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A Modelling Framework for Energy-Management and Eco-Driving Problems using Convex Relaxations

Y. J. J. Heuts, M. C. F. Donkers

TL;DR

The paper addresses eco-driving and vehicle energy management by formulating them as networks of energy buffers and converters with linear dynamics ($x_{k+1}=A x_k + B u_k + f_k$) and node constraints ($E x_k + F u_k + G y_k + s_k = v_k$). It shows that nonlinear converter equalities $h_m(x,u,y)=0$ can be relaxed to convex inequalities, and, under mild linear-independence and monotonicity conditions, the relaxed problem yields the same global optimum as the original nonconvex formulation; some converter models admit Second-Order Cone Program (SOCP) reformulations for efficient solving. A simulation study on an eco-driving example demonstrates exactness of the relaxation and highlights a regularization strategy to enforce equality when needed, while preserving the optimal cost. Overall, the framework provides a scalable, solver-friendly approach that unifies eco-driving and complete vehicle energy management with provable global optimality guarantees and practical SOCP implementations.

Abstract

This paper presents a convex optimization framework for eco-driving and vehicle energy management problems. We will first show that several types of eco-driving and vehicle energy management problems can be modelled using the same notions of energy storage buffers and energy storage converters that are connected to a power network. It will be shown that these problems can be formulated as optimization problems with linear cost functions and linear dynamics, and nonlinear constraints representing the power converters. We will show that under some mild conditions, the (non-convex) optimization problem has the same (globally) optimal solution as a convex relaxation. This means that the problems can be solved efficiently and that the solution is guaranteed to be globally optimal. Finally, a numerical example of the eco-driving problem is used to illustrate this claim.

A Modelling Framework for Energy-Management and Eco-Driving Problems using Convex Relaxations

TL;DR

The paper addresses eco-driving and vehicle energy management by formulating them as networks of energy buffers and converters with linear dynamics () and node constraints (). It shows that nonlinear converter equalities can be relaxed to convex inequalities, and, under mild linear-independence and monotonicity conditions, the relaxed problem yields the same global optimum as the original nonconvex formulation; some converter models admit Second-Order Cone Program (SOCP) reformulations for efficient solving. A simulation study on an eco-driving example demonstrates exactness of the relaxation and highlights a regularization strategy to enforce equality when needed, while preserving the optimal cost. Overall, the framework provides a scalable, solver-friendly approach that unifies eco-driving and complete vehicle energy management with provable global optimality guarantees and practical SOCP implementations.

Abstract

This paper presents a convex optimization framework for eco-driving and vehicle energy management problems. We will first show that several types of eco-driving and vehicle energy management problems can be modelled using the same notions of energy storage buffers and energy storage converters that are connected to a power network. It will be shown that these problems can be formulated as optimization problems with linear cost functions and linear dynamics, and nonlinear constraints representing the power converters. We will show that under some mild conditions, the (non-convex) optimization problem has the same (globally) optimal solution as a convex relaxation. This means that the problems can be solved efficiently and that the solution is guaranteed to be globally optimal. Finally, a numerical example of the eco-driving problem is used to illustrate this claim.
Paper Structure (11 sections, 2 theorems, 34 equations, 7 figures, 1 table)

This paper contains 11 sections, 2 theorems, 34 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Modelling framework
  3. General description
  4. Optimal Control Problem
  5. Examples
  6. Solution Framework
  7. Global Solutions using Convex Relaxations
  8. Enforcing Exact Relaxations
  9. Second-order Cone Formulation
  10. Simulation study showing the exactness of relaxations
  11. Conclusion

Key Result

Lemma 4

Optimization problem eq:NLLP_rel satisfies linear independence constraint qualification (LICQ), leading to $d_{\mathrm{CR}} = p_{\mathrm{CR}}$, if Requirements ii,iv,v) are satisfied.

Figures (7)

  • Figure 1: Schematic of a vehicle power network.
  • Figure 2: Block diagram of eco-driving problem
  • Figure 3: Block diagram power network series hybrid vehicle
  • Figure 4: Two common forms of convertor models $h_m = 0$. It can be seen that for $h_1$, we have $\frac{\partial y}{\partial u}\geqslant 0$ and $\frac{\partial y}{\partial u}\geqslant 0$ for $h_2$.
  • Figure 5: Velocity profile resulting from solving the eco-driving problem subject to a given height map. The velocity is given in $m/s$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof