## 1. Introduction

In this chapter, two permanent magnet synchronous motors which are supplied by two voltage inverters are used. The system of traction studied Fig.1, belongs to the category of the multi-machines multi-converters systems (MMS). The number of systems using several electrical machines and/or static converters is increasing in electromechanical applications. These systems are called multi-machines multi-converters systems (Bouscayrol, 2003). In such systems, common physical devices are shared between the different energetic conversion components. This induces couplings (electrical, mechanical or magnetic) which are quite difficult to solve (Bouscayrol, 2000).

The complexity of such systems requires a synthetic representation in which classical modeling tools cannot always be obtained. Then, a specific formalism for electromechanical system is presented based on a causal representation of the energetic exchanges between the different conversion structures which is called energetic macroscopic representation (EMR). It has developed to propose a synthetic description of electromechanical conversion systems. A maximum control structure (MCS) can be deduced by from EMR using inversion rules. The studied MMS is an electric vehicle. This system has a mechanical coupling, Fig.1. The main problem of the mechanical coupling is induced by the non-linear wheel-road adhesion characteristic. A specific control structure well adapted to the non-linear system (the Behaviour Model Control) is used to overcome this problem. The BMC has been applied to a non-linear process; therefore, the wheel-road contact law of a traction system can be solved by a linear model.

The control of the traction effort transmitted by each wheel is at the base of the command strategies aiming to improve the stability of a vehicle. Each wheel is controlled independently by using an electric motorization. However, the traditional thermal motorization always requires the use of a mechanical differential to ensure the distribution of power on each wheel. The mechanical differential usually imposes a balanced transmitted torques. For an electric traction system, this balance can be obtained by using a multi-motor structure which is shown in Fig.1. An identical torque on each motor is imposed using a fuzzy direct torque control (FDTC) (Miloudi, 2004; Tang, 2004; Sun, 2004, Miloudi, 2007). The difficulty of controlling such a system is its highly nonlinear character of the traction forces expressions. The loss of adherence of one of the two wheels which is likely to destabilize the vehicle needs to be solved in this chapter.

## 2. Presentation of the traction system proposed

The proposed traction system is an electric vehicle with two drives, Fig.1. Two machines thus replace the standard case with a single machine and a differential mechanical. The power structure in this paper is composed of two permanent magnet synchronous motors which are supplied by two three-phase inverters and driving the two rear wheels of a vehicle through gearboxes, Fig. 2. The traction system gives different torque to in-wheel-motor, in order to improve vehicle stability. However, the control method used in this chapter for the motors is the fuzzy direct torque control with will give the vehicle a dynamic behaviour similar to that imposed by a mechanical differential (Arnet, 1997; Hartani, 2005; Hartani, 2007). The objective of the structure is to reproduce at least the behaviour of a mechanical differential by adding to it a safety anti skid function.

### 2.1. Energetic macroscopic representation of the traction system

The energetic macroscopic representation is a synthetic graphical tool on the principle of the action and the reaction between elements connected (Merciera, 2004). The energetic macroscopic representation of the traction system proposed, Fig. 3, revealed the existence of only one coupling called overhead mechanical type which is on the mechanical part of the traction chain.

The energetic macroscopic representation of the mechanical part of the electric vehicle (EV) does not take into account the stated phenomenon. However, a fine modeling of the contact wheel-road is necessary and will be detailed in the following sections.

### 2.2. Traction motor model

The PMSM model can be described in the stator reference frame as follows (Pragasen, 1989):

and the electromagnetic torque equation

### 2.3. Inverter model

In this electric traction system, we use a voltage inverter to obtain three balanced phases of alternating current with variable frequency. The voltages generated by the inverter are given as follows:

### 2.4. Mechanical transmission modeling of an EV

#### 2.4.1. Modeling of the contact wheel-road

The traction force between the wheel and the road, Fig. 4(a), is given by

where

The wheel speed can be expressed as:

where

Principal non-linearity affecting the vehicle stability is the adhesion function which is given by Eq. (7) and represented on Fig. 4(b).

We now represent in the form of a COG (Guillaud, 2000) and an EMR, the mechanical conversion induced by the contact wheel-road, Fig. 5.

#### 2.4.2. Modeling of the transmission gearbox-wheel

Modeling of the transmission gearbox-wheel is carried out in a classical way which is given by:

where

#### 2.4.3. Modeling of the environment

The external environment is represented by a mechanical source (MS) on Fig. 3, leading to the resistance force of the vehicle motion

The rolling resistance is obtained by Eq. (10), where

The resistance of the air acting upon the vehicle is the aerodynamic drag, which is given by Eq. (11), where ρ is the air density, *C*
_{
D
} is the aerodynamic drag coefficient, *A*
_{
f
} is the vehicle frontal area and *v* is the vehicle speed (Wong, 1993).

The slope resistance and down grade is given by Eq. (12)

### 2.5. EMR of the mechanical coupling

The modeling of the phenomenon related to the contact wheel-road enables us to separate the energy accumulators of the process from the contact wheel-road. However, the energy accumulators which are given by the inertia moments of the elements in rotation can be represented by the total inertia moments of each shaft motor

and

The final modeling of the mechanical transmission is represented by the EMR, Fig. 7.

## 3. Control strategy

### 3.1. Fuzzy direct torque control

A fuzzy logic method was used in this chapter to improve the steady-state performance of a conventional DTC system. Fig. 8 depicts schematically a direct torque fuzzy control, in which the fuzzy controllers replace the flux linkage and torque hysteresis controllers and the switching table normally used in conventional DTC system (Takahachi, 1986; French, 1996; Vyncke, 2006; Vasudevan, 2004).

The proposed fuzzy DTC scheme uses the stator flux amplitude and the electromagnetic torque errors through two fuzzy logic controllers (i.e., FLC1 and FLC2) to generate a voltage space vector

The Mamdani and Sugeno methods were used for the fuzzy reasoning algorithms in the FLC1 and FLC2, respectively. Figs. 9 and 10 show the membership functions for the fuzzy logic controllers FLC1 and FLC2, respectively. Fig. 11 shows the fuzzy logic controller structure.

### 3.2. PI resistance estimator for PMSM-fuzzy DTC drive

The system diagram can be shown in Fig. 11. It is seen that the input to the PI resistance estimator is torque and flux reference, together with the stator current. Rotor position is not needed for the PI estimator (Mir, 1998; Hartani, 2010).

When the stator resistance changes, the compensation process will be applied. Therefore, the change of stator resistance will change the amplitude of the current vector. So, the error between the amplitude of current vector and that of the reference current vector will be used to compensate the change in stator resistance until the error becomes zero.

Based on the relationship between change of resistance and change of current, a PI resistance estimator can be constructed by Eq. (15), as shown in Fig. 13. Here,

#### 3.2.1. Amplitude of stator reference current

The PMS machine’s torque and flux vector are given by (16) and (17) in the vector reference frame. The amplitude of flux vector can be calculated using (19).

The reference torque and flux are given, respectively, by:

The stator current amplitude is obtained using the relation given by:

A Matlab/Simulink model was built in order to verify the performance of the PI resistance estimator. The system performances with and without the resistance estimator were then compared. Fig. 14 shows the result of flux, torque, and amplitude of current of the system with PI stator resistance estimator.

### 3.3. Maximum control structure

The maximum control structure (MCS) is obtained systematically by applying the principles of inversion to the model EMR of the process (Bouscayrol, 2002). We define by this method the control structure relating to the mechanical model of the vehicle. In our application, the variable to be controlled is the vehicle speed by acting on the motor torques of both wheels. The maximum control structure is shown in Fig. 15, by taking into account only one wheel for simplification.

This structure shows the problems involved in the inversions of the mechanical coupling by the accumulation element and of non-linearity relating to the contact wheel-road.

#### 3.3.1. Inversion of the coupling by accumulation elements

The inversion of the relation (15) shows that this control is done by controlling a traction effort (

#### 3.3.2. Inversion of the converter CR

We should now define the speed of each wheel in order to ensure the balance of the traction forces. However, the problems which we will face are the nonlinearity and the badly known of the relation to reverse which depends of the road conditions. In the following work, we propose two methods to solve these problems.

### 3.4. Anti skid strategy by slip control

The proposed solution will be obtained from the maximum control structure, Fig. 15. In this case, the solution to be used to reverse the converter CR is based on the principle of inversion of a badly known rigid relation. We can now apply with this type of relation, the principles of inversion of a causal relation which will minimize the difference between the output and its reference by using a classical controller. The principle of this strategy is based on Fig. 17.

#### 3.4.1. Inversion of relation CR1

Figure 18, shows the COG inversion of relation CR1. We need now the measurements or estimations of the variables speed and traction effort of each wheel.

The linear velocity estimation permits a true decoupling in the control of both wheels (Pierquin, 2001) and the definition of the reference speed which is given by the following equation:

In the control structure, the maximum slip is limited to 10% which constitutes the real function anti skidding.

### 3.5. Anti skid strategy by BMC

#### 3.5.1. The BMC structure

The behaviour model control (BMC) can be an alternative to other robust control strategies. It is based on a supplementary input of the process to make it follow the model (Hautier, 1997 ; Vulturescu, 2000; Pierquin, 2000).

The process block correspondens to the real plant, Fig. 19. It can be characterised by its input vector *u* and its output vector *y*.

The control block has to define an appropriated control variable *u*, in order to obtain the desired reference vector *y*
_{
ref
}.

The model block is a process simulation. This block can be a simplified model of the process.

The difference between the process output *y* and the model output *y*
_{
mod
} is taken into account by the adaptation block. The output of this block acts directly on the process by a supplementary input, Fig. 19. The adaptation mechanism can be a simple gain or a classical controller (Vulturescu, 2004).

#### 3.5.2. Application of the BMC control to the traction system

The first step to be made is to establish a behaviour model. In this case, we choose a mechanical model without slip, which will be equivalent to the contact wheel-road in the areas known as pseudo-slip. This model can be considered as an ideal model. However, the inertia moments of the elements in rotation and the vehicle mass can be represented by the total inertia moments *J*
_{
t_mod
} of each shaft motor which is given by:

The dynamic equation of the model is given by:

By taking into account the wheel slip, the total inertia moments will become:

We now apply the BMC structure for one wheel to solve the skid phenomenon described before. In Fig. 20, we have as an input the reference torque and as an output the speed of the motor which drives the wheel. However, the main goal of this structure of control is to force the speed Ω_{
m
} of the process to track the speed Ω_{
m_mod
} of the model by using a behaviour controller.

It was shown that the state variables of each accumulator are not affected with the same manner by the skid phenomenon. The speed wheel is more sensitive to this phenomenon than that of vehicle in a homogeneous ratio to the kinetic energies, Fig. 5(a). Hence, the motor speed is taken as the output variable of the model used in the BMC control. The proposed control structure is given by Fig. 20.

The influence of the disturbance on the wheel speeds in both controls is shown in Fig. 21. An error is used to compare the transient performances of the MCS and the BMC. This figure shows clearly that the perturbation effect is negligible in the case of BMC control and demonstrates again the robustness of this new control.

## 4. Simulation results

We have simulated by using different blocks of Matlab/Simulink the proposed traction system. This system is controlled by the behaviour model control (BMC) based on the DTC strategy applied to each motor, Fig. 20, for the various conditions of environment (skid phenomenon), Fig. 22.

### 4.1. Case 1

Initially, we suppose that the two wheels are not skidding and are not disturbed. Then, a 80 *km/h* step speed is applied to our system. We notice that the speeds of both wheels and vehicle are almost identical. These speeds are illustrated in the Fig. 24(a) and (b). Fig. 24(c) shows that the two motor speeds have the same behaviour to its model. The difference between these speeds is represented in the Fig. 24(d). From Fig. 24(e) we notice that the slips λ_{1} and λ_{2} of both wheels respectively, are maintained in the adhesive region and the traction forces which are illustrated by the Fig. 24(f) are identical, due to the same conditions taken of both contact wheel-road. The motor torques are represented in Fig. 24(g) and the imposed torques of the main controller and the behaviour controllers are shown in Fig. 24(h). The resistive force of the vehicle is shown by the Fig. 24(i).

### 4.2. Case 2

We simulate now the system by using the BMC control and then applying a skid phenomenon at

The BMC control has a great effect on the adaptation blocks and by using the behaviour controllers to maintain permanently the speed of the vehicle and those of the wheels close to their profiles, Fig. 25(a). However, both driving wheel speeds give similar responses as shown in Fig. 25(b).

Figure 25(c) shows that the two motor speeds have the same behaviour with the model during the loss of adherence. The difference between these speeds which is negligible is represented in the Fig. 25(d).

The loss of adherence imposed on wheel 1 results to a reduction in the load torque applied to this wheel, consequently its speed increases during the transient time which induces a small variation of the slip on wheel 2, Fig. 25(e). The effect of this variation, leads to a temporary increase in the traction force, Fig. 18(i). However, the BMC control establishes a self-regulation by reducing the electromagnetic torque

### 4.3. Case 3

In this case, the simulation is carried out by applying a skid phenomenon between

As shown in Fig. 26(i). and during the loss of adherence, the traction forces applied to both driving wheels have different values. At

### 4.4. Case 4

The simulation is carried out by applying a skid phenomenon to both wheels successively at different times. Figure 27(c) shows that the two motor speeds have the same behaviour to the model. The difference between these speeds is represented in the Fig. 27(g).

When the adherence of the wheel decreases, the slip increases which results to a reduction in the load torque applied to this wheel. However, the BMC control reduces significantly the speed errors which permits the re-adhesion of the skidding wheel. Therefore, it is confirmed that the anti-skid control could maintain the slip ratio around its optimal value, Fig. 27(h).

## 5. Abbreviations

COG: Causal Ordering Graph

DTFC: Direct Torque Fuzzy Control

EC: Electrical Coupling

EM: Electrical machine

EMR: Energetic Macroscopic Representation

ES: Electrical Source

EV: Electric Vehicle

MC: Mechanical Coupling

MCS: Maximum Control Structure

MMS: Multi-machine Multi-converter Systems

MS: Mechanical Source

PMSM: Permanent Magnet synchronous Machine

SC: Static converter

## 6. Appendix

## 7. Conclusion

In this chapter, a new anti-skid control for electric vehicle is proposed and discussed. This work contributes to the improvement of the electric vehicle stability using behaviour model control. According to the results obtained by simulations for all the cases, the proposed traction system shows a very stable behaviour of the electric vehicle during the various conditions of adherence.