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Simplified Battery Aging Model– Implications for Storage Valuation
12th Conference on Sustainable Development of Energy, Water and Environment Systems
Benjamin Böcker
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Battery in future energy systems
2
Future energy systems
▪ dominated by renewable resources (primary supply-dependent like wind and PV)
▪ novel flexibilities and interconnections are needed to ensure security of supply at any time and any place
Battery technologies as one promising flexibility option
▪ declining investment costs, proven technology
▪ already efficient in some applications (e.g. primary reserve) as well as home storage systems (+PV)
Open challenge
▪ adequate valuation approaches which consider aging effects respectively capacity fade
Motivation and methodology overview 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Valuation of battery systems
3
Existing valuation approaches which consider aging typically
▪ assuming exogenous lifetime (e.g. 10 years) classic NPV approaches with annualized investment costs and discounted average revenues
▪ adding linearized restrictions, e.g. counting full-cycles by using energy turnover changes the schedule when the storage operates not how
▪ adding penalty/cost function (linear or quadratic) gives direct incentives to change the actual operation (when and how) but in which amount? (actual the battery value it selves, therefore investment costs are often used as approximation)
Novel valuation approach
▪ endogenous tradeoff between actual battery operation (revenues) and corresponding lifetime(endogenous lifetime, consideration of non-linear battery aging and corresponding capacity fade)
▪ model objective: versatile in application, robust optimization (single optimum), acceptable running time
Motivation and methodology overview 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Agenda
Motivation and methodology overview 1
Simplified battery aging model 2
Novel valuation approach 3
Application 4
Conclusion 5
4
Battery aging and their implications for efficient operation and valuation 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Basic assumptions
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▪ battery aging in period 𝑃:
▪ aging:
▪ normalized capacity
Simplified battery aging model 1 2 3 4 5
𝑎 𝑃 =
𝑡∈𝑃
∆𝑎 𝑡
∆𝑎 𝑡 = ∆𝑎𝑐𝑎𝑙 𝑡 + ∆𝑎𝑐𝑦𝑐+ 𝑡 + ∆𝑎𝑐𝑦𝑐
− 𝑡
𝑣 𝑡 = 1 − 1 − 𝑣e 𝑎 𝑡
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Calendrical aging (lithium ion) – model approach
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▪ summary:
aging increases with higher state of charge
linear dependency
▪ calendrical aging model (adjusted to SAFT Data):
▪ parametrization:
Simplified battery aging model 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Quelle: Ecker et al.; Calendar and cycle life study of Li(NiMnCo)O2-based 18650 lithiumion batteries; Journal of Power Sources, 2013
Calendar life time (one year calendar life time at 50°C corresponds to approximately 5.6 years at 25°C)
∆𝑎𝑐𝑎𝑙 𝑡 = 𝛼1 𝑠(𝑡) + 0.5 𝑠+ 𝑡 − 𝑠− 𝑡 ∆𝑡 + 𝛼2∆𝑡
1 − 𝑣e
Mean state of charge in discrete time
𝛼1 7.2 ∙ 10−7 ℎ−2
𝛼2 1.4 ∙ 10−7 ℎ−1
Source: Saft 2014
Cyclical aging (lithium ion) – model approach
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▪ Summary: Aging increases with higher depth of discharge (DOD)
▪ Basic dependency (Number of cycles for different DOD)
▪ Integrative approach, aging while charging and discharging
charging:
discharging:
Simplified battery aging model 1 2 3 4 5
∆𝑎𝑐𝑦𝑐+ 𝑡 =
1
2𝛽1∙ 1 − 𝑠 𝑡
𝛽2− 1 − 𝑠 𝑡 − 𝑠+ 𝑡 ∆𝑡 𝛽2
𝑁 𝐷𝑂𝐷 = 𝛽1 ∙ 𝐷𝑂𝐷−𝛽2
𝑡1 𝑡2
1
𝑠
𝑡3
𝑠 𝑡1
𝑠 𝑡2
𝑠 𝑡3
∆𝑎𝑐𝑦𝑐(𝐷𝑂𝐷) =1
𝑁 𝐷𝑂𝐷=
1
𝛽1∙ 𝐷𝑂𝐷𝛽2
∆𝑎𝑐𝑦𝑐− 𝑡 =
1
2𝛽1∙ 1 − 𝑠 𝑡 + 𝑠− 𝑡 ∆𝑡 𝛽2 − 1 − 𝑠 𝑡
𝛽2
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
𝛽1 5000 −
𝛽2 2 −
Source: Saft 2014
Agenda
Motivation and methodology overview 1
Simplified battery aging model 2
Novel valuation approach 3
Application 4
Conclusion 5
8
Battery aging and their implications for efficient operation and valuation 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Discount factor:
Optimization problem: Objective function
9
Objective function
▪ revenues spot market (initial)
▪ revenue multiplier 𝑀𝑅 = 𝑓 ∆𝑎 𝑡 , 𝑞𝑎 as adjusted NPV for
linear decreasing revenues (no constant annuity) due to the capacity fade of the storage volume
increasing revenues due to changes in the price pattern (driven by more RES or competitors) – optional
corresponding lifetime (given by aging)
Novel valuation approach 1 2 3 4 5
max 𝑅0 ∙ 𝑀𝑅
Par. Unit Description
𝑁 − Number of periods
𝜂 − Storage charging and discharging efficiency
𝑉𝑆 𝑀𝑊ℎ Storage volume
𝐾𝑆 𝑀𝑊 Storage power
𝑖𝑎 − Interest rate (annual)
𝑔𝑎 − Revenue changing rate (annual)
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
𝑅0 = ∆𝑡 ∙ 𝑉𝑆𝑡∈𝑃
𝑝𝑠𝑚 𝑡 𝜂𝑠− 𝑡 −1
𝜂𝑠+ 𝑡 ∙ 𝑞𝑎
𝑇−𝑡 ∆𝑡8760
𝑞𝑎 =1 + 𝑖𝑎1 + 𝑔𝑎
Extension of classic present value of annuity
Optimization problem: Revenue multiplier
10
▪ idea of the revenue multiplier
initial equation
with
𝑁𝑃 =1
𝑎 𝑃𝑞𝑃 = 𝑞𝑎
𝑇
8760
𝑚𝑅 =1−𝑣𝑒
𝑁𝑃𝑚0 = 1 +
1
2𝑚𝑅
▪ derived revenue multiplier
Novel valuation approach 1 2 3 4 5
𝑀𝑅 =1 − 𝑞−𝑁
𝑞 − 1
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
𝑀𝑅 =𝑚0 −𝑚𝑅 𝑞𝑃 −𝑚0 + 𝑚𝑅 𝑁𝑃 + 1 −𝑚0 𝑞𝑃
−𝑁𝑃+1 + 𝑚0 −𝑚𝑅𝑁𝑃 𝑞𝑃−𝑁𝑃
𝑞𝑃 − 1 2
𝑛1 𝑛2
1
𝑟𝑚 𝑛
𝑛3 = 𝑁𝑃
1−𝑣 𝑒
𝑎𝑚 1 = 0.95𝑣𝑒
𝑎𝑚 2 = 0.85
𝑎𝑚 3 = 0.75
Example
𝑣𝑒 = 0.7𝑁𝑃 = 3
𝑀𝑅 =𝑛∈𝑁𝑃
𝑚0 −𝑚𝑅 ∙ 𝑛 ∙ 𝑞𝑃−𝑛
Optimization Problem: Restrictions
11
▪ State of charge (storage level)
▪ Power restriction
▪ Aging (part of the objective function)
calendrical
cyclical
𝑠 𝑡 + 1 = 𝑠 𝑡 + 𝑠+ 𝑡 − 𝑠− 𝑡 ∆𝑡
𝑠+ 𝑡 + 𝑠− 𝑡 ≤𝐾𝑆𝑉𝑆
𝑠 1 = 𝑠 𝑇 + 𝑠+ 𝑇 − 𝑠− 𝑇 ∆𝑡
𝑠 𝑡 ≤ 1
∆𝑎𝑐𝑎𝑙 𝑡 = 𝛼1 𝑠(𝑡) + 0.5 𝑠+ 𝑡 − 𝑠− 𝑡 ∆𝑡 + 𝛼2𝛼3∆𝑡
1 − 𝑣e
∆𝑎𝑐𝑦𝑐+ 𝑡 =
1
2𝛽1∙ 1 − 𝑠 𝑡
𝛽2− 1 − 𝑠 𝑡 − 𝑠+ 𝑡 ∆𝑡 𝛽2 ∆𝑎𝑐𝑦𝑐
− 𝑡 =1
2𝛽1∙ 1 − 𝑠 𝑡 + 𝑠− 𝑡 ∆𝑡 𝛽2 − 1 − 𝑠 𝑡
𝛽2
∆𝑎 𝑡 = ∆𝑎𝑐𝑎𝑙 𝑡 + ∆𝑎𝑐𝑦𝑐+ 𝑡 + ∆𝑎𝑐𝑦𝑐
− 𝑡
Novel valuation approach 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Agenda
Motivation and methodology overview 1
Simplified battery aging model 2
Novel valuation approach 3
Application 4
Conclusion 5
12
Battery aging and their implications for efficient operation and valuation 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Application Data
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Base Case
▪ lithium ion battery (cf. right table)
▪ investigation year – 2016, hourly resolution (mean: 29.0 €/MWh, std: 14.0 €/MWh)
▪ comparison of three approaches:
exogenous calendrical aging assumptions (10 years), no cyclical (ExoCal)
exogenous calendrical aging (10 years) and counting full-cycles (𝑁(1) = 5000) (ExoCalFullCyc)
endogenous calendrical and cyclical aging – novel approach (EndCalCyc)
Sensitivities
▪ time-step resolution (1,1/4 h) with different C-Rates (1,2,4) assuming same 𝑉𝑆
▪ cyclical aging parameter 𝛽1 (2000, 4000, 5000, 6000, 8000)
▪ interest rate – rate of return
▪ investigation period (weekly, monthly, annual)
▪ investigation years (2016, 2020, 2025, 2030)
Application 1 2 3 4 5
Unit Value
𝑉𝑆 𝑀𝑊ℎ 1
𝐾𝑆 𝑀𝑊 1
𝜂∗ − 0.9
𝑖𝑦𝑒𝑎𝑟 − 0.05
𝒈 − 0
𝑣e − 0.7
𝛼1 ℎ−1 6.8e-4
𝛼2 ℎ−1 1.3e-4
𝛼3 − 1.1e-3
𝛽1 − 5000
𝛽2 − 2
∗ for charging and discharging, 0.81 roundtrip efficiency
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Base case – General results
14
Application 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Considering aging (EndCalCyc to ExoCal)
▪ increases the value of the battery by 75%, while
▪ decreases first year spot revenues by 36%
▪ more than triple the corresponding lifetime
Value of the battery system as target costs
▪ 45 €/kWh/kW respectively 76 €/kWh/kW, already volume-based costs are significantly higher (>>100€/kWh)
▪ as expected inefficient investment in a lithium ion battery system (application, arbitrage hourly spot-market, 2016)
2016 - hourly
Mean (rStd)
Value Lifetime R_0 M_R
realistic alleged realistic alleged realistic / alleged realistic alleged
ExoCal 45.4k€ 85.4k€ 4.7a 10.0a 13.0k€ 3.5 6.5
ExoCalFullCyc 68.3k€ 74.7k€ 8.8a 10.0a 11.4k€ 6 6.5
EndCalCyc 79.4k€ 54.7k€ 16.3a 10.0a 8.4k€ 9.5 6.5
bold, assumed/calculated during optimization
realistic – end. lifetime, decreasing capacity and revenues – novel approachalleged – exo. lifetime, average capacity and revenues
Base Case – Storage operation
15
Approach ExoCal and ExoCalFullCyc
▪ bang-bang strategy ~17 resp. 11 full cycles
▪ maximizing current profit
▪ ExoCalFullCyc only reduced cycle frequency
EndCalCyc (noval approach)
▪ depth of discharging depends on prices
▪ avoid high depth of discharge, only 4 full cycles and 3 additional below 0.5
▪ trade off between generating revenues and expected lifetime
Application 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Results – Sensitivities – Time-step resolution and C-Rates
16
Increasing time-step resolution (incl. prices)
▪ increase cycle frequency
▪ increase annual initial revenues
▪ decrease depth of discharge (1C)
Increasing C-rate
▪ increase depth of discharge and initial revenues
▪ leads to additional aging effects (not considered)
Application 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
C-Rates 1 (hourly) 1 (1/4 hourly) 2 (1/4 hourly) 4 (1/4 hourly)
Value 79.4k€ 148.0k€ 160.8k€ 163.7k€
Lifetime 16.3a 16.8a 14.6a 14.0a
R_0 8.36k€ 15.22k€ 18.18k€ 19.13k€
M_R 9.5 9.7 8.8 8.6
Results – Sensitivities – Other
17
Application 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
running time ExoCal ExoCalFullCyc EndCalCyc
LP 0.6 s 1 s - s
NLP 4 min 17 s 5 min 49 s 11 min 44 s
annual monthly weekly
Value 79.4k€ (0.0%) 77.9k€ (20.9%) 78.0k€ (36.7%)
Lifetime 16.3a (0.0%) 15.9a (5.0%) 15.2a (10.1%)
R_0 8.36k€ (0.0%) 0.68k€ (19.7%) 0.16k€ (32.5%)
M_R 9.5 (0.0%) 114.6 (3.7%) 505.1 (30.8%)
Different investigation period:Annual, hourly 2000 4000 5000 6000 8000
Value 47.4k€ 70.4k€ 79.4k€ 86.2k€ 98.9k€
Lifetime 13.4a 15.5a 16.2a 16.8a 17.9a
R_0 5.7k€ 7.7k€ 8.4k€ 8.9k€ 9.7k€
M_R 8.3 9.2 9.5 9.7 10.2
Annual, hourly 0.025 0.05 0.075 0.1
Value 94.0k€ 79.4k€ 68.4k€ 60.8k€
Lifetime 19.2a 16.2a 13.9a 12.3a
R_0 7.2k€ 8.4k€ 9.3k€ 10.0k€
M_R 13.0 9.5 7.4 6.1
Annual, hourly 2016 2020 2025 2030
Value 79.4k€ 119.5k€ 127.2k€ 129.5k€
Lifetime 16.3a 14.8a 15.7a 15.7a
R_0 8.36k€ 13.42k€ 13.70k€ 13.93k€
M_R 9.5 8.9 9.3 9.3
Running time, annual, hourly
Different cyclical aging parameter 𝛽1:
Different investigation year:
Different interest rate :
Agenda
Motivation and methodology overview 1
Simplified battery aging model 2
Novel valuation approach 3
Application 4
Conclusion 5
18
Battery aging and their implications for efficient operation and valuation 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Conclusion
19
▪ considering battery aging adequately improve valuation
▪ simplified battery aging model also usable for other applications
▪ introducing the aging induced revenue multiplier allows
considering capacity fade driven by calendrical and cyclical aging as well as
related endogenous lifetime
▪ robust algorithm (despite of non-linear aging effects) due to single optimum
▪ further sensitivities / next steps
second-Life application (two stage aging process, e.g. double aging after 𝐸𝑂𝐿) additional value
multi-purpose application (e.g. reserve market, system services)
application in energy system model
Conclusion 1 2 3 4 5
Energy system analysis 3 | 07-Oct-17 | SDEWES 2017 | Dubrovnik
Many Thanks
Related paper: Böcker, B.; Battery aging and their implications for adequate valuation –novel approach of an endogenous lifetime consideration; mimeo; 2017.
Contact: Benjamin BöckerChair for Management Science and Energy Economics, University of Duisburg-Essen
email: [email protected]
phone: +49 201/183-7306