P Richard

E R Masilya

A Mudakikwa

Jean Muzana

Guillard

A Tessier
email: atessier@marepolis.fr

A Richard

P Masilya
email: pascalmasilya@yahoo.fr

E R Mudakikwa

A Muzana

J Guillard
email: jean.guillard@inrae.fr

Spatial and temporal variations of Limnothrissa miodon stocks and their stability in Lake Kivu

Keywords: hydroacoustics, large African lake, clupeid, biomass, nutrient, phytoplankton

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

Introduction

Inland fisheries are a main livelihood resource for the human population in the African Great Lakes. They are currently dominated by small pelagic fish species, which are endemic or introduced [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF]. Over the last three decades, stocks of small pelagic clupeids fisheries have increased rapidly in many African lakes resulting in a shift of fisheries from large species to smaller ones [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF]. The contribution of small pelagic fishes to commercial catches in many African lakes increased since the 1970s to 45% in Victoria, 35 % in Malawi, 65 % in Tanganyika, 60 % in Kivu, 80 % in Lake Albert, and 17 % in Kyoga [START_REF] Ogutu-Ohwayo | Variations and changes in habitat, productivity, composition of aquatic biota and fisheries of the Kyoga lake system: lessons for management[END_REF]Ogutu-Ohwayo et al., 2000) due to a shift, for example, to Rastrineobola argentea from Lates niloticus in Lake Victoria (CIFA 1984), and in Lake Malawi from Oreochromis lidole to Engraulicypris sardella [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF].

Several factors are possibly contributing to this shift, such as overexploitation of large species or increased eutrophication of the ecosystem [START_REF] Kolding | Are the Lake Victoria fisheries threatened by exploitation or eutrophication? Towards an ecosystembased approach to management[END_REF]. [START_REF] Ogutu-Ohwayo | Implications of climate variability and change for African lake ecosystems, fisheries productivity, and livelihoods[END_REF] mentioned that FAO (2010) has predicted that fisheries will shift to small pelagic opportunistic species and the increases in many African lakes may be associated with the increasing variability and change in climate of these lakes and its effects on lake productivity processes.

Small pelagic clupeids represent a more sustainable resource than large species because of their better resilience to natural and anthropic disturbances [START_REF] Jul-Larsen | Management, co-management or no management? Major dilemmas in southern African freshwater fisheries[END_REF][START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF]. Clupeid stocks are known to fluctuate strongly according to environmental conditions due to their low trophic level in the food web and rapid maturity [START_REF] Karenge | Inshore fish population changes in Lake Kariba, Zimbabwe[END_REF][START_REF] Marshall | The influence of river flow on pelagic sardine catches in Lake Kariba[END_REF][START_REF] Trochta | The highs and lows of herring: A meta-analysis of patterns and factors in herring collapse and recovery[END_REF]. As clupeids are mainly zooplanktivorous, their stock level is mainly driven by nutrient inputs and primary production, so it can vary greatly within and between years [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF]. Clupeid stocks are resilient to high fishing intensity because they mature much faster than large fish species [START_REF] Mangeni-Sande | Spatial and temporal differences in life history parameters of Rastrineobola argentea (Pellegrin, 1904) in the Lake Victoria basin in relation to fishing intensity[END_REF] and have multiple spawnings per year in tropical areas like Lake Kivu [START_REF] Marshall | Variations saisonnières et annuelles de l'abondance du Limnothrissa miodon du lac Kivu[END_REF].

In the African Great Lakes regions, small pelagic clupeid fisheries make important contributions to reducing poverty and improving food security and nutrition. For example, in 2005, 30 to 45 million people in Africa depended on fish for their livelihoods [START_REF] Béné | Fish and food security in Africa[END_REF]. These small fishes also contribute to gender equality with women and youth dominating in the processing and marketing [START_REF] Funge-Smith | A fresh look at inland fisheries and their role in food security and livelihoods[END_REF]. Limnothrissa miodion is one of the three predominant fish species of Lake Tanganyika [START_REF] Kimirei | Influence of environmental factors on seasonal changes in clupeid catches in the Kigoma area of Lake Tanganyika[END_REF][START_REF] Plisnier | Limnological variability and pelagic fish abundance (Stolothrissa tanganicae and Lates stappersii) in Lake Tanganyika[END_REF], where it is endemic. It also dominates the fisheries of lakes and reservoirs where it has been introduced, like the Itezhi-tezhi reservoir (where it was introduced in the beginning of the 1990s) [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF], Lake Kariba and Cahora Bassa (at the beginning of the 1960s) [START_REF] Balon | Lake Kariba: a man-made tropical ecosystem in Central Africa[END_REF], and Lake Kivu (at the end of the 1950s) [START_REF] Kaningini | Pêche du Sambaza au filet maillant dans le lac Kivu[END_REF].

Located between the Democratic Republic of the Congo (DRC) and Rwanda, Lake Kivu is in a region where the annual climate consists of two dry and two rainy seasons [START_REF] Thiery | Understanding the performance of the FLake model over two African Great Lakes[END_REF]. It is a meromictic lake with mixolimnion (oxygenated layer) mixing until a maximum depth of around 60 m during the dry season [START_REF] Reusens | Programme de recherche halieutique au lac Kivu en particulier sur le Limnothrissa miodon[END_REF][START_REF] Schmid | Stratification, mixing and transport processes in Lake Kivu[END_REF]. The environmental conditions of Lake Kivu can respond quickly to atmospheric conditions linked to climate change [START_REF] Schmid | Stratification, mixing and transport processes in Lake Kivu[END_REF]. In 2012-2014, the mixing of the mixolimnion was modified by changes in wind patterns, which led to a significant change in the composition of the phytoplankton communities, with green algae supplanting diatoms and cyanobacteria [START_REF] Rugema | Long-term change of phytoplankton in Lake Kivu: the rise of the greens[END_REF].

L. miodon was introduced to establish a pelagic fishery in what was originally an impoverished fishery by colonizing the pelagic zone, which was a vacant ecological niche [START_REF] Collart | L'introduction du Stolothrissa tanganicae (Ndagala) au lac Kivu[END_REF]. The introduction has been considered a successful one [START_REF] Gozlan | Introduction of non-native freshwater fish: is it all bad?[END_REF][START_REF] Marshall | Why is Limnothrissa miodon such a successful introduced species and is there anywhere else we should put it?[END_REF]. The catches of L. miodon may represent up to 75% of total catches [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF]. This fishery supports more than 6,500 fishermen [START_REF] Guillard | Fish population in Lake Kivu: review of recent advances on management and knowledge[END_REF] and provides nutrition for the 2 million people around Lake Kivu [START_REF] Wüest | Methane formation and future extraction in Lake Kivu[END_REF].

The pelagic stock of L. miodon in Lake Kivu was assessed by FAO using hydroacoustic methodology during the daytime over different seasons each year from the early 1980s to the beginning of the 1990s [START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF][START_REF] Reusens | Programme de recherche halieutique au lac Kivu en particulier sur le Limnothrissa miodon[END_REF]. The stock of L.

miodon was estimated at 7,000 t (35 kg/ha) before 1989 and 4,000 t (18 kg/ha) in the 1990s, with high seasonal fluctuations. Fish were mainly scattered between depths of 15 and 45 m without a clear spatial pattern [START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF]. Around 80% of the detected fish sizes were estimated between 4 and 10 cm.

In 2008, two hydroacoustic surveys led to similar conclusions on the vertical distribution of fish in the water column, the size distribution, and the absence of a clear geographical pattern [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]. The stock levels were estimated at 4,400 t to 5,000 t, and it was concluded the fisheries had been sustainable over the intervening two decades [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]. This conclusion is contrary to the possible collapse proposed by [START_REF] Dumont | The Tanganyika sardine in Lake Kivu: another ecodisater for Africa[END_REF] based on changes in the assemblages and decline in biomass of mesozooplankton. The mixolimnion zone of Lake Kivu is not entirely mixed every year, which is probably due to climate change, and the phytoplankton show modification of the assemblage [START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF][START_REF] Rugema | Long-term change of phytoplankton in Lake Kivu: the rise of the greens[END_REF]. These modifications of the environmental conditions and functioning can affect the stock of L. miodon.

Stock estimates were carried out from 2012 to 2018, and in 2013 and 2014 using hydroacoustic methods. The surveys were carried out during the daytime in the pelagic zone using the same sampling method (vertical echo-sounding) and the same water layers as previous surveys (2-15 m, 15-30 m, 30-45 m and 45-60 m). The population size, structure, and fish density were analyzed with special emphasis on spatial variations (between basins) and temporal variations (seasonal and inter-annual). From theses analyses, the most appropriate season to evaluate the stocks of L. miodon in Lake Kivu was established in relation to the reproductive behavior pattern of the species [START_REF] Kaningini | Etude de la croissance, de la reproduction et de l[END_REF]. Because environmental conditions had changed between the 2000s and 2010s [START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF][START_REF] Rugema | Long-term change of phytoplankton in Lake Kivu: the rise of the greens[END_REF], the fish densities and size structures were compared with previous assessments, and spatial geographical patterns were analyzed. Finally, the evolution of the fish stock at the scale of the whole lake was analyzed relative to the historical time series.

Material and methods

Study area

Lake Kivu (1°34'30''-2°30'S; 28° 50'-29°23' E, Fig. 1) has a surface area of 2370 km² and is located in a geographical region with four seasons: a short dry season (January to February), a long rainy season (March to May), a long dry season (June to September), and a short rainy season (October to December) [START_REF] Thiery | Understanding the performance of the FLake model over two African Great Lakes[END_REF]. The lake can be divided into five basins (Fig. 1), each with different wind patterns of, precipitation, fishing effort, and governance [START_REF] Kwetuenda | Climatic conditions in the hydrological basin of Lake Kivu during 12 years[END_REF]. Due to its high altitude (1463 m above sea level), the water's surface temperature is lower than in other African great lakes and usually ranges from 23.5 to 25°C [START_REF] Katsev | Recent warming of Lake Kivu[END_REF].

Below around 60 m, the deep monimolimnion is anoxic with high concentrations of dissolved methane in the deeper layers [START_REF] Tassi | Water and gas chemistry at Lake Kivu (DRC): Geochemicalevidence of vertical and horizontal heterogeneities in amultibasin structure[END_REF]. The maximum depth is 485 m in the northern part (Data Environnement, 2011;[START_REF] Descy | Lake Kivu -Limnology and biogeochemistry of a tropical great lake[END_REF]. Lake Kivu is oligotrophic [START_REF] Isumbisho | Zooplankton of Lake Kivu, half a century after the Tanganyika introduction[END_REF], and it has only 29 fish species (of which 5 were introduced). This is due to profound geologic disturbances that occurred around 11,000 years BP and because it is at the upper part the basin of species-rich Lake Tanganyika [START_REF] Haberyan | The late Pleistocene and Holocene stratigrphy and paleolimnology of Lake Kivu and Tanganyika[END_REF].

Presently, the pelagic zone of the mixolimnion is dominated by L. miodon [START_REF] Collart | L'introduction du Stolothrissa tanganicae (Ndagala) au lac Kivu[END_REF] and fish are not present at depths below 60 m [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF][START_REF] Lamboeuf | Estimation de l'abondance du stock d'Isambaza (Limnothrissa miodon), résultats de la prospection acoustique de septembre 1989[END_REF].

Acoustic sampling

Hydroacoustic surveys were performed on a small craft using the same equipment and protocol as the previous surveys performed by [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]. They were performed using a SIMRAD (Kongsberg Maritime AS, Horten, Norway) EK60 scientific echosounder equipped with a 70-kHz split-beam transducer (SIMRAD ES70-11) with a half-power beam angle of 11° (at -3dB) beaming vertically. The transducer was regularly calibrated in situ using a copper sphere with 23-mm diameter according to procedures recommended by [START_REF] Foote | Calibration of acoustic instruments for fish density estimation: a practicle guide[END_REF] and the manufacturer. The transducer was attached to a pole along the boat and positioned at 0.7 m below the water surface to sample the water column in the range of 2 m below the surface to 65 m. The echosounder's ping rate was set to 5 pings per second, and the pulse duration was fixed at 0.256 ms [START_REF] Godlewska | How pulse lenght impact fish stock estimations during hydroacoustic measurements at 70 kHz[END_REF]. A Global Positioning System (GPS) was coupled to the echosounder, and the boat speed was around 8 km/h. To identify whether results from hydroacoustic surveys were representative of the pelagic stock of the lake, the degree of coverage was computed according to [START_REF] Aglen | Empirical results on precision-effort relationship for acoustic surveys[END_REF]. The degree of coverage was obtained by dividing the total length of the path (km) by the square root of the studied surface (km²). The result should be at least 3.0 and preferably above 6.0 to obtain a reliable assessment [START_REF] Emmrich | Strong correspondence between gillnet catch per unit effort and hydroacoustically derived fish biomass in stratified lakes[END_REF].

The hydroacoustic surveys were carried out in the pelagic areas (areas with bottom depth deeper than 50 m) for each of the five basins (North Rwanda, North DRC, East, West, and South) using zigzag tracks (Fig. 1) [START_REF] Guillard | The repeatability of fish biomass and size distribution estimates obtained by hydroacoustic surveys using various sampling strategies and statistical analyses[END_REF]. Seasonal surveys were performed in 2013 and 2014 to identify the best season to perform stock estimation [START_REF] Rudstam | Fisheries acoustic[END_REF].

The annual surveys from the best season regarding stock estimation were then analyzed from 2012 to 2018 to provide inter-annual and spatial variations of fish density, size, structure, and total biomass. Finally, annual stock estimates were compared with historical data using the same season [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]Lamboeuf, 1989a[START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF]. Hydroacoustic surveys were not carried out for some basins or periods due to staff availability, failure to obtain legal authorizations, boat issues, or bad weather (Table 1).

Hydroacoustic sampling is usually carried out at night due to fish behavior because fish are scattered and individual size estimations are possible, in contrast to the daytime, when fish are mainly in schools or close to the bottom [START_REF] Coll | Use and limits of three methods for assessing fish size spectra and fish abundance in two tropical man-made lakes[END_REF][START_REF] Girard | Comparisons of day-time and night-time hydroacoustic surveys in temperate lakes[END_REF][START_REF] Prchalová | Acoustic study of fish and invertebrate behavior in a tropical reservoir[END_REF]. However, as the historical surveys were performed during the daytime [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF][START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF], the hydroacoustic surveys in the present study were performed during this photoperiod between 05:00 and 16:00. Moreover, both safety and legal reasons necessitated that the hydroacoustic surveys be performed during the daytime.

Data analyses

The software Sonar5-Pro [START_REF] Balk | Sonar4 and Sonar5-Pro post processing system[END_REF] was used to process the data. The processing thresholds were set at -60 dB for single targets detection and -66 dB for echo integration. A difference of 6 dB between the threshold intensity and the detection thresholds is recommended [START_REF] Parker-Stetter | Standard operating procedures for fisheries acoustic surveys in great lakes[END_REF]. The data were cleared of different types of noise (e.g., bottom structure, false bottom, bubbles, debris) at the appropriate thresholds by using manual deletions to conserve only fish echoes for analyses [START_REF] Draštik | Fish hydroacoustic survey standardization: A step forward based on comparisons of methods and systems from vertical surveys of a large deep lake[END_REF][START_REF] Emmrich | Strong correspondence between gillnet catch per unit effort and hydroacoustically derived fish biomass in stratified lakes[END_REF]. The analyses were stopped at 60 m because fish are not present below this depth [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]. However, echograms were checked to verify that this was always true.

The elementary sampling distance unit (ESDU) was fixed a priori at 1000 m according to [START_REF] Muzana | Stock estimation of Limnothrissa miodon in Lake Kivu: assessment of seasonal and spatial variation of fish abundance[END_REF] and served as replicates [START_REF] Yule | Comparing two fish sampling standards over time: largely congruent results but with caveats[END_REF].

For each ESDU, we used the hectare acoustic scattering strength (sA, with units m 2 /ha) as a proxy for fish density [START_REF] Boswell | Biomass, density, and size distribution of fishes associated with large-scale artificial reef complex in the Gulf of Mexico[END_REF]:

sA = 10 (Sv/10) x T x 4 π x 10000
Sv is the volume backscattering strength in decibels (dB) [START_REF] Maclennan | A consistent approach to definitions and symbols in fisheries acoustics[END_REF], T is the mean thickness of the domain being integrated, 4 π is the steradians in a sphere that converts a "backscattering" cross section to "scattering" cross section, and 10,000 is a conversion factor (m 2 /ha) [START_REF] Yule | Comparing two fish sampling standards over time: largely congruent results but with caveats[END_REF]. sA can be considered as a proxy of the density in this study because pelagic areas of Lake Kivu are dominated by a single species with homogeneous sizes. Seasonal variations of sA were analyzed using the Sheirer Ray-Hare (SRH) test on the mean values for 2013 and 2014 and for each basin while considering the seasons and the water layers as factors. These years were chosen because at least 3 or 4 seasons were sampled in these years, unlike other years (Table 1).

The water column was split into four layers (2-15 m, 15-30 m, 30-45 m, and 45-60 m), as performed by [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]. The SRH test was followed by the Conover-Iman test with Bonferroni correction to identify the modalities of the factors that induce a difference and how the mean sA varies. The inter-annual and spatial variations of sA were examined by applying a two-way ANOVA test to the mean values with the basins and the years considered as factors. Afterwards, a post-hoc Tukey's honestly significant difference (HSD) test was performed to identify the modalities of the factors that induce a difference and how the mean sA varied.

For each tracked fish and each transect, the mean individual target strength (TS) was calculated in the linear domain in dB [START_REF] Maclennan | A consistent approach to definitions and symbols in fisheries acoustics[END_REF] as a proxy for total individual length [START_REF] Simmonds | Fisheries acoustics. Theory and practice[END_REF]. The parameters used in software Sonar5-Pro the same mention by [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF] .To estimate fish size in cm (TL = Total Length), Love's equation [START_REF] Love | Measurements of fish target strength: a review[END_REF]) was used [START_REF] Emmrich | Strong correspondence between gillnet catch per unit effort and hydroacoustically derived fish biomass in stratified lakes[END_REF][START_REF] Love | Measurements of fish target strength: a review[END_REF]

: = 10 . × .
The equation for TS is still relevant and is usually used in freshwater studies [START_REF] Macnamara | Bigheaded carps (Hypophthalmichthys spp.) at the edge of their invaded range: using hydroacoustics to assess population parameters and the efficacy of harvest as a control strategy in a large North American river[END_REF][START_REF] Morrissey-Mccaffrey | Effects of differing ground-truth data, transect design and statistical analysis on the repeatability of hydroacoustic assessments of pollan Coregonus autumnalis pollan[END_REF][START_REF] Ye | Daynight differences in hydroacoustic estimates of fish abundance and distribution in Lake Laojianghe, China[END_REF].

To consider L. miodon exclusively, which presents a maximal standard length of 17.5 cm in Lake Kivu [START_REF] De Iongh | The impact of an introduction of sardine into lake Kivu[END_REF][START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF] only individual fish with TS less than -39 dB according to [START_REF] Love | Measurements of fish target strength: a review[END_REF] were considered. The proportion of individuals in three size classes (small, medium, and large) were determined. The small fish correspond to individuals smaller than 5 cm. The upper boundary for the medium size was set to 12 cm, which is two-thirds of the maximum total length [START_REF] Harmelin-Vivien | Evaluation visuelle des peuplements et populations de poissons: méthodes et problèmes[END_REF].

Seasonal variations were addressed using the frequency distribution of fish sizes, which were computed for each season and basin for 2013 and 2014. The fraction of small fish (< -50 dB, around 5 cm according to Love) was then compared between seasons and basins as independent variables using logistic models with logit transformation [START_REF] Peng | An introduction to logistic regression analysis and reporting[END_REF], which was performed separately for each year (2013 and 2014). The significance of the variables was tested using likelihood-ratio (LR) tests compared to a χ 2 distribution, followed by Wald tests to compare coefficients between seasons. The inter-annual and spatial variations of the mean total size were analyzed using a two-way ANOVA test while considering the basins and years as factors. Post-hoc Tukey's HSD tests were then performed to identify the differences of mean sizes and how they varied in space and time.

From sA values and TS, the fish biomass (ρa; kg/ha) per basin was estimated using [START_REF] Forbes | Manuel des méthodes de prospection et d'évaluation des ressources halieutiques[END_REF] equation:

ρ a = sA 4π * 10 TS/10
Furthermore, the average individual fish weight (kg) was calculated from individual sizes based on [START_REF] Minns | Allometry of home-range size in lake and river fishes[END_REF] equation:

W = 0.01 (TL) 3
W is the weight of a fish in grams and TL in cm. The total biomass for each basin and for the whole lake was calculated by multiplying the biomass per kg/ha by the surface of the pelagic zone of the basin or the total area of 2217 km² of Lake Kivu (2217 km², [START_REF] Kaningini | Pêche du Sambaza au filet maillant dans le lac Kivu[END_REF]. The surface of the pelagic zone for the five basins was estimated using the software ArcGIS 10.3 (ESRI, Redlands, CA). For the estimated biomass, confidence intervals were generated by a bootstrap procedure with 100 iterations.

In 2012, the total biomass for the whole lake was estimated from only three sampled basins (North Rwanda, East, and South) instead of five. To determine whether this value was comparable with other data, a Wilcoxon signed-rank test was performed to check whether the biomass extrapolations from all 5 basins differed from those obtained using only three basins (North Rwanda, East, and South). The statistical analyses were conducted using the software R 3.5.1 (R Development Core Team, 2009). All tests were considered significant for p-values less than 0.05.

Results

Degree of coverage

The degrees of coverage by basins mostly ranged between 3 and 6 (Table 1). About 15% of the ratios were less than 3, which were concentrated among the surveys performed in 2013 and 2014. Only two basins showed values less than 2 (4%). For 2012, 2015, 2016, and 2018 (Table 1), all coverage ratios were greater than 3. When Lake Kivu was considered as a whole, the result was always greater than 6. Thus, the present results can be considered as representative of the pelagic zone when considering Lake Kivu as a whole or by basin.

sA values as a proxy of density of L. miodon stock

In the study of seasonal variations in 2013 and 2014, the SRH tests showed a significant interaction between the season factor and the water-layer factor (Table 2) regarding the mean sA values. The Conover-Iman tests indicated that the mean sA was always lower at 45-60 m than in the other layers (Fig. 2). These tests also showed that the mean sA was usually higher at 15-30 m than in the other layers. During the long dry season, the mean sA was higher at 15-30 m and 30-45 m than at 2-15 m (Fig. 2).

Concerning the inter-annual sA variation, the two-way ANOVA showed no significant interaction between the year and basin (F = 1.397; p = 0.128), but there was a significant difference between years (F = 22.288; p < 0.001) and between basins (F = 28.922; p < 0.001).

The post-hoc Tukey's HSD test indicated that the mean sA was lower in 2012 and higher in 2018 than in other years (Table 3). Furthermore, the mean sA was lower in 2014 than in 2015 and lower in 2016 than in 2015 (Table 3). The post-hoc Tukey's HSD tests indicated that the mean sA of North Rwanda, North-DRC, and East basins were lower than those of the West and South basins (Table 3).

Size distribution of L. miodon

The frequency distribution of predictive fish sizes showed that the medium class size represented 50 to 68% of the total sample, irrespective of basin, season, and year (Fig. 3).

Generally, the fish size distribution was unimodal with a mode at ≈ 6 cm during the short dry, long rainy, and short rainy seasons. However, during the long dry season, the fish size distribution was bimodal with modes at 3 cm in 2013 and 2 cm in 2014, and a second mode at 6 cm in 2013 and one at 5 cm in 2014 (Fig. 3). For both years, the proportion of small fish (< 5 cm) differed between seasons and basins, and there was significant interaction between the two factors (LR tests, p < 0.001). Higher contributions of small fish were observed during the long dry season (between 36% and 46% depending on the basin in 2013 and between 27% and 48% in 2014) compared to the other seasons (Wald z tests, p < 0.001).

Considering only the long dry seasons from 2012 to 2018, the two-way ANOVA showed significant interaction between the year factor and the basin factor on fish size (F = 92.61; p < 0.001). The post-hoc Tukey's HSD test indicated that in 2012, the mean sizes were similar between basins (5.5 or 5.6 cm). In this year, the basins presented the same percentages regarding each size category (Table 4). In 2013 and 2014, the mean sizes were lowest in North DRC (5.8 and 5.4 cm, respectively; p < 0.001), followed by North Rwanda (6.1 and 6.3 cm, respectively; 0.03 > p < 0.001). In other basins, they were between 6.2 and 6.4 cm in 2013 and between 5.8 and 6.2 cm in 2014.

In 2013 and 2014, the percentages of large fish were lower in the two North basins (2.0 to 4.5% vs. 4.3 to 5.5% for other basins), and the percentages of small fish were higher for these two basins (41.7 to 47.3% vs. 35.2 to 44.5% for other basins; Table 3). Since 2014, the mean sizes were always higher in the West basin than in the other basins (between 6.2 and 7.5 cm; p < 0.001). This basin always presented lower percentages of the small category (Table 4). The post-hoc Tukey's HSD test indicated that for all the basins, the mean sizes were always higher in 2018 than other years (p < 0.001), except for the East basin, where it was highest in 2015, followed by 2018 (p < 0.001). Furthermore, for each basin in 2018, the percentages of small fish were always lower than in other years.

Biomass of L. miodon stock

Regarding seasonal variation, the estimated fish biomass was higher during the long rainy season than the long dry and short rainy seasons in 2013 for each basin (Table 5). In 2014, the fish biomass was generally higher during the short dry season, while similar fish biomass was estimated for the long rainy and long dry surveys of all basins. Regarding interannual variations, the fish biomass increased over time, although the biomass showed a slight drop in 2014 and 2016 in almost all basins (Table 5). In 2012, the biomass ranged between 3.8 and 10.3 kg/ha, whereas in 2018, it ranged between 11.9 and 22.7 kg/ha (Table 5).

The South and West basins usually had biomass greater than 11 kg/ha, while the other basins had biomass less than 10 kg/ha. The total biomass estimated by basin ranged between 100 and 389 t in 2012 and between 204 and 1,069 t in 2018.The total biomass estimated by basin ranged between 100 and 389 t in 2012 and between 204 and 1,069 t in 2018. The total biomass of pelagic fish in all of Lake Kivu was estimated at 1,056 t in 2012 and reached 4,067 t in 2018 (Fig. 4). The estimates of total biomass based on either all sampled basins or only three basins (North Rwanda, East, and South) for the years 2013 to 2018 indicated no significant difference (Wilcox signed-rank test; v = 12; p = 0.3125). This result confirms that the lack of one or two sampled basins during the survey (in 2012 and 2015) is not likely to change the estimates of total fish biomass.

Discussion

In this study, a seasonal pattern was observed for the frequency distribution of predictive fish sizes. Most distributions were unimodal, except during the long dry season, when they were bimodal, corresponding to two cohorts. During the first two months of life, the mean growth rate of L. miodon is 1.23 cm per month [START_REF] Kaningini | Etude de la croissance, de la reproduction et de l[END_REF], while between 3 and 6 months, they grow at 1 cm per month, reaching around 10 cm after 1 year [START_REF] Snoeks | Fishes in lake Kivu: diversity and fisheries[END_REF]. It follows that individuals belonging to the size class of 2-3 cm should be 2-3 months old, and those in the class of 6-7 cm should be 6-7 months old in Lake Kivu [START_REF] Mannini | Parametres de la population de Limnothrissa miodon du Lac Kivu[END_REF][START_REF] Spliethoff | Success of the introduction of the freshwater clupeid Limnothrissa miodon in Lac Kivu[END_REF].

L. miodon showed two reproduction peaks in Lake Kivu [START_REF] Kaningini | Etude de la croissance, de la reproduction et de l[END_REF] although its reproduction is continuous over time [START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF]. The first peak occurred from March to May (long rainy season), and the second was from September to October (short rainy season) [START_REF] Kaningini | Etude de la croissance, de la reproduction et de l[END_REF]. Thus, the smallest mode observed during the long dry season should represent the cohort arising from March-May spawning, whereas the second mode represents the cohort resulting from the short rainy-season spawning.

Mature males and females of L. miodon are around 6.0 cm in Lake Kivu [START_REF] Marshall | Biology of the African clupeid Limnothrissa miodon with reference to its small size in artificial lakes[END_REF][START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF]. Reproduction takes place in shallow waters of littoral areas [START_REF] Kaningini | Etude de la croissance, de la reproduction et de l[END_REF][START_REF] Marshall | Biology of the African clupeid Limnothrissa miodon with reference to its small size in artificial lakes[END_REF][START_REF] Snoeks | Fishes in lake Kivu: diversity and fisheries[END_REF], and the juveniles remain in these areas before moving to the pelagic areas when their sizes is close to 4 cm [START_REF] Marshall | Biology of the African clupeid Limnothrissa miodon with reference to its small size in artificial lakes[END_REF].

The present study indicates that in Lake Kivu, this migration could start earlier when juveniles reach 2-3 cm. Thus, the present study highlights that the long dry season seems to be the most appropriate season to carry out hydroacoustic surveys for L. miodon in the pelagic zone of Lake Kivu, because it gives access to both adults and recruitment at the beginning of the year.

The density and biomass of L. miodon showed seasonal fluctuations without any clear seasonal pattern. This was also observed by [START_REF] Marshall | Variations saisonnières et annuelles de l'abondance du Limnothrissa miodon du lac Kivu[END_REF], who considered catches and fishing effort. These results were not surprising because clupeid recruitment can fluctuate according to seasonal environmental conditions (e.g., nutrient inputs, light), which drives phytoplankton availability which in turn affects the availability of zooplankton that are the main prey of L. miodon [START_REF] Karenge | Inshore fish population changes in Lake Kariba, Zimbabwe[END_REF][START_REF] Marshall | The influence of river flow on pelagic sardine catches in Lake Kariba[END_REF][START_REF] Trochta | The highs and lows of herring: A meta-analysis of patterns and factors in herring collapse and recovery[END_REF].

Furthermore, the environmental conditions for a season can be highly variable [START_REF] Kolding | Freshwater small pelagic fish and fisheries in major African lakes and reservoirs in relation to food security and nutrition[END_REF].

Although density did not present a seasonal pattern, it varied by depth. L. miodon was scarce below 45 m and was not present below 60 m. This result is supported by those of previous studies, which observed the same vertical distribution (Lamboeuf,1989a;[START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF]; [START_REF] Guillard | Is the fishery of the introduced Tanganyika sardine (Limnothrissa miodon) in Lake Kivu (East Africa) sustainable?[END_REF]. The layer at 45-60 m is less favorable to L. miodon because dissolved oxygen is often limited in this deeper layer [START_REF] Schmid | Stratification, mixing and transport processes in Lake Kivu[END_REF]. Indeed, the mixolimnion is only mixed to ~60 m during the dry season [START_REF] Reusens | Programme de recherche halieutique au lac Kivu en particulier sur le Limnothrissa miodon[END_REF][START_REF] Schmid | Stratification, mixing and transport processes in Lake Kivu[END_REF]. Increases in temperature due to climate warming is reported to have been accompanied by decreased wind speed, relative humidity and water levels of Lake Kivu over the last part of the 20 th century may have led to reductions in nutrient recycling and lake productivity and contributed to decline in stocks especially of the introduced clupeid (L.

miodon) [START_REF] Akonkwa | Climate change and its impact on the fisheries in Lake Kivu, East Africa[END_REF]. Similarly, gas exploitation could lead to changes in the lake's parameters that might change lake productivity processes and vertical distribution of L.

miodon. Hydroacoustic surveys would be useful in monitoring vertical distributions of fish and relating it to environmental factors in future.

The present results showed an inter-annual spatial pattern in the distributions of L.

miodon. The South and West basins had higher densities, mean sizes, and biomass than the North-Rwanda and North-DRC basins. These results are consistent with those of Guillard et al. (2012), who also found higher concentrations and biomass in these basins, but not those of [START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF], who found no spatial pattern. The conclusions of [START_REF] Lamboeuf | Abondance et répartition du Limnothrissa miodon du lac Kivu, résultat des prospections acoustiques d'avril[END_REF] are linked to the fact that spatiality was coupled with seasonality, meaning that the seasonal variation could have hidden the spatial variation.

It seems that environmental conditions are more favorable for L. miodon in the southern basin. For example, [START_REF] Sarmento | Phytoplankton ecology of Lake Kivu (eastern Africa)[END_REF] showed that phytoplankton concentrations were higher in these regions. Thus, the main food resource of L. miodon, zooplankton, could be more available in these basins because zooplankton dynamics has been shown to be mostly bottom-up controlled (i.e., mostly driven by the primary production), at least during the period of [START_REF] Ruth | Impact of fishing pressure on mean length of fish[END_REF]-2004[START_REF] Isumbisho | Zooplankton of Lake Kivu, half a century after the Tanganyika introduction[END_REF]. This spatial pattern could also be due to lower fishing effort in these basins, thus explaining the larger mean size as well. Indeed, fishing pressure can decrease the mean length of individuals [START_REF] Ruth | Impact of fishing pressure on mean length of fish[END_REF], but this was not demonstrated for L. miodon. Testing these hypotheses will require additional studies on the spatial distributions of the zooplankton communities and fishing effort as this information is currently incomplete [START_REF] Descy | East African Great Lake Ecosystem Sensitivity to changes[END_REF][START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF].

The density, mean size, and biomass of L. miodon were all higher in 2018 than 2012.

The lower values in 2012 could come from underestimation due to a partial survey design because sampling was only carried out in three out of five basins. However, this hypothesis is unlikely because for the other surveys, the total biomass estimated from these three basins gave comparable values to those estimated from all five basins. The low 2012 values could be due to particular environmental conditions that occurred at this time. [START_REF] Rugema | Long-term change of phytoplankton in Lake Kivu: the rise of the greens[END_REF] identified a considerable change in the phytoplankton community of Lake Kivu between 2012 and 2014 with a shift of diatoms and cyanobacteria to green algae. They also reported a difference in nutrients between 2002-2008 and 2012-2014 (an increase of nitrogen and phosphorus) due to a modification of the mixing of the mixolimnion linked to climate variability. Recent analyses of the phytoplankton community revealed a stable concentration of chlorophyll a in 2017-2018 compared to 2012-2014, but there were some changes in phytoplankton composition, with a decrease of green algae and an increase of cyanobacteria in recent years [START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF]. These changes in the phytoplankton community induced shifts in the zooplankton community, leading to a return to conditions that are closer to the community of 2002-2008.

Zooplankton constitute the only food of juvenile L. miodon [START_REF] De Iongh | Feeding habits of the clupeid Limnothrissa miodon (Boulenger), in Lake Kivu[END_REF].Thus, a modification of the zooplankton community could have affected the population, particularly the juveniles. However, it was not possible to test this hypothesis because information regarding variation of the zooplankton community in the pelagic areas was not available for the period of the present study and some years before. It is possible that a change of abiotic factors that occurred between 2011 and 2012 could have led to the decline of L. miodon stock by negatively impacting their recruitment, reproductive cycle, or natural mortality rate. [START_REF] Marshall | Variations saisonnières et annuelles de l'abondance du Limnothrissa miodon du lac Kivu[END_REF] highlighted that abiotic factors (e.g., wind, rainfall) impact the density of L.

miodon in Lake Kivu. High rainfall resulting in high water levels positively influenced the catch per unit effort, while high surface water temperature had a negative influence [START_REF] Akonkwa | Climate change and its impact on the fisheries in Lake Kivu, East Africa[END_REF][START_REF] Marshall | Variations saisonnières et annuelles de l'abondance du Limnothrissa miodon du lac Kivu[END_REF]. This was also observed in Lake Kariba [START_REF] Chifamba | The relationship of temperature and hydrological factors to catch per unit effort, condition and size of the freshwater sardine, Limnothrissa miodon (Boulenger), in Lake Kariba[END_REF].

However, [START_REF] Kaningini | Reproduction et activité migratoire de larves du poisson clupeide Limnothrissa miodon Boulenger[END_REF] pointed out a lack of knowledge concerning the influence of abiotic factors on the physiology of L. miodon, which is still true today. Even if it is not possible to identify the mechanisms leading to an increase of the L. miodon stock, the increase was highlighted.

In the beginning of the 1980s, the biomass of L. miodon were estimated at around 6,000 and 7,000 t (Fig. 4), and in the late 1990s and 2000s, they were around 3,500 and 5,500 (Fig. 4). In 2018, the stock estimate was similar to the 1990s. This long time series of biomass clearly shows that a change occurred in the beginning of the 2010s, which is concurrent with the change in the phytoplankton community detected by [START_REF] Rugema | Long-term change of phytoplankton in Lake Kivu: the rise of the greens[END_REF] but has since then showed an upward trend. This is possible because clupeids have a huge annual regenerative capacity [START_REF] Marshall | Variations saisonnières et annuelles de l'abondance du Limnothrissa miodon du lac Kivu[END_REF]. Overall, the L. miodon stocks have been stable over time. There is need to monitors the stocks of L. miodon and explore the factors that determine stability in the stocks this species.

Conclusions

The present study indicated that the long dry season was the most appropriate season to assess the L. miodon stock in the pelagic zone of Lake Kivu. The stock was concentrated above depths of 45 m and was more abundant it the South and West basins, at least during the long dry season. Even though the L. miodon stock showed downward trends between 1985 up to 2012, the stocks have shown an upward trend since then up to the current estimate of 2018 and have over this whole period been stable at around 4000 tons.

This evaluation showed that the L. miodon stocks of Lake Kivu have been stable over a long period. It has not experienced a decline predicted by [START_REF] Dumont | The Tanganyika sardine in Lake Kivu: another ecodisater for Africa[END_REF] or more recently [START_REF] Akonkwa | Climate change and its impact on the fisheries in Lake Kivu, East Africa[END_REF], which were respectively based on a depletion of prey and climate change (increased water surface temperature and decreased water level due to lower rainfall).

The introduction of L. miodon can still be considered successful as qualified by [START_REF] Gozlan | Introduction of non-native freshwater fish: is it all bad?[END_REF]. The low stock recorded in 2012 on the basis of which [START_REF] Akonkwa | Climate change and its impact on the fisheries in Lake Kivu, East Africa[END_REF] predicted a decline has been reversed.

Thus, as Lake Kivu Monitoring Program (LKMP) established monitoring programs of the main limnological parameters [START_REF] Richard | Conducting fish studies on Lake Kivu and reinforcement of plankton capacities -Final report[END_REF], including hydroacoustic surveys, it is recommended that this program continues periodically. These surveys should relate fish stocks to environmental parameters. Protocols have to be conducted with accuracy to preserve the data quality of long temporal series, including calibration of the equipment before each survey [START_REF] Parker-Stetter | Standard operating procedures for fisheries acoustic surveys in great lakes[END_REF]. As indicated by [START_REF] Robarts | Limnology and the future of African inland waters[END_REF] and highlighted by [START_REF] Rugema | Long-term change of phytoplankton in Lake Kivu: the rise of the greens[END_REF], the present study demonstrates the need of combining high-quality limnological and meteorological data when evaluating freshwater communities, conservation, and governance of the African Great Lakes. For future assessments, it will be necessary to carry out a transdisciplinary study by integrating additional data on fishing effort, the zooplankton community, and climatic factors into analyses to identify the main parameters regulating the L. miodon population in Lake Kivu.

Fig. 2 .

 2 Fig. 2. Mean sA values of the four-vertical water layers per basin in Lake Kivu in 2013 and 2014 in short dry season (black solid line), long rainy season (grey solid line), long dry season (black dashed line), and short rainy season (grey dashed line). The horizontal bars are 95% confidence intervals.

Fig. 3 .

 3 Fig. 3. Length frequency distribution of fish converted from target strength using Love's equation (1971) data in 1-cm classes per basin of Lake Kivu in 2013 in the short dry season (black solid bar), short rainy season (grey solid bar), long dry season (black hatched bar), and long rainy season (grey hatched bar). Arrows indicate peaks.

Fig. 4 .

 4 Fig. 4. Total biomass (tonnes) for all of Lake Kivu by hydroacoustic sampling during different years (present study and historical surveys). The vertical bars are 95% confidence intervals.

Table 3

 3 Acoutsic biomass (sA in m²/ha; proxy for density) of Limnothrissa miodon of Lake Kivu by year and basin. IC is the 95% confidence intervals, *p<0.05. **p<0.01. ***p<0.001 were issued from the post-hoc Tukey's HSD test performed on the year and basin factors

		Mean	IC	Tukey's HSD test results
	A -Year
	2012	0.40	0.05	2012/2013 *, 2012/2014 *, 2012/2015 ***, 2012/2016 **, 2012/2018 ***
	2013	0.79	0.11	2013/2018 ***
	2014	0.68	0.06	2014/2015 **, 2014/2018 ***
	2015	0.98	0.09	2015/2016 *, 2015/2018 ***
	2016	0.72	0.09	2016/2018 ***
	2018	1.33	0.17
	B -Basin
	North-Rwanda	0.77	0.10	North-Rwanda/West ***, North-Rwanda/South ***
	North-DRC	0.62	0.10	North-DRC/West ***, North-DRC/South ***
	East	0.65	0.05	East/West ***, East/South ***
	West	1.23	0.15
	South	1.23	0.14

Table 4

 4 Percentage of Limnothrissa miodon per size class in the different basins of Lake Kivu according to year. Small < 5 cm, 5 cm ≤ medium > 12 cm and large ≥ 12 cm. Map of Lake Kivu showing the transects sampled. The black lines represent the hydroacoustic survey performed between July 30 th and August 14 th 2014 and September 7 th and 27 th 2016 over the five defined regions.

	Year	Size category North-Rwanda	North-DRC	East	West	South	Lake Kivu

Acknowledgments

This research was supported by the Rwanda Energy Group -Energy Development Corporation Limited (REG -EDCL) and supervised by the Lake Kivu Monitoring Programme (LKMP) under the project, "Fish studies on Lake Kivu and reinforcement of plankton capacities," n° 11.07.053/Cons. Firm/578/RFP/2014-2015/PROCEDCL/MD/EK/jan. Previous data collected in the framework of the baseline and EAGLES project (project SD/AR/02A, BELSPO) were used. We are grateful to the LKMP and ISP Bukavu teams for their field assistance. We also thank Jean-Pierre Descy (University of Liège) for his scientific and technical support during the project.

Tables

 Table 1Degree of coverage by hydroacoustic surveys per basin and for the whole lake during different years and seasons. The degree of coverage was obtained by dividing the total length of the path (km) by the square root of the studied surface (km²). The result should be at least 3.0 and preferably above 6.0 to obtain a reliable assessment [START_REF] Emmrich | Strong correspondence between gillnet catch per unit effort and hydroacoustically derived fish biomass in stratified lakes[END_REF].


W U Peizhe
email: peizhe.wu@univ-lorraine.fr

Touhami Salah
email: salah.touhami@univ-lorraine.fr

Julia Mainka
email: julia.mainka@univ-lorraine.fr

Daoudi Meriem
email: meriem.daoudi@univ-lorraine.fr

Lottin Olivier
email: olivier.lottin@univ-lorraine.fr

Jean-Christophe Perrin
email: jean-christophe.perrin@univ-lorraine.fr

A hybrid method combining genetic and Nelder-Mead algorithms for the interpretation of electrochemical impedance dataapplication to Proton Exchange Membrane Fuel Cells

Keywords: electrochemical impedance spectroscopy, equivalent electrical circuit modeling, proton exchange membrane fuel cells, genetic algorithm, Nelder-Mead algorithm, least-square fitting

Electrochemical Impedance Spectroscopy (EIS) is a technique commonly used for characterizing electrochemical systems such as fuel cells, supercapacitors or batteries [1]-[3]. Modeling of the experimental results is mostly done under the form of equivalent electrical circuits (EEC) aiming to represent the charge and mass transport phenomena and electrochemical reactions occurring at the material scale inside the individual components of the electrical components, such as resistors, capacitances, etc. It is necessary to use a numerical solver to estimate the values of the model parameters starting from experimental data. Traditional methods use deterministic algorithms, which makes the results highly dependent on the initial values chosen as starting points. Obtaining reliable and consistent results even requires sometimes to analyze separately the low, intermediate and high frequencies regions of the impedance spectra [4], [5]. In this work, we propose to use a hybrid method combining a genetic algorithm (GA) with a deterministic Nelder-Mead (NM) algorithm. The GA is used for a first estimate because of the main advantages it offers as a global optimization method: it does not require initial values for parameter estimation and is also robust and well adapted for problems with -possibly-multiple solutions. However, GA are well known to require many iterations and are therefore rather slow when some of the optimization parameters are -even partially-correlated [6]. This is the reason why the GA was combined with a NM algorithm to accelerate convergence. Both types of algorithms were taken from MATLAB standard libraries. This hybrid GA-NM method is used mainly to interpret the data measured on Proton Exchange Membrane Fuel Cells (PEMFC), which are generally modeled with Randles-type circuits or Transmission Line Models [7], depending on the shape of the spectra at high frequencies. Such EEC include at least 5 and sometimes up to 12 parameters.

The results show that the combined GA-NM method converges toward satisfying solutions in most of the cases, with low least-square residuals, including when the range of parameter values is unknown and/or when the sensitivity of some of them is low.

I. INTRODUCTION

EIS is based on the application of small amplitude sinusoidal variations of the electrical current (potentiostatic mode) or voltage (galvanostatic mode) around a given operation point. The response, assumed linear, is used to determine a transfer function, i.e. the impedance. The impedance depends on the frequency at which the perturbation is applied, as well as on the system operating point. It is a complex number, often represented in a Nyquist plot (negative of the imaginary part vs. real part), allowing to analyze the phenomena occurring in the system in the frequency domain according to the shape, or other typical characteristics (inflexion points, radii, etc.) of the spectra.

Modeling of the experimental results, i.e. of the impedance spectra, is mostly done under the form of equivalent electrical circuits (EEC) aiming to represent the charge and mass transport phenomena as well as the electrochemical reactions occurring in the components of the system using -more or less-simple electrical components, such as resistors, capacitances, etc. The most used EEC with PEMFC are the Randles circuit and the Transmission Line Model (TLM) suitable for thin and thick electrodes, respectively. The TLM is a generalization of the Randles circuit [START_REF] Touhami | Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells[END_REF], taking into account the electrical potential variation through the electrode thickness. Then, mass transport limitations at low frequencies, are usually accounted for by adding a finite Warburg element in the EEC. The Warburg impedance is based on the hypothesis of Fickian diffusion, surface reaction and constant gas concentration at the interface with the gas channel [START_REF] Warburg | Ueber das Verhalten sogenannter unpolarisirbarer Elektroden gegen Wechselstrom[END_REF]. This element can be placed in series with the charge transfer resistance (in the case of the Randles circuit) or in series with the Cathode Catalyst Layer (CCL) circuit, depending on which is the most limiting layer for gas diffusion: the CCL or the Gas Diffusion Layer (GDL) [START_REF] Touhami | Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells[END_REF].

The identification of EEC parameters requires the use of Least-Square (LS) fitting algorithms, minimizing the global difference between the experimental data and the EEC impedance. One of the most common LS fitting algorithms is the Nelder-Mead algorithm using the concept of simplexes [START_REF] Lagarias | Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions[END_REF]: a determinist and local method, which always converges to the same, and nearest minimum, starting from input (initial) values. On the other hand, Genetic Algorithms are global stochastic algorithms inspired by the theory of natural evolution of Charles Darwin. A group of individuals is generated randomly at the beginning of each execution, each one being associated with a fitness value calculated using the function defined by the user. At each iteration, only a part of individuals, including those with higher fitness value (lower error function) are authorized to reproduce, following a certain way of crossover and mutation. Hence, the population evolves to improve their fitness score from one generation to the other. GA allow thus to explore the entire space of solutions to identify possible multiple local minima, but are therefore much slower 1 The EEC model parameters sent to the algorithms are considered as "variables" of the error function. than deterministic algorithms. Another difference between GA and NM is that GA only require a range of variables 1 instead of starting points (NM algorithm can be adapted to look for a minimum over a given variable range but their performances are not very good in these conditions). Both kinds of algorithm can be executed directly by calling the MATLAB functions "ga" and "fminsearch". Taking figure 1 as an example, NM algorithms will converge rapidly towards the nearest "bottom" of the function, while GA can analyze the entire area and converge to one of the lowest "bottoms" (deep blue ones) [START_REF] Diegotorquemada | English: Rastrigin Function[END_REF]. In such a case, if the GA is repeated many times, more than one solution, i.e. some of the best solutions illustrated by the blue circle may be found.

In our work, we mainly focused on the validation of GA-NM algorithm: a combination of these two algorithms, applied to different PEMFC Membrane-Electrode Assemblies (AME) and cells, which will be explained in details in the following paragraphs.

II. METHOLOGY

Both NM and GA algorithms have their advantages and weaknesses. The NM algorithm is rapid and stable in convergence but is very sensitive to the starting point: a tiny change in one parameter initial value could lead to a completely different result. On the other hand, GA are powerful tools to identify a global minimum and do not require starting point but a search range for each parameter. However, they are much slower and less stable in convergence. These weaknesses are more evident when the problem dimension (number of parameters to fit) is high. To combine the advantages of both and limit their weaknesses, we considered the possibility of a hybrid GA-NM algorithm: in a first step, the GA is used to search in given ranges of the parameters and returns an intermediary result x1, which is then used as a second step through a loop of successive executions of fminsearchbnd to accelerate the convergence to the final result. The "fminsearchbnd" 2 function is repeated until the results between 2 successive calls are close enough to each other (variation of the residuals below 1%). The entire method is illustrated in figure 2.

The idea of combing a global stochastic algorithm and a local deterministic algorithm has already been tested in different domains. Zhou Jian [START_REF] Zhou | A hybrid least-squares genetic algorithm-based algorithm for simultaneous identification of geometric and compliance errors in industrial robots[END_REF] used a hybrid GA algorithm to simultaneously identify the geometric parameters, joint values, and the transition joint torques for industrial robots. Li Wang [START_REF] Wang | A hybrid Genetic Algorithm and Levenberg-Marquardt (GA-LM) method for cell suspension measurement with electrical impedance spectroscopy[END_REF] tested a hybrid method of GA coupled with Levenberg-Marquardt method for the interpretation of EIS data applied on a biological cell suspension model interpretation. G.Cai applied GA combined with COMSOL to optimize the structure of their PEMFC [START_REF] Cai | Design and optimization of bioinspired wave-like channel for a PEM fuel cell applying genetic algorithm[END_REF]. Researches being little developed on electrochemical systems, our work will be focused on the application of this concept to solve the least-square problem of EIS parameter estimation in PEMFC modeling, with the final objective to extend later this method to other electrochemical systems such as supercapacitors and batteries.

As mentioned previously, the GA is a stochastic algorithm. So, when the problem is complex (often because of large number of parameters, correlations between them, or minor influence of some of them, i.e. some parameters influence the error function much more/less than the others), it is possible to obtain more than one result for different calls (figure 1). These results represent different local minima of the error function and could stand for different physical hypotheses. On the other hand, when the parameter search interval is too narrow, the function may miss a better local minimum i.e. a result that minimizes better the error function. Therefore, the parameter interval sometimes needs to be reduced for a more stable convergence, or extended to make sure to explore more physically consistent possibilities.

III. EXPERIMENTAL SETUP

The experimental data we studied cover a wide range of different electrochemical systems (mainly PEMFC single cells and to a lower extend supercapacitors) who differ mainly by the type of electrolyte membrane and cell geometry. We present here the most representative cases: the results and tendency obtained on these various systems are similar.

Cell A: The MEA used were manufactured in the laboratory using 25.6 cm 2 (64 × 40 mm 2 ) Nafion XL100 membranes and 7.22 cm 2 (19 × 38 mm 2 ) commercial Gas Diffusion Electrodes (GDE) with 0.5 mgPt.cm -2 catalyst layers and 235μm thick gas diffusion layers (5% PTFE and 80% porosity) coated with Sigracet 29BC Micro-Porous Layers (MPL). This PEMFC is evaluated under 2 The default fminsearch function of MATLAB does not admit to set bound constraints to variables of the error function. We thus used a slightly modified version "fminsearchbnd" which also uses the Nelder-Mead air supply, the other operating conditions are exactly the same as for "cell B" mentioned in [START_REF] Touhami | Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells[END_REF].

Cell B is fed with pure oxygen and hydrogen. The MEA performance was tested in a single cell made of two identical gold-coated stainless-steel plates on the anode and cathode sides, with single serpentine channel (19 passes) of 0.7 mm in depth and 1 mm in width. The cell was operated at atmospheric pressure and fed in counterflow with 80% Relative Humidity (RH) hydrogen and oxygen. The gas stoichiometries were set to 1.5 on both sides. The flow field plate temperature was controlled thanks to a water circulation loop and set to 70°C. To avoid any damage to the MEA during the start-up procedure, the temperature was slowly increased from 40°C to 70°C -with 600 seconds steps of 10°C-while algorithm but with bounds applied to the variables [START_REF]fminsearchbnd, fminsearchcon[END_REF].

For short, "fminsearch" always stand for "fminsearchbnd" in this paper.

both the anode and cathode compartments were flushed with 10 SLPH (standard liters per hour) of nitrogen.

IV. RESULTS

For a PEMFC supplied with air, mass transfer limitations are much more significant than with pure oxygen. Therefore, the mass transport loop in the EI spectra (at low frequencies) of the PEMFC fed with oxygen is much less pronounced or even sometimes negligible compared to a FC supplied with air (figure 4 -PEMFC fed with air, figure 5-PEMFC fed with oxygen), especially at low current densities (i.e. below 1 A/cm²). Therefore, numerical methods (GA and NM) may not be able to clearly identify the mass transport impedance parameters, resulting in many solutions at the output.

After several tests and comparisons, it was not possible to satisfyingly fit the high frequency part of the spectra of cell B using either a TLM EEC alone to represent the CCL, or by adding an anode EEC. This difficulty corresponds more likely to the influence of inductive phenomena [START_REF] Thompson | Electrochemical Impedance Spectroscopy Data from Solid Oxide Cells Undergoing Co-Electrolysis: The Influence of Rig Inductance[END_REF], which are pronounced mostly at high current densities (1.5 to 2 A/cm2). So, in this case, it was necessary to add an inductive component to the EEC to enhance the fitting quality, although this parameter adds some uncertainty to the parameter estimation without bringing any physical information regarding the MEA.

The EEC used for analyzing cell A and B are shown in figure 3. The EEC used for cell A (figure 3 left side) is composed of a modified TLM EEC and a Warburg element in series (tW), supposing the GDL responsible for mass transport [START_REF] Touhami | Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells[END_REF].The only difference with the EEC used for cell B (figure 3 right side) is the inductance element in series (tWL). The associated impedance expressions are hence (with or without Z L (𝜔)): The model parameters to fit are:

𝑍 Cell (𝜔) = R hf + 𝑍 TLM + Z W (𝜔)(+Z L (𝜔)) (1
• Rhf: high frequency resistance, representing the resistance phenomena at high frequencies.

• Rct:: charge transfer resistance, characterizing the rate of the electrochemical reaction of the FC

• Cdl: double layer capacity, an indicator of the electrode ability to separate the positive and negative charges.

• 𝑅𝑑: diffusion resistance.

• 𝜏: characteristic diffusion time.

• 𝑅ion: ion conduction resistance. Cell A 100 repetitions of GA-NM algorithm result in 3 solutions, corresponding to 3 numerically different local minima (marked as GA-NM 1, 2 and 3 in table 1), with the result GA-NM1 being the most frequent (66 occurrences) and the most coherent with the values obtained in the former work of Touhami et al. [START_REF] Touhami | Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells[END_REF]. In this previous work, the results were obtained using a NM algorithm alone with carefully chosen initial parameters It is difficult to conclude on the best solution only by looking at the shape of spectra (figure 4), but we can see from the value of the residuals that the GA-NM1 fits best to the experimental spectrum (table 1). The tiny differences between that solution and the results previously published in the article confirms the advantage of GA-NM because the residuals are slightly lower. This also confirms the sensitivity of the NM algorithms to the starting points.

Cell B

Due to the complexity of the multiple local minima are often observed if we repeat the GA-NM algorithm using the same variable bounds as cell A: the parameter values obtained vary drastically, giving sometimes values without any physical meaning. This could be because of correlations between some parameters and/or of the poorly developed mass transport loop at low frequencies.

To solve this problem of instability, we reduced the bounds for the most influent parameters: Rhf and Rct which stand for the intercept of the spectrum with the real axis at the highest frequencies and the diameter of cathode loop at intermediate frequencies, respectively. In fact, the approximative value of these 2 parameters can be evaluated graphically from the experimental spectrum, all values out of these magnitudes are physically meaningless. After the modification of bound limits, the combined algorithm resulted in one unique solution out of 100 repetitions for each tested EEC: From figure 5, we can see that the addition of an inductive element improves significantly the fitting quality although the inductance value is relatively small. The residual value of the tWL model is less than half of that of the tW model. The common parameters are also close to each other except for Rion, which means the algorithm ignored the influence of electrode thickness without the inductance element which is known to influence the EI spectrum at high frequencies [START_REF] Touhami | Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells[END_REF]. It is worth mentioning that even if the high slope typical of TLM behavior is only poorly developed for cell B, the use of TLM EEC clearly improved the fit quality compared to a Randles type circuit.

V. CONCLUSION

The results shown in the previous paragraphs are a part of the work we carried out and are the most representative ones. We tested a lot of cases of PEMFC and SC impedance spectra with different MEA/electrode materials or dimensions, under different operating conditions (current/voltage), the tendencies of the different cases are similar.

According to our results, the main conclusions are:

• Generally, our hybrid GA-NM algorithm can successfully converge to the right solution and explore more possible local minima and this without the need of a starting point.

• When the problem is complex, physical analysis will be necessary to evaluate the physical consistency of the results or limit parameter bounds for a more stable convergence.

Good performance is proven for different systems confirming the convenience and advantage of combined GA-NM algorithm. This method can be applied for further modeling of new configurations and systems in the future.

VI. ACKNOWLEDGMENT

This work has received funding from the region Grand Est and the European Fund for Economic and Regional Development FEDER.

Figure 1 :

 1 Figure 1: Top: Comparison of GA and NM algorithm, example for a 2-dimension function, bottom: concept of the GA-NM algorithm[10].

Figure 2 :

 2 Figure 2: Procedure of GA-NM algorithm.

  and Z L (𝜔) = 𝑖𝜔𝐿 , and 𝜔 = 2𝜋𝑓 the angular frequency.

Figure 3 :

 3 Figure 3: EEC of the two PEMFC. Left: Modified TLM + Warburg in series (tW) used for cell A. Right: Modified TLM + Warburg in series + inductance (tWL) used for cell B.

Figure 4 :

 4 Figure 4: Comparison of different fitting results of cell A, I = 0.5A/cm 2 , T = 70C°, RH = 80%.

Figure 5 :

 5 Figure 5: Comparison of the fitting curves obtained with different EEC models for cell B.

TABLE 1 :

 1 Comparison of estimated parameters of the three different local minima. Parameter bounds: 0 to 1 for all.

TABLE 1 :

 1 Impedance parameters estimated for different EEC. Modified parameter bounds: 0.06 < Rhf < 0.1 Ω.cm 2 , 0.04 < Rct < 0.06 Ω. cm 2 , and 0 to 0.1 for others.

	Model

Rhf (Ω.cm 2 ) Rct (Ω.cm 2 ) Cdl (F.cm -2 ) Rion (Ω.cm 2 ) Rds (Ω.cm 2 ) τs (s) L (H. cm 2 ) Residual

	Data	Rhf (Ω.cm 2 ) Rct (Ω.cm 2 )	Cdl (F.cm -2 )	Rion (Ω.cm 2 ) Rds (Ω.cm 2 )	τs (s)	Residuals	Occurrence
	Article of Touhami
	et al. [7]	0.0710	0.0740	0.0570	0.191	0.172	0.0730	0.124
	GA-NM1	0.0707	0.0754	0.0550	0.188	0.173	0.0739	0.117	66
	GA-NM2	0.0711	0.0164	0.316	1	0.174	0.0733	0.159	31
	GA-NM3	0.0716	0.131	0.234	0.389	0.0610	0.0035	0.209	3
	tW	7,77*10 -02	5,47*10 -02	1,98*10 -02	1,27*10 -09	7,43*10 -03	1,71*10 -01		5,23*10 -02
	tWL	6,19*10 -02	5,34*10 -02	2,08*10 -02	5,55*10 -02	7,34*10 -03	1,76*10 -01	9,04*10 -07	2,58*10 -02
					.