Truck Depot Charging: Grid, Battery Buffer and Schedule

Each truck and the BESS receive constant power commands for each scheduling slot, 15 minutes by default. SOC then changes linearly while a command is realizable. For the entered nonincreasing acceptance curve,

For truck energy $x_j$ in a slot of duration $\Delta t$, the LP imposes both

Here $S_{end,j}=S_{arrival}+\sum_{\ell\le j}x_\ell/C$. The second constraint is linear in cumulative delivered energy. Because acceptance decreases with SOC, satisfying it at the end proves that the constant command is accepted throughout the slot. This avoids combining a full-power charging integral with a smaller available grid power.

The full-horizon LP simultaneously enforces truck availability and target SOC, shared DC cabinet power, the AC grid limit and absolute peak, no export, BESS charge/discharge converter power, and cumulative stored-energy bounds. Since all these commands are constant and BESS energy is linear within a slot, the constraints apply throughout that slot. A first simplex solve maximizes total delivered truck energy. A second fixes that total and minimizes monthly energy and demand charges, with a USD 0.001/kWh AC-storage-throughput regularizer. If the fleet is infeasible, this minimizes total shortfall without promising equal or priority-weighted service.

The AC balance is $P_g=P_{base}+\sum P_{truck}/\eta_c+P_{B,c}-P_{B,d}$; storage evolves as $dE/dt=\eta_B P_{B,c}-P_{B,d}/\eta_B$. Cyclic mode restores at least the initial stored energy. One-off mode can use the BESS to supply a base load above the grid limit, provided the converter and initial usable energy can cover the entire base-load deficit. The initial LP basis explicitly includes that supporting discharge.

An independent RK4 replay integrates $dS/dt=\min(P_{command},P_a(S))/C$, checking the actual truck energy and grid, cabinet, converter and storage limits at its stages. A failed replay blocks results. The separate EDF controller uses the same realizable constant-command acceptance bound and reserves storage for future base-load deficits.

The 3 × 3 sweep warm-starts the same LP. Each option first checks base-load supply; an infeasible option is marked individually. The capacity search uses LP sensitivity to bracket the least nominal BESS energy for the entered command interval, power and terminal policy. The reported bracket is optimization precision, not a physical uncertainty margin.

All solver matrices are explicitly normalized by 1 h, 1 kW and 1 kWh. Independent SciPy 1.15.3/HiGHS record s7 replaces the former integral-envelope record s6, and independently integrates the review counterexample. Context ratings from bp pulse, ASKO and Alfredsson describe different grid, DC charger and BESS power limits; they do not determine storage energy.

The timetable is synthetic and editable, not a measured ASKO or bp pulse operating record. Every active truck has its own available plug; a shared cabinet limits aggregate DC output. Truck usable capacity is fixed. DC charger output is treated as energy entering the usable battery; additional pack losses, temperature limits and auxiliary loads require adjusted acceptance curves or site load.

Other site load and conversion efficiencies are constant. BESS charge and discharge share the stated converter rating. One-way BESS efficiency is applied separately in each direction. There is no PV, export, vehicle-to-grid, outage reserve beyond the minimum SOC, or integer plug assignment.

Default tariff assumptions are USD 0.15/kWh and USD 12/kW/month as a user-editable scenario dated 2026-11-02, not a quoted utility tariff. Monthly demand charges apply once to the modeled peak. The reference cases reproduce ICCT’s nominal 350/750 kW, 45-minute events with losses and taper disabled; they are ideal energy benchmarks, not measurements of MCS performance.

The optimization and EDF comparison both hold each DC setpoint for its entire slot. There are no within-slot power reallocations. The input charging curve is a static upper bound on vehicle acceptance; constant setpoints are constrained at the slot's end SOC.

The result is optimal for the explicitly modeled piecewise constant DC command policy. A continuously varying, acceptance-following controller can deliver more energy and need less storage. Refine the scheduling interval to assess that difference before procurement. In the review's 600 kWh, SOC 0.65→0.94, 15-minute, 700 kW-grid case, the continuous-control ceiling is 162.748 kWh; a held command can deliver only 140 kWh. The calculator reports the resulting 34 kWh shortfall, not a feasible departure.

Availability rounds inward to complete slots and can lose almost two intervals of dwell time. The acceptance constraint uses end SOC and the full entered taper curve; it does not approximate an unavailable full-power trajectory as an average power. Converter ramps, communications delays, cooling and temperature-dependent derating are not modeled. The site controller must implement the stated commands and actual equipment limits.

The default replay uses four RK4 substeps in each charging slot and one exact linear interval for idle slots. Valid constant commands give a constant vehicle derivative, so replay refinement should change only roundoff. Increase replay substeps to diagnose a failed or near-boundary replay; refine the scheduling interval to change the control policy itself. The model's default cost targets about three seconds; finer schedules take longer.

The BESS bracket applies to the entered SOC window, initial SOC, converter power, grid limit and terminal policy. It excludes ageing, forecast uncertainty, temperature derating and procurement margin. If the search ceiling cannot supply all departures, the minimum-capacity result is blank. Sweep cells whose base load cannot be supplied are blank for truck shortfall and explicitly labeled Base unmet; they do not block other options.

Large dimensionless LP tableaux exceed the engine's element-by-element unit audit; physical inputs and energy-accounting summaries retain units. The independent ODE replay and AC energy-balance check verify the recovered dispatch. The cost comparison excludes off-horizon loads/peaks, tariff ratchets, fixed charges, taxes, BESS capital and degradation costs. EDF terminal restoration is reported separately if it needs extra recharge time.

IEC TS 63379:2026, Edition 1.0, supersedes draft-status wording in older summaries. This is energy scheduling rather than connector compliance; check against the standard in force.