Littlewood’s Rule for a Home Battery

Ontario’s Ultra-Low Overnight electricity plan charges 3.9 cents a kilowatt-hour overnight and 39.1 cents between 4 and 9 PM on weekdays, and home batteries are sold on that tenfold spread. This paper asks how such a battery should be run when the price is set by the clock, the household’s own evening usage is uncertain and stored energy cannot be sold back, and when, if ever, a household should buy one. Running the battery is a revenue-management problem: the morning and evening windows are two fare classes for the same stored energy. The optimal policy fills the battery every night and during the morning mid-peak discharges only down to a reserve set by Littlewood’s rule, extended for a second mid-peak window after 9 PM and for the cost of refilling overnight. On 462 days of one house’s hourly usage, reconciled to its twelve bills, the reserve for a 13.5 kWh battery is 11.9 kWh in summer, 7.4 in spring and autumn and 11.4 in winter. It earns $757 a year off the bill, 94% of what a battery that knew each day’s usage could earn and 92% when each month’s reserve is set without that month, against $567 for discharging whenever the house draws. Buying the battery is McDonald and Siegel’s problem of investing when both the project’s value and its cost are random. Its savings over a ten-year warranted life are worth about $5,500 against an installed cost of $19,000 to $24,000, and on the regulator’s prices since 2006 and the market’s storage prices since 2021 the HJB variational inequality puts the installation boundary at an installed cost of $430 to $2,700, depending on the drifts one believes, decades away on present trends.