Yann Algan
email: ayann.algan@sciencespo.fr

Elizabeth Beasley
email: belizabeth.beasley@cepremap.org

Florian Guyot
email: cflorian.guyot@sciencespo.fr

Kazuhito Higad

Fabrice Murtin
email: efabrice.murtin@oecd.org

Claudia Senik
email: fsenik@pse.ens.fr

Kazuhito Higa
email: dkazuhito.higa@oecd.fr

Big Data Measures of Well-Being: Evidence from a Google Well-Being Index in the US

Keywords: Subjective Well-Being, Big Data, Bayesian statistics

come

Introduction

There is a growing interest in social sciences in going beyond the income-based approach of human development by using new measures of wellbeing [START_REF] Stiglitz | Rapport de la Commission sur la mesure des performances économiques et du progrès social[END_REF]. In particular, GDP does not measure non-market social interactions, such as friendship, family happiness, moral values or the sense of purpose in life. This motivates the recourse to subjective self-reported measures of wellbeing, such as Life Satisfaction (i.e. answers to the question: "All things considered, how satisfied are you with your life as a whole those days?"), which economists increasingly use as a direct measure of utility. 1 Political leaders have embraced this move by calling for representative surveys of well-being to guide their policy, as illustrated by the Cameron's commission of well-being in UK. In spite of these achievements, subjective well-being measures still raise a number of challenges and concerns among economists both in measurement and interpretation [START_REF] Kahneman | High income improves evaluation of life but not emotional well-being[END_REF][START_REF] Deaton | The Financial crisis and the well-being of Americans[END_REF][START_REF] Deaton | Do context effects limit the usefulness of self-reported well-being measures?[END_REF]Krueger and Stone, 2014); and alternative approaches such as the Day Reconstruction Method (Kahneman et al., 2004, Stone andMackie, 2014) and Time Use Surveys (Krueger et al., 2015) have been developed to address some of these issues.

This paper contributes to this new research agenda by showing how Big Data can improve our understanding of the foundations of wellbeing. A major consequence of the accelerated digitization of social life is the traceability of attitudes, feelings, and social relations, embedded in large datascapes, such as Google, Facebook, Twitters or the Blogosphere. The quantification of those social traces should be of considerable interest for social scientists. However, as stated by Lazer et al. (1999), while the capacity to collect and analyze massive amount of data has transformed the fields of physics and biology, such attempts have been much slower in social sciences. This paper illustrates the potential use of Big Data for both the measure and the analysis of human wellbeing.

The first advantage of these data is that they offer social scientists the possibility to observe people's behavior, such as searches on Google, to make inferences about attitudes and feelings, rather than statements about attitudes and feelings. Second, rather than relying on the answers to pre-defined questions, social scientists can listen to what people say. This approach of revealed preferences unveils a reflexive picture of society because it allows the main concerns of citizens and the priority ranking of those concerns, to emerge spontaneously, and it complements as such the information captured by GDP. In a nutshell, these data are based on the actual behavior of people when they search for information and they endogenously elicit the relative importance of people's concerns.

The second main advantage of these data is their timeliness, as they offer an immediate source of information for policy-makers, who are often confronted with short-term horizons and data scarcity in the midst of the decision-making process. Moreover, they are available at a local level -as long as internet penetration and use is sufficient to obtain statistical representativeness. Finally, Big Data are often made available for free.

However, the volume of Big Data to be treated is potentially enormous and it is a statistical challenge to disentangle signal from noise and to identify the relevant piece of information while avoiding cherry picking. We identify several issues with the dataset that we use (Google Search volumes) and propose several solutions, in particular the construction of categories reflecting different dimensions of life as well as the use of Bayesian techniques to select the most robust determinants of subjective wellbeing. Our methodology allows us to construct a model that has four important qualities: it is grounded in theory and the existing literature on well-being, it is testable and has strong out of sample performance, it is reasonably transparent, and it is adaptable and can potentially be used to predict well-being on a continuous and recurrent basis. This paper builds on the growing literature that seeks to exploit search engine data. The early contribution of [START_REF] Ettredge | Using Web-based Search Data to Predict Macroeconomic Statistics[END_REF] used internet search data to forecast the unemployment rate in the US. The same idea was explored by [START_REF] Askitas | Google Econometrics and Unemployment Forecasting[END_REF]Zimmermann (2010), D'Amuri and[START_REF] D'amuri | Google it! Forecasting the US Unemployment Rate with a Google Job Search Index[END_REF] and [START_REF] Suhoy | Query Indices and a 2008 Downturn: Israeli Data[END_REF], while [START_REF] Baker | The Impact of Unemployment Insurance on Job Search: Evidence from Google Search Data[END_REF] use a measure of job search based on Google search data to study the effects of unemployment insurance and job finding. Choi and[START_REF] Choi | Predicting Initial Claims for Unemployment Insurance Using Google Trends[END_REF] have explained how to use search engine data for forecasting macroeconomic indicators of unemployment, automobile demand, and vacation destinations, while several papers have analyzed consumer sentiment [START_REF] Radinsky | Predicting the News of Tomorrow Using Patterns in Web Search Queries[END_REF][START_REF] Penna | Constructing Consumer Sentiment Index for U.S. Using Google Searches[END_REF][START_REF] Preis | Complex Dynamics of Our Economic Life on Different Scales: Insights from Search Engine Query Data[END_REF]. Regarding subjective well-being, Stephens-Davidowitz (2013) used Google data to study trends of depression. [START_REF] Schwartz | Characterizing Geographic Variation in Well-Being using Tweets[END_REF] used tweets and found that words related to outdoor activities, spiritual meaning, exercise and good jobs correlate with increased life satisfaction (controlling for socioeconomic variables). To the best of our knowledge, our paper is the first to use search data to identify the nature of life dimensions that best predict subjective well-being. This paper demonstrates the capacity of search engine data to track and replicate the trends in subjective affects that are traditionally captured by surveys and to elicit and identify the type of activities that predict subjective well-being. We construct robust predictors of subjective well-being measures in the United States using a very large amount of search engine data covering the years 2008-2013. We measure the life dimensions whose search intensity is robustly associated with selfreported wellbeing collected by the Gallup Healthways Wellbeing survey, such as life evaluation (Cantril ladder) or the percentage of people who declare that they have experienced happiness, stress or worry "during a lot of the day yesterday". With our composite categories, we predict a time-series that tightly fits the Gallup survey trends in subjective wellbeing, and also behaves nicely out of sample. This method allows identifying the type of behaviors, activities and experiences that are associated with higher or lower wellbeing. In practice, we run a simple variance decomposition to quantify the contributions of each dimension to predict well-being. For all subjective wellbeing variables, material conditions are the most important family of predictors, followed by social factors and health/wellness categories. At the category level, we find that keywords related to job search, financial security, family life and leisure are the most important predictors of subjective wellbeing.

Results

A Model of SWB in the United States

To facilitate estimation, interpretation, and the robustness of the model, Google search terms were grouped into twelve domains that can be organized into three aspects of life: Material Conditions (Job Search, Job Market, Financial Security and Home Finance), Social (Family Stress, Family Time, Civic Engagement and Personal Security), and Health and Wellness (Healthy Habits, Health Conditions, Summer Activities and Education and Ideals) 2 . The composition of each category is described in the Supporting Information. Note that the search terms in Job Market and Job Search do not group together, and the types of words in each category give some intuition as to why: Job Search seems to be related to searching for a job (any job) from unemployment, while Job Market seems to be related to job quality, which might reflect searching in a looser job market. The lowest Cronbach's alpha3 (for Healthy Habits) is 0.84, which is still reassuringly high. A commonly accepted rule of thumb sets 0.7 as a threshold for an acceptable degree of internal consistency [START_REF] Nunnally | Assessment of Reliability[END_REF][START_REF] George | SPSS for Windows step by step: A simple guide and reference[END_REF]. Table 1 shows the composite categories and their components.

( TABLE 1 ABOUT HERE)

Table 2 presents the results from the model for each SWB variable4 . The coefficients are generally consistent for positive and negative affects. Categories of search terms that are consistently associated with higher wellbeing (excluding the Learn and Respect SWB variables, for which our model does not perform well, as discussed below) are Job Market, Civic Engagement, Healthy Habits, Summer Leisure, and Education and Ideals. Categories that are consistently associated with lower wellbeing are Job Search, Financial Security, Health Conditions, and Family Stress.

( TABLE 2 ABOUT HERE)
Overall, these findings are in line with the literature on subjective wellbeing. The consistent negative relationship of Job Search (which seems to relate to searching for a job from unemployment) confirms the importance of employment as a foundation of subjective wellbeing: having a job is one of the strongest correlates of life satisfaction and happiness, while, conversely, being unemployed is highly detrimental to life satisfaction, notwithstanding the loss of income that this entails (see, for example, Clark and [START_REF] Oswald | Unhappiness and Unemployment[END_REF], and is most difficult to adapt to [START_REF] Diener | Lags and Leads in Life Satisfaction: A Test of the Baseline Hypothesis[END_REF]. There is evidence that objective health shocks (such as heart attacks) are negatively correlated with SWB (Shields and Wheatley Price, 2005), though this relationship is more generally difficult to disentangle with confidence as reported health status may be a proxy of subjective wellbeing and as such are highly correlated (see [START_REF] Deaton | Income, Health and Well-Being around the World: Evidence from the Gallup World Poll[END_REF], for instance). In addition, there is evidence that higher levels of well-being themselves lead to better health outcomes [START_REF] Howell | Health benefits: Meta-analytically determining the impact of well-being on objective health outcomes[END_REF].

Civic Engagement is related to the importance of social capital, which has been amply demonstrated to be strongly associated with subjective wellbeing (see for instance Helliwell andWang 2011 or Helliwell et al. 2010). Healthy Habits, notably physical exercise, are associated with less depression and anxiety and improved mood [START_REF] Biddle | Physically active lifestyles and well-being, The science of wellbeing[END_REF]. Education and Ideals is a grouping of words that includes references to religion, political goals, philosophy, and education, and which is associated with higher SWB, consistent with the literature that finds higher well-being associated with religious activity (Clark and Lelkes, 2005, Helliwell 2003). Finally, family, health and security are identified, through choices, as extremely important in terms of people's happiness by [START_REF] Benjamin | Beyond Happiness and Satisfaction: Toward Well-Being Indices Based on Stated Preference[END_REF] and these categories coincide with the "satisfaction domains" that have been explored by the Leyden school (van Praag and Ferrer-i-Carbonell, 2008). Many of these patterns of subjective wellbeing are summarized in The World Happiness Report (2013,2015).

Two categories have inconsistent signs. Family Life is associated with more Happiness and Laughter, and less Sadness, but also with more Anger and Stress. The finding of inconsistent associations of Family Life may reflect the complicated nature of interactions with children, consistent with the finding in Deaton and Stone (2014) that parents experience both more daily joy and more daily stress than non-parents. See also [START_REF] Buddelmeyer | The Stress Cost of Children[END_REF]. Personal Security is positively associated with Life Evaluation (today), but negatively associated with Life Evaluation in 5 years and Laughter, and positively associated with Sadness. We cannot explain this difference (note that generally, living in a high crime area is associated with lower SWB as argued by [START_REF] Lelkes | Knowing what is good for you: Empirical analysis of personal preferences and the objective good[END_REF].

Variance Decomposition

Simple variance decomposition as described can help to quantify the contributions of each covariate to the explanatory power of the model (see Supporting Information for details). Table 3 reports the results. Overall, it appears that, for all subjective wellbeing variables except stress, material conditions are the most important family of predictors, followed by social factors and health/wellness categories. At the category level, the most important variables are job search, financial security, summer leisure and family life. Regarding the stress variable, it appears to be mostly explained by family life, summer leisure and healthy habits.

(TABLE 3 ABOUT HERE)

Predicting SWB in the United States and Reliability of the Model

The model is quite reliable in out of sample tests. Table 4 displays the correlations between the predicted values and the actual values of SWB variables for both the training and the test sub-periods, while Figure 1 depicts the predicted and observed SWB variables. During the training sub-period (that is, the dataset with which the model was estimated) correlations between the predicted and actual series are generally high, ranging from 0.87 for Anger to 0.97 for Learn and Stress. A major finding of this paper is that the correlations remain high in out-of-sample testing periods for most subjective well-being variables. Out of 20 tests, 16 yield correlations over 0.60, 13 over 0.70, and 11 over 0.80. Two of the ten affects stand out as being particularly difficult to predict (out of sample): Learn (0.47 in 2008 and 0.55 in 2013) and Respect (-0.58 in 2008 but a respectable 0.80 in 2013). One possible reason for this is that these affects are not well defined or understood.

(TABLE 4 AND FIGURE 1 ABOUT HERE)

Note that a high correlation does not necessarily imply an accurate prediction, since the correlation measures the degree to which the two data move together, rather than whether they are equal. For example, the 2008 test period has a very high correlation for Life evaluation, but visual inspection of the graph shows that while the series move together, the predicted series is much higher. In the case of Life evaluation this may be due to the change in the ordering of questions in the Gallup survey that took place at about this time, and is thought to depress the overall Life evaluation measure, as reported by Deaton (2011). The objective of this exercise is to obtain a combination of keywords that is able to predict the evolution -and not the level-of subjective wellbeing, both cognitive and emotional. With respect to this objective, it seems that such an exercise is not out of reach, at least over a 12month period. Big Data creates a new avenue for constructing a high-frequency indicator of wellbeing beyond GDP

Discussion

This paper proposes an original methodology to construct robust predictors of subjective well-being variables in the United States from a very large amount of search engine data covering the 2008-2014 period. Our framework is simple and transparent, has strong validity, is grounded in theory and the existing literature on subjective well-being, and allows for continuous updating of the predicted series. In particular, we make use of a sensible statistical arsenal to filter out the relevant information among a large number of noisy measures, which is often an important concern when working with Big Data. As a result, we find that keywords related to job search, financial security, family life and summer leisure are the most important predictors of subjective wellbeing. Moreover, the model successfully predicts the out-of-sample evolution of most subjective wellbeing measures at a one-year horizon. Regarding future research, this paper presents an original methodology that lays the groundwork to construct well-being indices at the local level (state or metropolitan area), which might then be used to measure the impact of local shocks or policy reforms on well-being in the United States.

Materials and Methods

Data

The SWB data is taken from Gallup Analytics, which is a daily telephone survey of at least 500 Americans aged 18 and older. More than 175,000 respondents are interviewed each year, and over 2 million interviews have been conducted to date since the start of the survey in 2008. The survey includes 6 measures of self-reported positive emotions (happiness, learn, life evaluation today and in 5 years, laugh, being respected) as well as 4 measures of negative emotions (anger, sadness, stress, worry). The time span covers 300 weeks from January 6, 2008 to January 4, 2014. Additional detail is given in Supporting Information

The 'Big Data' in this paper is the search frequency of keywords on Google, which are available from Google Trends. The data on search volume present many challenges for empirical estimation which are discussed in detail in Supporting Information. The initial list of keywords is selected as follows. We extract two long lists of keywords potentially linked to SWB outcomes. The first one comes from the Better Life Index Online Database, which records answers from data users to the question "What does a Better Life mean to you?" The second one is based on the American Time Use Survey that records the daily activities undertaken by US citizens as well as the positive or negative emotions that are associated with these episodes. This selection method allows us to avoid a cherry picking of a limited set of search queries on Google. On the other hand, survey-based keywords may be disconnected from the day-to-day life of Americans if they do not include their usual internet queries or do not reflect their practical living conditions. As a consequence, we have added a set of keywords that were likely to be relevant to different life experiences related to subjective well-being: this ranges from job concerns (e.g. the website 'getajob'), poverty ('coupons') or family stress ('women shelter'). In total, the initial database contains 845 keywords, split into 144 for the BLI, 95 for the US time-use data and 606 from our own judgment.

Empirical Framework

Our empirical goal is to build a model that accurately predicts the evolution subjective well-being using a type of revealed information, that is, searches on Google. We first aim to avoid overfitting of the dependent variable. For instance, using all of the available Google keywords and month dummies would result in a model with extremely high explanatory power and a R-squared statistics near 1, but a very low predictive power, as the model would essentially fit random variations in the sample rather than actual relationships. Conversely, using too few Google keywords creates a risk of underfitting of the dependent variable, which also would yield poor predictions, given that it is likely that all Google keyword variables are noisy signals of an underlying latent good predictor of SWB. This dual problem is pervasive in the world of 'Big Data', which is often characterized by the availability of a lot of information (i.e., in our setting, a large number of potential explanatory variables) and a lot of noise (each variable being a poor predictor of the dependent variable).

To cope with the first problem of high dimensionality, Bayesian Model Averaging (BMA) is commonly viewed as a powerful means of selecting the most robust determinants (e.g. Sala-i-Martin et al., 1997;Fernandez et al., 2001). Formally, let X denote the set of all possible categories and 𝑋 ! ∈ 𝑋 a given subset. We use the BMS package for R [START_REF] Zeugner | Bayesian Model Averaging Employing Fixed and Flexible Priors: The BMS Package for R[END_REF], where the approach simply consists of evaluating a very large number of models 𝑀 ! of the following form:

𝑦 = 𝛼 ! + 𝛽 ! 𝑋 ! + 𝜀 𝜀 ~ 𝑁 0, 𝜎 ! 𝐼
with y being the SWB outcome, 𝛼 ! a constant, 𝛽 ! the coefficients and 𝜀 a normally distributed error term of variance 𝜎 ! . Asa matter of fact, most of SWB dependent variables surveyed by Gallup are approximately normally distributed. Evaluating the model 𝑀 ! with regressors 𝑋 ! can be done through the posterior model probability (PMP) which is from Bayes' rule:

𝑃𝑀𝑃 ! = 𝑝 𝑀 ! 𝑦, 𝑋 = 𝑝 𝑦 𝑀 ! , 𝑋 𝑝 𝑀 ! 𝑝 𝑦 𝑋
The quantity 𝑝 𝑦 𝑋 is independent of the model considered and can be viewed as a constant. As a result, the PMP can be calculated as the model prior 𝑝 𝑀 ! times the conditional marginal likelihood 𝑝 𝑦 𝑀 ! , 𝑋 . In a second step, these PMPs are used to infer the Posterior Inclusion Probability (PIP) of each category variable, which is simply the sum of the PMPs of all models in which a given category variable is included5 .

However, BMA analysis can be distorted when there are many redundant or highly correlated covariates in the sense that the detection of robust and distinct predictors gets diluted away from covariates that are not highly correlated with other predictors. This well-known problem is related to the independence of irrelevant alternatives problem in discrete choice models (see George, 1999, Brock and Durlauf, 2001, and Durlauf, Kourtellos and Tan, 2012), and it can be addressed by using a preliminary factor analysis in order to reduce the number of potential explanatory variables. In practice, we combine individual search terms into composite categories that are used as predictors of SWB, which has the added advantage of limiting the noise due to any individual variable. This approach opens up the possibility of continuous and ongoing prediction of subjective well-being, as it will allow us to remove any search term that may become unusable in the future due to internet 'cascades' or cultural change, without greatly altering the significance of its category as a predictor of SWB. In addition, constructing categories offers more visibility on the nature of correlates of SWB variables, and allows disentangling the aspects of life (e.g. housing, employment, health, leisure…) that correlate most with different types of SWB variables, such as short-run emotional affects (e.g. feelings of happiness, stress and worry) and cognitive variables such as life evaluation.

The grouping of words into categories must be coherent both logically and statistically. The words grouped together must meet a common sense test, and they must also pass a statistical test, which implies first conducting factor analysis (using only the training data) and then calculating the Cronbach alpha, which measures the cross-correlation of the components and is an estimate of the internal consistency and reliability of the constructed category. As many keywords exhibit seasonality (as discussed above), and different keywords may exhibit similar patterns of seasonality without sharing the same meaning, we used the residuals of a regression over month and week dummies (to remove seasonal effects) in order to test the coherence of the word grouping. However, we used the raw data (without the removal of season variations) in order to construct the categories. Search terms were excluded if the factor loading was negative or less than 0.3, and many search terms were not used because they did not fit consistently with any category grouping. We use 215 keywords of the 554 words available after cleaning. We only used search terms with a positive factor loading.

Categories are constructed on the basis of a simple average of the z-scores. This is to avoid the structure of a category from depending on the inclusion of a single word, and to facilitate future construction and revision of the categories, in case one of the components needs to be dropped due to an unexpected peak. Using an estimate of a latent variable calculated from the factor loadings produces substantially similar results. Further detail on the categories is given in Supporting Information, and here we note only that many of the constructed categories match trends observed in administrative data linked to the underlying construct (though the frequency of the administrative data makes a detailed empirical analysis of this relationship difficult). For example, the Financial Stress category matches the yearly trend of Chapter 11 filings in the United States fairly well, and Family Stress is decreasing over the period along with reports of Intimate Crime as provide by the FBI.

Observing the trends of the composite categories and their relationship to social trends and other data series gives us confidence that our categories are not random associations, but reflect actual trends and social phenomena which correspond to our interpretation of the categories.

Regarding

Sadness

Big Data Measures of Well--being: Evidence from a Google Well--Being Index in the United States

SUPPORTING INFORMATION

Data on Subjective Well-Being

Figure S1 depicts the ten indicators over the period. The consequences of the Great Recession are visible on most SWB indices: life evaluation today and in 5 years, happiness and laugh have dropped significantly in 2008-2009, while the percentages of people experiencing worry, anger, stress and sadness have increased at the same time. A second observation concerns the cyclicality of these variables, which all display large seasonal swings.

( FIGURE S1 ABOUT HERE)

Data challenges with search volumes

The search volumes obtained from Google Trends pose several challenges for estimation. The Google Trends data on search volume is not the raw search volume; rather it is the proportion of total searches over a given period that included that keyword, normalized so that the highest volume over the period is equal to 100. This has several consequences: first, the value of the series obtained directly from Google Trends is difficult to interpret, as it depends not only on the volume of searches for a given word but also on the volume of other searches. Second, the value of the series on any given day cannot be compared between terms, since they are normalized to the maximum value by term. To deal with this issue, we normalize all search volumes so that they have a mean of zero and a standard deviation of one, since we are interested in how volume changes within a given term (rather than which terms have the highest search volumes overall).

Spikes

There may be sharp spikes in the popularity of a word. While some of these spikes are surely related to the degree to which the concept represented by this word is important in people's lives, others are less directly related. The example of the spike in "divorce" searches induced by the divorce of Kim Kardashian (an American celebrity) from Kris Humphries in October 2010 is shown in Figure S2. This is a concern for estimation as it creates a risk of over-fitting: if a sufficient number of search terms have a sufficient number of spikes, one could predict almost any series perfectly (though with poor out of sample performance). We address this by smoothing the data using a simple three period moving average and by creating composite category indicators to dampen the importance of a shock in any individual keyword.

(FIGURE S2 ABOUT HERE)

Cliffs

Other search terms show "cliffs", where volume is at or near zero for some substantial period (see Figure S3), and it is difficult to know whether it is because volume was zero or because there is an issue with the way the Google trends data is compiled. These cliffs pose an issue similar to that of the spikes, especially since words have cliffs at different points (that is, it is not a uniform discontinuity). However, we do not wish to exclude all zeros, because some zeros reflect zero volume. To address this issue, we dropped any search term with more than five zeros during the period (changing the number of allowable zeros does not substantially change the results). This results in a loss of information, as we have to exclude many terms that are potentially salient and important (such as mace spray).

(FIGURE S3 ABOUT HERE)

The January 2011 Discontinuity

We observed an unexplained discontinuity in many series from the last week of December 2010, to the first week of January 2011. An example for the word "pregnancy" is provided in Figure S4. We believe this discontinuity to be related to the change in the algorithm used by Google to localize the searches in January 2011. To adjust for this discontinuity, we calculate the average index in December and January for the unaffected years, we take the average change during the unaffected years, subtract this unaffected average change from the observed change from December 2010 to January 2011, and adjust all data from 2011 onwards using this difference. That is, we assume that the change from December 2010 to January 2011 should be the same as in the other years, and we adjust accordingly. While we are undoubtedly losing some information with this adjustment, there should not be any bias introduced.

Time Trends

Many of the search terms have a strong time trend. The example given in Figure S5 is "teeth hurt", where the time trend from 2008 to 2014 explains 89% of the variance in frequency. The consistent relative increase in the search volume of "pain" may be due to at least two possibilities: people are feeling more pain, or people are feeling the same amount of pain but are turning towards the internet for medical care as a general cultural shift. We would like to capture the first, but we have no way to distinguish it from the second. In this case we chose to drop all words where the adjusted R 2 (using training data only, as described in the Methods section) from a regression of time on the keyword is greater than 0.6, and to visually investigate words between 0.5 and 0.6. This process reduces the number of available keywords from 845 to 554. We may be losing some important information in this step, but we feel the danger posed by conflating shifts in the way internet is used with how people are actually feeling is more severe.

(FIGURE S5 ABOUT HERE)

Finally, many search terms exhibit extreme seasonality (particularly those that have to do with leisure). Since some of the subjective well-being variables also exhibit seasonality, this is a major concern, as search terms might be well-correlated with a given subjective well-being variable merely because they follow the same seasonal trend. We guard against this by using month dummies in all specification with one small modification: the months of December and January exhibit consistent and dramatic intra-month patterns, presumably due to the Christmas holidays and New Year's Eve. We thus also construct additional dummy controls for the each of the four weeks of those two months (and so the December and January month dummies are dropped).

Additional Information on Categories

To group the search terms into coherent categories, we took the following steps. First, we removed seasonality by regressing month and, for January and December, week dummies on the search volume, to avoid grouping terms together only because they shared seasonal patterns. Second, we roughly grouped the words into a priori related categories (such as jobs or family). Third, we ran a factor analysis, removing any terms with a factor loading of less than 0.3. These removed terms might also form a separate category (so jobs was divided into Job Search and Job Market). The factor loadings from the final categories are given below. In order to preserve the option of future application of the categories, we use a z-score average to compute the actual category rather than the factor loadings. Using the factors rather than the zscore averages produces almost identical results. The factor loadings for each of the categories are given in Tables S1-S3.

(TABLES S1-S3 ABOUT HERE)

Figure S6 shows the evolution of the category variables over time, and Figure S7 provides some comparison of those trends to other social trends reflected in administrative data. Job Search and Job Market both show the severity of the crisis in 2008-2009 and the subsequent improvement of labor market conditions. Note that Job Search peaks in 2009, when the unemployment rate was increasing the most quickly, and Job Market peaks in early 2010, when the unemployment rate was stabilizing and starting to drop, and Job Search shows less of a seasonal drop around Christmas than Job Market. Similarly, the declining trend in Financial Security and Home Finance seem to indicate that Americans have been less and less preoccupied by housing conditions and their financial conditions over the period. Financial Security also closely tracks bankruptcy (Chapter 11) petitions in US courts. Personal Security shows a slow decrease from 2009 to 2012 but a marked jump around December 2012 -one possibility is that this jump shows the fears and grief of the public following the Sandy Hook Elementary School shooting on December 14, 2012. Family Life shows an increasing trend over the period, whereas Family Stress decreases after the financial crisis, and the decrease in Family Stress maps onto the decrease in Intimate Crime incidents reported by the FBI. Civic Engagement is somewhat higher during the financial crisis but not markedly so, as are Health Problems and Education and Ideals. Healthy Habits showed a rebound as the economy began to recover; its sharp discontinuities every January are remarkable and probably reflect New Year's Eve resolutions. Finally, Summer Leisure exhibits a slight downward trend with a high seasonality, and the smoothed and seasonally adjusted series maps onto consumer spending on entertainment.

FigureFigure S7 .

 S7 Figure S1. Subjective Well Being Variables over time

Table 1 . Category Components

 1 model selection and out-of-sample prediction, we divide the sample into a "training" and "test" sample, and use the training sample to build the model, and evaluate its performance on the test sample. Data from 2009 to 2013 is used for the training set, while data from 2008 to 2009 is used for one test set and from 2013 to 2014 for the other. The reason for the symmetrical test sets is that we would like to construct an index that predicts as well for periods of crises (i.e. the 2008 economic crisis) as for periods of relative stability.In practice, we first identify the most robust categories by applying BMA to the training data. All categories with a PIP larger than 0.7 are pre-selected as a potential determinant of the subjective wellbeing variable. The PIPs of the various categories found in the exploratory analysis are reported in the Supporting Information. On a second step, we refine the model by simply regressing the dependent SWB variable on selected categories while using the training dataset, and we remove any category with a non-significant coefficient. Finally, we use the coefficients from the final model (estimated over the training period) to predict SWB over the whole period, namely both training and test subperiods.6

	Figure Legends
			Cronbach's
	Life Aspects	Category	Alpha		Component Keywords
	Material			part_time_job	layoffs	jobfair	apprenticeships
	Conditions	Job Search	0.8720	severance_pay unemployment_rate	careerfair jobs
				unemployment_benefits
				jobbenefits employmentcontract careercenter coverletter
		Job Market	0.9315	pension work_experience certification_program entrylevel	retirement qualifications
				discrimination employeebenefits
				401k banking familybudget housingauthority section8
		Financial Security	0.9101	inflation studentloans school_loans interestrate financial_crisis loans eitc socialsecurity coupons medicaid fired
				fileforbankruptcy shelter eviction foodbank homeless
		Home Finance	0.9618	mortgagerates mortgagecalculator mortgage refinancing houseprice homeloan housing_crisis mortgagepayment
	Social	Family Stress	0.8818	domesticabuse marriagehelp marriagecounseling familysupport womensshelter marriageprob custody
				snacksforkids	weekends adhd	daycare
		Family Time	0.9140	child_care_center play_football reading recipe housework laundry toddler pta kidsparty volleyball
				babyshower bullying kids_books ideasforkids tuition
				volunteering blood_donation	homeownersassociation
		Civic Engagement	0.9148	boyscouts teaparty lionsclub kiwanis	citycouncil freemasons bingo club communitymeeting
				civic_engagement	rotaryclub towncouncil
				firearm	victims gun	gunsafety violent_crime
		Personal Security	0.8657	crime_rate assault securitycamera murder selfdefense
				aggression risks	homealarm mugging
	Health and			stress hypertension diabetes obesity panicdisorder illness
	Wellness	Health Conditions	0.9051	tobacco_use obsessivecompulsive cancer relax_tech lung_cancer health_status fracture arthritis	heartdisease antidepressant asthma relaxation
				self_care sleepprob mayoclinic depression_symptoms
				symptomchecker	suicide_rates	drug_use

Note: Chronbach alpha calculated using seasonally adjusted search volumes.

Table 2 . Regression of Categories on Subjective Well-Being Data

 2

				Positive Affects				Negative Affects
		Life Evaluation	Life 5 Years Evaluation	Happiness	Laugh	Learn	Respect	Anger	Stress	Worry	Sadness
		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
	Material conditions
	Job Search	-0.772***			-0.293*** -0.466*** -0.452***	0.571*** 0.387*** 0.751*** 0.619***
		(0.075)			(0.058)	(0.061)	(0.070)	(0.067)	(0.051)	(0.042)	(0.069)
	Job Market	0.552***		0.294**		-0.297*** 1.261***
		(0.121)		(0.145)		(0.101)	(0.150)
	Financial Security	-0.327*** -0.712*** -0.596*** -0.481*** -0.415*** -0.657***	1.234*** 0.402*** 0.438*** 0.440***
		(0.112)	(0.066)	(0.114)	(0.079)	(0.088)	(0.110)	(0.180)	(0.078)	(0.057)	(0.109)
	Social
	Family Life			0.251*** 0.266*** 0.403***		0.416*** 0.478***		-0.367***
				(0.080)	(0.065)	(0.057)		(0.082)	(0.049)		(0.077)
	Family Stress	-0.381***		-0.269***
		(0.091)		(0.096)
	Civic Engagement	0.477***							-0.182**
		(0.115)							(0.088)
	Personal Security	0.199*** -0.410***		-0.266***						0.282***
		(0.049)	(0.067)		(0.055)						(0.049)
	Health and Wellness
	Healthy Habits	0.203** 0.785*** 0.460*** 0.551*** 0.267***	-0.203*		-0.373*** -0.402*** -0.280***
		(0.079)	(0.076)	(0.103)	(0.059)	(0.066)	(0.110)		(0.057)	(0.048)	(0.078)
	Summer Leisure							-1.521*** -0.881*** -0.786*** -0.906***
								(0.308)	(0.143)	(0.135)	(0.212)
	Health Conditions					0.478***			0.262**
						(0.113)			(0.116)
	Education and Ideals							-1.092*** -0.393***		-0.654***
								(0.182)	(0.139)		(0.150)
	Month dummies	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	N	200	200	200	200	200	200	200	200	200	200
	R 2	0.798	0.652	0.688	0.828	0.861	0.662	0.606	0.904	0.868	0.816
	Adj. R 2	0.770	0.614	0.650	0.806	0.843	0.623	0.557	0.890	0.853	0.791

note: Robust standard errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Week dummies included for weeks in December and January, month dummies included for all other months.

Table 3 . Decomposition of the Explained Variance of SWB variables Table 4. Correlation between Predicted and Observed SWB values (Smoothed)

 3

			Life
		Life	Evaluation 5
		Evaluation	Years	Happiness	Laugh	Learn	Respect	Anger	Stress	Worry Sadness
	Training	0.93	0.87	0.90	0.96	0.96	0.92	0.91	0.97	0.96	0.94
	Test 2008	0.85	0.76	0.63	0.85	0.47	-0.58	0.60	0.92	0.94	0.89
	Test 2014	0.92	0.86	0.47	0.86	0.55	0.80	0.61	0.82	0.84	0.70

Predictions and Observed SWB (Training Data inside the red lines, Test Data outside)

		Life Evaluation			Life Evaluation 5 years
	7				7.8
	6.9				7.7
	6.8
					7.6
	6.7
	6.6				7.5
	01jan2008	01jan2010	01jan2012	01jan2014	01jan2008	01jan2010	01jan2012	01jan2014
		Life Evaluation	Life Evaluation (Big Data)			Life Evaluation 5 years	Life Evaluation 5 years (Big Data)
		Happiness				Laugh
	89				83
	88.5
					82
	88
	87.5				81
	87
					80
	01jan2008	01jan2010	01jan2012	01jan2014	01jan2008	01jan2010	01jan2012	01jan2014
		Happiness	Happiness(Big Data)			Laugh	Laugh(Big Data)
	92.5	Respect			66	Learn
					64
	92
					62
	91.5
					60
	91				58
	01jan2008	01jan2010	01jan2012	01jan2014	01jan2008	01jan2010	01jan2012	01jan2014
		Respect	Respect(Big Data)			Learn	Learn(Big Data)
	36	Worry			14.5	Anger
					14
	34
					13.5
	32
					13
	30				12.5
	01jan2008	01jan2010	01jan2012	01jan2014	01jan2008	01jan2010	01jan2012	01jan2014
		Worry	Worry(Big Data)			Anger	Anger(Big Data)
		Stress
	42				20
	41
					19
	40
	39				18
	38
	37				17
	01jan2008	01jan2010	01jan2012	01jan2014	01jan2008	01jan2010	01jan2012	01jan2014
		Stress	Stress(Big Data)			Sadness	Sadness(Big Data)

Table S1 . Factor Loading for Material Conditions Variables

 S1

	Job Search	Factor	Job Market	Factor	Financial	Factor	Home Finance	Factor
		Loadings		Loadings	Security	Loadings		Loadings
	part time job	0,734	jobbenefits	0,528	401K	0,478	mortgage rates	0,962
	layoffs	0,879	employment	0,864	banking	0,790	mortgage	0,913
			contract				calculator
	jobfair	0,874	careercenter	0,871	family budget	0,374	mortgage	0,995
	apprenticeships	0,825	coverletter	0,888	housing	0,712	refinancing	0,934
					authority
	severance pay	0,817	pension	0,557	section8	0,500	houseprice	0,780
	unemployment	0,798	cert prog	0,482	inflation	0,770	homeloan	0,927
	rate
	careerfair	0,626	retirement	0,901	studentloans	0,764	housing crisis	0,849
	jobs	0,408	work	0,568	school loans	0,858	mortgage	0,764
			experience				payments
	unemployment	0,473	entrylevel	0,755	interestrate	0,839
	benefits
			qualifications	0,905	fired	0,562
			discrimination	0,806	financial crisis	0,782
			employee	0,903	loans	0,954
			benefits
					eitc	0,837
					socialsecurity	0,692
					coupons	0,492
					medicaid	0,413
					file for	0,622
					bankruptcy
					shelter	0,726
					eviction	0,623
					foodbank	0,755
					homelessshelters	0,716

Table S2 . Factor Loading for Social Variables Family Stress Factor Loadings Family Time Factor Loadings Civic Engagement Factor Loadings Personal Security Factor Loadings

 S2

	domestic	0,889	snacksforkids	0,821	volunteering	0,855	firearm	0,845
	abuse
	marriagehelp	0,763	weekends	0,380	blood donation	0,656	victims	0,771
	marriage	0,776	adhd	0,558	homeowners	0,817	gun	0,728
	problems				association
	custody	0,690	daycare	0,543	boyscouts	0,645	gunsafety	0,754
	marriage	0,822	child care	0,555	kiwanis	0,920	violent crimes	0,858
	counseling		center
	familysupport	0,870	pta	0,758	citycouncil	0,767	crime rate	0,870
	womensshelter	0,432	kidsparty	0,402	freemasons	0,642	assault	0,647
			volleyball	0,359	bingo	0,548	securitycamera	0,558
			play football	0,792	teaparty	0,496	murder	0,481
			reading	0,470	lionsclub	0,653	selfdefense	0,552
			recipe	0,602	club	0,779	aggression	0,303
			housework	0,397	community	0,439	risks	0,245
					meeting
			laundry	0,820	civic	0,364	homealarm	0,246
					engagement
			toddler	0,810	rotaryclub	0,889	mugging	0,314
			babyshower	0,820	towncouncil	0,559
			bullying	0,637
			kids books	0,881
			ideasforkids	0,919
			tuition	0,782

Table S3 . Factor Loading for Health and Wellness Variables Health Conditions Factor Loadings Healthy Habits Factor Loadings Summer Leisure Factor Loadings Education and Ideals

 S3

								Factor
								Loadings
	stress	0,772	exercise	0,824	golf	0,934	middleschool	0,493
	hypertension	0,583	weights	0,679	fishing	0,952	juniorhigh	0,798
	diabetes	0,841	healthydiet	0,814	motorcycle	0,905	economics	0,866
	obesity	0,834	dental care	0,735	ponds	0,921	homework	0,690
	Panic disorder illness	0,748 0,559	fruits and vegetables	0,591	hiking biking	0,925 0,939	State university moral	0,872 0,879
	tobacco use	0,832	life	0,800	water sports	0,610	individualism	0,893
			expectancy
	lung cancer	0,776	lose weight	0,355	boating	0,878	billofrights	0,825
	heartdisease	0,776	health care	0,773	tours	0,880	human capital	0,633
	obsessive		quitsmoking	0,383	beachcottage	0,588	constitution	0,880
	compulsive disorder	0,885			sightseeing	0,874	politicalaction	0,587
					bedandbreakfast	0,839	social justice	0,530
	cancer	0,429			baseball	0,323	dropout	0,419
	relax tech	0,642			softball	0,420	reading books	0,681
	antidepressant	0,871			playpool	0,617	grammar	0,909
	health status	0,339					tutor	0,861
	fracture	0,316					buddhist	0,774
	arthritis	0,625					school	0,866
	asthma	0,866					highschool	0,906
	relaxation	0,615					ethics	0,944
	self care	0,506					philosophy	0,941
	sleepproblems	0,638					onlinecourses	0,476
	mayoclinic	0,775					literature	0,976
	depression symptoms	0,620					degree language	0,598 0,940
	symptomchecker	0,543					science	0,942
	suicide rates	0,772					literacy	0,942
	drug use	0,776					studies	0,941
	chronicfatigue	0,638					study	0,625
							mathematics	0,953
							learning	0,966
							skills	0,725
							secondary	0,876
							education
							writing	0,952
							worship	0,812
							creativity	0,523
							psychologist	0,896
							therapy	0,850
							selfesteem	0,894
							morality	0,809
							poverty	0,931
							feminist	0,547
							rights	0,951
							relations	0,933
							freedomofspeech	0,731
							civilrights	0,830
							religion	0,509
							religious	0,932
							propertyrights	0,672

Subjective well-being measures have been used to test a variety of potential determinants of wellbeing such as income, unemployment, inflation, health status, income, inequality and income comparisons (see, for example,Helliwell et al.

2015).

As explained below, Home Finance is excluded in the analysis that follows but we present the results on the category.

Cronbach's alpha is a normalized index of correlation that indicates the degree to which a set of variables measures a single latent variable. It is often viewed as an indicator of internal consistency.

The dependent and explanatory variables have been standardized to allow for a comparison of the magnitude of the effects at stake.

We choose the model prior most commonly used in applied studies, namely the uniform distribution that assigns equal prior probability 𝑝 𝑀 ! = 2 !! to all 2 ! possible models. The use of the uniform model prior was first suggested by[START_REF] Raftery | Approximate Bayes Factors for Generalized Linear Models[END_REF] while[START_REF] Hoeting | Bayesian Model Averaging: A Tutorial[END_REF] reviewed the evidence supporting the good performance of the uniform model prior. It should be noted BMA does not compute the marginal likelihood and the posterior distribution for each possible model. Instead, it identifies the subset of models where the mass of the model posterior probability is concentrated. This is achieved with the help of a Metropolis-Hastings algorithm that walks randomly through the model space but visits its most important part. In practice, we use the "birth-death" sampler that randomly chooses one of the K variables at each step of the algorithm, and excludes it from the model if it is already included in it, or includes it otherwise.

We intentionally exclude Home Finance from the model. While we recognize that financing a home is an important life event for many Americans, the predominance of words related to home finance ("mortgage", for example) during this period is also critically linked to the financial crisis, and so the importance of these words in predicting SWB is likely to be highly time-specific. As such, this category might dominate the prediction in the training data, and risk yielding poor results in the out of sample tests, and having poor power to extrapolate to the future.

ACKNOWLEDGEMENTS

Algan, Beasley and Senik are grateful to CEPREMAP for financial support. Yann Algan has received the financial support of the ERC Advanced Grant. This paper has received the technical support from the Medialab at Sciences Po and the authors are especially grateful to Paul Girard, Guillaume Plique and Mathieu Jacomy. We also thank Melissa Leroux for excellent research assistance. This document expresses the views of the authors and does not reflect the official views of the institutions with which the authors are affiliated. All errors are our own.


Irem Demir-Yilmaz

Ines Lüchtefeld

Claude Lemen

Etienne Dague

Pascal Guiraud

Tomaso Zambelli

Cécile Formosa-Dague

Probing the interactions between bubbles (gas / liquid) and (bio)interfaces at the molecular scale using FluidFM technology

Understanding the molecular mechanisms underlying bubble-(bio)surfaces interactions is currently a challenge that if overcame, would allow to understand and control the various processes in which they are involved. Atomic force microscopy is a valuable tool to measure such interactions, but it is limited by the large size and instability of bubbles. To overcome these challenges, we here develop a new method to probe more accurately the interactions between bubbles and (bio)-interfaces by using the fluidic force microscopy technology (FluidFM) that combines AFM with microfluidics. In this study, we use FluidFM in an original manner, to produce microsized bubbles. We characterized the bubbles produced using this method, and their interactions with hydrophobic surfaces were probed, showing that bubbles behave like hydrophobic surfaces. Thus they can be used to measure the hydrophobic properties of microorganisms' surfaces. Finally we developed a strategy to functionalize their surface, thereby modulating their interactions with microorganisms' surfaces. This new method provides a valuable tool to understand bubble-(bio)surfaces interactions but also to engineer them.

INTRODUCTION

P. aeruginosa

Interactions of cells with bubbles are controlled by bubble functionalization

CONCLUSIONS

Here we provide the possibility to (i) modify easily the surface of the bubbles produced using FluidFM, and (ii) to show in what way the modification of the bubble surface influences the nature and strength of the interaction with cells. In future projects, this strategy could also be used to specifically separate cell populations from each other; for example to separate bacterial cells from human blood cells in the case of sepsis, but many other applications can now be envisioned.

FluidFM

  Technology to produce bubbles and measure interactionsTechnological challenges:• Produce stable microbubbles • Functionalize bubbles with surfactants • Measure interactions with cells at the molecular levelWe developed an original process where the interactions of cells with bubbles

RESULTS [3] Pressure : 20 mbar No bubble Bubble probe method can be used with reliability to measure the hydrophobicity of complex surfaces


Thi Thu

Nga Vu

Gilbert Teyssedre
email: gilbert.teyssedre@laplace.univ-tlse.fr

Séverine Le Roy
email: severine.leroy@laplace.univ-tlse.fr

Electric Field Distribution in HVDC Cable Joint in Non-Stationary Conditions

Keywords: HVDC cable system, accessories, space charge, field distribution

Accessories such as joints and terminations represent weak points in HVDC cable systems. Indeed, the DC field distribution is intimately dependent on the thermal conditions of the accessory and on material properties. Moreover, there is no available method to probe charge distribution in those conditions. In this work, the field distribution in non-stationary conditions, both thermally and electrically, is computed considering crosslinked polyethylene (XLPE) as cable insulation and different insulating materials (silicone, rubber, XLPE) for a 200kV joint assembled in a same geometry. In the conditions used, i.e. temperatures up to 70°C, and with the material properties considered, the dielectric time constant appears of the same order or even longer than the thermal one and is of several hours. It means that both physical phenomena need to be considered for modelling the electric field distribution. Both the radial and the tangential field distributions are analysed, and focus is given on the field distribution under the stress cone on the ground side and near the central deflector on the high voltage side of the joint. We show that the position of the maximum field varies in time in a way that is not easy to anticipate. Under the cone, the smallest tangential field is obtained with the joint insulating material having the highest electrical conductivity. It results from a shift of the field towards the cable insulation in which the geometrical features produce less axial component of the field. At the level of the central deflector, it is clear that the tangential field is higher when the mismatch between the conductivity of the two insulations is larger. In addition, the field grows as a function of time under stress. The work points to the need of precise data on materials conductivity and to the need of probing field distribution in 3D.

Introduction

Accessories may represent a weak point in HVDC cable links, especially when going to ever-higher voltages where the feedback on in-service behaviour is lacking [START_REF] Mazzanti | The insulation of HVDC extruded cable system joints. Part 1: Review of materials, design and testing procedures[END_REF][START_REF] Ye | Review on HVDC cable terminations[END_REF]. Compared to bulk cables where many research works are carried out, both experimentally and in modelling, for assessing insulation endurance, anticipating field distribution in accessories is more difficult to tackle. Especially, methods for probing charge and field distributions in localized areas are lacking and the fact that different insulating materials coexist brings further difficulty. For this reason, thermal and electrical modelling is necessary. Various degrees of sophistication can be achieved for DC field modelling of insulations [START_REF] Jörgens | A review about the modeling and simulation of electro-quasistatic fields in HVDC cable systems[END_REF], ranging from bipolar charge transport model with identification of generation, transport and trapping processes of charges, to macroscopic models based on conductivity expression. Resorting to macroscopic modelling, i.e. based on field and temperature dependencies of conductivity and/or permittivity, the numerical resolution of the problem is not a real difficulty. However, it must be based on reliable experimental data characterizing the materials, especially conductivity, and on the exploration of different practical combinations of thermal/electrical stresses that may be encountered.

In this contribution, we report mainly on the modelling of the electric field distribution in unsteady situations from the electrical and thermal point of view in cable accessories comprising an association of insulators of different nature in a specific geometry. In many situations, simulations have been carried out on admittedly complex objects, but in a stationary situation [START_REF] Frobin | A generic approach for HVDC cable accessories modelling[END_REF]. However, one must wonder about the redistribution of the electric field in the case where the transmitted power suddenly increases, implying to have a thermal transient in the accessory, and also to have an idea of the thermal equilibrium time of the system. The transient in electrical stress obviously involves the passage from a capacitive distribution to a resistive distribution of the electric field after energizing the cable and joint, with an 'electric' time constant depending on the temperature. It is also important to estimate the maximum field values that can be obtained during operation, particularly in the case of polarity inversion of the applied voltage, for inverting the power flow using Line-commutated converters, LCC [START_REF] Albertini | HPTE Extruded cables polarity reversals performance in LCC HVDC systems[END_REF]. Modelling may help anticipating the 'hot spots' of electrical stress during operations on the system, such as these polarity reversals or temperature variations linked to fluctuations in the transmitted power. In addition, 'type' and pre-qualification tests at 1.85 × Uo and 1.45 × Uo, where Uo is the nominal service voltage, also involving variations in the current over several hours as well as polarity reversals or pulse tests, must be taken into account in the design [START_REF] Saltzer | Electrothermal simulations of HVDC cable joints[END_REF][START_REF] Fälth | Robustness analysis of classical high voltage joint design under high voltage DC stress[END_REF].

As stated previously, accessories represent weak points of the transmission system. Cable joints are mainly of two types for cables with extruded insulation [START_REF] Mazzanti | The insulation of HVDC extruded cable system joints. Part 1: Review of materials, design and testing procedures[END_REF], generally made of cross-linked polyethylene (XLPE). The 'factory' joints apply mainly to submarine cables, and are processed at the time of extrusion, for example for jointing two cables of several tens of km each, for the purpose of production lines maintenance. They are compact and reform the cable almost identically. The materials and processes used are the same as those of the cable insulation. Prefabricated -or pre-moulded joints, on the other hand, are made using a material different from that of the cable insulation. They are generally made of elastomers as silicone rubber (SiR) or ethylene-propylene-diene monomer terpolymer (EPDM). They are manufactured during the laying of cables, in particular at the junction of cable sections of limited length for buried cables for logistical reasons. Rebuilding the insulation and all cable components takes more space than for factory junctions. On part of the joint, the extruded insulation and material joint coexist.

It is recognized that the dielectric/dielectric interface represents a threat for the accessory reliability and the tangential field can be a driving mode for failure [START_REF] Mazzanti | The insulation of HVDC extruded cable system joints. Part 1: Review of materials, design and testing procedures[END_REF][START_REF] Ghorbani | HVDC cable systems-Highlighting extruded technology[END_REF][START_REF] Ye | Design aspects on HVDC cable joints[END_REF][START_REF] Mauseth | Electric field simulations of high voltage DC extruded cable systems[END_REF]. Whereas the behaviour of the radial field distribution in a bilayer dielectric can be reasonably anticipated based on the conductivity behaviour, the tangential component is more difficult to tackle, especially the respective role of geometry and thermal stress.

In this work, based on a conductivity law established from experiments on typical materials that are XLPE, SiR, and EPDM, the field distribution in a 200 kV cable joint is computed. Polarity reversal of the DC stress after long charging time is considered, as well as different thermal conditions: isothermal at 30 °C and non-stationary thermal gradient when energizing the cable. The tangential and radial electric fields are computed in these transient electrical and thermal conditions. To evaluate how far the nature of material imparts the field distribution, the case of a joint fully made of XLPE, as could be the case for factory joints [START_REF] Mazzanti | The insulation of HVDC extruded cable system joints. Part 1: Review of materials, design and testing procedures[END_REF], is considered. We also modelled the association of silicone rubber (SiR) with XLPE.

Joint Characteristics and Model

Geometry

The object of the present study is a 200 kV, 1 kA HVDC joint with general design and geometry as given in Figure 1 [START_REF] Stancu | Space charge and electric field in thermally aged multilayer joints model[END_REF]. The cable insulation is made of XLPE. The pre-moulded sleeve is initially made of EPDM. The nature of this material and corresponding properties will be changed in the modelling to investigate the impact on the field distribution. The cones and the deflector are made of semiconductor material (semicon), i.e. a carbon black charged polymer. The semiconducting layer is also present on the conductor. In the model described here, the outer layer, which normally contains shielding layers, is simplified by using a 5 mm thick semicon layer ensuring electrical continuity and providing thermal resistance at the surface of the joint. The field distribution in the insulation layers comprised between the outer cone and the central deflector is investigated.

Materials properties

The data on conductivity versus field and temperature of XLPE and EPDM materials are detailed elsewhere [START_REF] Vu | Correlating conductivity and space charge measurements in multi-dielectrics under various electrical and thermal stresses[END_REF][START_REF] Vu | Field distribution under temperature gradient in polymeric MV-HVDC model cable: simulation and space charge measurements[END_REF]. The following equation has been parameterized using experimental data on current obtained as a function of electric field on plane samples.

𝜎(𝑇, 𝐸) = 𝐴. 𝑒𝑥𝑝 ( -𝐸 𝑎 𝑘 𝐵 𝑇 ) . 𝑠𝑖𝑛ℎ(𝛽(𝑇). 𝐸) . 𝐸 𝛼 (1)

where the pre-exponential factor A, activation energy Ea, field coefficients β(T)=aT+b and α are given in Table 1 [13] and kB is the Boltzmann's constant. The parameters are listed for conductivity given in S/m, the field in V/m, the temperature in K.

a (m/V/K) 0 (T <313 K); -1.3×10 -9 (T≥313 K) 4.8×10 -10 b (m/V) 1.38×10 -7 (T <313 K); 5.45×10 -7 (T≥313 K) -5.1×10 -8  0.15 -1.42
For the silicone rubber, the expression of conductivity given by Baferani et al [START_REF] Baferani | Electric field tailoring in HVDC cable joints utilizing electro-thermal simulation: effect of field grading materials[END_REF] was adopted:

𝜎(𝑇, 𝐸) = 𝐴. 𝑒𝑥𝑝(𝛾𝑇). 𝑒𝑥𝑝(𝛽𝐸) (2)
with A = 2.9 × 10 -17 S/m, the temperature coefficient is γ = 0.019 K -1 and β= 4.1 × 10 -16 mm/kV. Figure 2 shows a comparison of conductivity as a function of the electric field for the different materials at different temperatures. The behaviour for XLPE is in line with previous report [START_REF] Boggs | Effect of insulation properties on the field grading of solid dielectric DC cable[END_REF]. For EPDM, a decrease of the conductivity with field appears, which may appear not common. This behaviour was discussed in a previous work [START_REF] Vu | Correlating conductivity and space charge measurements in multi-dielectrics under various electrical and thermal stresses[END_REF] and it was suggested that some form of ionic processes might be at play in the sub-linear behaviour, as may happen in insulating liquids. Not many reports on DC conductivity measurements using reasonable charging time are available in the literature for EPDM. In a recent work, Z.Y. Li et al [START_REF] Li | Significantly Improved electrical properties of photo-initiated auxiliary crosslinking EPDM used for cable termination[END_REF] reported on quasi field-independent conductivity of EPDM for fields up to 10 to 20kV/mm that contrasted with the strong field dependence of conductivity of XLPE. Besides, EPDM is a complex material and its behaviour may change with compounding: D. Li et al [START_REF] Li | Research on nonlinear EPDM for ±525kV HVDC cable accessories[END_REF] found power law relation between conductivity and field with an exponent nin the range from 0.2 to 0.9 depending on EPDM grade (with using only 1min charging time!). . The conductivity of SiR exhibits very weak field dependence and relatively mild temperature dependence. Another expression proposed by Qin et al [START_REF] Qin | Impacts of temperature on the distribution of electric-field in HVDC cable joint[END_REF] follows the same trend.

Comparing XLPE and EPDM, the conductivity is sometimes higher in one material or the other, depending on the field and temperature conditions, which will result in a transfer of the DC field in one or the other of the materials according to these same conditions

The other physical quantities used in the model are reported in Table 2. The heat input by the Joule effect in the copper conductor takes into account a reference resistivity of 1.7 × 10 -8 Ω.m for copper at 20 °C and a temperature coefficient for the resistivity of 3.9 × 10 -3 K -1 .

Thermal-electrical model

The resolution of the field distribution was carried out using the Comsol® tool using the thermal and electrical modules. Owing to the cylindrical symmetry of cable joints, the considered geometries are built up as 2D axisymmetric models. The general equations to be solved are the following:

-Maxwell equation reduced to the DC case [START_REF] Fälth | Robustness analysis of classical high voltage joint design under high voltage DC stress[END_REF]:

which becomes in local form and in continuous medium, the continuity equation:

𝜕𝜌 𝜕𝑡 + ∇𝐽 = 0 (4)
-Equation of thermal dynamics:

ρ 𝑚 c 𝑝 𝜕𝑇 𝜕𝑡 -∇(𝜆∇𝑇) = 𝑄 ( 5
)
where ρm is the density, cp the specific heat, λ the thermal conductivity, Q the heat source.

The electric stress is applied to the conductor, using a nominal voltage of ±200 kV. The electric field distribution is calculated under unsteady conditions with 24 h energizing, short-circuiting, and during polarity reversals. Switching from one polarity to another is done with an intermediate short-circuit of 3 min. A duration of 24 h was chosen to approach a stationary situation both from the thermal and electrical point of view. Indeed, in a previous work on the association of EPDM and XLPE, the dielectric time constant was of the order of 24h at 20°C [START_REF] Vu | Maxwell-Wagner effect in multi-layered dielectrics: interfacial charge measurement and modelling[END_REF]. As the minimum temperature is 30°C in the present work, it was considered that 24h was enough for reaching the steady state. For the thermal response, the first simulations showed that 24h was actually enough for obtaining a steady state (see below).

Thermal modelling is carried out by considering a heat input by the Joule effect in the conductor and external losses by convection phenomena in the air. The considered conductor cross section is 50 mm².

The power dissipated in the conductor is of the order of 350 W/m under 1 kA at 30 °C. Heat exchange with the surrounding environment is assumed to occur by natural convection:

𝑞 𝑐 = ℎ(𝑇 𝑠 -𝑇 𝑎𝑚𝑏 ) (6)
where h represents the convective transfer coefficient, linked to the Nusselt number (h of the order of 5 W/m²/K taking into account the geometry of the cable is rigorously determined in the resolution software). Ts and Tamb are the joint surface temperature and the ambient temperature, respectively.

The heat flux exchanged by radiation (which remains low under our thermal conditions) is also taken into account by considering a surface emissivity coefficient of 0.8.

Results

Figure 3 shows the dimensions of the modelled object, of external radius 92 mm. For the representation of the radial distribution of the field in the form of profiles, an axial position at z = 800 mm was chosen, such to be halfway between the deflector cone and the internal semiconductor. The axial distribution of the field is taken at the interface between the XLPE insulation and the joint material as this region is critical and constitutes a weak point of the joint.

The applied stresses are those previously mentioned (±200 kV). We have considered three thermal conditions: a homogeneous temperature at T = 30 °C, a stationary condition with a thermal gradient due to the injection of a current of 1 kA in the conductor, and an unsteady condition both from an electrical and thermal point of view, keeping the same current flowing in the conductor and considering a homogeneous initial temperature of 30 °C. The limit conditions are a constant potential applied to the conductor and welding, and reference potential to the outer surface of the joint.

Thermal simulation

The thermal modelling of the XLPE/EPDM joint was carried out assuming that the joint was initially at an isothermal temperature of 30°C. At t = 0 s, a current of 1 kA is imposed on the conductor, which has the effect of slowly heating the joint and producing a thermal gradient within the insulation. The final temperature gradient is of the order of 40 °C, with a temperature maximum situated between the end of the internal deflector and the end of the joint. It corresponds to the region where the insulation is the thickest, due to the low thermal conductivity.

Electric Field Distribution

Examples of electric field distributions as a function of time under applied voltage and polarity reversals, obtained for the joint with EPDM insulation on XLPE-insulated cable are shown in Figure 5. The voltage is considered applied when the current is applied to the conductor. One can notice that over In order to dissociate the thermal effects from the nonlinear electrical conduction effects on the field distribution, we considered the field distributions obtained in a transient manner, under a voltage of 200 kV applied for 24 hours, followed by a polarity reversal and again by a stressing under -200 kV for 24 h. This protocol was applied for 3 different thermal conditions: an isotherm at 30°C, a stationary thermal gradient condition (pre-set current of 1 kA), and finally the case illustrated above where the electrical and thermal stresses are applied simultaneously. Details of the field distribution evolution are presented considering the radial and tangential components.

Distribution of the radial component of the electric field

The results obtained for the radial distribution of the field at the position z = 800 mm are shown in Figure 6 for XLPE/EPDM joint. As will be shown in section 3.2.2, the axial component of the field is almost zero at this position. The field distribution under isothermal conditions (Figure 6a) is clearly distinguished from the other cases. At 100 s, i.e. short after voltage application, the field is capacitively distributed. Hence, EXLPE > EEPDM due to the lower dielectric permittivity in XLPE. As expected, this initial field profile is independent from the thermal conditions, since no temperature dependence of the dielectric permittivity was introduced in the model.

For the isothermal case at 30 °C, the field gradually evolves in time with further field reinforcement in XLPE, without any pronounced change in the general shape. This evolution corresponds to two phenomena: on the one hand, it is the consequence of a resistivity at 30 °C of XLPE greater than that of EPDM, for fields < 10 kV/mm, cf. Figure 2. The other feature is that the field strengths remain moderate (< 10 kV/mm) so that the nonlinear phenomena, tending to a homogenization of the field over time, are barely perceptible in the two materials. followed by polarity inversion) for different thermal conditions. Distribution taken at z = 800 mm, cf. Figure 3.

Vertical lines indicate the boundaries of the insulators. The bold black curve corresponds to the reset to 0 V after 24 h at +200 kV. In the legend, P stands for positive voltage; N for negative. The arrows provide a guide to the eyes for the variation of the field as a function of time.

Under the conditions of a stationary thermal gradient, the electric field rapidly evolves towards a situation of equilibrium; one can consider that after 8 h this equilibrium is almost reached, cf. Figure 6b. Here within each of the materials the field tends to increase with the radius at long stressing time:

this is because the thermal gradient produces a conductivity gradient whose effects exceed the variation of the field due to the cylindrical geometry. The residual field, taken 100 s after resetting the potential to zero, presents a profile different from that of the isothermal case. This residual field corresponds to the distortion introduced under DC stressing as compared to the geometric field distribution. The lower value of the residual field near the XLPE/EPDM interface results from the fact that with a temperature at the dielectric / dielectric interface of 45 °C, and fields of the order of 3 kV/mm, the conductivities of the two materials are quite close. In addition, the comparison of the field jumps for voltage on and voltage off conditions reveals a decrease within 100 s after setting the potential to zero, due to the partial dissipation of the interface charge. With the reversal of polarity, the field near the conductor is strongly reinforced.

The combination of unsteady electrical and thermal conditions, Figure 6c, mainly produces a slower kinetics towards the steady state compared to Figure 6b. Indeed, the temperature is initially lower, and so is the dielectric time constant (τ = ε / σ with a combination of physical quantities and geometry for the two materials) while the temperature gradient is setting up. The field distribution after 24 h is identical to that in the case of the stabilized gradient. A fortiori, it is the same for the following profiles.

The field profiles obtained for XLPE/XLPE and XLPE/SiR joints under non-stationary electrical and thermal conditions are presented in Figure 7. The field profiles for XLPE/XLPE joint are not very different from the ones obtained in the case of XLPE/EPDM joint (Figure 7a vs. Figure 6c), except for the field step at the interface that has disappeared, due to permittivity effects. In the steady state situation, there is full stress inversion along the radius, i.e. a larger field at the outer diameter compared to inner diameter, as expected for a cable under thermal gradient [START_REF] Boggs | Effect of insulation properties on the field grading of solid dielectric DC cable[END_REF]. For the case of a SiR/XLPE joint (Figure 7b), the initial field step is quite large due to the high permittivity of SiR. In addition, when comparing the field at the dielectric/dielectric interface after 24 h and during the short-circuit, it can be seen that the field step is decreased within 100 s after resetting the potential, due to the partial dissipation of the interface charge. When the polarity is reversed, the field at the core is strongly reinforced. Here, the conductivity is higher in the SiR and therefore the field is higher in the XLPE part. The field redistribution is particularly fast after polarity inversion because of the high temperature in XLPE and high conductivity in SiR.

Tangential field distribution

The field distribution in the radial direction of the insulator can be relatively well anticipated and understood with knowledge of the behaviour of the materials. However, special attention should also be paid to the tangential electric field along the interface between the joint body and the cable under different electrical and thermal stresses. In fact, manufacturing the joint is a delicate step in which any imperfection such as a lack of adhesion or the presence of air bubble can be detrimental, even more if the electrical stress is significant. It has been reported that the tangential field at the interface of the two insulators is distributed with a strong non-uniformity, the field being strengthened in the insulator at the vicinity of the semiconductor connected to the ground [START_REF] Mazzanti | The insulation of HVDC extruded cable system joints. Part 1: Review of materials, design and testing procedures[END_REF][START_REF] Ghorbani | HVDC cable systems-Highlighting extruded technology[END_REF][START_REF] Ye | Design aspects on HVDC cable joints[END_REF][START_REF] Mauseth | Electric field simulations of high voltage DC extruded cable systems[END_REF]. For these reasons, solutions with field-grading materials (FGM) [START_REF] Donzel | Nonlinear resistive electric field grading Part 2: Materials and applications[END_REF] with strong non-linear properties have been adopted by some cable manufacturers [START_REF] Fälth | Robustness analysis of classical high voltage joint design under high voltage DC stress[END_REF]. However, this is not the rule, as reliability problems have arisen with this technology. The drawing at the top of (a) represents a part of the joint. The axial field profiles are taken along the red line at the interface between XLPE and EPDM. Vertical lines define insulator boundaries. The curve in black corresponds to the reset to 0 after 24 h at +200 kV. Ground on the left; HV on the right. Same conventions as in Figure 6.

Under thermal gradient condition (Figure 8b), the field increases with time on the cone side, by a factor 3. As a result, in the first moments of the polarity reversal, the sign of the field is not reversed in this zone. On the HV side, the field variations are milder. The presence of a negative residual field at grounding is noticeable. It is added to the applied field after stress polarity inversion, thereby producing an over-stress. In case of non-stationary thermal conditions, Figure 8c, only mild differences are observed compared to the case of steady gradient. Only a faster convergence to steady state appears without further field distortion. Globally, the maximum tangential fields remain lower than in the case of isothermal conditions. This is not necessarily an effect of the thermal gradient but of the fact that, taking into account the average applied stresses, the conductivity values of the two insulators become closer by heating the joint. They are identical at 60 °C for a field of 4 kV/mm, while at 30 °C the equivalence is obtained under a field of 15 kV/mm (Figure 2) which is never reached here.

Using a single material, XLPE (Figure 9a), under non-stationary thermal and electrical conditions, the evolution of the field distribution is similar to the XLPE/EPDM one in the same conditions (Figure 8c).

It means that, for the cases considered here, the temperature conditions are more important than the material nature. With the SiR joint material (Figure 9b) having a much larger conductivity than XLPE, there is practically no field distortion on the ground side (the residual field at voltage removal is nearly zero) and the field is small compared to the other cases. On the HV side, there is also a mild variation of the field with time, and hence weak residual field at grounding. Here the field reaches the highest values at about 7.5 kV/mm in steady state. The profiles are in fact similar to the ones of Figure 8a: the common feature here is a higher conductivity in the joint material than in XLPE. and current application (non-stationary electrical and thermal conditions). Vertical lines define insulator boundaries. Same conventions as in Figure 6.

Discussion

Besides differences in the field dependence of conductivity between materials, the temperature distribution may affect the field distribution. Figure 10a represents the temperature distribution in the radial direction in quasi steady state for the three couples of materials considered. It shows that due to lower thermal conductivity in SiR, the temperature is significantly higher in the joint in this case. The temperature gradient is about 30°C with pure XLPE and it increases to 38°C in XLPE/SiR joint. At the dielectric/dielectric interface, the temperature is about 10°C higher in case of XLPE/SiR joint. This is a direct consequence of a lower thermal conductivity. This higher temperature in the insulation explains why the radial field redistributes faster in XLPE in Figure 7b than in Figure 6c for example once the thermal equilibrium is reached (negative polarity) since conductivity in XLPE increases by roughly a factor 3 in 10°C temperature variation considering the activation energy of 1 eV. where the temperature variation along the interface is represented. Thermal gradients along the axial direction are milder than under radial direction: the rise is by less than 10°C over a distance of ≈100 mm from the colder part, under the cone, to the central region. On the cone side, the temperature gradient is clearly larger with SiR than with EPDM joint material. On the deflector side, there is nearly no temperature variation along the z-axis. Above considerations show that the thermal properties of the joint material may significantly affect the temperature distribution in this joint design, and therefore, contribute to the actual field distribution. Managing large temperature changes will go with more difficulty in optimising designs and selecting materials. Questions to discuss are on the expected joint material properties to get an optimum field distribution. At least three features are to be considered as criteria on the field for optimizing both design and materials: -get a minimum capacitive field strength: this aspect concerns the field distribution short after DC voltage application, and, as importantly, field distributions in transient situations such as with over-voltages. In this case, the permittivity, and the geometric design, drive the field;

-get a minimum steady state DC field, notably the tangential part of the electric field;

-get a minimum field redistribution at polarity reversal, meaning that the residual field at grounding should be minimum, which implies that the capacitive and resistive field distributions are close.

The chosen design permits to appreciate, besides temperature effects and material effects, the dependence of field strengthening on geometrical features, with the cases of acute and obtuse angles respectively for the cone side and the deflector side, at the contact between the semicon and the joint materials.

On the cone side, the tangential field at short time is lowest for the case of SiR joint material, Figure 9b. The effect is purely associated to the value of permittivity, which is higher in the SiR. The highest field values (5 kV/mm) are obtained with EPDM under thermal gradient, cf. Figures 8b and8c, whereas for the isothermal case at 30 °C, Figure 8a, the field remains low. All these results tend to show that the tangential electrical field is lower when the conditions are such that the field is moved to the cable (XLPE) insulation, either because of higher permittivity or high conductivity in the joint material: moved to a region of lower divergence, the field stays more concentrated in the radial direction. In order to confirm this, we have plotted in Figure 11 the tangential field for an isotherm at 70 °C with EPDM joint material. At this temperature, the electrical conductivity is lower in the EPDM than in XLPE. Here the tangential field under the cone is clearly enhanced compared to the case at 30 °C.

With SiR joint material, large contrast in conductivities appears whatever the temperature and electric field values. The same happens also for EPDM at 30 °C, which explains the low intensity of the tangential field. For EPDM joint under thermal gradient, high fields are obtained at the tip of the cone (Figures 8b and8c). This feature is presumably due to the thermal gradient that tends to move the field in this colder region. The circulation of the field being conservative, it can be stated that:

∫ E 𝑧 𝑑 𝑐 𝑑𝑧 = V 𝑐 -V 𝑑 (5)
where c represents the deflecting cone and d the central deflector. Therefore, a field strengthening somewhere along the interface is compensated elsewhere along the interface, and the integral under the tangential field component should be zero during the grounding step. Based on this rule, the tangential field near the central deflector would be the largest in case of SiR joint material and for the EPDM at 30 °C, i.e. in cases where a minimization of the tangential field under the cone is obtained. A better control of the field at the central deflector would be reached with shaping it as is done for the cone [START_REF] Frobin | A generic approach for HVDC cable accessories modelling[END_REF][START_REF] Ye | Design aspects on HVDC cable joints[END_REF].

According to Figure 2, XLPE is the most 'non-linear' material. XLPE joint material obviously gives perfect match of the conductivities and permittivities of the two insulations. Referring to Figure 9a, it can be considered that it corresponds to the case where the tangential field is the most homogeneously distributed. However, one main notice that the residual field near the central deflector is negative after positive voltage application. As a result, the field is strengthened after polarity reversal, which may represent a threat.

Figure 12 shows results for the potential and for the radial and tangential field distributions using different representations obtained with XLPE and with SiR joint materials just after polarity inversion.

The tangential field around the central deflector is slightly higher in case of XLPE joint material. In both cases, it is quite low under the cone. Considering the potential or radial field maps, it is clear that the stress is more homogeneously distributed in case of SiR joint material. This behaviour results from the residual field set-up at the end of the previously applied positive voltage. After 24 h under stress, Figure 13, the situation has clearly evolved: the axial field under the cone has substantially increased for the XLPE joint, the radial field is more homogeneous, and the potential variations become smoother in this case. To obtain an equilibrated tangential field distribution along the interface, targeting continuity in the conductivities of the two insulations seems to be a good option. However, the field is not homogeneous in the joint, neither is the temperature and experimental data reveal that only in particular combinations (field-temperature), equal values of conductivity are obtained. It is in fact difficult to have similar conductivities from materials so different as silicones or EPDM and XLPE. The use of an elastomer remains essential to apply a homogeneous pressure and to match the shape of the insulated cable. This avoids surface tracking, surface discharges and the resulting failures. The design and modelling of joints on the standpoint of mechanical aspects was addressed recently by Lu et al [START_REF] Luo | A Sophisticated method of the mechanical design of cable accessories focusing on interface contact pressure[END_REF]. However, the switch from XLPE to an elastomer leads to different dielectric behaviours, whether it be the conduction processes or the field and temperature dependencies, which result therefrom. Further, it is complicated to produce materials having a predefined electrical conductivity.

Besides continuity in electrical conductivity, the shape of deflecting pieces, the compatibility with the capacitive field distribution and the possibility of having resistive field grading are worth considering.

The resistive field grading, since it is with slow reaction, does not solve the problem of transient stresses obtained during operation such as polarity reversal or during lightning impulse [START_REF] Baferani | Electric field tailoring in HVDC cable joints utilizing electro-thermal simulation: effect of field grading materials[END_REF]. A target for materials resistivity could be to reach the same field redistribution as obtained in the ac case with the capacitive distribution. An immediate consequence is that the residual field at grounding would be minimum. As regards the field grading due to field dependence of conductivity, it is not easy to control here as the average fields are relatively low, of the order of 5 kV/mm, and in general for insulating materials, the threshold for non-linearity is not much lower. Resistive field grading materials have comparatively lower threshold fields, typically 1 kV/mm. An advantage of FGM layer is to decouple the stresses in the insulation material of the cable and the joint body effectively. The design remains a tactful exercise, with trade-off to be done between losses, fields, and operating temperature [START_REF] Späck-Leigsnering | Towards electrothermal optimization of a HVDC cable joint based on field simulation[END_REF]. Finally, an important point is the thermal characteristics of the materials. In the examples shown XLPE/XLPE XLPE/SiR Ez (V/m) Er (V/m) XLPE/XLPE XLPE/SiR here, the insulation under the cone tends to be colder than the body of the joint, meaning that a counter-effect of field grading is obtained, i.e. field enhancement in the cold parts. Therefore, improvement could be brought by implementing joint material with higher thermal conductivity.

Compared to other studies on field distributions in joints, the work presented here considers the transient conditions of field establishment whereas very often only a direct resolution in stationary state is proposed. Multiple stress conditions can be tested. This is quite easily achievable, but it is important to stress that the collection of experimental data on conductivity, representative of materials in their operating environment, is a major preliminary step in all these modelling and design tasks.

Conclusions

The field distribution in HVDC cable joints has been investigated in non-stationary electrical and thermal conditions, considering XLPE as cable insulation and different materials as insulation joint:

EPDM, XLPE and SiR. The joint materials differed by their field and temperature dependencies of electrical conductivity and by their thermal conductivity. The radial distribution of the field follows predictable trends with temperature, either in isothermal or thermal gradient conditions, depending on materials electrical conductivity. The temperature gradient, and the nonlinear conduction, play a role of field grading. However, this effect is annihilated after polarity reversals: a transient overstress appears just after voltage polarity inversion in all cases involving a thermal gradient.

The tangential distributions of the field along the interface between the insulator of the joint and that of the cable have temporal and temperature behaviours that are not deduced in a simple way from the stress conditions. Under the deflecting cone, the smallest field values and mildest field redistribution are obtained with the joint insulation having the highest electrical conductivity (SiR). This feature has been explained by a shift of the field towards the cable insulation in which the geometrical features of the system produce less axial component. At the level of the high voltage semicon (central deflector), it is clear that the tangential field is higher when the mismatch between the conductivity of the two insulations is larger. In addition, the field grows as a function of time under stress. This is verified whether the system is in isothermal or thermal gradient conditions. Thermal gradient effects are still to be analysed.

The thermal conductivity of the joint material has a substantial impact on the temperature in the core of the joint and at the interface. This temperature gradient is obvious along the radial direction and is effective along the axial direction too. The joint being cooler under the deflecting cone, field strengthening appears in this region.

Figure 1 .

 1 Figure 1. Scheme of the modelled joint with main approximate dimensions: length 800 mm, conductor radius 22.8 mm, insulation thickness 20 mm in the cable and 70 mm in the joint.

Figure 2 .

 2 Figure 2. Field dependence of the electrical conductivity of XLPE, EPDM and SiR at 30 and 60 °C.

Figure 3 .

 3 Figure 3. Dimensions of the joint and positions of the simulated field profiles indicated by arrows: (1) Radially at a position z = 800 nm at mid-distance between the outer cone and central deflector; (2) axially: along the interface between the two insulators.

  Figure4a) shows the temperature distribution in the joint after 3 min and 8 h of heating, while Figure4b) shows the change in temperature as a function of the heating time for a cut at z = 800 mm. The temperature profile resulting from a resolution in stationary condition is also represented. It can be seen that several hours are necessary to reach a steady state. Even after 16 h, the thermal equilibrium is not completely reached. The interface between the two insulators, XLPE and EPDM in this case, is not very marked because the thermal conductivities are close, see Table2

  .

Figure 4 .

 4 Figure 4. a) Temperature distribution maps in the joint with XLPE/EPDM insulations at different times after applying the current of 1 kA in the conductor; b) Radial temperature distribution at different times and stationary solution, taken at z = 800 mm. Vertical lines indicate the boundaries of the insulators.

Figure 5 .

 5 Figure 5. Some examples of radial field maps in the joint after application of a current of 1 kA and voltage of +200 kV (t = 0) switched to -200 kV after 24 h.

Figure 6 .

 6 Figure 6. Radial field distribution in the XLPE/EPDM joint at different times after voltage application (+200 kV

Figure 7 .

 7 Figure 7. Radial field distribution at different times after voltage application for a) XLPE/XLPE joint and b) XLPE/SiR joint under non-stationary thermal gradient. Same conventions as in Figure 6.

FieldFigure 8 Figure 8 .

 88 Figure8represents the tangential field profiles in the joint material (i.e. EPDM), near the interface between XLPE and EPDM (see arrow n°2 in Figure3), as a function of time under stress, for the different thermal conditions previously explored. The tangential field values are the same in XLPE near the interface since the continuity rule imposes a continuity in tangential electric fields. Under isothermal conditions (Figure8a), it is clear that the tangential field is significantly greater on the right of the figure, corresponding to the deflector region. This can be explained by a geometry that does not include a field-grading cone, unlike the potential reference side of the joint. In fact, some designs use a deflector cone also for the central part of the joint set to HV[START_REF] Ye | Design aspects on HVDC cable joints[END_REF]. The quasi-constant axial field on the ground side corresponds precisely to the region (z = 20-160 mm) where the cone produces a non-radial component to the field. Over time, the tangential field tends to decrease on the ground side and to strengthen significantly on the HV side. These trends reflect the non-linear nature of the conductivity as well as the conductivity gradient due to the presence of two insulators. It is difficult to anticipate / explain given the divergence in geometry.

Figure 9 .

 9 Figure 9. Tangential field distribution at a) XLPE/XLPE and b) XLPE/SiR interface at different times after voltage

  temperature with a SiR joint material compared to EPDM is reflected in Figures 10b and 10c

Figure 10 .

 10 Figure 10. a) Temperature profiles along the radial direction in quasi-steady state (24 h) for the considered three couples of materials and profiles during thermal transient for XLPE/SiR at 2 h and 8 h. (b, c): Temperature distributions along the interface, for XLPE/EPDM and XLPE/SiR joints at different times. In all cases, 1 kA is injected to the conductor and the initial temperature is 30 °C.

Figure 11 .

 11 Figure 11. Tangential field distribution at XLPE/EPDM interface under isotherm condition at 70 °C.

Figure 12 .Figure 13 .

 1213 Figure 12. Potential distribution 3' after polarity reversal to -200 kV for (a) XLPE/XLPE and (b) XLPE/SiR joints. Corresponding radial (surface) and tangential (contour lines) field distributions for (c) XLPE/XLPE and (d) XLPE/SiR joints.

Table 1 .

 1 Parameters of the conductivity equation used for XLPE and EPDM materials.

		XLPE	EPDM
	A (S.I.)	0.8	97
	Ea(eV)	1.0	0.44

Table 2 .

 2 Material parameters used in the model.

		XLPE	EPDM	SiR	Semicon
	Relative permittivity εr	2.30	2.90	3.50	2.30
	Thermal cond. λ (W/m/K)	0.38	0.30	0.20	0.34
	Specific heat cp (J/g/K)	1.90	0.73	2.25	1.90
	Electrical cond. σ (S/m)	cf. Eq. (1)	cf. Eq. (2)	6.0×10 3

Data Availability Statement: Experimental data on conductivity are more completely described in [START_REF] Vu | Correlating conductivity and space charge measurements in multi-dielectrics under various electrical and thermal stresses[END_REF]. Specific simulation data can be available upon request to the corresponding authors.

Conflicts of Interest:

The authors declare no conflict of interest.