Lappeenraman teknillinen korkeakoulu Lappeenranta University of Technology
011 i Pyrh on en
ANALYSIS AND CONTROL OF EXCITATION, FIELD WEAHENING AND STABILITY IN DIRECT TORQUE CONTROLLED ELECTRICALLY EXCITED SYNCHRONOUS MOTOR DRIVES
Thesis for the degree of Doctor of Science (Technology) to be presented with due permission for public examination and criticism in the Auditorium in the Students’ Union Building at Lappeenranta University of Technology, Lappeenranta, Finland on the I Ithof December, 1998, at noon
Tieteellisia julkaisuja Research papers 74 ISBN 95 1-764-274-1 ISSN 0356-8210
Lappeenrannan teknillinen korkeakoulu Monistamo 1998 DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. 1
ABSTRACT
Lappeenranta University of Technology Research papers 74
Olli Pyrhonen
Analysis and Control of Excitation, Field Weakening and Stability in Direct Torque Controlled Electrically Excited Synchronous Motor Drives
Lappeenranta 1998
ISBN 95 1-764-274-1 UDK 621.313.32 : 621.3.077
Key words: synchronous machines, synchronous motor drives, direct torque control, excitation, field weakening, stability control
Direct torque control (DTC) is a new control method for rotating field electrical machines. DTC controls directly the motor stator flux linkage with the stator voltage, and no stator current controllers are used. With the DTC method very good torque dynamics can be achieved. Until now, DTC has been applied to asynchronous motor drives.
The purpose of this work is to analyse the applicability of DTC to electrically excited synchronous motor drives. Compared with asynchronous motor drives, electrically excited synchronous motor drives require an additional control for the rotor field current. The field current control is called excitation control in this study. The dependence of the static and dynamic performance of DTC synchronous motor drives on the excitation control has been analysed and a straightforward excitation control method has been developed and tested.
In the field weakening range the stator flux linkage modulus must be reduced in order to keep the electro motive force of the synchronous motor smaller than the stator voltage and in order to maintain a sufficient voltage reserve. The dynamic performance of the DTC synchronous motor drive depends on the stator flux linkage modulus. Another important factor for the dynamic performance in the field weakening range is the excitation control. The field weakening analysis considers both dependencies. A modified excitation control method, which maximises the dynamic performance in the field weakening range, has been developed.
In synchronous motor drives the load angle must be kept in a stabile working area in order to avoid loss of synchronism. The traditional vector control methods allow to adjust the load angle of the synchronous motor directly by the stator current control. In the DTC synchronous motor drive the load angle is not a directly controllable variable, but it is formed freely according to the motor’s electromagnetic state and load. The load angle can be limited indirectly by limiting the torque reference. This method is however parameter sensitive and requires a safety margin between the theoretical torque maximum and the actual torque limit. The DTC modulation principle allows however a direct load angle adjustment without any current control. In this work a direct load angle control method has been developed. The method keeps the drive stabile and allows the maximal utilisation of the drive without a safety margin in the torque limitation. 11
ACKNOWLEDGEMENTS
I took special interest in electrical drives and in the DTC method already during the years 1990 and 1993, when I was working at the research department in ABB Industry Oy. These years of working experience turned out to be very valuable for all later research work, because during that time the DTC method has been intensively developed for asynchronous motor drives by the company’s research team. Additionally, my working in the industry branch gave me a good general survey of the large application area of electrical drives.
The research work of this study has been camed out during the years 1995 and 1998 in the Laboratory of Electrical Engineering at Lappeenranta University of Technology, where I have worked as a laboratory manager and senior researcher. The research work introduced in this thesis is part of a larger research project, in which the different aspects of the DTC method for synchronous motor drives are studied.
I would lie to thank professor Jarmo Partanen, the head of the Institute of Electrical Engineering and the supervisor of this work. It was his activity and enthusiasm, which made this research project possible in the first place. His has encouraged and helped throughout the work.
Special thanks are due to professor Juha Pyrhonen, my brother, whose knowledge of electrical machines has been extremely valuable also for this project. His contribution to the revision of the manuscript has been of immense importance.
I wish to express my sincere thanks to the pre-examiners of this work, professor Mats Al&la, Lund Institute of Technology, Sweden and docent Janne Vglsniinen, Tellabs Oy, for their valuable comments and corrections.
The “red pen boutique” of professor Pekka Eskelinen has given an important contribution to the scientific style of the manuscript. He revised the text not only once, but twice. I am most gratehl to him for his valuable help as well as for his encouragement. I also would like to thank Mrs Julia Parkkila for her contribution to improve the English language of the manuscript.
The good working atmosphere in the synchronous motor drive research team has been of great importance to me. I would like to thank the members of the team as well as the other staff in the Laboratory of Electrical Engineering. I wish also to express my thanks to my former boss, professor Martti Harmoinen, and to my former colleagues at the research team in ABB Industry Oy, for giving me an interesting career start.
I am obliged to the NO Foundation, the Lauri ja Lahja Hotisen Siizitio and the Sahkoinsinooriliitto for the financial support. I also thank ABB Industry Oy for helping our research team to construct the test equipments for the laboratory tests.
My parents, Raili and Jorma, have given me all the best for a good start in life, they have encouraged and supported me also during the recent years. They deserve my special thanks.
Most of all, I am indebted to my wife Kaisa for her love and patience, and to my children, Aino, Lauri and Eeva, for giving me strength and motivation for this work.
Lappeenranta, December 1998 OIli Pyrhonen ... u1
CONTENTS
ABSTRACT ...... _._...... 1 ACKNOWLEDGEMENTS ...... , ...... 11... CONTENTS ...... 111 NOMENCLATURE ...... V
1 INTRODUCTION ...... 1 1.1 History of speed controlled AC motor drives...... 1 1.2 Functional principle of DTC ...... 2 1.3 DTC for electrically excited synchronous mot 8 1.4 Outline of the thesis..._...... 12
2 ANALYSIS OF STATIC AND DYNAMIC PERFORMANCE ...... 13 2.1 Maximal static torque...... 13 2.2 Maximal dynamic torq 17 2.2.1 Stator current, stator flux linkage and fie1 18 2.2.2 Maximal dynamic torque in the theoretic 21 2.2.3 Transient analysis with derived operator inductances ...... 24
3 EXCITATION CONTROL OF DTC SYNCHRONOUS MOTOR DRIVES ... . ___.___. . __ 35 3.1 Reactive power compensation ...... 35 3.2 Effect of magnetic saturation on the calculation of the excitation curve...... 36 3.3 Combined open loop and feedback control ...... 39 3.4 Reaction excitation control in DTC ...... 42
4 FIELD WEAKENING CONTROL OF DTC SYNCHRONOUS MOTOR DRIVES ...... 46 4.1 Voltage Reserve in field weakening ...... 47 4.2 Relation between voltage reserve and excitation voltage ...... 50 4.3 DTC modulation in field weakening range ...... 53 4.4 DTC stability control in field weakening range ...... 61 4.4.1 Indirect load angle control ...... 61 4.4.2 Direct load angle control ...... 64
5 SIMULATION AND TEST RESULTS ...... 67 5.1 Description of the simulation method ...... 67 5.2 Simulation results of excitation control ...... 69 5.2.1 Combined excitation control in the n eed range ...... 69 5.2.2 Excitation control in the field weakening range ...... 76 5.3 Simulation results of flux control in the field weakening range ...... 83 5.3.1 Voltage reserve and dynamic performance...... 83 5.3.2 Results of modulation modification in the field weakening range...... 86 5.4 Simulation results of stability control in the field weakening range ...... 88 5.5 Results of laboratory tests ...... 91 5.5.1 Description of the laboratory test drive ...... 91 5.5.2 Measurements of excitation control ...... f...... 93 5.5.3 Measurements of field weakening control ...... 95 5.5.4 Measurements of stability control ...... __.__...... 97 5.6 Discussion of the results ...... 98 iv
6 SUMMARY ...... 100
7 REFERENCES ...... 102
APPENDIX A, Reference motor parameters used in the simulation...... 105
APPENDIX B, Excitation curve calculation for the salient pole synchronous
APPENDIX C, Description of the PC-based C-language DTC-Simulator ...... V
NOMENCLATURE
value of cost function electro motive force vector, pu electro motive force component produced by yf,pu basic frequency for per unit system cost function motor modelling function stator voltage vector index d-axis stator current, pu d-axis damper winding current, pu d-axis current sum, pu field current, pu field current reference for power factor correction, pu field current corresponding saturated excitation curve, pu field current corresponding unsaturated excitation curve, pu magnetising current of non salient pole machine, pu rotor component of magnetising current im, pu stator component of magnetising current im, pu current vector, pu active stator current component, pu reactive stator current component, pu x-axis stator current component in stator reference frame, pu y-axis stator current component in stator reference frame, pu q-axis stator current component in rotor reference frame, pu q-axis damper winding current, pu q-axis current sum, pu torque stator current component, pu phase current, pu flux linkage stator current component, pu Measured field current field current used by digital control system phase current complex operator, time step index discrete time instant over excitation coefficient d-axis transient coefficient in Eq.(2.12) inverse inductance matrix component, pu inverse inductance matrix component, pu inverse inductance matrix component, pu inverse inductance matrix component, pu inverse inductance matrix component, pu d-axis transient coefficient in Eq.(2.12) q-axis transient coefficient in Eq.(2.12) q-axis inductance coefficient in Eq.(2.12) inverse inductance matrix component, pu cost hnction coefficient for flux linkage amplitude error inductance coefficient for d-axis damper winding inductance coefficient for q-axis damper winding voltage reserve coefficient, which includes resistive voltage drop vi voltage reserve coefficient, which excludes resistive voltage drop inverse inductance matrix component, pu inverse inductance matrix component, pu inverse inductance matrix component, pu inverse inductance matrix component, pu coefficient in general d-axis damper winding inductance, pu d-axis transient inductance, pu d-axis damper winding leakage inductance, pu field winding inductance, pu mutual inductance between field winding and d-axis damper winding, pu field winding leakage inductance, pu field winding and d-axis damper winding common leakage inductance, pu magnetising inductance of non salient pole machine, pu d-axis magnetising inductance, pu saturated d-axis magnetising inductance, pu q-axis magnetising inductance, pu saturated q-axis magnetising inductance, pu q-axis damper winding inductance, pu q-axis transient inductance, pu q-axis damper winding leakage inductance, pu d-axis stator inductance, pu q-axis stator inductance, pu stator leakage inductance, pu rotating speed, pu rotating speed proportional gain for feedback excitation controller field winding resistance, pu stator resistance, pu d-axis damper winding resistance, pu q-axis damper winding resistance, pu Laplace operator switching commands for three phase inverter time electrical torque vector, pu electrical torque vector modulus, positive (+lie[)for anticlockwice torque, negative (-Ifel) for clockwice torque, referred as torque in text, pu torque for unity power factor, pu maximal dynamic torque, pu maximal torque, pu simulation step time electrical torque vector (see also (?,I for modulus and sign definition) d-axis transient time constant d-axis subtransient time constant field winding time constant d-axis damper winding time constant ramp time constant for excitation filter time constant mechanical torque vector (see also Jt,l for modulus and sign definition) vii
q-axis damper winding time constant q-axis transient time constant ramp time constant load transient time constant Ud d-axis stator voltage component, pu d-axis stator voltage component produced by DTC modulation, pu d-axis voltage reserve component, pu excitation voltage, pu excitation unit voltage reserve, pu discrete voltage vector of a two level inverter, pu instantaneous phase voltage, pu q-axis stator voltage component, pu q-axis stator voltage component produced by DTC modulation, pu q-axis voltage reserve component, pu voltage reserve, pu internal voltage reserve which does not include resistive voltage drop, pu voltage reserve vector, pu Voltage reserve vector in rotor reference frame, pu stator voltage vector, pu internal stator voltage vector which does not include resistive voltage drop, pu x-axis stator voltage component in stator reference frame, pu y-axis stator voltage component in stator reference frame, pu Discrete voltage vectors of three phase inverter, pu DC link voltage of voltage source inverter stator voltage vector instantaneous phase voltage d-axis synchronous reactance d-axis transient reactance d-axis subtransient reactance q-axis synchronous reactance q-axis transient reactance q-axis subtransient reactance
direction of voltage vector in switching sector angles between stator flux linkage vector and discrete voltage vector angle between air gap and stator flux linkage vectors stator flux linkage load angle load angle for unity power factor load angle for maximal dynamic torque higher load angle limit in load angle hysteresis control lower load angle limit in load angle hysteresis control air gap flux linkage load angle load angle, for which triangle in Fig. (2.1) is orthogonal load angle for maximal torque error signal for feedback excitation control modulation modification angles rotor position angle in stator reference frame position angle of stator flux linkage vector in stator reference frame swithing sectors of DTC ... vlll
dimensionless time, 7 = clast load transient time constant, dimensionless phase shift angle between stator current and stator voltage d-axis damper winding flux linkage component, pu flux linkage component created by field current, =if lmd, pu air gap flux linkage vector, pu stator flux linkage vector, pu q-axis damper winding flux linkage component, pu d-axis stator flux linkage component, pu stator flux linkage vector estimate, pu real stator flux linkage in motor, pu q-axis stator flux linkage component, pu x-axis stator flux linkage component in stator reference frame, pu y-axis stator flux linkage component in stator reference frame, pu stator flux linkage vector angular speed, pu basic angular speed for per unit system mechanical angular speed, pu
Symbols, Laplace domain
transfer function between q-axis stator current and voltage reserve d-axis stator current d-axis damper winding current field current q-axis stator current q-axis damper winding current d-axis operator inductance q-axis operator inductance q-axis voltage reserve d-axis stator flux linkage reference q-axis stator flux linkage
Subscripts act actual value av avarage value b basic value of per unit system cos9 1 unity power factor condition d,max dynamic maximal torque condition d,q rotor reference frame, stator quantities QQ rotor reference frame, damper winding quantities est estimated value 1 current model max maximal min minimal opt optimal ix
Pr previous ref reference value ref,d dynamic reference S stator quantity sat magnetic saturation tr transient U voltage model %Y Stator reference fiame 0 Initial value
Superscript
(4 Order of derivative r Rotor reference frame R slope limited with ramp * Final working point
Operators
RE{ } Redpart IM{ } Imaginarypart
Acronyms
emf electromotive force mmf magneto motive force PU per unit value AC Alternate Current DFLC Direct Flux Linkage Control DC Direct Current DTC Direct Torque Control IGBT Insulated Gate Bipolar Transistor IGCT Integrated Gate Commutated Thyristor LPF Low Pass Filter LUT Lappeenranta University of Technology PWM Pulse Width Modulation VSI Voltage Source Inverter 1
1 INTRODUCTION
1.1 History of speed controlled AC motor dnves
The idea of vector controlled AC motor drives is based on the excellent control properties of DC motors, where torque and motor magnetic flux can be separately controlled. In the development of vector controlled AC technology the first theoretical problem was to model a three phase AC coil system in a way, that separation between torque and flux control could be done.
Extensive research work of synchronous machine theory was done in the 1920s mainly in the United States. Motivation for synchronous motor modelling that time was not vector control development, but the increasing importance of synchronous machines in power systems. Improved models for synchronous machines were required particularly for the analysis of abnormal situations in power networks.
One of the most thorough researches on synchronous machine theory has been done by R.E.Doherty and C.A.Nickle, who represented their results in a four paper series at the end of 1920s (Doherty & Nickle 1926, 1927, 1928 and 1930). These papers already included the two-axis modelling of synchronous machine. The more detailed mathematical analysis of the two-axis model was presented by Park (1929). Park’s two axis mathematical analysis was a success, and that is the reason why the synchronous motor two-axis model carries his name. Park’s two-axis model includes indirectly one basic element of vector control; the actual measurable phase quantities were replaced by calculatory elements in a different reference frame.
The next major steps in AC machine modelling were taken in the 1950s, when the space vector theory was developed for multi-phase AC machines in Hungary. The theory was published in German at the end of 1950s by Rovacs & Racz (1959). The space vector theory made it possible to combine motor phase quantities into a single complex vector variable in any reference frame. This was a breakthrough, which made the final vector control innovation possible.
An important factor, which made controlled electrical drives more interesting on a practical level, was the intensive development of semiconductor devices in the middle of this century. The first breakthrough was the development of the transistor in 1948. The introduction of the first commercial thyristor ten years later was the second discovery. This thyristor was the first controllable semiconductor power switch, which made the true electronic control of power electric circuits possible.
In the 1960s, the new semiconductor technology was intensively applied in controlled DC motor drives, and thyristor bridge supplied drives gained popularity in industrial and traction applications. At the same time, intensive research work has been done to develop AC drive systems with variable frequency. The first variable frequency AC drives were based on the pulsewidth modulation (PWM), (Stemmler 1994).
At the end of 1960s, German engineer Felix Blaschke made an innovation, which lead to the development of the first field oriented vector controlled AC motor drive. He represented the principle of field orientation and the separate control of motor magnetic flux linkage and torque, the so called transvector control, (Blaschke 1972). This method made it possible for the first time to 2 control AC motors like DC motors. Siemens applied the transvector control for large synchronous motors (Bayer et al. 1971).
At the same time, intensive research work was going on in the Finnish company Stromberg to develop the asynchronous motor speed control. Asynchronous motors were one of the company’s main products, and they suited very well for traction drives and industrial applications due to their robustness and competitive price. The solution was not a vector control, but a PWh4 based variable frequency control, which was named scalar control. The main designer of the new technology was Finnish engineer Martti Harmoinen. As a result, the first world wide AC motor traction drive was introduced in the underground of Helsinki at the beginning of 1980s. Another important application area, where Stromberg used for the first time world wide AC motor technology, was the sector of the paper machine speed controlled drives.
In 1980s, AC motor drive technology was getting more popular in the different application areas. Because of the growing demand for high dynamic performance the Blaschke’s idea of field oriented vector control was introduced into the field of asynchronous machines as well. During the whole decade the vector controlled asynchronous motor has been an object for intensive research work as well as product development. The wide survey of the vector control methods for AC drive systems has been represented e.g. by Leonhard (1 996) and Vas (1 992).
In the middle of 1980s, a new principle for asynchronous motor control has been developed in Germany and in Japan almost simultaneously by Depenbrock (1985) and by Takahashi and Noguchi (1 986). Depenbrock named his new method the Direct Self Control. A name, which better describes the method, is Direct Flux Linkage Control - DFLC. The name points out, that the stator flux linkage of the AC motor is directly controlled with the stator voltage vector and no current vector control is necessary. Again, the essential tool for the new control method was the space vector theory. The first industrial application, which used the DFLC method, was introduced in Finland by the company ABB Industry Oy, previously Strornberg Oy (Tiitinen et a1 1995). There DFLC is combined with a current model, which keeps the stator flux linkage estimate accurate also at low frequencies. The method was named Direct Torque Control, called later DTC.
1.2 Functional principle of DTC
The understanding of DTC or other field oriented vector control methods requires the presentation of the space vector. If a three phase symmetrical coil system is assumed, scalar valued electrical quantities in different coils, like current, voltage or flux linkage, can be combined into one complex vector variable, where different coil magneto motive force (mmf) directions are defining the directions for each coil quantities. The stator voltage vector Usis defined as
In Eq. (1.1) the voltage space vector is introduced. The voltages Ul(t), U2(r)and U3(t) represent the instantaneous phase voltage values and the angles 0, 2d3 and 4x/3 represent the mmf directions of the respective coils in a symmetrical three phase system. The length of the voltage vector is reduced by the factor 2/3 in order to make the vector length equal to the amplitude of the sinusoidally varying phase voltage of the three phase symmetrical system. Per unit (pu) quantities will be used this work 3 throughout and thus pu stator voltage vector usis defined using pu phase voltages ul(t),u2(t) and 4t)as
The stator current pu space vector is is defined respectively using pu phase currents il(t), i2(t) and k(t) as
With a three phase two level inverter in conjunction with a three phase winding six different non zero voltage vectors can be produced. Faraday's induction law gives a connection between the stator voltage vector us and the stator flux linkage vector v/. When the basic angular speed of the pu system ua and the stator resistance r, are known, the relation between y, and usis
ys= obI (us- isrs)dt. (1.4)
This will later be called the voltage model. The pu electromagnetic torque t, can be calculated as the cross product of the flux linkage space vector and the current space vector
t, = y, x is. (1.5)
Later torque moduluslt,I will be named torque for simplicity. According to Eq. (1.4) it is possible to drive the stator flux linkage to any position with the available six voltage vectors of a three phase two level inverter. It has been proved e.g. by Takahashi and Noguchi (1986), that the increase of slip frequency immediately increases the motor torque in an asynchronous motor. From the motor point of view, it is advantageous to keep the stator flux linkage modulus at its nominal value. These two conditions together with the field orientation give all necessary information to control the power stage transistors in order to meet the required flux linkage modulus and the requested torque.
Besides the voltage model, Eq. (1.4), and the torque estimation law, Eq. (1.5), the third essential part of DFLC is the optimal switching table. The name optimal switching table is given by the Japanese inventors Takahashi and Noguchi (1986). The optimal switching table selects at every modulation instant the most suitable voltage vector in order to meet the flux linkage and the torque control requirements. The selection is done according to the stator flux linkage orientation.
I I I
Figure 1.1 Three phase two level voltage source inverter (VSI) in conjunction with a three phase winding. Six non zero voltage vectors ul..u6 and two zero voltage vectors uo and u7 are available. DC link voltage UDC. 4
The optimal switching table is a logic array, which has three logical input variables. The logical input for the torque and flux linkage modulus is done by supplying the error signals lyslref- Iy,/,,, and (reIrd - Ir,lxt into hysteresis comparators. The third input is the position angle 8 for w, which is converted to a discrete variable K The discrete variable K describes six different switching sectors q..~,where different voltage vectors are used for the certain combination of the flux linkage and torque logical variables, Figs. (1.2) and (1.3).
optimal switching
I I I I I I
Figure 1.2 The hnctional principle of the optimal switching table of DFLC is based on the three multi valued logical variables; the torque error Ifelref-lrelx,, the flux linkage modulus error lyslrd-lyslxt and the position angle Bfor w. The actual switching commands SI,SZ, S3 are available in the output. K - discrete valued field orientation, ? - increase lysl,& - decrease/y,l, k’-.\ - increase/t,/to positive direction, /Y - increase/i,/to negative direction, 0 - no torque or flux linkage modulus chance.
The optimal switching table is an ideal modulator. Every switching, that occurs, transfers the stator flux linkage towards the desired direction. No unnecessary switching takes place and the dynamic response is good. Since DFLC uses the voltage model, Eq. (1.4), for the stator flux linkage estimation, the only necessary parameter is the stator resistance. DFLC controls directly the torque of the motor by using the voltage and no stator current controllers are required. In principle DFLC suits well for the different types of rotating field machines. The represented parts of DFLC - the voltage model, the torque estimation law and the optimal switching table - are valid without modifications for asynchronous and synchronous motors.
The principal assumption of DFLC is, that the stator flux linkage can be estimated by using the voltage model alone. In practice this is not possible. The transistor switches cause nonlinearly current dependent voltage drops. The power switches require a certain time to switch on or off, and during that time the voltage is difficult to estimate. The resistive voltage drop in the motor cable and in the 5 stator winding causes further uncertainty. The voltage model is based on the integration, and errors are integrated, too. Even with quite accurate estimates for all these voltage drops the pure voltage integral based estimate is erroneous. The DFLC method keeps the stator flux linkage estimate origin centred. If it contains a cumulative error, the real motor stator flux linkage drifts erroneous, as is showninFig. (1.4).
Figure 1.3 A stator flux linkage vector trajectory as the result of DFLC. No unnecessary switching occurs in the DELC method. The voltage vector is selected according to the hysteresis limits of torque (Itc[-, Itclmax)and stator flux linkage modulus (I wl-, 1~1~).DFLC works in the x-y stator reference frame. Also rotor oriented d-q reference frame is shown. It is used later by the Park’s two axis model.
Symbols in the figure: is - Stator current vector vs - Stator flux linkage vector vm - Air gap flux linkage vector 8 - Position angle of w in stator reference frame y - Angle between vm and vs 6 - Stator flux linkage load angle o - Angular speed of stator flux linkage vector 6
Figure 1.4 The controlled stator flux linkage estimate yqatand the real flux linkage in the motor K,,,,~~,when the voltage model is used alone.
The voltage model requires some feedback to be able to estimate the stator flux linkage correctly. Since current measurements are required for the torque estimation, they are available also for the stator flux linkage estimation. The stator flux linkage can be calculated with the so called current model, which uses the measured stator currents and motor inductance parameters. Here DTC is defined as a vector control method, which uses the DFLC principle, but includes also some current feedback correction to keep the stator flux linkage estimate accurate.
Fig. (1.5) compares the control structures of DTC and a traditional field oriented control method. Most traditional vector control methods are based on the idea of field oriented control (Blaschke 1972), where torque and flux linkage can be separately controlled like in DC motors. Thus the basic difference between DTC and earlier methods is, that DTC combines torque and flux linkage control, whereas previous methods have separate control paths for torque and flux linkage. Further the control structure of DTC is simpler, since stator current controllers are not required. It is possible to control the flux linkage directly also in the traditional field oriented concept, like proposed by Alakula (1993). However, when compared to DTC, control structure is more complicated also in that case.
The structural simplicity of DTC core, where torque and flux linkage control are combined directly with the voltage modulation, enables better dynamic response than traditional field oriented control methods. The voltage model can estimate the stator flux linkage with good accuracy in torque steps. The long term accuracy for the stator flux linkage estimate is achieved by using feedback currents and some current model. Since every switching in DTC requires a controller decision, high computation power is required. Modern microcontrollers, however, offer good alternatives for high speed real time control. Figure 1.5 a
I' A
Figure 1.5 b
Figure 1.5 Control structure in DTC (a) and one example of traditional field oriented control of an AC motor (b). DTC combines torque and flux linkage control with voltage modulation and no current controllers are needed. In traditional field oriented control methods separate control paths are used for flux linkage control and torque control. Also separate current controllers are often used. The field current control has been left intentionally away from the figure.
Symbols in figure: It&f, Itelact - Torque reference and actual value I vlnf,I ySlact - Stator flux linkage modulus reference and actual value it13 - Phase current actual values il..3pf - Phase current reference values ~,Pfii, - Reference and actual values for flux current component imf, it - Reference and actual values for torque current component
9 I Rotor position 8
1.3 DTC for electrically excited synchronous motors
DTC was originally designed for asynchronous motors, but its robustness and simplicity has encouraged researchers to investigate other motor drives with DTC as well. The papers at the EPE conference in Norway 1997 were among the first international publications about the general features of DTC electrically excited synchronous motor drives (Pyrhonen, J. et a1 1997), (Zolghardi, M.R. et al, 1997).
In an asynchronous motor, there is a natural feedback between the rotor current and the load of the machine due to the increasing slip frequency. In a synchronous motor, the stator generates currents into rotor only during transients, when the damper windings are resisting the change of the air gap flux linkage. Despite of this functional difference, the torque can be changed in both machine types in the same way, by accelerating the stator flux linkage vector rotating speed. The idea of torque production by using the stator flux linkage vector acceleration becomes more evident, if the torque is expressed with the stator flux linkage vector w, and the air gap flux linkage vector y,. When the stator leakage inductance I,, is known, the torque can be expressed as
The air gap flux linkage ym has quite a long time constant due to different damping effects in the air gap region, typically in the range of 10.. 100 ms. The angular acceleration of the stator flux linkage is, according to Eq. (1.4), dependent only on the voltage available. A large torque step can be achieved by accelerating the rotational speed of the stator flux linkage y, in a fast way, so that the angle y between y, and the air gap flux linkage y, increases rapidly (Pyrhonen, J. et al 1997).
The fknctional difference of DTC applied to asynchronous or synchronous motors becomes evident during a load step. While in asynchronous motors the increase of slip frequency is permanent, in synchronous motors the angular speed difference between the stator flux linkage vector and the rotor disappears after the transient, and the induced damper winding currents decay to zero.
The electrically excited synchronous motor has typically quite large synchronous inductance values. Therefore, the armature reaction of the stator current is large as well. The large armature reaction and the lack of the natural rotor current increase will cause a large increase in the load angle and possible loss of synchronism, if the rotor field current is not increased along with the electrical torque. The most important differences between the asynchronous motor and synchronous motor DTC control appear thus in the field current control, later called the excitation control, and in the stability control. The term excitation control, familiar in generator applications, is used here, even though it is known, that in the case of a synchronous motor the stator voltage defines the machine’s magnetic condition.
Synchronous motors have, in principle, a stable working area approximately within the load angle 6 between -7d2...x/2. According to the basic theory of DTC the angular acceleration of y, is assumed to increase the torque in all cases. DTC does not recognise the unstable working area, but tries to accelerate the stator flux linkage vector y, as long as the actual value less than the torque reference. It is thus obvious, that DTC causes a very fast torque breakdown for the synchronous machine, if the load angle leaves the stable working area. 9
Variable frequency rotating field AC motor drives have a linear frequency dependent connection between the produced electro motive force e (em and the stator flux linkage modulus, defined by the angular speed w of the stator flux linkage vector
If the motor uses the nominal flux linkage, the maximal available voltage will be reached with a certain angular speed. This particular speed is called the fieZd weakening point, since the rotating speed cannot be increased above it without decreasing the flux linkage modulus. The speed range above the field weakening point is called the field weakening range, whereas the speed range below it is called the nominal speed range. The difference between the maximal available voltage modulus Ius/and the electromotive force e is defined as the voltage reserve u,,,
u,,= Is1u -e Isl-Is/ - u -my,.I I (1.8)
In the field weakening range only a small voltage reserve is available, and thus the drive dynamics is reduced. Also the armature reaction is stronger, since the currents are larger in relation to the stator flux linkage, than in the nominal flux range. Field weakening is thus a special drive mode, which should be examined separately.
It was found earlier, that DTC requires feedback for the stator flux linkage estimate correction, since the voltage model alone is not stable. The current model of the salient pole synchronous machine is based on the Park's two axis model. The salient pole motor is unsymmetric, and thus it is advantageous to examine the motor separately on the direct axis (d-axis) and on the quadrature axis (q-axis). The dependence between the currents and the flux linkages is then
where wd - d-axis stator flux linkage component, yq - q-axis stator flux linkage component,
cy^ - d-axis damper winding flux linkage component, YQ - q-axis damper winding flux linkage component, - field winding flux linkage component,
id - d-axis stator current component, i, - q-axis stator current component, iD - d-axis damper winding current, iQ - q-axis damper winding current, if - field winding current. 10
The inductance parameters in Eq. (1.8) are
where inductance components are Ima - d-axis magnetising inductance, lmq - q-axis magnetising inductance, I,, - stator leakage inductance, ID, - d-axis damper winding leakage inductance, IQa - q-axis damper winding leakage inductance, Ifa - field winding leakage inductance, lka - common leakage inductance for field winding and d-axis damper winding.
The voltage equations in the two-axis model are
O=iDrD+-- 1 4% (1.10) Wb at ' 1 avo O=iQrQ+-- Wb at ' 1 uf = ifrf + --, awf Wb at where ud - d-axis stator voltage component, uq - q-axis stator voltage component, rD - d-axis damper winding resistance, rQ - q-axis damper winding resistance, uf - excitation voltage.
The two-axis mode of a salien pole synchronous machine has been shown in Fig. (1.6).
Figure 1.6 Equivalent circuits for the salient pole synchronous machine two-axis model 11
With the known inductance parameters and measured currents and rotor position the current model of Eq. (1.9) can produce an estimate for the stator flux linkage. DTC for a synchronous motor requires thus the rotor position feedback and the knowledge of the motor inductance parameters and does not differ in this sense from traditional synchronous motor field oriented control methods. However, the essential difference between DTC and earlier control methods is still valid; the DTC control is based on the accurate physical connection between the voltage and the flux linkage, Eq. (1,4), and the control principle is simple and avoids unnecessary switching events. The drifting of the voltage integral is slow, and the current model of the motor is just needed to prevent that slow drifting. In fast transients, the voltage model is superior compared to the current model, where more specific the accurate damper winding modelling is the most demanding task. By combining the best parts of the voltage and the current model, a good stator flux linkage estimate can be achieved (Pyrhonen, J. et al, 1997).
The block diagram of a DTC synchronous motor drive is shown in Fig. (1.7). The main control blocks are: the torque and flux linkage hysteresis control including the optimal switching table, the motor model calculating the actual values for the stator flux linkage and for the torque, the flux linkage controller for the field weakening range, the load angle limitter to keep the drive in the stable working area and the excitation control, which controls the power factor and reacts fast to the load changes in order to improve the drive stability and dynamic performance.
Figure 1.7 Functional control block diagram of DTC electrically excited synchronous motor drive. Following symbols are introduced in the figure for the first time:
yqu,wY," - stator flux components of the voltage model in xy-reference frame w&i,yv,i - stator flux components of the current model in xy-reference frame isx,isy - stator current components in xy-reference frame if - rotor field current 12
1.4 Outline of the thesis
The research work introduced here is a part of a larger research project, where the different aspects of the DTC method for salient pole synchronous motors have been studied. The starting point for the project was a DTC drive for asynchronous motors, and the goal was to convert and enlarge it to be suitable for the control of salient pole synchronous motors. The main topics in the project have been
- flux linkage modelling - parameter identification and estimation - sensorless control without rotor position feedback - field current control (later called excitation control) - field weakening control - stability of the drive
The last three topics were studied by the author and are introduced in this work.
In traditional synchronous motor field oriented control methods, excitation control is usually related to stator current controllers. Thus the electromagnetic state of the machine is actually defined by the stator current controllers as well. In DTC the primary control variable is the stator flux linkage vector instead of the stator current. Since DTC lets the stator current to be formed freely, excitation control can not be related to stator current controllers. On the contrary, the field current control is needed in DTC to be able to adjust the stator current, like in synchronous generators. Thus a new concept for the excitation control is needed in DTC synchronous motor drives.
In the field weakening control there is a contradiction between the dynamic performance, motor losses and stability. A sufficient voltage reserve is essential for the dynamic performance, since the voltage reserve is the actual means of increasing the motor torque. On the other hand a reduced flux linkage increases the stator and rotor currents as well as the load angle for the certain level of torque. Also the maximal available torque will be reduced. The effects of the voltage reserve to the drive dynamics and stability have been studied in the field weakening range. Also an enhanced excitation method has been developed and analysed to improve both dynamic performance and stability.
The stability of a DTC synchronous motor drive is an important question because of the basic control principle, where it is assumed, that the rotational acceleration of the stator flux linkage vector always increases torque. This assumption is not valid in the unstable working area of the synchronous motor. The traditional method to assure synchronous motor drive stability is to limit the maximal torque below the assumed breakdown torque. The drawback is, that the utilisation of the motor is reduced, since some safety margin must be left between the maximal torque and the breakdown torque. In current controlled drives, the instantaneous overswing of the maximal load angle in a transient can be accepted, since the current controllers will reduce the load angle to the stable value, when the transient has been dampened. In DTC, such a recovery is not possible. Thus, if safety margin is used for DTC, it should be selected larger than in the previous case. In this work, an alternative method for DTC stability control was studied. In the alternative method the load angle can be controlled directly and no safety margin is necessary. This improves the usability and reliability of DTC synchronous motor drive especially in the field weakening range. 13
2 ANALYSIS OF STATIC: AND DYNAMIC PERFORMANCE
2.1 Maximal static torque
The field current magnitude and the stator flux linkage modulus define the maximal static torque of an electrically excited synchronous machine. The maximal static torque can thus be affected by the excitation control. Fig. (2.1) shows the envelope curve of the stator current vector is and the stator flux linkage vector vsfor the load angle LYE[ 0, XI, when the field current has a certain fixed value. The triangle surface area, which is formed by is and ys is proportional to the machine torque.
Figure 2.1 Envelope curves for the stator current vector is and the stator flux linkage vector vs, when the field current if and the stator flux linkage modulus I y/sI are ked. The dots are shown with I0"deg load angle intervals. Vectors is and y/s correspond to the maximal torque point. The magnitude of if is not in scale in the figure. Salient pole synchronous motor.
The maximal torque producing load angle 6- can be solved, if the field current if and the stator flux linkage modulus I cy1 are given. The stator current modulus is assumed to be unlimited here. The stator flux linkage components and the stator current components in the rotor reference frame can be expressed as a hnction of the stator flux linkage load angle 6
The flux linkage component Wf is not a physical quantity, but represents the product of the field current if and the d-axis magnetising inductance
Yf = I,, .if 14
The torque can be expressed as a function of the load angle:
which corresponds to the salient pole synchronous machine power equation divided by the stator flux linkage angular speed w.The more traditional expression is the one for the pu power p
The maximal torque producing load angle can be found by differentiating Eq. (2.3)with respect to the load angle 6 and finding the zero for the derivative
The zero of Eq. (2.5)can be obtained by rewriting it into a polynomial form, where the variable of the polynomial function is cos(&)
Eq. (2.6)represents a quadratic equation, which can be solved by the standard method
(2.7) 15
The maximal static torque modulus of an electrically excited synchronous machine will be reached using a load angle 6 E]-x/~,~/2[. That means, that from Eq. (2.7) such a root must be selected, which gives a positive value to the load angle cosinus
By substituting Eq. (2.8) into Eq. (2.3) the maximal torque as a function of the field current if and the stator flux linkage modulus I is obtained
L J
t
The numerical example below shows more clearly the dependence between the field current and the maximal torque of the motor. The motor parameters used also in further numerical examples are shown in Appendix A. Fig. (2.2) represents the maximal torque ltelmavand the load angle of the maximal torque working point as a function of the field current if. For comparison the torque and the load angle for unity power factor, liclcwcpl and i%wcpl,have been shown. Fig. (2.2) points out, that the maximal torque with a limited field current has been achieved with a lagging power factor instead of unity power factor. With a higher field current the difference of different torque curves is negligible. It is still to be noticed in Fig. (2.2), that the significance of the reluctance torque is quite large, when the field current and the stator flux linkage are both small. 16
0 1 2 3
Figure 2.2 The maximal torque producing load angle 4,- and the maximal torque (t,l,,, as a function of field current if in the field weakening range. The same quantities for unity power factor, &oq, and (fe~coslpl,are also shown. I = 0.3, reference machine inductances (see Appendix A).
In Fig. (2.1) the triangle formed by the current and the stator flux linkage vectors in the maximal torque point seems to be orthogonal. This, however, is not valid. When the line segment C and the stator flux linkage vector ysin Fig. (2.1) are orthogonal, the following condition is valid
(2.10)
Again, like in the case of Eq. (2.7), a load angle with a positive cosinus is relevant and the root must be derived according to it
cos6 = I (2.11)
Fig. (2.3) shows the difference between the load angle values given by the maximal torque condition, Eq. (2.8), and the orthogonality condition, Eq. (2.11). There is only one value for the field current, which fulfills both conditions. Figure 2.3. Load angle at the maximal torque working point & and at the orthogonal working point &, as a finction of field current if. 1 lys(=0.3,reference machine inductances (see Appendix A).
2.2 Maximal dynamic torque
Synchronous machines produce the maximal torque with the load angle 6~]-x/2, x/2[ in steady state. However, it can be proved, that the maximal torque during the load transient can be achieved with a load angle 6bigger than x/2. This is based on the fact, that due to damping effects air gap flux linkage vmcan not follow immediately the fast rotating v/s. Thus the torque in a fast transient can reach higher values than in a steady state with the same certain power angle. Niiranen (1993) has presented following equation for the torque of a synchronous machine during a torque step
lvs(2 kd - kq It, I = -[ (k, - kd) sin 6.t- sin26+(kq - kq,)sinsOcos6 , 4, 2 1 (2.12)
A Vmd A Vmq Vmd 0 'mq kd =- , kq=--, k, = - , k,, =-, A Vsd A vsq Vsd0 Io +Imq where
Avmd - Change in the air gap flux linkage d-axis component, Avmq - Change in the air gap flux linkage q-axis component, AY/,~ - Change in the stator flux linkage d-axis component, Avo, - Change in the stator flux linkage q-axis component, vmd0- Initial value for the air gap flux linkage d-axis component, vlld0 - Initial value for the stator flux linkage d-axis component, 6 - Initial stator load angle. 18
Eq. (2.12) shows, that the torque response of a synchronous machine is affected by the stator flux linkage modulus 1 , the stator flux linkage load angle 6,the initial state of the machine (6,VmdO, WdO), the machine parameters and the dynamic dependence between the stator flux linkage ys and the air gap flux linkage vi. The dynamic dependence between vsand has been defined with two time variant coefficients kd and kq. The coefficients describe the change in the air gap flux linkage components Aymd and Aymqas a fhction of change in the stator flux linkage components A'y,d and A Wq. Alternatively to Eq. (2.12) the dynamic response of a synchronous motor can be analysed by examining the changes in the stator current and in the stator flux linkage. In a DTC synchronous motor drive the air gap flux linkage vmis not a primary control variable. On the other hand the stator current is a true feedback signal in DTC. This is the reason for replacing the air gap flux linkage by the stator current in the transient analysis.
2.2.1 Stator current, stator flux linkage and field current during a torque step
Fig. (2.4) represents stator flux linkage vector and stator current vector trajectory curves in a torque step. In a DTC synchronous motor drive a torque step is produced by driving the stator flux linkage vector with over synchronous angular speed in the stator reference frame, until the new torque reference is achieved. After the transient the stator flux linkage can be driven with the synchronous angular speed again and the new torque will be maintained. In the rotor reference frame this means, that the stator flux linkage vector will be turned towards a new position with the available voltage reserve. Assuming a constant voltage reserve and an ideal modulator and also by neglecting the stator resistive voltage drop, the stator flux linkage can be rotated in the rotor reference frame with a constant angular speed
(2.13)
where ris dimensionless time. The stator flux linkage in the rotor reference frame is then
(2.14) where 6 is the initial load angle before the torque step. The accelerating voltage reserve can be expressed in a vector form as well
(2.15)
The transition of the stator current is caused by the same voltage, which produces the transition in the stator flux linkage. The stator flux linkage rotates in the rotor reference frame according to Eq. (2.14) with a constant angular speed during a torque step. In this situation the stator current vector trajectory is dependent on the motor parameters. These can be seen in the two-axis inductance model of the salient pole machine in Fig. (2.5). In the simplified model of Fig. (2.5) only inductance parameters are presented. An approximative transient analysis can be done by neglecting the rotor 19 resistances, if the transient is fast compared to the time constants of the damper windings. The damper winding time constants for the d-axis, TD,and for the q-axis, TQ,are defined as
TD =-, ID wb ‘D (2.16) q =-. Ob ‘Q
Both TDand TQare scaled to the real time with the factor 1/m, where wb is the basic angular speed of the pu system. In large synchronous machines damper winding time constants are typically more than 100 ms, and a fast transient can be then analysed by considering only the rotor inductances.
Figure 2.4 Stator flux linkage vector transition w0-+ yS1and stator current vector transition i,O -+ isl in the rotor reference frame during a load step. The stator flux linkage change in the rotor reference frame is proportional to the stator voltage time integral u,-At. The stator current change is proportional to the voltage time integral u,-At and is also dependent on the motor inductance parameters. A constant excitation voltage in the field winding is assumed.
Figure 2.5 Two-axis inductance model of a salient pole synchronous machine for transient analysis. The transient is assumed to be fast compared to the damper winding time constants, and the rotor resistances can be neglected in the transient analysis. 20
The machine model in Fig. (2.5) reacts unsymmetrically in different machine directions to a supplied stator voltage transient due to the saliency and the field winding on the d-axis. In the case of an asynchronous motor a parameter called transient inductance is often used to describe the dependence between a stator voltage transient and the respective change in a stator current e.g. by Vas (1990). Respective transient inductances can be defined for a salient pole synchronous machine. However, different machine directions must be analysed separately due to rotor unsymmetry. Besides excitation control has an effect on the transient behaviour on the d-axis.
On the q-axis the transient inductance Iq,& is formed by the stator leakage inductance I,, and the parallel connected q-axis damper winding leakage inductance IQ, and the magnetising inductance Imq
(2.17)
On the d-axis there is one additional path for the stator current transient, because the field winding leakage inductance If, is coupled parallel to the magnetising inductance Imq and the damper winding leakage inductance ID,
(2.18)
The influence of the field current on the d-axis transient can be analysed by comparing different excitation methods during a transient. These different methods might be constant excitation voltage, constant field current and d-axis damper winding compensation control. The d-axis compensation control keeps the d-axis air gap flux linkage constant thus compensating totally the d-axis damper winding current. This excitation control method is proposed by MLd et a1 (1990-1) and is also called the reaction control, Mlrd (1993).
In the case of constant excitation voltage the d-axis stator current uses all three inductance paths in the transient, Fig. (2.6), and the d-axis current time derivative is inversely proportional to the transient inductance in Eq. (2.18). --?n(?I uf - constant Figure 2.6 The d-axis stator current transient path, when the excitation voltage is kept constant.
In the second case the field current is kept constant during the transient, Fig. (2.7). Then there is no transient in the field winding and the leakage inductance does not have any effect on the transient. The effective transient inductance is then
(2.19) 21
Figure 2.7 The d-axis stator current transient path, when the field current is kept constant
In the third case the reaction excitation control is used. The method makes it possible to examine the synchronous motor d-axis transient behaviour with the steady state equations, since there is no current in the d-axis damper winding. The equivalent circuit for this case is presented in Fig. (2.8).
Figure 2.8 The d-axis stator current and field current transient path, when the d-axis damper winding current is filly compensated.
In the transient equivalent circuit the zero damper winding current means, that the damper winding leakage inductance path can be removed. The damper winding reacts to the changes in the air gap flux linkage. The zero damper winding current on the d-axis means, that the air gap flux linkage d- axis component does not change. In that case no voltage drop appears on the magnetising inductance Imd, and the only stator current limiting inductance is the stator leakage inductance lso. The effective transient inductance is then equal to it, = I,, .
2.2.2 Maximal dynamic torque in the theoretical case of a fast transient
In the idealised case, where only an inductive voltage drop in rotor is assumed due to a transient, which is faster than the damper winding time constants, the dependence between the stator flux linkage change and the stator current change can be expressed separately on the different axis
(2.20) 22
Using Eq. (2.20) the torque during the transient can be solved:
(2.21)
For transient analysis initially a zero load and the excitation alone of the rotor origin are assumed. The initial stator current component and the initial stator flux linkage component on the q-axis are equal to zero and Eq. (2.21) can be simplified
(2.22)
The changes in the stator flux linkage components can be expressed as a function of the load angle
(2.23)
Substitution of Eq. (2.23) into Eq. (2.22) gives
(2.24) It, = 1v.r sin {t+ (e- +-) cos6) '
The maximal value for the torque and the load angle corresponding to the maximal torque can be found by differentiating Eq. (2.24) with respect to the stator flux linkage load angle 6
(2.25)
The load angle cosinus, which corresponds to the maximal dynamic torque, is obtained by solving the root for Eq. (2.25)
_- cos6 = (2.26) 4
Table (2.1) shows the effective transient inductances, the optimal load angles and the maximal torque for different excitation methods in the case of a reduced stator flux linkage I y/,l=0.3 pu. 23
Table 2.1. Effective transient inductances, maximal dynamic torque (teId,- and corresponding load angle &mx with different excitation methods. Stator flux linkage modulus I y/,l=0.3, reference machine inductances. The motor load is initially zero and excitation is alone of the rotor origin.
Excitation method 4,~[pu] I,,,& [pu] Ifelci,max [pu] &JMX ['deg] A. Constant voltage 0.1724 0.2268 0.5361 102.5 B. Constant current 0.1856 0.2268 0.4925 99.8 C. Damper compensation 0.1200 0.2268 0.8184 110.7
In Fig. (2.9) the trajectory curves during the transients for the three different excitation methods corresponding to Table (2.1) are presented. The load angle 6goes from 0 to 7~ rad. In Fig. (2.10) the torque for different cases has been shown as a hnction of the load angle 6:
The comparison of different excitation methods shows the meaning of the field current response. During a fast transient good dynamics in an excitation system gives a much higher peak value for the dynamic torque and this is obtained with a higher load angle compared to an excitation system, where constant current or constant voltage is used. The comparison shows also, that a constant field current gives a smaller dynamic torque response than a constant excitation voltage.
1 d
Figure 2.9. Stator flux linkage and stator current trajectory curves in a fast transient with different excitation methods, when the load angle 6 E [0,7c]. A - constant excitation voltage, B - constant field current, C - reaction excitation control, D - stator flux linkage vector trajectory curve. The current vectors in each case correspond to the respective maximal dynamic torque point. Reference motor parameters (see Appendix A). 24
I tel
"0 50 100 150 200 6 ["degl
Figure 2.10 Torque response Itc/as a function of the load angle Swith different excitation methods. A - constant excitation voltage, both the d-axis damper winding and the field winding act as damping windings. B - constant field current, the field winding does not take part into the transient. C - the d-axis damper winding is fully compensated, and the d-axis air gap flux linkage v/md keeps constant. The highest dynamic torque is achieved in the case C. Reference motor parameters (see Appendix A).
2.2.3 Transient analysis with derived operator inductances
The previous maximal dynamic torque analysis neglected the resistive voltage drop in the rotor. If a low voltage reserve is used in the field weakening area, the transient is not necessarily fast compared to the damper winding time constants. Then both resistive and inductive voltage drops must be considered. In that case the transient inductances are not constants, but frequency dependent transfer functions.
Frequency domain representation has been used e.g. by Buhler (1977-1) and by Luomi (1982). Frequency dependent, stator flux linkage and stator current dependency describing coefficients are presented in the Laplace domain as a function of the complex variable s. The coefficients are named either as reactance operators or operator inductances. In this work the latter name operator inductances is used. Traditionally these coefficients are represented as functions of the standard synchronous machine time constants, because those can be defined by standard measurements. For a vector controlled drive it might be more illustrative to use the resistance and inductance parameters of the two-axis equivalent circuit instead. These are shown in Fig. (2.1 1). 25
Figure 2.11 Two-axis parameter model of a salient pole synchronous machine for the transient analysis. Resistance parameters are included into the transient analysis. The stator resistance is assumed to be small and has been neglected in the following analysis.
The voltage, which changes the stator flux linkage component on the d- or q-axis, consists of the stator voltage component, the resistive voltage drop and the electro motive force component in that particular direction,
(2.27)
The angular speed a in Eq. (2.27) represents the initial stator flux linkage vector rotation speed before the transient. It is assumed again, that the voltage drop in the stator resistance can be neglected and thus the voltage reserve is kept constant. The transfer hnction G,(s) between the q- axis current i, and the q-axis voltage reserve u,,~is defined by the q-axis magnetising inductance Imq and the q-axis damper winding parameters l~,,and rQ
(2.28)
Eq. (2.28) shows, that the transient voltage is zero, if the current time derivative is zero. On the other hand the stator flux linkage is the time integral of the stator voltage,
(2.29)
The dependence between the q-axis stator flux linkage and the q-axis stator current can be solved by using Eqs. (2.28) and (2.29). This dependence defines the q-axis operator inductance as
(2.30)
With a rapid transient the conditions sIQ0lmq>> rQ1 mq and s(~Q,,+lmq) >> rQare valid and Eq. (2.30) approaches the same value as the earlier defined transient inductance in Eq. (2.17). In steady state, where all derivatives are zero, Eq. (2.30) produces the value of the synchronous stator inductance Iq. 26
The analysis of the d-axis by using different excitation methods can be done in the same way as for the transient inductances in the previous chapter. The only difference is, that the resistive components in the rotor will be included in the operator inductances and they will thus become frequency dependent.
If the excitation voltage is kept constant during the transient, the stator current transient will use all
(2.3 1)
where
If the field current is kept constant, Fig.(2.7), the operator inductance looks like the operator inductance on the q-axis, because no transient takes place in the field winding
(2.32)
In the third case, Fig.(2.8), the reaction excitation control has been used. Therefore no resistive path is used by the transient stator current, and the effective operator inductance gets the same value as in the idealised case, Ld(s)='s,,. The d-axis transient analysis is then similar with the previous analysis, where the transient inductance Id,@ was used.
In the idealised case, a linear dependence between the changes of the stator flux linkage and the stator current could be applied, Eq. (2.20). When the operator inductances are used, the relation between those two depends on the transient frequency. The difference between the analysis with the transient inductances and the analysis using the operator inductances depends thus on the speed of the transient. In the field weakening range, the voltage reserve is small and a transient can not be as fast as it is in the nominal speed range.
Angular speed for the stator flux linkage vector during a transient in the rotor reference frame has been defined in Eq. (2.13). The respective dimensionless time constant would describe the time for the full rotation of a stator flux linkage vector. However, the full rotation does not correspond to the real transient, since the maximal torque is achieved approximately with the load angle n/2. For this reason, it is more convenient to define the dimensionless transient time constant to be the time corresponding to the stator flux linkage vector rotation from zero to load angle x/2 27
(2.33)
Since motor time constants are usually represented with a dimension, it is usehl to convert the transient time constant as well into real time scale
(2.34)
As shown in Eq. (2.34), the transient time constant does not depend on the stator flux linkage modulus, if the relation between 1 ry,l and the voltage reserve u,, is kept constant.
The transient analysis can not be done by simple analytical calculations, as has been applied in the analysis with the transient inductances. Instead, a numerical simulation is used. The simulation is based on the idea, that a change in the stator flux linkage component causes a change in the stator current component, which can be expressed in Laplace domain
(2.35)
The change in the stator flux linkage is caused by the voltage reserve, and the transient speed is dependent on the relative voltage reserve, Eq (2.34). The simulation has been done in the SimulinkTM environment. Fig. (2.12) shows the symbolic block diagram of the simulation.
Figure 2.12 Symbolic block diagram for the transient simulation algorithm in the rotor reference frame.
Since the damper winding time constants define mainly the transient behaviour of the machine, it makes sense to compare the transient time constant to the damper winding time constants. Simulation results with two different ratios between the transient time constants and the damper winding time constants are shown in Fig. (2.13). 28
Itel [PUI I I I I I I I I
0.8
0.6
0.4
0.2
0 40 80 120 160 G[Odeg] Figure 2.13 Torque development during a transient while using different excitation methods and different damper winding time constants with a reduced stator flux linkage modulus ly,l=O.3. Transient time constant T,= 40 ms. A: constant excitation voltage, B: constant field current, C: reaction excitation control, 1: TD=TQ=Tb, ~:TD=TQ=~'T~.Reference motor parameters, only damper winding resistances are modified.
In the first case the transient time constant is equal to the damper winding time constants, Td=Tq=Ttr. The curves Al, B1 and C1 in Fig. (2.13) correspond to these time constants. The comparison between torque peak values produced by the operator inductance analysis, Fig. (2.13), and those of the transient inductance analysis, Fig. (2.10), shows clearly, that they are smaller in the case of operator inductances, if the field current or the excitation voltage is kept constant. This is caused by the large damper winding resistances, which cancel the damper winding currents fast.
In the second case, corresponding curves A2, B2 and C2 in Fig. (2.13), the damper winding resistances have been reduced to one fifth of the original. The relation between different time constants is then TD=TQ=5-T,. Here constant excitation voltage or field current gives about 20% smaller peak value (A2,B2) than in the case of transient inductances in Fig. (2.10).
If the d-axis damper winding current is compensated, the dynamic torque curves (Cl,C2) are close to each other and thus quite independent on the time constants. The simulation with the rotor resistive voltage drops is then close to the transient curve in Fig. (2.10) in both cases. Actually, the analysis with transient inductances gives a smaller peak torque than the analysis with the operator inductances. This is due to the fact, that the q-axis air gap flux linkage does not stay constant, but increases during the transient, when the q-axis damper winding current decreases.
The above analysis shows, that the effect of the transient time constant on the dynamic peak torque of the motor is small, if the d-axis damper winding current is compensated. However, the torque rise time in this case is directly dependent on the transient time constant. Fig. (2.14) shows two different torque transients with different transient time constants, T,=20 ms (A) and T,=40 ms (B). The damper winding time constants are the same in both cases, TD=TQ=40 ms. The torque rise time occurs to be approximately same as the transient time constant. 29
Figure 2.14. Torque development during the transient using different transient time constants Tk. A: T,= 20 ms. B: T,= 40 ms. Compensated damper winding current. TD=T,=40 ms. Reduced stator flux linkage modulus (cy1 = 0.3. Reference motor parameters, only damper winding resistances are modified (see Appendix A).
The simulated results show, that the dynamic torque peak value is achieved in the unstable working area, where 14 > x/2. In DTC, Gmust be kept in the stable working area, 14 < x/2, and the maximal torque should also be obtained there.
In the field weakening range, where the stability problems mostly occur, the limitation for the torque is often caused by the available inverter current. The maximal torque is achieved then with unity power factor. The maximal dynamic performance thus requires, that unity power factor is obtained immediately at the end of a torque step.
When the reaction excitation control (MArd 1990-1) is used, the ratio of changes in the d-axis stator current component and in the field current is fixed,
AIf= -AId. (2.36)
Let the d-axis currents in the final working point be if' and id. and the initial values before the transient iw and idO.The changes in the d-axis current components are thus
ai,= i; - i,, . .* . (2.37) AId=ld -Ido.
The changes in the d-axis current components should have the same modulus and opposite sign
i; - i, = -(.*Id - Id0 ), (2.38) 30
The initial d-axis stator flux linkage component is v/aO, which is formed by the initial d-axis current components
(2.39) WdO = ‘md . ifO “sd . idO
The initial field current value, which produces the final working point (i:, id’) when using the reaction excitation control, can be solved from Eqs. (2.38) and (2.39)
(2.40)
The steady state field current reference, for which unity power factor and thus the minimum value for the stator current are obtained, can be solved according to Pyrhonen 0. et a1 (1997) as
(2.41)
The reaction control law, Eq. (2.36) and unity power factor condition, Eq. (2.41), are in contradiction during a torque step. Considering the maximal performance of a drive, it is useful, that the ratio stator currenthorque obtains minimum with the maximal stator current. This is of importance specially when using a reduced stator flux linkage, where the maximal torque of a drive is defined by the maximal inverter current. Thus the initial field current reference should be selected in a way, that the reaction control law, Eq. (2.36) fblfills Eq. (2.41) at the end of a torque step. In practice this means, that the field current reference should be above the value given by Eq. (2.41), when the motor is working with a partial load, where lisl < lis,-l. This is later called the partial loud over excitation.
A numerical example further clarifies the principle of the partial load over excitation. In the previous example in Fig. (2.13) dynamic peak torque 0.9 pu (curve C2) has been achieved by using the reaction excitation control. Let the same value be the torque reference in this example. Buhler (1 977- 11) has showed, that the load angle value of a salient pole synchronous machine with unity power factor is
(2.42)
With the numerical values the field current (and the stator flux linkage load angle6*in the final working point are 31
teI 0.9 .0.57 6' = atan [', = atan (7)= 80.05 "deg. 4-f
The d-axis stator flux linkage component can be solved according to the load angle and the stator flux linkage modulus
Finally, the d-axis current components can be solved
0.052 1.05 3.34 = -2.95, Id Isd If - 1.17 1.17
1.17.(3.34+(-2.95)) - 0.3 i, = Id (Z; +id.) - ydo--_ = 1.27. Ird -'md 1.17 - 1.05
In Fig. (2.15) the curve B shows the torque during a transient as a function of the load angle for the above presented initial field current and using the reaction excitation control. For comparison, curve A shows the static torque as a fbnction of the load angle, when the stator flux linkage modulus and the stator current modulus are fixed, (lv,l=O.3and lisl=3. These values can produce the required torque, when unity power factor is obtained.
As can be noticed, the required torque reference in the transient will be obtained with approximately the same load angle, which corresponds to the unity power factor working point in the steady state. Curve C shows the dynamic torque for the case, where q-axis components are assumed to follow the steady state equations also during the transient (no damping effect on the q-axis). In this case, the desired torque is reached exactly with the unity power factor load angle, aCw,= 80.05"deg.
The solution for the initial value of the field current can be found, when the maximal current of the drive and the stator flux linkage modulus are known. The maximal torque is then
(2.43) 32
Figure 2.15 Static and dynamic torque as a function of the load angle S. A: Static torque, when the stator flux linkage modulus and the stator current modulus are fixed, I ry,l=0.3 and lisl=3. B: Dynamic torque, when the initial value of the field current has the value of the above numerical example. C: The same as B, but the q-axis is assumed to follow the steady state equation. D: Dynamic torque, when the excitation is initially of the rotor origin alone. T,=TD=TQ=~Oms. Reference motor parameters (see Appendix A).
The required field current i;,,and the d-axis current id for unity power factor can be calculated using Eqs. (2.41) and (2.42). The reaction excitation control requires, that the d-axis air gap flux linkage will have a constant value, and thus the change in the d-axis current will be totally compensated by the change in the field current according to Eq. (2.36). Starting from the final working point ( y; ,id ), which represents the maximal available torque and the maximal stator current, the values of the d-axis stator current and the field current can be calculated as a function of the decreasing torque by keeping Eq. (2.36) valid. This can be done by using the following equations
(2.44)
a polynomial equation form of fourth order can be found for Aid
(2.45) 33
where
k 4 = -I2xu (l-?)’,
k, = -2ys:Is0( 1 - t)- 21; (1 - t)(il- t),
Again, numerical example is used. The quantities of a maximal torque working point in the field weakening range are represented in Table (2.2).
Table 2.2 A maximal torque representing final working point in the field weakening range with a given stator flux Iinkage modulus and maximal stator current.
~ ~~ _____ Variable I Du-value * I DescriDtion 1 I w-1 I 0.3 I Stator flux linkage reference I
11s1, 3.33 Maximal inverter current modulus
l&lax 1.0 Maximal torque with unity power factor 6’ 81.03 “deg Load angle with the maximal torque and unity power factor Y:, 0.0468 d-axis stator flux linkage component in maximal torque point .* Id -3.29 d-axis stator current component in maximal torque point .I I, 3.71 field current component in maximal torque point *) Load angle unit in degrees
The field current can then be solved at every working point, when the final value i; and the required change Aif = -Ai, are known. Fig. (2.16) represents the field current value for the constant d-axis air gap flux linkage component. For a comparison the unity power factor field current as a function of the torque is also shown.
Transition from the working point bl to the working point ba in Fig (2.16) is represented with the space vectors in the rotor reference frame in Fig (2.17). Over excitation requires a large capasitive stator current even with a small load. However, this must be accepted, if the maximal torque defined by the maximal stator current is to be achieved immediately afier a load transient. 34
0 0.2 0.4 0.6 0.8 1
ItCl[PUI
Figure 2.16. Unity power factor producing field current (A). Damper winding compensating field current (B), which moves the working point of the motor directly into maximal torque producing working point in transient, when the reaction excitation control is used. Point bl represents the initial working point with a small load, point bZrepresents the maximal torque and current working point. Tq
Figure 2.17. The initial and final stator flux linkage vector (wso,y/:),air gap flux linkage vector ( ymo, w:), stator current vector ( is, ,is*) and field current modulus (i,,, i; ) before and after torque step from Jte(=0.05to lt,l=l.O corresponding to points b, and bZin Fig. (2.16). The reaction excitation control keeps the d-axis air gap flux linkage component Ymd constant. The maximal torque point with unity power factor is achieved immediately after the transient. The current vectors shown in the figure have been reduced.
As a conclusion, the excitation control has an important role for the DTC synchronous motor drive, when the drive is working with high load angles in the field weakening range. By using the partial load over excitation and the reaction excitation control the available stator current can be hlly utilised also in the torque step and the maximal torque is achieved without stability problems or an unnecessary stator current limitation. 35
3 EXCITATION CONTROL OF DTC SYNCHRONOUS MOTOR DRIVES
The dynamic performance of DTC synchronous motor drives has been found to be very much dependent on the field winding hehaviour. The unity power factor is advantageous for the static performance, since the stator current can then be filly utilised for the torque production. Especially in the field weakening range the drive stability is dependent on the excitation control. Unlike most traditional vector control methods DTC cannot directly adjust the load’ angle but it is formed freely, according to the load and the magnetic state of the machine. For the reasons mentioned, both the static and dynamic performance as well as the stability of the drive should be considered, when the excitation control is analysed.
3.1 Reactive power compensation
The power factor cosp defines the reactive power of a synchronous motor. Unity power factor means, that all current supplied to the stator will produce active power and the torque production with the given current and voltage modulus is at its maximum. When considering synchronous motors with a field winding in the rotor, it is optimal for the dimensioning of the stator inverter to drive the stator with unity power factor.
The field current, which produces unity power factor, is dependent both on the motor parameters and the loading condition and can be calculated, if they are known. The vector diagram of a synchronous motor has been shown in Fig. (3.1). The voltage vector usis leading the current vector is and the phase shift angle cp is negative. In this case reactive power must be supplied to the stator and the stator current does not attain its minimum.
Figure 3.1 Vector diagram of an electrically excited synchronous motor. The stator current is formed by the active stator current component iS,p and the reactive stator current component is,o.
In a voltage source inverter connected to a motor there is only a limited amount of selectable voltage vectors, depending on the phase number of the motor and the type of the inverter. For this reason, it is profitable to examine the angle between the current vector is and the stator flux linkage vector ys. The condition for the unity power factor is found to be the orthogonality between the current vector 36 and the respective stator flux linkage vector. The field current, which produces unity power factor, can be solved, ifthe machine inductance parameters are known (Buhler 1977-11):
DTC does not control the stator current directly. Therefore the excitation control cannot be related to the stator current control, as in the traditional field oriented control methods of synchronous motor drives. If Eq. (3.1) with the actual stator current modulus would be used for the field current reference, magnetic energy oscillations could take place between the stator and the rotor.
Instead, the excitation control should be related to the torque and the flux linkage modulus control of DTC. The calculation of the field current reference based on the DTC control variables I ysl and Ifel can be done by eliminating the stator current in Eq. (3.1). Assuming unity power factor, the stator current modulus is:
With Eqs. (3.1) and (3.2) the field current reference calculation can be related to the DTC controlled quantities Iysl and lfJ like shown in Eq. (2.41). The full derivation of Eq. (2.41) has been shown in Appendix B .
3.2 Effect of magnetic saturation on the calculation of the excitation curve
The two axis synchronous machine model assumes zero magnetic coupling between the d- and q- axes. However, if magnetic saturation is considered, the cross coupling must be taken into account. It is an important issue specifically for the current vector controlled drives. Alaktda and Vas (1989) have shown, that if the magnetic saturation is neglected in the current vector control, the drive performance will suffer due to the erroneous estimation of the flux linkage. In the case of DTC the inductance model is necessary for the excitation control and for the low speed operation, where the voltage model is not able to estimate the stator flux linkage.
A method for measuring and modelling the saturated inductances in the case of a DTC drive has been analysed by Kaukonen et a1 (1997) and Kaukonen (1998). The saturation has been analysed from the excitation control point of view by Pyrhonen, 0. et a1 (1998). The saturated magnetic inductances Zmd,sat and must then be modelled as a function of the currents on both magnetic axis in each case:
(3.3)
The current sum on a magnetic axis is the sum of all the currents on that axis: 37
i,,, =id+i, -kif, .. (3.4) i,,, = iq +I~.
The stator leakage inductance saturation can be modelled as a function of the stator current modulus
The saturation must be taken into account when using Eq. (2.41) for the unity power factor. Kaukonen (1998) has measured and modelled saturated inductance values for a 14.5 kVA test motor. Those results have been used also here to compare the excitation curve for saturated and unsaturated parameters. Note, that the working point of the machine must be solved iteratively, when saturated inductances are modelled as functions of the current sums.
In Fig. (3.2a) saturated inductance values Irnd,=,and Irnq,=,for the test motor have been shown as a function of the torque. The nominal stator flux linkage modulus is assumed. Fig. (3.2b) shows the excitation curves corresponding to the unsaturated and saturated inductances, respectively if, and if,=,.The corresponding curves for the reduced stator flux linkage I vsv,l=0.5pu has been shown in Fig. (3.3). The reference working point for the unsaturated inductance values have been selected to correspond the nominal stator current and unity power factor in both cases. With such inductance values, the unsaturated curves give a slight over excitation above and below the reference working point decreasing thus the stator current/torque ratio. With high loads in the field weakening the over excitation is useful, since the load angle will be reduced and the drive stability accordingly will be improved. Over excitation is not necessary, when a nominal stator flux linkage modulus is used. Instead, the inverter current should be minimised. In some cases the available inductance parameters are erroneous and the minimal stator current is not obtained by using the field current reference according to Eq. (2.41). The excitation control can be further improved by a feedback algorithm.
Stator flux linkage lY/,l=1.0. Unity 0.8 ...... ______..;.....______..~.....,...... ; .....__.._....; 38
if [pul Figure 3.2b
The unsaturated and saturated excitation curves, icw and it=, as a function of the torque. Stator flux linkage I wc/,J=l.O. Unity power factor is assumed.
0.5 0 0.5 1 1.5 2 2.5
te [PUI
Figure 3.3a
The magnetic inductances Zmd,sat and Zmq,sat as a function of the torque. Stator flux linkage 1%/,1=0.5. Unity power factor is assumed.
Figure 3.3b
The unsaturated and saturated excitation curves, if,w and iCwtas a function of the torque. Stator flux linkage Jyl,)=0.5.Unity power factor is assumed. 39
3.3 Combined open loop and feedback control
The excitation control in DTC synchronous machine drives has formally the same features as the excitation control in network connected synchronous generators. There is no stator current control, but stator currents and load angle are formed according to the load and the controlled field current. Additionally the frequency and the modulus of the stator flux linkage vector are controlled in DTC. The excitation control has the same goals in both cases. The dynamic stability is one of the key issues from the excitation control point of view. Further the power factor is controlled by adjusting the field current.
The main goal in synchronous generator excitation control is to prevent electromechanical oscillations in the generator and to secure the generator and the power system stability during the load transients. The main difficulties in the feedback control are the nonlinear character of a synchronous machine and the parameter variations due to saturation and temperature variation. Hsu and Wu (1988) have proposed an adaptive method, where the process is linearised piece wise and the controller variables are updated according to the working point. Also the feedback can be linearised, as e.g. in the excitation control method represented by Mielczarski and Zajaczkowski (1991). Nonlinear methods have been applied for the synchronous generator excitation control as well, e.g. by Xianrong et a1 (1993). Adaptive state feedback control has been proposed by Fork and Schreurs (1988). Even hzzy sets (Handschin et a1 1993) and neural networks (Zhang and El-Hawary 1994) have been proposed for the excitation control of synchronous generators.
Despite of the formal similarities the same methods are not necessarily applicable for the excitation control of synchronous generators and DTC synchronous motor drives. The operating range of DTC controlled synchronous motor, including the field weakening range, is large compared to that of a synchronous generator and thus the parameter variation is larger. The linearisation in the case of DTC is difficult due to the wide working area. On the other hand, the stator voltage is a controlled variable in the DTC system unlike in the case of synchronous generators. It is thus obvious, that the excitation control has different roles in the DTC synchronous motor drives and in synchronous generator control.
The dynamic behaviour of the machine is an important issue in both cases. In the case of DTC the main control variables, which are the stator flux linkage modulus and the torque, can be controlled accurately to a certain extent even without any field current adjustment. However, since the maximal static and dynamic performance of the DTC synchronous motor drive was found to be dependent on the field current adjustment, the excitation control should be able to follow the main control loop of DTC in load transients.
The open loop control of a synchronous motor magnetic state has been widely used in current vector controlled synchronous motor drives. In the transvector control of a smooth air gap synchronous motor (Bayer et a1 1971) the magnetising current reference z-fis calculated according to the air gap flux linkage modulus I yml.It is hrther divided into the rotor magnetising current reference i,,,f,refand the stator magnetising current reference imnGf.The stator field current zmn,=f occurs only in the dynamic changes 40
The synchronous motor control method proposed by Mird et al (1990-1) calculates the field current reference value as follows
which keeps the load angle 6 constant. Niiranen (1993) proposes an excitation control method according to the reaction excitation control combined with the power factor correction term if,,o,
vmd ifJef = -- Zd,ref + 'f,con ' (3.8) 'md
The reaction control part of the field current reference compensates the d-axis damper winding current and requires fast dynamics for the excitation unit. The power factor correction works with a slow time constant.
Good dynamics for the excitation unit is required also in the control method proposed by Yamamoto et al (1993), where the inverse dynamic model of a synchronous motor is used. The field current reference is then -S
Eq. (3.9) is presented in the Laplace domain. The given examples show, that in current vector controlled synchronous motor drives the excitation control is normally related to the stator current control and no fast feedback is utilised as in the excitation control methods for synchronous generators. Due to the missing feedback of the machine magnetic state, the drive performance is dependent on the accuracy of the saturation model. DTC is less dependent on the saturation model, since the voltage model is not inductance parameter dependent. For this reason, the open loop excitation control method can very well be used for the electrically excited DTC synchronous motor drive.
Eq. (2.41) makes it possible to relate the excitation control to the DTC main control and it can be implemented as an open loop control. This is usefbl, since the magnetic energy balance between the stator and the rotor will be found rapidly after a load change without oscillations, which might occur in the case of an unoptimally tuned feedback control. Furthermore, this method allows the excitation control with the same speed as the torque is controlled. That is important for both the stability and the dynamic performance of a drive, especially in the field weakening range. 41
The drawback of the open loop excitation control is, that the control method is not able to compensate erroneous inductance values. Excitation curve calculation can be improved by using inductance parameters given by a saturation model. The saturation model is required also for the current model, which is needed specifically in the low speed operations, where the voltage model is no more reliable.
The minimum for the stator current and unity power factor is obtained, when the stator current vector i, and the stator flux linkage vector are orthogonal to each other. The orthogonality of two vectors can be investigated with the scalar product. If unity power factor is desired, the scalar product can be used to calculate an error signal €for the excitation control
E = w, . is = y,,i, + ~/.~i,, (3.10) where the stator reference frame oriented current and stator flux linkage components are used. The DTC modulation principle causes modulation noise in the stator current. The magnitude of the ripple is both switching frequency and transient inductance dependent. For that reason the error signal in Eq. (3.10) is also disturbed. A further source of error in Eq. (3.10) occurs to be the stator flux linkage estimate. Even a small error in the stator flux linkage estimate causes eccentricity in the real stator flux linkage. As a result, the stator current will be eccentric too, and the error signal in Eq. (3.10) will oscillate with the basic electrical frequency.
Because of inaccuracies in the feedback signal of Eq. (3.10) the feedback signal should be filtered with a low pass filter before using it for control purposes. The power factor control is then slow compared to the torque control dynamic requirements and the control method is not sufficient to handle fast load transients.
By combining the fast open loop control implemented by the excitation curve calculation, Eq. (2.41), and the slow feedback control for power factor correction with error signal from Eq. (3.10), a good excitation control method for DTC synchronous machine drive can be achieved. This control method fulfills two main features, which are; a quick response in load transients in order to ensure the drive stability and a stator current minimisation with the slow power factor control. The block diagram of the excitation system is represented in Fig. (3.3). 42
Open Loop Control
Feedback Control
ws j, jY,tqI
Figure 3.3 DTC synchronous motor drive excitation system with a fast open loop control and a feedback power factor control, which uses filtered values wsf and is$. The feedback uses P-controller with a gain P,. The output of the feedback controller has been limited.
3.4 Reaction excitation control in DTC
Electrically excited synchronous machines are in most of the cases equipped with damper windings, which allow sudden changes for the stator current without immediate influence on the air gap flux linkage. The damper windings are thus essential for the dynamics of vector controlled synchronous motor drives. The rapid increase of the stator current is necessary in load transients. This is made possible by the leakage paths of the rotor.
The d-axis damper winding protects the field winding against overvoltages in transients and gives the excitation control time to react to load changes. However, slow excitation control can cause stability problems with higher load angles and the increasing inaccuracy of the current model due to erroneous damper winding parameters.
The damper winding currents are not measurable as the other current components are in synchronous machines. According to Mdrd et a1 (1990-1) the estimates for the damper winding currents can be calculated as follows
(3.11) 43
where
ID(s) - d-axis damper winding current in Laplace domain, Id(s) - d-axis stator current in Laplace domain, I&) - field current in Laplace domain, IQ(s) - q-axis damper winding time current in Laplace domain, I&) - q-axis stator current in Laplace domain.
The time constants TDand TQhave been defined in Eq. (2.16). The inductance coefficients kD and kQ are defined as
(3.12)
For example the standard IEC 34-4 specifies, how to determine the synchronous machine parameters. For large machines these are given by the manufacturer as well. However, such parameters include inaccuracies and describe the machine only in the nominal working point. The saturation effects and the changing temperature have some effect on the coefficients in Eq. (3.11). As a consequence, the damper winding current estimates and also the current model based stator flux linkage estimate will be erroneous during transients. This is not a problem for DTC, when the frequency is above a few hertz and the voltage model can be used. At low speeds, the accuracy of the current model is important also for DTC.
The field current is a measurable quantity unlike the d-axis damper winding current. From the modelling point of view, it would be most useful to have a measured quantity instead of an estimated one as an input variable in the flux linkage model. Actually, the d-axis damper winding current compensation improves the current model accuracy. Considering a DTC drive the reaction excitation control can not be implemented with the same way as Mkd et a1 (1 990-1) has presented, since the stator current is not directly controlled. Instead, the DTC adjusts only the torque producing current component, not the magnetising current component of the stator current, which is formed according to the magnetic state ofthe machine, Fig. (3.4).
The rise time of the stator current during a load step, when defined by DTC, is not constant, but depends on the voltage reserve and the transient inductances of the machine. It is also dependent on the load angle. If the d-axis damper winding current is to be compensated during transients, the changes in the field current and in the d-axis stator current should be the same, Eq. (2.36). This can be done by limiting the rate of change for both the torque reference and for the field current reference to a level, which both the excitation unit and the stator inverter unit can fulfill. Additionally, the partial load over excitation was found to be necessary. 44
Figure 3.4. The variation of the stator current vector produced by DTC, when the stator flux linkage modulus reference and the torque reference are fixed and the field current is varying between 80% (is], ysl)and 130% (is6, (y6) of that particular field current, which corresponds to unity power factor. The stator flux linkage modulus and the torque producing current component are defined by DTC, while the load angle and the stator current modulus are defined by the excitation control.
In the case of the reaction excitation control the maximal field current rise time is determined by the available excitation voltage and the field winding leakage inductance Z,. At low speeds the stator voltage reserve is large and DTC changes stator current very fast. If the d-axis damper winding current is to be compensated, the torque rise time should be limited with a suitable ramp, since otherwise the excitation system may not able to compensate the fast change in the d-axis stator current component. In the field weakening range the stator voltage reserve can be selected corresponding to the excitation voltage reserve, which can be defined as
(3.13)
The equal ramp times in the torque and excitation control, according to Fig. (3.5), do not hlly compensate the d-axis damper winding current, if the field current reference is calculated according to Eq. (2.41). This is due to the fact, that unity power factor does not keep the d-axis air gap flux linkage component constant, but it changes as a fbnction of the torque as well, as shown in Fig. (3.6). However, the reduction of the damper winding current peak value can be achieved, which leads to a better current model accuracy and the reduction of the rotor losses.
When working in the field weakening range with high loads, the reaction excitation control can be done in a way, that optimal working point will be reached simultaneously with the maximal inverter current. The solution for this was found to be the partial load over excitation, where the unity power factor is achieved only with the maximal ratings of the drive. In that case, the feedback control for the power factor correction can not be used. Instead, the field current reference value is to be calculated with a sufficient approximation based on the solution of Eq. (2.45). 45
Figure 3.5 An algorithm for the reaction excitation control in DTC. The torque reference IteI:f is limited with a ramp time constant TmmP.The time constant is kept on such a level, that both the stator torque response and the field current response can be achieved.
- WdPUI < i Iy4V,=l.O : 0.8 ......
0.6 -...... : IvsV,=0.5 : 0.4 _......
0.2 ...... 46
4 FIELD WEAKENING CONTROL OF DTC SYNCHRONOUS MOTOR DRIVES
A voltage source inverter has a certain maximum value for the available voltage modulus, determined by the DC intermediate voltage. When the angular speed increases, the electro motive force modulus increases correspondingly, Eq. (1.7). Above a certain angular speed the full stator flux linkage can no more be produced with the available voltage, but it must be reduced to maintain a sufficient voltage reserve, Eq. (1.8). This speed range was defined as the field weakening range.
The field weakening range is an important working area for speed controlled electrically excited synchronous motor drives. Theoretically the maximal torque of a synchronous machine does not depend on the available voltage, but on the available stator and field currents. If a synchronous motor is driven with unity power factor, the maximal torque is
The large loadability of a synchronous motor is a clear advantage compared to an asynchronous motor. The maximal torque of an asynchronous motor is approximately proportional to the square of the stator flux linkage modulus
Fig. (4.1) compares the maximal torque of a synchronous and an asynchronous motor in relation to a maximal stator current. According to Eq. (4.2) the functional principle of an asynchronous motor limits the torque unlike that of a synchronous motors, where only stator and rotor current ratings are limiting the maximal available torque. This feature makes synchronous motors more suitable for such applications, where a wide speed range is required.
In the field weakening range the load angle of a synchronous motor rises fast as a function of the load due to the reduced stator flux linkage and stability problems may thus arise. The stability can be affected with the excitation control and with the active load angle limitation. The dynamic performance of the drive with a small voltage reserve is another important question, which is handled in the following chapter. 47
I 0- 0- 0 1 2 3 n [PUI
Figure 4.1 Maximal available torque for a synchronous motor and an asynchronous motor as a function of the rotating speed n, when stator current is assumed to be limited to 1 pu. A - synchronous motor maximal torque based on the current limit. B - asynchronous motor maximal torque based on the current limit. C - breakdown torque curve for asynchronous motor. D - combined asynchronous motor maximal torque curve according to the current limit and the breakdown torque. All curves have been scaled according to the maximal torque of the synchronous motor in the nominal speed range. Power factor cosp = 0.9 is assumed for the asynchronous motor. The breakdown torque of the asynchronous motor is assumed to be 160% of the nominal torque in the nominal speed range.
4.1 Voltage reserve in field weakening
A three phase voltage source inverter can generate six different voltage vectors into a three phase motor, according to Fig. (4.2). The task of a modulator is to switch the different voltage vectors so, that the motor current vector rotates smoothly.
An internal stator voltage vector usi, which does not include the resistive voltage drop, is defined here as
(4.3)
In steady state the internal stator voltage vector us,imust be able to compensate the electro motive force e. During a torque step this is not enough, but it should be able to accelerate the stator flux linkage above the synchronous frequency to increase the torque.
By switching the six vector system in Fig. (4.2) it is possible to create an average voltage vector in an arbitrary direction. The maximal modulus of this vector depends however on the direction. The modulus obtains its minimum exactly between two vectors, and the value is then 48
Figure 4.2 Discrete voltage vectors of a three phase inverter u,..u6and the trajectory curve of the maximal rotating voltage vector (hexagon), which the discrete voltage vectors can produce. The electromotive force e must be kept inside the outer circle to keep the stator flux linkage path circular.
If a circular stator flux path is to be maintained, the modulus of the electro motive force e should not increase above the minimum value of the internal stator voltage modulus
The voltage vector modulus can be determined as a function of the discrete voltage vector modulus tun[and the direction angle p shown in the Fig. (4.3) and in a following way 49
Figure 4.3 The orthogonal elements of the stator voltage u, formed by the discrete voltage vectors u2and ug.
The average voltage vector modulus can be defined by integrating Eq (4.6) with respect to the angle /? using the interval [O,n/6] and by dividing the result by the integration interval,
0.85 I 1 0 .x 2.x e [rad]
Figure 4.4Maximal voltage vector modulus Ius[of the modulated stator voltage as a hnction of the stator flux linkage vector direction 8,where the voltage vector usis assumed orthogonal to the stator flux linkage vector y,. The average value is luslaV= 0.909 . lunl.
Eq. (4.7) gives the theoretical maximum for the average modulus of the electromotive force e. Thus a condition 50
defines a theoretical maximum average value for the stator flux linkage modulus. However, the modulation cannot produce a circular flux linkage path, if the maximal flux linkage reference is selected according to Eq. (4.8). In practice, the voltage reserve is kept on a level, for which Eq. (4.5) is valid thus allowing a circular flux linkage path. If the stator current vector and the voltage vector are assumed to be parallel (unity power factor), the condition for the reference of the stator flux linkage modulus is then
(4.9)
4.2 Relation between voltage reserve and excitation voltage
The angular acceleration of the stator flux linkage vector, which is needed for the torque increase in DTC, is found to be dependent on the available voltage reserve u,. In the case of synchronous machines, the fast acceleration of the stator flux linkage vector does not assure a good torque response, especially in the field weakening range. The transient analysis showed, that also good field current response is required.
There are two aspects in the selection of the voltage reserve. The stator flux linkage modulus should be kept as high as possible to keep the stator current modulus small. On the other hand, a small voltage reserve means also a poor dynamic response. A good solution can be found by considering the voltage reserve together with the available excitation voltage.
It has been shown earlier, that the reaction excitation control combined with the partial load over excitation gives the maximal torque utilisation and a good dynamic response for the drive. The reaction control requires equal changes with different signs in both the d-axis stator current and the field current, Eq. (2.36). That means a constant d-axis air gap flux linkage and an inductive voltage drop appears only in the leakage inductances.
The load angle angular speed has been found to be voltage reserve dependent, Eq. (2.13). From Fig.(4.5) and assuming the voltage vector and the stator current vector parallel we get
(4. IO)
The stator flux linkage angular speed in the rotor reference frame is then
(4.11) where the internal voltage reserve u,i includes the resistive voltage drop in the stator winding, which is assumed to be parallel to the stator voltage vector. 51
Figure 4.5 Load angle change due to the difference between the modulus of the electromotive force e and the modulus of the internal stator voltage vector uri.
If the reaction excitation control. is applied, the change in the d-axis stator current during a load transient depends on the leakage inductance I,, and on the internal voltage reserve on the d-axis
(4.12)
Eq. (4.12) is an approximation, since the condition for the internal voltage reserve, where the resistive voltage drop in the stator winding is assumed to be parallel to the stator voltage, is not valid for the reaction excitation control.
The change in the field current, when the reaction excitation control is assumed, depends on the maximal available excitation voltage uf-, the excitation winding resistance rf and the leakage inductance If,
1 aif = -(uf - ifrf)8r (4.13) If*
Eqs. (4.12) and (4.13) show, that the time derivatives for the stator current component id and for the field current if are not constant, but change during the transient along with the load angle and the resistive voltage drop. For this reason it is more profitable to consider the average current changes during the transient instead of investigating instantaneous values. By using Eq. (4.10) the transient time 7,- from zero load angle to the value corresponding to the maximal torque, ab,- can be solved
(4.14) 52
Solving z,, in Eq. (4.14) gives
(4.15)
The equality of the total changes for the current components id and if can be investigated by integrating Eq. (4.12) with respect to the load angle S, S~[0,4,-],and the Eq. (4.13) with respect to time z, 7 E [0, rmx]
(4.16)
(4.17) where the excitation voltage has been approximated by using the average excitation voltage during the transient
(4.18)
In Eq. (4.18) i; is the field current corresponding to the maximal torque and im is the field current initial value before the transient. In the field weakening the maximal torque is achieved with a large load angle and it is approximated here with the value 4,mx=7c/2. The equality of Eqs. (4.16) and (4.17) gives then a necessary internal stator voltage reserve urqi, when the reaction excitation control is used,
(4.19)
An important conclusion of this theoretical result is, that the stator voltage reserve should be selected not only as a function of the resistive stator winding voltage drop, but also as a function of the excitation voltage reserve. This means also, that the dynamic torque requirements concern both the stator dynamics and the excitation dynamics. In some cases the excitation system does not correspond to these requirements. In that case the torque response can be improved by keeping the initial field current level high. However, such an over excitation must be compensated by the stator current, and the resistive losses are increased in the motor. 53
4.3 DTC modulation in field weakening range
The optimal switching table of DTC should select an optimal voltage vector as a hnction of the input values. The input variables introduced in Fig. (1.2) are the error signal of the stator flux linkage modulus, the torque and the geometrical position of the stator flux linkage vector in the stator reference frame.
A certain minimum voltage reserve has been found necessary, if a’circular flux linkage path is required. The voltage reserve in a two level inverter depends on the selected voltage vector and the position of the electromotive force e. In the modulation analysis, the vector form of the voltage reserve is advantageous. Instantaneous voltage reserve vector u, is defined as the vector sum of the stator voltage vector and the electro motive force vector
u,, = u, +e, n E { 1,2,3,4,5,6}. (4.20)
The angle between the voltage reserve vector and the stator flux linkage vector is not constant, but it varies, when the stator flux linkage vector rotates. This causes some unideality in the optimal switching table. Fig. (4.6) shows two non optimal situations in DTC modulation. In the first case ( vsl,el) the voltage vector u3 is selected to decrease the stator flux linkage modulus and to increase the torque. In the second case ( vSz,ez) the voltage vector uz is selected to increase both the stator flux linkage modulus and the torque. In both cases the voltage reserve vector reduces the load angle. Due to this the torque is reduced even though the increase of the torque was required. In the first example ( vsl,el) the reduction of the stator flux linkage simultaneously with the decreasing load angle makes the situation even worse. In the second case ( vSz,e*) the increasing stator flux linkage modulus and the decreasing load angle are compensating each other to some extent. However, the torque reduction is possible also in this case.
Both cases have been illustrated in more detail in the rotor reference frame, Fig. (4.7). As shown, the voltage reserve vector turns the stator flux linkage vector in the rotor reference frame into the direction of decreasing load angle, which causes a decreasing torque. 54
A B
Figure 4.6 Working points near the both limits of sector K~.In the case A, (W1,el) illustrates the situation, where the voltage vector u3 is selected to decrease the stator flux linkage modulus and to increase the torque. In the case B, ( s2,ez) illustrates the situation, where the voltage vector u2 is selected to increase both the stator flux linkage modulus and the torque. In both cases the voltage reserve vector reduces the load angle. The angles €1 and €2 define the switching sector areas, where the modulation rules should be modified.
A B
Figure 4.7 Non optimal switching near the sector limits. In the case A, (t.1 is too small and I fil is too large. The angle is the angle between the voltage vector and the stator flux linkage vector in this situation. In the case B, both lt,l and I ysI are too small. The angle p2 is the angle between the voltage vector and the stator flux linkage vector in this situation.
This non optimal switching causes the reduction of dynamic performance in the field weakening range, where the voltage reserve is small. Especially by a low switching fiequency below 1 IcHz even a single torque reducing switching can be hadl.
The optimal switching table adjusts two control variables, the stator flux linkage modulus and the torque. For the drive control the torque is the primary control variable. Keeping this in mind, the optimal switching table should be modified so that the torque control has a higher priority than the stator flux linkage control. 55
In Fig. (4.7A) the torque is too small, the flux linkage modulus is too large and the stator flux linkage vector has just arrived into a switching sector. The relation between the discrete stator voltage vector un and the electromotive force e is defined with the voltage reserve coefficient k,
n E{ 1...6}. (4.21)
The resistive voltage drop in the stator has been neglegted also here. The limit value for the angle p1, which fulfills the condition of non decreasing load angle, can be found in the working point, where the voltage reserve vector u3+e and the stator flux linkage vector v/s have the opposite directions, Fig. (4.8).
Figure 4.8 The working point, where the voltage reserve vector and the stator flux linkage vector have opposite directions. This working point represents the limit value of the angle pl, for which the load angle does not decrease.
From the right angled triangle formed by u3 and e a limit value for fll can be defined,
le1 sin(x -pi) = -= k,, , (4.22) 1% I fiom which fll can be solved as a hnction of the voltage reserve coefficient ks
p, = x - arcsin(k,) . (4.23)
In Fig. (4.7B) both torque and flux linkage modulus are too small and the stator flux linkage vector approaches a switching sector limit. Fig. (4.9) shows the limit value for the angle b, which again hlfills a condition for the non decreasing load angle. 56 7’’, e
Figure 4.9 The working point, where the voltage reserve vector u2+e and the stator flux linkage vector w are parallel. This working point represents the limit value of the angle b, for which the load angle does not decrease.
Again, the limit value for the angle pZ can be solved as a hnction of voltage reserve coefficient kres
p, = arcsin(k,,). (4.24)
The unideality of DTC modulation can be investigated with an optimisation algorithm, where every switching is optimised step by step during the load transient. There are two quantities to be controlled; the electrical torque Itel and the stator flux linkage modulus Iv/JI.In a load transient the torque ltel should achieve a new value Itclref. On the other hand the stator flux linkage modulus should maintain its reference value (v/&f. This leads to the cost function for the stator voltage vector optimisation, where the total cost C is defined as
(4.25)
where the cost of the stator flux linkage error can be tuned with the cost coefficient kpsi. The optimisation can be described with the following flow chart, Fig. (4.10). According to the simulation flow chart a voltage vector for every switching is selected optimally in order to minimise the cost function in Eq. (4.25). This kind of optimisation does not necessarily find the global minimum for the whole transient. However, by tuning the coefficient kps, experimentally, a converging solution is possible.
Figs. (4,11), (4.12) and (4.13) compare the torque, the load angle and the stator flux linkage modulus during the optimised torque step using the different values for the coefficient kpsi. A comparison between the results with two different values of kpsi shows, that a smaller coefficient (bSi=50)gives better performance at the beginning of the transient, and a larger coefficient (kpsi=lOO) improves the performance more with larger load angles near 7d2.
The global minimisation of the torque and flux linkage error during a load step would require dynamic optimisation, where the final working point should be known. The implementation in a practical modulation system is not feasible, since in most cases the final working point after a load transient is not known, but is defined by the speed controller. Initial values of the machine: ,% initial rotor angle yso initial stator flux linkage vector ltelo . initial torque urn initial excitation voltage
Stator voltage vector matrix US:
10 --143 2 _-lk - 22 c=c,,, iop=i I- I! I 'I us = -1 0
_-1J5 - - 2 --1& 2 2
00 uf selection I ~
Functions: fM Motor modelling fbnction f, Cost fbnction of optimisation
Optimisation variables: i Stator voltage vector index io,, Optimal stator voltage index C Cost of a stator voltage vector / \\ j Time step index j,, Amount of time steps A t Length of time step
Figure 4.10 Flow chart for the dynamically optimised stator voltage vector selection in the load transient. Subscript "pr" means the values corresponding to previous optimisation. The voltage matrix us describes the x- and y-components for different voltage vectors Ul..U7 in the stator reference frame.
The comparison between the presented stepwise modulation optimisation and basic DTC modulation in Figs. (4.14) and (4.15) shows, that improvement can be achieved even with this method. Fig. (4.16) still shows, how the optimisation selects voltage vectors during the transient. There is a certain margin on the side of a switching sector, where stator flux linkage modulus is not actively 58
decreased or respectively increased. These margins are shown in Fig. (4.16) as the angles €1 and €2. These limit angles are not equal, but €1 > c2. This is due to the fact, that in the range of €1 the reduction of the stator flux linkage has a double effect, the torque will be reduced by both the stator flux linkage modulus reduction and the load angle reduction, while in the range of €2 the increasing stator flux linkage modulus and the decreasing load angle partially compensates each other.
Figure. 4.11 Comparison of torque responses for the optimised modulation during a nominal torque step with two different values for the coefficient kpai. Torque Itell represents the optimised torque step for hi=IOO and (tC2l the optimised torque step for hi=50. Speed ~2.0pu, voltage reserve k~0.8.
t [ml 0 5 10 15
Figure. 4.12 Comparison of stator flux linkage load angles for the optimised modulation during a nominal torque step with two different values for the coefficient kpai. Load angle 61 represents the optimised torque step for kpai=lOO and b; the optimised torque step for kpai=50. Speed ~2.0pu, voltage reserve km=0.8. 59
0.44
0.42
0.4
V 0.38 t Io I 0 5 10 15
Figure. 4.13 Comparison of stator flux linkage moduli for the optimised modulation during a nominal torque step with two different values for the coefficient kpsi. 1 wll represents the optimised torque step for t$,si=lOOand 1 w21 the optimised torque step for b;=50.Speed n=2.0 pu, voltage reserve kc,=0.8.
I 0 5 1 15
Figure. 4.14 Comparison of torque responses during a nominal torque step with optimised DTC modulation licll and basic DTC modulation ltcl. 60
Figure. 4.15 Comparison of load angle development during nominal torque step with optimised DTC modulation 61and basic DTC modulation 6.
Figure 4.16 Voltage vector selection during the load transient, when the modulation has been optimised. Voltage vector index, stator flux linkage modulus IfiI and the stator flux linkage load angle 6are presented as a fhction of the stator flux linkage angle 8. Also switching sectors are shown. Speed n=2.0 pu, voltage reserve k,,=O.S, cost coefficient k,,i=lOO. The areas of a switching sector K~,where the stator flux linkage modulus is not actively decreased, [-x/6, -x/6+~1], or increased, [x/6-~2,d6], are also shown. See also Fig. (4.6). 61
4.4 DTC stability control in field weakening range
Synchronous machines have a steady state stable working area, where the torque and the load angle depend on each other with a positive coefficient. If the load angle leaves the stable working area, the coefficient is negative. This causes a fast torque break down and loss of synchronism, if the load angle is not actively limited. Stability problems occur especially in the field weakening range, where a large load angle is necessary to produce a high torque.
It is possible to drive vector controlled synchronous motors with high load angles temporarily outside the stable working area. It has been shown by MIrd et a1 (1990-11), that in a current vector controlled synchronous motor drive the stator flux angle can achieve a value bigger than 160" in e load transient without loss of synchronism or torque oscillation. In that case the stability is based on the air gap flux linkage angle limitation, and the current vector control according to it.
In a fast transient, the torque may indeed increase even with load angle values bigger than 7c/2 as it appears to be in the transient analysis. In traditional field oriented control methods, where the control is often based on the current vector adjustment corresponding a certain load angle value, unstable values for the load angle can be temporarily allowed in transients. The current controllers will force the drive into a stable working point after the transient, since the load angle is one of the control variables in a current vector controlled synchronous motor drive.
In basic DTC method direct load angle adjustment is not possible. Instead, it is formed freely, while the torque and the stator flux linkage modulus are controlled. This principle allows very high dynamics in the torque control due to its simplicity and optimality in the limited load angle range. However, with a certain maximal load angle value the acceleration of the stator flux linkage vector is no more capable of increasing the torque. In that situation DTC has no natural mechanism to return the load angle inside the stable working area. Instead, the drive acts like a network connected synchronous motor, which has lost the synchronism.
The DTC synchronous motor drive stability can be affected either by indirect or direct load angle control. In indirect methods the torque and the field current are adjusted so, that the load angle is assumed to stay on the stable area. The indirect load angle control requires accurate knowledge of the motor inductance parameters and is thus parameter sensitive. For that reason a certain margin must be left between the theoretical maximal torque and the actual torque limit.
In the direct method the load angle is controlled directly by the modulator. This allows larger load angles and thus a better utilisation of the maximal motor torque.
4.4.1 Indirect load angle control
In a current vector controlled drive the load angle control is based on the adjustment of the current references so, that the maximal load angle is not exceeded. A method for controlling the load angle in a current vector controlled synchronous motor has been represented e.g. by Mird et a1 (1990-1). The torque control and the load angle control are done separately. The stator flux linkage components are in the desired working point 62
(4.26)
The torque can be calculated according to Eq. (1.5). The current components can be calculated, when the stator flux linkage modulus and the load angle are selected. The q-axis current does not depend on the torque reference but on the load angle and the q-axis component of the stator flux linkage,
(4.27)
The d-axis current depends on the torque, the load angle and the d- and q-axis components of the stator flux linkage
(4.28)
The stator flux linkage d-axis component is kept in the reference value by compensating the change of the d-axis stator current with the field current control
(4.29)
The presented method is straightforward and suits well for the load angle control of a current vector controlled synchronous motor. The d-axis current vector control and the field current control can be tuned so, that a balanced transition is achieved during a torque step.
The above method is not directly applicable to DTC because the stator current is not a controlled variable. However, the method can be used indirectly by adjusting the torque reference, if a relation between the torque and the stator flux linkage load angle 6is known.
The dependence between the DTC stator control variables and the field current can be derived, if the load angle di, is known. The torque expressed with the scalar components of the stator flux linkage and the stator current is
(4.30)
The q-axis stator flux linkage component is defined by the q-axis inductance and the q-axis stator current component
(4.31)
Using Eqs. (4.30), (4.3 1) and (4.26) an expression for the torque can be obtained 63
(4.32)
The field current is on the other hand
If. = -Ysd - (1 + k)id . (4.33) Imd
From Eqs. (4.32) and (4.33) the field current for the maximum load angle can be solved
(4.34)
The required torque limit can then be solved fiom Eq. (4.34)
(4.35)
The indirect load angle control for DTC electrically excited synchronous motor can be done according to Eq. (4.35). The torque limit of Eq. (4.35) takes the possible slowness of the excitation system into account by using the actual value of the field current.
The maximal torque in relation to the available stator current is achieved with a stator flux linkage angle less than x/2, Eq. (2.42). A small reduction in the stator flux angle fiom the maximal torque point causes only a small reduction in the torque, Fig. (4.17). Indirect load angle control methods are not able to bring the drive back to the stable working area in DTC, if the maximal load angle has been exceeded. For this reason, a safety margin is required in the torque limitation. Since the stator flux linkage and the motor inductance parameters include errors, the stator flux linkage angle limit value should be selected 5..10% smaller than a theoretical maximal point. 64
P,Q bul 6 ["degl \ .. . - .. . 6 80" -......
3 ...... 40"
......
0 / 0" - 4.4 4.5 4.6 4.7 if [PUI
Figure 4.17 Capasitive reactive power (Icap,active power P and load angle 6 as a function of field current if. Flux linkage lylsl = 0.33, maximal stator current modulus lisl-=4.0, speed ~3.0,reference motor parameters (see Appendix A). The maximal torque working point (unity power factor) lies on the load angle St, = 81.8"deg.
4.4.2 Direct load angle control
DTC requires high speed control algorithms for the stator flux linkage estimation and for the modulation control. Tiitinen et a1 (1995) have reported the control cycle of 25 ps. Such a speed in the controller gives possibilities for the direct control of the stator flux linkage angle as well.
The basic functional principle of DTC should be considered again. The modulation is based on three variables, which are the torque error, the flux linkage modulus error and the stator flux linkage vector location in the stator reference frame. As a result of the torque comparison the modulator selects a voltage vector, which increases or decreases the stator flux linkage vector rotational speed in the stator reference frame. When the torque is increased, the load angle changes to the same direction, but with a lower speed. This observation gives the idea of the direct load angle control, where load angle is directly controlled by means of the torque error variable.
The direct stator flux linkage angle control is very unsymmetrical in different directions. Fig. (4.18) shows a situation, where the stator flux linkage angle has reached its maximal value. If the modulator tries to increase the torque, the stator voltage is used mainly for the compensation of the electro motive force and the change in the load angle is slow
(4.36) 65
When a too high load angle value is observed, such a voltage vector can be selected, which reduces the load angle as fast as possible. The change in the load angle is much faster than in Eq. (4.36), because the electro motive force and the voltage vector have approximately the same direction
(4.37)
If the maximal speed in the load angle reduction is used according to Eq. (4.37), the load angle controller must work with a very short control cycle, and a high switching frequency is required. Another possibility is to select a voltage vector parallel to the stator flux linkage vector. In that case the DTC adjusts the stator flux linkage modulus and only the electro motive force e reduces the load angle. This type of load angle reduction avoids large changes in the stator flux linkage angle and in the torque, and the load angle controller can work with approximately half a speed compared to the previous case. The different cases are shown in Fig. (4.18).
'AS,A4I__ -AS,
l \ e(%)t
Figure 4.18 Change of the stator flux linkage load angle in the rotor reference frame from the initial value yo when the positive torque producing voltage vector is used (A yl),the load angle is reduced by the electro motive force e alone (A y2)or the load angle is reduced by the maximal speed (A y3).
DTC works in the stator reference frame, and thus the fastest control loops work in the stator reference frame as well. The presented load angle control method works in the rotor reference frame, where fast estimates are not available for the load angle. Co-ordination transformations require a lot of computation power due to the trigonometric functions included. This method requires, however, fast estimates of the stator flux linkage components in the rotor reference frame. A shortcut for the co-ordination transformation equations can be found with the Taylor series. The stator flux linkage components in the rotor reference frame are
(4.38)
If the values of a function f and its k. derivatives f "(x) are known in a certain point a, the Taylor series gives an approximation for the function value in the neighbourhood of the known function valuefTu),
m rlkl (4.39) 66
In the case of the co-ordination transformation according to Eq. (4.39) Taylor series is very useful, because the transformation itself contains the values of the hnction and its derivative. Thus time consuming trigonometrical calculations can be avoided in the fast control cycles. The estimates for the trigonometric hnctions are
sin(9(t + d))= sin(9(t)) + cos(9(t))wb .w, d (4.40) cos(Q(t + d))= cos(9(t)) - sin(9(t))wb . 0, . d
Mechanical time constants are large compared to those of the electrical system, and the angular speed w. can be assumed to be constant during the approximation in Eq. (4.40). A typical value for the maximal value for At in Eq. (4.40) is 1 ms. With Eqs. (4.38) and (4.40) fast estimates for the stator flux linkage components can be calculated in the rotor reference fiame
vs,(t + At) = (t + A~)cos(~(~+ d)) + v, (t + at) sin($(t + ~t)) (4.41) vSq(t + d)= v/, (t + at) cos(~(t+ .t))- vsx(t + at) sin(G(t + .t))
Eq. (4.41) can be calculated with high speed in a real time control, e.g. within a few hundred microseconds cycle, since it includes only a few DSP processor operations.
In the field weakening range another possibility for the stability control is to increase the stator flux linkage modulus reference. If the voltage reserve is reduced to the theoretical minimum, the stator flux linkage vector can no more rotate with an over synchronous speed, and thus the load angle can no more increase. Further comparison between these two methods is presented in the simulations.
The presented methods assume, that the synchronous machine is working in the motor mode, where the electrical power is converted to the mechanical power. If the machine is working in the generator mode, the previous assumption is not valid, and the methods require some modifications. The generator mode is however not analysed in this work. 5 SIMULATION AND TEST RESULTS
The calculation power of PC computers has been increasing rapidly, and is sufficient for quite accurate simulation with a detailed model of an electrical drive. The real challenge of producing a simulation is the accuracy of the model. The space vector theory assumes the flux linkage to be sinusoidally distributed along the stator and rotor surfaces as well as in the air gap. This is however not true, since the motor has only a limited amount of slots and thus the flux linkage changes stepwise along the air gap. In many cases it is also assumed, that the electrical motor model is based on a group of time invariant differential equations. In a real motor the magnetic saturation and temperature have a remarkable effect on the model parameters. In a computer simulation the differential equations must be solved with time stepping methods. This causes fbrther inaccuracy in the simulation model.
Despite of some deficiencies, computer simulation tests are a very powerful tool for testing control algorithms. Signals inside the system are accessible and algorithm modifications are easy. Estimation errors of the control system can be detected by comparing the estimated signals to the “real” signals of the motor model. This is not necessarily the case in the laboratory test drive, where all the real world limitations are present. For example a synchronous motor model verification is a difficult task for a real drive system, since rotor damper winding currents can not be measured and the rotor parameters are time variant. Also the stator flux linkage is a quantity, which can not directly be measured.
5.1 Description of the simulation method
A complete simulation model of a drive system includes both a DTC frequency converter and a synchronous motor model. A suitable electrical drive simulation structure for a PC-computer has been proposed by Burchanovski and Pohjalainen (1 990), on the basis of which the DTC synchronous motor drive simulator has been developed by the LUT synchronous motor drive research group. The simulator has been coded in C-language. Since C-language is a pure compiled language, the simulation is fast. This allows a very short 5 ps time step for the discrete simulation, thus improving the discrete model accuracy.
The main parts of the simulator are the power electronics model, the synchronous motor model and the tested DTC method. It is important to notice, that the simulator includes actually two motor models, one for modelling the real motor and another corresponding to the motor model included in the control system.
The power electronics part consists of a six pulse diode rectifier, a DC link with a necessary capacitor and a throttle coil, a six pulse transistor inverter and an excitation bridge. Unidealities of particular switches, like threshold voltages, finite switching times and conducting resistances are included. Also commutations in the diode rectifier and transistor inverter are included into the system model.
The synchronous motor part is based on a discrete time synchronous motor model, where the stator voltage and the excitation voltage are inputs and flux linkages and currents are outputs. DTC works 68
in the stator reference frame, but the salient pole synchronous motor model is based on the rotor reference frame. Thus it is usefil to have the stator voltage equations in the stator reference frame and the rotor voltage equations as well as all current equations in the rotor reference frame.
The derivatives of the stator flux linkage components at the discrete time instant k expressed in the stator reference frame are
-(k)d WSy = .(u, (k)- i, (k). rs). dt w,
The rotor flux linkage components are calculated in a same way, but in the rotor reference frame
*-(k) = -0,.ZQ(k)TQ. dt
The Eqs. (5.1) and (5.2) have been converted to a real time scale by multiplying with the basic angular speed ob. When the derivatives of the flux components are and the time step' T are known, the flux linkage components for the next simulation step can be calculated
(5.3)
Wf(k + 1) = Wf(k) + -(k).d Wf T. dt
The current components can be calculated, when the inverse of the inductance matrix and the flux linkage components are known. The inductance matrix is shown in Eq. (1.9). This conversion must be made totally in the rotor reference frame, since the Park's (1929) two axis model is used -id(k + 1)' i, (k + 1) i,(k + 1) i, (k + 1) - if(k + 1)
The stator flux linkage components and yq have been converted to the rotor reference frame according to Eq. (4.38). If a saturated motor model is used, the inverse inductance matrix in Eq. (5.4) must be updated at every simulation step. Finally the electrical torque of the motor can be calculated
Eq. (5.3) shows, that the motor model assumes the flux linkage components to be constant during a simulation step. Also the rate of saturation, which is defined by the k-coefficients in Eq. (5.4),is assumed to be constant during a simulation step. However, if a small enough time step, here T=5 ps, has been selected, the error is acceptable small. The machine parameters used in the simulations are shown in Appendix A. A closer description of the whole simulator can be found in Appendix C.
5.2 Simulation results of excitation control
5.2.1 Combined excitation control in the nominal speed range
The dominant problems with a pure feedback excitation should be examined first. The dependence between the feedback signal based on the Eq. (3.10) and the controlled field current if is not linear and the feedback signal is disturbed due to modulation noise and model errors in the stator flux linkage estimate. The feedback controller used in the simulations has the structure shown in Fig.(3.3). In the case of the pure feedback control, however, the field current reference produced by the open loop part has been replaced by a constant reference value corresponding to a nominal stator flux linkage with zero load.
Fig. (5.1) shows how the pure feedback excitation control functions. A linear P-controller has been used with a proportional gain P,. The feedback signal has been filtered by a first order low pass filter using time constant Tf= 25 ms. The torque response is fast and accurate. However, the magnetic energy oscillates significantly in all d-axis windings. When the filter time constant has been increased to Tf= 50 ms, as in Fig. (5.2), the oscillations are reduced to an acceptable level. Using a slower filter will reduce the dynamics. A large time constant increases also phase shift in the feedback control loop, which may cause stability problems. 70
2-
I I 1-___I I
0- 5t
-1
Figure 5.1 A torque step at the time instant P600 ms with a feedback excitation ontrol. Speed n=0.5 pu. Field current if, feedback signal E, d-axis damper winding current iD, d-axis stator current id and torque Itel. Feedback gain P,,=6, feedback filter time constant Tf=25 ms. Magnetic energy oscillations occurs between the stator and the rotor.
[PUI
1 --
590 \ 620 660 700
id
-1 ' I
Figure 5.2 A torque step at time instant P600 ms with a feedback excitation control. Speed n=0.5 pu. Field current if, feedback signal E, d-axis damper winding current iD, d-axis stator current id and torque Ite].Feedback gain Pe,=6, feedback filter time constant T~50ms. Compared to previous case magnetic energy oscillations have decreased significantly, due to increased feedback signal filter time constant.
Fig. (5.3) shows a torque step using feedback excitation control in the field weakening range with a speed n =1.1 pu. The dynamic dependence between the torque and the feedback signal has changed significantly; it now looks like a non minimum phase linear system, where in the beginning of a transient the system response time derivative has a different sign than the system input. This uncertainty makes it difficult to design a linear feedback control for the system. 71
311pylI
Figure 5.3 A torque step at time instant e600 ms with a pure feedback excitation control. Speed n=l . 1 pu. Field current if, feedback signal E, d-axis damper winding current iD,d-axis stator current id and torque Itel.Feedback gain Pex=6,feedback filter time constant T~25ms. The feedback signal E responses to the torque step like a non minimum phase linear system.
Saturation has a significant role for the system characteristics during the transient. Fig. (5.4) illustrates the behaviour of the magnetising inductances during the torque step shown in Fig. (5.3). The saturation turns out to be a reason for the non minimum phase character of the torque and the feedback signal. In Fig. (5.5) the torque step corresponding to Fig. (5.3) has been shown again, but this time using a linear motor model with constant inductances. The non minimum phase character is not present, also the torque response is much slower compared to that in the previous case, where saturation model has been used.
...... o.8 t 0.6 ......
0.4 -;.'--I^-~ 580 620 660 700 740 t[rnsl 780
Figure 5.4 The behaviour of the magnetising inductances during the torque step shown in Fig. (5.3) 72
I [pul
&
660 700 740 f[msl
-2 ' I
Figure 5.5 A torque step at time instant s600 ms with pure feedback excitation control. Speed n=l . 1 pu. Field current if, feedback signal E, d-axis damper winding current iD, d-axis stator current id and torque Itel.Feedback gain Pex=6,feedback filter time constant Tf=25 ms. A linear motor model has been used. The non minimum phase character shown in Fig. (5.4) is not present due to linear motor model.
The shown simulation examples, where a simple feedback control has been used, point out some problems involved with a linear excitation control design. The system is both non linear and time variant, and a robust and simple feedback control design for such a system is difficult to find.
The state feedback and non linear control design principles could be one possible way to improve the feedback excitation control. This would however not be in line with the simplicity of the primary control system, which DTC presents. Also the system reliability would decrease, if the excitation controller should be carefully tuned for every drive case by case. Another problem in the control design would be the wide working area of the drive. The excitation controller would require adaptivity to work well with zero speed as well as in the field weakening range.
Due to difficulties in a pure feedback excitation control a combined open loop and feedback control has been selected. Figs. (3.2b) and (3.3b) showed, that the inductance values of the nominal stator current point used in Eq. (2.41) give a good estimate for the field current needed to achieve unity power factor. For this reason inductance values in that working point are used. The feedback signal has been filtered by using time constant Tf= 100 ms and the feedback correction in the field current reference has been limited between [-15 %, 15 %].
Fig. (5.6) shows a torque step, when this excitation control method has been used. The torque response does not differ from the previous case, where a feedback excitation control has been used, Figs. (5.1) and (5.2). The benefit of this control method is a stabilised behaviour of the magnetic energy. The magnetic energy oscillations are small compared to those of the feedback design. 73
“-.pI1--_-
-1 Figure 5.6 A torque step at the time instant e600 ms with the combined excitation control. Speed n=0.5 pu. Field current if, feedback signal E, d-axis damper winding current iD,d-axis stator current id and torque lre1. Feedback gain PeX=2,feedback filter time constant Tf =lo0 ms. The excitation control works without significant oscillations, but due to its slowness the large d-axis damper winding current is induced during the torque slope.
The d-axis damper winding current can be reduced during the transient with synchronised excitation and torque control slopes, as shown in Fig. (5.7). Also in this case some current is induced into the damper winding, but the peak amplitude is significantly smaller than without a synchronisation. This may improve the current model accuracy, especially if there are large uncertainties in the d-axis damper winding parameters. This is often the case, since an accurate estimation of the damper winding parameters is difficult and the rotor temperature changes the damper winding resistance values. It would be then usehl to keep the damper winding currents small.
2
1
iD lo& 0 -, -1 5 660 700 740 f[ ms] 7 0
-1
Figure 5.7 A torque step at the time instant e600 ms with the combined excitation control. Speed n=0.5 pu. Field current if, feedback signal E, d-axis damper winding current iD, d-axis stator current id and torque Ire/. Feedback gain Pex=2,feedback filter time constant Tf=lOO ms. The excitation control and the torque control have been synchronised with an equal slope thus reducing significantly the d-axis damper winding peak amplitude. 74
With speed n=l.1 pu the stator flux linkage is somewhat reduced and the armature reaction is more significant. Fig. (5.8) shows a torque step, when n=l.l pu. The magnetic energy oscillates on the d- axis between the damper winding and the stator winding. The field current is, however, steady during the oscillation. The reason for the oscillation is not the feedback correction in the field current. This is proved in Fig. (5.9), where only the open loop part of the excitation control is in use. The magnetic energy oscillation is still the same, as when using the feedback correction. Instead, DTC seems to cause the magnetic energy oscillation. Further simulations will show, that a potential reason ' for these oscillations are the fluctuations in the DC link voltage.
The torque controller and the excitation controller have been synchronised by using approximately the same slope in Fig. (5.8). In Fig. (5.9) the excitation controller has a faster slope than the torque controller. In both cases a large d-axis damper winding current occurs. This is caused by the armature reaction, which is larger in the field weakening range than in the nominal speed range. The behaviour of the d-axis air gap flux linkage modulus and the stator flux linkage modulus are also shown in Fig. (5.8). The armature reaction can be reduced by an over excitation, which is dependent on the torque. In Fig. (5.10) the over excitation has been defined by the over excitation factor k,,,
The d-axis air gap flux linkage reduction is smaller compared to that shown in Fig. (5.8) and therefore also the damper winding current is reduced. The drawbacks of this solution are the increasing losses in both the stator and the rotor because of the increasing currents. jlll -'t -2 Figure 5.8 A torque step at the time instant F600 ms with the combined excitation control. Speed ~1.1pu. Field current if, feedback signal E, d-axis damper winding current iD, d-axis stator current id, torque Itel, stator flux linkage modulus IvsJand d-axis air gap flux linkage Vmd. Feedback gain Pex=2,feedback filter time constant Tf =lo0 ms. Despite of the synchronised excitation control and the torque control a significant damper winding current is obtained due to a large armature reaction. Also the magnetic energy oscillates during the transient. Note, that the limitation in the power factor correction occurs during the time period G625ms. ,760ms. 75
3
Itel
620 660 700 740 tIms1 7fO
-3 LL Figure 5.9 A torque step at the time instant P600 ms with the open loop excitation control. Speed n=l.l pu. Field current if, d-axis damper winding current zD, d-axis stator current id and torque Itel. The field current slope is faster than the torque controller slope. Magnetic energy oscillates during the transient also without the feedback part of the excitation control.
-3 ' I Figure 5.10 A torque step at the time instant MOOms with the open loop excitation control. Speed n=l.1 pu. Field current if, d-axis damper winding current iD, d-axis stator current id, torque Ite\, stator flux linkage modulus I and d-axis air gap flux linkage y&. The d-axis damper winding current has been reduced by using the over excitation according to Eq.(5.6), k,,=0.2. 76
5.2.2 Excitation control in the field weakening range
In the field weakening range the flux must be reduced as a function of the rotating speed and a larger armature reaction, than in the nominal speed range, is obtained. Considering the excitation controller, this means, that it takes some more effort to respond to a certain level torque step in the field weakening range.
In the both simulation and laboratory tests a modified voltage reserv; coefficient kSlhas been used. Unlike the coefficient k, defined in Eq. (4.21), the modified voltage reserve coefficient defines that particular voltage reserve, which can be used totally for the stator flux linkage acceleration by excluding the resistive voltage drop in the stator winding fiom the available stator voltage,
n E{ 1...6} (5.7)
The approximation shown by Eq. (5.7) has bee used. It assumes, that the voltage vector and the current vector have approximately the same direction. The stator flux linkage reference in the field weakening range is calculated as
In order to see the effect of the excitation control during a torque transient in the field weakening, three different excitation control methods has been simulated. Also the dynamic performance of the excitation unit has been varied by limiting the field current reference value with a ramp. In following examples rotating speed ~2.0pu has been used, which requires a stator flux linkage modulus reduction larger than 50 % from the nominal value. The voltage reserve coefficient k~0.84has been used.
In the first case an excitation control method with combined open loop and feedback control has been used. The field current dynamics has been adjusted by limiting the current slope with a ramp, where the slope has been defined by the ramp time Tcx.Three different ramp times 0.2 pu/ms, 0.1 pu/ms and 0.05 pdms has been used. Fig.(5. 11) shows the field current in different cases, Fig. (5.12) shows the corresponding torque responses and Fig. (5.13) the corresponding load angle behaviour, when different field current slopes are used. The difference of the torque response between case A (Tcx=0.2pdms) and case B (T,=O. 1 pdms) is not significant. This is due to the fact, that the d-axis damper winding is able to keep the load angle below the limit value 85"deg, as illustrated in Fig. (5.13). If the excitation system dynamics is further reduced, case C (Tcx=0.05pdms), the load angle is getting too high and it must be limited in order to maintain the synchronism. A load angle limitation reduces significantly the torque response. In this case the direct load angle limitation based on the fast torque reference adjustment has been used (see also Fig. (5.30)). 77
01 I 580 620 660 t[msl 700
Figure 5.11 A field current response in a 150 % torque step at the time instant P600ms with three different slopes Tex=0.2 pdms (A), Tex=0.1 pdms (B) and T,= 0.05 pdms (C).Speed n = 2.0 pu, keS1=0.84
580 620 660
Figure 5.12 The torque response in a 150 % torque step at the time instant F600ms with three different field current slopes corresponding to Fig. (5.11). Speed n = 2.0 pu, k1=0.84.Also the stator flux linkage modulus in case A, I& has been shown. Iv/lA must be reduced due to the increasing voltage drop in the stator resistance. The fast field current response (A) gives the best torque response. 78
2- [PUI
1 --
0 580 620 660 t[msl 700
Figure 5.13 The load angle development in a 150 % torque step at the time instant e600ms with three different field current slopes corresponding to Fig. (5.11). Speed 7~2.0pu, k1=0.84. A fast field current response keeps the load angle smallest (A), in case of a the slowest field current response the load angle must be reduced actively (C).
A fast excitation control unit may not always be available. If a fast torque response in the field weakening is however required, the excitation slowness must be taken into account, when selecting the excitation method. One possibility is to use the partial load over excitation introduced before. If the maximal stator current value and the stator flux linkage modulus are known, the required excitation curve can be solved using Eq. (2.45). Fig. (5.14) shows the excitation curves for unity power factor, curve A, and for the constant d-axis air gap flux linkage corresponding to the maximal working point with ltel,=1.55 and lw;l=O.33, curve B. The curve B has been solved from Eq. (2.45) with the given values and fUrther approximated by using the polyfit hnction of h4ATLABm,
One drawback of the excitation according to curve B occurs to be the high currents both in the stator and the rotor with small loads and thus increased losses. Another drawback of curve B is the numerical complexity involved with the iterative calculation. Note, that the approximation given in Eq. (5.9) is valid only for the given values lt,lm=1.55 and l&v,l=0.33. Curve C represents a compromise between the unity power factor excitation and the constant d-axis air gap flux excitation, and is calculated by using the over excitation coefficient k,,
(5. IO) 19
The excitation according to Eq. (5.10) does not significantly increase losses with small loads compared to unity power factor excitation, but improves stability with high loads. Also the numerical implementation is simple and it does not require lots of calculation power. In Fig. (5.14) over excitation coefficient kex=0.6 has been used. ne relative over excitation below refers to this excitation method.
'0 '0 -0.2 0.4 016 018 1 1:2 1.4 ll6 L[PUI Figure 5.14 Different excitation curves with ~te~mx=1.55and lvs1=0.33.A nominal excitation curve (A), constant d-axis air gap flux linkage (B), Eq. (5.9) and relative over excitation (C), Eq. (5.10). All curves minimise the stator current in the maximal working point, where lisl=4.65.
Figs. (5.15) and (5.16) show a torque step when the field current reference according to Eq. (5.9) has been used. The torque and the excitation curve slopes have been synchronised. The d-axis damper winding peak value during the transient is small and the magnetic energy oscillates only with a very small amplitude. Since the d-axis air gap flux component is kept approximately constant, the rate of saturation is also kept constant, which partially reduces oscillations and improves dynamic current model accuracy on the d-axis. The torque response is very good despite of a small voltage reserve on the stator side, and the load angle reacts well during the transient.
6- IPUI
4 --
2 --
;I, I - - 0 \ , Sflo 620 660 t[ ms1 700 -2 --
-4 --
-6 1 I Figure 5.15 A torque step 150% at the time instant i=600ms with approximately constant d-axis air gap flux linkage, speed ~2.0pu, k1=0.84.The field current reference has been calculated using Eq. (5.9).The field current reference and the torque reference have been synchronised. Field current if, d-axis damper winding current iD and stator current id. 80
2- [PUI [rad1 \ "Dc 1.5 --
1 --
0.5 --
0- ~
When using the relative over excitation according to Eq. (5.10), the torque dynamics is reduced and a larger d-axis damper winding current is generated. However, the same well behaving load angle is obtained. The torque step with this excitation method has been shown in Figs. (5.17) and (5.18).
[PUI if 4 --
2 I /
-6 ' Figure 5.17 A torque step 150% at the time instant t=600ms with the relative over excitation according to Eq. (5.10), k,,=0.6, speed ~2.0pu, &1=0.84. Field current reference and the torque reference have been synchronised. Field current if, d-axis damper winding current iD and stator current id. 81
2 - [PUI [rad1
1.5 --
I --
580 620 660 t[msl 700
Figure 5.18 The torque step corresponding to Fig. (5.17). Stator flux linkage modulus I d-axis ai gap flux linkage v,d, load angle 4 torque response It,[ and DC link voltage UDC. Torque response is reduced compared to the previous case, Fig. (5.16), where the d-axis air gap flux linkage has been kept constant. Also a larger DC link voltage oscillation can be observed. The load angle is kept small due to the relative over excitation.
In Figs. (5.16) and (5.18) also the DC link voltage has been shown. In both cases significant oscillation are seen, which is one reason also for the magnetic energy oscillation. Sudhoff et a1 (1998) has shown a method for damping DC link oscillations. DC link control was not a subject in this work and damping methods were not tested. However, this is a typical real world limitation, which must be taken into consideration in the control design. The electrical drive should be considered as a system, where single controllers should not be sensitive to each others unideal behaviour.
Fig. (5.19) and (5.20) still compares the torque response and load angle behaviour during the transient with the described three different excitation methods. The large initial air gap flux linkage gives a fast torque response and keeps the drive stable. If the relative over excitation is used according to Eq. (5.lo), the load angle is kept smaller than when the unity power factor excitation is used, but the torque response remains almost equal. 82
1.5
1
0.5
0 580 620 660 f[ msl 700
Figure 5.19 A comparison of torque responses with different excitation methods. Speed n=2.0 pu, k,l=O.84. Constant d-axis air gap flux linkage (A), relative over excitation according to Eq. (5.10) with excitation coefficient k,,=0.6 (B) and the unity power factor excitation with T,=0.2 pu/ms (C).
580 620 660 f[ msl 700
Figure 5.20 A comparison of load angle behaviour by different excitation methods during a torque step corresponding Fig. (5.19). Speed ~2.0pu, k=0.84.Constant d-axis air gap flux linkage (A), relative over excitation according to Eq. (5.10) with excitation coefficient kc,=0.6 (B) and unity power factor excitation (C). 83
5.3. Simulation results of flux control in the field weakening range
5.3.1 Voltage reserve and dynamic performance
The torque response in the field weakening range was found to depend on the excitation control. Another findamental condition for good torque dynamics is a sufficient voltage reserve in the stator inverter. Both conditions must be valid in order to gain a good dynamic response. A limit value for the voltage reserve, which allows maintaining a synchronous speed and a circular stator flux linkage path, is given by Eq. (4.9). An increase of torque requires a higher voltage reserve. In the following simulations the flux linkage reference has been calculated according to Eq. (5.8). Fig. (5.21) compares the torque dynamics with different voltage reserve values. It shows, that the torque dynamics is improved by using a higher voltage reserve to a certain extent. However, a larger voltage reserve requires larger stator and rotor currents in order to obtain a certain torque level.
Fig. (5.22) shows the stator flux linkage modulus with different values for the voltage reserve and Fig. (5.23) shows correspondingly the d-axis currents in two particular cases. A rather large stator flux linkage reduction is necessary due to increasing resistive voltage drop and decreasing DC link voltage. Therefore the actual voltage reserve is not constant, but is changing during the transient. This can also be noticed in the torque responses in Fig. (5.21), where there is some oscillation during the torque increase. The large oscillation at the end of the torque step is because of the torque limitation algorithm, which is activated due to a DC link undervoltage.
1.6
1.2
0.8
0.4
0 580 600 620 640 660 t[ms1 Figure 5.21 A comparison of dynamic performance in torque step at the time instant F602 ms with different values for the voltage reserve. Speed n=2.0 pu. A) k1=0.78,B) ksl=O.80, C) kS1=0.82, D) k~0.84.Excitation method relative over excitation according to Eq. (5. IO), ke,=0.4. 84
0.5 I
0.4
0.3-.- -!- , 580 600 620 640 660 t[ms1
Figure 5.22 A comparison of stator flux linkage modulus in the torque step at the time instant t=602ms with different voltage reserve. Speed ~2.0pu. A) k1=0.78,B) k1=0.80, C) k1=0.82, D) k~0.84.Excitation method relative over excitation according to Eq. (5. lo), kex=0.4.The stator flux linkage must me reduced due to increasing resistive voltage drop.
The dimensioning of the stator voltage reserve also depends on the excitation unit voltage reserve and excitation controller dynamics. Fig. (5.23) shows d-axis currents in cases A (kcs~=0.78)and D (k1=0.84). The field current has the same ramp speed in both cases, Tex=0.2pu/ms. In case A the torque control dynamics is faster than the excitation control dynamics, and the damper winding tries to prevent the reduction of d-axis air gap flux linkage. In case D the damper winding tries to reduce the increase of the d-axis air gap flux linkage. Comparison shows, that in the case A the voltage reserve in the stator side and the excitation control dynamics correspond better to each other, and the damper winding current peak amplitude is smaller.
6
4
2
0 5 -2
-4
-6 Figure 5.23 Comparison of d-axis currents with different voltage reserve. Torque step at the time instant F602 ms. A) kl=0.78, D) k1=0.84.Excitation method relative over excitation according to Eq. (5.10), kex=0.4,speed n=2.0 pu. At the end of the transient the stator current id and the field current if are reduced due to the DC link undervoltage limitter. 85
In principle the limit for the voltage reserve given by Eq. (4.9) is sufficient to maintain a constant motor torque in a steady state. The advantage of stator flux linkage being as large as possible is the minimisation of the stator current. A larger voltage reserve is necessary only, when the torque step is to be performed. This can be taken into account by increasing the voltage reserve during the load step and by decreasing it again after the transient in order to achieve a minimum stator currendtorque-ratio in steady state. The previous flux control method is defined here as the dynumic voZfuge reserve. The flux linkage modulus reference in the transient, Ifilref,d, is calculated as a function of a dynamic voltage reserve coefficient ks,d
(5.11)
where
An improvement of the torque response can be gained by using the dynamic voltage reserve, as shown in Fig. (5.24). In the simulated case the fast transient in the stator creates a voltage drop in the DC link voltage, and thus the torque must be limited. With a stiffer DC link this could be avoided.
2
1.6
1.2
0.8
0.4
0 580 600 620 640 660 t[ms1
Figure 5.24 Torque dynamics with (B) and without (A) dynamic voltage reserve. Torque step at the time instant P602 ms. Also DC link voltage has been shown in case B. Excitation method relative over excitation according to Eq. (5.10), k,,=0.4, speed ~2.0pu, kl=0.82. At the time instants t=630ms, ... the torque is limited due to DC link undervoltage limitter. 86
5.3.2 Results of modulation modification in the field weakening range
Optimisation of the DTC modulation showed, that a basic DTC optimisation table does not give the best possible torque response with a small voltage reserve. Instead, in a certain range of the stator flux linkage vector angle it seems to be appropriate to modify the modulation rules. In a real system the modulator can not be implemented by finding the best voltage vector before every switching event. Those stator flux linkage directions, where modification, of the modulation rules are advantageous during a torque step, can however be defined approximately according to Eqs. (4.23) and (4.24). The angle range g1 from the switching sector border illustrated in Fig. (4.6A), where decrease of the stator flux linkage modulus is not allowed, can be determined according to Eq. (4.23) and Fig. (4.7A)
5rt x E, I -- P, = arcsin(k,) - - (5.12) 6 6
The optimisation showed, that the angle range 82 on the other switching sector border is not as large as since reduction in the torque production, caused by a decreasing load angle, is partially compensated by the increasing stator flux linkage modulus, Fig. (4.16). In a real modulator symmetrical sectors are useful because of the single implementation. Thus an equal value has been used for both g1 and E~,which can be determined also according to Eq. (4.24)
z2 I p, = arcsin(k,,). (5.13)
When the load angle is close to the stability limit, it cannot be increased further. Therefore the reduced limit sectors and EZ* has been defined as a function of the load angle
(5.14)
In Fig. (5.25) a torque step obtained by a modified modulation is represented. The used voltage reserve coefficient here is k1=0.84.Also the respective torque step is shown for the standard DTC modulation. A clear enhancement particularly in the beginning of the transient can be observed. With higher load angles the average torque response is almost equal in both cases. Fig. (5.26) shows the same transient with a closer resolution. The sector limits can be observed clearly in the figure. 87
1.6 1 ...... , ...... [PUI -
1.2 --
0.8 --
0.4 --
I 01 580 600 620 640 660 t[ms1
Figure 5.25 A torque step with (B) and without (A) modulator modification. Torque step at the time instant e602 ms. Also the torque reference liclrcf,Bis represented to show, why the torque response has a "notch" at the end of the transient. This is due to DC link undervoltage limitter. Speed n=2.0 pu, k1=0.84, excitation method relative over excitation according to Eq. (5.IO), k,,=0.4, ltel,,=1.55. Clear improvement in the dynamic response can be achieved by applying the modified modulation.
If the voltage reserve is increased, the difference between the standard modulation and the modified modulation becomes negligible, as can be noticed in Fig. (5.27), where voltage reserve k~O.8has been used.
1.5
1
0.5
0 600 610 620 630 t[msl 640
Figure 5.26 The transient of Fig. (5.25) curve B with a higher time resolution. Electrical torque Ire/, load angle Gand stator flux linkage modulus I ysI.The sector limits can be observed in all curves. 88
580 600 620 640 6604 msl Figure 5.27 A torque step with (B) and without (A) modulator modification. Torque step at the time instant F602 ms. Speed ~2.0pu, &1=0.80, excitation method relative over excitation according to Eq. (5.10), kcx=0.4,~tC~-=1.55. The modified modulation can provide only a small improvement, if the voltage reserve is sufficient.
The torque dynamics can be improved in the field weakening range by applying the method presented above. An additional advantage is, that it is relatively easy to add the method into the standard DTC modulator by rejecting certain control actions with certain stator flux linkage vector angles. The method suits for both two level inverter and three level inverter.
5.4. Simulation results of stability control in the field weakening range
DTC modulation principle was found to be suitable for direct load angle control. The earlier presented method based on the direct voltage vector selection causes high switching frequency, when the limitation occurs. For that reason, a hysteresis interval for the load angle was used. Additionally, the voltage vector was not selected directly. Instead the torque reference was adjusted with a high speed algorithm, in this case cycle time for the algorithm was 100 ps. The necessary co-ordination transformations were calculated according to Eq. (4.41). The principle of the modified load angle control method is shown in Fig. (5.28).
A load angle limitation achieved by this method is shown in Fig. (5.29). The stability limit is achieved by limiting the field current to a value if-4 pu and by setting the torque reference above the theoretical maximum, Eq. (2.9). As a result, the load angle limitation works as assumed.
A corresponding load angle limitation is illustrated in Fig. (5.30) with a smaller field current maximum value iw=3 pu. Also in this case the stability is maintained without problem. However, the load angle oscillates with a larger amplitude as in the previous case.
If the load angle is limited by adjusting the stator flux linkage modulus, a smoother effect can be achieved. Another advantage of this method compared to the previous is, that the torque reduction is smaller, since the stator flux linkage modulus increases during the limitation. The drawback of the 89 method is, that it is dependent on the voltage reserve and is therefore not necessarily suitable for low speed operations. Also a larger torque oscillation is obtained.
LSynchronous motor model Figure 5.28 A modified load angle control method based on the fast adjustment of the torque reference. The hysteresis limits for the load angle are &,,,high, &,,,lo. The load angle controller keeps the estimated load angle inside the hysteresis interval during the limitation by adjusting the change in the torque reference Al&.
The simulation algorithm for the second method was similar to the algorithm shown in Fig. (5.28). However, instead of the torque reference reduction the reference of the stator flux linkage modulus was increased to decrease the load angle and visa versa. Also a narrower hysteresis interval had to be used. Figs. (5.31) and (5.32) two load angle limitations achieved by applying this method with different field current maximal values, zf-4 pu and ifs11~[=3pu.
1.5
1
0.5
0 6 IO
-0.5 '
Figure 5.29 The load angle limitation with a fast 100 ps torque control loop. Speed ~2.0pu. Load angle hysteresis limits &[80°deg, 85"degl. Field current maximal value if=4 pu. 90
2
1.5
1
0.5
0 6 0
-0.5 ' I
Figure 5.30 The load angle limitation with a fast 100 ps torque control loop. Speed n=2.0 pu. Load angle hysteresis limits 6 E [80"deg, 85"degI. Field current maximal value if=3 pu.
-0.5 2
Figure 5.31 The load angle limitation with a stator flux linkage modulus control. Speed ~2.0pu. Load angle hysteresis limits 6 E [84"deg, 85"degl. Field current maximal value if+ pu. 91
-0.5 I
Figure 5.32 The load angle limitation with a stator flux linkage modulus control. Speed n=2.0 pu. Load angle hysteresis Kits 6~[84"deg,85"degI. Field current maximal value if=3 pu.
5.5 Results of laboratory tests
5.5.1 Description of the laboratory test dnve
The laboratory test drive consists of a 14.5 kVA salient pole synchronous motor, a torque transducer, a DC-generator and DSP-controlled power electronic units. The power electronic controller boards included measurement fimctions and thus it was possible to make current measurements by the units themselves.
The parameters of the tested salient pole synchronous motor have been determined by applying the methods in standards IEC 34-4 and IEEE 115-1983. Also the parameters given by the manufacturer are included in the following tables. Some deviations between those two parameter sets are visible.
The field winding was supplied by a two-quadrant DC-chopper with a current hysteresis control. The communication delay time between the DTC stator inverter and the excitation unit was quite large. Fig. (5.33) compares the field current measured by a digital oscilloscope and measured by the digital control system during a torque step. The comparison shows, that the delay between the real measured field current value if,,,, and the field current value used by the digital control system if,eis about 7 ms. In the measured field current if,,,, can also been observed, that the delay of the excitation controller is about 8 ms. In order to have a fast responding excitation control in the torque step tests this delay was compensated by adding corresponding delay to the torque reference. The major drawback of the delay in the actual. value is the fact, that the damper winding current model on the d- axis is inaccurate due to erroneous field current estimate. 92
Table 5.1 Nominal values of the test motor given by the manufacturer
Power S,, 14.5 kVA Frequency& 50 Hz Voltage U, 400 V Speed n. 1500 rpm Current I,, 21 A Power factor cosp 0.8 cap. Field current If 10.5 A Reduction factor k, 4.637
Table 5.2 Measured parameters of the test motor and parameters given by the manufacturer
Parameter Measured Manufacturer Data Stator resistance r, 0.048 pu 0.048 pu Excitation winding resistance rf 0.0083 pu - Reduction factor k, 4 4.63
The laboratory test drive had also other limitations, which did not allow as comprehensive tests as presented in the simulation examples. The main limitations were
- a long communication delay (7-8 ms) between the stator inverter and the excitation unit
- a long time delay (7-8 ms) in the field current measurement
- an excitation unit, which was not able to produce negative voltage
- a time delay (1-2 ms) in the rotor angle measurement
- the limited computation power in the control electronics and limited access to the internal control software
Because of uncertainties in the excitation control and in the field current measurement it was insignificant to fine tune the excitation controller by ramp times or feedback parameter variation. The delay time in the field current measurement caused also a large error to the damper winding estimate and therefore investigation of the damper winding current compensation was not possible.
The delay time in the rotor angle measurement was the main limitation for the field weakening range tests. With rotating speed N=3000 rpm, which corresponds to a double nominal speed in the laboratory test drive, the delay time of one milliseconds causes an angle error of 36' electrical degrees! Since the measured rotor angle is the essential part of this control system, such a large rotor angle error could not be accepted. 93
For reasons mentioned, only limited laboratory tests are presented here. The maximal rotating speed in the tests was 2000 rpm, by which the stator flux linkage is reduced approximately to 60 % of the nominal value. Nevertheless, although limited, the application of laboratory tests is appropriate in order to test and decide, if the proposed methods are hnctional in real drive systems.
-4 0 0.02 0.04 f[s] 0.06
Figure 5.33 Measured field current If,,,,and field current used by the digital control system during a nominal torque step at the time instant P0.02 s. The phase current ZI shows clearly the torque step instant. The field current reference should follow the torque reference immediately. However, significant delays in the excitation control and in the field current estimate used by the digital control system can be observed. Time delay in the measured field current is approximately 7 ms and the delay time in the excitation control is approximately 8 ms.
5 S.2 Measurements of excitation control
Excitation control was tested in the torque step both in the nominal speed range and in the field weakening range. Fig. (5.34) shows a torque step by the combined excitation control in the nominal speed range using frequency 25 Hz. A good torque response and a magnetic energy balance without oscillation is achieved.
In Fig. (5.35) respective torque step in the field weakening range using frequency 60 Hz is represented. In this case the uncertainties of the test system come out more clearly. The excitation unit time delay can be seen be comparing the field current reference value to the actual value. The excitation controller is not able to produce a negative voltage. As a consequence, the excitation unit is unable to keep magnetic energy balance and some oscillation occurs between stator and rotor. The torque control works also in this case without any problem. The field current reference has been increased after the torque step by both the decreasing stator flux linkage modulus and the feedback correction of the excitation controller. 94
2
1 --
00
-1 ' Figure 5.34 A measured torque step from 0.5 pu to 1.5 pu at the time instant F0.03 s in the nominal speed range. Speed n = 0.5 pu. Combined open loop and feedback excitation control. Feedback gain Pcx=1.6,feedback filter time constant T'lO ms. A good torque response and a balanced magnetic energy behaviour without oscillations have been achieved. Stator currents id and i, are reconstructed from the measured phase currents, field current if is directly measured, torque Itc[ is calculated according to the measured stator currents and the estimated stator flux linkage.
-2 '
Figure 5.35 A measured torque step from 0.5 pu to 1.5 pu at the time instant F0.03 s in the field weakening range. Speed n = 1.2 pu. Combined open loop and feedback excitation control. Feedback gain Pex=1.6,feedback filter time constant T'lO ms. Voltage reserve k1=0.82. Some magnetic energy oscillation occurs during the transient.
Fig. (5.36) still compares the effect of the relative over excitation in a torque step. A voltage reserve coefficient kl=0.78 has been used. The comparison shows, that the effect of over excitation on the torque response is negligible in this case. However, stability of the drive can be improved, since the 95 load angle increases slower in the case of over excitation. The meaning of over excitation for the torque response would be more important when operating at higher speeds, like shown in the simulation results. In the laboratory drive high speeds could not be used due to the limitations presented before. 3 [PUI
0.04 0.06 0.08 0.1 f[sl 0.12 Figure 5.36 Comparison of torque steps from 0.5 pu to 1.5 pu with (B) and without (A) over excitation at the time instant f=0.055 s in the field weakening range. Speed ~1.2pu. Over excitation coefficient kex=0.35,ItCl-=l .55. The torque responses are almost identical, the load angle increases slower in the case of over excitation.
5.5.3 Measurements of field weakening control
The torque response has been tested using two different voltage reserve values in the field weakening. The same over excitation method as above has been used. Fig. (5.37) represents a comparison of the torque responses with different voltage reserves. Also the stator flux linkage moduli are shown. The voltage reserve has a significant effect on the torque dynamics. In case B can be observed, that the increasing resistive stator voltage losses use all the voltage reserve in the torque step, and thus the required torque value can not be maintained. The stator flux linkage modulus is changing slowly because of the ramped output of the flux linkage controller. With a faster ramp the torque lack in case B could have been avoided. The drawback of bigger voltage reserve is higher stator current modulus, as represented in Fig. (5.38). 96
27
1 -.
0 , I
Figure 5.37 A comparison of the torque responses Itel and stator flux linkage moduli Ivsl,when different voltage reserve in the field weakening range has been used. Speed n = 1.2 pu. A) k1=0.78, B) k~0.82.In case B voltage reserve is too small to maintain the required torque. Torque step at the time instant t~0.057s.
0.04 0.06 0.08 0.1 t[s] 0.12
Figure 5.38 A comparison of the stator current modulus and the load angle response by using different voltage reserves in the field weakening range. Speed n = 1.2 pu. and 8Ausing voltage reserve k~0.78,lisl~ and 6~ using voltage reserve k1=0.82. Torque step at the time instant t=0.057s. 97
5.5.4 Measurements of stability control
The stability controller based on the torque reference adjustment according to Fig. (5.28) has been tested in the laboratory drive. Fig. (5.39) shows the functioning of the load angle controller. The drive stability has been maintained during the limitation and average torque has been kept on a high level. The unity power factor producing excitation control has been used.
The values corresponding the situation in Fig. (5.39) are; stator q-axis inductance 1,=0.5 pu (saturated), stator flux linkage modulus I wl=0.61 pu and torque ltel=1.5 pu. The corresponding theoretical load angle in steady state can be calculated according to Eq. (2.42),
1.5.0.5 6 = arctan 7= 63.6" deg . (5.15) 0.6 1
The load angle estimate in Fig. (5.39) exceeds the theoretical value, and activates the load angle limitation. Large dif€erence between the theoretical and practical value could be caused by the delayed rotor angle measurement and the accordingly erroneous co-ordination transformation, which produces the stator flux linkage components and the load angle in the rotor reference frame. Also the erroneous stator flux linkage estimate increases the error in the load angle.
Fig. (5.40) shows the torque measured from the test motor shaft corresponding to the load step in Fig. (5.39). The estimated electrical torque in Fig. (5.39) and the measured shaft torque in Fig. (5.40) correspond to each other well (torque value ltel=1.5 pu corresponds to the torque lTel=lll Nm in the test drive, which is shown with a broken line in Fig(5.40)). Thus the stator flux linkage estimate seems to be exact. For this reason, errors in the co-ordination transformation and erroneous current model apparently seems to produce too high load angle estimate in Fig. (5.39). Despite of the offset error the load angle controller works as expected.
2
1.5
1
0.5
0
Figure 5.39 The functioning of the load angle limitation. The load angle is limited between [8O0deg,85"deg]. Speed n = 1.2 pu. Unity power factor excitation. Voltage reserve ke,,1=0.78. Torque step at the time instant td1.057 s. 98
sorpml INm1
150
...... 100
50
0 500 600 700 800 900 I000 1100 1200 1300 1400 t[ms1
Figure 5.40 The measured shaft torque IT,,,] of the test motor in the torque step corresponding to Fig. (5.39). Also the rotating speed N is represented. The speed fluctuation is based on the slow speed control of the braking DC-motor. ITelref=l11 Nm corresponds to the Irel,F1.5 pu torque reference in the test drive. The average error between the measured torque and the estimated torque in Fig. (5.39) is small. The time instant P750 ms corresponds approximately to the time instant F57 ms in Fig. (5.39).
5.6 Discussion of the results
The excitation control method represented seems to be robust and easy to implement in the nominal speed range. The armature reaction is smaller than in the field weakening, and thus the excitation control is not as significant for the drive performance as in the field weakening range. However, synchronised excitation and torque control can be useful at very slow speeds, when the voltage model can no more be used and the d-axis damper winding parameters are not exactly known. In that case a small d-axis damper winding current improves the dynamic accuracy. However, the q-axis damper winding current will be present and its contribution should be estimated with the necessary accuracy.
In the field weakening both the excitation control and the field weakening control have a significant effect on the dynamic performance of the drive. The high initial air gap flux linkage seems to be the most important factor for the fast torque response. By using a large voltage reserve the DTC dynamics can be improved. If a good torque response is to be obtained, both DTC and the excitation control should have a good dynamic performance and should be balanced with each other. When a small voltage is available, further improvement for the torque dynamics can be achieved by modifying the modulation rules of the optimal switching table.
The stability control method based on the fast torque reference adjustment seems to work well according to results. Stability control is of major importance for a synchronous motor drive, since this drive types are normally very large, a few megawatts, and the loss of synchronism means often a very dangerous situation for both the mechanical system and the electrical drive itself. A reliable 99 stability control method allows maximal torque utilisation of the drive without safety margin for the load angle. This way the overdimensioning of a DTC synchronous motor drive can be avoided.
The question of the field weakening point is very challenging. Where should it lay? From the DTC point of view, the best and simplest solution would be to avoid the field weakening by selecting a small enough motor voltage compared to the inverter voltage. In the case of a large synchronous motor this is often not possible. The construction of the motor would suffer and motor losses would increase, if the nominal voltage is reduced significantly. Some imprbvement in this field may be obtained, when new power switches for higher voltages, as high voltage IGBT or IGCT, will be available. But even then, the field weakening can not be avoided, since requirements for higher speeds are increasing as well.
Already many years DTC has been applied with success for the asynchronous motor control, and it seems to be a promising alternative to control the electrically excited synchronous motor drive as well. The excitation control principles must be then converted, but this can be done with success, as shown in this work. Good dynamic performance can be reached in the field weakening range with a sufficient excitation and a stator flux linkage control. The results in the stability control makes the DTC an attractive control method also for synchronous motor drives. 100
6 SUMMARY
In this work three different aspects of the DTC electrically excited synchronous motor drive have been investigated; the excitation control, the flux linkage control in the field weakening and the stability control of the drive.
The excitation control and excitation unit set certain limits for the mkimal performance of the drive. These have been analysed in the static and dynamic case. The maximal static torque working point has been solved as a function of the field current. The dynamic performance of the drive depends on the excitation control specially in the field weakening range. The transient analysis has been carried out in an idealised case with pure transient inductances, where resistive voltage drops in the rotor have been neglected. A comparative analysis has been done with operator inductances, where the resistive parts of the equivalent circuits are taken into account. A theoretical solution for the torque maximisation during the torque step with a given maximal stator current has been proposed.
In traditional field oriented control methods for synchronous motors the stator current components and the load angle are adjusted directly by means of the stator current controller and the field current controller. In DTC, stator currents are not controlled. In the basic DTC method the stator currents and the load angle are formed freely according to the magnetic state and the torque of the machine. The stator current and the load angle can, however, be controlled indirectly by means of the excitation control. The magnetic energy balance between the stator and the rotor must be maintained by the excitation control in DTC synchronous motor drive.
An excitation control method, which is related to DTC primary control variables, the stator flux linkage modulus and the torque, is developed and tested. The method prevents magnetic energy oscillations between the stator and the field winding. In the nominal speed range, unity power factor is used in order to minimise the stator current. The inductance parameters are required in the method for the field current reference calculation. If the parameters are erroneous, unity power factor is not achieved. The error can be compensated by using an additional slow feedback control.
In the field weakening, high initial value for the d-axis air gap flux linkage has found to be essential for a good torque response. The excitation method has been modified for the field weakening so, that the d-axis air gap flux linkage is kept constant independent on the torque reference. Such a constant value can be selected, which minimises the stator current in the maximal torque working point thus allowing both the high torque dynamics and maximal torque. This has been combined with the reaction excitation control (MBrd 1990-I), where d-axis damper winding current has been compensated.
In the field weakening, sufficient voltage reserve is needed to be able to increase motor torque. The minimal voltage reserve has been shown for a two level inverter. A balanced transition in the field weakening requires a certain voltage reserve in both the stator inverter and in the excitation unit. A short mathematical analysis of this has been given.
The DTC modulation principle causes some unideality in the load transient, if there is only a small voltage reserve. This has been investigated by a step wise optimisation, which selects at every switching instant that particular voltage vector, which gives the best performance according to the selected cost function. Since a step wise optimisation is not possible in a real modulator, an approximative method has also been developed, which suits well for the real time application. Simulation results prove, that torque dynamics can be improved by applying this method. 101
An important part of synchronous motor control is the stability control, which guarantees, that the loss of synchronism can be avoided in all cases. In vector controlled synchronous motor drives load angle can be directly adjusted with the current controllers. In DTC this is not possible, since there is no stator current control. However, DTC is a very fast control method and the stator flux linkage is the primary control variable there. These two facts make it possible to develop methods, which allow to control directly the load angle with the stator voltage. Two methods for direct stability control have been proposed and tested. One method accesses the torque reference in a very fast control loop. This method is independent on the voltage reserve and can be udd in the nominal speed range as well as in the field weakening. The second method controls the load angle by increasing or decreasing the stator flux linkage modulus. This method suits for the field weakening range, where the voltage reserve is small. Both methods have been considered only for the situation, where the synchronous motor acts as a motor. In the generator mode, the methods should be modified. 102
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APPENDIX A
Reference motor parameters used in the simulation
The motor model parameters used in the simulations are based on the test motor parameters. The simulator works with the pu-values. The parameters have been converted to pu-values using the nominal values of the test motor parameters shown table A. 1 as the basic pu reference values. The parameters used in the linear motor mode are shown in table A.2. Note, that the magnetising inductances have not been constant, when the saturation model in the simulation has been used. Also the damper winding resistances have been modified, when the effect of different damper winding time constants have bee studied.
Power S, 14.5 kVA Frequency& 50 Hz Voltage U, 400 V Speed n, 1500 rpm Current I, 21 A Power factor cosp 0.8 cap. Excitation current If 10.5 A Reduction factor k,., 4.637
Parameter Value Note d-axis magnetising inductance l,d 1.05 saturation dependent 106
APPENDIX B
Excitation curve calculation for the salient pole synchronous motor
Unity power faxtor is assumed.
Q-axis condition:
ly,(sins=Iqiq = I,/~,(COSS
Eq. (B. 1) gives equation for the load angle tangent
tang= -1, lisI
IYSI
D-axis condition:
lyslcos~=Imdi, +Zdid = lmdif-1~1iJsin~
Excitation curve can be solved fiom Eq. p.3)
Using trigonometric functions
1 cos2 6 = 1 +tan2S tan’ 6 sin2 6= 1 +tan’ 6 in Eq. (B.4) gives 107
Substituting Eq. (B.2) into Eq. (B.7) gives
Modlfyng Eq. (B.9) an expression for the excitation current of a salient pole synchronous motor is obtained
The same expression has been shown by Buhler (1977-11). Due to the assumed unity power factor stator current amplitude can be expressed with stator flux linkage amplitude and torque
(B. 11)
Substituting Eq. (B. 11) into Eq. (13.10) the open loop control law for the excitation current is finally obtained
(B. 12) 108
APPENDIX C purzanowska and Pohjalainen 19901
Description of the PC-based C-language DTC-Simulator
Simulations of the DTC-control and motor models are executed in C-language. In Fig. (C. 1) block diagram of the simulation program is shown and in Fig. (C.2) flowchart is presented. During the simulation differential equations of the motor and process are solved numerically step by step. The time step for numerical integration is 5 ps.
-- I n= I I I I J rf i
Fig. C. 1. Variable-speed DTC inverter-motor drive simulator. 109 Gstart ) binput data.csv I 1 initialisations I
I I
break simulation
function (- end ,) eneration u
process
a\no +!! counter = ~tstep+ L \ 9 inverter control I
increment
no
Fig. C.2. Flowchart of the variable-speed DTC inverter-motor drive simulator.
LAPPEENRANNAN TEKNILLINEN KORKEAKOULU LAPPEENRANTA UNIVERSITY OF TECHNOLOGY TlETEELLlSlA JULKAISUJA RESEARCHPAPERS
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ISBN 951 - 764-274-1 ISSN O.356-8210 Lappeenranta 1998