# Analytical solution for battery cell discharge characteristic

• Battery cell discharge characteristic
• Discharge curves generation process
• Challenges of discharge curves generation
• Analytical solution
• Scilab example

### Battery cell discharge characteristic

The battery cell discharge characteristics represents the behaviour of a battery cell voltage as it discharges at different rates. The rate is typically given in terms of “C-rate,” which is a measure of the current relative to the battery’s capacity. A 1C rate means that the discharge current will empty the entire battery in 1 hour. For example, if the battery cell has the nominal capacity of 2600 mAh, at 1C rate, the battery will be discharged in 1 hour with a 2.6 A current. At 2C rate the same battery cell will be discharged with a double current of 5.2 A but in half the time, 30 minutes. During the discharge process the battery cell is kept at constant current.

Image: LIR18650 Li-ion battery cell discharge characteristic
Credit: EEMB

The image above shows a graph with the vertical axis representing battery voltage and the horizontal axis representing discharge capacity as a percentage of total capacity, for the LIR18650 2600mAh battery cell [1].

• Each curve represents the voltage of the battery as it discharges from fully charged (0% discharged) to fully discharged (100% discharged) at different constant current rates.
• The coloured lines represent different C-rates, with red being the slowest discharge (0.2C) and purple being the fastest (3C).
• As the discharge rate increases, the battery voltage drops more quickly. This is due to the increased internal resistance and the reduced efficiency of the battery at higher discharge rates.
• The curves start at around 4.2 V, which is a common full charge voltage for Li-ion batteries.
• As the battery discharges, the voltage remains relatively stable for a while before beginning to fall off more rapidly as it approaches complete discharge.
• The “knee” of the curves, where the voltage drop becomes more pronounced, occurs later for lower discharge rates. At higher discharge rates, the knee occurs earlier, and the voltage falls off more sharply.

Image: LIR18650 Li-ion battery cell
Credit: EEMB

### Discharge curves generation process

The discharge curves are generated through a series of discharge tests on a battery cell or a battery pack. The process of generating the discharge curves is summarized in the steps below:

1. Setup: the battery starts at a full charge, typically standardized to a specific voltage such as 4.2V for Li-ion cells.
2. Discharge process: the battery is then discharged at a constant current rate until it reaches the cut-off voltage, which is the limit for which the battery is considered empty (3.0V).
3. Data collection: during the discharge process, the voltage of the battery is measured and recorded at regular intervals (for example 1 s).
4. Curve plotting: the recorded data points are then plotted on a graph to create the discharge curves. The horizontal axis is the percentage of the battery’s capacity that has been discharged, and the vertical axis is the voltage at that point.
5. Repetition for different C-rates: this process is repeated for various constant current rates to observe the effect of discharge rate on the voltage profile of the battery.

In more detail, the entire set of curves can be used to understand how a battery will perform under different loads. This information is crucial for the design of electronic devices and electric vehicles, where battery life and performance under different usage patterns are key considerations. The tests are typically performed under controlled temperature conditions to ensure consistency, as temperature can significantly affect battery performance.

Image: LIR18650 2600mAh battery cell datasheet
Credit: EEMB

### Challenges of discharge curves generation

Performing rate discharge characteristic tests for batteries involves a set of challenges that make them complex and demanding. Here’s a summary of why these tests can be difficult to perform:

1. Controlled environment: battery testing must be conducted in a temperature-controlled environment to ensure accurate and consistent results, as temperature variations can significantly impact battery performance.
2. Precision equipment: accurate testing requires precision equipment to maintain constant current levels and to accurately measure voltage drops over time. This equipment can be expensive and requires regular calibration.
3. Long duration: especially at lower C-rates, tests can be time-consuming, lasting several hours or even days to fully discharge a battery, which requires reliable equipment that can operate continuously without failing or causing errors.
4. Monitoring and safety: batteries need to be monitored closely for overheating, voltage drops, and other potential issues that could lead to battery damage or safety hazards. This often means that tests cannot be left unattended, especially in the final stages of discharge.
5. Data management: tests generate a large amount of data that must be collected, stored, and analysed. Managing this data requires robust software and can be complex, especially when dealing with multiple cells or battery configurations.
6. Battery variation: no two batteries are exactly alike, even from the same production batch. Accounting for these variations to ensure the reliability of the test results adds another layer of complexity.
7. Cycling and conditioning: before the tests, batteries often need to be cycled (charged and discharged multiple times) to condition them, which is a process that can take additional time and resources.
8. Multiple rates and repetition: to obtain a comprehensive profile, each battery must be tested at multiple discharge rates. This process needs to be repeated several times to ensure the accuracy and repeatability of the results.
9. Regulatory compliance: battery testing must often comply with various industry and safety standards, which can include specific test procedures and performance benchmarks.

Due to these complexities, rate discharge testing of batteries is typically carried out in specialized labs with trained personnel. The detailed insights gained from these tests are critical for understanding battery performance and are essential for the safe and efficient design of battery-powered devices and systems.

### Analytic solution

Having an analytical solution for the discharge curves is highly beneficial, especially for battery modelling and simulation activities, particularly considering the challenges of physical testing. While physical tests are essential for validating battery models and ensuring that batteries meet required standards, analytical solutions are indispensable tools for efficient design, optimization, and understanding of battery systems.

We are often in the situation in which we need to use the discharge curves of battery cells for testing Battery Management System, without having the physical battery or measurement data. In this context having a analytical solution to the discharge curves, would be highly beneficial for control engineers, modeling and simulation engineer or HiL engineers, which are dealing with simplified battery cell mathematical models.

Image: Battery cell general discharge characteristic

The image presents a typical discharge curve for a battery cell, illustrating how the voltage changes as the battery discharges from full to empty. The curve is divided into three distinct regions, each with a different characteristic shape in relation to the cell’s depth of discharge (DoD).

• Initial nonlinear region: this part of the curve is steep, showing a rapid drop in voltage as the cell starts to discharge from its maximum charged state.
• Linear region: the voltage decreases at a much slower, constant rate over a significant portion of the discharge cycle.
• Final nonlinear region: as the battery approaches full discharge, the voltage drops off sharply again.

The parameters describing the discharge curves are the following:

• Umax [V] – maximum voltage: the highest voltage level of the cell when it is fully charged, it is the starting point of the discharge curve
• Umin [V] – minimum voltage: the voltage level at which the cell is considered fully discharged; this voltage is also known as cut-off voltage, the point where the battery should no longer be discharged to prevent damage
• Unom [V] – nominal voltage: typical expected voltage under normal conditions, crucial factor for the design of electronic circuits that are powered by the battery
• Ua [V] – voltage at the end of the initial nonlinear region; not typically standard battery parameter, and represents the voltage at DoDa [%]
• Ub [V] – voltage at the start of the final nonlinear region; not typically standard battery parameter, and represents the voltage at DoDb [%]
• DoDa [%] – depth of discharge for the beginning of the linear region: the point at which the initial nonlinear voltage drop transitions to the linear phase
• DoDnom [%] – nominal depth of discharge: typically represents a standard or nominal value where the battery is often operated
• DODb [%] – depth of discharge for the end of the linear region: the point at which the linear voltage begins to transition to the final nonlinear voltage drop

The curve is typical for Li-ion cells, which have a relatively flat discharge profile, meaning the voltage stays consistent over a large range of the discharge cycle. This is beneficial for electronic devices because it provides a stable voltage supply until the battery is nearly depleted.

Problem definition: How can we generate the discharge curve for different battery cells, taking as inputs all the parameters above?

Solution: Develop a piece-wise analytical solution (equation) for the discharge curve, function of the above parameters. Using ChatGPT 4, this analytic solution was found:

Initial nonlinear region

For the initial nonlinear part (Equation 1), we can use an exponential decay function:

$U_{1}(DoD) = U_{max} – (U_{max}-U_{a}) \cdot \left ( 1-e^{-k_{1} \cdot DoD} \right) \tag{1}$

where k1 is a constant that determines the rate of decay.

Linear region

For the middle linear part (Equation 2), the equation of a line can be used:

$U_{2}(DoD) = U_{a} + \left ( \frac{U_{b} – U_{a}}{DoD_{b} – DoD_{a}} \right ) \cdot (DoD – DoD_{a}) \tag{2}$

Final nonlinear region

For the final nonlinear part (Equation 3), we can use a power law equation:

$U_{3}(DoD) = U_{min} + \left ( U_{b} – U_{min} \right ) \cdot \left ( 1- \left ( \frac{DoD – DoD_{b}}{DoD_{a}} \right )^{n} \right ) \tag{3}$

where n is a positive constant that determines the curvature of the function. If n > 1, the curve will be convex. The greater the value of n, the sharper the curvature.

Observation: in all of the equations above, DoD is the independent variable, the voltages in the 3 regions are calculated function of DoD and input parameters. Also, DoDnom was not included as a parameter but set as a constant at 50 % DoD.

### Scilab example

All the equations above were implemented in Scilab, for the following use cases:

Use case 1:

Umax = 4.2 V
Unom = 3.6 V
Umin = 3.0 V
Ua and Ub were set as 2% higher and respectively lower than Unom
DoDa = 20 %
DoDb = 80 %
k1 = 0.25
n = 2
DoD was generated as an array between 0 and 100 %

The generated discharge curve for this use case is depicted in the image below.

Image: Battery cell discharge characteristic – use case 1

Use case 2:

Umax = 4.1 V
Unom = 3.6 V
Umin = 2.5 V
Ua and Ub were set as 1% higher and respectively lower than Unom
DoDa = 10 %
DoDb = 90 %
k1 = 0.6
n = 3
DoD was generated as an array between 0 and 100 %

The generated discharge curve for this use case is depicted in the image below.

Image: Battery cell discharge characteristic – use case 2

### Conclusions

• ChatGPT 4 is a powerful tool to solve engineering problems quickly
• you can find the right solution only after several iterations and when you are able to critically asses the results
• not knowing what to expect will not lead to a suitable solution, you need to steer the AI solutions along the way