# Distributed Systematic Grid-Connected Inverter Using IGBT Junction Temperature Predictive Control Method: An Optimization Approach

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## Abstract

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## 1. Introduction

- To improve the search performance of the intelligent optimizer, so as to achieve the best convergence;
- To achieve the best SVM performance model using the SVM parameters optimized by an intelligent optimizer;
- To address the IGBT junction temperature prediction accuracy by optimizing the model parameters.

## 2. Literature Review

#### 2.1. The Forecasting Methods of Junction Temperature

#### 2.2. The Proposed Method

## 3. Method

#### 3.1. Power Cycle Test

_{1}. U

_{G}is 15 V, which ensures that the IGBT is always triggered. FWD is the freewheeling diode. The process of the power cycle test is presented according to IEC 60747-9:2007.

- The S
_{1}is closed. The output current of U_{dc}is set to 70 A. Further, the IGBT is driven. In this process, the junction temperature and case temperature of IGBT gradually rise from the ambient temperature. - When the case temperature detected by the K-type thermocouple reaches 125 °C, the S1 is turned off. The junction temperature and the case temperature are rapidly decreased until the temperature drops to 25 °C. So far, a power cycle is completed.
- Step (2) and step (3) are repeated until the IGBT reaches the failure standard. The junction-to-case thermal resistance of IGBT increases by 20% compared with the initial value.

#### 3.2. Single Pulse Test

_{L}is the power resistance. The specific test process is as follows:

- The IGBT is placed in the incubator and the temperature of the incubator is adjusted point by point according to the above description.
- When the IGBT reaches thermal equilibrium, the case temperature and junction temperature are equal to the incubator temperature. Then the selected current point is set in turn. The IGBT is triggered by single pulse. The saturation voltage drop value is measured by using voltage probe.

#### 3.3. Test Results

_{ce}, current I

_{c}and junction temperature T

_{j}of the IGBT under different power cycles are obtained. The test data at currents 65 A, 70 A, 75 A and 80 A are selected as modeling data. Based on the modelling data, the relationships between current, junction temperature and saturation voltage drop under different power cycles is obtained, which are shown in Figure 5.

_{c}, T

_{j}and V

_{ce}is moved up overall, which indicates that T

_{j}of the IGBT is closely related to the power cycles. The influence of the number of power cycles on the T

_{j}should be fully considered in the process of establishing the IGBT junction temperature prediction mode.

## 4. The Establishment Process of Junction Temperature Prediction Model

#### 4.1. Support Vector Machine

#### 4.2. The CSO Algorithm and the ICSO Algorithm

#### 4.2.1. The CSO Algorithm

- The whole chicken swarm is divided into several subgroups, and each subgroup consists of a rooster, multiple hens and chicks.
- The rooster is the leader in each population, and each population has only one rooster which has the strongest search ability. The search ability of hen is worse than of cock. The chicks follow the hens to search food, and the chicks have the worst search ability.
- The mother-child relationship between the hens and the chicks is updated at each iteration.

#### 4.2.2. The ICSO Algorithm

- (1)
- The weight is adjusted by the dynamic inertia learning strategy. The reduction strategy is adopted for improving the convergence speed of algorithm.
- (2)
- To solve the problem that the populations gradually decrease in the iterative process, in the later stage of algorithm, the Gaussian mutation process is changed to the Cauchy mutation process, which improves the diversity of the population.
- (3)
- The part of learning from the global optimal individual is added.

- $w$ is added to the position update equation to enhance the self-learning ability of the chick.
- The part of learning from the global optimal individual is added.
- When the fitness value of the chick remains unchanged during multiple iterations, the position of the chick is reset to avoid premature convergence.
- The improved position update equation of the chick is as follows.$${x}_{ij}(t+1)=w*{x}_{ij}(t)+{H}_{i}*({x}_{mj}(t)-{x}_{ij}(t))+S*({x}_{bestj}(t)-{x}_{ij}(t))$$$$S=\mathrm{exp}({f}_{\mathrm{min}}-{f}_{i})$$

#### 4.3. The Establishment Process of ICSO-SVM Model

_{c}and V

_{ce}are used as input variables of the proposed model to forecast the T

_{j}.

- (1)
- The training samples and test samples are selected and normalized.
- (2)
- The number of roosters, hens and chicks in the chicken swarm and the number of iterations are set.
- (3)
- The population is initialized and the fitness value of the individual is calculated. The chicken swarm is divided into ${r}_{n}$ population. The hens randomly enter different populations.
- (4)
- The position of the rooster, hen and chick is updated according to Equations (17), (19) and (20). The fitness values of the swarm are calculated.
- (5)
- If the current individual fitness is better after the position update, the individual position is updated. Otherwise, the original individual position is maintained.
- (6)
- Determine whether the fitness of the chick remains unchanged after multiple iterations (five times). If the fitness value remains unchanged. The position of the chick is reset.
- (7)
- The fitness of each individual is calculated after each position update.
- (8)
- Determine whether the maximum number of iterations reaches. If it is satisfied, the optimal parameters are output. Otherwise, the parameters continue to be optimized.
- (9)
- The optimal parameters are input into the ICSO-SVM model. The junction temperature is forecasted by the ICSO-SVM model. The predictive results are inverse normalized.

## 5. Results

_{c}and V

_{ce}are selected as the input parameters. T

_{j}is selected as the output parameter. The initial setting parameters of the PSO-SVM model are shown in Table 1. The initial setting parameters of the CSO-SVM model and the ICSO-SVM model are shown in Table 2.

_{j}. The absolute error (AE), relative error (RE), mean relative error (MRE), root mean square error (RMSE) and mean absolute error (MAE) are used to evaluate the test results of the ICSO-SVM model.

^{−3}. The iterative curve of the CSO-SVM model tends to be stable when the number of iterations is 18, and the fitness value is 2.89 × 10

^{−3}. The iterative curve of the ICSO-SVM model is basically stable when the number of iterations is 7, and the fitness is 2.89 × 10

^{−3}. The iterative stability times and fitness value of the ICSO-SVM model are better than other models.

_{j}and predictive errors of the PSO-SVM, CSO-SVM and ICSO-SVM models are presented in Figure 8, Figure 9 and Figure 10.

## 6. Implications

_{ce}, I

_{c}and T

_{j}of IGBT under different power cycles are measured by power cycle test and single pulse test. Following this, the database of junction temperature is established. Based on the database, this study establishes the ICSO-SVM prediction model. The trajectories of rooster, hen and chick are improved in the ICSO algorithm. The prediction results of the ICSO-SVM model are closer to the true value than that of the CSO-SVM model and PSO-SVM model. The ICSO-SVM model does not require the complex circuits, and the model only needs to set the input and output. In this study, the methods of increasing the diversity of rooster and the improved methods of chick and hen is applied to other optimization algorithms. In the process of model evaluation, this study uses the mean error to evaluate the prediction accuracy of CSO-SVM model and ICSO-SVM model, which provides a new idea for the visualization of junction temperature prediction.

## 7. Concluding Remarks

- The V
_{ce}, I_{c}and T_{j}data of IGBT under different power cycles are measured by the power cycle test and the single pulse test. Based on the test data, a three-dimensional relationship between I_{c}, T_{j}, and V_{ce}under different power cycles is constructed to directly reflect the IGBT aging effect. - The ICSO algorithm is proposed. The self-learning ability of rooster, hen and chick is improved by dynamic inertia reduction strategy. Cauchy mutation is introduced in the position of rooster, which increases the diversity of roosters in the later stage of iterations. The convergence ability of ICSO algorithm is strengthened. In this paper, ICSO algorithm is applied to distributed grid-connected inverter.
- Through the comparison of the three models, the RMSE of the ICSO-SVM model is only 1.1836, which is lower than the 1.3121 of the CSO-SVM model. The results prove effectiveness of the ICSO-SVM model.
- The MAE value of the ICSO-SVM model is 0.7989, which is lower than that of the PSO-SVM model and CSO-SVM model. The prediction accuracy is higher compared with CSO-SVM model. The error curve of the ICSO-SVM model is stable, which proves the superiority of the proposed model. The model presented in this paper shows high prediction accuracy.
- Accurate life prediction of ICBT has a positive impact on promoting the development of the distributed systematic grid-connected inverter industry and the use of new energy.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Li, L.L.; Qi, F.D.; Tseng, M.L.; Tan, K. Predicting the Remaining Lifetime for Insulated Gate Bipolar Transistor Power Module using the Aging State Evaluation. Microelectron. Reliab.
**2019**, 102, 113476. [Google Scholar] [CrossRef] - Dedeban, G.; Mitchell, P.; Dossou, P.-E. Energy Audit Methodology and Energy Savings Plan in the Nautical Industry. Biotechnol. Bus. Concept Deliv.
**2014**, 425–437. [Google Scholar] [CrossRef] - Tseng, M.-L.; Tan, K.; Lim, M.K.; Lin, R.-J.; Geng, Y. Benchmarking eco-efficiency in green supply chain practices in uncertainty. Prod. Plan. Control.
**2013**, 25, 1079–1090. [Google Scholar] [CrossRef] - Liu, Z.-F.; Li, L.-L.; Tseng, M.-L.; Lim, M.K. Prediction short-term photovoltaic power using improved chicken swarm optimizer—Extreme learning machine model. J. Clean. Prod.
**2020**, 248, 119272. [Google Scholar] [CrossRef] - Muralikrishna, I.V.; Manickam, V. Energy Management and Audit. Environ. Manag.
**2017**, 9, 153–175. [Google Scholar] [CrossRef] - Lalvani, J.I.J.; Parthasarathy, M.; Dhinesh, B.; Annamalai, K.; Ashok, B. Experimental investigation of combustion, performance and emission characteristics of a modified piston. J. Mech. Sci. Technol.
**2015**, 29, 4519–4525. [Google Scholar] [CrossRef] - Preda, S.; Oprea, S.-V.; Bara, A.; Velicanu, A.B. PV Forecasting Using Support Vector Machine Learning in a Big Data Analytics Context. Symmetry
**2018**, 10, 748. [Google Scholar] [CrossRef] [Green Version] - Malik, M.Z.; Chen, H.; Nazir, M.S.; Khan, I.A.; Abdalla, A.; Ali, A.; Chen, W. A New Efficient Step-Up Boost Converter with CLD Cell for Electric Vehicle and New Energy Systems. Energies
**2020**, 13, 1791. [Google Scholar] [CrossRef] [Green Version] - Anwar, M.A.; Abbas, G.; Khan, I.; Awan, A.B.; Farooq, U.; Khan, S.S.; Majeed, R. An Impedance Network-Based Three Level Quasi Neutral Point Clamped Inverter with High Voltage Gain. Energies
**2020**, 13, 1261. [Google Scholar] [CrossRef] [Green Version] - Wang, Q.; Yao, W.; Fang, J.; Ai, X.; Wen, J.; Yang, X.; Xie, H.; Huang, X. Dynamic modeling and small signal stability analysis of distributedphotovoltaic grid-connected system with large scale of panel level DCoptimizers. Appl. Energy
**2020**, 259, 114132. [Google Scholar] [CrossRef] - Vinnikov, D.; Chub, A.; Liivik, E.; Kosenko, R.; Korkh, O.; Liivik, L. Solar Optiverter—A Novel Hybrid Approach to the Photovoltaic Module Level Power Electronics. IEEE Trans. Ind. Electron.
**2018**, 66, 3869–3880. [Google Scholar] [CrossRef] - Bielskis, E.; Baskys, A.; Valiulis, G. Controller for the Grid-Connected Microinverter Output Current Tracking. Symmetry
**2020**, 12, 112. [Google Scholar] [CrossRef] [Green Version] - Chang, F.; Ilina, O.; Lienkamp, M.; Voss, L. Improving the Overall Efficiency of Automotive Inverters Using a Multilevel Converter Composed of Low Voltage Si mosfets. IEEE Trans. Power Electron.
**2019**, 34, 3586–3602. [Google Scholar] [CrossRef] - Moosavi, S.; Kazemi, A.; Akbari, H. A comparison of various open-circuit fault detection methods in the IGBT-based DC/AC inverter used in electric vehicle. Eng. Fail. Anal.
**2019**, 96, 223–235. [Google Scholar] [CrossRef] - Rannestad, B.; Maarbjerg, A.E.; Frederiksen, K.; Munk-Nielsen, S.; Gadgaard, K. Converter Monitoring Unit for Retrofit of Wind Power Converters. IEEE Trans. Power Electron.
**2017**, 33, 4342–4351. [Google Scholar] [CrossRef] - Yang, X.; Lin, Z.; Ding, J.; Long, Z. Lifetime Prediction of IGBT Modules in Suspension Choppers of Medium/Low-Speed Maglev Train Using an Energy-Based Approach. IEEE Trans. Power Electron.
**2018**, 34, 738–747. [Google Scholar] [CrossRef] - Li, L.-L.; Lv, C.-M.; Tseng, M.-L.; Song, M. Renewable energy utilization method: A novel Insulated Gate Bipolar Transistor switching losses prediction model. J. Clean. Prod.
**2018**, 176, 852–863. [Google Scholar] [CrossRef] - Li, L.-L.; Zhang, X.-B.; Tseng, M.-L.; Lim, M.K.; Han, Y. Sustainable energy saving: A junction temperature numerical calculation method for power insulated gate bipolar transistor module. J. Clean. Prod.
**2018**, 185, 198–210. [Google Scholar] [CrossRef] - Chen, C.; Pickert, V.; Al-Greer, M.; Jia, C.; Ng, C. Localization and Detection of Bond Wire Faults in Multi-chip IGBT Power Modules. IEEE Trans. Power Electron.
**2020**. [Google Scholar] [CrossRef] [Green Version] - Yu, L.; Zhang, Y.; Huang, W.; Teffah, K. A Fast-Acting Diagnostic Algorithm of Insulated Gate Bipolar Transistor Open Circuit Faults for Power Inverters in Electric Vehicles. Energies
**2017**, 10, 552. [Google Scholar] [CrossRef] [Green Version] - Busca, C.; Teodorescu, R.; Blaabjerg, F. An overview of the reliability prediction related aspects of high power IGBT in wind power applications. Microelectron. Reliab.
**2011**, 51, 1903–1907. [Google Scholar] [CrossRef] [Green Version] - Lin, C.-W.; Jeng, S.-Y.; Tseng, M.-L.; Tan, R. A cradle-to-cradle analysis in the toner cartridge supply chain using fuzzy recycling production approach. Manag. Environ. Qual. Int. J.
**2019**, 30, 329–345. [Google Scholar] [CrossRef] - Ouhab, M.; Khatir, Z.; Ibrahim, A. New Analytical Model for Real-Time Junction Temperature Estimation of Multi-Chip Power Module Used in a Motor Drive. IEEE Trans. Power Electron.
**2018**, 33, 5292–5301. [Google Scholar] [CrossRef] - Fabis, P.M.; Shum, D.; Windischmann, H. Thermal modeling of diamond-based power electronics packaging. In Proceedings of the Fifteenth Annual IEEE Semiconductor Thermal Measurement and Management Symposium, San Diego, CA, USA, 9–11 March 1999. [Google Scholar]
- Liu, B.; Xiao, F.; Luo, Y.; Huang, Y.; Xiong, Y. A Multi-timescale Prediction Model of IGBT Junction Temperature. IEEE J. Emerg. Sel. Top. Power Electron.
**2019**, 7, 1593–1603. [Google Scholar] [CrossRef] - Chen, H.; Yang, J.; Xu, S. Electro-thermal-Based Junction Temperature Estimation Model for Converter of Switched Reluctance Motor Drive System. IEEE Trans. Power Electron.
**2020**, 67, 874–883. [Google Scholar] - Bazzo, J.P.; Lukasievicz, T.; Vogt, M.; De Oliveira, V.; Kalinowski, H.J.; Da Silva, J.C.C. Thermal characteristics analysis of an IGBT using a fiber Bragg grating. Opt. Lasers Eng.
**2012**, 50, 99–103. [Google Scholar] [CrossRef] - Liu, B.-Y.; Wang, G.-S.; Tseng, M.-L.; Wu, K.-J.; Li, Z.-G. Exploring the Electro-Thermal Parameters of Reliable Power Modules: Insulated Gate Bipolar Transistor Junction and Case Temperature. Energies
**2018**, 11, 2371. [Google Scholar] [CrossRef] [Green Version] - Tang, Y.; Lin, L.; Ma, H. An Improved Transient Electro-Thermal Model for Paralleled IGBT Modules. Trans. China Electrotech. Soc.
**2017**, 32, 88–96. [Google Scholar] - Xie, K.; Jiang, Z.; Li, W. Effect of Wind Speed on Wind Turbine Power Converter Reliability. IEEE Trans. Energy Convers.
**2012**, 27, 96–104. [Google Scholar] [CrossRef] - Li, L.; Xu, Y.; Li, Z.; Wang, P.; Wang, B. The effect of electro-thermal parameters on IGBT junction temperature with the aging of module. Microelectron. Reliab.
**2016**, 66, 58–63. [Google Scholar] [CrossRef] - Eleffendi, M.A.; Johnson, C.M.; Eleffendi, A.; Johnson, C.M. Application of Kalman Filter to Estimate Junction Temperature in IGBT Power Modules. IEEE Trans. Power Electron.
**2016**, 31, 1576–1587. [Google Scholar] [CrossRef] - Cortes, C.; Vapnik, V. Support-Vector Networks. Mach. Learn.
**1995**, 20, 273–297. [Google Scholar] [CrossRef] - Gao, D.; Huang, M. Prediction of Remaining Useful Life of Lithium-ion Battery based on Multi-kernel Support Vector Machine with Particle Swarm Optimization. J. Power Electron.
**2017**, 17, 1288–1297. [Google Scholar] - Wang, C.; Yang, Q.; Wang, J.; Zhao, J.; Wan, X.; Guo, Z.; Yang, Y. Application of support vector machine on controlling the silanol groups of silica xerogel with the aid of segmented continuous flow reactor. Chem. Eng. Sci.
**2019**, 199, 486–495. [Google Scholar] [CrossRef] - Meng, X.-B.; Liu, Y.; Gao, X.; Zhang, H. A New Bio-Inspired Algorithm: Chicken Swarm Optimization; International Conference in Swarm Intelligence; Springer: Cham, Switzerland, 2014; Volume 8794, pp. 86–94. [Google Scholar]
- Zouache, D.; Arby, Y.O.; Nouioua, F.; Ben Abdelaziz, F. Multi-objective chicken swarm optimization: A novel algorithm for solving multi-objective optimization problems. Comput. Ind. Eng.
**2019**, 129, 377–391. [Google Scholar] [CrossRef] - Lai, W.; Chen, M.Y.; Li, R. Analysis of IGBT failure mechanism based on ageing experiments. Proc. CSEE
**2015**, 35, 5293–5300. [Google Scholar] - Ghimire, P.; De Vega, A.R.; Beczkowski, S.; Rannestad, B.; Munk-Nielsen, S.; Thogersen, P. Improving Power Converter Reliability: Online Monitoring of High-Power IGBT Modules. IEEE Ind. Electron. Mag.
**2014**, 8, 40–50. [Google Scholar] [CrossRef] - Smet, V.; Forest, F.; Huselstein, J.-J.; Richardeau, F.; Khatir, Z.; Lefebvre, S.; Berkani, M. Ageing and Failure Modes of IGBT Modules in High-Temperature Power Cycling. IEEE Trans. Ind. Electron.
**2011**, 58, 4931–4941. [Google Scholar] [CrossRef] - Ji, B.; Pickert, V.; Cao, W.; Zahawi, B. In Situ Diagnostics and Prognostics of Wire Bonding Faults in IGBT Modules for Electric Vehicle Drives. IEEE Trans. Power Electron.
**2013**, 28, 5568–5577. [Google Scholar] [CrossRef] [Green Version] - Huang, Y.; Luo, Y.; Xiao, F.; Liu, B. Failure Mechanism of Die-Attach Solder Joints in IGBT Modules Under Pulse High-Current Power Cycling. IEEE J. Emerg. Sel. Top. Power Electron.
**2018**, 7, 99–107. [Google Scholar] [CrossRef] - Xiang, D.; Ran, L.; Tavner, P.; Yang, S.; Bryant, A.; Mawby, P. Condition Monitoring Power Module Solder Fatigue Using Inverter Harmonic Identification. IEEE Trans. Power Electron.
**2011**, 27, 235–247. [Google Scholar] [CrossRef] [Green Version] - Trentin, A.; Wheeler, P.; Zanchetta, P.; Clare, J. Performance evaluation of high-voltage 1.2 kV silicon carbide metal oxide semi-conductor field effect transistors for three-phase buck-type PWM rectifiers in aircraft applications. IET Power Electron.
**2012**, 5, 1873–1881. [Google Scholar] [CrossRef] - Vapnik, V.N. The Nature of Satistical Learning Theory; Springer: New York, NY, USA, 1999; Volume 10, pp. 988–999. [Google Scholar]
- Lipu, M.S.H.; Hannan, M.A.; Hussain, A.; Hoque, M.; Ker, P.J.; Saad, M.; Ayob, A. A review of state of health and remaining useful life estimation methods for lithium-ion battery in electric vehicles: Challenges and recommendations. J. Clean. Prod.
**2018**, 205, 115–133. [Google Scholar] [CrossRef] - Morais, C.L.M.; Lima, K.M.G.; Martin-Hirsch, P. Uncertainty estimation and misclassification probability for classification models based on discriminant analysis and support vector machines. Anal. Chim. Acta
**2018**, 1063, 40–46. [Google Scholar] [CrossRef] [PubMed] - Gashteroodkhani, O.; Majidi, M.; Etezadi-Amoli, M.; Nematollahi, A.; Vahidi, B. A hybrid SVM-TT transform-based method for fault location in hybrid transmission lines with underground cables. Electr. Power Syst. Res.
**2019**, 170, 205–214. [Google Scholar] [CrossRef] - Das, S.; Suganthan, P.N. Differential Evolution: A Survey of the State-of-the-Art. IEEE Trans. Evol. Comput.
**2011**, 15, 4–31. [Google Scholar] [CrossRef] - Shayokh, A.; Shin, S.Y. Bio Inspired Distributed WSN Localization Based on Chicken Swarm Optimization. Wirel. Pers. Commun.
**2017**, 97, 5691–5706. [Google Scholar] [CrossRef]

**Figure 3.**The relationship between the IGBT junction-to-case thermal resistance and the number of power cycles.

**Figure 6.**The modeling process of the improved chicken swarm optimization-support vector machine (ICSO-SVM) junction temperature prediction model.

The Maximum Number of Iterations | 100 |
---|---|

Number of population | 20 |

Global search level c1 | 1.5 |

Local search level c2 | 1.5 |

Particle velocity range | [−50, 50] |

Particle position range | [−500, 500] |

Range of parameter C | [0.1, 100] |

Range of parameter σ | [0.01, 1000] |

**Table 2.**Initial setting parameters of chicken swarm optimization (CSO)-SVM model and ICSO-SVM model.

The Maximum Number of Iterations | 100 |
---|---|

Number of population | 20 |

Dimension | 2 |

Ratio of rooster | 0.2 |

Ratio of hen | 0.6 |

Ratio of hen with chicks | 0.3 |

Range of parameter C | [0.1, 100] |

Range of parameter σ | [0.01, 1000] |

Models | MAE (℃) | MRE (%) | RMSE (℃) |
---|---|---|---|

PSO-SVM | 2.8695 | 4.7003 | 3.5308 |

CSO-SVM | 0.9316 | 1.4068 | 1.3121 |

ICSO-SVM | 0.7989 | 1.0560 | 1.1836 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Li, G.; Tseng, M.-L.; Wong, W.-P.; Liu, B.
Distributed Systematic Grid-Connected Inverter Using IGBT Junction Temperature Predictive Control Method: An Optimization Approach. *Symmetry* **2020**, *12*, 825.
https://doi.org/10.3390/sym12050825

**AMA Style**

Wang Z, Li G, Tseng M-L, Wong W-P, Liu B.
Distributed Systematic Grid-Connected Inverter Using IGBT Junction Temperature Predictive Control Method: An Optimization Approach. *Symmetry*. 2020; 12(5):825.
https://doi.org/10.3390/sym12050825

**Chicago/Turabian Style**

Wang, Zhengping, Guoyi Li, Ming-Lang Tseng, Wai-Peng Wong, and Boying Liu.
2020. "Distributed Systematic Grid-Connected Inverter Using IGBT Junction Temperature Predictive Control Method: An Optimization Approach" *Symmetry* 12, no. 5: 825.
https://doi.org/10.3390/sym12050825