Abstract: Australian households rely on heat-pump air conditioners for summer cooling and winter heating. Air conditioners are often operated during periods of peak electricity demand in the morning and evening, when prices are highest. Thermal energy storage technologies can shift energy consumption to off-peak periods, but optimal scheduling must also account for temperature’s impact on heat-pump efficiency. I developed a mathematical model of an air-conditioning system with thermal storage and used Pontryagin’s principle from 1952 to determine optimal ‘on/off/partial’ operating modes with precise switching times to minimise operating cost while ensuring comfort.
Many Australian households need cooling in summer and heating in winter. Most people use heat-pump air conditioners and switch them on when uncomfortable. The problem is timing—this often happens in the morning or evening when electricity is most expensive.
For many households, electricity prices follow a daily cycle:
- moderate overnight when demand on the grid is low
- lower from 10 am until 4 pm, with plentiful solar
- highest in the morning and evening when demand peaks.
Unfortunately, the times we most want heating or cooling often line up with those peak prices, so the bill can be higher than it needs to be.
Thermal storage can help break this link. Thermal storage is a `heat/cold battery’. When electricity is cheap, a heat pump can charge the store by heating or cooling water. Later, when prices rise, stored energy can warm the home or help remove heat. This turns one question—when to turn on the air conditioner? —into three practical decisions:
- how much energy to store
- when to store it
- when to release it into the house.
There is a trade-off. Heat-pump efficiency depends on outdoor temperature. For heating, it often works better around midday when it is warmer outside, which coincides with lower prices. For cooling, midday can be less efficient because the outdoor temperature is higher. So should we use cheaper electricity with lower efficiency, or pay more when the heat pump runs better?
To answer this, we model the system and formulate an optimal control problem. We control the heat-pump power, and how fast energy is transferred from the store into the home. The aim is to minimise cost while keeping temperature comfortable. Pontryagin’s Principle (1952) shows the best strategy uses simple control modes:
- the heat pump should be fully on or fully off, unless the store is full or empty,
- heat or cold transfer to the house should be either no transfer or maximum transfer, unless the house is at the desired temperature.
By analysing optimality conditions, we can determine the sequence of operating modes for the heat pump and the timing of mode transitions, while establishing the sequence and timing for switching between heat transfer modes.
This project is a demonstration of how mathematics can be put into practice. For me, elegant theories are merely the starting point—what truly matters is transforming them into tools that can explain how things work, answer questions, and provide real-world benefits.
Rong Xu
Adelaide University
