# HPPC Test Methodology Using LFP Battery Cell Identification Tests as an Example

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## Abstract

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## 1. Introduction

- An optimization-based battery cell time constant identification algorithm is implemented in software written by the authors.
- An HPPC-based method for OCV vs. SOC characteristic determination is established.
- Other contributions of the article are as follows:
- This paper gives the values of all parameters necessary to build a fully parameterized mathematical model of the cell.
- The paper explains the HPPC test development methodology step by step. In the literature, usually only the results of HPPC are given, but the process of obtaining them is not described. This paper fills that gap.
- The paper discusses potential flaws in the HPPC test results. Not every HPPC pulse recorded during measurements is suitable for further analysis and must be omitted. In the literature, this problem is hardly commented on. This paper fills that gap.
- The paper applies edge detection techniques in the analysis of the HPPC test results.
- The paper remarks on battery cell true capacity experimental estimation.

## 2. Materials and Methods

_{4}) battery cell with the rated parameters given in Table 1. The following laboratory tests were carried out: discharge characteristics to estimate the actual cell capacity, HPPC tests to identify the equivalent circuit parameters, and a charge-depleting cycle (CDC) test [64] to verify the identified mathematical model.

## 3. Results

#### 3.1. Battery Cell Equivalent Circuit

_{1}= R

_{1}C

_{1}, τ

_{2}= R

_{2}C

_{2},

_{OC}in Figure 3) depend on the SOC of the cell [31], which is estimated on the basis of the cell current [22,37,51,57,67,68,70]:

_{0}is the initial SOC of the cell, and Q is the cell capacity. Note that the actual cell capacity depends on many factors, such as temperature and SOH of the cell, and is usually different from the rated one, Q

_{n}. Here, it was estimated based on the measurement results as described in Section 3.2.

#### 3.2. Capacity and State of Charge Estimation

_{CC}(Figure 4).

_{CV}. The results obtained during the tests are summarized in Table 2, and the recorded transients are shown in Figure 5.

_{CC}and Q

_{CV}values changes. The higher the discharge current, the lower the Q

_{CC}and the higher the Q

_{CV}.

_{n}, but the differences between them were significant. Therefore, the question as to which of them should be treated as the final one (the total capacity of the cell) should be asked, which will be used in the created mathematical model. In order to find the answer, a number of simulations were carried out for all the values obtained and the results of the selected values are presented in Section 3.4. The best result was obtained for the value of Q = 45.7 Ah, calculated as the average of the Q

_{CC}values for all four current values (average of the values from the third column of Table 2).

#### 3.3. HPPC Tests

_{n}was used for large SOC values, and then the interval was increased to 0.1 Q

_{n}to return to 0.05 Q

_{n}for small SOC values.

_{n}), which resulted from the fact that the discharge pulses in the profile were cut off by the CC/CV mechanism due to reaching the minimum voltage. All the performed tests are summarized in Table 3, where ΔQ is the charge taken from the cell during the whole HPPC test (including final discharge by 0.05/0.1 Q

_{n}). For the last two tests (17 and 18), this value drops significantly, which means that the cell is already discharged. In Table 3, Q is the total value of the charge taken from the cell at the end of the given test, taking into account the charge taken in the preceding tests.

#### 3.3.1. Filtering and Slope Detection

_{i}is the i-th sample of the current waveform, r

_{i}is the i-th sample of the exponential moving average, α is the weight coefficient, and Δ

_{i}is the i-th sample of the waveform difference. The principle of operation of the method is shown in Figure 7. The difference Δ between two waveforms averaged with different weight values α (α

_{slow}= 0.02, α

_{fast}= 0.1) contains peaks at moments when there is a rapid change in the trend of the source waveform.

_{filtered i}filtered voltage sample. The filtration consists of calculating the average for N samples preceding and following the sample with the number i.

#### 3.3.2. OCV vs. SOC Characteristic

_{OC}(i.e., the voltage source in the Thevenin equivalent circuit), is identified by measurement. The averaged charging and discharging characteristics may be used here [44,48,49]. However, this method has some disadvantages. The measured cell voltage contains not only the OCV but also the voltage drop at the impedance, which also depends on the SOC. Moreover, the measured charge and discharge capacities differ due to power losses. This makes it difficult to correlate them before the averaging.

#### 3.3.3. Impulse Evaluation and Selection

_{OC}of the cell, which translates into the shape of the recorded waveform U (Figure 10). The shape of the waveform ceases to depend only on the time constants τ

_{1}and τ

_{2}, which is a necessary assumption to make the identification of these constants possible. In the extreme case, the recorded waveform bends in a direction opposite (Figure 11) so that it results from (4), assuming that the time constants τ

_{1}and τ

_{2}are positive.

#### 3.3.4. Impulse Waveform Approximation

_{1}and τ

_{2}and the resistances R

_{0}, R

_{1}, and R

_{2}. Then, on the basis of Formula (1), capacities C

_{1}and C

_{2}were calculated. The approximation was carried out using the PSO optimization method. At this stage of the research, a configuration of the PSO algorithm was found that guaranteed high repeatability of the obtained results. The fully informed particle swarm cognition method and the 8th order ring lattice swarm topology were used. The cognition factor was 4.1, the swarm consisted of 64 particles, and the number of iterations of the algorithm was set to 180.

#### 3.3.5. R and C vs. SOC Characteristics Approximation

_{1}and τ

_{2}are the product of the approximating functions, respectively R

_{1}and C

_{1}for τ

_{1}, R

_{2}and C

_{2}for τ

_{2}, according to (1).

#### 3.4. Model Verification

_{0}, R

_{1}, R

_{2}, C

_{1}, and C

_{2}(Section 3.3.5) characteristics fully describe the Thevenin equivalent circuit shown in Figure 3. This circuit, together with functions describing its parameters, was implemented in the MATLAB/Simulink environment by creating a simulation model of the cell, in a similar way as in [31]. The last parameter describing the model is the charge value Q corresponding to SOC = 1. Due to the problems with determining the actual capacity of the cell described in Section 3.2, this value was found by performing a series of simulations of the cell operating in model conditions and comparing their results with the transients recorded in the laboratory.

_{n}, so it was repeated over 30 times until the cell was fully discharged.

## 4. Discussion

_{n}. It should be noted that according to the discharge characteristics provided in the cell data sheet by the cell manufacturer, the cell capacity at normal temperature (i.e., the temperature at which the tests described herein were performed) varied with the discharge current from about 1.04 Q

_{n}(3 C) to 1.15 Q

_{n}(0.5 C).

_{CC}values in Table 2. Particularly significant here was the value of the discharge current. This is why the capacity determined from the HPPC tests was the largest. This was because, during these tests, the charge was taken from the cell in small increments separated by long relaxation times. Thus, the cell had a lot of time to regenerate and rebuild the voltage lowered by the discharge.

_{0}values are arranged in a narrow, regular band, which proves good quality of identification. In the case of the time constant τ

_{1}, the obtained band is much wider and the dispersion of values is greater, but some regularity is still visible. In the case of the time constant τ

_{2}, the dispersion of the results is very large, and their arrangement on the graph does not show any regularity. Note that the values of the time constant τ

_{2}in Figure 12 changed in the interval from 30 s to 120 s, i.e., by 400%. Probably, in individual cases, values greater than 120 s would have been obtained, if not for the fact that such a value was set as a limitation of the search space in the applied PSO algorithm. It should be noted that, as stated in Section 3.3, to ensure good quality identification of the exponential waveform time constants, the length of its recorded fragment should be several times greater than the length of its time constants. However, with the applied HPPC pulse length equal to 60 s, more than half of the identified τ

_{2}values were greater, even up to two times. Increasing the duration of the HPPC pulses would be undesirable, because it would cause changes too large in the SOC during the pulse duration. In the case of the tested LFP type cell, resignation from determining two time constants in favor of only one should be considered, as well as shortening the duration of the HPPC pulse. Let us also pay attention to the obtained resistance and capacitance values, given in Figure 12. The resistances are of the order of mΩ, which results in high short-circuit currents of lithium-ion cells. Capacitances are of the order of kF. Similar values were obtained, for example, in [32].

_{2}graph in Figure 12. This problem, however, requires confirmation and further analysis.

_{OC}(OCV) voltages.

## 5. Conclusions

- Among the various cell capacity values obtained as measurements, the best performance of the mathematical model was obtained for the averaged charge taken from the cell during discharge in the CC mode for different current values. Therefore, this method is recommended for determining the actual capacity of the cell.
- The OCV characteristics of the LFP cell are best approximated by the LEE function.
- Identification of the second time constant of the LFP cell is difficult, because of its large value, greater than a typical HPPC impulse duration.
- Suggestions for further research:
- It would be advisable to develop methods for automatic quality evaluation of HPPC impulses, based on the criteria given in Section 3.3.3, which would enable full automation of the HPPC test results processing.
- A method should be developed to detect the occurrence of distortion of HPPC pulses in cases where the distortion is small and does not significantly change the shape of the voltage waveform yet, but already overestimates the obtained values of time constants.
- Simulation model accuracy may be improved by better OCV characteristic approximation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 8.**Exemplary HPPC impulse before and after data filtering. Near t = 10 s, a false peak generated by the control system of the active power supply is visible.

**Figure 12.**Thevenin equivalent circuit R and C parameter characteristics approximated with 3rd order polynomial.

**Figure 13.**Comparison of simulation and measurement CDC test results. Simulations performed for various values of cell capacity.

Parameter | Value |
---|---|

Capacity Q_{n} | 40 Ah |

Energy density | 82.5 Wh/kg |

Voltage (min./nominal/max.) | 2.5/3.3/4.0 |

Current (typical/max. discharge) | 20 A (0.5C ^{1})/400 A (10C ^{1}) |

^{1}Battery cell C-rating, based on nominal capacity: 1C = 40 A.

Relative Discharge Current | Total Discharge Q [Ah] | Discharge in CC Mode Q _{CC} [Ah] | Discharge in CV Mode Q _{CV} [Ah] |
---|---|---|---|

0.5C | 47.71 | 46.30 | 1.407 |

1C | 47.71 | 45.78 | 1.934 |

2C | 47.70 | 45.21 | 2.495 |

3C | 47.62 | 45.41 | 2.212 |

HPPC Test No. | Impulse No., Type and Relative Current Value | ΔQ [Ah] | ΔQ/Q_{n}[%] | Q [Ah] | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 0.5 C | 2 0.5 C | 3 1 C | 4 1 C | 5 2 C | 6 2 C | 7 3 C | 8 3 C | ||||

1 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 2.41 | 6.03 | 2.41 |

2 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 2.21 | 5.53 | 4.63 |

3 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 4.21 | 10.53 | 8.84 |

4 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 4.21 | 10.53 | 13.05 |

5 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 4.22 | 10.54 | 17.27 |

6 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 4.23 | 10.57 | 21.50 |

7 | (−) | (+) | (−) | (+) | (−) | (+) | (−) | (+) | 4.22 | 10.56 | 25.72 |

8 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 4.21 | 10.54 | 29.93 |

9 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 4.22 | 10.55 | 34.16 |

10 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.16 | 5.40 | 36.31 |

11 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.18 | 5.45 | 38.49 |

12 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.20 | 5.51 | 40.70 |

13 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.21 | 5.52 | 42.90 |

14 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.22 | 5.54 | 45.12 |

15 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.20 | 5.49 | 47.32 |

16 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 2.22 | 5.55 | 49.54 |

17 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 0.92 | 2.31 | 50.46 |

18 | (+) | (−) | (+) | (−) | (+) | (−) | (+) | (−) | 0.25 | 0.62 | 50.71 |

a | b | c | d | |
---|---|---|---|---|

R_{0} | 3.551 × 10^{−3} | −6.172 × 10^{−3} | 8.993 × 10^{−3} | −4.267 × 10^{−3} |

R_{1} | 9.601 × 10^{−4} | −1.154 × 10^{−3} | 1.611 × 10^{−3} | −5.716 × 10^{−4} |

R_{2} | 6.169 × 10^{−3} | −2.678 × 10^{−2} | 4.690 × 10^{−2} | −2.485 × 10^{−2} |

C_{1} | 5549 | −1.359 × 10^{4} | 5.058 × 10^{4} | −3.397 × 10^{4} |

C_{2} | 1.712 × 10^{4} | 8.510 × 10^{4} | −2.850 × 10^{4} | −4.243 × 10^{4} |

Voltage RMS Error | Cell Capacity Q [Ah] | Comment |
---|---|---|

0.0432 | 45.7 | Average for discharge characteristics, CC mode only |

0.0487 | 47.7 | Average for discharge characteristics, CC + CV |

0.120 | 50.7 | HPPC tests total discharge |

0.167 | 40.0 | Q_{n}—nominal cell capacity |

Cell Capacity Q [Ah] | Average Error |δU| [%] | Average Error for t from 5 min to 180 min |δU| [%] | Peak Error |δU| [%] | Peak Error for t from 5 min to 180 min |δU| [%] |
---|---|---|---|---|

45.7 | 0.977 | 0.751 | 14.9 | 9.62 |

47.7 | 1.07 | 0.805 | 14.6 | 9.83 |

50.7 | 2.44 | 0.873 | 22.9 | 10.1 |

40 | 2.73 | 0.579 | 20.4 | 9.04 |

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**MDPI and ACS Style**

Białoń, T.; Niestrój, R.; Skarka, W.; Korski, W.
HPPC Test Methodology Using LFP Battery Cell Identification Tests as an Example. *Energies* **2023**, *16*, 6239.
https://doi.org/10.3390/en16176239

**AMA Style**

Białoń T, Niestrój R, Skarka W, Korski W.
HPPC Test Methodology Using LFP Battery Cell Identification Tests as an Example. *Energies*. 2023; 16(17):6239.
https://doi.org/10.3390/en16176239

**Chicago/Turabian Style**

Białoń, Tadeusz, Roman Niestrój, Wojciech Skarka, and Wojciech Korski.
2023. "HPPC Test Methodology Using LFP Battery Cell Identification Tests as an Example" *Energies* 16, no. 17: 6239.
https://doi.org/10.3390/en16176239