Convex Optimization Framework for Eco-Driving and Vehicle Energy Management Problems

Core Concepts

A convex optimization framework is presented for solving eco-driving and vehicle energy management problems, which can be formulated as optimization problems with linear cost functions, linear dynamics, and nonlinear converter constraints. Under mild conditions, the non-convex optimization problem has the same globally optimal solution as a convex relaxation.

Abstract

The paper presents a general modeling framework for eco-driving and vehicle energy management problems, where the system is represented as a network of energy buffers and energy converters. The optimization problem aims to minimize the total energy usage over a time horizon, subject to the input-output behavior of the energy converters and the dynamics of the energy buffers.
The key insights are:

The eco-driving and energy management problems can be formulated using the same modeling framework, consisting of energy buffers, energy converters, and a power network.
Under mild conditions on the converter models and the power network, the non-convex optimization problem can be relaxed into a convex optimization problem, which can be solved efficiently and guarantees a globally optimal solution.
The conditions for the convex relaxation to be exact are related to the monotonicity of the converter models and the linear independence of the constraints.
For certain classes of converter models, the convex relaxation can be formulated as a second-order cone program, which can be solved using off-the-shelf solvers.
A numerical example of the eco-driving problem is provided, demonstrating the effectiveness of the proposed convex optimization framework.

Stats

The vehicle mass is 13,400 kg.
The sampling distance is 5 m.
The discretized system matrices are A = 0.9981 and B = 0.005.

Quotes

"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."
"We will show that under some mild conditions, the (non-convex) optimization problem has the same (globally) optimal solution as a convex relaxation."

Key Insights Distilled From

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

by Y.J.J. Heuts... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00447.pdf

Deeper Inquiries

How can the proposed framework be extended to handle more complex vehicle dynamics, such as nonlinear longitudinal models or multi-dimensional vehicle models?

The proposed framework can be extended to handle more complex vehicle dynamics by incorporating higher-order dynamics and additional state variables into the optimization problem. For nonlinear longitudinal models, the converter models can be adapted to capture the nonlinear relationships between inputs and outputs. This may involve using higher-order terms or non-polynomial functions in the converter models to represent the dynamics more accurately. Additionally, for multi-dimensional vehicle models, the framework can be expanded to include multiple subsystems representing different aspects of the vehicle's behavior, such as propulsion, energy storage, and auxiliary systems. By defining appropriate energy storage buffers and converters for each subsystem and connecting them through a power network, the framework can accommodate multi-dimensional vehicle models.

What are the potential challenges and limitations of the convex relaxation approach when dealing with larger-scale energy management problems involving multiple vehicles or complex transportation networks?

When dealing with larger-scale energy management problems involving multiple vehicles or complex transportation networks, there are several challenges and limitations associated with the convex relaxation approach. One challenge is the computational complexity of solving large-scale convex optimization problems, which can increase significantly as the number of vehicles or network nodes grows. This can lead to longer computation times and resource-intensive calculations, especially when considering real-time applications or dynamic scenarios.
Another limitation is the scalability of the convex relaxation approach, as the relaxation may not always guarantee the exact solution to the original non-convex problem. In complex transportation networks with diverse vehicle types and interactions, the convex relaxation may oversimplify the problem, leading to suboptimal solutions or inaccuracies in modeling the system dynamics. Additionally, incorporating constraints specific to each vehicle or network component can introduce additional complexity and may require customized formulations for each scenario.
Furthermore, the convex relaxation approach may struggle to capture the full complexity of interactions between multiple vehicles or network components, especially in scenarios where non-linearities or uncertainties play a significant role. Ensuring the convexity of the optimization problem while addressing these complexities can be a challenging task, requiring careful consideration of the model assumptions and constraints.

How can the framework be adapted to incorporate additional objectives, such as emissions reduction or travel time minimization, while maintaining the convexity and global optimality properties?

To incorporate additional objectives such as emissions reduction or travel time minimization into the framework while maintaining convexity and global optimality properties, these objectives can be included as part of the cost function in the optimization problem. By formulating the objectives as linear or convex functions of the decision variables, the overall optimization problem remains convex and can be solved efficiently using convex optimization techniques.
For emissions reduction, emission models can be integrated into the converter models or included as separate constraints in the optimization problem. These models can capture the relationship between energy consumption and emissions, allowing for the optimization of energy management strategies that minimize emissions while meeting performance requirements.
Similarly, for travel time minimization, additional constraints related to travel time or vehicle speed can be incorporated into the optimization problem. By defining the trade-off between energy efficiency, emissions, and travel time within the cost function, the framework can simultaneously optimize multiple objectives while ensuring convexity and global optimality.
Regularization techniques can also be employed to balance the importance of different objectives and prevent overfitting to specific objectives. By adjusting the regularization parameters, the framework can adapt to varying priorities and objectives while maintaining the overall convexity and optimality of the solution.