⚡ electrical engineering

SymbolicPhasor: Power System Phasor Estimation via Deep Symbolic Regression

This paper introduces SymbolicPhasor, a deep symbolic regression framework that accurately estimates power system phasors within one cycle by learning interpretable analytical expressions for distorted fault currents containing decaying DC offsets, harmonics, and frequency deviations, thereby outperforming conventional DFT-based methods for protective relaying applications.

Original authors: Sina Mohammadi, Wencong Su

Published 2026-08-11

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CC BY 4.0

Original authors: Sina Mohammadi, Wencong Su

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 trying to listen to a single, clear voice singing a melody in a crowded, chaotic room. The singer is the "fundamental" frequency—the pure, steady note that tells you the song's true pitch. But the room is filled with other noises: the bass thumping from a speaker (harmonics), a sudden burst of static (noise), and a weird, wobbly echo that starts loud and slowly fades away (the decaying DC offset). In the world of electricity, this "room" is the power grid, and the "voice" is the current flowing through wires. When a fault happens—like a short circuit or a lightning strike—the current goes wild. It's no longer a smooth, clean wave; it's a messy, distorted mess.

Engineers need to know exactly what that "pure voice" is doing, right now, to keep the lights on and prevent blackouts. They use special tools called "phasor estimators" to filter out the noise and find the true signal. The most common tool is like a very fast, very strict librarian who only knows how to read a specific type of book (the Discrete Fourier Transform, or DFT). But when the "book" gets messy with that weird, fading echo, the librarian gets confused and starts making mistakes. If the librarian guesses wrong, the safety systems might trip the power off unnecessarily, or worse, fail to trip when they should. So, the big question is: How do we build a smarter listener that can ignore the chaos and find the true song, even when the room is a disaster zone?

This paper introduces a new, clever listener called SymbolicPhasor. Instead of just trying to guess the answer or filter out the noise, this new method uses a technique called Deep Symbolic Regression. Think of this as a detective who doesn't just look at the clues; the detective actually writes out the mathematical recipe for the entire messy sound wave.

Here is how it works: The system takes a snapshot of the messy electrical signal and slices it into tiny, overlapping windows, like looking at a movie one frame at a time. Inside each window, the AI tries to write a simple math equation that perfectly describes the entire chaotic wave, including the fading echo, the bass thumps, and the static. To help the AI, the researchers provide it with a special "reference sheet" of symbols. They tell the AI, "Hey, we know there's a main song, a third-harmonic echo, and a fifth-harmonic echo, so please look for math formulas that use those specific rhythms."

Once the AI has written this perfect mathematical recipe for the messy wave, it doesn't just stop there. It uses that recipe to mathematically "project" the signal onto a clean, pure sine wave. It's like taking a distorted, wobbly drawing and tracing over it with a ruler to find the perfect, straight line underneath. This allows the system to instantly calculate the true strength (magnitude) and timing (phase) of the fundamental signal, even while the chaos is still happening.

The researchers tested this idea in a simulated environment, creating digital "faults" with all kinds of trouble: single fading echoes, double fading echoes, and even when the power grid's frequency drifted slightly off its normal tune. They compared their new method against the old, standard librarian (the DFT). The results were impressive. In these simulations, SymbolicPhasor managed to reconstruct the messy signals with incredible accuracy, achieving a "coefficient of determination" (a score of how well the math fits the data) as high as 0.985. This means the math recipe it wrote was almost a perfect match for the real signal.

Even when the signal had two different fading echoes at once, or when the frequency was slightly off (like 59.5 Hz or 60.5 Hz instead of the standard 60 Hz), the method held up well. It suggested that by forcing the AI to look for physically meaningful patterns (like the specific frequencies of the harmonics), it could recover the true signal within just one cycle of the wave. This is fast enough to be useful for real-world safety systems.

However, the authors are careful to note that this is currently a simulation. While the math works beautifully on the computer, the main hurdle is that this "detective" takes a bit of computing power to solve the equations in every single window. The paper suggests that while the method is highly effective at filtering out the noise in these tests, future work will need to focus on making it faster and more efficient so it can run on actual hardware in the real world. For now, it stands as a promising new way to listen to the power grid, even when the room is absolutely chaotic.

Technical Summary: SymbolicPhasor

Problem Statement

Accurate phasor estimation during power system faults is a critical challenge due to the complex nature of fault currents. When a fault occurs, the current deviates from a pure sinusoidal waveform primarily due to the presence of a decaying DC offset (DCO), which arises from the inductive nature of the system and the inability of current to change instantaneously. Additionally, fault currents often contain harmonics, measurement noise, and potential frequency deviations.

Conventional estimation techniques, particularly the Discrete Fourier Transform (DFT), struggle under these conditions. While DFT is robust against higher-order harmonics, it is highly sensitive to DCOs, which it fails to eliminate or attenuate. This sensitivity can lead to phasor estimation errors of up to 15%, causing transient oscillations in magnitude estimates and potentially leading to the maloperation of protection and measurement devices. Existing mitigation strategies (pre-DFT, post-DFT, least-squares, and AI-based methods) often suffer from limitations such as requiring full-cycle data (introducing latency), dependence on specific system parameters (e.g., X/R ratios), high sensitivity to noise, or a lack of generalization regarding multiple DCO components and frequency variations.

Methodology

The paper proposes SymbolicPhasor, a dynamic framework based on Deep Symbolic Regression (DSR) designed to estimate the fundamental component of distorted fault current signals. The methodology operates through the following stages:

Dynamic Windowing Strategy:
Instead of relying on a single static fitting interval, the method processes the signal using overlapping moving windows. In the implementation, each window spans one cycle with a step size of a quarter cycle. This ensures that windows around the fault inception instant ( $t_f$ ) capture both pre-fault and post-fault behaviors.
Full-Signal Reconstruction via DSR:
Within each window, the DSR model does not force a direct fit to a pure sinusoid. Instead, it learns an interpretable analytical expression for the entire distorted waveform.
- Input: The model takes only the time variable $t$ as input.
- Symbolic Constraints: To guide the search toward physically meaningful expressions, the operator library is restricted. It includes standard arithmetic operators ( $+, -, \times$ $+, -, \times$ ) and specific unary functions: $\sin(\cdot)$ $sin (\cdot)$ , $\exp(\cdot)$ $exp (\cdot)$ , and predefined symbolic tokens representing angular frequencies:
  - $w_{377}(t) = 377t$ (Nominal fundamental frequency, 60 Hz).
  - $w_{1131}(t) = 1131t$ (Third harmonic).
  - $w_{1885}(t) = 1885t$ (Fifth harmonic).
- Output: The model automatically discovers an expression approximating the sum of the fundamental, odd harmonics, and the decaying exponential term (DCO).
Fundamental Component Extraction:
Once the full waveform is reconstructed within a window, the fundamental component is isolated by projecting the reconstructed signal onto the nominal sine and cosine bases ( $\sin(\omega t)$ and $\cos(\omega t)$ ) using a least-squares approach. This yields the instantaneous amplitude and phase for that window.
Signal Merging:
Local estimates from overlapping windows are combined (averaged) to generate a continuous, dynamic time-varying estimate of the fundamental phasor over the entire signal duration.

Key Contributions

DSR Framework for Phasor Estimation: The paper introduces a novel application of Deep Symbolic Regression to power system protection, moving beyond "black-box" neural networks to discover explicit, interpretable mathematical expressions for fault currents.
Physics-Informed Symbolic Tokens: By embedding tokens corresponding to the nominal fundamental, third, and fifth harmonic frequencies directly into the search space, the method is biased toward discovering expressions that reflect the physical reality of power system faults without requiring extensive offline training datasets.
Dynamic Windowing: The use of overlapping moving windows allows the framework to capture the transient evolution of the signal around the fault inception instant, addressing the delay issues common in methods requiring full-cycle data.
Robustness to Complex Distortions: The method is explicitly designed to handle single and multiple decaying DC offsets, harmonics, and noise simultaneously within a single analytical framework.

Experimental Results

The proposed method was evaluated using a PySR-based symbolic regression engine under three distinct scenarios, comparing performance against the standard DFT method. The signal model included a nominal 60 Hz system (with off-nominal tests at 59.5 Hz and 60.5 Hz), Gaussian noise (SNR = 30 dB), and various harmonic contents.

Single DCO Scenarios: The method achieved high reconstruction accuracy with $R^2$ values ranging from 0.945 to 0.955 across different time constants (10ms, 20ms, 40ms). Magnitude overshoot (MO) varied, with the lowest being 3.63% and the highest 17.02%.
Multiple DCO Scenarios: In cases simulating current transformer saturation with two decaying DC components, the framework maintained high accuracy, achieving $R^2$ values between 0.966 and 0.979.
Off-Nominal Frequency: When tested at 59.5 Hz and 60.5 Hz (while the model remained constrained to the nominal 60 Hz operator), the method demonstrated robustness. The $R^2$ values were 0.985 (at 59.5 Hz) and 0.951 (at 60.5 Hz), indicating that the approach can approximate the fundamental component effectively even with moderate frequency mismatches.

Significance and Claims

The authors claim that SymbolicPhasor offers a viable alternative for protective relaying and measurement applications by recovering the fundamental component within one cycle with high accuracy. The primary significance of the work lies in its ability to:

Recover the fundamental component within a single cycle despite severe waveform distortions (DCO, harmonics, noise).
Provide interpretability, as the output is an analytical expression rather than a black-box prediction.
Maintain flexibility through data-driven learning while being guided by physical constraints (symbolic tokens).

The paper acknowledges a limitation regarding the computational burden of performing symbolic regression in every window, noting that future work will focus on improving computational efficiency. The authors do not claim the method eliminates the need for all traditional techniques but rather present it as a robust framework for scenarios where conventional DFT-based estimators degrade significantly.