Low-level Online Control of the Formula 1 Power Unit with Feedforward Cylinder Deactivation

TL;DR

This paper introduces a computationally efficient, multi-layer control architecture for optimizing Formula 1 hybrid engine performance online, effectively managing complex thermal and electrical interactions under various track conditions.

Contribution

It presents a novel control framework combining supervisory, feedforward, and model predictive controllers tailored for real-time engine management in F1 cars.

Findings

The control system performs well under simulated realistic scenarios.

Cylinder deactivation reduces suboptimality by 7-8 ms.

The architecture handles unexpected disturbances with minimal performance loss.

Abstract

Since 2014, the F\'ed\'eration Internationale de l'Automobile has prescribed a parallel hybrid powertrain for the Formula 1 race cars. The complex low-level interactions between the thermal and the electrical part represent a non-trivial and challenging system to be controlled online. We present a novel controller architecture composed of a supervisory controller for the energy management, a feedforward cylinder deactivation controller, and a track region-dependent low-level nonlinear model predictive controller to optimize the engine actuators. Except for the nonlinear model predictive controller, the proposed controller subsystems are computationally inexpensive and are real time capable. The framework is tested and validated in a simulation environment for several realistic scenarios disturbed by driver actions or grip conditions on the track. In particular, we analyze how the control architecture deals with an unexpected gearshift trajectory during an acceleration phase. Further, we demonstrate how an increased maximum velocity trajectory impacts the online low-level controller. Our results show a suboptimality over an entire lap with respect to the benchmark solution of 49 ms and 64 ms, respectively, which we deem acceptable. Compared to the same control architecture with full knowledge of the disturbances, the suboptimality amounted to only 2 ms and 17 ms. For all case studies we show that the cylinder deactivation capability decreases the suboptimality by 7 to 8 ms.

Equations

$$\mathbf{x}(s)={\left[\begin{array}{ccccc}v(s)& p_{\mathrm{im}}(s)& E_{\mathrm{f}}(s)& E_{\mathrm{b}}(s)& E_{\mathrm{tc}}(s)\end{array}\right]}^{\top },$$

$$\begin{split}\mathbf{u}(s)=\big{[}u_{\mathrm{th}}(s)\quad&\dot{m}_{\mathrm{f,cyl}}(s)\quad P_{\mathrm{k}}(s)\quad P_{\mathrm{h}}(s)\quad\dots\\ \dots\quad&u_{\mathrm{wg}}(s)\quad u_{\mathrm{sa}}(s)\quad P_{\mathrm{brk}}(s)\big{]}^{\top},\end{split}$$

$$\frac{\text{d}}{\text{d}s}{p}_{\mathrm{im}}(s)=\frac{1}{v(s)}\cdot \frac{R_{\mathrm{air}}\cdot {\vartheta }_{\mathrm{im}}}{V_{\mathrm{im}}}\cdot \left({\dot{m}}_{\mathrm{c}}(s)-{\dot{m}}_{\beta }(s)\right),$$

$$\dot{m}_{\beta}(s)=\frac{p_{\mathrm{im}}(s)\cdot V_{\mathrm{d}}}{R_{\mathrm{air}}\cdot\vartheta_{\mathrm{im}}}\cdot\frac{\omega_{\mathrm{e}}(s)}{4\pi}\cdot\lambda_{\mathrm{vol}}\big{(}\omega_{\mathrm{e}}(s)\big{)}\cdot u_{\mathrm{th}}(s),$$

$${\Psi }_{\mathrm{e}}(s)\in \{0,\dots ,N_{\mathrm{cyl}}\},$$

$${\lambda }_{\mathrm{af}}(s)=\frac{{\dot{m}}_{\beta }(s)/N_{\mathrm{cyl}}}{{\dot{m}}_{\mathrm{f},\mathrm{cyl}}(s)}\cdot \frac{1}{{\sigma }_0},$$

$${\lambda }_{\mathrm{af}}^{\min }⩽{\lambda }_{\mathrm{af}}(s)⩽{\lambda }_{\mathrm{af}}^{\max },$$

$$0⩽{\dot{m}}_{\mathrm{f},\mathrm{cyl}}(s)⩽{\dot{m}}_{\mathrm{f},\mathrm{cyl}}^{\max }\left({\omega }_{\mathrm{e}}(s)\right):={\dot{m}}_{\mathrm{f}}^{\max }\left({\omega }_{\mathrm{e}}(s)\right)/N_{\mathrm{cyl}},$$

$$\dot{m}_{\mathrm{f}}^{\mathrm{max}}\big{(}\omega_{\mathrm{e}}(s)\big{)}=\begin{cases}f_{\omega_{\mathrm{e}}}^{\mathrm{FIA}}\big{(}\omega_{\mathrm{e}}(s)\big{)}&\text{if }\omega_{\mathrm{e}}(s)\leqslant 10.5\text{ krpm},\\ f^{\mathrm{FIA}}_{\mathrm{const}}&\text{if }\omega_{\mathrm{e}}(s)>10.5\text{ krpm},\end{cases}$$

$${\dot{m}}_{\mathrm{f}}(s)={\dot{m}}_{\mathrm{f},\mathrm{cyl}}(s)\cdot {\Psi }_{\mathrm{e}}(s).$$

$$P_{\mathrm{f}}(s)={\dot{m}}_{\mathrm{f}}(s)\cdot H_l,$$

$$\frac{\text{d}}{\text{d}s}{E}_{\mathrm{f}}(s)=\frac{1}{v(s)}\cdot P_{\mathrm{f}}(s).$$

$$P_{\mathrm{e},\mathrm{comb}}(s)=\eta ({\omega }_{\mathrm{e}}(s),{\lambda }_{\mathrm{af}}(s))\cdot u_{\mathrm{sa}}(s)\cdot P_{\mathrm{f}}(s),$$

$$P_{\mathrm{e}}(s)=P_{\mathrm{e},\mathrm{comb}}(s)+P_{\mathrm{e},\mathrm{fp}}(s),$$

$$\frac{\text{d}}{\text{d}s}{E}_{\mathrm{tc}}(s)=\frac{1}{v(s)}\cdot \left(P_{\mathrm{t}}(s)-P_{\mathrm{c}}(s)+P_{\mathrm{h}}(s)\right),$$

$$\begin{split}\dot{m}_{\mathrm{c}}(s)&=\mathcal{M}_{\dot{m}_{\mathrm{c}}}\big{(}E_{\mathrm{tc}}(s),p_{\mathrm{im}}(s)\big{)},\\ P_{\mathrm{c}}(s)&=\mathcal{M}_{P_{\mathrm{c}}}\big{(}E_{\mathrm{tc}}(s),p_{\mathrm{im}}(s)\big{)},\\ \dot{m}_{\mathrm{t}}(s)&=\mathcal{M}_{\dot{m}_{\mathrm{t}}}\big{(}E_{\mathrm{tc}}(s),p_{\mathrm{em}}(s),\vartheta_{\mathrm{em}}(s)\big{)},\\ P_{\mathrm{t}}(s)&=\mathcal{M}_{P_{\mathrm{t}}}\big{(}E_{\mathrm{tc}}(s),p_{\mathrm{em}}(s),\vartheta_{\mathrm{em}}(s)\big{)},\end{split}$$

$$\frac{\text{d}}{\text{d}s}{E}_{\mathrm{b}}=\frac{1}{v(s)}\cdot f_{\mathrm{b}}\left(P_{\mathrm{k}}(s),P_{\mathrm{h}}(s)\right),$$

$$P_{\mathrm{trac}}(s)=f_{\mathrm{trac}}\left(P_{\mathrm{e}}(s),P_{\mathrm{k}}(s),P_{\mathrm{brk}}(s)\right),$$

$$m_{\mathrm{car}}\frac{\text{d}}{\text{d}s}v(s)=\frac{1}{v(s)}\left(P_{\mathrm{trac}}-P_{\mathrm{res}}(v(s),s)\right),$$

$$v(s)⩽v_{\max }(s),\forall s.$$

$${\omega }_{\mathrm{e}}(s)=\left(\frac{\Gamma (s)\cdot {\Gamma }_{\mathrm{d}}}{r_{\mathrm{w}}}\right)\cdot v(s)=\tilde{\Gamma }(s)\cdot v(s),$$

$${\omega }_{\mathrm{e}}^{\min }⩽{\omega }_{\mathrm{e}}(s)⩽{\omega }_{\mathrm{e}}^{\max },$$

$$\underset{\mathbf{u}}{\min }{\int }_0^S\frac{\text{d}s}{v(s)},$$

$$\text{eqref}eq:pIM-\text{eqref}eq:Pe,\text{eqref}eq:omegaenginelimits$$

$$\text{eqref}eq:Etc,\text{eqref}eq:TCmaps,$$

$$\text{eqref}eq:Eb,$$

$$\text{eqref}eq:tracpower,\text{eqref}eq:eqdiffvelocity,\text{eqref}eq:vmaxconstraint,\text{eqref}eq:vtoomega,$$

$$E_{\mathrm{f}}(S)⩽E_{\mathrm{f}}(0)+\Delta E_{\mathrm{f},\mathrm{target}},$$

$$E_{\mathrm{b}}(S)⩾E_{\mathrm{b}}(0)+\Delta E_{\mathrm{b},\mathrm{target}},$$

$$\begin{split}\mathbf{r}^{\mathrm{nom}}(s)=\big{[}v^{\mathrm{nom}}_{\mathrm{max}}(s)\quad\Gamma^{\mathrm{nom}}(s)\quad&p^{\mathrm{nom}}_{\mathrm{im}}(s)\quad E^{\mathrm{nom}}_{\mathrm{f}}(s)\quad\dots\\ \dots\quad&E^{\mathrm{nom}}_{\mathrm{b}}(s)\quad E^{\mathrm{nom}}_{\mathrm{tc}}(s)\big{]}^{\top},\end{split}$$

$$P_{\mathrm{trac}}(s)⩽P_{\mathrm{req}}(s).$$

$$P_{\mathrm{trac}}(s)=P_{\mathrm{req}}(s).$$

$${\mathbf{r}}^{\mathrm{act}}(s)={\left[\begin{array}{cc}{v}_{\max }^{\mathrm{act}}(s)& {\Gamma }^{\mathrm{act}}(s)\end{array}\right]}^{\top }.$$

Full text

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DOI 10.1109/TVT.2023.3246130, © 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

I Introduction

Formula 1 (F1) cars belong to the fastest circuit racing vehicles in the world, competing in the most prestigious racing championship. Since its foundation in the early 1950s, the Fédération Internationale de l’Automobile (FIA) released multiple rule updates concerning safety, racing regulations, as well as technical requirements [1, 2]. One revolutionary technical update was introduced in the 2014 season, featuring the parallel electric hybrid powertrain architecture shown in Fig. 1.

Since then, a downsized $1.6\text{\,}\mathrm{L}$ V6 internal combustion engine (ICE) with individual cylinder deactivation capability is boosted by a turbocharger, which is further electrified by means of an electric motor-generator unit (MGU-H, H for heat). This electric machine is exploited to compensate for suboptimal transient phenomena such as the turbo lag, and also recuperates energy from the exhaust gases thanks to an oversized turbine producing more power than used by the compressor. A second motor, called MGU-K (K for kinetic), is mounted on the crankshaft of the engine, providing extra torque when accelerating and recuperating energy in braking phases. It is restricted by the FIA with a power constraint of $\pm$120\text{,}\mathrm{k}\mathrm{W}$$. Henceforth denoted as power unit, this propulsive package is connected to the wheels through an eight-speed sequential gearbox and a limited slip differential. Together with the friction brakes actuated by the driver and controlled via a brake-by-wire system, this setup represents the complete powertrain of the considered F1 racing car. To incentivize the efficient operation of this power unit, the regulations allow the use of an energy management controller. To understand its degrees of freedom, we need to distinguish between two regions that depend on the driving behavior of the driver: the power-limited region (PL) and the grip-limited region (GL). In the power-limited region the velocity of the car is only limited by the maximum power output of the power unit: This usually occurs on the straights where the driver is fully pressing the throttle pedal and the maximum velocity profile is not reached. Here, the electronic control unit (ECU) is allowed to overwrite the driver’s request and provide a different amount of traction power depending on the selected strategy. In the grip-limited region, however, the car is limited by the grip of the tires: This usually occurs in the corners where the driver is not requesting full power. Since the regulations do not allow active traction control, the power requested by the driver needs to be fulfilled and only the split between ICE, MGU-K and friction brakes can be chosen. Additionally, refueling during pit stops is prohibited, the battery capacity is limited to $4\text{\,}\mathrm{M}\mathrm{J}$, and electrical boosting and recuperating is restricted to $4\text{\,}\mathrm{M}\mathrm{J}$ and $2\text{\,}\mathrm{M}\mathrm{J}$ per lap, respectively.

The lap time optimization of this complex system is of paramount importance to ensure that the driver has a car that makes optimal use of its propulsive capabilities within a predefined regulatory framework. The setup of this hybrid power unit introduces highly complex interactions between the thermal system and the electric motors, e.g., we do not only have to consider trivial scenarios such as pure electric boost or recuperation, but also need to account for nonlinear turbocharger transients in the energy balance. Coupled to the numerous FIA rules that constrain energy availability, this problem statement calls for model-based optimization of the powertrain’s operation. In this research paper, we present a two-level online control architecture combining a computationally inexpensive supervisory energy management with a model-based low-level power unit controller. Furthermore, we include our controller setup in a simulation framework to assess the online solution and compare it to offline optimizations.

I-A Literature Review

Owing to the power unit topology of modern F1 cars, the relevant methodologies investigated are found in the hybrid electric vehicle research field. Therefore, we identify three different streams of research related to our topic.

The first one deals with the offline optimization of the energy management of hybrid electric vehicles. Given the steadily increasing restrictions on CO2 emissions, many non-causal fuel optimal control strategies using dynamic programming (DP) [3, 4, 5], convex optimization [6, 7], and Pontryagin’s minimum principle (PMP) [8, 9, 10] were investigated. Further, models with integer variables such as engine on/off and gearshifts were considered and optimized by means of iterative linear programming [11], via mixed-integer linear programs (MILP) [12] and PMP [13]. However, given the racing application, our main interest lies in lap time optimizations. For F1 racing vehicles the focus has been put on the optimal energy recovery system control [14], whilst modeling the internal combustion engine from a high-level perspective. Additionally, a three-dimensional track was identified from GPS data, and subsequently used for joint optimization of lap time and driven line [15, 16, 17]. In our research group, time-optimal control strategies have been investigated, leveraging convex approximations and relaxations [18]. Finally, to the best of the authors’ knowledge, in [19] the joint optimization of a detailed low-level model of the F1 powertrain and the energy management from a lap time perspective was presented for the first time.

In the second stream we consider the online control of hybrid electric powertrains. In [20] the authors compare the non-causal DP and PMP methods to the real time feasible Equivalent Consumption Minimization Strategy (ECMS) [21]. A comparison between rule-based control, adaptive ECMS (A-ECMS) and $\mathcal{H}_{\infty}$ control has been pursued in [22], concluding that the A-ECMS strategy is the best performing one. These strategies have been augmented to consider additional constraints such as pollutant emissions [23] or the battery state-of-health [24]. Moreover, other tuning approaches such as the fuzzy-tuned ECMS were proposed as real-time solution [25]. Various model-based approaches were investigated to tackle the optimal energy management: In [26, 27] the authors linearized the model at each sample time and employed model predictive control (MPC), while in [28] a supervisory controller was merged with an $\mathcal{H}_{\infty}$ for electrified turbocharger diesel engines. Further, non-causal trajectories of the battery’s state of charge generated with DP optimizations were tracked by means of an MPC in [29]. In [30] the author also considered the optimal trajectory optimization. In our research group, the ECMS was adapted for the goal of lap time optimization by defining the optimal control policy through PMP [31], resulting in the Equivalent Lap Time Minimization Strategies (ELTMS) [32, 33]. The online control of a linearized model of an F1 car was tackled also by means of a two-level MPC scheme [34]. Most recently, research on lap time optimal control gained traction in the field of pure electric race cars, particularly focusing on the transmission design and control [35, 36], as well as thermal limitations [37].

Finally, in the third research stream we looked at state-of-the-art cylinder deactivation online control. In [38] the authors examine the engine efficiency improvement due to cylinder deactivation introducing various engine modes that employ specific control actions. However, in literature the investigated operation modes mostly include binary decision variables for the state of the ICE: either on or off [39, 40]. In [41] the authors incremented the modes with a “half” running engine and propose a probabilistic approach using Markov’s chains to determine the engine operation mode. Further, additional variations such as three, four, and six running cylinders were analyzed [42]. In [43] the authors discuss the case of fully individual cylinder deactivation. They used engine speed vs. torque maps to determine the optimal amount of cylinders. To include the possibility of shutting off the engine in an optimization framework and avoid mixed-integer programs, partial outer convexification [44] and iterative schemes [11] have been investigated. Finally, in racing applications without refueling possibility, the crucial influence of the fuel quantity at the race start on the vehicle’s weight and thus on the lap time was shown in [45, 46, 47].

I-B Research Statement

To the best of the authors’ knowledge, there exists a gap in the low-level online control of hybrid electric racing vehicles. In particular, low-level actuators of the power unit are not considered in the state-of-the-art lap time optimal energy management. The literature focuses either on fuel optimality which is irrelevant in racing applications, or online controllers that rely on high-level models. The latter allow for computationally efficient implementations. However, the underlying assumption is that a requested power is always achievable. This entirely neglects the actual actuation that allows for its delivery. Therefore, in this paper we focus on a model-based online control architecture that aims for lap time optimal disturbance rejection by considering actual power unit actuators and accounting for fast changing system dynamics. Such an architecture allows us to react in a lap time optimal fashion to energy budget deviations resulting from various disturbances by controlling low-level actuators, such as the air-to-fuel ratio or the spark advance efficiency. Due to confidentiality reasons, a comparison with the online controller currently employed in our specific application field is not possible. We therefore focus on the analysis of the suboptimality under given disturbances compared to a benchmark solution, commenting on the origins of the lost lap time and on the strength of our model-based approach. Finally, we want to provide a tool to test online controllers and infer potential heuristics that can be applied on a race weekend. Due to the immense cost of testing on a physical system, we therefore design a simulation environment.

I-C Contribution

The scientific value of this paper is threefold: The main contribution is the implementable online control architecture. Its novelty is the combination of a supervisory high-level controller ELTMS taking care of slow changing dynamics, a detailed low-level nonlinear MPC (NMPC) accounting for all relevant system states, and an estimator providing predictive data on the driver actions. Second, we devise a feedforward cylinder deactivation strategy relying on look-up tables that are derived from behavioral patterns of various lap time optimizations. This novel architecture avoids computationally expensive mixed-integer programming and allows the deactivation of single cylinders during transients rather than just an on/off behavior. Finally, we augment the online controller with a simulation environment. The resulting framework provides the possibility to run the controller without expensive test bench sessions and assess its suboptimality under given disturbances with respect to a benchmark. Furthermore, control heuristics for the race can be inferred and other controllers can be compared.

II Framework Overview

To facilitate the understanding of this paper, in this section we introduce the structure of our work and highlight the interconnections between the elements. Fig. 2 illustrates the three main components of our framework: the driver model, the controller, and the race car model. To test the performance of the controller in a simulation environment we need a driver model. This component is responsible for providing driver inputs that are not subject to optimization and act as a disturbance to the controller. Next, we dive into the main contribution outlined in Section I-C: the control architecture. The heart of this controller is an NMPC, which performs reference tracking by optimizing low-level powertrain actuators. The reference trajectories are obtained by solving the offline optimization problem presented in [19] for the race car model. Additionally, the controller includes an estimator that is designed to generate future driver actions relying on the same offline optimization results and feedback from the race car. To cope with disturbances from an energetic point of view, we also include a supervisory controller, which needs the optimal reference trajectories to recognize discrepancies and react accordingly. To compute the suboptimality with respect to a non-causal benchmark, we recompute the optimal solution offline with the energy consumption obtained during online control and known disturbances. Finally, the driver’s gear choice and the control inputs are fed to the race car model.

II-A Paper Structure

In Section III we introduce the low-level race car model that we aim to control. In Section IV, we illustrate the offline optimization of such a system that has been carried out previously in our research group. Section V addresses the driver model needed for the simulation of the online control loop. Then, in Section VI, we outline the estimator that provides the predictive information for the NMPC. Furthermore, we include the supervisory ELTMS controller to comply with slow dynamical disturbances in energy trajectories, and a feedforward cylinder deactivation controller to tackle the mixed-integer nature of such a system. We conclude the section with the definition of the optimal control problems solved by the NMPC. In Section VII we showcase the performance and robustness of our framework by means of two case studies where we analyze multiple disturbances generated by an unforeseen driver behavior. Finally, in Section VIII we draw the conclusions, comment on the relevant insights gained by means of our framework, and give an outlook on future research.

III Race Car Model

As introduced in Section I, the system that we control is an F1 hybrid electric race car. Its modeling relies on [48] and [19], to which the reader is referred for the detailed model validation. The race car model is embedded in our framework as shown in Fig. 2. The state vector $\mathbf{x}$ of the race car model reads as

[TABLE]

where $v$ is the velocity of the car, $p_{\mathrm{im}}$ is the intake manifold pressure, $E_{\mathrm{f}}$ is the fuel energy, $E_{\mathrm{b}}$ is the battery energy and $E_{\mathrm{tc}}$ is the kinetic energy of the turbocharger. The control input vector $\mathbf{u}$ is

[TABLE]

where $u_{\mathrm{th}}$ is the throttle position, $\dot{m}_{\mathrm{f,cyl}}$ is the cylinder fuel mass flow, $P_{\mathrm{k}}$ is the MGU-K power, $P_{\mathrm{h}}$ is the MGU-H power, $u_{\mathrm{wg}}$ is the waste-gate position, $u_{\mathrm{sa}}$ is the spark advance efficiency and $P_{\mathrm{brk}}$ is the brake power. Additionally, also the engaged gear $\Gamma$ is an input to the plant, which is not controlled by the controller. The nominal reference vector $\mathbf{r}^{\mathrm{nom}}$ will be introduced in Section IV, whilst the actual reference vector $\mathbf{r}^{\mathrm{act}}$, the fed back controller information $\mathbf{u}_{\mathrm{dr}}$, and the requested power $P_{\mathrm{req}}$ will be defined in Section V. As we include track dependent parameters and optimize over the lap time, we write all equations as a function of the continuous path variable $s\in[0,\dots,S]$, where $S$ is the length of the track. This variable denotes the distance covered along the racing line on track.

III-A Internal Combustion Engine

The heart of the power unit is the internal combustion engine. The dynamics of the air path are essentially determined by the dynamics of the intake manifold pressure $p_{\mathrm{im}}$, namely

[TABLE]

where $R_{\mathrm{air}}$ is the specific gas constant of air, $\vartheta_{\mathrm{im}}$ is the assumed to be constant intake manifold temperature, $V_{\mathrm{im}}$ is the intake manifold volume, $\dot{m}_{\mathrm{c}}$ is the mass flow through the compressor and $\dot{m}_{\beta}$ is the air mass flow entering the cylinders. The division by $v$ stems from the fact that the spatial derivative is considered. By approximating the engine as a volumetric pump, $\dot{m}_{\beta}$ can be modeled as

[TABLE]

where $V_{\mathrm{d}}$ is the displacement volume, $\omega_{\mathrm{e}}$ is the engine speed, and $\lambda_{\mathrm{vol}}$ is the engine speed dependent volumetric efficiency. By design, we can shut off single cylinders, i.e., no fuel injection and no ignition occurs. The number of active cylinders is

[TABLE]

where $N_{\mathrm{cyl}}=6$ is the number of the ICE’s cylinders. Only in those cylinders we inject the cylinder fuel mass flow $\dot{m}_{\mathrm{f,cyl}}$, that we assume to be equal for each active cylinder. The normalized air-to-fuel ratio $\lambda_{\mathrm{af}}$, defined as

[TABLE]

with $\sigma_{0}$ as the stoichiometric constant, must lie in its bounds at all times

[TABLE]

where $\lambda_{\mathrm{af}}^{\mathrm{min}}$ and $\lambda_{\mathrm{af}}^{\mathrm{max}}$ are operational bounds outside of which a proper combustion in a gasoline engine can no longer occur. The FIA regulations constrain the control input $\dot{m}_{\mathrm{f,cyl}}$ with an engine speed dependent limit, i.e.,

[TABLE]

where $\dot{m}_{\mathrm{f}}^{\mathrm{max}}\left(\omega_{\mathrm{e}}\right)$ is the speed dependent maximum fuel mass flow imposed:

[TABLE]

with $f_{\omega_{\mathrm{e}}}^{\mathrm{FIA}}$ being an affine function decreasing with engine speed, and $f^{\mathrm{FIA}}_{\mathrm{const}}$ a constant limit. The resulting total fuel mass flow reads as

[TABLE]

The amount of fuel that we inject gives us the fuel power $P_{\mathrm{f}}$ according to

[TABLE]

where $H_{l}$ is the fuel’s lower heating value, which leads to the state equation of the fuel energy $E_{\mathrm{f}}$, i.e.,

[TABLE]

The engine power due to combustion is

[TABLE]

where $\eta$ is a lumped efficiency depending on various engine quantities, and $u_{\mathrm{sa}}$ is the efficiency due to the spark advance angle retardation. Finally, we obtain the total engine power according to

[TABLE]

where $P_{\mathrm{e,fp}}$ contains the friction and pumping powers, derived in [19].

III-B Turbocharger

The turbocharger of the F1 vehicle consists of a turbine, a compressor and an electric motor (MGU-H) mounted on its shaft. The state equation according to which the rotational kinetic energy $E_{\mathrm{tc}}$ evolves reads as

[TABLE]

where $P_{\mathrm{t}}$ is the turbine power, $P_{\mathrm{c}}$ is the compressor power, and $P_{\mathrm{h}}$ is the MGU-H control input power. The turbine and compressor powers and mass flows result from experimental turbocharger maps denoted by $\mathcal{M}$, i.e.,

[TABLE]

where $\dot{m}_{\mathrm{t}}$ is the mass flow through the turbine, and $p_{\mathrm{em}}$ and $\vartheta_{\mathrm{em}}$ are the exhaust manifold pressure and temperature, respectively. As shown in Fig. 1, the turbine can be bypassed by the waste-gate, actuated by $u_{\mathrm{wg}}\in[0,1]$, with $u_{\mathrm{wg}}=0$ representing the closed position and $u_{\mathrm{wg}}=1$ the opened one.

III-C Energy Recovery System (ERS)

The ERS is composed of the MGU-K, the MGU-H and the battery. The battery energy $E_{\mathrm{b}}$ evolves according to

[TABLE]

where $f_{\mathrm{b}}$ accounts for all mechanical-to-electrical and electrical-to-mechanical losses.

III-D Vehicle Dynamics

To increase the power of the ICE or recuperate kinetic energy, we use the MGU-K, which is mounted on the crankshaft of the engine. In combination with the braking power $P_{\mathrm{brk}}$ generated by the friction brakes we obtain the final traction power

[TABLE]

where $f_{\mathrm{trac}}$ considers all friction losses and the tires’ slip, and $P_{\mathrm{k}}$ is the MGU-K power. The longitudinal dynamics of the car evolve according to the state equation

[TABLE]

where $m_{\mathrm{car}}$ is the mass of the car, which is assumed to be constant, and $P_{\mathrm{res}}$ is a lumped power considering all external powers opposing the motion (e.g., aerodynamic drag and rolling resistance). To account for the lateral dynamics, we integrate the complex interactions between tire and road into a track dependent maximum velocity $v_{\mathrm{max}}$ profile shown in the upper plot of Fig. 3 [18]. The resulting constraint reads as

[TABLE]

The maximum velocity profile is obtained by considering the throttle pedal position of the telemetry data acquired during a representative lap. At those indices where the value is below $100\text{\,}\mathrm{\char 37\relax}$, the driver is in a corner and we assume that the vehicle is at the limit of the tire’s grip, implying that the telemetry velocity represents the maximum velocity achievable. Starting from those grip-limited regions, the vehicle dynamics are simulated backwards and forwards subject to tire adhesion limits only. An alternative was presented in [49], where the tire adhesion is described through velocity dependent maximal forces. However, to keep the simulation algorithms simple, we opted for the description by means of maximum velocity. The link between the velocity of the car and the engine speed is

[TABLE]

where the overall transmission ratio $\tilde{\Gamma}$ includes the gear ratio input $\Gamma\in\{1,\dots,8\}$, the constant differential transmission ratio $\Gamma_{\mathrm{d}}$, and the wheel radius $r_{\mathrm{w}}$. This conversion needs to account for the engine speed limits

[TABLE]

where $\omega_{\mathrm{e}}^{\mathrm{min}}$ and $\omega_{\mathrm{e}}^{\mathrm{max}}$ are chosen for mechanical and regulatory reasons [2].

IV Offline Optimization

The objective of this study is to control the F1 powertrain in an optimal way to minimize the lap time. To obtain the reference trajectories we solve the lap time optimal control problem (OCP) that was presented in Appendix 1 of [19], which was formulated for the above mentioned plant. In our paper, we limit the definition of the OCP to the constraints and model equations introduced in Section III.

Problem 1**.**

The lap-time-optimal low-level control strategy satisfying the fuel and battery targets is the solution of

[TABLE]

subject to the following constraints:

[TABLE]

where $\Delta E_{\mathrm{f,target}}$ and $\Delta E_{\mathrm{b,target}}$ are design parameters determined by the race strategy representing the desired fuel and battery targets, respectively.

1 contains two integer variables, resulting in an undesired mixed-integer problem: the engaged gear and the number of active cylinders. To remove the first one, the model is convexified with respect to the gear selection using the outer convexification method [50, 51]. Next, the result is rounded applying a rounding strategy that does not violate the SOS-1 property according to the methodology proposed in [52]. A detailed explanation of this adaptation can be found in Section 3.2 of [19]. The number of active cylinders, however, is treated as continuous variable for the task of reference trajectory generation. To compute the benchmark solution used to assess the suboptimality of the online controller, we round the continuous solution and reoptimize the OCP with given integers. The OCP is implemented in Matlab using the symbolic framework CasADi [53] and discretized into a nonlinear program (NLP) using the multiple shooting method and Euler forward integration. The optimization problem is then solved with the interior point optimizer IPOPT. Computation times for a lap interval of $1000\text{\,}\mathrm{m}$ are up to 20 minutes while for a complete lap they reach 4 hours. As it will be shown in Section VI, various inputs and states resulting from the non-causal solution of 1 with a step size of $\Delta s=$2\text{,}\mathrm{m}$$ are needed in our controller architecture. In particular, we need the gear trajectory for the estimator, while we use the states as reference trajectories for the high-level (ELTMS) and the low-level (NMPC) controllers. Therefore, to comply with the schematic in Fig. 2 we define the nominal reference vector $\mathbf{r}^{\mathrm{nom}}$ as

[TABLE]

where all trajectories are not guaranteed to be globally optimal because of the nonlinear problem statement. Finally, the solution of 1 is recomputed with the energy consumption achieved by the online controller and known disturbances to obtain the benchmark solution shown in Section VII.

IV-A Notation

In the following, we consider two discrete space reference frames shown in Fig. 4.

The track reference frame with the index $i\in\{0,\dots,M\}$, where $i=M$ at the finish line, represents the discretization of the continuous path variable $s$ with a step size of $\Delta s=$2\text{,}\mathrm{m}$$. Further, in the model-predictive control setting, we use the reference frame with index $k\in\{0,\dots,N-1\}$ indicating the step inside the optimization horizon of length $N$. To link the two frames and be able to compute the position on track, we introduce $i=\tilde{i}$, being the index in the track frame at which the optimization begins (i.e., $k=0$). This instant is also called the current step.

V Driver Model

The real driver is a trained professional always acting to be as fast as possible by exploiting the full grip limits of the tires in corners, while requesting full power on straights. Since in this paper we focus on the powertrain operation, we assume the steering behavior to be optimal and the ideal racing line to be followed at all times.

In particular, the relevant human decisions are the power request $P_{\mathrm{req}}$, set as input for the controller, and the engaged gear $\Gamma$, which is known by the controller and set as input for the plant. In order to understand the behavior of the driver, we recall the two distinct regions introduced in Section I. In power-limited regions, the requested power can be overwritten by the ECU, translating to

[TABLE]

In grip-limited regions, however, the velocity of the car matches the maximum velocity profile, i.e., (20) holds with equality. The fulfillment of the power requested by the driver reads as

[TABLE]

To distinguish between those two different track regions, we introduce the binary variable $b_{\mathrm{GL}}$, which is set to 0 in power-limited regions and to 1 in grip-limited regions.

In the present paper, we model the driver with a decision making component with input/output relationship shown in Fig. 5. We need it in our framework to simulate a causal behavior of the system under given disturbances. In fact, we provide the driver model with actual reference trajectories $\mathbf{r}^{\mathrm{act}}$ that are unknown to the controller, i.e.,

[TABLE]

Additionally, we provide the driver model with the velocity $v$ of the car, and two internal variables of the controller lumped into the vector $\mathbf{u}_{\mathrm{dr}}$ defined as

[TABLE]

with $P_{\mathrm{trac}}^{\mathrm{NMPC}}$ being the traction power optimized by the NMPC, representing the driver perception. Fig. 6 illustrates the decision-making inside the driver model, adapted from [31]. In a first step, we need to check whether we were in a grip-limited region at the previous space index $i-1$. Depending on the outcome of this assessment, we determine the traction power with which we compute the velocity at space index $i+1$ by means of (19). If we come from a grip-limited region, we use the maximum traction power achievable by the power unit $P_{\mathrm{trac}}^{\mathrm{max}}$. By setting such a high value we account for the eventuality that we are exiting the grip-limited region. If we come from a power-limited region, however, we use the previously delivered traction power $P_{\mathrm{trac}}^{\mathrm{NMPC}}[i-1]$. Since in those regions the ECU can overwrite the power request of the driver, its current decisions need to be based on that effectively perceived traction power and not prior requests. The resulting velocity $v[i+1]$ is compared to the actual maximum velocity achievable on the track $v_{\mathrm{max}}^{\mathrm{act}}[i+1]$, in order to determine whether at the current step $i$ the vehicle is in grip- or power-limited region. In the former case, we need to compute the power request such that (20) holds with equality at step $i+1$. Otherwise, we set the requested power to the maximum traction power achievable. As previously mentioned, the ECU can overwrite the request, and the power output of the driver model does not influence the optimized traction power determined by the controller.

The last output of the driver is the engaged gear $\Gamma[i]$. We determine it by considering the vehicle’s velocity and an actual reference gear $\Gamma^{\mathrm{act}}[i]$ that we assume to be followed as indicated below:

In a disturbance-free scenario, the actual reference gear is equal to the nominal reference gear $\Gamma^{\mathrm{nom}}[i]$ known by the controller, whereas otherwise (as analyzed in Section VII-A) they differ.

VI Controller

In this section we present the online control framework. Fig. 7 shows an overview of the components and their interaction.

Given that we employ a nonlinear model predictive controller for the low-level actuation, we need an estimator to provide predictive information such as track properties and the driver’s possible future intentions. Also, we include a high-level supervisory controller, in the form of ELTMS, to take care of slowly changing energy budget dynamics, i.e., the battery state of charge and fuel consumption. Furthermore, since the internal combustion engine allows for individual cylinder deactivation, we need to handle integer values in our control problem. Since solving mixed-integer nonlinear problems in a receding horizon fashion is particularly computationally expensive, we determine the number of active cylinders in a feedforward manner with the feedforward cylinder controller (FCC). In the upcoming sections we dive into the details of each of those subsystems.

VI-A Estimator

Given the predictive nature of the controller, we need the driver’s future intentions and the velocity evolution for the complete NMPC horizon of length $N$. This results in the inputs and outputs shown in Fig. 8. Accordingly, the output signals consist of trajectories where deterministic information at step $i$ is augmented with predictions computed by the estimator subsystem.

In particular, the first entries of the output vectors, i.e., at the current step, are the driver’s decisions (outputs of Fig. 5) and the measured velocity, while the remaining entries are predictions denoted with a hat $\hat{(\cdot)}$, i.e.,

[TABLE]

The predictions of the requested power and the GL index are determined by looping the algorithm depicted in Fig. 6 over the NMPC horizon. The velocity prediction arises from its evolution according to (19) and the predicted requested power. In Fig. 9 we illustrate quantitatively how such a velocity prediction evolves when encountering the $v_{\mathrm{max}}^{\mathrm{act}}$ profile within the horizon.

In the corners, the predictions can be performed without loss of generality because the car is traveling at the maximum velocity of the car. On the straights, however, we have no knowledge of the NMPC’s optimized traction power in the future. Therefore, we assume the delivered traction power $P_{\mathrm{trac}}^{\mathrm{NMPC}}[i-1]$ at the previous step to be constant over the entire horizon. Given that in reality the power rather decreases towards the corner, this assumption is sensible and robust. In fact, this possibly leads to an overestimation of the future velocity and thus might result in a deviating corner entry point at the end of the straight. However, this mismatch disappears within the proposed framework due to the receding horizon property of the controller.

The last component to be predicted is the gearshift trajectory, which depends on the difference between current nominal reference gear $\Gamma^{\mathrm{nom}}[i]$ and the driver’s decision $\Gamma[i]$. If both are equal, the complete sequence for the predicted trajectory is set equivalent to the nominal one. If they differ, this could lead to the three possible scenarios shown in Fig. 10 and described below:

The driver performed a gearshift although the nominal trajectory does not show one: The future gear trajectory is set equal to the driver’s unexpected new gear, as we assume that in the short NMPC horizon of $8\text{\,}\mathrm{m}$ (see Section VI-D) not more than one shift occurs.

• 2.

The nominal trajectory shows an upshift although the driver has not shifted: We assume the driver to have missed the ideal shifting point and set the predicted upshift to occur once the engine speed has increased by a further $200\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}$.

• 3.

The nominal trajectory shows a downshift although the driver has not shifted: As before, we assume the driver to have missed the ideal shifting point. Since this scenario usually only occurs at the entry of a corner while braking, we keep the engaged gear for the entire horizon.

As a final step, we verify that the engine speed limits in (22) are satisfied: If at some point the estimated gear trajectory leads to a constraint violation, the appropriate up- or downshift is introduced and kept for the subsequent steps.

VI-B Equivalent Lap Time Minimization Strategies (ELTMS)

From an energetic point of view, the goal of our online framework is optimizing the low-level powertrain actuators while minimizing lap time and meeting the energy consumption targets introduced in Problem 1, i.e., $\Delta E_{\mathrm{f,target}}$ and $\Delta E_{\mathrm{b,target}}$.

Those energy state variables display much slower dynamics compared to the ones of the internal combustion engine [19]. Therefore, while the intake manifold pressure and the turbocharger’s kinetic energy are controlled by a short-sighted NMPC, for the slowly changing states, we employ a dedicated feedback controller structure, which can handle them in a computationally inexpensive and accurate manner. We opted for the ELTMS controller, which was previously developed in our research group. The main advantage of this supervisory controller lies in its simplicity. It relies on computationally inexpensive one-dimensional look-up tables derived with PMP [33]. Its major disadvantage is that this derivation is based on a simplified high-level model, where the engine is static, hence differs from the plant. Therefore, to use the high-level ELTMS outputs shown in Fig. 11 as reference trajectories for the NMPC, further assumptions need to be made (see Section VI-D). The outputs of the look-up tables are the fuel power $P_{\mathrm{f}}$, the MGU-K power $P_{\mathrm{k}}$ and the waste-gate position $u_{\mathrm{wg}}$. In the grip-limited regions, where the total amount of delivered power needs to match the driver’s power request, we can directly read the power split shown on the right side of Fig. 12.

In the power-limited regions instead, where we can control also the total traction power, we need to quantify where an increase in power leads to the highest time saving capabilities. Such a quantification can be defined by the kinetic costate $\lambda_{\mathrm{kin}}$ shown in the lower plot of Fig. 3, which is the position-dependent dual variable of the kinetic energy of the car. This costate reaches its negative peak at the beginning of a straight, where the lap time sensitivity with respect to propulsive power is high, and increases towards the corner where the car is already at a very high speed. As a result, on the left side of Fig. 12 we show the $\lambda_{\mathrm{kin}}$-dependent look-up table used in power-limited regions.

In [33], it was demonstrated that the power- and grip-limited look-up tables, characterized by their nodes, are fully defined by the value of the costate variables $\lambda_{\mathrm{f}}$ and $\lambda_{\mathrm{b}}$ associated with the states $E_{\mathrm{f}}$ and $E_{\mathrm{b}}$, respectively. Therefore, with changing costates, the resulting look-up tables vary accordingly, as illustrated in Fig. 12 for different values of $\lambda_{\mathrm{b}}$. Via linear interpolation between nodes it is possible to obtain a unique representation of the ELTMS outputs according to the costate evolution along the optimization. As a consequence, to track the trajectories of the energy budgets and counteract possible disturbances, we adjust the costate values by means of proportional–integral (PI) control loops, as shown in the upper part of Fig. 13. For example, if the battery trajectory differs from the nominal one, the PI controller adapts the battery costate such that the look-up tables change to compensate for the drift. Finally, to compute the predictions of the fuel power, the MGU-K power, and the waste-gate position, we assume the look-up tables to stay constant over the optimization horizon and feed them with the requested power prediction or the kinetic costate, depending on the track region (see lower part of Fig. 13).

VI-C Feedforward Cylinder Controller (FCC)

The V6 internal combustion engine of the F1 car features the ability of deactivating single cylinders individually. As introduced in (5) we include this degree of freedom by defining the number of active cylinders with the integer variable $\Psi_{\mathrm{e}}$. However, since the online computation of a mixed-integer NMPC is not tractable in a sensible time frame, we employ an algebraic subsystem that computes the number of active cylinders from a range of feasible solutions in a feedforward fashion. The inputs and outputs of the FCC are shown in Fig. 14.

In order to define a range of feasible $\Psi_{\mathrm{e}}$, we recall the linear relationship between cylinder fuel mass flow and total fuel mass flow shown in (10). The total fuel mass flow $\dot{m}_{\mathrm{f}}$ indicates how much fuel power $P_{\mathrm{f}}$ can be supplied to the ICE, as modeled in (11). In Fig. 15 we plot this linear dependency augmenting it with optimization results stemming from Problem 1, solved with many different energy budget targets and over various track intervals. We recognize that there exists a clearly defined range of $\dot{m}_{\mathrm{f}}$, and therefore of $P_{\mathrm{f}}$, for each number of active cylinders (indicated with the colored bars). This implies that a set of active cylinders can be inferred from a given $P_{\mathrm{f}}$ requested by the ELTMS. As an example, $\dot{m}_{\mathrm{f}}=0.62$ leads to three possible active cylinders, i.e., ${\Psi_{\mathrm{e}}=\{4,5,6\}}$. However, from Section III-A we know that the control input $\dot{m}_{\mathrm{f,cyl}}$ is limited by (8). This additional constraint decreases the possible number of active cylinders for $\dot{m}_{\mathrm{f}}=0.62$ at a given velocity and a certain gear ratio to ${\Psi_{\mathrm{e}}=\{5,6\}}$. By repeating this procedure over the whole prediction we obtain various feasible trajectories.

In Fig. 16 we show two possibilities for a portion of lap: the trajectory of the maximum $\Psi^{\mathrm{max}}_{\mathrm{e}}$ and minimum $\Psi^{\mathrm{min}}_{\mathrm{e}}$ number of feasible active cylinders. We see how at some indices in the acceleration phase up to three possible values arise.

For the prediction we opt for the maximum strategy, i.e.,

[TABLE]

because it provides robustness advantages whilst keeping the computational effort very low. In particular, choosing more active cylinders than optimal results in a higher fuel power according to (10) and (11). However, this can be compensated by decreasing the spark advance efficiency $u_{\mathrm{sa}}$ in the NMPC, allowing to regulate the amount of traction power according to (13). The resulting wasted power can be partially recuperated by the MGU-H. On the other hand, less than optimal active cylinders might lead to feasibility issues. In fact, given that $\Psi_{\mathrm{e}}$ directly influences the achievable traction power, less active cylinders might prevent the power matching in grip-limited regions described by (25) in cases where the MGU-K is already operating at its maximum bound.

VI-D Nonlinear Model Predictive Control (NMPC)

With the number of active cylinders and the engaged gear having already been determined, in this section we formulate the NMPC that optimizes the remaining inputs to the race car model gathered in the input vector $\mathbf{u}$ defined in (2). To correctly set up the objective of the optimization, we need to address the two track regions introduced in Section V differently. Therefore, in the following we present two distinct structures and point out their main assumptions. The power-limited NMPC is used as long as there is at least one power-limited point (i.e., at least one GL index is 0) in the NMPC horizon. This means that on a straight, at a corner entrance, and at a corner exit, the power-limited NMPC is used, while the grip-limited NMPC is employed only during a corner, i.e., when all GL indices are equal to 1. Recall Section IV-A for the index convention throughout the online control.

VI-D1 Power-limited Region – Time-optimal Control

The NMPC for the power-limited regions reads as follows:

Problem 2**.**

In power-limited regions, the lap time optimal low-level control sequence $\mathbf{u}^{\star}[\tilde{i}]$ defined by the entries of $\mathbf{u}$ at the current step $i=\tilde{i}$ is the solution of

[TABLE]

subject to the following constraints:

[TABLE]

In power-limited regions the car is solely limited by the maximum power output. These sections represent the only intervals of the track where it is possible to directly gain lap time through the operation of the power unit because we are not hitting the maximum velocity profile. From a pure lap time point of view, it is therefore optimal to maximize the velocity at each step $k$, translating into

[TABLE]

where $\Delta s$ is the step length. However, given the importance of meeting the energy targets over the complete lap, we augment the NMPC with further components. In addition to the time minimization, the objective also includes the slack variables

[TABLE]

and their weights

[TABLE]

The slack variables, which can take on only positive values, are needed to introduce the unilateral soft constraints on the system’s energy reservoirs shown in (30) $-$ (33). For example, if we have consumed more battery energy than the nominal trajectory at the end of the horizon, we penalize the deviation $\varepsilon_{E_{\mathrm{b}}}=E_{\mathrm{b}}^{\mathrm{nom}}[\tilde{i}+N-1]-{E}_{\mathrm{b}}[N-1]$ with $w_{E_{\mathrm{b}}}$. If instead we saved some battery energy, we do not influence the objective as $\varepsilon_{E_{\mathrm{b}}}$ takes on the value 0.

Given the model mismatch between NMPC and ELTMS, it is not possible to impose an equality constraint on the fuel power. Therefore, we introduce a fuel power reference tracking component in the last term of the objective, weighted by $w_{E_{\mathrm{f}}}$. Next, in (34) and (35) we see that the MGU-K power and the waste-gate position are taken from the ELTMS. This is done to avoid aggressive action by the short-sighted low-level NMPC. As an example, at the beginning of a straight, changing the MGU-K power from positive (boosting) to negative (recuperating) could fulfill the desired battery target at the end of the horizon, although being highly suboptimal from a lap time point of view. Finally, we need to account for the possibility that some points inside the horizon are already in the grip-limited region. To correctly model this effect we include (36).

VI-D2 Grip-limited Region – Power Request Realization

The nonlinear model predictive controller for the grip-limited regions reads as follows:

Problem 3**.**

In grip-limited regions, the lap time optimal low-level control sequence $\mathbf{u}^{\star}[\tilde{i}]$ defined by the entries of $\mathbf{u}$ at the current step $i=\tilde{i}$ is the solution of

[TABLE]

subject to the following constraints:

[TABLE]

While cornering we cannot directly improve the lap time since we are following the maximum velocity constraint, i.e., (20) holds with equality. Accordingly, the time component (37) is not included in the objective of the grip-limited NMPC anymore. The soft constraints in (40) $-$ (43) stay the same, as well as the waste-gate operation. In contrast to the power-limited NMPC, the MGU-K operation is now freely optimized by the low-level controller to avoid infeasibilities. This is done for the following reason: Since in grip-limited regions we have to fulfill the power requested by the driver (see (45)) and the fuel power command $\hat{P}_{\mathrm{f}}$ from the ELTMS might not respect the FIA engine-speed dependent fuel limit or the ICE dynamics, the MGU-K power can differ from the desired ELTMS value $\hat{P}_{\mathrm{k}}$. The MGU-K works as a buffer that can be adapted quickly to respond to situations where the internal combustion engine power is limited (e.g., by the fuel flow limit, the maximum achievable efficiency or the number of active cylinders).

Finally, we discuss the choice of $\Delta s$ and the NMPC horizon length. To properly capture the relevant low-level dynamics, a minimum update frequency of $10\text{\,}\mathrm{H}\mathrm{z}$ is necessary. In space domain this translates to a maximum step length determined at the critical point of lowest corner velocity. With $v^{\mathrm{min}}\approx$70\text{,}\mathrm{k}\mathrm{p}\mathrm{h} we derive a fixed step length of $\Delta s=$2\text{\,}\mathrm{m}. Moreover, to keep the average computational time within these $0.1\text{\,}\mathrm{s}$, the horizon length is set to 5 steps (resulting in an $8\text{\,}\mathrm{m}$ horizon).

VII Results

In this section we present the simulation results and assess the performance of the control architecture presented in Section VI in the presence of disturbances. First, in Section VII-A, we consider a modified gear trajectory representing a driver that shifts differently than expected. Such an investigation has high practical relevance due to the fact that the driver can never exactly reproduce a precomputed gearshift strategy. Second, in Section VII-B, we increase the maximum velocity profile in the corner by $5\text{\,}\mathrm{\char 37\relax}$ to emulate new tires with better grip. Both disturbances were introduced only over a portion of lap on the Bahrain International Circuit. To facilitate the analysis, also the evaluation of the low-level trajectories is done only over the same portion of lap. However, the optimization and the online simulation have been performed over the complete lap. The online control results are obtained with the simulation environment presented in Fig. 2 and compared to the benchmark introduced in Section IV. These trajectories are computed a posteriori to exactly match the energy consumption obtained online. The lap time difference between this non-causal solution and the online simulation represents our performance metric: the lap time suboptimality over a full lap. For both case studies we illustrate the suboptimalities, where we also included an online scenario with full knowledge of the disturbance and an equivalent case study without cylinder deactivation capability.

VII-A Differing Gearshift Scenario

In this case study we want to assess the robustness of the low-level NMPC by modifying one of the driver’s actuators.

In particular, the driver’s gear choice $\bm{\Gamma}^{\mathrm{act}}$ in Algorithm 1 is modified without altering the nominal trajectory $\bm{\Gamma}^{\mathrm{nom}}$ known by the estimator. As shown in Fig. 17, we introduce two advanced and two retarded gearshifts. These disturbances occur at the beginning of the straight, where the impact on lap time is greater. The advances and delays are set at variable spacing, reaching from $6\text{\,}\mathrm{m}$ too early to $10\text{\,}\mathrm{m}$ too late. As anticipated in the introduction to the case studies, they capture the imperfection of the human agent in the control loop. Naturally, a person cannot perfectly match a precomputed gearshift strategy and thus the timing of the shifts can differ by some tenths of a second.

Fig. 18 shows the state and input trajectories for the online control and the benchmark. The first analysis concerns the advanced upshifts, resulting in a sudden and unexpected change of scenario. As explained in Section VI-A, the estimator handles this kind of disturbance by simply adapting future shifts. A quantitative illustration of this scenario is shown in the first plot of Fig. 10 (advanced up- and downshifts are treated equally). In the case of an advanced gearshift, it keeps the newly engaged gear constant for the rest of the horizon of the controller. From a low-level perspective, the NMPC needs to handle a more complex chain of events. The disturbance introduces an unexpected drop in engine speed $\omega_{\mathrm{e}}$, leading to a sudden decrease of air mass flow entering the cylinders according to (4). To compensate for this deficit, the low-level controller tries to quickly increase the intake manifold pressure $p_{\mathrm{im}}$ by boosting with the MGU-H connected to the turbocharger, resulting in the visible peaks. Unfortunately, due to the inertia of the intake manifold this pressure increase does not entirely compensate for the lack of air mass flow instantly. Additionally, the sudden changing operating conditions of the compressor lead to potential surge issues, meaning that a further intake manifold pressure increase would cause unwanted flow instabilities. Because in a gasoline engine we cannot differ from the optimal air-to-fuel ratio significantly, the lack of air means that less fuel can be injected, resulting in less engine power $P_{\mathrm{e}}$. In the grip-limited region the lack of power is compensated by the MGU-K power $P_{\mathrm{k}}$. In the power-limited region where the MGU-K is already at its maximum power, the deficit directly affects the lap time suboptimality $\Delta T$. In addition to the previously explained pressure deficit, there is another effect decreasing $\dot{m}_{\mathrm{f}}$. As can be seen in the benchmark solution, at the instants of advanced upshift even the non-causal solution experiences a small lack of fuel mass flow. In fact, the engine speed drops below the crucial value of $10\,500\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}$, allowing for even less fuel to be injected according to (8). It is important to point out that this repercussion on $\dot{m}_{\mathrm{f}}$ cannot be avoided by means of control action in this scenario. The opposite scenario of retarded upshifts poses a different challenge for the controller. As soon as the estimator realizes that the shift does not occur in the nominal instant but rather later, we enter the retarded upshift scenario. This implies that the prediction delays the upshift by a fixed amount of engine revolutions, as quantitatively shown in the second plot of Fig. 10. From a low-level perspective, these disturbances are easier to handle, since the FIA fuel mass flow limit is not an issue and all energy adaptations needed at the gearshifts (e.g., increase of the kinetic energy of the turbocharger) are already performed. The challenge arising is the optimal usage of the energy stored in the reservoirs at the instants where a nominal gearshift is missed. For example, in the second retarded upshift interval, the turbocharger speed $\omega_{\mathrm{tc}}$ was raised to increase the intake manifold pressure and react to the gearshift. However, since the nominal shift does not occur, the rotational kinetic energy in that instant is too high for that unexpected operating condition, resulting in a suboptimal energy surplus. The NMPC stores this excess in the battery by recuperating with the MGU-H. A similar scenario can be observed in the intake manifold, where the pressure level was raised too early. The benchmark solution avoids this suboptimal scenario, by retarding the energy adaptations accordingly, representing the optimal solution (without global optimality guarantees since it is an NLP). In Fig. 19 we summarize the suboptimalities in terms of lap time over an entire lap lasting about $94\text{\,}\mathrm{s}$. The presented control architecture is $49\text{\,}\mathrm{m}\mathrm{s}$ suboptimal compared to the benchmark solution. However, this value also includes the suboptimality due to the tuning of the PI controllers inside the ELTMS, responsible for the MGU-K and ICE power cuts (see VI-B). By tuning them very aggressively, we would perform power cuts at highly suboptimal track positions to quickly compensate for energy drifts. Likewise, when relaxing the supervisory controller we could diminish the suboptimality at the cost of losing the ability of drift compensation in a reasonable amount of time. Therefore, in the presented results, the ELTMS tuning is set such that energy drifts are almost completely compensated after one lap at the latest ($\Delta E_{\mathrm{b}}\approx$0.002\text{,}\mathrm{M}\mathrm{J} and $\Delta E_{\mathrm{f}}\approx$0.5\text{\,}\mathrm{\char 37\relax}), given that we assume each lap to repeat itself. To assess the effective suboptimality due to the gearshift disturbance, we consider the online simulation with full knowledge, meaning that the first two entries of $\mathbf{r}^{\mathrm{nom}}$ in (23) are set equal to $\mathbf{r}^{\mathrm{act}}$ from (26). The difference in suboptimality between the two online simulations is $2\text{\,}\mathrm{m}\mathrm{s}$. When removing the single cylinder deactivation capability, i.e., either all 6 cylinders are active or the entire engine is switched off (as shown in Fig. 22 for the second case study), the suboptimality rises above $55\text{\,}\mathrm{m}\mathrm{s}$. Since the suboptimality difference does not increase, this suggests that the disturbance rejection in this scenario is not affected by the cylinder deactivation.

VII-B Race Trajectory Deviation

In the second case study we increase the $v_{\mathrm{max}}$ profile of a corner by $5\text{\,}\mathrm{\char 37\relax}$ to emulate higher achievable lateral accelerations, e.g., due to newer tires or more favorable track conditions. In Fig. 20 we show the velocity trajectories over another interval, including a cornering maneuver. In gray we depict the online control and in black the benchmark. Additionally, we include the nominal maximum velocity profile $v_{\mathrm{max}}^{\mathrm{nom}}$ fed to the estimator and its increased counterpart $v_{\mathrm{max}}^{\mathrm{act}}$, known and followed by the driver.

In the analysis we focus on the handling of corrupt predictive information given by the estimator and its effects on the supervisory control action. Moreover, we consider the repercussions of the cylinder deactivation on the low-level NMPC.

We can observe how the grip-limited region of the online scenario (in light gray) begins slightly later compared to the benchmark (in darker gray). This can be affiliated to the estimator, which is only aware of the lower maximum velocity trajectory $v_{\mathrm{max}}^{\mathrm{nom}}$. Given that its velocity prediction fulfills (20) with equality at an earlier index, the grip-limited region is predicted to start sooner. This leads to a lower requested power and therefore a lower velocity. As a result, the effective $v_{\mathrm{max}}^{\mathrm{act}}$ trajectory is reached later.

Nevertheless, once the grip-limited region is reached, the corrupted prediction made by the estimator is rejected in a receding horizon fashion, since the requested power $P_{\mathrm{req}}[i]$ is defined by the driver. Therefore, the increased maximum velocity profile $v_{\mathrm{max}}^{\mathrm{act}}$ is followed even though $v_{\mathrm{max}}^{\mathrm{nom}}$ is off. In Fig. 21 we show additional state and input trajectories. To explain the MGU-H power peaks optimized by the low-level controller, we need to consider the driver model. In fact, as it can be observed in the gear trajectory $\Gamma$, from a low-level perspective the disturbance is twofold: Not only the maximum velocity profile, but also the engaged gear differ from nominal. Accordingly, there is an unexpected decrease in the engine speed, influencing the air path as stated in (3). As shown in the previous case study, the electric motor on the turbocharger shaft is able to swiftly compensate such a lack of air in a close-to-optimal fashion. In the online cylinder activation phase we can observe the influence of the maximum cylinder strategy introduced in (29) and the robustness of this approach. When more cylinders are active (between $1500\text{\,}\mathrm{m}$ and $1580\text{\,}\mathrm{m}$), the MGU-K and the spark advance efficiency are used to match the requested power. Whenever one of these two actuators is at its saturation, the other one takes over and either increases or decreases the overall power unit power. It can be noticed that in the regions where the ignition is retarded, the waste-heat recuperation capability by the ERS is increased. In Fig. 19 we expose the suboptimalities in terms of lap time of this case study. With respect to the benchmark, the online simulation is $64\text{\,}\mathrm{m}\mathrm{s}$ slower, although only $17\text{\,}\mathrm{m}\mathrm{s}$ are incurred due to the introduced disturbance. This aligns with the fact that the disturbance introduced in this case study has a higher impact on the energy budget than the one introduced in Section VII-A owing to its repercussions on MGU-K and MGU-H recuperation. In fact, even after the disturbance has occurred, the lap time that is lost compared to the benchmark keeps increasing until the end of the lap. As expected, removing the cylinder deactivation as shown in Fig. 22 results in a suboptimality increase of 7-$8\text{\,}\mathrm{m}\mathrm{s}$ in both online scenarios.

VIII Conclusion

In this paper, we presented a low-level online control structure for the F1 hybrid electric vehicle. First, we proposed a mathematical model of the powertrain and introduced its offline optimization needed for reference trajectory generation and as benchmark. Second, we devised a control framework in which we split the high-level control of the slow changing energy budgets, i.e., the fuel and the battery, from the fast changing low-level power unit state variables, i.e., the intake manifold pressure and turbocharger dynamics. The high-level supervisory controller is based on PI controllers that translate online energy budget deviations into time-varying one-dimensional look-up tables, while the low-level controller is based on a track region-dependent nonlinear model predictive controller. Third, to cope with the computationally expensive and highly undesirable online control of integer decision variables, we define the cylinder deactivation strategy of the engine by means of heuristic look-up tables generated through the analysis of a high number of benchmark solutions. Finally, we integrated the control framework in a suitable simulation environment to provide a tool to assess online controllers without incurring major testing costs. In a first case study, our results showed that the online controller is able to handle disturbances introduced by the driver’s gear choice. The lap time lost over an entire lap including four differing gear shifts is $49\text{\,}\mathrm{m}\mathrm{s}$ compared to the benchmark solution. However, the nature of this suboptimality lies mainly in the tuning of the high-level ELTMS controller. When comparing the online simulation with the same control architecture, but with full knowledge of the differing gearshifts, the suboptimality is decreased to only $2\text{\,}\mathrm{m}\mathrm{s}$ over the full lap. In a second case study we analyzed a cornering scenario, where the actual maximum velocity achievable is above the predicted one, e.g., due to varying track conditions. We were able to show that the online control of the low-level actuators and the cylinder deactivation heuristic were robust and compensated for sudden unexpected changes within the system’s constraints. The suboptimality reached over a full lap is of $64\text{\,}\mathrm{m}\mathrm{s}$ compared to the benchmark, and $17\text{\,}\mathrm{m}\mathrm{s}$ compared to the online solution with perfect knowledge. For both case studies we also included the suboptimalities incurred in a scenario without single cylinder deactivation. We showed that this degree of freedom decreases the suboptimality by between 7 and $8\text{\,}\mathrm{m}\mathrm{s}$ in all scenarios.

Since the prediction of future gearshifts is of crucial importance for the control architecture, further research could focus on its improvement. Machine learning techniques considering track- and driver-dependent quantities could significantly improve the quality of the overall framework.

Acknowledgment

We would like to thank Ferrari S.p.A. for supporting this project. Moreover, we would like to express our gratitude to Dr. Ilse New for her helpful and valuable comments during the proofreading phase.