SULEMENTARY INFORMATION Article ttp://doi.org/10.1038/41560-018-0129-9 In te format provided by te autor and unedited. An intertemporal deciion framework for electrocemical energy torage management Guannan He 1,2, Qixin Cen 3, anayioti Mouti 4, Soummya Kar 4 and Jay F. Witacre 1,2,5 1 Department of Engineering and ublic olicy, Carnegie Mellon Univerity, ittburg, A, USA. 2 Wilton E. Scott Intitute for Energy Innovation, Carnegie Mellon Univerity, ittburg, A, USA. 3 Department of Electrical Engineering, Tingua Univerity, Beijing, Cina. 4 Department of Electrical and Computer Engineering, Carnegie Mellon Univerity, ittburg, A, USA. 5 Department of Material Science and Engineering, Carnegie Mellon Univerity, ittburg, A, USA. e-mail: witacre@andrew.cmu.edu Nature Energy www.nature.com/natureenergy 2018 Macmillan ublier Limited, part of Springer Nature. All rigt reerved.
Supplementary Note 1. Comparion of ti Work wit a reviou Work In a previou work 1, overimplified aumption are made in te propoed model. For example, te model i optimizing te life-cycle profit by multiplying te profit over one repreentative day by te EES lifetime. Becaue of ti poor aumption on objective, te model in te previou work cannot reflect te trade-off on different ort-term profit opportunitie in te long-term, e.g., weekday v. weekend/oliday, ummer v. winter, and year wit increaing penetration of renewable (o te arbitraging opportunitie may be increaing over year). One contribution of te previou work i tat it ow tat tere i ignificant difference between te etimated profit made wit and witout conideration for EES degradation, o it i eential to incorporate EES degradation in operational deciion making. However, te model dicloed in te previou work fail to provide te rigorou optimal olution for operational deciion regarding te tradeoff between ort-term profit and lifetime and between ort-term profit over time. In ti work, we can conclude from Fig. 2 in te main manucript tat if te ort-term marginal degradation cot i not et properly, conidering degradation, e.g., LCOD, could be even wore tan not conidering degradation. Te propoed intertemporal deciion framework in ti work tand at a iger level tan te contribution in te previou work. In contrat to aumption made in te previou work on EES cemitry, degradation calculation model, and application, plu te poor aumption about te objective function, te intertemporal deciion framework i baed on te canonical Intertemporal Coice teory in microeconomic and make a few aumption a poible to enance te general applicability of te framework. Ti i alo wy we name it a framework rater tan a model. Under te framework, any ort-term deciion model wit different application/revenue tream and variou degradation repreentation approace can be implemented. Moreover, we rigorouly proved te optimality of te MBU metod in term of life-cycle profit maximization, wic i abent in te previou work. Te oter key contribution of te framework propoed in ti work i te ABU (average benefit of uage) metod. Nearly all economic valuation on EES in te literature are baed on unit capacity metric, e.g., cot/kw-intalled capacity, wile we believe unit uage metric merit attention in many cae, e.g., comparing te relative invetment attractivene of variou EES cemitrie in certain application, EES tecnology learning tudie, and environmental impact analyi.
Supplementary Note 2. rice Uncertainty Ti ection prove tat te propoed framework in IDF paper i guaranteed to provide te optimal deciion tat maximize te expected life-cycle revenue of EES, and ort-term price uncertainty a little effect on te comparative advantage of our propoed framework over exiting metod in term of te expected life-cycle revenue maximization. For implicity, we firt conider a point-etimate etting for price ing. We denote te price ing error at our (or any ort-term deciion time interval) by ε and aume it i identically ditributed; denote te optimal ort-term EES output at our given te price λ by ; denote te price at our by λ ; Te relationip between te and fore- cated price i: λ = (1 + ε ) λ (1) Operational deciion are made baed on ed price, wic are fixed input in our model, becaue te price are alway unknown wen one i making operational deciion in electricity market and will only be available after te deciion ave been made in time. Ti implie tat i only dependent on te λ rater tan λ are determinitic, o do te output deciion λ, te price. Once we ave ed te price,. Terefore, only te price λ and te price ing error ε are random at te tage of deciion making. Witout lo of generality, we aume te ort-term revenue of EES at our i λ. Literally, we only need to aume tat te ort-term revenue i proportional to market price (or oter benefit rate) wen te output of EES are fixed, wic i atified in mot EES application. Ten te expectation of te life-cycle benefit of EES can be expreed a: E[LB ] = E t t T [, t t+δt] δ ( λ ) (2) λ = Ε δt (3) t T [, t t+δt] 1+ ε = δ λ Ε t t T [, t t+δt] 1+ ε 1 (4) 1 =Ε δt λ 1+ ε t T [, t t+δt] (5) = K LB (6) were LB i te life-cycle benefit, δ t i te dicount factor for time t, and LB i te etimated life-cycle benefit baed on ed price. Equation (2) to (3) are baed on Equation
(1), and Equation (3) to (4) are valid becaue λ and are not random. Te above equation old for our framework and exiting metod (LCOD and not conidering degradation), wic implie tat te price uncertainty a te ame effect on te expected life-cycle revenue produced by different metod, a te expected revenue are jut te ed revenue multiplied by a ame factor K, given te ame ing error. A long a we are maximizing te ed revenue, we are maximizing te expectation of te life-cycle revenue. Terefore, regardle of te magnitude of ing error, our framework provide te optimal olution tat maximize te expectation of te life-cycle revenue. A reported in te literature 2,3, te mean of electricity price error, m, i typically below 10% (te literature i uggeting te mean of abolute error i around or below 10%, and te abolute of mean i no larger tan te mean of abolute) and te variance i below 0.01. In ti context, wit a normally ditributed error wit a mean of 0.1 or -0.1 and a variance of 0.01, we ave: 1 1 K =Ε 1+ ε 1+ m (7) wic can be verified numerically (we ave verified error mean from 0.01 to 1 given a variance of 0.01). Terefore, we ave: 1 1+ m E[LB ] LB We ave imulated te daily revenue of EES baed on te ame price data a te main manucript (CAISO 2016 price) wit different metod to addreing degradation (te propoed MBU metod, exiting LCOD metod, and not conidering degradation). We conider an identically and normally ditributed price ing error wit mean 0.1 and variance 0.01. Te ed and daily revenue in Year 1 are plotted in Supplementary Figure 1-3. For te cae uing MBU metod, te annual revenue of Year 1 i $1.207 million, wile te ed annual revenue i $1.328 million, wic give a revenue etimation error of 10.02%. For te cae uing LCOD metod (auming capital cot i $200/kW), te annual revenue of Year 1 i $0.1374 million, wile te ed annual revenue i $0.1512 million, wic give a revenue etimation error of 10.04%. For te cae not conidering degradation, te annual revenue of Year 1 i $1.386 million, wile te ed annual revenue i $1.524 million, wic give a revenue etimation error of 9.95%. Te imulation reult exactly agree wit te previou derivation tat te price uncertainty a te ame effect on te expected life-cycle revenue produced by different metod. Te effect of price uncertainty on te life-cycle revenue in energy arbitrage application i plotted in Supplementary Figure 4, auming alo tat te mean price ing error (bia) i 10%. We can oberve from te figure tat te profitability difference between our framework and exiting metod are te ame (in percentage) wit or witout price ing bia (non-zero or zero mean of price ing error). Te profitability difference will not cange for any level of price ing error. For cae wit non-identical error acro different our, te relation between te revenue and te ed revenue i a little different: E[LB ] = K LB (8) (9) Becaue te daily carging/dicarging pattern for different metod are imilar (carging wen te price i low and dicarging wen te price i ig), te ratio of te ed revenue of all our,
LB, to te total ed revenue, LB, i alo approximately te ame over different metod. Tu, te price uncertainty alo a imilar effect on te expected life-cycle revenue produced by different metod, for cae wit non-identical error acro different our. For a tocatic multi-cenario etting, te expected revenue i jut a linear combination or an average of ed revenue in eac cenario. Let, π denote te probability of price cenario, λ denote te ed price of our and cenario, and ε, denote te price ing error of our and cenario and i aumed to be identically ditributed acro different our for implicity. Te expected ed revenue LB mean = π LB, were LB denote te ed revenue at cenario. In ti etting, torage operator i determining it carging-dicarging cedule to maximize te expected ed revenue mean LB. We can replace (2)-(6) wit (10)-(14), wic decribe te relationip between te expected revenue and te ed revenue at cenario. E[LB ] = E t t T [, t t+δt] δ ( λ ) (10) = λ Ε, δ t t T [, t t+δt] 1+ ε, (11) = δ λ Ε t, t T [, t t+δt] 1+ ε, 1 (12) 1 =Ε δt λ, 1 ε (13) +, t T [, t t+δt] Since LB mean = π LB, we ave: LB mean = (14) K LB = π LB E[LB ] = E[LB ] K = E[LB ] = π K 1 K ' π Similar wit (6), we can derive te relation between te expected revenue and te average ed revenue a: mean E[LB ] K ' LB (15) = (16) Terefore, we ave te ame concluion in a tocatic multi-cenario etting a tat in a point-
etimate etting: te price uncertainty a te ame effect on te expected life-cycle revenue produced by different metod.
Supplementary Note 3. Computational Complexity Te MBU metod add a contraint to te ort-term optimization model (ee Equation (12) in te Metod ection of te main manucript) to guarantee tat te long-term profit i maximized wen te ort-term operational deciion are made. Te ort-term optimization model wit te MBU contraint a te ame computational complexity a te original model witout te contraint, watever form te model take, e.g., linear programming (L) or non-linear programming (NL), and algoritm it i olved by, e.g., exact olution, euritic, or meta-euritic 4. Weter te model i L or not depend on ow te relationip between te EES uage (power output) and te incurred degradation i repreented and formulated. Baically, tere exit a trade-off between accuracy and modelling/computational complexity in EES degradation repreentation model. Among electro-cemical, empirical, and emi-empirical model, te electro-cemical model ave te iget accuracy but entail te larget computational complexity, wile emi-empirical model are te mot computationally-tractable but rik incurring etimation error if te training data are incomplete 5. Statitical metod can be furter applied for pecific application, e.g. frequency regulation to etimate a degradation function tat i eaier to incorporate into optimization tan emi-empirical model 6. Moreover, NL model, in general, can be piecewie approximated by mixed-integer linear programming (MIL) model to reduce modelling/computational complexity 7,8. In te cae tudie preented in te main manucript, we make reaonable aumption for approximation purpoe, e.g. EES take one or two cycle per day in energy arbitrage, and etimate a degradation function for te combined application of energy arbitrage and frequency regulation. We alo analyze te robutne of te MBU metod againt te degradation etimation error.
Supplementary Figure Supplementary Figure 1. Daily ed and revenue of Year 1 for energy arbitrage application baed on te propoed MBU metod and CAISO 2016 price cenario. Supplementary Figure 2. Daily ed and revenue of Year 1 for energy arbitrage application baed on te LCOD metod and CAISO 2016 price cenario.
Supplementary Figure 3. Daily ed and revenue of Year 1 for energy arbitrage application baed on CAISO 2016 price cenario, not conidering degradation. Supplementary Figure 4. Life-cycle revenue of a 50MW-200MW litium-ion EES ytem for energy arbitrage/olar integration in CAISO wit MBU and LCOD metod, conidering cae wit and witout price ing bia.
Supplementary Reference 1. He, G., Cen, Q., Kang, C., inon,. & Xia, Q. Optimal bidding trategy of battery torage in power market conidering performance-baed regulation and battery cycle life. IEEE Tran. Smart Grid 7, 2359-2367 (2016). 2. Abedinia, O., Amjady, N. & Zareipour, H. A new feature election tecnique for load and price of electrical power ytem. IEEE Tran. ower Syt. 32, 62-74 (2017). 3. Conejo, A. J., laza, M. A., Epinola, R. & Molina, A. B. Day-aead electricity price ing uing te wavelet tranform and ARIMA model. IEEE Tran. ower Syt. 20, 1035-1042 (2005). 4. Weitzel, T. & Glock, C. H. Energy management for tationary electric energy torage ytem: A ytematic literature review. Eur. J. Oper. Re. 264, 582-606 (2018). 5. Jin, X. et al. Comparion of Li-ion battery degradation model for ytem deign and control algoritm development. 2017 American Control Conference (ACC) 4, 74-79 (2017). 6. He, G., Cen, Q., Kang, C., Xia, Q. & oolla, K. Cooperation of wind power and battery torage to provide frequency regulation in power market. IEEE Tran. ower Syt. 32, 3559-3568 (2017). 7. Tan, X., Qu, G., Sun, B., Li, N. & Tang, D. H. K. Optimal ceduling of battery carging tation erving electric veicle baed on battery wapping. IEEE Tran. Smart Grid, in pre; ttp://doi.org/10.1109/tsg.2017.2764484 8. Xu, B., Zao, J., Zeng, T., Litvinov, E. & Kircen, D. S. Factoring te cycle aging cot of batterie participating in electricity market. IEEE Tran. ower Syt. 33, 2248-2259 (2018).