⚡ electrical engineering

Bid Lattices and the Value of Flexibility:A Granularity Ratio for Capacity Markets

This paper introduces the dimensionless "granularity ratio" to quantify how the discrete nature of capacity market awards creates size-dependent valuation errors and structural sellability constraints, demonstrating that ignoring lattice effects leads to significant overestimation of asset value and distorted market participation strategies.

Original authors: Brieuc Le roux tardif

Published 2026-08-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Brieuc Le roux tardif

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Grid's Ruler: Why Size Matters in the Energy World

Imagine the electricity grid as a giant, humming marketplace where power is bought and sold every second. In this world, there are two main types of transactions: selling the actual electricity you generate (like water flowing from a tap) and selling the promise to be ready to generate electricity if needed (like holding a fire extinguisher ready on the wall). This "readiness" is called capacity or reserve.

For decades, scientists and engineers have tried to calculate how much money a battery or a power plant can make in this market. To do this, they usually use math models that treat electricity like water: smooth, continuous, and infinitely divisible. You can imagine selling 0.999 liters of water or 1.001 liters with no problem. But in the real world, electricity markets don't work like a smooth stream of water; they work like a ruler with tick marks. You can only sell power in specific, fixed steps, like 1 MW, 2 MW, or 3 MW. You can't sell 1.5 MW if the market only accepts whole numbers. This paper explores what happens when you try to fit a smooth, continuous idea of value into a market that only speaks in "chunks." It asks a simple but tricky question: How much money do we lose because the market's ruler is too coarse?

The Paper's Big Discovery: The "Chunkiness" Tax

This paper, written by Brieuc Le Roux-Tardif, dives into the messy reality of electricity markets, specifically looking at how the "chunkiness" of bidding rules affects the value of energy storage, like giant batteries. The author introduces a clever new way to measure this problem using a single number called the granularity ratio (let's call it $\rho$ ). Think of $\rho$ as a measure of how "big" your battery is compared to the market's smallest step size.

If the market says, "We only buy in 1 MW blocks," and you have a 1 MW battery, your ratio is 1. If you have a 100 MW battery, your ratio is tiny (0.01), meaning the 1 MW step is just a speck on your giant asset. The paper proves that the error in valuing your battery depends entirely on this ratio.

Here is the twist: The paper finds that the common belief—that bigger batteries are always "safer" and closer to the perfect math model—is only true on average. In reality, the relationship is a jagged, sawtooth pattern. Sometimes, a battery that is just slightly smaller than a perfect multiple of the market step gets hit with a massive penalty. For example, if the market step is 1 MW, a battery sized at 1.99 MW is in deep trouble. It can't sell its full 1.99 MW because the market only takes whole numbers. It has to round down to 1 MW, leaving nearly half of its power "structurally unsellable." The paper shows that the worst-case scenario happens when an asset is just under twice the size of the market step, where up to half of the connection can be wasted.

The French Experiment: When "Ready" Kills "Trading"

To test these ideas, the author ran a massive simulation using 943 days of real market data from France (covering 2024 to 2026). They looked at a standard 1 MW, 2-hour battery.

The results were surprising and counter-intuitive. Most people assume that if a market has "chunky" rules, the main loss comes from not being able to sell enough capacity (the promise to be ready). But the paper found the opposite. At the smallest asset size (where the ratio $\rho = 1$ ), the continuous math model overestimated the battery's value by 4.2%. However, 77% of that lost value didn't come from missing out on capacity sales. It came from displaced arbitrage.

Here is the metaphor: Imagine you are a taxi driver who can either drive passengers (energy trading) or hold a spot in a VIP line (capacity reserve). The market rules say if you take the VIP spot, you must park your car and stop driving for the whole shift. If your car is small (1 MW), taking the VIP spot means you can't drive at all. You lose all your passenger fares. The paper shows that for small batteries, the "chunky" rules force them to choose between being a taxi or a VIP, rather than doing both. The continuous model assumes you can do both smoothly, but the real market forces a hard stop, costing the battery owner a lot of potential trading profit.

What This Means for the Future

The paper doesn't just point out a problem; it offers a toolkit for three different groups:

For the Math Geeks (Modellers): Stop pretending electricity is perfectly smooth. If you are valuing a battery, you need to report the "chunkiness ratio" ( $\rho$ ). If the battery isn't a perfect multiple of the market step, your math is likely wrong, and you should run a second, more realistic calculation to see the error bar.
For the Builders (Developers): If you are building a battery, the size matters more than you think. Building a battery that is exactly 1 MW, 2 MW, or 3 MW (perfect multiples of the market step) is smarter than building one that is 1.9 MW or 2.1 MW. The latter sizes leave you with "unsellable" power that you paid for but can't use.
For the Rule Makers (Market Designers): The way the market sets its step size (the increment) acts like a tax on small players. A coarse step size (like 1 MW) hurts small batteries much more than big ones. The paper suggests that making the steps smaller or allowing small batteries to team up (aggregate) into a bigger group could fix this unfairness.

The Bottom Line

The paper proves that the "granularity ratio" is the key to understanding how much value is lost in electricity markets due to rigid rules. It shows that the error isn't just a small rounding issue; it's a structural flaw that can make small assets significantly less valuable than models predict. The biggest surprise is that the cost of these rigid rules isn't just losing capacity revenue; it's often losing the ability to trade energy at all. The author measured this gap at 4.2% for a standard small battery in France, with the vast majority of that loss coming from the inability to trade energy while holding a reserve spot.

In short, the market's ruler is too coarse for small players, and until the rules change or the players get bigger, they are leaving money on the table—not because they aren't smart, but because the math of the real world doesn't match the smooth curves of the textbooks.

Technical Summary: Bid Lattices and the Value of Flexibility

Problem Statement
Capacity markets for flexible assets, such as energy storage, typically award reserve in discrete increments (e.g., 1 MW). However, standard valuation models almost universally treat the awarded quantity as a continuous decision variable to simplify optimization. This paper identifies a systematic modelling error arising from this discrepancy. A standalone asset cannot offer a fractional increment (e.g., a 1.99 MW battery cannot offer 1.99 MW if the minimum increment is 1 MW). The paper argues that the magnitude of the resulting valuation error is not merely a function of asset size but is organized by a specific dimensionless parameter: the granularity ratio $\rho = \delta/P$ , where $\delta$ is the award increment and $P$ is the installed power of the asset.

Methodology
The author formulates a day-ahead co-optimization problem for a price-taking storage asset, maximizing revenue from energy arbitrage and symmetric capacity reserves (specifically Frequency Containment Reserve, FCR). The model includes standard constraints for power limits, energy balance, efficiency, and endurance.

The core methodological innovation is the re-parametrization of the problem. Instead of sweeping installed power $P$ while holding $\delta$ fixed, the author analyzes the problem through the lens of $\rho$ . They leverage the positive homogeneity of the optimization problem: scaling $P$ , energy capacity $E$ , and the increment $\delta$ by the same factor $\alpha$ scales the optimal revenue by $\alpha$ . This implies that per-megawatt revenue depends on asset size only through $\rho$ and storage duration.

The study utilizes 943 days of French day-ahead and FCR settlement prices (2024–2026). The experimental design involves:

Granularity Sweep: Varying $\delta$ directly on a fixed 1 MW/2 MWh asset to create a dense grid of $\rho$ .
Homogeneity Test: Verifying that different absolute asset sizes with the same $\rho$ yield identical per-MW revenues.
Comparison: Solving the problem with a continuous award domain ( $\delta=0$ ) versus a lattice-constrained domain ( $\delta > 0$ ) to measure the valuation bias $\beta(\rho)$ .
Decomposition: Separating the revenue loss into "foregone capacity" and "displaced arbitrage."

Key Structural Results
The paper derives three primary structural results, verified numerically:

Error Bounds via Rounding:
Rounding a continuous optimal award down to the nearest lattice point always yields a feasible schedule. This establishes an upper bound on the valuation error:
$R(0) - R(\rho) \leq \rho \Lambda$
where $\Lambda$ is the revenue of selling 1 MW in every product over the horizon. The error is $O(\rho)$ and vanishes at least as fast as the inverse of asset size.
Structural Unsellability (The Sawtooth Effect):
The largest award a lattice permits is $\delta \lfloor P/\delta \rfloor$ . Consequently, a fraction of the connection, $u(\rho) = 1 - \rho \lfloor 1/\rho \rfloor$ , is structurally unsellable. This fraction forms a sawtooth function where the supremum is 0.5, approached when the asset size is just below twice the increment (e.g., an asset sized just under 2 MW in a 1 MW increment market).
Exclusion of Energy Arbitrage at $\rho = 1$ :
When an asset pledges its entire connection power ( $P = \delta$ ), the capacity constraint forces the net grid position to zero for the duration of the product. This converts a partial capacity commitment into a total exclusion from the energy market. At $\rho=1$ , the bidder must choose between capacity and energy; they cannot combine them.

Empirical Results
Using French market data, the paper reports the following findings for a 1 MW, 2-hour battery ( $\rho=1$ ):

Valuation Bias: The continuous model overstates revenue by 4.2% compared to the lattice-constrained reality.
Source of Error: Contrary to the intuition that lattice constraints primarily cost capacity revenue, 77% of the 4.2% gap is due to displaced arbitrage. The lattice forces the asset to forego energy trading to maintain a full capacity pledge.
Maximum Bias: The bias is not monotone. The maximum measured bias on the grid is 21.5% at $\rho \approx 0.505$ (an asset sized just below twice the increment), confirming the theoretical prediction of the "structural unsellability" peak.
Homogeneity: The numerical experiments confirm that per-MW revenue is invariant under joint scaling of $P$ , $E$ , and $\delta$ , validating the $\rho$ parametrization.

Significance and Claims
The paper positions its contribution as a narrow but rigorous identification of a widespread modelling error. It claims three specific utilities for the granularity ratio $\rho$ :

For Modellers: $\rho$ serves as a diagnostic for the magnitude of valuation error. Modellers should report $\rho$ and the bound $\rho\Lambda$ alongside valuations. If the connection is not an integer multiple of the increment, both continuous and lattice formulations should be solved.
For Developers: The results provide a sizing rule. Connecting an asset at an integer multiple of the increment eliminates the structurally unsellable tail, though it does not eliminate the residual integrality gap. The measured gap also serves as an upper bound on the gross value of aggregation (the revenue a standalone asset loses by not being part of a portfolio that can access finer lattices).
For Market Designers: Bid granularity acts as a distributional instrument. The incidence of the error falls most heavily on small assets. The paper suggests that reducing the increment $\delta$ or allowing aggregation are the primary levers to mitigate this, noting that the magnitude of the impact is computable ex-ante using only price series and product rules.

The author explicitly states that they propose no new dispatch algorithms or market designs. Instead, they provide a dimensionless framework to quantify the cost of indivisibility, distinguishing between modelling error, sizing inefficiency, and the distributional impact of market rules.

The Grid's Ruler: Why Size Matters in the Energy World

The Paper's Big Discovery: The "Chunkiness" Tax

The French Experiment: When "Ready" Kills "Trading"

What This Means for the Future

The Bottom Line

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