Energy supply and dense energy storage

An hourly simulation of rooftop solar plus a battery on an 8,000 kWh/yr home, over 20 seeded synthetic weather years (Haurwitz clear sky, a three-state cloud chain, an invented load) with assumed round-number prices. A 6 kW array alone meets 35% of the load; 5, 10 and 15 kWh of battery lift that to 49%, 59% and 62%, then returns diminish fast. With import at $0.30 and export at $0.05 per kWh a 10 kWh battery pays only below about $451/kWh installed (assumed $800), $143 at $0.20 export, and never when export is paid at the import price. Stylised: synthetic weather, flat tariff, no incentives or degradation; not installation advice.

Adds an hourly time-of-use import price to energy-home's solar-plus-battery model (10 seeded synthetic weather years, assumed three-band tariff with the same $0.296 average as a flat $0.30). A 10 kWh battery behind 6 kW of PV breaks even at $641/kWh installed with peak-only discharge and off-peak grid charging, against $444 on the flat tariff, still under the assumed $800. Only small batteries clear $800 ($856 at 2.5 kWh, $847 at 5 kWh); 15 and 20 kWh fall to $427 and $320. Half-filling from the grid ($674) beats a full fill ($641). Break-even tracks the peak price ($307, $641, $1,208 at $0.28, $0.45, $0.70). Stylised: assumed tariff, synthetic weather, fixed dispatch rule, no degradation.

A single-node PyPSA model of Germany's 2019 hourly demand (ENTSO-E data via Open Power System Data) with solar, wind, gas, a battery and hydrogen storage sized together under an emissions cap. Gas alone costs $79/MWh; a 50% cut adds $5.6 using wind only; 80%, 90%, 95% and 100% cuts cost $113, $131, $144 and $177/MWh, the last needing 10.6 TWh of hydrogen storage. Implied abatement cost rises from $30/t to $1,765/t for the last 5%. At zero emissions the hydrogen tank price moves the answer from $152 to $254/MWh; without hydrogen, batteries alone give $287. Stylised: assumed costs, one weather year, one node, no grid.

Theoretical ceilings from first principles for batteries, hydrogen, flywheels, pumped hydro and heat, reproducing five published values to within 0.3% (lead-acid 170, Zn-air 1,086, Li-S 2,567, Li-O₂ 3,458 Wh/kg; Al-air 8,100 on Al). No battery couple releases more than 1.65 eV per reactant atom, so the spread comes from atomic mass: Wh/kg = 26,801 × eV per atom / mean g/mol. A carbon-fibre flywheel holds at most 756 Wh/kg (0.34 eV per atom), sand at 500 °C 107 Wh/kg of heat (41% recoverable as electricity) and water lifted 300 m 0.82 Wh/kg. Ceilings for the active material only, with some voltages and material properties assumed.

Volumetric ceilings (Wh/L) for the couples of energy-density, from the sum of reactant mass over assumed handbook density. With oxygen taken from air, zinc-air reaches 9,659 Wh/L, lithium-oxygen 6,104 and aluminium-air 5,887, against 1,382 for graphite/LiCoO₂ and 724 for lead-acid. The ranking changes with the basis: lithium-sulfur beats graphite/LiCoO₂ by 6.8 times per kg but only 2.1 times per litre (lithium is light and bulky), while zinc-air beats lead-acid by 6.4 times per kg and 13 times per litre. Hydrogen holds 1,314 Wh/L at 700 bar and 2,314 as a 20 K liquid. Sand at 500 °C holds 171 Wh/L as heat. Active material only; densities are assumed, not checked.

A small sourced dataset set beside the ceilings of energy-density. A Panasonic NCR18650B cell (data sheet) stores 243 Wh/kg, 0.64 of the 380 Wh/kg graphite/LiCoO₂ reactants could release; commercial LiFePO₄ reaches 0.25–0.45 and lead-acid 0.21–0.24. DOE’s projected 700 bar hydrogen system (1.71 kWh/kg) is 5.1% hydrogen by mass against 23.8–41.3% for an ideal carbon-fibre sphere. Round-trip efficiency runs from 50% (compressed air) to 85% (advanced lead-acid). Limited: five families, five of seven data rows read through secondary pages, gap not split into causes.

A transient finite-volume model of a spherical sand store in mineral wool (a hot-water tank run through the same code), checked against analytic cases. A store sized to still give 10 MWh after 120 days needs 222 m³ (against 70 if lossless) and returns only 26% of the electricity put in as heat, so a COP-4 heat pump wins unless its electricity costs 15× more, before capital cost. After a day or a week the same store returns 79% and 73%: a days-scale store at house size, seasonal only near 1,000 m³ (43% left after 180 days). Stylised: assumed properties, a sphere, instant uniform charging, no pipes or ground.

Levelised cost of storage for lithium-ion, vanadium flow, pumped hydro, cavern compressed air and cavern hydrogen, from capital cost, round-trip efficiency, life, fade and charging power. With round assumed inputs Li-ion is cheapest to about 4 hours ($116/MWh at 4 h) and costs about 3× a cavern store at 24 hours ($337 vs $109); cavern hydrogen beats Li-ion above 11 hours but cavern compressed air only above 308 hours. Li-ion would need an energy cost of $43/kWh instead of $200 to win at 24 hours. The 8–10 hour winner flips with the discount rate and charging price. Stylised: every input is assumed and unsourced, caverns need geology, one fixed use pattern.

A PyBaMM single-particle model with electrolyte of an NMC811 / graphite-SiOx cell, scored on discharge energy per kilogram of electrode stack after 100 cycles at a fixed current. Four design numbers (positive thickness, two porosities, negative-to-positive capacity ratio) are searched with 30 evaluations by random search, Gaussian-process Bayesian optimisation (5 seeds each) and the session agent proposing by hand (one run). The reference cell scores 109 Wh/kg; all three reach 141–143 (+30%) with a thicker, more porous positive electrode and about 30% more negative capacity, and their seed ranges overlap, so no proposer is shown to be better. Limits: one aggressively ageing parameter set, constant-current charge only, stack not finished cell, optimum at the search bounds. Stylised and simulated, not a tested cell.

A Monte Carlo over the energy-lcos model: 4,000 scenarios draw each technology's energy cost, power cost and efficiency from assumed ranges, plus the discount rate (3–10%) and charging price ($10–100/MWh), and count how often each is cheapest. Lithium-ion wins 96% of scenarios at 2 hours, 64% at 4 and 18% at 8; between 6 and 12 hours no technology wins more than 48%, so the point-estimate winner is a coin toss there. Caverns win most scenarios from 24 hours (compressed air 61% at 24 h, 81% at 100 h) and hydrogen 69% at 500 hours. Charging price is the largest single swing in cost. Stylised: every range is assumed, draws are independent and uniform, and the use pattern is fixed.

Takes the first next step of energy-cell: the 49 designs it logged (baseline, each search run's best, 36 spread over the SPMe ranking, 3 SPMe failures) are re-scored with the same parameters, protocol and mass model under the full Doyle-Fuller-Newman model. Over the 45 designs both models cycled, the cycle-100 Wh/kg rank correlation is 0.86 (0.93 at beginning of life); the SPMe winner (143.4 Wh/kg) is 3rd of 45 under DFN (160.9 vs 163.0 for the DFN best) and beats the reference cell by 38% under DFN against 31% under SPMe. DFN gives more energy for nearly every design (mean +18.6 Wh/kg), with gaps up to 127 Wh/kg, and the SPMe plateau is not flat under DFN. Limits: one cell, one parameter set, 49 designs chosen to spread over the SPMe ranking; both models share the same degradation parameters and neither is validated against measured data; stack-level mass only. Every input comes from energy-cell and PyBaMM's OKane2022 set.

Reads the actual dispatch of the sourced one-node PyPSA German system (energy-mix-sourced, 2030 costs, 2019 weather) and re-prices storage at that use instead of the assumed 2,000 discharge hours and $10–100/MWh charging. Li-ion is used close to the assumption (10–14 h long, 122–180 cycles, 1,717–1,865 h a year, EUR 31–38/MWh charging), so its LCOS rises only 9–19% ($197 to $231/MWh at 14 h). The cavern hydrogen store runs 719 h a year (2.6 cycles, 280 h long), so its LCOS is $864/MWh against $383 at 2,000 h; LCOS falls about tenfold between 300 and 4,000 h a year. The ranking of technologies at each store's duration is unchanged. Limitations: one weather year, one node, planner shadow prices rather than market prices, cycle life and fade still the assumed energy-lcos values.

Re-runs energy-lcos and energy-lcos-mc with every cost, efficiency and lifetime read from PyPSA technology-data (Danish Energy Agency; Viswanathan et al. 2022) and NREL ATB 2024, converted at the ECB 2025 rate. Point estimates keep the assumed map's order (Li-ion up to 4 h at $121/MWh, cavern compressed air from 6 h up) except at 500 h, where hydrogen no longer wins ($425 against $258). Over the sourced spread (ATB scenarios, pumped-hydro site classes, cost years 2020 to 2050, rate 3–10%, charging $10–100/MWh) 8 h is a three-way contest: pumped hydro 52%, compressed air 27%, Li-ion 22%; cavern air wins at least 95% from 48 h. Cavern air has a single sourced value, so the cost multiple that would cost it the win is reported (1.33 at 8 h, 2.2 at 24 h). Limitations: cycle life, fade, discount rate and charging price are not sourced; the spread is the range of two sources, not a probability.

Re-runs energy-mix (one-node PyPSA model of Germany's 2019 hourly load with solar, onshore wind, gas, Li-ion and hydrogen storage under an emissions cap) with every cost, lifetime, O&M, efficiency, fuel price and CO₂ intensity read from PyPSA technology-data at pinned revisions (Danish Energy Agency and other primary sources named per row). On 2030 costs in 2025 euros the unconstrained optimum costs EUR 75/MWh and zero emissions EUR 222/MWh with a steel hydrogen tank (1.3 TWh of hydrogen, 1.4 TWh of batteries) or EUR 144/MWh with underground storage (24.9 TWh); the last 5% of emissions costs about EUR 6,100/t with the tank and EUR 1,700/t with caverns. With caverns, technology-data's older 2030 and 2050 sets give EUR 114.5 and 85.5/MWh. Without batteries zero emissions costs EUR 254 (tank) or 162 (cavern); a discount rate of 3% or 10% gives EUR 152 or 282 (tank). Not from a source: the 7% discount rate (swept). Limitations: cavern storage is a geological resource that the model does not site or limit, one weather year, one node, no imports, central cost values only, and no check against a published whole-system figure.

Re-solves energy-mix-sourced (one-node PyPSA model, solar, onshore wind, gas, Li-ion and hydrogen storage, PyPSA technology-data 2030 costs in 2025 euros) on five weather and demand years of Open Power System Data / ENTSO-E (2015 to 2019) instead of only 2019; 20 deterministic LPs. At zero emissions the cost ranges EUR 203 to 271/MWh with a steel hydrogen tank (2018 cheapest, 2017 and 2015 dearest) and EUR 141 to 155 with caverns, so the tank case moves by about EUR 68 between years and the cavern case by about EUR 14. The hydrogen store is set by the year: 1.1 to 5.6 TWh with a tank (2015 largest) and 23.7 to 35.9 TWh with caverns (2016 largest), against 24.9 TWh in 2019 alone; the tank case needs 1.0 to 1.7 TWh of batteries. At a 95% cut the tank case buys no hydrogen in any year and costs EUR 118 to 133. The 2019 results reproduce the sibling project (EUR 222 and 144). Limitations: five years, each solved separately with perfect foresight (a store sized for all five years together was not solved), one node, no imports or offshore wind, capacity factors from actual generation so they include curtailment, caverns not sited or limited, and the 7% discount rate is not from a source.

Fits price as a non-decreasing function of residual load (load minus solar minus onshore wind) by isotonic regression on hourly German day-ahead prices 2015–2019 (OPSD/ENTSO-E), and scores it on a held-out year. Trained on 2015–2018 and tested on 2019: correlation 0.87 and RMSE 9.6 EUR/MWh, against 14.5 for an hour-of-day mean, 15.7 for the training mean and 23.0 for the flat EUR 54.6/MWh merit-order price of energy-thermal-hourly. Leave-one-year-out over all five years gives correlation 0.78–0.90 and beats every non-residual baseline in every year; the flat model is worse than the training mean in four of five. The predicted cheapest-quarter mean, which is what a storage buyer pays, is within EUR 3.7–9.2/MWh of the actual one. Most of the remaining error is the yearly price level (fuel and carbon), not shape: an oracle level cuts RMSE to 6.2–11.2. No data after 2019 (hosts unreachable from the run container); one held-out year per score, no confidence interval; realised rather than forecast load and renewables; onshore wind and solar only. All inputs are from sources.

The earlier cost comparisons charge the air-source heat pump a flat November–March price divided by one table COP of 3.6. Here the pump's electricity cost is the heat-weighted mean of price / COP(T) over every hour, with hourly temperatures from the ERA5 reanalysis (five German cities, population-weighted, 2015–2019) and the Staffell et al. (2012) COP fit used by when2heat and PyPSA-Eur. The table's 3.6 is what a 35 °C floor-heating sink gives (seasonal COP 3.52–3.67); a 45 °C sink gives 2.79–2.91 and a 55 °C radiator sink 2.18–2.28. The store's weekly-cycling flip multiplier rises from 1.16/1.20/1.20/0.59/1.02 (table) to 1.25/1.29/1.33/0.71/1.15 at 45 °C and 1.37/1.40/1.48/0.86/1.30 at 55 °C, so a realistic COP widens the store's margin in four years but 2018 still favours the pump. Daily cycling stays robust (flip 2.4–4.1×). The heating base temperature (12–18 °C) moves the flip by at most 0.06. Inputs not from a source: the 15 °C base, city weights and the sink temperatures, which are swept; the COP fit is a published regression, not a measurement of a specific unit.

Scales the PyPSA technology-data capital costs in the cost model of energy-thermal-real and bisects for the multiplier at which a house-sized sand store and an air-source heat pump cost the same, on real German day-ahead prices 2015–2019. Weekly cycling flips at 1.16, 1.20, 1.20, 0.59 and 1.02 times the store’s table capex, so the table’s own vintage step (1.257×) alone flips four of five years; daily cycling holds to 2.1–3.5×, monthly and seasonal never come close. With capex drawn uniformly within ±h (assumed, swept 0.10–0.50), the spread across price years (sd 0.0063 EUR/kWh) exceeds the cost spread at h = 0.10 (0.0039) and falls below it at h = 0.25 (0.0098). Five years, capex only, wholesale prices; h is an assumption, labelled; all other inputs are sourced.

The sand-store versus heat-pump comparisons leave out the equipment that takes heat out of the sand. No sourced sand-to-water exchanger price is in the cost table, so the price (EUR per kW of peak discharge) is solved for rather than assumed: the break-even at which the store stops beating the air-source pump on German prices and ERA5 weather 2015–2019, house scale, 45 °C sink. Weekly cycling survives an exchanger of 211, 241 and 274 EUR/kW in 2015–2017 and 123 in 2019; 2018 loses even with a free exchanger. Daily cycling survives 460–901 EUR/kW. At 100 EUR/kW the exchanger adds 0.0032 EUR/kWh. The break-even halves if the house is sized for 1,000 full-load hours. Not from a source: the exchanger price (swept), the 2000-hour peak rule, and the exchanger's lifetime and O&M, borrowed from the sand charger row; the sand-side discharge limit is not modelled.

Feeds the residual-load price model of energy-price-residual (isotonic fit, leave-one-year-out) into the cost model of energy-thermal-real for German day-ahead prices 2015–2019 and compares the house- and district-scale verdicts, sand store against air-source heat pump, with the real-price verdict. The model gives the same verdict as real prices in 32 of 40 scale×cycle×year cells, against 21 of 40 for the flat merit-order price. Daily cycling and monthly and seasonal storage are right in every house-scale cell; weekly cycling, whose real margin is only −0.005 to +0.011 EUR/kWh, is right in 0 of 5 years because the fit misstates each year’s price level (an oracle level restores 4 of 5). Using the model only to pick charge hours while paying the real price costs EUR 0.0004–0.0011/kWh over perfect foresight, against 0.003–0.020 for the flat price. Five years only, realised not forecast inputs, wholesale prices, no data after 2019; the tie band and the charge fraction are assumptions, labelled. All other inputs are from sources.

Costs the sand store of energy-thermal with every capital cost, lifetime, O&M and efficiency read from PyPSA technology-data (Danish Energy Agency; Viswanathan et al. 2022, PNNL-33283) at pinned revisions, sizing it with the transient model and comparing levelised cost of heat against an air-source heat pump. A sand store charged at EUR 0.05/kWh beats a heat pump when the pump's electricity costs about 2 times as much if the store cycles daily, 5 times weekly, 16 times monthly and 120 times seasonally (house, 10 MWh a year; 222 m³ and EUR 1.53 of capital per kWh for the seasonal store); at district size (5 GWh a year) 2.2, 2.8, 5.3 and 38 times. At 3 times the store can stand 2 days (house) or 7 (district); at 10 times, 14 or 60. A district hot-water tank beats sand for monthly and seasonal storage. Not from a source: electricity prices (swept), 2,000 full-load hours, 0.5 m insulation, cycle counts. Limitations: the sand-store cost is a utility-scale figure, no heat exchanger is costed (favours the store), flat prices, constant COP, instant discharge.

Replaces the flat electricity price of energy-thermal-cost with a modelled hourly price built from real German hourly load, solar and wind (OPSD/ENTSO-E, 2015–2019) and a one-fuel merit order (gas at EUR 54.6/MWh from PyPSA technology-data; surplus hours at a floor of 0, swept to 40). The store buys the cheapest quarter of each cycle; the heat pump pays the November–March mean. With the 189 GW solar and 137 GW wind of the 80%-emissions-cut optimum of energy-mix-sourced, a house-sized sand store cycled daily or weekly beats the pump in all five years (EUR 0.027 and 0.032 against 0.071 per kWh of heat in 2019); monthly and seasonal storage lose. With a fleet like today's there are no surplus hours and the store loses; a levy of EUR 0.10/kWh on both prices makes it lose in every case; a surplus-hour price above about EUR 20–35/MWh ends the advantage. Not from a source: the surplus-hour floor (swept), no storage or grid in the price model, flat winter heat demand, the quarter-cycle charge window. The price is modelled, not a market price, and no taxes or network charges are included in the base case.

Re-runs energy-thermal-hourly on the actual hourly German day-ahead price, 2015–2019 (OPSD/ENTSO-E; annual means 29–44 EUR/MWh, 66–211 negative-price hours a year), with the store and heat-pump costs of energy-thermal-cost unchanged. At wholesale prices a house-sized sand store cycled daily beats an air-source heat pump in all five years (EUR 0.045–0.062 against 0.070–0.074 per kWh of heat) and weekly in four of five; monthly and seasonal storage lose even at a negative break-even price. The earlier one-fuel merit-order model, rebuilt with each year's installed solar and wind, has no surplus hour and prices every hour at EUR 54.6/MWh, so its conclusion that the store loses with today's fleet came from the model, not the market. A levy of EUR 0.10/kWh on both prices removes the win in every case. Not from a source: the charge window (a quarter of each cycle, swept), flat winter heat demand, the levy (swept), perfect foresight within each cycle. Wholesale prices only; no taxes, network charges or retail margin.

Replaces the fixed dispatch rule of energy-tou with a linear program (HiGHS) over 10 seeded synthetic weather years and the same assumed three-band tariff. A 10 kWh battery behind 6 kW of PV breaks even at $867/kWh installed under optimal dispatch with perfect foresight, against $674 for the best fixed rule and $444 on a flat tariff; with life limited to 4,000 full cycles it is $742, 10% above the rule and still under the assumed $800. A rolling 36 h plan with a perfect forecast matches the whole-year optimum to $1. The optimiser adds $33 to $108/kWh across 2.5–20 kWh, and a wear-aware optimiser adds $43–58 when cycle life binds. Stylised: perfect foresight, assumed tariff, synthetic weather, crude cycle-count degradation.

Re-runs the rolling 36 h linear program of energy-tou-lp with forecasts built only from hours already observed (yesterday's weather, or the mean of the last seven days) and executes the plan on the actual PV and load, over 10 seeded synthetic weather years and the same assumed tariff. For 10 kWh behind 6 kW PV the perfect-forecast optimiser saves $892 a year against the best fixed rule's $694; with yesterday's weather it saves $726 and with a seven-day mean $691, so 84–101% of the optimiser's gain over the rule is lost. Life-limited break-even is $667 and $663/kWh against the rule's $674 and the perfect $733. Forecasts of the truth with 10% and 25% multiplicative noise keep 75% and 39% of the gain; at 100% noise the optimiser is worse than the rule. Stylised: assumed tariff, synthetic weather and load, forecast methods naive, noise abstract, crude cycle-count life.

Reruns the rolling linear program of energy-tou-lp-tmy on PVGIS typical-year irradiance (Golden CO, Phoenix AZ, Seattle WA) but re-plans every 24, 12, 6, 3 or 1 hours from the actual state of charge, with a nowcast forecast added. For 10 kWh behind 6 kW PV, re-planning every 3–12 h adds $10–36 a year to the realistic controllers (Golden clear-sky-index forecast $725 → $754; Seattle $678 → $713; Phoenix $780 → $797) against perfect-forecast savings of $930, $881 and $853, so it closes at most 17 percentage points of the gap between the best fixed rule and the perfect forecast. Most of the loss is forecast loss, not execution loss. A three-hour nowcast does not beat the plain clear-sky index and hourly re-planning is no better than every 3 h; it adds 20–30 cycles a year and lowers life-limited break-even price. Differences under about $10 are inside the noise of three load seeds and one typical year. Not from a source: the tariff, synthetic load, battery cost and 4,000-cycle life (inherited, assumed), the nowcast time constant (3 h, swept at one site) and the sunlit threshold.

Replaces the synthetic weather of energy-tou-lp-forecast with hourly irradiance from PVGIS typical-meteorological-year files (ERA5-based) for Golden CO, Phoenix AZ and Seattle WA, and adds the standard cheap PV forecast: yesterday's clear-sky index applied to the known clear-sky curve (“smart persistence”). The same rolling 36 h linear program, tariff and execution rule are used. The clear-sky forecast cuts hourly day-ahead nRMSE by 4–13% against yesterday's weather (0.63→0.55 at Golden, 0.68→0.60 at Seattle) but changes the annual saving by at most $5 in dispatch: for 10 kWh behind 6 kW PV the perfect-forecast optimiser saves $929, $880 and $852 a year at Golden, Phoenix and Seattle against the best fixed rule's $734, $668 and $618, while the realistic forecasts save $722–733, $779–785 and $677–691. They keep −6% to −1% of the gain at Golden (the fixed rule wins), 52–55% at sunny Phoenix and 25–31% at cloudy Seattle, and on life-limited break-even price they sit below the rule at Golden ($672–677/kWh against $713). So the forecast, not the optimiser, sets the value, and a better PV forecast alone does not buy it: a perfect load forecast adds only $15–18 a year. Real daily clear-sky index has lag-1 autocorrelation 0.29–0.36 against 0.50 for the synthetic chain. Not from a source: the tariff, load shape, battery cost and the 4,000-cycle life (inherited, assumed); the typical year stitches months from different years, and its irradiance is a reanalysis estimate, not a pyranometer record.

The DC optimal power flow is the workhorse of dispatch and price studies; how much does it lose against the AC OPF? On pandapower's MATPOWER cases with their own quadratic costs, load scaled 0.6 to 1.1, both OPFs are solved and the DC set-points are run through an AC power flow (slack pays the losses). IEEE 118-bus (no branch limit binds): DC cost is 1.9–2.9 % below the AC optimum and its dispatch costs 0.7–1.7 % above it, with no violations. IEEE 39-bus: DC cost 1.0–1.5 % below, dispatch within 0.06 % of the optimum but 15, 5 and 1 buses out of voltage limits at 0.6, 0.8 and 1.0 of load. IEEE 30-bus: DC 1.3–2.3 % below; at full load the DC dispatch overloads a branch to 111 % and is 0.31 % cheaper than the AC optimum, i.e. a violated constraint. Checks: a two-bus closed form and agreement of the cost function with pandapower. Limits: tightened-limit sweeps frequently returned no solution from the solver and non-convergence is not separated from infeasibility; 118-bus branch limits never bind so it is a losses-only comparison; no prices (LMPs), unit commitment or N-1; costs are the cases' own, not market prices. Modelled benchmark cases, not measured systems. Loss-adjusted DC (AC losses spread uniformly over loads, iterated 2–4 passes): the 118-bus cost estimate falls within 0.6–1.2 % of the AC optimum, but the AC cost of its dispatch is still 0.58–1.18 % above it (13–32 % of plain DC's excess recovered) and case39's voltage violations are unchanged; a uniform loss share fixes the price level, not the merit order.

A size-by-size ball-mill model (selection function S_i = S₁(x_i/1 mm)^α, Austin breakage function, three perfectly mixed tanks in series) with a Whiten classifier loop, solved by forward substitution and fixed-point iteration in the standard library. At the same 100 µm P80 from a 1500 µm feed, a closed circuit with the cut at P80 and 278 % circulating load uses 0.47 of the open-circuit energy (53 % saving); a coarser cut saves less (0.78 at cut = 2×P80, 35 % load). The saving is 1.8 to 3.7 times in energy ratio over tank number 1 to 10, classifier sharpness 2 to 8, bypass 0 to 0.3 and the breakage parameters; it does not collapse to the published figures: Bond’s open-circuit factor EF2 is 1.2 at 80 % passing and a vendor guide gives about 10 % for cement mills. That is reported as a discrepancy, with likely causes (slowing of fines, tested, does not close it; a mill flow limit of 1.5 to 2 times the open circuit’s feed, with that limit assumed rather than sourced, brings the fixed-power capacity advantage to 1.2 to 1.4; auxiliary power) rather than tuned away. The local size exponent of energy (E ∝ P80^−m, 50 to 200 µm) runs from 0.60 to 1.27 over α 0.4 to 1.2 against Bond’s 0.68, so Bond’s law corresponds to α near 0.5 in this model. Absolute kWh/t is in model units (S₁ was not fitted to a mill); every shape parameter is a typical value from memory of the Austin literature, swept, not measured.

OpenDSS on the IEEE 13-node test feeder (3,466 kW peak): total PV is ramped from zero over 300 random placements (1–8 customer nodes, seed 1) until the first service voltage above 1.05 pu, line above its ampacity or transformer above its kVA rating, with every measure scored on the same placements. At 30 % of peak load the median placement takes 1,015 kW (10th–90th percentile 510–1,893) and voltage binds first in 224 of 300. Lowering the regulator set-point by 2 V adds the most (+310 kW median, better on 74 % of placements, worse on 10 %, at 0.007 pu less margin at peak). A fixed power factor of 0.95 changes the median by −60 kW and the IEEE 1547-2018 Category B volt-var curve by 0, because the regulators’ line-drop compensation holds the voltage up; with taps frozen they add +110 and +60 kW, and with line-drop compensation off +50 and +10. Midday EV charging adds +10 kW because the load lands on the same 500 kVA transformer. The unmodified feeder reproduces the published IEEE voltages to 0.003 pu. Limits: one small feeder, snapshots not a day, conductor ampacities and the 1547 curve points recalled and not verified, EV load a rounded estimate, utility regulator settings not known.

Does the 13-node result of infra-hosting (a lower regulator set-point beats inverter reactive power) survive on larger feeders? The same OpenDSS ramp (PV added at 1–8 random customer loads until the first service voltage above 1.05 pu or transformer above its kVA rating; 200 placements, seed 1, every measure on identical placements) is run on the IEEE 123-node feeder (3,490 kW) and the 34-node feeder (1,769 kW). 123-node, load 0.3 of peak: baseline median 1,160 kW (10th–90th percentile 380–2,440); a 2 V lower set-point on all regulators +790 kW (better on 100 % of placements), fixed power factor 0.95 +180 kW (better 64 %, worse 34 %), the IEEE 1547-2018 Category B volt-var curve +100 kW (64 % / 28 %); the lowest service voltage at peak with no PV stays 0.960 pu at −2 V and falls to 0.940 at −4 V. The ranking holds, but inverter reactive power does help here, unlike on the 13-node feeder. The published 34-node case sets its source to 1.05 pu, so a regulated bus is already above 1.05 with no PV and hosting capacity is 0 for every placement and measure; with the source at 1.00 (an edit) the median is 590 kW and fixed power factor (+180) edges out the set-point (+140). Limits: snapshots, not a day; no conductor ampacities (the cases give none; worst line current at capacity is reported instead); the volt-var curve points are recalled, and the 34-node source edit is mine. Quasi-static day (added): the 123-node feeder run for a day at one-minute steps with the PV and load shapes shipped with the OpenDSS case (40 placements, 2,000 kW of PV, peak-load day): the baseline spends a median 247 min above 1.05 pu and 401 regulator tap operations; a 2 V lower set-point removes the overvoltage (0 min) but adds 193 min below 0.95 pu (baseline 11) and no fewer tap operations (412); fixed power factor 0.95 and volt-var cut overvoltage to 145 and 134 min and tap operations by about 30 % (292, 280) with no undervoltage. So the snapshot ranking reverses once undervoltage is counted. Limits: one PV shape (a cloudy day of unstated origin), regulators settle fully each minute, and some volt-var runs at 3,000 kW did not converge.

A two-state RC house (ASHRAE 140 Case 600 style envelope, recalled and not checked) with a heat pump (Staffell et al. 2012 COP fit) on NREL TMY3 hourly weather for Greensboro NC and Sand Point AK, a PI thermostat, and a receding-horizon LP controller (HiGHS, 24 h) on an assumed three-band tariff. The controller's model is wrong by a stated amount (true loss +15 to 30%, lighter, sunnier) and its forecast is yesterday's weather. Greensboro: $783 a year against $877 for a 20/24 thermostat (−10.7%), with 2,383 against 7,621 K h outside the 20–24 °C band; a thermostat tuned to 2,839 K h pays $1,300. Flat tariff 4.8%; Sand Point 4.1% and 2.4%. The saving grows with peak/off-peak ratio (4.9% to 14.7%) and thermal mass (6.5% to 25%), and vanishes at 8 K forecast error. Not sourced: heat capacities (swept), cooling COP, solar factor, tariff and 6 kW capacity (assumed); no measured building data was reachable, so nothing is fitted. Simulation only.

Operators screen N-1 security with a linear DC power flow. On pandapower's MATPOWER cases 14, 30, 39, 57 and 118, every line and transformer outage is solved as an AC power flow (truth: any other branch over its rating, a bus newly out of the 0.94–1.06 band, or non-convergence) and as a DC flow (screen). Ratings are a swept headroom over base-case flow, not taken from the cases. At 25 % headroom the DC screen misses 25, 17, 13, 40 and 14 % of insecure outages with almost no false alarms; flagging at 0.8 of rating misses none on any case but flags all 186 outages on case118 (84 truly insecure). Judging AC on MW alone cuts the miss rate (e.g. 14 to 12 % on case118, 40 to 20 % on case57), so reactive flow and voltage explain part. Checks: a closed-form 3-bus LODF case. Limits: base dispatch only (no redispatch), single outages, swept ratings (assumed), non-converged AC solves count as insecure, 20–186 outages per case. Modelled benchmark cases, not measured systems.

A lossless DC OPF orders generators by raw marginal cost; the classical fix scales each generator's cost by a penalty factor 1/(1−∂Loss/∂P_i). Sensitivities come from an AC power flow by finite difference, the DC OPF is re-solved with scaled costs and losses added to load, and the dispatch is scored by AC power flow at the original costs against the AC OPF (pandapower, MATPOWER cases, load 0.6–1.1). IEEE 118-bus: excess cost over the AC optimum falls from 0.73, 0.75, 1.48 % (plain DC) to 0.56, 0.46, 0.97 % at 0.6, 0.8, 1.0 of load, versus 0.64, 0.58, 1.18 % for a uniform loss share; at 1.1 it is 2.45 %, worse than both. The damped fixed-point iteration hit its 20-pass cap at 0.8, 1.0 and 1.1 (dispatch still moving 2.5–75 MW per pass), so those rows are not fixed points. Case30 and case39 differences are within 0.1 % and change sign; voltage violations are identical to plain DC. Checks: a three-bus case and two case118 inequalities. Limits: non-converged rows, no AC OPF solution at 1.1 of load on case30/39, no binding branch limits, no prices. Modelled benchmark cases, not measured systems.

A force-balance model of four freight modes (truck rolling plus air drag, Davis train resistance, Admiralty-law ship, aircraft at L/D) gives the cruise-only floor of 0.50, 0.10, 0.035 and 4.4 MJ of fuel per net tonne-km. Swapping the store at fixed gross mass and range: a 250 Wh/kg pack leaves a truck (800 km) 79 % of its diesel payload, a unit train (1,500 km) 93 %, a Panamax bulk carrier (20,000 km) 55 %, and a 747-400F cannot reach 8,250 km (669 km at zero payload). Keeping 90 % of the payload needs about 510, 180, 990 and 5,600 Wh/kg; 700-bar hydrogen keeps 95, 98 and 90 % on the first three, liquid hydrogen 90 % on the aircraft. Checks: a Semi-like truck gives 1.76 against a rated 1.7 kWh/mile (+4 %, assumed inputs); the diesel truck floor is 7.6 mpg against 5.5–6.5 for fleets; the train model reaches 0.46–0.61 of the FHWA rail figure because grades, stops and idling are not modelled. Limits: most drag, efficiency and tank-fraction inputs are assumed and only partly swept; vehicle structure mass is held fixed; no reserves, volume limits, cost or charging.

infra-pumpdemand billed Net3 pump schedules on PG&E Schedule B-19 with every day identical and found that schedules idling the 310 kW booster cut the bill 56–67 %. A demand charge is set by the highest 15-minute kW of the month, so this project re-simulates all 21,806 schedules with at most two daily on-runs at a 900 s step on days with every Net3 demand scaled by 0.8, 1.0, 1.2, 1.4 and 1.6, and bills 30-day months of ordinary days plus 1, 3 or 10 hot days. With demand 1.5 % above Net3’s shipped level the lowest feasible peak among one-run schedules (tanks above minimum, 20 psi, tanks back to within 0.1 % of the day’s demand) jumps from 63 kW to 310 kW, the booster; at +20 % it is 372 kW, the same as the shipped schedule. A month with three days at +20 % saves 43.5 % (summer) and 18.7–19.2 % (winter, spring) against the shipped schedule on the same month, against 56, 66 and 67 % with identical days; at +40 % 1at +40 % 3–14 %ndash;15 %; at +60 % no schedule meets the periodic tank test. Choosing a different schedule for the hot days than for ordinary days adds at most 4.2 points (summer; under 1 point in winter and spring) over one schedule feasible on every day. The day multiplier is a sweep, not a measured demand distribution; each day is simulated as periodic (tanks return to where they started), so a hot day cannot borrow from its neighbours; the threshold is for one-run schedules and a 0.1 % tolerance; B-19 is not the tariff a utility of Net3’s size is on, and it is one teaching network.

The 21,806 pump-10 schedules with at most two daily on-runs are simulated on EPANET’s Net3 (WNTR) at a 900 s step and billed on PG&E Schedule B-19 (secondary voltage, published rates effective March 2026): seasonal energy prices plus the peak, part-peak and any-time demand charges on the highest 15-minute kW. The shipped schedule peaks at 373 kW because booster pump 335 (about 310 kW) runs when tank 1 is low; its demand charges are 60–64 % of the shipped bill. Long lake-pump runs idle the booster and cut the peak to 63 kW, so the bill falls 56 % (summer), 66 % (winter) and 67 % (spring) with the tanks returned to within 0.1 % of daily demand (63–65 % with a 1 % tolerance), 62.5 % over a year. Pricing demand changes nothing about which schedule wins: the energy-only and full-bill optima are the same schedule, and stay so with demand rates scaled from 0 to 20 times. Time-of-use energy prices add under 1.5 points beyond the flat-energy optimum. The tariff is a real list price but not a tariff Net3’s size would be on (B-19 is mandatory above 499 kW), Net3’s demand is not seasonal, riders and taxes are left out, and it is one teaching network with a gravity river source.

WNTR simulates Net3 for three days with pump 10 (the lake source) on a repeating 24-hour schedule, scoring the last day on pump energy (Net3’s 75 % efficiency) and time-of-use cost under three assumed tariff shapes with the peak/off-peak ratio swept 1.5–10. Feasible schedules keep the tanks in range and the daily balance within 1 %; pressure and tank range never bind. At a flat tariff the best window (01:00, 21 h) uses 1,307 kWh a day against 2,728 as shipped, 52 % less, because Net3’s river supply is mostly gravity and the booster pump 335 only runs when tank 1 is low, so more lake pumping idles it: a saving that needs no tariff. Beyond that, knowing the tariff adds 3.3 % of shipped cost at ratio 1.5, 15.8 % at 3 and 95 % at 10 for a 16:00–21:00 peak, and 3.1/12.1/39.2 % for a daytime peak. An exhaustive search of all 21,806 schedules with at most two on-runs a day finds a second window worth 0–0.02 % of cost for 8 of 10 tariffs (9.5 % and 1.2 % for the other two); a genetic algorithm and a local search over all 2²⁴ hourly schedules beat that by up to 4 points, but most of it is tank drawdown within the 1 % tolerance, and priced it falls to about 0–1.3 points. A first version mis-simulated windows that cross midnight; corrected here. One network, assumed tariffs (none sourced), no demand charge, a 1 % mass-balance gap not traced.

An hourly IEEE C57.91-form top-oil and hot-spot model (ageing factor exp(15000/383 − 15000/(θ_H+273)), 180,000 h normal life) driven by 10 homes: the BDEW H0 profile at 10,800 kWh/yr each, NREL TMY ambient, a heat pump with a temperature-dependent COP, and EV charging either on arrival or valley-filled overnight. On a transformer sized at 0.5 per unit of baseline peak in Greensboro weather, loss of life is 0.0019 %/yr at baseline, 0.0075 at 50 % of homes with a heat pump and an EV, and 5.6 %/yr at 100 % with uncoordinated charging (about 18 years to end of life) against 1.07 %/yr with smart charging. Smart charging removes an EV-only load entirely (0.035 to 0.0026 %/yr) but not coincident heat-pump cold-snap peaks; at 0.7 per unit every heat-pump case fails. Uniform growth of 1.16–1.28× doubles loss of life but 2.2–3.1× is needed to reach the rated-life rate. Limits: the standard’s constants (55 K top-oil rise, R 3.2, n=m=0.8, τ 3 h) are recalled, not verified, and are swept; heat pumps are fully coincident with no setback; TMY weather rather than extremes; a German load shape on US weather; smart charging has perfect foresight.