Estimating Risk-Aware Flexibility Areas for EV Charging Pools via Stochastic AC-OPF

Abstract: This paper introduces a stochastic AC-OPF (SOPF) for the flexibility management of electric vehicle (EV) charging pools in distribution networks under uncertainty. The SOPF considers discrete utility functions from charging pools as a compensation mechanism for eventual energy not served to their charging tasks. An application of the proposed SOPF is described where a distribution system operator (DSO) requires flexibility to each charging pool in a day-ahead time frame, minimizing the cost for flexibility while guaranteeing technical limits. Flexibility areas are defined for each charging pool and calculated as a function of a risk parameter involving the solution's uncertainty. Results show that all players can benefit from this approach, i.e., the DSO obtains a risk-aware solution, while charging pools/tasks perceive a reduction in the total energy payment due to flexibility services.

• The paper introduces a two-stage stochastic MISOCP framework integrating AC-OPF to estimate risk-aware flexibility areas for EV charging pools.
• The methodology employs discrete utility functions and scenario-based optimization to manage uncertainty in EV charging behavior and network constraints.
• Empirical results show improved cost efficiency and grid security via calibrated risk parameters within a realistic distribution test system.

Estimating Risk-Aware Flexibility Areas for EV Charging Pools via Stochastic AC-OPF

Overview

This paper presents a stochastic AC optimal power flow (AC-OPF) formulation specifically targeted at estimating risk-aware flexibility areas for electric vehicle (EV) charging pools within distribution networks. The approach systematically addresses uncertainty arising from stochastic EV user behavior and introduces discrete utility functions for charging pools, facilitating compensation-based demand flexibility procurement. The methodology allows distribution system operators (DSOs) to specify, in a day-ahead context, flexibility requirements for each charging pool. The main technical contributions include the integration of scenario-based stochastic programming with nonlinear/nonconvex prosumer preferences, and the introduction of a risk parameter to quantify and manage the trade-off between operational limits and economic outcomes in flexibility provision.

Methodological Framework

Stochastic Two-Stage Optimization

The core problem is formulated as a two-stage stochastic mixed-integer second-order cone program (MISOCP), distinguishing between first-stage (day-ahead commitments) and second-stage (realization-dependent) variables. The first stage determines the reserved power bounds for each charging pool, while the second stage manages actual power allocations and computes deviations, under numerous stochastic scenarios representing the aggregate uncertainty of arrival times, energy requirements, and departure times for EV charging tasks. The statistical properties of these parameters are modeled via Poisson, exponential, and uniform distributions, respectively, and can accommodate empirical or synthetic data (2301.00564).

Discrete Utility Function Representation

Charging pool flexibility valuation is formalized through piecewise-linear, possibly nonconvex and non-monotonic utility functions. This circumvent the limitations of quadratic assumptions common in prior work and allows precise modeling of arbitrary compensation schemes for energy-not-served. The mathematical model enforces semantics via convex combination constraints and binary variables to ensure segment-wise activation, supporting both convex and nonconvex profiles.

AC Power Flow with Uncertainty

The AC-OPF model incorporates the full nonlinear power flow constraints (using convex relaxations where appropriate), explicitly enforcing voltage and current magnitude limits at all network nodes and branches. This ensures that the flexibility estimation respects actual grid operational boundaries in all scenario realizations, a feature overlooked in most existing flexibility quantification methodologies. The stochastic representation is solved using a Sample Average Approximation (SAA) approach with a high number of Monte Carlo samples, providing robust statistical validity.

Definition of Risk-Aware Flexibility Areas

A key contribution is the operational definition of flexibility areas per charging pool and period, parameterized by a DSO-chosen risk tolerance $\beta_{s,t}$. The flexibility area is computed as the sum of the day-ahead reserved power and a scenario quantile (determined by $\beta_{s,t}$) of the positive deviation in actual consumption. This construct allows DSOs to finely tune the trade-off between strict adherence to network limits and the economic benefits of aggressive flexibility utilization, quantifying the operational risk in a statistically meaningful manner.

Experimental Validation

The methodology was validated on a realistic 34-node, 11kV radial distribution test system featuring four charging pools. The optimization employed 500 representative scenarios, with detailed network and EV pool parameters, and utility functions with three-segment discretizations per pool. Several experiments were performed to contrast base-case (no flexibility, no grid constraints) and flexibility-enabled operations.

Key Numerical Results

• Cost Impact: Total expected payment due to flexibility services (energy not served) decreased by 3.7% when network constraints and flexibility procurement were enabled versus the base case. The flexibility cost was distributed among pools proportionally to their location and network impact.
• Operational Risk Management: With high-risk tolerance ($\beta_{s,t}=0.99$), 88% of Monte Carlo scenarios violated voltage limits at some periods, versus zero violations for $\beta_{s,t}=0.57$. This empirically demonstrates the effectiveness of risk-aware flexibility specification in preventing operational constraint violations.
• Payments to Aggregators: Charging pools with greater network influence (those on critical feeders) obtained higher compensation, and their flexibility areas exhibited greater sensitivity to $\beta_{s,t}$. For risk parameters below 0.57, revenue inadequacy was observed, underscoring the importance of well-calibrated compensation mechanisms.

Theoretical and Practical Implications

The proposed SOPF-based flexibility quantification not only ensures rigorous compliance with non-linear AC network constraints but also provides a general framework for incorporating arbitrary prosumer utility structures. This supports the deployment of nuanced market mechanisms, risk-adjusted flexibility services, and can directly inform aggregator-DSO contractual structures. Flexibility areas become actionable operational envelopes, within which charging pools can autonomously optimize local objectives while guaranteeing grid security under uncertainty.

The use of risk parameters to define flexibility quantiles integrates the stochastic programming viewpoint directly into operational dispatch and market-clearing, enabling higher granularity in DSO-aggregator coordination. It also highlights node-specific and time-specific differences in flexibility valuation, which is relevant for future distribution-level ancillary service design.

Future Research Directions

Potential avenues for further development include:

• Incorporation of vehicle-to-grid (V2G) capability to explicitly enable bidirectional flexibility.
• Extension of the framework to account for reactive power management and DER provision beyond EVs.
• Investigation of decentralized or distributed solution algorithms to enhance scalability and privacy.
• Integration of real-time flexibility activation and settlement mechanisms enabling dynamic market environments.
• Adaptation to multi-level market settings, such as TSO-DSO interactions with overlapping flexibility products.

Conclusion

This work presents a comprehensive, risk-aware methodology for managing and quantifying EV charging pool flexibility using stochastic, network-constrained optimization with expressive prosumer utility functions. The empirical evidence confirms that network-aware, scenario-based flexibility provisioning results in both improved economic outcomes for aggregators and robust satisfaction of grid operational limits, provided risk parameters are appropriately calibrated. The approach directly addresses previously unfilled gaps in the flexibility quantification literature and lays a foundation for practical implementation of probabilistic flexibility markets at the distribution level (2301.00564).