Pol Duhr

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Duhr

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Pol

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0000-0002-7708-9846

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Publications 1 - 10 of 14

Convex Performance Envelope for Minimum Lap Time Energy Management of Race Cars
Duhr, Pol; Sandeep, Ashwin; Cerofolini, Alberto; et al. (2022)
IEEE Transactions on Vehicular Technology
The optimization of the energy management of modern hybrid-electric or fully electric race cars for minimum lap time requires a description of the vehicle dynamics performance envelope, that is, of the tires' grip limit in corners, braking zones and during acceleration. In this paper, we present a computationally efficient performance envelope model in the form of convex constraints on the achievable longitudinal and lateral acceleration, on the assumption that the path on the track is given. The proposed acceleration limits are modeled velocity-dependent to take into account the effect of aerodynamic downforce present in many circuit race cars. The formulation as linear equality, inequality and second-order cone constraints allows to embed the model in a convex energy management optimization framework. To showcase the approach, we identify the model with data obtained from a state-of-the-art hybrid-electric Formula 1 car and present results for the Silverstone and Spa-Francorchamps circuits. The optimal energy management strategies can be evaluated with a computational time of less than 1 s. The optimal velocity profile subject to the performance envelope constraints is close to the measured one. The good agreement between the optimal solution and the measurement data shows that the proposed model captures the vehicle dynamics accurately enough for the purposes of energy management optimization.
Time-optimal Energy Management of the Formula 1 Power Unit with Active Battery Path Constraints
Duhr, Pol; Schaller, Maximilian; Arzilli, Luca; et al. (2021)
2021 European Control Conference (ECC)
Modern Formula 1 racing cars are highperformance hybrid-electric vehicles whose battery acts as an energy storage. When the powertrain is operated close to the lower or upper state-of-charge bound of the battery, its finite size limits the electric boosting and recuperation capacity, respectively. Given the detrimental effect on the achievable lap time, such scenarios call for a careful optimization of the energy management strategies. Based on a convex model of the car’s powertrain, we first study the impact of battery path constraints on the optimal control policy analytically, using a non-smooth version of Pontryagin’s minimum principle. We then corroborate the derivations with the numerical solution obtained from a convex optimization framework and discuss the time-optimal energy management strategy when the lower bound on the battery state-of-charge is active. Finally, we leverage the non-causal results to improve an existing online controller in the case of an overtake maneuver. Our simulations yield a lap time gain of about 370 ms over three laps. ©2021 EUCA
Time-Optimal Low-Level Control and Gearshift Strategies for the Formula 1 Hybrid Electric Powertrain
Balerna, Camillo; Neumann, Marc-Philippe; Robuschi, Nicolò; et al. (2021)
Energies
Today, Formula 1 race cars are equipped with complex hybrid electric powertrains that display significant cross-couplings between the internal combustion engine and the electrical energy recovery system. Given that a large number of these phenomena are strongly engine-speed dependent, not only the energy management but also the gearshift strategy significantly influence the achievable lap time for a given fuel and battery budget. Therefore, in this paper we propose a detailed low-level mathematical model of the Formula 1 powertrain suited for numerical optimization, and solve the time-optimal control problem in a computationally efficient way. First, we describe the powertrain dynamics by means of first principle modeling approaches and neural network techniques, with a strong focus on the low-level actuation of the internal combustion engine and its coupling with the energy recovery system. Next, we relax the integer decision variable related to the gearbox by applying outer convexification and solve the resulting optimization problem. Our results show that the energy consumption budgets not only influence the fuel mass flow and electric boosting operation, but also the gearshift strategy and the low-level engine operation, e.g., the intake manifold pressure evolution, the air-to-fuel ratio or the turbine waste-gate position.
Long-term stochastic model predictive control for the energy management of hybrid electric vehicles using Pontryagin’s minimum principle and scenario-based optimization
Ritter, Andreas; Widmer, Fabio; Duhr, Pol; et al. (2022)
Applied Energy
This paper presents a new approach to efficiently integrate long prediction horizons subject to uncertainty into a stochastic model predictive control (MPC) framework for the energy management of hybrid electric vehicles. By exploiting Pontryagin’s minimum principle, we show that the energy supply required to obtain a certain change in the state of charge (SOC) of the battery can be approximated using a quadratic equation. The parameters of these mappings depend on the power request imposed by the driving mission and thus allow to compress the time-resolved power profile into only three scalar variables. Having a driving mission divided into several segments of arbitrary length, the corresponding sequence of quadratic approximations allows to reformulate the original energy management problem as a quadratic program, which offers an efficient way to include a large number of future scenarios. The resulting scenario-based stochastic MPC approach prevents SOC boundary violations with a certain probability, which can be controlled by the number of scenarios considered. To validate the quadratic approximation, we study two numerical examples using two different vehicles, a series hybrid electric passenger car and a battery-assisted trolley bus. Finally, a case study based on the operation of the latter is provided, which demonstrates the expected behavior and the real-time capability of the stochastic MPC approach. While the SOC is maintained in predefined boundaries with high probability, the required energy supply is only increased by 1.41% compared to the non-causal optimum.
Minimum-fuel Engine On/Off Control for the Energy Management of a Hybrid Electric Vehicle via Iterative Linear Programming
Robuschi, Nicolò; Salazar, Mauro; Duhr, Pol; et al. (2019)
IFAC-PapersOnLine ~ 9th IFAC Symposium on Advances in Automotive Control, AAC 2019. Proceedings
Analysis of optimal energy management strategies for the hybrid electric Formula 1 car under consideration of the finite battery size
Duhr, Pol; Schaller, Maximilian; Arzilli, Luca; et al. (2021)
Contemporary Formula 1 racing cars feature a high-performance hybrid-electric powertrain. Beside the fuel tank, the battery is a second on board energy storage. The energy management strategy in terms of battery deployment must be carefully optimized in order to minimize the lap time on a given race circuit. In particular, the finite size of the battery must be taken into account, since the electric boosting and recuperation capabilities of the powertrain are restricted when the battery is depleted and fully charged, respectively. So far, this problem has scarcely been investigated in a time-optimal racing context. Using a previously developed convex optimization framework, we study the optimal solution to the energy management problem when either the lower or the upper bound on the battery state-of-charge is attained. First, we show that the operating strategy differs substantially for these two cases: Whilst it is optimal to hit the upper bound only in one particular time instant and then discharge the battery again, it can be kept at the lower bound for prolonged sections of the lap. We highlight that these differences are related to the interaction between the two electric motor-generator units of the powertrain. Second, based on Pontryagin’s minimum principle, we analyze the trajectory of the costate variable associated with the battery energy, which in such scenarios is crucial for the parameterization of an optimal control policy. The results underline the importance of correctly considering the cross-couplings between the battery deployment and the limit on electric recuperation imposed by the technical regulations.
Minimum-Race-Time Energy Allocation Strategies for the Hybrid-Electric Formula 1 Power Unit
Duhr, Pol; Buccheri, Daniele; Balerna, Camillo; et al. (2023)
IEEE Transactions on Vehicular Technology
The hybrid-electric powertrain currently used in Formula 1 race cars draws its energy from the car's fuel tank and battery. The usable battery size is limited, and refueling during a race is forbidden by the regulations of the Formula 1 race series. From a strategic point of view, lap-by-lap targets for the fuel and battery consumption must be chosen and imposed on the energy management controller of the car. This task is non-trivial due to the influence of the on-board fuel mass on the achievable lap time, as well as the cross-couplings between the electric and the combustion part of the powertrain. A systematic approach is thus required to compute the energy allocation strategy that minimizes the total race time. In this paper, we devise an optimization framework in the form of a non-linear program, yielding the optimal battery and fuel consumption targets for each lap of the race. The approach is based on maps that capture the achievable lap time as a function of car mass and allocated battery and fuel energy. These maps are generated beforehand with a model-based single-lap optimization framework and fitted using artificial neural network techniques. To showcase the approach, we present three case studies: First, we compare the optimal strategy to a heuristic method. The improvement of 2s over the entire race is substantial, given that the difference only lies in the energy allocation, but not in the overall consumption. It underlines the importance of optimizing the energy allocation. Second, we leverage the framework to compute the optimal fuel load at the beginning of the race. Finally, we apply the developed non-linear program in a shrinking-horizon fashion. Our simulation results show that the resulting model predictive controller correctly reacts to disturbances that frequently occur during a race.
Time-optimal gearshift and energy management strategies for a hybrid electric race car
Duhr, Pol; Christodoulou, Grigorios; Balerna, Camillo; et al. (2021)
Applied Energy
Modern Formula 1 race cars are hybrid electric vehicles equipped with an internal combustion engine and an electric energy recovery system. In order to achieve the fastest possible lap time, the components’ operation must be carefully optimized, and the energy management must account for the impact of the gearshift strategy on the overall performance. This paper presents an algorithm to calculate the time-optimal energy management and gearshift strategies for the Formula 1 race car. First, we leverage a convex modeling approach to formulate a mathematical description of the powertrain including the gearbox, preserving convexity for a given engine speed trajectory. Second, we devise a computationally efficient algorithm to compute the energy management and gearshift strategies for minimum lap time, under consideration of given fuel and battery consumption targets. In particular, we combine convex optimization, dynamic programming and Pontryagin’s minimum principle in an iterative scheme to solve the arising mixed-integer optimization problem. We showcase our algorithm with a case study for the Bahrain racetrack, underlining the interactions between energy management and gear selection. Finally, we use our approach as a benchmark to evaluate the sub-optimality of a heuristic gearshift rule. Our results show that using an optimized engine speed threshold for upshifts can yield close-to-optimal results. However, already deviations smaller than 4% from the best possible threshold can increase lap time by more than 100 ms, highlighting the importance of jointly optimizing energy management and gearshift strategies.
Convex modelling for ship speed optimisation
van Dooren, Stijn; Duhr, Pol; Onder, Christopher H. (2023)
Ocean Engineering
Optimising the ship speed depending on the forecast oceanic and atmospheric conditions is an effective operational measure to reduce both fuel consumption and carbon dioxide emissions. In the literature, this optimal control problem (OCP) has been formulated using different types of mathematical models and solved using various optimisation methods. In this paper, we use convex functions to reformulate the models and cast the OCP as a convex optimisation problem. This type of problem can be solved very efficiently and yields the globally optimal solution. As a first step, we assume that the environmental conditions do not vary with time and only depend on the distance along the route. For this case, we show that the convex reformulation of the OCP is accurate and the computation time to solve it is low. Second, we introduce an iterative method to solve the problem under time-varying conditions, drawing inspiration from the literature on hybrid electric vehicles. Whilst this method converges quickly, it does not converge to the global optimum, which was calculated using dynamic programming. We present an artificial scenario to explain why convergence to the global optimum is not guaranteed. Finally, we discuss the implications of our findings for potential future research on this topic.
Minimum Lap Time Control of Hybrid Electric Race Cars in Qualifying Scenarios
Salazar Villalon, Mauro; Duhr, Pol; Balerna, Camillo; et al. (2019)
IEEE Transactions on Vehicular Technology

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