1 Introduction

Hybrid electric vehicles (HEVs) have attracted the interest of many original equipment manufacturers and researchers due to their low fuel consumption advantages [1–4]. HEVs can combine the advantages of fuel engines and electric motors, and can achieve braking energy recovery through a regenerative braking system, which further improves fuel economy [5]. In addition, combined with X-by-wire technologies such as steer-by-wire (SBW) and brake-by-wire (BBW), HEVs can achieve more precise vehicle chassis dynamics control. Among them, the brake-by-wire system is the main research content of this paper. Through the coordinated distribution of braking force by BBW, the braking safety and energy recovery performance of HEVs can be significantly optimized.

In recent years, many studies focus on maximizing the braking energy recovery efficiency and ensuring braking stays for HEVs. Since the regenerative braking system can only provide a limited electric braking torque, the mechanical braking system should be considered simultaneously. Therefore, designing an appropriate distribution strategy is a crucial technology for HEVs [6–9]. Xu et al. proposed a fuzzy control regenerative braking strategy to distribute the braking force between front and rear wheels, and the braking force on the driving wheels should be within the range allowed by the economic commission of Europe (ECE) regulations as much as possible. Meanwhile, a battery temperature influence factor is designed to modify the calculated regenerative braking force [10]. To recover more braking energy, Lv et al. proposed a synthetic power distribution control strategy considering the braking intension, the wheel speed, and the battery state of charge (SOC) [11]. Gao et al. analyzed the working status and motor characteristics under different braking conditions and proposed three motor dynamic distribution strategies based on different targets. The results of the urban cycle driving simulations showed that the braking energy could be recovered, and the braking system has not been changed too much [12–14].

Vehicle mass and road slope are closely related to the braking performance. Lack of this information may lead to insufficient braking force or an unstable deceleration process. Sun et al. put forward a hybrid estimation method for simultaneously estimating the vehicle mass and road grade in the driving process of hybrid electric buses, but they did not consider the braking process and the energy recovery system [15]. Winstead et al. used an extended Kalman filter (EKF) based method to estimate key parameters online, and then used a model predictive control (MPC) method to track the desired vehicle speed based on these parameters. However, the vehicle mass is assumed to be a constant value without any change [16].

In reality, the total mass of the vehicle varies with the number of passengers, and different roads have different slopes. Due to these uncertain factors, it is difficult to design an optimal braking control strategy [17]. For hybrid electric buses (HEBs), the braking system usually adopts a pneumatic braking scheme, and the control accuracy and response time are not as good as the hydraulic braking system of passenger vehicles. In addition, the driving motor of the bus is usually located on the rear axle, so it is of great significance to study the braking force distribution strategy of different axles and the regenerative braking control strategy [18, 19].

Considering the impact of braking force on the vehicle dynamics system, this paper estimates the vehicle mass and road slope based on a hybrid algorithm of EKF and recursive least squares (RLS). Subsequently, a sliding mode controller is designed to calculate the braking force, while a coordination method of regenerative braking and pneumatic braking is adopted under the conditions allowed by ECE regulations to improve the energy recovery performance.

The rest of this paper is organized as follows: the structure of the braking system and the estimation algorithm are described in Section 2. The sliding mode controller and the braking force distribution strategy are proposed in Section 3. Simulations and experiment results are presented in Section 4. Conclusions are given in Section 5.

2 Vehicle System Modelling

2.1 HEB’s Chassis Structure and Whole Control Architecture

The configuration of the HEB braking system is shown in Figure 1. The braking system mainly includes regenerative braking and pneumatic braking. Regenerative braking only acts on the driving wheels (the driving wheels in this paper are the rear wheels), and pneumatic braking can act on all wheels [20–23]. Regenerative braking and pneumatic braking can freely distribute braking torque. When the driver depresses the brake pedal, the micro control unit (MCU) can identify the braking intention and distribute the torque relationship between regenerative braking and pneumatic braking.

Figure 1
figure 1

Chassis structure of the HEB

Full size image

By collecting the pedal stroke signal, the total braking force is distributed to the front and rear wheels. After that, the rear wheel braking force is divided into electric motor (EM) and pneumatic braking based on wheel speed information and EM status.

Parameters of the HEB model given by Zhongtong Bus Holding Co., Ltd. are listed in Table 1.

Table 1 Main parameters of the HEB
Full size table

The structure of the control algorithm for the braking force of the HEB is shown in Figure 2. First of all, the mass and the road slope are estimated by EKF and RLS. After that, the total braking torque is calculated from the desired deceleration. Then the braking torque is distributed to achieve the goal of tracking the desired deceleration while ensuring energy recovery.

Figure 2
figure 2

Overall architecture of the proposed method

Full size image

2.2 HEB’s Longitudinal Dynamics Model

As shown in Figure 3, the HEB’s longitudinal dynamics model considering the influence of braking force is as follows [23]:

$$\delta m\frac{{{\text{d}}v}}{{{\text{d}}t}} = F_{{\text{t}}} - F_{{\text{f}}} - F_{{\text{w}}} - F_{{\text{i}}} - F_{{\text{b}}} ,$$
(1)

where \(\delta\), \(m\), and \(v\) are the rotational mass conversion factor acceleration, the vehicle mass, and the longitudinal speed, respectively. \(F_{{\text{t}}}\) is the driving force of the vehicle, \(F_{{\text{f}}}\), \(F_{{\text{i}}}\), \(F_{{\text{w}}}\), and \(F_{{\text{b}}}\) are the rolling resistance, the slope resistance, the air resistance, and the braking force, respectively. In Figure 3, \(a\), \(b\) are the distances from the vehicle gravity center to the front and rear axles, respectively. \(l\) is the wheelbase and \(h_{{\text{g}}}\) is the height of the vehicle gravity center.

Figure 3
figure 3

Longitudinal dynamics model on a ramp

Full size image

After expanding the expression of Eq. (1), the vehicle longitudinal dynamics equation can be expressed as:

$$\begin{aligned} \delta m\frac{{{\text{d}}v}}{{{\text{d}}t}} = & \frac{{T_{{{\text{tq}}}} i_{{\text{g}}} i_{{0}} \eta }}{r} - mgf\cos i \\ & - \frac{{C_{{\text{D}}} A\rho v^{2} }}{2} - mg\sin i - \frac{{k_{{\text{p}}} P_{{{\text{brk}}}} }}{r}, \\ \end{aligned}$$
(2)

where \(T_{{{\text{tq}}}}\) is the driving torque, and it can be the braking torque generated by the EM when the vehicle needs, \(i_{{\text{g}}}\) is the gearbox ratio, \(i_{{0}}\) is the final drive ratio, \(\eta\) is the mechanical transmission efficiency, \(r\) is the wheel rolling radius, \(f\) is the coefficient of the rolling resistance, \(i\) is the road gradient, \(C_{{\text{D}}}\) is the drag coefficient, \(A\) is the frontal area, \(\rho\) is the air density, \(k_{{\text{p}}}\) is the torque conversion factor and \(P_{{{\text{brk}}}}\) is the brake chamber pressure.

In the normal driving process of urban buses, the longitudinal speed will not change significantly due to the small braking force and wheel slip ratio [21]. Therefore, the vehicle speed can be equivalently obtained by the wheel speed. At this point, the mass and road slope can be estimated by EKF and modified by RLS based on our previous research [15].

2.3 Hybrid Estimation Method of Mass and Slope

The longitudinal speed v, the whole vehicle mass m, and the road gradient i are selected as the state variables of EKF. Hence, the state vector of the system is defined as \(x(k) = [v(k),\, m(k),\,i(k)]^{{\text{T}}} .\) The mass of the vehicle can be regarded as constant when driving in a sampling interval (0.1 s), and the road gradient changes slowly under one sampling interval. The derivative of both is approximated to zero, and the differential equations can be expressed as:

$$\left\{ \begin{aligned} \dot{v}(k) & = [\frac{{T_{{{\text{tq}}}} (k)i_{{\text{g}}} i_{0} \eta }}{m(k)r} - g\sin i(k) - gf\cos i(k) \\ & \quad - \frac{{C_{{\text{D}}} A\rho v^{2} (k)}}{2m(k)} - \frac{{k_{{\text{p}}} P_{{{\text{brk}}}} }}{m(k)r}]\text{/}\delta , \\ \dot{m}(k) & = 0, \\ \dot{i}(k) & = 0. \\ \end{aligned} \right.$$
(3)

After time \(\Delta t\), the discrete state-space model of the system can be obtained by:

$$\left\{ \begin{aligned} v_{k} & = v_{k - 1} + \Delta t[\frac{{T_{{{\text{tq}}}} (t_{k - 1} )i_{{\text{g}}} i_{0} \eta }}{{m_{k - 1} r}} - g\sin i_{k - 1} \\ & \quad - gf\cos i_{k - 1} - \frac{{C_{{\text{D}}} A\rho v_{k - 1}^{2} }}{{2m_{k - 1} }} - \frac{{k_{{\text{p}}} P_{{{\text{brk}}}} }}{{m_{k - 1} r}}\text{ ]}/\delta , \\ m_{k} & = m_{k - 1} , \\ i_{k} & = i_{k - 1} , \\ \end{aligned} \right.$$
(4)

where \(\Delta t\) represents the sampling interval as 0.1 s.

It is assumed that the process noise and measurement noise of the system are additive noise. The process noise vector and measurement noise vector are \({\varvec{W}}_{k}\) and \({\varvec{V}}_{k}\) respectively, which are mutually independent and have a mean Gaussian white noise.

The state-space expression of the EKF system is as follows:

$$\left\{ \begin{gathered} \hat{x}_{k} = \hat{x}^{ - }_{k} + {\varvec{K}}_{k} (z_{k} - {\varvec{H}}\hat{x}^{ - }_{k} ) + {\varvec{W}}_{k - 1} , \hfill \\ z_{k} = {\varvec{H}}\hat{x}^{ - }_{k} + {\varvec{V}}_{k} , \hfill \\ \end{gathered} \right.$$
(5)

where \({\varvec{H}} = [1,\,0,\,0]\) is the chosen observation matrix.

The vehicle mass \(m\) and the road slope \(i\) are estimated according to EKF, which includes two calculation processes: time update and measurement update. The flow chart of the EKF algorithm is shown in Figure 4.

Figure 4
figure 4

Flow chart of the EKF algorithm

Full size image

Equations of the time update process can be written as:

$$\hat{x}^{ - }_{k} = f(\hat{x}_{k - 1} ),$$
(6)
$${\varvec{P}}_{k}^{ - } = {\varvec{J}}_{f} (\hat{x}_{k - 1} ){\varvec{P}}_{k - 1} {\varvec{J}}_{f}^{{\text{T}}} (\hat{x}_{k - 1} ) + {\varvec{Q}}_{k - 1} ,$$
(7)

where \({\varvec{P}}_{k}^{ - }\), \({\varvec{J}}_{f}\), \(\hat{x}_{k}\), \({\varvec{P}}_{k - 1}\), \({\varvec{Q}}_{k - 1}\) represent the error covariance matrix of the last step, Jacobian matrix obtained by partial derivative of process equation vector function on state variables, optimal prediction of observer estimation, last moment error covariance, and covariance matrix of noise process, respectively.

In this estimation algorithm, it is available in Eq. (8):

$${\varvec{J}}_{f} = \left( {\begin{array}{*{20}c} {\frac{{\partial f_{{1}} }}{\partial v}} & {\frac{{\partial f_{{1}} }}{\partial m}} & {\frac{{\partial f_{{1}} }}{\partial i}} \\ {\frac{{\partial f_{{2}} }}{\partial v}} & {\frac{{\partial f_{{2}} }}{\partial m}} & {\frac{{\partial f_{{2}} }}{\partial i}} \\ {\frac{{\partial f_{{3}} }}{\partial v}} & {\frac{{\partial f_{{3}} }}{\partial m}} & {\frac{{\partial f_{{3}} }}{\partial i}} \\ \end{array} } \right)\text{ } = \left( {\begin{array}{*{20}c} \user2{\varvec A} & \user2{\varvec B} & \user2{\varvec X} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right),$$
(8)

where,

$$\user2{\varvec A} = \left( { - gf_{2} \cos i - C_{{\text{D}}} A\rho v/m} \right){{\Delta t} \mathord{\left/ {\vphantom {{\Delta t} \delta }} \right. \kern-\nulldelimiterspace} \delta },$$
$$\user2{\varvec B} = {{\left( {C_{{\text{D}}} A\rho v^{2} r - 2T_{{{\text{tq}}}} i_{{\text{g}}} i_{0} \eta } \right)\Delta t} \mathord{\left/ {\vphantom {{\left( {C_{{\text{D}}} A\rho v^{2} r - 2T_{{{\text{tq}}}} i_{{\text{g}}} i_{0} \eta } \right)\Delta t} {2m^{2} r\delta }}} \right. \kern-\nulldelimiterspace} {2m^{2} r\delta }},$$
$$\user2{\varvec X} = {{( - g\cos i + gf\sin i)\Delta t} \mathord{\left/ {\vphantom {{( - g\cos i + gf\sin i)\Delta t} \delta }} \right. \kern-\nulldelimiterspace} \delta }.$$

The measurement update process of the algorithm can be written as:

$${\varvec{K}}_{k} = {\varvec{P}}_{k}^{ - } {\varvec{H}}^{{\text{T}}} ({\varvec{HP}}_{k}^{ - } {\varvec{H}}^{{\text{T}}} + {\varvec{R}})^{ - 1} ,$$
(9)
$$\hat{x}_{k} = \hat{x}^{ - }_{k} + {\varvec{K}}_{k} (z_{k} - {\varvec{H}}\hat{x}^{ - }_{k} ),$$
(10)
$${\varvec{P}}_{k} = ({\varvec{I}} - {\varvec{K}}_{k} {\varvec{H}}){\varvec{P}}_{k}^{ - } ,$$
(11)

where \({\varvec{K}}_{k}\), \(\hat{x}_{k}\), \({\varvec{P}}_{k}\), and I represent the gains of Kalman filter, optimal estimation of observed variables, error covariance of observer estimation, and identity matrix, respectively.

2.4 Modified Mass Estimation Algorithm Using RLS

In this section, the recursive least square (RLS) method with a forgetting factor is used to establish a modified model of mass estimation based on the longitudinal dynamics model.

Rewrite Eq. (2) into the form that satisfies the requirement of RLS:

$$\begin{gathered} \underbrace {{{{T_{{{\text{tq}}}} i_{{\text{g}}} i_{0} \eta } \mathord{\left/ {\vphantom {{T_{{{\text{tq}}}} i_{{\text{g}}} i_{0} \eta } r}} \right. \kern-\nulldelimiterspace} r} - 0.5C_{{\text{D}}} A\rho v^{2} + {{T_{{\text{b}}} } \mathord{\left/ {\vphantom {{T_{{\text{b}}} } r}} \right. \kern-\nulldelimiterspace} r}}}_{{y_{k} }} = \hfill \\ \text{ }m\text{ }\underbrace {{\left[ {gf\cos i + g\sin i + \dot{v}} \right]}}_{{H_{k} }}, \hfill \\ \end{gathered}$$
(12)

where \(y_{k}\), \({\varvec{H}}_{k}\), \(m\) are the system output, observable data vector, and system parameters to be estimated. Therefore, the recursive format of vehicle mass can be given as follows:

$$\hat{m}_{k} = \hat{m}_{k - 1} + {\varvec{K}}_{k} (y_{k} - {\varvec{H}}\hat{m}_{k - 1} ).$$
(13)

The gain matrix \({\varvec{K}}_{k}\) and error covariance matrix \({\varvec{P}}_{k}\) are:

$${\varvec{K}}_{k} = \frac{{{\varvec{P}}_{k - 1} {\varvec{H}}_{k}^{{}} }}{{{\varvec{H}}_{k}^{{\text{T}}} {\varvec{PH}}_{k}^{{}} + \lambda }},$$
(14)
$${\varvec{P}}_{k} = \frac{{({\varvec{I}} - {\varvec{K}}_{k} {\varvec{H}}_{{_{k} }}^{{\text{T}}} ){\varvec{P}}_{k - 1} }}{\lambda },$$
(15)

where \(\lambda\) is the forgetting factor of the RLS model. To decrease the instability of the estimation, the arithmetic uses the lesser forgetting factor. Because of the changing mass of the HEB, the forgetting factor is adjusted, and the error covariance matrix is increased when the velocity of the HEB is 0 m/s. The flow chart of RLS is shown in Figure 5.

Figure 5
figure 5

Flow chart of the RLS algorithm

Full size image

3 Braking Force Distribution Strategies

After estimating the mass and road slope, the total braking force is calculated based on the desired acceleration from the brake pedal signal input by a sliding mode controller. Then, the total braking force is distributed to the front and rear wheels according to the ECE regulations, and a regenerative braking system is adopted on the driving wheels (rear wheels) to recover energy.

3.1 Calculation of the Total Braking Torque

Define the sliding surface as:

$$S(\dot{v}) = \dot{v}^{ * } - \dot{v},$$
(16)

where \(\dot{v}^{ * }\) is the desired acceleration, and \(\dot{v}\) is the actual acceleration.

An exponential approach rate method is used, which is proposed as follows:

$$\dot{S} = - \varepsilon {\text{sgn}} (S) - kS,\text{ }\varepsilon > 0,K > 0.$$
(17)

The acceleration of the vehicle is defined as:

$$\dot{v} = \frac{{ - g\sin i - gf\cos i - {{C_{{\text{D}}} A\rho v^{2} } \mathord{\left/ {\vphantom {{C_{{\text{D}}} A\rho v^{2} } {2m}}} \right. \kern-\nulldelimiterspace} {2m}} - {{T_{{\text{b}}} } \mathord{\left/ {\vphantom {{T_{{\text{b}}} } {mr}}} \right. \kern-\nulldelimiterspace} {mr}}}}{\delta },$$
(18)

where \(T_{{\text{b}}}\) is the sum of regenerative braking torque and friction braking torque. Take the derivative of Eq. (16), and combine Eq. (17):

$$\dot{S} = - \ddot{v} = - \varepsilon {\text{sgn}} (S) - kS.$$
(19)

Finally, the calculated output braking torque is:

$$T_{{\text{b}}} = \left[ - g\sin i - gf\cos i - \frac{1}{2m}C_{{\text{D}}} A\rho v^{2} - \frac{{(\varepsilon {\text{sgn}} (S) - kS)\delta^{2} }}{{gf\cos i - {{C_{{\text{D}}} A\rho v} \mathord{\left/ {\vphantom {{C_{{\text{D}}} A\rho v} m}} \right. \kern-\nulldelimiterspace} m}}}\right]mr.$$
(20)

3.2 Ideal Braking Force Distribution (I Curve)

According to the force analysis of the HEB in Figure 3, the vertical forces of the front and rear axles are as follows:

$$\left\{ \begin{gathered} F_{{{\text{zF}}}} = {{mg(b\cos i + \varphi h\cos i + \sin i)} \mathord{\left/ {\vphantom {{mg(b\cos i + \varphi h\cos i + \sin i)} l}} \right. \kern-\nulldelimiterspace} l}, \hfill \\ F_{{{\text{zR}}}} = {{mg(a\cos i - \varphi h\cos i - \sin i)} \mathord{\left/ {\vphantom {{mg(a\cos i - \varphi h\cos i - \sin i)} l}} \right. \kern-\nulldelimiterspace} l}. \hfill \\ \end{gathered} \right.$$
(21)

When the front and rear axles are locked simultaneously, the use of road adhesion and braking stability is the most advantageous [24–26]. At this time, the ideal braking force relationship between the front and rear axles can be expressed as:

$$\left\{ \begin{gathered} F_{{{\text{xR}}}} + F_{{{\text{xF}}}} = \varphi G, \hfill \\ F_{{{\text{xF}}}} = \varphi F_{{{\text{zF}}}} , \hfill \\ F_{{{\text{xR}}}} = \varphi F_{{{\text{zR}}}} , \hfill \\ \end{gathered} \right.$$
(22)
$$\frac{{F_{{{\text{xF}}}} }}{{F_{{{\text{xR}}}} }} = \frac{b\cos i + \varphi h\cos i + \sin i}{{a\cos i - \varphi h\cos i - \sin i}},$$
(23)

where \(F_{{{\text{xR}}}}\), \(F_{{{\text{xF}}}}\) are the braking force of the front and rear axle, \(\varphi\) represents the adhesion coefficient of the road surface. If the road gradient is zero, the rear wheel braking force is shown in Eq. (24):

$$F_{{{\text{xR}}}} = \frac{1}{2} \left[\frac{G}{h}\sqrt {b^{2} + \frac{4hl}{G}F_{{{\text{xF}}}} } - \left(\frac{Gb}{h} + 2F_{{{\text{xF}}}}\right)\right].$$
(24)

3.3 Front and Rear Axle Braking Force Distribution

The braking force distribution coefficient \(\beta\) of this paper can be expressed as:

$$\beta = \frac{{F_{{{\text{xF}}}} }}{{F_{{{\text{xF}}}} + F_{{{\text{xR}}}} }}.$$
(25)

In order to ensure the driving safety of the vehicle, a series of requirements are put forward for the braking force of the front and rear axles [27–29]. When the braking intensity z is between 0.1 and 0.61, the utilization adhesion coefficients of the front and rear axles must meet the requirements of Eq. (26):

$$\left\{ {\begin{array}{*{20}l} {\varphi_{{\text{f}}} \le {{\left( {z + 0.07} \right)} \mathord{\left/ {\vphantom {{\left( {z + 0.07} \right)} {0.85}}} \right. \kern-\nulldelimiterspace} {0.85}},} \hfill \\ {\varphi_{{\text{r}}} \le {{\left( {z + 0.07} \right)} \mathord{\left/ {\vphantom {{\left( {z + 0.07} \right)} {0.85}}} \right. \kern-\nulldelimiterspace} {0.85}},} \hfill \\ {\varphi_{{\text{f}}} \ge \varphi_{{\text{r}}} ,} \hfill \\ \end{array} } \right.$$
(26)

where \(\varphi_{{\text{f}}}\) is the front axle utilization adhesion coefficient, \(\varphi_{{\text{r}}}\) represents the rear axle utilization adhesion coefficient, \(z\) represents the braking intensity.

$$\left\{ \begin{gathered} \varphi_{{\text{f}}} = \frac{{F_{{{\text{xF}}}} }}{{F_{{{\text{zF}}}} }} = \frac{\beta zl}{{b\cos i + zh\cos i + \sin i}}, \hfill \\ \varphi_{{\text{r}}} = \frac{{F_{{{\text{xR}}}} }}{{F_{{{\text{zR}}}} }} = \frac{(1 - \beta )zl}{{a\cos i - zh\cos i + \sin i}}. \hfill \\ \end{gathered} \right.$$
(27)

According to Eqs. (25)‒(27), the distribution coefficient \(\beta\) for the ECE regulation can be calculated:

$$\left\{ \begin{gathered} \beta \le {{(z + 0.07)(b\cos i + zh\cos i + \sin i)} \mathord{\left/ {\vphantom {{(z + 0.07)(b\cos i + zh\cos i + \sin i)} {0.85zl}}} \right. \kern-\nulldelimiterspace} {0.85zl}}, \hfill \\ \beta \ge 1 - {{(z + 0.07)(a\cos i - zh\cos i + \sin i)} \mathord{\left/ {\vphantom {{(z + 0.07)(a\cos i - zh\cos i + \sin i)} {0.85zl}}} \right. \kern-\nulldelimiterspace} {0.85zl}}, \hfill \\ \beta \ge {{b\cos i + zh\cos i + \sin i} \mathord{\left/ {\vphantom {{b\cos i + zh\cos i + \sin i} l}} \right. \kern-\nulldelimiterspace} l}. \hfill \\ \end{gathered} \right.$$
(28)

As shown in Figure 6, the relationship between the braking intensity \(z\) and the braking force distribution coefficient \(\beta\) can be obtained by taking the vehicle parameters into Eq. (28) when the road slope is 0. The green region in the figure is the feasible region of the distribution coefficient [30].

Figure 6
figure 6

The feasible region of the distribution coefficient

Full size image

Because the HEB in this paper uses the rear axle drive, the braking force is distributed to the rear axle under the premise of ensuring safety as far as possible to achieve the goal of maximizing the recovery of braking energy while meeting the requirements of the braking regulations.

When the braking intensity is under 0.1, all the braking force is distributed to the rear axle, and the braking force distribution coefficient is 0. When the braking intensity is between 0.1 and 0.61, to guarantee braking safety while maximizing braking energy recovery, the distribution coefficient of the front and rear axle braking forces is calculated using the lower bound of the region in Figure 6. When the road slope is 0, the maximum braking force of the rear axle is as follows:

$${F_{{\text{xR}} \max }} = \left\{ \begin{array}{*{20}l} {Gz,} & {z \leqslant 0.1,} \\ {{{Gz(a - zh)} \mathord{\left/ {\vphantom {{Gz(a - zh)} l}} \right. \kern-\nulldelimiterspace} l},} & {0.1 < z < \varphi,} \\ {{{G\varphi (a - \varphi h)} \mathord{\left/ {\vphantom {{G\varphi (a - \varphi h)} l}} \right. \kern-\nulldelimiterspace} l},} & {z > \varphi,} \\ \end{array} \right.$$
(29)

where \(F_{{\text{xR max}}}\) represents the maximum braking force of the rear axle.

3.4 Regenerative Braking and Pneumatic Braking Force Distribution

After determining the front-to-rear braking axle braking force distribution strategy, the second part of the regenerative braking control strategy is the distribution of the rear axle pneumatic braking force and the regenerative braking force. The coordination between regenerative braking and pneumatic braking uses a coordinated control strategy. The target of the regenerative braking system is to recover the maximum braking energy under the premise of meeting the requirements shown in Eq. (28) and Eq. (29), so the rear axle braking force adopts the following distribution strategy.

If the maximum braking force of EM exceeds the rear axle braking force requirement, the braking force will all be provided by the EM. If the EM’s braking force cannot meet the rear axle braking force requirements, the maximum EM braking force and the rear axle pneumatic braking force will participate in braking together. The distribution strategy is given in Figure 7.

Figure 7
figure 7

Distribution of regenerative braking and friction braking

Full size image

For the maximum braking torque that the motor can provide, it needs to be calculated based on the characteristics of the EM, current velocity, motor efficiency, and battery status [27–30]. When the vehicle velocity is low, the motor’s torque characteristics are unstable, and when the motor is engaged in braking at a low velocity, it consumes electric energy. Therefore, to ensure stable braking performance and reduce energy consumption, regenerative braking is stopped when the motor speed is less than 300 r/min.

4 Simulations and Experiments

To verify the accuracy of the estimation algorithm and the control performance of the braking system, simulations are carried out using MATLAB/Simulink, in which continuously changing road slope and time-varying braking intensity are set. In addition, the mass of the HEB is designed to change when the HEB stops to simulate passengers getting on and off the bus. Finally, the influence of the parameters in the control algorithm and the experimental results are analyzed.

4.1 Simulation of the Hybrid Estimation Algorithm

Figure 8 shows the estimation results of the vehicle mass and road slope by the method proposed in Section 2.

Figure 8
figure 8

Results of the mass and slope estimation algorithm: (a) speed calculation; (b) whole vehicle mass estimation; (c) road slope estimation, and (d) relative error of mass estimation

Full size image

In terms of mass estimation, the error of using the hybrid estimation algorithm is within 3%‒4%. Compared with the EKF algorithm alone, the hybrid algorithm has higher stability and precision.

During the whole driving cycle, the slope error estimated by the EKF algorithm is between ± 1.5°, and the responding speed of the changing slope is fast.

4.2 Simulation of the Proposed Braking Force Control Method

In order to compare the effects of the two braking force control algorithms, the same braking intensity is used, as shown in Figure 9.

Figure 9
figure 9

Desired acceleration

Full size image

Actual accelerations under different methods obtained by the simulation and the desired acceleration are shown in Figure 10.

Figure 10
figure 10

Comparison between actual accelerations and the desired acceleration

Full size image

In Figure 10, the acceleration represented by the orange curve is obtained under a fixed vehicle mass and road slope input, while the acceleration represented by the blue curve uses the real-time estimation values as the input of the controller. It can be seen from Figure 10 that when the slope changes and the desired acceleration is small, the actual output acceleration without real-time estimation of vehicle states has obvious chatter compared with the control method proposed in this paper. This will seriously affect the comfort and stability of the HEB. In addition, the actual acceleration under control with vehicle states estimation is closer to the desired value than the actual acceleration without estimation in the region of 120‒140 s and 260‒280 s.

4.3 Simulation of the Braking Force Distribution Strategy

After determining the total braking force, the braking force of the front and rear wheels and the distribution of regenerative braking and pneumatic braking are performed according to the distribution strategy mentioned. The simulation results of the distribution of braking force are shown in Figure 11.

Figure 11
figure 11

Simulation results of braking force distribution

Full size image

To compare the results of the battery SoC, three different braking force distribution strategies are set for simulations: the braking force of front and rear axles are distributed according to a fixed ratio, the ideal braking distribution strategy, and the distribution strategy designed in this study. Simulation results of the battery SoC are shown in Figure 12.

Figure 12
figure 12

Simulation results of the battery SoC

Full size image

At the end of the simulation, the battery SoC using the maximum energy recovery strategy is 0.852, while the battery SoC using the ideal braking force distribution strategy is 0.835. The strategy of maximizing energy recovery can relatively increase electrical energy by 2%.

4.4 Experimental Verification

A hardware-in-the-loop (HIL) test bench of the braking system shown in Figure 13 is established to verify the proposed control strategy.

Figure 13
figure 13

Test bench of the braking system

Full size image

A personal computer (PC) is used as the host computer to display system operating information and download programs to the lower-layer controller. The PXI system of National Instruments serves as a slave computer to run vehicle and motor models, which can collect signals from the test bench and transmit them to the control system in the host computer. The brake-by-wire ECU uses the NXP MPC5744P MCU as the core processor and executes the proposed braking control strategy together with the motor controller through CAN communication.

The pressure curve of the braking chamber and the speed tracking process results are shown in Figures 14 and 15.

Figure 14
figure 14

Braking chamber pressure: (a) front axle braking chamber pressure; (b) rear axle braking chamber pressure

Full size image
Figure 15
figure 15

Results of the speed tracking process

Full size image

The pressure error between the front axle braking chamber and the expected pressure is smaller than that of the rear axle. This is because the pneumatic braking intensity of the rear wheels is low, and there is a delay in the distribution of regenerative braking and pneumatic braking, so the pressure response rate of the rear axle braking chamber is slow.

Speed tracking errors under the same SMC controller can be obtained through the HIL simulation, as shown in Figure 16.

Figure 16
figure 16

Errors of the speed tracking process

Full size image

When the actual speed is greater than the desired speed, the speed error is negative, which indicates that the braking performance is not good. In region II, when the time region is within 180‒210 s, the braking performance without vehicle mass estimation is worse than the result with the mass estimation algorithm. This is because the whole vehicle mass of the HEB has changed at the beginning of region II. In the time region of 300‒400 s, the speed control errors are generally large due to the big braking intensity.

Results show that the proposed real-time control strategy can be implemented and used in actual working conditions and the braking distance of the vehicle can be shortened under the control strategy.

5 Conclusions

This paper proposed an overall control architecture based on the electronically controlled braking system. Firstly, a hybrid estimation method that can estimate both vehicle mass and road slope online is adopted, and the steady-state error of mass estimation can be controlled within 4%. Secondly, a sliding mode controller is used to calculate the total braking force demand based on the braking intensity. Finally, the distribution strategy is proposed to distribute the braking force of the front and rear axles. This strategy can achieve the coordinated control of regenerative braking and pneumatic braking, and improve the energy recovery value by about 2% while optimizing the essential braking ability. In addition, HIL tests indicate that the control architecture has practical application value.

The driving conditions in the real world are very complicated, and curved roads or changing ramps limit the application scenarios of this method. Therefore, future studies will focus on optimizing the estimation performance and considering the impact of road friction coefficient on the braking process.