Florida State University Libraries

Transcription

1 Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2014 A Particle Swarm Optimization Based Maximum Torque Per Ampere Control for a Switched Reluctance Motor Lee Griffin Follow this and additional works at the FSU Digital Library. For more information, please contact lib-ir@fsu.edu

2 FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING A PARTICLE SWARM OPTIMIZATION BASED MAXIMUM TORQUE PER AMPERE CONTROL FOR A SWITCHED RELUCTANCE MOTOR By LEE GRIFFIN A Thesis submitted to the Department of Electrical and Computer Engineering Department in partial fulfillment of the requirements for the degree of Master of Science Degree Awarded: Summer Semester, 2014 Copyright 2014 Lee Griffin. All Rights Reserved.

3 Lee Griffin defended this thesis on July 17, The members of the supervisory committee were: Chris S. Edrington Professor Directing Thesis Petru Andrei Committee Member Pedro Moss Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with university requirements. ii

4 ACKNOWLEDGMENTS First, I would like to express my gratitude to my thesis director, Chris S. Edrington PhD. He provided support, guidance, a vast knowledge of the switched reluctance motor, exceptional material on various subjects, and a spectacular lab. He also supplied me with an office, a computer, and engineering software, all of which proved to be indisposable. Ultimately, it was his suggestions that shaped the direction of this thesis. Without his advise, I believe this thesis would have been less substantial. Finally, I would like to thank him for encouraging me to become a more independent engineer. I would also like to thank Fletcher Fleming for consistently supporting me and my work. He provided useful knowledge of the test bed and of the simulation and was invaluable when editing papers. He was always willing and able to assist, despite his busy schedule. I would also like to thank the SMART program, who funded my degree entirely, and the Electrical and Computer Engineering professors at Florida State Universities Panama City campus: Jerome Barnes PhD, Geoffrey Brooks PhD, and Clay Hughes PhD, who taught me the fundamentals of engineering. iii

5 TABLE OF CONTENTS List of Tables List of Figures List of Abbreviations Abstract vi vii ix x 1 Introduction Motivation Switched Reluctance Machine Maximum Torque Per Ampere and Firing Angles Particle Swarm Optimization Literature Review Fundamentals of Switched Reluctance Motors Introduction Principles of Operation Principles of Control Asymmetric Bridge Converter Hysteresis Current Control Switching Strategy Conventional Control Computational Intelligence Based Methods of Control Particle Swarm Optimization Methods Introduction to Particle Swarm Optimization Optimizing the Firing Angles Optimizing the Current Waveform Genetic Algorithm Method Modeling of the SRM Basics of Modeling Mathematics Obtaining the Inductance Finite Element Analysis Experimental Implementing the Inductance Function Simulation Simulation Results Open-Loop Start Up Closed-Loop Steady State iv

6 6 Experimentation Experimental Set Up Experimental Results Open-Loop Start Up Closed-Loop Steady State Conclusion Discussion Future Work Appendix A The Particle Swarm Optimization Algorithm 44 B Switched Reluctance Machine Specifications 48 C Results of Finite Element Analysis 49 D Inductance Measurement Experiment 51 E Optimal Angles 56 Bibliography Biographical Sketch v

7 LIST OF TABLES 2.1 Comparison of synchronous and switched reluctance machines RMS current, torque, and torque per ampere for the open-loop test RMS current, torque, and torque per ampere for the closed-loop test RMS current, torque, and torque per ampere for the open-loop test RMS current, torque, and torque per ampere for the closed-loop test B.1 4-phase 8/6 SRM specifications D.1 Values used in experiment vi

8 LIST OF FIGURES 1.1 Example of the rotor of an SRM Example of the effects of back EMF Examples of a simple 2/2 SRM Plot of the inductance and torque vs rotor position for the 2/2 SRM of Figure Cross section of a typical 8/6 SRM Inductance functions for an 8/6 SRM for a constant current An asymmetric bridge converter for a 4 phase SRM A block diagram representing control of an SRM motor An example of the current waveform and the firing angles in conventional SRM motor control An example of particle movement in a PSO An example plot of the function given in (3.9) FEA of the flux at various rotor positions and various currents for the A phase FEA analysis of the inductance at various rotor positions and various currents for the A phase The experimental set up used to determine inductance Flux linkage determined experimentally Inductance determined experimentally Open-loop test for conventional angles Open-loop test for optimal angles Closed-loop test total current draw at 500 rpms Experimental set up of the SRM Output speed of the motor in the start up test DC link current drawn in the start up test Torque output of the motor in the start up test vii

9 6.5 DC link current drawn in the closed-loop test for the optimal strategy DC link current drawn in the closed-loop test for the conventional strategy Torque output of the SRM in the closed-loop test C.1 FEA analysis of the torque output at various rotor positions and various currents for the A phase C.2 FEA analysis of the flux at various rotor positions and various currents for the A phase 50 C.3 FEA analysis of the inductance at various rotor positions and various currents for the A phase D.1 A phase flux linkage results from the inductance test D.2 The inductance of the A phase as determined experimentally D.3 B phase flux linkage results from the inductance test D.4 The inductance of the B phase as determined experimentally D.5 C phase flux linkage results from the inductance test D.6 The inductance of the C phase as determined experimentally D.7 D phase flux linkage results from the inductance test D.8 The inductance of the D phase as determined experimentally E.1 Optimal on angle as generated by the PSO in Appendix A E.2 Optimal off angle as generated by the PSO in Appendix A viii

10 LIST OF ABBREVIATIONS SRM: MTA: PSO: FEA: EMF: HCC: GA: ADC: DAC: IGBT: Switched Reluctance Machine Maximum Torque per Ampere Particle Swarm Optimization Finite Element Analysis Electromotive Force Hysteresis Current Control Genetic Algorithm Analog to Digital Converter Digital to Analog Converter Insulated Gate Bipolar Transistor ix

11 ABSTRACT The Switched Reluctance Machine (SRM) is known for being one of the oldest electric machine designs. Unfortunately, it is usually assumed that this implies that the machine is outdated. However with the advent of microprocessors, the SRM has become a suitable option for a number of applications because the shortcomings of the machine can be mitigated with control. Compared to other machines, the SRM is more rugged, has a simpler structure, and is less expensive to manufacture. The machine has two control regions: when the speed of the machine is beneath a value called the base speed and when the speed is above the base speed. The base speed is the speed at which the back electromotive force (EMF) of the motor becomes substantial when compared to the source voltage. In both regions, the turn-on and turn-off angles of the machine can be used to control the machine. This thesis proposes a method of generating optimal turn-on and turn-off angles. The method presented in this thesis is concerned with finding the turn-on and turn-off angles needed to generate maximum torque per ampere (MTA). The strategy applies a particle swarm optimization (PSO) technique that searches for the angles that maximize the inductance of the SRM in order to achieve MTA. The inductance function was obtained via Finite Element Analysis (FEA) and experimentally. The method was applied to a 4-phase 8/6 SRM. The proposed strategy was found to be effective at both low speeds (beneath the base speed) and high speeds (above the base speed), but MTA could only be asserted for low speeds. x

12 CHAPTER 1 INTRODUCTION 1.1 Motivation Switched Reluctance Machine The SRM is one of the oldest designs for an electric machine, dating back to the 19 th century; it is also one of the simplest designs. Similar to the more popular designs, the SRM has a salient stator with windings. However, the rotor of an SRM is much more simplistic. The rotor contains no windings, brushes, or magnets; it is simply a salient piece of iron, Figure 1.1. This simple rotor design makes the machine significantly cheaper to fabricate. Another advantage of the SRM is the fact that the machine s phases operate independently. Torque is generated by pulsing a particular phase which generates a magnetic field. The rotor, attempting to minimize reluctance, generates torque to align with the pulsed phase. Torque can be maintained by independently pulsing the phases. Since the phases are operated independently, they can be controlled independently, and if one phase happens to fail the others can still operate. Furthermore, the SRM can reach higher speeds compared to other common machine designs [1]. While the most common machines in industry are the Induction Machine, the Synchronous Machine, and the Permanent Magnet Machine, the SRM is experiencing an increase in popularity [2] due to the fact that it is cheaper, can obtain higher speeds, and each phase can be controlled independently Maximum Torque Per Ampere and Firing Angles Control of the SRM is dependent on the speed of the rotor. When the rotor reaches a speed defined as the base speed the back electromotive force (EMF) becomes so substantial that the desired current waveform cannot be properly produced, as seen in Figure 1.2. This figure shows the current input to each of the phases of a 4-phase 8/6 SRM at speeds below and above the base speed. Below the base speed the desired current waveform is synthesized. Above the base speed, the waveform cannot be properly synthesized due to the back EMF. In these two speed regions, the control of the machine differs. When the speed is below the base speed, the on and off angles, or firing angles, 1

13 Figure 1.1: Example of the rotor of an SRM for each phase and the shape of the current waveform can be controlled. However, above the base speed only the firing angles for each phase can be controlled. There are many effective methods for controlling an SRM by shaping the current waveform [3], but due to the back EMF issue these strategies are restricted to low speeds applications. However, strategies based off of the firing angles can potentially be applied to all speeds. Controlling the firing angles can affect the efficiency of an SRM generator [4], the power factor of an SRM motor [5], or a number of other performance characteristics. In particular, the strategy presented in this thesis attempts to achieve MTA by optimizing the firing angles. MTA and strategies designed to achieve it have been investigated and developed for a number of machine designs [6]-[7]-[8]. MTA is highly desirable in any application where current draw is an issue. One such example would be a battery powered machine. Applying an MTA strategy would reduce the necessary current input and subsequently extend the battery life with minimal reduction in torque output. Clearly, an effective MTA strategy would be very useful in the battery powered vehicle industry. 2

14 40 Current Input: Below the Base Speed 30 Current (A) time (s) 40 Current Input: Above th Base Speed 30 Current (A) time (s) Figure 1.2: Example of the effects of back EMF Particle Swarm Optimization A drawback of the SRM is its high degree of non-linearity, which makes developing analytical solutions very difficult. A popular alternative to solving problems analytically is to apply an optimization technique. These techniques are search-based methods that seek local or global extremes. There are a number of optimization techniques including evolutionary algorithms, particle swarm optimization (PSO) algorithms, and ant algorithms. When compared, the PSO technique maintains a number of advantages over the other techniques. Namely, PSO algorithms: Converge faster Are simpler to implement Manage problem constraints faster Do not require derivatives 3

15 Have fewer parameters that need to be controlled Due to the many advantages of the PSO algorithm, the technique has become very popular and has been applied in many fields. For example, PSO algorithms have been used in route planning in unmanned aerial vehicles [9], neural network design [10], robotics [11], speech recognition [12], and power systems [13]. In particular, this thesis uses a PSO algorithm to optimize the firing angles of an SRM to achieve maximum torque per ampere. 1.2 Literature Review The SRM is a machine design that has existed since the mid 1800s, although it was not until recently that the machine design began to gain popularity. The machine was originally invented in Scotland to propel a locomotive [14]. However, before the rise of semiconductors the SRM was limited because the switching aspect of the SRM s control had to be achieved with mechanical switches. The modern semiconductor based structure of the SRM was introduced by S.A. Nasar in the 1960s [15]. Then in the 1970s interest in SRMs began to increase which can be attributed to the invention of IGBTs, MOSFETs, and digital signal processors. Maximum torque per ampere (MTA) strategies have been investigated for most machine designs. For the SRM, many of these strategies attempt to generate an optimal current waveform. In [16], it was shown that controlling the current waveform can increase torque production. However, the current waveform also increased torque ripple, and the approach did not consider the possibility of two phases conducting at the same instant, which will occur in practical SRM applications. In [17], a PSO was applied to determine the relationship between current waveforms of the leading and trailing phases, where the leading phase is defined as the phase already conducting and the trailing phase is defined as the phase about to be turned on. This strategy successfully achieved MTA while reducing torque ripple. Regardless, the primary drawback of both these strategies is their speed limitation. Due to the back EMF issue previously discussed, the waveforms generated by both strategies cannot be synthesized properly at high speeds. Strategies based on firing angles, however, can theoretically operate at any speed. Operation of the SRM is highly dependent on the firing angles, which can be controlled to affect a number of parameters, from efficiency [18] to torque ripple [19]. In [20], look-up tables for the firing 4

16 angles were generated with a genetic algorithm in an attempt to maximize torque and efficiency. The tables were speed and current input dependent. The tables were effective but generating the tables required the genetic algorithm to run for each pair of current input and speed. A particle swarm optimization (PSO), [21], algorithm is a strong candidate for determining the optimal firing angles. Due to the advantages previously discussed, PSO algorithms have been applied to many different fields including electrical machines [22]. This thesis uses a PSO algorithm to optimize the firing angles of an SRM in order to maximize torque per ampere at low speeds. 5

17 CHAPTER 2 FUNDAMENTALS OF SWITCHED RELUCTANCE MOTORS 2.1 Introduction A reluctance motor can be defined as an electric motor in which torque is produced by the tendency of its movable part to move to a position where the inductance of the excited winding is maximized" [23]. Typically, the movable part of this machine is made only of iron, shaped to maximize inductance variation with position. The structural simplicity of the machine is one of its most attractive traits; without windings or permanent magnets, the manufacturing cost is lower, while reliability is improved. There are two types of reluctance machines: synchronous and switched. A brief comparison of the two types is shown in Table 2.1. Table 2.1: Comparison of synchronous and switched reluctance machines Switched Reluctance Synchronous Reluctance 1. Stator and rotor have salient poles 1. Stator has a smooth bore, except for slotting 2. Stator winding comprises of a set of coils each of which is wound on one pole 2. Stator has a polyphase winding with sinedistributed coils 3. Excitation is a sequence of current pulses applied to each phase individually 3. Excitation is a set of polyphase balanced sinewave currents 4. As the rotor rotates, the phase flux-linkage should have a sawtooth waveform 4. The phase self-inductance should vary sinusoidally Physically, a switched reluctance machine (SRM) is doubly salient, in that both the rotor and stator have salient poles, with windings on the stator poles. The windings for each phase are independent and are individually excited to generate torque. Each phase of the SRM consists of two stator poles that are opposite of each other. One pole of a phase is excited with some voltage and the opposite pole is simultaneously excited with the opposite voltage. An SRM can have different combinations of rotor and stator poles. For example, this thesis considers an 8/6 SRM, i.e. it has 8 stator poles and 6 rotor poles. As previously stated, each phase consists of two stator poles. Therefore, an 8/6 SRM has 4 phases. 6

18 (a) An aligned 2/2 SRM (b) An unaligned 2/2 SRM Figure 2.1: Examples of a simple 2/2 SRM 2.2 Principles of Operation In an SRM the inductance of the stator windings, or phase inductances, and the output torque will vary with rotor position, θ r. In Figure 2.1, a simple one-phase 2/2 SRM is shown at the aligned position and the unaligned position. Assuming positive rotation is in the counterclockwise direction, the variation of the phase inductance and the torque output with rotor position for the 2/2 SRM with a constant current input is plotted in Figure 2.2. The inductance rises as the rotor moves from the unaligned to the aligned positions and then falls as it moves from aligned to unaligned. Note that the phase inductance will reach a maximum at the aligned position and a minimum at the unaligned position. Due to the fact that an SRM s rotor moves in an attempt to maximize the inductance of the excited winding, the rotor will attempt to align with the stator pole, Figure 2.1a, when the phase is excited. As a result, the torque output will be positive when inductance is on the rise and negative when inductance is falling, as in the torque plot of Figure 2.2. If the current through the phase is constant across all angles, then the positive and negative torque impulses will 7

19 cancel. Thus, to generate positive torque, current must be switched off at or before the aligned position. From Figure 2.1 it can be determined that the plots in Figure 2.2 will be periodic and that the period will be 180. For example, if the aligned position, Figure 2.1a, is defined as θ r = 0 then the rotor would return to the aligned position every 180n, where n is any integer. However, according to torque plot in Figure 2.2, if positive torque is desired, current can only flow in the phase for the 90 of the rotation in which inductance is not falling. Figure 2.2: Plot of the inductance and torque vs rotor position for the 2/2 SRM of Figure 2.1 An equation for torque can be derived by applying Faraday s law to a single phase. First, assume the machine is linear, i.e. the inductance does not vary with current, and that mutual coupling between phases is negligible. Then, by Faraday s law, the terminal voltage of a phase can be expressed as in (2.1). v =Ri+ dλ dt = Ri+ω dλ r dθ r d(li) =Ri+ω r = Ri+L di dθ r dt +ω ri dl dθ r Note that v is the terminal voltage, i is the current, λ is the flux linkage, R is the phase resistance, L is the phase inductance, θ r is the rotor position, and ω r is the angular velocity. Notice the last (2.1) 8

20 term in (2.1). This term is defined as the back emf, given in (2.2). Back EMF and its impact on current synthesis and control were previously discussed in Chapter 1. e = ω r i dl dθ r (2.2) Continuing from (2.1), notice that multiplying by the current will yield the total power input, given in (2.3). P total = vi = Ri 2 +Li di dt +ω ri 2 dl dθ r (2.3) By the law of conservation of energy, mechanical power, given by P m = ω r T e, is what remains after resistive loss and the derivative of magnetic stored energy are subtracted from the total input power. Note that T e is the instantaneous electromagnetic torque. Also recall that resistive loss is given by P r = Ri 2 and magnetic stored energy is given by U m = 1 2 Li2. The derivative of magnetic stored energy can easily be found with the chain rule, see (2.4). ( ) d 1 dt 2 Li2 = 1 2 i2dl dt +Lidi dt = Lidi dt ω ri 2 dl (2.4) dθ r Subtracting resistive losses and (2.4) from (2.3) and solving for torque yields (2.5). T e = 1 dl i2 (2.5) 2 dθ r Although (2.5) and the functions in Figure 2.2 are for a 2/2 SRM, they can easily be expanded to any combination of stator and rotor poles. In particular, consider the 8/6 SRM in Figure 2.3. Recall that a 8/6 SRM has four phases, which will be denoted as the A phase, B phase, C phase, and D phase, each of which will have its own inductance function. However, the inductance function for each phase should have the same shape as the inductance function for the 2/2 SRM. The difference being that the 8/6 SRM s inductance functions will have a shorter period due to the increase in rotor poles. Namely, the period will be T 86 = 360 N r = = 60, where N r is the number of rotor poles. Also, there will be a phase shift between the 4 phases inductance functions due to the physical separation. The phase shift will be equal to the stroke angle, which is given by (2.6). Where m is the number of phases. For an 8/6 SRM note that the stroke angle is 15. Thus the inductance 9

21 functions for a 8/6 SRM will be given by Figure 2.4, assuming that aligned with the A phase is θ r = 0. ǫ = 360 mn r (2.6) Figure 2.3: Cross section of a typical 8/6 SRM Figure 2.4: Inductance functions for an 8/6 SRM for a constant current In order to expand (2.5) to a multi-phase machine, the torque contributions from each individual phase can be linearly added. However, in Figure 2.4 it can be seen that positive torque can only be generated by a maximum of two phases at a time (notice the slopes). Thus, only two phases need 10

22 to be accounted for in the torque equation and these phases can be defined as the leading phase, subscript l, and the trailing phase, subscript t, (2.7). T e = 1 2 i2 l dl l + 1 dl t dθ r,l 2 i2 t (2.7) dθ r,t 2.3 Principles of Control As shown in Figure 2.2, motoring with an SRM requires that current be pulsed when the phase inductance is on the rise and current be zero when the phase inductance is falling. Clearly unlike other machines, the SRM cannot operate as a motor without some amount of control. Specifically, the current through each phase must be controlled. While there are a number of methods to achieve current control, this thesis uses an asymmetric bridge converter, hysteresis current control, and soft chopping. Figure 2.5: An asymmetric bridge converter for a 4 phase SRM Asymmetric Bridge Converter Asymmetric bridge converters are commonly used in machine control. As seen in Figure 2.5, the converter consists of two switch-diode pairs per phase. The switches are controlled to synthesize desired voltage waveforms or current waveforms. Because the SRM is a current driven machine, a current control strategy is typically applied to the converter. In this thesis, a current control strategy known as hysteresis current control is applied. 11

23 2.3.2 Hysteresis Current Control Hysteresis current control (HCC) is relatively easy to implement, although it requires a current feedback. First, an allowable error, called h, and a desired current waveform, called i, must be provided. Then current is measured at the output of the converter and compared to the error bands (i ±hi ) of the desired waveform. If the measured current is below the top error band and rising or below the bottom error band, the converter s switches are set to apply positive voltage, which raises the current. If the measured current is above the top error band the converter s switches are set to apply zero or negative voltage, depending on the switching approach. As a result, this strategy restricts the current to the space in between the error bands, called the hysteresis band. In practice, the current will resemble Figure 1.2 at speeds below the base speed. For typical SRM motoring operations the waveform is a square pulse whenever the inductance is rising Switching Strategy There are two different methods of switching. They are referred to as hard chopping and soft chopping. The two methods differ in how they turn off a phase. An asymmetric bridge converter has the ability to apply positive, negative, or zero voltage to a phase. When a phase reaches its off angle, the current in the phase must be reduced to zero, or negative torque will be generated. Due to the inductance of the coils, the current in a phase cannot be reduced to zero instantaneously. However, negative voltage can be applied to cause the current to fall faster. If negative voltage is applied to reduce the current in a coil, the switching strategy is called hard chopping. The other option is to merely apply zero voltage and let the current die naturally, called soft chopping. In this thesis, soft chopping is applied due to its simplicity Conventional Control Figure 2.6 shows the general control structure of an SRM motor with speed control. Speed control of an SRM motor can be easily achieved with a PI controller. While there are a number of error control methods, PI control is easy to implement, cheap, and sufficient for most motoring applications. Depending on the approach, the PI can generate a torque or current command. In conventional control, a current command is typically generated. The command is then used to generate commanded current waveforms for each phase. The commanded waveforms are then implemented via HCC, which generates the switching logic for the asymmetric bridge converter. 12

24 Figure 2.6: A block diagram representing control of an SRM motor The current waveform generator in Figure 2.6 is the critical piece of the control. The shape of the waveform and the firing angles used can greatly affect the performance of the machine. In conventional control, the shape is a square pulse of height i. The firing angles in conventional control are based off of the inductance function. Specifically, the turn-on angle is set equal to the unaligned point and the turn-off angle is set equal to the aligned point. This way positive torque is guaranteed. Typically, by convention aligned with the A phase is θ r = 0 and considering that the inductance function will be periodic by 60 then the unaligned position should be at θ r = 30. Thus the firing angles for the A phase are θ on = 30 and θ off = 60. The firing angles for the other phases are merely the A phase angles shifted by 15, 30, and 45 for the B, C, and D phase respectively. The ideal current waveforms for conventional control can be seen in Figure

25 Figure 2.7: An example of the current waveform and the firing angles in conventional SRM motor control 14

26 CHAPTER 3 COMPUTATIONAL INTELLIGENCE BASED METHODS OF CONTROL As discussed in Chapter 2, conventional control of an SRM motor applies HCC with firing angles of θ on = 30 and θ off = 60, with respect to the A phase, and a square demand current. However, conventional control is not guaranteed to be optimal but is guaranteed to generate some amount of positive torque. The firing angles of an SRM can be controlled to affect efficiency, torque production, torque per ampere, and more. For example, in [20] a genetic algorithm is applied to determine the firing angle that will generate maximum torque and efficiency, and in this thesis a PSO is applied to determine firing angles that will generate MTA. An alternative to adjusting the firing angles, the current waveform can also be optimized. In [3] a PSO is used to generate the optimal current waveform for MTA and minimal torque ripple. 3.1 Particle Swarm Optimization Methods Introduction to Particle Swarm Optimization Particle swarm optimization (PSO) was originally developed by James Kennedy and Russel Eberhart [24], [25]. The method models the ability of some groups of animals to behave more intelligently than the individual animal. This perceived increase in intelligence is called emergence. Some examples of emergence are a flock of birds, a school of fish, or a pod of humpback whales. Studies into the behavior of these animal groups have shown that their behavior can be attributed to each individual animal following a simple behavior [26]. Similarly in a PSO, the individuals in the swarm, called particles, follow a simple behavior: emulate the success of neighboring particles. A PSO begins by initializing a number of particles to random or defined locations and then the particles move through the given search space with some velocity. The basic behavior of the particles are governed by (3.1) and (3.2). Where V i is the velocity of the ith particle, x i is the position of the ith particle, pb i is the best solution found by the ith particle, w is the inertial weight, nb is the neighborhood best, c 1, c 2, r 1, and r 2 are constants, and n is the generation. Notice that the 15

27 Figure 3.1: An example of particle movement in a PSO velocity depends on the past velocity, the location of the best solution a particle has found, and the best solution found in its neighborhood of particles. As a result, the particles will converge on the optimal solution after a number of generations. The inertial weight is included to increase the particle s exploration, which is its ability to thoroughly explore the search space. The weight acts as mass and causes the particle to resist changes in its velocity. V i (n+1) =w(n)v i (n)+c 1 r 1 (pb i (n) x i (n)) +c 2 v 2 (nb(n) x i (n)) (3.1) x i (n+1) = x i (n)+v i (n+1) (3.2) A particle s neighborhood is the group of particles that the particle shares information with. It is between a particle and its neighborhood that the neighborhood best, nb, is determined. The simplest neighborhood is one where the entire swarm is the neighborhood. Such a neighborhood has a high degree of connectivity between particles and will converge on a solution faster. However, this neighborhood is also more susceptible to local extremes. Other neighborhood types have been proposed and investigated [27], [28]. 16

28 f(x 1,x 2 ) = x x cos(3πx 1 ) 0.4cos(4πx 2 )+0.7 (3.3) Figure 3.1 shows an example of particle movement in a PSO. The algorithm is attempting to minimize (3.3), which is a common benchmark problem referred to as Bohachevsky 1. The minimum value of the function is know to be zero which is located at the origin. This can easily be confirmed by substituting x 1 = 0 and x 2 = 0 into (3.3). In this case, the particles are initialized at defined locations and after a few iterations the particles converge around the global minima at (0,0) Optimizing the Firing Angles This thesis applies a PSO algorithm in order to optimize the variables θ on,t and θ off,l to achieve MTA. These variables refer to the on angle for the trailing phase and the off angle for the leading phase. The leading phase is defined as the phase that has been on longer and the trailing phase is defined as the phase that has more recently been turned on. For example, in Figure 2.7, at 30 the A phase turns on and is considered the trailing phase until the D phase turns off at 45. At this point the B phase is considered trailing and the A phase is leading. In order to apply a PSO algorithm, a function to optimize must be derived. In this case, torque per ampere as a function of the firing angles is desired. This function can be derived by considering the equation for the torque output of the motor. Recall the equation for torque derived in Chapter 2, given again in (3.4). T e = 1 2 i2 l dl l + 1 dl t dθ l 2 i2 t (3.4) dθ t Where subscripts t and l denote the trailing and leading phases, respectively. Recall that for a 8/6 SR, θ t = θ l +15 and consequently dθ t = dθ l. Also, if it is assumed that i l = i t = i, then (3.4) simplifies to (3.5). 2T i 2 = dl l dθ l + dl t dθ l (3.5) Notice that (3.5) shows that the sum of the derivatives, dl l dθ l and dlt dθ l, corresponds to the torque per ampere. Hence, if the right hand side of (3.5) is maximized, then torque per ampere will be maximized. f(θ l ) = dl l dθ l + dl t dθ l (3.6) 17

29 However, (3.6) does not depend on the desired values, θ on,t and θ off,l. Therefore instead of maximizing the instantaneous value, the average value over the range θ on,l to θ off,l can be maximized. This corresponds to the integral of f(θ l ), yielding (3.7) and (3.8). θoff,l θ on,l f(θ l )dθ l = θoff,l θ on,l dl θoff,l l dl t dθ l + dθ l (3.7) dθ l θ on,l dθ l ˆf(θ l ) = L l (θ off,l ) L l (θ on,l )+L t (θ off,l ) L t (θ on,l ) (3.8) Notice the fourth term on the right hand side of (3.8), L t (θ on,l ). Recall that by definition θ on,t will occur after θ on,l. There is no need to consider the inductance contribution from the trailing phase until it is turned on. Thus the term L t (θ on,l ) can be replaced with L t (θ on,t ). A critical assumption made is that the optimization does not optimize the angles for each individual phase, but optimizes the relationship between each phase and its respective trailing phase. That is to say that for each phase, it is assumed that the turn-on angle is known and the optimization is ran to determine the optimal off angle for the current phase, θ off,l, and the on angle for that phase s corresponding trailing phase, θ on,t. For clarification, consider Figure 2.7. Assume that the C phase is being optimized, which means that the C phase is the leading phase and the D phase is the trailing phase. Essentially, the algorithm will only consider the range of 0 to the off angle generated for C; in Figure is shown. Once the optimal off angle for C and on angle for D are determined, the algorithm would then move to optimizing the D phase, i.e. the D phase is leading and the A phase is trailing. Thus, each run of the algorithm only needs to consider from the leading phase s on angle to the leading phase s off angle. This is why in (3.7) the function is integrated over θ on,l to θ off,l. ˆf(θ l,off,θ t,on ) = L l (θ off,l ) L l (θ on,l )+L t (θ off,l ) L t (θ on,t ) (3.9) Thanks to the simplicity of (3.9), the PSO algorithm applied in this thesis is relatively straightforward. In the algorithm: Particles are randomly initialized The neighborhood includes the entire swarm 18

30 Velocity is clamped, or limited, and the max velocity is non-linearly decreased over iterations Inertial weight is applied as a randomly generated number Both of the variables are limited to θ min and θ max, which are determined as described below The search space bounds, θ min and θ max, are implemented to prevent the algorithm from returning angles that are not feasible. First, the search space is restricted to positive angles, such that θ min = 0. The second constraint is not as simple. For discussion consider Figure 3.2. It represents a possible plot of the sum of L l (θ off,l ) and L t (θ off,l ), the portion of (3.9) that depends on θ off,l. Since these terms are positive the PSO algorithm will attempt to maximize this function. Figure 3.2 shows that mathematically the maximum angle is around 51. Unfortunately, this is not a physically feasible solution. At this point the slope of the curve, dl dθ l, shifts from positive to negative and according to (2.5) negative dl dθ l indicates negative torque. Current cannot be reduced to zero instantaneously and thus if 51 is set as the off angle negative torque will be generated. To avoid this situation, the algorithm first finds the zero crossings of the derivative of the inductance function, L l (θ off,l ) + L t (θ off,l ). Then the angle associated with the zero crossing that transitions from positive to negative is selected. Finally, this angle minus one degree is set as the upper bound of the search space, θ max. In summary, this thesis applies a PSO to the function given by (3.9) in order to determine the optimal firing angles for MTA. The PSO algorithm is given in Appendix A for reference. Due to the fact that (3.5) is integrated to obtain the objective function (3.9), the resulting angles will not necessarily generate instantaneous MTA but will generate average or rms MTA. Also, note that this approach requires that the inductance function for at least one phase is known and assumes that the currents through the leading and trailing phase are equal, namely the demanded current waveform must be a square wave which is only achievable at speeds below the base speed. Also, if it can not be assumed that the phases are balanced, the inductance function for all of the phases must be obtained or approximated. The inductance function used for (3.9) is found as discussed in Chapter Optimizing the Current Waveform A PSO can also be used to optimize the current waveform, as opposed to optimizing the firing angles. In [3], an algorithm similar to the one used in this thesis is used to obtain MTA, however, 19

31 4 x 10 3 Sum of Leading and Trailing Intuctance vs Off Angle Inductance (L) Off Angle (Degrees) Figure 3.2: An example plot of the function given in (3.9) the algorithm generates an optimal current waveform instead of optimal firing angles. This leads to a difference in objective functions. While the angles based algorithm maximizes (3.9), the waveform based algorithm attempts to minimize copper losses. Recall that the power dissipated in a resistor is given by P r = I 2 R and that at some angles two phases can be firing in an SRM motor. Thus, total copper loss is given by (3.10), which is the function minimized by the waveform based algorithm. P copper = I 2 tr+i 2 l R (3.10) Other than the difference in objective functions, there are a few key differences between the waveform based algorithm and the angles based algorithm. While the angles based algorithm is relatively basic, the waveform based algorithm requires some additional components. First, the waveform based algorithm requires more constraints. Both algorithms apply bounds to the search space, essentially limiting the variables to a range of values, however, the waveform based algorithm requires a torque based constraint and a location based constraint as well. The torque based constraint requires that potential solutions generate a certain amount of torque. For example, on each iteration the particles of the waveform algorithm evaluate (3.4). If a particle s location does 20

32 not generate a certain amount of torque or more, then the particle s location is removed as a viable solution. The location based constraint requires that the optimization problem be divided into two parts, namely, below and above the critical angle, or the angle where the leading phase begins to fall. At angles below the critical angle, the problem is constrained to only allow solutions where the trailing phase is increasing. At angles above the critical angle, the problem is constrained to only allow solutions where the leading phase is decreasing. Finally, the waveform algorithm requires a randomized particle swarm (RPS). This simply randomly resets some of the particle positions after a number of iterations. This encourages particles to explore new locations in the search space. 3.2 Genetic Algorithm Method A genetic algorithm is a type of evolutionary algorithm, which are algorithms that model the process of evolution. The process of modeling evolution involves applying evolutionary concepts such as inheritance, mutation, selection, and crossover. For example, in a GA the best solutions are encouraged to reproduce over generations, while the poorer solutions are less likely to reproduce, mimicking the concept of selection. In [20] a genetic algorithm (GA) was applied in order to optimize the firing angles for maximum torque. The algorithm was ran for various currents and speeds, resulting in look-up tables for both of the firing angles. It was found that a range of values were acceptable for the turn-on angles. So the algorithm was re-applied in order to maximize both torque and efficiency. Once again, angles were generated for various currents and speeds. While the algorithm successfully generated optimal angles, the algorithm took hours to run. 21

33 CHAPTER 4 MODELING OF THE SRM 4.1 Basics of Modeling The SRM is modeled in Simulink. The characteristics of the machine that are required to model the SRM are the phase resistance, R x, the self inductance of the phase, L x, and the moment of inertia of the machine, J. Typically, the quantities R x and L x are considered to be uniform across the phases, i.e. each phase has equal resistance and inductance. However, this is not necessarily true in practical applications. The variables that are desired to be modeled by the simulation are the voltages across each of the phases, v x, the current through each of the phases, i x, the flux linkage of each phase, λ x, the total torque output, T e, the speed of the rotor, ω r, and the position of the rotor, θ r. Note that x represents any phase (e.g i A or i B ) and that L x, v x, i x, λ x, and T e will all be functions of rotor position, θ r, but for simplicity of notation are written as f as opposed to f(θ r ) Mathematics There are three equations that govern the behavior of an SRM: Faraday s Law, the torque equation, and Newton s 2nd law. These equations can be used to determine all of the desired variables from the characteristics of the machine. Modeling begins with the voltage input to the phases, v x. The voltage input will be determined by some HCC type control given a demanded current waveform and thus can be assumed to be known. Faraday s law, (4.1), can then be applied to determine the current in the phases, i x. Solving for current in (4.1) gives (4.2). v x = R x i x + dλ x dt = R x i x +L x di x dt +ω ri x dl x dθ r (4.1) 1 i x = L x ( ) dl x v x ω r i x R x i x dt (4.2) dθ r Once current is obtained via (4.2) it can be used to solve for torque and flux linkage. Also, notice that current must be fed back to (4.2) and the HCC. Flux linkage can be calculated with (4.1) or 22

34 (4.3). The torque generated by each phase, T x, can be found with (2.5) and the total torque, T e, is given by the sum of the individual torques. λ x = L x i x (4.3) ΣT = Jα r (4.4) ΣT = T e T l Bω r (4.5) Torque can then be used to solve for the speed and the position by applying Newton s second law. Recall that Newton s second law for rotational frames can be given as (4.4). Where ΣT is the sum of all the torques present, α is the rotational acceleration, and J is the moment of inertia. The sum of the torques can be given as (4.5). Where T l is the torque load, ω r is the derivative of the position, θ r, and B is the viscous coefficient. Thus (4.6) and (4.7) can be applied to solve for the last of the desired variables. Note that ω r must be fed back to (4.2). Te T l Bω r ω r = dt (4.6) J θ r = ω r dt (4.7) Notice that this approach to modeling requires that R x, J, and L x be known. R x can easily be found with a multi-meter or an LCR meter and J can be estimated or found experimentally. However, the phase inductances, L x, are more difficult. In most machines, L x will vary with current. However, this variance is typically ignored. In an SRM the variance is too significant to be ignored. Thus, L x must be found not only as a function of position but also as a function of current, L x (θ r,i). Two different methods of obtaining inductance were applied. The first method was to apply an FEA software to determine the inductance. The second method was to experimentally determine inductance. 23

35 4.2 Obtaining the Inductance Finite Element Analysis Finite element analysis (FEA) is a powerful tool that can be used to analyze systems. The process works by generating a mesh that divides the problem into a number of smaller problems. Each point in the mesh contains the properties of the system at that point and the system is solved numerically at each of these points. Meshes can be tuned to be more dense at problematic locations and less dense at locations of minimal interest. The result is a highly adaptive method for solving complex and nonlinear systems. A software produced by Infolytica called MagNet was used to apply FEA to the SRM. The SRM built in MagNet was built according to the specifications given in Appendix B, which are the specifications of the machine used in experimentation. The machine is a 4-phase 8/6 SRM rated for 1 kw of power. Due to the fact that the functions are periodic, sweeps were performed over 60 for various currents. Torque and flux linkage were calculated for one phase, the A phase. All of the data calculated via FEA can be found in Appendix C. Note that aligned with the A-phase is defined as θ r = 0. Figure 4.1: FEA of the flux at various rotor positions and various currents for the A phase 24

36 In Figure 4.1, the flux as calculated by MagNet is shown for various currents. The plot is as expected with flux linkage peaking at the aligned position, 0, and then reaching a minimum at the unaligned position, 30. In order to determine inductance, recall that flux linkage is given by (4.3). Thus, the values seen in Figure 4.1 were divided by the respective currents. This yielded Figure 4.2. Figure 4.2: FEA analysis of the inductance at various rotor positions and various currents for the A phase Due to the fact that Figure 4.1 was only divided by constants, the inductance maintains a shape similar to the flux linkage, which is as expected. Also, notice that while Figure 4.2 is not identical to the ideal inductance function in Figure 2.4, they share the same general shape and maximum and minimum locations. Furthermore, notice the inductance is on the rise from 30 to 60 and thus will be producing positive torque, which is as expected Experimental While the FEA approach is quick and capable of determining a large number of variables, experimental data is more likely to accurately describe the machine and does not require software. The experimental approach is based on Faraday s equation, (4.1), which can be rearranged in order to find flux linkage, (4.8). According to this equation, flux can be determined experimentally by 25

37 allowing current, i x, to rise due to a voltage, v x, over some time, t. These three values can then be used to calculate flux linkage. λ x = (v x R x i x ) dt (4.8) It would be preferable if current rose naturally without any control. This way each instant in time could be correlated to a single current. Due to the small resistance of the phase windings, R x 71 mω, uncontrolled current would quickly reach excessive amounts. Thus, in order to limit current to values appropriate for the machine, a series resistance must be introduced, R s. In particular, a 2 kw 1 Ω resistor was used. The acutal resistance was measured as 1.16 Ω. The experimental set up for such a test is given in Figure 4.3. Figure 4.3: The experimental set up used to determine inductance Recall that when current flows through a phase the rotor will attempt to align with that phase. This is undesirable because flux linkage is needed at specific angles, θ r. Thus, the rotor needs to be locked but it also needs to be swept through various angles. The most convenient method of doing this is by attaching an index head. The head will lock the rotor but still allow for the rotor to be moved to the desired angle. The process of the experiment was as follows: 1. A 1 Ω resistor was added in series with a phase 2. Current was allowed to flow through the phase in order to align the rotor and the encoder was set to zero 26

38 (a) Phase A (b) Phase B (c) Phase C (d) Phase D Figure 4.4: Flux linkage determined experimentally 3. The index head was tightened onto the rotor 4. The switch was opened and current was allowed to fall to zero 5. The switch was closed for 0.01 seconds and current rose according to Ohm s law 6. Voltage, current, time, and rotor position were captured Steps 5 and 6 were repeated for the mid-aligned position, θ r = 15, and the unaligned position, θ r = 30, for all phases. Once all data was obtained, flux linkage for each phase at the various positions was calculated with (4.8). Exact parameters of the test are given in Table D.1. Figure 4.4 shows the flux versus current for each phase at three positions: aligned, mid-aligned, and unaligned. Notice there is a significant difference between the flux of each phase at low current. This confirms that the machine is not perfectly balanced and that each phase will have slightly different inductance. Also, compare Figure 4.4a and Figure 4.1. Notice that the two agree until the higher currents, 20 A to 25 A. This implies that the inductance begins to saturate sooner than the FEA model predicts. 27

39 (a) Phase A (b) Phase B (c) Phase C (d) Phase D Figure 4.5: Inductance determined experimentally The inductance was calculated from the flux with (4.3). The results are given in Figure 4.5. Clearly, the machine is not perfectly balanced. Notice that at lower currents the inductance of each phase is not equal. However, as current increases, the inductance functions become more comparable. In fact, at 25 A and 30 A the plots become indistinguishable. All data from experimental analysis of the flux linkages is given again in Appendix D Implementing the Inductance Function Implementing the inductance function in simulation can be done many different ways. One method is to perform a sweep from 0 to 60 using one of the previously described methods. Then a look-up table could be generated with the rotor positions and current values. While this method is effective, it is time consuming to obtain entire sweeps. Another option is to use a truncated Fourier series approach. 28

40 L k (θ,i) = L 0 (i)+l 1 (i)cos(6θ +2φ k )+L 2 (i)cos(12θ +2φ k ) (4.9) In [29], it is shown that the phase inductance of an SRM can be approximated with a truncated Fourier series, as in (4.9). Where k represents the phase, i.e. A, B, C, D, etc. The term φ k is the shift between phases, which depends on the geometry of the machine, specifically the shift depends on the stroke angle, (2.6). Each phase of the machine will be out of phase by the stroke angle. For a 8/6 SRM the shifts will be 0, 15, 30, and 45 for the A, B, C, and D phases respectively. In order to apply this approach the Fourier coefficients, L 0 (i), L 1 (i), and L 2 (i) must be determined. These terms can be determined by substituting θ values into (4.9). For example, allowing θ to equal 0, 15, and 30 yields (4.10). Note that these are the aligned, mid-aligned, and unaligned positions for the A phase. Converting (4.10) to matrix notation and solving for the Fourier coefficients yields (4.11). Thus, the Fourier method of approximating the inductance requires that the inductance only be found at three points. L a (i) = L 0 (i)+l 1 (i)cos(φ k )+L 2 (i)cos(2φ k ) L m (i) = L 0 (i) L 1 (i)sin(φ k ) L 2 (i)sin(2φ k ) (4.10) L u (i) = L 0 (i) L 1 (i)cos(φ k ) L 2 (i)cos(2φ k ) L 0 (i) 1 cos(φ k ) cos(2φ k ) L 1 (i) = 1 sin(φ k ) sin(2φ k ) L 2 (i) 1 cos(φ k ) cos(2φ k ) 1 L a (i) L m (i) (4.11) L u (i) In this thesis, the inductance function was implemented with the Fourier series method. The inductance data was based off of experimental data taken from the machine as described in 4.2.2, the results of which are given in Appendix D. The algorithm described in and given in Appendix A was used to generate the optimal angles for the SRM used in experimentation and simulation. Due to the fact that the inductance varies with current, the optimal angles vary with current. Thus, the algorithm was run for various currents. The angles resulting from the algorithm are given in Appendix E. Note that for these angles θ r = 0 is when the respective phase is aligned. 29

41 CHAPTER 5 SIMULATION 5.1 Simulation Results In order to determine the capabilities of the proposed control strategy, two tests are performed. The first test is in order to determine the capability of the proposed strategy at start up. The second test is intended to analyze the performance of the strategy at steady state. Each test is performed for the conventional angles of 30 and 60 and for the optimal angles generated by the PSO, Appendix E. Note that the DC link current refers to the sum of the currents in each phase or the total current that would be drawn out of the DC link Open-Loop Start Up This test is an open loop test and does not use any speed control. Instead, the SRM is allowed to run up to speed uncontrolled. The test is performed at 40 V and for 10 A, 15 A, and 25 A square current waveforms. Also, the machine is loaded with 0.3 N m. The results of this test for the conventional strategy are shown in Figure 5.1 Curiously, the steady state speed of the machine tends to fall as the size of the current waveform rises. This is because it is not the back EMF that is limiting the speed but the negative torque being generated. The conventional strategy has acceptable performance at low speeds. However, as the speed of the machine increases the conventional strategy begins to generate negative torque. This is due to the fact that the turn-off angle is so late that the current does not have enough time to fall to zero before the negative torque region is reached. Eventually, the amount of negative torque will equal the amount of positive and the machine will cease to accelerate. Depending on the voltage, current, and torque load this may be before or after base speed is reached. Notice that as the current increases so does the amount of negative torque, and as a result the negative torque will equal the positive torque sooner. Thus, as the current is increased in Figure 5.1 the steady state speed falls. 30

42 The other quantities in Figure 5.1 are as expected. Torque peaks at an initial start up torque but eventually settles at a steady state value. The total current draw rises over time because of the issue of the turn-off angle. When the machine begins to move faster, the phases begin to fire faster and there tends to be residual current in the phases that should be off. In Figure 5.2 the results for the optimal strategy are shown. The first notable difference is the speed. The optimal strategy reaches much higher speeds than the conventional strategy. This can be attributed to the lack of negative torque. Furthermore, the total current begins to fall over time. This is due to the machine reaching the base speed. As discussed, at the base speed the back EMF becomes so large that the current waveform cannot be synthesized properly. As shown in Figure 1.2, this causes current to fall, which will continue to fall as the machine speeds up. This results in the total current draw plots seen in Figure 5.2. Comparing Figure 5.1 and Figure 5.2, the initial torque production of the optimal strategy is slightly lower than the torque seen in the conventional strategy. However, the optimal strategy draws significantly less current than the conventional strategy, both initially and as the machine begins to reach steady state. This indicates that the optimal strategy generates higher torque per ampere. Table 5.1 shows rms values calculated with the MATLAB command rms. Table 5.1: RMS current, torque, and torque per ampere for the open-loop test At 10 A Value Optimal Conventional DC Link Current (RMS) A A Torque Output (RMS) Torque per Ampere (RMS) N m N m N m A N m A At 15 A Value Optimal Conventional DC Link Current (RMS) A A Torque Output (RMS) Torque per Ampere (RMS) N m N m N m A N m A At 25 A Value Optimal Conventional DC Link Current (RMS) A A Torque Output (RMS) Torque per Ampere (RMS) N m N m N m A N m A 31

43 (a) Speed for 10 A (b) Torque for 10 A (c) Current for 10 A (d) Speed for 15 A (e) Torque for 15 A (f) Current for 15 A (g) Speed for 25 A (h) Torque for 25 A (i) Current for 25 A Figure 5.1: Open-loop test for conventional angles 32

44 (a) Speed for 10 A (b) Torque for 10 A (c) Current for 10 A (d) Speed for 15 A (e) Torque for 15 A (f) Current for 15 A (g) Speed for 25 A (h) Torque for 25 A (i) Current for 25 A Figure 5.2: Open-loop test for optimal angles 33

45 (a) Conventional at 0.3 N m (b) Conventional at 0.5 N m (c) Conventional at 1 N m (d) Optimal at 0.3 N m (e) Optimal at 0.5 N m (f) Optimal at 1 N m Figure 5.3: Closed-loop test total current draw at 500 rpms Closed-Loop Steady State This test maintains the SRM at 500 rpms with a PI loop in order to analyze the machines performance at steady state. The machine is given 40 V and is loaded with 0.3 N m, 0.5 N m, and 1 N m. Speeds above the base speed are not considered because the strategy cannot guarantee MTA at high speeds. This is because the assumption that i t = i l is not necessarily true at high speeds. Figure 5.3 shows the results. It is important to note that the data shown is at steady state, i.e. the velocity is 500 rpms plus or minus 1 percent. The time it took the machine to accelerate to 500 rpms is ignored because this test is only concerned with the performance at steady state. It is very apparent from Figure 5.3 that the optimal strategy draws less current. Also, because this is at steady state, the torque output is the same for both strategies. Namely, the torque output is equal to the torque load, otherwise the machine would be accelerating. Thus, the optimal strategy produces more torque per ampere because it draws less total current. This can be confirmed by finding the rms values of the current and torque. 34

46 Table 5.2: RMS current, torque, and torque per ampere for the closed-loop test At T l = 0.3 N m Value Optimal Conventional DC Link Current (RMS) A A Torque Output (RMS) Torque per Ampere (RMS) N m N m N m A N m A At T l = 0.5 N m Value Optimal Conventional DC Link Current (RMS) A A Torque Output (RMS) Torque per Ampere (RMS) N m N m N m A N m A At T l = 1 N m Value Optimal Conventional DC Link Current (RMS) A A Torque Output (RMS) 1.02 N m 1.02 N m Torque per Ampere (RMS) N m A N m A Table 5.1 and Table 5.2 show the rms torque per ampere generated in the two tests. In both situations it is quite clear that the proposed strategy generates higher torque per ampere than the conventional strategy. 35

47 CHAPTER 6 EXPERIMENTATION 6.1 Experimental Set Up The primary means of controlling the SRM is through the Digital Signal Processing and Control Engineering (dspace TM ) The DS1104 is one of a group of boards designed to test control structures on the intended hardware. A control can be built in MATLAB/Simulink and can then be compiled onto the DS1104, which uses many different forms of I/O to interface with the hardware. The simulated control structure is then applied and tested on the hardware and any changes necessary can easily be corrected in MATLAB/Simulink and then recompiled onto the board. dspace also provides an interface software, dspace Control Desk. The environment is used to control variables in the MATLAB/Simulink model during run time and to capture data such as speed, torque, current, etc. The DS1104 has 8 ADCs, 8 DACs, and 20 parallel I/Os with a 60 bit processor. The DS1104 is used to perform all necessary calculations. Referring back to Figure 2.6, the dspace board encompasses the adder, the PI, the current waveform generator, and the HCC. Consider Figure 6.1. This figure depicts the set up of the SRM test bed used in experimentation. The encoder, on the very left, is a H25 Incremental Encoder from BEI which has a resolution of 0.1. The encoder measures the rotor position and the speed of the rotor. The position is fed back to the DS1104 and is used in HCC, while the speed is fed back to be used in the PI loop and to be captured. On the right of the encoder is the SRM, the specifications of the SRM are given in Appendix B. The current fed into the SRM comes from the IGBTs that are controlled by the DS1104, note 2SD316EI IGBT drivers from Concept are used. Furthermore, the current flowing in the phases are measured with Hall effect based current sensors LA55-P from LEM. The measured current is fed back to the DS1104 for use in HCC and to be captured. The torquemeter shown is a MCRT 48201V non-contact DC operated torquemeter from S. Himmelsteing and Company. The torque is not used in control but is fed back to the DS1104 to be captured. The servo motor, on the very right, is a SVL-210 motor provided by Sureservo. The servo motor is used to provide a torque load and receives torque commands from the DS1104. Finally, a DCS40-75f DC power source from 36

48 Sorenson is used to supply power. The source charges a capacitor bank which in turn feeds the IGBTs. The power supply and the capacitor bank are not shown in Figure 6.1. Current drawn from the Sorenson is plotted in this chapter as DC link current. Figure 6.1: Experimental set up of the SRM 6.2 Experimental Results As in simulation, there are two tests performed. The first test is the open-loop start up test and the second test is the closed-loop steady state test. In simulation, these tests were performed at various currents and torque loads. However, the experimental tests only consider the tests at 15 A and at a torque load of 0.3 N m Open-Loop Start Up As in simulation, this test consists of allowing the SRM to run without any speed control. The machine is allowed to run for 3 seconds, then the rms torque and current values are divided to 37

49 determine the torque per ampere over that time. The test is ran with a DC voltage of 40 V and with a commanded current of 15 A square waves. Also, the machine is loaded with 0.3 N m. Note that conventional control uses firing angles of 30 and 60 and the optimal control uses the angles generated by the PSO algorithm, Appendix E. Figure 6.2: Output speed of the motor in the start up test In Figure 6.2 the speed of the SRM over time is given. At low speeds the optimal and conventional angles rise at nearly the same rate. This implies that torque production is equal because torque is related to acceleration. At high speeds, the conventional strategy saturates at around 2000 rpms. This is exactly what was seen in simulation and just as in simulation, this is not due to the back EMF. As discussed, this is caused by the conventional turn-off angle being too late for the current to fall to zero before the negative torque region is reached. Figure 6.3 shows the total current drawn by the SRM over time. It is apparent that the optimal strategy significantly reduces the total current draw. In Figure 6.4 the torque output of the motor is plotted. The optimal strategy produces more start up torque and more steady state torque. In both the optimal and conventional strategies the machine is loaded with 0.3 N m. Both strategies should settle at this value as it approaches steady state, however, the optimal strategy does not. The difference in the steady state torque is due to air resistance, friction, etc., which usually vary with speed in machine applications, for example T air = Bω. Regardless of the reason, the optimal strategy still produces more torque with less current, which results in higher torque per ampere. 38

50 Figure 6.3: DC link current drawn in the start up test Figure 6.4: Torque output of the motor in the start up test Closed-Loop Steady State In this test, the machine is maintained at a specific speed by using a PI speed loop. This allows the performance of the strategies to be examined while operating at specific speeds. In order to maintain consistency, this PI loop is the same as the one used in simulation. Namely, the P and I gains are 1 and 0.1 respectively. For this test, 40 V, a speed demand of 500 rpms, and a load of 0.3 N m is used. Also, the time it took the machine to accelerate to 500 rpms is ignored because 39

51 Figure 6.5: DC link current drawn in the closed-loop test for the optimal strategy Figure 6.6: DC link current drawn in the closed-loop test for the conventional strategy the performance of the strategies at start up were already considered in the last test. In Figures 6.5 and 6.6 the total current draw of the two strategies are shown. Clearly, the optimal strategy draws as little as 5 A at times, while the average current draw of the conventional strategy appears to be significantly higher. In order to make the comparison easier, the rms values of the DC link currents are given in Table 6.2. As seen, the optimal angles reduce the current draw by approximately 1 A. 40