Improving landfill monitoring programs with the aid of geoelectrical - imaging techniques and geographical information systems Master’s Thesis in the Master Degree Programme, Civil Engineering KEVIN HINE Department of Civil and Environmental Engineering Division of GeoEngineering Engineering Geology Research Group CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2005 Master’s Thesis 2005:22 . Sensorless scalar and vector control of a subsea PMSM Master of Science Thesis Dimitrios Stellas Department of Energy and Environment Division of Electric Power Engineering Chalmers University of Technology Göteborg, Sweden, 2013 THESIS FOR THE DEGREE OF MASTER OF SCIENCE Sensorless scalar and vector control of a subsea PMSM Dimitrios Stellas Performed at: Chalmers University of Technology, Göteborg, Sweden, FMC Kongsberg Subsea, Asker, Norway Examiner: Professor Torbjörn Thiringer Department of Energy and Environment Division of Electric Power Engineering Chalmers University of Technology Göteborg, Sweden Supervisor: Torbjørn Strømsvik FMC Kongsberg Subsea Asker, Norway Department of Energy and Environment Division of Electric Power Engineering Chalmers University of Technology Göteborg, Sweden, 2013 i Sensorless scalar and vector control of a subsea PMSM Dimitrios Stellas c©Dimitrios Stellas, 2013 Department of Energy and Environment Division of Electric Power Engineering Chalmers University of Technology SE - 412 96 Göteborg, Sweden Telephone: +46 (0) 31-772 1000 Chalmers Bibliotek, Reproservice Göteborg, Sweden, 2013 ii Abstract This thesis deals with the position-sensorless control of a subsea PMSM, which is fed by a remote VSD. Two V/f control alternatives are implemented for the startup of the PMSM and a sensorless vector controller is designed for operation at higher speeds. Several simulations are performed, in order to investigate the performance of the imple- mented models and to determine the optimal control settings. The conclusions drawn from the simulation results are validated with measurements on a lab model. The obtained results demonstrate that the implemented V/f control schemes can provide secure startup for the PMSM. The smoothness of the startup depends on the initial rotor position and on the load of the motor. One of the two V/f controllers accelerates the PMSM with significantly lower current, thanks to its ability to produce a more precise voltage reference. During vector-controlled operation, maximum efficiency can be achieved and the response of the system to load disturbances is almost ideal. Furthermore, the implemented field- weakening algorithm can extend the speed range of the PMSM by up to 17.6% for the considered load. Index terms: PMSM, sensorless, scalar control, vector control, V/f . iii Acknowledgement First of all, I would like to express my gratitude for the workplace and the equipment that were provided to me by the Division of Electric Power Engineering at Chalmers University of Technology. I am grateful to my examiner at Chalmers, Professor Torbjörn Thiringer, for his valuable guidance and his insightful suggestions throughout this work. Furthermore, I would like to thank Magnus Ellsén and Alvaro Bermejo Fernández for their help with the data acquisition system and Tarik Abdulahovic for his support with software issues. I also wish to express my sincere thanks to Georgios Stamatiou for his supportive attitude and his knowledgeable advice throughout my master studies. The financial support provided by FMC Kongsberg Subsea and the opportunity to con- duct a part of this thesis in the company’s facilities are gratefully acknowledged. Many thanks go to my supervisor at FMC, Torbjørn Strømsvik, for his valuable support and guidance during this thesis. I am also thankful to Morten Thule Hansen, whose contribution in the experimental part of this work was crucial, and Harald Bjørn Ulvestad, who kindly shared his knowledge and previous experience on the investigated system. I also wish to thank Ola Jemtland, Vidar Kragset and Ragnar Eretveit for some useful and interesting technical discussions during this thesis. Finally, I am deeply indepted to my family for their unceasing support and encourage- ment. Dimitrios Stellas Göteborg, 2013 . iv Contents 1 Introduction 1 1.1 Problem background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Purpose and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Theoretical background 3 2.1 Motors for subsea applications . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Drives for submersible motors . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Placement of the drive . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Effects of PWM voltage . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Control of permanent magnet motors . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Scalar control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.2 Vector control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.3 Combination of scalar and vector control . . . . . . . . . . . . . . 10 3 System description 11 3.1 Permanent magnet synchronous motor . . . . . . . . . . . . . . . . . . . 12 3.1.1 Electrical equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 Mechanical equations . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Transmission system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Simplifying assumptions . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 Considerations on transformer saturation . . . . . . . . . . . . . . 14 3.2.3 Introduction of equivalent quantities . . . . . . . . . . . . . . . . 15 3.3 Open-loop V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.1 Basic idea of the V/f controller . . . . . . . . . . . . . . . . . . . 16 3.3.2 Necessity of low initial frequency . . . . . . . . . . . . . . . . . . 17 3.3.3 Voltage boosting at low speeds . . . . . . . . . . . . . . . . . . . 17 3.3.4 Simplifying assumption . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.5 Implementation of the controller . . . . . . . . . . . . . . . . . . . 18 3.4 Closed-loop V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4.1 Voltage reference calculation . . . . . . . . . . . . . . . . . . . . . 20 3.4.2 Need for stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4.3 Basic idea of the stabilizer . . . . . . . . . . . . . . . . . . . . . . 22 3.4.4 Implementation of the stabilizer . . . . . . . . . . . . . . . . . . . 22 3.4.5 Structure of the controller . . . . . . . . . . . . . . . . . . . . . . 23 v 3.4.6 Comments on the controller . . . . . . . . . . . . . . . . . . . . . 25 3.5 Vector control with position sensor . . . . . . . . . . . . . . . . . . . . . 25 3.5.1 Structure of the controller . . . . . . . . . . . . . . . . . . . . . . 25 3.5.2 Need for constant torque reference . . . . . . . . . . . . . . . . . 27 3.5.3 Current reference calculation . . . . . . . . . . . . . . . . . . . . . 27 3.5.4 Transfer function of the controlled system . . . . . . . . . . . . . 28 3.5.5 Inclusion of compensating terms . . . . . . . . . . . . . . . . . . . 28 3.5.6 Design of the PI regulator . . . . . . . . . . . . . . . . . . . . . . 30 3.5.7 Implementation of voltage and current limiters . . . . . . . . . . . 31 3.6 Position-sensorless vector control . . . . . . . . . . . . . . . . . . . . . . 32 3.6.1 Necessity of eliminating the position sensor . . . . . . . . . . . . . 33 3.6.2 Review of position estimation methods . . . . . . . . . . . . . . . 34 3.6.3 Structure of the controller . . . . . . . . . . . . . . . . . . . . . . 36 3.6.4 Implementation of the position estimator . . . . . . . . . . . . . . 36 3.6.5 Basic idea of field-weakening . . . . . . . . . . . . . . . . . . . . . 40 3.6.6 Overvoltage risk during field-weakening . . . . . . . . . . . . . . . 41 3.6.7 Field-weakening capability of the system . . . . . . . . . . . . . . 41 3.6.8 Derivation of the field-weakening algorithm . . . . . . . . . . . . . 43 4 Simulation results 45 4.1 Open-loop V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 Different critical frequency values . . . . . . . . . . . . . . . . . . 46 4.1.2 Different frequency reference slopes . . . . . . . . . . . . . . . . . 55 4.1.3 Different initial rotor positions . . . . . . . . . . . . . . . . . . . . 60 4.1.4 Response to load steps . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Closed-loop V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.1 Different frequency reference slopes . . . . . . . . . . . . . . . . . 65 4.2.2 Different initial rotor positions . . . . . . . . . . . . . . . . . . . . 69 4.2.3 Response to load steps . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Vector control with position sensor . . . . . . . . . . . . . . . . . . . . . 75 4.3.1 Control transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.2 Response to load steps . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.3 Startup performance . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Position-sensorless vector control . . . . . . . . . . . . . . . . . . . . . . 83 4.4.1 Control transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4.2 Response to load steps . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4.3 Field-weakening performance . . . . . . . . . . . . . . . . . . . . 91 5 Experimental results 97 5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.1.1 Small-scale laboratory model . . . . . . . . . . . . . . . . . . . . . 98 5.1.2 Control and monitoring system . . . . . . . . . . . . . . . . . . . 100 5.2 V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.1 Startup performance . . . . . . . . . . . . . . . . . . . . . . . . . 101 vi 5.2.2 Steady-state operation . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Position-sensorless vector control . . . . . . . . . . . . . . . . . . . . . . 106 5.3.1 Control transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.2 Response to load steps . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3.3 Reversal of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 Conclusions 115 6.1 Open-loop V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Closed-loop V/f control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3 Vector control with position sensor . . . . . . . . . . . . . . . . . . . . . 117 6.4 Position-sensorless vector control . . . . . . . . . . . . . . . . . . . . . . 117 6.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 vii List of symbols List of subscripts ac Bandwidth of current controller 0 Initial value cl Stabilizer gain constant a Phase-a quantity e Back-EMF aw Anti-windup f Electrical frequency b Phase-b quantity Fb Voltage-boosting factor c Phase-c quantity i Current C Cable parameter J Moment of inertia comp Compensating value k Gain cor Corrected value kl Proportional gain of stabilizer cr Critical value L Inductance d Direct-axis quantity nr Mechanical speed (rpm) e Electrical quantity np Number of pole pairs est Estimated value pe Electric power gen Generator R Resistance i Integral Ra Active-damping resistance lim Limited quantity t Time m Mechanical quantity tr Rise time max Maximum value Te Electromagnetic torque p Proportional TL Load torque pr Predicted value Ts Sampling period q Quadrature-axis quantity u Voltage r Rotor quantity θ Transformation angle rated Rated value κ Sampling number s Stator quantity φ Angular position T Transformer parameter φ0 Power factor angle tr Transmission system parameter ψ Flux linkage upd Updated value Ψm Flux linkage of magnets ω Electrical speed Ω Mechanical speed List of abbreviations List of superscripts AC Alternating Current ∗ Reference quantity DC Direct Current ′ Equivalent quantity EMF Electromotive Force s Stationary-frame vector HPF High-Pass Filter IMC Internal Model Control LC Inductive-Capacitive List of accents LPF Low-Pass Filter MTPA Maximum Torque Per Ampere ˆ Peak value PI Proportional-Integral ¯ Space vector PMSG Permanent Magnet Synchronous Generator PMSM Permanent Magnet Synchronous Motor PWM Pulse-Width Modulation RL Resistive-Inductive V SD Variable Speed Drive viii Chapter 1 Introduction This chapter introduces the problem that is addressed in this thesis, presents some pre- vious work that has been conducted on the investigated system and states the scope and the objectives of this thesis. 1.1 Problem background Induction motors are the most traditional option for driving submersible pumps. How- ever, as the oil production moves to deeper waters, it becomes necessary to consider alternatives that can facilitate higher performance. PMSMs have the potential to meet the emerging challenges in subsea applications, thanks to positive features such as high efficiency, high power density and high-speed capabilities. The VSDs that are used for the control of submersible PMSMs are often installed on oil platforms, which can be located several kilometers away from the subsea pumping activities. The long eletrical connection between the motor and the drive in these cases introduces new challenges in the control of the PMSM. Although the control requirements of pump drives are usually low, the recent develop- ment of advanced pumping solutions calls for more efficient and precise motor regulation. The new performance demands can be met by vector control, a control type which is usually applied in advanced drive systems, which require fast and accurate current and speed regulation. Typical examples of applications that use vector control can be found within the automotive industry. The harsh operating conditions imposed by the deepwater environment of submersible PMSMs necessitate the sensorless operation of the applied control schemes. This means that the used vector controllers must not rely on mechanical measurements, but should have position-estimating capabilities. Since the performance of position-sensorless vector control is usually problematic during the startup of the PMSM, a more traditional control type, called scalar control, can be used during the low-speed operation of the system. 1 1.2 Previous work The offshore system considered in this thesis consists of a submersible motor, which is remotely controlled by a VSD through a transmission system. The transmission system consists of a step-up transformer, a long cable and a step-down transformer. A small-scale lab model of this system was designed and implemented as part of a pre- vious work [1]. Computer simulations were also performed during the same work, in order to study the startup performance of the downscaled system under scalar control. A separate task included the setup of the scalar and vector control configurations in the VSD of the lab model. Since the measurements that are presented in this thesis were taken on the small-scale lab model and since the aforementioned drive configurations were used to control the PMSM during the performed experiments, it is clear that the experimental tasks performed in this thesis are based on previously conducted work. The control schemes implemented in this thesis are mainly based on concepts that have been found in literature. The designed models have resulted from the combination of ideas from different papers and the adaptation of these ideas to the peculiarities of the investigated system. 1.3 Purpose and scope The first objective of this thesis is to develop sensorless scalar and vector control models for the studied system and to simulate their performance both during the startup and during high-speed operation of the PMSM. The modeling and the simulations are per- formed with MATLAB and Simulink. Since this work focuses on the development of control schemes, the modeling of the physical components is not handled in detail. The PMSM model is obtained from the SimPowerSystems toolbox and the transmission system parameters are integrated into the motor parameters. The VSD and its output filter are represented by a simplified interface between the control circuit and the power circuit. The second objective of this thesis is to verify the results of the simulations with measure- ments on the lab model of the system. The preparation for these measurements includes the development of a data-acquisition system, which can be used to monitor different quantities during the conducted experiments. The implemented monitoring system is based on LabVIEW and CompactRIO. 2 Chapter 2 Theoretical background This chapter presents some theoretical considerations that are related to the studied sys- tem. The increasingly important role of PMSMs for subsea operations is discussed and different options for the placement of drives in offshore applications are presented. In cases where the submersible motor is controlled by a remote VSD, the combination of the PWM voltage output of the drive and the long cable connection between the motor and the drive introduces certain challenges, which are discussed in this chapter. Finally, some theoretical background about the sensorless scalar and vector control of PMSMs is presented and the combination of these two control types is introduced, as a way to achieve safe motor startup and high control performance during high-speed operation. 2.1 Motors for subsea applications PMSMs have positive features that make them popular in a variety of applications. When it comes to deepwater pumping operations, the advantages of PMSMs become even more significant, as explained in this section. Due to the high deepwater pressures involved, subsea motors typically operate with fluid- filled mechanical gaps. The presence of pressurized fluid between the rotor and the stator helps the motor withstand the high pressure of its subsea surroundings, but also results in increased friction and causes higher drag losses, compared to the case of air-filled gaps. Whereas the electrical losses are dominant in conventional motors with air-filled gaps, this is not the case for motors that operate subsea. The highly effective cooling that is available in subsea applications makes the impact of the electrical losses less significant, compared to that of frictional losses [2]. A typical breakdown of the total losses in this case shows that approximately 70% of them are drag losses, almost 20% are losses from the impeller which circulates the fluid and only 10% are electrical losses [3]. Apparently, the high drag losses pose severe limitations in the operating speed of motors 3 with fluid-filled gaps. Since high speeds are increasingly important for subsea pumping applications [2], it is crucial to decrease the frictional losses in submersible motors. Induction motors have been traditionally used in subsea applications, thanks to their sim- plicity, robustness and low cost. However, as the oil production moves to deeper waters [4], it becomes harder for them to meet the increased speed and efficiency requirements, mainly because their need for a small mechanical gap results in high frictional losses. PMSMs on the other hand, can be constructed with much larger mechanical gaps, which results in higher efficiency and greater speed capabilities. Their decreased sensitivity to larger gaps allows smaller rotor diameters and extra space for the placement of a sleeve in the fluid-filled gap of the motor [2]. Additionally, the absence of rotor windings or conductive bars in PMSMs eliminates the resistive rotor losses. This allows even higher efficiencies and superior thermal perfor- mance compared to induction motors. Apart from their high efficiency and high speed capabilities, PMSMs are characterized by high power density, compact size and low weight [5, 6]. Their construction is simple and the absence of brushes results in higher robustness and lower maintenance needs [7]. On the negative side, the high cost of the permanent magnets increases the overall cost of PMSMs [8]. Moreover, the risk of magnetic property loss makes the reliability of this motor type questionable under certain circumstances [6]. 2.2 Drives for submersible motors The use of VSDs for the control of submersible motors improves the reliability and the efficiency of pumping operations. This section presents different options regarding the placement of the drive and discusses some problematic issues that arise from the combination of the PWM voltage generated by the VSD and the long cable that connects the motor and the drive. 2.2.1 Placement of the drive The VSDs that are used for the control of submersible motors can be placed either on the seabed, in the vicinity of the controlled motor [9], or on a platform, which might be located several kilometers away from the actual pumping activity [10, 11]. Placement in low-pressure subsea enclosures In the first case, the drive is enclosed in a vessel, which is installed subsea. Since power electronic converters that are based on present technology cannot withstand high-pressure environments, the interior of the drive vessel needs to be at low pressure [12]. 4 Due to the high deepwater pressure, the drive enclosure in this case must be able to with- stand a significant pressure difference between its interior and its subsea surroundings. For this reason, the walls of the used vessel need to be thick and heavy. Naturally, the bulky design of the enclosure results in an increase in the size and cost of the drive system, but also in problematic heat dissipation [12]. Due to the thick walls of the vessel, the heat conduction from the power electronic components to the sea water becomes more difficult, which results in higher operating temperatures for the drive. Placement on the platform The second option is to install the converter on a platform. Although the topside envi- ronment eliminates the necessity of a large and costly enclosure, it introduces the need for a long electrical connection between the VSD and the controlled PMSM [10, 11]. In order to decrease the transferred current and thus reduce the transmission losses and the voltage drop in the cable connection, higher transmission voltage levels are needed for longer step-out lengths. For this reason, a topside transformer can be used to step up the voltage output of the VSD. In this case, either the submersible PMSM needs to operate at the voltage levels of the transmission system [1], or an additional transformer needs to be installed on the seabed, so that the transmitted voltage can be stepped down before reaching the motor. In the framework of this thesis, the latter alternative is considered. A potential drawback of this solution is that the control of the PMSM through the transmission system could introduce new filtering requirements [13]. More specifically, the transmission of the PWM pulses of the drive through the long cable could subject the windings of the PMSM to significant electrical stress, if the necessary measures are not taken. This issue is discussed in greater detail in Section 2.2.2. Development of pressure-tolerant power electronics The development of pressure-tolerant power electronics could facilitate the possibility of placing the drive in subsea enclosures of reasonable size and cost. The vessel in this case could be filled with a pressurized insulating liquid, which would improve the heat dissipation and would eliminate the differential pressure that the vessel would otherwise have to withstand. The insulating liquid must protect the converter components from mechanical damage and electrical flashovers. It should have good dielectric and thermal properties, in order to provide reliable insulation and effective heat removal. It should be uncompressible and should have a uniform and stable structure, which would not cause any chemical reactions or change its character for any reason [12, 14]. A dielectric liquid, which has been tested for the described purpose and appears to have several desirable features, is Midel 7131. Its properties include high breakdown strength, low thermal expansion, high temperature stability, biodegradability and non-toxicity [14]. 5 A strategy that has been proposed for the development of power electronic components that could withstand the high pressure of the insulating liquid is the removal of the in- sulating gas that originally surrounds the converter chips and its replacement with an uncompressible liquid. This liquid should ideally fill every void in the converter [12]. Although the development of pressure-tolerant power electronics could enable the place- ment of the VSD in high-pressure submersible enclosures in the future, such a solution is not an option for present applications. 2.2.2 Effects of PWM voltage The PWM voltage produced by the topside VSD is transmitted through the cable to the submersible PMSM. The high-frequency voltage pulses generated by the inverter could have detrimental effects on the motor and the cable. As discussed in this section, the effects of the PWM voltage may be quite severe in the in- vestigated application, due to the long transmission distance and the subsea environment of the cable. Impact of high-order harmonics Compared to a purely sinusoidal voltage excitation, the PWM pulses that are supplied to the motor result in increased losses, mechanical vibrations, noise generation, higher electromagnetic emission, more severe insulation stress, and undesirable currents induced to the shaft and bearings of the motor [15, 16]. The high-order harmonics of the PWM voltage induce high-frequency currents and thus create minor hysteresis loops in the steel of the motor. These currents are the cause of additional losses, which reduce the efficiency of the system and result in higher operating temperatures. Furthermore, the interaction of the induced currents with flux harmonics may result in stray forces that cause mechanical vibrations and increase the generated noise [15]. Increasing the switching frequency of the drive is a way to reduce the harmonic content of the motor currents and thus mitigate the negative effects of the high-frequency current components. However, this also results in higher switching losses in the drive and increases the possibility of voltage overshoots in the motor terminals [17]. Voltage overshoots at the terminals of the motor The high-frequency voltage pulses generated by the VSD cause travelling waves in the transmission cable between the drive and motor. When each travelling wave reaches the PMSM, voltage reflection occurs, due to the mismatch between the cable impedance and the motor impedance [18]. The magnitude of the reflected voltage depends on the cable length, the rise time of the pulses, the characteristic impedances of the motor and the cable, the propagation velocity 6 of the waves and the dielectric medium surrounding the cable [13]. Shielded cables and cables that are submersed in water have significantly higher capaci- tance than cables in air. This results in lower characteristic impedances for submersible cables and therefore a more significant mismatch between the cable impedance and the motor impedance [13]. Since the impedance of the motor is much higher than the one of the cable, the reflected wave is expected to be almost equal in magnitude to the incident wave. Since the voltage at the motor terminals is equal to the sum of the two waves, its magnitude is expected to reach twice the magnitude of the VSD voltage [19]. Due to the occuring voltage reflection phenomena, when the distance between the motor and the drive is long, as in the case of the investigated application, the voltage pulses at the motor terminals may be different than the ones generated by the drive. More specifically, the distributed inductance and capacitance of the cable result in high- frequency voltage oscillations at the motor end of the cable. These oscillations are usually referred to as ’voltage ringing’ [18]. Moreover, the high voltage derivatives experienced by the PMSM during PWM operation cause the voltage peaks to be unevenly distributed across the motor windings. In this case, the largest portion of the supplied voltage may appear between the first turns of the windings, thus causing the insulation of these turns to experience higher stress and faster degradation [17]. It can be concluded that the combination of voltage reflection and voltage ringing can cause the motor to experience voltages higher than twice the DC-link voltage of the VSD [17], while the high voltage derivatives can subject the motor windings to additional elec- trical stress. If these phenomena are not taken into account during the design of the system, the repeated overvoltages can cause significant stress to the insulation of the motor and eventually reduce its lifetime. Cable-charging current pulses Every time the voltage output of the VSD changes, the distributed cable capacitance must be charged or discharged. This results in pulses of charging current that not only increase the losses of the system, but can also cause overcurrent problems in the inverter and affect the control performance of the drive [18]. The magnitude of the charging current is proportional to the derivative of the voltage supplied by the drive and to the cable capacitance. Since PWM pulses are characterized by rapid voltage variations and since the capacitance of submersible cables is large, the charging currents are expected to be high in the investigated application. 7 The frequency of the cable-charging current pulses depends on the frequency of the voltage pulses generated by the inverter. Therefore, for higher switching frequencies of the VSD, the distributed cable capacitance is charged and discharged more often, which increases the transmission losses of the system. Mitigation of the PWM effects Based on the aforementioned considerations, the combination of PWM operation and the long cable connection between the VSD and the PMSM can potentially cause significant stress to the insulation of the motor and decrease the efficiency of the system. These negative issues can be mitigated by filtering the voltage that is generated by the VSD, or by eliminating the impedance mismatch at the end of the cable. There are dif- ferent types of filters that can be used for these purposes, the most common ones being output line inductors, output limit filters, sine wave output filters and motor termination filters [18]. The cost and the effectiveness of the aforementioned solutions vary. The use of sine wave filters has been suggested in several papers [13, 16, 19], as a way to eliminate the high-order harmonics of the PWM pulses and therefore supply the motor with almost sinusoidal voltage. The advantages of this solution include the absence of transient overvoltages at the termi- nals of the motor, the elimination of power losses due to harmonic currents, the reduction of motor noise and the decrease of electromagnetic emission [19]. Moreover, when applications with submersible motors and long step-out distances are considered, the use of sine wave filters is expected to eliminate the negative phenomena that are associated with voltage reflection and high voltage derivatives [13]. A sine wave filter is essentially a LC filter, whose resonance frequency is much lower than the lowest harmonic frequency of the inverter voltage and much higher than the fundamental frequency of the system [16]. An alternative solution, which has been introduced as an effort to eliminate the require- ment for filters, is the use of a multilevel inverter in the VSD [10]. The special topology of this inverter results in a significant reduction in the high-order harmonics at the out- put of the drive and eliminates several problems that are associated with common PWM voltage pulses. 2.3 Control of permanent magnet motors Depending on the requirements of each application, different methods can be used to control PMSMs. This section introduces scalar control as a simple control method, which is suitable for low-cost drive systems, and vector control as a more advanced option, which is well-suited for applications that demand higher dynamic performance. 8 2.3.1 Scalar control In drive systems where simple, low-cost control is desired and where reduced dynamic performance is acceptable, open-loop control methods can be used. Typical applications of such systems include pump and fan drives [20]. Open-loop control methods (or scalar control methods, as they are often called) exist in different variations, which include V/f schemes [21] and I-f schemes [22]. Despite their simplicity and their ability to operate over a wide speed range, it has been found that the performance of open-loop methods often depends on the motor parameters and the load conditions of the system. Such methods can experience power swings within specific speed ranges, which might cause the motor to lose synchronism [5]. Furthermore, the behaviour of some open-loop schemes is heavily dependent on the se- lected parameters of the controller. The selection of the control settings for these schemes is often based on a trial-and-error approach and is therefore quite time-consuming [21]. The term ’open-loop’ often refers to the fact that no speed or position feedback is needed for the operation of these schemes. In this thesis however, this term is used to denote that neither electrical nor mechanical feedback is required by a controller. For instance, a scalar control method which uses current feedback has been presented in [23]. Although this method does not require any position or speed measurements, it is called ’closed-loop’ in this thesis, in order to differentiate it from schemes with no feedback at all. 2.3.2 Vector control For more advanced drive systems, which require higher dynamic performances, vector control is a more appropriate option than scalar control. Demanding applications that need vector control can be found, for instance, within the automotive industry [20]. Vector control allows the torque and the flux of the PMSM to be controlled separately from each other, through a control structure which is similar to that of a separately excited DC machine [20]. This decoupled control results in the precise and efficient reg- ulation of the motor. However, a major issue with vector controllers is that their operation requires informa- tion about the rotor position and the speed of the PMSM. The most direct approach for obtaining this information, is the use of mechanical sensors on the shaft of the PMSM [24]. In many applications however, the presence of mechanical sensors is undesirable or un- acceptable, since it increases the cost and the complexity of the system [25]. Numerous position-estimating techinques have been developed, as an effort to eliminate the need for mechanical sensors. The position estimation can be based, among others, on the flux linkage, the back-EMF or the inductance of the PMSM [26, 27]. The effec- 9 tiveness of these techniques is not universal, but depends on the motor topology and the application requirements. Several proposed vector control schemes utilize flux-linkage-based estimation techniques [5, 21]. Such a technique has been described in [26] and is implemented in this thesis, after being adapted to the peculiarities of the investigated system. Variations of the implemented estimating algorithm have also been presented in [25, 28, 29]. Field-weakening algorithms are often integrated into vector control schemes, in order to allow the PMSM to operate above its rated speed. Several papers have presented theoretical considerations on field-weakening [30, 31] and have applied different field- weakening strategies [32, 33]. A field-weakening algorithm, which has been derived in [34], is implemented and tested in this thesis. 2.3.3 Combination of scalar and vector control A problem with most position-estimating techniques is their inability to produce accu- rate estimates at zero speed. Due to this problem, the startup of PMSMs with position- sensorless vector controllers is often problematic, unless the initial rotor position is known [29]. A solution would be to bring the rotor to a known position before the actual accelera- tion [25]. This could be achieved by injecting a proper DC current into the windings of the PMSM, thus forcing the rotor to align in the desired direction. For the investigated system however, such a solution is not acceptable, since the presence of the transformers does not allow the injection of DC currents. Due to the startup issue of position-sensorless vector controllers, scalar control schemes are often used for the initial acceleration of PMSMs [5, 22]. Provided that their control parameters are properly set, these schemes should be able to accelerate the PMSM for every initial rotor position [29]. After a certain speed is reached, the scalar controller can be dismissed and the position-sensorless vector controller can be deployed. This well-known strategy is also applied in this thesis. Two V/f control alternatives are presented for the startup of the PMSM and a position-sensorless vector controller is implemented and tested for higher speeds of the motor. A vector control scheme with mechanical sensors is also designed, as an intermediate step before the implementation of the sensorless controller. 10 Chapter 3 System description This chapter describes the models that have been implemented in this thesis and the considerations that determined their design. The described models correspond both to the power components and the drive controller of the studied system. The considered power system consists of a PMSM, which drives a multiphase pump and which is fed by a remote VSD through a transmission system. The transmission system consists of a step-up transformer, a long cable and a step-down transformer. The topol- ogy of the system is shown in Fig. 3.1. Step-up transformer M P VSD Power source PMSM Multiphase pump Cable Step-down transformer SubseaTopside Figure 3.1: Topology of the investigated power system, consisting of a PMSM, which is con- nected to a pump and is fed by a remote VSD through a transmission system. The controller of the VSD determines the voltage output of the drive and therefore the voltage input of the motor. The application of proper control schemes is essential, in order to achieve the required performance during different types of operation. Two different V/f control models, an open-loop and a closed-loop model, are implemented for the initial acceleration of the PMSM and are presented in this chapter. The former one is simpler but the latter one provides higher control performance. Moreover, two different vector control models are designed for higher operating speeds of the PMSM. In the first model a position sensor is present, while in the second model this sensor is eliminated, in order to decrease the cost and increase the robustness of the 11 system. 3.1 Permanent magnet synchronous motor This section presents the equations which dictate the electrical and mechanical perfor- mance of the PMSM and provide the basis, not only for the implementation of the motor model, but also for the design of its control. 3.1.1 Electrical equations It is generally convenient to express the electrical equations of the PMSM in the dq ref- erence frame [35]. By using this representation, the steady-state AC quantities of the motor are transformed into DC values, which are easier to analyze and control. The space vector of the stator voltage is denoted as ūs and is given by ūs = Rsīs + dψ̄s dt + jωrψ̄s (3.1) where Rs is the stator resistance, īs is the space vector of the stator current, ψ̄s is the space vector of the stator flux linkage and ωr is the electrical speed of the motor. The stator flux linkage ψ̄s is obtained from ψ̄s = Lsdisd + jLsqisq + Ψm (3.2) where Ψm is the amplitude of the flux linkage of the permanent magnets, Lsd and Lsq are the d-axis and q-axis stator inductances and isd and isq are the d-axis and q-axis currents respectively. It can be observed from (3.2) that Ψm lies in the d direction, thus the magnetic axis of the PMSM is the same with the d axis of the selected dq frame. This is achieved by using a proper angle θr for the dq tranformations. The back-EMF of the PMSM depends on the flux linkage Ψm and the speed ωr. Its space vector ē is given by ē = jωrΨm (3.3) By combining (3.1) and (3.2), the d-axis and q-axis stator voltages, denoted as usd and usq respectively, can be obtained as usd = Rsisd + Lsd disd dt − ωrLsqisq (3.4) usq = Rsisq + Lsq disq dt + ωrLsdisd + ωrΨm (3.5) where the term ωrΨm represents the magnitude of the back-EMF vector ē, or equiva- lently, the amplitude of the three-phase back-EMF. It is important to note that, since 12 amplitude-invariant transformations are used in this thesis, the magnitudes of space vec- tors correspond to the amplitudes of three-phase quantities. Next, the electromagnetic torque Te of the motor can be calculated from Te = 3np 2 Im[ψ̄∗s īs] = 3np 2 [Ψmisq + (Lsd − Lsq)isdisq] (3.6) where np is the number of pole pairs. The values of the inductances Lsd and Lsq depend on the geometry of the motor and the placement of the permanent magnets [8, 36]. PMSM topologies with inset-mounted or interior-radial magnets are characterized by dq saliency, which results in a difference between Lsd and Lsq. On the other hand, in motors with surface-mounted magnets, the saliency is often negligible and the two inductances can be considered to be equal, thus Lsd = Lsq = Ls. In the latter case, (3.6) is simplified to Te = 3np 2 Ψmisq (3.7) The inductances Lsd and Lsq of the PMSM considered in this thesis differ slightly from each other. However, for the sake of simplicity, the difference between them is neglected and the simplified expression (3.7) is used for the calculation of the electromagnetic torque. It is interesting to observe that the torque in (3.7) is proportional to the current isq and independent of isd. The current isd on the other hand, has a more significant effect on the stator flux linkage magnitude than isq according to (3.2), since the first one lies on the same axis with the main flux linkage component Ψm, while the second one lies on a perpendicular axis. 3.1.2 Mechanical equations The equation that dictates the mechanical performance of the PMSM, by relating the electromagnetic torque Te, the load torque TL and the electrical speed ωr, is J np dωr dt = Te − TL (3.8) where J is the moment of inertia of the system. The relation between the electrical speed and the electrical rotor position is given by dφr dt = ωr (3.9) where φr corresponds to the angle between the magnetic axis of the motor and the a axis of the three-phase system. In order to achieve perfect-field orientation, thus in order to align the magnetic axis of the PMSM with the d axis of the dq frame, the electrical rotor angle φr needs to be equal 13 to the selected dq-transformation angle θr, thus φr = θr. The motor model which is used in this thesis is based on the electrical equations (3.4) - (3.6) and the mechanical equations (3.8) - (3.9). 3.2 Transmission system The transmission system in the studied application consists of a step-up transformer, a cable and a step-down transformer. In order to reduce the computational complexity of the simulations and to simplify the selection of the control settings, the electrical parameters of the transmission components are integrated into the respective parameters of the PMSM. 3.2.1 Simplifying assumptions The aforementioned treatment of the transmission system is facilitated by two basic as- sumptions. Firstly, it is assumed that the short-line model can represent the cable of the system with adequate accuracy, thus the shunt capacitance of the cable can be ignored, without introducing any significant error. Under this assumption, which is generally valid for lines with length up to 80 km [37], the transmission cable can be modelled as an inductance LC connected in series with a resistance RC . Secondly, it is assumed that none of the two transformers experience saturation, thus they are both considered to remain in the linear magnetic region under all operating conditions of the system. Under this assumption, the magnetizing branch of their equivalent circuit can be ignored and each transformer can be modelled as an inductance LT connected in series with a resistance RT . 3.2.2 Considerations on transformer saturation In general, the core of a transformer gets saturated when the applied V/f ratio becomes too high. In such a case, the rise in the magnetic flux linkage of the transformer is accom- panied by a disproportionally large increase in its magnetizing current and, therefore, a significant decrease in its magnetizing inductance. Clearly, saturation is an undesirable condition for transformers and should be avoided by dimensioning their core properly, so that they remain in the linear magnetic region even for the maximum applied V/f ratio. The aforementioned requirement bears particular significance for the studied application. As discussed in Section 3.3.3, the system has to operate at a high V/f ratio during the initial stage of the PMSM startup. This may lead to the inconvenient necessity of over- sizing the transformer core for the sake of a few seconds of low-frequency operation. 14 The problem applies mainly to the topside step-up transformer, since the V/f ratio that is experienced by the subsea step-down transformer is lower, due to the voltage drop in the cable. In order to minimize the size and cost of the topside transformer, it is necessary to optimize the control of the PMSM during the early stages of its startup, thus to find the minimum V/f ratio that can safely accelerate the motor. 3.2.3 Introduction of equivalent quantities By neglecting the shunt capacitance of the cable and the magnetizing branch of the transformers, the transmission system can be simply modelled as a series RL circuit. Its resistance Rtr and and its inductance Ltr can be written as Rtr = RC + 2RT (3.10) Ltr = LC + 2LT (3.11) respectively. Since the transmission system is connected in series with the stator windings of the PMSM, its parameters can be integrated into the motor parameters. In this case, the combination of the actual PMSM and the transmission system can be regarded as an equivalent PMSM, the parameters of which can be written as R′s = Rs +Rtr (3.12) L′s = Ls + Ltr (3.13) where R′s is the equivalent stator resistance and L′s is the equivalent stator inductance. The control of the actual motor through the transmission system can then be regarded as direct control of the equivalent motor. The implemented controllers compensate for the voltage drop in the transmission system, by considering the equivalent stator parameters R′s and L′s, an equivalent stator voltage ū′s and an equivalent stator flux linkage ψ̄′s, whenever needed. Having integrated the transmission system parameters into the motor parameters, the actual voltage ūs of the PMSM can be calculated by subtracting the voltage drop in the transmission impedance from the equivalent voltage ū′s, which is the voltage produced by the VSD. The voltage drop, in turn, can be easily obtained when the transmission parameters Rtr and Ltr and the stator current īs are known. The system should be designed in a proper way, so that a specified steady-state voltage drop occurs in the transmission system. In the actual application, where the transmission distance is fixed, this could be achieved by selecting an appropriate level for the operating voltage and a cable with suitable parameters. 15 In the performed simulations on the other hand, the voltage level and the transmission system parameters are considered to be fixed and the desired voltage drop is obtained by selecting the proper cable length. 3.3 Open-loop V/f control This section discusses the basic idea of V/f control, explains the need for a low startup frequency and introduces a voltage-boosting factor for low-speed operation of the PMSM. It also describes the implemented open-loop V/f regulator, which is deployed to accelerate the motor from standstill up to a certain speed level. The term ’open-loop’ refers to the fact that no feedback is received by the controller and therefore no current or speed measurements are needed for its operation. 3.3.1 Basic idea of the V/f controller The fundamental idea of V/f control can be demonstrated by considering (3.1) under the assumptions that the stator flux linkage is constant and that the resistive term is negligible. This gives ūs ' jωrψ̄s ⇒ ψ̂s ' ûs ωr ⇒ ψ̂s ' ûs 2πf (3.14) where f is the electrical frequency and ψ̂s and ûs are the magnitudes of the flux linkage vector and the voltage vector respectively, thus the amplitudes of the respective three- phase quantities. Equation (3.14) shows that in order to keep the stator flux linkage approximately con- stant, the amplitude of the supplied stator voltage must be varied proportionally to the electrical frequency. With the exception of field-weakening operation, it is generally desirable to maintain the value of the stator flux linkage around its nominal level, which is approximately equal to the rated voltage over the rated speed [20]. For high values of the V/f ratio, the motor becomes overexcited and a rise in the current isd occurs, according to (3.2). A low V/f ratio on the other hand, causes the motor to experience underexcitation, which is associated with a negative isd. Both overexcited and underexcited states are accompanied by a rise in the stator current, as a result of the increased magnitude of its d-axis component. Since the current isd does not contribute to any torque production, according to (3.7), the increase of its magnitude is translated into a rise in power losses. Although this might be tolerated in the special case of field-weakening operation, it is in general clearly undesirable. 16 3.3.2 Necessity of low initial frequency During the initial stage of its startup, the PMSM tries to establish synchronism with the supplied magnetic field. For high values of the applied frequency however, the rotor might be unable to follow the fast rotation of the field, leading to an unsuccessful startup with rotor vibrations. In order to avoid scenarios of unsuccessful startup, it is necessary to supply the motor with low frequency in the beginning. Then, as the rotor accelerates, the supplied frequency can be gradually increased up to its steady-state value. In order to safeguard the stability of the system, the rate at which the frequency is increased should be kept adequately low [7]. According to the discussion in Section 3.3.1, in order to achieve approximately constant stator flux linkage and therefore avoid the undesirable states of overexcitation and under- excitation, the increasing frequency must be accompanied by an almost proportionally increasing voltage. 3.3.3 Voltage boosting at low speeds According to (3.14), the applied V/f ratio must be approximately equal to the rated voltage over the rated speed of the motor. However, it must be borne in mind that (3.14) was derived from (3.1) under the assump- tion that the resistive term of the equation is negligible. This assumption is not valid for very low speeds, when the speed-dependent term is comparable to the resistive voltage drop in the stator. This issue is handled by using a voltage-boosting factor for electrical speeds below a low critical value ωr,cr. This factor, whose mission is to compensate for the resistive drop at low frequencies, is denoted as Fb and is defined as Fb = îs,ratedR ′ s + ωr,crΨm ωr,crΨm (3.15) where îs,rated is the peak value of the rated stator current. According to (3.3), the term ωr,crΨm represents the back-EMF of the motor at ωr = ωr,cr. The numerator of (3.15), consisting of the back-EMF term and the resistive drop in the equivalent PMSM (calculated for the rated stator current), is a rough approximation of the needed voltage amplitude at ωr = ωr,cr. When the electrical speed ωr is below its critical value ωr,cr, the voltage-boosting factor Fb is included in the calculation of the stator voltage amplitude reference u∗s according to û∗s = ω∗rFbΨm ⇒ û∗s = 2πf ∗FbΨm (3.16) where f ∗ is the frequency reference and ω∗r is the corresponding electrical speed reference. 17 Clearly, this method of compensating for the resistive voltage drop at low speeds is ap- proximate, mainly because the stator current used in (3.15) is considered to be constant and equal to îs,rated. More accurate compensation can be achieved by measuring the actual stator currents and including them in the calculation of the stator voltage. Such an improved method is discussed in the section of the closed-loop V/f controller. 3.3.4 Simplifying assumption For the sake of simplicity, it has been assumed during the design of the different models in this thesis that the voltage at the output of the VSD is sinusoidal and equal to the three-phase voltage reference produced by the implemented controllers. In practice, the three-phase reference signal at the output of the controller enters a PWM stage, where it is compared with a carrier wave. The result of the comparison determines the form of the voltage pulses that are generated by the VSD. In the designed models however, the PWM stage is not taken into account. Instead, a simplified interface between the control circuit and the power circuit is used to transform the voltage reference produced by the controller into a power-level voltage of the same form. The aforementioned simplification can be justified by the following considerations. By assuming that the voltage output of the VSD is properly filtered, according to the discussion in Section 2.2.2, the high-order harmonics of the PWM pulses produced by the inverter can be neglected and only the fundamental component of the VSD output can be taken into account during the design of the control models. Since the form of the fundamental component for each phase is expected to match the form of the respective voltage reference provided by the controller [38], the latter one can be transformed into a power-level signal, which can be applied directly at the output of the VSD. 3.3.5 Implementation of the controller The mission of the open-loop V/f controller is to supply the PMSM with a voltage of proper magnitude and frequency, so that successful acceleration from standstill up to a desired speed level is achieved. Since no feedback is received by the controller in the open-loop scheme, the voltage out- put of the VSD is pre-determined and is not affected by the actual response of the PMSM. As shown in Fig. 3.2, the calculation of the output voltage of the controller is a straight- forward process, which consists of three steps. 18 f * t f * f * fcr f * y=sinx, x∊[0,2π] y=-sinx, x∊[0,2π] O 1 -1 π/2 π 2 π 3π/2 y x t 1/f * sû * sû * sû * su * su * Figure 3.2: Block diagram of the open-loop V/f controller The first step is the generation of a frequency reference ramp, the slope of which is deter- mined by the desired acceleration of the PMSM, but also by the stability requirements of the system. It is important to ensure that the increase-rate of the frequency reference curve is not too high, otherwise the synchronism of the PMSM may be at risk. The second step is the calculation of the voltage amplitude reference, according to pre- defined V/f ratios. When the frequency reference f ∗ is below a critical value fcr (corresponding to the critical electrical speed ωr,cr), voltage boosting is necessary and the voltage amplitude reference is calculated according to (3.16). On the other hand, when the frequency reference has exceeded the critical value fcr, the V/f controller calculates the voltage amplitude reference û∗s from û∗s = û′s,rated − û∗s(f∗=fcr) frated − fcr f ∗ (3.17) where û′s,rated is the peak value of the equivalent rated stator voltage, frated is the rated frequency and û∗s(f∗=fcr) is the voltage amplitude reference at the critical frequency (cal- culated from (3.16)). The value û′s,rated represents the voltage output of the VSD which corresponds to the rated voltage ûs,rated of the PMSM, increased by the desired voltage drop in the transmission system. The third step for the calculation of the output voltage of the controller is to insert the generated frequency reference and the calculated voltage amplitude reference into the sinusoidal equations that produce the three-phase voltage output of the controller. These equations are written as u∗sa(t) = û∗s(t) cos θ∗r(t) u∗sb(t) = û∗s(t) cos[θ∗r(t)− 120o] u∗sc(t) = û∗s(t) cos[θ∗r(t) + 120o] (3.18) where u∗sa, u ∗ sb and u∗sc are the reference voltages for the three phases and θ∗r is the electrical angle reference, which is given by 19 θ∗r(t) = ∫ ω∗r(t)dt = ∫ 2πf ∗(t)dt (3.19) The implemented open-loop V/f control scheme is based on (3.15) - (3.19). Its most important advantages are the simplicity of its control algorithm and the absence of feed- back, which eliminates the need for current, speed and position sensors. On the negative side, the control algorithm is based on rough approximations, which may have a negative effect on the performance of the controller. 3.4 Closed-loop V/f control A second V/f control scheme has been implemented, based on a method presented in [23], and is discussed in this section. Unlike the open-loop method which was presented in Section 3.3, this scheme uses current feedback to determine the voltage reference with higher accuracy. A built-in stabilizer is included in the closed-loop controller, in order to ensure that the PMSM does not lose synchronism. 3.4.1 Voltage reference calculation The calculation of the voltage reference of the closed-loop V/f controller is based on (3.1). Considering steady-state operation, the space vector ū′s of the equivalent stator voltage is given by ū′s = R′sīs + jωrψ̄′s (3.20) where ψ̄′s is the space vector of the equivalent stator flux linkage. The magnitude û′s of the voltage vector can be obtained by algebraically adding the projections of the terms R′sī and jωrψ̄′s in the direction of ū′s [23]. This gives û′s = R′sîs cosφ0 + √ (ωrψ̂′s) 2 + (R′sîs cosφ0)2 − (R′sîs) 2 (3.21) where îs is the peak value of the stator current and cosφ0 is the power factor. As was discussed in Section 3.1, the d-axis current isd usually has a more significant effect on the magnitude of the stator flux linkage than the q-axis current isq. By neglecting the q-axis current term in (3.2), it can be observed that when the d-axis current isd is set to zero, the magnitude ψ̂′s of the equivalent stator flux linkage vector is equal to the flux linkage Ψm of the permanent magnets. Since isd does not contribute to any torque production, the aforementioned condition yields the most efficient operating point of the PMSM, thus the point at which the torque-to-current ratio is maximum. Based on this statement, the stator flux linkage ψ̂′s in (3.21) can be set equal to Ψm, so that approximately optimal efficiency is achieved. The voltage amplitude reference û∗s can then be calculated from û∗s = R′s(̂is cosφ0) + √ (ω∗rΨm)2 + [R′s(̂is cosφ0)]2 − (R′sîs) 2 (3.22) 20 It should be borne in mind that the accuracy of (3.22) depends on the validity of the assumption that the effect of isq on the stator flux linkage is negligible. This assumption is weaker for motors with high q-axis inductance Lsq or high torque output (therefore high isq according to (3.7)). Even though, according to the presented derivation, the current amplitude îs and the power factor cosφ0 in (3.22) are steady-state quantities, their instantaneous values can be considered during the calculation of the voltage amplitude reference û∗s [23]. Assuming balanced operation of the system, these values can be obtained by perform- ing current measurements in two of the three phases of the PMSM. The stator current magnitude îs is then given by îs = √ 1 3 (isa + 2isb)2 + i2sa (3.23) where isa and isb are the measured phase currents. The term îs cosφ0 is calculated as îs cosφ0 = 2 3 [isa cos θ∗r + isb cos(θ∗r − 120o)− (isa + isb) cos(θ∗r + 120o)] (3.24) where θ∗r is the electrical angle reference, the calculation of which is discussed in Section 3.4.4. 3.4.2 Need for stabilization A problematic issue that is commonly associated with V/f control schemes is their prone- ness to instability, thus the tendency of the controlled PMSMs to lose synchronism within specific speed ranges [24, 26]. The instability phenomena are accompanied by power and speed oscillations, which result in the inability of the motor to stay synchronized with the rotating magnetic field of its stator. In order to safeguard the stability of the system, the settings and the parameters of the V/f controller must be selected carefully. For instance, it is important to choose a proper slope for the speed reference curve, since too high increase rates may prevent the PMSM from establishing synchronism. Furthermore, the transitions between the different intervals of the speed reference curve should be as smooth as possible, since the existence of sharp edges might cause overcur- rents and loss of synchronism [21]. However, even if the control parameters are selected properly, the performance of the system is still dependent on the motor parameters and on the load conditions [5]. For the closed-loop V/f control scheme which has been implemented in this thesis, it has been previously found that when a certain applied frequency is exceeded, the control 21 poles of the rotor pass into the instability region of the s-plane and synchronism is lost [23]. In order to improve the stability of the system, PMSMs are sometimes designed with damper windings in their rotor. However, since this solution increases the manufacturing costs and complicates the motor construction, a more convenient way to stabilize the motor is needed. A more flexible solution is to add damping to the system, not by modifying its physical topology, but by including a stabilizing algorithm in the V/f controller. 3.4.3 Basic idea of the stabilizer A stabilizing loop has been implemented according to [23] and has been included in the closed-loop V/f controller, in order to prevent the PMSM from losing synchronism. The mission of the stabilizing loop is to provide extra damping to the system, so that the con- trol poles of the PMSM are kept in the stable region for the whole applied frequency range. If a control scheme with speed sensors was considered, the stabilizing loop would coun- teract the perturbations in the measured speed, by modulating the applied frequency. By adjusting the electrical excitation of the motor according to the mechanical response of the PMSM, the stabilizer would contribute to the attenuation of the mechanical oscil- lations and would help the motor stay in synchronism. Of course, since the implemented control scheme is position- and speed-sensorless, the operation of the stabilizing loop cannot depend on speed measurements. However, based on the observed relation between the speed perturbations and the result- ing power oscillations, the stabilizer can utilize the available current measurements, in order to calculate the power perturbations and modulate the applied frequency accord- ingly. 3.4.4 Implementation of the stabilizer The operation of the implemented stabilizing loop relies on the approximately linear re- lation between the speed perturbations of the PMSM and the resulting power oscillations [23]. Using the term îs cosφ0, which is calculated from (3.24), and the voltage amplitude reference û∗s, which is given by (3.22), the electric power p′e of the equivalent motor can be obtained from p′e = 3 2 û∗s îs cosφ0 (3.25) In order to extract the power perturbations ∆p′e from the calculated power p′e, a high-pass filter is used. Based on the obtained value of ∆p′e, the stabilizer produces a frequency modulation signal ∆ω∗r , which is obtained from 22 ∆ω∗r = −kl∆p′e (3.26) where kl is the speed-dependent gain of the stabilizer, given by kl = cl ω∗r (3.27) where cl is a constant, which is determined by trial and error. The block diagram of the stabilizer is shown in Fig. 3.3. Electric power calculator HPF Gain calculator Calculator of the modulation signal sû * s 0î cosφ rω * eΔp ep  lk rΔω * Figure 3.3: Block diagram of the stabilizer During the early stages of the startup, thus when the PMSM operates at very low speeds, (3.27) gives a large value for the gain kl, which might result in problematic operation of the stabilizing loop. Considering that the implemented control method experiences stability issues only when a certain frequency is exceeded (as mentioned in Section 3.4.2), the stabilizer can be disabled at very low speeds, without any risks for the synchronism of the PMSM. The frequency modulation signal ∆ω∗r , which is calculated from (3.26), is included in the calculation of the electrical angle reference θ∗r according to θ∗r(t) = ∫ [ω∗r(t) + ∆ω∗r(t)]dt = ∫ [2πf ∗(t) + ∆ω∗r(t)]dt (3.28) To summarize, the stabilizer detects possible oscillations in the electric power of the system and modulates the frequency reference f ∗ (or, equivalently, the electrical angle reference θ∗r) of the controller, by producing a signal which opposes the detected oscilla- tions. 3.4.5 Structure of the controller Compared to the structure of the open-loop controller, which was described in 3.3.5, the implementation of the closed-loop V/f control scheme is significantly more complex, as can be observed in Fig. 3.4. 23 f * t f * Position calculator calculator calculator Voltage amplitude calculator y=sinx, x∊[0,2π] O 1 -1 π/2 π 2 π 3π/2 y x t Stabilizer f * f * rΔω * rθ * s 0î cosφs 0î cosφ sî sî si rθ * s 0î cosφ sû * rΔω * sû * su * su * Figure 3.4: Block diagram of the closed-loop V/f controller Its operation relies on stator current measurements and includes an increased amount of calculations, the aim of which is to produce an accurate voltage reference and to safe- guard the stability of the system. Similarly to the open-loop scheme, a frequency reference ramp is initially generated by the controller. In order to eliminate sharp edges from the transition intervals of the ramp, a low-pass filter is used to smoothen the curve. According to the discussion in Section 3.4.2, the generation of a smooth frequency reference f ∗(t) is expected to have a positive effect on the performance of the controller. Based on the generated frequency signal f ∗, the electrical angle reference θ∗r is obtained. Together with the measured stator currents isa and isb, the angle θ∗r is used to calculate the current terms îs and îs cosφ0 from (3.23) and (3.24) respectively. In order to eliminate the high-frequency ripple in îs and îs cosφ0, two low-pass filters are needed. Next, the current terms are used for the determination of the voltage amplitude reference û∗s from (3.22). A voltage limiter has been included in the controller, in order to ensure that the calculated value of the voltage amplitude reference is within acceptable limits. Eventually, the electrical angle reference θ∗r and the voltage amplitude reference û∗s are inserted into (3.18), so that the three-phase voltage reference of the controller is obtained. By using the generated frequency reference f ∗, the calculated term îs cosφ0 and the produced voltage amplitude reference û∗s, the stabilizer generates a frequency modulation signal ∆ω∗r from (3.26). This signal is added on top of the generated speed reference when calculating the angle θ∗r according to (3.28). A stabilizing loop is therefore formed, the mission of which is to protect the PMSM from losing synchronism. 24 3.4.6 Comments on the controller Despite its increased complexity, the designed closed-loop V/f controller is expected to provide higher control precision, compared to the open-loop scheme. The need for current measurements is not a problem, since the availability of current sensors is necessary for the vector controller anyway. Having the measured stator currents at its disposal, the controller is capable of taking the resistive voltage drop of the PMSM stator into account, when calculating the voltage reference. This eliminates the need for a voltage-boosting factor for the initial stage of the PMSM startup, in other words, voltage boosting is inherent in the closed-loop control method. In contrast to the open-loop scheme, where the frequency reference is fixed and indepen- dent of the motor response, the closed-loop V/f controller can indirectly detect speed oscillations and modify its frequency reference, so that the PMSM does not lose synchro- nism. This is expected to result in higher reliability and lower sensitivity to load changes. On the negative side, the voltage reference calculation in the closed-loop V/f controller is based on equation (3.22), which has been derived for steady-state operation. Considering that the V/f control method is deployed during the startup of the PMSM, this assumption could slightly affect the performance of the controller. 3.5 Vector control with position sensor Although the presence of position or speed sensors is undesirable for the studied applica- tion, a vector control scheme with such sensors has been implemented, as an intermediate step towards the development of a position-sensorless method. The design of this scheme is discussed in this section. 3.5.1 Structure of the controller The structure of the implemented vector controller is presented in Fig. 3.5. The currents of the system are measured and are transformed into the dq reference frame. The rotor position, which is necessary for the dq transformation of the currents, is mea- sured with a mechanical sensor. The vector controller receives an external torque command and calculates the current 25 references in the dq reference frame. The calculated references, together with the trans- formed stator currents, enter the current controller. The knowledge of the rotor speed is necessary for the operation of the current controller. The speed is assumed to be measured directly from the shaft of the PMSM, although it can also be calculated as the derivative of the measured rotor position. The current controller produces a reference voltage in the dq reference frame. By using the measured rotor position, this voltage is transformed into the three-phase system and is used to determine the output of the VSD. In the actual application, the three reference voltage waveforms, which are generated by the controller, enter a PWM stage, where they are compared with a carrier wave. The result of the comparison determines the state of the switches of the inverter and, there- fore, the form of the voltage pulses at the output of the VSD. However, as in the case of the V/f control models, the PWM stage is omitted and the reference voltage generated by the vector controller is assumed to be equal to the voltage output of the drive. Therefore, it is assumed that the VSD produces sinusoidal voltage waveforms, instead of PWM pulses. Equivalent PMSM r rθ , ω Inverter abc dq * * * sa sb scu ,u ,u sa sb scu ' , u ' , u ' abc dq sa sb sci , i , i Current controller Current reference calculation Si S *isu * rθ rθ rω su * Vector controller of the VSD * eT Figure 3.5: Vector control topology, including mechanical sensors. 26 3.5.2 Need for constant torque reference The PMSM in the studied application needs to operate under constant-torque control. The multiphase pump accommodates fluid stream of varying composition. This stream consists of oil, gas and water, the analogy of which changes with time. Due to the varying mass density of the stream mixture, the load of the PMSM also changes with time. By applying a constant torque reference in the vector controller, the speed of the motor- pump assembly can be regulated rapidly in response to load variations. This behaviour corresponds to the desired operation of the system. 3.5.3 Current reference calculation As was discussed in Section 3.1.1, the only torque-producing current component in PMSMs with surface-mounted magnets is the q-axis current. The reference i∗sq for this component can be calculated from (3.7) as i∗sq = 2 3npΨm T ∗e (3.29) where T ∗e is the electromagnetic torque reference. Regarding the reference i∗sd of the d-axis stator current, it is usually selected in such a way, that the efficiency of the PMSM is maximized. In general, its optimal value can be obtained by the MTPA method [34, 36] according to i∗sd = Ψm − √ Ψ2 m + 8(Lsq − Lsd)2(̂i∗s)2 4(Lsq − Lsd) (3.30) where î∗s is the reference of the stator current amplitude. In PMSMs with surface-mounted magnets, the d-axis current does not contribute to any torque production and the maxi- mum torque-to-current ratio is obtained by simply setting i∗sd = 0 [30, 39]. Since no dq saliency is considered for the PMSM in this thesis, the d-axis current reference is set to zero for speeds lower than the rated speed ωr,rated. However, when operation above ωr,rated is desired, the reference i∗sd needs to be modified by applying a proper field- weakening strategy. Such a strategy is discussed in the section of the position-sensorless vector controller. 27 3.5.4 Transfer function of the controlled system Before the vector controller can be designed, the transfer function of the controlled sys- tem must be derived. This system includes the PMSM and the transmission components, namely the cable and the two transformers. For PMSMs with surface-mounted magnets, the d-axis and q-axis inductances are ap- proximately equal. Considering equivalent motor quantities, the flux-linkage equation (3.2) can then be written as ψ̄′s = L′sīs + Ψm (3.31) By substituting (3.31) into (3.1), the equivalent stator voltage ū′s is given by ū′s = R′sīs + L′s d̄is dt + jωrL ′ sīs + jωrΨm (3.32) where the term jωrΨm is the back-EMF ē of the PMSM (according to (3.3)) and jωrL ′ sīs is the cross-coupling term. The cross-coupling effect, thus the inherent interaction between the d-axis and q-axis quantities of the PMSM, prevents the independent control of isd and isq, as can be demon- strated through the d-axis and q-axis voltage equations of the motor. According to (3.4), a variation in usd results in a desirable change in isd. However, due to the d-axis component of the cross-coupling term in (3.5), the voltage usq is also forced to change, which in turn causes an undesirable variation in isq. In effect, a change in isd is inevitably accompanied by a change in isq and vice versa. By applying the Laplace transform and rearranging terms, (3.32) becomes īs = 1 sL′s +R′s + jωrL′s (ū′s − ē) (3.33) Equation (3.33) represents the transfer function of the physical system that consists of the PMSM and the transmission components. 3.5.5 Inclusion of compensating terms In order to improve the performance of the current regulation, it is necessary to modify the transfer function of the physical system (obtained from (3.33)) by including certain 28 compensating terms in the vector controller [35]. As was demonstrated in Section 3.5.4, the inherent cross-coupling in the PMSM prevents the independent regulation of the d-axis and q-axis currents, thus the separate control of the flux and the torque of the motor. The negative impact of the cross-coupling effect on the performance of the controller can be mitigated by compensating for the term jωrL ′ sīs in (3.32). Clearly, the compensation demands the knowledge of the stator current and the motor speed and its precision de- pends on the accuracy of the estimation of the equivalent stator inductance L′s. The control performance can be further improved by cancelling out the effect of the back- EMF of the PMSM, thus by compensating for the term jωrΨm in (3.32). By feeding- forward this term in the vector controller, the interaction between the electrical and the mechanical dynamics of the motor is eliminated. Finally, the term Raīs can be added on the right-hand side of (3.32). The value Ra is called active-damping resistance and is used to effectively increase the resistance R′s of the system. Its presence in the controller increases the damping of the system and therefore improves its response to disturbances. Of course, since Ra is a control parameter and not a physical resistance, it does not contribute to any losses. The block diagram of the physical system with the added compensating terms is shown in Fig. 3.6. s s r s 1 sL R j L     +_su r me j   Si+ + + comp r me j   r sj L R   Physical system Figure 3.6: Block diagram of the physical system with the added compensating terms. After cancelling out the back-EMF and the cross-coupling term of the PMSM and after 29 introducing the active-damping resistance in the vector controller, (3.33) becomes īs = ū′s sL′s +R′s +Ra (3.34) Equation (3.34) represents the modified transfer function of the system, thus the transfer function that is considered for the design of the controller. 3.5.6 Design of the PI regulator The use of the dq reference frame, which results in the transformation of steady-state AC quantities into DC values, makes it possible to use PI regulators for current control. The reason is that integrators are well-suited for the control of DC quantities, because they provide infinite gain at zero frequency and can therefore ensure the absence of steady- state errors. The relation between the input (̄i∗s − īs) and the output ū∗s of the PI current regulator is given by ū∗s = (kp + ki s )(̄i∗s − īs) (3.35) where kp is the proportional gain, ki is the integral gain and ī∗s is the space vector of the stator current reference. The mission of the PI regulator is to provide current control to the modified system, which was shown in Fig. 3.6. The current-controlled system is presented in Fig. 3.7. The PI regulator is designed according to the IMC method [35]. The closed-loop system, which consists of the modified transfer function of the controlled system (given by (3.34)), the transfer function of the PI regulator (given by (3.35)) and unit feedback, is designed to be a first-order system with bandwidth ac. The gains of the PI regulator can then be readily obtained as kp = acL ′ s (3.36) ki = ac(R ′ s +Ra) (3.37) where the active-damping resistance Ra can be calculated from Ra = acL ′ s −R′s (3.38) 30 +_+ + + Physical system i p k k s  Modified system + _ PI regulator comp r me j   r me j   s s r s 1 sL R j L     su * Si Si S *i r sj L R   Figure 3.7: Block diagram of the current-controlled system, consisting of the physical system, the compensating terms and the PI regulator. The bandwidth ac of the current regulator corresponds to a frequency which is selected to be several times lower than the switching frequency of the VSD. The value of ac determines the response speed of the controller according to tr = ln9 ac (3.39) where tr is the rise time, thus the time needed by the controlled currents to change from 10% to 90% of their final value. 3.5.7 Implementation of voltage and current limiters The design of the vector controller must ensure that the electrical limits of the PMSM and the VSD are not exceeded [31]. The value of the stator current reference must not be greater than the rated current of the PMSM, while the the commanded voltage must not exceed the maximum voltage capability of the VSD. The limits of the current references i∗sd and i∗sq can be expessed as i∗sd ≤ is,max (3.40) i∗sq ≤ √ (is,max)2 − (i∗sd) 2 (3.41) where is,max is the maximum acceptable stator current, which is equal to the peak value of the rated current of the PMSM, thus is,max = îs,rated. The limit of the voltage amplitude reference û∗s can be written as 31 û∗s ≤ us,max (3.42) where us,max is the maximum acceptable voltage at the output of the VSD. The purpose of the implemented voltage and current limiters is to protect the PMSM and the VSD. However, it should be noted that when the current or voltage limits are hit, the intervention of the limiters causes the system to enter a practically uncontrolled state, which includes distorted current responses and higher rise times. One particular issue concerning the voltage limiter is integrator windup. When the volt- age limit is hit, the current error (̄i∗s− īs) becomes high, since the controller is not allowed to provide the voltage that is required to achieve ī∗s = īs. Integrating this error would result in a large ouput for the integrator, which should be decreased later, causing a substantial current overshoot [26]. In order to prevent integrator windup, an anti-windup function is implemented in the controller. Its idea is to cancel out the current error that results from the difference between the unlimited voltage ū∗s and the limited voltage ū∗s,lim. The output of the anti- windup function is given by īaw = 1 kp (ū∗s,lim − ū∗s) (3.43) where īaw represents the error term due to the action of the voltage limiter. This term is added to the current error (̄i∗s − īs) which enters the integrator. The complete current- controlled system, with the current limiters and the anti-windup function included, is presented in Fig. 3.8. 3.6 Position-sensorless vector control The operation of the vector control scheme which was described in Section 3.5 is based on the presence of mechanical sensors. This section explains why mechanical sensors need to be eliminated and describes differ- ent position-estimating methods which can facilitate sensorless operation of the PMSM. Based on one of these methods, a position-estimating algorithm has been implemented and is presented. In order to extend the speed range of the PMSM, the option of applying a field-weakening 32 +_+ + + Physical system Modified system PI regulator + _ Voltage limiterCurrent limiter p 1 k + _ Anti-windup function S *iS *i S,limi * i p k k s  su * su * s,limu * comp r me j   r me j   s s r s 1 sL R j L     Si r sj L R   Si Figure 3.8: Block diagram of the current-controlled system, consisting of the modified system, the PI regulator, the current and voltage limiters and the anti-windup function. strategy is investigated for the position-sensorless vector controller. After evaluating the suitability of the studied system for the application of such a strategy, a field-weakening algorithm is derived and is integrated into the vector controller. Apart from the position estimator and the field-weakening algorithm, the other parts of the controller are the same as the ones presented in Section 3.5. 3.6.1 Necessity of eliminating the position sensor The presence of position sensors in PMSMs is usually undesirable, since it increases the cost and the complexity of the system. When it comes to harsh subsea environments, it is particularly important for the system to be as simple and robust as possible. The use of position-sensorless control schemes is therefore a necessity. The drawbacks of position sensors include an increased number of connections between the motor and the control system, increased electromagnetic interference due to the pres- ence of connecting leads and limited measuring accuracy due to environmental factors, such as temperature, humidity and vibrations [25]. Moreover, the cost of the position-encoding device increases the overall cost of the system, while the static and dynamic friction of the sensors are added on top of the mechanical load of the motor [25, 27]. 33 3.6.2 Review of position estimation methods The operation of sensorless vector control schemes is facilitated by the development of techniques that estimate the rotor position θr. Different methods have been proposed for this purpose, some of which are shortly discussed in the section. Back-EMF-based methods By measuring the stator currents and voltages of the motor, the space vector of the back- EMF in the αβ reference frame can be calculated. The obtained back-EMF vector can then be used to determine the rotor position. Back-EMF-based methods cannot be used at a standstill or during low-speed operation of the PMSM, mainly because the back-EMF becomes too low in this range of speeds [27]. Furthermore, in order to calculate the back-EMF of the PMSM, the derivative of the measured stator current is used, which makes these methods prone to noise [28]. The accuracy of the estimation is also affected by errors in the values of the motor parameters [26]. Flux linkage-based methods If the stator currents and voltages are known, the stator flux linkage in the αβ reference frame can be calculated and used for the determination of the rotor position. Due to the integration process by which the flux linkage is obtained, flux linkage-based methods often suffer from the effects of integrator drift, which are compensated by analog electronics or software techniques [28]. The accuracy of the position estimation is also affected by motor parameter variations [25]. Inductance-based methods Inductance-based techniques can be used to estimate the rotor position in PMSM topolo- gies in which the stator inductances vary significantly over an electrical cycle [28]. The estimation in this case can be based on look-up tables, which contain information about the relation between the stator inductances and the rotor position [26]. 34 Problems with these methods may include the requirement of high switching frequency for the accurate calculation of the inductances, as well as sensitivity to certain parameter variations [26]. Observer-based methods In observer-based techniques, a mathematical model of the system is created and is sup- plied with the same excitation as the actual system. This model produces estimated outputs, which are compared with the measured outputs of the real system. The errors between the estimated and the measured quantities are used to correct the outputs of the observer [27]. Since the PMSM model is nonlinear, the design of a state observer is generally quite complex. The used algorithms may often be robust against parameter variations and measurement noise, but they have high computational requirements [26]. Selection of estimation method The estimation scheme which has been implemented in this thesis is flux linkage-based. It uses a thoroughly tested algorithm, which provides accurate position estimation within a wide speed range. The issue of integrator drift, which is often considered to be the main drawback of flux linkage-based methods, is resolved by constantly correcting and updating the estimated quantities, by using the available current measurements. Other methods could also provide adequate control performance for the studied applica- tion. Back-EMF-based methods, for instance, have proved to be quite popular and would probably be suitable for the studied system. Their inability to provide accurate position estimation at low speeds would not be a problem, since low-speed operation is handled by the V/f controller anyway. Inductance-based methods however, would not be a sensible choice, since they are only well-suited to motors with significant inductance variations over one electrical period. As the considered motor has negligible dq saliency, the resulting variations would be too low to produce a proper position estimation. 35 3.6.3 Structure of the controller The structure of the designed position-sensorless vector controller is presented in Fig. 3.9. In contrast to the layout presented in Fig. 3.5, no mechanical sensors are present in the new topology. Although this increases the robustness of the system, it also complicates the design of the vector controller. The elimination of the mechanical sensors has been achieved by adding a position esti- mator in the controller. The mission of the estimator is to provide position estimates to the dq transformation blocks and speed estimates to the current controller. The in- puts required by the estimator are two phase currents and a previously generated voltage reference. Equivalent PMSMInverter abc dq * * * sa sb scu ,u ,u sa sb scu ' , u ' , u ' abc dq sa sb sci , i , i Current controller Current reference calculation Si S *isu * rθ rθ rω su * Vector controller of the VSD * eT Position estimator rθ rω su * Figure 3.9: Position-sensorless vector control topology. 3.6.4 Implementation of the position estimator The implemented estimator estimates the rotor position θr, by executing a multi-step discretized algorithm. Its operation is based on a method described in [26], although variations of this method have been earlier presented in [25, 28, 29]. For the design of the position estimator, it is convenient to use the αβ reference frame. In 36 contrast to the dq frame, which is synchronized to the rotation of the PMSM rotor, the αβ frame is stationary and its α axis is aligned with the a axis of the three-phase system [35]. Figure 3.10 provides an overview of the functions executed by the position estimator. The steps of the discretized estimating algorithm are discussed in more detail below. Flux linkage estimation  s su 1    s si   s s,est  Current estimation  r,pr   s s,esti  Position correction  r,pr   r,cor  Position prediction  r,pr 1    s s,upd 1   r,est Speed estimation Position compensation  r,est Flux linkage updating  s si   r,cor   s s,upd  Figure 3.10: Block diagram of the position estimator. Flux linkage estimation The first step of the algorithm is the estimation of the equivalent stator flux linkage. By considering equivalent motor quantities, (3.1) can be represented in the αβ frame as ū′ s s = Rsī s s + dψ̄′ s s dt (3.44) The representation of the αβ vectors is similar to the one of the dq vectors. The only difference is the additional superscript ’s’, which denotes that αβ vectors are stationary. By using discretized quantities and integrating, (3.44) can be written as ψ̄′ s s,est(κ) = Ts[ū′ s s(κ− 1)−R′sīss(κ)] + ψ̄′ s s,upd(κ− 1) (3.45) where Ts is the sampling period, κ is the sampling number and the subscripts ’est’ and ’upd’ denote estimated and updated values respectively. For the discrete integration of the difference between the stator voltage and the resistive drop, the rectangular rule has been applied. Equation (3.45) estimates the equivalent stator flux linkage, based on the generated volt- age reference and the updated flux linkage of the previous sampling period κ− 1, as well as on the stator current, which is measured in the current period κ. The flux linkage and stator voltage values of the previous sampling period are obtained 37 by including unit delay blocks in the controller. By storing signals for a specified amount of sampling periods, these blocks allow the controller to access previous values of its variables. Current estimation The second step of the algorithm is the estimation of the stator current, based on the estimated flux linkage ψ̄′ s s,est. Equation (3.31) can be represented in the αβ frame as ψ̄′ s s = L′sī s s + Ψme jθr (3.46) The stator current can be estimated from īss,est(κ) = ψ̄′ s s,est(κ)−Ψme jθr,pr(κ) L′s (3.47) where the subscript ’pr’ denotes predicted quantities. Equation (3.47) estimates the stator currents by using the flux linkage obtained from the previous step and the position predicted for the current sampling period. Position correction and speed estimation The third step of the algorithm is the correction of the previously predicted rotor posi- tion. The current error, thus the difference ∆īss between the measured and the estimated currents, is calculated and is transformed into the dq reference frame by using the pre- dicted rotor angle θr,pr. The presence of a current error indicates the need for position correction. It can be proven, as in [26], that only the q-axis component ∆iq of the dq-transformed current error ∆īs is needed for the calculation of the position correction term ∆θr according to ∆θr(κ) = −L ′ s∆iq(κ) Ψm (3.48) The corrected position is then obtained from the predicted position θr,pr and the correction term ∆θr according to θr,cor(κ) = θr,pr(κ) + ∆θr(κ) (3.49) where the subscript ’cor’ denotes corrected values. The estimated speed ωr,est of the PMSM is easily obtained from the discrete derivative of the corrected position according to 38 ωr,est(κ) = θr,cor(κ)− θr,cor(κ− 1) Ts (3.50) The estimated speed passes through a low-pass filter, which eliminates the high-frequency ripple that might be present, and is then directed to the current regulator of the vector controller. Flux linkage updating The next step is to update the previously estimated flux linkage, so that the updated value ψ̄′ s s,upd can be used for the flux linkage estimation in the next sampling period. Equation (3.46) was used earlier to estimate the stator currents from the estimated flux linkage and the predicted position. By applying the same equation, this time using the measured currents and the corrected position, the updated flux linkage can be obtained as ψ̄′ s s,upd(κ) = L′sī s s(κ) + Ψme jθr,cor(κ) (3.51) By using the updated flux linkage ψ̄′ s s,upd in the next sampling period, the negative effects of integrator drift are avoided. Position prediction The next step of the algorithm is the prediction of the rotor position for the next sampling period. A common assumption is that the position varies with time as a second-order polynomial [25, 28]. The predicted position θr,pr can be calculated from θr,pr(κ+ 1) = 3θr,cor(κ)− 3θr,cor(κ− 1) + θr,cor(κ− 2) (3.52) where the previously estimated positions θr,cor(κ − 1) and θr,cor(κ − 2) are obtained by using a single-period unit delay and a double-period unit delay respectively. Position compensation Tests on the implemented model revealed the presence of steady-state differences between the estimated currents and the real currents of the system. Furthermore, it was noticed that the magnitude of the estimation errors increased as the steady-state speed of the PMSM increased. In order to eliminate these estimation errors, a speed-dependent position compensator 39 has been included in the estimator. Its mission is to generate a compensating term θr,comp, which increases linearly with the estimated motor speed ωr,est and which is added to the corrected position θr,cor. In order to determine the linear function θr,comp = f(ωr,est) of the compensating block, the required compensation θr,comp for two different speeds is obtained by trial and error. The straight line which passes from the two position-speed pairs corresponds to the transfer function of the position compensator. 3.6.5 Basic idea of field-weakening A field-weakening strategy has been applied in the sensorless vector controller, in order to extend the operating speed range of the PMSM. Before the implemented algorithm is presented, the fundamental principles of field-weakening are briefly discussed. Assuming that the stator flux linkage of the PMSM is constant and that the resistive stator voltage drop is negligible, the stator voltage equation is given by (3.14). By solving for the electrical speed ωr, this equation yields ωr ' ûs ψ̂s (3.53) Equation (3.53) demonstrates that the speed of the motor can be increased either by increasing the supplied voltage amplitude, or by decreasing the stator flux linkage. When the PMSM operates below its rated speed, the flux linkage should be kept around its nominal value, so that optimal efficiency is achieved, according to the discussion in Section 3.3.1. The speed in this case can be adjusted by varying the supplied voltage amplitude. Speed control through the voltage amplitude is possible up to the rated speed of the mo- tor. In order to increase the speed even further, without exceeding the rated voltage of the PMSM, it is necessary to decrease the amplitude of the stator flux linkage. Therefore, for high-speed operation of the motor, field-weakening is needed. According to (3.2), a reduction of the flux linkage amplitude can be achieved by injecting a negative d-axis current into the motor. Since the d-axis current does not contribute to any torque production, field-weakening results in a decreased torque-to-current ratio. Moreover, since the overall stator current cannot exceed its rated value, an increase of 40 the d-axis current restricts the maximum acceptable q-axis current and therefore the maximum torque that the PMSM can produce. Based on the previous discussion, two control regions can be identified for the PMSM. The ’voltage control’ region corresponds to constant maximum torque and power which increases with speed, while the ’flux control’ region corresponds to constant power and maximum torque which decreases with speed [32, 40]. 3.6.6 Overvoltage risk during field-weakening When it comes to field-weakening, an issue that needs special attention is the risk of exceeding the voltage withstand limits of the VSD [36]. According to (3.3), the magnitude of the back-EMF is proportional to the speed of the motor. Therefore, the high speeds which are reached during the field-weakening opera- tion result in high back-EMF values. If the stator current is suddenly lost for some reason, the voltage drop in the transmission system and the stator impedance becomes zero and the inverter of the VSD experiences a voltage equal to the back-EMF of the PMSM. In order to avoid damaging the inverter, it is necessary to design the VSD so that it can withstand the maximum back-EMF that can be reached during field-weakening [33]. For a VSD with a specified voltage output capability, it is necessary to limit the maximum speed during the field-weakening operation, so that the resulting back-EMF does not exceed the maximum voltage of the inverter [34]. 3.6.7 Field-weakening capability of the system Some systems provide favourable conditions for field-weakening operation, while others do not. This section evaluates the possibility of applying a field-weakening strategy in the studied system. The evaluation is based on considerations on the mechanical load and the system inductances. In order to increase the motor speed above its rated value, a negative d-axis current is required, as was discus