ELECTRICITY: Power System Analysis

Contents Preface .................................................................................................................. (vii) Acknowledgment ................................................................................................... (ix) 1 Introduction I. I The Electrical Power System .................................................................. I 1.2 Network Models ...................................................................................... 3 1.3 Faults and Analysis .................................................................................. 3 1.4 The Primitive Network ............................................................................ 4 1.5 Power System Stability ........................................................................... 5 2 Graph Theory 2.1 Introduction ............................................................................................. 6 2.2 Definitions ................................................................................................ 6 2.3 Tree and Co-Tree .................................................................................... 8 2.5 Cut-Set .................................................................................................... 9 2.4 Basic Loops ............................................................................................. 9 2.6 Basic Cut-Sets ....................................................................................... 10 vVorked Examples .. ' .... .. II Prohlems ....... IS Questions 16 (xii) Content~ 3 Incidence Matrices 3.1 Element Node Incidence Matrix ............................................................ 18 3.2 Bus Incidence Matrix ............................................................................ 18 3.3 Branch - Path Incidence Matrix K ....................................................... 19 3.4 Basic Cut-Set Incidence Matrix ............................................................ 20 3.5 Basic Loop Incidence Matrix ................................................................. 21 3.6 Network Performance Equations .......................................................... 22 3.7 Network Matrices .................................................................................. 24 3.8 Bus Admittance Matrix and Bus Impedance Matrix ............................. 25 3.9 Branch Admittance and Branch Impedance Matrices ........................... 26 3.10 Loop Impedance and Loop Admittance Matrices ................................. 28 3.11 Bus Admittance Matrix by Direct Inspection ........................................ 29 Worked Examples. ...... ... ..... ........ ....... ............ . ................................ 33 Problems ........................... : ..................................................... 51 Questions .......... ........................................... '" .......................... 52 4 Building of Network Matrices 4.1 Partial Network ...................................................................................... 53 4.2 Addition ofa Branch .............................................................................. 55 4.3 Addition ofa Link .................................................................................. 60 4.4 Removal or Change in Impedance of Elements with Mutual Impedance ................................................................................. 66 Worked Examples. ........ " .. . 70 Problems .. . .............. : ............ . .... 96 Questions. . ............. . . ............. 97 5 Power Flow Studies 5.1 Necessity for Power Flow Studies ....................................................... 98 5.2 Conditions for Successful Operation of a Power System .................... 99 5.3 The Power Flow Equations ................................................................... 99 5.4 Classification of Buses ........................................................................ 101 5.5 Bus Admittance Formation .................................................................. 102 5.6 System Model for Load Flow Studies ................................................ 104 5.7 Gauss-Seidel Iterative Method ............................................................ 105 5.8 Gauss - Seidel Iterative Method of Load Flow Solution .................... 106 5.9 Newton-Raphson Method ................................................................... 109 5.9.1 The Rectangular Coordinates Method .................................. 110 5.9.2 The Polar Coordinates Method ............................................ 112 Contents (xiii) 5.10 Sparsity of Network Admittance Matrices .......................................... 115 5.11 Triangular Decompostion .................................................................... 116 5.12 Optimal Ordering ................................................................................. 118 5.13 Decoupled Methods ............................................................................. 119 5.14 Fast Decoupled Methods ..................................................................... 120 5.15 Load Flow Solution Using Z Bus ..................................................... 121 5.15.1 Bus Impedance Formation ................................................... 121 5.15.2 Addition of a Line to the Reference Bus .............................. 122 5.15.3 Addition of a Radial Line and New Bus ............................... 122 5.15.4 Addition of a Loop Closing Two Existing Buses in the System ............................................................. 123 5.15.5 Gauss - Seidel Method Using Z-bus for Load Flow Solution .............................................................. 124 5.16 Convergence Characteristics ............................................................... 124 Worked Examples ..... .... . . 126 Problems .............. 161 Questions ................................. . . ... . ........................ 175 6 Short Circuit Analysis 6.1 Per Unit Quantities ............................................................................... 176 6.2 Advantages of Per Unit System .......................................................... 178 6.3 Three Phase Short Circuits ................................................................. 178 6.4 Reactance Diagrams ............................................................................ 181 6.5 Percentage Values ................................................................................ 181 6.6 Short Circuit KVA ................................................................................ 182 6.7 Importance of Short Circuit Currents ................................................. 183 6.8 Analysis ofR-L Circuit ........................................................................ 184 6.9 Three Phase Short Circuit on Unloaded Synchronous Generator ...... 185 6.10 Effect of Load Current or Prefault Current ........................................ 185 6.11 Reactors ............................................................................................... 186 6.12 Construction of Reactors .................................................................... 186 6.13 Classification of Reactors .................................................................... 187 Worked Examples. ..... ....... ....... ..... . .. . . ........... 189 Problems . ................................... . . ............ 216 Questions........ ... ... ....... ......... .. .......... .. .............. ................ .. 2 I 6 (xiv) Contents 7 Unbalanced Fault Analysis 7. I The Operator "'a'" ................................................................................. 218 7.2 Symmetrical Components of Unsymmetrical Phases ......................... 219 7.3 Power in Sequence Components ........................................................ 221 7.4 Unitary Transformation for Power Invariance ................................... 222 7.5 Sequence Impedances ......................................................................... 224 7.6 Balanced Star Connected Load ........................................................... 224 7.7 Transmission Lines .............................................................................. 226 7.8 Sequence Impedances of Transformer ............................................... 227 7.9 Sequence Reactances of Synchronous Machine ................................ 228 7.10 Sequence Networks of Synchronous Machines ................................. 228 7.10.1 Positive Sequence Network .................................................. 228 7.10.2 Negative Sequence Network ................................................ 229 7.10.3 Zero Sequence Network ....................................................... 230 7.11 Unsymmetrical Faults .......................................................................... 231 7.12 Assumptions for System Representation ............................................ 232 7.13 Unsymmetrical Faults on an Unloaded Generator ............................... 232 7.14 Llne-to-Line Fault ................................................................................ 235 7.15 Double Line to Ground Fault ............................................................... 238 7.16 Single-Line to Ground Fault with Fault Impedance ............................ 241 7.17 Line-to-Line Fault with Fault Impedence ............................................ 242 7.18 Double Line-to-Ground Fault with Fault Impedence .......................... 243 Worked Examples............................. . ................ ............ .. .............. 245 Problems ............... .. .............. ............ ............ ...... ........ .. ........ 257 Questions . ...................................................................................... 257 8 Power System Stability 8.1 Elementary Concepts ........................................................................... 259 8.2 Illustration of Steady State Stability Concept ..................................... 260 8.3 Methods for Improcessing Steady State Stability Limit ..................... 261 8.4 Synchronizing Power Coefficient ....................................................... 262 8.5 Transient Stability ................................................................................ 262 8.6 Stability ofa Single Machine Connected to lnfinite Bus .................... 262 8.7 The Swing Equation ............................................................................ 263 8.8 Equal Area Criterion and Swing Equation ........................................... 267 Contents (xv) 8.9 Transient Stability Limit ...................................................................... 269 8.10 Frequency of Oscillations .................................................................... 270 8.11 Critical Clearing Time and Critical Clearing Angle .............................. 272 8.12 Fault on a Double-Circuit Line ............................................................ 274 8.13 Transient Stability When Power is Transmitted During the Fault ...... 275 8.14 Fault Clearance and Reclosure in Double-Circuit System .................. 277 8.15 Solution to Swing Equation Step-by-Step Method ............................. 277 8.16 Factors Affecting Transient Stability .................................................. 279 8.17 Dynamic Stability ................................................................................ 280 8.18 Node Elimination Methods ................................................................... 282 Worked Examples ............. . , ........ ....... ...... ... . ........................ 285 Problems ..... , ........................ ............................ ................ . ... 303 Questions. ........................ .......... ...... . ...... .................. ....... . ...... 304 Objective Questions ................................................................................. 305 Answers to Objective Questions ............................................................ 317 Index ........................................................................................................... 319