domain: spi info spi.auto

text:
Valentina Mattioni
email: valentina.mattioni@unibo.it

Edoardo Ida'
email: edoardo.ida2@unibo.it

Marc Gouttefarde
email: marc.gouttefarde@lirmm.fr

Marco Carricato
email: marco.carricato@unibo.it

A Practical Approach for the Hybrid Joint-Space Control of Overconstrained Cable-Driven Parallel Robots

Keywords: Overconstrained robots, cable-driven parallel robots, force distribution, workspace computation, sensitivity analysis, hybrid joint-space control

Hybrid joint-space control of overconstrained cable-driven parallel robots requires regulating the length of as many cables as the degrees of freedom of the end-effector, and controlling the tension of the remaining ones. Previous studies showed how to choose which cables to length control, and which ones to tension control, according to minimum tension error propagation techniques. The sets of tension-controlled and length-controlled cables achieving minimum tension error propagation typically vary throughout the workspace, which is an undesirable outcome. This paper aims to achieve practicality in hybrid joint-space control, by investigating which cables to be tension-and length-controlled in the whole workspace, without needing to change such sets. The HRPCable robot at LIRMM is analyzed for this application, and experiments are carried out on it.

Introduction

Overconstrained Cable-Driven Parallel Robots (OCDPRs) employ more cables than those strictly needed to control the end-effector EE degrees of freedom (DoFs). In this way, cables pull against each other, keeping the system under tension and control over a wide range of externally applied loads [START_REF] Gouttefarde | Geometry selection of a redundantly actuated cable-suspended parallel robot[END_REF].

For an overconstrained system, determining cable forces is crucial for effectively controlling the EE motion. An optimal set of tensions is usually computed beforehand, solving the force (or tension) distribution (FD) problem based on a chosen criterion, such as minimum tension, maximum stiffness, or others [START_REF] Müller | Analysis of a real-time capable cable force computation method[END_REF]. Lastly, the tensions are converted to servo-drive set-points [START_REF] Santos | A simple and efficient non-model based cable tension control[END_REF]. If the final objective is the EE pose control, most existing control strategies for OCDPRs imply cascaded control loops [START_REF] Oh | Cable suspended planar robots with redundant cables: Controllers with positive tensions[END_REF].

A different strategy was proposed as a simple feed-forward alternative to more complex closed-loop controllers [START_REF] Bouchard | A simple control strategy for overconstrained parallel cable mechanisms[END_REF][START_REF] Bruckmann | A new force calculation algorithm for tendon-based parallel manipulators[END_REF], and it was later called hybrid force-position control in joint space (hybrid joint-space control, in short) [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors in overconstrained cable-driven parallel robots[END_REF]. According to this strategy, the robot inputs, namely the joint-space controlled variables, are divided into position-and force-controlled ones, similar to hybrid control techniques in the Cartesian space, where EE controlled variables are similarly divided [START_REF] Kraus | Hybrid position-force control of a cabledriven parallel robot with experimental evaluation[END_REF]. In practice, hybrid joint-space control results in several cables equal to the robot degree of redundancy being tension controlled, while the others are length controlled. Since the number of length-controlled cables is equal to the EE DoFs, the pose of the latter is completely determined (if elasticity is negligible).

A selection criterion for force-controlled cables in hybrid joint-space control was initially proposed in [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors: A preliminary investigation[END_REF] and later extended in [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors in overconstrained cable-driven parallel robots[END_REF]. This criterion is based on evaluating a performance index, called force-distribution sensitivity to cable tension error (FD sensitivity, in short), which allows quantifying how the FD is influenced by the tension error in a chosen force-controlled cable set. From a practical point of view, the FD sensitivity allows identifying which cable set is best to be force controlled to ensure the lowest error in the overall FD. Usually, there are different zones in the robot workspace characterized by a different optimal cable set. A theoretical minimal cable-tension error would require switching the force-controlled cable set in real time throughout the robot workspace. This feature is unpractical, especially for 8-cable robots, which can have up to 28 different zones [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors in overconstrained cable-driven parallel robots[END_REF]; additionally, the switching procedure could cause a discontinuity in the control action and, thus, it could make the robot operation unsafe or jerky.

For hybrid joint-space control purposes, this paper presents a heuristic approach for selecting the force-controlled cable set of an 8-cable OCDPR to keep the cabletension error as low as possible throughout the WS, without changing the forcecontrolled cable set. Two practical methods are proposed based on this approach, exploiting the results obtained in controlling a planar 4-cable case [START_REF] Mattioni | Design of a planar cable-driven parallel robot for noncontact tasks[END_REF]. Then, the hybrid joint-space strategy is tested on the HRPCable Robot, an 8-cable OCDPR. The paper is structured as follows. Section 2 introduces the kinetostatic model of a generic OCDPR, and recalls elements of force-distribution sensitivity analysis. In Section 3, the wrench-feasible workspace of the HRPCable robot is characterized, and its forcedistribution sensitivity is analyzed. In Section 4, a practical method for selecting the force-controlled cables to be used in hybrid joint-space control is proposed and applied to the HRPCable robot. Section 5 reports the results of some experiments on the application of the proposed methodology on the HRPCable robot. Finally, conclusions are drawn in Sec.6, and a perspective on future research developments is given.

Kinetostatic Modeling

An OCDPR consists of a mobile platform (the EE) connected to a fixed base by n > n d cables, which are always under tension. n d is the number of EE DoFs, and the degree of redundancy [START_REF] Ming | Study on multiple degree-of-freedom positioning mechanism using wires. i: Concept, design and control[END_REF] is defined as µ = nn d . Cables are coiled and uncoiled by servocontrolled winches, characterized by a nominally constant transmission ratio. Their length varies proportionally to the actuator rotations if elasticity and hysteresis are neglected [START_REF] Idà | Cable-driven parallel robot actuators: State of the art and novel servowinch concept[END_REF].

In this work, cables are considered ideal unilateral constraints, thus no mass and elasticity effects are taken into account. According to Fig. 1, Ox y z is an inertial frame, Fig. 1: Geometric model of a cable transmission.

whereas P x ′ y ′ z ′ is a mobile frame attached to the EE center of mass. The EE pose is described by the position vector p of P , and a rotation matrix R = R(ϵ), which describes the orientation of the mobile platform with respect to the base of the mechanism. ϵ is a minimal set of orientation parameters, i.e., three Euler's angles. The EE pose is defined as ζ = p T ϵ T T .

Cables are attached to the platform, and the fixed base at distal and proximal anchor points A i and B i , respectively, for i = 1, ..., n. a i and a ′ i are the position vectors of A i with respect to O and P in the inertial frame (Fig. 1). The constant position vector of A i in the mobile frame is denoted by a ′ i ,P . b i denotes the position of B i with respect to O. The latter is a constant vector if the cable exit point is an eyelet; otherwise, it depends on the cable transmission model [START_REF] Idá | Automatic self-calibration of suspended under-actuated cable-driven parallel robot using incremental measurements[END_REF], and, ultimately, the EE pose.

The i -th cable is modeled as the line segment between points A i and B i and, thus, the i -th cable vector can be expressed as [START_REF] Pott | Cable-driven Parallel Robots: Theory and Application[END_REF]:

ρ i = a i -b i = p + R a ′ i ,P -b i (1)
Considering l i as the i -th cable length, the constraint imposed by the i -th cable, and the unit vector of the i -th cable, pointing from the base towards the platform, are:

ρ T i ρ i -l 2 i = 0, t i = ρ i l i (2)
The static equilibrium of the mobile platform can be formulated as follows:

-J T τ -W = 0, J i = t i T -t T i ã′ i ( 3
)
where τ ∈ R n is the array of cable tensions, W ∈ R n d is the external wrench acting on the platform, J T ∈ R n d ×n , usually referred to as the structure matrix, is the transpose of the inverse kinematics Jacobian matrix, and J i is the i -th row of J. The symbol ∼ over a vector denotes its skew-symmetric representation. For OCDPRs (µ > 0), the solution of Eq.3 is underdetermined if all bodies are considered rigid, and infinitely many solutions exist for a given wrench and EE pose. The solution to this problem consists in computing a tension distribution according to a given criterion [START_REF] Müller | Analysis of a real-time capable cable force computation method[END_REF].

If a hybrid joint-space controller is used, tension errors in force-controlled cables influence the tension in length-controlled cables, and, ultimately, the FD and the robot performances. Such influence can be analyzed by computing the forcedistribution sensitivity to cable-tension error. Assuming that the tension of the last µ cables, denoted by τ c , is to be controlled, the array τ of cable tensions and the structure matrix can be partitioned as:

τ ≜ τ d τ c , J T = J d J c ( 4
)
where

J d ∈ R n d ×n d and J c ∈ R n d ×µ
, and τ d ∈ R n d denotes the tension of the lengthcontrolled cables. The force-distribution sensitivity σ is defined as the maximum tension variation in the position-controlled cables generated by a unit tension variation (or error) of the force-controlled cables, namely [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors in overconstrained cable-driven parallel robots[END_REF]:

σ ≜ max ∥τ c ∥ ∞ =1 ∥∆τ∥ ∞ = ∥ -J -1 d J c ∥ ∞ (5)
The FD sensitivity index can be computed for every set j of force-controlled cables, thus obtaining C values, σ j , for j = 1, . . . ,C . C is the maximum number of ways of selecting µ cables out of n, without repetition. After computing all possible C values of the FD sensitivity, the minimum value can be identified as:

σ ⋆ = min σ j , j = 1, . . . ,C , C = n µ = n! (n -µ)!µ! (6)
The index σ ⋆ points out the cable set that, if force-controlled, propagates the least tension control errors in the other cables. That cable set is denoted by j ⋆ .

HRPCable Workspace Characterization

In this section, the constant-orientation wrench-feasible WS of an 8-cable OCDPR prototype at LIRMM is characterized with respect to its force-distribution sensitivity to cable-tension error. The prototype, known as HRPCable, was used as an experimental set-up during the EU project Hephaestus [START_REF] Taghavi | Cable-driven parallel robot for curtain wall modules automatic installation[END_REF]. It can be installed in a suspended or overconstrained configuration. For this work, the robot is configured as an OCDPR (Fig. 2). In general, an OCDPR is in a wrench-feasible pose ζ if at least one FD (τ ∈ R n ) exists, for a given wrench W ∈ R n d , such that the EE equilibrium in Eq. ( 3) is satisfied with bounded tensions:

∃τ : τ mi n ⪯ τ ⪯ τ max , -J T τ -W = 0 (7)
where τ mi n and τ max are the tension limits, and the symbol ⪯ denotes element-wise inequality between a scalar and a vector quantity.

The inertial frame Ox y z is located in the center of the base, and the moving frame P x ′ y ′ z ′ is at the center of the mobile platform, coinciding with its center of mass. The only external load applied to the robot EE is gravity, and its mass is m = 23kg. Table 1: Geometrical properties of the HPRCable OCDPR. The geometrical properties of the robot are summarised in Tab. 1, and the tension limits are set to τ mi n = 30 N, and τ max = 2000 N. The robot has two degrees of redundancy (µ = 2), and thus 2 cables have to be force-controlled, whereas the others may be length-controlled when a hybrid joint-space strategy is adopted for its control. According to Eq.( 6), the number of possible cable combinations is C = 28.

The minimum FD sensitivity σ ⋆ in the constant-orientation wrench-feasible WS (with R = I 3 , where I 3 is the identity matrix of order 3) is shown in Fig. 3, whereas Fig. 4 highlights the volumes where the optimal cable set j ⋆ remains constant. Each color is associated with a certain j ⋆ pair. The 28 distinct constantj ⋆ zones cannot be easily pointed out, and their boundaries are not linear and not easily identifiable.

The analysis of the FD-sensitivity variation for different choices of forcecontrolled cable pairs is conducted to show how a poor choice of force-controlled cable pair can have negative consequences on robot control. To quantitatively assess how specific cable pairs can be an issue when force-controlled, two thresholds for the sensitivity values are selected: σ = 2N is a practically acceptable value of the minimum sensitivity σ ⋆ [10], whereas σ = 10N is a value where control performances are expected to be appreciably degraded. As an example, if we choose cable pair 2-4 as the force-controlled one (Fig. 5), only in 4% of the WS, the sensitivity value σ is lower or equal to 2N, 30% of the workspace is characterized by 2N < σ < 10N, and Fig. 3: Variation of the minimum FD sensitivity σ ⋆ throughout the wrench-feasible WS of HRPCable OCDPR. Fig. 4: Wrench-feasible WS of HRPCable characterized by 28 distinct constant j ⋆ zones, where j ⋆ identifies the cable pair to be force-controlled to have the lowest error in the tension distribution in each zone. Each color is associated with a certain j ⋆ pair, and a detailed description of coloring is omitted since it would not add descriptive value.

66% by σ ≥ 10N (Fig. 5b), with peak values higher than 1000N (Fig. 5a) 3 . On the contrary, considering the pair 7-8 (Fig. 6), 68% of the WS poses have sensitivity values lower than 2N, while the others are between 2N and 4.41N, which is the absolute maximum value of σ in this case (Fig. 6a). This means that by choosing the pair 7-8 to be force-controlled, the tension error throughout the whole WS would be, at most, around 4.4 times the error in the force-controlled cables.

A practical cable-selection strategy

As highlighted in Sec.3, the constantj ⋆ map (see Fig. 4) derived from the FDsensitivity analysis is not straightforward to interpret for control purposes. In fact, the constantj ⋆ zones are numerous (C = 28) and they have a highly irregular shape: switching between force-and length-controlled cables while crossing the borders of constantj ⋆ zones could be very frequent, impractical, and possibly leading to various kinds of control problems, such as control signal discontinuities, vibrations, or unjustifiably large control efforts. A solution to these problems can be found, such as smoothing the control signal with additional low-pass filters to be activated during switching. We propose an alternative that does not require the addition of such control modules, which ultimately may limit the performances due to the induced control lag.

A heuristic method to keep the tension error near the theoretical smallest possible value during the EE motion throughout the WS is introduced. A single cable set can be chosen to be force-controlled in the whole WS. The analysis of several cases showed that to select this cable set, one should evaluate how many points over the totality of the WS have an acceptable value of σ. We define a value of σ as acceptable, if it is less than or equal to the maximum value of the minimum FD sensitivity σ ⋆ (called σ ⋆ max ) in the WS. σ ⋆ is by definition, the minimum value of σ for every j -th cable set (Eq. [START_REF] Idà | Cable-driven parallel robot actuators: State of the art and novel servowinch concept[END_REF]. As an example, for the HRPCable σ ⋆ max is equal to 2N. Considering this criterion, the force-controlled cable set can be chosen by comparing the percentage of WS configurations whose sensitivity is within the acceptable limit (σ ≤ σ ⋆ max ) for each cable combination. The best candidate to be forcecontrolled would be the cable set having the highest percentage of WS configurations within the acceptable sensitivity value. For example, comparing all the cable combinations for the HRPCable robot (the complete analysis is not here reported for the sake of brevity), the pair 7-8 is chosen, as the percentage of FD sensitivity values lower or equal to σ ⋆ max = 2N is the highest (68%). If this rationale is followed to keep the tension error as low as possible, during the EE motion, one should take into account that in certain WS zones the cable-tension error could be excessively amplified where σ has its peaks (e.g., the bottom right corner of the WS, in Fig. 6a), which could limit the robot WS. In other words, the EE should work far from the peaks to avoid tension-error amplification, unless these peaks have a reasonably small value.

Furthermore, for a given force-controlled cable set, the percentage of WS configurations whose sensitivity is within the acceptable limit could be enlarged, optimizing the robot geometry. In fact, from the analysis of the 8-cable cases, it was noticed that a lack of symmetry in the robot structure makes some cable sets more suitable to be force-controlled with respect to others. In a symmetric case, such as the 8-cable IPAnema 3 OCDPR reported in [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors in overconstrained cable-driven parallel robots[END_REF], there is no pair significantly better than the others. On the contrary, in the HRPCable case, whose cable attachment points are not symmetric to avoid cable interference (see Tab.1 and Fig. 2b), the results obtained with the pair 7-8 are clearly better than the results obtained with other pairs, and could be further optimized, for example by allowing the EE to rotate as demonstrated in [START_REF] Mattioni | Force-distribution sensitivity to cable-tension errors: A preliminary investigation[END_REF]. Pair 7-8 not only has the highest percentage of configurations with an acceptable σ among all cable combinations, but it also shows the lowest maximum value of σ in the entire WS (4.41N). Taking this observation into account, it should be possible to optimize the cable anchor point positions in order to enlarge the volume with σ < 2N for this cable set, thus completely removing the need for switching tensionand length-controlled cables 4 .

In general, the analysis reported in this Section highlights that some robot geometries are more effective at being hybrid-controlled than others, and considering this control option at design phase could be more effective than an a-posteriori analysis.

Experimental Validation

The practical hybrid-input control strategy proposed in this paper is tested on the HRPCable prototype (Fig. 7). Geometrical and inertial parameters of the robot are summarized in Sec. 3, and gravity is the only external wrench acting on the EE. The trajectory planning, the computation of the tension distribution, and the hybridinput control strategy run on a Beckhoff real-time IPC at 0.5 kHz rate. At the servodrive level, for ease of implementation, position set-points are always assigned. The k-th control update on the j-th motor, p j ,k , is obtained as follows:

p j ,k = p j ,k-1 + ∆p j ,k ,
∀k, for j = 1, . . . , 8

where ∆p j ,k is computed differently, according to the cable being length-of tensioncontrolled. In the former case, ∆p j ,k is calculated through inverse kinematics, whereas in the latter case ∆p j ,k = v j ,k ∆t , with ∆t = 2 ms and v j ,k is assigned, as in Eq. 1 of [START_REF] Santos | A simple and efficient non-model based cable tension control[END_REF], as the nominal cable velocity to follow the prescribed path. While the EE orientation is kept such that R = I 3×3 , the same trajectory of the EE reference point is tracked by first controlling the tensions of the cable pair 7-8, which is the best overall, and then by controlling the tensions of the pair 2-4, the worst overall. In both cases, the EE is moved from the home position, p H = [0, 0, 0] T m, to the final position p F = [-0.2, 0.2, -0.05] T m in 25s. Then, the robot is halted for 10s, and finally goes back to the home position in 25s. Please note that the HRPCable prototype is intended for quasi-static operations, and faster trajectories could not be tested. The tension distribution is computed by applying a minimal 2-norm algorithm with τ mi n = 30N and τ max = 500N, that is [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF]:

min τ ∥τ∥ 2 with τ mi n ⪯ τ ⪯ τ max , -J T τ -W = 0 (9)
Tension error norms, denoted by ∥e i ∥ = ∥τ ⋆ i -τ i ∥, with i = 1, . . . , 8, and computed by subtracting the results of Eq. ( 9) τ i to the experimentally obtained values τ ⋆ i , are shown in Fig. 8; the first trajectory from p H to p F is executed from t = 25s to t = 50s, while the second one from p F to p H is executed from t = 60s to t = 85s. Errors are generally fairly large due to the robot's need for maintenance and re-calibration, and the limited time available for experiments was not sufficient for statistical analysis of more trajectories. Nonetheless, Fig. 8a shows that tension-controlling cable pair 7-8 performs generally better, with lower maximum tension errors in several cables, with respect to tension-controlling cable pair 2-4 (Fig. 8b). This result is further confirmed in Fig. 9 by looking at the 2-norm of the FD distribution errors, denoted by e x y = ∥τ x y -τ ⋆ x y ∥ with x y indicating the experiment where the x and y cables were tension-controlled, τ is the result of Eq. ( 9), and τ ⋆ is the array of the experimental tensions: tension-controlling cable pair 7-8 always results in a smaller 2-norm error than tension-controlling cable pair 2-4. Figure 9 shows an additional result, which was expected in this study: there may be a better cable pair to be locally tension controlled (cable pair 2-7 for the selected trajectory), and the larger error in cable pair 7-8 is the trade-off for not needing to switch the tension-controlled cables throughout the workspace.

Conclusions and Outlook

This paper proposed a practical approach for the selection of tension-controlled cables when a hybrid joint-space control technique is used in overconstrained cabledriven parallel robots. The proposed approach is based on the computation of the force-distribution sensitivity to cable-tension errors (FD sensitivity). In general, selecting tension-controlled cable pairs through a FD sensitivity analysis would require switching the selected cables throughout the workspace, which may be inconvenient. Thus, in this paper, a "best overall" selection strategy that would allow to tension-control always the same cable pair was introduced. This strategy is based on evaluating which tension-controlled cable pair has the largest workspace volume with a force distribution sensitivity below an acceptable threshold. The method was successfully validated through experiments, also showing that the best cable pair overall may not be locally better than other choices. Future studies will focus on optimizing a manipulator EE geometry so that the best cable pair overall is also locally better than any other cable, a result that was hinted at by the analysis proposed in this paper.

  (a) Robot layout.(b) Close-up view of the mobile platform.

Fig. 2 :

 2 Fig. 2: HRPCable prototype at LIRMM (CNRS -University of Montpellier).

3

  Void zones in the top right corner of Fig. 5a are due to the contour plot nature of the illustration: the FD sensitivity change negligibly in said volume with respect to the rest of the workspace, and contour lines are not depicted. (a) Variation of the FD sensitivity when the cable pair 2-4 is force-controlled. (b) σ values percentage.

Fig. 5 :

 5 Fig. 5: Variation of the FD sensitivity σ for the cable pair 2-4 of HRPCable robot.

( a )

 a Variation of the FD sensitivity when the cable pair 7-8 is force-controlled. (b) σ values percentage.

Fig. 6 :

 6 Fig. 6: Variation of the FD sensitivity σ for the cable pair 7-8 of HRPCable robot, which gives the best results above all the possible combinations.

Fig. 7 :

 7 Fig. 7: HRPCable prototype.

( a )

 a Cable pair 7-8 is tension controlled. (b) Cable pair 2-4 is tension controlled.

Fig. 8 :

 8 Fig. 8: Experimental tension distribution errors.

Fig. 9 :

 9 Fig. 9: FD error 2-norm when cable pairs 7-8, 2-4, and 2-7 are tension controlled.

This option will be further analyzed in future studies, but preliminary simulations showed its theoretical feasibility.

Acknowledgments:

The authors greatly acknowledge support of the European Union through FEDER grant n°49793. This work has also been partially supported by ROBOTEX 2.0 (Grants ROBOTEX ANR-10-EQPX-44-01 and TIRREX ANR-21-ESRE-0015) funded by the French program Investissements d'avenir.


Gu Énol É Ch

Roman Le Goff Latimier

Fabien Sanchez
email: fabien.sanchez@ens-rennes.fr

Hamid Ben
email: benahmed@ens-rennes.fr

Ahmed Satie

Misestimation of Impedance Values Within a Distribution Network Optimal Power Flow

Keywords: Optimal Power Flow, distribution networks, impedance estimation, voltage limits, social welfare

Distribution networks are likely to be significantly impacted by the energy transition as numerous flexible agents will be connected there. Management algorithms within these networks are therefore becoming more complex and must consider the physical constraints of the network. However, line impedances in these networks cannot be assumed to be perfectly known, leading to suboptimal management and constraint violations. Considering the Optimal Power Flow as a reference for optimal management, the consequences of these misestimations are here outlined on an illustrative test case. The performed sensitivity studies indicate that the statistical distribution used to describe impedance errors does not influence the results. A combination of decentralized generators and flexible consumers within the same neighborhood limits the impact of misestimated impedances.

INTRODUCTION

The energy transition requires the generalization of renewable productions [START_REF] Satie | An integrated review and analysis of multienergy transition from fossil fuels to renewables[END_REF]. These are mostly decentralized and of small unitary power. Most of these installations are therefore likely to be connected to distribution networks -in terms of the number of installations, if not in terms of connected power [START_REF] Humayd | Distribution system planning to accommodate distributed energy resources and pevs[END_REF]. Besides, the issues of predictability and intermittency raised by renewable energies require the implementation of demand side management [START_REF] Strbac | Demand side management: Benefits and challenges[END_REF]. The joint management of these decentralized productions and flexible consumptions -both comprising many small power agents -is a major challenge that leads to a rich and dynamic literature [START_REF] Goff Latimier | Stochastic optimization of an electric vehicle fleet charging with uncertain photovoltaic production[END_REF]- [START_REF] Goff Latimier | Online learning for distributed optimal control of an electric vehicle fleet[END_REF].

These investigations are mainly carried out in the context of distribution networks. The modeling of physical quantities in each line and each bus is thus not usually taken into account [START_REF] Goff Latimier | Distributed optimisation with restricted exchanges of information: Charging of en electric vehicle fleet[END_REF], [START_REF] Ghazvini | Congestion management in active distribution networks through demand response implementation[END_REF]. Indeed, the design capacity of these networks is currently much higher than the actual flows. Congestion is therefore currently uncommon. The limiting operational constraint is most often voltage limits at the furthest buses from the connection point to a highervoltage network [START_REF] Sun | Review of challenges and research opportunities for voltage control in smart grids[END_REF]. But in the future, downstream flows will strongly increase due to electric mobility [START_REF] Venegas | Active integration of electric vehicles into distribution grids: Barriers and frameworks for flexibility services[END_REF]. Upstream flows will also grow because of distributed generators. In such a context, the challenge will not only be to maintain the power quality, but also to overcome congestion. According to current state-of-the-art approaches, this evolution should lead to a strengthening of distribution networks. This would result in a considerable consumption of raw material, but also in significant deployment costs. Moreover, such a revision of distribution networks would require extensive periods of work. An alternative to this consolidation would be to apply mechanisms of distributed management that would take into account the physical state of the network [START_REF] Bouloumpasis | Congestion management using local flexibility markets: Recent development and challenges[END_REF]. The aim would then be not only to match flexibilities with renewable generation but also to limit network congestion and voltage violations [START_REF] Chérot | A real-time congestion control strategy in distribution networks[END_REF]. These mechanisms would then be comparable to optimal power flow (OPF) problems [START_REF] Gill | Dynamic optimal power flow for active distribution networks[END_REF], [START_REF] Soares | Active distribution grid management based on robust ac optimal power flow[END_REF].

Distribution networks are yet far more complex than transmission networks in the sense that they include a vast number of connected lines and agents. Moreover, their topology is not perfectly known since they are constantly evolving: it is not always possible to know with certainty on which phase a device is connected. Wear and tear, degradations or weather variations can cause significant fluctuations in the actual impedance of lines compared to their nominal impedance. Furthermore, distribution networks are only very sparsely monitored, compared to transmission networks [START_REF] Gholami | Detecting the location of shortcircuit faults in active distribution network using pmu-based state estimation[END_REF]. This low observability is legitimate in the current situation, as their design capacity is significantly larger than the effective flows and because of the almost complete lack of flexible resources in distribution neighborhoods. Research on the observability of these networks is being developed [START_REF] Zhang | Topology identification and line parameter estimation for non-pmu distribution network: A numerical method[END_REF]. Nevertheless, it would appear to be certain that line impedances cannot be considered as known for sure in a distribution network: each node cannot be monitored in a near future and estimators would still commit a residual error [START_REF] Brouillon | Bayesian error-in-variables models for the identification of distribution grids[END_REF].

However this lack of knowledge of line impedances cannot but have repercussions on power flow (PF) and OPF results. Consequences may include, in increasing order of criticality, a suboptimality of the proposed management, an increase in losses, non-compliance with voltage limits, or violations of line capacities. The purpose of the present contribution is therefore to assess the consequences of impedance estimation errors on the results of the OPF in distribution networks. This publication is organized as follows: Section 2 will present the methodology and the formulation of the OPF used. Then section 3 will present the impact of impedance estimation errors on an illustrative distribution network test case. Section 4 will examine the sensitivity of our previous findings by considering the impact of the statistical distribution chosen to describe the impedance estimation errors. It will also explore how the results vary between predominantly consuming and predominantly producing distribution areas.

METHODOLOGY AND TEST CASE

This section first introduces the methodology adopted to evaluate the impact of line impedance estimation errors on OPF results. Then, the test case will be presented.

Misestimation Impact Method

The chosen methodology is summarized in Figure 1. 1 ⃝ With Z being the estimated and inaccurate impedances, the first step is to perform an OPF as a DSO might operationally do. Only the active P -and reactive Q when applicable -power injection plans are considered. Indeed, voltages and line flows calculated by this first OPF are incorrect, since they were based on incorrect impedances values. 2 ⃝ The actual values of the network impedances Z are assumed to deviate from Z by an estimation error ∆Z. Since the estimated values can only worsen during the lifetime of the network, ∆Z are strictly positive. They are randomly drawn according to a probability distribution with support in R + . The consequences of the choice of this distribution and its parameters will be evaluated in section 4. 3 ⃝ These actual impedances Z are then used to assess the physical state of the network, when it is operated under the power injection plan P , Q of the first OPF. This physical state is referred to as V Figure 1 in a concise manner, but refers to both the voltages and the line flows. Social Welfare value SW is jointly assessed. Since step 3 ⃝ gathers impedance values Z and an injection plan calculated separately, there is no guarantee that the physical values V comply with network limitations. Hence constraint violations may occur. 4 ⃝ In parallel, these actual impedances Z are used to perform an OPF. The latter being carried out without bias on the estimation of the impedances, it jointly establishes the optimal power injection plan P, Q and the associated physical configuration V and social welfare SW . These results are regarded as a reference since they relate to the optimal management of the real network. Although it cannot be performed operationally, it provides a reference to evaluate the deviations of the injection plan P , Q and the suboptimality of the solution SW . Since the methodology presented in Figure 1 involves drawing random variables, a Monte Carlo approach is mandatory. Steps 2 ⃝, 3 ⃝ and 4 ⃝ are therefore repeated several times until observed deviations are statistically reproducible.

During this study, only distribution networks will be considered. Indeed, transmission networks are sufficiently instrumented and monitored so that impedance estimation errors can be considered negligible. Therefore, the following formulation is used for the OPF problem [START_REF] Jabr | Radial distribution load flow using conic programming[END_REF].

min s,S,v,l,s0 i∈N f i (Re(s i )) (1a) s.t. S ij = s i + h:h→i (S hi -z hi l hi ), ∀(i, j) ∈ E; (1b) 0 = s 0 + h:h→0 (S h0 -z h0 l h0 ); (1c) v i -v j = 2Re(z ij S ij ) -|z ij | 2 l ij , ∀(i, j) ∈ E; (1d) l ij = |S ij | 2 /v i , ∀(i, j) ∈ E; (1e)
s i ∈ S i , i ∈ N + ; (1f) v i ≤ v i ≤ v i , i ∈ N + . ( 1g
)
where s i is the complex injected power at the bus i, S ij is the complex reverse power flow between bus i and j, z ij is the impedance, v i is the square of the magnitude of the voltage and l ij is the square of the magnitude of the current. The reader is invited to refer to [START_REF] Huang | A sufficient condition on convex relaxation of ac optimal power flow in distribution networks[END_REF] for a complete analysis of this formulation. Its significant features are to allow for a relaxation of the voltage angle (1d) and of the equality constraint determining the flow in the lines (1e) [START_REF] Gan | Exact convex relaxation of optimal power flow in radial networks[END_REF]:

l ij ≥ |S ij | 2 /v i , ∀(i, j) ∈ E; (2)
Substituting it by an inequality constitutes a relaxation on a second order cone (SOC). Theses relaxations allow us to compute the exact solution of the AC-OPF problem in the case of non-mesh networks and when the reverse power flow is only active or reactive, or none [START_REF] Huang | A sufficient condition on convex relaxation of ac optimal power flow in distribution networks[END_REF]. This happens in most distribution networks and especially in our test case -except when explicitly noted. This formulation will be referred to as AC-OPF-SOC in the rest of this paper.

Test Case Description

The European Low Voltage Feeder (ELVF) test case [START_REF] Schneider | Analytic considerations and design basis for the ieee distribution test feeders[END_REF] has been selected to illustrate the consequences of impedance misestimations. It is representative of European distribution networks and is the framework for many studies [START_REF] Wang | Coordinating multiple sources for service restoration to enhance resilience of distribution systems[END_REF], [START_REF] Caballero-Pe Ña | Distributed energy resources on distribution networks: A systematic review of modelling, simulation, metrics, and impacts[END_REF]. Although its adoption here does not allow a direct generalization of the results presented, it is a relevant first step to highlight the risks related to impedance estimation errors.

To achieve OPF resolutions, objective functions must be specified for both producers and flexible consumers. These only serve the purpose of creating solicitations on the network used as an example. The remainder of this section describes them for the sake of transparency and reproducibility. The results presented in sections 3 and 4 are not affected by these chosen objective functions.

Energy can be exchanged with the external network, considered of infinite power, either positive or negative. It is connected to the slack bus, whose voltage is fixed to 1 p.u. A history of the EPEXSpot market prices 1 is used for its selling price to the distribution neighborhood. It is assumed to purchase at a price arbitrarily set at 90% of the EPEXSpot price. Not only is this discount representative of current feed-in tariffs, but it also avoids 'speculative' buying and selling phenomena within the OPF resolution. The network is randomly populated with production and consumption profiles from the following database [START_REF] Barker | Smart*: An open data set and tools for enabling research in sustainable homes[END_REF]. Selling prices of distributed producers are randomly and individually assigned according to a normal distribution with mean value 6 c€/kWh and standard deviation 2 c€/kWh. Since the database provides the power demand of each consumer without any further differentiation between usages, we choose to split the consumption profiles into a non-flexible part and a flexible one. The non-flexible part is set to 90% of the required power P 0 . Below this limit, the consumer is assumed to be willing to pay threefold the grid price. Flexibility bounds are then set at [0.9P 0 ; 1.5P 0 ] between which the consumer assigns a purchase price drawn at random according to a normal distribution of mean 15 c€/kWh and standard deviation 5 c€/kWh. Two configurations will be discussed regarding reactive power control. A first case will consider that no reactive power is produced or consumed on the network under study. This configuration is close to the current operation of distribution networks. This will be the configuration adopted in most of the studies presented here, unless otherwise stated. In a second case, all distributed generators will be considered capable of exchanging positive or negative reactive power, within the same limits as their active power. No cost will be associated with this exchanged reactive power. This would correspond to the systematic control of reactive power by the inverters of the distributed generators. This configuration does not correspond to current networks, but is likely to become more widespread, in particular to better handle voltage constraints far away from the slack bus.

The resolution of OPFs and PFs in this study is performed using the implementations proposed by PowerModels [START_REF] Coffrin | Powermodels. jl: An open-source framework for exploring power flow formulations[END_REF] in Julia and PandaPower [START_REF] Thurner | pandapower-an opensource python tool for convenient modeling, analysis, and optimization of electric power systems[END_REF] in Python. In accordance with an open and reproducible science approach, all sources can be accessed on Gitlab 2 .

ASSESSMENT OF MISESTIMATION IMPACT

The methodology described in section 2.1 is applied in this section to the test case described section 2.2. Initially, estimation errors are assumed to follow a truncated normal distribution. Effective impedances Z are then deduced from a random variable ε,

Z = Z + ∆Z = (1 + ε) • Z (3)
This random variable being only positive, it will be characterized by its distribution and its expectation, the standard deviation being its consequence. Different draws are made while shifting the mean of this law by steps of 10%. For each considered mean value, 10 networks are randomly generated, each of 130 lines, that is 1300 random variables in total. For each of these networks, the OPF is simulated for 12 time steps over 36 days evenly distributed over the year. The statistical variations of the obtained electrical quantities are then represented Figure 3.

On the left end of Figure 3, impedances are perfectly known. Thus, the OPF resolution allows setting voltages of the most solicited buses to the admissible limit values -the test case being designed to activate these limits. In contrast, moving to the right means impedances are more and more poorly estimated. More and more buses violate voltage limits, with several deciles passing one after the other the tolerated limits, indicated in red. Simultaneously, a global decrease in network voltages can be noticed while impedances increase. This is due to the test case including more consumers than producers: the energy flow being then overall consumed, the impedance increase leads to a voltage decrease at the end of lines.

Besides the allocation between consumers and decentralized producers adopted for this study, the voltage decrease is also the consequence of the choice to consider here only impedance errors causing degradations rather than improvements. Typically, this refers to the case of a DSO estimating its impedances Z by their nominal values before wear. In contrast, impedance values provided by a data assimilating estimator would generate a positively and negatively distributed error. Here, when the impedances are incorrectly estimated by an average of 40%, the average voltage is reduced to 0.97 p.u. and a third of the buses violate the specified limits. It should be noted that extreme values can be observed in the experimental statistical distribution, both positively and negatively.

All voltages are thus affected by fluctuations of varying magnitude. The observation is, however, very different concerning the injected powers -middle panel of Figure 3. Here are shown the deviations between the injected powers calculated by an OPF using the true impedance values Z (block 4 ⃝ on Figure 1) and those from an OPF using erroneous values Z (block 1 ⃝). Only a few percent of the injected powers show deviations, although these are rapidly very significant. Actually, the only agents likely to modify their injection plans between these two OPFs are those concerned by the activation of a voltage constraint. In particular, these limits may compel distributed generators not to inject all their available power. This must then be compensated by the external network, whose purchase price is significantly more expensive. The impact of these few percentiles of deviation is therefore critical on the fluctuations of the social welfare -visible on the third panel. Comparing social welfare SW -SW (according to Figure 1 notations) does not make sense when the constraints of a problem are not fully respected. Here with misestimated impedances, no configuration resulted in an improved social welfare taking advantage of constraint violation. Under this clarification, the suboptimality is represented to illustrate that severe decreases come along with impedance errors. With minimal deviations in injected power leading to large shifts in voltages and social welfare, a misestimated network appears to behave as an uncertainty amplifier that magnifies errors. Figure 3 does not display line flows for two reasons. First, the limiting constraint within the ELVF case is the voltage limit, which is consistent with current distribution networks. Regardless of the load imposed on the network, power constraints on the lines cannot be activated because the power flows would be beforehand limited to comply with voltage constraints. Besides, the network has been completely controlled by power set points. Therefore, impedance fluctuations cannot cause any modification of the flows, which are imposed by end-of-line producers and consumers. Only the value of the losses can be impacted: it is assumed here that the external network compensates for any disequilibrium.

Four modifications are therefore made to the test case for the rest of this section. First, consumers are no longer specified by cost functions depending on their power consumption, but rather by load impedances (wards according to PandaPower formalism). These impedances are set so that the rated power consumed by each load is equal to the previous case. Second, decentralized generators are allowed to exchange reactive power. Third, all power exchanges are multiplied by 30. Finally, AC-OPF-SOC may lead to unfeasible solution since reverse active and reactive PF of may occur. Thus, we will use the AC-OPF formulation. It is then possible to reach line capacity constraints, as illustrated on the left end of Figure 4. When consumers are specified by impedances, the deviation of the power consumed will vary with the squared voltage. Voltage fluctuations then affect the line flows. This effect is measurable but statistically very marginal. Nonetheless, the risk of a line being in overcapacity is real, as illustrated by the most heavily loaded line. We notice that the evolution of the flow in this line is not monotonous with the increase of the impedance estimation error. Indeed, when using a Monte Carlo approach, it is much more difficult to estimate extreme events than average behaviors. Due to the high computational cost, the number of draws performed here cannot ensure the accurate estimation of extreme events.

SENSITIVITY AND DISCUSSIONS

The previous section highlighted the risks associated with incorrect estimation of line impedances when using an OPF. The purpose of this section is to discuss the various parameters which have been used arbitrarily so far. Specifically section 4.1 will consider the sensitivity to the probability density describing the estimation errors. Section 4.2 will describe the differences between high-consuming and highproducing distribution networks.

Impact of the Probability Distribution

With no real large-scale experimental data, it is relatively difficult to motivate the adoption of one distribution rather than another. Hence the adopted approach is to compare results obtained using several different distributions. A truncated normal distribution, a Weibull distribution and an exponential distribution are used here. They are adjusted to have the same mean -160% of nominal impedance values, corresponding to the right end of Figure 3. The top panel of Figure 5 shows voltages experimental densityhistogram normalized to make its integral unitary. On all configurations, a peak is observed at 1 p.u.: in the time series used for this test case, some periods present little to no power exchanges, at night for instance. Voltages are thereby not disturbed at all. The distribution based on perfectly known impedances -in purple -reads on the right side as the density reaches much higher values. As explained for Figure 3, the test case involves more consumption than production. Voltages are therefore pulled down. An OPF with perfect knowledge of the impedances manages to activate the voltage constraint of numerous buses. However, when impedances become poorly known, this distribution is spread over a wider voltage range. Peaks corresponding to initially active constraints can still be identified, around 0.9 p.u. and 1.1 p.u. On the bottom panel, the distribution of social welfare suboptimalities is represented in logarithmic scale for better visualization. A significant part of the tested configurations do not show any suboptimality, although significant values can be obtained infrequently. It can be observed that the three probability laws produce very similar outcomes. The choice of a particular distribution does not seem to have any impact on the state of the network, characterizing the average error is enough.

Producers versus consumers share

To obtain the previous results, the ELVF test case has been populated with production and consumption time series. However, these results are significantly impacted by the ratio between producers and consumers. These impacts are depicted Figure 6: with a 160% mean impedance error, a scaling factor is applied on all productions of the test casescale gen -and another one on all consumptions -scale load.

The point (0,0) means that no production or consumption is exchanged. The point (1,1) refers to the configuration used in all previous studies. 400 points constitute this grid which is represented Figure 6 as pixels without interpolations. 15 random networks were sampled for each pixel. On the top panel, the frequency of voltage constraint violation is displayed in color scale. Under full load without any production -(1,0) bottom right corner -70 % of all studied buses have a voltage violating the power quality constraints. Since these results are obtained using a Monte Carlo approach, non-monotonic fluctuations can be observed. Although, trends are clearly marked. Starting from the origin point (0,0), an increase in consumption (right shift) or in production (up shift) are both associated with voltage violations. The increase is sharper for consumption due to the selected time series: the max to average ratio is greater for the consumption profiles, resulting in more frequent violations. The coexistence of generation and consumption appears not to be the worst configuration with respect to the frequency of voltage violations. This is an uplifting indication as this configuration is the most likely to become widespread in the future. For each scale load value, there is an optimal scale gen value for which voltage violations are less frequent -although this needs to be confirmed as this case study may not represent all distribution networks. The bottom panel represents the average suboptimality of the obtained solution. Similar trends can be observed, although the comparison of social welfare cannot be straightforward when constraints are not respected as previously mentioned. Yet configurations with only generators -the left side of the figure -show significant suboptimalities that diminish as soon as some consumers are involved. Indeed, voltage constraints restrict exports from the distribution neighborhood. Local consumption prevents producers from being compelled to curtail their production. Here again, the effects of the lack of knowledge regarding impedances seem to be limited by the coexistence of producers and consumers in the same distribution neighborhood.

CONCLUSION AND PERSPECTIVES

This contribution sought to estimate the impact of misestimated impedances on the results of an OPF in distribution networks. Indeed, such networks include a large number of lines and are sparsely instrumented, while their production and consumption flows are likely to increase in the future. The proposed methodology has shown on a case study that non-compliance with voltage constraints could appear across the network while only a few agents modify their injection plan. Lines as a whole are much less affected, although dangerous configurations can quickly appear on the most stressed lines. The statistical distribution chosen to describe the impedance errors does not seem to impact the observed results, characterizing the average error is enough. However, the proportion of producers and consumers in the distribution neighborhood significantly affects the frequency of constraint violations and suboptimalities. The coexistence of distributed consumers and producers seems to be an element to mitigate the hazards of impedance misestimations in distribution networks.

This contribution needs to be pursued in several directions. First, it is crucial to overcome the dependence of the obtained results to a specific case. A generalization to any distribution network -characterized by metrics to be identified -would be a powerful tool to anticipate the sensitivity of a neighborhood to impedance misestimations. The local control, or the distributed instrumentation, could then be adapted. Furthermore, impedance estimation errors are an additional source of uncertainty in the OPF literature. Already investigated in areas where the stochastic distribution of uncertainties is available -renewable generation, line failure-, the development of a robust or chance constraint approach on line impedances would allow to optimally define the margins necessary to respect the operating limits of the network. If necessary, a generalization to a distributionally robust optimization approach could be implemented.

Fig. 1 .

 1 Fig. 1. Methodology used to assess the impact of impedance misestimation. Notations . refers to inaccurate impedances or, any value based upon it.

Fig. 2 .

 2 Fig. 2. Power flow example in the European Low Voltage Feeder with distributed generation and flexible consumption.

Fig. 3 .

 3 Fig.3. Sensitivity of AC-OPF-SOC results depending on the average of impedance estimation errors. Top: repartition of voltages at each bus; middle: repartition of power injection deviation; bottom: suboptimality of social welfare. Permissible limits in red, deciles and percentiles in colored areas, extreme values observed in dotted blue line. Results observed on ELVF under a truncated normal distribution.

Fig. 4 .

 4 Fig. 4. Sensitivity of line flows computed by an AC-OPF depending on the average of impedance estimation errors.

Fig. 5 .

 5 Fig. 5. Experimental distributions of bus voltages (top) and suboptimality of the social welfare (bottom) according to the probability distribution used to describe impedance misestimations. Mean impedance errors are set at 160%. AC-OPF-SOC formulation is used.

Fig. 6 .

 6 Fig. 6. Impact of the production-consumption ratio -x-axis: consumption, y-axis production -on the frequency of voltage violation (top) and on the welfare suboptimality (bottom) using an AC-OPF-SOC.


G L J A Rikken
email: geert.rikken@lncmi.cnrs.fr

Comparing Electrical Magnetochiral Anisotropy And Chirality-Induced Spin Selectivity

The combination of chirality and magnetism has steadily grown over the last decennia into an area of intense research which evolves around two distinct manifestations and in two non-overlapping communities: electrical magnetochiral anisotropy (eMChA) and chirality-induced spin-selectivity (CISS). Here we discuss the similarities and differences of these two effects. Whereas the original CISS reports suggest an intimate relation with eMChA, magneto-resistance (MR) results on two-terminal chiral devices attributed to CISS, have symmetry properties that are different from those of eMChA. At the same time, the magnitudes of CISS MR and eM-ChA turn out to be similar when normalized to current density and spin polarization, suggesting a common underlying mechanism.

TOC Graphic

Chirality is vital in many areas of physics, chemistry and biology, where entities exist in two non-superimposable forms (enantiomers), one being the mirror image of the other. Since the time of Pasteur, the interplay between chirality and magnetism has been attracting much attention as a source of emergent phenomena. Fundamental symmetry arguments show that when a chiral system is placed in a magnetic field, a whole new family of effects, called magnetochiral anisotropies (MChA), becomes allowed (for a recent review, see [START_REF] Atzori | Magnetochiral anisotropy: from fundamentals to perspectives[END_REF] ). The first member of this family to be experimentally reported, optical MChA, corresponds to a difference in the absorption and refraction of unpolarized light propagating parallel or anti-parallel to the field, 2 . [START_REF] Barron | Magnetochiral birefringence and dichroism[END_REF] Initially observed in the visible wavelength range, [START_REF] Rikken | Observation of magneto-chiral dichroism[END_REF] , 5 , 6 its existence was later confirmed across the entire electromagnetic spectrum, from microwaves, 7 8 to X-rays, 9 10 and in photochemistry. [START_REF] Rikken | Enantioselective magnetochiral photochemistry[END_REF] Electrical MChA (eMChA), was subsequently experimentally observed in the electrical resistance of bismuth helices, [START_REF] Rikken | Electrical Magnetochiral Anisotropy[END_REF] carbon nano tubes, [START_REF] Krstić | Magnetochiral anisotropy in charge transport through single-walled carbon nanotubes[END_REF] bulk organic conductors, [START_REF] Pop | Electrical magnetochiral anisotropy in a bulk chiral molecular conductor[END_REF] metals, [START_REF] Yokouchi | Electrical magnetochiral effect induced by chiral spin fluctuations[END_REF] , [START_REF] Aoki | Anomalous nonreciprocal electrical transport on chiral magnetic order[END_REF] semiconductors [START_REF] Rikken | Strong electrical magnetochiral anisotropy in tellurium[END_REF] and superconductors [START_REF] Qin | Superconductivity in a chiral nanotube[END_REF] as a resistance  that depends on the handedness of the conductor and on the relative orientation of electrical current I and magnetic field B given to first order by   (B I)

=  0 (1 +   B • I) (1)
where   = -  refers to the right/left-handed enantiomer of the conductor.

In the case of a ferromagnetic material, it is more meaningful to express eM-ChA in terms of the magnetization M or the spin polarization S instead of the magnetic field B. [START_REF] Aoki | Anomalous nonreciprocal electrical transport on chiral magnetic order[END_REF] A special case of electrical MChA is dielectric MChA that was observed in the displacement current in chiral ferroelectrics. [START_REF] Rikken | Dielectric Magnetochiral Anisotropy[END_REF] The latest addition to the MChA family, phonon MChA, was recently observed in the propagation of ultrasound, [START_REF] Nomura | Phonon magnetochiral effect[END_REF] further illustrating the universality of the phenomenon. MChA has become a prominent representative of the wider class of non-reciprocal transport phenomena in broken-symmetry systems, that play an important role in topological quantum systems and in Berry phase physics, [START_REF] Tokura | Nonreciprocal responses from noncentrosymmetric quantum materials[END_REF] an area that is currently intensively studied by condensed matter physicists.

The other main manifestation of the combined effect of chirality and magnetism is the so-called chirality-induced spin selectivity (CISS) (for a recent review see [START_REF] Naaman | Chiral molecules and the electron spin[END_REF] ). The first indication of CISS was observed in the transmission of spin-polarized photoelectrons from a gold substrate through chiral molecular layers. [START_REF] Ray | Asymmetric Scattering of Polarized Electrons by Organized Organic Films of Chiral Molecules[END_REF] CISS was explicitly established to depend on the handedness of the molecules [START_REF] Ray | Asymmetric Scattering of Polarized Electrons by Organized Organic Films of Chiral Molecules[END_REF] and on the spin polarization S. [START_REF] Göhler | Spin Selectivity in Electron Transmission Through Self-Assembled Monolayers of Double-Stranded DNA[END_REF] The model presented for CISS is that of spin-filtering, where the spin-and chirality-dependent electron transmission coefficient  through the chiral layer is expressed to first order as

  (S v) =  0 ³ 1 +   S • v ´(2)
where   = -  describes the chirality of the molecular layer and v is the charge velocity. Note that the dependence on v was not explicitly verified in the original CISS experiments, but is generally assumed in the explanations of the CISS effect. [START_REF] Naaman | Chiral molecules and the electron spin[END_REF] The underlying physical picture for eMChA and for CISS is therefore the same: a charge carrier that moves through a medium and that has a longitudinal spin or orbital magnetic moment is chiral, and its chirality relative to that of the medium determines its interaction with that medium. (Strictly speaking a translating and spinning particle has helicity, but when considered in the molecular rest frame, chirality and helicity are equivalent.) Eq. 2 is clearly analogous to Eq. 1 as a magnetic field and spin polarization have the same parity-time (P-T) symmetry, and likewise for a current and a velocity. The second terms on the right-hand sides of both equations are therefore invariant under time-reversal and parity, i.e. fully symmetry allowed. CISS can therefore be regarded as a form of eMChA that is limited to the spin magnetic moment of charge carriers. Whereas MChA manifestations are generally quite weak, very strong CISS was observed, but so far no convincing quantitative theory has been put forward that explains the strength of the observed effect, the obvious coupling between chirality and spin through spin-orbit coupling being very weak in organic molecules (for a recent review see [START_REF] Evers | Theory of Chirality Induced Spin Selectivity: Progress and Challenges[END_REF] ). CISS has become a very active research topic in physical chemistry and biology, and more recently several other fascinating and strong manifestations of CISS were reported, ranging from magnetization reversal by the absorption of chiral molecules on a ferromagnet, [START_REF] Ben Dor | Magnetization switching in ferromagnets by adsorbed chiral molecules without current or external magnetic field[END_REF] to chiral separations by flowing a chiral solution over a ferromagnetic substrate. [START_REF] Banerjee-Ghosh | Separation of enantiomers by their enantiospecific interaction with achiral magnetic substrates[END_REF] The potential application of CISS in spintronics has also spurred a lot of activity (for a recent review see [START_REF] Yang | Chiral spintronics[END_REF] ). In particular, two-terminal devices consisting of chiral molecular layers between ferromagnetic contacts have shown quite strong effects, 29 , 30 , 31 , 32 , 33 , [START_REF] Amsallem | Spin polarization through axially chiral linkers: Length dependence and correlation with the dissymmetry factor[END_REF] . [START_REF] Aizawa | Enantioselectivity of discretized helical supramolecule consisting of achiral cobalt phthalocyanines via chiralinduced spin selectivity effect[END_REF] Some discussion as to the possible role of CISS in such observations has arisen, 36 37 but most experimental reports agree on the phenomenology of anti-symmetric I-V curves, where the current is an odd function of the voltage (i.e. I(V) = -I(-V)), that are different for the two directions of longitudinal magnetization of the ferromagnetic contacts and that are different for the two handednesses of the chiral layer. It should be noted that a few diverging reports exist, both in junction devices [START_REF] Abendroth | Spin-Dependent Ionization of Chiral Molecular Films[END_REF] and in field effect transistors. 39 Some theoretical microscopic models also predict I-V curves that are strictly odd in V, [START_REF] Huisman | Chirality-Induced Spin Selectivity (CISS) Effect: Magnetocurrent-Voltage Characteristics with Coulomb Interactions[END_REF] but other models predict 'eMChA-like' I-V curves that contain V-even terms [START_REF] Yang | Detecting Chirality in Two-Terminal Electronic Nanodevices[END_REF] or both. [START_REF] Huisman | CISS effect: Magnetocurrent-voltage characteristics with Coulomb interactions. II[END_REF] One recent model is based on eMChA to explain CISS-MR, giving also anti-symmetric I-V curves. [START_REF] Zhao | Nonreciprocal Nature and induced Tunneling Barrier Modulation in Chiral Molecular Devices[END_REF] Fig. 1 schematically summarizes and compares eMChA and the dominant experimental CISS magneto-resistance (CISS-MR) results. As pointed out above, the original CISS reports and eMChA share the same P-T symmetry properties, but as can be seen from Fig. 1, the more recent CISS-MR reports on two-terminal devices clearly differ from eMChA, showing different symmetry properties and suggesting different microscopic mechanisms. The microscopic origin of CISS-MR is still under intense debate and an analysis of its symmetry properties can provide powerful constraints on any microscopic model. In the remainder we perform such an analysis and compare the intrinsic orders of magnitude of both effects.

Firstly it should be noted that for symmetrical devices, magnetic field dependent antisymmetric I-V curves as in Fig. 1 C and D are unphysical, as a simple rotation over 180 • around an axis perpendicular to the current leaves the device structure and its I-V curves unchanged, but inverts the direction of the magnetization. We must therefore conclude that device asymmetry is a necessary condition for anti-symmetric I-V curves like in Fig. 1 C. Note that such an asymmetry can result from two different electrodes, either different in composition or in geometry, as is often the case in CISS-MR experiments, but also from a polar molecular orientation in the chiral layer which may result unintentionally from the deposition process. Paradoxically, asymmetric tunnel structures generally lead to asymmetric I-V curves [START_REF] Choi | A Focused Asymmetric Metal-Insulator-Metal Tunneling Diode: Fabrication, DC Characteristics and RF Rectification[END_REF] which is at odds with Fig. 1 C. However such a deviation from anti-symmetric I-V curves may be below the experimental detection limit. To elucidate the role of device asymmetry in the CISS-MR results, CISS-MR experiments with two identical ferromagnetic contacts should be performed and very careful comparisons between the results for the two enantiomers should be performed, excluding all other differences than the chirality of the central layer.

Next we will consider the C-P-T (C charge conjugation) symmetries of both effects. For eMChA, C-symmetry is satisfied if the material parameter  in Eq. 1 is even under charge inversion. Indeed, the microscopic model for eMChA in Te finds such behavior for . [START_REF] Rikken | Strong electrical magnetochiral anisotropy in tellurium[END_REF] Clearly CISS-MR breaks time-reversal symmetry, as changing both the current direction and the magnetic field/spin polarization direction for a given enantiomer does not yield the same  . This is not altogether surprising, because generally time-reversal symmetry is not preserved in macroscopic dissipative processes, as Ohm's law clearly demonstrates. For such a situation, one should resort to the Onsager reciprocal relation which states that, close to thermodynamic equilibrium, a diagonal transport coefficient, like an electrical conductivity, should be even under time-reversal, which means that for eMChA and for CISS-MR the resistance for up and down magnetization should be identical at very small bias. This is the case for eMChA, however, as can be seen in Fig. 1 C and D, CISS-MR does not obey the Onsager relation as for a given enantiomer  ( = 0 ) 6 =  ( = 0 ) (see the inset in Fig. 1 D) as can be seen in Refs., [START_REF] Alam | Spin Filtering through Single-Wall Carbon Nanotubes Functionalized with Single-Stranded DNA[END_REF] , 30 , 33 , 34 whilst most other publications do not allow to judge whether the Onsager relation is violated or not. One clear observation of Onsager invariance in a CISS-MR experiment has been reported, albeit with the magnetization perpendicular to the current, which is not the normal CISS-MR configuration. [START_REF] Hossain | Transverse magnetoconductance in two-terminal chiral spin-selective devices[END_REF] The Onsager reciprocal relation has been experimentally verified for diffusive transport, [START_REF] Miller | The Onsager Relations; Experimental Evidence[END_REF] but to our knowledge no theoretical justification nor any experimental verification for the case of tunneling or any other form of charge transport has been reported. Therefore, as an explanation for the apparent violation of Onsager non-reciprocity by CISS-MR close to thermal equilibrium, one must conclude that the Onsager relation does not apply to the electrical transport in CISS-MR devices with thin chiral layers, implying that this transport is not diffusive. Such an explanation would be consistent with recent observations of CISS-MR in macroscopic chiral field-effect devices, which are clearly in the diffusive transport regime, and where 'eMChA-like' asymmetry in the I-V curves is reported. 39 Apart from an effect of the electroweak force, which we can safely ignore here, no uncertainty exists for the requirement of parity invariance for dissipative processes and CISS-MR, like it is the case for eMChA, should be invariant under parity. If the device structure is symmetrical, i.e. consists of a chiral layer sandwiched between two identical ferromagnetic layers, parity invariance requires that  ( +) =  ( -) for each magnetic field/spin polarization direction. It can be easily seen from Fig. 1 C and D that for CISS-MR, this relation is not obeyed, whereas it is obeyed by eMChA in Fig. 1 A andB. If the CISS-MR device has any polar asymmetry along the current direction, e.g. two different contacts or a polar chiral layer, the above condition no longer holds, the I-V curves in Fig. 1 C and D do not necessarily break parity. Again, device asymmetry appears as a necessary condition for the predominant CISS-MR results.

Whereas eMChA is fully C-P-T symmetry allowed, as is CISS as expressed by Eq. 2, the symmetry analysis above shows that the experimental CISS-MR claims are not conform with the original CISS concept, and they raise questions as to their origin. The violation of the Onsager reciprocity is suggestive of an intrinsic absence of thermodynamic equilibrium. One possibility could therefore be that the CISS-MR is the result of the falsely chiral action of  •  away from equilibrium, where  is the electric field across the junction, proportional to the voltage V, and S the electron spin, proportional to the magnetization M of the contacts, and not of the truly chiral action of  • . [START_REF] Barron | Reactions of chiral molecules in the presence of a timenoninvariant enantiomorphous influence[END_REF] The intrinsic out-ofequilibrium character of false chirality would then automatically explain why the Onsager relation does not apply to CISS-MR. However, in the case of false chirality, for a given chiral barrier, the response to  •  should be the same as that to (-) • (-) which is clearly not the case in Fig. 1 C or D. Although our translation of false chirality, as developed for chemical reactivity, into electrical transport may be subject to discussion, we arrive at the conclusion that false chirality does not seem able to explain these CISS-MR observations, unless a strong built-in electric field is present in these devices. The presence of such a field would however not be compatible with anti-symmetric I-V curves. The enantioseparation at ferromagnetic surfaces, which is also considered a manifestation of CISS, [START_REF] Banerjee-Ghosh | Separation of enantiomers by their enantiospecific interaction with achiral magnetic substrates[END_REF] is however fully compatible with false chirality. [START_REF] Barron | Symmetry and Chirality: Where Physics Shakes Hands with Chemistry and Biology[END_REF] Another apparent difference between eMChA and CISS-MR lies in the magnitudes of the respective reported relative non-reciprocities   ≡ ((+)-(-))( = 0) and   ≡ ((+)-(-))( = 0), the values for eMChA being several orders of magnitude smaller than the CISS-MR values. However, for a meaningful quantitative comparison, both should be normalized to the same current density  and to the same average magnetic polarization  of the charge carriers, i.e. one should compare e  ≡ . Assuming simple Pauli paramagnetism for the conduction electrons, i.e.  = (   )  ' 4 1 10 -3   in an external field of 1  at room temperature, the eMChA results on [DM-EDT-TTF] 2 ClO 4 [START_REF] Pop | Electrical magnetochiral anisotropy in a bulk chiral molecular conductor[END_REF] and on Te 17 correspond to normalized relative non-reciprocity values e   of 5 10 -8  2   and 1 10 -6  2   . respectively. Alternatively one could assume Landau dia-magnetism for the conduction electrons, which would yield normalized nonreciprocity values that are a factor 3 larger, the Landau susceptibility being 3 times smaller than the Pauli susceptibility. For a typical CISS-MR result with ferromagnetic nickel contacts, a current density of 2 10 5  2 and a relative non-reciprocity of 1 5%, 30 and using  ' 0 23   49 one finds a normalized e   = 3 10 -7  2   , i.e. of the same order of magnitude as e   . In junctions using an atomic force microscope, much higher current densities can occur, resulting in very large   observed in such devices. Therefore we can conclude thateMChA and CISS-MR have very similar intrinsic magnitudes, and the apparent differences in strength only result from the different experimental configurations in which they are measured. Despite their different symmetry properties, identified above, this does point to a common underlying mechanism which would be consistent with the similarity in symmetry between eMChA and the original CISS observations. In particular, no exceptional spin-orbit coupling strength is required for CISS-MR.

Summarizing the analysis above, we can say that i) the commonly observed magnetic field direction dependent antisymmetric CISS-MR I-V curves cannot solely be an expression of chirality but also require a polar asymmetry in the device, which is not compatible with the simplest possible interpretation of CISS-MR being a purely spin filtering effect, only dependent on the charge carrier's helicity as compared to the central layer's chirality. As most CISS-MR reports use different metals for the two contacts, this suggest that an interface effect could be important.

ii) the predominantly observed violation of the Onsager reciprocity relation excludes a diffusive transport mechanism and it is currently unknown how other transport mechanisms (tunneling, hopping, field emission,...) relate to Onsager reciprocity.

iii) the magneto-chiral material/device parameter should be odd in the charge of the charge carriers iv) false chirality cannot be the explanation for the observed I-V curves v) the intrinsic magnitudes of eMChA and CISS-MR, normalized for current density and magnetic polarization, are comparable

Although at a fundamental level, eMChA and CISS are very close, the first three points listed above for CISS-MR are at odds with eMChA, and require an enhanced model for CISS-MR, beyond the original CISS concept. Clearly there are basic issues to address for CISS-MR and we hope that the new insights presented here will contribute to developing a detailed understanding and identifying a microscopic model for CISS-MR.

Corresponding author G.L.J.A. Rikken, geert.rikken@lncmi.cnrs.fr

Figure 1 :

 1 Figure 1: Schematic I-V characteristics of eMChA (A and B, strength exagerated) and CISS MR (C and D) for both enantiomers (L/D) and magnetic field/spin polarization directions (up/down). Insets zoom on zero crossings.

Fig. 1 C

 1 and D however require the material/device parameter of CISS-MR to be odd under charge conjugation. So far, no microscopic model has adressed this aspect but it suggests a different origin for CISS-MR as compared to eMChA, or an additional necessary component.

Acknowledgements

This work was supported by the French National Agency for Research project SECRETS (ANR PRC 20-CE06-0023-01). We gratefully acknowledge helpful discussions with L. Barron.