Mathematical Models and Algorithms for Power System Optimization

Description

A number of mathematical models and algorithms are presented in this book for solving the practical problems in planning, operation, control, and marketing decisions for power systems. It focuses on economic dispatching, generator maintenance scheduling, load flow, optimal load flow, load optimization, reactive optimization, load frequency control, transient stability, and electricity marketing where mathematical models are transformed into relatively standard optimization models to make optimization applications possible. The optimization models discussed include linear (0–1, integer, mixed-integer), nonlinear, mixed integer, and nonlinear mixed integer models. Both numerical and non-numerical optimization algorithms are used in this book, the former (mathematical programming approachs) includes linear programming, nonlinear programming, mixed integer programming and dynamic programming, the latter (rules based approaches) includes Genetic Algorithm (GA), Simulated Annealing (SA), and Expert System (ES). Based on the authors’ extensive research experience in developing models and algorithms for power system optimization, this book also provides an in-depth analysis of some practical modeling techniques which are seldom explained comprehensively in the existing textbooks, both from theoretical and practical standpoints, for example, validity testing of data, type setting of variables, special setting of limit values of variables, special setting of constraints, and preprocessing of parameter and data. These techniques can be effectively applied to the modeling of power system optimization problems. Therefore, the readers of Mathematical Models and Algorithms for Power System Optimization will gain important insights into: how to transform the practical problems into mathematical models, how to develop the standard optimal mathematical models and utilize commercially available and reliable programming software, how to deal with various issues that affect the performance of a model, and how to evaluate the effectiveness of the models. Cover......Page 1 Mathematical Models and Algorithms for Power System Optimization: Modeling Technology for Practical Engineering Problems......Page 3 Copyright......Page 4 Foreword......Page 5 Preface......Page 6 General Ideas about Modeling......Page 11 Ideas about the Setting of the Variable and Function......Page 13 Ideas about the Selection of the Model Type......Page 14 Ideas about the Selection of the Algorithm......Page 15 Ideas about the Applications of Artificial Intelligence Technology......Page 17 2 Daily Economic Dispatch Optimization With Pumped Storage Plant for a Multiarea System......Page 19 Outline of the Problem......Page 20 Basic Requirements of Pumped Storage Plant Operation......Page 22 Overview of This Chapter......Page 24 Basic Ideas of Developing an Optimization Model......Page 25 Way of Processing Objective Function......Page 26 Way of Processing Variables and Constraints......Page 27 Formulation of the Problem......Page 29 Basic Expression of the Optimization Model......Page 30 Preprocessing of the Optimization Calculation......Page 33 Validity for the Rationality of the Constraints......Page 35 Forming the Virtual Cost Function for a Pumped Storage Plant......Page 38 Computation Procedure for Optimization......Page 39 Description of the Input Data......Page 40 Special Settings to Meet the Calculation Requirements......Page 42 Concrete Expression of Objective and Constraint Functions for a Small Scale System......Page 43 Scale of the Practical System......Page 45 Analysis of Peak-Valley Difference in the Load Curve......Page 46 Constraints of Pumped Storage Plant and Related Conversion Calculation......Page 48 Optimization Calculation Results......Page 49 Conclusion......Page 57 3 Optimization of Annual Generator Maintenance Scheduling......Page 59 Description of the Problem......Page 60 Basic Ideas of Developing an GMS Model......Page 62 Way of Handling Unit Maintenance Intervals......Page 63 Way of Processing the Objective Function......Page 64 Way of Processing the Variable Settings and Constraints......Page 65 Notations......Page 66 Constraints......Page 67 Selection of Fuzzy Membership Function......Page 69 Formation of Fuzzy Objective Index of GMS......Page 70 Formation of Fuzzy Constraints for GMS Problem......Page 71 Selecting of Time Units......Page 74 Introducing of Operation Index......Page 75 Rules of verifying GMS capability......Page 77 Search Paths and Recursive Formulas of Fuzzy Dynamic Programming......Page 78 Main Calculation Procedure......Page 79 Description of the Input Data......Page 81 Implementation......Page 85 Input Data of a Real Scale System......Page 86 Part of Output Results......Page 87 Conclusion......Page 89 4 New Algorithms Related to Power Flow......Page 91 Way of Processing Variables in Traditional Power Flow Equation......Page 92 Overview of Unconstrained Power Flow with Objective Function (based on SA Method)......Page 93 Overview of Constrained Power Flow with Objective Function (based on OPF Method)......Page 94 Description of Ill-Conditioned Power Flow Problem......Page 95 Outline of Simulated Annealing Method......Page 97 Notation......Page 98 Calculation Procedure of SA based N-R Method......Page 99 Initial Conditions of 197-Bus System......Page 102 Conditions and Results of Four Cases......Page 103 Similarities and Differences Between LF and OPF......Page 109 Description of the Problem......Page 110 Features of the Problem......Page 111 Mathematical Model......Page 113 Main Solution Procedure of Discrete OPF......Page 115 SLP-based Solution Procedure for Linear MIP Problem......Page 117 Verification by the Concrete Formulation of 5-Bus System......Page 120 Conditions and Results of Four Cases for 135-Bus Large-scale System......Page 124 Conclusion......Page 128 5 Load Optimization for Power Network......Page 130 Description of Maximizing Load Supply Capability......Page 131 Overview of This Chapter......Page 132 Way of Processing the Variable Settings and Constraints......Page 133 Notations......Page 134 Model of Load Supply Capability (LSC)......Page 135 The Derivation Process of LP Model for LCO......Page 136 The Derivation Process of LP Model for LSC......Page 137 Calculation Procedure of Minimizing LCO......Page 138 Step One: Input Data......Page 139 Step Four: Result Output......Page 140 Example of 5-bus system......Page 141 Concrete expression of the LP model......Page 142 Result of 5-bus system for state 1......Page 146 Results of 5-bus system for three states......Page 147 Results of the Real-Scale System......Page 149 Description of the Test System......Page 153 Results Analysis......Page 154 Conclusion......Page 157 6 Discrete Optimization for Reactive Power Planning......Page 158 Introduction......Page 159 Practical Method for Discrete VAR Optimization......Page 160 Overview of This Chapter......Page 161 Way of Processing Discreteness......Page 162 Way of Processing Multiple States......Page 163 Consideration to Obtain Global Optimization......Page 164 Verification of the Correctness of Discrete Solutions......Page 165 Special Treatments for Practical Problems......Page 166 Outline......Page 167 Description of the problem......Page 168 Mathematical model......Page 169 Characteristics of the problem......Page 173 The main computational procedure......Page 174 Improved algorithm for integer solutions......Page 175 Numerical results of five cases......Page 178 Overview......Page 183 Description of the problem......Page 186 Formulation of multistate problem......Page 187 Characteristics of multistate problem......Page 189 Main calculation procedure......Page 190 Decomposition and coordination within main procedure......Page 192 Implementation......Page 196 Overview......Page 201 Necessity of Introducing Expert Rules......Page 203 Basic procedure of the algorithm......Page 204 Way of processing transformer tap ratio......Page 205 Rules of making LP solution feasible by relaxing integers......Page 206 Rules to determine integer variables for tap ratio......Page 208 Verification by 5-Node test system......Page 209 230-Node practical system......Page 210 Summary......Page 214 Overview......Page 216 Necessity of Applying Artificial Intelligence Algorithms......Page 217 GA-based Model for Discrete VAR Optimization......Page 218 String performance of integer variables......Page 219 Calculation procedure......Page 220 Numerical Results......Page 223 Summary......Page 225 Conclusion......Page 227 7 Optimization Method for Load Frequency Feed Forward Control......Page 229 Descriptions of the Problem......Page 230 Overview of this Chapter......Page 232 Basic Ideas of Modeling......Page 233 Way of Constructing Estimator at All Levels......Page 234 Way of Setting Up the Load Frequency Controller based on the Invariance Principle......Page 235 Identification of Load Disturbance Model DeltaPL......Page 236 Time series analysis......Page 237 Stationary time series analysis......Page 238 AR, MA, and ARMA models......Page 239 Autocorrelation functions and partial autocorrelation functions of AR, MA, ARMA process......Page 240 Identification of the Model DeltaPL......Page 243 Parameter Estimation of the Model DeltaPL......Page 244 Generator Model......Page 245 Dynamic characteristics of a steam turbine governor......Page 246 Dynamic characteristics of thermal generating unit......Page 247 Dynamic characteristics of hydroturbine speed governing system......Page 248 Dynamic characteristics of hydrogenerating unit......Page 250 Equivalent Generator Model of the Power System......Page 252 Hierarchical Estimation for the Power System......Page 253 Local Estimator......Page 255 Estimation and Forecasting of Load Disturbance DeltaPL......Page 256 Invariance Principle......Page 257 Load Frequency Control based on Invariance Principle......Page 258 Simulation Procedure of Tracking Control......Page 260 Transformation Methods of Linear Models......Page 261 Eigenvalue method......Page 262 Logarithmic matrix expansion method......Page 264 Mutual Transformation Method Between Difference Equations (Successive Approximation Method)......Page 266 Method of direct Z transform......Page 268 Method of state equation solution......Page 270 Parameters From Figs. 7.4 and 7.7......Page 273 Simulation Results of Identification for Load Disturbance Model DeltaPL......Page 274 Simulation Results of Local Estimator and Central Estimator......Page 275 Difference T=1s......Page 279 Simulation Results of Tracking Control for Five-Unit Test System......Page 282 Results of Example 1......Page 284 Results of Example 2......Page 285 Results of Example 3......Page 287 Conclusion......Page 290 8 Local Decoupling Control Method for Transient Stability of a Power System......Page 291 Description of the Problem......Page 292 Overview of This Chapter......Page 294 Analysis of Two Scenarios in Power System Instability......Page 295 Purposes of Introducing an Observation Decoupled State Space......Page 296 Two Stage Countermeasures in Local Stability Controls......Page 297 Simplified Model and Typical Network of the Power System......Page 298 Basic Concept of the First Stage Control Criterion (Energy Equilibrium)......Page 300 Basic Concept of the Second Stage Control Criterion (Norm Reduction)......Page 302 The concept of observation decoupled reference state vector......Page 303 Composition and characteristics of observation decoupled state space......Page 306 Formulation of observation decoupled state space in the power system......Page 307 The dynamic relationship in observation decoupled state space......Page 309 Formulation of norm reduction control criterion......Page 310 Formulation of norm reduction control criterion in other observation decoupled state space......Page 311 Formulation and Proof of the First Stage Control Criterion (Energy Equilibrium)......Page 313 Mathematical Model of the Observation Decoupled State Space......Page 315 Simplification of the network......Page 318 The unique existence of the decoupled criterion......Page 320 The topological property of the observation decoupled state space......Page 323 Several Necessary Lemmas and Definitions......Page 325 Mapping is a homeomorphism within the specified region......Page 332 Proof of Topology Equivalence Between Different Forms of Observation Decoupled State Space and Original System Sta .........Page 335 Origin of Observation Decoupled State Space......Page 337 General Simulation Calculation Procedure in Two-Stage Control......Page 340 Simplified Assumptions and Network Diagram......Page 341 Variables of Measurement and Calculation in Online Control......Page 342 Preprocessing of Calculations......Page 343 Way of processing generation bus (to provide internal potential and power angle)......Page 344 Main Steps of Simulation Calculation Procedure......Page 345 Generator rotor motion equation......Page 346 Bus voltage equation of power network......Page 347 Calculation of Fault Equivalent Impedance......Page 349 Calculation of excess kinetic energy DeltaWi......Page 351 Calculation of the first stage control criterion and optimal clearing time......Page 352 Calculation of decoupling reference δei......Page 353 Calculation of critical control power for norm reduction control criterion......Page 360 Network Structure and Parameters......Page 361 Prefault power flow......Page 362 Results under different control cases......Page 363 Fault type......Page 365 Results under different control cases......Page 366 Conclusion......Page 372 Problem Description......Page 375 Competitive equilibrium......Page 376 The definition of a power market......Page 377 Power market transaction mode......Page 378 Cournot simulation......Page 379 Overview of This Chapter......Page 380 Constraint Conditions......Page 381 Model of Active Power Transaction......Page 382 Network loss allocation......Page 386 Congestion dispatch......Page 390 Power market local equilibrium......Page 393 Model Considering Both Active and Reactive Transactions......Page 395 A modified equal bidding method considering active power transaction......Page 399 A modified equal quotation method considering active and reactive power transaction......Page 400 Example Analysis Considering Active Power Transaction......Page 402 Example Analysis Considering Active and Reactive Power Transaction......Page 404 Conclusion......Page 405 Basic Algorithm......Page 406 Phase 2......Page 407 Comparisons Between the Algorithm and Branch-and-Bound Algorithm......Page 409 Appendix B: The Differential Expressions for Transformer Tap and Shunt Capacitor Unit......Page 411 Appendix C: A DC Load Flow Method for Calculating Generation Angle......Page 416 References......Page 420 B......Page 425 D......Page 426 G......Page 427 L......Page 428 N......Page 430 P......Page 431 U......Page 432 Y......Page 433 Back Cover......Page 434