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Balancing the equations of a low carbon energy network

Rachel Thomas

The UK is aiming for a decarbonised electricity supply by 2035. Generating electricity without the use of fossil fuels is not just an engineering and industrial challenge, it is also a huge mathematical challenge. "We need to force the pace on research and innovation if we are going to achieve the challenging target of decarbonising the electricity supply by the mid 2030s," says Chris Dent, Professor of Industrial Mathematics at the University of Edinburgh. "For such a radical paradigm shift 2035 is not far away."

Dent organised an intensive two week "deep dive" workshop on the Mathematics and statistics for low carbon energy systems earlier this year as part of a longer research programme at the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge. The typical rule of thumb for a technology to go from basic research to wide rollout in the real world is about 20 years. But the decarbonisation targets mean this timeline is reduced to a decade or less. What is more, this research needs to be applied in places such as the control rooms of power stations, which are of critical importance to our infrastructure. "There are extremely tight targets for achieving this in the energy system."

The calculations of power

A series of electricity pylons linked together by overhead wires

One of the most complex systems we've ever created.

When you flip a switch in your home you are tapping into one of the most complex systems humans have ever created – the electricity network. As well as an energy supply, a lot of maths is required to keep the network running smoothly, in particular to ensure the right amount of power is supplied to all parts of the system at all times.

In particular, maths is needed when you want to move things around a network, to make sure that they go where they are supposed to go. In a communications network, the messages (say traditional phone calls or the data packets of digital communication) need to be routed from individual senders to the specified recipients. In a power network it's electrons that need to be routed – this makes things a little simpler because, unlike phone calls, electrons are all identical.

Although you, as a household user, have a contract with a specific energy supplier, in reality all the electrons generated by all the suppliers go into one big pool across the electricity network. "Then you 'just' have to deal with what the 20 million or so end users are doing and what that implies physically for the flow of those electrons through the wires. You don't have the additional complexity of tracing specific electrons destined for specific end users." However this doesn't mean that running power networks is an easy task, far from it.

We expect the lights in our home to go on whenever we want, and we expect them to stay on as long as we want. But the demand in each part of the whole system varies over time and varies across the system. For example less power is drawn overnight than during the day, and demand for industrial use is very different from the power needs of a residential home. But unexpected events can test a power network, such as when a problem at a large generating unit causes it to suddenly stop contributing power to the network.

To manage this challenge, the scheduling of power supplied to the network happens centrally – essentially the power operators decide when to generate electricity at a power station to supply it to the network to match the forecasted demands. Forecasting this demand and scheduling of supply was already difficult enough for a traditional electricity network which relied mainly on coal, gas and nuclear power stations. As we move to renewable and low carbon energy sources, these questions of balancing the supply and demand for electricity become much trickier.

Historically in Britain there were just a few hundred large generating units (each power station would have multiple such units) interacting with the electricity network. Today there are thousands of generating units in the system, including some generating power through wind turbines and solar panels. The biggest increase in these numbers comes from renewable sources acting at a much more local level, with industrial users, and even households, generating electricity that can meet local demands, or even feed back into the grid.

wind farm

We're currently still able to centrally balance the supply of energy from power stations with the demand from users across the network over time. But integrating increasing amounts of renewable energy into the system means managing additional uncertainty: When will the wind blow and the sun shine? How can we make the best use of the electricity whenever it is generated? And how can we ensure that electricity will always be available when we need it?

"As we're looking to the future of energy systems we're looking to a much greater complexity," says Dent. "And besides, even without these new real world challenges, a better scheduling algorithm would mean we could manage our existing network better."

Solving the problem of a low carbon power network

Mathematics offers well developed ways of managing this sort of uncertainty. One area of maths that is particularly useful is optimisation, where you are trying to minimise or maximise some quantity, technically known as a cost function. Applying optimisation techniques allows you to develop algorithms that can reliably take you to an optimal solution of your problem every time. We currently apply this optimisation approach to a power network centrally, but we will need to move to a decentralised approach as we move to a more complex future.

One way a decentralised approach could help is it could more flexibly manage the demand from users. We are pretty inflexible about when we want to turn on the lights or cook dinner. But an opportunity is arising with the significant shift to electric vehicles and electric heating in the coming years. Electric cars can be charged anytime they are not in use. And as it is relatively cheap to locally store heat (for example as hot water in a large tank) the heat from an electric boiler could be stored at any time for use when needed. But if we don't coordinate the demands of all these different small users, we could get enormous extra demands at some peak time when you add up the demands across the whole country.

"[Electric cars and electric heating] are ultimately demands on the system but they are naturally more flexible in terms of when they can take energy from the grid," says Dent. "For example they could take energy overnight or at a time that makes better use of renewable generation." Scheduling of these local demands could be influenced through flexible pricing algorithms, for example making the price of power cheaper when there is less demand. Such a decentralised solution would work across the power network at different scales: aggregating the demands at different points in the network, comparing that with the energy being generated, and cascading changes in the way energy is priced down through the network to the users drawing energy from the grid. "It's important to emphasise that we not only need to balance supply and demand across this large number of entities, but we also need to respect the similarly large number of constraints that local network capacity imposes," says Dent.

Lightbulb with filament formed into a network or brain like structure

Can maths help solve the problem?

Solving such a decentralised optimisation problem is one challenge, but making sure the solution you have is physically sensible is another. "You not only need a solution that works at a single point in time – the system also needs to remain a good solution over time," says Dent. "Also, we need to make sure the algorithm reaches the good solution that it's intended to rather than wandering off into some kind of crazy behaviour. Not all approaches that look sensible at first glance actually deliver good outcomes in practical deployment, and mathematics can help provide guarantees that algorithms will actually perform well in practice."

Another difficulty Dent and his colleagues are looking at is how to keep the amount of data exchanged in such a decentralised algorithm relatively low. "Many such algorithms involve everything exchanging information with everything else many times in a single calculation, in which case you'd have to overlay so much communication infrastructure over the energy infrastructure it wouldn't be practically viable." Instead, ingenious methods have to be developed to optimise the amount of information shared across the network. (You can see an example of how such a distributed communication model works for distributed computing here.)

Collaboration is power network of maths

The deep dive workshop continues a series of events that have allowed Dent and his colleagues to build up this area of mathematics. It all started with the INI programme, Stochastic processes in communication sciences, held in 2010, which included an energy systems week. "That was the first big gathering of people working on the maths of energy in the UK, which identified common areas of potential mathematical interest," says Dent. (You can read more about that event here.)

There have been many events over time since then, both at the INI (such as the workshop on Maths and Public Policy for Cities & Infrastructure in 2015 and the longer research programme on the mathematics of energy systems in 2019), and elsewhere (such as the 2018 event on the management of energy networks at the International Centre for Mathematical Sciences in Edinburgh).

As such, this deep dive workshop, and the longer research programme it is a part of, demonstrate that this type of scientific meeting powers mathematical progress. "Bringing people together for these sorts of discussion is the infrastructure of the mathematical sciences," says Dent. "The deep dive [workshop] is a catalyst for various conversations, drawing people from mathematics, computer science and [other areas] into energy innovation. They all have much to contribute."

All parties involved in energy innovation, particularly those that fund it, are keen to accelerate these conversations between disciplines to meet the challenges we face. This deep dive workshop was a chance to bring people together who might develop entirely new collaborations. And the longer research program gives researchers an unrivalled opportunity of time and space to work on the area. "It blew the minds of some colleagues from outside mathematics that an institute would bring together people for as long as a week, let alone six months, without a very specific ask of them," says Dent. "However over its 30 years of existence, the Isaac Newton Institute has demonstrated how productive these kinds of visitor programmes are in bringing together key people to advance an area of science."

About this article

Chris Dent is the Professor of Industrial Mathematics at the University of Edinburgh.

Rachel Thomas, Editor of Plus, interviewed Dent in April 2023.

This content was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.

The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit to find out more.

INI logo

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