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Scheduling Battery-Electric Bus Charging under Stochasticity using a Receding-Horizon Approach

Justin Whitaker, Derek Redmond, Greg Droge, Jacob Gunther

TL;DR

The paper tackles cost-effective charging of battery-electric bus fleets under stochastic operations by designing a two-layer receding-horizon framework that integrates a static day-long plan with a reactive, short-horizon controller. It advances a network-flow MILP formulation to model charger allocation, SOC dynamics, and a non-linear CC-CV charging profile via a discrete-time, piecewise-linear approximation, including a variable-rate charging mechanism. A two-stage hierarchical strategy uses a terminal-cost reference to guide the receding-horizon planner, ensuring buses follow the day plan while adapting to disturbances and uncontrolled loads. Monte-Carlo experiments across multiple deployment scenarios demonstrate substantial savings from TOU-aware costs and robust feasibility relative to open-loop and simple threshold-based strategies. Overall, the work combines realistic pricing, partial charging fidelity, stochastic awareness, and receding-horizon control to enable practical, cost-effective BEB fleet charging.

Abstract

A significant challenge of adopting battery electric buses into fleets lies in scheduling the charging, which in turn is complicated by considerations such as timing constraints imposed by routes, long charging times, limited numbers of chargers, and utility cost structures. This work builds on previous network-flow-based charge scheduling approaches and includes both consumption and demand time-of-use costs while accounting for uncontrolled loads on the same meter. Additionally, a variable-rate, non-linear partial charging model compatible with the mixed-integer linear program (MILP) is developed for increased charging fidelity. To respond to feedback in an uncertain environment, the resulting MILP is adapted to a hierarchical receding horizon planner that utilizes a static plan for the day as a reference to follow while reacting to stochasticity on a regular basis. This receding horizon planner is analyzed with Monte-Carlo techniques alongside two other possible planning methods. It is found to provide up to 52\% cost savings compared to a non-time-of-use aware method and significant robustness benefits compared to an optimal open-loop method.

Scheduling Battery-Electric Bus Charging under Stochasticity using a Receding-Horizon Approach

TL;DR

The paper tackles cost-effective charging of battery-electric bus fleets under stochastic operations by designing a two-layer receding-horizon framework that integrates a static day-long plan with a reactive, short-horizon controller. It advances a network-flow MILP formulation to model charger allocation, SOC dynamics, and a non-linear CC-CV charging profile via a discrete-time, piecewise-linear approximation, including a variable-rate charging mechanism. A two-stage hierarchical strategy uses a terminal-cost reference to guide the receding-horizon planner, ensuring buses follow the day plan while adapting to disturbances and uncontrolled loads. Monte-Carlo experiments across multiple deployment scenarios demonstrate substantial savings from TOU-aware costs and robust feasibility relative to open-loop and simple threshold-based strategies. Overall, the work combines realistic pricing, partial charging fidelity, stochastic awareness, and receding-horizon control to enable practical, cost-effective BEB fleet charging.

Abstract

A significant challenge of adopting battery electric buses into fleets lies in scheduling the charging, which in turn is complicated by considerations such as timing constraints imposed by routes, long charging times, limited numbers of chargers, and utility cost structures. This work builds on previous network-flow-based charge scheduling approaches and includes both consumption and demand time-of-use costs while accounting for uncontrolled loads on the same meter. Additionally, a variable-rate, non-linear partial charging model compatible with the mixed-integer linear program (MILP) is developed for increased charging fidelity. To respond to feedback in an uncertain environment, the resulting MILP is adapted to a hierarchical receding horizon planner that utilizes a static plan for the day as a reference to follow while reacting to stochasticity on a regular basis. This receding horizon planner is analyzed with Monte-Carlo techniques alongside two other possible planning methods. It is found to provide up to 52\% cost savings compared to a non-time-of-use aware method and significant robustness benefits compared to an optimal open-loop method.
Paper Structure (26 sections, 3 theorems, 54 equations, 5 figures, 4 tables)

This paper contains 26 sections, 3 theorems, 54 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A Network Flow Representation of the BEB Charging Schedule
  3. Graph and Network Flow From Previous Work
  4. Constraints from Previous Work
  5. Flow balance
  6. Vertex Grouping Constraint
  7. Non-Linear, Variable-Rate Charge Profile
  8. Modelling a CC-CV Charging Profile in Continuous Time
  9. Variable-rate Mixed Integer Charging Model
  10. Formulating the Charging Profile Model for Optimization
  11. Minimization of Utility Costs
  12. Consumption Cost
  13. Baseline Demand Cost
  14. TOU Demand Cost
  15. Base Model
  16. ...and 11 more sections

Key Result

Lemma 1

The CC-CV charging profile with switching SOC of $\eta_j$ for a bus, $j$, with battery capacity $E_j$ can be modeled in continuous-time by the piecewise-linear time-invariant continuous dynamic system in the charge level, $s_j$: where and $\alpha_{j,l}$ and ${p}^{cc}_{l}$ are parameters of the model derived from the charger characteristics.

Figures (5)

  • Figure 1: A graph depicting the possible state and transitions for a single charger type over time for a simple bus charging example. The bottom row corresponds to the charger at rest while the other rows each correspond to charging a particular bus. Yellow ovals show the vertex groups that are placed around charging windows.
  • Figure 2: The two-level hierarchical planning method and its interactions with the environment
  • Figure 3: The schedules for buses in the Salt Lake City, Utah area that serve routes 2, 209, 220, and 509.
  • Figure 4: The solid lines are the average charge levels (kWh) across Monte-Carlo runs and buses for each scenario. The dotted lines denote the three-$\sigma$ variation across Monte-Carlo runs and buses (with each bus's charge level zeroed to its mean across Monte-Carlo runs) Note that the Qin strategy deviates from the nominal plan significantly during the day (as expected). Significantly, the three-$\sigma$ variations of the Open-Loop strategy continue to grow over time, while the variations of the other two strategies do not.
  • Figure 5: A visualization of the two parts of the approximate gain upper bound constraint (not to scale). The blue constant line corresponds to the linear portion of the CC/CV charging profile, while the red corresponds to the exponential portion. The intersection point of the two lines is the switching point for the approximation, while the vertical dashed line is the true switching point. Thus, the approximation introduces error in between the intersection point and the true switching point, with the most error occurring in the limit approaching the true switching point.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof