Stéphane Ranfaing

Lucas De Zorzi

Jacques Honoré

Hugo Critchley

Henrique Sequeira
email: henrique.sequeira@univ-lille.fr

Attention Orientation to Pleasantness and Depressive Symptomatology Predict Autonomic Reactivity

Keywords: Emotion, Depression, Attention bias, Eye movements, Autonomic responses, skin 34 conductance. 35

Depression is characterised by attentional bias to emotional information and dysregulated autonomic reactivity. Despite its relevance to understanding depressive mechanisms, the association between attentional bias and autonomic reactivity to emotional information remains poorly characterised. This study compared behavioural and autonomic responses to emotional images in 32 participants in whom subclinical depressive symptomatology was quantified using the Beck Depression Inventory. Pairs of emotional and neutral images (unpleasant-neutral, U-N; pleasant-neutral, P-N; neutral-neutral, N-N) were presented while attentional indices (eye movements) and autonomic activity (skin conductance responses, SCRs; heart rate, HR) were recorded. Results showed that all recorded ocular parameters indicated a preferential orientation and maintenance of attention to emotional images. SCRs were associated with a valence effect on fixation latency: lower fixation latency to pleasant stimuli leads to lower SCRs whereas the opposite was observed for unpleasant stimuli. Finally, stepwise linear regression analysis revealed that latency of fixation to pleasant images and scores of depression predicted SCRs of participants. Thus, our research reveals an association between autonomic reactivity and attentional bias to pleasant information, on the one hand, and depressive symptomatology on the other. Present findings therefore suggest that depressive individuals may benefit from attention training towards pleasant information in association with autonomic biofeedback procedures.

Introduction

Healthy individuals mobilise privileged attentional resources toward emotional information [START_REF] Vuilleumier | Affective and motivational control of vision[END_REF] and express associated autonomic and behavioural responses [START_REF] Damasio | A Second Chance for Emotion[END_REF]. However, abnormalities in the amount of attention dedicated to emotional information are implicated in the etiology and maintenance of depressive symptomatology [START_REF] Gotlib | Cognition and depression : Current status and future directions[END_REF]. For mood-congruent information, depressed individuals are characterised by biases at all stages of attentional processing [START_REF] Ingram | Cognitive vulnerability to depression[END_REF]. Metaanalysis reveals that, in depression, attentional bias toward negative information is observed for verbal and non-verbal stimuli, in both clinical and subclinical populations [START_REF] Peckham | A meta-analysis of the magnitude of biased attention in depression[END_REF]. It is also reported that, in depression, less attention is dedicated to pleasant information [START_REF] Duque | Double attention bias for positive and negative emotional faces in clinical depression : Evidence from an eye-tracking study[END_REF]. This may reflect an absence of the 'protective bias' toward positive stimuli that is usually observed in healthy or non-dysphoric individuals [START_REF] Shane | An evaluation of early and late stage attentional processing of positive and negative information in dysphoria[END_REF]. Furthermore, attentional bias to pleasant information is found to correlate negatively with the onset of depressive symptomatology, and is associated with greater trait resilience.

Hence, attentional bias to pleasant information can be considered as an index of adaptive emotion regulation [START_REF] Thoern | Attentional Bias towards Positive Emotion Predicts Stress Resilience[END_REF]. Conversely, reduced attentional bias to pleasant material increases vulnerability to stress-related psychopathology [START_REF] Fox | Preconscious Processing Biases Predict Emotional Reactivity to Stress[END_REF]. In brief, individuals with depressive symptomatology show greater orientation and maintenance of attention towards unpleasant stimuli and reduced orientation towards pleasant stimuli [START_REF] Gotlib | Cognition and depression : Current status and future directions[END_REF].

The dot probe task is one established approach for measuring the time course of attentional processes in depression [START_REF] Macleod | Attentional bias in emotional disorders[END_REF]. Here, the participant views a pair of stimuli, usually one emotional and one neutral image, which are immediately followed by a stimulus ('probe'), which appears at the location of one of two images. The participant is instructed to make a reaction time response to the probe. Emotional attentional bias to it expresses faster responses on trials when the probe is presented in the location of the emotional compared to the neutral image. This task and variations of it has become a gold standard for investigating attentional bias and its time course. However, the use of reaction times does not allow direct measurement of the attention span and has recently been criticised regarding its psychometric properties [START_REF] Chapman | Fleeting reliability in the dot-probe task[END_REF]. As an alternative, measures of eye movements can provide a more direct index of attentional deployment. In depressed individuals, they showed an increased attention to negative stimuli and decreased attention to positive stimulation when compared to nondepressed individuals [START_REF] Armstrong | Eye tracking of attention in the affective disorders : A meta-analytic review and synthesis[END_REF]. Eye movements can reflect both orienting (e.g. initial orientation and latency to first fixation) and maintenance (e.g. number of fixations, or total duration of fixation; Duque & Vazquez, 2015) components of attention, as well as attentional reorientation to stimuli. Indeed, unlike reaction times, eye movements allow the continuous measurement of attentional processes and can thus better characterise attention biases to emotional stimuli. A few studies in the literature have investigated the psychometric properties of eye movement indices and most of them used different stimuli, paradigm or sample characteristics [START_REF] Waechter | Measuring Attentional Bias to Threat : Reliability of Dot Probe and Eye Movement Indices[END_REF][START_REF] Lazarov | Social anxiety is related to increased dwell time on socially threatening faces[END_REF][START_REF] Skinner | The reliability of eyetracking to assess attentional bias to threatening words in healthy individuals[END_REF][START_REF] Sears | The reliability of attentional biases for emotional images measured using a free-viewing eye-tracking paradigm[END_REF]. Although there are still mixed results for the early attention cues [START_REF] Skinner | The reliability of eyetracking to assess attentional bias to threatening words in healthy individuals[END_REF], the results for indices of maintenance of attention such as total fixation time and number of fixations appear encouraging concerning psychometrics properties [START_REF] Sears | The reliability of attentional biases for emotional images measured using a free-viewing eye-tracking paradigm[END_REF]. Consequently, some authors suggest that eye-tracking measures of attentional bias may have better overall psychometric properties as compared than traditional RT measures of attentional bias for children and adults [START_REF] Chong | Psychometric properties of threat-related attentional bias in young children using eyetracking[END_REF].

Depressive symptoms are associated with altered patterns of autonomic activity, which has been related to a disengagement from emotional information [START_REF] Bylsma | A meta-analysis of emotional reactivity in major depressive disorder[END_REF].

Increases in electrodermal activity [START_REF] Branković | System identification of skin conductance response in depression-An attempt to probe the neurochemistry of limbic system[END_REF], skin temperature and respiratory frequency [START_REF] Wenzler | Intensified emotion perception in depression : Differences in physiological arousal and subjective perceptions[END_REF] and decreases in heart rate variability (HRV; [START_REF] Kemp | Impact of depression and antidepressant treatment on heart rate variability : A review and meta-analysis[END_REF] are reported. However, across studies of depression, there is heterogeneity in autonomic reactivity to emotive stimuli, which remains to be clarified. We propose this heterogeneity may reflect individual differences in attentional focus [START_REF] De Zorzi | Autonomic reactivity to emotion : A marker of sub-clinical anxiety and depression symptoms[END_REF]. For example, depressed individuals show heightened electrodermal reactivity only for stimuli in direct attentional focus (i.e., central vision) and not for stimulation presented in peripheral vision [START_REF] De Zorzi | Depression restricts visual capture and promotes the perception of negative information[END_REF]. More generally, attentional processes are linked to the modulation of autonomic activity. Thus, electrodermal activity, notably the amplitude of sympathetic skin conductance responses (SCRs), is related to attention-orientation behaviours and reflects focused attention to new stimuli and their salience [START_REF] Boucsein | Principles of Electrodermal Phenomena[END_REF]. Similarly, HRV, reflecting both parasympathetic and sympathetic influences on heart rate, is also considered as an objective indicator of attentional processes. In this context, superior selective or sustained attention is associated with increased HRV, particularly in the dominant highfrequency range (HF-HRV, indexing vagal parasympathetic autonomic activity) [START_REF] Suess | Cardiac vagal tone and sustained attention in school-age children[END_REF]. Interoceptive feedback of autonomic bodily signals also influences emotional and attentional processes [START_REF] Critchley | Visceral influences on brain and behavior[END_REF]. Altered sensitivity to bodily arousal is observed in depression [START_REF] Paulus | Interoception in anxiety and depression[END_REF]. As depressive individuals suffer from attentional disturbance when emotional information is involved, the use of a task allowing to measure attentional processes appears relevant in the study of their autonomic reactivity to emotion.

The present study aimed to characterise the association of attentional bias and autonomic reactivity to emotional information in individuals expressing different levels of depression.

To this end, in an original paradigm, we presented pairs of emotional and neutral images, at near eccentricities within left (-12°) and right (+12°) visual fields. Eye movements were recorded to enable the tracking of attentional deployment toward one or other images, while we simultaneously measured autonomic variables (SCR and HR). Accordingly, we hypothesised that: 1) Attention will be preferentially directed to emotional images in all participants; 2) attentional bias to emotional information will be associated with autonomic reactivity (SCR and HR responses); 3) depressive symptoms will be associated with greater attention towards unpleasant stimuli and reduced orientation towards pleasant stimuli on the one hand, and with autonomic reactivity on the other hand.

Method

Participants

Thirty-four healthy unmedicated participants were recruited through an online questionnaire. All were French speakers, right-handed and had normal or corrected-to-normal vision. Individuals with a history of neurological disorders or regular and/or recent illicit drug consumption were not included. Two participants were excluded due to recording problems, giving a sample of 32 participants (24 females and 8 males; Table 1). Each participant provided an informed consent statement and received a 20 € compensation for his or her participation.

This study was approved by the Ethics Committee of Université de Lille [Référence: 2019-352-S73], and conducted in accordance with the Declaration of Helsinki at Faculté de Medecine, Pôle Recherche, Université de Lille, France.

[Insert Table 1 about here]

Stimuli and Apparatus

The stimuli used were pairs of images of emotional or neutral scenes selected from the International Affective Pictures System (IAPS; [START_REF] Lang | International affective picture system[END_REF], which provides standardised a priori values for each image on valence and arousal dimensions. One value is provided for men, and another for women. Given the recognised differences in gender-based emotional assessments [START_REF] Bradley | Emotion and motivation I : Defensive and appetitive reactions in picture processing[END_REF], we performed two image selections adapted to each gender, but resulting in equivalent valence and arousal values. Ninety-six images were selected and used to build three kinds of pairs: 16 unpleasant-neutral pairs (|UN|), 16 pleasant-neutral pairs (|PN|) and 16 neutral-neutral pairs (|NN|). In order to control the salience of the two images that made up each pair, the difference of valence and arousal between the two images constituting each pair were calculated. These within-pairs differences significantly differed between the three kinds of pairs on their a prioi but there were no differences between |UN| and |PN| pairs. For each pair of images, no withinpairs differences of valence or arousal was observed between the selections for men and women (all Fs < 0.275 and ps > 0.609). For each image, the angular size (12°x8°), the energy across spatial frequencies [START_REF] Delplanque | Spatial frequencies or emotional effects? A systematic measure of spatial frequencies for IAPS pictures by a discrete wavelet analysis[END_REF] and the main physical properties were extracted (ImageJ v1.50 software), including the luminance and contrasts for the greyscale version and the RGB (red, green and blue) layers. No significant differences were observed between the three sets of images for both genders (all ps > .20).

Thus, the image pairs differed only in terms of their emotional dimensions (see table in Appendix Table S1).

Participants were seated at a fixed viewing distance of 60 cm from the projection screen (30 inches, 256 x 160 ppi, DELL 3007WFP HC), which was connected to a computer (DELL Optiplex 9020, Windows 7 Professional) that managed the presentation of the pairs of images.

The images were displayed on a black background and each pair of images was presented pseudo-randomly, based on a Latin squares design, at near visual eccentricities (-12°, + 12 °).

The presentation of the 48 trials lasted approximately 15 minutes.

Recordings

Anxiety state, trait and depressive scores were measured using French language versions of the State-Trait Anxiety Inventory (STAI-A & B; Spielberger, 1983) and Beck Depressive Inventory (BDI-II; Beck, Steer, & Brown, 1996) respectively.

Regarding behavioural data, the eye movements were recorded using an eye tracker (SMI RED-m Eye Tracking System) connected to the projection computer with SMI iView RED-m 2.11 software for an acquisition at 120 Hz. The skin conductance (SC) and electrocardiogram (ECG) were recorded during two minutes of baseline, during the task and over 2-minute recovery periods, using a BIOPAC MP35 system connected to a second computer (running BIOPAC Student Pro 3.7 software) for an acquisition at 200 Hz. SC was recorded using bipolar Ag/AgCl surface electrodes (BIOPAC EL507) pre-gelled with an isotonic electrolyte (0.05 molar NaCl) and attached to the palmar side of the middle phalanges of the index and middle fingers of the participant's non-dominant hand. SC was measured with a gain of 5 µS/V and a 10 Hz low-pass filter. The ECG was recorded using a DI modified bypass placing the Ag/AgCl pre-gelled (BIOPAC EL503, 7% NaCl) surface electrodes on the participant's left and right wrists and with a band-pass filter set between 0.5 and 66.5 Hz. At the end of the experiment, the participant was required to review each of the images and to rate them individually for valence and arousal values using two nine-point SAM scales (Self-Assessment Manikin; [START_REF] Bradley | Measuring emotion : The self-assessment manikin and the semantic differential[END_REF], ranging from 1, very unpleasant, to 9, very pleasant, and from 1, very calm, to 9, very arousing. Ratings were recorded with OpenSesame [START_REF] Mathôt | OpenSesame : An open-source, graphical experiment builder for the social sciences[END_REF].

Procedure

The experimental procedure was divided into three steps. First, the SCR and ECG electrodes were attached and the participant was acclimatised to the experimental environment.

The participant completed a psychometric measure of anxiety state (State Anxiety Inventory, STAI-A; [START_REF] Spielberger | STAI-Y : Inventaire d'anxiété état-trait forme Y. Éditions du centre de psychologie appliquée[END_REF], then the task was again explained orally in full.

Next, autonomic responses were recorded over a task-free 2-minute baseline period.

This period was then followed by the main task, with recording of behavioural (eye-tracking) and autonomic responses together. The participant saw 48 pairs of images; 16 pairs of unpleasant-neutral |UN| images, 16 pairs of pleasant-neutral |PN| images and 16 pairs of neutral-neutral |NN| images presented in a pseudo-random order. Each trial had the following sequence: First, a central fixation cross was projected for a duration of 0.5 s then this was replaced by the probe; a digit number (between 1 and 9) replacing the fixing cross for 1 s. The participant was instructed to say this number as quickly as possible. As shown by Duque and Vazquez (2015), this procedure ensured that the participant watched the centre of the screen before the presentation of the stimuli. The procedure also helped maintain the participants' attention during the experimental phase. Following the presentation of the number, a pair of images was presented simultaneously, at -12° and +12° on the projection screen for 3.5 s, followed by a black screen for a random duration between 9 to 13 s. Participants were invited to view the images naturally without any further requirements. The inter-stimulus interval (ISI) varied from 10.5 to 14.5 s; an ideal interval to avoid habituation inherent to autonomic responses, particularly electrodermal activity. After the task, we recorded autonomic activity during a 2minute recovery period.

Finally, to validate our selection of images, the participant was asked to rate the valence and arousal dimensions using the two 9-point SAM scales. These subjective ratings correspond to a posteriori values of images.

Data and Statistical Analyses

All trials containing eye movements before the onset of the image were rejected to exclude trials for which gaze is already directed to a image's side. Across all participants and conditions, 5.14% of the trials were rejected.

Eye Movements

Attentional deployment was assessed based on eye movements (OpenGazeAndMouseAnalyzer; [START_REF] Voßkühler | OGAMA (Open Gaze and Mouse Analyzer) : Opensource software designed to analyze eye and mouse movements in slideshow study designs[END_REF]. For initial fixation, the percentage of first fixation for each type of image (unpleasant, pleasant and neutral) in pairs (|UN| and |PN|)

was calculated for each participant. In addition, total fixation duration, number and latency of fixations were recorded.

Conductance and Cardiac Activities

SC and ECG variations initially sampled at 200 Hz, were down-sampled at 10 Hz using LabChart7. For the SC variations to the presentation of pairs of images, phasic waveforms were derived from the tonic signals with an offline 0.05 Hz high-pass filter using AcqKnowledge 4.1 software. SCRs were analysed by computing the integrals of SC amplitude variations over time for each condition and participant. One participant with no SCR was excluded from the analysis.

For the ECG, the instantaneous heart rate in beats per minute (BPM) was calculated from the R wave intervals and smoothed using the triangular Bartlett window with a 1 s width using LabChart7. SC variations in response to stimulation were obtained by subtracting the average over a 3 s pre-stimulus period from the 10 s post-stimulus period data. For each condition and participant, after the baseline correction (-3 to 0 s), we averaged the epochs (-3 to 10 s), timelocked to the stimulus onset. Finally, we analysed the heart rate variability (HRV) during the 2 minutes of baseline and recovery periods. HRV quantification was computed with an in-house customised program MATLAB program referring to HRV guidelines [START_REF] Berntson | Heart rate variability : Origins, methods, and interpretive caveats[END_REF].

The R-R intervals were detrended with a smoothness-prior method in order to remove the slow (< 0.04 Hz) non-stationary trends from the HRV signal. For the frequency domain method, a power spectrum density analysis was performed for the RR interval series using fast Fourier transform method with the low frequency (LF) band set at 0.04-0.15 Hz and a high frequency (HF) band set at 0.15-0.4 Hz. The LF/HF ratio was also computed. For the time domain method, we computed the mean heart rate (HR), the standard deviation of HR (namely the variability of the HR) as well as the root mean square of successive RR intervals differences (RMSSD).

Statistical Analyses

Regarding emotion and in accordance with its dimensional theory (Lang et al., 1993)

Results

Concerning psychometry, STAI-B (trait anxiety) scores correlated with BDI (depression) (r33 = 0.810; p < 0.001) and STAI-A (state anxiety) scores (r33 = 0.631; p < 0.001). BDI scores correlated with STAI-A scores (r33 = 0.651; p < 0.001).

Eye Movements and Emotional Arousal

Analysis of the contrasts revealed an emotional arousal effect on the initial fixation (QC: [Insert FIGURE 1 about here]

Psychometry, Eye Movements and Autonomic Activity

Mean SCRs to pairs of images correlated with depression (r32 = 0.475; p = 0.005). Thus, higher SCRs to images were associated with higher depression scores even when age or sex were controlled (respectively r30 = 0.476; p = 0.006 and r30 = 0.495; p = 0.004). HF-HRV at baseline correlated with the duration of fixation on pleasant images. Thus, higher HF-HRV at baseline was associated with a longer duration of fixation on pleasant images during the task (r31 = 0.374; p = 0.035). This association was still observed when age was controlled (r29 = 0.375; p = 0.037) and was still marginally significant when controlled for sex (r29 = 0.320; p = 0.079). During the task, a correlation was observed between ocular parameters and autonomic reactivity: the difference of fixation latency between pleasant and unpleasant images (valence effect) correlated with mean SCRs (r30 = -0.381; p = 0.034) even when age or sex were controlled (respectively r28 = -0.380; p = 0.038 and r28 = -0.438; p = 0.015). No other associations were found between ocular and autonomic variables (rs > 0.281; ps > 0.120) nor between ocular variables and depression or anxiety (rs < 0.249; ps > 0.170).

Depression, Eye Movements and Autonomic Reactivity

As SCRs correlated with depression scores and with the valence effect on the fixation latency, the stepwise linear regression model was performed by including depression (BDI scores) and the latency of fixing pleasant and unpleasant images, the age of participants, and their state and trait anxiety (STAI-A and STAI-B scores).

On the basis of the corrected Akaike information criterion (AICc), the selected model for predicting SCRs was found to explain 21% of the variance in the integral of SCR (F3,29 = 3.60, p = 0.027) and include the depression scores (BDI), the latency to fix pleasant images and the trait-anxiety scores (STAI-B) as predictors. The measures found did not point to the existence of significant collinearity between the predictors, minimal tolerance = 0.429 and maximal variance inflation factor (VIF) = 2.33 (Figure 2 A1).

[Insert FIGURE 2 about here]

In the model, depression scores (t = 2.22, p = 0.035) and the latency to fix pleasant images (t = 2.33, p = 0.028) significantly contributed to a better prediction of the SCRs while the trait-anxiety scores (t = -1.92, p = 0.066) did not significantly contribute to the predictive power of the model. Hence, greater SCRs were associated with higher depression, with a coefficient of 1.30, and higher latency to fix pleasant images, with a coefficient of 0.05.

Interestingly, contrary to what has been observed for depression scores and the latency to fix pleasant images, the coefficient describing the direction of the relation between STAI-B scores and SCRs was negative.

In sum, the best model to predict SCRs integrate latency of fixation to pleasant images, depression and anxious scores such higher depression level and lower orientation to pleasant stimuli were expected to predict higher SCRs, after controlling for the other variables.

Discussion

The aim of this study was to investigate potential links between attentional bias and autonomic reactivity to emotional information and the implication of depressive symptomatology on these potential associations. Firstly, all recorded parameters of ocular behaviour indicated a preferential orientation of attention to emotional images. Secondly, higher orientation and maintenance of attention towards pleasant images were associated with lower values of autonomic arousal during baseline (HRV) and during the task (SC). Thirdly, the best model to predict SCRs of participants includes latency of fixation to pleasant images, and scores of depression and anxiety.

The first result, showing a preferential orientation and maintenance of attention toward emotional contents, reflected by the initial fixation and the shorter latency to fixate upon emotional images, extends data from previous eye-tracking studies demonstrating that emotional stimuli benefit from enhanced perceptual processing, with more fixations, especially during the first saccades [START_REF] Niu | Visual and emotional salience influence eye movements[END_REF]. In terms of unpleasant and pleasant value of images, no attentional preference was observed for one over the other, suggesting that attentional bias depends only on the arousal dimension of emotion.

Depressive symptomatology was not correlated with any ocular parameter which is consistent with few studies that failed to report bias for emotion in people with depression or dysphoria [START_REF] Koster | Cueing of visual attention by emotional facial expressions : The influence of individual differences in anxiety and depression[END_REF][START_REF] Elgersma | Attentional bias for negative, positive, and threat words in current and remitted depression[END_REF]. Moreover, data from the literature report an attentional bias in depression towards "mood-congruent" or "dysphoric" information, using discrete emotional stimuli, and mainly for attention maintenance indicators [START_REF] Armstrong | Eye tracking of attention in the affective disorders : A meta-analytic review and synthesis[END_REF]. The present study did not use mood-congruent stimuli but natural scene based on dimensional theory of emotion, which could explain that such correlation was not observed.

Besides, in the present study, the images' selection was carried out in order to ensure homogenised arousal differences between the two images constituting the pairs. Thereby, attention may have been captured by the emotional image regardless of symptomatology for all participants.

The second result supports specific links between attentional bias to pleasant information and the expression of autonomic arousal both at baseline and during the task.

During the baseline, greater HF-HRV was associated with longer fixation on pleasant images.

Hence, higher parasympathetic influence, and thus increased HF-HRV, appears linked to maintenance of attention toward pleasant information. The polyvagal theory [START_REF] Porges | The Polyvagal Perspective[END_REF] proposes that baseline cardiac measures of parasympathetic activity can index the capacity to adapt to the environment. More precisely, parasympathetic HRV activity at rest and reactivity are associated with adaptive expression of emotion and self-regulatory skills. Therefore, our observed association between baseline autonomic activity and preferential attention to pleasant information reveals a positive impact of a more parasympathetic psychophysiological state on an upcoming emotional task in the domain of emotional regulation [START_REF] Beauchaine | Vagal tone, development, and Gray's motivational theory : Toward an integrated model of autonomic nervous system functioning in psychopathology[END_REF].

Individuals with higher parasympathetic activation at baseline may be better able to employ the best strategies to respond to stressful emotional challenges by focusing on pleasant information.

The present results also reinforce the neurovisceral integration model [START_REF] Thayer | A model of neurovisceral integration in emotion regulation and dysregulation[END_REF], which postulates that cardiac activity, through HRV, is informative about the integrity of brain networks supporting interaction between emotion and cognition. However, this association became marginally significant when sex is controlled. Indeed, sex differences on HRV have been reported with higher HRV in women characterised by a relative dominance of vagal activity [START_REF] Koenig | Sex differences in healthy human heart rate variability : A meta-analysis[END_REF]. Due to the sample size, it seems difficult to conclude on sex differences for HRV and it is therefore advisable to remain cautious about this result.

Additional data on the association between attentional and autonomic measurement in relation to sex are necessary and may point out different associations for these parameters for men and women.

During the task, a valence effect on the fixation latency was associated with autonomic reactivity. Individuals who fixated upon pleasant stimuli more quickly showed lower SC to the pairs of images whereas the reverse was observed for fixation on unpleasant stimuli. This positivity bias, described in the attentional literature [START_REF] Troller-Renfree | The beneficial effects of a positive attention bias amongst children with a history of psychosocial deprivation[END_REF] bears witness to a regulatory interaction between the capture of information and autonomic adaptation.

The third result showed that the best model to predict SCRs integrate latency of fixation to pleasant information with depression and trait-anxiety scores. At psychometric level, higher depression predicted greater SCRs in response to pair of images while trait anxiety did not significantly contribute to the prediction of SCRs. At attentional level, measured by images' fixation latency, a lower attentional orientation towards the pleasant images was associated with greater SCRs. This association between pleasant orientation and autonomic reactivity is particularly interesting since depressed individuals appear to lack positive attentional bias (Duque & Vazquez, 2015) and are characterised by autonomic dysfunctions sometimes reported as increased autonomic activity [START_REF] Branković | System identification of skin conductance response in depression-An attempt to probe the neurochemistry of limbic system[END_REF][START_REF] Wenzler | Intensified emotion perception in depression : Differences in physiological arousal and subjective perceptions[END_REF]. Consequently, attempts to reinforce such bias could help to attenuate autonomic activation and potentially serve as a protective homeostatic adaptation or a coping strategy. In the same vein, similar autonomic hyporeactivity to emotional challenges has already been reported, and interpreted as the expression of a coping strategy engaged to improve performance of a behavioural task [START_REF] Naveteur | The influence of anxiety on electrodermal responses to distractors[END_REF]. In this context, an intervention to attenuate autonomic reactivity could increase capacity to orient attention towards pleasant information which may be beneficial beneficial, in part by fostering enhanced attentional bias toward pleasant information. Finally, our research supports an approach integrating both cognition and physiology to better understand their interdependence in healthy and pathological expressions of emotion.

  valence values (women: | U -N | = 2.56, | P -N | = 2.47, | N -N | = 0.54, F1,15 = 290.196; p < 0.001; men: | U -N | = 2.65, | P -N | = 2.28, | N -N | = 0.40, F1,15 = 640.406; p < 0.001), and on their arousal a prioi values (women : | U -N | = 2.74, | P -N | = 2.80, | N -N | = 0.50, F1,15 = 263.916; p < 0.001 ; and men: | U -N | = 2.48, | P -N | = 2.67, | N -N | = 0.54, F1,15 = 232.177; p < 0.001) with higher within-pairs difference for |UN| and |PN| pairs than |NN| ones,

  , we tested for two emotional effects: 1) A valence effect (Unpleasant vs. Pleasant), being modelled by a first degree polynomial contrast (Linear Contrast, LC = | PN | -| UN | ; and 2) an arousal effect (Emotion vs. Neutral) being modelled by a second degree polynomial contrast (Quadratic Contrast, QC = (| UN |+ | PN |) / 2 -| NN |). These contrasts were assessed with a repeated measure analysis of variance (ANOVA), applied to the individual subjective image assessments, eye movements, electrodermal and cardiac measurements with emotion (type of pairs: | UN |, | NN |, | PN |) as intra-subject factors. The analysis of the factors associated with the autonomic variables was performed by calculating the Pearson correlation coefficient given linear relationships were expected between variables and after inspection of scatterplots. Partial correlations were then assessed controlling for age and gender. The search for predictors of autonomic variables was performed with stepwise linear regression analyses. The multivariate model includes variables for which associations were observed between ocular and autonomic parameters. Thus, the model was constructed by including variables associated with SCRs, but also including covariates such as the age of participants, and their state and trait anxiety (STAI-A and STAI-B scores), regardless of their degree of significance in the univariate analyses. The model selection was based on considerations of the corrected Akaike information criterion (AICc). The validity of the multivariate model was established by a study of the residuals.

F

  (1.31) = 7.144; p = 0.012 η² = 0.187), first fixation latency (QC: F(1.31) = 75.624; p < 0.001; η² = 0.709), fixation duration (QC: F(1.31) = 107.966; p <0.001; η² = 0.777) and number of fixation (QC: F(1.31) = 125,181; p <0.001 ; η² = 0.802; Figure1). However, no differences were observed between unpleasant and pleasant for these parameters (initial fixation: LC: F(1.31) = 2.827; p = 0.103 η² = 0.084; first fixation latency : LC: F(1.31) = 0.040; p = 0.843; η² = 0.001; fixation duration : LC: F(1.31) = 0.92; p = 0.764; η² = 0.003 ; number of fixation : LC: F(1.31) = 0.049; p = 0.826 η² = 0.002). In sum, eye movements were initially oriented and engaged by emotional images.

Figure 1 .

 1 Figure 1. Initial fixation and fixation latency (A-B) and total fixation duration and number of fixations (C-D) to image emotion in |UN| and |PN| pairs. U: Unpleasant; N: Neutral; P: Pleasant; QC: Quadratic contrast.

Figure 2 .

 2 Figure 2. A. Multiple linear regression model to predict SCRs. VIF: maximal variance inflation factor, index allowing to verify the premise of multicollinearity (A1) and representation of the model (A2) B. Partial trace regression with intervals of confidence for BDI scores (B1) and for latency of fixation to pleasant images (B2).

Table and Figures

 and

			Sample (N = 32)
		Mean	SD	(Min -Max)
	Age	21	2	(18 -24)
	STAI-A	30.4	9.7	(20 -58)
	STAI-B	47.1	12.8	(25 -71)
	BDI-II	13.3	10.4	(1 -41)

Table 1 . Demographic and psychometric characteristics of participants.

 1

Acknowledgements

We would like to acknowledge the organisations which support this project: CNRS, Pôle Recherche de la Faculté de Médecine and Université de Lille. We are also grateful to Clinical Imaging Sciences Centre, Brighton and Sussex Medical School, University of Sussex (UK) for stay facilities to S. Ranfaing.

Funding Source

The present study received no specific grant from any funding agency, commercial or not-for-profit sectors and was supported by the annual financial support of University of Lille and CNRS to the DEEP Team of SCALab UMR 9193 (HS and LDZ), the Ministère de la Recherche, France (SR), the CNRS, the Ministère de la Recherche and the University of Lille, France (HS and JH). The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

especially for depressed individuals. Therefore, these results suggest that the pleasantness bias could constitute a cognitive marker of behavioural and autonomic adaptations to emotion.

Finally, this study has several methodological strengths. First, we carefully considered influences of physical saliency of images on visual search and attention [START_REF] Lucas | Effects of emotional and non-emotional cues on visual search in neglect patients : Evidence for distinct sources of attentional guidance[END_REF]) and gender differences in emotional assessment and reactivity [START_REF] Bradley | Emotion and motivation I : Defensive and appetitive reactions in picture processing[END_REF]. Thus, we carried out a rigorous selection of stimuli for each type of pair of images (see Appendix 1; Table S1). Second, this study was enriched by taking into account both facets of the autonomic nervous system, sympathetic and parasympathetic, which potentially have distinct contributions to attentional and emotional processes. Third, the integration of behavioural and autonomic parameters, encompassing their reciprocal influences, allowed us to clarify the relationship between attentional bias and autonomic reactivity and the relation with depressive symptomatology. The study also had some limitations, notably the sex-ratio of participants. Indeed, even if images selection was homogenised between women and men, sex of participants seems to influence some of the association observed (e.g. HRV) and the sample size for men do not allow to examine potential sex differences properly on theses associations.

Therefore, additional researches are needed to determine potential implication of this variable on interaction between attention and autonomic expression. Moreover, the choice of a nonclinical population constrains the scope of our results to moderate levels of depression. Indeed, participants are not clinically depressed in our sample. Nonetheless, even when considering lower depression levels, prominent effects were observed. Moreover, findings remain relevant since attentional bias is considered to be implicated in the development of clinical depression (Gotlib and Joorman, 2010).

In conclusion, our characterisation of the association between attentional processes and specific patterns of autonomic reactivity extends a growing literature, and points in the direction of reciprocal influences of both systems that can lead to the personalisation of clinical remediation and rehabilitation procedures, including attentional training and/or biofeedback therapies. Furthermore, the data obtained in this work argue in favour of evidence-based interventions using attentional training towards pleasant information. Consequently, considering the lack of orientation to pleasant information and autonomic dysfunction in depression, two strategies can emerge from these results. On the one hand, in depressed individuals who lack strong attentional bias toward pleasantness, attentional training towards emotionally positive information may be beneficial in reducing autonomic hyperarousal and reactivity to emotional information. On the other hand, in depressed individuals who show autonomic hyperarousal, biofeedback procedures based on HRV may be therapeutically

Disclosure statement

No potential conflicts of interest were reported by the author(s).


Jiali Liang
email: liang@ipno.in2p3.fr

Marc Ernoult
email: ernoult@ipno.in2p3.fr

Xavier Doligez

Sylvain David

Sandra Bouneau

Nicolas Thiolliere

Guillaume Krivtchik

Fanny Courtin

Weifeng Zhou

Stéphanie Tillement

Assessment of strategy robustness under disruption of objective in dynamic fuel cycle studies

Keywords: Nuclear fuel cycle, Robustness, Disruption of objective, Status of plutonium, Temporality of adaptation

The future of nuclear power depends on the interests and decisions. To take the relevant deep uncertainty into account, a new methodology of robustness analysis of fuel cycle strategies has been developed. The method has been applied to the French cycle, considering the future deployment of fast reactor as the preselected objective and an uncertain change, called disruption, towards a new objective: the minimization of transuranic inventories without fast reactor. The status of plutonium is contradictory in two cases. Two approaches of identifying robust strategies were tested, which correspond respectively to the static and adaptive robustness assessment. One identifies static strategies in a pre-disruption scenario, which achieve acceptable outcomes for both objectives. The other takes a trajectory pursuing the pre-selected objective and, in case of disruption, adapts it for the new objective. The comparison of two approaches indicates the temporality of adaptation relative to immediate actions under the uncertain disruption.

I. Introduction

Deployment of Sodium-cooled Fast Reactors (SFRs) of GEN IV used to be one of the referential perspectives of nuclear power development in France [START_REF]Nuclear Fuel Cycle Transition Scenario Studies -Status Report[END_REF] [START_REF]Des Rercheches et Etudes Relatives à la Gestion des Matières et des Déchets Radioactifs[END_REF]. Some studies of nuclear fuel cycle show that the integration of SFRs can reduce the pressure on the interim storages of spent fuels (SFs) [START_REF] Chen | Prospects in china for nuclear development up to 2050[END_REF] as well as the consumption of natural uranium resources [START_REF]Nuclear Fuel Cycle Transition Scenario Studies -Status Report[END_REF] [START_REF]Transition Towards a Sustainable Nuclear Fuel Cycle[END_REF], and it is possible to stabilise the plutonium inventories in the cycle [START_REF] Tiphine | Simulations of Progressive potential scenarios of Pu multi-recycling in SFR and associated phase-out in the French nuclear power fleet[END_REF]. If SFRs are deployed, plutonium (Pu) inventories in the interim storages of SFs will be strategic resources because plutonium is one of the main element of SFR fuels in European design [START_REF]Benchmark for Neutronic Analysis of Sodium-cooled Fast Reactor Cores with Various Fuel Types and Core Sizes[END_REF]. However, given the cancellation of ASTRID project [START_REF]Stratégie Française pour l'Energie et le Climat : Programmation Pluriannuelle de l'Energie 2019-2023 2024-2028[END_REF], the SFR deployment in France seems to be uncertain. If the planning of SFR deployment is given up, plutonium inventories can be considered as potential nuclear wastes due to the high toxicity during long time. In particular, the minimization of all transuranic inventories (TRU) in the fuel cycle is emphasised in this case. Concerning the status of plutonium, these two states of future are contradictory. Owing to the deep uncertainty of future and objectives, the relevant strategy specific to the nuclear power development cannot be simply decided.

Optimisation methods of dynamic fuel cycle simulations in nuclear scenario studies can identify optimal strategies with respect to given objectives which are not changed over time. Given a single objective of stabilization of TRU inventories, [START_REF] Courtin | Etude de l'incinération du plutonium en REP MOX sur support d'uranium enrichi avec le code de simulation dynamique du cycle CLASS[END_REF] identify appropriate parameters of the fuel cycle using UOX and MOXEUS fuels in Pressurised Water Reactors (PWRs). Another study analyses the strategies of SFR deployment quantity in different phases, which can minimise both the consumption of natural uranium resources and the nuclear waste packages [START_REF] Freynet | Multiobjective optimization for nuclear fleet evolution scenarios using cosi[END_REF]. But these methods of optimisation cannot provide sufficient information of strategies or related consequences if the objective is deeply uncertain, especially when the possible choices are contradictory. In this situation, the robustness assessment of alternative strategies is more important than the optimisation.

This work develops a method of robustness assessment on possible strategies subject to such uncertainty. The concept of robustness is commonly employed in sociological studies and policymaking; but in practice, lots of metrics introduced in robustness assessment study are sometimes different or even inconsistent [START_REF] Mcphail | Robustness metrics: How are they calculated, when should they be used and why do they give different results?[END_REF], and there is no standard mathematical definition. In this work, we will present our definition of robustness adapted for nuclear fuel cycle analysis. This paper is divided in four parts. Firstly, based on the example of objectives mentioned above, definition of disruption and the two approaches relevant respectively to static robustness and adaptive robustness are introduced. In the second step, a pre-disruption scenario is presented, in which the robust static strategy is identified. The static robustness of possible strategies is analyzed. In consideration of the information updating upon the uncertainty, adaptive strategies for the change of objective are studied, which is related to the adaptive robustness. Finally, in addition to concluding the strategy robustness and the temporality of adaptation in this example, we discuss the validity of our method as well as the relevant implicit assumptions.

II. Method of robustness assessment

In dynamic fuel cycle simulations of nuclear scenario studies, trajectories of nuclear system are simulated under a set of assumptions. A trajectory denotes a single evolution of the nuclear system, with parameters and operation states explicitly set at each step of time. In the simulation, a trajectory is one deterministic run, with material evolutions well defined in all facilities simulated. A strategy is usually represented by a group of variables, which are some specifically controllable parameters. For example, nuclear industry can evaluate and then decide the burn-up of UOX fuel in current PWRs. Specified by these controllable parameters, a strategy leads to a trajectory when other parameters are well defined (and it is the case in our work, because the uncertain factor in this work is assumed to be the objective in the scenario, and all parameters of the fuel cycle can be well decided if a strategy is given). It seems that a trajectory is equivalent to a strategy, but they are conceptually different: a trajectory describes the evolution, whereas a strategy formalises a trajectory into variables and links the outputs to the objective in scenario studies. Involving the performance related to a given objective, a strategy can be assessed, but it does not make sense to evaluate a trajectory.

i. Simulation of the French fleet by CLASS

The dynamic fuel cycle simulation is performed by the simulator CLASS, which employs artificial neuron networks to build simulation models [START_REF] Leniau | A neural network approach for burn-up calculation and its application to the dynamic fuel cycle code CLASS[END_REF]. For example, the equivalence model predicts the fissile content of fresh fuel for a given burn-up according to the isotopic composition of resources; the irradiation model simulates the evolution of material in the reactor. To carry out a fuel cycle simulation, facilities such as fabrication plant, reactor and cooling pool are simulated; the material flows among facilities are also taken into account. Take the French fleet of PWR UOX and PWR mono-MOX as example, which is simulated in this study. For a required burn-up of UOX fuel depletion, UOX fresh fuel is fabricated in the UOX fabrication plant for given time of fabrication. The fresh UOX fuel is then transported (instantly) to the PWR UOX. After the depletion, UOX SFs are put in the pool for a given time span of cooling. The UOX SFs is then available for reprocessing and MOX fabrication if it is demanded. To highlight the availability, the storage of SFs after necessary cooling is simulated separately from the pool. The MOX fuel follows the same steps in the cycle, except that the MOX SFs are not reprocessed because the Pu is mono-recycled in this example. Descriptions in detail of physics model building and facility simulation by CLASS can be found in [START_REF] Leniau | A neural network approach for burn-up calculation and its application to the dynamic fuel cycle code CLASS[END_REF].

Based on the French fleet, PWRs are normalized into 2 macro-reactors in this work. In other words, instead of 58 individual reactors, only 2 are simulated: one is a PWR UOX that contributes to 90% of the fleet total power, being 169.2 GWth at nominal state; the other is a PWR MOX with 18.8 GWth. The loading factors, burn-ups and heavy metal mass of 2 reactors are normalized according to the information of 58 reactors, summarized in [START_REF] Courtin | Etude de l'incinération du plutonium en REP MOX sur support d'uranium enrichi avec le code de simulation dynamique du cycle CLASS[END_REF].

ii. Objectives, criteria and disruption

To illustrate the assessment method, the French fleet of PWR UOX and PWR mono-MOX aforementioned is considered. The objective of SFR deployment for 100% of fleet in 2090 is pre-selected. One of the most important constraints of deploying SFRs is the availability of plutonium inventory. To quantify the capacity of substituting the nuclear fleet of PWRs with SFRs at a given time, the ratio of substitution, denoted as R Subs , is defined:

R Subs (t) = P deployable,SFR (Pu in (t)) P tot,elec (t) (1)
where P deployable,SFR denotes the deployable electric capacity by break-even SFRs only depending on the in-cycle inventory of plutonium Pu in ; P tot,elec denotes the total electric capacity of the nuclear fleet. In the definition, P deployable,SFR only depends on the in-cycle Pu inventory, which means that SFR is not simulated. In this work, the specific thermal power of a SFR is supposed to be 48 MW/tHM as the model built in [START_REF] Ernoult | Global and flexible models for Sodium-cooled Fast Reactors in fuel cycle simulations[END_REF]; the core-average Pu content in the initial heavy metal of fresh fuel is supposed to be 17%, approximative to the design in [START_REF]Benchmark for Neutronic Analysis of Sodium-cooled Fast Reactor Cores with Various Fuel Types and Core Sizes[END_REF]. It means that the content fluctuation due to isotopic effects and other operation requirements are neglected (for instance, the fresh fuel richer in 239 Pu+ 241 Pu will require smaller content of Pu than the one richer in other Pu isotopes for the same burn-up). Besides, the dependency on the inventory of in-cycle plutonium Pu in means that the plutonium in the cooling pools and even in the operating PWRs are required to achieve such ratio of substitution. Time of Pu recycling for SFRs, including two years of fuel fabrication and five years of SF cooling, is also taken into account, which means that the quantity of Pu in allows to deploy SFRs with a capacity indicated by R Subs in a closed cycle. Since the nuclear reactors are defined with thermal power in CLASS [START_REF] Leniau | A neural network approach for burn-up calculation and its application to the dynamic fuel cycle code CLASS[END_REF], the yield of a PWR and an SFR are supposed to be 34% and 40% respectively. Value of R Subs indicates that at the given time t, a fraction R Subs of this PWR-fleet with P tot,elec can be replaced by SFRs according to the material availability. Theoretically, SFR deployment for 100% of fleet requires a strategy leading to R Subs (t = 2090) = 1. However, because of those assumptions on the estimation of P deployable,SFR without simulation of SFR, a margin on the R Subs should be taken to achieve the objective of SFR deployment, and thus a criterion of R Subs (t = 2090) > 1 is required. Moreover, the future of SFR deployment will not evolve to a phase-out state. For the first step, we suppose that the capacity of the nuclear fleet will finally evolve to a level between 50% and 150% of the current one. The objective pre-selected in this study, called "objective A" in the following, and the relevant criterion C A can be summarized as:

• Obj. A: SFR deployment for 100% of fleet in 2090 with a capacity evolving finally to [50%, 150%] of the current one (64 GWe/188 GWth currently). • Criterion C A : R Subs (t = 2090) > 1 and maximization of R Subs (t = 2090) One may consider that the criterion C A is not sufficient to choose the strategies achieving objective A. The deployment of such new reactors is complicated, involving the development of SF reprocessing and fresh fuel fabrication plants, the requirement of new technology and nuclear safety, the engineering complexity, etc. In this work, these practical constraints are neglected, so that we can focus on the single factor of material availability.

It is possible that the objective pre-selected today will be changed before 2090. The change can be triggered by an accident, by economic and political interests of stakeholders, or by other factors that are not well previewed. Neither the probability nor the time of change is well estimated. The change of objective in such deep uncertainty is called the disruption of objective. If the preselected objective of SFRs deployment is given up, one may consider primarily the minimization of potential nuclear wastes in total cycle. We take the TRU inventory in the total cycle, noted as TRU tot , as the output of interest for the new objective. In case of the disruption, new strategies are explored in order to minimize the TRU. Different from the context of objective A, a phase-out strategy is possible. Thus, the new objective and the relevant criterion C B linked to the disruption can be summarized as:

• Obj. B: Minimisation of nuclear wastes without deployment of SFR; phase-out strategy is acceptable.

• Criterion C B : Minimization of TRU tot (t = 2090)
Two foregoing objectives correspond to different states of future, indicating different criteria and strategy spaces. Nonetheless, it does not mean that the TRU cannot be minimized if the objective A is pursued; neither does it mean that the minimization of TRU inventory is the sole choice if SFR deployment is abandoned. Other objectives and criteria can also be suggested. For example, the minimization of nuclear wastes normalized by the electricity produced by reactors, or the minimization of natural uranium consumption can be considered after disruption. The objectives and the related criterion aforementioned are selected to simplify the analysis, and at the same time, to keep the contradiction between them.

iii. Pre-disruption precaution, post-disruption adaptation and the robustness of strategy

In the context of deep uncertainty, a robust strategy is often considered to be insensitive to violated assumptions and to have acceptable performance within a wide range of future states [START_REF] Lempert | Managing the risk of uncertain threshold responses: comparison of robust, optimum, and precautionary approaches[END_REF]. In our study, the possible violation of assumption is associated with the disruption of objective. To study the robustness of possible strategies under deep uncertainty in this work, two approaches are suggested, which are respectively related to static robustness and adaptive robustness described in [START_REF] Walker | Deep Uncertainty[END_REF]. One approach starts from the principle of precaution before disruption: the strategy is decided in a precautionary manner that assures an acceptable level of performance for any possible objective finally determined. Strategy studied in this approach is considered static, which means that parameters of system at each time step are decided at the beginning, and the decision of evolution of these parameters is not changed in case of the disruption. Since the disruption is deeply uncertain, the analysis of static strategy allows to estimate preliminarily the consequences on different states of future. The static robustness is assessed for static strategy. Technically, it is equivalent to study strategy performance in a multi-objective problem, in which the optima of problem are regarded as robust static strategies. Due to the contradiction of possible objectives, robust static strategies in the Pareto front trade some performance on one objective to another. In this work, a criterion C B|A is defined to give a single optimum, called robust static optimum:

• Criterion C B|A : Minimization of TRU tot (t = 2090) under the condition of R Subs (t = 2090) > 1
By definition, the robust static optimum achieves the lowest TRU tot (t = 2090) among all static strategies which keep theoretically the Pu availability for complete SFR deployment by 2090.

The other approach is based on the post-disruption adaptation: a strategy dedicated to the pre-selected objective (the A in our example) is firstly implemented; once the disruption takes place, the related trajectory will be adapted with readjustments for the new objective. Based on the given prior trajectory, the new adaptive strategy that minimizes TRU tot (t = 2090) is called the adaptive optimum (under the disruption from objective A to B). Since only one trajectory is adapted in this work, the notation of "based on the given prior trajectory" is omitted in the rest of paper. If several trajectories are adapted, the link between the adaptation scenario and the prior trajectory should be highlighted. Adaptive robustness is assessed for adaptive strategies. In this work, an adaptive strategy is considered robust and called robust adaptive strategy if it has better performance (thus lower TRU tot (t = 2090)) than the robust static optimum. According to the assessment criterion, the adaptive optimum can be robust or non-robust over the time of adaptation, which indicates the temporality of adaptive robustness: if the adaptive strategy is implemented before the time limit, the prior trajectory can be readjusted to accumulate TRU inventories as low as the pre-disruption strategy with sufficient precaution; otherwise, there is regret in comparison to this optimal strategy decided before disruption. The temporality may impact the decision-making vis-à-vis the deep uncertainty of disruption.

To simplify the notations, the time "t=2090" is omitted when the outputs of interest R Subs (t = 2090) and TRU tot (t = 2090) are mentioned. Nevertheless, the year 2090 is so important that the subsequent analysis depend directly on it.

III. Pre-disruption scenario i. Design of experiment of the pre-disruption scenario

First of all, the pre-disruption scenario is studied: the objective A aiming at deploying SFRs in 100% of fleet by 2090 is pre-selected, while the stakeholders may remain a doubt on the disruption towards the objective B, the minimization of TRU in total cycle without SFR deployment in the future. Subject to the deadlock between two contradictory objectives without a definitive decision, strategies are explored by varying the parameters of the fuel cycle, which requires the simulation of a large quantity of trajectories. Two outputs of interest, R Subs and TRU tot , are calculated for these trajectories.

The simulation of fuel cycle starts from the current French fleet, in which the PWRs are normalized to two macro-reactors, a PWR UOX and a PWR mono-MOX. To explore possible strategies, a transition is considered, starting from the time T start (between 2030 and 2050). The transition will last for more than 10 years (duration denoted as D) and end before 2090. In T start , the burn-ups of UOX and MOX fuels (denoted respectively as BU UOX and BU MOX ), the cooling time of UOX SF (denoted as TC UOX ) and the UOX SF reprocessing management for MOX fuel fabrication (denoted as MPu) will be redetermined. The UOX SF reprocessing management for MOX fuel fabrication is either LiFo, "Last in First out", or FiFo, "First in First out". During the transition, the total thermal power as well as the MOX fraction of fleet changes linearly towards their final value (denoted as P tot, f and FrMOX f ); after the transition, they will keep constants until the end (year 2090). Under these hypotheses, possible strategies and the related trajectories are characterized by eight variables. The ranges of variation of these eight variables are presented in table 1. These strategies explored in the pre-disruption scenario are the so-called static strategies, since they are decided at the beginning of scenario, and the performance is evaluated at the end of scenario without adaptation. Depending on the interest of stakeholders, some of these eight parameters are controllable, while others are not. In our study, we suppose that all of them are controllable so that the strategy space can be reasonably large for exploration. Their ranges of variation are not completely arbitrary: some of them are based on industrial and economic constraints (the ranges of FrMOX f , BU UOX , BU MOX , TC UOX and MPu), while others are associated with the assumptions of the future nuclear power development in France. The range of T start supposes that the transition will start after the commissioning of the first European Pressurised Reactor (EPR) in France (around 2030 as supposed), and some old reactors will be shut down. The range of D cannot be too short because a period of adjustment is necessary for the change of electricity supply by nuclear power and the change of MOX fraction (related to licenses for MOX, fabrication capacity, etc.). Because the deployment of SFR is regarded as the pre-selected objective, the future installed capacity of fleet cannot be too low; the range of P tot, f supposes therefore a wide range of possibility. Nonetheless, the control of total power of fleet for a given objective seems unreasonable, since it is related to the commissioning and shut-down of reactors; under the technical and the economic constraints, the controllability of total capacity of fleet by stakeholders is not obvious. One may also reckon that the evolution of total capacity should match the electricity demand and it is not controlled in the same way as other fuel cycle parameters. Anyway, the discussion on the nature of P tot, f , to be controllable or not, is out of the scope of study. For the development of method, all groups of stakeholders who may "virtually control" the total capacity of fleet in order to achieve a given objective, are included in this work.

ii. Preliminary study of the nuclear system by Principal Component Analysis According to the space of strategy aforementioned, 6400 strategies are selected by Latin Hypercube Sampling (LHS) [START_REF] Mckay | A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer code[END_REF] and the related trajectories are simulated. 3177 are valid without the Pu shortage for MOX fabrication before 2090. The problem of Pu shortage is well studied in [START_REF] Thiollière | A methodology for performing sensitivity analysis in dynamic fuel cycle simulation studies applied to a pwr fleet simulated with the class tool[END_REF], and in our work those invalid trajectories are not of interest. For each valid trajectory characterized by eight input variables, the outputs R Subs and TRU tot in 2090 are calculated. To study the impacts of input variables on a given output in a valid space a-posteriori defined, principal component analysis (PCA) is performed.

PCA identifies an orthonormal transformation of original variables to create new ones, called principal components (PCs), such that: the first created variable maximizes the variance among all possible linear combinations of original variables; the second one maximizes the variance among all possible linear combinations uncorrelated to the first one; the iterative steps are successively performed, subject to the condition that the new created variable is uncorrelated to all previously created ones. Mathematically, the eigenvalue-eigenvector pairs of the covariance/correlation matrix of original variables are respectively the variances of PCs and the corresponding linear combinations. In summary, the relation between an n-dimension vector of original variables X and the vector of principal components Y is Y = A t X or X = AY, where the columns of the n × n orthonormal matrix A are composed of the normalized eigenvectors of the covariance/correlation matrix of X.

PCA is often used to reduce the dimensionality of variables, identifying a smaller number of variables (PCs) which account for the major variability of original variables; in this work, it is used to highlight the importance of an input variable on a given output. The output variable is considered to be a component of the original variable, so that the original variable vector is under the form of X = (T start , D, ..., MPu, X out ) where X out denotes the output of interest R Subs or TRU tot . The square of correlation coefficient between X out and the j-th PC Y j is calculated, denoted as ρ 2 (X out , Y j ). Letting α ij be the (i, j)-th element of matrix A aforementioned, a brief proof is given in [START_REF] Mardia | Multivariate Analysis[END_REF] that

ρ 2 (X out , Y j ) = α 2 out,j V(Y j ) V(X out ) (2)
where V denotes the variance, and the index out of α out,j denotes the row with respect to the position of X out in the variable vector. Given the orthonormality of A revealing that

V(X out ) = ∑ j α 2 out,j V(Y j ) (3)
ρ 2 (X out , Y j ) can be interpreted as the fraction of the variance of X out explained by Y j . In other words, if Y j is highly correlated with X out , the majority of variation of X out can be explained by Y j , and thus X out can be sensitive to Y j . In this case, because the coefficients of linear combination can indicate the weights of original variables on Y j , they indicate the impacts of corresponding original variables on X out as well.

One should distinguish the variability of original variables from the one of output X out . The variability of all original variables include the ones of inputs sampled in a large exploration space, whereas the output of interest is a function of these input variables. Even though PCs are originally defined and used to represent the original variables with variability as large as possible (indicated by the corresponding eigenvalues of covariance/correlation matrix), we focus principally on the variance of X out explained by PCs (indicated by ρ 2 (X out , Y j )) which allows to analyze the influence of the input on the output X out .

In practice, the analysis with the output R Subs and the one with TRU tot are separate. Valid observations of 9-dimension vector (8 inputs + 1 output) are considered. PCs of the dataset are identified after being centred and scaled in the software R [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF]. After that, the squares of correlation coefficients ρ 2 (R Subs , Y j ) (resp. ρ 2 (TRU tot , Y j )) are calculated, indicating the fractional variation of R Subs (resp. TRU tot ) explained by PCs. Once the most correlated PC is found, the coefficients that transform original variables to this PC are investigated; due to the high correlation between the PC and R Subs (resp. TRU tot ), the coefficients of original variables indicate their importance on R Subs (resp. TRU tot ).

The correlation between a given PC and the output of interest in two analyses are shown in table 2. The PCs are ordered by the corresponding eigenvalues of the correlation matrix of original variable data: the PC Y 1 has the largest variance and thus it explains most of the variability of original variables among all PCs, whereas the PC Y 9 has the smallest variance. The variability of PCs are also presented by the ranges of axis in figure 1a and1b.

Table 2: Correlation squares between the given output X out and PCs in the corresponding PCA

ρ 2 (X out , Y j ) Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9
X out = R Subs .88 .02 .04 .00 .00 .00 .03 .01 .02 X out = T RU tot .95 .01 .02 .00 .00 .00 .00 .00 .02

Regarding R Subs , Y 1 can explain about 88% of the variance of R Subs . It means that as a linear combination of original variables, Y 1 has a strong impact on the output R Subs , which can be observed in the scatter plot of Y 1 and R Subs in figure 1a. Table 3 shows the coefficients of original variables with respect to Y 1 . Besides R Subs itself, the coefficient of P tot, f , which means that P tot, f also has a large weight in the component Y 1 . Given the high fraction of R Subs variance explained by Y 1 , it indicates the strong impact of P tot, f on R Subs . The coefficients of FrMOX f and BU UOX seem non-negligible. Other input variables contribute to the variability of other PCs on which R Subs has little dependence, as shown by scatter plots in figure 1a, and thus they have little effect on the variation of R Subs .

One may argue that the weak correlation between a PC and R Subs does not imply that they are weakly related, such as the special pattern of Y 9 over R Subs in figure 1a; and thus we cannot conclude that other inputs, possibly with high weights in the PC Y 9 , have negligible influence on R Subs . To answer this question, we should first note that the variation of Y 9 is actually much smaller than other PCs. We can also verify the composition of Y 9 , indicated by the coefficients of original variables in Y 9 as presented in table 3. Similar to the weights in Y 1 , the coefficients of P tot, f and R Subs are also large in Y 9 ; the coefficients of FrMOX f and BU UOX seem non-negligible as well. Actually, Y 9 is the last PC and explains the least of variability of original variable vector; Johnson and Wichern commented that [START_REF] Johnson | Applied Multivariate Statistical Analysis -sixth edition[END_REF] this last PC indicate an unnoticed linear dependency. As shown by the axis of PC, Y 9 can be approximately regarded as constant due to its limited variation in comparison to other PCs. The coefficients of original variables in Y 9 reveal therefore the linear dependency between P tot, f and R Subs . Again, the importance of P tot, f on R Subs is emphasized.

The same analysis is carried out with the output TRU tot (the PCs deduced from the analysis with TRU tot are different from the ones with R Subs , because the analyses are separate). The PC Y 1 shown in table 2 of PCA with TRU tot can nearly explain the total variance of TRU tot . The composition of Y 1 presented in table 3 shows important weights of P tot, f and FrMOX f in addition to TRU tot . As a result, TRU tot is sensitive to P tot, f and FrMOX f . Other variables have non-measurable impacts In conclusion, PCA and the correlation squares between the output and PCs help identify important input variables with regard to given outputs of interest in a simple but quantitative manner, and thus help understand the behaviour of system: the input variable P tot, f dominates the behaviour of two outputs of interest R Subs and TRU tot ; FrMOX f has measurable impact on TRU tot ; FrMOX f and BU UOX may have some effect on R Subs as well. Other parameters have non-measurable effects on two outputs of interest. This result should not be over-interpreted, because the importance of an input variable depends not only on the system and output of interest, but also on the assumptions of the design of experiment, such as the variation range. The non-measurable impact of an input on the output does not prove the independence between them; it only indicates that the contribution of this input within the given variation range is much smaller than the most impacting one.

iii. Static robustness and robust static optimum

Result of two outputs of valid strategies without Pu shortage for MOX fabrication is shown in figure 2, coloured by the value of P tot, f . Among them, the lowest TRU tot achieved in 2090 is about 848 tons, and the maximal R Subs is 1.68. The dispersion of points in figure 2 shows the complexity of the relation between two outputs of interest for a valid strategy. Actually, within a small variation of P tot, f (which means that the final level of nuclear capacity in France is well decided), two outputs show a strong correlation, plotted with the same colour. That is because the TRU in total cycle is principally composed of the in-cycle Pu inventory, which is proportional to R Subs if P tot, f is constant; when P tot, f varies, the slope of the linearity changes as well, according to the definition of R Subs . Trajectories with low TRU tot and high R Subs result from low P tot, f , whereas a high level of power will finally yield high inventory of TRU and a relatively low material availability for future substitution of PWRs with SFRs. The complexity of the relation between two outputs of interest indicates the contradiction of two objectives and the existence of the Pareto front, shown as stars in figure 2 (in consideration of objective A, their R Subs should be larger than 1). They are robust static strategies if two objectives are considered at once, and present the competition between the risk of failing objective A and the regret on the TRU inventory production regarding the disruption towards objective B. According to the criterion C B|A , the robust static optimum is shown as purple star in figure 2. The value of its inputs and outputs of interest are summarized in table 4. Particularly, it is also the global optimum for TRU minimization in the pre-disruption scenario, which means that the optimal strategy dedicated to TRU minimisation in the pre-disruption scenario can theoretically achieve objective A.

The robust static optimum identified by the criterion C B|A in the pre-disruption scenario leads to TRU tot (t = 2090) as low as 848 tons, but to a margin less than 1% for objective A (R Subs = 1.01). Although the objective finally adopted in 2090 is unknown, this precautionary strategy sets a bound of optimal performance for TRU minimization among valid static strategies that achieves theoretically objective A.

IV. Post-disruption: scenario of adaptation

The strategy represented as a red star in figure 2, called reference strategy in the subsequent analysis, maximizes R Subs among all possible static strategies. In conventional scenario study without consideration of disruption, the reference strategy can be chosen for the pre-selected objective A according to the criterion C A . Its inputs and outputs are shown in table 4. The reference strategy leads to 68% of margin of Pu availability for objective A (R Subs = 1.68), and more than 1100 tons of TRU is accumulated in the cycle by 2090. FrMOX f in this strategy is low so that Pu can be accumulated effectively, and the total thermal power evolves to a relatively low level so that R Subs can be maximised.

In the analysis of adaptive robustness, readjustment by new adaptive strategies for the new objective B is allowed in case of disruption. Since the reference strategy is optimal for objective A before disruption, the related trajectory is adapted in this work. In other words, the reference strategy is supposed to be implemented specifically for the objective A before any disruption of objective; in case of the disruption towards objective B, new strategies is explored in order to minimize the TRU inventories.

The minimization of TRU in this adaptation scenario problem is based on a given prior choice of strategy before disruption (the reference strategy in this work); the global minimum that minimizes TRU over any possible trajectory before disruption is not studied. A-priori, if a trajectory of a static strategy presents fewer than 1000 tons of TRU by 2090, it is highly possible that a TRU-reduction adaptive strategy that adapts it will lead to lower TRU inventories than that the trajectory of reference strategy is adapted. Nevertheless, for the first step of applying the assessment method, two prior choices studied that respect the objective(s) before disruption should have "large difference" (from the viewpoint of the consequences of interest, e.g. indicated by R Subs and TRU tot ), in order to highlight the difference of consequence. The reference strategy and the robust static optimum are two extreme cases of preference in the pre-disruption scenario. The assessment of adaptive robustness depending on these two strategies may give a relatively wide range of information on the difference between a precautionary strategy (represented by the robust static optimum) and post-disruption adaptation (represented by the readjustment on the trajectory of reference strategy). Hence, the trajectory of reference strategy will be adapted in the following adaptation scenario study, and the results will be compared with the ones of robust static optimum.

i. Design of experiment of post-disruption adaptation

In the scenario of post-disruption adaptation, the reference strategy is supposed to be implemented before disruption, and adaptation is allowed in case of disruption. To analyse the temporality of adaptive robustness, the adaptation time T ad is sampled, which must be later than the disruption. To simplify, year 2040, 2050, 2060 and 2070 are considered for T ad .

According to the objective B after disruption, the planning of SFR deployment is abandoned, and the minimization of TRU inventories is required. The new context different from the one with objective A implies different strategy space from the one of pre-disruption scenario. A new transition of adaptation is supposed after T ad , and the duration as short as 1 year is considered acceptable. It means that an abrupt change of the fuel cycle is possible in order to adapt for objective B. Similar to the pre-disruption scenario, burn-ups BU UOX and BU MOX , UOX SF cooling time TC UOX and UOX SF management for MOX fuel fabrication MPu will change after T ad ; total capacity and MOX fraction of the fleet will change linearly during the transition. To minimise TRU in a PWR UOX and mono-MOX fleet, the final total capacity P tot, f is supposed to be at a level between 0 and the one just before adaptation P tot (t = T ad ). It means that a phase-out strategy is possible for objective B. It is also supposed that the full MOX in PWRs can be realized (FrMOX f can be 100%). The new ranges of input variables in the adaptation scenario are summarized in table 5. ii. Assessment of adaptive robustness over adaptation time

For each T ad considered, 2000 adaptive strategies are sampled by LHS of the corresponding design of experiment (hence, 8000 trajectories have been simulated for this part of study). Some trajectories have Pu shortages for MOX fabrication and they are considered invalid. Figure 3a shows the approximative percentage of valid strategies in the total sampling with different T ad . It shows that earlier is the adaptation, fewer strategies are valid. It can be explained by the fact that the reference trajectory accumulates progressively a large inventory of Pu in the interim storage of UOX SF due to decreasing MOX fraction. Earlier is T ad , earlier this accumulation is stopped and lower is the Pu inventory presented in the storage. Furthermore, an earlier T ad is equivalent to a longer scenario of adaptation, because the output of interest TRU tot in the adaptation scenario is also evaluated in 2090. For a given nuclear fuel cycle characterised by the same parameters (total capacity, MOX fraction, burn-ups of fuels, etc) and a given stock of Pu, the risk of Pu shortage for MOX fabrication is evidently higher with a longer time horizon of scenario. However, the percentage of valid adaptive strategies does not indicate well the information about adaptive robustness. We recall that an adaptive strategy is considered robust if it leads to smaller TRU tot (t = 2090) than the one of the robust static optimum identified in pre-disruption scenario, being 848 tons in this study. As shown by approximative distributions of TRU tot (t = 2090) of valid trajectories with different T ad in figure 3b, a larger part of valid trajectories is adaptively robust when T ad is earlier. The evolution of TRU inventory resulting from valid adaptive strategies is presented in figure 4. Without any adaptation, the evolution of TRU tot of the (static) reference strategy is presented by the bold curve in red, and the one of the robust static optimum is presented by the bold curve in purple. The grey thin curves present the results of all valid static strategies in the pre-disruption scenario. The zone of light red curves shows the TRU tot evolution of valid adaptive strategies when T ad = 2040; by definition, the corresponding adaptive optimum is presented by the bold curve in brown. Similarly, the thin curves in light blue, light green and light magenta present the TRU tot evolution of adaptive strategies with respect to T ad of year 2050, 2060 and 2070; the adaptive optima are respectively the bold curve in deep blue, the curve in deep green and the one in deep magenta.

When T ad is early, such as in 2040, TRU tot (t = T ad ) is much lower than 848 tons as the "threshold", so that the margin to be robust for those adaptive strategies is relatively large. If the adaptation is late, like in 2060, the margin becomes small (or even negative if T ad = 2070, which means that the TRU tot should strictly decrease with adaptation in order to be robust). Figure 4 also shows that the TRU tot of reference trajectory will go over 848 tons around year 2063 if no adaptive strategy is implemented. Otherwise, some adaptive strategies before 2060 will lead to lower TRU tot (t = 2090) than 848 tonnes by 2090. Therefore, some adaptive strategies upon the reference trajectory can be robust if the objective is disrupted and the readjustments are taken before 2060. However, none of the adaptations in our scenario is robust if T ad = 2070: the adaptive optimum presented by the bold curve in magenta is always above the robust static optimum.

Value of inputs and the output TRU tot (t = 2090) of adaptive optima over different adaptation time T ad are presented in table 6. The respective evolution of TRU tot are presented by the bold curves in brown, deep blue, deep green and magenta in figure 4. All adaptive optima show an abrupt transition towards a fleet with very high MOX fraction and low installed capacity of fleet. It is coherent with the variable analysis that P tot, f and FrMOX f are the most impacting factors on TRU inventory in total cycle (even though it is not exactly the same design of experiment). Low level of nuclear capacity limits the net production of TRU; high MOX fraction allows to incinerate lots of Pu in UOX SF interim storage; and moreover, a short transition amplifies these effects.

Another remark can be observed from the evolutions of TRU tot of adaptive optima in figure 4: the later is T ad , the more negative is the slope of the TRU evolution of the adaptive optimum. Again, the accumulation of Pu in UOX SFs in the reference trajectory and different time length Last but not least, the year 2090 as observation time should be highlighted for the assessment of robustness. If the adaptive optimum with T ad = 2070 is extrapolated, we may expect that its TRU tot (t = end) can be lower than the one of the robust static optimum after a certain time (so that it can be adaptively robust). However, the threshold of being adaptively robust should be modified (as the time horizon of scenario should be also extended in this case). Moreover, the validity with regard to Pu shortage after 2090 of these two trajectories are not a-priori assured. Analysis with other time of interest requires actually a new study and new simulations. Hence, the analysis of robustness in this work depends on the assumptions and constraints of this scenario study, including the method of robustness evaluation, time horizon of scenarios studied, etc. It is dangerous to extrapolate any result on the domain out of the scope of study.

V. Conclusion and discussion

i. Conclusion on the example

In this work we develop a method of strategy robustness assessment vis-à-vis the deep uncertainty of disruption of objectives based on the nuclear fuel cycle in France. The pre-selected objective is to substitute the PWR-based French fleet with SFRs by 2090, and the one linked to disruption is to minimize TRU inventories in the total cycle without SFR deployment in the future. The disruption of objective before 2090 is so uncertain that the choice of strategy for nuclear development is problematic. Starting from the principle of immediate precaution which assures acceptable outcomes for possible objectives, a conservative strategy, which considers two objectives at once, is identified. By applying this strategy, 848 tonnes of TRU will be accumulated in 2090 in total cycle, and less than 1% of margin of material availability will be kept to achieve the pre-selected plan of SFR deployment. Since it leads to the lowest TRU inventory among all possible static strategies with R Subs > 1, it is identified as the robust static optimum in the pre-disruption scenario. Another approach suggests post-disruption adaptation, according to the updating information on the uncertainty of objective. In this approach, an optimal strategy dedicated to the pre-selected objective about SFR deployment is implemented before disruption. If without adaptation, it leads to 68% of margin on the Pu availability for SFR deployment by 2090. In case of the disruption, adaptive strategies are explored based on this trajectory. It is shown that some adaptive strategies before 2060 lead to lower TRU inventories in the cycle by 2090 than the TRU inventory of robust static optimum. However, if it is not adapted until 2070, all valid adaptive strategies lead to higher TRU inventories by 2090 than the robust static optimum. Hence, subject to the uncertain disruption of objective analysed in this work, the time limit for adaptive robustness related to the reference trajectory is around the year 2060. Whether to make immediate decision in a conservative manner as the robust static optimum, or to pursue the objective currently determined with a preparation plan for the future adaptation, depends on the estimation of disruption and the relevant temporality. Is year 2060 the time when stakeholders can determine their objective for year 2090? It is an open question for different audiences regarding the information updating in the future.

ii. Conclusion on the method of robustness assessment

This work develops a method of strategy robustness analysis applied in dynamic nuclear fuel cycle study. In the analysis of static robustness, the robust static strategy is identified in the predisruption scenario, which can theoretically achieve the pre-selected objective, and it is optimised for another objective linked to the disruption. It is called "static" because the results of relevant trajectory are evaluated without adaptation. In terms of adaptive robustness, the outcomes of adaptive strategies is analyzed based on the trajectory initially pursuing the pre-selected objective. As the disruption is deeply uncertain, robust adaptive strategies are analyzed over different adaptation time. The time limit for adaptive robustness is deduced by comparing two approaches. Robust strategies suggested by two approaches can animate the discussion among stakeholders over their estimation of disruption.

As it is shown in [START_REF] Mcphail | Robustness metrics: How are they calculated, when should they be used and why do they give different results?[END_REF], a single strict mathematical definition of robustness and relevant metrics does not exist, and therefore the results of robustness assessment can be strongly dependent on the specific definition of robustness, the assumptions of scenario study, the context of problem and the interests of stakeholders. The criteria of robustness are adapted for the scenario study of nuclear fuel cycle in this work; it is also possible to apply other definitions to different problems. Nonetheless, the comparison of analyses with diverse robustness methods is out of our study category.

Moreover, the results of robustness assessment in this work is directly linked to the reference strategy and the robust static optimum identified in the pre-disruption scenario. These two strategies are studied in this work because they are two extreme cases in the Pareto front, which helps develop the methodology and highlight the impact of uncertain disruption. But depending on the evaluation upon the disruption and the interest of stakeholders, other strategies, e.g. other strategies on the Pareto front in figure 2, can be chosen as reference strategy or robust static optimum, which indicate different hypotheses of scenario study. In this case, other trajectories can be adapted and the threshold deduced from the robust static optimum with regard to adaptive robustness may vary. The relevant results are a-priori different from the ones in this work. Again, the results of robustness assessment in this work are not absolute for decision-making; instead, with different assumptions, the method help stakeholders discuss and manage their strategies involving disruption of objectives.

Besides the specific assumptions made in this scenario study, one may have different interpretations of static and adaptive robustness. The robust static optimum in this work can be simply regarded as a role of threshold, which indicates the most conservative performance among all possible strategies subject to the deep uncertainty of disruption. In this respect, the adaptive robustness indicates the possibility of post-disruption adaptations with better performance than any historical choice made before disruption. It can be also interpreted as the optimal strategy under the assumption of staticity, which means that the post-disruption adaptation is not allowed, possibly due to technological and economic concerns. Staticity or autorization of adaptation depends on the evaluation of future industrial and political decisions, which are different assumptions of scenario study. From this angle, these two relevant approaches can be interpreted as the robustness assessment associated with different assumptions of staticity.

iii. Perspectives

As it is mentioned, the nature of total capacity of fleet seems different from other input variables of strategy. The change of total capacity of fleet may involve many stakeholders, and it is not obvious that it can be decided as easily as others. To be more conservative, several similar analysis with different assumptions of total capacity evolution should be carried out.

This kind of problem can be generalized to other variables as well, because the difficulty of modifying the parameters of fleet depends on the agreement and the interests of stakeholders. Some loss of performance is possibly more acceptable than a huge modification of a special parameter of fleet. This kind of study involves the preference on the gain of performance as well as the inertia or the difficulty of modification, which in practice, constrains the exploration space of strategy in the adaptation scenario. For instance, we can suppose that the constraints of modifying operation parameters for adaptation in reality are much stricter than the ones assumed in this work:

• Ct(P): Nuclear industry does not accept the phase-out strategy; the lowest acceptable level of total capacity is the half of P tot (t = T ad ). The constraint becomes P tot, f ≥ 1 2 P tot (t = T ad ) • Ct(D): Due to the demand of stability of electricity supply, the new transition should be more than 10 years. The constraint becomes D ≥ 10.

• Ct(PMOX): Even though the full MOX core is considered possible after disruption, the reprocessing and fabrication capacity is not infinite. Since the MOX fuel demand is proportional to the capacity contributed by MOX (for the first degree of approximation), one may suppose that the final capacity contributed by MOX cannot exceed the double of current level (it is a choice depending on industrial and sociological estimation; here it serves as an example).

The constraint may be P tot, f × FrMOX f ≤ 188 × 10% × 2 = 37.6 GWth. • Ct(TC): Due to the high requirement for nuclear safety, it is possible that all UOX SF should be cooled at least 5 years in the future. The constraint may become TC UOX ≥ 5.

Ct denotes the different constraints from the adaptation scenario in our work. To see the impacts of these constraints on the adaptation, we can accumulate them in the order of Ct(P), Ct(D), Ct(PMOX) and Ct(TC) and check the lowest TRU tot (t = 2090) achieved in our sampling. Without launching new calculations and without losing generality, we take the valid trajectories in the adaptation scenario with T ad = 2070 as example. By accumulating constraints in the order aforementioned, the lowest TRU tot achieved by adapting the reference trajectory are presented in table 7. Since new constraints are stricter than the ones in our work, the lowest TRU tot achieved becomes higher than 894 tons that is identified with original constraints. The richness and the degree of freedom of strategies characterized by the strategy space, as discussed in [START_REF] Lempert | Managing the risk of uncertain threshold responses: comparison of robust, optimum, and precautionary approaches[END_REF], affect evidently the space of result and therefore the conclusion of a scenario study. However, the sensitivity of output (in this example it is the lowest TRU tot ) on different constraints is still not obvious. The preference of stakeholders, the role of constraints and the normalization of gain should be investigated in order to answer these questions. But on one hand, each sub-scenario of this analysis may require large quantity of calculations (problems of optimisation); on the other hand, the determination of constraints involve further research in the field of economy and sociology, which are completely out of the scope of this work.

Another problem related to constraints may be involving some new variables and technology that are not considered in the pre-disruption scenario. In this work, reactors and fuels of the French fleet are still PWR UOX and PWR MOX after the disruption. In consideration of TRU minimization, is it possible to employ Pu multi-recycling fuels in PWRs such as MIX or CORAIL [START_REF]Stratégie Française pour l'Energie et le Climat : Programmation Pluriannuelle de l'Energie 2019-2023 2024-2028[END_REF]? The gain of performance by employing new technology can be studied in the future.

  (a) Scatter plots of PCs vs R Subs in the PCA with X out = R Subs (b) Scatter plots of PCs vs TRU tot in the PCA with X out = TRU tot

Figure 1 :

 1 Figure 1: Scatter plots of PCs vs X out in two PCA

Figure 2 :

 2 Figure 2: T RU tot versus R Subs of valid trajectories in pre-disruption scenario, coloured by P tot, f

Figure 3 :

 3 Figure 3: Global analysis of post-disruption scenario of adaptation

Figure 4 :

 4 Figure 4: Evolutions of T RU tot of valid adaptations from the referential trajectory

Table 7 :

 7 Effects of accumulation of stricter constraints than the original study of adaptation in a given order on the lowest achievable TRU tot , when T ad = 2070

Table 1 :

 1 Initial values and variation ranges of variables of the pre-disruption scenario

	Var. T start	D	P tot,

f FrMOX f BU UOX BU MOX TC UOX MPu

	Init.	-	-	188	10%	45.3	45	5	LiFo
	Min. 2030	10	94	0%	30	30	3	1/LiFo
	Max. 2050 2090-T start	272	30%	60	60	10	2/FiFo
	Unit	-	year	GWth	-	GWd/t GWd/t	year	-

Table 3 :

 3 Coefficients of original variables in the component Y 1 and Y 9 in two PCA

	Coef. of	T start	D	P tot,

f FrMOX f BU UOX BU MOX TC UOX MPu X out

	in Y 1 , X out = R Subs	-.02	-.05	.68	-.24	.17	.04	.03	.00	-.67
	in Y 9 , X out = R Subs	.00	.01	-.69	-.19	-.17	-.01	-.01	-.02	-.68
	in Y 1 , X out = T RU tot	.07	.08	-.60	.41	.09	.06	-.05	-.01	-.67
	in Y 9 , X out = T RU tot	.02	.01	-.59	.25	.25	.03	.00	.00	.73

Table 4 :

 4 Variables and outputs in 2090 of the robust static optimum and the reference strategy in pre-disruption scenario

	Traject./Strategy T start	D	P tot, f	FrMOX f BU

UOX BU MOX TC UOX MPu R Subs T RU tot

	Rob. stat. op.	2031.0 14.2 109.4	13.6%	52.5	56.2	8.2	FiFo	1.01	848
	Ref.	2032.7 56.3	94.6	1.9%	32.4	41.9	3.6	LiFo	1.68	1116
	(Unit)	-	year GWth	-	GWd/t GWd/t	year	-	-	tonne

Table 5 :

 5 Variation ranges of variables in the adaptation scenario

	Var.	T ad	D	P tot, f	FrMOX f BU UOX BU MOX TC UOX	MPu
	Min. 2040	1	0	0%	30	30	3	1/LiFo
	Max. 2070 2090-T ad P tot (t = T ad )	100%	60	60	10	2/FiFo
	Unit	-	year	GWth	-	GWd/t GWd/t	year	-

Table 6 :

 6 Variables and the outputs TRU tot (t = 2090) of adaptive optima over different T ad in the adaptation scenario

T ad of the Op. D P tot, f FrMOX f BU UOX BU MOX TC UOX MPu T RU tot

  due to T ad should be emphasized: the accumulation of Pu in the storage of UOX SF in the reference trajectory allows to feed higher and higher MOX fraction if needed; later T ad yields a larger Pu inventory in UOX SFs as well as a shorter time length of adaptation scenario, so that such a higher incineration rate of Pu by MOX can be realized. Nevertheless, such post-disruption incineration is always less efficient than the previous accumulation. Adaptive robustness can be no longer achieved if readjustment is too late.

	2040	1.4	8.0	83.2%	54.8	49.0	7.4	LiFo	599
	2050	2.5	20.2	80.0%	46.1	44.7	7.7	FiFo	724
	2060	3.3	26.9	90.2%	39.5	37.7	3.5	FiFo	818
	2070	2.9	37.3	98.3%	41.1	58.8	4.1	LiFo	894
	(Unit)	year GWth	-	GWd/t GWd/t	year	-	tonne
	of adaptation scenario

Acknowledgement

The authors acknowledge the French Programme NEEDS for its financial support to the project.