Direct Torque Control for Induction Motor Drives
基于静态补偿电压模型的改进转子磁链观测器
基于静态补偿电压模型的改进转子磁链观测器宋文祥;阮智勇;尹赟【摘要】为解决纯电压模型磁链观测器存在的积分漂移和饱和问题,常采用低通滤波器代替纯积分器.针对传统低通滤波器磁链观测方案的不足,本文提出一种改进的转子磁链观测方案,采用串联低通滤波器提取直流偏置得到理想的转子反电势,然后用可编程低通滤波器代替纯积分器,并在反电势低通滤波前补偿磁链误差.所提出的观测器可以有效消除直流偏置的影响,提高磁链观测的动态精度并改善系统的动态性能.在一台2.2kW感应电机无速度传感器矢量控制系统上对本文提出的改进转子磁链观测器方案进行了仿真和实验研究,结果验证了其正确性和有效性.%In the pure voltage model based flux observer, a LPF is normally used to replace the pure integrator to a-void integration drift and saturation problems. In order to eliminate the DC offset efficiently and compensate the error brought about by LPF as well as improve the dynamic performance, a modified rotor flux observer is proposed in this paper. In the proposed scheme, series LPF is used to remove the DC drift firstly, then a programmable LPF is used instead of the pure integrator, and the amplitude and phase error is compensated before the back EMF filtered for the flux estimation. Simulation and experiment based on induction motor speed sensor-less vector control systems verified its correctness and effectiveness.【期刊名称】《电工电能新技术》【年(卷),期】2012(031)004【总页数】5页(P19-23)【关键词】磁链观测器;电压模型;低通滤波器;直流偏置;矢量控制【作者】宋文祥;阮智勇;尹赟【作者单位】上海大学机电工程与自动化学院,上海200072;上海大学机电工程与自动化学院,上海200072;上海大学机电工程与自动化学院,上海200072【正文语种】中文【中图分类】TM343感应电机矢量控制和直接转矩控制系统中,准确观测磁链是获得高性能控制的关键。
基于矢量控制的感应电机弱磁控制算法研究
基于矢量控制的感应电机弱磁控制算法研究陶华堂;李强【摘要】变频调速控制系统要求电机具有宽范围的恒功率弱磁调速能力,并能输出较大的转矩.提出一种感应电动机弱磁状态下励磁电流和转矩电流轨迹控制的新方法.在满足电机和驱动器最大电压和电流约束条件的前提下,对电机励磁电流轨迹和转矩电流轨迹分别独立控制,实现全速度范围内的最大转矩输出.设计了该弱磁控制算法的实现策略,并在7.5 kW感应电机上进行实验研究,与传统弱磁控制方法相比,提出的弱磁控制方法可以输出更大的转矩,电流波动小,系统更稳定.【期刊名称】《电气传动》【年(卷),期】2016(046)003【总页数】5页(P7-11)【关键词】感应电机;矢量控制;弱磁;最大转矩电流比控制【作者】陶华堂;李强【作者单位】中国卫星海上测控部,江苏江阴214400;中国卫星海上测控部,江苏江阴214400【正文语种】中文【中图分类】TM30感应电机具有转子结构坚固、可靠性高、成本低、转矩波动小和噪声小等优点。
基于矢量控制的感应电机变频调速系统被广泛应用于家用电器、电梯曳引、电动汽车、数控机床、船舶动力等领域。
采用电压源逆变器驱动电机时,由于受到逆变器最大输出电压和最大输出电流的限制,需要采用弱磁调速等方法使电机输出最大转速,且高速时仍能最大限度输出电磁转矩。
传统的弱磁控制方法是在基速以上,控制电机磁链和电机转速成反比[1-2],这种方法简单易实现,但是没有输出最大转矩电流比,即没有最大限度输出转矩;文献[3]提出了一种过调制算法,用来实现永磁同步电机的弱磁调速,但是该方法实现起来较困难;文献[4-11]提出的查表修正方法目前较为流行,主要是根据电机的转速通过查表修正电机励磁电流iM和转矩电流iT,此类方法简单易实现,应用也较为广泛,但是受电机本身参数影响较大;文献[12]提出了一种通过控制电机电压轨迹的方法实现电机弱磁调速控制,该方法不受电机参数影响,但前提是要获得电机励磁电流的大小。
基于TMS320F2808+DSP的直接转矩控制系统的实现
基于TMS320F2808 DSP的直接转矩控制系统的实现
TMS320F2808 DSP Based on Implementation of Direct Torque Control
赵俊梅 任一峰(中北大学信息与通信工程学院,山西太原030051)
参考文献 [1]李夙.异步电动机直接转矩控制[M].北京:机械工业出版社,1998 [2]R.Toufouti S.Meziane。H.Benalla.Direct Torque Control For In.
duction Motor Using Intelligent Techniques[J].Journal of Th争 oretical and Applied Information Technology,2007,1:35—44 [3]苏奎峰。等.TMS320F2812原理与开发[M].北京:电子工业出版社, 2005 [4]陈伯时.电力拖动自动控制系统[M].北京:机械工业出版社,2006: 216-218 [5]王泳.宣接转矩控制在交流电气传动中的应用研究[J].平顶山师专 学报,2001,16(4):33—36
赵敏
(北京茨浮测控技术研究所,北京lOll01)
摘要 详细介绍了直接转矩控制算法在TMS320F2808 DSP芯片上的实现方法,完成了交流异步电动机的直接转矩控制系 统的仿真研究,结果表明交流异步电动机的直接转矩控制系统具有良好的调速特性。 关键词:DSP,直接转矩控制,仿真研究
Abstract This paper introduces Direct torque control algorithm implementation On TMS320F2808 DSP in detail,and completes the exchange of asynchronous motor direct torque control system simulation studies.The results Show that the exchange of asynchronous motor direct torque control system has good speed regulation characteristics. Keywords:DSP.direct torque control,simulation research
基于模糊控制的感应电机直接转矩控制系统
基于模糊控制的感应电机直接转矩控制系统作者:林辉来源:《现代电子技术》2010年第21期摘要:根据直接转矩控制理论,在Matlab 6.5/Simulink下构造了一个感应电机直接转矩控制系统的仿真模型。
为改善感应电机系统的动、静态品质,设计了模糊自适应PI速度调节器,根据速度偏差与偏差变化率,通过模糊推理在线调整PI参数,提高系统的调速性能。
仿真结果表明,这种模糊控制器具有比常规PID控制器更好的控制效果。
关键词:模糊控制; 直接转矩控制; 感应电机; 速度调节器中图分类号:TN919-34文献标识码:A文章编号:1004-373X(2010)21-0151-03Direct Torque Control System of Induction Motor Based on Fuzzy ControlLIN Hui(Xi’an Railway VocationalTechnical Institute, Xi’an 710014, China)Abstract: A simulation model of the direct torque control (DTC) system for induction motors wasto improve the static and dynamic perfor-mances of induction motors, a novel fuzzy adaptive PI regulator is proposed, which adopts a fuzzy controller to modify the PI parameter according to the speed error and its vary rate, and improves the speed control performance effectively. The simulation results show that the fuzzy controller has better control effect in comparison with the conventional PID controller.Keywords: fuzzy control; direct torque control; induction motor; speed regulator0 引言直接转矩控制 (DTC) 是继矢量控制技术之后又一先进电机控制技术,其结构简单、对电机参数不敏感、转矩响应迅速而被广泛应用[1]。
异步电机模型预测直接转矩控制
异步电机模型预测直接转矩控制2.西华大学电气与电子信息学院,成都市 610039摘要:直接转矩控制存在输出定子磁链和转矩脉动大以及开关频率不稳定等问题。
针对该问题,提出了一种考虑延迟补偿的模型预测直接转矩控制方法。
本文在三相异步电机数学模型基础上,建立异步电机离散时间预测模型,以转矩和定子磁链差值作为目标函数,通过在线评估开关矢量对电机的作用效果,选择最优目标函数,优化系统结构。
并通过仿真模型验证了该算法的可行性与有效性。
关键词:异步电机;模型预测;延迟补偿0引言直接转矩控制(Direct Torque Control, DTC)自20世纪80年代提出以来,得到了逐步完善和发展,并成为具有代表性的高性能控制策略之一[1]。
直接转矩控制因能实现高性能的动态响应而有广泛的应用,但其存在输出转矩脉动大和开关频率不稳定等问题[2]。
近年来,模型预测控制(Model Predictive Control,MPC)作为新型预测控制方式在电力电子领域受到广泛关注[3]。
模型预测控制是一种非线性预测控制策略,处理非线性约束的适应力强,利用给定的目标函数作为优化准则,让控制具有灵活性。
文献[4]提出模型预测直接转矩控制(Model Predictive Direct Torque Control,MPDTC)技术,其原理简单,易于处理非线性约束条件,已被广泛应用于电机控制领域。
但是MPDTC在控制过程中存在大量计算,系统存在延迟问题[5]。
本文利用MPDTC思想,提出一种考虑延迟补偿的MPDTC方法,有效降低转矩脉动。
首先建立异步电机数学模型,用两步预测法,推算出定子转矩和磁链的预测值;最后通过目标函数选择电压矢量控制异步电机,从而确立考虑延迟补偿的MPDTC方法。
1逆变器驱动异步电机模型图1所示是一个三相电压型逆变器驱动异步电机等效图。
图1 三相电压型逆变器驱动异步电机等效图定义开关函数为:(1)其中,x=a,b,c 。
外文翻译---动态建模与驱动的双馈风力发电机直接供电网络的电压不平衡条件下的控制
附录A 英文参考文献Dynamic modeling and direct power control of wind turbine driven DFIG under unbalanced network voltage condition INTRODUCTION,Wind farms based on the doubly-fed inductiongenerators(DFIG)with converters rated at 25%~30%ofthe generator rating for a given rotor speedvariation range of±25%are becoming pared with the wind turbines using fixedspeed induction generators or fully-fed synchronousgenerators with full-size convertersthe DFIG-basedwind turbines offer not only theadvantages of variable speed operation and four-quadrant active andreactive power capabilities,but also lower convertercost and power losses(Pena et al.,1996).However,both transmission and distribution networks couldusually have small steady state and large transientvoltage unbalance.If voltage unbalance is not considered by the DFIG control system,the stator currentcould become highly unbalanced even with a smallunbalanced stator voltage.The unbalanced currentscreate unequal heating on the stator windings,andpulsations in the electromagnetic torque and statoroutput active and reactive powers(Chomatetal.,2002;Jang et al.,2006;Zhou et al.,2007;Pena et al.,2007;Hu et al.,2007;Xu and Wang,2007;Hu andHe,2008).Control and operation of DFIG wind turbinesystems under unbalanced network conditions istraditionally based on either stator-flux-oriented(SFO)(Xuand Wang,2007)or stator-voltage-oriented(SVO)vector control(Jang et al.,2006;Zhou et al.,2007;Hu et al.,2007;Hu and He,2008).The schemein(Jang et al.,2006;Zhou et al.,2007;Xu and Wang,2007;Hu et al.,2007)employs dual-PI(proportionalintegral)current regulators implemented in thepositive and negative synchronously rotating referenceframes,respectively,which has to decompose themeasured rotor current into positive and negative sequence components to control them individually.One main drawback of this approach is that,the timedelays introduced by decomposing the sequentialcomponents of rotor current can affect the overallsystem stability and dynamic response.Thus,acurrent control scheme based on a proportionalresonant(PR)regulator in the stator stationaryreference frame was proposed in(Hu and He,2008),which can directly control the rotor current withoutthe need of sequential decomposition.Whereas,theperformance of the vector control scheme highlydepends on the accurate machine parameters such asstator/rotor inductances and resistances used in thecontrol system.Similar to direct torque control(DTC)ofinduction machines presented a few decades ago,which behaves as an alternative to vector control,direct power control(DPC)of DFIG-based windturbine systems has been proposed recently(Gokhaleet al.,2002;Xu and Cartwright,2006;Zhi and Xu,2007).In(Gokhale et al.,2002),the control schemewas based on the estimated rotor flux.Switchingvectors were selected from the optimal switchingtable using the estimated rotor flux position and theerrors of rotor flux and active power.The rotor fluxreference was calculatedusing the reactive powerreference.Since the rotor supply frequency,equal tothe DFIG slip frequency,might be very low,the rotorflux estimation could be significantly affected by themachine parameter variations.In(Xu and Cartwright,2006),a DPC strategy based on the estimated statorflux was proposed.As the stator voltage is relativelyharmonics-free and fixed in frequency,a DFIGestimated stator flux accuracy can then be guaranteed.Switching vectors were selected from the optimalswitching table using the estimated stator fluxposition and the errors of the active and reactivepowers.Thus,the control system was simple and themachine parameters’impact on the systemperformance was found to be negli able.However,like a conventional DTC,DPC has the problem ofunfixed switching frequency,due to the significantinfluence of the active and reactive power variations,generator speed,and power controllers’hysteresisbandwidth.More recently,a modified DPC strategyhas been proposed in(Zhi and Xu,2007)based onSFO vector control in the synchronous referenceframe for DFIG-based wind power generationsystems with a constant switching frequency.The control method directly calculates the required rotorcontrol voltage within each switching period,basedon the estimated stator flux,the active and reactivepowers and their errors.The control strategy providesimproved transient performance with the assumptionof the stator(supply)voltage being strictly balanced.However,the operation could be deteriorated duringthe supply voltage unbalance and there is no reportyet on DFIGDPC under unbalanced networkvoltage conditions.This paper investigates an improved DPCscheme for a DFIG wind power generation systemunder unbalanced network conditions.In the SVO dqreference frame,a mathematical DPC model of aDFIG system with balanced supply is presented,which is referred to as the conventional model in thispaper.Then during network unbalance,a modifiedDFIG DPC model in the SVO positive dqandnegative dqreference frames is developed.Based onthe developed model,a system control strategy isproposed by eliminating the stator output activepower oscillations under unbalanced network conditions.Finally,simulation results on a 2-MW DFIGwind generation system are presented to demonstratethe correctness and feasibility of the proposed controlstrategy.SIMULATION STUDIES,Simulations of the proposed DPC strategy for aDFIG-based wind power generation system wereconducted using PSCAD/EMTDC.A single-phase load at the primary side of the coupling transformer was used to generate the voltageunbalance.The nominal DC link voltage was set at1 200 V and the switching frequencies for both converters were 2 000 Hz.The main target of the grid sideconverter was assigned to control the DC link voltagewith the similar method used in(Song and Nam,1999;Hu et al.,2007).As shown in Fig.7,a high frequencyAC filter is shunt-connected to the stator side to absorb the switching harmonics generated by the twoconverters.Initial studies with various active and reactivepower steps were carried out to test the dynamic response using the conventional control scheme shownin Fig.4 in the conditions of balanced supply voltage.First,the DFIG was assumed to be in speed control,viz.,the rotor speed was set externally,as thelargeinertia of the wind turbine resulting in a slow changeof the rotor speed.The activeand reactive powers were initially set at 0 MW and0.5 MV·A,respectively,whererefers to absorbing reactive power.Various power steps were applied,viz.,active and reactive power references werechanged from 0 to2 MW at the instant of 1.3 s.CONCLUSION,This paper has proposed an analysis and an improved DPC design for a DFIG-based wind powergeneration system during network voltage unbalance.Simulation results were presented to demonstrate thefeasibility of the proposed control scheme.Conclusions can be drawn as follows:(1)The conventional DPC scheme without net-work unbalance considered can provide pretty gooddynamic system performance when the supplyvoltage is strictly balanced.However,once the net-work is slightly unbalanced,the performance deteriorates with high stator/rotor current unbalances andsignificant oscillations in the stator active/reactivepower and electromagnetic torque.(2)The proposed DPC scheme,which is implemented in the SVO positive dq+ and negative dqreference frames,gets rid of the decompositionprocess of positive and negative sequence rotor currents in the vector control scheme using dual-PI rotorenhanced by the elimination of the stator output activepower oscillations and the reduction of the electro-magnetic torque pulsations during network unbalance.附录B 中文参考文献动态建模与驱动的双馈风力发电机直接供电网络的电压不平衡条件下的控制风力发电场的双馈感应发电机与转换器在25%〜30给定转子速度变化范围± 25%额定发电机(双馈)的基础正在变得越来越受欢迎。
矩阵变换器_异步电机矢量控制系统仿真研究
Abstract: Space vector pulse width modulation of matrix converter and rotor field oriented vector control of asynchronous motor are combined,and the paper researchs combination strategy for vector control of the matrix converter and rotor field oriented vector control of asynchronous motor. Input voltage and current of matrix converter are simulated by MATLAB ,at the same time,noload starting torque and speed waveform and the electrical load torque waveform of motor are simulated. The simulation results show that the combined control strategy for matrix converterasynchronous vector control system has good speed performance,and has the advantage of a ACDCAC voltage type PWM variable frequency speed regulation system. Key words: matrix converter; vector control; composite control; asynchronous motor; control strategy
感应电机 Super-twisting 算法定子磁链观测器设计
感应电机 Super-twisting 算法定子磁链观测器设计潘月斗;陈涛;陈泽平【摘要】In order to improve the observation accuracy of stator flux of induction motor,a stator flux esti-mation method based on Super-twisting algorithm was proposed.A stator flux observer was designed and applied for direct torque control of induction motor.According to the robustness of sliding mode variable structure control,the disturbance of the multiple input multiple output stator flux observer system was re-strained.By using the advantages of Super-twisting algorithm which require less information to design a simple control law,and thus more suitable for practical engineering applications.The speed and amount of coupling were regarded as disturbances in the analysis of the stability of observer,and the sufficient condi-tions of the system uniformly asymptotically stable was pared with the u-i model observer, the proposed observer based on Super-twisting algorithm is more accurate and has better robustness to the change of stator resistance.Simulation and experiment results validate the proposed method.%为了提高感应电机定子磁链的观测精确度,提出了一种基于Super-twisting算法的磁链观测方法,设计了定子磁链观测器,并应用到感应电机直接转矩控制中。
直接转矩控制(DTC)技术概述
直接转矩控制(DTC)技术概述作者:同济大学电气工程系袁登科陶生桂王志鹏刘洪1 引言交流电机传动系统中的直接转矩控制技术是基于定子两相静止参考坐标系,一方面维持转矩在给定值附近,另一方面维持定子磁链沿着给定轨迹(预先设定的轨迹,如六边形或圆形等)运动,对交流电机的电磁转矩与定子磁链直接进行闭环控制。
最早提出的经典控制结构是采用bang-bang控制器对定子磁链与电磁转矩实施砰砰控制,分别将它们的脉动限制在预先设定的范围内。
bang-bang调节器是进行比较与量化的环节,当实际值超过调节范围的上、下限时,它就产生动作,输出的数字控制量就会发生变化。
然后由该控制量直接决定出电压型逆变器输出的电压空间向量。
这种经典的直接转矩控制技术具有:(1) 非常简单的控制结构;(2) 非常快速的动态性能;(3) 无需专门的pwm技术;(4) 把交流电机与逆变器结合在一起, 对电机的控制最为直接,且能最大限度发挥逆变器的能力;(5) 前面叙述的实际被控量必须发生脉动才能产生合适的数字控制量,所以它不可避免地存在着一种与其特有的pwm技术密切相关的定子磁链与电磁转矩的脉动。
2 传统的直接转矩控制(dtc)方案直接转矩控制技术于上世纪80年代中期提出, 当时的控制系统有两种典型的控制结构:德国学者的直接转矩自控制方案与日本学者的直接转矩与磁链控制方案。
两者都属于直接转矩控制的范围,但仍有着较大的不同。
下面对各种方案进行介绍与分析。
2.1 德国depenbrock教授的直接自控制(dsc)方案[1]直接自控制方案是针对大功率交流传动系统电压型逆变器驱动感应电机提出来的控制方案。
由于当时采用大功率gto半导体开关器件,考虑到器件本身的开通、关断比较慢,还有开关损耗和散热等实际问题,gto器件的开关频率不能太高。
当时的开关频率要小于1khz,通常只有500~600hz。
而即便到现在,大功率交流传动应用场合中开关频率也只能有几khz。
基于占空比调节的无刷直流电机直接转矩控制
基于占空比调节的无刷直流电机直接转矩控制潘峰;周运杰;卢沁雄;闫庚龙;韩如成【摘要】For the traditional direct torque control (DTC) system of Brushless DC Motors (BLDCM),the motor was imposed on one fixed voltage vector during a control cycle,which lead to great torque ripples and inconstant switching frequency.In order to suppress the torque ripples and keep the switching frequency constant so that the reliability of the system could be improved,the duty ratio control method was introduced into the DTC system of BLDCM and the effect of four duty ratio determination method for suppressing the torque ripples were studied.By using the duty ratio control method,the acting time of the active voltage vector and zero voltage vector during one control cycle were decided by the duty ratio,so the torque could be controlled more accurately and the torque ripples could be suppressed.The results of the simulation and experiments highlight the validation of the method proposed.%传统直接转矩控制(DTC)在每一个控制周期对电机施加一个不变的电压矢量,从而导致较大的转矩脉动,而且开关管的开关频率不固定.为减小转矩脉动同时使开关管的开关频率恒定从而提升系统的稳定性,将占空比调制技术引入到无刷直流电机(BLDCM)DTC系统当中,并研究了4种占空比生成方法对转矩脉动的抑制效果.由于引入了占空比调制技术,零电压矢量和非零电压矢量作用的时间随占空比的改变而改变,因此能够对转矩进行更精细的控制从而抑制转矩脉动.仿真和试验结果验证了所提方法的有效性.【期刊名称】《电机与控制应用》【年(卷),期】2017(044)011【总页数】8页(P42-49)【关键词】无刷直流电机;直接转矩控制;占空比;转矩脉动【作者】潘峰;周运杰;卢沁雄;闫庚龙;韩如成【作者单位】太原科技大学电子信息工程学院,山西太原 030024;太原科技大学电子信息工程学院,山西太原 030024;太原科技大学电子信息工程学院,山西太原030024;太原科技大学电子信息工程学院,山西太原 030024;太原科技大学电子信息工程学院,山西太原 030024【正文语种】中文【中图分类】TM301.2直接转矩控制(Direct Torque Control,DTC)以电磁转矩为主要控制变量使系统具有较高的动态响应性能,但是在每一个固定的控制周期内一直施加不变的电压矢量会引起较大的转矩脉动,降低了系统的稳态性能且开关频率不恒定[1-3]。
直接转矩控制的研究现状和应用现状
Research and Application of Direct Torque Control in AC MotorJames Abin HillCollege of Automation Science and Engineering, South China University of TechnologyI. INTRODUCTIONDirect Torque Control (DTC) is one method used in variable frequency drives to control the torque (and thus the speed) of 3-pahse AC electric motors. It involves calculating an estimate of the motor’s magnet flux and torque based on the voltage and current measured from the motor. Three kinds of DTC schemes are presented as following: a, DTC scheme in Chinese books as shown in Figure 1; b, DTC scheme in English book as shown in Figure 2; c, DTC scheme in ABB technical guide as shown in Figure 3/4/5. In spite of some differences among three kinds of DTC scheme, DTC consists of a stator flux and torque (and speed for speed-sensorless) estimator, two hysteresis controllers for magnet flux and torque and a voltage vector selector. In this paper, both research and application of DTC in AC motor are summarized.Chinese Books,“异步电动机的控制”李鹤轩、李杨译,119页;“电力拖动自动控制系统”陈伯时著,214-216页;“交流调速控制系统”李华德主编,192-219页:Figure 1. DTC scheme in Chinese booksEnglish Book, “Power Electronics and Motor Drive”, 2006 Edition, page 412:Figure 2. DTC scheme in English bookABB, Drivers of Change Embedded DSP-based motor control, page 2, 2/2006:Figure 3. DTC scheme in ABB technical guide ABB, Technical Guide No.1 – Direct Torque Control, page 26, 8/2002:Figure 4. DTC scheme in ABB technical guideABB, Direct Torque Control Principle:Figure 5. DTC scheme in ABB technical guide Emotron, Direct Torque Control:Figure 6. Comparison of anti-interference between VC and DTCII. Status of Research on DTCThis paper investigates 33 papers about DTC from journals embodied by ISI, EI and IEEE of 2008 to 2010. The study shows that recent research on DTC comes from 4 perspectives as following: 1. torque and flux (if sensorless and speed) estimation; 2, torque and flux ripple reduction; 3, motor types; 4, torque, flux and speed controllers.1. 14 from 33 papers are research on torque and flux (if sensorless and speed) estimation (16-20, 22-30). They propose many estimation methods mostly to improve the estimation accuracy at low-speed (standstill included sometimes), such as adaptive estimation (MRAS included), Extended Kalman Filter (EKF) based estimation, non-linear estimation (Sliding Mode included), High-Frequency Signal Injection (HFSI) Algorithm, stator resistance compensator based estimation and so on.2. 11 from 33 papers are research on torque and flux ripple reduction (1, 6-15). They propose several methods to reduce the torque and flux ripple, such as improving torque and flux controllers (predict control and neuro-fuzzy control, for instance), reforming the switching patterns (symmetry switching patterns of the applied voltage vectors and closed-loop switching frequency control, for instance), increasing the number of inverter states or degrees of freedom (matrix-converter and five-phase inverter, for instance) and so on.3. Most of 33 papers study DTC in Induction Motor (IM) and Permanent Magnet Synchronous Motor (PMSM), while others study Double Fed IM (DFIM), Brushless DC Motor (BLDCM), Multilevel-Inverter IM, Matrix-Converter-Fed PMSM, Three-level Inverter, Synchronous Reluctance Machine (SynRM), brushless doubly fed reluctance machine (BDFRM), Switched Reluctance (SR) motor and Five Phase Induction Motor.4. 4 from 33 papers are research on torque, flux and speed controllers. (20, 30) apply PI controller and fuzzy controller to replace the hysteresis controller in conventional DTC for torque and flux ripple reduction. When torque and flux hysteresis controllers are changed to continuous controllers, voltage vector selector (switching table) should be replaced by a space vector modulation (SVM) at the same time. (32, 33) compare different speed controllers such as conventional PI controllers, fuzzy logic controller and hybrid fuzzy sliding mode controller.III. Application Status of DTCDTC AC drive has already come into our daily life since several years ago. However, only two companies (ABB of Switzerland, Emotron of Sweden) have put it into production. Product:1)ABB ACS 600 AC DRIVES2)ABB ACS 800 AC DRIVES3)Emotron VFX 2.0 AC DRIVEReferences1. Abad, G., Rodriguez, M. A. & Poza, J. 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(2010) Rotor Position and Speed Estimation of a Variable Structure Direct-Torque-Controlled IPM Synchronous Motor Drive at Very Low Speeds Including Standstill|, , 57|, 3715-3723|.18. Zhifeng, Z., Renyuan, T., Baodong, B. & Dexin, X. (2010) Novel Direct Torque Control Based on Space Vector Modulation With Adaptive Stator Flux Observer for Induction Motors|, , 46|, 3133-3136|.19. Foo, G. H. B. & Rahman, M. F. (2010) Direct Torque Control of an IPM-Synchronous Motor Drive at Very Low Speed Using a Sliding-Mode Stator Flux Observer|, , 25|, 933-942|. 20. Foo, G., Sayeef, S. & Rahman, M. F. (2010) Low-Speed and Standstill Operation of a Sensorless Direct Torque and Flux Controlled IPM Synchronous Motor Drive|, , 25|, 25-33|. 21. Ozturk, S. B., Alexander, W. C. & Toliyat, H. A. (2010) Direct Torque Control ofFour-Switch Brushless DC Motor With Non-Sinusoidal Back EMF|, , 25|, 263-271|.22. Hajian, M., Soltani, J., Markadeh, G. A. & Hosseinnia, S. 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(2010) Improved Direct Torque Control of Permanent Magnet Synchronous Electrical Vehicle Motor with Proportional-Integral Resistance Estimator, JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY, 5, 451-461.28. Barut, M. (2010) Bi Input-extended Kalman filter based estimation technique forspeed-sensorless control of induction motors, ENERGY CONVERSION AND MANAGEMENT, 51, 2032-2040.29. Khedher, A. & Mimouni, M. F. (2010) Sensorless-adaptive DTC of double star induction motor, ENERGY CONVERSION AND MANAGEMENT, 51, 2878-2892.30. Abbou, A. & Mahmoudi, H. (2008) Sensorless speed control of induction motor using DTFC based fuzzy logic, Journal of Electrical Engineering, 8 pp.31. West, N. T. & Lorenz, R. D. (2009) Digital Implementation of Stator and RotorFlux-Linkage Observers and a Stator-Current Observer for Deadbeat Direct Torque Control of Induction Machines|, , 45|, 729-736|.32. Gadoue, S. M., Giaouris, D. & Finch, J. W. 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永磁同步电动机直接转矩控制方法的比较研究
第25卷第16期中国电机工程学报V ol.25 No.16 Aug. 2005 2005年8月Proceedings of the CSEE ©2005 Chin.Soc.for Elec.Eng.文章编号:0258-8013(2005)16-0141-06 中图分类号:TM351 文献标识码:A 学科分类号:470⋅40永磁同步电动机直接转矩控制方法的比较研究史涔溦1,邱建琪1,金孟加1,Friedrich W. Fuchs2(1.浙江大学电气工程学院,浙江省杭州市310027;2.Faculty of Engineering, Christian-Albrechts-University of Kiel, 24143 Kiel, Germany)STUDY ON THE PERFORMANCE OF DIFFERENT DIRECT TORQUE CONTROL METHODS FOR PERMANENT MAGNET SYNCHRONOUS MACHINESSHI Cen-wei1, QIU Jian-qi1, JIN Meng-jia1, Friedrich W. Fuchs2(1. College of Electrical Engineering, Hangzhou 310027, Zhejiang University, China;2. Faculty of Engineering, Christian-Albrechts-University of Kiel, 24143 Kiel, Germany)ABSTRACT: A modified direct torque control strategy based on error flux linkage vector compensation (EFVC-DTC) for permanent magnet synchronous machines (PMSM) is presented. The theoretical background of EFVC-DTC is introduced and an algorithm to estimate the flux linkage error is proposed. The steady state and dynamic performances of EFVC-DTC have been compared with those of the conventional direct torque control (DTC). The simulation and experimental results confirm that both flux linkage and torque ripples are significantly reduced in EFVC-DTC with a fixed switching frequency while the dynamic torque response is almost as good as the basic DTC.KEY WORDS:Permanent magnet synchronous machine (PMSM); Direct torque control (DTC); Error flux linkage vector compensation; Space vector pulse-width-modulation (SVM)摘要:提出一种用于永磁同步电动机的基于磁链误差矢量补偿的直接转矩控制(EFVC-DTC)策略,给出了磁链误差矢量的估计算法,并将该控制策略下的稳态和动态运行性能与常规DTC进行比较。
A modified direct torque control for induction motor sensorless drive
A Modified Direct Torque Control for InductionMotor Sensorless DriveCristian Lascu,Ion Boldea,Fellow,IEEE,and Frede Blaabjerg,Senior Member,IEEE Abstract—Direct torque control(DTC)is known to producequick and robust response in ac drives.However,during steadystate,notable torque,flux,and current pulsations occur.They arereflected in speed estimation,speed response,and also in increasedacoustical noise.This paper introduces a new direct torque andflux control based on space-vector modulation(DTC-SVM)forinduction motor sensorless drives.It is able to reduce the acous-tical noise,the torque,flux,current,and speed pulsations duringsteady state.DTC transient merits are preserved,whilebetter quality steady-state performance is produced in sensorlessimplementation for a wide speed range.The flux and torqueestimator is presented and an improved voltage–current modelspeed observer is introduced.The proposed control topologies,simulations,implementation data,and test results with DTCand DTC-SVM are given and discussed.It is concluded that theproposed control topology produces better results for steady-stateoperation than the classical DTC.Index Terms—Direct torque control,estimators,sensorless.I.I NTRODUCTIONR ESEARCH interest in induction motor(IM)sensorlessdrives has grown significantly over the past few years dueto some of their advantages,such as mechanical robustness,simple construction,and maintenance.Present efforts are de-voted to improve the sensorless operation,especially for lowspeed and to develop robust control strategies.Since its introduction in1985,the direct torque control(DTC)[1](or direct self control(DSC)[2])principle waswidely used for IM drives with fast dynamics.Despite its sim-plicity,DTC is able to produce very fast torque and flux controland,if the torque and flux are correctly estimated,is robustwith respect to motor parameters and perturbations.during steady-state operation,notable torque,flux,and currentpulsations occur.They are reflected in speed estimation and inincreased acoustical noise.Paper IPCSD99–46,presented at the1998Industry Applications Society An-nual Meeting,St.Louis,MO,October12–16,and approved for publication inthe IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS by the Industrial DrivesCommittee of the IEEE Industry Applications Society.This work was supportedby the Danfoss Professor Programme and the Institute of Energy Technology,Aalborg University,Aalborg East,Denmark.Manuscript submitted for reviewOctober15,1998and released for publication August23,1999.scu is with the Department of Electrical Machines and Drives,University Politehnica of Timisoara,RO-1900Timisoara,Romania(e-mail:cristi@et.utt.ro).I.Boldea is with the Department of Electrical Machines and Drives,University Politehnica of Timisoara,RO-1900Timisoara,Romania(e-mail:boldea@lselinux.utt.ro).F.Blaabjerg is with the Institute of Energy Technology,Aalborg University,DK-9220Aalborg East,Denmark(e-mail:fbl@iet.auc.dk).Publisher Item Identifier S0093-9994(00)00036-0.Several solutions with modified DTC are presented in the lit-erature.Due to its simple structure,DTC can be easily integratedwith an artificial intelligence control strategy.The fuzzy logicsolution for flux and torque control is shown in[3].A different approach is to combine the voltage vector selec-tion with an adequate pulsewidth modulation(PWM)strategy inorder to obtain a smooth operation.The closed-loop stator fluxpredictive control,open-loop torque control using space-vectormodulation(SVM)implementation is shown in[4].The SVMis a performant open-loop vector modulation strategy[5].This paper introduces a new direct torque and flux controlbased on SVM(DTC-SVM)for IM sensorless drives.It imple-ments closed-loop digital control for both flux and torque in asimilar manner as DTC,but the voltage is produced by an SVMunit.This way,the DTC transient performance and robustnessare preserved and the steady-state torque ripple is reduced.Ad-ditionally,the switching frequency is constant and totally con-trollable.Another important issue for a sensorless drive is the flux,torque,and speed estimation.Both open-loop and closed-loopspeed and position estimators are widely analyzed in the litera-ture.The most promising speed observers seem to be the adap-tive ones,either with linear or nonlinear structures[6],[7].How-ever,the low-speed range estimation still remains an unsolvedproblem.This is not the case for flux and torque observers whichare able to generate accurate estimation for the whole speedrange[8]–[10].An improved voltage–current model speed ob-server based on a model reference adaptive controller(MRAC)structure is proposed herewith.The paper presents the complete sensorless solution based ona DTC-SVM strategy.The proposed control topologies,digitalsimulations,implementation data,and test results with DTC andDTC-SVM are given and discussed.II.P ROPOSED S ENSORLESS IM D RIVEThe proposed sensorless IM drive block diagram is shown inFig.1.It operates with constant rotor flux,direct stator flux,and torque control.The speed controller is a classical propor-tional-integral-derivative(PID)regulator,which produces thereference torque.Only the dc-link voltage and two line currentsare measured.The IM model isFig.1.The DTC-SVM sensorless ac drive.the derivation operator.The electromagnetic torque isthe number of pole pairs.The stator flux and torque closed-loop control is achieved bythe DTC-SVM unit.In order to reduce the torque and flux pulsa-tions and,implicitly,the current harmonics content,in contrastto the standard DTC,we do use decoupled PI flux and torquecontrollers and SVM.III.F LUX AND S PEED E STIMATORThe estimator calculates the stator fluxrotor flux components are(7)”)is the stator fluxis the estimated rotor flux from(7)and(8)in a sta-tionary reference frame(see Fig.2).The voltage model is based on(1)and uses the stator voltageand current measurement.For the stator reference frame,thestator flux(12)Values such as20–30rad/s for the twopoles(13)The detailed parameter sensitivity analysis of this observer canbe found in[9].Fig.2.The flux estimator for the DTC-SVM drive.Fig.3.The MRAC speed estimator.The speed estimator has the structure of a model referenceadaptive controller(MRAC)[6],[7].In order to achieve a widespeed range,an improved solution,which uses the full-orderflux estimator,is proposed(see Fig.3).The reference model is the rotor flux estimator presented sofar(13).It is supposed to operate accurately for a wide fre-quency band(1–100Hz).The adaptive model is a current modelbased on(2)for a stationary reference frame(”)–(17)(18)From(1),for a stator flux reference frame(If the stator flux is constant,it is evident that the torque can becontrolled by the imaginary component—the torque com-ponent—of the voltage vector(22)The stator flux speedand as—the flux component—of the voltage vector.For each sampling periodvoltage asvoltage drop can be neglected andthe voltage becomes proportional with the flux change andwith the switching frequency1/termis not negligible.The current–flux relations are rather compli-cated(in stator flux coordinates)(25)(26)where(27)It is evident that a cross coupling is present in terms ofand currents.The simplest way to realize the decouplingis to add the(28)and angleandor(30)andandFig.6.The classical DTCcontroller.Fig.7.The real and estimated speed (!,!)and the real and estimated torque (M ,M )with the tuned estimator—simulationresults.Fig.8.The estimated speed and torque with detuned estimator when R =0:4R (!;M )—simulation results.The proposed strategy was called DTC-SVM because it re-alizes the direct torque and flux voltage control combined with SVM and uses DTC when the errors are large.The two methods are compatible since DTC is a high-gain voltage control.The classical DTC topology is presented in Fig.6.Fig.9.The estimated speed and torque with detuned estimator when R =0:4R(!;M )—simulationresults.Fig.10.The estimated speed and torque with detuned estimator when T =0:4T (!;M )—simulationresults.Fig.11.The experimental setup.The DTC strategy can be simply expressed:each sampling period the adequate voltage vector is selected in order to rapidly decrease,in the same time,the torque and flux errors.The convenient voltage vector is selected in accordance with the signals produced by two hysteresis comparators and the stator flux vector position.Fig.12.DTC-SVM—1Hz(30rpm)no load steady state—experimental results.Fig.13.Classical DTC—1Hz(30rpm)no load steady state—experimental results.Fig.14.DTC-SVM no load starting transients—experimental results.V .S IMULATION R ESULTSThe simulation results with DTC-SVM are presented next.The induction motor used for experiments and simulations has the ratedvaluespolepairsandtheparameters,,and astep from 50to 1Hz is appliedats.Fig.7shows the real and estimated speed and torque with tuned estimator.A correct estimation can be observed.Fig.8shows the estimated speed and torque when the stator resis-tance used for estimation is under and overestimated(s andthe switching frequency 8kHz.Deadtime compensation was in-cluded.Both DTC-SVM and classical DTC sensorless strategies were implemented.The design of the two PI controllers is based on (22)and (24).The torque controller gain should equal,at least,the first term in(22):kHz,but the overall system’sstability is improved,even if the flux controller is not a very fast one.The integrator term in both controllers introduces a unitary discretepoleandcompensatesforthecross-couplingerrors.The controllers’parameters used for experiments are the fol-lowing.•The PI compensator for the flux estimator in Fig.2uses thevaluesandFig.16.DTC-SVM speed and torque transients zoom during no load acceleration from 5–50Hz—experimentalresults.Fig.17.Classical DTC speed and torque transients zoom during no load acceleration from 5–50Hz—experimental results.Fig.18.DTC-SVM speed reversal transients (from 25Hz to −25Hz)—experimental results.Comparative experimental results with low-speed no-load operation are presented first.Fig.12shows the estimated speed,torque,stator,and rotor flux,and the measured current for steady-state 1–Hz DTC-SVM operation.Fig.13shows the estimated speed,torque,stator,and rotor flux for steady-state 1–Hz DTC operation.An improved operation in terms of high-frequency ripple can be noticed with DTC-SVM.The no-load starting transient performance is presented in Fig.14—estimated speed and torque—for DTC-SVM and inFig.15—the same quantities—for DTC.Again,the torque ripple isdrasticallyreduced,whilethefastresponseispreserved.The same conclusions are evident for the no-load speed tran-sients—from5to50Hz—presented in Fig.16for DTC-SVM and in Fig.17for DTC.A zoom of torque proves the fast torque response of the proposed strategy.Fig.18shows the speed reversal from25to−25Hz—speed, flux,and current—for DTC-SVM.Some small flux oscillations can be observed when the flux changes due to the absence of the decoupling term in the flux controller.The system’s stability is influenced by the precision and the speed of convergence of the flux and speed estimation.The speed estimator is not a very fast one,and this can be seen from Fig.18where some speed oscillations occur.The DTC-SVM controller does not depend on motor parameters and is relatively robust as was proved by simulation.VII.C ONCLUSIONSThis paper has introduced a new direct torque and flux control strategy based on two PI controllers and a voltage space-vector modulator.The complete sensorless solution was presented. The main conclusions are as follows.•DTC-SVM strategy realizes almost ripple-free operation for the entire speed range.Consequently,the flux,torque, and speed estimation is improved.•The fast response and robustness merits of the classical DTC are entirely preserved.•The switching frequency is constant and controllable.In fact,the better results are due to the increasing of the switching frequency.While for DTC a single voltage vector is applied during one sampling time,for DTC-SVMa sequence of six vectors is applied during the same time.This is the merit of SVM strategy.•An improved MRAC speed estimator based on a full-order rotor flux estimator as reference model was proposed and tested at high and low speeds.It can be stated that,using the DTC-SVM topology,the overall system performance is increased.R EFERENCES[1]I.Takahashi and T.Noguchi,“A new quick response and high efficiencystrategy of an induction motor,”in Conf.Rec.IEEE-IAS Annu.Meeting, 1985,pp.495–502.[2]M.Depenbrock,“Direct self control for high dynamics performance ofinverter feed AC machines,”ETZ Arch..,vol.7,no.7,pp.211–218,1985.[3] A.Mir,M.E.Elbuluk,and D.S.Zinger,“Fuzzy implementation of directself control of induction motors,”IEEE Trans.Ind.Applicat.,vol.30,pp.729–735,May/June1994.[4] D.Casadei,G.Sera,and A.Tani,“Stator flux vector control for highperformance induction motor drives using space vector modulation,”in Proc.OPTIM’96,1996,pp.1413–1422.[5]P.Thoegersen and J.K.Pedersen,“Stator flux oriented asynchronousvector modulation for AC-drives,”in Proc.IEEE PESC’90,1990,pp.641–648.[6] C.Schauder,“Adaptive speed identification for vector control of induc-tion motors without rotational transducers,”IEEE Trans.Ind.Applicat., vol.28,pp.1054–1061,Sept./Oct.1992.[7]H.Tajima and Y.Hori,“Speed sensorless field-oriented control of theinduction machine,”IEEE Trans.Ind.Applicat.,vol.29,pp.175–180, Jan./Feb.1993.[8]P.L.Jansen,R.D.Lorenz,and D.W.Novotny,“Observer-based di-rect field orientation:Analysis and comparison of alternative methods,”IEEE Trans.Ind.Applicat.,vol.30,pp.945–953,July/Aug.1994.[9]P.L.Jansen and R.D.Lorenz,“A physically insightful approach to thedesign and accuracy assessment of flux observers for field oriented I.M.drives,”IEEE Trans.Ind.Applicat.,vol.30,pp.101–110,Jan./Feb.1994.[10]H.Kubota,K.Matsuse,and T.Nakano,“DTC-based speed adaptive fluxobserver of induction motor,”IEEE Trans.Ind.Applicat.,vol.29,pp.344–348,Mar./Apr.1993.Cristian Lascu received the M.Sc.degree in elec-trical engineering from the University Politehnica ofTimisoara,Timisoara,Romania,in1995.He became an Assistant Professor in1995at theUniversity Politehnica of Timisoara.His researchareas are ac drives,power electronics,and staticpower converters.In1997,he was involved in theDanfoss Professor Programme in Power Electronicsand Drives at the Institute of Energy Technology,Aalborg University,Denmark.He is currently aVisiting Research Scholar at the University of Nevada,Reno.scu was the recipient of a Prize Paper Award at the IEEE Industry Applications Society Annual Meeting in1998.Ion Boldea(M’77–SM’81–F’96)is a Professor ofElectrical Engineering at the University Politehnicaof Timisoara,Timisoara,Romania.He has alsorepeatedly been a Visiting Professor with theUniversity of Kentucky,Lexington,Oregon StateUniversity,Corvallis,the University of Glasgow,U.K.,and Aalborg University,Aaalborg,Denmark.He has worked and published extensively onlinear and rotary machines and drives,mainly onlinear motor Maglevs and linear oscilomotors andgenerators,vector control(direct torque and flux control of both induction and synchronous motors),reluctance synchronous machines,and drives and automotive new alternator systems.He has authored and coauthored11books in English,the latest,with S.A.Nasar,being Linear Electric Actuators and Generators(Cambridge,U.K.:Cambridge Univ.Press, 1997)and Electric Drives(Boca Raton,FL:CRC Press,1998).Frede Blaabjerg(S’86–M’88–SM’97)was born inErslev,Denmark,in1963.He received the Msc.EE.degree from Aalborg University,Aalborg,Denmark,in1987and the Ph.D.degree from the Institute ofEnergy Technology,Aalborg University,in1995.He was with ABB—Scandia,Randers,Denmark,from1987to1988.He joined Aalborg University in1992as an Assistant Professor and became an Asso-ciate Professor in1996and a Full Professor in powerelectronics and drives in1998.His research areas arepower electronics,static power converters,ac drives, switched reluctance drives,modeling,characterization of power semiconductor devices,and simulation.He is involved in more than ten research projects with industry.Among them is the Danfoss Professor Programme in Power Elec-tronics and Drives.Dr.Blaabjerg is a member of the Industrial Drives,the Industrial Power Converter,and the Power Electronics Devices and Components Committees of the IEEE Industry Applications Society,as well as being the Paper Review Chairman of the Industrial Power Converter Committee.He is a member of the European Power Electronics and Drives Association and the Danish Technical Research Council and a Member of the Board of the Danish Space Research Institute.In1995,he received the Angelos Award for his contribution in modulation technique and control of electric drives and an Annual Teacher Prize from Aalborg University.In1998,he received the Outstanding Young Power Electronics Engineer Award from the IEEE Power Electronics Society and an IEEE T RANSACTION ON P OWER E LECTRONICS Prize Paper Award for the best paper published in1997.He also received two Prize Paper Awards at the1998IEEE Industry Applications Society Annual Meeting.。
毕业设计外文翻译
基于自抗扰控制器的永磁同步电机控制系统刘德俊电气与信息工程学院北华大学吉林,中国电子邮件:dejunliu@车长进周振雄电气与信息工程学院北华大学吉林,中国电子邮件:bhchecj@zzx701111@摘要-磁场定向控制技术被广泛应用于运动控制的感应电动机。
然而,在实时执行,这需要准确的电机参数的精确的去耦不能充分认识到由于显著的的植物不确定性,如外部干扰,参数变化和植物非线性动力学,这可能磁通和转矩的动态性能显着恶化。
为了实现高动力驱动系统的性能,非线性自抗扰控制器控制交流永磁同步伺服电机(AC-PMSM)直接转矩控制调速系统.关键词:交流永磁同步电机,自动抗扰,直接转矩控制系统仿真1.简介通常用在高性能交流调速系统矢量控制,动态性能取决于型号。
但交流调速系统具有的特性是非线性的强耦合,多变量[1],它是难以得到精确的数学模型,这使得在改善的动态性能的矢量控制的限制。
学者提出了异步电机直接转矩控制(DTC),去耦矢量控制的想法被放弃DTC,复杂的坐标变换,避免磁场定向,定子磁链的估计仅涉及定子电阻,它被削弱依赖电机参数。
这个方法很简单,它具有的扭矩响应速度快和良好的动态性能。
直接转矩控制的感应电机控制系统,在目前的这些优点,一些学者致力于发展同步电机直接转矩控制模式,初步实现了永磁同步,反映电机直接转矩控制,直接转矩控制是一个简单的控制方法,并具有优良的动态性能[2]。
自抗扰控制器应用到DTC速度控制系统,在本文中,观察到系统的状态和系统的内部和外部扰动扩张状态观测器,获得干扰被添加到由前馈控制系统的输入,使系统作为一个简单的线性系统。
基于传统的自抗扰控制器的非线性函数进行了分析,提出了改进的自抗扰控制器仿真结果表明,该控制方案具有良好的实时性能和较强的鲁棒性。
2. 永磁同步电动机直接转矩控制理论在转子运动的参考坐标,di和qi和sω是状态变量,3相交流永久磁铁的同步伺服电机非线性动态数学模型可以表示为以下:qd s dd q rd d dq q fs dq d r sq q q dLdi R Ui idt L L Ldi LR Ui idt L L L Lωψωω=-++=---+⎧⎪⎨⎪⎩通过坐标变换,在dq坐标轴的转矩是:()32sin sin2 4s f q s q dd qpT L L LL Lψψδψδ⎡⎤=--⎣⎦当定子交链磁通是恒定的,转矩可以表示为以下:()32cos2cos2 4s f q s q dd qdT pL L Ldt L Lψψδψδδ⎡⎤=--⎣⎦从方程(3)中,我们可以得出结论,当定子磁链是一个恒定值,电磁转矩电机决定的交链磁通的分离角δ。
03 Analysis of Direct Torque Control in Permanent Magnet Synchronous Motor Drives
528IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL.12,NO.3,MAY 1997Analysis of Direct Torque Control in Permanent Magnet Synchronous Motor DrivesL.Zhong,M.F.Rahman,Senior Member,IEEE,W.Y.Hu,and K.W.Lim,Senior Member,IEEEAbstract—This paper describes an investigation of direct torque control (DTC)for permanent magnet synchronous motor (PMSM)drives.It is mathematically proven that the increase of electromagnetic torque in a permanent magnet motor is proportional to the increase of the angle between the stator and rotor flux linkages,and,therefore,the fast torque response can be obtained by adjusting the rotating speed of the stator flux linkage as fast as possible.It is also shown that the zero voltage vectors should not be used,and stator flux linkage should be kept moving with respect to the rotor flux linkage all the time.The implementation of DTC in the permanent magnet motor is discussed,and it is found that for DTC using currently available digital signal processors (DSP’s),it is advantageous to have a motor with a high ratio of the rated stator flux linkage to stator voltage.The simulation results verify the proposed control and also show that the torque response under DTC is much faster than the one under current control.Index Terms—Direct torque control,permanent magnet syn-chronous motor,saliency,sensorless control,stator flux linkage.I.I NTRODUCTIONPERMANENT MAGNET synchronous motors (PMSM’s)are used in many applications that require rapid torque response and high-performance operation.The torque in PMSM’s is usually controlled by controlling the armature current based on the fact that the electromagnetic torque is proportional to the armature current.For high performance,the current control is normally executed in therotorPublisZHONG et al.:ANALYSIS OF DIRECT TORQUE CONTROL IN MOTOR DRIVES529Fig.1.The stator and rotorflux linkages in different reference frames.follows:(3)whereand(5)where(6)where represents the amplitude of the statorflux linkage.Substituting(5)and(6)for current into(2)gives-axis component of the stator current if the amplitudeof the statorflux linkage is constant.B.The Flux Linkage Equations inthe Reference FrameEquation(3)can be rewritten into matrix form asfollows:(12)or-axis isfixed at the statorflux linkage.Then,can be solved from the second equation of(13)if the amplitude of the statorflux linkage is keptconstantandThe maximum torque occurswhen530IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL.12,NO.3,MAY1997 is considered to be a step change corresponding to achange of voltage vector.Then,the derivative of(15)becomesiswithin the range of This equation implies thatthe increase of torque is proportional to the increase of theanglecan be obtained by solvingfrom(12),with(18)Then,the torque equation is as follows:(19)Equation(19)consists of two terms.Thefirst is the excita-tion torque,which is produced by the permanent magnetflux,and the second term is the reluctance torque.For each statorflux linkage,there exists the maximum in this equation.It willnot be discussed how to control the amplitude of statorfluxlinkage and load angle to get maximum torque in this paper.However,it is necessary to discuss the relationship betweenthe amplitude of statorflux linkage and the derivative of thetorque.Figs.2–5show the torque-and,which implies that DTC cannot be applied in this case.Therefore,for a PMSM with pole saliency,the amplitude ofthe statorflux linkage should be changed,with the change ofactual torque even for constant torque operation.The derivative of torque in(20)is as shown in(21),withconstant statorflux and:(20)At(21)The condition for for positive isZHONG et al.:ANALYSIS OF DIRECT TORQUE CONTROL IN MOTOR DRIVES531Fig.5.Torque with respect to :j's j=2'f:III.C ONTROL OF S TATOR F LUX L INKAGE BY S ELECTING THE P ROPER S TATOR V OLTAGE V ECTORIn the previous section,it has been proven that the change of torque can be controlled by keeping the amplitude of the statorflux linkage constant and increasing the rotating speed of the statorflux linkage as fast as possible.It will be shown in this section that both the amplitude and rotating speed of the statorflux linkage can be controlled by selecting the proper stator voltage vectors.The primary voltagevector is defined by the followingequation:is connectedtoTherefore,there are six nonzero voltagevectors:apart from each other as in Fig.7.These eightvoltage vectors can be expressedas(24)whereisthe initial statorflux linkage at the instant of switching.Toselect the voltage vectors for controlling the amplitude of thestatorflux linkage,the voltage vector plane is divided intosix regions,as shown in Fig.8.In each region,two adjacentvoltage vectors,which give the minimum switching frequency,are selected to increase or decrease the amplitudeofand532IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL.12,NO.3,MAY 1997TABLE IT HES WITCHING T ABLE FOR INVERTERFig.8.The control of the stator flux linkage.According to the torque (17)and (19),the electromag-netic torque can be controlled effectively by controlling the amplitude and rotational speedof For counter-clockwise operation,if the actual torque is smaller than the reference,the voltage vectors thatkeepincreases as fast as it can,and the actualtorque increases as well.Once the actual torque is greater than the reference,the voltage vectors thatkeepdecreases,and the torque decreases also.By selecting the voltage vectors in thisway,and,then the actual flux linkage is smaller than the reference value.The same is true for thetorque.and ,can be obtained fromthe measured three-phase currents,andvoltagesand are calculated from dc-link voltage since the voltage vectors determined by the switching table are known.The fluxlinkagesth sampling instant are calculated from the integration of the stator voltages asfollows:are the previous samples.The initial valuesofaheador behind the rotor flux linkage.The torque in (3)can be rewritten in the stationary reference frameasZHONG et al .:ANALYSIS OF DIRECT TORQUE CONTROL IN MOTOR DRIVES533Fig.10.Dynamic responses of a PMSM drive with DTC:T s =10 s :TABLE IIDQ A XES VOLTAGESThe reference torque is obtained from the output of the speed controller and is limited at a certain value,with respect to a given reference flux linkage,which guarantees the stator current not to exceed the limit value.For a PMSM with no pole saliency,the stator flux linkage can be kept at its rated value for constant torque operation,while for a PMSM with pole saliency,the reference flux linkage should increase with the actual torque for positive slope with respectto3Nmat3to 3Nmata nd s ,re s p e c t i v e l y .I t i sF i g .10t h a t t h e s t a t o r flu x l i n k a g e i s v a l u e q u i t e w e l l .T h e t r a j e c t o r y of534IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL.12,NO.3,MAY1997Fig.11.Dynamic responses of a PMSM drive with DTC:T s=100s.s or less.For standard induction motors,this problem[40]does not arise because of the sufficiently large value of theflux linkage for the standard voltages and speeds of these motors.With DTC, the stator voltage vector changes every sampling time instead of every switching time as in a PWM current-controlled drive. According to(26),the change of statorflux linkage is equal to the product of dc-link voltage and sampling time.The bandwidth of theflux linkage hysteresis controller is normally set at5%of the rated value.Therefore,the sampling time should be very small for controlling theflux linkage properly, as shown in the simulation results of Fig.10for which the sampling interval is1010),which is2times of that of PMSM I in Table III and13times of that of PMSM II in Table IV.For applying the DTC in a PMSM drive,one PMSM,with the desirable ratio offlux linkage to dc-link voltage,has been built in which a standard induction motor stator is used,and the stator flux linkage is designed to have the same rated value as the induction motor.The data for this motor(PMSM III)is in Table V.TABLE IIID ATA OF PMSMITABLE IVD ATA OF PMSMIIC.The Comparison of the Torque ResponseBetween DTC and PWM Current ControlTo examine the performance of DTC,simulations on a PMSM with no saliency in Table VI(PMSM IV)under DTC and under PWM current control have been carried out.For DTC,the statorflux linkage is kept at its rated value,while for current control,is kept at zero.In both cases,the referenceZHONG et al.:ANALYSIS OF DIRECT TORQUE CONTROL IN MOTOR DRIVES535 TABLE VD ATA OF PMSM IIITABLE VID ATA OF PMSM IV(a)(b)Fig.12.Torque responses with DTC and current control:(a)torque responseunder DTC and(b)torque response under PWM current control in the rotorreference frame.torque is changed abruptly from3.0to536IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL.12,NO.3,MAY1997W.Y.Hu received the Master’s degree from Jianxi University,Nanjing,China,in1966and the Mas-ter’s degree from Nanjing Aeronautical Institute, Nanjing,China,in1981.He worked at the Jinaxi Machine Tools Factory from1967to1977.He later joined the staff of the Nanjing Aeronautical Institute,undertaking re-search in the areas of induction motor drives and switched-mode power supplies.He is currently a Professor at the Nanjing University of Aeronautics andAstronautics.K.W.Lim(M’83–SM’92)received the B.Eng.and D.Phil.degrees from the University of Malaysia, Kuala Lumpur,Malaysia,and Oxford University, Oxford,U.K.,respectively.He is currently an Associate Professor at the University of New South Wales,Australia.His current research interests are in control of machines and multirate systems.Dr.Lim has been active in IEEE activities and is a Member of the Administrative Committee of the Industrial Electronics Society.。
TORQUE CONTROLLER OF INDUCTION MOTOR FOR VARIABLE
专利名称:TORQUE CONTROLLER OF INDUCTIONMOTOR FOR VARIABLE VOLTAGE VARIABLEFREQUENCY INVERTER-CONTROLLEDELECTRIC CAR发明人:TSUJI MICHIHIKO,IMAI ISATO申请号:JP14147689申请日:19890603公开号:JPH037085A公开日:19910114专利内容由知识产权出版社提供摘要:PURPOSE:To reduce an electromagnetic noise at the time of a small load by making the current pattern of an induction motor driven by VVVF constant irrespective of the magnitude of the load of an electric car and by changing the pattern of the ratio of voltage to frequency relative to said load. CONSTITUTION:DC taken from an aerial line 4 via pantograph 3 is inverted 2 so that VVVF AC is generated to drive a motor 1. A frequency controller 3 is formed by a pattern generator part 3b and a control voltage generator part 3a. Said pattern generator part 3b generates slip pattern Fs (3b1), V/f pattern (3b2), and motor current pattern (3b3) and inputs them to said control voltage generator part 3a. On the other hand, the outputs of DC current transformer 7, CT9 and tachometer 10 is also inputted to the control voltage generator part 3a. Said frequency voltage controller 3 changes the pattern of voltage and frequency according to the size of a variable load and outputs said pattern to control the ignition of an inverter 2. Thus, an electromagnetic noise is reduced effectively even at the time of a small load.申请人:HITACHI LTD更多信息请下载全文后查看。
基于Super-twisting滑模永磁同步电机驱动的转速和转矩控制
基于Super-twisting滑模永磁同步电机驱动的转速和转矩控制万东灵;赵朝会;王飞宇;孙强【期刊名称】《电机与控制应用》【年(卷),期】2017(044)010【总页数】6页(P42-47)【作者】万东灵;赵朝会;王飞宇;孙强【作者单位】【正文语种】中文直接转矩控制(Direct Torque Control,DTC)技术是继矢量控制技术之后,由德国和日本学者提出的一种具有高性能的交流变频调速技术[1-2]。
传统DTC技术在转矩环和磁链环采用滞环控制,具有动态响应快速、控制结构简单和外干扰鲁棒性强等优点,但也存在着转矩和磁链脉动大、开关频率不恒定等缺陷[3-4]。
为了解决这些问题,国内外学者提出了许多改进方法[5-8]。
其中,文献[8]中提出了一种基于SVPWM的DTC系统方案。
该方案采用预期电压矢量计算单元取代传统直接转矩系统中滞环比较器,与以前的SVPWM控制相比仅需要转速和转矩两个PI调节器,优化了控制结构。
但特定的PI调节器参数往往会对电机参数、转速和负载变化敏感,存在系统鲁棒性不强等问题[9]。
滑模变结构控制通过不断的切换控制量来实现快速的动态响应,这种控制方案拥有很强的鲁棒性[10-11]。
滑模变结构控制适合系统非线性化程度高,参数可变或者说存在大的扰动。
抖振现象一直是滑模变结构控制需要解决的一大难题,抖振的发生会影响控制系统的性能,严重时甚至会造成系统失稳。
因此,国内外学者对于该问题提出了滤波、降低切换增益等方法来降低系统抖振[12-13]。
Super-twisting滑模变结构的控制理论是在高阶滑模控制的基础上发展而来,其使用控制误差及其误差的积分来构造滑模控制器。
这种控制方案具有很好的鲁棒性和动态特性[14-15]。
目前Super-twisting滑模控制方案在电机控制中的文献并不多,主要应用在磁链观测器和控制器上[15-17]。
结合PI控制器和滑模控制器各自的优缺点,本文将Super-twisting滑模控制引入基于空间矢量控制的PMSM的DTC方案中去,将仅有的两个PI调节器替换成Super-twisting滑模变结构控制器,并通过理论推导证明这个新的控制系统能够在有限时间内收敛,期望能够解决超调频繁、动态响应时间慢的问题,且希望进一步减少转矩脉动。
基于高通谐振滤波器的高频脉振电流注入法研究
基于高通谐振滤波器的高频脉振电流注入法研究关振宏;杜平;王涛;张羽【摘要】当高频脉振电流注入法利用电机的凸极效应实现无位置传感器控制时,为了避免注入信号幅值选取的不当,导致提取转子位置信息的高频响应信号解调失败和转矩脉动,提出一种将高通谐振滤波器引入转子位置提取通道的方法。
利用高通谐振滤波器在低频段的高度衰减性,在谐振点对解调信号放大增幅的特性,提高了含有转子初始位置信息的信噪比。
对转子位置信号的滤波优化,使得电机无位置算法在解调响应信号的难度降低,转子位置和转速的估算可以实现零误差。
最后通过仿真对比研究验证了方法的正确性和有效性。
%When sensorless control is realized in the high-frequency pulsating current injection method by means of saliency effect of the motor, to avoid demodulation failure of the HF response signal for extraction of rotor position information and torque ripple due to improper selection of the amplitude of injection signal,this paper presents a method for importing the high-pass resonator filter into the motor position extraction channel.Its principle is to take advantage of high attenuation of the high-pass resonator filter in the low frequency range as well as amplification of demodulated signal at the resonance point to raise the signal-to-noise ratio of the information containing original rotor position.This approach optimizes filtering of the rotor position signal so that it is not so difficult to demodulate the response signal by sensorless motor algorithm,and zero error can be realized in the estimation of rotor position and speed.Finally, simulation and comparison verify the correctness and effectiveness of the proposed approach.【期刊名称】《电气自动化》【年(卷),期】2016(038)006【总页数】3页(P11-13)【关键词】高频脉振;凸极效应;无位置传感器;高频注入;转子位置【作者】关振宏;杜平;王涛;张羽【作者单位】西南交通大学电气工程学院,四川成都 610031;西南交通大学电气工程学院,四川成都 610031;西南交通大学电气工程学院,四川成都 610031;西南交通大学电气工程学院,四川成都 610031【正文语种】中文【中图分类】TM351;TM341由于铷铁硼等稀土材料应用和电机控制理论的不断进步,使永磁同步电机具有优越调速性能,在工业、交通和航天领域获得广泛的关注[1-3]。
高速动车组在直接转矩控制下的机电耦合模型
高速动车组在直接转矩控制下的机电耦合模型陈双喜;邓小军【摘要】In order to study the dynamic performance of high -speed EMU from traction drive angle,a electro-mechanical coupling model of traction drive system for high -speed EMU with direct torque control was devel-oped.Firstly,the vehicle dynamics model of EMU was built in Simpack and the direct torque control simulation model of traction drive system was built in Matlab /Simulink.Then they were linked by Simpack co -simulation interface model.Consequently,a full electromechanical coupling model of EMU was developed.A simulation ex-ample was also given.Result indicates that this model can effectively reveal the change of mechanical and char-acteristics of running EMV.%从牵引角度对高速动车组车辆动力学性能开展多学科研究,建立牵引系统在直接转矩控制(Direct torque control)下动车组运行的机械电气耦合模型。
建立动车组的车辆动力学 Simpack 模型和牵引传动系统的直接转矩控制 Matlab /Simu-link 仿真模型,通过接口模块将2个系统连接起来,从而实现机械电气系统耦合,并且给出仿真算例。
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Direct Torque Control for Induction Motor Drives:A Model Predictive Control ApproachBased on FeasibilityTobias Geyer and Georgios PapafotiouAutomatic Control Laboratory,Swiss Federal Institute of Technology(ETH),CH-8092Zurich,Switzerland{geyer,papafotiou}@control.ee.ethz.chAbstract.In this paper,we present a new approach to the DirectTorque Control(DTC)problem of three-phase induction motor drives.This approach is based on Model Predictive Control(MPC)exploitingthe specific structure of the DTC problem and using a systematic designprocedure.Specifically,by observing that the DTC objectives,which re-quire the controlled variables to remain within certain bounds,are relatedto feasibility rather than optimality,and by using a blocking control in-puts regime for the whole prediction horizon we derive a low complexitycontroller.The derived controller is an explicit state-feedback control lawthat can be implemented as a look-up table.Even though the controlleris derived here for a DTC drive featuring a two-level inverter,the controlscheme can be extended to also tackle three-level inverters.Simulationresults demonstrate that the proposed controller leads to performanceimprovements despite its simple structure.1IntroductionEnabled by significant technological developments in the area of power electron-ics,variable speed induction motor drives have evolved to a state of the art technology within the last decades.These systems,in which DC-AC inverters are used to drive induction motors as variable frequency three-phase voltage or current sources,are used in a wide spectrum of industrial applications.One of the methods for controlling the induction motor’s torque and speed is Di-rect Torque Control(DTC),which wasfirst introduced in1985by Takahashi and Noguchi[13]and is nowadays a industrial standard for induction motor drives[14,11].The basic characteristic of DTC is that the positions of the inverter switches are directly determined rather than indirectly,thus refraining from using a mod-ulation technique like Pulse Width(PWM)or Space Vector(SVM)modulation. In the generic scheme,the control objective is to keep the motor’s torque and the amplitude of the statorflux within pre-specified bounds.The inverter is triggered by hysteresis controllers to switch whenever these bounds are violated. M.Morari and L.Thiele(Eds.):HSCC2005,LNCS3414,pp.274–290,2005.c Springer-Verlag Berlin Heidelberg2005Direct Torque Control for Induction Motor Drives275 The choice of the new switch positions is made using a pre-designed look-up ta-ble that has been derived using geometric insight in the problem and additional heuristics.The main reason that makes the design of the switching table difficult is the fact that the DTC drive constitutes a hybrid system,i.e.a system incor-porating both continuous and discrete dynamics-in particular discrete-valued manipulated variables.Additionally,constraints on states,inputs and outputs are present imposing further complications on the controller design,since the underlying mathematical problems are intrinsically complex and hard to solve.Recently,we have proposed in[9,10]a systematic procedure for the design of the DTC switching table by reformulating the control problem as a Model Predictive Control(MPC)[8]problem for a two-and a three-level inverter. Modelling the DTC drive as a hybrid system,introducing integer variables for the inverter switch positions that represent the manipulated variables of the control problem and expressing the control objectives in a cost function led to a constrainedfinite time optimal control problem.By solving the underlying optimization problem on-line and comparing the results with the behavior of ABB’s ACS6000drive[1]featuring a three-level inverter,we have demonstrated a potential performance improvement in the range of20%.Subsequently,moving towards the practical implementation of the method,we have pre-computed off-line the optimal control problem for all feasible states and thus derived the explicit state-feedback control law.The latter was done for a DTC drive featuring a two-level inverter and for a specific operating point.Nevertheless,the complexity of the derived state-feedback controller prohibits the practical implementation on the currently employed controller hardware. On the other hand,two observations suggest the existence of a low complexity controller resulting from a systematic design procedure.Firstly,albeit their very simple controller structure,the existing DTC schemes have proven to yield a satisfactory control performance.Secondly,the post analysis of the derived state-feedback control law reveals a simple and robust pattern in the solution to the optimal control problem.These observations have motivated the control scheme presented in this paper which is based on the following fundamental property of DTC.The control objectives only weakly relate to optimality but rather to feasibility,in the sense that the main objective is tofind a control input that keeps the controlled variables within their bounds,i.e.a control input that is feasible.The second, weaker objective is to select among the set of feasible control inputs the one that minimizes the average switching frequency.The latter can be approximated by the number of switch transitions over the(short)horizon.We therefore propose an MPC scheme based on feasibility with a prediction horizon N and an internal model of the DTC drive for the predictions.We propose to switch only at the current time-step and to disregard switching within the prediction horizon,which is equivalent to a move blocking strategy.This greatly reduces the number of control input sequences from8N to8and allows us to evaluate a small number of input sequences by moving forward in time.For276T.Geyer and G.Papafotioueach input sequence,we determine the number of steps the controlled variables are kept within their bounds,i.e.remain feasible.Next we define the number of switch transitions divided by the number of predicted time-steps an input remains feasible as a cost function emulating the switching frequency.In a last step,the control input is chosen that minimizes the cost function.We refer to this concept as the Feasibility Approach.The simplicity of the control methodology (with the only design parameter N)translates into a state-feedback control law with a complexity that is of an order of magnitude lower than the one of its counterpart obtained through solving the optimal control problem[9].The paper is organized as follows.Starting with the derivation of a low com-plexity piecewise affine model for the DTC drive in Section2,we pose in Section3 the control objectives.In Section4,wefirst present the Feasibility Approach as a control scheme that is evaluated on-line,and subsequently,we show how the control problem can be pre-solved off-line and translated into a state-feedback control law.Simulation results for the case of a two-level inverter are shown in Section5,while Section6summarizes the results and discusses the extendability of the control approach to DTC drives featuring three-level inverters.Due to the page limitation the paper had to be shortened by a few pages (mostly Section2).The full paper is available as technical report[4].2Modelling2.1Physical SetupFor the modelling of the DTC drive,all variables are transformed from the three-phase system(abc)to an orthogonal dq0reference frame with a direct(d),a quadrature(q)and a zero(0)axis,that can be either stationary or rotating[6]. For the needs of this paper,the transformation of a vectorξabc=[ξaξbξc]T from the three-phase system to the vectorξdq0=[ξdξqξ0]T in the dq0frame is(a)The equivalent representation of a three-phase two-level inverter driving an inductionmotor(((b)The voltage vectors onthe dq plane with switchpositionsFig.1.Physical setup and voltage vectorsDirect Torque Control for Induction Motor Drives277 carried out throughξdq0=P(ϕ)ξabc,whereϕis the angle between the a-axis of the three-phase system and the d-axis of the reference frame,and P(ϕ)is the Park transformation[6].An equivalent representation of a three-phase two-level inverter driving an induction motor is shown in Fig.1(a).At each phase,the inverter can producetwo different voltages−V dc2,V dc2,where V dc denotes the voltage of the dc-link.The switch positions of the inverter can therefore be fully described using the three integer variables u a,u b,u c∈{−1,1},where each variable corresponds to one phase of the inverter,and the values−1,1correspond to the phase potentials−V dc2,V dc2,respectively.There are23=8different vectors of the form u abc=[u a u b u c]ing the Park transformation these vectors can be transformed into the dq0frame resulting in vectors of the form u dq0=[u d u q u0]T.The latter are shown in Fig.1(b),where they are mapped into the(two-dimensional)dq plane.Even though they are commonly referred to as voltage vectors,this term describes the switch positions rather than the actual voltages applied to the machine terminals.The dynamics of the squirrel-cage rotor induction motor are commonly mod-elled in a dq0reference frame that can be either stationary or rotating.The standard modelling approach,which can be found in detail in[6],yields a5-dimensional nonlinear state-space model,that uses as state variables the d-and q-components of the stator and rotorflux linkages per secondψds,ψqs,ψdr and ψqr,respectively,and the rotor’s rotational speedωr.The0-axis components are neglected,since they do not contribute to the electromagnetic torque and are decoupled from the dynamics in the d-and q-axis.The model parameters are the stator and rotor resistances r s and r r,the stator,rotor and mutual induc-tive reactances x ls,x lr and x m,respectively,the inertia constant H expressed in seconds,and the mechanical load torque T .In this standard dynamical model of the induction motor,the saturation of the machine’s magnetic material,the changes of the rotor resistance due to the skin effect and the temperature changes of the stator resistance are ignored.A more elaborate presentation of the induction motor’s modelling procedure is out of the scope of this paper.For details,the reader is referred to[6].2.2Low Complexity ModellingIn[9,10],we have derived a low-complexity model of the DTC drive taking into account that the statorflux dynamics are significantly faster than the dynamics of the rotorflux and the rotational speed,and that the length of the statorflux vector and the electromagnetic torque are invariant under a rotation of theflux vectors.This model has the state vectorx(k)=ψϑds(k)ψϑqs(k)cos(ϕ(k))T,(1)whereψϑds (k)andψϑqs(k)denote the d-and q-component of the rotated andmapped statorflux vector,andϕ(k)captures the position of the rotating refer-ence frame withϕ(k+1)=ϕ(k)+ωr T s.The output vector278T.Geyer and G.Papafotiouy (k )= T e (k )Ψ2s (k ) T (2)comprises the electromagnetic torque and the squared length of the stator flux vector,and the input vector is composed of the integer variables u a ,u b and u c u (k )=u abc (k )= u a (k )u b (k )u c (k ) T ∈{−1,1}3.(3)For a summary of the low-complexity modelling,the reader is referred to [4],whereas the complete modelling can be found in [9].2.3Piecewise Affine ModelIn a subsequent step,we have computed in [9]a piecewise affine (PWA)model for a DTC drive featuring a two-level inverter.PWA models [12]are defined by partitioning the state-space into polyhedra and associating with each polyhedron an affine state-update and output functionx (k +1)=f j (k )(x (k ),u (k ))(4a)y (k )=g j (k )(x (k ))(4b)with j (k )such that x (k )u (k ) ∈P j (k ),(4c)where x (k ),u (k ),y (k )denote at time k the real and binary states,inputs and outputs,respectively,the polyhedra P j (k )define a set of polyhedra {P j }j ∈J on the state-input space,and the real time-invariant functions f j (k )and g j (k )are affine in the states and inputs,with j (k )∈J ,J finite.For simplicity,we will later drop the index j (k )and (4c),and use x (k +1)=f (x (k ),u (k ))and y (k )=g (x (k ))to denote the PWA system (4).Note that the PWA system (4)has no throughput,i.e.y (k )is independent of u (k ).To derive such a PWA model,all nonlinearities need to be replaced by PWA approximation over a bounded set of (feasible)states X 0.The set X 0can be easily determined by translating the output hysteresis bounds imposed by the control objectives into constraints on the state-space.Introducing the lower and upper bounds on the electromagnetic torque T e,min and T e,max ,respectively,and noting that in the low-complexity model the torque is a linear expression of the second state,the torque bounds can be directly translated into linear bounds on x 2(k )D m dr T e,min ≤x 2(k )≤D m drT e,max ,(5)where ψϑdr is equal to the length of the rotor flux,which is treated as a parameter in the low-complexity model.Similarly for the stator flux,its lower and upperbounds Ψ2s,min and Ψ2s,max turn into the quadratic state constraintΨ2s,min ≤x 21(k )+x 22(k )≤Ψ2s,max .(6)To account for measurement noise and small disturbances causing the torque or the stator flux to slightly violate the imposed bounds,we relax (5)and (6)by 20%of the corresponding bound width.Direct Torque Control for Induction Motor Drives279 The bounds on the third state are derived from the angleϕ(k).To ensure that the model remains feasible for at least N time-steps when starting with aϕ(k)close toπ3,the bounds onϕ(k)are set to0≤ϕ(k)≤π3+Nωr T s,whichtranslate into the following bounds on x3(k)cos(π3+Nωr T s)≤x3(k)≤1.(7)Summing up,the constraints(5),(6)and(7)define the set of states X0for which the PWA model is to be defined.Thus,the nonlinearities of the DTC drive need to be approximated for x∈X0as shown in[9,10].Starting from a model description in the HYbrid Systems DEscription Lan-guage Hysdel[15],andfixing the operating point,namely the parametersωrandψϑdr ,the model can be transformed into PWA form with the mode enumera-tion algorithm[5].This procedure yields a PWA model defined on a polyhedral partition with48polyhedra in the six-dimensional state-input space.3Control ProblemThe most prominent control objective concerning the induction motor is to keep the electromechanical torque within bounds around its reference.In order to avoid the saturation or demagnetization of the motor,the amplitude of the stator flux has to be kept between certain pre-specified bounds around the reference which are in general time-invariant.The control objective concerning the inverter is to minimize the average switching frequency.4Feasibility ApproachTraditionally,based on the imposed bounds,the next voltage vector to be applied to the induction motor is selected by evaluating a look-up table every T s=25µs. The goal of this paper is to replace the look-up table by a new DTC scheme that is based on a systematic design procedure.This controller needs to address the above formulated objectives,i.e.to minimize the average switching frequency while keeping the controlled variables(torque and length of the statorflux) within the given bounds.Similar to[9,10],this controller is based on predictive control with a receding horizon policy.Minimizing the average switching frequency leads to a prediction horizon with an infinite number of steps.As such a problem in the context of hybrid systems is computationally not tractable,we need to approximate this objective.In[9,10],we have done this by restricting the prediction horizon N to a small number of steps(three or four)and by formulating an objective function that postpones switching and penalizes the violation of the bounds using soft constraints.In particular,we have allowed for switch transitions within the prediction interval.Dynamic programming[3]allowed us to compute off-line the explicit state-feedback control law for the whole state-space.280T.Geyer and G.Papafotiou4.1On-line Computation of the Control InputOn the other hand,the underlying optimization problem of the above stated control problem is not so much based on optimality but rather on feasibility,meaning that the controlled variables have to be kept within their bounds,i.e.feasible.This insight greatly simplifies the control problem.Furthermore,we propose to switch only at the current time-step k and to disregard switching within the prediction horizon,which is equivalent to a move blocking strategy.This greatly reduces the number of control input sequences from 8N to 8and allows us to evaluate a small number of control sequences by moving forward in time.As a result,dynamic programming moving backwards in time becomes obsolete.More formally,let u (k −1)denote the last voltage vector.If u (k −1)is also feasible at time-instant k ,i.e.all controlled variables are predicted to lie within their bounds at time-instant k +1,a reasonable choice is to apply it again,i.e.u (k )=u (k −1).If not,however,the controller must choose another voltage vector.For each of the remaining seven voltage vectors,one can easily com-pute through open-loop predictions the number of time-steps this voltage vector would keep the controlled variables within their bounds.This step reduces the op-timal control problem to a feasibility problem.The voltage vector is chosen that minimizes the average switching frequency over the prediction interval,i.e.the number of switch transitions over the number of time-steps,thus re-introducing the notion of optimality.This control concept,to which we refer as the Feasibil-ity Approach,is summarized in Algorithm 1,where f and g refer to the PWA model (4).An output vector y (k )is said to be feasible,if the corresponding bounds are met,and U ={−1,1}3denotes the set of available voltage vectors.Algorithm 1function u (k )=Algo1(x (k ),u (k −1))x (k +1)=f (x (k ),u (k −1))if y (k +1)=g (x (k +1))feasibleu (k )=u (k −1)elsefor all u (k )∈U \u (k −1)n u =−1repeatn u =n u +1x (k +n u +1)=f (x (k +n u ),u (k ))until y (k +n u +1)=g (x (k +n u +1))infeasible or n u =Nendfor u (k )=arg min u (k )||u (k )−u (k −1)||n u endifDirect Torque Control for Induction Motor Drives281 Compared to MPC,this control policy is by definition significantly simpler, as only eight control sequences(or control strategies)need to be compared with each other.Unlike in MPC,switch transitions within the prediction interval are not considered,and can only be performed at the current time-instant k.Fur-thermore,the length of the prediction horizon is time-varying,ranging from one step to10or even20steps.As the next section will show,an explicit form of the proposed controller can be computed easily.Even more important,the explicit form has a low complexity but maintains or improves the control performance with respect to MPC.4.2Off-line Computation of the State-Feedback Control LawWe restrict the computation of the explicit state-feedback control law to the set of states X0,which we have obtained by relaxing the bounds on the torque and theflux by20%.Furthermore,wefix the operating point,namely the rotor speedωr and the length of the rotorfluxψϑdr ,and set the lower and upper bounds onthe outputs(torque and statorflux).Next,we derive the PWA model defined on X0.Rewriting(5)and(6),let C denote the set of states whose corresponding outputs are feasibleC={x∈X0|T e,minΨ2s,min≤g(x)≤T e,maxΨ2s,max,(8)where we have replaced the quadratic expression in(6)by the PWA approxima-tion for the statorflux.Before presenting the computation of the state-feedback control law in three stages,we introduce the following notation.Let n denote the time-step withinthe prediction horizon N,X nfeas the set of states at time-step k+n correspondingto feasible outputs y(k+ )for all ∈{1,...,n},X ninfs the set of states at time-step k+n with feasible outputs y(k+ )for all ∈{1,...,n−1},but infeasible outputs y(k+n),and Q n u the set of states at time-step k that keep the outputs for n time-steps feasible when applying the voltage vector u.Stage I.First,we determine the set of states x(k)∈X0for which the controlled variables are feasible at time-step k+1when applying u(k)=u(k−1).We denote this set of states as the coreQ c u={x∈X0|f(x,u)∈C},(9) and its complement in X0as the ringQ r u=X0\Q c u.(10) Example1.To visualize the algorithm,consider as an example a two-level in-verter driving an induction machine with the rated voltage3.3kV and the rated real power1.587MW.All parameters can be found in[9]in Tables3and4. The operating point is given by the rotor speedωr=0.8p.u.,the load torque282T.Geyer and G.Papafotiou0.820.860.940.981.020.200.220.240.260.280.50.60.70.80.90.91.0x 1(k )x 2(k )x 3(k )(a)The core 0.820.860.940.981.020.200.220.240.260.280.50.60.70.80.90.91.0x 1(k )x 2(k )x 3(k )(b)The ringFig.2.Core and ring for u (k −1)=[1−1−1]T =0.8p.u.,the torque bounds T e,min =0.72p.u.and T e,max =0.88p.u.,andthe flux bounds Ψ2s,min =0.82p.u.and Ψ2s,max =1.04p.u..After deriving thePWA model on X 0(enlarged by 20%as in Section 2.3),and determining theset C ,the core and the ring can be easily computed as shown in Fig.2for the voltage vector u (k −1)=[1−1−1].This operation takes on a Pentium IV roughly 1s. Stage II.For each new voltage vector u (k )∈U \u (k −1),the following procedure is performed for the initial set 1X 0.Initially,we set n =0.Next,we map the polyhedra X n from time-step k +n to k +n +1yielding X n +1.The states corresponding to infeasible outputs form the set X n +1infs .Consequently,we map X n +1infs back to the time-step k and associate with them the number of time-steps n .We denote these polyhedra by Q n u ,where u corresponds to the chosen voltage vector u (k ),and n denotes the number of time-steps this voltage vector u (k )can be applied to the set of states before any of the outputs violates a bound.If there remain any feasible states,we move one time-step forward in the future by increasing n by one and repeat the above procedure.This yields for each new voltage vector a polyhedral partition of the ring {Q n u }N n =0,where each polyhedron is associated with a unique number indicating for how many time-steps the respective voltage vector can be applied before any of the controlled variables violates a bound.1Conceptually,this stage of the algorithm should be initialized with the ring Q r u rather than X 0.Let us note though that since the facets of the initial set are mapped forward and backward in time,in the worst case,the complexity of the algorithm both in terms of the computation time and the number of resulting polyhedra {Q n u }N n =0is exponential in the number of facets of the initial set.Therefore,as X 0is by definition a very simple polytopic set with only a few facets,whereas the ring is a non-convex set with possibly many facets,we initialize Algorithm 2with X 0rather than the ring.Direct Torque Control for Induction Motor Drives283(a)Initial set X 0atstepk (b)Set X 1at step k+11x 2(k )(c)Set Q 0u at step kthat yields infeasibleoutputs at k +1Fig.3.First step of Algorithm 2in the x 1x 2plane for u (k )=[1−1−1]Next,the algorithm is summarized,where the two subfunctions mapForw and mapBack are affine transformations of polyhedra using the PWA model (4)for a fixed voltage vector u (k ).Specifically,mapForw yields X n +1={f (x,u )|x ∈X n ,u =u (k )},and mapBack yields Q n u ={x |(f u ◦...◦f u )(x )∈X n +1infs },where we have set f u (x )=f (x,u )and concatenated f u n times.Note that mapForw maps a set of states by one time-step forward in time,whereas mapBack maps a set of states by n time-steps backwards.The subscript feas (infs)refers to sets of states corresponding to feasible (infeasible)outputs.Algorithm 2function {Q n u }N n =0=Algo2(C ,X 0,u ,N )n =0while X n =∅and n <NX n +1=mapForw (X n ,u )X n +1feas =X n +1∩CX n +1infs =X n +1\X n +1feas Q n u =mapBack (X n +1infs ,u )X n +1=X n +1feasn =n +1endwhileQ n u =mapBack (X n ,u )Example 1(continued ).Setting N =4,we proceed with Example 1.Fig.3visualizes the first step (n =0)of Algorithm 2in the x 1x 2plane,where the same scaling is used for all three figures.Starting with the initial set of states X 0in Fig.3(a),the voltage vector u (k )=[1−1−1]maps X 0from time-stepk to k +1as shown in Fig.3(b).The set X 1comprises two parts.X 1feas (X 1infs )contains the states corresponding to feasible (infeasible)outputs.This set X 1infs(a)Set X1at step k+1(b)Set X2at step k+2(c)Set Q1u at stepk that yields feasibleoutputs at step k+1,but infeasible outputsat k+2Fig.4.Second step of Algorithm2in the x1x2plane for u(k)=[1−1−1]is consequently mapped back from time-step k+1to k resulting in Q0u and indicating that this set is zero-step feasible for the chosen u(k).Furthermore,weset X1=X1feas .The second step(n=1)is shown in Fig.4starting from the set X1at time-step k+1in Fig.4(a).Applying u(k)=[1−1−1]to this set maps it fromtime-step k+1to k+2as shown in Fig.4(b).Again,X2feas (X2infs)containsthe states corresponding to feasible(infeasible)outputs.The states in X2infs aremapped back for two steps from k+2to k yielding Q1u which is shown in Fig.4(c) and refers to states which are one-step feasible for u(k).Repeating the above procedure for n=2,3,4and collecting the sets Q n u yields the polyhedral partition{Q n u}4n=0shown in Fig.5.The outer polyhedra correspond to outputs that are feasible for zero time-steps when applying u(k)= [1−1−1],while the inner polyhedra are feasible for one,two,three and four time-steps as x2(k)is increasing.Note that{Q n u}4n=0is by construction a polyhedral partition of the set X0.The computation time for the second stage for the given example is approx-imately2min on a Pentium IV.Summing up,Stages I and II yield a semi-explicit control law that is eval-uated by following Algorithm1,with the main difference that the number of steps n u is not calculated by mapping operations but rather by set membership tests evaluating if the given state lies in the respective polyhedron.Specifically, if for the given u(k−1),the state x(k)lies in the core,reapply the last volt-age vector again.Else determine for each new voltage vector the polyhedron in {Q n u}N n=0containing x(k),evaluate the associated number of time-steps n u,and find the voltage vector u(k)with the lowest cost as defined in Algorithm1.This is formalized in Algorithm3.(a)Polyhedral partition in the x1x2 plane for x3=0.950.90.210.820.860.940.981.020.220.240.260.280.50.60.70.80.9x1(k)x2(k)x3(k)(b)Polyhedral partition in the three-dimensional state spaceFig.5.The resulting polyhedral partition{Q n u}N n=0of Algorithm2for u(k)=[1−1−1] and N=4,where the colors correspond to the number of steps nAlgorithm3function u(k)=Algo3(x(k),u(k−1))if x(k)∈Q c uu(k)=u(k−1)elsefor all u(k)∈U\u(k−1)determine n u such that x(k)∈Q n u uendforu(k)=arg min u(k)||u(k)−u(k−1)||n uendifRegarding the computational burden for the on-line computation of the con-trol input,in the worst case,one core needs to be evaluated and the seven polyhedral partitions of U\u(k−1)which feature in general a low number of polyhedra.Stage III.In the third stage we pre-compute Algorithm3and derive the fully ex-plicit control law as a function of the last voltage vector u(k−1)and the current state x(k).For u(k−1)∈U,we evaluate for each polyhedron in{Q n u}N n=0the cost and associate with it the voltage vector u(k).Next,the core Q c u is added with zero cost and the voltage vector u(k)=u(k−1).Finally,we compare the cost expressions and iteratively remove(parts of)polyhedra with inferior。