AGGREGATION OF VOLTAGE AND FREQUENCY DEPENDENT ELECTRICAL LOADS by Kwok-Wai Louie B.Sc., Simon Fraser University, 1989 B.A.Sc., The University of British Columbia, 1993 M.A.Sc., The University of British Columbia, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1999 © Kwok-Wai Louie, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of jlX^UAL AUV CdAA?(Jj&l 'SllfaH&t&H^ The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ABSTRACT Electrical loads play a very important role in the behaviour of an electric power sys-tem. Since there is a tremendous number of different loads in the system, representing each load with its own model becomes impractical for system level studies. This thesis deals with the issue of aggregating loads to simplify system level studies. Six new and ac-curate aggregate static load models, a novel EMTP based load model, and four very accu-rate aggregate induction machine models have been developed. The proposed aggregate load models are voltage and frequency dependent and ac-commodate the different data formats of individual loads. By including the information of voltage and frequency dependence, the models can be used in larger ranges of studies than the conventional aggregate static load models, thus resulting in more accurate repre-sentations. The valid voltage range of the models is about 75% to 125% of rated voltage and, the valid range of frequency of the models is about 85% to 115% of rated frequency. The proposed EMTP load model represents a load with basic circuit elements. The model consists of two varied turns ratio transformers and two varied admittance RLC cir-cuits. It represents the voltage dependence and the frequency dependence of a load sepa-rately, resulting in a much simpler load representation than a conventional load model. The model not only improves the accuracy of load representations, bus also broadens the EMTP application in studies other than the transient analyses, such as power flow studies. The proposed aggregate induction machine models have been developed based on the specifications and circuit parameters of individual machines. Since the specification of the machines are the most basic information of the devices, they provide a natural and accurate representation of the machines. The circuit parameters of the machines reflect the behaviour of the devices, they can be used to compose the machines under high and low frequencies, resulting in a simple and accurate machine representation. These two ii different aggregate models accommodate the data availability of individual machines. To verify the validity of the proposed load models, field tests from published litera-ture are compared with computer simulations. The simulation results of some test systems obtained i) by the proposed load models, ii) by the conventional load models, and iii) by solving the original systems without aggregating their circuit components are also com-pared. The results of these comparisons prove that the proposed load models can repre-sent loads in power systems more accurately than the existing load models. The proposed load models have been developed into computer software packages with the high level computer programming language Ada 95. iii TABLE OF CONTENTS ABSTRACT • ii TABLE OF CONTENTS iv LIST OF FIGURES viii LIST OF TABLES xvii ACKNOWLEDGEMENT xix Chapter One INTRODUCTION 1 1.1 Overview of Load Modelling 1 1.1.1 Need for Accurate Load Modelling 1 1.1.2 Importance of Aggregating Loads 4 1.1.3 Previous Work 5 1.1.3.1 Static Load Models 5 1.1.3.2 Dynamic Load Models 8 1.2 Summary of this Thesis Work 10 1.3 Thesis Organization 12 Chapter Two AGGREGATION OF STATIC LOADS 13 2.1 Basic Terminologies in Load Modelling 13 2.2 Aggregation of Static Loads at a Same Bus Bar 14 2.2.1 Aggregate Model with Exponential Functions 15 2.2.2 Aggregate Model with IEEE Recommended Representation 21 2.2.3 Aggregate Model with Polynomials of Voltage 28 2.2.4 Generalization of the Modelling Techniques 30 iv 2.3 Aggregation of Static Loads at Different Bus Bars 35 2.3.1 Calculation of Load Terminal Voltages 3 5 2.3.2 Aggregation of Static Loads 37 2.3.3 Generalization of the Modelling Technique 41 2.4 Aggregate Static Load Computer Program 46 2.5 General Discussion of the Results 46 Chapter Three EMTP BASED LOAD REPRESENTATION 49 3.1 EMTP Basic Concepts 49 3.2 Conversion of Load Data 51 3.3 Representation of a Function by its Periodic Extension 55 3.3.1 Basic Concepts of the Fourier Method 55 3.3.2 Periodic Extension of a Function 57 3.3.3 Locations of the Extreme Points 58 3.3.4 Varied-Width Moving Window in the Time Frame 62 3.4 Calculations of Network Quantities 63 3.5 EMTP Based Load Computer Program 66 3.6 Network Simulations 67 3.7 General Discussion of the Results 79 Chapter Four AGGREGATION OF MACHINES WITH SPECIFICATIONS 81 4.1 Basic Concepts of Machines 81 4.2 Aggregation of Machines at a Same Bus Bar 85 4.2.1 Manufacturer Specification of a Machine 85 4.2.2 Electrical Specification of the Aggregate Machine 86 • v 4.2.3 Mechanical Specification of the Aggregate Machine 88 4.2.4 Additional Specification of the Aggregate Machine 91 4.2.5 Mechanical Load of the Aggregate Machine 92 4.3 Aggregation of Machines at Different Bus Bars 93 4.3.1 Solution of a Network 93 4.3.2 Machine Specification Adjustment 98 4.3.3 Specification of the Aggregate Machine 100 4.3.4 Mechanical Load of the Aggregate Machine 102 4.4 Aggregation of Network Impedances 103 4.5 Aggregate Machine Data Conversion Computer Program 104 4.6 Network Simulations 105 4.6.1 Simulation Test 4.1 105 4.6.2 Simulation Test 4.2 110 4.6.3 Simulation Test 4.3 116 4.6.4 Simulation Test 4.4 121 4.7 General Discussion of the Results 127 Chapter Five AGGREGATION OF MACHINES WITH PARAMETERS 128 5.1 Aggregation of Machines at a Same Bus Bar 128 5.2 Aggregation of Machines at Different Bus Bars 134 5.3 Mechanical Parameters of the Aggregate Machine 143 5.3.1 Machines at a Same Bus Bar 143 5.3.2 Machines at Different Bus Bars 143 5.4 Aggregation of Network Impedances 143 vi 5.5 Aggregate Machine Data Conversion Computer Program 144 5.6 Network Simulations 145 5.6.1 Simulation Test 5.1 145 5.6.2 Simulation Test 5.2 151 5.6.3 Simulation Test 5.3 156 5.6.4 Simulation Test 5.4 162 5.6.5 Simulation Test 5.5 . 167 5.6.6 Simulation Test 5.6 172 5.7 General Discussion of the Results 178 Chapter Six CONCLUSIONS AND RECOMMENDATIONS 179 REFERENCES 181 APPENDIX A Generation of the Input Data for the Induction Machine Model in the EMTP 186 vii LIST OF FIGURES Figure 1.1 Voltage dependence of the current and apparent power of a static load Figure 1.2 A static load model including network effects Figure 1.3 A network with static loads Figure 1.4 Two port network for a composite load Figure 2.1 A group of static loads connected to a same bus bar and their aggregate static load model Figure 2.2 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.1 Figure 2.3 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.2 Figure 2.4 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.2 Figure 2.5 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.3 Figure 2.6 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.4 Figure 2.7 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.4 Figure 2.8 Network with static loads connected to different bus bars and the aggregate network model Figure 2.9 Network with static loads at different bus bars in Test 2.5 Figure 2.10 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.5 Figure 2.11 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.5 Figure 2.12 Network with static loads at different bus bars in Test 2.6 viii Figure 2.13 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.6 45 Figure 2.14 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.6 46 Figure 2.15 Flowchart of the aggregate static load program 47 Figure 3.1 Models of the basic elements in different time domains 52 Figure 3.2 EMTP based representation of a static load 54 Figure 3.3 Voltage with purely sinusoidal wave-form 60 Figure 3.4 Voltage with sinusoidal wave-form of the variable frequency 60 Figure 3.5 Voltage of sinusoidal shape with many extreme points 61 Figure 3.6 Voltage with the variable amplitude and many extreme points 61 Figure 3.7 Varied-width moving window used to determine the subsections of a voltage with variable voltage amplitude and frequency 64 Figure 3.8 Flowchart of the EMTP based load program 66 Figure 3.9 Real and reactive powers absorbed by the fluorescent lighting unit with the variation of the voltage amplitude in Test 3.1 68 Figure 3.10 Real and reactive powers absorbed by the induction motor unit with the variation of the voltage amplitude in Test 3.2 68 Figure 3.11 Network with static loads at different bus bars in Test 3.3 69 Figure 3.12 Source voltage and current in a steady state in Test 3.3 72 Figure 3.13 Terminal voltage and current of the loads at bus bar 1 in a steady state in Test 3.3 72 Figure 3.14 Terminal voltage and current of the loads at bus bar 2 in a steady state in Test 3.3 73 Figure 3.15 Real and reactive powers absorbed by the loads at bus bar 1 in a steady state in Test 3.3 73 ix Figure 3.16 Real and reactive powers absorbed by the loads at bus bar 2 in a steady state in Test 3.3 74 Figure 3.17 Source voltage and current with the variation of voltage amplitude in Test 3.3 74 Figure 3.18 Terminal voltage and current of the loads at bus bar 1 with the variation of voltage amplitude in Test 3.3 75 Figure 3.19 Terminal voltage and current of the loads at bus bar 2 with the variation of voltage amplitude in Test 3.3 75 Figure 3.20 Real and reactive powers absorbed by the loads at bus bar 1 with the variation of voltage amplitude in Test 3.3 76 Figure 3.21 Real and reactive powers absorbed by the loads at bus bar 2 with the variation of voltage amplitude in Test 3.3 76 Figure 3.22 Source voltage and current with the variation of frequency in Test 3.3 77 Figure 3.23 Terminal voltage and current of the loads at bus bar 1 with the variation of frequency in Test 3.3 77 Figure 3.24 Terminal voltage and current of the loads at bus bar 2 with the variation of frequency in Test 3.3 78 Figure 3.25 Real and reactive powers absorbed by the loads at bus bar 1 with the variation of frequency in Test 3.3 78 Figure 3.26 Real and reactive powers absorbed by the loads at bus bar 2 with the variation of frequency in Test 3.3 79 Figure 4.1 Electrical representation of an induction machine 82 Figure 4.2 A group of induction machines connected to a same bus bar and their aggregate model 87 Figure 4.3 Network with induction machines connected to different bus bars and the aggregate network model 95 Figure 4.4 Torque-speed characteristics of an induction machine at some selected terminal voltages 97 Figure 4.5 Flowchart of aggregate machine data conversion program 104 x Figure 4.6 Network with induction machines connected to a same bus bar in Test 4.1 Figure 4.7 Terminal voltage and current of phase-a of the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.8 Load and short circuit currents of phase-a of the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.9 Load currents of phase-b and phase-c of the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.10 Power flowing into phase-a of the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.11 Power flowing into phase-b of the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.12 Power flowing into phase-c of the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.13 Total power flowing into the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs Figure 4.14 Network with induction machines at two different bus bars in Test 4.2 Figure 4.15 Terminal voltage and current of phase-b of the network in Test 4.2 when a two-phase-to-ground fault occurs Figure 4.16 Load and short circuit currents of phase-b of the network in Test 4.2 when a two-phase-to-ground fault occurs Figure 4.17 Load currents of phase-a and phase-c of the network in Test 4.2 when a two-phase-to-ground fault occurs Figure 4.18 Power flowing into phase-a of the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs Figure 4.19 Power flowing into phase-b of the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs Figure 4.20 Power flowing into phase-c of the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs xi Figure 4.21 Total power flowing into the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs Figure 4.22 Network with induction machines at three different bus bars in Test 4.3 Figure 4.23 Terminal voltage and current of phase-c of the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.24 Load and short circuit currents of phase-c of the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.25 Load currents of phase-a and phase-b of the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.26 Power flowing into phase-a of the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.27 Power flowing into phase-b of the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.28 Power flowing into phase-c of the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.29 Total power flowing into the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs Figure 4.30 Network with induction machines at four different bus bars in Test 4.4 Figure 4.31 Terminal voltage and current of phase-b of the network in Test 4.4 when a single-phase-to-ground fault occurs Figure 4.32 Load and short circuit currents of phase-b of the network in Test 4.4 when a single-phase-to-ground fault occurs Figure 4.33 Load currents of phase-a and phase-c of the network in Test 4.4 when a single-phase-to-ground fault occurs Figure 4.34 Power flowing into phase-a of the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs Figure 4.35 Power flowing into phase-b of the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs xii Figure 4.36 Power flowing into phase-c of the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs 126 Figure 4.37 Total power flowing into the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs 126 Figure 5.1 A group of single-and double-cage-rotor induction machines at a same bus bar \ 129 i Figure 5.2 Circuit representations of single-and double-cage rotor induction machines at both high and low frequencies 129 Figure 5.3 Circuit representations of the aggregate induction machine at different frequencies 130 Figure 5.4 Network with single- and double-cage-rotor induction machines a different bus bars and the aggregate network 135 Figure 5.5 Representations of the network with single- and double-cage- rotor induction machines at different bus bars at high and low frequencies 136 Figure 5.6 Circuit representations of the aggregate network at high and low frequencies 136 Figure 5.7 Flowchart of the aggregate machine data conversion program 144 Figure 5.8 Network with single- and double-cage-rotor induction machines connected to a same bus bar in Test 5.1 147 Figure 5.9 Terminal voltage and current of phase-a of the network in Test 5.1 when a two-phase-to-ground fault occurs 147 Figure 5.10 Load and short circuit currents of phase-a of the network in Test 5.1 when a two-phase-to-ground fault occurs 148 Figure 5.11 Load currents of phase-b and phase-c of the network in Test 5.1 when a two-phase-to-ground fault occurs 148 Figure 5.12 Power flowing into phase-a of the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs 149 Figure 5.13 Power flowing into phase-b of the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs 149 xiii Figure 5.14 Power flowing into phase-c of the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs Figure 5.15 Total power flowing into the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs Figure 5.16 Network with single-and double-cage-rotor induction machines at four different bus bars in Test 5.2 Figure 5.17 Terminal voltage and current of phase-a of the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.18 Load and short circuit currents of phase-a of the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.19 Load currents of phase-b and phase-c of the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.20 Power flowing into phase-a of the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.21 Power flowing into phase-b of the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.22 Power flowing into phase-c of the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.23 Total power flowing into the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs Figure 5.24 Network with single-cage-rotor induction machines at a same bus bar in Test 5.3 Figure 5.25 Terminal voltage and current of phase-b of the network in Test 5.3 when a three-phase-to-ground fault occurs Figure 5.26 Load and short circuit currents of phase-b of the network in Test 5.3 when a three-phase-to-ground fault occurs Figure 5.27 Load currents of phase-a and phase-c of the network in Test 5.3 when a three-phase-to-ground fault occurs Figure 5.28 Power flowing into phase-a of the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs xiv Figure 5.29 Power flowing into phase-b of the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs Figure 5.30 Power flowing into phase-c of the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs Figure 5.31 Total power flowing into the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs Figure 5.32 Network with single-cage-rotor induction machines at four different bus bars in Test 5.4 Figure 5.33 Terminal voltage and current of phase-a of the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.34 Load and short circuit currents of phase-a of the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.35 Load currents of phase-b and phase-c of the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.36 Power flowing into phase-a of the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.37 Power flowing into phase-b of the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.38 Power flowing into phase-c of the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.39 Total power flowing into the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs Figure 5.40 Network with double-cage-rotor induction machines connected to a same bus bar in Test 5.5 Figure 5.41 Terminal voltage and current of phase-a of the network in Test 5.5 when a single-phase-to-ground fault occurs Figure 5.42 Load and short circuit currents of phase-a of the network in Test 5.5 when a single-phase-to-ground fault occurs Figure 5.43 Load currents of phase-b and phase-c of the network in Test 5.5 when a single-phase-to-ground fault occurs xv Figure 5.44 Power flowing into phase-a of the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs Figure 5.45 Power flowing into phase-b of the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs Figure 5.46 Power flowing into phase-c of the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs Figure 5.47 Total power flowing into the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs Figure 5.48 Network with the double-cage-rotor induction machines at four different bus bars in Test 5.6 Figure 5.49 Terminal voltage and current of phase-c of the network in Test 5.6 when a single-phase—to-ground fault occurs Figure 5.50 Load and short circuit currents of phase-c of the network in Test 5.6 when a single-phase-to-ground fault occurs Figure 5.51 Load currents of phase-a and phase-b of the network in Test 5.6 when a sirigle-phase-to-ground fault occurs Figure 5.52 Power flowing into phase-a of the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs Figure 5.53 Power flowing into phase-b of the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs Figure 5.54 Power flowing into phase-c of the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs Figure 5.55 Total power flowing into the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs xvi LIST OF TABLES Table 2.1 Data of test system in RGEC in Test 2.1 20 Table 2.2 Data of static loads in Test 2.2 24 Table 2.3 Data of static loads in Test 2.2 25 Table 2.4 Data of static loads in Test 2.3 29 Table 2.5 Data of static loads in Test 2.4 33 Table 2.6 Data of static loads in Test 2.4 33 Table 2.7 Data of static loads in Test 2.5 39 Table 2.8 Data of network impedances in Test 2.5 39 Table 2.9 Data of static loads in Test 2.6 44 Table 2.10 Data of static loads in Test 2.6 44 Table 2.11 Data of network impedances in Test 2.6 44 Table 3.1 Data of static loads in Tests 3.1 and 3.2 67 Table 3.2 Data of static loads in Test 3.3 70 Table 4.1 Data of induction machines in Test 4.1 106 Table 4.2 Data of network impedances in Test 4.1 106 Table 4.3 Data of induction machines in Test 4.2 111 Table 4.4 Data of network impedances in Test 4.2 111 Table 4.5 Data of induction machines in Test 4.3 117 Table 4.6 Data of network impedances in Test 4.3 117 Table 4.7 Data of induction machines in Test 4.4 122 Table 4.8 Data of network impedances in Test 4.4 123 xvii Table 5.1 Data of single- and double-cage-rotor induction machines in Test 5.1 146 Table 5.2 Data of network impedances in Test 5.1 146 Table 5.3 Data of single-and double-cage-rotor induction machines in Test 5.2 152 Table 5.4 Data of network impedances in Test 5.2 152 Table 5.5 Data of single-cage-rotor induction machines in Test 5.3 157 Table 5.6 Data of network impedances in Test 5.3 157 Table 5.7 Data of single-cage-rotor induction machines in Test 5.4 163 Table 5.8 Data of network impedances in Test 5.4 163 Table 5.9 Data of double-cage-rotor induction machines in Test 5.5 168 Table 5.10 Data of network impedances in Test 5.5 168 Table 5.11 Data of double-cage-rotor induction machines in Test 5.6 173 Table 5.12 Data of network impedances in Test 5.6 173 xviii ACKNOWLEDGEMENT Thanks are deeply expressed to all the people who have assisted me in the accom-plishment of this thesis work. Particularly, I would like to thank those who have made the most valuable contributions to the success of this thesis. I deeply express my thanks to my dear wife Sally Choo for her tremendous love, ex-traordinary patience, constant and strong support, and continuous encouragement. I would like to dedicate all my achievements to my dear parents for their love and support. I am indebted to my supervisor, Dr. J. R. Marti, whom I respect and admire tremen-dously, for his constant guidance, constant encouragement, strong support, and indispen-sable role in the success of this thesis. I am very thankful to Dr. H. W. Dommel, whom I respect and admire tremendously, for all his valuable time in patiently teaching me how to use the induction machine model and the machine data conversion routine in Micro-tran, the U.B.C version of the EMTP. I am also very grateful for his encouragement and strong support in the project. I am very grateful to Dr. H. Jin for his efforts in guiding me on how to use his commercial version of Psim, his encouragement, and his strong sup-port. I would like to thank Dr. T. Niimura for his encouragement and strong support. I am also very thankful to Dr. W.G. Dunford for his continuous support in my teaching assistantships in the undergraduate power laboratories during these years. I express my gratitude to Dr. L. M. Wedepohl who was the first person suggesting and encouraging me to pursue the Ph.D. program. Great thanks are expressed to the Natural Sciences and Engineering Research Coun-cil (NSERC) for the Postgraduate Scholarships Award. The University of British Co-lumbia is gratefully acknowledged for the Graduate Fellowships Award. xix I would also like to thank the fellow students in the power group of Electrical and Computer Engineering Department at U.B.C. for their warm and durable friendships. Thanks are also expressed to all the people in the Electrical and Computer Engineering Department at U.B.C. for their hard work to create and maintain an excellent academic environment. xx CHAPTER ONE INTRODUCTION Accurate modelling of electric power networks is very important in power system planning, operation, and control. Basically, an electric power system consists of genera-tion units, transmission units, distribution units, and electrical loads. Models for the gen-eration and transmission units are well developed [1, 2, 3]. However, not much attention has been paid to the modelling of distribution systems and electrical loads, which have millions of different devices such as motors, lighting, and electrical appliances [3]. The performance of these loads plays a major role in the power system's behaviour [4, 5, 6]. 1.1 Overview of Load Modelling Because of the wide variety and large number of loads, aggregate load modelling is an important and challenging task in electric power system studies. In order to have a bet-ter understanding of load modelling, it is worth reviewing the related basic background. 1.1.1 Need for Accurate Load Modelling When accurate load data are not available, pessimistic load representations are com-monly used in order to provide some safety margin in system designs or operating limits [6]. However, this practice can be dangerous because it is impossible to select representa-tions that will be pessimistic for all parts of the system or all test conditions [6]. The im-pact of different load models on the results of power system studies can be very significant. For instance, it has been shown that designs in which the heating load of the residential part was represented by a constant impedance during winter resulted in over expenditures in transfer limit capacity in the Pacific Northwest System [6]. On the other hand, optimistic load representations may push power systems to operate beyond their ac-tual limits, resulting in vulnerable major collapses. For example, in 1987 the Tokyo 1 power system collapse was partly caused by the underestimation of the influence of the reactive power consumed by the air conditioning loads [6]. The influence of loads on transient stability is one of the major concerns of power system designs and operations [4, 7]. Usually, the load characteristics have a strong im-pact on the system's stability limits, such as maximum power transfer and maximum clearing time, thus showing that the stability limits are closely related to load representa-tions [4, 7, 8, 9, 10, 11, 12, 13, 14]. Traditionally, the real and reactive powers of static loads are represented by algebraic functions of voltage and frequency, such as exponentials or polynomials. In an exponen-tial representation, if the exponent is two, the model describes a constant impedance load; if the exponent is one, the model gives the constant current load characteristic; if the ex-ponent is zero, the model is called the constant power load representation. When a static load representation changes from a constant impedance type to a constant power type, for example, the stability limits can decrease dramatically in cases where major loads are far away from the generation units, while these limits can actually increase when the loads are near the generating plants [3]. When short circuits occur, large and rapid voltage excursions are generated during the initial fault inception and slower voltage excursions are produced during the first power angle swing [3]. The power consumed by the loads during this period is affected by the load responses to the voltage variations. These changes influence the generation-load balance and, therefore, affect the magnitude of the angular excursion and the first swing stability of the system [3]. Consider, for example, a case where the loads have a constant current characteristic. A constant impedance load model gives a power con-sumption which varies with the square of the voltage, indicating a lower value than the actual power during the voltage depression period [3]. As a result, the loads near 2 accelerating machines give pessimistic results [3]. Loads remote from the accelerating machines (near decelerating machines); however, give an optimistic impact on the results [3]. On the other hand, since the constant power load model holds the power at a higher value during the voltage depression, the model has the opposite impact on the results [3]. Since short circuits are the most severe phenomena in power system operations, the load characteristics during a fault should deserve special attention. During a short circuit, bus voltages are depressed, power balance is lost, and the generators accelerate or decelerate depending on the nature of the network [3, 14]. Load characteristics can have a signifi-cant impact on the behaviour of individual generators during the fault and, therefore, on the eventual stability of the system [3]. Voltage stability analyses require more accurate load models and simple static load representations are not suitable for these kinds of studies. For instance, in the 1983 black-out of the Swedish power system, the Swedish State Power Board had to use the detailed load models to study the problem [6]. At first, the attempt to use simple static load mod-els could not explain the voltage collapse scenario [6]. It was then realized that at volt-ages lower than 0.8 p.u. the load would not follow the characteristics that are traditionally used in stability studies [6]. As another example, voltage stability studies done at Ontario Hydro for the Ottawa area emphasized the importance of accurate dynamic load model-ling [6]. In a study of the impact of losing one of the transmission lines feeding the area, a substantial difference between the results obtained with static and dynamic models was found [6]. Load characteristics also depend on the bus frequency. When a fault occurs, a brief frequency excursion during the power angle swing makes the frequency characteristics of loads close to generators very important to the behaviour of the machines. In a power sys-tem, both load frequency characteristics and generator amortisseurs are the major sources of damping [3]. However, the amortisseurs influence the oscillations among local 3 generators because their effects rapidly decrease with increasing system impedance be-tween the generators. Consequently, load frequency characteristics provide an important damping during a large disturbance [3]. It is, therefore, important to consider the fre-quency dependence of loads in transient stability studies. Since inaccurate load models may produce results that miss significant phenomena, closely representing loads will be beneficial to power system planning and operation [6]. In planning studies, having accurate load models can avoid the expense of system modifi-cations and equipment additions, and can also avoid system inadequacies that may be re-sulted in costly operating limitations [6]. In operating studies, having excellent load models may prevent system emergencies resulting from overly optimistic operating limits [6]. Efforts directed at accurate load modelling are, therefore, of high importance in sys-tem planning and operation. 1.1.2 Importance of Aggregating Loads Normally, millions of different load components are consuming energy in an electric power system, resulting in a large sized and complicated network. Some load components such as induction machines require detailed models in some studies of the system, mak-ing a rather complex modelling process. Even if each load had a rather simple mathemati-cal representation, it would be very difficult to model each individual load component in the network separately due to the heavy computational burden. In reality, loads are usu-ally different and their mathematical models are not similar. Representing all the load de-vices with their own models, therefore, becomes impractical in large power systems [3]. In order to overcome this dilemma in power system modelling, aggregating the tremen-dous number of load components must be undertaken in order to simplify the overall sys-tem model. Since static and dynamic loads behave differently, they should be aggregated sepa-rately in order to obtain a more natural representations of the loads. Static loads, which consume a large portion of energy in a power system and have a significant impact on the system's behaviour, should be modelled more accurately in wider ranges of voltage and frequency than done in conventional power system studies. Since the EMTP technique is an excellent method for modelling detailed electric power system behaviour, EMTP based load models should also be developed. Induction motors are the major dynamic loads in an electric power system and have a great dynamic impact on the network's be-haviour during major disturbances. These machines must be aggregated accurately. 1.1.3 Previous Work In the past, some work has been done on load modelling by researchers. There are different load representations which can be classified as static and dynamic load models depending on the load characteristics. 1.1.3.1 Static Load Models A static load model is a load representation that expresses the active and reactive powers of the load at any instant of time as functions of voltage magnitude and frequency at that time instant. To model the voltage effects, the real and reactive powers of the load are expressed as exponentials or polynomials of voltage amplitude [4, 5, 6, 14, 15]. In an exponential representation, the exponent is a real number and its value determines the na-ture of the model. Figure 1.1 shows the voltage dependence of the current and the appar-ent power of a static load represented by the constant impedance, constant current, and constant power load models, respectively [16]. Combining the constant impedance, constant current, and constant power load mod-els with a polynomial results in a so called "ZIP" load model [4, 5, 6, 15, 17]. The 5 • 140 1 1 1 1 1 1 r load current (percent) load apparent power (percent) go i- 1 ••—« ' « 1 1 1 - 1 80 85 90 95 100 105 110 115 120 voltage (percent) Figure 1.1 Voltage dependence of the current and apparent power of a static load coefficients of the voltage in the "ZIP" load model depend on the nature of the loads. In order to include the effects of both the voltage and the frequency, the real and reactive powers flowing into the loads are represented by products of voltage and frequency func-tions [4, 5, 6, 15, 18]. According to reference [5], conventional static load models are constructed with the following assumptions: 1. All load components are treated as if they were connected directly in parallel at a bus, with no intervening distribution system. 2. The net reactive losses and compensation of the distribution system are treated as an-other device in parallel with the load components at the bus. 3. Each load component is characterized by a set of static parameters which consist of the power factor and the partial derivatives of the real and reactive powers with respect to the voltage magnitude and frequency at their terminals. 6 4. Single phase loads are assumed to be uniformly distributed and are represented as a balanced three phase load. However, in reality, each component sees different changes due to the voltage drops in impedances of distribution systems. In addition, the static load models are based on the power partial derivatives near the normal voltage and frequency and, therefore, the mod-els can only be presumed to be accurate for fairly narrow ranges of voltage and fre-quency. Furthermore, since the resistive portion of physical loads is not frequency dependent, the load models with the product of bus voltage and frequency functions in the "ZIP" load representation is not physically based [15]. In addition, the "ZIP" load model is limited in its flexibility because the reactive power is highly voltage sensitive [15]. It is difficult to model such a high and non-linear voltage dependence over the volt-age range of interest with the above model [15]. The characteristics of a system depend not only on the load components, but also on the network impedances. The load characteristics and the network effects primarily influ-ence the system and utilization voltages, which in turn affect the power flow. To include the network effects, some researchers, such as S. Ihara, M. Tani and K. Tomiyama used an equivalent circuit of an inductor in series with the parallel combination of an inductor and a capacitor connected to the static load as shown in Figure 1.2 [19]. However, since the loads are separately located at different bus bars by the network impedances which in-clude the resistive components, the effects of distribution system thus cannot be accu-rately represented by the above circuit structure. Researchers have developed some aggregate static load models to reduce the network sizes in power system studies. In 1972, G.J. Berg introduced an aggregate static load model which combines the effects of both network and load components as shown in Fig-ure 1.3 [18]. In this model, all the network elements are combined into a single load 7 xt V(t) Qs Qc Load PI, Ql Figure 1.2 A static load model including network effects component. Since the actual load components are connected to different bus bars with network impedances in between, Berg's model does not naturally represent this physical structure. In addition, the model treats the partial derivatives of the real and reactive pow-ers with respect to the voltage and frequency as constants. This is true only under the rated operating condition. To further reduce the work in aggregating static loads, J.R. Ribeiro and F.J. Lange developed a simplified aggregate static load model in 1982 [20]. In their model, the aggregate static load representation has the same format as those of the individual loads. That is, the real and reactive powers are the products of a voltage expo-nential function and a linear frequency function. The exponent in the exponential function is the weighted average value of the exponents of the individual loads. This model, how-ever, does not include the effects of network impedances. Omitting the influence of the network in the model can cause severe errors in some power system studies. Moreover, Ribeiro and Lange's model also treats the partial power derivatives as constant, thus re-sulting in an inaccurate representation of the loads. 1.1.3.2 Dynamic Load Models A dynamic load model expresses the real and reactive powers of a load at any instant of time as functions of the voltage amplitude and frequency at the past instant of time, and usually including the present instant [6]. Induction motors are the main work-horse in 8 electric power systems and contribute the most dynamic influence to the networks' behav-iour. Hence, these machines need special attention when load modelling is concerned. Traditionally, the electrical part of an aggregate induction motor is represented by a stan-dard equivalent circuit [21, 22, 23, 24, 25]. The circuit parameters of the aggregate ma-chine are determined by weighted functions of the apparent powers of individual machines [25]. The mechanical torque of the aggregate machine is represented by a poly-nomial of the rotor's angular speed. Dynamic loads could also be represented by transfer functions. For example, at Ontario Hydro, S.A.Y. Sabir and D.C. Lee represented a group of dynamic loads by a two port network as shown in Figure 1.4 [26]. By analyzing the system system feeder aggregate load transformer components | | | | | | | (a) original network (b) aggregate network Figure 1.3 A network with static loads P1,Q1 P2,Q2 4 VI V2 Figure 1.4 Two port network for a composite load 9 input-output relationship of the loads during system disturbances, a transfer function of the network is derived and the real and reactive powers flowing into the loads are ob-tained. This approach does not need any prior knowledge of the actual loads. However, Sabir and Lee's load model is a quasi-steady state model since it gives acceptable simula-tion results only when the changes of power systems vary very slowly. In other words, the model does not provide satisfactory simulation results of a power system under some operating conditions. 1.2 Summary of this Thesis Work In this thesis work six aggregate static load models, one EMTP based load model, and four aggregate induction machine models have been developed. The proposed models are aimed at both quasi-steady state network solutions and EMTP studies [27]. These load models improve the accuracy of traditional load representations and greatly reduce the sizes of electric power networks in system simulations. The models also accelerate the network solution process and provide added flexibility for detailed level of modelling cir-cuit elements and subsystems. The thesis focuses on the aggregation of electrical loads and the studies of load harmonics are not considered. According to the available data, aggregating static loads can be done in different ways. If the accuracy of the system study is a major issue, a more sophisticated model can be used. However, when the computational speed is more important than the precision, a simpler load representation can be applied. The general formats of the different aggregate static load models are similar and the differences among them are used to accommodate the available load data. To enlarge the voltage and frequency ranges, some correction schemes are applied in the models. Further improvement of the models' accuracy is done by making the exponents of the aggregate loads' power functions voltage and frequency dependent. The models give accurate load representations in the voltage range from about 10 75% to 125% of the rated voltage and in the frequency range from about 85% to 115% of the rated frequency. The EMTP based load model is generated by two varied RLC-networks, two varied turns ratio transformers, and a varied resistor, resulting in a voltage and frequency de-pendent load representation. The values of the RLC-network elements depend only on the frequency and are determined by admittance calculations at each time instance [4, 8, 9, 10]. The value of the varied resistor is voltage dependent and is not sensitive to the fre-quency. The transformers have varied turns ratios which are only voltage dependent [21, 22, 23, 24]. Therefore, the load model represents the voltage dependence and the fre-quency dependence separately by different circuit elements. The computation of the bus frequency is done by a varied-width moving window along the time axis. The resistance of the varied resistor and the varied turns ratios of the transformers are obtained by com-puting the voltage amplitude with the Fourier method [28, 29, 30, 31, 32, 33, 34]. Both the varied resistance and the varied turns ratio transformers are evaluated at each time step in the simulation to accommodate the variations in voltage amplitude. When aggregating induction machines, the availability of the machine data deter-mines the aggregation method. The combined induction machines are generated from ei-ther the machine specifications or the known equivalent circuit parameters. If the specifications of the individual machines are the only available information, the machines are aggregated by composing their specifications. The circuit representation of the aggre-gate machine is then obtained by converting the specifications of the aggregate machine into circuit parameters with the well established data conversion routine in Microtran (U.B.C. version of the EMTP). When the equivalent circuit parameters of the individual machines are known, the machines can be aggregated under high and low system fre-quency conditions. The aggregate induction machine models can represent the behaviour of both single- and double-cage-rotor machines. 11 1.3 Thesis Organization The derivation of the aggregate static load models for stability studies is discussed in Chapter Two. The EMTP based load model is presented in Chapter Three. In Chapter Four, the aggregation of induction machines from manufacture's specifications is dis-cussed. Chapter Five presents the aggregation of single- and double-cage-rotor induction machines using known equivalent circuit parameters. The conclusions and recommenda-tions of this thesis work are presented in Chapter Six. 12 CHAPTER TWO AGGREGATION OF STATIC LOADS A static load is a power-consuming unit whose characteristics can be represented by algebraic equations of the voltage and frequency. There are different methods to represent a static load and, therefore, different models may be developed to compose a group of static loads. In this chapter, six static load models for aggregating static loads at a same bus bar or connected to different bus bars are presented. 2.1 Basic Terminologies in Load Modelling Since there are many terminologies in load modelling, clearly defining the essential terms is of importance. In reference [6], when describing load representations, the follow-ing common terms are applied: 1. LOAD COMPONENT: A load component is an equivalent of all devices of similar type. 2. LOAD CLASS: A load class is a category of loads, such as residential, commercial, and industrial loads. 3. LOAD CHARACTERISTICS: Load characteristics that describe the behaviour of a specified load unit are a set of parameters, such as the power factor and the partial deriva-tives of the real and reactive powers of the load with respect to the voltage and frequency. 4. STATIC LOAD MODEL: A static load model is a load representation that expresses the real and reactive powers of the load at any instant of time as functions of the voltage magnitude and frequency at the same instant. 5. CONSTANT IMPEDANCE LOAD MODEL: A constant impedance load model is a 13 static load model in which the real and reactive powers of a load vary directly with the square of the voltage magnitude. 6. CONSTANT CURRENT LOAD MODEL: A constant current load model is a static load model in which the real and reactive powers of a load vary directly with the voltage magnitude. 7. CONSTANT POWER LOAD MODEL: A constant power load model is a static load model in which the real and reactive powers of a load do not vary with the voltage magnitude. 8. POLYNOMIAL LOAD MODEL: A polynomial load model is a static load model that represents the relationship between the real power of the load and the voltage magnitude and the relationship between the reactive power of the load and the voltage magnitude by polynomials. 9. EXPONENTIAL LOAD MODEL: An exponential load model is a static load model which represents the relationship between the real power of the load and the voltage mag-nitude (or frequency) and the relationship between the reactive power of the load and the voltage magnitude (or frequency) by exponential equations. 2.2 Aggregation of Static Loads at a Same Bus Bar When static loads are connected to a common bus bar in an electric power system, all the loads have the same terminal voltage as shown in Figure 2.1. As indicated in the fig-ure, the circuit representation is greatly simplified by composing the individual loads with a single equivalent. In accordance with the available data of the individual load compo-nents in the network, the loads can be aggregated in different ways. 14 loadl load2 loadn aggregate load bus point bus point Figure 2.1 A group of static loads connected to a same bus bar and their aggregate static load model 2.2.1 Aggregate Model with Exponential Functions Sometimes, the only available information of a static load component is its power factor and the partial derivatives of its real and reactive powers with respect to the voltage and frequency. Traditionally, static load i in a group of static loads is represented by its real and reactive powers as [4, 15, 35, 36] P , . = W w ) = P o i . r x ) - f e ) - (2.1.1) (2.1.2) where Poi is the real power of static load i under the rated condition; Qoi is the reactive power of static load i under the rated condition; V0 and co 0 are the rms voltage amplitude and the angular frequency under the rated con-dition, respectively; V and co are the rms voltage amplitude and the angular frequency under any operating condition, respectively; 15 Pvt, Pai, qvi, and q^ are the partial derivatives of the real and reactive powers of static load i with respect to the voltage and frequency under or very near the rated condition, respectively. On the per unit bases of voltage and frequency, equations (2.1.1) and (2.1.2) become Pi=Pi(V,(o) = PoiVP*i®P»i (2.2.1) Qi = Qi(V,co) = Q o i V ^ ^ (2.2.2) The aggregate static load model is assumed to have the similar mathematical expres-sions as those of the individual static load components. That is, the real and reactive pow-ers of the aggregate load are represented by P = P(V,(o) = P0VP*aP° (2.3.1) Q = Q(V,(o) = Q0V<>*m<"° (2.3.2) where P and Q are the real and reactive powers of the aggregate load under any operating condition, respectively; P0 and Q0 are the real and reactive powers of the aggregate load under the rated condi-tion, respectively; V and co are the per unit voltage and angular frequency, respectively; pv, pa, qv, and q^ are the partial derivatives of the real and reactive powers of the load with respect to the voltage and frequency, respectively. The real and reactive powers absorbed by the aggregate load are the sums of those of the individual loads. At the rated frequency, the real and reactive powers of the aggregate load are given by P = P(F,co0)=lf^fW (2.4.1) ,= 1 V r 0 J Q = Q(V,(»0)=2Z\%L\V^ (2.4.2) 16 where n is the number of the static loads connected to the common bus bar. The values of the exponents pvi, pai, qvi, and q m i in the conventional load model of static load i are obtained under or very near the rated operating condition. However, for each individual load, the derivatives of the real and reactive powers with respect to the voltage and frequency are functions of the voltage and frequency. In order to have the ag-gregate load model valid under other than the rated condition, the individual values of pvi, pmi, qvi, and q m i must be first adjusted. Since the adjustment of pvi is similar to those of pmi, qvi, and qm, only the modification of pvi is considered. pvi is first modified to accommodate a slightly larger voltage range and then adjusted according to the procedure outlined below to include more voltage information. The pre-liminary modification is to obtain a weighted average value of pvi based on the rates of power changes. In the voltage range from 1 to 1 - MAV p.u. with M being an integer, the partial derivatives of the real power with respect to the voltage are (/>„,) (S K ) v=v„=\ = Pvi V=Vi = l-AV = Pvi(\-L\V)P«-X (2.5) POP* V=Vm=\-MAV =pVi(l - MAV)' L e t Pvi-ad D e the weighted average value of the partial derivatives of the real power. Then the adjusted value of pvi is given by M Pvi-ad ~S J=0 t^PoAdV - i - i v=vk] Vpoi)\dVJ y=v-(2.6) Applying the binomial theorem and the series formulae on equation (2.6) yields Pvi-ad ' 1 + MAV {MAV)pV{ \+MAV(\-pvi) + (l-pviy M(2M+ l)(AV)2 6 (2.7) 17 Similarly, in the voltage range from 1 to 1 + MAV p.u., pvi.ad becomes Pvi-ad Pvi 1 MAV (MAV)pVl 1 2 + 2 l-MAV(l-pvi) + (l-pviy M(2M+ 1)(AF)2 6 (2.8) Since in the calculation of pvi.ad, the constant exponent pvj is used to compute the par-tial derivatives, errors are introduced. Further correction is needed to reduce the deviation of pvi.ad from the correct value. Let the real power functions with the exponent pvi.ad and the corrected exponent pvi.ad + Apvi.ad be Pi-ad = PoiVP^ vi-ad 1 i-cor ~ 1 oi v (2.9.1) (2.9.2) From equations (2.9.1) and (2.9.2), at a voltage of 1 - NAV p.u. with N being an inte-ger, the difference between (jf~J (^fj/^) m d i.P~J (^dJ^O i s c o m P u t e c * a s \ p j \ dV , v=l_mv. '(±\( 5P i-cor {pol)y dV V=\-NAV * Apvi-ad[-2pvi-ad(NAV) + NAV+ 1] (jA(dPj-ad (2.10) The partial derivative of the corrected real power with respect to the voltage is then given by ( 1 Vd/Vc,, VpoiJ\ 8V J v = l-NAV * (NAV)[(pvi -pvi-ad)(2pvi-ad - 1) + Pvi-ad(-Pvi-ad + 1)] + Q-Pvi-ad ~Pvi) (2.11) From equations (2.6) and (2.11), applying the binomial theorem and the series for-mulae gives the corrected value for pvi.ad as Pvi- \){NAV)b a+(X)(NAV)b (2.12) where a = 2pvi-ad ~Pvt; b = (Pvi ~Pvi-ad)(2pvi-ad ~ 1) + Pvi-ad(~P'vi-ad + !)• 18 The value of pvi.cor can be computed in the voltage range'from 1 to 1 + NAV p.u. by a similar procedure. After the exponents in the power functions of the individual loads are adjusted, the new load parameters are used to compute the parameters of the aggregate load. From equations (2.2.1), (2.2.2), (2.3.1), (2.3.2) and (2.12), we have pvVP^ =t (^Pvi-corVr-™-1 (2.13) /=1 ^r0J At V = 1 - NAV p.u., applying the binomial theorem to equation (2.13) yields (NAV)p2v - (1 + NAV)pv +MNAV) « 0 (2.14) where/ifWAF) * (NAV) ± ( ^ ) (-A2 +Ai+B^j+i [^lAr, ,=1 V^o J v y /=1 v - r 0 J Ai = 1pvi-ad —p\i\ (jj (2pvi-ad-pvi)[(pvi -Pvi-ad)(2pvi-ad ~ 1) +Pvi-ad(-pvi-ad+ 1)] (2pvi-ad -Pvi) + ( ^ ) [(Pvi -Pvi-ad)(2pvi-ad ~ 1) + Pvi-ad(~Pvi-ad + 1)] When V = 1 - NAV p.u., pv, the exponent of the real power function of the aggregate load, is obtained by solving equation (2.14). Similarly, at V = 1 + NAV p.u., pv is ob-tained from (NAV)p2v + (1 -N A V ) p v +fR(NAV) « 0 (2.15). where fR(NAV) » (NAV) ± (a2-Ai + B^+t ( ^ A , . i=i v r 0 / v ) = i \ro / To verify the aggregate static load model, Test 2.1 was performed on a test system of the Rochester Gas and Electric Corporation (RGEC) with the data given in Table 2.1 [20]. In addition, the breaking down consumer types are 30% industrial, 38% commer-cial, and 32% residential. The partial derivatives in Table 2.1 are the original values of the exponents of the power functions of the individual loads. In the simulation, these data for the individual loads are first adjusted with the correction schemes indicated above, 19 and then used to calculate the aggregate load exponents. In the test, the bus voltage was varied from 0.95 to 1.05 p.u, and the loads' real and reactive powers shown in Figure 2.2 were computed. The field test was conducted by General Electric Co. for EPRI under pro-ject 849-1 [20]. The results obtained using i) the EPRI's static load model [20], ii) the simplified load aggregate model (SLA) developed by Niagara Mohawk Power Corpora-tion [20], and iii) the load model proposed in this thesis clearly show that the proposed model can more accurately aggregate a group of static loads at a common bus bar than other models. Table 2.1 Data of test system in R G E C in Test 2.1 L O A D C O M P O N E N T S PARTIAL DERIVATIVES FRACTION OF T O T A L D E M A N D P v i qv i Pcoi industrial commercial residential incandescent light 1.55 0 0 0 0.03* 0.13 0** fluorescent light 0.96 7.38 1 -2.66 0.49 0.39 0 air conditioner 0.2 2.3 0.9 -2.67 0.21 0.4 0.31 dryer, fore, air heater 2.04 3.27 0 -2.63 0 0 0.23 refrigerator, freezer 0.77 2.5 0.53 -1.46 0 0 0.13 elect, range cooking 2 0 0 0 0 0 0.08 pump. fan. ind.motors 0.08 1.6 2.9 1.8 0 0.08 0 heatenhotwater, space 2 0 0 0 0 0 0.17 T V , computer E T C . 2 5.2 0 -4.6 0 0 0 0.03 is expected to be 0.3; ** 0 is expected to be 0.08. 20 1.05 P(V) 1 0 (P.u.) 1 0 0.95 1.40 1.20 1.00 0.80 1 1 1 _ ^ - « — field test o o o S L A + + + E P R I / U T A + . . . . . the program Q(V) (p-u.) field test 1 1 o o o S L A + + + E P R I / U T A the program 1 1 1 0.94 0.96 0.98 1.00 1.02 1.04 1.06 V(p .u . ) Figure 2.2 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.1 2.2.2 Aggregate Model with IEEE Recommended Representation According to the IEEE recommended standard static load model, the real and reac-tive powers of an individual load have the following mathematical forms [15]: Pi< = P,(V,(o) = k0l,+kuV+k2iV2 + * 3 i? P v"(l +suA(o) + k4iVP^(\ + s2iAco) (2.16.1) Qt = Qi(V,<o) = l0i + luV+l2iV2+l3iV^(\ +n,Aco) + / 4 , ^ 2 ' ( l + r2,Aco) (2.16.2) where Aco = co -co 0; koi, k,j, k2j, k3i, k4i, sli5 s2i, l o i , 1H, l 2 i , l 3 i , l 4 i , r1( and r2i are coefficients. Assuming that the aggregate static load model has the similar form as those of the in-dividual static loads results P = P(V, (o) = k0 + kx V+ k2 V2 + k3 (1 + si Aco) + h P" 2(l + ^ Aco) (2.17.1) Q = QiY, co) = l0 + h V+12 V2 + h V* (1 + r i Aco) + U Vq«(\ + r2 Aco) (2.17.2) 21 where k0, k„ k2, k3, k4, s,, s2,1 0, li, 1 2,13,14, ii and r 2 are coefficients. The real and reactive powers of the aggregate load must be the sums of those of the individual loads. At the rated frequency, equations (2.16.1), (2.16.2), (2.17.1) and (2.17.2) give ka + k\ V+ k2V2 + k 3 + k 4 = S [kol + kuV+ k2iV2 + k3iJ^"" + k4iVPv21"\ (2.18.1) i=i lo + hV+l2V2+ hW« + UVq* =E [/„( + hiV+ hiV2 + hV9vU + l4iVqv2i] (2.18.2) Since the voltage dependence of the real power is similar to that of the reactive power, only the voltage dependence coefficients in the real power function are considered in the model derivation. Comparing the terms on the left side with those on the right side of equation (2.18.1), respectively, generates k0=£koi (2.19.1) ;'=1 * i = E * u (2.19.2) i=i k2=£k2i (2.19.3) ;=i Also, comparing the terms on the left side with those on the right side of equation (2.18.1), we assume the following relations hold: k3VP*1 =£ k3iVP>» (2.20.1) i=\ k4VP* =£ k4iVP*> (2.20.2) ;'=1 At the rated voltage, equations (2.20.1) and (2.20.2) become k3=£hi (2.21.1) 22 k4=£k4i (2.21.2) J=I At voltage V, p v l and p v 2 can be obtained from equations (2.20.1) and (2.20.2) by ap-plying the solution technique described in Section 2.2.1. The frequency dependence coefficients of the aggregate load model can also be cal-culated from equations, (2.16.1), (2.16.2), (2.17.1), (2.17.2), (2.18.1), and (2.18.2). Once again, since the frequency dependence of the real power is similar to that of the reactive power, only the frequency dependence coefficients in the real power function are consid-ered. At the rated voltage, from equations (2.16.1), (2.17.1), and (2.18.1) we have n foD +si(co -C0oYJ £3;[1 +5n(co -co0)] (2.22.1) (=1 n ka\\ +^2(00 - CO0)] =S &4/[l +^2/(co -co0)] (2.22.2) '=1 From equations (2.22.1) and (2.22.2), the coefficients of the frequency variation are *i=2j(^)*» (2.23.1) S2=t{jjs2i (2.23.2) To verify the proposed load model, Test 2.2 was performed. The test network had twenty-eight static loads connected to a same bus bar with the data summarized in Tables 2.2 and 2.3 [15, 35]. To test the voltage dependence of the aggregate load model, in the simulation the voltage was changed from 0.9 to 1.1 p.u at the rated frequency, and the real and reactive powers flowing into the loads were calculated. Then the frequency was varied from 0.9 to 1.1 p.u. at the rated voltage to test the frequency dependence of the ag-gregate load model, and the real and reactive powers flowing into the loads were com-puted. The real and reactive powers of the static loads obtained i) by solving the original system without aggregating the loads, ii) by the conventional load model, and iii) by the 23 proposed aggregate load model are shown in Figures 2.3 and 2.4. The results firmly prove that the proposed load model can closely represent aggregations of static loads at a com-mon bus bar and is more accurate than the conventional load model. Table 2.2 Data of static loads in Test 2.2 load i load % Pfi k 0 i k „ k 2 i loi In la l 3.57 1 0 0 l 0 0 0 2 3.57 1 0 0 l 0 0 0 3 3.57 1 0 0 l 0 0 0 4 3.57 0.89 0 0 0.3 0 0 -0.15 5 3.57 1 0 0 0.6 0 0 0.03 6 3.57 1 0.33 -0.44 0.22 0.23 -0.49 0.27 7 3.57 0.84 0 0 0 0 0 0 8 3.57 0.81 0 0 0 0 0 0 9 3.57 0.81 0 0 0 0 0 0 10 3.57 0.75 0 0 0 0 0 0 11 3.57 0.84 0 0 0 0 0 0 12 3.57 0.65 0 0 0 0 0 0 13 3.57 0.73 0 0 0 0 0 0 14 3.57 0.84 0 0 0 0 0 0 15 3.57 0.81 0 0 0 0 0 0 16 3.57 0.75 0 0 0 0 0 0 17 3.57 0.75 0 0 0 0 0 0 24 18 3.57 0.87 0 0 0 0 0 0 19 3.57 0.83 0 0 0 0 0 0 20 3.57 0.98 0 0 0 0 0 0 21 3.57 0.8 0 0 0 0 0 0 22 3.57 0.99 0 0 0 0 0 0 23 3.57 0.99 0 0 0 0 0 0 24 3.57 0.77 0 0 0 0 0 0 25 3.57 0.9 0 0 0 0 0 0 26 3.57 0.9 0 0 0 0 0 0 27 3.57 0.72 0 0 0 0 0 0 28 3.57 0.85 0 0 0 0 0 0 Table 2.3 Data of static loads in Test 2.2 load i k 3 i P v l i Sii k 4 i Pv2i s2> l 3 i q v i i hi l 4 i qV2i r 2 i 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 7 0.84 0.2 0.9 0 0 0 0.54 2.5 -1.3 0 0 0 8 0.81 0.2 0.9 0 0 0 0.59 2.5 -2.7 0 0 0 9 0.81 0.2 0.9 0 0 0 0.59 2.2 • -2.7 0 0 0 25 10 0.75 0.5 0.6 0 0 0 0.66 2.5 -2.8 0 0 0 11 0.84 0.8 0.5 0 0 0 0.54 2.5 -1.4 0 0 0 12 0.65 0.08 2.9 0 0 0 0.75 1.6 1.8 0 0 0 13 0.73 0.08 2.9 0 0 0 0.68 1.6 1.8 0 0 0 14 0.84 0.1 1 0 0 0 0.54 2.5 -1.3 0 0 0 15 0.81 0.1 1 0 0 0 0.59 2.5 -1.3 0 0 0 16 0.75 0.1 1 0 0 0 0.66 2.5 -1.3 0 0 0 17 0.75 0.5 0.6 0 0 0 0.66 2.5 -2.8 0 0 0 18 0.87 0.08 2.9 0 0 0 0.49 1.6 1.8 0 0 0 19 0.83 0.1 2.9 0 0 0 0.56 0.6 -1.8 0 0 0 20 0.89 0.05 1.9 0 0 0 0.2 0.5 1.2 0 0 0 21 0.8 0.08 2.9 0 0 0 0.6 1.6 1.8 0 0 0 22 0 0 0 0.99 1.8 0 0 0 0 0.14 3.5 -1.4 23 0 0 0 0.99 2 0 0 0 0 0.14 3.3 -2.6 24 0 0 0 0.77 2 0 0 0 0 0.64 5.2 -4.6 25 0 0 0 0.9 1 1 0 0 0 0.43 3 -2.8 26 0 0 0 0.9 1.8 -0.3 0 0 0 0.43 2.2 0.6 27 0 0 0 0.72 2.3 -1 0 0 0 0.69 1.61 -1 28 0 0 0 0.85 1.4 5.6 0 0 0 0.53 1.4 4.2 26 1.00 0.95 P(V) 0 90 (p.u.) 0.85 0.80 solving original network o o o o o conventional model the program Q(V) (P.u.) 0.30 solving original network o o o o o conventional model the program 0.90 0.95 1.00 1.05 V(p .u . ) 1.10 1.15 Figure 2.3 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in test 2.2 P(co) (P.u.) 1.00 0.95 0.90 0.85 0.80 1 1 1 1 ^^ -0 - "® '^ solving original o ^ e - © " ^ ^ network er^^ o o o o o conventional model the program Q(co) (P.u.) 0.55 0.50 0.45 0.40 1 solving original network o o o o o conventional model "^"©-e^o^, the program 1 0.90 0.95 1.00 1.05 co 'p.u.) 1.10 1.15 Figure 2.4 Real and reactive powers of a group of static loads with the variation of the frequency in test 2.2 27 2.2.3 Aggregate Model with Polynomials of Voltage When the coefficients of power polynomials of voltage are the only available static load data, the aggregate static load model is again assumed to have the similar form as those of the individual loads [ 3 7 ] . In addition, the real and reactive power expressions of the aggregate load will have all the terms in those of the individual loads. The real and re-active powers of static load i in a group of static loads can be expressed as P, = P,(V) = k-miV-m + ••;+*_„ r 1 +koi + kuV+ -+kMiVM (2.24.1) Qi = Qi(V) = s-mV-" + -+S-UV-1 +soi + suV+ -+smVN (2.24.2) Similarly, the real and reactive powers of the aggregate load can be assumed to be P = P(V) = k-mV-m + -+k-iV-1 +k0 + kiV+-+kMVM (2.25.1) Q = Q(V) = s-nV-n + -+s-iV-1 +s0+siV+-+sNVN (2.25.2) The real and reactive powers of the aggregate load are equivalent to the totals of those of the individual loads. Comparing the similar terms in equations (2.24.1), (2.24.2), (2.25.1), and (2.25.2), the coefficients in equations (2.25.1) and (2.25.2) are calculated as k-m = ^ k-tni k-\ = S k-u 1=1 k 0 = i k o i (2.26.1) i=i k\f=^ kMi i=i 28 s0 s0i (2.26.2) ;'=1 S\ = £ Su i=\ Sn=& SNi i=\ where q is the number of loads. To verify the aggregate load model, Test 2.3 was performed on a network having three static loads with known polynomials of voltage at a same bus bar. The data of the static loads in the test are given in Table 2.4 [37]. In the simulation, the voltage amplitude was varied from 0.9 to 1.1 p.u., and the real and reactive powers absorbed by the loads were calculated. The simulation results obtained by the proposed load model and by solv-ing the original network without aggregating the loads were compared. The total real and reactive powers of the static loads obtained by the two different methods are shown in Figure 2.5, clearly showing the high accuracy of the proposed load model. Table 2.4 Data of static loads in Test 2.3 load i load % Pfi k0i k H k2i S-li Soi S 2i S 3i l 33.33 0.7 0 2.97 -4 2.02 0 12.9 -26.8 14.9 0 2 33.33 0.71 -1.45 2.18 0.29 0 0 6.31 -15.6 10.3 0 3 33.33 0.71 0.17 0.72 0.11 0 0 2.08 1.63 -7.6 4.89 29 0.75 P(V) o 7 0 (p.u.) 0 . 6 5 solving original system o o o o the program 1.00 Q(V) 0 .80 ( P U ) nen 0.60 0.40 solving original system o o o o the program 0.90 0.95 1.00 1.05 V(p .u . ) 1.10 1.15 Figure 2.5 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.3 2.2.4 Generalization of the Modelling Techniques In general, the real and reactive powers of static load i in a group of static loads can be represented by the sums of polynomials and exponential functions as Pi(V,co) = [k„miV-m + -+k-UV-1 +koi + kuV+ -+kMiVM] "p\i "p2i L/=i *=i Qi(V,a) = [s-niV-" + -+s-UV-1 +soi +suV+ -+sNiVN] "pit np2i H i (2.27.1) + pl nguL, Qo\ijVqvUJ(Qq^+zZ Qo2ikVq*2>k(isq<°2ik (2.27.2) where npli is the number of the exponential functions in which the exponents are less than the pre-set exponent-limit of p v l and np2i is the number of the exponential functions in which the exponents are greater than or equal to the pre-set exponent-limit pvl. 30 To generalize the previous three modelling techniques, the aggregate load model is assumed to have the following mathematical expressions: P(V,(o) = [k-mV~m + -+k-x V~l +k0 + kiV+ -+kMVM] + [hFP"©^1 + l 2 W * ^ \ (2.28.1) Q(V,(o) = [s-„V-" + -+s-iV-i +s0 + siV+-+kNVN] + [riV^(a^ +r2V9ia(oq'2] (2.28.2) With the same reasoning discussed in the previous sections, the coefficients in equa-tions (2.28.1) and (2.28.2) are given by k~m = £ k-mi k-\ = £ k-u ko ~ £ koi 1=1 k\ =l\ ku & M = £ kbti ii =£ /•=i i 2 = i i=i /=i '«pi 'pi; t PoUj 7=1 £ Polik S-n — Li S-ni /=1 (2.29.1) S-i =£ S-U /=i f 0 = £ So — 2J 7=1 •Si = £ 5 1 / i=l S/V = £ Sm /=1 "pi >*2 (=i ;=1 <pli £ <3oly 7=1 "p2; S Qolik k=\ (2.29.2) where q is the number of loads. 31 At any voltage, the values of pvl, pv2, p 0„ pra2, qvi» ^ 2, q mi, and qffl2 can be obtained by solving the following equations with the technique discussed in Section 2.2.1: 7=1 "all PoUj (Po2... n VTT)p Pvlij-corV^"1'' 1 "f (^iPvm-corVP**™-1 .k=\ P a , ! ® " - ' - 1 =L n 1 •• rfiPviHj-cor 1 P(i)\ij-corU> 1=1 t KIT)" Pm2ik-cor~l *=1 U=l co2//t-cort0 ;'=1 "f (Q02A '^jqvlik-corVI™-™-<3,(oiCO (=1 2 J ^"TT-)<]<alij-cor<0 .*=1 >"2 • J <7a>2/A:- rC0 Qo>2ik-cor—l (2.30.1) (2.30.2) (2.30.3) (2.30.4) (2.30.5) (2.30.6) (2.30.7) (2.30.8) To verify the aggregate static load model, Test 2.4 in which the test network had eight static loads at a same bus was conducted. The data of the static loads in the test are given in Tables 2.5 and 2.6 [35, 37]. In the simulation, at the rated frequency the voltage amplitude was varied from 0.9 to 1.1 p.u. and the real and reactive powers flowing into the loads were calculated. Similarly, the frequency was varied from 0.9 to 1.1 p.u. at the rated voltage and the real and reactive powers of the loads were computed. The simula-tion results were obtained by the proposed load model, by the conventional load model, and by solving the original network without aggregating the loads. The total real and re-active powers of the static loads when the voltage and frequency were varied are shown in Figures 2.6 and 2.7. The test proves that the proposed load model can provide more 32 accurate simulation results than the conventional load model. Table 2.5 Data of static loads in Test 2.4 load i load % k.n k 0i k„ k2l S-li Soi Sli S 2i s3i l 12.5 0.7 0 2.97 -4 2.02 0 12.9 -26.8 14.9 0 2 12.5 0.71 -1.45 2.18 0.29 0 0 6.31 -15.6 10.3 0 3 12.5 0.71 0.17 0.72 0.11 0 0 2.08 1.63 -7.6 4.89 4 12.5 0.65 0 0 0 0 0 0 0 0 0 5 12.5 0.73 0 0 0 0 0 0 0 0 0 6 12.5 0.87 0 0 0 0 0 0 0 0 0 7 12.5 0.8 0 0 0 0 0 0 0 0 0 8 12.5 0.89 0 0 0 0 0 0 0 0 0 Table 2.6 Data of static loads in Test 2.4 load i Poli Pvli Pal i Po2i Pv 2 i P » 2 i Q o i , Qo2i q v 2i Qco2i 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 4 0.65 0.08 2.9 0 0 0 0.76 1.6 1.8 0 0 0 5 0.73 0.08 2.9 0 0 0 0.68 1.6 1.8 0 0 0 6 0.87 0.08 2.9 0 0 0 0.49 1.6 1.8 0 0 0 7 0.8 0.08 2.9 0 0 ,0 0.6 1.6 1.8 0 0 0 8 0 0 0 0.89 0.05 1.9 0 0 0 0.46 0.5 1.2 33 Q(V) (p.u.) 0.90 0.80 0.70 0.60 0.50 1 1 1 g - C 0 snh/ing original arja^f*^ network j©-®"®"^ o o o o o conventional model 1 the program • i i — 0.90 0.95 1.00 1.05 V(p .u . ) 1.10 1.15 Figure 2.6 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.4 p(co) (P-U.) 0.90 0.80 0.70 0.60 ^rf<<*5 ^ solvina onainal network o o o o o conventional . model the program Q » (p.u.) 0.75 0.70 0.65 0.60 0.55 0.90 0.95 ---rrcr0 solvina oriainal j®-©-^r^^ network o o o o o conventional model the program • • i 1.10 1.00 1.05 <o(p.u) Figure 2.7 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.4 1.15 34 2.3 Aggregation of Static Loads at Different Bus Bars When static loads are connected to different bus bars in an electric power system, the terminal voltages of the loads are different from their rated values because of the voltage drops in the network impedances. To aggregate static loads at different bus bars accu-rately, the method used to compose loads at the same bus bar must be modified. 2.3.1 Calculation of Load Terminal Voltages A simple network with static loads connected to different bus bars and its composed equivalent representation is shown in Figure 2.8. z i Z2 load groupl load group2 load groupn Zbus aggregate load bus point (a) original network bus point (b) aggregate network Figure 2.8 Network with static loads connected to different bus bars and the aggregate network model In order to aggregate the static loads at different bus bars, the terminal voltages of the loads must be first calculated. Although the static loads can have the different representa-tions as those mentioned in the previous sections, for the simplicity in the derivation of the aggregate load model, the real and reactive powers of the loads in Figure 2.8 are all assumed to be 35 P,=W/,co) = P 0 , F > ^ (2.31.1) Qt = Qi(yhri) = QoiV1*^ (2.31.2) where Vj is the terminal voltage of the ith static load. From equations (2.31.1) and (2.31.2), at the rated frequency, the admittance of static load i is given by Y^PoiVr^-jQoiVr2 (2.32) The nodal matrix equation of the system, therefore, can be formed from the admit-tances of both the static loads and the network as [YsystemWbus] = [Ibus] (2.33) Since the static load admittances are functions of the bus voltages, equation (2.33) is a non-linear matrix equation. However, the bus voltages can be obtained by solving equa-tion (2.33) with the Newton-Raphson's iterative method which can be summarized as fol-lows [8]: (1) Calculate [AI b u s] k at state k as Wbus\k = [hus]-[Ysystem\k[Vbus\k (2.34) (2) Calculate the Jacobian Matrix [J]k at state k by ( — — represents — or ^ ) U\k = ^[[Ysys^UVbush] (2.35) (3) Solve the following matrix equation for [AV b u s] k at state k [J]k[AVbus]k = [Mbus]k (2.36) (4) Calculate [V b u s] k + 1 in the next state k + 1 as [Vbus]k+l=[Vbus]k + [AVbus]k (2.37) 36 2.3.2 Aggregation of Static Loads When the bus voltages become available, the total apparent power absorbed by the network impedances in the steady state is given by ( \ 1 (2.38) where Vzsj is the complex voltage across network impedance Zsj . The aggregate network impedance in Figure 2.8 can then be calculated as ;=1 (2.39) The terminal voltage of the aggregate load is, therefore, given by VL =VS-ZS S/ , /•=1 (2.40) where Vs is the complex voltage at the bus point. Since the real and reactive powers of the aggregate load must be the sums of those flowing into the individual loads, at the rated frequency, we have QoViv = s Qo^r i=l (2.41.1) (2.41.2) From equations (2.41.1) and (2.41.2), pv and qv can be computed. Once again, since the derivation of qv is similar to that of pv, only the calculation of pv is considered. After pvi is adjusted by the technique discussed in Section 2.2.1, equation (2.41.1) can be re-written as AVL, Vd = P 0 ^ - I PoiVr"' = 0 (2.42) /•=i 37 Taking the total derivative of f(VL,V;) in equation (2.42) gives PvKv~'dVL-t {^Wvi-coyr^dVi = 0 (2.43.1) ;=l ^ r0 / or pvK~l-£ Iji) {^-jP^-corK^ * 0 (2.43.2) It is reasonable to assume that when V L changes, the change of each individual bus bar voltage is approximately according to the ratio of Voi/VoL at the rated condition. For a voltage range from V L = 1 to V L = 1 - NAV L p.u, with N being an integer, pv is obtained by solving (NAVL)p2v - (1 + NAVL)pv+± [^j V^r^Pvi-cor * 0 (2.44) Similarly, for a voltage range from V L = 1 to V L = 1 + NAV L p.u., with N being an in-teger, pv is then obtained from (NAVL)pl + (1 -NAVL)pv-± { ^ ] r^Pvi-cor * 0 (2.45) A similar procedure can be applied to obtain the values of qv, p^ , and qra in the aggre-gate static load model. Test 2.5 was used to verify the proposed load model. The test network is shown in Figure 2.9 and its data are given in Tables 2.7 and 2.8 [35]. In the test, the real and reac-tive powers of the loads were calculated when the voltage was changed from 0.9 to 1.1 p.u. at the rated frequency. Also, the frequency was varied from 0.9 to 1.1 p.u. at the rated voltage and the real and reactive powers absorbed by the loads were computed. The re-sults were obtained i) by solving the original network without aggregating the loads, ii) by the conventional load model which is generated by the weighted average parameters of individual loads, and iii) by the proposed load model. The real and reactive powers of the 38 loads are shown in Figures 2.10 and 2.11. The proposed load model is proved to be a load representation providing better results than the conventional load model. Table 2.7 Data of static loads in Test 2.5 bus bar i load % Pfi pcOj l 20 0.65 0.08 1.6 2.9 1.8 l 20 0.89 0.05 0.5 1.9 1.2 l 20 0.8 0.08 1.6 2.9 1.8 2 20 0.89 0.05 0.5 1.9 1.2 2 20 0.8 • 0.08 1.6 2.9 1.8 Table 2.8 Data of network impedances in Test 2.5 Zbusl (ohm.) Zbus2 (ohm) 0.03+J0.03 0.03+J0.03 bus point Zbusl loads Zbus2 loads Figure 2.9 Network with static loads at different bus bars in Test 2.5 39 P(V) (p.u.) 0.820 0.815 0.810 0.805 1 1 1 TR ^fQr^ solving original «^ r ° r ^ system o o o o conventional model • the program Q(V) (p.u.) 0.600 0.500 <fe 1 1 1 j ^ © - 0 " ^ solving original system o o o o conventional model • the program 0.90 0.95 1.00 1.05 V(p.u.) 1.10 1.15 Figure 2.10 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.5 P(co) (p-u.) 1.00 0.50 solving original system o o o o conventional model the program Q(co) (P-U.) 0.20 solving original system o o o o conventional model the program 0.90 0.95 1.10 1.00 1.05 co (p.u.) Figure 2.11 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.5 1.15 40 2.3.3 Generalization of the Modelling Technique Similarly to the generalization of the modelling techniques for static loads at a same bus bar, the modelling method for static loads at different bus bars can be generalized. Once again, the real and reactive powers'of an individual static load are assumed to be P,(7/,co) =MV,)+f2,(Vh<o) (2.46.1) Q,(Vh co) = gu(Vd + g2i(Vh co) (2.46.2) where AM) = k-niVJ" + •••+*_,, J 7 1 +koi + kuV, + -+kMiVff; j=\ k=\ gu(Yi) = s-njVJ" + -+s-UV~l +s0i + suVi + -+smVll; g 2/(F,-,co)="f Qouj^^'J+f QMV9,"*®""*; j=\ k=\ npli is the number of the exponential functions in which the exponents are less than the pre-set exponent-limit p v l and n^j is the number of the exponential functions in which the exponents are equal to or greater than the pre-set exponent-limit pvl. The aggregate static load model is assumed to be P(VL,<o) = Fl(VL) + F2(.VL,<o) (2.47.1) 6J(F i,co) = G 1 ( F t ) + G 2 ( ^ , c o ) (2.47.2) where Fi(VL) = k.m VT + ...+*_, V? + k0 + kx VL + -+kMVt; F2(VL, co) = / , V^' co"-' + h K2®"*2; Gi(VL) = S-nVT + -+S-1 V? +s0 + si VL + -+sNVNL; G 2 ( n , c o ) = nJ>t v l co^' + r 2 F l v 2 c o ^ ; 41 Vi is the terminal voltage of the aggregate load. By the same reasoning discussed in the previous sections the coefficients in equa-tions (2.47.1) and (2.47.2) are given by "•-m ~ "—mi \ v ho ~ 2 ho h / „ \ M h ;=1 r \ Inli Po\ij "pi; . 2J Po\ik v *=l / f \ "p2i 1 7=1 P oVj , 2 Polik vt=l y s-„ «S 5_„,(K™ - l 5_i 5JV i=i r 2 * E <=i (2.48.1) (2.48.2) where h is the number of loads. 42 Once again, the values of pvlij, pv2ij, polij, po2ij, qvlij, qv2ij, q m l ir and qco2ij should be cor-rected with the technique discussed in Section 2.2.1 before being used in the aggregate load model. The exponents pvl, pv2, p m l, p^, qvl, qv2, qml, and qra2 are calculated from those of the individual loads. Since the derivation of p v ! is similar to those of pv2, p^i, p(o2, qv), qv2, qml, and q m 2 , only the computation of pv, is presented here. By the similar reasoning as discussed in Section 2.2.1 and the previous section, p v l of the aggregate load is ob-tained from the following equations: (NAVL)Pvi -(1 +NAVL)pvl+t"f (NAVL)PV\ +(1 -NAVL)pvl-J:f ;=ly'=l where pv]i.cor is the exponent which has been adjusted by the technique discussed in Sec-tion 2.2.1. Equations (2.49.1) and (2.49.2) are appropriate for the computations of pv, in the voltage range from 1 to 1 - NAV p.u. and that from 1 to 1 + NAV p.u., with N being an integer, respectively. Test 2.6 was conducted to verify the modelling technique. The test network is shown in Figure 2.12 and the test data are given in Tables 2.9, 2.10, and 2.11 [35, 37]. In the test, the voltage was changed from 0.9 to 1.1 p.u. at the rated frequency and the real and reactive powers of the loads were calculated. Once again, the frequency was varied from 0.9 to 1.1 p.u. at the rated voltage and the real and reactive powers absorbed by the loads were computed. The results were obtained i) by solving the original network without ag-gregating the loads, ii) by the conventional load model, and iii) by the proposed load model. The real and reactive powers are shown in Figures 2.13 and 2.14. The results clearly show that the proposed load model can accurately represent power systems with Polij i' p0 k=l lik V Oij jyPvMj-cor-i y LJ oij Pvlij-cor « 0 (2.49.1) olij "pli 'nil £ Polik k=l \voL oy | ^ a , r - l p ^ _ c g r ^ Q ( 2 ^ 2 ) 43 static loads connected to different bus bars, and is superior to the conventional model. Table 2.9 Data of static loads in Test 2.6 bus bar i load % Pfi k.i. k0i k2i S-li Soi Sn S2i S3i 1 8.33 0.7 0 2.97 -4 2.02 0 12.9 -26.8 14.9 0 1 8.33 0.71 -1.45 2.18 0.29 0 0 6.31 -15.6 10.3 0 1 8.33 0.71 0.17 0.72 0.11 0 0 2.08 1.63 -7.6 4.89 2 8.33 0.7 0 2.97 -4 2.02 0 12.9 -26.8 14.9 0 2 8.33 0.71 0.17 0.72 0.11 0 0 2.08 1.63 -7.6 4.89 Table 2.10 Data of static loads in Test 2.6 bus bar i load % Pfi pv( qVi pCOi qcOi 1 8.33 0.65 0.08 1.6 2.9 1.8 1 8.33 0.73 0.08 1.6 2.9 1.8 1 8.33 0.87 0.08 1.6 2.9 1.8 1 8.33 0.89 0.05 0.5 1.9 1.2 1 8.33 0.8 0.08 1.6 2.9 1.8 2 8.33 0.89 0.05 0.5 1.9 1.2 2 8.33 0.8 0.08 1.6 2.9 1.8 Table 2.11 Data of network impedances in Test 2.6 Zbusl (ohm) Zbus2 (ohm) 0.03 +J0.03 0.03+J0.03 44 Zbusl Zbus2 loads loads bus point Figure 2.12 Network with static loads at different bus bars in Test 2.6 P(V) (p.u.) 0.78 0.76 1 1 1 lrrJ^' solving original e - ^ ro r ^ system o o o o conventional model 1 the program Q(V) (p.u.) 0.80 0.70 0.60 0.50 1 1 r solving original 1 system • o o o o conventional model the program ^ ^ ^ ^ 1 1 L 0.90 0.95 1.00 1.05 V(p.u.) 1.10 1.15 Figure 2.13 Real and reactive powers of a group of static loads with the variation of the voltage amplitude in Test 2.6 45 p(co) (P.u.) 1.00 0.90 0.80 0.70 0.60 solving original system o o o o conventional model the program Q ( * o ) (P.u.) 1.00 0.80 0.60 0.40 solving original 1 1 system o o o o conventional model the program • 1 1 i i 0.90 0.95 1.00 1.05 ©(p.u.) 1.10 1.15 Figure 2.14 Real and reactive powers of a group of static loads with the variation of the frequency in Test 2.6 2.4 Aggregate Static Load Computer Program The static load models have been developed into a computer software package with the high level computer programming language Ada 95. The flowchart of the program is shown in Figure 2.15. 2.5 General Discussion of the Results The comparison between the tests presented in this chapter shows that in small ranges of voltage and frequency, both the proposed models and the conventional models gives almost identical simulation results. However, as the deviation of voltage and fre-quency from their rated values becomes larger, the proposed models stay much closer to the field test results or the simulation results obtained by solving the original test systems without aggregating their circuit components. The improved accuracy of the proposed 46 START GET DATA INITIALIZATIONS 3 I LOAD DATA CORRECTIONS AND AGGREGATION OF LOAD PARAMETERS 1 r NETWORK SOLUTIONS POWER CALC ULATIONS LOAD PARAMETERS UPDATED S LOAD PARAMETERS UPDATED YES OUTPUTS c END INITIALIZATIONS NETWORK SOLUTION FOR LOAD TERMINAL VOLTAGES UNDER RATED CONDITION AGGREGATION OF NETWORK IMPEDANCES 1 r LOAD DATA CORRECTIONS AND AGGREGATION OF LOAD PARAMETERS I NETWORK SOLUTIONS I POWER CALCULATIONS NO Figure 2.15 Flowchart of the aggregate static load program 47 models is due to their correction schemes. Making the exponents of the power functions of aggregate loads voltage and frequency dependent also improves the accuracy of the new models. 48 CHAPTER THREE EMTP BASED LOAD REPRESENTATION In an electric power system, some static loads are both voltage and frequency de-pendent. In order to represent these components in the time domain simulations, their data structures need be converted into some proper forms. In this chapter, modelling static loads with the EMTP representation is presented. 3.1 EMTP Basic Concepts The EMTP method is an excellent modelling technique for electric power systems in the time domain. With this method, each component in a system is represented in the dis-crete time domain by some circuit elements [27]. The basic circuit elements in the model-ling process considered in this chapter are resistors, inductors, and capacitors. A resistor in both the continuous and discrete time domains is represented by where is the current through the resistor; v/{(f) is the voltage across the resistor; R is the resistance of the resistor. An inductor in the continuous time domain is represented by where VL(U) is the voltage across the inductor; iiif) is the current through the inductor; L is the inductance of the inductor. However, the representation of the inductor in the discrete time domain becomes (3.2.1) 49 idt) = (^)vL(t) + iLh(t) (3.2.2) where L is the inductance of the inductor; At is the time step; Vi(/) is the voltage across the inductor; I'IO) is the current through the inductor; tLhif) = [}-f0 v & ~ AO + hit - At) is the history current. Comparing equations (3.2.1) and (3.2.2) shows that the inductor in the continuous time domain can be represented by a resistor and a history current source in the discrete time domain, resulting in a greatly simplified representation of the element. Similarly, a capacitor in the continuous time domain can be modelled as / C ( A = C ^ > (3.3.1) at where ic(t) is the current through the capacitor; vc(t) is the voltage across the capacitor; C is the capacitance of the capacitor. Similarly to the representation of an inductor, the capacitor can be represented in the discrete time domain as icit) = {^)vcit) + ich(t) (3.3.2) where C is the capacitance of the capacitor; At is the time step; vc(t) is the voltage across the capacitor; ic(t) is the current through the capacitor; ichit) = vc(t - At) - ic{t - At) is the history current. 50 Similarly to the modelling of an inductor, equations (3.3.1) and (3.3.2) reveal that a capacitor in the continuous time domain can also be represented by a resistor and a cur-rent source in the discrete time domain. In the development of the inductor and capacitor models, a proper rule of integration must be carefully chosen [27]. Since the trapezoidal rule of integration provides the ex-cellent numerical stability which is essential in power system simulations, this rule of in-tegration has been applied to the load modelling in the discrete time domain. The circuit representations of a resistor, an inductor, and a capacitor in the continuous and discrete time domains are summarized in Figure 3.1. 3.2 Conversion of Load Data Conventional static load models represent static loads with the real and reactive power functions of voltage and frequency. Most of the available load data are the expo-nents and coefficients of these functions. To model static loads with the EMTP technique, their data must be converted into proper forms such that the loads can be represented by basic circuit elements. In general, the real and reactive powers of a static load can be ex-pressed as P = P(V,<o)=i,P0(fu(V)gu(<i>) (3.4.1) <=i Q = QiV,co) =t QoMV)g2i(fo) (3.4.2) where Poi and Qoi are coefficients; fu(V) is a voltage dependent function of the real power; gw(co) is a frequency dependent function of the real power; fniV) is a voltage dependent function of the reactive power; g2i(co) is a frequency dependent function of the reactive power; 51 iR(t) iR(t) + VR(t) -(a) R in continuous time domain + VR(t) -(b) R in discrete time domain iLh(t) iL(t) iL(t) + VL(t) (c) L in continuous time domain + VL(t) At /2L (d) L in discrete time domain ich(t) ic(t) ic(t) Vc(t) (e) C in continuous time domain + Vc(t) 2C/At (f) C in discrete time domain Figure 3.1 Models of the basic elements in different time domains V is the terminal voltage of the load; n is the number of terms in a real or reactive power function. The apparent power absorbed by the load, therefore, is given by [4, 8, 9,10] S (F,co) =S PoMV)gu((o)+j I QoMV)g2i(fo) i=\ i=\ From equation (3.5), the admittance of the load can be arrived at (3.5) YL {V, co) =1 atuiY)YLU{^)+L at2i(V)YL2i(.(o)+zZ aaM -f- (3.6) ;=1 i=\ i=\ v 52 where a,\i(V) = Poi at2i(V) = Pc V2 v2 . YLU(<O) = gu((o) +7'g2,(co) Ruifo) +ycoCi/(eo); i? 0 is the resistance of a resistor. Since a^fy") and a,2i(V) are functions of voltage amplitude only, they may be viewed as the turns ratios of transformers. However, these values are not constant and vary with the voltage. Hence, they can be realized with varied turns ratio transformers [ 23, 24]. As we can see, the functions YLli(a>) and YL2i(co) are only sensitive to the frequency and thus they can be represented with frequency dependent admittances [38, 39, 40, 41, 42, 43, 44]. Overall, equation (3.6) can be realized with the circuit representation shown in Figure 3.2. The voltage dependence is represented by the varied turns ratio transformers and the frequency dependence is modelled by variable admittances. Consequently, the voltage dependence and the frequency dependence of the load are separated and modelled independently, resulting in a much simpler representation of the original static load. Since the EMTP based load model is the combination of the varied turns ratio trans-formers and the variable admittances, the terminal voltage of the load and the frequency must be known in order to generate the model. As a result, the period of the static load's terminal voltage has to be established in order to obtain the voltage amplitude at each time step, making the frequency also available. The calculated frequency can then be re-applied to produce the circuit parameters of the EMTP based load model. 53 m i 1112 I' R31(V) bus point Figure 3.2 EMTP based representation of a static load 54 3.3 Representation of a Function by its Periodic Extension As already mentioned in the previous section, since some static loads in an electric power system are both voltage and frequency dependent, the voltage amplitude and fre-quency influence the loads' admittances which in turn determine the system's admittance matrix. Thus the voltage amplitude and frequency must be known before solving the en-tire power network. In addition, since in a transient state the rms voltage amplitude and frequency in the EMTP solution are not generated, the real power, the reactive power, and the apparent power are not known. However, these values are also needed to be calcu-lated at each time step in the load representations. To compute these quantities, the period of the voltage which can be obtained by some special technique must be known. 3.3.1 Basic Concepts of the Fourier Method The Fourier method is a very powerful tool in power system studies. The technique can be used in different aspects in power system modelling such as load representations. In electric power system studies, when the harmonic contents of voltages and currents are available, other electrical quantities, such as the current and voltage amplitudes, the real power, the reactive power, and the apparent power can be calculated. A signal g(t) is said to be periodic if there exists a time interval T such that g(t) = g(t + T) for all time t, where T is called the period of the signal. The exponentials used to form sin(nco0t), which is periodic, are the members of a family of signals, called the pha-sor signal set having the form of qn(t) = ejno)0t with n being an integer. Each member in the phasor signal set is a complex periodic signal with |ein<001| = 1 and Ze j n o ) 0 t = nco0t. The sig-nals in the phasor signal set are said to be harmonically related because their frequencies are integer multiples of the fundamental frequency co0. The signals d1"00' with n being an integer greater than one are called the harmonics of e^01 [32, 33]. A periodic signal g(t) can be represented by a complex Fourier series as [32, 33] 55 —» # = E G „ ^ » ' (3.7) n=-°o where G„ is the Fourier coefficient and is defined as G„ = — jT git)e Jn(0a'dt. If g(t) is real valued, the complex Fourier coefficients for +n and -n are a conjugate pair. Therefore, the complex Fourier series is equivalent to cosine Fourier series and we have [32, 33] g(t) = Go + 22 n=l i(rao0 cos I na0t + Z Gn (3.8) Applying Euler's law to equation (3.7) gives the following trigonometric Fourier se-ries [32, 33] g(t) = a G +2 [ancos(nco00 + b„sm(n(£>0t)] 71=1 (3.9) where a ° = i h Sit)dt; •t 2 2 f an = -yT git)cos(n(o0t)dt; 2 f 1 bn = - \\g(t)sm(n(Q0t)dt; I 2 T is the period of the function. If g(t) is an odd function, b\ is zero for all n and [32] H" = T Jo «<0sin T Jo On the other hand, if g(t) is an even function, bn is zero for all n and [32] (3.10) A" = T Jo ^ )cos(nco 0 0^ T Jo (3.11) 56 3.3.2 Periodic Extension of a Function In a state other than the steady state of a power system, a bus voltage may not be purely sinusoidal and its period is not well defined. The calculation of the amplitude of a purely sinusoidal function may not be applicable in a transient state. To obtain the voltage amplitude and frequency at each time step, other tools must be used. For a function, a discontinuity at which the one-sided limits exist, but do not agree, is called a jump discontinuity [32]. It is also possible that at some point both limits exist and agree, but that the function is not defined at that point. In such a case the function is said to have a removable discontinuity. A function is sectionally continuous in an interval a<t<b , if it is continuous, except possibly for a finite number of jumps and removable discontinuities. A non-periodic sectionally continuous function f(t) defined in a finite in-terval a<t<b can be represented by a periodic extension of period equal to the length of the interval. This periodic extension can be approximated by a truncated Fourier series as follows [32, 33, 34, 45, 46]: T is the length of the time interval; N is an integer. Equation (3.12) can also have the following form which is more convenience in am-plitude computations: (3.12) where 57 N J{t)» c0+E c„cos n (3.13) where In order to obtain the voltage amplitude and frequency, the period of the voltage must be determined. Let us consider a voltage having the wave-form as shown in Figure 3.3. The period of the voltage is defined as the distance between two adjacent maximum points or two adjacent minimum points. Since the wave-form in the figure is purely sinu-soidal, the period is constant [47]. However, when the wave-form is not purely sinusoi-dal, the period needs be redefined. Although the definition for the period of a function in a transient state is still an ongoing issue, the period of its periodic extension is well de-fined. The period of the periodic extension of a non-periodic function in a finite interval is defined as the width of the interval. Based on the extreme points, a voltage in a tran-sient state can, therefore, be divided into subsections which are treated as the voltage de-fined in different finite intervals. The subsections of the voltage can then be represented by their periodic extensions. 3.3.3 Locations of the Extreme Points Since the extreme points of a voltage affect the determination of its subdivisions, the important maximum and minimum points are needed to be located accurately. It is rea-sonable to divide a voltage wave-form into subsections according to the distances be-tween the adjacent maximum points or the adjacent minimum points as shown in Figure 3.4. Sometimes, a wave-form may have some adjacent maximum or minimum points 58 having very small amplitude differences as shown in Figure 3.5. The influence of the un-important extreme points must be eliminated. A maximum point of a voltage has the fol-lowing properties: dv(t-&t) ft dt U Jf = 0 (3.14) Mt+At) n dt U And those of a minimum point are given by dvit-M) dt <0 ^> = 0 (3.15) dv(t+At) ft dt U The appropriate maximum or minimum points used in the determination of the sub-divisions should satisfy the following constraint: | |vmax-a<#acen/| — |vmin-a4r'acenf |1 >Vd (3-16) where Vd is the pre-set voltage difference between two adjacent extreme points. When the extreme points are not well defined, the changes in the voltage amplitude are also used to assist in locating the proper maximum and minimum points as shown in Figure 3.6. If the horizontally symmetrical line of a voltage wave-form is changing, the average amplitude of the voltage is first calculated. Then the maximum and minimum points are determined according to the following constraints: Vmax - VaverageQ - At) > 0 Vmax ~ Vaverage (t+At)>0 Vmin - VaverageQ - At) < 0 Vmin-Vaverage(t+At)<0 In equations (3.14), (3.15), (3.17.1), and (3.17.2), the voltage in the next time step is unknown and must be predicted with some prediction schemes. The simplest technique 59 V(t) (p.u.) -1 .0 0.00 0.01 0.04 0.02 0.03 t (second) Figure 3.3 Voltage with purely sinusoidal wave-form 0.05 V(t) (P-U.) 0.0 0.04 0.05 0.02 0.03 t (second) Figure 3.4 Voltage with sinusoidal wave-form of the variable frequency 60 V(t) (p.u.) -0 .5 -1.5 previous minimum point \ present minimum ponit 0.00 0.01 0.02 0.03 t (second) 0.04 0.05 Figure 3.5 Voltage of sinusoidal shape with many extreme points V(t) (P-U.) -3 .0 0.00 0.01 0.02 0.03 0.04 0.05 t (second) Figure 3.6 Voltage with the variable amplitude and many extreme points 61 used in predicting the voltage in the next time step is the interpolation method as v(t + At) = 2v(f) - v(t -At) (3.18) 3.3.4 Varied-Width Moving Window in the Time Frame In order to properly determine the width of a voltage subsection which is represented by its periodic extension, a varied-width moving window along the time axis can be used. The width of the window is the period of the section and is initially assigned the period of the voltage in a steady state. The power system solution for one period time is then calcu-lated under the steady state condition. Consequently, there are a maximum point and a minimum point in the solution data in the initial period of time. When the network simu-lation starts, the simulation time increases and the window is moving to the right along the time axis accordingly. The maximum and minimum points are detected by the proce-dure discussed in Section 3.3.3. Before reaching a maximum or a minimum point, the ini-tially assigned period is used to calculate the coefficients of the Fourier series of the voltage. As shown in Figure 3.7, after arriving at a maximum or a minimum point, the width of the window or the period of the periodic extension will be updated as T = dprevious + dpresent (3.19) where dprevious is the distance between the previous adjacent maximum and minimum points, and d p r e s e n t is the distance between the previous minimum and present maximum points or that between the previous maximum and present minimum points. If the present extreme point is a maximum point, for example, the next extreme point is expected to be a minimum point, or vice versa. However, if the window has been moved to the right along the time axis from the present extreme point by an expected dis-tance dpresent.expected and the next expected extreme point has not yet been reached, the width of the window is updated by 62 T=d, 'previous "F (*present-expected (3.20) where n is the number of the increments of time steps and nAt is the time from the point Once the next extreme point has been reached, the window width is updated by equa-tion (3.19) again as shown in Figure 3.7. After the period of the considered section of the voltage becomes available, the frequency can be calculated. The varied-width moving window in the time frame is used to obtain the period of a segment of a function; however, the window also provides the information of the instan-taneous value of the function. The information contained in the window is updated at each time step when solving an entire power network. 3.4 Calculations of Network Quantities As discussed before, when solving an electric power system having static loads, the amplitudes of the load terminal voltages are required to construct the nodal matrix equa-tion of the system. Once the period of the periodic extension of a considered voltage seg-ment is defined, its amplitude can be computed. Also, the amplitude of the current flowing into a load can be calculated. Consequently, other electrical quantities of the power network, such as the different powers can also be obtained accordingly. The voltage and current can be obtained with the following finite Fourier series [33]: where the window has just moved by dpresent.expected from the present extreme point. (3.21.1) (3.21.2) where 63 1.0 V(t) (p.u.) 0.0 -0.5 -1.0 -1.5 -2.0 - / _ jr-\ 1 — 1 —i 1 1 — i -_win j _> wjn2. \ / _ _win3 _ _ ^ w dprevious dpresent 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 t (second) Figure 3.7 Varied-width moving window used to determine the subsections of a voltage with variable amplitude and frequency Vo=^;;Tvit)df, 0 v n = tan 1 f;o+Tv(t)sm{^f)dt 2 r K * W ¥ K + !::+Tv(t)cos(if)dt i0 = ^;;;Tmdt; 0 , „ =tan - 1 ffr/(()sin(2f)rfr ~\2 + \'tfiif)cos^f)dt 64 The voltage amplitude is used to construct the power network admittance matrix; however, the instantaneous value of each bus voltage is not available at the moment when the network nodal equation is being solved. Thus the bus voltage needs to be predicted at each time step. As mentioned in Section 3.3.3, the interpolation prediction scheme can be applied to predict the voltage at the time instant t. For the purpose of completeness the prediction method is rewritten here: v(0 = 2v(/ - At) - v(t - 2At) (3.22) Once the Fourier coefficients of the voltage and current are available, the amplitudes of the voltage and current can be calculated as [33] V* jvi + l ^ j i v l (3.23) J*lll+(dp2" (3-24) These voltage and current amplitudes are then used to compute the real power, the reactive power, and the apparent power of the load. The real power dissipated in the load is given by P«V0I0+t ^cos ( e v „ - e , „ ) (3.25) From equations (3.23) and (3.24), the apparent power absorbed by the load is com-puted as S=VI (3.26) Once the real power and the apparent power are known, the reactive power through the load is calculated as Q=Js2-P2 (3.27) 65 3.5 EMTP Based Load Computer Program The EMTP based load model has been developed into a computer software package with the high level computer programming language Ada 95. The flowchart of the load model program is shown in Figure 3.8. L O A D P A R A M E T E R U P D A T E D START G E T DATA L O A D D A T A CONVERSIONS YES INITIALIZATIONS pi PERIOD C A L C U L A T I O N N E T W O R K SOLUTIONS P O W E R C A L C U L A T I O N S 1 NO r OUTPUTS END v. J Figure 3.8 Flowchart of the EMTP based load program 66 3.6 Network Simulations To verify the EMTP based load model, Test 3.1, Test 3.2, and Test 3.3 were con-ducted. The first two tests were performed on single device units and the last test in-volved a network with different static loads at different bus bars. Tests 3.1 and 3.2 were performed on a fluorescent lighting unit and an induction mo-tor unit, respectively. The data of the load devices are given in Table 3.1 [37]. Table 3.1 Data of static loads in tests 3.1 and 3.2 test device name representations of real and reactive powers test 3.1 flu.lighting unit P(V) =2.18 + 0.286V - 1.45V"1 Q(V) = 6.31 - 15.60V + 10.30V2 test 3.2 ind. motor unit P(V) = 0.72 + 0.109V + 0.172 V"1 Q(V) = 2.08 + 1.63V - 7.60V2 + 4.89V 3 In each simulation, the power representation of a load was first transformed into its circuit representation as shown in Figure 3.2 and, the circuit elements were discretized with the trapezoidal rule of integration. The system equation was then formed and solved in the time domain. In the generation of the circuit representation of the test system, a varied-width moving window was applied to calculate the period of the load voltage. Fi-nally, the voltage and current amplitudes, the real power, the reactive power, and the ap-parent power were computed with the Fourier method. In each test, the amplitude of the load terminal voltage was varied from 0.9.to 1.1 p.u., and the real and reactive powers ab-sorbed by the load was computed. Figures 3.9 and 3.10 show the real and reactive powers of the load devices in Tests 3.1 and 3.2 obtained i) in the field tests which were conducted by R. Barnett Adler and Clifford C. Mosher at Drexel University in Philadelphia [37], and ii) by the proposed EMTP based load model. In each simulation, the amplitude of the 67 1.5 P(V) 1 o (p-u.) 0.5 E M T P b a s e d / ' " ~ ~ • • / f ield test 2.0 1.5 Q(V) (p.u.) 1.0 0.5 F 0.0 0.90 1.10 Figure 3.9 Real and reactive powers absorbed by the fluorescent lighting unit with the variation of the voltage amplitude in Test 3.1 P(V) (p.u.) Q(V) (p.u.) 1.4 1.2 1.0 0.8 0.6 1.4 1.2 1.0 0.8 0.6 • , 1 E M T P based field test . 1 field test • E M T P based 0.90 0.95 1.00 V (p .u . ) 1.05 1.10 Figure 3.10 Real and reactive powers absorbed by the induction motor unit with the variation of the voltage amplitude in Test 3.2 68 terminal voltage of the load was increased according to V = V, + nAV as time increased by t = nAt with n being an integer, V, being the initial voltage amplitude, AV being the voltage amplitude increment, and At being the time step, respectively. Consequently, the real and reactive powers flowing into the loads versus voltage can be plotted. The simula-tion results are closely in agreement with the field test results, firmly proving the validity of the new model. Test 3.3 was conducted on a system with a voltage source supplying eight static loads connected to two different bus bars. The test system is shown in Figure 3.11 and the data of the loads are given in Table 3.2 [35]. Both the proposed EMTP based load model and the phasor solution technique were used to solve the test network. In the time domain simulations, the loads at bus bars 1 and 2 were first composed into two single loads, respectively. The power representation of each single load was then transformed into its circuit representation as shown in Figure 3.2 and, the circuit elements were discretized with the trapezoidal rule of integration. The system equation was next formed using the load and network impedances. In the generation of the circuit represen-tation of the test system, a varied-width moving window was applied to calculate the peri-ods of the bus voltages. The instantaneous voltages and currents of the network were obtained by solving the system equation. Finally, the voltage and current amplitudes, the Z s l Zs2 Figure 3.11 Network with static loads at different bus bars in Test 3.3 69 real power, and the reactive power of the loads were computed with the Fourier method. In the phasor solution, the power representation of each load was first converted into its phasor admittance representation. Then, in the phasor domain, the system equation of the test network was formed using the load and network impedances. Finally, the voltages and currents were obtained by solving the system equation and, the real and reactive pow-ers were computed accordingly. Table 3.2 Data of static loads in Test 3.3 bus bar i load % Pfi Pvi Pcoi qVi <L»i 12.5 0.84 0.2 0.9 2.5 -1.3 12.5 0.84 0.8 0.5 2.5 -1.4 12.5 0.75 0.1 1 2.5 -1.3 12.5 0.83 0.1 2.9 0.6 -1.8 12.5 0.83 0.1 2.9 • 0.6 -1.8 2 12.5 0.84 0.2 0.9 2.5 -1.3 2 12.5 0.84 0.8 0.5 2.5 -1.4 2 12.5 0.75 0.1 1 2.5 -1.3 In Figure 3.11, the network impedances Zsl and Z s 2 are both equal to 0.03 + J0.01 ohm. In order to test the proposed load modelling technique, a simulation on the test net-work under the rated condition was conducted. That is, the voltage amplitudes and the system frequency were constant throughout the test. The source voltage and current in a steady state are shown in Figure 3.12. The terminal voltages and currents of the static loads at bus bars 1 and 2 in the steady state are shown in Figures 3.13 and 3.14. The real and reactive powers absorbed by the static loads at bus bars 1 and 2 in the steady state are 70 given in Figures 3.15 and 3.16. Once again, the simulation results indicate that under the steady state operating condition, both the phasor and the EMTP solution approaches give the identical network solutions, indicating that under the steady state condition the pro-posed model provides very accurate load representations. In order to verify the proposed model's capability in representing the voltage depend-ence of loads, using the same network a simulation in which the source voltage amplitude was varied from 0.75 to 1.25 p.u at the rated frequency was also performed. The source voltage and current are given in Figure 3.17 and, the terminal voltages and currents of the static loads at bus bars 1 and 2 with the variation of voltage amplitude are shown in Fig-ures 3.18 and 3.19. The real and reactive powers absorbed by the static loads at bus bars 1 and 2 while changing the voltage amplitude are indicated in Figures 3.20 and 3.21. From the simulation results, it is clear that the EMTP based load model can closely represent the voltage dependence of the static loads at different bus bars. To test the ability of the new model in representing the frequency dependence of loads, using the same test network another simulation in which the system frequency was varied from 0.85 to 1.15 p.u at the rated voltage was also conducted. The source voltage and current are provided in Figure 3.22. The terminal voltages and currents of the static loads at bus bars 1 and 2 with the variation of frequency are given in Figures 3.23 and 3.24. The real and reactive powers absorbed by the static loads at bus bars 1 and 2 while changing the frequency are shown in Figures 3.25 and 3.26. Once again, when the fre-quency is varying, the EMTP based load model provides accurate simulation results, firmly proving that the proposed load model can accurately represent the frequency de-pendence of static loads connected to different bus bars. 71 Vs(t) (p-u.) Is(t) (P-u.) -1 .0 .0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.12 Source voltage and current in a steady state in Test 3.3 V1(t) (P-u.) I1(t) (p.u.) -1 .0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.13 Terminal voltage and current of the loads at bus bar 1 in a steady state in Test 3.3 72 . 0 .5 I • — • • 1 1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.14 Terminal voltage and current of the loads at bus bar 2 in a steady-state in Test 3.3 1.0 P1(t) (p.u.) 0.5 -0.0 _i i i i_ 1.0 Q i ( t ) (p.u.) 0.5 -i r~ 0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.15 Real and reactive powers absorbed by the loads at bus bar 1 in a steady state in Test 3.3 73 1.0 P2(t) (p.u.) 0.5 0.0 1.0 Q2(t) (p.u.) 0.5 0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.16 Real and reactive powers absorbed by the loads at bus bar 2 in a steady state in Test 3.3 2.0 1.0 S5>00 -1 .0 -2 .0 ls(t) (P.u.) -1 .0 -2 .0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.17 Source voltage and current with the variation of voltage amplitude in Test 3.3 74 V1(t) (P-u.) I1(t) (p.u.) -1 .0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.18 Terminal voltage and current of the loads at bus bar 1 with the variation of voltage amplitude in Test 3.3 V2(t) (P-u.) I2(t) (p.u.) -0 .5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.19 Terminal voltage and current of the loads at bus bar 2 with the variation of voltage amplitude in Test 3.3 75 0.45 P1(t) 0 .40 (p.u.) 0 .35 0.30 Q1(t) (p.u.) 0.50 0.40 0.30 0.20 1 1 1 1 1 1 phasor ' 1 ^ — ~ " EMTP based . 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.20 Real and reactive powers absorbed by the loads at bus bar 1 with the variation of voltage amplitude in Test 3.3 0.25 P2(t) (p.u.) 0.20 0.15 Q2(t) (P-U.) 0.28 0.20 0.10 0.00 1 1 phasor ^ ~~ EMTP based 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.21 Real and reactive powers absorbed by the loads at bus bar 2 with the variation of voltage amplitude in Test 3.3 7 6 Vs(t) (p.u.) ls(t) (p.u.) 0.00 0.01 0.02 0.03 0.04 0.05 t (second) 0.06 0.07 Figure 3.22 Source voltage and current with the variation of frequency in Test 3.3 2.0 1.0 0.0 (p.u.) -1 .0 -2 .0 0.00 0.01 0.02 0.03 0.04 0.05 t (second) 0.06 0.07 Figure 3.23 Terminal voltage and current of the loads at bus bar 1 with the variation of frequency in Test 3.3 7 7 V2(t) (P.u.) I2(t) (P.u.) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.24 Terminal voltage and current of the loads at bus bar 2 with the variation of frequency in Test 3.3 P1(t) (pu . ) 0.45 0.40 0.35 0.30 — I I I phasor — i . . , . _ _ — : 1 i i i EMTP based 0.40 Q1(t) 0-30 (P-U.) 0.20 0.10 1 1 1 1 EMTP based ^ — \ 1 ' phasor — ^ i i i i 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.25 Real and reactive powers absorbed by the loads at bus bar 1 with the variation of frequency in Test 3.3 78 T 1 1 1 T phasor EMTP based 0.20 0.15 Q2(t) (p.u.) 0 .10 0.05 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (second) Figure 3.26 Real and reactive powers absorbed by the loads at bus bar 2 with the variation of frequency in Test 3.3 3.7 General Discussion of the Results The simulation results obtained with the EMTP load model were accurate compared with those obtained in the field tests and by the phasor solution technique. When there was only one device unit in a test system, the proposed model closely represented the load. However, if different loads were connected to different bus bars, the simulation re-sults deviated from their true values in a certain degree due to the errors introduced in the procedure of the load aggregation. Under the steady state condition, both the proposed model and the phasor method could represent the test system identically. In a quasi-steady state of voltage, the difference between the simulation results obtained with the proposed model and those with the phasor method was very small, indicating the high ac-curacy of the EMTP load modelling technique. Also, in quasi-steady state, when fre-quency changed very slowly, the simulation results obtained with the proposed model and 79 P2(t) (p.u.) 0 .20 the phasor technique deviated more from each other as the frequency differed more from its rated value. The small ripples in the power output wave-forms were caused by the er-rors incurred with the truncated Fourier series representation of the voltages and currents. However, the errors were small and could be reduced by increasing the number of terms in the Fourier series. The EMTP modelling technique is more suitable in the simulations when the variations of voltage and frequency are large. On the other hand, the phasor method is only good for very small and slow changes of voltage and frequency. As a re-sult, the proposed model provides more accurate load representation than the phasor method. 80 CHAPTER FOUR AGGREGATION OF MACHINES WITH SPECIFICATIONS During disturbances, dynamic loads may have a significant contribution on the be-haviour of an electric power system. Induction motors constitute the major dynamic loads in electric power systems and should be modelled accurately. Since the specification of an induction machine reflects its characteristics, its circuit parameters can be derived from these data using some well established data conversion programs, such as the induc-tion machine data conversion routine in Microtran, the U.B.C. version of the EMTP. There is a large number of induction machines in electric power systems and simplifying their representations is of great importance in power system modelling. When the specifi-cations of individual machines are the only available data, the machines can be composed by a single equivalent using these data. In this chapter, the methods of aggregating induc-tion machines using their specifications are presented. 4.1 Basic Concepts of Machines Understanding the basic theories of induction machines are essential in modelling these devices. A three phase induction machine has a stator in which stator windings re-side and a rotor where rotor windings are held. Different rotor features classify different types of induction machines, such as wound-rotor induction machines and squirrel-cage induction machines [23, 24, 48, 49, 50]. A squirrel-cage induction motor consists of a sta-tor having copper wound windings connected to voltage sources and a rotor having metal solid bars short circuited at each end. The rotor bars may be cylindrical (single-cage-rotor), rectangular with greater depth than width (deep-bar-rotor), or two separate bars lo-cated one above the other with separate end rings (double-cage-rotor). The characteristics of an induction machine are determined by its specification, such as the rated output power, the starting and breakdown torques, and the rated machine slip. 81 The three phase stator windings are displaced 120 degrees with respect to one an-other in the slots of the stator core. The rotor has a symmetrical structure and its speed is both voltage and frequency dependent. Figure 4.1 shows the representation of the elec-trical circuit of an induction machine [4]. If slot effects are neglected, only the mutual in-ductances between the stator and rotor windings are functions of the rotor position [4, 23, 24]. isa(t) irb(t) isb(t) isc(t) STATOR ROTOR Figure 4.1 Electrical representation of an induction machine When the stator windings are excited with a three phase voltage source, alternating currents are produced in the rotor windings and the voltage-current-flux relationships of the magnetically coupled windings are given by [1,2, 51] [v(t)] = [R][Kt)] + jt[Ut)} (4-1) where 82 [ v « ] = v*(0 Vsbif) Vscif) 0 0 0 Rs 0 0 0 0 0 0 Rs 0 0 0 0 0 0 Rs 0 0 0 0 0 0 Rr 0 0 ' 0 0 0 0 Rr 0 0 0 0 0 0 Rr ' isa(t) ' isb(t) . _ iscit) irbif) ircif) _ n M O A . r*(f) _ M O _ All the flux linkages in equation (4.1) are defined as [X(t)] = [L(t)][i(t)] (4.2) where [Lif)] = LAA(S) LAB® LBAQ) L B B i t ) [LAAW = 83 cos(9) cos(0+12O°) cos(0-12O°) [LAB(f)] = LSR\ cos (0-120°) cos(0) cos (0+120°) cos(0 +120°) cos(0 - 120°) cos(0) cos(0) cos(0-120°) cos(0+120°) [LBA(t)] = LSr\ cos (0+120°) cos(0) cos (0-120°) cos(0-12O°) cos(0 + 12O°) cos(0) LRR LR LR [LBB(/)] — LR LRR LR L f L f L ff vsa(t), vsb(t), and vsc(t) are the terminal voltages of phase-a, phase-b, and phase-c windings, respectively; isa(t), isb(t), and isc(t) are the currents passing through phase-a, phase-b, and phase-c wind-ings, respectively; R s and R , are the resistances of the stator and rotor windings, respectively; ^-sa(t), s^b(t)> and A,sc(t) are the flux linkages of the stator windings, respectively; ^•ra(t)j r^b(t)> and X.rc(t) are the flux linkages of the rotor windings, respectively; L s s and L s are the self and mutual inductances of the stator windings, respectively; L„ and L r are the self and mutual inductances of the rotor windings, respectively; L s r is the amplitude of the mutual inductance between a stator winding and a rotor winding. The mechanical part of the induction machine is represented by [27, 51, 52, 53] j^m+Dm+m)=TM dt dt if) (4.3) where 84 J is the moment of inertia of the rotor; 9(t) is the angle of the rotor position; D is the damping coefficient of the fluid around the rotor; K is the stiffness coefficient of the rotor; Tnet(t) is the net torque exerted on the rotor shaft. 4.2 Aggregation of Machines at a Same Bus Bar In an electric power system, many induction machines may be connected to a same bus bar. The first step in simplifying the network representation is to combine these ma-chines together into a single equivalent [54, 55, 56, 57, 58, 59]. 4.2.1 Manufacturer Specification of a Machine Normally, the specification of an induction machine is provided by the manufacturer. The most important data specifying the machine's behaviour usually are [48] (1) electrical data: rated terminal voltage V0; rated frequency f0; rated real output power P0; rated current I0; rated power factor pf0; rated efficiency r]0; 85 locked rotor current Ilro. (2) mechanical data: angular moment of inertia of the rotor J; number of poles N; locked rotor torque T,ro; break down torque T m a x o; rated machine speed como. 4.2.2 Electrical Specification of the Aggregate Machine Figure 4.2 shows a group of induction machines connected to a same bus bar and their aggregate equivalent. The apparent power absorbed by the aggregate machine is the sum of those absorbed by the individual machines. Thus we have where V is the terminal voltage of the machines; / is the current flowing into the aggre-—> gate machine; / 1 is the current flowing into machine i; and n is the number of machines connected to the common bus bar. Equation (4.4) can be rewritten as where I R and I[ are the real and imaginary parts of the current flowing into the aggregate machine, respectively; 1^ and I,j are the real and imaginary parts of the current through machine i, respectively. (4.4) h -jli =£ Im -j £ In i=l /=1 (4.5) 86 - ^ M n ^ 1 bus bar (a) a group of machines Rs Ls L r l L m j§ bus bar Lr2 Rrl / s (b) aggregate machine Figure 4.2 A group of induction machines connected to a same bus bar and their aggregate model Rr2/s Equating the real and imaginary parts in equation (4.5) gives IR =2 h i ii=iin i=l (4.6.1) (4.6.2) For the given power factor pfoi of machine i, the phase angle of the input impedance of the machine is computed as cp0,=cos l(pfoi) (4.7) From equations (4.6.1), (4.6.2), and (4.7), the real and imaginary parts of the current through the aggregate machine can be calculated using the currents passing through the individual machines. The rated current of the aggregate machine is then given by I0 = J(s 70,coscp0() + (E 70,sincpc where 70, is the given rated current of induction machine i. (4.8) 87 Again, from equations (4.6.1) and (4.7) we have n I0cos cp0 =S /0,cos cpo/ (4.9) 1=1 The combination of equations (4.8) and (4.9) gives the power factor of the aggregate machine as coscpQ = — = / = 1 (4.10) £ 70,coscpc 7oicoscp0/j + 70isin(p0/) The output power of the aggregate machine must be the sum of those of the individ-ual machines, making the following relation valid: n0(y3 r o/ 0coscp 0) =T,r\oi(j3V0iIoicosyoi) (4.11) ;'=1 where n 0 and r|oi are the efficiency of the aggregate machine and that of machine i, respectively. Since the voltage applied to the aggregate machine is the same as the voltage applied to the individual machines, from equations (4.9) and (4.11) the efficiency of the aggregate machine is given by n 2 r)0,70,cos9 oi no^^h (4.12) X /O,C0S(p o, /•=1 4.2.3 Mechanical Specification of the Aggregate Machine Since the mechanical part of an induction machine affects the machine's perform-ance, the manufacturer also provides this information. From the available mechanical data of each machine, the mechanical specification of the aggregate machine can also be derived. 88 The relationship between the rated output power and the developed electrical torque of machine i is given by Poi=<*moiToi (4.13) where como, is the angular speed of the rotor shaft of machine i and Toi is the shaft torque which machine i can develop under the rated condition. The slip of machine i can be calculated from the system frequency and the rotor speed as co0 -co moi V 2 J = ^ (4-1 4) where coc is the rated angular frequency of the system and N; is the number of poles of machine i. In order to derive other mechanical specifications of the aggregate machine, the num-ber of poles of the equivalent must be known. This parameter (an integer) of the aggre-gate machine can be approximated by the following weighted average value: 1=1 (4.15) Since the output power from the aggregate machine must be the sum of those deliv-ered by the individual machines, the following relationship holds: TQQJQ _ S0(T0G)o) _ y T0j(Q o y S0j(T0j(£>0) 1 (!) " (f) • " (?) " (?) ( ' where T 0 is the rated rotor shaft torque of the aggregate machine and s0 is the rated slip of the aggregate machine. T0coQ/(N/2) and s0(T0co0)/(N/2) are the air gap power and the power lost in the rotor 89 resistance of the aggregate machine, respectively. Similarly, Toico0/(Ni/2) and soi(Toico0)/(Ni/2) are the air gap power and the rotor power loss of machine i. The air gap power of the aggregate machine must be equivalent to the sum of those of the individual machines. Also, the power loss in the rotor of the aggregate machine must be the total ro-tor power losses in the individual machines. We arrive, therefore, at the following relationships: T0=NtIrf (4-17.1) ,=i Ni s°=it¥y&-tL) (4-i7-2) When the rotors of the machines are locked, all the machine slips are equal to one and the input powers are equivalent to both the core and copper losses. Consequently, if the rotors are locked, the air gap power in the aggregate machine and that in machine i are Plro = {^)<OoT,ro . (4.18.1) Plrol = (j£)<OoTtoi (4.18.2) where T|ro is the shaft torque of the aggregate machine when its rotor is locked; Ti r o i is the shaft torque of machine i when its rotor is locked. Combining equations (4.18.1) and (4.18.2) yields Tlro = N ± ^ (4.19) By the same reasoning as in deriving the rated torque of the aggregate machine, its maximum torque is computed by T t m o = N i ^ L (4.20) 7=1 -<M 90 where T m a x o is the maximum torque which the aggregate machine can develop and T m a x o i is the maximum torque which machine i can develop. Under the rated operating condition, the kinetic energy stored in the rotor of the ag-gregate machine and that stored in machine i are given by W k = M^o (4.21.1) W k . J j ^ o L ( 4 2 1 2 ) where J is the angular moment of inertia of the aggregate machine; como is the angular ro-tor speed of the aggregate machine; J; is the angular moment of inertia of machine i; and comoi is the angular rotor speed of machine i. The kinetic energy stored in the rotor of the aggregate machine must be the sum of those stored in the individual machines. Thus from equations (4.21.1) and (4.21.2) the an-gular moment of inertia of the aggregate machine can be computed as f \ - S 0 i \ 1=1 (4.22) 4.2.4 Additional Specification of the Aggregate Machine After other electrical specifications and the slip of the aggregate machine become available, its locked rotor current can be computed from those of the individual machines. When the rotors are locked, the air gap power of the aggregate machine and that of ma-chine i can also be approximated by Plro = 3ll0Rr (4.23.1) Plroi = 5i]roiRri (4.23.2) where 91 Rr Rri « Rr\R r Kr2 Rr\ + Rr2 ' RrXiRrli Rr\i + Rrli ' Ii r o is the locked rotor current of the aggregate machine and Ilroi is the locked rotor current of machine i. When the rotors are locked, the air gap power dissipated in the aggregate machine is the sum of those absorbed by the individual machines. From equations (4.23.1) and (4.23.2) we have i 2 hroiRri Rr (4.24) The rotor resistance of the aggregate machine and that of machine i can be approxi mated as [54, 60]: •S oT| 0 R r Rri ' (1 -So)COS(p0 5 ojY\ oi (1 -Soi)cOS<p0j Combining equations (4.24), (4.25.1), and (4.25.2) gives (4.25.1) (4.25.2) Ilro ~ T(l -v)cos(p 0 V S0iT] oJlroi s0r\o ,=1 (1 -50,)C0S(p0,_ (4.26) 4.2.5 Mechanical Load of the Aggregate Machine Following [15, 51], the applied mechanical load torque of the aggregate machine is assumed to be Tmo = aaymo + boimo+c (4.27) where a, b, and c are constants. 92 Similarly, the applied mechanical load torque of machine i is supposed to have the similar form as that of the aggregate machine, resulting in Tmoi = cii(i>moi + bi(ssmoi + ci (4.28) where a^, bf, and c; are constants. The mechanical load applied to the aggregate machine must be the sum of those ap-plied to the individual machines. From equation (4.28) the coefficients in equation (4.27) are given by a = £ » 2 f l ~ S ° ? (4-29.1) M N 2 ( \ - s 0 ) 2 b - i ^ ^ i (4.29.2) ; = 1 Nj(\-S0) c =t Ci (4.29.3) 4.3 Aggregation of Machines at Different Bus Bars A network with induction machines connected to different bus bars is shown in Fig-ure 4.3. The terminal voltages of the machines are different from their rated values due to the voltage drops in the network impedances. However, a similar procedure discussed in Section 4.2 for the aggregation of machines at a same bus bar can be applied to obtain the aggregate machine. In the derivation, the voltages at the bus bars are first computed by the Newton-Raphson's iterative technique. Then the specifications of the individual ma-chines are adjusted to accommodate the actual terminal voltages. And finally the spe-cification of the aggregate machine are deduced from the available data. 4.3.1 Solution of a Network The network impedances, such as those of feeders and transformers, cause a 93 deviation of the terminal voltages of the machines from their rated values. In order to ob-tain an accurate specification of the aggregate machine, those of the individual machines must be modified according to the variations of their terminal voltages. At the rated fre-quency, the slip which affects the input impedance of each machine is a function of the machine's terminal voltage. As a result, the network admittance matrix is voltage depend-ent and the system's nodal matrix equation is non-linear. However, the Newton-Raphson's iterative method can be used to solve the non-linear network matrix equation for the volt-ages at the bus bars. To form the network admittance matrix, the input admittance of each machine must first be found. For machine i, the slip at which the maximum electrical torque is generated can be approximated by srmsxoi * — j T - — + Soi^{-f—J " I (4-30) where Tmaxoi is the maximum electrical torque which machine i can develop; T0j is the rated electrical torque which machine i can develop; s0i is the rated slip of machine i. It is important to note that the slip at which a maximum electrical torque occurs is in-dependent of the machine's terminal voltage. That is, at voltage V, we have ST -ST . (4.31) 1 max/ ' maxo/ v ' where STmaxl is the slip at which the maximum electrical torque that machine i can develop at Vh The developed electrical torque of machine i near the rated operating point and that under the rated condition can be approximated as V2X2 • (4.32.1) 94 Z l Z2 groupl) ( group2] (groupn bus point (a) original system Zbus R s Ls L r l L r2 K L m Rr l /s < ^ Rr2/s M bus point (b) aggregate system Figure 4.3 Network with induction machines connected to different bus bars and the aggregate network model T (m\ (soA °'*l2co0AicnJ V2JC2-' or mi (4.32.2) The combination of equations (4.32.1) and (4.32.2) produces T,*>Tt \SoiJ (4.33) Similarly, the maximum torque generated by machine i near the rated voltage and that at the rated voltage are approximated as T (wA 7max'*UcoJ J3NA ~ Uco0) X2V2 •''•mi' i R M + JR2M + (XTHI +XRI)2 ][RI + (XSI +XMI)2] (4.34.1) X2 V2-v l mi' oi \[R2si + (Xsi+Xmi)2] (4.34.2) where Rthi = RsiXm Rsi + (Xsi +Xmi) 95 V SI Xsi = CO oLsi j Xmi = © oLmi , Xrj = (d0Lrj. The combination of equations (4.34.1) and (4.34.2) gives (Vi V Tmaxi * T/'maxo/'^"^r-J (4-35) Some typical torque-speed curves of an induction machine at some selected terminal voltages are shown in Figure 4.4. Each curve in the figure clearly shows that the portion between the maximum torque and the operating torque can be linearized. Under a steady state operating condition, the deviation of the terminal voltage of induction machine i from the rated voltage is not large. It is, therefore, reasonable to assume that at small slips the slopes of the torque-speed characteristics at operating voltage Vj and rated voltage V 0 are close to each other. Thus we have ST - S t ~ K c i { s T -s 1 max/ * >- 1 maxo/ where kci is a correction factor. ~ M — r r r ( 4- 3 6) From equations (4.33) and (4.35), equation (4.36) becomes kciSTmni(Tmaxoi ~ T0i) — TmsK0j(STmixoi ~soi)[y.) Si * (4.37) fTAfv'Y kciiTmaxoi ~ T0i) ~ (S7"m a x o, _ • s o i ) l ^ J {JT. J Again, and co0Lmi can be approximated with the method given in [54, 60]. For the purpose of completeness, the formulae are rewritten here: Rn « ^ f " (4.38.1) (1 -s0/)coscp0, 96 where Rr\i + Rr2i At the rated frequency, the input admittance of machine i is given by Yt (yi)« x — r - R Y ( 4 3 9 ) Rsi +j(o0Lsi + (jto0L„ayi\j^ +jd)0Lri) where T ~ T J-T ( ^ R L I ^ 2 Ljri ~ Lir\i * *->riA\ D D I • V - K r i ; + Kr2i' To simplify the .derivation process, near the rated operating point the input admit-tance of machine i can be approximated as st / 1 ^ Kri \(Q0Lmi/ (4.40) 97 From equations (4.37), (4.38.1), (4.38.2), and (4.40), the input admittance of machine i is approximately given by Y,(V,) +jYn(V,) (4.41) where YRi(Vi) = YIi(Vi) = (1 - S 0;)C0S ( D c T| oiS oi (1 -50,)sincpc r\oi kciS r m a x , (Tmax oi ~T0i) TmXn 0i(S Tm kci(Tmaxj T0i) (STmmoi S0i) I°L)(R Soi)\VHi The network admittance matrix can then be constructed using the network and ma-chine admittances. At the rated system frequency, the following matrix equation is solved by the Newton-Raphson's iterative solution technique for the machine terminal voltages: [Ysyste^mV] = [/] (4.42) 4.3.2 Machine Specification Adjustment When the voltages at the bus bars are available, the original specification of machine i can be adjusted. The current flowing into the machine is given by ViZyvi l ( rfz<py< {VoiZ<p vol A T0,Z(p YoiJ (4.43.1) or j rfViVYA (4.43.2) and q>/, »<p/o,. + ( 9 v , - co Voi) + (9r, - 9r„,) (4.43.3) Neglecting 9 V 0 , , the power factor of the induction machine is then computed as pfi = cos 9 , » cos ( - 9 / o i - 9 Y l + 9 K „ , ) (4.44) 98 Assuming that the circuit parameters do not change under different operating condi-tions, again according to [54, 60] the rotor resistance can be approximated by r\iSj (1 -5,)C0SCp, (4.45.1) or Rri « r\oiSc (1 -5 0 , )cOS(p c (4.45.2) Combining equations (4.45.1) and (4.45.2) gives r), « r ] 0 , Soijl -5,)coscp, Si(\ -50()coscp0/ (4.46) The locked rotor current near the rated operating point and that under the rated condi-tion can be approximated by Uri = hr, Iri I Iroi Vi Rsi +j(00Lsi + (j(OoLmi)//(Rri +j(00Lri) ' oi Rsi +j(o0LSi + (j(D0Lmj)//(Rri +j(o0Lri) (4.47.1) (4.47.2) From equations (4.47.1) and (4.47.2) we have T ~ 7 Uri * llroi\~y~ I (4.48) At voltage Vj, when the rotor is locked, the torque of machine i is given by l n ^2©,J V]XmiRri I [(Rihi + Rri)2 + (XM +Xri)2J_R2si + (Xsi +Xmi)2]\ (4.49.1) where Xri ~ co0 r2\Rr\t+ Rr2j) 99 2 2 Rsi^ mi R2si + (D20(Lsi+Lmiy v si Rsj(£) oLmi + ®oLsiLmi(Lsi + Lmi) R2+(o20(Lsi + Lmiy Similarly, at the rated voltage the locked rotor torque is approximately given by T (3NA Tiroi * y^j V20jX2miRri [(RM + Rr,)2 + (XM +Xri)2J_R2si + (Xsi +Xmi)2] (4.49.2) The combination of equations (4.49.1) and (4.49.2) gives the locked rotor torque at voltage as rr, rr, ( Vi^\ Tlri*.Tlroi[—j (4.50) 4.3.3 Specification of the Aggregate Machine Once the specifications of the individual machines have been adjusted, those of the aggregate machine can be computed. Since the current through the aggregate machine must be the sum of those through the individual machines, we have /cos cp + y7sin cp =zZ //cos cp, +jzZ //sin cp, i=l ;=1 The amplitude of the current through the aggregate machine is then given by (4.51) I = J(E 7*cos9/J + /,sincp, (4.52) Equating the real parts in equation (4.51) and applying equation (4.52) gives the power factor of the aggregate machine as coscp S //coscp, /=i l(s //coscp,-) + Zjsincp,-) (4.53) 100 Since the input power delivered to the aggregate machine is the total input power ab-sorbed by the individual machines, as shown in Figure 4.3 the rated terminal voltage of the aggregate machine is computed as n S FV/Coscp, V M = ^ - (4.54) /cos 9 From equations (4.52) and (4.53), equation (4.54) can be rewritten as n S F;/;COSCpi VM=—n (4.55) 2 /,coscp, 7=1 Also, the output power from the aggregate machine is the total output power from the individual machines, making the following equation hold: ri(TJ FM / C O S c p ) =S TI,(73 F,/,coscp,-) (4.56) From equations (4.54), and (4.56), the efficiency of the aggregate machine can be computed by n E r|,F,7,coscp, tl = ^ (4.57) £ JV/coscp, i=i By the same reasoning given in Section 4.2 when dealing with machines at a same bus bar, the developed electrical torque of the aggregate machine is calculated as T = M (4.58) Similarly, the slip of the aggregate machine is given by s=tiJ^- (4-59) i=\ N> 101 The locked rotor current of the aggregate machine is then given by Ilr r (i - s)cos op ;=1 (1 -S,)cOS(p/ (4.60) Also, the locked rotor and maximum torques of the aggregate machine are r - NY MAX' J max — La . , 1=1 M And the angular moment of inertia of the aggregate machine is given by (4.61) (4.62) (4.63) 4 . 3 . 4 Mechanical Load of the Aggregate Machine Once again, the applied mechanical load to the aggregate machine must be the total of those applied to the individual machines. By the same reasoning for dealing with ma-chines connected to a same bus bar, the mechanical load torque of machine i is assumed to be Tmi = aicarni + bj&mi + ci (4.64) Then the mechanical load torque of the aggregate machine is computed as (4.65) Tm = aam + b(am + c where ;=i N2(l-s)2 b~h mis)' 102 C =£ Ci. (=1 4.4 Aggregation of Network Impedances The network impedances also affect the behaviour of the system. They should be rep-resented accurately in the aggregate system model. As shown in Figure 4.3, the network impedances can be simply aggregated with an impedance Zbus. The apparent power ab-sorbed by the network impedances is ( \ * - » / » - > 2 j 5 ' =2J Vbus-j ^> 7 = 1 \Zbus-j J -> -» where Vbus-j is the complex voltage across network admittance Zbus-j • (4.66) The apparent power of the aggregate network impedance is also given by S= I, 2 -> Zbus (4.67) where /, is the terminal current of the aggregate induction machine. The terminal current of the aggregate machine is the sum of those through the indi-vidual machines, resulting in Therefore, from equations (4.66), (4.67), and (4.68) we have (4.68) m Zbus — -> 2 f \ Vbus-j 1 -* hus-j / (4.69) /=i 103 4.5 Aggregate Machine Data Conversion Computer Program The data conversion for aggregate induction machine has been developed into a com-puter software package with the high level computer programming language Ada 95. The flowchart of the aggregate machine data conversion program is shown in Figure 4.5. AGGREGATION OF MECHANICAL SPECIFICATIONS NO AT DIFFERENT BUS BARS • YES NETWORK SOLUTION FOR TERMINAL V O L T A G E S AT RATED CONDITION AGGREGATION OF ELECTRICAL SPECIFICATIONS ADJUSTMENTS O F SPECIFICATIONS O F INDIVIDUAL MACHINES A G G R E G A M E C H A N I SPECIFIC* TION O F C A L HTONS A G G R E G A T I O N O F E L E C T R I C A L SPECIFICATIONS AGGREGATION OF N E T W O R K IMPEDANCES Figure 4.5 Flowchart of aggregate machine data conversion program 104 4.6 Network Simulations The validity of the aggregate induction machine models need be verified. Some sys-tems have been chosen in the simulation tests, Test 4.1, Test 4.2, Test 4.3, and Test 4.4, performed by the proposed aggregate model, the power computation routine discussed in Chapter Three, and Microtran, the U.B.C. version of the EMTP. In the tests, the specifi-cations of the individual machines were first aggregated by the proposed model. Then with Microtran the voltages and currents of the test systems were obtained by solving the original networks with and without composing the induction machines and the network impedances. Once the voltages and currents were available, the real power, the reactive power, and the apparent power flowing into the machines were calculated by the power computation routine. 4.6.1 Simulation Test 4.1 To verify the aggregate model for induction machines connected to a common bus bar, simulations of a network having five induction machines at a same bus bar were con-ducted. The test network in Test 4.1 is shown in Figure 4.6 and its data are given in Ta-bles 4.1 and 4.2 [48]. In the test, a three-phase-to-ground short circuit was applied to the network at t = 0.02 second by closing switches sw; the fault was then cleared at t = 0.09 second by opening the switches. The terminal voltage and current of phase-a of the net-work are indicated in Figure 4.7. The load and short circuit current of phase-a of the net-work are shown in Figure 4.8. Also the load currents of phase-b and phase-c of the network are exhibited in Figure 4.9. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are indicated in Figures 4.10, 4.11, and 4.12. The total power absorbed by the induction machines are shown in Figure 4.13. The simulation re-sults clearly indicate that the proposed model can accurately aggregate induction ma-chines connected to a common bus bar. 105 Figure 4.6 Network with induction machines connected to a same bus bar in Test 4.1 Table 4.1 Data of induction machines in Test 4.1 motor i Po T|ro T * maxo I|ro Pfo Tie v0 fo N S 0 J load (hp) (p.u.) (p.u.) (p.u.) (volt) (Hz) (%) (kg.m2) coeffic M l 6.3 1.9 2.6 6.2 0.9 0.86 380 50 2 2.33 1.4e-2 bl=1.0 M2 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.17 1.9e-2 b2=1.25 M3 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-l b3=1.5 M4 10 2.6 3 7.3 0.83 0.89 380 50 2 1.67 3.3e-2 b4=1.65 M5 12.9 2 2.3 5.8 0.86 0.88 380 50 2 2.33 3.3e-2 b5=1.8 Table 4.2 Data of network impedances in Test 4.1 Zsw (ohm) Zso (ohm) 1.0e-6 (0.8+j0.314)e-2 106 400.0 t (second) Figure 4.7 Terminal voltage and current of phase-a of the network in Test 4.1 when a three-phase-to-ground fault occurs 1000.0 i 1 1 • t (second) Figure 4.8 Load and short circuit currents of phase-a of the network in Test 4.1 when a three-phase-to-ground fault occurs 107 iLc(t) A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.9 Load currents of phase-b and phase-c of the network in Test 4.1 when a three-phase-to-ground fault occurs Pa(t) x 1 0 5 W a t t 1.0 Qa(t) 5 0.5 x 1 0 V a r 0.0 2.0 Sa(t) x 1 0 5 V A 1.0 0.0 1 1 1 1 —v\ . J aggregate^"^"^—-^ 1 — i 1 1 1 — original - / -A , 1^/aggregate 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.10 Power flowing into phase-a of the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs 108 Pb(t) x 1 0 5 W a t t Qb(t) x 1 0 5 V a r Sb(t) x 1 0 5 V A n L 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.11 Power flowing into phase-b of the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs Pc(t) x 1 0 5 W a t t Qc(t) x 1 0 5 V a r Sc(t) x 1 0 5 V A 0. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 4.12 Power flowing into phase-c of the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs 109 P(t) x 1 0 5 W a t t Q(t) x 1 0 5 V a r S(t) x 1 0 5 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.13 Total power flowing into the machines in the network in Test 4.1 when a three-phase-to-ground fault occurs 4.6.2 Simulation Test 4.2 To test the aggregate model for induction machines connected to different bus bars, a network having nine induction machines connected to two different bus bars was simu-lated. The network is shown in Figure 4.14, and the test data are given in Tables 4.3 and 4.4 [48]. In the test, a two-phase-to-ground short circuit in phase-a and phase-b occurred at t = 0.02 second by closing switches sw in the respective phases; the fault was then cleared at t = 0.09 second by opening the switches. The terminal voltage and current of phase-b of the network are shown in Figure 4.15. The load and short circuit currents of phase-b of the network are indicated in Figure 4.16. The load currents of phase-a and phase-c of the network are exhibited in Figure 4.17. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are shown in Figures 4.18, 4.19, and 4.20. The total power absorbed by the induction machines are indicated in Figure 4.21. The simulation results clearly show that the proposed model can accurately represent 110 aggregations of induction machines at different bus bars. Table 4.3 Data of induction machines in Test 4.2 motor i Po (hp) T|ro (p.u.) T * maxo (p-u.) I|ro (p.u.) Pfo n 0 v 0 (volt) to (Hz) N S 0 (%) J (kg.m2) load coeffic M l 6.3 1.9 2.6 6.2 0.9 0.86 380 50 2 2.33 1.4e-2 bl=1.0 M2 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.17 1.9e-2 b2=l.25 M3 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-l b3=1.5 M4 6.3 1.9 2.6 6.2 0.9 0.86 380 50 2 2.33 1.4e-2 b4=1.0 M5 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.17 1.9e-2 b5=1.25 M6 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-l b6=1.5 M7 10 2.6 3 7.3 0.83 0.89 380 50 2 1.67 3.3e-2 b7=1.55 M8 12.9 2 2.3 5.8 0.86 0.88 380 50 2 2.33 3.3e-2 b8=1.65 M9 15 1.92 2.18 6.25 8.19 8.17 380 50 2 2.62 3.3e-2 b9=1.75 Table 4.4 Data of network impedances in Test 4.2 Zsw (ohm) Zso (ohm) Zbusl (ohm) Z b u s 2 (ohm) 1.0e-6 (0.8+j0.314)e-2 (5.16+j4.87)e-2 (5.16+j4.87)e-2 111 Zso Z s l Zs2 Figure 4.14 Network with induction machines at two different bus bars in Test 4.2 400.0 t ( s e c o n d ) Figure 4.15 Terminal voltage and current of phase-b of the network in Test 4.2 when a two-phase-to-ground fault occurs 112 1000.0 500.0 iLb(t) A isb(t) 4 x 1 0 A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.16 Load and short circuit currents of phase-b of the network in Test 4.2 when a two-phase-to-ground fault occurs iLa(t) A 2000 .0 1000.0 [ iLc(t) A -1000 .0 -2000 .0 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 4.17 Load currents of phase-a and phase-c of the network in Test 4.2 when a two-phase-to-ground fault occurs 113 2.0 Pa(t) x 1 0 5 W a t t 1.0 0.0 1 1 1 1— 1 aggregate - J V . 1.0 Qa(t) 5 0.5 x10 V a r 0.0 1 1 1 original *W . A J ^ aggregaTe r a ^^ ' < °^°^ ^ i i . . i 2.0 Sa(t) x 1 0 5 V A 1.0 r 0.0 -1 1 1 1 aggregate '—f\\ , at original . I I I 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.18 Power flowing into phase-a of the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs Pb(t) x 1 0 5 W a t t 1.0 Qb(t) 5 0.5 x 1 0 V a r Sb(t) x 1 0 5 V A 0.0 2.0 1.0 0.0 1 1 original —A\ . / \ a g g r e g a t e ^ ^ ^ " * * ^ ^ - ^ • i i 1 1 1 1 original — A , . y ^ ^ ^ ^ g r e g a t e ' 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 4.19 Power flowing into phase-b of the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs 114 Pc(t) x 1 0 5 W a t t x10 V a r Sc(t) x 1 0 5 V A n _ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.20 Power flowing into phase-c of the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs P(t) x 1 0 5 W a t t Q(t) x 1 0 5 V a r S(t) x 1 0 5 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 4.21 Total power flowing into the machines in the network in Test 4.2 when a two-phase-to-ground fault occurs 115 4.6.3 Simulation Test 4.3 To further verify the aggregate model for induction machines at different bus bars, an-other network having eight induction machines at three different bus bars was simulated. The network is shown in Figure 4.22 and its data are given in Tables 4.5 and 4.6 [48]. In the test, a single-phase-to-ground short circuit occurred by closing switch sw in phase-c at t = 0.02 second and, the fault was cleared at t = 0.09 second by opening the switch. The load terminal voltage and current of phase-c of the network are shown in Figure 4.23. The load and short circuit currents of phase-c of the network are indicated in Figure 4.24. The load currents of phase-a and phase-b of the network are exhibited in Figure 4.25. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are shown in Figures 4.26, 4.27, and 4.28. The total power absorbed by the induction machines are indicated in Figure 4.29. Although the machines are more spread out in the system than in Test 4.2, the aggregate machine model still gives accurate simulation results under the unbalanced fault. Again, the simulation results firmly prove that the proposed model can closely represent aggregations of induction machines at different bus bars. Vs Figure 4.22 Network with induction machines at three different bus bars in Test 4.3 116 Table 4.5 Data of induction machines in Test 4.3 motor i P o (hp) Tiro (p.u.) T A maxo (p.u.) I|ro (p.u.) P f o ™o v0 (volt fo (Hz) N S 0 ( % ) J (kg.m2) load coeffic M l 6.3 1.9 2.6 6.2 0.9 0.86 380 50 2 2.33 1.4e-2 bl=1.0 M2 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.17 1.9e-2 b2=1.25 M3 6.3 1.9 2.6 6.2 0.9 0.86 380 50 2 2.33 1.4e-2 b3=1.0 M4 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.17 1.9e-2 b4=1.25 M5 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-2 b5=1.5 M6 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.17 1.9e-2 b6=1.25 M7 10 2.6 3 7.3 0.83 0.89 380 50 2 1.67 3.3e-2 b7=1.55 M8 12.9 2 2.3 5.8 0.86 0.88 380 50 2 2.33 3.3e-2 b8=1.65 Table 4.6 Data of network impedances in Test 4.3 Zsw (ohm) Zso (ohm) Zbusl (ohm) Zbus2 (ohm) Zbus3 (ohm) 1.0e-6 (0.8+j0.314)e-2 (4.38+j6.19)e-2 (4.38+j6.19)e-2 (4.38+j6.19)e-2 117 400.0 t ( s e c o n d ) Figure 4.23 Terminal voltage and current of phase-c of the network in Test 4.3 when a single-phase-to-ground fault occurs 1000.0 0 1 1 1 1 1 1 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.24 Load and short circuit currents of phase-c of the network in Test 4.3 when a single-phase-to-ground fault occurs 118 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.25 Load currents of phase-a and phase-b of the network in Test 4.3 when a single-phase-to-ground fault occurs 1.0 Pa(t) x 1 0 5 W a t t 0.5 0.0 aggregate original • i i_ i i i Sa(t) x 1 0 5 V A 0._ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.26 Power flowing into phase-a of the machines in the network , in Test 4.3 when a single-phase-to-ground fault occurs 119 x10 V a r Sb(t) x 1 0 5 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.27 Power flowing into phase-b of the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs Pc(t) x 1 0 5 W a t t x 1 0 V a r Sc(t) x 1 0 5 V A 0._ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.28 Power flowing into phase-c of the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs 120 o.o 1 1 1 1 1 1 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 4.29 Total power flowing into the machines in the network in Test 4.3 when a single-phase-to-ground fault occurs 4.6.4 Simulation Test 4.4 An additional test was performed to verify the aggregate model for induction ma-chines at different bus bars. The test system had ten induction machines at four different bus bars as shown in Figure 4.30 with the data given in Tables 4.7 and 4.8 [48]. In the test, a single-phase-to-ground short circuit was initiated by closing switch sw in phase-b at t = 0.02 second and, the fault was cleared by opening the switch at t = 0.09 second. The load terminal voltage and current of phase-b of the network are shown in Figure 4.31. The load and short circuit currents of phase-b of the network are exhibited in Figure 4.32. The load currents of phase-a and phase-c of the network are indicated in Figure 4.33. The powers flowing into phase-a, phase-b, and phase-c of the machines are shown in Figures 4.34, 4.35, and 4.36. The total power absorbed by the machines are indicated in Figure 4.37. The machines are much different and more diverged in the system than in Test 4.2 and Test 4.3. However, the aggregate model still provides accurate simulation results 121 under the unbalanced fault, clearly verifying that the proposed model can accurately rep-resent groups of different induction motors which are more diverged at different bus bars. Zso Zbusl Zbus2 Zbus3 Zbus4 Figure 4.30 Network with induction machines at four different bus bars in Test 4.4 Table 4.7 Data of induction machines in Test 4.4 motor i Po (hp) Tiro (P-u.) T A maxo (P-u.) Ilro (P-u.) Pfo 1o V 0 (volt fo (Hz) N S 0 (%) J (kg.m2) load coeffic M l 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.16 1.9e-2 bl=1.25 M2 20 2.3 2.8 7.1 0.85 0.9 380 50 2 2 4.0e-2 b2=2.65 M3 75 1.6 2.4 6.3 0.88 0.93 380 50 2 1.16 4.8e-l b3=4.8 M4 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.16 1.9e-2 b4=1.25 M5 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-2 b5=1.5 M6 50 2.5 2.8 6.6 0.87 0.93 380 50 2 1.66 1.5e-l b6=3.25 M7 7.5 2 2.6 6.5 0.91 0.88 380 50 2 2.16 1.9e-2 b7=1.25 M8 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-2 b8=1.5 M9 6.3 1.9 2.6 6.2 0.9 0.86 380 50 2 2.33 1.4e-2 b9=1.0 M10 8.8 2.3 2.7 7.1 0.91 0.88 380 50 2 2 1.9e-2 bl0=1.5 122 Table 4.8 Data of network impedances in Test 4.4 Zsw (ohm) Zso (ohm) Zbusl (ohm) Zbus2 (ohm) Zbus3 (ohm) Zbus4 (ohm) 1.0e-6 (0.8+j0.314)e-2 (2.26+jl.0)e-2 (2.26+jl.0)e-2 (2.26+jl.0)e-2 (2.26+jl.0)e-2 400.0 t (second) Figure 4.31 Terminal voltage and current of phase-b of the network in Test 4.4 when a single-phase-to-ground fault occurs 123 4000 .0 2000 .0 iLb(t) A isb(t) 4 x 1 0 A -4 .0 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 4.32 Load and short circuit currents of phase-b of the network in Test 4.4 when a single-phase-to-ground fault occurs 2000 .0 1000.0 iLa(t) A 0.0 -1000.0 -2000 .0 4000 .0 2000.0 iLc(t) A -4000 .0 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 4.33 Load currents of phase-a and phase-c of the network in Test 4.4 when a single-phase-to-ground fault occurs 124 Pa(t) x 1 0 5 W a t t Qa(t) 5 1 0 x 1 0 V a r Sa(t) x 1 0 5 V A n „ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.34 Power flowing into phase-a of the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs 4.0 Pb(t) x 1 0 5 W a t t 2.0 0.0 1 1 1 1 1 original - a , y a ^ g r e g a ^ ^ - ^ < Sb(t) x 1 0 5 V A 0. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.35 Power flowing into phase-b of the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs 125 Pc(t) x 1 0 5 W a t t 4 .0 2.0 1 1 - i 1 1 original ' aggregate 2^®= -J 1 . — i Qc(t) 5 1.0 x 1 0 V a r x 1 0 V A 0. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.36 Power flowing into phase-c of the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs P(t) x 1 0 6 W a t t Q(t) x 1 0 6 V a r S(t) x 1 0 6 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 4.37 Total power flowing into the machines in the network in Test 4.4 when a single-phase-to-ground fault occurs 126 4.7 General Discussion of the Results The simulation results obtained with the proposed models agreed well with those ob-tained by solving the original test systems without composing the circuit elements. In general, the manufacturer's machine specifications are given only for stalled rotor and rated operating conditions and, the short circuit tests produce large deviations from steady state. Despite these large deviations in the validity of the data operating conditions, the models still produced reasonable results. Due to the effects of network impedances, the proposed model for induction machines at a same bus bar represented the machines more closely than the proposed model for machines at different bus bars. When machines are located at different bus bars, their operating states are different from their rated operating points due to the deviation of their terminal voltages from the rated values. The farther the machines are spread out in a power system, the more difficult their aggregation will become due to the greater influence of the network impedances. 127 CHAPTER FIVE AGGREGATION OF MACHINES WITH PARAMETERS In addition to the methods presented in Chapter Four to aggregate induction ma-chines from their specifications provided by the manufacturer, these machines can also be aggregated from their circuit parameters. If the machines' equivalent circuit parameters are known, aggregating these devices can be done at high and low system frequencies, re-sulting in both high accuracy and great simplification in load representations. In this chapter, accurate methods of composing single- and double-cage-rotor induction ma-chines from their circuit parameters at high and low system frequencies are presented. 5.1 Aggregation of Machines at a Same Bus Bar The circuit representations of single- and double-cage-rotor induction machines con-nected to a common bus bar are shown in Figure 5.1. When the circuit parameters of the individual machines are known, the circuit parameters of their aggregate equivalent can be generated. The slips which are functions of the system frequency affect the input ad-mittances of the machines [23, 24, 49, 50]. At high system frequency, an induction ma-chine's air gap flux is not strong and the developed electrical torque is consequently small. Assuming the mechanical load applied to the machine does not change, the rotor of the machine will turn very slowly compared to the system frequency and its slip will be close to unity. On the other hand, when the system frequency is low, the air gap flux of the machine is strong, resulting in a large developed electrical torque. Again, if the ap-plied mechanical load is assumed not to be varied, the machine rotor will rotate fast com-pared to the system frequency and the slip will be small. The circuit representations of the machines under these two extreme conditions are indicated in Figure 5.2. The aggregate machine can have the similar equivalent circuits as those of the individual machines un-der the same conditions, as shown in Figure 5.3. 128 bus point R'sl L ' s l S ^ L ' r l R"sl >L"sl • m i g RV1 ^ ~sT" L ' V l l K=iL"ml -e • L"r21 R " r l l R-V21 s"l I s'*l R'sr L ' s m ^ L'rm Vmmpi, R'rm R"sn ; L"sn L"r2n L"rln jlg,L''mn R'Vln R»r2n Figure 5.1 A group of single- and double-cage-rotor induction machines at a same bus bar R'sl L'sl L ' r l R'rl R"sl L"sl L ' V l l R - r l l R'sm L'sm L'rm R'rm R"sn L"sn L"rln R'Vln bus point (a) at high frequency R'sl L'sl L'ml R"sl L"sl L"ml R'sm L'sm L'mm R " s n L"sn L"mn bus point (b) at low frequency s"n Figure 5.2 Circuit representations of single- and double-cage-rotor induction machines at both high and low frequencies At high frequency the current flowing into the ith single-cage-rotor machine is given by hi « — 7 -V„ (5.1) (R^+R'rd+jOihiL^+L'r,) where subscript h stands for the condition at high frequency and superscript"'" indicates a single-cage-rotor machine. 129 Rs Ls L r l Lr2 Lm ^ e Rrl/s ^ Rr2/s bus point (a) operating at normal frequency Rs Ls L r l Rr i bus point (b) operating at high frequency Rs Ls Lm bus point (c) operating at low frequency Figure 5.3 Circuit representations of the aggregate induction machine at different frequencies Similarly, the current flowing into the kth double-cage-rotor machine is - > / / I hk (R'lk + R'!Xk)+j®h(L'!k + C k ) where superscript""" indicates a double-cage-rotor machine. Operating at high frequency, the current through the aggregate machine is (5.2) (Rs+RrX)+jayh(Ls+Lri) (5.3) The current flowing into the aggregate machine must be the sum of those flowing into the individual machines. Combining equations (5.1), (5.2), and (5.3) gives 1 (Rs + Rri)+j(i>h(Ls+Lrl) = Yy,R +jYhi (5.4) where YhR =£J K + K =i (R'si + R'rif + t»2n(L'si + L'ri)2 *=i (R'lk + R'r\k)z + (ofa^ + L"rXk) +1 K'k+K\k " . u'l x 2 . „ 2 , r / / , rll x 2 ' 130 n z=i (R'si + < ) 2 + ^ ( 4 + L'rif ' ~> (R'!k + R'r\k)2 + <o2h(L% + Lr\k)2 ' m is the number of the single-cage-rotor machines and n is the number of the double-cage-rotor machines. Separating the real and imaginary parts in equation (5.4) produces YhR -(OhYhi ' Rs + Rrl ' ' 1 " Yu G>hYhR Ls +Lr{ 0 (5.5) Solving equation (5.5) for (R, + R,,) and (L s + Lrl) gives YhR Rs+Rr\ = Ls+Lrl -YhR + YJj -Yu (Oh{Y2hR + Y2h^j (5.6.1) (5.6.2) At low frequency the current flowing into the ith single-cage-rotor machine is In «—, —, — (5.7) Rsi +J<ai(LSi+Lmi) where subscript 1 indicates the condition at low frequency. Also, the current through the kth double-cage-rotor machine is given by Vi I l k ~ R^+j^L'i + L^) ( 5 ' 8 ) As shown in Figure 5.3 the current through the aggregate machine is -» ~* Vi J, « U C5 9) Rs+j(Oi(Ls+Lm) Similarly to the case when the frequency is high, equations (5.7), (5.8) and (5.9) give 131 where R *=! (<)2 + co?(Z:,+lL)2 (*£)2+a>?(Z£ + I / i ) 2 ' -co/(Z^ + Z^) M ( < ) 2 + " > / ( 4 + ^ 4 ) 2 M ( < * ) 2 + o f ( ^S + L'Lkf ' Once again, equating the real and imaginary parts in equation (5.10) generates Ym -co/ Yu Rs ' 1 " Y„ (OIYIR Ls "T* Lm 0 (5.11) Solving equation (5.11) for and (Ls + Lm) gives p - Y l R Ks - - ; 7 -Yu Ls+Lr, toAY^ + Y i) (5.12.1) (5.12.2) The ratio L r l / L s of the aggregate machine can be approximated from the ratios L^/L's and L" r l k /L" s k of the individual single- and double-cage-rotor machines as Ls i=\ s' TTl f ft it E S + E S'l ,i=i ' k=\ k *=1 E S + E S" ,i=i *=i . us; (5.13) The values of R ^ , L s , L m , R , , , and L r l can then be calculated by solving the following system of equations (these equations have been derived previously and they are rewritten here for the purpose of completeness): R Y,R YIR+Y), (5.14.1) 132 Rs+Rri = Ls + Lrl = YhR YhR + YhI -Yu -Yu <»h(Y2hR + Y2M) T . m Ls i=l (s'YT' ^ J'. -r\k where g -ui(L'si + L'mi) ^ -co / ( I^ + Z ^ ) "M ( < ) 2 + c o 2 ( 4 + Z L ) 2 w (<*) 2 + c o 2 ( / ^ + 1 ' ^ ) 2 ' YhR =£ Afflt + Krlk • ( < + < ) + C O ^ , + Z / r / ) + < U ) + 0 ) ^ + 0 Y u = £ -®h(L'sl + L'rl) ^ - c o / , ( Z ^ + Z ^ ) <=i (<• + R'ri)2 + <o2k(L'sl + L'ri)2 *=i (Rl + R'r\k)2 + G>2h(L's'k + L'f]k)2 ' m n i=i *=i The impedance of the aggregate machine under the rated condition is Z= Rs +j(o0Ls + (j(o0Lmyi jv>oLr\ + (jjfj +j&0Lrl) The total impedance of the individual machines at the bus bar is 1 'total m , M i — 1 . v 1 X — + S — 1=1 2 *=1 (5.14.2) (5.14.3) (5.14.4) (5.14.5) (5.15) (5.16) where 133 Zi = R'si +/co 0L'si + (/co oL'mi)ll\ -f +j& 0 l ' Zk =R/s,k+j(ooL/s'k + 0(ooL/mk)// j&0Lrlk + ( nil ^ R r\k II ll (R" r 2 k _ i _ T" — +JG>0LR2K \sok J V sok The impedance of the aggregate machine must be the total impedance of the individ-ual machines. That is, we have Z—Z total (5.17) By the similar reasoning given in Chapter Four, the slip of the aggregate machine is computed by m s' P'• " S"lP"l V 1 "oiroi |_ J 'ok1 ok So = n -+E (5.18) Knowing the values of P^ , L s, L m, R,.,, and L r l, R.2 and L r 2 can be obtained by solving equations (5.15), (5.16)^ (5.17), and (5.18) simultaneously. 5.2 Aggregation of Machines at Different Bus Bars A network with single- and double-cage-rotor induction machines connected to dif-ferent bus bars is shown in Figure 5.4 (a). To simplify the network representation, the machines can be aggregated into an equivalent machine and the network impedances can be modelled by an equivalent impedance as well, thus resulting in an aggregate network as shown in Figure 5.4 (b). Again, special operating conditions can be applied to calculate the circuit parameters of the aggregate machine. The network representations at both low and high system frequencies are exhibited in Figure 5.5 and, the aggregate network rep-resentations under the same operating conditions are shown in Figure 5.6. At high frequency, the terminal voltages of the individual machines are computed 134 from where Yh is the admittance matrix of the network at high frequency. (5.19) From Figures 5.5 and 5.6, at high frequency we have (Rs + Rri)-j(oh(Ls + Lri) =Sh _> „ vM[yM) „ vhk[yhk where S„ =E ^ ^, _ r/^+£ x n„ _ n„ \ (R'si + R'ri)-j<oh(L'si + L'ri) k=x (Kk + R"xk) -7<o*(Zji + L"rXk)' (5.20) Zbusl Zbus2 Zbusk L " m l R " r l l / s " l hNv— R ' s l ; L " s l ; L " r l l L" r21 R"r21/s" l (a) original network Zbus Rs Ls L r l L r2 vM L m i bus point Rr l /s Rr2/s (b) aggregate network Figure 5.4 Network with single- and double-cage-rotor induction machines at different bus bars and the aggregate network 135 Zbusl Zbus2 Zbusk _ i IjJ (a) at high frequency Zbusl Zbus2 Zbus3 (b) at low frequency Figure 5.5 Representations of the network with single- and double-cage-rotor induction machines at different bus bars at high and low frequencies Zbus Rs Ls L r Rr bus point ' Mh Zbus Rs Ls Lm (a) at high frequency bus V m i point (b) at low frequency Figure 5.6 Circuit representations of the aggregate network at high and low frequencies 136 Again the current through the aggregate machine must be the total of those through the individual machines, resulting in h) =Z (/*, +E [ikk (5.21) where /->\* {VMYI ) /y y ) V ») " (R'si+R'ri)-j^hiL'si+L'ri) ~ ( < , + < u ) - ; c o ^ + Z ^ ) ' Equations (5.20) and (5.21) give the terminal voltage of the aggregate machine as •I |2 n . // I 2 '=1 k=\ {R^+R'r'u)-j^k+L'rQ (5.22) When the frequency is low, the terminal voltages of the individual machines can be calculated by solving Y, F'H Similarly, at low frequency the following relations are valid: (5.23) VMI{VMI) m Vii [Vii ) n V,k [Vlk ) Rs-j(o,(Ls+Lm) " § 7?' - y W l ' +Z' ) + £ R'L eo/(i;, ^) A=I "k -yeo i(L"k + L"mk) (5.24.1) 137 (VMI) =y ^ II Vm (5.24.2) Rs-jv>i(Ls+Lm) % R ^ - j a ^ + L^'Zi ic^-yco/(Z^ + Z^) Combining equations (5.24.1) and (5.24.2) gives the terminal voltage of the aggre-gate machine as m E n r+Z \Vr, ,11 |2 VMI = n r+E r„ J (5.25) fel ^ -^ /(ii+C) *=1 R^-ju^+L^) Again, the current through the aggregate machine is the sum of those through the in-dividual machines. At high frequency the following relationship is valid: FAjftCOSCpM, +jVMhSJn<PMh (Rs + Rri)+jah(Ls + Lri) = IhR +jhi (5.26.1) or IhR -co hihi Rs+Rrl Vmcos (pm hi (OhlhR Ls + Lr\ FM,sincpAff, (5.26.2) where (< + <X^< C 0 S <P*/) + ^ hjL'si + ^ XK/Sin cp^) - i + + co2,(Z/+Z^) *=I (^ + <u) 2+a>5(^ + ^ u ) 2 f ( < +i?/w)(K)sincp/fa.)-co^(Z^ +Z;,)(Pvfa.coscp/,) - i ( < + < ) +co^ (Z:,. + Z/n.) f + <u)(K*sincpfo - co„(Z^ + I^X^coscpfo *=1 (i?^ + i?f1/t)2 + <Q2h(L's'k + L"rXk) <?Mh, cpw> m& ty'L ^ voltage phase angles. 138 Solving equation (5.26.2) for (Ft. + R,,) and (L s + LrI) gives IhRJVMhCOS q>Mh) + Ihl(VMhSin<PMh) Rs+Rrl = J2 + /2 1hR+1hI Ls+Lr\ = (5.27.1) (5.27.2) Similarly, at low frequency the current through the aggregate machine is the sum of those through the individual machines and we have FM/COS ( D M + jVMisin 9 M Rs +j(tii(Ls+Lm) = IlR or " IlR - C O / / / / Rs FM/COS 9 M / hi CO///* Ls + Lm VMism<pMi where T f RsijKcos 9/,) + co/(4 + L,mi)(V/lism<p'li) w ( < ) 2 + co2(Z:,.+zL)2 , g A ^ ( ^ c o s 9 a + co/(ZS + Z ^ ) ( ^ s i n 9 ^ ) (*£)2+o,(i2t+J&2 f, A ^ ( ^ s m 9;,)-co/(Z^+Z^X^ , c o s 9 ; ; ) • (<)2+co2(z:;+zL) ^ *£(PftSin <p%) - co + l l ) ( ^ c o s <p%) *=> (<,) 2 + co2(Z^ + Z ^ ) 2 9 M , 9/ , , and 9 ^ are voltage phase angles. ;=i Solving equation (5.28.2) for Rj and (L s + L m) gives IIR(VMICOS 9 M ) + ///(J^M/sin 9 M ) R* = I2 +J2 (5.28.1) (5.28.2) (5.29.1) 139 -IU(VMICOS (?MI) + IiRJVMisin yMi) (5.29.2) The parameters R., Ls, L m, R,!, and L r l can be obtained by solving the following equa-tions simultaneously (these equations have been derived previously, but they are rewritten here in a more appropriate order): IIR(VMICOS CPM) + / / / ( F M / S U I c p M ) I2 +T2 Rs'+Rr\ = IhRJVMhCOS (pA/ft) + Ihi(VMhsinq>Mh) T2 +J2 1hR+1hI Ls H" Lm — -IU(VMICOS CPM) + IiRJVMism c p M / ) Ls +Lr\ = co/(4+/2/) -IhijVMhCQS cpM>) + IhRJVmsin cpMi) T ™ (=1 U ' J (rll \ Ik k=\ V O J L"r ^L"sk) where f, (< + <)(K;cos q>L) + Q>*(4 + Z/w)(FJsincp/,) l h R ~ ^ ' / — n — 2 — j — n <=i ( < + < ) +©^(1^+1;,.) | £ (<, + <uxK>scp^) + c o ^ (i?s + i?fu) +©^(1^ + 1 )^ k=\ h , = L £ (Ki + KdiKi^Whd - <*h(L'si + Z ' Q ^ C O S c p L ) (<• + <•) +co^Z .^+Z:,.) A + R'nkXKkSin 9 M ) ~ <o*(^S* + Q ( ^ c o s 9 * 0 . fc=i (i?^ + R"lk)2 + (o2n(Ls'k + L"Xk)2 (5.30.1) (5.30.2) (5.30.3) (5.30.4) (5.30.5) 140 ft ^ ( ^ c o s c p ; , ) + m / ( 4 +ZV)(F;,sin(p; ;.) l l R = ^ 7 1 — — 2 w ( ^ ) 2 + co2(Z^+zS)2 w (<) 2 + <o2(L'si + Lmi)2 + £ A ^ ( ^ C O S <p%) - co/(Z^ + Z ^ ) ( ^ s i n cp%) w ( ^ ) 2 + c o 2 ( Z ^ + Z ^ ) 2 In order to obtain the values of and L r 2 of the aggregate machine, the slips of the aggregate and individual machines near the rated condition must be known. In addition, the terminal voltages of these machines are needed to calculate R^ and Lr2. Furthermore, the terminal voltages and slips depend on each other, and the admittances of these devices are also functions of slips. However, the dilemma can be overcome by the similar proce-dure discussed in Chapter Four. The admittance of an individual machine is given by (5.31) where s„ « Rthn + 141 R rn Rrln + Rr2n for a double-cage-rotor induction machine; Lrn * Lr\„ + Lr2n[ K r l " D — j f°r a double-cage-rotor induction machine; \Rr\n+Rr2n' Gimon is the rated speed of the rotor of machine n. The admittance matrix of the network can be formed from the machine and network admittances. The non-linear nodal matrix equation of the system can be solved for the ter-minal voltages of the machines by the Newton-Raphson's iterative solution technique. Once these terminal voltages become available, the slips and output powers of the indi-vidual machines can be adjusted with the similar procedures discussed in Chapter Four. The slip of the aggregate machine can then be calculated as s ~ (5.32) The impedance of the aggregate machine at the rated frequency is given by Z= Rs + j(Q0Ls + j a 0 L m / / j(£>0Lr\ + ( —y )//( -jr- +j(£>0Lr2 ) (5.33) 142 Also, the total impedance of the individual machines can be calculated as •Z total — 1 (5.34) -*• R' where Z-= R'SI,+j(a0L'si+jm0Lmi// -f+ju)oL'ri ; Zk = R"k +J®oL"k +j(a0L'mkll j<S>oL"r\k + v s" ) V s" +j(o0L II rlk J The impedance of the aggregate machine must be the total impedance of the individ-ual machines. Solving equations (5.32), (5.33), and (5.34) simultaneously gives the val-ues of Rjj and L r 2 of the aggregate machine. 5.3 Mechanical Parameters of the Aggregate Machine As mentioned in Chapter Four, since mechanical parameters of a machine affect its behaviour, these parameters of an aggregate machine need to be represented accurately. 5.3.1 Machines at a Same Bus Bar The same procedure given in Chapter Four can be used to obtain the parameters of the aggregate machine for machines connected to a same bus bar. 5.3.2 Machines at Different Bus Bars The mechanical parameters of the aggregate machine for machines connected to dif-ferent bus bars can be derived by the same procedure provided in Chapters Four. 5.4 Aggregation of Network Impedances The network admittances can also be aggregated by the same procedure given in Chapter Four. 143 5.5 Aggregate Machine Data Conversion Computer Program The data conversion models of the aggregate machine have been developed into a computer software package with the high level computer programming language Ada 95. The flowchart of the aggregate machine data conversion program is shown in Figure 5.7. AGGREGATION OF SLIPS AGGREGATION OF ELECTRICAL AND MECHANICAL PARAMETERS START V ' GET DATA K OUTPUTS END J YES NETWORK SOLUTION FOR TERMINAL VOLTAGES AT HIGH AND LOW FREQUENCIES YES NETWORK SOLUTION FOR TERMINAL VOLTAGES AT RATED FREQUENCY ADJUSTMENT OF SLIPS AND POWERS AGGREGATION OF SLIPS AGGREGATION OF ELECTRICAL AND MECHANICAL PARAMETERS AGGREGATION OF NETWORK IMPEDANCES Figure 5.7 Flowchart of the aggregate machine data conversion program 144 5.6 Network Simulations To verify the aggregate induction machine data conversion models, six simulation tests, Test 5.1, Test 5.2, Test 5.3, Test 5.4, Test 5.5, and Test 5.6 were conducted. First, the aggregate representations of the test networks were obtained by the proposed models. Then with Microtran, the U.B.C. version of the EMTP, the voltages and currents of the networks were obtained by solving the test networks with and without composing the in-duction machines and network impedances. After the voltages and currents became avail-able, the real powers, the reactive powers, and the apparent powers were calculated by the power computation routine discussed in Chapter Three. 5.6.1 Simulation Test 5.1 Test 5.1 was performed on a network with eight different two-pole single- and double-cage-rotor induction machines connected to a same bus bar, as shown in Figure 5.8. The data of the test are given in Tables 5.1 and 5.2 [48, 51]. In the test, a two-phase-to-ground fault in phase-a and phase-b was applied to the circuit by closing switches sw in these phases at t = 0.02 second and, the fault was cleared by opening the switches at t = 0.09 second. The load terminal voltage and current of phase-a of the network are shown in Figure 5.9. The load and short circuit currents of phase-a of the network are exhibited in Figure 5.10. The load currents of phase-b and phase-c of the network are indicated in Figure 5.11. The powers flowing into phase-a, phase-b, and phase-c of the induction ma-chines are exhibited in Figures 5.12, 5.13, and 5.14. The total powers absorbed by the in-duction machines are shown in Figure 5.15. The simulation results clearly indicate that the proposed model can accurately represent aggregations of different kinds of induction machines at a same bus bar. 145 Table 5.1 Data of single- and double-cage-rotor induction machines in Test 5.1 motor i R s R r . Rr2 L r 2 P o So J load (ohm) (H) (H) (ohm) (H) (ohm) (H) (hp) ( % ) (kg.m2) coeffic M l 1.75e0 2.83e-3 1.57e-l 1.2e0 2.83e-3 6.3 6.7 1.4e-2 bl=1.0 M2 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 1.9e-2 b2=1.25 M3 1.12e0 1.85e-3 1.25e-l 8.09e-l 1.85e-3 8.8 5.8 1.9e-2 b3=1.5 M4 1.79e0 2.13e-3 1.69e-l 1.94e0 2.13e-3 7.98e-l 8.72e-3 6.3 3.1 1.4e-2 b4=1.0 M5 1.31e0 2.00e-3 1.73e-l 1.60e0 2.00e-3 6.41e-l 7.14e-3 7.5 2.8 1.9e-2 b5=1.25 M6 1.14e0 1.34e-3 1.34e-l 1.33eO 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 b6=1.5 M7 7.95e-l 1.01e-3 7.38e-2 1.25e0 1.01e-3 3.41e-l 5.05e-3 10 2.3 3.3e-2 b7=1.65 M8 6.67e-l 1.15e-3 7.39e-2 1.20e0 1.15e-3 3.78e-l 5.02e-3 12.9 3.2 3.3e-2 b8=1.8 Table 5.2 Data of network impedances in Test 5.1 Zsw (ohm) Zso (ohm) 1.0e-6 (0.8+j0.314)e-2 146 Zso |L Figure 5.8 Network with single- and double-cage-rotor induction machines connected to a same bus bar in Test 5.1 400.0 A Q I 1 1 1 1 1 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 5.9 Terminal voltage and current of phase-a of the network in Test 5.1 when a two-phase-to-ground fault occurs 147 isa(t) 4 x10 A -4 .0 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 5.10 Load and short circuit currents of phase-a of the network in Test 5.1 when a two-phase-to-ground fault occurs iLb(t) A 2000 .0 1000.0 [ iLc(t) A 0 0 - 1000 .0 -2000 .0 0.00 0.05 0.10 0.15 0.20 time (second) 0.25 0.30 Figure 5.11 Load currents of phase-b and phase-c of the network in Test 5.1 when a two-phase-to-ground fault occurs 148 Pa(t) x 1 0 5 W a t t i i — i 1 original — A . A/aggregate — l—i i • • 1.0 Qa(t) 5 0.5 x10 V a r 0.0 2.0 Sa(t) x 1 0 5 V A 0,_ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 time (second) Figure 5.12 Power flowing into phase-a of the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs 1.0 Pb(t) x 1 0 5 W a t t 0.5 0.0 ^/original - A . iff aaareaate^-^ 1.0 Qb(t) 5 0.5 x10 V a r 1 i i i i original -A, J a g g r e g a t e ^ ^ - * ^ , . . 0.0 2.0 Sb(t) x 1 0 5 V A 0 . . 0.00 0.05 0.10 0.15 0.20 0.25 0.30 time (second) Figure 5.13 Power flowing into phase-b of the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs 149 4.0 0 • = 1 1 1 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ime (second) Figure 5.14 Power flowing into phase-c of the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs 4.0 I 1 1 • r 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ime (second) Figure 5.15 Total power flowing into the machines in the network in Test 5.1 when a two-phase-to-ground fault occurs 150 5.6.2 Simulation Test 5.2 Test 5.2 was performed on a network having ten different two-pole single- and double-cage-rotor induction machines connected to four different bus bars, as shown in Figure 5.16. The data of the test are given in Tables 5.3 and 5.4 [48, 51]. In the test, a three-phase-to-ground fault was applied to the circuit by closing switches sw in the phases at t = 0.02 second and, the fault was cleared by opening the switches at t = 0.09 second. The load terminal voltage and current of phase-a of the network are shown in Figure 5.17. The load and short circuit currents of phase-a of the network are indicated in Figure 5.18. The load currents of phase-b and phase-c of the network are exhibited in Fig-ure 5.19. The powers flowing into phase-a, phase-b, and phase-c of the induction ma-chines are shown in Figures 5.20, 5.21, and 5.22. The total power absorbed by the induction machines are indicated in Figure 5.23. The results clearly prove that the pro-posed model can accurately represent aggregations of different single- and double-cage-rotor induction machines at different bus bars. Zso Zbusl Zbus2 Zbus3 Zbus4 Figure 5.16 Network with single- and double-cage-rotor induction machines in four different bus bars in Test 5.2 151 Table 5.3 Data of single- and double-cage-rotor induction machines in Test 5.2 motor i R s (ohm) (H) L m (H) R r . (ohm) L r l (H) Rr2 (ohm) L r 2 (H) P0 (hp) So ( % ) J (kg.m2) load coeffic M l 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 1.9e-2 bl=1.25 M2 7.95e-l 1.01e-3 7.38e-2 1.25e0 1.01e-3 3.41e-l 5.05e-3 10 2.3 3.3e-2 b2=1.65 M3 7.65e-2 2.71e-4 1.78e-2 1.58e-l 2.71e-4 3.22e-2 6.07e-4 70 3.2 4.8e-l b3=4.8 M4 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 1.9e-2 b4=1.25 M5 1.14e0 1.34e-3 1.34e-l 1.33e0 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 b5=1.5 M6 1.09e-l 2.92e-4 2.29e-2 3.28e-l 2.92e-4 6.92e-2 1.27e-3 50 2.2 1.5e-l b6=3.25 M7 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 1.9e-2 b7=1.25 M8 1.14e0 1.34e-3 1.34e-l 1.33eO 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 b8=1.5 M9 1.75e0 2.83e-3 1.57e-l 1.20e0 2.83e-3 6.3 6.7 1.4e-2 b9=1.0 M10 1.14e0 1.34e-3 1.34e-l 1.33e0 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 bl0=1.5 Table 5.4 Data of network impedances in Test 5.2 Zsw (ohm) Zso (ohm) Zbusl (ohm) Zbus2 (ohm) Zbus3 (ohm) Zbus4 (ohm) 1.0e-6 (1.0+j0.81)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 152 400.0 t ime (second) Figure 5.17 Terminal voltage and current of phase-a of the network in Test 5.2 when a three-phase-to-ground fault occurs t ime (second) Figure 5.18 Load and short circuit currents of phase-a of the network in Test 5.2 when a three-phase-to-ground fault occurs 153 4000 .0 2000.0 iLb(t) A 0.0 -2000.0 -4000 .0 4000 .0 2000.0 iLc(t) A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 time (second) Figure 5.19 Load currents of phase-b and phase-c of the network in Test 5.2 when a three-phase-to-ground fault occurs Pa(t) x 1 0 5 W a t t 4.0 2.0 0.0 1 r 1 1 1 ^original - A . A / \ aggregate — - — 4.0 Sa(t) x 1 0 5 V A 2.0 [ 1 —i 1 1 1 A ^ original - A . Jy aggregate " it i i • 0.00 0.05 0.10 0.15 0.20 0.25 0.30 time (second) Figure 5.20 Power flowing into phase-a of the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs 154 Pb(t) x 1 0 5 W a t t Qb(t) x 1 0 5 V a r 4 .0 2.0 0.0 1 T 1 1 1 aggregate - A . / origiridl ' ^ r , i 1 Sb(t) x 1 0 5 V A 4.0 2.0 0.0 1 1 1 1 original J aggregate " — » • i • 0.00 0.05 0.10 0.15 0.20 0.25 0.30 time (second) Figure 5.21 Power flowing into phase-b of the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs Pc(t) x 1 0 5 W a t t Qc(t) x 1 0 5 V a r Sc(t) x 1 0 5 V A 4.0 2.0 0.0 1 i i i original —*A . i / aggregate ^ — ^ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 time (second) Figure 5.22 Power flowing into phase-c of the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs 155 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ime (second) Figure 5.23 Total power flowing into the machines in the network in Test 5.2 when a three-phase-to-ground fault occurs 5.6.3 Simulation Test 5.3 To test the aggregate model for single-cage-rotor induction machines connected to a common bus bar, test 5.3 was performed on a network having ten different single-cage-rotor induction machines connected to a same bus bar, as shown in Figure 5.24. The data of the test are given in Tables 5.5 and 5.6 [48, 51]. In the test, a three-phase-to-ground fault was applied to the circuit at t = 0.02 second by closing switches sw in the phases and, the fault was cleared at t = 0.09 second by opening the switches. The terminal volt-age and current of phase-b of the network are indicated in Figure 5.25. The load and short circuit currents of phase-b of the network are shown in Figure 5.26. The load currents of phase-a and phase-c of the network are exhibited in Figure 5.27. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are shown in Figures 5.28, 5.29, and 5.30. The total power absorbed by the induction machines are shown in Figure 5.31. 156 The results clearly show that the proposed machine model can accurately represent aggre-gations of single-cage-rotor induction machines connected to a common bus bar. Table 5.5 Data of single-cage-rotor induction machines in Test 5.3 motor i R s L s L m R r P o So N J load (ohm) (H) (H) (ohm) (H) (hp) ( % ) (kg.m2) coeffic M l 1.75e0 2.83e-3 1.56e-l 1.20e-0 2.83e-3 6.3 6.7 2 1.4e-2 bl=1.0 M2 1.28e0 2.56e-3 1.61e-l 1.01e-0 2.56e-3 7.5 6.1 2 1.9e-2 b2=1.25 M3 1.12e0 1.86e-3 1.24e-l 8.09e-l 1.86e-3 8.8 5.8 2 1.9e-2 b3=1.5 M4 7.89e-l 1.56e-3 7.34e-2 7.37e-l 1.56e-3 10 6.3 2 3.3e-2 b4=1.65 M5 6.36e-l 1.56e-3 6.39e-2 6.95e-l 1.56e-3 12.9 8.1 2 3.3e-2 b5=1.8 M6 1.12e0 1.86e-3 1.24e-l 8.09e-l 1.86e-3 8.8 5.8 2 1.9e-2 b6=1.5 M7 7.89e-l 1.56e-3 7.34e-2 7.37e-l 1.56e-3 10 6.3 2 3.3e-2 b7=1.65 M8 6.36e-l 1.56e-3 6.39e-2 6.95e-l 1.56e-3 12.9 8.1 2 3.3e-2 b8=1.8 M9 7.89e-l 1.56e-3 7.34e-2 7.37e-l 1.56e-3 10 6.3 2 3.3e-2 b9=1.65 M10 6.36e-l 1.56e-3 6.39e-2 6.95e-l 1.56e-3 12.9 8.1 2 3.3e-2 bl0=1.8 Table 5.6 Data of network impedances in Test 5.3 Zsw (ohm) Zso (ohm) 1.0e-6 (0.8+j0.314)e-2 157 Zso IL *llrl I I 1 1 I I Figvire 5.24 Network with single-cage-rotor induction machines at a same bus bar in Test 5.3 itb(t) 4 . x10 A -4.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.25 Terminal voltage and current of phase-b of the network in Test 5.3 when a three-phase-to-ground fault occurs 158 2000 .0 1000.0 iLb(t) A isb(t) 4" x 1 0 A -4 .0 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 5.26 Load and short circuit currents of phase-b of the network in Test 5.3 when a three-phase-to-ground fault occurs 2000 .0 1000.0 iLa(t) A iLc(t) A -2000 .0 0.25 0.30 0.00 0.05 0.10 0.15 0.20 t (second) Figure 5.27 Load currents of phase-a and phase-c of the network in Test 5.3 when a three-phase-to-ground fault occurs 159 Pa(t) x 1 0 5 W a t t x10 V a r Sa(t) x 1 0 5 V A 0._ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.28 Power flowing into phase-a of the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs Pb(t) x 1 0 5 W a t t Sb(t) x 1 0 5 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 5.29 Power flowing into phase-b of the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs 160 Pc(t) x 1 0 5 W a t t Qc(t) x 1 0 5 V a r Sc(t) x 1 0 5 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.30 Power flowing into phase-c of the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs P(t) x 1 0 6 W a t t Q(t) x 1 0 6 V a r S(t) 6 x10 V A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t ( s e c o n d ) Figure 5.31 Total power flowing into the machines in the network in Test 5.3 when a three-phase-to-ground fault occurs 161 5.6.4 Simulation Test 5.4 To verify the aggregate model for single-cage-rotor induction machines at different bus bars, Test 5.4 was performed on a network with eight different single-cage-rotor in-duction machines connected to four different bus bars as shown in Figure 5.32. The test data are summarized in Tables 5.7 and 5.8 [48, 51]. Again, in the test a three-phase-to-ground fault was initiated in the circuit at t = 0.02 second by closing switches sw in the phases and, the fault was cleared at t = 0.09 second by opening the switches. The terminal voltage and current of phase-a of the network are indicated in Figure 5.33. The load and short circuit currents of phase-a of the network are shown in Figure 5.34. The load cur-rents of phase-b and phase-c of the network are exhibited in Figure 5.35. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are indicated in Figures 5.36, 5.37, and 5.38. The total power absorbed by the induction machines are shown in Figure 5.39. Again, the results clearly prove that the proposed machine model can accurately represent aggregations of different single-cage-rotor induction machines at different bus bars. Zso Zbusl Zbus2 Zbus3 Zbus4 Figure 5.32 Network with single-cage-rotor induction machines at four different bus bars in Test 5.4 162 Table 5.7 Data of single-cage-rotor induction machines in Test 5.4 motor i R s (ohm) L s (H) (H) R , (ohm) L r (H) P o (hp) So ( % ) N J (kg.m2) load coeffic M l 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 2 1.9e-2 bl=1.25 M2 1.12e-l 1.86e-3 1.25e-l 8.09e-l 1.86e-3 8.8 5.8 2 1.9e-2 b2=1.5 M3 7.89e-l 1.56e-3 7.34e-2 7.37e-l 1.56e-3 10 6.3 2 3.3e-2 b3=1.65 M4 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 2 1.9e-2 b4=1.25 M5 1.12e-l 1.86e-3 1.25e-l 8.09e-l 1.86e-3 8.8 5.8 2 1.9e-2 b5=1.5 M6 1.75e0 2.83e-3 1.57e-l 1.20e-0 2.83e-3 6.3 6.7 2 1.4e-2 b6=1.0 M7 1.28e0 2.56e-3 1.61e-l l.OleO 2.56e-3 7.5 6.1 2 1.9e-2 b7=1.25 M8 1.75e0 2.83e-3 1.57e-l 1.20e-0 2.83e-3 6.3 6.7 2 1.4e-2 b8=1.0 Table 5.8 Data of network impedances in Test 5.4 Zsw (ohm) Zso (ohm) Zbusl (ohm) Zbus2 (ohm) Zbus3 (ohm) Zbus4 (ohm) 1.0e-6 (0.8+j0.31)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 163 ita(t) 4 x10 A -4.0 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 5.33 Terminal voltage and current of phase-a of the network in Test 5.4 when a three-phase-to-ground fault occurs isa(t) 4 x10 A 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 5.34 Load and short circuit currents of phase-a of the network in Test 5.4 when a three-phase-to-ground fault occurs 164 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.35 Load currents of phase-b and phase-c of the network in Test 5.4 when a three-phase-to-ground fault occurs Pa(t) x105Watt Qa(t) x105Var Sa(t) x105VA 0.0 1.0 0.5 0.0 1 —"A , / o r i g l n a ^ J ^ : ^ : : ; > , ^ ^ • r - -/ i i i i 1 r \ ^ * \ ^ ^ ^ original — A . aggregate ' " " ^ ^ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t(second) Figure 5.36 Power flowing into phase-a of the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs 165 Pb(t) x105Watt Qb(t) x105Var 1.0 0.5 — i 1 r : 1 ^ original f aggregate L i i_ Sb(t) x105 VA 0.5 0. 1 • — i 1 original r aggregate 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.37 Power flowing into phase-b of the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs Pc(t) x105Watt Qc(t) x105Var 1.0 0.5 0.0 0.5 1 1 1 i original j aggregate — > • — - i —1 1 1 1 " • /v aggregate Sc(t) x105VA 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.38 Power flowing into phase-c of the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs 166 P(t) x105Watt 4.0 2.0 0.0 1 1 — i 1 original J aggregate^ JL i _ j i Q(t) x105Var S(t) x105VA 4.0 2.0 0.0 0.00 1 1 — i — i original J aggregate ^ i • 0.05. 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 5.39 Total power flowing into the machines in the network in Test 5.4 when a three-phase-to-ground fault occurs 5.6.5 Simulation Test 5.5 To test the aggregate model for double-cage-rotor induction machines at a same bus bar, Test 5.5 was performed on a network with five different two-pole double-cage-rotor induction machines at a same bus bar as shown in Figure 5.40. The data of the test are given in Tables 5.9 and 5.10 [48]. In the test, a single-phase-to-ground fault was initiated in the circuit by closing switch sw in phase-a at t = 0.02 second and, the fault was cleared at t = 0.09 second by opening the switch. The terminal voltage and current of phase-a of the network are exhibited in Figure 5.41. The load and short circuit currents of phase-a of the network are shown in Figure 5.42. The load currents of phase-b and phase-c of the network are indicated in Figure 5.43. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are shown in Figures 5.44, 5.45, and 5.46. The total power absorbed by the induction machines are exhibited in Figure 5.47. The results 167 clearly show that the proposed model can accurately represent aggregations of different double-cage-rotor induction machines at a same bus bar. Zso IL Figure 5.40 Network with double-cage-rotor induction machines connected to a same bus bar in Test 5.5 Table 5.9 Data of double-cage-rotor induction machines in Test 5.5 motor i L s L m L r l R,2 L r 2 P o s0 J load (ohm) (H) (H) (ohm) (H) (ohm) (H) (hp) ( % ) (kg.m2) coeffic M l 1.79e0 2.13e-3 1.69e-l 1.94e0 2.13e-3 7.98e-l 8.72e-3 6.3 3.1 1.4e-2 bl=1.0 M2 1.31 eO 2.00e-3 1.73e-l 1.60e0 2.00e-3 6.41e-l 7.14e-3 7.5 2.8 1.9e-2 b2=1.25 M3 1.14e0 1.34e-3 1.34e-l 1.33e0 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 b3=1.5 M4 7.95e-l 1.01e-3 7.38e-2 1.25e0 1.01e-3 3.41e-l 5.05e-3 10 2.3 3.3e-2 b4=1.65 M5 6.67e-l 1.15e-3 7.39e-2 1.20e0 1.15e-3 3.78e-l 5.02e-3 12.9 3.2 3.3e-2 b5=1.8 Table 5.10 Data of network impedances in Test 5.5 Zsw (ohm) Zso (ohm) 1.0e-6 (0.8+j0.314)e-2 168 Vta(t) V 400.0 200.0 ita(t) 4 x10 A 4.0 2.0 0.0 -2.0 -4.0 J I f I [ 1 1 — i 1 1 1 — • u i i i 1 1— 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.41 Terminal voltage and current of phase-a of the network in Test 5.5 when a single-phase-to-ground fault occurs 1000.0 500.0 iLa(t) A isa(t) 4 x10 A 0.00 0.05 0.10 0.15 0.20 t (second) 0.25 0.30 Figure 5.42 Load and short circuit currents of phase-a of the network in Test 5.5 when a single-phase-to-ground fault occurs 169 1000.0 500.0 i L b ( t ) A 0 0 -500.0 •1000.0 1000.0 500.0 iLc(t) A 0.0 -500.0 -1000.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.43 Load currents of phase-b and phase-c of the network in Test 5.5 when a single-phase-to-ground fault occurs Pa(t) x105Watt Qa(t) x105Var Sa(t) x105VA 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.44 Power flowing into phase-a of the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs 170 Pb(t) x105Watt 1.0 Qb(t) 5 0.5 x10 Var 0.0 Sb(t) x105VA 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t(second) Figure 5.45 Power flowing into phase-b of the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs Pc(t) x105Watt Qc(t) x105Var Sc(t) x105VA 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t(second) Figure 5.46 Power flowing into phase-c of the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs 171 0.0 o.o I • • • • • 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.47 Total power flowing into the machines in the network in Test 5.5 when a single-phase-to-ground fault occurs 5.6.6 Simulation Test 5.6 To test the aggregate model for double-cage-rotor induction machines at different bus bars, Test 5.6 was performed on a network with nine different two-pole double-cage-rotor induction machines at four different bus bars as shown in Figure 5.48. The test data are given in Tables 5.11 and 5.12 [48]. In the test, a single-phase-to-ground fault was ap-plied to the circuit by closing switch sw in phase-c at t = 0.02 second and, the fault was cleared at t = 0.09 second by opening the switch. The terminal voltage and current of phase-c of the network are indicated in Figure 5.49. The load and short circuit currents of phase-c of the network are shown in Figure 5.50. The load currents of phase-b and phase-c of the network are exhibited in Figure 5.51. The powers flowing into phase-a, phase-b, and phase-c of the induction machines are shown in Figures 5.52, 5.53, and 5.54. The to-tal power absorbed by the induction machines are indicated in Figure 5.55. Again, the 172 the results clearly indicate that the proposed model can accurately represent the aggrega-tions of different double-cage-rotor induction machines at different bus bars. Table 5.11 Data of double-cage-rotor induction machines in Test 5.6 motor i (ohm) L s (H) L m (H) R r . (ohm) L r l (H) (ohm) L r 2 (H) P o (hp) So ( % ) J (kg.m2) load coeffic M l 1.14e0 1.34e-3 1.34e-l 1.33e0 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 bl=1.5 M2 7.95e-l 1.01e-3 7.38e-2 1.25e0 1.01e-3 3.41e-l 5.05e-3 10 2.3 3.3e-2 b2=1.65 M3 6.67e-l 1.15e-3 7.39e-2 1.20e0 1.15e-3 3.78e-l 5.02e-3 12.9 3.2 3.3e-2 b3=1.8 M4 1.79e0 2.13e-3 1.69e-l 1.94e0 2.13e-3 7.98e-l 8.72e-3 6.3 3.1 1.4e-2 b4=1.0 M5 1.31e0 2.00e-3 1.73e-l 1.60e0 2.00e-3 6.41e-l 7.14e-3 7.5 2.8 1.9e-2 b5=1.25 M6 1.14e0 1.34e-3 1.34e-l 1.33e0 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 b6=1.5 M7 1.14e0 1.34e-3 1.34e-l 1.33e0 1.33e-3 4.93e-l 5.80e-3 8.8 2.5 1.9e-2 b7=1.5 M8 1.79e0 2.13e-3 1.69e-l 1.94e0 2.13e-3 7.98e-l 8.72e-3 6.3 3.1 1.4e-2 b8=1.0 M9 1.31e0 2.00e-3 1.73e-l 1.60e0 2.00e-3 6.41e-l 7.14e-3 7.5 2.8 1.9e-2 b9=1.25 Table 5.12 Data of network impedances in Test 5.6 Zsw (ohm) Zso (ohm) Zbusl (ohm) Zbus2 (ohm) Zbus3 (ohm) Zbus4 (ohm) 1.0e-6 (1.0+j0.81)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 (3.0+j3.0)e-2 173 Zso Zbusl Zbus2 Zbus3 Zbus4 Figure 5.48 Network with the double-cage-rotor induction machines at four different bus bars in Test 5.6 400.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.49 Terminal voltage and current of phase-c of the network in Test 5.6 when a single-phase-to-ground fault occurs 174 1000.0 500.0 iLc(t) A isc(t) 4 x10 A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.50 Load and short circuit currents of phase-c of the network in Test 5.6 when a single-phase-to-ground fault occurs 1000.0 500.0 tt-a(t) o.o A -500.0 .1000.0 —i 1 1 1 aggregate M A A A A A A A A A j\j V V v W V V V v I original • i • • — iLb(t) 1000.0 500.0 0.0 -500.0 -1000.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.51 Load currents of phase-a and phase-b of the network in Test 5.6 when a single-phase-to-ground fault occurs 175 Pa(t) x105Watt x10 Var Sa(t) x105 VA 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.52 Power flowing into phase-a of the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs Pb(t) x105Watt Sb(t) x105VA 0. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.53 Power flowing into phase-b of the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs 176 1.0 Pc(t) x105Watt 0.5 0.0 1 1 1 i rs. ^ original -— J ^ 1 aggregate x10 Var Sc(t) x105 VA 0 _ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.54 Power flowing into phase-c of the machines in the network P(t) x105Watt Q(t) x105Var S(t) x105VA 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (second) Figure 5.55 Total power flowing into the machines in the network in Test 5.6 when a single-phase-to-ground fault occurs 177 5.7 General Discussion of the Results Similarly to the aggregate machine models from the manufacturer's data, the simula-tion results obtained with the proposed machine models from circuit parameters closely agreed with those obtained by solving the original systems without composing the circuit components. Due to the effects of network impedances, the proposed model for induction machines at different bus bars provided less accurate representations than the proposed model for machines at the same bus bar. For machines at different bus bars, the terminal voltages are different from their rated values, thus making their slips and output powers deviate more noticeable from their rated values. When machines are spread out farther in a power system, their aggregation becomes more difficult due to the greater influence of the network impedances. 178 CHAPTER SIX CONCLUSIONS AND RECOMMENDATIONS Electrical loads greatly influence the behaviour of an electric power system and accu-rately modelling these devices deserves great attention. Usually, there are thousands of electrical loads in an electric power network and a simplifying representation is necessary for fast system level studies. In this thesis work, aggregate models have been developed for static and dynamic loads. The developed aggregate static load models are superior to the existing aggregate load models by including more information of both voltage and frequency. The proposed models represent static loads in much wider voltage and frequency ranges than the con-ventional aggregate static load models, thus improving the accuracy of network simula-tions. In addition, the proposed models accommodate different formats of load data. The EMTP based load model is formulated based on the load data transformation from the power representation to the circuit representation. The proposed model repre-sents the voltage dependence and the frequency dependence separately. The proposed model makes it possible to apply the EMTP technique to represent loads in the time do-main, resulting in high accuracy in load representations. The developed aggregate induction machine models with known machine specifica-tions compose machines from their available manufacturer data. The proposed models not only relieve the computational burden in data conversion for equivalent circuit parame-ters, but also provide the most important information of aggregations of induction ma-chines in an electric power system. This basic knowledge of aggregations of the induction machines which are not available in the existing aggregate induction machine models can be used to assist in power system operation and planning. 179 The developed aggregate induction machine models with known circuit parameters allow induction machines to be aggregated when the machines' circuit parameters are available. The models include the most popular induction machines, such as single- and double-cage-rotor machines. The models break away from the traditional weighted aver-age modelling technique, improving the accuracy of machine representations. The developed aggregate load models were coded into computer programs using the high level language Ada 95. The simulation results obtained with the proposed models and the field test results agree well with each other. In addition, the results obtained with the new models and those obtained by solving the original systems without aggregating the loads are in agreement. Furthermore, The results obtained with the proposed models are more accurate than those obtained with the conventional load models. It can then be concluded that the proposed models can closely represent the behaviour of electrical loads in power systems. The following list gives some aspects that can be recommended for future developments: 1. Methods for accurate measurements of load characteristics. 2. Models for new and redesigned load devices. 3. Accurate representation of the non-linearity of network impedances. 4. Methods for modelling the impact of weather, time, day, and season on load composition. This concludes this thesis work. The research work is expected to be useful in the fu-ture development of power system computer programs, such as real-time power system simulators. 180 REFERENCES [I] J.R. Marti and K.W. Louie, "A Phase-Domain Synchronous Generator Model Including Saturation Effects", IEEE Trans, on Power Systems, Vol. 12. No. 1 February 1997. [2] K.W. Louie and J.R. Marti, "Effects of Non-Uniform Air-Gap Saturation During Unbalanced Faults in Synchronous Generators", Canadian Electricity Association, Conference of Supply & Delivery of Electricity and Power System Planning & Operation, April 1996. [3] C. Concordia and S. Ihara, "Load Representation in Power System Stability Studies", IEEE Trans, on PAS, Vol. PAS-101, No. 4, April 1982. [4] Kundur, P, Power System Stability And Control, McGraw-Hill, Inc., New York, 1994. [5] W.W. Price, K.A. Wirgau, A. Murdoch, J.V. Mitsche, E. Vaahdi, M.A. Ei-Kady "Load Modelling for Power Flow and Transient Stability Computer Studies", IEEE Trans. On Power System, Vol. 3, No. 1, February 1988. [6] IEEE Task Force on Load Representation for Dynamic Performance, "Load Representation for Dynamic Performance Analysis", IEEE Trans, on Power System, Vol. 8, No. 2, May 1993. [7] Dommel, H. W, State of the Art of Transient Stability Simulations for Electric Power Systems, The University of British Columbia, Vancouver, 1976. [8] Glover, J. D, Power System Analysis and Design, PWS-KENT Publishing Company, Boston, 1989. [9] Bergen, R, Power System Analysis, Prentice-Hall, Inc., New Jersey, 1986. [10] Anderson, P. M and Fouad, A. A, Power System Control and Stability, The Iowa State University Press, Ames, 1977. [II] Yu, Y. N, Electric Power System Dynamics, Academic Press, New York, 1983. [12] O'Leary, H. R, "Report to the President: The Electric Power Outages in the Western United States, July 2-3, 1996", Washington, August, 2, 1996. [13] Byerly, R. T and Kimbark, E, W, Stability of Large Electric Power Systems, IEEE Press, New York, 1974. 181 [14] P.W. Sauer, M.A. Pai, Power System Dynamics And Stability, Prentice Hall, New Jersey, 1998. [15] IEEE Task Force on Load Representation for Dynamic Performance, "Standard Load Models for Power Flow and Dynamic Performance Simulation", IEEE Trans, on Power Systems, Vol. 10, No. 3, August 1995. [16] M.H. Kent, W.R. Schmus, F.A. McCrackin, "Dynamic Modelling of Loads in Stability Studies", IEEE Trans, on PAS, Vol. PAS-88, No. 5, May 1969. [17] H.D. Chiang, J.C. Wang, C.T. Huang, Y.T. Chen, CH. Huang, "Development of a Dynamic ZIP-Motor Load Model from On-Line Field Measurements", Electrical Power & Energy System, Vol. 19, No. 7, 1997. [18] G.J. Berg, "Power-System Load Representation", Proc. IEE, Vol. 120, No. 3, March 1973. [19] S. Ihara, M. Tani, K. Tomiyama, "Residential Load Characteristics Observed at KEPCO Power System", IEEE Trans, on Power Systems, Vol. 9, No. 2 May 1994. [20] J.R. Ribeiro, F.J. Lange, "A New Aggregation Method for Determining Composite Load Characteristics", IEEE Trans, on Power Systems, Vol. PAS-101, No. 8, August 1982. [21] Adkins, B. and Harley, R. G., The General Theory of Alternating Current Machines, John Wiley & Sons, Inc., New York, 1975. [22] Lyon, W. V., Transient Analysis of Alternating Current Machinery, John Wiley & Sons, Inc., New York, 1954. [23] Slemon, G. R and Straughen, A, Electric Machines, Addison-Wesley Publishing Company, 1982. [24] Sen, P. C, Principles of Electric Machines and Power Electronics, John Wiley & Sons Inc., New York, 1989. [25] F. Nozari, M.D. Kankam, W.W. Price, "Aggregation of Induction Motors for Transient Stability Load Modeling", IEEE Trans, on Power Systems, Vol. PWRS-2, No. 4, November 1987. [26] S.A.Y. Sabir, D.C. Lee, "Dynamic Load Models Derived from Data Acouired During System Transients", IEEE Trans, on PAS, Vol. PAS-101, No. 9, Septermber 1982. 182 [27] Dommel, H. W, EMTP Theory Book, Microtran Power System Analysis Corporation, Vancouver, 1992. i [28] Arrillaga, J and Bradley, D. A and Bodger, P. S, Power System Harmonics, John Wiley & Sons, New York, 1985. [29] Lander, C. W, Power Electronics, McGraw-Hill Book Company, New York, 1987. [30] A.S. Sedra, K.C. Smith, Microelectronic Circuits, Holt, Rinehart And Winston, New York, 1987. [31] S.J.S. Jr. M.S. Roden, G.L. Carpenter, Electric Circuit Design, The Benjanin/ Commings Publishing Company, Inc., Santiago, 1987. [32] D.L. Powers, Boundary Value Problems, Academic Press, Inc., New York, 1979. [33] J.W. Nilsson, Electric Circuits, Addison-Wesley Publishing Company, New York, 1990. [34] J.W. Dettman, Applied Complex Variables, Dover Publications, Inc., New York, 1984. [35] C.W. Taylor, Power System Voltage Stability, McGraw-Hill, Inc., New York, 1994. [36] C.Y. Chiou, C.H. Huang, H.D. Chiang, J.L. Yuan, "Development of a Micro-Processor-Based Transient Data Recording System for Load Behavior Analysis", IEEE Trans, on Power Systems, Vol. 8, No. 1, February 1993. [37] R.B. Adler, C.C. Mosher, "Steady-State Voltage Power Characteristics for Power System Loads", IEEE paper 70 CP 706-PWR. [38] E.W. Kimbark, "Power System Stability", John Wiley & Sons, Inc., New York, 1976. [39] L.O. Chua, Nonlinear Network Theory, McGraw-Hill Book Company, New York, 969. [40] F.A. Benson, D. Harrison, Electric-Circuit Theory, Edward Arnold Ltd., London, 1975. [41] C.W. Cox, W.L. Reuter, Circuits,Signals,andNetworks, The Macmillan Copany, New York, 1969. 183 [42] A. Papoulis, Circuits and Systems: A Modern Approach, Holt, Rinehart and Winston, Inc., New York, 1980. [43] J.D. Irwin, Basic Engineering Circuit Analysis, Macmillan Publishing Company, New York, 1989. [44] M.H. Rashid, Power Electronics: Circuits, Devices, and Applications, Prentice Hall, Inc., New Jersey, 1993. [45] N. Mohan, T.M. Undeland, W.P. Robbins, Power Electronics: Converters, Applications, and Design, John Wiley & Sons, New York, 1989. [46] S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications, Inc., New York, 1993. [47] H. Flanders, Calculus, W.H. Freeman and Company, New York, 1985. [48] Power Engineering Guide, Siemens, 1980. [49] P.L. Alger, Induction Machines: Their Behavior and Uses, Gordon and Breach Science Publishers, New York, 1970. [50] M.G. Say, The Performance and Design of Alternating Current Machines, Sir Isaac Pitman & Sons, Ltd., London, 1949. [51] R.P.K. Hung, Synchronous Machine Models for Simulation of Induction Motor Transients, A Master Thesis, The University of British Columbia, Vancouver, 1995. [52] G.R. Fowles, Analytical Mechanics, Saunders College Publishing, New York, 1986. [53] V.L. Streeter, E.B. Wylie, Fluid Mechanics, McGraw-Hill Book Company, New York, 1985. [54] J.D. Manno, R.T.H. Alden, "An Aggregate Induction Motor Model for Industrial Plants", IEEE Trans, on PAS, Vol. PAS-103, No. 4, April 1984. [55] A.H.M. A. Rahim, A.R. Laldin, "Aggregation of Induction Motor Loads for Transient Stability Stability Studies", IEEE Trans, on Energy Conversion, Vol. EC-2, No. 1 March 1987. [56] M.M.A. Hakim, G.J. Berg, "Dynamic Single-Unit Representation of Induction Motor Groups", IEEE Trans, on PAS, Vol. PAS-95, No. 1, January 1976. 184 J.C. Wang, H.D. Chiang, C.L. Chang, A.H. Liu, C.H. Huang, C.Y. Huang, "Development of a Frequency-Dependent Composite Load Model Using the Measurement Approach", IEEE Trans, on Power Systems, Vol. 9, No. 3, August 1994. D.Q. Ma, P. Ju," A Novel Approach to Dynamic Load Modelling", IEEE Trans, on Power Systems, Vol. 4, No. 2, May 1989. R.J. Frowd, R. Podmore, M. Waldron, "Synthesis of Dynamic Load Models for Stability Stadies", IEEE Trans, on PAS , Vol. PAS-101, No. 1, January 1982. G.J. Rogers, D. Shirmohammadi, "Induction Machine Modelling for Electromagnetic Transient Program", IEEE Trans, on Energy Conversion, Vol. EC-2, No. 4, December 1987. 185 APPENDIX A Generation of the Input Data for the Induction Machine Model in the EMTP Before using the induction machine model in Microtran, the U.B.C. version of the EMTP, appropriate input data must be available. As shown in Figure A l , the induction machine model in Microtran needs the following input data which are calculated from the equivalent circuit parameters of an induction machine L*o — Ls (A.1) Ld = Ls + Lm (A.2) L/f = Lr\ +Lm (A.3) LDD — Lr\ + LR2 + Lfn (A.4) Mfo = Lr\ +Lm (A.5) Mdf = MDD = Lm (A.6) Lq - Ls + Lm (A.7) LQQ = Lri +Lri +Lm (A.8) MgQ - Lm (A.9) where L 0 , L D , L Q , L F F , L D D , LQQ, Mfu Mfd, and M q Q are the circuit parameters of a double-cage-rotor synchronous machine which is used to represent an induction machine in Microtran. Figure A l Equivalent circuit of a double-cage-rotor induction machine 186
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Aggregation of voltage and frequency dependent electrical loads Louie, Kwok Wai 1999
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Title | Aggregation of voltage and frequency dependent electrical loads |
Creator |
Louie, Kwok Wai |
Date Issued | 1999 |
Description | Electrical loads play a very important role in the behaviour of an electric power system. Since there is a tremendous number of different loads in the system, representing each load with its own model becomes impractical for system level studies. This thesis deals with the issue of aggregating loads to simplify system level studies. Six new and accurate aggregate static load models, a novel EMTP based load model, and four very accurate aggregate induction machine models have been developed. The proposed aggregate load models are voltage and frequency dependent and accommodate the different data formats of individual loads. By including the information of voltage and frequency dependence, the models can be used in larger ranges of studies than the conventional aggregate static load models, thus resulting in more accurate representations. The valid voltage range of the models is about 75% to 125% of rated voltage and, the valid range of frequency of the models is about 85% to 115% of rated frequency. The proposed EMTP load model represents a load with basic circuit elements. The model consists of two varied turns ratio transformers and two varied admittance RLC circuits. It represents the voltage dependence and the frequency dependence of a load separately, resulting in a much simpler load representation than a conventional load model. The model not only improves the accuracy of load representations, bus also broadens the EMTP application in studies other than the transient analyses, such as power flow studies. The proposed aggregate induction machine models have been developed based on the specifications and circuit parameters of individual machines. Since the specification of the machines are the most basic information of the devices, they provide a natural and accurate representation of the machines. The circuit parameters of the machines reflect the behaviour of the devices, they can be used to compose the machines under high and low frequencies, resulting in a simple and accurate machine representation. These two different aggregate models accommodate the data availability of individual machines. To verify the validity of the proposed load models, field tests from published literature are compared with computer simulations. The simulation results of some test systems obtained i) by the proposed load models, ii) by the conventional load models, and iii) by solving the original systems without aggregating their circuit components are also compared. The results of these comparisons prove that the proposed load models can represent loads in power systems more accurately than the existing load models. The proposed load models have been developed into computer software packages with the high level computer programming language Ada 95. |
Extent | 6756780 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-07-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065171 |
URI | http://hdl.handle.net/2429/10008 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1999-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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