⚡ electrical engineering

A Gray-Box Approach for Decentralized Grid-Equivalent Model Identification

This paper proposes a decentralized, frequency-domain gray-box approach that decouples equivalent impedance and voltage effects to accurately identify grid-equivalent models for local converters in multi-converter systems, even under non-ideal operating conditions.

Original authors: Sanjay Chandrasekaran, Florian Dörfler, Silvia Mastellone

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sanjay Chandrasekaran, Florian Dörfler, Silvia Mastellone

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 bustling city where every house has its own solar panel and battery (these are the converters). In the past, these houses just plugged into a giant, unchanging power grid (like a massive, rigid wall). But now, the grid is made up of all these houses talking to each other. They are all trying to manage their own power while also helping the neighborhood.

The problem? If you are one of these houses trying to figure out how the rest of the neighborhood is behaving, it's incredibly confusing. When you push a little bit of power to test the system, your neighbors push back at the exact same time. It's like trying to hear a whisper in a room where everyone is shouting at once. If you try to measure the "resistance" of the neighborhood (the grid impedance) while everyone is shouting, you get a messy, wrong answer.

This paper proposes a clever new way to listen to the neighborhood without getting confused by the noise. Here is the breakdown in simple terms:

The Core Problem: The "Shouting Neighbors" Effect

In the old days, engineers could treat the power grid like a solid, unchanging wall. They could poke it, measure the reaction, and know exactly how stiff it was.

The Reality: Today, the grid is full of smart devices (converters) that react instantly. When one device pokes the system to test it, the others react immediately.
The Mistake: If you try to measure the grid's "stiffness" (impedance) while ignoring these reactions, your math gets biased. It's like trying to weigh yourself while someone else is pushing you up and down; the scale gives you the wrong number.

The Solution: A "Gray-Box" Detective

The authors call their method a "Gray-Box" approach. Think of it like this:

White Box: You know everything inside the machine (too hard here because we don't know what every neighbor is doing).
Black Box: You know nothing inside (too vague).
Gray Box: You know the shape of the machine (it's a wire with some resistance and inductance), but you don't know the active parts (what the neighbors are doing).

The algorithm splits the problem into two distinct parts:

The Passive Part (The Wire): This is the physical wire connecting the houses. It's boring, predictable, and follows simple laws of physics.
The Active Part (The Neighbors): This is the voltage coming from the other houses. It's chaotic, changing, and unpredictable.

How the Algorithm Works (The 3-Step Dance)

Step 1: The "Instrumental Variable" Trick (Filtering the Noise)
Imagine you are trying to hear a specific song in a crowded bar. Everyone is talking, but you have a special earpiece that only picks up the sound of your voice when you hum a specific tune.

The algorithm tells each converter to hum a specific, random "tune" (a wide-band excitation signal).
Because every converter hums a slightly different tune, the algorithm can mathematically separate "my hum" from "the neighbor's hum."
This allows it to ignore the noise coming from the other houses and focus only on how the wire reacts to its own hum.

Step 2: The "Frequency Detective" (Ignoring Bad Data)
The algorithm looks at the data across different frequencies (like listening to different musical notes).

Low Notes: It ignores these because they are dominated by the normal rhythm of the grid (like the hum of a refrigerator).
High Notes: It ignores these because they are just static noise.
The "Coherence" Check: If the data at a certain note looks like it's being driven by the neighbors rather than the local hum, the algorithm throws that data point in the trash. It only keeps the "clean" data where the local house is clearly in control.

Step 3: The Two-Step Estimation
Once the clean data is isolated, the algorithm does two things in order:

First, it calculates the Wire: Using a special math technique (Constrained Least Squares), it figures out the exact resistance and inductance of the connection. It treats this as a fixed, physical fact.
Second, it calculates the Neighbors: Now that it knows the wire's properties, it uses a "Kalman Filter" (a smart prediction tool) to figure out what the neighbors are actually doing. It treats the neighbors' voltage as a moving target that changes from note to note.

The Results: A Clearer Picture

The authors tested this on a simulation of 5 converters talking to each other.

The Test: They made the system noisy, with different line lengths and control settings, and had all 5 converters trying to measure at the same time.
The Outcome: Their method successfully separated the "wire" from the "neighbors."
- They measured the "stiffness" of the connection with extremely high accuracy (less than 3% error).
- They were able to reconstruct the "voice" of the neighbors (the equivalent voltage) almost perfectly.
The Key Win: They did this decentralized. This means every converter did the math on its own, using only its own local measurements. They didn't need to talk to each other, sync their clocks, or wait for a central computer.

Summary

This paper presents a way for smart power devices to figure out how the rest of the grid is behaving, even when everyone is acting at the same time. By treating the physical wires as one thing and the active neighbors as another, and by using a clever "humming" trick to filter out the noise, the algorithm allows each device to see the grid clearly without needing to coordinate with anyone else. It's like learning to hear your own voice in a choir without needing the conductor to tell you when to sing.

Technical Summary: A Gray-Box Approach for Decentralized Grid-Equivalent Model Identification

Problem Statement
The integration of renewable energy sources and power electronics into electric grids has necessitated robust data-driven modeling and stability analysis. A critical challenge in this domain is the identification of grid-impedance models. Traditional centralized approaches struggle with time-scale synchronization between low-inertia renewables and high-inertia synchronous generators. Furthermore, existing decentralized methods often rely on restrictive assumptions that fail in dynamic, multi-agent networks. Specifically, many current algorithms require coordinated system identification (where converters excite sequentially), multiple operating points, or the unrealistic assumption that the grid behaves as a "stiff" infinite bus with a perfectly constant equivalent voltage.

In a multi-converter setup, local signals at a Point of Common Coupling (PCC) are influenced by voltage inputs from other converters acting as the "grid." Attempting to estimate a single apparent impedance without decoupling these active influences leads to biased and inaccurate results. The core problem addressed is the need for a minimally invasive, decentralized algorithm capable of estimating both the passive equivalent grid impedance and the active equivalent grid voltage simultaneously, without interrupting grid operations or requiring inter-converter communication.

Methodology
The authors propose a gray-box identification algorithm operating in the frequency domain. The approach models the grid from the perspective of a local Voltage Source Converter (VSC) using a Thevenin equivalent circuit, where the grid is represented by a passive equivalent admittance ( $Y_i(s)$ ) and an active equivalent voltage ( $\Delta \tilde{v}_i(s)$ ).

The methodology proceeds through the following steps:

System Modeling: The network is modeled as interconnected VSCs via resistive-inductive lines. The small-signal dynamics are expressed in complex coordinates (derived from $dq$ frames). The relationship between local current ( $\Delta i_i$ ) and voltage ( $\Delta v_i$ ) is formulated as a bilinear map involving the unknown parameter vector $\theta_i$ (representing line resistance and inductance ratios) and the unknown dynamic coupling factor $\tilde{h}_i$ (representing the grid voltage influence).
Gray-Box Framework: The algorithm distinguishes between the "passive" grid impedance, which is modeled parametrically (low-order, resistive-inductive structure), and the "active" grid voltage, which is modeled non-parametrically. This distinction acknowledges that the active grid voltage arises from the closed-loop reactions of other converters and loads, which are unknown and potentially complex.
Instrumental Variables (IV) and Frequency Discrimination: To address the correlation between the local voltage and the equivalent grid voltage (which causes bias in standard least-squares estimation), the authors employ Instrumental Variables. They utilize the local wide-band excitation signal (a pseudo-random binary sequence) as an instrument.
- Frequency Discrimination (FD): A filtering process identifies and discards unreliable frequency points. This includes removing frequencies with low coherence between excitation and response, frequencies dominated by fundamental grid dynamics (low-frequency) or aliasing/harmonics (high-frequency), and frequencies where the estimated admittance violates passivity constraints.
Two-Step Estimation:
- Step 1 (Parameter Estimation): A constrained least-squares algorithm estimates the parametric equivalent impedance ( $\theta_i$ ) using the filtered frequency data and instrumental variables.
- Step 2 (Voltage Estimation): With the parameters fixed, a Kalman filter (formulated as an unknown input observer) estimates the non-parametric equivalent grid voltage ( $\Delta \tilde{v}_i$ ) across the frequency spectrum.

Key Contributions
The paper highlights three primary contributions:

Fully Decentralized Execution: The algorithm operates independently at each VSC using strictly local PCC measurements. It requires zero communication or coordinated timing with other converters, making it scalable to grids with arbitrary sources and loads.
Separation of Active and Passive Entities: By estimating the parametric impedance and non-parametric voltage separately in the frequency domain, the method decouples physical transmission line dynamics from the active control reactions of other VSCs. This resolves severe cross-coupling errors that typically corrupt time-domain identification.
Robustness: The method is validated under non-ideal conditions, including low-inertia grids (non-rigid buses), simultaneous multi-agent excitations, deviations in nominal amplitude and frequency, heterogeneous line parameters, and harmonic noise.

Results
The algorithm was tested via numerical simulations on an interconnected 5-converter system operating in grid-forming mode.

Setup: 20 Monte-Carlo experiments were conducted with varying droop control parameters and line impedances. All converters injected minimal wide-band excitations (0.001 p.u.) adhering to IEEE 519 harmonic standards.
Impedance Estimation: The estimated equivalent admittance ( $Y_1(s)$ ) showed high accuracy, with an average magnitude error of 0.26 dB (3.13%) and a phase error of 0.07 degrees (0.08%).
Voltage Estimation: The algorithm successfully reconstructed the magnitude and phase of the small-signal equivalent voltage. Time-domain results (via inverse Fourier transform) demonstrated that the estimated voltage closely matched the true values in both $d$ and $q$ coordinates.
Comparison: The results were compared against raw Empirical Transfer Function Estimates (ETFE), which showed significant bias at low frequencies due to closed-loop interactions, validating the necessity of the proposed frequency-domain pre-processing and IV approach.

Significance and Claims
The authors claim that this work enables robust grid-equivalent model identification under active grid conditions where previous decentralized methods fail. By removing the requirement for coordinated excitation or the assumption of a stiff infinite bus, the proposed gray-box approach allows for accurate stability analysis and adaptive control design in modern, dynamic power systems. The paper concludes that the identified model can serve as a foundation for future decentralized adaptive control algorithms and stability analysis, though it notes that applying the algorithm to real-world hardware and extending it to include synchronous generators and diverse control schemes are directions for future work.