📐 Methodology

This document describes the analytical methods, estimators, and definitions used in this study.

Data Pipeline
Definitions
Drift Estimation
MA-60s Noise Reduction
Temperature-Voltage Regression
Architectural Immunity Assessment
References

1. Data Pipeline

The following diagram shows the data flow from sensor to analysis outputs:

flowchart TD
    subgraph Hardware
        A[🔋 Battery Bank<br>12V 500Ah LiFePO₄]
        B[📡 Shelly Plus Uni<br>Voltage Sensor]
        C[🌡️ Temperature Sensor]
    end

    subgraph Collection
        D[🏠 Home Assistant<br>State-change logging]
        E[(💾 Database<br>SQLite/InfluxDB)]
    end

    subgraph Export
        F[📤 CSV Export<br>Manual periodic]
    end

    subgraph Processing
        G[🐍 Python Analysis<br>lifepo4_analysis.py]
    end

    subgraph Outputs
        H[📊 Hourly Aggregates<br>combined_output.csv]
        I[📈 Figures<br>fig1–fig7]
        J[📉 Statistics<br>Drift, MA-60s, etc.]
        K[📄 Reports<br>Technical analysis]
    end

    A --> B
    A -.-> C
    B -->|~3s samples| D
    C -->|~3s samples| D
    D --> E
    E --> F
    F --> G
    G --> H
    G --> I
    G --> J
    J --> K

1.1 Sampling Characteristics

Data Type	Method	Cadence	Notes
High-frequency voltage	State-change logging	~3s median	Gaps when stable
Hourly voltage	Aggregated min/max	1 hour	Continuous
Temperature	State-change logging	~3s median	Started Dec 29, 2025

1.2 Known Regime Changes

Date	Time (Local)	Event	Impact
Dec 23, 2025	~15:40	Eco Mode enabled	Spread measurement change
Dec 29, 2025	—	Temperature sensor added	Enables temp correlation

2. Definitions

2.1 MA-60s (Moving Average, 60 Seconds)

A trailing, time-based 60-second rolling mean applied to high-frequency voltage data:

df['MA60'] = df['voltage'].rolling('60s', min_periods=1).mean()

[!IMPORTANT] This is a time-window average, not a fixed-sample-count average. It adapts to variable sampling cadence, ensuring consistent smoothing regardless of data density.

2.2 Mid-Voltage

The arithmetic mean of hourly minimum and maximum voltage:

Mid = (Min + Max) / 2

2.3 Spread

The difference between hourly maximum and minimum voltage:

Spread = Max - Min

[!NOTE] This is a single-channel bus measurement. It reflects measurement noise and ADC behavior, not per-cell or per-block voltage divergence. To claim cell-level divergence, per-unit sensing would be required.

2.4 Effective Draw vs. System Draw

Term	Definition
Effective draw	Parasitic current inferred from voltage drift during stasis (constant-load model)
System draw	Actual instantaneous current, which varies with telemetry bursts (Wi-Fi, polling cycles)

The effective draw (~13–20 mA) is lower than peak system draw because it averages over duty cycles.

3. Drift Estimation

3.1 Method: OLS on Daily Mean Mid-Voltage

Compute daily mean of hourly mid-voltage
Fit Ordinary Least Squares (OLS) regression: V = a + b·t
Report slope b in mV/day

from scipy import stats

# Compute daily means
daily_mid = hourly_df.groupby('date')['Mid'].mean()

# Days from start
days = (daily_mid.index - daily_mid.index[0]).days

# OLS regression
slope, intercept, r, p, se = stats.linregress(days, daily_mid)
drift_mv_per_day = slope * 1000  # Convert V to mV

3.2 Window Dependence

[!WARNING] Drift rates are window- and estimator-dependent on a non-linear relaxation curve. The voltage decay follows an exponential-like approach to equilibrium, not a linear decline.

Window	Period	Drift Rate	R²	Interpretation
Full stasis	Nov 22 → Jan 31	−0.665 mV/day	0.876	Long-term average
Last 30 days	Jan 2 → Jan 31	−0.165 mV/day	0.132	Near-equilibrium

The 75% rate reduction indicates the system is approaching a stable storage state.

3.3 Why Multiple Slopes?

Depending on:

Window start/end dates
Whether you use daily means, MA-60s means, or raw samples
OLS vs. robust regression

…you may compute drift rates anywhere from ~0.16 to ~0.30 mV/day for late January.

This is not a contradiction—it’s expected behavior for a flattening curve.

[!TIP] Best practice: Always state the window and estimator explicitly when reporting drift rates.

4. MA-60s Noise Reduction

4.1 Method

Apply time-based 60-second rolling mean to high-frequency voltage
Compute standard deviation of raw and smoothed series
Report reduction: (1 - σ_MA60 / σ_raw) × 100%

# Apply MA-60s (time-based)
hf_df['MA60'] = hf_df['voltage'].rolling('60s', min_periods=1).mean()

# Compute noise reduction
raw_std = hf_df['voltage'].std() * 1000      # mV
ma60_std = hf_df['MA60'].std() * 1000        # mV
reduction = (1 - ma60_std / raw_std) * 100   # %

4.2 Results

Scope	Raw σ	MA-60s σ	Reduction
Global (328k samples)	10.38 mV	5.98 mV	42.5%
Segment range	—	—	42–50%

4.3 Why Report a Band (42–50%)?

Noise reduction varies with:

Sampling regularity — Gaps reduce smoothing effectiveness
Interference environment — EMI varies over time
Time of day — Different noise profiles

Reporting as a band (42–50%) rather than a single number provides a more honest representation.

5. Temperature-Voltage Regression

5.1 Two-Factor Model

To isolate temperature effects from monotonic drift:

V = a + b₁·t + b₂·T + ε

Where:

t = days from start (controls for drift)
T = temperature (°F)
b₁ = residual drift rate after temperature control
b₂ = temperature coefficient

import statsmodels.api as sm

# Prepare features
X = merged_df[['days', 'temperature']]
X = sm.add_constant(X)
y = merged_df['mid_voltage']

# Fit model
model = sm.OLS(y, X).fit()
temp_coef = model.params['temperature'] * 1000  # mV/°F

5.2 Results

Coefficient	Value	SE	Interpretation
b₂ (temperature)	+1.01 mV/°F	0.27	System-level sensitivity
b₁ (residual drift)	−0.115 mV/day	0.026	After temperature control

5.3 Interpretation Caveats

[!NOTE] This coefficient is system-level, not pure LiFePO₄ OCV temperature behavior.

It includes:

Pack electrochemistry
ADC reference drift
Wiring/contact resistance changes
Enclosure thermal gradients

The temperature effect is second-order relative to monotonic drift for endurance inference, but matters for:

Seasonal extrapolation
Residual fitting quality
Comparing measurements across different ambient conditions

6. Architectural Immunity Assessment

6.1 What We Can Claim

From bus-level voltage monitoring, we observe:

✅ No growing instability signatures ✅ Trendless anomalies (no systematic pattern) ✅ Stable detrended variance over 94+ days ✅ Spread increase correlates with measurement regime, not electrochemistry

6.2 What We Cannot Claim (Without Additional Sensing)

❌ Per-cell SOC equality ❌ Per-block current sharing ❌ Individual cell degradation rates

6.3 Evidence Quality

The “architectural immunity” hypothesis is supported but not proven by this data.

Evidence Type	Available	Status
Bus voltage stability	Yes	✅ Supports hypothesis
No divergence trend	Yes	✅ Supports hypothesis
Per-cell voltage	No	❓ Would strengthen claim
Per-block current	No	❓ Would strengthen claim

Definitive confirmation would require per-unit voltage or current sensing.

7. References

Wang, Y., et al. (2023). “State of Charge Estimation of LiFePO₄ in Various Temperature Scenarios.” Batteries, 9(1), 43. DOI: 10.3390/batteries9010043
Espressif Developer Portal (2025). “Comparing ADC Performance of Espressif SoCs.” Link: developer.espressif.com
ESP-IDF Programming Guide. “ESP32-S2 ADC Calibration.” Link: docs.espressif.com
NIST/SEMATECH e-Handbook of Statistical Methods. “OLS Regression.” Link: itl.nist.gov

📐 Methodology

Architectural Immunity & Long-Term Storage Analysis: 94+ days of continuous monitoring of a DIY 12V 500Ah LiFePO₄ battery bank.

📐 Methodology

Contents

1. Data Pipeline

1.1 Sampling Characteristics

1.2 Known Regime Changes

2. Definitions

2.1 MA-60s (Moving Average, 60 Seconds)

2.2 Mid-Voltage

2.3 Spread

2.4 Effective Draw vs. System Draw

3. Drift Estimation

3.1 Method: OLS on Daily Mean Mid-Voltage

3.2 Window Dependence

3.3 Why Multiple Slopes?

4. MA-60s Noise Reduction

4.1 Method

4.2 Results

4.3 Why Report a Band (42–50%)?

5. Temperature-Voltage Regression

5.1 Two-Factor Model

5.2 Results

5.3 Interpretation Caveats

6. Architectural Immunity Assessment

6.1 What We Can Claim

6.2 What We Cannot Claim (Without Additional Sensing)

6.3 Evidence Quality

7. References

See Also