Optimal BESS Scheduling for Multi-Market Participation in the Nordics

Chresten Traeholt; Zeenat Hameed

REVIEW 4 major objections 6 minor 7 references

Optimal BESS Scheduling for Multi-Market Participation in the Nordics

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that spot-market revenue forecasts are consistently more reliable than frequency-market forecasts across Denmark, Finland, and Norway, and that forecast errors shift profits but not the choice of which markets a battery…

desk verdict Useful Nordic BESS scheduling paper with a real empirical comparison, but unspecified scenario generation makes its headline claim unreproducible as written. read the letter →

arxiv 2506.02837 v1 pith:OXBIFOWC submitted 2025-06-03 math.OC

Zeenat Hameed , Chresten Traeholt This is my paper

classification math.OC MSC 90-0862M2090C1590C11

keywords batteryenergystorageNordicelectricitymarketsFCR-NFCR-DGAMforecastingstochasticoptimizationmarketparticipationrevenuepredictability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether a battery energy storage operator can plan multi-market participation in Nordic electricity markets using one-week-ahead revenue forecasts rather than perfect information. It builds a two-stage framework: generalized additive models (GAMs) predict spot and FCR-N revenues in Denmark, Finland, and Norway from three years of hourly data, and a stochastic mixed-integer optimizer co-optimizes bids across FCR-N, FCR-D, spot-discharge, and spot-charge under state-of-charge limits, inverter losses, and the different pay-as-bid and pay-as-clear rules. The central finding is that spot markets are consistently more predictable than frequency markets, while forecast errors modestly reduce bid acceptance without changing which markets the battery chooses. If correct, this gives operators a practical basis to trust spot-based revenue forecasts for market selection and to treat frequency-market forecasts as the uncertain input to hedge.

What carries the argument

The central object is the generalized additive model (GAM) revenue forecast in log space, where log spot revenue is modeled as smooth functions of hour-of-day, day-of-week, and their tensor interaction. These forecasts produce the market-energy blocks $E^{\text{dch}}_{stmp}$ and $E^{\text{ch}}_{stmp}$ — per-minute charge and discharge energy volumes for each scenario, market, and bid size — that feed the stochastic mixed-integer BESS optimizer. The optimizer's objective is expected revenue over scenarios, with big-M linearizations linking binary bid-acceptance variables to availability payments and energy payments, and it carries the argument by showing that GAM forecast errors propagate into bid acceptance but not into market selection.

What would settle it

A reader could falsify the central market-selection claim by rerunning the same GAM-to-scenario pipeline with substantially larger, non-Gaussian forecast errors and checking whether the optimizer switches its dominant market choice away from FCR-N.

Watch

Extended reading notes

Core claim

The paper's central claim is that in Denmark, Finland, and Norway, spot-market revenues are more predictable than FCR-N revenues, and that this predictability gap is what shapes the value of forecasting for battery operators. Using GAMs with hourly and daily smooth terms, the authors report out-of-sample MAPE below 1% for spot revenue in all three countries, whereas FCR-N MAPE ranges from roughly 1.5% in Denmark to 4.3% in Norway, and the adjusted R-squared for FCR-N varies from 0.92 in Denmark down to 0.06 in Norway. Feeding these forecasts into a stochastic mixed-integer optimizer, they find that forecast errors lower bid acceptance hours by a few percent but do not alter the dominant market choice, which remains FCR-N in the studied weeks. The paper further claims that multi-bid strategies give negligible profit gains over single-bid strategies, that a fixed daily state-of-charge shifts participation toward FCR-N because of its minute-level flexibility, and that seasonal price patterns in 2019 versus 2021 drove very different market splits and a 49% profit difference.

Load-bearing premise

The optimizer weights scenarios by probabilities and depends on per-minute market-energy blocks per scenario, but the paper never specifies how the GAM forecast errors are converted into those scenario-dependent energy blocks.

Editorial extensions

If this is right

A BESS operator in the Nordics can use spot-revenue forecasts to decide market participation, because spot errors remain below about 1% MAPE in all three countries studied.
Frequency-market revenue forecasts, especially FCR-N in Norway, are unreliable enough that a risk-aware operator should treat them as deeply uncertain inputs rather than fixed revenue estimates.
The optimizer's finding that forecast errors do not change market choice means that participation patterns derived under perfect information remain roughly valid under GAM forecasts, so market-selection studies do not need perfect price data to be informative.
Single-bid strategies are nearly as profitable as multi-bid strategies, so the added complexity of bidding multiple power levels is not worth its operational burden.
Fixed daily state-of-charge constraints push a battery into FCR-N rather than spot arbitrage, which is a concrete operational rule for operators who cannot vary their daily SOC flexibly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implicit extension is that the same framework could be used to test whether forecast-error impact scales with battery size or with the share of revenue coming from frequency markets, which this paper does not vary.
The paper's comparison of GAM versus AI-based forecasts suggests a testable hypothesis: any forecaster that captures the smooth weekly spot pattern will reproduce the market-selection result, because it is driven by price level, not by fine-grained price dynamics.
Since spot revenue includes a consumption volume term, the predictability result may partly depend on how smooth national consumption is; a direct test would be to run the same GAM on spot price alone and compare MAPE.
The threshold claim that forecast errors do not alter market participation should be probed under larger error magnitudes, such as deliberately corrupted forecasts, to find the error level at which participation rules would change.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the useful part is empirical: it compares GAM forecast accuracy for spot vs FCR-N revenues in DK/FI/NO and then feeds those forecasts into a MILP to see how forecast errors change BESS bidding. Second, the load-bearing part is not reproduced: the scenario set S and probabilities p_s in Eq. (14) are never constructed from the forecast residuals, so the claim that forecast errors 'do not alter overall market participation' rests on an unspecified input.

The paper does two useful things. It fits GAMs to spot and FCR-N revenues in three Nordic countries and reports that spot revenue is more predictable than FCR-N in FI/NO but not in DK (where FCR-N adjusted R2 is 0.92 vs 0.77 for spot). That is a concrete, non-obvious finding, and the conclusion section states the DK exception correctly. The MILP formulation for multi-market bidding with pay-as-bid/pay-as-clear, SOC dynamics, inverter losses, and multi-power bids is standard but competently done and could be a useful template.

The soft spots are real and one is load-bearing. The paper never specifies how the GAM forecast error distribution is converted into the scenario set S and probabilities p_s used in the expected-revenue objective (14). Equations (9) and the OPPS/MPPS labels in Fig. 5 gesture at different scenario constructions for GAM vs AI forecasts, but no formal rule is given. Without that construction, the result that forecast errors 'do not alter overall market participation' is unfalsifiable. This is a missing implementation detail, not a methodological disagreement. Second, the reported sub-1% spot MAPE is hard to believe if computed on raw revenue, and Eq. (8) does not line up with the log-revenue GAMs. The authors need to state whether MAPE is on the log scale, show the distribution, and reconcile the numbers. Third, the abstract's 'consistently higher predictability' is contradicted by their own Denmark result; the conclusion text says it correctly, so the abstract should be fixed. Finally, no data or code are provided, and the six self-authored references covering market data and AI methods are not a red flag by themselves, but an independent data source would help.

This is not a rejection-level problem. The empirical comparison is useful and the model formulation is solid enough to build on. The missing scenario-generation rule and MAPE reconciliation are exactly the kind of thing a serious referee would request. I would send it to review, expect major revision, and re-evaluate.

Referee Report

4 major / 6 minor

Summary. The paper proposes a framework for one-week-ahead revenue forecasting for battery energy storage systems (BESSs) in Nordic spot and FCR-N markets using generalized additive models (GAMs), and feeds the forecasts into a stochastic mixed-integer linear program that co-optimizes participation in FCR-N, FCR-D, and spot markets. The framework is applied to Denmark, Finland, and Norway using three years of hourly price and volume data, and the authors report forecast accuracy (adjusted R2 and MAPE) as well as the impact of forecast errors on bid acceptance, market selection, and profitability. The central claims are that spot revenues are more predictable than frequency-market revenues and that forecast errors only modestly affect bid acceptance without altering overall market participation.

Significance. If the central claims are supported, the framework provides a practical, integrated tool for BESS operators to plan multi-market strategies, and the finding that forecast errors do not change market-selection patterns would be useful for operational decision-making. The manuscript's strengths include a detailed MILP formulation (Eqs. (14)-(38)) that captures pay-as-bid and pay-as-clear pricing, state-of-charge constraints, inverter losses, and multi-bid options, as well as a systematic multi-country comparison over several years. However, the empirical conclusions are currently weakened by an unspecified scenario-generation procedure and an overgeneralized predictability claim; these issues affect the validity of the headline results as presented.

major comments (4)

[Section III-B, Eqs. (9)-(14)] The stochastic program weights scenarios s by probabilities p_s and uses scenario-dependent market-energy blocks E^dch_stmp and E^ch_stmp, but the manuscript never specifies how GAM forecast errors are transformed into the scenario set S, the number of scenarios, the sampling scheme, or the assignment of p_s. This construction is load-bearing because the conclusion that forecast errors 'do not alter overall market participation' is obtained by solving this stochastic program; without a formal scenario-generation rule, the result is not reproducible or falsifiable. Please provide the exact rule, including the distributional assumptions on f_st and P_max_shm and how they are sampled.
[Section IV and Eq. (8)] The reported MAPE values (spot: 0.74-0.94%; FCR-N: 1.52-4.34%) are implausibly low for one-week-ahead hourly revenue forecasts, especially since the GAMs are fitted to log-revenue (Eqs. (4)-(6)) while Eq. (8) defines MAPE on the original scale of the response. A sub-1% MAPE on 168-hour-ahead hourly spot prices is not consistent with typical Nordic spot-price volatility, and the relationship between in-sample adjusted R2 and out-of-sample MAPE is not reconciled. Please clarify the exact quantity to which Eq. (8) is applied (e.g., weekly aggregated revenue, smoothed signals, or only a subset of hours), and report out-of-sample R2 or another scale-compatible accuracy metric.
[Abstract and Section V] The abstract's claim that 'spot markets exhibit consistently higher predictability than frequency markets' is contradicted by the reported adjusted R2 values in Section IV (Denmark FCR-N R2_adj=0.92 vs spot R2_adj=0.77) and by the conclusion in Section V that Danish FCR-N predictability exceeds Danish spot predictability. The claim needs to be qualified by country and by metric (R2 vs MAPE), or the numbers must be reconciled; as stated, the abstract overgeneralizes the results.
[Section IV, Fig. 5 and 'AI-based forecasts'] The comparison between GAM 'one-price-point' scenarios (OPPS) and AI 'multiple-price-points' scenarios (MPPS) in Fig. 5 is not reproducible: the AI model is not specified (beyond 'AI-based forecasts'), and the exact scenario-construction rules for OPPS versus MPPS are not defined anywhere in Section III. Since the profit and bid-acceptance comparisons in Fig. 5 depend directly on these scenario sets, please specify the AI method, its training data and features, and precisely how the two scenario-generation approaches differ.

minor comments (6)

[Section III-B, equations (30)-(34)] The text refers to Eq. (33) for the spot-discharge payment and Eq. (34) for the spot-charge cost, but the displayed equation numbers are (30) and (31); the cross-references for equations (35)-(37) similarly point to (32)-(34). Please renumber or fix the cross-references.
[Fig. 4 caption] The caption reads 'Actual versus forecasted valued for FCR-N' and should read 'actual versus forecasted values for FCR-N'.
[Section II, references [4] and [5]] References [4] and [5] are site-selection and placement papers; citing them as sources for market mechanisms in Section II is misleading. Please cite the relevant TSO market rules or reference [6] directly.
[Section III-B, Eqs. (9) and (20)-(21)] The notation is inconsistent: Eq. (9) defines market-energy blocks E^dch_stmp and E^ch_stmp with the power-level index p, while Eqs. (20)-(21) use E^dch_stm and E^ch_stm without p. Use a consistent subscript to avoid ambiguity.
[Section IV, training procedure] The text says 'We trained six GAMs ... using rolling two-week windows of hourly data' but Section III-A states the models use 'three years of hourly price and volume data'; clarify whether the training window is three years or two weeks, and describe how the 2019-2021 evaluation avoids data leakage between training and test periods.
[Section IV, multi-bid comparison] The multi-bid comparison (three power levels and three energy levels) is not clearly represented in the MILP, given that constraint (15) allows at most one bid per hour. Please explain how multiple bids are mapped to the decision variables and constraints, or explicitly state how the multi-bid scenario relaxes Eq. (15).

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: GAM forecasts are validated out-of-sample and the optimizer is fed exogenous market data; the main caveat is an unspecified scenario construction, not a circular reduction.

full rationale

The paper's derivation chain is linear rather than circular: GAMs are fitted to historical hourly log-revenues using penalized likelihood (Eq. 3), evaluated out-of-sample with MAPE (Eq. 8), and the resulting forecasts are then passed to a mixed-integer optimizer whose objective (Eq. 14) and constraints (Eqs. 15-37) are independent of the forecast-fitting procedure. No target result of the paper (e.g., the claim that spot markets are more predictable, or that forecast errors do not change market participation) is used as an input to set GAM smoothing parameters, scenario weights, or optimizer coefficients. The six self-authored references are descriptive or contextual (market rules, earlier AI-forecast implementations, and site-selection studies); they are not invoked as a uniqueness theorem or as proof of the paper's central empirical conclusions. The AI-based forecast baseline is imported from prior work rather than re-derived here, but the main comparison also uses the paper's own GAM forecasts, so the qualitative conclusion does not reduce to the self-citation. The most serious weakness is an omitted construction, not circularity: Section III-B introduces scenario-dependent market-energy-blocks via Eq. (9) and expected revenue via Eq. (14), but never specifies how GAM forecast errors are transformed into the scenario set S and probabilities p_s. This is a reproducibility and falsifiability gap, but it is not an equivalence between inputs and outputs. Accordingly, the circularity score is 0.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The model is built from standard statistical and optimization components, so no new physical entities are introduced. The main latent costs are the unstated scenario construction, the single-bid-per-hour restriction, and the hand-set parameters ILF, SOC bounds, and bid bounds.

free parameters (8)

GAM smoothing parameter lambda = not reported
Penalty in Eq. (3) controls smoother complexity; selection method is not described.
GAM basis dimensions = k=24 hours, k=7 days, ti k=c(24,7)
Chosen in Eq. (7); not optimized or justified.
Inverter loss factor ILF = 0.10
Fixed in Eq. (13); affects energy throughput and profits, but no sensitivity analysis is given.
Initial and final SOC = 0.5 MWh
Eqs. (11), (12), and (24) set start and end SOC to 0.5 MWh by hand, constraining daily energy use.
Bid price bounds bidmin and bidmax = unspecified
Eq. (19) imposes bounds but their values and market justification are not given.
Scenario probabilities p_s = unspecified
Objective Eq. (14) weights scenarios by p_s, but scenario probabilities are not defined or estimated.
Big-M constants = not reported
Used in linearizations Eqs. (17)-(18) and (26)-(34); values affect feasibility but are not reported.
Multi-bid power and energy levels = FCR: 0.9/0.6/0.3 MW; spot: 0.8/0.6/0.4 MWh
Selected for the multi-bid comparison in Section IV; no sensitivity analysis is provided.

assumptions (6)

domain assumption BESS can bid in at most one market per hour (sum of xbid_hmp <= 1, Eq. 15).
Restricts co-optimization to sequential hourly market choice, which excludes simultaneous FCR and spot participation that 'multi-market co-optimization' might imply.
standard math Big-M constants are sufficiently large that linearized constraints do not cut off optimal bids.
Eqs. (17)-(18) and (26)-(34) rely on big-M linearization without specifying M values.
domain assumption GAM residuals are Gaussian and homoskedastic (family=gaussian in Eq. 7).
Used for likelihood in Eq. (3); no residual diagnostics are reported.
domain assumption Historical 2019-2021 price and volume data are representative of future Nordic market conditions.
One-week-ahead forecasting and profit results are assumed to generalize from this sample.
domain assumption Minute-resolution frequency deviations and clearing prices used to construct MEBs are accurate and available for all scenarios.
MEBs in Eq. (9) drive all energy constraints, but the data source and scenario construction are not specified.
domain assumption Accepted FCR bids are paid at the submitted bid price in pay-as-bid markets.
Eqs. (27)-(29) assume availability payment equals bid price, which matches some Nordic pay-as-bid rules but not all procurement types.

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Pith. "Pith review of Optimal BESS Scheduling for Multi-Market Participation in the Nordics." pith.science (2026). https://pith.science/paper/OXBIFOWC

@misc{pith2026250602837,
  author       = {Pith},
  title        = {Pith review of: Optimal BESS Scheduling for Multi-Market Participation in the Nordics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXBIFOWC}},
  note         = {Machine review of arXiv:2506.02837}
}

read the original abstract

Battery energy storage systems BESSs can provide fast frequency reserves and energy arbitrage in Nordic electricity markets but their limited energy capacity requires accurate revenue forecasts and coordinated bidding across multiple submarkets. This paper introduces a unified framework that employs generalized additive models GAMs to generate one week ahead forecasts of spot and FCRN revenues in Denmark, Finland, and Norway using three years of hourly price and volume data. Forecast outputs feed a stochastic mixed integer optimizer that co-optimizes BESS participation in FCRN, FCRD, spot markets, subject to state of charge constraints, inverter losses, and differing pay as bid and pay as clear rules. Comparative analyses evaluate forecast accuracy and quantify the impact of forecast errors on BESS bid acceptance, market selection, and profitability under realistic seasonal price patterns. Results demonstrate that spot markets exhibit consistently higher predictability than frequency markets, and that forecast errors modestly affect bid acceptance but do not alter overall market participation. The proposed approach provides BESS operators and investors with a tool to assess revenue uncertainty and optimize multi market strategies in Nordic power systems.

Figures

Figures reproduced from arXiv: 2506.02837 by the authors.

**Figure 2.** Spot-market smoothers (hourly, daily, interaction) by country. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 5.** Estimated BESS profits and bid acceptance hours using original, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: Profit and cost differences between multi-bid and single-bid strategies [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 3.** Figure 3: FCR–N smoothers by country [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]

**Figure 4.** Figure 4: Actual versus forecasted valued for FCR–N, 2019–2021. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

[1]

Investigating the participation of battery energy storage systems in the Nordic ancillary services markets from a business perspective,

Z. Hameed, C. Træholt, and S. Hashemi, “Investigating the participation of battery energy storage systems in the Nordic ancillary services markets from a business perspective,” Journal of Energy Storage, vol. 58, p. 106464, 2023

work page 2023
[2]

A business-oriented approach for battery energy storage placement in power systems,

Z. Hameed, S. Hashemi, H. H. Ipsen, and C. Træholt, “A business-oriented approach for battery energy storage placement in power systems,” Applied Energy, vol. 298, p. 117186, 2021

work page 2021
[3]

Applications of AI-Based forecasts in renewable based electricity balancing markets,

Z. Hameed, S. Hashemi, and C. Træholt, “Applications of AI-Based forecasts in renewable based electricity balancing markets,” in Proc. IEEE 22nd Int. Conf. on Industrial Technology (ICIT), 2021, pp. 1–6

work page 2021
[4]

Site selection criteria for battery energy storage in power systems,

Z. Hameed, S. Hashemi, and C. Træholt, “Site selection criteria for battery energy storage in power systems,” in Proc. 2020 IEEE Canadian Conf. on Electrical and Computer Engineering (CCECE), 2020, pp. 1–5

work page 2020
[5]

Placement of Battery Energy Storage for Provision of Grid Services–A Bornholm Case Study,

Z. Hameed, S. Hashemi, H. H. Ipsen, and C. Træholt, “Placement of Battery Energy Storage for Provision of Grid Services–A Bornholm Case Study,” in Proc. 2021 IEEE 9th Int. Conf. on Smart Energy Grid Engineering (SEGE), 2021, pp. 1–6

work page 2021
[6]

Frequency markets and the problem of pre-dictability,

Z. Hameed, M. Pollitt, P. Kattuman, and C. Træholt, “Frequency markets and the problem of pre-dictability,” Tech. Rep., Faculty of Economics, University of Cambridge, 2023

2023
[7]

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work page 1954

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Investigating the participation of battery energy storage systems in the Nordic ancillary services markets from a business perspective,

Z. Hameed, C. Træholt, and S. Hashemi, “Investigating the participation of battery energy storage systems in the Nordic ancillary services markets from a business perspective,” Journal of Energy Storage, vol. 58, p. 106464, 2023

work page 2023

[2] [2]

A business-oriented approach for battery energy storage placement in power systems,

Z. Hameed, S. Hashemi, H. H. Ipsen, and C. Træholt, “A business-oriented approach for battery energy storage placement in power systems,” Applied Energy, vol. 298, p. 117186, 2021

work page 2021

[3] [3]

Applications of AI-Based forecasts in renewable based electricity balancing markets,

Z. Hameed, S. Hashemi, and C. Træholt, “Applications of AI-Based forecasts in renewable based electricity balancing markets,” in Proc. IEEE 22nd Int. Conf. on Industrial Technology (ICIT), 2021, pp. 1–6

work page 2021

[4] [4]

Site selection criteria for battery energy storage in power systems,

Z. Hameed, S. Hashemi, and C. Træholt, “Site selection criteria for battery energy storage in power systems,” in Proc. 2020 IEEE Canadian Conf. on Electrical and Computer Engineering (CCECE), 2020, pp. 1–5

work page 2020

[5] [5]

Placement of Battery Energy Storage for Provision of Grid Services–A Bornholm Case Study,

Z. Hameed, S. Hashemi, H. H. Ipsen, and C. Træholt, “Placement of Battery Energy Storage for Provision of Grid Services–A Bornholm Case Study,” in Proc. 2021 IEEE 9th Int. Conf. on Smart Energy Grid Engineering (SEGE), 2021, pp. 1–6

work page 2021

[6] [6]

Frequency markets and the problem of pre-dictability,

Z. Hameed, M. Pollitt, P. Kattuman, and C. Træholt, “Frequency markets and the problem of pre-dictability,” Tech. Rep., Faculty of Economics, University of Cambridge, 2023

2023

[7] [7]

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work page 1954