Bidding in Ancillary Service Markets: An Analytical Approach Using Extreme Value Theory

Abstract

To enable the participation of stochastic distributed energy resources in ancillary service markets, the Danish transmission system operator, Energinet, mandates that flexibility providers satisfy a minimum 90% reliability requirement for reserve bids. This paper examines the bidding strategy of an electric vehicle aggregator under this regulation and develops a chance-constrained optimization model. In contrast to conventional sample-based approaches that demand large datasets to capture uncertainty, we propose an analytical reformulation that leverages extreme value theory to characterize the tail behavior of flexibility distributions. A case study with real-world charging data from 1400 residential electric vehicles in Denmark demonstrates that the analytical solution improves out-of-sample reliability, reducing bid violation rates by up to 8% relative to a sample-based benchmark. The method is also computationally more efficient, solving optimization problems up to 4.8 times faster while requiring substantially fewer samples to ensure compliance. Moreover, the proposed approach enables the construction of feasible bids with reliability levels as high as 99.95%, which would otherwise require prohibitively large scenario sets under the sample-based method. Beyond its computational and reliability advantages, the framework also provides actionable insights into how reliability thresholds influence aggregator bidding behavior and market participation. This study establishes a regulation-compliant, tractable, and risk-aware bidding methodology for stochastic flexibility aggregators, enhancing both market efficiency and power system security.

Figures (9)

• Figure 1: Illustrative examples of flexibility estimation: Figure (a) illustrates how downwards and upwards flexibility for the aggregated case are calculated. Downwards flexibility ($r^\downarrow$) is determined as the difference between the lowest point of the maximum consumption of the EVs and the highest actual consumption across the hour, while upwards flexibility ($r^\uparrow$) is the minimum of the actual consumption values during the hour. The figures in (b) show how energy flexibility ($r^{\rm{\rm{E_{20}}}}$) is calculated for a single EV and for the aggregated case. The single EV case represents a complete charging session, while the aggregated case captures a time interval in the day with large fluctuations in available flexibility, typically during the morning when many charging sessions end. Energy flexibility is calculated based on the current state of charge (SoC) and the amount of power that can be applied to the EV over the next 20 minutes without exceeding the battery's capacity. For instance, at hour 1 in the single EV case, the SoC is 30 kWh, and the battery capacity is 75 kWh, meaning that, theoretically, the charge box (CB) could apply a maximum of 45 kWh in the next 20 minutes to fully charge the EV. However, due to the CB's physical constraint of 12 kW, the actual available energy flexibility is much lower. Similarly, in the aggregated case, the theoretical and realistic energy flexibilities are on vastly different scales, reflecting the system's capacity and constraints.
• Figure 2: Scatter plots illustrating the estimated upward (top), downward (middle), and energy (bottom) flexibilities for a representative hour across all days of the year. Blue samples represent the in-sample data used for optimization in a given run, while pink samples correspond to the ex-post out-of-sample validation in the same run. The dashed line marks the empirical 10th percentile of the blue dots. Consistent with extreme value theory, only the blue samples below this percentile are used for estimation in the analytical method.
• Figure 3: Distribution of the downwards flexibility at the arbitrarily selected hour 13. The blue distribution is constructed using in-sample data, while the pink part represents the out-of-sample data. A Weibull distribution with the shape parameter $\hat{\gamma}=0.9265$ is fitted to the lower tail of the blue distribution. The vertical dashed line indicates the empirical 10th percentile value of the in-sample data, below which the Weibull distribution is fitted.
• Figure 4: Best (left) and worst (right) distribution fits (grey) compared to all data below the 10th percentile (pink). The $p$-values for the best fit range from 0.871 to 0.999, while for the worst fit they range from 0.218 to 0.871.
• Figure 5: A simple two-dimensional illustrative example of a joint distribution.