq enter their manifolds i * l,dq rapidly.

V. SIMULATION In this section, numerical simulation results are presented to verify the theoretical analysis and Conditions 1-2 derived from the previous section. The parameters of the VSC terminal and the base quantities used in the per-unit system are listed in Table I and Table II, respectively.

For all the simulations, the VSC terminal is supposed to be connected to a controlled current source with i c = -0.7 p.u.. The setpoints u * c and Q * l provided by the secondary control are set to 1 p.u. and 0 p.u., respectively. The converter starts from the initial point (i ld0 = 0.5 p.u., i lq0 = 0.1 p.u., u c0 = 0.95 p.u.). Taking Condition 1 into consideration, K u must be greater than 0.2075 in order to ensure that the converter acts in its domain of operation, i.e. u c ∈ [0.9, 1.1] p.u.. In the following section, K u = 0.3 is applied. As seen in the previous analysis, K d can be determined once K u is fixed. Hence, according to Condition 2, a feasible region for K d is given, i.e. K d > 985. [START_REF] Prieto-Araujo | Methodology for droop control dynamic analysis of multiterminal VSC-HVDC grids for offshore wind farms[END_REF]. In order to demonstrate the main results derived from the theoretical analysis, several simulation scenarios are carried out.

Case 1: We choose K d = 23.5 and K i = 221.6. Obviously, K d is beyond the given region. is then equal to 0.043. Figures 2(a) and 2(b) illustrate the behaviors of u c and i l,dq , respectively. We find that these state variables diverge with time. It is pointed out that in this case, the current controller cannot follow the reference trajectory given by DC voltage droop control. As seen in ( 25)-(26), the dynamics of i * ld can be considered as a perturbation to its boundary-layer model (32). The divergence of the system variables indicates that the current controller is not strong enough to tolerate this perturbation. It also reveals that not all current controllers can separate the system dynamics.

Case 2: The control gains are selected as K d = 36.2 and K i = 340.9. In this case, K d is sill less than 985.7, and is equal to 0.028. Simulation results are displayed in Figs. 3(a) and3(b). The trajectories of all the system variables converge to their equilibrium values. It implies that the asymptotic stability of the system can be achieved despite a K d that does not satisfy Condition 2. This is true because our condition is derived from a specific Lyapunov function and that it is not a necessary condition but a sufficient condition. Therefore, it is possible that there exists a K d not satisfying Condition 2 but still ensuring the stability of the system. However, the transient performances of the system variables are not good. Looking at the response of u c in Fig. 3(a), the VSC terminal operates beyond its region of normal operation ([0.9, 1.1] p.u.) during the interval t ∈(0, 1.5) s, which may damage electronic devices. On the other hand, the response time of the system is shown to be unsatisfactory.

Case 3: The current control gains are chosen as K d = 986 and K i = 9.28 × 10 3 satisfying Condition 2. In this case, we have = 0.001.

The responses of i l,d and u c are shown in Fig. 4. Let us first focus on the response of i ld in Fig. 4(a). The trajectory of i ld displays a two-time-scale behavior. In contrast to i ld starting from i ld0 = 0.5 p.u., its quasi-steady state i * ld does not start from the same initial value but from i * ld0 = K u (u c0 -u * c ). Hence, a large discrepancy between i ld0 and i * ld0 exists at the initial instant, as depicted in Fig. 4(a). It is found that the trajectory of i ld starts with a very fast transient from i ld0 to i * ld . After this transient, i ld remains close to i * ld . For example, the error between i ld and i * ld remains less than 0.114 p.u. after t = 0.005 s. Subsequently, i ld and i * ld converge to the equilibrium value (the red curve) together. As illustrated in Fig. 4 here) shows that i lq converges to its reference value, i.e. 0 p.u., after a short transient. Case 4: As mentioned in Remark 12, K d has a great effect on the convergence speed of the system. In order to evaluate this influence, K d = 1972 ( = 5 × 10 -4 ) and K i = 1.86 × 10 4 are selected by comparison with Case 3. By making the comparison between Case 3 and Case 4, we point out that K d ( ) can heavily influence the behaviors of the system. The simulation results show that we can improve the system transient performances by increasing K d . This confirms our theoretical analysis since the convergence speed of the system can be accelerated by a large K d .

VI. CONCLUSIONS

In this paper, we consider the problem of regulating the DC-bus voltage with DC voltage droop control. First, we present the conventional vector current control and indicate that the control strategy ensures the stability of the system only when the dynamics between the current and the DC voltage are well distinguished. We then analyze the steady-state condition of the system and find that the current control gain K d has no effect on it. In addition, the droop gain K u and the setpoint u * c determine the steadystate values together. Furthermore, a necessary condition for K u is deduced such that the VSC terminal acts in the region of safe operation. Subsequently, singular perturbation theory is applied to analyze the dynamic limitations of the system. Then, we prove that there exist K u and K d such that the system variables exhibit a multi-time-scale behavior. Finally, we derive a sufficient condition for the control gains by means of Lyapunov theory to guarantee the asymptotic stability of the system. We establish the Lyapunov function based on the boundary-layer model and the reduced model.

It is pointed out that a large K d can improve the system transient performances.

The sufficient condition is verified by carrying out four sets of simulations. It is shown that the derived conditions indeed guarantee the asymptotic stability and make the behaviors of the system well regulated. In order to understand the effects of K d on the dynamics of the system, a comparison is made and the simulation results confirms the theoretical analysis.

In the present paper, the studied VSC is connected to a controlled current source. In the future, it is proposed to consider the VSC terminal in the context of an MT-HVDC system and analyze the effects of the control gains on the problem of stability.

Fig. 1 .

 1 Fig. 1. A VSC terminal connected to AC system.

2 V

 2 I b ≈ 19.67 A DC side S dc,b = 10 kVA V dc,b = 730 V I dc,b ≈ 13.70 A

  p.u.) (a) uc response.

  p.u.) i lq (p.u.) (b) i l,dq response.

Fig. 2 .

 2 Fig. 2. Simulation results with = 0.043, K d = 23.5 and K i = 221.6.

  p.u.) (a) uc response.

  p.u.) i lq (p.u.) (b) i l,dq response.

Fig. 3 .

 3 Fig. 3. Simulation results with = 0.028, K d = 36.2 and K i = 340.9.

Fig. 4 .

 4 Fig. 4. Simulation results with = 0.001, K i = 9.28 × 10 3 and K d = 986 corresponding to Condition 2.

  .u.) (a) i ld response.

  uc response.

Fig. 5 .

 5 Fig. 5. Simulation results with a large K d = 1972, K i = 1.86 × 10 4 and a small = 5 × 10 -4 .

Fig. 6 .

 6 Fig. 6. Comparison between Case 3 and Case 4.

TABLE I THE

 I VSC TERMINAL PARAMETERS.

	R l	L l	V line	C	f
	9.9 mΩ	3.2 mH 415 V 680 µF	50 Hz
			TABLE II
	BASE QUANTITIES APPLIED IN THE PER-UNIT SYSTEM.

This work is supported by WINPOWER project (ANR-10-SEGI-016).