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Multi-Domain Graph-Based Modeling of Energy Systems with Applications to Lithium-Ion Batteries
Mahsa Hemmat, Andrew G. Alleyne
TL;DR
Existing graph-based energy-system formulations have difficulty representing non-adjacent state dependencies and composite edges with multiple inputs. The paper introduces recursive state-to-input feedback and parallel edge decomposition, then applies them to an electro-thermal Li-ion battery module. The resulting model tracks measured module temperatures closely, while the framework preserves local energy conservation and modular interconnection.
Problem
Existing graph formulations cannot generally represent edge flows dependent on non-adjacent states or composite flows driven by different inputs while retaining the graph formulation.
Method
The paper uses recursive state-to-input feedback for non-adjacent dependencies and parallel edge decomposition for composite interactions with distinct power-flow inputs.
Results
The methods produce an electro-thermal graph-based model whose simulated module temperatures track sensor measurements closely over a discharge session.
Takeaways & Limitations
Both methods preserve vertex energy conservation and operate locally, allowing extended components to interconnect directly with components using prior graph-based formulations.
Takeaways & Limitations
Future work is needed to apply the framework to larger systems such as actively cooled battery packs and to integrate it with multi-state graph-based formulations.
Abstract
from arXiv · showhide
Graph-based models have been shown to provide a structured representation for complex multi-domain energy systems but face limitations when edge power flows depend on non-adjacent states or when a single edge carries multiple power-flow types driven by different inputs. This paper proposes two general extensions to address these limitations: a recursive state-to-input feedback scheme that embeds non-adjacent state dependencies into edge inputs without altering the graph structure, and a parallel edge decomposition method that represents composite interactions using sets of single-input edges while preserving energy conservation at the vertices. The extended framework is demonstrated on a lithium-ion battery module consisting of 36 parallel cells, and the resulting model predicts module temperatures with errors below 1°C. Validation on this electro-thermal battery system demonstrates the effectiveness of the extended framework for multi-domain systems that cannot be represented by previously established graph-based formulations, and indicates its potential for broader application to complex energy systems in control and design studies.
1. INTRODUCTION
Graph-based models offer modular, scalable representations for interconnected multi-domain energy systems, but existing formulations cannot generally represent non-adjacent state dependencies or composite edges with multiple inputs. The paper introduces recursive feedback and parallel edge decomposition methods to address these limitations while preserving graph-based modeling structure and energy balance.
- Motivation: Multi-domain energy systems couple electrical, thermal, and mechanical subsystems, creating many interconnected states that models must represent for estimation, monitoring, and control.Examples include battery packs, motors, power electronics, and thermal-management subsystems.
- Existing framework: Graph-based models represent components as vertices and physical interactions as energy-conserving edges, enabling modular composition across domains.Components can be modeled and validated separately before being interconnected into a full system model.
- Limitations: Existing formulations restrict each edge power flow to adjacent vertex states and explicit edge inputs, excluding dependencies on non-adjacent states.Such dependencies arise when an edge parameter varies with a thermal state elsewhere in a multi-domain graph.
- Limitations: A single edge typically has one input, although practical interactions may combine distinct power-flow types governed by different inputs.The needed extension must preserve the original dynamics, exact vertex energy balance, and consistency with the graph formulation.
- Proposed extensions: The paper introduces recursive state-to-input feedback for non-adjacent dependencies and parallel edge decomposition for composite interactions with multiple inputs.The methods extend graph-based modeling without requiring a general change to the established graph structure.
- Application: The study applies the proposed methods to a Li-ion battery module and compares graph-based model predictions with experimental temperature measurements.The paper presents the application and validation in Sections 3 and 4 after introducing the framework and extensions.
2. GRAPH-BASED MODELING FRAMEWORK
The framework represents interconnected multi-domain energy systems as oriented graphs whose vertices store energy and whose edges encode conserved power exchanges. It extends this formulation with recursive state-to-input feedback for non-adjacent dependencies and parallel edge decomposition for interactions driven by multiple inputs.
- Graph structure: Dynamic vertices store energy in scalar states, while external sink/source vertices provide boundary interfaces without storing energy.The state and capacitance interpretation depends on the physical domain; external quantities are specified by neighboring subsystems or the environment.
- Graph structure: Internal edges carry power flows determined by adjacent vertex states and one scalar edge input, whereas external edges represent exchanges specified by the surroundings.Edge flows are collected separately from external power flows in the graph-based formulation.
- Graph structure: The incidence matrix encodes edge orientation and maps internal edge flows to dynamic-vertex state dynamics and sink/source connections.Its entries are 1 at a tail, −1 at a head, and 0 otherwise, with dynamic and sink/source rows arranged in blocks.
- Recursive state-to-input feedback: Non-adjacent state dependencies are routed through recursive state-to-input feedback, preserving the original graph structure and leaving state dynamics unchanged.A selected remote state is exposed, mapped through g(·), and supplied as the affected edge input.
- Parallel edge decomposition: Composite interactions with multiple independently driven power-flow types are represented as parallel edges sharing endpoints, preserving the vertex balances.This decomposition restores the single-input edge structure when the total flow cannot be represented with one effective input.
3. VALIDATION OF GRAPH-BASED MODELING METHODS ON A LI-ION BATTERY MODULE
The proposed graph-based extensions are applied to a 36-parallel-cell Li-ion battery module with electro-thermal dynamics and validated against temperature measurements during a 30 A discharge.
- Module architecture: A 36P36S battery pack is represented through a single 36-parallel-cell module with lumped module and side thermal nodes.The validation focuses on one module, modeled with SOC, module temperature, and side temperature states.
- Electro-thermal dynamics: The module thermal model uses two first-order energy balances, with SOC modeled by Coulomb counting and heating represented through electrical power terms.The model includes irreversible Joule heating, reversible entropic heating, thermal capacitances, and thermal resistances to the side node and ambient.
- Graph-based formulation: Non-adjacent SOC dependence is handled by exposing SOC as an output and mapping it to edge inputs, preserving the established edge structure.The open-circuit voltage and entropic coefficient in the heat-generation interaction depend on SOC.
- Graph-based formulation: The composite heat-generation interaction is decomposed into two parallel single-input edges whose summed contributions recover the original heat generation.The decomposition preserves the original vertex dynamics and allows each structurally different power-flow type to conform to the single-input edge form.
- Validation results: During a 30 A constant-current discharge from near 100% to about 20% SOC, simulated module and side temperatures track experimental measurements closely.The experiment uses passive ambient cooling, with measured current, power load, and ambient temperature supplied as disturbances.
4. CONCLUSION
The paper extends graph-based modeling with recursive feedback and parallel edge decomposition, then applies these methods to an electro-thermal Li-ion battery module. The methods preserve local energy conservation and system-level modularity, while future work targets larger, more complex systems.
- Methods: The paper introduces recursive state-to-input feedback for non-adjacent state-dependent power flows and parallel edge decomposition for composite interactions with multiple inputs.Both methods address limitations in prior graph-based formulations while retaining the established graph structure or representing composite connections through parallel edges.
- Application: The proposed methods are demonstrated by constructing an electro-thermal graph-based model of a Li-ion battery module.The battery module provides a concrete application of the extended framework.
- Methods: Both methods preserve energy conservation at vertices and operate locally at the component level.This allows components using the new constructions to connect directly with components modeled using prior graph-based formulations.
- Implications: The extended framework maintains modularity and scalability for larger energy systems without altering the system-level representation.The result follows from local component-level constructions that remain compatible with earlier graph-based formulations.
- Future work: Future work will apply the framework to larger systems, including actively cooled battery packs, and integrate it with multi-state graph-based formulations.These directions are intended to further expand the applicability of graph-based models.