Đang tải... (xem toàn văn)
Learn to apply optimization methods to solve power system operation problemsOptimization of Power System Operation applies the latest applications of new technologies to power system operation and analysis, including several new and important content areas that are not covered in existing books: uncertainty analysis in power systems; steadystate security regions; optimal load shedding; and optimal reconfiguration of electric distribution networks.The book covers both traditional and modern technologies, including power flow analysis, steadystate security region analysis, securityconstrained economic dispatch, multiarea system economic dispatch, unit commitment, optimal power flow, reactive power (VAR) optimization, optimal load shed, optimal reconfiguration of distribution network, power system uncertainty analysis, power system sensitivity analysis, analytic hierarchical process, neural network, fuzzy set theory, genetic algorithm, evolutionary programming, and particle swarm optimization, among others. Additionally, new topics such as the wheeling model, multiarea wheeling, the total transfer capability computation in multiple areas, reactive power pricing calculation, and others are also addressed.Power system engineers, operators, and planners will benefit from this insightful resource. It is also of great interest to advanced undergraduate and graduate students in electrical and power engineering.
OPTIMIZATION OF POWER SYSTEM OPERATION IEEE Press 445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board Lajos Hanzo, Editor in Chief R Abari J Anderson S Basu A Chatterjee T Chen T G Croda M El-Hawary S Farshchi B M Hammerli O Malik S Nahavandi W Reeve Kenneth Moore, Director of IEEE Book and Information Services (BIS) Jeanne Audino, Project Editor Technical Reviewers Ali Chowdhury, California Independent System Operator Loi Lei Lai, City University, UK Ruben Romero, Universidad Estadual Paulista, Brazil Kit Po Wong, The Hong Kong Polytechnic University, Hong Kong OPTIMIZATION OF POWER SYSTEM OPERATION Jizhong Zhu, Ph.D Principal Engineer, AREVA T&D Inc Redmond, WA, USA Advisory Professor, Chongqing University, Chongqing, China A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2009 by Institute of Electrical and Electronics Engineers All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data is available ISBN: 978-0-470-29888-6 Printed in the United States of America 10 To My Wife and Son TABLE OF CONTENTS Preface xvii Introduction 1.1 1.2 1.3 Conventional Methods / 1.1.1 Unconstrained Optimization Approaches / 1.1.2 Linear Programming / 1.1.3 Nonlinear Programming / 1.1.4 Quadratic Programming / 1.1.5 Newton’s Method / 1.1.6 Interior Point Methods / 1.1.7 Mixed-Integer Programming / 1.1.8 Network Flow Programming / Intelligent Search Methods / 1.2.1 Optimization Neural Network / 1.2.2 Evolutionary Algorithms / 1.2.3 Tabu Search / 1.2.4 Particle Swarm Optimization / Application of Fuzzy Set Theory / References / Power Flow Analysis 2.1 2.2 2.3 2.4 Mathematical Model of Power Flow / Newton–Raphson Method / 12 2.2.1 Principle of Newton–Raphson Method / 12 2.2.2 Power Flow Solution with Polar Coordinate System / 14 2.2.3 Power Flow Solution with Rectangular Coordinate System / 19 Gauss–Seidel Method / 27 P-Q decoupling Method / 29 vii viii TABLE OF CONTENTS 2.4.1 2.4.2 2.5 Fast Decoupled Power Flow / 29 Decoupled Power Flow Without Major Approximation / 37 DC Power Flow / 39 References / 41 Sensitivity Calculation 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Introduction / 43 Loss Sensitivity Calculation / 45 Calculation of Constrained Shift Sensitivity Factors / 49 3.3.1 Definition of Constraint Shift Factors / 49 3.3.2 Computation of Constraint Shift Factors / 51 3.3.3 Constraint Shift Factors with Different References / 59 3.3.4 Sensitivities for the Transfer Path / 60 Perturbation Method for Sensitivity Analysis / 62 3.4.1 Loss Sensitivity / 62 3.4.2 Generator Shift Factor Sensitivity / 62 3.4.3 Shift Factor Sensitivity for the Phase Shifter / 63 3.4.4 Line Outage Distribution Factor / 63 3.4.5 Outage Transfer Distribution Factor / 64 Voltage Sensitivity Analysis / 65 Real-Time Application of Sensitivity Factors / 67 Simulation Results / 68 3.7.1 Sample Computation for Loss Sensitivity Factors / 68 3.7.2 Sample Computation for Constrained Shift Factors / 77 3.7.3 Sample Computation for Voltage Sensitivity Analysis / 80 Conclusion / 80 References / 83 Classic Economic Dispatch 4.1 4.2 43 Introduction / 85 Input-Output Characteristic of Generator Units / 85 4.2.1 Input-Output Characteristic of Thermal Units / 85 4.2.2 Calculation of Input-Output Characteristic Parameters / 87 4.2.3 Input-Output Characteristic of Hydroelectric Units / 90 85 PROBABILISTIC OPTIMAL POWER FLOW 589 a series composed of a standard normal distribution and its derivatives As a part of the proposed P-OPF method, distributions are reconstructed with the use of the Gram–Charlier A Series The series can be stated as follows: ∞ f ( x ) = ∑ c j He j ( x )α ( x ) (13.183) j =0 where f(x) is the PDF for the random variable X, cj is the jth series coefficient, Hej(x) is the jth Tchebycheff–Hermite, or Hermite, polynomial, and α(x) is the standard normal distribution function The Gram–Charlier form uses moments to compute series coefficients, while the Edgeworth form uses cumulants, which is discussed here Since the PDF for a normal distribution is an exponential term, taking derivatives successively returns the original function with a polynomial coefficient multiplier These coefficients are referred to as Tchebycheff–Hermite, or Hermite, polynomials To illustrate how the Hermite polynomials are generated, the first four derivatives of the standard unit normal distribution are taken as follows: D0α ( x ) = D0 e − x2 =e − x2 (13.184) −2x = − xe 2 −2x = ( x − 1) e D α( x) = D e D2 α ( x ) = D e D α( x) = D e 3 − x2 −2x D4 α ( x ) = D e − x2 (13.185) − x2 = ( 3x − x ) e (13.186) − x2 = ( x − x + 3)e (13.187) − x2 (13.188) where Dn is the nth derivative The Tchebycheff–Hermite polynomials are the polynomial coefficients in the derivatives Using the results of the first four derivatives in equations (13.184)–(13.188), the first five Tchebycheff–Hermite polynomials are written as follows: He0 ( x ) = (13.189) He1 ( x ) = x (13.190) He2 ( x ) = x − (13.191) He3 ( x ) = x − x (13.192) He4 ( x ) = x − x + (13.193) Because of the structure of equations (13.184)–(13.188), the highest-power coefficients of the odd derivatives, i.e., the third, fifth, seventh, etc., are negative Equations (13.189)–(13.193) have been formed following the convention 590 UNCERTAINTY ANALYSIS IN POWER SYSTEMS that the equations relating to the odd derivatives are multiplied by negative one, such that the coefficient of the highest power is positive [33] Therefore, the nth Tchebycheff–Hermite polynomial can be symbolically written as n Hen ( x )α ( x ) = ( − D) α ( x ) (13.194) In addition, a recursive relationship is available to determine third-order and higher polynomials Hen ( x ) = xHen−1 ( x ) − ( n − 1) Hen− (13.195) 13.9.3.2 Edgeworth A Series Coefficients Given the cumulants for a distribution in standard form, i.e., zero mean and unit variance, the coefficients for the Edgeworth form of the A series can be computed To find the equations for the A series coefficients, an exponential representation of the PDF is broken into its series representation and equated with the Gram–Charlier A series in equation (13.183) The PDF, as an exponential, is written in the following form using cumulants [9]: ⎛− K D3 + K D4 − K D5 + ⎜ 3! 4! 5! f ( x ) = e⎝ )α( x) (13.196) where Dn is the nth derivative of the unit normal distribution, Kn is the nth cumulant, and α(x) is the standard unit normal PDF Expanding equation (13.196) as an exponential series yields ( K K K ⎡ − D3 + D4 − D5 + ⎢ 3! 4! 5! f ( x ) = ⎢1 + 1! ⎢ ⎣ K K K − D3 + D4 − D5 + 3! 4! 5! + 2! K3 K4 K5 D + D − D + − 3! 4! 5! + 3! ) ( ) ( ) + ⎤ ⎥ ⎥α( x) ⎥ ⎥ ⎦ (13.196) If each of the terms is expanded individually and grouped based on powers of D, the following result is obtained: PROBABILISTIC OPTIMAL POWER FLOW Table 13.5 591 A Series Coefficient Equation Coefficient Equation 0 K3 K4 24 K5 120 (K6 + 10K 32 ) 720 ( K + 35K K ) 5040 K K K3 ⎞ ⎡ K ⎛K f ( x ) = ⎢ − D3 + D4 − D5 + ⎜ + D ⎝ ! ! 3!2 ⎟ ⎠ 3! 4! 5! ⎣ 2K3 K4 ⎞ ⎛K ⎤ D + ⎥α( x ) +⎜ + ⎝ ! ! 3! ! ⎟ ⎠ ⎦ (13.197) Returning to the definition for the Gram–Charlier A series in equation (13.183) and expanding the summation yields f ( x ) = c0 He0 ( x )α ( x ) + c1 He1 ( x )α ( x ) + c2 He2 ( x )α ( x ) + (13.198) Comparing equations (13.197) and (13.198), the values for the coefficients can be determined Based on the equations presented, the first seven terms of the Edgeworth form of the A series are presented in Table 13.5 13.9.3.3 Adaptation of the Cumulant Method to the P-OPF Problem The cumulant method relies on the behavior of random variables and their associated cumulants when they are combined in a linear fashion This section discusses the formation of random variables from a linear combination of others and the role cumulants play in this combination Given a new random variable z, which is the linear combination of independent random variables, c1, c1, … , cn z = a1c1 + a2 c2 + + an cn (13.199) the moment generating function Φz(s) for the random variable z can be written as 592 UNCERTAINTY ANALYSIS IN POWER SYSTEMS Φ z ( s) = E[ e sz ] = E[ e s(a1c1 + a2 c2 + + an cn ) ] = E[ e s(a1c1 ) e s(a2 c2 ) e s(an cn ) ] (13.200a) Since c1, c2, … , cn are independent, the above equation can be written as Φ z ( s) = E[ e s(a1c1 ) ]E[ e s(a2 c2 ) ]…… E[ e s(an cn ) ] = Φ c1 ( a1 s)Φ c2 ( a2 s)……Φ cn ( an s) (13.200b) The cumulants for the variable z can be computed with the cumulant generating function, in terms of the component variables as follows: Ψ z ( s) = ln( Φ z ( s)) = ln(Φ c1 ( a1 s) Φ c2 ( a2 s) Φ cn ( an s)) = ln(Φ c1 ( a1 s)) + ln(Φ c2 ( a2 s)) + + ln(Φ cn ( an s)) = Ψ c1 ( a1 s) + Ψ c2 ( a2 s) + + Ψ cn ( an s) (13.201) To compute the second-order cumulant, the first- and second-order derivatives of the cumulant generating function for the random variable z are computed as Ψz ( s) = a1 Ψ c1 ( a1 s) + a2 Ψ c2 ( a2 s) + ′ ′ ′ + an Ψ cn ( an s) ′ (13.202) Ψz ( s) = a1 Ψc1 ( a1 s) + a2 Ψc2 ( a2 s) + ′′ ′′ ′′ + an Ψcn ( an s) ′′ (13.203) 2 Evaluating equation (13.203) at s = gives Ψz ( 0) = a1 Ψc1 ( 0) + a2 Ψc2 ( 0) + ′′ ′′ ′′ + a2 Ψcn ( 0) ′′ n (13.204) Similarly, the nth-order cumulant for z, a linear combination of independent random variables, can be determined with the following equation: n λ n = Ψ(zn) ( 0) = a1 Ψ(cn) ( 0) + an Ψ(cn) ( 0) + 2 + an Ψ(cn) ( 0) n n (13.205) where the exponent (n) denotes the nth derivative with respect to s The cumulant method is adapted from the basic derivation above to accommodate the P-OPF problem when a logarithmic barrier interior point method (LBIPM)-type solution is used The Hessian of the Lagrange function is necessary for the computation of the Newton step in the LBIPM The inverse of the Hessian, however, can be used as the coefficients for the linear combination of random bus loading variables The pure Newton step is computed at iteration k of the LBIPM with the following equation: yk +1 = yk − H −1 ( yk ) G( yk ) (13.206) where y is the vector of variables G(yk) is the gradient of the Lagrange function H−1(yk) is the inverse Hessian matrix, which contains the multipliers for a linear combination of PDFs for random bus loads COMPARISON OF DETERMINISTIC AND PROBABILISTIC METHODS 593 It is necessary to introduce the cumulants related to the random loads into the system in such a way that the cumulants for all other system variables can be computed Some characteristics of the gradient of the Lagrangian are used to accomplish this When the gradient of the Lagrangian is taken, the power flow equations appear unmodified in this vector Therefore, cumulant models in the bus loads map directly into the gradient of the Lagrangian For the purposes of mapping, the mismatch vector, in equation (13.206), is replaced by a new vector containing the cumulants of the random loads in the rows corresponding to their associated power flow equations The linear mapping information contained in the inverse Hessian can be used to determine cumulants for other variables when bus loading is treated as a random variable If −H−1(yk) is written in the following form ⎡ a1,1 a1, … a1, n ⎤ ⎢ a a … a2, n ⎥ − H = ⎢ 2,1 2,1 … ………⎥ ⎢ ⎥ ⎣an,1 an, … an, n ⎦ −1 (13.207) then the nth cumulant for the ith variables in y is computed with the following equation: λ yi , n = ain,1λ x 1, n + ain, λ x 2, n + , … , + ain, n λ xn, n (13.208) where yi is the ith element in y, and λxj,n is the nth cumulant for the jth component variable For the cumulant method used for P-OPF, the cumulants for unknown random variables are computed from known random variables, and PDFs are reconstructed with the Gram–Charlier/Edgeworth expansion theory 13.10 COMPARISON OF DETERMINISTIC AND PROBABILISTIC METHODS As we have analyzed in this chapter, it is impossible to obtain all available data in the real-time operation because of the above-mentioned uncertainties of power systems and competitive environment Nevertheless, it is important to select an appropriate technique to handle these uncertainties The existing deterministic methods and tools are not adequate to handle them The probabilistic methods, gray mathematics, fuzzy theory, and analytic hierarchy process (AHP) [34–37] are very useful to compute the unavailable or uncertain data; so that power system operation problems such as the economic dispatch, optimal power flow, and state estimation can be solvable even some data are not available A comparison of the deterministic method and the probabilistic method is shown in Table 13.6 594 UNCERTAINTY ANALYSIS IN POWER SYSTEMS Table 13.6 Deterministic vs Probabilistic Methods Methods Comparison Deterministic Method Probabilistic Method Contingency selection Typically a few probable and extreme contingencies Contingency probabilistic Based on judgment Load levels (forecast) Typically seasonal peaks and selected off-peak loads Traditional optimization technology Deterministic security region Well established More exhaustive list of contingencies; Ranking based on Fuzzy/AHP methods Based on inadequate or uncertain data (ANN, Fuzzy and AHP methods) Multiple levels with uncertain factors (Fuzzy, ANN) Optimization technology & AHP/Fuzzy/ANN Variable security regions Unit commitment Security regions Criteria for decision Need a suitable method/ criteria to make decision (ANN, Fuzzy and AHP methods) Through comparing the various approaches, the following methods to handle uncertainties are recommended: • • • • • Characterization and probabilistic methods Probabilistic methods/tools for evaluating the contingencies Fuzzy/ANN/AHP methods to handle uncertainties (e.g., contingency ranking) Risk management tools to optimize energy utilization while maintaining the required levels of reliability Cost-benefit analysis (CBA) for quantifying the impact of uncertainty REFERENCES [1] A.S Merlin, “Latest developments and future prospects of power system operation and control,” International Journal of Electrical Power and Energy System, Vol 16, No 3, 1994, pp 137–139 [2] N Rau, C.C Fong, C.H Grigg, and B Silverstein, “Living with uncertainty,” IEEE Power Engineering Review, November, 1994, pp 24–26 [3] K.H Abdul-Rahman, S.M Shahidehpour, and N.I Deeb, “Effect of EMF on minimum cost power transmission,” IEEE Trans On Power Systems, Vol 10, No 1, 1995, pp 347–353 REFERENCES 595 [4] M Ivey, “Accommodating uncertainty in planning and operation,” Workshop on Electric Transmission Reliability, Washington, DC, Sept 17, 1999 [5] A.M Leite da Silva, V.L Arienti, and R.N Allan, “Probabilistic load flow considering dependence between input nodal powers,” IEEE Trans on PAS, Vol 103, No 6, 1984, pp 1524–1530 [6] M.E El-Hawary and G.A.N Mbamalu, “A comparison: probabilistic perturbation and deterministic based optimal power flow solutions,” IEEE Trans On Power Systems, Vol 6, No 3, 1991, pp 1099–1105 [7] P.W Sauer and B Hoveida, “Constrained stochastic power flow analysis,” Electric Power Systems Research, Vol 5, 1982, pp 87–95 [8] T.S Karakatsanis and N.D Hatziargyriou, “Probabilistic constrained load flow based on sensitivity analysis,” IEEE Trans On Power Systems, Vol 9, No 4, 1994, pp 1853–1860 [9] K.H Abdul-Rahman, S.M Shahidehpour, and N.I Deeb, “AI approach to optimal VAR control with fuzzy reactive loads,” IEEE Trans On Power Systems, Vol 10, No 1, 1995, pp 88–97 [10] V Miranda and J.T Saraiva, “Fuzzy modeling of power system optimal load flow,” IEEE Trans On Power Systems, Vol 7, No 2, 1992, pp 843–849 [11] Z Wang and F.L Alvarado, “Interval arithmetic in power flow analysis,” IEEE Trans On Power Systems, Vol 7, No 3, 1992, pp 1341–1349 [12] NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/ div898/handbook [13] R.E More, Interval Analysis, Prentice-Hall, Englewood Gliffs, NJ, 1996 [14] F.N Ris, Interval Analysis and Applications to Linear Algebra, D Phil, Thesis, Oxford, 1972 [15] E Hansen and R Smith, “Interval arithmetic in matrix computations, Part II,” SIAM Journal of Numerical Analysis, Vol 4, 1967, pp 1–9 [16] E Hansen and S Sengupta, “Bounding solutions of systems of equations using interval analysis,” BIT, Vol 21, 1981, pp 203–211 [17] M Valdma, M Keel, O Liik, and H Tammoja, “Method for Minimax Optimization of Power System Operation,” Proc of IEEE Bologna PowerTech 2003, 23–26 June 2003, Bologna, Italy Paper 252: pp 1–6 [18] M Valdma, M Keel, and O Liik, “Optimization of active power generation in electric power system under incomplete information,” Proc of Tenth Power Systems Computation Conference, 1990, Graz, Austria, pp 1171–1176 [19] K Selvi, N Ramaraj, and S.P Umayal, “Genetic Algorithm Applications to Stochastic Thermal Power Dispatch,” IE(I) Journal EL, Vol 85, June 2004, pp 43–48 [20] K.H Abdul-Rahman and S.M Shahidehpour, “Static Security in Power System Operation with Fuzzy Real Load Constraints,” IEEE Trans On Power Systems, Vol 10, No 1, 1995, pp 77–87 [21] A Kaufmann and M.M Gupta, Fuzzy Mathematical Models in Engineering and Management Science, North-Holland Publishing Company, Amsterdam, 1988 [22] T.S Dillion and T Tun, “Integration of the sub-problems involved in the optimal economic operation of hydro-thermal system,” Proc IFAC Symp Contr And Manag of Integ Indust France, Sept 1977, pp 171–180 596 UNCERTAINTY ANALYSIS IN POWER SYSTEMS [23] T Tun and T.S Dillion, “Sensitivity analysis of the problem of economic dispatch of hydro-thermal system,” Proc IFAC Symp Autom Contr and Prot of Elec Power System, Melb 1977 [24] U.A Ozturk, M Mazumdar, and B.A Norman, “A Solution to the Stochastic Unit Commitment Problem Using Chance Constrained Programming,” IEEE Trans on Power Systems, Vol 19, No 3, 2004, pp 1589–1598 [25] J Qiu and S.M Shahidehpour, “A New Approach for Mini- mizing Power Losses and Improving Voltage Profile,” IEEE Trans on Power Systems, Vol 2, No 2, pp 287–295, May 1987 [26] K.H Abdul-Rahman and S.M Shahidehpour, “Application of Fuzzy Sets to Optimal Reactive Power Planning with Security Constraints,” IEEE Trans On Power Systems, Vol 9, No 2, 1994, pp 589–597 [27] G.B Dantzig and P Wolfe, “The Decomposition Algorithm for Linear Programs,” Econometrica, Vol 29, No 4, pp 767–778, Oct 1961 [28] M Madrigal, K Ponnambalam, and V.H Quintana, “Probabilistic optimal power flow,” in Proc IEEE Can Conf Electrical Computer Engineering, Waterloo, ON, Canada, May 1998, pp 385–388 [29] G Verbic and C.A Cañizares, “Probabilistic Optimal Power Flow in Electricity ˇ Markets Based on a Two-Point Estimate Method,” IEEE Trans on Power Systems, Vol 21, No 4, 2006, pp 1883–1993 [30] H.P Hong, “An ef?cient point estimate method for probabilistic analysis,” Reliab Eng Syst Saf., Vol 59, pp 261–267, 1998 [31] E Rosenblueth, “Point estimation for probability moments,” Proc Nat Acad Sci U S A., Vol 72, No 10, pp 3812–3814, Oct 1975 [32] A Schellenberg, W Rosehart, and J Aguado, “Cumulant based probabilistic optimal power flow (P-OPF),” IEEE Trans on Power Systems, Vol 20, No 2, 2005, pp 773–781 [33] M.G Kendall and A Stuart, The Advanced Theory of Statistics, 4th ed New York: Macmillan, 1977 [34] T.L Satty, The Analytic Hierarchy Process, McGraw Hill, Inc., New York, 1980 [35] J.Z Zhu and M.R Irving, “Combined Active and Reactive Dispatch with Multiple Objectives using an Analytic Hierarchical Process,” IEE Proceedings, Part C, Vol 143, pp 344–352, 1996 [36] J.Z Zhu, M.R Irving, and G.Y Xu, “Automatic contingency selection and ranking using an analytic hierarchical process,” Electrical Machines and Power Systems Journal, No 4, 1998 [37] J.Z Zhu and J.A Momoh, “Optimal VAR pricing and VAR placement using analytic hierarchy process,” Electric Power Systems Research, Vol 48, No 1, 1998, pp 11–17 AUTHOR BIOGRAPHY Jizhong Zhu is currently working at AREVA T&D Inc as a principal power systems engineer He received his Ph.D degree from Chongqing University, P.R China, in Feb 1990 Dr Zhu was a full-time professor in Chongqing University He won the “Science and Technology Progress Award of State Education Committee of China” in 1992 and 1995, respectively, “Sichuan Provincial Science and Technology Advancement Award” in 1992, 1993 and 1994, respectively, as well as the “Science and Technology Invention Prize of Sichuan Province Science & Technology Association” in 1992 Dr Zhu was awarded as an Excellent Young Teacher by Chongqing City Government in 1992; selected as an Outstanding Science & Technology Researcher and won annual Science & Technology Medal of Sichuan Province in 1993 He was also selected as one of four outstanding young scientists working in China by The Royal Society of UK and China Science & Technology Association and awarded Royal Society Fellowship in 1994 and the national research prize “Fok Ying-Tong Young Teacher Research Medal” in 1996 He worked in a variety of places all over the world, including Chongqing University in China, Brunel University in the UK, the National University of Singapore, Howard University in the US, and AREVA T&D Inc (since 2000) He is also an advisory professor at Chongqing University His research interest is in the analysis, operation, planning and control of power systems He has published four books as an author and coauthor, and has published over 100 papers in international journals and conferences Optimization of Power System Operation, by Jizhong Zhu, Ph.D Copyright © 2009 Institute of Electrical and Electronics Engineers 597 INDEX AC power flow, 29 Admittance matrix, 10 AGC, 44 AHP, 273, 345, 420, 463 Analysis Contingency, 336 Sensitivity, 43, 420 Analytic hierarchy process, 273, 345, 420, 463 Eigenvalue, 279 Eigenvector, 279 Hierarchical model, 278 Judgment matrix, 278 Performance index, 285 Root method, 279 Scaling method, 286 Available transfer capability ATC, 218 Congestion management, 492 Total transfer capability, 242 Average production cost, 253 B′ matrix Decoupled power flow, 32 Sensitivity analysis, 52 B coefficients, losses, 103 Benefit cost ratio: BCR, 448 CBA, 448 Cost benefit analysis, 448 Beta distribution, 549 Bus Generation, 11 Load, 11, 348 PQ, 11, 312 PV, 11 Reference, 11 Slack, 11 Capability, ATC, 218 Capacity, generation, 86 Chance constrained optimization, 574 Chi-square distribution, 553 Classic economic dispatch, 85 Combined active and reactive dispatch, 339 Complementary slackness conditions, 155, 161 Congestion management, 492 Constraints Active power, 341 Dynamic, 345 Import and export, 342 OPF, 316 Reactive power, 342 Shift factor, 49 Spinning reserve, 343 Contingency analysis, 49, 336 Continuation power flow method, 243 Control, VAR, 66, 420 Controller, 487 Convergence, Power flow, 16 Coordination equation, 104 Cost Decremental, 212 Fuel, 86, 340, 546 Optimization of Power System Operation, by Jizhong Zhu, Ph.D Copyright © 2009 Institute of Electrical and Electronics Engineers 599 600 INDEX Cost (cont’d) Incremental, 212 Cost benefit analysis, 448 Cost function Linear, 142, 340 Piecewise linear, 149, 523 Quadratic, 152, 340 Dantzig-Wolf decomposition, 570 DC power flow, 39 Decoupled power flow, 29 Density function, probability, 547 Deterministic method, 573, 593 Dispatch Economic, 85, 141, 562 Multiple areas, 211 Reactive power, 410 Secure, 141 Distributed interruptible load shedding, 479 Distribution factor, line outage, 63 Distribution network Load flow, 509 Reconfiguration, 503 DNRC, 503 Dual Optimization, 260 Theory, 161, 260 Variables, 162 Duality gap, 263 Dynamic programming, 254 Economic dispatch, 85, 141, 562 Economic operation, 95 Eigenvalue, 279 Eigenvector, 279 EMS, 44 Energy Control center, 44 Function, 236 Management system, EMS Market, 44, 546 Equal incremental rate, 91 Everett method, 471 Evolutionary algorithm, 265, 433 Evolution programming, 264, 530 Expansion method, security regions, 372 Exponential distribution, 548 Fast decoupled power flow, 29 Fitness function, 269, 272, 527, 533 Frequency drop, 457 Fuel cost, 86, 340, 546 Fuzzy Numbers, 379, 554 Power flow, 560 Set, 379, 386, 554 GA, 120, 199, 524 Gamma distribution, 550 Gauss-Seidel method, 27 Generator Bus, 11 Input-output characteristic, 85 Shift factor, 52 Genetic algorithm, 120, 199, 524 Chromosomes, 120, 536 Crossover, 120, 524 Fitness function, 121, 527 Mutation, 120, 525 Selection, 524 Gradient Economic dispatch, 112 Method, 112, 131 OPF, 307 Search, 112 Graph theory, 503 Gumbel distribution, 552 Heuristic algorithm, 508 Hierarchical model, 278 Hopfield neural network, 124 Hydro Input output, 105 Scheduling, 105 Unit, 90 Hydro characteristic, unit, 90 Hydrothermal system, 104, 573 IEEE test systems, 147 Incremental Cost, 212 Power loss, 100 Rate, 91 Input output characteristic, 85 Intelligence search methods, Intelligent load shedding, 459 INDEX Interchange, 214 Interconnected area, 230 Interior point algorithm, 318 IPOPF, 318 IQIP, 323 Iteration, Power flow, 13 Jacobian matrix, 14, 309 Judgment matrix, 279, 345, 421, 463 KCL, 182, 505 Kuhn-Tucker conditions, 138 KVL, 183, 505 Labeling algorithm, OKA, 164 Labeling rules, NFP, 164 Lagrange Equation, 137 Function, 137, 236, 259, 308, 411 Multiplier, 137, 236, 259, 309, 319 Relaxation, 258 Line outage distribution factor, 54 Line overload, 333 Linear programming Constraints, 142 Economic dispatch, 141 Objective function, 142 OPF, 346 Security regions, 386 VAR optimization, 415 Load Bus, 475 Flow, 9, 323, 507 Damping coefficient, 458 Probability function, 547 Reference, 497 Shedding, 455 LODF, 54 Lognormal distribution, 547 Loss Factor, 45 Minimization, 316 Network, 100 Power, 506, 530 Sensitivity, 45 Sensitivity calculation, 45 Transmission, 168 LP, 3, 14, 141, 386, 393, 415 601 MAED, 211 Marginal cost, 220 Market, energy, 44 Matrix B′, 32 B″, 32 Jacobian, 14, 309 Matroid theory, 535 Mean value, 547 Min-max optimal, 562 Mix-integer linear programming, 4, 513 Modified interior point OPF, 315 Monte Carlo simulation, 588 Multiarea Economic dispatch, 211 Interconnection, 212 VAR pricing, 444 Wheeling, 223 Multiplier, Lagrange, 236, 259, 309, 319 Multiobjective optimization, 321, 434, 530 Network flow programming, 5, 159, 201 Network Limitation, 276 Losses, 100, 565 Security, 141 Sensitivity factors, 49 Neural network, 124, 233 Newton’s method OPF, 298 Power flow, 19 Newton–Raphson method, 12, 132 NFP, 5, 159, 201 NLCNFP, 180, 211, 244 NLONN, 233, 426 Nonlinear convex network flow programming, 180, 211, 244 Nonlinear optimization neural network, 233, 426 Normal distribution, 547 N − security constraints, 171, 378 OKA, 159, 465 Operating, cost, 85 602 INDEX OPF Gradient method, 307 Interior point method, 315 Linear programming method, 313 Modified interior point, 315 Multiple objective, 339 Newton method, 298 Optimal power flow, 297 Particle swarm optimization, 347 Phase shifter, 330 Quadratic programming, 357 Optimal load shedding, 455 Optimal power flow, 297 Optimal reconfiguration, distribution network, 503 OTDF, 64 Outage, 63 Outage transfer distribution factor, 64 Out-of-kilter algorithm, 159, 465 Pareto-optimal, 435 Participation factors, 61 Particle swarm optimization, 268, 438 Peak load, 338 Penalty factor, 229, 299 Perturbation method, 243 Phase shifter, 50, 330 Polar coordinate system, power flow, 14 Pool Operation, 214 Savings, 215 Post contingency, 44 Power balance, 113 Power, pools, 214 Power flow AC, 29 Analysis, Convergence, 16 DC, 39 Decoupled, 29 Equation, 15 Gauss-Seidel, 27 Newton-Raphson, 12 Optimal, 297 Power output, unit, 86 P-Q decoupling method, power flow, 29 Pricing, Reactive power, 440 Principle, equal incremental rate, 91 Priority list, unit commitment, 253 Probabilistic optimal power flow, 581 Probabilistic power flow, 559 Probability density function, 547 Probability method, 593 Probability theory, 576 PSO, 268 Quadratic function, unit fuel cost, 152, 306 Quadratic interior point method, 323 Quadratic programming, 3, 152 Radial network, 507 Random variable, 565, 591 Reactive power Balance, 410 Dispatch, 410 Optimization, 409 Pricing, 440 Reserve margins, 340 Reconfiguration, distribution network, 503 Rectangular coordinate system, power flow, 19 Reduced gradient method, 189 Reference bus, 11 Root method, 279 Savings, pool, 215 SCED, 141 Scheduling, 275 Search, gradient, 112 Security constrained economic dispatch, 141 Security analysis, 378 Security corridor, 366 Security region, 365 Sensitivity analysis, 43, 420 Sensitivity factor, 48 Shift factor, generation, 52 Simplex method, 399 Slack bus, 11 Slack variable, 399 Spinning reserve, 265 Stability, 426 Standard deviation, 547 INDEX Static VAR compensators, 344 Steady-state security regions, 365 Stochastic model, 564 Stochastic programming, 573 Sum method, 282 Tabu search method, 264 Taylor series, 12, 584 Tie-line, 212 Total transfer capability, 242 Transfer, power, 244 Transfer path, 60 Transmission Access open, 217 Losses, 581 Services, 220 System, 60 Transportation problem, 201 TTC, 245 Two-point estimate method, 582 UC, 251, 573 Uncertainty, 546 Uncertainty analysis, 545 Uncertain load, 547 Unit commitment Analytic hierarchy process, 273 Dynamic programming, 254 Evolutionary programming, 264 Lagrange relaxation, 258 Particle swarm optimization, 268 Priority list method, 253 Tabu search, 264 VAR Compensation, 66, 426 Control, 66, 420 Optimization, 409, 579 Planning, 316 Pricing, 444 Support, 316 Variables, Slack, 399 Variance, 560 Voltage Collapse, 341 Sensitivity analysis, 45, 65, 426 Stability margin, 426 Weibull distribution, 553 Wheeling, 217 603 ... gives us the electrical response of the transmission system to a particular set of loads and generator power outputs Power flows are an important part of power system operation and planning Generally,... in the development of this book, especially Professor Kit Po Wong of the Hong Kong Polytechnic University, Professor Loi Lei Lai of City University, UK, Professor Ruben Romero of Universidad Estadual... bus real power P, reactive power Q, voltage V, and angle θ To solve the power flow equations, two of these should be known for each bus According to the practical conditions of the power system