Keywords

1 Introduction

As the transition from fossil fuels to hydrogen gains momentum for carbon neutrality [1], the development of eco-friendly fuel cell heavy-duty trucks has become a focal point [2,3,4]. Predicting the braking performance of these trucks is a critical design challenge, usually assessed through costly and time-consuming field and dynamometer tests. A need exists for a method to forecast braking performance beforehand amid changing design specifications. Accessing vehicle data such as states of the powertrain, braking, and control systems is crucial for accurate performance prediction [5]. However, obtaining comprehensive data, especially for new vehicle development, is often challenging. This requires a modeling technique that can forecast braking performance using field test data. This paper proposes a modeling approach using experimental data to forecast braking system limits in fuel cell heavy-duty trucks. The method enables deriving algorithms for complex components and diverse systems, even with minimal knowledge of the underlying mechanism and control algorithms.

2 Braking System of Fuel Cell Heavy-Duty Trucks

The braking system of a fuel cell heavy-duty truck can be classified into two main components: the primary brake and two auxiliary brakes. The primary brake utilizes air disc brakes with the operational limit dependent on the temperature of the disc. The two auxiliary brakes, namely the regenerative brake and retarder, operate through distinct mechanisms. The regenerative brake converts kinetic energy into electrical energy using a generator during deceleration, and its operational limit is influenced by the battery’s state of charge (SOC) [6]. Fuel cell and cooling fan operations play crucial roles in SOC changes during deceleration, and the control algorithms for fuel cell operation and the cooling fan significantly impact both SOC and the operational limit of regenerative braking. On the other hand, the retarder generates braking force by slowing down the output shaft of the transmission, with the braking torque produced by fluid friction [7]. The operational limit of the retarder is determined by the retarder oil temperature.

3 Brake System Modeling

To construct a brake system model, diverse vehicle states during braking were obtained through a series of field tests. The tests were conducted on long downhill descents, with the truck endeavoring to maintain a consistent speed range as much as possible. The dataset includes 52 signals. The dataset was collected from 15 distinct driving scenarios (see Table 1). These scenarios were executed by a 6x4 tractor, specifically a fuel cell heavy-duty truck.

Table 1. Details of field test driving scenarios
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The model structure for the fuel cell heavy-duty truck is shown in Fig. 1. Model of each component is described below.

The vehicle is modeled as a simple lumped mass model as follows:

$$ m_{eq} \dot{v}_{x} = \tau_{traction} /R_{wheel} - F_{brake} - m_{eq} g\sin \theta - k_{rr} m_{eq} g\cos \theta - 0.5C_{d} A_{f} \rho v_{x}^{2} . $$
(1)

The structure of the retarder oil temperature model is illustrated in Fig. 2a, and the temperature dynamics are defined as follows:

$$ m_{rtd} c_{p,rtd} \frac{{dT_{rtd} }}{dt} = P_{rtd} - \dot{m}_{clnt} c_{p,clnt} \left( {T_{rtd} - T_{clnt,rtd\_in} } \right) - h_{rtd} A_{rtd} \left( {T_{rtd} - T_{air} } \right), $$
(2)
Fig. 1.
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Model structure for brake system modeling of a fuel cell heavy-duty truck

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Fig. 2.
figure 2

(a) Structure of retarder oil temperature model, (b) Structure of neural network (NN) for equivalent heat transfer coefficient, (c) Validation of the parameter modeled by NN

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Fig. 3.
figure 3

Validation of models for (a) retarder oil temperature, (b) disc temperature and (c) SOC

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where mrtd, cp,rtd, Trtd, Prtd, hrtd and Artd are mass, specific heat, temperature, power, convective heat transfer coefficient and convection area of retarder, ṁclnt and cp,clnt are mass flow rate and specific heat of coolant, Tclnt,rtd_in is coolant temperature at retarder inlet and Tair is ambient temperature.

The equation for the temperature difference through the radiator is

$$ \dot{m}_{clnt} c_{p,clnt} \left( {T_{clnt,rad\_in} - T_{clnt,rad\_out} } \right) = h_{rad} A_{rad} \left( {\left( {T_{clnt,rad\_in} + T_{clnt,rad\_out} } \right)/2 - T_{air} } \right), $$
(3)

where Tclnt,rad_in and Tclnt,rad_out are coolant temperature at radiator inlet and outlet, hrad and Arad are convective heat transfer coefficient and convection area of radiator. Assuming that the amount of heat generated by the transmission is the same as the energy change in the coolant, the following equation is derived:

$$ \dot{m}_{clnt} c_{p,clnt} \left( {T_{clnt,TM\_out} - T_{clnt,TM\_in} } \right) = \dot{Q}_{TM} , $$
(4)

where Tclnt,TM_out is coolant temperature at transmission outlet, Q̇TM is heat generated by the transmission. Combining Eqs. (2), (3) and (4), the following temperature equation can be derived:

$$ m_{rtd} c_{p,rtd} \frac{{dT_{rtd} }}{dt} \approx P_{rtd} - K_{eq} (T_{rtd} - T_{air} ), $$
(5)

where Keq is equivalent heat transfer coefficient of retarder.

The temperature model for the retarder oil is intricately linked to the transmission, both of which are cooled by the same coolant. Accounting for this, modeling the transmission temperature becomes imperative, necessitating the inclusion of several other unknown variables. Additionally, the mass flow rate of the coolant is influenced by an internal algorithm whose specifics are unknown. To address this challenge, this paper introduces the utilization of a neural network to model the equivalent heat transfer coefficient described in Eq. (5). Inputs (vehicle speed, motor speed, retarder torque, coolant and ambient temperatures, gear ratio) obtained from the model feed into the neural network shown in Fig. 2b. This model, operating as a two-hidden-layer feedforward neural network with 64 nodes in each layer and utilizing the Rectified Linear Unit (ReLU) activation function, outputs the equivalent heat transfer coefficient. Figure 2c shows the validation results of the equivalent heat transfer coefficient model.

$$ K_{eq} = f_{NN} (x),\,\,\,\,x = [v_{vehicle} ,\,\omega_{motor} ,\,\,\tau_{rtd} ,\,T_{coolant} ,T_{ambient} \,,\,R_{gear} ]. $$
(6)
Fig. 4.
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Comparison of results between simulation and field test data in 3 scenarios

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Fig. 5.
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Comparison of brake operational limit points caused by temperature or SOC constraints in simulations and field tests

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Figure 3a shows the validation results of the retarder oil temperature model.

The differential equation for the disc temperature is designed as follows:

$$ m_{brk} c_{p,brk} \frac{{dT_{brk} }}{dt} = P_{brk} - \dot{Q}_{conv} = P_{brk} - h_{brk} A_{brk} (T_{brk} - T_{air} ), $$
(7)

where Pbrk is the power generated by the air disc brake, mbrk, cp,brk, hbrk, and Abrk are the mass, specific heat, convective heat transfer coefficient, and convection area of the air disc brake, respectively. The experimental formula for the convective heat transfer coefficient of a disc brake in turbulent flow [8] is given by

$$ h_{brk} = 0.04\left( {k/D} \right)Re^{0.8} = \left( {0.04k\rho^{0.8} L^{0.8} /D\mu^{0.8} } \right)v^{0.8} . $$
(8)

Figure 3b shows the validation results of the disc temperature model.

The battery model is designed based on the equivalent circuit model [9] by open circuit voltage, internal resistance, battery capacity, and battery charge/discharge power as follows:

$$ S\dot{O}C = \left( {V_{oc} - \sqrt {V_{oc}^{2} - 4P_{bat} R_{i} } } \right)/2Q_{bat} R_{i} . $$
(9)

Figure 3c shows the performance validation results of the battery model.

4 Model Validation

To validate the braking system model, we systematically compare field test data and simulation results for Scenarios #1, #2, and #14, as illustrated in Fig. 4. Directly comparing field test data with simulation is challenging due to variations in drivers; the truck was operated by a human driver in the field test, while a driver model controlled the truck in the simulation. However, our primary focus is on ensuring the consistency of the braking limit point during downhill driving under the same gradient road conditions.

In both the field tests and simulation, the order of brake demand distribution was regenerative brake first, followed by the retarder, and then the disc brake. If a brake in priority reaches its torque limit, the next one is engaged. In Scenarios #1 and #2, all three brake systems were activated, whereas in Scenario #14, the retarder was intentionally deactivated to simulate a retarder failure case. Consequently, Scenario #14 underscores the predominant engagement of the primary braking system, surpassing the braking limit point.

Figure 5 demonstrates the consistency of the proposed model in terms of the distance to the brake operational limit caused by temperature or SOC constraints. Across several scenarios, the model’s distance to the brake limit exhibits a very high correlation. These robust findings affirm the effectiveness of the proposed modeling approach.

5 Conclusion

This paper presents an approach utilizing field test data to model the braking system of fuel cell heavy-duty trucks when internal details are unavailable. It models the primary braking system (air disc brake) and auxiliary systems (regenerative braking and retarder). Specific models for disc temperature, SOC, and retarder oil temperature, representing braking limit points, are developed. Validation against field test data confirms the simulation model’s accuracy in forecasting braking limit points to be over 99%. The proposed method allows for advanced forecasting of braking performance in fuel cell heavy-duty trucks based on design specification changes, optimizing vehicle stability and efficiency through simulation.