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Research Papers

# Model Order Reduction of 1D Diffusion Systems Via Residue Grouping

[+] Author and Article Information
Kandler A. Smith

Center for Transportation Technologies and Systems, National Renewable Energy Laboratory, Golden, CO 80401

Christopher D. Rahn1

Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA 16802cdrahn@psu.edu

Chao-Yang Wang

Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA 16802

1

Corresponding author.

J. Dyn. Sys., Meas., Control 130(1), 011012 (Jan 11, 2008) (8 pages) doi:10.1115/1.2807068 History: Received May 23, 2006; Revised April 24, 2007; Published January 11, 2008

## Abstract

A model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues. Spatially distributed residues are found either analytically (from a transcendental transfer function) or numerically (from a finite element or finite difference state space model), and residues with similar eigenvalues are grouped together to reduce the model order. Two examples are presented from a model of a lithium ion electrochemical cell. Reduced order grouped models are compared to full order models and models of the same order in which optimal eigenvalues and residues are found numerically. The grouped models give near-optimal performance with roughly $1∕20$ the computation time of the full order models and require 1000–5000 times less CPU time for numerical identification compared to the optimization procedure.

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## Figures

Figure 1

Solid state diffusion problem for Li-ion cell

Figure 2

Solid state diffusion poles and residues: Analytical (∙), fifth order grouped (○), and fifth order optimal (×). Fifth order grouping brackets are shown with vertical dotted lines.

Figure 3

Truncated and grouped solid state diffusion ROMs versus exact frequency response: Exact (○), 180th order truncated (∙), 3rd order grouped (-∙), and 5th order grouped (—). (a) Magnitude, ∣ΔCs,e(s)∕JLi(s)∣. (b) Phase angle, ∠(ΔCs,e(s)∕JLi(s)).

Figure 4

Grouped and optimal solid state diffusion ROMs versus higher order model unit step response: 1000th order truncated (○), 5th order grouped (-), and 5th order optimal (-∙)

Figure 5

Electrolyte phase diffusion problem for Li-ion cell with uniform reaction current distribution

Figure 6

Electrolyte phase diffusion eigenvalues and residues at x=Lcell: 45th order finite element rm,k>0 (○) and rm,k<0 (∙) and 3rd order grouped r¯m,k>0 (◻) and r¯m,k<0 (×). Third order grouping brackets are shown with vertical dotted lines.

Figure 7

Truncated and grouped electrolyte phase diffusion ROMs versus higher order model frequency response at x=Lcell: 45th order finite element (○), 28th order truncated (∙), and 3rd order grouped (-). (a) Magnitude, ∣ΔCe(Lcell,s)∕I(s)∣. (b) Phase angle, ∠(ΔCe(Lcell,s)∕I(s)).

Figure 8

Grouped and optimal electrolyte phase diffusion ROMs versus higher order model unit step response: 45th order finite element (○), 3rd order grouped (-), and 3rd order optimal (∙)

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