I . 2 fig... an Ad. 9 3‘. 2 1.. . .. . I L . r 21.15.“.5: 531.3. km}? .23.. .3. 2% w u... a... .23. 2:34., . 3.4 . smut. a i... . s 4.5? .- hulk: it I #4.... 13" x ”pm“. .z 2.. 2...? .5. c . . : . .-.:LL:H.V2 LI. :2. firfiyga .. . F. {Ex-t... 2 :2: «3:2... 2%.. L ‘. 1 . 1’4" 2 x .2223. . e: ”a :3: ('9 Nod. Licfinmyg min. I: .2 .5. . “have“ .1: lint (\an 3 A», ‘ .V Ii Bissau-«mnflg’ngfiru in}: luv-NB .d .5? 11.1.5 $1 .Vh : . v.2 .H , 241...... . 5x..¥.2-Cxe: \f tviianuhuli.‘ ll 5. ‘ 15.... 1 ..u . . 1..) .215... » s 7:“ lIIIHHJIIHlllHll(UH)IJIIHIHUIIHllHlHllHllHNlHl 293 01812 7146 r a a a This is to certify that the dissertation entitled DISPATCHING POWER SYSTEM FOR PREVENTIVE AND CORRECTIVE VOLTAGE COLLAPSE PROBLEM IN A DEREGULATED POWER SYSTEM presented by Nasser Ahmed Alemadi has been accepted towards fulfillment of the requirements for Ph. D. degree in Electrical Eng Ciréd A Main Major professor Date Augwf/zo, @962 MSUI’: an Affirmative Action/Equal Opportunity Institution 042771 ‘ \ _ v“-W—T~"vw— 1— v— v 7" ‘v— ' v~—: ‘ a LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1M www.mu Dispatching Power System for Preventive and Corrective Voltage Collapse Problem in a Deregulated Power System By Nasser Ahmed Alemadz' A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical and Computer Engineering 1999 ABSTRACT Dispatching Power System for Preventive and Corrective Voltage Collapse Problem in a Deregulated Power System By Nasser Ahmed Alemadz' Deregulation has brought opportunities for increasing efficiency of production and delivery and reduced costs to customers. Deregulation has also bought great chal- lenges to provide the reliability and security customers have come to expect and demand from the electrical delivery system. One of the challenges in the deregulated power system is voltage instability. Voltage instability has become the principal con- straint on power system operation for many utilities. Voltage instability is a unique problem because it can produce an uncontrollable, cascading instability that results in blackout for a large region or an entire country. In this work we define a system of advanced analytical methods and tools for secure and efficient operation of the power system in the deregulated environment. The work consists of two modules; (a) contingency selection module and (b) a Security Constrained Optimization module. The contingency selection module to be used for voltage instability is the Voltage Stability Security Assessment and Diagnosis (VSSAD). VSSAD shows that each volt- age control area and its reactive reserve basin describe a subsystem or agent that has a unique voltage instability problem. VSSAD identifies each such agent. VS SAD is to assess proximity to voltage instability for each agent and rank voltage instability agents for each contingency simulated. Contingency selection and ranking for each agent is also performed. Diagnosis of where, why, when, and what can be done to cure voltage instability for each equipment outage and transaction change combina- tion that has no load flow solution is also performed. A security constrained optimization module developed solves a minimum control solvability problem. A minimum control solvability problem obtains the reactive re- serves through action of voltage control devices that VSSAD determines are needed in each agent to obtain solution of the load flow. VSSAD makes a physically impossi- ble recommendation of adding reactive generation capability to specific generators to allow a load flow solution to be obtained. The minimum control solvability problem can also obtain solution of the load flow without curtailing transactions that shed load and generation as recommended by VSSAD. A minimum control solvability problem will be implemented as a corrective control, that will achieve the above objectives by using minimum control changes. The control includes; (1) voltage setpoint on gener- ator bus voltage terminals; (2) under load tap changer tap positions and switchable shunt capacitors; and (3) active generation at generator buses. The minimum control solvability problem uses the VSSAD recommendation to obtain the feasible stable starting point but completely eliminates the impossible or onerous recommendation made by VSSAD. This thesis reviews the capabilities of Voltage Stability Security Assessment and Diagnosis and how it can be used to implement a contingency selection module for the Open Access System Dispatch (OASYDIS). The OASYDIS will also use the corrective control computed by Security Constrained Dispatch. The corrective control would be computed off line and stored for each contingency that produces voltage instability. The control is triggered and implemented to correct the voltage instability in the agent experiencing voltage instability only after the equipment outage or operating changes predicted to produce voltage instability have occurred. The advantages and the requirements to implement the corrective control are also discussed. This work is dedicated to my parents, my wife, and my children iv ACKNOWLEDGMENTS All thanks, Prayers and glories are due to ALLAH, the most gracious and most merciful, who taught humankind every thing they knew not. I am deeply grateful and indebted to several people who helped by their effort and support in the development of this dissertation. My sincere appreciation and thanks to my advisor, Professor Robert A. Schlueter, for his scientific counsel, commitment, support, continuous advice and encouragement throughout this research, who gave me unlimited time and support. without the willing of “ALLAH” in the first place and then the help of Prof. Robert A. Schlueter this dissertation would not have been completed. Special thanks also go to the members of my doctoral committee, Prof. H. Khalil, Prof. E. Strangas, and Prof. J. Schuur for their intellectual contributions to this dissertation. I would like to extend my sincere appreciation and gratitude to the State of Qatar government, represented by Qatar University for their financial support throughout my studies in the USA. This acknowledgment would be incomplete without recognizing the love and en— couragement of my family. Great thanks are extended to my parents, brothers, and sisters for their continuous moral support, help, and prayers. Most importantly, I would like to express my profound gratitude and greatest thanks to my great wife, Fawziya who stood by me and was a trusted friend and a compassionate partner in all my ups and downs during my graduate study in the USA. Her emotional support, understanding, love and constant encouragement made my life much happier and my work much easier. I am also grateful to my daughters, Maha and Bashaer, and my son, Muhammad for their love, concern and patience. TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES 1 Introduction 1.1 Description of the problem ........................ 1.2 Motivation and Objective ........................ 1.3 Literature Review ............................. 1.3.1 Optimal Power Flow ....................... 1.3.2 Interior Point Method ...................... 1.3.3 Voltage Instability ........................ Interior Point Method 2- 1 Linear Programming ........................... 2 - 2 Nonlinear Programming ......................... 2.2.1 Primal Method .......................... 2.2.2 Dual Method ........................... 2.2.3 Penalty Function Method .................... 2.2.4 Barrier Function Method ..................... 2.2.5 Interior Point Method ...................... 2- 3 Primal-Dual Interior Point Method ................... 2.3.1 Predictor-Corrector Interior Point Algorithm .......... 2-4 Computational Implementation ..................... VOItage Stability Security Assessment and Diagnosis 3 - 1 Introduction ................................ 3-2 Voltage Stability Overview ........................ 3-3 Types of Voltage Instability ....................... 3.3.1 Loss of Control Voltage Instability ............... 3.3.2 Clogging Voltage Instability ................... 3-4 Knowledge Development ......................... vi ix toxlOMI—II—I 10 13 14 15 16 17 18 20 22 25 29 31 37 37 39 45 45 46 46 3.5 Intelligence Development ......................... 50 3.6 Diagnosis of Voltage Instability ..................... 51 3.6.1 Loss of voltage control voltage instability ............ 52 3.6.2 Clogging Voltage Instability ................... 53 3.7 Numerical Results ............................. 56 3.7.1 Loss of Control Voltage Instability ............... 59 3.7.2 Clogging Voltage Instability ................... 61 Optimization Applications in Dispatch of Power System 75 4.1 Introduction ................................ 75 4.2 Active Power Dispatch .......................... 76 4.3 Reactive Power Dispatch ......................... 79 4.4 Security Constrained Dispatch ...................... 84 4.4.1 Preventive Control Formulation ................. 85 4.4.2 Corrective Control Formulation ................. 88 4.5 Voltage Collapse Constrained Optimal Reactive Dispatch Problems . 92 4.6 Bender Decomposition .......................... 97 4- 7 Open Access System Dispatch (OASYDIS) ............... 100 4.7.1 Power System Dispatch Function ................ 102 4-8 State Estimation Modeling and Measurement ............. 104 4- 9 Secondary Voltage Control ........................ 106 4- 1 0 Optimization Requirements ....................... 108 Open Access System Dispatch 111 5- 1 Load Flow Modeling ........................... 112 5.1.1 Power System Components .................... 112 5.1.2 Load Flow Equation ....................... 116 5 - 2 Constraints ................................ 122 5.2.1 Equality Constraints ....................... 123 5.2.2 Inequality Constraints ...................... 123 5- 3 Open Access System Dispatch Security Constrained Optimization . . 125 5.3.1 Minimum Control Solvability Problem ............. 129 5.3.2 Minimum Ancillary Services Cost Problem ........... 139 5.3.3 Master Problem Formulation ................... 142 5-4 Numerical Results ............................. 143 5.4.1 Loss of control voltage instability ................ 145 5.4.2 Clogging Voltage Instability ................... 151 vii 6 Conclusion and Future Work 6.1 Summary of the Work Completed .................... 6.2 Conclusions oooooooooooooooooooooooooooooooo 6.3 Future Work ................................ APPENDIX BIBLIOGRAPHY viii 173 173 176 180 182 197 LIST OF FIGURES 4.1 Structure, Modules and Interfaces of the OASYDIS Application. . . . 5.1 Two Bus Model. ooooooooooooooooooooooooooooo 3.1 3.2 3.3 3.4 3.5 3.6 3.8 3-9 3- 10 3- 11 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5.10 5.11 5-12 5.13 5.14 LIST OF TABLES Single Contingency Ranking for Voltage Collapse Region# 35. . . . . 64 Double Contingency Ranking for Voltage Collapse Region# 35. . . . 65 Single Outage that had the most affect on Voltage Collapse Region. . 66 Double Outage that had the most affect on Voltage Collapse Region. 67 Double Outage with negative reserves when the gen. limits are ignored. 68 Voltage Stability Assessment Results for Voltage Collapse Regions that are Vulnerable to Loss of Control Voltage Instability .......... 69 Additional Reactive Supply Needed to Cure Loss of Control Voltage Instability .................................. 70 Unsolved Double Contingency Cases. .................. 71 Load Shed for Case# 8. ......................... 72 Load and Generation Shedding for Case# 11. ............. 73 Load and Generation Shedding for Case# 12. ............. 74 Selected Double Contingency Cases for Solvability problems. ..... 157 Reactive Generation Supply by Fictitious Generators for Case# 1. . . 158 Reactive Generation Supply by Fictitious Generators for Case# 2. . . 159 Reactive Generation Supply by Fictitious Generators for Case# 3. . . 160 Reactive Generation Supply by Fictitious Generators for Case# 4..161 Voltage Set Point, Shunt Cap., and Transformer Tap Control Changes. 162 Voltage Set Point, and Shunt Cap. Control Changes. ......... 163 Voltage Set Point, Shunt Cap., and Transformer Tap Control Changes. 164 Active Generation, and Transformer Tap Control Changes. ...... 165 Active Generation Supply by Fictitious Generators for Case# 5. . . . 166 Active Generation Supply by Fictitious Generators for Case# 6. . . . 167 Active Generation Supply by Fictitious Generators for Case# 7. . . . 168 Voltage Set Point, Active Generation, and Transformer Tap Control Changes ................................... 169 Voltage Set Point, Active Generation, and Transformer Tap Control Changes ................................... 170 5.15 Active Generation and Transformer Tap Control Changes. ...... 171 5.16 Reactive Generation Before and After Optimization ........... 172 1 Voltage Collapse Region, Voltage Control Areas, and Reactive Reserves Basin. ................................... 182 2 Voltage Comparison, Before and After Contingency for case# 1. . . . 191 xi CHAPTER 1 Introduction 1 - 1 Description of the problem Deregulation has brought great opportunities for increased efficiency of production and delivery and reduced cost to customers. Deregulation has also brought great Challenges to provide the operating reliability and security that customers have come to expect and demand from electrical delivery system. In a deregulated environment, the transmission system will provide open access to all suppliers and electric energy CUStOIners. This competitive environment will force the generation and transmission companies to provide and sell their services under market conditions. one of the challenges in a deregulated power system is voltage instability. System failure and blackouts already have been observed in Europe, Japan, Ontario Hydro, New York Power Pool, and lately two blackouts in the west of USA due to voltage inSt ability. Voltage instability is one of the biggest concerns in operating and planning eleCtl'iC power systems before deregulation occurs. In a deregulated environment I Volt age collapse will become much more common. One of the reasons is that power is being transferred, wheeled, and interchanged through hundreds if not thousands of transactions. Other reasons for the voltage collapse are (1) real POW"r is shipped 1 along diflerent paths in different directions than what they were designed for, (2) the rapid changes in power dispatch due to competition of selling power to different customers and the ability of these customers to change their generating company at their discretion, (3) the transmission and subtransmission system were built and compensated to provide stability and security for flow of power supplied from a known set of generators and delivered to a known set of loads, and (4) the absence of the knowledge that there is sufficient reactive reserve in each reactive reserve basins [1] due to the lack of knowledge that additional reactive supply may be necessary. 1 - 2 Motivation and Objective Voltage instability is caused by exhaustion of the reactive supply on one or more gen- erat ors in a subregion, that causes loss of control voltage instability in that subregion. Clogging voltage instability occurs when some subregion in the system can’t obtain needed reactive supply because the network absorbs all the reactive power flowing to that subregion. Increasing transfer, wheeling, interchange of power on transmission lines , and the increasing demand for power can cause both clogging voltage instability and loss of control voltage instability. The power flow problem does not solve (no solution), or one or more eigenvalues of the Jacobian become negative as an indica- tion of voltage instability. Q-V and P-V curves [2] are some of the traditional way of asSessing proximity to voltage instability. The Q-V curve is used to test for voltage i“St-atfility since it determines the maximum amount of reactive supply that can be added to a bus in order for the load flow to still have a solution. The P-V curve assesses the maximum real power transfer, wheeling, and interchange transactions in the s3'stem before the load flow no longer has a solution. Neither of the methods asseSSes the effects of equipment outage nor the kind of voltage instability ( loss of contra or clogging ) that occurs, the cause of instability, and the cure for any voltage ‘rability 1 iarian. Sec literati: eigemiue in 0pc the come? we: syst Acres 533 for irpiex 0 conti 9‘ ' bear instability problem for any particular equipment outage or operating change combi— nation. Section 3.2 provides a review on what voltage instability is, when it occurs mathematically, and the P — V , Q — V curve and minimum distance, minimum eigenvalue and minimum singular value proximity measures for voltage instability. An Open Access System Dispatch is a controller proposed for implementation in the control centers, called Independent System Operators (ISO), of a deregulated power system. The objective, capabilities, and structure for this proposed Open Access System Dispatch are given in [15] but no concrete methodology is suggested for implementing it. The Open Access System Dispatch, as proposed, has a 0 contingency selection module 0 Security Constrained Optimization module This thesis develops a set of optimization problems that will meet the requirements Set forth for the Security Constrained Optimization module and use the Voltage Stability Security Assessment and Diagnosis (VSSAD), developed at Michigan State University, for the contingency selection module. The contingency selection module is a very significant extension of current P ‘ V, Q - V curve, and minimum distance proximity measure tools because these IIleasures only use continuous parameter changes and not the discontinuous changes in Parameter changes. Discontinuous parameter changes are associated with equip- Ialellt outage, large transactions of power between generating companies, or between gelaerating companies and customers. The P — V curves and minimum eigenvalue or Ila‘itlimum singular value proximity measures only assess one mode of instability for one particular eigenvalue at a time where several can experience instability and each is vulnerable to different equipment outage and transaction combinations. The contin- getIcy selection module indicates where, when, why, and corrective action ( operating cha-11ges ) for every mode of instability and for every equipment outage and operat- 3 mg change 1 should also The Voltage {be above c rear The pro re: techno prm’ded b) iris asses pic-aides fiC if accomm tee shoul age- and 0;) Elle. The warrants 33th Hem '35 also be Voltage 1m {manic i1 ill-12mm In this be investig "1312.194 e M in ”Ufa-3; . 0- dflt ing change that produces that particular mode of voltage instability. The analysis should also provide operating and security constraints for each mode of instability. The Voltage Stability Security Assessment and Diagnosis programs can provide all the above capabilities desired for the contingency selection module. The VSSAD is discussed in chapter 3 where its capabilities and advantages are discussed. The proposed Security Constrained Optimization module is even further from cur- rent technology. The Security Constrained Optimization module utilizes constraints provided by a voltage stability analysis for voltage stability problems and dynamic se curity assessment for transient instability problems. The dynamic security assessment provides flow constraints on particular paths or interfaces and these constraints could be accommodated easily. Voltage stability constraints are not easy to obtain because there should be one for each mode of instability and possibly for each equipment out- age and operating change combination that produces or threatens instability for that mOde. The voltage stability assessment in VSSAD provides the structure for these cOnstraints. There are also operating constraints that prevent thermal overload on each network branch and bus voltage limit operating constraints on every bus. There can also be security constraints associated with thermal overload on each branch, bus Volt age limit violation on each bus, voltage instability of each mode of instability, and dynamic instability for each transient stability problem for every equipment outage and transaction combination that can cause any of these problems. Finding a feasible Sollltion to the power system load flow model that satisfies all of these constraints is daunting. Optimization given all these constraints is even more difficult. In this thesis the voltage instability in the deregulated environment system will be investigated. We propose a secondary corrective control as being computed and 111)tiated every 5-30 minutes as part of the Open Access System Dispatch as dis- c118Sed in [15]. The secondary control is actually precomputed for each equipment out age and operating change predicted to produce voltage instability at that update m hi it mud-2113' C0} :izatlon. w0‘ Seem lsse rate mm: 1ijments. mopeimal fa sneer mores large combi mmhgi liftihlilli {Ii :2' the gym 302‘ is {um mid is: M 1‘19 0 ‘filhagE‘ ]n m Oper, “as: e . K l W7] [in ,1. . F. l L§Jf$~ ' I'M. . “If": . interval by the Voltage Stability and Security Assessment and Diagnosis [1]. This secondary control, called Open Access System Dispatch Security Constrained Opti- mization, would correct a specific mode of voltage instability (loss of control voltage or clogging) that is developing because the equipment outage and operating change combination predicted to produce the voltage collapse in the Voltage Stability and Security Assessment and Diagnosis (VSSAD) has occurred and been detected via the state estimators. The switching of capacitors, under load tap changer tap position adjustments, and generator excitation voltage set point changes are determined in an optimal fashion to eliminate the loss of control voltage or clogging instability on one or more subregions in a system for each specific equipment outage and operating change combination. This set of optimized control changes not only prevent or cor- rect voltage instability in the subregion experiencing it but also prevent a cascading inst ability from producing loss of control voltage or clogging instability in the rest of the system. The Open Access System Dispatch Security Constrained Optimiza- tion is formulated to obtain a minimum set of control changes to achieve corrective control for any particular equipment outage and operating change, and also to pos- ture the operating state and control settings on the whole system to help prevent voltage instability from occurring for any of the VSSAD predicted equipment outage and Operating change combination. The minimum set of control changes for each Specific equipment outage and operating change combination predicted by VSSAD to canSe voltage instability would be stored, triggered, and implemented once the state efitiInator detects the occurrence of that equipment outage and operating change com- bination predicted to produce voltage instability by VSSAD. These control changes must be implemented by a local security controller with a sampling and control com— Ina-Dd update rate of 5-10 seconds. The emergency secondary voltage control would be Used to insure stability and security of the system in case the control change cannot be determined for the secondary voltage control using switchable capacitors, under load tap changer tap position, and generator excitation control voltage set points as controls. An emergency secondary voltage control will change or curtail transaction and even curtail load if the secondary voltage control could not achieve stability and security of the system. A review of the literature on the optimization used in dispatch of power systems is given in chapter 4. It discusses the security constrained economic dispatch and the reactive power dispatch problems that simplify the optimal power dispatch into two separate but coupled optimization problems. The difficulties in handling operating and security constraints is discussed. The Benders decomposition for solving a sepa- rate optimization problem for each equipment outage and operating change to correct all thermal, voltage, and voltage instability problems is discussed. Finally, the objec— tives, constraints, controls, and capability of the Security Constrained Optimization module developed in this thesis is given in section 4.4 . A discussion of the exact for- mulation is give in chapter 5 of this proposal and the development and testing of this module will be the principal subject of this thesis. The algorithms required to solve the security constrained optimization problems must be the most capable yet devel- Oped, There have been several developments over the last 15 years that have greatly improved the convergence rate and convergence robustness of algorithms. A review 0f Optimization theory is given in chapter 2 to justify use of the Primal-Dual Loga- rithmic Barrier Interior Point Method to solve the security constrained optimization Problems. 1-3 Literature Review In the last three decades, a number of studies in electric power utilities have laid the groundwork for solving optimal power flow problems. The following is a brief liteI‘a.ture review of some of the approaches which have been developed to solve the he subject. mkhmc 1.3.1 0 The dessiea where the c :er: simple ref to use remark mo hem .OPF] form the use 01 he] of eq' [49.11110 21 Variables 0] the OFF p Three ’33 optimal dispatch problems, and may represent a general overview of the research on this subject. Also, a comprehensive literature review of the optimal dispatch problems can be found in [29] 1.3.1 Optimal Power Flow The classical economic dispatching problem first appeared in the early 50’s [30, 31], where the objective and the nonlinear load flow constraints were approximated by some simpler equality linear constraints on total generation and load to avoid the need to use an iterative approach. These methods are simple and fast because the network model was limited to its simplest form. In early 1960, the work of [32, 33] laid the groundwork of Optimal Power Flow (OPF) formulation. The work of Carpentier attempted a solution method which made use of Kuhn-"flicker necessary condition from nonlinear programming to obtain a. set of equations that provide candidates for an optimal solution. The equations take into account the load flow equations and constraints on the state and control variables of the load flow model. In subsequent work, Carpentier [34] tries to solve the OPF problem by using the generalized reduced gradient method. Dommel and Tinney [35] attempted to solve the Kuhn-Tucker conditions using the gradient method With a penalty function to handle nonlinear inequality constraints. This work has the a’dVEEI-ntage of a fixed formulation. The work of Carpentier’s and Dommel et al. was considered to be the most popular in the OPF research area for many years. In [36, 37] linear models were developed to approximate the first and the second order infol‘Iliation of the objective function and constraints. These model approximation lnet110d were applied in several papers in the 1970’s [38, 39, 40]. In [41], important improvement were proposed to make use of the fast decoupled load flow model and the sparsity technique. In the above methods, the convergence behavior, however, proVed to be much more difficult and erratic than was initially anticipated. Other 7 [ankles tttelUC ashram?“ restraints SO l method :41 45' rethetl provide 3'35 still not g1“ adiEeult and u muld n0t be in; New methods These Quasi Nev at the Hessian it useful only for pt updated at each el. 3%. l9". \1'. tithing the 0 smug t‘n Wheat c0] will] BUIC: 1‘51 and se lllfiftlon. C Epdated a: hmme 51m 8/ a. Lima/)2. a {1 c hwy/4666.? 0p 554“. "h ' imtt} tech», ht. ' «he \s her that difficulties include; (a) the need for solving the load flow solution at each iteration which required significant computation and (b) ineffective handling of the inequality constraints so that convergence was problematic. In 1970’s a Newton’s technique method [42, 48, 49], was applied to the optimal power flow problem. This Newton method provided excellent local convergence properties, but its global convergence was still not guaranteed. The difficulty in handling inequality constraints was still a difficult and unsolved problem. The computation time remained high and thus it could not be implemented on large scale power system problems. New methods proposed in 1980’s [45, 46, 47], were based on Newton method. These Quasi Newton methods use an iterative scheme based on an approximation of the Hessian matrix, which is calculated at each iteration. These methods are useful only for problems of limited size because the reduced Hessian matrix must be updated at each iteration, and because they form a dense Hessian matrix. Burchett, et al. [48, 49] have reported the formulation and implementation of several methods of solving the optimal power flow problem. In [48], a Quasi Newton method is used for optimizing the subproblems which are transformed from the original problem. The nonlinear constraints are linearized by using the Newton Raphson J acobian matrix. In [49] Burchett creates a sequence of quadratic subproblems from the exact analytical first and second derivative of the power flow equations and the nonlinear objective futleizion. The dimension of the Hessian matrix was not fixed in these method and was updated at each iteration which makes the algorithm require significant computation for On-line implementation application in a power system control center. Sm et a1. [50] and later in [51] solve the classical OPF by decoupling the problem into active and reactive power problem using a Newton approach. The method uses Kuhn—Thicker optimality conditions, produces quadratic programming problems and Uses Sparsity techniques. The methods converge to the Kuhn-Tucker optimality con- ditions in few iterations if a set of binding inequality constraints is predetermined. The major cl rants. Sort to solve the 0 method. :53: 1111954: were first in: the early 70' ttnlhteat 0p 1.3.2 In [he interior 3133M pow the linear Pitt lletht SMil-C01 Pmmbala scheduling ( ’hg Studies 1 ’E’IW'BOn ’0 all)? the The major challenge in Sun’s algorithm is in identifying the binding inequality con- straints. Some other authors have also used real-reactive decompositions of the OPF to solve the optimization problem using approaches that use [52] a linear programming method, [53] a quadratic programming method, or [54] a gradient method. In 1984 a new method, called the Interior Point Method [27] was introduced for solving a linear programming problem. The Interior Point Method have been applied to solve large scale linear optimization problems [55]. Although, these method were first introduced into nonlinear programming by Fiacco and McCormick [56] in the early 70’s, only recently has the theory matured to provide methods for solving nonlinear optimization problems [57 , 18, 76]. 1.3.2 Interior Point Method The Interior Point Method has proven to be a feasible alternative for the solution of optimal power flow problems. In the last five years several papers were proposed to solve linear and nonlinear programming problems in power system using an Interior Point Method. Vargas, et at. [71] used a dual-affine scaling algorithm to solve a Security-Constrained Economic Dispatch problem by sequential linear programming. P annambalam, et al. [72] used a dual-affine algorithm for the optimization of hydro scheduling Operation which is a large scale linear programming problem. Both of the Studies showed that the computational results favor the dual-affine algorithm in comparison to the MINOS simplex code. Lu,et a1. [73] applied Karmarkar’s algorithm to Solve the linear contingency constrained security dispatch problem. Clements, et al. [74] applied a primal-dual logarithmic barrier interior point method to solve a power system state estimation problem using the Lagrangian function and the Hessian Inat‘el‘ix. Momoh, et al. [14] presented an extended quadratic interior point method, baSGd on an algorithm for improvement of initial point for solving linear and quadratic proSlamming problems. The applt let problem the s'atlt tat atc‘ the use slatita'tahlt ltt op: miza‘ nthizattor hates the p Battle: to s hlillflé‘ar un :22 and dua ‘56 the inte h 57 p: fi‘ aptinal lit: uses .\'e l“ '3‘ also littlem, BC l’e alghtith jig‘36le 3p] Elle r‘321Ctit' “Her alggr L 333% “title ; The application of the interior point method algorithm to nonlinear optimal power flow problem consists of three crucial steps [75]. The first step consist of introducing the slack variables to transform the inequality constraints in to equality constraints and the use of Fiacco and McCormick’s logarithmic barrier method [56] to add the slack variables to the objective function as soft constraints. Using Lagrangian ftmction for optimization with equality constraints in the second step converts the constrained optimization problem to an unconstrained optimization problem. This almost elim- inates the problem of handling inequality constraints. Finally, applying Newton’s method to solve the Karush-Kuhn-Tucker (KKT) first optimality condition of the nonlinear unconstrained optimization problem provides quadratic convergence in pri- mal and dual variables. Application of nonlinear programming worked by [57, 18, 76] uses the interior point method to solve optimal power flow problems. In [57] primal-dual logarithmic barrier algorithm is directly applied to a nonlin- ear optimal power flow problem by using Pure Primal-Dual interior point algorithm that uses Newton’s method to solve the Karush—Kuhn-"Ilucker optimality condition. Wu [57] also used Predictor-Corrector interior point algorithm to solve the nonlinear problem. Both methods were based on a method suggested by Mehrotra [77]. A sim- ilax algorithm was developed in parallel with the Wu’s one by Granville [18] with a different application. Granville uses a Primal- Dual logarithmic barrier algorithm to SOIVe reactive dispatch problem. Torres, et a1. [76] applied Primal-Dual logarithmic barrier algorithm to solve a large scale nonlinear programming problem using both a 1"3Ct8ongular and polar variables. 13.?» Voltage Instability Voltage instability has become the principle constraint on power system oper- ation for many utilities [58]. Many blackouts have affected the Pennsylvania, New Jersey, Maryland Interconnection, the Western System Coordinating Council 10 tt‘SCCt 53’ 5'1 50 61. [11066911 5C hg voltage safer horn Recentl) ml; tn the he have b histentittl z grates 0m 3th 69 at the h‘h “cat't he; that s s a htttttea and hfiete‘ (WSCC) system, Florida, France, Sweden and Japan. In the 1980’s several author [59, 60, 61, 62, 87] investigated the voltage instability problems. These investigation provided some knowledge of the development, propagation, and some factors caus- ing voltage instability. Despite the knowledge gained, voltage collapse scenarios still suffer from a lack of knowledge of modelling and understanding of the problem. Recently, voltage instability has received an increasing attention [63, 64, 65]. The work in these articles and in the report [66] have been done to study the bifurcations that have been found to be one of the primary causes for voltage instability in a differential algebraic power system model. It has been shown that bifurcation se- quences occur in a differential algebraic model that can include saddle- node [67, 68], H0pf [69], and chaotic[70] bifurcation. Instability in the dynamics can occur before the bifurcation occurs in the algebraic model [3]. Furthermore, Schlueter at al. [9] Show that saddle-node bifurcation in a differential algebraic model at equilibrium is a bifurcation in the load flow model that includes both the algebraic submodel and differential submodel at equilibrium. In [11] a bifurcation subsystem method is defined that identifies the subsystem that not only experiences but produces the voltage instability observed in the load flow model in a differential algebraic model. Sdlllleter in [11] determines the conditions for bifurcation to occur in each bifurcation subsystem that can experience voltage instability in load flow model. Two eigenvalue eStimates that bound the bifurcating eigenvalue associated with the bifurcation sub- System were derived. The two conditions for a bifurcation subsystem to exist are that bifurcation occurs nearly simultaneously in the subsystem and full system mod- 6:13. The two eigenvalue estimates are shown to respectively measure satisfaction of these bifurcation subsystem conditions. The theory provides theoretical justification 0f the diagnostic procedures in the voltage stability security assessment and diagnostic (VSSAD) methods. There are several books that discuss voltage stability. Kundur [23] is the most 11 mmhem assome 0ft Sawyf whimd uflyusu iflsbnd i producing complete in describing the modeling required to perform voltage stability as well as some of the algebraic model based methods for assessing proximity to voltage instability. Taylor [24] provides a tutorial review of voltage stability, the modeling needed, and simulation tools required to perform a planning study on a particular utility or system. Van Cutsen and Vournas’. [25] provide the only dynamical system discussion of voltage instability and also, show the various dynamics that play a role in producing voltage instability. 12 .5. :exiew of inflation i 9‘91“ in Ch mmMm- interior poi 2.191th (p. ilgarithm 59]] USE il 90939316111 9:19.13], 50999011 1 59999991119 the Dual a measure to: [36911 a 99313 :0 be 93: T", " ‘9 [5 CHAPTER 2 Interior Point Method A review of optimization theory in general and the interior point algorithms since op— timization is central to the development of the preventive and corrective controls pro- POSed in chapter 4 and 5 of the thesis. The focus of this discussion is to justify use of an interior point algorithm as well as the particular algorithm used in the thesis. The interior point algorithm can be divided in to three main methods: the affine-scaling method (Primal affine and Dual affine); the projective method such as Karmarkar’s algorithm; and the path-following method and the potential-reduction method, which bOth use the Primal-Dual algorithms. We use in this thesis the Primal-Dual interior point method that is based on use of a barrier function in the performance index. The P rinlaJ-Dual interior point method has been particularly successful in practice. The bound on the number of iterations is on the order of 05/771), whereas in the affine- Scaling method the bound on the number of iterations for both the Primal affine and the Dual affine is on the order of 0(nlz) [26], where n is the number of nodes and l is a measure of the length of the input data for the problem. The projective method has not been as successful as the other two interior point method. Furthermore, it ap- Pears to be slower and less robust in computational tests. Computational experiments [75’ 77, 78, 79] showed that Primal-Dual algorithms also performed better than the 13 other interio performed b [ems This < the Primal-l 2.1 Li A linea: prt: an: inequah other interior point methods on theoretical and practical problems. The Primal Dual performed better than the Simplex method on large-scale linear programming prob- lems. This chapter briefly reviews the linear and nonlinear programming and then the Primal-Dual algorithm of the Interior Point Method will explained in detailed. 2. 1 Linear Programming A Linear programming problem has a linear objective function with the linear equality and inequality constraints. The linear programming problem has the form [26]. Minimize F(a:) = cTa: subject to A2: = b P (2-1) a: 2 0 J with b 2 O . Here a: and c are vectors of length n, b is vector of length m, and A is an m x n matrix called the constraint matrix. The feasible region of a linear programming problem is defined by its linear Performance index and its linear constraints that forms a convex set. A point x is a SOlution to the problem if it satisfies the equality constraints, and the columns 0f the constraint matrix corresponding to the linear components of a: are linearly independent [26]. The point a: it is a feasible solution (extreme point) if it satisfies the equality constraints and non negativity constraints, and it is an optimal solution if it minimizes F(:1:) over all feasible 3:. The Simplex method is a classical method for s01ving a linear programming written in the standard form. It is an iterative mehhod that moves from one feasible solution (extreme point) to another as long as the objective function improves. At each iteration the components of feasible Selution a: are separated into two vectors, one consisting of all zero components which 14 are when} 1h exponents. hthen pert; tween son: phce when a paint. The aim} to another firemenze the procedu 5110 direct 2.2 N 9 XOthea: Einstraints. are called the n — m nonbasic variables (EN , and the other consisting of nonzero components, which are called the m basic variables :1: B. The test for optimality is then performed to see if there exist any feasible decent direction. An exchange between some components of basic variables 23 and nonbasic variables 131v will take place when a feasible solution moves from one extreme point to an adjacent extreme point. The simplex method moves from one set of binding constraints 2:,- = 0 i E I,- to another 2:,- = O i E I,“ looking for an optimal set of binding constraints that characterize the optimal solution. The difficulty with the simplex algorithm is that the procedure does not guarantee convergence to the optimal solution because there is no direct convergent search for the set of binding constraints. 2.2 Nonlinear Programming A Nonlinear programming problem has a nonlinear objective function and nonlinear constraints. The problem that can be written in the general form [26] Minimize F (:13) A N [\D V subject to G,(a‘) = 0;i E E H,(a:) Z 0;i E I where: a: E Rmxnis the vector of decision variable that include both control and state variables, that is a: = [U X] U E Rm is the vector of control variables X E R" is the vector of state variables E is a set of equality constraints I is a set of inequality constraints 15 lhesoh solution is I .~ ,. IEEé‘tDlE ttua 213:”; a pot: ta! (359 W] Mann 1. Prtn: ‘3. Dual 3. Pena 9 Bart 5- Intel 999 are d} 211 nipmb Chitin a [ thre A The solution to the constrained problem is a local solution. However, the local solution is also global solution if the objective fimction is convex function and the feasible region is convex. Unlike a linear programming where the feasible movement 60111 a point to a nearby point along a feasible direction, the movements of a nonlin- ear case will be made along a feasible curve. Four important methods of nonlinear programming solution techniques are commonly used: 1. Primal Method 2. Dual Method 3. Penalty Hmction Method 4. Barrier Function Method 5. Interior Point Method and are discussed in the following subsection of this section. 2.2.1 Primal Method The problem( 2.2) is known as the primal problem since it directly searches for :13. Often a Lagrangian is formed £03, AM“) 2 F03) - Z AiGi(m) — ZMHKQ» (23) iEE iEI Where ,\ and [1. are Lagrange multiplies. Kuhn Tucker conditions for the optimal solution requires 8£(m,/\,u) _ T _ o (2.4) Bax, Mt) ___ 0(3) = 0 (2.5) 6A 16 p.,-H,(a:)=0; i=1,2,---,I (2.6) Mi 2 0 (2-7) The condition ( 2.6) is the complementary slackness condition and states either p,- or H,(z) must be zero. The determination of a: and A can be determined for any value of u that satisfies ( 2.7) either by solving the gradient equations ( 2.4 , 2.5 ) analytically or by a Newton Method that requires finding a Hessian matrix £2z($1 A, u). The difficulty with this primal method is that there is no direct convergent search for u and no assurance of feasibility[x; G,=0Vi€E, H,ZOViEI]. 2.2.2 Dual Method The dual problem is a min — max problem most,“‘ [mint [£(m, A, u)]] (2.8) that optimizes on both A, u and 3;. Thus there can be convergence in u as well as a: and A. If the performance index F(:c, u) is convex and the constraints H,(:c, u) are concave, the solution to the dual is the solution to the primal [26]. There can be a duality gap between the primal solution and dual solution if the problem is not a convex programming problem. Another condition for lack of a duality gap requires the Hessian 325(3. A, 11) W (2.9) to be defined at all value of (2:, A, u) and be positive definite at (:v‘, A“, ’2‘) which is dual feasible. There are two popular algorithms for solving the primal problem. The gradient 17 leiéffllh gafien‘. attain“ «9 9c _N‘r‘ .-¢_--. ‘ Jain: LA Cl determines a gradient of L(:c, A, u) with respect to a; and A. The generalized reduced gradient method (GRG) reduces the dimension of the primal problem by using the equality constraints C(23) = O to solve for the state variables and thus eliminate these equality constraints and the Lagrange multipliers associated with the equality constraints 3—5 that contain A. The Newton Method solves ( 2.4, 2.5 ) using the Newton Rapheson Method for solving nonlinear equations is a second method for solving the primal problem. The Newton Method requires computing the Hessian 95%- where 2T = (xT, AT) and thus assures quadratic convergence in z to a solution. The reduced gradient method and Newton Method do not optimally search for the subset of optimal binding constraints that either satisfy rim," 2 2:,- or 113i...“ = 1:,- and the vector u as the dual algorithm does. There is no assurance of convergence to an optimal set of binding constraints and the GRG and Newton Method often do not converge on large nonlinear problems such as the Reactive Dispatch Problem. It is desired that one should find feasibility and quadratic convergence in selecting the binding constraints and parameters u. There are several approaches to assuring feasibility. The penalty function method and the barrier function method are two popular approaches and they are now discussed. 2.2.3 Penalty Function Method The penalty function method have been used over the past three decades to solve the nonlinear constrained problem. The main idea behind the penalty ftmction method is to transfer the constrained problem into a single unconstrained problem or a sequence of unconstrained problems, with the introduction of a penalty whenever a constraint is violated. The penalty function is only applied when the solution is infeasible. The penalty ftmction penalizes the lack of satisfaction of a particular inequality constraint, but has no value if the constraint is satisfied. The penalty function is placed into the objective function via a penalty parameter that can be used to insure that the penalty 18 .42.; I: L is 5.... humor again The per Mischa is suficiently large to correct violation of any inequality constraint. A suitable penalty function must also incur a penalty for violation that increases dramatically with the magnitude of the violation, which forces the solution toward the feasible region. To motivate use of penalty function, consider the following problem (here we repeat problem of eq: 2.2 ): Minimize f (x) subject to G,(z) = O;i E E > (2-10) H,(a:) Z O;i E I J The performance index function with penalty for nonlinear problem( 2.10) is f($) = FUD) + 111/1(2) (2-11) and ib(m) is referred to as the penalty function. The following unconstrained problem is expressed as Minimize FCC) + mph”) 1 (2,12) subject to a: E R" where u is a large positive penalty parameter, and 12(3) is continuous penalty func- tion. 1/J(.’B) is a continuous function that penalizes any violation of constraints, with the property that 30(3) 2 0 if a: is feasible (2.13) ib(:v) > 0 otherwise and is defined in the general form: ’l/J($) = 1/3 g(a:)Tg(:1:) + Zminimum {0,h,(.v)} (2.14) £61 19 The snltt preh'ten bl ah iteratic penalty fnnt Onthe other set bounds; penalty funt panels par; of the amtin tntanstraint The pens nth a huge ha simple . guarantee qt 9"? I to G mud the 1 MES it Yer ‘Lfi‘Cllle fur [littlon pro. 112‘? feasible 1 2.2.4 Be The banter f l... . ‘3‘ [0 a Seri thel Willie) l‘. uh; Under funCl l The solution can be obtained to be arbitrarily close to feasible region of the original problem by choosing u to be sufficiently large. The parameter [2 is increased after each iteration until the resulting solution is feasible. The main advantage of using the penalty function method is that each iteration is not required to be strictly feasible. On the other hand, the penalty function method will force the iteration toward feasible set boundary, but not necessarily and not generally toward the optimal solution. The penalty function method also suffers from a problem of ill conditioning. As the penalty parameter increases ( u —> 00) to enforce feasibility, the Hessian matrix of the auxiliary function will become ill-conditioned near the solution, causing the unconstrained problem to become increasingly difficult to solve. The penalty function method at least achieves feasibility of solution for problems with a large number of equality and inequality constraints. Adding a penalty function is a simple method for guaranteeing feasibility and for some penalty function can guarantee quadratic convergence to a feasible solution when u is large. However, if p. —> co to enforce feasibility the penalty function appears as a infinitely high wall around the feasible set that assures feasibility. The ill conditioning of the Hessian makes it very difficult to find an optimal solution within the feasible set since the objective function appears nearly flat over the feasible region compared to the penalty flmction produced wall. There can be some searching for the optimal solution within the feasible region but often no convergence to an optimal solution. 2.2.4 Barrier Function Method The barrier function method uses a barrier function to transform a constrained prob— lem to a series of unconstrained problems just as a penalty function method does. The barrier function method requires starting from a feasible solution and adding the barrier functions to prevent leaving the feasible region. The barrier function method 20 Minimize 6(a) (2.15) subject to u > 0 (2.16) where so.) = in 12,, { f(a:) — ATG,(1:) + 23(3)} and a: 6 {11(3) 2 o ,G(a:) = 0}. The barrier function is one that approaches infinity as the boundary of {3: ; H,(a:) Z 0} is approached from the interior. Possible barrier function are N 1 81(2)) = 2111411?) (2.17) an) = —§i1n{H.-x( )} (2.18) cl". L The optimization of @(u) occurs for values of decreasing u that allow the solution at), to approach the boundary of the set {x ; H,(:c) Z 0}. The barrier function cause the Hessian to experience serious ill conditioning and round off error when u is small but when u is large these problems disappear. The barrier method finds a feasible solution at every iteration rather than at the final iteration. The barrier function ill conditioning problem increases as [2 decreases with a barrier ftmction method just like penalty function ill conditioning problem in- crease as feasibility is starting to be assured. The barrier function can optimize within the feasible solution region that grows as u decreases. The ill conditioning of the Hessian matrix rules out use of an unconstrained method whose convergence depends on the condition number of the Hessian matrix. Therefore Newton type methods are usually the choice. The Newton equations are also sensitive to the ill conditioning of the Hessian matrix. The numerical errors can result in poor search directions. The ill conditioning of the barrier led to their abandonment in the early 19703. Interest in 21 terrier ft method I see of :l nqne s $123!: 6‘ barrier function method was renewed in 1984 with the announcement of Karmarkar’s method [27]for linear programming and the discovery that this method is a special case of the barrier method (No ill conditioning occurs in a linear program which has a unique solution.) Recently specialized algebraic techniques have been developed that compute a numerically stable approximate solution to the Newton equations. 2.2.5 Interior Point Method The interior point algorithm for nonlinear programming problems are motivated out of Karmarkar’s algorithm for linear programming problems. The interior point algo— rithm in contrast to simplex algorithm (and nonlinear programming algorithms such as Newton and GRG) do not move from one set of binding constraints to another. The interior point algorithm moves from point to point interior to a feasible region. A primal-dual interior point algorithm is now formulated for the problem Minimize F(z) (2.19) s.t. C(z) = 0 (2.20) l g 2 S u (2.21) The problem is first reformulated by introducing the slack variables 31 and 32 and the barrier function Minimize F(z) — u E": ln(slj) — [22“: ln(82j) (2.22) subject to G(z) = 0]:1 3:1 (2.23) z — 31 = l (2.24) z + 32 = u (2.25) 31,32 2 0 (2.26) 22 The flat" DAG ., : .Q‘I“ The Lagrangian is L = F(z) — ATG(z) —I11(z —51 —l) —H2(z+32 —u) —u 2 ln(sl,~) ——,u 2 111(32j) (2.27) j=l j=l where A, H1 and H2 are the dual variables. The first order necessary conditions are VF(z) — JT(z)A — H1 — 112 = 0 (2.28) G(z) = 0 (2.29) 2 — 81 —l = O (2.30) z + 52 — u = 0 (2.31) [1.6 —' 31111 = 0 (2.32) [1.8 — 52112 = O (2.33) where: VF(z) is gradient of F(z) .7(z) is Jacobian of C(z) A are the Lagrange multipliers E R'" where m is the number of equality constraints H1 and H2 are the Lagrange multipliers E R" where n is the number of state and control variables 51 and 52 are diagonal matrices whose diagonal elements are 31, and 32,- respectively and they are 6 RM" 6: [1,1,---,1]T E R” The Newton equations are generated by taking derivatives of the equations ( 2.28 - 2.33 ) with respect to z, A, II1,II2,s1, and 32 to produce a solution for these variables. These conditions are given in chapter 4 when the interior point 23 algorithm for the OASYDIS is given. This primal- dual problem is thus quadratically convergence in both primal and dual variables. The value of u is chosen based on the duality gap which in this case is equal to $le — sg‘II2 (2.34) and [28] selects u _ Serl — 527.112 n2 (2.35) where n is number of variables. In [18] u is selected as u _ 5311. — 3311. an? (2.36) where ,6 > 1 specified by the user. The control path or barrier trajectory is z(u) : u > 0, along this barrier trajectory. If an affine search algorithm were used and the problem was linear, the primal z and dual II,- and S,- search direction are orthogonal along the barrier trajectory. This linear programming primal-dual can converge in \fli— iteration. It is not as clear that the convex nonlinear programming barrier trajectory has such desirable quadratic convergent properties in both the primal and dual orthogonal directions. It is certainly anticipated that the primal-dual logarithmic barrier interior point algorithm is rapidly convergent in both the primal and dual directions. The primal-dual logarithmic barrier interior point algorithm is used in this thesis. 24 lcl tjwl .l 2.3 Primal-Dual Interior Point Method To apply the Primal-Dual algorithm we consider the problem stated here in the following form [80]: Minimize f(a:) Sub 'ect to : :1: = 0 J g( ) > (2.37) hmin S h($) _<. h'maa: xmin _<_ (I: S xmaz Using slack variables to transform the inequality constraints into equality constraints, the problem of ( 2.37) can be transformed to: Minimize f(a:) Subject to: g(a:) = 0 h(:r) + Shl = hmaa: Shl + 5212 = hmaz — hmin l (2-38) :1: + 3,1 = 93mm, 5:1 + 5:32 = 33mm: — xmin 51:1) 52:2) Sh], 5’12 Z 0 These nonnegative slack variables 8,1, 5,2,Sh1,3h2 in ( 2.38) are eliminated by adding the barrier penalties to the objective function (Fiacco and McCormick’s method). The 25 L] ‘ ‘ it re ’3‘ ‘ hf] \ .'°“j A ' 31 a 7 27.17 f a resulting problem with the barrier penalties is defined as: Minimize f(:c) — ,u 221 ln(S;,1) — u 2311n(3h2) ‘ ‘2‘ 233:1 111(321) “ ll 2:3;1 111(Sx2l Subject to : g(a:) = 0 (a) hm —— h(:c) — shl = 0 (b) l (239) hm, — hm,-n — SM — Shz = 0 (c) 23m” — a: -— le = 0 (d) mm“; - 33min — le — 5,2 = 0 (e) , where m and n are the number of inequality constrained function and the number of the primal variable that have lower and upper bound respectively, p is a positive interior point barrier parameter that decreases to zero iteratively. Based on Fiacco and McCormick’s theorem [56], the solution of ( 2.39) .v(uk) approaches the local optimal solution 3“ of ( 2.37) as u decreases towards zero. We now consider the Lagrangian function to transform the constrained prob- lem ( 2.39) into unconstrained problem. The Lagrangian function is given as: 503, A) = f(1=) — A3903) -)‘hllhma:l: — h(1‘7) — Shll “ Ah2lhmaa= — hmin — Shl — Sh2l —A:1[a:m - :1: —— le] — Affirm“ —- mm," — Sfl -— 322] (2-40) -u 2211 1n(Sh1) - u Z3211 ln(Sh2) —H 231.4 1n(le) - Ii 2}; 1D(Sx2) where Ag 6 R" are the Lagrangian multipliers of constraints ( 2.3.9-a). Ah1,Ah2 E R” and A31,A,,2 E R" are the Lagrangian multipliers of constraints ( 2.39-b), ( 2.39-c) and ( 2.3.9-d), ( 2.39—e) respectively. 26 lhe L a 8‘3. ‘ l first '" "JJJ J33. . The Lagrange multipliers are the dual variables. The dual problem can be formed using the Lagrangian duality concept [80]. Therefore we can state the dual problem in the following form: Minimize L(:c, A) Subject to : V3L(a:, A) = Vsquw A) = VSh2£(x’ Al = V5.15“, A) = V5,,L(r, A) OOOOO l (2.41) The relationship of the primal problem and the dual problem can also be found in [80]. The local minimizer (x‘, A, S ‘) of ( 2.37— 2.41) is given in terms of the stationary point of L, which satisfies the KKT conditions, also known as the first-order necessary conditions. The KKT conditions are defined as following : v.5 v,,c VAML VAML vmc was V5,,L VSML vsflc V53,L Vf(a:) — \7g(;I;)TAg + Vh(a:)TAh1 + A31 = “9(3) —hm + h(a:) + SM —hmaz + hmin + Shl + Shz firm“. + a: + 5,1 —$maz + 33min + 5:1 + 5:2 AM + Ahg — uSglle Ahz — ungle A31 + Azg — [1536 A32 — #536 27 (2.42) | oooooooooo rl. Ton-"‘7'. L J.“— rm ‘7! ,:.JI.:I (53:98 995?? a A I Edi \ ‘k \‘. where V f(x) is the gradient of the performance index and Vg(x), Vh(x) are the gradient of the equality and inequality constraints respectively, A are the Lagrangian multipliers of constraints ( 2.3.9), Sh1,Sh2,S,1, and 532 are diagonal matrices in 6 RM" whose diagonal elements are shlj,sh2j,s,,1j, and 822, respectively. 6 E R", e = [1, l,....,1]T The above set of equation can be solved using Newton’s method since it is inher- ently nonlinear. The solution of f (x) is usually approximated by a single iteration of Newton’s method, since the Newton’s direction is the only means to follow the central path parameterized by [76]. The following iterative equation is obtained ’ Ax ' ”vxc ’ AAg VAQL AAM mac Aim v.,,c [W] AA" =-— V*“£ (2.43) AA... vmc ash, VSML AS,2 V5,,L AS...1 V5,,L _Asfl‘ _Vsfia where [W] is an augmented matrix which will be defined later. at each iteration ( k, we solve the system of equations ( 2.43) for determining the Newton’s search direction Ax,AAg,AA“,AAhg,AA31,AA32,ASM,AS;,2,AS$1 28 and A! I"! .- 29le ' :m " ' :1 =1 . t-m‘afill. 'll 8Y9: lie to and A532), then a new approximation to all variables is obtained as follows : xlkfl) = x(’°)+an Agk“) = A[,")+aAAg Alf?” = A[,’°,)+aAA,,1 All? 1) = AEQ+aAAh2 ASS“) = Ag-l-aAAzl > (2.44) AS?” = A£§’+aAA.2 39;“) = 3,9? +aASh1 5,95“) = 5,93) +425,2 55;“) = affine/5.3,,l 523*” = Sh? +aAS.2, where the sealer a 6 [0,1] is the step length parameter chosen to preserve the feasibility of all the variables. At every iteration step we reduce the barrier parameter ,u and solve the prob- lem to insure fast convergence instead of taking several iteration steps with fixed u. 2.3.1 Predictor-Corrector Interior Point Algorithm Mehrotra [77]developed another procedure called the Predictor - Corrector Primal Dual Interior Point Method. In this procedure he generates correction terms to the current estimate, the new corrected point can then be substitute into KKT conditions 29 GK “.1 in t 2.45 ( 2.42) directly, to obtain; - Ax AAg Ahm AAhz AA21 AA22 A5211 A522 A521 [W] . A532 .. 47,1: —v,,c —v,,,,c —v,,,c —v,,,c —V,,,c uSglle — AM — SgllAShlAAhl p.536 — AM - Ahg - nglAShfiAAhl + AA“) uSglle — A31 — SgllASzlAAfl _ #5342 — A31 — Azz — S;21A5.,2(AA,.1 + AM) ] (2.45) The main difference between the Predictor—Corrector Primal Dual and the Pure Pri- mal Dual algorithms is the presence of the nonlinear terms in the right hand side of ( 2.45). The Predictor-Corrector method take an affine step to approximately solve ( 2.45) where the barrier parameter is set to zero and the nonlinear terms are dropped. 30 .223 The affine step is consists of the solution of the system: Ax VxL AA, VA; AA“ VAMC AAhg VAML'. [W] AA“ = -— VEL (2.46) AAzg VAfll: A5121 Am A522 AM + An A521 A21 _ A522 . _ A21 + M2 ] The solution of the affine step is then used to estimate the barrier parameter and to approximate the nonlinear terms in ( 2.45). Finally, the actual new search direction can be solved for using ( 2.45). 2.4 Computational Implementation The outline for the Optimal Power Flow algorithm may be summarized as the fol- lowing: Step 1: Initialization. In this step we solve for the initial point (starting point) of the OPF problem. The starting point in IPM need to strictly satisfy the nonnegativity condition. However, a strictly feasible starting point is not required. An interior point algorithm will per- form better if the starting point is defined in systematic way. In this study we will estimate the starting point as given by the load flow solution for the primal variables x0. Starting with the load flow solution not only will insure the feasibility and solv- 31 this ‘21: t 'L “ I’n .tr it. fit a ability of the power balance equations, but also the nonnegativity conditions. The slack variables of the primal problem can be chosen arbitrarily; so that 521 + 522 = hm... — hm... (2.47) 521 + 5:22 : xmax — 37min (2.48) the dual variables of the Lagrange multiplier of the equality constraints, Ag, can be set to zero, the other dual variables can be solved for using the following equations, A21 = #5133 (2-49) A22 = #35218 — #3133 (2-50) )‘21 = H 2:113 (2-51) A2. = uSa‘e - “33.. (2.52) Step 2: Forming the Newton’s system. The process of forming the Newton’s system of equations ( 2.43) involves evaluation ’ of the gradient vectors, and the Hessian and Jacobian matrices. The elements of these vectors and matrices are computed and can be found in [76]. However in practice implementation of these vectors and matrices are not actually formed. The augmented Hessian matrix W and the primal and dual variables are rearranged in a way described by the rearranged Hessian matrix, incremental variable vector, and 32 the mismatch vector. 23;: 0 0 0 0 0 I 0 0 0 q 0 )2ng 0 0 0 0 I I 0 0 0 psfi 0 I 0 0 0 0 0 0 0 0 )2ng I I 0 0 0 0 0 0 I I 0 0 0 0 0 0 (2.53) 0 0 0 I 0 0 0 0 I 0 I I 0 0 0 0 0 0 0 o 0 I 0 0 0 0 0 0 Vh(x) 0 0 0 0 0 o I 0 Vh(x)T H, —Vg(x)T 0 0 0 0 0 0 0 0 -—Vg(x) 0 J Step 3: Computing the Newton’s search direction. Good algorithm performance requires an efficient computation of the Newton’s sys- tem. The major computational effort in this algorithm is to solve large, sparse, and symmetrical system of equations. Most of the work in the primal dual algorithm is in the solution of system of this form H, —-Vg(x)T ’ x V —Vg(x) 0 y W (2.54) Symbolic factorization and optimal ordering schemes need to be performed only once at the beginning and can then be used for all iterations [80], since the sparse structure of the system of ( 2.54) can always be preserved. Step 4: Barrier parameter and determining the step length. A critical step in the primal dual algorithm is the choice of the barrier parameter, [1. The value of u is estimated based on the predicted decrease of the duality gap 33 CCflslllhitI he then: for the linear programming problems [75, 77, 79]. The duality gap is defined as the difi'erence between the primal and dual objective functions. In nonlinear programming problems the complementary gap is used to estimate the duality gap instead of the the real duality gap because of the inability of having some of the nonnegativity conditions satisfied and because of some infeasibility of the primal and dual variables. We choose the barrier parameter as given by Wu et a1. [57] for the predictor-corrector primal-dual interior point algorithm, .. 2 .. gap gap 2 _ __ 2.55 ,2 (gap) (20% +m)l ( ) where gdp is the complementary gap when we consider updating the variables in ( 2.45) and gap is also a complementary gap that approximates the duality gap. The variables gap and grip are given as the following: gap = (M1 + Ah2)TSh1 + A£25h2 + (A21 + Az2lTSzl + A2322 (2-56) ~ ~ T ~ gap = [Am + An + (3(AAh1 'l' AAh2)] (SM '9' EYASM) + (AM + E!A)\;,2)T(Sh2 + (iASja) + [A,1 + A,2 + 52(AA;1L + AA;2)]T (5.1 + aAs;1) + (A,2 + 5AA;2)T(S,2 + aAs;2) (2.57) where, . 2 M..{. we.) ... elwz A4 ’(AA,.,+AA,,2)’AA,,2’(AA,,+AA,2)’AA,2’ 5"} , Sh? , 5’3 , S“? } (2.58) AShr A5112 A521 A522 for those (AA;1 + AAis). Axis, (AA; + AA;2), AA;2,AS';.1.ASI.2, Ash, ASL. s o 34 Ll ‘li’ifig‘, :‘Dlen ~1 The step length a is chosen to preserve the feasibility of all the problem variables and is determined as 0(/\h1 + )‘hzl 0M2 0021 + 3:2) 0&2 (AAhl + AA”), AA”, (AAz-l 'l' AAz-g) , AA”, O'Shl USh2 O'le 0532 AS)“ ’ AShg ’ A531, A532} a 2 Min {1, (2.59) for those (AAM + AAhg),AAh2,(AA,,1 + AA”),AA32,ASM,AS),2,AS,,1,AS,2 g 0 and a is chosen to be less than 1. A typical value is a = 0.9995. Step 5: Update variables and check for convergence. The new approximation value to the primal and dual variables are then estimated using ( 2.44) and then the convergence check is performed. The convergence check and the stopping criteria for linear programming problems are usually defined in terms of the relative duality gap [76]. For nonlinear problems the iteration procedures are terminated as both the relative complementary gap and the mismatches of the KKT conditions are sufficiently small[57]. The stopping criteria for the nonlinear problems are as follows; 3 gap 1 + [dobj] El ( ) and [the largest mismatch of KKK] <52 (2.61) where dobj is the dual objective function value and 61, and 52 are the tolerance values. The problem solution is said to have converged when ( 2.60) and ( 2.61) reach their tolerance values, the optimal solution is found, and the algorithm stops. 35 Step 6: If the solution is not found then set the iteration index k = k + 1 and start a new iteration from step 3. 36 ‘5 .i ir‘ \,§ In CHAPTER 3 Voltage Stability Security Assessment and Diagnosis 3.1 Introduction Voltage instability is a very complex phenomena. Use of mid-term transient stability models and simulation tools are required if a reasonably accurate simulation of equip- ment outage or operating change induced voltage instability events is to be possible. These models require differential equation models of turbine energy systems, genera- tors, excitation systems, network controls, and load as well as algebraic equation of the network [3]. The network models must include the transmission network, sub- transmission network, and some aggregated representation of the distribution network over a fairly large geographical region to accurately simulate such events. Finally, one must have an excellent mid-term simulation tool that can accommodate such a large model. Screening for all the subregions that can experience voltage instability as well as the operating changes, equipment outages, and equipment outage and operating changes combinations that can produce voltage instability in each region requires use 37 of a computationally fast simulation tool. A simpler model and a computationally fast simulation tool is needed since the computation per equipment outage and operating change combination using a mid-term transient simulation tool can be quite large and since there are a huge number of equipment outages and operating changes to be studied. Load flow has been found to be a remarkably accurate tool for assessing voltage instability despite its many modeling, algorithmic, and control shortcomings. Voltage Stability Security Assessment and Diagnosis (VSSAD) should determine most if not all equipment outage, operating changes, and all the contingencies that cause voltage instability. VSSAD can also determine the cause of the voltage insta- bility in terms of lack of reactive supply on specific reactive sources or an inability to deliver reactive supply to the specific region experiencing voltage collapse. It can also indicate what operating condition changes and control changes could be made in order to prevent the voltage instability from occurring when contingency and operat- ing condition changes combination predicted by the state estimator to cause collapse occurs. Voltage Stability Security Assessment and Diagnosis (VSSAD) method is used to find voltage collapse regions and subregions that have unique voltage collapse prob- lem, the reactive reserve basins protecting each voltage collapse region and subregion from voltage collapse. The VSSAD method simulates all the contingencies that are most likely to occur and than find out all the single and double equipment outage contingencies, that are responsible for voltage collapse in each voltage collapse region and subregion, and the voltage collapse region that are most vulnerable to voltage instability for contingency. 38 3.2 Voltage Stability Overview Voltage stability is the ability of a power system to preserve the voltage of an operation equilibrium under normal condition and to maintain an acceptable voltage at all buses after being subjected to a disturbance. A system will start to lose stability and enters the state of instability when a disturbance, changes in system operating condition, or increase in load demand causes a progressive and spreading drop in voltage. The incapability of the power system to meet the reactive power demand is the main cause of voltage instability. The drop in voltage results in (a) reducing shunt capacitive reactive supply and (b) increasing magnetic field due to increased current flow that together increase the network reactive losses. The increased network losses result in ( 1) reducing reactive power flow to the region that needs the most reactive supply and (2) exhaustion of the reactive reserves on generators, synchronous condensers, or SVC’s causing loss of voltage control that result in further voltage drop and further increase in network reactive losses. Voltage collapse has been studied in a load flow (algebraic) and in a differential algebraic model. It has been shown that bifurcation sequences occur in a differential algebraic model that can include saddle node, Hopf, singularity induced, or algebraic bifurcation. Instability in the dynamics can occur before the bifurcation affects the algebraic model. It has been shown that saddle node bifurcation in a differential algebraic model at equilibrium is a bifurcation in the load flow model that includes the algebraic model and differential model at equilibrium [81]. In other cases, the bifurcation solely in the algebraic model has no affects on generator dynamics (algebraic bifurcation) or alternately in the algebraic model that produces very rapid changes in generator dynamics(singularity induced bifurcation). The bifurcation in the algebraic equations is almost always associated with the ultimate blackout even when saddle node or Hopf bifurcation initiates the instability that results in 39 blackout [3] This thesis will only discuss the voltage stability problems in an algebraic model, f (x, B) where x is the n dimension state of the model and is of the same dimension as £(x, P) and p is an m vector of parameters that can change and produce bifurcation or instability if p changes continuously. The implicit function theorem can indicate when solutions exists and the solutions are unique. Theorem (Implicit function theorem) [4] Let f = (fl, - - - , f") be a vector valued function defined on an open set S in Rn'l'k with values in R“. Suppose f 6 C1 on S. Let (x0; p0) be a point in S for which f(xo;po) = 0 and for which the n x n determinant det[fx(x0;p0)] 75 0. Then there exists a k-dimensional open set Po containing p0 and one, and only one, vector-valued function g, defined on P0 and having values in R“, such that a) g 6 C1 on P0, h) 8(Po) = X0, c) f(g(p);p) = 0 for every p in P0. When the J acobian is nonsingular the implicit function theorem indicates there exist solutions that are unique for all pg 6 Po. When a solution exists, the system may be stable or unstable depending on whether there are any non positive eigenvalues of the J acobian f,(xo, p0). However, when no solution exists the system is considered unstable and this singularity of load flow J acobian can be used to detect voltage instability. When the det[f,(xo,po)] is zero or the Jacobian f,,(xo, p0) is singular, the implicit function theorem does not provide any information but it may imply no solution x0 = x(po) exists at these values of po or there are multiple solutions x0, (p0), Both can be true as will be noted later. A number of indices have been developed to 40 r—Q I) —Q n: {he and s; test for load flow bifurcation when p is changed in some direction n p=po+kn (3.1) via change in 1:: until det[f,,(x*,p‘)] = 0. The indices come from tracking the minimum eigenvalue A:(k) using f.(a:(k).p(k))_w(k) = Mkludk) (32) where A;(k) = min,[A,-(k)] or the minimum singular value 0;?(k) obtained using fz($(k),p(k))sz(-’I=(k),1006)) = Wa(k)h(k)VaT(k) (33) where 2(k) = diag[01(k),og(k), . - - ,on(k)] and 0:09) 2 min,[o,~(k)] The singular values o,(k) are the eigenvalues of fz($(klgF(kllff($(kl,P(kll (3-4) and satisfy fz($(kl:P(kllvi(kl = a,(k)w,~(k) (3-5) w3’(k)fz($(klrp(kll = 02(klv:T(kl (3-6) where v,-(k) and w,-(k) are the right and the left singular vectors of o,(k) and are columns of matrices V,(k) and W,(k) above. 41 and l The minimum singular value and minimum eigenvalue are just two of many sensitivity based indices or measures of proximity to voltage instability. The Q — V and P —- V curve are particular scalar ( m = 1) proximity measures where (1) for a Q — V curve the direction 1; vector is a unit vector where the voltage at a bus i is the only nonzero element, It is the real valued negative number that starts at zero and decreases, and —Q,-(V,-) is the reactive load that is added at bus i for each value of V,. The curve Q,(V,~) is the reactive injection at bus i obtained by changing the bus type from a load bus to a generator bus reducing the voltage V,- until Q,(V}) reaches a minimum at Km," with maximum added load —Q,(V§m,n) = ‘Qim... Z 0. The value of (Vim...) Qim...) defines the minimum of the Q — V curve when ‘23,? = 0, that corresponds to the bifurcation point (x‘, p“). (2) The P — V curve can add active power load at a bus i or at several load buses simultaneously P = P0 + knload (3.7) and pick up that power at several generators g = .90 + kngen (3.8) where n is made up of rig”, and mm and both are participation vectors where ZVini = 1- A P — V curve can also result in transfer power from one set of generators g“ to another set of generators g g" = ga+kn“ (3.9) g, — m (3.10) ta) || 42 art" Q: 1’ £5 t). Note ngm, mead, n“, and ii are unit vectors where one or more elements are nonzero and E n,- = 1. The P—V curve plots voltage at some bus i for change in k = Pmtem where it represents system power load change or k = Ptmmfe, represents the total power transfer change. Optimization based methods have been used to calculate Q -— V and P — V curves in [5]. These scalar optimization based methods optimize performance index Q,- to produce a Q — V curve with load flow equality constraints f(:c.p.U) = 0 (3.11) and inequality constraints on voltage controls u. These controls can include under load tap changer tap position, switchable shunt capacitor susceptance, and possibly generator excitation control set points. The P — V curve computed by load flow for varying k = Pmtem or Bram," has all or most of these controls fixed. The P — V curve would optimize P for a particular transfer or wheeling transaction with the same load flow model, same controls u and inequality constraints on controls u. The particular transfer or wheeling transaction is defined via specification of ngm,nzmd,n‘ and ii. In [6], a scalar optimization based method to maximize the reactive power margin when n can be a unit vector with several nonzero elements. The generalized Q — V curve allows added reactive load at several buses in the participation factor normal direction rather than just one as in a typical load flow based Q — V. The approach used in [6] eliminates the active power and phase angle relationship, using active power generation as control, and imposes reactive power limits on the generators. Dobson[7] is first to develop a vector optimization based method that optimizes the normal direction vector n and the loading factor k, which are assumed to be positive real numbers. This paper computes load power at which saddle node bifurcation occurs, that represent the worst case load power parameter 43 (ll-55011 first) (1 _lo variation. The proximity measure IP“ — P0] to saddle node bifurcation, where Po and P‘ represent the current load power and the critical load power respectively, was first noted in this paper [7]. All of these methods [5, 6, 7] assess bifurcation in a single mode due to continuous scalar or vector parameter variation. The methods are thus not applicable to assess- ing proximity to collapse for equipment outage or transactions that are modeled by discontinuous parameter change. These methods also do not take into account the discontinuity in eigenvalues that occur for continuous parameter and discontinuous parameter changes. In many cases the eigenvalue changes due to discontinuities is virtually all the change that occurs in an eigenvalue that approaches bifurcation [10]. The Voltage Stability Security Assessment and Diagnoses [1] 1. determines the number of discontinuities in any eigenvalue that have already occurred due to generator PV to load PQ bus type changes that are associated with an eigenvalue. The eigenvalue is associated with a coherent bus group (voltage control area). The set of generators that experience PV — PQ bus type changes (reactive reserve basin) for computing a Q — V curve at any bus in that bus group are proven to capture the number of discontinuities in that eigenvalue before bifurcation. The reactive reserves on generators in each voltage control area of a reactive reserve basin measure proximity to each of the remaining discontinuities in the eigenvalue required for bifurcation in the agent composed of the test voltage control area and its reactive reserve basin; 2. can handle discontinuous (equipment outage or large transfer or wheeling transaction changes) or continuous change (load increase, transfer increases, and wheeling increases) where the above methods are restricted to continuous changes; 3. can simultaneously assess proximity to voltage instability for all bifurcation 44 7;- 3.3 lift“) res - ‘Ul’Jl ‘ 10. 9‘69. t modes in a system by assessing percentage of generators in a reactive reserve basin with zero reserves and the reactive reserves remaining on reactive reserve basin voltage control areas that have not yet exhausted reserves; . can provide operating constraints or security constraints on reactive reserve basin reserves that prevent voltage instability in each reactive reserve basin in a manner identical to how thermal constraints prevent thermal overload on each branch and voltage constraints prevent bus voltage limit violation at each bus; . the reactive reserve basin operating constraints allow optimization that assures that correcting one voltage instability problem will not produce other voltage stability problems in the rest of the system; . the reactive reserve basin constraints after an equipment outage and operating change combination allows optimization of transmission capacity that specifi- cally corrects that particular equipment outage and transaction change induced voltage collapse; 3.3 Types of Voltage Instability Two kinds of voltage instability have been associated with a load flow model [8]: 3.3.1 Loss of Control Voltage Instability Loss of control voltage instability is caused by exhaustion of reactive supply with resultant loss of voltage control on a particular set of generators, SVC’s, or synchro- nous condensers. The loss of control voltage not only cuts off the reactive supply to a region requiring reactive power supply, but also increases the network reactive losses that choke the network and blocks reactive power supply from reaching that region 45 needing reactive power. Loss of control voltage instability occurs in the transmis- sion and sub transmission system due to equipment outages and operating changes combination, such as 0 line, transformer, and generator outages 0 generator outage with a particular real power generation pickup pattern 0 increase in load and generation a change in wheeling, transfer, and interchange transactions 3.3.2 Clogging Voltage Instability Clogging voltage instability occurs due to reactive power series I 2X losses, tap chang- ers reaching tap limits, switchable capacitor reaching susceptance limits, and shunt capacitive withdrawal due to decreasing voltage. These network reactive losses can completely block the reactive power flow from reaching the region needing the reac- tive power supply without even exhausting any reactive reserve and loss of voltage control on generators, SVC’s, and synchronous condensers. This effect can occur in distribution, subtransmission, and even in transmission system. This effect occur due to increased wheeling, transfer, and interchange transaction changes or loading and generation pattern change 3.4 Knowledge Development There are off-line and on-line aspects of the Voltage Stability Security Assessment. The off-line task is knowledge development via learning through applying stress tests. The knowledge gained from the off-task is then used on the on-line task to assess severity and diagnose the voltage instability problem. The knowledge development aspects are to identify the following [8]: 46 The tan}: 1‘.) (D t)“; that ‘. i 0 Parameters that make a particular region and subregion vulnerable to voltage collapse. 0 The structural cause of voltage collapse. 0 A proximity measure for identifying how close the region and subregion is to voltage collapse. The On-Line aspects of Voltage Stability Security Assessment are; (a) to identify and rank the most insecure region and subregion, (b) to find the equipment outages that cause voltage collapse in any region, (c) and finally to rank the equipment outage that brings the region and subregion closer to face voltage collapse [8]. The first step in the knowledge development aspect of VSSAD is the stress test. In this stress test we acquire knowledge about each agent or subsystem that can experience voltage instability and in each agent the structural cause of voltage collapse in that agent. The stress test will explain why any equipment outage or operating change will cause voltage collapse to occur at any bus or group of buses. Since loss of control voltage and clogging voltage instabilities are both due to shortage of the reactive power supply to a bus or group of buses in the region or subregion, the stress test must determine why and when voltage collapse occurs due to shortage of reactive power supply. Thus, a Q — V curve is used as the stress test for the knowledge development aspect of voltage stability security assessment and diagnosis since it determines the maximum amount of reactive load that can be added to the bus before voltage instability occurs and the load flow no longer have a solution. A P — V curve, although quite useful in assessing maximum transfer, wheeling, and interchange before voltage instability does not relate to shortage of reactive power supply. Another reason why P — V curve is not as effective as Q — V curve is that P — V curve does not effectively pinpoint the region and subregion where reactive power supply shortage occur in the system for generator outages or line outages. This 47 information is easily identified using Q — V curve. A final reason for not using P — V curve is that the minimum singular value of the reactive power J acobian approximates the changes in the minimum singular value of the full load flow Jacobian [8]. The second step of the knowledge development aspects of voltage stability se- curity assessment is to acquire knowledge about each agent or subsystem composed of the test voltage control area and their reactive reserve basin. Computing Q — V curves at every bus in a region and determining the set of generators, that exhausted their reactive reserves (the reactive reserve basin) in the process of reaching the Q — V curve minima at each bus, is needed to identify the agents. The agent is a coherent bus group where all the Q — V curve minima are identical and the reactive reserve basin for every bus in that coherent group is identical. Voltage control areas are the coherent bus groups that have the same Q — V curve minima and the same reactive reserve basin. The algorithm [83, 84, 85] for selecting the size of each non overlapping coherent bus group so all the buses in each group have the same voltage collapse requires the connection between buses in that group have very low impedance. The most important attribute of voltage control areas and their reactive reserve basins is that they don’t change when severe contingencies and operating changes occur that cause voltage instability. Another important fact is that more than one voltage control area can have the same reactive reserve basin. Exhausting all of the reactive reserve in a reactive reserve basin in an agent and thus losing voltage control at all these generators will cause voltage collapse or near collapse in every agent (voltage control area and associated reactive reserve basin). The reactive reserve basin provides the reactive supply needed to prevent every agent with that reactive reserve basin from experiencing voltage collapse. The subset of agents with the same reactive reserve basin is called a voltage collapse region. The exhaustion of all the supply in the reactive reserve basin causes voltage instability in the associated voltage collapse region. Reactive reserve basins and their agents that 48 (EEC 1'th ' Eeze V .jf ’h L’b'u contain them can be classified as global, local, or locally most vulnerable. Global reactive reserve basins are associated with test voltage control areas of the EH V transmission grid encircling different load centers. This global reactive reserve basins generally overlap but usually belong to electrically and geographically distinct region of the transmission system. A global reactive reserve basin can contain one or more nested sets of smaller reactive reserve basins. These are called local reactive reserve basins and contain fewer generators than global reactive reserve basins and their test voltage control areas are either electrically or geographically more remote from gen- erators than the voltage control areas associated with global reactive reserve basins. One or more of these nested sets of progressively smaller local reactive reserve basins can not only cause voltage collapse in the’associated test voltage control area or it’s agent, but also can cause voltage collapse in test voltage control areas of agents asso- ciated with larger reactive reserve basins in which it’s local reactive reserve basin is nested. Such local reactive reserve basins are called locally most vulnerable or critical reactive reserves basins and are more often electrically remote from the larger of the nested set of reactive reserve basins it belongs to and in which its reactive supply ex- haustion causes reactive reserve exhaustion in all of the larger reactive reserve basins it belongs to. The locally most vulnerable reactive reserve basin usually exhausts supply at minima of the Q — V curve computed in the voltage collapse region of the larger reactive reserve basins in the nested set [8] The final step of the knowledge development aspects of VSSAD is to select a proximity measure for voltage collapse in voltage collapse region and subregion. There are two measures of proximity to voltage collapse. The most obvious measure is the percentage of reactive reserve available after contingency has occurred in the reactive reserve basin compared to that of the base case, which often is the peak load case with no contingency. The other measure of proximity to voltage collapse requires the list of generators in the reactive reserve basin that belongs to voltage 49 The cent the : of tl collapse region be grouped by the voltage control area they belong to. This measure is the percentage of a reactive reserve basin voltage control areas that are unexhausted after a contingency. 3.5 Intelligence Development The knowledge development or the learning activity of finding voltage collapse regions and their associated reactive reserve basins must be completed before applying the on- line contingency selection and ranking aspects of VSSAD. The contingency selection and ranking procedure is an intelligence development task because it ranks the worst contingencies that have a load flow solution for each voltage collapse region as well the most insecure voltage collapse region for each contingency. The outline procedure of the On-Line process of VSSAD is as follows [8]: 1. Rank the worst single line and generator outage contingencies for each volt- age collapse region in terms of the smallest percentage of unexhausted reactive reserve of that reactive reserve basin after the contingency; 2. Find and list all of the single line outage and generator outage contingencies that will exhaust more than P% of the reactive reserve in any reactive reserve basin; 3. Simulate a list of double line outages, double generator outages, and a com- bination of line and generator outages contingencies from the list of line and generator outages found in step (2); 4. Rank the worst contingencies produced in step (3) for each reactive reserve basin in terms of the the smallest percentage of reactive reserve remaining in the reactive reserve basin; 50 The contingencies with zero reserves for any voltage collapse region may experience voltage collapse in that voltage collapse region and its reactive reserve basin. If is not possible to say that with zero reserves that the voltage collapse region is experiencing voltage collapse because it may still be obtaining sufficient reactive supply to survive and not experience blackout. With zero reserves in several nested reactive reserve basins, one is virtually certain voltage collapse has occurred. This information can be obtained by observing the ranking of reactive reserve basins for any contingency (step 4). If the percentage of voltage control area with zero reserves is small but not zero for a particular contingency for several nested reactive reserve basins, the system is very near voltage collapse since as exhaustion of the reserves on voltage control areas in a reactive reserve basin occurs, the network reactive losses rise exponentially for each subsequent exhaustion of reserves in yet unexhausted voltage control areas. If is also known that when the reactive reserves in all voltage control areas of a critical reactive reserve basin in a nested set occurs, many if not all larger reactive reserve basins in that nested set also exhaust reserves. This explains why several reactive reserve basins will approach exhaustion and experience exhaustion of reactive reserves simultaneously. 3.6 Diagnosis of Voltage Instability The use of the physical structural knowledge developed of voltage collapse regions and agents (voltage control areas and their reactive reserve basins) provides a basis for performing diagnostics of the location, cause, and remedial action for each equipment outage and operating change combination that cause voltage instability. The physical structural knowledge developed and the diagnostic capability far exceeds that can be accomplished by; (a) doing trial and error effort to obtain a load flow solution by adjusting operating conditions, load and generation reduction, or adding new reactive 51 and gene gEIlf supply sources; (b) finding the last best iteration for a specific equipment outage and operating change combination; or (3) by ignoring the reactive limits on the generators. The diagnosis would suggest whether lack of load flow solution is an algorithmic convergence problem or whether it is voltage instability. Diagnosis can also indicate if voltage voltage stability margin is increased following the control action changes. The diagnostic capability also indicate where, why, and what to do about loss of control voltage instability or clogging voltage instability. Two diagnosis methods has been identified in voltage stability security assessment and diagnosis (VSSAD). One diagnostic method is for loss of voltage control voltage instability, and the other is for clogging voltage instability. The diagnosis for loss of voltage control voltage instability is performed first and if the method fails to obtain the load flow solution, then the diagnosis for clogging voltage instability is applied. There are two cases where the diagnosis will not work; (a) outage of all generators in a reactive reserve basin since additional reactive generation is needed in that reactive reserve basin to obtain a solution; (b) outage of branches that cause isolation of a bus or subsystem. This often requires action that are above and beyond system dispatch of generation and voltage control devices, and require system commitment of additional lines and generators. The diagnosis is thus solely for equipment outages where dispatch of generation or voltage control devices can provide a load flow solution. 3.6.1 Loss of voltage control voltage instability Loss of control voltage instability occurs because the reactive reserves in a one or more agent’s reactive reserve basins are all exhausted due to the contingency. Exhaustion of an agent’s reactive reserves and thus loss of voltage control on all generators in its reactive reserve basin produces dramatic increase in network reactive losses as well as terminating reactive supply from these generators. Ignoring reactive limits in all 52 O "”55? The lack . hi n e the generators in the network Q62...“ S QGi S Q02...” (3.12) and solving the load flow would indicate which generators and which reactive reserve basins exceed their reactive limit causing the reactive reserve basin to be negative. RRUD) = Z (6202...... — Q00 (313) iERRB(P) where: RR(P) is the reactive reserve in reactive reserve basin P, and i E RRB(P) is the set of all the generators in reactive reserve basin P. The reactive reserve basin with the most negative reserve generally causes the lack of a load flow solution. Adding reactive reserves to that reactive reserve basin by: 0 switching in shunt capacitors; o changing voltage setpoints on generator’s exciters; 0 adding additional generators (Peakers) can help the load flow to have a solution. On the other hand if reactive limits are ignored and no load flow solution exists then a lack of load flow solution is due to clogging voltage instability. 3.6.2 Clogging Voltage Instability If the solution to the load flow does not exist even though that the reactive limit on all of the generators are ignored, then the lack of solution is because of the reactive losses choke off the reactive supply flow to a region or subregion needing reactive supply. 53 to re: mail fittiti area hours luster PG The calla] in :h. Methods for obtaining a solution for clogging voltage instability involve (a) trying to reduce the loads and generation (transfer or wheeling transaction reductions or modifications) in agents where collapse occurs until the load flow solves, or (b) adding fictitious reactive power supply (generators) at buses in the agent’s test voltage control area until the load flow has a solution. These procedures can take an average of 15 hours per contingency to find a solution. A diagnostic procedure for clogging voltage instability is as follow: 1. for the double contingency that has no load flow solution find the reactive reserve basin where each single contingency exhausts 50% or more of the reserves. 2. the reduction of real and reactive load at all buses in voltage collapse region and reduction of generation on the generators in that reactive reserve basin will generally obtain a load flow solution. The above procedure works because adding reactive load at buses in the voltage collapse region causes increased reactive losses that together exhausts reactive reserves in the reactive reserve basin, and the contingencies causes increased reactive losses that also exhausts reserves in these same reactive reserve basins. Reducing load should reduce the reactive losses produced by the double contingency that causes clogging voltage instability. In a deregulated environment voltage collapse problems will become much more common. Deregulation of power industry will start to; (a) bring many additional generating station on to the network, (b) allow shipping of real power along different paths and in different directions than what they were designed for, (c) allow a rapid change in power dispatch due to competition of selling power to difierent customers as they change generating companies as often as every hour, ((1) let power be transferred, wheeled, and interchanged. The absence of the knowledge that there is sufficient reactive reserve in each reactive reserve basin due to the lack of knowledge that 54 additional reactive power supply may be necessary will also contribute to voltage collapse in a deregulated power industry. System failure and black outs due to voltage instability already have been observed in Europe, Japan, Ontario Hydro, New York Power Pool, and lately three blackouts in western USA. The changes (a-d) above brought on by deregulation will only make voltage instability a more common and frequent event. In this research, the problem of corrective control for voltage collapse problems in a deregulated power system is investigated. The study of the problems is carried out by computing the minimum set of control devices changes and the most effective corrective control changes for each equipment outage and operating changes combi- nation predicted to cause voltage instability problem by Voltage Stability Security Assessment and Diagnosis (VSSAD). The set of control changes would also posture to some extent against the cumulative threats presented by all of the voltage in- stability causing equipment outage and operating changes. This research is divided into two stages. The first is the development procedure of Voltage Stability Security Assessment and Diagnosis discussed in [8]. The second stage is the development of Secondary and Tertiary control described in [1, 9, 82] using optimal power flow. A revised form of the Transmission Dispatch and Congestion Management [15] could utilize VSSAD and the secondary and tertiary control, but it’s proposed sam- pling rate is far too slow to even detect occurrence of the voltage instability caused by equipment outage and operating condition change combinations until after voltage collapse has already occurred. Enthermore it’s control update rate is far too slow to correct for the voltage instability problem that is developing due to occurrence of a particular equipment outage and operating condition changes combination. Fi- nally, there is no proposed method within the Transmission Dispatch and Congestion Management [15] that could predict all single and multiple contingencies that cause voltage instability and an associated corrective control for each contingency. Selecting 55 asin 9055‘ sen tram whet ‘Ar . U] 01 {I state for t a single control uo as proposed to correct every possible voltage instability, thermal and low voltage problems caused by equipment outage and operating changes as pro- posed by [15] (1) could have no feasible solution, (2) may be impossible to determine even if such a feasible solution exists, (3) may require large significant and costly transaction changes and (4) may require load shedding as a continuous precaution whenever voltage instability is even remotely possible. A VSSAD based Transmission Dispatch and Congestion Management requires 5 second sampling and control update rate, a fast state estimator using the fast 5 second sampling rate to detect contingencies quickly, a fast 5 second control rate to implement corrective control computed proposed by a Tertiary Control for the equipment outage or transaction change predicted by VSSAD to cause instability and detected by the state estimator. The corrective control would come from a set of all corrective control computed and stored using the Tertiary Control to be used by the dispatcher for possible later implementation via secondary control when the state estimator detects the occurrence of equipment outage and operating change combinations predicted to produce voltage instability by VS SAD. The corrective control computed by the Tertiary Control is an optimal version of that proposed by the diagnostics of VSSAD for that contingency and operating change condition. 3.7 Numerical Results The Voltage Stability Security Assessment and Diagnosis is carried out using the following; (a) AVCASP program to compute the voltage collapse region, voltage con- trol area, and reactive reserves basin; (b) CONRES program to perform the single and double contingency analysis; and (c) a PTI load flow to perform the diagnostic analysis in a non-automated fashion. The test is done on a 162 bus system. This system represent a reduced model of 56 Iowa, Nebraska, Minnesota, and S. Dakota system and was obtained from a University of Washington data base. This model was the only model where a subset of generator had finite reactive limits and thus could experience loss of control voltage instability. To obtain alpha for identifying voltage collapse region and reactive reserve basin Q — V curves were computed at every bus in the model with the voltage rating above 20 K V. The test value of alpha (1 obtained from AVCASP, that would make every bus in each voltage control area have the same Q — V curve minima and the same set of generators in their reactive reserves basin, was found to be a = 0.2224. The knowledge development task of finding voltage collapse regions, voltage control areas, and reactive reserves basin associated with each voltage collapse region is provided as outputs of AVCASP. The results are shown in Table 1 in the Appendix. The voltage collapse region number is given in column 1, voltage control areas in each voltage collapse region are given in column 2 ( bus # and bus name ), and column 3 represents the reactive reserves basin associated with each voltage collapse region. Having knowledge about the coherent bus groups or voltage control areas that experience unique voltage collapse problem will help developing the single and double contingency analysis. The contingency analysis is performed by ranking N = 5 worst contingencies for each reactive reserve basin. The contingencies are ordered and presented in term of the largest percentage of the base case reactive reserves exhausted or by each contingency. The contingency ranking for voltage collapse regions# 35 for single contingencies is given in Table 3.1. The voltage control areas that belongs to the voltage collapse region are given at the top left of the table. Each voltage control area is specified by the number and the name of the bus given. Below the voltage control area is the reactive reserve basin. The reactive reserve basin is specified by specifying each of the generators that belong to it in terms of the generator bus number and name as well as their continuous rating reactive capacity in MVAR’s, their base case reactive reserves in MVAR’s, and the voltage control area number 57 the generator belong to. These voltage control areas are connected by 161 KV lines and the reactive reserve basin at buses 121, 118, 73, and 101 are generating stations that surrounded the voltage collapse region in the 162 system. At the end of reactive reserves basin information, the total base case reactive reserves in the specified voltage collapse region is given. The contingency ranking results for that reactive reserves basin follows the reactive reserves basin information. The contingencies are ordered and presented in term of the largest percentages of the base case reactive reserves exhausted by each contingency. The first column indicates the contingency case number along with description of the contingency showing whether it is a line outage or generator outage. The second column gives the percentage of the reactive reserves of the base case reserves available after the contingency. The following two columns give the generator bus name and its reserves after the contingency. The final column contains the percentage of the reactive reserves in that voltage control area of base case reserves available in that voltage control area after the contingency had occurred. The last column is the most important to indicate how close this voltage collapse region is to loss of control voltage instability caused by that contingency since percentage of voltage control areas with zero reserves is the best proximity measure for assessing the proximity to loss of control voltage instability. There were 291 single contingencies simulated. These 291 contingencies consist of all single line and generator outage on a 162 bus system. Only 90 double contingencies were identified from combinations of single generator and line outages that exhausted all but 25% of the reserves in some reactive reserve basin based on results similar to Table 3.1. Similar results to Table 3.1 are given in Table 3.2 for the double contingency analysis for the same voltage collapse region and the same reactive reserve basin. The double contingencies consist of double generator outages, double line outages, and a combination of generator outages and line outages. Tables 3.3 and 3.4 shows the single and double contingencies respectively that had the most affect on the largest number 58 of voltage collapse regions . In both tables, the first column shows the contingency case number, the second column represent the kind of the contingency (line outage or generator outage), and the third column shows the number of voltage collapse regions where the contingency was among the worst five contingencies. Table 3.3 shows that the single contingencies identified as being among the five worst single contingencies in the most reactive reserve basins make up each of the worst double contingencies in Table 3.4. This is expected from studies of large power system models 3.7 .1 Loss of Control Voltage Instability Twenty seven out of the ninety double contingencies simulated did not have a load flow solution. These twenty seven double contingencies were resimulated and eighteen of them solved when the reactive limits on all the generators in the network were ignored. These eighteen double contingencies were associated with loss of control voltage instability. The remaining nine of these twenty seven contingencies were associated with clogging voltage instability. Nine out of eighteen loss of control voltage instability contingencies caused at least one or more voltage collapse regions to have negative reserves in its associated reactive reserves basins. The results are presented in Table 3.5. The first two columns of the table give the contingency case number and the double contingency description, the last column gives the number of voltage collapse regions with negative reserves in its reactive reserves basin. A summary results of loss of control voltage instability for the one voltage collapse region found to be most affected by each contingency in Table 3.5 is given in Table 3.6. The voltage collapse region number is given in column 1. Columns 2,3 represents the voltage control areas, and the reactive reserve basin associated with that voltage collapse region. The base case reactive reserves in the reactive reserves basin is given in column 4, and the last two columns indicates the percentage of the reactive reserves after the contingency and the contingency case number that caused the instability. 59 The contingency associated with the case number can be found from Table 3.5. The reactive reserve basins in Table 3.6 are nested reactive reserve basin and they form a root of tooth as can be observed by observing the size of the reactive reserve basins and the generators that belong to each as one proceeds down the table. The smallest reactive reserve basins were the most affected reactive reserve basin for the less severe contingencies that requires smaller amounts of additional reactive reserves. Contingencies that require more reactive reserves to be added most severely affected the larger reactive reserve basins in the tooth. The six voltage collapse regions shown in Table 3.6 all had negative reactive reserves in their reactive reserves basins for the nine associated contingencies shown when generator reactive limits are ignored. A diagnostic study is now performed using the PTI load flow. The result is presented in Table 3.7. The first column shows the generator bus numbers in all of the reactive reserve basin most severely affected by these nine contingencies in Table 3.6. The next nine columns represent each of the nine contingencies by case number and the required additional reactive supply needed by each of the generators and finally the total additional supply required to solve the non converged double contingency. The last two rows of the table represents the most affected voltage collapse region numbers and the generators bus number in the associated reactive reserve basin decided as most affected by that contingency. Note that the generators where reserves were added agree exactly with the generators in the most affected reactive reserve basin except that generator 99 was not in any of the reactive reserve basins. Contingencies are thus shown to sometimes requires reserves outside the agents reactive reserve basin that causes voltage instability and sometimes require adding supply at generators in two reactive reserve basins. The load flow was shown to solve each of the nine contingencies if reactive reserves equal to the value of its negative reserves are added to each generator in the voltage collapse region’s reactive reserve basins given in Table 3.7 for each of the above contingencies. Four of the 60 nine diagnosed contingencies were chosen to perform the solvability problem (the corrective control problem) in Chapter 5. The result shows the worst contingencies that affect the larger reactive reserve basins outage two large generators in these reactive reserve basins. Generators 6, 121 and 6, 131 seems to be the most severe double generator outage contingencies. 121 and 131 are large generators with significant reactive capacity and base case reserve and therefore it makes sense that outage of these generators would have sig- nificant affect on loss of control voltage instability. These results suggest that severe contingencies cause increase network reactive losses and or significant reduction in reserves of the affected reactive reserve basin. The results also shows that the double line outage contingencies affected smaller reactive reserve basin and double generator outage contingencies affected the larger reactive reserve basin. Generator outages that affected these large reactive reserve basin were generators within these reactive reserve basins. The results also shows that generator bus number 121 is the most critical bus in the network. This bus needed an additional reactive supply in each one of the contingencies above, except in the case 2 and case 14. In case 2 the gener- ator buses number 6 and 121 were outaged. In case 14 that affected voltage collapse region number 3, generator bus 121 does not belong to its reactive reserves basin. 3.7.2 Clogging Voltage Instability Nine out of the twenty seven double contingencies did not solve the load flow even when infinite reactive supply were provided to every generator in the network. Some of these nine contingencies, listed in Table 3.8 produced clogging voltage instability. The PTI load flow package is used to study these double contingency cases. Reducing real and reactive power flow on paths with large reactive losses and voltage decline that supply real and reactive power to appropriate voltage collapse region will eventu— ally obtain a load fiow solution. The voltage collapse regions, that are most severely 61 impacted by both single contingency components of a double contingency, are voltage collapse regions where all real and reactive load is shed and where an equal amount of active generation is shed. This load and generation shedding is performed one at a time for each of the voltage collapse regions ranked as most severely impacted by the double contingency (based on the results from simulation of its single contin- gency components). This load and generation shedding is continued until a load flow solution is obtained when reactive limits on all generation are ignored. If reactive power on any generators exceed reactive capability, then reactive capability is added until a solution is obtained when reactive limits are enforced. This procedure was applied to three contingencies in Table 3.8 where the load shedding and generation shedding cured the problem. The amount of load shedding required at various buses and generation shedding required at the generators in various voltage collapse regions is shown in Tables 3.9, 3.10, and 3.11. The voltage collapse regions in the order that load and generation was shed is also shown. In case 8, Table 3.9 shows the load shed equals the generation level on the outaged generators. In other cases, Tables 3.10 and 3.9 shows the total load and generation shed is equal. The load flow solution are obtained in each case after the amount of load and generation is shed. The procedure did not work for case 5 that outaged generators 73 and 76. Generators 73 and 76 are the only two generators in a reactive reserve basin that causes a cascading voltage collapse problem. Reducing generation and load was required in such a large number of voltage collapse regions that the action was so drastic that it was felt that adding an SVC or synchronous condenser to its agent was necessary rather than load and generation shedding. Since Chapter 5 results will obtain corrective controls that elim- inate the need to perform load and generation shedding, and since these results in this chapter seeks contingencies where such action might be successful, the results for case 5 were never obtained. For the remaining two cases of Table 3.8 the procedure did not work because these double line outaged caused the network to be split into 62 two networks. 63 Table 3.1. Single Contingency Ranking for Voltage Collapse Region# 35. 74 LEHIH 3 345 : 55 PLYMH 5 161 : 27 WILMRT3 345 : Reac. Reset. Basin CAPACITY RESERVE VCA 121 C.BL 3G 24 250.00 99.11 35 118 DPS 57G 14 100.00 40.25 37 73 NEAL12G 20 267.00 181.16 69 101 MTOW 3G 14 38.60 8.26 147 TOTAL BASE RESERVE 328.78 CONTINGENCY RANKING RRB% GEN GEN VCA CONTINGENCY of BASE NAME RES % Case:9 L. OUTAGE: 55 149 0.00% C.BL 3G 24 0.00 0.00% DPS 57G 14 0.00 0.00% NEAL12G 20 0.00 0.00% MTOW 3G 14 0.00 0.00% Case:31 L. OUTAGE: 161 162 0.30% C.BL 3G 24 0.00 0.00% DPS 57G 14 0.00 0.00% NEAL12G 20 1.00 0.55% MTOW 3G 14 0.00 0.00% Casez39 L. OUTAGE: 68 69 0.33% C.BL 3G 24 0.00 0.00% DPS 57G 14 0.00 0.00% NEAL12G 20 1.10 0.61% MTOW 3G 14 0.00 0.00% Case:15 L. OUTAGE: 71 85 1.61% C.BL 3G 24 0.00 0.00% DPS 57G 14 0.00 0.00% NEAL12G 20 5.30 2.93% MTOW 3G 14 0.00 0.00% Case:12 L. OUTAGE: 69 77 2.19% C.BL 3G 24 0.00 0.00% DPS 57G 14 0.00 0.00% NEAL12G 20 7.20 3.97% MTOW 3G 14 0.00 0.00% 64 Table 3.2. Double Contingency Ranking for Voltage Collapse Region# 35. 74 LEHIH 3 345 : 55 PLYMH 5 161 : 27 WILMRT3 345 : Reac. Reser. Basin CAPACITY RESERVE VCA 121 C.BL 3G 24 250.00 99.11 35 118 DPS 57G 14 100.00 40.25 37 73 NEAL12G 20 267.00 181.16 69 101 MTOW 3G 14 38.60 8.26 147 TOTAL BASE RESERVE 328.78 CONTINGENCY RANKING RRB% GEN GEN VCA CONTINGENCY of BASE NAME RES % Case:4 G. OUTAGE: 6 -57.85% G. OUTAGE: 131 C.BL 3G 24 -127.3 -128.44% DPS 57G 14 -38.4 -95.40% NEAL12G 20 14.5 8.00% MTOW 3G 14 -39.0 20.50% Casez27 L. OUTAGE: 68 69 -39.78% L. OUTAGE: 69 77 C.BL 3G 24 -19.3 -19.47% DPS 57G 14 -32.3 -80.25% NEAL12G 20 -44.0 -24.29% MTOW 3G 14 -35.2 26.91% Case:10 L. OUTAGE: 55 149 -39.69% L. OUTAGE: 71 85 C.BL 3G 24 -39.8 -40.16% DPS 57G 14 -53.9 -133.91% NEAL12G 20 -10.1 -5.58% MTOW 3G 14 -26.7 20.46% Case:16 L. OUTAGE: 71 85 -33.58% L. OUTAGE: 161 162 C.BL 3G 24 -36.5 -36.83% DPS 57G 14 -51.7 -128.45% NEAL12G 20 3.6 1.99% MTOW 3G 14 -25.8 23.37% Case:13 L. OUTAGE: 55 149 -32.27% L. OUTAGE: 68 69 C.BL 3G 24 -24.8 -25.02% DPS 57G 14 -34.5 -85.71% NEAL12G 20 -27.5 -15.18% MTOW 3G 14 -19.3 18.19% 65 Table 3.3. Single Outage that had the most affect on Voltage Collapse Region. Cont. # of Case Single Contingency VCR 9 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 48 10 Line Outage: 55 PLYMH 5 161 162 LEEDS 5 161 43 12 Line Outage: 69 HOPET 5 161 77 WRJGT 5 161 10 15 Line Outage: 71 MONOA 5 161 85 CARRLL5 161 48 31 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 48 39 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 43 2 Gen. Outage: 73 NEAL12G 20 1 66 Table 3.4. Double Outage that had the most affect on Voltage Collapse Region. Cont. # of Case Double Contingency VCR 8 Gen. Outage: 6 6R1G 22 19 Gen. Outage: 73 NEAL12G 20 14 Gen. Outage: 73 NEAL12G 20 3 Gen. Outage: 130 FT.CL1G 22 15 Gen. Outage: 73 NEAL12G 20 2 Gen. Outage: 121 C.BL 3G 24 18 Gen. Outage: 73 NEAL12G 20 1 Gen. Outage: 99 PRARK4G 18 21 Gen. Outage: 130 FT.CL1G 22 1 Gen. Outage: 131 NEBCYlG 18 24 Gen. Outage: 121 C.BL 1G 24 31 Gen. Outage: 131 NEBCY 1G 18 57 Gen. Outage: 76 NEAL34G 24 6 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 59 Gen. Outage: 76 NEAL34G 24 3 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 61 Gen. Outage: 76 NEAL34G 24 3 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 71 Gen. Outage: 99 PRARK4G 18 47 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 72 Gen. Outage: 99 PRARK4G 18 38 Line Outage: 71 MONOA 5 161 85 CARRLL5 161 73 Gen. Outage: 99 PRARK4G 18 32 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 75 Gen. Outage: 99 PRARK4G 18 4 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 95 Line Outage: 55 PLYMH 5 161 162 LEEDS 5 161 40 Line Outage: 85 CARRLL5 161 86 GR JT 5 161 97 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 6 Line Outage: 85 CARRLL5 161 86 GR JT 5 161 67 Table 3.5. Double Outage with negative reserves when the gen. limits are ignored. Cont. Double Contingency with # of Case Ignoring Reactive Limits VCR 1 Gen. Outage: 6 6R1G 22 9 Gen. Outage: 130 FT.CL1G 22 2 Gen. Outage: 6 6R1G 22 29 Gen. Outage: 121 C.BL 3G 24 4 Gen. Outage: 6 6R1G 22 38 Gen. Outage: 131 NEBCYlG 18 10 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 26 Line Outage: 71 MONOA 5 161 85 CARRLL5 161 13 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 13 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 14 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 1 Line Outage: 69 HOPET 5 161 77 WRIGT 5 161 16 Line Outage: 71 MONOA 5 161 85 CARRLL5 161 13 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 22 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 2 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 27 Line Outage: 68 HOPE 5 161 69 HOPET 5 161 28 Line Outage: 69 HOPET 5 161 77 WRIGT 5 161 68 Table 3.6. Voltage Stability Assessment Results for Voltage Collapse Regions that are Vulnerable to Loss of Control Voltage Instability VCR Voltage Reactive B. Case R of Cont. # Control Area Reserve Basin Reserves B. Case Case 3 125 PALM710 345 73 NEAL12G 20 181.16 —6.18% 14 156 E SIDE8 69 5 161 KELOG 5 161 73 NEAL12G 20 280.27 -l3.74% 22 45 TRIBJIS 161 121 C.BL 3G 24 23 HRN K 5 161 35 74 LEHIH 3 345 73 NEAL12G 20 328.78 -39.78% 27 55 PLYMH 5 161 101 MTOW 3G 14 27 WILMRT3 345 118 DPS 57G 14 -32.27% 13 121 C.BL 3G 24 34 77 WRIGT 5 161 73 NEAL12G 20 493.41 -25.03% 10 39 HAZLON3 345 101 MTOW 3G 14 37 ADAM 3 345 118 DPS 57G 14 -20.25% 16 121 C.BL 3G 24 . 130 FT.CL1G 22 22 119 SYCAOR3 345 6 6R1G 22 712.60 -21.37% 2 106 MONRE 5 161 73 NEAL12G 20 52 D.MON 5 161 101 MTOW 3G 14 4 BOONIL3 345 118 DPS 57G 14 -11.65% 1 86 GR JT 5 161 121 C.BL 3G 24 130 FT.CL1G 22 23 110 CBLUFS5 161 6 6R1G 22 900.29 -26.88% 4 73 NEAL12G 20 114 C.BL12G 14 121 C.BL 3G 24 130 FT.CL1G 22 131 NEchlG 18 69 Table 3.7. Additional Reactive Supply Needed to Cure Loss of Control Voltage In- stability. Gen. Double Contingrncy Case # B.# 14 22 13 27 16 10 1 2 4 73 11.20 15.90 27.50 44.00 10.10 121 22.60 24.80 19.30 36.50 39.80 53.00 127.30 101 19.30 35.20 25.80 26.70 39.10 43.20 39.00 118 34.50 23.30 51.70 53.90 22.90 43.10 38.40 114 40.40 41.30 42.60 73.60 59.70 99 20.30 23.40 15.80 130 77.10 69.50 Tot. 11.20 38.50 106.10 130.80 154.40 171.80 177.90 260.40 349.70 VCR 3 5 35 35 34 34 22 22 23 RRB 73 73 73 73 73 73 73 73 73 121 121 121 121 121 121 121‘ 121 101 101 101 101 101 101 114 118 118 118 118 118 118 131* 130 130 130‘ 130 130 6" 6" 6" * means the generator bus # is part of the double contingency. 70 Table 3.8. Unsolved Double Contingency Cases. conting. Double Contingency Case: 5 Gen. Outage: 73 NEAL12G 20 Gen. Outage: 76 NEAL34G 24 Case: 8 Gen. Outage: 76 NEAL34G 24 Gen. Outage: 131 NEBCYIG 18 Case: 11 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 Case: 12 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 Line Outage: 55 PLYMH 5 161 162 LEEDS 5 161 Case: 20 Line Outage: 71 MONOA 5 161 85 CARRLL5 161 Line Outage: 85 CARRLL5 161 86 GR JT 5 161 Case: 21 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 Line Outage: 55 PLYMH 5 161 162 LEEDS 5 161 71 Table 3.9. Load Shed for Case# 8. Case: 8 Gen. Outage: 76 NEAL34G 24 Gen. Outage: 131 NEBCY 1G 18 Load Bus# Bus Name Load Reduction 20 HINTON8 69 -40.90 40 BLKHK 5 161 -52.88 87 GUTHIE7 115 -16.91 103 DAVNRT5 161 -322.00 111 AVOC 5 161 -65.41 113 $1211 5 161 -32.70 139 S706 8 69 -10.10 142 CLRNDA8 69 -27.09 157 PLYMTH8 69 -32.00 160 SC WST8 69 -14.40 30 HAYWD 5 161 -190.20 38 DUNDE 5 161 -14.76 46 DENIN 5 161 -65.31 59 EAGL 4 230 -104.43 91 CDRPS 5 161 -51.25 94 HILL 5 161 -162.00 105 DUNDE 7 115 -24.84 Total = -1227.18 72 Sl#l w" I ‘olF'lEJ Table 3.10. Load and Generation Shedding for Case# 11. Case: 11 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 Line Outage: 161 KELOG 5 161 162 LEEDS 5 161 Load Bus# Bus Name Load Reduction 27 WILMRT3 345 -324.00 151 INTRCG5 161 -24.00 80 POMEOY5 161 -15.76 161 KELOG 5 161 -42.00 45 TRIBJI5 161 -20.00 162 LEEDS 5 161 -30.00 54 WISDM 5 161 -94.04 57 SAC 5 161 -48.48 56 OSGOD 5 161 -25.29 29 WINBGO5 161 -28.31 28 FOX R5 161 -38.47 18 ADAM 5 161 -40.40 15 FTRAD 4 230 -160.00 Total = -890.75 Gen. Bus# Bus Name Gen. Reduction 73 NEAL12G 20 —235.59 76 NEAL34G 24 -133.98 101 MTOW 3G 14 -81.00 118 DPS 57G 14 -81.00 121 C.BL 3G 24 -235.59 130 FT.CL1G 22 -123.59 Total = —890.75 73 Table 3.11. Load and Generation Shedding for Case# 12. Case: 12 Line Outage: 55 PLYMH 5 161 149 RAUN 5 161 Line Outage: 55 PLYMH 5 161 162 LEEDS 5 161 Load Bus# Bus Name Load Reduction 27 WILMRT3 345 —324.00 151 INTRCG5 161 -24.00 80 POMEOY5 161 -15.76 162 LEEDS 5 161 -30.00 54 WISDM 5 161 -94.04 57 SAC 5 161 -48.48 56 OSGOD 5 161 -25.29 29 WINBGO5 161 -28.31 28 FOX RS 161 -38.47 18 ADAM 5 161 —40.40 15 FTRAD 4 230 -160.00 Total = -828.75 Gen. Bus# Bus Name Gen. Reduction 73 NEAL12G 20 -204.59 76 NEAL34G 24 -133.98 101 MTOW 3G 14 -81.00 118 DPS 57G 14 -81.00 121 C.BL 3G 24 -204.59 130 FT.CL1G 22 -123.59 Total = -828.75 74 4.1 This ch 16mg. 0. pOWer 5 reactive di‘cusse COUecti afld the CHAPTER 4 Optimization Applications in Dispatch of Power System 4.1 Introduction This chapter reviews the active power dispatch and reactive power dispatch prob- lems, that are the most common optimal power dispatch problems used in operating power systems. The need for including reactive reserve basin constraints on each reactive reserve basin as a means of preventing loss of control voltage instability is discussed. Security constrained dispatch and the tradeoffs between preventive and corrective control formulations is then addressed for thermal and voltage problems and then voltage stability problems. A corrective control is formulated and developed as an alternative to the preventive control currently proposed for the Open Access System Dispatch. The special state estimation, secondary control, and optimization algorithm requirements for implementing the corrective control are discussed. The results obtained from performing corrective control for specific contingencies where VSSAD has provided corrective action are discussed in Chapter 5. 75 4.2 15 The classi dispatch c output of generatior for produ‘ when Pg mefiicien The line include p branches 4.2 Active Power Dispatch The classical approach to the economic operation of power system is called economic dispatch or active power dispatch. Total production cost is minimized by varying the output of the generators. Normally in active power dispatch problem, the real power generation cost is approximated by a quadratic polynomial, representing the fuel cost for producing power level Pg, on generator 2' [12] f,(Pg,) = a,- + 12,199, + an); (4.1) where P9, is the MW (per unit) output of the generatori and a,,b,-,c,~ are constant coefficients. The line limits and equipment unit limits are typically used as constraints which include power balance equations, real power generation limits, and thermal limits on branches. The active power dispatch problem formulation is given as follows [13]. Minimize f(Pg) = Zil‘da. + b.Pg. + cin.) SUbjBt t0: fg’k(6) — Pgi _ Pdi = 0; Z: 1: "“v Nb (4 2) Pgimin S P91 S Pgimflzl i 6 N9 Ii 3min S Iij S Iijmu; Li 5 Nb where: a, b, and c are system dependent parameters which aflect the cost of real power generation; and P9,- is the real power generation at bus i Pd, is the real power demand at bus i [,3- is the flow of current in branch ij Pgim.” /Pg,-m“ is the minimum/maximum real power for generator i I.-:,-m,,,/I,c,-mz is the minimum/maximum limit on flow of current on the branches 76 connecting 0, is the pl N, is the 1 “b is lllf 7 Active p0 schedule t constraint The 01 power bal; list liters found tha the perfo: equations Utilities t patch do. also 1mm Central Open Ac index Wi System l plbllde in an 0] A si: marhet. entails ( Dfiund l connecting bus i and j 9,- is the phase angle of the voltage at bus i N9 is the number of generators in the system Nb is the number of buses in the system Active power dispatch minimizes the fuel cost by determining the generation schedule that minimizes the operating cost and does not violate any of operating constraints of the system. The optimal active power dispatch as formulated above does not include reactive power balance equation or bus voltage limit violation constraints even though the ear- liest literature on optimal power flow included such constraints. Subsequent research found that decoupling active power dispatch allowed use of linear programming since the performance index used is quadratic or piecewise linear and linearized network equations could be used. The optimal active power dispatch is implemented on many utilities to dispatch generation by minimizing fuel costs while guaranteeing the dis- patch does not violate thermal overload limits. The optimal active power dispatch is also known as Security Constrained Economic Dispatch (SCED) or the Transmission Constrained Economic Dispatch (TCED). SCED is proposed as a component of the Open Access System Dispatch as discussed in Section 4.6. The fuel cost performance index will be replaced by one that is price based in the SCED used in Open Access System Dispatch. The formulation should account for transmission losses, and would provide secure and efficient participation factors for the automatic generation control in an Open Access System Dispatch. A similar optimal active dispatch would also be developed as part of the power market administration function of an independent system operator [15]. This function entails developing a schedule for all loads and energy supplies for an entire scheduling Period based on results of an energy auction. The optimal active dispatch has a quite 77 difetent per nuplus in t well as the dipatch pr: Security C c set the acti which gene The act any reactii collapse or stem. If to include each react each conti been formi ll“? active collapse. ‘ fIOI11 prod Constraint We Volt 305888“ Solution ‘ of the Op, in Part tl CDHilnuOU different performance index than what is given in 4.2 because it models the consumer surplus in the energy auction market and possibly the spinning reserve market as well as the cost of transmission losses and spinning reserve[16]. This optimal active dispatch problem is an auction resolution optimization that is solved separately from Security Constrained Optimization in a Open Access System Dispatch and the results set the active generation levels on each generator in the auction as well as resolving which generating company will serve each customer through the auction process. The active power dispatch can drive a system into voltage collapse. The lack of any reactive reserve basin or transfer constraints to prevent loss of control voltage collapse or clogging voltage collapse implies that voltage collapse is possible on this system. If voltage collapse is possible, then active power dispatch must be modified to include reactive power balance equations, reactive reserve basin constraints for each reactive reserve basin in the system, and transfer or wheeling constraints for each contingency that can produce clogging voltage instability. This problem has been formulated and solved and shows that without reactive reserve basin constraints the active power dispatch on a 162 bus model always drives the system into voltage collapse. Adding reactive reserve basin constraints prevents the optimal power flow from producing a solution that has no load flow solution. These reactive reserve basin constraints continue to prevent the optimal power flow from producing solutions that cause voltage collapse and no solution in a load flow model as reactive reserves are progressively reduced in all reactive reserve basins. The optimal power flow has solution when reactive reserve basin constraints are all at limit because the Hessian of the optimal power flow is of larger dimension than the load flow J acobian because in part the optimal power flow treats tap changers and shunt capacitor controls as continuous variables and thus does not become singular when the load flow J acobian, a submatrix, becomes singular. 78 4.3 I In the de number 01 on transn power he» the circul profiles. '. and bus V minimizii Will. there, ( set of all OI mini] ailng in Where, SCUSilr k! Ab : The as 91103 eq Teal am 4.3 Reactive Power Dispatch In the deregulated environment power utilities will be interconnected to a larger number of other utilities in order to facilitate transactions between them. The stress on transmission system will increase and the ability to control voltage and reactive power flow will be very crucial. The use of reactive power dispatch will help reduce the circulation of reactive power in the network and also maintain acceptable voltage profiles. This reactive dispatch problem assures generator reactive power supply limits and bus voltage limits are not violated. The reactive dispatch problem usually involves minimizing total real power loss P, in the transmission network which is represented byl17l. Minimize H = 2 0.5042 + V,-2 — 2V,-VJ- cos(6,- - 6,-)) (4.3) ijEI where, Gij is the conductance of the line connected between buses i and j, I is the set of all branches in the system or minimizing reactive power injection costs required to keep the system oper- ating in feasible range. This performance index can be written as[18], Nb Minimize 2(Cck3ck + CikSik) (4.4) k=l where, (Yak/Cg, are capacitive/inductive injection installation costs at bus k, Sch/5,7, are the amount of reactive injection of the capacitive/inductive type at bus k, Nb is the number of buses in the system. The associated equality constraints typically are real and reactive power bal- ance equations. The inequality constraints includes the limits of all of the following; real and reactive power generation, voltage at all bus, thermal limited flow on all 79 branches. reactive 1 arch as a capacitor The ' problem I015? 1 branches, transformer tap position, and switchable shunt capacitor susceptance. A reactive power dispatch will determine the optimal value of all the control variables such as all generation levels, all under tap changer positions, and all switchable shunt capacitor susceptance levels assuming all imposed constraints limits are met. The problem described above can be formulated mathematically as a separate problem or both of the problem combined together as described in [18]: Minimize Pr + 25:1(03561. + 0,-1.5“) Subjet to: fk(V,0,T) — P9,. - Pd], = 0; k = 1, ...., Nb gk(V10)T) — ng _ Qdk _ V2036]: — Sik) : O; k = 11"") Nb View... 5 VI: 5 Vic...“ k E Nb ngmin SngSngm", kENg > ngmin S ng S ngm“, k 6 N9 Iijm... S Iij S 1.3m“; i,j 6 Nb Ti-.. 5 T,- 5 TM 0 S SC S Scma: OSSiSSi ma: J (4.5) where: ng/ng is the real/reactive power generation at bus k Pdk/Qdk is the real/reactive power demand at bus I: ngmm /ngM, is the minimum/maximum real power produced by generator k ngmm /ngm“ is the minimum/maximum reactive power produced by generator k: Vkmin/Vkm. is the minimum/maximum voltage at bus I: These set of voltage limits on generator terminal voltage k 6 N9 can actual specify generator terminal voltage or constrain the control change of these voltage set points Ii,- is the flow of current on branch connecting buses i and j 80 The opt ol the 6 it has i overloa active 3. cl ll'ith d all of t all? imp [awn/1,3,,” is the minimum/maximum flow of current in all the branches 0;. is the phase angle of the voltage at bus I: T,- is transformer tap ratio for the j-th in-phase controllable transformer N9 is the number of generators in the system Nb is the number of buses in the system The optimization tool, that performs optimal reactive dispatch, exists in most of the electric power utility control centers in the U. S., but industry surveys indicate it has not been used even though it helps correct voltage limits violation, thermal overload violation, and can save the utility some money in fuel cost by minimizing active losses B, It has not been widely used because: 1. all of the fuel cost savings are returned to customers through fuel adjustment clauses and thus there is no economic incentive for a utility to use it; 2. operators desire to supervise and control adjustments in excitation system volt- age set points 13,-, capacitor insertions Sc, and tap positions T,- on under load tap changers, and are reluctant to allow an optimization program to carry out these adjustments. The coordination of tap changers, the coordination of capacitor insertions, and the coordination of both tap changers and capacitors together has not been acceptably modeled and included in the above formula- tion; 3. the state estimator is not reliable or consistently accurate enough to allow it to perform loss minimization and thermal and voltage correction. With deregulation, the operator of the transmission network would accrue some if not all of the I 2R loss savings. Many utilities that hope to be associated with the ISO are implementing this optimal reactive dispatch. Consumer Energy is the first utility to implement optimal reactive dispatch. A difficulty with optimal reactive dispatch 81 as well as For utilit the react loss perfr Qt vas usec series so shunt in tapacito at bus 2'. security. The reserve llOD C01 as well as optimal active dispatch is that the solution could result in voltage collapse. For utilities with voltage collapse problems, the current optimal active dispatch and the reactive dispatch are useless. Replacing the real power losses by a reactive power loss performance measure Q1: 2 8,5 (V,2 + VJ-2 — 2V,-V,- cos (6,- - 6,-)) + Z |V,-|2 B”; V i andj (4.6) ijEI i was used as a means of maximizing reactive supply on generation, where B,,- is the series susceptance of line i j and B,-, is the shunt susceptance at bus i from load, shunt inductors, tap changing transformer shunts (3,. < 0, or 3,. > 0), shunt capacitors (Bi. < 0), and line charging of transmission lines (Bi, < 0) components at bus i. Although this performance index would appear to improve voltage stability security, it still could steer a system into voltage collapse. The maximization of reactive reserve is not intelligent enough to add reactive reserve in reactive reserve basins that suffer severe equipment outage and transac- tion combinations that bring it close to voltage collapse. Maximizing system reactive reserve would also not borrow reactive reserve from reactive reserve basins with sig- nificant reactive supply compared to the worst equipment outage and transaction combinations that it must endure. Use of the performance measure ( 4.6) did not prevent voltage instability and in fact could result in voltage instability.The voltage stability security assessment and diagnosis can assess whether any reactive reserve basin will experience voltage instability and the quantity of additional reactive reserve basin reactive reserve necessary to survive any loss of control voltage instability. The maximum added reactive reserve AR), needed to prevent voltage instability in that reactive reserve basin over all equipment outages and transaction combinations found to produce loss of control voltage instability in that reactive reserve basin can be obtained from VSSAD diagnostic results. If RR;co is the reactive reserve in reactive 82 reserve l reserve l where: HEB), i QC, is 1 HR; is conditic this active QXhau Stabil Opiiin algori These in 0m reserve basin k in the base case, the reactive reserve basin constraint for reactive reserve basin k is Z QGima: - QGg' > RRko + ARI: (4.7) {612”}, where: BBB,: is the generators in reactive reserve basin k; Qcim, is the continuous rating upper reactive limit on generator i; Q0, is the reactive generation on generator i; RR]. is the level of reactive reserve in reactive reserve basin k at base case operating conditions; ARkis the amount of reactive added to reactive reserve basin k Adding ARI, to each reactive reserve basin It: would prevent loss of control volt- age instability detected by VSSAD [1]. This result is obtained when RR), = 0 and AR]. = 0.10 pu in each reactive reserve basin. It was not surprising to find that the reactive dispatch problem consistently had no solution when the reactive reserve basin constraints were omitted because the reactive capabilities on generator in all reactive reserve basins should prevent violation of generator limits. The optimal re- active dispatch problem with reactive reserve basin constraints prevents simultaneous exhaustion of reserves on all generators and thus prevent loss of control voltage in- stability in every agent as reactive capability on every generator is reduced. The optimal power flow and load flow had solutions until the interior point optimization algorithm could no longer find a feasible solution to the reactive dispatch problem. These constraints ( 4.7) also assure that preventing loss of control voltage instability in one location does not cause loss of control voltage instability in another location. 83 4.4 St hsecmitvt 1.3 and con ll an equip corntraint real activ thermal limit vic position or an e: ll an e reserve reactiv has so PM equalit probler PIObler Eration Positior lSOptitr EXists a diniatcl mizatior 5€€urity OlPEIatir 4.4 Security Constrained Dispatch A security constraint places an additional inequality constraint on the operating state so and control no for each equipment outage that violates that inequality constraint. If an equipment outage causes a thermal limit violation on an element, the security constraint on flow on that branch could cause a reduction in flow on that branch via real active generation dispatch change so that when the equipment outage occurs the thermal limit violation does not occurs. If the equipment outage causes a voltage limit violation at a bus, the security constraint on the bus voltage could cause a tap position change on an under load tap changer, a switchable shunt capacitor insertion, or an excitation control set point change that would avoid bus voltage limit violation. If an equipment outage or operating change caused a reactive reserve basin reactive reserve constraint( 4.7), a security constraint would add the needed reserves in that reactive reserve basin so that if equipment outage occurs that reactive reserve basin has sufficient reserves to prevent voltage instability. Preventive security constrained dispatch is frequently carried out by adding in- equality constraints to the active power dispatch problem or reactive power dispatch problem known as security constraints. The preventive security constrained dispatch problem consists of finding optimal value of all control variables such as active gen- eration output, generator excitation control system set point, and tap changer tap position of the under load tap changer transformer such that the system operation is optimal for some objective function, and making sure that no constraints violation exists after any disturbance or equipment outage. Preventive security constrained dispatch tries to optimize some performance specification such as fuel cost, loss mini- mization or VAR injection[20] subject to equality constraints, operating constraints, security constraints and control constraints for a list of possible N contingencies and operating change combinations. 84 4.4.1 Preventive Control Formulation Preventive security constrained dispatch produces a preventive control in such a way that the controls are adjusted to satisfy the equality and inequality constraints before any of the contingency and operating change combinations have occurred. A preven- tive control formulation has the form [21]: Minimize F (330, no) subject to Go(mo,uo) = 0 Ho($o,uo) S 0 l (4'8) G,-(:r,-,uo) = 0, i = 0,1,2,....,N H,(:r:,-,uo) S 0, 2: 1,2, ....,N J where: i = 0 is the base case, and i > 0 represents the it" post-contingency configuration. N is the number of contingencies F(xo,uo) represent the performance index to be minimized. Go(zo,uo),G,-(a:,-,uo) are the real and reactive load flow equation for the system operating constraints and for the ith post-contingency configuration respectively. Ho(:ro, v.0) represent the system operating constraints limits on reactive reserve basin reactive reserves, thermal overload, bus voltage, tap setting, capacitor insertion, and bound on Var generations before contingency and H,-(:r,-,uo) represent these same inequalities as security constraints state on m,- and control uo for the ith contingency respectively. In most cases Gg(x,-, uo) is never included explicitly in the formulation ( 4.7) but used to determine Am,- and the security constraints are then H,(:ro + Arm, uo) S 0. This dramatically reduces the dimension of the optimization problem by N n since 85 A3,- are no longer variables and G,(:c,, uo) are no longer constraints. The dimension of the optimization problem is then n + m but the number of inequality constraints is so large that obtaining feasible and / or optimal solution may be dificult because there are only m controls no to satisfy all these inequality constraints. The security constrained optimal reactive dispatch would place a very heavy bur- den on most optimization solution algorithms. In a stressed system, there can be several branches with thermal limit violations and yet generally they are not inde- pendent. Thus correcting only one or possibly a few of the thermal limit violation would correct all of the operating constraint and security constraint thermal limit vio- lations not only for one equipment outage and operating change but for all equipment outage and operating change combinations. Bus voltage limit violations (reactive re- serve basin reserve violation) are not independent and correcting only a few of the very large number of the voltage limit violations (reactive reserve basin reserve vio- lations produced for all VSSAD determined equipment outage and operating change combinations) could correct all such violations. Similarly the reduction of flow on boundaries and interfaces of a voltage control area that solve clogging voltage insta- bility for a specific equipment outage and operating change are not independent and thus correcting a few such violation could correct all such clogging related operat- ing and security constraint violations produced for all VSSAD determined clogging voltage instability problems. Every optimization algorithm searches for the subset of inequality constraints that should be binding at the optimal solution and act as binding equality constraints. The convergence of any optimization algorithm depends heavily on optimally searching for and finding the correct set of binding inequality constraints that correct all thermal limit operating and security constraint violations, all voltage limit operating and security constraint violations, all reactive reserve basin operating and security constraint violations, and all clogging operating and security transfer or wheeling constraints. Experience with VSSAD [1] indicates there can 86 be a relatively large number of reactive reserve basins that need reactive reserves for any equipment outage and operating change combination predicted to produce voltage instability. This would require adding security constraints for each reactive reserve basin. The VSSAD process attempts to determine critical reactive reserve basin that contains relatively few generators and is nested in successively large reac- tive reserve basins. The smaller reactive reactive reserve basins are associated with voltage control areas in the distribution system or in a electrically remote part of the subtransmission system. Exhaustion of all reactive reserve on a critical reactive reserve basin causes large series 12X losses and shunt capacitive supply withdrawal that exhausts reactive reserve in the successively larger reactive reserve basins causing voltage collapse to spread uncontrollably throughout a system. VSSAD [1] may help the optimization algorithm in finding the optimal binding constraints by recognizing the binding constraint always is on reactive reserve in the critical reactive reserve basin for any equipment outage and operating change combination. Experience with VSSAD suggests that there are only a few critical reactive reserve basins in a system that are critical and they generate cascading voltage collapse for all equipment out— age and operating change combinations that produce voltage collapse in a system. This implies adding reserve to a relatively few critical reactive reserve basins in the reactive reserve basin constrained optimal reactive dispatch problem ( 4.5, 4.7) or adding security constraints for each contingency and operating change on just one or a few critical reactive reserve basins can prevent voltage instability for all equipment outage and operating change combinations found to experience voltage instability on a system. The critical reactive reserve basin for loss of control voltage instability may in some cases also be critical for clogging voltage instability. Placing transfer or wheeling constraints on flow on one or more interfaces or on boundaries of criti- cal voltage control areas or on generation and load in the critical agent for clogging voltage instability may cure most clogging voltage instability problems. 87 The major difficulty with security constrained optimization problems (security constrained active power dispatch and security constrained reactive power dispatch) is that trying to prevent all possible thermal limit violations, all possible voltage limit violations, and all possible voltage instability problems that can occur on all equipment outage and transfer and wheeling transaction combinations adds so many conflicting constraints on selecting the operating state 2:0 and control no that there is no solution (2:0, uo) to the optimization problem ( 4.8). Even if there are feasible solutions (2:0 6 X0, no 6 U0), they are so constrained that the performance measured by the objective function is poor and the optimization algorithm may not be able to converge to it due to inability in finding the set of binding constraints that produce an optimal solution. This inability to find feasible and then find optimal solutions for security constrained dispatch problems was already recognized and caused the corrective control problem ( 4.9 ) to be formulated. 4.4.2 Corrective Control Formulation Adding security constraints on reactive reserve basins for each possible loss of con- trol voltage instability contingency affecting each reactive reserve basin, and adding interface and boundary flow constraints for each clogging voltage instability on each distribution voltage control area just makes the determination of a feasible set or an optimal solution that much more difficult. A corrective control based formulation for optimization with security constraints has been developed for thermal and voltage limit violation that assigns a control change u,- — uo from operating state (smug) for each equipment outage and operating change combination that has thermal or voltage limit violations H,(a:,-,u,-) > 0 rather than satisfaction H,-(a:,-, u,-) S 0. This corrective control formulation is reasonable because thermal and voltage limit viola- tion can be endured for up to fifteen minutes, which is sufficient time to detect the violation H,(:c,-,u,) > 0, from a state estimator, compute a control u, to correct 88 it, and implement a corrective control change within 15 minutes after the equipment outage and operating change is detected to have occurred via a 5 second updated state estimator. The 15 minutes update of the security constrained optimization corrected the thermal or voltage problem before equipment damage can occur. A corrective control for voltage instability is also possible because the sequence of events before a voltage instability develops after an equipment outage and operating change combi- nation takes minimum of 2 - 5 minutes to develop. The sequence of actions after the equipment outage and operating change occurs includes 1. distribution level tap changer and switchable shunt capacitor controls that incur delays of 10 seconds to a minute between each step change in tap position and each switching in of an additional capacitor bank segment. These control changes raise distribution level voltage and load, that fell after the contingency, back to precontingency levels over 2-5 minutes . The first change in tap position or switchable shunt capacitors controls requires far greater delay than every subsequent change; 2. generator continuous field current limit violation due to the equipment outage and operating change combination and the control action of distribution level tap changes and switchable shunt capacitors that increase reactive load to pre- contingency levels. The field current can violate thermal limit for a duration of between 10 seconds and 2 minutes depending on the magnitude of the current limit violation before the maximum excitation limiter reduces field current and reactive power output back to continuous rating limit. The maximum excita- tion limiter causes the exciter to lose control of voltage because the maximum exciter limiter adjusts the voltage set point of the excitation control system on that generator once the maximum excitation limiter over rides the excitation control after the 10 second to 2 minute delay; 89 3. the progressive loss of voltage control on generators via maximum excitation limiters that reduce field current and reactive power to continuous rating levels after a delay that can range from 10 to 120 seconds [23] depending on the magnitude of the field current limit violation. The reduction of reactive power on generators due to maximum excitation limiter action and the reduction of voltage and increase in network reactive losses due to voltage reduction can cause or increase magnitude of field current and reactive power limit violation on other generators that ultimately no longer supply reactive and lose control of voltage until voltage collapse occurs. Loss of control voltage instability requires all three control related difficulties to develop. Clogging only requires action (1) which occurs due to restoration of distribution voltage and load. This sequence of control actions can take a minimum of 2 - 5 minutes to occur. Using a Wide Area Measurement System (WAMS) with a sampling period of five seconds to provide state estimation and detection of the equipment outage and operating change, there is an ample time to implement a precomputed computed corrective control for voltage instability. However, if the sampling period is fifteen minutes or up to an hour as proposed for the Transmission Congestion Management System [15] to be used in Independent System Operator (ISO) control center, there is no opportunity to correct thermal, voltage or voltage collapse problems. The corrective control problem formulation is now given assuming that corrective control is needed and will be implemented in local security based control centers outside the Independent System Operator as proposed in [1]; 90 Minimize F ((170,110) subject to Go(zo, uo) = 0 Ho($oiuo) S 0 G,(:c,,u,~) = 0, i = 1,2, ....,N H,(.i:,-,u,-) g o, i = 1,2,....,N cp,(u,- — uo) g 6,, i = 0, 1, 2, ...., N r where: u0,u,- E Rmare the vector of control variables for the base-case and post-contingency configuration i respectively. arms,- 6 3?"are the vector of state variables for the base-case and post-contingency configuration i respectively. G.- : Rm“ —) ER“ is the vector function representing equality constraints for the ith configuration. H,- : 52m“ -> 3?" is the vector function representing inequality constraints for the 2"" configuration. 90,-(o) is the distance metric( the Euclidean norm). 9, is the vector of upper bounds reflecting ramp-rate limits The dimension of the corrective optimization problem is (N + 1)(n + m) and is nearly the same as the preventive control( 4.8) but now has m control to satisfy the inequality constraints for the base case and m controls for each contingency rather than just m controls for satisfying the inequality constraints for the base case and all N contingencies. This corrective control problem is quite diffith to solve due to the large number of equality and inequality constraints and the large dimension (N + 1)(n + m) of the optimization problem. Making G,(.r,-,u,-) and H;(:c,,u,-) functions of u,- and not uo lets this complex optimization problem be decomposed 91 / 1 .in in to N slave optimization problems that each minimize H u,- — uo || subject to the last three constraints in ( 4.9) that depend on u,- and m,- and a master problem that optimizes F(zo, uo) subject to the first two constraints and last constraint in ( 4.9) that depend on :50 and uo. A Benders decomposition algorithm, described in this section is used in this master-slave set of optimization problems. 4.5 Voltage Collapse Constrained Optimal Reac- tive Dispatch Problems A preventive voltage collapse constrained optimal reactive dispatch (referred as prob- lem 1) could be formulated that corrects thermal overloads on any branch and voltage limit violations on any bus, via operating constraints Ho(:ro, uo) S 0 and prevents and corrects voltage collapse in any reactive reserve basin. The preventive control for solely loss of control voltage collapse corrects voltage collapse not only for the base case with operating state 2:0 but also prevents voltage collapse after each equipment outage and operating change combination with operating state 2:, using security con- straints H;(:r,-, uo) S 0. A revision of the corrective optimal reactive dispatch ( 4.5) is possible by adding sufficient reactive reserve basin reactive reserve ARI, above base case reactive reserve RR],0 for all the equipment outage and transaction combinations that would cause loss of control voltage instability. This optimization problem does not add equality constraint G,(a:,-,uo) or inequality constraints H,(a:,-,uo) but just uses VSSAD information to modify the operating constraints Ho(xo,uo) to incor- porate the reactive reserve basin constraint( 4.7) for each contingency. This control corrects and prevents voltage collapse but also corrects thermal or voltage limit vi- olations for each contingency. The operating constraints would correct for thermal and voltage limit violations as they develop and would avoid equipment damage if 92 the update of this control was sufficiently often (10 - 15 minutes). A second preventive control problem ( 4.5 ) would not only add the reactive reserve constraint via Ho(a:o, no) but also inequality constraints H,(:r,-, uo) for specific contingencies. The inequality constraint in H0(a:o, uo) assure there is finite reserves in each reactive reserve basin at solution so that satisfies Go(zo,uo) = 0 as well as correcting thermal and voltage limit violations for solution 930 . The constraints H,(:c,, uo) would cause ac to change to prevent thermal limit violation, and voltage limit violation, and reactive reserve basin constraint violations after the equipment outage and transaction combination that produced load flow model G,(a:,~, uo) = 0. The combination of equality constraints G,(:c,~, uo) = 0 and H,(a:,, uo) S 0 for each thermal, voltage, or voltage instability insecure equipment outage and transaction would place terrible burden on selecting no and may make the optimization problem infeasible as mentioned earlier. If a feasible solution could be found, the performance would be unacceptable. Preventive control (referred as problems 1 and 2) has been replaced in terms of a strategy for alleviating thermal and voltage limit violation by a corrective con- trol problem formulation ( 4.9 ) that allows control change u,- - uo from base case control no. Selecting u,- for equipment outage and operating change combinations i = 1, 2, ...., N and no for the base case implies there is control change u,- — uo for each equipment outage and transaction combination to be used to correct reactive reserve basin constraint violation( 4.7) and voltage collapse only when that equip- ment outage and transaction combination occurs. The constraint H,(:c,-, u.) would include thermal or voltage limit constraints. The control uo would posture the sys- tem based on the performance index F(rco, no) and constraint go,(u,~ - no) in ( 4.9 ) and satisfy thermal, voltage and reactive reserve basin constraints on $0 imposed by H (:00, ac) S 0 satisfying Go(a:o, uo) = 0. The control change u,- —uo makes satisfying G,(:c,-, u.) = 0 and H,(a:,-, u.) S 0 possible through the tailored reactive reserve basin 93 constraints for one or more reactive reserve basin k". The reactive reserve basin k” would add the reactive reserve ARk.(a:,-) recommended by VSSAD Z Qgima: — QG.($i) S BBQ,“ + 1331:4118) (4.10) iERRBk. for z,- satisfying G,(:r,-, u,) = 0 and H,(:c,-, u,). All other reactive reserve basin would only have minimal level of reactive reserves 2 QC."m — 520.0130) 2 Rka,,, (4.11) iERRBh The corrective control problem formulation ( 4.9 ) can also correct for clogging voltage instability problems for equipment outage and transaction combination deter- mined by VSSAD [1]. The constraints H,(.v,-, u.) S 0 associated with load flow model G,(a:,~, u.) = 0 after equipment outage and transaction combination would restrict the flow —V;V,- le'jl Sin (9.- — 9i — 5:5) S Rim... (4-12) or reduce load ZPDi< _ Pin...( (xii- — 0 (4.13) ZQDi S Qimin ($5) = 0 (4.14) as suggested by VSSAD [1]. These corrective control problem is demonstrated in Chapter 5, where we discuses the solvability problem. Perhaps the most significant advantages of corrective control are 1. the reactive resources needed to prevent a loss of control voltage instability for a particular equipment outage and transaction combination that impacts a re- 94 active reserve basin need not reside on that reactive reserve basin but can be borrowed from other reactive reserve basins, that at the moment when that equipment outage and transaction combination occurs have large reactive sup- ply or reserve surpluses. The small level of RR]. needed for all reactive reserve basins k other than k“ assures voltage collapse does not occur in these reactive reserve basins due to this borrowing on reserves for k‘. How small RR,c can be is a subject for research since it would be quite small if reactive reserve basin 1:: 76 k" could also borrow reserve it needs also suffers a change that requires reactive reserve in its reactive reserve basin. The probability that reactive reserve basins (a) have totally different generators in their reactive re- serve basins and (b) are vulnerable to totally independent set of equipment outages requiring additional reactive reserve experience voltage collapse simul- taneously is so small that there is no need to deal with this case in operations. Thus borrowing reserves from reactive reserve basins not in the same nested set is only limited by the significant physical limitation in shipping reactive power over any significant distance. Borrowing reactive reserves from large reactive reserve basins that lie in the same nested set as smaller reactive reserve basins that need reserves is not as limited by physical constraints on shipping reac- tive power. Such borrowing strategy is problematic because it imposes security difficulties in each reactive reserve basin in the nested set that may all cascade into voltage collapse if one of the reactive reserve basins in the nested set, not necessarily the smallest one exhausts reserves. Borrowing reserves from larger reactive reserve basins for smaller ones in the nested set, will likely consume reactive supply in the larger critical reactive reserve basin thus bringing on the voltage collapse. The ability to quickly change exciter set point, tap position on underload tap changer, switchable shunt capacitors, and active power genera- tion set points in the 2 - 5 minutes interval is needed. Such a control structure 95 is discussed in [9] and in section 4.7 and 4.8 along with changes in state es- timation; the OASYDIS dispatcher of reserves; and OASYCOM scheduler of reserves needed to implement it; . The active and reactive constraints on power flow or load needed to prevent a specific clogging voltage instability for a specific equipment outage and transac- tion combination need not be imposed on base case operation (2:0, uo) but is only implemented through constraints H,(x,-,u,-) S 0 on operation G,(a:,~,u,) = 0 after the equipment outage and transaction combination occurred. Imposing load reduction or transfer constraints on Ho(a:0,uo) S 0 would cause large economic penalties on operational costs for the ISO due to penalty payments for curtailed transactions and curtailed customers load. When the probability of the specific equipment outage and transaction combination is small and the cost of prevention is high, imposing such constraints as ( 4.12, 4.13, 4.14) in H0030, uo) S 0 is unthinkable. This is especially true on systems there numer- ous agents are one highly probable contingency away from interruption of load to a small set of distribution level customers due to the possibility of voltage instability. In such cases, it would be impossible to find a feasible (11:0, uo) that satisfies all constraints in H0(a:o, an) S 0; . the ancillary services required by the control change (u,- —uo) would only be paid for and thus implemented when it is absolutely required to prevent blackout for a specific transaction and equipment outage predicted by VSSAD to produce a clogging or loss of control voltage instability; . the control change u,- — no required to prevent clogging may not require load shedding for clogging voltage instability even though VSSAD indicates such change will obtain a load flow solution; 96 5. the increase in transmission capacity and system reliability (security and ade- quacy) could be quite large for the cost of implementing an improved control using a Wide Area Measurement System (WAMS). Electricite de France (EDF) claims a 20% transmission capacity enhancement via secondary voltage control that just adds reserves via switching of capacitors and reallocation of reactive reserves in a reactive reserve basin so that no generator in a reactive reserve basin exhausts reactive reserve. The EDF control (a) does not coordinate tap changer, switching of capacitor, generator excitation controls in several reactive reserve basins simultaneously; (b) does not optimally allocate reserves on gen- erators in a stressed reactive reserve basins to maximize security; (c) does not optimally borrow reserves from other reactive reserve basins; ((1) does not redis- patch active power generation to achieve optimal allocation and borrowing of reactive reserve after a contingency and transaction. Adding capabilities (a-d) could add very large additions to transmission capacity. The corrective control formulation ( 4.9 ) of the optimal reactive dispatch can be solved by a decomposition algorithm. The decomposition methods, discussed in [21] break this corrective control into optimization subproblems that are much more easily solved. The Benders decomposition will be used in this thesis. 4.6 Bender Decomposition Benders decomposition is a technique usually used for large nonlinear programming to partition the system state and control variables (2:, u) by holding some variables fixed while the other are being solved for . The fixed variables are adjusted by the master problem while the other variables are solved by the slave problems. The iteration between the master and the slave problems continue until the optimal solution is obtained. The application of this technique to optimal power flow is discussed in [22]. 97 The master problem can be formulated as follows [21]: Minimize F(mo, uo) sub'ectto G z,uo =0 J o( o i (4.15) Ho($oiuo) S 0 wi(?.lo) S 0, Z: 0,1,2,....,N where: the Benders cut function for the it“ contingency is w.- = w: + Aflu‘,‘ — US) a? is the optimal value of the slave problem control variable for the it" contingency. u; is the optimal value of the master problem control variable obtained from previous iteration. A,- is the Lagrangian multipliers associated with the it“ coupling constraint, ll “5 "' “i ll ‘3i SH 95 ll w: is the minimum value of the slave objective for the ith contingency. Suppose now for the slave problem that the penalty associated with the con- trol change (as — u?) have added to the slave problem for each equipment outage and operating change combination for a given us. The performance index of the slave problem is to minimize the control changes and therefore, minimize the cost associated with the control changes. The slave problem is given as [21] Minimize w: 2 d3,- subject to G,(:1:,-, u.) = 0 H,(a:,-,u,-) S 0 i (4-16) ll ““6 " u,- ll “3i Sll 9i ll 5520 98 where d is a positive constant and s,-, u,, 2:,- are the variables to be optimized. The algorithm for this decomposition is Step 1: we start with an approximation of Bender cut function w, for i = 1, 2, - . - N Step 2: solve the master problem ( 4.15 ) and obtain a new $5 and u; Step 3: solve the slave problems ( 4.16 ) for u: for i = 1, 2, - - - ,N given us Step 4: check if w:(uo) is zero for all i = 1,2, - . - ,N ; if yes, m3 and US are the optimal solution, if not, use the results found in Step 3 to build a more accurate Bender cut function, and go to Step 2 The dimension of the master and each slave problem is n + m which reduces the extremely large dimension of the corrective control problem to manageable levels but at the cost of solving N + 1 optimization problems rather than just one. Each of the N + 1 optimization problems have a large feasible region compared to the protective control problem( 4.8). The master and slave corrective control are easy to solve with excellent performance because feasible region is large and only satisfies inequality constraints for one operating condition whereas the protective control problem has poor control performance may be very costly to implement, and will be difficult if not impossible to compute since it is attempting to meet the control, operating, and the security constraints for N contingencies. The preventive control provides protection against security violations for N simultaneous contingencies that would never be required but are being provided and paid for continually. These difficulties with preventive control have been discussed previously and have led to rejection of preventive control for thermal and voltage problems and adaption of corrective control with just operating and control constraints Ho(:c0, no) and a sufficiently fast corrective update rate so that thermal overloads are not left on the system long enough to cause 99 equipment damage. The corrective control for clogging and loss of control deve10p quickly (2-5 minutes) compared to the duration needed to avoid equipment damage (15 minutes) for thermal violation. A special secondary control will be needed to implement the specific controls developed for each contingency. 4.7 Open Access System Dispatch (OASYDIS) This section is abstracted from [15]. The application of OASYDIS in a deregulated power system is to ensure the security and efficiency of power system operation. The OASYDIS, shown in figure 4.1 is proposed as the principal dispatch control for correcting and preventing thermal, voltage, voltage collapse, and transient stability problems. In this application the normal power system operation is studied as well as the post contingencies. OASYDIS consists of two major modules: 0 Contingency analysis module, which identifies the worst contingencies that makes the system insecure. We can use VSSAD to detect, rank, and diagnose contingencies and operating change cause voltage instability. The advantages of using VSSAD are mentioned in section 3.2 . 0 Security Constrained Optimization module (SCO), which responsible for dis- patching all of the system resources and control. A modified reactive power dispatch corrective control could be used with a different performance index to implement this module. The Security Constrained Optimization problem is formulated in section 5.3 of this thesis. The Security Constrained Optimization module can be used to run twice for each hour in a control center. The Security Constrained Optimization module consist of two action runs 100 a Run 1: in this action run the SCO attempts to dispatch only dispatchable system resources and control such as power generations, loads ,transaction, and transmission controls which includes voltage set point on generators, switchable capacitors, and tap position on underload tap changer. a Run 2: this action run is put into effect immediately if control action run 1 fails to make the system operation secure. In this action run the SCO attempts to reduce generations, loads, and wheeling and transfer transaction plus the transmission controls. The major features of the OASYDIS Security Constrained Optimization module, formulated in detail in section 5.3, are: 1) Objective The performance index of OASYDIS security constrained optimization module is to minimize the operation cost of the system which includes, (a) payments to energy gen- eration companies (or cost of energy if the generation supply is owned or leased by the ISO); (b) cost of transmission losses, reactive power supply, and voltage control; (c) penalty costs for changing bilateral transaction (wheeling and transfer) schedules by the ISO; (d) cost of operating transmission controls including underload tap changer, capacitors, generator AVR set points and (e) costs for curtailing supplies, Pg,, load, PL, , and wheeling and transfer transaction reduction. 2) Controls The control of the OASYDIS includes: (a) all dispatchable energy supplies and loads (P3,, P14), (b) dispatchable supplies and load via bilateral transactions ( Pwheeung, Ptmnder), (c) transmission control (tap position on underload tap changers, capacitor susceptances, voltage setpoint on generator’s AVRs) and (d) curtailment of loads and supplies and bilateral transactions ( P0,, PL,,Pwhee1,-,,g and Brand”) 3) Constraints 101 Constraints Ho(a:o, uo) and H,-(:r,-, vi) include: (a) static network operating con- straints (bus voltage limits on buses, thermal limits and angular difference limits on branches, reactive reserve limits on each reactive reserve basin, and transfer and wheeling limits for each contingency that cause clogging voltage instability or some agent), (b) constraints on individual control variable including limits on generators active power Pg, and reactive power 620,, and limits on changes in transmission controls (tap position on underload tap changers, capacitor susceptances, and volt- age setpoints on generator AVRs). This module should be able to solve very large size networks with more than 10000 buses with all relevant controls, constraints and contingencies. The control in [15] shown in Figure 4.1 is not in this form of the OASYDIS, are price sensitive loads and price sensitive bilateral transactions. This dispatch could be added to an extended formulation. Ancillary services such as regulation, load follow- ing, network stability, real power loss, energy imbalance could also be handled in a security constrained economic dispatch. It may be possible to combine the Security Constrained Economic Dispatch and the OASYDIS Security Constrained Optimiza— tion module but efforts at this would produce such a large complex optimization problem that obtaining solution has been problematical. The auction market or bi- lateral transaction market and the ancillary services market will not be included in the power system dispatch to be formulated in section 5.3 . 4.7 .1 Power System Dispatch Function Power system dispatch shown in Figure 4.1 attempts to output a secure and reliable operating point for a power system. This power system dispatch fimction should also insure a least economic cost real time dispatch of all dispatchable system resources and controls. The power system dispatch function generally includes the following steps: 102 _+Tl Step 1: A State Estimation and Power Flow application to establish a consistent and accurate picture of the power system operating condition. This application will run every 5-30 seconds storing the system operating condition and making it available for use by all other applications of power system dispatch function; Step 2: A Transmission Constrained Economic Dispatch (TCED) application that updates the active power generation dispatch in response to load change and operating changes to the system in an economic fashion. TCED application would run frequently to reschedule the generation by adjusting the generators active power output P0,. Linear programming can be used to implement this application considering penalty for transmission loss and including transmission thermal and voltage constraints. Such a dispatch is corrective and does not require development of a preventive control; Step 3: An On-Line Open Access System Dispatch (On-Line OASYDIS) application that make the necessary modification to the system dispatch to ensure security and efficiency of the system. The application runs every 15-30 minutes or as needed. The On—Line OASYDIS will find the penalty factors and the binding constraints for the TCED application. This dispatch uses all the dispatch- able resources and the controls subject to their limits. It will also reduce the generations, loads, bilateral transactions (wheeling, transfer), add or remove switchable capacitors, change underload tap changer tap positions, and change generator, SVC and synchronous condenser voltage setpoints to ensure security of power system operation; Step 4: An On-Line Voltage Security Assessment (On-Line VSA) application that make sure all bus voltages are within their limits and the power system operation is voltage stability secure. If insecurity is detected via detection of a VSSAD predicted equipment outage and transaction combination has actually occurred 103 on the system, based on using a filter on the state estimator, a trigger would be sent for implementation of the corrective control computed by the Security Constrained Optimization module for that equipment outage and operating change combination. The On-Line VSA runs frequently (every 5 second or more) or as needed. Step 5: An On—Line Dynamic Security Assessment (On-Line DSA) application that makes sure the loss of transient stability after any disturbance or contingency will not occur and that the transiently stable operating is secure. If insecurity is detected than On-Line DSA will implement a set of operation constraints and send it to On—Line OASYDIS application. On—Line DSA runs infrequently or if it is needed. 4.8 State Estimation Modeling and Measurement The knowledge of the voltage control areas and their reactive reserve basins that experience loss of control or clogging voltage instability might require providing state estimation at the lowest subtransmission and higher level distribution network as well as geographically remote regions that contain the locally most vulnerable voltage control areas and reactive reserve basins. The monitoring of these regions is necessary to detect occurrence of equipment outage and operating change combinations that can initiate an uncontrollable spreading voltage collapse. The region monitored might need to be outside the region where the ISO has control responsibility if one or more of the locally most vulnerable reactive reserve basins have their bifurcation subsystems [1, 11] partially or totally outside the region of ISO control responsibility. The state estimator for any ISO is projected to have an update rate of 10-30 minutes and covers the region where the ISO has the Transmission Dispatch and Congestion Management responsibility [15]. Such a sampling rate requires imple- 104 mentation of preventive controls for thermal, voltage, and voltage instability detected by the contingency selection module in the OASYDIS as discussed earlier. A 10—15 minute update period for a state estimator can’t detect the equipment outage and operating changes in sufficient time to correct them to avoid thermal damage and insulation damage. This state estimator update period cannot detect the occurrence of the equipment outage and operating changes quickly enough (5-30 seconds) after the initiating events to allow implementation of precomputed controls to correct the voltage instability problem. State estimation is currently provided in a single util- ity with a 5-10 second update period when required. This type of sampling period is thus proposed for an ISO or auxiliary control center. This sampling and control update period would allow for detection of equipment outage and operating changes that cause thermal and voltage limit violations and update of the corrective tertiary (security constrained optimization) based control to correct the violations before they cause equipment damage. This sampling and control update period would allow im— plementation of corrective control for voltage instability via the scheduling or dispatch changes as described earlier in this chapter. A state estimator based on the Wide Area Measurement system’s 5 second update period would be sufficient to detect the equipment outage and operating change, trigger a precomputed and stored control, and implement the control before the 2-5 minute time frame of development of the classic voltage instability problem. The classic voltage instability is produced through (a) action of distribution level tap changers and switchable shunt capacitors to bring back the voltage and load in the distribution system and (b) a sufficient number of generator field current limit controllers to reduce field current back to continuous rating. This sequence generally requires between 2-5 minutes. The 5 second sam- pling period is short enough so that each voltage collapse initiating equipment outage and operating change as well as each of the above steps (a,b) in the development of the voltage instability will all occur between samples of the state estimator and thus 105 will be detected. The precomputed corrective control could overcome each of these changes if they were predicted properly and thus the precomputed corrective control could be self checking. Despite the capabilities of the secondary voltage just described, a state estimator based on Wide Area Measurement system and the secondary voltage control would not have sufficiently fast sampling rate to prevent transient voltage instability that can occur in seconds after the equipment outage. A faster synchronized measure- ment system and secondary control is required if precomputed control are to be implemented to correct for a transient voltage instability problem. A Wide Area Measurement system measurement system using a 5 second sampling period and a secondary voltage control system with a 2—3 minute response time would not be suf- ficient to detect via state estimator any VSSAD predicted equipment outage and operating change combination known to produce transient voltage instability in some bifurcation subsystem; trigger the precomputed remedial control; and implement this control. An emergency control, using a very fast measurement system and secondary control, would be needed to correct transient voltage instability. The measurement and control would be part of the primary control system. 4.9 Secondary Voltage Control A secondary voltage control as proposed by EDF [86] has the objective of producing a strong voltage profile against voltage collapse and assuring that reactive resources are put to better use. Producing a strong voltage profile implies maintaining voltage control in all reactive reserve basin voltage control area. Assuring reactive resources are put to better use could imply making sure that reactive resources are postured or used in a corrective control against voltage stability insecurity. The EDF sec- ondary voltage control [86] achieved these objective by distributing reactive reserves 106 by adjustment of generators excitation voltage set points in each zone so that all generators operate at an equal percentage of their reactive capability. If insufficient reactive reserves are available in a zone, capacitors can be switched in and under load tap changer that pump reactive out of the generators toward the load are blocked or reversed via secondary voltage control. The EDF secondary voltage control has a response time of 3 minutes and thus can sometimes correct a specific developing voltage collapse but certainly can always aid in this correction. In most cases, the EDF secondary voltage control would help prevent voltage collapse for a subsequent equipment outage and operating change. A far more capable secondary corrective control is proposed as being computed or updated every 10—15 minutes and implemented with a 5 second sampling rate as part of the Open Access System Dispatch (OASYDIS). This secondary control would be aimed at correcting a very specific voltage collapse that is developing because the equipment outage and operating change combination predicted to produce the voltage collapse by the VSSAD has occurred and has been detected via a filter on the Wide Area Measurement system state estimator. The switching of capacitors, under load tap changer position adjustments, and generator excitation voltage set point is determined in an optimal fashion in the Open Access System Dispatch using a set of optimization problems: 1. to eliminate the clogging or loss of control voltage instability on one or more bifurcation subsystems for a specific equipment outage and operating change combination; 2. to prevent loss of control or clogging voltage instability from developing in other reactive reserve basins and their test voltage control areas; 3. to utilize the fewest control change to achieve objective (1) and (2); 4. to minimize ancillary cost for the control changes used; 107 5. to posture the base case control setting on the entire system to make (1—3) possible for every equipment outage and operating change combination found by VSSAD to produce voltage instability. The secondary corrective control for each specific equipment outage and operat- ing change combination would be stored, triggered, and implemented once the state estimator with a 5 second sampling period detects the occurrence of that equipment outage through a specific filter that is attempting to detect the occurrence of each one of the equipment outages and operating change combinations predicted to pro- duce voltage instability by VSSAD. This secondary voltage control has available to it stored control changes that correct every developing voltage instability produced by an equipment outage and operating change predicted by VSSAD to produce volt- age instability. If a control change cannot be determined for this secondary voltage control using switchable capacitor, under load tap changer tap position, or generator excitation control voltage set points as controls then an emergency secondary voltage control could be used. An emergency secondary voltage control could also change or curtail transaction and even curtail load and generation if the secondary voltage control could not achieve stability and security. Such an emergency secondary voltage control is far beyond that used in EDF but would be necessary if the stability and security of the system was jeopardized and there was no other method for achieving stability and security for specific equipment outage and operating change combinations. 4.10 Optimization Requirements The algorithms to be used for solving the Open Access System Dispatch must be selected with care because 1. the power system model is large and nonlinear. The number of network nodes 108 or buses can be as large as 20000. The voltage and voltage collapse problems occur on stressed system where the nonlinear effects dominate; 2. the number of control constraints, operating constraints, and security con- straints can be huge. The algorithm must not only find a feasible solution that satisfies these constraints, but an optimal set of binding constraints that characterize the optimal solution; 3. convergence of the algorithm must be rapid and robust from any starting point; 4. there must be a rapid convergence to finding the set of binding constraints as there is to find the optimal control and state of the power system model; Nonlinear programming via interior point methods described in Chapter 2, has undergone rapid development over the last fifteen years that allow second order rather than linear convergence to the optimal solution and ability to assure such convergence in selecting the binding constraint set as well as selecting the control and state for the particular optimization based control problem. 109 can?» 2683. - v.83.— - 8:82 music “580% non-3.— - _ :5.— .e Sauna. 8035. 3:65 8:85 .. m : .52. __8 gang. 3:28 6.38 Sang 8:8:— §§s§ Egan as ass use...“ 3.55 . _ .. 52. é seams assess Essa as .82 5&3 5.25 > Tabasco maufloao 3.8: —l “55383.4 E05383 bgoom @508 282.3 “M ems—P» Ti 55 552 - ._..52§§a€2 3 a E as. 5&5 .. 2:52. 8.. £833 5298 I _ .52. r... .2 53%.“. l 88m; A 55 s v 822% 5:22 Figure 4.1. Structure, Modules and Interfaces of the OASYDIS Application. 110 CHAPTER 5 Open Access System Dispatch We begin this chapter by briefly describing the most important components of power system that are included in any load flow model. The load flow equations are derived for the general case, the constraints associated with the load flow balance and the physical limitation of the devices used in this load flow model are defined. The formulation of three optimization problems: (1) minimum control solvability problem, (2) the minimum ancillary services cost problem (slave problem) and (3) the master problem are developed. Finally numerical results of applying the minimum control solvability for loss of control voltage instability and the minimum control solvability for clogging to three examples of clogging voltage instability are presented. The cases solved are the worst contingencies that were found to produce loss of control and clogging voltage instability in Chapter 3 where VSSAD recommendation obtains a load flow solution. This VSSAD recommendation based solution is the starting point for the minimum control solvability problem that attempts to retain solvability of the solution without the impossible or onerous VSSAD recommendation of increasing generator reactive capability or performing load and generation shedding. 111 5.1 Load Flow Modeling Load flow has been found to be exceptional tool for assessing power system solvability and stability analysis. The load flow model consists of (a) a set of buses interconnected together by a transmission system consisting of transformers and transmission lines, and (b) generators and loads connected to the buses of the system. Real and reactive power is injected into the transmission system by generators. The basic load flow problem solves for the voltage at generator buses and both voltage and phase at load buses given the voltage and real power injection (generation) at generator buses and real and reactive load demands at load buses. The load flow problem also determines real and reactive power flow, line current, and transmission losses on transmission lines and transformers. The basic load flow problem involves a large number of nonlinear algebraic equations. In this load flow analysis, a balanced three-phase system operation is assumed. The purpose of a power system is to deliver power to meet customers demand in real time without any violation in voltage at any bus, thermal overload violation on any branch, and to assure that stability of the solution is preserved. 5.1.1 Power System Components The following are some of the most important components of power system that are included in any load flow model: 0 Power generators 0 Transformers 0 Transmission lines o Shunt capacitors and inductors 112 e Loads the components such as series capacitors,and DC transmission line with associated converter stations may occur in some load flow cases. Generators Generators i can usually generate specified amounts of real power Pg, at specified terminal voltage V,-. A generator can also produce positive or negative reactive power QC“ depending on the excitation level [12]. The constraints of the power generators for the load flow model, are: Poi..." S PGi S Pain... QGi....-n S QGi S QGim“ i (5-1) Vi S Vi S Vi min ma: J In a load flow, the terminal voltage V, and the real power generation Pg,- are specified at a valiie that satisfy the constraints. Reactive power is proportional to the field current. As reactive power generation Q0,- is increased such that field current exceeds the continuous rating limit and is ultimately reduced after a delay by the maximum excitation limiter, the terminal voltage starts to drops and voltage collapse may begin to develop due to the drop in reactive supply QG, from the generator. . Transformers The transformer branch can be represented by an equivalent II model. The derivation of H equivalent model that applies to fixed and variable tap as well as to under load tap changer. The tap ratio can be a real number or complex number indicating phase shifting characteristics. In this model [12] Im = tZVmK + t2VmYl — thK (5'2) 113 I,c = —tv.,,Y,+V,,Y, (5.3) this reduces to Im t2(}fs + Yl) —tYl Vm = (5.4) II: —th Y1 Vic In most practical cases, Y, is set to zero. Horn the above result we can easily show that Ysmk = —t(1 " 0Y1 (5-6) Yam = (1 — tle (5-7) Transmission lines The transmission line for each branch is represented by a resistance R and inductance L. The resistance represents the resistance of the aluminum cable in each line and the inductance represents the effects of the flux linkages set by current in each line. Both R and L depends on the size and the construction of the cable as well as it’s length. The transmission lines in the transmission system can be represented by it’s equivalent H model. In this model [12] 1 Ymk = —Z_k (5.8) R+ 'wL ”2 , _ G + ij “2 7d Km]: — (m) tanh (3 (5.10) 7 = E [(R+ij) (G+ij)]1/2 (5.11) 114 where: d is the line length in miles i = Fl R + ij is the series impedance per mile G + ij is the shunt admittance per mile The impedance ka and the admittance Ymk are defined as follows ka = Rmk+ijk (5.12) Ymk = Gmk-l-ijk (5.13) where Rm], and ka are the line series resistance and reactance respectively, Gm], and Bmk are the line series conductance and susceptance respectively Km). corresponds to shunt capacitance in parallel with shunt conductance, and is given by Kink : Gsmk + stmk (5.14) Shunt Capacitors and Inductors Shunt capacitors and inductors are devices used for voltage and reactive power control in the network system. These devices are turned ON or OFF depending on the current operating condition. Capacitors ( BC > 0) need to be turned ON to increase reactive production and hence increase bus voltage. This occurs when voltage tend to decrease to a point less than the acceptable level (typically 0.95 pu) due to high real and reactive demands specially in distribution system (near loads ). If real and reactive demand is low on transmission lines near the generators, line charging may 115 be sufficient to over come reactive power losses ( I 2X,,.,,,) and bus voltages increase. Inductors ( B; > 0) may he need to be turned ON to create an effective reactive load and reduce the bus voltages. In the general formulation of the load flow one does not know a priori whether inductors or capacitors banks need to be ON or OFF. This is determined as part of the iterative load flow solution process. Load A load is extremely complex and involves many (a) small devices such as appliances, light, and so on, and (b) large components like arc furnaces, large motors, large engines which are considered industrial loads. The demand power or load is produced by generation buses and delivered to these loads through the transmission system, subtransmission system, and finally the distribution system. The power generated goes through several level of step-down transformer before reaching these loads. The affect of distribution level voltage controls, under load tap changers and switchable shunt capacitors, keep steady state level voltage and load equal to precontingency levels. Thus, the loads are modeled as constant power. 5.1.2 Load Flow Equation A simple two-bus system representation will be analyzed first and then we generalize the load flow equation to an n-bus system [12]. Figure 5.1 shows a simple two-bus system. We have a generator and load at bus 1; at bus 2 we have only load. Bus 1 is connected to bus 2 through a transmission line, whose H-equivalent model is represented. There is also shunt capacitor bank connected to bus 1. We define the complex per phase injected power SI at bus 1 as follows 31 = Sc, — SD, (5.15) 116 where 50, and SD, are the complex power generated and load consumed at bus 1 respectively. The complex power injected at bus 1 can be written in rectangular form, 51 = P1 'l' jQ1 (5.16) where P1 and Q1 are the active and reactive power injected at bus 1 respectively and i=./——1. The net active and reactive power injected into bus 1 are defined as follows P1 = Pg,—PD, (5.17) Q1 Q01 — Q01 (5.18) where Pa, , Q0, are the active and reactive power generated at bus 1 respectively, and PD, , Q D, are the active and reactive power load at bus 1 respectively. The complex voltage and current at bus 1 are given as: v1 = Vlech(j51) (5.19) 1:1 2 II exp(jC1) (5.20) where V1, [1 are the voltage and current magnitude at bus 1 respectively, and 61 and (1 are the phase angle of voltage and current at bus 1 respectively. With these above definitions, we can obtain the complex power in term of 117 complex bus voltage and current, 5 = V11; (5.21) where * denotes a conjugate value. 51 51 V1 exp (761) [0312 +11le + 3312)] + (V1 6331? (3'51) - Va 8931? (152)) [012 +1312] (5-22) _ V1 6’31) (.751) [V1 8319 (‘j51l [0.912 — .71le + 3.12)” + V1 exp (j51)l(V1 69319 (“351) -' V2 €33? (“152” [012 — 1312]] V1 61‘? (351) V1 6131? (*151)le12 - 113.1 + 3:12)] + V1 6331? (151) V1 83319 (“151)lG12 —j312] — Vi 8931? (3'51) V2 833? (*j5zilG12 —jBlzl V12 [Gan — 31331 + 3:12)] + V12 [012 — jBrzl " V1172 6331? [.7151 — 52)] [012 — jB12] — V12 (0.912 + G'12) ‘71/12 (Bu + 8.912 + 312) ‘— V1V2 €171? W51 — 52)] [012 — 1312] (5-23) V12 (Gm + G'12) ‘- J'Vlz (Bel + 3.12 + 812) - 1/11/2 [(COS (61 — 62) +jSlIl (61 — 62)) (G12 — jBl2)] (5.24) 1/12 (0,12 + G12) — 1/11/2 [COS (61 — 62)G12 'i' sin (61 — (52)B12] —j[l/12(B.91 + 3.12 + 312) + V1V2[Sin (51 — 52)Glz — 118 COS (61 '— 62)B12] (5.25) The real and reactive power balance, obtained by separating the real and imaginary components of ( 5.25) and they are given as following: P1 = Pg1 -—- PD1 P1 = 1/12 (0512 + 012) — 1/11/2 [COS (61 — 62)Glz + sin (61 — 62)B12] (5.26) Q1 : Q01 _ Q01 Q1 = -V12 (831 + 3.12 + 312) — V1V2lSiD (51 — (5)012 — C08 (61 — 62)Bl2] (5.27) The above expressions can be easily extended to represent the n—bus system: P1 = VlzGu — V) Z Vm [cos (6; — 5m)G1m + sin (61 — 6m)Blm] (5.28) m€k(l) Q1 = —V12Bll — W Z Vm [sin (61 - 6m)G(m — COS (61 — 6m)Blm] (5.29) m€k(l) where n = number of buses in the system, k(l) is the set of all buses connected to bus l, and Cu = Z (Gslm‘l'Glm) (5.30) mEk(l) Bu = B,.+ 2 (B,,..,+B,,,,) (5.31) m€k(l) The branch complex power flow is computed as follows, 5... = VmI,* (5.32) 119 = Vm(Vm _ Villa/m1): = Vm exp (jdm)[Vm exp (jdm) — VI exp (j61)1(Gml - ijI) = Vm exp (jcim) Vm 69319 (-j5m)(sz " ijI) _ Vm 6931’ (3'5...) V, 8931? (“j5z)(sz - ijz) 5m! = VflGm, - jBnu) - VmVlKexp (j(5m - 51))(Gm1 — Bmt))l (5-33) 2 V,?,Gm1 — jViBmz — Vsz[cos (6m — 6)) + jsin (6m — 61)][Gm1 — ijz] Sm; = V,,2,sz — le/j[cos (6m — 60G...) + sin (6m — 603mg] + j (_VniBml _ VmW[Sin (6m "‘ 600ml "' COS (6m —' (”)BmlD (5.34) Once again we can separate ( 5.34), into real and reactive branch power flow: Pm, = Vij; — VmW[cos (6m — MG,“ + sin (6m - 603mg] (5.35) Qmj = —V,3,Bm( — Tani/([8111 (6m — 6l)Gm.l — COS (5m — (ll)Bmz] (5.36) Now we will represent all of the complex elements derived earlier by rectangular coordinates. The general form of the load flow equations for n—bus system earlier will be represented here by the rectangular form. The complex bus voltage V,-, can be replaced by rectangular form as follows Vi = 6i + jfi (5'37) 120 where j = \/—1. Given the model current injection vector I produced by converting all equiv- alent models of loads or generations sources into Norton equivalents, equations for n—bus system can be written in the form [[12qu = [Ybusl [Vbusl (5'38) 11 Yu Y12 Yln V1 12 = 1’21 Y22 16.. V2 (539) In J Ynl Yn2 ° ' ' ° ° ' Ynn Vn where: [me],-,- = sum of all admittance connected to bus i, (including the admittance of the H equivalent model for the transformer and the shunt capacitors connected to the bus i if any) [Ybu,],-j = - sum of all admittance connected between bus i and bus j ,(including the admittance of the II equivalent model for the transformer) and Ii : 2(1/01/1) ; Z: 1:21'°°rn (5'40) j=l where Yij = Gij + jBij. As a result, if the fact that a constant power load model is used we have i ‘ Pi — ' i 1,: (5) = ——’—Q— (5.41) Vi 6:“ "jfi 121 and thus, the nodal admittance matrix current equation can be written in the power form, a — is. = (e.- — if.) imam (542) 1:1 Substituting ( 5.37) into ( 5.42) and after some simplification, we have the real and the reactive power balance equation in rectangular form Pi — jQi = Gijej — B.,-f,-) + fi 2(Gijfj + Brier) '=1 2;, {f‘ ‘(i (Giiej — Bijfj) — 6.- imufi + Bijejl} (5-43) j=l i=1 and as a result we have R = e'. li(G¢j6j—B Bijfj) +fi [2(01jfj +B§j€j) (5.44) 13:1 . i=1 . Qi = fi i(Gijej—B Bijfj) _ei li(Gijfj+Bijej) (545) _j:1_ _jzl J 5.2 Constraints In normal and secure case with safety margin sense, the system must satisfy a set of algebraic constraints that can be written in the form of equality and inequality constraints as follows: G(:r, u) = 0 122 H(:1:,u) Z 0 When the system is insecure or sustains violation, the goal is to come up with a preventive or corrective control to eliminate the violation in the system as soon as possible. Some violation cause equipment damage if they persist. In other cases the system enters the state of instability when some equality or inequality constraints are violated. Both types of constraints are used in this proposed problem. 5.2.1 Equality Constraints The equality constraints corresponds to an AC power flow model. The equality constraints are ones that must be exactly satisfied to have a feasible solution. The power flow model ( 5.44) and ( 5.45) is written here again as follows: ' n ‘ ' n l P,- = e,- 2(Gijej_ Bijfj) + fi 2(Gijfj + B,,-e,-) (5-45) _J'=1d i=1 . [ n l ' n ' Q.- = f.- 2(Gijej— 3.4-fr) - e.- Z(G.-,-f,- + B,,e,-) (5.47) .jzl . J=1 .l where i = 1, 2, - - - ,n and n is the number of buses in the network. 5.2.2 Inequality Constraints In this proposal three kinds of inequality constraints are used. They are operating constraints, control constraints, and security constraints. The inequality constraints must satisfy either an upper or lower bound or both on the value of a variable or function for the system to have a feasible solution. Thermal limits, voltage limits, and reactive supply limits expressed as (5.48) 123 V,- SV, SV, 'i=l2... mas) ) 3 , L (5.49) min QGim... S QGi S QGimu; Gi = 1,2..., M (5-50) where ij is the current flow in branch ij, L is the number of load buses, and M is the number of generation buses in the system. These are usually the operating constraints, The control constraints are the tap position limit on under load tap changing transformer limits, the switchable capacitor susceptance limits, and voltage set points on generators Atmin S At S Atm‘n; t: 1,2, ...,T (5.51) Cshmm S 0,}, S Cmm“; Sh = 1, 2, ..., S (5.52) Vain,“ S VG,- S Vain"; Gt: = 1,2..., M (5.53) where T is the number of transformers in system, S' is the number of switchable capacitors, and M is the number of generation buses in the system. The security constraints are the thermal and voltage limits in the operating constraints plus the changes due to occurrence of contingency. The security constraints can be represented by 1%,, _<_ 13. + A13; _<_ 1,,m (5.54) “min S K0 + AW S mm“: (5.55) where ([3, V?) and (AIS, AVE) are the current and voltage of the base case operat- ing condition and the changes due to contingency p respectively, where p = 1, 2, ..., N and N is the number of contingencies considered. 124 5.3 Open Access System Dispatch Security Con- strained Optimization The Open Access System Dispatch problem is composed of Contingency Selection Module and a Constrained Optimization Module as described in chapter 4. The Contingency Selecting Module finds all combinations of all equipment outage and one or more transfer and wheeling transactions violation of minimum reactive reserve limits on some reactive reserve basin or cause clogging voltage instability for some agent. VSSAD can identify the combinations of an equipment outage with one or more transfer and wheeling transactions that either 1. violate minimum reactive reserve limits on one or more reactive reserve basins. VSSAD will automatically detect the reserve basins with reactive limit viola- tions as well as generators where added reactive supply would cause satisfaction of all violated reactive reserve basin minimum reactive limits. VSSAD would also determine the level of additional reactive reserve or reactive supply capa- bility needed on each of these generators to avoid reactive limit violations; 2. exhaust all reactive reserves on one or more reactive reserve basins, do not have a load flow solution when reactive limits are enforced, but have solution when reactive reserve level increases are added on specific generators, SVCs, or synchronous condensers. VSSAD would also determine the level of additional reactive reserve or reactive supply capability that must be added to each of these generators to obtain a load flow solution; 3. that do not have a load flow solution when reactive limits are enforced or when they are ignored, but have solution when one or more transactions are modified 125 or curtailed, or load and generation is shed at appropriate buses. VSSAD would determine the transaction to be curtailed and the level of curtailment, the transaction to be eliminated, and the level of load and generation to be shed. This development of VSSAD capability is a very large contribution because with— out VSSAD (a) it takes 15 hours engineering and computing time of trial and error by an engineer per equipment outage to obtain a solution when no solution exists, (b) there is no guarantee that the trial and error solution correctly diagnoses the cause, and (c) there is no guarantee the suggested action has the minimal affect on current power system operation. Even though VSSAD provided changes may be diagnostic in determining and verifying the actual cause, it is not clear how the needed reactive reserve can be obtained. It is also not clear (1) what is the minimum set of control actions that are most effective is providing the reactive reserve or reducing network losses in particular agents and (2) what are the control changes that both provide the reactive reserves or reduces the reactive losses and yet require a minimum cost for ancillary services for each equipment outage and transaction change combination identified by VSSAD. Additional reactive reserves are needed in particular agents either because a load flow solution can not be obtained due to loss of control volt- age instability without additional reserves on specific generators or because minimum reactive reserve levels on one or more reactive reserve basins are violated after an equipment outage and operating change combination. VSSAD recommends curtail- ing one or more transactions by cutting real and reactive load in a voltage control area and active generation in its reactive reserve basin generators to reduce the network losses that cause clogging voltage instability for that agent or voltage control area. If is not clear that such a drastic action as curtailing one or more transactions of power marketers and generators is actually necessary to obtain a solution. If transac- tion curtailment isn’t necessary or if other control actions can reduce the transaction 126 curtailments, what are the minimum set of controls, the most efiective set of control changes, and the minimum ancillary cost for the set of control actions that will totally or partially eliminate the transaction curtailment. Two optimization problems are posed; for (a) adding reactive reserves and (b) for minimizing transaction curtailments: 1) Minimum Control Solvability Problem. A minimum control solvability problem either (a) obtains the reactive reserves on generators in the amounts VSSAD determines are needed to obtain solution of the load flow or so that minimum reactive reserve basin reserve requirements are met or (b) obtains solution of the load flow without curtailment of the set of transac- tions determined as needed via VSSAD for a solution to exist. The controls include (1) adding switchable shunt capacitors, generators or synchronous condensers close to the generators needing additional reserves (loss of control voltage instability) or the voltage control area needing reactive supply (clogging voltage instability); (2) adjusting AVR voltage set points to reduce reactive power pickup on generators that need reactive reserves (loss of control voltage instability) or to increase reactive gen- eration supply rate out of generators, SVCs, or synchronous condensers outside and sometimes inside voltage control areas needing reactive supply (clogging voltage in- stability); (3) adjusting tap position on distribution and subtransmission level under load tap changers to reduce distribution level voltage and load (loss of control volt- age instability) or to adjust tap position to avoid tap position limits and allow more effective pumping of reactive into the voltage control area needing reactive supply (clogging voltage instability); and (4) reducing active generation out of generators that need reactive reserves (loss of control voltage instability) or redispatch active generation from one or more set of generators to another set of generators to relieve reactive clogging on particular paths to the voltage control area needing reserves. 127 The objective function for the minimum control solvability problem is to reduce reactive generation and thus provide the additional reactive reserves VSSAD decides are needed (loss of control voltage instability) or to totally eliminate the need to curtail the transactions VSSAD decides are necessary to obtain a load flow solution for a particular equipment outage and operating change combination (clogging voltage instability). The VSSAD recommendations provide a starting feasible solution for the interior point algorithm that is also a load flow solution. 2) Minimum Ancillary Services Cost Problem The minimum ancillary services cost problem has an objective of minimizing the cost of ancillary services required to correct a clogging or loss of control voltage instability problem while maintaining solvability for a particular VSSAD determined equipment outage and operating change combination. A minimum ancillary cost problem uses the solution of the minimum control solvability problem to achieve as a starting solution that is both feasible and solvable in terms of satisfying reactive reserve basin reactive reserve constraints, thermal constraints and voltage constraints. The objective is to adjust controls (1-4) to minimize ancillary services cost to the ISO. This minimum ancillary services cost can also further attempt to further reduce the number of controls used in the minimum control solvability problem solution. The minimum control solvability and the minimum ancillary services cost prob— lems are the slave problems in a Benders decomposition. There is a slave problem for each equipment outage and operating change combination that VSSAD determines needs reserves to meet minimum reactive reserve basin requirements, to avoid loss of control voltage instability or that experiences clogging voltage instability. The control determined out of the minimum control solvability and minimum ancillary services cost problem sequence is 11.". The slave problem is generally just one optimization problem not two as discussed above. One set of objectives is (a) to obtain some control change that obtains a so- 128 lution and (b) find the minimum control set that will obtain that solution when no solution exists without the control change. These two objective are met by the mini- mum control solvability problem. An additional objective is to find an even narrower set of controls that are effective in obtaining a load flow (solvability and feasibility) solution and minimize the cost of ancillary services. This third objective is met from the minimum ancillary services cost problem. It was thought that one optimization problem could not achieve all of these objectives, and therefore a sequence of opti- mization problems is proposed. The minimum control solvability problem is much like the optimizations used to obtain feasible starting solutions for an interior point algorithm. The minimum control solvability is seeking a feasible starting solution for the optimization but also one that has a load flow solution. The master problem attempts to adjust the control no to posture the system so it is not as vulnerable to the set of equipment outage and operating change combinations via minimizing the ancillary services cost increase and holding it under a certain maximum for each equipment outage and operating change combination identified as causing loss of control or clogging voltage instability in some voltage control area and its reactive reserve basin. 5.3.1 Minimum Control Solvability Problem The minimum control solvability problem depends on whether clogging or loss of control voltage instability occurs because 0 VSSAD recommends adding reactive reserves Q3, to each generator i E I ,- for loss of control voltage instability for the j t” equipment outage and operating change combination. 0 VSSAD recommends reducing a set of a transfer or wheeling transaction for clogging voltage instability for the 3"" equipment outage and operating change 129 combination. The procedure used in VSSAD to determine these precise recommendation is given in chapter 3 and omitted here. The objective of the minimum control solvability prob- lems for any VSSAD recommendation is to completely avoid having to take VSSAD recommended action by adjusting the voltage controls and possibly as a last resort adjusting active power dispatch. The minimum control solvability problem for loss of control voltage instability is now formulated. The minimum control solvability prob- lem for loss of control voltage instability must add reactive generation and generation capacity 623;, to each generator i by modifying Qgim and Qgi. at. = cams; (5.56) 625 = 62% + 3 (5.57) imm: "run: where Q3, is the reactive supply added at generator i in order to obtain a load flow solution for the j‘h equipment outage and operating changes. Q’G. is the generation capacity after Q3; is added to the base case. Q23"... and Qéim“ is the maximum reactive generation in the base case at generator i and the maximum reactive supply at generator i after Q3; is added as VSSAD recommends to obtain a solution for the jth equipment outage and operating change respectively. If Q0 i = QOG,,,,,,, is the reactive supplied by generator i before reactive supply capability is added to obtain a load flow solution, ( 5.56) is reactive generation after the equipment outage and operating change j occurred. Use Q5, as a starting point for the optimization problem. These modification are needed so that the load flow and optimal power flow has a solution to start the optimization that it would not otherwise have without adding reserves 623;. to generators i E I ,- for equipment outage and operating condition change j. The minimum control solvability problem 130 for loss of control voltage instability will minimize this added reactive generation Q3, 21%.. - 62?:.-,,..,,)2 = 2&3}? (5.58) iEIj iEIj subject to the reactive reserve basin, thermal,voltage and control constraints that specify three successively large control sets. The constraints are PGi — PDi = 65 2(Gijej - Bijfj) + fi 2(Gijfj + Bijej) (5-59) J=1 . a=1 . r n - n 1 QGi — Q0; = fi 22(93ij — Bijfj) - 6i l (Gijfj + Bijej) (5-60) i=1 4 1:1 . PGimm S Pg; S Pagm" ; 1:: 1,2, ...,M (5.61) QGimgn S QGi S QGimag ; i = 1: 2) "'7 M (5'62) V.2 < e?+f,? g V? - 2': 1,2,...,n (5.63) ‘min — ‘maz l 1?. s [(e.—e.-)2+(f.-—fj)212[02~+B.-2,1SI? -z‘j=1,2....,Br (5.64) 'Jmin i] ‘jmaz ’ Atmin S At S Atmaz ; t: 1,2, ...,T (5.65) Csh < Csh S Cs min — Ran g 2 1c}... —QG, ; r = 1,2,...,R (5.67) henna. .... ° sh = 1,2, ...,5' (5.66) hma: ’ where: M = number of generator buses in the network n = number of buses in the network Br = number of branches connected between buses in the network T = number of transformers connected between buses in the network S = number of shunt capacitors in the network RRM," is the minimum reactive reserve in each of the reactive reserve basins R = number of reactive reserve basins( RRB, ), and i, is the number of the generators in each reactive reserve basin. 131 The minimum control solvability problem is really a series of problems with successively different control sets. There are four control sets proposed: 1. generator AVR set point voltage on generators i E I,- and generators near 1: E 13'; 2. switchable shunt capacitors electrically close to generators i E I j ; 3. under load tap changer tap position; 4. active generation at generators PG.- ; i E I ,- and generators near set i E 1,. The control sets used are 1 and 2, 1-3, and 1-4 where if adding a different control didn’t result in a significant improvement in obtaining solution, it was no longer used in a large control set that added control of another set of control devices. There are six very major difference in this optimal power flow problem formulation from the previous literature 1. a set of reactive reserve basin constraints that assure load flow solution when loss of control voltage instability occurs. Since tap changers and capacitors are treated as continuous variables in an optimal power flow, the Hessian of the optimal power flow problem is not singular when the load flow model and the actual power system are experiencing instability. If tap changers and capacitors were treated as discontinuous variables, and the maximum excitation limiter action was more accurately reflected in the Optimal power flow, there would never be a solution to the optimal power flow that could not be a solution to the load flow when clogging voltage instability is assumed to be impossible on a particular system. The reactive reserve basin constraints with very modest levels of reactive reserve (10 MVAR or loss) on each helps guarantee that the optimal power flow has no solution when the load flow has no solution due to 132 loss of control voltage instability. This is true because if all generators on a reactive reserve basin are at reactive limits, no load flow solution may exist and the optimal power flow has no solution due to violation of the reactive reserve basin constraint; 2. when the optimal power flow has a solution since every reactive reserve basin has reactive reserves and thus loss of control voltage instability can not occur on this system, an optimization is possible that attempts to maintain a stable load flow solution while minimizing the VSSAD recommended addition of fictitious generators with fictitious reactive supply capability. This is impossible without reactive reserve basin constraints on all reactive reserve basins from (1) above since one could not be sure that optimal power flow solution would imply a load flow solution and vice versa for system that can only experience loss of control voltage instability. VSSAD recommended action of adding generators at particular busas i E I,- with starting generation Q3; assures the load flow has a solution and that the optimal power flow with reactive reserve basin constraints has a starting solution; 3. a performance index that is quite different from optimal active and optimal reactive dispatch (total active or reactive network losses). This performance index minimizes a quadratic performance measure Ema)? (5.68) iEIj 4. (23‘, i E I, were added fictitious generators in the model after the equipment outage to obtain solution to the load flow and starting point for the optimiza- tion model that also has the equipment outage. Fictitious generators were not added into reactive reserve basin constraints that quite possible could have in- 133 cluded this fictitious reactive generation. Using a modest level of 10MVARs on the reactive reserve basins that could but do not contain these fictitious genera- tors suggests that these reactive reserve basin reserve levels must be maintained as the optimization proceeds using the network model where the contingency has occurred so that load flow solvability in maintained. These reactive reserve constraints maintain these very modest reactive reserves as the fictitious gener- ators reactive generation is reduced to zero and the reserves on these reactive reserve basin generators would have been driven to zero and negative without the reactive capability constraints on each generator and these reactive reserve basin constraints on these reactive reserve basin; . the reactive reserve basin constraints on other reactive reserve basins are set at modest level of reserves to prevent loss of control voltage instability and thus voltage collapse on these other reactive reserve basins. These reactive re- serve basin constraints allow borrowing of reactive reserve on all reactive reserve basins not needing additional reserves but prevents the solution from exhaust- ing all reactive reserves in any reactive reserve basin causing it to produce a voltage collapse; . the use of four different control sets of increasing number of control rather than one that contains all four control sets allows the problem to select a minimum control set; This set of optimization problems attempts to reduce 62%., rather than add the reactive supply capability 623;. on a set of generators I ,- that will obtain a load flow solution. If the performance index is not zero with control set (1), then control set (1) and (2) is used; and if the performance index is still not zero using control sets (1) and (2), then control set (1), (2), and (3) is used since zeroing the performance index assures solvability with reactive limits Qgim“ enforced. Control set ( 1), (2), (3), and 134 (4) is used as a last resort. Control set (1), (2) and (3) has solely ancillary services cost. Control set (4) allows modification or curtailment of transfer and wheeling transactions which is an anathema to power marketer and other generation companies whose business depends on having unfettered and uninterrupted transmission service. These power marketer would receive the cost of the transaction interruption and the profit they would have received if their transaction had been completed, plus some possible penalty which can be a fairly high cost to an ISO. The power marketer may even sue the ISO for damages if the transaction interruptions are frequent. Control set (4) is considered the last choice. The minimum control solvability problem for loss of control voltage instability attempts to find the minimum set of control changes that provide a solution to the load flow and hopefully allow for stable operation if sufficient reactive reserve margin is achieved for the reactive reserve basins found deficient for a specific equipment outage and operating change combination. The minimum control solvability solution for loss of control voltage instability can also be used to add reactive reserves Q3, to generator i E I ,- to allow (1) reactive reserve basins to remain within minimum reserve requirements 1212,, = 10 M VAR after an equipment outage and operating change combination j, (2) allow the load flow to have a solution, (3) avoid loss of control voltage instability, and (4) obtain the reactive reserves from reactive reserve basins that had excess reactive reserves or more likely from capacitors that switch in or tap changers that do not pump reactive power into the distribution and subtransmission system. Clogging voltage instability occurs when a voltage control area can’t obtain needed reactive supply after an equipment outage or operating change occurs because the reactive supply flowing from generators, SVCs, synchronous condensers never reach the voltage control area. The line outage may open an important path for reactive flow to the voltage control area causing clogging voltage instability due to excessive 135 network reactive losses on the other paths to that voltage control area (agent). The operating change may add real and reactive flow on heavily loaded paths or interfaces with rapidly increasing network reactive losses that absorb reactive power that would otherwise flow to and supply reactive needs in the voltage control area ( agent for clogging voltage instability). Minimum control solvability for clogging voltage instability for a particular equip- ment outage and operating change combination has a different performance index than the minimum control solvability for loss of control voltage instability because VSSAD recommended remedy is different. VSSAD recommends cutting load at buses in voltage control area j and active generation at generators in RRBJ- associated with that voltage control area and thus curtail one or more wheeling transactions which would result in a modification of load flow data PL, = Pg—amwheeung z'eVCAj, j=1,2,---,J (5.69) 201,- = 1 (5.70) mlDG‘. = 193,— fl.Pwh..z.-ng 5612123,, j=1,2,---,J (5.71) Zfi. = 1 (5.72) i=1 where PB. and PL,- are the power at load bus i before and after the wheeling transaction is curtailed respectively and Pwheeling is the amount of power wheeling curtailed. P8.- and Pg, are the power at generation bus i before and after the wheeling transaction is curtailed respectively. a, and )6,- are the bus participation factors on the curtailed load and generation, respectively. VGA,- and RRBJ- are the voltage control area and its reactive reserve basin respectively, j represents the j equipment outage and operating change that produces voltage instability in VGA,- and RRijorj=1,2,---,J 136 The minimum control solvability performance index minimizes J 2|: 2 (PLi—nglz'l' Z (PCs—P8.)2 iEVCAJ- iERRB, J 2( Z all-l. Z fit?)P31heeling (573) j=1 ieVCA, iERRB, subject to the load flow equation equality constraints ( 5.59, 5.60) with generation and loads specified by voltage ( 5.63), thermal ( 5.64), and reactive reserve basin ( 5.67)constraints, and control constraint set ( 5.61, 5.62, 5.65, 5.66) corresponding to set (4), ( 1), (3) or (2). These constraints are the same as those in the case of loss of control voltage instability ( 5.59 - 5.67 ). This performance index for the minimum control solvability for wheeling reduction solved clogging voltage instability should be zero requiring Pwheeling = 0 if a solution to a minimum control solvability exists for a particular control set since without Pwheeung being zero there is no load flow solution for the 3"” equipment outage and operating change combination. Using successively larger control sets (1), (1 and 2), (1, 2, and 3) and (1, 2, 3 and 4) sequentially should obtain a solution. A second VSSAD recommended action is to curtail one or more transfer transac- tion PG; = Pgi—aiPmee, 5e15,, j=1,2,---,J (5.74) iEEj Pa, = ng—skp,m,,e, 761,-, j=1,2,-~-,J (5.76) 26,. = 1 (5.77) kEIj 137 where E, is the set of exporting generators and I, is the set of importing generators for the transfer of Brand” from E, with participation factor a, to I, with participation factor 6),. The minimum control solvability performance index that curtails the transfer transactions is Z i=1 2 (Pa.- - P8,.)2 + 2030.. - P8,.)’ {651' kEIj J 2 (Z a? + 2 fit) P3...” (5.78) j=1 iEEJ’ kEIj subject to the load flow equation constraints, reactive reserve basin, thermal, and voltage inequality constraints and the control constraints corresponding to sets (1), (1 and 2), (1, 2, and 3) and (1, 2, 3 and 4). These constraints are the same as in ( 5.59 - 5.67 ). The performance index for the minimum control solvability transfer reduction solved clogging voltage instability should be zero requiring Ptmufe, = 0 if a solution to the minimum control solvability for clogging voltage instability exists for a particular control set. If Ptmmfe, 76 0, VSSAD indicates there would be no load flow solution for equipment outage and operat- ing change combination. Using successively larger control sets, there should be a solution to this minimum control solvability for clogging voltage instability problem. The minimum control solvability problem for clogging voltage instability has unique attributes (1,5,6) of the minimum control solvability problem for loss of control voltage instability. Attributes (2,3) for the minimum control solvability problem for clogging voltage instability are similar to those for the minimum control solvability problem for loss of control voltage instability except that we have to shed generation 138 and load to obtain a load flow solution in attribute 2 rather than adding fictitious re- active generators so our load flow and optimal power flow have solution when reactive reserve basin constraints are imposed. The performance index, addressed in attribute 3, is again quite different than for optimal active and optimal reactive dispatch but is a quadratic measure of the shed generation and load for wheeling transaction or the curtailed transfer level for a transfer transaction. The solution to the optimal power flow occurs when the level of shed load and generation in a wheeling transaction or the level of the curtailed transfer for a transfer transaction is zero. The performance index require complete elimination of the VSSAD recommended actions while start- ing the optimization from the VSSAD recommended solution. Note that there are three minimum control solvability subproblems involving three suc- cessively larger control sets (1), (2), and (3) are (a) The minimum control solvability for loss of control voltage instability. (b) The minimum control solvability for wheeling reduction solved clogging voltage instability. (c) The minimum control solvability for transfer reduction solved clogging voltage instability. 5.3.2 Minimum Ancillary Services Cost Problem The minimum ancillary services cost problem starts with the solvability problem so- lution appropriate to the VSSAD recommended solution. It attempts to minimize the total system ancillary services cost for the ISO to correct any VSSAD determined voltage instability inducing equipment outage and operating change. The ancillary services cost for tap changers, switchable shunt capacitors, generators, SVCs, or syn- chronous condenser, and modification of reactive supply generation and change in 139 active generation are a... +b,, 7;, —Tf,’. +c.,. (1",, —T,.‘} )2 (5.79) ac, + qug + chf (5.80) a,, + b,,Qg,. + c.5623, (5.81) a,, + b_.,,PGi + 59,123, (5.82) The performance index for control set (3), (2), (1), and (4) in ( 5.79 - 5.82) is the sum of the cost, 0 (u), of all services used. The performance index for the minimum ancillary services cost problem should not be a measure of cost, C(ug), but a measure of incremental cost, C(u, - no), that measures the cost of the change in control from no to u,-. Such a measure can be obtained via Taylor series approximation of C (u) evaluated at no . The cost given in ( 5.79 - 5.82) are speculative since no such function have been derived or justified. It is anticipated that the price of any ancillary service provided rises with consumption levels or rises reciprocally with decrease in reserve levels. Since the quadratic function are analytically easier to handle they are used in formulating the problem. Note that any positive or negative movement of under tap position from normal tap position T}; has a cost for under load tap changers and cost rises quadratically with consumption of 0;, Qg, and Pg, in ( 5.80 - 5.82). These costs depend on location and how much demand there is for a particular device based on its effectiveness is resolving difficulties with voltage control and resolving difficulties in completing transactions of different power marketers and supplies. Active power generation cost must include the penalty for generators i E E, U I, that is involved transfer transaction reduction or in i E I, for wheeling transaction reduction. Generators i, that experience generation change outside of E, U I, for transfers and I, for wheeling must be on the spot market or provide energy backup for that ISO or for the generating companies involved 140 in that transaction. The cost of this power would likely be higher than could be arranged via bilateral transactions. The constraints include the load flow equality constraints, thermal, voltage, reactive reserve basin, operating constraints, and the control constraints set (1), (2), (3) or (4) found sufficient to obtain a solution to the solvability problem. The starting point for this minimization is the solution to the solvability problem The minimum ancillary services cost problem is a second stage of the slave prob- lem that finds a minimum control subset that not only solves a clogging or loss of control voltage instability problem for a particular equipment outage and operating change combination but also minimizes the ancillary services cost and further reduces the number of control changes of voltage control and reactive supply devices. The objective of the slave problem is so broad that breaking up the problem into two appears to be the only way of solving it. Operators at power system control centers will not implement controls that require changes in several control devices to solve a stability problem because the operators (a) believe they can achieve excellent control that corrects security or stability problems with relatively few control changes and believe that any computer determined control should also be able to obtain corrective control with few control changes; (b) they believe several control changes can cause deviations from nominal operation that result in dynamic instability and additional equipment outages that can result in blackout of their system. The minimum control constraint is firmly enforced by the use of control sets ( 1), (2), (3), and (4) for the solvability problem for each equipment outage and operating change induced insta- bility with no solution. The minimum ancillary services costs problem can be used to further reduce the number of controls used by the solution of the minimum control solvability problem by further restricting the number of controls used in set (1), (2), (3), or (4) selected via the solvability problem. The controls used in the minimum 141 ancillary cost problem would be those found to be most effective in the solvabil- ity problem with a secondary requirement of having minimum ancillary service cost ( 5.79 - 5.82) to correct any voltage instability problem detected by VSSAD. 5.3.3 Master Problem Formulation The optimal reactive dispatch is the master problem, it is the third level of opti- mization problem. The price of purchasing power to provide 12R losses may be the only performance index for selecting no and operating state x0 in the master prob— lem similar to the reactive dispatch performance index that minimize the PR losses ( 4.3). The performance index for the master problem would be F1(xo, no) 2 a + bPum + 613,20“ (5.83) where Pu,” represent the power losses in the network, and a, b, and c are constant coefficients. The price of ancillary services, C (uo), to provide control uo could be added to this performance measure since one is selecting these controls in this problem. The per- formance index for the optimal reactive dispatch for a deregulated power system may be of this form T,, ‘73 + 06(712' — 15;.)2} + 2: {aa + qui + chf} + 2 {am + b,,,Qgi + caiQéJ + 2: {agi + bgiPGi + cgepczh} = F1030, ”0) + C(U; — 11.0) (5.84) F2050; 11.0) = a + bPloss + CH1” + E: {at + bt, where C(u, -— uo) is the ancillary cost services for the control changes (u,- — uo). The master problem is stated formally in ( 4.9). It has equality ( 5.59- 5.60) 142 and operating and control inequality constraints ( 5.61- 5.67) on x0 and ac, and a normed constraint on the control change