Duc-Hoan Tran
email: duc.hoan.tran@laplace.univ-tlse.fr

Bruno Sareni
email: bruno.sareni@laplace.univ-tlse.fr

Member, IEEE Xavier Roboam
email: xavier.roboam@laplace.univ-tlse.fr

Christophe Espanet
email: christophe.espanet@univ-fcomte.fr

D.-H Tran

Integrated Optimal Design of a Passive Wind Turbine System: An Experimental Validation

Keywords: Genetic algorithms, integrated optimal design (IOD), multiobjective optimization, passive wind turbine Rotor/Stator yoke thickness, phase EMF. Wind energy. Wind turbine damping coefficient. Airgap width. Optimization constraint

This work presents design and experimentation of a full passive wind turbine system without the active electronic part (power and control). The efficiency of such a device can be obtained only if the system design parameters are mutually adapted through an integrated optimal design method. This approach, based on multiobjective optimization, aims at concurrently optimizing the wind power extraction and the global system losses for a given wind speed profile while reducing the weight of the wind turbine generator. It allows us to obtain the main characteristics (geometric and energetic features) of the optimal Permanent Magnet Synchronous Generator (PMSG) for the passive wind turbine. Finally, experiments on the PMSG prototype built from this work show a good agreement with theoretical predictions. This validates the design approach and confirms the effectiveness of such a passive device.

I. INTRODUCTION

C LOSE to high-power wind turbines for onshore or off- shore applications, small wind systems represent an interesting target for applications such as rural electrification and autonomous energy production networks for water pumping, desalination. Optimizing energy efficiency generally leads to adaptation of the load impedance and consequently the speed of the generator with the wind turbine operating conditions. Many active structures have been thus proposed [START_REF] Baroudi | A review of power converter topologies for wind generators[END_REF]- [START_REF] Mirecki | Architecture cost and energy efficiency of small wind turbines: Which system tradeoff?[END_REF] to allow tracking the maximum power operation through corresponding maximum power point tracking (MPPT) strategies.

However, for such an application frame, the system cost has to be drastically minimized, for instance, by simplifying the structure with PM synchronous generator feeding a diode rectifier associated with a battery bus. For grid-connected applications, impedance adaptation can be obtained through the grid inverter as in [START_REF] Stannard | Performance aspects of mains connected small scale wind turbines[END_REF].

In earlier studies [START_REF] Abdelli | Optimization of a small passive wind turbine generator using multiobjective genetic algorithms[END_REF], [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF], we proposed a very "low-cost" and more reliable structure for remote applications without an active control unit and with a minimum number of sensors (see Fig. 1). In fact, for such a device, a "natural" impedance adaptation can be achieved with the passive structure by optimizing the accordance between system parameters using an integrated optimal design (IOD) approach. This approach has been successfully applied in simulation for a small Savonius wind turbine of 300 W. In this paper, the same methodology is investigated on a passive system composed of 1.5-kW three-bladed wind turbine. The optimal Permanent Magnet Synchronous Generator (PMSG) resulting from the IOD process is validated from experiments on a real prototype.

II. MODELING OF THE PASSIVE WIND TURBINE SYSTEM

A. Wind Model

A typical wind cycle defined in terms of speed harmonic decomposition is taken for describing the wind speed behavior [START_REF] Mirecki | Architecture cost and energy efficiency of small wind turbines: Which system tradeoff?[END_REF] (1)

B. Wind Turbine Model

A three-bladed wind turbine of radius is considered in this study. It provides a wind power defined by [START_REF] Hau | Wind Turbines: Fundamentals, Technologies, Application, Economics[END_REF] (2)

where the power coefficient varies as a function of the tip speed ratio according to Fig. 2. In particular, it can be interpolated by a seventh-order polynomial as follows:

(3) Fig. 2. Three-bladed wind turbine with its power coefficient. where are, respectively, the wind and electromagnetic torques, and being the total wind turbine inertia and viscous damping coefficient [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF].

C. PMSG Model

Two analytical models have been developed in order to characterize the electromagnetic behavior of the PMSG from its geometric features defined in Fig. 3. [START_REF] Baroudi | A review of power converter topologies for wind generators[END_REF] 1st Model: This model has been derived from [START_REF] Slemon | Modeling and design optimisation of permanent magnet synschronous motors[END_REF] and is presented in more detail in [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF]. It considers a 1-D distribution of the magnetic field in the airgap and the permanent magnet represented by a first-harmonic approximation. Because of its simplicity and its low computational cost, it is used in the IOD process to optimize the PSMG. 2) 2nd Model: This model has been developed in [START_REF] Zhu | Improved analytical model for predicting the magnetic field distribution in brushless permanent magnet machines[END_REF]- [START_REF] Dubas | Exact analytical model of the No-load flux density in the air-gap the permanent magnets and the rotor yoke for the surface mounted permanent magnet motors[END_REF].

It takes into account 2-D space harmonics of the magnetic field distribution in the airgap and in the permanent magnet. Because of its better accuracy, this model is taken as a reference to validate the results obtained with the previous simplified model. Note that the electrical circuit parameters of the PMSG, i.e., phase flux, stator inductances, and stator resistances, can be obtained from both models as shown in Appendix A.

D. Circuit Model of The Passive Wind Turbine System

In order to reduce the computing time of electrical values, a simplified dc equivalent model (called mixed-reduced model in [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF]) has been proposed in [START_REF] Abdelli | Optimization of a small passive wind turbine generator using multiobjective genetic algorithms[END_REF] and [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF] for representing the PMSG-Diode rectifier association (see Fig. 4). This model includes the armature reaction in the generator and the diode overlapping during the commutation interval. The correspondence between ac [root mean square (rms)] values and dc ones, in the PMSG circuit model is given in Appendix B.

III. MULTIOBJECTIVE OPTIMIZATION OF THE PASSIVE WIND TURBINE SYSTEM

In this section, the IOD of the PMSG is presented, converting the IOD problem into a multiobjective optimization problem.

A. Design Variables, Objectives, and Constraints

1) Design Objectives: Because of the "naturally small" efficiency of the passive wind turbine, the first objective deals with the maximization of the average useful power during the wind cycle defined in (1). This power consists of the wind power extracted from the wind turbine reduced from all losses in the system. These losses include mechanical losses in the turbine, conduction losses in the diode rectifier, iron losses , and Joule losses in the PMSG. The power balance in the system is illustrated in Fig. 5. For more details about the calculation of these losses, we invite the reader to refer to [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF] (5)

It should be noted that maximizing the useful power implies the maximization of the extracted wind power as well as the minimization of all losses in the system.

The second objective consists in the minimization of the total system mass which includes the wind turbine mass , the PMSG mass , and the radiator mass associated with the diode rectifier (6) The PMSG mass is evaluated from the mass of each components (iron, copper, magnet) and geometric features similar to [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF] and [START_REF] Sareni | Integrated optimal design of heterogeneous electrical energetic systems using multiobjective genetic algorithms[END_REF]. The radiator mass of the diode rectifier is calculated from the radiator thermal resistance and the conduction losses according to [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF].

2) Design Variables: Design variables and their associated bounds are mentioned in Table I. These variables should be optimized in the IOD process in accordance with the design objectives defined in the last subsection.

3) Design Constraints: Five constraints are introduced to ensure the system feasibility in relation to the parametric variation of design variables in the optimization process. These constraints concern the number of copper winding per slots (higher than one and limited by the minimal section of winding conductors), the maximum temperature associated with the copper windings in the PMSG, the demagnetization limit of the magnets, and the maximum temperature in the semiconductor junctions of the diode rectifier. All details about the calculation of these constraints and the computation of thermal values in the PMSG and in the diode rectifier can be found in [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF] and [START_REF] Sareni | Integrated optimal design of heterogeneous electrical energetic systems using multiobjective genetic algorithms[END_REF].

B. IOD Process Based on Multiobjective Optimization

The IOD process based on multiobjective optimization is illustrated in Fig. 6. The nondominated sorting genetic algorithm (NSGA-II) [START_REF] Deb | Multi objective Optimisation using evolutionary evolutionary algorithms[END_REF] with a self-adaptive recombination procedure of genetic operators [START_REF] Sareni | Recombination and self-adaptation in multi-objective genetic algorithms[END_REF] is used for optimizing both objectives in compliance with all constraints. These constraints are integrated in the algorithm by modifying the dominance rule according to [START_REF] Sareni | Integrated optimal design of heterogeneous electrical energetic systems using multiobjective genetic algorithms[END_REF]. Five independent runs are performed with the control parameters defined in Table II, in order to take into account the NSGA-II stochastic nature. Note that optimization is performed for the particular wind cycle defined in (1). However, it has been shown in [START_REF] Roboam | Optimization of a passive small wind turbine based on mixed Weibull-turbulence statistics of wind[END_REF] that results obtained with this IOD process are relatively insensitive to wind conditions.

C. Results

Fig. 7 shows the global Pareto-optimal front obtained by merging all fronts associated with the five independent runs. In this figure, we present an optimal solution ("selected solution for design") whose parameters are mentioned in Appendix C and compare it with an initial nonoptimized PMSG configuration [START_REF] Sambatra | Modeling and test of a PM synchronous generator based small stand alone wind energy converter[END_REF]. This optimal solution is chosen in the Pareto-front elbow in order to ensure similar "weightings" between useful power and mass objectives. We also plot in Fig. 8 the wind power extraction of the passive wind turbine system for both solutions. It can be seen that wind power extraction of the optimized solution for various wind speeds is clearly better, nearby the ideal wind power extraction.

In order to validate the design of the optimal solution extracted from the Pareto-optimal front and obtained with the less accurate sizing, we compare the electromagnetic behavior of the PMSG with both analytical models presented in Section III and with the finite element method. Fig. 9 shows a comparison of the flux density in the PMSG airgap, obtained with the FEMM software [START_REF] Meeker | Analysis and control of a three pole radial magnetic bearing[END_REF], [START_REF] Meeker | Finite Element Method Magnetics Sep. 22[END_REF] and with the second analytical model. The main electromagnetic parameters of the PMSG determined with each III. We also mention in Table III the values obtained from measurements on the actual PMSG prototype built from this study. It can be noted that there is a remarkable agreement between all data, and especially between the PMSG design and the corresponding prototype.

IV. EXPERIMENTAL VALIDATION OF THE PASSIVE WIND TURBINE SYSTEM

A test bench has been developed in order to experimentally validate the passive wind turbine system (see Fig. 10). It allows a "physical emulation" of the passive wind turbine system, the turbine being "simulated" by a servomotor which reproduces torque and inertia with respect to rotation and wind speeds ( curve). The servomotor is coupled through a torquemeter with the optimal PMSG prototype issued from the IOD approach. The PMSG is itself connected through a three-phase diode rectifier with battery of dc voltage V. The system is controlled by a digital signal processor (DSP) processor (DS1102 board) which converts sensor information (voltage, current, position, and torque). We investigate the behavior of the passive wind turbine system by comparing experimental measurements with corresponding simulations. For that purpose, a complete three-phase model of the PMSG-diode rectifier and battery association has been developed. This model allows us to characterize phase voltage and phase current in the PMSG and in the diode rectifier as well as the dc voltage and current in the battery. We first compare experiment and simulation electrical values at the input and output of the diode rectifier for a constant PMSG rotation speed close to the PMSG base speed, i.e., rad.s . Fig. 11 presents the PMSG phase current and dc current. For the same conditions, Fig. 12 gives the phase-to-phase voltage value in the PMSG. In order to validate the passive wind turbine system in wind variable operating, the wind cycle defined by ( 1) is imposed to the servomotor which emulates the wind turbine. Fig. 13 compares the electromagnetic power and the associated energy during the simulated wind cycle. The maximal power and energy corresponding to the case of an ideal wind extraction are also drawn to assess the wind power extraction quality of the passive wind turbine system. Note that differences are not important which confirms the system's "natural" efficiency when its parameters have been optimized in relation to the wind turbine. Finally, the average useful power during the simulated wind cycle is given in Table IV and compared with average extracted wind power to estimate the global efficiency of the wind turbine system. We also mention in Table IV the average power associated with an ideal wind extraction. It can be seen from measurements that the wind power extraction efficiency related to the particular wind cycle is about 90%. The global efficiency

TABLE IV AVERAGE POWERS IN THE PASSIVE WIND TURBINE SYSTEM DURING

A WIND CYCLE of the wind turbine system is close to 70%. Note also the very good agreement between simulated and experimental results for all investigated tests.

V. CONCLUSION

In this paper, the design of a full passive wind power system based on a 1500-W three-bladed wind turbine has been investigated. Such a "low-cost" structure without active control and with a minimum number of sensors can be efficient only if the system design parameters are mutually adapted through an IOD approach which aims at optimizing the wind power extraction, the total system losses, and the total system mass. Issued from this study, an optimal PMSG prototype has been constructed and implanted in an experimental test bench. This test bench allows a "physical emulation" of the passive wind turbine system, the turbine being "simulated" by a servomotor which reproduces torque and inertia with respect to the rotation and wind speeds. All experimental tests have shown a very good agreement between the system simulations and corresponding measurements. Finally, a good global efficiency of the passive wind turbine system, close to 70%, has been noted from experiments, for the particular wind cycle profile used in the IOD process. All the results obtained in this study confirm the interest of such a "low-cost" passive structure (no control and power electronic devices, a minimum of sensors) sized from an IOD approach. In the outlooks of this work, the sensitivity analysis of the passive wind turbine efficiency according to design variable variations will be investigated in more detail. Robust design methods will be applied in order to take into account design variable uncertainties and their impact on design objectives or constraints.

APPENDIX

A. Analytical Models of the PMSG

This Appendix illustrates the calculation of the main electromagnetic parameters of the PMSG for both analytical models investigated in the paper (see Table V).

B. DC Equivalent Model

A simplified equivalent dc model is used to represent the behavior of the PMSG-diode rectifier association in transient state [START_REF] Abdelli | Optimization of a small passive wind turbine generator using multiobjective genetic algorithms[END_REF], [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF]. With this model, electrical values are determined from a quasi-static approach using analytical relations. Only thermal and mechanical modes (i.e., slow modes) are simulated by ODE equations. Electrical modes (i.e., fast modes) are neglected. Consequently, the computation time devoted to the simulation of all physical values (thermal, mechanical, and electric values) during a wind cycle is considerably reduced. With this model, the CPU time (on a 2-GHz standard PC computer) required for evaluating one individual (including geometric sizing, wind simulation, constraint, and objective computation) in the NSGA-II is about 0.6 s. This leads to a global CPU time of 10 hours for each NSGA-II run using 100 individuals and 600 generations. More details about the analysis of the CPU time devoted to the electrical simulation of the passive wind turbine can be found in [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF].

The correspondence between synchronous generator circuit and the equivalent dc model is given in Table VI. The voltage can be calculated from Fig. 4 as detailed in [START_REF] Sareni | Model simplification and optimization of a passive wind turbine generator[END_REF] (7)

where models the diode overlapping during the commutation interval. From the above relation, we obtain the rectified current

C. Wind Turbine System Parameters

The parameters of the wind turbine and of the optimized PMSG are listed below.

Wind turbine Radius of wind turbine rotor: m; Equivalent inertia: kg m ;

  Generator (PMSG) active length. Main/Leakage/Stator inductance. Tip speed ratio. Magnet permeability. Number of conductors per slot. Number of slots per pole per phase. Base/Wind turbine rotation speed. Base/Useful power. Conduction/Iron/Joule/Mechanical loss. Number of pole pairs. Air density. Slot depth bore radius/Radius length ratio. Equivalent armature bore radius. Blade radius. DC/Stator/Overlap resistance. Diode resistance. Bore radius. Electromagnetic/Wind turbine torque. Base/dc/phase voltage. Wind speed. DC equivalent/phase flux. Opening angle of a slot. Magnet/Slot/Tooth width.

Fig. 1 .

 1 Fig. 1. Fully "Passive" wind turbine system.

Fig. 3 .

 3 Fig. 3. PMSG geometric features.

Fig. 4 .

 4 Fig. 4. Equivalent dc model of the PMSG rectifier association.

Fig. 5 .

 5 Fig. 5. Power balance in the passive wind turbine system.

Fig. 6 .

 6 Fig. 6. IOD process based on multobjective NSGA-II optimization.

Fig. 7 .

 7 Fig. 7. Pareto-optimal configurations of the PMSG obtained from the IOD.

Fig. 8 .

 8 Fig. 8. Extracted wind power of two solutions (initial and optimal).

Fig. 9 .

 9 Fig. 9. Flux density in the PMSG airgap.

Fig. 10 .

 10 Fig. 10. Passive wind turbine system bench.

Fig. 11 .

 11 Fig. 11. Phase current in the PMSG and dc current in the battery.

Fig. 12 .

 12 Fig. 12. Phase-to-phase voltage in the PMSG.

Fig. 13 .

 13 Fig. 13. Electromagnetic PMSG Power and energy during a wind cycle.

TABLE I DESIGN VARIABLES

 IVARIABLES

TABLE II NSGA

 II

-II CONTROL PARAMETERS

TABLE III ELECTROMAGNETIC

 III PARAMETERS OF THE OPTIMAL PMSG

ACKNOWLEDGMENT

The authors would like to thank the Novelté Système Society for the production of the PMSG prototype.

CNRS Full-Time Researcher. Since 1998, he has been the Head of the team "GENESYS" whose objective is to process design problems at the "system level." He develops design methodologies specifically oriented toward multifield systems design for applications such as electrical embedded systems and renewable energy systems.

Christophe Espanet was born in Belfort, France, in 1972. From 1991 to 1995, he studied at the "Ecole Normale Supérieure" in Cachan in Paris, France. He received the Ph.D. degree from the University of Franche-Comté (UFC) in 1999. His doctoral research dealt with the design and the optimization of PM high torque in-wheel motors.

From 1999 to 2007, he has been an Associate Professor at the Laboratory of Electrical Engineering and Systems (L2ES). He is now Full Professor at the UFC and Head of the Energy Conversion Systems Design Research Team in the FEMTO-ST Laboratory, Belfort, France. His research interests include the modeling and the design of electrical systems and more particularly PM synchronous machines.


Nathalie Saint-Geours

Frédéric Grelot

Jean-Stéphane Bailly

Christian Lavergne

J.-S Bailly

Ranking sources of uncertainty in flood damage modelling: a case study on the cost-benefit analysis of a flood mitigation project in the Orb Delta, France

Keywords: Average annual damages, flood damages, cost-benefit analysis, uncertainty, sensitivity, Sobol' indices

Introduction

Flooding is recognised as one of the most damaging natural hazards, responsible for approximately one-third of the total economic losses due to natural hazards in Europe (EEA et al., 2008). Following the approval of the EU flood directive (2007/60/EC) in 2007, EU member states now have to establish flood risk management plans focused on prevention, protection and preparedness in all flood-prone river basins and coastal areas. To evaluate these plans, flood risk managers are advised to use cost-benefit analysis (CBA) as part of their appraisal (European Commission, 2008). In France since 2011, using CBA is mandatory for local managers who claim national subsidies. Despite known limitations (European Commission, 2009), CBA is a useful tool that provides significant rational information to the decision makers. To consider the expected benefits related to flood management, CBA requires an accurate estimate of the amount of flood damage that will be reduced yearly by the appraised measures. We will use the term CBA-AD to refer to this implementation of CBA based on avoided damage. This estimate relies on a complex modelling chain, including hydrological, hydraulic and economic modelling as well as GIS-based spatial analysis [START_REF] Messner | Guidelines for socio-economic flood damage evaluation[END_REF]. Two output indicators are commonly produced in such studies: the reduction of the expected annual flood damage costs [START_REF] Arnell | Expected annual damages and uncertainties in flood frequency estimation[END_REF] and the net present value of the appraised measures [START_REF] Erdlenbruch | Une analyse coût-bénéfice spatialisée de la protection contre des inondations: application de la méthode des dommages évités à la basse vallée de l'Orb[END_REF]. These two indicators may also take into account the benefits and costs that are not directly related to flood management, such as environmental impacts or landscape modification, but this issue is not in the scope of the present paper.

Meanwhile, there is a growing consensus [START_REF] Apel | Flood risk assessment and associated uncertainty[END_REF] that flood damage assessments are fraught with uncertainties, which arise from inaccurate or missing data, model assumptions, measurement errors, incomplete knowledge, etc. -see [START_REF] Refsgaard | Uncertainty in the environmental modelling process: a framework and guidance[END_REF] and [START_REF] Walker | Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support[END_REF] for an enlightening discussion on the nature of uncertainty. Uncertainty analysis is thus required to identify and quantify the impacts of uncertainties in the modelling chain to i) increase the reliability of flood damage assessments and related CBA-AD [START_REF] Mostert | The European flood risk directive: challenges for research[END_REF] and ii) inform relevant stakeholders with the best information possible for decision making [START_REF] Ascough | Future research challenges for incorporation of uncertainty in environmental and ecological decision-making[END_REF]. Over the last few years, various methods of quantitative uncertainty analysis have been used in flood damage assessment research (Pappenberger and Beven, 2006;Pappenberger et al., 2006). Nevertheless, many authors first focused on the uncertainty in a single component of the flood damage assessment chain: hydraulic modelling (e.g., [START_REF] Bernardara | Uncertainty analysis in flood hazard assessment: hydrological and hydraulic calibration[END_REF][START_REF] Gouldby | Development and application of efficient methods for the forward propagation of epistemic uncertainty and sensitivity analysis within complex broad-scale flood risk system models[END_REF][START_REF] De Rocquigny | An applied framework for uncertainty treatment and key challenges in hydrological and hydraulic modeling[END_REF], inundation mapping [START_REF] Bales | Sources of uncertainty in flood inundation maps[END_REF]Wagner, 2009, Stephens et al., 2012), damage functions [START_REF] Kutschera | What are the uncertainties when calculating flood damage potentials?[END_REF][START_REF] Merz | Estimation uncertainty of direct monetary flood damage to buildings[END_REF][START_REF] Merz | Flood risk curves and uncertainty bounds[END_REF][START_REF] Merz | Assessment of economic flood damage[END_REF] or land use (Te [START_REF] Linde | Future flood risk estimates along the river Rhine[END_REF]. To go further, a number of recent studies investigated how combinations of these uncertainty sources interact and propagate through flood damage assessments.

They differ by the components under study (extreme value statistics, hydraulic model, potential dyke breach, inundation mapping, exposure assessment, damage functions) and by the uncertainty analysis method used. In some studies [START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF][START_REF] Merz | Flood risk curves and uncertainty bounds[END_REF][START_REF] De Moel | Effect of uncertainty in land use, damage models and inundation depth on flood damage estimates[END_REF], the various components of the modelling chain were varied manually in a 'one-factor-at-a-time' (OAT) approach (Saltelli et al., 2008) to estimate the confidence bounds around the flood damage estimates. Other authors described uncertainty sources in a probabilistic setting and explored the space of input uncertainty within a Monte Carlo framework (Helton and Davis, 2006), which requires a large number of model evaluations [START_REF] Apel | Quantification of uncertainties in flood risk assessments[END_REF][START_REF] De Kort | Decision making under uncertainty in a decision support system for the Red River[END_REF][START_REF] De Moel | Uncertainty and sensitivity analysis of coastal flood damage estimates in the west of the Netherlands[END_REF][START_REF] Weichel | Sensitivity and uncertainty in flood inundation modelling -concept of an analysis framework[END_REF].

Another related but distinct issue is to identify, in the flood damage assessment process, the main sources of uncertainty that contribute the most to the variability of damage estimates and CBA-AD outcomes, which is the role of sensitivity analysis methods (SA). These methods aim to study how the uncertainty of a model output can be apportioned to different sources of uncertainty in its inputs (Saltelli et al., 2008). SA is recognised as an essential component of model building (European Commission, 2009;US Environmental Protection Agency 2009) and is widely used in different fields [START_REF] Cariboni | The role of sensitivity analysis in ecological modelling[END_REF][START_REF] Tarantola | Can global sensitivity analysis steer the implementation of models for environmental assessments and decision-making?[END_REF]. Ranking uncertainty sources, usually by so-called 'sensitivity indices' or 'importance measures' is useful to orientate further research, collect additional data on most influential inputs but also simplify the model under study by fixing non-influential inputs. While many quantitative SA approaches are available, most studies in the field of flood damage assessment used a 'one-at-a-time' and qualitative SA approach, manually comparing the separate effects of each uncertainty source on the damage estimates. They used the large number of model evaluations produced from uncertainty analysis either in a probabilistic setting or using various versions of input data [START_REF] Apel | Flood risk assessment and associated uncertainty[END_REF][START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF][START_REF] De Moel | Effect of uncertainty in land use, damage models and inundation depth on flood damage estimates[END_REF][START_REF] Pappenberger | Multi-method global sensitivity analysis of flood inundation models[END_REF]). To our knowledge, only the work of de [START_REF] De Moel | Uncertainty and sensitivity analysis of coastal flood damage estimates in the west of the Netherlands[END_REF] was based on a quantitative global sensitivity analysis method (GSA), in which i) quantitative sensitivity indices are estimated for each uncertainty source and ii) all uncertain model inputs are varied at the same time, which allows the effect of their interactions on the overall output variability to be discussed.

Nevertheless, to date, only the uncertainty on the flood damage assessments have been studied without questioning how this uncertainty may impact the robustness of the CBA-AD of flood management policies nor how to improve this robustness. Our paper is an attempt in this direction. We try to answer the following questions: how does uncertainty propagate through the CBA-AD of a flood management policy? What is the ranking of the uncertainty sources in such a CBA-AD?

We discuss these questions through a case study of a CBA-AD applied to a flood risk management plan on the Orb River delta, France, where only structural flood-control measures are considered. Our goal is to check the robustness of the CBA-AD results and assess the contribution of uncertainty sources to the overall output variability. A modelling chain named NOE (Section 2) is used to estimate the potential flood damages at the scale of individual assets and perform a cost-benefit analysis of the flood control measures. Uncertainty sources are then described in a probabilistic framework, propagated through the NOE modelling chain with pseudo Monte Carlo simulations, and variance-based sensitivity indices are computed for each of them (Section 3). Only epistemic uncertainty (Refsgaard et al., 2009) is considered here because aleatory uncertainty is already accounted for in the definition of output CBA-AD indicators (average annual flood damages). The description of uncertainty sources is based on the literature, expert opinion or measurements using univariate or bivariate probability density functions or more complex models for spatially distributed uncertainty. The results (Section 4) include confidence bounds and empirical pdf of CBA-AD outputs and a ranking of the uncertainty sources based on their sensitivity indices. We discuss the outcomes of our approach and its limits in Section 5.

Study site

As a study area, we selected the Lower Orb River fluvial plain, known as the Orb delta, located in the south of France. We focused on a 15 km reach from Béziers to the Mediterranean sea that is bounded by an area of 63 sq. km and includes the Draft Submitted to J Flood Risk Management 06/12/12 cities of Béziers, Portiragnes, Sauvian, Sérignan, Valras-Plage and Villeneuve-lès-Béziers (Figure 1). The Orb catchment has a typical Mediterranean sub-humid regime. The annual maximum discharge in Béziers (Tabarka gauge) varies from year to year between 100 and 1 500 m 3 /s (BCEOM and SMVO, 2000). The flood prone area in the Orb delta is home to approximately 16 290 permanent people (total population of the six localities: 90 000 people), 774 companies and 30 seaside campgrounds (which attract up to 100 000 tourists in summertime). Approximately one-third of the area is devoted to agriculture. The flood of December 1995 -January 1996, with a peak discharge of 1 700 m 3 /s at the Tabarka gauge, caused a total amount of damage of 53 M€ [START_REF] Smvol | Programme d'actions de prévention des inondations sur les bassins de l'Orb et du Libron pour les années 2011 à[END_REF]. In 2001, local authorities launched a flood risk management project, mainly based on various structural mitigation measures, including dyke strengthening around urban areas, restoration of sea outfalls and channel improvement. In 2011, to claim national subsidies, they completed a cost-benefit analysis of their project [START_REF] Grelot | Programme d'actions de prévention des inondations sur les bassins versants de l'Orb et du Libron[END_REF].

This study site was mainly chosen because it was a 'real' case-study, with a flood risk management plan under construction and a cost-benefit analysis produced by the local authorities. Moreover, the area was already documented with numerous available data. These data included aerial photographs, a high-resolution Digital Terrain Model (DTM) built from photogrammetry, the annual maximum flow series from 1967 to 2009 at the Tabarka gauge, and various spatial datasets on buildings, agricultural land and economic activities in the area [START_REF] Erdlenbruch | Une analyse coût-bénéfice spatialisée de la protection contre des inondations: application de la méthode des dommages évités à la basse vallée de l'Orb[END_REF].

Description of the NOE modelling chain

Cost-benefit analysis based on avoided damages (CBA-AD) was used to evaluate the flood risk management project launched on the Orb River. In the literature, flood damage assessments and related CBA-AD vary in their scope and scale Draft Submitted to J Flood Risk Management 06/12/12 as well as the data used and their outputs. Here, a complex modelling chain named NOE [START_REF] Erdlenbruch | Une analyse coût-bénéfice spatialisée de la protection contre des inondations: application de la méthode des dommages évités à la basse vallée de l'Orb[END_REF] combines hydrological, hydraulic, GIS and economic modelling to estimate the flood damages on individual assets and compute two output indicators: i) the Average Annual Avoided Damage (∆AAD [€/year]) over the study area, which is defined as the amount of annual expected damage costs that are reduced due to the flood mitigation measures; and ii) the Net Present Value (NPV [€]) of the flood mitigation measures. The modelling chain consists of seven steps that are further described in this section (Figure 2).

Flood scenarios

The calculation of the Average Annual Avoided Damage requires damage estimation for a number of relevant flood scenarios with different characteristics to represent the aleatory uncertainty associated with flood hazard in the study area.

The first step of the NOE modelling chain is thus to choose a range of potential flood events of various magnitudes. Six flood scenarios e 1 to e 6 were selected, characterised by a maximum discharge q i at Tabarka gauge (Table 1). e 1 is supposed to be the smallest flood event that induces damage (q 1 = 1 000 m 3 /s). Scenarios e 2 to e 5 include historical floods and design floods. Scenario e 6 is an extreme flood, which would result in an over-topping of all existing flood-control dykes.

Flood frequency analysis

The return intervals T 1 to T 6 and exceedance frequencies f 1 to f 6 associated with flood scenarios e 1 to e 6 were deduced (Table 1) from the discharge-frequency Gumbel curve (Q-f), which was fitted on the annual maximum flow series at the Tabarka gauging station available from 1967 to 2009 (AMFS 1967(AMFS -2009)).

Flood hazard modelling

For each flood scenario, a typical flood hydrograph was first generated based on expert opinion (BCEOM and SMVO, 2000). The 1D, step-backwater hydraulic ISIS Flow model (unsteady flow) was then used to propagate the hydrographs within the floodplain. The ISIS Flow model solves the full 2D, depth-averaged momentum and continuity equations for free-surface flow (ISIS, 2012). Two different floodplain flow simulations were produced for each flood scenario e 2 to e 6 : one describing the present situation and the other describing future situation with enforced mitigation measures. The floodplain flow simulations were then combined with a high-resolution DTM to produce two sets of raster surfaces with a 5 m cell size, giving spatially explicit maximum water depths for each flood event over the study area: these hazard maps are denoted by H(e 2 ) to H(e 6 ) (present situation) and H(e 2 ') to H(e 6 ') (future situation).

Flood exposure modelling

Four economic sectors were considered in the exposure analysis: private housing, agricultural land, campgrounds and other economic activities. Flood exposure was assessed at the scale of small individual assets (buildings, plots of cultivated land, etc.). Data from various sources were collected to build a land use geo-database (LU-GDB) over the study area, including digital cadastral maps, a dataset of the regional Chamber of Commerce and Industry ( 2009), and the national agricultural land use statistics (RPG dataset, 2009). An extensive field survey was also conducted to collect additional data on assets, such as ground floor elevation of buildings. In the end, the LU-GDB dataset describes private housing units (individual buildings), plots of cultivated land, campgrounds and other economic activities by individual polygonal features in a single GIS vector layer (Table 2). Plots of cultivated land were further characterised by a subtype (wheat, vineyard, etc.), while economic activities were classified into sixty categories following the French classification of economic activities NAF2008 [START_REF] Insee | French classification of activities -NAF Rev. 2[END_REF].

The flood exposure of assets was then assessed by confronting the LU-GDB dataset with water depth maps H(e 2 ) to H(e 6 ) and H(e 2 ') to H(e 6 '). For each exposed object (represented by a polygonal feature in LU-GDB dataset) and each inundation map, the average water depth over the object was extracted as an attribute column by a simple overlay analysis.

To compute meaningful average water depths for very large objects (e.g., large plots of cultivated land), we first divided all features into pieces of 40 000 sq. m max by intersecting features of the LU-GDB dataset with a regular square grid of 200 m cell size.

Damage estimation

The following module of the NOE modelling chain estimates the total damage costs (D) within the study area for each flood scenario e 1 to e 6 , for the present (D(e 1 ) to D(e 6 )) and future (D(e 1 ') to D(e 6 ')) situations. We will denote by ∆D 1 to ∆D 6 the damage reduction brought by the mitigation measures for each flood scenario: ∆D i = D(e i ) -D(e i '). As scenario e 1 was defined as the "flood event where damage to property begins", the damage estimates D(e 1 ) and D(e 1 ') are both assumed to be equal to zero. For scenarios e 2 to e 6 , as a coarse estimation, only direct and tangible monetary losses were considered- [START_REF] Merz | Assessment of economic flood damage[END_REF] present the other types of damages that should be estimated for a more complete analysis.

Damage functions were used (Table 3), which depend mainly on the following parameters: type and floor surface area of the exposed object, average water depth, and season of occurrence (campgrounds and agriculture). Flood velocity and flood duration were considered to be homogeneous. These damage functions were taken from the recommendations of the French State [START_REF] Meddtl | Programmes d'action de prévention des inondations (PAPI). De la stratégie aux programmes d'action[END_REF]. For a complete description, see the original study [START_REF] Grelot | Programme d'actions de prévention des inondations sur les bassins versants de l'Orb et du Libron[END_REF]. In the end, a total of 94 depth-damage relationships were used, one for each land use type and subtype.

Average Annual Avoided Damages

The average annual damage cost from flooding (AAD [€/year]) is a common performance indicator used to measure potential flood damages over a given territory [START_REF] Arnell | Expected annual damages and uncertainties in flood frequency estimation[END_REF][START_REF] Messner | Guidelines for socio-economic flood damage evaluation[END_REF]. It is equal to the area under the damage-frequency curve, which is the graph of damage D against exceedance frequency f = 1/T:

∫ 1 0 ) ( df f D = AAD (1)
To assess the benefits of the flood risk management project launched on the Orb River in 2001, we computed the potential reduction of the average annual damage costs brought by the mitigation measures, i.e., the variation ∆AAD = AAD -AAD' from the present to the future situation. This Average Annual Avoided Damage (∆AAD [€/year]) is also equal to:

∫ ∆ ∆ 1 0 ) ( df f D = AAD (2)
It can be computed from the range of flood scenarios e 1 to e 6 and corresponding avoided damages ∆D 1 to ∆D 6 by estimating the integral (Eqn 2) with a simple trapezoidal rule (Figure 3).

Net Present Value of the mitigation measures

The last step of the NOE modelling chain is the cost-benefit analysis, which evaluates the efficiency of the flood mitigation measures by comparing their costs with their expected benefits. The costs of the mitigation measures include the initial investment (CI = 35.2 M€) and maintenance costs (CM = 1.6 M€/year). The benefits of the project are measured by the ∆AAD indicator. The Net Present Value (NPV [€]) of the flood mitigation measures is then calculated by comparing the discounted costs and benefits over a time period of R = 30 years (Eqn 2).

( )

∑ = × - ∆ + - = R i i CM AAD CI NPV 0 τ (3)
where τ i is the discount coefficient for year i. A positive NPV value means that the benefits generated by the flood risk management project outweigh its costs. The larger the NPV value, the more efficient the flood mitigation measures are.

Uncertainty and sensitivity analysis

Uncertainty and sensitivity analyses of the NOE modelling chain (Figure 4) were performed using variance-based global sensitivity analysis approach (Saltelli et al., 2008). In the first step of the analysis, sources of uncertainty in the NOE modelling chain were identified and modelled in a probabilistic framework, and a set of random realisations was sampled for each uncertain modelled input. Next, pseudo-Monte Carlo simulations were used to explore the space of input

Modelling sources of uncertainty

Table 4 lists the epistemic uncertainty sources that we took into account in the uncertainty and sensitivity analyses of the NOE modelling chain. Each source of uncertainty was modelled in a probabilistic framework using measurement or expert opinion.

Flood frequency analysis

Uncertainty in the flood frequency analysis may arise from i) stream gauge measurement errors (Neppel et al., 2009); ii) possible non-stationarity of the series due to climate change [START_REF] Khaliq | Frequency analysis of a sequence of dependent and/or nonstationary hydro-meteorological observations: a review[END_REF] and iii) uncertain fitting of a dischargefrequency (Q-f) relationship to the AMFS 1967-2009 dataset [START_REF] Countryman | Flood frequency confidence bounds -Art, science or guess[END_REF]. Here, only the latter uncertainty was modelled and simulated. After a log transformation leading to the usual linear regression context, standard joint distributions [START_REF] Maidment | Handbook of Hydrology[END_REF] for the parameters of the fitted Gumbel curve were calculated (Figure 5). A set of n 1 = 10 3 Gumbel curves was then randomly sampled from the parameter joint distribution. From this set of curves, 10 3 exceedance frequencies f i and according return intervals T i = 1/f i (Figure 6) were generated for each discharge value q i (i = 1 to 6).

Flood hazard modelling

Another major source of uncertainty in the NOE modelling chain is the inundation mapping process, which includes hydraulic modelling and combination with a high-resolution DTM, to derive water depth maps H(e 2 ) to H(e 6 ) and H(e 2 ') to H(e 6 '). For the sake of simplicity, a restrictive choice was assumed in considering the error on the high-resolution Digital Terrain Model as the single uncertainty source in water depth maps. Including more detailed descriptions of the hydraulic uncertainties in this study was impossible as the ISIS hydraulic model used for initial flow simulations was not available to us. This choice may be partly justified by the findings of both [START_REF] Bales | Sources of uncertainty in flood inundation maps[END_REF] and [START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF], who investigated the various sources of error encountered in this process and conclude that high-resolution topographic data is the most important factor required for accurate inundation maps. The DTM-a raster surface of 5 m cell size-was initially built by stereo-photogrammetry. Both measurement errors and interpolation errors affect the quality of this input data [START_REF] Wechsler | Uncertainties associated with digital elevation models for hydrologic applications: a review[END_REF]. These errors were modelled by a Gaussian noise without spatial correlation, whose characteristics were determined from a set of 500 control field points (mean = 0 cm, s.d. = 17 cm). A set of n 2 = 100 random realisations of the Gaussian random error field was generated and added as « noise » to the initial water depth maps H(e 2 ) to H(e 6 ) and H(e 2 ') to H(e 6 '). We may note that this procedure induces independent variations of the water levels for each exposed asset; it differs from the study of de Moel and Aerts (2011), who described uncertainty in the water levels with a spatially uniform bias.

Flood exposure modelling

The third source of uncertainty is the location and attribute data errors in the LU-GDB dataset [START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF].

The error in the land use GIS layer may stem from the following: i) misclassification of polygonal features representing assets; ii) error on the ground floor elevation of buildings; and iii) error on the surface area of features. Other sources of uncertainty were identified: geometric errors [START_REF] Bonin | Sensitivity analysis and uncertainty analysis for vector geographical applications 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences[END_REF]Girres, 2010), errors in the asset locations, and the evolution of land use over time [START_REF] Linde | Future flood risk estimates along the river Rhine[END_REF]. Although in some studies (de Moel and Aerts, 2011) the uncertainty of the land use data was represented by using a small number of different datasets, here each uncertainty source was modelled in a probabilistic setting. To describe the misclassification of polygonal features, which may arise in the process of photo-interpretation, a confusion matrix [START_REF] Fisher | Modelling soil map-unit inclusions by Monte Carlo simulation[END_REF] was built based on expert opinion, giving a confusion probability p a,b for each pair of land use types (a,b) (Table 5). Then, to model the variability of the ground floor elevation of buildings, measurements were taken during a field survey on a sample of 100 buildings. The study area was divided into five homogeneous zones; in each zone, the distribution of ground floor elevation was described by an empirical histogram (Figure 7). Random ground floor elevation was also attributed to campgrounds, plots of agricultural land and other economic activities (Table 4). Next, the surface area of the buildings was also randomised, as the features area extracted from cadastral maps differ from the effective surface area of buildings that should be taken into account for flood damage estimation (e.g., wall width should be subtracted). To cope with this issue, the nominal surface of each building was multiplied by a corrective random coefficient drawn independently in a uniform pdf in [0.75; 0.85], considering a digitalising error of 0.3 mm at the map scale [START_REF] Hengl | Finding the right pixel size[END_REF]. Finally, from this probabilistic description of the uncertainty in the LU-GDB dataset (confusion matrix, empirical distribution of ground floor elevations, corrective coefficient for surface areas), a set of n 3 = 1 000 random LU-GDB datasets was sampled.

Damage estimation

The fourth uncertain model input is the set of 94 depth-damage curves (one for each land use type and subtype) used for the damage estimation. Uncertainty about the damage functions has been extensively discussed in previous studies [START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF][START_REF] Kutschera | What are the uncertainties when calculating flood damage potentials?[END_REF][START_REF] Merz | Estimation uncertainty of direct monetary flood damage to buildings[END_REF][START_REF] Merz | Assessment of economic flood damage[END_REF][START_REF] Merz | Flood risk curves and uncertainty bounds[END_REF][START_REF] De Moel | Effect of uncertainty in land use, damage models and inundation depth on flood damage estimates[END_REF]. In these papers, uncertainty was mainly represented by using two or three different sets of damage functions coming from various studies. Only de Moel and Aerts (2011) used a parametric uncertainty model (beta pdf) derived from [START_REF] Egorova | Uncertainty in flood damage estimation[END_REF]. Here, to treat all uncertainty sources in a probabilistic framework, we made the choice to use a single set of depth-damage curves and represent their uncertainty by a uniform pdf, defining a -50% to +50% uncertainty range around nominal curves (Figure 8). Depth-damage curves associated with each land use type and subtype were assumed to vary independently-contrary to de Moel and Aerts (2011), where they were sampled collectively from a single p-value. A total of n 4 = 1 000 random sets of depth-damage relations was sampled this way.

Project costs

Finally, the last source of uncertainty in the NOE modelling chain is related to the costs of the flood risk management project. Based on expert opinion, investment costs CI and maintenance costs CM were assumed to follow a triangular pdf with the parameters shown in Table 4.

Propagating uncertainty

Once uncertainty had been modelled for each source of uncertainty, it was propagated through the NOE modelling chain using Monte Carlo simulation and a specific sampling scheme following [START_REF] Lilburne | Sensitivity analysis of spatial models[END_REF]. It uses two independent quasi-random LP-τ matrices A and B [START_REF] Sobol | Distribution of points in a cube and approximate evaluation of integrals[END_REF], here of length N base =4 096-this sampling size was chosen to fill the necessary conditions for the LP-τ samples (N base must be a power of 2) and large enough to obtain a satisfactory level of accuracy for the sensitivity indices estimates. These two matrices were combined through several permutations to explore the uncertainty domain of the five model inputs considered, respectively: exceedance frequencies; inundation maps; LU-GDB dataset; depth-damage curves; and project costs. The i th line of sample A or B is a set (l 1 (i) , l 2 (i) , l 3 (i) , l 4 (i) ,l 5 (i) ) where each l i (i) is a random integer label sampled from {1, ..., n j } associated with a single random realisation of the j th model input (from the set of n j random realisations that was previously generated). One can note that the number n j of random realisations is not the same for each model input: these numbers were chosen under the constraints of CPU time and storage space. Next, the NOE modelling chain was run for each line of samples A, B and a number of combinations of A and B-more details on the procedure can be found in [START_REF] Lilburne | Sensitivity analysis of spatial models[END_REF]. The total number of model runs was N tot = 28 672 (this number depends on the initial sample size N base and the number of uncertain model inputs considered) for a total CPU time of 24 hours on a 6-nodes cluster computer.

Variance-based sensitivity indices

Uncertainty propagation results in a set of N tot = 28 672 values for the following outputs of interest:

avoided flood damages per scenario (∆D 1 to ∆D 6 )

-Average Annual Avoided Damages (∆AAD)

-Average Annual Avoided Damages per type of assets

-Net Present Value of mitigation measures (NPV)
Then, the variance-based total-order sensitivity indices of each source of uncertainty with respect to each output of interest were estimated using the expressions given by [START_REF] Lilburne | Sensitivity analysis of spatial models[END_REF]. These sensitivity indices, denoted by ST i , measure the contribution of a given source of uncertainty, denoted by U i , and all its interactions with other sources of uncertainty, denoted by U ~i, to the variance of a given model output, denoted by Y:

[ ] ( ) ( ) Y Var U Y Var E = ST i i ~ (3) Draft Submitted to J Flood Risk Management 06/12/12
ST i is the expected part of output variance Var(Y) that would remain if all sources of uncertainty but X i were fixed. Please refer to Saltelli et al. (2008) for more details on global sensitivity analysis and the estimation of sensitivity indices.

Results

Uncertainty analysis

Table 6 summarises the outcome of the uncertainty analysis. For each output of interest, it gives descriptive statistics over the N tot = 28 672 model runs. It displays mean values of avoided damages per flood scenario, ranging from 9.593 M€ to 111.5 M€. The largest avoided damage is reached for scenario e 5 (100-year design flood), with a total reduction ∆D 5 = 111.5 M€, whereas the mitigation measures performed worst for the extreme flood scenario e 6 , with a mean avoided damage ∆D 6 = 9.593 M€ and a negative minimum value of -4.695 M€, meaning that the damage costs might increase from the present to the future situation for this scenario. The Average Annual Avoided Damage indicator shows a mean value of ∆AAD = 5.459 M€/year. Table 6 clearly suggests that the contribution of the four types of assets to this total indicator is uneven: while the economic activities and private housing account respectively for 64% and 34% of the total ∆AAD, the share of campgrounds and cultivated land is only equal to 1.3% and 0.7%, respectively. Finally the effect of the flood mitigation measures appears to be heavily dependent on the type of assets considered. Despite all input uncertainties, the ∆AAD indicator on private housing and economic activities (other than agriculture and campgrounds) prove to be always positive in this uncertainty analysis. In contrast, the mitigation measures will most likely result in an increase in the average annual damage for agricultural land as ∆AAD is negative in this sector for all model runs. Regarding campgrounds, no conclusion can be drawn from the study as ∆AAD is positive in this sector for only 72.6% of the model runs. It can also be noted that all flood damage indicators display a coefficient of variation ranging from 11.76% to 44.80%. Finally, Figure 9 shows the empirical distribution of the Net Present Value of the flood mitigation measures over N tot = 28 672 simulations. With a mean value of +34.29 M€, the flood risk management project seems to be a sound investment.

The NPV indicator also appears to be positive for 96% of model runs, which we interpret to mean that despite all the uncertainty sources that were considered in the NOE modelling chain, the benefits of the flood mitigation measures still prove to almost certainly outweigh their costs.

Sensitivity analysis

The total order variance-based sensitivity indices were computed for each uncertain model input with respect to each output of interest (Table 7). First, the variance of the total ∆AAD indicator can be observed to be almost equally explained by the uncertainty in the exceedance frequencies, water depth maps, depth-damage curves and LU-GDB dataset, with sensitivity indices ranging from 0.18 to 0.33. No main source of uncertainty can be identified, meaning that they are in a sense "well-balanced".

However, sensitivity indices with respect to the partial ∆AAD indicator for each economic sector give a very different picture. The variance of ∆AAD on private housing appears to be mainly explained by the uncertainty of the depth-damage curves (sensitivity index: 0.78). For campgrounds and agricultural land, the depth-damage curves also prove to be the most important source of uncertainty (sensitivity index: 0.6 and 0.4, respectively), followed by the uncertainty of the LU-GDB dataset (sensitivity index: 0.38 for both sectors).

Figure 10

In addition, for private housing, agricultural land and campgrounds, the uncertainty in the water depth maps is almost noninfluential (sensitivity index < 0.02), while it is the second most important source of uncertainty for other economic activities (sensitivity index: 0.38). Finally, Table 7 also indicates that the variance of the Net Present Value of flood mitigation project is mainly due to the uncertainty on its benefits (measured by total ∆AAD) rather than the uncertainty on its costs, which contribute to only 12% of the NPV variance.

To further identify the factors that explain the variance of ∆AAD indicator, sensitivity analysis can also be based on a graphical analysis. Figure 10 displays the sampled flood exceedance frequencies f 1 to f 6 against the estimated avoided damages ∆D 1 to ∆D 6 for all N tot = 28 672 model runs, along with the curves associated with the nominal, minimum and maximum values of the ∆AAD indicator, which is equal to the area under the curve. It may be noted that as the damage and frequency estimates of each flood scenario are correlated, the extremum values of the ∆AAD indicator do not always correspond to the extremum points in the damage-frequency graph for all flood scenarios. Figure 9 supports the conclusion that the variance of the ∆AAD indicator is mainly due to the uncertain position of scenarios e 1 (first flood event where the damage to property begins) and e 2 (10-year design flood) on this damage-frequency graph. Flood scenarios e 3 to e 5 show a larger dispersion of estimated avoided damages, but their position on the x-axis (exceedance frequency) is less spread; hence their contribution to the total variance of the ∆AAD indicator is small.

Discussion

Assessing robustness of a flood risk CBA study Our first goal was to assess the robustness of the cost-benefit analysis of the structural flood mitigation measures on the Orb River through an uncertainty analysis of the NOE modelling chain. Our approach was strongly motivated by prior publications in which uncertainty analysis was used to evaluate the robustness of flood damage assessments. We completed these works by propagating uncertainty up to the cost-benefit analysis of the flood mitigation measures, which had not been performed before. We obtained empirical descriptive statistics for the two CBA-AD outcomes, the Average Annual Avoided Damages and the Net Present Value of flood mitigation measures, displaying quite large coefficients of variation of 20.33% and 61.27%, respectively. The visualisation of the Monte Carlo simulations on a damage-frequency graph (Figure 10) gave us a new insight on the robustness of the ∆AAD indicator, proving that its variance is mainly explained by the uncertain characterisation (in terms of exceedance frequency and estimated damages) of flood scenarios with small return intervals (5 and 10-year floods). We also found (Figure 9) that the probability of the project costs outweighing the project benefits appears to be lower than 5%. We are convinced that these results may prove useful to provide water managers and stakeholders with a more complete picture on the cost-benefit analysis and the associated uncertainty, even though we know that they are usually untrained in coping with the uncertainty related to scientific information in flood risk studies [START_REF] Morss | Flood risk, uncertainty, and scientific information for decision making[END_REF].

Improving the NOE modelling chain

Our research also sought to identify the main sources of uncertainty in the NOE modelling chain to find ways to improve it. Although variance-based sensitivity analysis is widely used for that purpose in many disciplines, it had never been applied, to our knowledge, to a CBA-AD of flood control measures. We demonstrated the use of Sobol' sensitivity indices (Table 7) to rank the sources of uncertainty, depending on their contribution to the variance of the ∆AAD and NPV indicators. A first conclusion is that, on a global scale, sources of uncertainty are well-balanced, meaning that they all explain a significant share of the variance of the ∆AAD and NPV indicators. Yet, more useful lessons can be learnt from looking at each economic sector separately. The uncertainty on the depth-damage curves proved to be the key factor that explains the variance of the average annual damages on private housing, campgrounds and agricultural land. This result is in line with those of [START_REF] Apel | Quantification of uncertainties in flood risk assessments[END_REF] and de Moel and Aerts (2011), who found that in damage assessment of a single flood event, the choice of damage functions is a much more important factor than the choice of a hazard model. It thus supports the conclusion that improving the depth-damage curves and more generally the damage functions are priorities to make more robust flood damage assessments in these sectors. Our results also indicate that there is almost a third of the variance of the ∆AAD and NPV indicators that cannot be reduced as it stems from the variance of the flood return interval estimates. This observation corroborates the conclusions of [START_REF] Apel | Flood risk assessment and associated uncertainty[END_REF], who stated that reliable extreme value statistics were crucially important for reducing the uncertainty of the risk assessment. Unfortunately, reducing this input uncertainty would require longer time series of maximum discharges at Tabarka gauging station, which are not available.

Finally, the uncertainty on the floor elevation of buildings proved to have a negligible contribution to the variability of the annual damage estimates for private housing. This finding fits well with the results of [START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF], who showed that adding a single elevation value per building was inadequate to obtain more accurate damage estimates. Of course, these findings are specific to the Orb River study site: in a different case-study, ranking of the various uncertainty sources may be significantly different.

Averaging-out effects

The sensitivity analysis also provided an interesting insight on how the uncertainty on inundation maps influence the variance of damage estimation. Our results offer evidence that improving water depth estimation would be of almost no use in reducing variance of ∆AAD estimates for campgrounds, agricultural land and private housing, while it is the second most important source of uncertainty for other economic activities. This finding is in apparent conflict with the conclusions of [START_REF] Apel | Flood risk assessment and associated uncertainty[END_REF][START_REF] Apel | Quantification of uncertainties in flood risk assessments[END_REF] or de Moel and Aerts (2011), who reported that uncertainty in the water depths is less important than other uncertainty sources without distinction of the economic sector considered. This discrepancy may be explained by two different "averaging-out effects": one based on the surface area of assets and the other based on the number of assets. On one hand, campgrounds and agricultural land have a large surface area compared with other types of assets (33 000 and 9 000 sq. m. in average, Table 2): as a result, the error on water depths, if unbiased, is reduced when it is averaged over the large surface area of these assets. Hence, for both sectors, the contribution of the water depth maps to the variance of the ∆AAD indicator is low. On the other hand, the polygonal features classified as "private housing" or "economic activities" have a rather small surface area (83 sq. m. and 904 sq. m. on average, respectively), and the uncertainty in the average water depth for each individual asset is thus large. Nevertheless, the number of features classified as "private housing" (16 436) is much larger than the number of assets classified as "other economic activities" (691), which results in a "number averaging-out effect": the dispersion of water depth errors is averaged over the large number of housing polygons scattered across the study area. A similar "number averaging-out effect" may partly explain why the uncertainty on depth-damage curves appears to be more influential on the private housing sector, which is described with only one depth-damage curve, than for the other economic activities, which are described by 60 damage curves that are assumed to vary independently. These findings support the conclusion that various averaging-out effects (related not only to the surface area of assets and the number of assets but also to the number of land use types considered, the number of damage functions used, etc.) control the ranking of the uncertainty sources in the NOE modelling chain. [START_REF] Saint-Geours | Change of support in variance-based sensitivity analysis[END_REF] discussed this issue from a theoretical point of view and showed that the ranking of the uncertainty sources is closely related to the spatial support (and thus to the scale) of the model output. This result is in agreement with the call of [START_REF] Koivumäki | Uncertainties in flood risk mapping: a case study on estimating building damages for a river flood in Finland[END_REF] for further research on what constitutes a reasonable 'scale-accuracy relationship' in flood damage assessments: our results suggest that the 'scale-accuracy-sensitivity' relationship must be further investigated.

Limits

It should be noted that our work is based on hypotheses that may limit the strength of some of its results. First, some sources of uncertainty were identified in the NOE modelling chain but not taken into account in the uncertainty and sensitivity analysis: the evolution of land use over the next thirty years, errors arising from uncertainty on friction coefficients in hydraulic modelling, errors on the location and shape of polygonal features in the LU-GDB dataset, etc.

Why were they ignored? Because the data required to rigorously characterise them in a probabilistic framework were not available. Even if we can assume that some of these uncertainty sources would prove to be negligible in the NOE modelling chain, (e.g., errors on the location of assets in the LU-GDB dataset), others are definitely not (e.g., biased errors in hydraulic modelling).

Moreover, even for those sources of uncertainty that were included in the study, the models of uncertainty are sometimes only based on expert opinion. The results of our study heavily depend on such uncertainty parameters. To cope with this issue, a second-level uncertainty analysis could be performed by exploring how the output uncertainty and the sensitivity indices change for given sets of uncertainty parameters.

Conclusion

This work was performed with a view toward promoting the use of Monte Carlo uncertainty analysis and variance-based sensitivity analysis in flood damage assessments and related CBA-AD through a case-study on the Orb Delta, France. For this case-study, we derive the following main conclusions:

(1) Monte Carlo uncertainty analysis allows empirical pdf and confidence bounds on the economic indicators of a cost-benefit analysis applied to flood mitigation measures to be computed.

(2) The variance of the Average Annual Avoided Damages is mainly due to the uncertain characterisation of flood scenarios with small return intervals.

(3) Approximately one-third of the variance of the ∆AAD and NPV indicators cannot be reduced as it stems from a flood frequency analysis based on short time series.

(4) The ranking of uncertainty sources depend on the economic sector considered (private housing, agricultural land, economic activities)

(5) Uncertainty in the depth-damage curves is a prominent factor for computing the ∆AAD for private housing and agricultural land.

(6) The ranking of uncertainty sources is influenced by various averaging-out effects that depend on the number and surface area of the assets considered, the number of land use types, the number of damage functions, etc.

Further research is now needed to extend the reach of our study by trying to reduce the uncertainty in the input data identified as being influential in the study and including in the analysis uncertainty sources that were ignored so far.

Captions of figures

Figure 1

 1 Figure1

Figure 2

 2 Figure2

Figure 3

 3 Figure3

  uncertainty and assess the resulting variance of model outputs (∆AAD and NPV indicators). Finally, variance-based sensitivity indices were computed to rank the sources of uncertainty, depending on their contribution to the variance of output indicators.

Figure 4 Draft

 4 Figure 4

  Figure 5

  Figure 7

  Figure 9

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10

 12345678910 Figure 1 The study site is located in Hérault département, south of France. The Orb River flows southward. Source: www.geoportail.fr Figure 2 Flowchart of the NOE modelling chain Figure 3 Computation of Average Annual Avoided Damage Figure 4 Flowchart for uncertainty and sensitivity analysis Figure 5 Annual maximum flow series AMFS 1967-2009 and fitted Gumbel distribution (red solid line) with 95% confidence bounds (dotted blue lines) Figure 6 Empirical distributions of return intervals T for flood scenarios e1 to e6

Table 6

 6 Descriptive statistics for each output of interest over N tot = 28672 simulations

	Output	mean	s.d.	2.5% perc.	97.5% perc.	c.var.
	Avoided damage per flood scenario (∆D [M€])
	∆D1	0	0	0	0	-
	∆D2	20.21	7.796	11.164	40.6443	38.57%
	∆D3	48.84	8.972	29.307	65.927	18.37%
	∆D4	58.28	10.63	39.491	84.334	18.24%
	∆D5	111.5	13.11	84.410	136.405	11.76%
	∆D6	9.593	4.298	1.967	20.149	44.80%
	Average Annual Avoided Damage (∆AAD [M€/year])
	Total	5.459	1.11	3.6109	7.9816	20.33%
	Eco. activities	3.506	0.9406	2.0995	5.77331	26.83%
	Private housing	1.887	0.4465	1.1002	2.8317	23.66%
	Camp grounds	0.071	0.129	-0.213	0.3116	18.23%
	Agricultural land	-5.274.10 -3 1.912.10 -3 -9.25.10 -3 -1.965.10 -3 36.25%
	Net Present Value (NPV [M€])
	NPV	34.29	21.01	-40.83	106.5	61.27%

Table 7

 7 Total order sensitivity indices with respect to the different outputs of interest. Grey cases indicate the most important sources of uncertainty

			Total-order sensitivity index of:
			Exceed. Prob.	Water depths	Depth-damage curves	LU-GDB dataset	Project costs
		Average Annual Avoided Damage
	With respect to:	Total Eco. activities Private housing Campgrounds Agricultural land Net Present Value	0.33 0.4 0.22 0.2 0.27	0.29 0.38 0.005 0.01 0.02	0.18 0.2 0.78 0.6 0.4	0.21 0.3 0.03 0.38 0.38	0 0 0 0 0
		NPV	0.25	0.18	0.23	0.21	0.12

Acknowledgements

The flood-related data have been kindly provided by the Syndicat Mixte de la Vallée de l'Orb (Laurent Rippert) and the Syndicat Béziers-la-Mer (Pierre Enjalbert).

Tables

Only DTM error is considered: Gaussian noise without spatial correlation (mean = 0 cm, s.d. 17 cm). 100

LU-GDB dataset

• Misclassification of land use types and subtypes

• Variability of ground floor elevation of buildings (lack of data)

• Shape and surface of polygonal features (measurement errors)

• Confusion matrix for land use misclassification.


François Artis
email: fartis@laas.fr

David Dubuc
email: dubuc@laas.fr

Charline Blatche

Katia Grenier
email: grenier@laas.fr

Biological Cells Proliferation in Microwave Microsystems

Keywords: Biological effects, Biosensor, Cell proliferation, Coplanar wave-guide, Microwaves

This paper presents the biological compatibility of a microwave analyzing microsystem of living cells through the indicator of cells proliferation. The cells under investigation correspond to adherent cells of Normal Rat Kidney (NRK). In a first time, both their adhesion and proliferation into the highfrequency-based micro-device have been successfully obtained. In a second step, microwave signals have been applied at different power levels. Experimental studies demonstrate that microwave power levels up to +8,6 dBm do not impact cells proliferation.

I. INTRODUCTION

For a decade, microtechnologies and microfluidics have enabled the development of biological chips at the cellular and molecular levels [START_REF] Whitesides | The origins and future of microfluidics[END_REF]. Towards the elaboration of lab-on-chips, the investigation of integrated sensors is mandatory. Among the various developed sensing techniques, optical, mechanical ones for instance, microwave signals may also be used, which confers to the analyzing technique the advantages of being contact-less and marker-less. The employment of such a detection scheme and the corresponding micro-devices nevertheless imply to evaluate their biocompatibility.

Biocompatibility is concerned in terms of:

(1) cells adhesion capabilities. These imply to develop technological process flow, which favors cell adhesion and proliferation.

(2) cells proliferation under and after a microwave power stress.

If the assessment of biocompatibility of various biomaterials (such as yeasts, blood red cells, algae, enzymes, cells [START_REF] Sajin | Effects of low power microwave radiation on human blood red cell[END_REF][START_REF] Monteiro | Effect of microwave fields on the growth rate and mitosis of a single cell in spirogyra algae[END_REF][START_REF] Vojisavljevic | Low intensity microwave radiation as modulator of the L-lactate dehydrogenase activity[END_REF][START_REF] Merola | Proliferation and Apoptosis in a Neuroblastoma Cell Line Exposed to 900MHz Modulated Radiofrequency Field[END_REF]) in microsystems and/or integrated circuits [START_REF] Prakash | Tracking cancer cell proliferation on a CMOS capacitance sensor chip[END_REF] is quite mature, the biocompatibility in microwave (i.e. under electromagnetic stresses) microsystems constitutes a really recent issue [START_REF] Merla | Real-Time RF Exposure Setup Based on a Multiple Electrode Array (MEA) for Electrophysiological Recording of Neuronal Networks[END_REF].

This paper presents the study of biocompatibility of living cells within a microfluidic reservoir placed over a coplanar waveguide.

The cells used in our case correspond to Normal Rat Kidney ones (NRK). Their choice has been driven by two reasons: [START_REF] Whitesides | The origins and future of microfluidics[END_REF] for their ability to resist to in-vitro culture conditions and (2) due to the fact that they are adherent [START_REF] Prakash | Tracking cancer cell proliferation on a CMOS capacitance sensor chip[END_REF]. These NRK cells are also immortalized ones, which means that they may not stop proliferating as normal cells do through apoptosis.

The indicator chosen in our study is the cells proliferation or more exactly the ability for cells to divide again after a microwave stress. To the best of our knowledge, we report, for the first time at the micro scale, the impact of the MW power excitation inside a biosensor dedicated to living cells probing.

II. DESCRIPTION OF THE HF-BASED MICROSENSOR

As presented with the picture of Fig. 1, the device consists in a coplanar waveguide with a microfluidic tank perpendicularly placed on the top, in order to well localize the adherent cells and prevent any liquid and pollution on the coplanar accesses. The waveguide is obtained through a lift-off process with the superposition of a titanium flash layer and a 300 nm-thick gold layer on a quartz substrate. The line exhibits widths of the central conductor and slots of 150 µm and 15 µm large respectively, in order to match a 50-Ω impedance in air.

The walls of the tank are made of SU8 photoresist. The width of the tank corresponds to 1,5 mm, with a height close to 250 µm. The fluidic volume is less than 2 µL. With such a volume and the open tank configuration, some cares are taken to avoid strong evaporation, which may be critical for the NRK cells in terms of dehydration and heat shock.

III. BIOLOGICAL PROTOCOL

A specific biological protocol to make the cells adhering on the device and to control the viability of the cells through their proliferation has been defined.

First, the manufactured micro-device is sterilized in a solution of 70°-ethanol. The sample is then placed into traditional bio-well plates with NRK cells in solution. The used biological medium corresponds to a mixture of Dulbecco's Modified Eagle's Medium (DMEM), which provides the necessary nutrients to cells, as well as 10% of fetal calf serum (FCS), which offers the growth factors, antibiotics to prevent bacterial contamination and finally a buffer employed to maintain the pH. The entire plate is then placed into an incubator, which provides the adequate condition of cells proliferation, i.e. a constant temperature of 37°C and 5% of CO 2 . This step permits the seeding of the cells on the surface of the microwave biosensor and the initialization of their proliferation. After incubation, which warrants the adhesion of few cells in the fluidic tank, the cells are firstly observed with a microscope at the day called D0. The cells are then observed twice a day during three days after the D0 day.

To evaluate the cells proliferation, the developed observation method is based on the determination of the surface occupied by the NRK cells, as presented in Fig. 2. We based our counting method on the assessments that the area occupied by the cells is proportional to their number and that the cells have enough space not to be squeezed. By considering the surface occupied by the cells (cf. Fig. 2) divided by the whole surface area on top of the CPW in the microfluidic zone, we have defined a proliferation ratio, which is normalized with respect to D0 and considered versus time.

IV. MICROWAVE EXPERIMENTS

A. Microwave application protocol and test setup

For the microwave test setup, an ANRITSU MG3694B signal generator is connected at the input of the micro-device through a coaxial cable ended by a coplanar probe. The generator is used to apply specific power levels on the device at a single frequency, 20 GHz in our case, as it corresponds to the maximum absorption of the biological medium. A power meter ANRITSU ML2437A is also connected at the output of the coplanar line to collect the output power level of the micro-device.

Three different power levels are tested in addition to a control sample, which is submitted to the measurement protocol without any HF signal application. The three levels correspond to -10 dBm, +5 dBm and +20 dBm respectively.

The microwave application protocol is as it follows. After the optical observation of the sample at D0, the samples are submitted to the microwave signal for 10 minutes. They are then placed again into the incubator with sufficient biological medium to prevent any lack. The samples are then optically observed periodically. All pictures are treated as previously indicated in the biological protocol.

B. EM Simulations

EM simulations have been performed with the 3D EM software HFSS in order to get a qualitative assessment of the wave attenuation in the medium. Fig. 4 presents a corresponding result. The electrical field rapidly attenuates in the lossy medium and dominates in the vicinity of the coplanar slots. With such simulations, we have determined that the attenuation P(x) of the power is position dependent in the tank and follows the equation ( 1) :

! ! = ! ! . ! !!,!".!
(1) where ! and ! ! correspond to the position in the width of the tank and the input power at the beginning of the fluidic reservoir, respectively. This formula and the simulations have allowed us to estimate the power at different locations in the biological medium. We would like to outline that the absorbed power is mainly located in the biological medium rather than in the cells, due to the high ratio: E.M. energy volume / cells' P TANK_IN ! P 0 P TANK_OUT volume. As far as the bio-compatibility of cells proliferation under microwave power stress is concerned, we did not look at the power absorbed by the bio sample and focused on the assessment of the absolute maximum rating in term of RF power to ensure cells proliferation. In this scope, in order to obtain a good accuracy, we have discretized the fluidic tank in four parts, which correspond to specific average power levels, as indicated in Fig. 5. V. RESULTS AND DISCUSSION Fig. 6 presents the observed normalized number of cells in part 1 of the micro-devices versus time, for the -10 dBm and +5 dBm applied power levels, as well as for the control sample. No significant impact of HF signals has been encountered, as all the curves are very similar and conform to the expected proliferation kinetic. This result proves the correct biocompatibility, regarding cell proliferation, under this range of frequency and power operation. As far as the +20 dBm level is used for the generator, the results obtained in the three first different parts of the microfluidic reservoir are indicated in Fig. 7.

From this figure, we can notice that cells submitted to +16,2 and +12,4 dBm totally stopped proliferating (the ratio remains flat for several hours), whereas those located in the third section of the channel are always growing. After 30 and 40 hours in the case of +12,4 and +16,2 dBm respectively, the cells seem to grow again (as indicated in dotted lines). This "regrown effect" corresponds in fact to a parasitic effect, due to peripheral cells, located at the edges of the device, which were not submitted to the microwave signal and migrate towards low cells concentrated areas. This also explains the time shift of this effect (close to 10 hours) between the second and first part of the reservoir.

Moreover, the proliferation behavior for the +8,6 dBm zone is in good agreement with our precedent measurements at lower power levels (cf. Fig. 6). It should be noticed that additional experiments have demonstrated that these results do not result from thermal effect. Microwave biosensing at the cells and molecules level constitutes a promising approach for new biological and medical analyzing tools. The corresponding micro-devices require consequently to be evaluated in terms of biocompatibility and limitations. We have assessed in this paper the compatibility of coplanar lines with a microfluidic reservoir placed on top, with respect to the proliferation indicator of adherent NRK cells. Both adhesion and proliferation of cells were well realized at the microscale and the application of HF signals exhibiting a power level equal to or lower than +8,6 dBm did not affect or stop the cells proliferation.

HF microsystems are thus in excellent adequacy for noninvasive biological analysis at very low power levels. Such devices may also be suitable for microdosimetry investigations at the cellular and molecular levels.

Fig. 1 .

 1 Fig. 1. Photography of the micro-CPW device with a SU8-tank placed on top

Fig. 2 .

 2 Fig. 2. Delimitations of cells clusters and the entire CPW areaThe normalized cells proliferation versus time is indicated in Fig.3and indicates a cell cycle of about 1,5 day (cells doubling at the end of the mitosis).

Fig. 3 .

 3 Fig. 3. Cell proliferation in the microwave biosensor without any applied HF signal The biological protocol successfully yields the correct adhesion of cells into the HF-based biosensor and their proliferation. Based on this protocol, High-Frequency signals were applied.

Fig. 4 .

 4 Fig. 4. Electromagnetic field distribution in the SU8-tank

Fig. 5 .

 5 Fig. 5. Microwave power levels distribution in the microfluidic tank with +20 dBm as an input power

Fig. 6 .

 6 Fig. 6. Cell proliferation for HF micro-devices submitted to -10, +5 dBm, as well as the control sample

Fig. 7 .

 7 Fig. 7. Cell proliferation into the micro-device at different locations, with +20 dBm as an input power VII. CONCLUSIONS