# Mathematical foundations for energy networks: buffering, storage and transmission

Lead Research Organisation:
University of Cambridge

Department Name: Pure Maths and Mathematical Statistics

### Abstract

Electrical power grids are complex networked systems. Demand andsupply must be balanced on a minute-by-minute basis and there arelimited opportunities for large-scale storage. Further, flows innetworks are subject to the laws of physics, so that there is verylittle control over the routing of flows; generating capacity cannotin general be instantly switched on or off; sources of generationcapacity, whether renewable or nuclear, are often located far from theurban and industrial areas they must serve. In today's market theprovision of generation capacity is typically determined by marketforces in which many competing operators each seek to optimize theirown returns.The need to reduce carbon emissions has led to new policy which willtransform the grid. Notably, renewable sources such as wind powerproduce supplies which are highly variable, and often unpredictableeven on relatively short time scales. To combat this variability theintroduction of demand response through dynamic prices has beenproposed. There is also significant future potential for thebuffering and storage of electrical energy over short time scales.These possibilities are integrated through the advent of smart gridtechnology, with the possibility of real-time price signalling towhich consumers may respond flexibly. Further, the availability tothe network of significant short-term buffering and storage, alongwith the ability to time-shift demand, should assist in the avoidanceof transient monopolies (localised in space or time) which isconsidered to be one of the reasons for the problems encountered inthe deregulated market in California in the last decade.The energy grid of the future thus poses formidable challenges forengineers and mathematicians. Among the questions to be answered are:- will geographic diversity of supply help to reduce volatility?- will demand response through pricing help to reduce the impact of volatility?- to what extent can buffering and storage assist in the balancing of supply and demand?- what is the effect of power system dynamics in a volatile network?- how may we schedule generation units and calculate efficient reserves for a reliable grid in this more complex setting?- how do we do better forecasting in this new world?We propose to develop mathematical techniques to assist in answeringthese questions, to measure the costs of addressing the volatilitiesin future networks, and to assess the comparative effectiveness of thevarious forms of time- and space-shifting of energy which may be used;this will then enable the benefits of such measures to be tradedagainst each other. We shall develop these techniques in the contextof the transmission and distribution networks: while buffering,storage and the time-shifting of demand all correspond to movingenergy through time, the ability of the network to move energy throughspace - determined by the capacities in its links and the laws ofphysics - is inextricably linked to the benefits of moving energythrough time.There are two major and interlinked themes: (a) the development of themathematics of volatility in energy networks: of particular importancehere is the creation of a calculus of effective capacities, formeasuring capacity required by flows exhibiting volatility on a rangeof time- and space-scales, and for determining those time- andspace-scales which are of critical importance in the operation of anetwork; and (b) the development of advanced probabilistic techniquesfor measuring the effects of extreme events in networks. These twothemes together provide the results necessary to assess, control andoptimize the performance of energy networks, and to devise the pricingand incentivisation schemes for competing suppliers, operators andconsumers so as to maximise economic efficiency.

### Publications

Buzna L

*Controlling Congestion in Complex Networks*in Scientific Reports
Buzna L
(2017)

*Controlling congestion on complex networks: fairness, efficiency and network structure.*in Scientific reports
Carvalho R
(2015)

*Critical behaviour in charging of electric vehicles*in New Journal of PhysicsDescription | The increasing penetration of electric vehicles over the coming decades, taken together with the high cost to upgrade local distribution networks and consumer demand for home charging, suggest that managing congestion on low voltage networks will be a crucial component of the electric vehicle revolution and the move away from fossil fuels in transportation. We have modelled protocols for the control of congestion caused by a fleet of vehicles charging on two real-world distribution networks. We show that the system undergoes a phase transition to a congested state as a function of the rate of vehicles plugging to the network to charge. We show that charging times are considerably more equitable in proportional fairness than in max-flow. |

Exploitation Route | Several questions are raised, and this is a rich field. |

Sectors | Energy,Transport |

URL | http://iopscience.iop.org/article/10.1088/1367-2630/17/9/095001 |

Description | The findings have contributed to ongoing discussions with government agencies and system operators on the impact of renewable sources of energy on the management of electricity grids - see e.g. "Analytic Research Foundations for the Next-Generation Electric Grid" - https://www.nap.edu/catalog/21919/analytic-research-foundations-for-the-next-generation-electric-grid |

First Year Of Impact | 2016 |

Sector | Energy |

Impact Types | Policy & public services |

Title | Large Scale Fast Response Energy Storage Model |

Description | The model represents a stochastic model for large scale fast response energy storage and slow-to-moderate ramping generators with high wind penetration. The model defines a strategy for operating the storage facility and allows one to investigate the system-wide long-term effects of fast response energy storage in reducing the amount of conventional power used. In particular, trade-offs between various system performance quantities, including wind spill and the loss of load probability can be analysed using this model. |

Type Of Material | Computer model/algorithm |

Year Produced | 2012 |

Provided To Others? | Yes |

Impact | We studied statistical properties of the above novel stochastic model applied to a large scale electricity system with aggregated storage (e.g. water pumped storage) and integrated renewables (wind power) when conventional power have to be scheduled some time in advance. This work is important as it looks at the interplay of renewables and energy storage as a trade-off between various system characteristics, in particular, the ability to curtail wind and energy lost to the system. This model has been further extended and its optimality studied and improved upon by the research community starting with the Sigmetrics 2012 paper 'Optimal Storage Policies with Wind Forecast Uncertainties' by Gast, N., Tomozei, D., and Le Boudec, J.-Y. The novelty of the introduced model was also emphasised in this latter paper. |

URL | http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6310769 |