Antoine Dazin
email: antoine.dazin@lille.ensam.fr

Guy Caignaert

Geneviève Dauphin-Tanguy
email: genevieve.dauphin-tanguy@ec-lille.fr

Model Based Analysis of the Time Scales associated to Pump Start-ups

Keywords: Transient, Pump start-up, Bond-graph model, time scales

Model based analysis of the time scales associated to pump start-ups

Antoine Dazin, Guy Caignaert, Geneviève Dauphin-

Introduction

Cooling systems are a crucial point of nuclear power plants safety. Transient operations are commonly encountered in such hydraulic systems, (Gao et al 2013, Farhadi et al 2007) and an accurate prediction of their behavior during these operations is necessary. Nevertheless, the performance of their components during these transient periods is rather different than during the steady one. Concerning pumps, this has been well established, through different studies dealing with :

-Starting periods in non cavitating (Tsukamoto and Ohashi (1982), Saito (1982), Ghelici (1993), Bolpaire (2000), Lefebvre and Barker (1995)) or cavitating conditions (Tanaka & Tsukamoto (1999), Duplaa et al (2010Duplaa et al ( , 2013)).

-Stopping periods (Tsukamoto et al, (1986)), -Valve opening and closure (Picavet, (1999) -Dazhuan et al ( 2010))

Quasi-steady models (using the steady performance of the pump) are still in use for the transient performance prediction of pumps during starting periods (Rizwan Uddin 1994). They give rather good results if the transient is relatively slow (i.e. if the transient effects are negligible compared with the steady one). However, to obtain reliable results compared with the experiments, in case of fast transient periods, more sophisticated models, including the effects due to the acceleration of the runner and of the flow, have been built (Tsukamoto et al (1982), Dazin et al (2007)).

*Manuscript

Besides, whereas the similarity laws for steady operation of hydraulic turbomachineries are well established, very few works are dealing with the non dimensional analysis of a fast start-up. Tsukamoto et al (1982) have introduced the non-dimensional parameter na f T ω (where ω f is the final rotation speed and T na is a time characteristic of the stat-up) as the most important parameter for the transient characteristics of a pump.

An attempt to determine which parameter are affecting the performance of a pump during a start-up has been proposed by Dazin et al (2006) through the study of experimental results: start-ups with various final operating points and acceleration were the support of an analysis of the performance of a pump operating in transient operations. Unfortunately, the analysis was limited because of some experimental constraints: the acceleration rate was not easily driven because the fast start-ups were obtained by an electromagnetic clutch which is not easy to control. It was also not possible to change the geometry of the test rig. On top of that, the relative uncertainty measurement is very high at the beginning of the start-up and the analysis of the results is thus difficult during the first period of the transient.

More recently, Elaoud et al (2011) have proposed a study based on the modelling of the governing equations of the transient flow in pipes to analyse the effect of pump starting times on the transient flow in pipes.

The purpose of the present work is to present a more universal analysis of pump start-ups. More precisely, it aims at giving the time scales associated to pumps start-up and to determine which parameters can affect them.

To do so, a model of a whole hydraulic test rig has been simulated using the bond graph methodology.

The prediction of the model proposed in the present paper is validated by comparison to experimental data from Bolpaire (2000). An analysis of the evolution with time of the terms contained in the instantaneous total pressure increase in the pump is proposed using non dimensional results with a log-log presentation allowing for a clear decomposition of a start-up into three periods whose duration can be estimated. Then, this analysis is used to investigate the impact of the variation of several parameters (acceleration rate, final rotation speed, final flow rate, pipe lengths) on the intensity and duration of the transient effects during a start-up. These analyses are made assuming non cavitating operation conditions. It is also assumed that the pressure wave propagation phenomena within the system and the pump are neglected, as the paper focuses on the interpretation of the instantaneous pump performance.

1/ BOND GRAPH MODEL

11/ Presentation of the model

The test rig modelled (Fig. 1) is a closed loop composed of a radial flow pump connected to a tank (at atmospheric pressure) through hydraulic pipes. A valve is also available on the test rig, in order to set the final operating point. The whole system is modelled using the bond-graph methodology.

A bond graph consists of subsystems linked together by half arrows, representing power bonds. They exchange instantaneous power at places called ports. The variables that are forced to be identical when two ports are connected are the power variables, considered as functions of time. A few basic types of elements are required in order to represent models in a variety of energy domains. For 1Dsystem modelling, basic 1-port elements represent power dissipation (R), energy storage (I, C) and power supply (effort and flow sources), changing of domains without power losses (TF and GY-elements). A causal stroke, placed perpendicularly to the bond, shows up the way the constitutive relations in an element have to be written.

The model bond graph can be directly simulated using software with a graphical bond graph interface, as 20Sim (Controllab Products) software used here. From the bond graph model can be derived dynamic mathematical models, under ODE (ordinary differential equation) or DAE (differential algebraic equation) equations, which can be simulated using software as Matlab/Simulink. Plenty of papers exist in the literature. For more details on this methodology, see for example Karnopp (1975).

Before describing the model, one can note that :

-all the pressures are expressed in terms of total pressure, -as the flow is considered to be incompressible, the volume flow rate q is the same in all the components that will be described in the following.

The synoptic of the bond graph model is presented Fig. 2. It is composed of: An effort source (a constant pressure boundary condition) used to model the tank.

A dissipative element (R) representing the hydraulic losses in the pipes and the valve. These losses are supposed to be proportional to the square of the volume flow rate.

( 1 )

Where are the total pressure losses in the pipe and the valve, q, the volume flow rate, ρ the fluid density, S the pipe cross-section area and k, the loss coefficient which is adjusted for each start-up in order to reach the desired final operating point (the adjustment of this parameter could be compared to the opening/closing of a valve to set an experimental operating point).

An inertial element (I) representing the inertial hydraulic effects in the pipes and depending on the geometry of the pipe:

( 2 )

Where L is the total pipe length and S, the pipe cross-section area.

The model of the pump is based on the one proposed by Dazin et al (2007) for radial flow pumps.

( 3 ) With:

ω, the instantaneous angular speed of the impeller, , the instantaneous pump total pressure rise, a steady term representing the steady total pressure rise of the pump (i.e.

the pressure rise obtained at a given velocity and flow rate in steady condition), supposed to be well described by the following polynomial approximation: The angular velocity evolution is imposed to the pump through a modulated flow source (MSf). This evolution is obtained with a first order transfer function whose equation is given by 9 :;

! " " " # ! " $ # ! " $ # % (
, where κ is the gain, τ, the time constant and s, the complex Laplace variable. In this function, the gain and time constant are set in order to fix the final angular velocity and start-up duration.

The integration method used to solve the equations system is a backward differentiation formula.

The geometrical parameters needed to model the pump are calculated using the geometry of the single volute radial flow pump used in the experiments realised by Bolpaire (2000) and reported by Dazin et al (2007).

The main geometrical characteristics of this pump are reminded in Table 1.

12/ Experimental validation

To check the validity of this model, its results are compared with experimental ones. To do so, the angular velocity input is replaced by an experimental velocity evolution (Fig. 3a) extracted from Bolpaire (2000). The pipe length and section are adjusted to reproduce the experimental test rig. The loss coefficient of the valve is adjusted to reach the final experimental operating point. The experimental evolution of the pump total pressure rise is compared with the evolution predicted by the model Fig. 3b. The differences between the experimental and model data are less than 1% during the first part of the start-up (t < 0.2 s) and are never exceeding 7%. This confirms the validity of the model (this validity has already been established in Dazin et al (2007) for a single pump model, but not for a complete loop).

13/ SIMULATION TEST PLAN

The test cases are summarized in Table 2. The first case (a reference case) corresponds to a start-up for which the final operating point corresponds to the pump nominal point. The time constant of the first order filter is set to

s 2 . 0 = τ
. The pipe length is set to L = 5m whereas the pipe diameter is 0.04 m. The second, third and fourth cases correspond to variation of the final steady operating point (final velocity for case 2 and final flow rate for cases 3 and 4), compared with the first case. Cases 5 and 6 correspond to start-ups ending at the same operating point as case 1 but with different acceleration rates. Case 7 corresponds to a variation of the pipe lengths (L = 10 m instead of 5 m). The pipe diameter is remained unchanged whatever the case considered is.

2/ REFERENCE CASE -TIME SCALES ASSOCIATED TO A START-UP

Fig. 4 presents typical simulation results for the reference case (case 1 in table 2). These are typical results obtained for a fast start-up. The acceleration rate is relatively high (it takes 0.32 s to reach 80% of the final impeller angular speed).

The pressure and flow rate are rising with a lag due to inertial effects in the pump and pipes. (Tsukamoto et Ohashi (1982), Saito (1982), Lefebvre et Baker (1995)). Two phenomena can be identified. At the beginning of the start-up, the pressure coefficient is higher than the steady one because of angular acceleration effect. In a second stage, the pressure coefficient is lower than the steady one, because of inertial effects in the pump. At the end of the start-up, the pump is reaching its final (steady) operating point.

The time evolutions of several pressure coefficients are presented in Fig. 7. On this figure can be found:

-< =>?= " ! " " , the instantaneous total pressure coefficient of the pump.

-< " ! " " : the steady term that seems to be nearly constant during a great part of the start-up. This is due to the fact that the flow coefficient remains at a very low value while t/τ is lower than 10 -1 (Fig. 8) because the angular velocity of the runner is increasing more rapidly than the volume flow rate (Fig. 5). The pump is then operating in the left part (δ < 0.001) of its steady characteristics curve where the pressure coefficient is nearly constant.

-< @ A B " ! " "

: the acceleration effects on the pump performance. It is the dominant term at the very beginning of the start-up because the acceleration rate is great and the velocity is low. It can be noticed that the evolution of this parameter is quasi-linear when plotted in a log-log scale (with a slope equal to -2). This is due to the fact that the impeller acceleration is nearly constant during a great part of the start-up. Thus, the pressure coefficient time-evolution is proportional to C D , that is to E D .

As the velocity increases rapidly, this term decreases rapidly and becomes negligible for E F G H6HI.

-

< J8 @ K B " ! " " , < J . *- L M N : M N O B " ! " "
:the inertial terms in the impeller and in the pump stator.

They are quite important during a great part of the start-up. It can also be noticed that, because of the geometry of the pump studied, the inertial effects are greater in the stator (diffuser and volute) than in the rotor. At the beginning of the start-up (t*<0.01), these terms are also proportional to t -2 . This can be explained easily from the equations governing the system. The one governing the pipes can be written as follows:

P 8 Q R ( 5a )
Where

S "
Which gives, in a dimensionless form :

=T=U " ! " " @ " " ! " " " ! " " (5b)
It can be noticed that the pressure delivered by the pump P is equal to the pressure P 8 Q required by the system. Consequently, combining equation ( 3) and ( 5) gives : R

Which gives, in a dimensionless form :

< < < +* Q < J 8 Q < J8 < J . *- (6b)
Where < +* Q @ " " ! " " , and represents the non-dimensional pressure losses in the pipes and in the valve, and < J 8 Q represents the dimensionless hydraulic inertial effects in the pipe.

At the beginning of the start-up, as pointed out above, the dominant term of this equation is the angular acceleration term. More precisely, the steady term P is negligible (see Fig. 7). The resistive term is also negligible because the flow rate is very low at the beginning of the start-up. Finally, equation (6a) becomes:

V (7)
This proves that, at the beginning of the start-up, the evolution of the flow acceleration is proportional to the angular acceleration.

In a second time period (from E F H6HI to E F I), the two terms < J8 @ K B " ! " " and < J . *-

L M N : M N O B " ! " "
appear to be nearly constant. This can also be explained through the analysis of equation ( 6): as pointed out above, in this equation, the angular acceleration term is negligible when E F G H6HI. Besides, the first term of the right hand side of equation (6a & b), that is the resistive term, is also negligible from E F H6HI to E F I because the flow rate is still very low.

Consequently, from E F H6HI to E F I, equation ( 6) can be written:

< V B ! " " " . (8)
As the steady pressure coefficient is constant during the beginning of the start-up, the term

B ! " " "
is also constant while the resistive effects in the test rig are negligible compared to the resistive effect in the pipe.

Consequently, is proportional to t 2 . This explains why the flow rate q is proportional to t 3 (as observed Fig. 6.)

Finally, the start-up of a pump can be decomposed into three periods. This period lasts as long as

W i.e. < W @ A A X Y " ! " " (9)
The order of magnitude of < is 1.

The order of magnitude of is V ! " " ./0 ?

, whereYZ is the mean value of the relative flow angle in the impeller, defined as the average of the outlet and inlet relative flow angle.

The order of magnitude of Y is [ ; , where C \ is the final speed of the impeller. And consequently, the order of magnitude of the instantaneous speed is C V [ ; E. Consequently, the first period of the start-up ends at a normalised time E F defined as follows:

E F 3 ; V ] ./0 ? [ ; (10)
It can be noticed that, in the previous expression, τ can be replaced by another time characteristic of the start-up.

Thus, in the expression of E F , the non-dimensional parameter C \ ^ introduced by Tsukamoto et al (1982) as a parameter characteristic of the start-up, can be identified.

For the first case of table 2, and using the expression of equation ( 10), one obtainYE F V H6HI, which matches very well with the duration of the first period observed in Fig. 7.

Second period: In the second period (0.01 < t* < 1), the start up is governed by the inertial effects in the test rig.

During that period the flow rate is evolving as t 3 if the acceleration of the impeller is constant.

This period lasts as long as the inertial effects in the loop are greater than the resistive effects in the pipes, i.e. as long as UK _ , where & Q is the equivalent length of the whole loop and is defined as :

& Q ` & ` & ` & )*+ `)*+ During this time period, UK V a C V a [ " ; " E And consequentlyYR V % UK a [ "
; " E % , and so:

V b ! " c UK " [ c ; c E d .
Consequently, the second period ends at a normalized time

E F E F " ; V b S 3 c ] UK ! " [ ; (11)
In the present case (first case of table 2), and using the expression of equation ( 11), one obtain E F V I6e. It can be noticed that these value gives the correct order of magnitude of the observed time duration of the second time period which was estimated to be around 1 by the observation of Fig 7.

Third period: At last, in a third period, the resistive effects in the pipes are increasing and the start up tends to a quasi steady start -up.

Using the analysis of the pressure coefficients evolution, the impact of the parameters having an influence on the start-up is easier, and is illustrated by the simulation of cases 2 to 7 in the following section. -The reference test case for which the acceleration time is short (^ H6fYg),

3/ PARAMETERS AFFECTING THE

-test case 5, for which the acceleration time is moderate (^ fYg),

-test case 6, for which the acceleration time is long (^ fHYg).

It appears that, as expected, an increase of the acceleration rate (i.e. a decrease of the time constant τ) leads to an increase of the pressure coefficient < (Fig. 9a). It can also be observed (fig 9b and9c) that the durations of the first and second periods of the start-up are increased. This is consistent with the definition of the normalized times E F and E F introduced in equations ( 10) and ( 11): a decrease of τ leads to an increase of the duration of the first and second periods. The effect of a diminution of the final speed (Fig. 10) is similar to the effect of an increase of the acceleration rate of the angular velocity as the parameters ^ and C \ play similar roles in equations ( 10) and ( 11).

32/ Effect of the pipe length and resistance

The evolution of < (fig 11a), < J8 (fig 11b) and < (fig 11c) is now plotted for 2 different pipe lengths.

As the angular acceleration is imposed, there is no influence of the pipe length: the angular acceleration coefficient pressure coefficient < is similar in both cases (fig 11a). On the contrary the hydraulic inertia pressure coefficients are plotted Fig. 11b. This term is obviously smaller when longer pipes are used. This is well explained by equations ( 7) and ( 8) showing that longer pipes lead to a decrease of the time derivative of the flow rate. On the contrary, the duration of the second time period increases when the pipe length is higher, which is coherent with the definition of t 2 * (Equation 11).

A change of the pipe resistance has also been tested. To do so, the pressure loss coefficient k of the pipes has been changed. It has been adjusted to reach the design flow rate of the pump for the reference case, 0.5 q n for case 3 of table 2 and 1.5 q n for case 4 of table 2. during the second time period (t* < 1) as the viscous effects are negligible during this period. Besides, this second time period is shorter when the final flow rate is greater, which is coherent with the estimation of the duration t 2 *, showing an increase when the loss coefficient is smaller.

CONCLUSION

In the present contribution, a detailed analysis of the start-up of a hydraulic system composed of a radial flow pump and its suction and delivery pipes is proposed. It is based upon simulations obtained using the bond-graph methodology.

The analysis of the results shows that the start-up of the pump can be decomposed into three main periods :

-In the first period, the effects of the angular acceleration are the dominant ones. During this phase, the flow acceleration is proportional to the angular acceleration. The order of magnitude of this stage duration is E I

F E I ^V ] I E#hZ i 6C j 6^ . -
In the second period, the angular acceleration effects become negligible and the hydraulic inertial effects are dominant compared with the viscous effects. During this second period, the parameter " is constant and consequently the flow rate is proportional to E % , if the acceleration rate of the runner is constant. This second time period ends at E f

F E f ^V Ik I l m & nR a f I C j ^.
-In the third period the viscous effects become dominant and the start-up is quasi-steady.

Consequently, concerning pump transient modelling it can be said that a quasi-steady simulation of a pump startup is sufficient:

-for the simulation third period identified above.

-for the whole simulation if the two first periods identified above are short (i.e. if E I F and E f F are very small compared with 1). This happens when C j ^is great.

Besides, an analysis of the variation of several parameters is proposed.

An increase of the acceleration rate or a decrease of the final speed have similar effects. They lead to an increase of the transient effect as well as an increase of the duration of the two first periods of the start-up described

above.

An increase of the pipe length leads to an increase of the duration of the second period of the start-up as well as a decrease of the inertial effects amplitude.

At last, an increase of the losses in the pipe does not affect the amplitude of the transient effects but leads to a decrease of the duration of the second period of the transient period.

Nomenclature

  4)& , ' ( & )*+ and ' )*+ are the equivalent lengths and sections of the diffuser and volute of the pump, and are used to estimate the hydraulic inertial effects in the pump stator. which allows estimating the impact of the angular acceleration effects on the pump instantaneous total pressure rise. which allows estimating the impact of the fluid inertia in the impeller on the pump instantaneous total pressure rise.In this two expressions, R 1 and R 2 are the inlet and outlet radii of the impeller, b the height of the blade passage and β is the relative flow angle.These steady and angular acceleration terms are modelled by a gyrator (MGY). The hydraulic inertial effects in the rotor and in the stator (diffuser and volute) are modelled separately by two inertial elements.

  Fig 5 shows what can be the interest of using log log scale to present such results : the flow rate and the pump rotation speed are normalized by their final values in steady operating conditions and time is normalized by τ : as expected, the evolution of the velocity is linear as it is imposed by the first order filter ; but it is also very clear on fig 5, that the evolution of the flow rate is proportional to t 3 during a large part of the start-up. This evolution will be explained in the next paragraphs of the present paper. The comparison of the steady non dimensional characteristics and of the non dimensional characteristics of the start-up is proposed in Fig 6. It is very similar to the ones observed experimentally

First

  period: During the very beginning of the start-up (t* < 0.01 on Fig 7) the dominant effects are the angular acceleration effects in the pump. During this period, the flow acceleration is proportional to the angular acceleration. It can be said that in this first period, the start-up is governed by the motor and the driving shaft characteristics.

  of the acceleration rate and final speed To compare the effect of the acceleration rate on the start-up, the evolution of < (fig 9a), < J8 (fig 9b) and < (fig 9c) are plotted for three different test cases :

  The evolution of < (fig 10a), < J8 (fig 10b) and < (fig 10c) is now plotted for two different rotation speeds (2900 rpm and 1450 rpm).

  It corresponds to what would have been obtained in an experimental test-rig by opening or closing of a valve. The evolution of < (fig 12a), < J8 (fig 12b) and < (fig 12c) is now plotted for 3 final flow rates (and consequently, three values of the pipe loss coefficient. It does not affect at all the beginning of the start-up as it can be seen Fig. 12. The angular acceleration coefficient is not affected by the final flow rate because the angular acceleration is imposed in the model. The flow acceleration is also not affected by the resistance of the pipe

":Figure 1 :Figure 2 :

 12 Figure 1 : Sketch of the hydraulic test rig

Figure 3 :Figure 4 :

 34 Figure 3 : Time evolution of the experimental angular velocity (a) and comparison of the pressure evolution (b)

Figure 5 :

 5 Figure 5 : Time-evolutions of the pump angular velocity and of the flow rate (log-log representation).

Figure 6 :

 6 Figure 6 : a. Pump pressure coefficient as a function of the pump flow coefficient during the start-up (in blue) compared with the pump steady characteristics (in red). b. zoom on the initial instants of the start-up.

Figure 7 :

 7 Figure 7: Time evolution of several pressure coefficients in a Log-log presentation.

Figure 8 :

 8 Figure 8: Time evolution of the flow coefficient.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure9: Comparison of the pressure coefficient evolution for three acceleration rates (cases 1, 5 and 6 of table2)

Figure 12 :

 12 Figure12: Effect of the pipe resistance (and consequently of the final flow rate -case 1, 3 and 4 from table 2)

Table 1 :

 1 Pump specifications

	Impeller geometric specifications	Diffuser geometric specifications Hydraulic Specifications
	Inlet vane angle β 1α	32.2°	Inlet Diameter R 3	101.5 mm	Nominal speed	2900 rpm
	Outlet vane angle β 2α	23°	Outlet Diameter R 4 120 mm	Nominal flow rate	23 m 3 /h
	Number of vanes	5	Constant width b 3 6 mm	Nominal total pump
					pressure	4.15 bar
	Inlet diameter R 1	19.25 mm
	Outer diameter R 2	101.25 mm
	Outer width b 2	7 mm

Table 2 :

 2 Test plan

		Final speed (rpm)	Final flow rate (m 3 /h)	Pressure loss coefficient k	Time constant	Pipe
	1	2900	23 (q n )	33.2	τ=0.2 s	Short (L = 5 m)
	2	1450	11.5 (q n )	33.2	τ=0.2 s	Short (L = 5 m)
	3	2900	34.5 (1.5 q n )	10	τ=0.2 s	Short (L = 5 m)
	4	2900	11.5 (0.5 qn)	157	τ=0.2 s	Short (L = 5 m)
	5	2900	23 (q n )	33.2	τ=2 s	Short (L = 5 m)
	6	2900	23 (q n )	33.2	τ=20 s	Short (L = 5 m)
	7	2900	23 (q n )	33.2	τ=0.2 s	Long (L = 10 m)


IEEE Duc Tan Vu

Student Member, Ngac Ky Nguyen

IEEE, Member Eric Semail
email: eric.semail@ensam.eu

An Overview of Methods using Reduced-Order Transformation Matrices for Fault-Tolerant Control of 5-phase Machines with an Open Phase

Keywords: reduced-order transformation matrices, 5-phase machines, open-phase faults, fault-tolerant control, back-EMF waveform, current references, torque control

This paper studies control strategies using modified transformation matrices when five-phase machines operate in oneopen-phase faults. The basic idea of these methods is to maintain the rotating field under asymmetrical conditions as the same as in healthy condition by determining new transformation matrices. The dimension of the new matrices is equal to the number of remaining healthy phases in post-fault conditions. There have been different ways to determine the new transformation matrices applied for different types of five-phase machines in recent decades. In this study, an overview and analyses on these methods will be presented. In addition, advantages and drawbacks of these control strategies are clarified by numerical results.

I. INTRODUCTION

Multiphase machines have been used in many applications in recent decades because of their advantages such as low torque pulsation, fault tolerance, adequacy to low voltage supply and reduced per phase power rating [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF], [START_REF] Huang | Multiphase machine theory and its applications[END_REF]. The design, modeling and control strategies in healthy mode of operation are analyzed in [START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines-Part I[END_REF]. The most common faults, which occur in electrical machine drives, are open-circuited faults of stator windings of the machine and power converters [START_REF] Duran | Recent Advances in the Design, Modeling, and Control of Multiphase Machines-Part II[END_REF]. Many strategies have been proposed to determine optimal current references to maximize torque with given copper losses when one or more phases of the machines are open-circuited. In [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], strategies for four, five, six and seven-phase induction machines have been proposed to determine optimal current references to maintain the magnetomotive force in degraded mode the same as in healthy condition. However, these methods in [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF] deal with only fundamental components of current.

In this paper, methods using reduced-order transformation matrices for fault-tolerant control of 5-phase machines in one open phase faults will be analyzed. Several papers [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF]- [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF] consider only sinusoidal back-EMF machines imposed by fundamental harmonic currents while paper [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] dealing with non-sinusoidal back-EMF machines proposes an injection of the third harmonic currents. In fact, [START_REF] Parsa | Fault-tolerant five-phase permanent magnet motor drives[END_REF] introduces the use of third harmonic components of current to five-phase permanent magnet motor drives with quasi-rectangular back EMFs in an open phase fault. However, high torque ripples still exist in [START_REF] Parsa | Fault-tolerant five-phase permanent magnet motor drives[END_REF] even when the speed is controlled perfectly. The present paper aims at giving a general view and verifying the applicability of methods using adjusted transformation matrices in fault-tolerant control.

It is organized as follows: section II describes the basic model of a 5-phase machine; in section III, the analysis of control strategies using reduced-order transformation matrices in a single open-circuit fault is demonstrated.

II. FIVE-PHASE MACHINE MODELING

To identify the machine model, these following assumptions are considered: 5-phase windings are equally shifted and wyeconnected; the machines with sinusoidal or trapezoidal back-EMFs are considered; the saturation of the magnetic circuits is not considered in the calculation of the back-EMFs and fluxes; and no reluctance effect.

Clarke and Park transformations are applied to convert the machine parameters from natural (𝑖 , 𝑖 , 𝑖 , 𝑖 , 𝑖 ) frame to α-β 𝑖 , 𝑖 , 𝑖 , 𝑖 , 𝑖 and d-q frames 𝑖 , 𝑖 , 𝑖 , 𝑖 , 𝑖 as expressed in equations [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF][START_REF] Huang | Multiphase machine theory and its applications[END_REF][START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines-Part I[END_REF][START_REF] Duran | Recent Advances in the Design, Modeling, and Control of Multiphase Machines-Part II[END_REF]. In α-β and d-q frames, the real fivephase machine is equivalent to three fictitious machines (the first machine, the second machine and the homopolar machine). Each fictitious machine consists of a given group of harmonic components [START_REF] Semail | Right harmonic spectrum for the back-electromotive force of an n-phase synchronous motor[END_REF]. It is assumed that the fundamental component is considered in the first fictitious machine while the third harmonic component is in the second fictitious machine. Other higher harmonic components are neglected. The zero-sequence current is zero (𝑖 = 0) because of the wye-connected winding; therefore, it cannot contribute to torque generation [START_REF] Kestelyn | Vectorial multi-machine modeling for a five-phase machine[END_REF]- [START_REF] Parsa | Five-phase permanent-magnet motor drives[END_REF].

⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑖 𝑖 𝑖 𝑖 ⎦ ⎥ ⎥ ⎥ ⎤ = [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] ⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑖 𝑖 𝑖 𝑖 ⎦ ⎥ ⎥ ⎥ ⎤ ; ⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑖 𝑖 𝑖 𝑖 ⎦ ⎥ ⎥ ⎥ ⎤ = [𝑻 𝑷𝒂𝒓𝒌 ][𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] ⎣ ⎢ ⎢ ⎢ ⎡ 𝑖 𝑖 𝑖 𝑖 𝑖 ⎦ ⎥ ⎥ ⎥ ⎤ (1-2) [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 1 0 1 0 1/√2 𝑐𝑜𝑠(𝛿) 𝑠𝑖𝑛 (𝛿) 𝑐𝑜𝑠(2𝛿) -𝑠𝑖𝑛 (2𝛿) 1/√2 𝑐𝑜𝑠(2𝛿) 𝑠𝑖𝑛 (2𝛿) 𝑐𝑜𝑠(𝛿) 𝑠𝑖𝑛 (𝛿) 1/√2 𝑐𝑜𝑠(2𝛿) -𝑠𝑖𝑛(2𝛿) 𝑐𝑜𝑠(𝛿) -𝑠𝑖𝑛 (𝛿) 1/√2 𝑐𝑜𝑠(𝛿) -𝑠𝑖𝑛 (𝛿) 𝑐𝑜𝑠(2𝛿) 𝑠𝑖𝑛 (2𝛿) 1/√2 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ (3) 2 [𝑻 𝑷𝒂𝒓𝒌 ] = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑐𝑜𝑠𝜃 -𝑠𝑖𝑛𝜃 0 0 0 sin 𝜃 𝑐𝑜𝑠𝜃 0 0 0 0 0 𝑐𝑜𝑠3𝜃 -𝑠𝑖𝑛3𝜃 0 0 0 𝑠𝑖𝑛3𝜃 𝑐𝑜𝑠3𝜃 0 0 0 0 0 1 ⎦ ⎥ ⎥ ⎥ ⎤ (4)
where [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] and [𝑻 𝑷𝒂𝒓𝒌 ] are the Clarke and Park transformation matrices in healthy operation; 𝛿 = 2𝜋/5 is the spatial phase shift angle; 𝜃 is the electrical angle of the machine.

III. FAULT-TOLERANT CONTROL METHODS USING REDUCED-ORDER TRANSFORMATION MATRICES IN DEGRADED MODE

Overview of the recent methods

In healthy mode, the three subspaces 1-1, 3-3 and zerosequence associated respectively to the first, third and fifth harmonics are orthogonal to each other, which makes the control currents in these subspaces independent. Therefore, constant torque is easy to produce even if the electromotive forces contain the first, third and fifth harmonics. When phase a is open-circuited (Fig. 1), the first harmonic of current interacts with all the harmonics of electromotive forces. As a result, an asymmetric condition in the rotating field appears and torque pulsations are produced. The imposed current references of healthy mode are often unsuitable in this case; hence, these current references need to be modified in degraded modes [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF]. The basic idea of these methods is to preserve the sinusoidal M.M.F under asymmetrical conditions. In [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF], H. Ryu et al. introduces new transformation matrices for five-phase PMSM machines according to the concept that every harmonic component of the current and voltage, which can contribute to the torque positively, can be equivalently represented as dc components even under asymmetrical fault conditions. Therefore, the fundamental currents in d-q frame are kept constant as in heathy case with the new transformation matrices. H. Guzmán et al. in [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF], [START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF] proposes new transformation matrices for 5-phase induction machines to preserve symmetrical models of resistive, leakage inductance and back-EMF with timeinvariant coefficients in asymmetrical fault conditions. In [START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF], [START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF], B. Tian et al. constructs a model based on the concept similar to the studies in [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF], [START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF] for 5-phase PMSM machines using sliding mode control to eliminate the speed pulsation. In [START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF], [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF], H. Zhou et al. develops new orthogonal reduced-order transformation matrices derived by the fault-tolerant references. These studies use a remedial field-oriented control (RFOC). In [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF], a control strategy is proposed by A. Seck et al. for a permanent magnet synchronous generator with single open phase by applying Particle Swarm Optimization to find elements of a new transformation matrix. All researches above have not considered the nonsinusoidal back-EMF machines as well as non-sinusoidal current references. In those methods, only fundamental currents cannot eliminate torque ripples with the non-sinusoidal back-EMFs. Thus, G. Liu et al. in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] proposes a new model for non-sinusoidal back-EMF machines. The torque ripples caused by the third harmonic air-gap flux is analyzed and eliminated by imposing third harmonic currents in the rotating frame. The next section analyzes these methods with more details and a five-phase machine described in Table 1 is used for validations. Categorization of reduced-order transformation matrix methods When phase a is opened, the current of phase a is equal to zero; hence, the Clarke transformation matrix from equation (3) becomes [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] as expressed in equation [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF]. The coupling between fundamental current and the third harmonic current is shown in equation [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF].

[𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑐𝑜𝑠(𝛿) 𝑠𝑖𝑛 (𝛿) 𝑐𝑜𝑠(2𝛿) -𝑠𝑖𝑛 (2𝛿) 1/√2 𝑐𝑜𝑠(2𝛿) 𝑠𝑖𝑛 (2𝛿) 𝑐𝑜𝑠(𝛿) 𝑠𝑖𝑛 (𝛿) 1/√2 𝑐𝑜𝑠(2𝛿) -𝑠𝑖𝑛(2𝛿) 𝑐𝑜𝑠(𝛿) -𝑠𝑖𝑛 (𝛿) 1/√2 𝑐𝑜𝑠(𝛿) -𝑠𝑖𝑛 (𝛿) 𝑐𝑜𝑠(2𝛿) 𝑠𝑖𝑛 (2𝛿) 1/√2 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ (5)
𝑖 = 0 → 𝑖 = -𝑖 (6)
The first, third and fifth row vectors of matrix [𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 ] in equation ( 5) are no longer orthogonal to each other. Consequently, the third harmonics of current cannot make a rotating field independently of the first ones; therefore, the third components are the consequences of the first harmonic currents, leading to uncontrolled current and torque ripples. A new transformation matrix 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 needs to be determined with its dimension equal to the number of remaining healthy phases (4 by 4) as shown in equation [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF]. By using matrix 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 , the currents of healthy phases in natural frame are projected onto the space αβz0 including fundamental stationary plane α-β (𝑖 , 𝑖 ), subspace z (𝑖 ) and zero-sequence subspace (𝑖 ). To control the currents in new subspaces independently (no coupling between these two where 𝑖 , 𝑖 , 𝑖 , 𝑖 are the new phase currents in natural frame of the five-phase machine.

There are several ways to find the new transformation matrix

𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏
to preserve sinusoidal currents in α-β frame, leading to constant currents in d-q frame in single open phase fault. It means that currents 𝑖 and 𝑖 form a circle as shown in Fig. 2. Five methods using new transformation matrices in faulty modes will be analyzed in this section and summarized in Table 2. In methods [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF][START_REF] Huang | Multiphase machine theory and its applications[END_REF][START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines-Part I[END_REF][START_REF] Duran | Recent Advances in the Design, Modeling, and Control of Multiphase Machines-Part II[END_REF], a new Park transformation matrix in equation ( 8) is used to convert currents and back EMFs into d-q frame, considering only fundamental components. Method [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF] applies an extra matrix in equation ( 20) for the third harmonic components.

𝑻 𝑷𝒂𝒓𝒌 𝟏 = cos 𝜃 -𝑠𝑖𝑛𝜃 0 0 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 0 0 0 0 1 0 0 0 0 1 (8)
(1) Method 1

The method in [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF] is proposed for a sinusoidal back-EMF fivephase PMSM with a new transformation matrix 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 as shown in equation ( 9) by eliminating the third row of equation [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF]. The reason for this elimination is the coupling between the first and third harmonic currents as expressed in equation [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF]. For the sake of simplicity, all elements of the fifth-row vector are set to 1 without loss of orthogonality. Afterwards, the first row is adjusted by adding a coefficient x with the aim of respecting the orthogonality of matrix 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 and adjusting the symmetry of back-EMFs in degraded modes. Indeed, the first three row vectors in equation ( 9) are always orthogonal to each other regardless of the value of x because their dot products are always equal to zero. Induction Machine in [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF], [START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF] and Non-sinusoidal back-EMF PMSM in [START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF], [START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF] No 3 [START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF], [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF] --Sinusoidal back-EMF PMSM Yes However, the first and fourth row vectors of 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 are perpendicular only when 𝑥 equals to 0.25. In addition, the expression of current 𝑖 in equation [START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF] shows that coefficient 𝑥 is unable to affect the projection of current onto fundamental α-β frame since the sum of remaining healthy phase currents is always zero. However, there is an asymmetrical waveform of back EMFs in the new frame α-β as shown in Fig. 3a. The reason is that the sum of back EMFs in healthy phases is not equal to zero. Therefore, from equation [START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF], the value of 𝑥 affects the symmetrical characteristic of the back EMFs. (2) Method 2

The method in [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF]- [START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF], developed from [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF], is applied for five-phase induction and permanent magnet machines with coefficient 𝑥 in equation ( 9) equal to -1. In [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF], [START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF], for an induction machine, the projections of currents, voltages and back EMFs onto the new fundamental frame α-β are obtained with the symmetrical characteristic. The back EMFs in the subspace α-β form a circle with 𝑥 = -1 as plotted in Fig. 3b. However, the fourth-row vector is not orthogonal to the first-row vector anymore, but it does not affect the control of current in the new fundamental subspace due to the wye connection. Authors in [START_REF] Tian | Decoupled Modeling and Nonlinear Speed Control for Five-Phase PM Motor Under Single-Phase Open Fault[END_REF], [START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF] apply the same transformation matrices as in [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF], [START_REF] Guzman | Comparative Study of Predictive and Resonant Controllers in Fault-Tolerant Five-Phase Induction Motor Drives[END_REF] but for a non-sinusoidal back-EMF PMSM. Because the fundamental components of currents interact with the third harmonic back-EMFs, resulting in torque ripples. Therefore, a sliding mode control for speed is used. Nevertheless, the torque ripples as well as speed pulsation still exist.

(3) Method 3 In [START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF], [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF], the new transformation matrix is proposed for a sinusoidal back-EMF five-phase PMSM according to (5) without an additional coefficient x as in equation [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF]. All row vectors in equation ( 12) are orthogonal to one another when every row of matrix in equation ( 5) is modified. The first two row vectors are derived based on the current references in natural frame in [START_REF] Liu | Comparison of Two SVPWM Control Strategies of Five-Phase Fault-Tolerant Permanent-Magnet Motor[END_REF] as shown in Fig. 4. The phasor angles of ccurrent and d-current are unchanged while those of phase b and phase e are shifted 0.5𝛿 compared to the healthy case to compensate the loss of current in phase a. The same torque as in healthy mode can be preserved by increasing the amplitude of new current 1.382 times higher than the one in healthy mode.

In equation [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF], the third-row vector is defined to guarantee its orthogonality to the first two row vectors and make sure that current 𝑖 is equal to zero. Constant coefficients in the denominators of all elements of 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑 are determined to change the amplitude of the current references in natural frame. This adjusting enables the machine to generate a torque in faulty mode equal to healthy mode. Additionally, in [START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF], [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF], the authors tackle the problem of the asymmetrical back EMFs in α-β frame as presented in [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF] by changing the way to calculate the projection of back EMFs onto the new stationary frame α-β as shown in equations [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF][START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF][START_REF] Parsa | Fault-tolerant five-phase permanent magnet motor drives[END_REF]. Specifically, the transpose of the inversion of the new transformation matrix is applied instead of the transformation matrix. Indeed, the inversion of 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟑 is different form its transpose. In addition, equations [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF][START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] prove that the product of current and back-EMF vector in the new stationary subspaces is the same as in natural frame with a transformation in equation [START_REF] Parsa | Fault-tolerant five-phase permanent magnet motor drives[END_REF]. However, the experimental result in [START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF], [START_REF] Zhou | Dynamic Performance Improvement of Five-Phase Permanent-Magnet Motor With Short-Circuit Fault[END_REF] contains high torque ripples since the back EMFs of the experimental machine include high harmonic components. (4) Method 4

In [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF], a new transformation matrix applied for a sinusoidal back-EMF five-phase PMSM is expressed in equation ( 16). The principle is that the first two row vectors of the matrix in equation ( 5) are unchanged while its third-row vector is eliminated. In addition, elements of the fourth-row vector of (5) become 4 scalar variables (𝑘 , 𝑘 , 𝑘 , 𝑘 ). These variables need to be estimated by using Particle Swarm Optimization, according to two optimization criteria. The first criterion, copper loss minimization resulting in unbalanced phase currents, is derived with (𝑘 , 𝑘 , 𝑘 , 𝑘 ) = (0.6328 , -1, 1, -0.6328). The other criterion with balanced phase currents is obtained when (𝑘 , 𝑘 , 𝑘 , 𝑘 ) = ( -1, 1, -1, 1).

𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟒 = 𝑐𝑜𝑠𝛿 𝑠𝑖𝑛𝛿 𝑘 1 cos(2𝛿) sin (2𝛿) 𝑘 1 cos(2𝛿) -𝑠𝑖𝑛 (2𝛿) 𝑘 1 𝑐𝑜𝑠(𝛿) -𝑠𝑖𝑛 (𝛿) 𝑘 1 (16)
Simulation results in [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF] show that this method works with good performance when being applied to a purely sinusoidal machine.

(5) Method 5 In [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF], the authors propose new transformation matrices for a non-sinusoidal back-EMF five-phase PMSM with an open phase. The healthy phases of the machine are imposed with nonsinusoidal current references to eliminate torque ripples. The transformation matrices for the fundamental components are the same as in [START_REF] Guzmán | A comprehensive fault analysis of a five-phase induction motor drive with an open phase[END_REF]- [START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF] by using the matrix in equation ( 9) with 𝑥 = -1. The new space for the fundamental currents and back EMFs is denoted as α1β1z101 to differ from the space for third harmonic components. The fundamental current and back EMFs in the new space are presented in Figs. 5a and5b respectively.

The principle to find a new transformation matrix for the third harmonic components is like the fundamental ones. Thus, the MMF and back EMFs of the third harmonic components are expected to be the same as in healthy condition. Equations [START_REF] Kestelyn | Vectorial multi-machine modeling for a five-phase machine[END_REF][START_REF] Semail | Vectorial formalism for analysis and design of polyphase synchronous machines[END_REF] present the conversions of the third harmonic currents and the third harmonic back EMFs from natural frame into the new space α3β3z303 including the third harmonic plane α3-β3, subspace z3 (𝑖 ) and zero-sequence subspace (𝑖 ). Indeed, the new Clarke transformation matrix 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟓 in equation ( 19) is obtained by removing the first row of the matrix in equation [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF]. This row vector is associated to the first harmonic current 𝑖 which is coupled with the third harmonic current 𝑖 . The second row of the matrix in [START_REF] Fu | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], related to 𝑖 , is remained and associated to a The third harmonic components of currents and back EMFs in the new space α3β3z303 are symmetrical as shown in Fig. 6. Accordingly, the fundamental and third harmonic currents can be controlled independently, and their waveforms in natural frame are shown in Fig. 7. The total phase currents in natural frame are the sum of the corresponding first and third harmonic currents as presented in Figs. 8a. The non-sinusoidal back EMFs, with the third harmonic component equal to 32% of the first harmonic one, are presented in 8b. The ripple-free torque, when phase a is opencircuited, is the same as the torque in healthy operation as plotted in Fig. 9a.

Summary of studied methods

When phase a is open-circuited, types of the machines (related to back-EMF waveforms) have a strong effect on torque pulsations of the machines. The methods (1-4) in [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF]- [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF] propose only new transformation matrices for only fundamental components of currents and back EMFs. pulsations if the experimental back EMFs are not sinusoidal. However, the torque ripples still exist in these systems.

Only method 5 in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] is able to solve the problem of third harmonic components theoretically by using new transformation matrices for the third harmonic components as discussed in the previous subsection. However, the experimental results show that torque ripples are produced in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] with single open phase fault. The presence of torque ripples can be explained by the control of non-constant current references in d-q frame at high speed or the existence of other high harmonic components of back EMFs in the experimental machine.

In all studied methods above, the current in subspace z (iz) is used for optimization criteria or torque ripple elimination. Criterion Lowest Jules Losses (LJL) results in minimizing copper losses in which the amplitudes of phase currents are different in healthy phases. With criterion Equal Jules Losses (EJL), copper losses are distributed equally in every phase and healthy phase currents have the same amplitude. Using these methods, the new transformation matrices need to be determined for each case of open phase fault. For example, a new transformation matrix for open phase fault in phase a is unable to be applied for an open phase fault in phase b, c, d and e, and vice versa. As a result, the calculations and data storage will be challenges in practical applications.

IV.

CONCLUSIONS The paper has given an overview of several methods using reduced-order transformation matrices for fault-tolerant control for a five-phase machine in single open phase fault. The paper has analyzed the basic ideas and strategies to find the new transformation matrices. With methods (1 to 4) in [START_REF] Ryu | Synchronous-frame current control of multiphase synchronous motor under asymmetric fault condition due to open phases[END_REF]- [START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF], the fundamental components of current and back EMFs are considered to obtain constant torque. However, these methods are no longer suitable for non-sinusoidal back EMF machines due to torque ripples. The fifth method in [START_REF] Liu | Third Harmonic Current Injection in Fault-Tolerant Five-Phase Permanent-Magnet Motor Drive[END_REF] proposes the new transformation matrices for non-sinusoidal back EMFs. Thus, non-sinusoidal current references are taken into account to produce a ripple-free torque. This method has high potential to apply to all non-sinusoidal back-EMF multiphase machines with a certain number of phases. However, a drawback of all methods above is that a new matrice need to be defined for a specific open phase fault.

Fig. 1 .

 1 Fig. 1. Asymmetrical condition of the rotating field of a five-phase machine when phase a is open-circuited TABLE I. ELECTRICAL PARAMETERS OF CONVERTER AND MACHINE Parameter Value Stator resistance R (Ω) 1.4 Phase inductance L (mH) 10.1 Mutual inductance 𝑀 (mH) 3.1 Mutual inductance 𝑀 (mH) -1.05 Mutual inductance 𝑀 (mH) -5.3 Number of pole pairs p 3 Speed-normalized amplitude of 1 st harmonic back-EMF 𝐸 (V/rad/s) 1Speed-normalized amplitude of 3 rd harmonic back-EMF 𝐸 (V/rad/s) 0.32

  Axis of phase aAxis of phase b Axis of phase cAxis of phase dAxis of phase e ω subspaces), the new fundamental plane α-β needs to be orthogonal to subspace 𝑧 even when 𝑖 is imposed to zero. It means that the first 3 row vectors of the new transformation matrix 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 must be perpendicular to one another. The orthogonality of the fourthrow vector of 𝑻 𝑪𝒍𝒂𝒓𝒌𝒆 𝟏 to other row vectors is not mandatory since the zero-sequence current (𝑖 ) is obviously zero due to the wyeconnected winding.

Fig. 2 . 1 Fig. 3 .

 213 Fig. 2. The current references in fundamental α-β frame in post-fault conditions with any value of x in methods 1 and 2

1 ( 9 )

 19 𝑖 = {𝑖 𝑐𝑜𝑠𝛿 + 𝑖 𝑐𝑜𝑠2𝛿 + 𝑖 𝑐𝑜𝑠2𝛿 + 𝑖 𝑐𝑜𝑠𝛿 + 𝑥(𝑖 + 𝑖 + 𝑖 + 𝑖 )}[START_REF] Tian | Cancellation of Torque Ripples With FOC Strategy Under Two-Phase Failures of the Five-Phase PM Motor[END_REF] 𝑒 = {𝑒 𝑐𝑜𝑠𝛿 + 𝑒 𝑐𝑜𝑠2𝛿 + 𝑒 𝑐𝑜𝑠2𝛿 + 𝑒 𝑐𝑜𝑠𝛿 + 𝑥(𝑒 + 𝑒 + 𝑒 + 𝑒 )}[START_REF] Zhou | Remedial Field-Oriented Control of Five-Phase Fault-Tolerant Permanent-Magnet Motor by Using Reduced-Order Transformation Matrices[END_REF]

  𝑒 ⃗ = 𝑒 𝑒 𝑒 𝑒 and 𝚤 ⃗ = 𝑖 𝑖 𝑖 𝑖 are the back EMF and current vectors in the new stationary subspaces respectively; 𝑒 ⃗ = [𝑒 𝑒 𝑒 𝑒 ] and 𝚤 ⃗ = [𝑖 𝑖 𝑖 𝑖 ] are the back EMF and current vectors in natural frame respectively.

Fig. 4 .

 4 Fig. 4. New current references when phase a is opened in method 3

  subspace called 𝑧 . The third and fourth rows of the matrix in (5) associated to the third harmonic subspace α3-β3 are reused. In equation[START_REF] Parsa | Five-phase permanent-magnet motor drives[END_REF], a coefficient x is added to the row vector related to 𝛼 to ensure the symmetrical characteristic of the third harmonic back-EMFs (𝑥 is equal to -1).

  , 𝑖 , 𝑖 , 𝑖 ) are new third harmonic phase currents in natural frame; (𝑒 ,𝑒 ,𝑒 ,𝑒 ) are the third harmonic components of back EMFs in natural frame; (𝑖 , 𝑖 , 𝑖 , 𝑖 ) and (𝑒 , 𝑒 , 𝑒 , 𝑒 ) are currents and back EMFs in the new space α3β3z303 respectively.

Fig. 5 .Fig. 9 .

 59 Fig. 9. Torques in healthy case and in single open phase fault with a nonsinusoidal back-EMF machine at 10 rad/s using method 5 (a) and using methods (1-4) (b)If methods (1-4) in references [6]-[START_REF] Seck | Automatic generation of optimal phase currents for five-phase PMSG control under open phase condition[END_REF] are applied for machines with non-sinusoidal back EMFs, the interactions between the fundamental harmonics of currents and the third and other high harmonic components of back EMFs will cause torque ripples as shown in Fig.9b. Thus, in those papers, different control strategies are applied such as sliding mode control, remedial fieldoriented control and model predictive control to reduce the torque

TABLE 2 .

 2 SUMMARY OF METHODS USING REDUCED-ORDER TRANSFORMATION MATRICES

	Method References	x	Applied Machines	Orthog-onality
	1	[6]	0.25	Sinusoidal back-EMF PMSM	Yes
	2	[7]-[10]	-1


C Caredda

L Mahieu-Williame

R Sablong

M Sdika

J Guyotat

B Montcel

Real time intraoperative functional brain mapping

come   L'archive ouverte pluridisciplinaire

Introduction

Real-time intraoperative functional brain mapping using a RGB camera

The correction of the collected intensity slope is not optimal for the first maps (from C 1 to C 6 ). This problem will be investigated in a future work.

The functional quantitative maps processing time depends on the amount of pixels contained in the region of interest. CPU and GPU optimization code is required.

-Correction of the collected intensity slope

The slow drift of the collected intensity (cortical dessication) was corrected through a linear regression substracted to the original data.

-Modified Beer-Lambert law [4]

with Experimental paradigm: After the patient craniotomy and before the brain tumor resection surgery, the patient was awakened and functional tests have been realized. The stimulation of the motor cortex was achieved through a repetitive and alternative hand opening and closing at ≈1Hz. The paradigm consisted of 3 steps : 30s of rest, 30s of stimulation and 30s of rest. The neurosurgeon realized electrical brain stimulations to localize the patient brain motor and sensory areas.

Real-time architecture

Functional model

-Selection of activated areas

For each camera pixel, the Hb or HbO 2 concentration changes time courses are obtained by matrix inversion. The time courses which best fit the expected patient hemodynamic response [6] are considered (the Pearson correlation coefficient threshold is set to 0,5). charly.caredda@creatis.insa-lyon.fr Background: Optical imaging can be used to monitor the brain activity [1] whith the measure of hemoglobin concentration changes (∆[HbO 2 ] and ∆[Hb]). A device consisting of a continuous wave white light illumination and a RGB camera is a suitable approach [2].

-Functional quantitative maps

Objectives:

We propose to realize a real time intraoperative localization of brain functional areas using a RGB camera with the analysis of ∆[HbO 2 ] and ∆[Hb] time courses which is closely linked with the Bold contrast in f-MRI studies.

  D ; HOSPICES CIVILS DE LYON ; Bron, France.

  Hb or HbO 2 concentration changes are averaged over the duration of the patient cortical activity. C represents either [Hb] or [HbO 2 ]. (x,y) is the position of an image point. A global mean path length L is calculated with Monte-Carlo simulations [5]. D i : spectral sensitivity of the camera detector i (R,G,B). S: light source spectrum є n : chromophore n(Hb, HbO 2 ) molar absorption coeffient (L. Mol -1 .cm -1 ) Discussion & conclusion Hb real-time functional quantitative maps A global mean path length is calculated for all camera pixels. A pixel-wise computation of the mean path length is required. The Hb functional quantitative maps correspond to the motor and sensory areas validated by the neurosurgeon. Relevance of the functional quantitative maps as soon as the patient activity is established. Time (s) Context: Non invasive functional brain mapping is an imaging technique used before brain tumor resection to localize the functional areas of the patient brain. f-MRI and electrical brain stimulation are two complementary clinical tests realized before neurosurgery. f-MRI tests are realized before the patient craniotomy. Electrical brain stimulation tests are realized once the patient cortex is exposed.


David Gaudrie
email: david.gaudrie@mpsa.com

Rodolphe Le Riche

Victor Picheny

Benoit Enaux

Vincent Herbert

Benoît Enaux

From CAD to Eigenshapes for Surrogate-based Optimization

Keywords: Dimension Reduction, Principal Component Analysis, Parametric Shape Optimization, Gaussian Processes, Bayesian Optimization

come   L'archive ouverte pluridisciplinaire

Introduction

The most frequent approach to shape optimization is to describe the shape by a vector of d Computer Aided Design (CAD) parameters, x ∈ X ⊂ R d and to search for the parameters that minimize the objective function, x * = arg min x∈X f (x). In the CAD modeling process, the set of all possible shapes has been reduced to the space of parameterized shapes, Ω Ω Ω := {Ω x , x ∈ X}.

It is common for d to be large, d 50. Optimization in such a high-dimensional design space is difficult, especially when f (•) is the output of a high fidelity numerical simulator that can only be run a restricted number of times. In computational fluid dynamics for example, simulations easily take 12 to 24 hours and evaluation budgets range between 100 and 200 calls. Surrogate-based approaches [START_REF] Jones D R | Efficient global optimization of expensive black-box functions[END_REF] have proven their effectiveness to tackle optimization problems in a few calls to f (•). They rely on a surrogate model (or metamodel, e.g., Gaussian Processes [START_REF] Rasmussen | Gaussian processes for machine learning[END_REF]) built upon n past observations of y i = f (x (i) ). The metamodel is utilized by an acquisition function such as the Expected Improvement [START_REF] Jones D R | Efficient global optimization of expensive black-box functions[END_REF] to decide what will be the next parametric design x (n+1) evaluated. However, such techniques suffer from the curse of dimensionality when d is large. The budget is also typically too narrow to perform sensitivity analysis and select variables prior to optimizing. A further issue is that the CAD parameters x commonly have heterogeneous impacts on the shapes Ω x : many of them are intended to refine the shape locally whereas others have a global influence so that shapes of practical interest involve interactions between all the parameters. Most often, the set of all CAD generated shapes, Ω Ω Ω, can be approximated in a δ-dimensional manifold, δ d. In [START_REF] Raghavan | Towards a space reduction approach for efficient structural shape optimization[END_REF][START_REF] Raghavan | Numerical assessment of springback for the deep drawing process by level set interpolation using shape manifolds[END_REF] this manifold is accessed through an auxiliary description of the shape, φ(Ω), φ being either its characteristic function or the signed distance to its contour. The authors aim at minimizing an objective function using diffuse approximation and gradient-based techniques, while staying on the manifold of admissible shapes. Another way to handle shapes is the Point Distribution Model, in which the contour of the shape is discretized [START_REF] Stegmann | A brief introduction to statistical shape analysis[END_REF][START_REF] Wang | Kernel principal component analysis and its applications in face recognition and active shape models[END_REF]. In [START_REF] Bouhlel | Improving kriging surrogates of high-dimensional design models by Partial Least Squares dimension reduction[END_REF], a kriging surrogate is built in the space of the first Partial Least Squares axes. The consideration of modes of discretized shapes is further pursued in [START_REF] Li | A data-based approach for fast airfoil analysis and optimization[END_REF] where bounds on the modes coefficients are enforced and the gradient of the metamodel used for optimization purposes.

Following the same route, in Section 2, we retrieve a shape manifold with dimension δ d by principal component analysis of shapes described in an ad hoc manner. Section 3 is devoted to the construction of a kriging surrogate model in reduced dimension (but the remaining dimensions are not completely omitted). In Section 4, we employ the metamodel to perform global optimization [START_REF] Jones D R | Efficient global optimization of expensive black-box functions[END_REF] via the maximization of the Expected Improvement. A reduction of the space dimension is achieved through a random embedding technique [START_REF] Wang | Bayesian optimization in high dimensions via random embeddings[END_REF] and a pre-image problem is solved to keep the correspondence between the eigenshapes and the CAD parameters.

2 From CAD description to shape eigenbasis CAD parameters are usually set up by engineers to automate shape generation. These parameters may be Bézier or Spline control points which locally readjust the shape. Other CAD parameters, such as the overall width or the length of a component, have a more global impact on the shape. While these parameters are intuitive to a designer, they are not chosen to achieve any specific mathematical property and in particular do not let themselves interpret to reduce dimensionality.

In order to define a better behaved description of the shapes that will help in reducing dimensionality, we exploit the fact that the time to generate a shape Ω x is negligible in comparison with the evaluation time of f (x).

In the spirit of kernel methods [START_REF] Schölkopf | Kernel principal component analysis[END_REF][START_REF] Vapnik | The nature of statistical learning theory[END_REF], we analyze the designs x in a high-dimensional feature space Φ ⊂ R D , D d (potentially infinite dimensional) that is defined via a mapping φ(x), φ : X → Φ. With an appropriate φ, it is possible to distinguish a lower dimensional manifold embedded in Φ. As we deal with shapes, natural candidates for φ are shape representations. In the literature, shapes have been described by the characteristic function [START_REF] Raghavan | Towards a space reduction approach for efficient structural shape optimization[END_REF] (a grid of 0's or 1's whether we are inside or outside the shape, cast as a vector), the signed distance to its contour ∂Ω x [START_REF] Raghavan | Numerical assessment of springback for the deep drawing process by level set interpolation using shape manifolds[END_REF] (again on a spatial grid which is transformed into a vector), or the discretization of ∂Ω x [START_REF] Stegmann | A brief introduction to statistical shape analysis[END_REF] (the coordinates of nodes on the contour in a given order and transformed into a vector). We map a large number (N ) of uniformly sampled designs x (i) ∈ X to Φ and build the matrix Φ Φ Φ ∈ R N ×D which contains the centered φ(x (i) ) ∈ R D in rows. Next, we perform a Principal Component Analysis (PCA) on Φ Φ Φ: the eigenvectors of Φ Φ Φ Φ Φ Φ, written v j , form an ordered orthonormal basis of Φ with decreasing importance as measured by the PCA's eigenvalues λ j , j = 1, . . . , D. The v j 's are orthonormal directions in Φ that explain the most the dispersion of the high-dimensional representations of the shapes, φ(x (i) ). Any design x can now be expressed in the eigenbasis

1 V := (v 1 , . . . , v D ) since φ(x) = φ φ φ + D j=1 α j v j , where (α 1 , . . . , α D ) =: α α α = V (φ(x) -φ φ φ)
are the coordinates in the V basis (principal components). α j is a deviation from the mean shape, φ φ φ ∈ Φ, in the direction of the eigenshape v j .

In experiments that are partially reported here due to space limitations, we have generated shapes of known low intrinsic dimension: in the example of Fig. 1, the shapes are a set of circular holes of varying centers and radii, therefore described by 1, 2 or 3 parameters. PCAs were then carried out on the Φ Φ Φ's associated to the three mappings (characteristic function, signed contour distance and contour discretization). We have observed the drop in eigenvalues λ j and plotted the α α α's manifolds.

The discretization of ∂Ω x was the only mapping φ for which the intrinsic shape dimension was equal to the number of non-zero eigenvalues, i.e., the shape intrinsic dimension was recovered. With this φ, the manifold of α α α (i) 's was better unfolded, which is advantageous for metamodeling and the eigenshapes v j were physically meaningful. For these reasons in the following, we will only consider the α α α's obtained using the contour discretization as φ mapping.

The three first eigenshapes v 1 , v 2 and v 3 as well as the mean shape φ φ φ of the NACA 22 airfoil can be seen in Figure 2. The NACA 22 benchmark is a data set made of airfoils described by d = 22 CAD parameters with the associated lift and drag coefficients as outputs [START_REF] Gaudrie | Budgeted Multi-Objective Optimization with a Focus on the Central Part of the Pareto Front -Extended Version[END_REF]. Notice that even though the NACA 22 shapes contain bumps, the three first eigenvectors do not include them. Bumps, which correspond to local refinements of the shapes, appear from the 4th eigenshape on.

GP model for reduced eigenspace

Building a surrogate-model in the space of principal components has already been investigated to construct reduced order models [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF]. In most applications, the dimension reduction is carried out in the output space, which has large dimension when it corresponds to values on a finite element mesh. The response is approximated by a linear combination of a small number of modes, and the metamodel is a function of the modes coefficients.

Here, we aim at reducing the dimension of the input space by building a surrogate of the design principal components, α α α. A first idea to reduce the dimension of the problem is to conserve the δ first eigenvectors v j according to some reconstruction quality criterion measured by the eigenvalues. Given a threshold T (e.g., 0.99), only the first δ modes such that δ j=1 λj D j=1 λj > T are retained for the surrogate-based optimization process because they contribute for 100 × T % of the variance in Φ.

But such an approach relies only on considerations about the shape geometry. The output y is not taken into account for the dimension reduction even though some v j , j ∈ {1, . . . , δ} may influence y or not. Two 1 In a slight abuse of notation, we use V for both the basis and the matrix made of the v's. shapes which differ in the α j components with j ≤ δ may behave similarly in terms of output y, so that further dimension reduction is possible. Vice versa, eigencomponents that have a small geometrical effect and were neglected may be reintroduced because they matter for y.

A second idea is thus to select the eigencomponents that impact y the most. This is done by maximizing the penalized log-likelihood [START_REF] Yi | Penalized Gaussian Process Regression and Classication for High-Dimensional Nonlinear Data[END_REF] of a Gaussian process (GP) in the high (D) dimensional space of α α α's,

max ϑ pl λ (α α α (1:n) , y 1:n ; ϑ) where pl λ (α α α (1:n) , y 1:n ; ϑ) := l(α α α (1:n) , y 1:n ; ϑ) -λ θ θ θ -1 1 (1)
The ϑ are the GP's hyper-parameters which include the length-scales of the GP, θ j . l stands for the loglikelihood of the GP and θ θ θ -1 := (1/θ 1 , . . . , 1/θ D ) is the vector containing the inverse length-scales of the GP. It is indeed known [START_REF] Salem | Sequential dimension reduction for learning features of expensive black-box functions[END_REF] that if θ j → +∞, the direction v j has no influence on y. The L 1 penalty term applied to the θ j 's performs variable selection: this Lasso-like procedure promotes zeros in the vector of inverse length-scales, hence sets many θ j 's to +∞. In the end, the directions with small enough θ j , denoted α α α a ∈ R δ , are declared to be active with regard to y, and have to be emphasized during metamodeling and optimization: if θ j ≤ 10 × min i=1,...,D θ i , then α j is a component of the active α α α a . Even if the maximization of pl λ is carried out in a D-dimensional space, the problem is tractable since the gradients of pl λ are analytically known, and because the L 1 penalty convexifies the problem. Most local optima to this problem solely differ in θ j 's that are already too large to be relevant and efficient local optimizers consistently yield the same set of active variables α α α a . In our implementation, we have modified the likelihood maximization of the kergp package [START_REF] Deville | Package 'kergp[END_REF] to include the penalization term. After a dimensional anaysis of pl λ , we have chosen to take λ = n D to balance both terms.

Other techniques such as cross-validation or the use of different λ's for obtaining a pre-defined number of active components can also be considered. On the NACA 22 benchmark with few observations of f (•) (n = 15 here), Figure 2 shows that only few active components among the D = 600 are selected by the penalized maximum likelihood procedure. The three first principal axes, v 1 , v 2 and v 3 are retained when considering the drag (top). Indeed, these are the eigenshapes that globally impact the shape the most and change its drag. When the output y is the lift (bottom), only the second principal axis is selected. This eigenshape modifies the camber of the shape, which is known to highly impact the lift. The other eigenvectors are detected to be less critical for y's variations. When n grows, more eigenshapes get selected because they also slightly impact the output. For instance when n = 50, some eigenshapes that contain bumps (the 4th, the 5th, the 8th, etc.) are selected for modeling the lift. They also contribute to changing the camber of the airfoil, hence its lift.

Figure 2: Variable selection on the NACA 22 benchmark by penalized maximum likelihood. For the drag (top), the three first eigenshapes that act on the shape, hence on its drag, are selected (red coefficients). For the lift, only the second eigencomponent (v 2 ) is selected (bottom). Indeed v 2 modifies the camber of the airfoil, hence it plays a major role on the lift. The other eigenbasis vectors (green coefficients) are estimated to be less influent on y.

Completely omitting the "non-active" dimensions, α α α a ∈ R D-δ , and building the surrogate model Y (•) in the sole α α α a space is not satisfactory however, as it is equivalent to erasing some geometric patterns of the shapes which may contribute to small variations of y. For this reason, an additive GP with zonal anisotropy [START_REF] Allard | Anisotropy models for spatial data[END_REF] between the active eigenshapes and the residual ones is considered: Y (α α α) = Y a (α α α a ) + Y a (α α α a ). Y a (α α α a ) is the anisotropic main-effect GP which works in the reduced space of active variables. It requires the estimation of δ + 1 hyper-parameters (the length-scales θ j and a GP variance) and aims at capturing most of y's variation, related to α α α a 's effect. Y a (α α α a ) is a GP over the large space of inactive components. It is a GP which solely aims at taking residual effects into account. Thus, a modeling assumption to keep it tractable is to consider it isotropic, i.e., it only has 2 hyper-parameters, a unique length-scale and a variance. In the end, even though Y (α α α) operates with α α α's ∈ R D and there are fewer observations than dimensions, n D, it remains tractable since only a total of δ + 3 hyper-parameters have to be learned, which remains affordable even when the number of observations is small. This additive model can be interpreted as a GP in the α α α a space, with an inhomogeneous noise fitted by the Y a GP.

The effect of this refined surrogate model is a better predictive capacity, demonstrated by its increased performance on the NACA 22 test set shown in Figure 3. The additive GP over the space of active and inactive variables (left "Selection" boxes) achieves better performance than the GP which only uses the δ = 3 first principal components (central "3 first" boxes). It also outperforms the classical approach of building the metamodel in the space of CAD parameters (right "CAD params" boxes). An interesting phenomenon is visible in the two rightmost plots of Figure 3 where lift is the output: even though the three first eigenvectors V 1:3 = (v 1 , v 2 , v 3 ) contribute to 98.5% of the shape discretizations variability, metamodeling in the space they span leads to a deteriorated performance, the prediction being even worse than in the 22-dimensional CAD parameters space. This is explained by the fact that none of these eigenshapes contains bumps, which appear from the 4th one. Since the bumps contribute to the airfoils camber hence to the lift, critical information is lost and performance is degraded. This highlights the benefits of selecting the variables that impact y for the main effect GP, and of not completely disregarding the remaining variables.

Optimization in reduced dimension

We now turn to the problem of finding the shape that minimizes an expensive objective function f (•). To this aim, we employ the previous additive GP, which works in the space of eigencomponents α α α, in an Efficient Global Optimization procedure [START_REF] Jones D R | Efficient global optimization of expensive black-box functions[END_REF]: at each iteration, a new shape is determined given the previous observations {(α α α (1) , y 1 ), . . . , (α α α (n) , y n )} by maximizing the Expected Improvement (EI) as calculated with the GP Y (α α α): α α α (n+1) = arg max α α α∈R D EI(Y (α α α)). Although no call to the expensive f (•) is needed at this step, optimizing the EI is tricky as it is a multi-modal and high (D) dimensional function.

We can however take advantage of the dimension reduction beyond the construction of Y : α α α a ∈ R δ are the variables that affect y the most and should be prioritized for the optimization of f (•). Maximizing the EI solely with respect to Y a (α α α a ) is nonetheless not an option, as the full GP Y (•) requires the knowledge of α α α = [α α α a , α α α a ]. Moreover, α α α a act as local refinements to the shape that contribute a little to y, and should be optimized also.

For these reasons, instead of maximizing EI([α α α a , α α α a ]) , we maximize the EI with respect to α α α a and use a random embedding [START_REF] Wang | Bayesian optimization in high dimensions via random embeddings[END_REF] to coarsely optimize the components α α α a : we maximize EI([α α α a , α A]), where α ∈ R is the coordinate along a random line in the α α α a space, A. Since α α α a have been classified as inactive, it is not necessary to make a large effort for their optimization. The EI maximization is hence carried out in a much more tractable δ + 1 -dimensional space and still has analytical gradients. From its optimum α α α * = (α α α a * , α * ) ∈ R δ+1 arises a D-dimensional vector, α α α (n+1) = (α α α a * , α * A 1 , . . . , α * A D-δ ) to be evaluated by the true function.

A last point needs however to be fixed: the expensive simulator neither works with shapes nor with α α α's but with CAD parameters x. We have to solve the pre-image problem, that is to say to find the CAD parameter vector x whose representation in the V basis equals α α α (n+1) . Because there are more α α α's than x's, D d, a strict equality may not be feasible and the pre-image problem is relaxed into:

x (n+1) = arg min x∈X V (φ(x) -φ φ φ) - α α α (n+1) 2 .
To complete an iteration, x (n+1) , the parametric shape that resembles α α α (n+1) the most, is evaluated by the computer code, which returns y n+1 = f (x (n+1) ). Notice that the surrogate model is updated with y n+1 and α α α (n+1) and not with the x (n+1) .

Figure 3 shows an optimization run for the minimization of the NACA 22's drag. The main advantage of our approach (bottom left) is that it enables an early search for low drag airfoils. The standard approach (bottom center) needs much more function evaluations for building the initial surrogate model (black dots) because the inputs live in a space of higher dimension. Furthermore, the approach introduced in this paper would gain in importance in problems with much more than d = 22 CAD parameters, where it would almost be impossible to build a large enough initial design of experiments. In this paper we have proposed a new methodology to apply Bayesian optimization techniques to parametric shapes. Instead of working directly with the CAD parameters, which are too numerous for an efficient optimization and may not be the best representation of the underlying shape, we unveil the lower dimensional manifold of shapes through an auxiliary mapping. The dimensions of this manifold that contribute the most to the variation of the output are prioritized for building a surrogate model, which is utilized for Bayesian optimization. A significant dimension reduction and speed-up is achieved on an aerodynamic CAD benchmark.

Figure 1 :

 1 Figure 1: Three first eigencomponents of the α α α (i) 's for three parametric test cases (columns) with low effective dimension equal to 1 (left), 2 (center) and 3 (right). The rows correspond to different φ's which are the characteristic function (top), the signed distance to the contour (middle) and the discretization of the contour (bottom).

Figure 3 :

 3 Figure 3: Top row: mean absolute prediction errors in drag (left) and lift (right) of the additive GP with zonal anisotropy and mode selection based on penalized likelihood ("Selection"), of an anisotropic GP using the first 3 eigenshapes ("3 first") and of an anisotropic GP built in the space of CAD parameters ("CAD params"). Bottom row: Drag optimization of the NACA 22 airfoil in the reduced eigenbasis (left) or carried out in the CAD parameters space (center). Low-drag airfoils are found while the classical method still evaluates the airfoils of the initial design of experiments (right).

Acknowledgments

This research was performed within the framework of a CIFRE grant (convention #2016/0690) established between the ANRT and the Groupe PSA for the doctoral work of David Gaudrie.


Charly Caredda

Laurent Mahieu-Williame

Rapha El Sablong

Micha El Sdika

Jacques Guyotat

Bruno Montcel

Real time intraoperative functional brain mapping using a RGB camera

Keywords:

e demonstrte tht qf mer nd ontinuous wve white light illumintion is suitle pproh to intropertively lolize in rel time the sensory nd motor res of the ptient rinF he nlysis of the reetne spetr of the quired light nd wonte grlo simultions llow us to mesure the onentrtion hnges of oxygented nd deoxygented hemogloin during the video quisitionF hese mesures re ompred to the expeted hemodynmi response to preisely lolize the funtionl res of the ptient rinF

INTRODUCTION

xonEinvsive funtionl rin mpping is n imging tehnique used to lolize the funtionl res of the rinF his tehnique is used efore rin tumor resetion surgery to indite to the neurosurgeon the ortil tissues whih should not e removed without ognitive impirmentF wsEf is widely used to lolize ptient sensory or motor resF roweverD fter ptient rniotomyD rin shift invlidtes the relevne of neuroEnvigtion to intropertively lolize the funtionl res of the ptient rinF por this resonD the loliztion of ortil motor or sensory res is hieved through eletril stimultion ut this tehnique suers from limittionsF Chance et al. hve demonstrted tht optil imging n e used to monitor rin tivity with the determintion of onentrtion hnges of oxygented hemogloin ¡HbO 2 D deoxygented hemogloin ¡HbD lood volume ¡Hb C ¡HbO 2 nd oxygented ytohromeEoxidseF 1 st lso hs een demonstrted tht devie onsisting of ontinuous wve white light illumintion nd qf mer is suitle pprohF 2,3 he ojetive of this work is to supply the methodologil tools for the rel time onstrution of quntittive funtionl rin mps sed on the nlysis of the ortil reetne spetr quired y digitl qf merF he motivtion for this pper is derived from the neessity to nlyze the hemogloin time ourses in rin tissue following neuronl tivtion whih re losely linked to the fyvh @lood oxygention level dependentA ontrst used in funtionl ws studiesF 4 gonentrtion hnges of oxygented hemogloin nd deoxygented hemogloin n e mesured for eh mer pixel using the modied feerEvmert lwF 5 his lw is omputed with n estimted men pth length lulted y wonteEgrlo simultionsF 6 pinllyD the onentrtion hnges time ourses re ompred to the expeted hemodynmi response of the ptient rin whih llow us to preisely lolize the funtionl tissues of the ptient rinF 2. MATERIAL AND METHODS 2.1 Experimental setup he instrumenttion onsisted of digitl qf gwy mer @fevi ePHHHE ITSuAD ontinuous wve white light soure @yew glssi IIT PQH vight ulA nd lptop @proessorX sntel gore iS UPHHUD P:SHGHz¢RD rmX IS:QGiBAF he qf mer ws used with n idmund yptis mer lens @f a ITmmf5 a I X I X RAF he sensor provided V its qf imges every QQ msF Further author information: (Send correspondence to Charly Caredda) Charly Caredda: E-mail: charly.caredda@creatis.insa-lyon.fr 2.2 Patient inclusion and experimental paradigm he video quisition ws relized efore rin tumor resetion surgeryF he ptient ws wkened nd funtionl tests hve een relizedF timultion of the motor ortex ws hieved through suessive st lenhing nd relxingF he prdigm onsisted of Q stepsX QHs of restD followed y QHs of stimultion nd QHs of restF he neurosurgeon relized eletril rin stimultion to lolize ptient rin motor nd sensory resF 2.3 Data pre-processing yne n imge is quiredD severl steps re proessed in prllelF pirstD the imge is registered to ompenste the rin repetitive motionF he lgorithm developed y Sdika et al. 7,8 ws implemented to ensure tht eh mer pixel orresponds to the sme ortil re ll long dt quisitionF he qf intensities of the quired imge re then independently ltered with lowEpss ss fessel temporl lterF 1 2.4 Model e funtionl model is pplied to ssess H b nd H bO 2 onentrtion hnges for eh mer pixelF Data correctionX he slow drift of the olleted intensity due to ortil tissue desition during the video quisition 9 ws orreted through liner regression sutrted to the originl dtF Functional modelX he ssessment of the onentrtion hnges through the modied feerEvmert lw 5 depends on the determintion of the men pth length of the deteted photonsF wonteErlo simultions 6 were proessed to estimte glol men pth length whih ws pplied to ll mer pixelsF he omputed onentrtion hnges times ourses re ompred to the expeted hemodynmi response with the erson orreltion oeientF e quntittive funtionl mp is then omputed for eh pixel whose orreltion oeient is superior to threshold vlueF hese mps orrespond to the H b or H bO 2 onentrtion hnges verged over the ptient ortil tivityF 2.5 Real time processing hese proessing steps re implemented in soft relEtime rhitetureF he preEproessing steps @see eF PFQA nd the model @see eF PFRA re exeuted in dierent prllel thredsF en intermedite uer is pled etween eh omputtionl step llowing the dt storgeF hese uers re required in our soft relEtime rhiteture sine the proessing times re dierent for eh omputtionl stepF he thred orresponding to the model is omputed s soon s the ortil tivity strtedF his thred is reursively lled while imges re preEproessedF 3. RESULTS AND DISCUSSION sn pigF ID the rel time H b quntittive funtionl mps @see eF PFRA whih re est orrelted with the expeted hemodynmi response re plotted @orreltion threshold sets to H:SAF he highest H b men onentrtions re lolized on lood vessels surrounding motor nd sensory resF he highest orreltion oeient re lolized on these lood vessels nd on motor nd sensory resF he omputed funtionl mps eome signint s soon s the ptient tivity is estlishedD see pigF IEgF Pichette et al. 10 proposed to msk the lood vessels in order to render more lerly the smller hemodynmi hnges in grey mtterF he mesurement of the orreltion oeient etween the onentrtion hnge time ourses nd the expeted hemodynmi response llows us to visulize the tivted ortil res without msking the lood vesselsF e glol men pth length is used for eh mer pixels in the modied feer vmert lwF 5 e pixelEwise determintion of the men pth length ould e implemented to ssess more preisely the H b onentrtion hngesF rowever it would require lot of wonte grlo simultions nd use n inrese of the proessing time of the model @see eF PFRAF he slow drift of the olleted intensity is orreted eh time the model is exeuted @see eF PFRA whih use n inrese of the proessing timeF sn dditionD the estimtion of the olleted intensity slope is not optiml for the rst mps @from pigF IEe to pigF IEpAF his prolem will e investigted in future workF 4. CONCLUSION sn onlusionD qf mer nd ontinuous wve white light illumintion seem to e suitle setup to intropertively monitor the sensory nd motor res of the ptient rin in rel timeF hese results indite tht our setup ould omplement the tul intropertive gold stndrd for the loliztion of ptient funtionl resF ACKNOWLEDGMENTS his work ws supported y the vefi swi @exEIIEvefEHHTQA of niversit e de vyonD within the progrm snvestissements devenir @exE IIEshiEHHHUA operted y the prenh xtionl eserh egeny nd prne vife smgingF REFERENCES I ghneD fFD hungD FD nehD gFD elterD gFD nd viptonD vFD gognitionEtivted lowEfrequeny modultion of light sorption in humn rinFD4 Proceedings of the National Academy of Sciences 90D QUUH{QUUR @eprF IWWQAF P wlonekD hF nd qrinvldD eFD sntertions etween eletril tivity nd ortil miroirultion reveled y imging spetrosopyX smplitions for funtionl rin mppingD4 272D SSI{R @HS IWWTAF Q teimersD eFD qrmerD wFD iertD fFD p uhtemeierD wFD oylD qFD veithnerD gFD hreierD tF FD vinduerD FD nd uohlEfreisD wFD smging of ortil hemogloin onentrtion with qf reetometryD4 UQTVIQ @tuly PHHWAF R eldsmnD wFD gummingD FD nd frodtmnnD eFD feyond fyvhX yptimizing funtionl imging in stroke popultionsX yptimizing fyvh smging in trokeD4 Human Brain Mapping 36D ITPH{ITQT @eprF PHISAF S uohlEfreisD wFD iertD fFD hreierD tF FD veithnerD gFD vinduerD FD nd oylD qFD epprtus for mesuring lood prmetersD4 @II PHIPAF T pngD F nd fosD hF eFD wonte grlo simultion of photon migrtion in Qd turid medi elerted y grphis proessing unitsD4 Optics express 17@PPAD PHIUV{PHIWH @PHHWAF U dikD wFD elstonD vFD ousseuD hFD quyottD tFD whieuEillimeD vFD nd wontelD fFD epetitive motion ompenstion for rel time intropertive video proessingD4 Medical Image Analysis 53D I { IH @PHIWAF V dikD wFD elstonD vFD whieuEillimeD vFD quyottD tFD ousseuD hFD nd wontelD fFD oust rel time motion ompenstion for intropertive video proessing during neurosurgeryD4 in 2016 IEEE 13th International Symposium on Biomedical Imaging D IHRT{IHRWD siiiD rgueD gzeh epuli @eprF PHITAF W yelshl gelD wFD weyerD FD hlD rFD oottkD F fFD uirshD wFD hkertD qFD nd worgensternD FD ivlution of intropertive optil imging nlysis methods y phntom nd ptient mesurementsD4 Biomedizinische Technik/Biomedical Engineering 58 @tnF PHIQAF IH ihetteD tFD vureneD eFD enguloD vFD vesgeD pFD fouthillierD eFD xguyenD hF uFD nd velondD pFD snE tropertive videoErte hemodynmi response ssessment in humn ortex using snpshot hyperspetrl optil imgingD4 Neurophotonics 3D HRSHHQ @ytF PHITAF

Figure 1 .

 1 Figure 1. Hb concentration changes maps emitted during the video acquisition. The correlation threshold is set to 0:5. The horizontal axis represents the acquisition time of the RGB camera. The experimental paradigm is represented under this axis. The vertical lines represent the display time of the computed functional maps (images from A to H). The double black arrows represent the processing time of the model, see Sec. 2.4. The markers M and S designate respectively the motor and the sensory areas which have been validated by the neurosurgeon through electrical stimulation. The colorbar represents the scale of variation of Hb mean concentration in Mol=L.