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Optimal Battery Control Under Cycle Aging Mechanisms in Pay for Performance Settings

Yuanyuan Shi, Bolun Xu, Yushi Tan, Daniel Kirschen, Baosen Zhang

arXiv:1709.05715v4math.OCcs.SYeess.SY

TL;DR

Battery operators must trade off following uncertain instructions against degradation, while existing approaches do not jointly address cycle aging and missing future information. This paper combines a rainflow cycle-based degradation model with an online threshold controller. It reports a bounded, time-invariant optimality gap and improved operation and lifetime in PJM regulation simulations.

  • Problem

    Battery control must balance following random instruction signals against degradation costs in real time, while decisions are coupled across time by finite battery capacity.

    Source: §I · p. 1

  • Method

    The paper models degradation with rainflow cycle counting, proves convexity, and uses a simple threshold policy that limits state of charge.

    Source: §I · p. 2, §IV · p. 6

  • Results

    Up to 30% higher operational revenue and a battery lifetime up to 4 times longer are reported in a PJM frequency-regulation case study.

    Source: §IV · p. 6

  • Takeaways & Limitations

    The online policy has a bounded optimality gap independent of operation duration and is strictly optimal under certain market scenarios.

    Source: §IV · p. 6, §VIII · p. 12

    The threshold controller’s stated optimality applies to pay-for-performance settings where penalty prices are determined ahead of time.

    Source: §I · p. 2, §VIII · p. 12

Abstract

from arXiv · show

We study the optimal control of battery energy storage under a general "pay-for-performance" setup such as providing frequency regulation and renewable integration. In these settings, batteries need to carefully balance the trade-off between following the instruction signals and their degradation costs in real-time. Existing battery control strategies either do not consider the uncertainty of future signals, or cannot accurately account for battery cycle aging mechanism during operation. In this work, we take a different approach to the optimal battery control problem. Instead of attacking the complexity of battery degradation function or the lack of future information one at a time, we address these two challenges together in a joint fashion. In particular, we present an electrochemically accurate and trackable battery degradation model called the rainflow cycle-based model. We prove the degradation cost is convex. Then we propose an online control policy with a simple threshold structure and show it achieve near-optimal performance with respect to an offline controller that has complete future information. We explicitly characterize the optimality gap and show it is independent to the duration of operation. Simulation results with both synthetic and real regulation traces are conducted to illustrate the theoretical results.

I. INTRODUCTION

The paper addresses battery control when operators must balance following uncertain instructions against cycle-related degradation without knowing future signals. It combines a cycle-based aging model with an online threshold policy, while situating the approach in prior work and a constrained battery-operation model.

II. BACKGROUND AND PRIOR WORKS

  • Earlier approaches either simplify degradation as power-based cost or assume full future information when using accurate electrochemical models, leaving online control challenging.Power- or throughput-based models can miss cycle-depth effects; a cited NMC example has ten times more degradation near 100% DoD than at 10% DoD for the same charged power or energy throughput.Source: §I · p. 2, §II · p. 3

I. INTRODUCTION

  • The paper jointly addresses cycle-aging complexity and missing future signals, proposing a rainflow-based degradation model and an online threshold policy with a constant gap to the offline optimum.The gap is characterized with respect to possible future signals; the policy is designed for stochastic-signal applications with prices constant over a period.Source: §I · p. 2

II. BACKGROUND AND PRIOR WORKS

  • The paper models capacity fading as cycle-dependent fatigue, counts cycles with rainflow, and proves the resulting degradation model convex.Each cycle contributes independent stress, and total life loss accumulates across cycles; the model focuses on cycle depth, while other factors such as temperature are treated as controlled or outside the focus.Source: §II · p. 3

A. Battery Operations

  • Battery operation is formulated as maximizing revenue from following an instruction signal minus degradation cost, subject to state-of-charge and charge/discharge power constraints.The signal is normalized between -1 and 1, with positive values indicating a power shortage and nonpositive values an excess; the state of charge evolves over finite control steps.Source: A. Battery Operations · pp. 3–4
  • The resulting control problem is difficult because realistic cycle-based degradation is not generally known to be convex and random signals must be handled with coupled state constraints.The paper’s convexity result and cycle-based structure set up its proposed online strategy.Source: A. Battery Operations · p. 4

B. Cycle Counting via Rainflow

Rainflow counting extracts cycles from local extrema in the state-of-charge profile, and cycle depth determines the modeled battery life loss. The paper also specifies the pay-for-performance revenue setting in which the battery operates.

B. Cycle Counting via Rainflow

  • Rainflow identifies cycles by testing four successive local extrema and extracts the middle range when it is no greater than either adjacent range.The removed extrema are discarded and the remaining sequence is connected; unextracted points form a residue whose consecutive pairs are half cycles.Source: B. Cycle Counting via Rainflow · p. 4

C. Battery Degradation Cost

D. Revenue Model

  • The generalized two-stage market model gives a known ahead-of-dispatch payment, then penalizes real-time deviations from an instruction signal.It covers frequency regulation and demand shaping; the paper identifies random penalty prices, as in real-time arbitrage, as a future extension.Source: D. Revenue Model · p. 5

E. Optimization Problem

The paper formulates offline and online battery-control settings and establishes convexity and a duration-independent constant gap for its online policy. It also sketches the convexity proof using step-function decompositions of SoC profiles.

IV. MAIN RESULTS

  • The offline problem is convex when the cycle-aging stress function Φ is convex, while the online threshold policy has a constant worst-case gap independent of operation duration T.The bound applies for all initial states and instruction sequences; the paper also reports zero gap when πηd = θ/ηc.Source: §IV · p. 6
  • The PJM frequency-regulation case study reports operational revenue improvements of up to 30% and battery life up to four times longer.These are case-study outcomes for the proposed control algorithm.Source: §IV · p. 6

V. CONVEXITY AND SUBGRADIENT ALGORITHM

A. Proof of Theorem 1

  • The proof reduces convexity to showing that the rainflow cycle-based cost is convex in the SoC profile when Φ is convex.It uses the symmetry of charging and discharging variables and analyzes the mapping from profiles to cycle costs.Source: A. Proof of Theorem 1 · p. 6

1) Unit step decomposition:

  • Any SoC series can be represented as a finite sum of unit step functions, providing the decomposition used in the convexity proof.Profiles of different lengths can be compared by appending zeros to the shorter profile.Source: 1) Unit step decomposition · p. 7

2) Initial case:

The convexity proof begins with a single-step profile, extends by induction, and motivates an analytical subgradient method for the offline problem.

2) Initial case:

  • The initial case establishes convexity when one profile has a single non-zero step change.The proof accounts for the effect of that step on all cycle depths.Source: 2) Initial case · p. 7

3) Induction Steps:

  • The induction extends convexity from K to K + 1 step changes by analyzing whether successive changes align or oppose one another and comparing their magnitudes.The proof divides the opposing-direction cases according to whether |ZK| is at least |ZK+1| or smaller.Source: 3) Induction Steps · p. 7

B. Subgradient Algorithm

  • Because cycle cost is convex but not continuously differentiable at cycle junctions, the offline problem uses a subgradient method with analytical subgradients.With an appropriate step size, the method converges to the optimal solution at user-defined precision; the log-barrier formulation approaches the constrained problem as λ tends to +∞.Source: B. Subgradient Algorithm · pp. 7–8

VI. ONLINE POLICY

The online policy balances instruction mismatch penalties against cycle-aging costs while respecting operating constraints. It limits the battery’s SoC range with a threshold and achieves an optimality gap that does not grow with operation time.

VI. ONLINE POLICY

  • The policy balances instruction-deviation and cycle-aging costs while satisfying operating constraints, with an optimality gap independent of operation duration.The section assumes regulation capacity is fixed during the preceding capacity-settlement stage.Source: §VI · p. 8

A. Control Policy Formulation

  • Threshold û depends on deviation penalties and degradation cost: lower replacement cost relative to penalties permits a wider SoC range, while higher replacement cost narrows it.The threshold calculation uses the inverse derivative of the cycle-stress function.Source: A. Control Policy Formulation · pp. 8–9
  • The controller tracks the battery’s maximum and minimum SoC and follows instructions until their distance reaches threshold û, then constrains responses to avoid deeper cycles.The policy uses observed SoC and regulation signals to determine its dispatch.Source: A. Control Policy Formulation · p. 9, Figure 7 · p. 9

B. Optimality Gap to Offline Problem

The policy’s worst-case gap to an offline optimum is explicitly characterized under strict convexity of the cycle-depth stress function. Full-cycle depths explain its performance, while half-cycle differences account for near-optimal rather than exact control in some price settings.

B. Optimality Gap to Offline Problem

  • The gap’s characterization uses costs for full cycles and charge or discharge half cycles, with Φ’s strict convexity supplying the condition for the theorem.The cycle costs are denoted Ju, Jv, and Jw, respectively.Source: B. Optimality Gap to Offline Problem · p. 9
  • Under strictly convex cycle stress Φ, the policy’s worst-case optimality gap is εw when πηd > θ/ηc, zero when equal, and εv when lower.The gap is characterized as a constant relative to the offline optimum.Source: B. Optimality Gap to Offline Problem · p. 9
  • The policy is optimal when û is also the optimal half-cycle depth; otherwise, differing optimal half-cycle depths create a bounded gap because incomplete half cycles occur only a limited number of times.Battery operation consists mostly of full cycles, for which enforcing û ensures optimal responses.Source: B. Optimality Gap to Offline Problem · p. 10
  • Fig. 8 holds θ + π constant and assumes unit round-trip efficiency, so û stays the same across price settings despite differences in policy optimality.The offline controller selectively responds to instructions with zero penalty price to return the battery to a shallower cycle depth.Source: Figure 8 · p. 10, B. Optimality Gap to Offline Problem · p. 10

VII. SIMULATION RESULTS

The simulations compare rainflow-based degradation with a linear throughput model and test the proposed online controller against theoretical bounds and baseline controllers. They report lower operating cost under PJM-like penalty prices and a worst-case optimality gap that does not grow with operation duration.

A. Comparison between the Rainflow Cycle-based Model and Benchmark Energy Throughput Model

  • The rainflow model’s response stays within a moderate, smooth SoC range, unlike the linear model’s wilder SoC changes under the tested regulation signals.The figure compares battery responses for mismatch penalties of 100 and 50.Source: Figure 9 · p. 11, A. Comparison between the Rainflow… · p. 11
  • At π = θ = 50 < λe, the rainflow model earns 14.1$/hour while the linear model leaves the battery idle and earns no regulation service revenue.At π = θ = 100 > λe, the linear model follows the signal completely, whereas the rainflow model avoids costly deep cycles.Source: A. Comparison between the Rainflow… · p. 11, Figure 9 · p. 11

B. Online Controller and Time-invariant Optimality Gap

  • Doubling the regulation-signal duration leaves the proposed policy’s worst-case optimality gap unchanged, matching the theoretical bound across the tested cases.Cases 1–3 have zero gap when θ/ηc = πηd; Cases 4–9 have nonzero gaps when η < 1.Source: B. Online Controller and… · pp. 11–12, Table I · p. 11

C. Comparison with Baseline Online Control Algorithms

  • Below 50$/MWh mismatch penalties, the proposed policy saves more than 30% and can extend battery life 3–4 times versus greedy and MPC controllers.Across the tested penalty prices, it has the lowest operating cost; the gap between policies narrows as penalty prices increase.Source: C. Comparison with Baseline Online… · p. 12, Figure 10 · p. 12
  • The proposed controller’s threshold structure limits battery response to the optimal cycle depth, while MPC can act overaggressively by underestimating cycle depth.The MPC comparison assumes perfect prediction of the next 60 signal steps, a condition the text says does not hold in practice.Source: C. Comparison with Baseline Online… · p. 12, Figure 11 · p. 12

VIII. CONCLUSION AND FUTURE WORK

  • The PJM case study concludes that the degradation model and online policy reduce operating cost and extend battery lifetime.The study also identifies random penalty prices and online estimation of degradation parameters as future directions.Source: §VIII · pp. 12–13

A. Single step change convexity

This subsection establishes convexity for a SoC profile with a single step change, then extends the argument to arbitrary step functions by induction. The resulting rainflow cycle-based degradation cost is convex under the stated convexity condition on the cycle stress function.

A. Single step change convexity

  • Theorem 1 states that if the battery cycle-aging stress function Φ is convex, then degradation cost f(x) is convex in the SoC profile x.Convexity is established for a single step change as a base case before the general proof by induction.Source: A. Single step change convexity · pp. 14, 16
  • For any x, y ∈ R^T and λ ∈ [0, 1], the cost satisfies f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y).The proof decomposes profiles into step functions and uses the single-step result as its base case.Source: A. Single step change convexity · p. 14
  • Adding a step change to a SoC profile changes the rainflow cycle depths in a controlled way, supporting the single-step convexity argument.The proof considers a step of amplitude Qt and bounds changes in individual cycle depths by Qt.Source: A. Single step change convexity · pp. 14–16
  • Propositions 1–6 and Lemma 1 establish the base-case convexity needed to prove convexity for profiles with multiple step changes.The proof then proceeds by induction over the number of steps.Source: A. Single step change convexity · pp. 14, 16

B. General rainflow cycle life loss convexity

The proof extends rainflow degradation-cost convexity from profiles with K step changes to profiles with K + 1. It handles cases based on step directions, magnitudes, and whether adjacent changes form a separate cycle.

B. General rainflow cycle life loss convexity

  • When adjacent unit changes are combined, the cycle-stress cost does not increase, supporting the induction argument for rainflow convexity.The proof considers whether the adjacent changes have the same or opposite directions and analyzes the resulting local extrema and cycles.Source: B. General rainflow cycle life loss… · p. 17
  • The induction proof treats three cases for the final step: same-direction changes, opposite-direction changes with the earlier step larger, and opposite-direction changes with the later step larger.The third case is further divided according to the preceding step directions and whether a separate cycle forms.Source: B. General rainflow cycle life loss… · p. 17
  • When the final changes do not form a separate cycle, the proof compares four cycle-depth configurations and establishes convexity for each considered charging-cycle situation.One difficult situation is when consecutive changes differ in direction and the later change is larger; the proof analyzes subcases to derive the induction relation.Source: Figure 13 · p. 17, B. General rainflow cycle life loss… · p. 18
  • Proposition 8 supplies an inequality for a convex increasing function that the proof applies to the remaining induction cases.It compares g(r1) + g(r2) with g(r1 − δ) + g(r2 + δ) when r2 + δ < r1.Source: B. General rainflow cycle life loss… · p. 18

APPENDIX B PROOF OF THEOREM 3

The theorem bounds the proposed online policy’s worst-case gap from the offline optimum by a constant independent of operation duration. The gap is zero when the specified efficiency-adjusted penalty prices are equal.

APPENDIX B PROOF OF THEOREM 3

  • The proposed policy has a constant worst-case optimality gap independent of operation duration T, with r(Jg − J*) ≤ ϵ for all initial states and signal sequences.The theorem assumes a strictly convex battery cycle-aging stress function Φ.Source: Appendix B · p. 19
  • When πηd = θ/ηc, the proposed online algorithm has zero optimality gap from the offline optimum.Theorem 3 gives gap ϵw when πηd > θ/ηc and ϵv when πηd < θ/ηc.Source: Appendix B · p. 19

A. Model Reformulation

The reformulation expresses battery operation through rainflow cycles, with degradation cost defined over those cycles. It replaces charge and discharge quantities with cycle depths in the objective.

A. Model Reformulation

  • The optimization is rewritten using the rainflow cycle-based degradation cost f(c, d), with actions constrained not to exceed regulation signals.Source: A. Model Reformulation · p. 19
  • The rainflow method maps the entire operation uniquely to cycles, allowing charge and discharge power to be represented by cycle depths.A full cycle has symmetric depth for charge and discharge.Source: A. Model Reformulation · p. 19
  • Substituting the cycle-depth expressions into the reformulated objective replaces charge and discharge variables with cycle depths.Source: A. Model Reformulation · p. 19

B. Proof of Theorem 3

The proof relates offline and online cycle depths to operation constraints, then uses the rainflow residue's half-cycle structure to characterize the optimality gap. It identifies full cycles as optimal under the policy and locates the gap in half-cycle results.

B. Proof of Theorem 3

  • Offline cycle depths either reach the optimal cycle depth or are bounded by operation constraints, including regulation signals and battery energy limits.Source: B. Proof of Theorem 3 · p. 19
  • For the control action of g(·), a cycle depth either reaches ˆu or is bounded by operation constraints.Source: B. Proof of Theorem 3 · p. 19
  • A rainflow residue has one unique largest half cycle, with other half cycles strictly decreasing away from it on either side.Source: B. Proof of Theorem 3 · p. 20, Figure 14 · p. 20
  • The proposed policy achieves optimal control for all full cycles, while the optimality gap is caused by half-cycle results.The proof analyzes a rainflow residue profile whose largest half cycle is in the discharging direction.Source: B. Proof of Theorem 3 · p. 20
  • The proof bounds the remaining half-cycle depths by operation constraints and uses their conditions to calculate the worst-case gap.The half-cycle depth conditions are summarized in Table IV; the policy sets ˆu as the depth of cycles unbounded by operation.Source: B. Proof of Theorem 3 · p. 20, Table IV · p. 20
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