Brieuc Le roux tardif

arXiv2026

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

Capacity markets award reserve in discrete increments, yet valuation models almost always treat the awarded quantity as continuous. We show that the size dependence of the resulting error is organised by one dimensionless number, the granularity ratio rho = delta/P of award increment to installed power: the joint energy-and-reserve problem is positively homogeneous, so, at fixed market data and operating parameters, asset size enters per-megawatt revenue only through rho and storage duration. Three structural results follow. Rounding a continuous award down to the lattice is always feasible, so the valuation error is O(rho) and its upper bound falls as the inverse of asset size. The largest feasible award is P rho floor(1/rho), so up to half of a connection can be structurally unsellable, worst just below twice the increment. Pledging the whole connection forces a zero net grid position for the entire product, converting a partial capacity commitment into exclusion from the energy market. On 943 days of French day-ahead and frequency containment reserve settlement prices, a 1 MW two-hour battery priced with a continuous award is worth 4.2% more than the same asset on the 1 MW French lattice, and 77% of that difference is displaced arbitrage rather than foregone capacity. The bias is not monotone in size: the largest value on our grid is 21.5% at rho = 0.505. Bid granularity is therefore a market-design variable with a size-dependent incidence, and the measured gap is at once a modelling error, a sizing rule, and a ceiling on aggregation value.

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arXiv2026

Anonymous sharing is pairwise phase-blind

Independent training jobs sharing a storage system write their checkpoints through the same finite bandwidth, and the resulting bursts of correlated I/O are commonly described as a self-reinforcing "checkpoint storm". We formalise the self-reinforcement as phase locking in a population of integrate-and-fire oscillators coupled through a shared resource, and show that within that model it fails. Call a resource anonymous if the rate it delivers to an active user depends on how many users are active and not on which. For identical jobs whose write is shorter than their compute interval, an anonymous resource produces no pairwise coupling at all: the two-job return map of the phase gap is the identity, under storage contention, under a shared power cap and under both, so the two-body interaction on which the Kuramoto and Mirollo-Strogatz frameworks are built is not weak here but absent. Anonymity also freezes the firing order, for any fleet size and any cap, so no trajectory reaches the synchronous state from outside it. What survives is a third-order effect: where all $N$ write windows overlap and the cap does not bind, the map is diagonal in the intervals between consecutive write starts, $a_j \mapsto ((N-j)/j)a_j$, with reciprocal spectrum and unit determinant, making synchrony a fixed point with $\lceil N/2\rceil-1$ expanding directions rather than an attractor. That determinant follows from anonymity and not from fairness: for any anonymous throughput $f$ with $f(n)\le n$ the spectrum becomes $(N-j)f(j)/(j f(N-j))$, whose product is still one. Numerically, a fleet launched at random neither locks nor clusters, and absence of locking is not absence of bursts: the upper tail of the number of concurrent writers stays above its independent-phase value. Heterogeneous jobs behind a binding cap do acquire a genuine pairwise coupling, which is where the statement stops generalising.

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