State of Charge

What state of charge (SoC) means and how to estimate it using coulomb counting and voltage-based methods.

State of Charge (SoC) is a dynamic measure of how much charge remains in a battery at any given moment, expressed as a percentage of its total capacity. It’s analogous to a fuel gauge—telling you how much capacity is available before the battery needs recharging.

SoC is an essential parameter for:

• Estimating runtime (e.g. smartphone battery life, electric vehicle range)
• Protecting batteries from overcharging or over-discharging

Definition

State of Charge is often expressed as a percentage, with 100% representing a fully charged battery and 0% representing an empty battery. Sometimes, it is expressed as a fraction of the battery’s total capacity, with 1 representing a fully charged battery and 0 representing an empty battery.

While this definition seems intuitive, accurately determining SoC can be complex.

Estimation Methods

Coulomb Counting

Coulomb counting (or current integration) is one of the most widely used methods for SoC estimation. This approach calculates SoC by measuring the current flowing into or out of the battery over time.

$\text{SoC}(t) = \text{SoC}_0 + \frac{1}{Q} \int_0^t I(\tau) \, d\tau$

where $Q$ is the total capacity and $I$ is the current.

While precise over short periods, Coulomb counting is prone to cumulative errors over time due to measurement inaccuracies and side reactions (e.g., self-discharge).

Open-Circuit Voltage Method

Open-circuit voltage (OCV) is another commonly used approach to estimating SoC. The OCV of a battery correlates directly with its SoC.

Many manufacturers provide OCV-SoC curves that map the relationship between voltage and SoC for a specific battery chemistry. By measuring the battery’s OCV after it has rested (with no current flowing), the SoC can be estimated accurately.

Voltage Under Load

In some cases, the actual voltage of a battery under load is used to estimate SoC. However, this requires caution:

Measurement	Relationship to SoC
OCV (at rest)	Well-defined relationship
Voltage under load	Affected by transient effects (Ohmic losses, overpotentials)

After a discharge, voltage often rises slightly as lithium concentrations equilibrate in the electrodes and electrolyte. If this operational voltage is used naively to estimate SoC, it may appear that SoC increases after discharge stops—an incorrect result since no additional charge has entered the battery.

Kalman Filtering

Kalman filtering is an advanced estimation technique that combines multiple information sources to produce more accurate SoC estimates than any single method alone. It’s widely used in modern battery management systems.

The key insight is that each estimation method has different strengths:

Method	Strength	Weakness
Coulomb counting	Good short-term tracking	Drifts over time
OCV lookup	Accurate absolute reference	Requires rest periods

A Kalman filter fuses these approaches by:

1. Predicting the next SoC using a model (e.g., Coulomb counting)
2. Updating the prediction when a measurement becomes available (e.g., voltage)
3. Weighting each source based on its uncertainty

The filter maintains an estimate of both the SoC and its uncertainty. When measurements are noisy or the model is uncertain, the filter automatically adjusts how much to trust each source.

The Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are variants that handle the nonlinear relationship between SoC and voltage, making them particularly well-suited for battery applications.

The mathematical formulation involves a state-space model:

$x_{k+1} = f(x_k, u_k) + w_k$ $y_k = h(x_k) + v_k$

where $x_k$ is the state (SoC), $u_k$ is the input (current), $y_k$ is the measurement (voltage), and $w_k$, $v_k$ represent process and measurement noise.

Kalman filters can estimate not just SoC but also other hidden states like internal resistance and capacity, enabling joint SoC-SoH estimation.

Impact of Battery Aging

As a battery ages, its total capacity decreases. This can cause a mismatch between estimated and actual SoC if capacity fade is not accounted for.

Advanced battery management systems (BMS) dynamically adjust the total capacity over time, improving SoC accuracy throughout the battery’s lifespan.

State of Power (SoP)

While SoC tells us how much charge remains, State of Power (SoP) tells us the maximum power the battery can deliver or accept at any given moment. SoP is calculated in real-time and depends on:

Factor	Effect on SoP
SoC	At very high SoC, charge power is limited to prevent overvoltage; at very low SoC, discharge power is limited
Temperature	Cold batteries have reduced power capability due to increased resistance
SoH	Aged batteries with higher resistance have lower power limits

For example, if a battery is at 95% SoC, the SoP for charging will be very low to prevent overvoltage, even though the battery isn’t technically “full.”

SoP is critical for applications like electric vehicles, where the BMS must communicate power limits to the motor controller in real-time.

State of Energy (SoE)

State of Energy (SoE) estimates the remaining usable energy in the battery, accounting for:

• The varying voltage during discharge
• Efficiency losses at different power levels
• Temperature effects on available energy

While SoC is based on charge (Ah), SoE is based on energy (Wh):

$\text{SoE} = \frac{\text{Remaining energy (Wh)}}{\text{Total energy capacity (Wh)}} \times 100\%$

SoE provides a more accurate estimate of remaining range or runtime than SoC alone, since:

• A battery at 50% SoC doesn’t necessarily have 50% of its energy remaining
• Higher discharge rates reduce the usable energy due to voltage drop and thermal losses

Related Concepts

State of Charge, together with SoP and SoE, forms the foundation of battery state estimation in modern battery management systems. Another key factor is internal resistance, which plays a critical role in efficiency, heat generation, and power output.