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Local voltage stability assessment for variable load characteristics Vargas Rios, Leon Maximino 2009

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LOCAL VOLTAGE STABILITY ASSESSMENT FOR VARIABLE LOAD CHARACTERISTICS  by  Leon Maximino Vargas Rios  B.Sc., Tecnologico de Monterrey, campus Ciudad de Mexico, 2003  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)  May 2009  © Leon Maximino Vargas Rios, 2009  Abstract Voltage stability problems originate when a generation-transmission system is not able to supply a load connected remotely from the generation centers. Knowledge of the load characteristics and the load composition are necessary tools for voltage stability assessment (VSA). This thesis presents methods and results to evaluate local voltage stability conditions, considering the actual static load characteristics. Simulation of a modern load, verified by experimental tests, such as a variable frequency drive feeding an induction motor (IM), shows that the real power characteristic is very similar to the directly-connected motor, while the reactive power characteristic is different. The effect on voltage stability is described. A small-scale voltage stability test is performed for a single IM under increasing mechanical load, fed by a source and series reactance. A set of slip dependent PV curves, which show variable power factor (PF) behavior, is obtained and compared with the assumption of constant PF loads for VSA. A comprehensive case study is performed, which presents an experimentally obtained IM load characteristic, resulting in variable PF as expected from the equivalent circuit model. The IM is aggregated by simulation in a bus fed by a Thevenin network, and a numerical method is proposed to compute a local PQV curve that considers the actual load characteristic. It is demonstrated that traditional PV curves for constant PF loads do not describe properly the aggregation of induction motor loads in a bus. A graphical approach of network and load PV characteristic intersections for this variable PF load, confirmed with a time domain simulation, shows that the point of matching impedance, typically assumed as the voltage stability limit, is not the power transmission limit (“nose”), and the latter is not the static voltage stability (loadability) limit. The methods and results developed in the case study are extended to other motor, heating and lighting types of loads. Their experimental characteristics are later combined by simulation in one distribution bus of a university building to perform static local VSA. Finally, some implementation ideas for on-line load characteristic estimation and PQV curve computation are described as part of the tools for local VSA.  ii  Table of Contents Abstract  ..................................................................................................................... ii  Table of Contents ............................................................................................................. iii List of Tables .................................................................................................................... vi List of Figures.................................................................................................................. vii Acknowledgements .......................................................................................................... ix Dedication  ..................................................................................................................... x  Statement of Co-Authorship ........................................................................................... xi Chapter 1.  Introduction............................................................................................... 1  1.1.  Background and Motivation ....................................................................... 1  1.2.  Thesis Organization .................................................................................... 4  1.3.  References................................................................................................... 6  Chapter 2.  Load Modeling of an Induction Motor Operated with a Variable Frequency Drive ....................................................................................... 8  2.1.  Introduction................................................................................................. 8  2.2. 2.2.1. 2.2.2. 2.2.3.  Variable Frequency Drive Circuit and Expected Waveforms .................. 10 Simplified Equivalent Circuit ................................................................... 10 Three Phase Inverter ................................................................................. 10 Three Phase Rectifier................................................................................ 12  2.3. Experimental Results ................................................................................ 15 2.3.1. Experimental Setup................................................................................... 15 2.3.2. Motor Drive Output Waveforms............................................................... 15 2.3.3. Motor Drive Input Waveforms ................................................................. 16 2.3.4. Power-Voltage Steady State Characteristic .............................................. 17 2.3.4.1. Real and Reactive Power .................................................................. 17 2.3.4.2. Apparent Power ................................................................................ 18 2.3.5. Exponential Load Model PV and QV Characteristics .............................. 19 2.4.  Harmonics and Power Factor.................................................................... 20  2.5.  Relation to the Voltage Stability Problem ................................................ 22  2.6.  Conclusions............................................................................................... 24  2.7.  References................................................................................................. 25  iii  Chapter 3.  Induction Motor Loads and Voltage Stability Assessment Using PV Curves ..................................................................................................... 26  3.1.  Introduction............................................................................................... 26  3.2. 3.2.1. 3.2.2.  Motor Load and System PV Curves ......................................................... 30 IM Directly Connected to the Network .................................................... 30 IM Fed by a VFD Connected to the Network........................................... 33  3.3.  Experimental Setup................................................................................... 34  3.4. 3.4.1. 3.4.2.  Results....................................................................................................... 36 Measured PV Curves for Source and Impedance Fed IM ........................ 36 Measured Static Load PV Characteristics................................................. 37  3.5.  Analysis of Static Load PV Characteristics .............................................. 39  3.6.  Conclusions............................................................................................... 40  3.7.  References................................................................................................. 41  Chapter 4.  Local Voltage Stability Assessment for Variable Load Characteristics ................................................................................................................... 43  4.1.  Introduction............................................................................................... 43  4.2. 4.2.1. 4.2.2.  Loads with Variable Power Factor Characteristics................................... 46 Load Characteristics from the Induction Motor Equivalent Circuit ......... 46 Variable Slip and Power Factor due to Aggregate Loads......................... 47  4.3. 4.3.1. 4.3.2.  Load Aggregation for Analysis in PQV Space ......................................... 49 Single Load Characteristics and Power Factor Angle .............................. 49 Load Aggregation ..................................................................................... 50  4.4. Case Study for Induction Machine Type of Loads ................................... 53 4.4.1. Induction Motor Experimental Load Characteristic ................................. 53 4.4.1.1. Equipment Setup............................................................................... 53 4.4.1.2. Experimental Procedure.................................................................... 54 4.4.1.3. Data Post-Processing ........................................................................ 55 4.4.1.4. Results............................................................................................... 55 4.4.2. Aggregation of Induction Motor Loads .................................................... 56 4.4.3. Power Transmission Limits and Impedance Matching............................. 58 4.4.4. Voltage Stability Limits............................................................................ 59 4.4.4.1. Dynamic Simulation Setup ............................................................... 60 4.4.4.2. Dynamic Simulation Results............................................................. 61 4.5. 4.5.1. 4.5.2.  Characteristics and Aggregation of Other Commonly Used Loads.......... 65 Single Loads.............................................................................................. 65 Aggregated Loads in a Building ............................................................... 69  4.6. 4.6.1. 4.6.2.  Proposed Implementation Methods .......................................................... 73 Curve Fitting and Extrapolation of Power Factor Angle .......................... 74 Database Curve Matching......................................................................... 74  4.7.  Conclusions............................................................................................... 75 iv  4.8. Chapter 5.  References................................................................................................. 76 Conclusions and Future work................................................................ 81  5.1.  Summary ................................................................................................... 81  5.2.  Conclusions............................................................................................... 81  5.3.  Future Work .............................................................................................. 83  5.4.  References................................................................................................. 85  Appendix A. Contributions.......................................................................................... 86 Appendix B. Measurement Box Modifications.......................................................... 87 Appendix C. Additional Results from Chapter 3 ...................................................... 88 Additional Experimental Results for Motor Type Load Characteristics ...................... 88 Additional Experimental Results for Network PV Curves of Single IM...................... 90 Appendix D. Additional Results from Chapter 4 ...................................................... 91 Miscellaneous Load Combinations............................................................................... 91 Theoretical IM Load Characteristic .............................................................................. 93 Building Local Voltage Stability Assessment with Exponential Model ...................... 94 Appendix E. Specifications of Equipment from Chapter 2...................................... 95 Induction Motor Nameplate Ratings............................................................................. 95 Variable Frequency Drive Nameplate Ratings ............................................................. 95 Appendix F. Specifications of Equipment from Chapter 3...................................... 96 Induction Motors Parameters........................................................................................ 96 Iron Core Inductors (each) ............................................................................................ 96 Variable Frequency Drive Nameplate Ratings ............................................................. 96 Appendix G. Specifications of Equipment from Chapter 4...................................... 97 Induction Motor 3-Phase Parameters............................................................................ 97 Other Loads Nominal Parameters ................................................................................. 97 Appendix H. Vita .......................................................................................................... 98  v  List of Tables Table 2-1. Diodes and current flow in a 3-phase rectifier. ............................................... 13 Table 2-2. Measured load characteristics.......................................................................... 19 Table 3-1. Measured load characteristics, full load. ......................................................... 39 Table 3-2. Measured load characteristics, half load. ........................................................ 39 Table 4-1. Induction motor exponential load model......................................................... 56 Table 4-2. Load demand time sequence. .......................................................................... 61 Table 4-3. Experimental single load characteristics, exponential (V0 = 120Vrms). ........... 65 Table 4-4. Experimental single load characteristics, polynomial (same P0, Q0, V0). ....... 66 Table 4-5. University building load composition. ............................................................ 70 Table 4-6. Building combined load characteristic, polynomial. ....................................... 71 Table C-1. Exponential static load model for additional motor type loads. ..................... 88 Table C-2. Polynomial static load model for additional motor type loads. ...................... 89 Table D-1. Experimental miscellaneous combinations load characteristics, exponential.91 Table D-2. Experimental miscellaneous combinations load characteristics, polynomial. 91 Table D-3. Calculated and experimental IM load characteristic, exponential.................. 93 Table F-1. Induction machines parameters. ...................................................................... 96  vi  List of Figures Fig. 2-1. Phase voltage and currents of an IM, showing lagging power factor. ................ 9 Fig. 2-2. Block diagram of VFD-fed IM system. .............................................................. 9 Fig. 2-3. Equivalent circuit of the variable frequency drive. ........................................... 10 Fig. 2-4. Inverter model output voltage, detailed and fast averaged................................ 11 Fig. 2-5. IM stator phase voltage and current, coming from VFD output. ...................... 11 Fig. 2-6. Simulated 3- phase currents seen in the AC line............................................... 12 Fig. 2-7. Peak detection characteristic of 3-phase rectifier.............................................. 13 Fig. 2-8. Capacitor current, DC voltage ripple, rectifier 3-phase input current............... 14 Fig. 2-9. Fast averaged line-to-ground inverter output voltage. ...................................... 16 Fig. 2-10. Measured, IM stator phase voltage and current. ............................................. 16 Fig. 2-11. VFD input terminals 3-phase measured voltages and current......................... 17 Fig. 2-12. Real and reactive power of IM and VFD-IM, experimental results................ 18 Fig. 2-13. Apparent power calculation for different voltage levels. ................................ 19 Fig. 2-14. Distortion factor as a function of THD. .......................................................... 21 Fig. 2-15. Harmonic spectrum of the line current for the VFD-fed IM........................... 21 Fig. 2-16. Basic power transmission circuit with resistive losses neglected. .................. 22 Fig. 2-17. Phasor relationships for inductive (top) and resistive loads (bottom)............. 22 Fig. 2-18. Normalized PV curves for different values of load phase angle..................... 23 Fig. 2-19. Operating point for loads with different PF and α values ............................. 23 Fig. 3-1. Simplified one-line diagram of power transmission equivalent circuit. ........... 27 Fig. 3-2. Constant power factor PV curves...................................................................... 27 Fig. 3-3. Typical load PV characteristics at different demands. ...................................... 28 Fig. 3-4. Intersections of system and loads PV curves, before and after contingency. ... 29 Fig. 3-5. Induction machine per phase equivalent circuit................................................ 31 Fig. 3-6. IM slip dependent and constant power factor PV curves comparison. ............. 32 Fig. 3-7. Torque-slip characteristics for constant and slip dependent voltages. .............. 33 Fig. 3-8. Test setup for network PV curve when load is a single IM. ............................. 35 Fig. 3-9. Test setup for PV load characteristic of directly connected IM. ....................... 35 Fig. 3-10. Test setup for PV load characteristic of open loop VFD-fed IM. ................... 35 Fig. 3-11. Test setup for PV load characteristic of closed loop VFD-fed IM.................. 35 Fig. 3-12. PV curve theoretical and experimental, induction motor #1........................... 36 Fig. 3-13. PV curve theoretical and experimental, induction motor #2........................... 36 Fig. 3-14. Measured speed when IM1, in series with 6 Xth, stalls.................................. 37 Fig. 3-15. Experimental load PV characteristic, induction motor #1. ............................. 38 Fig. 3-16. Experimental load PV characteristic, induction motor #2. ............................. 38 Fig. 4-1. Basic power transmission circuit for lossless line............................................. 44 Fig. 4-2. Family of normalized PV curves for constant power factor loads.................... 44 Fig. 4-3. Induction machine equivalent circuit. ............................................................... 47  vii  Fig. 4-4. Simplified induction machine equivalent circuit. ............................................. 47 Fig. 4-5. Terminal voltage and operating slip.................................................................. 48 Fig. 4-6. Load PV characteristic and network PV curves................................................ 50 Fig. 4-7. Modified power transmission circuit for variable load characteristics. ............ 51 Fig. 4-8. Flow diagram of PQV curve computation for variable load characteristics. .... 52 Fig. 4-9. Experimental setup one-line block diagram...................................................... 54 Fig. 4-10. Induction motor exponential load characteristics (real and reactive power). . 56 Fig. 4-11. Power factor angle of single and aggregated network load. ........................... 57 Fig. 4-12. Network PQV curve for aggregation of IM’s. ................................................ 57 Fig. 4-13. PQ plane projection for aggregation of IM’s. ................................................. 58 Fig. 4-14. PQ and QV projections for aggregation of IM type of load............................ 59 Fig. 4-15. Time results of real and reactive powers, voltage and impedance.................. 61 Fig. 4-16. Load characteristic, network and dynamic model PV curves. ........................ 62 Fig. 4-17. Load characteristic, network and dynamic model QV curves. ....................... 62 Fig. 4-18. Dynamic model results. Power transfer and stability limits............................ 64 Fig. 4-19. Points of maximum power transfer and stability (loadability) limit. .............. 64 Fig. 4-20. Network PQV curves for a variety of single-type loads. ................................ 67 Fig. 4-21. Network PV curves for motor and heating type loads. ................................... 67 Fig. 4-22. Network PV curves for lighting type loads only............................................. 68 Fig. 4-23. Power factor and power characteristic comparison......................................... 69 Fig. 4-24. Building aggregated load characteristic and power factor angle. ................... 72 Fig. 4-25. Building network PQV curve, polynomial load characteristic. ...................... 72 Fig. 4-26. Building network PV curve and polynomial load characteristics for VSA. ... 73 Fig. B-1. Schematic of laboratory measurement box modifications. .............................. 87 Fig. C-1. PV curve theoretical and experimental, induction motor #3. ........................... 90 Fig. C-2. PV curve theoretical and experimental, induction motor #4. ........................... 90 Fig. D-1. Network PQV curves for miscellaneous combinations of loads...................... 92 Fig. D-2. Network PV curves for miscellaneous combinations of loads......................... 92 Fig. D-3. Network QV curves for miscellaneous combinations of loads. ....................... 93 Fig. D-4. Calculated and experimental single IM load characteristic.............................. 94 Fig. D-5. Building network PV curve and exponential load characteristics for VSA. .... 94  viii  Acknowledgements I would like to thank my supervisor Dr. Jose Marti for providing me the opportunity of coming to Vancouver to join his worldwide recognized research group at UBC, for sharing with me some of his vast knowledge and ideas, for his help to select and follow on a good research topic and for his financial support. I feel privileged for having had the opportunity of holding technical discussions with someone of his caliber. I want to express my very deep appreciation to my supervisor Dr. Juri Jatskevich, for all his clear direction, encouragement and continuous feedback during this research work, for providing me all the facilities and equipment that I needed to produce results, for all his time and overtime stays reviewing and helping me improve my manuscripts and presentations product of this research, and for all his support towards the next step of my professional career back in Mexico. I appreciate the time and feedback from Dr. Hermann Dommell and Dr. K.D. Srivastava for the final review and oral examination of the present graduate thesis. I am also very grateful to Thomas Chelliah. He contributed, professionally and personally, to make my experimental work in the electrical machines laboratory much safer, reliable and enjoyable. The financial support of The University of British Columbia and the government of British Columbia through the Ministry of Advanced Education, by means of the University / Pacific Century Graduate Fellowship, is greatly appreciated too. I want to recognize Tom de Rybel, for ensuring that we have a nice facility to work where everything runs as it is supposed to, and for sharing his practical experience to solve tiny problems for which books do not help. I am also grateful to many other of my colleagues in the Power Lab, who have helped me in very different moments and ways along my master program, special thanks to Sina, Siva, Prasad, Mike and Marcelo. Thank you very much to my girlfriend, Alexandra, for her unconditional support and patience, especially during the toughest times of this work when I needed it the most. Last, but not least, I also want to say thank very much my parents, Imelda and Max, and my brother Adrian, for all their encouragement and help during a long time. My parents should share a significant portion of my professional achievements.  ix  Dedication  Para mis papas, Imelda y Max  x  Statement of Co-Authorship I am the first author and principal contributor of all the manuscript chapters. These are co-authored with Dr. Jose R. Marti and Dr. Juri Jatskevich, both of them as research supervisors for my M.A.Sc. program.  xi  Chapter 1. Introduction 1.1.  Background and Motivation Voltage stability refers to the ability of a power system to maintain steady  voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition [1]. It is strongly related to the equilibrium between load power demand and network power supply. A major contributor to voltage instability is the voltage drop that occurs when active and reactive power flow through a line from the generation areas to the load areas. This drop in voltage limits the power that can be delivered to the load. Depending on the load sensitivity to the voltage, also known as load characteristic, the load may try to restore the original power, by means of slip adjustment in the case of motors, automatic load tap-changers, regulated power supplies, etc. But, once the loadability limit [2], [21] is reached, the network may not be able to transfer the power required by the loads, resulting in voltage collapse; which may lead to loss of load, tripping of protective devices, operation in abnormally low voltages for loads that remain on, and eventually a blackout. In North America, the demand for the electrical energy has increased at a rate of 2% yearly for the last 25 years [3]. The electrical transmission networks that carry power from the generation centers to the load centers, usually far away, have not grown in the same proportion, due to lack of investment, as well as public opposition to build new infrastructure [4]. Transmission and distribution systems that were originally overdesigned are now operating in stressed mode. Other problems of stability, such as rotor angle stability, originated from loss of synchronism between machines, are now of less concern due to the existence of faster fault clearing devices; while problems such as voltage stability have become one of the main concerns for the electrical utilities (North American Electric Reliability Corporation (NERC) surveys [5]). Not only the lack of construction of infrastructure for power transmission, but also the aging of the existing one is a factor which increases the risk of voltage collapse scenarios. This in turn increases the possibility of contingencies that stress even more an 1  already congested grid. Recent large outages, such as the one in Eastern US / Canada in August 2003 [6], have emphasized the importance of adequate monitoring and policies to operate the power grid reliably. Operating the system to the point where voltage stability problems are a concern requires to carefully monitor the operating conditions of feeding networks and loads on critical buses, in order to evaluate how far the system is from the voltage stability limits, how it will react in case of a contingency, and how much more load can be supplied without going to unsafe conditions. Energy infrastructure defense systems need to be developed to avoid compromising the quality of life of people and cause global disruptions [7]. Due to cost and feasibility considerations, or due to failures in communication systems, not all the buses in the power grid can be monitored on-line from a utility control center at all times. However, there is a strong need to monitor voltage stability conditions on buses which are known to be “weak”, that is, buses which can be prone to voltage stability problems due to heavy loads, feeding lines operating close to design limits or increasing load demand. This is especially true at distribution levels, where most of the buses are not monitored. The need to monitor buses locally and to take self-corrective actions in case of contingencies has become the basis for initiatives such as the Smart-Grid [8] and the concept of Self-Healing-Networks. A necessary component of these systems is the capability of local on-line voltage stability assessment (VSA). Voltage stability is a highly complex, multi-dimensional problem, which includes long-term scenarios (slow change of steady state operating conditions), as well as dynamic phenomena of fast and slow transients [9]. Several researchers have worked on voltage stability assessment at a system level, where modifications to conventional power flow routines have allowed computing power-voltage (PV) curves for all buses. A relay for local VSA must be capable of monitoring the operating conditions, calculate the margin with respect to estimated voltage stability limits at all times, and deciding if, when and which loads to shed [10] in case that there exists an imminent risk of voltage instability. All of this relies only on the measurements of local 3-phase voltage and current waveforms.  2  Work performed by the UBC Electrical Power and Energy Systems Research Group on the voltage stability area has included the estimation of the equivalent Thevenin impedance on a bus, based on local, single-time measurements [11]-[12]. The focus of this thesis is on the load side of the bus, during steady state conditions. The maximum power transfer theorem states that the maximum power transfer between source and load through a transmission line occurs when the load impedance is the conjugate of the network impedance. For power systems applications, though, the load impedance magnitude and angle is not an independent controllable variable. Also the transmission lines are usually simplified as purely inductive (at high voltage levels) so that maximum power transfer would occur on a purely capacitive load, which by definition, does not consume real (active) power [2]. As a result, a modification of this theorem that has been traditionally used in Power Systems Applications [2], assumes constant power factor loads, so that the power is maximized when the load impedance magnitude equals the network impedance magnitude. The previous modified power transfer theorem has been traditionally used in local voltage stability analysis [13]-[14] to define the voltage stability limit. Some loads, as observed from analytical derivation, experimental tests and literature review [9], [15]-[20], behave in such a way that their power factor is not constant under variable voltage. Then traditional tools for VSA, such as families of network PV curves for constant power factors, may not accurately describe the combination and aggregation of variable PF loads in a bus. The objective of this thesis is to perform local VSA considering actual load characteristics. The methodology is to carry out experimental tests to find typical single load characteristics, and then run simulations to aggregate them in a fixed known network and obtain improved calculations for the same traditional tools of PV curves of network and load, to perform local VSA. The scope of this work is to determine off-line the methods and assumptions valid for local steady state VSA; in order to provide the tools to combine the load characteristic estimation with the network estimation. The implementation of these results on-line and their integration with transient characteristics will be future work of our research group in the field of voltage stability analysis.  3  1.2.  Thesis Organization The present manuscript based thesis consists of five chapters, including this  introductory one which comprises the background, motivation and organization of this work. Chapter 2 presents a manuscript mostly focused on load-modeling, in particular for the variable frequency drive fed induction motor (VFD-IM); in order to understand whether an IM controlled by VFD shows characteristics relevant to VSA similar to the directly-connected IM. Chapter 3 extends the load modeling to VFD-IM’s and IM’s operating under different conditions of speed feedback and mechanical load. Additionally it presents a first approach of PV curves for variable power factor loads, for the particular case of IM’s under variable mechanical load, which are calculated and experimentally verified. Chapter 4, the core chapter of this thesis, gives a much more comprehensive view of the local steady state voltage stability analysis considering load characteristics, starting from the experimental characterization of an IM load, explaining the cause and effects of variable PF when this type of load is aggregated in a network fed bus. This is followed by the development of a method to compute steady state PQV curves for variable load characteristics, not constrained by constant power factor or for a specific load model; the method is tested for the load aggregation previously mentioned, and then combined with a time domain simulation, in order to understand the differences between impedance matching condition, power transfer limit and load-driven voltage stability limit (loadability limit), again not limited to constant power factor, as traditionally assumed, but considering actual load characteristic. The experimental methods and results are extended to a diversity of types of loads, and an estimated combination of those in a known building is applied by simulation to make a local off-line voltage stability assessment on a low voltage distribution bus. Finally, this manuscript chapter also provides preliminary ideas about the on-line implementation of the load characteristic estimation and the PQV curve computation applied to VSA. The general thesis summary and conclusions are included in Chapter 5, as well as suggested improvements on the static load characterization methods presented here; based on the experimental and simulation results during this 18-month work.  4  To wrap-up this thesis, the appendices show additional experimental and simulation results which were not included in the original manuscripts due to space or logical organization reasons, but the author believes they are important to complement this thesis to provide more information for the students of the group following on this research topic. Appendices include also the specifications of the equipment used for the experimental work in each chapter.  5  1.3. [1]  References IEEE/CIGRE Joint Task Force on Stability Terms and Definitions. “Definition and Classification of Power System Stability”. IEEE Transactions on Power Systems, vol. 19, no. 2, May 2004, pp. 1387-1401.  [2]  T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. Kluwer Academic Publishers, 1998, 378p.  [3]  UBC Electric Power and Energy Systems Group, Group Website 2009. Online [http://power.ece.ubc.ca].  [4]  D. Novosel, “Defense Plan Against Cascading Outages (CIGRE TF38.02.24), IEEE T&D Conference, Dallas TX, May 2006, Presentation.  [5]  North American Electric Reliability Corporation (NERC), “Results of the 2007 Survey of Reliability Issues”, rev. 1, October 24th, 2007, 18p.  [6]  US – Canada Power System Outage Task Force, “Final Report on the August 14th 2003 Blackout in the United States and Canada: Causes and Recommendations”, April 2004. Online [https://reports.energy.gov/BlackoutFinal-Web.pdf], 228p.  [7]  M. Amin. “Scanning the Technology: Energy Infrastructure Defense Systems”, proceedings of the IEEE, vol. 93, no. 5, May 1993, pp. 861-874.  [8]  Power Technology Task Group, “A Vision for Growing a World-Class Power Technology Cluster in a Smart, Sustainable, British Columbia”, Report to the Premier’s Technology Council, March 2005, 127p.  [9]  C.W. Taylor, Power System Voltage Stability. McGraw-Hill, 1994, 273p.  [10] C.W. Taylor, “Concepts of Undervoltage Load Shedding for Voltage Stability”, IEEE Transactions on Power Delivery, vol. 7, no. 2, Apr 1992, pp. 480-488. [11] M. Wrinch, “Negative Sequence Impedance Measurement for Distributed Generator Islanding Detection”, University of British Columbia, PhD Thesis, 2008. [12] K. Foo, “Real Time Voltage Stability Monitoring by Thevenin Impedance Estimation with Local Measurement”, University of British Columbia, MASc Thesis, 2009.  6  [13] K.T. Vu and D. Novosel, “Voltage Instability Predictor (VIP) Method and System for Performing Adaptive Control to Improve Voltage Stability in Power Systems”, United States Patent, number 6 219 591, April 17th 2001. [14] C. Pinzon, J. Bertsch and C. Rehtanz, “Method and Device for Assessing the Stability of an Electric Power Transmission Network”, United States Patent, number 6 690 175, Feb. 10th 2004. [15] P. Kundur, Power System Stability and Control. McGraw-Hill, 1994, Ch. 7. [16] L.M. Korunovic, D.P. Stojanovic and J.V. Milanovic, “Identification of Static Load Characteristics Based  on Measurements in Medium-Voltage Distribution  Network”, Generation, Transmission & Distribution IET, vol. 2, no. 2, Mar 2008, pp. 227 – 234. [17] IEEE Task Force on Load Representation for Dynamic Performance, “Standard Load Models for Power Flow and Dynamic Performace Simulation”, IEEE Transactions on Power Systems, vol. 10, no. 3, August 1995, pp. 1302-1312. [18] S. Ihara, M. Tani and K. Tomiyama, “Residential Load Characteristics Observed at KEPCO Power System”, IEEE Transactions on Power Systems, vol. 9, no. 2, May 1994, pp. 1092–1101. [19] M.L. Coker and H. Kgasoane, “Load Modeling”, IEEE AFRICON 1999, vol. 2, pp. 663-668. [20] L. M. Hajagos and B. Danai, "Laboratory Measurements and Models of Modern Loads and Their Effect on Voltage Stability Studies", IEEE Transactions on Power Systems, vol. 13, no. 2, May 1998, pp. 584-592. [21] IEEE/PES Power System Stability Subcommittee, “Voltage Stability Assessment: Concepts, Practices and Tools”, IEEE Special Publication, August 2002, 29p.  7  Chapter 2. Load Modeling of an Induction Motor Operated with a Variable Frequency Drive 1 2.1.  Introduction Variable-frequency-drive fed induction machines (VFD-IM) are being used more  frequently in industry, due to their decreasing cost, increasing power ratings, and the advantages of robustness, size and maintenance of induction machines over DC ones. As it is well known, the induction machine (IM), when operated directly from the AC line, draws a sinusoidal current from the grid that lags the phase voltage, thus resulting in the consumption of both real and reactive power. Typical voltage and current waveforms corresponding to an IM are shown in Fig. 2-1, where one can see a lagging power factor. A variable frequency drive (VFD) comprises a rectifier, a DC link and a controlled 3 phase inverter [4]. The rectifier converts the 3-phase AC line current, with fixed amplitude and frequency, into a DC voltage. Small ripple is desired in the DC stage, so a large parallel capacitor of adequate value is added to the rectifier. The inverter is a bridge of controlled power electronic switches (IGBT and diodes packed into a single IC for this particular VFD). It sends DC voltage pulses to the output terminals; in which frequency and duty cycle for each phase depend on the control strategy and modulation scheme. The inverter output terminals are then connected to the IM stator connected in Y, resulting in an AC line to neutral voltage, comprising a sinusoidal fundamental component and some harmonics due to switching. This voltage has controlled amplitude and fundamental frequency, as shown in Fig. 2-2.  1  A version of this chapter has been published. Vargas, L.M.; Jatskevich, J. and Marti, J.R. (2008) Load Modeling of an Induction Motor Operated with a Variable Frequency Drive, IEEE Electric Power Conference EPEC 2008, Vancouver BC, Oct 6-7th, 2008, pp. 1-7.  8  Phase voltage and current for small industrial IM, full-load  Voltage (V) and Current (A)  200  Va Ia*100  100  0  -100  -200 0  0.01  0.02  0.03  0.04  0.05  time (s)  Fig. 2-1. Phase voltage and currents of an IM, showing lagging power factor.  Fig. 2-2. Block diagram of VFD-fed IM system.  The static voltage stability concepts have been well described in the literature [2][3], as well as the main causes and effects of power system harmonics in variable frequency drives [4], [10]. However, there are no well-established static load models for IM’s with VFD’s [9]. Some authors have suggested a static load model [1], without details on how the drive works and its effects on the power grid. There is a need to better understand how the modern loads of increasing proportion affect the voltage stability assessment (VSA). Therefore, it is important to know if the assumptions of standard induction machine load models [1]-[3], [5], [9], which assume directly-connected IM’s, are still valid when VFD’s are being used.  9  2.2.  Variable Frequency Drive Circuit and Expected Waveforms  2.2.1.  Simplified Equivalent Circuit  Fig. 2-3 shows the equivalent circuit of the VFD. As one can see, the power input is connected to the AC line, and its output is fed to the IM. A detailed model with ideal switches and simplifying the DC bus load was built. The drive’s control and interfaces use DC power, but since its power consumption is very low these circuits were not included. The power source is modeled as 3-phase ideal source at the rated amplitude and frequency of the VFD. The motor is represented by its steady state equivalent circuit [2] at full-load speed.  Fig. 2-3. Equivalent circuit of the variable frequency drive.  2.2.2.  Three Phase Inverter  In order to control the firings of IGBTs (T1 to T6), the modulation scheme and the control strategy needed to be specified. From [6], the drive was configured with a 6 kHz carrier signal frequency and constant volt per hertz control. The sine triangle modulation with third harmonic injection was chosen as it does not require any position feedback and low frequency harmonics are not generated in the voltage going into the IM. The details of the modulation and control for VFD dynamic simulation models can be found in [4]. Fig. 2-4 shows one phase voltage for the inverter output voltage Vag , corresponding to a commanded frequency of 60Hz (nominal frequency for the IM). On the same figure, the fast average [4] of this signal over one switching cycle of 6 kHz is superimposed.  10  Inverter output, DC voltage pulses, Vag  300  Vag pulses  Voltage (V)  Vag averaged  200  100  0  0  0.005  0.01  time (s)  0.015  0.02  0.025  Fig. 2-4. Inverter model output voltage, detailed and fast averaged.  The voltage Vag in Fig. 2-4 is still single polarity, with pulse width modulation (PWM). The stator is Y-connected with floating neutral and zero sequence neglected [4]. Therefore, the line-to-neutral voltages V as ,Vbs ,Vcs seen by IM stator are calculated as in (2-1).  Vas   2 / 3 − 1 / 3 − 1 / 3 Vag  Vbs  = − 1 / 3 2 / 3 − 1 / 3 ⋅ Vbg   V  − 1 / 3 − 1 / 3 2 / 3   Vcg   cs   (2-1)  The stator steady state AC voltage and current waveforms for a single phase, and  Voltage (V) and current (A)  the voltage fundamental signal are shown in Fig. 2-5. IM stator phase voltage and current, when fed by the VFD 250 Vas Vas fundamental  150  Ias*100 50  -50  -150  -250  0  0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016  time (s)  Fig. 2-5. IM stator phase voltage and current, coming from VFD output.  11  2.2.3.  Three Phase Rectifier  As shown in Fig. 2-3, the inverter input voltage Vdc comes from a three phase diode bridge rectifier and a capacitor filter [10]. The input for the diode bridge is the AC voltages from the source. These AC voltages and currents seen by the source are of interest to develop a load model. The simulated 3-phase currents for the AC line are plotted in Fig. 2-6. It can be seen that the currents drawn by the VFD are discontinuous with narrow pulses going in positive and negative directions. Phase currents from power source to VFD 10 Ia  Current (A)  Ib 5 Ic  0  -5  -10 0  0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016  0.018  0.02  time (s)  Fig. 2-6. Simulated 3- phase currents seen in the AC line.  The rectifier bridge has 6 diodes connected as in Fig. 2-3. During one cycle of the power source, the possible states of the diodes are shown in Table 2-1 as well as the corresponding phase and direction of the current for each state. The positive current pulses of phase A occur in the vicinity of the V ab and Vbc peaks and the negative current pulses are close to Vba and Vca peaks. The midpoint between the positive and negative current pulses will be very close in time to the positive and negative peaks of the phase voltage Va . So, after extracting the 60Hz component of the phase current I a , it will be nearly in phase with Va . Analogously for the other phases. Fig. 2-7 shows this peak detection mode [10] graphically, the DC ripple was intentionally exaggerated and the current scaled.  12  Table 2-1. Diodes and current flow in a 3-phase rectifier. Voltage First diode ON and Second diode ON relationships current path and current path Vab > Vdc  D1 → I a +  D 4 → Ib −  Vac > Vdc Vbc > Vdc  D1 → I a + D3 → Ib +  D6 → I c − D6 → I c −  Vba > Vdc  D3 → Ib +  D2 → I a −  Vca > Vdc  D5 → I c +  D2 → I a −  Vcb > Vdc None > Vdc  D5 → I c +  D 4 → Ib −  None → I a = I b = I c = 0  Explanation of peak detection mode for 3 phase diode bridge rectifier 400  Voltage (V)  350  Vdc  300 250 200  Vab  Vac  Vbc  Vba  Vca  Vcb  150 100 50  Voltage (V) and current (A)  200 150 Diode turns on.  Diode turns off  100 50 0 -50 Va -100  Ia * 20  Phase displacement is very small  Ia * 20 fund  -150 -200 0  0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016  0.018  0.02  time (s)  Fig. 2-7. Peak detection characteristic of 3-phase rectifier.  In real hardware, the exact waveforms will be slightly different. Because components, loads and power sources are not ideal, the actual pulses may not be identical and symmetric. In any case, the model suggests that the peak detection characteristic with small phase displacement φ is not dependent on the nature of the load connected to the DC bus. However, the fact that the fundamental component of the current is nearly in phase with the voltage, implies that only real power P is taken from the AC line, while the reactive power Q may be quite small and even negligible for practical purposes.  13  It is known that an IM requires both real and reactive power. As shown in Fig. 2-5, the stator phase current lags the respective phase voltage, which implies that reactive power is being consumed by the motor. However, in the case of VFD, this reactive power is not taken from the AC source. The phase current feeding the IM is also continuous, whereas the phase current drawn from the AC line is discontinuous as shown in Fig. 2-7 and Fig. 2-8. In this case, the DC link filter capacitor provides the energy to the IM when all the AC line currents are zero. In particular, Fig. 2-8 shows one 60Hz cycle-window for the capacitor current, capacitor voltage ripple and the 3-phase AC line current. As one can see, the capacitor is alternately charged and discharged by six pulses per one AC input cycle. This results in the pulses at 360 Hz rate. It can also be seen in Fig. 2-8 that capacitor current has high frequency component corresponding to the 6 kHz switching frequency of the inverter. Capacitor current, DC voltage ripple and rectifier 3-phase input current  Current (A)  10  5  0  -5  Voltage (V)  340  336  332  Current (A)  10  0  -10 0  0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016  time(s)  Fig. 2-8. Capacitor current, DC voltage ripple, rectifier 3-phase input current.  14  2.3.  Experimental Results  2.3.1.  Experimental Setup  The experimental system set in our laboratory included a 3-phase VARIAC in order to change the amplitude of the AC voltage supplied to the VFD. The base phase voltage V0 was chosen to be the nominal line voltage, 120V. The IM shaft was mechanically coupled to a permanent magnet DC machine mounted in a dynamometer cradle. The DC machine acted as generator loaded with a bank of parallel resistors in order to emulate a desirable mechanical load torque. For the purpose of this paper, the VFD was configured to operate in open loop constant V/Hz control without feedback on speed or torque. In this case, the mechanical load would impact the IM speed due to the torque-dependent slip. The motor and drive specifications appear in Appendix E. A customized data acquisition system, with a signal conditioning stage (op-amps for the voltage and transducers for the current) and a NI card interfaced with a PC, was used to acquire and record the voltage, current, speed and mechanical torque waveforms. To capture the voltages with the inverter modulation at high frequency, a commercial oscilloscope was used with the sampling rate of 1 MHz. The recorded data was subsequently imported into MATLAB / Simulink and the noise was filtered using moving average window filters [5] before further post-processing.  2.3.2.  Motor Drive Output Waveforms  The phase-to-ground output voltage of the inverter Vag was measured and fast averaged [4] over one switching cycle. To make an accurate comparison with the inverter model, the simulation used the actual measured and recorded voltage Vdc (not shown) as an input. As one can see, an excellent match between the model and the hardware is observed in Fig. 2-9.  15  Voltage modulation, averaged over one switching cycle  Voltage (V)  400 300 200 100 Vag, simulated 0  Vag, measured  -100 0.005  0.01  0.015  0.02  0.025  0.03  0.035  time (s)  Fig. 2-9. Fast averaged line-to-ground inverter output voltage.  The actual measured phase voltage that the VFD applies to the IM is shown in Fig. 2-10, wherein one can clearly observe the inverter switching and the resulting modulated fundamental component. For comparison purpose, the motor phase current is also shown in Fig. 2-10 on the same plot. To make the current waveform appear visibly comparable, it has been scaled by a factor of 100. As expected, the motor phase current contains some small ripple due to switching and it is lagging the applied phase voltage.  Voltage (V) and cuurrent (A)  IM stator phase voltage and current when connected to a VFD 200 Vas, measured Vas, fundamental  100  Ias * 100  0  -100  -200  0  0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016  0.018  0.02  time(s)  Fig. 2-10. Measured, IM stator phase voltage and current.  2.3.3.  Motor Drive Input Waveforms  The measured input terminals voltages fed to the VFD and the resulting phase currents corresponding to the nominal operating conditions are shown in Fig. 2-11. As one can see in Fig. 2-11 (top plot), the input voltages are not perfectly sinusoidal and/or balanced, as was the case for the actual AC voltages supplied in our building due to various non-linear loads. As a result of that, the current pulses shown in Fig. 2-11 (bottom plot) are also different from the ideal case depicted in Fig. 2-7 (bottom plot). 16  Voltage (V)  VFD input, 3 phase voltages and currents 200  Va Vb Vc  0  Current (A)  -200 10  Ia Ib Ic  0  -10 0  0.005  0.01  0.015  0.02  0.025  0.03  0.035  0.04  0.045  0.05  time(s)  Fig. 2-11. VFD input terminals 3-phase measured voltages and current.  2.3.4.  Power-Voltage Steady State Characteristic  2.3.4.1. Real and Reactive Power The real and reactive power P and Q are calculated according to (2-2) and (2-3); for each i -th harmonic order frequency.  Pi = Virms ⋅ I i rms ⋅ cos φi  (2-2)  Qi = Vi rms ⋅ I i rms ⋅ sin φi  (2-3)  The net real power in one fundamental cycle of period T is:  1 P = T  T  ∫ [v( t ) ⋅ i( t )]dt  (2-4)  0  From (2-2), that power is transferred in a cycle only if there are voltage and current components at the same frequency. If the harmonics are negligible, then:  P1 ≈ P  (2-5)  In the following test, the constant torque was kept by adjusting the DC generator load bank while the AC voltage applied to the subject IM and VFD-fed-IM was varied in a wide range. The load torque was set to nominal full load for the given motor. First, the IM was directly connected to the VARIAC and the phase voltage was changed in 5V steps. Then, same test was repeated with the IM fed by the VFD, which was configured to operate in open loop constant V/Hz control without the feedback on speed or torque.  17  Fig. 2-12 shows the steady state P and Q being consumed by the IM and the VFD-IM combination for every input voltage. The 60 Hz fundamental of every voltage and current signal was extracted by software using the Fourier analysis, and the power was calculated using (2-2)-(2-3).  Phase Voltage (V)  140  Real power (P), VFD-IM Reactive power(Q), VFD-IM  130  Real power (P), IM only Reactive power (Q), IM only  120 110 100 90 80 0  50  100  150  200  250  300  350  400  450  500  Real and Reactive 3-phase power (W) (VAR)  Fig. 2-12. Real and reactive power of IM and VFD-IM, experimental results.  As it is observed in Fig. 2-12, the real power profile does not change significantly between the IM directly connected to the AC line and the IM fed by the VFD. A slightly higher real power was consumed when the VFD was used, which is attributed to the internal losses therein. Below 85 V, equivalent to 0.71 p.u., the VFD shuts off. But above that voltage, the two PV characteristics look very similar. However, the QV characteristic is completely different when the VFD is used, wherein the reactive power consumption is nearly zero for all voltages. This experimental result was expected. From the previous discussion in Section 2.2, the phase displacement between measured current fundamental and voltage is close to zero. 2.3.4.2. Apparent Power The magnitude of apparent power S was also calculated from the experimental waveforms, and the results are shown in Fig. 2-13. Here, as one can see, there exists an appreciable difference, the VFD-fed IM demonstrating significantly larger apparent power than the IM alone, despite that its reactive power is negligible. The motivation and discussion for this additional calculation will be later explained in Section 2.4.  18  Phase Voltage (V)  140 130 Apparent power |S|, VFD-IM Apparent power |S|, IM only  120 110 100 90 80 0  100  200  300  400  500  600  700  3-phase apparent power (VA)  Fig. 2-13. Apparent power calculation for different voltage levels.  2.3.5.  Exponential Load Model PV and QV Characteristics  The exponential load model [1]-[3], [9] is often used for voltage stability assessment (VSA). This model has the following form: P = P0 ⋅ ( V / V0 )α  (2-6)  Q = Q0 ⋅ ( V / V0 ) β  (2-7)  where V0 is the nominal value of the AC input voltage, P0 , Q0 are the powers at V0 , and α , β  are the characteristic exponents. The experimental results from Fig. 2-12  were used together with the curve-fitting technique to identify the best fit of (2-6) and (2-7) into the measured PV and QV load characteristics. A summary of the resulting coefficients is in Table 2-2. Table 2-2. Measured load characteristics. β α Load IM, constant load torque  0.11  VFD-IM, constant load torque, 60Hz, constant V/Hz, open loop mode Pure ideal resistor  0.32, if V > 0.7pu 2.0  19  cos φ → V = V0  2.09  0.71  0.0  1.0  0.0  1.0  2.4.  Harmonics and Power Factor Generally, when the voltages and currents are sinusoidal, the power factor (PF)  may be calculated using either one of the following expressions: PF = cos φ  (2-8)  PF = P / | S |  (2-9)  However, when the VFD was used, the experimental results between (2-8) and (2-9) showed appreciable difference due to the presence of harmonics [10]. The RMS value for any signal X and the apparent power magnitude may be calculated as: T  X RMS =  1 ⋅ ∫ x 2 ( t )dt T  (2-10)  0  | S |= Vrms ⋅ I rms  (2-11)  These expressions are valid for any shape of waveform. If it is purely sinusoidal without offset, the fundamental component RMS and the full signal RMS have identical values. When the signal has harmonics, the relationship between the full signal RMS and the RMS values of each frequency component i is:  X RMS =  2  2  X 0 + X 1rms +  ∞  ∑ ( X irms 2 )  (2-12)  i=2  For the VFD-fed IM studied, the phase voltage is close to pure sinusoidal, but the current has significant distortion, then (2-5) applies and: P / | S |=  V1rms ⋅ I 1rms ⋅ cos φ1 Vrms ⋅ I rms  Vrms ≈ V1rms  (2-13) (2-14)  ∞  ∑ ( I irms 2 ) = I harmonicsrms  (2-15)  i =2  A common indicator of the harmonic content in a waveform is the total harmonic distortion THD calculated by (2-16). By combining (2-14)-(2-16) into (2-13) and rearranging, (2-17) and (2-18) apply.  20  THD( X ) = X harmonics RMS / X 1RMS  (2-16)  1  P / | S |= cos φ ⋅  (2-17)  1 + THD 2 ( I )  PF = Displacement factor ⋅ Distortion factor  (2-18)  For low THD values, distortion factor is close to unity, as it is evident in Fig. 2-14. Then, (2-8) and (2-17) are equivalents only if THD is not high. With the VFD-fed IM operated at nominal conditions, the current, which has the harmonic spectrum from Fig. 2-15, has a THD of 1.8, resulting in a distortion factor of 0.49. Therefore, the power factor is reduced by more than one half.  Distortion factor  Distortion factor as a function of total harmonic distortion 1  0.8  0.6  0.4 20%  40%  60%  80%  100%  120%  140%  160%  180%  200%  THD  Fig. 2-14. Distortion factor as a function of THD. Fundamental magnitude (60 Hz) = 1.133 A  Current (A)  1  THD = 180% 0.8 0.6 0.4 0.2  0  5  10  15  20  25  30  35  40  45  50  Harmonic order  Fig. 2-15. Harmonic spectrum of the line current for the VFD-fed IM.  The fact that harmonics are not considered in voltage stability analysis does not imply that they should be simply ignored. Harmonics create other problems in the power network, such as increasing the total RMS and increasing the peak value (for the measured current: I rms = 2.06 ⋅ I 1 rms and I peak ≈ 10 ⋅ I 1rms ). This affects the thermal operating margin of all the involved components. Unbalance can also be created in the voltages due to current peaks (which produce voltage drops at frequency components  21  above the fundamental), EMI problems are also increased. Active and passive circuits that can be found in [10]-[11], help to reduce the harmonic content while keeping the phase displacement angle φ  close to zero.  2.5.  Relation to the Voltage Stability Problem The basic equivalent circuit and phasor relationships for power transmission are  shown in Fig. 2-16 and Fig. 2-17. The network behind the load is represented by an equivalent Thevenin voltage V 1∠δ  and Thevenin impedance jX . The Thevenin  resistance is neglected because it is relatively small compared to the reactance. The Thevenin equivalent values can be known or estimated as in [7]-[8] in order to build the system of PV curves such as the one shown in Fig. 2-18. To find the actual operating point for a certain load, one has to know the power and the power factor. Knowing the operating point in PV curve in Fig. 2-18, one can see how close the system is to the maximum power transfer limit, which occurs when the operating point approaches the “nose” of the system PV curve [2]-[3].  Fig. 2-16. Basic power transmission circuit with resistive losses neglected. V1 φ  δ  V2  I  V1 V2  δ φ  jXI  jXI  I  Fig. 2-17. Phasor relationships for inductive (top) and resistive loads (bottom).  It can be seen in Fig. 2-17 and Fig. 2-18 that if the load is mostly inductive ( φ ≈ 90° ), there is a significant voltage drop in V 2 for increasing powers before the 22  nose. If the load is mostly resistive ( φ ≈ 0° ), then the voltage drop is significantly reduced. This shows that reactive power transmission is closely linked to voltage drop, and real power to the displacement angle δ between source and load voltages. The measured results for the VFD-fed IM load show that φ ≈ 0° , and that this load behaves as unity power factor for the 60Hz power transmission frequency. Therefore, it may reduce the voltage drop or the amount of reactive compensation required for that load bus, as shown in the normalized PV curves [2]-[3] of Fig. 2-18. Fig. 2-19 shows the difference in operating points between the VFD-IM load model, the IM model and a constant resistive load. Both VFD-IM and IM load curves are close to constant power load model; but for a same reference power at 1 p.u. voltage, the IM operating point is closer to the nose of its PV curve, due to its lagging PF. Normalized PV curves  v = |V2| / |V1|  1.2  φ = 30 deg  φ = 15 deg + (lead)  1  PF = 0.97+  VFD-IM  φ = 0 deg  0.8  PF = 1 IM  0.6  φ = 45 deg- (lag) PF = 0.71-  0.4  0.2  0 0  0.1  0.2  0.3  0.4  0.5  p = P * X /  0.6 |V1|  0.7  0.8  0.9  2  Fig. 2-18. Normalized PV curves for different values of load phase angle. VFD-IM or resistor operating point 1.2  α = 2: Constant impedance exponential model  v = |V2| / |V1|  1  IM operating point  0.8  α = 0.11: IM exponential model  0.6  α = 0.32: VFD-IM exponential model  0.4  0.2  α = 0: Constant power exponential model 0 0  0.1  0.2  0.3  p = P * X /  0.4 2  |V1|  Fig. 2-19. Operating point for loads with different PF and α values  23  0.5  2.6.  Conclusions It has been shown that a variable frequency drive feeding an induction machine  does not draw significant reactive power from the grid at any voltage. Moreover, when the VFD is configured to operate in constant V/Hz open-loop control, the real power to voltage relationships remain very similar to the motor directly connected to the line, nearly constant power. This load is seen in the system PV curves as unity power factor. So, if it is a significant proportion of the total bus load, it may reduce the voltage drop (at 60Hz) and increase the maximum power transfer limit compared to an equivalent load power of IM without the VFD. Although the input current is discontinuous with extremely high harmonic distortions, if the feeding bus is sufficiently strong, the voltage harmonics there would remain small. In this case, the use of 60 Hz phasors remains valid for all power calculations and the harmonic content does not have an effect on static VSA analysis, wherein only the load characteristic at the fundamental frequency may be considered.  24  2.7. [1]  References L. M. Hajagos and B. Danai, "Laboratory Measurements and Models of Modern Loads and Their Effect on Voltage Stability Studies", IEEE Transactions on Power Systems, vol. 13, no. 2, May 1998, pp. 584-592.  [2]  P. Kundur, Power System Stability and Control. McGraw-Hill, 1994, 1176p.  [3]  C.W. Taylor, Power System Voltage Stability. McGraw-Hill, 1994, 273p.  [4]  P.C. Krause, O. Wasynczulk and S.D. Sudhoff, Analysis of Electric Machinery and Drive Systems. Second Edition, IEEE Press, 2004, 613p.  [5]  D. Karlsson and T. Pehrsson, "A Dynamic Power System Load Model and Methods for Load Model Parameter Estimation", School of Electrical and Computer Engineering, Chalmers University of Technology, Göteborg, Sweden, Technical Report No. 22L, 1985, 97p.  [6]  Lenze AC Tech, “SMVector - Frequency Inverter Operating Instructions”. AC Technology Corporation, 2006  [7]  M. Wrinch, “New Methods for Power System Islanding Detection”, PhD Thesis (in progress), Dept. of Electrical and Computer Engineering, The University of British Columbia, 2008.  [8]  K.T. Vu and D. Novosel, “Voltage Instability Predictor (VIP) – Method and System for Performing Adaptive Control to Improve Voltage Stability in Power Systems”, United States Patent, number 6 219 591, Apr 17th, 2001.  [9]  IEEE Task Force on Load Representation for Dynamic Performance, “Standard Load Models for Power Flow and Dynamic Performace Simulation”, IEEE Transactions on Power Systems, vol. 10, no. 3, August 1995, pp. 1302-1312.  [10] R. Erickson and D. Maksimovic, Fundamentals of Power Electronics. Second Edition, Kluwer Academic Publishers, 2001, 883p. [11] M. Brown, Power Supply Cookbook. Second Edition, Newnes, 2001, 261p.  25  Chapter 3. Induction Motor Loads and Voltage Stability Assessment Using PV Curves 2 3.1.  Introduction Voltage stability assessment is one of the challenges for electrical utilities today.  In the past, when the transmission systems were over designed and the majority of the loads were simple in composition, voltage stability assessment represented a much easier problem. However, reduced investment in generation and transmission infrastructure in the 80’s, while the electric power demand has been continuously increasing, brought the attention of researchers and power utilities to this problem. According to NERC surveys [1], operating closer to the limits of the system, as well as limited construction of new transmission, significantly contributes to the technical challenges being faced by electric industry today. Major blackouts described in [2] have been related to voltage stability issues. The ability to transfer power from generation sources to loads during steady state conditions is a major aspect of voltage stability [2]. Static load models that describe the power dependency on the voltage, and system PV curves of the network that feeds the load are typically used to evaluate the operating conditions of the system and how it is expected to move under disturbances. For the purpose of discussion in this Section, we consider a simplified electrical network shown in Fig. 3-1. Here, the source is represented by a Thevenin equivalent voltage source E behind equivalent series impedance. The load is fed from the bus with the voltage V . The well-known system or network PV curves show the voltage magnitude in a given bus for variation of load power. These network characteristics graphically show how the voltage on the load bus changes depending on the real power going into the load. The maximum power transmission limit is seen as the “nose” of the curve. Depending on the load power factor angle φ , a family of such PV characteristics can be produced as depicted in Fig. 3-2, which also shows that loads with higher power factor angle will generally result in increased output voltage and improved voltage 2  A version of this chapter has been accepted for publication. Vargas, L.M.; Jatskevich, J. and Marti, J.R. (2009) Induction Motor Loads and Voltage Stability Assessment Using PV Curves. IEEE Power Engineering Society General Meeting 2009, Calgary AB, Jul 26-30th, 2009.  26  stability margin by extending the “nose” of the PV curve to the right-hand side. Based on classical derivations [2]–[4], it can be easily demonstrated that the maximum power transfer happens when Z load = Z th . If the load impedance is further reduced (implying more load), the power transmitted will now decrease together with the bus voltage, and this can cause an unstable condition which may lead to a progressive voltage collapse in the bus load and propagate through the power grid as explained in [2]–[6].  th  j X  V  th  P, Q  +  R  I R  E  j X  load  load  _  Fig. 3-1. Simplified one-line diagram of power transmission equivalent circuit. Normalized network PV curves for constant power factor loads  1  φ = 15 + (lead)  v = |V| /  |E|  φ=0 0.8  φ = 15 - (lag) φ = 30 -  0.6  φ = 45 ( Rth / Xth = 0 )  0.4  0.2 p = P * |Zth| / |E| 0  0  0.1  0.2  0.3  0.4  0.5  2  0.6  Fig. 3-2. Constant power factor PV curves.  The most known network PV characteristics are the family of curves for constant power factor loads such as the ones shown in Fig. 3-2, where the power factor is cos φ . It is evident that higher load power factor allows for increased power transmission. Due to  27  this property, solutions such as shunt capacitors [2]-[3] and Static VAR Compensators (SVC’s) [7] have been used to provide local reactive power support. The static load model shows how the power required by the load changes with the applied voltage. The so called exponential load model [2], [4], [8] is widely used for voltage stability studies. This load model has the following general form: P = z ⋅ P0 ⋅ ( V / V0 )α  (3-1)  Q = z ⋅ Q0 ⋅ ( V / V0 ) β  (3-2)  where V0 is the nominal value of the load input voltage, P0 is the nominal real power at V0 , and coefficients α , β define the load characteristic of real and reactive power. The variable z represents the quantity (demand [5]) of a given type of load. Special cases of the load characteristic are: constant impedance load ( α = 2 ); constant current load ( α = 1 ); and constant power load ( α = 0 ). A combination of these three specific characteristics is commonly referred to as the Z.I.P. or polynomial model. In general, the coefficient α is not limited to only these discrete values. A model with combination of load characteristics is also possible. The shape for some typical PV load characteristics is shown in Fig. 3-3. Examples of common load PV characteristics 1.2  z = 0.5  α=0  1  z = 1  α=1 0.8  V / V  0  α=2 0.6  0.4  α = 0.1 0.2  0 0  0.2  0.4  0.6  0.8  1  1.2  P / P  0  Fig. 3-3. Typical load PV characteristics at different demands.  Characteristics such as those shown in Fig. 3-3 are very useful for determining the system’s operating condition and its proximity to the voltage stability limits. In particular, the intersection between the load and the system PV curve defines a possible operating  28  point. If this operating point is above the “nose” of the PV curve, the system will be stable, whereas if the load keeps increasing until it operates on a point below the nose, the dynamics of the system can take the system down to a total voltage collapse. The system is considered to be unstable when the load PV curve does not intersect the system PV curve [2]-[3], [5]-[6]. This can happen due to overload and/or a contingency such as the loss of a line, as shown in Fig. 3-4. As depicted in Fig. 3-4, after a contingency, the Thevenin impedance increases, which in turn results in a new network PV curve with smaller power transfer capability – the nose of the curve is shifted to the left. Under this new condition, the exemplified load with characteristic α = 0.5 will remain in the limit of stability as the constant impedance load will never cause instability. However, the constant power load will bring the system to a voltage collapse.  1  α=2 0.6  V / V  0  = V / E  0.8  Original PV curve,  0.4  PF=1 Same load PF, after line  0.2  α = 0.5 0  0  0.2  0.4  loss (increased Z  0.6  0.8  )  th  α=0 1  1.2  1.4  P / P  0  Fig. 3-4. Intersections of system and loads PV curves, before and after contingency.  Intersections of short and long-term load characteristic with a variety of system PV curves that show different conditions on the generation and/or load side (i.e. reactive power limits on generators, load shedding, tap changing transformers) are also widely used. Detailed examples of PV curves that include those conditions can be found in [2][5], [9]. In some instances, corrective actions such as voltage control with SVC’s and LTC’s or thermostatic effects may restore the power demand to the original values, thus making a non-constant power load behave as constant power in the long term [2]-[4]. So, on the absence of detailed information about the load, it has been a common conservative  29  practice to assume a constant power, constant PF load ( α = β = 0 ), such that no intersection exists below the “nose”. The network PV curves have been well described [2]-[6] for constant power factor loads. The approach described in [5], [9]-[10] includes some load characteristics within the network PV curves calculations from power flow solutions, so that no additional load characteristic curve and its intersection is needed to find an operating point (the stability limit is when a power-flow solution does not exist anymore). However, the examples considered there assume that the real and reactive power characteristics are the same, meaning a constant power factor load. In practice, there are loads for which the power factor is not constant under expected operating conditions. For example, one of the most common loads, an induction motor (IM) operating under variable mechanical load will exhibit variable power factor. This phenomenon is easily explained by the means of its steady state equivalent circuit depicted in Fig. 3-5. Many other real loads also show different P and Q characteristics, as it can be seen on [2]-[8]. Investigating the voltage stability for this type of variable PF loads has significant practical application.  3.2.  Motor Load and System PV Curves  3.2.1.  IM Directly Connected to the Network  The power factor at the stator terminals of an IM is the cosine of the angle of the equivalent impedance seen at this point. Looking at the IM steady state equivalent circuit in Fig. 3-5, it is evident that the change in slip s will change both the magnitude and angle of the equivalent impedance seen at the stator terminals [11].  30  Rs  Rr  Xr  +  Xs  Xm  V  R r 1-s s  _ Fig. 3-5. Induction machine per phase equivalent circuit.  When the system PV curves are based only on local measurements at the load bus, the Thevenin equivalent that represents the network behind the bus can be estimated by methods like negative sequence [12] or two-time readings [13]. When the solution is performed system-wide, power flow solution is run for continuously increasing load level in a bus. Therein, continuation methodologies are used to obtain the power flow solution when the bus voltage approaches the nose, so that numerical convergence can be achieved [3], [14]. Here, it will be assumed that the network is balanced, so that a singlephase phasor representation at fundamental frequency (60Hz) is sufficient. Using phasor notations, the source is given as: E = E∠0° (reference angle)  (3-3)  Z th = Rth + jX th  (3-4)  If the PF is constant, the load impedance Z load may be expressed as: Zload = Zload ⋅ (cos φ + j sin φ )  (3-5)  However, for the IM based on equivalent circuit of Fig. 3-5, the equivalent load impedance becomes: Z load = R s + jX s + ( jX m || [Rr + Rr ( 1 − s ) / s + jX s ])  (3-6)  Once the equivalent load impedance and the Thevenin equivalent are known, each point of the network PV curve for decreasing load impedance magnitude Z load is calculated using the following equations: I = E /( Z th + Z load )  (3-7)  V = E − Z th ⋅ I  (3-8)  31  S = V ⋅ I * = P + jQ  (3-9)  A comparison between PV curves for constant PF loads and slip dependent IM load is depicted in Fig. 3-6. As expected, the IM-network PV curve passes along different constant PF curves as the slip changes from 0 (synchronous speed) to 1 (stall). This means that at low slip, the voltage is almost constant due to increasing PF. The rate of change of voltage with respect to power for the IM shows very different behavior than for constant PF loads, it is almost flat first, and decreases abruptly close to the “nose”, as observed from the IM curve. At s = 0 (no load torque, no slip, no mechanical power delivered) the electrical power demand is not zero, because of IM losses, and V < E because there is still significant reactive power transmission at a very low power factor. At s = 1 (machine stalled), the PV curve does not reach neither zero power, nor zero voltage. In this case, machine impedance is low, but not zero, and thus significant active and reactive power is consumed due to losses at high current, and a constant low voltage will remain at the machine terminals. Extrapolating the curve to values outside 0 to 1 makes no sense, as the IM is operating in motor mode only. Constant power factor PV curves and IM slip dependant PV curve comparison, normalized 1  φ = 15 (lag) φ = 75  0.8  φ = 45  |V| / |E|  IM PV curve s=0 0.6  0.4 s=1  0.2  0 0  0.1  0.2  0.3  0.4  2  P * Xth / |E|  Fig. 3-6. IM slip dependent and constant power factor PV curves comparison.  In order to understand what happens at the “nose” of the IM PV curve, Fig. 3-7 is helpful. The thinner curves show the torque-slip characteristics of an IM at different  32  constant stator voltages, the thicker curve shows the torque-slip characteristic for this variable input voltage. The dotted horizontal line shows a constant torque. The maximum torque that can be developed by the line-fed IM occurs at much lower slip and torque than if constant voltage were to the machine stator, so the IM could stall at loads within its normal region of operation. The right plot is a zoom of the left one. Comparison of IM torque-slip at constant voltages and Thevenin impedance-fed IM Te (Nm)  Te (Nm)  see right-side plot  2  2  1.5  1.5  1  1  0.5  0.5  0  0 1  0.8  0.6  0.4 slip  0.2  0  Constant V curves  0.25  0.2  I  0.15 slip  0.1  0.05  0  I  Fig. 3-7. Torque-slip characteristics for constant and slip dependent voltages.  It is known that more load increases the slip s . Above the nose, more slip will provide more electrical power to match the mechanical load power, so that the system is stable; but when this happens below the nose, the point is unstable because more slip will now reduce the electrical power, thus increasing the power mismatch and producing more slip until the machine stalls.  3.2.2.  IM Fed by a VFD Connected to the Network  There is also a need to understand how modern electronic loads, such as variable frequency drives for IM’s affect the computation of the PV curve. It has been shown in [15] that the VFD-IM does not take any reactive power from the source, no matter the voltage and mechanical load levels, due to the 3-phase peak detection rectifier characteristics. Thus, the conventional PV curve for cos φ = 0° can be employed. For VSA purposes and power calculations, harmonic distortion in the current is neglected if the voltage is sinusoidal, so that only the fundamental component of the current is considered, as explained in [15].  33  Furthermore, large VFD’s or other large rectifiers, which are the ones that could pollute the voltage, always have harmonic filtering.  3.3.  Experimental Setup An experimental verification was performed to get a series of network PV curves  when the load is a single induction motor under variable mechanical load. Other tests were also performed to obtain the static load model of the VFD-fed IM in closed loop and open loop speed control, and compare against an IM without VFD control. The experimental system set in our laboratory included a 3-phase VARIAC to set the amplitude of the AC voltage required, a set of iron core inductors to emulate the equivalent Thevenin impedances, an off-the-shelf fractional horsepower VFD (configured in constant V/Hz control [16]), and a couple of small industrial induction motors. The motor, drive and inductors are specified in Appendix F. The IM shaft was mechanically coupled to a permanent magnet DC machine mounted in a dynamometer cradle. The DC machine acted as generator loaded with a bank of resistors parallel to the terminals in order to emulate a mechanical load of adjustable magnitude. A customized data acquisition system, with a signal conditioning stage (op-amps for the voltage and transducers for the current) and a NI card interfaced with a PC, was used to acquire and record the voltage, current, speed and mechanical torque waveforms. The recorded data was subsequently imported into MATLAB/Simulink and the noise was filtered using moving average window filters [17]-[18] before further post-processing. An encoder mounted in the cradle counted the number of turns of the mechanical load, which is the same as the IM; the output of the encoder was connected to a frequency to voltage converter chip in the measurement box, in order to obtain an analog DC voltage proportional to the shaft speed. A shielded cable was necessary to bring the analog voltage from the measurement box to the VFD feedback terminals. By using conventional wires, the noise picked up by the long cable made the feedback signal unusable, activating the over speed protections of the drive. The block diagrams of the setup configurations for each of the experimental tests are shown in Fig. 3-8 to Fig. 3-11.  34  Fig. 3-8. Test setup for network PV curve when load is a single IM.  Fig. 3-9. Test setup for PV load characteristic of directly connected IM.  Fig. 3-10. Test setup for PV load characteristic of open loop VFD-fed IM.  Fig. 3-11. Test setup for PV load characteristic of closed loop VFD-fed IM.  35  3.4.  Results  3.4.1.  Measured PV Curves for Source and Impedance Fed IM  The system PV curves for a single IM load, both calculated and measured are shown in Fig. 3-12 and Fig. 3-13 for two different motors. Each curve corresponds to a different number of (ideally) identical inductors in series, and each point marker in the curve to a different measured value of mechanical load and therefore a different slip. PV curve theoretical and experimental. Induction Motor #1 (IM1) 150  V (Volts)  100  Calculated 1 Xth in series  π60(Lth)  Xth = 2  50  2 Xth in series  E = 160V  3 Xth in series  Lth = 51.2mH at 1A,  4 Xth in series 5 Xth in series  for each inductor in series 0 0  50  100  150  200  6 Xth in series 250  300  350  400  P (W)  Fig. 3-12. PV curve theoretical and experimental, induction motor #1.  PV curve theoretical and experimental. Induction Motor #2 (IM2)  160  V (Volts)  140  120 1 Xth in series 100  2 Xth in series 3 Xth in series  80  4 Xth in series 5 Xth in series  60  6 Xth in series Calculated  E = 160V 40  Lth = 51.2mH at 1A,  π60(Lth)  Xth = 2  20 0  20  40  60  80  for each inductor in series 100  120  140  160  180  P (W)  Fig. 3-13. PV curve theoretical and experimental, induction motor #2.  36  200  As it can be seen in the previous plots, there is a good correlation between the calculated slip dependent and the experimental PV curves along the stable zone. Once the system was close to the nose for the higher X th cases, the next step increment of load made the system go to the unstable region, thus decelerating the IM very fast as seen in Fig. 3-14 until it completely stalled. The instantaneous PV values during stalling are not shown in the experimental plots, as the system or network PV curves are meant to represent steady state operating points only. 2000  Stall of induction machine #1 (IM1) with 6 Xth in series  n (RPM)  1500  1000  500  0  1  2  3  4  5  t (seconds)  Fig. 3-14. Measured speed when IM1, in series with 6 Xth, stalls after it reaches the nose of its corresponding network PV curve.  3.4.2.  Measured Static Load PV Characteristics  The voltage fed to the IM or the VFD-IM load was changed in approximately 5V steps. The mechanical load torque to the IM shaft was kept constant. Two motors were tested for the cases on Fig. 3-9, Fig. 3-10 and Fig. 3-11 at rated mechanical load first, and then at 50% of the rated load. The mechanical load can be speed-dependent, such as fan or conveyor loads, where a reduced voltage in the IM will cause the mechanical torque to drop, thus requiring significantly less electrical power. From a voltage stability point of view, the constant mechanical torque load, such as in compressor/air-conditioning systems, is the one with most risk as it demands more power under all voltage conditions [4]-[5]. Fig. 3-15 and Fig. 3-16 show the experimental results, after extracting the fundamental signals and calculating the real power using MATLAB/Simulink. This  37  additional step was required because the VFD used does not have input harmonic filter. In this case, the current THD was high enough so that the 60Hz RMS value is significantly smaller than the full-signal RMS. However, the current harmonics were not considered to be significant in terms of contributing to the real power [19]. Interested readers may find a good discussion regarding the current harmonics and their impact on the voltage stability in [15]. Static load PV characteristic of Induction Machine #1 (IM1) VFD closed loop full load  140  VFD closed loop 50%load VFD open loop full load  130  VFD open loop 50% load Directly connected full load Directly connected 50%load  V (Volts)  120  110  100  90  80 80  100  120  140  160  180  200  220  240  P (W)  Fig. 3-15. Experimental load PV characteristic, induction motor #1. Static load PV characteristic of Induction Machine #2 (IM2)  140  130  V (Volts)  120  110  VFD closed loop full load  100  VFD closed loop 50%load VFD open loop full load VFD open loop 50% load  90  Directly connected full load Directly connected 50%load 80 50  60  70  80  90  100  110  P (W)  Fig. 3-16. Experimental load PV characteristic, induction motor #2.  38  3.5.  Analysis of Static Load PV Characteristics Curve fitting was employed from the results in Fig. 3-15 and Fig. 3-16 to obtain  the exponential model parameter α from (3-1), as summarized in Table 3-1 and Table 3-2. As it can be observed here, the directly-connected IM and VFD-fed IM in open loop show similar behavior, close to constant power. Although it is noticed that at low voltages the power reduces slightly in one motor and increases on the other. If load torque is kept constant, machine slip increases at lower voltages, so the mechanical speed and power are reduced. But, the efficiency of the VFD-fed-IM also should change with the voltage and slip. If the efficiency decreases faster than the mechanical power, the electrical power may increase, as it was seen. Table 3-1. Measured load characteristics, full load. IM model With VFD closed loop With VFD open loop Directly connected  α_IM1 -0.43 -0.18 -0.14  α_IM2 +0.01 +0.18 +0.16  Table 3-2. Measured load characteristics, half load. IM model With VFD closed loop With VFD open loop Directly connected  α_IM1 +0.08 +0.12 +0.09  α_IM2 +0.45 +0.50 +0.45  The closed loop VFD-fed-IM at full load shows the worst behavior from static VSA point of view. At reduced voltage, the power demand increases, that is why α < 0 , although there is a minimum electrical power at about 110V. The same tests with only half of the rated mechanical load torque showed that the power decreases at low voltages, either with or without VFD in open or closed loop. An induction machine operating at lower loading factor is more efficient at a voltage lower than the nominal, so that the condition of α < 0 was not reached. In closed loop, the synchronous speed (frequency) changes automatically in order to keep the mechanical speed constant. The torque depends on the ratio of air-gap voltage squared to stator frequency, according to the following relationship:  39  Te _ 3φ =  3 ⋅ Vag 2 ⋅ Rr / s 2πf s ⋅ ( Rag + Rr / s )2 + ( X ag + X r )2  (3-10)  The V/Hz ratio of the stator is kept constant as per the VFD control strategy, to keep the torque characteristic at lower speeds. But when the desired speed (frequency) is higher than the base frequency, the V/Hz reaches a limit in the voltage magnitude as limited by the DC bus and the modulation limits [20]. It is also important for stability purposes to know at which voltage and power levels the VFD will shed the IM load, due to its incorporated under-voltage and/or over-current protection. For the tested VFD’s, the lower operating limit was around 0.7 to 0.75 p.u. Therefore, the model shown for VFD-IM is valid at V > 0.7 pu , otherwise P ≈ 0 .  3.6.  Conclusions It has been shown that a system or network PV curve for an IM under variable  mechanical load shows significantly different behavior than a load of variable magnitude and constant power factor (i.e.: incandescent lighting, heating loads). This is explained by the equivalent circuit of an IM, which makes the PF of the IM to change with the slip. The PV load characteristics of an IM with constant load when directly connected to the grid and when fed with a VFD in open and closed loop speed control configuration have been investigated. It is observed that the conservative assumption that the IM behaves close to a constant power load is accurate for most of the times. However, in some cases, at lower voltage levels the power could increase so that the constant power model may not be sufficiently conservative. It is also shown that in order to have an accurate load model, it is important to include the load factor (which is a frequently overlooked parameter [2]), the control strategy of the drive, the losses in the motor-drive system, and the drive protection.  40  3.7. [1]  References North American Electric Reliability Corporation (NERC), “Results of the 2007 Survey of Reliability Issues”, rev. 1, October 2007, 18p.  [2]  C.W. Taylor, Power System Voltage Stability. McGraw-Hill, 1994, 273p.  [3]  IEEE/PES Power System Stability Subcommittee, “Voltage Stability Assessment: Concepts, Practices and Tools”, IEEE Special Publication, August 2002.  [4]  P. Kundur, Power System Stability and Control. McGraw-Hill, 1994, 1176p.  [5]  T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. Kluwer Academic Publishers, 1998, 378p.  [6]  B.C.  Lesieutre, P.W. Sauer and M.A. Pai, “Existence of Solutions for the  Network/Load Equations in Power Systems”, IEEE Transactions on Circuits and Systems, vol. 46, no.8, August 1999, pp. 1003-1011. [7]  M.H. Haque, “Determination of Steady State Voltage Stability Limit of a Power System on the Presence of SVC”, IEEE Porto Power Tech Conference, Portugal, September 2001, vol. 2, pp. 1-6.  [8]  IEEE Task Force on Load Representation for Dynamic Performance, “Standard Load Models for Power Flow and Dynamic Performance Simulation”, IEEE Transactions on Power Systems, vol. 10, no. 3, August 1995, pp. 1302-1312.  [9]  S. Corsi, G.N. Taranto “Voltage Instability – The Different Shapes of the ‘Nose’”, i-REP Symposium – Bulk Power System Dynamics and Control, Charleston SC, USA, August 2007, pp. 1-16.  [10] L. Hua, C. Hsiao-Dong; H. Yoshida, Y. Fukuyama and Y. Nakanishi, “The Generation of ZIP-V Curves for Tracing Power System Steady State Stationary Behavior due to Load and Generation Variations”, IEEE Power Engineering Society Summer Meeting 1999, vol. 2, July 1999, pp. 647 - 651. [11] T. Gonen, Electrical Machines, Power International Press, 1998, 532p. [12] M. Wrinch, J.R. Marti and M. Nagpal, “Negative Sequence Impedance Based Islanding Detection for Distributed Generation (NSIID)”, to be published in Proc. of IEEE Electric Power Conference EPEC 2008, Vancouver BC, Canada.  41  [13] K.T. Vu and D. Novosel, “Voltage Instability Predictor (VIP) – Method and System for Performing Adaptive Control to Improve Voltage Stability in Power Systems”, United States Patent, number 6 219 591, Apr 17th 2001. [14] F. Milano, “Quick Reference Manual for PSAT v2.1.2”, 2008. Online [http://www.power.uwaterloo.ca/~fmilano/archive/psat-2.1.2-ref.pdf], 99p. [15] L.M. Vargas, J. Jatskevich and J.R. Marti, “Load Modeling of an Induction Motor Operated with a Variable Frequency Drive”, to be published in Proc. of IEEE Electrical Power Conference EPEC 2008, Vancouver BC, Canada. [16] Lenze AC Tech, “SMVector – Frequency Inverter Operating Instructions”. AC Technology Corporation, 2006. [17] S.W. Smith, "The Scientist and Engineer’s Guide to Digital Signal Processing ", 1998. Online [http://www.dspguide.com/]. [18] D. Karlsson and T. Pehrsson, "A Dynamic Power System Load Model and Methods for Load Model Parameter Estimation", School of Electrical and Computer Engineering, Chalmers University of Technology, Göteborg, Sweden, Technical Report No. 22L, 1985, 97p. [19] R. Erickson and D. Maksimovic, Fundamentals of Power Electronics. Second Edition, Springer, 2001, 883p. [20] P.C. Krause, O. Wasynczulk and S.D. Sudhoff, Analysis of Electric Machinery and Drive Systems. Second Edition, IEEE Press, 2004, 613p.  42  Chapter 4. Local Voltage Stability Assessment for Variable Load Characteristics 3 4.1.  Introduction Voltage stability is one of the challenges for electrical utilities today. In the past,  when transmission systems were over designed and most of the loads were simple in composition, voltage stability assessment represented a much easier problem. However, reduced investment in generation and transmission infrastructure, while the electric power demand has been continuously increasing, brought the attention of researchers and power utilities to this problem. According to NERC surveys [1], operating closer to the limits of the system, as well as limited construction of new transmission, significantly contributes to the technical challenges being faced by the electric industry today. Major blackouts described in [2] have been related to voltage stability issues. The ability to transfer power from generation sources to loads during steady state conditions is a major aspect of voltage stability [2]. Regions such as British Columbia, Canada, have more than 70% of the total load in regions where the generation is only 15% of the total; the rest of the power has to be transmitted from the main generation centers located hundreds of kilometers away [3]. To introduce the topic, we can consider the simplified electrical network shown in Fig. 4-1. Here, the source is represented by a Thevenin equivalent voltage source E behind equivalent series impedance Z th . The load is fed at the bus with the voltage V . The network PV curves show the voltage magnitude in a given bus for gradually increasing values of the load power (decreasing values of load impedance). The maximum power transmission limit is seen as the “nose” of the curve. Depending on the load’s power factor angle φ , a family of such PV curves can be produced, as depicted in Fig. 4-2. The normalized plots show that loads with better power factors (higher power factor angle) have higher voltage stability margins by extending the “nose” of the PV curve to the right-hand side. This is consistent with the fact that Q transmission is the 3  A version of this chapter will be submitted for publication. Vargas, L.M.; Marti, J.R. and Jatskevich, J. (2009) Local Voltage Stability Assessment for Variable Load Characteristics.  43  main factor in voltage drop on inductive feeders. Local Q generation, such as shunt capacitors and Static VAR Compensators (SVC’s) [4]-[6], avoid Q transmission and, therefore, improve voltage stability. Z  th  =  jX  V  th  P + jQ =  +  P (1+j tan  φ)  I  Z  = R  load  E  load  R  load  + jX  load =  (1+j tan  φ)  _ Fig. 4-1. Basic power transmission circuit for lossless line. 1 φ=0  |V| / |E|  0.8 φ = 75− φ = 60−  0.6  φ = 45−  φ = 30−  φ = 15−  0.4 0.2  Maximum power transfer for each  φ  0 0  0.1  0.2 0.3 2 P |Zth| / |E|  0.4  0.5  Fig. 4-2. Family of normalized PV curves for constant power factor loads.  Based on classical derivations [8], it can be demonstrated that, under constant power factor, the maximum power transfer (the “nose”) happens when Z load = Z th . If the load is increased (reduced load impedance) beyond that point, the power transmitted will decrease simultaneously with the bus voltage and, depending on the load response, this can lead to a progressive voltage collapse in the bus load and propagate through the power grid [6]–[8]. Several utilities and researchers have worked on the computation of voltage stability conditions under a number of operating conditions, such as generators reaching excitation limits [2], [7], [9]-[10], effects of transformer tap changers [6], [11],  44  interaction of transient load characteristics [6],[12], line contingencies [13], networks where the bus load is fed by more than one equivalent Thevenin network [14]-[15], consideration of transmission losses ( Rth ≠ 0 ), and the effects of the line X R ratio [16]. However, there is still a need for on-line local voltage stability assessment that considers the true load characteristics, instead of the assumption of constant power factor. In order to know the actual power that is consumed for a load at any voltage, we need to know the amount of load and its characteristic. Load characteristics for typical individual loads have been well studied [2], [7], [17]-[19], as well as combinations based on residential/industrial/commercial proportions of load composition [20]-[22], and combinations based on the season of the year and on the time of day [20], [22]-[23], [46]. Those results show how real and reactive power change with respect to the magnitude of the voltage. A variety of common loads, such as fluorescent lighting, induction machines and aggregated load combinations do not show the same characteristic for real and reactive power respect to voltage. As a consequence, the ratio of real to reactive power changes with the voltage and the power factor specified for that load may be only valid at nominal voltage. For system level voltage stability assessment (VSA), conventional power flow solutions have difficulties to converge when a bus is reaching the nose of the PV curve, and there is no solution exactly at the “nose”. Methods such as the continuation power flow [24]-[25] have been developed to be able to compute complete PV curves. Furthermore, recent methods such as continuation three phase power flow [26]-[27] have been developed to account for the unbalance between phases (e.g.: neutral currents), which is more significant at lower voltage buses. For local VSA, which is the focus of this work, different methods such as the Stability Monitoring and Reference Tuning Device (SMART) [28], Voltage Instability Predictor (VIP) [29]-[30] and derivations like VIP++ [31], and the Sequence Components Voltage Instability Relay (SVIR) [32] have been developed to estimate the Thevenin equivalent network behind the load bus. Traditional local PV curve methods [28]-[31] have assumed that the maximum power transfer and the voltage stability limit occur at | Z load |=| Z th | .  45  It is demonstrated in this paper that the former maximum power transfer condition is only valid under constant power factor loads. It is also shown that the nose is the stability limit only for loads which are constant power type. For loads that behave mostly or purely as constant impedance, the load by itself can not drive the system to a point of voltage instability [48]. Single and aggregated induction motors (IM), which are usually a significant portion of the total load in a bus, exhibit a power factor that changes with the terminal voltage. A simple numerical method is presented for obtaining PQV curves for general type of load characteristics. A case study is presented in which the load characteristic is experimentally obtained for a 3-phase induction motor in our lab. Then the method to compute the local network PQV curves is tested for the aggregation of that single type of load in a bus fed by a given network. The results obtained are combined with a dynamic simulation test case to find and explain the relationship between load characteristics, power transmission limits and load-driven static voltage stability limit (also known as loadability limit). The experimental tests to get the load characteristic and the computation of a local PQV curve for the same Thevenin network are extended to other loads, such as lighting and heating types. A simulation example for local VSA was studied by aggregating a combination of those loads into an existing bus of a building. Finally, we present some preliminary ideas on how the local load characteristic estimation and the methods presented here to compute local PQV curves and assess local voltage stability conditions could be obtained on-line in an actual bus of the power grid.  4.2.  Loads with Variable Power Factor Characteristics  4.2.1.  Load Characteristics from the Induction Motor Equivalent Circuit  If all the losses in the induction motor (IM) are neglected, its equivalent circuit model [34] from Fig. 4-3 simplifies to the one in Fig. 4-4, where it is clear that the reactive power Q( V ) is quadratic with respect to the terminal voltage magnitude V because the magnetizing impedance is constant (saturation neglected). On the other hand,  46  the real power P( V ) can be approximated as constant when V is reduced due to the increase of slip s that occurs to match the mechanical load.  Rs  Rr  jXr  +  jXs  Rr 1-s s  jXm  V _  +  Fig. 4-3. Induction machine equivalent circuit.  V  P  jQ Rr 1-s s  jXm  _  Fig. 4-4. Simplified induction machine equivalent circuit.  Because P and Q do not maintain the same ratio for changes in V , the power factor angle φ is a function of the voltage magnitude V .  4.2.2.  Variable Slip and Power Factor due to Aggregate Loads  A network PV curve is computed analytically in [35] for a single IM fed by a Thevenin network, where the parameter is the slip (due to an increasing mechanical load). Although this approach clearly shows how a slip dependent PV curve does not follow the PV curves for constant PF, it might not be the best option to represent the aggregated motor load connected in a bus of the power grid, since trying to define an equivalent slip for a single equivalent motor may be inaccurate. For the load-driven voltage stability analysis, the value of interest is the total bus load power due to all the motors connected to the bus. For example, the torque and slip required by the IM of one air conditioning equipment can be assumed as constant under normal operating conditions; even though, when the ambient temperature becomes warmer, more air  47  conditioners will be turned on at the same time, so that the load bus will see a larger quantity of IM’s turned on (a smaller equivalent Z load ), but with the mechanical load and slip corresponding to each of them. Due to the drop in Z th , the voltage V received by each of the IM’s in the bus drops whenever additional power is consumed there, then the slip s of each machine will adjust accordingly, and so its PF angle, not due to a change in the mechanical load of individual IM’s, but due to a voltage decrease that requires more slip in each IM to match the electromagnetic torque to the mechanical load torque, as observed in Fig. 4-5. To analytically find the exact operating slip (and from there the PF angle) at different voltages, the IM torque characteristic and the mechanical load characteristic intersection would be 4.5 needed; for this paper though, we use experimentally obtained load characteristics for the induction motor. 4  T  e  or T  load  (Nm)  3.5  IM torque-slip characteristic at constant  V = 1.0 V0 = Vnominal  3  V = 0.8 V0  2.5  2  Linear Quadratic  V = 0.6 V0  1.5  Constant mechanical load  1  0.5  0  1  0.9  0.8  0.7  0.6  0.5 slip  s  0.4  0.3  0.2  0.1  0  Fig. 4-5. Terminal voltage and operating slip.  Then, PV curves of constant power factor (PF) loads, such as the family depicted in Fig. 4-2, can not describe the steady state operating points when loads such as the IM are aggregated in a bus fed by an equivalent Thevenin network.  48  4.3.  Load Aggregation for Analysis in PQV Space  4.3.1.  Single Load Characteristics and Power Factor Angle  Due to the variable power factor angle, the ratio of real to reactive power in the load is not a constant, but changes with the voltage. It is then more meaningful to work in the PQV space than in the traditional PV plane only. The method we propose to calculate the network PQV curves for variable power factor loads requires knowledge of the characteristics of the load so that the load power factor angle φ can be estimated as a function of the voltage magnitude φ = φ ( V ) . For a certain amount k of any type of load, where k is also known as demand [8], the real P and reactive Q power consumed by the load as a function of the variable voltage V is expressed in (4-1)-(4-2), where subscript “ 0 ” refers to known reference values at a reference voltage (usually chosen as the nominal voltage). P( V ) = k ⋅ P0 ⋅ f ( V )  (4-1)  Q( V ) = k ⋅ Q0 ⋅ g ( V )  (4-2)  Commonly used models to define the load characteristics f ( V ) and g ( V ) are the exponential and polynomial models [8]. f exp onential ( V ) = (V V0 )α  (4-3)  g exp onentail ( V ) = (V V0 )β  (4-4)  f polynomial ( V ) = z P ⋅ (V V0 )2 + i P ⋅ (V V0 ) + c P  (4-5)  g polynomial ( V ) = z Q ⋅ (V V0 )2 + iQ ⋅ (V V0 ) + cQ  (4-6)  z P + i P + c P = z Q + iQ + c Q = 1  (4-7)  It is useful to understand conceptually how the stable operating points of the network PV curve relate to the load PV characteristics for continuously increasing load demand k . This is depicted in Fig. 4-6.  49  1  Ideal network PV curve  0.8  |V|=|E|, no power transmission  |V| / |E|  limit if |Zth|=0 0.6  Real network PV curve V changes for increasing load demand k due to |Zth| > 0  0.4  Load characteristics for increasing demand k  0.2  Load real power characteristic Increase in V, same demand k 0 0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2  P / P  0  Fig. 4-6. Load PV characteristic and network PV curves.  From conventional circuit theory, it is known that: PF = cos φ = P S = P  P 2 +Q 2  Q = P ⋅ tan φ  (4-8) (4-9)  Then, the load power factor angle φ can be expressed as:  φ ( V ) = a tan P( V )  = a tan P0 ⋅ Q  Q( V )    Q0     0  g( V )  f ( V )   P0 = tan( φ0 )  (4-10) (4-11)  The load power factor angle characteristic is not dependent on the value of the load demand because k from (4-1) and (4-2) cancels out in (4-10), but only on the type of the load, as determined by its voltage characteristics f ( V ) and g ( V ) .  4.3.2.  Load Aggregation  A modified version of the basic circuit of Fig. 4-1 is shown in Fig. 4-7. Since we are interested in local voltage stability assessment (VSA), we do not need to use power flow methodologies and the two bus system depicted here is enough to get the PV or PQV curves. Throughout this paper, the Thevenin equivalent of the network behind the bus E , Z th is assumed to be known or estimated.  50  Z  = R  +  th  th  + jX  V = V  th  I = |I|  Z E = |E| /_0  δ−φ  load  |Z  |  load  |V|/|I|  δ P(V) + j Q(V)  =  φ(V) = φ(V)  _  Fig. 4-7. Modified power transmission circuit for variable load characteristics.  The load power factor angle characteristic φ ( V ) is independent of the network behind the bus. Although the operating load voltage V and the individual powers P , Q depend on the network feeding the load, the ratio of the reactive to real power Q( V ) P( V ) depends solely on the aggregated load characteristics. For an increasing load demand k , the load impedance magnitude | Z load | seen from the bus will decrease, and the load bus voltage magnitude V will change. Because the load power factor angle φ is not constant, the circuit can not be directly solved analytically for V , since transcendental equation (4-12) appears for the voltage divider formula. The power factor angle φ ( V ) is defined in (4-10).   Z th V = E ⋅  1 − Z th + | Z load | ⋅(cos φ ( V ) + j ⋅ sin φ ( V     )   (4-12)  An iterative numerical procedure is needed to solve the circuit for each value of decreasing load impedance | Zload | . The proposed method is explained step by step in the flow diagram of Fig. 4-8.  51  START Zth and E = |E|+0j = |E| 0o Thevenin network, known Q0 / P0, f(V) and g(V) Load characteristics, known |Z load(m)@m=1| = Infinity open circuit Initialize load angle: φ(m,1) @m=1 = 0o Initialize load voltage: V(m,1) @m=1 = E m = m+1 |Zload(m)| < |Zload(m-1)|  Increasing amount of load  φ (m,n)@n=1 = φ PQV(m-1)  V(m,n)@n=1 = V PQV(m-1) n = n+1  V(m,n) = E - Zth I = E - Zth (E / (Zth + Zload(m,n-1)) = = V(m,n) = E (1 - Zth / (Zth + |Zload(m)| φ(m,n-1) )) φ(m,n)  NO  = atan ( Q(|V(m,n)|) / P(|V(m,n)|) ) | |V(m,n)| - |V(m,n-1)| | < εV ? | φ(m,n) - φ(m,n-1) | < εφ? YES  φPQV(m) = φ(m,n)  ZPQV(m) = |Zload(m,n)| VPQV(m) = V(m,n) IPQV(m) = (E - V(m,n)) / Zth SPQV(m) = PPQV(m) + j QPQV(m) = VPQV(m) IPQVm* NO  ZPQV(m)= 0 ? end of PQV curve YES save and plot3D (PPQV, QPQV, |VPQV|) END  Fig. 4-8. Flow diagram of PQV curve computation for variable load characteristics.  52  In the algorithm of Fig. 4-8, there is an outer loop, defined by the counter index “m”. Each execution of this loop corresponds to the solution of the circuit on Fig. 4-7 for a single value of | Z load | (which will span from (∞ ,0 ] ) and to one point on the PQV curve (from top to bottom). The step size for the decreasing load impedance does not need to be constant, it can be larger when | Z load |>>| Z th | (far away from the expected “nose”), but gradually smaller when the order of magnitude of | Z load | approaches | Z th | . This allows to have a good computational efficiency without sacrificing the shape  and precision of the PQV curve closer to and below the “nose”, where voltage stability assessment becomes more critical. The inner loop with index “n” alternatively solves (4-10) and (4-12), until the difference of both load voltage magnitude V and power factor angle φ respect to the previous iteration ( n − 1 ) is less than specified errors ε V ,ε φ . Since the PQV curve is expected to be smooth, the initial values each time circuit is solved ( n = 1 ) can be taken from the previous converged solution ( m − 1 , previous point of the PQV curve) to accelerate the convergence of the solution. The proposed method calculates a local PQV curve which tracks the operating point for continuously increasing load demand k with a known characteristic and is not constrained to constant PF like traditional PV curves. The procedure would work for any type of load characteristic functions f ( V ), g ( V ) if available and is not restricted to exponential or polynomial load models.  4.4.  Case Study for Induction Machine Type of Loads  4.4.1.  Induction Motor Experimental Load Characteristic  4.4.1.1. Equipment Setup The experimental system set in our laboratory and depicted in Fig. 4-9 included a 3-phase VARIAC to set the amplitude of the AC voltage required, and a fractional horsepower, off-the-shelf, 3-phase induction motor. The IM shaft was mechanically  53  coupled to a permanent magnet DC machine mounted in a dynamometer cradle. The DC machine acted as generator loaded with a bank of resistors parallel to the terminals in order to emulate a mechanical load for the IM. The value of the resistors bank was set to represent a nominal mechanical load torque at the nominal speed of the IM. A customized data acquisition system with a signal conditioning stage (op-amps for the voltage, detailed in Appendix B, and transducers for the current) and a NI card interfaced with a PC using Lab-View software was used to monitor in real time, acquire, and record the voltage, current, speed and mechanical torque waveforms data as a text file.  PC-based LabView Data Acquisition System  3-phase voltages  Speed and  and currents  torque  3-phase motor  IM VFD  3-phase Y source  Mechanical load emulation  VARIAC  Speed feedback  208V  L-L  Other electrical loads  Fig. 4-9. Experimental setup one-line block diagram.  The switches are used to represent in a single block diagram the various setups required for different single and combined types of loads to be tested; which are detailed in Appendix G. 4.4.1.2. Experimental Procedure The phase voltage applied to the load started from about 140Vrms and was reduced progressively in steps of 5V each. For the band between 110-130V, the voltage was changed in 1V steps to increase the accuracy of the curve fitting within the normal operating voltage range. For each test voltage, once the readings showed steady state, the phase voltages and currents waveforms were recorded for a fixed window size of 6 cycles (0.1 seconds).  54  4.4.1.3. Data Post-Processing MATLAB was used to import the data from the experiments, and the SimPowerSystems [36] library used to make the post-processing of the waveforms data and get steady state values for P , Q , V . Because frequency response was not of interest, moving average window filters [37]-[38] were the choice to reduce the noise in the measurements. The Ezy-Fit open source library for MATLAB [39] was used to automatically load the power-voltage matrices data, average the phase values to neglect slight unbalances, identify the powers P0 , Q0 at nominal phase voltage V0 ≈ 120Vrms , apply least squares curve fitting and find two exponential models that describe the load active and reactive power characteristics from (4-3) and (4-4) for the electrical loads of interest. An undesirable effect of experimenting with very small scale loads was that the harmonic content present in the current feeding the electronic loads (PC, compact fluorescent lamps, VFD) was very large, as standards for small loads in the market do not specify harmonic filtering requirements [40]. Large current THD is not representative of a typical bus, as single large loads or buildings must have some type of harmonic filtering. For our data analysis, Fast Fourier Transform (FFT) blocks were employed to extract the fundamental 60Hz signals for the measured window. Furthermore, in non-linear loads, the PF is not only a function of φ , but also of the total harmonic distortion THD [41]. For voltage stability analysis, the power relationships are based only on the fundamental 60Hz signals since no power is transmitted at other frequencies if the applied voltage V is sinusoidal. Therefore, (4-8) and (4-9) remain valid after harmonics are filtered out. The reader is referred to [42] for a more detailed explanation. 4.4.1.4. Results The experimental plot of the load characteristics P( V ) and Q( V ) for a single ( k = 1 ) IM are depicted in Fig. 4-10. As expected from Section 4.2, the real power characteristic is close to constant power ( α ≈ 0 ), while the reactive power is similar to a constant reactance ( β ≈ 2 ), as it can be seen in Table 4-1.  55  Table 4-1. Induction motor exponential load model. P0 Q0 V0 Load α β (W) (VAR) (Vrms) IM-3ph experimental 98.3 102.7 120 0.30 2.41 (IM)  100  P  (W)  200  0 0.6  0.8  0.9  1  1.1  1.2  0.9  1  1.1  1.2  Fitting-function Data-points  100  Q  (VAR)  200  0.7  0 0.6  0.7  0.8  V / V  0  Fig. 4-10. Induction motor exponential load characteristics (real and reactive power).  4.4.2.  Aggregation of Induction Motor Loads  The network PQV curve was computed using the method on Fig. 4-8 for a load bus having the IM experimental characteristic, in series with a fixed Thevenin network E = 140 ∠0°Vrms , Z th = 10 ∠90°ohm . Without loss of generality, the method works for Z th with losses ( Rth ≠ 0 ). However, one must be careful when using normalized values  in the P or Q axis as in Fig. 4-2 to specify the X th R ratio at which the curve was th calculated. To corroborate that the aggregation of the same type of loads effectively tracks the single load power factor angle characteristic, Fig. 4-11 shows a plot of φ ( V ) coming from a single load characteristic, superimposed to φ ( V ) from an aggregation in the load bus.  56  φ (degrees)  90 From single load - experimental Aggregated - PQV curve calculated  60  30  0  0.6  0.7  0.8  0.9  1  1.1  1.2  V / V  0  Fig. 4-11. Power factor angle of single and aggregated network load.  Fig. 4-12 and Fig. 4-13 show the tri-dimensional PQV curve of the induction motor aggregation with the experimental load characteristic, as well as the PQ plane projection to better visualize the actual IM power factor angle. For comparison, those curves are plotted together with the PQV surface generated by the given Thevenin network with iso-voltage contours, sweeping the load angle φ from 0 to 90 in 5 degree steps. The equator or “cliff” of the surface is a locus of all the points at which | Z th |=| Z load | .  φ = 0o  φ = 90o  |V| (Vrms)  PQV curve for aggregation of IM type loads  140  120  100  80  60  "cliff" of  40  Thevenin surface  Q (VAR)  20  0  P (W)  200  200  400  400 600  800 1000  Fig. 4-12. Network PQV curve for aggregation of IM’s.  57  0  PQ curve for aggregation of IM type loads  500  Q  maximum  400  |Z  | = |Z  Q (VAR)  load  300  |  th  P  maximum  200  "cliff"  constant PF  100  φ  0 0  200  400  600  800  1000  P (W)  Fig. 4-13. PQ plane projection for aggregation of IM’s.  As expected, the PQV coming from the aggregation of individual IM loads crosses the planes of constant power factor, while always running along the surface defined by the fixed E , Z th . To this point, the load characteristic of an IM and its aggregation on a networkfed bus have been described without further analysis of the power transmission or the stability limits. These considerations will be performed in the following section.  4.4.3.  Power Transmission Limits and Impedance Matching  Three relevant points are highlighted in Fig. 4-13, the point of maximum real power  P , the point of maximum reactive power Q , and the point where  Z load = Z th = 10 Ω , which touches the “cliff” of the PQV surface.  For the IM load aggregation, these three conditions do not happen at the same operating point of the bus. In fact, the “nose” of the reactive power QV curve happens at Z load > Z th (above the “cliff”), while the “nose” of the real power PV curve occurs at Z load < Z th (below the “cliff”). When the power factor is constant, the relationship  between P and Q from (4-9) is linear, departing from the origin, as observed in Fig. 4-13, so that the “cliff” is the same point as the PV “nose” and the QV “nose”.  58  Projections on the PV and QV planes are shown together in Fig. 4-14 to facilitate the understanding of the results. The “cliff”, where Z load = Z th = 10 Ω , clearly is a “nose” neither of the PV curve, nor of the QV curve. The impedance matching condition then does not imply maximum power transfer, as typically assumed in local VSA. This is a consequence of accounting for the load not to be PF constant.  140 120  "cliff" intersection "nose" of QV  |V| (Vrms)  100 80 60 "nose" of PV  40  QV curve PV curve  20 0 0  100  200  300  400  P (W) or Q (VAR)  500  600  700  Fig. 4-14. PQ and QV projections for aggregation of IM type of load.  Fig. 4-14 also depicts that the PV and QV curve projections have different shapes; they are not simply scaled one to each other as in the case of constant PF, making the calculation in the tridimensional PQV space more relevant. Note that what we call the reactive power transmission QV curve in this paper is not the same as the VQ curve commonly used for system level voltage stability studies [2], [6]-[8]; the latter is related to the reactive power support required in a bus. We have not demonstrated yet whether any of the power transmission limits or the cliff intersection corresponds to the voltage stability limit. This is discussed next.  4.4.4.  Voltage Stability Limits  Previous research for local VSA [28]-[30], [43]-[44], relies on the assumption that the limit of power transfer occurs when Z load = Z th . As we have explained, this is not valid if the load PF is not constant, as in the case of aggregation of induction motors on a load bus (which could represent a typical industrial bus). 59  Furthermore, it has been demonstrated rigorously in [6], [8], [33], [45] that the maximum power transmission point (“nose” of the PV curve) is not necessarily the maximum loadability point, where the network can not longer meet the load power demand and voltage instability starts. The loadability limit -or static voltage stability limit- is graphically seen as the point where the load characteristic curve has a demand k that it is tangent to the network PQV curve. Then, unless the load characteristic is vertical (meaning purely constant power α = 0 ), the PV “nose” and the loadability limit points are not coincident. In addition, passive loads, which show a load characteristic close to constant impedance α ≈ 2 , will always intersect the network PQV curve for any value of demand k and, therefore, they can not cause voltage collapse because no loadability limit exists [33]. Next, the relationships between load characteristics and network PQV curves for voltage stability assessment are plotted and were confirmed with a time-domain simulation.  4.4.4.1. Dynamic Simulation Setup A study was performed, comprising the network steady state PV and QV curves, the static load characteristics, and a SimPowerSystems [36] dynamic model of the load and network for our case study. A 3-phase dynamic load block was set to the exponential model coefficients obtained in Table 4-1 (which are in per phase), and connected in series to the same perphase Thevenin equivalent network ( E = 140 ∠0°Vrms , Z th = 10 ∠90°ohm ). As the focus of this work is on static load and network characteristics, the time constant of the dynamic load used was set to τ ≈ 0 to neglect transient effects, such as the mechanical time constant for a motor load. The voltage at which the load exponential model changes to a constant impedance ( α = β = 2 ) model was set to Vmin ≈ 0 , to force the voltage collapse process not to stop at any voltage once it had started. The load demand k was increased in steps to the values shown in Table 4-2 at every second of time.  60  t (s) k  Table 4-2. Load demand time sequence. 0-1 1-2 2-3 3-4 4-5 5-6 0 5.37 7.02 8.05 8.13 8.21  6-7 8.26  4.4.4.2. Dynamic Simulation Results Fig. 4-15 shows how the voltage, real and reactive power and load impedance magnitude change with respect to time, due to the increase in load demand k . The time domain results are superimposed onto the steady state network and load PV and QV curves in Fig. 4-16 and Fig. 4-17, respectively, in order to perform the  P (W)  voltage stability assessment of the load bus.  600 400  see Fig. 18  200 0  Q (VAR)  1  2  3  4  5  6  5  6  600 maximum Q transmission (QV "nose")  400 200 0  |V| (Vrms)  1  2  3  4  150 100 50 voltage collapse (instability)  |Z  load  | (ohms)  0 1  2  3  4  5  6  20 |Z  15  |=|Z  th  | ("cliff" intersection)  load  10 5 0 1  2  3  4  5  t (s)  Fig. 4-15. Time results of real and reactive powers, voltage and impedance.  61  6  140 dynamic model "nose"  120  of QV "cliff"  steady state  100  |V| (Vrms)  PV curve k = 5.37  80  k = 7.02 60 see Fig. 19 40  20  0 0  100  200  300  400  500  600  700  800  P (W)  Fig. 4-16. Load characteristic, network and dynamic model PV curves.  140 dynamic model k = 5.37 120  steady state  k = 7.02  QV curve  |V| (Vrms)  100 "nose" 80  of QV "cliff"  60  "nose"  40  of PV  20  0 0  100  200  300  400  500  Q (VAR)  Fig. 4-17. Load characteristic, network and dynamic model QV curves.  Every time the dynamic model touches the network PQV curve corresponds to a steady state operating point, as confirmed by the intersection of the static load characteristic at the corresponding demand k . The operating point moves every second after a step change in k . After t = 6 s , the network is not longer able to supply the load and a new steady state point is not reached. The system is no longer stable and the load bus voltage has collapsed.  62  An inspection of some important points in Fig. 4-15, Fig. 4-16 and Fig. 4-17 is necessary to fully understand the process. The first point is the maximum reactive power transfer point. When the limit of reactive power transmission Q was reached at t = 1s , the system was stable. The load demand k was later increased, and the system reached a stable operating point at a lower voltage and lower reactive power. This means that the “nose” of the QV curve was not the voltage stability limit. At t = 2 s , the amount of load was such that Z load ≈ Z th = 10 Ω , reaching the “cliff” of the PQV surface. The load could still be increased, making the load impedance Z load < Z th , as it is shown at t = 3 s seconds. The system remained stable with more  real power still delivered to the load, confirming that the “cliff” was neither the limit of real power transfer nor the steady state voltage stability (loadability) limit. A zoom on the time-domain and on the PV plane plots of Fig. 4-18 and Fig. 4-19 shows that with a load demand k = 8.13 at t = 4 s , the maximum real power transmission P limit was reached, seen as the “nose” in the PV plane projection. Even though, more  demand ( k = 8.21 ) could be added to the load bus at t = 5 s and it reached a steady state stable operating point at a lower real power and voltage. This was expected because the static load characteristic still intersects the network PV curve a little below the “nose”, as can be seen in Fig. 4-19. Then the PV “nose” represents the real power transfer limit, but not the steady state voltage stability limit. The load was increased even more at t = 6 s seconds, to reach a load demand k = 8.26 . Then, the system became unstable and did not reach steady state anymore;  voltage collapse occurred. This is confirmed from Fig. 4-19, which shows that at this load demand value, the static load PV characteristic does not longer intersect the network steady state PV curve. From a quick inspection of Fig. 4-16 and Fig. 4-17, one can see that the static voltage stability limit is independent of the projection we use to visualize the intersection of the network and load characteristic curves. From the proceeding analysis we conclude that the network PQV (or PV) curves are not sufficient to assess the voltage stability limit, only the power transmission limits. The intersection of the load characteristics with the system PQV (or PV) curves is  63  necessary for accurate static local VSA. The traditional assumption that the PV “nose” is the static voltage stability limit is not accurate, although it is conservative and, therefore, on the safe side.  2 (W)  680  Slightly above the PV "nose"  670  660  Maximum P transmission limit  No steady state  ("nose" of PV curve)  Slightly below the "nose",  reached after  the system is still stable  loadability limit  650  3.5  4  4.5  J (s)  5  5.5  6  Fig. 4-18. Dynamic model results. Power transfer and stability limits.  dynamic model steady  70  state "nose" of  PV curve  PV  |V| (Vrms)  68  k = 8.05 66  k = 8.21  k = 8.13  k = 8.26  64  no intersection of network and  62  load characteristics  60 650  660  670  680  P (W)  Fig. 4-19. Points of maximum power transfer and stability (loadability) limit.  In typical power systems, the load bus will never be operated at those extreme limits because small contingencies or small load variations may take the system to voltage instability conditions. Nonetheless, it is useful to know more precisely (than assuming constant PF loads) the theoretical static limits of weak buses, so that the real operating limits, the  64  corrective actions, and the transmission growth planning do not drive the system too close to the theoretical limits.  4.5.  Characteristics and Aggregation of Other Commonly Used Loads  4.5.1.  Single Loads  To determine the characteristics of other typical loads, the same experimental procedure performed on the 3-phase IM was repeated for a few other motor, lighting and heating types of loads, described in Table 4-3. The experimental load characteristic coefficients (exponential and polynomial models) and the simulated PQV network curves (exponential model only) for aggregating those loads in the (previously used) network-fed bus with E = 140 ∠0°Vrms and Z th = 10 ∠90°ohm are summarized in Table 4-3, Table 4-4 and Fig. 4-20. Table 4-3. Experimental single load characteristics, exponential (V0 = 120Vrms). P0 Q0 tan φ0 = Type of load α β (W) (VAR) Q0 / P0 Induction motor – 3ph 98.3 102.7 1.045 0.30 2.41 (IM-3PH) or (IM) Fluorescent lamp, 14.9 31.5 2.114 2.80 4.58 magnetic ballast (FLM) Incandescent lamp 82.8 0.03 0.002 1.55 0.32 (INC) Power resistor-heater 289.3 8.5 0.029 2.02 1.98 (RES) Electric stove -4.9 -0.005 1.97 1.97 933.5 (STOVE) Personal computer 156.9 -13.3 -0.085 -0.01 -0.18 (PC) Induction motor – single 329.4 480.6 1.459 0.38 2.31 phase (IM-1PH) Fluorescent lamp, 80.7 -18.4 0.228 1.34 0.90 electronic ballast (FLE) Compact fluorescent 52.0 -27.8 0.535 0.98 0.42 lamp (CFL) VFD closed loop + 107.0 2.2 0.021 -0.06 0.17 IM3ph (VFD-CL) VFD open loop + 108.7 2.17 0.020 0.29 0.28 IM3ph (VFD-OL)  65  Table 4-4. Experimental single load characteristics, polynomial (same P0, Q0, V0). Type of load Induction motor – 3ph (IM-3PH) or (IM) Fluorescent lamp, magnetic ballast (FLM) Incandescent lamp (INC) Power resistor-heater (RES) Electric stove (STOVE) Personal computer (PC) Induction motor – single phase (IM-1PH) Fluorescent lamp, electronic ballast (FLE) Compact fluorescent lamp (CFL) VFD closed loop + IM3ph (VFD-CL) VFD open loop + IM3ph (VFD-OL)  zP  iP  cP  zQ  iQ  cQ  1.208  -1.758  1.551  3.689  -4.528  1.839  2.780  -2.696  0.916  6.857  -9.044  3.187  0.535  0.509  -0.045  -6.667  9.333  -1.667  1.025  -0.030  0.005  1.009  -0.034  0.025  0.952  0.063  -0.015  0.953  0.061  -0.014  0.132  -0.254  1.121  -0.822  1.290  0.532  2.524  -4.068  2.544  4.341  -5.850  2.509  0.342  0.691  -0.032  -0.046  1.001  0.046  -0.652  1.913  -0.262  -2.222  3.589  -0.367  3.251  -6.151  3.900  44.10  -82.26  39.16  1.441  -2.389  1.948  53.44  -99.76  47.31  It is relevant to notice that, although loads operating at higher power factor (e.g.: load buses with reactive power compensation) allow higher power transmission limits, they can reach these limits within the range of normal operating voltages. Therefore, simple under-voltage detection will not protect against voltage instability (collapse) if the loadability limit happens to be close to that high voltage “nose”.  66  φ = 0o  VFD-CL  φ = 15o (lead)  φ = 90o  |V| (Vrms)  VFD-OL  INC  140  120  100 "cliff" 80  RES PC  60 STOVE IM-3PH  40 CFL (out of scale  Q<<0)  IM-1PH  FLE  P (W) 1000  1200  800  600  400  20  FLM 0  200  -200  0  200  φ  0  Q (VAR)  Q>0)  Inductive loads ( lagging  400  Fig. 4-20. Network PQV curves for a variety of single-type loads.  For the PV curve projections on Fig. 4-21 and Fig. 4-22, lighting loads are plotted separately from the other types to show that classifying and combining different types of lighting loads generically might be inaccurate because there are significant differences in the characteristics and PQV curves among sub-types (e.g.: incandescent or fluorescents with different types of ballasts). 160  140  STOVE  |V| (Vrms)  120  PC  RES  100  IM-3PH  VFD-OL  IM-1PH  VFD-CL  80  60  40  20  0  200  400  600  800  P (W)  1000  1200  1400  Fig. 4-21. Network PV curves for motor and heating type loads.  67  1600  160  140 FLE  |V| (Vrms)  120  INC  100 CFL 80  FLM  60  40  20  0 200  400  600  800  1000  P (W)  1200  1400  1600  Fig. 4-22. Network PV curves for lighting type loads only.  It is also important to note in Fig. 4-21 and Fig. 4-22, that there are several loads with nearly identical PV curves (RES, STOVE, INC, VFD’s), however, their load characteristic α is significantly different. Because in all these cases Q0 ≈ 0 (nearly purely resistive loads), the solution from (4-10) becomes trivial φ ≈ 0° (unity constant power factor). An intersection of network and load characteristics in Fig. 4-23 helps to clarify the apparent ambiguity between constant power factor loads and constant power characteristic loads, and their effect on the voltage stability of the load bus. Again, the steady state PQV curve alone was sufficient to find power transfer limits, which depend on power factor only, but was not sufficient to assess local voltage stability. Loads with same power factor characteristics and different power characteristics will have the same PQV curve but different static voltage stability (loadability) limits, which can be observed in Fig. 4-23. This emphasizes the fact that the load power characteristic intersection is also needed for accurate local VSA.  68  RES:  α = 2.02 ~ Constant impedance α = -0.06 ~ Constant power  VFD-CL: 140  Steady state PV  120  |V| (Vrms)  curve for 100  k = 2: Only  φ = 0o  RES stable  k = 1: Both stable Same power  80  transfer limit  60  k >= 4: RES loads will be  40  stable for all demands, 20  P  0  due to its load characteristic  = 500W  @ k=1 0 0  200  400  600  800  1000  1200  P (W)  Fig. 4-23. Power factor and power characteristic comparison.  4.5.2.  Aggregated Loads in a Building  A voltage stability assessment of a building in our university was made based on a combination of simulation and previous experimental results from this paper and from [47]. Some assumptions were required to complete the data required by the analysis. After converting the values reported in [47] from p.u. to absolute; the 208VL-L bus on the third floor of this building (120VL-N) had a load which fluctuated on a winter business day between S = 11kVA and S = 19 kVA per phase. The measured series Thevenin impedance per phase measured at that point was Z th = 0.058 ∠90°Ω . The Thevenin equivalent phase voltage seen from the bus was not specified; thus, it was estimated as | E |= 1.05 ⋅ V0 = 126V . It was also assumed that the total load in the building and the various floors maintained the same type composition all day. The load composition of the building at nominal phase voltage V0 = 120V was estimated to be as shown in Table 4-5, where the characteristics for each type of load were obtained from our previous experimental results in Table 4-3.  69  Table 4-5. University building load composition. % of total MVA load Individual type of load (i) (w) at V=V0 Heating loads (RES) 10 % Personal computers (PC) 45 % Fluorescent lamps magnetic (FLM) 40 % Induction motors 3-ph (IM) 5%  Because only the apparent power magnitude S is known, the load composition is used to calculate the combined load power factor angle at V0 , from the weighted average  w of each individual i type of load, as in (4-13).  (  )  n   Q0   Q0   = 0.857 = tan φ = w ⋅    ∑ 0 i  combined P0  combined P0  i    i = 1,2 ,...,n   (4-13)  The separate values of individual reference powers P0 and Q0 can be misleading in the percentage aggregation of the building. As their ratios have already been considered in the combined power factor angle from (4-13), what only remains to be solved is the aggregation of the individual load characteristics f ( V ) and g ( V ) from (4-1) and (4-2), by using again weighted average w of each i load characteristic as in (4-14) and (4-15).  f ( V )combined =  g ( V )combined =  n  ∑ [wi ⋅ f ( V )i ]  (4-14)  i = 1,2 ,...,n n  ∑ [wi ⋅ g( V )i ]  (4-15)  i = 1,2 ,...,n  For the estimated building load composition, the characteristic using the exponential models from (4-3) and (4-4) becomes:  f exp ( V ) = 0.10(V V0 )2.02 + 0.45(V V0 )−0.01 + 0.40(V V0 )2.80 + 0.05(V V0 )0.30  70  (4-16)  g exp ( V ) = 0.10 (V V0 )1.98 + 0.45 (V V0 )−0.18 + 0.40 (V V0 )4.58 + 0.05 (V V0 )2.41  (4-17)  There is no easy analytical way to simplify (4-16) or (4-17) due to different exponents in each term. This would be disadvantageous for on-line implementation and for understanding the combined load from inspection of its load characteristic coefficients. Using a combination of polynomial models coefficients from Table 4-4, it is easier to factorize the terms, and obtain a combined polynomial model as in (4-5) and (4-6). The combined load coefficients of the building are shown in Table 4-6 and the combined load characteristics and power factor angle are plotted in Fig. 4-24. The network PQV curve for the building and its PV projection intersected with three relevant load demand values k are shown in Fig. 4-25 and Fig. 4-26, respectively. From the results, it is observed that the operating condition on the third floor bus of this building is very far from the voltage stability or from the power transmission limits. No significant voltage drop is expected at the normal daily peak operating conditions. If local steady state voltage stability were the only constraint, up to 7 times ( k = 133 ) the current peak demand, keeping the load composition, could be supplied by the bus. In this particular case, voltage drops and thermal limits would be the constraints to set operating limits, not VSA.  Table 4-6. Building combined load characteristic, polynomial. S0 @ k=1 Q0 /P0 zP iP cP zQ iQ cQ 1000 VA 0.857 1.334 -1.284 0.949 2.658 -3.267 1.609  The reference building demand ( k = 1 ), was set to correspond to a load at nominal voltage S ( V0 ) = 1kVA .  71  P (W)  P  0  800  = 760W  S  600  0  = 1kVA  400 0.6  0.7  0.8  0.9  1  1.1  Q (VAR)  800 Q  0  = 650VAR  600  400  0.6  0.7  0.8  0.9  1  1.1  1  1.1  φ (deg)  45  φ0 = 40.6o  40  35  30 0.6  0.7  0.8  0.9 V / V  0  Fig. 4-24. Building aggregated load characteristic and power factor angle.  φ = 0o  |V| (Vrms)  φ = 90o (lag) 120  100  80  "cliff" of 60  Thevenin surface E = 126 Vrms Z  th  = j 0.058  Ω  40  Building PQV curve, with aggregated  P (kW)  load characteristic  120  80  0  40  20  Q (kVAR) 40 60  20  0  Fig. 4-25. Building network PQV curve, polynomial load characteristic.  72  140 Steady-state PV curve with combined load characteristic 120  k = 11  |V| (Vrms)  100  (Minimum daily demand)  80  Voltage stability limit  k = 19  60  (Maximum daily demand)  40  k = 133  20  0 0  10  20  30  40  P (kW)  50  60  70  Fig. 4-26. Building network PV curve and polynomial load characteristics for VSA.  4.6.  Proposed Implementation Methods If a relay for local VSA were implemented on a grid bus, one of the tools it would  need is an estimator of the load characteristics and network PQV curves, in order to decide if and when some load must be shed [49] to avoid voltage instability. In this application, the estimator will acquire voltage and current values in real time so that the active and reactive load powers can also be computed at those time points. The change in load demand during the day will make the voltage in the load bus change within some percentage; in fact, weaker buses, which are more prone to voltage stability problems, will have more voltage variation, so that this band will be larger. The estimator will then be able to capture an actual network PQV curve for the portion of existing operating voltages around the nominal value. If the equivalent of the network behind the load E , Z th is known from the same measurements, two approaches are possible in order to estimate the load power factor angle characteristic φ ( V ) and build the entire PQV curve.  73  4.6.1.  Curve Fitting and Extrapolation of Power Factor Angle  If the values of P and Q are known for every voltage within, for example, a 10% band, a plot of φ ( V ) can be computed. Curve fitting methods can then be employed to find a function for h( V ) in (4-18).  φ ( V ) = a tan( Q0 P ⋅ h( V ))  (4-18)  0  The function φ ( V ) is the one incorporated in the flow chart of Fig. 4-8. Implementing this function into the computation of the entire PQV curve extrapolates the load characteristic of the measured portion of the curve to all possible voltages. The accuracy of this method is limited by how good the curve fitting of a limited band can be with respect to the real load characteristic at all voltages. If the function and constraints are selected reasonably, this approach should be more precise than simply assuming constant power factor φ ( V ) = φ0 . This method could compute the PQV curve, but does not give information about the separate load characteristics f ( V ) and g ( V ) and, therefore, the problem from Fig. 4-23 may arise for the voltage stability limit estimation, thus forcing the use of the PV “nose” criteria as the stability limit in order to be on the safe side.  4.6.2.  Database Curve Matching  The estimator could store a large database of different types of loads with their steady state power-voltage characteristics  Q0  P0 ,  f (V ) ,  g ( V ) . Those load  characteristics could be based on utility historical data, laboratory tests, accepted standard models, etc. Using the Thevenin voltage and impedance estimated by the device, the network PQV curves can be computed on-line with the method of Fig. 4-8 for each of the database load characteristics. Then, the measured portion of the actual network PQV/PV/QV curve (within the bus voltage variation band) can be compared against the complete PQV/PV/QV curves from the stored load characteristics, so that it can be matched or interpolated to the one best fit of the stored curves.  74  A large number of load combinations and comparison criteria could be employed, and an approach with artificial intelligence methods (e.g., neural networks [51]) could be one of the options to automate matching of the network PQV curve with the stored data. The relay could also store information on the transient load characteristics or in possible contingencies that affect E and Z th . This additional information would be helpful to improve the load characteristic matching when more than one close fit of the aggregated PQV curves is identified.  4.7.  Conclusions It has been shown experimentally, and explained analytically for the 3-phase  induction motor case, that a variety of typical loads have a characteristic such that their power factor is not constant, but is a function of the voltage magnitude. A numerical method was developed to compute local PQV curves when those types of loads are aggregated in a network-fed bus and to find power transfer and voltage stability limits that consider actual load characteristics. The results show that under non-constant PF, the criteria of load impedance magnitude matching the Thevenin impedance magnitude does not correspond to maximum power transfer. It was demonstrated that the “nose” of the PV curve (maximum power transfer point) is not necessarily the static voltage stability limit. The load characteristic intersection is needed to determine if, and when, the load may take the bus to a voltage instability condition. In fact, different unity PF loads may have the same network PQV curve, but different static voltage stability (loadability) limits due to different real power characteristics. A composition of loads was assumed for a building in which the Thevenin equivalent impedance and the total load demand were known, in order to assess local voltage stability conditions off-line in an actual bus. Preliminary ideas of on-line implementation methods for the local VSA with load characteristic considerations were presented.  75  4.8. [1]  References North American Electric Reliability Corporation (NERC), “Results of the 2007 Survey of Reliability Issues”, rev. 1, October 24th, 2007, 18p.  [2]  C.W. Taylor, Power System Voltage Stability. McGraw-Hill, 1994, 273p.  [3]  BC Hydro, “Making the Connection: the BC Hydro Electric System and How it is Operated”, BC Hydro, April 2000, 56p.  [4]  M.H. 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This was done to compute PV or PQV curves that consider the load characteristics, and improve our understanding of the load characteristic effect on power transmission limits and local voltage stability limits. The first manuscript [1] was focused on the modeling of a modern motor type load and its effect on voltage stability. The second manuscript [2] extended the load modeling for a variety of operating conditions. It also included an experimental voltage stability verification of a single IM load under variable slip. The third manuscript [3] was a much more complete approach to the static voltage stability problem. Experimental load modeling of diverse loads, and their simulated aggregation in a network-fed bus was performed, in order to find and analyze local power transmission and voltage stability limits. Significant differences with the traditional methods, that assume loads with constant power factor and constant power characteristic, were found and explained.  5.2.  Conclusions Laboratory tests were performed to find static load characteristics for active and  reactive power of typical loads, including motor type with and without variable frequency drive control, lighting and heating. For motor type loads, the characteristics were previously predicted from their equivalent circuit models [4]-[5]. A discussion about current harmonics was presented, which confirmed that those should generally be neglected for static voltage stability assessment (VSA). The experimental results and the literature coefficients [6]-[8] for common load models showed that some loads, such as induction motors, which represent a significant portion of the total power delivered in a typical power grid, do not have constant power factor (PF) under variable voltage. Thus, the traditional families of PV curves that 81  assume constant power factor angle [7], [9] do not describe the operating points when the demand of variable power factor loads increases in a bus. A small-scale voltage stability test was performed for a 3-phase IM under increasing mechanical load, fed by a voltage source and series reactance, to obtain a family of experimental PV curves with the IM slip as the parameter for a set of different network impedance values, which were successfully compared to the calculated curves. Although this case was of interest to represent a case of voltage stability for an available variable PF load, it may not be the best option to represent load aggregation in an actual bus. A comprehensive case study was shown for a 3-phase IM, where it was simulated how the load demand in a bus changes, not as the mechanical load of a single IM, but as the aggregation of similar IM’s with a given mechanical load. Since variable PF implies that the reactive and real power are not scaled to a constant ratio, the need for a tridimensional PQV space representation arose. Due to the variable load power factor angle, the basic power transmission circuit of a fixed Thevenin equivalent and decreasing load impedance magnitude could not be solved analytically. Instead, a simple, yet effective, numerical iteration method was developed to solve the circuit and calculate each point on the PQV curve. It was proven that this method actually tracked the power factor angle characteristic of the load when the voltage on the load bus dropped due to increased load power, following the surface determined by the fixed Thevenin equivalent network. Analysis of the results showed that the maximum power transfer point (the “nose”) does not occur when the load impedance magnitude equals the Thevenin magnitude. The latter condition was valid only if assuming constant PF load. Furthermore, neither the impedance matching, nor the power transfer limits represent the local voltage stability limit. A time domain simulation confirmed that the static voltage stability limit (also known as loadability limit [9]) is reached when the demand takes the load characteristic of the bus to a point where it is tangent to the network PQV curve (or PV or QV curve, the projection was irrelevant). The network PQV curve was sufficient to graphically find power transfer limits (the “nose”), but not the voltage stability limit. The “nose” represents the stability limit only for pure constant power characteristic loads  82  [10], which is a conservative assumption and, therefore, safe. Loads with characteristics similar to constant impedance are not able to produce voltage instability [10]. The simulated aggregation of loads (PQV curves and projections) in a fixed network was extended to a diversity of experimentally obtained load characteristics. These showed important differences between subtypes of lighting loads, which are generally grouped into a single category. A local voltage stability assessment, that included the combination of loads with variable load characteristics, was performed for a known distribution bus [11] in a building of the university, by using the methods and findings from the previous sections of the thesis.  5.3.  Future Work While all the work of this thesis was performed with off-line analysis; ideas for  on-line implementation of the PQV curve computation and local voltage stability assessment considering load characteristics were presented. This, in order to provide a frame for future work towards a device that helps to assess local on-line voltage stability. For the off-line part developed, the methods and results presented assumed that the exponential or polynomial curve-fitting models for the loads were continuous. This could be inaccurate to describe the load characteristics at low voltages. For example, electronic loads can simply behave as a zero power load as they will turn off when voltage is too low. Another example, a completely stalled induction motor behaves as constant impedance if the voltage keeps decreasing, because slip will not go above 1, and it also has a discontinuity between the breakdown slip and stall where all the intermediate slips can not be steady state points; maybe motor protections will disconnect it. Although loss of load is unacceptable from the customer point of view, loads turning off or becoming constant impedance will provide relief in terms of voltage stability and may help to avoid or stop an ongoing voltage collapse due to high load demand. Static load characteristics can be refined to be piecewise functions, depending on the change of load characteristic at non typical operating voltages. Thermostatic effects, not considered in this paper, may cause some aggregations of constant impedance loads such as electric heaters, to become close to constant power  83  in longer time frames. This is because at sustained low voltages, on average, more heaters will be turned on in order to provide the thermal energy for the spaces where they are installed [12]. Some distribution buses also have automatic transformer tap changers, which will try to restore the voltages on their secondary when they drop below some level; those tap changes may drive the system to a voltage collapse if they are not controlled during contingencies [12]-[13]. Dynamic effects that include load mechanical and thermal time constants, transient load characteristics for the corresponding transient timeframes, and changes on the Thevenin side due to dynamic equipment on the electrical network, might be necessary to take into account in the design of an accurate relay for on-line local voltage stability assessment, to complement the steady state approach used throughout this thesis.  84  5.4. [1]  References L.M. Vargas, J. Jatskevich and J.R. Marti, “Load Modeling of an Induction Motor Operated with a Variable Frequency Drive”, published in Proceedings of IEEE Electric Power Conference EPEC 2008, Vancouver BC, Oct 6-7th 2008, pp. 1-6  [2]  L.M. Vargas, J. Jatskevich and J.R. Marti, “Induction Motor Loads and Voltage Stability Assessment Using PV Curves”, accepted for IEEE Power Engineering Society General Meeting PES-GM 2009, Calgary AB, Jul 26-30th, 2009.  [3]  L.M. Vargas, J.R. Marti and J. Jatskevich, “Local Voltage Stability Assessment for Variable Load Characteristics”, to be submitted for IEEE Transactions on Power Delivery.  [4]  T. Gonen, Electrical Machines, Power International Press, 1998, 532p.  [5]  P.C. Krause, O. Wasynczulk and S.D. Sudhoff, Analysis of Electric Machinery and Drive Systems. Second Edition, IEEE Press, 2004, 613p.  [6]  P. Kundur, Power System Stability and Control. McGraw-Hill, 1994, Ch. 7.  [7]  C.W. Taylor, Power System Voltage Stability. McGraw-Hill, 1994, 273p.  [8]  L. M. Hajagos and B. Danai, "Laboratory Measurements and Models of Modern Loads and Their Effect on Voltage Stability Studies", IEEE Transactions on Power Systems, vol. 13, no. 2, May 1998, pp. 584-592.  [9]  T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. Kluwer Academic Publishers, 1998, 378p.  [10] M.K. Pal, “Voltage Stability Conditions Considering Load Characteristics”, IEEE Transactions on Power Systems, vol. 7, no. 1, Feb. 1992, pp. 243 – 249. [11] M. Wrinch, J.R. Marti and M. Nagpal, “Negative Sequence Impedance based Islanding Detection for Distributed Generation (NSIID)”, IEEE Electric Power Conference EPEC 2008, Vancouver BC, Canada, October 2008, pp. 1-6. [12] IEEE/PES Power System Stability Subcommittee, “Voltage Stability Assessment: Concepts, Practices and Tools”, IEEE Special Publication, August 2002, 29p. [13] H. Ohtsuki, A. Yokohama and Y. Sekine, “Reverse Action of On-Load Tap Changer in Association with Voltage Collapse”, IEEE transactions on Power Systems, vol. 6, No. 1, Feb. 1991, pp. 300-305.  85  Appendix A. Contributions The following publications are the result of research work performed for this M.A.Sc. thesis:  [1]  L.M. Vargas, J. Jatskevich and J.R. Marti, “Load Modeling of an Induction Motor Operated with a Variable Frequency Drive”, published in Proceedings of IEEE Electric Power Conference EPEC 2008, Vancouver BC, Oct 6-7th 2008, pp. 1-6.  [2]  L.M. Vargas, J. Jatskevich and J.R. Marti, “Induction Motor Loads and Voltage Stability Assessment Using PV Curves”, accepted for IEEE Power Engineering Society General Meeting PES-GM 2009, Calgary AB, Jul 26-30th, 2009.  [3]  L.M. Vargas, J.R. Marti and J. Jatskevich, “Local Voltage Stability Assessment for Variable Load Characteristics”, to be submitted for IEEE Transactions on Power Delivery.  86  Appendix B. Measurement Box Modifications A standard measurement box from the Electrical Machines Laboratory (MCLD130) was slightly modified to be able to read values within the nominal building voltage range. An op-amp output voltage is limited by its supply voltage, causing that the original configuration (designed for very low voltage experiments) saturated for our range of test voltages. The use of a 10X probe did not solve the problem, because its 10X compensation is only on the positive side, not working for “floating” voltages, only for ground referenced ones. It was inconvenient to modify the equipment substantially, so, the solution was to replace the input resistor stage and recalibrate, to change the amplification ratio of the op-amp from 1/10 to 1/100, as shown in Fig. B-1. Although the values shown there do not appear to give exactly 1/100 when adjusting the variable resistor to either side, the calibration was successful with those components; we assume this small mismatch was due to tolerances in the internal chip and the external resistors. Given that modification, and the use of fuses in the VARIAC secondary for safety, we were able to read three phase or line voltages up to 280Vrms without problem.  Fig. B-1. Schematic of laboratory measurement box modifications.  87  Appendix C. Additional Results from Chapter 3 Additional Experimental Results for Motor Type Load Characteristics Several other motors under different operating conditions were tested during the work for Chapter 3, but not included in the manuscript. All the original waveforms were post-processed again with the automated tools of Chapter 4, and the results summarized in Table C-1 and Table C-2. In general, polynomial models showed better curve-fittings. Table C-1. Exponential static load model for additional motor type loads. Case  Motor  % of nominal torque load  VFD configuration  Q0 / P0  α  β  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  IM1 IM1 IM1 IM1 IM1 IM1 IM1 IM1 IM1 IM2 IM2 IM2 IM2 IM2 IM2 IM3 IM3 IM3 IM3 IM3 IM3 IM3 IM3 IM3 IM4 IM4 IM4  25 25 25 50 50 50 100 100 100 50 50 50 100 100 100 25 25 25 50 50 50 100 100 100 50 50 50  closed loop not used open loop closed loop not used open loop closed loop not used open loop closed loop not used open loop closed loop not used open loop closed loop not used open loop closed loop not used open loop closed loop not used open loop closed loop not used open loop  0.031 1.916 0.027 0.023 1.237 0.020 0.001 0.808 0.001 0.032 1.690 0.032 0.021 1.055 0.020 0.027 2.531 0.028 0.017 1.619 0.018 0.001 0.960 0.001 0.031 1.621 0.024  0.35 0.26 0.34 0.09 0.07 0.13 -0.46 -0.32 -0.18 0.46 0.15 0.52 0.01 -0.03 0.18 0.60 0.38 0.62 0.22 0.08 0.31 -0.38 -0.17 -0.06 0.78 0.30 0.85  0.81 1.91 0.85 1.22 1.67 1.53 23.99 0.57 21.99 0.71 1.97 0.52 -0.09 1.66 -0.12 0.63 2.13 0.70 0.12 1.77 -0.08 22.87 1.25 23.64 0.78 2.51 0.94  88  Case  Motor  % of nominal torque load  VFD configuration  α  Q0 / P0  β  28 IM4 100 closed loop 0.021 -0.07 0.17 29 IM4 100 not used 1.040 0.29 2.57 30 IM4 100 open loop 0.020 0.29 0.28 Note: For Q0 / P0 ≈ 0, the value of β is irrelevant. Q(V) = 0 constant, may be used. Table C-2. Polynomial static load model for additional motor type loads. Case  zP  iP  cP  zQ  iQ  cQ  1 0.078 0.226 0.697 16.395 -29.849 14.454 2 0.379 -0.407 1.028 1.387 -0.776 0.389 3 0.258 -0.123 0.865 20.235 -36.991 17.757 4 0.815 -1.433 1.617 34.328 -62.992 29.664 5 0.394 -0.640 1.246 1.569 -1.305 0.736 6 0.468 -0.742 1.274 46.242 -85.055 39.813 7 2.528 -5.322 3.794 -87.040 158.800 -70.755 8 1.737 -3.467 2.730 4.214 -6.921 3.707 9 1.046 -2.163 2.117 -93.502 170.710 -76.211 10 0.538 -0.526 0.988 15.663 -28.545 13.881 11 1.012 -1.384 1.372 1.780 -1.290 0.510 12 0.699 -0.771 1.073 14.812 -27.157 13.345 13 1.294 -2.408 2.114 35.363 -66.192 31.829 14 1.103 -1.893 1.790 2.430 -2.684 1.255 15 0.587 -0.901 1.314 38.504 -72.146 34.642 16 0.774 -0.826 1.052 18.992 -34.864 16.872 17 0.718 -0.758 1.040 1.916 -1.485 0.569 18 0.582 -0.453 0.871 20.612 -37.836 18.224 19 0.893 -1.434 1.541 62.603 -116.990 55.382 20 0.973 -1.509 1.537 2.378 -2.473 1.095 21 0.596 -0.790 1.194 50.356 -94.280 44.923 22 3.204 -6.657 4.454 -93.692 173.480 -78.791 23 1.407 -2.755 2.347 3.604 -5.373 2.769 24 0.981 -1.939 1.958 -105.370 196.900 -90.532 25 1.359 -1.750 1.391 13.009 -23.519 11.510 26 1.693 -2.370 1.677 2.928 -2.975 1.047 27 1.185 -1.352 1.167 21.742 -39.716 18.973 28 3.251 -6.151 3.900 44.193 -82.440 39.247 29 1.604 -2.696 2.092 4.347 -5.818 2.470 30 1.440 -2.389 1.948 53.489 -99.844 47.355 Note: For Q0 / P0 ≈ 0, the values of zQ, iQ, cQ are irrelevant. Q(V) = 0 constant, may be used.  89  Additional Experimental Results for Network PV Curves of Single IM The experimental and calculated results from Fig. 3-12 and Fig. 3-13 were also obtained for two other IM’s specified in Appendix F. They are depicted in Fig. C-1 and Fig. C-2. PV curve theoretical and experimental. Induction Motor #3 (IM3)  160  140  120  80 Calculated  60  1 Xth  π60(Lth)  Xth = 2 40  E = 160V Lth = 51.2mH at 1A,  20  for each inductor in series  in series  2 Xth  in series  3 Xth  in series  4 Xth  in series  5 Xth  in series  6 Xth  in series  0 0  50  100  150  200  P (W)  Fig. C-1. PV curve theoretical and experimental, induction motor #3.  PV curve theoretical and experimental. Induction Motor #4 (IM4)  160  140  120  V (Volts)  V (Volts)  100  100  80 Calculated  60  π60(Lth)  Xth = 2 40  E = 160V Lth = 51.2mH at 1A,  20  for each inductor in series  1 Xth  in series  2 Xth  in series  3 Xth  in series  4 Xth  in series  5 Xth  in series  6 Xth  in series  0 0  50  100  150  200  P (W)  Fig. C-2. PV curve theoretical and experimental, induction motor #4.  90  Appendix D. Additional Results from Chapter 4 Miscellaneous Load Combinations The 3-phase IM and some of the previous experimental loads were connected in parallel to get combined loads characteristics, as well as simulated network PQV curves, PV curves and QV curves shown in Fig. D-1, Fig. D-2 and Fig. D-3, respectively. The percentages indicated on the load combination are with respect to the sum of nominal real powers of each load. Results for exponential and polynomial models are summarized in Table D-1 and Table D-2. The reference voltage is the nominal per-phase V0 = 120Vrms . Table D-1. Experimental miscellaneous combinations load characteristics, exponential. No. 1 2 3 4 5 6 7 8 9 10  % of total P0 at V=V0 IM 79% + CFL 21% IM 65% + CFL 35% IM 80% + INC 20% IM 62.5% + INC 37.5% IM 50% + INC 50% IM 33.3% + INC 66.7% IM 58% + RES 42% IM 33.3% + RES 66.7% IM 77% + FLE 23% IM 52% + FLE 48%  P0 (W) 120.2 150.7 122.4 164.6 196.4 296.2 173.2 304.7 131.6 196.3  tan φ0 = Q0 / P0 0.721 0.479 0.842 0.639 0.525 0.350 0.613 0.354 0.710 0.406  α  β  0.46 0.54 0.56 0.72 0.92 1.13 0.94 1.40 0.60 0.90  3.03 3.60 2.50 2.38 2.49 2.50 2.47 2.46 2.66 3.02  Table D-2. Experimental miscellaneous combinations load characteristics, polynomial. Type of load 1 2 3 4 5 6 7 8 9 10  tan φ0 = Q0 / P0 0.721 0.479 0.842 0.639 0.525 0.350 0.613 0.354 0.710 0.406  zP  iP  cP  zQ  iQ  cQ  0.997 0.880 1.066 0.968 0.850 0.728 1.129 1.07 0.722 0.547  -1.273 -0.989 -1.307 -0.938 -0.590 -0.187 -1.068 -0.588 -0.650 -0.062  1.277 1.110 1.240 0.970 0.740 0.459 0.939 0.517 0.929 0.516  4.369 5.034 3.872 3.829 3.789 3.735 3.903 3.773 4.306 5.150  -5.437 -6.289 -4.870 -4.791 -4.721 -4.620 -4.946 -4.718 -5.560 -6.878  2.067 2.255 1.997 1.962 1.932 1.886 2.043 1.945 2.254 2.728  91  φ = 0o  7  φ = 90o (lag)  3  5  |V| (Vrms)  1 8  9  140  120  100  6 10  80  2  60  4  40 "cliff" of  Q (VAR)  Thevenin  P (W)  surface  0 200  200  400  400  600 800 1000  Fig. D-1. Network PQV curves for miscellaneous combinations of loads.  140  5 120  9  81  3  |V| (Vrms)  100  6  80  7  QV "nose" point  60  10 4  40  2  20  0 0  100  20  200  300  400  500  600  700  800  900  P (W) Fig. D-2. Network PV curves for miscellaneous combinations of loads.  92  0  140  5  120  2  |V| (Vrms)  100  9  8 6 10  80  3  60  40  1 7 4  PV "nose" point  20  0 0  100  200  300  400  Q (VAR)  Fig. D-3. Network QV curves for miscellaneous combinations of loads.  Theoretical IM Load Characteristic The static load characteristic of an IM was also calculated using the equivalent circuit parameters of Fig. 4-3 and the curves of Fig. 4-5, where the machine friction from Appendix G is included as part of the mechanical load. The exponential model curvefitting coefficients were compared with the experimental results from Section 4.4.1.4, and shown in Table D-3 and Fig. D-4. A very good match can be observed.  Table D-3. Calculated and experimental IM load characteristic, exponential. IM 3-ph static load characteristic Experimental Calculated  P0 (W) 98.3 90.2  Q0 (VAR) 102.7 106.5  93  V0 (Vrms) 120 120  α  β  0.30 2.41 0.18 1.97  P (W)  200  100  0 0.6  0.7  0.8  0.9  1  1.1  1.2  1.1  1.2  Q (VAR)  200 Theoretical points and curve fitting Experimental points and curve fitting 100  0 0.6  0.7  0.8  0.9  1  V / V 0  Fig. D-4. Calculated and experimental single IM load characteristic  Building Local Voltage Stability Assessment with Exponential Model The local VSA of the building from Section 4.5.2 was performed again, but using the exponential model coefficients in (4-13), (4-16) and (4-17). The results do not change significantly. The voltage stability limit would occur at a demand value k = 143 , instead of k = 133 with the polynomial model prediction. The exponential and polynomial models diverged more when going farther from the reference voltage V0 = 120Vrms .  Zth|  Network PV curve if doubling | 120  Network PV curve for  Zth = j0.058ohm, E = 126V  k = 11  100  "nose" of PV  |V| (Vrms)  (Minimum daily demand)  80  k = 19 60  (Maximum daily demand)  k = 143  40  (Voltage stability -loadability- limit)  20  0 0  20  40  P (kW)  60  80  Fig. D-5. Building network PV curve and exponential load characteristics for VSA.  94  Appendix E. Specifications of Equipment from Chapter 2 Induction Motor Nameplate Ratings - Marathon Electric G044; 208-230/460VL-L, 3 phase, 1 / 4 HP; 1725 RPM.  Variable Frequency Drive Nameplate Ratings - Lenze-Ac Tech ESV371N02YXB; 240VL-L, 1 / 2 HP, 3 phase input and output, 6 kHz carrier signal frequency, constant V/Hz control with no speed and/or torque feedback.  95  Appendix F. Specifications of Equipment from Chapter 3 Induction Motors Parameters Table F-1. Induction machines parameters. Marathon Y534 Y592 G063 G044 Electric (IM1) (IM2) (IM3) (IM4) 5.3 9.9 4.50 6.30 Rs (Ω) 7.6 6.5 7.37 5.82 Xs (Ω) 4.7 8.3 0.86 5.74 Rr (Ω) 7.6 6.5 7.37 5.82 Xr (Ω) 96.6 122 124 137 Xm (Ω) Pnom (HP) 1/2 1/4 1/2 1/4 (All machines: 208-230VL-L RMS, 3 phase, 60 Hz, 4 poles)  Iron Core Inductors (each) Nominal rating:  L = 40 mH @ 3 A,60 Hz Real rating, saturation effects accounted: L( I )average = 51.2 ⋅ I RMS @ 60 Hz −0.24 mH X 60 Hz ≈ 15 R60 Hz ⇒ X th ≈ Z th  Variable Frequency Drive Nameplate Ratings - Lenze-Ac Tech ESV371N02YXB. 240VL-L, 3 phase input and output, 1 / 2 H.P. 6 kHz modulation frequency. Constant V/Hz control (60Hz base).  96  Appendix G. Specifications of Equipment from Chapter 4  Induction Motor 3-Phase Parameters - Marathon Electric G044; 3-phase, 208-230VL-L, 1/4 HP, 1725RPM, 60% efficiency (internal friction torque ≈ 0.20Nm), Rs = 9.9, Xs = 6.5, Rr = 8.3, Xr = 6.5, Xm = 122 (ohms).  (  )  - Mechanical load torque: 1.0 Nm ⋅ ω rm ω syn .  Other Loads Nominal Parameters - Induction motor single phase (IM-1PH): Marathon Electric G114; 1-phase, 120V, 1/4 HP, 1735RPM. - Power resistor (RES): Biddle Lubritact Rheostat, 1250W max. - Electric stove (STOVE): Toastess International THP-11S, 1000W. - variable frequency drive (VFD - CL / OL): Lenze-AcTech ESV371N02YXB; 3-phase, 208-240V, 0.50HP, 6 kHz modulation frequency, constant V/Hz. - Incandescent lamps (INC): Phillips Duramax, 25W, 60W, 100W. - Compact fluorescent lamps (CFL): Phillips Marathon, 27W. - Magnetic ballast fluorescent lamp (FLM): GE UCF28 18W (2005). - Electronic ballast fluorescent lamps (FLE): GE UCF28 (2008), 18W; and GE UCF18P (2008), 15W. - Personal computer (PC): Generic, Pentium IV, 3GHz, 400W max. power supply.  97  Appendix H. Vita Leon Max Vargas received the B.Sc. degree in Mechanical Engineering from Tecnologico de Monterrey, campus Mexico City, in 2003. From 2004 to 2006, he worked for Delphi Automotive Systems, Juarez, Mexico, as a Product Design Engineer in the Actuators Competency Group. He is currently pursuing the M.A.Sc. degree in Electrical Engineering at the University of British Columbia - Vancouver, Canada. His research interest is in the field of load modeling for voltage stability. He will join Schneider Electric R&D at Monterrey, Mexico, in the summer of 2009, as an Electromagnetic Specialist in the Analytics and Innovation Group.  98  

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