# American Institute of Mathematical Sciences

February  2019, 2(1): 55-71. doi: 10.3934/mfc.2019005

## Online optimization for residential PV-ESS energy system scheduling

 1 1400 Washington Ave, Mathematics and Statistics Department, Albany, NY 12222, USA 2 251 Fuller Road, Atmospheric Sciences Research Center, Albany, NY 12203, USA

* Corresponding author: Yiming Ying

Published  March 2019

Fund Project: The second author is supported by NSF grant 1816227

This paper studies a residential PV-ESS energy system scheduling problem with electricity purchase cost, storage degradation cost and surplus PV generated cost [2]. This problem can be viewed as an online optimization problem in time $t \in [1, T]$ with switching costs between decision at $t-1$ and $t$. We reformulate the problem into a single variable problem with ${\bf{s}} = (s_1, ..., s_T)^T$, which denotes the storage energy content. We then propose a new algorithm, named Average Receding Horizon Control (ARHC) to solve the PV-ESS energy system scheduling problem. ARHC is an online control algorithm exploiting the prediction information with $W$-steps look-ahead. We proved an upper bound on the dynamic regret for ARHC of order $O(nT/W)$, where $n$ is the dimension of decision space. This bound can be converted to a competitive ratio of order $1+O(1/W)$. This result overcomes the drawback of the classical algorithm Receding Horizon Control (RHC), which has been proved [11] that it may perform bad even with large look ahead $W$. We also provide a lower bound for ARHC of order $O(nT/W^2)$ on the dynamic regret. ARHC is then used to study a real world case in residential PV-ESS energy system scheduling.

Citation: Zhenhuan Yang, Yiming Ying, Qilong Min. Online optimization for residential PV-ESS energy system scheduling. Mathematical Foundations of Computing, 2019, 2 (1) : 55-71. doi: 10.3934/mfc.2019005
##### References:
 [1] L. Andrew and S. Barman and K. Ligett and M. Lin and A. Meyerson and A. Roytman and A. Wierman, A tale of two metrics: Simultaneous bounds on competitiveness and regret, SIGMETRICS '13 Proceedings of the ACM SIGMETRICS/International Conference on Measurement and Modeling of Computer Systems, 2015, 329-330, arXiv: 1508.03769. doi: 10.1145/2465529.2465533. [2] A. Chis and J. Lunden and V. Koivunen, Coalitional game based cost optimization of energy portfolio in smart grid communities, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017. [3] O. Babacan, E. L. Ratnam, V. R. Disfani and J. Kleissl, Distributed energy storage system scheduling considering tariff structure, energy arbitrage and solar PV penetration, Applied Energy, 205 (2017), 1384-1393. doi: 10.1016/j.apenergy.2017.08.025. [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press New York, 2004. doi: 10.1017/CBO9780511804441. [5] M. Dubarry, C. Truchot and B. Y. Liaw, Synthesize battery degradation modes via a diagnostic and prognostic model, Journal of Power Sources, 219 (2012), 204-216. doi: 10.1016/j.jpowsour.2012.07.016. [6] M. A. Epelman, Barrier Methods for Constrained Optimization, 2012. Available from: http://www-personal.umich.edu/ mepelman/teaching/NLP/Handouts/NLPnotes12_89.pdf. [7] C. Graves, New York Mandatory Hourly Pricing Program Case 03-E-0641, 2009. Available from: https://www.nyiso.com/documents/20142/1404435/PRLWG_MHP_Overview.pdf/0d12b88b-2921-23c5-4c0f-2edb1d16257a. [8] E. Hazan, Introduction to online convex optimization, Foundations and Trends in Optimization, 2 (2015), 157-325. doi: 10.1561/2400000013. [9] P. L. Joskow and C. D. Wolfram, Dynamic pricing of electricity, American Economic Review, 102 (2012), 381-385. doi: 10.1257/aer.102.3.381. [10] Y. Li, G. Qu and N. Li, Online optimization with predictions and switching costs: Fast algorithms and the fundamental limit, 2018 Annual American Control Conference (ACC), 2018, arXiv: 1801.07780. doi: 10.23919/ACC.2018.8431296. [11] M. Lin, Z. Liu, A. Wierman and L. L. H. Andrew, Online Algorithms for Geographical Load Balancing, 2012 International Green Computing Conference (IGCC), 2012. doi: 10.1109/IGCC.2012.6322266. [12] A. C. Luna, N. L. Diaz, M. Graells, J. C. Vasquez and J. M. Guerrero, Online energy management system for distributed generators in a grid-connected microgrid, 2015 IEEE Energy Conversion Congress and Exposition (ECCE), 2015. doi: 10.1109/ECCE.2015.7310313. [13] X. Luo, J. Wang, M. Dooner and J. Clarke, Overview of current development in electrical energy storage technologies and the application potential in power system operation, Applied Energy, 137 (2015), 511-536. doi: 10.1016/j.apenergy.2014.09.081. [14] M. Morari and H. H. Lee, Model predictive control: Past, present and future, Computers and Chemical Engineering, 23 (1999), 667-682. doi: 10.1016/S0098-1354(98)00301-9. [15] National Grid, Hourly Electric Supply Charges, Available from: https://www9.nationalgridus.com/niagaramohawk/business/rates/5_hour_charge.asp. [16] National Solar Radiation Data Base, Hourly Solar Data, Available from: https://rredc.nrel.gov/solar/old_data/nsrdb/. [17] Office of Energy Efficiency & Renewable Energy (EERE), Commercial and Residential Hourly Load Profiles for all TMY3 Locations in the United States, Available from: http://openei.org/datasets/dataset. [18] I. Prodan and E. Zio, A model predictive control framework for reliable microgrid energy management, International Journal of Electrical Power & Energy Systems, 61 (2014), 399-409. doi: 10.1016/j.ijepes.2014.03.017. [19] A. Vaghefi, M. A. Jafari, E. Bisse, Y. Lu and J. Brouwer, Modeling and forecasting of cooling and electricity load demand, Applied Energy, 136 (2014), 186-196. doi: 10.1016/j.apenergy.2014.09.004. [20] C. Voyant, G. Notton, S. Kalogirou, M. L. Nivet, C. Paoli, F. Motte and A. Fouilloy, Machine learning methods for solar radiation forecasting: A review, Renewable Energy, 105 (2017), 569-582. doi: 10.1016/j.renene.2016.12.095. [21] R. Weron, Electricity price forecasting: A review of the state-of-the-art with a look into the future, International Journal of Forecasting, 30 (2014), 1030-1081. doi: 10.1016/j.ijforecast.2014.08.008. [22] L. Xie and M. D. Llic, Model predictive dispatch in electric energy systems with intermittent resources, 2008 IEEE International Conference on Systems, Man and Cybernetics, 2008, 42-47. doi: 10.1109/ICSMC.2008.4811248. [23] L. Yao, J. Y. Shen and W. H. Lim, Real-time energy management optimization for smart household, 2016 IEEE International Conference on Internet of Things (iThings) and IEEE Green Computing and Communications (GreenCom) and IEEE Cyber, Physical and Social Computing (CPSCom) and IEEE Smart Data (SmartData), 2016. doi: 10.1109/iThings-GreenCom-CPSCom-SmartData.2016.31. [24] Y. Yoon and Y. Kim, Charge Scheduling of an Energy Storage System under Time-of-Use Pricing and a Demand Charge, The Scientific World Journal, 2014 (2014), Article ID 937329, 9 pages. doi: 10.1155/2014/937329.

show all references

##### References:
 [1] L. Andrew and S. Barman and K. Ligett and M. Lin and A. Meyerson and A. Roytman and A. Wierman, A tale of two metrics: Simultaneous bounds on competitiveness and regret, SIGMETRICS '13 Proceedings of the ACM SIGMETRICS/International Conference on Measurement and Modeling of Computer Systems, 2015, 329-330, arXiv: 1508.03769. doi: 10.1145/2465529.2465533. [2] A. Chis and J. Lunden and V. Koivunen, Coalitional game based cost optimization of energy portfolio in smart grid communities, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017. [3] O. Babacan, E. L. Ratnam, V. R. Disfani and J. Kleissl, Distributed energy storage system scheduling considering tariff structure, energy arbitrage and solar PV penetration, Applied Energy, 205 (2017), 1384-1393. doi: 10.1016/j.apenergy.2017.08.025. [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press New York, 2004. doi: 10.1017/CBO9780511804441. [5] M. Dubarry, C. Truchot and B. Y. Liaw, Synthesize battery degradation modes via a diagnostic and prognostic model, Journal of Power Sources, 219 (2012), 204-216. doi: 10.1016/j.jpowsour.2012.07.016. [6] M. A. Epelman, Barrier Methods for Constrained Optimization, 2012. Available from: http://www-personal.umich.edu/ mepelman/teaching/NLP/Handouts/NLPnotes12_89.pdf. [7] C. Graves, New York Mandatory Hourly Pricing Program Case 03-E-0641, 2009. Available from: https://www.nyiso.com/documents/20142/1404435/PRLWG_MHP_Overview.pdf/0d12b88b-2921-23c5-4c0f-2edb1d16257a. [8] E. Hazan, Introduction to online convex optimization, Foundations and Trends in Optimization, 2 (2015), 157-325. doi: 10.1561/2400000013. [9] P. L. Joskow and C. D. Wolfram, Dynamic pricing of electricity, American Economic Review, 102 (2012), 381-385. doi: 10.1257/aer.102.3.381. [10] Y. Li, G. Qu and N. Li, Online optimization with predictions and switching costs: Fast algorithms and the fundamental limit, 2018 Annual American Control Conference (ACC), 2018, arXiv: 1801.07780. doi: 10.23919/ACC.2018.8431296. [11] M. Lin, Z. Liu, A. Wierman and L. L. H. Andrew, Online Algorithms for Geographical Load Balancing, 2012 International Green Computing Conference (IGCC), 2012. doi: 10.1109/IGCC.2012.6322266. [12] A. C. Luna, N. L. Diaz, M. Graells, J. C. Vasquez and J. M. Guerrero, Online energy management system for distributed generators in a grid-connected microgrid, 2015 IEEE Energy Conversion Congress and Exposition (ECCE), 2015. doi: 10.1109/ECCE.2015.7310313. [13] X. Luo, J. Wang, M. Dooner and J. Clarke, Overview of current development in electrical energy storage technologies and the application potential in power system operation, Applied Energy, 137 (2015), 511-536. doi: 10.1016/j.apenergy.2014.09.081. [14] M. Morari and H. H. Lee, Model predictive control: Past, present and future, Computers and Chemical Engineering, 23 (1999), 667-682. doi: 10.1016/S0098-1354(98)00301-9. [15] National Grid, Hourly Electric Supply Charges, Available from: https://www9.nationalgridus.com/niagaramohawk/business/rates/5_hour_charge.asp. [16] National Solar Radiation Data Base, Hourly Solar Data, Available from: https://rredc.nrel.gov/solar/old_data/nsrdb/. [17] Office of Energy Efficiency & Renewable Energy (EERE), Commercial and Residential Hourly Load Profiles for all TMY3 Locations in the United States, Available from: http://openei.org/datasets/dataset. [18] I. Prodan and E. Zio, A model predictive control framework for reliable microgrid energy management, International Journal of Electrical Power & Energy Systems, 61 (2014), 399-409. doi: 10.1016/j.ijepes.2014.03.017. [19] A. Vaghefi, M. A. Jafari, E. Bisse, Y. Lu and J. Brouwer, Modeling and forecasting of cooling and electricity load demand, Applied Energy, 136 (2014), 186-196. doi: 10.1016/j.apenergy.2014.09.004. [20] C. Voyant, G. Notton, S. Kalogirou, M. L. Nivet, C. Paoli, F. Motte and A. Fouilloy, Machine learning methods for solar radiation forecasting: A review, Renewable Energy, 105 (2017), 569-582. doi: 10.1016/j.renene.2016.12.095. [21] R. Weron, Electricity price forecasting: A review of the state-of-the-art with a look into the future, International Journal of Forecasting, 30 (2014), 1030-1081. doi: 10.1016/j.ijforecast.2014.08.008. [22] L. Xie and M. D. Llic, Model predictive dispatch in electric energy systems with intermittent resources, 2008 IEEE International Conference on Systems, Man and Cybernetics, 2008, 42-47. doi: 10.1109/ICSMC.2008.4811248. [23] L. Yao, J. Y. Shen and W. H. Lim, Real-time energy management optimization for smart household, 2016 IEEE International Conference on Internet of Things (iThings) and IEEE Green Computing and Communications (GreenCom) and IEEE Cyber, Physical and Social Computing (CPSCom) and IEEE Smart Data (SmartData), 2016. doi: 10.1109/iThings-GreenCom-CPSCom-SmartData.2016.31. [24] Y. Yoon and Y. Kim, Charge Scheduling of an Energy Storage System under Time-of-Use Pricing and a Demand Charge, The Scientific World Journal, 2014 (2014), Article ID 937329, 9 pages. doi: 10.1155/2014/937329.
48 hours experimental setup
Performance of ARHC
48 hours solution comparison of ARHC and offline optimal
Storage size vs cost
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