Field Oriented Sliding Mode Control of Surface-Mounted Permanent Magnet AC Motors: Theory and Applications to Electrified Vehicles

Permanent magnet ac motors have been extensively utilized for adjustable-speed traction motor drives, due to their inherent advantages including higher power density, superior efficiency and reliability, more precise and rapid torque control, larger power factor, longer bearing, and insulation life-time. Without any proportional-and-integral (PI) controllers, this paper introduces novel first- and higher-order field-oriented sliding mode control schemes. Compared with the traditional PI-based vector control techniques, it is shown that the proposed field oriented sliding mode control methods improve the dynamic torque and speed response, and enhance the robustness to parameter variations, modeling uncertainties, and external load perturbations. While both first- and higher-order controllers display excellent performance, computer simulations show that the higher-order field-oriented sliding mode scheme offers better performance by reducing the chattering phenomenon, which is presented in the first-order scheme. The higher-order field-oriented sliding mode controller, based on the hierarchical use of supertwisting algorithm, is then implemented with a Texas Instruments TMS320F28335 DSP hardware platform to prototype the surface-mounted permanent magnet ac motor drive. Last, computer simulation studies demonstrate that the proposed field-oriented sliding mode control approach is able to effectively meet the speed and torque requirements of a heavy-duty electrified vehicle during the EPA urban driving schedule

Modified PSO based PID Sliding Mode Control using Improved Reaching Law for Nonlinear systems

In this paper, a new model based nonlinear control technique, called PID (Proportional-Integral-Derivative) type sliding surface based sliding mode control is designed using improved reaching law. To improve the performance of the second order nonlinear differential equations with unknown parameters modified particle swarm intelligent optimization (MPSO) is used for the optimized parameters. This paper throws light on the sliding surface design, on the proposed power rate exponential reaching law, parameters optimization using modified particle swarm optimization and highlights the important features of adding an integral term in the sliding mode such as robustness and higher convergence, through extensive mathematical modeling. Siding mode control law is derived using Lyapunov stability approach and its asymptotic stability is proved mathematically and simulations showing its validity. MPSO PID-type Sliding mode control will stabilize the highly nonlinear systems, will compensate disturbances and uncertainty and reduces tracking errors. Simulations and experimental application is done on the non-linear systems and are presented to make a quantitative comparison.Comment: arXiv admin note: substantial text overlap with arXiv:2207.1112

Sliding mode control for robust and smooth reference tracking in robot visual servoing

[EN] An approach based on sliding mode is proposed in this work for reference tracking in robot visual servoing. In particular, 2 sliding mode controls are obtained depending on whether joint accelerations or joint jerks are considered as the discontinuous control action. Both sliding mode controls are extensively compared in a 3D-simulated environment with their equivalent well-known continuous controls, which can be found in the literature, to highlight their similarities and differences. The main advantages of the proposed method are smoothness, robustness, and low computational cost. The applicability and robustness of the proposed approach are substantiated by experimental results using a conventional 6R industrial manipulator (KUKA KR 6 R900 sixx [AGILUS]) for positioning and tracking tasks.Spanish Government, Grant/Award Number: BES-2010-038486; Generalitat Valenciana, Grant/Award Number: BEST/2017/029 and APOSTD/2016/044Muñoz-Benavent, P.; Gracia, L.; Solanes, JE.; Esparza, A.; Tornero, J. (2018). Sliding mode control for robust and smooth reference tracking in robot visual servoing. International Journal of Robust and Nonlinear Control. 28(5):1728-1756. https://doi.org/10.1002/rnc.3981S17281756285Hutchinson, S., Hager, G. D., & Corke, P. I. (1996). A tutorial on visual servo control. IEEE Transactions on Robotics and Automation, 12(5), 651-670. doi:10.1109/70.538972Chaumette, F., & Hutchinson, S. (2008). Visual Servoing and Visual Tracking. Springer Handbook of Robotics, 563-583. doi:10.1007/978-3-540-30301-5_25Corke, P. (2011). Robotics, Vision and Control. Springer Tracts in Advanced Robotics. doi:10.1007/978-3-642-20144-8RYAN, E. P., & CORLESS, M. (1984). Ultimate Boundedness and Asymptotic Stability of a Class of Uncertain Dynamical Systems via Continuous and Discontinuous Feedback Control. IMA Journal of Mathematical Control and Information, 1(3), 223-242. doi:10.1093/imamci/1.3.223Chaumette, F., & Hutchinson, S. (2006). Visual servo control. I. Basic approaches. IEEE Robotics & Automation Magazine, 13(4), 82-90. doi:10.1109/mra.2006.250573Chaumette, F., & Hutchinson, S. (2007). Visual servo control. II. Advanced approaches [Tutorial]. IEEE Robotics & Automation Magazine, 14(1), 109-118. doi:10.1109/mra.2007.339609Bonfe M Mainardi E Fantuzzi C Variable structure PID based visual servoing for robotic tracking and manipulation 2002 Lausanne, Switzerland https://doi.org/10.1109/IRDS.2002.1041421Solanes, J. E., Muñoz-Benavent, P., Girbés, V., Armesto, L., & Tornero, J. (2015). On improving robot image-based visual servoing based on dual-rate reference filtering control strategy. Robotica, 34(12), 2842-2859. doi:10.1017/s0263574715000454Elena M Cristiano M Damiano F Bonfe M Variable structure PID controller for cooperative eye-in-hand/eye-to-hand visual servoing 2003 Istanbul, Turkey https://doi.org/10.1109/CCA.2003.1223145Hashimoto, K., Ebine, T., & Kimura, H. (1996). Visual servoing with hand-eye manipulator-optimal control approach. IEEE Transactions on Robotics and Automation, 12(5), 766-774. doi:10.1109/70.538981Chan A Leonard S Croft EA Little JJ Collision-free visual servoing of an eye-in-hand manipulator via constraint-aware planning and control 2011 San Francisco, CA, USA https://doi.org/10.1109/ACC.2011.5991008Allibert, G., Courtial, E., & Chaumette, F. (2010). Visual Servoing via Nonlinear Predictive Control. Lecture Notes in Control and Information Sciences, 375-393. doi:10.1007/978-1-84996-089-2_20Kragic, D., & Christensen, H. I. (2003). Robust Visual Servoing. The International Journal of Robotics Research, 22(10-11), 923-939. doi:10.1177/027836490302210009Mezouar Y Chaumette F Path planning in image space for robust visual servoing 2000 San Francisco, CA, USA https://doi.org/10.1109/ROBOT.2000.846445Morel, G., Zanne, P., & Plestan, F. (2005). Robust visual servoing: bounding the task function tracking errors. IEEE Transactions on Control Systems Technology, 13(6), 998-1009. doi:10.1109/tcst.2005.857409Hammouda, L., Kaaniche, K., Mekki, H., & Chtourou, M. (2015). Robust visual servoing using global features based on random process. International Journal of Computational Vision and Robotics, 5(2), 138. doi:10.1504/ijcvr.2015.068803Yang YX Liu D Liu H Robot-self-learning visual servoing algorithm using neural networks 2002 Beijing, China https://doi.org/10.1109/ICMLC.2002.1174473Sadeghzadeh, M., Calvert, D., & Abdullah, H. A. (2014). Self-Learning Visual Servoing of Robot Manipulator Using Explanation-Based Fuzzy Neural Networks and Q-Learning. Journal of Intelligent & Robotic Systems, 78(1), 83-104. doi:10.1007/s10846-014-0151-5Lee AX Levine S Abbeel P Learning Visual Servoing With Deep Features and Fitted Q-Iteration 2017Fakhry, H. H., & Wilson, W. J. (1996). A modified resolved acceleration controller for position-based visual servoing. Mathematical and Computer Modelling, 24(5-6), 1-9. doi:10.1016/0895-7177(96)00112-4Keshmiri, M., Wen-Fang Xie, & Mohebbi, A. (2014). Augmented Image-Based Visual Servoing of a Manipulator Using Acceleration Command. IEEE Transactions on Industrial Electronics, 61(10), 5444-5452. doi:10.1109/tie.2014.2300048Edwards, C., & Spurgeon, S. (1998). Sliding Mode Control. doi:10.1201/9781498701822Zanne P Morel G Piestan F Robust vision based 3D trajectory tracking using sliding mode control 2000 San Francisco, CA, USAOliveira TR Peixoto AJ Leite AC Hsu L Sliding mode control of uncertain multivariable nonlinear systems applied to uncalibrated robotics visual servoing 2009 St. Louis, MO, USAOliveira, T. R., Leite, A. C., Peixoto, A. J., & Hsu, L. (2014). Overcoming Limitations of Uncalibrated Robotics Visual Servoing by means of Sliding Mode Control and Switching Monitoring Scheme. Asian Journal of Control, 16(3), 752-764. doi:10.1002/asjc.899Li, F., & Xie, H.-L. (2010). Sliding mode variable structure control for visual servoing system. International Journal of Automation and Computing, 7(3), 317-323. doi:10.1007/s11633-010-0509-5Kim J Kim D Choi S Won S Image-based visual servoing using sliding mode control 2006 Busan, South KoreaBurger W Dean-Leon E Cheng G Robust second order sliding mode control for 6D position based visual servoing with a redundant mobile manipulator 2015 Seoul, South KoreaBecerra, H. M., López-Nicolás, G., & Sagüés, C. (2011). A Sliding-Mode-Control Law for Mobile Robots Based on Epipolar Visual Servoing From Three Views. IEEE Transactions on Robotics, 27(1), 175-183. doi:10.1109/tro.2010.2091750Parsapour, M., & Taghirad, H. D. (2015). Kernel-based sliding mode control for visual servoing system. IET Computer Vision, 9(3), 309-320. doi:10.1049/iet-cvi.2013.0310Xin J Ran BJ Ma XM Robot visual sliding mode servoing using SIFT features 2016 Chengdu, ChinaZhao, Y. M., Lin, Y., Xi, F., Guo, S., & Ouyang, P. (2016). Switch-Based Sliding Mode Control for Position-Based Visual Servoing of Robotic Riveting System. Journal of Manufacturing Science and Engineering, 139(4). doi:10.1115/1.4034681Moosavian, S. A. A., & Papadopoulos, E. (2007). Modified transpose Jacobian control of robotic systems. Automatica, 43(7), 1226-1233. doi:10.1016/j.automatica.2006.12.029Sagara, S., & Taira, Y. (2008). Digital control of space robot manipulators with velocity type joint controller using transpose of generalized Jacobian matrix. Artificial Life and Robotics, 13(1), 355-358. doi:10.1007/s10015-008-0584-7Khalaji, A. K., & Moosavian, S. A. A. (2015). Modified transpose Jacobian control of a tractor-trailer wheeled robot. Journal of Mechanical Science and Technology, 29(9), 3961-3969. doi:10.1007/s12206-015-0841-3Utkin, V., Guldner, J., & Shi, J. (2017). Sliding Mode Control in Electro-Mechanical Systems. doi:10.1201/9781420065619Utkin, V. (2016). Discussion Aspects of High-Order Sliding Mode Control. IEEE Transactions on Automatic Control, 61(3), 829-833. doi:10.1109/tac.2015.2450571Romdhane, H., Dehri, K., & Nouri, A. S. (2016). Discrete second-order sliding mode control based on optimal sliding function vector for multivariable systems with input-output representation. International Journal of Robust and Nonlinear Control, 26(17), 3806-3830. doi:10.1002/rnc.3536Sharma, N. K., & Janardhanan, S. (2017). Optimal discrete higher-order sliding mode control of uncertain LTI systems with partial state information. International Journal of Robust and Nonlinear Control. doi:10.1002/rnc.3785LEVANT, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247-1263. doi:10.1080/00207179308923053Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76(9-10), 924-941. doi:10.1080/0020717031000099029Bartolini, G., Ferrara, A., & Usai, E. (1998). Chattering avoidance by second-order sliding mode control. IEEE Transactions on Automatic Control, 43(2), 241-246. doi:10.1109/9.661074Siciliano, B., Sciavicco, L., Villani, L., & Oriolo, G. (2009). Robotics. Advanced Textbooks in Control and Signal Processing. doi:10.1007/978-1-84628-642-1Deo, A. S., & Walker, I. D. (1995). Overview of damped least-squares methods for inverse kinematics of robot manipulators. Journal of Intelligent & Robotic Systems, 14(1), 43-68. doi:10.1007/bf01254007WHEELER, G., SU, C.-Y., & STEPANENKO, Y. (1998). A Sliding Mode Controller with Improved Adaptation Laws for the Upper Bounds on the Norm of Uncertainties. Automatica, 34(12), 1657-1661. doi:10.1016/s0005-1098(98)80024-1Yu-Sheng Lu. (2009). Sliding-Mode Disturbance Observer With Switching-Gain Adaptation and Its Application to Optical Disk Drives. IEEE Transactions on Industrial Electronics, 56(9), 3743-3750. doi:10.1109/tie.2009.2025719Chen, X., Shen, W., Cao, Z., & Kapoor, A. (2014). A novel approach for state of charge estimation based on adaptive switching gain sliding mode observer in electric vehicles. Journal of Power Sources, 246, 667-678. doi:10.1016/j.jpowsour.2013.08.039Cong, B. L., Chen, Z., & Liu, X. D. (2012). On adaptive sliding mode control without switching gain overestimation. International Journal of Robust and Nonlinear Control, 24(3), 515-531. doi:10.1002/rnc.2902Taleb, M., Plestan, F., & Bououlid, B. (2014). An adaptive solution for robust control based on integral high-order sliding mode concept. International Journal of Robust and Nonlinear Control, 25(8), 1201-1213. doi:10.1002/rnc.3135Zhu, J., & Khayati, K. (2016). On a new adaptive sliding mode control for MIMO nonlinear systems with uncertainties of unknown bounds. International Journal of Robust and Nonlinear Control, 27(6), 942-962. doi:10.1002/rnc.3608Hafez AHA Cervera E Jawahar CV Hybrid visual servoing by boosting IBVS and PBVS 2008 Damascus, SyriaKermorgant O Chaumette F Combining IBVS and PBVS to ensure the visibility constraint 2011 San Francisco, CA, USACorke, P. I., & Hutchinson, S. A. (2001). A new partitioned approach to image-based visual servo control. IEEE Transactions on Robotics and Automation, 17(4), 507-515. doi:10.1109/70.954764Yang, Z., & Shen, S. (2017). Monocular Visual–Inertial State Estimation With Online Initialization and Camera–IMU Extrinsic Calibration. IEEE Transactions on Automation Science and Engineering, 14(1), 39-51. doi:10.1109/tase.2016.2550621Chesi G Hashimoto K Static-eye against hand-eye visual servoing 2002 Las Vegas, NV, USABourdis N Marraud D Sahbi H Camera pose estimation using visual servoing for aerial video change detection 2012 Munich, GermanyShademan A Janabi-Sharifi F Sensitivity analysis of EKF and iterated EKF pose estimation for position-based visual servoing 2005 USAMalis, E., Mezouar, Y., & Rives, P. (2010). Robustness of Image-Based Visual Servoing With a Calibrated Camera in the Presence of Uncertainties in the Three-Dimensional Structure. IEEE Transactions on Robotics, 26(1), 112-120. doi:10.1109/tro.2009.2033332Chen J Behal A Dawson D Dixon W Adaptive visual servoing in the presence of intrinsic calibration uncertainty 2003 USAMezouar Y Malis E Robustness of central catadioptric image-based visual servoing to uncertainties on 3D parameters 2004 Sendai, JapanMarchand, E., Spindler, F., & Chaumette, F. (2005). ViSP for visual servoing: a generic software platform with a wide class of robot control skills. IEEE Robotics & Automation Magazine, 12(4), 40-52. doi:10.1109/mra.2005.157702

Experimental Validation of a Sliding Mode Control for a Stewart Platform Used in Aerospace Inspection Applications

The authors introduce a new controller, aimed at industrial domains, that improves the performance and accuracy of positioning systems based on Stewart platforms. More specifically, this paper presents, and validates experimentally, a sliding mode control for precisely positioning a Stewart platform used as a mobile platform in non-destructive inspection (NDI) applications. The NDI application involves exploring the specimen surface of aeronautical coupons at different heights. In order to avoid defocusing and blurred images, the platform must be positioned accurately to keep a uniform distance between the camera and the surface of the specimen. This operation requires the coordinated control of the six electro mechanic actuators (EMAs). The platform trajectory and the EMA lengths can be calculated by means of the forward and inverse kinematics of the Stewart platform. Typically, a proportional integral (PI) control approach is used for this purpose but unfortunately this control scheme is unable to position the platform accurately enough. For this reason, a sliding mode control (SMC) strategy is proposed. The SMC requires: (1) a priori knowledge of the bounds on system uncertainties, and (2) the analysis of the system stability in order to ensure that the strategy executes adequately. The results of this work show a higher performance of the SMC when compared with the PI control strategy: the average absolute error is reduced from 3.45 mm in PI to 0.78 mm in the SMC. Additionally, the duty cycle analysis shows that although PI control demands a smoother actuator response, the power consumption is similar.This research was funded by the Basque Government through the project SMAR3NAK (ELKARTEK KK-2019/00051), by the Ministerio de Economía y Competitividad (RTI2018-094669-B-C31) and by Aernnova and the Diputación Foral de Álava (DFA) through the project CONAVAUTIN 2 (Collaboration Agreement)

Induction Motor Performance Improvement using Super Twisting SMC and Twelve Sector DTC

Induction motor (IM) direct torque control (DTC) is prone to a number of weaknesses, including uncertainty, external disturbances, and non-linear dynamics. Hysteresis controllers are used in the inner loops of this control method, whereas traditional proportional-integral (PI) controllers are used in the outer loop. A high-performance torque and speed system is consequently needed to assure a stable and reliable command that can tolerate such unsettled effects. This paper treats the design of a robust sensorless twelve-sector DTC of a three-phase IM. The speed controller is conceived based on high-order super-twisting sliding mode control with integral action (iSTSMC). The goal is to decrease the flux, torque, the current ripples that constitute the major conventional DTC drawbacks. The phase current ripples have been effectively reduced from 76.92% to 45.30% with a difference of 31.62%. A robust adaptive flux and speed observer-based fuzzy logic mechanism are inserted to get rid of the mechanical sensor. Satisfactory results have been got through simulations in MATLAB/Simulink under load disturbance. In comparison to a conventional six-sector DTC, the suggested technique has a higher performance and lower distortion rate

Switching frequency regulation in sliding mode control by a hysteresis band controller

© 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksFixing the switching frequency is a key issue in sliding mode control implementations. This paper presents a hysteresis band controller capable of setting a constant value for the steady-state switching frequency of a sliding mode controller in regulation and tracking tasks. The proposed architecture relies on a piecewise linear modeling of the switching function behavior within the hysteresis band, and consists of a discrete-time integral-type controller that modifies the amplitude of the hysteresis band of the comparator in accordance with the error between the desired and the actually measured switching period. For tracking purposes, an additional feedforward action is introduced to compensate the time variation of the switching function derivatives at either sides of the switching hyperplane in the steady state. Stability proofs are provided, and a design criterion for the control parameters to guarantee closed-loop stability is subsequently derived. Numerical simulations and experimental results validate the proposal.Accepted versio

Modified Nonlinear Integral Sliding Mode Control for Satellite Attitude Stabilization Using Magnetically Suspended Gimbaled Momentum Wheel

This paper treats the attitude stabilization problem for satellite using only one MSGMW (Magnetically Suspended Gimbaled Momentum Wheel). To start, the coupled dynamic model of satellite and MSGMW is defined and simplified based on the fact that the attitude errors are small during the mission mode that the MSGMW services. In order to improve the dynamic performance, reduce the steady state error and avoid the chattering phenomenon, a modified integral chattering-free sliding mode controller with a nonlinear integral function and a saturation function is introduced. Lyapunov theory is employed to prove the convergence characteristic outside the boundary layer and the terminal convergence characteristic inside the boundary layer. A numerical simulation example is employed to show the effectiveness and suitability of the proposed controller

Lyapunov approach on a homogeneous family of controllers for robotic manipulator

Second order sliding mode has been successfully implemented for solution of real problems for its inherent features such as finite-time convergence and robustness to disturbances. For the first order sliding modes, it is common to deal with the issues of stability, robustness, and convergence rate of the equilibrium by means of a Lyapunov approach. For higher order sliding modes, however, a similar treatment has not been developed until recently. The focus of this thesis is the construction of strong Lyapunov function, i.e. for which its time derivative can be upper bounded by negative-definite functions, for the design of control strategies for robotic manipulator, which is a nonlinear system, subject to combinations of parametric uncertainty, bounded disturbances, actuator saturation, and output feedback. The first contribution of this work is the development of a strong Lyapunov function for a parameterized family of homogeneous sliding mode based controller comprising twisting algorithm, continuous finite time control, linear PD control law, and uniformly stable control law, all of which belongs to a general homogeneous family of control algorithms. The strict locally Lipschitz homogeneous Lyapunov function proposed permits the estimation of convergence time for the trajectories of the system to the origin, in finite-time, exponentially, or uniformly asymptotically, even in the case when it is affected by bounded non-vanishing or growth bounded vanishing external perturbations. Moreover, the relationship between the control gains and its convergence performance can be analyzed. Leveraging on these results, a strong Lyapunov function is developed for a closely related second order sliding mode algorithm, the super-twisting algorithm based controller. In particular, the construction of these strong homogeneous Lyapunov function is able to show the relationship between the twisting and super-twisting algorithms and allows linear combination of two homogeneous control of different degree. Extending the results for MIMO robot manipulator, a type of Euler-Lagrange dynamic systems, a family of integral sliding mode-based controller is introduced for trajectory tracking. In particular, the homogeneous dynamics is employed as the desired error dynamics for the controller. Additionally, the conventional PID control is shown to be a special case and the present formulation presents the relationship between the gains of the controller and the desired performance, which provides a systematic method for gain selection for a robust PID control. In addition, for the special problem of regulation, employing the results of homogeneous control, finite-time regulation of the robot manipulator is achieved. Since actuator saturation is a phenomenon that affects the performance of dynamic systems under closed-loop control, a saturated version of the controller is also developed that achieved global stability while maintaining the features of the unbounded version of the controller in terms of trajectory tracking and finite time regulation. Extending the results for system with position measurements only, a saturated output feedback version of the controller is introduced that can achieved global stability as well. Each of the proposed controllers provides advantages over the previous literature in their ability to design desired error dynamics and the time derivative of the disturbance is not required in the stability analyses. Throughout the work, Lyapunov-based stability, in particular the nonsmooth Lyapunov analysis techniques, and numerical experiments are provided to highlight the performance of each controller design

Fuzzy second order sliding mode control of a unified power flow controller

Purpose. This paper presents an advanced control scheme based on fuzzy logic and second order sliding mode of a unified power flow controller. This controller offers advantages in terms of static and dynamic operation of the power system such as the control law is synthesized using three types of controllers: proportional integral, and sliding mode controller and Fuzzy logic second order sliding mode controller. Their respective performances are compared in terms of reference tracking, sensitivity to perturbations and robustness. We have to study the problem of controlling power in electric system by UPFC. The simulation results show the effectiveness of the proposed method especiallyin chattering-free behavior, response to sudden load variations and robustness. All the simulations for the above work have been carried out using MATLAB / Simulink. Various simulations have given very satisfactory results and we have successfully improved the real and reactive power flows on a transmission lineas well as to regulate voltage at the bus where it is connected, the studies and illustrate the effectiveness and capability of UPFC in improving power.В настоящей статье представлена усовершенствованная схема управления, основанная на нечеткой логике и режиме скольжения второго порядка унифицированного контроллера потока мощности. Данный контроллер обладает преимуществами с точки зрения статической и динамической работы энергосистемы, например, закон управления синтезируется с использованием трех типов контроллеров: пропорционально-интегрального, контроллера скользящего режима и контроллера скользящего режима нечеткой логики второго порядка. Их соответствующие характеристики сравниваются с точки зрения отслеживания эталонов, чувствительности к возмущениям и надежности. Необходимо изучить проблему управления мощностью в энергосистеме с помощью унифицированного контроллера потока мощности (UPFC). Результаты моделирования показывают эффективность предложенного метода, особенно в отношении отсутствия вибрации, реакции на внезапные изменения нагрузки и устойчивости. Все расчеты для вышеуказанной работы были выполнены с использованием MATLAB/Simulink. Различные расчетные исследования дали весьма удовлетворительные результаты, и мы успешно улучшили потоки реальной и реактивной мощности на линии электропередачи, а также регулирование напряжения на шине, к которой она подключена, что позволяет изучить и проиллюстрировать эффективность и возможности UPFC для увеличения мощности