Clementine Straub

Sorin Olaru

Jean Maeght

Patrick Panciatici

Clémentine Straub

Olaru ˚jean Maeght

Panciatici ˚l

Robust MPC for temperature management on electrical transmission lines

Keywords: Robust MPC, power systems, congestion management, dynamic line rating, batteries

INTRODUCTION

As renewable energies become more efficient and cheaper, an expansion of the sector is observed worldwide. The report RTE (2016) on electricity supply and demand for the French territory foresees a dramatic increase in renewable production, which may lead to congestions. The traditional lever to face this issue was network development. However, due to heavy costs and difficulties in obtaining authorizations, TSOs are looking for new solutions that maximize the utilization of the existing network instead of building new infrastructures.

In general the capacity of a line is defined by its Ampacity, which is the maximal constant current meeting security and safety criteria, maintaining a sufficient clearance and avoiding thermal damage to the conductor. This value has been traditionally determined based on the conductor maximal allowable temperature using a worstcase scenario on weather conditions. However, conductor temperatures, and hence its power transfer capability, are affected by wind and solar conditions. Moreover, there is a correlation between high wind power production and high convective cooling. When the wind is strong, power flows are larger, but the transmission capacity is increased in the same time, see [START_REF] Stephen | Guide for Application of direct real-time monitoring systems[END_REF].

Therefore, the use of Dynamic Line Rating which considers dynamic limits evolving according to weather conditions has been proposed in literature, see [START_REF] Foss | Dynamic line rating in the operating environment[END_REF]. Economic DLR gains are highlighted in several papers, as [START_REF] Schell | Using dynamic line rating to minimize curtailment of wind power connected to rural power networks[END_REF]. Some solutions, giving realtime information on lines ampacities are already available: Ampacimon is described in [START_REF] Cloet | Uprating transmission lines through the use of an innovative real-time monitoring system[END_REF]. Generation redispatch and operation of large battery systems have also been proposed as an alternative to network reinforcement to deal with congestions in [START_REF] Wen | Enhanced security-constrained opf with distributed battery energy storage[END_REF].

The French TSO defined an operating policy where limitations are evolving seasonally and where overloads are tolerated for short amounts of time as an approximation of the conductor heating, as explained in [START_REF] Rte | Memento sur la surete du systeme electrique[END_REF]. For example, an overload of 15% of the worst-case scenario ampacity can be tolerated for 1 minute and an overload of 10% can be tolerated for 5 minutes. [START_REF] Straub | Zonal congestion management mixing large batteries storage systems, production curtailment and network reconfiguration[END_REF] designs a real-time controller managing congestion in this operating context. The controller is based on a Model Predictive Control approach combining generation curtailment and large batteries to enforce the limitations.

The aim of the present paper is to go a step further and develop a new management temperature framework acting on the same quick levers (batteries and renewable production curtailment) and monitoring lines temperatures in real-time. The goal of the designed controller is to enforce temperature constraints. The controller thus includes a model of transmission line heating, needs predictions on temperature evolution and has to handle delays on control. It is natural to design a MPC controller [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF] as operational constraints are included. The method also benefits from the feedback mechanism as linear approximations are considered in the model. It is possible to design a robust controller by means of the Tube MPC strategy [START_REF] Rakovic | Parameterized tube model predictive control[END_REF], [START_REF] Mayne | Robust model predictive control of constrained linear systems with bounded disturbances[END_REF] and this is a crucial issue as wind is highly unpredictable.

The main contribution of this paper is to develop a robust MPC controller to manage the real-time transmission lines temperature. A non-linear receding-optimization formulation is considered based on the temperature prediction model. The recursive feasibility will be enforced by widening uncertainties bounds to include the model mismatch and performances will be checked with representative simulations on a test case of the French network.

MODELING

Electrical transmission network modeling

The DC load flow principles are used in this article to model power flows evolution [START_REF] Zimmerman | Matpower: Steady-state operations, planning, and analysis tools for power systems research and education[END_REF]. DC load flow consists in a linearization of AC network equations with three assumptions: flat voltage profile, small differences in voltage angles and transmission without losses. The DC modeling enables us to subsequently define a model-based control problem and exploit the linearity in order to obtain a convex optimization formulation that can be solved very efficiently. The three assumptions made are not major restrictions.The feedback controller will be able to handle them. In order to determine power flows, the knowledge of injections at each node of the network is needed with the DC load flow. Such knowledge on the whole network is an ideal feature but is not necessary: congestion are a local phenomenon and can be handled locally. Power Transfer Distribution Factors (PTDF), described in [START_REF] Bart | Network modelling for congestion management: zonal representation versus nodal representation[END_REF], can help representing local power flows, without solving DC load flow on the whole network. The flow representation by PTDF means and DC load flows are strictly equivalent [START_REF] Zimmerman | Matpower: Steady-state operations, planning, and analysis tools for power systems research and education[END_REF]. Using PTDF, power flows evolution can be described through linear equations. A PTDF is a factor dependent on a line and a bus which gives the amount of power going through the line if the power injection at the bus increases of 1. In other words, let F ij denotes the power flow on line ij. Then a change in the flow of line ij ∆F ij takes place if the injection at bus n increases of ∆P n , and leads to the formal definition of the factor:

P T DF pij, nq " ∆F ij ∆P n
The variation ∆F ij depends on the choice of the 'slack bus'. In a network, the balance between load and generation must be maintained. If the power injection varies at bus n, the injection at another bus must vary accordingly and maintain the balance. This second bus is called the slack bus. For a fixed slack bus and a fixed network topology, the rapport ∆Fij ∆Pn is constant. Power flows evolutions between k and k `1 time instants can thus be written as:

F k`1 ij " F k ij `ÿ n P T DF pij, nqp∆u k n `wk n q (1)
where F k ij is the power flow on line ij at time k, w k ij the natural change in generation at node n and ∆u k n is the injection variation due to control (generation curtailment or battery charge). The sampling time is a model parameter. We consider here a sampling time of approximately 1 minute: as limitations concern temperature and not power flows, 1 minute is precise enough for power flows evolution. 1 minute is also an appropriate sampling time for temperature evolution, as it will be detailed next.

Temperature evolution

Traditional Dynamic Line Rating algorithms give the lines ampacities as a function of weather conditions. The ampacity is updated frequently (every 10 minutes for instance) when new measurements on solar radiation, wind and temperatures (ambient and of the lines) are available. The controller developed here does not have limitations on intensity but only on lines temperatures, as it includes a model of line heating. Equations similar to DLR are used to model this heating. The work in [START_REF] Iglesias | Guide for thermal rating calculations of overhead lines[END_REF] gives a general method to calculate the thermal rating of overhead lines and presents in particular an algorithm to track conductor temperatures with time considering that weather data and current are provided every 10 minutes. Data are considered constant on the 10 minutes interval and calculations are performed every minute. One minute is small enough compared to the time constants of the line heating and provides accurate result. The algorithm described in [START_REF] Iglesias | Guide for thermal rating calculations of overhead lines[END_REF] is adapted to give an equation describing the temperature evolution. The time step used is also one minute. Measurements both on power flows and line heating are available every minute. Equations are based on the discretized heat balance equation for transient state:

∆T " P J `PS ´PC m ¨c ∆t (2)
m is the mass per unit (kg.m ´1), c is the heat capacity (J.kg ´1.J ´1), T the line temperature, ∆t the time step, P J , P S and P C respectively the Joule and solar heating, and the convective cooling. P J " R ¨I2 P S " α S ¨D ¨IT P C " π ¨λf ¨Nu ¨pT c ´Ta q I is the total direct current, R the direct current resistance per unit length (Ω.m ´1), α S P r0.2, 0.9s the surface absorbance which depends on the conductor age, D the conductor outer diameter (m), I T the global radiation intensity (W.m ´2), λ f the air thermal conductivity (W.K ´1.m ´1). T c / T a are the conductor / air temperatures and N u is the dimensionless Nusselt number calculated with the wind speed and direction. The model makes several assumptions: R and λ f are constant with temperature, the conductor temperature is constant across section and the magnetic heating is neglected. The term coming from the Joule heating needs the current intensity I and the DC modeling is based on F, the three phase active power. The relation between I and F is given by: I " a F 2 `Q2 ? 3V with V the phase to phase voltage and Q the reactive power. Q can be measured and V taken to its nominal value. The equation ( 2), replacing I by its expression with F , is non-linear due to the square on the active power term F .

The evolution of temperature in an electrical line can thus be written as:

T k`1 " p1 ´π ¨λf ¨Nu m ¨c ∆tqT k `R ¨∆t 3V 2 0 ¨m ¨c pF k q 2 `∆t m ¨c pα S ¨D ¨IT `π ¨λf ¨Nu ¨Ta `R Q 2 0 3V 2 0 q (3)
with λ f , R, V 0 , α S , D, m and c constants. N u , Q 0 , T a and I T can be regularly updated. Coefficients are timedependant (variations of weather and electrical values: N u , Q 0 , T a , I T , F 0 ). However they can be considered constant for prediction purposes as long as their variation is slow with respect to the chosen sampling time.

MPC FORMULATION FOR TEMPERATURE MANAGEMENT

MPC formulation

Curtailment actions present time-delay while the battery action is immediate. It is considered that a 1 minute delay is necessary for curtailment (time needed for order transmission). As ∆t " 1min, the delay for curtailment is one time step. If the delay is larger or the time step smaller, an extended state space representation can be constructed. The control variables are the power injected in the battery and generation curtailment. Several sites curtailment are considered, each with a different impact on power flows. This impact is given by the PTDF. The battery is considered strategically situated, at a point where congestion are important. In eq. ( 1), power flows are described with variations in generation curtailment and power injected in the battery. The control needs to be expressed in term of differences: ∆u k batt " u k batt ´uk´1 batt for battery injections and ∆u k curt,i " u k curt,i ´uk´1 curt,i for curtailment. u k batt P R is the power injected in the battery at time k and u k curt,i P R ng the production curtailed in site i at time k. n g is the number of curtailment sites. The control is defined as:

u k " `∆u k batt ∆u k curt,i ˘P R ng`1
The state x thus includes u batt and u curt,i , as well as the energy stored in the battery E P R, the power flows on controlled lines F ij P R n l with n l the number of controlled lines, and their temperature T ij P R n l . There are n state variables, with n " 2pn l `1q `ng

x k " `F k ij T k ij u k batt E k batt u k curt,n ˘T P R n
The dynamical model includes the power flow equations (1), an integrator to relate ∆u and u, the evolution of the battery state of charge:

E k`1 batt " E k batt `∆t ¨uk batt ,
and the temperature evolution. A term is added compared to (3) for temperature evolution to take into account the immediate battery action.

T k`1 " p1 ´βqT k `αpF k `Lbatt ¨∆u k batt q 2 `γ (4) with α " R¨∆t 3V 2 0 ¨m¨c and β " π¨λ f ¨N u m¨c ∆t and γ " ∆t m ¨c pα S ¨D ¨IT `π ¨λf ¨Nu ¨Ta `R Q 2 0 3V 2 0 q.
The matrix L batt contains the Power Transfer Distribution Factors from equation (1). The signal w k represents the disturbances, as for instance the variations in power flows and temperatures due to wind evolution (variation in generation and cooling effect). It is assumed that w k P W and W is bounded.

The aggregated state dynamics can be expressed in the following non-linear compact form:

x k`1 " f px k , u k , w k q
(5) The constraints are constituted by limitations on temperature, as well as bounds on battery capacities E min,max and u min,max batt , and bounds on generation curtailment. The state and control constraints can be written in the form:

H x ¨x `Hu ¨u `Hw ¨w ď H (6) Disturbances w are unknown but bounded. Eq. ( 6) can be written to consider the worst case as:

H x ¨x `Hu ¨u ď H (7) We define P , the mix admissible set for control and state:

P " "" x u  P R n ˆRng`1 : H x ¨x `Hu ¨u ď H* (8)
The goal of the controller is to stay within admissible temperatures while minimizing control costs. Only control terms are penalized in the value function: the state constraints are taken into account explicitly and doesn't represent an operational objective:

V N px, uq " ÿ tPt0,N ´1u ||∆u|| Q
with Q a diagonal matrix containing the costs for curtailment and battery charge. The general MPC formulation for temperature management on electrical transmission lines using batteries and generation curtailment is:

V 0 N pxq " min u V N px, uq s.t. x k`1 " f px k , u k , w k q, k P t0, N ´1u, H x ¨xk`t `Hu ¨uk ď H, k P t0, N u, (9)

ROBUST PROBLEM FORMULATION BASED ON TUBE MPC

The formulation (9) contains weather data, as the Nusselt number related to the wind power and direction and the ambiant temperature. If these data are updated periodically (measurements on wind and power flows), it is possible to define several functions f k corresponding to the updates in terms of a parameter (or time) dependant model. The strategy employed here is to consider only one function f , and to extend the disturbance set W to cope with the model mismatch. The set W will handle errors related to weather variations. The simultaneous presence of constraints and disturbances can lead to infeasibility.

The Tube MPC method, described in [START_REF] Mayne | Robust model predictive control of constrained linear systems with bounded disturbances[END_REF], enforces constraints satisfaction while maintaining a manageable computational complexity. The method relies on inserting suitable constraints restrictions. The tightened constraints are defined through the help of a disturbance invariant set. The method is built on linear time-invariant models of the dynamics. In order to apply this approach, a linearization of f will be used next.

Linearization of the state dynamics

The state dynamic equation is non-linear, the nonlinearity caused by the square on the active power term F anc control ∆ubatt in (4). Let F 0 be the average power flow on the line. We linearize the equation ( 4) around F 0 using the approximation F " F 0 `F « F 0 . The variations F around F 0 can be considered small as they are usually less than 10% of the F 0 value.

pF `Lbatt ∆u batt q 2 " pF 0 `F `Lbatt ∆u batt q 2 « F 2 0 `2F 0 F `2F 0 L batt ∆u batt `pL batt ∆u batt q 2 « F 2 0 `2F 0 F `ũ batt ( 10
)
with ũbatt " 2F 0 L batt ∆u batt `pL batt ∆u batt q 2 , the new control. It can be checked that ũbatt realizes a bijection with u batt on its domain r´P , P s if F 0 ě 1M W . The assumption on F 0 is not strong since the usual flows are around 60 ´90M W . The variations of ũbatt must belong to the interval r P 2 ´2F 0 P , P 2 `2F 0 P s.

We denote Tij the variations of the real temperature around T 0 , the equilibrum point for temperature corresponding to a flow F " F 0 .

The evolution of temperature in an electrical line can thus be written as:

T k`1 " p1 ´βq T k `2F 0 α F k `αũ batt `wk (11) with α " R 3¨m¨c¨V 2 0 ∆t and β " π¨λ f ¨N u m¨c ∆t. The dynamic of the linearized system is: xk`1 " A ¨x k`1 `B ¨ũ k `wk (12) with xk " `F k ij T k ij ũk batt E k batt u k curt,n ˘T P R n and A " ¨1 0 0 0 0 L 2F 0 ¨MF M T 0 0 0 0 0 0 1 0 0 0 0 0 ∆t 1 0 0 0 0 0 0 0 1 ‹ ‹ ‹ ' , B " ¨Lbatt L curt M F 0 1 0 ∆t 0 0 1 ‹ ‹ ‹ '
The matrices M F and M T are diagonal and contain the coefficients described in (11).

Robust Invariant Set Calculation

Let Ω a positive disturbance invariant set for the system:

x k`1 " Āx k `w, w P W (13) Ω satisfies:

Āx `w P Ω, @x P Ω, x P W Olaru et al. ( 2010) revisits the construction of robust positive invariant sets (RPI) for linear systems with additive bounded disturbances. The following theorem from [START_REF] Kofman | A systematic method to obtain ultimate bounds for perturbed systems[END_REF] provides a simple choice for the RPI set:

Theorem: Considering (13), let Ā " V ´1∆V be the Jordan decomposition of Ā, a strictly stable matrix. If W is described by w such that |w| ď w, @w P W , then the set:

Ω " tx P R : |V ´1x| ď |I ´∆| ´1|V ´1B| w `θu (14)
is a RPI set with θ any arbitrary small vector with positive elements.

In the particular case of temperature management controller, we consider a closed-loop with a stabilizing gain K. The matrix Ā from the theorem represents A `BK with A and B the matrices from (9). The gain K can be obtained with the pole placement method, the condition being A`BK strictly stable with real eigenvalues. Obtaining a small disturbance invariant set is an important issue: the smaller the set is, the less important is the constraints tightening procedure for the tube MPC algorithm. Getting the minimum disturbance invariant set enables us to be less conservative. [START_REF] Olaru | Positive invariant sets for fault tolerant multisensor control schemes[END_REF] proposes an algorithm to refine the disturbance invariant set obtained in the theorem.

Nominal problem and algorithm

The robust model predictive controller from [START_REF] Mayne | Robust model predictive control of constrained linear systems with bounded disturbances[END_REF] is used to guarantee the recursive feasibility for linear time-invariant prediction models and will be adapted to the temperature management problem. The controller is based on the resolution of a nominal MPC problem. This problem is (9) with the tightened constraints: " x u

 P P a " 1 1 K  Ω (15)
X a Y denotes the Pontryagin difference, defined by X a Y " tx|x'Y P Xu, with X 'Y " tx`y|x P X, y P Y u the Minkowsky set addition. The robust constraint satisfaction is guaranteed if the optimization-based control problem P N pxq (16) is deemed feasible. The control law (17) leads to a robust closed loop functioning.

V ˚0 N pxq " min u,x0 V N px k , uq s.t. x k`1 " Ax k `Bu k `wk , k P t0, N ´1u, u k P U a KΩ, k P t0, N ´1u
x k P X a Ω, k P t0, N ´1u

x k P x 0 ' Ω (16)
where x k is the state of the system at time k. The solution of P N pxq gives the optimal control sequence u ˚pxq " u 0 pxq, ...u N ´1pxq and the optimal state sequence x ˚pxq " x 0 pxq, ...x N pxq. x 0 pxq is not necessary equal to x. The robust law applied to the system is: κ 0 N px k q " u 0 px k q `Kpx k ´x0 pxqq (17)

The following algorithm guarantees the recursive feasibility of the MPC problem and the satisfaction of the temperature constraints.

Algorithm:

(1) Define a controller K such that the matrix A K " A BK is strictly stable and calculate the associated robust invariant set Ω.

(2) Measure x k , the power flow and temperature on the controlled lines.

(3) Solve the nominal problem P N px k q giving the nominal control u k and the optimal initial state x k . (4) Apply the control κ k N pxq " u k pxq `Kpx ´xk pxqq. (5) k " k `1 and go back to (2) 5. SIMULATIONS

Description of the zone

The robust model predictive controller for temperature management has been simulated on a zone of the French transmission network in the West area of France, see Fig. 1. Congestion are expected between Laitier and Maureix due to upcoming wind farm connections. A battery is planned to be installed in Isle Jourdain in 2020 for congestion management. Convergence, a RTE simulation tool for network analysis, provides data to perform the simulations, see [START_REF] Josz | Ac power flow data in matpower and qcqp format: itesla, rte snapshots, and pegase[END_REF] for a short presentation of this tool. We consider two lines in the zone: Isle Jourdain -Bellac and Bellac -Maureix. The conductors for both lines are Aster 288. Their characteristics are presented in [START_REF] Rte | Referentiel technique chapitre 6 -catalogue de materiels[END_REF] and summarized in Table 1. The control is constituted by a battery and two curtailment sites: Isle Jourdain and Bellac, as their impact on the flow on the controlled lines is the biggest. Table 2 presents the PTDF used. It is considered that 30M W can be curtailed in Bellac and the battery has a p15M W ´30M W hq capacity. The maximal disturbance for power flows is 0.1M W and 0.05˚C for temperature.

Results

It is important to notice that the controller contains a linear model for temperature evolution, but simulations are performed with a non linear model. The point F 0 chosen for the Joule heating linearization is 78M W . The first step is to define the controller K given a small disturbance invariant set Ω while maintaining the control set big enough. The tightened constraints (15) couples constraints on control and state. For simplicity, we assume the constraints defined independently on input and state: u P U a KΩ, x P X a Ω We consider a controller whose poles are placed at p0.7, 0.9, 0.45, 0.21q. Fig. 2 shows the temperature on both lines. The set in grey is the projection of the invariant set Ω on the temperatures subspace. The maximal allowed temperature is 55.7˚C. The temperature on Bellac-Maureix is close to its limit, and the controller acts to reduce it. The margin required by the constraint x k P X a Ω in ( 16) is represented by the dashed line. The horizon length is 10 prediction steps. Blue points represent the evolution of the temperature, while red points are the optimal initial state x ˚given by the resolution of the nominal MPC problem. Fig. 3 enables us to compare the temperature evolution Fig. 2. Temperature evolution on the two controlled lines on the free system and the controlled system. Without any corrective control action, the temperature exceeds the limit. The simulation also emphasizes the benefits of the robust controller with respect to the classical MPC one. The controller without the robust part also exceeds slightly the limit. The robust controller maintains the temperature below 55.7˚C, with a margin, related to the shape of the disturbance invariant set. Fig. 4 shows the controls applied to the battery and generation curtailment. They are of two types: U batå nd U curt˚are the solution of ( 16). U bat ˚and U curt increases to maintain the maximal temperature below limitation in the controlled system and compensate the increase of the temperature in the free system. U bat and U curt are the controls applied to the system taking into account the robust controller K part. Despite the model approximation concerning the linearization of the Joule heating in the MPC formulation, the problem remains feasible and limits are not crossed, this being related to the constraints tightening approach. However, the disturbance set W considered in these simulations is not truly representative of the power flows variations: 1M W would be a more accurate value for disturbances on power flows, especially because of errors in measurements, but considering such value results in a conservative disturbance invariant set Ω, and eventually the design of a linear controller K corresponding to a non-empty set U a KΩ becomes challenging.

CONCLUSION

This work presents a robust MPC controller for temperature management. The controller maximizes lines capacities as it is based on real-time lines heating, and allows therefore more renewable integration. In the model developed, the matrices A and B are not evolving through time, meaning that the weather conditions and the linearized point for the Joule heating are considered constant. The increase of the disturbance set is used to deal with these errors. Simulations show that the framework is robust to these modeling approximations.

  M F " ˜α 0 ... 0 α 0 ... 0 ... ¸MT " ˜1 ´β 0 ... 0 1 ´β 0 ... 0 ... Ţhe two matrices L contain the Power Transfer Distribution Factors from equation (1) in the correct order. L batt contains the PTDF for the battery node, L curt contains the PTDF for the curtailment nodes.

Fig. 1 .

 1 Fig. 1. Map of 90 kV Isle Jourdain zone m pkg.m ´1q c pJ.kg ´1.C ´1q D pmmq R pΩq 0.627 909 19.6 1.15e´4

Fig. 3 .

 3 Fig. 3. Evolution of Bellac-Maureix line temperature

Table 1 .

 1 Aster 288 conductor characteristics

	Isle Jourdain on Isle Jourdain -Bellac	0.36
	Isle Jourdain on Bellac -Maureix	0.36
	Bellac on Isle Jourdain -Bellac	0.38
	Bellac onBellac -Maureix	0.62

Table 2 .

 2 Power Transfer Distribution Factors

	Nu	Ta p˚Cq I T pW.m ´2q	V pkV q QpV arq	F 0 pM W q
	34	20	10	90	5	70

Table 3 .

 3 Weather and electrical dataWeather and electrical data considered in the simulations are presented in Table3. The method used to calculate the Nusselt number comes from[START_REF] Iglesias | Guide for thermal rating calculations of overhead lines[END_REF]. The air thermal conductivity considered is λ f " 2.61 ¨10 ´2.

‹ This work has been supported by the RTE-CentraleSupélec

Chair "The Digital Transformations of Electrical Networks"(https://rtechair.fr/)


Cécile R Scotto
email: cecile.scotto@univ-poitiers.fr

Van Hoan 1

Vu

Géry Casiez

Laure Fernandez

Sensorimotor control and linear visuohaptic gain

Introduction

Our direct interactions with the environment are performed through the sense of haptics. Touch and kinesthesia are used to extract objects properties' as well as to control our motion relative to them. The effectiveness of this sensorimotor control is a key question in the field of Human-Computer Interaction to enhance user performance. This is notably the case when we use a touchpad to control a visual cursor on a separate screen (e.g., a laptop). The control of these graphical interfaces is configured through a mapping between the motor space (e.g., the touchpad) and the visual space (e.g., the screen) called Transfer Function (TF). When we use the touchpad to control the cursor on the screen, the motion is compounded of a preprogrammed phase performed at high speed, and a following homing phase, performed at low speed and based on visuohaptic feedbacks [START_REF] Elliott | Goal-directed aiming: Two components but multiple processes[END_REF]). Some operating system TFs (e.g., Windows, OS X) are based on this principle with a visuomotor gain during the preprogrammed phase which is high while it is low during the homing phase to reduce the Movement Time (MT). Such TFs have been shown to enhance performance with this increasing visuomotor gain [START_REF] Casiez | The Impact of Control-Display Gain on User Performance in Pointing Tasks[END_REF][START_REF] Casiez | No more Bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF]. However, the reasons of this improvement are not totally elucidated, notably because the prescribed gains are non-linear. Here we analyzed the kinematics of a pointing task with a linear velocitybased TFs to assess how we plan and control our movement based on vision and haptics (i.e., touch and kinesthesia involved in motion perception). We compared two non-linear increasing and decreasing TF with constant gain TFs.

Methods

Eleven participants performed a Fitts' task in the horizontal plan on a MacBook Pro Retina (OS X 10.11.6 El Capitan) with a screen of 13.3 inches (1280 x 1024 pixels) refreshed at 60 Hz and a touchpad with a resolution of 400 CPI sampling at 125 Hz. Raw data from the pointing device and transfer functions were handled using libpointing [START_REF] Casiez | No more Bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF] while we used an application written using C++/Qt to provide the instructions and stimuli. Participants were asked to point the target as precisely and as accurately as possible with a smooth movement: they had to avoid stopping outside the target. The target distance was 8 cm and the width 2 cm, 0.5 cm or 0.125 cm, corresponding to the index of difficulty (ID; [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF])) 3, 5 and 7, respectively. Four TF were applied from the pad velocity to the visual cursor velocity: i) a constant gain of 1 (CG1), ii) a constant gain of 3 (CG3), iii) an increasing gain (IG) from 1 (at motor velocity of 0 cm.s -1 ) to 3 (at motor velocity of 10 cm.s -1 ), iii) a decreasing gain (DG) from 3 (at motor velocity of 0 cm.s -1 ) to 1 (at motor velocity of 10 cm.s -1 ). Twelve blocks with thirty successive trials of each TF and ID were performed, and only the last 20 were analyzed to avoid post-effect of TF change.

Results and Discussion

A repeated-measures Anova on the pointing movement time (MT) showed a main effect of TF (F3,20=18.1 p<.001), and ID (F2,20=125.4, p<.001) as well as an interaction between both factors (F6,60=2.5, p<.05; Fig. 1). Post-hoc Newman-Keuls further revealed the following statistical differences: The CG1 induced a longer MT than the CG3, regardless the ID. In addition, participants were as fast with CG3 than with the IG. Finally, while the DG induced longer MT for ID3, it was no longer the case for ID7 where CG3, IG and DG were not different. Overall, a high visuomotor gain appeared necessary to be fast even for the smallest target (ID7). Surprisingly, MT were not shorter for the CG3 than for the IG even at ID7.Therefore, the visuomotor gain of IG was reduced at low motor velocity and would have been helpful for an accurate sensorimotor control near the target ((Woodworth 1899)). A possible explanation of such absence of MT shortening might be that even if the low gain at low motor velocity allowed a better ending control, it might also be detrimental at the beginning of the motion because it delayed the time to reach the peak of visual velocity. This hypothesis is supported by the profile of the operating system TF: starting with a relative high gain and exponential at low speed. Further kinematic analyses will be conducted to assess how we use vision and haptics to control our motion regarding the manipulated visuomotor gains and task complexity.

Figure 1 :

 1 Figure 1: Movement Time as a function of ID and TF. CG1: Constant Gain of 1; CG3: Constant Gain of 3; IG: Increasing Gain; DG: Decreasing Gain. Error bars denote standard error.

Acknowledgements This work was funded by the ANR 'TurboTouch' (ANR-14-CE24-0009).


Quentin Schoen
email: quentin.schoen@mines-albi.fr

Sébastien Truptil
email: sebastien.truptil@mines-albi.fr

Matthieu Lauras
email: matthieu.lauras@mines-albi.fr

Franck Fontanili
email: franck.fontanili@mines-albi.fr

Aurélie Conges
email: aurelie.conges@mines-albi.fr

A new approach for Supply Chain Management monitoring systems adapted to crisis

Keywords: Monitoring system, Supply Chain, Sensitive products, Collaboration, Real Time

Sensitive products supply chain and supply chain facing crisis management share several aspects. In both cases, several decision makers have to choose the best options most of the time under pressure, often in emergency and need to access numerous information from the field. This shared monitoring aspect put forward the visualization need to consider in each decision all the crisis potential impacts. Unfortunately, for the transportation steps we focus on, the current transport management systems do not reach these requirements. In this paper, focusing on supply chains during crisis situations, we present a new monitoring system with adapted functionalities. The added value is a connection in real time and relevant way the data from the field to the information on a shared model used to make reliable decisions. We use the French Blood Establishment supply chain to illustrate the proposition.

Introduction

The supply chain management, as any other field, may have to face with crisis situations. The Webster's New World College Dictionary defines it as "a turning point in the course of anything; decisive or crucial time, stage, or event". We find here the idea that a crisis is decisive for the future of the given environment [START_REF] Lagadec | La gestion des crises: outils de réflexion à l'usage des décideurs[END_REF] and must have consequences. This breaking point of a stable environment may affect all the actors in it, whatever their role is. Thus, according to [START_REF] Devlin | Crisis management planning and execution[END_REF] the crisis nature may be military, politic, economic, social, technical or humanitarian.

Another aspect to consider is the time. Indeed, a crisis may occur suddenly or progressively [START_REF] Tomasini | Genetically modified food donations and the cost of neutrality: logistics response to the 2002 food crisis in Southern Africa[END_REF], depending on the triggering event(s). These/This event(s) may have severe impacts on the considering system/environment, depending on the environment's vulnerability and the consequences of these/this event(s). For instance, a truck crash is a sudden event whereas a vehicle stuck in traffic jam is more progressive. Depending on the products inside (well protected or not, urgent or not), impacts may be important (valuable and/or sensitive products) or not (empty truck, not valuable products).

Being able to detect these events as soon as they happen and understand their impacts (current and potential) is one of the key elements to limit the crisis spreading and cascade effect. In order to reach this objective it is necessary to monitor the processes in real time and understand what is happening on the field. A decision support system fed with frequently updated, relevant and reliable information, would be able to help crisis units providing Common Operational Pictures (COP). The most reliable and relevant are the information provided quickly to the crisis unit, the highest are the chance to limit the impacts.

Proposal: In this situation, considering the supply chains transportation processes, our objective is to define a new way of monitoring them in crisis situation and in a larger extent a new way of monitoring sensitive products transports in general. In parallel, we describe a potential information system (IS) able to support this approach based on data collection and treatment. Our study perimeter is limited to the collection, treatment and decision support system abilities, focusing on transportation steps. The decisions broadcast to actors on the field and/or people able to concretely change parameters of the environment concerned by the crisis is out of our scope.

To reach this objective, we describe in the first section the concrete needs to respond to a crisis in supply chain. Focusing on the transportation steps, we describe in which extent Transport Management Systems could contribute to the crisis response in the second section. In the third one, a proposal is made in order to meet the described needs. Finally, we present a real use case based on our experience in the French Blood Establishment that deals with sensitive products 2 Crisis Response from Supply Chain

A Response to the Crisis

Whatever are the prevention measures to reduce the system vulnerability and the preparation plans to anticipate the crisis management, a triggering event may happen. Thus, if we did not succeed in avoiding the crisis, it is necessary to be able to respond to it.

First, [START_REF] Devlin | Crisis management planning and execution[END_REF] defines the crisis response as "special measures taken to solve problems caused by a crisis". The objective is to stop the system deviation from the "normal" or "expected" state in order to implement a new stable one. One of the main problems concerns the variety of potential actors involved in a crisis response. In order to respond to it, they may have to collaborate, each one being responsible of its area of expertise and competencies. As [START_REF] Lagadec | Cellules de crise: les conditions d'une conduite efficace: gouvernements, ministères, entreprises, préfectures, administrations, municipalités, régions, médias, organisations internationales, organisations non gouvernementales, associations, syndicats[END_REF] put forward, being able to make relevant decisions, under external and internal pressure, collaborating with people we are not used to work with is not an easy task. In order to achieve this objective of collaboration and relevant decision-making, a crisis unit may be defined during crisis preparation steps. The role of this crisis unit is to create a decision sharing and making environment. Each person involved is responsible of its domain (vertical aspect) and of the interface with the other ones (horizontal aspect), exchanging relevant information to coordinate the response.

Thus, one of the key issues is to collect data and share between stakeholders relevant and reliable information from the crisis state. As [START_REF] Truptil | Etude de l'approche de l'interopérabilité par médiation dans le cadre d'une dynamique de collaboration appliquée à la gestion de crise[END_REF] put forward, many decisions are experience and knowledge based in each domain and not shared in a formalized way. In a dynamic and unstable situation, with potential emergency pressures, this must be a challenge to meet. In the following parts, we focus on this ability to obtain relevant and reliable information in the supply chain management.

Supply Chain Management Needs to Support Collaborative Crisis Management

In order to make relevant decisions based on reliable information about the supply chain, the stakeholders have different needs, described below, to make decisions.

Collect relevant data

The main objective of the supply chain management is to provide the required product, in the best state, at the right place, at the right time. During a crisis, the organization scheduled has to be reconsidered and the expected events may not happen. The environment evolves quickly in unexpected ways, the system is not stable and the consequences may be important. Thus, the decision makers may become blind quickly on their own processes. The first step to find a solution is to collect data from the field in order to follow the evolutions. These data are internal, from our own processes ("Where are my vehicles? In which state are my products?"), and external from the other factors which must affect our environment (weather, traffic, emergency services, lambda individuals, etc.).

Transmit and exploit reliable data

After collecting data from the crisis state the next challenge is to transmit them to an appropriate IS and exploit them. In fact, the crisis IS has to deal with data from different sources, with different formats, putting forward different aspects of the crisis. The main challenge here is to be able to deal with these amounts of heterogeneous data, understand their meaning and build automatically relevant and reliable information. This step of data treatments allows reducing their volume and classifying them in order not to drown the stakeholders with raw data but provide required information based on them.

Encourage exchanges around a shared picture

Gathering all these information is most of the time not sufficient. Even if the information flow is understandable, these information are separated in lists, tables, and schemes, focusing on a domain. It is useful for each crisis unit stakeholders independently but does not encourage them to discuss. The information flow is reduced compared with the data flow but there is still a conceptual step to carry out in order to build a crisis situation overview. Creating a Common Operational Picture (COP) fosters the crisis unit members to work together from the same visualization of the crisis state; thus they understand others problematic and objectives. Moreover, this COP must be dynamic, updated frequently with new data based information, which contribute to follow and anticipate the situation.

Anticipate and foresee to support the decision-making

One of the key questions each stakeholder of a crisis unit is supposed to answer is: "knowing the current situation, what is the expected one in a near future". The need to foresee the potential next events is a key element to make decisions and evaluate their impact before the implementation.

Inabilities of Transport Management Systems to Support Collaborative Crisis Management

In order to meet the needs we described above we could try to use existing systems that deal with transportation processes. These systems manage every day thousands of containers through the world and handle anomalies, consequences of unexpected and/or unwanted transportation events. We describe below their main functionalities, failures, and some logistic trends that should contribute to prevent Transport Management Systems (TMS) to meet this challenges.

Main Functionalities

The Transport Management systems are applications developed from the nineties which aim to provide « "an enabling tool'' for the safe and efficient operation of Freight Transport systems » [START_REF] Giannopoulos | The application of information and communication technologies in transport[END_REF]. Progressively the TMS evolved due to the complexification of logistics operations on one part [START_REF] Marchet | An exploratory study of ICT adoption in the Italian freight transportation industry[END_REF] and abilities of the computing systems and the Internet on the other part. Nowadays, these applications are most of the time proposed as Software as a Service (SaaS) through the Internet and functionalities have been split in blocks activated or not. Companies, either carrier or sender, which need TMS functionalities activate the ones they want in order to fulfill their needs. They pay a rent to access them and store their data in datacenters. Cloud computing technologies and SaaS approach offer the ability to Small and Medium Size companies to access to these technologies [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF]. Thus, whatever is the number of actors involved in a supply chain, the interconnectivity becomes easier to reach through normalized systems like Application Program Interface (API) and Electronic Data Interchange (EDI).

The main functionalities TMS providers offer are: (i) pick-up and Delivery areas setup, (ii) rounds setup, (iii) clients and/or subcontractors setup, (iv) packages shipment, (v) transportation and customs documents creation, (vi) rounds optimization and simulation, (vii) packages transportation steps tracking, (viii) vehicles monitoring, (ix) freight market place, quotes and order, (x) finances, transportation billing, (xi) reverse logistic management, (xii) performance and anomaly management, KPI, (xiii) transportation fleet management, (xiv) mobile devices setup.

These functionalities cover a broad scope of the most common needs and are split in details in different blocks activated or not. However, in case of sensitive products transportation or crisis, these systems are not relevant enough.

Identified Problems in Case of Crisis

For almost 30 years, the logical approach used to design TMS was to consider containers as boxes defined by a list of parameters (sender, recipient, required transportation environment, weight, permissible transportation time, etc.) and gather them in vehicles. Next, tracking the vehicles (location, state) we track the containers. Nevertheless, this approach is not fully adapted and precise enough to crisis transportation management neither sensitive products supply chain.

In fact, these products are too important to be damaged/stolen/lost, because a priori necessary to reduce the damages and contribute to recover. We should be able to know exactly where is located each product and what its state is (temperature, humidity, shocks, etc.). Consequently, the TMS real time vehicles' tracking is not precise enough and agility is limited. Indeed, during a crisis, we have to be able to make relevant decisions based on reliable facts from the field. Seeing a vehicle as a black box do not usually allow to deviate one container among the others in case of unexpected event (urgent need on a site, container damage).

Moreover, we are not able to take advantages of the multimodality opportunities that still represent a complex step because of physical and computational normalization lacks. Thus, most of the time we do not take advantages of all the transportation means opportunities [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF] but use one vehicle from the sender to the recipient whereas it might not be the most effective and efficient option.

Finally, optimization rounds and allocation, which aim to increase the efficiency, cannot be fed with other data than the available means from companies the sender knows or has a contract with. Car sharing and common transport availability are not tracked. Thus, these potential loads displacement are not used in the optimization algorithms. In case of emergency, TMS are able to take advantages of just a part of all the available ability.

These for the crisis and sensitive products supply chain management issues should be exacerbated, considering the upcoming logistic environment.

Upcoming environment

In order to complete this analysis and anticipate future development we develop here some future logistic trends. Some of them may reinforce the TMS failures whereas some others could offer new opportunities in crisis management.

First, the product customization is increasing as [9] foresees it. Product configurations to reach the client needs improve the capability to customize the products. The problem is the potential uniqueness of each product. In case of damage on the container or theft during the transportation steps, it would be more complicated to replace the products by identical one. In case of crisis, it would be impossible to find quickly adapted and effective solutions without gathering reliable information in real time about the locations and states of every product and container. Moreover, the additive manufacturing development encourages this customization and last miles deliveries should increase [START_REF] Boon | Influence of 3D printing on transport: a theory and experts judgment based conceptual model[END_REF]. Finally, traceability requirements in particular for cold supply chains would require too this somewhat real time monitoring systems.

Meantime, several technologies are about to offer new interesting possibilities in order to deal with crisis management. The development these last years of connected devices, the Internet of Things, allows to connect any object to the Internet [START_REF] Harris | ICT in multimodal transport and technological trends: Unleashing potential for the future[END_REF]. The development of these powerful small devices represents a remarkable opportunity to collect and transmit data from the field easily and in real time. Cloud computing development has simplified the data storage and exchanges between these devices and people. The Software as a Service [START_REF] O'sullivan | Software as a service: developments in supply chain IT[END_REF] way of thinking permits to offer an easy, flexible and relatively cheap access to interesting functionalities to any company. Combining these technologies, we are about to create a hyper-connected world in which the problem is not how to collect data but how not to be drown by them. As we describe it in the following sections, ability to deal with big data [START_REF] Wamba | How 'big data' can make big impact: Findings from a systematic review and a longitudinal case study[END_REF] issues, powerful data treatments like machine learning, process mining… allow exploiting these data in real time to extract relevant information. As for them, autonomous vehicle development [START_REF] Van Meldert | Introducing autonomous vehicles in logistics: a review from a broad perspective[END_REF] and « anytime, anywhere delivery model » should create in few years a capillary distribution network, connected to monitoring systems.

This state of the art of current TMS and the expected logistic environment shows these systems would not be relevant enough to deal effectively with supply chains, which are expected to be more and more sensitive to unexpected events and should require more and more collaboration. Therefore, crisis management needs are spreading and new opportunities to meet them are about to be available. A new supply chain management IS, able to deal with them, is proposed in the next section.

Proposal

Monitoring System

The approach we present here focuses on the containers (and products) to be transported, considering the vehicles as "potential load displacement". In fact, considering the hyper-connected and more collaborative world we are about to live in, it seems relevant to focus on the goods to be transported and not on the means as TMS do. As data are encapsulated and send on the network with switches and connections on the Internet, we can imagine that the normalized physical network would move physically and computationally normalized containers. This idea is very close from the Physical Internet philosophy developed by [START_REF] Montreuil | Toward a Physical Internet: meeting the global logistics sustainability grand challenge[END_REF]. We describe below the main functionalities to address, summarized in the figure 1: -Container surveillance: It is the ability to collect data about the container state, with tags in it or in its near environment, and to deduce relevant information from them. Topics like the physical and computational normalization, the container autonomy level and in which extent the decision support system is centralized or multi-agent based have still to be discussed.

-Network analysis and surveillance: While we gather data from the containers we have to gather data about all the available transportation means that are willing to load containers and their capabilities (destination, loading capacity, cost, etc.). Exploiting raw data from the field, we will be able to create reliable information about the network. On the other part, the surveillance must provide information about the network state and prevent unwanted events. Delivery center expected capacity, weather, roadwork, are some of the parameters interesting to control in real time.

-Container steering: Gathering the previous functions it would be possible to route automatically the containers on the network, step-by-step. Defining containers allocation rules to the available transportation means around them would orientate the containers to the most suitable way. In this hyperconnected environment, the ability to take advantage of the best solutions would allow efficiency and effectiveness rise. Thus, the supply chain becomes able to respond almost automatically to any unexpected event. Even if this kind of hyper-connected system may have problem in case of inability to collect enough relevant data from the field, it would meet the needs described above.

BIG PICTURE

A New IS to Treat Big Data Issues

In order to meet the needs we described in the first section we have to design a IS able to cope with this upcoming technological environment. Whatever are the tags and technologies used to collect these data, if we want to be reactive and agile enough to reach the challenges described we need frequent updates from all of them. The Internet of Things development should allow data collection easily, from autonomous tags and sensors, mobile phones, camera, etc. Finally, we may produce too many data compared with what we really need. Thus, it brings big data constraints. In fact, as [START_REF] Wamba | How 'big data' can make big impact: Findings from a systematic review and a longitudinal case study[END_REF] define it big data may be characterized by five variables: Volume, Velocity, Variety, Veracity, and Value. As the figure 2 shows it, the data flow has to be filtered through a homogeneity layer that deals with data variety [START_REF] Wang | An automatic model-to-model mapping and transformation methodology to serve model-based systems engineering[END_REF]. After that, the data flow is split in two parts. The first one consists in data storage for a posteriori treatment and the second one in real time treatments. The database is used for two purposes. First, these data represent the raw material to define the rules used to understand the data flow. For instance, combining data values from a vehicle we can deduce its state and if it represents a problem. Moreover, with a machine-learning module complex rules based on experience will be deduced in order to feed the Complex Event-Processing (CEP) module, which aims to treat in real time the data flow. Secondly, registering crisis responses these data may provide proposition in order to respond to a new one whom aspects and parameters are very close from a previous one. Machine learning analysis will enrich reference models, which contribute to help for expected model simulations. The second part concerns real time data treatment in which the CEP will provide reliable and relevant information based on the data from the field. Thanks to the rules, activities and warnings will be modeled and feed the field model. This model is the Common Operational Picture the crisis unit is looking for.

Need 4: Anticipate and foresee to help the decision-making

Based on the previous crises and the current field situation it is possible to foresee the next steps. Databases register how the previous crises were solved with the potential solutions, their risks and impacts. Considering the pressure of crisis decision making this ability to foresee and simulate a decision would be a major improvement.

Use Case -The Blood Supply Chain

Based on the French Blood Supply Chain, we describe below the main advantages and benefits of the defined system. The advantage of this example is that the blood supply chain is itself a sensitive one, which implies a high level of surveillance in real time. Considering the products sensitiveness and the numerous transportation steps, any unexpected event may have impacts on the products and people who need them.

French Blood Establishment Products

The French Blood Establishment (FBE) is the only company in France allowed dealing with the blood supply chain. The main objective of the FBE is to be able to provide anywhere, anytime, any blood product required to cure a receiver. This objective implies to collect blood from donors, check its innocuousness, separate the 3 components (plasma, red blood cells and platelets) and distribute the blood products bags the nearest to the hospitals. Due to the variety of blood products (groups and rhesus) and incompatibilities between them, it is necessary to collect the highest variety of blood types and distribute them, the closest to all the hospitals.

In France, around 10.000 pouches have to be collected every day. Consequently, every year around 40.000 one day collection sites are set up and 130 distribution sites are located the nearest to the hospitals. This variety of locations implies significant transportation steps between them. Moreover, the preparation and checking processes require high cost equipment. They have to be highly controlled and to follow strict procedures. Thus, the FBE settled 14 preparation sites and 4 innocuousness-checking sites in France in which the pouches and samples have to pass through. In this environment, blood products transportation is required, at least from the collection site to the preparation site and checking site, then from the preparation site to the distribution site.

Finally, the blood supply chain is sensitive because the distributed products are vital for thousands of people and because of product constraints. The storage temperature and the lifecycle are the two major constraints, which affect transportation steps. We summarize in the table 1 these constraints:

This Supply Chain is a sensitive one due to the products value, their temperature and lifecycle constraints and the potential need as a matter of emergency of any product anywhere at any time. Any unexpected event may affect directly this supply chain; we develop briefly some of them in the following sub-section.

Unexpected Events

We can consider 4 main unexpected events that will affect the SCM. In the following situations, a crisis unit must be set up to find adapted solutions:

-A massive accident / terrorist attack raises suddenly the blood products needs in an area. This implies unexpected transports to be quickly organized.

-Traffic jam and/or bad weather conditions affect directly the transports (rounds and urgent ones). The temperature might be difficult to control if the transportation time is longer than expected.

-A major problem on a site (fire, flood, snow, inaccessibility, etc.) where high cost equipment shared between several sites and regions (samples checking) is located. This implies quick, adapted and efficient rerouting.

-In case of accident or attacks in which people are injured, the other people usually want to help and give their blood. In this situation, an unexpected rise of blood donation may affect the whole supply chain.

In order to respond to these crises, we describe in the next sub-section what would be the major improvements of the crisis management system described before.

Illustration

Considering this use case and the described unexpected events the figure 3 illustrates an ideal Common Operational Picture that would foster and support collaboration and help the FBE to make the best decision. Truck deviation to deliver products to the hospitals the nearest from the accident, urgent need on a site and real time truck stuck in traffic jam location are examples of this kind of abilities (cf. figure 3). The major improvement of this kind of map is its ability to indicate on the same picture information used by all the stakeholders. This kind of visualization contributes to share an understanding of the situation and make collaborative decision.

Moreover, the truck on the left enlightened the added value of simulation to foresee the future system state. Based on machine learning analysis that feed reference models, the IS may provide an expected situation if we do not act and simulate impacts from a potential action.

Conclusion

To conclude, considering the current and foreseeable inability of Transport Management Systems to meet the needs of sensitive products supply chain and crisis management, we propose a new way of designing a transport monitoring system, data based, and putting forward the visualization. This approach would represent an important added value for supply chain management. Considering the future deployments, we have to consider several challenges. First, in this kind of collaborative platform, keeping a clear visualization for stakeholders with enough relevant information without drowning them with unnecessary ones may be difficult. Secondly, the stakeholders may be reluctant to use this kind of systems because they may feel it reduces their autonomy. Finally, the whole system is data-based, consequently if some data are not available for any reason (communication issues, tags breakdown, etc.) the whole system may create an inaccurate picture.

Our next tasks are now to develop concretely this kind of IS and feed it with real time transportation data. These data may come from tracking systems and devices we will use directly inside the FBE containers. The main objective is now to experiment this proposition on the field, estimate its benefits in crisis situation and discuss its failures.

Fig. 1 .

 1 Fig. 1. Transportation monitoring systems evolution

Need 1 :

 1 Collect relevant data.

Fig. 2 .

 2 Fig. 2. Information system to support supply chain management during a crisis

Need 3 :

 3 Encourage exchanges around a shared picture

Fig. 3 .

 3 Fig. 3. Use case example of crisis management visualization

Table 1 .

 1 Lifecycles and storage temperature of the most common blood products

	Products	Lifetime	Temperature
	Whole blood	Less than 48h	18-24°C (0-24h post swab)
	Blood samples	Less than 48h	2-10°C (6-48h post swab)
	Platelets	5 days	20-24°C shaken
	Red blood cells	42 days	2-10°C
	Plasma	1 year	< -25°C


Rémi Cousson

Quentin Leclère

Marie-Agnès Pallas

Michel Bérengier

BeBeC-2018-D17 IDENTIFICATION OF ACOUSTIC MOVING SOURCES USING A TIME-DOMAIN METHOD

The issue of the identification of acoustic sources has been widely treated in static or quasi-static source contexts, more recently for moving sources, in a wide range of applications. The beamforming method is a reference in that situation, but it has well-known limitations, namely a poor resolution at low frequencies which makes it difficult to discriminate close sources or with too different levels, and to quantify acoustic power levels. Deconvolution methods are often used to overcome those limitations. Performed in the frequency domain, they are effective in static contexts or with quite restrictive conditions if considering moving sources. In the present investigation, an original method is presented to address those problems. It is a matching pursuit-type method, performed exclusively in the time-domain and combining the sweeping focus point beamforming approach to an iterative cleaning algorithm, with the noticeable advantage to be adapted to non-stationary signals in pass-by situations. Experimental validation setups in both linear and rotational motions are presented to prove the potential and versatility of this method.

INTRODUCTION

The present study's goal is to find an effective method to localize and quantify acoustic sources in both linear movement, with a road vehicle pass-by application, and rotating movement. Beamforming is well suited to the localization of moving sources by steering the focus point of the microphone array with the moving source-point, by means of time-varying delays [START_REF] Barsikow | Experiences with various configurations of microphone arrays used to locate sound sources on railway trains operated by the DB AG[END_REF]. However, it has a poor resolution in low frequency and is not very accurate to quantify sources.

To overcome these issues, deconvolution methods have been developed to improve the localization and the estimation of the power level radiated by static sources [START_REF] Brooks | A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays[END_REF][START_REF] Högbom | Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines[END_REF]. Several of them have been adapted to the moving source problem. Fleury and Bulté [START_REF] Fleury | Extension of deconvolution algorithms for the mapping of moving acoustic sources[END_REF] proposed a solution which was tested in simulation and real experiment in the case of an aicraft flying over. A variant in the context of underwater acoustics was implemented in the case of a ship, studied by Oudompheng [START_REF] Oudompheng | Array processing for the localization and the contribution of acoustic sources of a passing-by ship using a short antenna[END_REF] and Lamotte [START_REF] Lamotte | A theoretical and experimental comparison of the deconvolution methods for moving sources[END_REF], involving far lower vehicle speeds. The performance of this method has been numerically assessed in the case of road vehicle pass-by conditions and with an academic experimental set-up [START_REF] Cousson | Identification of acoustic moving sources in the context of a road vehicle at pass-by[END_REF]. Although it showed a clear improvement from conventional beamforming, some limitations were also pointed out in terms of separation power and in the frequency range. Road vehicle pass-by conditions differ from the cases studied in [START_REF] Fleury | Extension of deconvolution algorithms for the mapping of moving acoustic sources[END_REF][START_REF] Lamotte | A theoretical and experimental comparison of the deconvolution methods for moving sources[END_REF][START_REF] Oudompheng | Array processing for the localization and the contribution of acoustic sources of a passing-by ship using a short antenna[END_REF] in the vehicle speed and distance from the microphone array, ultimately resulting in a great difference in the pass-by duration.

Another domain concerned with the localization and quantification of moving sources is the characterization of rotating sources such as fans. Time domain beamforming using a rotating focusing point is the topic of several works [START_REF] Geng | Identification of rotating sound source based on microphone array measurements[END_REF][START_REF] Minck | Fan noise analysis using a microphone array[END_REF][START_REF] Sijtsma | Location of Rotating Sources by Phased Array Measurements[END_REF], major drawbacks being identified as the quite heavy computational costs and the difficulty to implement standard deconvolution techniques. Other approaches are based on the construction of a virtual rotating array based on the fixed array measurements [START_REF]Rotating beamforming -motion-compensation in the frequency domain and application of high-resolution beamforming algorithms[END_REF][START_REF] Herold | Frequency domain deconvolution for rotating sources on an axial fan[END_REF]. Using signals reconstructed on this virtual rotating array, standard frequency domain methods can be implemented as well as deconvolution. However, these techniques require the use of specific arrays, such as ring arrays, and can be limited by the spatial sampling that is required to correctly interpolate the sound field.

The present paper proposes an original method, exclusively performed in the time domain, that takes advantage of the fact that time domain beamforming actually reconstructs the signal emitted from the focus point. The non-stationary nature of a pass-by measurement makes it interesting to operate in time domain, especially to take into account the Doppler effect. In a first section, the method is described and discussed. Then, an academic experimental setup designed to test the method and compare it with moving source conventional beamforming followed by a deconvolution algorithm (noted CBF from this point) is presented. A third section is dedicated to an experimental application of the method to rotating sources.

PROBLEM FORMULATION

Identification of static sources

Conventional beamforming

The reference method to identify static sources with a phased array of microphones is conventional beamforming. It can be performed either in time-domain or frequency-domain. In either case, the location and the power of sources in a scan area are estimated thanks to the signals received by an array of sensors. In the frequency-domain, the power estimate from the i th point in the sampled scan area is, at each frequency f :

B i ( f ) = w * i ( f )C( f )w i ( f ) (1)
where w i is a steering vector depending on the source and propagation models, * denotes the complex conjugate transpose and C is the cross-spectral matrix of microphone signals.

Conventional beamforming works as a spatial filter but has a poor resolution in low frequency and is not very accurate to quantify the power of extended sources or in a multi-source environment. Deconvolution methods have been developed to overcome those limitations and are briefly presented in the next section.

Deconvolution

Most deconvolution methods are applied to a beamforming output that has been performed in the frequency domain. For time-invariant conditions and decorrelated sources, the deconvolution consists in inversing the following system:

B( f ) = H( f )S( f ) with S ≥ 0 ( 2
)
where B is the vector of quadratic beamforming outputs for the set of focus points on the grid representing the sampled scan area, H is an energetic transfer function matrix and S is a vector representing the unknown quadratic source strengths. H depends on the model of sources and propagation chosen according to the physical context. When considering point sources, the columns of H are called Point Spread Functions (PSF).

Various algorithms are available in the literature and can be used to find a solution to Eq. ( 2), for instance DAMAS [START_REF] Brooks | A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays[END_REF], CLEAN [START_REF] Högbom | Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines[END_REF] or NNLS (non-negative least square) [START_REF] Lawson | Solving Least Squares Problems[END_REF]. While deconvolution can be very effective for static sources, its application to moving sources is more challenging. This problem must be addressed in a different way, as it is presented in section 2.2.

Identification of moving sources

Sources with a linear motion

The signal received by a microphone m of the array from a source s linearly moving at the constant speed V is:

p m t + R sm (t) c = Q(t) R sm (t)(1 -M cos θ sm (t)) 2 (3)
where Q(t) = q (t) 4π , q (t) is the derivative of the source mass flow, M is the Mach number (M = V c ), R sm is the propagation distance between the position of source s and sensor m, θ sm is the angle between the source movement direction and the source-sensor direction and c is the speed of sound. Beamforming focuses the array on a point F at position [X F ,Y F , Z F ] on the source-grid con-stituted of N F points, which samples the area scanned on the vehicle. With the hypothesis that the source is moving along the x axis, the beamforming output at this point is:

b(t, F, {p m }) = W (t + X F V ) N m N m ∑ m=1 R Fm (t)(1 -M cos θ Fm (t)) 2 p m t + R Fm (t) c (4)
b(t, F, {p m }) can be considered as an estimate of the time signal emitted from the moving focus point F. The focus point moves along with the vehicle, so knowledge of the vehicle kinematics is essential. To avoid an unacceptable broadening of the beamforming response main lobe when focusing at high angles from the normal to the array, a monitoring area in front of the array is defined (Fig. 1). W (t) is a centred apodization window which is null for all |t| > L/V where L is half the length of the monitoring area (Fig 1).

The expression (4) corresponds to inversing Eq. ( 3) for each microphone and averaging the result over all the sensors.

Sources with a rotating motion

The time domain focusing method described in the previous section for a source traveling with a linear motion is directly applicable to a source traveling with a rotating motion. Equation ( 3) is still valid and only windowing parameters have to be adapted in Eq. ( 4). Of course, geometrical parameters R Fm (t) and θ Fm (t) have to take into account the rotating motion, and the Mach number depends on the distance between the source point and the rotation axis. This formulation of the beamforming applied to rotating source has been used in several papers (see for instance [START_REF] Geng | Identification of rotating sound source based on microphone array measurements[END_REF][START_REF] Minck | Fan noise analysis using a microphone array[END_REF][START_REF] Sijtsma | Location of Rotating Sources by Phased Array Measurements[END_REF]), the originality of this work is to implement it in the frame of the CLEANT methodology, which is detailed in the next section.

CLEANT procedure

CLEANT is an iterative deconvolution approach which operates in the time domain. Using conventional moving source beamforming, it gradually subtracts the source time contributions from the microphone signals. The "dirty map" and "clean map" specific to the CLEAN method [START_REF] Högbom | Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines[END_REF] are of a different nature in this CLEANT procedure: they are arrays of source time signals associated with the main source locations. The term "map", although inappropriate, is used in the following for these arrays so as to highlight the parallels with CLEAN.

The beamforming mentioned in the algorithm corresponds to the moving source beamforming as described in 2.2. The output is composed of the signals b(t, F, {p m }) reconstructed for each point F of the source-grid during the time interval T F , completed with zero-padding over the recording interval T .

The algorithm performs the following steps:

1. Initialization of the dirty "map"D and the clean "map" Γ at iteration i = 0:

D (0) (t, F) = b(t, F, {p m }) ∀F ∈ 1; N F Γ (0) (t, F) = 0 p res(0) m (t) = p m (t)
2. At iteration i (i ≥ 1), detection of the signal D (i-1) (t, F) of the dirty "map", having the maximum energy:

F = argmax F T D (i-1) (t, F) 2 dt 3.
The dominant source contribution is removed from the microphone signals:

p res(i) m t + R Fm c = p res(i-1) m t + R Fm c -γ D (i-1) t, F R Fm (t) 1 -M cos θ Fm (t) 2 ∀m ∈ 1; N m
where γ ∈ [0, 1] is the CLEANT loop gain and p res m is the signal on microphone m cleaned from the dominant source time signal contribution. 4. Clean "map" is updated:

Γ (i) (t, F) = Γ (i-1) (t, F) + γD (i-1) (t, F) 5.
Dirty "map" is updated:

D (i) (t, F) = b(t, F, {p res(i) m }) ∀F ∈ 1; N F

A new iteration starts back at step 2 unless one of the stopping criteria is

The stopping criteria are defined as follows:

• At each iteration, the total amount of energy left in the dirty map should decrease. However, when all the significant sources have been removed, only noise is left and this energy can increase from this point: the algorithm can stop when the energy at an iteration is higher than at the previous one.

• A maximum number of iteration can be set by the user: depending on the gain factor chosen and the possible knowledge of the number of sources expected, the user can shorten the duration of the method without waiting for the first criterion to be reached.

The CLEANT loop gain γ ∈ [0, 1] is chosen to remove only a portion of the dominant source estimate found at each iteration. Indeed, the estimate of the source at this location is likely to include contributions from the beamforming sidelobes of nearby sources. For an optimum robustness of the algorithm, one should choose a quite small value for γ. However, the smaller the loop factor, the longer the algorithm, and values around γ = 1/2 are usually good enough in most cases. Even γ = 1 gives good results in the simplest cases (e.g. simulation of a single source).

In the end, all the non null reconstructed source time signals are stored, as well as the associated positions, and become available for any analysis, such as a power spectral density estimate for instance. Then, using the clean "map" Γ(t, F), we can create a deconvoluted display of the source locations and of their levels on the source-grid, at any frequency.

EXPERIMENTAL APPLICATION OF THE METHOD: CASE OF A PENDULUM

This section presents an academic experimental set-up used to test the method and compare it to CBF in real conditions. In order to carry out a scaled experiment of a vehicle in pass-by condition, involving sources in linear motion, at an approximately constant speed, a pendulum with a long rod carrying a noise source has been used. The circular movement can be locally approximated by a linear movement with the condition that the monitoring area is small with respect to the rod length. The parameters of the experiment are described below.

Experimental parameters and configurations

The pendulum is made with a three meter-long rigid rod that revolves around an axis supported by a weighted structure (Fig. 2). An optical system is used to get information on kinematics (position, speed). The microphone array has 30 microphones, regularly spaced by 2 cm. Measurement is in direct proximity of the pendulum position at rest, which is used as a reference. Thus, the horizontal array is centered on this reference position and located at the height of the rod end, which supports the sources. A microphone, providing a reference signal for further research, is also fixed to the rod end and placed in direct proximity of one of the sources. The sound device, located at 0.5 m from the array, is a smartphone equipped with two loudspeakers 128 mm apart, which has several advantages for this experiment. First, the signals are amplified inside the phone, so there is no need for a cable which would be impractical. Second, the two loudspeakers are independent and can play two different signals at the same time: two tone noises at different frequencies, or two uncorrelated white noises for instance. The last advantage of the device is its small dimensions which make it easy to fix at the rod end, and its lightness with no risk to modify the movement.

Finally, acoustic panels have been placed on the floor between the pendulum and the array to remove ground reflections.

The test procedure consisted in dropping the pendulum without initial speed from a variable height. Two distinct heights were chosen to obtain movements with various speeds. Two types of source signals were used : i) a tone at 6 kHz on one speaker and another at 8 kHz on the other one; ii) two uncorrelated white noises, one per speaker. One measurement was made with the pendulum at rest to get a reference absolute position of the sources and an accurate estimate of the actual sound source levels.

In all of the measurements, both CLEANT and CBF have been implemented on a single snapshot that lasted 200 ms. For CLEANT, a gain factor of γ = 0.7 has been chosen.

Experimental results and comparison with beamforming method

Only the results corresponding to the highest source speed tested are presented here, informing on the methods performance and flagrant differences in the most difficult conditions investigated. The maximal speed of the sources in this configuration is 3.5 m/s (speed of the pendulum when reaching the lowest position) and corresponds to approximatively 30 km/h at road pass-by vehicle scale. We recall that the CBF applied here is adapted to moving sources, with a sweeping focus point steered on the sources (time domain) and that the PSF for the deconvolution (frequency domain) is the same as in the static case. A first comparison is made for tone sound sources, with a 6 kHz (resp. 8 kHz) tone on the right (resp. left) speaker. Beamforming output without deconvolution points out both sources, with slightly disymetrical and flattened sidelobe patterns (Fig. 3). The deconvolution involved in CBF gives several significant source contributions instead of one expected, within the beamforming main lobe extent (Fig. 3b). All of them must be taken into account to get the overall level found in the static case (not presented here), but the position of the real source is then ambiguous. On the other hand, CLEANT yields two main sources, with spurious sidelobes for the right source but at noticeably lower levels. This removes the ambiguity of the position compared to CBF, but the level is slightly underestimated in this moving source context. The source characteristics estimated are given in Table 1 (weighted averaged source position and overall level are given in the case of important secondary source contribution). The beamforming map for moving white noise sources is blurrier than in the case of static sources. This results in very dispersed source contributions in the deconvolution map obtained with CBF (Fig. 4b). The moving source context degrades the result so much that it is not possible to determine a clear location of any of the two sources. CLEANT gives a much clearer map with two well localized sources (-74 mm and +50 mm) and almost no dispersion, the gap between sources being 124 mm (1 mm narrower than in the static context). The total sound level over the frequency range amounts to 82 dB for the right source with CLEANT and to 78 dB for the left source. The levels are slightly underestimated compared to the static case (83 dB and 79 dB), probably because of the proximity of the two sources in the moving context. This level is not calculated for CBF because of the excessive dispersion. This experiment with moving sources shows the improvement brought by CLEANT compared to CBF. The deconvolution map is precise and gives clear positions for the sources. The sources levels are estimated within 1 dB lower than in a static context.

EXPERIMENTAL APPLICATION ON ROTATING SOUND SOURCE

A validation experiment is also conducted for the case of a rotating source. A special prototype of rotating source is used, already presented in the frame of a previous study [START_REF] Geng | Identification of rotating sound source based on microphone array measurements[END_REF]. A 2-channels generator, an amplifier and two loudspeakers are embedded on an equilibrated rotating disc (see close-up view in Fig. 5, right). The disc is mounted on an electric motor and can be operated up to a rotation speed of 900 rpm. A tachometer is used to determine the angular location of the disc as a function of time. A 30 microphone array is placed parallel to the source plane, at a distance of 15 cm (see Fig. 5,left). The CLEANT algorithm proposed in this work is tested for 6 different source configurations: 3 different rotation speeds (0, 300 and 900 rpm), for either 1 or 2 sources activated. The results obtained for the configuration (2 sources, 900 rpm) are given in Fig. 6 at different iterations of the algorithm, in order to illustrate the efficiency of CLEANT in these conditions. After only 2 iterations, the two sources are correctly localized and quantified. The position of the sources are perfectly localized (within the limitation of the source grid resolution) for all source configurations.

The ability of the method to correctly quantify source strength in any operating condition is illustrated in Table 2, in which the acoustic power level identified for each source is given for all source configurations. The identified acoustic powers for each of the two sources for different rotation speeds show very little dispersion, less than 1dB. For configurations with 2 active sources, source number 2 is systematically found about 1 dB louder than source number 1, which shows the ability of the method to correctly rank the noise sources even in the case of a very small difference of level.

CONCLUSIONS

CLEANT is a new approach for the deconvolution of a beamforming output using a matching pursuit inspired algorithm, operating exclusively in time domain and available for moving sources as well. Its efficiency has been proved in various experimental situations presented in this paper. In the case of sources in linear motion, it has been compared to conventional beamforming followed by a deconvolution based on the static point spread function: CLEANT showed a great improvement, either for tone or white noise sources. In the case of rotating sources, it provided accurate results in only few iterations, with a very satisfactory quantification and source ranking ability. The strength of this method is to be applicable to any beamforming output performed in time domain which is well adapted to the identification of moving sources, whether in linear or rotating motion.

Figure 1 :

 1 Figure 1: Pass-by configuration in the fixed (green) and moving (red) coordinate systems.

Figure 2 :

 2 Figure 2: Global view of the experimental set-up.

Figure 3 :

 3 Figure 3: Comparison of deconvolution output for tone noise moving sources.

Figure 4 :

 4 Figure 4: Comparison of deconvolution output for white noise moving sources.

Figure 5 :

 5 Figure 5: Experimental setup : rotating source in front of a 30 microphone array (left). Close up view of the rotating source with the two loudspeakers (left).

Figure 6 :

 6 Figure 6: Source rotation speed: 900 rpm. Evolution of Dirty (left) and Clean (right) maps, iteration # 1 (top) to 3 (bottom). Wideband integration [0.5-5] kHz. Loop gain = 1. Percentage values above dirty maps (left) represent the ratio of residual acoustic pressure that is used to obtain the map. Percentage above clean maps (right) represents the part of the measured pressure that is recovered by the identified sources.

Table 1 :

 1 Source position and level estimates with moving tone sources.

	Method	Source 1 (8 kHz)	Source 2 (6 kHz)	Source gap
		Position (mm) Level (dB) Position (mm) Level (dB)	(mm)
	BF	-74	109	+53	114	127
	CBF	-73	109	+53	115	126
	CLEANT	-72	108	+52	113	124

Table 2 :

 2 Source acoustic power levels identified for all source configurations and rotation speeds. Levels are integrated in the frequency range [0.5 -5]kHz

	Speed		0 rpm	300 rpm	900 rpm
	Active source S1 only S1 and S2 S1 only S1 and S2 S1 only S1 and S2
	S1 Level (dB)	59.4	59.2	58.8	58.7	59.4	59.1
	S2 Level (dB)	n/a	60.3	n/a	59.8	n/a	59.7

th Berlin Beamforming Conference 2018 Cousson et al.

ACKNOWLEDGEMENTS

This work was performed within the framework of the Labex CeLyA of Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ ANR-11-IDEX-0007).