🤖 machine learning

Learning Without Adversarial Training: A Physics-Informed Neural Network for Secure Power System State Estimation under False Data Injection Attacks

This paper proposes a Physics-Informed Neural Network (PINN) for secure power system state estimation that achieves robustness against stealthy False Data Injection Attacks (FDIAs) without using adversarial training, instead employing a dynamic loss-weighting formulation based on homoscedastic uncertainty to automatically balance data-fit and physics-residual terms.

Original authors: Solon Falas, Markos Asprou, Charalambos Konstantinou, Maria K. Michael

Published 2026-04-28

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CC BY 4.0

Original authors: Solon Falas, Markos Asprou, Charalambos Konstantinou, Maria K. Michael

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 you are a high-tech air traffic controller. To keep planes from crashing, you rely on sensors that tell you exactly where every plane is, how fast it’s going, and which direction it’s heading.

But there is a problem: Cyber-attackers.

A sophisticated hacker doesn't just turn the sensors off (that would trigger an alarm). Instead, they perform a "stealth attack." They subtly tweak the data—maybe making a plane look like it’s 50 feet higher than it actually is—in a way that looks perfectly normal to your standard equipment. If you trust the fake data blindly, you might give a pilot instructions that lead to a collision.

This paper describes a new "Smart Digital Controller" (a Physics-Informed Neural Network) designed to spot these lies and keep the power grid safe.

1. The Problem: The "Lying" Sensors

In a power grid, we use "State Estimation" to figure out the health of the system. We take measurements (like voltage) and turn them into a map of the grid.

The danger is the False Data Injection Attack (FDIA). Think of this like a "Deepfake" for electricity. The attacker knows the "rules of the road" (the physics of the grid) and crafts fake data that follows those rules perfectly. Because the fake data looks "physically possible," the old security systems think everything is fine, even though the grid is actually being pushed toward a blackout.

2. The Solution: The "Physics-Savvy" Brain

Most AI models are like students who only learn by reading textbooks (the data). If the textbook contains a lie, the student believes the lie.

The researchers created a Physics-Informed Neural Network (PINN). This student doesn't just read the textbook; they also have a built-in "Common Sense" engine based on the laws of physics.

Even if an attacker sends fake data saying, "The power is flowing perfectly through this wire," the AI checks its internal physics engine and says, "Wait a minute. According to the laws of electricity, if that much power were flowing there, the voltage at the next station would have dropped. This data is a lie."

3. The Secret Sauce: The "Adaptive Volume Knob"

The hardest part of training this AI is balance. You have two competing goals:

Listen to the sensors (Data).
Follow the laws of physics (Physics).

If you listen to the sensors too much, you fall for the hacker. If you follow physics too strictly, you might ignore real, unexpected changes in the grid.

In the past, engineers had to manually turn a "volume knob" to balance these two. But if the grid changes (like during a storm), the manual setting might be wrong.

The researchers introduced "Dynamic Weighting." Imagine a smart volume knob that turns itself in real-time. During training, the AI learns exactly how much to trust the data versus how much to trust the physics. It’s like a chef who automatically adjusts the salt in a soup based on how much water is in the pot—it adapts to the situation automatically.

4. The Results: A Much Tougher Guard

The researchers tested this "Smart Controller" on a massive simulated power grid (the IEEE 118-bus system) and threw four different types of "Deepfake" attacks at it:

The Distorter: Trying to trick the system about voltage levels.
The Shifter: Trying to trick the system about where electricity is being used.
The Overloader: Trying to trick the system into stressing out specific power lines.
The Mimic: Trying to make the fake data look exactly like the "noise" of a normal system.

The outcome? The new AI was a superstar. It was 82% more accurate than previous versions. Even when the attacks were at their most clever and "stealthy," the AI’s ability to cross-reference the data with the laws of physics allowed it to see through the deception and maintain a clear, truthful picture of the grid.

Summary in a Nutshell

Instead of just trusting what the sensors say, this new AI uses the Laws of Physics as a lie detector. By automatically balancing "what we see" with "what we know must be true," it creates a shield that can protect our electricity from even the most sophisticated digital hackers.

Technical Summary: Learning Without Adversarial Training for Secure Power System State Estimation

1. Problem Statement

Power System State Estimation (PSSE) is critical for the real-time monitoring and control of electrical grids. As grids become increasingly digitalized, they become vulnerable to False Data Injection Attacks (FDIAs). The most dangerous attacks are "stealthy," meaning they are coordinated with the network's topology and physics so that they bypass traditional Bad-Data Detectors (BDDs) by remaining within physically plausible bounds.

While Physics-Informed Neural Networks (PINNs) have emerged as a promising alternative to conventional model-based methods by embedding power-flow equations into the learning process, they face a significant practical challenge: loss-weight balancing. PINNs require balancing a "data-fit" loss (matching measurements) and a "physics-residual" loss (satisfying Kirchhoff’s laws). In traditional PINNs, these weights are fixed manually, which often leads to brittle performance when the system faces varying operating conditions or different types of adversarial attacks.

2. Methodology

The authors propose a robust, attack-agnostic PINN formulation that does not rely on adversarial training (training on attacked data). Instead, the model is trained exclusively on clean, steady-state data but uses an adaptive mechanism to ensure physical consistency.

Key Methodological Components:

Homoscedastic Uncertainty-Based Weighting: Rather than using fixed coefficients, the model treats the weighting of each loss component as a learnable parameter. It uses a multi-task learning approach where trainable log-uncertainty scalars ( $s_m$ ) automatically rescale the supervised data loss and the physics-residual loss. This allows the model to learn the optimal relative importance of physics vs. data during training.
One-Sided Log-Ratio Safeguard: To prevent the "physics collapse" problem (where the network ignores physics to minimize data error), the authors introduce a regularization term ( $P_{ratio}$ ). This term acts as a "hinge" penalty that only activates if the influence of the physics loss falls below a specific target ratio relative to the data loss.
Stealth-Constrained AC-FDIA Framework: To rigorously test the model, the authors developed a systematic optimization pipeline to generate four families of sophisticated, AC-consistent attacks:
1. Simple FDIA: Direct state distortion (voltage and angle).
2. Load Redistribution: Economic-driven manipulation of demand and generation.
3. Line Overload: Targeted attacks on electrically strong corridors to induce congestion.
4. State Estimation Corruption: The most stringent attack, which forces the injected data to mimic the residual patterns of clean data to remain invisible to BDDs.

3. Key Contributions

Adaptive PINN Formulation: A novel training objective that uses uncertainty-based dynamic weighting and a log-ratio safeguard to balance data fidelity and physical consistency without manual tuning.
Attack-Agnostic Robustness: Demonstrating that a model trained only on clean data can resist sophisticated, model-aware attacks by relying on its learned "physical intuition."
Comprehensive FDIA Benchmarking: A standardized, optimization-based framework for generating diverse, stealthy, and AC-consistent attack scenarios for evaluating state estimators.

4. Results

The model was evaluated on the IEEE 118-bus system and compared against fixed-weight PINNs and prior state-of-the-art baselines.

Superior Accuracy: The proposed dynamic PINN reduced the average Mean Absolute Error (MAE) by 82% compared to fixed-weight PINN variants.
Attack Resilience:
- Against Simple FDIA, it outperformed the strongest prior baseline by a significant margin.
- Against Load Redistribution, MAE dropped from $9.46 \times 10^{-2}$ to $1.85 \times 10^{-2}$ .
Stability: The dynamic weighting approach showed much smoother convergence (Fig. 2) and significantly lower worst-case errors (95th and 99th percentiles) compared to fixed or "frozen" weight models.
Scalability: The model maintained robustness even when the number of attacked buses increased, showing only a mild increase in error.

5. Significance

This research shifts the paradigm of securing AI-based power system tools. Instead of the computationally expensive and complex process of adversarial training (trying to predict every possible attack), the authors demonstrate that improving the physical integrity of the estimator through adaptive weighting is a highly effective defense. This provides a more stable, interpretable, and deployable solution for protecting modern smart grids against sophisticated cyber-physical threats.