Risk-Aware Hosting Capacity Analysis for Flexible Load Interconnection in Distribution Networks

1. Problem Statement

The rapid growth of electricity demand, driven by flexible loads such as Electric Vehicles (EVs) and AI data centers, challenges the capacity of existing distribution networks. Traditional grid expansion is often economically and operationally unfeasible. While Demand Response (DR) and flexible connection agreements offer a solution by allowing load curtailment during stress periods, existing methods face two critical limitations:

1. Lack of Reliability Guarantees: They often fail to explicitly bound the severity (depth) or frequency of curtailment events, risking unacceptable service interruptions.
2. Scalability Issues: Previous approaches to modeling intervention frequency often rely on combinatorial or non-convex formulations, making them difficult to solve for large-scale systems.

The core research question is: How can flexible loads be integrated into distribution networks to maximize hosting capacity while ensuring that load curtailments remain sparse (infrequent) and bounded (limited severity), even during rare, high-impact stress events?

2. Methodology

The authors propose a convex, risk-aware optimization framework that explicitly models both the magnitude and frequency of curtailment. The methodology consists of the following components:

A. System Model

• Network: A single-phase radial distribution feeder modeled using Linearized DistFlow (LinDistFlow) equations.
• Constraints: The model enforces voltage limits at all buses and thermal limits at the substation feeder.
• Flexible Load: A new load (e.g., EV station) is connected at a specific bus with a normalized temporal profile $\hat{l}(t)$. The actual served load is the requested load minus a curtailment amount $p_{curt}(t)$.
• Curtailment Depth: A hard constraint limits the curtailment at any single time step to a fraction $\rho$ of the requested load ($0 \le p_{curt}(t) \le \rho P \hat{l}(t)$).

B. Risk-Aware Formulation (CVaR)

To manage the risk of extreme curtailment events exceeding the agreed limit $\rho$, the authors introduce a Conditional Value-at-Risk (CVaR) constraint.

• Risk Variable: $\zeta(t) = p_{curt}(t) - \rho P \hat{l}(t)$ represents the violation of the depth limit.
• Constraint: $CVaR_\alpha[\zeta(t)] \le \epsilon$.
  • This ensures that the average of the worst $(1-\alpha)$ fraction of curtailment violations remains below a threshold $\epsilon$.
  • This is reformulated into a convex epigraph form using auxiliary variables, making the problem tractable.

C. Sparse Intervention Scheduling ($\ell_1$ Regularization)

To control the frequency of interventions (i.e., how many time steps involve curtailment), the authors introduce a sparsity-inducing penalty:

• Objective Modification: The objective function is augmented with a weighted $\ell_1$ regularization term: $-\lambda \sum w_{norm}(t) p_{curt}(t)$.
• Weighting: Weights are proportional to the residual feeder capacity ($\Delta P(t)$) to discourage curtailment when the grid is already tight.
• Tuning: A regularization parameter $\lambda$ is tuned via a bisection algorithm to meet a target intervention budget ($N_{int}$), ensuring the number of curtailment periods does not exceed a desired limit.

D. Optimization Problem

The final formulation is a convex optimization problem that maximizes the connectable load capacity ($P$) subject to:

1. Power flow and voltage constraints.
2. CVaR constraints (controlling tail risk of severity).
3. Sparsity constraints (controlling intervention frequency via $\lambda$).

3. Key Contributions

1. Convex Risk-Aware Framework: Unlike previous non-convex approaches, this study provides a convex formulation that simultaneously handles load capacity maximization, tail-risk control (CVaR), and intervention frequency.
2. Explicit Control of Severity and Frequency: The method decouples the control of how much load is cut (via CVaR and depth limits) from how often it is cut (via $\ell_1$ regularization), offering grid operators precise tuning capabilities.
3. Scalability: The convex nature of the problem allows for efficient, scalable implementation suitable for real-world distribution networks.
4. Open Source: The authors provide the code and dataset (IEEE 123-bus feeder with NREL load data) to facilitate reproducibility.

4. Numerical Results

Experiments were conducted on the IEEE 123-bus test feeder using NREL residential load data and EV charging profiles.

• Capacity Gain vs. Intervention Count: Even a very small number of interventions (e.g., 1% of the planning horizon) unlocks significant hosting capacity. The capacity gain saturates once the most critical time steps (peak violations) are addressed.
• Impact of Risk Level ($\alpha$): Stricter risk constraints (higher $\alpha$) reduce the achievable hosting capacity, as the system becomes more conservative. However, allowing deeper curtailment ($\rho$) consistently yields higher capacity gains.
• Curtailment Behavior:
  • The optimized curtailment is sparse, targeting only critical peak periods.
  • The CVaR constraint effectively minimizes tail violations. The empirical Cumulative Distribution Function (CDF) of violations shows that most violations are near zero, and the average of the worst-case tail remains within the safety bound.
  • The total load profile after curtailment remains within transformer and voltage limits, whereas the uncurtailed profile would violate them.

5. Significance and Conclusion

This paper presents a practical bridge between grid reliability and the integration of flexible loads. By quantifying hosting capacity under risk-aware constraints, the study demonstrates that:

• Grid constraints are often driven by a few critical time periods; addressing these with targeted, sparse interventions can significantly defer infrastructure upgrades.
• The proposed framework offers stronger reliability guarantees than traditional DR programs by mathematically bounding both the frequency and severity of service interruptions.
• The approach is computationally efficient, enabling utilities to plan for high-penetration flexible loads (like AI data centers and EVs) without compromising system stability.

Future Work: The authors suggest extending the framework to network-wide multi-bus optimization and integrating backup resources (e.g., generators) to achieve zero-curtailment scenarios.