Guilhem Cassan
email: guilhem.cassan@unamur.be.

Marc Sangnier
email: marc.sangnier@unamur.be.

Jérémie Decalf

Romain Lutaud

Vincenzo Verardi

François Woitrin

Klaus F Zimmermann

The impact of 2020 French municipal elections on the spread of COVID-19 *

Keywords: COVID-19, Hospitalizations, Electoral turnout, Municipal elections, Prediction errors

Soon after the onset of the COVID-19 pandemic, the French government decided to still hold the first round of the 2020 municipal elections as scheduled on March 15. What was the impact of these elections on the spread of COVID-19 in France? Answering this question leads to intricate econometric issues as omitted variables may drive both epidemiological dynamics and electoral turnout, and as a national lockdown was imposed at almost the same time as the elections. In order to disentangle the effect of the elections from that of confounding factors, we first predict each département's epidemiological dynamics using information up to the election. We then take advantage of differences in electoral turnout across départements to identify the impact of the election on prediction errors in hospitalizations. We report a detrimental effect of the first round of the election on hospitalizations in locations that were already at relatively advanced stages of the epidemic. Estimates suggest that the elections accounted for at least 3,000 hospitalizations, or 11% of all hospitalizations by the end of March. Given the sizable health cost of holding elections during an epidemic, promoting ways of voting that reduce exposure to COVID-19 is key until the pandemic shows signs of abating.

Introduction

The two rounds of 2020 French municipal elections were planned to take place on March 15 and 22, 2020. By the beginning of March, the early spread of the COVID-19 epidemic led to a debate in France about whether the first round should be postponed. This option was finally rejected and Emmanuel Macron-the French President-announced on the evening of March 12 that the election would take place as planned. This decision was accompanied by the announcement of the closing of all schools and universities by March 16 and was followed by an announcement by Edouard Philippe-the then-Prime Minister-on March 14 about the closing of all non-essential public spaces by the next day to prevent the spread of COVID-19. This marks the start of anti-contagion policies in France.

According to an Odoxa opinion poll published on March 12, 64% of French people approved of the decision to maintain the election and 61% of voters reported that the epidemic would not change their decision to vote. On March 15,19,863,660 out of 44,650,472 voters in metropolitan France cast their vote, with no alternative but to go to the voting booth in order to do so. Although people were advised to maintain distance while voting and to use hand sanitizer, there was little availability of masks at the time and the recommendation for the general public was to not wear them since it was then believed that COVID-19 mainly spread via droplets rather than via aerosols, so that maintaining distance and hand hygiene would be enough to prevent contamination. On March 16, Emmanuel Macron announced that strict lockdown measures would be put in place from March 17 onwards and that the second round of the municipal elections was postponed sine die. The second round eventually took place on June 28, 2020, after anti-contagion policies had drastically reduced the circulation of In this paper, we show that the first round of 2020 municipal elections caused an acceleration of the COVID-19 epidemic in metropolitan France. Our estimates suggest that elections accounted for at least 3, 000 excess hospitalizations by the end of March, which represents 11% of all hospitalizations by this time.

Our methodology takes advantage of electoral turnout differences between départementsthe third-highest administrative level-to distinguish the impact of the election on hospitalizations from that of simultaneously implemented anti-contagion policies. Our approach builds on methods from the abnormal financial returns and public policies evaluation literatures. 1 We proceed in two steps. We first fit for each département a simple epidemic model of hospitalizations for COVID-19 suspicion over the period that excludes hospitalizations that might relate to events that took place by March 15 or in the following days. We then use these models to predict the evolution of the epidemic in each département if propagation conditions were held constant and compute daily predictions errors as the difference between the realized and predicted cumulative number of hospitalizations in each département.

Second, we relate prediction errors to turnout and to differences in the epidemic stage across départements by that date. This approach allows us to assess the causal effect of elections on hospitalizations while accounting for other contemporaneous events, such as anti-contagion policies, that were a priori uniform throughout the country. This approach explicitly accounts for different dynamics at the local level and builds on the assumption that prediction errors should not be related to turnout and epidemic stage on March 15 in the absence of an effect of the election on hospitalizations. We show that post-calibration errors are increasing with turnout in départements where COVID-19 was actively circulating on the day of the election. In contrast, turnout is not related to post-calibration errors in locations with low COVID-19 activity by March 15. These results reveal the impact of the first round of municipal elections on the COVID-19 epidemic. Applying the same methodology to the second round of the election, held on June 28 after a severe lockdown was implemented, we do not find a detrimental effect of the second round.

Our identification strategy is akin to a quadruple-differences method, effectively taking advantage of the following differences: the within-département difference between realized and predicted hospitalizations; the within-département difference between periods before and after the election; the between-départements difference in electoral turnout; and the between-départements difference in epidemic stage on election day. This combination of differences allows us to asses the causal impact of the elections on hospitalizations related to COVID-19. Importantly, our estimation strategy also allows us to effectively remove the first-order effect of factors that might explain differences in the dynamics of the epidemic, such as population density or the population structure.

As highlighted by [START_REF] Hsiang | The effect of large-scale anti-contagion policies on the COVID-19 pandemic[END_REF], most studies that analyze the impact of policies on COVID-19 rely on complex epidemiological models that require a detailed knowledge of the fundamental epidemiological parameters of the epidemic. Our approach, taken from the standard methods of reduced-form econometrics commonly used to assess the impact of public polices [START_REF] Angrist | Mostly Harmless Econometrics[END_REF], does not require such detailed information. It allows us to disentangle the impact of the election from others confounding shocks that may have hidden it without requiring much information about mechanisms of the epidemic.

The structure of the paper is as follows. Section 2 reviews the literature on the link between elections and the COVID-19 spread. Section 3 presents and discusses our methodology. Section 4 presents the results of our analysis of March 15 elections. Section 5 retrospectively discusses the expected impact of the second round of the municipal elections. Finally, Section 6 concludes.

Literature review

A large and growing literature aims at evaluating the impact of various events on the COVID-19 spread.2 In this literature, numerous studies investigate the role of elections on the spread of COVID-19.3 They can broadly be categorized based on the two types of approaches they adopt.

A first strand of papers relies on epidemiological methods. Berry et al. (2020) report no impact of the Wisconsin primary elections, comparing the epidemic trajectory of Wisconsin to that of the rest of the US, i.e., assuming that, absent the primary elections, Wisconsin would have followed the same epidemic trajectory as other US states. [START_REF] Leung | No Detectable Surge in SARS-CoV-2 Transmission due to the April 7, 2020 Wisconsin Election[END_REF] also study the Wisconsin primary elections, with a simple before-after epidemiological approach, showing that the number of cases and the effective reproduction number did not increase after the primary. Due to its before-after methodology, this paper does not use a counterfactual and relies on the strong assumption that the impact of the elections on the epidemic would have been so strong so as to be detectable directly by looking at changes of the fundamental parameters of the epidemic. Zeitoun et al. (forthcoming) take a similar approach to study the French municipal elections. They compare the post-election epidemic trajectories of départements with high turnout and low turnout. While this method improves on the simple before-after difference used by [START_REF] Leung | No Detectable Surge in SARS-CoV-2 Transmission due to the April 7, 2020 Wisconsin Election[END_REF], it does not allow départements to follow idiosyncratic pre-and post-elections trajectories. It amounts to assume that départements would have followed similar epidemic dynamics if elections had not taken place, no matter how different their underlying characteristics (e.g., population density or share of elderly population) are. It leads Zeitoun et al. (forthcoming) to report no effect of the French local elections on the spread of the epidemic in France. Finally, [START_REF] Duchemin | Bayesian investigation of SARS-CoV-2-related mortality in France[END_REF] present a Bayesian investigation of an epidemiological model that uses the number of deaths at the regional level to asses the effect of a variety of events on the COVID-19 epidemic in France. They report no effect of the first round of municipal elections. However, [START_REF] Duchemin | Bayesian investigation of SARS-CoV-2-related mortality in France[END_REF] make clear that their approach would be able to uncover such an effect only if massive.

A second strand of papers uses econometrics methods. [START_REF] Feltham | No increase in COVID-19 mortality after the 2020 primary elections in the USA[END_REF] study the presidential primary elections in the USA. They find no evidence of effect of these elections on the COVID-19 dynamics by using two methods: a matching difference-in-differences between counties with elections and counties without, and an epidemiological model. This paper relies on the assumption that the effect of holding an election would be the same no matter the turnout and no matter the local spread of the epidemic, making the findings potentially biased downwards by elections taking place in area with low COVID-19 circulation or with low turnout. Palguta et al. ( 2022) is an ambitious paper that studies the October 2020 Czech elections. These elections provide a interesting natural experiment as, for a random subset of constituencies, a one round local election was combined with a two rounds national election. For this random subset of constituencies, turnout was higher in the local elections (which combined both the one round local election and the first round of the national election) and a second round took place. They compare the evolution of the epidemic in constituencies with two elections and two rounds of elections to that in constituencies with only one election and one round. [START_REF] Palguta | Do Elections Accelerate the COVID-19 Pandemic? Evidence from a Natural Experiment[END_REF] find that the former group experienced an increase in the number of cases and hospitalizations, but no increase in tests positivity rate. This interesting finding is hard to interpret as it is impossible to disentangle how much of the change in epidemiological trajectories is due to the higher turnout in the first round of the election relatively to the holding of a second round. [START_REF] Bertoli | Turnout in the Municipal Elections of March 2020 and Excess Mortality during the COVID-19 Epidemic in France[END_REF] study the effect of the French municipal elections on excess mortality at home in the subset of French municipalities that have no hospital. Using an instrumental variable approach to predict turnout at the very local level, they report a qualitatively strong impact of the election on excess mortality. However, [START_REF] Bach | Does holding elections during a Covid-19 pandemic put the lives of politicians at risk?[END_REF] provide evidence that results reported by [START_REF] Bertoli | Turnout in the Municipal Elections of March 2020 and Excess Mortality during the COVID-19 Epidemic in France[END_REF] are driven by measurement error. Using individual level data and various econometrics methods, [START_REF] Bach | Does holding elections during a Covid-19 pandemic put the lives of politicians at risk?[END_REF] show that local politicians who participated in the 2020 French municipal elections did not face a higher mortality risk after the elections. This finding can be interpreted in two rather different fashions. Either that the excess hospitalizations we uncover did not predominantly concern this very specific population or, that candidates, being more healthy than the general population of the same age, may have been infected and hospitalized because of the election, but did not die. 4Our approach differs from these studies in several dimensions. As we combine an epidemiological approach with econometric methods, we take advantage of both approaches while avoiding to rely on unrealistic assumptions that each approach taken alone requires: we explicitly model each départment's counterfactual epidemiological evolution, absent both the election and the containment measures, which prevents us from comparing very different areas with one another as most of the epidemiological literature does and which may prove problematic given the highly non linear nature of epidemiological spread. From the econometrics approach, we take advantage of quadruple differences methods that avoid the unrealistic assumption behind the before-after approaches taken in the epidemiological literature, which can not account for other shocks, such as changes in containment policies for example. In addition, we explicitly look into how the holding of the election may have different consequences as a function of the initial level of circulation of the virus.

This allows us to present findings that are both methodologically sound and easily interpretable. In addition, the context of the French municipal elections allows for a precise interpretation of our findings. Indeed, since a severe lockdown started almost at the same time as the elections, and that only one round of the election took place in March, we precisely know what our estimates are capturing: the causal effect on hospitalizations of having gone voting on March 15 . Because of the lockdown, chains of contagions were almost entirely cut, except within household. That is, the hospitalizations that we capture can only be caused by direct contamination while voting or indirect contamination within the household by a person contaminated while voting. This is as close as one can realistically be to measuring the direct effect of the election on contaminations.

Data and methodology

This section presents the data used in this paper as well as the methodology we rely on to assess the impact of the March 15 elections on the spread of the COVID-19 epidemic.

Data

Our analysis relies on two main datasets: hospitalizations for COVID-19 suspicion and electoral turnout at the 2020 French municipal elections.

In the absence of systematic testing policy by spring 2020, incidence rates cannot be used to measure the epidemic spread. Hospitalization data are the best data that allow to accurately observe the epidemic by this time. French hospitalization data are open access data published by a governmental agency. Data are based on hospitals' reports and present the daily counts of hospitalization decisions for COVID-19 suspicion at département level from February 24 onwards. These widely trusted data are the official source for COVID-19 related hospitalizations in France. 2020 electoral turnout data for the first round of municipal elections are official electoral records available at the city-level. We aggregated turnout data at the département level.

Methodology

We use the daily cumulated number of hospitalizations for COVID-19 suspicion to fit a series of département-level epidemic trajectories up to the date at which individuals contaminated on March 15 start being hospitalized. We separately estimate the following standard logistic model of epidemiological trajectory for each département d:

Cumulated hospitalizations d,t = a d 1 + exp(-b d (t -c d )) , ( 1
)
where a d , b d and c d capture the asymptotic level, the inflection date and the scale of the epidemic trajectory in département d, respectively. We estimate equation ( 1) using all dates t until March 26, i.e. 11 days after the elections took place. This 11-day lag is one day shorter than the median estimate of the number of days from infection to hospitalization suggested by the clinical studies literature. 5, 6 As a result, the model's forecasts can be interpreted as départements trajectories in the absence of any event that took place since March 15.

We estimated model (1) for each of the 96 départements of metropolitan France. The model was successfully estimated for 91 départements. The 5 départements for which we are not able to calibrate the model are départements that do not exhibit sufficient variation in hospitalizations until March 26 to allow for parameters' estimation. These départements account for 1.6% of the total French population.

Following insights from the literature on short term epidemiological forecast (see Chowell et al. 2019 andRoosa et al. 2020a,b among others), we use the series of estimated parameters âd , bd and ĉd to predict for each département the daily cumulated number of hospitalizations up to 7 days after the end of the calibration period, i.e. up to April 2. 2020) estimate that the time from symptoms to hospitalization is between 5 and 12 days. The French Institut Pasteur relies on these estimates to announce a 5-day period from infection to onset of symptoms, followed by a 7-day period from symptoms to hospitalization. 6 We show in Section 4.3 that results are robust to the use of 10 or 12 days instead of 11.

7 The literature on forecasting the COVID-19 spread in the very early days of the epidemic tends to focus on relatively short term predictions, typically between 5 and 15 days. See for example Roosa et al. (2020a,b) and [START_REF] Read | Novel coronavirus 2019-nCoV (COVID-19): early estimation of epidemiological parameters and epidemic size estimates[END_REF]. Forecasts made at later stages of epidemic can use a longer horizon as they benefit from a long time span on which to fit their model, making the prediction more precise over a longer period of time. See for example [START_REF] Tariq | Transmission dynamics and forecasts of the COVID-19 pandemic in Mexico, March-December 2020[END_REF]. Also see [START_REF] Chowell | Fitting dynamic models to epidemic outbreaks with quantified uncertainty: A primer for parameter uncertainty, identifiability, and forecasts[END_REF] about the link between the length of the period of fit and the quality of the forecast and [START_REF] Chowell | Is it growing exponentially fast? -Impact of assuming exponential growth for characterizing and forecasting epidemics with initial nearexponential growth dynamics[END_REF] about the unprecision of forecasts in the early days of an epidemic. 8 We show in Section 4.3 that doubling the forecasting period to 2 weeks days does not alter the results.

Predicted trajectories proxy the evolution of the epidemic in each département in the absence of the election and of any other shock contemporary or posterior to the election, such as lockdown policies. We next use the actual number of hospitalizations for COVID-19 suspicion in each département to construct prediction errors in hospitalizations per 100, 000 inhabitants. As shown by Appendix Figures A1(a) and (b), predictions errors are generally positive over the post-calibration period, which suggests that most départements surpass their predicted epidemic trajectory after March 15. Our interest is however not to assess whether it is possible to correctly predict the evolution of the epidemic, nor to estimate whether policies implemented after this date were able to twist trajectories. In contrast, our interest lies in whether deviations of epidemic trajectories depend on the March 15 elections.

We take advantage of two sources of variations to assess whether the March 15 elections impacted the spread of the COVID-19 epidemic. First, we distinguish between départements depending on the local stage of the epidemic by the day of the election. Second, we use differences in electoral turnout to proxy for difference in exposure across départements at comparable stages of the epidemic. Accordingly, we estimate the following expression:

Prediction error d,t = T t=1 β t × Turnout d × (1 -High COVID-19 circulation d ) × τ t + T t=1 γ t × Turnout d × High COVID-19 circulation d × τ t + T t=1 δ t × τ t + T t=1 ζ t × High COVID-19 circulation d × τ t + Y d + Z d,t + α + d,t ,
(2) where Prediction error d,t is the difference between actual and predicted cumulated hospitalizations per 100, 000 inhabitants in département d on day t, Turnout d is electoral turnout on March 15 in département d, τ t is a variable equal to 1 on day t, High COVID-19 circulation d is a variable equal to 1 for départements at advanced stages of the epidemic on March 15, series of δ and ζ coefficients account for daily patterns in prediction errors across départements in both groups, Y d is a vector of département fixed effects which account for départementspecific patterns, α is a constant term, and dt is the error term. Z d,t is a vector of interactions between day fixed effects and two sets of départements' characteristics. We first include the population density and the share of population aged above 60 as time invariant characteristics that might affect the dynamics of the epidemic at the département-level. We collect official total population and population aged above 60 in each département from official population records and construct population density at the département level using départements area information.

Second, we control for two département-specific shocks that might be associated with both turnout and the local dynamics of the epidemic after the election: local weather conditions on March 15 and population compliance with lockdown restrictions. We obtain meteorological information from Météo-France, average station-level data into départementlevel series, construct an index of weather conditions from the first principal component of daily precipitations and mean temperature in each département, and save the value of this index on March 15.9 Changes in behavior may also problematic for our estimation is they turned out to be correlated with our prediction error, turnout and epidemic spread on election day. As the French government implemented a national lockdown, we expect this changes in behavior to vary homogeneously across departments. However, one may worry that the compliance with the lockdown varies across departments. No official information allow to measure population compliance with anti-contagion policies. To overcome this lack of information, we proxy for compliance with lockdown restrictions at the département-level using the ratio from mobility during the lockdown period to mobility in normal times, as originally constructed from smartphones geo-located data by GEO4CAST 's Covimoov application and made available for March 26 and April 2 in a Le Journal du Dimanche's newspaper article. We construct an index of compliance with lockdown restrictions from the average across these two dates in each département. 10 As the effect of these shocks will likely differ depending on the extent of epidemic spread on the election day-which also corresponds to the eve of the start of strict lockdown-, we further interact their daily coefficients with the High COVID-19 circulation variable.

Consistent with the aforementioned 11-day lag between infection and hospitalization, we use cumulated hospitalizations per 100,000 inhabitants on March 26 to construct the High COVID-19 circulation variable that distinguishes between départements depending on the stage of the epidemic by the day of the election. We arbitrarily distinguish between départments in the bottom third of the COVID-19 epidemic according to this measure and others.11 The latter are considered as locations at relatively more advanced stages of the epidemic. 12As shown by Appendix Figure A1(b), some départements exhibit very large prediction errors compared to others as we move away from the end of the calibration period. This feature is likely to let outliers drive the estimation of coefficients of interest for these days. We mitigate this threat by weighting post-calibration observations by the inverse of the standard errors of daily predictions returned by model (1).

We estimate expression (2) using the above mentioned weights in a linear regression and two-way standard errors clustered at the départment and day levels. The sample is made of the 91 départments for which model model (1) was successfully estimated and of all days from March 1 to April 2, 2020.

In expression (2), the main parameters of interest are the estimated series of β t and γ t . These coefficients indicate the impact of electoral turnout on hospitalizations for départements with low and high COVID-19 circulation by the day of the elections, respectively. Under the assumption that the March 15 elections impacted epidemic trajectories only in locations that were at advanced stages of the epidemic by that day, we expect βs to be close to zero and γs to be positive in the post-calibration period.

Threat to identification

A key assumption for the above presented approach to allow us to safely assess the impact of municipal elections on the dynamics of the COVID-19 epidemic is that electoral turnout on March 15 is unrelated to the stage of the epidemic by that date. Namely, turnout was low as only 45% of voters cast their vote, compared to 64% at the 2014 municipal elections. There is a wide consensus in the French society that this low turnout was mainly caused by the fear of contagion. This might actually be the case but would be a threat to identification only if differences in turnout across départements ended up being related to differences in the epidemic across départements. We find no evidence of such a correlation between the level of turnout in a départment and the information on the spread of the epidemic in that département on the day of the election. This is best illustrated by Figure 1(a) which plots turnout against publicly known cumulated hospitalizations on March 15. Turnout appears evenly distributed at each stage of the epidemic.

Figure 1(b) further accounts for the 11-day lag from infection to hospitalization to better capture the underlying stage of the epidemic in each département and only reveals a weakly decreasing link between turnout and hospitalizations. In contrast, turnout at the 2020 municipal elections is strongly correlated with turnout at the preceding municipal elections that took place in 2014 as shown by Figure 1(c) which we constructed by supplementing 2020 data with 2014 electoral turnout data. It shows that the shift in turnout was uniform across départements. Figure 1 All in all, while the COVID-19 epidemic might have impacted turnout at the 2020 municipal elections-a question that is beyond the scope of this paper-, differences in the spread of the epidemic by March 15 did not translate into differences in turnout across départements, thereby allowing us to confidently interpret estimates that will be delivered by our identification strategy.

Discussion of the methodology

The outcome of interest of our approach is the extent to which the first-step predictive model fails to predict the evolution of hospitalizations. In the absence of an effect of the election on hospitalization, our model should make similar errors of predictions across départements, no matter their turnout.

However, if the election had an effect on the epidemic, the prediction errors should be relatively larger in départements with relatively higher turnout. Indeed, if elections did contribute to spread the epidemic, the predictive model should underestimate by a larger amount the number of cases in départements with high turnout compared to départements with low electoral turnout. And this stronger underestimation should start only when individuals infected on the election day are hospitalized, not before. Similarly, the effect of turnout on the epidemic should only exist in départements in which contagious individuals are indeed present: a high turnout in a département with no or few contagious individuals should result in 0 additional contagions.

We therefore analyse prediction errors via a triple-difference approach: not only do we compare départements with high and low turnout before and after the elections, but we study how this double-difference varies between départements with very low infection rates around the election date and other départements. We would expect turnout to only have an effect on the epidemic in départements already affected by the epidemic at the time of the election. Since our outcome is a difference, and we analyse it using a triple difference, our methodology is akin to a quadruple differences approach. This alleviates concerns about omitted variables that one may have in the context of a double difference approach. Indeed, with this approach, an omitted variable is a threat to identification if and only if it is correlated : (i) with the difference between actual and predicted epidemic spread (not just the actual epidemic spread); (ii) with turnout; (iii) with time; (iv) with the level of epidemic spread on election day. Note in particular that many variables may be correlated with actual epidemic spread (think for example of the share of elderly). However, these variables would also directly contribute to our model's prediction. Therefore, while they may be correlated with actual epidemic spread, they will also be correlated with predicted epidemic spread, but it is unlikely that they would be correlated with the difference between actual and predicted epidemic spread, together with the other 3 dimensions of correlation required for an omitted variable to be a threat to identification. Note however that it is still possible that changes in behavior post election may be both correlated with turnout, prediction error and epidemic spread on election day. Given the implementation of a full lockdown 24h after the election, we believe this is unlikely to have happened in practice. However, we explicitly controls for differences in the enforcement of lockdown across départments.

This approach has several advantages. First, it does not require blind faith in the ability of the predictive model to deliver accurate predictions. In fact, it does rely on the model's predictions being wrong while a priori uncorrelated with turnout under the null assumption that elections had no impact on the spread of the COVID-19 epidemic. Second, the event study aspect of the approach allows us to exactly observe when the prediction errors become correlated with turnout: predictions error should start being correlated with turnout only when people infected on the election day start showing up at hospital, that is, only when enough (but not too much) time has passed since the election for the symptoms to be severe enough to lead to hospitalization. This approach therefore automatically implements a sanity check as the correlation between the model's prediction errors and turnout should emerge with a lag compared to the election date, but not too long a lag.

A drawback of our approach is however that these type of simple predictive models are typically precise in the short run only, so that predictions are likely to become more and more noisy the further away we move from the end of the calibration period, which should result in imprecise estimates. This is the reason why we stop the analysis 7 days after the end of the model's fit. This time span is however likely to cover most of the additional hospitalizations that could be related to the March 15 elections as severe lockdown policies were implemented in the days that immediately followed the elections, thereby limiting further transmission by people who would have been contaminated on that day. In a robustness check, we extend the analysis up to 15 days after the election.

Results

In this section, we first present and interpret the results of the study. We next investigate their robustness.

Relationship between electoral turnout and hospitalizations

Figure 2 presents the series of β t and γ t coefficients estimated from equation ( 2). The series of β t coefficients stays small and insignificant over both the calibration and prediction periods. This shows that turnout did not have any impact on hospitalizations in départements with very low infection rates on the day of the election. Similarly, the series of γ t coefficients is close to zero and statistically insignificant over the calibration period. In contrast, this series starts to increase by March 27 and becomes unambiguously statistically significant. This suggests that turnout is positively associated with hospitalizations in départements in which there were a relatively high number of contagious individuals by the election day exactly 12 days after the day of the election, in line with the 12-day lag between infection and hospitalization estimated by the literature.

As discussed in section 3, the uncovered positive relationship can be interpreted as evidence of a causal relationship from the election to hospitalizations. However, beyond the increasing pattern of the series of γ t coefficients after March 27, Figure 2 also displays increasing standard errors of the estimates as close as 3 days from the end of the calibration period. This feature calls for caution in the interpretation of the point estimates.

Quantification of the total effect

As shown by Appendix Table A1, γ coefficients estimated for March 27, 28 and 29 correspond to 23.9 (p-value = 0.000), 39.4 (0.000) and 48.0 (0.000) excess cumulated hospitalizations, respectively, for an hypothetical change in turnout from 0% to 100% in départements at relatively advanced stages of the epidemic by the election day. This contrasts with coefficients estimated on the following days that are larger and less precisely estimated. For instance, coefficients estimated for April 1 and 2 correspond to 66.7 (p-value = 0.005) and 70.7 (0.005) excess cumulated hospitalizations, respectively, for the same hypothetical change.

Actual electoral turnout data can help us quantify the contribution of the March 15 elections to the COVID-19 epidemic. To this end, we use estimated coefficients of equation ( 2) and compute turnout-related excess cumulated hospitalizations per 100,000 inhabitants on each day from the end of the calibration period to April 2 in départements that were at advanced stages of the epidemic on March 15 as:

Excess d,t = (γ t -γ0 ) × Turnout d , ( 3
)
where γ0 is the average of γ estimates over the calibration period. We then multiply these figures by each département population to obtain absolute figures, set excess hospitalizations to zero in départements with low COVID-19 activity on the election day, and sum daily excess hospitalizations across départements. We proceed identically with the bounds of 95% confidence intervals of γ estimates. Daily election-related excess hospitalizations are computed using equation (3). Short dashed lines are bounds of 95% confidence intervals. See the text for more details.

Robustness checks

We conducted a series of tests to demonstrate the robustness of reported results.

or 12 days as time from infection to hospitalization

One of the key feature of the estimation framework we use is the fact that hospitalizations reflect the epidemic situation with some delay. In order to test the sensitivity of reported results to this feature, we replicated the data construction and estimation steps using 10 or 12 days, rather than 11 days, as lag from infection to hospitalization. In the first case, March 25 is thus used in lieu of March 26 as the date at which the calibration period ends and as the day at which we distinguish between départements with low or high COVID-19 circulation by the time of the municipal elections. As the prediction model is calibrated on a shorter period, model ( 1) is successfully estimated for only 88 out of the 96 départements. The 8 left-aside départements account for 4.0% of the French population. Figure 4(a) displays coefficients of interest when estimating equation (2) using March 25 in lieu of March 26. Although less precisely estimated, the patterns of coefficients over the days after the end of the calibration period is similar to that found using March 26. Importantly, the coefficient Figures 4(c) and (d) plot results we obtain when using 12 days as time from infection to hospitalization. March 27 is thus used in lieu of March 26. In this case, model (1) was successfuly estimated for the same 91 départements as when using March 26. The patterns of the series of estimates of interest is qualitatively similar to baseline results. However, estimates of the effect of the elections become very large as we move away from the end of the calibration period. So do their standard errors. These observations suggest that these estimates are less reliable than baseline ones, presumably because March 27 is now included in the calibration period while this date is likely to already include the first excess hospitalizations linked to the March 15 municipal elections.

Alternative definitions of advanced epidemic stage

One of the source of variation we exploit to identify the effect of March 15 elections on the COVID-19 epidemic is the circulation of the virus in départements on the day of the elections. We arbitrarily distinguished between départements using the first tercile of the distribution of cumulated hospitalizations per inhabitant by this date. We use two alternative definitions of high virus circulation to study whether reported results hold when modifying the above mentioned arbitrary choice.

We first use the 25th percentile of the distribution of cumulated hospitalizations per 100,000 inhabitants across départements on March 26 to construct the group of départements considered as at advanced stage of the epidemic by March 15. Figure 5(a) presents the estimated coefficients of equation ( 2) when using this alternative cut-off. Figure 5(b) plots the corresponding total excess hospitalizations associated with the elections.

Second, we identify départements that experienced more than 7 days of increase in hospitalization until March 26 as départements with high COVID-19 circulation. Figure 5(c) and (d) display results obtained when using this categorization.

As show by Figures 5(a)-(d), both alternative definitions lead to results that are qualitatively and quantitatively similar to baseline ones.

Removing départements one by one

To test the sensitivity of results to a particular département, we re-estimate equation ( 2), but omitting each département one by one. Figure 6 displays the series of estimated coefficients. It allows us to visualize how the inclusion of each département affects estimates. While some series are actually distinct from others, thereby showing the large influence of some départements, the overall patterns are consistent with previously reported results.

14-day forecasts

Our estimation framework builds on a simple logistic model that we calibrate for each French département and whose predictions are then used in a second step. Figures 7(a display results obtained when extending the prediction and estimation period up to 14 days after the end of the calibration period. As shown by Figure 7(a), standard errors of estimated coefficients that build on more than 1-week ahead forecasts becomes very large. This best illustrates the weak capacity of the estimation framework to apply to longer time horizons and highlights again that only short-term estimates can reasonably be trusted. However, results obtained for this longer period calls for two comments. First, the structures of estimated series is qualitatively similar to that obtained previously. Second, estimates of the relationship between turnout and excess hospitalizations reaches a plateau about one week after the end of the calibration period. This latter observation is consistent with electionsrelated excess hospitalizations we capture being circumvented to exposed individuals as strict lockdown was enforced after the elections.

Analysis of the second round

According to our analysis, measures implemented on March 15 to prevent contamination in voting stations by the first round of the 2020 French municipal elections were not fully effective and resulted in thousands additional hospitalizations.

On May 22, the French government announced that the second round of the municipal elections would take place on June 28 in municipalities in which no list gained majority in the first round. About 15, 400, 000 voters-mostly in the largest municipalities-were called to vote again. In [START_REF] Cassan | Contaminé? Estimating the impact of French municipal elections on COVID-19 spread in France[END_REF]-an earlier version of this paper made public on June 22, 2020-we argued that it was very unlikely that holding elections on June 28 would cause a statistically detectable number of new contaminations. This prediction built on the fact that on June 19, when we last accessed available data, most départements qualified as safe places using the threshold above which we detected a worsening of epidemic trajectories because of the March 15 election. In addition, mask wearing and social distancing were much more widespread in the second round than they were during the first round.

Updated data allow us to actually assess the situation of French départements after the second round of municipal elections. To this end, we reconstruct the 32-day cumulated number of hospitalizations per 100,000 inhabitants by July 9-i.e., 11 days after the second round has taken place-in each département and compare it to the March 26 distribution and the above mentioned threshold. 13 As shown by Figure 8, only 4 départements were still, by the date of the second round, above the threshold of hospitalizations that we use to classify departments as unsafe. After more than two months of lockdown and severe anti-contagion policies, the epidemic situation in June was not comparable to that in March. While the first round took place at the beginning of the exponential part of epidemic curve, the lockdown essentially amounted to a reset of infections. Infection levels were thus much lower on June 28 than they were on March 15. As a consequence, it is likely that holding elections on June 28 did not cause a statistically detectable number of hospitalizations. In Appendix B, we further illustrate this claim by replicating the analysis around the second round of municipal elections. Results show that the approach used for the first round does not allow to uncover a significant effect of June 28 turnout on subsequent hospitalizations.

Conclusion

Combining simple epidemiological modelling with a quadruple differences flavoured econometric methods, we estimated the impact of the first round of the 2020 French municipal elections, held on March 15, on the spread of COVID-19. While evaluating the impact of policies on an epidemic spread poses various challenges, the methodology we propose makes possible to causally link events to the subsequent evolution of the COVID-19 epidemic. As discussed above, only very peculiar omitted factors, could threaten the estimation we propose and the interpretation of estimates. For example, behavioural changes in voting or shielding that would differ across locations depending on early exposure to the epidemic could bias estimates in a a priori unknown direction. However, limited information on the epidemic by the time of the elections and the documented absence of correlation between turnout and the spread of the epidemic suggest that such specific effects are unlikely to be at play.

We show that March 15 elections resulted in at least 3, 000 hospitalizations, which repre-sents 11% of all cumulated hospitalizations for COVID-19 suspicion in metropolitan France by the end of March. This contrast with the absence of impact of the second round, held on June 28 in a context where the circulation of the virus was considerably lower after a severe lockdown was implemented and with much more knowledge of anti-contagion gestures. From a policy perspective, our results match findings by [START_REF] Palguta | Do Elections Accelerate the COVID-19 Pandemic? Evidence from a Natural Experiment[END_REF] and inform us about the health cost of holding elections without proper anti-contagion measures in times of active virus circulation

Statements and declarations B Results around the second round of municipal elections

We replicated the main analysis around the second round of municipal elections that took place on June 28, 2020. More precisely, we estimated model (1) for each département using 32 days-the exact length of the period used to calibrate the model for the main analysis-to July 9, i.e. 11 days after the date of the second round, and compute prediction errors up to 7 days after this date. Because of lower epidemic activity by that time, the model was successfully calibrated for 82 départements only (the 14 left-aside départements account for 6.0% of the French population).

We then use expression (2) to relate these prediction errors to differences in June 28 turnout and to the stage of the epidemics by that date. The estimated expression only differs from equation ( 2) by the exclusion from Z d,t of a measure of population compliance with lockdown restrictions as no comparable measure is available for late June. Figure B1 displays the series estimated coefficients. Both series are close to zero and precisely estimated. This suggests that the method used for the first round does not allow to uncover a significant effect of June 28 turnout on subsequent hospitalizations, ot that the effect is precisely zero.

5

  Using Chinese data, Li et al. (2020), Chan et al. (2020) and Guan et al. (2020) estimate that the time from infection to onset of symptoms is between 4 and 5 days. Li et al. (2020), Huang et al. (2020), Wang et al. (2020), Cai et al. (2020), Chan et al. (2020), Chen et al. (2020) and Guan et al. (

  (d) and (e) further illustrate this claim by plotting the 2014 to 2020 turnout difference against cumulated hospitalizations on March 15 and 26, respectively.

Figure 1 :

 1 Figure 1: Electoral turnout and the COVID-19 epidemic.

  (a) 2020 turnout and hospitalizations on March 15. (b) 2020 turnout and hospitalizations on March 26. (c) 2020 and 2014 turnout.

  (d) 2014 to 2020 turnout difference and hospitalizations on March 15. (e) 2014 to 2020 turnout difference and hospitalizations on March 26.

Figure 2 :

 2 Figure 2: Relationship between electoral turnout and excess hospitalizations.

  Figure 3 plots elections-related excess and actual cumulated hospitalizations at the national level. Our point estimates suggest that the March 15 municipal elections accounted for about 10, 000 cumulated hospitalizations by the end of March. This figure represents about 40% of cumulated hospitalizations by that time. Estimates are however imprecisely estimated as discussed above. Namely, upper bounds of confidence intervals suggest that March 15 elections resulted in about 20, 000 hospitalizations. More conservative figures conveyed by estimates' lower bounds suggest that elections resulted in at least 3, 000 hospitalizations, which represents 11% of all cumulated hospitalizations for COVID-19 suspicion in metropolitan France by March 31. All in all results indicate that elections accounted for thousands hospitalizations.

Figure 3 :

 3 Figure 3: Election-related excess hospitalizations.

Figure 4 :

 4 Figure 4: Estimates using 10 or 12 days as time from infection to hospitalization.

  (a) Relationship between electoral turnout and excess hospitalizations, 10-day lag from infection to hospitalization. (b) Election-related excess hospitalizations, 10-day lag from infection to hospitalization. (c) Relationship between electoral turnout and excess hospitalizations, 12-day lag from infection to hospitalization. (d) Election-related excess hospitalizations, 12-day lag from infection to hospitalization. Sub-figures (a) and (c) mimic Figure 2. Vertical lines are 95% confidence intervals. Sub-figures (b) and (d) mimic Figure 3. Short dashed lines are bounds of 95% confidence intervals. Sub-figures (a) and (b) use March 25 in lieu of March 26. Sub-figures (c) and (d) use March 27 in lieu of March 26. estimated for March 26 remains insignificant while this date is now included in the postcalibration period. This is consistent with effects of the elections showing up only after 11 days. Figure 4(b) displays the corresponding total excess hospitalizations associated with the elections.

  ) and (b)

Figure 5 :

 5 Figure 5: Estimates using alternative definitions of advanced epidemic stage by March 15.

  (a) Relationship between electoral turnout and excess hospitalizations, 25th percentile cut-off. (b) Election-related excess hospitalizations, 25th percentile cut-off. (c) Relationship between electoral turnout and excess hospitalizations, 7-days of hospitalizations increase. (d) Election-related excess hospitalizations, 7-days of hospitalizations increase. Sub-figures (a) and (c) mimic Figure 2. Vertical lines are 95% confidence intervals. Sub-figures (b) and (d) mimic Figure 3. Short dashed lines are bounds of 95% confidence intervals. Sub-figures (a) and (b) use the 25th percentile of the distribution of cumulated hospitalizations per 100,000 inhabitants across départements on March 26 to construct the group of départements considered as at advanced stage of the epidemic by March 15. Sub-figures (c) and (d) identify départements that experienced more than 7 days of increase in hospitalization until March 26 as départements with high COVID-19 circulation.

Figure 6 :

 6 Figure 6: Estimates removing départements one by one.

Figure 7 :

 7 Figure 7: Estimates using up to 14 days after the end of the calibration period.

  (a) Relationship between electoral turnout and excess hospitalizations, 14-day forecast. (b) Election-related excess hospitalizations, 10-day lag from infection to hospitalization. Sub-figures (a) and (b) mimic Figures 2 and 3, but use observations up to 14 days after March 26. Vertical lines are 95% confidence intervals. Short dashed lines are bounds of 95% confidence intervals.

Figure 8 :

 8 Figure 8: 32-day cumulated hospitalizations per 100, 000 inhabitants on March 26 and July 9.

Figure B1 :

 B1 Figure B1: Relationship between electoral turnout and excess hospitalizations around the second round of municipal elections.

  Estimates of βt and γt from equation (2). See section 3 for more details. Vertical lines are 95% confidence intervals. Départements with high (low) COVID-19 circulation by June 28 are départements in the top two thirds (bottom third) of the distribution of 32-day cumulated hospitalizations for COVID-19 suspicion by July 9.

For example,[START_REF] Fang | Human mobility restrictions and the spread of the Novel Coronavirus (2019-nCoV) in China[END_REF],[START_REF] Qiu | Impacts of social and economic factors on the transmission of coronavirus disease 2019 (COVID-19) in China[END_REF]), Brauner et al. (2021[START_REF] Amuedo-Dorantes | Timing of social distancing policies and COVID-19 mortality: County-level evidence from the U.S[END_REF],[START_REF] Bonacini | Identifying policy challenges of COVID-19 in hardly reliable data and judging the success of lockdown measures[END_REF],[START_REF] Deopa | Coronagraben in Switzerland: culture and social distancing in times of COVID-19[END_REF],[START_REF] Kahanec | The impact of repeated mass antigen testing for COVID-19 on the prevalence of the disease[END_REF] and[START_REF] Juranek | The effect of non-pharmaceutical interventions on the demand for health care and on mortality: evidence from COVID-19 in Scandinavia[END_REF] look at the effect of policies against the COVID-19 spread.[START_REF] Bernheim | The Effects of Large Group Meetings on the Spread ofCOVID-19: The Case of Trump Rallies[END_REF],Dave et al. (2020aDave et al. ( ,b, 2021)),[START_REF] Harris | The Subways Seeded the Massive Coronavirus Epidemic in New York City[END_REF] study the role of various events in the COVID-19 spread. See also[START_REF] Adda | Economic Activity and the Spread of Viral Diseases: Evidence from High Frequency Data[END_REF] for evidence about the role of economic activities on viral spread.

There is also a literature that studies the impact of the COVID-19 epidemic on electoral outcomes. See for example[START_REF] Baccini | The COVID-19 pandemic and the 2020 US presidential election[END_REF],[START_REF] Leromain | Voting Under Threat: Evidence from the 2020 French Local Elections[END_REF] and[START_REF] Pulejo | Electoral concerns reduce restrictive measures during the COVID-19 pandemic[END_REF].

Using only mortality data,[START_REF] Bach | Does holding elections during a Covid-19 pandemic put the lives of politicians at risk?[END_REF] can not distinguish between these two opposite interpretations.

There are no meteorological station in 5 départements of metropolitan France. For each of these départements, we reconstructed daily meteorological information from the average of values observed in immediately neighbouring départements.

Maps available from Le Journal du Dimanche display mobility ratio in 5% intervals. We randomly draw a value within each interval to construct averages within départements across dates.

As shown by Appendix FigureA2, this threshold correspond to 14 hospitalizations per 100, 000 inhabitants.

 12 We show in Section 4.3 that results are robust to alternative definitions of the threshold used to distinguish between départements with low or high COVID-19 circulation on the elections day.

Sources: Authors' calculation using Santé publique France, Ministère de l'intérieur and Institut national de la statistique et des études économiques data. Regression lines of each subfigure represent the linear relationship between the variable represented on the y-axis and the variable represented in the xaxis. Associated equations displays estimated coefficients and their standards errors (in parentheses).

32 days is the exact length of the period-February 24 to March 26-used to construct the advanced COVID-19 epidemics indicator for the analysis of the first round.

and three anonymous reviewers for helpful comments. This work was financially supported by the Fonds Wetenschappelijk Onderzoek -Vlaanderen (FWO) and the Fonds de la Recherche Scientifique -FNRS under EOS Project O020918F (EOS ID 30784531), by French National Research Agency grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -A*MIDEX.

Data and code availability

All the data used in this article is publicly available. Replication codes are available from the authors' websites.

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix A Supplementary figures and table


Daniel Gomez
email: daniel.gomez-acero@supergrid-institute.com

Kosei Shinoda

Juan Paez Alvarez

Florent Morel

Marc Cheah-Mane

Oriol Gomis-Bellmunt

Piotr Dworakowski

J D Páez

Case study of dc-MMC interconnecting two HVDC lines with different grid topologies

Keywords: dc-dc HVDC converters, dc-MMC, grid topologies, multi-terminal dc grid, modular multilevel converters

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

INTRODUCTION

High voltage direct current (HVDC) transmission lines are currently used to transmit high power over long distances. The interest in interconnecting two or more HVDC lines has risen in the latest years inspiring multiple publications and a CIGRE working group (B4.76) dedicated to the study of dc-dc converters for HVDC applications.

HVDC transmission system does not have an established standard yet, hence, the installation features vary from one project to another, such as the operation voltage, the technology (line commutated converter or voltage source converter), the line topology (e.g. bipole, monopole), the grounding strategy (e.g. start point reactor), the protection scheme or the conductor type used (cable or overhead lines). Dc-dc converters are important devices for the development of future dc grids [START_REF]HVDC Grid Feasibility Study[END_REF] as they provide the interface needed to interconnect HVDC systems with different characteristics. Several HVDC dc-dc converters have been proposed in the literature to interconnect two dc systems [START_REF] Páez | Overview of DC-DC Converters Dedicated to HVdc Grids[END_REF], but the literature focuses mainly on the interconnections between two dc systems with different voltage levels [START_REF] Jovcic | Case Study of Non-Isolated MMC DC-DC Converter in HVDC Grids[END_REF], while the impact of their grid topology is not often considered.

There are four different grid topologies for HVDC transmission systems: asymmetric monopole (Fig. 1a), symmetric monopole (Fig. 1b), bipole (Fig. 1c) and rigid bipole (Fig. 1d) [START_REF] Gómez | Requirements for interconnection of HVDC links with DC-DC converters[END_REF]. Each topology behaves differently in case of a fault. There are different possible faults: faults pole-to-ground, pole-to-pole, or internal faults in a converter. Both monopoles and the rigid bipole must stop the power transmission during any fault and they remain inoperative until the fault is cleared. Compared to the monopoles, the rigid bipole can be reconfigured to isolate a fault on a converter. Then it can use the healthy conductors and converters to exchange, up to, half of the rated power. However, the rigid bipole loses the complete power exchange capacity in case of a fault in a conductor. In case of a pole-to-ground fault, the symmetric monopole and the rigid bipole have a pole displacement leading to an increased voltage (pole-to-ground) on the healthy pole. The symmetric monopole can experience voltages up to 2 p.u., whereas, voltages around 1.4 p.u. have been reported for a given rigid bipole application [START_REF] Goertz | Analysis of overvoltage levels in the rigid bipolar MMC-HVDC configuration[END_REF]. The main advantage of the bipole is its inherent redundancy. Contrary to the rigid bipole, the bipole continues the power transmission, without interruption, for any fault except the pole-to-pole fault. The bipole use the healthy pole and the metallic/ground return to exchange up to half the rated power without being subject to voltage or currents disturbances. To interconnect two dc systems with different grid topologies, a dc-dc converter is needed. The dc-dc converter must respect the operation limits of each system throughout any operation condition, including faults. These operation limits can vary depending on the installed equipment, the protection strategy and the fault detection algorithm. This paper focuses on the operation limits given by a basic fault detection algorithm. The interconnection of two dc systems with different grid topologies (rigid bipole and symmetric monopole) is considered in this paper. To interconnect the lines, a non-isolated dc-dc converter is considered, here called flexible dc-MMC. The converter is based on the one presented in [START_REF] Kish | Modeling techniques for dynamic and steady-state analysis of modular multilevel DC/DC converters[END_REF], [START_REF] Kish | A Modular Multilevel DC/DC Converter With Fault Blocking Capability for HVDC Interconnects[END_REF]. It offers lower losses and cost compared to the isolated dc-dc converter front-to-front [START_REF] Lüth | High-frequency operation of a DC/AC/DC system for HVDC applications[END_REF], [START_REF] Sasongko | A front-to-front (FTF) system consisting of two modular multilevel cascade converters based on double-star chopper-cells[END_REF]. The study evaluates the dc-dc converter behavior and its impact on the HVDC systems in normal operation and during faults.

3

The paper starts by presenting the main characteristics of the HVDC lines to interconnect. Section 3 presents the topology and control of power converters (ac-dc and dc-dc). Section 4 presents the parameters of the simulation model. The simulation results are shown in section 5 and the conclusions are given in section 6.

PROPOSED CASE STUDY

This paper investigates the use of a dc-dc converter interconnecting two lines with different grid topologies based on a possible case proposed in [START_REF] Gómez | Requirements for interconnection of HVDC links with DC-DC converters[END_REF]: the interconnection between the NordLink and Cobra Cable projects. The NordLink project has 54 km of underground cable and 516 km of submarine cable connecting Norway and Germany. This HVDC link is based on a rigid bipole (RB) rated at 1400 MW with ±525 kV [START_REF]hitachi ABB Power Grid[END_REF]. The Cobra Cable project has 325 km of submarine cable interconnecting Denmark and the Netherlands. This HVDC link is based on a symmetric monopole (SyM) rated at 700 MW with ±320 kV [START_REF] Tourgoutian | Design considerations for the COBRAcable HVDC interconnector[END_REF]. The approximated geographical layout is presented in Fig. 2. The interconnection demands a dc-dc converter able to adjust the dc voltage and the grid topology between the two projects.

The possible benefit of this interconnection is the increased power flow flexibility between all the interconnected countries: Netherlands, Germany, Denmark and Norway. The installation of a dc-dc converter enables the direct power exchange between Norway and Netherlands, not possible in the initial system. The dc-dc converter can also increase the redundancy of the interconnection. After a line or converter fault, the interconnected system can reconfigure itself to isolate the damaged segment allowing the power exchange between the healthy sections. For example, in case of a fault between the dc/dc converter and Denmark (or in the ac/dc converter in Denmark), a power exchange with Netherlands-Norway-Germany could be possible after the fault isolation. In the initial system, the same fault will isolate the Netherlands from the Norway.

CONVERTER TOPOLOGIES AND CONTROL

The proposed case study considers two types of HVDC converters: ac-dc and dc-dc converters. The considered ac-dc converters are modular multilevel converters (MMC). The dc-dc converter is a variation of the dc-MMC presented in [START_REF] Kish | Modeling techniques for dynamic and steady-state analysis of modular multilevel DC/DC converters[END_REF]. Some details are presented in this section.

Ac-dc modular multilevel converter

The MMC is shown in Fig. 3, it has two arms per phase (upper and lower). Each arm is composed of a series connection of half bridge submodules (HBSMs) and an inductor. At the connection point between the upper and lower arm, there is an ac output port. The converters used here have 3 legs for a three-phase connection to the ac grid.

The MMC controller is designed in line with [START_REF] Shinoda | Virtual capacitor control: mitigation of DC voltage fluctuations in MMC-based HVdc systems[END_REF]. A general control diagram is presented in Fig. 4a. Three energy controllers (total, horizontal and vertical) manage the energy by setting the current references (I ref diff and I ref ac), which are a linear combination of the upper and lower arm currents Iup and Ilow. The control changes the ac current reference depending on the control mode, either voltage control or power control. The voltage controllers were tuned to have a second order response with a time constant of 100 ms and an overshoot of 10%. The converter 4 control adopts a virtual capacitor controller (VCC) as presented in [START_REF] Shinoda | Virtual capacitor control: mitigation of DC voltage fluctuations in MMC-based HVdc systems[END_REF], which is a proportional control that helps to support the dc voltage variations using the energy in the converters: energy stored in the SMs capacitors. The energy in the converter varies proportionally to the dc voltage fluctuation and the virtual capacitor coefficient value, here set to 1. This coefficient is used to calculate the voltage and energy controllers. The power control in the rigid bipole is modified (Fig. 4b): both MMCs (positive and negative pole) control the total energy with the ac current and have a common dc terminal voltage ΔVdc to control the dc power [START_REF] Hirsching | On control and balancing of MMC-HVdc links in rigid bipolar configuration[END_REF]. The station controlling the voltage implements the control presented in Fig. 4a including VCC.

Flexible dc-dc modular multilevel converter

The dc-dc converter used for the proposed case study is a modular structure presented in Fig. 5. Each phase of the converter is composed of 3 arms (upper, middle and lower). These arms contain a series connection of submodules (SMs) either half bridge (HBSM) or full bridge (FBSM) and an arm inductor (L). The FBSMs are used to have negative voltage outputs and to block the dc fault currents. The dc-dc converter needs enough FBSMs to block the fault interactions between the two interconnected dc systems. In this study, the upper and lower arms use 400 FBSMs each. The converter has an output passive filter (Lout) allowing the ac power to circulate between phases and other arms. These interactions are needed to keep the internal energy balanced [START_REF] Kish | A Modular Multilevel DC/DC Converter With Fault Blocking Capability for HVDC Interconnects[END_REF], [START_REF] Cheah-Mane | Energy-based control of a DC Modular Multilevel Converter for HVDC grids[END_REF]. Although Lout works as an ac filter, the control strategy keeps the phase currents balanced to avoid ac components on the dc side.

The high voltage side (left side in Fig. 5) is connected to the rigid bipole while the low voltage side (right side in Fig. 5) is connected to the symmetric monopole line. The number of phases can change depending on the converter rated power and the semiconductors current rating. A dc-dc converter with three phases is considered in this paper.

The converter is controlled to keep the internal energy constant, while a power is transferred between dc systems. The control strategy is a cascaded structure with the high level control following the energy references and the low level controlling the currents. There are three energy controllers per converter phase: total energy (Σ), total difference (ΔT) and the difference between the upper and lower arm (Δu-l). The internal currents are controlled to follow the references given by the energy controllers, the pole imbalance (on the monopole side) and the power reference, as shown in Fig. 6. In this work, a power control mode is implemented, but additional controllers can be added to control the voltage on the dc sides.

SIMULATION MODEL

Figure 7 shows the schematic of the simulation model implemented in Matlab/Simulink. The 7 converters were implemented using the parameters presented in Table 1. The converters use average arm models, including the blocked state, depending if they are composed by HBSM [START_REF] Zhang | Average value MMC model with accurate blocked state and cell charging/discharging dynamics[END_REF] or FBSM [START_REF] Xinkai | Fast electromagnetic transient simulation models of full-bridge modular multilevel converter[END_REF]. The cables are modelled using the wideband model from the Best Paths project [START_REF] Paths | Best Paths, Transmission for sustainability[END_REF]. The model considers two reference points: the first on the dc side of the RB, which is grounded through a resistance (RG) of 0.7 Ω as presented in [START_REF] Goertz | Analysis of overvoltage levels in the rigid bipolar MMC-HVDC configuration[END_REF]; the second, on the ac side of the SyM line through a star point reactor as shown in Fig. 7 [18, p. 291]. The five converter stations implement a fault detection algorithm, which uses local voltage and current measurements to detect a fault. The fault detection algorithm blocks the converter before the arm current reaches 3 kA, as the complete case study was designed using IGBT valves rated at 1.5 kA and 3 kV. After blocking the converter, it is assumed that the fault current circulates through the by-pass thyristors integrated within the SMs [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF]. In addition, the algorithm trips the station ac circuit breakers (ACCB) to stop the fault current contribution from the ac side. The fault detection algorithm has a 300μs delay [START_REF] Dennetière | Modeling of modular multilevel converters for the France-Spain link[END_REF] to emulate the detection, communication and tripping time inside a converter station. An additional 100ms delay is added to the ACCB opening signal. With regard to the voltage measurements, the algorithm triggers the block signal if the measurements deviate 20% from their nominal values. The voltage thresholds in the dc-dc converter are set to 15% of deviation, to block the converter before the tripping of the protections in the lines. This last feature allows the dc-dc converter to act as a firewall i.e. preventing the spreading of fault disturbances from one side to the other.

SIMULATION RESULTS

The proposed case study interconnecting two HVDC lines using a dc-dc converter may have multiple power flow distributions around the system. Figure 7 shows the positive convention for the dc currents, which are also used for the power direction. The positive power direction in the lines are from the dc side towards the power controlled stations, whereas the positive convention in the dc-dc converter is from the RB to the SyM line. Eight power flow variations are proposed to test the system during normal operation, i.e. without faults. Based on these eight power flow scenarios, the fault behavior is analyzed considering 2 different fault locations per case, which gives a total of 16 fault simulations. The considered fault locations are: a fault on the positive pole of the rigid bipole (P-Ctrl RB+) and a fault on the positive pole of the V-Ctrl SyM station, as shown in Fig. 7. The details of the power flow directions and fault locations are organized in Table 2.

Table 2: Variations of the case study, where the positive conventions (+) are shown in Fig. 7. In the RB and SyM lines from right to left (towards the P-Ctrl stations) and in the dc-dc converter from the RB to the SyM. The simulation starts with nominal voltage and power reference on the RB and SyM lines, 1000 MW and 300 MW respectively. After reaching the steady state, a power reference is sent to the dc-dc converter to exchange power (400 MW) between dc systems in both directions. The simulation results in normal operation are shown in Fig. 8, for case 1: positive power flow in the lines and bidirectional exchange through the dc-dc converter. The results obtained for the other 7 cases are similar: the fluctuations presented in Fig. 8 are representative for all tested normal operation cases. The power exchange measured at the five converter stations are presented in Fig. 8a. It can be seen that the stations in voltage control mode change their power exchange to compensate the power through the dc-dc converter (magenta line), while the stations in power control keep their power exchange invariant as their references have not been changed (solid lines).

Parameter
Figure 8b shows the voltage variations pole-to-ground on each converter. These perturbations depend on the voltage control tuning on each line (time response of 100 ms) and the rate of change in dc-dc converter power reference (2.3 GW/s, i.e. 3.3 p.u./s). In this case study, the voltage perturbations do not exceed 6% of the nominal line voltage, but they can be reduced if a decreased slope in the dc-dc converter power references is used.

Figure 8c shows the variations in the average total energy stored in every converter. In this case, the power controlled station in the SyM line has the higher energy variation (11.4%) followed by the voltage control station in the same line. These energy variations are the consequence of the VCC, which helps to keep the dc voltage regulation using the converter internal energy. For the converters in the RB line, the VCC is implemented only in the V-Ctrl station. The P-Ctrl station controls the total converter energy with the ac power (Fig. 4b) therefore, P-Ctrl in RB keeps its energy constant. Similar to the voltage variation, these energy disturbances can be attenuated by reducing the slope in the dc-dc converter power reference, but for this paper the slope remains unchanged to test the systems responses to fast power transients. Regarding the fault simulations, the starting sequence is the same, but the dc-dc converter only exchanges power in one direction: positive from RB to SyM and negative in the opposite direction. After reaching the steady state, a fault of 1 Ω to ground is simulated (at 2s), at the terminals of the P-Ctrl RB+ or V-Ctrl SyM as shown in Fig. 7.

The results presented in Fig. 9 are the worst cases per station found when a fault in the RB line is simulated (odd cases of Table 2). During the fault, the voltage in the faulted pole drops to zero as shown in Fig. 9a. At the fault instant (t=2s), the negative pole in the RB has an increased voltage reaching -880 kV for the 15 th case (see Fig. 9b).

The steady state voltage for this pole stabilizes at -762 kV, which represents 1.45 p.u., similar to the results found in [START_REF] Goertz | Analysis of overvoltage levels in the rigid bipolar MMC-HVDC configuration[END_REF]. On the SyM side (Fig. 9b-c), the voltages do not exceed ±10% of the nominal voltage for any case, which agrees with the initial control design. Figure 10 presents the higher dc currents measured at the RB (Fig. 10a), the SyM (Fig. 10b) and the dc-dc converter (Fig. 10c) terminals for the cases when the fault is on the RB side. Figure 10a shows a peak of 8 kA for the currents measured in V-Ctrl RB+ converter. It is worth mentioning that the converter protection strategy blocks the MMC under 400μs after the fault. The currents observed in V-Ctrl RB+ continue to rise until the ACCB is opened: after 2.1 s, which illustrates how the ac currents feed the fault in that pole. On the SyM side (Fig. 10b) the currents have a perturbation due to the loss of power from/to the dc-dc converter (t = 2s). After the transient, the currents reach the operation point where the voltage controlled stations exchange the same power as the power controlled stations Figure 11 shows the worst voltages measured at the stations, for the simulations when the fault is on the SyM line (even cases in Table 2). The voltages on the RB exceed the 10% of variation with respect to the nominal value for the cases 8 (Fig. 11b), 10 and 12 (Fig. 11a). These fluctuations are consequence of the RB control scheme, which is sensitive to the variations in the dc side due to the terminal control in the P-Ctrl station. The voltage fluctuations depend on the rate of change of the power delivered by the dc-dc converter. These variations can be reduced if the dc-dc converter reduce the disturbances during the fault. Reference [START_REF] Cheah-Mane | Energy-based control of a DC Modular Multilevel Converter for HVDC grids[END_REF] present a control strategy that allows the converter to control the fault current, decreasing the magnitude of the disturbances. On the SyM side, the positive pole voltage drops to zero (Fig. 11c), while the negative pole presents an overvoltage of 2 p.u. (Fig. 11d). The results were obtained without surge arresters, but the implementation of overvoltage protections can reduce the voltage peak and steady state values. Figure 12a shows the currents in the RB and how they recover steady state values after the fault. Similar to the results presented in Fig. 10, the power and voltage controlled stations find the operation point where the dc-dc is absent. Figure 12b presents the dc currents in the SyM and how they are blocked in a short time (less than 50 ms). Figure 13 presents the energy in the dc-dc converter during and after the 16 faults cases of Table 2. Figure 13a presents the cases for a fault on the RB side. In these scenarios the energy variations are near 1% of the nominal value (30.24 MJ). For the cases where the fault is located on the SyM (Fig. 13b), the energy in the converter increases up to 110% as a consequence of the FBSMs charging as they stop the fault current.

(a) (b)

Figure 13: Energy in the dc-dc converter for (a) a fault in the RB (odd cases of Table 2) and (b) a fault in the SyM (even cases of Table 2).

CONCLUSIONS

Dc-dc converters are needed for the interconnection between dc systems with different characteristics. The present publication explores the hypothetical interconnection between the NordLink (rigid bipole -RB) and Cobra Cable (symmetric monopole -SyM), using a non-isolated dc-dc converter.

The case study was modelled and simulated in Matlab/Simulink. The power flow in the lines and through the dc-dc converter was changed to simulate 8 possible normal conditions. These initial simulations validated that the exchange of power between the two lines does not trip any protection. After validating the converter behavior in normal operation, a set of 16 fault cases were simulated. Satisfactory results were obtained for the dc-dc converter firewall capability: in case of a fault in a line, the dc-dc converter is able to stop the fault propagation to the healthy system. This firewall capability was validated independently of the power flow direction or the fault location. If enough FBSMs are installed in the dc-dc converter, the firewall capability can be ensured without the presence of dc circuit breakers.

The interconnected lines do not require major adjustments to operate in an interconnected system using a dc-dc converter. The control strategy implemented in the dc-dc converter can be adapted to guarantee the normal operation of the initial lines, and its protection strategy can be design to operate before the line protections. A fast power exchange between dc lines was tested in this paper disturbing the operation of the lines without triggering the fault protections. A slower rate of change can be implemented to reduce the disturbances on the system. The case study proposed in this paper did not change the grounding strategy and reference points of the initial links. A dc reference is found on the RB and an ac reference on the SyM line. In this particular case, no problems were found, but further studies must be carried to check the impact of multiple reference points on the protection strategy and isolation coordination. Further studies should evaluate the impact of the ac faults on the dc-dc converter control strategy and fault detection algorithm. Additional fault studies should be done to assess the dc-dc converter behavior under different fault conditions. Different types of faults and locations should be tested. Supplementary studies can be done for the degraded operation, where a fault in the interconnected system leads to the isolation of a portion of a line (e.g. a three-terminal system or the RB reconfiguration into an asymmetric monopole -isolating the faulted converter).

Figure 1 :

 1 Figure 1: HVDC grid topologies. (a) asymmetric monopole (b) symmetric monopole (SyM) (c)bipole with metallic or ground return and (d) rigid bipole (RB) [4].

Figure 2 :

 2 Figure 2: Map of the North Sea region presenting NordLink and Cobra cable. The dc-dc converter is proposed at the crossing point.

Figure 3 :

 3 Figure 3: Circuit diagram of the ac-dc modular multilevel converter: submodule (SM), arm inductor (L) and half bridge submodules (HBSM).

Figure 4 :

 4 Figure 4: MMC control diagrams. (a) The general control strategy for power or voltage control mode. (b) The terminal control for the rigid bipole (ungrounded station).

5

 5

Figure 5 :

 5 Figure 5: Flexible dc-MMC for interconnections between different grid topologies.

Figure 6 :

 6 Figure 6: Control strategy used for the dc-dc converter.

Figure 7 :

 7 Figure 7: Proposed case study interconnecting a rigid bipole (upper side) with a symmetric monopole (lower side). The currents present the positive power flow direction in the lines and through the dc-dc converter. The squares adjacent to each ac-dc converter represent the ac circuit breakers.

Figure 8 :

 8 Figure 8: Simulation results in normal operation for the case 1. (a) Power per station. (b) Pole-to-ground voltage, positive (+) or negative (-) measured at the different converter terminals in the system (the negative poles were inversed to simplify the visualization). (c) Total energy in the converters.

Figure 9 :

 9 Figure 9: Dc voltages pole-to-ground during and after the fault on the RB side (worst cases). (a) positive pole in RB. (b) Negative pole RB. (c) Positive pole SyM. (d) Negative pole SyM.

9 (

 9 behavior expected without the use of dc-dc). Figures9b-cand 10b illustrate that the dc-dc converter can act as a firewall during a fault in a RB, as the healthy link continues normal operation once the dc-dc converter is blocked, irrespectively of the state of the other link.

Figure 10 :

 10 Figure 10: Dc currents during and after the fault on the RB side (worst cases). (a) currents in all four MMCs in the RB line. (b) Currents measured in the SyM. (c) Currents measured in the dc-dc.

Figure 11 :

 11 Figure 11: Dc voltages pole-to-ground during and after the fault on the RB side (worst cases). (a) positive pole in RB. (b) Negative pole RB. (c) Positive pole SyM. (d) Negative pole SyM.

Figure 12 :

 12 Figure 12: Dc currents during and after the fault on the RB side (all even cases on Table 2). (a) currents in all four MMCs in the RB line. (b) Currents measured in the SyM. (c) Currents measured in the dc-dc.

  Energy in dc-dc converter for a fault in RB Energy in dc-dc converter for a fault in SyM +10% Case 811

Table 1 :

 1 Converters parameters

	Parameter	MMC Rigid bipole	MMC monopole	dc-dc
	Rated power	700 MW per pole	700 MW	700 MW
	Dc voltage	525 kV	640 kV	±525 kV -±320 kV
	Frequency	50 Hz	50 Hz	150 Hz
	Arm inductance	47 mH	69.8 mH	15 mH
	Number of phases	3	3	3
	Average SM voltage	1.6 kV	1.6 kV	1.6 kV
	Upper arm
	 Number of HBSMs	 329	 400	 0
	 Number of FBSMs	 0	 0	 400
	 SM Capacitance	 11.1mF	 9.1 mF	 7 mF
	Middle arm
	 Number of HBSMs			 650
	 Number of FBSMs	Not present	Not present	 0
	 SM Capacitance			 7mF
	Lower arm
	 Number of HBSMs	 329	 400	 0
	 Number of FBSMs	 0	 0	 400
	 SM Capacitance	 11.1mF	 9.1 mF	 7 mF