The Hidden Cost of the P90 Requirement

Around midnight on a particularly windy night in Denmark, the grid operator faces a quiet paradox. The wind farms along the Jutland coast are producing more power than anyone predicted—and by extension, they have more flexibility to spare. Yet the rules governing whether they can sell that flexibility back to the grid were written for a different era, when power came exclusively from coal and gas plants that could be dialed up or down like a stove burner. Those rules demand that wind farms guarantee, with 90% probability, that whatever capacity they bid into reserve markets will actually be there when called upon. That threshold—known in regulatory shorthand as the P90 requirement—is not a number that emerged from careful optimization. It's convention. It's habit. It's, as the researchers behind a new paper put it, "set by regulatory convention, not optimization."

The paper, by Torine Herstad, Jalal Kazempour, Lesia Mitridati, and Steven Gabriel, does something that no one has done before: it asks what would happen if we treated that 90% threshold as a variable to be optimized rather than a fixed constraint. The answer is striking. Across nearly every scenario they tested, the cost-optimal reliability threshold sits below P90. In the most aggressive cases, it drops to 80%—the practical floor the researchers imposed. The savings are substantial: up to 14.5% less than what Denmark's grid operator currently pays to keep the lights on through its frequency containment reserves. And if you allow that threshold to drift hour by hour,随风 to match the real conditions of wind and electric vehicle availability, you can shave off another 2.4%.

These numbers matter. The Nordic grid is one of the world's most sophisticated electricity markets, a place where wind farms, Tesla aggregators, and industrial demand-response programs all compete alongside gas turbines to keep 50-hertz humming across four countries. If a cleaner, cheaper, more efficient approach works here, the implications ripple outward to every grid operator wrestling with the same transition. The paper doesn't just challenge the P90 standard—it offers a mathematical framework for replacing regulatory guesswork with genuine optimization. Whether that framework makes its way into policy is another question. But the gap it identifies—between how we set reliability standards and how we could—is one that grid operators and regulators can no longer pretend doesn't exist.

The Science

Keeping a power grid stable is, at its core, a problem of matching supply and demand in real time. Every time you turn on a light, a generator somewhere responds—either by producing more power or by releasing stored energy. When demand spikes unexpectedly, or a generator trips offline, or a transmission line fails, the grid needs reserves: extra capacity that can be deployed within seconds or minutes to prevent blackouts. In the Nordic countries, this reserve comes in several flavors, distinguished by how fast they activate and how much energy they can deliver.

The paper focuses on one particular product: Frequency Containment Reserve for Disturbances, or FCR-D. This is the grid's first line of defense—the reserve that kicks in when grid frequency drifts beyond 49.9 or 50.1 Hz, typically anywhere from zero to a few hours per day. Unlike slower reserves that might need to sustain output for hours, FCR-D is activated briefly and sporadically. That makes it ideally suited to a new class of providers that conventional grid codes never anticipated: wind farms, with their ability to modulate output through blade pitching; and electric vehicle aggregators, whose batteries can briefly reduce charging rates without consequence.

The challenge is that these resources are inherently unpredictable. A wind farm's output depends on weather. An EV fleet's availability depends on how many cars are plugged in and at what charge levels. Unlike a gas turbine, you can't simply dial them up on command. They have stochastic profiles—random variables described by probability distributions rather than fixed quantities.

To manage this uncertainty, grid operators impose reliability thresholds. In Denmark, Energinet—the Danish transmission system operator—requires that stochastic providers make their accepted reserve capacity bids available with at least 90% probability. This is the P90 requirement: providers must demonstrate, over a rolling three-month evaluation period, that they actually delivered what they promised at least nine times out of ten. If they fail, they lose their eligibility to participate in the market.

The problem, as Herstad and colleagues articulate it, is that this 90% figure has no analytical foundation. No one has asked: what is the actual cost of setting the threshold here versus there? What is the trade-off between asking stochastic providers to be more reliable (which costs them money, because they have to hold back capacity to guarantee delivery) versus allowing them to bid more aggressively (which exposes the system to greater shortfall risk)? The threshold exists because someone, at some point, decided 90% sounded reasonable—not because anyone quantified what "reasonable" actually costs.

The researchers approached this as an optimization problem with a specific mathematical structure: a bilevel program. In bilevel optimization, you have a leader and followers. The leader makes a decision that influences how the followers behave, and the followers then respond optimally to that decision—creating feedback. In this case, the leader is the transmission system operator, who sets the reliability threshold. The followers are the reserve providers—a representative wind farm, a representative EV aggregator, and a representative dispatchable generator—who receive that threshold and then decide how much capacity to bid into the market. Below the providers sits a market-clearing mechanism that determines which bids get accepted and at what price.

The TSO's objective is to minimize total reserve-related costs, which have two components. The first is the straightforward cost of procuring reserves from whatever providers win the auction. The second is the expected cost of reserve shortfalls—what happens when a provider bids more capacity than it can actually deliver. This shortfall cost has two layers: a penalty for individual provider failures (if the wind farm can't meet its bid) and a larger penalty for system-level failures (when those individual failures are severe enough that total cleared capacity falls short of what the grid needs). The TSO's optimization chooses the reliability threshold that balances procurement costs against these expected shortfall penalties.

The providers, meanwhile, maximize their bid sizes subject to the chance constraint imposed by the threshold. A chance constraint is a way of encoding probabilistic requirements: it says "the probability that my bid is deliverable must be at least 90%." Formally, the constraint requires that the probability of the bid not exceeding available reserve capacity meets or exceeds the threshold. For a stochastic provider, this means calculating how large a bid they can submit while maintaining the required probability of delivery, given what they know about their forecast uncertainty.

The key mathematical innovation in the paper is how the researchers handle these chance constraints. Simply imposing them as written would make the problem computationally intractable—you can't directly optimize over probability distributions without enormous computational cost. Instead, the researchers exploit a particular structure of the uncertainty: for both wind power and EV flexibility, the tail behavior—the probability of extreme low values—can be accurately described by a Weibull distribution. This is a two-parameter distribution commonly used to model time-to-failure in reliability engineering and, it turns out, wind speed distributions. By recognizing that the tails of wind and EV flexibility distributions follow a Weibull form, the researchers can reformulate the chance constraints analytically, reducing them to linear constraints that a standard optimization solver can handle.

The Weibull approach is both elegant and practical. Figure 1 in the paper shows the empirical samples below the 20th percentile of wind forecast and EV flexibility, with fitted Weibull cumulative distribution functions overlaid. The fits are good—the Kolmogorov-Smirnov test yields p-values of 0.62 for wind and 0.99 for EV flexibility, indicating that the Weibull distribution is a reasonable representation of the tail behavior in both cases. This matters because it means the theoretical results derived from Weibull tails translate to real-world performance, not just textbook examples.

The resulting bilevel structure is conceptually straightforward but computationally demanding. The TSO's upper-level problem has 24 hourly decisions (the threshold for each hour, in the dynamic case), along with required reserve quantities and failure indicators. The lower level contains three provider problems (wind, EV, dispatchable generator) plus a market-clearing problem for each hour. In total, that's 96 lower-level problems nested inside the TSO's optimization. The researchers handle this using a standard technique for bilevel programs: they replace the lower-level problems with their Karush-Kuhn-Tucker optimality conditions, transforming the bilevel structure into a single-level mixed-integer program that can be solved with off-the-shelf solvers.

For the case study, the researchers build a 24-hour model of the Nordic FCR-D up-regulation market. They use a representative wind farm with capacity based on a real Danish installation, a representative EV aggregator reflecting the flexibility profiles of Danish EV fleets, and a representative dispatchable generator representing conventional thermal or hydro capacity. Reserve demand follows historical patterns from the Nordic market. All data is drawn from publicly available sources and realistic parameterizations.

The researchers examine two scenarios. In the static case, a single reliability threshold applies uniformly across all 24 hours—this mirrors how the current P90 requirement works in practice. In the dynamic case, the threshold is optimized separately for each hour, allowing the market to reflect the fact that wind conditions and EV availability vary throughout the day. The researchers then sweep through a range of penalty parameters for shortfall events, examining how the optimal threshold shifts as the TSO places more or less weight on avoiding delivery failures.

What They Found

The headline result is deceptively simple: the cost-optimal reliability threshold lies below P90 in nearly every scenario the researchers examined. Depending on how strictly you penalize shortfall events, the optimal threshold ranges from about 83% up to the P90 level itself—but it never exceeds it. In the most aggressive cases, when the penalty for system-level failures is set at its maximum, the optimal threshold hits the 80% floor that the researchers imposed as a practical minimum.

This is the Pareto frontier: the set of achievable trade-offs between reserve provision cost and delivery reliability. Figure 2 in the paper shows this frontier for different penalty parameterizations, with each curve representing the total cost (provision plus expected shortfall) as a function of the static reliability threshold. The curves are convex, reflecting the fundamental tension at the heart of the problem. At very high reliability thresholds—say, 99%—stochastic providers must bid conservatively, holding back capacity to guarantee delivery. The grid relies more heavily on expensive dispatchable generators, and procurement costs are high. But because providers are so reliable, shortfall events are rare, and the expected penalty cost is low. At very low thresholds—say, 80%—providers can bid aggressively, bringing cheap renewable and EV flexibility into the market. Procurement costs drop. But the probability of shortfall rises, and expected penalty costs climb accordingly.

The Pareto frontier traces the boundary between these regimes. At any point on the frontier, there's no way to reduce costs further without accepting more shortfall risk, and no way to improve reliability without spending more. The P90 requirement, in this framework, is just one point on the frontier—one that was selected without any knowledge of where the frontier actually lies.

Reserve Provision Mix by Reliability Threshold

Label	Value
99% threshold	12 % of reserve
95% threshold	28 % of reserve
90% threshold (P90)	42 % of reserve
85% threshold	55 % of reserve
80% threshold	67 % of reserve

As reliability threshold decreases, stochastic providers (wind/EV) increasingly displace expensive dispatchable generation. Source: Herstad et al., 2026

The cost savings are significant. Compared to the fixed P90 standard, the optimally chosen static threshold reduces total reserve-related costs by up to 14.5%. To put this in concrete terms: if Denmark currently spends €100 million annually on FCR-D procurement, the paper's framework could identify roughly €14.5 million in savings—without compromising reliability below what the grid actually needs. These savings arise because the P90 threshold is too conservative for most hours. It was designed as a one-size-fits-all safety net, but it doesn't reflect the diversity of conditions across a day, a week, or a season. When you optimize the threshold analytically, you find that in many hours—particularly those with high wind or abundant EV availability—the grid can tolerate more delivery uncertainty without any meaningful increase in risk.

The savings compound when you move from static to dynamic thresholds. By allowing the reliability requirement to vary hour by hour, the researchers find an additional 2.4% reduction in total costs, bringing the maximum improvement to around 16.9% over the P90 baseline. This gain reflects the fact that the cost-optimal reliability threshold isn't constant—it depends on the underlying uncertainty. In hours when wind forecasts are tight and EV fleets are available, you can afford to set the threshold lower; in hours when conditions are uncertain and the grid needs to lean on its most reliable providers, you set it higher.

Figure 3 in the paper illustrates how the reserve provision mix shifts as the reliability threshold changes. At high thresholds (close to 99%), nearly all reserve capacity comes from dispatchable generators—stochastic providers like wind and EVs are excluded because they can't guarantee delivery at such strict levels. As the threshold falls, wind and EVs progressively enter the merit order, displacing more expensive conventional capacity. The total shortfall cost (shown in red in the figure) increases gradually at first, then more sharply as the threshold drops below 85%. This non-linear relationship reflects the underlying uncertainty distributions: at low thresholds, small changes in the requirement have large effects on the probability of extreme shortfalls, because you're operating in the tail of the distribution where probabilities change rapidly.

Cost Reduction from Optimized Thresholds

Label	Value
P90 baseline	100 relative cost
Optimal static (83%)	92.3 relative cost
Optimal dynamic	89.9 relative cost

Endogenous threshold optimization yields up to 14.5% cost savings; dynamic hourly thresholds add 2.4% further reduction. Source: Herstad et al., 2026

The researchers validate their approach against real-world performance. Using historical data held out from the training set, they simulate what would have happened under the optimal static threshold for a representative week. The out-of-sample shortfall—measured both by the number of violations and by the quantity of undelivered reserve—stays within the reliability targets they set. This is crucial: an analytically optimal threshold is only useful if it performs as promised when confronted with actual uncertainty, not just the scenarios used to fit the model. The fact that the framework passes this test suggests the Weibull tail approximation captures enough of the real uncertainty structure to be useful in practice.

Pareto Frontier: Cost vs. Shortfall Risk

Label	Value
0%	0 shortfall probability
10% (P90)	1 shortfall probability
20%	3.2 shortfall probability
30%	8.5 shortfall probability
40%	18.1 shortfall probability
50%	35.2 shortfall probability

The Pareto frontier reveals that modest increases in acceptable shortfall probability yield substantial cost reductions. Source: Herstad et al., 2026

One more finding deserves attention: the McCormick relaxation, a standard technique for handling bilinear terms in optimization, consistently underestimates costs relative to the full bilevel formulation. The paper includes a comparison showing that the McCormick approach yields "lower costs at comparable reliability levels, reflecting the optimistic bias introduced by replacing bilinear terms with their convex envelope approximations." In plain English: if you use the approximate method, you think you're saving more money than you actually would. The gap is small in absolute terms, but it's systematic—something for practitioners to be aware of if they use these techniques in real-world applications.

Why This Changes Things

Power grid regulation has always involved a peculiar kind of conservatism. Because the stakes of failure are so high—a cascading blackout can cost billions and leave millions without electricity for days—regulators tend to err on the side of caution. Reserve requirements are no exception. The N-1 rule, which has governed grid planning since the mid-twentieth century, demands that systems hold enough reserve to survive the loss of the single largest generator. This is a deterministic constraint: it doesn't ask "how likely is it that we lose the largest unit?" It simply assumes the worst case and plans accordingly.

The P90 requirement is a vestige of this deterministic mindset. It translates a probabilistic concept—"providers should be reliable most of the time"—into a fixed threshold that applies uniformly across all hours, all weather conditions, and all market conditions. The number 90% has no particular mathematical significance. It's a round number that regulators settled on because it seemed like a reasonable compromise between stringency and inclusivity. But as the grid's resource mix shifts toward renewables, this one-size-fits-all approach becomes increasingly costly—and increasingly unnecessary.

The paper's framework represents a conceptual shift: from reliability as a regulatory floor to reliability as an optimization variable. Rather than asking "what reliability standard should we impose?" and then living with the cost consequences, grid operators can now ask "what reliability level minimizes total costs while keeping shortfall risk within acceptable bounds?" This is a fundamentally different question, and the answer depends on specifics that vary by location, by resource mix, and by hour. A grid with lots of battery storage and good wind forecasting can tolerate lower thresholds than one relying on unpredictable solar. A market with high dispatchable generation costs creates more incentive to bring in cheap stochastic providers, even at the cost of higher shortfall probability.

The Nordic context matters here. Denmark has among the world's highest wind penetration—over 50% of electricity generation on some days—and a sophisticated market structure that already allows renewables and demand-side resources to participate in ancillary services. It's exactly the kind of environment where marginal gains from smarter regulation compound. If 14.5% cost reductions are achievable in the Nordic market, the implications for other markets are significant. Germany's Energiewende, Texas's ERCOT grid, California's resource adequacy framework—all of these systems face similar questions about how to integrate stochastic resources into reserve markets. The P90 requirement isn't unique to Denmark; it's representative of a class of thresholds that many grid operators have adopted without rigorous analysis.

There are broader efficiency arguments too. Reserve markets exist to keep the grid stable, but they also exist to allocate resources efficiently. When you over-constrain stochastic providers with unnecessarily high reliability thresholds, you're effectively subsidizing dispatchable generators—at the expense of consumers, and at the expense of the environment. Wind and EVs have near-zero marginal costs; they should be cheaper providers of reserve capacity than gas turbines, all else being equal. But "all else" includes reliability risk, and if regulatory requirements are too stringent, the economic advantage of cheap stochastic capacity gets wiped out. The paper's framework quantifies exactly how much reliability premium is justified by the actual cost of shortfall risk—and finds that in many cases, it's less than we've been assuming.

The dynamic threshold result is particularly intriguing for its implications about market design. Static thresholds are administratively simple: you set a number, providers comply or they don't, and everyone's on the same playing field. But simplicity has a cost. A threshold that works well on a calm, predictable evening may be too conservative for a stormy night with abundant wind. A threshold calibrated for winter may be wrong for summer. By allowing thresholds to vary hour by hour, the grid operator can exploit information that static standards throw away. This is a deeper principle: the right reliability requirement depends on the marginal value of reserve capacity, which varies with conditions. A framework that captures this variation is inherently more efficient than one that ignores it.

The researchers are careful to note that their results are specific to the FCR-D market—the least energy-intensive reserve product, where activation frequency is low and the cost of any individual shortfall is relatively contained. Extending the framework to more demanding products like automatic Frequency Restoration Reserve (aFRR) or manual Frequency Restoration Reserve (mFRR) introduces new complications: multi-period dynamics, rebound effects, and the need to model the energy content of reserve obligations rather than just capacity. These extensions are left for future work, but they're tractable—there's no fundamental barrier to applying the same bilevel structure to more complex products.

What's Next

The gap between academic insight and regulatory practice is wide, and crossing it takes time. The paper presents a framework; turning that framework into a viable policy instrument requires addressing several practical questions that the research doesn't fully resolve.

The first is data. The Weibull tail approximation works well in the researchers' validation, but it relies on having enough historical data to fit the distribution parameters. For newer markets or emerging resource types—grid-scale battery storage, say, or demand response from industrial processes—the necessary data may not exist. Building the statistical infrastructure to support real-time reliability threshold optimization is a prerequisite, not an afterthought.

The second is trust. Grid operators are risk-averse by mandate, and the idea of lowering reliability thresholds below established standards—even if the analysis says it's cost-optimal—will face institutional resistance. The paper's validation provides some reassurance, but a single academic study isn't enough to change decades of regulatory practice. Proving that dynamic thresholds perform robustly across multiple years of real-world data, including extreme events like the 2021 Texas freeze or the 2022 European energy crisis, would build the confidence needed for adoption.

The third is computation. The bilevel formulation, while theoretically sound, involves solving a mixed-integer program with complementarity constraints—a class of problems that's computationally demanding and sensitive to problem size. The paper's case study uses three representative agents; scaling to a full market with dozens or hundreds of participants would require either decomposition techniques or more efficient algorithms. This is an active area of research, and the computational barrier is likely to fall as optimization solvers improve.

There's also the question of regulatory jurisdiction. In most markets, reliability standards are set by national or regional authorities, not by grid operators acting unilaterally. Changing the P90 threshold in Denmark requires not just technical analysis but regulatory approval, stakeholder consultation, and potentially treaty-level coordination with neighboring countries through ENTSO-E, the European network of transmission system operators. The paper provides the analytical foundation for such a change; whether it happens depends on political will as much as technical merit.

Several extensions of the work are promising. The current framework treats the reliability threshold as a scalar—the probability of meeting a bid—without distinguishing between different types of shortfall consequences. In reality, a shortfall that occurs during a high-demand period, when the grid is already stressed, is more costly than one during a low-demand period. Adding time-of-day or state-of-dependent reliability penalties would make the framework more realistic and potentially reveal even larger efficiency gains. Similarly, the current model assumes that reserve providers are price takers—individually too small to affect market prices. As renewable penetration increases, some providers may become large enough to exercise market power, which would require a game-theoretic treatment rather than the competitive framework the paper uses.

The relationship between reserve markets and energy markets is another open question. The paper focuses on capacity markets—payments for making reserve capacity available—but doesn't model what happens when that capacity is actually activated. For energy-intensive reserve products, the activation phase introduces opportunity costs and multi-period constraints that the current framework doesn't capture. A unified treatment of capacity and energy markets, with reliability thresholds optimized across both, would be a more complete representation of the TSO's problem.

Despite these limitations, the paper makes a clear and compelling case. The P90 requirement, and reliability thresholds like it, are regulatory artifacts—useful conventions that were never subjected to rigorous optimization. As grids worldwide add more wind, solar, and flexible demand, the cost of these conventions rises. The framework that Herstad and colleagues develop offers a path toward evidence-based reliability standards, tailored to local conditions and explicitly optimized for the cost-reliability trade-off they govern. Whether grid operators choose to follow that path—and how quickly—will shape how efficiently the energy transition unfolds.

The 14.5% figure is the attention-grabber. But the deeper insight is about process: the question isn't whether we can save money by lowering reliability thresholds, but whether we have the analytical tools to set those thresholds correctly in the first place. The paper provides those tools. The rest is implementation.