Energy-efficient torque allocation for straight-line driving of electric vehicles based on pseudoconvex polynomials

TL;DR

The paper addresses energy-efficient torque allocation for a four-motor EV by modeling motor losses with high-order pseudoconvex polynomials constrained via sum of squares to enforce positivity and monotonicity, enabling real-time gradient-based optimization. The approach combines SOS-based polynomial fitting, a gradient-driven torque-distribution strategy, and comparisons with KKT-based and grid-search methods. Key findings show that pseudoconvex polynomial loss models maintain robust accuracy with noisy or sparse data, yield real-time performance (typical runtimes in the millisecond range), and achieve only modest energy increases relative to grid-search baselines on standard driving cycles, while offering better integration potential with other vehicle controls. The work demonstrates practical impact by enabling flexible, online energy optimization for EV powertrains with both equal and unequal motors, and it outlines pathways for further improvements via KKT-based pruning and multivariate loss modeling.

Abstract

Electric vehicles with multiple motors provide a flexibility in meeting the driver torque demand, which calls for minimizing the battery energy consumption through torque allocation. In this paper, we present an approach to this problem based on approximating electric motor losses using higher-order polynomials with specific properties. To ensure a well-behaved optimization landscape, monotonicity and positivity constraints are imposed on the polynomial models using sum of squares programming. This methodology provides robustness against noisy or sparse data, while retaining the computational efficiency of a polynomial function approximation. The torque allocation problem based on such polynomials is formulated as a constrained nonlinear optimization problem and solved efficiently using readily available solvers. In the nominal case, the first-order necessary conditions for optimality can also be used to obtain a global solution. The performance of the proposed method is evaluated on several certification driving cycles against a grid search-based benchmark. Results show a modest influence on electric energy consumption, while enabling real-time optimization and integration with other vehicle control systems.

Figures (8)

• Figure 1: The powertrain configuration considered in this paper
• Figure 2: Electric loss maps for the two considered motors
• Figure 3: Polynomial fits of 10th degree, using loss map 1 at $\omega_\mathrm{m}=300$ rad/s
• Figure 4: Polynomial fitting results using data from Fig. 4 in dizqah2016, at 105 km/h
• Figure 5: Total powertrain losses for varying torque demands at $\omega_\mathrm{w}=50$ rad/s; the optimal torque distribution is marked with a star