Simplified Battery Aging Model Implications for Storage ... · PDF fileExtension of classic present value of annuity Optimization problem: Revenue multiplier 10 idea of the revenue

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


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


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


Basic assumptions

5

▪ 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


Calendrical aging (lithium ion) – model approach

6

▪ summary:

aging increases with higher state of charge

linear dependency

▪ calendrical aging model (adjusted to SAFT Data):

▪ parametrization:



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

7

▪ 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:


∆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


𝛽1 5000 −

𝛽2 2 −

Source: Saft 2014

Agenda




Application 4

Conclusion 5

8



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)


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


𝑀𝑅 =1 − 𝑞−𝑁

𝑞 − 1


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



Agenda




Application 4

Conclusion 5

12



Application Data

13

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


Base case – General results

14



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



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)



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



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




Application 4

Conclusion 5

18



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


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

mailto:[email protected]

Documents

Simplified Battery Aging Model Implications for Storage ... · PDF fileExtension of classic present value of annuity Optimization problem: Revenue multiplier 10 idea of the revenue