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A framework for power system restoration Lindenmeyer, Daniel 2000

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A FRAMEWORK FOR POWER  SYSTEM  RESTORATION  by DANIEL L I N D E N M E Y E R Diplorri-Ingenieur, Universitat Karlsruhe (TH), Germany, 1996  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF  D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Electrical and Computer Engineering)  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A September 2000 © Daniel Lindenmeyer, 2000  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my w r i t t e n p e r m i s s i o n .  Department  of  The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada  Date  <T,  200/9  Abstract The problem of restoring power systems after a complete or partial blackout is as old as the power industry itself. In recent years, due to economic competition and deregulation, power systems are operated closer and closer to their limits. At the same time, power systems have increased in size and complexity. Both factors increase the risk of major power outages. After a blackout, power needs to be restored as quickly and reliably as possible, and consequently, detailed restoration plans are necessary. In recent years, there has also been an increasing demand in the power industry for the automation and integration of tools for power system planning and operation. This is particularly true for studies in power system restoration where a great number of simulations, taking into account different system configurations, have to be carried out. In the past, these simulations were mostly performed using power-flow analysis, in order to find a suitable restoration sequence. However, several problems encountered during practical restoration procedures were found to be related to dynamic effects. O n the one hand, simple rules that help to quickly assess these problems are needed. On the other hand, accurate modeling techniques are necessary in order to carry out time-domain simulations of restoration studies. In this work, the concept of a new framework for black start and power system restoration is presented. Its purpose is to quickly evaluate the feasibility of restoration steps, and if necessary, to suggest remedial actions. This limits the number of time-consuming time-domain simulations, based on the trial-and-error principle, and helps to efficiently find feasible restoration paths. The framework's principle is to subdivide the problem of assessing the multitude of different phenomena encountered during a restoration procedure into subproblems. These can be assessed by simple rules formulated in the frequency and Laplace domain. This thesis concentrates on the initial stages of restoration where three major problem areas are identified. The system frequency behavior after the energization of loads is assessed using analysis in the Laplace domain and simplified generator control system models. The occurrence of overvoltages is assessed in the frequency domain. Sensitivity analysis is used in order to find the most efficient network change that can be applied to limit overvoltages. In case time-domain simulations need to be carried out, a method based on Prony analysis and fuzzy logic helps to limit the overall calculation time. Problems related to motor starts are evaluated by rules formulated in the frequency and Laplace domain. For these studies, a new induction motor parameter estimation method is developed that helps to build more accurate motor models. All the proposed rules are validated using time-domain simulations based on actual system data. Of crucial importance in the restoration process is the black and emergency start of large thermal power plants with small hydro or gas turbines. These cases represent islanded ii  ABSTRACT  iii  system conditions with large frequency and voltage excursions that need careful investigation. It is shown how they can be simulated using the Electromagnetic Transients Program ( E M T P ) by means of an emergency motor start case whose simulation results are compared to measurements.  Contents Abstract  ii  List of Tables  . viii  List of Figures  ix  Acknowledgement  xiii  Quote  xiv  1 Introduction  1  1.1  General Background  1  1.2  Goals and Steps in Restoration  2  1.3  Problems in Restoration  4  1.4  Thesis Motivation and Objective  6  2 Literature Overview  8  2.1  Overview of Restoration Process  8  2.2  Active Power System Characteristics and Frequency Control .  9  2.3  Reactive Power System Characteristics and Voltage Control  10  2.4  Switching Strategies  10  2.5  Protective Systems and Local Control  11  2.6  Power System Restoration Planning  11  2.7  Power System Restoration Training  12  2.8  Power System Restoration Case Studies  12  2.9  Analytical Tools  12  2.10 Expert Systems .  13  iv  CONTENTS  y_  3 Structure of Framework  14  3.1  On-line and Off-Line Restoration Planning  15  3.2  Framework Database  15  3.3  Framework Rules  16  3.4  Framework Analytical Tools  17  3.4.1  Power Flow Analysis  18  3.4.2  Steady-State and Harmonic Analysis  19  3.4.3  Electromagnetic Transient Analysis  20  3.4.4  Stability Analysis  21  4 Frequency-Response Analysis 4.1  4.2  4.3  Overview of Frequency-Response Estimation Methods  23  4.1.1  Simple Frequency-Response Estimation Methods  23  4.1.2  Other Frequency-Response Estimation Methods  24  Frequency Behavior of Hydro Units for Static Load Pick-up  26  4.2.1  Assumptions for Model Development  26  4:2.2  Derivation and Verification of Equations  28  4.2.3  Example for Load Pick-up  31  4.2.4  Example for Load Rejection  32  4.2.5  Remedial Action  32  Summary  5 Voltage Response Analysis 5.1  5.2  5.3  22  34  35  Frequency-dependent Thevenin Impedance  37  5.1.1  Impedance Change between Bus and Neutral  39  5.1.2  Impedance Change between two Buses  40  5.1.3  Example  40  Sustained Overvoltages  41  5.2.1  Sensitivity Analysis  42  5.2.2  Example  45  Harmonic Overvoltages  46  5.3.1  Analysis and Assessment of Harmonic Overvoltages  47  5.3.2  Harmonic Characteristic of Transformers  51  5.3.3  Sensitivity Analysis  53  CONTENTS 5.4  6  E M T P Time-Domain Result Evaluation  60  5.4.1  Prony Analysis  61  5.4.2  Filter for Modes  63  5.4.3  Prony Analysis Results Evaluation  65  5.4.4  Reasoning Process  71  5.4.5  Example  73  5.5  Switching Transient Overvoltages  75  5.6  Summary  76  A u x i l i a r y System Analysis  77  6.1  Motor Parameter Estimation  77  6.1.1  Overview of Parameter Estimation Methods  77  6.1.2  Input Data  79  6.1.3  Nonlinear Optimization Procedure  79  6.1.4  Motor Model without Saturation  80  6.1.5  Motor Model with Saturation  86  6.1.6  Results  90  6.2  6.3 7  vi  Rules for Motor Start-up  92  6.2.1  Thevenin Equivalent  92  6.2.2  Estimation of Voltage Drop  94  6.2.3  Estimation of Inrush Current  95  6.2.4  Estimation of Start-up Time  95  6.2.5  Estimation of Thermal Behavior  98  6.2.6  Estimation of Frequency Drops  100  6.2.7  Remedial Actions  103  Summary  103  E m e r g e n c y Start Case S t u d y  105  7.1  Modeling of the System  105  7.1.1  Electrical System  105  7.1.2  Synchronous Generator  106  7.1.3  Excitation System Model  106  7.1.4  Governor System Model  108  7.1.5  Turbine Model  108  CONTENTS 7.1.6 7.1.7  vii Initialization of Excitation System, Governor System, and Turbine Models  109  Induction Motor Model  Ill  7.2  Simulation Results  Ill  7.3  Summary  114  8 Conclusions and Recommendations for Future Work  116  Bibliography  118  A Determination of Motor Load Characteristics from Measurements  137  B Soft Start of Motors  139  List of Tables 1.1  Frequency thresholds for Cyprus power system  5  1.2  Voltage limits during restoration for Hydro-Quebec power system  5  4.1  Governor data  24  4.2  Comparison of time constants  29  5.1  Total sensitivity vector  45  5.2  Index for harmonic overvoltage assessment  51  5.3  Sensitivity vectors for /=120 Hz  58  5.4  Total sensitivity vectors  58  6.1  Induction motor nameplate data  85  6.2  Induction motor characteristics  86  6.3  Induction motor parameters in per unit  87  6.4  Deviation of induction motor parameters  88  6.5  Induction motor parameters in per unit  90  viii  List of Figures 1.1  Power System Operating States  1  1.2  Power system restoration goals  3  3.1  On-line database for system restoration framework  16  3.2  Off-line database for system restoration framework  17  3.3  Assessment of feasibility of restoration steps  18  3.4  Overview of analytical tools  19  4.1  Generator supplying isolated load  23  4.2  Governor block diagram  25  4.3  Frequency deviation for 14 M W load pick-up  25  4.4  Overview of algorithm  27  4.5  Control block diagram of hydro unit  28  4.6  Simplified control block diagram of hydro unit  29  4.7  Comparison between full and simplified model  30  4.8  Frequency deviation for 14 M W load pick-up  32  4.9  Frequency deviation for 14 M W load rejection  33  4.10 Minimum frequency as a function of load  33  4.11 Time at which minimum frequency occurs as a function of load  34  5.1  Overview of algorithm  36  5.2  Overvoltage-versus-time capabilities of typical 230 kV system equipment  5.3  Thevenin equivalent circuit with variable parameters  38  5.4  System at the beginning of a restoration procedure  41  5.5  60 Hz impedance at bus B3 as function of number of generators  41  5.6  540 Hz impedance at bus B3 as function of number of generators  42  5.7  Assessment whether power flow is needed  43  5.8  System during restoration procedure  45  ix  . .  37  LIST O F F I G U R E S 5.9  x  Voltage at bus B7 as function of &Y /Y k  where Y - (2 - j • 0.884) mS . . . .  5.10 Impedance at bus B3 for Cases 1 to 4  46 47  5.11 Harmonic content of transformer inrush current at bus B3 for Cases 1 to 3  .  48  5.12 Voltage at bus B3 for Case 1  48  5.13 Voltages at bus B3 for Case 1 and Case 2  49  5.14 Voltages at bus B3 for Case 3 and Case 4  49  5.15 Harmonic content of transformer inrush current at bus B3 for Case 4  . . . .  50  5.16 Ideal saturation characteristic  52  5.17 Transformer saturation characteristic  53  5.18 Harmonics of transformer inrush current  54  5.19 System during restoration procedure  57  5.20 Impedance Z (120Hz) as a function of resistive changes  58  5.21 Impedance Z#i(120Hz) as a function of inductive changes  59  5.22 Harmonic characteristic of transformer  59  5.23 Weighting function  60  5.24 Impedance ZB\ for resistive changes as a function of frequency  60  5.25 Impedance Z \  61  B1  B  for inductive changes as a function of frequency  5.26 Overview of algorithm  62  5.27 Reasoning function for filter  64  5.28 Membership function for relative appearance  65  5.29 Membership function for signal-to-noise ratio (SNR)  65  5.30 Monotonic reasoning in order to determine total relevance of mode  66  5.31 Amplitude membership function  67  5.32 Damping membership function  ;  5.33 Membership function for tendency of damping  67 68  5.34 Monotonic reasoning in order to determine total "closeness to steady state" of mode  69  5.35 Amplitude membership function  70  5.36 Damping membership function .  70  5.37 Tendency of damping membership function  71  5.38 Membership function for variance of damping  72  5.39 Final reasoning process  72  5.40 E M T P signal and predicted signal .  73  5.41 E M T P signal and predicted signal for m> 15  73  LIST O F F I G U R E S  xi  5.42 Amplitude of 180 Hz mode .  74  5.43 Damping of 180 Hz mode  74  6.1  Induction motor model without saturation  80  6.2  Induction motor model with stator part replaced by Thevenin equivalent circuit 83  6.3  Values for objective function  86  6.4  Number of iterations  87  6.5  Induction motor model with saturation  88  6.6  Calculation of currents and describing functions  89  6.7  Current characteristic  91  6.8  Powerfactor characteristic  91  6.9  Torque characteristic  92  6.10 Overview of algorithm  93  6.11 Sample system for motor start  94  6.12 Excitation system of sample system  ; . .  94  6.13 Thevenin equivalent circuit of sample system  94  6.14 Voltage drop during motor start-up  95  6.15 Inrush current during motor start-up  .  96  6.16 Mechanical and electrical torque for / = 1.0 p.u. and V = 0.5 p.u  98  6.17 Current for / = 1.0 p.u. and V = 0.5 p.u  98  6.18 Rotor speed during motor start-up  99  6.19 Motor relay characteristic compared to averaged starting current  99  6.20 Active power during motor start-up compared to approximations  102  6.21 Frequency deviation for motor load pick-up  103  7.1  One-line diagram of the Oconee emergency electrical power system (ESF) . .  106  7.2  Block diagram of the Keowee generator excitation system  107  7.3  Block diagram of the Keowee governor system  108  7.4  Block diagram of the Keowee hydro turbine  109  7.5  Transient current characteristic, HPI motor  112  7.6  Transient voltage characteristic, HPI motor  112  7.7  E S F test: Comparison between measured and simulated frequency  113  7.8  E S F test: Comparison between measured and simulated voltage  113  7.9  E S F test: Comparison between measured and simulated current  114  LIST O F F I G U R E S  xii  7.10 E S F test: Comparison between measured and simulated field voltage  . . . .  114  7.11 E S F test: Comparison between measured and simulated field current  . . . .  115  7.12 E S F test: Comparison between measured and simulated power  115  A.l  138  Thevenin equivalent for locked-rotor operation (t = 0+)  A. 2 Thevenin equivalent for full-load operation ( £ - > o o )  138  B. l  Start-up time at voltage V = 1.0 p.u  140  B.2  Start-up time at frequency / = 1.0 p.u  140  B.3  Start-up current at voltage V = 1.0 p.u  141  B.4  Start-up current at frequency / = 1.0 p.u  141  Acknowledgement I wish to express my sincere gratitude to all the people who have assisted me in the accomplishment of this thesis work: To my extraordinary, beautiful, and magnificent wife Lise. Thank you very much for your love, understanding, never-ending patience, and constant encouragement. Thank you very much for the great time and memories we share. To my parents for the infinite love, support, and encouragement through all my dreams, adventures, and life. To my parents in-law, and all my relatives and friends who supported me with their love, kindness, and happiness. I would like to express my deepest gratitude to my thesis supervisors, Dr. Hermann W. Dommel and Dr. Prabha Kundur, whom I respect and admire tremendously. In particular, I would like to thank Dr. Prabha Kundur for the great opportunity to carry out a part of my work at Powertech Labs. Thank you both very much for your encouragement and support. Special thanks to Dr. A l i Moshref for helping me to find the topic for this thesis, for his friendship, encouragement, support, and technical guidance. His expertise and background in the area of power systems not only made this work possible but also tremendously enriched my professional life. Many thanks to Dr. Jose Marti for introducing me to the principle of the E M T P in his outstanding course at U B C . To Dr. Tak Niimura whose expertise in artificial intelligence and fuzzy logic helped me enormously during the development of the results evaluation algorithm. To all the other former and current members of the U B C Power Group for their friendship, support, encouragement, and cooperation. I am much indebted to Dr. Amir M . Miri, whom I admire and respect tremendously, for his support and encouragement to go to Canada and pursue a Ph.D. program. The financial assistance of the Natural Science and Engineering Council of Canada, and of B.C. Hydro & Power Authority, through funding provided for the N S E R C - B . C . Hydro Industrial Chair in Advanced Techniques for Electric Power System Analysis, Simulation and Control, are gratefully acknowledged. Many thanks to the engineers at Powertech Labs for providing me with the system data and technical expertise needed for my studies, and to Chris Schaeffer and Aldean Benge at Duke Energy for their cooperation and for letting me use the results of our emergency start case study in this thesis. Vancouver, B C , Canada September 7, 2000  Daniel Lindenmeyer  xiii  Everything should be made as simple as possible, but not simpler." Albert  Einstein  xiv  Chapter 1 Introduction 1.1  General Background  The operating conditions of power systems can be classified as five different states: normal, alert, emergency, in extremis, and restorative (see Figure 1.1) [73, 143]. In the normal  "E": "I":  Equality constraints Inequality constraints Negation  Figure 1.1: Power System Operating States operating state, all system variables are within the normal range and no equipment is being overloaded. The system can withstand a contingency without its security being threatened, and both equality and inequality constraints are satisfied. In the alert operating state, the system's security level is reduced, but the equality and inequality constraints are still 1  1.2. Goals and Steps in Restoration  2  satisfied. Even though the system is still operated within allowable limits, a contingency might lead to an emergency state, or, in case of a severe disturbance, to an in extremis state. In the emergency state, the inequality constraints are violated when the system operates at a lower frequency with abnormal voltages, or with some portion of the equipment overloaded. After the system has entered an emergency state as a consequence of a severe disturbance, it may be led back to an alert state by applying emergency control actions. If these measures are not applied successfully, the system may enter an in extremis state where both inequality and equality constraints are violated. Cascading outages might lead to partial or complete shut-downs of the system [143]. The system condition where control action is being taken to reconnect all the facilities, to restore system load, and to eventually bring the system back to its normal state, is called the restorative state. In this state the inequality constraints are satisfied and the equality constraints are violated. Due to the deregulation of the power industry worldwide, and the almost revolutionary changes in the industry structure, power systems are operated closer and closer to their limits. Furthermore, in recent years, they have grown considerably in size and complexity. This has led to an increasing number of major blackouts, such as the large power outages on the West Coast of North America in 1996 or the Brazilian blackout in 1999. Disturbances that can cause such power blackouts are natural disasters, line overloads, system instabilities, etc. Furthermore, temporary faults such as lightning, even if cleared immediately, can initiate a "domino effect" that might lead to a partial or complete outage, involving network separation into several subsystems, and load shedding. After power blackouts, the system has to be restored as quickly and efficiently as possible. In this restoration, the initial cause of the outage is of secondary importance and it might be futile to investigate it [2]. Although power outages differ in cause and scale, virtually every utility has experienced blackouts and gone through restoration procedures. As a consequence, there is an increasing interest in systematic restoration procedures, tools, and models for on-line restoration, as well as for restoration planning. A speedy, effective, and orderly restoration process reduces the impact of a power outage on the public and the economy, while reducing the probability of equipment damage [2].  1.2  Goals and Steps in Restoration  Even though each power blackout and restoration scenario is a unique event, there are certain goals and steps that are common in all restoration procedures. The goals in restoration, as generally defined in [75, 153, 155], are shown in Figure 1.2. They involve almost all aspects of power system operation and planning. Each restoration procedure that follows a complete or partial blackout of a power system can be subdivided into the following steps [2, 36]: 1. Determination  of System Status. In this stage, the boundaries of energized areas are  1.2. Goals and Steps in Restoration  3  Figure 1.2: Power system restoration goals  identified, and frequencies and voltages within these areas are assessed. Furthermore, in cases where no connections to neighboring systems exist, black start (or cranking) sources are identified in each subsystem and critical loads are located. 2. Black Start of Large Thermal Power Plants. Large thermal power plants have to be restarted within a certain period of time. For example, hot restart of drum type boilers is only possible within thirty minutes. If it cannot be accomplished and the boiler is not available for four to six hours, a cold restart has then to be performed. Thermal power plants can be restarted by means of smaller units with black start capability, i. e. power plants that can be started and brought online without external help and within a short period of time. Power plants with black start capability are hydro, gas, or diesel power plants. After such a power plant has been brought to full operation, a high voltage path to a large thermal power station is built and the thermal unit's auxiliaries which are driven by large induction motors are started. Along the path, "ballast" loads have to be supplied to maintain the voltage profile within acceptable limits and to prepare a load base for the thermal units. Additional smaller units can also be brought online through the path to improve system stability [46]. 3. Energization of Subsystems. In case of a large power blackout, it is advantageous in most cases to section the power system into subsystems in order to allow parallel restoration of islands, and to reduce the overall restoration time. Within each subsystem, starting from a large thermal power station, the skeleton of the bulk power system is energized. Paths to other power plants and to the major load centers are built, and loads are energized to firm up the transmission system. At the end of this step, the network has sufficient power and stability to withstand transients as a result of further load pick-up and addition of large generating units.  1.3. Problems in Restoration  4  4. Interconnection of Subsystems. In this stage of the power system restoration process, the subsystems are interconnected. Eventually, remaining loads are picked up and the system performs its transition to the alarm or normal state. Among the above four general steps of the restoration process, the second and third step are the most critical ones. Mistakes in these stages can lead to unwanted tripping of generators and load shedding due to extensive frequency and voltage deviations, and consequently to a recurrence of the system outage. Because of time-critical boiler-turbine start-up characteristics and possible further equipment damage, extensively prolonged restoration times may occur, resulting in a much higher impact on the public and industry, and an increased damage to the economy.  1.3  Problems in Restoration  During power system restoration, a multitude of different phenomena and abnormal conditions may occur. The problems encountered during restoration can be subdivided into three general areas [2, 36, 90, 91, 96]: 1. Active Power Balance and Frequency Response. During the restoration process, two different aspects of this type of problem can be identified. The first one is the black start of large thermal power plants, where large auxiliary motor loads are picked up, using relatively small hydro generators, diesel, or gas turbines. This can result in large frequency excursions and consequently in an activation of underfrequency load shedding relays and, in the worst case, in the loss of already restored load and a recurrence of the blackout [36]. Due to the importance of this problem, it is defined as a problem area in its own, as discussed further below. The second aspect is the pick-up of cold loads. When the network is extended, power plants are added to the generation, and loads are picked up, it is necessary to preserve a balance between active load and generation.. In the case this balance is disturbed, frequency deviations result. If these are extensive, an unwanted activation of loadshedding schemes can occur, and newly connected loads can be lost again. In the worst case, the frequency decline may reach levels that can lead to the tripping of steam turbine generating units as a consequence of the operation of underfrequency protective relays. This is due to the fact that the operation of steam turbines below a frequency of 58.8 Hz is severely restricted as a result of vibratory stress on the long low-pressure turbine blades [143]. Thus, in order keep the frequency deviations within allowable limits, the load increments should not exceed a certain level. However, if the load increments are too small, the overall restoration time will be unnecessarily prolonged [5, 46, 90]. As an example for an islanded system, Table 1.1 shows typical load shedding frequency limits for the (50 Hz) Cyprus power system [39]. The table indicates that at a frequency decline of 3 Hz almost all of the load will be disconnected.  1.3. Problems in Restoration  5  Table 1.1: Frequency thresholds for Cyprus power system Frequency / [Hz] Load shedding / [ % Delay / [s] Cumulative shed / [ % ]  49.0 15 0.2 15  48.8  48.4  47.8  47.0  20 0.2 35  25  10 0.35 70  10 0.35 80  0.35 60  2. Reactive Power Balance and Voltage Response. Analogous to the active power balance it is necessary to maintain a balance in reactive power. High charging currents, originating from lightly loaded transmission lines, can lead to the violation of generator reactive capability limits and to the occurrence of sustained (power frequency) overvoltages. These may cause underexciation, selfexcitation, and instability. Sustained overvoltages can also cause the overexcitation of transformers and the generation of harmonic distortions. Transient overvoltages are a consequence of switching operations on long transmission lines, or of switching of capacitive devices, and may result in arrester failures. Harmonic resonance overvoltages are a result of system resonance frequencies close to multiples of the fundamental frequency in combination with the injection of harmonics, mainly caused by transformer switching. They may lead to long-lasting overvoltages, resulting in arrester failures and system faults. Due to the small amount of load connected to the system, especially at the beginning of a restoration process, the voltage oscillations are lightly damped and can last for a long time, reaching very high amplitudes. This effect can be aggravated by transformer overexcitation as a result of sustained overvoltages, and power electronics [90]. As an example, the tolerable voltage deviations during restoration for the Hydro Quebec power system are shown in Table 1.2 [172]. It shows that in the case of temporary overvoltages the overvoltage duration has to be taken into account in addition to the amplitude. Table 1.2: Voltage limits during restoration for Hydro-Quebec power system Steady-state voltages / [p.u.] Switching overvoltages / [p.u.] Temporary overvoltages / [p.u.]  0.9 < V < 1.05  V<1.8 V < 1.5 (8 cycles)  3. Auxiliary Systems. The auxiliaries of large thermal power plants are essentially large induction motors driving pumps and fans. When they are energized during black or emergency starts, several problems can occur. The high reactive currents that are drawn during start-up may lead to voltage depressions which can result in overheating and permanent damage of machine windings. Furthermore, the high reactive inrush  1.4. Thesis Motivation and Objective  6  currents can exceed the reactive capability limits of generators which may trip due to the operation of underexcitation limiters [8, 95, 102]. In severe cases, the decrease in accelerating torque as a result of reduced voltage or frequency may lead to a failure in bringing the motor up to its rated speed. The startup of large auxiliary motors may also lead to frequency dips, causing unwanted load shedding, or generator tripping [36, 102].  1.4  Thesis Motivation and Objective  We can conclude from the previous sections, that a number of complex and serious problems need to be resolved during power system restoration. Detailed analysis during both the system planning stage and during on-line restoration is therefore necessary. As will be shown in the literature survey in Chapter 2, most of the research work that has been done so far in the area of power system restoration focuses on the application of artificial intelligence techniques in restoration path development. Expert systems or similar approaches, however, are not always easily generalized since circumstances and philosophies are different for each utility. The evaluation of the feasibility of restoration steps has been mainly assessed by means of (quasi-)steady-state analysis while ignoring dynamic effects. However, there is a trend in industry to include dynamic effects in power system restoration planning. In order to take these effects into consideration, a number of additional analytical tools, such as transient stability and electromagnetic transient programs have to be utilized, and new models and modeling techniques need to be developed to allow for the simulation of extreme off-nominal frequency and voltage conditions [35, 36]. The multitude of different phenomena and abnormal conditions during restoration makes it impractical to solve all problems at once, and the combinatorial nature of restoration problems makes it difficult or even impossible to investigate the feasibility of every restoration step combination. Applying the above mentioned simulation tools results in a large amount of time necessary to build suitable sequences during restoration planning, and makes their application in on-line restoration almost infeasible due to time limitations. The objective of this thesis is to give simple and approximate rules to assess the feasibility of restoration steps, and to provide modeling techniques for the simulation of abnormal voltage and frequency conditions. The rules provide operators with a simple and speedy methodology during on-line restoration, and limit the number of time-consuming simulations based on the trial-and-error principle during restoration planning. In Chapter 2, a comprehensive overview of the literature published in the area of restoration during the last two decades is given. Chapter 3 presents the general structure of the proposed framework. In Chapter 4 rules that deal with the estimation of frequency responses of prime movers are introduced. Chapter 5 describes methods that help to estimate and control overvoltages, and to shorten calculation times during time-domain simulations. Chapter, 6 introduces rules for the start-up of induction motors and a technique that helps to build more accurate induction motor models from manufacturer data. All rules and methods are  7  1.4. Thesis Motivation and Objective  validated using examples based on practical system data . Chapter 7 explains the model development for E M T P time-domain simulations of abnormal system conditions, by means of a practical emergency start case study whose results have been verified by measurements . Chapter 8 finally summarizes the contributions of this thesis and gives an outlook on future work to be done. 1  2  1  2  System data has been kindly provided by Powertech Labs Inc., Surrey, B C , Canada.  System data and measurements have been kindly provided by Duke Energy, North Carolina, USA.  Chapter 2 Literature Overview This chapter gives a literature survey that provides an overview of the relevant areas in restoration. It is subdivided into ten different topics. The first topic provides a general overview of the restoration process. It is followed by a discussion of active and reactive power system characteristics, and control of frequency and voltages during restoration. Then, topics covering the basic restoration switching strategies are explained, and protective system and local control problems are discussed. After overviews of power system restoration planning and of case studies, the training of operators in restoration is reviewed. The final two topics cover the applications of analytical tools and expert systems in power system restoration. The restoration of distribution systems is not specifically addressed in this survey.  2.1  Overview of Restoration Process  This section covers publications that give a general introduction to the restoration process, as well as reports by technical committees dealing with system restoration. In a typical restoration procedure, the stages of the restoration process are summarized as follows [2, 3, 15, 36, 73, 110, 119, 189]: in the first stage, the system status is assessed, initial cranking sources are identified, and critical loads are located. In the following stage, restoration paths are identified and subsystems are energized. These subsystems are then interconnected to provide a more stable system. In the final stage, the bulk of unserved loads is restored. In 1986 the Power System Restoration Task Force was established by the I E E E PES System Operation Subcommittee in order to review current operating practices, and to promote information exchange. Its first two reports [90, 91] give a general overview and a comprehensive introduction to power system restoration. Restoration plans, active and reactive power system characteristics, and various restoration strategies are reviewed. Furthermore, a survey on selected power disturbances and their restoration issues are given. These restoration issues are further discussed by the same authors in [3]. An international survey by C I G R E Study Committee 38.02.02 (modeling of abnormal conditions) on black start and power system restoration in 1990 [35], and a paper based 8  2.2. Active Power System Characteristics and Frequency Control  9  on this survey in 1993 [36], identified the needs of the power industry for system restoration planning. These papers give an overview of black start and restoration methods. In addition, a number of examples based on actual operating experiences are given, and requirements for modeling and simulation are recommended. Another C I G R E Task Force 38.06.04 investigated expert system applications for power system restoration [37]. Other publications that provide a general overview of power system restoration can be found in [2, 15, 92, 94, 96].  2.2  Active Power System Characteristics and Frequency Control  An overview of different types of generating units, and their characteristics relevant to restoration is provided in [90]. Special emphasis is placed on the treatment of steam units in [45], whereas [100] focuses on nuclear power plants and provision for off-site power during restoration. Depending on the available cranking power, thermal units are either restarted hot or cold. Hot restarts allow for start-up with hot turbine metal temperature, whereas cold restarts require a slow start-up in order to keep the turbine metal temperature changes within given limits [96]. These characteristics result in different start-up times for different types of generating units that have to be estimated and taken into account during restoration [2, 7, 49, 90, 96, 116]. The black start of generating units and subsequent cranking of other large thermal units is an important topic in restoration [35, 36], since mistakes in this early stage can lead to prolonged start-up times of thermal power plants, and consequently to a significant delay in the restoration process. A number of publications deal with this topic: a method for the identification of black start sources is described in [155], and the proper start-up sequence of power plants is discussed in [3, 96, 151, 155]. Several case studies show how gas turbines [102, 160, 203] or hydro units [46, 80, 149, 150, 160] were utilized as black start sources. In addition, a number of papers deal with the treatment of nuclear power plants after a blackout, using on-site diesel engines [74, 76, 232] or combustion turbines [102, 177] as emergency supply sources. The pick-up of cold loads is discussed in a recently published doctoral thesis [12], and in a number of other papers [28, 31, 96, 103, 205, 233]. Picking up heavy loads during the initial stages of restoration can lead to large frequency dips beyond a point of no return. The prediction of frequency dips and the optimal distribution of generator reserves are subject of [5, 49, 133]. Load shedding schemes that can be utilized during restoration are introduced in [4, 39, 152, 151, 198, 226].  2.3. Reactive Power System Characteristics and Voltage Control  2.3  10  Reactive Power System Characteristics and Voltage Control  The overvoltages of concern during power system restoration are classified as: sustained power frequency overvoltages, switching transients, and harmonic resonance overvoltages. They may lead to failure of equipment, such as transformers, breakers, arresters, etc. [95], and thus need to be analyzed thorougly. Systematic procedures dealing with the control of sustained overvoltages by means of optimal power flow programs can be found in [87, 88, 114, 172], and methods that deal with harmonic overvoltages in [147, 148, 172, 176]. Simple and approximate methods that deal with the evaluation of transient and sustained overvoltages, and asymmetry issues during transmission line energization are discussed in [1, 9]. The preparation of the network for reenergization and the energization of high voltage transmission lines during restoration are discussed in [2, 3, 80], and the special treatment of cables in underground transmission systems in [96]. The application of shunt reactors for overvoltage control, particularly in the cases of lightly loaded transmission lines in the beginning of the restoration procedures, is treated in [11, 12, 13, 80, 148, 172]. Another issue of importance, especially in the early stages of the restoration process, is the lead and lag reactive power capability limits of synchronous machines. These limits are important for high charging current requirements of lightly loaded transmission lines, or for the high reactive currents drawn by the start-up of power plant auxiliary motors [6, 8, 49, 102]. The generator reactive power resources must therefore be optimized in such cases [10, 155].  2.4  Switching Strategies  Two different switching strategies can be applied during restoration. For the "all open" strategy, all breakers are opened immediately after the loss of voltage, whereas for the "controlled operation" strategy only selected breakers are opened [3, 96]. A great amount of research work that has been done in power system restoration concentrates on finding suitable restoration paths. Most of these approaches are based on artificial intelligence techniques that try to capture an operator's knowledge [68, 77, 86, 106, 123, 130, 131, 159, 181, 195]. They are mostly restricted to specific systems and restoration philosophies. A general approach to this problem that subdivides each restoration process into generic restoration actions common to all utilities is proposed in [66, 151, 228]. Restoration paths have to be validated with respect to different phenomena. A n overview of steady-state and time-domain based tools that can be utilized for this purpose can be found in [2, 15, 98, 227]. Cases where simulation tools were actually applied to the validation of restoration paths can be found in [82, 169, 207].  2.5. Protective Systems and Local Control  2.5  11  Protective Systems and Local Control  The continual change in power system configurations and their operating conditions during restoration might lead to undesired operation of relays, since their settings are optimized for normal operating conditions. A n overview of resulting delays, possible modifications of relays, and other protective system issues is provided in [101]. Relay schemes that allow for low frequency isolation, whereby local generation and local load are matched in order to avoid extensive delays due to the otherwise necessary complete generator shutdowns, and controlled islanding schemes are treated in [3, 90]. A network protection scheme that focuses on stability issues is introduced in [194]. In cases where a dramatic decline in frequency occurs during the restoration process, it is necessary to reduce the amount of load that is connected, which can be accomplished by the application of underfrequency load shedding schemes [39, 134]. In closing network loops, e.g., reconnecting two subsystems, sometimes a significant standing phase angle difference (SPA) appears across the circuit breakers. This difference has to be reduced in order to close the circuit breaker without causing instability of the system [81, 98, 229].  2.6  Power System Restoration Planning  The organization and implementation of a restoration plan will substantially determine its success. Reports that deal with the actual deployment of restoration plans can be found in [73, 184]. A n overview of how restoration tasks can be shared most efficiently between operator and supporting computer systems is given in [94, 234]. Telecommunication issues during restoration are discussed in [65, 96, 184], and alarm issues in [92, 96]. One can distinguish between two different restoration approaches: the "bottom-up", and the "top-down" restoration strategies [2, 90]. For the "top-down" approach, the bulk power transmission system is established first, using interconnection assistance or hydro plants with large reactive absorbing capability. Subsequently, transmission stations and the required substations are energized, generators are resynchronized, and loads are picked up [2, 68, 87, 88, 90, 119, 172, 204]. For the "bottom-up" strategy, the system is first divided into subsystems, each with black-start capability. Then, each subsystem is stabilized, and eventually the subsystems are interconnected [2, 73, 90, 110, 111, 160]. The verification of steady-state models used for power system restoration planning is discussed in [111]. A number of publications describe the modeling of boilers, turbines, generators, controls, motors, etc. for black start studies in the time-domain, using stability programs [46, 74, 177, 203], or the Electromagnetic Transients Program (EMTP) [76, 232]. Special emphasis on the modeling of transmission lines for black start studies is given in [26], and the modeling of steam plants is treated in [45]. More information on analytical tools for restoration planning is given in Section 2.9.  2.7. Power System Restoration Training  12  2.7 Power System Restoration Training In order to provide operators with the necessary experience to confidently deal with timecritical restoration problems, thorough training is essential. A general overview of operator training techniques for power system restoration is given in [92, 97], and methods showing how restoration drills can be conducted and evaluated in practice are described in [224, 225]. Interactive and realistic operator training can be accomplished by means of an operator training simulator (OTS) [3, 55, 94, 167, 192, 208, 209, 210, 219, 236]. O T S can also be used to verify restoration plans [124], or in combination with knowledge-based systems for power system restoration planning [75, 132, 189]. Information on simulator-expert-system combinations that are specifically designed to train operators at utilities and to replace human instructors, can be found in [32, 33, 34, 93, 113, 126, 138, 193].  2.8  Power System Restoration Case Studies  Most of the published case studies come from North America. A number of black start studies can be found in [74, 76, 150, 203, 232]. The restoration of large power systems in North America for Pacific Northwest, Ontario-Hydro and Hydro-Quebec is covered in [18, 87, 88, 172, 196, 204], and a report dealing with the restoration of a metropolitan electrical system in [110]. A Mexican study of restoration policies and their application is treated in [73]. European case studies describe restoration experiences in French, Greek, and Swedish Systems [12, 40, 69, 119], Italy [46, 160], Slovenia [177], and Germany [218].  2.9 Analytical Tools An overview of analytical tools and their application for solutions of power system restoration problems is given in [36, 98]. The analytical tools can be subdivided into different categories depending on the frequency range of their application. The first type of tool is based on steady-state analysis: (optimum) power flow programs [87, 88, 114, 125] are the most basic tools used in system restoration. A n overview of how they can be applied to different types of restoration problems is provided in [227], and applications to the control of sustained overvoltages can be found in [95, 172]. Other tools, based on steady-state models, allow frequency scans that can be used to investigate and control resonance conditions during power system restoration [95, 148, 172]. Operator training simulators (OTS) can be regarded as quasi steady-state tools, since in addition to analyzing the power system's electrical behavior using power flow methods, they allow taking into account long-term dynamics, i.e. the electromechanical and mechanical aspects of power systems. Analytical tools that are applied to study frequency transients of short-, mid-, and long-term range are introduced in [69, 207]. Electromagnetic transients during restoration are investigated using the E M T P [26, 95] or similar programs [169].  2.10. Expert Systems  13  Methods that apply stability and security methods to restoration can be found in [65, 185, 186]. Other analytical tools that are of importance in restoration are short-circuit programs [98], tools for the reduction of standing phase angles (SPA) [81, 229], or tools that help to identify system islands [145]. There is an increasing demand in the power industry to integrate existing tools and to develop combination tools that allow studying different phenomena over a broad range of frequency [36, 98]. Attempts in integration or combination of analytical tools, and their applications to practical problems, are described in [69, 75, 82, 207].  2.10  Expert Systems  Power system restoration problems are of a combinatorial nature, and their solution is often based on the operator's knowledge and experience. Consequently, it is not surprising that most of the research that has been done in the area of system restoration has concentrated on artificial intelligence applications. A general bibliographical survey on the application of expert systems to electric power systems can be found in [235]. References [197, 237] give an overview of expert system applications in power system operation. A n international survey among utilities that investigated restoration expert systems and their application is presented in [37]. A number of publications address the requirements for knowledge-based systems, and give an overview of how expert systems can be applied during different stages of power system restoration [94, 99, 164]. The development of expert systems for restoration requires the transfer of operator knowledge into heuristic rules. That process has been the topic of a great number of publications: [25, 32, 43, 47, 51, 66, 70, 71, 72, 104,106,108, 109,112,113,114,115, 116,117,118, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 137, 138, 139, 140, 141, 152, 154, 155, 158, 159, 165, 166, 173, 174, 180, 181, 188, 195, 198, 199, 212, 214, 215, 217, 220, 221, 226, 227, 237]. Practical implementations of prototype knowledge-based systems in energy management systems can be found in [37, 48, 56, 58, 75, 86, 120, 122, 123, 128, 132, 146, 156, 159, 168, 170, 189, 202]. To reduce the number of rules of an expert system, mathematical programming [131, 174, 200] or other analytical optimization methods [173, 175, 180, 181, 200, 201] can be applied in combination with knowledge-based systems. The verification of restoration solutions provided by expert systems is accomplished using an integration with time-domain simulators [47, 75, 124, 132, 189]. Other artificial intelligence methods that are applied to black start and power system restoration problems are the use of petri nets [68, 77, 228], the application of fuzzy logic to restoration automation [147], the utilization of genetic algorithms for load restoration [133], the generation of switching sequences [135, 179], or as a hybrid approach of an expert system, the application of neural networks [108, 237].  Chapter 3 Structure of Framework Each restoration case is highly complex and unique, which makes it difficult to develop systematic restoration procedures, and to find methods that can be generalized in order to be applied to other systems and scenarios. However, restoration cases can be considered as a succession of simple restoration steps that are common in all restoration procedures, as described in more detail in [228]. The same principle is true when the feasibility of restoration steps needs to be assessed: the investigation of an aggregation of complex phenomena can be simplified by defining different problem areas that are common in all restoration scenarios and by developing rules that address specific problems during restoration. In this thesis project we focus mainly on problems that are related to the initial phases of a restoration process. However, the same principles can be extended to include other problem areas, such as the analysis of the reintegration of subsystems. Three different areas are identified that are considered to be important in the initial stages of a restoration process: 1. Voltage-response analysis • Steady-state overvoltages after changes in network • Harmonic overvoltages during transformer / line switching • Transient overvoltages during line switching 2. Frequency-response analysis • Frequency drops due to static load pick-up 3. Auxiliary system analysis • Voltage drops during motor starts • Inrush currents during motor starts • Thermal overload behavior during motor starts • Frequency drops during motor starts 14  3.1. On-line and Off-Line Restoration Planning  15  In the following, we explain the assumptions that underly the proposed framework, describe the general idea behind the rules that deal with the above problems, and give an overview of the analytical tools that are used within the framework.  3.1  On-line and Off-Line Restoration Planning  We deal with two different aspects of restoration: on-line restoration, and off-line restoration or restoration planning. In the former the pressure on operators is very high and decisions have to be made very quickly. These are mainly based on predefined restoration plans, operator experience, data given by state estimation or other data sources. Predefined restoration plans have the advantage to allow for a detailed study of the system behavior. However, since each blackout is of a different nature, and consequently, each restoration procedure will start from different initial conditions, these predefined plans can only give the operator a general idea on how to behave during restoration. This means that predefined restoration plans will give a strategy rather than a detailed set of instructions on how to proceed, since setting up restoration plans for each possible combination of restoration actions and for each system condition would be infeasible. Consequently, the operator has still to decide at each step whether his action is feasible or not, and to plan future actions. The rules developed in this thesis can help operators to assess the feasibility of restoration steps and to plan restoration sequences, using no or only a small number of costly time-domain simulations. During system restoration, planning engineers are faced with a large number of theoretical possibilities of how a system can be restored. These have to be explored based on the engineers' experience and on computer simulations carried out with power flow software, stability tools, electromagnetic transient programs, and harmonic analysis software. Simple rules that help to assess restoration steps with respect to their feasibility help to limit the number of trial-and-error simulations, and consequently, to shorten the overall time necessary to develop restoration strategies and restoration plans. Furthermore, the rules can be integrated into a tool that aids automated restoration planning, or they can be added to already available tools, such as the one described in [75]. The proposed framework has therefore two modes: an on-line restoration mode and a restoration planning mode. In the on-line mode, the rules support the operator in his decision making process. In the off-line mode, they help system planning engineers to find restoration plans in a faster and more efficient way.  3.2  Framework Database  As a basis of our analysis we assume that all necessary data is readily available in a database. This is schematically shown in Figure 3.1 for on-line restoration and in Figure 3.2 for restoration planning. In the case of on-line restoration, we assume that we have state-estimator data available, which give us bus voltages, generator output powers, switching status, etc. at the beginning of the restoration process. We further assume that we have sufficient information about the power system devices, such as generator data, data on the generator controls,  16  3.3. Framework Rules  line- and cable data, etc. Since the start-up behavior of motors is particularly important in the early stages of a restoration procedure, the estimation of motor parameters is depicted in an extra block and explained in more detail in a subsequent section. The data sources for system restoration planning are usually power flow, short-circuit, and dynamic data. In the case of single-phase calculations, the data for the electrical network for dynamic simulations can be extracted from power flow data. Saturation characteristics, overvoltage-versus-time capability characteristics, etc. have to be provided separately in form of tables, given by the manufacturer. In the case of three-phase simulations, zero-sequence data can be extracted from short-circuit data.  Rules  State estimator data Dynamic data Equipment data  Simulation tools Motor parameter estimation Motor data  Figure 3.1: On-line database for system restoration framework  3.3 Framework Rules Time-domain simulations give the most accurate assessment of whether a restoration step is feasible or not. However, they require a great amount of time, particularly when different combinations of restoration actions need to be assessed. Therefore, the rules that are described in more detail in the following chapters are based on analysis in the frequency and Laplace domain, and time-domain simulations are only performed for verification. The general principle that is the basis for the rules is schematically shown in Figure 3.3. The objective is to decrease the overall calculation time while still keeping a reasonable accuracy. As a basis for the analysis in the frequency domain, a software tool with harmonic analysis features is used. After an initial frequency scan is performed, new network conditions can be assessed by applying matrix calculations. Transfer functions for control systems, formulated in the Laplace domain, can be calculated and if necessary, simplified, for each control system type of interest, leaving only the inverse Laplace transform to be performed during the restoration process.  3.4. Framework Analytical Tools  17  Power flow data  Rules  Short-circuit data Dynamic data Equipment data  Simulation tools  Motor data  Motor parameter estimation  Figure 3.2: Off-line database for system restoration framework  Two different ways to avoid problems during restoration actions can be distinguished. In case of e. g. a load pick-up, only a single parameter, namely the amount of load can be changed. Depending on the operating guidelines of each utility, a maximum allowable frequency drop is given. Based on this parameter, the corresponding maximum allowable load that can be safely picked up can be estimated. In cases where a number of different parameters can be changed in order to remedy a given problem, e. g. in the case of sustained overvoltages, the most effective network change can be found by means of sensitivity analysis. If sensitivity analysis cannot be applied, the most effective network change has to be determined based on trial-and-error.  3.4 Framework Analytical Tools This section gives an overview of the analytical tools that are used within the framework. In the following, only a brief introduction into the theoretical aspects of the analytical tools is given since the underlying theory cannot be covered in detail in this thesis. The software packages that are used for the framework and that are listed subsequently are all commercially available products and have been in use in industry for a number of years. The methods that are described subsequently are not limited to specific software products but can be used with any software package that can perform the tasks described in the following sections. Examples for other products that can fulfill the same tasks are P O W E R F A C T O R Y [50] and N E T O M A C [142]. A schematic overview of the tools and software packages used for this project is provided in Figure 3.4. Additional tools that can be used for restoration studies are operator training  3.4. Framework Analytical Tools  18  Frequency scan Laplace transform Thevenin equivalent  Remedial action / Network changes  Yes  Figure 3.3: Assessment of feasibility of restoration steps  simulators (OTS), optimal power flow (OPF) programs, or short-circuit programs. Literature that deals more thoroughly with analytical tools in power system restoration can be found in Section 2.9.  3.4.1  Power Flow Analysis  Power flow analysis is the most common analysis method for power systems. The system is assumed to be balanced, which allows for single-phase positive-sequence calculation. When writing the nodal equations of the network we obtain the matrix equation I = Y-V  (3.1)  where I stands for the phasor currents flowing into the network, V for the phasor voltages to ground, and Y for the node admittance matrix. The effects of generators, nonlinear loads, and other devices are reflected in the node current, and constant impedance loads appear in  3.4. Framework Analytical Tools  19  Power flow IPFLOW  Rules  Steady state (Frequency scan) EMTP  Electromechanical Transients ETMSP  Electromagnetic Transients EMTP  Figure 3.4: Overview of analytical tools  the node admittance matrix. The current at node fc can be written as T  _ Pk - jQk  s  0  where P and Q stand for the active and reactive power, respectively, fed into the network at node k. Substituting Equation (3.2) into Equation (3.1) gives k  k  ^  ^  = Y V -rJ2YkiV kk  k  k  (3.3)  i  i=i  where fc = 1,..., n and n stands for the number of nodes in the system. Equation (3.3) is nonlinear and has to be solved iteratively using numerical algorithms such as the NewtonRaphson Method. More details on the theory that underlies power flow analysis can be found in [64, 143]. In this work, the software Interactive Power Flow ( I P F L O W ) [61] is used in order to solve power flow problems.  3.4.2  Steady-State and Harmonic Analysis  The basis for this type of analysis is Equation (3.1). When the currents I that flow into the network are readily given, this equation can be solved non-iteratively to obtain the node voltages V . The calculations can be either carried out for a three-phase system or—in case the system is balanced—as a positive sequence calculation.  3.4. Framework Analytical Tools  20  A harmonic analysis is a steady-state analysis that covers a whole frequency range with frequency steps A / . Since the network elements are frequency dependent, the node admittance matrix changes for each frequency and we obtain [20, 54]:  I(/)=Y(/)-V(/).  (3.4)  where /  fmini fmin ~f" Af,  fi  + 2 • Af,  m n  , fmax  In order to find the frequency-dependent impedance of a network seen from a particular location, all voltage sources are short-circuited, and all current sources are removed. Then, a current of value 1 A is injected into the node where the impedance is to be determined. The branch voltage will be equal to the impedance [54, 161]. In order to carry out harmonic (or steady-state) analysis, the frequency scan feature of the Electromagnetic Transients Program ( E M T P ) [54, 63] is utilized.  3.4.3  Electromagnetic Transient Analysis  When analysis in the time domain is performed, the differential equations that describe the dynamic behavior of the system have to be integrated with respect to the time. For the analysis of electromagnetic transients, the trapezoidal is usually used, because of its very good numerical stability and accuracy [161]. In case of e. g. a simple inductance L , the relationship between voltage and current is expressed by the differential equation  v(t) = L-j  (3.5)  t  It can be integrated using the trapezoidal rule and we obtain [54, 161]  v(t) + v(t - At)  =  %{t) - i(t - At)  At  2 Transmission lines can be described by the equations  iu{t) = | • hist  n  = •  + hist {t - r)  (3.7)  v (t - T) - i i{t - T)  (3.8)  12  2  2  where r stands for the transmission line traveling time and the subscripts "1" and "2" for the nodes the line is connected to. Similar operations can be carried out for all network elements and the system matrix equation is obtained as  Gv(t) = i(i) - hist  (3.9)  where hist stands for known "history" terms. For this project, the E M T P is used in order to simulate the electromagnetic transients. It allows for single- or three-phase calculations [63].  3.4. Framework Analytical Tools  3.4.4  21  Stability Analysis  The generating units and other dynamic devices of the system can be expressed in the state-space as x  =  f(x,V)  (3.10)  where x stands for the state vectors of devices such as generators, motors, exciters, or governors. For stability studies the network and generator stator transients can be neglected. The electrical network with the currents being a function of the state variables x and the node voltages V can be expressed by the relationship I(x,V)  =  Y  V  (3:11)  The differential Equation (3.10) can be solved using explicit integration rules such as RungeKutta, or implicit integration rules using the trapezoidal rule together with the Newton method [143]. Stability programs are based on balanced network conditions and therefore Equation (3.11) represents the electrical network single-phase with positive-sequence parameters. In this thesis we use the Extended Transient Midterm Stability Program ( E T M S P ) [60, 62] to perform stability analysis.  Chapter 4 Frequency-Response Analysis During power system restoration, it is desirable to reconnect loads as quickly and reliably as possible. At the very beginning of a restoration process, the loads that help to maintain a reasonable voltage profile in the transmission system, and help to stabilize the generating units, and the induction motors that drive the auxiliary systems of thermal power plants, are energized. Power plants that are in operation at this stage are small generating units with black start capability, such as hydro power plants. Frequency deviations that occur as a consequence of load pick-ups are of major concern in early stages of a restoration procedure since the ratio between available generating power and loads to be energized is the lowest. This ratio increases in later stages when thermal power plants are dominant, and the system is more stabilized. Consequently, frequency deviations as a result of load pick-ups are of less and less concern as the restoration of the system progresses [5]. When frequency deviations exceed certain limits, relay operations may be triggered and— in the worst case—a recurrence of the outage may result. On the one hand, the load that is picked up should not be too large in order to not exceed these limits. On the other hand, if the amount of load is too small, the restoration duration will be unnecessarily prolonged. Hence, it is desirable to find a way to estimate the frequency drops that are the result of load energization. A n assessment of the maximum load that can be picked up safely at each time shortens the time during on-line restoration as well as during restoration planning [5]. This chapter introduces methods that deal with the prediction of frequency variations due to load changes. The considerations are limited to the pick-up of static loads that can be approximated by simple steps in the electric energy output. In a later chapter that focuses on the analysis of auxiliary systems, the algorithm is extended to allow for the assessment of motor pick-ups. Since a load rejection can be considered as a negative load pick-up, the rules that are demonstrated in the following can also be used in order to estimate frequency swings during load shedding. The latter can play an important role during restoration, when system islanding situations occur as a result of a separation of the system into subsystems. The system frequencies in this case can be below the allowable limits and consequently, the load needs to be reduced in an efficient and reliable way [4].  22  4.1. Overview of Frequency-Response Estimation Methods  4.1 4.1.1  23  Overview of Frequency-Response Estimation Methods Simple Frequency-Response Estimation Methods  Operators often rely on their intuition and experience acquired during normal system operation or on fairly crude rules to predict the system frequency behavior when loads are reconnected. The frequency behavior of an islanded system, such as the one shown in Figure 4.1, is controlled by the governors. In case of a load pick-up they detect the decline of frequency and as a consequence, increase the gate or valve position in order to bring up the mechanical output power of the turbine, until a new equilibrium with an acceptable system frequency is reached.  Steam / water  Valve / gate •^ •!  Generator Turbine  Speed  Governor  1  Load  Figure 4.1: Generator supplying isolated load  The two most common rules used by operators in order to estimate the frequency response of prime movers ignore the action of the governing system and give a reasonable prediction only for the initial frequency behavior. Both methods are based on the fact that shortly after a load change the frequency behavior is solely determined by the change in load, the turbine's inertia constant, and the damping constant, and is not influenced by any governor control action. The equation that governs the behavior of an electro-mechanical system is the equation of motion. For small changes in power it can be written in per unit as [143]  2 H - ^ at  = AP -AP -Dm  e  Acu  (4.1)  where H stands for the inertia constant, D for the damping constant, and Au> for the deviation from the nominal frequency. AP represents changes in the electrical and AP changes in the mechanical power. e  m  According to Equation (4.1), a linear function, that defines the initial rate of frequency decay, following a load change of —AP , can be written as: e  f(t) = (1.0-—^-t)-60  Hz  (4.2)  4.1. Overview of Frequency-Response Estimation Methods  24  A more elaborate form for the estimation of the frequency behavior can be developed when Equation (4.1) is transformed into the Laplace domain: AUJ(S)  =  A  P  1  E  2H  s-(s +  (4.3)  &)  where —AP represents a load change. This equation can be transformed back into the time domain, giving the exponential function [143] E  /(*)  AP  1.0  [l-exp(-t/2>.)]  fe  D  60 Hz  (4.4)  where  T  Pe  =  2H D  (4.5)  Example A 14 M W static load pick-up at the terminals of a generator, whose governor block diagram is displayed in Figure 4.2 and whose data are given in Table 4.1, is simulated using the stability program E T M S P , and compared to above rules. The results are shown in Figure 4.3. The linear method allows one to assess the initial rate of decay only, whereas the exponential method allows for a reliable frequency prediction up to around 3 s. Although Equations (4.2) and (4.4) allow for a fast assessment of the frequency behavior, their usefulness is limited when the minimum frequency that is reached is of interest as well. Consequently, both methods should be only applied if solely the initial rate of decay is of interest. Table 4.1: Governor data  S  gen  [MVA] 87.5  4.1.2  V  gen  [kV] T  13.8  s 1.0 s  T  s 15.0 s D  s 0.035 s  Rp  Rt  0.02  0.7  v  V  0.0559  -0.04781  0  1.1  c  Other Frequency-Response Estimation Methods  In the past, the minimum frequency during load pick-up has been mostly determined by means of time-domain simulations using stability programs. This, however, can lead to long overall simulation times, particularly in cases where a multitude of scenarios has to be studied. Hence, there is a need to find the minimum frequency by simpler rules that are not based on time-domain simulations. A general overview of relevant publications in the area of frequency response analysis in power system restoration was given in Chapter 2. Simple guidelines for the estimation of the frequency response of prime movers during restoration are listed in [5]. They are  4.1. Overview of Frequency-Response Estimation Methods  25  1.0  to.'ref +  1+0-5SL  1+sT„  A  0.0  o sT  n  1+sT  n  Figure 4.2: Governor block diagram  ETMSP simulation Linear method Exponential method  Figure 4.3: Frequency deviation for 14 M W load pick-up  based on look-up tables that relate frequency dips to sudden load increases and that are produced based on a number of time-domain simulations. Other methods, outlined in [52, 133], are based on neural networks that are trained by running a large number of time-domain simulations. An alternative to above methods is the application of the Laplace transform. All methods, based on Laplace-domain calculations, are average frequency models, i. e. oscillations between generators are filtered out and only an average system frequency is retained. This means that  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  26  in case more than one generator is in operation, the generators are lumped into one dynamic equivalent. A n approach that belongs to this group of methods and that is based on strongly simplified governor and turbine models is presented in [22, 23]. It gives a simple function for the system's frequency behavior after a load pick-up. A low-order system frequency response model and the analysis of the frequency-response behavior of steam reheat turbines in an islanded condition is described in [16]. Other similar methods that are based on dynamic equivalents, and extended application of the Laplace transform are outlined in [30, 42]. A n extended analytical analysis of frequency decay rates can be found in [21].  4.2  Frequency Behavior of Hydro Units for Static Load Pick-up  This section introduces a new Laplace transform method, based on [16], to determine the frequency drops caused by load pick-up during the early stages of system restoration when hydro units are in operation. A n overview of the method is given in Figure 4.4. At the beginning of power system restoration we mainly deal with hydro units that provide cranking power for larger thermal units. Since [16] provides a method that can be readily applied to estimate frequency drops in the case of steam turbine units, thermal turbines are not considered further. Furthermore, the aggregation of generators is not discussed in this thesis. A n overview of dynamic aggregation methods can e. g. be found in [144]. In case only a small number of hydro units with similar parameters is considered, these can be aggregated according to the method outlined in [16].  4.2.1  Assumptions for Model Development  The development of the equations in this section is based on the following assumptions: • The operation of the governor-turbine system is unconstrained, i. e. limiters are not effective. This restriction is necessary since nonlinearities cannot be represented using the Laplace transform [16, 22, 23, 67]. • If more than one hydro generator is in operation, the generators are lumped into a single machine. Topologically, this replacement can be regarded as a single generator that is connected to the individual generator buses by ideal phase shifters [16]. • Oscillations between hydro generators are neglected and only an average frequency is taken into consideration [16]. That means that all generators remain in synchronism and do not fall out of step [30]. • Only the mechanical part of the system is taken into account—voltage effects are ignored [30].  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  27  Start  Determine load P to be picked up  Remedial Action  L  Determine P . =F(f„„J liln  t  Yes ETMSP simulation  Yes  No Next restoration step  • The transformer and generator impedances place a limitation on the amount of load the generating units can absorb. It is assumed that the disturbance is small enough and that the equivalent machine model is able to absorb this change [16]. • Governor and turbine models are chosen in a way that allows for a compromise between effort and efficiency. The governor model represents a typical proportional control circuit with transient droop [62, 143, 178]. The turbine is represented by its classical transfer function [143, 178] which implies an inelastic water column, which is of sufficient accuracy for our purpose [5].  28  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  4.2.2  Derivation and Verification of Equations  Simplified Governor and Turbine Control Block Diagram With the assumptions made in the previous section, we obtain the system in Figure 4.5. Based on this diagram, the function Aw(s) can be determined and then transformed back  Turbine  Generator  A  1-sT„ A,  U0.5SX,  x  2Hs  o  i  gate speed  gate  0) :  1+sT„  o 1+ST  n  Governor Figure 4.5: Control block diagram of hydro unit  into the time domain. However, the attempt to develop such a function, taking into account the full block diagram in Figure 4.5, leads to an unnecessarily complicated equation. In the following, the block diagram is therefore simplified.  Additional Simplifications The first simplification can be accomplished by comparing the time constants listed in Table 4.2: the time constant T is considerably smaller than the constants T ,TD,TL, P  W  and it can  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  29  therefore be set to zero. This reduces the respective block to a constant of value 1 which can be omitted from the block diagram. Table 4.2: Comparison of time constants Time constant s]  T  T -1.0w 15.0 D  1  T  T  0.035  9.8  L  In the next step, we consider the transfer function of the feedback loop which can be formulated in the Laplace domain as:  X2  —  (R  P  + Rt) • \*_  +s  T  • Xi  (4.6)  where ^ Since T  pt  = ^ - ^ • ' ^ Hp  = 178.6 8  (4.7)  ^> To, we can simplify Equation (4.6) to X2 — (Rp + Rt) • -j—;— + S  • Xi  (4.8)  The simplifications result in the new simplified control block diagram displayed in Figure 4.6.  Figure 4.6: Simplified control block diagram of hydro unit  30  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  Example In order to prove the validity of above simplifications, a comparison between SlMULlNK [163] simulations of the full and reduced model are carried out. The same 14 M W load pick-up that was described earlier is simulated. Figure 4.7 shows that noticable deviations between the results for the full and simplified model only occur from around 30 s onwards. Since we are only interested in the initial behavior of the frequency until around the time when the minimum is reached, our simplifications are justified. 60  59.5  59  X.  58.5  58  a <u cr  <D  57.5  57  56 56.5 0  5  10  15  20  25  30  35  40  45  50  Time t [s]  Figure 4.7: Comparison between full and simplified model  Development of Transfer Function In the Laplace domain, a sudden load change of A P can be represented as e  P (s) = - A P • e  e  s  (4.9)  where AP AP  e  e  > 0 < 0  for load pick-ups for load rejections  After some algebraic operations, the transfer function for the model displayed in Figure 4.6 can be formulated as:  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  31  where K  =  AP  (4.11)  " 2H 2  N(s)  D(s)  e  f~  +  (4.12)  + S  S  IN D  H  + +  -T  \H  •T  W  D  D  2H  +H  • TN  1  D 2H  f-  +  T  +  +  •T  W  N  D  A  -T  H  • [T  T  D  H  w  ]  3  S  3  -  T  D  2  T  2 .  N  + Rt +  T  A  +T  W  • TN  T) W  T,  s+  s + 2  (4.13)  4 +  S  W  (4.14)  1/T  D  Whereas the roots of the numerator of Equation (4.10) are readily available, the roots of the denominator are not readily given. They are determined using the numerical routine R O O T S [162]. Once the roots are available, Equations (4.13) can be formulated in a factored form as  D(s)  H(s- )  =  (4.15)  Zj  where Zj are the roots of Equation (4.13). Equation (4.10) can then be expanded in partial fractions and we obtain [67, 143] Rj 3=1  (4.16)  S — Zi  where Rj represents the residues at the poles Zj which can be computed by Rj -  [Aw(s) • (5 -  Z j  (4.17)  )]  Eventually, following [67], Equation (4.16) can be transformed back into the time-domain, and we finally obtain the system frequency:  /(*) =  4.2.3  i -  (Rj •  ex  P (i z  • *))  60 Hz  (4.18)  Example for Load Pick-up  The result is verified using the same 14 M W static-load pick-up example as before. Figure 4.8 shows the results obtained with the new method as compared to results obtained with  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  32  the stability program E T M S P . The results are very close and prove the validity of the above approach. It can be further seen that for a load pick-up of 14 M W (0.16 in per unit of the generator rating) the limits given for the gate position and speed have no effect on the result. For this case, the calculation time when using the new method is approximately 1% of the simulation time using E T M S P . The minimum frequency of 56.15 Hz at 7.8 s is obtained from Equation (4.18) using the numerical routine F M I N [162].  4.2.4  Example for Load Rejection  Analogous to the previous example we perform a 14 M W load rejection. The result of the analytical calculation, as compared to an E T M S P simulation, is depicted in Figure 4.9. Using the same numerical routines as above we obtain the maximum frequency 63.86 Hz at 7.8 s. The analytical results agree well with the E T M S P time-domain simulation results and prove that the method can be applied for both load pick-up and rejection.  4.2.5  Remedial Action  In the following, it is investigated whether a simple rule for hydro turbine systems, indicated in Figure 4.4, can be developed that relates the minimum frequency and the time where this minimum occurs to the amount of load that is energized. Figure 4.10 shows the minimum frequency and Figure 4.11 the corresponding time as a function of the amount of load being picked up (in the range of 0.02 to 0.2 per unit).  4.2. Frequency Behavior of Hydro Units for Static Load Pick-up  ETMSP simulation New method  64.5 h  O  a O*  33  62.5  62  61.5  Time t [s]  Figure 4.9: Frequency deviation for 14 M W load rejection  The time where the minimum occurs is not a function of the load. The minimum frequency is a linear function of the amount of load. This result can be used in order to determine the maximum amount of load that can be picked up safely at once, when a minimum allowable frequency is given. Therefore, the minimum frequencies /z,i,/z,2 for two different arbitrary loads P and P are calculated. The maximum load that can be picked Li  L2  4.3. Summary  34  a  ETMSP simulation New method  7.801  a  0.02  0.04  0.06  0.08  0.12  0.1  Load AP  e  0.14  0.16  0.18  0.2  [p.u.]  Figure 4.11: Time at which minimum frequency occurs as a function of load  up safely can then be calculated by P  limit  r  where  fumit  _ — L \ P  r  (PLI ( [JL1 f  ~ PLJ) I t \ \Jlimit — JL2) E  F  x JL2)  (4.19)  stands for the minimum allowable frequency.  4.3 Summary In this chapter a method was introduced that predicts the frequency behavior of hydro prime movers after the rejection or pick-up of static loads. The method is based on calculations in the Laplace domain and simplified governor and turbine models. Comparisons between the stability program E T M S P and the new method show good agreement and prove the validity of the rule.  Chapter 5 Voltage Response Analysis This chapter introduces a concept for overvoltage control during restoration. Overvoltages can be classified as switching transient, sustained, and harmonic resonance overvoltages. Steady-state overvoltages occur at the receiving end of lightly loaded transmission lines as a consequence of reactive power imbalance. Excessive sustained overvoltages may lead to damage of transformers and other power system equipment. Transient overvoltages are a consequence of switching operations on long transmission lines, or switching of capacitive devices, and may result in arrester failures. Harmonic resonance overvoltages are a result of system resonance frequencies close to multiples of the fundamental frequency, which are excited by switching of transformers, or switching of transformer-terminated transmission lines. They may last for a very long time and result in arrester failures and system faults [95]. The overvoltages of major concern during power system restoration are sustained and harmonic overvoltages [95, 172]. Therefore, this chapter focuses mainly on their analysis and control. The analysis and control of switching transient overvoltages is described only briefly for the sake of completeness. When energizing the transmission system, two conflicting issues arise. It is to energize line sections as large as possible. This, however, involves the risk overvoltages, and consequently of equipment damage. Bringing more generators energizing only small sections of transmission lines might reduce these overvoltages, to an increase in restoration time [1].  desirable of higher online or but leads  Traditionally, sustained overvoltages have been studied using power flow programs. Switching transient and harmonic overvoltages have been mainly investigated using the E M T P . The application of E M T P , however, requires long simulation times. This is especially true during system restoration when a large number of simulations need to be carried out in order to assess all possible restoration scenarios. In this chapter a method for fast and approximate assessment of the feasibility of switching restoration steps, using simple rules and the frequency scan feature of E M T P , is introduced. For the purpose of an approximate assessment, a single-phase representation of the system is chosen. The frequency scan has to be performed only once. Then, using matrix manipulation, changes to the system can be accommodated in a fast and efficient way. 35  36 Start  Determine network matrix [ Z ]  Remedial Action  Estimate bus voltage  Yes  Yes  Yes No EMTP simulation  Yes  Next restoration step  Figure 5.1: Overview of algorithm  A schematic overview of the proposed method is given in Figure 5.1: Once a suitable restoration path is found by approximate rules, it can be verified by an E M T P time-domain simulation, together with overvoltage-versus-time-capability-curves to assess whether a de-  37  5.1. Frequency-dependent Thevenin Impedance  vice can withstand the applied voltage. Typical characteristics of transformers, shunt reactors, and MOV arresters are displayed in Figure 5.2. When time-domain simulations are carried out, signal processing techniques and rules based on the application of artificial intelligence, are applied to terminate the simulation after a minimum amount of time. If a switching situation is found to be infeasible, sensitivity analysis is applied in order to find the most efficient network changes that remedy the problem. 420  ! ! I  :  :  :  :  :  : :  180 kV MOV arrester 253 kV shunt reactor 230 kV transformer  400  380  >  360 N.  <U  |0  340  320  300  280  _ i  10'  10  ; 2  1Q  3  Time t [s]  Figure 5.2: Overvoltage-versus-time capabilities of typical 230 kV system equipment  5.1  Frequency-dependent Thevenin Impedance  The combinatorial nature of a restoration procedure and the many resulting system configurations could require a large number of frequency scans using the E M T P . Furthermore, in each case the admittance matrix Y would need to be build, factored, and solved, resulting in large calculation times. A method is therefore derived which requires only one E M T P frequency scan. The results are then used as a basis for quickly calculating the impedance for any arbitrary set of parameters using simple matrix operations. In the first step of this procedure the multi-port Thevenin equivalent circuit is determined. The terminals of this circuit include the bus where the next switching operation, or restoration step in general, is to be performed, and buses where network element changes are possible. Examples are buses where the number of generators can be varied, where loads can be changed, or shunt reactances can be added. Such a multi-port Thevenin equivalent circuit is schematically shown in Figure 5.3. In order to find the matrix Z° = ( Y ) representing the Thevenin equivalent, currents of 0  - 1  5.1. Frequency-dependent Thevenin Impedance  38  magnitude 1 A are successively injected into each of its terminals. For this, voltage sources in the system are short circuited, and current sources are open circuited. The voltages at the nodes of the Thevenin equivalent then give—column by column—the matrix elements of Z°. More details about multi-port Thevenin equivalents can be found in [54, 161]. Variable elements  Switching bus  [Z] Figure 5.3: Thevenin equivalent circuit with variable parameters  If an element is added at a certain node, one possibility for calculating the new resulting matrix Z (besides carrying out another E M T P frequency scan) is to invert the matrix Z ° , then to add the new network element AYJt and to invert the resulting matrix Y again, obtaining the new matrix Z . Although this method is much faster than using the E M T P frequency scan for each different system configuration, it is still too time consuming if a great number of different system configurations for a large system needs to be examined. 1  1  1  Therefore, a new method is derived in the following, based on the application of the Sherman-Morrison [187] or Woodbury Formula [57], or the inverse matrix inversion lemma [14]. It allows the direct calculation of a modified matrix Z from the matrix Z ° and a given network change AY*,, without using time-consuming matrix inversions, by providing an explicit expression 1  Z  1  = /(Z°,AF ) f c  (5.1)  where AY*, stands for a network change at node k. In the following sections, this method is described for elements added from one of the nodes to neutral and for elements added between two nodes.  5.1. Frequency-dependent Thevenin Impedance  5.1.1  39  Impedance Change between Bus and Neutral  Examples of impedance changes from bus to ground are the addition (or omission) of generators, loads, and shunt reactors. If we add an element from bus k to ground we obtain the new admittance matrix • ••  n Y  =  Y& + A Y  In  1  /0  \  0\  Y° +  fc  (5.2)  AY,  This matrix can be written in a more convenient way as Y  = (Y°+ Ay .e .e^)  1  f c  (5.3)  k  where e stands for the k-th unit vector. Inverting the matrix Y Morrison Formula [187] we get  1  k  =  (Y )1  z  o  1  = ( Y ° + AF  AF  fc  fc  •e  • ( Z ° • e ) • (e  and applying the Sherman-  • e*)-  k  (5.4)  1  • Z°)  T  k  1 + A n • e-T • Z ° • e  (5.5)  k  This equation can be simplified to Z where Z °  x  n ) k  1 _r 0 1  _  = Z° 7  Z  (*i...n)k • k(l...n) • Z  A y  fc  (5.6)  i + ^VAn  describes the kth column and Z ^  ( 1  the kth row of Z°.  n )  The impedance at bus j can then be obtained as 7 _ 1 _ 0 _ % ' kj Z  7  j  ~  '  Z  7  j j  "  j j  A 1  fc  (5.7)  1 + 2SL • A n  This equation can be extended to the case where an arbitrary number of elements from different nodes to ground are changed:  7  \  _  £ An-A  o  7  i=fc,i,m,...  /  E t,i,m,...  where k,l,m,...  E /  (5.8)  \  r=k,/,m,...  E **V i=k,l,m,...  represents the nodes where the changes are possible.  t  (5.9)  40  5.1. Frequency-dependent Thevenin Impedance  5.1.2  Impedance Change between two Buses  The above method can be extended to impedance changes between arbitrary buses. Examples are the addition of parallel transformers, series capacitances, etc. In the case elements are added between two nodes k and I, the impedance matrix can be formulated as  /0 AY  . . . - A K *kl  kl  Y  =  1  Y° +  (5.10)  AY  ...  kl  \  kl  ...  0  (5.11) This equation can be again written in a more convenient way as Y where Aeki =  =  1  ( Y ° + AY  kt  • Aeki • Aejjj)  (5.12)  — ei. Following the same steps as in the previous section, we obtain  z  ,0  1  AZ  i +  (*i...n)kl ' k l ( l . . . n ) ' AZ  (zi + K -  (22, +  A  1  M  (5.13)  ZD) • AYkl  k  where 7O  7O  A7» ^^(l-.njkl  —  A7° ^^kl(l...n)  — 7 ° 7° — ^k(l...n) "~ ^1(1...n)  —  ^(l...n)k  _  (5.14)  ^(l...n)l  From Equation (5.13) the impedance at node j is obtained as 7  =  3  5.1.3  7l  \  j  =  7» _  "  ~ S ) • ( kj ~ lj) Z  l + (Zl  k  +  Z  Zl-{Zl  l  Z  +  ' fc' A F  (5.15)  Zl)).AY  kl  Example  The above method is tested using a simple two-port Thevenin equivalent, created from the system displayed in Figure 5.4. It represents a power system at the early stages of a restoration procedure in which, starting from the generators connected to bus B4, a path to a large power station is built. The impedance seen from bus B3 is calculated as a function of the impedance connected to the left of bus B4, which'could represent the number of parallel generators if Zi is constant. The above equations as well as E M T P frequency scans are used for this calculation. The results for the frequencies 60 Hz and 540 Hz are shown in Figures 5.5 and 5.6. They are so close that they are hardly distinguishable, which proves the validity of our approach. oad  5.2. Sustained Overvoltages  41  ^load  Gl G2 G3 Figure 5.4: System at the beginning of a restoration procedure  600  i  + *  1  1  1  1  + E M T P frequency scan * Matrix method  G  K o  CO  c ou  a  O) a  a  i  >  2.5  Number of generators at bus B4  Figure 5.5: 60 Hz impedance at bus B3 as function of number of generators  5.2  Sustained Overvoltages  Sustained overvoltages are caused by the charging currents of lightly loaded transmission lines. If not controlled, they may cause serious reactive power imbalances resulting in phenomena such as generator self-excitation and runaway voltage rise. They may further lead to overexcitation of transformers and generation of harmonic distortions which can lead to the excitation of harmonic resonant overvoltages. Furthermore, sustained overvoltages can lead to high transient overvoltages when line switching procedures are performed [95]. The occurrence of sustained overvoltages in the system can be assessed using the results  5.2. Sustained Overvoltages  42  1  + * G  880  ?  840  1  l  .  l  1  + E M T P frequency scan * Matrix method  iIf  iIf 4  1  1.5  2  2.5  3  3.5  4  4.5  5  5.5  6  Number of generators at bus B4  Figure 5.6: 540 Hz impedance at bus B3 as function of number of generators  of the frequency scan at the base frequency 60 Hz. As outlined in the previous sections, the system can be extended step by step by basic matrix operations, and therefore an assessment of the voltage level at the switching bus can be performed without the necessity for additional E M T P frequency scans, provided that the loads are constant impedances and the generators do not exceed their reactive power output limits [1]. Therefore, additional calculations are added at the end of the block "sustained overvoltage" in Figure 5.1: In case the bus voltage limit or the generator reactive power are exceeded, a more time-consuming load flow calculation is performed, as indicated in the schematic shown in Figure 5.7.  5.2.1  Sensitivity Analysis  When a steady-state overvoltage occurs, it is desirable to find the bus where network changes for reducing the voltage level are most effective. This can be accomplished by means of sensitivity analysis. This produces the sensitivity of the voltage magnitude at the bus where an overvoltage occurs with respect to network changes. The sensitivity at the switching bus j with respect to shunt admittance changes at bus k is defined as ^(AF j k  f c  )l  ~ ~~d\A~Yk\~^  AYk=d{AYk)  '  where Vj stands for the voltage at bus j, AY for a change in admittance at bus k, and d(AYfc) for a very small number representing an incremental perturbation. k  5.2. Sustained Overvoltages  43 Start  Steady state generator voltage V and current I Bus voltage V gen  gcn  bus  Compute generator reactive power Q =Imag [ V I * ] gen  V >v T  bus^  T  V <V  n u a ' ' b u s *mi v  No  Run powerflowcalculation  End  Figure 5.7: Assessment whether power flow is needed  The voltage magnitude at bus j can be formulated as:  = \V + jV \ jR  (5.18)  jr  Substituting Equation (5.17) into Equation (5.16) then yields  c  1  (v dVrR ,  dV \  w  *  s  = W\\  V i R  '^\  _ ^  (dz]  +  ,  T{  V i I  '9m)  .  ( 5 1 9 )  where  dv  jR  d\AY \ ~ k  dV  3l  «  rR  §  dz]  T  (aj^ff •  IrR ~ ^  _ "(dZ)  |  •  In)  (5-20)  dZj  r  rI  " ^VaJA^•  / r / +  aJAiv"  I t r ]  ( 5 > 2 1 )  The impedances are separated into real and imaginary parts according to  Z] = Z} + j • Z) T  rR  \Z) \ = ^(Z} ) + (Z) ) 2  rl  r  rR  2  rI  (5.22)  5.2. Sustained Overvoltages  44  The real and imaginary parts are defined as 7  i jrR  o  =  A • \AY \ + A • \AY \ \ + ft . \AY \ + B • \AY \ 2  7  r  jrR  7 ^jrl  2  k  (5.23)  2  x  k  A -\AY \  — 7° — ^jrl  1  k  3  2  +  k  l +  k  A -\AY f 3  k  (5.24)  B -\AY \+B -\AY \  2  3  k  4  k  and their partial derivatives are given by  dZ]  -A -2-A -\AY \  rR  1  2  + (A •B -B -  k  d\AY \  1  (l + B -\AY \  k  1  3  4  3  (l + B -\AY \  k  A ) • \AY \  2  l  2  3  2  2  4  +  k  k  (5.25)  B -\AY \ )  2  k  + (A -B -B -  k  d\AY \  +  k  -A -2-A -\AY \  2  A ) • \AY \  2  3  4  k  (5.26)  B -\AY \ ) 2  4  2  k  where  At  =  A  =  2  1 jk 1 • 1 kr 1 •  (<Pjk + Vkr + (PAY ) \Z%\ • \Zl\ • \Z° \ • [cos (<p + ip + <p ) • cos ((p + PAY ) + + sin (ip + (p ) • sin (<p + ip )] Z  Z  C 0 S  k  (  kk  kk  kr  jk  AYk  kk  kk  k  AYk  A  =  1 jk 1 • 1 kr 1 •  A  =  COS \ jk\ • \ kr\ • \ kk\ • [Sin {(Pjk + <Pkr + <PAY ) (<p + <PAY ) + ~ COS (ipkk + (pAY ) • Sin (tp + <PAY )]  Bi  =  B  3  = 2- \Z%k\ • cos (cp  B  =  B  4  =  3  4  Z  Z  Z  Z  S  m  2  k  Z  k  kk  kk  with (p , ip , and (p kr  (5.27)  (Vjfc + H>kr + <PAY )  k  jk  AYk  AYk  kk  k  k  + (p Y ) A  k  \Zl \  2  k  being the angles of Z? , Z\ , and AY , k  T  k  respectively.  The sensitivities defined in Equation (5.16) are calculated for all nodes of the Thevenin equivalent at which a remedial action can be performed. They are then arranged in a sensitivity vector Sji  (5.28)  Sjk  Sjn J  This vector.can be normalized by referring its elements to the maximum element:  I  Sji  (5.29)  maxi1 (Sji) Sjn  J  5.2. Sustained Overvoltages  45  Since we deal with either resistive, inductive or resistive-inductive changes depending on the available devices, two different sensitivity vectors can be defined—one for resistive (S ) and one for inductive (S/) changes. This is also reflected in the incremental change d(AY ), which in the case of resistive changes can be defined as a real number and in the case of inductive changes as an imaginary number. Capacitive changes can be considered as negative inductive changes. In case only loads are to be connected to the buses, the resistive and inductive sensitivity vectors can be combined to: R  k  Stotai = | S + j • tan (Tp) • S i |  (5.30)  R  where tan Tp takes into account the average powerfactor pf for the loads to be added which is defined as:  pf = cos (Tp)  5.2.2  (5.31)  Example  As an example for the application of the above method, the system in Figure 5.8 is used. It represents the same system as the one in Figure 5.4, in a later stage of the restoration procedure. The most effective load changes for reducing an overvoltage at bus B7 are inves-  G4«-®-=-&  B  7  G5 « - ( ? > = - $ -  Figure 5.8: System during restoration procedure  tigated. Figure 5.9 shows the voltage behavior at bus B7 with respect to load changes at buses B2, B3, B4, and B7, obtained from full solutions. The load changes AY are referred to a typical bus load Y during normal operation. The results from sensitivity analysis with the sensitivity vector of Equation (5.30), for an average power factor of pf = 0.9, are displayed in Table 5.1. It can be seen that the higher the values of elements of the sensitivity vector, the more efficient the changes are at the corresponding buses for reducing the voltage level at bus B7. k  Table 5.1: Total sensitivity vector Bus k  2  3  4  7  Stotal(7k)  0.2908  1.0000  0.1553  0.4578  5.3. Harmonic Overvoltages  5.3  46  Harmonic Overvoltages  During the restoration of power systems after a complete or partial blackout, resonance conditions different from the ones during normal operation are encountered. If the frequency characteristic shows resonance peaks around multiples of the fundamental frequency, high overvoltages of long duration may occur when the system is excited by a harmonic disturbance. Such a disturbance can originate from the saturation of transformers, from power electronics, etc. The major cause of harmonic resonance overvoltage problems during restoration is the switching of lightly loaded transformers at the end of transmission lines or the switching of transmission lines, to which transformers are connected. When a transformer is switched, inrush currents with significant harmonic content up to frequencies around 600 Hz are created. They can be represented by a harmonic current source connected to the transformer bus [172]. The relation between nodal voltages, network matrix, and current injections can be represented by V(h • 60 Hz) = Z{h • 60 Hz) • I(h • 60 Hz)  (5.32)  where h stands for the order of the harmonic frequencies / = 120,180,... Hz. The harmonic components of the same frequency as a system resonance frequency are amplified in case of parallel resonance, thereby creating higher voltages at the transformer terminals. This leads to a higher level of saturation resulting in higher harmonic components of the inrush current which again results in increased voltages. This can happen particularly in lightly damped systems, common at the beginning of a restoration procedure when a path from a black start source to a large power plant is being established and only a few loads  5 . 3 . Harmonic Overvoltages  47  are restored yet [90, 1 7 2 ] .  5.3.1  Analysis and Assessment of Harmonic Overvoltages  In the following, four different system configurations of the sample system shown in Figure 5 . 4 are used in order to study conditions leading to harmonic resonance overvoltages. In Case 1 the impedance at bus B 4 is omitted and an E M T P frequency scan is carried out. Its result is displayed in Figure 5 . 1 0 . It shows a resonance peak at the third harmonic. The steady-state voltage at bus B 3 at the moment of switching is 1 9 7 . 2 kV ( 1 . 0 5 p.u.). The generation of harmonics from the transformer is shown in Figure 5 . 1 1 . It has significant harmonic content up to a frequency of around 6 0 0 Hz. When the transformer is energized,  1  Case 1 &  Case  Case  2  Case  3  4  .....  G i  1 \ j i  u  ii ii ii !i ii ii  CD  a,  a 20  — J  J  fi i  200  -•  i  Frequency  i1 /  L  [Hz]  Figure 5 . 1 0 : Impedance at bus B 3 for Cases 1 to 4  the resonance condition results in the overvoltage Vjg (t) shown in Figure 5 . 1 2 . If there were no overvoltage limiting equipment, the voltage would reach a level of 2 7 9 . 5 kV ( 1 . 4 9 p.u.). Figure 5 . 1 2 also shows the Fast Fourier Transform ( F F T ) of the signal for the period where the voltage reaches its maximum. It reflects the amplifying effect of the system resonance frequency: the magnitude of the third harmonic is almost half of the fundamental. 3  There are several methods for preventing harmonic overvoltages. In Case 2 , a highly resistive load can be added to bus B 4 , leading to a decrease in the magnitude of the impedance Zj33, and consequently to a reduced amplification. The frequency-dependent impedance and the voltage at bus B 3 are shown in Figures 5 . 1 0 and 5 . 1 3 , respectively. Another method that can be used in order to prevent harmonic resonant overvoltages is to bring additional generators online: a higher number of generators results in a lower overall  5.3. Harmonic Overvoltages  ^  48  5  120  180  240  300  360  420  480  540  600  Frequency / [Hz]  Figure 5.11: Harmonic content of transformer inrush current at bus B 3 for Cases 1 to 3 V ( 0 - ) = 1.05 p.u. B3  S  100  0  — ' — i — i — i  0.05  0.1  0.15  0.2  i  i  0.25  i  i_  0.35  0.3  0.4  0.45  0.5  Time t [s]  >  150  c;  ioo  CO  ^?  50 _J  0  60  120  160  240  300  360  I  I  420  480  .  L_  540  600  Frequency / [Hz]  Figure 5.12: Voltage at bus B3 for Case 1  inductance, and consequently in a higher resonance frequency, according to the equation 1 resonance  27TVT^C  This means that if generators are added to bus B4, the resonance peak is shifted to higher frequencies—if generators are omitted, it is shifted to lower frequencies. The frequency-  5.3. Harmonic Overvoltages  49 Case 1  0  0.02  0.04  0.06  0.08  Case 2  0.1  0.12  0.14  0.16  0.18  0.2  Time t [s]  Figure 5.13: Voltages at bus B3 for Case 1 and Case 2 — Case 3  0  0.02  0.04  0.06  0.08  Case 4  0.1  0.12  0.14  0.16  0.18  0.2  Time t [s]  Figure 5.14: Voltages at bus B3 for Case 3 and Case 4  dependent impedance for Case 3, where another generator is added to the system, is shown in Figure 5.10, and the resulting voltage in Figure 5.14. Another possibility for reducing harmonic overvoltages is represented by Case 4: a decrease of the generators' scheduled voltage leads to a proportional decrease of the preswitching steady-state voltages. This effect results in a change of the transformer inrush current. Figure 5.15 gives the harmonic content of the transformer inrush current for a pre-  5.3. Harmonic Overvoltages  50  switching voltage at bus B3 of 178.4 kV (0.95 p.u.). The comparison to Figure 5.11 shows that the harmonic content of the inrush current is significantly reduced.  From the previous results it can be concluded that three different criteria for the occurrence of harmonic resonance overvoltages have to be considered: the impedance at the bus where the switching operation is performed, and the harmonic content of the transformer inrush current which is a function of the pre-switching steady-state voltage level at the switching bus, and the transformer saturation characteristic. Based on the above observations we can define a rule for the assessment of harmonic overvoltages for switching operations at bus j that is similar to the total harmonic distortion (THD) [20] known from power system harmonic analysis: 10  H =  [Zjj(h • 60 Hz) • Ij (h • 60 Hz, Vj ( 0 - ) , Lj (A))] > H  \h=2  2  limit  (5.33)  Ij stands for the transformer inrush current which is a function of the frequency h • 60 Hz, the pre-switching voltage level Vj(Q—), and the transformer saturation characteristic Lj. In Section 5.3.2, it is shown how the harmonic characteristic of transformers can be estimated. The constant Hn stands for the maximum allowable limit. It is defined by the system operator or restoration planner. The results for H for Cases 1 to 4 in Table 5.2 reflect the voltage behavior at bus B3 displayed in Figures 5.13 and 5.14. mit  The above rule can only provide an approximate assessment of whether a restoration step is feasible or not with respect to the occurrence of harmonic overvoltages. It is therefore recommended to always carry out time-domain simulations when a certain level of H is  5.3. Harmonic Overvoltages  51  Table 5.2: Index for harmonic overvoltage assessment Case H  1  2  3  4  6.33  0.22  0.15  0.35  exceeded. In order to keep the simulation time as short as possible in that case, a method for the reduction of the simulation time of E M T P simulations is introduced in Section 5.4.  5.3.2  Harmonic Characteristic of Transformers  For an assessment of harmonic overvoltages during transformer switching, the harmonic characteristic, i. e. the harmonics of the inrush current as a function of the voltage of the transformer that is energized, is needed. In case the harmonic characteristic is not given by the manufacturer, it can be determined from the saturation characteristic and the transformer terminal voltage. In the following, analytical equations are introduced to determine the harmonic content of the transformer inrush current.  Analytical Calculation of Transformer Inrush Current The development of the equations for the calculation of transformer inrush currents is described in detail in [27, 211] and, is therefore omitted from this section. When applying a sinusoidal voltage v{t)  = Vmax •  COs(ut + ip)  directly to a saturated inductance, the harmonics of the transformer inrush current can be calculated as Onrush(0 Hz)  hnrush{§Q Hz)  i.[2cos(  a )  +  ( 2 a - , ) ] . ^  — • [TT — 2a — sin(2ai)] • ^  Z7T  Iinrush{h • 60 Hz)  (h  (5.34)  (5.35)  m a x  tOL*  _1_ 'sin (fc - 1) (a + 7r/2)  hn  h-  sin  1  (h + 1) (a + h+ 1  TT/2)  V  m  LOLe  (5.36)  2,3,4,...)  where  a  arcsin (1 _  V ' max  i  m  ip  T  to  sin ip  u  L  s  (5.37)  52  5.3. Harmonic Overvoltages  and where ip stands for the saturation flux linkage, V v for the remanent flux, and L for the saturated inductance. s  S  The simplified saturation characteristic, on which Equations (5.34) to (5.36) are based, is shown in Figure 5.16. It consists of two slopes—the first one with an infinitely high value and the second one with value L . S  Current j  Figure 5.16: Ideal saturation characteristic  Saturation characteristics are usually given as n pairs of numbers 1/0(1), i(l)j , [^(2), i(2)] , ... , [tp(n),i(n)}. The magnetizing inductance and the saturation flux linkage can then be calculated from the equations  l, =  fl")-?"- )  (5.38)  1  i(n) — x(n — 1) ip  s  =  ip{n)-l -i{n)  (5.39)  s  In addition to the nonlinear magnetizing inductance, transformers have a leakage inductance Leakage, resistances Ri and R representing the I R-\osses, and resistance R representing also the core losses. For our approximation, the resistances R\ and R can be neglected since R\,R <C u)L . The core loss resistance R can be neglected as well since R » OJL . We can then include the inductance Li in the magnetizing inductance L and obtain: 2  2  c  2  2  c  LEAKAGE  eakage  Ls — Ls + Lieakage  c  S  S  (5.40)  Using this value in Equations (5.34) to (5.36), we obtain the harmonic characteristic of the transformer inrush current.  5.3. Harmonic Overvoltages  53  Example As an example for the above method, the harmonic characteristic of the transformer that is switched at bus B3 in Figure 5.4 is determined. Its saturation characteristic is shown in Figure 5.17. The harmonics for a transformer terminal voltage of 1.05 per unit calculated with above equations are compared to an E M T P time-domain simulation of the inrush current, followed by F F T , in Figure 5.18. The results show good agreement and therefore prove the validity of the approach.  600 h 500 'co' > 400 ~5>J2 300 Cm  200;  100 0' 0  : 1  20  1 40  : i  60 Current i [A]  :  :  -  i  i  80  100  120  Figure 5.17: Transformer saturation characteristic  5.3.3  Sensitivity Analysis  In cases where the power system during restoration has already grown to a considerable size, and a harmonic overvoltage condition has been detected, it is not always clear at which nodes admittance changes should be made in order to obtain an optimum impedance-frequency characteristic at the switching node. Therefore, a method is developed which determines the nodes where changes are most effective. The method is based on sensitivity analysis for the harmonic frequencies [147]. Only harmonic frequencies are of interest since the major components of the frequency spectrum of the voltages and currents during transformer saturation are multiples of the fundamental frequency 60 Hz. The algorithm consists of four steps. Steps 2 to 4 are described in the following, while the first step has already been dealt with in Chapter 5.1. 1. Calculate the Thevenin equivalent circuit for the system.  5.3. Harmonic Overvoltages  54 FFT of EMTP Simulation  — I  •—r-  120  i  180  240  300  1  /[Hz]  1  420  1  1  480  540  r~  600  Analytical Calculation I  t  r  —  5  120  180  240  300  /[Hz]  420  480  540  600  Figure 5.18: Harmonics of transformer inrush current  2. Determine the sensitivity for each harmonic frequency. 3. Determine the weighting function for the inrush current. 4. Calculate the total sensitivity.  Individual Sensitivity We define the sensitivity of an impedance as its magnitude's derivative with respect to an admittance change at each node of the Thevenin equivalent circuit. For a harmonic frequency of order h, the sensitivity with respect to a change at node k can be written as: , _aiSa(ft-60 Hz,An)l, M  ^  W-  d\AY \  (5.41)  lAn=d(An)  k  where Zjj stands for the self impedance at bus j, AY for a change in admittance at bus k, and d (AY ) for a very small number representing an incremental perturbation. Equation (5.41) can be formulated as k  k  dZj  jT  Z  Z l j j R  h  '  d\AY  j j I k  '  (5.42)  d\AY  k  where h  Z  ~ jjR Z  3 ' jji  +  Z  5  \ )j\ z  ~ \j )jR Z  +  jji  Z  The values for Z , Z}, °&$ \, and ^y \ can be calculated with the equations developed in Section 5.2.1. R  9  k  d  k  5.3. Harmonic Overvoltages  55  Calculating the sensitivities for all the nodes of the Thevenin equivalent circuit gives the sensitivity vector for the harmonic frequencies:  S {h) fl  S(/i) =  S (h)  (5.43)  jk  I S (h) J jn  If only sensitivities for single frequencies are considered, the vectors can be normalized by referring the vector S(h) to the value of its maximum element, i. e. we get  S (h) n  S(h) =  maxi (Sji(h))  Sjk(h)  (5.44)  I S (h) J jn  Since we deal with either resistive, inductive or resistive-inductive changes depending on the available devices, two different sensitivity vectors can be defined—one for resistive (S ) and one for inductive (S/) changes. This is also reflected in the incremental change d(Ay ), which in the case of resistive changes can be defined as a real number and in the case of inductive changes as an imaginary number. Capacitive changes can be considered as negative inductive changes. R  fc  Total Sensitivity Calculating the sensitivities for each of the harmonics results in a sensitivity matrix, defined as  S -  £11,120  511,180  Si 1,600  5l2,120  5i2,180  5l2,600  Sin,120  <~>ln,180  >ln,600  (5.45)  An overall sensitivity taking into account the sensitivities for all the harmonics up to a frequency of 600 Hz can then be calculated using a weighting vector 'w,120 ^,180  'w,600  (5.46)  5.3. Harmonic Overvoltages  56  which is described in more detail in the following section. Multiplying the weighting vector and the sensitivity matrix (5.45) results in X)h=2 (5ll,/i-60 ' Iw,h-m)  S•I  'total  Ylh=2  (5l2,/i-60 ' Iw,h-w)  v  (5.47)  L J2h=2 {Sln,h-60 • Iw,h-m) J  where n stands for the number of buses where network changes are possible.  Weighting Vector The lower harmonics of transformer inrush currents are of higher magnitude than the higher ones. Therefore a resonance peak at a lower harmonic is worse with respect to resonant overvoltage conditions than one at a higher harmonic. Consequently, changes at lower harmonics are considered more important than changes at higher harmonics. This is taken into account by a weighting vector I which is based on the harmonic characteristic of the transformer inrush current. As pointed out in Section 5.3.2, this characteristic is either given by the manufacturer or can be obtained by simulating the energization of an unloaded transformer using the E M T P and taking the F F T of the inrush current, or by the application of the method outlined in Section 5.3.2. w  In order to find a suitable weighting vector I , a normalized current for each transformer terminal voltage V is calculated as w  T  7( .60,V )= f t  (5.48)  r  maxn (I (h • 60, V )) T  where h stands for the harmonic frequency and V for the pre-switching steady-state transformer voltage. We then obtain the normalized average by T  I (h • 60) =  E  w  /  max  h  ^ "  f  (  ^ , , (Zvl- J(h-W)) 6  0  )  (5.49)  Tmi  Example As an example, the system in Figure 5.19 is examined. It represents the same system as the one in Figure 5.4 and Figure 5.8, but a few restoration steps later. The impedance sensitivity at bus B l is investigated with respect to changes at buses B2, B3, B4, B5, B6 and B7. 1. Thevenin Equivalent Circuit  As mentioned earlier, the E M T P frequency scan in combination with matrix calculations can be utilized to calculate the Thevenin equivalent circuit for the desired range of harmonics. In this, only the frequencies from 60 to 600 Hz are of interest.  5.3. Harmonic Overvoltages  57  G l G2 G3 Figure 5.19: System during restoration procedure  2. Individual Sensitivity Representative for all harmonics up to a frequency of 600 Hz, the determination of the sensitivity for 120 Hz is presented in the following. The elements of the resistive sensitivity vector SR and of the inductive sensitivity vector S/ can be found in Table 5.3. In order to investigate the validity of these results, the impedance at bus B l as function of admittance changes at the other nodes is calculated. They are varied in the range AY = (0.2... 2) Y. For the inductive case, the value Y = —j • 4.421 • 1 0 S is chosen, which corresponds to an inductance L = 3000 mH at the frequency / = 120Hz. For the resistive case Y — 2 • 1 0 S is chosen, which corresponds to a resistance of R — 500 f l The results are shown in Figure 5.20 and Figure 5.21. The numbers in the graphs refer to the number k in Tables 5.3. k  _4  _3  The results show good agreement with the corresponding sensitivities in Table 5.3. The most effective impedance changes for the frequency / = 120 Hz at bus B l are achieved by changing the admittance values at either bus B l or B3.  58  5.3. Harmonic Overvoltages Table 5.3: Sensitivity vectors for /=120 Hz Bus k  Si (ik)  1  2  3  4  5  6  7 .  -1.0000  0.1174  -0.3297  0.0140  -0.0034  -0.0022  0.6388  0.4081  1.0000  0.1966  0.0022  -0.0033 0.0022  0.0006  3. Weighting Function Figure 5.22 shows the frequency components of the transformer inrush current, for terminal voltages between 0.95 per unit and 1.05 per unit, and for frequencies from 120 Hz to 600 Hz. The result for the weighting function is depicted in Figure 5.23.  4- Total Sensitivity The results for the total resistive and inductive sensitivities can be found in Table 5.4. In orTable 5.4: Total sensitivity vectors Bus k  1  2  3  4  5  6  7  SR(U)  -1.0000  -0.0306  -0.1406  -0.2630  -0.0034  -0.0034  -0.0001  SI (ik)  1.0000  0.0455  0.2051  0.2636  0.0035  0.0034  0.0002  der to verify their validity, the impedance at the switching bus B2 as a function of frequency is calculated. Figure 5.24 shows the results for resistive changes at the buses of the Thevenin  5.3. Harmonic Overvoltages  59  Figure 5.22: Harmonic characteristic of transformer  equivalent circuit. The index 0 stands for the case where no change is made and 1, 3 and 4 stand for resistive changes oi R — 100 at the respective nodes listed in Table 5.4. Only the sensitivities with a magnitude of relevance are shown. The same procedure was carried out for inductive changes of L = 2000 mH and its outcome is depicted in Figure 5.25. The results correlate well with their total sensitivities. The most effective changes for reaching an optimum resonance avoidance are changes of the resistances at bus B l . Inductive changes should be avoided at all of the buses.  5.4. E M T P Time-Domain Result Evaluation  60  Frequency / [Hz]  Figure 5.23: Weighting function  Frequency / [Hz]  Figure 5.24: Impedance Z  BX  5.4  for resistive changes as a function of frequency  EMTP Time-Domain Result Evaluation  Although the methods introduced in the previous section allow for a reduction of the number of E M T P time-domain simulations, it will be still necessary to perform a number of such simulations. This chapter investigates the possibilities of finding criteria in order to terminate  61  5.4. E M T P Time-Domain Result Evaluation  8I  N  4-  —i  1  1  ;  :  :....;...,\  1  1—  r  Frequency / [Hz]  Figure 5.25: Impedance Z \ B  for inductive changes as a function of frequency  E M T P transformer energization studies after a minimum amount of time, to limit the overall simulation time [148]. The method that is described in the following sections is based on Prony analysis combined with fuzzy logic rules. A n overview of the method is shown in Figure 5.26.  5.4.1  Prony Analysis  Prony analysis is a fast and effective method for obtaining modal information from timedomain signals. It has been mainly used for the analysis of time-domain results of stability simulations [61, 78], mostly as a complement to small signal stability analysis [143]. However, only a few attempts have been made to use this method for the processing of results from electromagnetic studies (see e. g. [85]). Prony analysis extends the Fourier Transform, which gives the frequency, amplitude and phase of modal components, by providing explicit damping information. In the following, the theory behind Prony analysis is briefly described. More details can be found in [61, 62, 78, 79, 85, 143, 183, 216].  5.4. E M T P Time-Domain Result Evaluation  62  Prony Analysis on the last 2 cycles of E M T P signal Mode filter  Relevance of mode  Closeness to steady state  Fuzzy Reasoning  Yes  Continue E M T P simulation for 1 cylce  No  End  Figure 5.26: Overview of algorithm  Theoretical Background If a signal y(t) is given as a record of N evenly spaced samples y(tk) =y(k), fc = 0,1, the Prony algorithm fits a function of the form [78, 61]: M =  £  i  •  A  e x  P  fot)  (5.50)  • C O S ^ t + 4>i)  This is accomplished with the following procedure [79]: • Fit the function y(t) with the discrete linear prediction model y(t ) n  = a -i n  • y(t -i) n  + ... + a ' Q  y{t ) 0  (5.51)  • Find the roots of the characteristic polynomial associated with the fitting. • Using the roots as complex modal frequencies for the signal, determine the amplitude and initial phase of the signal's modal components. As a result of this calculation we get the modal frequency, amplitude, phase angle, and damping.  5.4. EMTP Time-Domain Result Evaluation  63  Practical Issues It is not possible to identify the modes of a system using Prony analysis when it is subjected to a large disturbance such as network switching or system faults. After such a disturbance, the nonlinearities in the system-e. g. transformer saturation characteristics-are reflected in the current and voltage signals obtained by time-domain simulation. However, Prony analysis assumes linear system behavior. As a consequence, when Prony analysis is applied successively using a sliding window, the analysis results will change as a function of time [59, 79]. The purpose of the application of Prony analysis in our case is not to identify system modes and investigate system properties. Instead, it is utilized to find the frequency components of the signal in each window, in a way similar to the Short-Time Fourier Transform (STFT) (see e. g. [190]) and the case presented in [27]. A single sample window is chosen to cover two 60 Hz cycles, which allows the capture of the fundamental frequency component and leads to a sufficient resolution of the harmonic components of the signal. The first analysis is performed after two 60 Hz cycles of an E M T P simulation have passed. Then the sample window is moved by one cycle, allowing for a one-cycle overlap of two successive windows. This leads to more reliable results since each cycle is covered twice by an analysis window.  5.4.2  Filter for Modes  The results obtained by Prony analysis are sometimes ambiguous and therefore not easily interpreted. Modes, needed for correct fitting of the signal in one window may disappear in the next one. Furthermore, the signal-to-noise ratio (SNR), which is a measure for the quality of fitting [79], may be too small to make a correct statement about the spectral content of the signal analyzed. A number of modes can therefore be discarded from further evaluation using approximate reasoning [213]. All of the fuzzy logic techniques described in the following are based on Monotonic (Proportional) Reasoning [41]. This method allows for a fast and efficient decision making process, since the rules described in the following can be implemented with a few simple equations. It is well suited for problems using fuzzy rules with the same consequent fuzzy set, such as ours, since it avoids the saturation of the solution fuzzy set. Details about fuzzy logic in general and Monotonic (Proportional) Reasoning in particular can be found in [41, 213]. In addition, an application similar to ours, where the same method is used for the step-size control of a time-domain power system simulator can be found in [82, 83].  Approximate Reasoning for Filter The underlying rule for this reasoning process can be expressed as: "// the mode has recently appeared and if the signal-to-noise ratio (SNR) of the analysis is sufficiently high, then the mode is kept for further evaluation, else it is discarded."  5.4. E M T P Time-Domain Result Evaluation  64  This process is depicted in Figure 5.27 and can be expressed by the following equation: P filter  (5.52)  (m)  where m stands for the number of the current Prony analysis and ip allows for the adjustment of the filter sensitivity. total appearance  <  m.  A  filter function u , . ^ signal-to-noise ratio | ^  N R  .1  1 yes keep mode  Ii  discard mode  8  Figure 5.27: Reasoning function for filter  Membership Function for Total Appearance The total appearance assigns to each mode a membership function which depends on how long ago and how often it has appeared in the past. This is shown in Figure 5.28 where the relative appearance membership function p, A is displayed for the last N — 5 analyses. The modes are weighed the higher the more recent their appearance has been. From the relative appearance we can then determine the total appearance using the equation R  N =  fi A{m,h) T  v  £ T ( ™ - i,h) •  p {i) RA  (5.53)  i=l  where  /  . j\  f 1 if mode appears at (m — i)th analysis  and h stands for harmonic order, and rj normalizes the maximum of the function to one.  Membership Function for Signal-to-Noise Ratio The Signal-to-Noise Ratio (SNR) is a measure of how well the original signal is fitted by the modes as obtained by Prony analysis, i.e. a high SNR indicates an accurate analysis result. The SNR is defined as SNR  =  20-lo yrffi l|y(  8  )l1  dB  (5.54)  Wk)\\ where | j . . . || stands for the root-mean-square norm [79]. The membership function for the SNR, P-SNR, is shown in Figure 5.29.  5.4. E M T P Time-Domain Result Evaluation  m-5  65  m-1  Figure 5.28: Membership function for relative appearance  S  N  R  Figure 5.29: Membership function for signal-to-noise ratio (SNR)  5.4.3  Prony Analysis Results Evaluation  After a mode passes the filter it is evaluated with respect to two different criteria: its importance and its "closeness to steady state". The number of Prony analyses N, on which the statement whether an E M T P simulation should be finished or not is based, results in the following conflicting matters: if N is smaller, the E M T P calculation time will be smaller and the statement will be less reliable, and vice versa. A n analysis depth N between 5 and 8 has shown to be a good compromise between reliability and effectiveness of the determination of the termination criteria. In the following, the various membership functions used for this process are explained, and the reasoning process leading to a conclusion whether the E M T P simulation can be terminated, is described.  5.4. EMTP Time-Domain Result Evaluation  66  Relevance of Modes  The underlying rules for the evaluation of the relevance of a particular mode can be summarized as: 1. "If the amplitude of the mode is large compared to the fundamental frequency mode, then the relevance of the mode is high" 2. "If the damping is small, then the relevance of the mode is high" 3. "If the average damping changes quickly, then the relevance of the mode is high"  This reasoning process, which is based on monotonic chaining and leads to the membership function UTR, is depicted schematically in Figure 5.30.  tendency  relevance level  Figure 5.30: Monotonic reasoning in order to determine total relevance of mode  1. Membership Function for Amplitude  This function, shown in Figure 5.31, can be obtained by referring the amplitudes of each mode to the 60 Hz steady-state value as obtained with the initial E M T P steady-state solution. It means that the higher the amplitude of a mode, the higher its relevance:  5.4. E M T P Time-Domain Result Evaluation  67  A(60 Hz)  A(m,h)  Figure 5.31: Amplitude membership function  2. Membership Function for Damping The smaller the damping of a mode, the more important it is. The membership function for damping 1 ^dampingn  {jTlj h) —  for d(m, h) < d  min  1- n 0  —i  (d(m, h) - d ) min  for d  min  < d(m, h) < d  max  (5.56)  for d(m, h) > d,  is depicted in Figure 5.32.  Figure 5.32: Damping membership function  3. Membership Function for Tendency of Damping Besides the value for damping it is also necessary to take into account its tendency. This is accomplished by using a statistical method called linear regression [182]. With this, the  5.4. E M T P Time-Domain Result Evaluation  68  slope of a linear interpolation through the last N values of d is defined as  12Ef  p(m,h)  =1  (i-^)'-(m-(JV + l ) + t )  (5.57)  -N + N  3  The higher the value of the tendency, the more important its change. function, which is displayed in Figure 5.33 is defined as  Htend {m, h) R  1  for p(m, h) < p  0  for 0  B  ainl  m a x 2  -8  , f3{m, h) >  < /5(m, h) <  minl  1 -  m i n 2  Pmirtl  (/5(m, h) -  (^( ' ) m  m a x l  Pmin2  h  (3  max2  B  maxl  /3 in2) m  for  p 2  - Pmaxl) for p l  Pmn il  Its membership  max  Pmaxl  P ax2  min  <  /3(m, h) <  < P{m, k) <  (5.58)  /3  minl  f3  max2  P(m,h)  m  Figure 5.33: Membership function for tendency of damping  Closeness to Steady State In this section each mode is assigned a membership function for "closeness to steady state". The rules can be summarized as: 1. " / / the amplitude of the mode is small, then steady state is close" 2. " / / the damping is large and positive, then steady state is close" 3. "// the tendency of damping is moderate and negative, then steady state is close" 4. "// the variance of damping is small, then steady state is close" The result of these rules is the "closeness to steady state" membership function PSST- The underlying reasoning process is displayed in Figure 5.34 and the membership functions are  5.4. E M T P Time-Domain Result Evaluation  69  Figure 5.34: Monotonic reasoning in order to determine total "closeness to steady state" of mode  described subsequently. 1. Membership Function for  Amplitude  This function, displayed in Figure 5.35, is obtained by inverting the amplitude membership function in Equation (5.55): (^amplitudes  ~  1  A''amplituden  (5.59)  It means that the mode is the closer to steady state the smaller its amplitude. 2. Membership Function for  Damping  Here, the inverse of the membership function (5.56) is used, since the higher the damping, the closer we are to steady state: ^dampings  ~  ^  l^dampingn  The membership function is displayed in Figure 5.36.  (5.60)  5.4. EMTP Time-Domain Result Evaluation  70  A(60 Hz)  A(m,h)  Figure 5.35: Amplitude membership function  d(m,h)  Figure 5.36: Damping membership function  3. Membership Function for Tendency of Damping This function is shown in Figure 5.37 and is defined as:  {  0  for p(m, h) < j3 ,  P(m, h) > /3  8  .. \-0 • , (/^( ' ) ~  minS  1  m  ~~ B  *-B  7  max3  for  h  (^( > ) m  h  P in3 m  < P{m, h) <  ~ Pmediumz) for fimediumZ <  (5.61)  fimediumZ  P{m, h) < /3  max3  4- Membership for Variance of Damping It is not enough to only consider the damping by its tendency, since the latter may be small although there is still a high fluctuation. As a measure for thisfluctuationwe use the  5.4. E M T P Time-Domain Result Evaluation  71  Figure 5.37: Tendency of damping membership function  variance of damping [182]: m—1  1 h-) — , \ | JV  V mping(m, da  ' : (d{i,h)-di (i)f  (- ) 5  inear  62  i=m-N i=m-l  \  where d (i) stands for the linear regression through the last N samples of damping d. The variance is normalized by dividing by di (m): Hnear  inear  V  (i)  dampm9  =  .J*™*"  (5.63)  (^linear \ m)  The membership function, displayed if Figure 5.38 can then be defined by:  1,  •  (m  Til -  I  1 1  for Vdampingim, K) < Vdmin ~  y<tmaJ-V  lOr Vdmin  0 for V mping(rn,  5.4.4  g  *damping\m, da  h) ~  ( dampin (m, V  dmin  <  V  )  dmin  Vdmax  h) > V  dmax  Reasoning Process  Based on above, we can conduct a final decision making process, giving us a statement whether we are close to steady state and whether the E M T P simulation in progress can be terminated or not. The underlying rule can be formulated as: "If each important mode is close to steady state, we are overall close to steady state and the E M T P simulation can be terminated."  72  5.4. E M T P Time-Domain Result Evaluation  ,(m,h)  Figure 5.38: Membership function for variance of damping  Mathematically this can be defined by: Vtotai(m)  El=o* (t*ssr{m, h) • u, (m, h)) TR  (5.65)  where h = 0 stands for the fundamental frequency, / i = l for the first harmonic, etc. This process is depicted in Figure 5.39. A signal consisting purely of the fundamental frequency would lead to a total membership function value of (itotai = 1- The value of 0.95 in Figure 5.39 can be increased, depending on the specific requirements concerning the "closeness to steady state". 60 Hz total steady •-l.iii-  60 Hz total relevance UTR (™.0)  120 Hz total steady state H (m,l) SST  120 Hz total relevance (m.l)  total membership function  no 60Hz total steady state  stop EMTP simulation  continue EMTP simulation  ±1 60Hz total relevance  Figure 5.39: Final reasoning process  5.4. E M T P Time-Domain Result Evaluation  73  E M T P signal  predicted signal  Time t [s]  Figure 5.40: E M T P signal and predicted signal E M T P signal  >  predicted signal  ,00  0  a "3  -100  >  |  V f I IK 0.35  0.4  ? 0.45  Time £ [s]  Figure 5.41: E M T P signal and predicted signal for m> 15  5.4.5  Example  The system under study is shown in Figure 5.4. It has a resonance frequency at 180 Hz. The restoration step under consideration is the switching of the transformer at bus B3. For comparison a full E M T P simulation for 35 cycles has been carried out. Both Figures 5.40 and 5.41 show the E M T P simulation result. Since the resonance frequency is 180 Hz, the  5.4. E M T P Time-Domain Result Evaluation  74  rjl  1  1  i  i  I  i  I  0  5  10  15  20  25  30  35  Number of Prony analysis m  Figure 5.42: Amplitude of 180 Hz mode  ;l  0  1  5  —  l  10  l  15  l  20  I  25  l  30  i  35  Number of Prony analysis m  Figure 5.43: Damping of 180 Hz mode  only harmonic of relevance is the third. Its amplitude and damping are shown in Figures 5.42 and 5.43, respectively. The fuzzy reasoning algorithm terminates the E M T P simulation after m = 15 Prony analyses. Subsequently, the signal can be predicted using Equation (5.50). The predicted signal is also displayed in Figures 5.40 and 5.41. It is so close to the E M T P simulation result that the differences are barely noticable.  5.5. Switching Transient Overvoltages  75  The application of the routine for this example shortens the overall calculation time by approximately 3 7 %. This percentage increases for larger networks, since the E M T P simulation time increases with increasing network size, whereas the total overhead time for the Prony analysis and fuzzy logic algorithms stays constant.  5.5  Switching Transient Overvoltages  Switching surges are fast transients with fundamental frequencies between 100 Hz and 1000 Hz. They occur when transmission lines are energized, and consist of a transient component that is superimposed on a power frequency component. In the worst case, switching surges can lead to flashovers, and to serious damage to equipment. However, they are usually not of particular concern during the reenergization of the transmission system during restoration. As long as the steady-state voltages remain below 1.2 per unit they can be easily limited by surge arresters [95]. In cases where no surge arresters are available, line flashovers do not impose a problem on the restoration process as well, since line insulators are usually designed conservatively to withstand steadystate overvoltages during heavy fog conditions, resulting in very high impulse strength. A n exception is the case where transformer terminated lines are energized. That case may lead to the occurrence of harmonic resonance overvoltages [95] and has been investigated in Section 5.3.  As a consequence of the above, the control of transient switching voltages is not investigated in this work, and only general observations are made. The parameters of most influence on the line switching overvoltages are [38]: • Line parameters: — line length — degree of parallel compensation — line termination (open or transformer terminated) — trapped charges • Circuit breaker parameters: — Closing resistors — Phase angle at moment of switching • Supply side parameters: — Total short-circuit power — Inductive / complex network In the case where underground cables are energized, the transients are dominated by the cable capacitances. When switching cables, traveling wave effects are not prevalent but there  5.6. Summary  76  is a higher potential for sustained overvoltage problems. The latter is dealt with in Section 5.2. The influence of different parameters on switching overvoltages is currently investigated in detail as part of a Ph.D. thesis research project of Awad Ibrahim at the University of British Columbia.  5.6 Summary In this chapter, a methodology has been outlined that assesses the feasibility of restoration steps with respect to overvoltages. It is based on analysis in the frequency domain using the frequency scan feature of the E M T P , and analytical matrix manipulations. Using approximate rules allows one to estimate whether an overvoltage occurs and whether time-domain simulations or power-flow calculations need to be carried out. In the case of time-domain simulations, a method is introduced that allows one to shorten the simulation time using a method based on Prony analysis and fuzzy logic. When overvoltage problems occur, the most efficient remedial action is determined by sensitivity analysis. The validity of the methods has been verified using several examples based on actual system data.  Chapter 6 Auxiliary System Analysis After a widespread blackout, the power of large thermal generating units has to be restored quickly and reliably to bring major loads back into service as soon as possible. If no assistance from other utilities is available, a restoration path from a relatively small hydro generator or gas turbine has to be built to the large thermal power station to start up the thermal power plant's auxiliaries. These auxiliaries are mostly driven by large induction motors, drawing high currents and requiring long starting times. To ensure a successful start of a thermal power plant, it is crucial to investigate the feasibility of such auxiliary motor startups [36]. This is usually accomplished using time-domain simulation programs such as the Electromagnetic Transients Program ( E M T P ) , or stability programs (see Chapter 3.4). The implementation of the induction motor differential equations in most simulation programs can be considered as mature, and in most cases no further improvement is required [36]. However, currently available methods for determining induction motor parameters often do not take into account all available motor data. This may lead to inaccuracies in the motor model. Therefore, a new parameter estimation method for induction motors is introduced. It creates induction motor models from manufacturer nameplate data and from motor performance characteristics. The induction motor models can then be used for black and emergency start studies with the Electromagnetic Transients Program ( E M T P ) , or with stability programs, and for the rules outlined in this chapter. Time-domain simulations of motor start-ups require a vast amount of calculation time. Therefore, rules are developed in the following that estimate motor start-up times, currents, voltage drops, and frequency drops during motor starts.  6.1 6.1.1  Motor Parameter Estimation Overview of Parameter Estimation Methods  Generally, induction motor parameter estimation methods can be classified into five different categories, depending on what data is available, and what the data is used for.  77  6.1. Motor Parameter Estimation  78  1. Parameter calculation from motor construction data. This method requires a detailed knowledge of the machine's construction, such as geometry and material parameters. On the one hand, it is the most accurate procedure, since it is most closely related to the physical reality. On the other hand, it is the most costly one since it is based on field calculation methods, such as the finite element method [24, 53]. This method is mainly applied in induction motor design. 2. Parameter estimation based on steady-state motor models. The methods in this category use iterative solutions based on induction motor steady-state network equations and given manufacturer data [89, 105, 191, 222]. This is the most common type of parameter estimation for system studies since the data needed for it is usually available. 3. Frequency-domain parameter estimation. The stand-still frequency response (SSFR) method is based on measurements that are performed at standstill. The motor parameters are estimated from the resulting transfer function [223]. The major advantage of this method is its accuracy. However, stand-still tests are not common industry practice, and this method can therefore not be used very often. 4. Time-domain parameter estimation. For this type of method, time-domain motor measurements are performed and model parameters are adjusted to match the measurements [44, 107, 171]. Since not all parameters can be observed using measurable quantities, the motor models need to be simplified [107]. The method is costly, and the required data is usually not available. 5. Real-time parameter estimation. This type of parameter estimation is used to tune the controllers of induction motor drive systems. This requires real-time parameter estimation techniques, using simplified induction motor models, that are fast enough to continuously update the motor parameters and therefore prevent the "detuning" of induction machine controllers [84, 206]. The motor parameter estimation method proposed in this section belongs to the second group of methods. It is suitable for system studies since sufficient data is usually available to determine a motor model of sufficient accuracy. Most methods in this category have the disadvantage that they only accept nameplate data [89, 191], even when more data, such as performance characteristics, are available. Furthermore, they often ignore constraints on the machine parameters or saturation effects [105, 222]. This paper proposes a new approach to overcome these drawbacks and to make it possible to flexibly determine motor models for any given combination of manufacturer data. The new motor parameter estimation routine can be considered as a generalization and combination of the methods introduced in [105, 191]. In the following, the input data to the motor parameter estimation routine is described. Then, a parameter estimation algorithm neglecting saturation is developed, tested and extended to allow for saturation effects. Finally, the results for a typical 600 HP pump motor, such as commonly used in large thermal power plants, are given.  6.1. Motor Parameter Estimation  6.1.2  79  Input Data  The proposed parameter estimation routine allows the input of different types of data, depending on their availability. The first type of data is the one given on the motor nameplate, where subscript " F L " refers to full load, and " L R " to locked rotor test. • Rated power PFL • Rated voltage  V d rate  • Efficiency T\FL • Power factor P/FL • Rated slip SFL • Starting current I^R • Starting torque T  LR  • Breakdown torque T  max  This set of data is not always sufficient to obtain a motor model which behaves accurately over the entire speed range. The new parameter estimation routine can therefore use extra motor performance data such as: • Current-slip characteristic I .f.( ) s  m  • Torque-slip characteristic T / . ( s ) m  • Power factor-slip characteristic p/ ./.(s) m  where "m./." stands for "manufacturer data". Additional data, such as rotational losses P tFL, can also be taken into account, which create constraints and boundaries for the nonlinear optimization method of the following section. ro  6.1.3  Nonlinear Optimization Procedure  Our algorithm is based on the nonlinear optimization routine SOLNP [230, 231], which solves nonlinear programming problems of the general form:  minimize subject to:  (6.1)  F(X) G(X) = 0  (6.2)  L < H(X) < U L <X<U h  x  x  h  (6.3) (6.4)  6.1. Motor Parameter Estimation where XeR ,  F(X) : R  n  n  80  —> R, G(X) : R  —>• R \  n  H(X) : #  m  n  —• # , m2  L ,U ei? , m 2  h  h  L < U , L , U eR and L < U , L < X < U , XeR . n  h  h  x  n  x  x  x  x  0  x  The function F(X) represents the objective function. Equation (6.2) stands for the equality constraints, and Equation (6.3) for the inequality constraints, and Equation (6.4) represents the boundaries on the variables. X is the initial estimate for the solution. The underlying mathematical algorithm uses major iterations, in which a linearly constrained optimization problem is solved based on a Lagrangian objective function. Within each major iteration, minor iterations are carried out, based on linear and quadratic programming. The functions (6.1) to (6.4), as used for our parameter estimation routine, will be defined in the following sections. 0  6.1.4  Motor Model without Saturation  Objective Function The objective function F(X) is developed starting from the steady-state equations for a double-cage induction motor. These equations can also be used for induction motors with deep-bar rotors and with a single-cage [143]. For the derivation of the steady-state equations the motor equivalent circuit of Figure 6.1 is used [54, 143]. All parameters and equations are given in per unit. The impedances I,  *  i  R  x  i  *2 ^27  hi  M  X  22  l  V  s  Figure 6.1: Induction motor model without saturation  for the two rotor circuits in Figure 6.1 are #21 . •  ry  (6.5)  v  s  #22  ry  •  A-2 =  r-  (6.6)  v  jX  22  This results in the total rotor impedance Z  r  = Rr + jX  r  = jX  2  +  Z\ r  Z\ r  • Zr2 +  Z2 r  (6.7)  6.1. Motor Parameter Estimation  81  and the total motor impedance as seen from the motor terminals  Z  mot  = R +jX + 1  J  1  *  ' :  m  (6.8)  Z  The motor current as a function of the slip follows as  h =  (6.9)  where Vi represents the motor terminal voltage. Using current divider equations we then get  h  J  =  2 „  X m  hi  =  h  + z  jX  „ 2  h  (6:11)  •h  (6-12)  Z  7  Z/  (6.10)  r  L2  -|-  R L  2y  T  Zr  =  hi  „  From the above equations we finally obtain the equations for current magnitude, power factor and torque: Us)  =  \h\  (6.13)  pf {s) = cos Z(Z ) c  (6.14)  mot  Us)  =  ^l-|/  2  1  |  2  + ^-|/  s  2  2  |  (6.15)  2  s  where the index "c" stands for "calculated". Equations (6.13) to (6.15) form the basis for the objective function F ( X ) in Equation (6.1), which is defined as a quadratic error function: nj  pf  n  TIT  = W r - J ^ ^ + WT-^AUsJ i=l i=l  F(X)  + Wpf-^Al^) i=l  (6.16) (6.17)  where X  =  [Ri,X ,X ,X ,R2i,X2uR22,K,m\ 1  M  (6.18)  2  with K and m defined in Section 6.1.4, and  i{si)  A  A  s  i  =  )  =  — ^  , (  T(si) — T ,j_{si) m  *= l , . . . , n /  i =  1 ;  >  n  (6.19)  r  ( 62  0  )  J-m.f. A  /  \  Pf{Si)-pfm.f.{Si) =  A (Si) pf  7 *r  . i = l,...,n  ,  ,  (6.21)  p f  PJm.f.ySi) The quantity Si represents discrete values for the induction motor slip, and ri[, n , n j are the total number of data points available for current, torque, and power factor, respectively. The factors Wj, W , and W f are weighting factors described next. T  T  p  p  6.1. Motor Parameter Estimation  82  Weighting Factors The choice of the weighting factors for the objective function -F(X) is of great importance for the optimization process. It is desirable to give each of the torque-slip, powerfactorslip, or current-slip characteristics equal weight. Otherwise, if e. g. a large number of data points for the torque-slip characteristic and only a small number of points for the current-slip characteristic is provided, the latter would only have a negligible influence on the value of the objective function if each point had equal weight. This would lead to significant differences between manufacturer and calculated values in the current-slip characteristic. Generally, nameplate parameters are more reliable than motor performance characteristics. The latter do not always give exact numbers, but rather indication of the generic performance behavior [17]. Based on these observations we define the following weighting factors: W  r  =  TiT + n f  (6.22)  W  T  =  n + n  p /  (6.23)  W  =  ni + riT  (6.24)  pf  p  7  The importance of nameplate parameters is reflected by multiplying its weighting factors by an additional factor W  n  p  «>+«r  =  +  n  "  (  6  2  g  )  Constraints and Boundaries If neither constraints nor boundaries are used for the optimization, the result vector X may contain "non-physical" values, such as negative values for resistances. Consequently, we define the following boundaries: R ,X ,X ,X ,R ,X ,R l  1  M  2  21  2l  > 0  22  (6.26)  We further define a boundary condition for the design factor m [191]: m  =  R  2i  +R  A i + 2  0.4  <  22  ,  .  (6.27)  A 2 2  m < 1.1  (6.28)  when neither the rotor type nor m are known. In cases where the rotor type is known, the boundaries in Equation (6.28) are adjusted. The following boundary conditions are chosen, allowing for a ± 1 0 % variation of the values given in [191]: 0.45 <  m < 0.65  for deep bar rotors  (6.29)  0.90 <  m < 1.10  for double cage rotors  (6.30)  If the rotor is of type single cage, the constraint for m is removed since a design factor in this case is not defined.  6.1. Motor Parameter Estimation  83  Based on nameplate data, four constraints G ( X ) , . . . , G^X.) are defined: 1  X  2  - K - X  l  X -Xi r  Tmax m.f. PFL~  =  0  (6.31)  =  0  (6.32)  T (X.)  (6.33)  — 0  max  (#1 • If FL + #21 • I$i FL + #22 ' I$2 FL + P.rot FL)  PiFL  -T]FL  =  0  (6.34)  The factor K in Equation (6.31) is subject to another boundary, whose value is chosen according to [17, 29, 191]: 0 < K < 0.4  (6.35)  Equation (6.32) comes from the assumption that the rotor and stator reactance are equal [136, 143]. The maximum or breakdown torque T is reached when the power flow from stator to rotor is a maximum, i. e. when the following condition is fulfilled: max  ^  \J(#Lt +  =  X (s )) )  (6.36)  2  (Xstat +  r  max  The values for R and X are obtained from the Thevenin equivalent impedance that replaces the stator part of the induction motor (see Figure 6.2): stat  stat  Zstat — Rstat + j X  —  s t a t  j X  - ( # l + j * l )  M  (6.37)  From Equation (6.7) we can further calculate the resistance R and reactance X r  ^tat  X  r  of the  st  *21  *22 12,  Stator  Rotor  Figure 6.2: Induction motor model with stator part replaced by Thevenin equivalent circuit  rotor as: 1  #21 • #22 • (#21 + #22) + (#21 + #22) + 2  S '  1  X  r  S '  # | i • ^22  + #22 ' X21  2  2  2  (#21+#2 )  +  S  • (R  2  +  2  •Xi 2  2  - ( X  •X  22  2  +R  • X\  21  • (X i  S  + S 2  2  S  1  + X  • X ,) 2  22  (6.38)  X)  2  22  • (X j 2  2  2  )  2  + X) 22  + X  2  (6.39)  6.1. Motor Parameter Estimation  84  Equation (6.36) can be solved analytically for s determined from Equation (6.15).  and the maximum torque T  max  max  is then  Initialization It is important to start the optimization process with initial estimates X as close as possible to the values that lead to a minimum for the objective function F(K). This is accomplished using approximate equations based on [191]: 0  V'FL  =  0-25 + 0.75 •  RI  =  P/FL  RFL  =  s -  y  _  FL  • (l V  1-  F  6.40)  VFL  6.41)  -  1 -  SFLJ  ;6.42)  -P/FL  L  SFL  VFL  ;6.43)  (1 - SFL) • smcpFL RLR  =  6.44)  T -rf -pf l-^1- sL FL  FL  FL  F  m  =  •R21 = 7?  RLR  JX22  —  P ri2i  X  =  0  21  A2 v  2  6.45)  0.7 •  (1 + m ) - R 2  R  Ri  f  —  6.46) ;6.47)  l  RFL  ;6.48)  R21 + R22 —  • m  2  FL  m  A  v 2  ;6.49)  i  ;6.50) 6.51) X  2  K  —  X\ — RFL •  X2  m R22 m + l R21  2  ;6.52) 6.53)  Verification of Algorithm In case a set of adverse initial estimates is chosen, the optimization routine may converge to a local minimum for the objective function that does not represent the best possible solution. In order to ensure with a high probability that the parameter estimation algorithm leads to an absolute minimum, the initial estimates are varied randomly between 0% and 200% of the values calculated according to Equations (6.40) to (6.53), except for the parameters m and K (Equations (6.29) and (6.35)), which are randomly varied between their boundaries. If the initial estimates are chosen close enough to the set of parameters that leads to the  6.1. Motor Parameter Estimation  85  absolute minimum of the objective function, the objective function value should always be equal or smaller than the values we obtain for random initial estimates. Different rotor parameter sets may lead to the same value for the objective function. To compare between the results for different initial estimates, one of the rotor parameters is fixed for this test. Otherwise the objective function may lead to the same value although the motor parameters are different. As suggested in [143, 191], the rotor reactance X i is therefore limited to a small value: 2  0 <X  21  < IO  (6.54)  - 3  For the test, we use the motor data of a 600 H P pump motor, which is currently in use in a large thermal power plant. Its nameplate data is listed in Table 6.1 and the torque-slip, current-slip, and powerfactor-slip characteristics are displayed in Table 6.2. Table 6.1: Induction motor nameplate data PFL [HP] V  [kV]  RATED  Synchr. speed [RPM] T  FL  [Nm]  IFL [A] PIFL VFL SFL [%]  [Nm]  600 4 . 900 4812 75.5 0.914 0.936 1.333  ILR [A]  4861 480.8  PILR  0.29  T  LR  Tmax [Nm]  13810  For 100 different sets of initial estimates, the algorithm converged in 81 cases to the same objective function values and delivered the same motor parameters. The values for the other cases were significantly higher and therefore belong to local minima that do not represent the best possible solution. The objective function values with F ( X ) < 0.2 are displayed in Figure 6.3 and the number of major iterations in Figure 6.4. The first set of initial estimates is the one calculated with Equations (6.40) to (6.53). In Table 6.3 the mean of the motor parameters X and the percentage deviation A X from the mean are given. The deviations are negligible, despite the variation in the number of major iterations between 3 and 10. As a second test, the algorithm's robustness is investigated. "Correct" current-, torque-, and powerfactor-slip characteristics are created from the motor parameters in Table 6.3. These are then fed back into the algorithm. The difference A X . f . _ . between the parameters obtained from the "correct" characteristics and from the manufacturer data, given in Tables 6.1 and 6.2, is shown in Table 6.4. The only significant difference can be observed for variable X . However, since its maximum value is limited to the small value of 10~ , it is of no relevance. m  c  o  r  r  3  2X  6.1. Motor Parameter Estimation  86  Table 6.2: Induction motor characteristics  Speed [RPM]  I [A]  0.0  480.8 -  150 180 400 405 600 630 750 765 850 888  75.5  T [Nm]  Pf  0.290  4861.0  -  5288.0 -  0.306 0.339 0.416 0.570 0.813 0.914  6454.0 8243.0 12202.0 12609.0 4812.0  0.0616  X  0.0614  fc.  a. o  S£  JO  O  0.061  0.0608  0.0606 h  0.0604  10  20  30  40  50  Set of initial estimates  60  70  80  Figure 6.3: Values for objective function  6.1.5  Motor Model with Saturation  In this section, we discuss how saturation effects can be taken into account. Only the saturation of the stator leakage reactance X\ and the rotor mutual leakage reactance X are considered, which is a good approximation for most motor start-up studies [191]. The equivalent circuit for a motor including saturation effects is depicted in Figure 6.5 [143]. In the following, only the changes needed for the saturation effects are outlined. 2  6.1. Motor Parameter Estimation  87  a o CP  CD  S sf--  20  30  40  50  70  Set of initial estimates  80  Figure 6.4: Number of iterations  Table 6.3: Induction motor parameters in per unit X  X / [p.u.]  A X / [%]  Ri  0.024111  x  0.076779 0.027005  0.0005 0.0006 0.0257  Xi  0.00099999 0.027389  0.0416 0.0271  0.070127  0.0415 0.0293  x  R21 2  R22  x  2  XM  3.2316 0.60206 0.91336  m  K  0.0996 0.5426  Objective Function To represent saturation of the leakage reactances, we use the "describing function" DF defined in [191]: DF  =  1  DF  =  - • (a + 0.5 • sin (2a))  7  a  =  for 7 > 1  hsL  — arcsin (7)  (6.55) for  7 < 1  (6.56) (6.57) (6.58)  6.1. Motor Parameter Estimation  88  Table 6.4: Deviation of induction motor parameters X  AX .f._ m  c o r r  . / [%]  0.0459 0.0263 0.9137 25.606 0.8115 0.4028 0.2592 0.5889 0.4292  Ri #21 X  2X  #22  x  2  XM  m K  \  Figure 6.5: Induction motor model with saturation  where I stands for the saturation threshold current and / for the current flowing through the reactance. sat  It is assumed that both stator and rotor leakage reactance have the same percentage part saturated, and define the modified reactances as: X  x  = Xi + DFi • X  = (1 - SAT) • X  X  2  = X  = (l-SAT)-X  u  2u  ls  + DF -X 2  2s  w  20  + DF\ • SAT • X  w  + DF -SAT-X 2  (6.59) (6.60)  20  The factor SAT represents the per unit value of the saturable parts of the leakage reactances Xi - and X with respect to the unsaturated leakage reactances X i and X o's  2s  0  SAT  Xu  x  Xw  Xo  2s  2  (6.61)  2  The values of the currents and the describing functions are now determined iteratively, since the currents depend on the values of the reactances, and vice versa. This is shown in the diagram in Figure 6.6, where the superscript "n" stands for the nth iteration. The objective function F ( X ) for the saturated induction motor model has the same form as the one without saturation, with two more variables added: X  =  [Ri,Xi,X ,X ,R i,X i,R ,K,m,I ,SAT] M  2  2  2  22  sat  (6.62)  6.1. Motor Parameter Estimation  Xj - X n  1 0  89  ; X j " - X.  I," ;  20  V  DF, ; DF  2  X , ; Xj ¥ n  —  T  n+1  .  T  n_  T  n+1  No  Figure 6.6: Calculation of currents and describing functions  If detailed saturation data is available, the saturation threshold current can be fixed, introducing an additional boundary.  Constraints, Boundaries, Weighting Factors, and Initialization In addition to the boundaries described earlier, we add [191]: 0.2 < SAT < 0.8 1.5 < I  sat  < 3.0  This allows a deviation of ±60% from the value SAT = 0.b suggested in [191].  (6.63) (6.64)  6.1. Motor Parameter Estimation  90  The breakdown torque for the saturated case can no longer be calculated analytically since, as outlined earlier, the currents have to be determined iteratively. Instead, it is found with an additional maximum finding routine. Since the torque function may have another (local) maximum, the range where the maximum torque is located is limited to: 0.0 <  s  max  < 0.2  (6.65)  For the initialization procedure, the values calculated in Equations (6.40) to (6.53) remain the same, and two initial values for the new variables are added [191]: hot  =  2.0  (6.66)  SAT  =  0.5  (6.67)  Verification of Algorithm For the verification of the algorithm that includes saturation of leakage reactances, the same motor is investigated and the same procedures are carried out as for the case without saturation. For 100 different initial estimates, the algorithm converged in 78 cases to the same objective function values. The objective function values and the number of iterations as a function of the set of initial estimates look similar to Figure 6.3 and 6.4. Table 6.5 shows the mean of the motor parameters and the percentage deviation from the mean. Table 6.5: Induction motor parameters in per unit  X  X / [p.u.]  AX- / [%]  Ri  0.02369  0.00009  0.11078 0.02965 0.00002  0.00060 0.00190 0.00030  0.02372 0.10257 0.20314  0.00120 0.00080  1.10000  0.03970  0.92592  0.00950  0.49999 1.50000  0.00020 0.00080  x  l  R21 X21 R22  x  2  X-M  m K SAT hat  6.1.6  0.00690  Results  Figures 6.7 to 6.9 show the results obtained from the algorithm for the 600 HP motor. Good agreement between the manufacturer's and model characteristics can be observed. The match for the model with saturation is slightly better than the one for the model without  6.1. Motor Parameter Estimation  91  saturation. This difference is not significant for this particular motor. It can be explained by the additional degree of freedom that is introduced by the inclusion of saturation, which leads to a better overall fitting. The results also show that close to the rated voltage, the model without saturation gives a reasonably good fit, and can therefore be used for cases where voltage variations are small. 600  1  i  i  ...;V....  1-  o  ° >  )  No saturation Saturation Manufacturer data 1  1  1  1  1  1  200  300  400  500  600  700  1  Speed [RPM]  Figure 6.7: Current characteristic  i  i  No saturation  i  0.2 -  0.1  o' 0  i 1 100  • 1— 200  1  1  i  •:  r  ;  •  i  •:  1 300  I  I  I  400  500  600  .  -  ;  i  1 700  1 800  Speed [RPM]  Figure 6.8: Powerfactor characteristic  -  900  6.2. Rules for Motor Start-up  92  Figure 6.9: Torque characteristic  6.2  Rules for Motor Start-up  A general overview of the methodology described in this section is given in Figure 6.10. For the formulation of the rules, it is assumed that only one motor is started at a time. In case several motors are started at once, they are lumped into a single machine. Methods that deal with the aggregation of static and motor loads were investigated in a Ph.D. thesis that was recently published at the University of British Columbia [157]. Thus, motor aggregation methods are not discussed in this report. In order to evaluate the following rules [149], the sample system shown in Figure 6 . 1 1 is used. It represents the worst-case scenario in a power system where a number of induction motors are emergency-started by a hydro generator. The turbine, governor system, and generator of the system have already been used in the example in Section 4 . 1 . 1 . The block diagram that represents the excitation system is displayed in Figure 6.12. All induction motors are lumped into a single machine of power 14.4 MW. Data for this aggregated load model has been provided by the utility.  6.2.1  Thevenin Equivalent  A multi-port Thevenin equivalent impedance Z with terminals for the motor and generator is determined by E M T P frequency scans. This is done only once, since changing network conditions during the extension of the system can be calculated using the matrix calculations outlined in Section 5.1. All variable elements, such as loads, reactances, capacitances, and motors are assumed to be included in the impedance matrix Z. To one of the terminals we connect the generator subtransient reactance X' ' and to the other the locked rotor motor impedance Z , calculated according to Equation (6.8) (with s = l). The Thevenin d  LR  6.2. Rules for Motor Start-up  93  C ^ ^ ^  Start  ~^^>  Build equivalent motor model  No  Determine network matrix [Z]  Remedial action  Yes  Determine I ^ , Q  sl  Yes  Yes  Yes  EMTP Simulation  Yes  Next restoration step  Yes  End  Figure 6.10: Overview of algorithm  6.2. Rules for Motor Start-up  Generator  94  Transformer  Cable  Cable/ circuit breaker Induction motor  Figure 6.11: Sample system for motor start V.ref V,IMAX V,  1+sT  c  1+sT  n  1+sT,  V,IMIN  V  t RMIN" ^ ' f d V  K, 1+sT,  Figure 6.12: Excitation system of sample system  equivalent circuit for the sample system is displayed in Figure 6.13. A V  Generator  X'  Motor  d  [Z(f=60Hz)]  1 Z ,(s=l) mn  J  Figure 6.13: Thevenin equivalent circuit of sample system  6.2.2  Estimation of Voltage Drop  The start-up of large induction motors or a group of induction motors can lead to significant voltage drops. These can be determined using the voltage divider equation  AV[p.u.] = l - -  Z  l  LR  + Z (60 2  Uz)+j-X'J  (6.68) l  where Z represents the Thevenin equivalent impedance between terminals 1 and 2. It is important to notice that the initial voltage drop is always independent of the generator exci2  6.2. Rules for Motor Start-up  95  tation system since every exciter has a delayed response to disturbances due to its inherent time constants. For the validation of Equation (6.68) we use our sample system and assume that initially both frequency and voltage are equal to their rated values. This results in a voltage drop of 0.38 per unit. The voltage behavior obtained by an E M T P simulation of the system is shown in Figure 6.14. Its value of 0.4 per unit is very close to the estimated one.  Time t [s]  Figure 6.14: Voltage drop during motor start-up  6.2.3  Estimation of Inrush Current  The inrush current is calculated according to  I m o t  =  z  l  LR  + z (mriz) 2  +  J  -x»  ]  -  (6  69)  In case of our sample system it gives a value of 6147.2 A . It agrees well with 6200 A obtained from the E M T P time-domain simulation shown in Figure 6.15. As for the voltage drop calculation, the initial value of the current is independent of the action of the generator excitation system. In cases where the motor is started at frequencies different from the nominal frequency, the Thevenin equivalent impedance can be determined by an E M T P steady-state solution at that particular frequency.  6.2.4  Estimation of Start-up Time  The start-up time of induction motors can be predicted, using a quasi-steady-state calculation. For this, we use the equation of motion that governs the mechanical behavior of  96  6.2. Rules for Motor Start-up 1  a  1  1  1  i  5000  cu Si  o  v  1  2  3  4  i  .  1  1  1  5  6  7  Time t [s]  Figure 6.15: Inrush current during motor start-up  the induction motor, together with its electrical steady-state equations. Following the basic physical law that relates induction motor speed u and acceleration a we get a  ( ) U  =  ~TT at  =>• tstart = J/ a{UJ) —7-T " -  W  (6.70)  0  The equation of motion is J ~  = T {u)-T (u)  at  e  r a  T  m  {  u  )  =  (6.71)  m  «M  ~ W  (6.72)  represents the mechanical torque which can be generally formulated as [19] T  = a + (Xi • s + a • s  (6.73)  2  m  0  3  where a , o;i, a are constants that are either given by the manufacturer or, can be determined as outlined in Appendix A. T represents the electrical torque of the motor calculated with Equation (6.15). Combining Equation (6.70) and Equation (6.71), and considering the relationships 0  2  e  UJ = (1 - S) • U)  m  base  du = — UJ base " d"  s  m  gives us the integral  tstart  —  J • W base ' m  J • UJ. m base /  j  -  Tm(s)l _\ e  m l _\ T {s)  (6-74)  m  so j(s)ds  (6.75)  6.2. Rules for Motor Start-up  97  The operating slip So is reached when the condition T {s ) - T (s ) e  0  m  = 0  0  (6.76)  is fulfilled. It is determined numerically by the zero-finding routine F Z E R O [162]. Equation (6.74) can be solved by writing the sum "As  t start = J  'Urn base  ' As • ^  7  (6.77)  {  i=l where A s = ^  SO  e final  ;  H ^  - ^ ^ -  1  =  {(-final  =  0.002  * (1 -  (6:78)  S ) + S ) 0  0  (6.79) (6.80)  Since the actual final motor speed is an asymptote, the numerical calculation of the startup time may result in different values for different slip increments A s . Therefore, we define the operating slip as Equation (6.79) and the associated speed as w = (1 - s ) • io 0  0  m b a s e  (6.81)  When calculating the start-up time, the question arises whether a feasible operating condition exists at all. We define an infeasible operating condition when the current in the operating point is larger than the limit provided by the manufacturer. The torque behavior for such a condition is shown in Figure 6.16. It may occur e. g. during the soft start of motors. As shown in Figure 6.17, the motor draws very high currents continuously, leading to possible overheating and machine damage. The soft start of motors is discussed in more detail in Appendix B. For the validation of the above equations we again use our sample system. Due to the control action of the excitation system, the duration of the initial voltage drop is much smaller than the motor start-up time. Therefore, the generator terminal voltage can be approximated to be constant during start-up. By adding the Thevenin impedance of the system to Equation (6.13) we determine the current for the calculation of the electrical torque T , as e  Imot = | ~  i 7 tan TJ ^ I  (6.82)  ZLR + Z {60 Hz) 2  If a more conservative start-up time estimation is desired, Equation (6.69) instead of Equation (6.82) can be used for the electrical torque calculation. The rotor speed characteristic as calculated with the E M T P is given in Figure 6.18. Its start-up time of 3.7 s agrees well with the estimated time of 3.82 s.  6.2. Rules for Motor Start-up  98 !  1  \  1  t  o  I  \ ^  ; V  14000  \ : \ :  fc  \  : •  ^ x  \ \  cd  \  V:  1 \ J \  a  m  L  \  W  -a  l  e  \ 1 0  1  Operating Point E l . torque T Mech. torque T  o  : \  \  x  :  N  : \  '• \  :  V N.  CD  — a  _  v  & 0  1  1  1  0.1  0.2  0.3  v..  1i  0.4  • — — ' • — — — — — ^  1  1  0.5  0.6  0.7  0.8  0.9  1  Slip  Figure 6.16: Mechanical and electrical torque for / = 1.0 p.u. and V = 0.5 p.u.  1  1  i  i  i  <  4  o 1  0  0.1  0.2  0.3  0.4  o  Operating Point Current  i  i  0.5  0.6  i  0.7  i  i  0.8  0.9  1  Slip  Figure 6.17: Current for / = 1.0 p.u. and V = 0.5 p.u.  6.2.5  Estimation of Thermal Behavior  During emergency or black start, any unwanted trip of motors should be avoided. To examine the motor protection coordination, the motor current is integrated in the time range where its value is above the relay current pickup. This time range can be approximated by the motor start-up time calculated according to Equation (6.74). A conservative estimate of the average motor inrush current is given by Equation (6.69). As an example for this principle,  6.2. Rules for Motor Start-up  99  Figure 6.19 shows the comparison of the relay characteristic of the 600 H P induction motor of Table 6.1 to the averaged and estimated currents. The averaged current has been obtained by integration of the current given by an E M T P motor-start simulation.  |  1  -•• -  !  ,  Relay characteristic  Averaged starting current  Estimated starting current  0  200  i I 400  600  600  1000  ,  "  1200  Current [A]  Figure 6.19: Motor relay characteristic compared to averaged starting current  6.2. Rules for Motor Start-up  6.2.6  100  Estimation of Frequency Drops  In this section the behavior of prime movers under static load pick-up, discussed in Chapter 4, is extended to allow for the estimation of frequency drops that occur as a consequence of the pick-up of motor loads. For this, only a minimum amount of effort needs to be added to the method for static load pick-up, as will be shown subsequently.  Approximation of Motor Power For the determination of prime mover frequency deviations it is only necessary to consider the active power drawn by a load. As compared to static loads which can be approximated by single step functions, motor loads need to be represented by two successive steps: P (t) = - [Pel • 0{t) + (P e  - P ) • a(t - tstart)]  e2  el  (6.83)  where a(t) and a(t — t t) stand for unity step functions, starting at times 0 and t , respectively. The parameter P is the steady-state active power drawn by the motor (t—>oo), and P i the average inrush active power during the motor start (t-*0+). In this section, in order to avoid confusion with the Laplace operator "s", we use the variable "slip" when we refer to the motor slip. start  star  e2  e  The power P Equation (6.69)  el  is obtained by averaging the motor active power that is derived from  P (slip) = real (y/3 •  Y°  e  .  )  (6.84)  over the slip range from 1 to the steady-state slip : 0  slipo  P {slip) • dslip  (6.85)  e  /  When Equation (6.83) is transformed into the Laplace-domain we obtain: Pel  P (s) = -  ,  Pe2  — Pel  ,  V  e  s  .  exp(-t  \ atart  • s)  (6.86)  s  If this equation (as a replacement of Equation (4.9)) is used for the development of the transfer function, a complicated expression is obtained that cannot be easily transferred back into the time-domain. Therefore, we replace Equation (6.83) by an exponential function Pe{t) = (Pel ~  P e 2 )  • exp (-^—)  J- mot J  \  +  P2 e  (6.87)  When this function is transferred into the Laplace domain we obtain the rational function P ( ) e  s  Pel — Pe2 =  ,  I f fPe2  ;"  S + 7f^—  +  s  mot  7^7 •*• mot  •  Pe2  +  Pel ' S  (6.88)  6.2. Rules for Motor Start-up  101  that can be transformed back into the time-domain more easily. The parameter P simply equals P . However, the values of the variables P i-,Pe2,t , and T have still to be determined. This can be accomplished by an optimization routine that determines the parameters in a way to satisfy the condition e2  e2  e  x  mot  [' Jo  P (t)-dt e  = [' Jo  P (t)-dt  (6.89)  e  Integrating Equations (6.83) and (6.87) results in the objective function: F(P tx  | tstart (Pel ~ Pe2) +  ; Tmot)  eU  (6.90)  -Tmot • (Pel ~ Pel) ' ( l " exp (~^)  )  I= 0  Furthermore, the power P (t ) needs to be equal to P . Since we are dealing with an exponential function, P can only be an asymptote for P (i), and we therefore add the tolerance e to obtain an additional constraint for the optimization: e  x  e2  e2  e  G(P l,t ,Tmot) e  x  = Pe(t ) - £ ~ Pel = 0 x  (6.91)  where e = 0.1 • P . The optimization problem can be solved in a few iterations with the mathematical optimization routine S O L N P described in Section 6.1.3. e2  The exponential function, the step function calculated according to Equation (6.83), and the active power obtained by an E M T P simulation, are displayed in Figure 6.20. Since the calculation of the steady-state power P is accomplished by assuming rated voltage and frequency, it is higher then the actual power calculated with E M T P , which adds some conservatism to the frequency drop estimation. e2  Frequency Drop Estimation For the estimation of frequency drops during motor starts we follow the same steps outlined in Chapter 4 and obtain the transfer function  D{s)  6.2. Rules for Motor Start-up  102  i  1  1  1  1  E M T P simulation Step function El-function  1 1  ^  r l\  0.1  \ \  [ :  .;  I  0  5  10  15  •  20  25  :  :  i  i  i  30  35  40  45  50  Time t [s]  Figure 6.20: Active power during motor start-up compared to approximations  where K  D .) (  = -w = [  '  <6 93)  >  (  A  [  H • T • TQ  \H -T • Tyv  w  \^2H-T  D H-T  N  T  N  =  T:  T  T  N  p  Pe  =  t  W  W  )  ^ (  ,  2  N  ^  t  T -^ mot  ^  T ot) m  7—  R + R + 1/T T  w  A , 2 H TT)  w  +  H • To • T  w  ,  D  ' \\T,H  +*-vo-T-y< .,  D  +  (- ) 6  96  v D  (6.97)  The only difference to Equations (4.12) and (4.13) is the addition of one root to both the numerator and denominator. This root, however, is easily found. As in the case of static load pick-up, we only have to solve a fourth order polynomial numerically to obtain the roots, and obtain the transform back into the time-domain the same way as described in Section 4.2.2.  6.3. Summary  103  Example The E M T P result for the frequency obtained for our sample system as compared to the estimated result is displayed in Figure 6.21. Although the frequency obtained by the analytical equation gives only an approximation of the value obtained by the time-domain simulation, it is still accurate enough to allow for an assessment of the frequency drop.  o  1=1 QJ  3  cr  Time t [s]  Figure 6.21: Frequency deviation for motor load pick-up  6.2.7  Remedial Actions  In case the reactive power capability limits are exceeded due to the high reactive power of the motors during start-up, additional generating units can be brought on-line. Furthermore, the auxiliary transformer tap positions can be changed [102]. In this case and in the case the bus voltage needs to be increased as a consequence of excessive voltage drops, the voltages can be controlled following the method outlined in Section 5.2. In order to decrease the frequency drops during motor load pick-up, the number of generators online can be increased.  6.3 Summary This chapter deals with the analysis of auxiliary systems during restoration. A new induction motor parameter estimation technique is introduced that produces motor models by flexibly taking into account different types of data, such as nameplate data and motor performance characteristics. Rules for the feasibility of motor start-ups taking into account  6.3. Summary  104  inrush currents, voltage drops, thermal behavior, and frequency drops during the energization of motors are introduced. The induction motor parameter estimation as well as the rules are validated using actual system data.  Chapter 7 Emergency Start Case Study This chapter describes how the E M T P can be used for time-domain simulations of emergency or black start studies. The simulation of a hydro generator start and the subsequent energization of a power system, are described. The power system represents the emergency safeguard functions (ESF) of a nuclear power plant under off-nominal frequency and voltage conditions. The chapter describes the modeling of the generator, turbine, generator controls, and induction motors that drive the power plant's auxiliaries, and compares the results with measurements. Among the different types of power plants, nuclear stations require special treatment, since, following a system blackout, there always has to be sufficient energy available for their safe shutdown. Emergency supply sources for nuclear plants, in case of a loss of off-site power (LOOP), are black start combustion turbines, conventional or pumped hydro units, or diesel generators [100]. L O O P cases and the subsequent use of emergency power sources lead to an islanded system operation that can result in abnormal conditions with respect to frequency and voltage behavior. This situation requires careful analysis, using accurate system models and simulation tools. Studies similar to ours can be found in [74, 76, 102, 177, 232]. A l l the cases have in common that they investigate the energization of power plant auxiliary motors with generators running at nominal speed. In the following, we describe the individual models and give indications of how they were verified by measurements. We focus on modeling issues and refer to nuclear technical aspects only when necessary. A n overall model validation is performed, using a system test and comparing simulations to measurements.  7.1 7.1.1  Modeling of the System Electrical System  A single-line diagram of the emergency electrical system for the Oconee nuclear power plant that is simulated in our study is shown in Figure 7.1. Since the cables in the system are electrically short and traveling wave effects can be neglected, they are modeled using lumped  105  7.1. Modeling of the System  106  elements. Keowee Hydro Plant o  r\  Standby Bus  O O O O JL 0  RBCF  I  J  ~3r  "Jr ~cT~cT o  RBS  LPSW  o  o  HPI  LPI EFW  o  3F JL  LPI  d) d) HPI  EFW  C7  "cT  O RBCF  c5  c5  Figure 7.1: One-line diagram of the Oconee emergency electrical power system (ESF)  7.1.2  Synchronous Generator  The synchronous generator to be modeled is a hydro generator of 87.5 M V A and 13.8 kV. It is implemented using the E M T P synchronous machine model SM59 [63]. This model includes the effects of the amortisseur windings as well as saliency and saturation effects [54, 143]. The generator parameters are verified by an open circuit magnetization test for the validation of saturation parameters and D-axis and Q-axis parameter tests in order to identify sub-transient, transient and steady state parameters.  7.1.3  Excitation System Model  The excitation system, and the other controllers described subsequently, are implemented in the Transient Analysis of Control Systems ( T A C S ) module [63] of the E M T P . A control block diagram of the excitation system model used for our study is shown in Figure 7.2. It is a static exciter that is represented by the I E E E S T 1 exciter model, modified with the addition of a V / H z limiter, I E E E droop compensator, and the replacement of the field voltage by a field current feedback. In reality, the static exciter is a rectifier bridge, whose firing angle is controlled by a firing circuit logic. Since we are dealing with a generator start-up from zero initial voltage and speed, field flashing is applied until the generator terminal voltage is sufficiently high. This  7.1. Modeling of the System  107  is accomplished by modeling a battery in parallel to the rectifier bridge, which delivers the field current in the beginning.  Figure 7.2: Block diagram of the Keowee generator excitation system Modeling the whole rectifier bridge would be too detailed for the purpose of our study, since the accuracy gained with it is negligible as compared to the added complexity. However, modeling of the rectifier bridge might be justified if the objective of the study is to examine the behavior of the rectifier and its control circuits. For our study, we model the field flashing separately, using a simple equation in that gives the field voltage, as provided by the battery: Efd(FieldFlash)  —  VB — RB  TACS  ' Ifd  (7-1)  where Vg stands for the battery voltage, RB for the battery's internal resistance, and Ifd for the field current. After a certain V / H z level is reached, the field voltage output is switched from the battery voltage to the exciter voltage output, as indicated in Figure 7.2. This results in a small discontinuity in the field voltage when switching from battery to rectifier dc output. The excitation system model is verified using a reactive droop compensation test and step tests to identify the individual elements of the block diagram in Figure 7.2. The reactive droop compensation is implemented as an algebraic function. The output voltage of a droop compensator as a function of the terminal voltage V , the current I , and the compensator's resistance R and reactance X are given as T  C  c  \V \ = \V + (R C  t  T  C  + j-X )-I \ c  (7.2)  t  Writing the generator terminal voltage and current as complex equations, we get V  =  V + j-V  (7.3)  It  =  Id + j-Ia  (7-4)  t  D  Q  Substituting Equations (7.3) and (7.4) into (7.2) then leads to  I K I = \J (v  d  + Rjd  -  xj y + (V + RJ + q  Q  V  XJd)  2  (7.5)  7.1. Modeling of the System The quantities Vd, V , I , output! q  7.1.4  d  108  and I  q  are directly obtained from the E M T P generator model  Governor System M o d e l  The hydro governor system model used for this study is shown in Figure 7.3. It is a proportional control with transient droop and is described in more detail in [143, 178]. For the generator start-up, the model has to be modified since it is only valid for a certain speed range. Outside that range the upper and lower speed limits are used as integrator inputs in Figure 7.3, and in that case the feedback loop is inoperative. Furthermore, the gate closing speed is a function of the gate position, as indicated schematically in Figure 7.3.  o  if R P M <= R P M ,  cif R P M < R P M < R P M 2  1  Go  s  1 sT +  Q  GATE  3 G  min  if GATE >= G^ff then X<=V  C1  if GATE < G ^ then Xf=V  C2  Figure 7.3: Block diagram of the Keowee governor system  7.1.5  Turbine M o d e l  For the turbine a non-linear model including water column traveling wave effects [143, 178] is chosen. Its block diagram is depicted in Figure 7.4. A crucial factor in the modeling of hydro turbines for generator start-up studies and other abnormal system conditions was found to be the damping term  DAMP  = D-Aui-  GATE  (7.6)  Although Equation (7.6) led to a good agreement between simulation and measurement for the cases investigated in our study, it cannot be considered valid for all possible operating conditions.  7.1. Modeling of the System  109  -2T.S  t-  e  +  1  FLOW from other unit  HEAD  GATE  D-Aeo-  Figure 7.4: Block diagram of the Keowee hydro turbine  The turbine model time constants are calculated from manufacturer data. They are verified by applying step changes to the governor set point, and measuring gate position and active power output.  7.1.6  Initialization of Excitation System, Governor System, and Turbine Models  A very important aspect, discussed subsequently, is the initialization of control system models in T A C S . Although T A C S initializes some of the control blocks automatically, others have to be initialized manually by the user. We recommend to initialize all state variables manually in order to avoid faulty initializations and to allow for a better verification of the model. This section gives hints on how a proper control system initialization can be accomplished for practical case studies. One way is to run a time-domain simulation without applying a disturbance at the control's inputs and without prior initialization of the state variables. The simulation is continued until the state variables settle down to their steady-state values.  110  7.1. Modeling of the System  These can then be used to initialize the control and to perform another non-disturbance simulation. If the model implementation and initialization is done correctly, the control's state variables should remain constant at all times. One disadvantage of this method is that it doesn't give any indication of whether the control itself has been set up properly in TACS. Even when this is not the case, the state variables may settle down to steady-state values. Furthermore, state variables may not settle down to final values at all. Instead, the state variables' values may increase continuously to very high numbers, resulting in numerical problems and an unwanted early termination of the simulation. Also, control systems with large time constants might need a long time in order to settle down, resulting in overly long simulation times. Therefore, it is better to calculate the initial values prior to the simulation and perform a non-disturbance simulation only in order to ensure that the control is initialized correctly, and that all control parameters are entered properly. In that case, all state variables remain constant. As an example for such a procedure, we focus on the initialization of the synchronous machine and the most important state variables of the turbine model as shown in Figure 7.4. Since a start from exactly zero frequency and voltage cannot be simulated with the E M T P , due to the implementation of its generator model, we approximate such a zero condition by choosing a small generator base voltage (V = 10~ • V i) and frequency (f = 10~ • fnominal)The E M T P generator parameters must then be modified as well since they have to be entered in per unit. As a consequence of the change of the base voltage, the base impedance decreases by a factor of 10~ . Therefore, the resistances entered into E M T P are 10 times higher. The reactances increase by a factor 10 /10 = 100, since they have to be corrected as well with respect to the modified base frequency. 2  base  nomina  base  2  4  4  4  2  Furthermore, E M T P requires to load the generator with a small initial load, which is accomplished by a high resistance Ro at the generator terminals. This is removed immediately after the simulation is started, since with growing voltage, it would represent a significant load, giving erroneous results. Since in steady state, the mechanical power PM equals the electrical power PEL, calculate the initial value for PM as:  W  E C  A  N  (7.7)  PM — PEL = VQ/RQ  We can now go "backwards" in the block diagram in Figure 7.4. The initial value for the head can be calculated as  HEAD  = H, ref,  (7.8)  The other inputs to the respective summer are zero since the factors f and f are negligible, and the initial output of the summers, representing the traveling wave effects, are zero. This results in the following values for flow and gate: t  FLOW  PM  A -H, t  GATE  ref  FLOW VHEAD  + QNL  p  (7.9)  (7.10)  7.2. Simulation Results  111  Starting from these equations, the initialization of the other variables is then straightforward since the values only have to be multiplied by constant factors, e. g. the output of the block with proportionality factor Z is FLOW • Z , etc. p  p  The exciter model is initialized as well, even though initially, its output is not connected to the generator, since the field voltage is given by the field flashing battery. A n initialization as if the controls were already in full operation is performed since initial transients in the control systems, caused by non- or insufficient initialization, may not settle down before the field voltage is switched from field flashing mode to exciter control, and may therefore lead to erroneous results.  7.1.7  Induction Motor Model  For our study we modeled six different motor loads: • Reactor building cooling fans (RBCF) • Emergency feedwater pumps (EFW) • High pressure injection pumps (HPI) • Low pressure service water pumps (LPS) • Reactor building spray pumps (RBS) The induction motor model selected for this study is the universal machine model UM3 of the E M T P [63]. Although the UM3 model allows for an arbitrary number of rotor circuits, experience has shown that a model with two rotor circuits is sufficient over the whole range of operation for motor start-up studies (see e. g. [105, 191]). The model parameters are derived from manufacturer nameplate data and current-speed, torque-speed, and powerfactor-speed performance characteristics of the motor, following the principles outlined in Section 6.1. The load characteristics are modeled by quadratic functions [19] according to Appendix A , based on measurements. Representatively, Figures 7.5 and 7.6 show the E M T P simulation results for a motor start-up (signature test) of a high pressure injection pump (HPI) motor as compared to measurements. Measurement and simulation results agree well.  7.2 Simulation Results The test carried out for validating the E M T P model of the Oconee emergency electrical system is an engineered safeguards functions (ESF) test during a simulated loss of coolant accident (LOCA) concurrent with a loss of off-site power (LOOP). The Keowee hydro generator is used for this to provide emergency power. The nuclear plant's auxiliaries are energized by closing the switch shown in the single line diagram in Figure 7.1, when the voltage at the main feeder reaches 0.43 per unit, provided  7.2. Simulation Results  112 f  1  1  ——-  1  Simulation Measurements  ft 1 1 \  H  l\  *A  0.5  0  1.5  1  2.5  2  3  Time [s]  Figure 7.5: Transient current characteristic, HPI motor  /  \  — 0  t  i  i  0.5  1  1.5  Simulation Measurements 2  2.5  3  Time [s]  Figure 7.6: Transient voltage characteristic, HPI motor  11 seconds have elapsed since the loss of off-site power to the standby bus. This delay gives enough time for the generator to build voltage and frequency up to around 0.6 per unit This condition represents a motor soft-start which is analyzed in more detail in Appendix B. The most relevant simulation results are shown in Figures 7.7 to 7.12. The shape of the generator current suggests that some of the motors (with lower inertia) have locked to the system frequency (drop-off of the start-up current) sooner than others. The zero field current  7.2. Simulation Results  113  Time [s]  Figure 7.7: E S F test: Comparison between measured and simulated frequency  •  i  i  ft  h it {f t  h It (vI  1  It Ii i  >  I  Measurement Simulation h  If ft  /t  ft  Ii  // r  •  0  5  10  15  20  25  30  35  40  Time [s]  Figure 7.8: E S F test: Comparison between measured and simulated voltage  of the measurement result as compared to a continuously rising current in the simulation is a result of the measurement not taking into account the initial current delivered by the field flashing circuit. Overall, the simulation results agree well with the measurements.  7.3. Summary  114 1  1  1  1  -.-  1  1  Measurement Simulation  -  \  1 1 1 1 1 1 1 1  _  0  5  _-~  -•  10  15  20  25  30  35  40  Time [s]  Figure 7.9: E S F test: Comparison between measured and simulated current  Measurement Simulation  o< 0  1  1  I  l  l  I  5  10  15  Time [s]  20  25  30  i _  35  I  40  Figure 7.10: E S F test: Comparison between measured and simulated field voltage  7.3 Summary In this chapter we describe the modeling of a hydro generator and its controls in E M T P and T A C S . The successive development and test of each of the models is outlined. As an important aspect of building control system models for system studies in E M T P , we identified the need for proper initialization, and the correct modeling of induction motors and of the  115  7.3. Summary  1  Measurement Simulation  .. \\.  u  •.  —  •  Time [s]  Figure 7.12: E S F test: Comparison between measured and simulated power  generator damping. Field tests and simulations of an engineered safeguards test show good correlation. Our case study has demonstrated the potential of E M T P to be utilized as a system stability tool for the simulation of abnormal conditions, such as islanded operation during black start and power system restoration procedures.  Chapter 8 Conclusions and Recommendations for Future Work This dissertation presents a new framework for power system restoration. The results of an extensive bibliographieal research showed that there is a need for a methodology that allows one to assess restoration steps quickly and efficiently, and that supports operators during on-line restoration, and system planners during off-line restoration planning. Moreover, it revealed that models and modeling techniques are needed that help to take into account the abnormal voltage and frequency conditions during restoration. The new method is based on a subdivision of the aggregation of complex phenomena encountered during restoration into simpler problems that can be assessed using simple and efficient rules formulated in the frequency and Laplace domain. The rules allow for fast screening of the large number of possible combinations of restoration steps, and support the selection of the most promising restoration paths. The time-domain modeling techniques that have been developed for this thesis allow E M T P simulations of the large deviations in frequency and voltage that are of particular concern during the black or emergency start phase of a system restoration procedure. Special emphasis is thereby given to the modeling of large induction motor loads. The major conclusions and contributions of this work are: • A new method for the assessment of the frequency response behavior of hydro power plants has been developed. By using calculations in the Laplace domain and simplifying governor control circuits, the initial rate of frequency decline, the minimum frequency, and the time when this minimum occurs can be calculated with a minimum number of iterations. • Matrix manipulation techniques have been developed for determining the matrix changes of the network impedance matrix directly from the network changes of single network elements, without the need for additional frequency scans or matrix inversions. In the case of overvoltages, it permits a quick assessment of different network conditions with respect to overvoltages. A sensitivity analysis method for sustained and harmonic overvoltages helps to find the most efficient network changes. When time-domain sim-  116  117  ulations need to be carried out, an algorithm based on Prony analysis in combination with fuzzy logic shortens the overall simulation time. • A new induction motor parameter estimation algorithm helps to build accurate induction motor models for restoration studies. It flexibly takes into account the different data sets. Frequency-domain based rules help to assess the feasibility of motor starts with respect to overcurrents, voltage drops, and thermal behavior. • Using a practical emergency-start case study, it is shown how the E M T P can be utilized as a stability tool. It is demonstrated how generator controls, turbines, etc. can be modeled. A n overall system test is carried out and compared to measurements that confirm the validity of the simulation results. In summary, the new methodology gives an approximate assessment of the feasibility of restoration steps. It can be used in conjunction with commercially available analytical tools, such as electromagnetic transient programs, stability programs, harmonic analysis tools, power flows, and operator training simulators, or combination tools. By eliminating infeasible restoration sequences, the overall restoration time can be reduced significantly. The following studies are suggested for future research: • The frequency-response analysis procedure could be extended to other types of power plants, such as gas turbines, and the feasibility of approximations taking into account constrained generator operation could be investigated. • For cases where more than one generator is in operation, a simplified method for the aggregation of generators should be developed, to obtain a single-generator model suitable for frequency-response analysis. • Laplace-domain based frequency-response methods could be investigated for the design of load shedding schemes that are in use, e. g. when the system disintegrates into several islands with the system frequencies below the allowable limits. • Rules that help to assess transient switching overvoltages could be investigated. • The possibility of replacing Prony analysis by a method such as wavelets that is more suited for time-frequency analysis should be explored. • A simplified motor aggregation method could be developed, and the interaction of individual motors when several large motors are started in sequence should be investigated. • Rules that cover other parts of the restoration process, such as the integration of subsystems or the protection system, could be developed. • A more detailed representation for the generator damping for E M T P time-domain simulations should be developed.  Bibliography M . 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Typical factors k and load types are: m  m  m  k  m  = 1  for fan-type loads  (A.2)  k  m  — 2  for centrifugal pumps  (A.3)  Often, during motor start-ups, the rotor speed is not recorded due to higher technical requirements and costs. Therefore we determine C using the measurements of motor current and voltage. The network that drives the motor can be approximated by a simple circuit, as outlined in the following. m  Starting from the measurements, we determine the initial voltage drop V (t = 0+) and the steady-state voltage V (t -> oo). From Equation (6.8) we obtain the locked-rotor impedance Z ( s = l) and the motor impedance at full load Z (s = s ). Applying voltage divider equations to the circuits shown in Figure A . l and A.2, for locked-rotor and full-load operation, we then obtain: m  m  m o (  0  mot  vg +^  ^ ( ^ l f  _ [v (t=o+)¥ m  +U ^ l ) )  2  K  Rmot{s = S )  2  0  + (X  th  ~  \Z t(s = l)\  =  [v (t^™)}  2  m  + X (s mot  = S ))  2  0  {  m0  \Z {s mot  ' '  2 ( A 5 )  = S )\  2  0  where Z — Rm t + j • X and where the system driving the motor is assumed to be purely inductive. Using Equations (A.4) and (A.5) we get the following parameters for the mot  0  moU  137  138  V(t->0+)  Z t(s=1) mo  Zmot(S=S )  V(tH»oo)  0  Figure A . l : Thevenin equivalent for lockedrotor operation (t = 0+)  Figure A.2: Thevenin equivalent for full-load operation ( £ - » o o )  Thevenin equivalent circuit:  =  X  th  V (t = 0+) m  V  * =  7  /  g  _ i  - - + \J~-c b  (A.6)  I V l ^ m ( s = l)l + X 2  ^ + 2X ,  2  0 t  T  e v  X  m o t  ( = l) S  (A.7)  where  y ( t = 0+)  V {t->oo)  m  m  .-2mot(s=l)  2 • (a • X (s  Z  m o f  (A.8)  (s = S )J 0  = l) - X (s  mot  12  = s ))  mot  Q  a-l  (A.9)  Thus, the motor current at full load follows as: Imot(s = s ) = —  —  0  T  (A.10)  The operating slip so is not known a priori and can be determined by a simple iteration using the zero-finding routine FZERO [162] to solve: Imot{s = s )-I (t^oo)  = 0  m  Q  (A.ll)  This permits to determine the associated torque T (s ) using Equation (6.15). In steady state, we have e  2H • ^  Q  = T (s ) - T (so) = 0  and therefore we eventually obtain C  e  m  as  0  m  (A.12)  Appendix B Soft Start of Motors When auxiliaries are started on reduced voltage, this is referred to as "soft start" condition. In that case, one expects to observe lower starting currents and longer start-up times. However, during emergency starts of power plants (see Chapter 7), the motors are picked up under lower frequency as well. It is expected that the motors will lock to the system frequency sooner, and hence draw start-up currents for a shorter time. Since in this case we deal with both an under-voltage and under-frequency condition, the analysis is more complex and less intuitive. In order to analyze the starting current and time as a function of frequency and voltage, both are calculated using the equations developed in Section 6.2. Values of current and startup time are normalized to their values at / = V = 1.0 per unit. The start-up time at rated voltage as a function of frequency calculated according to Equation (6.74) and compared to the start-up time obtained with E M T P are shown in Figure B . l . The start-up time as a function of voltage at rated frequency is displayed in Figure B.2. The currents as function of voltage and frequency as compared to the E M T P results are shown in Figures B.3 and B.4. Only the range where a feasible operating condition can be reached is displayed. The estimated start-up times and currents agree well with the E M T P results. The results show that the motor start-up time increases exponentially with increasing frequency, and decreasing voltage. The relationship between motor start-up current and voltage is linear, and the relationship between start-up current and frequency is approximately linear.  139  2.5  Estimated time E M T P time  0' 0.5  1  1  I  I  I  0.6  0.7  0.8  0.9  1  I 1.1  Frequency / [p.u.]  Figure B . l : Start-up time at voltage V = 1.0 p.u-.  Estimated time E M T P time  Voltage V [p.u.]  Figure B.2: Start-up time at frequency / = 1.0 p.u.  141  1  >  °  •  °  "I  T  Estimated current EMTP current  <  0.5  0.6  0.7  0.8  0.9  1  1.1  Frequency / [p.u.] Figure B.3: Start-up current at voltage V = 1.0 p.u.  

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