M Rouxel

L Ruiz

Y Hamon

G Chirié

D Michot

Agrocampus Ouest

J Molénat

Original Research

Experimental Determina on of Hydrodynamic Proper es of Weathered Granite

Weathering processes transform hard fresh rock into regolith materials, composed of weathered bedrock and soils. The thickness of the weathered bedrock layer varies greatly, a aining several meters under certain clima c and lithologic condi ons. The hydrology of catchments on bedrock is strongly controlled by the hydrologic func oning of these weathered bedrock layers, and the understanding and modeling of this func oning requires a knowledge of their hydraulic proper es, such as their water reten on curves. There are few references available on the water reten on curve of weathered bedrock. To fi ll this gap, we measured the water reten on characteris c of highly weathered Plomelin leucogranodiorite in replicate core samples. The study of weathered granite samples showed that their mean water reten on curves diff ered signifi cantly from those es mated for soil samples of the same grain size distribu on using a pedotransfer func on and for other granites with the same weathering index. Even if one can generate hypotheses to explain the observed diff erences, the three approaches we used to compare our results cannot replace the experimental approach to generate the water reten on curve of a material such as weathered granite. In addi on to enriching the body of work on water reten on curves of weathered bedrocks, the results of this work suggest that there is a need for developing a database of reten on proper es of weathered bedrocks in parallel with the development of a model based on factors more appropriate than the weathering index. Weathering processes transform hard fresh rock into regolith materials, composed of weathered bedrock (saprolite, saprock) and soils [START_REF] Graham | Rock to regolith conversion: Producing hospitable substrates for terrestrial ecosystems[END_REF]. h e thickness of the weathered bedrock layer varies greatly, attaining several meters under certain climatic and lithologic conditions. Recent studies on catchment hydrology have shown that weathered bedrocks play an important role in hydrologic and hydrochemical processes [START_REF] Haria | Evidence for deep sub-surface fl ow rou ng in forested upland Wales: Implica ons for contaminant transport and stream fl ow genera on[END_REF][START_REF] Legout | Étude des mécanismes hydrologiques et biogéochimiques de la recharge des nappes à surface libre[END_REF]Martin et al., 2004;[START_REF] Molénat | Hydrologie et transfert de substances dissoutes dans les aquifères superfi ciels[END_REF][START_REF] Rouxel | Season-al and spa al varia on in groundwater quality along the hillslope of an agricultural research catchment (western France)[END_REF] and thus in the transport of dissolved substances. In fact, weathered bedrock can store large quantities of water, inl uencing l ows during low-water periods and constituting, along with the soil, a water reserve for tree growth [START_REF] Graham | Rock to regolith conversion: Producing hospitable substrates for terrestrial ecosystems[END_REF]Hubbert et al., 2001a;[START_REF] Kew | Classifi ca on, strength and water reten on characteris cs of later-i c regolith[END_REF][START_REF] Ruiz | Water balance modelling in a tropical watershed under deciduous forest (Mule Hole, India): Regolith matric storage buff ers the groundwater recharge process[END_REF][START_REF] Schwinning | The ecohydrology of roots in rocks[END_REF]. h e groundwater that develops in weathered bedrock aquifers responds strongly to rainfall and has annual water-level amplitudes of several meters [START_REF] Legout | Solute transfer in the unsaturated zone-groundwater con nuum of a headwater catch-ment[END_REF]Martin et al., 2006;[START_REF] Molénat | How to model shallow water-table depth varia ons: The case of the Kervidy-Naizin catchment, France[END_REF][START_REF] Rouxel | Season-al and spa al varia on in groundwater quality along the hillslope of an agricultural research catchment (western France)[END_REF]. Because of this, the water contents of weathered bedrock aquifers are highly variable, from saturation to contents below water-holding capacity. Consequently, understanding and modeling the hydrologic functioning of weathered bedrock requires knowing their hydrodynamic properties both when completely saturated (hydraulic conductivity, total porosity) and when partially saturated (retention curve, curve of the decrease in hydraulic conductivity). Few studies (and therefore few references) exist regarding the unsaturated hydrodynamic properties of weathered bedrock (Hubbert et al., 2001b;[START_REF] Johnson-Maynard | Physical and hydraulic proper es of weathered grani c rock in southern California[END_REF][START_REF] Jones | Water-holding characteris cs of weathered grani c rock in chaparral and forest ecosystems[END_REF][START_REF] Katsura | Saturated and unsaturated hy-draulic conduc vi es and water reten on characteriscs of weathered grani c bedrock[END_REF][START_REF] Katsura | Hydraulic proper es of variously weathered grani c bedrock in headwater catchments[END_REF][START_REF] Kew | Classifi ca on, strength and water reten on characteris cs of later-i c regolith[END_REF].

To measure the hydrodynamic properties of weathered bedrock, two solutions are theoretically possible. h e i rst is based on methods specii c to samples of solid rock, i.e., the transient-pulse method [START_REF] Brace | Permeability of granite under high pressure[END_REF]), gas permeability (reviewed by [START_REF] Neuzil | Groundwater fl ow in low-permeability environments[END_REF], or Hg intrusion porosimetry [START_REF] Lin | Aquivalent channel models for perme-ability es ma on and their applica on to sedimentary rocks[END_REF], cited by Katsura et al., 2006) to measure their hydrodynamic properties. But these methods are not suitable for samples of highly weathered bedrock. h e second solution is to use soil science methods. From a textural and structural point of view, a highly weathered bedrock behaves much like a soil. Consequently, soil science methods This paper presents water-reten on curves of weathered granite. They differ from those estimated with pedotransfer functions and from those reported for other granites with the same weathering index. There is a need to develop a database of retention properties of weathered bedrocks and a classifica on more appropriate than the weathering index.

(gravimetric water content measurement associated with pressure plate equipment, pressure chamber equipment, or microtensiometers under drying conditions) appear better able to take the hydrodynamic properties of these materials into account. Nonetheless, the ability of soil science methods to measure the hydrodynamic properties of weathered bedrock has, to our knowledge, been little studied.

To avoid applying direct measures on samples, pedotransfer functions constitute an approach also used in soil science to derive the properties of a soil, such as its hydrodynamic properties under partially saturated conditions, from its basic properties. h ese pedotransfer functions have been constructed from databases containing i eld and laboratory measurements of dif erent types of soil. Gravelly soils, or in general those with coarse fragments, are rarely studied because measuring their water-retention properties is methodologically dii cult. Nonetheless, currently developed pedotransfer functions could be suitable for estimating the hydrodynamic properties of weathered bedrock.

Weathering processes transform hard fresh rock (e.g., granite) into weathered bedrock with greatly increased porosity (Fig. 1). [START_REF] Graham | Rock to regolith conversion: Producing hospitable substrates for terrestrial ecosystems[END_REF] described the mechanisms and implications of transforming nonporous hard rock into porous regolith.

A variety of methods have been proposed to quantify the degree of weathering of igneous rocks using quantitative criteria [START_REF] Gokceoglu | A compara ve study on indi-rect determina on of degree of weathering of granites from some physical and strength parameters by two so compu ng techniques[END_REF]. To our knowledge, however, these approaches have never been used to relate the degree of weathering to hydrodynamic properties. To date, studies of water retention curves of weathered granite have used a qualitative approach proposed in the classii cation system of [START_REF] Clayton | Prac cal grain size, fracturing density and weathering clas-sifi ca on of intrusive rocks of Idaho batholith[END_REF].

h e objective of this study was to determine experimentally the mean retention properties (retention curve) of a weathered granite. With a view toward deriving the retention properties of weathered granite from properties more easily measured, we studied the suitability of pedotransfer functions classically used for soils and the pertinence of a weathering index as auxiliary measures to estimate the retention properties of weathered granite. Ultimately, it is a matter of better understanding the properties of weathered bedrocks and the means to estimate them.

Materials and Methods

Site Descrip on and Weathered Granite Characteris cs

h e site is located in southwestern Brittany (2347.35 N, 117.52 E). It has belonged to the French network of environmental observatories (Observatoire de Recherche en Environnement) since 2002 (http:// www.inra.fr/ore_agrhys_eng/). The site has an elevation of 33 m above sea level. h e climate is oceanic. h e mean annual temperature is 11.4°C, with a minimum of 6.1°C in January and a maximum of 17.6°C in July. h e mean annual rainfall and potential evapotranspiration for the last decade are 1167 ± 195 and 616 ± 71 mm, respectively. h e soils at the site are sandy loam (distric cambisol, FAO classii cation). h e upper horizon (0-20 cm) is rich in organic matter (4.5-6%). h e soil depth was surveyed throughout the site, and the average value is 0.8 m. Soils are well drained except in the relatively narrow bottomlands, where hydromorphic soils are found. h e bedrock is a i ssured and fractured granite (leucogranodiorite of Plomelin, Paleozoic; [START_REF] Béchennec | No ce explica ve, Carte géologique de France (1/50000[END_REF], overlain by weathered granite. Weathering of minerals in the granite has resulted in the formation of secondary minerals consisting mostly of clay. Chemical alteration of minerals involving disaggregation of quartz grains and the redistribution of clay and silt has produced a characteristic sandy to clayed layer.

From a pedologic pit in the same site, [START_REF] Legout | Étude des mécanismes hydrologiques et biogéochimiques de la recharge des nappes à surface libre[END_REF] described five weathered granite types with different degrees of weathering and found that dif erent types can be intermingled across short distances at the same depth. At depth, based on an electrical resistivity tomography survey, [START_REF] Cosenza | Tomographie de résis vité electrique 3D sur le bassin versant de Kerrien (Finistère, France): Comparaison avec d'autres méthodes géophysiques[END_REF] proposed a three-dimensional model including several structures, in particular a network of sealed fractures and unweathered granite boulders.

h e average depth of the weathered granite layer was estimated to be 20 m [START_REF] Legchenko | Magne c resonance sounding applied to aquifer characteriza on[END_REF], and the unconfined aquifer corresponds to this layer. The water table reacts quickly to recharge events, within a few hours a er rain events of more than several millimeters [START_REF] Legout | Solute transfer in the unsaturated zone-groundwater con nuum of a headwater catch-ment[END_REF]Martin et al., 2004). Upslope, the permanently unsaturated zone is 2 to 3 m deep, and [START_REF] Legout | Invesga on of bio-geochemical ac vi es in the soil and unsaturated zone of weathered granite[END_REF][START_REF] Legout | Solute transfer in the unsaturated zone-groundwater con nuum of a headwater catch-ment[END_REF] showed that this layer determines time transfer and is a medium for biogeochemical reactions.

Observations have shown that the weathered granite is permeated by a continuous network of mesofractures [START_REF] Legout | Étude des mécanismes hydrologiques et biogéochimiques de la recharge des nappes à surface libre[END_REF][START_REF] Legout | Invesga on of bio-geochemical ac vi es in the soil and unsaturated zone of weathered granite[END_REF]. It maintains the original rock texture but is friable and can be crumbled by hand into its individual grains. Feldspars are powdery and weathered to clay minerals, while biotite appears silver and has weakly visible joints. h e scheme developed by [START_REF] Clayton | Prac cal grain size, fracturing density and weathering clas-sifi ca on of intrusive rocks of Idaho batholith[END_REF] was used to classify the weathered granite samples by weathering index. h is qualitative description corresponds to Index 6 in the Clayton and Arnold classii cation or very well weathered granite. h e mesofracture network in weathered granite opens up the rock mass to extensive percolation of water.

Weathered Granite Sampling

Ten samples were extracted using stainless steel cylinders 15 cm in diameter and 7 cm tall (Fig. 2). Samples were taken from depths of 130 to 170 cm in a pit 2 m deep on a plateau area (Fig. 3). h e cylinders were inserted vertically in a l at zone thanks to a cylinder holder with a hammering head and using a no-rebound hammer provided by the Wind equipment supplier (SDEC France). Samples were taken from a zone in which the granite was highly weathered and sandy. Several attempts were necessary to get each sample because all attempts where insertion was dii cult, due the presence of gravels or stones or when sample compaction was suspected, were systematically discarded.

As shown in Fig. 3, the trench cross-section showed a zone where the weathered granite appeared visually heterogeneous, with decimetric unweathered granite blocks (Geh). All 10 core samples were extracted from the zone (Ge) where the texture and color of the weathered granite appeared homogeneous. h e mean grain size of the dif erent weathered granite types was determined by [START_REF] Legout | Étude des mécanismes hydrologiques et biogéochimiques de la recharge des nappes à surface libre[END_REF] from four replicates. h e type corresponding to our samples had bulk densities ranging from 1.4 to 1.7 g cm -3 , with 5 to 9% clay, 11 to 13% silt, and 70 to 82% sand.

Laboratory Experiments

h e method of [START_REF] Wind | Capillary conduc vity data es mated by a simple method[END_REF] was used to estimate the retention curves of undisturbed samples of the weathered granite. h is method of experimentation is based on monitoring free evaporation from the surface of a presaturated undisturbed sample. h e total mass of the sample and the matric potential (h) at several depths are continually measured and recorded every 20 min by microtensiometers inserted at i ve depths (Fig. 4). h e method is based on the principle of a dynamic measurement (Fig. 5). h e principle is to use, at each measurement step, a function that approximates the volumetric moisture proi le from the mean water content derived from the sample mass. h e retention curve is thus constituted by the set of data pairs [h(z,t), θ(z,t)], where h(z,t) is the matric potential and θ(z,t) is the moisture estimated at depth z and time t from the approximation function of the moisture proi le. h is method was chosen because it does not require waiting for an equilibrium between the pressure applied and the water volume of the sample. In fact, [START_REF] Jones | Water-holding characteris cs of weathered grani c rock in chaparral and forest ecosystems[END_REF] reported that one must wait several weeks to reach a state of equilibrium as a function of the degree of weathering and the potential applied to the sample.

h e samples were progressively saturated with water from the bottom up in a bath of an aqueous solution of KCl to minimize air entrapment. Once placed on the experimental bench, the samples were equipped with i ve microtensiometers connected to pressure sensors to record pressure variability at dif erent depths (z) during the evaporation of water contained in the sample (Fig. 4). h e microtensiometers were numbered 1 to 5 from the surface to the bottom of the sample, respectively. A balance (XS6001S Mettler Toledo) was used to follow the decrease in sample mass during the experiment. A datalogger connected to a computer allowed calibration of the pressure sensors before mounting the sample on the experiment bench and then recording the sample mass and the tension at the terminals of the pressure sensors during the experiment. h e calibration consisted in building a linear relationship between water pressure and electric tension measured at the sensor terminal. Pressure was applied in a continuous pneumatic network (the real pressure was measured with precision sensors calibrated by an accredited laboratory) connected to all pressure sensors. h e point control series were obtained every 5 kPa from 0 to 80 kPa.

Moisture Profi le Approxima on

A moisture proi le of the sample was estimated at each measurement step by i tting a van Genuchten function:

() ( ) () 1 1 11 / bot surf bot 1 1 1 n n z az - ⎡⎤ ⎢⎥ θ = θ + θ-θ ⎢⎥ ⎢⎥ + ⎣⎦ [1]
where θ surf is the water content at the level of the microtensiometer located at the surface and θ bot is the water content of the deepest microtensiometer, z is the depth from the surface of the sample, and n 1 and a 1 are i t parameters.

A numeric inversion method was used to determine the parameters of the retention curve from the measurements performed during the experiment. We applied this method with ESPAS 1.0 so ware [START_REF] Mohrath | Méthode de Wind: Guide d'u lisa on du logi-ciel, version1[END_REF]. h e quality of the i t depended on the ability of the model and the i tted parameters to reproduce the change in the mean water content of the sample. h e mean water content was obtained from the mass of the sample recorded by the balance and the bulk density of the sample.

We used two criteria to evaluate the quality of the i t. h e i rst criterion verii ed that the relation between the mean water content measured with the equipment and the water content estimated by the equation was linear and unambiguous. h e second criterion verii ed that the dif erences between the observed and predicted water content values were <0.0025 cm 3 cm -3 . To satisfy these two criteria, some values were excluded from the analysis, especially when the hydraulic gradient was not signii cant given the accuracy of the sensors (±3 cm). h is was frequently the case at the early stages of the experiment, as illustrated in Fig. 5c and5d.

Numerical Es ma on of the Water Reten on Curve

h e set of data pairs [h(z,t), θ(z,t)] dei ned the retention curve of the sample. We i t the curves with the van Genuchten-Mualem equation:

() ( ) ( ) rs r 1 m n hh - ⎡⎤ θ = θ + θ-θ + α ⎢⎥ ⎣⎦ [2]
where θ is soil water content, θ s is the saturated water content, θ r is the residual water content, α (>0 cm -1 ) is an empirical parameter whose inverse is o en referred to as the air-entry value or bubbling pressure, n is a pore-size distribution parameter af ecting the slope of the retention function, and m is a parameter related to the asymmetry of the retention curve that equals 1 -1/n (Mualem's approximation).

h e moisture value at saturation was i xed in Eq. [2] from the saturated sample mass before beginning the experiment; the other parameters were i tted by the [START_REF] Marquardt | An algorithm for least-squares es ma on of nonlinear parameters[END_REF] algorithm with the ESPAS 1.0 so ware [START_REF] Mohrath | Méthode de Wind: Guide d'u lisa on du logi-ciel, version1[END_REF]. To calculate a mean retention curve representative of the sampled weathered granite, we used the computer code RETC [START_REF] Van Genuchten | The RETC code for quanfying the hy-draulic func ons of unsaturated soils, version 1.0[END_REF]. All parameters were optimized: moisture at saturation (θ s ), residual moisture (θ r ), and the terms α and n.

To compare the retention curves of the weathered granite samples with one another and with the various approaches used to estimate them, we calculated the quantity of free water (θ e ). Free water is that which lies outside the i eld of attraction of solid particles and which is susceptible to move with gravity or pressure gradients [START_REF] De Marsily | Hydrogéologie quan ta ve[END_REF]. h us, θ e was calculated as the dif erence between θ s and θ r . For the same reasons, we determined the value of the matric potential (ψ m ) when the ef ective saturation (S e ) was equal to 0.5:

r e sr S θ-θ = θ-θ [3]

Soil Par cle Size Approaches

We used the UNSODA database [START_REF] Nemes | Descrip on of the unsaturated soil hydraulic database UNSODA version 2.0[END_REF] to select water content measurements coupled with matric potential measurements for soil horizons with a particle size distribution equivalent to the weathered granite material that we studied. h e experimental results extracted from the UNSODA database concerned only undisturbed samples with bulk densities ranging from 1.41 to 1.56 g cm -3 , which correspond to the range of bulk densities of weathered granite types estimated by [START_REF] Legout | Solute transfer in the unsaturated zone-groundwater con nuum of a headwater catch-ment[END_REF]. Also, only data from laboratory soildrying methods were compared with the mean retention curves of the weathered granite samples to eliminate dif erences due to hysteresis ef ects on the water retention curve. Finally, the results of i ve samples from the studies of [START_REF] Bruce | Physical char-acteris cs of soils in the southern region: Cecil series[END_REF], [START_REF] Dane | Physical characteris cs of soils of the south-ern region: Troup and Lakeland series[END_REF], and [START_REF] Cassel | In situ soil water holding capaci es of selected North Dakota soils[END_REF] were matched with the criteria.

Hierarchical Pedotransfer Func on

We used the ROSETTA computer program [START_REF] Schaap | ROSETTA: A computer program for es ma ng soil hydraulic parameters with hierarchical pedotransfer func ons[END_REF], which implements i ve hierarchical pedotransfer functions and allows the estimation of the van Genuchten (1980) water-retention parameters using texture and bulk density input data [START_REF] Schaap | ROSETTA: A computer program for es ma ng soil hydraulic parameters with hierarchical pedotransfer func ons[END_REF].

To determine pedotransfer functions, Schaap and Leij (1998) used three databases, those of [START_REF] Ahuja | Es ma ng soil water characteris cs from simpler proper es or limited data[END_REF], de Rawls et al. (1982), and UNSODA. We used this program to generate the parameters of the retention curves of two materials having textures and bulk densities surrounding the values characterizing the weathered granite type we studied. From these two retention curves, we used the calculation code RETC again to calculate the mean retention curve and the retention curves delimiting the coni dence interval of this mean curve.

Weathering Index

We collected from the literature the results of water-retention measurements for granite samples belonging to Index 6 in the Clayton and Arnold (1972) classii cation system. h e results came from the studies of Hubbert et al. (2001b) and [START_REF] Jones | Water-holding characteris cs of weathered grani c rock in chaparral and forest ecosystems[END_REF] on samples of granite rock from central California and [START_REF] Katsura | Hydraulic proper es of variously weathered grani c bedrock in headwater catchments[END_REF] on samples of granite rock from northeastern Osaka, Japan. In the i rst two studies, samples were extracted on an outcrop and an excavated trench. In the last study, samples came from deep coring from the surface to a depth of 38 m. To apply a matric potential to the samples, all three studies used a pressure plate according to the method of either [START_REF] Klute | Water reten on: Laboratory methods[END_REF] or [START_REF] Dane | Water reten on and storage: Laboratory[END_REF].

Results

Of the 10 samples analyzed by the Wind method, only eight allowed description of a part of the retention curve that met the two i t criteria. h e curve parameters are presented in Table 1. h e decrease in water content with the matric potential was monitored until the matric potential reached nearly -580 cm for one sample (Fig. 5) and on average -430 cm. h e value of the minimum potential reached during the experiment depended on the moment when the hydraulic continuity between the microtensiometers and the media was broken. h e variability in the matric potential in the samples coincided with the range of values measured in situ by the tensiometers and at depths equal to those from which the samples were extracted. Nonetheless, the eight partial retention curves (Fig. 6) show that the range of validity is narrower for some of the retention curves obtained with the Marquardt algorithm.

h e data retained for analysis allowed calculation of the parameter values for Eq. [2] that met the two i t criteria (Table 1). Moisture at saturation (θ s ) was close to total porosity values from the literature for Index 6 (Fig. 1), indicating that nearly all the pore space is accessible to water. Excluding Sample 3a, the mean water content at saturation was 34% (SD = 2.5%), the mean residual water content was 7% (SD = 2.6%), and the mean free water content was 27% (SD = 3.9%).

h e quantity of free water in the weathered granite samples fell by 50% as soon as the matric potential approached approximately -40 cm. h is ψ m value increased sharply with increasing sampling depth: -49 cm (SD = 11.3 cm) for samples extracted at -135 cm below ground level, -37 cm (SD = 7.4 cm) for those extracted at -155 cm, and -9 cm for the single sample extracted at -167 cm. h is trend is not signii cant, however, given the small number of samples at each depth. h e high ψ m values indicate that a large proportion of the water in the weathered granite moves before the matric potential changes signii cantly. In other words, most water in the weathered granite samples was weakly connected to particles and could move easily.

On average, the percentage of free water (M) that moves in the partial retention curves is only 40%. h e method applied to the weathered granite samples is therefore less robust for the part of the curve in which the water content is near saturation. Nearly all the curves intersect or are highly similar, except for Sample 3a, which dif ers noticeably from the others. h is dif erence can be explained by a higher value for parameter α and lower value for θ s for this sample than for the others. When the matric potential falls below -150 cm or when only an average of 25% of free water remains, the water content decreases little as the matric potential decreases.

h e curve of Sample 3a was not included in the calculation of the parameters of the mean retention curve of the sampled weathered granite. Fitting was performed from 226 pairs of h and θ values that came from the curves at regular intervals of 10 cm. Figure 6 shows that the coni dence interval of the mean retention curve does not cover the partial retention curve of Sample 3a. h is coni rms that the retention properties of Sample 3a dif er signii cantly from those of the other samples.

h e i tted parameter values and their 95% coni dence intervals are given in Table 2. At the two limits of the coni dence intervals, the free water content is relatively dif erent (±4%). h e ψ m varies by a factor of nearly 5 from the lower to the upper limit.

Table 1. Values of the van Genuchten model parameters (saturated water content θ s , residual water content θ r , and i t parameters α and n) describing the water retention curves of weathered granite samples according to the Wind method and characteristic values of the free volumetric water content, dei ned as the dif erence between θ s and θ r (θ e ), the matric potential when ef ective saturation is 0.5 (ψ m ), and the free-water proportion, which moves along the water retention curve portion described with the Wind method (M) at depths of 135 (four samples), 155 (three samples), and 167 cm (one sample) below the surface. Fig. 6. Experimental water retention curves from weathered granite samples (gray lines). Mean water retention curve and coni dence interval curves were i tted using the RETC computer code.

Table 2. Mean values and 95% coni dence intervals of the van Genuchten model parameters (saturated water content θ s , residual water content θ r , and i t parameters α and n) and characteristic values of the free volumetric water content, dei ned as the dif erence between θ s and θ r (θ e ) and the matric potential when ef ective saturation is 0.5 (ψ m ) describing the mean water retention curve and 95% coni dence interval curves of weathered granite samples using the Wind method. Values were i tted with the RETC computer code [START_REF] Van Genuchten | The RETC code for quanfying the hy-draulic func ons of unsaturated soils, version 1.0[END_REF] from retention curves of weathered granite samples.

Soil Par cle Size Distribu on Approaches

Although we selected undisturbed samples with restrictive criteria for particle size distribution (PSD), measuring methods, and bulk density, the cloud of points formed by the measurements of moisture and matric potential is stretched along the matric potential axis (Fig. 7a). h is implies that the retention parameters vary among the samples. h e means and upper and lower quartiles of water content were calculated for every matric potential, considering the i ve soil horizon samples collected from the UNSODA database. By overlaying these statistics with the mean retention curve of the weathered granite samples (Fig. 7b), we observed that the coni dence interval of the mean retention curve of the weathered granite samples had the same order of magnitude as the scattering of data from the UNSODA database. For matric potentials smaller than -80 cm, water content values of samples from the UNSODA database almost completely cover the coni dence interval of the mean retention curve of the weathered granite. Consequently, for matric potential values smaller than -80 cm, the water content mean values are not signii cantly dif erent from those predicted by the mean retention curve of the weathered granite samples. For matric potentials larger than -80 cm, the mean water content of the soil samples from the UNSODA database was consistently larger than the upper limit of the coni dence interval of the experimental mean retention curve for weathered granite.

Pedotransfer Func on Approaches

h e θ s value predicted by the pedotransfer function (Table 3) was signii cantly larger than those measured for the weathered granite samples (Table 2). For the θ r , the weathered granite samples had larger values. h ese dif erences were signii cant because there was no overlap in the ranges of predicted vs. observed water content values (Table 3). Consequently, the free water contents measured in the weathered granite samples were signii cantly smaller than those predicted by the pedotransfer function. Moreover, the shape of the retention curve predicted by the pedotransfer function showed strong dif erences with the mean curve of the weathered granite samples (Fig. 8). As for the PSD approaches, the water content for the retention curve predicted by the pedotransfer function is close to saturation and for matric potential values smaller than those for the mean retention curve of the weathered granite samples. h is dif erence illustrates for the values of α and n the deviation between those predicted by the pedotransfer function and those measured in the weathered granite samples (Table 3).

Reten on Proper es of Similarly Weathered Granite

Retention properties from the literature for weathered granite with the same weathering index vary greatly (Table 4; Fig. 9). For θ s , values range from 15 to 37%, while those for θ r range from 6 to 24%. h e mean water retention curve of the weathered granite studied here lies in the upper part of the range of θ s and the lower part of the range of θ r . For θ e , the range of the values of the weathered granite Table 3. Mean values and 95% coni dence intervals of the van Genuchten model parameters (saturated water content θ s , residual water content θ r , and i t parameters α and n) and characteristic values of the free volumetric water content, dei ned as the dif erence between θ s and θ r (θ e ) and the matric potential when ef ective saturation is 0.5 (ψ m ) describing the mean water retention curve and 95% coni dence interval curves of weathered granite samples using a pedotransfer function based on soil with an equivalent particle size distribution. Values were i tted with the RETC computer code [START_REF] Van Genuchten | The RETC code for quanfying the hy-draulic func ons of unsaturated soils, version 1.0[END_REF]. samples studied here (Table 1) has the same order of magnitude as that from the literature. h e value of ψ m from the literature is lower than that of the weathered granite samples studied. Unlike for the water content values, for the values of ψ m , the study of [START_REF] Katsura | Hydraulic proper es of variously weathered grani c bedrock in headwater catchments[END_REF] had values closest to our results.

h e results we obtained for the weathered granite samples were overall of the same order of magnitude as those obtained from samples extracted using the same method (Hubbert et al., 2001b;[START_REF] Jones | Water-holding characteris cs of weathered grani c rock in chaparral and forest ecosystems[END_REF]. Nonetheless, there is a large disparity in the results, even though the weathered granite samples belonged to the same weathering class.

Discussion Water Reten on Characteris cs

For undisturbed weathered granite samples, the Wind (1968) method appears interesting to describe the portion of the retention curve in which the water content is the lowest (Fig. 6). In addition, examination of an apparently homogenous weathered granite (Fig. 3) revealed that the shapes of the retention curves from samples extracted from dif erent locations were similar and nearly parallel to each other. h e mean water retention curve of the weathered granite exhibited a large change in θ for h values from 0 to -100 cm but little change below -100 cm (Fig. 6). We can infer that the weathered granite contained distinguishable macropores and micropores. Variability can be observed, however, between the measured water retention curves. h e variability may have many origins. Dif erences in the relative degree of weathering might be one of them. Even if all the collected samples were classii ed as Index 6 of Clayton and Arnold's weathering classii cation, slight dif erences in weathering degree cannot be excluded. h is could mean that this index is not accurate enough to discriminate water retention properties. Another origin of this variability might be the dif erences in the compaction of the weathered granite by overburden forces, depending on sampling depth. h e rest of the discussion will focus on comparison of our results with soils of the same texture and with results obtained for similar materials (weathered granite).

U lity of Pedotransfer Func ons

Indirect estimation methods (PSD and pedotransfer function approaches) used for soils did not appear adequate to predict the retention curve of the weathered granite samples in this study.

Comparison of the mean retention curve of the weathered granite Table 4. Characteristic values of the free volumetric water content, dei ned as the dif erence between θ s and θ r (θ e ) and the matric potential when ef ective saturation is 0.5 (ψ m ) and specii c water contents (saturated, θ s , and residual, θ r ) of weathered granite from this and other studies. Fig. 9. Comparison between the mean water retention curve for weathered granite samples and water content-matric potential, θ(h), pair measurements from other weathered granite bedrocks with the same weathering index (Index 6). In [START_REF] Jones | Water-holding characteris cs of weathered grani c rock in chaparral and forest ecosystems[END_REF], available values were averages from the San Jacinto (SJ) or San Bernardino (SB) sites, and one set of values from one individual sample from the SB site is also given. derived from the Wind experiment and the curves obtained from the PSD approach and pedotransfer functions reveal strong differences (Fig. 7b and8). h e value of θ s predicted by the pedotransfer function (Table 3) is larger than that measured in the weathered granite samples (Table 2). h is dif erence can be explained by the degree of weathering of the weathered granite, which is a little higher than that of a soil. Porosity being highly correlated with the degree of weathering for weathered granite (Fig. 1), for a given grain size, soil contains more water than weathered granite. h e relation between θ e and θ s indicates that 72% of the water contained in a sample of weathered granite can be mobilized. For a soil of equivalent texture, this relation is 88%. h is dif erence suggests that the micropore volume containing immobile water is proportionally larger in a weathered granite than in a soil of equivalent texture, which has a larger volume of macropores that contribute more to the quantity of free water. h e dif erence between the values of n in Eq. [2] for the mean retention curve of the weathered granite samples (Table 2) and that predicted by the pedotransfer function (Table 3) is not signii cant because the coni dence intervals of the values of n overlap. Nonetheless, the value of α (Tables 2 and3), corresponding to the inverse of the air-entry pressure, is signii cantly larger for the weathered granite samples because the coni dence intervals do not overlap. Comparing the retention curves linking the ef ective saturation (S e ) with the matric potential (h), shows the ef ect of α and the shape of the retention curves (Fig. 8). With the pedotransfer function, the water content remains close to saturation for matric potential values between -1 and -10 cm, while for a matric potential of -10 cm, the retention curve of weathered granite indicated that the volume of free water in the sample represented no more than 75% of the volume contained in the sample when it was saturated. h e dif erence between the water content values predicted by the two curves remained signii cant until the matric potential fell below -30 cm.

Can the Weathering Index be Used to Derive the Water Reten on Curve?

h e weathering class did not distinguish weathered granite samples sui ciently to infer their retention properties. Our results showed relatively large dif erences between the retention properties of the weathered granite samples in this study (Table 2) and those reported in the few published studies on the subject (Table 4). h ese dif erences could be explained by variations in the mineral composition and size of crystals that make up the original rock. In fact, the minerals do not weather at the same speed nor in the same way, some producing clays and others producing sandy or silty particles. h e mineral composition of the original substrate certainly has an important inl uence on the texture, in particular on the clay content, of the weathered material and thus an inl uence on the retention properties. h e mineral composition of the original rock determines the ability of weathering agents to modify the rock's porosity and retention capacity. h e values obtained for the sample studied by [START_REF] Katsura | Hydraulic proper es of variously weathered grani c bedrock in headwater catchments[END_REF] clearly dif er from those obtained for all the other samples (Fig. 9), in particular for matric potential values below -100 cm. h e coring technique used is particular to the samples extracted by [START_REF] Katsura | Hydraulic proper es of variously weathered grani c bedrock in headwater catchments[END_REF] and could have modii ed the sample structure. h us, the method used to extract samples could lead to modii cations of the material that have consequences on the retention curve of the sample.

Conclusions

h e study showed that the Wind method can be used to measure the retention curve of undisturbed samples of weathered granite. h is method describes the portion of the retention curve in which water contents are relatively low. Regardless of the approach used to compare our results to those obtained for soils with similar texture, we noted that the weathered granite had a retention curve notably dif erent from that of a soil. h is dif erence was larger at high matric potentials.

h e moisture values predicted by the retention curve for the weathered granite samples approached values that were obtained for samples of granite with the same degree of weathering and dif erent origins as long as the samples were extracted with the same method. h e weathering classii cation of [START_REF] Clayton | Prac cal grain size, fracturing density and weathering clas-sifi ca on of intrusive rocks of Idaho batholith[END_REF], however, is not accurate enough to predict hydrodynamic parameters. h us, the experimental approach remains, for the moment, the only way to obtain a reliable estimate of the hydrodynamic properties of these materials; however, this approach is exacting regarding both extraction of samples and laboratory procedure. Consequently, given the importance of weathered granite layers in water storage and transport processes, it is necessary to develop databases that bring together the experimental results of measurements of the hydrodynamic properties of this type of material in order to assess if there is a need to develop specii c functions for this kind of material. To develop predictive relations, the samples studied must be dei ned with variables that are simple to measure, suitable, and contain sufi cient criteria to distinguish dif erent types of weathered granite. An extension of this work would be to characterize the hydrodynamic properties of all weathered bedrocks, not only granitic. If the relations found for weathered bedrocks appears specii c compared with that found for soils, the results of these studies could be regrouped into a specii c database similar to that already existing for soils, which currently has no equivalent for weathered bedrocks.

  Abbrevia ons: PSD, par cle size distribu on.

Fig. 1 .

 1 Fig. 1. Relation between the porosity of some granites and weathering class. Porosity of an undisturbed granite ranges from 0.02 to 1.80%.

Fig. 3 .

 3 Fig.3. Coni guration of the sampling trench section: R, the shallow root zone; P, plow layer; S, structural horizon with deep roots; Ge, weathered granite with both homogeneous color and structure; Geh, weathered granite with heterogeneous blocks. Cylinders show the core sample positions. Letters a to e correspond to the pit crosssections. h e bent vertical lines suggest the width of the trench, with the horizontal zone where sampling cylinders were inserted.

Fig. 4 .

 4 Fig. 4. Experimental device for matric potential and weighing humidity measurements (Wind method).

Fig. 5 .

 5 Fig. 5. Example of time series of (a) sample mass and (b) matric potential for the i ve tensiometers during a Wind experiment; (c) and (d) illustrate portions of the experiment where data were discarded.

Fig. 7 .

 7 Fig. 7. (a) Scatterplot of data selected using particle size criteria in the UNSODA database; (b) mean water retention curve (red line) with coni dence interval (dashed blue line) obtained with an experimental approach from weathered granite samples and compared with predictions of particle size approaches (box plots). Box plots show lower (bottom of box) and upper (top of box) quartiles, medians (gray band), means (black band), and possible extreme values (ends of whiskers).

  Sampling by deep coring. ‡ Sampling from a soil pit. § Values of an individual sample from San Bernardino. ¶ ND, not determined.

Fig. 8 .

 8 Fig. 8. Weathered granite samples mean retention curve (red line) and coni dence interval (dashed blue lines) compared with the water retention curve predicted by a ROSETTA pedotransfer function (black line).

Acknowledgments

h is research was supported by grants from the following French programs: ORE AgrHyS, SOERE RBV (Allenvi), and ANR ACASSYA (ANR-08-STRA-01).


Borjan Tchakaloff

Sébastien Saudrais

Jean-Philippe Babau

Efficient models configuration for an electric vehicle energy management software

Energy management of electric vehicles has been the focus of recent research to allow optimal engine and battery usage. Many efforts have been realised to use the trip knowledge -or a prediction of it -to provide the best vehicle efficiency. Yet few works take into account the embedded devices and the vehicle global Quality of Service. The ORQA framework has a vehicle systemic approach, its purpose is to generate an architecture to counter the range anxiety and offer the best quality effort based on the driver preferences. The work described in this paper is about the off-line configuration of the ORQA framework to match a target vehicle characteristics and abilities. With a more precise configuration, the on-line execution of ORQA is optimised. Two leads are presented to reduce the computation time needed to explore the solution space on-line. The final result is an energy management software tuned for a specific targeted vehicle which offers a driving strategy and a control of the embedded devices matching the driver destination and preferences.

I. INTRODUCTION

Though they still represent a small part on the global market, the electric vehicles have reached the worldwide market. The main issue raised against the technology is the short autonomy this kind of vehicle provide, hence a certain reluctance about their usage (commonly known as range anxiety). Current vehicles can handle a few hundred kilometres. One way to deal with the embedded energy issue of the electric vehicles is through software. An Energy Management System (EMS) is a high level software monitoring and managing an environment through specific-purposed components. EMS are commonly used in (full-and hybrid-) electrical vehicles, though they mainly manage only the engine and ignore the end-user provided Quality of Service (QoS). In order to offer an efficient energy management and to take into account the user-related QoS, an electric vehicle EMS has to consider every embedded devices and the user expectations.

The ORQA framework [START_REF] Tchakaloff | ORQA: Modeling Energy and Quality of Service within AUTOSAR Models[END_REF] offers to tackle the global energy management while providing a QoS as good as possible. In ORQA, each embedded device is characterised at design phase by its energy consumption(s) and its quality(ies), if applicable. The framework offers to realise a components architecture which will elaborate on-line a solution to achieve the driver request (to reach a destination) while providing the best possible vehicle QoS. The main idea is to limit by software the devices and engine usage. The framework architecture and process are presented in [START_REF] Tchakaloff | ORQA: Modeling Energy and Quality of Service within AUTOSAR Models[END_REF]. But the solution space of an optimal solution in which operates ORQA exponentially increases with the routes amount and the number of devices. To reduce the solution space, the challenge is to propose efficient models and an associate configuration for the framework.

This paper presents the decision models of ORQA, how they can reduce the solution space when correctly configured, and how it impacts the on-line computation. An extensive offline search is realised on generated data to identify possible solution space reductions. Two approaches are presented, one which reduces the search input dimension, and the other which approximates the solution space. The designer is invited to browse through the results and to select the final configuration. In the end, the solution domain is tuned according to the target vehicle characteristics and abilities, while still providing various viable solutions.

The rest of the paper is structured as follows. Section II is the ORQA background, what the framework is and how it operates. Section III details the decision models of the framework. Section IV presents the two reduction approaches. Section V evaluates the approaches with a use-case and compares the obtained results. The related works are compared in Section VI, and we finally conclude the paper in Section VII.

II. ORQA STRUCTURE AND USAGE

ORQA is a framework to set up an Energy Management System (EMS) for electrical vehicles. ORQA proposes specific models and a methodology which leads to an energy manager connected to the vehicle embedded software at run-time. The idea behind ORQA is to offer a global service to the driver for a specific journey while maximising the device services. Also, the destination point should be reached quickly while saving the battery as much as possible.

First, ORQA provides a library of predefined models to characterise the energy consumption of different physical devices embedded in an electric vehicle. The library contains energy models for the electric engine, the climate control, the lighting system, and the entertainment system. For each journey, the possible routes are modelled: depending on the route characteristics (like speed constraints and duration), one can limit or not engine usage to reach the destination point. The driver preferences are introduced to integrate user priorities between devices usage. They belong in the system memory and can be updated at need. The on-line usage of ORQA results in a driving strategy to reach the destination under constraints over the vehicle velocity and the devices usage. Another result is the corresponding device controller configuration (see Fig. 1 which illustrates the global overview of ORQA EMS), according to the driving strategy.

The vehicle devices model characterises the devices embedded in the target vehicle. ORQA considers the target vehicle as a system composed of a set of known devices. A device is a software function that controls a physical device. A device may function at different operating states to realise its service. An operating state has a certain power requirement to operate, that is a power function that can be constant or parameterised by environmental data. The energy consumption of an operating state basically depends on the duration the device operates in it. The whole energy consumption of a device is directly given by the sum of its operating states energy consumptions. ORQA differentiates devices that are mandatory to operate the vehicle (they are handled automatically by the vehicle) and those that are not. One can control the latter devices by selecting their operating states, thus changing the way they realise their service and hence the Quality of Service they offer. Each operating state of these controlled devices (the controlled devices) has a quality attribute denoting its QoS. A devices combination is a map such as each controlled device has an operating state. ORQA defines two extreme combinations: 1) the least consuming combination and 2) the nominal combination. The least consuming combination corresponds to every controlled devices operating at their least consuming operating state (in most cases, switched off). Whereas the nominal combination is the combination of the controlled devices operating at their best quality operating state. ORQA does not interfere with the mandatory devices except for the engine. It manages the qualified devices at runtime by constraining their operating states.

The routes model represents the different paths between the departure point and the arrival point, divided into steps. A route is a collection of steps and belongs to an environment. A step is defined by its initial and final velocities, its distance, and its slope. The initial velocity of a step is the final velocity of the preceding step, or 0 if there is none (that is, the step is the first of the route). An environment defines usual initial and final velocity ranges, and common step distances. Default environments are urban, rural, and motorway as defined by the Common Artemis Driving Cycles (CADC) [START_REF] André | The ARTEMIS European driving cycles for measuring car pollutant emissions[END_REF] European project.

At run-time, the ORQA Energy Management System handles the driver request. The EMS operates as follow:

1) The driver defines a destination point;

2) The possible routes are retrieved or generated from the system (e.g. from the GPS unit);

3) Each possible driving strategy is evaluated (consumption, duration, quality): a driving strategy is composed of a route and a devices combination; 4) A driving strategy is chosen and the driver is informed; 5) The embedded energy manager controls the devices online following the proposed strategy. Dedicated broking components control the devices behaviour. For the fourth step (the driving strategy selection), the ORQA framework has to define an efficient ranking of different driving strategies. We propose in this paper a simple ranking based on scoring functions. And because ORQA is executed on-line, the ranking computation has to be efficient. To limit the search space, we propose to reduce off-line the range of the different parameters to evaluate in the third step. We now present the different necessary models to evaluate the driving strategies.

III. ORQA MODELS FOR DECISION

The logic to select the driving strategy is embedded in the EMS. It is used on-line to determine which driving strategy to choose. The parameters of the choice are the velocity coefficients, the driver preferences and the scoring functions.

A. The velocity coefficients

The possible routes of a journey are retrieved from the system. A route models the nominal driving conditions of a journey: nominal velocities and no traffic congestion. A route is refined by applying a velocity coefficient to its nominal velocities. A velocity coefficient is a percentage and represents how much a vehicle is slowed down compared to a nominal driving. The new route is a variation of the nominal route and represents a different driving condition. This refinement allows the EMS to explore different driving conditions of the retrieved routes.

B. The driver preferences

The triggering event of the ORQA process is the driver request to reach a destination. Aside from the destination point itself, two sets of parameters can be set: the controlled devices ranking and the consumption policy. The devices ranking is a distribution of a finite amount over each controlled device reflecting the driver preferences. The default devices ranking is set at design phase but the driver must be able to update it for his own preferences. For instance, if there is not enough energy to operate the controlled devices in the nominal combination, then another less consuming combination is selected according to the importance of the devices for the driver. The consumption policy is the driver policy to match for the proposed solution. It is a named policy with optional constraints over the duration and the energy consumption. A consumption policy defines weights for the main results of a solution: duration, consumption, and quality. The consumption policy weights are used to rank the computed solutions as described later in the scoring functions. The duration constraint is expressed as a maximal delay to the nominal solution (e.g. 2×). The consumption constraint is expressed as a minimum level of energy left in the battery (e.g. 20%).

C. The scoring functions

The driving strategies are created by mapping each route to each devices combination. We introduce a ranking for the driving strategies to select the best one according to the consumption policy defined by the driver. The score of a driving strategy is defined by its main results: duration, consumption and quality. These results come from the route evaluation against devices combinations. Each one of the main results is passed to a corresponding scoring function. A scoring function normalises a main result over every driving strategies and vehicle capacities (e.g. the consumption of a route against the maximum level of energy stored in the battery). Depending on which main result is involved, the scoring function returns the best score with the minimum (duration and consumption) or the maximum (quality) value. The consumption policy constraints are taken into account by the scoring functions, an out-of-range result discards its driving strategy. For example, if the driver has set a duration constraint of 1.5×, any route that has a route duration nominal route duration ratio over 1.5 is discarded. The scoring functions results are combined by a weighted arithmetic mean. The meaning weights are defined by the consumption policy from the driver request. So the score S of a route is:

S = S T • w T + S E • w E + S Q • w Q feasible strategy S = 0 discarded strategy (1)
where S i is a scoring function, w i is a weight defined by the current consumption policy, and the indices T , E , Q respectively represent the duration, the consumption, and the quality results. A score has a null value if at least one of its results is discarded (i.e. it does not satisfy the constraints). The best scoring strategy is given back to the driver as the proposed driving strategy.

The ORQA framework has to take into account different parameters to choose the best strategy on-line: the velocity coefficients, the driver preferences and the scoring functions. In order to optimise the search, the framework has to be configured according to the vehicle capacities. We now propose two approaches to reduce the solution space during design time which lead to an on-line reduced complexity.

IV. OFF-LINE CONFIGURATION

The configuration of ORQA is based on the exploration of the velocity coefficients domain applied to each route. As the solution space is composed of the routes and their variations, the amount of velocity coefficients directly impacts the online search process. We define two approaches to accelerate the search process: a reduction of the search input dimension and an approximation of the route variations. We present both approaches in the remainder of this section. The former approach lies on the reduction of the velocity coefficients amount. It helps the designers determine an adequate group of velocity coefficients to represent the whole coefficients domain. The latter approach explores approximations of the route variations so as to avoid the whole variations computation. These approaches can be used separately or combined together. It is possible to apply the approaches on each environment independently and embed on-line different reductions.

For each route, one can compute the duration and consumption which characterise the route with the ORQA devices models. We use the vehicle models defined in [START_REF] Tchakaloff | ORQA: Modeling Energy and Quality of Service within AUTOSAR Models[END_REF] (presented in the Evaluation section) which represent an urban electric car. To understand the approaches and their interests, we first present how duration and consumption evolve along the velocity coefficients. One hundred routes are generated per environment. Each route is refined a hundred times by velocity coefficients from 1% to 100%. This assures a good coverage of the velocity coefficients domain which is the complete input domain of the solution space. The duration and consumption are represented by relative values, called ratios, compared to the nominal routes (velocity coefficient of 100%). The duration ratio is in fact the delay compared to the nominal duration. Fig. 2.a illustrates the ratios variation for the generated routes with only one device in the vehicle, the engine. In the context of transportation, a delay of twice the nominal duration seems to be a reasonable limit for a journey. So, in the remainder of the paper, we limit the velocity coefficients domain under study to [40%; 100%] as the example variations under 40% exceed a duration ratio of 2×. Fig. 2.b is a focus of Fig. 2.a on this restricted velocity coefficients domain.

A. Velocity coefficients of grouped results

The first approach is based on data clustering. It groups the variations to limit the on-line route exploration. The idea is to define a limited number of representative groups of velocity coefficients instead of using the whole domain. The partitioning is based on the k-medoids partitioning algorithm [START_REF] Kaufman | Clustering by means of medoids[END_REF], [START_REF] Theodoridis | Pattern Recognition[END_REF] initialised with the k-means++ algorithm1 [START_REF] Arthur | k-means++: The advantages of careful seeding[END_REF]. The input of the algorithm is a set of vectors. Each vector represents a velocity coefficient, it is characterised by the ratios of every routes for that coefficient. It is important to notice that duration and consumption ratios evolve independently (see Fig. 2), so they both have to be considered independently for partitioning. The partitioning algorithm yields a set of k (from 1 to the number of coefficients) groups of vectors. A set of routes are in the same group if they minimise the distortion metric. The error rate measures the relative error (the standard deviation) between a vector and one particular vector of the group, called the representative vector. As a vector represents a velocity coefficient, the k groups of vectors returned by the partitioning algorithm lead to k groups of velocity coefficients. Also, the representative vector of a group leads to the representative velocity coefficient of the group. If k is not fixed, a full range of clustering has to be realised to find the "best" groups possible.

B. Ratios approximation

In the second approach, we consider the fact that ratios, for different routes, evolve in a same way (see Fig. 2). We propose to approximate a set of ratios evolutions with a representative evolution, called the approximation function. So at each velocity coefficient (the evolution step), the ratios are represented by one ratio called the approximated ratio. The approximation functions resulting of this approach are given to the designers as hints to optimise the variation computation. Indeed, the complete evaluation of the variations is replaced by approximating the nominal route results, so the on-line complexity is greatly reduced. The duration and the consumption results evolve differently, their ratios are approximated independently. So it is possible that there is one approximation for the duration ratios and three for the consumption ratios (one per environment).

Both of the two approaches rely on approximation, so we introduce a metric (the error rate) to assess an approximation reliability. The error rate relies on the standard deviation of the results and is defined as follows:

error = n i ((x i -x ′ i )/x i ) 2 n (2)
where x i is a computed result and x ′ i is its corresponding estimation.

We present two different approaches to reduce the on-line search process complexity based on an adequate off-line configuration. Both approaches rely on a set of data gathered from generated routes and both consider the similarities between the routes variations. On the one hand, a specific amount of velocity coefficients is computed off-line to be later embedded in the EMS. On the other hand, ratios are approximated offline to provide an on-line look-up table of ratios.

V. EVALUATION

We apply the two approaches on a use-case to evaluate their benefits over the complete solution space. The use-case contains three different routes, one per environment: urban (3.4km), rural (28.7km), and motorway (47.8km). The usecase vehicle is basically a city car with two optional devices: an air-conditioning unit and an entertainment system. The vehicle models are thoroughly described in [START_REF] Tchakaloff | ORQA: Modeling Energy and Quality of Service within AUTOSAR Models[END_REF]. The vehicle main characteristics are the following: a mass of 1200kg, a frontal area of 2.75m 2 , an engine nominal power of 35kW, and a battery rated energy of 20kWh. We assume an outer temperature of 30 • C and an inner objective temperature of 20 • C. The driver preferences are equally distributed among the two optional devices. The consumption policy chosen by the driver for this use-case is conservative, it is a trade-off between high energy savings and good comfort. The weights of the duration, the consumption and the quality used to compute the route scores are respectively 0.2, 0.5, and 0.3. As the same vehicle is used to illustrate the previous section, we use the same generated routes to serve as the base data to the two reduction approaches. The scoring functions are defined as follows:

S T = 100 + 99 • (T min -T ) T max -T min (3)
S E = 100 + 99 • (E min -E) E max -E min (4)
S Q = 101 -100 + 99 • (Q min -Q) Q max -Q min (5)
with the min and max exponents representing the minimal and maximal valid values (i.e. not discarded) found for the results. T , E, Q respectively represent the route duration, consumption, and quality. The scoring functions allow each result score to be in [1; 100], or < 1 if discarded. The driving strategy scores follow [START_REF] Tchakaloff | ORQA: Modeling Energy and Quality of Service within AUTOSAR Models[END_REF].

A. Initial configuration

The initial configuration explores the complete solution space, one case per velocity coefficient and per devices combination. The two optional devices have respectively three and two operating states, so there are six possible devices combinations. And as there are sixty-one velocity coefficients in the studied domain ([40%; 100%]), the complete solution space is composed of 366 cases. For each case, the results are evaluated and the corresponding scores are computed. Fig. 5 shows the scores evolution for the three different route environments. For the urban route, the fifteen velocity coefficients in [79%; 93%] yield the best scores. For the rural route, it is the thirteen ones in [63%; 75%], while for the motorway route it is the fifteen ones in [59%; 74%]. The optimal driving strategies results are listed in Table I along their relative deviations.

B. Application of the reduction approaches

We now apply the two approaches to the data coming from the generated routes. The explored approaches are applied to five configurations: 1) the urban routes, 2) the rural routes, 3) the motorway routes, 4) both the rural and the motorway routes, and 5) every routes (all of the environments). The other environments combinations (i.e. urban-rural and urbanmotorway) are not shown here because of their low value for the study.

a) Velocity coefficients of grouped results: The first approach is now applied to the data from the generated routes. Fig. 3 displays the evolution of the global error rate for the five configurations. The global error rate is the combination of the duration error rate and of the consumption error rate. The elbow criterion is a visual method to determine an adequate configuration based on the evolution of the error metric. In this method, the searched solution is visible when the results plot forms an important angle (the elbow). The external edge of the elbow points out the limit configuration from which the results look alike. We identify in each configuration the adequate number k of clusters with the elbow method, which is four for every configuration. The representative values of the configurations are a) {46%, 60%, 75%, 92%}, b) {46%, 60%, 76%, 93%} and {46%, 60%, 77%, 93%} for the c), d), and e) configuration. We notice that, in this example, the representative values of velocity coefficients are very similar for the five configurations. Also, though the ratios evolution looks linear in Fig. 2.b, both duration and consumption evolutions are taken into account to cluster the velocity coefficients. Moreover, compared to a regular discretisation of the velocity coefficients domain, the proposed approach provides the sufficient number of groups and more representative values. A naive by-4 discretisation would be {40%, 60%, 80%, 100%} instead.

b) Ratios approximation: The second approach is now applied to the data from the generated routes. Fig. 4 displays the error rates of the approximation ratios along the velocity coefficients for each configuration. On the one hand, the duration approximations have a low error rate of at most 2.15% for the all-routes configuration. It means that the duration of a route variation can be approximated with less than 2.15% with a single approximation function on the duration. The "all routes" configuration is thus chosen to estimate on-line every duration ratios. On the other hand, the consumption approximations have more disparate error rates as can be seen on Fig. 4. The configurations respectively reach an error rate of 11.3%, 8.6%, 4.3%, 7.4%, and 20.4% at the last velocity coefficient (40%). It is up to the designers to act on these results as they are highly subjective. For instance, if we were to consider that a configuration should have a maximal consumption error rate of 10% at the last velocity coefficient, then the "ruralmotorway routes" configuration would be acceptable for rural and motorway routes. But there would be no configuration for the urban routes. Hence no approximation function would be available on-line to approximate urban routes and their variations should be computed instead.

C. Configuration I: velocity coefficients of grouped results

The different groups found by the first approach are close enough to allow one representative groups for every environment: {46%, 60%, 77%, 93%}. The six devices combinations are still explored, so there are 24 cases to evaluate per route. Fig. 6 shows the three different environments for the twenty-four cases. For the urban route, the 77% coefficient corresponds to the best score. For both of the rural and motorway routes, the 60% coefficient yields the best scores. We can see that for both the urban and the rural routes, suboptimal driving strategies have been selected. The motorway route has an optimal score that is also one of those found in the complete evaluation. Their results are summed up in Table II. In summary, the urban route driving strategy differs from the mean optimal of 36 seconds (10%), 25Wh (11%), and 4.5 percent of quality. The chosen rural route differs from the mean optimal of 6 minutes and 24 seconds (14%), 287Wh (16%), and 4.5 percent of quality. Finally, the motorway route differs from the mean optimal of 6 minutes and 28 seconds (11%), 511Wh (15%), and 4 percent of quality.

D. Configuration II: ratios approximation

We choose to use the ratios approximations of each environment. Each nominal route is evaluated by ORQA and its variations results are approximated. So there are only 6 evaluations per route (one per device combination), and 360 approximations. Fig. 7 shows the scores resulting from these approximations. For the urban route, the thirteen velocity coefficients in [79%; 99%] yield the best scores. For the rural route, it is the nine coefficients in [68%; 76%], while for the motorway route it is the twelve ones in [64%; 75%]. The optimal scores of the approximated results share common coefficients with the complete evaluation scores. The mean results of the driving strategies are summed up in Table III. In summary, the urban route mean driving strategy differs from the mean optimal of 8 seconds (2%), 11Wh (4%), and 1.25 percent of quality. The chosen mean rural route differs from the mean optimal of 1 minute and 47 seconds (14%), 287Wh (5%), and 5.5 percent of quality. Finally, the mean motorway route differs from the mean optimal of 1 minute and 17 seconds (3%), 568Wh (13%), and 10.5 percent of quality. Both the urban and rural routes are within the relative deviations of the optimal results, aside from the quality level of the rural route. The motorway mean driving strategy has a duration in the optimal range but its high quality devices combination leads to a higher consumption (18% more).

E. Configuration III: compound approach

We now combine the approaches by approximating the ratios of the chosen coefficients. Each route has 6 evaluations and 24 approximations. Fig. 8 shows the scores resulting from these approximations. For the urban route, the 77% coefficient corresponds to the best score. For both of the rural and motorway routes, the 60% coefficient yields the best scores. We can see that for both the urban and the rural routes, suboptimal driving strategies have been selected. The motorway route has an optimal score that is also one of those found in the complete evaluation. This is coherent with the first approach results, the four chosen coefficients give close results but not in the range of the optimal results. The results of the three routes are summed up in Table IV. We evaluate the two reduction approaches against three different routes. We see that the new solution spaces are effectively reduced (24 evaluations against 366 in the first approach) or less complex to compute (6 evaluations and 360 approximations in the second approach). On the three examples, the obtained driving strategies are quite close to the optimal ones for the two proposed approaches. For the urban and the rural routes, we see that the error is within the relative deviation range when relying on the ratios approximation approach but not for the motorway route. The first approach, grouping the velocity coefficients, produces less accurate solutions which results variate more than 10% from the optimal ones. On the other hand, the compound approach (6 evaluations and 24 approximations) gives mixed results. They are coherent with the two approaches but too much loss is noted. This composition is not adequate in this use-case. The next step of this evaluation is a run-time monitoring of a real executing platform to tune the two approaches.

VI. RELATED WORKS

Energy consumption optimisation for and by software is a main-stream research domain, from data-centres to embedded systems or vehicles. In the domain of computing platforms, Snowdon et al. [START_REF] Snowdon | Koala -A Platform for OS-Level Power Management[END_REF] offers a platform to trade performance and energy consumption of applications at an operating system (OS) level. They rely on an off-line characterised model to predict software performance and energy consumption. The consumption modelling is specific to the hardware-platform (CPU/bus/memory clock frequencies, CPU voltage, temperature sensors). They collect each process behaviour at run-time based on statistics provided by the OS kernel. A subjective policy is then dynamically applied to perform the trade-off for each process. The authors approach is similar to a systemic dynamic power management (DPM). In [START_REF] Benini | A survey of design techniques for system-level dynamic power management[END_REF], several systemlevel DPM techniques are detailed. They advocate it is best to implement DPM at the higher software system-level (the OS). Such DPM benefits from the global view of the OS over on-going operations, and software-based management is more flexible than hardware-based. Druilhe et al. [START_REF] Druilhe | Energy-driven consolidation in digital home[END_REF] presents an energy optimisation in digital homes by using consolidation over IT devices. They take into account the devices appearance, heterogeneity and the services QoS. Devices and services are characterised off-line. The system optimisation is triggered by the execution of new services and the devices appearance updates. ORQA shares the same philosophy of these approaches. As them, ORQA takes into account the vehicle controllers consumption and also considers every other devices consumption, including the engine which is the main consumer of vehicles. Thus, it enhances the vehicle manager to perform an overall optimisation at run-time.

Work has been done to optimise the energy consumption of vehicles, whether focused on the engine, the embedded devices, and even their controllers. Optimising the driving process to minimise the trip energy consumption can be related to optimal trajectory with constraints on the engine power [START_REF] Mensing | Vehicle trajectory optimization for application in ECO-driving[END_REF], [START_REF] Petit | Optimal Drive of Electric Vehicles Using an Inversion-Based Trajectory Generation Approach[END_REF] and trip duration [START_REF] Dib | Evaluation of the Energy Efficiency of a Fleet of Electric Vehicle for Eco-Driving Application[END_REF]. These approaches focus on the engine and its optimal operation. Several techniques are used such as backward search (Mensing et al.), optimal control problem (Petit et al.), and inversion-based approach (Dib et al.). In [START_REF] Katoen | Model-Based Energy Optimization of Automotive Control Systems[END_REF], Katoen et al. presents a model-based approach to determine an energy-optimal software/hardware deployment found off-line. The authors approach is to systematically explore the design space of possible mappings and rank them along their energy consumption. The result is a hardware deployment with a minimal energy consumption which is the embedded devices controllers have an energy-optimal consumption. ORQA offers to take into account both the engine operation and the embedded devices to perform a better trip. As the control is realised on-line, the search must be executed fast. So ORQA relies on a complete off-line overview of the solution space to fasten the results computation.

VII. CONCLUSION

The paper presents techniques to model and configure an electric vehicle Energy Management System. We outline the ORQA framework based on a systemic view of the vehicle. It generates a dedicated EMS featuring the global energy management while providing a QoS as good as possible. We introduce the decision models defining the evaluation part of ORQA. We present two different approaches to effectively configure the decision models off-line. They are based on extensive results matching the targeted vehicle capacities. One offers to reduce the search input dimension (6% of the complete solution space evaluated) while the other proposes to approximate the search space (2% evaluated with a whole coverage, but with less precision). The approaches can be used exclusively or in combination. The compound approach is even less complex than the second one but also cumulates the loss of both approaches. We see on the three route examples that the compound approach is not interesting for this use-case vehicle. The approaches application leads to a suitable configuration optimised for the vehicle EMS.

The current framework approach to reduce the vehicle velocity is to apply one coefficient throughout the trip. A variable coefficient approach offers a more realistic driving but introduces a new dimension to the search space. This enhancement therefore requires a new configuration setting to be validated. Also, we plan to realise a complete phase of measures on a real vehicle controller for the different approaches. Then the two proposed approaches can be evaluated against the initial solution space on the execution time.

Fig. 1 .

 1 Fig. 1. Overview of the ORQA Energy Management System.

Fig. 2 .

 2 Fig. 2. Evolution of the routes ratios on (a) the complete domain of velocity coefficients and on (b) a domain restricted to [40%; 100%].

Fig. 3 .Fig. 4 .

 34 Fig. 3. Evolution of the global error rate per number of clusters (amount of velocity coefficients) for each configuration in the first approach.

TABLE I OPTIMAL

 I DRIVING STRATEGIES FOUND FOR EACH ROUTE IN THE COMPLETE SOLUTION SPACE. MEANS OF THE OPTIMAL RESULTS ARE DISPLAYED ALONG THEIR RELATIVE DEVIATIONS.

	Route	Optimal	Route results
		velocity coefficient	duration	consumption	quality
	urban	∈ [79%; 93%]	5"35' ±16'	243Wh ±14Wh	68% ±2%
	rural	∈ [63%; 75%]	40"28' ±2"09'	2113Wh ±132Wh 59.5% ±2%
	motorway ∈ [59%; 74%]	52"24' ±3"40'	3720Wh ±312Wh	58% ±2%

TABLE II RESULTS

 II OF THE BEST DRIVING STRATEGIES FOUND FOR EACH ROUTE BASED ON THE REDUCED INPUT DOMAIN (FIRST APPROACH).

	Route	Optimal	Route results
		velocity coefficient	duration consumption	quality
	urban	77%	6"11'	218Wh	63.5%
	rural	60%	46"04'	1826Wh	55%
	motorway 60%	57"40'	3305Wh	55%

TABLE III MEAN

 III RESULTS OF THE BEST DRIVING STRATEGIES FOUND FOR EACH ROUTE BASED ON THE APPROXIMATED SOLUTION SPACE.

	Route	Mean optimal	Mean route results
		velocity coefficient	duration consumption	quality
	urban	89%	5"27'	254Wh 66.75%
	rural	72%	38"41'	2142Wh	54%
	motorway 69%	49"55'	4384Wh	69.5%

The k-medoids algorithm output depends on the initial input. The k-means++ is a classical initialisation method that maximise the initial input coverage.