Control and Modeling of Electromagnetic Converter Dedicated to Electric Traction

Abstract: This paper describes a parameterized dynamic model of an electromagnetic converter inserted into a model of the power train of an electric vehicle in the goal to improve its dynamic characteristic. Generally for electromechanical interrupters, the stability of the contact during the closing and opening control is not studied. In this context, this paper presents the dynamic behavior of the interrupter in order to analyze the performances and to improve the dynamic characteristic of the electric vehicles. The parameters can be optimized by genetic algorithms method.


Introduction
A determining factor is the choice of components of the electrical systems power train such as electric vehicle regarding its influence in the production cost and the energy consumption of the global system. A basic component in this type of energy conversion chain is the electric converter. Several research studies have shown that the three-phase converter with two voltage level allows to expect good performance for industrial application as the motorization of electric cars [1][2][3][4]. Two types of configurations can be presented to knowledge are: 1. Converter with two voltage levels with Insolated Bipolar Transistors (IGBTs) having multiple disadvantages, such as the floating voltage problem of an arm, tail current, energy loss by conduction and switching, need to integrate a cooling system for the majority of cases and presence of capacities: Trigger-Emitter, Trigger-Collector and Collector-Emitter especially at high frequencies [1][2][3][4]. 2. Converter with two voltage levels to electromagnetic switches having several advantages such as mechanical isolation between the power circuit and the control circuit without any of the disadvantages mentioned above for the converter with IGBTs [5][6][7][8][9][10][11][12][13][14]. Hence, the choice fell on an innovated structure of electromagnetic converter having the disadvantage of a low switching frequency. Several sizing and modeling studies of this type of converter encouraging its integration at low frequencies electrical systems are developed and presented in the literature [1][2][3][4][5].
In this context, firstly this paper presents the structure and the modeling approach of the electromagnetic converter dedicated to electric vehicles motorization application. Secondly, this paper presents the principle of coupling of this type converter to electric vehicles power chain. To finish with a presentation and descriptions of simulation results completed by a conclusion.

Objectives and Methodology
The primary the objective of this study is to argue the choice of electromagnetic converter compared to its equivalent with IGBTs to push the multiple disadvantages of the converters with IGBTs already mentioned. An essential advantage of the electromagnetic converters is their reduced cost relating to IGBTs converter.
The design methodology of this converter type is based on the coupling of design and modeling programs of the various components of the electric vehicle power chain. Design problem solving of the power chain takes into account the systemic interactions between the various components. For example the power chain is adapted to operate at low frequency, the outside diameter of the motor is delimited by the space reserved for the motor etc. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].
The method chosen for the design of the power chain components is the analytical method view it brings the following benefits: 1. Fast and thereafter consistent with optimizations approaches. 2. It takes into account the interactions between the components and permit to evaluate the performance of the power chain by simple equations. 3. The resolution of an electrical device design problem is direct (without iterations). 4. This method is adjusted and supplemented by the finite elements method .

Structure of Electromagnetic Converter Arm
An arm of the electromagnetic converter (Figure1) comprise tow left electromagnets and two right electromagnets to increase the switching frequency regardind the movable stem strength is multiplied by two.
When the coils of the electromagnet 1 are energized, the moving stem is attracted until the value of the displacement D co related to the opening of the power contacts. In this case the power contact open at repot are closed and the other closed at repot are opened. The duration of opening and closing is imposed by the generator of the supply voltages of the coils and of the dynamics of two electromagnets. Indeed, the displacement of the movable stem may exceed the value D co , since the return spring are attached to the yoke of the electromagnet to damp the collusion between the movable stem and the yoke. In this case the amortization springs are compressed. The movement of the movable stem toward the other direction to change the power contacts state is ensured by canceling the coil voltage of the electromagnet 1 (freewheeling phase) and supplyng the coils of the electromagnets 2 to change the sign of the attraction strength of the stem and subsequently the direction of movement of the moving stem [1][2][3][4][5][6][7][8][9][10][11][12].   The control circuit model of the an electromagnets arm ( Figure 2) is based on the comparison of the reference voltage delivred from the speed and current regulator of the electric vehicles power chain. The comparator output drives a hysteresis varying between 0 and 1 to reproduce the waveform of the control signal 1. The output of the hytérisis is converted into a voltage varying between the values + Vcc when the signal is at 1 and -Vcc when the signal is at zero. This signal is related to the voltage supplying the coils of the electromagnet 1 (V1). The voltage supplying the coils of the electromagnet 2 (V2) is the inverse of the voltage V1 [1][2][3][4][5][6][7][8][9][10][11][12].
( ) Where E b is the length of the main tooth (or length of the stem), H cu is the height of the coil and L dc is the distance between the coil and the right tooth.
The simulink model of the two electromagnets inductance is illustrated in the Figure 4.

Electromagnetic Strengnth Modelling
The coils phase's voltages are expressed by the following relationships: ( ) where R is the coil resistance, L 1 and L 2 are respectively the inductance of the coil 1 and 2 and i 1 , i 2 are respectively the current of the coil 1 and 2. The strength developed by the two electro-magnets is expressed by the following equation: The electrical-mechanical model of left and right electromagnets is implanted under the simulation environment Matlab-Simulink according to Figure 5.

Motion Equation of the Movable Stem
The movable stem motion equation is derived from the fundamental relation of dynamic.  where m t is the movable stem mass, x t and V t are respectively the movable stem position and speed, D co is the power contacts opening, s is the dry friction coefficient, n u is the viscous friction coefficient, x si is the fluid friction coefficient, F R is the total resistance strength, k is stiffness of the spring and F em is the movable stem strength.
The movable stem motion equation is implemented under Matlab-Simulink simulation environment according the

Global Model of the Electromagnetic Converter Arm
The coupling of various sub model of the electromagnetic converter leads to the global model of an arm of the converter (Figure 7) having as an input the reference voltage and as outputs the two control signals of an arm S1 and S2.   The speed regulator generates the amplitude of the reference currents minimizing the error between the reference speed and the response speed. Indeed, the reference speed is compared to the response speed. The comparator output drives a proportional / integral controller type (PI) to provide the amplitude of the reference currents minimizing the speed error.

Speed and Currents Regulations
Regulators currents allow the imposition of currents having the same shape and in phase with the back electromotive forces. Indeed, the reference currents are compared to the phase's current of the motor. The outputs of the three comparators attack three regulators of the type proportional / integral (PI) to provide three reference voltages necessary to impose ideal currents in phase with the back electromotive forces and to minimize the error between the reference speed and the response speed of the electric vehicle [4].
The Simulink model of the current regulator is illustrated in Figure 8:

Back Electromotive Forces Model
The back electromotive forces are expressed by the three following equations [1][2][3][4][5][6]: where ke is the electric constant of the motor and Ω is the angular velocity of the motor and p is the number of pole pairs. These equations are implanted under the Matlab-Simulink environment according to Figure 9:

Motion Equation
The vehicle motion equation is derived from the fundamental relationship of dynamics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]: Where F r is the rolling resistance force, Fa is the aerodynamic force, F c is the force of gravity, r d is the amplification ratio, T em is the electromagnetic torque, T f is the iron losses torque, T mec is the mechanical losses torque, R r is the radius of the wheel of the car, and v the speed of the car and M v and the mass of the car. The vehicle motion equation is implanted under the environment Matlab / Simulink according to Figure 10

Battery-DC/DC Elevetor Converter-Motor Model
Regarding the dynamics of the electromagnetic converter, it must be powered by a high voltage equal to 14500 volts. Therefore, the voltage delivered by the batteries is elevated to a value equal to 14500 Volt via a DC-DC elevator converter kept constant by pulse width modulation technique. This voltage is filtered by a super-capacity. The super-capacity also increases the vehicle's autonomy, since it offers additional stored energy.
Each phase of the motor is equivalent to a resistor in series with an inductance and a back electromotive force.
The three phase model of the motor is described by the following equations. where R, L, M and k e are respectively the resistance, inductance, and mutual inductance and the electric motor constant, i i and u i are the current and the voltage of the phase i. The electromagnetic torque is given by the following relationship: where ei is the electromotive force against the i phase.
The model of the Battery-DC/DC elevator convertermotor-AC converter is implanted under the Matlab-Simulink according to Figure 11:

Power Chain Global Model
The control model is based on the scalar control with compensation of the electromotive forces. Four control loops are used to minimize the error between the reference speed and the speed of response and impose phase currents in phase with the back electromotive forces to minimize consumption.
The first control loop enables to provide the amplitude of the reference current and the other three control loops allow the imposition of the phase currents in phase with the back electromotive forces and having a sinusoidal shape. This model is implanted under the simulation environment Matlab-Simulink accordind to the Figure 12

Simulations Results and Discussion
The battery voltage and the DC bus elevated volatge are illustrated in Figure 13.  The speed, the displacement of the movable stem and the two control signals are illustrated in Figure 14. This figure shows that the S1 and S2 control signals are at 1 when the movable stem position is respectively greater than 0.001 m and less than or equal to zero. This figure shows that changes in the position and speed of the moving stem is stabilized after two seconds, which shows the importance of dampers. Figure 14 shows that the switching frequency of the movable stem can be acheive 30 Hz.      The Figure 18 illustrates the evolutions over time of the back electromotive force and the current of the Phase 1. The control algorithm imposes backs electromotive forces in phase with current in order to reduce consumption. Figure 18 shows that this property is not actually reached next to the dynamics of the electromagnetic converter.    Changes in the reference and the response speeds are shown in Figure 21. This figure shows the existence of fluctuations in the response speed, which can be explained by the poor dynamics of the electromagnetic converter.
The simulation results show the importance of this innovative study and open the avenue of research to the work of power chain parameters optimization to improve the dynamics of the electric car equipped by the innovated structure of electromagnetic converter.

Conclusion
In this paper is presented a parameterized modelling approach of an electromagnetic converter based on a design program. The performance of this converter is analyzed by coupling the model developed to