Human-in-the-loop Energy and Thermal Management for Electric Racing Cars through Optimization-based Control

Imagine you are driving a high-performance electric race car. You have a full tank of "electric juice" (battery), but you can't stop to refuel often because charging takes too long. Your goal is to drive as fast as possible for a set number of laps without running out of power or overheating your engine.

This paper is about a smart co-pilot designed to help the human driver do exactly that.

Here is the breakdown of how it works, using simple analogies:

1. The Problem: The "Perfect" vs. The "Real" Driver

In a perfect world, a computer could calculate the exact speed you should be going at every single meter of the track to finish the race in the shortest time possible. It would tell you to accelerate, slow down, and coast in a perfectly smooth, flowing curve.

The Analogy: Imagine a dance instructor telling you to move your arms in a perfectly smooth, continuous wave. It looks beautiful on paper, but in a real race, a human driver can't follow those tiny, constant adjustments. It's too distracting and unsafe. Also, racing rules often say, "You can't just override the driver's foot on the pedal."

The Solution: The system simplifies the instructions. Instead of a smooth wave, it gives the driver a simple, binary command: "Floor it!" (Full throttle) or "Let it roll!" (Coast/No throttle). It's like a traffic light for the driver: Green means go, Red means glide.

2. The Brain: The "Crystal Ball" (Optimization)

Before the race starts (or during a pit stop), the computer acts like a crystal ball. It looks at the entire track ahead (about 47 kilometers) and calculates the mathematically perfect plan.

The Magic Number: It calculates a "sensitivity score" (called a co-state) for every part of the track. Think of this score as a "Value of Speed."
- If the score is high, it means slowing down now will save you a lot of time later (maybe because you are about to hit a long straight where you can regain speed easily).
- If the score is low, it means you should keep your foot down because slowing down now would cost you too much time.

3. The Tuning Knob: The "Bisection Algorithm"

The computer knows the perfect plan, but it needs to translate that into a simple "Coast" signal for the driver. It does this by finding a specific Threshold.

The Analogy: Imagine you are trying to find the perfect temperature for a shower. You turn the knob a little bit, test it, turn it a little more, and test again.

The computer uses a method called Bisection (splitting the difference) to find the exact "Value of Speed" threshold where the driver should start coasting.
It asks: "If I tell the driver to coast when the score is above 50, do we finish too slow? What if I say 40? What if 45?"
It quickly narrows down the perfect number so the driver gets the signal at the exact right moment.

4. The Safety Net: The "Feedback Loop"

Races are messy. You might get a draft from another car (wind resistance drops), your tires might get slippery, or a safety car might slow everyone down. The perfect plan from the start might no longer be perfect.

The Analogy: Think of this like a cruise control system that learns.

If the car is using too much energy compared to the plan, the system says, "Hey, we need to coast a bit earlier next time."
If the car is using too little, it says, "We can push a bit harder."
It uses a PI Controller (Proportional-Integral), which is just a fancy way of saying it corrects small mistakes immediately and fixes big, lingering mistakes over time.

5. The Results: How Good is it?

The researchers tested this system on a real electric race car simulation (InMotion) on the Zandvoort track.

The "Perfect" Computer vs. The "Human-Friendly" System: The simplified "Floor it or Coast" system was only 0.05% to 0.22% slower than the impossible, perfect computer plan. That is a difference of a few hundredths of a second over a whole race!
Robustness: Even when they threw "disturbances" at it (like a safety car slowing the race down or tires getting worn out), the system adapted and kept the car on track.

Summary

This paper presents a smart assistant for electric race car drivers. It takes a complex, super-smart mathematical plan and translates it into simple, safe, and easy-to-follow instructions ("Go" or "Glide").

It ensures the car doesn't run out of battery or overheat, while keeping the driver in control. It's like having a genius navigator who doesn't try to steer the car for you, but simply taps you on the shoulder at the exact right moment to say, "Time to save some juice," ensuring you cross the finish line as fast as possible.

Here is a detailed technical summary of the paper "Human-in-the-loop Energy and Thermal Management for Electric Racing Cars through Optimization-based Control."

1. Problem Statement

Electric racing cars face a critical trade-off between maximizing lap speed and adhering to strict energy budgets and thermal constraints (battery and motor temperatures). Unlike autonomous systems, electric race cars must be driven by humans, which introduces specific challenges:

Driver Constraints: Racing regulations often prohibit the direct overwriting of driver power requests. Drivers cannot be forced to follow complex, smooth power trajectories generated by optimal control algorithms. Instead, they require simple, actionable signals (e.g., "lift-off" or "coast" indicators).
Real-Time Execution: Energy management strategies must run in real-time on the car's Electronic Control Unit (ECU) to adapt to disturbances (e.g., tire degradation, drafting, safety cars) and model errors.
Thermal Management: Efficient driving requires managing battery and motor temperatures to allow for rapid charging during pit stops, yet many existing optimization methods ignore thermal limits.
Computational Complexity: Global optimal control for an entire race stint (approx. 47 km) is computationally expensive, often taking minutes to solve, which is too slow for real-time application.

2. Methodology

The authors propose a robust, real-time capable control framework that bridges the gap between global optimization and human drivability. The methodology consists of three main components:

A. Convex Optimization (Offline/Slow Loop)

The core of the system relies on a simplified convex model of the electric powertrain (battery, inverter, motor, and vehicle dynamics) formulated in the space domain.

Objective: Minimize stint time (integral of $dt/ds$ ) subject to energy, thermal, and kinetic energy constraints.
Solver: Uses Second-Order Conic Programming (SOCP) to find a globally optimal solution in approximately 2.5 seconds for a 47-km horizon.
Output: This yields a smooth, optimal power trajectory and the kinetic energy co-state ( $\lambda_{kin}$ ), which represents the sensitivity of the total stint time to changes in kinetic energy.

B. Lift-and-Coast Adaptation (Human-in-the-Loop)

To make the solution drivable, the authors enforce a "Maximum-Power-or-Coast" constraint. The driver is either at 100% throttle or coasting (0% throttle) when not grip-limited.

Threshold Tuning: Instead of solving a complex mixed-integer problem, the system uses the kinetic co-state trajectory ( $\lambda_{kin}$ ) from the convex solution. A threshold value ( $\lambda^*$ ) is tuned such that if $\lambda_{kin}(s) \geq \lambda^*$ , the driver is signaled to coast.
Bisection Algorithm: An online bisection algorithm iteratively adjusts $\lambda^*$ to find the optimal coasting points that satisfy the energy budget while minimizing time loss.
Robustification: The co-state trajectory is smoothed between corner apexes and straights to prevent the driver from receiving conflicting signals (e.g., being told to coast while still accelerating).

C. Model Predictive Control (MPC) & Feedback

The framework operates as a shrinking-horizon MPC:

Re-optimization: Every few seconds, the system re-solves the convex problem based on current sensor measurements (battery state, temperature, position).
PI Feedback: A Proportional-Integral (PI) controller adjusts the threshold $\lambda^*$ in real-time based on the error between the predicted and actual state trajectories (energy and temperature usage). This compensates for disturbances like drafting or tire wear.
Safety Logic: Coasting signals are only issued when the driver is at full throttle (not grip-limited) to prevent instability (e.g., lift-off oversteer).

D. Implementation Variants

The paper evaluates three implementation variants to assess computational feasibility:

Fully Online: Runs the convex solver and bisection algorithm on the car's ECU.
Fixed Co-state: Loads a pre-computed co-state trajectory; only the bisection threshold is tuned online.
Fixed Co-state & Threshold: No online optimization; relies entirely on feedback control based on energy usage deviations.

3. Key Contributions

Real-Time Human-in-the-Loop Control: A novel framework that translates complex global optimal control solutions into simple, safe, and actionable "lift-and-coast" signals for human drivers.
Thermal-Aware Optimization: Unlike many racing energy management studies, this approach explicitly models and optimizes for battery and motor temperatures to maximize charging efficiency.
Computational Efficiency: Demonstrates the ability to solve a 47-km horizon optimization in 2.5 seconds, making on-board implementation feasible.
Robustness Analysis: Provides a rigorous comparison of three implementation strategies against typical racing disturbances (drafting, tire degradation, Full-Course Yellow).

4. Results

The system was validated using an electric endurance race car (InMotion) on the Zandvoort circuit.

Optimality Loss: The "Lift-and-Coast" adaptation (enforcing binary throttle) results in a time loss of only 0.3% to 0.4% compared to the unconstrained global optimum. This confirms that the driver constraint is not a significant performance bottleneck.
Disturbance Rejection:
- Fully Online Approach: Performed best, losing only 0.056% to 0.22% of stint time compared to an offline optimization with perfect knowledge of disturbances.
- Fixed Trajectory Approaches: Performed slightly worse (0.1%–0.2% loss) but remained competitive. They struggled slightly with large velocity deviations (e.g., during Full-Course Yellow) where fixed thresholds suggested coasting at unsafe low speeds, though this can be mitigated with heuristics.
Consistency: The optimized coasting points were consistent lap-to-lap, allowing drivers to anticipate signals effectively.
Thermal Management: The system successfully managed temperatures to ensure the battery was hot enough for fast charging at the start of the stint but cooled down sufficiently to accept charge at the pit stop.

5. Significance

This paper represents a significant step toward the practical deployment of advanced energy management systems in human-driven electric motorsport.

Bridging the Gap: It solves the critical problem of how to apply complex mathematical optimization to human drivers without violating regulations or causing distraction.
Scalability: The computational efficiency opens the door for higher-level strategy optimization (e.g., optimizing pit stop timing and stint lengths) and real-time parameter sensitivity analysis.
Future Applications: The framework is not limited to racing; the principles of thermal-aware, real-time energy management could be adapted for electric road cars, particularly as in-race charging (or fast charging) becomes more prevalent.
Validation: By demonstrating that the performance loss is negligible compared to typical racing variations, the authors prove that on-board, real-time optimization is a viable and superior alternative to static, pre-race strategies.