Research on the State of Charge of Lithium-Ion Battery Based on the Fractional Order Model
Round 1
Reviewer 1 Report
1. The article deals with applying fractional-order calculus to estimate the state of charge for lithium batteries more precisely. As a result, the stable operation of a battery pack is ensured. The introduction provides sufficient background. However, I recommend authors to pay more attention to discuss the following publication in the literature review: “Kanagasabai, L. FCC algorithm for power loss diminution. Journal of Engineering Sciences 2021, 8(1), E29-E38, https://doi.org/10.21272/jes.2021.8(1).e5” that provides a general approach to solving power problems using different optimization algorithms.
2. Additionally, in my opinion, if the authors use the fraction-order formulation of the proposed capacitance model, it is mandatory to explain to the readers about changing dimensions for the capacity parameter “C”.
3. What about misprints, in Line 149 (“ü” has been lost), “Grnwald-Letnikov (GL) definition[19]” should be fixed as “Grünwald-Letnikov (GL) definition [24]” with the following relevant source for it: “Pavlenko, I.; Ochowiak, M.; Agarwal, P.; Olszewski, R.; Michalek, B.; Krupinska, A. Improvement of mathematical model for sedimentation process. Energies 2021, 14(15), 4561, https://doi.org/10.3390/en14154561”.
4. Finally, Figure 6 presents an interpolating curve for experimentally obtained dependence between S0C and OCV. However, it is better to use approximating curve considering peculiarities of the physical process. Particularly, the following dependence can be used: y(x) = a*(1 – exp(b*x)) + c*x. Parameters a, b, and c can be determined to ensure minimal least-square error between experimental points and the theoretical curve. However, it is rather a recommendation than a requirement because, on the whole, the presented research looks like a complete article.
5. Finally, it seems the References chapter is presented not according to the MDPI template.
Overall, the article can be recommended for publication after considering the recommendations, mainly 1, 3, and 5.
Author Response
Response to Reviewer 1 Comments
Thank you very much for the comments and suggestions of the experts, which enabled me to have a further understanding of my research content.
Point 1: The article deals with applying fractional-order calculus to estimate the state of charge for lithium batteries more precisely. As a result, the stable operation of a battery pack is ensured. The introduction provides sufficient background. However, I recommend authors to pay more attention to discuss the following publication in the literature review: “Kanagasabai, L. FCC algorithm for power loss diminution. Journal of Engineering Sciences 2021, 8(1), E29-E38, https://doi.org/10.21272/jes.2021.8(1).e5” that provides a general approach to solving power problems using different optimization algorithms.
Response 1: Thank you very much for the paper you mentioned, which gave me a more comprehensive understanding of lithium battery identification algorithm. I have revised the relevant content of the paper, which is in line 201 to line 204.
The modifications are as follows：In addition to the above two methods, genetic algorithm ，ant colony algorithm and so on can also identify battery parameters. The main challenge is balancing the exploration and exploitation in the algorithm process .
Point 2: Additionally, in my opinion, if the authors use the fraction-order formulation of the proposed capacitance model, it is mandatory to explain to the readers about changing dimensions for the capacity parameter “C”.
Response 2: Thank you very much for your proposal. For the change of capacitance value, I added the following explanation: It can be seen from the identification results that the parameters of the fractional order equivalent circuit model are significantly different from those of the integer order equivalent circuit model, especially the "capacitance value". The main reason for this is that the integer order equivalent circuit model is too simple, which only reflects the ohmic polarization, concentration polarization and electrochemical polarization in lithium battery, while the fractional order model not only reflects the three chemical reactions, but also reflects the double electric layer effect and the transfer reaction between electrolyte and solid phase interface. Lines 512 through 519 in the text.
Point 3: What about misprints, in Line 149 (“ü” has been lost), “Grnwald-Letnikov (GL) definition[19]” should be fixed as “Grünwald-Letnikov (GL) definition [24]” with the following relevant source for it: “Pavlenko, I.; Ochowiak, M.; Agarwal, P.; Olszewski, R.; Michalek, B.; Krupinska, A. Improvement of mathematical model for sedimentation process. Energies 2021, 14(15), 4561, https://doi.org/10.3390/en14154561”.
Response 3: Thank you very much for your correction. I have modified the relevant content in the article, which is in line 146.
Point 4: Finally, Figure 6 presents an interpolating curve for experimentally obtained dependence between S0C and OCV. However, it is better to use approximating curve considering peculiarities of the physical process. Particularly, the following dependence can be used: y(x) = a*(1 – exp(b*x)) + c*x. Parameters a, b, and c can be determined to ensure minimal least-square error between experimental points and the theoretical curve. However, it is rather a recommendation than a requirement because, on the whole, the presented research looks like a complete article.
Response 4: Thank you very much for your advice! I compared the formula you suggested with the formula in the original text, and found that there was little difference between them, as shown in Figure 1. In addition, the calculation complexity of the formula you proposed is less than that in the original text, which is very suitable for MCU. I will apply your suggestions in my subsequent work.
Figure 1 Comparison of two kinds of fitting equations
Point 5: Finally, it seems the References chapter is presented not according to the MDPI template. Overall, the article can be recommended for publication after considering the recommendations, mainly 1, 3, and 5.
Response 4: Thank you very much for your correction. The format of the references has been corrected.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear Authors,
General comment:
Best State of Charge estimation is one of the main tasks of lithium-ion battery management systems. The subject is under intense debate, and the research effort is welcome.
Generally, the paper has several disadvantages. Here are the more important:
1st Problem - The Filter
My Comment:
1.1 EKF – (Extended Kalman Filter)
The EKF is a nonlinear the Kalman Filter. It is solves the nonlinear estimation problem by linearising measurement equations.
The problem is: It is linearize on very unstable (bad) places and the Extended Kalman Filter's error estimates tend to underestimate state uncertainties.
1.2 UKF – (Unscented Kalman Filter)
The UKF picks so called sigma point samples from the filtering distribution. The weighted set of sigma points represents a Gaussian distribution. Has this assumption been verified? Explain, please.
Generally, this method has less uncertainty (See 3rd Problem).
1.3 HIF – (H-Infinity Filter)
The Kalman Filter gives better performance than the HIF but it requires more tuning. It’s a problem. How was this problem solved? Explain, please.
1.4 FOUHIF – (A Fractional Order model based Unscented Kalman filter and H-Infinity Filter)
The combination of these two methods is a good idea. All the advantages and disadvantages add up. Is there any analysis done? Explain, please.
Finally, the FOUHIF method is a good choice, but it is not revolutionary. The improvement is low.
2nd Problem - Singular Value Decomposition
„Furthermore, some problems in the two algorithms were also improved:
firstly, singular value decomposition (SVD) method was used to solve the problem of ill-conditioned matrix generated by Cholesky decomposition of the covariance matrix;
secondly, self-adaption method was taken to solve the problem of uncertain initial values of the state noise covariance matrix and the measurement noise covariance matrix.”
My Comment:
SVD is not a method; it's matrix decomposition.
3rd Problem - The Uncertainty
My Comment:
The Authors used an indirect measurement. In order to enable the comparison of these results of measurements, they should be reported (in accordance with the Guide to the Expression of Uncertainty in Measurement) along with the values of their uncertainty. In the discussion section, I would expect to see an analysis of the possibilities to reduce the measurement uncertainty.
(See:
- Guide to the Expression of Uncertainty in Measurement, ISO (International Standardization Organization) TAG 4, corrected and reprinted, 1995.)
4th Problem - The Validation
My Comment:
Each new measurement method must be verified. It must be validated. Explain, please.
5th Problem - The Mistakes
My Comment:
The paper has a lot of errors. I suggest reviewing the text again. Native Speaker help is needed.
E.g.
- „Grnwald-Letnikov (GL)”; the correct form: Grunwald-Letnikov
- „Hybrid Pluse Power”; the correct form: Hybrid Pulse Power
- „(2)Priori value of state estimation, Priori estimate value of variance”; the correct form: A priori
- „Nominal capacity 27AH”; the correct form: 27Ah
...
- „Figure 5. Flowchart of the algorithm”; error in the electrical diagram
....
6th Problem – Missing in the Text
Reference
16. Li, L. , Hu, M. , Xu, Y. , Fu, C. , & Li, Z. . (2020). State of charge estimation for lithium-ion power battery based on 642 h-infinity filter algorithm. Applied Sciences, 10(18), 6371.
And
Figure 5.
And
TABLE 4. Statistical data of terminal voltages under NEDC cycles
TABLE 5. Statistical data of terminal voltages under UDDS cycles
Other
All the figures are not legible and need to be corrected. I suggest using a larger font.
Author Response
Response to Reviewer 2 Comments
Point 1: Best State of Charge estimation is one of the main tasks of lithium-ion battery management systems. The subject is under intense debate, and the research effort is welcome.
Generally, the paper has several disadvantages. Here are the more important.
Response 1: Thank you very much for the comments and opinions of the experts. It is your comments that make me have a deeper understanding of the filtering algorithm.
Point 2: 1st Problem - The Filter
My Comment:
1.1 EKF – (Extended Kalman Filter)
The EKF is a nonlinear the Kalman Filter. It is solves the nonlinear estimation problem by linearising measurement equations.
The problem is: It is linearize on very unstable (bad) places and the Extended Kalman Filter's error estimates tend to underestimate state uncertainties
Response 2: Thank you for your summary of the EKF algorithm. Based on your summary and the characteristics of lithium batteries, I have the following understanding: the lithium battery is a nonlinear time-varying system. Although the EKF linearization for nonlinear section, its two characteristics will lead to inaccurate state estimation, (1) the EKF algorithm in Taylor expansion link will omit higher order term, which will inevitably lead to precision loss after multiple calculations; (2) The probability distribution of noise must be white noise, but in the actual situation, lithium battery system is often colored noise, which will also lead to the reduction of estimation accuracy. The position of this summary in the text is from line 249 to line 256.
Point 3: 1.2 UKF – (Unscented Kalman Filter)
The UKF picks so called sigma point samples from the filtering distribution. The weighted set of sigma points represents a Gaussian distribution. Has this assumption been verified? Explain, please.
Generally, this method has less uncertainty (See 3rd Problem).
Response 3: Thank you for commenting on the UKF algorithm section. Since the premise of the use of UKF is that the noise in the system is Gaussian white noise, but in practice, the noise of lithium battery system is often colored noise，in this case, using the UKF algorithm for estimation will result in inaccurate results. In this paper, the UKF algorithm is a control experiment with poor estimation performance. Aiming at this problem, this paper combines UKF with H infinity to filter the colored noise in the system by using the cost function of H infinity algorithm principle to improve the robustness of the algorithm. For details, see formula (55)-formula (61) in the original text.
Point 4: 1.3 HIF – (H-Infinity Filter)
The Kalman Filter gives better performance than the HIF but it requires more tuning. It’s a problem. How was this problem solved? Explain, please.
Response 4: Thank you very much for your comments on H infinite algorithm. However, there is no comparison between H-infinity and Kalman filter in this study. The function of H-infinity algorithm in this study is to help UKF algorithm to filter colored noise. At present, most studies show that H infinity algorithm performs better than Kalman filter algorithm. Please refer to the following two documents:
[1] Li, L. , Hu, M. , Xu, Y. , Fu, C. , & Li, Z. . (2020). State of charge estimation for lithium-ion power battery based on h-infinity filter algorithm. Applied Sciences, 10(18), 6371.
[2] Qiao, Z. , Xiong, N. , Yang, M. L. , Huang, R. S. , & Hu, G. D. . (2017). State of charge estimation for lithium-ion battery based on nonlinear observer: an h∞ method. Energies, 10(5), 679.
Point 4: FOUHIF – (A Fractional Order model based Unscented Kalman filter and H-Infinity Filter)
The combination of these two methods is a good idea. All the advantages and disadvantages add up. Is there any analysis done? Explain, please.
Finally, the FOUHIF method is a good choice, but it is not revolutionary. The improvement is low.
Response 4: Thank you very much for your comments on the FOUHIF algorithm section. The analysis of FOUHIF algorithm is as follows: Based on the fractional model of lithium battery, this algorithm combines UKF and H infinity together. The main frame is still the UKF algorithm, and its UT change can improve the estimation accuracy and speed.Its main disadvantage is that it needs to assume that the noise is white noise, thus introducing H infinite algorithm, mainly in pursuing the status value of covariance of parts was improved, by minimizing the cost function to filter out the color noise, which greatly improves the robustness of the algorithm. The analysis of the advantages of this approach is given in the original text on lines 376 to 388.
Point 5: 2nd Problem - Singular Value Decomposition
„Furthermore, some problems in the two algorithms were also improved:
firstly, singular value decomposition (SVD) method was used to solve the problem of ill-conditioned matrix generated by Cholesky decomposition of the covariance matrix;
secondly, self-adaption method was taken to solve the problem of uncertain initial values of the state noise covariance matrix and the measurement noise covariance matrix.”
My Comment:
SVD is not a method; it's matrix decomposition.
Response 5: Thank you very much for your reminder and correction. The original text has been modified. After modification, it is " the singular value decomposition of the covariance matrix was carried out to solve the original problem of there being an ill-conditioned matrix generated during Cholesky decomposition". Line 390 through 392 in the text.
Point 6: 3rd Problem - The Uncertainty
My Comment:
The Authors used an indirect measurement. In order to enable the comparison of these results of measurements, they should be reported (in accordance with the Guide to the Expression of Uncertainty in Measurement) along with the values of their uncertainty. In the discussion section, I would expect to see an analysis of the possibilities to reduce the measurement uncertainty.
(See:- Guide to the Expression of Uncertainty in Measurement, ISO (International Standardization Organization) TAG 4, corrected and reprinted, 1995.)
Response 6: Thank you very much for your proposal. I have conducted uncertainty analysis on three estimation measurement methods under two working conditions, and the details are as follows: The test was carried out in a laboratory environment. The lithium battery cycle test equipment was EVTS produced by Arbin Company in the United States. The voltage acquisition accuracy is , the current acquisition accuracy is , and the thermostat was produced by Giant Force Company, with the temperature control error within 1℃. I intercepted a section of data during each working condition test to calculate the uncertainty of the three measurement methods and selected the time period when the battery stood still. The main factors that significantly affect the measurement uncertainty of lithium battery SOC are as follows: the uncertainty caused by the measurement repeatability of the terminal voltage and the measurement error of the battery cycle test equipment itself. The two use the A and B evaluation methods respectively. Under the NEDC working condition, the uncertainty analysis of the three estimation methods was conducted to select the time period when the battery was standing, ranging from 11,711 to 11,741 seconds. The uncertainty components caused by repeated measurements in UKF, FOUKF and FOUHIF estimation methods are , and , respectively, and the uncertainties caused by equipment indication error are , and respectively. The synthetic standard uncertainties of the three methods are , and . In the selected measurement time interval, the measurement results of the three measurement methods are , and respectively. Therefore, the FOUHIF algorithm has the highest reliability when analyzing the three measurement methods from the angle of uncertainty. Line 560 through 575 in the text.
Under the UDDS operating condition test, the uncertainty analysis of the three estimation methods was conducted to select the time period when the battery was standing, ranging from 30,147 to 30,500 seconds. The uncertainty components of UKF, FOUKF and FOUHIF caused by repeated measurement are , and respectively, and the uncertainty caused by equipment indication error are , and respectively. The combined standard uncertainty of the three methods is , and . Within the selected measurement time interval, the measurement results of the three methods are as follows: , , , the test results of this condition also show that FOUHIF algorithm has the highest reliability and the best effect. Line 603 through 613 in the text.
Point 7 :4th Problem - The Validation
My Comment:
Each new measurement method must be verified. It must be validated. Explain, please.
Response 7: In this paper, each estimation algorithm is verified as follows: UKF, FOUKF and FOUHIF algorithms are verified by NEDC and UDDS conditions respectively. These two working conditions are complex, which can effectively verify the robustness of the algorithm. Under NEDC working condition, set the SOC initial value of the three algorithms to 70% and the initial error to 30%. After analyzing the SOC estimation effect and the terminal voltage prediction effect, the RMSE,EMA and RMSE of the SOC were calculated. The specific results are shown in Table 1. The results show that FOUKF is better than FOUKF, and FOUKF is better than UKF(Line 544 through 559 in the text). Under UDDS working condition, the initial value of the three algorithms was set as 50%, and the initial error was 50%. The analysis content was the same as that of NEDC condition experiment. The results were as follows: It can be seen that FOUHIF is better than foukf and foukf is better than UKF. (Line 591 through 602 in the text)
Point 8: 5th Problem - The Mistakes
My Comment:
The paper has a lot of errors. I suggest reviewing the text again. Native Speaker help is needed.
E.g.
- „Grnwald-Letnikov (GL)”; the correct form: Grunwald-Letnikov
- „Hybrid Pluse Power”; the correct form: Hybrid Pulse Power
- „(2)Priori value of state estimation, Priori estimate value of variance”; the correct form: A priori
- „Nominal capacity 27AH”; the correct form: 27Ah
...
- „Figure 5. Flowchart of the algorithm”; error in the electrical diagram
Response 8: Thank you very much for your correction. All the five questions you raised have been revised, and the positions are line 146, line 198, line351,line353 ,page 13 and page 15 respectively. This article has been revised by native speakers.
Point 9: 6th Problem – Missing in the Text
Reference
16. Li, L. , Hu, M. , Xu, Y. , Fu, C. , & Li, Z. . (2020). State of charge estimation for lithium-ion power battery based on 642 h-infinity filter algorithm. Applied Sciences, 10(18), 6371.
And
Figure 5.
And
TABLE 4. Statistical data of terminal voltages under NEDC cycles
TABLE 5. Statistical data of terminal voltages under UDDS cycles
Response 9: For the four items in the sixth question, explanations have been added one by one in the paper, and the specific positions are respectively in line 86, line 476, lin558 and line 601. And all the pictures in the article have been formatted.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Dear Authors,
Thank you for your response. The introduced corrections have improved the paper.
It is not a radical method, but the direction is right. It is certainly the development of science.
Thank you.
General comment:
The analysis of the measurement uncertainty is very sketchy. The final presentation of the results should have units (seconds). It sounds better.
I suggest reviewing references in the text again.
The order:
Line39 [1-2]
Line80 [9-10]
Line97 [16]
Line111 [1-20]
Line116 [22]
Line117 [21]
Line219 [24]
Missing in the Text
Reference
No 23.
My Conclusion
After these additions, I recommend this paper.
Author Response
Response to Reviewer 2 Comments
Point 1: Thank you for your response. The introduced corrections have improved the paper.
It is not a radical method, but the direction is right. It is certainly the development of science.
Thank you.
Response 1: Thank you very much for your help in revising the article. Your suggestions and methods have made me make great progress. Thank you for your help and dedication.
Point 2: The analysis of the measurement uncertainty is very sketchy.
Response 2: Thank you very much for your proposal. I have added a detailed description to the “analysis of the measurement uncertainty”. Lines 573 through 598 in the text.
Point 3: The final presentation of the results should have units (seconds). It sounds better.
Response 3: Thank you very much for your comments. But the final result of our measurement is SOC, SOC is defined as follows: the ratio of the remaining battery power to the rated capacity. The formula for this definition is as follows, represents the current remaining power, represents rated capacity, so the final presentation of the results don’t have unit.
Point 4: I suggest reviewing references in the text again.
The order:
Line39 [1-2]
Line80 [9-10]
Line97 [16]
Line111 [1-20]
Line116 [22]
Line117 [21]
Line219 [24]
Response 4: Thank you very much for your inspection. I have corrected some of my mistakes.
[1-2]: Reference[1] has been replaced. Reference [2] has not changed.
[1] Imran, R. M. , Li, Q. , & Flaih, F. . (2020). An enhanced lithium-ion battery model for estimating the state of charge and degraded capacity using an optimized extended kalman filter. IEEE Access, 8, 208322-208336.
[9-10]: Reference[9]and[10] have been replaced.
[9] Huang, Z. , Fang, Y. , & Xu, J. . (2021). Soc estimation of li-ion battery based on improved ekf algorithm. International Journal of Automotive Technology, 22(2), 335-340.
[10] Sun, D. , Yu, X. , Zhang, C. , Wang, C. , & Huang, R. . (2020). State of charge estimation for lithium‐ion battery based on an intelligent adaptive unscented kalman filter. International Journal of Energy Research, 44(14).
[16]: Reference [16] has not changed.
[1-20]: Reference[19] has been replaced.
[19] Mawonou, K. , Eddahech, A. , Dumur, D. , Beauvois, D. , & Godoy, E. . (2019). Improved state of charge estimation for li-ion batteries using fractional order extended kalman filter. Journal of Power Sources, 435(SEP.30), 226710.
[22][21]: The interpretation of reference [22] and [21] in the text have been revised. Lines 97 through 105 in the text.
Reference [24][25]are renumbered.(Page 23)
Reference [24] is in line 150 of the text. Reference [25] is in line 210 of the text.
[22]: Reference[22] has not changed.
At the same time, I made some modifications to the grammar of the paper, the positions are line 204-210,259-262,272,293,397,398.
Point 5: Missing in the Text Reference No.23
Response 5: Reference 23 is in line 121 of the text.
Author Response File: Author Response.pdf
