Validating Lithium-Polymer Battery Discharge Models to Ensure Uav Flight Safety and Performance

To verify developed models of battery operation – such as those presented in [1, 2] – it is essential to verify their accuracy against real-world conditions. Achieving this verification requires systematically evaluating the quality of experimental data obtained from testing actual LPABs under various temperatures and operational scenarios. By comparing the measured discharge characteristics against the outcomes predicted by these previously developed models, we can identify permissible deviations and refine regression-based representations of battery behavior. This verification process is guided by ISO 5725-1:2023 standards [3], which define accuracy as encompassing both trueness and precision (in terms of repeatability and reproducibility).

Table 1 summarizes the measured discharge characteristics of a lithium-polymer battery across a wide range of external temperatures. These results serve as key reference points for validating the accuracy and consistency of the developed battery discharge models under diverse environmental conditions.

In the context of ISO 5725-1:2023, the term accuracy encompasses both trueness and precision. Trueness refers to the closeness of agreement between the average value obtained from a large series of test results and the accepted reference value, indicating the absence of systematic error. Precision denotes the closeness of agreement between independent test results under stipulated conditions, reflecting the extent of random errors. Precision is further divided into repeatability (conditions where independent test results are obtained with the same method on identical test items in the same laboratory by the same operator using the same equipment within short intervals of time) and reproducibility (conditions where test results are obtained with the same method on identical test items in different laboratories with different operators using different equipment).

In this study, repeatability was assessed by conducting multiple charge/discharge cycles on a single LPAB sample, allowing for the evaluation of variability in discharge characteristics and surface temperature within the same laboratory setting. Reproducibility, in turn, was evaluated by testing multiple LPAB samples with identical passport data across different laboratories, accounting for variations due to manufacturing processes and different testing environments.

By adhering to the definitions and guidelines outlined in ISO 5725-1:2023, this study ensures a rigorous assessment of the trueness and precision of the experimental data, thereby validating the reliability of the developed battery models.

To investigate the discharge characteristics and surface temperature behavior of lithium-polymer batteries (LPABs) under controlled laboratory conditions, experimental studies were conducted on a single battery sample at a fixed ambient temperature of +25°C. Each sample was tested 10 times using a mock-up scheme in a climatic chamber. The collected data provide insight into discharge behavior and thermal dynamics during operation (Tables 2, 3).

The experimental data are visually represented in Figures 1 and 2.

Fig. 1.

Family of AB discharge characteristics at +25°C.

Fig. 2.

Algorithm for analyzing results under repeatability conditions.

The analysis of test results under repeatability conditions is a critical step in ensuring the accuracy and reliability of experimental data. The process adheres to ISO 5725-1:2002 guidelines and includes the following steps, as illustrated in Figure 2.

According to ISO 3534-1, reproducibility and reproducibility conditions are formulated as follows:

Reproducibility – the precision in terms of reproducibility.

In this study, AB discharge characteristics are measured in certain climatic conditions (at a given temperature). Given the dynamic nature of the discharge process, with voltage measurement at fixed time intervals, it is not feasible to evaluate for a static point over a certain period of time. Hence, the reproducibility conditions for this study will be formulated as follows:

Reproducibility conditions – the conditions under which independent results of multiple repetitions in the discharge process are obtained using the same method, on several research objects identical according to their passport data, in the same laboratory, by the same operator, using the same equipment, at the same time.

To verify the developed battery models under reproducibility conditions, three sample lithium-polymer ABs with the same passport data were used and the reproducibility conditions established above were observed. The voltmeter readings were taken at intervals of 1 minute until the engine stopped, 10 times in a row. As an example, a comparison of the results for two sample LPAB are given (Table 3, Table 7).

To assess the trueness of the results, it is necessary to estimate the difference between the average values of each group of measurements (bit characteristics for one sample) and the ratio of the difference between the average scores of sample MSDs. For this, we use data arrays (Table 2, Table 6).

Next, to assess the reproducibility, we take the data of the second column of Table 2 and Table 6 (the 0 min. column) and perform a check for the presence of a systematically component error. 1)

The average value xj¯ by row in each table is calculated according to the formula: 8xj¯=∑i=1nxijn, where x_ij represents the voltage values from the table, n is the number of measurements, and j is the sample AB number.

If the obtained criterion value F_emp is less than the critical value F_kr for a certain level of significance and the corresponding number of degrees of freedom, the variances can be considered equal to each other, and the considered aggregates of measurement results can be combined into one general aggregate. In this way, the correctness of the measurement method will be proven in terms of reproducibility.

According to the results of the experiment, we obttain: Femp=σjmaxσjmin.

The critical value of the criterion F_kr is chosen from the table, taking into account that the number of degrees of freedom for each row will be n – 1 = 10 – 1 = 9, and the chosen significance level of 0.05.Fkr=3.18Femp=2.83⇒Fkr>Femp

Therefore, we can conclude that the correctness of the measurement method in terms of reproducibility, based on the considered experimental data, has been proven.

Having carried out the calculations described above for each column of Table 2 and Table 4 and for each set temperature value, we can conclude that the measurement method is correct in terms of reproducibility throughout the entire experiment.

Since the precision of the obtained results is evaluated for reproducibility at each point in time, here we begin with a series of data (voltage measurements at the same point in time). The repeatability limit is used as a measure for evaluating R – the difference between two measurement results. Since we have several series of data (measurement results of two samples), which according to the calculations of the previous points can be combined into one general population, the difference between the data will be defined as the difference between each of the results and the average value for the general population. The standard deviation will be σ · ✓2.

To consider the difference between two random variables, the critical range coefficient f is again used, which depends on the probability level and the distribution law of the random variable, i.e f · σ · ✓2. For repeatability and reproducibility limits, the confidence probability is again 95% and the underlying distribution is considered approximately normal. With this in mind, f = 1.96, and for statistical evaluation we will again adopt f · σ · ✓2 = 1.96 σ · ✓2 = 2.8 · σ.

Thus, the criterion for assessing the precision of test results under reproducibility conditions is defined as the ratio of the difference between the measurement results and the general average b_ij in all possible combinations with a repeatability limit R = 2.8 · σ_R. If b_ij ≤ R, then the results are considered to exhibit precision under the reproducibility conditions, otherwise they are deemed to lack precision under the reproducibility conditions with the chosen confidence probability.

Combining the obtained results into a set and calculating the average characteristic and its uncertainty

In the previous section, the criteria for the possibility of combining the obtained results into one set were given. If there are no systematic errors in several series, that is, the difference between the arithmetic mean and the difference in MSD estimates is permissible, then the series of measurement results can be considered series with the same dispersion [8]. The general average is determined by formula (8). The general scattering of results, i.e. the general variance S² is determined by the formula [9, 10]: 17S2=(n−1)(n·k−1)·∑j=1kSj2+nn·k−1·∑j=1k(xj¯−x¯)2, where S²_j is the variance of the sample of results for each sample.

The dispersion limit of the discharge characteristic results is determined by the formula: C=2·S2, and the scattering limits x¯±C $$C = 2\cdot\sqrt {{S^2}} ,\,{\rm{and}}\,{\rm{the}}\,{\rm{scattering}}\,{\rm{limits}}\,\bar x \pm C$$

Thus, after performing the calculations for all values by time (Table 2, Table 6), we will obtain the average value, the variance of the total sample and the general dispersion of the results of the discharge characteristics for a temperature of +25°C (Table 8).

These results are graphically represented in Fig. 4.

Fig. 4.

Plot of overall average value and overall dispersion of discharge characteristic measurements at a temperature of +25°C.

Having carried out similar calculations for the characteristics of the surface temperature of LPAB [11], we will obtain the average value, the variance of the total sample and the general dispersion of the results of the characteristics of the surface temperature of LPAB for an ambient temperature of +25°C (Table 9).

Table 9.

Overall average value and the overall dispersion of LPAB surface temperature measurements at an ambient temperature of +25°C.

	0 m.	1 m.	2 m.	3 m.	4 m.	5 m.	6 m.	7 m.	8 m.
x¯	22.595	25.595	31.085	33.365	35.520	36.745	36.010	35.200	34.670
S2	0.00155	0.00682	0.47818	0.00555	0.00800	0.00997	0.10832	0.02105	0.00642
C	0.079	0.165	1.383	0.149	0.179	0.199	0.658	0.290	0.160

These results are graphically represented in Fig. 5.

Fig. 5.

Plot of overall average value and the overall dispersion of LPAB temperature characteristic measurements at an ambient temperature of +25°C.

After performing the above calculations for other temperature values, we obtain estimated discharge characteristics for the entire specified temperature range [12], graphically illustrated in Fig. 6.

Fig. 6.

Plot of overall average value of LPAB discharge characteristic measurements for the temperature range from -20°C to +50°C.

Estimating the characteristics of the surface temperature according to the given algorithm makes it possible to estimate the LPAB surface temperature characteristics for the entire specified temperature range. The results of the calculations are shown in Fig. 7.

Fig. 7.

Plot of the overall average value of LPAB surface temperature characteristic measurements of LPAB for the range of ambient temperatures from -20°C to +50°C.

The obtained results indicate that when a specific battery is tested under varying environmental conditions, using measuring devices of the same accuracy class and identical load and temperature parameters, the results remain consistent in terms of accuracy. The reproducibility evaluation of the experimental data confirms that the developed mathematical models of LPAB discharge characteristics are reliable for any battery of this type. This enables the broad application of the proposed method for monitoring the state of the rechargeable autonomous energy source (RAES), tailored to the conditions and operational features of UAVs These findings validate the mathematical and simulation models for lithium-polymer battery discharge, ensuring their practical usability in UAV operations.