Study and implementation of a PMSM & Study of a sensorless control method

Bachelor’s Thesis Grau en Enginyeria en Tecnologies Industrials Study and implementation of a PMSM & Study of a sensorless control method REPORT Author: Helena Beltran Feliu Director: Daniel Montesinos Miracle Deadline: June 2017 Escola Tècnica Superior d’Enginyeria Industrial de Barcelona 2 Report Abstract The procedure followed in the project begins with a brief introduction of the features that the studied motor, a permanent magnet synchronous motor (PMSM), has. The fact that the motor is synchronous permanent magnet has to do with its greater efficiency in comparison with the induction motor, which is the most used nowadays. Then, the project is conducted in two steps. The first one is the study of the PMSM mathematical modelling and the subsequent control method applied. The second one is the study of a sensorless control algorithm. Traditionally for speed dependent applications, some kind of sensor is used to read the motor speed and position and feed the value back to the controller. However, extra sensors require extra physical space in the application and it also introduces another source of failure in the system. Thus, with the additional purposes of reducing cost and maintenance needs, the sensor can be replaced by an estimator that mathematically estimates the speed or position of the rotor. All these implementations have been simulated with MATLAB/Simulink based on the mathematical models. To design a controlled drive, the stability characteristics of PMSM under open-loop control (without having any feedback for speed) are analysed. The analysis shows that the PMSM becomes unstable after exceeding a certain applied speed. After tuning the controllers, it has been analysed that the maximum speed that the closed-loop can control with a reasonable settling time is 750 rpm. The more speed that the motor achieves, the more settling time appears. Thus, there is an upper limit for the speed. For all the simulations, an optimal speed of 550 rpm has been used. The control structure and the design of the controllers are described. A rotor position estimation technique for sensorless operation is studied. The estimator uses predictor-corrector method where the difference between the estimated current and the measured current (current error) is used to correct a predicted rotor position. More investigations are still required for accurate rotor position estimation. Study of a PMSM 3 4 Report Summary ABSTRACT ___________________________________________________ 2 SUMMARY ___________________________________________________ 4 1. GLOSSARY ______________________________________________ 5 1.1. List of figures .................................................................................................. 5 1.2. Symbols ......................................................................................................... 6 1.3. Abbreviations ................................................................................................. 8 2. PREFACE ________________________________________________ 9 3. INTRODUCTION __________________________________________ 11 3.1. Objectives of the project .............................................................................. 11 3.2. Scope of the project ..................................................................................... 12 4. PMSM MODEL ___________________________________________ 13 4.1. Introduction .................................................................................................. 13 4.2. Mathematical Model ..................................................................................... 15 4.3. Implementation and results .......................................................................... 20 4.3.1. Open-loop system ............................................................................................ 20 4.3.2. Closed-loop system ......................................................................................... 23 5. SENSORLESS CONTROL __________________________________ 34 5.1. Introduction .................................................................................................. 34 5.2. Flux-linkage method ..................................................................................... 35 5.3. Problems encountered ................................................................................. 39 CONCLUSIONS ______________________________________________ 41 Future work............................................................................................................ 42 THANKS ____________________________________________________ 43 BIBLIOGRAPHY ______________________________________________ 44 APPENDICES ________________________________________________ 46 A) Data for the IPMSM ......................................................................................... 46 B) MATLAB file ..................................................................................................... 47 C) Model implementation with MATLAB/Simulink ................................................ 47 Study of a PMSM 5 1. Glossary 1.1. List of figures Figure 1: Different rotor configurations for PMSM. (a) SPMSM (b) IPMSM ..................... 14 Figure 2: PM Synchronous Motor .................................................................................... 15 Figure 3: Equivalent circuit of a PMSM ........................................................................... 20 Figure 4: Open-loop system ............................................................................................ 21 Figure 5: Motor speed during open-loop system ............................................................. 22 Figure 6: Direct current and quadrature current............................................................... 22 Figure 7: Block diagram of a PID controller in a feedback loop ....................................... 24 Figure 8: Block diagram of the example below ................................................................ 25 Figure 9: Block diagram of the three functions 𝐺1𝑠, 𝐺2𝑠 and 𝐺𝐶𝐿𝑠 used to see the derivative influence .................................................................................................. 26 Figure 10: Effect of the derivative term in an external loop .............................................. 26 Figure 11: Effect of the derivative term in an internal loop ............................................... 27 Figure 12: Implementation of two PI controllers with two respective functions ................. 28 Figure 13: Bode Diagram comparing the internal loop (blue) and the external loop (orange) ................................................................................................................... 28 Figure 14: Speed response with the controlled system ................................................... 31 Figure 15: Final torque after the control implemented (12 Nm) ........................................ 32 Figure 16: Direct current and quadrature current after the control implemented .............. 32 Figure 17: Direct voltage and quadrature voltage with the control ................................... 33 Figure 18: Block diagram of the rotor position and velocity estimation algorithm ............. 35 Figure 19: Rotor position prediction using polynomial curve fitting .................................. 38 Figure 20: Battery block .................................................................................................. 47 Figure 21: Power converter ............................................................................................. 48 Figure 22: Electric block of the open-loop system ........................................................... 48 Figure 23: Electromechanical conversion block of the open-loop system ........................ 48 Figure 24: Mechanical block of the open-loop system ..................................................... 49 Figure 25: Mechanical block of the control ...................................................................... 49 Figure 26: Mechanical-Electrical conversion block of the control ..................................... 49 Figure 27: Electric block of the control ............................................................................ 49 Figure 28: Alpha modulator block .................................................................................... 50 6 Report Figure 29: Closed-loop system with the control implemented .......................................... 50 Figure 30: Flux-linkage algorithm .................................................................................... 50 Figure 31: Step 1 of the algorithm ................................................................................... 51 Figure 32: Step 2 of the algorithm ................................................................................... 51 Figure 33: Step 3 of the algorithm ................................................................................... 51 Figure 34: Step 4 of the algorithm ................................................................................... 52 Figure 35: Step 5 of the algorithm ................................................................................... 52 1.2. Symbols Symbol Unity Meaning 𝐵𝑚 [Nm s rad-1] Viscous friction coefficient 𝐼𝑎, 𝐼𝑏, 𝐼𝑐 [A] Phase a, b, c instantaneous stator current 𝐼𝑑,𝐼𝑞 [A] d- and qaxis components of stator current 𝐽 [kg m2] Inertia of the rotating system 𝐾𝑑 [-] Derivative constant 𝐾𝑖 [-] Integral constant 𝐾𝑝 [-] Proportional constant 𝐿𝑑,𝐿𝑞 [H] d- and qaxis stator self-inductances 𝐿𝑠 [H] Average inductance 𝐿𝑥 [H] Inductance fluctuation 𝑃 [-] Number of poles of the motor 𝑅𝑠 [Ω] Stator resistance 𝑠 [-] Laplace operator Study of a PMSM 7 T [Nm] Torque T𝐿 [Nm] Load torque 𝑡𝑟 [s] Settling time 𝑇𝑠 [s] Sampling time 𝑉𝑎,𝑉𝑏,𝑉𝑐 [V] Phase a, b, c instantaneous stator voltage. 𝑉𝑑,𝑉𝑞 [V] d and qaxis components of stator phase voltage α [s-1] Modulator constant θ [º] Electrical angle between a-axis and q-axis θ𝑟 [rad] Electrical rotor position λ𝑑𝑠, λ𝑞𝑠 [V s rad-1] Stator flux linkage in rotor fixed d,q frame λ𝑚 [V s rad-1] Peak flux linkage due to permanent magnet ω [rad/s] Angular velocity of rotation ω𝑚 [rad/s] Machine’s mechanical speed ω𝑟 [rad/s] Machine’s electrical rotor speed 8 Report 1.3. Abbreviations AC Alternating Current DC Direct Current EMF Electromotive Force EV Electric Vehicle FOC Field Orientated Control IPMSM Interior Permanent-Magnet Synchronous Motor PMSM Permanent-Magnet Synchronous Motor SC Scalar Control SPMSM Surface Permanent-Magnet Synchronous Motor VC Vector Control Study of a PMSM 9 2. Preface Many types of electric motors have been used in the industry for different purposes: cranes, spinning machines, public transportation and so on. AC motors are widely used and AC drives are subject of study for many researchers. Recently, AC drives in vehicle applications are gaining attention due to pollution and fuel price problems.[13] A large part of these applications are triggered by induction motors for their ease of use and their presence throughout history. But with the technology currently available these can be replaced by permanent magnet synchronous motors in order to improve their efficiency and performance. [13] Apart from replacing induction motors for permanent magnet synchronous motor, more improvements can be done. The rotational speed of a PMSM is commonly measured by sensors. High precision sensors are expensive, occupy space and the total complexity of the motor drive is increased. Sensorless control provides the option of reducing the size and complexity of the motor drive, and consequently, the price is also reduced. [5] 16 Report The following parameters are defined: 𝑃 Number of poles of the motor 𝐼𝑎, 𝐼𝑏, 𝐼𝑐 Phase a, b, c instantaneous stator current 𝑉𝑎,𝑉𝑏,𝑉𝑐 Phase a, b, c instantaneous stator voltage. 𝐼𝑑,𝐼𝑞 d- and qaxis components of stator current 𝑉𝑑,𝑉𝑞 d and qaxis components of stator phase voltage 𝑅𝑠 Stator resistance 𝑝 𝑑 𝑑𝑡 𝐿𝑑,𝐿𝑞 d- and qaxis stator self-inductances 𝐿𝑠 Average inductance 𝐿𝑠=1 2·(𝐿𝑞+𝐿𝑑) 𝐿𝑥 Inductance fluctuation 𝐿𝑥=1 2·(𝐿𝑞−𝐿𝑑) λ𝑚 Peak flux linkage due to permanent magnet θ Electrical angle between a-axis and q-axis in degrees ω=p·θ Angular velocity of rotation (in electrical rad/sec) Figure 2 illustrates a conceptual cross-sectional view of a 3-phase, 2-pole IPM synchronous motor along with two reference frames. To illustrate the inductance difference (𝐿𝑞>𝐿𝑑), rotor is drawn with saliency although actual rotor structure is more likely a cylinder. The stator reference axis for the a-phase is chosen to the direction of maximum magneto-motive force when a positive a-phase current is supplied at its maximum level. Reference axis for b- and cstator frame are chosen 120° and 240° (electrical angle) ahead of the a-axis, respectively. Following the convention of choosing the rotor reference frame, the direction of permanent magnet flux is chosen as the d-axis, while the q-axis is 90 degrees ahead of the d-axis. The angle of the rotor q-axis with respect to the stator a-axis is defined as θ. Note that as the machine turns, the d-q reference frame is rotating at a speed of ω=𝑑θ 𝑑𝑡, while the stator a-,b- ,caxes are fixed in space. [9] Study of a PMSM 17 The electrical dynamic equation in terms of phase variables can be written as: 𝑉𝑎=𝑅𝑠·𝐼𝑎+𝑝·λ𝑎 (4.1) 𝑉𝑏=𝑅𝑠·𝐼𝑏+𝑝·λ𝑏 (4.2) 𝑉𝑐=𝑅𝑠·𝐼𝑐+𝑝·λ𝑐 (4.3) while the flux linkage equations are λ𝑎=𝐿𝑎𝑎·𝐼𝑎+𝐿𝑎𝑏·I𝑏+𝐿𝑎𝑐·𝐼𝑐+λ𝑚𝑎 (4.4) λ𝑏=𝐿𝑎𝑏·𝐼𝑎+𝐿𝑏𝑏·I𝑏+𝐿𝑏𝑐·𝐼𝑐+λ𝑚𝑏 (4.5) λ𝑐=𝐿𝑎𝑐·𝐼𝑎+𝐿𝑏𝑐·I𝑏+𝐿𝑐𝑐·𝐼𝑐+λ𝑚𝑐 (4.6) Considering symmetry of mutual inductances such as 𝐿𝑎𝑏=𝐿𝑏𝑎. Note that in the above equations, inductances are functions of the angle θ. Since stator self-inductances are maximum when the rotor q-axis is aligned with the phase, while mutual inductances are maximum when the rotor q-axis is in the midway between two phases. Also, note that the effects of saliency appeared in stator self and mutual inductances are indicated by the term 2θ. [9] 𝐿𝑎𝑎=𝐿𝑠𝑜+𝐿𝑠𝑙+𝐿𝑥· cos (2θ) (4.4) 𝐿𝑏𝑏=𝐿𝑠𝑜+𝐿𝑠𝑙+𝐿𝑥·cos (2θ + 120) (4.5) 𝐿𝑐𝑐=𝐿𝑠𝑜+𝐿𝑠𝑙+𝐿𝑥·cos (2θ − 120) (4.6) 𝐿𝑎𝑏=−1 2𝐿𝑠𝑜+𝐿𝑥· cos (2θ − 120) (4.7) 𝐿𝑏𝑐=−1 2𝐿𝑠𝑜+𝐿𝑥· cos (2θ) (4.8) 𝐿𝑎𝑐=−1 2𝐿𝑠𝑜+𝐿𝑥· cos (2θ + 120) (4.9) For mutual inductances in the above equations, the coefficient −1 2 comes due to the fact that stator phases are displaces by 120º, and cos(120)=−1 2. Meanwhile, flux-linkages at the 18 Report stator windings due to the permanent magnet are λ𝑚𝑎=λ𝑚·cos (θ) (4.10) λ𝑚𝑏=λ𝑚· cos (θ − 120) (4.11) λ𝑚𝑐=λ𝑚· cos (θ + 120) (4.12) For this model, input power 𝑃𝑖 can be represented as 𝑃𝑖=𝑉𝑎·𝐼𝑎+𝑉𝑏·𝐼𝑏+𝑉𝑐·𝐼𝑐 (4.13) Unfortunately, the output power 𝑃𝑜 and the output torque 𝑇=(𝑃 2)·𝑃𝑂 𝜔 cannot be simply derived in this 3-phase model. The torque can be expressed as 𝑇=(𝑃 6)·(𝐿𝑞−𝐿𝑑)·[{(𝐼𝑎2−0,5𝐼𝑏2−0,5𝐼𝑐2−𝐼𝑎·𝐼𝑏−𝐼𝑎·𝐼𝑐+2·𝐼𝑏·𝐼𝑐)·sin(2θ) +√3 2·(𝐼𝑏2+𝐼𝑐2−2·𝐼𝑎·𝐼𝑏+2·𝐼𝑎·𝐼𝑐)·cos(2θ)}+λ𝑚 ·{(𝐼𝑎−0,5·𝐼𝑏−0,5·𝐼𝑐)·cos(θ)+√3 2·(𝐼𝑏−𝐼𝑐)·sin(θ)}] (4.14) Now, let S represent any of the variables (current, voltage and flux linkage) to be transformed from the a-b-c frame to d-q frame. The transformation in matrix form is given by [𝑆𝑞 𝑆𝑑 𝑆0]=2 3[cos (θ) cos(θ−120)cos(θ+120) 𝑠𝑖𝑛(θ)sin(θ−120)sin(θ+120) 0,5 0,5 0,5 ][𝑆𝑎 𝑆𝑏 𝑆𝑐] (4.15) Here 𝑆0 component is called the zero sequence component, and under balanced 3-phase system this component is always zero. Since it is a linear transformation, its inverse transformation exists and is [𝑆𝑎 𝑆𝑏 𝑆𝑐]=2 3[cos (θ) 𝑠𝑖𝑛(θ)1 cos(θ−120)sin(θ−120)1 cos(θ+120)sin(θ+120)1][𝑆𝑞 𝑆𝑑 𝑆0] (4.15) Now, by applying the transform of Equation 4.15 to voltages, flux-linkages and currents of Equations 4.1-4.6, we get a set of simple equations as Study of a PMSM 19 𝑉𝑞=𝑅𝑠·𝐼𝑞+𝑝·λ𝑞+ω·λ𝑑 (4.16) 𝑉𝑑=𝑅𝑠·𝐼𝑑+𝑝·λ𝑑−ω·λ𝑞 (4.17) where λ𝑞=𝐿𝑞·𝐼𝑞 (4.18) λ𝑑=𝐿𝑑·𝐼𝑑+λ𝑚 (4.19) Here, 𝐿𝑞 and 𝐿𝑑 are called q- and d-axis synchronous inductances, respectively, and are defined as 𝐿𝑞=3 2·(𝐿𝑠0+𝐿𝑥)+𝐿𝑠𝑙 (4.20) 𝐿𝑑=3 2·(𝐿𝑠0−𝐿𝑥)+𝐿𝑠𝑙 (4.21) As noticed in the above equation, synchronous inductances are effective inductances under balanced 3 phase conditions. Each synchronous inductance is made up of self-inductance (which includes leakage inductance) and contributions from other 2 phase currents. Now, a more convenient equation may result by eliminating the flux-linkage terms from equations 4.16-4.17 as 𝑉𝑞=(𝑅𝑠+𝐿𝑞·𝑝)·𝐼𝑞+ 𝜔· 𝐿𝑑·𝐼𝑑+𝜔·λ𝑚 (4.22) 𝑉𝑑=(𝑅𝑠+𝐿𝑑·𝑝)·𝐼𝑑− 𝜔 ·𝐿𝑞·𝐼𝑞 (4.23) These two equations are the final electric equations of the model. Figure 3 shows a dynamic equivalent circuit of an IPMSM based on equations 4.22 and 4.23. Note that in practice, magnetic circuits are subject to saturation as current increases. Especially, when 𝐼𝑞 is increased, the value of 𝐿𝑞 is decreased and λ𝑚 and 𝐿𝑑 is subject to armature reaction. Since 𝐼𝑑 is maintained to zero or negative value (demagnetizing) in most operation conditions, saturation of 𝐿𝑑 rarely occurs. [9] 20 Report Figure 3: Equivalent circuit of a PMSM The torque produced 𝑇, which is power divided by mechanical speed, can be represented as 𝑇=(3 2)·(𝑃 2)(λ𝑚·𝐼𝑞+(𝐿𝑑−𝐿𝑞)·𝐼𝑞·𝐼𝑑) (4.24) and defines the electromechanical conversion of the model. It is apparent from the above equation that the produced torque is composed of two distinct mechanisms. The first term corresponds to “the mutual reaction torque” occurring between 𝐼𝑞 and the permanent magnet, while the second term corresponds to “the reluctance torque” due to the differences in d-axis and q-axis reluctance (or inductance). Note that in order to produce additive reluctance torque, 𝐼𝑑 must be negative since 𝐿𝑞>𝐿𝑑. [9] The relationship among the machine produced electromagnetic torque 𝑇𝑒, load torque 𝑇𝑙 and the machine’s electrical speed 𝜔𝑟, gives the mechanical equation of the machine and it can be expressed as 𝑇𝑒=𝐽·2 𝑛·𝑝·𝜔𝑟+𝐵𝑚·2 𝑛·𝜔𝑟+𝑇𝑙 (4.25) where 𝐽 is the inertia of the rotor and the connected load, and 𝐵𝑚 is the viscous friction coefficient. 4.3. Implementation and results 4.3.1. Open-loop system Equations 4.22, 4.23, 4.24 and 4.25 have been implemented on MATLAB/Simulink. Apart from these equations, it is clear that the open-loop system needs energy to work. Thus, a battery that provides the enough voltage has been added. This voltage is a DC voltage and its value is 400√3 𝑉. As the motor works in AC, a power converter DC/AC must be added Study of a PMSM 21 (see second block of Figure 4). The alpha parameter will be later implemented in the control and it is defined as 𝛼=𝑈𝑎 ∗ 𝑈𝑒, where 𝑈𝑒 is the battery voltage and 𝑈𝑎∗ is the voltage of the motor after the whole closed-loop system. The power converter is connected to the electric motor and the combination of the two is referred to as an electric drive. Figure 4: Open-loop system See Appendix C to see the inside of each block. Nevertheless, with only these five blocks, the speed depends directly from the introduced voltage (as well as the fixed parameters of the motor and the load torque) and as a consequence it cannot be controlled. That is why a close-loop system is added: the control. Before proceeding to implement the control in the model, it was proved that from fixed voltages 𝑉𝑑=−178 𝑉 , 𝑉𝑞=284 𝑉 and with the load torque 𝑇𝑙=12 𝑁𝑚 the motor speed was the expected: 𝜔𝑟=550 𝑟𝑝𝑚; see Figure 5 below. 22 Report Figure 5: Motor speed during open-loop system Figure 6: Direct current and quadrature current The currents values when stabilized are 𝑖𝑑=0 𝐴 𝑖𝑞=5,7 𝐴 The 𝑖𝑑-reference has been set to zero, since there is no flux weakening operation. Study of a PMSM 23 4.3.2. Closed-loop system In the realm of motion control, the task of controlling the speed of a moving object or tool is frequently encountered. The actual speed should be made equal to the set speed. The difference between the actual and set speed is known as the speed error. It is the task of the speed controller to keep the speed error as small as possible, preferably equal to zero. To achieve this result, the controller generates the torque reference. [2] After having analysed the open-loop system, the control part which will close the system will be then implemented. The control algorithms are divided into two general groups. The first group is called scalar control (SC). The other group is called vector control (VC), or field-oriented control (FOC). The FOC technique brings over all improvements in drive performance when compared to the scalar control (higher efficiency, full torque control from zero to nominal motor speed, decoupled control of flux and torque, improved dynamics, etc.). [16] Therefore, high performance motor control is characterized by smooth rotation over the entire speed range of the motor, full torque control at zero speed, and fast accelerations and decelerations. To achieve such control, the Field-Orientated Control (FOC) technique for 3- phase AC motors is used. The basic idea of the FOC algorithm is to decompose the stator current into the magnetic field-generating part and the torque-generating part. Both components can be controlled separately after the decomposition. The structure of the motor controller is then as simple as that for separately excited DC motors. [16] FOC technique is able to control the field and torque of the motor separately. The aim of the control is to regulate motor speed. The control structure of the motor is directly deduced from the yet implemented mathematical equations of the motor. PID controllers will be used. A PID controller continuously calculates an error value e(t) as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms. The general structure of a PID controller is shown in Figure 7. [12] 24 Report Figure 7: Block diagram of a PID controller in a feedback loop where 𝑟(𝑡) is the desired process value, 𝑦(𝑡) is the measured process value and 𝑒(𝑡) is the error 𝑒(𝑡)=𝑟(𝑡)−𝑦(𝑡). In this model, three PI controllers will be used, two in the electrical part (d and q currents) and the other one in the mechanical system (speed). In proportional control, adjustments are based on the current difference between the actual and desired speed, while in integral control, adjustments are based on recent errors. [11] Note that there is no derivative part. The reason is that in the derivative part of a controller, the noise affects the system, making that these small variations of the input influence in a bigger scale in the output. [11] The function of the derivative action is to maintain minimum error correcting it proportionally at the same speed that occurs, thus avoiding the error increase. Nevertheless the derivative action occurs when there is a change in the absolute value of the error (if the error is constant, as in this case, only the proportional and the integral part act). [11] The derivative action, therefore, is appropriate when there is delay between the movement of the control valve and its impact on the controlled variable. When the time derivative action is large, there is instability in the process. It is usually little used because of the sensitivity to noise that manifests and the complications that entails. [11] In order to prove this fact, some graphics have been made. A first order system has been taken; see Figure 8. Study of a PMSM 25 Figure 8: Block diagram of the example below In this case, 𝐻(𝑠) is equal to one. 𝐺𝑃𝐼(𝑠) and 𝐺𝑝(𝑠) are defined as: 𝐺𝑃𝐼(𝑠)=𝐾𝑝𝑠+𝐾𝑖 𝑠 (4.26) 𝐺𝑝(𝑠)=1 𝐿𝑠+𝑅 (4.27) The global function 𝐺𝐶𝐿(𝑠) is defined as: 𝐺𝐶𝐿(𝑠)=𝐺𝑃𝐼(𝑠)·𝐺𝑝(𝑠) 1+𝐺𝑃𝐼(𝑠)·𝐺𝑝(𝑠)=𝐾𝑝𝑠+𝐾𝑖 𝑠·1 𝐿𝑠+𝑅 1+𝐾𝑝𝑠+𝐾𝑖 𝑠·1 𝐿𝑠+𝑅=⋯= 𝐾𝑝𝑠+𝐾𝑖 𝐿𝑠2+(𝐾𝑝+𝑅)𝑠+𝐾𝑖 =𝐾𝑝𝑠 𝐿𝑠2+(𝐾𝑝+𝑅)𝑠+𝐾𝑖+𝐾𝑖 𝐿𝑠2+(𝐾𝑝+𝑅)𝑠+𝐾𝑖 (4.28) Now, if 𝐺1(𝑠)=𝐾𝑝𝑠 𝐿𝑠2+(𝐾𝑝+𝑅)𝑠+𝐾𝑖 and 𝐺2(𝑠)=𝐾𝑖 𝐿𝑠2+(𝐾𝑝+𝑅)𝑠+𝐾𝑖, it is easier to see if the derivative part affects the response, being 𝐺1(𝑠) the derivative part of the function. 32 Report Figure 15: Final torque after the control implemented (12 Nm) Figure 16: Direct current and quadrature current after the control implemented The fast control loop executes these two independent current-control loops: the direct-axis current (𝑖𝑑) is used to control the rotor magnetizing flux, while the quadrature-axis current (𝑖𝑞) corresponds to the motor torque. Study of a PMSM 33 Figure 17: Direct voltage and quadrature voltage with the control The values when the system is stabilized are the same as in the open-loop system, so it can be affirmed that the control is good implemented. 34 Report 5. Sensorless control 5.1. Introduction The use of sensorless control for PMSM is widely used in industrial and offshore applications. By removing the traditional speed sensor, the entire complexity and costs of the motor drive is reduced. A traditional sensor obtains disturbances in the signals at different load conditions for the motor drive. Additionally, the sensor is exposed to the environment. [1] In order to eliminate the rotor position sensor in the rotor permanent-magnet flux oriented controlled drive system, is of great interest to use a rotor position and velocity estimation technique. The method that will be used is called flux linkage method. The method estimates the fluxlinkage, based on stator currents and voltage signals from the previous control of the motor. A prediction-correction approach is used. [3] The rotor position and velocity estimation algorithm require the estimation of the stator flux linkage. Using the estimated stator flux, the stator currents are estimated at a predicted rotor position. The difference between the estimated stator currents and the measured currents are used to correct the error in the predicted rotor position. The block diagram of the algorithm is shown in Figure 18. The number in each block of Figure 18 indicates the execution order of the estimation algorithm. [3] Each step of the algorithm will be discussed. Thus, the aim of this method is not to measure the speed/position directly, but to employ some indirect techniques to estimate the rotor position instead. Study of a PMSM 35 5.2. Flux-linkage method Figure 18: Block diagram of the rotor position and velocity estimation algorithm 36 Report Step 1: Stator flux linkage estimation [1] The first step of the algorithm is to estimate the flux-linkage integrating the difference between the stator voltage and the ohmic voltage drop. As it is in discrete time, a sampling time 𝑇𝑠 is used for the integration. Thus, the stator flux-linkages are obtained as 𝜆𝑞𝑠 𝑒𝑠𝑡(𝑘)=𝑇𝑠·[𝑣𝑞𝑠 ∗(𝑘−1)−𝑟𝑠·𝑖𝑞𝑠(𝑘)]+𝜆𝑞𝑠(𝑘−1) (5.1) 𝜆𝑑𝑠 𝑒𝑠𝑡(𝑘)=𝑇𝑠·[𝑣𝑑𝑠 ∗(𝑘−1)−𝑟𝑠·𝑖𝑑𝑠(𝑘)]+𝜆𝑑𝑠(𝑘−1) (5.2) The phase voltages are not measured and the controller commanded voltages to the machine in the previous sampling period (note that 𝑣𝑞𝑠 ∗(𝑘−1) and 𝑣𝑑𝑠 ∗(𝑘−1) are used). The last term of the equations is the updated value of the estimated flux-linkage (see Step 4). So, those updated values (𝜆𝑞𝑠(𝑘−1) and 𝜆𝑑𝑠(𝑘−1)) in the previous sampling period are used to calculate the stator flux values in the present sampling period. Step 2: Stator current estimation [1] Stator currents can be estimated by analysing the change in rotor angle and the updated flux-linkages. As the inductance varies with frequency, an estimated value of the stator currents can be calculated. 𝑖𝑞𝑠 𝑒𝑠𝑡(𝑘)= [𝐿−𝛥𝐿𝑐𝑜𝑠(2𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))]·𝜆𝑞𝑠 𝑒𝑠𝑡(𝑘)+𝛥𝐿𝑠𝑖𝑛(2𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))·𝜆𝑑𝑠 𝑒𝑠𝑡(𝑘)−(𝐿+𝛥𝐿)·𝜆𝑚sin(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)) 𝐿2−𝛥𝐿2 (5.3) 𝑖𝑑𝑠 𝑒𝑠𝑡(𝑘)= [𝐿+𝛥𝐿𝑐𝑜𝑠(2𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))]·𝜆𝑑𝑠 𝑒𝑠𝑡(𝑘)+𝛥𝐿𝑠𝑖𝑛(2𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))·𝜆𝑞𝑠 𝑒𝑠𝑡(𝑘)−(𝐿+𝛥𝐿)·𝜆𝑚cos(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)) 𝐿2−𝛥𝐿2 (5.4) Where 𝐿=𝐿𝑞+𝐿𝑑 2 and 𝛥𝐿=𝐿𝑞−𝐿𝑑 2. It is noted that the estimated stator currents are dependent on correct estimation of fluxlinkages and inductance. Study of a PMSM 37 Step 3: Position correction [1] The most important part of the algorithm is the correction of the predicted rotor position. By comparing estimated and measured (actual) stator currents, the rotor position can be calculated based on the currents errors. 𝛥𝑖𝑞𝑠 𝑠(𝑘)=𝑖𝑞𝑠(𝑘)−𝑖𝑞𝑠 𝑒𝑠𝑡(𝑘) (5.5) 𝛥𝑖𝑑𝑠 𝑠(𝑘)=𝑖𝑑𝑠(𝑘)−𝑖𝑑𝑠 𝑒𝑠𝑡(𝑘) (5.6) The current errors 𝛥𝑖𝑞 and 𝛥𝑖𝑑 can be obtained by transforming the stationary frame current errors 𝛥𝑖𝑞𝑠 𝑠 and 𝛥𝑖𝑑𝑠 𝑠 to the predicted rotor reference frame as 𝛥𝑖𝑞=𝛥𝑖𝑞𝑠 𝑠·cos(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))−𝛥𝑖𝑑𝑠 𝑠·sin(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)) (5.7) 𝛥𝑖𝑑=𝛥𝑖𝑞𝑠 𝑠·sin(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))+𝛥𝑖𝑑𝑠 𝑠·cos(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)) (5.8) And 𝑖𝑞𝑠 𝑝𝑟𝑒 and 𝑖𝑑𝑠 𝑝𝑟𝑒 as 𝑖𝑞𝑠 𝑝𝑟𝑒=𝑖𝑞𝑠 𝑠·cos(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))−𝑖𝑑𝑠 𝑠·sin(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)) (5.9) 𝑖𝑑𝑠 𝑝𝑟𝑒=𝑖𝑞𝑠 𝑠·sin(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘))+𝑖𝑑𝑠 𝑠·cos(𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)) (5.10) Two position errors 𝛥𝜃𝑞 and 𝛥𝜃𝑑 exist and the position correction term 𝛥𝜃 can be obtained by taking the average of the two position errors: 𝛥𝜃𝑞(𝑘)=𝐿𝑞·𝛥𝑖𝑞 −𝜆𝑚+2𝛥𝐿·𝑖𝑑𝑠 𝑝𝑟𝑒𝑑 (5.11) 𝛥𝜃𝑑(𝑘)=𝐿𝑑·𝛥𝑖𝑑 2𝛥𝐿·𝑖𝑞𝑠 𝑝𝑟𝑒𝑑 (5.12) 𝛥𝜃(𝑘)=𝛥𝜃𝑞(𝑘)+𝛥𝜃𝑑(𝑘) 2 (5.13) The correct rotor position is obtained as 𝛥𝜃𝑟𝑒𝑠𝑡(𝑘)=𝛥𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘)+𝛥𝜃(𝑘) (5.14) 38 Report Step 4: Updating of flux linkages [1] In Step 4, the flux is recalculated using the correct rotor position and the measured stator currents. 𝜆𝑞𝑠(𝑘)=[𝐿+𝛥𝐿𝑐𝑜𝑠(2𝜃𝑟𝑒𝑠𝑡)]·𝑖𝑞𝑠(𝑘)−𝛥𝐿𝑠𝑖𝑛(2𝜃𝑟𝑒𝑠𝑡)·𝑖𝑑𝑠(𝑘)+𝜆𝑚sin(𝜃𝑟𝑒𝑠𝑡) (5.15) 𝜆𝑑𝑠(𝑘)=−𝛥𝐿𝑠𝑖𝑛(2𝜃𝑟𝑒𝑠𝑡)·𝑖𝑞𝑠(𝑘)+[𝐿−𝛥𝐿𝑐𝑜𝑠(2𝜃𝑟𝑒𝑠𝑡)]·𝑖𝑑𝑠(𝑘)+𝜆𝑚cos(𝜃𝑟𝑒𝑠𝑡) (5.16) These updated flux values are used in step 1 of the algorithm in the next sampling interval to estimate the flux. In this way, the integrator drift problems can be avoided in the flux estimation in step 1. Step 5: Prediction of rotor position [1] The position is predicted assuming the position varies with time as a second-order polynomial 𝜃𝑟=𝐴𝑡2+𝐵𝑡+𝐶 (5.17) Using three estimated previous positions, the position at (k+1) sampling instant is predicted using second-order polynomial curve fitting. Figure 19: Rotor position prediction using polynomial curve fitting Study of a PMSM 39 Assuming 𝑡=0 at (𝑘−2) sampling instant, the rotor position can be obtained as 𝜃𝑟𝑒𝑠𝑡(𝑘−2)=𝐶 (5.18) At (𝑘−1) sampling instant 𝜃𝑟𝑒𝑠𝑡(𝑘−1)=𝐴𝑇2+𝐵𝑇+𝐶 (5.19) At (𝑘) sampling instant 𝜃𝑟𝑒𝑠𝑡(𝑘)=𝐴(2𝑇)2+𝐵(2𝑇)+𝐶 (5.20) At (𝑘+1) sampling instant the prediction position is 𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘+1)=𝐴(3𝑇)2+𝐵(3𝑇)+𝐶 (5.21) Solving these equations for A, B and C and substituting on the predicted position equation at (𝑘+1) sampling time, the equation is obtained as 𝜃𝑟𝑝 𝑒𝑠𝑡(𝑘+1)=3𝜃𝑟𝑒𝑠𝑡(𝑘)−3𝜃𝑟𝑒𝑠𝑡(𝑘−1)+𝜃𝑟𝑒𝑠𝑡(𝑘−2) (5.22) In step 5 of the algorithm, using three previously estimated positions, the position in next sampling instant is predicted using the final equation above. 5.3. Problems encountered This algorithm was implemented in MATLAB/Simulink (see Appendix C to see the whole implementation with the program). The difficulty appears when calculating 𝛥𝜃𝑑 in step 3. For small values of 𝑖𝑞𝑠 𝑝𝑟𝑒𝑑, 𝛥𝜃𝑑 can become very large leading to wrong position correction, and therefore 𝛥𝜃. Even though at high loads the problem does not appear, when the machine is operated under low loads the position correction becomes inaccurate leading to fail the position estimation. [1] In order to avoid wrong position correction from very large values of 𝛥𝜃𝑑, limits to the 𝛥𝜃𝑑 can be added. The rotor position variation during one sampling period depends on the speed of the rotor and is is given by  𝑟𝑇, where 𝑇 is the sampling period. Therefore, the maximum value of 𝛥𝜃 can be limited to  𝑟𝑇. However, at no-load the rotor position does not still track the actual rotor position accurately. [1] For non-salient pole machines (SPMSM) the denominator of 𝛥𝜃 does not include any time varying variable. Therefore, position correction of non-salient pole machines does not face 40 Report difficulties as it is seen for salient pole machines. [1] Because of this problem, the position estimation algorithm was investigated neglecting the saliency in the machine. This entails assuming 𝛥𝐿=𝐿𝑞−𝐿𝑑 2  0 and the position error appears only in q-axis equation. This is different from previously discussed salient pole machine (IPMSM) case, where the position error appears in both q and d equations. [1] The new equation for the position error is 𝛥𝜃(𝑘)=−𝐿𝑞·𝛥𝑖𝑞 𝜆𝑚 (5.23) The reason for the position error when the load is increased is due to the assumptions made in the position estimation algorithm. These assumptions are neglecting the saliency when the currents are estimated in step 2 of the algorithm and when updating the flux linkages in step 4 of the algorithm. This makes errors in the current estimation and the updating of flux linkages. The combination of these facts makes increased flux errors, increased current errors and inaccurate position correction when the load is increased in the machine. [1] Thus, the position error is dependent on the load of the machine and when the load is increased (bigger than the rated load) the position error is also increased. Another interesting fact is the effect of the speed. When the speed decreases, the estimation is worse. The reference signal is considered as a position measurement from an ideal sensor. In order to reduce the oscillations, additional filtering or different tuning methods for the controller gains can be applied. This is highly dependent on the criteria given for the operation of the motor drive. One option to avoid noise is by adding low pass filters with a natural frequency of 50 Hz and a damping ratio of 0.707. This will influence the flux linkage estimation in Step 1, and the error will remain until the sampled current loops. [3] The main problem with sensorless control lies in the low speed operations. The position algorithm is based on motor parameters and is sensitive to changes in the signal processing. Study of a PMSM 41 Conclusions PMSM are good candidates due to their attractive efficiency characteristics. Firstly, the whole drive system is simulated by the use of MATLAB/Simulink. With the motor equations, a model for the machine has been developed. The control system requires rotor position feedback in order to perform the self-synchronization function continuously. The basic PI controllers are sufficient for current and speed control in the drive system. The results of the simulation show the good response of the model when tracking a command speed. The design of the controllers has been validated with MATLAB/Simulink. For an IPMSM, the rotor d,q model is the most convenient, since the position dependent inductances disappear in that model. The electromagnetic torque of the IPMSM is not only produced by the permanent-magnet flux but also by the reluctance difference in rotor d- and q-axes. This is different from SPMSMs where the electromagnetic torque is only produced by the permanent flux. In order to achieve sensorless operation of the drive, the position and velocity estimation is required and it is important to consider the type of the machine (SPMSM or IPMSM) and the application of the drive (to know the convenient value of the speed and torque) when investigating a rotor position and velocity estimation technique. The saliency in IPMSMs increases the complexity in the algorithm compared to the SPMSMs. For position sensorless operation of the system a rotor position estimation technique has been investigated. The investigated rotor position estimation technique uses predictorcorrector method, which uses current errors to correct the predicted rotor position. It has shown that, in theory, the expressions exist for both SPMSM and IPMSM to correct the predicted rotor position using current errors. Nevertheless, the correction of the predicted rotor position is difficult for salient pole machines (IPMSM) using the predicted d-q reference frame current errors. For non-salient pole machines (SPMSM), position correction using current errors in the predicted d-q reference frame seems more convenient. The difficult of position correction in the position estimation algorithm for all operating conditions and the relatively low saliency that exists in the IPMSM used for the analysis has led to make assumptions to the position estimation algorithm. In order to improve the performance of the drive, more investigations are still required for accurate rotor position estimation of the IPMSM. 48 Report Figure 21: Power converter Figure 22: Electric block of the open-loop system Figure 23: Electromechanical conversion block of the open-loop system Study of a PMSM 49 Figure 24: Mechanical block of the open-loop system Figure 25: Mechanical block of the control Figure 26: Mechanical-Electrical conversion block of the control Figure 27: Electric block of the control 50 Report Figure 28: Alpha modulator block Figure 29: Closed-loop system with the control implemented Figure 30: Flux-linkage algorithm Study of a PMSM 51 Figure 31: Step 1 of the algorithm Figure 32: Step 2 of the algorithm Figure 33: Step 3 of the algorithm 52 Report Figure 34: Step 4 of the algorithm Figure 35: Step 5 of the algorithm