Balancing electricity supply and demand
R.S.MacKay@warwick.ac.uk, Mathematics

Each device, whether production, consumption, interconnector or storage, has its own operating characteristics in terms of response-timescales, limits and costs.  Ignoring spatial distribution, which at least for the UK is considered secondary, the problem is to match supply and demand.  It is generally considered that this should be achieved by market mechanisms, i.e. price.  

At present, the National Grid market operates in half-hour blocks (see http://www.bmreports.com/), leaving National Grid to fine-tune the balance on shorter timescales by various forms of direct intervention.  But it would seem reasonable to propose a move to continuous pricing.

To achieve continuous pricing, account needs to be taken of the temporal characteristics of the various supplies, demands and storage facilities.

We propose that each device d has a cost functional C_d[f] for the cost of providing power f as a function of t.  For a device that consumes power, in general the cost is negative, representing a benefit.  For an unbounded time domain, the cost may well be undefined, but the important aspect will be its variations with respect to variations of f over bounded time intervals.

Given a public price p, as a function of time, each device owner wishes to maximise the value V_d[f] = \int f_d(t)p(t)dt - C_d[f_d] of the power f_d they provide.

The price p is determined by the requirement to match supply and demand.  \sum_d f_d(t) = 0 for all t.

This is a multi-player game, of "mean-field" type, but dynamic and with uncertainty, as for example, wind turbines can supply power only up to a maximum function of time that depends on the wind, which may be known in advance at a statistical level but not in detail.

The challenge is to solve this game.  Is there a unique solution?  How close to a social optimum is it?  Are there antisocial behaviours it would encourage?  

An advantage of this approach is that storage is treated on the same footing as any other balancing mechanism, like elastic demand or supply.  Simply it has the option of both signs of f, but the disadvantage in general of losses.  This will allow for an objective comparison of storage solutions with any other solutions.

Some references:
P.Joskow, Dynamic Pricing of Electricity, 29 Nov 2011, www.aeaweb.org/aea/2012conference/program
M.C. Caramanis, J.M. Foster “Management of Electric Vehicle Charging to Mitigate Renewable Generation Intermittency and Distribution Network Congestion,” 48th IEEE CDC, Dec. 2009, pp. 4717-4722
R.D. Tabors, G. Parker, M.C. Caramanis, “Development of the Smart Grid: Missing Elements in the Policy Process,” IEEE Hawaii International Conference on System Sciences, (HICSS-43) Jan. 2010, pp. 978-0-7695-3869-3/10-1-6.
J.M. Foster, M.C. Caramanis, Energy Reserves and Clearing in Stochastic Power Markets: The Case of Plug-In-Hybrid Electric Vehicle Battery Charging, 49th IEEE Conference on Decision and Control, Dec 2010, , pp.1037-1044
MC Caramanis & JM Foster, Uniform and Complex Bids for Demand Response and Wind Generation Scheduling in Multi-Period Linked Transmission and Distribution Markets CDC2011 conference

The project has huge potential end-user impact on National Grid, DECC, electricity storage technologies…

There is strong potential to develop it into a PhD.  Possible co-supervisors would be Vassili Kolokoltsov (Stats), Mike Waterson (Economics).