Battery Modeling and the
Goldilocks Zone

Prashant Srivastava

Co- founder & CTO oorja

Battery Modeling and the Goldilocks Zone


With the global shift towards electrification as a primary energy storage solution, there is an escalating need to better understand the behaviour of batteries. As battery technologies advance, incorporating several rare earth elements and achieving higher energy densities, safety considerations become paramount. Hence, understanding the behaviour and performance of batteries over a diverse range of operating conditions is critical to the large-scale adoption of batteries. Predictive modelling that can accurately forecast safety, performance,  and the life of batteries is therefore indispensable. 


The conventional method for such analysis involves physics-based modeling, which, while powerful— presents considerable challenges. Batteries are intricately engineered devices composed of multiple layers of materials layered together. Their functionality depends on a complex interaction of diverse phenomena— electrochemistry, heat transfer, and fluid dynamics, to name a few— that govern ion and electron transport and heat generation within the battery.  


To effectively model these phenomena within the composite structure of an electrochemical cell, accurate design, transport, and degradation parameters are required to get physically relevant results from a physics-based model. A critical challenge during the digital twinning of batteries is identifying the ‘Goldilocks zone’ for parameters. For predictions to be reliable, the entire set of parameters must be precisely calibrated throughout the battery’s lifecycle. However, obtaining and validating the correct parameter set is far from straightforward. Inaccuracies in parameter estimation can lead to significant errors in model predictions, undermining the reliability of safety assessments and performance predictions. This complexity underscores the necessity for sophisticated tools capable of robust multi-parameter optimization, like our hybrid software solution, oorja, which extracts the necessary information from HPPC data to simulate real-world battery behavior under various operating conditions.


At oorja, we have pioneered a hybrid approach that combines data and physics-based models to perform multi-parameter optimization with a limited data set to provide insights into the transport and degradation parameters for Li-ion cells. The multi-parameter optimization technique to estimate parameters using HPPC data involves adjusting the model parameters until a satisfactory fit is achieved between the model and the experimental data.  The objective function for optimization is defined based on the sum of the squares of the error between the predicted and experimental values. The error is split into two parts.
  1. At the beginning of the step change in the pulse (estep).
  2. During the pulse (epulse).
During the step change in the pulse, the loss function is computed using the following formula:

where, ti corresponds to the beginning of the pulses where step change in the current occurs, and during the pulse, the loss function is given by:

where ti is the time during the pulse. Then, the total loss function is computed as the weighted sum of the estep and epulse,

where, wstep and wpulse are the respective weights.

Among all the electrochemical parameters associated with the Single Particle Model (SPM), a five-parameter set consisting of cell internal resistance, Li-ion diffusion coefficient of positive and negative electrodes, and reaction rate constant of positive and negative electrodes is found to be sensitive to operating conditions, specifically when temperature changes are significant. Optimization algorithms adjust these parameters to minimize the difference between the model predictions and the experimental data.

After optimizing the parameters set and obtaining the prediction fit, users can adjust these parameters to improve the prediction fit further based on their informed judgment.


In HPPC tests, the battery is subjected to a sequence of charge and discharge pulses of different magnitudes and durations at different states of charge (SoC). Its voltage response is observed and recorded. Using the technique mentioned above, the parameter optimization simulations were carried out on experimentally obtained HPPC data sets of cylindrical 21700 commercial cells (LGM50) with a rated capacity of 3Ah at 0, 25, and 40 degrees Celsius temperatures. The optimization was carried out for 40 minutes. The resulting values of parameters at various temperatures are shown in Table 1.

The prediction voltage responses obtained from the simulations and the percentage error plots are shown in Figure 1 ((a) – (c)) and ((d) – (f)), respectively, at different temperatures. 

The prediction responses obtained from simulations match quite well with the experimental data, as seen in Figure 1. The error is limited to 7% over most parts of the test, except in a few instances in between and towards the last part of the test, where the error spikes above 7%.  Table 1 clearly shows the sensitivity of the model parameters to the varying operating conditions, specifically temperature. With the increase in temperature, cell resistance decreases, whereas the reaction rate constants of positive and negative electrodes increases. Due to the strong interplay with the other factors, solid-state diffusion coefficients of positive and negative electrodes do not show any correlations with the temperature.


Internal cell resistance and reaction rate constants (of the anode and the cathode) are expected to decrease and increase with temperature, respectively. These appropriate trends are captured through our simulations. Generally, the diffusion coefficient for both positive and negative electrodes is expected to decrease with temperature, which is not seen in our simulations, thus requiring further investigation. Also, larger deviations in the fitted voltage are observed towards the lower SoC. This is due to the sensitivity of the open circuit voltage (OCV) at low SoC. During the regularization, a higher weightage can be used at low SoC to improve the quality of the fit. 


The present work demonstrates the successful identification of the Goldilocks Zone for electrochemical parameters using the HPPC data. The optimization parameters show very good physical trends with respect to temperature sensitivity. Also, the comparison of voltage responses of prediction and experiment data shows a good match. The accuracy of the parameters are also sensitive to the weights on the different parts of the HPPC pulses. Further improvements can be made by adjusting the weights of the individual error terms. In the present method, we regularize the parameters obtained at different SoCs to get a single set of parameters representing the cell for the SoC range. The weight of the regularization can be controlled to make the parameters SoC-dependent. This will improve the fit at different states of charge. Moreover, In the current exercise, the design parameters of the cell are taken from the literature, and the accuracy of fit can be further improved by using the design parameters obtained from the cell measurements.

To explore how you can use oorja, contact us at

Latest Blog Posts

Design safer and more reliable
batteries with oorja