Initialization & Electrode State of Health

Before running a simulation, you need to specify how lithium is distributed in each electrode. This determines the starting state of charge and affects the entire simulation. The Electrode State of Health (ESOH) algorithm finds the stoichiometry windows that map cell SOC to electrode lithiation states.

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Why Initialization Matters

The initial concentration distribution affects:

• Starting voltage: The cell begins at the OCV corresponding to the initial stoichiometry
• Available capacity: How much charge can be extracted before hitting voltage limits
• Simulation accuracy: Incorrect initialization leads to incorrect predictions throughout the simulation

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The ESOH Problem

When you measure a full-cell OCV curve, you observe the difference between electrode potentials: $$V_{\text{cell}}(z) = U_p(\theta_p) - U_n(\theta_n)$$ But the cell voltage alone doesn’t tell you the individual electrode stoichiometries. The ESOH algorithm solves this inverse problem by finding the stoichiometry windows that explain the full-cell OCV.

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What ESOH Determines

These define how each electrode is utilized across the SOC range—essential for accurate simulation and degradation tracking.

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Calculation of Minimum and Maximum Stoichiometries

First, the minimum and maximum stoichiometries in each electrode (based on the voltage range) must be calculated.

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Required Parameters

• Initial concentration in <electrode> electrode [mol.m-3] ($c_{s,t=0}^{+,-}$)
• Voltage at 100% SOC [V] ($V_{100}$)
• Voltage at 0% SOC [V] ($V_0$)

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System of Equations

The stoichiometries are found by solving: $$U^+(x_{100}, T) - U^-(y_{100}, T) - V_{100} = 0$$ $$U^+(x_0, T) - U^-(y_0, T) - V_0 = 0$$ where $x$ and $y$ are the stoichiometries of the positive and negative electrodes respectively, the subscript indicates the SOC percentage, $T$ is temperature, and $U$ is the open-circuit potential function.

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Cyclable Lithium Constraint

By default, cyclable lithium inventory ($Q_{Li}$) is used as the constraint to fully define the system: $$y_{100} = \frac{Q_{Li} - x_{100} \cdot Q^-}{Q^+}$$ $$y_0 = y_{100} + \frac{Q}{Q^+}$$ where $Q$ is given by: $$Q = Q^- \cdot (x_{100} - x_0)$$

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Electrode Capacity Calculation

Each electrode capacity is calculated as: $$Q^{+,-} = A^{+,-} \cdot c_{s,0}^{+,-} \cdot T^{+,-} \cdot \varepsilon_{am}^{+,-}$$ where $A$ is electrode area, $T$ is electrode thickness, and $\varepsilon_{am}$ is the active material volume fraction.

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Initialization Options

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Practical guidance

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References

1. Mohtat, Peyman, et al. “Towards better estimability of electrode-specific state of health: Decoding the cell expansion.” Journal of Power Sources 427 (2019): 101-111.
2. Weng, Andrew, et al. “Modeling battery formation: Boosted sei growth, multi-species reactions, and irreversible expansion.” Journal of The Electrochemical Society 170.9 (2023): 090523.