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Factoring the Cycle Aging Cost of Batteries Participating in Electricity Markets
Bolun Xu, Jinye Zhao, Tongxin Zheng, Eugene Litvinov, Daniel S. Kirschen
TL;DR
Battery degradation costs must be represented in market bids, yet existing models either do not fit dispatch calculations or fail to reflect electrochemical aging. The paper uses a piecewise linear cycle-aging cost model with marginal aging costs, showing close benchmark agreement and applicability to ISO New England market dispatch.
Problem
Existing battery degradation models either cannot be integrated into dispatch calculations or do not reflect the actual electrochemical aging mechanism.
Method
The paper constructs a piecewise linear approximation of cycle-aging cost and defines marginal aging costs for economic dispatch and market bids.
Results
The model’s accuracy improves with more linearization segments and approaches the benchmark result with sufficiently many segments.
Takeaways & Limitations
The approach allows battery degradation costs to be represented in market clearing and bids while supporting assessment of operating profitability.
Abstract
from arXiv · showhide
When participating in electricity markets, owners of battery energy storage systems must bid in such a way that their revenues will at least cover their true cost of operation. Since cycle aging of battery cells represents a substantial part of this operating cost, the cost of battery degradation must be factored in these bids. However, existing models of battery degradation either do not fit market clearing software or do not reflect the actual battery aging mechanism. In this paper we model battery cycle aging using a piecewise linear cost function, an approach that provides a close approximation of the cycle aging mechanism of electrochemical batteries and can be incorporated easily into existing market dispatch programs. By defining the marginal aging cost of each battery cycle, we can assess the actual operating profitability of batteries. A case study demonstrates the effectiveness of the proposed model in maximizing the operating profit of a battery energy storage system taking part in the ISO New England energy and reserve markets.
I. INTRODUCTION
Battery cycle aging is a substantial operating cost, but existing degradation models either cannot fit dispatch calculations or do not reflect electrochemical aging. The paper proposes a piecewise linear aging-cost model that supports market dispatch and bidding, with accuracy demonstrated against a benchmark and using ISO New England data.
- Motivation: Battery degradation must be included in operating costs because cell life is sensitive to charge and discharge cycles.Frequent cycling can accelerate degradation and battery replacement relative to fixed-lifetime assumptions.
- Motivation: Existing degradation models either do not fit dispatch calculations or fail to reflect electrochemical battery aging.Traditional generator heat-rate curves cannot represent electrochemical cycle aging.
- Contributions: The paper proposes a piecewise linear cycle-aging cost function that closely approximates electrochemical aging and integrates into economic dispatch.The formulation is designed to resemble cost functions already used in market dispatch programs.
- Contributions: Defining marginal cycle-aging costs enables system operators to reflect battery operating costs and owners to submit bids that recover battery life lost through dispatch.The model is intended for market clearing calculations and wholesale-market participation.
- Validation: Accuracy improves with more linearization segments, with error relative to the benchmark approaching zero when sufficiently many segments are used.Effectiveness is demonstrated using a full year of ISO New England energy-market price data.
- Motivation: Cycle depth produces strongly nonlinear aging: a 7 Wh NMC cell exceeds 50,000 cycles at 10% depth but only 500 cycles at 100% depth.The corresponding lifetime energy throughput is 35 kWh versus 3.5 kWh.
1) Cycle depth:
The paper treats cycle depth as the central operational aging factor while simplifying or constraining other degradation effects. Rainflow counting supplies the benchmark cycle decomposition, but its non-analytic form prevents direct optimization, motivating an approximating formulation.
- 1) Cycle depth:: Current-rate effects are omitted because laboratory results indicate they are small for grid-scale batteries with capacities above 15 minutes.The paper notes that current-rate stress could instead be represented with a piecewise linear power-output cost curve.
- 1) Cycle depth:: Over-charging and over-discharging are controlled through upper and lower state-of-charge limits in dispatch or by the battery controller.Extreme state-of-charge levels otherwise reduce battery life.
- 1) Cycle depth:: Average state of charge is excluded because its effect on cycle aging is highly nonlinear but slight.The proposed model therefore focuses on cycle depth rather than this stress factor.
- C. The Rainflow Counting Algorithm: Rainflow counting identifies full cycles from local extrema and leaves a residue containing only half cycles.A decreasing residue segment is a discharging half cycle, while an increasing segment is a charging half cycle.
- C. The Rainflow Counting Algorithm: The rainflow algorithm cannot be integrated directly into an optimization problem because it lacks an analytical mathematical expression.Simplified cycle-depth formulations enable optimization but introduce additional degradation-model errors.
- C. The Rainflow Counting Algorithm: The benchmark computes total life loss by summing the stress-function loss Φ(δ) over all rainflow-identified cycles.The proposed model is evaluated against this rainflow-based ex-post benchmark.
III. MARGINAL COST OF BATTERY CYCLING
The paper models battery cycle aging through marginal costs that can be embedded in dispatch optimization. A piecewise linear upper approximation tracks cycle-depth-dependent aging while supporting energy and reserve co-optimization.
- Aging-cost formulation: Cycle aging is assumed to occur only during discharge, with a discharging half cycle assigned the aging of a full cycle at equal depth.Charging half cycles cause no cycle aging under the simplifying assumption that daily charged and discharged energy are nearly identical.
- Aging-cost formulation: The model computes incremental aging from cycle depth and derives marginal aging with respect to discharge power.Cycle depth is updated from battery output power and discharge efficiency.
- Aging-cost formulation: A piecewise linear upper approximation divides the 0–100% cycle-depth range into J segments and assigns each segment a marginal aging cost.The segment costs are prorated from battery cell replacement cost.
- Dispatch optimization: The dispatch optimization maximizes energy and reserve-market revenue minus cycle aging cost over the optimization horizon.Decision variables track charging, discharging, reserve provision, and energy stored in each cycle-depth segment.
- Dispatch optimization: Segment-level constraints enforce power ratings, state-of-charge limits, energy evolution, initial and final storage conditions, and reserve requirements.Reserve provision must satisfy a one-hour sustainability requirement under the stated NERC constraint.
- Market integration: The model can be used by BES owners for bids and offers or incorporated by ISOs into market clearing.Owners provide cycle-aging parameters, while ISOs manage state-of-charge and charge-limit parameters.
V. CASE STUDY
The case study tests the proposed battery aging-cost model with ISO New England data. Simulations use GAMS with CPLEX over 24-hour optimization periods.
- Case-study design: The proposed model is tested with ISO New England data to evaluate BES profitability and longevity in market participation.All simulations use GAMS with the CPLEX solver and a 24-hour optimization period.
A. BES Test Parameters
The simulated BES represents a 20 MW / 12.5 MWh lithium-ion system with bounded state of charge and specified efficiency, lifetime, and operating assumptions. Perfect forecasts and identical cell aging make the reported profitability an upper bound.
- BES configuration: The BES has 20 MW charging and discharging ratings, 12.5 MWh capacity, 95% efficiency, and a 15%–95% state-of-charge range.These parameters define the simulated system’s operating limits.
- Battery assumptions: The battery pack replacement cost is 300,000 $/MWh, with a stated life of 3000 cycles at 80% depth and a 10-year shelf life.The simulated cells are Li(NiMnCo)O2-based 18650 lithium-ion cells.
- Battery assumptions: Cells are assumed identical, ideally managed, and maintained at 25°C, so all cells age at the same rate.The cycle-depth stress function is described as near-quadratic.
- Study scope: Because dispatch uses perfectly accurate price forecasts, the case-study results provide an upper bound on BES profitability in the market.This is an explicit scope condition of the simulation design.
B. Market Data
The study compares day-ahead and real-time ISO New England markets and examines how cycle-aging cost linearization changes BES dispatch. Increasing the number of segments makes dispatch more responsive to price fluctuations, while 16 segments make predictive aging-cost error negligible.
- Market scenarios: The study uses 2015 SE-MASS zonal prices from ISO New England across day-ahead and two real-time market settlement scenarios.The real-time scenarios use one-hour and 5-minute settlement periods, with a one-hour reserve sustainability requirement.
- Dispatch comparison: With more segments, dispatch becomes more sensitive to price fluctuations and its state of charge tracks market prices more closely.The 16-segment model limits power during small price fluctuations so marginal aging cost does not exceed marginal arbitrage income.
- Dispatch comparison: The zero-cost dispatch frequently switches between charging and discharging and can produce negative profits across all market scenarios.It maximizes market revenue without accounting for cycle-aging cost, so it does not maximize lifetime profit.
- Dispatch comparison: A single-segment aging model produces the most conservative response, with the BES idle unless price deviations are very large.It collects the smallest market revenues but does not lose money because predicted aging exceeds actual aging.
D. BES Market Profitability Analysis
The 16-segment aging-cost model produced the highest profit across market scenarios, while accounting for cycle aging changed dispatch behavior and improved profitability and life expectancy.
- The 16-segment model generates the largest profit in all market scenarios.
- The no-cost model aggressively arbitrages all price differences, producing very large negative profit and very short battery life expectancy.
- The 1-segment model is more conservative and arbitrages only during large price deviations.
- The BES achieves the largest profits in the 5-minute RTM because this market has the largest price fluctuations.
- In the hourly RTM, reserve provides about 74% of market revenue and 90% of prorated profits for this BES.
- Simulations using a full year of actual market price data show that the proposed model improves BES profitability and life expectancy.
APPENDIX
The appendix establishes that the piecewise linear aging-cost model simulates battery cycle operations under a convex cost curve and converges to the rainflow benchmark as segmentation becomes arbitrarily fine.
- Model accuracy: The proposed piecewise linear model closely approximates the benchmark rainflow-based battery cycle aging model.
- Evaluation: The appendix evaluates the model's accuracy against benchmark aging cost for a fixed feasible battery dispatch profile.
- Cost structure: A convex aging-cost curve has non-decreasing marginal cycle-aging costs, with c1 ≤ c2 ≤ . . . ≤ cJ across increasingly deep segments.
- Segment dispatch: The model assigns battery energy across cycle-depth segments and prioritizes shallower segments when the aging cost curve is convex.
- Model accuracy: With infinitely many linearization segments, the proposed model yields the same result as the benchmark rainflow-based cost model.
- Rainflow equivalence: The proposed model and rainflow method produce the same cycle-counting result for any cycles under the appendix construction.
A. Numerical example
The numerical example compares the proposed piecewise linear aging-cost model with the benchmark rainflow-based model on an example SoC profile. Under the stated assumptions, both models produce a total aging cost of 43.
- Setup: The example uses a perfect-efficiency battery, a 100δ2 aging-cost function, and 10 linearization segments covering 10% cycle-depth ranges.These assumptions define the numerical setup for the comparison.
- Benchmark calculation: Rainflow counting identifies two 10%-depth full cycles, one 40%-depth full cycle, one 50%-depth discharge half cycle, and one zero-cost charge half cycle.The corresponding costs are 1, 16, 25, and 0, respectively.
- Benchmark calculation: 43 is the total aging cost from the benchmark rainflow-based calculation for the example profile.The total is obtained by summing the identified cycle and half-cycle costs.
- Proposed model: The proposed model tracks each linearization segment's normalized energy level and records marginal costs across time intervals.The segment vector is sorted from shallower to deeper depths, with one denoting a full segment and zero an empty segment.
- Comparison: The proposed model and benchmark model both produce a cost of 43 for this example profile.This agreement is presented as an illustration of the approximation result established by Theorem 2.