State of Charge
Introduction
Determining the state of charge of a battery is an often underestimated task. This document seeks to provide a starting to point to such a task from an electrical first principles perspective.
What is state of charge? You know how your phone gives you a percentage of how charged it is? That’s a state of charge. The big question this document will introduce us to is how does your phone calculate that number? And consequently how we at WARG can use a similar, simplified, approach to get a value with the same meaning.
Related Documents
LiPo Battery Safety and Procedure
focus on safety
LiPo Battery Voltage and State of Charge
LiPo specific and lacks modelling information
covers the battery specific terminology
Background
Physics
Before we get to the calculations lets cover some relevant physical, mostly electrical, quantities:
Energy
the capacity to perform work, in this case electrical work
The integration of power with respect to time gives us energy
measured in joules
this is equal to a watt-second
Power
rate of energy
For electrical circuits power equals voltage times current
measured in watts for electronics
horsepower is more common for mechanisms
1 W = 746 Hp
power stretches far beyond just electronics as a concept
power from electronic circuits is often dissipated as heat, noise, or mechanical power
Voltage
electric potential
Mathematically voltage is the negative gradient of electric potential energy per unit charge
measured in volts
Current
rate of flow of electric charge
measured in amperes
amps for short
Current is equivalent to the charge through a point per unit time
Charge
the unit of charge is the coulomb
one coulomb represents 6.242 x 10^18 electrons
property responsible for creating electric fields
Energy Density
given as watt hour per gram
this is used to describes how much chemical energy is stored within the weight of those chemicals (and their housing)
Batteries
Now for some battery specifications which are also introduced in LiPo Battery Safety and Procedure. These are also used as marketing terms and ratings which are often exaggerated by manufacturers. Metrics cited here however are all measurable in the lab and so for a given battery you can likely find ratings that users have measured. Please note batteries do degrade over use. Anyways here’s some metrics:
Discharge Curve:
A discharge curve describes voltage over time for a constant load current assuming you start at a fully charged battery
The charge curve is extremely similar just in reverse
The shape of the discharge curve varies significantly based on battery chemistry, load current, & temperature
Most load profiles are not a constant current and consequently this discharge curve will not be a valid model your batteries use case discharge
An example:
Capacity
this is generally given commercially as a unit of ampere-hour or milliampere-hour when we need a bigger number for marketing purposes (business people out here manipulating us engineers gg)
describes how much constant current the battery can provide for one hour while remaining within a given safe voltage range assuming you start at the full given safe voltage and end at the lowest safe voltage at a reasonable temperature
Note that this value does not describe energy capacity of the battery because power is a function of voltage AND current.
Chemistry
batteries are made of different chemistries.
batteries convert between chemical energy and electrical energy making them pretty swag
some chemistries are better than others for specific load profiles, there is not a best chemistry
some chemistries are safer than others
This affects the discharge curve and the safe voltage range of the battery significantly
C-Rating
The maximum current that the battery is rated to draw continuously without damaging itself is given by the C-rating value multiplied by the capacity value
This is semi-arbitrary since of course under high currents battery voltage sag can be significant and generally heavy loading accelerates battery degradation so where the safe region lies is arbitrarily defined by manufacturers based on their testing, but hey, you can do your own testing if you care!
Cell
an individual small unit of chemical things
cells are used to make battery packs
are small and designed to be arranged in custom configurations
has no protections and is not recommended for use by itself
Pack
a group of cells in an xSyP configuration where x and y are whole numbers if one is zero then the letter is often left out for simplicity
#S Cells in a pack
Series cells in a battery pack
#P Cells in a pack
parallel cells in a battery pack
Internal Resistance
Fully Discharged
A fully discharged battery cannot safely provide more power
safe in this case means causing permanent degradation of the batteries physical properties
consuming power beyond this safety threshold could lead to fire
Note that a battery that is fully discharged may (and in most cases will) have a non-zero output voltage and non-zero energy within
This is why a fully discharged battery still has enough power to catch itself on fire
For most chemistries the threshold for when a battery is fully discharged is defined by a threshold in voltage do not be fooled however, voltage is not analogous to state of charge
Fully Charged
A battery is fully charged when a battery cannot safely accept more power
safe in this case means causing permanent degradation of the batteries physical properties
consuming power beyond this safety threshold could lead to fire
For most chemistries the threshold for when a battery is fully charged is defined by a threshold in voltage do not be fooled however, voltage is not analogous to state of charge
State of Charge
the quantity of usable energy remaining within a battery generally normalized and given as a percentage for human readability
Note that the usable energy is always less than the same as total energy of battery
Electronics
Power Module
A hobbyist term for a device that measure’s battery voltage and current and reports the data to a flight controller
Load Profile
A given electrical load on a battery or any other power supply is profiled by its voltage and current requirements with respect to time
While most loads are considered resistive in the sense that they dissipate electrical power into another form of power
Resistance
Resistors dissipate electrical power as thermal power
V = IR
Combining with P = IV we see P = V*I^2 where this is the power dissipated as heat
All conductors have some parasitic resistance
Capacitance
Inductance
Series
Parallel
Calculating SoC
Because a state of charge calculation is highly specific to the types of electrical loading and charging profiles as well as the battery chemistry this problem does not have a unilateral solution for every system with a battery. There are two general approaches to solving problems like this for a given system: modelling and testing. The best solution is a healthy combination of as much of both as possible.
There are a few things we can measured with respect to time from the the system that are relevant to the batteries state of charge:
Voltage
The voltage of the entire battery pack can be measured easily and is on WARG
The voltage of individual pack cells can be measured but on WARG often isnt
Current
The current of the entire battery pack output can be measured easily
Temperature
The temperature of a battery pack can be measured in various positions but on WARG it often isnt
temperature affect the chemical performance of the battery
Power Train Commands
Power train commands can be used to predict currents when a current sensor is not present or has an insufficient sample rate
i.e. If the flight controller command the motors to a higher throttle we can assume this corresponds to more current draw at a point in time
System Response
The response of the system to a given control input can be used to estimate battery parameters
i.e. a fully charged battery will cause more acceleration of the drone for a given throttle input as compared to a more discharged battery due to changes in internal resistance and voltage throughout the discharge curve.
Wikipedia introduces a few primitive methods using this aggregated data well via https://en.wikipedia.org/wiki/State_of_charge and I will break it down into modelling methods:
Chemical State
A model of a chemical reaction can be performed and integrated into the model of the load profile to calculate a state of charge
Data from the pack can be compared with the chemical model to guess the state of charge
Voltage Position
output voltage of a battery is affected by the output current and temperature in addition to the state of charge making this a noisy measurement
the raw voltage value can be used to approximate state of charge via a theoretical discharge curve
Because voltage can sag significantly due to internal resistance or spike due to inductance, raw voltage should be used with caution, especially for “threshold” based algorithms
Current Integration
integrated current measurements with respect to time is called coulomb counting and can be used to guess discharge state
error accumulates over time and a current sensor with a low sample rate can cause severe inaccuracies with this method of estimation
current may also experience lots of transients during flight
hovering consumes very little current
jerking upwards quickly consumes lots of current
Voltage & Current Combined
Using both values an algorithm can be formed that is more accurate than both
This boils down to taking the raw voltage vale and scaling it based on load current then using a curve to guess state of charge. Can also mix in coulomb counting.
generally for human (non-computer) SoC calculations (done during flight) a pilot will look at the coulomb count, current, and voltage at a given time to make a judgement calculation
Filtering
Generally Kalman filters
can be applied to improve accuracy due to the noise of sensors
Now for some commercial implementations which may be tangible/easy to use. Note that Mavlink has message structures for this data https://mavlink.io/en/messages/common.html#BATTERY_STATUS :
A guide for PX4 with some general information as well: https://docs.px4.io/main/en/config/battery.html
Solid open source guide
Solid open source codebase
Implemented approximation methods:
Raw Voltage
Voltage with load compensation
This does not require current sensing, it simply assumes thr current is higher when the controller is commanding more motor output power from the ESCs.
voltage with current integration
We do not use PX4 at WARG at the time of writing so of course we cannot out of box use these implementations
Gas Gauge ICs
Some integrated circuits have their own built in algorithms
Ardupilot
At the time of writing F23 and to my best knowledge Ardupilot does not feature a state of charge calculator?
Though it may support raw coulomb counting approach? This can be setup with some battery parameters through their interface
Ardupilot does support numerous power modules for relevant data acquisition
You have to use external code if you want to use that raw data to calculate a state of charge with a more complex and accurate algorithm
The reason cited for doing this is that SoC is complex, requires multiple accurate sensors, and based on numerous factors so they would rather just present the simplest form of the data to users and allow them to do their own analysis. This approach is extremely sensical for such a project.
Ardupilot is used at WARG extensively.
Source: https://github.com/ArduPilot/ardupilot/issues/1434 & https://github.com/ArduPilot/ardupilot/issues/12897
My source for this is a few Google searches so please correct me if I’m wrong.
Please note that people write PhDs on SoC algorithms. This is not a simple topic. However, getting a valid approximation that’s good enough for WARG purposes is absolutely achievable!