Alexandre Mathieu

Gilles Fraisse

Martin Thebault

Simon Thebault

Simon Boddaert

Leon Gaillard

Failure Risk Analysis of Photovoltaic Systems Based on Literature Review

Keywords: PV system, Failure risk analysis, FMECA

In the context of the quick expansion of photovoltaic (PV) capacity, monitoring and correcting system failures are crucial to maximize performance. To reduce these sources of underperformance, a well-rounded knowledge of failures becomes essential. This paper highlights the most critical photovoltaic failure modes using the Failure Mode Effect and Criticality Analysis (FMECA) methodology. A review of the current knowledge of failures in PV systems, their detection methods, and their relationships is first performed. Then, this study ranks the failures based on their Risk Priority Number (RPN) deducted from their severity, occurrence, and detection score. The novelty of the approach relies on the adaptation and combination of assigned scores from different literature sources. The failure risk analysis especially outlines that critical failure modes occur in any component of the PV installation and every single part of the system needs special attention to manage underperformances.

Introduction

Acknowledging the devastating consequences of climate change such as the dramatic increase in temperature, the rising water levels, and the intensification of extreme meteorological events [START_REF] Pachauri | Climate change 2014: synthesis report[END_REF] strong incentives at the European level have led to a massive deployment of renewable energies to mitigate energy-related environmental impact. The International Renewable Energy Agency forecasts a striking expansion of the photovoltaic capacity to 14 TW in 2050 in the 1.5°C scenario (International Renewable Energy Agency, 2021) which would correspond to 15 times more than in 2020 [START_REF] Masson | Trends in Photovoltaic Applications[END_REF]. At the dawn of an energy paradigm shift, developing a PV performance guarantee becomes crucial to meet the objectives of climate plans and European directives such as the Energy Performance of Building Directive, EPBD (European Union, 2018).

In recent decades, the photovoltaic (PV) market has grown rapidly but the operation of PV installations continues to face significant challenges. With a 23% increase in capacity in 2020 [START_REF] Masson | Trends in Photovoltaic Applications[END_REF], the global photovoltaic market is experiencing unprecedented growth. On the other hand, substantial production losses subsist and around 5% of the total theoretical production could be recoverable [START_REF] Leloux | Performance analysis of 10,000 residential PV systems in France and Belgium[END_REF][START_REF] Raycatch | Solar Asset Optimization, Industry Benchmark Study[END_REF] if adequate operation and maintenance are ensured. Therefore, the investigation and alleviation of underperformance sources become essential to optimize the production of solar installations.

Mitigating failures reduce the Levelized Cost of Energy (LCOE) by increasing the operating lifetime of PV systems [START_REF] Aghaei | Review of degradation and failure phenomena in photovoltaic modules[END_REF] and making power plants financially more viable. As mentioned by Catelani [START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF], failures and their maintenance are rarely taken into account in the initial phases of PV projects. However, reducing underperformance effects would significantly improve the overall business cases of PV plants.

This study aims to map PV failures and rank them based on risk. Inspired by [START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF][START_REF] Colli | Failure mode and effect analysis for photovoltaic systems[END_REF][START_REF] Collins | Reliability and availability analysis of a fielded photovoltaic system[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF][START_REF] Rajput | Risk priority number for understanding the severity of photovoltaic failure modes and their impacts on performance degradation[END_REF][START_REF] Rongbin | Failure Risk Analysis of the Photovoltaic Modules Based on the Improved FMECA Method[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF], the failures Risk Priority Numbers (RPN) are computed from its performance reduction, occurrence, and detection ability as in a Failure Modes, Effects and Criticality Analysis (FMECA) process. A sampling method taking scores according to several data sources is then established to alleviate knowledge bias and take full benefit of the available information. Finally, the most critical failures are short-listed and described extensively.

State of the art on photovoltaic failure modes and detection methods

In this section, the failure scope is defined to further list failure modes. Then, failure losses are dissociated from normal photovoltaic losses, and detection methods are collected to identify and avoid significant power losses. Finally, the inhomogeneous relationship between failures and detection methods is highlighted.

Failure Scope

This study focuses on failure modes that directly impact system performance on the following components: PV array, cables, and inverter. Even though PV systems are defined by their integration, even sometimes on buildings, failure modes specific to these features are accounted for as changes to PV array performance. This choice keeps the failure list generic and exploitable to any photovoltaic installation. Similarly, auxiliary systems are excluded from consideration as in most cases they do not affect the production generation itself [START_REF] Bun | Détection et localisation de défauts dans un système photovoltaïque[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF], and the failure of protection systems may be assumed negligible [START_REF] Miquel | Dysfonctionnement électriques des installations photovoltaïques: points de vigilance[END_REF] except for fuse faults that can be included in the "Combiner box failure" and "Inverter defect" category. Each of those system sub-components is subject to failures along its life and might impact the system performance at different levels.

Power photovoltaic underperformances: expected losses vs failure

All photovoltaic systems are affected by underperformance which results from expected losses. Those can be estimated at the design phase and should not be considered as failures. These can be divided into six categories [START_REF] Bun | Détection et localisation de défauts dans un système photovoltaïque[END_REF]:

• Module temperature losses: the efficiency of PV cells varies with temperature, and temperature coefficients are commonly specified on manufacturer datasheets.

• Non-optimal irradiance collection: effects including spectral response [START_REF] Lindsay | Errors in PV power modelling due to the lack of spectral and angular details of solar irradiance inputs[END_REF] and shading.

• Non-optimal array production extraction: Mismatches between PV modules and strings due to the slight differences in electrical characteristics cause some inevitable losses.

• Natural system aging: materials and components naturally degrade over time and induce a degradation rate in the order of percentages per year. Regarding the PV modules themselves, the manufacturer warranty usually conforms with a 20 % reduction in 25 years.

• Joule losses in the cables: The intrinsic resistive nature of the cables leads to power being cleared up into joule losses.

• Inverter losses: The conversion from DC to AC in the inverter unavoidably involves some power losses.

All losses that come in addition to the previous ones are then considered as failure losses. For instance, unexpected shading or module under-ventilation leads to additional losses compared to the estimation and falls in the failure category.

Several projects [START_REF] Aghaei | Review of degradation and failure phenomena in photovoltaic modules[END_REF][START_REF] Bansal | Comparative investigation of performance evaluation, degradation causes, impact and corrective measures for ground mount and rooftop solar PV plants -A review[END_REF][START_REF] Bun | Détection et localisation de défauts dans un système photovoltaïque[END_REF][START_REF] Colli | Failure mode and effect analysis for photovoltaic systems[END_REF][START_REF] Köntges | Assessment of photovoltaic module failures in the field[END_REF][START_REF] Köntges | Review of Failures of Photovoltaic Modules[END_REF][START_REF] Miquel | Dysfonctionnement électriques des installations photovoltaïques: points de vigilance[END_REF][START_REF] Rongbin | Failure Risk Analysis of the Photovoltaic Modules Based on the Improved FMECA Method[END_REF] aim at listing and describing failures. For the Agence Qualité Construction (AQC), which lists 37 faults [START_REF] Miquel | Dysfonctionnement électriques des installations photovoltaïques: points de vigilance[END_REF], a failure is characterized by its phase of appearance (design, installation, operation) and its target component (module/field, inverter, system, connectors, protection system, sensor). The frequency (low/high) and the risk (electric shock, fire) must also be considered when it comes to prioritizing failures. The causes of failures are very diverse and complex to anticipate from environmental stress, poor quality in commissioning, unexpected events to accelerated aging.

A total of 26 failure modes was identified and Table 1 collects the failures per system component. The failure modes have been initially collected from the AQC report [START_REF] Miquel | Dysfonctionnement électriques des installations photovoltaïques: points de vigilance[END_REF] and, then, rearranged to find intersections with the failure lists from IEA [START_REF] Herz | Quantification of Technical Risks in PV power Systems[END_REF] and Long Bun [START_REF] Bun | Détection et localisation de défauts dans un système photovoltaïque[END_REF] in order to complete information and counter-check the relevance of the given occurrence and performance scores. Most of the failures are related to "Module/Array" because of its sophisticated photoelectric phenomenon. Then, the wiring category is comprised of three failure modes accounting for each of its elements. Inverters are at the crossroads of important PV array segments which might induce significant performance losses in case of failures. Finally, "Insulation failure and ground connection defect" can appear in any location of the installation.

Active research is undertaken to avoid the apparition of failures and the first step towards risk mitigations is the implementation of detection methods.

Detection methods

A detection method is defined as follows according to the AQC report [START_REF] Miquel | Méthodes de détection des dysfonctionnements électriques des installations photovoltaïques[END_REF]: "Method that is manual or automated, enabling the detection of one or more dysfunctions of a system, and alerting the user". In Table 2, the detection method list has been inspired by the AQC report. The "Visual inspection" category depicts the simple observation, ideally, of a specialist on the installation subcomponents. The "Imaging method" covers all methods processing images of PV modules under stressed or normal operating conditions. The "Electrical tests" are all measurements that can be performed during a maintenance visit where measured values are compared to theoretical ones. Finally, "Monitoring" methods track the performance of the system from a remote location. Each method detects a specific spectrum of failures according to its methodology and its area of operation [START_REF] Aghaei | Review of degradation and failure phenomena in photovoltaic modules[END_REF][START_REF] Herz | Quantification of Technical Risks in PV power Systems[END_REF][START_REF] Köntges | Review of Failures of Photovoltaic Modules[END_REF].

Failure mode -detection method correspondence

A detection method can notice underperformances from several failures and a failure can usually be noticed from several detection methods. More specifically, Figure 1 displays the relationships between all failures and detection methods of the datasets previously developed.

A disparity in the associations between the 26 failure modes (red squares) and 13 detection methods (blue dots) is brought to light. There are failures such as "Shading" or "Cell interconnection defect" that can be detected with up to 9 detection methods but "Anti-reflective coating degradation" can only be discovered with "Visual inspection". On the other hand, some detection methods like "Visual inspection" have a wide spectrum of detection with 21 detectable failures while some others such as "Isc measurement" detect only 4 failures. Figure 1 shows the relationship between "detection methods" and failure modes only with the "detection" signification that actually differs from "identification". A "detection method" notices underperformances without being able to associate them with a specific failure. It is usually the case with monitoring methods where a drop in performance can be easily observed, accounting for a failure although identifying the exact cause of the failure might require some more developed techniques. On the other hand, an "identification method" can locate exactly the failure and characterize it fully in detail for further correction purposes. In the rest of the paper, the "identification" signification is retained since those methods are mandatory to identify the exact root of the failure and suggest a correction. Each failure will be associated with at least one "identification" method.

Methodology to rank failures via FMECA based on different data sources

This paper introduces a new way to adapt score scales from different literature data sources. Compared to other PV FMECAs [START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF][START_REF] Colli | Failure mode and effect analysis for photovoltaic systems[END_REF][START_REF] Collins | Reliability and availability analysis of a fielded photovoltaic system[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF][START_REF] Rajput | Risk priority number for understanding the severity of photovoltaic failure modes and their impacts on performance degradation[END_REF][START_REF] Rongbin | Failure Risk Analysis of the Photovoltaic Modules Based on the Improved FMECA Method[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF], the detection score also takes into account the maturity of the associated identification method. Then, a new methodology to combine different literature sources through a random score generation is investigated.

In this section, the Failure Modes, Effects, and Criticality Analysis (FMECA) is first introduced and the scoresampling methodology for each failure is then explained.

1. Failure Modes, Effects, and Criticality Analysis Failure Modes and Effects Analysis (FMEA) is based on expert opinions and would coincide during the early phase of strategy implementation [START_REF] Herz | Quantification of Technical Risks in PV power Systems[END_REF] such as failure mitigation. FMEA is particularly suited since operational data is limited due to the relatively recent uprise of the photovoltaic sector, the lack of systematic failure reporting [START_REF] Moser | Identification of technical risks in the photovoltaic value chain and quantification of the economic impact: Economic impact of technical risks in photovoltaic plants[END_REF], and the large diversity of technologies and climates.

Failure modes may be prioritized according to their importance and this process is referred to as Failure Modes, Effects, and Criticality Analysis (FMECA). As defined by Milic [START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF] in Figure 2, the FMECA starts with defining the scope of the PV system, identifying the failures modes, characterizing its causes as well as effects, and assigning severity/occurrence/detection scores to calculate the Risk Priority Number (RPN) in order to diagnose corrective measures. The Risk Priority Number (RPN) in Equation 1 reveals the criticality of each failure and is calculated with the product of three decision criteria: severity, occurrence, and detection. The severity (S) is an estimate of how strongly the failure will affect the system [START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF]. The effect on the performance raised by that failure mode is specific to each site. The occurrence (O) represents the likelihood that the failure mode might occur and lead to the indicated severity of the consequence. It may be defined according to the computed frequency of each failure mode. Detection (D) approximates the ability to identify and prevent the failure before the system is impacted.

𝑅𝑃𝑁 = 𝑆 * 𝑂 * 𝐷 (eq. 1) The performance and occurrence scores from the AQC [START_REF] Miquel | Dysfonctionnement électriques des installations photovoltaïques: points de vigilance[END_REF], IEA [START_REF] Herz | Quantification of Technical Risks in PV power Systems[END_REF], and [START_REF] Bun | Détection et localisation de défauts dans un système photovoltaïque[END_REF] have been adapted to fit the presented scoring system in Table 3. The AQC performance scoring includes five possible categories comparable to the chosen ones and then, the scores have been moved up by 1 additional point from the AQC's 0-4 scale to fits the 1-5 given performance scoring range. Regarding the AQC's occurrence scores, they have been updated from "+"," ++"," +++" to 1,3,5 respectively to fill the whole score range. Similarly, L. Bun's occurrence and severity scores got revised from 1,2,3 to 1,3,5.

In the aim of following the same systematic method, the qualitative costs and maturity levels given from another AQC report [START_REF] Miquel | Méthodes de détection des dysfonctionnements électriques des installations photovoltaïques[END_REF] are used to compute scores for all identification methods. The qualitative costs given by the AQC are translated from (€, €€, €€€) to (1,2,3). Then some points are added depending on the maturity level: 2 points are added if the maturity is at "First prototypes", 1 point is added if the identification method is in between "First prototypes" and no point is added if the method is "Mature". The detection score D of each failure is then set by the lowest score of its associated "identification" methods.

𝐷 = 𝑀 + 𝐶 (eq. 2)

Where:

• D: detection score of the identification method, between 1 and 5

• M: Market maturity of the method, between 0 and 2

• C: Identification method intervention cost, between 1 and 3

Regarding the adopted identification method for each failure, the correspondence with failure has been inspired and completed thanks to the AQC [START_REF] Miquel | Méthodes de détection des dysfonctionnements électriques des installations photovoltaïques[END_REF] and IEA report [START_REF] Herz | Quantification of Technical Risks in PV power Systems[END_REF].

In some cases, multiple severity and occurrence scores can be assigned to a failure from Table 1. The IEA report generally lists several score "options" out of five for the same failure mode. Also, several sub-"associated" failure modes from a data source can sometimes match only one failure from Table 1 and would correspond to variants or alternatives. All those failures were retained to randomly generate scores in the sampling methodology presented later.

All severity, occurrence, and detection scores are collected and a sample of five failures is shown in Table 4. If there are several sub-"associated" failures from a data source to a failure "k" from Table 1, the scores are listed with a comma separator. In the case a data source gives a range of different score "options" for the same sub-"associated" failure, those are all informed with a semicolon separator. When the data source does not contain any info about severity or occurrence, no score is shown. To fully take advantage of the available information from the diverse data sources, a methodology has been developed to randomly generate scenarios. Each scenario especially represents a point of view on the failure risk at three different levels: data source, sub-"associated" failures, and score options. Since there is not any valuable argument to trust more any data source, any sub-"associated" failure, or score options, an even distribution is applied to pick each of the levels starting from the data source, through the sub-"associated" failures to the score options.

For each scenario of each failure, the severity and occurrence scores are randomly picked separately with the following steps:

i.

The data source is randomly picked with uniform weights.

ii. Since, sometimes, several sub-"associated" failures from a data source are associated with only one failure from the list from Table 1. An "associated" failure is uniformly randomly picked.

iii.

If the IEA source has been picked, the final score is randomly picked among the failure score options from IEA.

The example below illustrates the random pick of the severity score for the delamination case. A total of 𝑁 𝑠 = 5000 scenarios are further generated with the severity and occurrence score picked distinctly for each scenario. The severity, occurrence, and detection scores are then combined to get the RPN score for each scenario "n" as in Equation 3 which enables to create a RPN distribution for each failure "k".

𝑅𝑃𝑁 𝑘,𝑛 = 𝑆 𝑘,𝑛 * 𝑂 𝑘,𝑛 * 𝐷 𝑘 (eq. 3)

In order to compare failures to each other from their RPN distribution, a final score indicator 𝑅𝑃𝑁 𝑅𝑀𝑆,𝑘 for each failure k, is defined in Equation 4. The Root Mean Square (RMS) method penalizes more significantly the failures containing 𝑅𝑃𝑁 𝑘,𝑛 with higher scores in their distributions.

𝑅𝑃𝑁 𝑅𝑀𝑆,𝑘 = √ 1 𝑁 𝑠 ∑ 𝑅𝑃𝑁 𝑘,𝑛 2 𝑁 𝑠 𝑛=1 (eq. 4)
RMS has the advantage to grasp a whole distribution through one indicator. It assigns more weight to the highest scores and, ineluctably, is sensitive to any outlier. The regular mean of the distribution could have been instead applied but its uniform weights would not promote critical failures with distribution with a fat tail risk. The RMS indicator is thereafter applied to rank and shortlist the most critical failures.

Results of the FMECA applied to photovoltaic failures

Sample generation has been performed according to the presented methodology and the first noteworthy observations come from the Severity (S) and Occurrence (O) scores. In Figure 4, the S and O means have been collected for all failures. The focal position of the inverter conveys the consequences of any failure mode on a large section of the PV array and includes the most dangerous financial repercussions according to Tjengradwira [START_REF] Tjengdrawira | PV Investment Technical Risk Management[END_REF]. Legitimately getting severity scores over 3 and coupled with steady occurrence scores, the "Inverter" component needs to actively be supervised to mitigate risks.

Similarly crucial to the architecture, the "Wiring" category bundles high severity and occurrence results. "Insulation failures and ground connection defect" has also the potential to lead to serious consequences. The wiring and insulation failures have the highest safety-risk scores in [START_REF] Miquel | Dysfonctionnement électriques des installations photovoltaïques: points de vigilance[END_REF] and, from a safety and performance perspectives, are definitely to prioritize

The number of failures related to "Module/array", 19, does not necessarily reflect more severe consequences compared to other components but highlights rather the complexity of the PV technology. 12 out of 19 of the failures obtain a mean occurrence score lower than 2 and are rather negligible in terms of technical risks. On the other hand, still some failures such as "Corrosion", "Soiling" and "Shading" remain with very high scores and need special care during maintenance visits. The variations in AQC and RMS scores globally follow each other although the latter has less amplitude since it gets flattened with the inclusion of different score sources. The RPN score distribution for each failure stays wide because the IEA report usually suggests large score intervals. Indeed, failure modes can impact the performance at different degrees such as soiling of which impact is a function of its homogeneity [START_REF] Maghami | Power loss due to soiling on solar panel: A review[END_REF]. Some of the failures with a low RPN score need special consideration. The "LID / LeTID" and "Not conform power rating" failure modes that were not considered in the AQC dataset stay at the low end of the ranking. For LID, this low score is particularly valid for all technologies except for amorphous silicon where the module power degradation can reach 10 -30 % according to Gostein [START_REF] Gostein | Light Soaking Effects on PV Modules: Overview and Literature Review[END_REF]. "PID" is also a well-identified failure mode with significant impact [START_REF] Dhimish | Power loss and hotspot analysis for photovoltaic modules affected by potential induced degradation[END_REF][START_REF] Luo | Potential-induced degradation in photovoltaic modules: a critical review[END_REF] reaching sometimes 40 % of losses on real installations [START_REF] Libby | Literature Study and Risk Analysis for Potential Induced Degradation[END_REF] but its low occurrence keeps the failure away from the top 15.

Regarding the changes from AQC to the RMS method, Table 5 highlights the respective scores and rankings. The retained failure modes to further establish risk mitigation strategies are the RMS top 12. Those are nearly the same as the AQC top 12 and capture the agreement between all data sources on the failure batch to mitigate. The RMS score also steps down by 3,5 significant points between the 12 th and the 13 th place which separates the top 12 from the rest of the failures.

The failures contained in the Top 12 almost do not change between the AQC and RMS methods but the internal ranking gets modified. The "Cell Interconnection defect" is ejected from the top 12 to the 16 th place in the RMS ranking due to the IEA report and L. Bun assigning more moderate severity and occurrence scores. In addition, the IEA report and L. Bun consider realistically more different failure alternatives for "Inverter defect" compared to AQC which has high scores, and the RMS makes it drop from the 4 th to the 12 th place in the final ranking. Finally, L. Bun has more severe scores for "Unexpected inverter voltage input" and significantly increases its ranking to 4 th place.

The failures excluded from the top 12 are all categorized in the Module/Array category. It particularly underlines that any failure mode appearing in the inverter, wiring, or system component is considered critical. On the other hand, it also stresses that all components need attention when it comes to monitoring underperformances since they all contained at least one failure in the top 12.

Discussion

The outcome of the FMECA is confirmed by the literature. "Insulation failure and ground defect" is generally in the very top priorities [START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF][START_REF] Colli | Failure mode and effect analysis for photovoltaic systems[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF]. "Inverter defect" and "inverter overheating" are well advised, usually in the top 5 [START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF]. The "abnormal values of the voltage" [START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF] and "improper function" [START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF] could reflect the "Unexpected inverter voltage input" and has a more versatile ranking but generally in the first half of the ranking. "Shading" and "Soiling", often combined, are frequently in the most critical failures [START_REF] Basu | Failure Modes and Effects Analysis (FMEA) of a Rooftop PV System[END_REF][START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF] and "Hot spot" in the top 10 [START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF]. "DC cable defect", "Connector defect" and "Corrosion" defect are never in the top priorities but maintain consistent scores [START_REF] Basu | Failure Modes and Effects Analysis (FMEA) of a Rooftop PV System[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF]. "Module under-ventilation" has not been identified in the literature except by Basu [START_REF] Basu | Failure Modes and Effects Analysis (FMEA) of a Rooftop PV System[END_REF] considering rooftop PV applications where its score is relatively low. However, when considering building-mounted installations, it might be relevant to account for it due to the complex integration. "Combiner box defect" has also not been found cited as such in the literature and represents one of the specificities of the analysis.

The presented study showcases the advantages of looking into a whole system with well-defined and counterchecked scores from different literature sources. Some studies have already looked into the FMECA but at the PV module level only [START_REF] Catelani | Electrical performances optimization of Photovoltaic Modules with FMECA approach[END_REF][START_REF] Rajput | Risk priority number for understanding the severity of photovoltaic failure modes and their impacts on performance degradation[END_REF][START_REF] Rongbin | Failure Risk Analysis of the Photovoltaic Modules Based on the Improved FMECA Method[END_REF]. Some other studies analyze the whole system [START_REF] Colli | Failure mode and effect analysis for photovoltaic systems[END_REF][START_REF] Collins | Reliability and availability analysis of a fielded photovoltaic system[END_REF][START_REF] Milic | Failure Modes and Effects Analysis of Urban Rooftop PV Systems -Case Study[END_REF][START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF] without analyzing the sensitivity of the RPN coefficients. Also, a thorough methodology is developed in this study to calculate detection scores as of the qualitative cost/maturity detection scores from the literature [START_REF] Miquel | Méthodes de détection des dysfonctionnements électriques des installations photovoltaïques[END_REF]. Finally, as mentioned by Villarini [START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF] different climate conditions might affect the FMECA analysis, datasets, and using datasets from different sources would showcase a generic analysis covering most of the cases.

Moreover, some additional features might improve the presented methodology. As mentioned by [START_REF] Villarini | Optimization of photovoltaic maintenance plan by means of a FMEA approach based on real data[END_REF] the FMECA could eventually be updated with live data provided by reliability maintenance systems, and/or the occurrence score could be fed according to Mean Time Between Failure (MTBF) figures as examined in [START_REF] Colli | Failure mode and effect analysis for photovoltaic systems[END_REF][START_REF] Collins | Reliability and availability analysis of a fielded photovoltaic system[END_REF]. A complete data-driven FMECA from [START_REF] Filz | Data-driven failure mode and effect analysis (FMEA) to enhance maintenance planning[END_REF] with a detection score based on forecasting model accuracies could eventually be extended to the severity and occurrence scores.

Additionally, FMECA scores could be improved to consider a higher level of detail. For instance, shading might lead to additional failures such as hot spot and bypass diode defects. Also, the concomitance of "Soiling" and "Potential Induced Degradation (PID)" could lead to more serious power reduction than when each phenomenon is considered separately [START_REF] Luo | Potential-induced degradation in photovoltaic modules: a critical review[END_REF]. Modeling capabilities with empirical studies would additionally improve the understanding of performance impact such as the impact of PID [START_REF] Annigoni | Modeling Potential-Induced Degradation (PID) in crystalline silicon solar cells: from accelerated-aging laboratory testing to outdoor prediction[END_REF] and LID as mentioned by [START_REF] Woodhouse | LID and LeTID Impacts to PV Module Performance and System Economics DRAFT Analysis[END_REF].

Having identified the most critical failures, the next steps would logically be to optimize the inspection methodology along the life of any PV installation in terms of frequency and content. Modeling more precisely the failure performance impact with their evolution in time and the interactions between them would give indications on how and when to perform diagnostics with which identification methods. From those simulations, a failurecorrection decision support tool could also be envisioned to optimize the energetic, environmental, and financial value of the PV plant. All those suggestions together could come to fruition with a PV performance guarantee.

Conclusion

In this article, all photovoltaic failure modes are listed and associated with identification methods. The Failure Modes, Effects, and Criticality Analysis (FMECA) methodology assigns criticality scores to each failure based on its severity, occurrence, and detection ability. Based on the information from different data sources, score samples are randomly generated to take into account the different failure variants, eliminate knowledge bias, and obtain a generic overview. Those failures are finally ranked according to the Root Mean Square (RMS) of their respective score distribution.

Results demonstrate that all PV components from "Module/Array", "Wiring", "Inverter" to "System" contain critical failures in the top 12, and not any of those can be set aside to develop a strategy to decrease underperformances. In the first place, the "Insulation failure and ground defect" can occur in any part of the installation and has among the highest severity and occurrence scores because of its major safety and performance consequences. All failure modes collected in the inverter and wiring components are in the top 12 because of their central position in the PV architecture. Due to its technology complexity, "Module/Array" collects 19 failure modes while only 5 are in the top 12: "Hot spot", "Corrosion", "Shading", "Soiling" and "Module underventilation".

Figure 1 :

 1 Figure 1: Network maps of failure and detection method

Figure 2 :

 2 Figure 2: FMECA flow chart (Milic et al., 2018)

Figure 3 :

 3 Figure 3: Severity random sampling, "delamination" case

Figure 4 :

 4 Figure 4: Severity (S) and Occurrence (O) mean scores from the sample generation

Figure 5 :

 5 Figure 5: RPN scores from AQC and sampling RMS with 95% confidence interval Appending the detection score in the random sampling, RPN scores have been calculated in Figure 5. The AQC dataset in orange is introduced as a benchmark. The 𝑅𝑃𝑁 𝑅𝑀𝑆 calculated for each failure with Equation 4 from the sampling distribution associated with the 95% confidence interval is displayed in blue and the failures are according to their 𝑅𝑃𝑁 𝑅𝑀𝑆 score.

Table 1 : PV failure modes

 1

Table 2 : PV detection method characteristics Category Detection method Maturity Cost Accessibility

 2

	Electrical test	Insulation test	Mature	€€	On-site
	Electrical test	Isc measurement	Mature	€€	On-site
	Electrical test	IV tracer	First prototypes/Mature	€€	On-site
	Electrical test	Signal transmission method	Mature	€/€€	On-site
	Electrical test	Voc measurement	Mature	€€	On-site
	Imaging method	Electroluminescence	Mature	€€/€€€	On-site
	Imaging method	Infrared thermography	Mature	€€	On-site
	Imaging method	UV fluorescence imaging	First prototypes/Mature	€€	On-site
	Monitoring	Inverter monitoring	Mature	€	Remote
	Monitoring	IV monitoring	First prototypes	€€	Remote

Table 3 : Severity, Occurrence, and Detection scoring systems partially from (Milic et al., 2018)

 3

	Score	Severity (S)	Occurrence (O)	Detection (D)
	1	Insignificant, no effect on the performance	Remote, unlikely	Undemanding, mature identification method with low costs
	2	Low, slight impact on the system performance	Low, rare apparition	Straightforward, detection with low requirements
	3	Moderate, noticeable degradation of parts of the system performance	Moderate, occasional downtime	Intermediate, identification method with moderate cost and maturity
	4	High, high degradation, non-functionality/loss of performance	High, frequent apparition	Complex detection with high requirements
	5	Hazardous, system interruption or severe loss of performance	Very High, almost certain apparition	Highly complex, not mature, and high costs

Table 4 : Five PV failures with performance, occurrence, and detection scores

 4

	Failure mode	AQC (Miquel et al., 2018)	Severity (S) IEA (Herz et al., 2022)	(Bun, 2011)	Occurrence (O) AQC (Miquel (Bun, et al., 2018) 2011)	Associated identification methods	Detection score (D)
	Inverter defect	5	1;2;3;4;5	[5, 5, 5, 5, 5, 3, 1, 1, 1, 1]	3	3	Visual inspection	1.5
	Inverter overheating	5	[3;4;5, 1;2;3;4;5]	[5, 5]	3	3	Platform monitoring	2.0
	Unexpected inverter voltage input	2		[5, 5, 5, 3, 3, 3]	3	5	Platform monitoring	2.0
	Anti-reflective coating degradation	1		3	1	1	Visual inspection	1.5
	Backsheet degradation	4	[1;2;3;4;5, 1]		1	1	Visual inspection	1.5
	2. Random FMECA score-sampling

Table 5 : PV failure ranking

 5

	Failure mode	Score RPN_AQC	Ranking AQC Score RPN_RMS Ranking RMS
	Insulation failure and ground connection defect	40	1	29,8	1
	Inverter overheating	30	2	24,6	2
	Soiling	30	2	22,6	3

Acknowledgement

The authors of the article would like to thank the ANRT funding as well as the co-financing between Heliocity and CSTB for the project "Methodology development to guarantee building photovoltaic systems' performance including failure modeling". This work has been supported by the French National Research Agency, through the Investments for Future Program (ref. ANR-18-EURE-0016 -Solar Academy).


Renyu Hu
email: renyu.hu@jpl.nasa.gov@2023

Fabrice Gaillard

Typeset using L A T E X twocolumn style in AASTeX62 Narrow loophole for H 2 -dominated atmospheres on habitable rocky planets around M dwarfs

Keywords:

Habitable rocky planets around M dwarfs that have H 2 -dominated atmospheres, if they exist, would permit characterizing habitable exoplanets with detailed spectroscopy using JWST, owing to their extended atmospheres and small stars. However, the H 2 -dominated atmospheres that are consistent with habitable conditions cannot be too massive, and a moderate-size H 2 -dominated atmosphere will lose mass to irradiation-driven atmospheric escape on rocky planets around M dwarfs. We evaluate volcanic outgassing and serpentinization as two potential ways to supply H 2 and form a steady-state H 2 -dominated atmosphere. For rocky planets of 1 -7 M ⊕ and early, mid, and late M dwarfs, the expected volcanic outgassing rates from a reduced mantle fall short of the escape rates by >∼ 1 order of magnitude, and a generous upper limit of the serpentinization rate is still less than the escape rate by a factor of a few. Special mechanisms that may sustain the steady-state H 2 -dominated atmosphere include direct interaction between liquid water and mantle, heat-pipe volcanism from a reduced mantle, and hydrodynamic escape slowed down by efficient upper-atmospheric cooling. It is thus unlikely to find moderate-size, H 2 -dominated atmospheres on rocky planets of M dwarfs that would support habitable environments.

INTRODUCTION

Rocky planets with H 2 -dominated atmospheres would be ideal targets for atmospheric characterization via transmission spectroscopy, because of their large atmospheric scale height that causes large expected spectral features (e.g., Miller-Ricci et al. 2008;Seager & Deming 2010;Greene et al. 2016). A moderately irradiated rocky planet with a H 2 -dominated atmosphere may have surface pressure and temperature consistent with liquid water (Pierrehumbert & Gaidos 2011;Wordsworth 2012;Ramirez & Kaltenegger 2017;Koll & Cronin 2019). Such potentially habitable worlds sustained by H 2 -dominated atmospheres, if they exist around M dwarfs, would unlock the opportunity to study extrasolar habitability with spectroscopy (e.g., [START_REF] Seager | [END_REF], as TESS and ground-based surveys find temperate rocky planets around M dwarfs (e.g., Sebastian et al. 2021;Kunimoto et al. 2022), and JWST provides the sensitivity to analyze any H 2 -dominated atmospheres on them with a wide spectral coverage (e.g., Batalha et al. 2018). Here we ask: Are such worlds likely?

To have surface liquid water, the H 2 -dominated atmosphere cannot be much larger than ∼ 10 bar, because the surface temperature is primarily a function of the size of the atmosphere (this is valid for the stellar irradiation of 200-1400 W m -2 (Koll & Cronin 2019)). The exact size and irradiation limit depends on the cloud albedo effect (e.g., Popp et al. 2015). This moderate-size atmosphere is much smaller than the massive H 2 -dominated atmospheres proposed to explain the sub-Neptune-sized lowdensity planet population, which are typically 1% planet mass or > 10 4 bar (e.g., Rogers et al. 2023). A 10-bar H 2 atmosphere would only add <∼ 0.1 R ⊕ to the planetary radius, which can be accommodated by typical uncertainties in planetary mass, radius, and Fe content (Luque & Pallé 2022). Also, the temperate rocky planets will have a solid surface, as opposed to the sub-Neptunes that may have a permanent magma ocean (Kite & Barnett 2020). Because the permeability of the crust decreases dramatically with increasing depth (Manning & Ingebritsen 1999), any post-formation source of H 2 must come from shallow fresh crust via either volcanic outgassing or crustal alteration processes such as serpentinization.

Meanwhile, temperate planets around M dwarfs receive intense high-energy irradiation from host stars because of their close-in orbits, and this intense irradiation can drive hydrodynamic escape from a H 2 -dominated atmosphere (e.g., Salz et al. 2016;Kubyshkina et al. 2018a,b). The high-energy irradiation will be measured by a bevy of new spacecraft (Ardila et al. 2022;France et al. 2023). We are thus motivated to determine the lifetime of a moderate-size H 2 atmosphere -permitting surface liquid water -on a large rocky planet orbiting an M dwarf against hydrodynamic escape, and evaluate the geologic processes that could resupply the H 2 atmosphere.

ATMOSPHERIC ESCAPE

The hydrodynamic escape rate, f es (kg s -1 ), can be approximated by the energy-limited escape rate formula,

f es = η es (F X + F EUV )πR 3 p a 2 KGM p , (1)
where F X and F EUV are the stellar fluxes in X-ray (5-100 Å) and extreme ultraviolet (EUV, 100-1240 Å), a ≥ 1 is the ratio between the X-ray/EUV absorbing radius and the (optical) planetary radius, K ≤ 1 is a factor that accounts for the Roche lobe effect (Erkaev et al. 2007), and η es is the escape efficiency. Recent hydrodynamic escape models find the escape efficiency to be in a range of 0.1-0.25 for solar-abundance atmospheres (Salz et al. 2016) and Equation ( 1) is a good approximation of the full hydrodynamic calculations in the Jeans escape parameter regime for temperate rocky planets (Jeans escape parameter = 25 -60, Kubyshkina et al. 2018a,b). For a conservative estimate of the escape rate, we adopt η es = 0.1, a = 1, and K = 1. As shown in Table 1, we pick GJ 832, GJ 436, and TRAPPIST-1 as the representative stars for early M, mid M, and late M dwarfs. Their emission spectra in X-ray, Lyman-α, FUV, and NUV bands have been measured, and their emission spectra and fluxes in the EUV band can be inferred from these measurements (Peacock et al. 2019a,b). We find that the lifetime of a 10-bar H 2 atmosphere on a rocky planet that receives Earth-like insolation from these stars would be uniformly < 0.1 Ga. Thus, a source of H 2 would be needed to maintain such an atmosphere.

VOLCANIC OUTGASSING

We first consider volcanic outgassing as the source of H 2 (e.g., Liggins et al. 2020). The volcanic outgassing rate, f og (kg s -1 ), can be modeled by the following equation,

f og = η og V x H , ( 2
)
where V is the rate of magma generation, x H is the hydrogen content (wt %) of magma that degasses, and η og is the outgassing efficiency. We do not expect the outgassing efficiency to be close to unity because, even though extrusive volcanism (magma that reaches and degases at the planetary surface) can probably degas effectively (but often not completely), intrusive volcanism (magma that does not reach the surface) probably degases poorly, especially for H 2 (to be detailed later in this section). For Earth, the extrusive:intrusive ratio is typically 3:1 to 10:1 (White et al. 2006), and so as a fiducial value, we assume η og = 0.1. The rate of volcanism can be estimated for a rocky planet by modeling its thermal evolution history. We adopt the geodynamics model of Kite et al. (2009) for the rate of volcanism, which used a melting model from pMELTS (Ghiorso et al. 2002) for the plate tectonic mode and Katz et al. (2003) for the stagnant lid mode (Table 2). Focusing first on the planets around field M dwarfs, we take the 4-Gyr age values for the rate of volcanism. The values for the plate tectonic and stagnant lid modes are similar. Detailed models of mantle convection in the stagnant-lid regime predict that volcanism would cease much sooner than what Table 2 indicates (Noack et al. 2017;Dorn et al. 2018), but this model uncertainty does not impact the conclusion of this paper. For volcanic outgassing to sustain the atmosphere, it is required that f es = f og . With f es and V , we derive the required x H and list the values in Table 1.

Arc volcanoes on Earth, which are formed by flux melting caused by release of water from subduction of plates rich in hydrated materials, have magmas that contain 1 -7 wt % water (e.g., Rasmussen et al. 2022). The water content in the mid-ocean ridge basalt and the ocean island basalt is lower by 1 -2 orders of magnitude (Dixon et al. 2002). Complete outgassing of 1 -7 wt % water in the form of H 2 would provide an x H of 0.1 -0.8 wt %. We consider this to be a very generous upper limit; comparing it with Table 1 shows that it is very unlikely for volcanic outgassing to sustain the H 2 atmosphere.

The hydrogen content of the magma for degassing depends on the oxygen fugacity of the magma and the pres-

Star

Type F (erg s -1 cm -2 ) fes (10 4 kg s Table 1. Escape rates and lifetimes of a 10-bar H2 atmosphere, and the required hydrogen content in magma for degassing to sustain this atmosphere, on a hypothetical rocky planet of an M dwarf. The distance between the planet and the star results in a bolometric stellar flux the same as Earth's insolation (i.e., the 1-AU equivalent distance). The X-ray fluxes (5-100 Å) are measured by XMM-Newton (Loyd et al. 2016;Ehrenreich et al. 2015;Wheatley et al. 2017) sure at which degassing takes place. We use the magma degassing and speciation model of Gaillard & Scaillet (2014) to calculate x H for the typical volatile content of terrestrial magmas and a wide range of oxygen fugacities (Figure 1). The H 2 content is higher for a more reducing magma and when degassing at a lower pressure. In addition to counting the H 2 degassing, one may also include the potential for atmospheric photochemistry to post-process CO to form H 2 , via CO + H 2 O --→ CO 2 + H 2 . The complete post-processing means that degassing 1 mole CO would be equivalent to degassing 1 mole H 2 , and this situation is shown as dashed lines in Figure 1. However, x H predicted by the geochemical model, even when including the CO conversion, is at least one order of magnitude lower than the minimum required x H for a 7-M ⊕ planet around an early M dwarf (Table 1). This again indicates that volcanic outgassing is unlikely to sustain an H 2 atmosphere.

SERPENTINIZATION

We turn to serpentinization as an alternative source of H 2 . Serpentinization is water-rock reactions between warm water and mafic and ultramafic rocks (usually olivine-rich) in the fresh crust. This process probably occurs on all rocky planets with liquid water, and it may have produced H 2 -rich water on early Earth (Sleep et al. 2004) and H 2 and CH 4 on early Mars and on modern Figure 1. Degassing of H2 from magma calculated by the magma degassing and speciation model of Gaillard & Scaillet (2014). We consider mid-ocean ridge basalt (MORB) with bulk H2O content of 0.19 wt % and bulk CO2 content of 0.16 wt %, degassing at the oxygen fugacities of FMQ-6 (corresponding to an undifferentiated planet) and FMQ-4 (corresponding to modern Mars), as well as ocean island basalt (OIB) with bulk H2O content of 1 wt % and bulk CO2 content of 0.3 wt %, degassing at the oxygen fugacities of FMQ-2.5. The solid lines count the degassing of H2 and the dashed lines count the degassing of both H2 and CO (with CO expressed in terms of its indirect effect on atmospheric H2, see text).

Enceladus (Oze & Sharma 2005;Chassefière & Leblanc 2011;Batalha et al. 2015;Zandanel et al. 2021).

For an upper bound of the H 2 production rate from serpentinization, we assume that 1 mole H 2 is produced for every 3 moles Fe 2+ oxidized, as the process can be written as H 2 O + 3 FeO --→ H 2 + Fe 3 O 4 . We also assume that the fresh crust is entirely composed of olivine, (Mg 0.9 Fe 0.1 ) 2 SiO 4 , and all Fe 2+ is used by serpentinization to produce H 2 . The olivine has a molar mass of 146.9 g, and it contains 0.2 moles Fe 2+ , corresponding to 0.067 moles H 2 , or a mass of 0.13 g. The corresponding x H is thus 0.13/146.9 = 0.09 wt. %. In reality, the fresh crust may not be entirely composed of olivine, and the rate of serpentinization is limited by the rate of dissolution of olivine in water (Oze & Sharma 2007), which is a function of temperature, pH, water/rock ratio, and the Fe/Mg composition of olivine (Wogelius & Walther 1992;Allen & Seyfried Jr 2003), as well as by the extent of fracturing of the crust (Vance et al. 2007). We thus expect the actual x H provided by serpentinization to be much smaller than 0.09 wt. %. However, even this generous upper bound falls short of the required x H by at least a factor of 4 (Table 1). It is thus also unlikely that serpentinization would sustain a moderate-size H 2 atmosphere on rocky planets around M dwarfs.

AGE AND DISTANCE DEPENDENCY

So far we have assumed 4 Gyr for the planet age, which broadly corresponds to the field M dwarfs. Now we consider the age dependency of the sources and sinks of H 2 and see if a steady-state H 2 atmosphere would be possible on younger planets. Richey-Yowell et al. ( 2019) recently presented the NUV, FUV, and X-ray fluxes of M dwarfs in their habitable zones as a function of age, and meanwhile, the EUV fluxes follow a similar age dependency as their FUV fluxes (Peacock et al. 2020). We adopt an age dependency of t -0.9 for the X-ray fluxes and t -1.3 for the EUV fluxes. Meanwhile, the rate of volcanism can be much higher for young planets, when the heat flux from the planetary interior is higher. We explore an age dependency that varies between t -1 (based on the model for Earth in [START_REF] Schubert | Mantle convection in the Earth and planets[END_REF]) and t -2 (based on the model for large rocky planets of Kite et al. 2009). We consider an age as young as 1 Gyr. Before that, the planet could have a residual primordial H 2 atmosphere (Kite & Barnett 2020) and the stellar high-energy output may have different age dependencies (Richey-Yowell et al. 2019). As shown in Figure 2, it remains unlikely for volcanic outgassing or serpentinization to compensate for the intense atmospheric escape of H 2 experienced by rocky planets of M dwarfs from 1 to 5 Gyr.

How about a planet that is located further away from the star than the 1-AU equivalent distance? Moving the planets 2.6 times further away (or receiving 7 times less irradiation, or ∼ 200 W m -2 ) would still produce a potentially habitable climate (Koll & Cronin 2019), and this would reduce the escape rate by a factor of 7. In this case, the escape rate is comparable to the upper limit of serpentinization (Figure 2). However, the upper limit assumes complete oxidization of Fe 2+ in the fresh crust and is thus unlikely. The utility of these distant habitable worlds for observations is probably limited, as they are less likely to transit and harder to find than the closer-in planets.

POTENTIAL ALTERNATIVE MECHANISMS

The estimates above show that it is unlikely to sustain moderate-size H 2 -dominated atmospheres on rocky planets around M dwarfs through volcanic outgassing or serpentinization. Here we explore alternative mechanisms that could result in large source fluxes of H 2 .

First, the rate of hydrogen generation during serpentinization is controlled by the Fe content of olivine (Klein et al. 2013). In Section 4, we have assumed a Fe:Mg ratio of 1:9, corresponding to the terrestrial value. On Mars, however, the Fe:Mg ratio of crustal olivine can be ∼1:1 (Koeppen & Hamilton 2008;Morrison et al. 2018). Such Fe-rich olivine could result in higher fluxes of H 2 from serpentinization than our estimates by a factor of ∼ 5, making it more likely for serpentinization to meet the H 2 escape flux.

Second, on a planet with plate tectonics but amagmatic spreading, water-rock interaction near the ridge axis could produce H 2 . Our discussion of serpentinization so far assumes that water interacts with the products of volcanism/partial melting. However, water could interact directly with the mantle if it is exposed by amagmatic spreading. This mechanism occurs on Earth today at Gakkel Ridge and Southwest Indian Ridge and has the potential to generate more H 2 because the Fe content of mantle rock is greater than that of the crustal basalt. This mechanism would decouple serpentinization from volcanism and allow serpentinization to continue even after volcanism has shut down. The upper limit of H 2 production from this mechanism is then given by assuming (unrealistically) a 100% fayalite (Fe 2 SiO 4 ), which would give an equivalent x H of 0.6 wt. %. Suppose fractures, and therefore hot-water alteration, penetrate as far down into the subsurface on this amagmatic planet as the base of the oceanic crust on our planet, which is probably unrealistic because fractures should self-seal at shallower depths (Vance et al. 2007). Then the present-day terrestrial production of 20 km 3 /year of MORB, with full serpentinization, would correspond to 1×10 4 kg s -1 of H 2 output. This is on the same order of magnitude as the lower end of the escape rate (Table 1), and could be higher for younger or larger rocky planets.

Third, Earth's heat flux of ∼ 0.1 W m -2 is ∼ 10% in the form of advective cooling (magma moves upward and cools) and ∼ 90% conductive cooling. This implies that the rates of volcanism could be 10× higher without excessively cooling Earth's mantle. Indeed, it has been hypothesized that heat-pipe tectonics occurred early in Earth's history (e.g., Moore & Webb 2013). Comparison between sources and sinks of H2 on rocky planets around M dwarfs. The grey areas show the range of escape rates depending on the type of the host star. The dark grey area is for a planet that locates at the 1-AU equivalent distance, while the light grey area is for a planet that locates 2.6 times farther away (i.e., receiving 7 times less irradiation). The outgassing rates encapsulate the plausible range from a highly reduced mantle (informed by Figure 1), with the lower bound corresponding to xH = 0.005 wt % and t -1 scaling, and the higher bound corresponding to xH = 0.02 wt % and t -2 scaling. The rate of serpentinization is a very generous upper limit (Section 4).

Small variations in exoplanet mantle composition or exoplanet mantle volatile content, among other factors, could make melting easier at a given mantle temperature (Spaargaren et al. 2020), perhaps enabling heatpipe mode of planetary cooling even for planets that are as old as the Earth. If heat-pipe volcanism occurs then the amount of outgassing and serpentinization could be ∼ 10× greater than for an Earth-scaled plate-tectonics model, because ∼ 10× more eruptions would occur.

Fourth, N-, O-, and C-bearing molecules mixed in the H 2 -dominated atmosphere may substantially reduce the escape rate. The escape rate and efficiency calculations have been based on solar-abundance atmospheres (Kubyshkina et al. 2018a,b). However, H 2 -dominated atmospheres sustained by volcanism should also have H 2 O, CO/CH 4 , and N 2 /NH 3 at the levels that exceed the solar abundance (Liggins et al. 2022). Recently, Nakayama et al. (2022) show that with an N 2 -O 2 atmosphere, cooling from atomic line emissions (of N, N + , O, O + ) and radiative recombination can prevent rapid hydrodynamic escapes for XUV irradiation fluxes that are up to 1000× the modern Earth value. It is thus conceivable that an H 2 -dominated atmosphere richer in N, O, and C would be more stable. The challenge is that the N-, O-, and C-bearing molecules are separated from H 2 by diffusion (typically at ∼ 1 Pa) and can be largely depleted in the upper atmosphere. It is thus unclear whether 10% non-H 2 species (which would still allow for a low molecular weight atmosphere for transmission spectroscopy) would slow down the hydrodynamic escape sufficiently to achieve long-term stability.

Lastly, there could be transient episodes of high volcanism and serpentinization that support H 2 -dominated atmospheres. A leading hypothesis for why Early Mars sometimes had lakes is that a lot of H 2 was emitted transiently from the subsurface by volcanism or serpentinization (Wordsworth et al. 2021). The amount of H 2 needed to warm up Mars by H 2 -CO 2 collision-induced absorption is now well understood (Turbet et al. 2020), and the H 2 flux needed is approximately 10 4 kg s -1 . A large rocky planet could have 10× the surface area of Mars and thus plausibly 10× the amount of serpentinization. This process-agnostic (but model-dependent) scaling hints at short-term source fluxes sufficient for H 2dominated atmospheres on a 7-M ⊕ planet (Figure 2).

SUMMARY

From the analyses presented above, we conclude that rocky planets around M dwarfs rarely have potentially habitable conditions accompanied by H 2 -dominated atmospheres. This is because, forming a potentially habitable environment will require a moderate-size (∼ 10 bar) atmosphere, but such an atmosphere is removed quickly by stellar X-ray and EUV irradiation, and could only exist on the planet as a steady-state atmosphere with replenishment. However, neither volcanic outgassing nor serpentinization provides the required H 2 source that would maintain such a steady-state atmosphere. Small planets around M dwarfs could have massive H 2 atmospheres, but to have a stable and moderate-size H 2 atmosphere consistent with habitability would require special circumstances such as direct interaction between liquid water and mantle (e.g., near a ridge axis undergoing amagmatic spreading), heat-pipe volcanism from a highly reduced mantle, or hydrodynamic escape quenched by efficient atomic line cooling. These special mechanisms to sustain the moderate-size H 2 atmosphere are generally more effective on large rocky planets (e.g., ∼ 7-M ⊕ planets exemplified by LHS 1140 b) than on Earth-sized planets.

The finding here is consistent with the non-detection to date of clear H 2 -dominated atmospheres on rocky planets of M dwarfs via transmission spectroscopy (e.g., De Wit et al. 2018;Lustig-Yaeger et al. 2023), although these results could also be interpreted as widely occurring photochemical haze that mutes transmission spectral features of H 2 -dominated atmospheres. Swain et al. (2021) suggested an H 2 -dominated atmosphere on the rocky planet GJ 1132 b based on HST data, but independent data analyses could not confirm their result (Mugnai et al. 2021;Libby-Roberts et al. 2022). The ongoing HST and JWST transmission spectroscopy of small exoplanets of M dwarfs will further test our findings and, potentially, discover exceptional cases. Meanwhile, N 2 -CO 2 or other high-mean-molecular-weight atmospheres should probably be considered as the default assumption when planning for future spectroscopic observations of rocky planets around M dwarfs. This would require planning more repeated visits of preferred targets for transmission spectroscopy (e.g., Batalha et al. 2018), or turning to thermal emission spectroscopy and phase-curve mapping (e.g., Angelo & Hu 2017;Kreidberg et al. 2019;Mansfield et al. 2019;Koll et al. 2019;Whittaker et al. 2022).

  Figure2. Comparison between sources and sinks of H2 on rocky planets around M dwarfs. The grey areas show the range of escape rates depending on the type of the host star. The dark grey area is for a planet that locates at the 1-AU equivalent distance, while the light grey area is for a planet that locates 2.6 times farther away (i.e., receiving 7 times less irradiation). The outgassing rates encapsulate the plausible range from a highly reduced mantle (informed by Figure1), with the lower bound corresponding to xH = 0.005 wt % and t -1 scaling, and the higher bound corresponding to xH = 0.02 wt % and t -2 scaling. The rate of serpentinization is a very generous upper limit (Section 4).

Table 2 .

 2 and the EUV fluxes (100-1240 Å) are based on PHOENIX synthetic spectra guided by FUV and NUV observations(Peacock et al. 2019a,b).* Bourrier et al. (2017) reported a much lower EUV flux (126 erg s -1 cm -2 at TRAPPIST-1 e) based on Lyman-α measurements, but using this lower value does not change the conclusion of this paper. Rate of volcanism (the mass of magma production divided by the mass of planet, in unit of current Earth's value 3.7516×10

	Mode	Plate tectonics Stagnant lid
	Age (Gyr)	2 4	6	2 4	6
	1 M⊕, 1 R⊕	8 1.5 0.5	7 1.5 0
	3 M⊕, 1.3 R⊕ -2	0.7	-2.5 0
	7 M⊕, 1.7 R⊕ -4	1	-3.5 0.7

-19 s -1 ), based on the parameterized model of

Kite et al. (2009)

. Dashes correspond to the heat-pipe tectonic regime.

ACKNOWLEDGMENTS

We thank Evgenya Shkolnik and Tyler Richey-Yowell for helpful discussion of stellar evolution. This work was supported in part by the NASA Exoplanets Research Program grant #80NM0018F0612. E.S.K. acknowledges support from a Scialog grant, Heising-Simons Foundation 2021-3119. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.


Oxygen-rich melt in deep magma oceans

Fabrice Gaillard [START_REF] Righter | Redox processes in early Earth accretion and in terrestrial bodies[END_REF] 1: Institut des Sciences de la Terre d'Orléans, CNRS/Université d'Orléans/BRGM, 1a rue de la Férollerie, 45071 Orléans cedex 2, France.

High pressures may have enabled ferric iron-rich silicate melts to coexist with iron metal near the base of magma oceans early in the history of large rocky planets like Earth. This suggests a relatively oxygen-rich atmosphere after core formation on these planets.

Earth has had an oxygen-rich surface for the past 2.5 billion years (Gyr). This is related to oxygen photosynthesis and possibly Earth's mantle being more oxidized than other known mantles of smaller terrestrial planetary bodies, such as Mars 1 . Yet Earth, like most terrestrial planetary bodies, is thought to have formed with a highly reduced primordial mantle. Internal oxidation has thus taken place in Earth's mantle, but it remains unclear when and how this occurred. Writing in Nature Geoscience, Kuwahara et al. [START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] propose that mantle oxidation occurred very early on, when Earth had a magma ocean more than 1000 km deep, because the pressures near the magma ocean floor were sufficiently high that oxygen-rich silicate melts coexisted with reduced iron metal. Mixing the deep silicate melts into the rest of the magma ocean may have supported more oxidized surface conditions than the present-day.

The oxidation state of mantle rocks, magma and volcanic gases is gauged by their oxygen fugacity, fO2, the thermodynamic expression of the minute O2-pressure (<<10 -7 bar) prevailing in the planet's interior. Substantial spatial variations in the fO2 of Earth's mantle have been reported, highlighting the ongoing interplay between pressure, temperature and mixing of deep and surficial components due to plate tectonics [START_REF] Cottrell | Oxygen Fugacity Across Tectonic Settings[END_REF] . Nevertheless, the cause of Earth's relatively oxidized mantle must be very persistent, as reconstruction of the mantle's paleo-fO2 using mineralogical proxies indicates only marginal temporal variability back to 4.4 Gyr 4 . Before then, during the first 100 million years of our solar system, magma oceans prevailed on terrestrial planetesimals, planetary embryos and planets [START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF] . This state led to planetary differentiation, with dense metals segregating inwards forming a core [START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF] , molten silicate solidification forming a mantle [START_REF]Redox evolution of the crystallizing terrestrial magma ocean and its influence on atmosphere outgassing[END_REF] and outgassing creating an atmosphere [START_REF] Gaillard | Redox controls during magma ocean degassing[END_REF] (Fig. 1). The presence of iron metal in magma oceans implies reduced conditions, involving fO2 values 4 to 8 orders of magnitude lower than that of Earth's present-day mantle 1 .

Several models have been proposed to explain the increase in the fO2 of Earth's mantle from core formation to the present. Firstly, it has been argued that complete crystallization of the magma ocean at upper mantle conditions and a constant ratio of ferric (Fe 3+ ) to ferrous (Fe 2+ ) iron should increase mantle fO2 [START_REF] Sossi | Redox state of Earth's magma ocean and its Venus-like early atmosphere[END_REF] , though this alone is insufficient to account for the magnitude of the fO2 increase.

Secondly, crystallization of the lower mantle deep in the magma ocean could produce ferric iron-rich silicate minerals, such as perovskite, together with iron metal [START_REF] Frost | The redox state of the mantle during and just after core formation[END_REF] . If this metal were to migrate into the core, the overall mantle would become more oxidized proportionally, however it is unclear whether dispersed iron metal could travel through a solid mantle. Thirdly, a recent numerical modelling study [START_REF]Redox evolution of the crystallizing terrestrial magma ocean and its influence on atmosphere outgassing[END_REF] proposed that degassing accompanying magma ocean crystallization could increase mantle fO2, but it is uncertain whether the solid-liquid partitioning behaviour of iron supports this.

Experiments have shown enhanced ferric iron stability in andesitic melts at high pressure [START_REF] Armstrong | Deep magma ocean formation set the oxidation state of Earth's mantle[END_REF] , which could bridge the fO2 gap between Earth's magma ocean and its present-day mantle. Kuwahara et al. [START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] also propose this based on the results of very high-pressure (up to 28 GPa) experiments on peridotite melts. The composition and pressures used are directly relevant to the magma ocean and to the lower mantle, respectively. The results show much greater stability of ferric iron under high pressures than previously suggested [START_REF] Armstrong | Deep magma ocean formation set the oxidation state of Earth's mantle[END_REF][START_REF] Deng | A magma ocean origin to divergent redox evolutions of rocky planetary bodies and early atmospheres[END_REF] . Some technical issues remain, however, in particular that the experimental melts did not quench as glass because of the ultrafast crystallization of dendritic minerals upon freezing, which introduce uncertainty into the measured Fe 3+ :Fe 2+ ratios.

Assuming the results of Kuwahara et al. [START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] are correct, the source of the oxygen required for a ferric iron-rich silicate magma ocean should be considered. If the source was dissolved CO or CO2, this would cause deep diamond sequestration [START_REF] Armstrong | Deep magma ocean formation set the oxidation state of Earth's mantle[END_REF] (Fig. 1). Since the amount of carbon species in a magma ocean is probably small and might occur at depth as CH4 instead [START_REF] Gaillard | A speciation model linking the fate of carbon and hydrogen during core -magma ocean equilibration[END_REF] , this scenario is questionable.

Incorporation of some magma ocean oxides (e.g., FeO or SiO2) into the core (as Fe or Si) [START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF][START_REF] Badro | Core formation and core composition from coupled geochemical and geophysical constraints[END_REF] might be a more plausible oxygen source. However, this process could incorporate oxygen in the core [START_REF] Badro | Core formation and core composition from coupled geochemical and geophysical constraints[END_REF] , which would counteract the magma ocean self-oxidation model of Kuwahara et al. [START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] . Further work is necessary to address this early oxygen question. Kuwahara et al. [START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] suggest that the fO2 of the Earth's magma ocean was very high, potentially higher than that of the present-day mantle. The atmosphere derived from such an oxidized magma ocean would be dry and dominated by CO2, N2 and SO2 [START_REF] Gaillard | Redox controls during magma ocean degassing[END_REF] . High sulfur volatility under such oxygen-rich conditions may explain Earth's non-chondritic sulfur isotopic composition [START_REF] Labidi | Non-chondritic sulphur isotope composition of the terrestrial mantle[END_REF] , provided that early atmospheres are dispersed by processes such as impacts. The magma ocean and the resulting mantle would be sulfurpoor under these conditions, so a process is required to return sulfur to the mantle to account for its present-day sulfur content [START_REF] Gaillard | Redox controls during magma ocean degassing[END_REF] . The potential relationship between the size of terrestrial planets and their mantle fO2 and thus atmospheric composition (Fig. 1) may enable new insights on rocky exoplanets from measurements of their atmospheric compositions. Notably, this relationship implies that very large terrestrial exoplanets may have oxygen-rich atmospheres that are not indicative of photosynthesis. Kuwahara et al. [START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] propose that planetary magma oceans are born reduced and can become oxidized for planets approaching or exceeding Earth's size. Therefore, Earth and possibly Venus were oxidized when their mantle formed, whereas rocky bodies the size of Mars or smaller remained reduced. This implies fundamental differences between these two groups for properties influenced by mantle fO2, such as mantle convection regime, melting and degassing.

Figure 1 -

 1 Figure 1 -Effects of magma ocean size on planetary oxidation state and atmospheric composition.Small planetary bodies (left) remain reduced through their magma ocean stage, with an atmosphere comprising predominantly CO and H2. Kuwahara et al.[START_REF] Kuwahara | Hadean mantle oxidation inferred from melting of peridotite under lower mantle conditions[END_REF] suggest that the high pressures near the base of deep magma oceans on large bodies (right) may enable a relatively oxidized mantle, which would result in an atmosphere primarily composed of CO2, N2 and SO2[START_REF] Gaillard | Redox controls during magma ocean degassing[END_REF] .

Competing interests

The author declares no competing interests.