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Improving optimal control of grid-connected lithium-ion batteries through more accurate battery and degradation modelling

Jorn M. Reniers, Grietus Mulder, Sina Ober-Blobaum, David A. Howey

arXiv:1710.04552v3eess.SYcs.SY

TL;DR

Long-horizon optimization with nonlinear battery models had not yet been used to quantify battery performance. The paper compares three increasingly complex dynamic battery and degradation models and reports a 175% simulated-profit increase from the simplest to the most complex model.

  • Problem

    Nonlinear battery models had not been used for long-horizon optimization, although horizons such as a year are needed to quantify performance.

    Source: §1 · p. 2

  • Method

    The paper formulates three dynamic battery and degradation models with increasing complexity.

    Source: §6 · p. 18

  • Results

    175% total simulated profit increase from the simplest to the most complex model, while RMS capacity-prediction error fell from 11% to 5% with increased model accuracy.

    Source: §6 · p. 18

  • Takeaways & Limitations

    More accurate battery models allowed more capacity use through decreasing safety margins, increasing revenue by up to 42%.

    Source: §6 · p. 18

    Single-particle-model validation may slightly misestimate lost capacity because it represents only one degradation mode.

    Source: §5 · p. 17

Abstract

from arXiv · show

The increased deployment of intermittent renewable energy generators opens up opportunities for grid-connected energy storage. Batteries offer significant flexibility but are relatively expensive at present. Battery lifetime is a key factor in the business case, and it depends on usage, but most techno-economic analyses do not account for this. For the first time, this paper quantifies the annual benefits of grid-connected batteries including realistic physical dynamics and nonlinear electrochemical degradation. Three lithium-ion battery models of increasing realism are formulated, and the predicted degradation of each is compared with a large-scale experimental degradation data set (Mat4Bat). A respective improvement in RMS capacity prediction error from 11\% to 5\% is found by increasing the model accuracy. The three models are then used within an optimal control algorithm to perform price arbitrage over one year, including degradation. Results show that the revenue can be increased substantially while degradation can be reduced by using more realistic models. The estimated best case profit using a sophisticated model is a 175% improvement compared with the simplest model. This illustrates that using a simplistic battery model in a techno-economic assessment of grid-connected batteries might substantially underestimate the business case and lead to erroneous conclusions.

1. Introduction

Growing intermittent renewable deployment creates electricity-system challenges, while battery storage offers flexibility but depends economically on use-sensitive lifetime. The paper addresses the lack of long-horizon optimization with nonlinear battery models by evaluating realistic dynamics and degradation over a full year.

1. Introduction

  • Battery storage offers flexibility against electricity-system challenges from growing intermittent renewables, while capacity and power fade affect profitability.Source: §1 · p. 2
  • Lithium-ion battery lifetime depends on usage because distinct degradation mechanisms respond differently to usage patterns.Source: §1 · p. 2
  • Prior economic assessments often use empirical degradation correlations, which may not generalize beyond limited test conditions, account for aging changes, or apply to other cell types.The cited discussion also cautions against extrapolating to other load profiles without a theoretical basis.Source: §1 · p. 2
  • The paper identifies a gap in using nonlinear battery models for long-horizon optimization, although horizons such as a year are needed to quantify performance.Source: §1 · p. 2
  • The study evaluates economic performance gains using a nonlinear electrochemical model with realistic dynamics and degradation in a realistic grid application over a full year.Source: §1 · p. 2

2. Nomenclature

The nomenclature defines the battery model's state, control, electrical, capacity, and degradation quantities.

2. Nomenclature

  • Capacity and degradation notation includes EWh and EAh for battery capacity, Elost,Wh for lost energy capacity, and C for degradation cost.Source: §2 · p. 3
  • The state-space model is denoted f, with x(t) for state variables and ci(r,t) for lithium concentration in electrode i at radius r.Source: §2 · p. 3
  • Control and electrical quantities include u(t) for control, P(t) for battery power, I(t) for current, and V(t) for battery voltage.Source: §2 · p. 3

3. Methods

The methods optimize grid-battery price arbitrage while accounting for capacity fade, using three battery models and model-specific procedures on perfectly known Belgian 2014 day-ahead prices. They address long horizons with sliding-window optimization and post-process optimal currents to enforce battery safety and estimate degradation costs.

3.1. Problem setup

  • The operator’s task is to choose a battery load profile that maximizes revenue while minimizing lost capacity, using 2014 Belgian day-ahead prices assumed perfectly known.This makes the optimization deterministic.Source: §3.1 · p. 4

3.2. Battery models

  • Three models represent the battery as an energy bucket, an equivalent circuit, or an electrochemical single particle model with thermal dynamics and SEI growth.The bucket model tracks state of charge; the circuit model also tracks current through a parallel resistor, while the single particle model represents electrode diffusion and lithium-consuming SEI growth that reduces capacity.Source: §3.2 · pp. 5–6

3.3. Optimisation algorithm

  • The equivalent-circuit optimization was about 10 times faster than the single-particle model but over 1000 times slower than the bucket model.For the single-particle model, spatial Chebyshev discretization and multiple shooting supported nonlinear optimization with IPOPT.Source: §3.3 · p. 8

3.4. Sliding window optimisation

  • Using previous optimization techniques, the single-particle model took about 6 hours to optimize one week at 5-second resolution, involving 10,752 variables; window size affected outcomes minimally beyond one day, so a two-day window was chosen.Source: §3.4 · p. 8
  • Day 2 was optimized to account for end-effects but discarded; each subsequent window started from the previous day’s ending battery state.The method optimizes overlapping two-day windows, advancing one day at a time.Source: §3.4 · p. 8
  • Unlike MPC, the sliding-window approach does not update states with measurements after each window and is suboptimal because it excludes the entire time horizon.Source: §3.4 · pp. 8–9

3.5. Post processing

  • Optimal currents could cause under- or over-charging in a real battery, so the study reduced modeled capacity to provide safety margins.The bucket model’s high-power input until full charge would overcharge a real battery because of diffusion delays, overpotentials, and resistive voltage drops.Source: §3.5 · p. 9
  • Holding the cell at a voltage limit is an alternative to avoid over- and under-voltages; it changes quantitative results but not overall conclusions.Source: §3.5 · p. 9
  • For validation without experimental tests, the study used the single particle model (SPM) as a real-battery proxy and scaled down currents whenever predicted voltage exceeded safety limits.Starting at 50% SoC, current scaling kept the battery closer to that state and avoided under- and overcharging.Source: §3.5 · p. 10
  • Degradation cost was estimated by multiplying capacity loss, measured from a low-current full charge/discharge cycle, by λdegr,Ah.Source: §3.5 · p. 10

4. Results

Across degradation validation and year-long price arbitrage, more realistic battery models improved capacity prediction, while accounting for degradation changed optimal use and profit.

4.1. Accuracy of battery degradation models

  • The equivalent-circuit empirical formula fits calendar ageing for the same NMC chemistry, but its cycle predictions are less reliable under different testing conditions.It predicts similar degradation across cycles centred near 50% SoC because it uses mean cycle voltage; its initial degradation is overestimated.Source: §4.1 · p. 10
  • In battery capacity prediction, average root mean square error fell from 11% of initial capacity for the bucket model to 5% of initial capacity for the single particle model.Source: §4.1 · p. 12
  • The single particle model predicts major degradation trends better than the other models, but underestimates degradation at low SoC because it represents only one degradation process.The model's scope means its predictions are not fully accurate.Source: §4.1 · p. 12

4.2. Optimal battery utilisation for arbitrage

  • More accurate battery models access more capacity, while co-optimising for profit reduces battery utilisation relative to revenue optimisation.Figure 5 compares wholesale prices and state of charge for revenue- and profit-maximising profiles.Source: §4.2 · p. 12, Figure 5 · p. 13
  • The bucket model limits power and energy throughput; the equivalent circuit model favours SoC levels with less calendar ageing; the single particle model lowers SoC and avoids high-power charging.These behaviours reflect the different degradation formulations used in the optimisation.Source: §4.2 · p. 14
  • Although the bucket model cannot access the full capacity, its restriction avoids the high-SoC region where SEI-growth degradation is greatest, while also limiting revenue.The reported performance benefit is partly a side-effect of the model's inaccessible high-SoC region.Source: §4.2 · p. 14
  • The single-particle model accounting for degradation produced 72.41 AC in profit, a 175% improvement over the simplest model, the bucket model without degradation accounting.Source: §4.2 · p. 17
  • Including degradation lowers capacity loss and raises profit for the bucket and single particle models, but the equivalent circuit model's empirical formula reduces use too much.The authors attribute the equivalent circuit model's lower profit to its unrealistic degradation formula.Source: Table 2 · p. 17, §4.2 · p. 17

5. Discussion

The discussion estimates an approximately 80% degradation reduction when maximising profit rather than revenue with the single particle model. It also notes limits in the degradation model and optimisation horizon that qualify the comparisons.

5. Discussion

  • The single particle model includes only one degradation mode, so degradation at low SoC, high power, and low temperatures may be underestimated.Using the same model for optimisation and validation also gives it an artificial advantage over the other models.Source: §5 · p. 17
  • The two-day optimisation horizon cannot capture the equivalent circuit model's predicted benefit from early use reducing later degradation, potentially producing suboptimal outcomes.The authors suggest optimising a whole year at once could capture this effect.Source: §5 · p. 17
  • Maximising profit instead of revenue reduced degradation by about 80% in the single particle optimisation.The authors say this estimate is consistent with other reported results and regard it as realistic for full-scale optimisation.Source: §5 · pp. 17–18

6. Conclusions

The paper compares increasingly complex dynamic battery and degradation models, validating capacity-fade predictions and applying them to year-long wholesale price arbitrage. More realistic models improve simulated profit and degradation outcomes, but add computational demands and depend on degradation terms being represented adequately.

6. Conclusions

  • 11% to 5%: increasing model complexity reduced the root mean square error in predicted capacity fade.The models were compared with a large degradation data set.Source: §6 · p. 18
  • The optimal-control study used known-a-priori wholesale prices, validated utilization profiles with the single particle model, and introduced safety margins against over- or under-charge.Source: §6 · p. 18
  • More accurate battery models allowed more capacity use through reduced safety margins, increasing revenue by up to 42%.Source: §6 · p. 18
  • More complex degradation models reduced degradation by up to 80%, but could lower profit when they overestimated degradation in some scenarios.Source: §6 · p. 18
  • The optimizer minimizes only degradation terms explicitly included, so the model needs all necessary terms.Source: §6 · p. 18
  • More accurate nonlinear models require small time steps for stability and create substantial computational demands, with thousands of optimization variables even after reductions.The single-particle model uses Chebyshev discretization to reduce the number of state variables.Source: §6 · p. 18
  • 175%: total simulated profit increased from the simplest to the most complex model.The simplest model is the bucket model without degradation accounting.Source: §6 · p. 18
  • Actual battery degradation may differ from the simulated degradation, while techno-economic assessments using simple models could under- or overestimate battery potential.Source: §6 · pp. 18–19

Appendix A. Model equations of the single particle model

Appendix A lists the single-particle model's reaction, SEI, and electrode-diffusion components, alongside defined physical variables and temperature-dependent coefficients.

Appendix A. Model equations of the single particle model

  • Diffusion and reaction coefficients are temperature-dependent through Arrhenius relations, with reference values and activation energies listed.Source: Appendix A · pp. 24, 26
  • The listed single-particle model components include Butler–Volmer reaction kinetics, an SEI side-reaction current, and diffusion in both electrodes.Source: Appendix A · pp. 24–25
  • Defined quantities include electrode lithium concentration, battery current and voltage, temperature, and lost lithium content.Source: Appendix A · pp. 26–27
  • The listed model parameters also include particle radius, SEI resistance, electrode surface areas, total battery surface, and battery volume.Source: Appendix A · pp. 26–27

Appendix B. Post processing without safety margins

Appendix B describes post-processing without safety margins and reports its effects on battery use, degradation, revenue, and profit. It also outlines a validation approach that follows the optimized profile until voltage limits are reached.

Appendix B. Post processing without safety margins

  • The alternative validation follows the optimal profile until a voltage limit is reached, then holds voltage fixed; this allows the full battery capacity to be used.Source: Appendix B · p. 27
  • When degradation was ignored, higher state of charge increased degradation and sharply reduced profit, producing an overall loss for bucket- and equivalent-circuit-controlled batteries.Source: Appendix B · p. 27
  • With degradation-aware profit optimization, overall profit remained almost the same, while the bucket-model battery had 40% higher revenue and double the degradation.Source: Appendix B · p. 27
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