# Juggling batteries in the face of uncertainty

In this post, I discuss a toy problem that has recently been the focus of a number of studies. I will show how this problem has practical relevance and sits at the core of the energy transition - and how we might go about solving this problem.

Assume that you are faced with a shortage of electrical power generation, for example because the wind is not blowing and it is dark outside, or because a few large electricity generators have failed. You have at your disposal a stack of charged electrical batteries with different specifications. Some of the these batteries have a high power rating, and low energy rating (supercapacitors would be an extreme example of such a battery), and others have a low power rating but high energy rating (such as a large hydrogen tank with a fuel cell). Which batteries should you use first, if you know how much power is required right now, but do not know how much power is required in the future? Can you avoid getting to a situation where you can no longer supply all demand - but you could have if only you had used your batteries differently? This is the optimal battery dispatch problem.

Although it is simple to state, the problem combines in a nutshell three key challenges of the energy transition:

1. The increasing prevalence of renewable power generation from wind turbines and PV panels leads to growing uncertainty about supply and demand, from minutes to weeks.
2. Instead of a few large controllable resources, the power system is increasingly controlled through deliberate actions in multitudes of smaller devices with very significant differences in their capabilities.
3. Whereas traditional power plants are often assumed to have a limitless supply of energy (i.e. fuel), batteries and battery-like resources (hydro, demand response, etc.) have strict energy limits in addition to power limits.
It is important to stress that the optimal battery dispatch problem can be considered at the scale of a single household (e.g. a home battery), but also for a (trans)national energy system.

It turns out that the solution to the problem sketched above is quite intuitive - although proving it is a bit more challenging (Evans et al., 2018). The important quantity to consider is the time-to-go (TTG): the time a battery can run for at full power until it is empty. The optimal way to use the batteries ('policy') in this case is to preferentially discharge those batteries with the largest TTG. In practice, this means that you avoid - whenever possible - using those batteries with the lowest TTG. As a result, you keep them at hand for when you a sudden high demand for power occurs that requires all batteries to participate.

Perhaps surprisingly, this solution is optimal regardless of what happens in the future: it is a 'greedy' policy that cannot be improved by forecasting (Evans et al., 2018). Interestingly, it turned out the same mathematical problem was previously stated and solved in a different setting: the optimal control of water reservoirs in the UK water network (Nash and Weber, 1978). Moreover, it turns out that the largest-TTG-first policy has additional useful features: even if the batteries are unable to supply all demand, the policy does so in a way that minimizes the amount of energy that is not supplied (Evans et al., 2019; Cruise and Zachary, 2018). This has important applications for the use of batteries in capacity markets, which pay resources (including batteries) for the contribution they make to security of supply. This policy can be used to assign a well-defined number to this contribution, instead of relying on various heuristics (National Grid, 2017).

Unfortunately, the beguiling simplicity and generality of these solutions vanishes when we also consider recharging of the batteries. As a simple example, consider an electric vehicle with two batteries: one long-range battery with limited power and a high-power battery with limited energy for acceleration. If we simply wish to maximise the acceleration potential during a drive, we would use the long-range battery whenever possible, and only use the smaller high-power battery if the extra power is required. However, consider what happens if we also want to recharge as quickly as possible during a stopover. In that case, it may be a good idea to consume energy from both batteries, so that the two batteries can recharge in parallel. The optimal discharging and recharging policies will depend in complex ways on the expected usage pattern of the vehicle.

This generalised problem is the subject of ongoing research. Clearly, the elegant simplicity of the discharging-only case does not apply when charging is considered, but I have good hope that other simplifying principles can be discovered, which will help in creating scalable solutions for smart grids.

A video of a talk I gave on this subject is available here.

References:

Robustly Maximal Utilisation of Energy-Constrained Distributed Resources, M Evans, SH Tindemans, D Angeli, 2018, Power System Computation Conference (PSCC 2018), Dublin.
• A simple optimizing model for reservoir control, P Nash, RR Weber, 1978, Technical Report
Optimal scheduling of energy storage resources, J Cruise, S Zachary, 2018, arXiv:1808.05901
Minimizing Unserved Energy Using Heterogeneous Storage Units, M Evans, SH Tindemans, D Angeli, 2019, Transactions on Power Systems
Duration-Limited Storage De-Rating Factor Assessment Final Report, National Grid, 2017, Technical Report