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arxiv: 2307.07580 · v4 · submitted 2023-07-14 · 🧮 math.OC · cs.SY· eess.SY

Home Battery Dispatch under a Tiered Peak Power Tariff

David P\'erez-Pi\~neiro , Sigurd Skogestad , Stephen Boyd This is my paper

Pith reviewed 2026-05-24 07:31 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords home battery dispatchtiered peak power tariffmodel predictive controlmixed-integer linear programmingelectricity cost minimizationpeak shavingTrondheim home data
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The pith

An MPC policy with simple forecasts dispatches home batteries to within 1.7% of the minimum cost under tiered peak power tariffs.

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

The paper studies battery operation in a home to minimize a combined electricity bill consisting of an energy charge plus a tiered charge based on the average of the N largest daily peak powers each month. It first solves a mixed-integer linear program that gives the absolute lowest cost possible when loads and prices are known perfectly in advance, establishing a performance bound. It then proposes a model predictive control policy that repeatedly solves a smaller version of the same program using only basic forecasts. Experiments on a full year of measured data from a home in Trondheim, Norway, show that this policy stays close to the bound and produces substantially larger savings than rule-based alternatives.

Core claim

With perfect foresight the minimum cost solves a mixed-integer linear program that provides a lower bound on the cost of any implementable policy. The proposed model predictive control policy uses simple forecasts of loads and prices and solves a small mixed-integer linear program at each time step. Numerical experiments on one year of data from a home in Trondheim, Norway, show that the MPC policy attains a cost within 1.7% of the prescient bound and saves close to three times as much as the best rule-based policy considered.

What carries the argument

The model predictive control policy that repeatedly solves a small mixed-integer linear program using forecasts of loads and prices.

If this is right

  • The prescient minimum cost is the solution of a mixed-integer linear program.
  • The MPC policy attains costs within 1.7% of the prescient bound on real data.
  • Savings from the MPC policy are nearly three times those of the best rule-based policy tested.
  • The approach remains practical because it solves only a small MILP at each time step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same MILP-plus-MPC structure could be adapted to other peak-based tariff designs by changing only the objective coefficients.
  • Improving forecast accuracy beyond the simple methods used would likely shrink the remaining 1.7% gap to the bound.
  • The framework could be extended to include on-site solar generation or electric-vehicle charging by adding the corresponding variables to the MILP.
  • Commercial or multi-home settings with the same tariff structure would be a direct next application.

Load-bearing premise

Simple forecasts of loads and prices are accurate enough for the MPC policy to reach near-prescient performance.

What would settle it

If the MPC policy applied to the Trondheim year-long dataset produces a cost more than 3% above the prescient MILP bound, the claim of near-optimality would be falsified.

Figures

Figures reproduced from arXiv: 2307.07580 by David P\'erez-Pi\~neiro, Sigurd Skogestad, Stephen Boyd.

Figure 1
Figure 1. Figure 1: Grid-connected home with a storage device: lt is the (net) load, pt is the grid power, ct is the charging power, and dt is the storage power, at time period t. 2.1 Power balance We consider hourly values of various quantities, and denote the hour by a subscript t = 1, 2, . . . , T. The load in period t is given by lt . This is a net value, which can include for example PV (photovoltaic) generation, and is … view at source ↗
Figure 2
Figure 2. Figure 2: Hourly loads from a home in Trondheim, Norway. Top. Three-year period from 1 January 2020 to 31 December 2022. Bottom. One week in January 2022. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Day-ahead electricity prices λ da t for Trondheim, Norway. Top. Three-year period from 1 January 2020 to 31 December 2022. Bottom. One week in January 2022. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time-of-use electricity prices λ tou t with distinct rates for day (6:00–22:00) and night (22:00–6:00) as well as for the period of January–March and April–December. 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 zk (kW) 0 100 200 300 400 500 Peak power cost (NOK) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Peak power cost as a function of zk, the average of the N = 3 largest peak daily powers over a month. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Monthly costs without storage for 2022, broken down into time-of-use energy charges (blue), day-ahead energy charges (orange), and peak power charges (green). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Monthly costs over 2022 for no-storage baseline (left) and prescient policy (right). Each broken down into time-of-use charges (blue), day-ahead charges (orange), and peak power charges (green) [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Optimal power flows over 2022 assuming perfect foresight, with tier thresholds shown as dashed black lines. Top left. Power drawn from the grid. Top right. Load. Middle left. Charging storage power. Middle right. Discharging storage power. Bottom left. Charge level. Bottom right. Average of the N = 3 largest maximum daily power flows in each month, zk, with and without storage. 20 [PITH_FULL_IMAGE:figures… view at source ↗
Figure 9
Figure 9. Figure 9: Optimal power flows over one week in July 2022 assuming perfect foresight, with tier thresholds shown as dashed black lines. Hourly electricity prices, λ tou t + λ da t , are shown as dotted black lines. Top left. Power drawn from the grid. Top right. Load. Middle left. Charging storage power. Middle right. Discharging storage power. Bottom. Charge level. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Annual savings versus storage capacity for 2022. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Monthly costs over 2022 for no-storage baseline (left), prescient policy (mid￾dle), and MPC policy (right). Each broken down into time-of-use charges (blue), day-ahead charges (orange), and peak power charges (green). 25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Power flows executed over 2022 using MPC, with tier thresholds shown as dashed black lines. Top left. Power drawn from the grid. Top right. Load. Middle left. Charging storage power. Middle right. Discharging storage power. Bottom left. Charge level. Bottom right. Average of the N = 3 largest maximum daily power flows in each month, zk, with and without storage. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_… view at source ↗
Figure 13
Figure 13. Figure 13: Power flows executed over one week in July 2022 using MPC, with tier thresh￾olds shown as dashed black lines. Hourly electricity prices, λ tou t +λ da t , are shown as dotted black lines. Top left. Power drawn from the grid. Top right. Load. Middle left. Charging storage power. Middle right. Discharging storage power. Bottom. Charge level. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Seasonal baseline forecast for η = 0.5 (orange) and actual load (blue). Top. One week in January 2022. Bottom. One week in June 2022. We fit these 25 parameters on historical hourly data spanning two years (2020-2021) [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of forecasts for a test day in May 2022. The vertical line indicates the hour the forecast is made. The solid black line represents the load in the last 24 hours, while the dashed black line represents the load in the next 23 hours. The blue line represents a forecast that uses only a seasonal baseline model and the orange line corresponds to a forecast with both a seasonal baseline and an auto… view at source ↗
Figure 16
Figure 16. Figure 16: Seasonal baseline forecast (orange) and actual day-ahead prices (blue). Top. One week in January 2022. Bottom. One week in June 2022. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of forecasts for a test day in May 2022. The vertical line indicates the hour the forecast is made. The solid black line represents day-ahead prices in the last 24 hours, while the dashed black line represents day-ahead prices in the next 23 hours. The blue line represents a forecast that uses only a seasonal baseline model and the orange line corresponds to a forecast with both a seasonal base… view at source ↗
read the original abstract

We consider the problem of operating a battery in a home connected to the grid to minimize electricity cost, which combines an energy charge and a tiered peak power charge based on the average of the $N$ largest daily peak powers in each billing month. With perfect foresight of loads and prices, the minimum cost is the solution of a mixed-integer linear program (MILP), which provides a lower bound on the cost of any implementable policy. We propose a model predictive control (MPC) policy that uses simple forecasts of loads and prices and solves a small MILP at each time step. Numerical experiments on one year of data from a home in Trondheim, Norway, show that the MPC policy attains a cost within $1.7\%$ of the prescient bound, and saves close to three times as much as the best rule-based policy we consider.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript formulates the home battery dispatch problem under a tariff combining energy charges and a tiered peak power charge (based on the average of the N largest daily peak powers per billing month) as a mixed-integer linear program (MILP) when loads and prices are known in advance. This prescient MILP provides a lower bound on achievable cost. The authors propose a model predictive control (MPC) policy that solves a smaller MILP at each time step using simple forecasts of loads and prices. On one year of real load and price data from a home in Trondheim, Norway, the MPC policy achieves a cost within 1.7% of the prescient bound and saves approximately three times as much as the best rule-based policy considered.

Significance. The work demonstrates a practical, implementable policy that comes close to the theoretical optimum for battery operation under a realistic and complex tariff structure. The direct comparison to the prescient MILP bound on real data provides strong evidence of near-optimality without circularity or post-hoc fitting. This could be significant for the design of home energy management systems and for understanding the value of storage under peak-power tariffs.

minor comments (2)
  1. [Abstract / Problem formulation] The value of N (number of largest daily peaks averaged for the tiered charge) is not stated in the abstract or problem statement; it should be given explicitly with a citation to the relevant equation or section.
  2. [MPC policy description] The description of the 'simple forecasts' used inside the MPC (e.g., persistence, moving average, or other) is brief; adding one sentence or a short paragraph on their exact construction would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are grateful to the referee for their positive assessment of our manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is an empirical performance comparison on one year of real load/price data from Trondheim: the proposed MPC policy (small MILP per step with simple forecasts) achieves cost within 1.7% of an independently solved prescient MILP bound and outperforms rule-based policies by a factor of ~3. The prescient bound is obtained by direct MILP encoding of the tiered tariff with perfect foresight; no parameters are fitted to the reported outcomes, no predictions reduce to fitted inputs by construction, and no self-citation chains or uniqueness theorems are invoked to justify the modeling steps. The MILP formulations follow standard mixed-integer linear programming for battery dispatch and are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from optimization theory and the specific tariff structure provided by the utility; no free parameters are fitted by the authors and no new entities are postulated.

axioms (2)
  • standard math The optimization problems can be solved to optimality as MILPs for the relevant problem sizes
    Invoked for both the prescient lower bound and the repeated MPC solves.
  • domain assumption Simple forecasts of loads and prices are adequate inputs for the MPC policy
    Explicitly used in the proposed policy description.

pith-pipeline@v0.9.0 · 5686 in / 1315 out tokens · 61080 ms · 2026-05-24T07:31:16.438726+00:00 · methodology

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Reference graph

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