# American Institute of Mathematical Sciences

2016, 3(4): 371-398. doi: 10.3934/jdg.2016020

## Optimal strategies for operating energy storage in an arbitrage or smoothing market

 1 Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, United Kingdom, United Kingdom 2 Economics Department, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, United Kingdom

Received  March 2016 Revised  September 2016 Published  October 2016

We characterize cost-minimizing operating strategies for an energy store over a given interval of time $[0,T]$. The cost functional here can represent, for example, a traditional economic cost or a penalty for time-variation of the output from a storage-assisted wind farm or more general imbalance between supply and demand. Our analysis allows for leakage, operating inefficiencies and general cost functionals. In the case where the cost of a store depends only on its instantaneous power output (or input), we present an algorithm to determine the optimal strategies. A key feature is that this algorithm is localized in time, in the sense that the action of the store at a time $t\in[0,T]$ requires cost information over only some usually much shorter subinterval of time $[t,t_k]\subset[t,T].$
Citation: Lisa C Flatley, Robert S MacKay, Michael Waterson. Optimal strategies for operating energy storage in an arbitrage or smoothing market. Journal of Dynamics & Games, 2016, 3 (4) : 371-398. doi: 10.3934/jdg.2016020
##### References:
 [1] , Compressed Air Energy Storage Power Plants,, BINE Informationsdienst projektinfo 05/07., (). [2] , ADELE - Adiabatic Compressed-Air Energy Storage for Electricity Supply,, RWE Power., (). [3] , \url{https://www.elexonportal.co.uk/}, (). [4] , Energy Storage and Management Study,, AEA, (2010). [5] , Options for low-carbon power sector flexibility to 2050 - a report to the Committee on Climate Change,, Pöyry, (2010). [6] , Study of Compressed Air Energy Storage with Grid and Photovoltaic Energy Generation, Draft Final Report - for Arizona Public Service Company,, Arizona Research Institute for Solar Energy (AZTISE), (2010). [7] M. Aunedi, N. Brandon, D. Jackravut, D. Predrag, D. Pujianto, R. Sansom, G. Strbac, A. Sturt, F. Teng and V. Yufit, Strategic Assessment of the Role and Value of Energy Storage Systems in the UK Low Carbon Energy Future - Report for Carbon Trust,, (2012)., (2012). [8] M. Black and G. Strbac, Value of bulk energy storage for managing wind power fluctuations,, IEEE Transactions on Energy Conversion, 22 (2007), 197. doi: 10.1109/TEC.2006.889619. [9] J. P. Barton and D. G. Infield, Energy storage and its use with intermittent renewable energy,, IEEE Transactions on Energy Conversion, 19 (2004), 441. doi: 10.1109/TEC.2003.822305. [10] J. Cruise, L. C. Flatley, R. Gibbens and S. Zachary, Optimal Control of Storage Incorporating Market Impact and with Energy Applications,, (2015), (2015). [11] J. Cruise, L. C. Flatley and S. Zachary, Impact of Storage Competition on Energy Markets,, working paper, (2016). [12] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty,, Princeton UP, (1994). [13] M. Giulietti, L. Grossi and M. Waterson, Price transmission in the UK electricity market: Was NETA beneficial?,, Energy Economics, 32 (2010), 1165. doi: 10.1016/j.eneco.2010.01.008. [14] P. Grünewald, T. Cockerill, M. Contestabile and P. Pearson, The role of large scale storage in a GB low carbon energy future: Issues and policy challenges,, Energy Policy, 39 (2011), 4807. [15] N. Löhndorf and S. Minner, Optimal day-ahead trading and storage of renewable energies - an approximate dynamic programming approach,, Energy Syst., 1 (2010), 61. [16] D. J. C. MacKay, Sustainable Energy - without the hot air,, Am. J. Phys., 78 (2010). doi: 10.1119/1.3273852. [17] R. S. MacKay, S. Slijepčević and J. Stark, Optimal scheduling in a periodic environment,, Nonlinearity, 13 (2000), 257. doi: 10.1088/0951-7715/13/1/313. [18] A. J. Pimm and S. D. Garvey, The economics of hybrid energy storage plant,, International Journal of Environmental Studies, (2014), 787. doi: 10.1080/00207233.2014.948321.

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##### References:
 [1] , Compressed Air Energy Storage Power Plants,, BINE Informationsdienst projektinfo 05/07., (). [2] , ADELE - Adiabatic Compressed-Air Energy Storage for Electricity Supply,, RWE Power., (). [3] , \url{https://www.elexonportal.co.uk/}, (). [4] , Energy Storage and Management Study,, AEA, (2010). [5] , Options for low-carbon power sector flexibility to 2050 - a report to the Committee on Climate Change,, Pöyry, (2010). [6] , Study of Compressed Air Energy Storage with Grid and Photovoltaic Energy Generation, Draft Final Report - for Arizona Public Service Company,, Arizona Research Institute for Solar Energy (AZTISE), (2010). [7] M. Aunedi, N. Brandon, D. Jackravut, D. Predrag, D. Pujianto, R. Sansom, G. Strbac, A. Sturt, F. Teng and V. Yufit, Strategic Assessment of the Role and Value of Energy Storage Systems in the UK Low Carbon Energy Future - Report for Carbon Trust,, (2012)., (2012). [8] M. Black and G. Strbac, Value of bulk energy storage for managing wind power fluctuations,, IEEE Transactions on Energy Conversion, 22 (2007), 197. doi: 10.1109/TEC.2006.889619. [9] J. P. Barton and D. G. Infield, Energy storage and its use with intermittent renewable energy,, IEEE Transactions on Energy Conversion, 19 (2004), 441. doi: 10.1109/TEC.2003.822305. [10] J. Cruise, L. C. Flatley, R. Gibbens and S. Zachary, Optimal Control of Storage Incorporating Market Impact and with Energy Applications,, (2015), (2015). [11] J. Cruise, L. C. Flatley and S. Zachary, Impact of Storage Competition on Energy Markets,, working paper, (2016). [12] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty,, Princeton UP, (1994). [13] M. Giulietti, L. Grossi and M. Waterson, Price transmission in the UK electricity market: Was NETA beneficial?,, Energy Economics, 32 (2010), 1165. doi: 10.1016/j.eneco.2010.01.008. [14] P. Grünewald, T. Cockerill, M. Contestabile and P. Pearson, The role of large scale storage in a GB low carbon energy future: Issues and policy challenges,, Energy Policy, 39 (2011), 4807. [15] N. Löhndorf and S. Minner, Optimal day-ahead trading and storage of renewable energies - an approximate dynamic programming approach,, Energy Syst., 1 (2010), 61. [16] D. J. C. MacKay, Sustainable Energy - without the hot air,, Am. J. Phys., 78 (2010). doi: 10.1119/1.3273852. [17] R. S. MacKay, S. Slijepčević and J. Stark, Optimal scheduling in a periodic environment,, Nonlinearity, 13 (2000), 257. doi: 10.1088/0951-7715/13/1/313. [18] A. J. Pimm and S. D. Garvey, The economics of hybrid energy storage plant,, International Journal of Environmental Studies, (2014), 787. doi: 10.1080/00207233.2014.948321.
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