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

1. Problem Statement

The paper addresses the challenge of minimizing battery energy consumption in electric vehicles (EVs) equipped with multiple independent motors (specifically a four-motor configuration) during straight-line driving.

• The Core Issue: While offline optimization can generate lookup tables or rules for torque distribution, these lack adaptability to changing vehicle parameters, environmental conditions, or real-time constraints (e.g., varying torque limits or comfort requirements).
• The Optimization Challenge: Online optimization (e.g., Model Predictive Control) requires accurate loss models. However, standard convex relaxations often introduce large modeling errors, while non-convex models (like high-order polynomials or neural networks) risk getting stuck in local minima or overfitting noisy data, making real-time global optimization difficult.
• Goal: Develop a computationally efficient, real-time feasible torque allocation strategy that accurately models motor losses, ensures global optimality (or near-global), and maintains robustness against sparse or noisy data.

2. Methodology

The proposed approach combines Sum of Squares (SOS) programming with pseudoconvex polynomial regression to create a robust loss model, which is then used in a constrained nonlinear optimization framework.

A. Motor Loss Modeling via Pseudoconvex Polynomials

• Polynomial Approximation: Instead of using simple cubic polynomials or lookup tables, the authors approximate motor losses ($P_{loss}$) as a function of torque ($\tau$) using higher-order polynomials.
• Pseudoconvexity Constraints: To ensure the optimization landscape is well-behaved, the authors enforce that the loss function is pseudoconvex. For a univariate function, this implies the function is monotonically increasing for $x > x^*$ (where $x^*$ is the global minimizer). Since motor losses increase with torque, the function is constrained to be monotonically increasing for non-negative torques.
• Sum of Squares (SOS) Formulation: Enforcing monotonicity on arbitrary-order polynomials is mathematically difficult. The authors use SOS programming to relax the non-negativity constraint on the derivative of the polynomial.
  • The condition $p'(x) \geq 0$ is replaced by requiring $p'(x)$ to be representable as a sum of squares (or $p'(x) - x \cdot s(x)$ being SOS for the interval $[0, \infty)$).
  • This transforms the fitting problem into a Semidefinite Program (SDP), which can be solved efficiently using modern solvers (e.g., MOSEK via YALMIP).
• Benefits: This method guarantees the physical properties of the loss function (positivity and monotonicity) even with noisy or sparse experimental data, preventing the "oscillatory" behavior often seen in unconstrained high-order polynomial fits.

B. Torque Allocation Strategy

• Problem Formulation: The vehicle is modeled with four independent motors. For straight-line driving, the optimal left-right split is assumed to be equal, reducing the problem to finding the optimal front-rear torque distribution ratio ($\sigma$).
• Optimization: The objective is to minimize the total powertrain loss:
  $$\min_{\sigma} P_{loss,f}(\sigma \cdot \tau_{ref}) + P_{loss,r}((1-\sigma) \cdot \tau_{ref})$$
• Solution Methods:
  1. Gradient-Based Solver: The problem is solved as a constrained nonlinear optimization problem using the Fatrop solver.
  2. KKT-Based Approach: For cases where the cost function is polynomial, the authors utilize Karush-Kuhn-Tucker (KKT) conditions. By finding the roots of the gradient polynomial, they can identify all potential local minima and select the global optimum, avoiding the suboptimal solutions common in gradient-based methods with random initialization.

3. Key Contributions

1. SOS-Based Loss Approximation: A novel method for fitting electric motor losses using arbitrary-order polynomials constrained to be positive and monotonically increasing via Sum of Squares programming. This bridges the gap between the accuracy of high-order models and the robustness of convex methods.
2. Real-Time Torque Allocation: A gradient-based allocation strategy that utilizes these pseudoconvex models, enabling real-time execution (approx. 3ms) suitable for integration with other vehicle control systems.
3. Global Optimality via KKT: A demonstration that for polynomial cost functions, using KKT conditions to find roots can effectively eliminate suboptimal local minima, significantly improving performance in complex scenarios (e.g., unequal motors).
4. Handling Unequal Motors: The framework is extended to powertrains with different front and rear motors, a scenario where simple rule-based strategies often fail.

4. Results

The method was evaluated on a modified Chevrolet Volt EREV simulation using standard certification driving cycles (WLTP, UDDS, US06, HWFET, NEDC).

• Fitting Accuracy: The pseudoconvex polynomials (up to 10th order) successfully captured motor loss characteristics without the oscillations seen in unconstrained high-order fits, even with low-resolution data.
• Energy Consumption:
  • Equal Motors: The proposed methods showed a slight energy penalty (approx. 1.0% – 1.8%) compared to a global grid-search benchmark.
  • Unequal Motors: The penalty was slightly higher for unconstrained polynomials but remained low for the pseudoconvex approach.
  • KKT Enhancement: When using the KKT-based root-finding method, the energy consumption penalty dropped significantly, especially for unequal motors (reducing the gap to the benchmark to < 0.6%).
• Computational Efficiency: The gradient-based solver achieved a mean execution time of ~3 ms (max ~15 ms) on standard PC hardware, confirming real-time feasibility for a 1-second control loop.

5. Significance and Conclusion

This paper presents a robust solution for the "curse of dimensionality" and non-convexity in EV torque allocation. By leveraging Sum of Squares programming, the authors create a loss model that is both mathematically tractable (ensuring monotonicity and positivity) and computationally efficient.

The significance lies in the ability to move away from static, offline-generated lookup tables toward dynamic, online optimization that can adapt to varying constraints and vehicle configurations in real-time. The small energy penalty (approx. 1.5% average increase) is deemed acceptable given the substantial gains in flexibility, adaptability, and the ability to integrate with other advanced vehicle control systems. The use of KKT conditions further refines the solution, offering a pathway to near-global optimality without the computational cost of exhaustive grid searches.