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.
Author(s): Mingtian Fan, Zuping Zhang, Chengmin Wang
Publisher: China Electric Power Press
Year: 2019
Language: English
Pages: 434
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
