⚡ electrical engineering

Adversarially and Distributionally Robust Virtual Energy Storage Systems via the Scenario Approach

This paper proposes a convex, data-driven scheduling framework for virtual energy storage systems using aggregated EV batteries that provides finite-sample, distribution-free guarantees on constraint violations while offering robustness against both adversarial data corruption and distributional shifts.

Original authors: Georgios Pantazis, Nicola Mignoni, Raffaele Carli, Mariagrazia Dotoli, Sergio Grammatico

Published 2026-04-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Georgios Pantazis, Nicola Mignoni, Raffaele Carli, Mariagrazia Dotoli, Sergio Grammatico

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive parking lot filled with electric cars. Each car has a battery, and while they are parked, they aren't just sitting there; they are a giant, collective battery bank. This is the concept of a Virtual Energy Storage System (VESS).

The paper you shared is about a smart manager (let's call him "The Parking Lot Manager") who wants to use these parked car batteries to help the neighborhood save money and keep the lights on. However, there's a catch: Uncertainty.

Here is the story of how this paper solves the problem, explained through simple analogies.

1. The Setup: The "Car Battery Bank"

Think of the parking lot as a giant water tank.

The Water: The electricity in the car batteries.
The Inflow: Cars arriving and plugging in (adding water).
The Outflow: Cars leaving (taking water away).
The Goal: The Manager promises the neighborhood (the "Prosumers") that he can supply them with power whenever they need it. He also buys and sells electricity to the main grid (the "Retailer") to make a profit.

The Problem: The Manager doesn't know exactly when cars will leave or how much battery they will have left. If he promises too much power and a bunch of cars leave early, the "tank" runs dry, and he fails his promise.

2. The Old Way vs. The New Way

The Old Way (Deterministic): The Manager guesses, "Okay, usually 10 cars leave at 5 PM." He plans based on that average. But what if 20 cars leave? He fails.
The "Scenario" Way (This Paper): Instead of guessing one average, the Manager looks at thousands of past days (data). He says, "I need a plan that works for all these different scenarios I've seen."

This is called the Scenario Approach. It's like packing for a trip. Instead of packing for "average weather," you pack for "rain, snow, and heatwave" because you have a list of 1,000 past weather reports.

3. The Big Innovation: The "Safety vs. Profit" Dial

The paper introduces a brilliant feature: A Tunable Dial.

Imagine the Manager has a dial on his control panel labeled "Risk vs. Profit."

Turn it to "Super Safe": The Manager keeps a huge safety buffer of energy. He rarely fails, but he misses out on making money because he's too cautious.
Turn it to "Max Profit": He takes big risks to make money. He might make a lot of cash, but he might also run out of power and get fined.

The paper gives the Manager a mathematical "certificate" (a guarantee) that says: "If you turn the dial to this setting, you can be 99% sure you won't fail, even if the future looks different from the past."

4. The "Super Villains": Bad Data and Shifts

The paper goes a step further. It assumes the data the Manager uses might be corrupted or tricky.

Adversarial Perturbations (The "Glitch"): Imagine someone sneaks into the data and changes a few numbers slightly (like a hacker or just a bad sensor). The Manager's plan might break.
- The Solution: The Manager designs his plan to be "adversarially robust." It's like building a fortress that can withstand a few bricks being knocked out of the wall. Even if the data is slightly "poisoned," the plan still holds.
Distributional Shifts (The "New Normal"): Imagine the Manager trained his plan on data from 2023. But in 2024, people start working from home, so cars stay parked longer. The "rules of the game" have changed.
- The Solution: The paper uses something called Wasserstein Distance. Think of this as a "similarity meter." The Manager says, "I know the future might be different, but it won't be too different from what I've seen." He builds a "bubble" of possible futures around his data. His plan works for everything inside that bubble.

5. The "Trust Radius"

The paper introduces a concept called the Trust Radius.

If the Manager trusts his data completely, the radius is small. He assumes the data is perfect.
If the Manager is suspicious (maybe the sensors are old, or the data is noisy), he turns the radius up. He assumes the "real" data could be anywhere within a certain distance of his recorded data.

The paper provides a formula to help the Manager pick the perfect radius. It's a balancing act:

Too small a radius: You get a cheap plan, but it might fail if the data was actually noisy.
Too large a radius: You get a super-safe plan, but it's so expensive you make no profit.

6. The Result: A "Smart Contract" for Energy

The paper proves mathematically that if the Manager follows this method:

He can guarantee how often he might fail (e.g., "I will fail less than 1% of the time").
He can tune exactly how much money he is willing to risk to get that guarantee.
He remains safe even if the data is slightly broken or if the future is slightly different from the past.

Summary Analogy

Imagine you are a Captain of a Ship (The Manager).

The Cargo: The neighborhood's power needs.
The Fuel: The parked EV batteries.
The Storm: Uncertainty (cars leaving early, bad data).

Old captains sailed by guessing the weather. This paper gives the Captain a Magic Compass.

The Compass shows a "Safety Zone."
The Captain can choose to sail right on the edge of the zone for speed (profit) or stay in the middle for safety.
Even if the map is slightly smudged (noisy data) or the ocean currents change slightly (distribution shift), the Compass guarantees the ship won't sink, provided the Captain stays within the chosen safety zone.

The paper is essentially a rulebook for building a safety net that is mathematically proven to work, allowing the Parking Lot Manager to make money without gambling the neighborhood's power supply.

1. Problem Statement

The paper addresses the challenge of managing Virtual Energy Storage Systems (VESS) formed by aggregating Electric Vehicle (EV) batteries in parking lots. A Parking Lot Manager (PLM) acts as an intermediary, aggregating EV storage to provide services to a prosumer community while trading energy with a retailer.

The core challenges identified are:

Uncertainty: EV arrivals, departures, and State-of-Charge (SoC) limits are time-varying and uncertain.
Multi-Actor Complexity: The system must balance the needs of prosumers (energy requests), the PLM (profit maximization), and the retailer (energy trading).
Data Reliability: Standard stochastic methods assume perfect knowledge of data distributions. However, real-world data can be noisy, adversarially perturbed (cyber-attacks), or subject to distributional shifts (where future operating conditions differ from historical training data).
Trade-off: The PLM needs a principled way to trade off economic performance (profit) against operational safety (constraint satisfaction) without relying on overly conservative deterministic models.

2. Methodology

The authors propose a convex, data-driven scheduling framework based on the Scenario Approach. The methodology evolves through three stages of robustness:

A. Baseline Scenario Optimization

Formulation: The problem is modeled as a convex optimization where the PLM minimizes energy trading costs (buying from/selling to the retailer) subject to SoC dynamics and capacity constraints.
Uncertainty Handling: Instead of assuming a known probability distribution, the method uses a finite set of $N$ sampled scenarios (historical data of EV departures $\ell$ and capacity limits $\beta$ ).
Relaxation: To allow for a tunable trade-off between profit and safety, the constraints are relaxed using auxiliary variables ( $\xi$ ) and a penalty weight ( $\rho$ ). This allows the PLM to intentionally violate training constraints slightly to lower costs, with the violation magnitude controlled by $\rho$ .

B. Adversarial Robustness

Threat Model: The framework accounts for adversarial perturbations where training data might be corrupted or noisy.
Trust Region: The authors define an adversarial region $\mathcal{A}_{\ell, \beta}$ around each sample. The optimization ensures feasibility not just for the recorded data points, but for all points within a "trust region" (defined by a radius $R$ ) around them.
Adversarial Complexity: The theoretical guarantees are adjusted based on the number of adversarial support samples (samples that define the solution boundary under perturbation).

C. Distributional Robustness (Wasserstein)

Distributional Shift: The framework addresses the scenario where the test data distribution $P'$ differs from the training distribution $P$ .
Wasserstein Metric: The authors use the Wasserstein distance to define an ambiguity set $\mathcal{B}_\mu(P)$ containing all distributions within a radius $\mu$ of the empirical distribution.
Robust Certificates: The method derives finite-sample guarantees that hold for any distribution within this ambiguity set, ensuring the solution remains safe even if the underlying EV behavior shifts statistically.

3. Key Contributions

Unified Convex Model: A novel formulation for VESS that integrates EV aggregation, prosumer requests, and retailer trading into a single convex framework, explicitly modeling uncertain departures and time-varying capacity.
Tunable Profit-Risk Mechanism: Introduction of a relaxation parameter ( $\rho$ ) that allows the PLM to explicitly tune the trade-off between economic performance and constraint violation risk, backed by finite-sample, distribution-free guarantees.
Adversarial & Distributional Robustness: Extension of the scenario approach to handle corrupted data (adversarial perturbations) and out-of-distribution (OOD) uncertainty (Wasserstein shifts). This provides robustness certificates that remain valid even when training data is imperfect or future conditions differ.
Algorithmic Framework: Development of Algorithm 1, a tuning procedure that selects optimal trust radii ( $R$ ) and sample sizes ( $N$ ) to minimize the violation bound while meeting a target safety level.

4. Results

The paper validates the methodology through numerical studies:

Profit-Risk Trade-off: Simulations show that increasing the relaxation parameter $\rho$ reduces costs but increases the probability of constraint violation, confirming the theoretical trade-off.
Robustness to Noise: The adversarially robust policy maintains safety even when training data is perturbed, whereas a nominal (non-robust) policy fails under similar conditions.
Out-of-Distribution (OOD) Performance:
- The robust policy consistently keeps the empirical violation probability below the theoretical certificate derived from Theorem 1.
- In contrast, a nominal policy (trained without adversarial samples) exhibits significantly higher violation rates under distributional shifts, often exceeding the risk levels acceptable for safe operation.
Sample Efficiency: Increasing the number of training samples ( $N$ ) significantly tightens the violation bounds, though at a marginal cost increase.

5. Significance

This work is significant for the deployment of Virtual Power Plants (VPPs) and EV-based storage in real-world grids because:

Safety Guarantees: It moves beyond heuristic or deterministic approaches, providing mathematical guarantees on constraint satisfaction (e.g., ensuring EVs can leave with sufficient charge) even with limited or imperfect data.
Resilience: It addresses critical cybersecurity and data integrity concerns by ensuring the system remains robust against data corruption and statistical shifts, which are common in dynamic energy markets.
Operational Flexibility: It empowers managers to make informed decisions on risk tolerance, allowing them to optimize for profit without blindly sacrificing system reliability.
Scalability: The convex nature of the formulation ensures the problem remains computationally tractable, making it suitable for real-time scheduling in large-scale parking lot networks.

In summary, the paper provides a rigorous, data-driven framework that transforms EV parking lots into reliable, robust, and economically viable virtual energy storage assets, capable of withstanding the uncertainties and adversities of modern power systems.