⚡ electrical engineering

A Graph Theoretic Approach in Combination With Dynamic Mode Decomposition With Control (DMDc) to Analyze Battery Degradation

This paper proposes a novel data-driven framework that integrates Dynamic Mode Decomposition with Control (DMDc) and graph-theoretic analysis to effectively characterize and interpret lithium-ion battery degradation by modeling operational data as an evolving network that transitions from a coherent structure to a fragmented one as aging progresses.

Original authors: Khalid Mahmud Labib, Saad Waheed, Bakhtiar Nafis, Shabbir Ahmed

Published 2026-05-05

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CC BY 4.0

Original authors: Khalid Mahmud Labib, Saad Waheed, Bakhtiar Nafis, Shabbir Ahmed

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 lithium-ion battery not as a simple box of chemicals, but as a bustling city where millions of tiny "citizens" (electrons and ions) are constantly interacting, moving, and talking to one another. When the battery is new and healthy, this city is well-organized: the citizens have strong, clear lines of communication, and everyone works together in a smooth, coordinated rhythm.

This paper proposes a new way to watch this city age and fall apart, using a mix of two powerful tools: DMDc (a mathematical camera that takes snapshots of how the city moves) and Graph Theory (a way of drawing maps of connections).

Here is how the authors break it down:

1. The Problem: The Battery's "Black Box"

Batteries get old and lose power over time. We know that they degrade, but it's hard to see how they degrade inside. Traditional methods are like trying to understand a city by only looking at its electricity bill (voltage and current). They give you a number, but they don't show you the traffic jams, the broken bridges, or the neighborhoods that have stopped talking to each other.

2. The Solution: Taking a "Motion Picture" of the Battery

The researchers used a technique called Dynamic Mode Decomposition with Control (DMDc). Think of this as a high-speed camera that records the battery's voltage response to a specific test (like a quick pulse of power).

Instead of just looking at the raw video, DMDc breaks the video down into its "fundamental moves" or modes.

The Analogy: Imagine a dance troupe. DMDc doesn't just watch the whole show; it identifies the specific steps each dancer is doing and how they move in relation to the others. It creates a "mode matrix" (a list of these dance moves).

3. Turning Moves into a Map (The Graph)

This is the clever part. The authors took that list of dance moves (the mode matrix) and turned it into a network map (a graph).

The Nodes (Dots): Each dot on the map represents a specific part of the battery's internal state.
The Edges (Lines): The lines connecting the dots represent how strongly those parts are influencing each other. A thick, dark line means they are talking loudly and clearly. A thin or missing line means they are barely communicating.

4. Watching the City Decay

The researchers ran this test on a battery as it aged from "brand new" to "worn out" (after 360 charge cycles). They looked at two main things on their map:

A. Connectivity (How many lines are there?)

Healthy Battery: The map looks like a dense spiderweb. Almost every dot is connected to many others with strong lines. The city is unified; information flows everywhere easily.
Aging Battery: As the battery gets older, the lines start to fade and disappear. The web becomes sparse. The "citizens" stop talking to each other. The authors found that the number of strong connections dropped steadily as the battery degraded.
- Simple Takeaway: A healthy battery is a tight-knit community; an old battery is a group of isolated individuals.

B. Modularity (Are the groups becoming messy?)

Healthy Battery: The connections are spread out evenly. No single group is dominating, and no part of the city is left in the dark. It's a balanced, uniform neighborhood.
Aging Battery: The map starts to look chaotic. Some areas become super-connected (cliques), while others become completely disconnected. The balance is lost. The "heterogeneity" (messiness) increases.
- Simple Takeaway: A healthy battery is a well-planned city; an old battery is a city where some neighborhoods are chaotic and others are abandoned.

5. What They Found

By watching this map change over time, the researchers saw a clear story:

The "Smooth" Becomes "Jagged": In a new battery, the internal movements are smooth and coordinated. In an old one, the movements become erratic, with sudden spikes and irregular patterns.
The Network Fragments: The battery doesn't just get "weaker" all at once; it loses its internal structure. The strong, unified network breaks down into smaller, weaker, and disconnected pieces.

The Bottom Line

This paper doesn't just tell us that a battery is dying; it gives us a new way to visualize how it is dying. By turning complex electrical data into a simple "connectivity map," they showed that battery aging is essentially the breaking of connections and the loss of order within the system.

Instead of just guessing the battery's health based on a single number, this method looks at the "social structure" of the battery's internal physics to see if the community is still working together or falling apart.

Technical Summary: A Graph Theoretic Approach in Combination With Dynamic Mode Decomposition With Control (DMDc) to Analyze Battery Degradation

Problem Statement
Accurate monitoring of lithium-ion battery (LIB) degradation is critical for safety and performance but remains challenging due to the complex, nonlinear, and time-varying nature of electrochemical aging. Traditional physics-based models offer high fidelity but are computationally intensive and difficult to calibrate for real-time use. Conversely, data-driven approaches like Equivalent Circuit Models (ECMs) are efficient but rely on fixed topologies that fail to capture evolving internal degradation mechanisms and structural changes. While machine learning methods have improved state-of-health (SOH) estimation, there is a lack of frameworks that utilize graph-theoretic approaches to fundamentally understand the structural evolution of battery degradation from operational data alone.

Methodology
The authors propose a data-driven framework that integrates Dynamic Mode Decomposition with Control (DMDc) and graph theory to characterize battery degradation. The methodology proceeds as follows:

Data Acquisition: A 30 Ah LIB is subjected to Hybrid Pulse Power Characterization (HPPC) tests. The cell undergoes continuous charge-discharge cycling up to 360 cycles, with HPPC tests repeated every 20 cycles to capture voltage responses under controlled current inputs.
DMDc System Identification: The DMDc algorithm is applied to the operational data to identify a linear state-space representation ( $x_{k+1} \approx A x_k + B u_k$ ). Through Singular Value Decomposition (SVD) and eigen-decomposition, the method extracts a mode matrix ( $\Phi$ ). This matrix encodes the amplitude and phase relationships among dynamical modes, representing the system's spatiotemporal evolution.
Graph Construction: The mode matrix $\Phi$ is reinterpreted as a weighted adjacency matrix for a graph $G(V, E)$ . In this representation, nodes correspond to delay-embedded state coordinates (inferred from voltage responses), and edge weights quantify the strength and directionality of dynamic coupling between states.
Graph Metric Analysis: Two primary metrics are computed to quantify structural changes:
- Connectivity (Network Density): An adaptive threshold based on the mean magnitude of matrix entries is used to convert the weighted matrix into a binary adjacency matrix. Connectivity is defined as the fraction of active edges relative to total possible connections.
- Modularity Proxy (Network Heterogeneity): Since explicit community detection is not performed, a modularity proxy is defined as the normalized dispersion (standard deviation divided by mean) of node strengths. This metric captures the heterogeneity of the network structure.

Key Contributions

Novel Framework Integration: This work establishes a principled connection between data-driven dynamical systems theory (specifically DMDc) and graph-theoretic analysis to model battery degradation.
Topology-Free Representation: Unlike ECMs, this approach constructs a network directly from data without imposing a predefined circuit topology, allowing it to capture emergent coupling patterns associated with aging.
Interpretable Metrics: The study introduces specific graph-based measures (connectivity and modularity proxy) that provide system-level insights into how degradation propagates and reorganizes the battery's internal dynamics.

Results
The application of this framework to HPPC data from a degrading battery yielded the following observations:

Spatiotemporal Coherence: In the healthy state, DMD modes exhibit smooth, continuous magnitude variations and organized phase behavior, indicating coherent system dynamics. As degradation progresses, the spatial distribution becomes irregular and scattered, with energy redistributing from dominant modes to previously inactive ones, reflecting a loss of structural coherence.
Connectivity Decline: The network connectivity decreases progressively with aging. Starting at a high value of 0.68 in the healthy state, connectivity drops to 0.653 by the 40th cycle and further to 0.623 by the 340th cycle. This indicates a transition from a highly connected, stable network to a fragmented structure with weakened interactions.
Increased Heterogeneity: The modularity proxy increases with degradation. The healthy state shows a low value (~0.3), indicating a uniform distribution of interactions. As the battery degrades, the value rises, signifying that interactions become unevenly distributed and the network structure becomes more heterogeneous and dispersed.

Significance and Claims
The authors claim that this work demonstrates the effectiveness of graph-theoretic representations in capturing the evolving dynamics of battery degradation. By transforming the DMDc mode matrix into an evolving network, the approach provides interpretable insights into system-level aging behavior that traditional models may miss. The study posits that degradation manifests as a systematic shift from a coherent, highly connected network to a weaker, fragmented, and heterogeneous structure.

The paper concludes that this framework offers a new avenue for understanding degradation processes in LIBs and potentially other energy storage systems. However, the authors remain modest regarding the current limitations, noting that the results depend on modeling choices (e.g., rank truncation, embedding dimension) and that the adjacency matrix represents abstract modal interactions rather than direct physical connections. They suggest that future work should focus on rigorously linking these matrices to physical structures and exploring advanced DMD techniques to improve robustness against noise.