Słowa kluczowe:
electrified railway
energy storage system negative sequential governance
railway power conditioner
regenerative braking energy

Aiming at the problems of the negative sequence governance and regenerative braking energy utilization of electrified railways, a layered compensation optimization strategy considering the power flow of energy storage systems was proposed based on the railway power conditioner. The paper introduces the topology of the energy storage type railway power conditioner, and analyzes its negative sequence compensation and regenerative braking energy utilization mechanism. Considering the influence of equipment capacity and power flow of the energy storage system on railway power conditioner compensation effect, the objective function and constraint conditions of the layered compensation optimization of the energy storage type railway power conditioner were constructed, and the sequential quadratic programming method was used to solve the problem. The feasibility of the proposed strategy is verified by a multi-condition simulation test. The results show that the proposed optimization compensation strategy can realize negative sequence compensation and regenerative braking energy utilization, improve the power factor of traction substations when the system equipment capacity is limited, and it also has good real-time performance.

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Słowa kluczowe:
energy-stored quasi-Z-source inverter
constant frequency control
model predictive direct power control
space vector modulation module

In order to overcome the shortcoming of large switching losses caused by variable switching frequency appears in the conventional finite control set model predictive control (FCS-MPC) algorithm, a model predictive direct power control (MP-DPC) for an energy storage quasi-Z-source inverter (ES-qZSI) is proposed. Firstly, the power prediction model of the ES-qZSI is established based on the instantaneous power theory. Then the average voltage vector in the ���� coordinate system is optimized by the power cost function. Finally, the average voltage vector is used as the modulation signal, and the corresponding switching signal with fixed frequency is generated by the shoot-through segment space vector pulse width modulation (SVPWM) technology. The simulation results show that the ES-qZSI realizes six shoot-through actions per control cycle and achieves the constant frequency control of the system, which verifies the correctness of the proposed control strategy.

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Słowa kluczowe:
charging stations
electric vehicle
e-mobility
optimization
transformer efficiency
transformer regulation

Transformer efficiency and regulation, are to be maintained at maximum and minimum respectively by optimal loading, control, and compensation. Charging of electric vehicles at random charging stations will result in uncertain loading on the distribution transformer. The efficiency reduces and regulation increases as a consequence of this loading. In this work, a novel optimization strategy is proposed to map electric vehicles to a charging station, that is optimal with respect to the physical distance, traveling time, charging cost, the effect on transformer efficiency and regulation. Consumer and utility factors are considered for mapping electric vehicles to charging stations. An Internet of Things platform is used to fetch the dynamic location of electric vehicles. The dynamic locations are fed to a binary optimization problem to find an optimal routing table that maps electric vehicles to a charging station. A comparative study is carried out, with and without optimization, to validate the proposed methodology.

Przejdź do artykułu
Słowa kluczowe:
entropy weight method silicone rubber composite insulator
fuzzy comprehensive evaluation
improved analytic hierarchy process
level difference maximization method
membership function

The aging of composite insulators in outdoor operation for a long time has a direct impact on the safe and stable operation of the power grid. To solve this problem, fuzzy comprehensive evaluation of composite insulators based on level difference maximum is proposed. To verify the feasibility of this method, insulators in Xinjiang are sampled and the index evaluation system for composite insulators is established based on electrical, mechanical, hydrophobic and other properties, combined with operational years, geographical environment and other factors; Firstly, different membership functions are established according to index types. It is more likely to determine the grade of insulator by comparing measured data with the boundary value. Then, to solve the problem that weights cannot be effectively integrated in the combination weighting, level difference maximization is proposed (during the operation of insulators, the index which has a greater influence on the performance of insulators takes a higher proportion of the weight). Finally, on the basis of fully considering the clarity and ambiguity of grade division, the grade state of insulators is obtained by using the linear weighting method. The results show that compared with the traditional method, the improved method of the membership function and level difference maximum can realize the dynamic adjustment of the index based on the degree of information change. The method can better evaluate the insulator grade. The case study shows that the model can accurately and quickly judge the state of composite insulators, which can be used as a reference for manufacturing and maintenance departments.

Przejdź do artykułu
[1] Liang X.D., Gao Y.F., Wang J.F. *et al*., *Rapid development of silicone rubber composite insulators in China*, High Voltage Engineering, vol. 42, no. 9, pp. 2888–2896 (2016), DOI: 10.13336/j.1003- 6520.hve.20160907025.

[2] Yuan Z.K., *Degradation characteristics and mechanism of materials in composite insulator under high humidity*, PhD Thesis, North China Electric Power University, Beijing (2019).

[3] Xoa Y.F., Song X.M., He J.Z. *et al*., *Evaluation method of aging for silicone rubber of compos- ite insulator*, Transactions of China Electrotechnical Society, vol. 34, no. S1, pp. 440–448 (2019), DOI: 10.19595/j.cnki.10006753.tces.L80647.

[4] Tang J., Liu Q.S., Liu J.W. *et al*., *Evaluation of composite insulators based on fuzzy comprehen- sive evaluation*, Engineering Journal of Wuhan University, vol. 52, no. 5, pp. 451–456 (2019), DOI: 10.14188/j.1671-8844.2019-05-012.

[5] Huang X., Wang L.Y., Wang Q. *et al*., *Grey fuzzy comprehensive evaluation model of contamina- tion state for insulators based on IFAHP with bayesian modified method*, Science Technology and Engineering, vol. 20, no. 13, pp. 5135–5141 (2020), DOI: CNKI:SUN:KXJS.0.2020-13-018.

[6] Liu Y.P., Xu Z.Q., Fu H.C. *et al*., *Insulation Condition Assessment Method of Power Transformer Based on Improved Extension Cloud Theory With Optimal Cloud Entropy*, High Voltage Engineering, vol. 46, no. 2, pp. 397–405 (2020), DOI: 10.13336/j.1003-6520.hve.20190215004.

[7] Fan L., Xia F., Su H.Y. *et al*., *Risk assessment of high voltage insulator contamination condition by cloud theory*, Power System Protection and Control, vol. 40, no. 15, pp. 57–62 (2012), DOI: CNKI: SUN:JDQW.0.2012-15-014.

[8] Wang S.H., Jing H., *Study on method for predicting pollution Flashover insulators in contact network*, Journal of the China Railway Society, vol. 40, no. 3, pp. 58–67 (2018), DOI: CNKI:SUN:TDXB.0.2018- 03-011.

[9] Huai M.Q., *Research on prediction of contamination state of insulator on catenary based on fuzzy neural network*, Master Thesis, Lanzhou Jiaotong University, Gansu (2018).

[10] Yang Z.C., Zhang C.L., Ge L. *et al*., *Comprehensive fuzzy evaluation based on entropy weight method for insulator flashover pollution*, Electric Power Automation Equipment, vol. 34, no. 4, pp. 90–94 (2014), DOI: CNKI:SUN:DLZS.0.2014-04-016.

[11] Zhou Y.M., *Studies on the degradation depth of silicon composite insulator in service*, Master Thesis, Wuhan University, Wuhan (2018).

[12] Chen X.C., Li L.Q., Wu Z.G. *et al*., *Research on shed properties of network operating composite insulators*, Guangdong Electric Power, vol. 29, no. 6, pp. 104–108 (2016), DOI: 10.3969/j.issn.1007- 290X.2016.06.019.

[13] Yang L.G., Florian Pauli, Kay Hameyer, *Influence of thermal-mechanical stress on the insulation system of a low voltage electrical machine*, Archives of Electrical Engineering, vol. 70, no. 1, pp. 233–244 (2021), DOI: 10.24425/aee.2021.136064.

[14] Liu Y., Wang J.G., Han F. *et al*., *Electrical and mechanical properties of composite insulators af- ter different operation periods*, High Voltage Engineering, vol. 34, no. 5, pp. 1017–1021 (2008), DOI: 10.13336/j.1003-6520.hve.2008.05.027.

[15] Yao L.N., Wu Y.H., Wang S.H. *et al*., *Electrical and mechanical properties of on-line compos- ite insulators*, Insulating Materials, vol. 48, no. 8, pp. 23–27 (2015), DOI: 10.16790/j.cnki.1009- 9239.im.2015.08.005.

[16] Jia Z.D., Yang C.X., Wang X.L. *et al*., *Aging characteristics of composite insulators based on hydropho- bicity transfer test*, High Voltage Engineering, vol. 41, no. 6, pp. 1907–1914 (2015), DOI: 10.13336/ j.1003-6520.hve.2015.06.019.

[17] Zhang M.M., *Research on evaluation Method of Insulator pollution Status Assessment Based on UV Pulse Parameters*, Master Thesis, Southwest Jiaotong University, Sichuan (2019).

[18] Mao Y.K., Guan Z.C., Wang L.M. *et al*., *Evaluation of contamination levels of outdoor insulators based on the principal components analysis of leakage current Pulse*, Transactions of China Electrotechnical Society, vol. 24, no. 8, pp. 39–45 (2009), DOI: 10.19595/j.cnki.10006753.tces.2009.08.007.

[19] Ning G.T., Fang B., Qin D. *et al*., *Design and application of comprehensive evaluation index system of smart grid based on coordinated planning of major network and power distribution network*, Archives of Electrical Engineering, vol. 70, no. 1, pp. 103–113 (2021), DOI: 10.24425/aee.2021.136055.

[20] Zhou L.L., *Research of methods and their application of determining the weights of attributes in fuzzy comprehensive evaluation*, Master Thesis, Northeastern University, Liaoning (2014).

[21] Li G., Li J.P., Sun X.L. *et al*., *Research on a combined methods of subjective-objective weighting and its rationality*, Management Review, vol. 29, no. 12, pp. 17–26+61 (2017), DOI: 10.14120/j.cnki. cn115057/f.2017.12.002.

[22] Chen Y.C., Dai J.Y., Xie D., *Comprehensive evaluation of mine ventilation system based on combi- nation weighting cloud model*, Systems Engineering, vol. 38, no. 6, pp. 35–42 (2020), DOI: 1001- 4098(2020)06-0035-08.

[23] Wang L.L., *Research on cleaner production evaluation index system and grade comprehensive evalua- tion methodologies of wastewater treatment plants in cities and towns*, PhD Thesis, Dalian University of Technology, Dalian (2015).

Słowa kluczowe:
ARFTPMSM
axial excitation device
flux regulation capability
HEM

Due to the fixed rotor magnetic field, the main magnetic flux of conventional permanent magnet synchronous motors (PMSMs) cannot be flexibly adjusted. Recently, the axial-radial flux type permanent magnet synchronous machine (ARFTPMSM) based on the hybrid excitation concept is proposed, which provides a new method for the speed and magnetic field regulations for PMSMs. To analyze the mechanism of magnetic field variation inside the ARFTPMSM, in this paper, three – dimensional finite element models for electromagnetic field calculation of the ARFTPMSM are established. On this basis, the influence of the axial device on the motor is discussed, and the mechanism of flux regulation is explained. By the quantitative calculation of air-gap flux density and the noload back-electromotive force (EMF), the flux regulation capability of the ARFTPMSM is verified. In addition, the effect of the excitation magnetomotive force on the magnetic field harmonics is analyzed combined with the winding theory, and the influence of the axial magneto-motive force (MMF) on the torque fluctuation is obtained. The flux regulation performance of the motor and the validity of the numerical calculation analysis are verified by the experiments.

Przejdź do artykułu
Słowa kluczowe:
distributed power flow controller
efficiency
FACTS
maximum power transfer capability
optimal location

Among the FACTS device, the distributed power flow controller (DPFC) is a superior device. This can be evaluated after eliminating the dc capacitor between shunt and series convertors of the unified power flow controller (UPFC) and placing a number of low rating single phase type distributed series convertors in the line instant of using single large rating three phase series convertors as in the UPFC. The power flow through this dc capacitor as in the UPFC now takes place through the transmission line at a third harmonic frequency in the DPFC. The DPFC uses the D-FACTS that allows the replacement of a large three-phase converter as in the UPFC by several small-size series convertors present in the DPFC. The redundancy of several series convertors increases the system’s reliability of the power system. Also, there is no requirement for high voltage isolation as series convertors of the DPFC are hanging as well as single-phase types. Consequently, the DPFC system has a lower cost than the UPFC system. In this paper, the equivalent ABCD parameters of the latest FACTSdeviceDPFChave been formulated with the help of an equivalent circuit model of the DPFC at the fundamental frequency component. Further, the optimal location in the transmission line and maximum efficiency of the DPFC along with Thyristor Controlled Series Compensator (TCSC), Static Synchronous Shunt Compensator (STATCOM) and UPFC FACTS devices have been investigated using an iteration program developed in MATLAB under steady-state conditions. The results obtained depict that the DPFC when placed slightly off-center at 0.33 fraction distance from the sending end comes up with higher performance. Whereas, when the TCSC, STATCOM and UPFC are placed at 0.16, 0.2815, 0.32 fraction distances from sending end respectively give their best performance.

Przejdź do artykułu
[1] Edris A.A., Aapa R., Baker M.H., Bohman L., Clark K., *Proposed terms and definitions for flexible ac transmission system (FACTS)*, IEEE Trans. on Power Delivery, vol. 12, no. 4, pp. 1848–1853 (1997), DOI: 10.1109/61.634216.

[2] Das D., Divan D.M., Harley R.G., *Power flow control in networks using controllable network transformers*, IEEE Trans. Power Electron., vol. 25, no. 7, pp. 1753–1760 (2010), DOI: 10.1109/ TPEL.2010.2042076.

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[4] Hingorani N.G., Gyugyi L., *Understanding FACTS: Concepts and Technology of Flexible AC Transmission Systems*, New York: IEEE Press (2000).

[5] Nabavi-Niak A., Iravani M.R., *Steady-State and Dynamic Models of Unified Power Flow Con- troller (UPFC) for Power System Studies*, IEEE Trans. Power Systems, vol. 11, no. 4 (1996), DOI: 10.1109/59.544667.

[6] Harjeet Johal, Deepak Divan, *Design Considerations for Series-Connected Distributed FACTS Converters*, IEEE Transactions on Industry Applications, vol. 43, iss. 6, pp. 1609–1618 (2007), DOI: 10.1109/TIA.2007.908174.

[7] Amir Hamidi, Sajjad Golshannavaz, Daryoush Nazarpour, *D-FACTS Cooperation in Renewable Inte- grated Microgrid A Linear Multiobjective Approach*, IEEE Transactions on Sustainable Energy, vol. 10, iss. 1, pp. 355–363 (2019), DOI: 10.1109/TSTE.2017.2723163.

[8] Narasimha Rao D., Srinivasa Varma P., *Comparison of UPFC and DPFC*, Journal of Critical Reviews, vol. 7, iss. 6 (2020), DOI: 10.31838/jcr.07.06.153.

[9]Abhilash Sen, Atanu Banerjee, Haricharan Nannam, *A Comparative Analysis between UPFC and DPFC in a Grid Connected Photovoltaic System*, IEEE International Conference on Intelligent Techniques in Control, Optimization and Signal Processing (INCOS) (2019), DOI: 10.1109/IN- COS45849.2019.8951352.

[10] Song W., Zhou X., Zhigang Liang, Subhashish Bhattacharya, Huang Alex Q., *Modeling and Control Design of Distributed Power Flow Controller based-on Per-phase control*, IEEE Energy Conversion Congress and Exposition (2009), DOI: 10.1109/ECCE.2009.5316307.

[11] Chandrakar V.K., Kothari A.G., *Optimal Location for Line Compensation by Shunt Connected FACTS Controller*, IEEE Power Electronics and Drive Systems Conference, vol. 1, pp. 151–156 (2003), DOI: 10.1109/PEDS.2003.1282736.

[12] Vikash Anand, Sanjeev Kumar Mallik, *Power flow analysis and control of distributed FACTS de- vices in power system*, *Archives of Electrical Engineering*, vol. 67, no. 3, pp. 545–561 (2018), DOI: 10.24425/123662.

[13] Zhihui Yuan, de Haan Sjoerd W.H., Ferreira Jan A., *Construction and first result of a scaled transmis- sion system with the Distributed Power Flow Controller (DPFC)*, 13th European Conference on Power Electronics and Applications (2009).

[14] Zhihui Yuan, de Haan Sjoerd W.H., Braham Frreira, Dalibor Cevoric, *A FACTS Device: Dis- tributed Power Flow Controller (DPFC)*, IEEE Trans. Power Electronics, vol. 25, no. 10 (2010), DOI: 10.1109/TPEL.2010.2050494.

[15] Syona Chawla, Sheetal Garg, Bhawna Ahuja, *Optimal Location of Series-Shunt FACTS Device for Transmission Line Compensation*, IEEE Control, Automation, Communication and Energy Conservation Conference, pp. 1–6 (2009).

[16] Shehata Ahmed A., Refaat Ahmed, Mamdouh K. Ahmed, Korovkin Nikolay V., *Optimal placement and sizing of FACTS devices based on Autonomous Groups Particle Swarm Optimization technique*, AEE, vol. 70, no. 1, pp. 161–172 (2021), DOI: 10.24425/aee.2021.136059.

[17] Riadh Essaadali, Anwar Jarndal, Ammar B. Kouki, Fadhel M. Ghannouch, *Conversion Rules Between X-Parameters and Linearized Two-Port Network Parameters for Large-Signal Operating Conditions*, IEEE Transactions on Microwave Theory and Techniques, vol. 66, no. 11, pp. 4745 4756 (2018), DOI: 10.1109/TMTT.2018.2863227.

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[21] Mengmeng Xiao, Shaorong Wang, Yong Huang, Chao Zheng, Qiushi Xu, Hongsheng Zhao, Aihong Tang, Dichen Liu, *Two-Level Control Method for DPFC Series Units Based on PLC Communication*, 2nd IEEE Conference on Energy Internet and Energy System Integration (EI2), pp. 20–22 (2018), DOI: 10.1109/EI2.2018.8582109.

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[24] Seyed Abbas Taher, Muhammad Karim Amooshahi, *New approach for optimal UPFC placement using hybrid immune algorithm in electric power systems*, International Journal of Electrical Power and Energy Systems, vol. 43, iss. 1, pp. 899–909 (2012), DOI: 10.1016/j.ijepes.2012.05.064.

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 109-124
| DOI: 10.24425/aee.2022.140200

Słowa kluczowe:
additional damping controller
CSP
HMIDC
LFO
mixed H2/H∞ control
TLS-ESPRIT

A hybrid multi-infeed HVDC (HMIDC) system is composed of line-commutated converter-based high-voltage direct current (LCC-HVDC) and voltage-source converterbased high-voltage direct current (VSC-HVDC), whose receiving ends have electrical coupling. To solve the problem of low-frequency oscillation (LFO) caused by insufficient damping in the HMIDC system, according to the multi-objective mixed H2/H∞ output feedback control theory with regional pole assignment, an additional robust damping controller is designed in this paper, which not only has good robustness, but also has strong adaptability to the change of system operation mode. In the paper, the related oscillation modes and transfer function of the controlled plant are obtained, which are identified by the total least squares estimation of signal parameters via rotary invariance technology (TLS-ESPRIT). In addition, the control-sensitive point (CSP) for suppressing LFO based on the small disturbance test method is determined, which is suitable for determining the installation position of the controller. The results show that the control sensitivity factor of VSC-HVDC is greater than that of LCC-HVDC so that adding an additional damping controller to VSC-HVDC is better than LCC-HVDC in suppressing the LFO of HMIDC. Finally, a hybrid double infeed DC transmission system with three receiving terminals is built on PSCAD/EMTDC where the time-domain simulations are performed to verify the correctness of the CSP selection and the effectiveness of the controller.

Przejdź do artykułu
Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 125-138
| DOI: 10.24425/aee.2022.140201

Słowa kluczowe:
electromagnetic torque measurements
inductance measurements
simulations
switched reluctance motors

The performance of drives with switched reluctance motors (SRMs) depends on magnetic materials used in their construction which influence static parameters such as inductance and electromagnetic torque profiles. The paper deals with simulations of switched reluctance motors in the finite element method and their comparison with measurements. Two kinds of switched reluctance motors were analysed, the modified Emerson Electric motor with a laminated steel core and a prototype, the one with a magnetic core made of iron-based powder composite materials. In the first part of the research, magnetization curves of magnetic materials were measured for static and dynamic conditions with 50 Hz. Next, simulations and measurements of inductance and developed torque were compared and analysed. In the last part of the research, simulations of magnetic flux density in motors were conducted. As the result of the research, it occurred that the simulations and measurements are quite close and two kinds of motors exhibit similar performance.

Przejdź do artykułu
[1] Miller T.J.E., Brushless permanent-magnet and reluctance motor drives, Oxford University Press (1989).

[2] Krishnan R., Switched reluctance motor drives: modelling, simulation, analysis, design, and applications, CRC Press (2001).

[3] Ahn J.-W., Switched reluctance motor, in book Torque control Ed. Lamchich M.T., Intech (2011), DOI: 10.5772/10520.

[4] Lawrenson P.J., Stephenson J.M., Blenkinsop P.T., Corda J., Fulton N.N., Variable-speed switched reluctance motors, IEE Proceedings B. (Electric Power Applications), vol. 127, no. 4, pp. 253–265 (1980), DOI: 10.1049/ip-b.1980.0034.

[5] Widmer J.D., Martin R., Kimiabeigi M., Electric vehicle traction motors without rare earth magnets, Sustainable Materials and Technologies, vol. 3, pp. 7–13 (2015), DOI: 10.1016/j.susmat.2015.02.001.

[6] Riba J.-R., López-Torres C., Romeral L., Garcia A., Rare-earth-free propulsion motors for electric vehicles: A technology review, Renewable and Sustainable Energy Reviews, vol. 57, pp. 367–379 (2016), DOI: 10.1016/j.rser.2015.12.121.

[7] Nakamura H., The current and future status of rare earth permanent magnets, Scripta Materialia, vol. 154, pp. 273–276 (2018), DOI: 10.1016/j.scriptamat.2017.11.010.

[8] Coey J.M.D., Magnetism and Magnetic Materials, Cambridge University Press (2010).

[9] Shokrollahi H., Janghorban K., Soft magnetic composite materials (SMCs), Journal of Materials Processing Technology, vol. 189, no. 1–3, pp. 1–12 (2007), DOI: 10.1016/j.jmatprotec.2007.02.034.

[10] Périgo E.A.,Weidenfeller B., Kollár P., Füzer J., Past, present, and future of soft magnetic composites, Applied Physics Reviews, vol. 5, no. 3 (2018), DOI: 10.1063/1.5027045.

[11] Przybylski M., Modelling and analysis of the low-power 3-phase switched reluctance motor, Archives of Electrical Engineering, vol. 68, no. 2, pp. 443–454 (2019), DOI: 10.24425/aee.2019.128279.

[12] Przybylski M., Slusarek B., Di Barba P., Mognaschi M.E.,Wiak S., Temperature and torque measurements of switched reluctance actuator with composite or laminated magnetic cores, Sensors, vol. 20, no. 3065, pp. 1–14 (2020), DOI: 10.3390/s20113065.

[13] Meeker D., Finite element method magnetics – User’s manual, ver. 4.2 (2018).

[14] Miller T.J.E., Optimal design of switched reluctance motors, IEEE Transactions on Industrial Electronics, vol. 49, no. 1, pp. 15–27 (2002), DOI: 10.1109/41.982244.

Przejdź do artykułu
[2] Krishnan R., Switched reluctance motor drives: modelling, simulation, analysis, design, and applications, CRC Press (2001).

[3] Ahn J.-W., Switched reluctance motor, in book Torque control Ed. Lamchich M.T., Intech (2011), DOI: 10.5772/10520.

[4] Lawrenson P.J., Stephenson J.M., Blenkinsop P.T., Corda J., Fulton N.N., Variable-speed switched reluctance motors, IEE Proceedings B. (Electric Power Applications), vol. 127, no. 4, pp. 253–265 (1980), DOI: 10.1049/ip-b.1980.0034.

[5] Widmer J.D., Martin R., Kimiabeigi M., Electric vehicle traction motors without rare earth magnets, Sustainable Materials and Technologies, vol. 3, pp. 7–13 (2015), DOI: 10.1016/j.susmat.2015.02.001.

[6] Riba J.-R., López-Torres C., Romeral L., Garcia A., Rare-earth-free propulsion motors for electric vehicles: A technology review, Renewable and Sustainable Energy Reviews, vol. 57, pp. 367–379 (2016), DOI: 10.1016/j.rser.2015.12.121.

[7] Nakamura H., The current and future status of rare earth permanent magnets, Scripta Materialia, vol. 154, pp. 273–276 (2018), DOI: 10.1016/j.scriptamat.2017.11.010.

[8] Coey J.M.D., Magnetism and Magnetic Materials, Cambridge University Press (2010).

[9] Shokrollahi H., Janghorban K., Soft magnetic composite materials (SMCs), Journal of Materials Processing Technology, vol. 189, no. 1–3, pp. 1–12 (2007), DOI: 10.1016/j.jmatprotec.2007.02.034.

[10] Périgo E.A.,Weidenfeller B., Kollár P., Füzer J., Past, present, and future of soft magnetic composites, Applied Physics Reviews, vol. 5, no. 3 (2018), DOI: 10.1063/1.5027045.

[11] Przybylski M., Modelling and analysis of the low-power 3-phase switched reluctance motor, Archives of Electrical Engineering, vol. 68, no. 2, pp. 443–454 (2019), DOI: 10.24425/aee.2019.128279.

[12] Przybylski M., Slusarek B., Di Barba P., Mognaschi M.E.,Wiak S., Temperature and torque measurements of switched reluctance actuator with composite or laminated magnetic cores, Sensors, vol. 20, no. 3065, pp. 1–14 (2020), DOI: 10.3390/s20113065.

[13] Meeker D., Finite element method magnetics – User’s manual, ver. 4.2 (2018).

[14] Miller T.J.E., Optimal design of switched reluctance motors, IEEE Transactions on Industrial Electronics, vol. 49, no. 1, pp. 15–27 (2002), DOI: 10.1109/41.982244.

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 139-157
| DOI: 10.24425/aee.2022.140202

Słowa kluczowe:
battery modeling
equivalent circuit
estimation algorithm
lithium-ion battery energy storage
simulation
state of charge (SOC)

The use of lithium-ion battery energy storage (BES) has grown rapidly during the past year for both mobile and stationary applications. For mobile applications, BES units are used in the range of 10–120 kWh. Power grid applications of BES are characterized by much higher capacities (range of MWh) and this area particularly has great potential regarding the expected energy system transition in the next years. The optimal operation of BES by an energy storage management system is usually predictive and based strongly on the knowledge about the state of charge (SOC) of the battery. The SOC depends on many factors (e.g. material, electrical and thermal state of the battery), so that an accurate assessment of the battery SOC is complex. The SOC intermediate prediction methods are based on the battery models. The modeling of BES is divided into three types: fundamental (based on material issues), electrical equivalent circuit (based on electrical modeling) and balancing (based on a reservoir model). Each of these models requires parameterization based on measurements of input/output parameters. These models are used for SOC modelbased calculation and in battery system simulation for optimal battery sizing and planning. Empirical SOC assessment methods currently remain the most popular because they allow practical application, but the accuracy of the assessment, which is the key factor for optimal operation, must also be strongly considered. This scientific contribution is divided into two papers. Paper part I will present a holistic overview of the main methods of SOC assessment. Physical measurement methods, battery modeling and the methodology of using the model as a digital twin of a battery are addressed and discussed. Furthermore, adaptive methods and methods of artificial intelligence, which are important for the SOC calculation, are presented. In paper part II, examples of the application areas are presented and their accuracy is discussed.

Przejdź do artykułu
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Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 159-174
| DOI: 10.24425/aee.2022.140203

Słowa kluczowe:
interior permanent magnet brushless DC motor
quasi-three-dimensional electromagnetic-field circuit coupling analysis
stator slot skew

This paper takes a 50 kW interior permanent magnet brushless DC motor as an example, and explores the influence of the degree of stator slot skew on the overall motor magnetic density and air gap magnetic density; then reveals the influences of stator slot skewed structure on a series of key electromagnetic properties like no-load back electromotive force (B-EMF), cogging torque, electromagnetic torque, torque fluctuation, electromagnetic loss, input power, output power and operating efficiency. On this basis, a relatively best range of the skew degrees is obtained. The research work in this paper has direct reference value for the further improvement of design and manufacture, operation and maintenance, control and protection of such motors.

Przejdź do artykułu
[1] Zhang Chen, Principle and Application of Brushless DC Motor, China Machinery Industry Press, Beijing (1996).

[2] Tang Renyuan, Modern Permanent Magnet Motor Theory and Design, Mechanical Industry Press, Beijing (2005).

[3] LiWeiqi, LinRongwen, Tao Tao, Optimized design based on the air gap length of the built-in permanent magnet brushless DC motor, Electric Switchgear, vol. 58, no. 05, pp. 58–63 (2020).

[4] Parsa L., Hao L., Interior Permanent Magnet Motors with Reduced Torque Pulsation, IEEE Transactions on Industrial Electronics, vol. 55, no. 2, pp. 602–609 (2008), DOI: 10.1109/TIE.2007.911953.

[5] Ren Dejiang, Huang Qu, Li Jianjun, Wu Ning, Cogging torque optimization analysis of built-in permanent magnet synchronous motor, Explosion-Proof Electric Machine, vol. 54, no. 4, pp. 4–7+43 (2019).

[6] Zhao W., Lipo T.A., Kwon B., Torque Pulsation Minimization in Spoke-type Interior Permanent Magnet Motors with Skewing and Sinusoidal Permanent Magnet Configurations, IEEE Transactions on Magnetics, vol. 51, no. 11, pp. 1–4 (2015), DOI: 10.1109/TMAG.2015.2442977.

[7] AimengW., Heming L.,Weifu L., Haisen Z., Influence of skewed and segmented magnet rotor on IPM machine performance and ripple torque for electric traction, IEEE International Electric Machines and Drives Conference, pp. 305–310 (2009), DOI: 10.1109/IEMDC.2009.5075222.

[8] Adrian Młot, Marcin Kowol, Janusz Kołodziej, Andrzej Lechowicz, Piotr Skrobotowicz, Analysis of IPM motor parameters in an 80-kW traction motor, Archives of Electrical Engineering, vol. 69, no. 2 (2020), DOI: 10.24425/aee.2020.133038.

[9] Yang Zhihao, Yang Mengxue, Wang Sinuo, Bao Xiaohua, The influence of stator skew on the performance of permanent magnet synchronous motors, Transactions of the Chinese Society of Electrical Engineering, vol. 14, no. 3, pp. 97–102 (2019).

[10] Wang Dongliang, Chen Wei, Discussion on the electromagnetic design of concentrated winding permanent magnet motor from the perspective of torque fluctuation, Electric Tool, vol. 4, pp. 15–17 (2017), DOI: 10.16629/j.cnki.1674-2796.2017.04.004.

[11] Xiaodong S., Zhou S., Long C., Zebin Y., Skew Angle Optimization Analysis of a Permanent Magnet Synchronous Motor for EVs, IEEE International Conference on Applied Superconductivity and Electromagnetic Devices (ASEMD), pp. 1–2 (2018), DOI: 10.1109/ASEMD.2018.8558826.

[12] Wang Changcheng, Guo Hui, Sun Pei, Liu Ningning,Wang Yansong, Qin Yifei, A method for reducing cogging torque of permanent magnet synchronous motors, Light Industry Machinery, vol. 36, no. 6, pp. 62–66 (2018).

[13] He Qiang, Magnetic field analysis and cogging torque study of brushless DC permanent magnet motors, Hefei University of Technology (2016).

[14] Hongwei Fang, Hongxu Chen, Analysis and reduction of the cogging torque of flux-modulated generator for wave energy conversion, Energy Procedia, vol. 158, pp. 327–332 (2019), DOI: 10.1016/j.egypro.2019.01.097.

[15] Fu Lixin et al., GB/T 1029-2005 Three-phase synchronous motor test method, China Standard Press, Beijing (2006).

Przejdź do artykułu
[2] Tang Renyuan, Modern Permanent Magnet Motor Theory and Design, Mechanical Industry Press, Beijing (2005).

[3] LiWeiqi, LinRongwen, Tao Tao, Optimized design based on the air gap length of the built-in permanent magnet brushless DC motor, Electric Switchgear, vol. 58, no. 05, pp. 58–63 (2020).

[4] Parsa L., Hao L., Interior Permanent Magnet Motors with Reduced Torque Pulsation, IEEE Transactions on Industrial Electronics, vol. 55, no. 2, pp. 602–609 (2008), DOI: 10.1109/TIE.2007.911953.

[5] Ren Dejiang, Huang Qu, Li Jianjun, Wu Ning, Cogging torque optimization analysis of built-in permanent magnet synchronous motor, Explosion-Proof Electric Machine, vol. 54, no. 4, pp. 4–7+43 (2019).

[6] Zhao W., Lipo T.A., Kwon B., Torque Pulsation Minimization in Spoke-type Interior Permanent Magnet Motors with Skewing and Sinusoidal Permanent Magnet Configurations, IEEE Transactions on Magnetics, vol. 51, no. 11, pp. 1–4 (2015), DOI: 10.1109/TMAG.2015.2442977.

[7] AimengW., Heming L.,Weifu L., Haisen Z., Influence of skewed and segmented magnet rotor on IPM machine performance and ripple torque for electric traction, IEEE International Electric Machines and Drives Conference, pp. 305–310 (2009), DOI: 10.1109/IEMDC.2009.5075222.

[8] Adrian Młot, Marcin Kowol, Janusz Kołodziej, Andrzej Lechowicz, Piotr Skrobotowicz, Analysis of IPM motor parameters in an 80-kW traction motor, Archives of Electrical Engineering, vol. 69, no. 2 (2020), DOI: 10.24425/aee.2020.133038.

[9] Yang Zhihao, Yang Mengxue, Wang Sinuo, Bao Xiaohua, The influence of stator skew on the performance of permanent magnet synchronous motors, Transactions of the Chinese Society of Electrical Engineering, vol. 14, no. 3, pp. 97–102 (2019).

[10] Wang Dongliang, Chen Wei, Discussion on the electromagnetic design of concentrated winding permanent magnet motor from the perspective of torque fluctuation, Electric Tool, vol. 4, pp. 15–17 (2017), DOI: 10.16629/j.cnki.1674-2796.2017.04.004.

[11] Xiaodong S., Zhou S., Long C., Zebin Y., Skew Angle Optimization Analysis of a Permanent Magnet Synchronous Motor for EVs, IEEE International Conference on Applied Superconductivity and Electromagnetic Devices (ASEMD), pp. 1–2 (2018), DOI: 10.1109/ASEMD.2018.8558826.

[12] Wang Changcheng, Guo Hui, Sun Pei, Liu Ningning,Wang Yansong, Qin Yifei, A method for reducing cogging torque of permanent magnet synchronous motors, Light Industry Machinery, vol. 36, no. 6, pp. 62–66 (2018).

[13] He Qiang, Magnetic field analysis and cogging torque study of brushless DC permanent magnet motors, Hefei University of Technology (2016).

[14] Hongwei Fang, Hongxu Chen, Analysis and reduction of the cogging torque of flux-modulated generator for wave energy conversion, Energy Procedia, vol. 158, pp. 327–332 (2019), DOI: 10.1016/j.egypro.2019.01.097.

[15] Fu Lixin et al., GB/T 1029-2005 Three-phase synchronous motor test method, China Standard Press, Beijing (2006).

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 175-187
| DOI: 10.24425/aee.2022.140204

Słowa kluczowe:
future technology trends
high specific output electrical machines
new manufacturing techniques and materials
thermal management

The continuous drive towards electrified propulsion systems has been imposing ever more demanding performance and cost targets for the future power electronics, machines and drives (PEMDs). This is particularly evident when exploring various technology road mapping documents both for automotive and aerospace industries, e.g. Advanced Propulsion Centre (APC) UK, Aerospace Technology Institute (ATI) UK, National Aeronautics and Space Administration (NASA) USA and others. In that context, a significant improvement of the specific performance and cost measures, e.g. power density increase by a factor of 10 or more and/or cost per power unit reduction by 50% or better, is forecasted for the next 5 to 15 years. However, the existing PEMD solutions are already at their technological limits to some degree. Consequently, meeting the performance and cost step change would require a considerable development effort. This paper is focused on electrical machines and their thermal management, which has been recognised as one of key enabling factors for delivering high specific output solutions. The challenges associated with heat removal in electrical machines are discussed in detail, alongside with new concepts of thermal management systems. Several examples from the available literature are presented. These include manufacturing techniques, new materials and novel integrated designs in application to electrical machines.

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[31] Gai Y., Widmer J.D., Steven A., Chong Y.C., Kimiabeigi M., Goss J., Popescu M., Numerical and Experimental Calculations of CHTC in an Oil-Based Shaft Cooling System for a High-Speed High- Power PMSM, IEEE Transactions on Industrial Electronics, vol. 67, no. 6, pp. 4371–4380 (2020), DOI: 10.1109/TIE.2019.2922938.

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[33] Brown G.V., Cryogenic Electric Motor Tested, NASA report – propulsion and power (2005).

[34] Arndt T., Basic Considerations and Recent Results in HTS Device Developments for Electric Aircraft, Safran-Group h Scientific Day, Paris, France (2020).

[35] ASuMED – Deliverable System Topology Report, 2017.

Przejdź do artykułu
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[14] Wrobel R., Mellor P.H., Popescu M., Staton D.A., Power Loss Analysis in Thermal Design of Permanent-Magnet Machines - Review, IEEE Transactions on Industry Applications, vol. 52, no. 2, pp. 1359–1368 (2016), DOI: 10.1109/TIA.2015.2489599.

[15] Liu H., Ayat S.,Wrobel R., Zhang C., Comparative Study of Thermal Properties of ElectricalWindings Impregnated with Alternative Varnish Materials, IET Journal of Engineering, vol. 2019, no. 17, pp. 3736–3741 (2019), DOI: 10.1049/joe.2018.8198.

[16] Ayat S., Liu H., Kulan M., Wrobel R., Estimation of Equivalent Thermal Conductivity for Electrical Windings with High Conductor Fill Factor, IEEE Energy Conversion Congress and Exposition (ECCE), pp. 6529–6536 (2018).

[17] Wrobel R., Ayat S., Godbehere J., A Systematic Experimental Approach in Deriving Stator-Winding Heat Transfer, IEEE International Electric Machines and Drives Conference (IEMDC), pp. 1–8 (2017).

[18] Chiodetto N., Mecrow B.,Wrobel R., Lisle T., Elector-Mechanical Challenges in the Design of a High- Speed-High-Power-PMSM Rotor for an Aerospace Application, IEEE Energy Conversion Congress and Exposition (ECCE), Baltimore, MD, pp. 3944–3951 (2019).

[19] Gerada D., Mebarki A., Brown N.L., Gerada C., Cavagnino A., Boglietti A., High-Speed Electrical Machines: Technologies, Trends, and Developments, in IEEE Transactions on Industrial Electronics, vol. 61, no. 6, pp. 2946–2959 (2014), DOI: 10.1109/TIE.2013.2286777.

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[23] Wrobel R., Hussein A., A Feasibility Study of Additively Manufactured Heat Guides for Enhanced Heat Transfer in Electrical Machines, IEEE Transactions on Industry Applications, vol. 56, no. 1, pp. 205–215 (2020), DOI: 10.1109/TIA.2019.2949258.

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[25] Sixel W., Liu M., Nellis G., Sarlioglu B., Ceramic 3D Printed Direct Winding Heat Exchangers for Improving Electric Machine Thermal Management, IEEE Energy Conversion Congress and Exposition (ECCE), pp. 769–776 (2019).

[26] Lindh P., Petrov I., Pyrhonen J., Scherman E., Niemela M., Immonen P., Direct Liquid Cooling Method Verified with a Permanent-Magnet Traction Motor in a Bus, IEEE Transactions on Industry Applications, vol. 55, no. 4, pp. 4183–4191 (2019), DOI: 10.1109/TIA.2019.2908801.

[27] Lorenz F., Rudolph J.,Werner R., Design of 3D printed High Performance Windings for Switched Reluctance Machines, International Conference on Electrical Machines (ICEM), pp. 2451–2457 (2018).

[28] Pyrhonen J., Montonen J., Lindh P., Vauterin J., Otto M., Replacing Copper with New Carbon Nanomaterials in Electrical Machine Windings, International Review of Electrical Engineering, pp. 12–21 (2015), DOI: 10.15866/IREE.V10I1.5253.

[29] Wohlers C., Juris P., Kabelac S., Ponick B., Design and Direct Liquid Cooling of Tooth-Coil Winding, Electrical Engineering, Springer, vol. 100, no. 4, pp. 2299–2308 (2018), DOI: 10.1007/s00202-018-0704-x.

[30] Ayat S., Daguese B., Khazaka R., Design Considerations ofWindings Formed with Hollow Conductors Cooled with Phase Change Material, IEEE Energy Conversion Congress and Exposition (ECCE), Baltimore, MD, pp. 5539–5546 (2019).

[31] Gai Y., Widmer J.D., Steven A., Chong Y.C., Kimiabeigi M., Goss J., Popescu M., Numerical and Experimental Calculations of CHTC in an Oil-Based Shaft Cooling System for a High-Speed High- Power PMSM, IEEE Transactions on Industrial Electronics, vol. 67, no. 6, pp. 4371–4380 (2020), DOI: 10.1109/TIE.2019.2922938.

[32] Davin T., Pelle J., Harmand S., You R., Experimental Study of Oil Cooling System for Electric Motors, Applied Thermal Engineering, Elsevier, vol. 75, no. 2, pp. 1–13 (2015), DOI: 10.1016/j.applthermaleng.2014.10.060.

[33] Brown G.V., Cryogenic Electric Motor Tested, NASA report – propulsion and power (2005).

[34] Arndt T., Basic Considerations and Recent Results in HTS Device Developments for Electric Aircraft, Safran-Group h Scientific Day, Paris, France (2020).

[35] ASuMED – Deliverable System Topology Report, 2017.

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 189-209
| DOI: 10.24425/aee.2022.140205

Słowa kluczowe:
inter-harmonic
parameter identification
power system
synchrosqueezed transform,time-frequency analysis

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13
Dual-mode control magnetically-coupled energy storage inductor boost inverter for renewable energy

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 211-225
| DOI: 10.24425/aee.2022.140206

Słowa kluczowe:
inter-harmonic
parameter identification
power system
synchrosqueezed transform
time-frequency analysis

A novel magnetically-coupled energy storage inductor boost inverter circuit for renewable energy and the dual-mode control strategy with instantaneous value feedback of output voltage are proposed. In-depth research and analysis on the circuit, control strategy, voltage transmission characteristics, etc., providing the parameter design method of magnetically-coupled energy storage inductors and output filter. The circuit topology is cascaded by the input source ��in, the input filter ��in, a single-phase inverter bridge with a magnetically-coupled energy storage inductor, and a CL filter; The control strategy serves the output voltage as a reference to achieve the switch of step-down and step-up modes smoothly. The simulation results of a 1000 VA 100–200 VDC, 220 V 50 Hz AC inverter show that the proposed inverter can realize single-stage boost power conversion, which can adapt to resistive, capacitive and inductive loads, has high power density and low output waveform distortion. It has good application prospects in small and medium-capacity single-phase inverter occasions with low input voltage.

Przejdź do artykułu

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14
Numerical assessment of energy generation from photovoltaic cells using the CM-SAF PVGIS database

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 227-243
| DOI: 10.24425/aee.2022.140207

Słowa kluczowe:
energy generation
photovoltaic cell
renewable energy sources
solar radiation

The main objective of this article is to assess the legitimacy of using different tracking systems applied to the photovoltaic panels, for the city of Wroclaw (Poland), using 2 numerical tools: the CM SAF (Climate Monitoring Satellite Application Facility) and PVGIS (Photovoltaic Geographical Information System). In order to identify the solar irradiation, the CM-SAF database (based on the measurements of MFG – Meteosat First Generation – and MSG – Meteosat Second Generation – satellites) was utilised, while the PVGIS (Photovoltaic Geographical Information System) – to calculate the energy yield from PV panels. Particular attention was given to the optimisation of the annual tilt angle and the determination of the energy benefits from the implementation of the various sun tracking systems. Conducted studies showed that up to 30% more electricity yearly can be yielded after the replacement of PV cells with optimally fixed both azimuth and tilt angles by the 2-axis tracking system (179 kWh/m2 instead of 138 kWh/m2). Moreover, by the adequate decreasing of tilt angles in the summer time or obtaining the most favourable local solar exposure conditions, the supply curve of PV units may be significantly flattened, which may be beneficial when energy storage systems have low capacities.

Przejdź do artykułu
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Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 245-263
| DOI: 10.24425/aee.2022.140208

Słowa kluczowe:
barrier function
chattering
gains adaptation
induction motor drive
slidingmode control
super twisting

This paper proposes two high-order sliding mode algorithms to achieve highperformance control of induction motor drive. In the first approach, the super-twisting algorithm (STA) is used to reduce the chattering effect and to improve control accuracy. The second approach combines the super-twisting algorithm with a quasi-barrier function technique. While the super-twisting algorithm (STA) aims at the chattering reduction, the Barrier super-twisting algorithm (BSTA) aims to eliminate this phenomenon by providing continuous output control signals. The BSTA is designed to prevent the STA gain from being over-estimated by making these gains to decrease and increase according to system’s uncertainties. Stability and finite-time convergence are guaranteed using Lyapunov’s theory. In addition, the two controlled variables, rotor speed, and rotor flux modulus are estimated based on the second-order sliding mode (SOSM) observer. Finally, simulations are carried out to compare the performance and robustness of two control algorithms without adding the equivalent control. Tests are achieved under external load torque, varying reference speed, and parameter variations.

Przejdź do artykułu

[1] Senthilnathan N., *Comparative analysis of line-start permanent magnet synchronous motor and squirrel cage induction motor under customary power quality indices*, Electrical Engineering, vol. 102, no. 3, pp. 1339–1349 (2020), DOI: 10.1007/s00202-020-00955-2.

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[5] Steinberger M., Horn M., Fridman L., *Variable-Structure Systems and Sliding-Mode Control: From Theory to Practice*, Springer International Publishing (2020).

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[7] Siddique N., Rehman F.U., *Hybrid synchronization and parameter estimation of a complex chaotic network of permanent magnet synchronous motors using adaptive integral sliding mode control*, Archives of Electrical Engineering, pp. 137056–137056 (2021), DOI: 10.24425/bpasts.2021.137056.

[8] Quintero-Manriquez E., Sánchez E., Felix R., *Induction motor torque control via discrete-time sliding mode*, World Autom. Congr., WAC, pp. 1–5 (2016), DOI: 10.1109/WAC.2016.7582984.

[9] Martínez-Fuentes C.A., Ventura U.P., Fridman L., *Chattering analysis of Lipschitz continuous sliding-mode controllers*, ArXiv200400819 Cs Eess (2020).

[10] Utkin V., Poznyak A., Orlov Y.V., Polyakov A., *Chattering Problem in Road Map for Sliding Mode Control Design*, Springer International Publishing, pp. 73–82 (2020), DOI: 10.1007/978-3-030- 41709-3.

[11] Chaabane H., Djalal Eddine K., Salim C., *Sensorless back stepping control using a Luenberger observer for double-star induction motor*, Archives of Electrical Engineering, vol. 69, no. 1, (2020), DOI: 10.24425/aee.2020.131761.

[12] Swikir A., Utkin V., *Chattering analysis of conventional and super twisting sliding mode control algorithm*, in 2016 14th International Workshop on Variable Structure Systems (VSS), pp. 98–102 (2016), DOI: 10.1109/VSS.2016.7506898.

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[14] Sun X., Cao J., Lei G., Zhu J., *A Composite Sliding Mode Control for SPMSM Drives Based on a New Hybrid Reaching Law With Disturbance Compensation*, IEEE Transactions on Transportation Electrification, vol. 7, no. 3, pp. 1427–1436 (2021), DOI: 10.1109/TTE.2021.3052986.

[15] Jin Z., Sun X., Lei G., Zhu J., *Sliding Mode Direct Torque Control of SPMSMs Based on a Hybrid Wolf Optimization Algorithm*, IEEE Transactions on Industrial Electronics (2021), DOI: 10.1109/ TIE.2021.3080220.

[16] Pérez-Ventura U., Fridman L., *Design of super-twisting control gains: A describing function based methodology*, Automatica, vol. 99, pp. 175–180 (2019), DOI: 10.1016/j.automatica.2018.10.023.

[17] Lascu C., Argeseanu A., Blaabjerg F., *Super twisting Sliding-Mode Direct Torque and Flux Control of Induction Machine Drives*, IEEE Transactions on Power Electronics, vol. 35, no. 5, pp. 5057–5065 (2020), DOI: 10.1109/TPEL.2019.2944124.

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[19] Zhang L., Laghrouche S., Harmouche M., Cirrincione M., *Super twisting control of linear induction motor considering end effects with unknown load torque*, in 2017 American Control Conference (ACC), Seattle, USA, pp. 911–916 (2017), DOI: 10.23919/ACC.2017.7963069.

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[34] Dávila A., Moreno J.A., Fridman L., *Optimal Lyapunov function selection for reaching time estimation of Super Twisting algorithm*, in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, Shanghai, China, pp. 8405–8410 (2009), DOI: 10.1109/CDC.2009.5400466.

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[37] Rolek J., Utrata G., Kaplon A., *Robust speed estimation of an induction motor under the conditions of rotor time constant variability due to the rotor deep-bar effect*, Archives of Electrical Engineering, vol. 69, no. 2, pp. 319–333 (2020), DOI: 10.24425/aee.2020.133028.

[38] Kiani B., Mozafari B., Soleymani S., Mohammad Nezhad Shourkaei H., *Predictive torque control of induction motor drive with reduction of torque and flux ripple*, Archives of Electrical Engineering (2021), DOI: 10.24425/bpasts.2021.137727.

Archives of Electrical Engineering | 2022 | vol. 71 | No 1
| 265-275
| DOI: 10.24425/aee.2022.140209

Słowa kluczowe:
arbitrary meshes
finite-difference operators
partial finite difference operators
periodic functions
two-variable periodic functions

This paper presents novel discrete differential operators for periodic functions of one- and two-variables, which relate the values of the derivatives to the values of the function itself for a set of arbitrarily chosen points over the function’s area. It is very characteristic, that the values of the derivatives at each point depend on the function values at all points in that area. Such operators allow one to easily create finite-difference equations for boundaryvalue problems. The operators are addressed especially to nonlinear differential equations.

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[17] Sobczyk T.J., Radzik M., Tulicki J., Direct steady-state solutions for circuit models of nonlinear electromagnetic devices, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 40, no. 3, pp. 660–675 (2021), DOI: 10.1108/COMPEL-10-2020-0324.

[18] Sobczyk T.J., Jaraczewski M., Application of discrete differential operators of periodic functions to solve 1D boundary-value problems, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 39, no. 4, pp. 885–897 (2020).

[19] Sobczyk T.J., 2D discrete operators for periodic functions, Proceedings IEEE Conference Selected Issues of Electrical Engineering and Electronics (WZZE), Zakopane, Poland, pp. 1–5 (2019), https://ieeexplore.ieee.org/document/8979992.

[20] Jaraczewski M., Sobczyk T., Leakage Inductances of Transformers at Arbitrarily Located Windings, Energies, vol. 13, no. 23, 6464 (2020), DOI: 10.3390/en13236464.

Przejdź do artykułu
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[3] Taflove A., Computational electrodynamics: the finite-difference time-domain method, Artech House, Boston – London (1995).

[4] Strikwerda J.C., Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics, Second Edition, Philadelphia (2004).

[5] LeVeque R.J., Finite difference methods for ordinary and partial differential equations, Society for Industrial and Applied Mathematics, Second Edition, Philadelphia (2007).

[6] Fortuna Z., Macukow B., Wasowski J., Numerical methods, WNT (in Polish), Warsaw (2009).

[7] Esfandiari R.S., Numerical Methods for Engineers and Scientists Using MATLABr, CRC Press, Taylor & Francis Group (2017).

[8] Zakrzewski K., Łukaniszyn M., Application of 3-D finite difference method for inductance calculation of air-core coils system, COMPEL International Journal of Computations and Mathematics in Electrical Engineering, vol. 13, no. 1, pp. 89–92 (1994).

[9] Demenko A., Sykulski J., On the equivalence of finite difference and edge element formulations in magnetic field analysis using vector potential, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 33, no. 1/2, pp. 47–55 (2014).

[10] Huang J., LiaoW., Li Z., A multi-block finite difference method for seismic wave equation in auxiliary coordinate system with irregular fluid–solid interface, Engineering Computations, vol. 35, no. 1, pp. 334–362 (2018).

[11] Chapwanya M., Dozva R., Gift Muchatibaya G., A nonstandard finite difference technique for singular Lane-Emden type equations, Engineering Computations, vol. 36, no. 5, pp. 1566–1578 (2019).

[12] Mawlood M., Basri S., AsrarW., Omar A., Mokhtar A., Ahmad M., Solution of Navier-Stokes equations by fourth-order compact schemes and AUSM flux splitting, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 16, no. 1, pp. 107–120 (2006).

[13] Ivanovic M., Svicevic M., Savovic S., Numerical solution of Stefan problem with variable space grid method based on mixed finite element/finite difference approach, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 27, no. 12, pp. 2682–2695 (2017).

[14] Sobczyk T.J., Algorithm for determining two-periodic steady-states in AC machines directly in time domain, Archives of Electrical Engineering, Polish Academy of Science, Electrical Engineering Committee, vol. 65, no. 3, pp. 575–583 (2016), DOI: 10.1515/aee-2016-0041.

[15] Sobczyk T.J., Radzik M., Radwan-Pragłowska N., Discrete differential operators for periodic and two-periodic time functions, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 38, no. 1, pp. 325–347 (2019).

[16] Sobczyk T.J., Radzik M., A new approach to steady state analysis of nonlinear electrical circuits, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 37, no. 3, pp. 716–728 (2017).

[17] Sobczyk T.J., Radzik M., Tulicki J., Direct steady-state solutions for circuit models of nonlinear electromagnetic devices, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 40, no. 3, pp. 660–675 (2021), DOI: 10.1108/COMPEL-10-2020-0324.

[18] Sobczyk T.J., Jaraczewski M., Application of discrete differential operators of periodic functions to solve 1D boundary-value problems, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 39, no. 4, pp. 885–897 (2020).

[19] Sobczyk T.J., 2D discrete operators for periodic functions, Proceedings IEEE Conference Selected Issues of Electrical Engineering and Electronics (WZZE), Zakopane, Poland, pp. 1–5 (2019), https://ieeexplore.ieee.org/document/8979992.

[20] Jaraczewski M., Sobczyk T., Leakage Inductances of Transformers at Arbitrarily Located Windings, Energies, vol. 13, no. 23, 6464 (2020), DOI: 10.3390/en13236464.

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[1] Steentjes S., von Pfingsten G., Hombitzer M., Hameyer K., *Iron-loss model with consideration of minor loops applied to FE-simulations of electrical machines*, IEEE Transactions on Magnetics. vol. 49, no. 7, pp. 3945-3948 (2013).

[2] Idziak P., *Computer Investigation of Diagnostic Signals in Dynamic Torque of Damaged Induction Motor*, Electrical Review (in Polish), to be published.

[3] Cardwell W., *Finite element analysis of transient electromagnetic-thermal phenomena in a squirrel cage motor*, submitted for publication in IEEE Transactions on Magnetics.

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[4] Popescu M., Staton D.A., *Thermal aspects in power traction motors with permanent magnets*, Proceedings of XXIII Symposium Electromagnetic Phenomena in Nonlinear Circuits, Pilsen, Czech Republic, pp. 35-36 (2016).

*Book, book chapter and manual*

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*example*

[5] Zienkiewicz O., Taylor R.L., *Finite Element method*, McGraw-Hill Book Company (2000).

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[6] Author1 A., Author2 A., *Title of patent*, European Patent, EP xxx xxx (YEAR).

*example*

[6] Piech Z., Szelag W., Elevator brake with magneto-rheological fluid, European Patent, EP 2 197 774 B1 (2011).

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[7] Author A., *Title of thesis*, PhD Thesis, Department, University, City of Univ. (YEAR).

*example*

[7] Driesen J., *Coupled electromagnetic-thermal problems in electrical energy transducers*, PhD Thesis, Faculty of Applied Science, K.U. Leuven, Leuven (2000).

**For on electronic forms**

[8] Author A.,* Title of article*, in Title of Conference, record as it appears on the copyright page], © [applicable copyright holder of the Conference Record] (copyright year), doi: [DOI number].

*example*

[8] Kubo M., Yamamoto Y., Kondo T., Rajashekara K., Zhu B., *Zero-sequence current suppression for open-end winding induction motor drive with resonant controller,*in IEE*E* Applied Power Electronics Conference and Exposition (APEC), © APEC (2016), doi: 10.1109/APEC.2016.7468259

**Website**

[9] http://www.aee.put.poznan.pl, accessed April 2010.

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Preparation of manuscript for Archives of Electrical Engineering (AEE)