A perturbation approach for consistent initialization of index-1 explicit differential–algebraic equations arising from battery model simulations

Methekar, Ravi; Ramadesigan, Venkatasailanathan; Pirkle, J. Carl; Subramanian, Venkat R.

doi:10.1016/j.compchemeng.2011.01.003

Cited by 22 publications

(32 citation statements)

References 20 publications

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“…These ODEs can be solved simultaneously using efficient initial value problem (IVP) solvers, such as dsolve from Maple [31] or FORTRAN solvers such as DASKR or DASSL [32]. For a more complicated problem, the discretized system may be composed of differential algebraic equations (DAEs), which requires proper initialization to ensure consistent ICs of the algebraic variables before dsolve of DASKR could be applied [33].…”

Section: Resultsmentioning

confidence: 99%

Analytical solution for electrolyte concentration distribution in lithium-ion batteries

et al. 2012

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In this article, the method of separation of variables (SOV), as illustrated by Subramanian and White (J Power Sources 96:385, 2001), is applied to determine the concentration variations at any point within a three region simplified lithium-ion cell sandwich, undergoing constant current discharge. The primary objective is to obtain an analytical solution that accounts for transient diffusion inside the cell sandwich. The present work involves the application of the SOV method to each region (cathode, separator, and anode) of the lithium-ion cell. This approach can be used as the basis for developing analytical solutions for battery models of greater complexity. This is illustrated here for a case in which nonlinear diffusion is considered, but will be extended to full-order nonlinear pseudo-2D models in later work. The analytical expressions are derived in terms of the relevant system parameters. The system considered for this study has LiCoO 2 -LiC 6 battery chemistry. List of symbolsa Specific interfacial area (m 2 /m 3 ) B, Brugg Bruggeman coefficient c 0 Concentration at initial time t = 0 c i (x, t) Concentration in region i (mol/m 3 ) C i (X, s) Dimensionless concentration in region i D Diffusion coefficient of lithium ions in the electrolyte (cm 2 /s) D eff,i Effective diffusion coefficient of the Li-ion in region i (cm 2 /s) F Faraday's constant (C/mol) i app Applied current density (A/m 2 ) j i Flux density of the Li-ions into the electrode in region i (mol/m 2 s) J i Dimensionless flux density in region i l i Thickness of region i (m) K Ratio of dimensionless flux densities in the electrodes L Total thickness of cell (m) p Dimensionless position of positive electrode/ separator interface q Dimensionless position of separator/negative electrode interface t Time (s) t þ Transference number x Position (m) X Dimensionless position

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Section: Resultsmentioning

confidence: 99%

Analytical solution for electrolyte concentration distribution in lithium-ion batteries

et al. 2012

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“…More information on the method and solution procedure of several ODE/DAE solvers, are provided elsewhere (Cash, 2003;Cellier and Kofman, 2006;Methekar et al, 2011). Solvers based on backward difference formula (DASKR/DASSL/GEAR) (Brown et al, 1994(Brown et al, , 1998 or implicit Runge-Kutta methods (RADAU IIA) (Hairer and Wanner, 1999) are efficient in solving nonlinear DAEs.…”

Section: Introductionmentioning

confidence: 99%

Extending explicit and linearly implicit ODE solvers for index-1 DAEs

Lawder

Ramadesigan

Suthar

et al. 2015

Computers & Chemical Engineering

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a b s t r a c tNonlinear differential-algebraic equations (DAE) are typically solved using implicit stiff solvers based on backward difference formula or RADAU formula, requiring a Newton-Raphson approach for the nonlinear equations or using Rosenbrock methods specifically designed for DAEs. Consistent initial conditions are essential for determining numeric solutions for systems of DAEs. Very few systems of DAEs can be solved using explicit ODE solvers. This paper applies a single-step approach to system initialization and simulation allowing for systems of DAEs to be solved using explicit (and linearly implicit) ODE solvers without a priori knowledge of the exact initial conditions for the algebraic variables. Along with using a combined process for initialization and simulation, many physical systems represented through large systems of DAEs can be solved in a more robust and efficient manner without the need for nonlinear solvers. The proposed approach extends the usability of explicit and linearly implicit ODE solvers and removes the requirement of Newton-Raphson type iteration.Published by Elsevier Ltd.

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“…However, it would fail or need significantly high computation time due to the solvers' approach in finding the solution. 22 Here, we implement the DAE-based MPPT algorithm consisting of AEs only. In this case, one arbitrary Ordinary Differential Equation (ODE) (Equation 9), which does not affect any processes of the entire microgrid system, can be added to convert the …”

Section: Governing Equationmentioning

confidence: 99%

“…The DAE solvers in the computational software allow the entire microgrid system to be simulated simultaneously with high accuracy and low computation time under rapidly changing environmental conditions. 22,31 As more detailed physics-based battery models are involved in the entire microgrid, the DAE-based MPPT algorithm will be better suited for simultaneous simulations under real environmental conditions. Reformulated P2D battery model.-An ideal battery model would provide good prediction while maintaining minimal computational cost.…”

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Direct, Efficient, and Real-Time Simulation of Physics-Based Battery Models for Stand-Alone PV-Battery Microgrids

Lee

Pathak

Ramadesigan

et al. 2017

J. Electrochem. Soc.

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With renewable energy based electrical systems becoming more prevalent in homes across the globe, microgrids are becoming widespread and could pave the way for future energy distribution. Accurate and economical sizing of stand-alone power system components, including batteries, has been an active area of research, but current control methods do not make them economically feasible. Typically, batteries are treated as a black box that does not account for their internal states in current microgrid simulation and control algorithms. This might lead to under-utilization and over-stacking of batteries. In contrast, detailed physics-based battery models, accounting for internal states, can save a significant amount of energy and cost, utilizing batteries with maximized life and usability. It is important to identify how efficient physics-based models of batteries can be included and addressed in current grid control strategies. In this paper, we present simple examples for microgrids and the direct simulation of the same including physics-based battery models. A representative microgrid example, which integrates stand-alone PV arrays, a Maximum Power Point Tracking (MPPT) controller, batteries, and power electronics, is illustrated. Implementation of the MPPT controller algorithm and physics-based battery model along with other microgrid components as differential algebraic equations is presented. The results of the proposed approach are compared with the conventional control strategies and improvements in performance and speed are Batteries have been integrated in microgrids to mitigate intermittent characteristics of alternative energy sources such as solar, wind, and wave, thereby enhancing grid operation and reliability.1,2 They are well suited for microgrid applications due to their versatility, high energy density, and efficiency. 3 The cost of batteries continues to decrease while their performance and life have continued to increase. 4 However, lithium-ion batteries, which are the most widely used energy storage systems implemented in microgrids today, are still the most expensive component, accounting for about 60% of the overall Capital Expenditure (CapEx).5 Conservative operations in current microgrids cause high cost and low energy efficiency, underutilizing and overstacking batteries. The current microgrid controls cannot utilize batteries aggressively to achieve high penetration of renewables and maximize life and usability of batteries in the meantime. They implement empirical/equivalent-circuit battery models, treating batteries as just a black box, which does not account for its internal states, and place batteries in a small portion of the entire microgrid, which means current microgrids do not consider batteries principal components. [6][7][8][9][10] For example, if the internal temperature of the battery is not modeled, then the battery must be operated at very low rates to ensure that the internal temperature does not reach high enough values that reduce battery life and create unsafe operating c...

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A perturbation approach for consistent initialization of index-1 explicit differential–algebraic equations arising from battery model simulations

Cited by 22 publications

References 20 publications

Analytical solution for electrolyte concentration distribution in lithium-ion batteries

Analytical solution for electrolyte concentration distribution in lithium-ion batteries

Extending explicit and linearly implicit ODE solvers for index-1 DAEs

Direct, Efficient, and Real-Time Simulation of Physics-Based Battery Models for Stand-Alone PV-Battery Microgrids

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