State of Charge (SOC) Estimation

The State of Charge (SOC) is the "fuel gauge" of an EV. Unlike a petrol tank where you can physically measure the liquid level, measuring the charge remaining in a battery requires sophisticated algorithms.

Accurate SOC estimation is essential for:

• Range prediction: How far can you drive?
• Charging control: When to stop charging?
• Power management: How much power can you draw?
• Battery health: Avoiding over-discharge/overcharge

The Challenge

You cannot directly measure SOC. Unlike voltage or current, it's not a physical quantity you can sense with a meter. SOC must be estimated from measurable quantities:

• Terminal voltage
• Current (in/out)
• Temperature
• Internal impedance

Each method has tradeoffs between accuracy, computational cost, and robustness.

Method 1: Coulomb Counting

The simplest and most intuitive approach: track how much charge flows in and out.

$$SOC(t) = SOC(t_0) - \frac{1}{Q_n} \int_{t_0}^{t} i(t) \, dt$$

Where:

• $SOC(t_0)$ = initial SOC
• $Q_n$ = nominal capacity (Ah)
• $i(t)$ = current (positive = discharge)

In discrete form (implemented in BMS):

$$SOC_k = SOC_{k-1} - \frac{i_k \cdot \Delta t}{Q_n}$$

Implementation Example

class CoulombCounter:
    def __init__(self, capacity_ah, initial_soc=1.0):
        self.capacity = capacity_ah * 3600  # Convert to As
        self.soc = initial_soc

    def update(self, current_a, dt_s):
        # Positive current = discharge
        charge_delta = current_a * dt_s
        self.soc -= charge_delta / self.capacity
        self.soc = max(0, min(1, self.soc))
        return self.soc

The Drift Problem

Coulomb counting has a critical flaw: drift. Errors accumulate because:

• Current sensor offset: Even a 10 mA offset causes 36 mAh error per hour
• Sampling errors: Discrete integration misses some charge
• Unknown initial SOC: Error propagates forever
• Capacity variation: Actual capacity changes with temperature and age

After a few hours of driving, the estimate can be off by 5-10%.

Mitigation Strategies

• When vehicle is parked (zero current), measure OCV
• Look up SOC from OCV table
• Reset Coulomb counter

• Use shunt resistors with <0.1% accuracy
• Sample at high frequency (>100 Hz)
• Apply temperature compensation

Method 2: OCV-SOC Lookup

Use the relationship between Open Circuit Voltage and SOC (learned in Lesson 2).

• No drift (each measurement is independent)
• Simple lookup table

• Only works at rest (zero current)
• LFP has very flat curve (poor resolution)
• Temperature affects OCV

Method 3: Kalman Filter

The Kalman Filter is the gold standard for SOC estimation. It combines:

• Coulomb counting (predict)
• OCV measurement (correct)

The key insight: neither method is perfect, but together they're more accurate than either alone.

How Kalman Filter Works

Use the battery model to predict next SOC:

$$\hat{SOC}_{k|k-1} = \hat{SOC}_{k-1} - \frac{i_k \cdot \Delta t}{Q_n}$$

$$P_{k|k-1} = P_{k-1} + Q$$

Where:

• $\hat{SOC}$ = estimated SOC
• $P$ = estimation uncertainty (covariance)
• $Q$ = process noise (model uncertainty)

Correct prediction using voltage measurement:

$$y_k = V_{measured} - V_{predicted}$$

$$K_k = P_{k|k-1} H^T (H P_{k|k-1} H^T + R)^{-1}$$

$$\hat{SOC}_k = \hat{SOC}_{k|k-1} + K_k \cdot y_k$$

$$P_k = (1 - K_k H) P_{k|k-1}$$

Where:

• $y_k$ = innovation (measurement residual)
• $K_k$ = Kalman gain
• $R$ = measurement noise

Intuitive Understanding

Think of it as weighted averaging:

• High Kalman gain ($K \approx 1$): Trust measurement more
• Low Kalman gain ($K \approx 0$): Trust prediction more

The gain automatically adjusts based on:

• How uncertain is our prediction?
• How noisy is our measurement?

Extended Kalman Filter (EKF)

For batteries, the OCV-SOC relationship is nonlinear. The Extended Kalman Filter linearizes around the current estimate:

$$H_k = \frac{\partial V}{\partial SOC} \bigg|_{SOC=\hat{SOC}_k}$$

This requires computing the slope of the OCV curve at the current SOC estimate.

Battery Model for Kalman Filter

The Kalman filter needs a model to predict voltage from SOC:

Equivalent Circuit Model (ECM)

         R0          R1
   +----/\/\/\----+--/\/\/\--+
   |              |          |
  OCV            C1          V_terminal
   |              |          |
   +--------------+----------+

$$V_t = OCV(SOC) - i \cdot R_0 - V_{C1}$$

$$\frac{dV_{C1}}{dt} = \frac{i}{C_1} - \frac{V_{C1}}{R_1 C_1}$$

Where:

• $R_0$ = internal resistance (ohmic)
• $R_1, C_1$ = polarization (diffusion dynamics)

Parameter Identification

Model parameters vary with:

• SOC (higher resistance at low SOC)
• Temperature (higher resistance when cold)
• Age (resistance increases over life)

Parameters are typically identified using:

• Pulse testing (HPPC: Hybrid Pulse Power Characterization)
• EIS (Electrochemical Impedance Spectroscopy)
• Real-time recursive estimation

Practical BMS Implementation

A real BMS typically uses:

• Primary: Kalman filter with ECM model
• Fallback: Coulomb counting (if model fails)
• Calibration: OCV lookup when parked >2 hours
• Bounds check: Never exceed 0-100% range

class BMS_SOC_Estimator:
    def __init__(self):
        self.coulomb_counter = CoulombCounter(100)  # 100 Ah
        self.kalman = ExtendedKalmanFilter()
        self.rest_time = 0

    def update(self, voltage, current, dt):
        if abs(current) < 0.1:  # At rest
            self.rest_time += dt
            if self.rest_time > 3600:  # Parked >1 hour
                ocv_soc = self.ocv_to_soc(voltage)
                self.kalman.reset(ocv_soc)
        else:
            self.rest_time = 0
            self.kalman.update(voltage, current, dt)

        return self.kalman.get_soc()

Accuracy Comparison

Indian Market Considerations

• Reduced capacity at extremes
• Different OCV curves
• Model parameter variation

• First-order ECM + EKF is typical
• Lookup tables over complex functions
• Fixed-point math for speed

Key Takeaways

• Coulomb counting is simple but drifts due to accumulated errors
• OCV lookup works only at rest but provides no-drift reference
• Kalman filter combines both methods optimally
• Battery models (ECM) relate voltage to SOC for prediction
• Practical BMS uses multiple methods with fallbacks
• Accuracy of ±1-3% is achievable with modern algorithms

What's Next

In the next lesson, we'll learn about the Battery Management System (BMS) — the electronic system that implements SOC estimation, cell balancing, protection, and communication in a real EV.