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Dynamic phasor modeling of doubly-fed induction machines including saturation effects of main flux linkage 2012

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 DYNAMIC PHASOR MODELING OF DOUBLY-FED INDUCTION MACHINES INCLUDING SATURATION EFFECTS OF MAIN FLUX LINKAGE  by Benjamin Braconnier  B.Sc., The University of Alberta, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  September 2012  © Benjamin Braconnier, 2012   ii Abstract  Power system simulation has become an integral part of power system planning, design and study today. Due to the vast array of power system disturbances that occur and the speed of their respective dynamics, simulation tools have traditionally been split into two types; Transient Stability type programs (TSP) which generally make simplifying assumptions to simulate longer electromechanical transients and Electromagnetic Transient type programs (EMTP) which require much greater detail in their models as they are generally used to simulate much faster transients that occur during electromagnetic disturbances. Several papers have been written and solutions have been proposed to come up with a solution that would combine these two different types of programs into one simulation tool that can be used to simulate both the slow electromechanical transients and fast electromagnetic transients that occur on power systems. One such method that has been proposed and is being thoroughly studied and showing promising potential for unified power system simulation is the concept of dynamic phasors. The dynamic phasor concept extends the traditional concept of steady state phasor representation used extensively in TSP programs by allowing these phasor quantities to change with time during transient events.  This thesis focuses on the modeling of induction machines using the dynamic phasor concept. The induction machine is chosen as it represents a significant portion of power system loads and therefore plays an important role in power system stability and behavior. Several induction machine models have already been proposed using the dynamic phasor concept however none of which take into account the effects of saturation that often occurs   iii during induction machine operation. This thesis therefore focuses specifically on modeling the effects of main flux linkage saturation in induction machines using the concept of dynamic phasors. The proposed model is based on the flux correction method and extends this concept to representation in dynamic phasor quantities. This is done by first verifying the consistency of the unsaturated dynamic phasor induction machine model against well known conventional models and then implementing the proposed dynamic phasor saturation model and again verifying it’s consistency with conventional saturation model of the induction machine. The results obtained in this thesis show that the proposed saturation model using the dynamic phasor concept is consistent with conventional saturation models with reasonable accuracy and efficiency and therefore provides a good starting point for further research in machine saturation modeling using the dynamic phasor concept.    iv Table of Contents  Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iv List of Tables .......................................................................................................................... vi List of Figures ........................................................................................................................ vii Acknowledgements ................................................................................................................. x Dedication ............................................................................................................................... xi Chapter  1: Introduction ........................................................................................................ 1 1.1 Background ............................................................................................................... 1 1.2 Thesis Motivation ..................................................................................................... 6 1.3 Thesis Outline ........................................................................................................... 8 Chapter  2: Dynamic Phasor Analysis ................................................................................ 10 2.1 Concept and Definition ........................................................................................... 10 2.1.1 Single Phase Systems .......................................................................................... 10 2.1.2 Polyphase Systems .............................................................................................. 13 2.2 Dynamic Phasors and Complex Space Vectors ...................................................... 17 2.2.1 Complex Space Vector Representation of 3-Phase Variables ............................ 17 2.2.2 Dynamic Phasor Representation of Complex Space Vectors ............................. 20 Chapter  3: Induction Machine Modeling Based on Dynamic Phasors ........................... 22 3.1 Time Domain Model of Doubly-Fed Induction Machine ....................................... 22 3.1.1 abc Phase Model of Doubly-Fed Induction Machine ......................................... 22 3.1.2 qd0 Model of Doubly-Fed Induction Machine ................................................... 25   v 3.1.3 Complex Vector Model of Doubly-Fed Induction Machine ............................... 31 3.2 Dynamic Phasor Model of Doubly-Fed Induction Machine ................................... 34 3.3 Model Verification .................................................................................................. 40 3.3.1 Free Acceleration of Induction Machine ............................................................ 41 3.3.2 Mechanical Torque Change ................................................................................ 43 3.3.3 Balanced 3-phase Voltage Dip............................................................................ 46 3.3.4 Unbalanced Single Line-to-Ground Fault ........................................................... 48 Chapter  4: Modeling of Saturation in Induction Machines ............................................. 56 4.1 Time Domain Modeling of Induction Machine Saturation ..................................... 56 4.2 Dynamic Phasors Model of Induction Machine Saturation .................................... 59 4.3 Saturation Model Verification ................................................................................ 62 Chapter  5: Induction Machine Dynamic Phasor Model Performance ........................... 70 5.1 Fixed Time-Step Model Performance .................................................................... 70 5.2 Variable Time-Step Model Performance ................................................................ 83 Chapter  6: Conclusion ......................................................................................................... 89 6.1 Summary of Contributions ...................................................................................... 89 6.2 Future Research ...................................................................................................... 90 References .............................................................................................................................. 94 Appendices ........................................................................................................................... 102 Appendix A -  500 HP Induction Machine Parameters .................................................... 102 Appendix B -  Saturation Data of 500 HP Induction Machine ......................................... 103    vi List of Tables  Table 3.1  Maximum error introduced by dynamic phasor model ......................................... 55 Table 4.1  Maximum error introduced by saturated dynamic phasor model .......................... 67 Table 5.1  CPU times for test case simulation ........................................................................ 82 Table 5.2  Model performance using variable time-step solver .............................................. 88 Table A.1  500hp induction machine parameters [49] .......................................................... 102 Table B.1  Saturation data for 500hp induction machine [59] .............................................. 103    vii List of Figures  Figure 1.1  General time-frame for power system disturbances ............................................... 2 Figure 2.1  Illustration of “sliding” window of periodicity .................................................... 11 Figure 2.2  Resolution of unbalanced 3-phase quantities into balanced sequence components ................................................................................................................................................. 14 Figure 2.3  Illustration of the rotating complex space vector ................................................. 18 Figure 2.4  Representation of rotating complex phasor .......................................................... 21 Figure 3.1  Illustration of the magnetic axis of a 2-pole induction machine .......................... 22 Figure 3.2  qd0 magnetic axis ................................................................................................. 26 Figure 3.3  qd0 equivalent circuit of three phase induction machine ..................................... 29 Figure 3.4  Y-connection of stator and rotor circuits of doubly-fed induction machine ........ 30 Figure 3.5  Complex Space Vector equivalent circuit of 3-phase induction machine ............ 33 Figure 3.6  Torque and speed ripple during unbalanced machine operation .......................... 36 Figure 3.7  Dynamic phasor equivalent circuit of 3-phase induction machine ...................... 38 Figure 3.8  Stator current during free acceleration (Δt=50µs) ................................................ 41 Figure 3.9  Electrical torque during free acceleration (Δt=50µs) ........................................... 42 Figure 3.10  Rotor speed during free acceleration (Δt=50µs) ................................................ 43 Figure 3.11  Stator current during mechanical torque change (Δt=50µs) ............................... 44 Figure 3.12  Electrical torque during mechanical torque change (Δt=50µs) .......................... 45 Figure 3.13  Rotor speed during mechanical torque change (Δt=50µs) ................................. 45 Figure 3.14  Stator current during balanced 3-phase voltage dip (Δt=50µs) .......................... 46 Figure 3.15  Electrical torque during balanced 3-phase voltage dip (Δt=50µs) ..................... 47   viii Figure 3.16  Rotor speed during balanced 3-phase voltage dip (Δt=50µs) ............................ 47 Figure 3.17  Real and imaginary parts of positive sequence stator current dynamic phasor .. 48 Figure 3.18  Real and imaginary parts of negative sequence stator current dynamic phasor . 49 Figure 3.19  Stator current during unbalanced fault (Δt=50µs) .............................................. 50 Figure 3.20  DC component of electrical torque dynamic phasor .......................................... 51 Figure 3.21  2nd harmonic component of electrical torque dynamic phasor ........................... 52 Figure 3.22  Electrical torque during unbalanced fault (Δt=50µs) ......................................... 52 Figure 3.23  DC component of rotor speed dynamic phasor .................................................. 53 Figure 3.24  2nd harmonic component of rotor speed dynamic phasor ................................... 54 Figure 3.25  Rotor speed during unbalanced fault (Δt=50µs) ................................................ 54 Figure 4.1  Typical magnetization curve of an induction machine ......................................... 57 Figure 4.2  qd decomposition of flux correction term ............................................................ 58 Figure 4.3  Saturation of main magnetizing flux linkage in terms of dynamic phasors ......... 61 Figure 4.4  Stator current in saturation region ........................................................................ 63 Figure 4.5  Magnetizing flux linkage in saturation region ...................................................... 64 Figure 4.6  Zoomed view of stator current in saturation region ............................................. 65 Figure 4.7  Zoomed view of magnetizing flux linkage in saturation region ........................... 65 Figure 4.8  Stator current in saturation region during unbalanced fault ................................. 66 Figure 4.9  Magnetizing flux linkage in saturation region during unbalanced fault ............... 67 Figure 4.10  Zoomed view of stator current in saturation region during unbalanced fault .... 68 Figure 4.11  Zoomed view of magnetizing flux in saturation region during unbalanced fault ................................................................................................................................................. 68 Figure 5.1  Stator current during start-up for fixed Δt studies ................................................ 71   ix Figure 5.2  Stator current during load torque change for fixed Δt studies .............................. 72 Figure 5.3  Stator current during balanced 3-phase fault for fixed Δt studies ........................ 73 Figure 5.4  Stator current during unbalanced fault for fixed Δt studies .................................. 74 Figure 5.5  Positive sequence stator current dynamic phasor using different Δt .................... 75 Figure 5.6  Negative sequence stator current dynamic phasor using different Δt .................. 76 Figure 5.7  Electrical torque using different Δt ....................................................................... 77 Figure 5.8  DC component of electrical torque dynamic phasor using different Δt ............... 77 Figure 5.9  2nd harmonic component of electrical torque dynamic phasor using different Δt 78 Figure 5.10  Rotor speed using different Δt ............................................................................ 79 Figure 5.11  DC component of rotor speed dynamic phasor using different Δt ..................... 80 Figure 5.12  2nd harmonic component of rotor speed dynamic phasor using different Δt ...... 80 Figure 5.13  Load change results using dynamic phasor model ............................................. 82 Figure 5.14  Stator current during free acceleration using variable Δt solver ........................ 84 Figure 5.15  Time-step results during free acceleration using variable Δt solver .................. 84 Figure 5.16  Stator current during load torque change using variable Δt solver .................... 85 Figure 5.17  Time-step results during load torque change using variable Δt solver ............... 85 Figure 5.18  Stator current during balanced fault using variable Δt solver ............................ 86 Figure 5.19  Time-step results during balanced fault using variable Δt solver ...................... 86    x Acknowledgements For their patience, knowledge and probing questions that encouraged me to look deeper and gain a better understanding of my research topic I would like to thank my supervisors Dr. José Marti and Dr. Hermann Dommel. I would like to express my deep gratitude to Dr. Hermann Dommel for his continuous financial support throughout my research and for taking a chance on me and opening a spot in the M.A.Sc. program; allowing me to transfer to UBC. I would also like to thank Dr. Juri Jatskevich; who was not my supervisor but at times I’m sure felt like he was, for his expertise and insight into electric machine theory and modeling. I would also like to sincerely thank all of the friends and colleagues that I have gained throughout my studies here at the University of British Columbia; I hope the relationships that we have developed will stay with me for the rest of my life.  Finally, I would like to express my deepest gratitude to my loving wife and our families who have been extremely patient and provided so much support and encouragement to me throughout my post-secondary education. I will be forever grateful to them and I find it difficult to express in words how much their ongoing support has meant to me throughout the years; from the bottom of my heart, thank you.   xi Dedication The work done in this thesis is dedicated to my loving wife Blair. Her ongoing support in all aspects has meant more to me than I would ever be able to express in words. I can only hope that one day I will be able to repay her by providing that same support to her. She has been my rock, my confidant and sounding board but most importantly, she has been my best friend. I will always be grateful to you Blair and I love you more than words can express. From the deepest parts of my soul I thank you.    1 Chapter  1: Introduction  1.1 Background The electric power system is often referred to as one of the most important infrastructure systems built by humankind. So important and fundamental in fact that many other systems that have been developed to progress the human race are dependent on electric power systems to allow them to operate. For example; communication, transportation, water supply and many other infrastructure systems are completely dependent upon the continuous reliable operation of the electric power system. Due to growing electricity demand and increased load and stress being added to an aging electric power system infrastructure, power system stability and security has become a major concern and area of focus for research and improvement [1]. Major blackouts in the USA, Europe and most recently in north eastern India for example; which caused an estimated 670 million people [2], or roughly 10% of the world’s population to be without power, have forced researchers to look closely at ways to increase the reliability and security of electric power systems to mitigate such catastrophic failures in the future.  During planning and operation, power systems must be thoroughly analyzed and therefore be accurately modeled. Models and computer programs for power system analysis have become an essential part of power system planning and operation. Power system engineers perform a variety of system studies in order to plan effectively for system upgrades and changes and to reliably operate the power system. Some of the types of studies performed regularly are • short circuit (or fault) analysis   2 • electromagnetic transient analysis • power (or load) flow analysis, and • voltage and transient stability studies  One of the major challenges in modeling and simulating the power system for the above studies is the fact that the dynamics of the system react at very different speeds depending on the type of disturbance to the system and the type of study being conducted. For example, Figure 1.1 illustrates the general time-frame in which many of the major power system disturbances occur. These disturbances can be generally classified into two different categories; electromagnetic transients and electromechanical transients.   Figure 1.1  General time-frame for power system disturbances   3 Electromagnetic transient phenomena generally occur during sudden system topology changes. These changes can be attributed to many different types of disturbances such as lightning strokes, closing or opening of circuit breakers, high frequency switching of power electronic devices and faults that occur in the power system to name just a few. The effects of these types of disturbances are usually fairly localized in the surrounding system and usually have very fast transients associated with them. Electromagnetic transient disturbances therefore require a greater amount of detail in modeling the system components that are most affected and because of the speed of the transients require a much smaller time-step when performing simulations; usually of the order of tens of microseconds or smaller depending on the phenomena being studied.  Electromechanical transients are generally associated with much slower disturbance phenomena such as mismatches between the mechanical energy present in rotating electric machines and the electric energy present in the electrical system that it is attached to. These types of slow transients can also occur during a power imbalance that occurs between two different areas of the power system; known as inter-area oscillations. The dynamics of this type of disturbance phenomena are much slower than those that occur in electromagnetic transients and therefore many simplifying assumptions are made to increase the efficiency of the simulation of these types of transients. For one, because of the timescales associated with these types of transient studies the electromagnetic transients are generally neglected and a quasi-steady state is assumed for simulation. This is generally achieved using the concept of steady-state phasor representation of system quantities around the system operating frequency.   4 Due to the vast difference in dynamic transient speeds that occur in the power system, traditionally two different types of simulators have been used to simulate transients in the power system depending on which type of study was being done. For the fast electromagnetic transient simulations, Electromagnetic Transient Program or EMTP type programs are used. These types of programs require a great amount of detail in the system component models and very small time-steps to simulate the fast dynamics that occur in these types of studies. The much slower transient stability studies are usually performed by Transient Stability Programs which require much less detail in the system component models and make simplifying assumptions to increase the efficiency of the programs as mentioned above.  The Electromagnetic Transients Program (EMTP) [3] was first introduced in the late 1960’s by H. W. Dommel [4] and was quickly adopted as one of the standard industry tools used for the digital simulation of power system transients. Prior to EMTP, a system known as a Transient Network Analyzer (TNA) was used extensively to perform load flow, short-circuit analysis and stability studies on AC power systems.  The quick adoption of EMTP-type programs (PSCAD/EMTDC, ATP, MICROTRAN, etc.) [5-8] as the industry standard tools for transient analysis is partly due to several advantages that they have over the TNA. Firstly, these programs are fully digital simulators that are able to run on inexpensive desktop computers, whereas the TNA is an analog scale model of the system that is very expensive to build and requires a significant amount of space. Secondly, EMTP-type programs are able to provide much more accurate simulation results than the TNA at higher frequencies. This is   5 due to phenomena that are present at higher frequencies in the TNA model such as parasitic capacitances that can have significant impacts on the accuracy of the simulation results.  One drawback to these EMTP-type simulators is that they require very small integration time-steps in order to accurately recreate all of the transient variables during simulations. Therefore, these types of programs work well with transient and mid-term stability studies; which contain faster dynamic phenomena, but become inefficient for use with long term stability problems and slow dynamic simulations of large systems that occur when studying problems such as voltage stability or inter-area oscillations. Other types of methods are used in this case, that make simplifying assumptions to increase the speed and efficiency of the program during simulation [9-11].  One of the first attempts to unify the simulation of dynamic operating conditions in power systems is outlined in [12], in which the authors make use of an implicit variable-step algorithm to simulate the faster transients and slower long-term phenomena that occur in power system operation. The algorithm presented here is quite robust but is not capable of simulating the very fast electromagnetic transients. Another, more recent, attempt at unifying power system simulators using the concept of time varying phasors or dynamic phasors is presented by Venkatasubramanian in [13] and further extended in [14, 15]. The dynamic phasor concept as presented in these papers extends the concept of traditional phasor representation of sinusoidal signals; which are the standard tool for power system stability analysis and generally limited to quasi-stationary assumptions, to a time-varying phasor representation that can accommodate much faster transient speeds. The concept of dynamic   6 phasors allows for the efficient solution of both fast electromagnetic transients as well as slow electromechanical simulations due to the fact that their dynamics are much slower in time than the time-domain circuit quantities. During steady state dynamic phasor quantities do not change with time and reduce to the traditional phasor representation of power system quantities making them efficient for modeling longer term electromechanical transients.  Since their early applications to averaging power conversion circuit behaviour for simulations [16], dynamic phasors have been used to successfully model many power system components and simulation phenomena. For example, system components such as HVDC converters [17-19], static var compensators (SVC) [20], unified power flow controllers (UPFC) [21] and synchronous machines [22] among others. The dynamic phasor concept has also been used to perform various system studies including analysis of balanced power system transients [13-15, 23] as well as unbalanced transients [24]. The concept of dynamic phasor estimates have also been applied to studying power system oscillations in [25-28].  1.2 Thesis Motivation Electric motors typically consume 60 to 70% of the energy supplied by a power system and therefore encompass a major portion of power system loads. The dynamics attributed to motors are usually the most significant contribution to power system load dynamic characteristics. In particular induction motors are the workhorse of modern electrical industry. From large industrial motors used in mills and manufacturing facilities to home appliances such as refrigerators, washing machines, dishwashers etc., induction machines are extensively used throughout many power systems. Thus it is very important to be able to   7 accurately and efficiently model induction motors for power system simulation and stability studies.  Various dynamic phasor models of induction machines have been presented in literature. For example references [23] and [29] present a dynamic phasor model for an induction machine that is derived from dimension reduced (DR) induction machine model where the equations of stator, rotor and flux variables are reduced to those of only stator variables. The analysis in these papers however assumes only balanced machine operation and doesn’t take into account unbalanced conditions that may occur in induction machine operation. Reference [30] presents a dynamic phasor model of a doubly-fed induction machine that does take into account unbalanced machine operation. The model presented is derived from the conventional qd0 induction machine model in a synchronous reference frame. This model contains a dc component dynamic phasor that captures the positive sequence machine quantities as well as a 2nd harmonic dynamic phasor that captures the negative sequence machine quantities during unbalanced operation.   In reference [31] a dynamic phasor model for doubly-fed induction machines is derived from the complex space vector induction machine model with a stationary reference frame. This model contains a positive system frequency dynamic phasor to account for positive sequence machine quantities and balanced operation as well as a negative system frequency dynamic phasor that accounts for negative sequence quantities that arise during unbalanced machine operation.  Although induction machines have been modeled successfully several different ways using the concepts of dynamic phasors, none of the models presented in the literature mentioned   8 include the effects of machine saturation. The effects of machine saturation have been a topic of significant research and study [32-43]. During operation, induction machines routinely face a number of operating conditions where the effects of saturation cannot be ignored. Therefore, in order to produce more accurate and reliable results during simulations the effects of saturation must be included in the machine models. The motivation behind this thesis therefore is to extend the dynamic phasor model of the doubly-fed induction machine to include the effects of main flux-linkage saturation. The dynamic phasor saturation model developed in this thesis is intended to be a starting point for further dynamic phasor model developments and research.  1.3 Thesis Outline Following the introduction, the concepts of dynamic phasors are introduced to the reader in Chapter 2. In particular, Section 2.1 presents the concept and definition of the dynamic phasors and how they can be used to represent circuit quantities in single phase systems (Section 2.1.1). The concept of dynamic phasors is then generalized to include polyphase systems (Section 2.1.2), particularly the three-phase system that is commonly encountered in the modern day power system. Section 2.2 describes how dynamic phasors can be used to represent complex space vector quantities by first describing the concept of complex space vectors (Section 2.2.1) and how they are used to describe three phase systems and then describing how dynamic phasors can be used to express complex space vector quantities (Section 2.2.2).    9 Chapter 3 first presents some conventional time domain models that are used to model induction machines. These include the three-phase abc induction machine model (Section 3.1.1), the two-axis qd0 induction machine model (Section 3.1.2) and finally the complex space vector induction machine model (Section 3.1.3). The dynamic phasor induction machine model is then presented in Section 3.2, followed by a few system studies in Section 3.3 to illustrate the validity of the dynamic phasor induction machine model.  Chapter 4 presents the method used to implement the effects of main flux linkage saturation into the dynamic phasor model. First however, a conventional time domain model for implementing the effects of main flux saturation is presented in Section 4.1 followed by its extension to the dynamic phasor model in Section 4.2. Once the models have been presented the dynamic phasor model is then verified in the saturation region of operation for both balanced and unbalanced machine operation in Section 4.3.  In Chapter 5 the focus is on assessing the performance of the dynamic phasor induction machine model. Section 5.1 looks at the models performance using a fixed time-step and compares it to the performance of a conventional time domain model. Section 5.2 investigates the dynamic phasor model performance using a variable time-step solver and compares the results to those obtained using a conventional time domain model.  Finally, Chapter 6 concludes the thesis by highlighting the key contributions provided in this thesis along with recommendations for future research and model improvements.    10 Chapter  2: Dynamic Phasor Analysis  2.1  Concept and Definition The concept of phasors is based on the property that a periodic (possibly complex-valued) time-domain waveform 𝑥(𝜏) with period 𝑇, that is 𝑥(𝜏) = 𝑥(𝜏 − 𝑇), can be expressed as a complex Fourier series [44] of the form  𝑥(𝜏) = � 𝑋𝑘 ∙ 𝑒𝑗𝑘𝜔𝑠𝜏∞ 𝑘=−∞  (2.1) where 𝜔𝑠 = 2𝜋 𝑇⁄  and 𝑋𝑘 is the 𝑘𝑡ℎ Fourier coefficient in complex form. In this case, since 𝑥(𝜏) is periodic, the Fourier coefficients 𝑋𝑘 are time invariant and can be expressed as  𝑋𝑘 = 1𝑇� 𝑥(𝜏) ∙ 𝑒−𝑗𝑘𝜔𝑠𝜏𝑑𝜏𝑡𝑡−𝑇  (2.2)  In power system studies, during steady-state analysis, these coefficients 𝑋𝑘 are also known as phasors and are extensively used as a tool to greatly simplify the study of steady-state power flow and system behavior.  2.1.1 Single Phase Systems In power system dynamic situations, the waveforms are generally not strictly periodic but are nearly periodic. The idea now is to generalize equations (2.1) – (2.2) to accommodate arbitrary, nearly periodic types of waveforms. This is achieved by using the approach outlined in [45], where the Fourier coefficients are allowed to vary with time. Considering a window of length 𝑇 for a given waveform, and viewing that waveform to be periodic with a   11 duration of 𝑇, such that a Fourier analysis of the waveform 𝑥(𝜏) can be performed. The time evolution of the Fourier coefficients can then be calculated as the window of length 𝑇 “slides” along the actual waveform. Figure 2.1 for example depicts the stator phase a current of an induction machine during and immediately following an unbalanced single phase fault along with the “sliding window” of length 𝑇.   Figure 2.1  Illustration of “sliding” window of periodicity  Now, for an arbitrary, possibly complex-valued time-domain waveform 𝑥(𝜏) with period 𝑇, the method considers the Fourier coefficients of 𝑥(𝜏) on the interval 𝜏 ϵ (𝑡 − 𝑇, 𝑡]. The Fourier series is now expressed in the form   12  𝑥(𝜏) = � 𝑋𝑘(𝑡) ∙ 𝑒𝑗𝑘𝜔𝑠𝜏∞ 𝑘=−∞  (2.3) where the 𝑘𝑡ℎ Fourier coefficient 𝑋𝑘(𝑡) is now time-dependent and can shift in both magnitude and phase as the window of periodicity “slides” over the waveform. These time- dependent Fourier coefficients are referred to as dynamic phasors in many research papers and throughout this thesis and are expressed as  𝑋𝑘(𝑡) = 1𝑇� 𝑥(𝜏) ∙ 𝑒−𝑗𝑘𝜔𝑠𝜏𝑑𝜏𝑡𝑡−𝑇 = 〈𝑥〉𝑘(𝑡). (2.4)  In many cases, a reasonable approximation of the waveform 𝑥(𝜏) can be achieved by considering only the Fourier coefficients 𝑋𝑘(𝑡) that are dominant in equation (2.3). That is, if we consider a subset {𝐾} containing only the dominant Fourier coefficients, we can approximate (2.3) as  𝑥(𝜏) ≈ � 𝑋𝑘(𝑡) ∙ 𝑒𝑗𝑘𝜔𝑠𝜏 𝑘 𝜖 {𝐾}  (2.5)  For the purposes of this thesis, we will only be considering the subset {𝐾} = {1,−1}; which corresponds to plus and minus the system frequency. For the system that is studied in this thesis, the system frequency is assumed to be 60 Hz and the above approximation looks only at the +60 Hz and -60 Hz components that are present in the system operation.  Some useful properties of the Fourier coefficients that will be used in later chapters of this thesis will now be outlined. For example, the time derivative for the 𝑘𝑡ℎ Fourier coefficient, as expressed in equation (2.6), is obtained by differentiating equation (2.3) with respect to   13 time. This property is used extensively in the dynamic model of the doubly-fed induction machine model.  𝑑𝑋𝑘 𝑑𝑡 = �𝑑𝑥 𝑑𝑡 � 𝑘 − 𝑗𝑘𝜔𝑠𝑋𝑘. (2.6)  Another important property is shown in equation (2.7) which illustrates that a product of two time domain signals is equivalent to a discrete time convolution of the two signals’ sets of dynamic phasors.  〈𝑥𝑦〉𝑘 = � �𝑋𝑘−𝑖𝑌  𝑖�∞ 𝑖=−∞  (2.7)  If the signal 𝑥(𝜏) is complex valued, the relationship between 𝑋−𝑘 and 〈𝑥〉𝑘 is given below in equation (2.8).  𝑋−𝑘 = 〈𝑥〉−𝑘 = 〈𝑥∗〉𝑘∗  (2.8) where 𝑥∗ represents the complex conjugate of 𝑥. In the case where 𝑥(𝜏) is real valued, equation (2.8) reduces to the form shown below in equation (2.9).  𝑋−𝑘 = 𝑋𝑘∗ (2.9)  2.1.2 Polyphase Systems Our goal now is to extend the dynamic phasor concept to include polyphase systems. We begin by using the method of symmetrical components [46] which allows a set of 𝑛 unbalanced phasors to be expressed as the sum of 𝑛 symmetrical sets of balanced phasors. Therefore, in the common three-phase (abc) power system, a set of three unbalanced phasors can be broken down into three sets of balanced symmetrical phasors which consist of;   14 • Positive-sequence components which are equal in magnitude, have an angular displacement of 120° from each other and have the same phase sequence as the original unbalanced set; • Negative-sequence components which are equal in magnitude, have an angular displacement of 120° from each other and have a phase sequence that is opposite to that of the original unbalanced set; • Zero-sequence components which are equal in both magnitude and phase. Figure 2.2 illustrates the three sets of balanced sequence components that add up to form the original unbalanced set.  Figure 2.2  Resolution of unbalanced 3-phase quantities into balanced sequence components   15 The transformation from an unbalanced set of phasors to the sequence components is achieved by introducing the common operator 𝛼 = 𝑒𝑗(2𝜋 3⁄ ) which represents a phase of 2𝜋 3⁄  in the complex plane. Using the operator 𝛼, we can express the phase quantities as shown in equation (2.10).  � 𝑥𝑎 𝑥𝑏 𝑥𝑐 � = � 1 1 1𝛼2 𝛼 1 𝛼 𝛼2 1���������� 𝒜 � 𝑋𝑎,𝑝 𝑋𝑎,𝑛 𝑋𝑎,0� (2.10) Where 𝛼2 = 𝑒𝑗(4𝜋 3⁄ ) = 𝑒−𝑗(2𝜋 3⁄ ).  The inverse transformation is given as follows  � 𝑋𝑎,𝑝 𝑋𝑎,𝑛 𝑋𝑎,0� = 13 �1 𝛼 𝛼 21 𝛼2 𝛼1 1 1 ������������ 𝒜−1 � 𝑥𝑎 𝑥𝑏 𝑥𝑐 � (2.11)  Using the transformation in equation (2.10) and extending the single phase dynamic phasor principles outlined in Section 2.1.1 to the poly-phase system, we can express three phase time domain quantities in dynamic phasor form as shown in equation (2.12).  � 𝑥𝑎 𝑥𝑏 𝑥𝑐 � (𝜏) = � 𝑒𝑗𝑘𝜔𝑠𝜏∞ 𝑘=−∞ 𝒜 � 𝑋𝑝,𝑘 𝑋𝑛,𝑘 𝑋0,𝑘� (𝑡) (2.12)  Note here that the subscript 𝑎 has been dropped from the Fourier coefficients on the right hand side of equation (2.12) to avoid clutter. Similar to the single phase case, the Fourier coefficients in equation (2.12) can be expressed as   16  � 𝑋𝑝,𝑘 𝑋𝑛,𝑘 𝑋0,𝑘� (𝑡) = 1𝑇� 𝑒−𝑗𝑘𝜔𝑠𝜏𝒜−1 �𝑥𝑎𝑥𝑏𝑥𝑐� (𝜏)𝑑𝜏𝑡𝑡−𝑇 = � 〈𝑥〉𝑝,𝑘 〈𝑥〉𝑛,𝑘 〈𝑥〉0,𝑘� (𝑡) (2.13)  The Fourier coefficients in equation (2.13) represent the dynamic phasors of the positive 𝑋𝑝,𝑘, negative 𝑋𝑛,𝑘 and zero-sequence 𝑋0,𝑘 components of the time varying signals represented in equation (2.12) at a frequency of 𝑘𝜔𝑠.  The same properties as those outlined in Section 2.1.1 for the single phase dynamic phasors can also be applied here to the poly-phase case since equation (2.12) is just a vector generalization of equation (2.3) for the three-phase case [31]. For example the time derivative of the time varying Fourier coefficients for the positive, negative and zero sequence components can be expressed in vector form as  𝑑 𝑑𝑡 � 𝑋𝑝,𝑘 𝑋𝑛,𝑘 𝑋0,𝑘� (𝑡) = 𝒜−1 ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡� 𝑑 𝑑𝑡 𝑥𝑎(𝜏)� 𝑘 � 𝑑 𝑑𝑡 𝑥𝑏(𝜏)� 𝑘 � 𝑑 𝑑𝑡 𝑥𝑐(𝜏)� 𝑘⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ (𝑡) − 𝑗𝑘𝜔𝑠 �𝑋𝑝,𝑘𝑋𝑛,𝑘 𝑋0,𝑘� (𝑡) (2.14)  Also, for a complex valued signal we have the relationship between balanced sequence components as  � 〈𝑥〉𝑝,𝑘 〈𝑥〉𝑛,𝑘 〈𝑥〉0,𝑘� = � 〈𝑥∗〉𝑛 ,−𝑘∗ 〈𝑥∗〉𝑝,−𝑘∗ 〈𝑥∗〉0,−𝑘∗ � (2.15)  In the case of real valued signals equation (2.15) can be expressed as   17  � 𝑋𝑝,𝑘 𝑋𝑛,𝑘 𝑋0,𝑘� = � 𝑋𝑛 ,−𝑘∗ 𝑋𝑝,−𝑘∗ 𝑋0,−𝑘∗ � (2.16)  2.2 Dynamic Phasors and Complex Space Vectors 2.2.1 Complex Space Vector Representation of 3-Phase Variables In modern literature on electric machines and drive systems it is common to represent three phase variables as what are known as complex space vectors [47, 48]. The concept of complex space vectors involves a transformation of the three phase variables to a stationary complex reference frame. This transformation is almost identical to the common qd0 reference frame transformation that is used extensively in rotating machine modeling [46, 49], with the added abstraction that the variables are represented as complex vectors in the q- and d-axis. That is, the q-axis quantities represent the real part of the complex space vector and the d-axis quantities representing the negative imaginary part of the vector.  To illustrate the concept of complex space vectors, Figure 2.3 depicts a simplified representation of a three-phase rotating machine. The stator consists of three windings that are sinusoidally distributed around the stator of the machine such that they are 120° apart spatially from one another. When a three phase sinusoidal current is supplied to the stator windings in the machine with positive current flowing into the crossed terminals and out of the dotted terminals, each phase will produce its own magneto-motive force (mmf ) which lies on the magnetic axis of that particular phase. If we consider the mmf produced by each particular phase as a vector in space, they can be added up to produce the resultant total mmf   18 that is produced within the machine which rotates inside the machine at a frequency equal to the supply frequency of the currents.   Figure 2.3  Illustration of the rotating complex space vector  We can define space vectors for the currents in each phase in a similar manner to lie on their respective magnetic axis as depicted for fluxes in Figure 2.3; from which we get  𝚤𝑎𝑠 = 𝑖𝑎𝑠 𝚤𝑏𝑠 = 𝛼𝑖𝑏𝑠 𝚤𝑐𝑠 = 𝛼2𝑖𝑐𝑠 (2.17)  Note here that 𝛼 in equation (2.17) is the same as that defined previously in section 2.1.2 of this thesis, that is 𝛼 = 𝑒𝑗(2𝜋 3⁄ ). 𝜆𝑎𝑠 𝜆𝑏𝑠 𝜆𝑐𝑠 𝜆𝑎𝑏𝑐𝑠   19 Now, adding these currents and introducing a scaling factor of 2 3⁄ , we get the total current space vector that results from the three phase currents as follows.  𝚤𝑎𝑏𝑐𝑠 = 23 (𝑖𝑎𝑠 + 𝛼𝑖𝑏𝑠 + 𝛼2𝑖𝑐𝑠) (2.18)  It is not difficult to see that if a balanced set of sinusoidal currents is applied to the terminals of the machine that equation (2.18) describes a phasor of constant magnitude that rotates in the complex plane with the same frequency as the supply frequency.  It is important to note that the coefficient of 2 3⁄  is introduced here to conserve the magnitudes between the supply currents and the resultant space vector. Other scaling factors are also used for different purposes. For example, the scaling factor of �2 3⁄  is sometimes used which will equate power quantities between the two representations of the system [48].  There is an easy way to retrieve the phase currents from the space vector in the case where the sum of the phase currents is equal to zero.  𝑖𝑎𝑠 + 𝑖𝑏𝑠 + 𝑖𝑐𝑠 = 0 (2.19)  In this case there will be no zero-sequence currents and the phase currents can be retrieved from the complex space vector by expanding it out into its complex form as follows.  𝚤𝑎𝑏𝑐𝑠 = 23 �𝑖𝑎𝑠 − 12 (𝑖𝑏𝑠 + 𝑖𝑐𝑠) + 𝑗 √32 (𝑖𝑏𝑠 − 𝑖𝑐𝑠)� = 𝑖𝑎𝑠 + 𝑗 √32 (𝑖𝑏𝑠 − 𝑖𝑐𝑠) (2.20)   20 From equation (2.20) it can be seen that the phase 𝑎 current can be retrieved by considering only the real part of the complex space vector, 𝚤𝑎𝑏𝑐𝑠. So we have  𝑖𝑎𝑠 = ℛℯ{ 𝚤𝑎𝑏𝑐𝑠 } (2.21)  Similarly, we can retrieve the phase 𝑏 and phase 𝑐 currents by considering the real part of the complex space vector after it has been multiplied by 𝛼2 and by 𝛼, respectively. So we get  𝑖𝑏𝑠 = ℛℯ{ 𝛼2 𝚤𝑎𝑏𝑐𝑠 } (2.22) and  𝑖𝑐𝑠 = ℛℯ{ 𝛼 𝚤𝑎𝑏𝑐𝑠 } (2.23)  It is important to note here that when zero sequence currents are present there will be an extra term that needs to be added to equations (2.21) through (2.23).  2.2.2 Dynamic Phasor Representation of Complex Space Vectors We are now at a point where we can extend the dynamic phasor concept to represent quantities expressed in terms of complex space vectors as dynamic phasors. Using a similar procedure as that outlined by A. Stankovik [31], we utilize equations (2.12) and (2.18) from which we get  ?⃑?(𝜏) = � 𝑒𝑗𝑘𝜔𝑠𝜏∞ 𝑘=−∞ 𝑋𝑝,𝑘(𝜏) (2.24)  Note here that equation (2.18) has been generalized from current quantities ( 𝚤 ) to general quantities ( ?⃑? ) in order to include other circuit quantities such as voltage or flux. We also   21 note here that equation (2.24) contains all of the information for positive and negative sequence components since all phase quantities are real valued quantities and we have 𝑋𝑝,−𝑘 = 𝑋𝑛,𝑘∗  as outlined in section 2.1.2 of this thesis. In the special case when there are only positive and negative sequence components (𝑘 = ±1) and there are no zero sequence components present, equation (2.24) reduces to  ?⃑?(𝜏) = 𝑋𝑝,1(𝜏)𝑒𝑗𝜔𝑠𝜏 + 𝑋𝑛,1∗ (𝜏)𝑒−𝑗𝜔𝑠𝜏 (2.25)  The space vector in equation (2.25) is illustrated conceptually in Figure 2.4 which shows that it can be broken down into a positive sequence space vector rotating in the anti-clockwise direction at an angular speed of 𝜔𝑠 rad/s and a negative sequence space vector that rotates in the clockwise direction at an angular speed 𝜔𝑠 rad/s.  ωs ωs tj p seX ω tj n seX ω−∗ x R e I m  Figure 2.4  Representation of rotating complex phasor   22 Chapter  3: Induction Machine Modeling Based on Dynamic Phasors  3.1 Time Domain Model of Doubly-Fed Induction Machine 3.1.1 abc Phase Model of Doubly-Fed Induction Machine An induction machine is comprised of two three-phase circuits; a stationary stator circuit and a rotating rotor circuit that rotates at an angular velocity of 𝜔𝑟 rad/s with respect to the stator circuit as illustrated in Figure 3.1.  In this analysis, 𝜃𝑟 is defined as the angle by which the phase a axis of the rotor circuit leads that of the stator circuit in radians and is represented by equation (3.1).  𝜃𝑟 =  𝜔𝑟𝑡 (3.1)  Figure 3.1  Illustration of the magnetic axis of a 2-pole induction machine   23  The voltage equations for both the stator circuit and the rotor circuit can be defined as follows; Stator voltage equations:  𝑣𝑎𝑠 =  𝑟𝑠𝑖𝑎𝑠 + 𝑑𝑑𝑡 𝜆𝑎𝑠 𝑣𝑏𝑠 =  𝑟𝑠𝑖𝑏𝑠 + 𝑑𝑑𝑡 𝜆𝑏𝑠 𝑣𝑐𝑠 =  𝑟𝑠𝑖𝑐𝑠 + 𝑑𝑑𝑡 𝜆𝑐𝑠 (3.2) Rotor voltage equations:  𝑣𝑎𝑟 ′ =  𝑟𝑟′𝑖𝑎𝑟′ + 𝑑𝑑𝑡 𝜆𝑎𝑟′ 𝑣𝑏𝑟 ′ =  𝑟𝑟′𝑖𝑏𝑟′ + 𝑑𝑑𝑡 𝜆𝑏𝑟′ 𝑣𝑐𝑟 ′ =  𝑟𝑟′𝑖𝑐𝑟′ + 𝑑𝑑𝑡 𝜆𝑐𝑟′ (3.3)  In equations (3.2) and (3.3), λ represents the flux linkage of the winding denoted by its respective subscript, rs is the stator phase resistance, and 𝑟𝑟′ is the rotor phase resistance referred to the stator winding.  The flux linkage of stator phase a winding can be written as  𝜆𝑎𝑠 =  𝐿𝑠𝑖𝑎𝑠 + 𝐿𝑀 �𝑖𝑎𝑠𝑐𝑜𝑠 𝜃𝑟 + 𝑖𝑏𝑠 cos �𝜃𝑟 − 2𝜋3 � + 𝑖𝑐𝑠 cos �𝜃𝑟 + 2𝜋3 �� (3.4) In the above equation 𝐿𝑠 represents the total phase a stator inductance and LM represents the maximum value of mutual inductance between the stator and rotor windings.   24 Similarly, the referred flux linkage of rotor phase a winding can be written as  𝜆𝑎𝑟′ =  𝐿𝑟′ 𝑖𝑎𝑟′ + 𝐿𝑀 �𝑖𝑎𝑟′ 𝑐𝑜𝑠 𝜃𝑟 + 𝑖𝑏𝑟′ cos �𝜃𝑟 − 2𝜋3 � + 𝑖𝑐𝑟′ cos �𝜃𝑟 + 2𝜋3 �� (3.5) where 𝐿𝑟′  represents the total phase rotor inductance referred to the stator winding and LM again, represents the maximum value of mutual inductance between the stator and rotor windings. Similar equations can be written for phase b and c stator and rotor flux linkages.  To complete the induction machine model, the mechanical part of the system needs to be defined. We first do this by defining the electromagnetic torque that is developed within the induction machine in terms of the phase currents and inductances that are present in each of the windings. It can be shown that the electromagnetic torque developed in the machine can be expressed as [49]  𝑇𝑒 = −�𝑃2� 𝐿𝑀 ��𝑖𝑎𝑠 �𝑖𝑎𝑟′ − 12 𝑖𝑏𝑟′ − 12 𝑖𝑐𝑟′ � + 𝑖𝑏𝑠 �𝑖𝑏𝑟′ − 12 𝑖𝑎𝑟′ − 12 𝑖𝑐𝑟′ � + 𝑖𝑐𝑠 �𝑖𝑐𝑟′ − 12 𝑖𝑏𝑟′ − 12 𝑖𝑎𝑟′ �� sin(𝜃𝑟) + √32 [𝑖𝑎𝑠(𝑖𝑏𝑟′ − 𝑖𝑐𝑟′ ) + 𝑖𝑏𝑠(𝑖𝑐𝑟′ − 𝑖𝑎𝑟′ ) + 𝑖𝑐𝑠(𝑖𝑎𝑟′ − 𝑖𝑏𝑟′ )] cos(𝜃𝑟)� (3.6) Where 𝑃 is the number of poles in the doubly-fed induction machine. The torque and rotor mechanical speed can be related by  𝑇𝑒 − 𝑇𝑚 − 𝐾𝑓𝑟𝑖𝑐𝜔𝑟𝑚 = 𝐽 𝑑𝜔𝑟𝑚𝑑𝑡 = 𝐽 2𝑃 𝑑𝜔𝑟𝑑𝑡  (3.7) where 𝜔𝑟𝑚 = 2𝑃 𝜔𝑟, 𝐾𝑓𝑟𝑖𝑐 is the rotor friction coefficient and 𝐽 is the combined polar moment of inertia of the rotor and the connected load and 𝑇𝑚 is the mechanical torque applied to the shaft of the rotor.   25 It can be seen from equations (3.4) and (3.6) that a significant amount of coupling occurs between the phases due to the currents that flow in each of the three windings. This coupling complicates the analysis and simulation of the induction machine and can make it quite cumbersome.  3.1.2 qd0 Model of Doubly-Fed Induction Machine The equations presented in section 3.1.1 can be greatly simplified if they are transformed from the stationary abc reference frame to a two axis, qd0 reference frame as illustrated in Figure 3.2. By projecting the abc phase quantities onto the two axis reference frame, the coupling that occurs in the abc phase model of the induction machine can be eliminated. It is assumed that the q-axis of this reference frame is 90⁰ ahead of the d-axis in the direction of rotation. The q-axis is chosen to coincide with the phase a axis at time t = 0.  It is important to note here that in general, the qd0 reference frame is one that is defined as rotating synchronously with the rotor circuit of a rotating machine. For the sake of this thesis, the qd0 reference frame will be defined as a two axis reference frame that is rotating at an arbitrary speed which is not necessarily synchronous to the rotating circuit of an electric machine.    26  Figure 3.2  qd0 magnetic axis  To make this transformation we use the commonly known Park’s Transformation which utilizes a transformation matrix K such that  𝑥𝑞𝑑0 = 𝑲𝑥𝑎𝑏𝑐 𝑥𝑎𝑏𝑐 = 𝑲−1𝑥𝑞𝑑0  (3.8) Where 𝑥𝑎𝑏𝑐 = [𝑥𝑎 𝑥𝑏 𝑥𝑐]𝑇, 𝑥𝑞𝑑0 = [𝑥𝑞 𝑥𝑑  𝑥0]𝑇 and 𝑥 represents either voltage, current or flux linkage quantities for the stator or rotor windings.  The stator transformation matrix, Ks and rotor transformation matrix, Kr of a rotating reference frame are defined as follows.   27  𝑲𝑠 = 23 ⎣ ⎢ ⎢ ⎢ ⎡cos(𝜃) cos �𝜃 − 2𝜋3 � cos �𝜃 + 2𝜋3 �sin(𝜃) sin �𝜃 − 2𝜋 3 � sin �𝜃 + 2𝜋 3 �1 2� 1 2� 1 2� ⎦⎥⎥ ⎥ ⎤  (3.9)  𝑲𝑟 = 23 ⎣ ⎢ ⎢ ⎢ ⎡cos(𝜃 − 𝜃𝑟) cos �𝜃 − 𝜃𝑟 − 2𝜋3 � cos �𝜃 − 𝜃𝑟 + 2𝜋3 �sin(𝜃 − 𝜃𝑟) sin �𝜃 − 𝜃𝑟 − 2𝜋3 � sin �𝜃 − 𝜃𝑟 + 2𝜋3 �1 2� 1 2� 1 2� ⎦⎥⎥ ⎥ ⎤  (3.10)  It can be shown that  𝑲𝑠−1 = ⎣ ⎢ ⎢ ⎡ cos(𝜃) sin(𝜃) 1cos �𝜃 − 2𝜋 3 � sin �𝜃 − 2𝜋 3 � 1cos �𝜃 + 2𝜋 3 � sin �𝜃 + 2𝜋 3 � 1⎦⎥⎥ ⎤  (3.11)  In the above matrices, 𝜃 represents the angular displacement of the q-axis from its original position, 𝜃0 and is represented by the following equation.  𝜃 = 𝜔𝑡 + 𝜃0 (3.12) Where 𝜔 is the angular velocity of the qd0 reference frame in rad/s. The angle 𝜃𝑟 is the angular position of the rotor with respect to its original position 𝜃𝑟0 and is represented with the following equations.  𝜃𝑟 = 𝜔𝑟𝑡 + 𝜃𝑟0 (3.13)  Applying equations (3.9) and (3.10) directly to (3.2) and (3.3) respectively, and assuming the machine is operated in a balanced three phase manner, we get the voltage equations for the qd0 reference frame.    28 Stator qd0 voltage equations:  𝑣𝑞𝑠 = 𝑟𝑠𝑖𝑞𝑠 + 𝜔𝜆𝑑𝑠 + 𝑑𝑑𝑡 𝜆𝑞𝑠 𝑣𝑑𝑠 = 𝑟𝑠𝑖𝑑𝑠 − 𝜔𝜆𝑞𝑠 + 𝑑𝑑𝑡 𝜆𝑑𝑠 𝑣0𝑠 = 𝑟𝑠𝑖0𝑠 + 𝑑𝑑𝑡 𝜆0𝑠 (3.14)  Rotor qd0 voltage equations:  𝑣𝑞𝑟 ′ = 𝑟𝑟′𝑖𝑞𝑟′ + (𝜔 − 𝜔𝑟)𝜆𝑑𝑟′ + 𝑑𝑑𝑡 𝜆𝑞𝑟′ 𝑣𝑑𝑟 ′ = 𝑟𝑟′𝑖𝑑𝑟′ − (𝜔 − 𝜔𝑟)𝜆𝑞𝑟′ + 𝑑𝑑𝑡 𝜆𝑑𝑟′ 𝑣0𝑟 ′ = 𝑟𝑟′𝑖0𝑟′ + 𝑑𝑑𝑡 𝜆0𝑟′ (3.15)  In equations (3.14) and (3.15), 𝜆𝑞𝑠, 𝜆𝑑𝑠,  𝜆0𝑠, 𝜆𝑞𝑟′ , 𝜆𝑑𝑟′ , and 𝜆0𝑟′  represent the stator and rotor flux linkages in the qd0 reference frame and are represented by the equations below. Stator qd0 flux linkages:  𝜆𝑞𝑠 = 𝐿𝑠𝑖𝑞𝑠 + 𝐿𝑀�𝑖𝑞𝑠 + 𝑖𝑞𝑟′ � 𝜆𝑑𝑠 = 𝐿𝑠𝑖𝑑𝑠 + 𝐿𝑀(𝑖𝑑𝑠 + 𝑖𝑑𝑟′ ) 𝜆0𝑠 = 𝐿𝑠𝑖0𝑠 (3.16) Rotor qd0 flux linkages:  𝜆𝑞𝑟 ′ = 𝐿𝑟′ 𝑖𝑞𝑟′ + 𝐿𝑀�𝑖𝑞𝑠 + 𝑖𝑞𝑟′ � 𝜆𝑑𝑟 ′ = 𝐿𝑟′ 𝑖𝑑𝑟′ + 𝐿𝑀(𝑖𝑑𝑠 + 𝑖𝑑𝑟′ ) 𝜆0𝑟 ′ = 𝐿𝑟′ 𝑖0𝑟′  (3.17)   29 As you can see in equations (3.16) and (3.17), the large amount of coupling that was present between the phases in equations (3.4) and (3.5) is no longer present. This greatly simplifies the induction machine model and reduces the computational burden that is present in the abc phase model of the Induction Machine. Figure 3.3 depicts the equivalent qd0 circuit of a 3- phase symmetric induction machine in an arbitrary rotating reference frame.   Figure 3.3  qd0 equivalent circuit of three phase induction machine  To complete the transformation and develop all of the equations needed to implement the model, we need to define electrical torque in terms of the qd0 reference frame quantities. It can be shown that the electromagnetic torque developed in the qd0 reference frame can be expressed as follows.   30  𝑇𝑒 =  �32� �𝑃2� �𝜆𝑑𝑠𝑖𝑞𝑠 − 𝜆𝑞𝑠𝑖𝑑𝑠� (3.18)  The electromagnetic torque developed by the motor drives the mechanical load attached to the shaft of the motor. If there is a mismatch between the electromagnetic torque developed and the load torque applied to the rotor shaft, the torque will accelerate the rotor according to the following equation.  𝑇𝑒 − 𝑇𝑚 − 𝐾𝑓𝑟𝑖𝑐𝜔𝑟𝑚 = 𝐽 𝑑𝜔𝑟𝑚𝑑𝑡 = 𝐽 2𝑃 𝑑𝜔𝑟𝑑𝑡  (3.19)  In equation (3.19), 𝐽 is the combined polar moment of inertia of the rotor and the connected load and 𝑇𝑚 is the mechanical torque applied to the shaft of the rotor.   Figure 3.4  Y-connection of stator and rotor circuits of doubly-fed induction machine    31 In most cases, induction machines are wired in either a Y-connected configuration as illustrated in Figure 3.4 or a Δ-connected configuration. In either case, there is generally no access to the neutral point in the circuit. In such cases, there is no return path for zero- sequence currents to flow and as such these components are zero. Therefore the zero- sequence equations in equations (3.14) through (3.17) and the zero sequence circuit in Figure 3.3 can generally be neglected.  3.1.3 Complex Vector Model of Doubly-Fed Induction Machine We are now in a position to be able to express the induction machine model in terms of complex space vector. We do this by respectively letting the q-axis and d-axis in Figure 3.2 correspond to the positive real and negative imaginary axis of the arbitrarily rotating complex reference frame. By doing this, we can express the stator circuit quantities as  𝑓𝑞𝑑𝑠 = 𝑓𝑞𝑠 − 𝑗𝑓𝑑𝑠 = 23 �𝑓𝑎𝑠𝑒−𝑗𝜃 + 𝑓𝑏𝑠𝑒−𝑗�𝜃− 2𝜋3 � + 𝑓𝑐𝑠𝑒−𝑗�𝜃+ 2𝜋3 �� (3.20) which can be expressed as  𝑓𝑞𝑑𝑠 = 23 𝑒−𝑗𝜃[𝑓𝑎𝑠 + 𝛼𝑓𝑏𝑠 + 𝛼2𝑓𝑐𝑠]                                                          = 𝑒−𝑗𝜃𝑓𝑎𝑏𝑐𝑠  (3.21) Similarly, the rotor circuit quantities can be expressed in the arbitrarily rotating complex reference plane as  𝑓𝑞𝑑𝑟 = 23 𝑒−𝑗(𝜃−𝜃𝑟)[𝑓𝑎𝑟 + 𝛼𝑓𝑏𝑟 + 𝛼2𝑓𝑐𝑟]                                         = 𝑒−𝑗(𝜃−𝜃𝑟)𝑓𝑎𝑏𝑐𝑟 (3.22)    32 Using equations (3.20) through (3.22) the qd0 model equations in Section 3.1.2 can be combined to represent the induction machine quantities in terms of complex space vectors as follows. The qd voltage equations for the stator circuit can be combined into the complex space vector form  ?⃑?𝑞𝑑𝑠 = 𝑟𝑠𝚤𝑞𝑑𝑠 + 𝑑𝑑𝑡 𝜆𝑞𝑑𝑠 + 𝑗𝜔𝜆𝑞𝑑𝑠. (3.23) Similarly, for the rotor circuit qd voltage equations we get  ?⃑?𝑞𝑑𝑟′ = 𝑟𝑟′𝚤𝑞𝑑𝑟′ + 𝑑𝑑𝑡 𝜆𝑞𝑑𝑟′ + 𝑗(𝜔 − 𝜔𝑟)𝜆𝑞𝑑𝑟′  (3.24) where  𝜆𝑞𝑑𝑠 = 𝐿𝑠𝚤𝑞𝑑𝑠 + 𝐿𝑀�𝚤𝑞𝑑𝑠 + 𝚤𝑞𝑑𝑟′ � (3.25) and  𝜆𝑞𝑑𝑟′ = 𝐿𝑟′ 𝚤𝑞𝑑𝑟′ + 𝐿𝑀�𝚤𝑞𝑑𝑠 + 𝚤𝑞𝑑𝑟′ �. (3.26)  For completeness the zero sequence quantities can be included here but, as mentioned in the previous section they can generally be neglected due to the connection of the induction machine. The zero sequence components are exactly the same in the complex space vector representation as they are in the qd0 representation, equations (3.14) through (3.17) of the induction machine model. So for the zero sequence quantities we have  𝑣0𝑠 = 𝑟𝑠𝑖0𝑠 + 𝑑𝑑𝑡 𝜆0𝑠 (3.27) and  𝑣0𝑟′ = 𝑟𝑟′𝑖0𝑟′ + 𝑑𝑑𝑡 𝜆0𝑟′ . (3.28)   33 where 𝜆0𝑠 = 𝐿𝑠𝑖0𝑠 and 𝜆0𝑟′ = 𝐿𝑟′ 𝑖0𝑟′  are the zero sequence flux linkages in the stator and rotor circuits, respectively. It is important to note here that the zero sequence quantities are not complex vectors but strictly real quantities. Figure 3.5 shows the equivalent circuit of the complex space vector representation of the doubly-fed induction machine including the zero sequence components of the model.   Figure 3.5  Complex Space Vector equivalent circuit of 3-phase induction machine  We now need to express the electrical torque developed in the machine in terms of the complex space vectors that have been defined. We know that the electrical torque is a real quantity so we need to manipulate the complex space vector quantities involved to give us the electrical torque in the same form as equation (3.18) which is defined in terms of the qd axis quantities. This can be done by multiplying the complex conjugate of the stator flux space vector with the stator current space vector;  𝜆𝑞𝑑𝑠 ∗  𝚤𝑞𝑑𝑠 = �𝜆𝑞𝑠 + 𝑗𝜆𝑑𝑠��𝑖𝑞𝑠 − 𝑗𝑖𝑑𝑠�                                                                             = 𝜆𝑞𝑠𝑖𝑞𝑠 + 𝜆𝑑𝑠𝑖𝑑𝑠 + 𝑗�𝜆𝑑𝑠𝑖𝑞𝑠 − 𝜆𝑞𝑠𝑖𝑑𝑠� (3.29)   34  It can be seen that the imaginary part of equation (3.29) is of the form that we need to express the electrical torque developed in the machine. That is, we can express the electrical torque developed in the machine using the complex space vector quantities by taking only the imaginary part of equation (3.29) as follows.  𝑇𝑒 =  �32� �𝑃2� ℐ𝓂� 𝜆𝑞𝑑𝑠∗  𝚤𝑞𝑑𝑠 � (3.30)  It is important to note here that there are other ways to express the electromagnetic torque developed in the machine using different circuit quantities. Many of these are outline by Novotny [47]. The reason that the electromagnetic torque was chosen to be represented in terms of the stator circuit flux linkage and current is due to its independence of the magnetizing inductance 𝐿𝑀. The advantage of this will become more apparent in our discussion of induction machine saturation in Chapter 4.  The mechanical equation for the complex space vector model is identical to that of the previous models as the quantities involved are all real valued quantities and so again we can express the equation for the mechanical and electric torque coupling as  𝑇𝑒 − 𝑇𝑚 − 𝐾𝑓𝑟𝑖𝑐𝜔𝑟𝑚 = 𝐽 𝑑𝜔𝑟𝑚𝑑𝑡 = 𝐽 2𝑃 𝑑𝜔𝑟𝑑𝑡 . (3.31)  3.2 Dynamic Phasor Model of Doubly-Fed Induction Machine We are now at a point where we can apply the concepts of dynamic phasors that were presented in Chapter 2 to the induction machine model. The dynamic phasor model of the   35 doubly-fed induction machine can be developed by applying equation (2.25) to the complex space vector quantities presented in Section 3.1.3. It is again noted here that there are only positive and negative sequence components present and the zero sequence quantities are equal to zero. Applying equation (2.25) to the stator voltage equation (3.23) we get the stator circuit quantities expressed in terms of dynamic phasors.  𝑉  𝑝𝑠𝑒𝑗𝜔𝑠𝑡 + 𝑉  𝑛𝑠∗ 𝑒−𝑗𝜔𝑠𝑡 = 𝑟𝑠�𝐼 𝑝𝑠𝑒𝑗𝜔𝑠𝑡 + 𝐼  𝑛𝑠∗ 𝑒−𝑗𝜔𝑠𝑡� + 𝑑𝑑𝑡 �𝛬𝑝𝑠𝑒𝑗𝜔𝑠𝑡 + 𝛬𝑛𝑠∗ 𝑒−𝑗𝜔𝑠𝑡�+ 𝑗𝜔�𝛬𝑝𝑠𝑒𝑗𝜔𝑠𝑡 + 𝛬𝑛𝑠∗ 𝑒−𝑗𝜔𝑠𝑡� (3.32)  If the differentiation in equation (3.32) is carried out and like terms are separated, we get the dynamic phasor equations for the stator voltage of the induction machine dynamic phasor model.  𝑉  𝑝𝑠 = 𝑟𝑠𝐼 𝑝𝑠 + 𝑑𝑑𝑡 𝛬𝑝𝑠 + 𝑗(𝜔 + 𝜔𝑠)𝛬𝑝𝑠 𝑉  𝑛𝑠∗ = 𝑟𝑠𝐼  𝑛𝑠∗ + 𝑑𝑑𝑡 𝛬𝑛𝑠∗ + 𝑗(𝜔 −𝜔𝑠)𝛬𝑛𝑠∗  (3.33) where  𝛬𝑝𝑠 = 𝐿𝑠𝐼 𝑝𝑠 + 𝐿𝑀�𝐼 𝑝𝑠 + 𝐼 𝑝𝑟� 𝛬𝑛𝑠 ∗ = 𝐿𝑠𝐼  𝑛𝑠∗ + 𝐿𝑀�𝐼  𝑛𝑠∗ + 𝐼  𝑛𝑟∗ � (3.34)  For the rotor circuit we have to consider the second harmonic of the rotor speed due to the torque ripple that is introduce into the machine during unbalanced operating conditions. The second harmonic torque ripple that is introduced into the system can be easily verified by doing a simple observation during simulation of the time domain models. For example,   36 Figure 3.6 illustrates the simulation of a typical 500 hp induction machine under a single phase unbalanced fault condition. It can be observed that under these circumstances a second harmonic ripple is introduced into the electromagnetic torque produced in the machine which in turn causes the rotor speed to have a small second harmonic torque ripple as well.   Figure 3.6  Torque and speed ripple during unbalanced machine operation  The torque ripple caused by unbalanced operating conditions is taken into account by considering both the dc component of the rotor speed and its second harmonic dynamic phasor, which are expressed as  𝛺𝑟,0 and 𝛺𝑟,2 respectively. It is important to note here that in this case 𝛺𝑟,0is real valued but for the sake of consistency it is expressed as a dynamic phasor. Applying equation (2.25) to the rotor voltage space vector equation (3.24) and taking into account the second harmonic dynamic phasor of the rotor speed we get   37  𝑉  𝑝𝑟𝑒𝑗𝜔𝑠𝑡 + 𝑉  𝑛𝑟∗ 𝑒−𝑗𝜔𝑠𝑡 = 𝑟𝑟′�𝐼 𝑝𝑟𝑒𝑗𝜔𝑠𝑡 + 𝐼  𝑛𝑟∗ 𝑒−𝑗𝜔𝑠𝑡� + 𝑑𝑑𝑡 �𝛬𝑝𝑟𝑒𝑗𝜔𝑠𝑡 + 𝛬𝑛𝑟∗ 𝑒−𝑗𝜔𝑠𝑡�+ 𝑗�𝜔 − 𝛺𝑟,0 − 𝛺𝑟,2𝑒𝑗2𝜔𝑠𝑡 − 𝛺  𝑟,2∗ 𝑒−𝑗2𝜔𝑠𝑡��𝛬𝑝𝑟𝑒𝑗𝜔𝑠𝑡+ 𝛬𝑛𝑟∗ 𝑒−𝑗𝜔𝑠𝑡� (3.35) Which, after expanding and separating like terms we can get the dynamic phasor equations of the rotor circuit voltage.  𝑉  𝑝𝑟 = 𝑟𝑟′𝐼 𝑝𝑟 + 𝑑𝑑𝑡 𝛬𝑝𝑟 + 𝑗�𝜔 + 𝜔𝑠 − 𝛺𝑟,0�𝛬𝑝𝑟 − 𝑗2𝛺𝑟,2𝛬𝑛𝑟∗ 𝑉  𝑛𝑟∗ = 𝑟𝑟′𝐼  𝑛𝑟∗ + 𝑑𝑑𝑡 𝛬𝑛𝑟∗ + 𝑗�𝜔 − 𝜔𝑠 − 𝛺𝑟,0�𝛬𝑛𝑟∗ − 𝑗2𝛺𝑟,2∗ 𝛬𝑝𝑟 (3.36) where  𝛬𝑝𝑟 = 𝐿𝑟′ 𝐼 𝑝𝑟 + 𝐿𝑀�𝐼 𝑝𝑠 + 𝐼 𝑝𝑟� 𝛬𝑛𝑟 ∗ = 𝐿𝑟′ 𝐼  𝑛𝑟∗ + 𝐿𝑀�𝐼  𝑛𝑠∗ + 𝐼  𝑛𝑟∗ � (3.37)  Note here that the prime symbol has been removed from the dynamic phasor rotor quantities to avoid clutter in the equations. However, they still represent rotor quantities that have been referred to the stator side circuit.  Figure 3.7 shows the dynamic phasor equivalent circuit of the 3-phase doubly-fed induction machine. Notice here that the zero sequence components of the induction machine have been neglected for reasons outlined in the previous sections of this chapter.   38  Figure 3.7  Dynamic phasor equivalent circuit of 3-phase induction machine  For the electromagnetic torque developed in the machine we have to also consider the second harmonic dynamic phasor as explained above. If we now apply equation (2.25) to the complex space vector equation (3.30) and take into account the second harmonic dynamic phasor to account for torque ripple we get the following equation for the electromagnetic torque developed in the machine in terms of dynamic phasors.  𝑇 𝑒,0 + 𝑇 𝑒,2𝑒𝑗2𝜔𝑠𝑡 + 𝑇  𝑒,2∗ 𝑒−𝑗2𝜔𝑠𝑡 = �3𝑃4 � ℐ𝓂��𝛬𝑝𝑠∗ 𝑒−𝑗𝜔𝑠𝑡 + 𝛬𝑛𝑠𝑒𝑗𝜔𝑠𝑡��𝐼 𝑝𝑠𝑒𝑗𝜔𝑠𝑡 + 𝐼  𝑛𝑠∗ 𝑒−𝑗𝜔𝑠𝑡�� (3.38)  Expanding out the equation (3.38) and collecting like terms we get dynamic phasor equations for the dc and second harmonic components of the electromagnetic torque developed in the machine.   39  𝑇 𝑒,0 = �3𝑃4 � ℐ𝓂�𝛬𝑝𝑠∗ 𝐼 𝑝𝑠 + 𝛬𝑛𝑠𝐼  𝑛𝑠∗ � 𝑇 𝑒,2 = 3𝑃𝑗8 �𝛬𝑛𝑠𝐼 𝑝𝑠 − 𝛬𝑝𝑠𝐼 𝑛𝑠� (3.39)  To couple the electrical quantities to the mechanical quantities we can express the mechanical equation for the doubly-fed induction machine in terms of dynamic phasors as follows.  𝐽 𝑑 𝑑𝑡 �𝛺𝑟𝑚,0 + 𝛺𝑟𝑚,2𝑒𝑗2𝜔𝑠𝑡�= �𝑇 𝑒,0 + 𝑇 𝑒,2𝑒𝑗2𝜔𝑠𝑡� − 𝑇𝑚 − 𝐾𝑓𝑟𝑖𝑐�𝛺𝑟𝑚,0 + 𝛺𝑟𝑚,2𝑒𝑗2𝜔𝑠𝑡� (3.40)  Which can be expanded and separated into like terms and finally we get the dynamic phasor equations for the dc and second harmonic components of the mechanical rotor speed as follows.  𝐽 𝑑 𝑑𝑡 𝛺𝑟𝑚,0 = �3𝑃4 � ℐ𝓂�𝛬𝑝𝑠∗ 𝐼 𝑝𝑠 + 𝛬𝑛𝑠𝐼  𝑛𝑠∗ � − 𝑇𝑚 − 𝐾𝑓𝑟𝑖𝑐𝛺𝑟𝑚,0 𝐽 𝑑 𝑑𝑡 𝛺𝑟𝑚,2 = 3𝑃𝑗8 �𝛬𝑛𝑠𝐼 𝑝𝑠 − 𝛬𝑝𝑠𝐼 𝑛𝑠� − �𝐾𝑓𝑟𝑖𝑐 + 𝑗2𝐽𝜔𝑠�𝛺𝑟𝑚,2 (3.41)  As in the previous induction machine models the rotor mechanical speed and the rotor electrical speed are related by the number of electromagnetic poles in the machine.  𝛺𝑟,0 = 𝑃2 𝛺𝑟𝑚,0 𝛺𝑟,2 = 𝑃2 𝛺𝑟𝑚,2 (3.42)    40 We now have all of the dynamic phasor equations needed in order to implement the model and verify that it is representative of the induction machine dynamic behavior.  3.3 Model Verification A series of tests are carried out in this section to verify that the dynamic phasor model is representative of the time domain electromagnetic transient model with considerable accuracy. The studies are carried out to represent different operating conditions and transient events to illustrate the behavior of the doubly-fed induction machine. These studies are carried out on a typical 500 hp induction machine with the machine parameters listed in Appendix A.  The studies that are carried out cover a range of operating circumstances that might typically be faced by an induction machine. These studies include; machine start-up from standstill; a change in mechanical torque; balanced voltage dips; and unbalanced voltage dips. These operating studies are carried out in the following sub-sections.  For comparison, the dynamic phasor model of the induction machine is compared to the dq0 induction machine model to verify its accuracy. This is done by simulating both models using a very small time step to show that the dynamic phasor model results converge to the well know dq0 model results. The time-step used throughout the model verification studies in the following sub-sections is ∆𝑡 = 50 µs. Also, a stationary reference frame, 𝜔 = 0, is used in both the dq0 induction machine model and the dynamic phasor induction machine model.    41 3.3.1 Free Acceleration of Induction Machine The model verification begins at time t=0s when a rated voltage of 2300 VL-L is applied to the motor which is allowed to accelerate freely from standstill. The resulting waveforms for the stator phase a current 𝑖𝑎, electromagnetic torque 𝑇𝑒 and the mechanical rotor speed 𝜔𝑟𝑚 are illustrated below in Figure 3.8, Figure 3.9 and Figure 3.10 respectively. These figures contain the time domain dq0 model quantities (EMT), the magnitude of the dynamic phasor quantities (DP) as well as the time domain result obtained using the dynamic phasor quantities (DP-EMT).   Figure 3.8  Stator current during free acceleration (Δt=50µs)  In Figure 3.8 it is observed the time domain result obtained using the dynamic phasor quantities matches very accurately the results obtained using the time domain dq0 induction machine model. Also, the magnitude of the dynamic phasor quantity illustrates that it acts as   42 an envelope for the time domain waveform. This can be advantageous in model simulations where current magnitudes are being studied and the actual details of the time domain waveforms are not the main focus of the study. For example, when one wants to determine the starting current magnitude of the induction machine, the dynamic phasor model can provide such information without having to determine all of the information of the time domain waveform.   Figure 3.9  Electrical torque during free acceleration (Δt=50µs)    43  Figure 3.10  Rotor speed during free acceleration (Δt=50µs)  For the electromagnetic torque and the rotor mechanical speed in Figure 3.9 and Figure 3.10 respectively, the dynamic phasor quantity is identical to the time domain result obtained by the dynamic phasor quantity. This is because under balanced operating conditions only the dc component of the electromagnetic torque and rotor speed is present and as mentioned previously, the dc components of these quantities are strictly real and contain no imaginary component. It is also observed that these quantities match with considerable accuracy the results obtained using the dq0 induction machine model.  3.3.2 Mechanical Torque Change At time t=2.5s a rated motoring torque of 1980 Nm is applied to the motor and then at t=3s the torque is reversed to a rated load torque. The same quantities that were plotted in the   44 previous study are again plotted here in Figure 3.11, Figure 3.12 and Figure 3.13 depicting the dynamic phasor magnitudes and both of the time domain results obtained using the dynamic phasor quantities and the dq0 induction machine model. It is observed that again the time domain result obtained using the dynamic phasor quantities represents very well the results obtained using the dq0 induction machine model and that the magnitudes of the dynamic phasor quantity in Figure 3.11 acts as an envelope to the time domain waveform.   Figure 3.11  Stator current during mechanical torque change (Δt=50µs)  Again, the electromagnetic torque developed in the machine and the rotor mechanical speed contain only the dc component of the dynamic phasor as the machine is operating under balanced conditions where no second harmonic torque component is developed. The time domain result is also accurately represented as illustrated in Figure 3.12 and Figure 3.13 for both the torque and rotor speed respectively.   45  Figure 3.12  Electrical torque during mechanical torque change (Δt=50µs)  Figure 3.13  Rotor speed during mechanical torque change (Δt=50µs)   46 3.3.3 Balanced 3-phase Voltage Dip At time t=4s of the simulation a balanced 30% voltage dip is applied to the terminals of the machine for 6 cycles and then cleared. The resulting phase a current, electromagnetic torque and rotor mechanical speed quantities are plotted below in Figure 3.14, Figure 3.15 and Figure 3.16, respectively. Again, it is observed that the dynamic phasor model results match the dq0 model results with considerable accuracy and the magnitude of the dynamic phasor current quantity provides an accurate representation of the envelope of the time domain current waveform.   Figure 3.14  Stator current during balanced 3-phase voltage dip (Δt=50µs)  Under balanced operation again, the electromagnetic torque developed in the machine as well as the rotor mechanical speed contain only the dc component as mentioned previously.   47  Figure 3.15  Electrical torque during balanced 3-phase voltage dip (Δt=50µs)  Figure 3.16  Rotor speed during balanced 3-phase voltage dip (Δt=50µs)   48 3.3.4 Unbalanced Single Line-to-Ground Fault At time t=5s an unbalanced single phase-to-ground fault is applied to the terminals of the machine and cleared after 6 cycles. In the case of an unbalanced fault we have both positive and negative sequence quantities in the machine. With the dynamic phasor model the positive and negative sequence dynamic phasors are treated separately and can therefore be studied separately. For example, Figure 3.17 and Figure 3.18 show the real and imaginary parts of the positive and negative sequence stator current dynamic phasors respectively.   Figure 3.17  Real and imaginary parts of positive sequence stator current dynamic phasor    49  Figure 3.18  Real and imaginary parts of negative sequence stator current dynamic phasor  Using equations (2.21) and (2.25) the dynamic phasors can be combined to form the time domain result for the stator current as illustrated below in Figure 3.19. This figure also illustrates the comparison between the qd0 model results (EMT), the time domain results obtained from the dynamic phasor model (DP-EMT) and the magnitude of the current envelope using the dynamic phasor model (DP).  When an unbalanced fault is introduced to the system it brings with it a slight error that propagates through the model as well. This error is difficult to notice in Figure 3.19 but becomes more apparent in the plot of the mechanical rotor speed in Figure 3.25. It is believed that this error is introduced to the model not by the introduction of the negative sequence current dynamic phasor but from the lack of higher frequency components that are not taken   50 into account during the transient period of the simulation as mentioned by Stankovic [31] and explained in greater detail by Sanders [45]. The introduction of these higher harmonics during switching transients and other events would increase the accuracy of the dynamic phasor model however the complexity of the model would greatly increase as well. Further investigation is required however, as the errors do not occur in any of the balanced operating conditions and seem to be introduced into the model only when unbalanced operating conditions are introduced to the model. The source of the errors was not definitively located in the model and requires further investigation at the time that this thesis was written.   Figure 3.19  Stator current during unbalanced fault (Δt=50µs)  As in the case of current dynamic phasors, the electromagnetic torque produced in the model as well as the rotor mechanical speed are analyzed as separate harmonic dynamic phasors.   51 The difference in these cases is that these quantities in this particular model contain only the dc component and 2nd harmonic dynamic phasors as mentioned in the previous section.  Figure 3.20 and Figure 3.21 respectively show the dc component and the real and imaginary components of the 2nd harmonic electromagnetic torque dynamic phasor developed in the machine model. These quantities can be recombined using the dynamic phasor equation (2.5) with 𝐾 = {0,2}. Note here that electromagnetic torque is a real valued quantity so we have 𝑇 𝑒,−2 = 𝑇 𝑒,2∗  so we can obtain the time domain result as 𝑇𝑒 = 𝑇 𝑒,0 + 2𝑅𝑒�𝑇 𝑒,2𝑒𝑗2𝜔𝑠𝑡�. The time domain result for the electromagnetic torque is shown in Figure 3.22 along with the result obtained using the dq0 model result for comparison.   Figure 3.20  DC component of electrical torque dynamic phasor   52  Figure 3.21  2nd harmonic component of electrical torque dynamic phasor  Figure 3.22  Electrical torque during unbalanced fault (Δt=50µs)   53 Similarly, the rotor speed is illustrated as its separate dynamic phasor components (dc, and 2nd harmonic) in Figure 3.23 and Figure 3.24, respectively. Also, as in the case of the electromagnetic torque, the mechanical rotor speed dynamic phasors can be combined to obtain the real valued time domain result using equation (2.5) with 𝐾 = {0,2}. Doing this we get the time domain result for the mechanical rotor speed as 𝜔𝑟𝑚 = 𝛺 𝑒,0 + 2𝑅𝑒�𝛺 𝑒,2𝑒𝑗2𝜔𝑠𝑡�. The time domain result for the mechanical rotor speed is illustrated below in Figure 3.25 along with the results obtained using the dq0 model for comparison.   Figure 3.23  DC component of rotor speed dynamic phasor    54  Figure 3.24  2nd harmonic component of rotor speed dynamic phasor   Figure 3.25  Rotor speed during unbalanced fault (Δt=50µs)   55 As mentioned previously, there is a slight error that is introduced in the results of the dynamic phasor model during unbalanced operating conditions. These errors introduced into the model were studied and the results are presented here in Table 3.1 which illustrates the maximum errors introduced into the model quantities both in absolute values and in percent error from the results obtained using the dq0 induction machine model. The maximum errors that are introduced into the model are quite small and are within ± 1.9% and are well within range to validate the dynamic phasor model during unbalanced operating conditions.  Table 3.1  Maximum error introduced by dynamic phasor model Variable Maximum Absolute Error Maximum Percent Error (%) 𝒊𝒂𝒔 7.5162 (A) 1.8924 𝑻𝒆 64.5221 (Nm) 1.4021 𝝎𝒓𝒎 0.0987 (rad/s) 0.0541    56 Chapter  4: Modeling of Saturation in Induction Machines  4.1 Time Domain Modeling of Induction Machine Saturation The induction machine models that were presented in Chapter 3 were based on the assumption that the main magnetizing flux linkage produced in the machine changed linearly with respect to the magnetizing branch current. In reality, the magnetizing flux does not follow such a linear relationship to the magnetizing current but instead starts to deviate from the linear relationship as the magnetizing current increases due to the effects of magnetic saturation in the induction machine [32-43]. Thus, there are some inaccuracies that occur in the models based on the linear magnetizing relationship. Saturation effects also occur in the leakage inductances present in each of the windings but are much more difficult to model so the analysis here will focus on saturation effects of the main magnetizing flux path. In order to produce a more accurate model the effects of saturation should be taken into account.  In induction machine modeling, saturation is often applied to the magnetizing flux as it takes into account the cross-saturation effects between the q- and d-axis of the machine [36, 49]. One method of modeling main flux linkage saturation is to apply a flux correction term to the magnetizing flux quantity known as magnetizing flux correction [49, 50]. Figure 4.1 illustrates the concept of the flux correction method. It depicts a typical saturation curve which is defined in terms of the magnitude of the magnetizing flux versus the magnitude of the magnetizing current. Along with the saturation curve there is also the linear relationship depicted along with the flux correction term ∆(𝜆𝑚).    57  Figure 4.1  Typical magnetization curve of an induction machine  In the linear model the magnetizing flux is related to the magnetizing current by the magnetizing inductance 𝜆𝑚 = 𝐿𝑚𝑖𝑚. We can correct this term to take into account the effects of saturation by subtracting the flux correction term from the linear model as follows.  𝜆𝑚 = 𝐿𝑚𝑖𝑚 − ∆(𝜆𝑚) (4.1)  Solving the above equation for ∆(𝜆𝑚) we get  ∆(𝜆𝑚) = 𝐿𝑚𝑖𝑚 − 𝜆𝑚 (4.2)  We now need to express ∆(𝜆𝑚) in terms of qd quantities in order to implement it into the time domain qd0 induction machine model. Referring to Figure 4.2 and assuming again that   58 we are dealing with a symmetric induction machine we can express the magnetizing flux in the qd reference frame as follows.  𝜆𝑚 = �𝜆𝑚𝑞2 + 𝜆𝑚𝑑2  (4.3) And the qd quantities of ∆(𝜆𝑚) as  ∆(𝜆𝑚)𝑞 = 𝜆𝑚𝑞𝜆𝑚 ∆(𝜆𝑚) ∆(𝜆𝑚)𝑑 = 𝜆𝑚𝑑𝜆𝑚 ∆(𝜆𝑚) (4.4)   Figure 4.2  qd decomposition of flux correction term  Now, to include the effects of saturation in the induction machine model we can simply subtract the qd quantities of ∆(𝜆𝑚) from the linear values of the qd quantities of 𝜆𝑚 as follows.   59  𝜆𝑚𝑞 = 𝐿𝑚�𝑖𝑞𝑠 + 𝑖𝑞𝑟′ � − ∆(𝜆𝑚)𝑞 = 𝐿𝑚 �𝜆𝑞𝑠𝐿𝑠 − 𝜆𝑚𝑞𝐿𝑠 + 𝜆𝑞𝑟′𝐿𝑟′ − 𝜆𝑚𝑞𝐿𝑟′ � − ∆(𝜆𝑚)𝑞 = 𝐿𝑎𝑞 �𝜆𝑞𝑠𝐿𝑠 + 𝜆𝑞𝑟′𝐿𝑟′ � − 𝐿𝑎𝑞𝐿𝑚 ∆(𝜆𝑚)𝑞 (4.5) and  𝜆𝑚𝑑 = 𝐿𝑚(𝑖𝑑𝑠 + 𝑖𝑑𝑟′ ) − ∆(𝜆𝑚)𝑑 = 𝐿𝑚 �𝜆𝑑𝑠𝐿𝑠 − 𝜆𝑚𝑑𝐿𝑠 + 𝜆𝑑𝑟′𝐿𝑟′ − 𝜆𝑚𝑑𝐿𝑟′ � − ∆(𝜆𝑚)𝑑 = 𝐿𝑎𝑑 �𝜆𝑑𝑠𝐿𝑠 + 𝜆𝑑𝑟′𝐿𝑟′ � − 𝐿𝑎𝑑𝐿𝑚 ∆(𝜆𝑚)𝑑 (4.6)  In equations (4.5) and (4.6) the terms 𝐿𝑎𝑞 and 𝐿𝑎𝑑 are equivalent inductance terms that are expressed as follows.  𝐿𝑎𝑞 = 𝐿𝑎𝑑 = � 1𝐿𝑚 + 1𝐿𝑠 + 1𝐿𝑟′ �−1 (4.7)  Equations (4.5) through (4.7) can now be used to implement the effects of magnetic saturation of the induction machine into the induction machine model.  4.2 Dynamic Phasors Model of Induction Machine Saturation We can extend the concept of the flux correction method to dynamic phasors by considering that we can represent a complex space vector as a sum of dynamic phasors rotating at different harmonics as explained previously in Section 2.2.2. So, in the case of the induction   60 machine dynamic phasor model we can represent the magnetizing flux as the sum of the two positive and negative sequence magnetizing flux components. That is we have  𝜆𝑚 = 𝛬𝑚𝑝𝑒𝑗𝜔𝑠𝜏 + 𝛬𝑚𝑛∗ 𝑒−𝑗𝜔𝑠𝜏 (4.8)  We can also express the flux correction term in terms of its dynamic phasors as well which gives us  Δ(𝜆𝑚) = Δ�𝛬𝑚𝑝�𝑒𝑗𝜔𝑠𝜏 + Δ(𝛬𝑚𝑛∗ )𝑒−𝑗𝜔𝑠𝜏 (4.9) Note here that the vector arrow and dynamic phasor under-bar have been dropped to avoid clutter in equation (4.9) and the rest of the equations in this section.  In a similar fashion as in section 4.1, we can express the corrected magnetizing flux vector in terms of its positive and negative sequence dynamic phasors.   𝛬𝑚𝑝 = 𝐿𝑚𝐼 𝑚𝑝 − Δ�𝛬𝑚𝑝� 𝛬𝑚𝑛 ∗ = 𝐿𝑚𝐼 𝑚𝑛∗ − Δ(𝛬𝑚𝑛∗ ) (4.10) So we get  Δ�𝛬𝑚𝑝� = 𝐿𝑚𝐼 𝑚𝑝 − 𝛬𝑚𝑝 Δ(𝛬𝑚𝑛∗ ) = 𝐿𝑚𝐼 𝑚𝑛∗ − 𝛬𝑚𝑛∗  (4.11)  A problem that we run into here is the fact that saturation data is generally given in terms of the magnitudes of the magnetizing flux vector versus the magnitude of the magnetizing branch current as mention earlier. So we have to determine the total dynamic phasor magnitude as �𝜆𝑚�. This causes the model to lose some of its advantage as we have to   61 recombine the positive and negative dynamic phasors to determine the magnitude of the magnetizing flux vector.   Figure 4.3  Saturation of main magnetizing flux linkage in terms of dynamic phasors  Once we have done this however we can determine the correction values for the positive and negative sequence magnetizing flux dynamic phasors in a similar manner as before. Figure 4.3 illustrates how we can break the rotating magnetizing flux vector down into its positive and negative sequence dynamic phasors and use them to graphically determine what the correction terms should be to include the effects of saturation in the dynamic phasor model of the induction machine. Doing this we get  Δ�𝛬𝑚𝑝� = 𝛬𝑚𝑝𝜆𝑚 Δ(𝜆𝑚) Δ(𝛬𝑚𝑛∗ ) = 𝛬𝑚𝑛∗𝜆𝑚 Δ(𝜆𝑚) (4.12)    62 Once we have determined the correction terms we can implement them in the model by simply subtracting the correction terms from the linear model. So the positive and negative sequence dynamic phasors of the magnetizing flux can be expressed as  𝛬𝑚𝑝 = 𝐿𝑒𝑞 �𝛬𝑝𝑠𝐿𝑠 + 𝛬𝑝𝑟𝐿𝑟′ � − Δ�𝛬𝑚𝑝� 𝛬𝑚𝑛 ∗ = 𝐿𝑒𝑞 �𝛬𝑛𝑠∗𝐿𝑠 + 𝛬𝑛𝑟∗𝐿𝑟′ � − Δ(𝛬𝑚𝑛∗ ) (4.13) Where  𝐿𝑒𝑞 = � 1𝐿𝑚 + 1𝐿𝑠 + 1𝐿𝑟′ �−1 (4.14)  We now have all the pieces necessary to implement the effects of saturation into the dynamic phasor model.  4.3 Saturation Model Verification A few studies are carried out in this section to verify that the modeling technique used to model the effects of magnetizing flux saturation in the dynamic phasor model are representative of the saturated qd0 induction machine model. We start by using the same machine parameters for a 500 hp induction machine contained in Appendix A. We then implement the saturation models outlined in sections 4.1 and 4.2 using the saturation data listed in Appendix B using a look-up table in Matlab/Simulink with the interpolation- extrapolation look-up method enabled for data occurring between the data points listed in Table B.1.    63 To illustrate the effects of saturation in the induction machine the model is started from standstill at 70% of its rated voltage and is allowed to reach steady state. At steady state, the machine is operating in the linear region of the magnetization curve. At time t = 3.5s, the machine voltage is raised to its full rated voltage of 2300 VL-L which brings the machine into the saturation region of operation. The machine is then allowed to reach steady state in the saturation region. Figure 4.4 and Figure 4.5 illustrate the transition of the machine model from the linear region of operation to the saturation region of operation for the stator phase a current and the magnetizing flux respectively. Figure 4.4 and Figure 4.5 also provide a comparison between the unsaturated model quantities (EMT), the qd0 time domain saturated quantities (EMT Sat), the dynamic phasor saturated quantities (DP Sat) and the time domain results of the saturated dynamic phasor model (DP-EMT Sat).   Figure 4.4  Stator current in saturation region   64  Figure 4.5  Magnetizing flux linkage in saturation region  We can see from Figure 4.6 and Figure 4.7, which are zoomed in sections of Figure 4.4 and Figure 4.5 respectively, that the dynamic phasor model is quite accurate when compared to the time domain qd0 induction machine model during balanced operation. There is very little error between the time domain qd0 model results and the time domain results obtained using the dynamic phasor model. Also the magnitude of the dynamic phasor provides an accurate envelope curve to the time domain results obtained using both models.    65  Figure 4.6  Zoomed view of stator current in saturation region   Figure 4.7  Zoomed view of magnetizing flux linkage in saturation region   66 The next study demonstrates the validity of the saturated dynamic phasor model during unbalanced machine operation. At time t = 4 s, an unbalanced single phase to ground fault is applied to the terminals of the machine for six cycles, at which time the fault is cleared and the machine is allowed to reach steady state in normal balanced operating conditions. Figure 4.8 and Figure 4.9 illustrate the operation of the machine models in the saturation region during unbalanced supply conditions. They also provide a comparison between the unsaturated model quantities (EMT), the qd0 time domain saturated quantities (EMT Sat), the dynamic phasor saturated quantities (DP Sat) and the time domain results of the saturated dynamic phasor model (DP-EMT Sat).   Figure 4.8  Stator current in saturation region during unbalanced fault    67  Figure 4.9  Magnetizing flux linkage in saturation region during unbalanced fault  We can again zoom in to the curves, Figure 4.10 and Figure 4.11, to illustrate that again the results obtained using the dynamic phasor model closely match those obtained using the qd0 induction machine model with little error. The error that is introduced during unbalanced machine operation is shown below in Table 4.1. From this we can see that the dynamic phasor model is again fairly representative of the qd0 induction machine model.  Table 4.1  Maximum error introduced by saturated dynamic phasor model Variable Maximum Absolute Error Maximum Percent Error (%) 𝒊𝒂𝒔 3.7327 (A) 2.1459 𝝀𝒎 0.0136(Wb) 0.3162   68  Figure 4.10  Zoomed view of stator current in saturation region during unbalanced fault   Figure 4.11  Zoomed view of magnetizing flux in saturation region during unbalanced fault   69 The studies performed in this section have validated the saturated dynamic phasor model and shown that it is quite representative of the well know qd0 induction machine model with little error. We are now in a position to be able to study the performance of the dynamic phasor model in terms of the time step that can be used in the model simulation while still producing reasonable results. The next chapter looks at the dynamic phasor model performance by conducting various studies using different time steps and analyzing the results obtained from the model.    70 Chapter  5: Induction Machine Dynamic Phasor Model Performance  5.1 Fixed Time-Step Model Performance To test the performance of the dynamic phasor induction machine model we perform simulations using the same test cases that were used in Chapter 4 using different time steps. The simulations are performed in Matlab/Simulink using the ode4 (Runge-Kutta) fixed time- step solver [51] which utilizes a fourth order Runge-Kutta formula to calculate the model states at each time-step. Three different time-steps were used in this study to show the accuracy of the model using small to large time steps which are; a small time-step of Δt=1.0ms, a medium time-step of Δt=2.0ms and a large time-step of Δt=3.5ms. The machine parameters in Appendix A   are utilized again here for the following studies.  First, the machine is started from standstill by applying a rated voltage of 23 kV to the input of the machine and allowed to freely accelerate to steady state. At time t = 2.5s a rated generating torque of 1980 Nm is applied to the model input and then reversed to a rated load torque of 1980 Nm at time t = 3s. Once the machine has again reached a steady-state a balanced 3-phase to ground fault is applied to the input terminals of the model at time t = 4s and then cleared after 6 electrical cycles. Then finally, a single phase to ground fault is applied to the terminals of the machine at time t = 5s which is again cleared after 6 electrical cycles. Figure 5.1 through Figure 5.4 show the stator current results for the previous test cases respectively. They contain the stator current envelope results obtained using both the conventional qd0 and the dynamic phasor model at different time-steps along with the time   71 domain stator current results obtained using the conventional qd0 model with a very small time-step of 50 µs; which is assumed to be the accurate solution for these test cases.   Figure 5.1  Stator current during start-up for fixed Δt studies   72  Figure 5.2  Stator current during load torque change for fixed Δt studies   73  Figure 5.3  Stator current during balanced 3-phase fault for fixed Δt studies     74  Figure 5.4  Stator current during unbalanced fault for fixed Δt studies  It can be seen from Figure 5.1 through Figure 5.4 that the results obtained using the dynamic phasor induction machine model are virtually identical to those obtained using the   75 convention qd0 model; even at a large time-step of 3.5 ms. Looking at Figure 5.4 however, we can see that the results obtained using the dynamic phasor induction machine model begin to deviate slightly from the results obtained using the convention qd0 induction machine model during unbalanced operation. The discrepancy becomes more pronounced at higher time-steps, but still the results are quite similar to each other.  As before, the current results for the dynamic phasor induction machine model are obtained by combining the positive and negative sequence dynamic phasors which are shown below in Figure 5.5 and Figure 5.6, respectively. It can be seen from these two figures that the large time-step results are still fairly representative of the more accurate results obtained at smaller time-steps for both the positive and negative dynamic phasor stator current quantities.   Figure 5.5  Positive sequence stator current dynamic phasor using different Δt   76  Figure 5.6  Negative sequence stator current dynamic phasor using different Δt  Similar results can be obtained for the electrical torque developed in the machine. Figure 5.7 illustrates the results for the electrical torque obtained using the dynamic phasor model at different time-steps. It can be seen from this figure that the results obtained using the larger time-step do deviate from the more accurate result. This is to be expected as the transient of the electrical torque contains both the dc component and the much quicker 2nd harmonic component which is changing at a rate of twice the system frequency, so when the dynamic phasor results are combined to form the results shown in Figure 5.7 the waveform becomes more distorted due to the error that is introduced at the higher time-steps. If we look at only the dynamic phasors themselves however, we can see from Figure 5.8 and Figure 5.9 that the results at larger time-steps are still quite consistent with those at the smaller time-step.   77  Figure 5.7  Electrical torque using different Δt  Figure 5.8  DC component of electrical torque dynamic phasor using different Δt   78  Figure 5.9  2nd harmonic component of electrical torque dynamic phasor using different Δt  Looking at the mechanical rotor speed in Figure 5.10 we can again see that the overall time domain result starts to deviate from the reference result at higher time-steps. This is again due to the 2nd harmonic component in the rotor mechanical speed similar to the electromagnetic torque which is a component that changes at twice the system frequency.    79  Figure 5.10  Rotor speed using different Δt  If we look at the dc component and the 2nd harmonic dynamic phasor separately as shown in Figure 5.11 and Figure 5.12 we can again see that the dc component at larger time-steps is still quite representative of the results at the small time-step. However, looking at the 2nd harmonic positive and negative sequence dynamic phasors we can see that the transient in the waveform reaches a steady value much more quickly at larger time-steps than it does at the small time-step, this is the major contributing factor in the error present in the overall time- domain result in Figure 5.10.   80  Figure 5.11  DC component of rotor speed dynamic phasor using different Δt  Figure 5.12  2nd harmonic component of rotor speed dynamic phasor using different Δt   81 We can see from the above test cases that the dynamic phasor induction machine model provides a fairly accurate and comparable model to the conventional qd0 induction machine model and can be used to simulate the induction machine during fast transients that occur during balanced and unbalanced faults as well as slow mechanical transients that occur during load changes. For example, Figure 5.13 shows the stator current results from the dynamic phasor model and we can see that during the slow transients of a load change we can see that the dynamic phasor model can accurately reproduce the current envelope even at the very large time-step of 7.5 ms.  Table 5.1 shows the CPU times for the simulation up to the 5 second point of the test case. We can see from this table that the dynamic phasor model is quite comparable to the conventional qd0 induction machine model with slightly faster CPU times for the simulation. It should be noted here that the CPU times presented for the dynamic phasor model reflect simulation during balanced machine operation only, meaning that there are no negative sequence phasors present. This reduces the total number of equations present in the dynamic phasor induction machine model to equal the number of equations in the qd induction machine model. With negative sequence dynamic phasors present the number of equations is doubled in the dynamic phasor model as presented in Section 3.2 of this thesis and the CPU times presented in Table 5.1 would be greater.       82  Table 5.1  CPU times for test case simulation Time-step Simulation Time qd0 (saturated) Dynamic Phasor (saturated) Δt = 0.5 ms 5 0.5898 s 0.4636 s Δt = 1.0 ms 5 0.3620 s 0.2798 s Δt = 5.0 ms 5 0.1778 s 0.1584 s Δt = 7.5 ms 5 0.1650 s 0.1490 s    Figure 5.13  Load change results using dynamic phasor model    83 5.2 Variable Time-Step Model Performance Due to the nature of dynamic phasor quantities; how their transient behaviour changes much slower with time and they reach a constant value at steady state, a real advantage in simulation can be achieved if variable time-step algorithms are used during simulation. To illustrate this, the dynamic phasor induction machine model is simulated in Matlab/Simulink using the ode45 (Dormand-Prince) variable time-step solver [51] which computes the models state at the next time-step using an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair for numerical integration. The maximum time-step parameter is set to 10 ms while the relative and absolute tolerances are set to 0.1 ms. The 500 hp induction machine parameters are used to simulate a 5 second simulation in which the model is started from standstill and allowed to freely accelerate to steady state at which time (t=2.5s) a mechanical rated load torque is applied to the model. At time t=3s the load torque is reversed to a generating torque and the model is again allowed to reach steady state. Then at time t=4s a balanced 3-phase to ground fault is applied to the supply voltage which is then cleared after 6 electrical cycles. This 5 second simulation is performed using the saturated dynamic phasor model as well as the saturated time domain qd0 model and the model performance results are compared. A reference result is obtained using the conventional qd0 time domain model with a very small fixed time-step of 50 µs; which is assumed to be the accurate result for comparison. Figure 5.14 through Figure 5.19 below show the results obtained for both the stator phase a current and the time-step used by the variable time-step solver for both the dynamic phasor induction machine model and convention qd0 induction machine model along with the reference solution for comparison.    84  Figure 5.14  Stator current during free acceleration using variable Δt solver  Figure 5.15  Time-step results during free acceleration using variable Δt solver   85  Figure 5.16  Stator current during load torque change using variable Δt solver  Figure 5.17  Time-step results during load torque change using variable Δt solver   86  Figure 5.18  Stator current during balanced fault using variable Δt solver  Figure 5.19  Time-step results during balanced fault using variable Δt solver   87  Table 5.2 summarizes the CPU times and the average time-step needed for the dynamic phasor and the qd0 model simulations. It can be seen that the dynamic phasor induction machine model is approximately 13% faster than the conventional qd0 induction machine model to achieve comparable accuracy using the variable step solver in Matlab/Simulink. It is also observed that the average time-step used during the 5 s simulation time for the dynamic phasor induction machine model is approximately 16% larger than that of the conventional qd0 induction machine model. This is not a huge improvement in overall simulation efficiency but illustrates that some advantages can be gained when using the dynamic phasor model for simulation.  The advantages of the dynamic phasor model over the conventional qd0 model are most evident during the faster transient sections of the simulation. For example, if we look at the time-step results of the simulation in Figure 5.15, Figure 5.17 and Figure 5.19 we can see that the time-step used by the dynamic phasor model is consistently greater than that used by the conventional qd0 model during the periods where the waveform is changing rapidly. Only when the waveforms have reached somewhat of a steady-state and the transient have slowed are the time-steps used by each comparable. This demonstrates that the dynamic phasor model does improve the efficiency of the simulation during transient periods but is fairly comparable to the convention qd0 model in efficiency during steady-state simulation.       88  Table 5.2  Model performance using variable time-step solver Model Simulation Time (s) CPU Time (s) Average Δt (ms) qd0 (saturated) 5 0.1760 3.9463 Dynamic Phasor (saturated) 5 0.1539 4.5914     89 Chapter  6: Conclusion  6.1 Summary of Contributions The aim of this thesis was to extend the simulation functionality of the dynamic phasor model of doubly-fed induction machines by incorporating the effects of saturation of main flux linkage into the model. This was accomplished by extending the time-domain flux correction method into frequency domain dynamic phasor quantities and implementing them in the model using saturation data for a 500 hp induction machine and a look up table in Matlab/Simulink. The main contributions provided by this thesis are listed below. I. The theory and some key properties of dynamic phasors and frequency domain analysis are outlined and explained. II. Several conventional time-domain induction machine models are outlined and explained. The time-domain models being the 3-phase abc induction machine model, the two-axis qd0 model and the complex space vector induction machine model. III. The dynamic phasor model of doubly-fed induction machines is derived from the complex space vector induction machine model and then verified by comparing it to the conventional qd0 induction machine model. IV. An effective method for modeling the effects of saturation in the main flux linkage in terms of dynamic phasors is proposed; based on the flux correction method commonly used in time domain models of induction machines. The saturation model is also verified by comparing it to the convention time-domain flux correction method.   90 V. The dynamic phasor induction machine model performance is then studied for both fixed time-step and variable time-step solution algorithms and found to be quite comparable and slightly better in performance than the convention qd0 time-domain induction machine model. The dynamic phasor model was found to be quite accurate and efficient for simulation of slow transient situations of mechanical load changes as well as fast transient situations such as machine start-up and balanced and unbalanced faults that occur in the system.  6.2 Future Research The purpose of this thesis was to provide an initial starting point for research into modeling the effects of saturation in induction machines using the concept of dynamic phasors. There are many paths that can be investigated to further the research into dynamic phasor modeling of induction machines as well as modeling the effects of saturation using the dynamic phasor concept. The following list is definitely not exhaustive but provides a few topics that could be a good direction for further investigation and research in the area of AC machine modeling and saturation modeling using the dynamic phasor concept.  I. In modeling of the main flux linkage saturation in an induction machine it was assumed that the rotor is round and that the air-gap between stator and rotor is uniform. By making this assumption it is reasonably assumed that the saturation in the machine affects the q- and d-axis components in the same manner, making it fairly straightforward to model. For machines that do not have a uniform air-gap such as salient pole synchronous machine this assumption cannot be made as the saturation   91 will affect the q- and d-axis differently [52]. There are many papers on the modeling of saturation in salient pole machines and this would provide a good area of research to extend the dynamic phasor saturation model to include the effects of saturation in salient pole machines.  II. In this thesis, the method of flux correction is used to calculate the effects of saturation in the main magnetizing flux. This method requires first the calculation of the unsaturated main flux and then corrects it based on the saturation characteristics of the machine as outlined in Chapter 4. This formulation is implicit and introduces an algebraic loop which must be solved iteratively at every time-step and therefore adds computation costs to the simulation. By formulating the calculation of the effects of saturation in an explicit manner this algebraic loop can be avoided and make the model much more stable and efficient. One such method of explicit calculation of main flux linkage saturation is presented in [53]. This paper derives a simple algebraically exact formulation of main flux saturation for the general purpose qd induction machine model by introducing intermediate variables which provide an explicit relationship for the calculation of the main flux linkage. A similar process could be used to implement an explicit relationship for the calculation of main flux linkage saturation using the dynamic phasor concept. By doing this it would be possible to further extend the robustness and utility of the dynamic phasor induction machine model.    92 III. In modeling of saturation of induction machines the effects of leakage flux saturation is generally ignored but can play an important part in the operation of induction machines, especially during transient operation [41]. The effects of leakage flux saturation are often ignored due to the fact that their saturation characteristics are very hard to determine experimentally and that data is often not available. Research in the area of leakage flux modeling using the concepts of dynamic phasors could be another rich area of research.  IV. The dynamic phasor induction machine model that was presented in this thesis is based on the transformation of abc phase domain quantities to the stationary qd reference frame as it provides a much simpler starting point to base the dynamic phasor model off of. Despite the advantages of the reference frame transformation, when simulating the transformed qd induction machine model with an external an interface between the model and the external system is required as in many cases the external system is modeled in abc phase domain quantities. This interfacing can prove difficult and costly in many cases as a transformation back to abc phase domain quantities is required. Other difficulties can arise as well depending on which type of simulation tool you are using.  One method suggested for providing a direct interfacing induction machine model to external networks in simulation is the use of voltage-behind-reactance (VBR) induction machine model [54-57]. The VBR induction machine model presented in these papers has the stator of the induction machine modeled in abc direct-phase   93 coordinates as three-phase dependant emf sources behind an RL circuit, while the rotor is expressed in the qd coordinate reference frame. Modeling the machine in this manner provides a direct interface between the induction machine model and the external network. The VBR models could be extended to the dynamic phasor concept and would provide a good direction for research in extending the applications of dynamic phasors.  V. Extending from further research IV, implementing the effects of magnetic saturation in the VBR models using the concept of dynamic phasors would be a good topic of research. Wang et al. 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