C Nadal

F Pigache

Y Lefevre

First Approach for the Modelling of the Electric Field Surrounding a Piezoelectric Transformer in View of Plasma Generation

Keywords: Electro-quasi static field problem, electrostatic field model, forced vibration analysis, modal analysis, numerical model, piezoelectric transformer, plasma generation

This paper is about an open multi-physics modelling problem resulting from recent investigations into plasma generation by piezoelectric transformers. In this first approach, the electric field distribution surrounding the transformer is studied according to a weak coupling formulation. Electric potential distribution views obtained numerically are compared to real views of plasma generation observed experimentally.

I. INTRODUCTION

P IEZOELECTRIC TRANSFORMERS (PT) are com- monly used in numerous applications. Recently, several studies have underlined the capability to generate non-equilibrium plasma from the electrical potential on a piezoelectric element undergoing mechanical vibrations [START_REF] Teranishi | Dynamic behavior of light emissions generated by piezoelectric transformers[END_REF]. In this kind of application, the classical Rosen type transformer is used because of its high voltage step-up ratio. This structure consists of a single rectangular piece of piezoelectric ceramic material: the primary part is poled in the thickness direction whereas the secondary part is poled in the length direction as illustrated in Fig. 1. This device is driven by an AC voltage power supply applied to the primary part at a frequency close to the resonance frequency of a longitudinal mode.

Experimentally, it can be shown that plasma is generated at the secondary part of the transformer [START_REF] Teranishi | Dynamic behavior of light emissions generated by piezoelectric transformers[END_REF]. This plasma is mainly due to the electric field surrounding the transformer, and is issued from the electric potential distribution on the secondary part surface. Thus, in order to understand how the plasma is generated, the surrounding electric field distribution must be studied. This electric potential distribution produced on the surface of the secondary part is expressed analytically by a previous model developed in [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF]. This analytical result is used to compute the distribution of the surrounding electrical field by means of a numerical model. This paper is organised as follows. In Section II, it is first shown that Electro Quasi Static (EQS) assumptions can be made [START_REF] Badics | Charge density-scalar potential formulation for adaptive time-integrationof nonlinear electroquasistatic problems[END_REF] and the problem can be reduced to an electrostatic problem. The PT analytical model is recalled in Section III and Section IV presents the numerical 2D electric field model. Finally, results showing the electric potential distributions are compared with real views of plasma glow discharges observed experimentally with different vibratory modes of the PT.

II. ELECTRO QUASI STATIC FIELD APPROXIMATIONS

The electric potential distribution on the surface of the PT secondary part creates a time varying electric field. In this first approach, we study the surrounding electric field under atmospheric pressure without the presence of plasma. The equations of electromagnetic field surrounding the transformer are as follows:

(1)

(2)

(3) (4) (5) (6)
where is the magnetic field, is the magnetic flux density, is the electric field, is the electric flux density, is the current density, is the magnetic permeability and is the electric permittivity. Generally, in low frequency applications, one refers to the relaxation time which depends on the electric permittivity and the conductivity of the medium to neglect displacement current.

A Magneto Quasi Static (MQS) field approximation can then be done and the displacement current can be neglected. But here, the conductivity of free space is very low and the displacement current cannot be neglected. Equation (2) means that a magnetic field is created by the displacement and (1), so that this magnetic field may interact in return with the electric field. Equations ( 1) to (6) lead to the electromagnetic wave equation:

(7) where is the conductivity of the medium. For the time harmonic electromagnetic field, this wave equation may be associated with the typical term:

(8)
where is the typical length scale of the problem, the speed of light and the angular frequency of the field.

In our problem, the frequency of the AC power supply is less than 300 kHz [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF]. So it yields:

(9)
This value is very great in front of the typical length of a PT [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF]:

(10) Thus, the wave propagation can be neglected and the problem can be reasonably considered as an Electro Quasi-Static (EQS) field problem, and dynamic term can be neglected in (1) [START_REF] Badics | Charge density-scalar potential formulation for adaptive time-integrationof nonlinear electroquasistatic problems[END_REF], [START_REF] Steinmetz | Electro-quasistatic field simulations based on a discrete electromagnetism formulation[END_REF].

By admitting there is no dissipation in the surrounding gas, the EQS problem can be considered as a succession of static field problems: at each time step the electric scalar potential is computed by solving a Laplace's equation where the electric scalar potential is imposed on the surface of the secondary part. For a given vibratory mode of the transformer, the electric potential on a point of the surface is synchronized by a sinusoidal time function with a magnitude depending on its location. As a consequence, for each vibratory mode, only one solution of the Laplace's equation is needed to know the distribution in space and the evolution in time of the electric field. Furthermore, previous studies have shown that the magnitude of electric potential can be considered as invariant along the width direction [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF]. In first approximation, the problem can then be reduced to a 2D electrostatic field problem.

III. ANALYTICAL MODEL OF PIEZOELECTRIC TRANSFORMER

This section provides information about the studied PT and the employed analytical model [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF].

A. Studied PT

Fig. 1 shows the geometric parameters of a classical Rosen Type PT: and are the lengths of the primary and secondary parts respectively, the thickness and the width.

The origin of the coordinate system is chosen at the centre of the interface between the primary and secondary parts. The primary part is made of layers with thickness. In the secondary part , the output electrode at the end is connected at the load resistance . In the present study, the secondary part is electrically open and the load is with

B. Analytical Model of Piezoelectricity in PT

According to the commonly made assumptions in PT studies and expressed in [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF], PT undergoes an axial stress along the axis and is free of shear stress. The stress and strain tensors are reduced to one component along axis :

(11) Each laminated layer of the primary part is supplied by a sinusoidal voltage, producing an electrical field oriented parallel to the axis . Thus, the electric field and electric displacement vectors of the primary side are:

(12) This leads to the following reduced constitutive relationships between the electric field and mechanical tensors in the primary part:

(13)

where:

Concerning the secondary part, an ideal insulation medium is assumed. Hence the components of the electric field alongthe axes and are equal to zero. Thus, the electric field in the secondary part is parallel to the axis:

(15)

This leads to the following reduced constitutive relationships between the electric field and mechanical tensors in the secondary part:

(16) where (17) The analytical model of PT has been presented in [START_REF] Nadal | Multimodal electromechanical model of piezoelectric transformers by Hamilton's principle[END_REF]. This multimodal general approach is based on Hamilton's principle. Finally, the application of the modal superposition method leads

TABLE III MODAL FREQUENCIES OBTAINED BY ANALYTICAL MODEL,ANSYS AND MEASUREMENTS

to proceeding to a modal analysis relying on forced vibration analysis [START_REF] Javadi | Electro-magneto-mechanical characterizations of the vibration of magnetic origin OE electrical machines[END_REF].

C. Modal Analysis

The first four longitudinal vibratory modes have been calculated by this analytical model and compared to the results of modal analysis performed with ANSYS Academic Research, Release 11.0. An infinite value of the resistance is taken. Fig. 2 shows the four modal wave shapes along obtained analytically and numerically. Fig. 3 shows the corresponding normalized electric potential distribution.

These results show that in spite of the 1D model approximation, the analytical model calculates correctly the mechanical mode shapes and their corresponding normalized electric potential distributions. Table III gives the resonant frequencies analytically, numerically and experimentally obtained. It shows a good accuracy of theoretical results with experiments.

D. Forced Vibration Analysis

After calculating the mode wave shapes, forced vibration can be calculated. In this case, the electric potential at the primary part is imposed equal to the sinusoidal supply voltage (Fig. 1). Moreover, the secondary part is still electrically open. In order to avoid a theoretical infinite value of the calculated potential, and displacement when the supply frequency is equal to the modal frequency, mechanical losses have to be taken into account. Therefore, some parameters of the piezoelectric model in ( 13) and ( 16) are usually replaced by complex numbers:

(18) where is the mechanical factor given in Table II. The electric potential distributions along axis for supply frequencies close to the modal ones have been obtained by the analytical model and are illustrated in Fig. 4. It is worth noting that the electric potential corresponding to the mode is very low compared to the others, and it cannot be sufficient to produce a glow discharge.

IV. NUMERICAL MODEL OF THE ELECTRIC FIELD

In order to solve the steady state problem of electrical potential surrounding the piezoelectric device, the Finite Difference Method (FDM) is applied because it can be implemented easily for this first approach [START_REF] Campbell | Two-dimensional finite difference method for the analysis of piezoelectric devices[END_REF]. Obviously, this model is a weak coupling formulation relying on the previous analytical model and a numerical 2D model for the electric field equations.

Fig. 5 shows the rectangular study domain in which the upper-half part of a Rosen type transformer is considered. , , and are the surrounding medium, the primary and the secondary parts and the Dirichlet boundaries, respectively, where the electrical potential is vanishing. The Rosen type transformer is supposed to be surrounded by a free electric charge air with vacuum permittivity . The primary part is considered as a dielectric medium with a high relative permittivity ( 1000). A -invariance and -axis symmetry are supposed.

Consequently, the electrical potential produced on the secondary side follows the boundary problem shown in Fig. 5, where is the Laplace operator and and is the electrical potential calculated by the analytical model (Fig. 4). In order to discretize this elliptic PDE, a 5-point instead of a 9-point stencils method is sufficient. V. R ESULTS For each resonant longitudinal vibratory mode, the electric potential distribution calculated by the analytical model of PT (Fig. 4) is applied on the definitiondomain of . Cross-sectional views of the electric potential in the domain obtained for each of the first three modes are shown at the top of Fig. 6, Fig. 7 and Fig. 8. These views can be compared with front views of the experimentally observed glow discharges at the bottom of Fig. 6, Fig. 7 and Fig. 8. Table IV gives the experimental conditions to generate plasma glow discharges for the first three modes of vibration. It can be noticed that the frequencies are very close to the modal frequencies given in Table III.

The visual similarity between the high gradient electric potential areas and the luminescent areas for each mode is note-

Fig. 1 .

 1 Fig. 1. Geometric parameters of a Rosen type PT.

Fig. 2 .

 2 Fig. 2. First four mechanical mode wave shapes obtained by the analytical model (solid lines) and ANSYS (dashed lines).

Fig. 3 .

 3 Fig. 3. Normalized electric potential distribution corresponding to the first 4 mode waveshapes obtained by the analytical model (solid lines) and ANSYS (dashed lines).

Fig. 4 .

 4 Fig. 4. Distributions of the electric potential along the x-axis when vibration is forced with frequency supply near each modal frequency.

Fig. 5 .

 5 Fig. 5. Domain definition of the numerical problem of the electrical potential generated by a Rosen type PT.

Fig. 6 .

 6 Fig. 6. Top: cross-section view of numerical electrical potential (V) distribution (=2-mode); bottom: front view of experimental glow discharge (=2-mode).

Fig. 7 .

 7 Fig. 7. Top: cross-section view of numerical electrical potential (V) distribution (-mode); bottom: front view of experimental glow discharge (-mode).

Fig. 8 .

 8 Fig. 8. Top: cross-section view of numerical electrical potential (V) distribution (3=2-mode); bottom: front view of experimental glow discharge (3=2-mode).TABLE IV EXPERIMENTAL CONDITIONS OF PLASMA GLOW DISCHARGES FOR THE FIRST THREE LONGITUDINAL VIBRATORY MODES

TABLE I GEOMETRIC

 I PARAMETERS OF THE PT

TABLE II STRUCTURAL

 II PARAMETERS OF THE PT infinite value. The geometric and structural parameters are given in Tables I and II.


J Régnier
email: regnier@leei.enseeiht.fr

B Sareni
email: sareni@leei.enseeiht.fr

X Roboam
email: roboam@leei.enseeiht.fr

System Optimization by Multiobjective Genetic Algorithms and Analysis of the Coupling between Variables, Constraints and Objectives

Keywords: System Design, Multiobjective Optimization, Genetic Algorithms, Coupling Analysis

This paper presents a methodology based on Multiobjective Genetic Algorithms (MOGA's) for the design of electrical engineering systems. MOGA's allow to optimize multiple heterogeneous criteria in complex systems, but also simplify couplings and sensitivity analysis by determining the evolution of design variables along the Pareto-optimal front. A rather simplified case study dealing with the optimal dimensioning of an inverterpermanent magnet motorreducerload association is carried out to demonstrate the interest of the approach.

I. INTRODUCTION

The existence of strong coupling levels in complex heterogeneous electrical devices leads to study the system design as a whole. The best architecture and the corresponding dimensioning have to be determined in order to minimize (or maximize) number of performance criteria (global losses, harmonic distortion, masses, economical cost…) with respect to several constraints. The resulting mathematical optimization problem is usually difficult since it involves mixed variables (continuous variables related to the real dimensioning parameters and combinatorial variables associated with architecture characteristics and discrete dimensioning parameters), various constraints and multiple objectives. Using traditional optimization approaches like gradient based methods is not suitable because of these combinatorial features and the difficulty to obtain analytically constraint and objective derivatives in case of numerical simulation. Genetic Algorithms (GA's) are well suited to treat this kind of problem. Because of their ability to explore multiple solutions in parallel, standard GA's can be easily extended to solve multiobjective problems and find the set of best trade-offs. This paper illustrates the application of MOGA's to the optimal design of a "simple" electromechanical system based on an inverterpermanent magnet motorreducerload association.

II. MULTIOBJECTIVE OPTIMIZATION WITH GENETIC ALGORITHMS

II.1. Multiobjective Optimization Problem and Pareto Optimality

The multiobjective optimization seeks to simultaneously minimize n objectives where each of them is a function of a vector X of m parameters (or design variables). These parameters may also be subject to k inequality constraints, so that the optimization problem may be expressed as :

k i g f f f f i n ... 1 for 0 ) ( subject to )) ( , ), ( ), ( ( ) ( Minimize 2 1    X X X X X  (1)
For this kind of problem objectives are typically conflicting with each other. Thus, in most of cases, it is impossible to obtain the global minimum at the same point for all objectives. Therefore, the problem has no single optimal solution but a set of efficient solutions representing the best objective trade-offs. These solutions consist of all design variable vectors for which the corresponding objective vectors cannot be improved in any dimension without degradation in another. They are known as Pareto-optimal solutions in reference to the famous economist [START_REF] Pareto | Cours d'économie politique[END_REF]. Mathematically, Pareto-optimality can be expressed in terms of Pareto dominance. Consider two vectors X and Y from the design variable space. Then, X is said to dominate Y if and only if

[2]- [5] : ) ( ) ( ... 1 and ) ( ) ( ... 1 Y X Y X j j i i f f n j f f n i        (2)

II.2. Multiobjective Optimization Approaches and Decision Making

Multiobjective optimization methods aim at finding one or multiple Pareto-optimal solutions to a particular optimization problem. Various multiobjective approaches can be used to guide the Decision Maker towards a final solution among the Pareto-optimal set. A classification of these approaches is depicted in Fig. 1.

Multiobjective Optimization Approaches

Multiobjective Optimization Approaches

A posteriori A posteriori We choose to classify Multiobjective Optimization approaches as many researchers [START_REF] Van Veldhuizen | Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations[END_REF], [START_REF] Hwang | Multiple Objective Decision Making -Methods and Applications[END_REF] defining three variants of the decision making :

A priori approaches (Decide  Search) -The Decision Maker combines the differing objectives into a global quality function. Thus, the multiobjective problem is transformed into a standard scalar optimization problem which can be solved using traditional optimization methods. This approach includes aggregation based methods such as weighting-sum or fuzzy logic techniques, -constraint procedure and goal attainment method. Although they have been widely used in the past, a priori techniques suffer from various drawbacks. In particular, in one optimization run, they provide a single Pareto-optimal solution. Moreover, this investigated Pareto-optimal solution is very sensitive to the scalarization of the objectives and the choice of parameters (e.g. weighting coefficients, target values...) associated with the preferences of the Decision Maker.

Progressive and sequential approaches (Decide  Search) -The Optimization Process and the Decision Making are intertwined. The preferences of the Decision maker are sequentially updated in function of the result of the Optimization Process. Note that a priori approaches can be iteratively used as progressive approaches as well as traditional techniques such as lexicographic method. The main drawback of these approaches resides in the fact that they require multiple optimization runs to provide multiple Pareto-optimal solutions to the Decision Maker.

A posteriori approaches (Search  Decide) -These approaches provide in a single optimization run a set of Pareto-optimal solutions to the Decision Maker who can choose among that set. They essentially include population based optimization methods such as Multiobjective Evolutionary Algorithms (e.g. Genetic Algorithms) [START_REF] Van Veldhuizen | Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations[END_REF]- [START_REF] Corne | The Pareto Envelope-based Selection Algorithm for Multiobjective Optimization[END_REF] or Multiobjective Particle Swarm Optimizers [START_REF] Coello | MOPSO: A Proposal for Multiple Objective Particle Swarm Optimization[END_REF].

II.3. Multiobjective Genetic Algorithms

Since the mid-1990s, there has been a growing interest in solving multiobjective problems by Genetic Algorithms. In particular, elitist MOGA's based on Pareto approaches have become more and more popular because of their capabilities to approximate the set of optimal trade-offs in a single run [START_REF] Van Veldhuizen | Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations[END_REF]- [START_REF] Deb | A fast-elitist non-dominated sorting genetic algorithm for multiobjective optimization: NSGA-II[END_REF]. Elitist MOGA's use an external population, namely archive, which preserves non-dominated individuals in the population. At each generation, individuals (parents) selected from the archive (and/or from the population) following Pareto domination rules (typically expressed by ( 1)) are crossed and mutated to create new individuals (children). The population of children and the archive are merged to assess the non-dominated set of the next generation. If the number of non-dominated individuals is higher than the size of the archive, a clustering method is used to preserve most representative solutions and eliminate others in order to keep a constant archive size. Note that niching is used in the selection scheme when individuals involved in a tournament have the same Pareto domination rank. The structure of an elitist MOGA is displayed in Fig. 2.

The second version of the Non-dominated Sorting Genetic Algorithm (NSGA-II) [START_REF] Deb | A fast-elitist non-dominated sorting genetic algorithm for multiobjective optimization: NSGA-II[END_REF] is based on the principles previously exposed. NSGA-II determines all successive fronts in the population (the best front corresponding to the non-dominated set). Moreover, a crowding distance is used to estimate the density of solutions surrounding each individual on a given front. In a tournament, if individuals belong to the same front, the selected one is that with the greater crowding distance. This niching index is also used in the clustering operator to uniformly distribute the individuals on the Pareto front. All details of the algorithm can be found in [START_REF] Deb | A fast-elitist non-dominated sorting genetic algorithm for multiobjective optimization: NSGA-II[END_REF]. Note that some adaptations have been made with the introduction of a self-recombination procedure to increase the robustness of the NSGA-II [START_REF] Sareni | Recombination and Self-adaptation in Multi-objective Genetic Algorithms[END_REF] and by means of an extended Pareto-dominance criterion [START_REF] Régnier | Optimal design of electrical engineering systems using Pareto Genetic Algorithms[END_REF] to include all constraints. In our work, this algorithm is taken as reference for the design of heterogeneous systems in electrical engineering.

III. OPTIMIZATION AND ANALYSIS OF A SIMPLE ELECTROMECHANICAL SYSTEM

III.1. Problem statement and optimization results

To illustrate the use of MOGA's in electrical engineering, the optimal design of an electromechanical system has been investigated. The corresponding device is based on an inverter-permanent magnet motor-reducer-load association (see Fig. 3). The purpose is to simultaneously minimize two objectives: the global losses f 1 (X) and the mass of the system f 2 (X). Each objective is composed of partial criteria : f 1 (X) is the sum of the inverter losses P inv (switching losses and conduction losses) and motor losses P mot (iron and Joule losses) and f 2 (X) is the sum of the motor mass M mot and the inverter heat skin mass M inv . The whole system behavior is described by means of analytical models. All the details dealing with the associated modeling issues can be found in [START_REF] Régnier | Conception de systèmes hétérogènes en Génie Electrique par optimisation évolutionnaire multicritère[END_REF]- [START_REF] Slemon | Modeling and design optimization of permanent magnet motors[END_REF]. The resulting optimization problem can be expressed by (1) with 2 objectives (n=2), 8 design variables (m=8) and 7 constraints (k=7). The design variable vector X and the set of technological and working constraints g k (X) associated to the system are displayed in Table 1 and 2 respectively. Note that the IGBT family represented by a switch [0,5] corresponds to Six Pack modules of IXIS with different range of voltage and current (600V/16A-45A-90A and 1200V/30A-52A-90A). Finally, the whole optimization procedure is displayed in Fig. 4.

The optimization of the system is carried out using the NSGA-II run for 500 generations with a population size of 100. The archive size is also set to 100 individuals and the crossover and mutation rates are respectively 1.0 and 1/m. 10 independent runs are made to take into account the stochastic nature of the GA. The Pareto-optimal front resulting from these runs is displayed in Fig 5 . The global losses evolve from 628W to 984W which leads to a global efficiency, in relation to the operating point of the machine (15 kW), varying from 96% to 93.8%. The corresponding system mass is between 94.9 kg and 39.4 kg. The design variables of the boundary solutions shown in Fig. 5 illustrate the diversity of the system configurations along the front.

By providing to the designer a set of optimal solutions, this a posteriori multiobjective approach can help him to understand the main relationships between design variables, constraints, and objectives in the system. In particular, we will illustrated this point in the following sections, through the study of partial objectives, the analysis of couplings, the investigation of parametric sensitivity, and the choice of a final solution in the case of the previously exposed system.

III.2. Partial objective analysis

The evolution of the partial objectives along the Pareto-optimal front is displayed in Fig. 6 and Fig. 7. A symmetric variation between iron and joules losses of the permanent magnet machine can be shown in Fig. 6, which illustrates a compensation phenomenon. Concerning the variations of the system mass, we observe in Fig. 7 that the inverter heat skin mass is constant. Its part of the global mass does no exceed 10%. The most significant mass in the system is that of the electrical machine. According to these observations, the permanent magnet machine appears as the central element of the system. Indeed, it causes the main mass variations and includes the most important part of the global losses (over 85%). This conclusion was expectable with regard to the complexity of this sub-system which presents some important multi-physical field characteristics.

F 5  switch a V 798  E 1  N -2 A.mm 05 . 5  S J 3  p 1 . 0  rl R Hz 3391  sw F 5  switch a V 624  E

III.3. Coupling analysis

One of the main difficulties in the system design is to identify couplings between design variables, constraints and objectives in case of a complex multi-physical model. The knowledge of these interactions is valuable with regard to the system behavior understanding. In this context, the global formulation of the problem and the set of solutions obtained with the MOGA allow to characterize and underline these couplings. In order to help the designer in this way, we propose to use two methods : a qualitative graphical analysis of variables along the Pareto-optimal front and an original quantitative method based on correlation coefficients. A classical coupling analysis can be performed in a graphical way by studying design variable variations of Pareto-optimal solutions in relation to the considered objectives. For example, we show, in Fig. 8 and Fig. 9 a coupling between the number of pole pairs (discrete design variable) and the reducer ratio (continuous design variable) for non-dominated solutions of the Pareto-optimal front. It can be observed that each variation of the number of pole pairs is related to an inverse variation of the reducer ratio. This phenomenon results from the variations of the iron losses in the machine, which are strongly linked to the electrical pulsation pN load of the motor (iron losses increase when  increases). Increasing the number of pole pairs p leads to a growth of  which consequently increases iron losses. To avoid an excessive value of iron losses, the reducer ratio N reacts to the increase of p by a simultaneous decrease. Other interesting analysis can be performed by studying different couples of design variables in relation to constraints [START_REF] Régnier | Conception de systèmes hétérogènes en Génie Electrique par optimisation évolutionnaire multicritère[END_REF]. This graphical approach is interesting because of its simplicity but it only provides qualitative information about the coupling levels.

Because of this main drawback, we propose to use a correlation coefficient in order to assess quantitatively the influence of couplings. Consider for example two coupled design variables X 1 and X 2 (see Fig 10). The correlation coefficient between the corresponding variations of these variables X 1 and X 1 (see Fig. 11) is defined by (3) [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF].

    2 1 2 1 2 1 , cov , cor X X X X X X          with            ) ( 2 ) 1 ( 2 2 ) ( 1 ) 1 ( 1 1 i i i i X X X X X X and 1 ,..., 1   N i (3)
where N denotes the number of considered points,  Xi the standard deviation of X i, and cov(X 1 ,X 2 ) the covariance between X 1 and X 2 . Nevertheless, note that this coefficient has no signification with the discrete design variables. In the example of Fig. 10 and Fig. 11, the correlation coefficient equals -1 which signifies that each variation of X 1 is always related to an inverse variation of X 2 . Therefore, these two design variables are strongly coupled. Fig. 12 presents the correlation coefficients of the current density with the other design variables for Pareto-optimal solutions in the case of the previous investigated system. It shows that the correlation coefficient between the current density J s and the radius/length ratio R rl of the motor is about -0.9. Thus, the global system performances are particularly sensitive to these two parameters. Correlation coefficients can also be computed between design variables and partial objectives. For example, Fig. 13 illustrates the correlation coefficients between the current density J s and the partial objectives. In particular, as expected, it can be observed that J s affects Joule losses P j (the corresponding correlation coefficient is equal to 0.7) so that an increase of J s induces an increase of P j for Pareto-optimal solutions. Similarly, we verify that the inverter losses strongly depend on the inverter switching frequency since the corresponding correlation coefficient is equal to 0.83. We see here the interest of correlation coefficients which can help the designer to detect the main coupling mechanisms governing the whole system performances.

III.4. Parametric sensitivity analysis

This optimization approach could also be very useful to investigate the influence of parametric variations and especially the restrictions resulting from technological limits on the efficiency of optimal solutions. As an example, we choose to modify the cooling system of the permanent magnet machine (from natural convection to forced ventilation) in order to study the corresponding impact on the global system performances. The improvement of the cooling system allows to favor the decrease of the system mass. In fact, thanks to the forced ventilation, the size of the motor can be reduced without reaching the thermal limit of the stator winding. The effect of this technological modification can also be observed on the design variables (see Fig. 15). With a natural convection, the reducer ratio N does not exceed 2.2. Even if the increase of N favors the decrease of the motor mass, it involves in the same time an increase of the iron losses and the motor temperature. Therefore, the reducer ratio is limited by the maximum acceptable thermal constraint. The forced ventilation allows to extract more heat from the motor and permits the reducer ratio to grow without an excessive increase of the motor temperature.

III.5. Choice of a final solution

The final choice between all Pareto-optimal configurations can be a posteriori done in relation to other issues which have not been considered in the optimization process. In this paper, we illustrate this point by considering the cogging torque for the final decision.

The cogging torque is computed by using the finite element method from the selected solutions of the Paretooptimal front. In our example, solution 4 of Fig. 16 presents the lowest value of the cogging torque and can be extracted a posteriori in relation to this added criterion. Note that every solution can be locally optimized by a modification of slot or rotor magnet shape in order to improve the cogging torque value. Once again, the interest of multiobjective a posteriori approaches which provide a set of solutions instead of a single one, has been put forward. Results can be discussed with experts of each sub-system to explore the possible actions to enhance the performance of a particular element and to evaluate with the system designer the impact of the modification on the whole system efficiency.

IV. CONCLUSION

In this paper, we have presented a global approach based on MOGA's devoted to the design of heterogeneous devices in electrical engineering. It has been shown that MOGA's are well suited to improve global system efficiency and also to help the designer in the understanding of the relationships between design variables, constraints, and objectives. In particular, from the extraction of Pareto-optimal solutions, MOGA's facilitate the investigation of parametric sensitivity and the analysis of couplings in the system. For this purpose, we have proposed an original quantitative methodology based on correlation coefficients to characterize the system interactions. Through a simple but typical academic problem dealing with the optimal dimensioning of a inverter permanent magnet motorreducerload association, it has been shown that this multiobjective a posteriori approach could offer interesting outlooks in the global optimization and design of complex heterogeneous systems.

Figure 1 :

 1 Figure 1: Classification of Multiobjective Optimization Approaches

Figure 3 :

 3 Figure 3: Electromechanical system design problem Figure 4: Optimization procedure

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Pareto-optimal front of the problem and associated boundary solutions

Figure 8 :Figure 9 :

 89 Figure 8: Variation of the reducer ratio along the Pareto-optimal front

Figure 10 : 2 Figure 11 : 2 - 1 -Figure 12 :

 102112112 Figure 10: Evolution example of two coupled design variables X 1 and X 2

Figure 14 :Figure 16 :

 1416 Figure 14 : Influence of the cooling system on Pareto-optimal solutions

TABLE I

 I

		: DESIGN VARIABLES
	Design variables	Nature	Definition range
	Slot current density [A/mm²]	Continuous	1 < J	s < 10
	Number of pole pairs	Discrete		p  [1,10]
	Bore radius / length ratio	Continuous	0.1 < R	rl < 10
	Number of slots / pole /phase	Discrete	N epp [1,3]
	Switching frequency [Hz]	Continuous	1 < F	sw < 50 000
	IGBT family	Discrete	a	switch  [0,5]
	Voltage level [V]	Continuous	1 < E < 1 000
	Reduction ratio	Continuous	1 < N < 10

TABLE II :

 II CONSTRAINTS ASSOCIATED TO THE OPTIMIZATION PROBLEM

		g 1 (X)							Minimal number of conductor slots
		g 2 (X)							Maximal number of conductor slots
		g 3 (X)							Demagnetization magnet limit
		g 4 (X)							Maximal winding temperature
		g 5 (X)							Voltage supply harmonic distortion
		g 6 (X)							Fulfillment of IGBT Voltage range
		g 7 (X)							Maximal IGBT junction temperature
		100		J				. 2	15	A.mm	-2
				S
		90		p		2
				R			. 0	637
				rl
	mass (kg)	70 80		sw		972	Hz
	System	60
		50
		40
		600 30	650	700				750	800	850	900	950	1000
										Global losses (W)