> ## Documentation Index
> Fetch the complete documentation index at: https://docs.ionworks.com/llms.txt
> Use this file to discover all available pages before exploring further.

# 初期化と電極健康状態 (eSOH)

> Set initial lithium distributions and use the ESOH algorithm to find stoichiometry windows mapping cell SOC to electrode lithiation.

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.

## 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

## 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.

### What ESOH Determines

| Parameter        | Meaning                            |
| ---------------- | ---------------------------------- |
| $\theta_{n,0}$   | Negative stoichiometry at 0% SOC   |
| $\theta_{n,100}$ | Negative stoichiometry at 100% SOC |
| $\theta_{p,0}$   | Positive stoichiometry at 0% SOC   |
| $\theta_{p,100}$ | Positive stoichiometry at 100% SOC |

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

## Calculation of Minimum and Maximum Stoichiometries

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

### 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]` ($V_0$)

### 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.

### 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:

$$
Q = Q^- \cdot (x_{100} - x_0)
$$

### 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.

## Initialization Options

<Tabs>
  <Tab title="Option 1: Direct Concentration">
    ### Directly Specify Initial Concentration

    The solver uses the value corresponding to `Initial concentration in <electrode> electrode [mol.m-3]` in the parameters object. To change the value, simply modify this parameter.
  </Tab>

  <Tab title="Option 2: Specify SOC">
    ### Specify Initial State of Charge

    Initial concentrations are defined using the specified initial SOC and the min/max stoichiometries.

    For the positive electrode:

    $$
    c_{s,t=0}^+ = c_{s,max}^+ \left( x_0 + \frac{SOC}{100}(x_{100} - x_0) \right)
    $$

    For the negative electrode:

    $$
    c_{s,t=0}^- = c_{s,max}^- \left( y_0 + \frac{SOC}{100}(y_0 - y_{100}) \right)
    $$
  </Tab>

  <Tab title="Option 3: Specify OCV">
    ### Specify Initial Open-Circuit Voltage

    One additional equation is added using an initial voltage $V_{init}$:

    $$
    U^+(x, T) - U^-(y, T) - V_{init} = 0
    $$

    where $x$ and $y$ are related similarly to the constraint equations.
  </Tab>
</Tabs>

## Practical guidance

<Tip>
  データに観測される初期電圧に基づいて初期濃度を設定することを一般に推奨します。通常は、フィットする最初のステップの直前のステップにおける最後の電圧を使用します。
</Tip>

<Note>
  このアプローチは、フィットする最初のステップの直前のステップが長い休止ステップである場合に適しています。
</Note>

## 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.