Optimal Electric Bus Depot Charging: Cost Savings, Grid Limits, and Robustness Trade-Offs

Christopher Harald Onder; Fabio Widmer; Luca Pinter; Mohammad Hossein Moradi

REVIEW 2 major objections 4 minor 13 references

Optimized electric bus depot charging can cut total electricity costs by 25–60%, reduce the grid capacity needed for feasible operation by over 40%, and guard against 10% trip-energy uncertainty at a cost increase below 0.1%.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-03 09:33 UTC pith:HH3KGYGL

load-bearing objection A competent sensitivity study of depot charging, but the headline robustness cost (<0.1%) is not supported because the worst-case SOE is not cycled; add Ewc[N]>=Ewc[0] and rerun before believing it. the 2 major comments →

arxiv 2607.29304 v1 pith:HH3KGYGL submitted 2026-07-31 eess.SY cs.SY

Optimal Electric Bus Depot Charging: Cost Savings, Grid Limits, and Robustness Trade-Offs

Fabio Widmer , Luca Pinter , Mohammad Hossein Moradi , Christopher Harald Onder This is my paper

classification eess.SY cs.SY

keywords electric busdepot chargingrobust optimizationdemand uncertaintyconvex optimizationdemand chargeelectricity price volatilitygrid capacity

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that overnight charging of electric bus fleets is best seen as one convex robust optimization problem that simultaneously minimizes energy cost and demand charges while staying feasible under uncertain trip energy use. Using realistic schedules for four depots of 7 to 35 buses, it claims that optimizing instead of charging on arrival cuts total electricity cost by 25–60% as price volatility rises, and that the same optimization lowers the minimum grid connection capacity needed for feasibility by more than 40% in the larger depots. It also claims that protecting against a worst-case 10% deviation in energy demand costs less than 0.1% in extra electricity cost. These results matter because they quantify how much value smart depot charging creates and how cheap robustness can be, giving fleet operators concrete numbers for investment and control decisions.

Core claim

The paper argues that a single convex robust formulation of overnight depot charging — minimizing energy cost plus a demand charge under worst-case state-of-energy constraints — outperforms charge-on-arrival across all studied depots. In four realistic Swiss depot scenarios, total electricity cost reductions range from roughly 25% under flat prices to nearly 60% under high price volatility, with smaller depots benefiting most because optimization removes charging peaks that dominate their bills. The formulation also cuts the minimum grid utilization factor needed for feasible operation by over 40% for larger depots, meaning a smaller grid connection can support the same fleet, while tight gr

What carries the argument

The central object is a convex optimal control problem over each bus's charging power, battery state of energy, and a single peak-power variable. Bounded demand uncertainty (each trip's energy demand within ±ρ of nominal) is enforced by propagating a worst-case state-of-energy trajectory that must stay nonnegative, guaranteeing feasibility for every admissible realization. Two nonconvexities are relaxed into convex constraints: the quadratic charging-loss equality becomes an inequality, and the maximum over time in the demand charge becomes a slack variable with an upper bound; the paper argues these inequalities are tight at the optimum, making the problem tractable as a finite-dimensional

Load-bearing premise

The load-bearing premise is that two mathematical shortcuts used to make the problem convex — replacing the battery-loss equality with an inequality and replacing the peak-power term with a slack variable — are always exactly tight at the optimal solution; the paper states this is checked after solving but reports no such check, and if either shortcut is not exact the quoted savings and feasibility numbers do not describe a physically realizable charging plan.

What would settle it

Solve the same depot instances without the two convex relaxations — for example with a nonlinear solver on the original equality-constrained problem — and compare the exact optimum's cost and peak power with the relaxed solution. Any gap in the battery-loss equality or in the peak-power variable at the reported optimum would falsify the central numbers; alternatively, report the promised a-posteriori equality check by computing actual charging power from the relaxed solution and comparing it with the relaxed variables.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

For depots of 7–35 buses, optimized charging reduces total electricity cost by roughly 25% under flat prices and up to nearly 60% under high price volatility, compared with charge-on-arrival.
Smaller depots achieve the largest relative savings because optimization eliminates charging peaks that otherwise dominate their electricity bills.
Optimization lowers the minimum grid utilization factor required for feasible operation by more than 40% in larger depots, so a smaller grid connection can support the same fleet.
A worst-case 10% energy-demand uncertainty adds less than 0.1% to total electricity cost, making robust feasibility nearly free in the studied scenarios.
The optimized charging policy has an interpretable structure — idling above an implicit price threshold and drawing an approximately flat power profile below it — which can guide practical rule-based charging heuristics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the savings grow with price volatility, the value of smart depot charging should rise as electricity markets incorporate more renewables and intraday price swings widen; operators in such markets should expect savings near the upper end of the 25–60% range.
The negligible cost of robustness suggests that adopting a conservative uncertainty set (e.g., 10% above nominal demand) is almost always preferable to ignoring uncertainty, because it buys feasibility at essentially no cost — though this conclusion depends on the box uncertainty model and on how well nominal demand is known.
The grid-capacity result points toward a testable design rule: when planning a new depot, one could size the grid connection using the optimizer's minimum feasible capacity rather than charge-on-arrival peaks, potentially cutting infrastructure investment; the paper raises this joint fleet-and-grid sizing question as future work.
The near-rectangular profile below a price threshold hints that a simple bang-bang heuristic might recover most of the optimization value; a natural test is to run such a threshold rule against the full optimizer on the same depot data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A competent sensitivity study of depot charging, but the headline robustness cost (<0.1%) is not supported because the worst-case SOE is not cycled; add Ewc[N]>=Ewc[0] and rerun before believing it.

read the letter

Short version: the cost-savings and grid-capacity results are plausible and useful, but the headline robustness number is not supported by the model as written. The stress-test note is right. Eq. (20l) cycles only the nominal SOE, not the worst-case SOE. Ewc[0]=Eu[0] and eq. (20f) only require Ewc>=0, so the optimizer can let the worst-case battery drain over the 24h horizon and fund the extra 10% energy from the initial charge. That makes the price of robustness look nearly free. If you want a steady-state robust operating cycle, you need Ewc[N]>=Ewc[0], and the Fig. 3 curves need to be regenerated. This is a load-bearing flaw in one of the three advertised contributions, not a nitpick.

What is genuinely new is the systematic sensitivity analysis across price volatility and grid capacity, plus the explicit cost-robustness trade-off curves for overnight depot charging under a combined energy/demand tariff. The four GTFS-derived depots give the study a realistic scale. The convex model is standard, but competently formalized, and the worst-case lower-SOE reasoning is correct: only the upper demand bound matters for the lower SOE bound. I saw no curve fitting or self-citation games.

The reader's main concern is legitimate but secondary: the paper promises a posteriori checks of the convex relaxations in eqs. (14) and (16) and never reports them. Without slack values, there is no evidence the relaxed solution is tight and physically realizable. Also missing: the discretization time step and the demand-charge normalization factor. And the abstract's "over 40%" grid-capacity reduction is an overstatement: Table 2 shows reductions of 0%, 17%, 38%, and 43%. The Glarus depot gets nothing. The claim should be per-depot, not global.

Who this is for: engineers and operators wanting quantified guidance on when smart depot charging pays off, and researchers working on robust EV charging. The paper deserves a serious referee—it has the right questions and mostly honest handling of data—but it needs a major revision on the robustness claim, plus the missing reporting details, before the numbers can be trusted.

Recommendation: send to peer review. Require the authors to add the worst-case terminal constraint, rerun the robustness curves, report the relaxation slacks, state the discretization and normalization, and soften the capacity headline.

Referee Report

2 major / 4 minor

Summary. The paper proposes a convex robust optimization model for overnight depot charging of electric bus fleets under bounded trip-energy demand uncertainty. The model minimizes energy cost plus a demand charge subject to charger, grid, and state-of-energy constraints, with robustness enforced through a worst-case state trajectory. The formulation is tested on four Swiss depots (7–35 buses) against a charge-on-arrival baseline. The paper reports total electricity cost reductions of 25–60% as electricity price volatility increases, a reduction of the minimum feasible grid connection capacity of up to 43%, and an additional robustness cost of less than 0.1% for a 10% demand uncertainty. It also claims that the resulting charging profiles have a simple price-threshold structure.

Significance. If the results are correct, the paper would provide a practical, computationally tractable tool for depot charging with explicit robustness guarantees, supported by realistic GTFS-based case studies and a systematic sensitivity analysis over price volatility and grid capacity. The worst-case reasoning for the lower SOE bound is sound: only the upper demand bound matters for the lower SOE constraint, and the upper SOE bound can be handled by reducing charging power. The convex reformulations are standard, and the sensitivity experiments are clearly presented. However, the central robustness claim has a load-bearing modeling flaw: the periodic terminal constraint is imposed only on the nominal state, not on the worst-case state. This flaw directly affects the reported <0.1% cost of robustness and therefore the paper's headline conclusion.

major comments (2)

[§3.2, Eqs. (20c), (20f), (20k), (20l)] The formulation enforces a periodic terminal condition only on the nominal state Eu, through Eq. (20l), while the worst-case state Ewc is only required to be nonnegative in Eq. (20f). Combining (20b) and (20c) gives, for each bus, Ewc_u[N] = Eu[N] - ρ·Δt·Σ_k Pdem_u[k]. With Eq. (20l) stating Eu[N] ≥ Eu[0] = Ewc_u[0], the worst-case state can end at Ewc_u[0] - ρ·E_total,u, which may be far below the starting value as long as it is nonnegative. In a periodic operating cycle, this is not a sustainable robustness guarantee: if the next day is also worst-case, the bus starts lower and the guarantee erodes. The correct periodic robust constraint would be Ewc_u[N] ≥ Ewc_u[0]. Without it, the optimizer can meet the worst-case demand by drawing the extra energy ρ·E_total from the initial battery capacity instead of purchasing it, which explains why Fig. 3 reports a <0.1% robustness cost for ρ=10%
[§3.1, Eq. (14)] The convex relaxation of the battery-loss equality, Eq. (14), is asserted to be tight at the optimum and this is said to be 'checked a posteriori,' but no such check is reported anywhere in Section 4 or the figures. All computed cost savings, SOE trajectories, and robustness figures are produced by solving the relaxed problem. If the optimum has strict inequality for some k and u, then Pchg,i is not the physical battery input power, and the computed SOC dynamics and the resulting costs are not physically realizable. The authors should report the maximum slack of Eq. (14) over all buses and time steps for every scenario, or prove that strict inequality is suboptimal. This is a load-bearing verification, not a cosmetic detail.

minor comments (4)

[Abstract and §4.5, Table 2] The abstract states the optimization 'lowered the grid capacity required for feasible operation by over 40%,' but Table 2 reports reductions of 0.0%, 17%, 43%, and 38%. The claim is true only for one depot; please report the per-depot values and avoid the impression that the result holds broadly.
[§1.2.2] There is a grammatical error: 'No prior work was identified that quantifies the “price of robustness”, in the sense of Bertsimas and Sim [12], has not been quantified' — this should be rephrased.
[§2.3] The demand-charge rate is given as a monthly 12.1 $/kW and said to be 'normalized to the optimization horizon.' Please specify the normalization (e.g., whether the monthly charge is divided by 30 or by the number of days in the billing cycle), since this directly affects the reported total cost and the relative savings.
[§4.4 and Fig. 4] The y-axis label 'Saving potential [%]' should be defined precisely: is it total electricity cost savings of the optimized strategy relative to charge-on-arrival? Also clarify that the flat price profile uses the same mean as the other profiles, as stated in the text.

Circularity Check

0 steps flagged

No significant circularity: results are computed optima, not fits; the flagged robustness-constraint caveat is a correctness risk, not a circular reduction.

full rationale

The paper's central claims are generated by solving well-posed convex programs and comparing them with a deterministic charge-on-arrival baseline; no parameter is fitted to the reported savings, and the volatility and grid-capacity sensitivities are recomputed from the same model under systematically varied tariffs and power limits. The robustness trade-off is the difference between two optimization runs sharing the same objective, with the uncertainty set as an input; that is the standard, non-circular meaning of the price of robustness. All references are external; there is no load-bearing self-citation chain or imported uniqueness theorem. Two caveats are noted for correctness rather than circularity: the paper states that the convex relaxations (14) and (16) are tight at the optimum 'which we check a posteriori' but reports no such check, and the robust formulation (20f) imposes no periodic terminal constraint on the worst-case SOE, so the reported <0.1% robustness cost may understate the cost of a sustainable multi-day guarantee. These issues affect the validity of specific numerical claims, but they are not cases where a 'prediction' is equivalent to the model input by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The paper introduces no new physical entities. Its central contribution rests on routine modeling assumptions plus a small number of untested choices—relaxation tightness, periodic terminal condition, dedicated chargers, and the reported demand normalization—the most fragile of which is the unverified tightness of the convex relaxations.

free parameters (3)

Demand-charge normalization factor = not stated
A monthly demand-charge rate of 12.1 $/kW is 'normalized to the optimization horizon' (§2.3), but the conversion factor is not disclosed; all cost results scale with this factor.
Discretization time step Δt = not stated
Section 3.2 introduces a uniform discretization time step but never reports its value; the numerical results (costs, peaks, feasibility) depend on it.
Battery/charger loss parameters = η=0.95, α=3.51×10⁻⁷ W⁻¹
Stated as fixed constants in §2.1. They are not universal physical constants and are treated as inputs from prior technology; if the actual fleet's loss characteristics differ, the cost and feasibility results change.

axioms (6)

domain assumption The convex relaxations in eqs. (14) and (16) hold with equality at the optimum
Section 3.1 asserts this and says it is 'checked a posteriori', but no evidence is presented; if false, the computed plan is not physically realizable.
domain assumption The demand-uncertainty set is a symmetric multiplicative box (1±ρ)P_dem applied at every time step
This assumes the worst case is simultaneous maximum demand on every trip; a conservative assumption that may overstate robustness cost, but it is the paper's stated model (§2.2).
domain assumption The 24-hour horizon is approximately periodic, so the terminal constraint E[N]≥E[0] with free initial SOE is appropriate
Stated in §3.2 as appropriate for public transport; if the real depot is not at a periodic steady state, the single-day plan may require a different initial state.
domain assumption Each bus has a dedicated 150 kW charger and unidirectional power flow
Stated in §2.1; many real depots share chargers among buses, so the results do not directly transfer to shared-charger configurations.
standard math Standard convex optimization and forward-Euler discretization are valid
The transcription of the continuous-time OCP into a convex program using multiple shooting and forward Euler is routine.
domain assumption Nominal demand profiles derived from GTFS schedules and elevation-based estimates are accurate
The depot demand data are not published and the estimation pipeline is not described, so the numerical results depend on unverifiable inputs.

pith-pipeline@v1.3.0-daily-deepseek · 7046 in / 17484 out tokens · 164797 ms · 2026-08-03T09:33:34.471583+00:00 · methodology

0 comments

read the original abstract

Depot charging of electric bus fleets must minimize electricity costs, respect grid limits, and remain feasible despite uncertain trip energy demand. While cost-optimal charging is well studied, its value under different electricity prices and grid connection capacities, as well as the economic cost of robustness, remain poorly quantified. We address these gaps with a convex robust formulation in which bounded demand uncertainty is enforced through worst-case state-of-energy constraints. The formulation is evaluated against charge-on-arrival using realistic service schedules for four Swiss depots containing 7-35 buses. In the depots studied, smaller depots achieve the greatest relative benefit from optimization, with total electricity cost reductions exceeding 50%, because optimization mitigates charging peaks that strongly affect their costs. For all depot sizes, the savings from optimization increase with electricity price volatility. Optimization can also lower the grid capacity required for feasible operation by over 40%, although tight limits reduce peak shaving potential. Protection against energy-demand deviations of 10% increases total electricity cost by less than 0.1%. The resulting charging power profiles exhibit interpretable price-threshold and peak-shaping behavior, providing practical guidance for real-world implementations.

Figures

Figures reproduced from arXiv: 2607.29304 by Christopher Harald Onder, Fabio Widmer, Luca Pinter, Mohammad Hossein Moradi.

**Figure 1.** Figure 1: Optimized charging power profile for the Glarus depot with seven buses. The panels show the electricity price profile, each bus’s depot availability, the optimized SOE trajectories with dashed worst-case trajectories, and the charging power profiles. 0 50 100 0 4 8 12 16 20 24 0 100 200 300 400 Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 7 Time of day [h] SOE [%] Charging power [kW] [PITH_FULL_IMAGE:figure… view at source ↗

**Figure 2.** Figure 2: Charge-on-arrival strategy for the Glarus depot, for the same scenario as fig. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Trade-off between the assumed worst-case energy-demand uncertainty and the resulting increase in total electricity cost. adjustments to the nominal charging power profile. This behavior is consistent with the robustness constraint being inactive for many missions, as illustrated for instance in fig. 1. 4.4 Sensitivity of Price Volatility and Depot Size To assess the influence of electricity price dynamics… view at source ↗

**Figure 5.** Figure 5: Influence of the grid utilization factor (a lower factor represents a more stringent grid limitation) on the total electricity cost savings achieved through charging optimization. The vertical dashed lines indicate the minimum grid utilization factor for which the charging optimization remains feasible. nomic case for fleet electrification. 5. CONCLUSION Across four realistic depots, optimized charging re… view at source ↗

discussion (0)

Reference graph

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Real-time charging scheduling and optimization of electric buses in a depot,

B. Verbrugge, A. M. Rauf, H. Rasool, M. Abdel- Monem, T. Geury, M. El Baghdadi, and O. Hegazy, “Real-time charging scheduling and optimization of electric buses in a depot,”Energies, vol. 15, no. 14, 2022

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2022

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Z. Zeng, T. Wang, and X. Qu, “En-route charge scheduling for an electric bus network: Stochas- ticity and real-world practice,”Transportation Re- search Part E: Logistics and Transportation Review, vol. 185, p. 103498, 2024

2024

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2025

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2025