Stembile Msiteli-Shumba

Shakki Kativu
email: skativu@gmail.com

Beaven Utete

Edwin Makuwe
email: emakuwe@yahoo.com

Florence D Hulot
email: florence.hulot@u-psud.fr

Driving factors of temporary and permanent shallow lakes in and around Hwange National Park, Zimbabwe

Keywords: eutrophication, water level fluctuation, organic matter inputs, groundwater pumping

Small aquatic ecosystems in semi-arid environments are characterised by strong seasonal water level fluctuations. In addition, land use as well as artificial pumping of ground water to maintain water resources throughout the dry season may affect the functioning of aquatic ecosystems. In this study, we investigated pans situated in and around Hwange National Park, Zimbabwe, where certain waterholes are artificially maintained during the dry season for conservation purposes. We monitored 30 temporary and permanent waterholes for seven months across the wet and dry seasons in 2013 and analysed them for standard parameters to investigate seasonal variations, assess the effects of land use and pumping on lake functioning, and determine the driving factors of these aquatic systems. Results show an increase in conductivity, hardness, and turbidity when temporary pans dry up and permanent ones are filled with ground water. Prominent parameters explaining the diversity of aquatic ecosystems are water hardness, conductivity, turbidity, and the presence of vegetation.

Seasonality differences in certain parameters suggest the influence of water level fluctuations associated with rainfall, evaporation, and pumping activities. Further, the distinction between turbid pans and those with clear water and vegetation suggests the alternative functioning of pans. Land use had no significant effects, while the effects of pumping are discussed. In times of water scarcity, animals gather around artificially maintained waterholes and foul water with faeces and urine, thus inducing water eutrophication.

INTRODUCTION

Water level fluctuation (WLF) is an important driver of shallow lakes in temperate and tropical zones [START_REF] Beklioglu | State of the art in the functioning of shallow Mediterranean lakes: workshop conclusions[END_REF][START_REF] Blaustein | Why study ecology in temporary pools?[END_REF][START_REF] Coops | The role of water-level fluctuations in shallow lake ecosystemsworkshop conclusions[END_REF][START_REF] Hofmann | Temporal scales of water-level fluctuations in lakes and their ecological implications[END_REF][START_REF] Jeppesen | Ecological impacts of global warming and water abstraction on lakes and reservoirs due to changes in water level and related changes in salinity[END_REF][START_REF] Mustapha Mk | Assessment of the water quality of Oyun Reservoir, Offa, Nigeria, using selected physico-chemical parameters[END_REF][START_REF] Schwartz Ss | Temporary habitats: and opportunities[END_REF]. Although most research in aquatic ecology focuses on permanent water bodies [START_REF] Schwartz Ss | Temporary habitats: and opportunities[END_REF], studying the role of WLF in the drying up of ecosystems and their responses is of great need [START_REF] Coops | The role of water-level fluctuations in shallow lake ecosystemsworkshop conclusions[END_REF]. Water levels in shallow lakes depend on regional climatic conditions (from temperate to arid) as well as human activities such as water abstraction for irrigation [START_REF] Coops | The role of water-level fluctuations in shallow lake ecosystemsworkshop conclusions[END_REF][START_REF] Jeppesen | Ecological impacts of global warming and water abstraction on lakes and reservoirs due to changes in water level and related changes in salinity[END_REF]. WLF and drying up affect species living in the aquatic-terrestrial transition zones, in particular littoral helophytes [START_REF] Hawes I | Effect of wave exposure on vegetation abundance, richness and depth distribution of shallow water plants in a New Zealand lake[END_REF][START_REF] Thomas S | The different primary producers in a small African tropical reservoir during a drought: temporal changes and interactions[END_REF][START_REF] Thomaz Sm | Effect of reservoir drawdown on biomass of three species of aquatic macrophytes in a large sub-tropical reservoir (Itaipu, Brazil)[END_REF] and benthic invertebrates [START_REF] Bazzanti | Functional characteristics of macroinvertebrate communities in Mediterranean ponds (Central Italy): Influence of water permanence and mesohabitat type[END_REF][START_REF] Coops | The role of water-level fluctuations in shallow lake ecosystemsworkshop conclusions[END_REF][START_REF] Strachan Sr | Microrefuges from drying for invertebrates in a seasonal wetland[END_REF][START_REF] White Ms | Natural lake level fluctuation and associated concordance with water quality and aquatic communities within small lakes of the Laurentian Great Lakes region[END_REF] as well as organisms living in the pelagic zone [START_REF] Kelley R | A survey of the physiochemical characteristics and the zooplankton communities in an ephemeral karst lake[END_REF][START_REF] Medeiros Ldc | Is the future blue-green or brown? The effects of extreme events on phytoplankton dynamics in a semi-arid man-made lake[END_REF][START_REF] Michaloudi | Plankton Succession in the Temporary Lake Koronia after Intermittent Dry-Out[END_REF][START_REF] Si | Zooplankton and environmental characteristics of Shiroro Lake at the extremes of its hydrological cycle[END_REF][START_REF] Teferi | Strong effects of occasional drying on subsequent water clarity and cyanobacterial blooms in cool tropical reservoirs[END_REF]. Chemical properties and nutrient concentrations are also affected by WLF with the alternation of dilution and concentration [START_REF] Barson | Influence of drought and flooding on the colonisation of floodplain pans by riverine fish in the Zimbabwean lowveld[END_REF][START_REF] Coops | The role of water-level fluctuations in shallow lake ecosystemsworkshop conclusions[END_REF][START_REF] Coppens | Impact of alternating wet and dry periods on long-term seasonal phosphorus and nitrogen budgets of two shallow Mediterranean lakes[END_REF][START_REF] Jeppesen | Ecological impacts of global warming and water abstraction on lakes and reservoirs due to changes in water level and related changes in salinity[END_REF][START_REF] Mcculloch Gp | Hydrochemical fluctuations and crustacean community composition in an ephemeral saline lake (Sua Pan, Makgadikgadi Botswana)[END_REF][START_REF] Nhiwatiwa | The importance of landscape and habitat properties in explaining instantaneous and long-term distributions of large branchiopods in subtropical temporary pans[END_REF][START_REF] Ozen | Drought-induced changes in nutrient concentrations and retention in two shallow Mediterranean lakes subjected to different degrees of management[END_REF]. The effects will be increasingly amplified by climate change because of increased temperatures that enhance evaporation and evapotranspiration and decreased precipitation [START_REF] Jeppesen | Ecological impacts of global warming and water abstraction on lakes and reservoirs due to changes in water level and related changes in salinity[END_REF]. As a consequence, increased salinity and eutrophication poses a strong threat to shallow lakes, especially in semi-arid and arid regions [START_REF] Coppens | Impact of alternating wet and dry periods on long-term seasonal phosphorus and nitrogen budgets of two shallow Mediterranean lakes[END_REF][START_REF] Jeppesen | Ecological impacts of global warming and water abstraction on lakes and reservoirs due to changes in water level and related changes in salinity[END_REF]. Land use is another strong driver of aquatic ecosystems: it determines allochtonous substances and the concentrations that reach water, as well as transportation rates and fluxes between the terrestrial and aquatic habitats [START_REF] Brendonck | Global diversity of large branchiopods (Crustacea : Branchiopoda) in freshwater[END_REF], Davis et al., 2015[START_REF] Nhiwatiwa | The importance of landscape and habitat properties in explaining instantaneous and long-term distributions of large branchiopods in subtropical temporary pans[END_REF][START_REF] Pacheco Fal | Environmental land use conflicts in catchments: A major cause of amplified nitrate in river water[END_REF][START_REF] Williams Cj | Human activities cause distinct dissolved organic mattercomposition across freshwater ecosystems[END_REF]. As a consequence, land use also drives species diversity and abundance, and so any changes in land use such as deforestation and increased agriculture and urbanisation have and will have profound effects on freshwater communities [START_REF] Bernard C | Design and application of a stratified sampling strategy to study the regional distribution of cyanobacteria (Ile-de-France, France)[END_REF][START_REF] Kelley R | A survey of the physiochemical characteristics and the zooplankton communities in an ephemeral karst lake[END_REF][START_REF] Stomp | Large-scale biodiversity patterns in freshwater phytoplankton[END_REF].

WLF and water scarcity in arid and semi-arid ecosystems are a major issue for wildlife conservation and domestic herbivores [START_REF] Epaphras Am | Wildlife water utilization and importance of artificial waterholes during dry season at Ruaha National Park, Tanzania[END_REF][START_REF] Fynn Rws | African wetlands and their seasonal use by wild and domestic herbivores[END_REF][START_REF] James Cd | Provision of watering points in the Australian arid zone: a review of effects on biota[END_REF][START_REF] Pettit Ne | Seasonal changes in water quality and macrophytes and the impact of cattle on tropical floodplain waterholes[END_REF][START_REF]Water quantity and quality as the factors driving the Serengeti ecosystem, Tanzania[END_REF]. In addition to drinking water, WLF determine plant zonation in wetlands and, as a result, access to key sources of forage for wild and domestic herbivores during the dry season [START_REF] Fynn Rws | African wetlands and their seasonal use by wild and domestic herbivores[END_REF][START_REF] Pettit Ne | Seasonal changes in water quality and macrophytes and the impact of cattle on tropical floodplain waterholes[END_REF]. To support pastoralism and conservation objectives, artificial sources of water have been created by feeding natural waterholes with ground water for instance. As aquatic ecosystems provide valuable water and forage sources in semi-arid regions, it is therefore important to understand their structure and functioning for management and conservation purposes. Furthermore, they are major contributors to regional biodiversity [START_REF] Williams Dd | The biology of temporary waters[END_REF].

Hwange National Park (HNP), Zimbabwe, is a typical semi-arid savanna with a surface water deficit for the greater part of the year (Chamaille-Jammes et al., 2007). Due to this water deficit, 67 boreholes were installed at some natural waterholes and wetlands, allowing for the artificial pumping of ground water and maintenance of surface-water availability throughout the year (Chamaille-Jammes et al., 2007). According to Chammaille-Jammes (2007), without artificial pumping, only 19.6 % of the park remains within 5km of a water source under average climatic conditions. These artificial waterholes help alleviate the effects of water shortage and variability for wildlife conservation. However, modifying the hydroperiod of HNP aquatic ecosystems might have profound effects on their functioning and change the driving factors. HNP is surrounded by communal areas and a forest reserve. It is therefore a very interesting site to explore the effects of WLF and land use on the physico-chemical characteristics of water. The objectives of the present study were therefore (1) to monitor natural and artificial waterholes in HNP and the neighbouring forest reserve and dams in the communal area over the course of a year; (2) to investigate the underlying factors driving the physico-chemical characteristics of waterholes and dams; (3) to study whether these factors change with the seasons; and (4) to examine whether the physico-chemical characteristics of waterholes and dams differ according to their location (i.e., inside the park, in the communal or forestry areas) and their pumping status.

METHODS

Study area

HNP is situated in Matabeleland North Province, Zimbabwe (Fig. 1). It covers an area of about 14,651 km 2 [START_REF] Child G | Wankie National Park[END_REF] and is located in a semi-arid region with an average annual rainfall of 650 mm [START_REF] Jp | Drought mortality of bush elephants in Hwange National Park, Zimbabwe[END_REF]. The vegetation in most of the park consists of mixed woodland and bushland on Kalahari sands (Chamaille-Jammes et al., 2007), especially in the Hwange Main Camp area, the Sikumi Forest, and communal areas adjacent to the park where this study was carried out (Fig. 1, Table 1). In the National Park, there is no inhabitants and space is devoted to wildlife conservation. Domestic and wildlife coexist in the forest reserve and in the communal area but the former is uninhabited."The rainy season lasts from December to April, with Table 1. List of the pans sampled, whether they were natural (N) or artificial (A), their location, and the months when they were sampled. √ (tick) represents the months sampled and × (cross) the months not sampled.

Pans (abbreviations) Status Location Months sampled Fev Mar Apr Jul Sep Nov Dec

Ballaballa (Bal) N Park √ √ √ √ × × √ √ √ √ Dopi (Dop) A Park √ √ × √ √ √ × Foster (Fos) A Forestry √ √ √ × × × √ √ √ √ Jambile Artificial (JamArt) A Park √ √ √ √ × √ √ Jambile Natural (JamNat) N Park √ √ √ × × × × Jwapi N Forestry √ √ √ √ × × √ √ √ √ × × Magoli (Mag) N Communal √ √ √ × × × × Mahule (Mahu) N Forestry √ √ × × × × Park √ √ √ × × × ×
Waterholes in HNP and in the forestry and communal areas are depressions that are fed during the rainy season by rainwater runoff; we refer to them as pans [START_REF] Fynn Rws | African wetlands and their seasonal use by wild and domestic herbivores[END_REF].

Some natural pans in the communal area are dammed up to store water; we refer to them as dams. Artificial pans are fed by groundwater during the dry season and are found in the park and forestry area. The surface area of dams found in the communal area ranged from 21,903 m 2 during the rainy season to 531 m 2 during the dry season if they did not dry up completely, while the natural and artificial pans in the park and forestry area ranged from 26,788 m 2 during the rainy season to 453 m 2 during the dry season, or 0 m² for natural pans that dried up. Both the dams and water pans were very shallow, with a depth not exceeding 3 m during the rainy season, while the natural water bodies retained water for only 3 to 6 months.

Sampling and analysis procedures

Sampling was carried out over seven months during the wet (February, March, April, and December) and dry (July, September, and November) seasons in 2013. Thirty pans were selected to represent a diversity of artificial and natural pans with or without vegetation, and located inside or outside the park. These comprised 5 dams from the communal lands (Hwange communal area and private area), 5 pans in the forestry area (Sikumi Forest), and 20 pans in the Hwange Main Camp and Sinamatella area (Fig. 1, Table 1).

Water samples were taken before 12h00 so as to standardise sampling and minimise variability in the time of day. The following variables were measured in situ near the shoreline of the pans and dams with a YSI 6600 VZ Multiparameter Water Quality Sonde: dissolved oxygen (DO), pH, conductivity (Con), total dissolved solids (TDS), turbidity (Tur), chlorophyll a (Chl a), and phycocyanin (PC). Water samples for laboratory analysis were collected on site and stored in ice in one-litre polyethene bottles. An assessment of vegetation cover was made using the following arbitrary scale: 0% = absent, 25% = one-quarter coverage, 50% = half coverage, 75%

= three-quarter coverage, and 100% = full coverage of the waterhole. Suspended matter (SM) was measured by filtering a known volume of water over pre-weighed GF/F 47mm filters, drying the filter at 105°C for 24 hours, and then measuring the difference in mass before and after filtration. Chemical variables were determined with a HACH DR 3900 portable data logging spectrophotometer and reagents. Total nitrogen (TN) and total phosphorous (TP) were determined from unfiltered water samples, while filtered samples were used to determine total organic carbon (TOC), hardness, calcium, magnesium, nitrate, nitrite, and ammonia concentrations in accordance with the manufacturer's manual (HACH DR/3900 manual, 2011).

Statistical analysis

This study aimed to explore the factors driving the functioning of pans, determine whether these factors change with the seasons, and ascertain whether the pans' functioning differs according to their location (inside the park, in the communal or forestry areas) and pumping status (natural or artificial). To respond to these questions, multiple factorial analysis (MFA) (Escofier and Pages, 1994) was applied to the dataset comprising monthly variables in order to assess the general structure of the data and determine the relationships between the monthly data and variables structuring these relationships. MFA allows the use of qualitative and quantitative variables. Consequently, the location and pumping status of the pans were included in the MFA as qualitative variables and all other parameters as quantitative variables. In total, 30 pans were monitored. However, occasionally, some pans could not be sampled for technical reasons, while natural pans dried up during the dry season (for an overview, see Table 1). For these reasons, two MFAs were conducted: one on a dataset restricted to the wet season (February, March, and April) involving 27 pans and another on a dataset limited to the 10 pans that did not dry up during the dry season. The numerical analysis and preparation of figures were carried out using software R (R Development Core Team, 2008), version 3.1.1. MFA was conducted using the FactoMineR package [START_REF] Husson | FactoMineR: Multivariate Exploratory Data Analysis and Data Mining[END_REF], version 1.29. Due to missing values in our data, the missMDA package [START_REF] Husson | missMDA: Handling missing values with/in multivariate data analysis (principal component methods)[END_REF], version 1.7.2, was used to impute the missing values.

RESULTS

Seasonal variation in physico-chemical parameters

The tested water pH ranged from 6 to 9.94 (Fig. 2a). Conductivity increased from the rainy season to the dry season, while its levels at all sites were generally high. Variability between the pans was low during the wet season (0ca 1 mS/cm), later increasing from July to November (0ca 5 mS/cm). Similarly to conductivity, there was a corresponding increase in salinity (Fig. 2d) from the rainy to dry months of the year. Even though most values were within acceptable ranges of less than 1 ppt, some of the values were saline. There was a steady increase in water hardness from February to September 2013 (Fig. 2c). Water hardness was associated with calcium (Fig. 2e) and magnesium (Fig. 2f) ions. Calcium, however, mostly determines the trend of water hardness for the year 2013.

Turbidity levels were highly variable, ranging from 50 NTU to 600 NTU with values rising to 1200 NTU (Fig. 3a). Turbidity was also low during the rainy season, increasing from April to December. The high variability in turbidity between sites should be noted during the dry season. Similarly to turbidity, suspended matter (Fig. 3b) also increased during the dry season. Values generally ranged between 0 to 8 g/L, even though outliers reached up to 13 g/L in September. DO (Fig. 3c) values were highly variable despite a decrease from the rainy to the dry season. TOC (Fig. 3d) gradually increased from the rainy to the dry season with values ranging from 3 mg/L during the wet months to approximately 55 mg/L during the dry season. Variability between sites was greater during the dry season as compared to the rainy season. There was a very subtle increase in total nitrogen (Fig. 4a) with a minimum variability observed except for the month of September; all monthly values were within 20 mg/L, excluding September when total nitrogen went up to about 70 mg/L. Between September and December, there was a high variability in ammonium concentration, which reached very high values (up to 21 mg/L) (Fig. 4b). Nitrate concentrations were not predictable as shown in Figure 4c. In contrast, nitrite concentrations (Fig. 4d) were fairly constant as compared to nitrate concentrations. Phosphorous concentrations were between 0 to 2.5 mg/L for TP (Fig. 4e). There was no change (Fig. 5a) in chlorophyll a concentration between the different months in our study.

Chlorophyll a concentrations generally ranged from 0 to around 300 µg/L, even though there were a few outliers, which reveal phytoplankton blooms at five sites (Kennedy 1, Kennedy 2, Guvalala, Nyamandlovu and Makwa). Subtle variations were observed in PC concentrations as shown in Figure 5b, but very high densities indicated blooms of cyanobacteria in April, July, and December. Figure 5c shows the trend for aquatic macrophytes for the year 2013. The results reveal the diversity of water reservoirs during the wet season, either with or without vegetation. As pans dry out, vegetation consequently disappears, only remaining in a few pans that retain water throughout the year.

Multifactorial analysis of the wet season

MFA was conducted on data measured in 27 pans during the wet months (February, March, and April). The first two dimensions of the MFA explained 32.57% of the total variance. The parameters that significantly correlated with the first dimension (19.17% of the total variance) were suspended matter, chlorophyll a and total nitrogen concentrations, turbidity, and conductivity for the three months, phycocyanin concentration for February and March, and hardness, reactive phosphorus and nitrite concentrations for February alone, total organic carbon concentration for April (Fig. 6, Appendix 1). These parameters were negatively associated with the presence of vegetation in and around the pans during these three months and high DO concentration in February and March (Fig. 6). This homogeneity in parameters regarding the first dimension is summarised in the group representation (Fig. 7) in which the months have the same relative position and make a similar contribution to the first dimension in the MFA (Table 2, contribution to the dimension 1). Pumping status, included in the MFA as a qualitative value, was not significantly correlated with the first dimension. However, pans that were artificially maintained with pumped water during the dry months tend to be associated with positive values of the first dimension, while natural pans tend to be associated with negative values (Fig. 8). The location of the pans, also included in the MFA, was not significantly correlated with the first dimension. The first dimension of the MFA suggests a distinction between highly conductive, turbid pans with a high abundance of phytoplankton and soft, clear pans with macrophytes and a low abundance of phytoplankton during the wet season.

a. b. During the wet months, the parameters with a significant positive correlation to the second dimension (13.40 % of total variance) were turbidity for the three months, suspended matter and total nitrogen concentrations for March and April, total organic carbon concentration for March, and pH and DO concentration for April. These parameters were negatively associated with high conductivity for the three months and hardness for February and March (Fig. 6, Appendix 1).

With the second dimension, the group of pans with turbid and hard water is divided between turbid pans with high suspended matter and total nitrogen concentrations and those with hard water and a high abundance of phytoplankton. The contributions of these groups are heterogeneous, with a higher contribution in March and April compared to February (Table 2, Fig. 7a). The distinction between turbid and hard water pans relies more on the dynamics of the pans at the end of the rainy season (i.e., March and April) than the mid-season (i.e., February). The correlation coefficients (Table 2) measure the extent to which the months are related to each other and to the MFA. The results (Table 2) show that February and March are more correlated to each other than are March and April, and finally February and April, which is in line with the progression of the season.

The third dimension explained 9.22% of the variance and is not represented.

Multifactorial analysis of the 2013 data

MFA conducted on the 10 pans that retained water during the dry season shows that the first two dimensions explain 44.63% of total variance, while the third dimension 12.61% (not shown). Similarly to the MFA of the wet season, the main parameters that were significantly correlated with the first dimension (25.42% of total variance) were conductivity for the seven months (Fig. 9, Appendix 1). Other parameters that were significantly and positively correlated with the first dimension and shared by several months are hardness, TOC, and TN concentration. Nitrite, ammonium, SM, and PC concentrations as well as pH are also positively correlated with the first dimension but are less homogeneous between months (Fig. 9, Appendix 1). This second dimension differentiates pans that experienced cyanobacterial blooms that tend to be associated with hard water and turbid pans. The contribution of the different groups (months) to this dimension is more heterogeneous than for the first dimension, with April having the highest contribution and July the lowest (Fig. 8b, Table 3). The months' contributions to dimensions 1 and 2 and the correlations between the months do not follow the seasons or differ between wet and dry seasons.

DISCUSSON

Our results show a strong seasonal variability in conductivity, hardness, turbidity, as well as calcium, magnesium, suspended matter, total organic carbon, total nitrogen, and ammonium concentrations. Values are low during the rainy season and increase during the dry season. An increase in conductivity and salinity was also observed in pounds in Southern Africa [START_REF] De Klerk Ar | Seasonal variations of water and sediment quality parameters in endorheic reed pans on the Mpumalanga Highveld[END_REF][START_REF] Koning N | The continued influence of organic pollution on the water quality of the turbid Modder River[END_REF][START_REF] Pettit Ne | Seasonal changes in water quality and macrophytes and the impact of cattle on tropical floodplain waterholes[END_REF][START_REF] Jc | SALINITY AND DISSOLVED SUBSTANCES IN THE VAAL RIVER AT BALKFONTEIN, SOUTH-AFRICA[END_REF]. Low Con during the rainy season is due to the dilution effects of rainfall, while decreasing water levels during the dry season have a concentrating effect on dissolved ions in the water (de [START_REF] De Klerk Ar | Seasonal variations of water and sediment quality parameters in endorheic reed pans on the Mpumalanga Highveld[END_REF][START_REF] Pettit Ne | Seasonal changes in water quality and macrophytes and the impact of cattle on tropical floodplain waterholes[END_REF]. This concentration effect may be exacerbated by groundwater pumped into artificial pans. Measurements of ground water at the opening of the pump in October 2014 show high water Con for some pans (Dopi: 952 µS.cm -1 ; Safari Lodge: 1.365 mS.cm -1 ; Nyamandlovu: 1.369 mS.cm -1 ; Guvalala:

8.549 mS.cm -1 ). Water hardness shows the same trend as Con and is also due to the concentration of calcium ions with water evaporation. Unfortunately, we did not measure pumped water hardness or calcium and magnesium concentrations. It was previously suggested that pumping water into most arid protected areas has produced more negative than positive results [START_REF] Jv | Surface water availability and the management of herbivore distributions in an African savanna ecosystem[END_REF]. In addition to vegetation trampling throughout the year and increasing prey predation around waterholes [START_REF] Aw | Resource heterogeneity and ungulate population dynamics[END_REF], pumping may induce the deterioration of water quality through contamination from saltwater intrusion, a major concern in semi-arid and arid environments (Borrok and Engle, 2014).

Our results show high Tur levels with an increase during the dry season, which is contrary to the results of Koning and Roos (1999) who demonstrated higher Tur levels during the rainy season. Their findings were explained by the inflow of rainfall into the ecosystem. Rainfall discharge caused the resuspension of silt molecules in the water. In HNP, the increase in Tur seems to be related to the increase in mega fauna activity around the remaining water bodies during the dry season (Chamaille-Jammes et al., 2007[START_REF] Mukwashi | Impact of African elephants on Baikiaea plurijuga woodland around natural and artificial watering points in northern Hwange National Park, Zimbabwe[END_REF]. In addition to fouling with faeces and urine, domestic and wild fauna may also induce bioturbation, which results in organic matter and sediment resuspension, and reduces water clarity. Fouling with dung is also observed, for instance, with feral dromedary camels in Australia [START_REF] Mcburnie | The impacts of an invasive herbivore (Camelus dromedaries) on arid zone freshwater pools: An experimental investigation of the effects of dung on macroinvertebrate colonisation[END_REF]. Unsurprisingly, DO levels were highly variable between pans. Oxygen concentration increases with photosynthesis, lower temperatures, and water stirring, while it decreases with a high concentration of decomposing organic material.

In addition, Tur reduces light penetration into the water; hence, photosynthesis is reduced, resulting in low oxygen levels, while the presence of organic matter implies that oxygen is depleted in its breakdown. Pans with a high abundance of phytoplankton or vegetation cover had higher oxygen saturation levels than turbid pans.

The variability of oxygen levels between pans and throughout the year reflects the diversity of pans.

The concentrations of soluble reactive phosphorus and inorganic nitrogen (ammonium, nitrate, and nitrite) exceed 5 µg/L and 300-500 µg/L, respectively. Our results suggest an eutrophication of waterholes in HNP and its periphery, which are characterised by high densities of phytoplankton, including blooms of cyanobacteria in some pans (Msiteli-Shumba et al., submitted). Despite the presence of the nutrient-poor Kalahari sands that are distinctive of HNP and its surrounding areas (Chamaille-Jammes et al., 2007[START_REF] Mukwashi | Impact of African elephants on Baikiaea plurijuga woodland around natural and artificial watering points in northern Hwange National Park, Zimbabwe[END_REF], the observed results show very high nutrient values in water during the dry season. As suggested above, the high nutrient values may be due to the increase in the number of ungulates visiting the remaining water sources during the dry months (Chamaille-Jammes et al., 2007, Valeix et al., 2007). Our findings have also been observed in protected areas (Chaichana et al., 2010, Gereta and[START_REF]Wildlife-water quality interactions in the Serengeti National Park, Tanzania[END_REF] where excessive levels of nutrients in water indicate the contamination of the ecosystem.

MFA of the parameters during the wet season opposes the pans with clear, soft water and a high coverage of vegetation to the turbid pans with hard and electrolyte-rich water and a high abundance of phytoplankton. This distribution of pans along the axis defined by the abovementioned parameters tends to follow the separation of natural and artificial waterholes, suggesting an important effect of pumping on the diversity of pans'

functioning. This effect depends on the chemistry of the pumped water among multiple factors. MFA for the seven sampling months confirms the importance of Con, hardness, Tur, and vegetation cover in explaining the diversity of waterholes. Tur with high concentrations of free nutrients confirms the important role played by wild and domestic fauna in terms of nutrient inputs and bioturbation, which increase during the dry season.

Within the pans showing high Tur, we could distinguish a gradient from pans with high SM to those with high phytoplankton abundance. Overall, the analyses suggest alternating states between clear and turbid water in these semi-arid ecosystems [START_REF] Meerhoff | Environmental Warming in Shallow Lakes: A Review of Potential Changes in Community Structure as Evidenced from Space-for-Time Substitution Approaches[END_REF].

The effect of location or land use was not significant in our research, as it was in small dams in the Mzingwane catchment and Matopos National Park in Zimbabwe [START_REF] Basima Lb | Impacts of land and water use on plankton diversity and water quality in small man-made reservoirs in the Limpopo basin, Zimbabwe: A preliminary investigation[END_REF]. As land use is not intensive in the HNP periphery, this result is not surprising.

In this study, we addressed the importance of WLF among different drivers in aquatic ecosystems in a semi-arid region. The natural cycle of WLF in about half the studied pans is modified by the pumping of ground water.

The analyses do not show a strong distinction between natural, temporary pans on the one hand and artificial, permanent ones on the other. Instead, a distinction between pans with clear water and vegetation and those with turbid water and high phytoplankton abundance is revealed in accordance with the classical model of alternatives states in shallow lakes [START_REF] Scheffer M | Ecology of shallow lakes[END_REF]. One might expect that heavy trampling around the pans by megafauna, in particular elephants, will preclude macrophyte growth. However, permanent pans such as Ganda, Jwapi, and Jambile Artificial still have macrophytes during the dry season despite being frequented. Further studies should identify macrophyte species and their resistance to trampling. The history of pans especially in terms of pump installation and soil properties like compaction could also explain the development or local extinction of macrophytes. As expected, our results show that water becomes harder and more conductive in winter with the decreased water volume. The change in WLF might worsen this process with the filling of ionrich ground water and eutrophication due to the concentration of wild and domestic fauna. Hard and conductive water is also nutrient-rich according to our analyses. Therefore, measures of terrestrial wildlife conservation with the pumping of water in selected pans are also strong drivers of aquatic ecosystems in and around HNP.

However, the conversion of temporary pans into permanent ones is a risky situation for species adapted to these threatened ecosystems [START_REF] Blaustein | Why study ecology in temporary pools?[END_REF].

MFA on the 2013 dataset

Figure 1 .

 1 Figure 1. Map of Hwange National Park and its surroundings (insert: location of the park in Zimbabwe) and the pumped and unpumped, natural pans monitored in this study.

Figure

  Figure 2a-f. Boxplots of pH, conductivity, water hardness, salinity, calcium, and magnesium measured in the pans at different months. The horizontal line, the top, and the bottom of the box represent the median, the 25 th , and the 75 th percentiles, respectively. The vertical dashed lines (whiskers) represent the minimum or maximum value or, in presence of outliers (open circles), the interquartile range of the data multiplied by 1.5. The width of the box is proportional to the number of pans.

Figure

  Figure 3a-d. a. Turbidity (NTU: nephelometric turbidity unit); b. Suspended matter (SM); c. Dissolved oxygen (DO, in % saturation); d. Total organic carbon concentration (TOC) recorded between February and December 2013.Figure legend as in Figure 2.

Figure

  Figure 4a-f. Nutrient concentration recorded between February and December 2013. a. Total nitrogen; b. Ammonium; c. Nitrate; d. Nitrite; e. Total phosphorus (TP); f. Reactive phosphorus (RP). Figure legend as in Figure 2.

Figure 5a- c .

 c Figure 5a-c. Chlorophyll a and Phycocyanin (Chl a and PC, respectively) and aquatic macrophyte abundance (% of vegetation cover) recorded between February and December 2013. Figure legend as in Figure 2.

Figure 6 .

 6 Figure 6. Correlation circles of the MFA of the wet season. Group variables are in blue for February (F), red for March (M), and black for April (A). Only significant variables are represented. The abbreviations (see the text) are followed by the first letter of the corresponding month.

Figure 7 .

 7 Figure 7. Groups and the qualitative variable representation on dimensions 1 and 2 of a) the wet season MFA and b) the 2013 MFA.

Table 2 .

 2 MFA of the wet season. The table gives (1) the correlations between the months and qualitative (Quali) variables, and between the months and qualitative variables and the MFA for the wet season; (2) the contribution of the months to the MFA for dimensions 1 (Dim1) and 2 (Dim2).

				Correlations			Contributions
		February	March	April	Quali	MFA	Dim1	Dim2
	February	1.00	0.61	0.44	0.15	0.78	31.73	18.00
	March	0.61	1.00	0.55	0.18	0.86	36.68	40.24
	April	0.44	0.55	1.00	0.16	0.84	31.59	41.76
	Quali	0.15	0.18	0.16	1.00	0.20	-	-

Figure 8

Individual factor map of the MFA of the wet season with natural (N) and artificial (A) pans in blue and black, respectively. "N" and "A" are the barycenters of the natural and artificial pans.

Table 3 .

 3 MFA of the 2013 dataset. The table gives (1) the correlations between the months, and between the months and the MFA; (2) the contribution of the months to the MFA dimensions 1 (Dim1) and 2 (Dim2).

					Correlations				Contributions
		Feb	March	April	July	Sept	Nov	Dec	MFA	Dim 1	Dim 2
	Feb	1.00	0.60	0.74	0.66	0.65	0.71	0.73	0.84	12.94	14.27
	Mar	0.60	1.00	0.76	0.63	0.63	0.72	0.60	0.83	14.05	11.99
	April	0.74	0.76	1.00	0.65	0.66	0.71	0.61	0.87	13.26	19.48
	July	0.66	0.63	0.65	1.00	0.78	0.82	0.70	0.87	16.79	10.29
	Sept	0.65	0.63	0.66	0.78	1.00	0.76	0.59	0.84	13.07	12.09
	Nov	0.71	0.72	0.71	0.82	0.76	1.00	0.70	0.90	16.69	16.07
	Dec	0.73	0.60	0.61	0.70	0.59	0.70	1.00	0.83	13.22	15.81

Acknowledgements

This work was conducted within the framework of the Research Platform "Production and Conservation in Partnership" (RP-PCP). We thank the ANR for their financial support through the SAVARID 2011 CEPL-003 project. The Zimbabwe National Parks and Wildlife Management Authority, Forestry Commission, and Hwange traditional leadership granted access to the study sites. The Department of Biological Sciences, University of Zimbabwe, availed the working facilities. Special thanks go to the rangers, students, and volunteers who participated in the fieldwork.

 Figure 9. Correlation circles of the MFA of the 2013 dataset. Group variables are in orange for February (F), blue for March (M), red for April (A), green for July (J), grey for September (S), purple for November (N) and black for December (D). Not all of the variables are represented. See text for variables' abbreviations.

Vegetation, which was not present in all pans as it dies out with dryness, was negatively correlated with the first dimension in February, March, September, and November as was DO concentration in September (Fig. 9, Appendix 1). The results suggest again dissimilarities between pans with hard water rich in ions, TN, and TOC and those with vegetation. The contributions of the groups, i.e. months, were homogeneous (Fig. 7b, Table 3).

The second dimension (19.21% of the total variance) was mainly characterised by Tur and SM concentration for positive values, and hardness and PC concentrations for negative values.


Adrien Gauché
email: adrien.gauche@ls2n.fr

David Morin
email: david.morin@powidian.com

Raphaël Chenouard
email: raphael.chenouard@ls2n.fr

Malek Ghanes
email: malek.ghanes@ls2n.fr

Adrien Gauch

Algorithms comparison for hydrogen storage predictive control on an islanded microgrid

Keywords: Smart grids, Optimal operation and control of power systems, Predictive control, Energy management

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INTRODUCTION

Context & motivation

By relying on renewable generations and storage instead of diesel generators, it is possible to supply loads while reducing logistics, operating costs and greenhouse gas emissions on islanded grids. A microgrid aggregates generations, storage and consumption in the same architecture. It allows gathering information and control to consider a unified management strategy [START_REF] Farhangi | The path of the smart grid[END_REF]; [START_REF] Olivares | Trends in Microgrid Control[END_REF], this is smartgrid.

The main difficulty of islanding is autonomy and reliability. Renewable generation periods are decoupled from consumption periods with intermittency. Thanks to its high energy density, hydrogen is a promising energy vector for storage, but H2 chain control remains under study [START_REF] Arsad | Hydrogen energy storage integrated hybrid renewable energy systems: A review analysis for future research directions[END_REF]. This is why an Energy Management System (EMS) provides a strategy to store energy [START_REF] Bukar | A review on standalone photovoltaic-wind energy system with fuel cell: System optimization and energy management strategy[END_REF] while taking into account technical and economical objectives. Moreover, EMS framework, optimization techniques and computational approaches have impacts on efficient and reliable operations [START_REF] Salehi | A Comprehensive Review of Control Strategies and Optimization Methods for Individual and Community Microgrids[END_REF]. In addition, a low computational burden to determine controls is required to provide commands in a reasonable time at lower power consumption.

Literature review

Raya-Armenta et al. ( 2021) produced a review on isolated microgrids and their control by EMS systems, outlining the landscape of the techniques, future trends and possible improvements. The definition of microgrid is discussed and authors explore 6 control aspects : constraints, timeframe, objective functions, frameworks, uncertainty and formulation. The key findings reside in the highlighting of new metaheuristics (e.g. Jaya) and outline of the future challenges to improve EMS: microgrids modelling, forecasting and learning, lower level controllers, decentralized EMS, communication and proof of concept.

Indeed, there are two main possibilities of optimization formulation: a full analytical mathematical definition (white box) or a partial definition based on Black Box Optimization (BBO) [START_REF] Boukouvala | Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization[END_REF]. With BBO, solvers improve local solutions without any insight about information in the black box (like derivatives), thus no global optimum guarantee can be provided [START_REF] Boukouvala | Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization[END_REF]. With white boxes, the analytical definition of the problem allows to use its mathematical properties to converge and prove the global optimality of the computed result [START_REF] Conn | Introduction to Derivative-Free Optimization[END_REF]. Even non-linear behaviors (i.e. generation, unit commitment) can be expressed [START_REF] Knueven | On Mixed-Integer Programming Formulations for the Unit Commitment Problem[END_REF]. When dealing with the optimal control of complex systems, using a simulation model is often the preferred way to evaluate their behavior. Moreover, the BBO framework offers a flexible way to solve an optimization problem without eliciting the whole mathematical model, then it easily adapts to various microgrid structures.

Most recent works implement an EMS based on MPC, the most widespread management method [START_REF] Arsad | Hydrogen energy storage integrated hybrid renewable energy systems: A review analysis for future research directions[END_REF]. The EMS frameworks are often single-level, but they can be bi-level as in [START_REF] Ma | A bi-level optimal configuration model of isolated microgrid considering energy storage lifetime[END_REF], where the upper-level minimizes the overall cost, then the lower-level optimizes the storage operation strategy. We note that optimization problems are often economic or technical (operation cost, dumped energies, etc.). Usually uncertain values (solar irradiance, consumption) are forecasted and used as deterministic in MPC Morin et al. (2018, 2019), but some studies take into account more finely their randomness, e.g. IGTD [START_REF] Tostado-Véliz | Uncertainty-aware dayahead scheduling of microgrids considering response fatigue: An IGDT approach[END_REF]. [START_REF] Garrido | Energy Management System for Microgrids based on Deep Reinforcement Learning[END_REF] used a simulation model and then Reinforcement Learning to learn the best commands. The MPC remains better than the RL control when the horizon is correctly predicted, i.e. the prediction error is moderate. Here components considered for generation are solar panel (PV), Wind Turbine (WT) and Thermal Generators (TG), such as diesel engine and microturbine. Considered storage are Battery (BT) and H2 (hydrogen chain, at lease Hydrogen, ELectrolyzer EL or/and Fuel Cell FC). Some microgrids can be islanded [START_REF] Ma | A bi-level optimal configuration model of isolated microgrid considering energy storage lifetime[END_REF] or connected to another grid.

As inferred by numerous reviews [START_REF] Olivares | Trends in Microgrid Control[END_REF], MPC remains the most discussed and the simplest predictive EMS to implement. Among BBO techniques, Raya-Armenta et al. ( 2021) concludes that metaheuristics are very promising to find the EMS efficient solutions. However, a large part of these studies compare metaheuristics between them, as well as their underlying improvements, but do not consider the use of more classical BBO methods (e.g. Pattern Search) [START_REF] Das | Multi-objective optimization of hybrid renewable energy system by using novel autonomic soft computing techniques[END_REF]. In addition, optimizers may not find the optimum, but a reasonable optimal solution. Then, the stochastic aspect of metaheuristics leads to different results (even slightly) at each execution and a larger number of cost function evaluations, thus to a higher consumption of computational resources. [START_REF] Morin | Evaluation of performance improvement by model predictive control in a renewable energy system with hydrogen storage[END_REF][START_REF] Morin | Energy Management of isolated DC microgrids with hybrid batteries-hydrogen storage system using Model Predictive Control and Wavelet Neural Networks based forecasts[END_REF] have proposed a binary formulation of the decision tree to get EL and FC Unit Commitment (UC) in an islanded microgrid (Figure 1). The horizon's best path is determined first by a greedy algorithm, then by a Branch & Bound (BB). Compared to myopic control, MPC improves the grid exploitation by increasing energy storage, decreasing power losses and storage degradation (BT, EL, FC). These benefits mainly come from the objective function that aggregate such criteria in a single function. However, evaluated commands are binary (On/Off), leading to a large part of the possible solutions' space not being explored, and thus having potential for improvements. In addition, BB hardly scales on embedded systems when increasing the number of decisions variables (related to equipment quantity, time step and horizon) due to its exponential algorithm complexity.

Contributions

In this work, we implemented the same microgrid as in [START_REF] Morin | Energy Management of isolated DC microgrids with hybrid batteries-hydrogen storage system using Model Predictive Control and Wavelet Neural Networks based forecasts[END_REF], but investigating an extended formulation with different solvers. Previous publications tend to consider separately metaheuristics or deterministic methods for MPC solving.

This paper focuses on comparing some state of the art BBOs to control a microgrid using a whole year simulation based on a real dataset. We selected BBOs from different categories: stochastic (population-based and single point) and deterministic. Several criteria are used to compare their performances and their impact on the microgrid behavior.

The problem and its black-box formulation are presented in section 2. The section 3 details the experiments with a discussion about the results, before the conclusion in section 4.

PROBLEM FORMULATION

Microgrid description

The microgrid case study is depicted in Figure 1. It includes a PV (solar) generation P P V (kW), a load consumption P L (kW), a BT (battery) with P BT (kW) power in and out for storage, H2 chain with EL (electrolyzer) power consumption P EL (kW) to produce H2, HT (hydrogen tank) power-to-gas energy storage under p H2 (bar) pressure and FC (Fuel Cell) power generation P F C (kW) by consuming H2. The power balance in the bus is (1).

P P V + P F C + P BT -P L -P EL = 0
(1) The controllable equipments are EL and FC, i.e., represented by P F C and P EL variables. The model use intermediary variables, namely the State Of Charge (SOC) of BT (between 0 and 1) and the pressure (p H2 ) in HT (bars). These variables define the state S over the time horizon and t 0 defines the initial state. Uncertain parameters are represented by their expected values with : P P V and P L .

S = SOC(t 0 ), • • • , SOC(t N ) p H2 (t 0 ), • • • , p H2 (t N ) (2)
The energy storage over the horizon (characterized by S(t): SOC(t), p H2 (t)) is calculated in a simulation model (i.e. the black box) by recurrence according to the initial state and a power balance based on incoming power (P EL , P + BT (t)), outgoing power (P F C , P - BT (t)) and energy losses in the microgrid [START_REF] Dahmane | Decentralized Control of Electric Vehicle Smart Charging for Cost Minimization Considering Temperature and Battery Health[END_REF].

Model Predictive Control (MPC)

In this paper, the MPC is implemented as in Figure 2, expected values W (irradiance, temperature and load) are forecasted by a Wavelet Neural Network [START_REF] Morin | Energy Management of isolated DC microgrids with hybrid batteries-hydrogen storage system using Model Predictive Control and Wavelet Neural Networks based forecasts[END_REF] in the Predictions block. Then, from the applied control V and input W , the Simulation model block computes the state S (stored energies) and power flows of the microgrid for the whole horizon (N ). The Optimizer block uses the Simulation model to seek an optimal vector V * that maximizes the cost function evaluation (see Equation ( 5)) under constraints (see Equation ( 4)). Decision variables for the EL and FC take their values between 500W and 2500W (equipment On) or 0 (equipment Off). These constraints force remaining in a zone where the electricity / H2 conversion coefficient is assumed constant. The control vector V is defined over the optimization horizon N , as detailed in Equation ( 3):

V = P EL (t 1 ), • • • , P EL (t N ) P F C (t 1 ), • • • , P F C (t N ) (3)
Two types of constraints are considered to restrict the admissible commands. The logical ones forbid EL and FC to be active on the same slot t i . Then, the technical constraints apply minimal and maximal thresholds for SOC and p H2 as in Equation (4), from (2) and Fig. 2:

0.2 0.8 ≤ SOC(t i ) p H2 (t i ) ≤ 1 38 , ∀i ∈ {1, . . . , N } (4)
The objective function is defined as an aggregation of several criteria, as in [START_REF] Morin | Energy Management of isolated DC microgrids with hybrid batteries-hydrogen storage system using Model Predictive Control and Wavelet Neural Networks based forecasts[END_REF][START_REF] Morin | Evaluation of performance improvement by model predictive control in a renewable energy system with hydrogen storage[END_REF]. It can be formulated as a sum of transition functions C j between states with coefficient α j for criterion j:

max V ∈A 3 j=1 N i=1 α j C j (V (t i ), S(t i ), E( W (t i ))) (5)
Criterion C 1 relates to power flows while satisfying the load, avoiding low BT and HT states, and PV energy not being stored. C 2 aims at reducing power losses in converters and storage (BT, H2 chain), thus enhancing the efficiency of the microgrid. Finally, C 3 takes into account premature aging of equipments.

Model data

The studied microgrid corresponds to an isolated microgrid in Col du Palet (refuge in mountains) with 4.8kW of peak PV generation, BT of 14.8kW h, 7.5kg of H2 capacity in HT (30 bars 2.5m 3 ), EL and FC of 2.5kW . From W , we obtain the delta between generation and consumption, the power flow in the bus without any control. Figure 3 is an example of three days in October, we observe power consumption during the night around -0.5kW and PV generation during the day reaching 4kW with some drops in production depending on the weather. The simulation uses a one-year dataset from April 1, 2018 to March 31, 2019, with mean power of 0.32kW , median of -0.39kW , maximum power of 4.4kW and minimum of -3.5kW .

Solvers

A few representative Derivative Free Optimization (DFO) solvers are selected to investigate their global behavior applied to our context. A Particle Swarm Optimization (PSO) represent the behavior of population-based algorithms and a Simulated Annealing (SA) the path algorithms for metaheuristics. A Pattern Search (PS) algorithm represents deterministic DFO algorithms. SA and PS are initialized with a zero vector as a good initial guess in most cases. These optimizers are taken from the Global Optimization Toolbox of Matlab (Release 2021b)1 . To generalize the optimization over a whole year, we do not consider a fine-tuning adapted to a specific horizon, we use the recommended default settings. As an example, all optimizers use the same FunctionTolerance parameter (10 -6 ). We compare the results with the original Branchand-Bound from [START_REF] Morin | Evaluation of performance improvement by model predictive control in a renewable energy system with hydrogen storage[END_REF]. We configure the simulation to optimize the orders on a sliding horizon of 12 hours with a time step of 0.5 hour (N=24). Experiments are run on a laptop with i7-11850H on 1 thread and 16 GB of memory.

Typical command

Figure 4 shows EL (full bar) and FC (dashed bar) commands obtained from the optimal vector for a scenario of 12 hours. The net power P net (red line) defines the power balance between the load and the PV generation. We can see that the H2 chain is switched off most of the time. The FC is only active for 2 consecutive slots in the night due to the SOC lower limit constraint. The battery continues to discharge until 09:00 AM while the PV generation is not high enough. Then, P net is positive and the SOC of BT increase close to 70% until around 01:00 PM. The EL is then activated for 1 hour to store the excess energy in HT, it is activated a second time at 3:30 PM for the same reason. This figure is a typical example of the control produced by a continuous solver for the EL and the FC.

Optimization results

Fig. 5. H2 mass in the tank

Each one year simulation takes more than 24 hours to compute. From Figure 5, in terms of mass of H2 stored, none of the BB, PS, PSO stand out. We notice that H2 is mainly used in January and February, when the generation is the lowest and the battery is more difficult to recharge. For the SA, it fails to provide a consistent command to fill the tank.

In the Table 1, we find the median, mean and minimum values of stored energy over one year of simulation with the 4 algorithms. The differences in the median, mean and minimum of SOC and H2 mass between BB, PS and PSO are about 1%. SA in its default configuration is not able The Table 2 shows the power control characteristics of the H2 chain (EL and FC) as well as the activation time and the number of starts. The BB switches the EL and FC the shortest with the fewest start-stop cycles. This is due to the binary resolution of the problem that results in consuming/delivering all energy at once, EL and FC started less often and for a shorter time for the same amount of energy. The PSO is the second algorithm and the PS the third to put less strain on the H2 chain. On Table 3 the electrical losses summed over the year with the PS and PSO algorithms are very close (around 1%).

The BB does worse with 6% more losses. With SA, the cumulative losses are the highest of the benchmark, as are the average and median losses (over 1 kW). The Table 4 summarizes the evaluations of the objective function at the output of the optimizer, when it has converged. As this is a maximization of the objective function, the higher the score, the more efficient the optimization. The PS and PSO are the best, equal from the median, but in average value the PS does slightly better (4%). The worst evaluation found by the PS is 10% below that found by the PSO. According to the median, the BB finds less efficient solution (41% worse) than PS and PSO.

In the Figure 6, we display the sum of the objective function values for all best solutions proposed by the optimizers, but evaluated with the real data and not the predictions. We do not display SA because the result is not relevant. BB, PS and PSO provide very close results. BB is slightly better because it provides all-ornothing commands, and the cost sub-function representing wear and tear provides a lower value. The PS and PSO commands can be disadvantaged because they are very optimized, therefore much more sensitive to deviations between predictions and realizations.

The Figure 7 summarize the number of evaluations of the objective function. The lower the number of evaluations is, the fewer iterations the algorithm needs to converge to the optimal value. According to the median value, the BB often converge quickly, due to cases where no decision is made (H2 chain off). Compared to the BB, the PS and the PSO need respectively twice and a thousand times more evaluations to converge. On the other hand, on average, the PSO has the fewest evaluations, followed by BB and PS. BB is the best in terms of worst number of evaluations, followed by PS and PSO (twice).

Discussions and improvements

We do not observe any significant difference between the PS, PSO and BB according to the predictive controls provided (see Figure 5 and Figure 6). For energy management in this case study, metaheuristics do not seem to be more suitable than deterministic methods. In detail, we notice that the BB is slightly better on wear, the PS on storage and the PSO on power losses. This can be due to the ability of the BB to decrease the number of activations by the binary commitment, the PSO to explore the search space to minimize losses, and the PS to search around the initial zero vector of low operating points (local search).

To prioritize the objectives differently, the weighting can be changed.

About the objective function optimization, the PS is very slightly the best (see Table 4). Moreover, the PS is by far the least greedy in terms of calculation (see Figure 7). The stopping criteria of the metaheuristics could be better set, but the gap with PS is important and may not be filled with a slight improvement on this aspect. Furthermore, a large part of the criteria were common between the PS and the meta heuristics (default setting of the MATLAB toolbox). All optimizers returned the same exit flag (1) on all optimizations, i.e. the objective function cannot be improved with a tolerance greater than 10 -6 . Obviously, a dedicating tuning for each algorithm should provide better results, as for the SA which fails to find an efficient solution at several steps of the simulation. This tuning step is a drawback of most metaheuristics and the tuning may not fit all the diverse cases encountered in the whole activity period of a microgrid.

Beyond the optimization capacities and computational burden of each algorithm, we notice in Figure 6 that a perfectly optimized control on a predicted horizon may not be the optimal solution in real life. For instance, random events can occur, like cloud crossing at 12 a.m. (see Figure 4). The impact of these random events should be integrated in the optimization to provide a more relevant control. Probabilistic models can be used instead of fixed predicted values as in [START_REF] Knueven | On Mixed-Integer Programming Formulations for the Unit Commitment Problem[END_REF]. Another approach is to use robust optimization to get solutions with guarantees on performances for uncertainty intervals for these parameters [START_REF] Stein | How to solve a semi-infinite optimization problem[END_REF]. The main issue is to be able to keep a light computing burden as such algorithms have to be embedded in the microgrid controller. Moreover, some modeling parameters like the number of considered steps in the horizon can be adapted to keep the computation as light as possible, but it will impact the ability of the optimizer to anticipate events.

CONCLUSION

To control a hydrogen chain (Electrolyser and the Fuel Cell) of a hybrid storage on an islanded microgrid, one MPC must provide a command in a reasonable time with limited computing resources. A Black Box formulation is used to have a finer simulation rather than White Box. Then, a comparison of different types of optimizers to solve the MPC has been performed with Branch & Bound (BB), Pattern Search (PS), Particle Swarm Optimization (PSO) and Simulated Annealing (SA).

The PS can be considered as the best candidate for this application. In this case study, we do not notice any major difference on the relevance of the commands between the different algorithms over a whole year of simulation. But the metaheuristics perform many more calculations to find similar results than the deterministic algorithms. We also notice that a perfectly optimized control on a predicted horizon is not necessarily relevant when considering the prediction errors with reality.

Improvements can be done in a better tuning of this algorithm, especially concerning the initialization of the start vector. Further investigations can be necessary to make comparisons between such BBO and using a simplified analytical model. The tradeoff is between computing the exact solution of a simplified model or finding a good solution to a real system [START_REF] Conn | Introduction to Derivative-Free Optimization[END_REF]; [START_REF] Knueven | On Mixed-Integer Programming Formulations for the Unit Commitment Problem[END_REF]. This analysis should also focus on the computation burden and the quality of solutions, with their impact on a whole year scenario. Finally, a probabilistic formulation of the model can be considered to take into account more precisely the stochastic aspect of generation and consumption, in order to make the control more resilient to unforeseen.

⋆

  This work is supported by the National Association of Research and Technology (Association Nationale Recherche et Technologie) of France through the program CIFRE under grant 2020/0995.

Fig. 1 .

 1 Fig. 1. Microgrid schematic Morin et al. (2019)

Fig. 2 .

 2 Fig. 2. Predictive control[START_REF] Camacho | Model Predictive Control[END_REF]

Fig. 3 .

 3 Fig. 3. Power flow (P P V -P L ) in the electrical bus 3. EXPERIMENTS

Fig. 4 .

 4 Fig. 4. Example of command optimization

Fig. 6 .

 6 Fig. 6. Cumulative sum of objective function values from each simulation step

Fig. 7 .

 7 Fig. 7. Cumulative number of objective function evaluations

Table 1 .

 1 Energy storage over 1 year

		BB	PS	PSO	SA
	Median SOC (%)	76.0	77.6	77.5	64.4
	Mean SOC (%)	74.0	74.1	73.9	60.4
	Min SOC (%)	13.8	13.4	12.4	0
	Median H2 (kg)	7.404	7.430	7.420	0.264
	Mean H2 (kg)	6.838	6.858	6.860	0.720
	Min H2 (kg)	0.201	0.201	0.201	0.170
	to find the most efficient solutions with the diversity of
	scenarii in a whole year.
	Table 2. EL and FC activation
		BB	PS	PSO	SA
	EL median (kW)	2.5	1.875	2.496	1.325
	EL mean (kW)	2.5	1.777	2.116	1.397
	EL on (h)	405	519	430	2102
	EL starts	237	309	289	1457
	FC median (kW)	2.5	0.624	0.858	0.787
	FC mean (kW)	2.5	1.047	1.047	0.898
	FC on (h)	58	98	90	494
	FC starts	93	111	124	834

Table 3 .

 3 Electrical losses

		BB	PS	PSO	SA
	Median (kW)	0.381	0.362	0.360	1.233
	Mean (kW)	0.407	0.388	0.385	1.220
	Sum (kWh)	3569	3406	3374	10693

Table 4 .

 4 Optimizers output cost function evaluation

		BB	PS	PSO	SA
	Median evaluation	-8.01	-5.65	-5.65	-46.6
	Mean evaluation	-16.2	-8.89	-9.20	-48.5
	Worst evaluation	-105.2	-90.7	-100.9	-191.7

Preprints of the 22nd IFAC World Congress Yokohama, Japan, July 9-14, 2023 ©2023 the authors. Accepted by IFAC for publication under a Creative Commons Licence CC-BY-NC-ND

https://fr.mathworks.com/help/gads/, consulted on September