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A predictive distribution model for a series cooperative braking system of an electric vehicle is proposed, which can solve the real-time problem of the optimum braking force distribution. To get the predictive distribution model, firstly three disciplines of the maximum regenerative energy recovery capability, the maximum generating efficiency and the optimum braking stability are considered, then an off-line process optimization stream is designed, particularly the optimal Latin hypercube design (Opt LHD) method and radial basis function neural network (RBFNN) are utilized. In order to decouple the variables between different disciplines, a concurrent subspace design (CSD) algorithm is suggested. The established predictive distribution model is verified in a dynamic simulation. The off-line optimization results show that the proposed process optimization stream can improve the regenerative energy recovery efficiency, and optimize the braking stability simultaneously. Further simulation tests demonstrate that the predictive distribution model can achieve high prediction accuracy and is very beneficial for the cooperative braking system.

As a core technology in electric vehicles, cooperative braking with regenerative braking and mechanical braking can not only improve the fuel efficiency but also maintain a satisfying braking stability, as studied in the literature [

Generally, when the vehicle is decelerating, the normal deceleration process happens in most cases, which indicate that the braking force distribution strategy is vital for the cooperative braking system. With respect to this, how to maximize the regenerative energy recovery efficiency under the premise of braking stability is the focus of traditional study. Many researchers have made many efforts to develop different distribution strategies. Based on the braking theory, the references [

The cooperative braking systems can be classified into parallel and series types [

In this paper, a series cooperative braking system with two motor/generators is studied for the normal deceleration process. Different from the traditional methods, a braking force distribution strategy based on a predictive distribution model is proposed. The predictive distribution model is constituted by a predictive model and an additional condition. To get the predictive model, some key technologies have been studied. Firstly, a general mathematical model is established by three disciplines and a braking force analysis. Secondly, an off-line process optimization stream which is constituted by the optimal Latin hypercube design (Opt LHD) and a concurrent subspace design (CSD) method is designed to get an off-line optimization data. Thirdly, a predictive model based on a radial basis function neural network (RBFNN) by the off-line optimization data and the general mathematical model is presented. Finally, the predictive distribution model is verified in a dynamic simulation framework.

Figure

The cooperative braking system of the electric vehicle.

A force diagram of the cooperative braking system is shown in Figure

The force diagram of the cooperative braking system.

In Figure

A general cooperative braking mathematical model is established to induce

When the vehicle decelerates, the generators will work simultaneously.

According to (

Define

Usually,

According to the above equations,

Three disciplines are proposed in this study. The first one is the maximum regenerative energy recovery capability discipline, which the total regenerative braking torque should be close to the optimum charging torque. The second one is the maximum generating efficiency discipline, in which the charging torque efficiency will be maximized under the given total regenerative braking torque of the first discipline. The third one is the optimum braking stability discipline, in which can be classified into four levels as follows.

No wheels are locked, each wheel maintains rolling and sliding state simultaneously, with this state, ABS is usually used. In this study, no ABS is equipped.

No wheels are locked earlier, if locked, all the wheels will be locked simultaneously. In this case, vehicle will keep the optimum braking stability. It is also the optimum braking stability objective in this study, which the braking force distribution coefficient should be close to it.

Front wheels will be locked earlier than rear wheels. In this case, the steering capability will be lost but still is a stability state. In this study, it is viewed as a precondition for braking stability.

Rear wheels will be locked earlier than front wheels. In this case, rear wheels will spin and will be in an instability state. In this study, it is considered as an infeasible region.

A concurrent subspace design (CSD) is performed here and its optimization mathematical model can be set up as follows.

where

The efficiency map of the generators.

The regenerative torque versus speed curve.

Additionally, if

The optimization mathematical model can be expressed as follows:

As can be seen in Figure

The coupling relationship between different disciplines.

As shown in Figure

The off-line process optimization stream.

Then, the optimization mathematic model is

As shown in Figure

The braking force distribution coefficient.

Define

It can be seen from Figure

As shown in Figure

The above optimization results show that the optimization model with the CSD method can get good results, and lay a good foundation for a further high-presicison predictive model.

Considering the poor real-time control of the optimization, a predictive distribution model together with a predictive model and an additional condition is presented. For the predictive model, the multiple correlation coefficients (

To ensure that the predictive distribution model meets the cooperative braking requirement, some key predictive parameters’ scope should be verified as follows.

As stated above,

With respect to

With regard to

The predictive results of

The predictive results of

The predictive results of

The predictive results of

Given the above analysis, an additional condition should be added to the predictive distribution model. For the additional condition, the basic principle is that, if the predictive parameters fall into the infeasible region, the braking torque will be only provided by hydraulic brakes. The predictive distribution model is shown as follows:

Generally, for the series system, the optimum braking stability object can be realized easily through the coordination between the generators and the hydraulic brakes. So, with regard to the predictive distribution model, the only advantage is embodied in the regenerative braking energy recovery efficiency. To verify the advantages of the predictive distribution model, two simulation models of the vehicle are established in MATLAB/Simulink. One is established with the predictive distribution model, the other is based on the ideal braking force distribution strategy which is also striving to maximize the regenerative braking energy recovery and optimize the braking stability simultaneously. Additionally, no optimization method is used in the other simulation model and the distribution of the regenerative braking torque is based on 1 : 1.

The other braking force distribution strategy can be expressed as follows.

First, based on the requirement of the ideal braking force distribution, the required braking torque of the front and rear wheels under a required braking torque can be obtained as follows:

Then, the optimum charging torque

Finally, the distribution of

In the simulation models, the initial vehicle speed is 48 km/h. The battery SoC is 0.5. The road is assumed to be a dry pavement. Two cooperative braking processes are defined in Figures

The simulation results of SoC.

Define the increasing rate of SoC as follows:

As shown in Figure

It can be seen from Figure

The simulation results of

The simulation results of the predictive distribution model.

Figure

This paper carries out a systematic study for a predictive distribution model of a series cooperative braking system. Three disciplines of the maximum regenerative energy recovery capability, the maximum generating efficiency, and the optimum braking stability are considered with an off-line optimization method. In consideration of the poor real-time performance of optimization, a predictive model which is based on the off-line optimization data is presented. Finally, a predictive distribution model which is constituted by the predictive model and an additional condition is proposed. The off-line optimization data proves that the optimization method can meet every discipline and improve the cooperative braking performance. The dynamic simulation results show that the predictive distribution model is reasonable for real-time control.

The authors declare that there is no conflict of interests regarding the publication of this paper.