Regenerative Brake Blending in Electric Hypercars: Benchmarking and Implementation

Abstract

This paper presents a novel Regenerative Brake Blending (RBB) strategy for an electric hypercar, framing it as a multi-objective problem. These include thermal and lifespan management of various components plus maximizing energy recovery. We begin by mathematically modelling each subsystem of the RBB layout. Then, we define an acausal offline optimal control problem to establish a benchmark solution; subsequently we propose a causal real-time control strategy inspired by the Equivalent Consumption Minimization Strategy (ECMS). The proposed real-time strategy shows a performance loss of 1.6% compared to the benchmark underscoring the efficacy of the proposed RBB strategy in maximizing energy recuperation while considering for brake temperatures.

1 Introduction

Efficient energy management in electric vehicles (EVs) is essential for enhancing performance and extending range. Regenerative braking, which captures kinetic energy during deceleration and converts it into electrical energy, is a promising solution. This technology is especially critical for electric hypercars with large power outputs, requiring advanced strategies to manage energy recovery and optimize dynamical performance of the overall vehicle.

Regenerative Brake Blending (RBB) consists in the allocation of the braking force between electric motors and friction brakes. The allocation impacts several and conflicting aspects: energy recuperation, battery stress, and friction brakes temperature, to name the most important ones. RBB usually encompasses two primary allocation levels: first-level allocation, typically front/rear-based to comply with European ECE regulation 13H [1], and wheel-level allocation, aiming to optimize energy, thermals, degradation considerations (for both brake and battery) in a blended strategy.

RBB at wheel level can be categorized into two main types: static and dynamic allocation methods. Static strategies, such as the serial blending strategy, rely on instantaneous efficiency maps of motor, inverter, battery and consist in instantaneously allocating all braking force to the electric motor (to maximixe energy recuperation) and fullfilling the braking request (if necessary) with the friction brakes; these methods have been widely explored in literature and industrial applications [4, 7] and represent the state of the art. However, they are based on heuristic-driven approaches, often resulting in suboptimal outcomes.

Fig. 1.

Regenerative brake blending layout.

In contrast, dynamic strategies incorporate predictive aspects and multi-objective functions, allowing for global optimality and the integration of constraints, system dynamics, and predictive features. Dynamic strategies have found application in the dual problem of optimizing power split between internal combustion engine and electric motors in hybrid vehicles [2, 6, 8]: acasual methods (like dynamic programming or Pontryagin’s Minimum Principle (PMP)) or casual ones like Equivalent Consumption Minimization Strategy (ECMS) have yet to be explored for RBB optimization in both literature and industrial practice.

Section 2 analyses the benchmark solution based on an acasual offline optimal control problem. Section 3 introduces a real-time implementation based on the ECMS principle and a comparitive analysis with the acasual method and the baseline heuristic will be carried out. Finally, Sect. 4 draws conclusions about the aforementioned RBB strategy.

2 Benchmark Regenerative Brake Blending

Theis strategy is based on a control oriented model validated on real data. The model, schematically represented in Fig. 1, considers the main components of the powertrain: the battery, the power electronics, motor and friction brakes. The battery state of charge and thermal dynamics ($T_{batt}$) are modeled [10]. Friction brakes thermal dynamics are captured with a lumped model [3] which consists of two separate elements (disks - $T_d$ - and pads - $T_c$ - ). Last, the electric motor and transmission have been included in the formulation via efficiency maps based on test bench data. Since the RBB strategy does not influence the longitudinal dynamics, the problem can be studied with the backward facing model approach [5].

Given a predefined driving cycle, the brake blending problem can be framed as an constrained minimization problem of the following cost function:

$$\begin{aligned} J = \int _{t_0}^{t_f} \left( w_1(T_{batt}) + w_2(T_{\{c,d\}, f}) + w_3(T_{\{c,d\} r}) + w_4(P_{reg}) \right) dt \end{aligned}$$

(1)

where the optimization variables are the four braking torques (electrical front and rear and friction front and rear). The constraints are summarized as follow:

$$\begin{aligned} \dot{x} = f(x(t), u(t), t), \end{aligned}$$

(2)

$$\begin{aligned} x \in [X_{\min }, X_{\max }], \end{aligned}$$

(3)

$$\begin{aligned} u \in [U_{\min }, U_{\max }], \end{aligned}$$

(4)

$$\begin{aligned} T_{\text {req}}^{\text {front}} + T_{\text {req}}^{\text {rear}} = T_{\text {req}}^{total}, \end{aligned}$$

(5)

$$\begin{aligned} T_{\text {req,EM}}^{\text {front}} \ge T_{\text {min,EM}}^{\text {front}}, T_{\text {req,EM}}^{\text {rear}} \ge T_{\text {min,EM}}^{\text {rear}}, \end{aligned}$$

(6)

$$\begin{aligned} T_{\text {req,EM}}^{\text {front}} + T_{\text {req,FB}}^{\text {front}} = T_{\text {req}}^{\text {front}}, T_{\text {req,EM}}^{\text {rear}} + T_{\text {req,FB}}^{\text {rear}} = T_{\text {req}}^{\text {rear}}, \end{aligned}$$

(7)

$$\begin{aligned} 2 T_{\text {req,EM}}^{\text {front}} \cdot \omega _{\text {mot}} \cdot \eta _{\text {front}} + 2 T_{\text {req,EM}}^{\text {rear}} \cdot \omega _{\text {mot}} \cdot \eta _{\text {rear}} \ge P_{\text {min}}^{\text {batt}}(T^{\circ }_{\text {batt}}), \end{aligned}$$

(8)

where (2) represents the system dynamics, (3) and (4) are the constraints on the states and control variables, equation (5) satisfy the total negative torque request (front plus rear), (6) limits the regenerative torque based on motors mechanical limits, (7) fullfill the brake request at wheel level (friction brake plus electric motor) and finally (8) limits the maximum regen torque based on battery derating.

In J (1), the weighting functions w are used to balance the conflicting objectives and to consider the nonlinear effect that temperature has on the life of the components. By changing the shape of the weights one can give priority to some aspects rather than others. Figure 2 shows the state trajectories for a state-of-the-art baseline (sequential blending) and two possible tunings of the weights: recovered energy optimization priority and battery temperature minimization. The optimal control strategy outperforms the baseline serial blending (black dotted), even when the request is to maximize energy recovery and minimize brake and battery temperatures. Note that regenerative braking has a low impact on battery temperature and that the main tradeoff is between harvested energy and brake temperatures. This information will be used to simplify the optimal control problem and make it implementable in real-time.

3 Real Time Implementation

The previous optimal problem formulation assumes a known driving cycle and is not therefore implementable on the vehicle. To address this, we took inspiration from the Equivalent Consumption Minimization Strategy (ECMS) [9]. ECMS transforms a dynamic constrained optimization problem into a static constrained optimization problem by introducing equivalence factors, which represent a trade-off between different energy sources.

Fig. 2.

State trajectories comparing the results of the offline optimal problem, for different weights, and the state-of-the-art sequential blending.

In the dual problem of power split (between internal combustion engine and electric motors) in hybrid vehicles, the generic formulation of ECMS involves minimizing a cost function that combines fuel consumption and electrical energy usage. The cost function is defined as:

$$\begin{aligned} J = m_f(t) + \sum _i s_i \cdot P_i(t) \end{aligned}$$

(9)

where $m_f(t)$ is the instantaneous fuel consumption, $P_i(t)$ are the electrical power demands, and $s_i$ are the equivalence factors that convert electrical power into an equivalent fuel consumption. To make this approach implementable on a vehicle, we propose a method to tune the equivalence factors based on the co-states behavior in a Pontryagin’s Minimum Principle (PMP) formulation [6]. By analyzing the behavior of the co-states, we can adjust the equivalence factors to ensure that the ECMS closely approximates the optimal solution given by the PMP. This tuning process allows the ECMS to dynamically adapt to varying driving conditions without requiring a pre-defined driving cycle, making it practical for real-world vehicle implementation.

Starting from the PMP insights, a formulation of the ECMS for the regenerative brake blending problem can be derived:

$$\begin{aligned} \min _{u} J = & -P_{regen}(x,u) + s_2 \cdot \dot{T}_{c,front}(x,u) + s_3 \cdot \dot{T}_{d,front}(x,u) \nonumber \\ & + s_4 \cdot \dot{T}_{c,rear}(x,u) + s_5 \cdot \dot{T}_{d,rear}(x,u), \end{aligned}$$

(10)

subject to constraints (5, 6, 7, 8); $u$ are the control variables (i.e., $T_{req,FB}^{front}$ and $T_{req,FB}^{rear}$) and $s_i$ are the equivalence factors. The practical implication of the aforementioned ECMS formulation is that whenever the vehicle applies the friction brakes for slowing down, in subsequent instances, it will utilize the electric motors, and so the battery, to cool them instead and recharge the battery.

The qualitative results depicted in Fig. 3a show the ECMS can closely replicate the offline optimal solution. The dataset used consists in two flying laps of the Nordschleife circuit. In terms of harvested energy, the improvement with respect to the serial blending is just $1.6\%$. This is an expected outcome, since the sequential blending only aim is to maximize energy recovery. However, if we look at the brake thermal management, we can notice that the ECMS outperforms the state-of-the-art solution thanks to an optimal allocation of regenerative power between electric and friction brakes. This allows to maintain the front brakes (the most stressed ones) under the critical threshold of $750^\circ C$ even in two consecutive pushalaps of the Nordschleife. The pareto front illustrated in Fig. 3b shows the quantitative results of the aforementioned results.

Fig. 3.

Real-time implementation performance assesment.

4 Conclusions

In this paper, we introduced a novel Regenerative Brake Blending (RBB) strategy for electric hypercars, framing it as a multi-objective problem. The acausal offline optimal control problem provided a benchmark solution, against which the proposed real-time strategy, based on the Equivalent Consumption Minimization Strategy (ECMS), was evaluated. Despite a minimal performance loss of only $1.6\%$ compared to the benchmark, the real-time strategy demonstrated significant efficacy in maximizing energy recuperation while managing brake temperatures. This underscores the potential of advanced RBB strategies in enhancing energy efficiency and performance in high-performance electric vehicles.