TRANSFORMER PROTECTION BASED ON DYNAMIC STATE ESTIMATION

Transcription

1 TRANSFRMER PRTECTIN BASED N DYNAMIC STATE ESTIMATIN A Disseraion Presened o The Academic Faculy by Rui Fan In Parial Fulfillmen f he Requiremens for he Degree Docor of Philosophy in he School of Elecrical and Compuer Engineering Georgia Insiue of Technology Augus 06 Copyrigh 06 by Rui Fan

2 TRANSFRMER PRTECTIN BASED N DYNAMIC STATE ESTIMATIN Approved by: Dr. A.P. Meliopoulos, Advisor School of Elecrical and Compuer Engineering Georgia Insiue of Technology Dr. Maryam Saeedifard School of Elecrical and Compuer Engineering Georgia Insiue of Technology Dr. Ying Zhang, School of Elecrical and Compuer Engineering Georgia Insiue of Technology Dr. Andy Sun School of Indusrial and Sysems Engineering Georgia Insiue of Technology Dr. David G. Taylor School of Elecrical and Compuer Engineering Georgia Insiue of Technology Dae Approved: July 4, 06

3 To my beloved parens Xiancheng and Silun and my wife Xiaoxia

4 ACKNWLEDGEMENTS The docoral sudy a Georgia Tech is an exciing and rewarding journey. I would no have accomplished his grea achievemen wihou he suppor and help of counless people. I would like o express my sincere graiude o all of hem. Firs of all, I wan o specifically hank my professor, Dr. Sakis Meliopoulos, for no jus being my advisor, bu also a menor. He is an erudie professor who provide he guidance hrough my graduae program. His paience and endless suppor helped me overcome numerous difficulies and I could no have imagined having a beer advisor and menor for my docorial sudy. I would also like o hank Dr. George Cokkinides, who always help me wih he experimenal work during my research. His professional experise is always excellen and admirable. Addiionally, I would like specially give hanks and appreciaion o he members of my commiee, Dr. David Taylor, Dr. Maryam Saeedifard, Dr. Andy Sun, and Dr. Ying Zhang, for providing feedback and commenary o my disseraion. Their ime, commimen, and experise have played imporan roles in my research. Special hanks go o all my fellows of he Power Sysems Conrol and Auomaion Laboraory (PSCAL) a Georgia Tech. I would like o hank Bai Cui, Liangyi Sun, Yu Liu, Zhenyu Tan, Dr. Renke Huang, Dr. Dongbo Zhao, Dr. Aniemi Umana, Dr. Evangelos Polymeneas, Yi Du, Boqi Xie, Wenlu Fu and oher laboraory sudens. I is my honor o spend he wonderful years wih you. I am also graeful o oher power group sudens for heir friendship and suppor. iv

5 I am hankful o all my friends a Georgia Tech. They suppored me academically, personally, and especially during difficulies. Among hem special hanks go o Jiaming Li, Chong Han and Ke Li who have provided encouragemen, friendship, and suppor hroughou my graduae sudies. Las bu no leas, I would like o hank my family members especially my parens Xiancheng Fan, Silun Xu and my wife Xiaoxia Min. I is your love, suppor and encouragemen ha gives me he confidence and moivaion o come hrough hose mos difficul days and helps me achieve he docoral degree. v

6 Table of Conens ACKNWLEDGEMENTS... iv Lis of Tables... ix Lis of Figures... xi SUMMARY... xv CHAPTER INTRDUCTIN.... Problem Saemen.... Research bjecives Thesis uline... 6 CHAPTER LITERATURE SURVEY verview Legacy Proecion Mehods Percenage-differenial proecion Harmonic-resrain differenial proecion Negaive-sequence differenial proecion vercurren proecion Vols-over-herz proecion Thermal proecion Gas-and-pressure proecion Recenly Proposed Alernaive Proecion Mehods Frequency analysis, ANN, fuzzy logic, and wavele-based proecion Adapive differenial proecion Summary... 8 CHAPTER 3 THE VERALL APPRACH verview The Proposed Approach Laboraory Hardware Implemenaion for DSE-Based Proecion Scheme Summary... 5 CHAPTER 4 TRANSFRMER ELECTR-THERMAL MDEL verview... 6 vi

7 4. Transformer Physical Elecro-Thermal Quadraized Model Single-Phase Saurable-Core Transformer Physical Elecro-Thermal Quadraized Model Single-Phase Auo-Transformer Physical Elecro-Thermal Quadraized Model Transformer AQCF Device Model Transformer AQCF Measuremen Model Consrucing a Three-Phase Transformer Wye-Wye Conneced Transformer Wye-Dela Conneced Transformer Dela-Wye Conneced Transformer Dela-Dela Conneced Transformer Summary CHAPTER 5 TRANSFRMER PRTECTIN BASED N DYNAMIC STATE ESTIMATIN verview Three Mehods for he DSE Problem Approach ne: UCWLS Mehod Approach Two: CWLS Mehod Approach Three: Exended Kalman Filer Mehod Proposed DSE-Based Transformer Proecion Logic Summary CHAPTER 6 TRANSFRMER MDEL PARAMETER CALIBRATIN verview Parameers Calibraion Auoransformer Parameers Idenificaion Numerical Resuls Summary CHAPTER 7 DEMNSTRATING EXAMPLES: DSE-BASED TRANSFRMER PRTECTIN verview Even ne: Transformer Energizaion Even Two: Secondary Side Coil Faul, 5% from Neural Even Three: 5% Secondary Side Coil Faul during Energizaion... vii

8 7.5 Even Four: Transformer % Iner-urn Faul Even Five: Auo-Transformer % Faul near Neural Even Six: Auo-Transformer ver-exciaion Summary CHAPTER 8 CNCLUSIN AND FUTURE WRK DIRECTIN Conclusion Fuure Work Direcions PUBLICATINS APPENDICES Appendix A: Quadraic Inegraion REFERENCES viii

9 Lis of Tables Table 4 -. Exernal saes of he ransformer Table 4 -. Inernal saes of he ransformer Table 4-3. Through variables of he ransformer Table 4-4. Exernal saes of he auo-ransformer Table 4-5. Inernal saes of he auo-ransformer Table 4-6. Through variables of he auo-ransformer Table 4-7. Exernal Saes of wye-wye conneced ransformer Table 4-8. Correspondence beween he exernal phase and bank saes index. 67 Table 4-9. Exernal Saes of wye-dela conneced ransformer Table 4-0. Correspondence beween he exernal phase and bank saes index... 7 Table 4 -. Exernal Saes of dela-wye conneced ransformer Table 4 -. Correspondence beween he exernal phase and bank saes Index Table 4-3. Exernal Saes of dela-dela conneced ransformer Table 4-4. Correspondence beween he exernal phase and bank saes Index Table 6 -. Parameers o be calibraed Table 6 -. Auoransformer hermal conducance Table 6-3. Exernal sae variables of auo-ransformer ix

10 Table 6-4. Inernal sae variables of auo-ransformer Table 6-5. Through variables of auo-ransformer Table 6-6. Parameers calibraion resuls Table 7 -. Summary of Even ne: Energizaion... 5 Table 7 -. Summary of Even Two: Inernal Faul... Table 7-3. Summary of Even Three: Energizaion and Inernal Faul Table 7-4. Summary of Even Four: Inernal Faul Table 7-5. Summary of Even Five: Inernal Faul Table 7-6. Summary of Even Six: Transformer ver-exciaion Table 7-7. Summary of Even ~5: Faul Deecion Time Table 7-8. Summary of Even ~5: Faul Trip Time Table 7-9. Summary of Even 6: Time o Trip verexcied Transformer... 5 x

11 Lis of Figures Figure -. Differenial proecion demonsraion... Figure 3 -. verview of DSE-based ransformer proecion scheme... Figure 3 -. Laboraory hardware implemenaion for proposed scheme... 4 Figure 4 -. Equivalen elecro-hermal circui of a single-phase ransformer... 8 Figure 4 -. Equivalen elecro-hermal circui of a single-phase auoransformer wih eriary winding... 4 Figure 4-3. Derive he AQCF forma wih quadraic inegraion mehod Figure 4-4. Three-phase wye-wye conneced ransformer Figure 4-5. The indices relaionship of he hree-phase wye-wye conneced ransformer Figure 4-6. Three-phase wye-dela conneced ransformer Figure 4-7. The indices relaionship of he hree-phase wye-dela conneced ransformer Figure 4-8. Three-phase dela-wye conneced ransformer Figure 4-9. The indices relaionship of he hree-phase dela-wye conneced ransformer Figure 4-0. Three-phase dela-dela conneced ransformer Figure 4 -. The indices relaionship of he hree-phase dela-wye conneced ransformer Figure 5 -. Proposed DSE-based proecion logic xi

12 Figure 5 -. K-facor curve for chi-square es... 9 Figure 6 -. Auo-ransformer physical parameer configuraion Figure 6 -. Auo-ransformer parameer calibraion es sysem Figure 6-3. Auo-ransformer measuremens Figure 7 -. Transformer esing sysem Figure 7 -. Transformer ime characerisic relaed o vols-over-herz Figure 7-3. Transformer energizaion siuaion Figure 7-4. Terminal volages and currens for ransformer energizaion... 0 Figure 7-5. Percenage differenial proecion resuls for energizaion... Figure 7-6. nd harmonic level for ransformer energizaion... Figure 7-7. Negaive-sequence differenial proecion resuls... 3 Figure 7-8. Time-overcurren proecion for ransformer energizaion... 3 Figure 7-9. Proposed DSE-based proecion for ransformer energizaion... 4 Figure 7-0. Transformer 5% faul near neural siuaion... 6 Figure 7 -. Terminal volages and currens for ransformer inernal fauls... 7 Figure 7 -. Percenage differenial proecion resuls... 8 Figure 7-3. nd harmonic level for ransformer inernal fauls... 9 Figure 7-4. Negaive-sequence differenial proecion resuls... 0 Figure 7-5. Time-overcurren proecion for ransformer inernal fauls... 0 Figure 7-6. Proposed DSE-based proecion for ransformer inernal fauls. Figure 7-7. Transformer 5% faul during ransformer energizaion... 3 xii

13 Figure 7-8. Terminal volages and currens for ransformer inernal fauls under energizaion... 4 Figure 7-9. Percenage differenial proecion resuls for inernal fauls under energizaion... 5 Figure 7-0. nd harmonic level for inernal fauls under energizaion... 6 Figure 7 -. Negaive-sequence differenial proecion resuls... 7 Figure 7 -. Time-overcurren proecion for inernal fauls under energizaion... 8 Figure 7-3. Proposed DSE-based proecion for inernal fauls under energizaion... 9 Figure 7-4. Transformer % iner-urn faul siuaion... 3 Figure 7-5. Terminal volages and currens for ransformer inernal fauls... 3 Figure 7-6. Percenage differenial proecion resuls Figure 7-7. nd harmonic level for ransformer inernal fauls Figure 7-8. Negaive-sequence differenial proecion resuls Figure 7-9. Time-overcurren proecion for ransformer inernal fauls Figure Proposed DSE-based proecion for ransformer inernal fauls. 36 Figure 7-3. Auoransformer % iner-urn faul siuaion Figure 7-3. Terminal volages and currens for auo-ransformer inernal fauls Figure Percenage differenial proecion resuls xiii

14 Figure nd harmonic level for ransformer inernal fauls... 4 Figure Negaive-sequence differenial proecion resuls... 4 Figure Time-overcurren proecion for ransformer inernal fauls... 4 Figure Proposed DSE-based proecion for ransformer inernal fauls. 44 Figure Auo-ransformer over-exciaion Figure Auo-ransformer phase A erminal measuremens Figure Vols-over-herz proecion for auoransformer over-exciaion 47 Figure 7-4. Thermal proecion for auoransformer over-exciaion Figure 7-4. Proposed DSE-based proecion for auoransformer over-exciaion Figure A -. The quadraic inegraion mehod xiv

15 SUMMARY Power ransformers are key devices ha connec power sysems of differen volage levels. They are very expensive and criical equipmen ha impac he sabiliy and reliabiliy of he enire elecric power sysem. High capaciy power ransformers can be very large and heavy, cosing several million dollars. nce a ransformer is broken down and replacemen is required, i akes a long ime o purchase and insall a new one. From he prospecive of he uiliy companies, hey never wan o see a ransformer breaking down because i will cos hem large amouns of money and ime. Moreover, ransformer fauls may cause more serious problems such as cascade failures or largearea black ou. For hese reasons, reliable and secure proecion schemes for ransformers are exremely imporan. Nowadays he numerical relay is he cenerpiece of all ransformer proecion schemes. The numerical relay is a microprocessor-based sysem wih sofware-based proecion algorihms for he deecion of ransformer inernal fauls. Despie he advancemens of he numerical relays, he coordinaion and seings for he numerical relay funcions are very complex, and numerical relays canno ensure 00% proecion reliabiliy for many reasons. For some ransformer fauls, such as minor iner-urn fauls, he exising numerical relay funcions are no able o deec hem. The objecive of ransformer proecion is o deec ransformer inernal fauls or ransformer overheaing and rip he ransformer, wih immuniy o exernal fauls for which ripping of he ransformer is no required. The proposed research aims o xv

16 develop a new reliable scheme o achieve he proecion objecives for power ransformers. Tha is he dynamic sae esimaion-based proecion. This mehod has been inspired from differenial proecion, which does no require coordinaion wih oher proecion funcions. The DSE-based proecion mehod also requires no coordinaion wih oher funcions and i has only very few simple seings. This mehod is very sensiive, secure and reliable, i can deec almos any ransformer inernal faul, even some minor inernal fauls such as iner-urn fauls. The fundamenal idea of he proposed DSE-based mehod i o check he consisency beween he ransformer dynamic model and measuremens a he erminals and/or inside he ransformer. Any mismach beween he model and measuremens indicaes somehing wrong inside he ransformer, and proecion acion should be aken. In conras o presen approaches for numerical relays ha he rip decision is based on seings and coordinaed logics, he proposed mehod accuraely makes he proecion decision only based on he operaing condiion of he ransformer. In his case, some unnecessary relay failures due o improper coordinaion, or improper seings, or even human errors can be avoided. The ransformer elecro-hermal models are buil in a sandard manner, which is referred as he algebraic quadraic companion form (AQCF). The measuremens model is also expressed as an objec wih similar synax as he AQCF. The proposed DSEbased proecion algorihm direcly works wih he model and measuremens expressed in he above AQCF objecs, so he DSE-based scheme is objec-oriened. xvi

17 The proposed DSE-based scheme is a model-based scheme. Modeling accuracy of he ransformer is fundamenal in he DSE-based approach. Some independen parameers are included in he dynamic sae esimaion as sae variables for he purpose of calibraing he ransformer parameers. Therefore, he proposed mehod can also provide beer models wih validaed parameers compared o radiional approaches. In his disseraion, he proposed DSE-based proecion is esed and compared agains he legacy mehods for a number of hard-o-deec fauls, such as ransformer fauls near he neural, inernal fauls during energizaion, and iner-urn fauls. The resuls show much beer performance of he proposed mehod over legacy mehods. The proposed mehod is secure, reliable, more sensiive and faser han legacy proecion funcions xvii

18 CHAPTER INTRDUCTIN. Problem Saemen Power ransformers are expensive and criical equipmen ha impac he sabiliy and reliabiliy of he enire elecric power sysem. For his reason, reliable and secure proecion schemes for ransformers are exremely imporan. The objecive of ransformer proecion is o deec ransformer inernal fauls or ransformer overheaing and rip he ransformer, wih immuniy o exernal fauls for which ripping of he ransformer is no required []-[]. The proposed research aims o develop a reliable scheme o achieve he proecion objecives for power ransformers. Transformer failures may cause many problems. Firs, inernal fauls can evolve o fires or even explosions, which are very dangerous o he safey of personnel. Second, a sudden broken-down ransformer may cause serious sysem disurbances or even large-area black ou. Third, ransformers are very expensive and hey may cos several million dollars. Fourh, replacing a ransformer is very complicaed, expensive and ime-consuming. If he ransformer is broken down, i akes a long ime o rebuild and insall a new one. To proec he ransformer, legacy relaying proecion schemes wih high degree of sophisicaion have been designed. However, hese schemes canno ensure 00% proecion reliabiliy and securiy for hree main reasons: () improper coordinaion or improper seings of relays () here is he possibiliy of false rips during inrush or over-exciaion siuaions; (3) relays may no have enough sensiiviy

19 o deec cerain inernal fauls, such as he minor iner-urn fauls or fauls near he neural erminal. Today, commercial ransformer relaying schemes are implemened wih muliple proecive funcions, each funcion requiring complex seings and coordinaion among he funcion and wih relays for neighboring proecion zones. For example a modern numerical relay has an average of proecive funcions [3]. The coordinaion of hese proecive funcions are quie complex. This complexiy increases he possibiliy of human error, and many imes i leads o inconsisencies and he possibiliy of improper proecion acions [4]. In fac, according o Norh American Elecric Reliabiliy Corporaion (NERC), approximaely 65% of he relay failures are caused by he improper coordinaion or improper seings [5]. When he ransformer energizaion or over-exciaion happens, false differenial currens are generaed and hey resemble he condiions of inernal fauls [6]. The currens are disored currens because of he core sauraion. As a consequence, relays migh fail o differeniae he disored curren from inernal faul curren, causing nuisance rips of he ransformer. If iner-urn fauls or fauls near neural erminal happen inside a ransformer, he resuling differenial currens are very small. In conras, faul currens flowing hough he shored circui can be unexpecedly high. The high faul currens generae serious hea ha causes localized hermal overloading, which ulimaely evolves o caasrophic failures [7]. Therefore, iner-urn fauls or fauls near neural erminal should be

20 deeced in heir earlies sages before furher damages occur o he ransformers. However, hese kinds of fauls are very difficul o be deeced by legacy ransformer relays because of he small differenial currens.. Research bjecives The disseraion objecives are () o propose a ransformer proecion scheme which requires no coordinaion wih oher funcions, () o reduce or simplify he seings of relays as much as possible, (3) o deec ransformer inernal fauls, including minor inernal fauls such as he iner-urn fauls, wih high sensiiviy and cerainy, (4) o preven false ripping during ransformer energizaion or exernal fauls, (5) o validae he elecro-hermal model of ransformers, and (6) o develop high fideliy models by parameer esimaion mehods. To realize hese objecives, a new proecion scheme based on dynamic sae esimaion (DSE) is proposed in his research. This mehod has been inspired from differenial proecion, which does no require coordinaion wih oher proecion funcions. Specifically, he proposed scheme coninuously moniors ransformer erminal volages and currens and oher measurable quaniies such as ap seings, emperaures, ec. The measuremen daa are uilized in a dynamic sae esimaor of he ransformer proecion zone. A chi-square es is performed o deermine how well he measured daa fi he dynamic model of he ransformer. When he fi is wihin he accuracy of he meers by which he measuremens are aking, he dynamic sae esimaor provides he rue operaing condiion of he ransformer. Discrepancies 3

21 indicae an inernal abnormaliy. The scheme akes decisions based on he operaing condiions of he ransformer. This scheme does no require any coordinaion wih oher proecion funcions. The only seing in his case is he maximum permissible ho spo emperaure (ypically 05 Celsius). The compuaional process requires he dynamic model of he ransformer, he measuremens and he dynamic sae esimaion algorihm. The analyics have been implemened in an objec-oriened manner. Specifically, he dynamic model of he ransformer is expressed as an objec wih specific synax referred o as he algebraic quadraic companion form (AQCF). The measuremens, obained wih radiional relaying insrumenaion or via merging unis, are also expressed in an objec wih similar synax as he AQCF. The dynamic sae esimaion algorihm operaes direcly wih he measuremen models expressed in he above objecs. The feasibiliy of he proposed DSE-based proecion algorihm has been esed in he laboraory. Three dynamic sae esimaion mehods are implemened, namely he unconsrained weighed leas square (UCWLS), consrained weighed leas square (CWLS) and he exended Kalman filer (EKF) mehod. Usually, he UCWLS and CWLS mehods are ofen applied for he saic sae esimaion of power sysem, while he EKF mehod is used for he dynamic sae esimaion. However, he inroducion of numerical inegraion mehod ha convers ransformer dynamic models ino algebraic companion models makes he UCWLS and CWLS mehods suiable for dynamic sae esimaion, and renders he process equivalen o dynamic sae esimaion [8]. The 4

22 elecro-hermal model of ransformers is buil in he algebraic quadraic companion form wih he quadraic inegraion mehod, so ha all he hree mehods can be applied o solve he dynamic sae esimaion problem. UCWLS mehod is used o find he bes esimaes for he saes ha generae he minimum weighed squared error. I works well wih a measuremen se ha represens acual measuremens wih usual measuremen errors. However, when i is used o handle virual measuremens wihou uncerainy (noiseless), i may generae numerical insabiliies due o he large separaion beween he variances of he acual measuremens and he virual measuremens. This is he reason ha he CWLS mehod is also used. The CWLS mehod is very similar o he UCWLS mehod, excep ha virual measuremens are reaed as consrains. EKF mehod linearizes he nonlinear sysem o is firs-order so ha he radiional Kalman filer equaions can be applied. The EKF mehod is recursive and i works in a wo-sep process. In he predicion sep, i predics he esimaes of sae variables; in he correcion sep, he esimaes are updaed wih observed measuremens for higher accuracy. The proposed proecion scheme provides many advanages versus legacy ransformer relays. Firs, i requires no coordinaion wih oher proecive relaying funcions and has only very few, simple seings. Therefore i avoids unnecessary relay mis-operaions caused by improper coordinaion or seings, or human errors. Second, he proposed scheme is objec-oriened using he sandard AQCF forma, so i can be easily applied o any ype of ransformer. Third, he proposed scheme provides faser 5

23 speed han legacy proecion funcions. I can deec he exisence of fauls wihin a few samples (fracion of one ms) so ha his scheme can rip he ransformer a he earlies sage of he faul before furher damage happens. Forh, he proposed scheme is very secure: mis-operaions would no happen when an inrush curren or exernal faul occurs. Finally, he proposed scheme is dependable and sensiive. I can deec almos any ransformer inernal faul, including iner-urn fauls or fauls near he neural erminal, wih high sensiiviy and hen akes correc acions o proec he ransformer..3 Thesis uline The ouline of he remaining pars of his disseraion is as follows. In Chaper, background informaion is provided along wih presenly available ransformer proecion mehods ha are being used. In addiion, a horough lieraure survey is presened ha summarizes relaed research work effors. In paricular, his chaper sars wih a summary of legacy proecion mehods ha are available in numerical relays. The principles of hese legacy proecion mehods are presened. A lieraure review on he recenly alernaive ransformer proecion mehods ha have been proposed by oher researchers follows. Finally, a summary is provided on boh he legacy proecion mehods and he recenly proposed mehods. Chaper 3 presens an overview of he proposed DSE-based proecion mehod. The laboraory hardware implemenaion of he proposed DSE-based proecion algorihm is presened o mimic he acual field applicaion. 6

24 Chaper 4 presens he general mehod of deriving objec-oriened elecro-hermal models for power ransformers. The elecro-hermal models of power ransformer are wrien in sandard synax, he algebraic quadraic companion form (AQCF). Examples for how o derive he AQCF elecro-hermal models from he original algebraic differenial equaions are also given for boh a single-phase saurable ransformer and a single-phase auo-ransformer wih eriary winding. In he end of his chaper, he mehod of combining hree single-phase ransformers ino one hree-phase ransformer is also inroduced. Chaper 5 presens in deail hree differen mehods of solving he dynamic sae esimaion (DSE) problem, namely he unconsrained weighed leas square (UCWLS) mehod, he consrained weighed leas square (CWLS) mehod and he exended Kalman filer (EKF) mehod. This chaper also inroduces he proecion logic of proposed DSE-based mehod, wih he descripion of calculaing he values of chisquare and confidence level. Chaper 6 indicaes ha dynamic sae esimaion can be uilized o calibrae he parameers of he ransformer models wih grea accuracy. The basic approach is o expand he dynamic sae esimaion o include independen parameers as sae variables. The mahemaical formulaion of an auo-ransformer parameers idenificaion problem is described in his chaper. Demonsraion resuls of he auoransformer physical parameer idenificaion are also presened in his chaper. 7

25 Chaper 7 presens five differen evens for he power ransformer/ auoransformer. The proposed DSE-based proecion mehod and six legacy proecion mehods have been implemened o proec he ransformer/ auo-ransformer in five evens, and he corresponding proecion resuls are compared. Finally, Chaper 8 summarizes he research work and provides fuure research direcions. There is also one appendix in his disseraion. In Appendix A, he quadraic inegraion mehod is summarized. 8

26 CHAPTER LITERATURE SURVEY. verview This chaper provides he background informaion on exising ransformer proecion schemes relaed o he proposed research along wih a lieraure review of he research effors on hese opics. From basic fuses o he mos advanced numerical relays, various proecion schemes have been applied o ransformers [9]-[]. In general, he following legacy relaying mehods are applied o proec power ransformers:. Percenage differenial proecion. Harmonic-resrain differenial proecion 3. Negaive-sequence differenial proecion 4. vercurren proecion 5. Vol-over-herz proecion 6. Thermal proecion 7. Gas-and-pressure proecion The above proecion funcions are presenly provided wih numerical relays. Recenly, aemps have been repored o implemen hese funcions wih alernae analyics, noiceably:. Frequency analysis, ANN, fuzzy logic, and wavele-based proecion. Adapive differenial proecion The above relaying mehods are reviewed sequenially. 9

27 . Legacy Proecion Mehods.. Percenage-differenial proecion Among he ransformer proecion schemes, he mos popular (legacy) one is he differenial proecion scheme []-[7]. This scheme is based on a comparison of he sum of currens a primary and secondary sides of ransformer. Differenial relays are designed o see zero differenial currens under normal-operaing or exernal-faul condiions. If an inernal faul happens, he relay will deec a subsanial differenial curren and hen rip he ransformer. To illusrae is principle, he applicaion of differenial relay o ransformers is presened wih a single-phase wo-winding ransformer in Figure - (a) [8]. The differenial relay calculaes he operaing curren Iop Is Is and resraining curren Ires Is Is of he ransformer. Ideally, he operaing curren I op remains zero unless an inernal faul occurs. However, he exisence of variable-ap ransformers and insrumenaion errors make his simple crierion inappropriae for pracical applicaions. To overcome his problem, a minimum pickup curren I min and differenial raio K I / I are inroduced. The op res relay will rip he ransformer only if () Iop I and () he raio K exceeds a cerain min hreshold. In he indusry, he hreshold is ypically se o be 0% ~ 40%. 0

28 CT Power Transformer CT IS IS Differenial Relay (a) (b) Figure -. Differenial proecion demonsraion Typically, he differenial relay uilizes an increasing percenage as he faul-curren level increases, shown as Figure - (b). The slope characerisic provides a lower sensiiviy (desensiized) o avoid he risk of mal-operaions when low levels of curren are flowing ino ransformers [9]-[0]. However, if ransformer energizaion happens, he generaed inrush curren resembles he condiion of an inernal faul, and differenial relays canno differeniae i from inernal-faul curren. As a consequence, differenial relays end o falsely rip he ransformer and cause unnecessary sysem disurbances... Harmonic-resrain differenial proecion Harmonic-resrain differenial relays are inroduced o solve he problem caused by ransformer energizaion []-[7]. The mehods are based on an assumpion ha inernal-faul currens and inrush currens conain differen levels of second-harmonic componens. Researchers claim ha he second and fourh harmonic componens of he inrush currens are ypically above 5% of he fundamenal currens, while he levels are very low for inernal fauls. Therefore, he harmonic-resrain differenial relay moniors he second and fourh harmonic levels, and i will block any rip signal if he

29 levels are higher han he seings. However, echnology has changed his siuaion. Today, he levels of second-harmonic componens in inrush currens are subsanially lower in ransformers wih improved core seels [8]-[9]. I is difficul o deermine wheher an inernal faul exiss based only on he level of second-harmonic componens. Moreover, if a faul happens a he ime a ransformer is energized, harmonics in he magneizing curren could preven he relay from ripping...3 Negaive-sequence differenial proecion Tradiional differenial relays can deec mos inernal ransformer fauls, excep for iner-urn fauls and fauls near he neural, since ransformer urn-urn fauls do no generae differenial currens. To solve he problem, a negaive-sequence differenial relay has been inroduced [30]-[36]. When he iner-urn fauls happen, he ransformer erminal currens become asymmerical. Therefore, negaive-sequence currens a he primary and secondary sides are good indicaors of iner-urn fauls. The magniudes and angles of erminal negaive-sequence currens are compared, and he negaivesequence differenial relay would rip he ransformer if he angle or raio of magniudes exceeds cerain hresholds. However, he negaive-sequence differenial relay canno proec he ransformer if urn-urn fauls exis a he ime ransformer is energized or in case of CT sauraion. Because when he ransformer is energized, he relay has o be desensiized o avoid blocking possible rips from oher proecion funcions [37]. In addiion, if he CT is sauraed he differenial-resraining poin may move o he operaing boundary, causing unnecessary mis-operaions. Furhermore, he negaive-

30 sequence differenial proecion is no sensiive enough if he iner-urn faul is oo small (less han %)...4 vercurren proecion The overcurren relays are also widely used for ransformer proecion [38]-[44]. They would rip he ransformer if he values of erminal currens exceed he pick-up seings. vercurren relays are able o deec some obvious fauls of ransformers, while hey could no cover oher inernal fauls. For example, if he urn-urn or urnground fauls happen inside he ransformer, depending on he locaion of he faul, he erminal currens may no change enough o aler he relay abou he exisence of fauls. However, he currens in he faul loop can be exremely large and can cause significan damage if no deeced and isolaed in ime. In addiion, if energizaion happens, i ends o draw a high-magniude inrush curren from he supply ha can be ypically many imes he normal full-load curren. The inrush curren could cause nuisance ripping of he overcurren relay...5 Vols-over-herz proecion Vols-over-herz schemes are implemened o proec ransformers from harmful core sauraion ha can generae harmonics, increased heaing and increased inerlaminaion volages causing iron damage [45]-[50]. To proec he ransformer, he raio of vols-over-herz is coninuously moniored. A normal operaing condiions, he raio of vols-over-herz is consan and known. If he ransformer is over-excied, he raio 3

31 of vols-over-herz will increase. In oher words, he raio Vols-over-herz indicaes he level of he magneic flux linkage in he ransformer. Transformers are designed in such a way o operae near he magneizaion knee region under normal operaion. If he magneic flux linkage in he ransformer increases, he iron core of he ransformer will be driven ino sauraion. In his case, excessive losses in he iron core may increase he emperaure of he ransformer and damage he ransformer. The relay will proec ransformer according o a ime characerisic relaed o he raio of vols-over-herz. The more ha raio exceeds normal operaing seings, he faser ha relay will rip he ransformer. However, he vols-over-herz relay is limied o proec he ransformer from over- or under-exciaions, hus addiional funcions are required o proec he ransformer from damage caused by oher inernal fauls...6 Thermal proecion The ransformer should also be proeced agains high emperaures as hey will deeriorae insulaion leading o elecric fauls. The maximum emperaure anywhere in he ransformer is referred o as he ho-spo emperaure. In general he ho spo emperaure should no exceed abou 0 0 C [5]-[56]. When he ransformer is overloaded, overexcied or he cooling equipmen is broken, relaively high emperaures can be developed inside he ransformer, increasing he ho-spo emperaure. A ransformer hermal relay calculaes he ho-spo emperaure based on readings from hermocouples, he hermal model, erminal volages and currens and ambien emperaure. Thermal relays can proec ransformers from overloads, shor 4

32 circuis, cooling equipmen failure and caasrophic failures by ripping he ransformer when pre-se emperaure hresholds are exceeded. However, when an iner-urn faul occurs inside he ransformer, i will be oo lae o proec he ransformer if waiing for he hermal relay o operae...7 Gas-and-pressure proecion Gas-and-pressure relays are uilized as proecive devices for oil-filled ransformers. The accumulaion of gas and pressure changes inside he ransformer ank are good indicaors of inernal fauls [57]-[6]. A combined gas-accumulaor and pressure relay, called he Buchholz relay, has been in successful service for over 70 years [63]. However, gas-and-pressure relays can only deec he fauls below he oil level inside ransformer, while i could no rip he ransformer if fauls happen a he bushings or erminal connecions. Furhermore, gas-and-pressure relays are relaively vulnerable o ambien disurbances. If vibraion, leaking or corrosion happens, relays migh mal-operae and cause severe damage..3 Recenly Proposed Alernaive Proecion Mehods The aforemenioned proecion schemes are presenly provided wih commercial numerical relays. Apar from hese legacy proecion funcions, researchers have also proposed some new alernaive schemes o proec ransformers. 5

33 .3. Frequency analysis, ANN, fuzzy logic, and wavele-based proecion Frequency analysis-based proecive relaying funcions have been sudied in [64]- [69]. In [64], he differenial curren was analyzed in erms of is Fourier series, he peaks of fundamenal and second-harmonic conens were calculaed, as well as he second-harmonic level in he differenial curren. The second-harmonic level is moniored o block mis-operaions during ransformer energizaion (second-harmonic level is high) Similar research has been proposed based on he frequency response analysis o achieve fas compuaions o deec winding deformaion under he influence of shor circuis [65]-[66]. The algorihm generaes he Fourier coefficiens by addiion and subracion rouines only. The frequency responses were classified ino high-, medium- and low-frequency responses and hey corresponded o differen faul siuaions. Wih differen responses, researchers could figure ou wheher a faul happened inside he ransformer. Unforunaely, hese frequency analysis-based mehods are no reliable enough. Faul condiions inside ransformer could be complex so frequency responses are insufficien o make a convincing conclusion in some cases. Moreover, almos all hese schemes are based on he condiions of periodiciy and saionariy. While disurbances in power sysems are of non-periodic, non-saionary, shor duraion [67]. Arificial neural nework (ANN) and fuzzy logic-based proecion schemes have been proposed [70]-[80]. ANN can be represened as a parallel muli-layer informaion processing srucure ha enables he inclusion of exper knowledge ino he processing, 6

34 recogniion and classificaion of signals. A feed forward ANN-based raining mehod has been proposed o discriminae beween power ransformer inrush and faul currens [70]. The back-propagaion mehod was used for he raining. The ransfer funcions of unis were changed o hard limiers wih hresholds equal o he biases obained from he sigmoid during raining process o increase he compuaion speed of he nework. nce he nework was rained, he ANN-based mehod can quickly deec cerain inernal fauls (if he fauls have been included in he raining ses) by checking he ransformer erminal measuremens. Fuzzy logic-based schemes have also been sudied for ransformer proecion. In 995, a muli-crierion differenial relay based on fuzzy was inroduced [77]. I. The consequences of wrong proecion decisions are considered, and he relay more inclined o rip or more inclined o block depends on acual condiions. The use of he cos of wrong decision-making and he amoun of informaion inflow improves he reliabiliy of he proecion scheme. Wavele-based differenial proecion schemes for ransformers have also been inroduced [8]-[86]. These mehods focus on deecing he difference beween inernal faul currens and inrush currens, wih he fac ha heir energy disribuions in ime and frequency were very differen. However, all he above mehods inroduce high compuaional burdens and may requires addiional expensive apparaus. For example, he ANN and fuzzy logic-based mehods require long raining ime and large raining ses, which migh even no be inclusive of all he evens ha may occur in he real world. In general, hese mehods have no been used in field condiions and hey have no been acceped for 7

35 pracice..3. Adapive differenial proecion In he las few years, adapive differenial relays have been sudied o deec he inernal fauls of ransformers [87]-[93]. These relays are based on he percenage differenial proecion scheme, bu hey can adjus he characerisic auomaically according o he differenial currens and ransformer saus such as ap seings. A muli-region adapive differenial relay has been inroduced in [93]. Based on he curren rajecories and faul condiions, he operaional zone of relay characerisic is divided ino hree operaing regions. When he curren rajecory eners he relay operaional zone, a weighing facor is adoped based on is region. The relay will rip he ransformer when he summaion of he weighed-poins exceeds a cerain prespecified value. Proecion characerisics are self-regulaed according o power sysem condiions so ha he relay can achieve boh securiy and sensiiviy. However, adapive differenial proecion relays canno guaranee 00% proecion of he ransformer. The sensiiviies of adapive differenial relays are no high enough o deec minor inernal ransformer fauls such as iner-urn fauls of a ransformer..4 Summary Today, commercially available ransformer relays have been implemened wih muliple mehods lised above, while he complicaed coordinaion and seings of relays increase he risk of improper proecion acions. I is feasible o subsiue he 8

36 lised legacy mehods from one o six as well as he aforemenioned newly proposed schemes wih only one approach: he proposed dynamic sae esimaion-based proecion scheme. This mehod requires no coordinaion wih oher relays and has only very few and simple seings, while, as i will be shown, provides beer securiy han legacy mehods. The proposed mehod is no limied o deecion of fauls ha are wihin he capabiliies of legacy funcions. I is sensiive enough o deal wih he fauls ha are very difficul o deec by legacy mehods, such as he iner-urn fauls and fauls near he neural erminal. 9

37 CHAPTER 3 THE VERALL APPRACH 3. verview In his chaper, a dynamic sae esimaion-based ransformer proecion scheme is proposed. This scheme requires no coordinaion wih oher funcions and i has only very few simple seings. The DSE-based proecion scheme is an exension of differenial proecion. I measures volages, currens, emperaures, ec. and hen fis he real ime measuremen daa o he ransformer mahemaical model. This is achieved in a mahemaical rigorous way by he use of dynamic sae esimaion. DSE calculaes he degree of consisency beween measuremens and he ransformer dynamic model. If here is a mismach, somehing is wrong inside he ransformer and proecive acions should be aken. The proposed scheme has been implemened in an objec-oriened manner. The ransformer model and measuremens are expressed in an objec wih specific synax referred o as he algebraic quadraic companion form (AQCF) [94]. The dynamic sae esimaion algorihm operaes direcly wih he measuremen models expressed in above objecs. The laboraory hardware implemened wih he proposed DSE-based proecion algorihm is presened o mimic he acual field applicaion. 3. The Proposed Approach The proposed DSE-based proecion scheme is shown in Figure 3-. The procedure for DSE-based proecion has been sreamlined. Iniially, he ransformer device model 0

38 Proecion Logic is wrien in algebraic quadraic companion form (AQCF) and proposed algorihm auomaically formulaes he measuremen model in AQCF synax, as illusraed in Figure. AQCF Measuremen Model DSE-based Transformer Proecion Transformer AQCF Model Measuremen Definiion Creae Acual Measuremen Model Creae Virual Measuremen Model Creae Derived Measuremen Model Creae Pseudo Measuremen Model Dynamic Sae Esimaion Temperaure Monioring Transformer Process Bus Model Parameer Idenificaion Merging Uni Figure 3 -. verview of DSE-based ransformer proecion scheme The only seing for he proposed scheme is he maximum permissible operaing condiions such as maximum permissible emperaure. The DSE-based proecion approach has wo ypes of inpu daa. ne is he measuremen model of he ransformer; he oher is he real-ime measuremens daa coming from merging unis (process bus). The real-ime measuremens daa are uilized in a dynamic sae esimaion by fiing he measuremen daa o he model equaions of he ransformer. The dynamic sae esimaion problem can be solved by hree mehods, namely he unconsrained weighed leas square (UCWLS) mehod, he consrained weighed leas square (CWLS) mehod and he exended Kalman filer (EKF) mehod. The dynamic sae esimaion gives he bes esimaes of all he saes of he ransformer including emperaures and, if needed, parameer idenificaion. Following he esimaion of he saes, a Chi-square es is

39 applied o deermine he probabiliy ha he measuremens are consisen wih he ransformer dynamic model [95]. This probabiliy (or confidence level) indicaes wheher here are inernal abnormaliies in he ransformer, such as a ground faul, an iner-urn faul, ec. An inegral funcion is applied o accumulae he confidence level values and diminishes he effec of unnecessary ransiens. The proecion scheme will rip he ransformer if any inernal faul is deeced. To make sure ha he proposed DSE-based scheme can be applied o any ype of ransformer, he ransformer elecro-hermal models are buil in a sandard manner, which is referred o as he algebraic quadraic companion form (AQCF). Deails of he AQCF will be inroduced in he nex chaper. The measuremens model is also expressed in an objec wih similar synax as he AQCF. The proposed DSE-based proecion algorihm direcly works wih he measuremen models expressed in he above AQCF objecs, hus he DSE-based scheme is objec-oriened. The proposed DSE-based scheme is a model-based scheme, herefore, i relies on high-fideliy ransformer models. Modeling accuracy of he ransformer is fundamenal for he DSE-based approach. The acual ransformer parameers used for sae esimaion are ofen quie differen from he nameplae raings, hus reliable ransformer parameers calibraion mehod is necessary o ensure he feasibiliy and correcness of he proposed mehod. To calibrae he ransformer parameers, some independen parameers are included in he dynamic sae esimaion as sae variables. Wih enough redundancy, boh he ransformer saes and he key parameers can be esimaed wih

40 a high confidence. Therefore he proposed mehod can also provide beer models wih field-validaed parameers compared wih radiional approaches. There are many ransformer fauls hard o be deeced or correcly reaed by he legacy proecion funcions, such as ransformer fauls near he neural, inernal fauls during energizaion, iner-urn fauls, ec. In his disseraion, he proposed DSE-based proecion is compared agains he legacy mehods for hese hard-o-deec fauls. Anoher advanage of he proposed DSE-based mehod is ha he scheme does no require coordinaion wih oher relays. In addiion i only requires very few and simple seings. In conras o he numerical relays in which he rip decision is based on he seings or coordinaion logic, he proposed mehod accuraely makes he proecion decision only based on he operaing condiion of he ransformer. In his way, some unnecessary relay failures due o improper coordinaion, or improper seings, or even human errors can be avoided. 3.3 Laboraory Hardware Implemenaion for DSE-Based Proecion Scheme Laboraory esing is he mos effecive way o es any innovaive mehodologies. The proposed DSE-based proecion scheme is esed in he laboraory. The implemenaion uses merging unis, GPS signals and IEC-6850 communicaions. The laboraory hardware implemenaion for proposed scheme is shown Figure 3-. 3

41 Figure 3 -. Laboraory hardware implemenaion for proposed scheme The simulaion plaform (program WinXFM) generaes and sreams digial waveforms of ransformer evens o a Naional Insrumen D/A converer. The Naional Insrumen D/A converers send he analog signals o a bank of MICRN amplifiers ha amplify hem o sandard relay insrumenaion volages and currens. These signals are in he range of ypical oupus from CT/VTs so ha hey are used o mimic he acual field signals. These signals are fed o he merging unis. The merging uni acs as a bridge beween primary equipmen and proecion devices ha capures and ransmis signals [96]. I convers he analog signals from analog CT/VT signals o digial signals, which are hen ransmied o he process bus via sandard proocol (IEC-6850) [97]- [99]. The process bus offers he obvious possibiliy of bringing many measuremens (as a maer of fac all he measuremens) o he process bus. Daa are synchronized by he used of an Arbier 093 GPS clock. A personal compuer is conneced o he process bus and acs as a daa concenraor ha feeds he colleced daa o dynamic sae esimaor. 4

42 3.4 Summary A DSE-based ransformer proecion scheme is proposed in his chaper. The overall srucure of he proposed mehod has been inroduced. In general, he proposed mehod moniors he healh saus of he ransformer and i can idenify any inernal abnormaliy of he ransformer wihin a few samples (a fracion of one ms). This mehod does no degrade he securiy because i does no rip in he even of normal behavior of he ransformer, for example, he inrush currens or over exciaion currens. Because in hese cases, as long as he inrush currens are consisen wih he ransien behavior of he ransformer as dicaed by he dynamic model, he mehod will produce a high confidence level ha he ransiens are consisen wih he model of he componen. The laboraory hardware implemenaion is also presened o mimic he acual field applicaion in his chaper. 5

43 CHAPTER 4 TRANSFRMER ELECTR-THERMAL MDEL 4. verview In his chaper, he ransformer elecro-hermal model is presened. Firs, he ransformer physical model will be presened in he quadraized device model (QDM) wih boh elecric and hermal par. Then he physical model will be cas ino he sandard algebraic quadraic companion form (AQCF) forma using he quadraic inegraion mehod [00]-[0]. Finally, he measuremen model is auomaically generaed in he AQCF synax wih he inroducion of measuremen definiion. The DSE-based proecion scheme direcly works on he AQCF objecs. For simpliciy, a single-phase saurable-core ransformer and a single-phase auoransformer wih eriary winding are aken as examples in his secion. However, hey can be easily generalized o hree-phase muli-winding ransformers or he hree-phase auo-ransformer bank. An example of consrucing a hree-phase ransformer wih hree single-phase ransformers is also provided in his chaper. 6

44 4. Transformer Physical Elecro-Thermal Quadraized Model The quadraized models of a single-phase saurable-core ransformer and a single-phase auo-ransformer wih eriary winding are inroduced in his secion. The saurable cores of boh ransformers are modeled by high-fideliy equaions o represen he nonlinear magneizaion characerisics. Exra saes and equaions are added o decrease he highes order of he models back o wo. 4.. Single-Phase Saurable-Core Transformer Physical Elecro-Thermal Quadraized Model The elecro-hermal model of a single-phase saurable-core ransformer is inroduced in his par. The equivalen circui of a single-phase saurable-core ransformer wih boh elecric par and hermal par is shown in Figure 4-. v () i () r gs g s i () c ET i () L ET 3 ET L i () L L r v () 3 i () 3 ET g 9 ET c e () ET c 8 ET0 7 v () i () i () m ET 4 ET amb ET 6 ET v () 4 i () 4 7

45 ET 9 ET 7 ET 3 ET ET 0 ET 5 ET 4 ET 8 ET ET 6 ET amb Figure 4 -. Equivalen elecro-hermal circui of a single-phase ransformer The numerical sabilizers gs ~ gs4 are inroduced o eliminae possible numerical problems. The numerical sabilizers inroduce errors orders of magniude below he measuremen errors and herefore do no affec he overall accuracy of he proposed proecion mehod. For he hermal par, he red spos (ETX) are he hermal emperaure poins. ET and ET are he hermal emperaure poins a he core (op and boom), ET3 o ET4 are he hermal emperaure poins a he primary coil (op and boom), ET5 o ET6 are he hermal emperaure poins a he secondary coils (op and boom), ET7 and ET8 are he hermal emperaure poins a he oil (op and boom), ET9 and ET0 are he hermal emperaure poins a he ank (fron and rear), ETamb is he ambien emperaure poin. The ransformer losses are compued from he measured volage and curren waveforms a he ransformer erminals. Compued losses include winding coil losses (hmic losses) as well as magneic core losses. [0]. A firs, he ransformer compac model of he single-phase saurable-core ransformer is inroduced in he following differenial algebraic equaions: 8

46 dil () i ( ) il ( ) gs L d dil () i ( ) il ( ) gs L d dil () i3 ( ) il( ) gsl d dil () i4 ( ) il ( ) gsl d dil ( ) dil ( ) 0 v ( ) v( ) r il( ) gs L L ec ( ) d d di ( ) di ( ) N 0 v ( ) v ( ) r i ( ) g L L e ( ) L L 3 4 L s c d d N dil () 0 Nic ( ) N il( ) gsl d d() 0 ec ( ) d dil () 0 il ( ) gs L ic ( ) im( ) gcec ( ) d () 0 im ( ) i0 sign( ( )) 0 n det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n0, i, j j core, d i j det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n0, i, j j core, d i j det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 3 3, j j coil _ pri, d i3 j3 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 4 4, j j coil _ pri, d i4 j4 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 6 5, j j coil _sec, d i5 j5 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 7 6, j j coil _ sec, d i6 j6 9

47 det () 0 C g ET ( ) g ET ( ) n0 n0 i j j 7 7 7, 7 7, d i7 j7 det () 0 C g ET ( ) g ET ( ) n0 n0 i j j 8 8 8, 8 8, d i8 j8 det () 0 C g ET ( ) g ET ( ) g ET n9tamb n9tamb 9 9 9, i 9 9, j j 9, T amb d i9 j9 det () 0 C g ET ( ) g ET ( ) g ET n0tamb n0tamb 0 0 0, i 0 0, j j 0, T ambien d i j0 0 Qcore,( ) eh( ) 0 Qcore,( ) eh( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) where v () and i () are he erminal volages and currens, ET () ~4 ~4 ~0 are he emperaure poins, r, r, L, L are he corresponding resisances and inducances, C ~ C0, g i, j( wherei, j ~0, i j) are he corresponding hermal capaciance and conducance, N and N are he number of urns a he primary and secondary windings, gc is he exciaion conducance, im () is he magneizing curren and () is he flux linkage hrough he iron core, Qcore,( ), Qcore,( ) are he hea generaed a he core, Q ( ), Q ( ) are he hea generaed a he primary winding, core _ pri, core _ pri, Q ( ), Q ( ) are he hea generaed a he secondary winding. core _ sec, core _ sec, There are 6 sae variables and 6 equaions in he compac model. The model is 30

48 quadraized by inroducing addiional inernal sae variables so ha he n h exponen is replaced by equaions of a mos quadraic degree. Since he exac degree of nonlineariy is no known unil he user specifies i, he model performs auomaic quadraizaion of he equaions. A special procedure is used, so ha he model is quadraized using he minimum number of addiional inernal saes, which also minimizes he addiional equaions. The mehodology is based on expressing he exponen in binary form. The binary represenaion provides all he informaion abou he number of new variables and equaions ha need o be inroduced and added o he model. Specifically, he equaion: () 0 im ( ) i0 sign( ( )) 0 n is quadraized wih he newly inroduced saes as: 0 0 i ( ) i y ( ) sign( ( )) n m m () 0 y ( ) 0 0 y ( ) y ( ) 0 y ( ) y ( ) m m 0 y ( ) y ( ) y ( ) m i j 0 y ( ) y ( ) y ( ) m m j 3

49 0 ym( ) ym ( ) y jm( ), if n even () 0 ym( ) ym ( ), if n odd 0 where he saes y ( ) ~ ym( ) are inroduced o ensure ha ransformer mahemaical model consiss of equaions of a mos quadraic degree, and m m m, m in log ( n) and m # of ones in binary form of n. The quadraized device model (QDM) of he single-phase saurable-core ransformer is expressed as: dil () i ( ) il ( ) gs L d dil () i ( ) il ( ) gs L d dil () i3 ( ) il( ) gsl d dil () i4 ( ) il ( ) gsl d dil ( ) dil ( ) 0 v ( ) v( ) r il( ) gs L L ec ( ) d d di ( ) di ( ) N 0 v ( ) v ( ) r i ( ) g L L e ( ) L L 3 4 L s c d d N dil () 0 Nic ( ) N il( ) gsl d d() 0 ec ( ) d dil () 0 il ( ) gs L ic ( ) im( ) gcec ( ) d 0 0 i ( ) i y ( ) sign( ( )) n m m det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n0, i, j j core, d i j det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n0, i, j j core, d i j 3

50 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 3 3, j j coil _ pri, d i3 j3 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 4 4, j j coil _ pri, d i4 j4 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 6 5, j j coil _sec, d i5 j5 det () 0 C g ET ( ) g ET ( ) Q ( ) n0 n , i 7 6, j j coil _ sec, d i6 j6 det () 0 C g ET ( ) g ET ( ) n0 n0 i j j 7 7 7, 7 7, d i7 j7 det () 0 C g ET ( ) g ET ( ) n0 n0 i j j 8 8 8, 8 8, d i8 j8 det () 0 C g ET ( ) g ET ( ) g ET n9tamb n9tamb 9 9 9, i 9 9, j j 9, T amb d i9 j9 det () 0 C g ET ( ) g ET ( ) g ET n0tamb n0tamb 0 0 0, i 0 0, j j 0, T ambien d i j0 () 0 y ( ) 0 0 y ( ) y ( ) 0 y ( ) y ( ) m m 0 y ( ) y ( ) y ( ) m i j 0 y ( ) y ( ) y ( ) m m j 33

51 0 ym( ) ym ( ) y jm( ), if n even () 0 ym( ) ym ( ), if n odd 0 0 Qcore,( ) eh( ) 0 Qcore,( ) eh( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) A sandard forma is inroduced here o represen he above QDM of he ransformer as follows: i( ) Y x ( ) D dx() C d 0 Y x ( ) D dx() C d eqx eqxd eqc eqx eqxd eqc 0 Y x( ) x( ) F x ( ) C T i eqx3 eqxx3 eqc3 T i h( x, u) Y x x F x C feqx feqxxx feqc Conneciviy: TerminalNodeName subjec o : h h( x, u) h, x x x min max min max where i () is he hrough variables of he device model; x() is he exernal and inernal sae variables of he device model; Y is he marix defining he linear par of sae variables in linear hrough variable eqx 34

52 equaions; D is he marix defining he differenial par of sae variables in linear hrough eqxd variable equaions; C is he consan vecor of he device model in linear hrough variable equaions; eqc Y is he marix defining he linear par of sae variables in linear virual equaions; eqx D is he marix defining he differenial par for sae variables in linear virual eqxd equaions C is he consan vecor of he device model in linear virual equaions; eqc Y is he marix defining he linear par of sae variables in he remaining quadraic eqx3 equaions, C is he consan vecor of he device model in he remaining quadraic equaions; eqc3 F is he marix defining he quadraic par of sae variables in he remaining quadraic eqxx equaions; TerminalNodeName is he erminal names defining he conneciviy of he device model; Y is he consrain marix defining he linear par of sae variables; feqx F is he consrain marix defining he quadraic par of sae variables; feqx C feqc is he consrain hisory dependen vecor of he device model; h h( x, u) h is he funcional consrain; min max x, x are he lower and upper bounds of he sae variables. min max The ransformer QDM has hree ses of equaions. The firs se of equaions are exernal equaions and he lef sides are erminal currens. The second and hird ses of 35

53 equaions are boh inernal equaions of he ransformer. The second se of equaions is linear while he hird se is nonlinear. The marices coefficiens of he QDM are (assume he exponen n equals ): Y eqx D eqx Ceqc gs L gs L gsl gsl Y eqx 0 0 r N r N N 0 0 N gc i g g g g g 3 4 g g3 g g g g g g 5 6 g5 g6 g g g5 g g g g 5 8 g8 g g3 g6 g8 g g g g g g4 g7 g9 g0 g g g g g

54 D eqx r gs L L r gl L NgsL gs L C C C C g T eqc 0, T ambien T Y eqx F eqxx 3 0,(6,6) 0 F,(0,0) eqxx3 F,(,) eqxx3 F 3,(0,) eqxx3 F eqxx 3 F eqxx3 4,(6,3) 5,(6,6) 0 e h 37

55 F F F F F eqxx3 6,(6,6) 7,(4,4) r eqxx3 8,(4,4) r eqxx3 9,(5,5) r eqxx3 0,(5,5) e h r eqxx3 Ceqc3 0 There are 3 sae variables and 3 equaions in he model. The definiion of he exernal sae, inernal sae, and hrough variables are lised in Table 4- o Table 4-3, respecively. Table 4 -. Exernal saes of he ransformer. Index Variable Descripion 0 v ( ) erminal volage of ransformer high side (V) v ( ) erminal volage of ransformer high neural (V) v ( ) erminal volage of ransformer low side (V) 3 3 v () 4 erminal volage of ransformer low neural (V) Table 4 -. Inernal saes of he ransformer. Index Variable Descripion 4 i () L curren hrough he high side inducance (A) 5 i () L curren hrough he low side inducance (A) 6 () flux linkage (Web) 38

56 Table 4 - coninued Index Variable Descripion 7 ec () volage generaed by he flux (A) 8 i c () curren hrough high side windings (A) 9 i m () magneizing curren (A) 0 y ( ) inroduced sae (p.u.) y () y () 3 3 y () 4 4 y () 5 5 ET () inroduced sae (p.u.) inroduced sae (p.u.) inroduced sae (p.u.) inroduced sae (p.u.) emperaure a core up poin (Celsius) 6 ET ( ) emperaure a core down poin (Celsius) 7 ET () 3 8 ET () 4 9 ET () 5 0 ET () 6 emperaure a primary coil up poin (Celsius) emperaure a primary coil down poin (Celsius) emperaure a secondary coil up poin (Celsius) emperaure a secondary coil down poin (Celsius) ET ( ) emperaure a inernal oil up poin (Celsius) 7 ET () 8 3 ET () 9 emperaure a inernal oil down poin (Celsius) emperaure a ransformer ank fron poin (Celsius) 4 ET () 0 emperaure a ransformer ank rear poin (Celsius) 5 Q () core, hea generaed a core up poin (W) 39

57 Table 4 - coninued Index Variable Descripion 6 Q () core, hea generaed a core down poin (W) 7 Q () _, hea generaed a primary coil up poin (W) coil pri 8 Q () _, hea generaed a primary coil down poin (W) coil pri 9 Q () _, hea generaed a secondary coil up poin (W) coil sec 30 Q () _, hea generaed a secondary coil down poin (W) coil sec Table 4-3. Through variables of he ransformer. Index Variable Descripion 0 i ( ) curren hrough ransformer high side i () i () 3 3 i () 4 curren hrough ransformer high side neural curren hrough ransformer low side curren hrough ransformer low side neural 40

58 4.. Single-Phase Auo-Transformer Physical Elecro-Thermal Quadraized Model The elecro-hermal model of a single-phase auo-ransformer is inroduced in his par. The equivalen circui of a single-phase auo-ransformer wih boh elecric par and hermal par is shown in Figure 4-. I r ET L g s jx PT ET 3 E ET 5 ET 8 i r 3 ET 6 c 3 L ET 3 9 ET 7 r ET 3 ET 4 ET ET ET 0 g c i m L m g s3 I I 3 I 4 4 ET 4 ET g s jx PS L jx ST I 5 ET amb 5 4

59 ET ET ET 3 ET 4 ET 8 ET 5 ET 9 ET 6 ET 7 ET3 4 ET 0 ET ET ET ET amb Figure 4 -. Equivalen elecro-hermal circui of a single-phase auoransformer wih eriary winding The numerical sabilizers gs ~ g s3 are inroduced o eliminae possible numerical problems. The numerical sabilizers inroduce errors orders of magniude below he measuremen errors and herefore do no affec he overall accuracy of he proposed proecion mehod. The red spos (ETx) are he hermocouple poins. ET and ET are he hermal emperaure poins a he core (op and boom), ET3 o ET7 are he hermal emperaure poins a he primary and secondary coils (op, middle, and boom), ET8 o ET0 are he hermal emperaure poins a he eriary coils (op, middle, and boom), ET and ET are he hermal emperaure poins a he oil (op and boom), ET3 and ET4 are he hermal emperaure poins a he ank (fron and rear), ETamb is he ambien emperaure poin. The ransformer losses are compued from he measured volage and curren waveforms a he ransformer erminals. Compued losses include winding coil losses (hmic losses) as well as magneic core losses. A firs, he ransformer compac model of he single phase auoransformer wih 4

60 eriary winding is inroduced in he following differenial algebraic equaions: i ( ) i ( ) i ( ) L gs i ( ) i ( ) i ( ) i ( ) i ( ) L gs L gs i ( ) i ( ) i ( ) 3 L3 gs3 i ( ) i ( ) i ( ) 4 L3 gs3 i ( ) i ( ) i ( ) 5 L gs N d() 0 v ( ) v( ) r il( ) igs ( ) igs ( ) / gs N N d N d() 0 v5 ( ) v( ) r il( ) igs( ) igs( ) / gs N N d N3 d() 0 v3( ) v4( ) r3 il3( ) igs3( ) igs3( ) / gs3 N N d di ( ) di ( ) di () d d d L L L3 0 igs ( ) / gs L LPS LPT di ( ) di ( ) di () d d d L L L3 0 igs( ) / gs L LPS LST dil3() dil( ) dil ( ) 0 igs3( ) / gs3 L3 LST LPT d d d 3 0 il3( ) igs3( ) ic ( ) im( ) gc ec ( ) N N 0 N i ( ) i ( ) N i ( ) i ( ) N i ( ) L gs L gs 3 c () 0 im ( ) i0 sign( ) 0 n N n4 n4 det () 0 C g ET ( ) g ET ( ) Q ( ), i, j j core, d i j det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n4, i, j j core, d i j 43

61 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 3 3, j j coil _ pri, d i3 j3 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 4 4, j j coil _ pri, d i4 j4 det () 0 C g ET ( ) g ET ( ) Q ( ) Q ( ) n4 n4 i j j coil pri coil 5 5 5, 5 5, _, _sec, d i5 j5 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 6 6, j j coil _sec, d i6 j6 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 7 7, j j coil _ sec, d i7 j7 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 8 8, j j coil _ er, d i8 j8 det () 0 C g ET ( ) g ET ( ) Q ( ) Q ( ) n4 n4 i j j coil er coil er 9 9 9, 9 9, _, _, d i9 j9 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 0 0, j j coil _ er, d i0 j0 det () 0 C g ET ( ) g ET ( ) n4 n4 i j j,, d i j det () 0 C g ET ( ) g ET ( ) n4 n4 i j j,, d i j det () 0 C g ET ( ) g ET ( ) g ET n4tamb n4tamb 3 3 3, i 3 3, j j 3, T amb d i3 j3 det () 0 C g ET ( ) g ET ( ) g ET n4tamb n4tamb 4 4 4, i 4 4, j j 4, T amb d i j4 0 Qcore,( ) eh( ) 0 Qcore,( ) eh( ) 44

62 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ er,( ) r3i L3( ) 0 Qcoil _ er,( ) r3i L3( ) where v () and i () are he erminal volages and currens, ET () ~5 ~5 ~4 are he emperaure a he hospos, r, r, r3, L, L, L 3 are he corresponding resisances and inducances, LPS, LPT, L ST are he corresponding muual inducances, C ~ C, g ( wherei, j ~4, i j) are he corresponding hermal capaciance and 4 i, j conducance, N, N and N3 are he number of urns a he primary, secondary and eriary windings, gc is he exciaion conducance, i () is he magneizing curren and () is he flux linkage hrough he iron core, Qcore,( ), Qcore,( ) are he hea generaed a he core, Qcoil _ pri, ( ), Qcoil _ pri,( ) are he hea generaed a he primary winding, Qcoil _ sec, ( ), Qcoil _ sec,( ) are he hea generaed a he secondary winding and m Q ( ), Q ( ) are he hea generaed a he eriary winding. coil _ er, coil _ er, There are 36 sae variables and 36 equaions in he compac model. Similar o he single-phase saurable-core ransformer presened in las secion, he auo-ransformer model is also quadraized by inroducing addiional sae variables and equaions. Specifically, he equaion: 45

63 () 0 im ( ) i0 sign( ( )) 0 n is also quadraized wih he newly inroduced saes as: 0 0 i ( ) i y ( ) sign( ( )) n m m () 0 y ( ) 0 0 y ( ) y ( ) 0 y ( ) y ( ) m m 0 y ( ) y ( ) y ( ) m i j 0 y ( ) y ( ) y ( ) m m j 0 ym( ) ym ( ) y jm( ), if n even () 0 ym( ) ym ( ), if n odd 0 where he saes y ( ) ~ ym( ) are inroduced o ensure ha ransformer mahemaical model consiss of equaions of a mos quadraic degree, and m m m, m in log ( n) and m # of ones in binary form of n. The quadraized device model (QDM) of he single-phase auo-ransformer is expressed as: i ( ) i ( ) i ( ) L gs i ( ) i ( ) i ( ) i ( ) i ( ) L gs L gs i ( ) i ( ) i ( ) 3 L3 gs3 46

64 i ( ) i ( ) i ( ) 4 L3 gs3 i ( ) i ( ) i ( ) 5 L gs dil () 0 z( ) d dil () 0 z( ) d dil3() 0 z3( ) d d() 0 ec ( ) d 0 v ( ) v ( ) r i ( ) i ( ) i ( ) / g N e ( ) L gs gs s c N N N 0 v ( ) v ( ) r i ( ) i ( ) i ( ) / g e ( ) 5 L gs gs s c N N N 0 v ( ) v ( ) r i ( ) i ( ) i ( ) / g e ( ) L3 gs3 gs3 s3 c N N 0 i ( ) / g L z ( ) L z ( ) L z ( ) gs s PS PT 3 0 i ( ) / g L z ( ) L z ( ) L z ( ) gs s PS ST 3 0 i ( ) / g L z ( ) L z ( ) L z ( ) gs3 s3 3 3 ST PT 3 0 il3( ) igs3( ) ic ( ) im( ) gc ec ( ) N N 0 N i ( ) i ( ) N i ( ) i ( ) N i ( ) L gs L gs 3 c N m 0 0 i ( ) i y ( ) sign ( ) n m det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n4, i, j j core, d i j det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n4, i, j j core, d i j det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 3 3, j j coil _ pri, d i3 j3 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 4 4, j j coil _ pri, d i4 j4 47

65 det () 0 C g ET ( ) g ET ( ) Q ( ) Q ( ) n4 n4 i j j coil pri coil 5 5 5, 5 5, _, _sec, d i5 j5 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 6 6, j j coil _sec, d i6 j6 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 7 7, j j coil _ sec, d i7 j7 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 8 8, j j coil _ er, d i8 j8 det () 0 C g ET ( ) g ET ( ) Q ( ) Q ( ) n4 n4 i j j coil er coil er 9 9 9, 9 9, _, _, d i9 j9 det () 0 C g ET ( ) g ET ( ) Q ( ) n4 n , i 0 0, j j coil _ er, d i0 j0 det () 0 C g ET ( ) g ET ( ) n4 n4 i j j,, d i j det () 0 C g ET ( ) g ET ( ) n4 n4 i j j,, d i j det () 0 C g ET ( ) g ET ( ) g ET n4tamb n4tamb 3 3 3, i 3 3, j j 3, T amb d i3 j3 det () 0 C g ET ( ) g ET ( ) g ET n4tamb n4tamb 4 4 4, i 4 4, j j 4, T amb d i j4 () 0 y ( ) 0 0 y ( ) y ( )... 0 y ( ) y ( ) m m 0 y ( ) y ( ) y ( ) m i j 48

66 0 y ( ) y ( ) y ( )... m m j 0 ym( ) ym ( ) y jm( ), if n even () 0 ym( ) ym ( ), if n odd 0 0 Qcore,( ) eh( ) 0 Qcore,( ) eh( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ er,( ) r3i L3( ) 0 Qcoil _ er,( ) r3i L3( ) Similar o he single-phase saurable-core ransformer, a sandard forma is also inroduced here o represen he above QDM of he single-phase auo-ransformer as follows: i( ) Y x ( ) D dx() C d 0 Y x ( ) D dx() C d eqx eqxd eqc eqx eqxd eqc 0 Y x( ) x( ) F x ( ) C T i eqx3 eqxx3 eqc3 T i h( x, u) Y x x F x C feqx feqxxx feqc 49

67 Conneciviy: TerminalNodeName subjec o : h h( x, u) h, x x x min max min max where he marices coefficiens of he QDM are (assume he exponen n equals ): Y eqx Deqx 0 Ceqc 0 Y Y eqx A Y Y 33 Y 30 A A 3 Y 0 A A N r 0 0 r gs N N N r 0 0 r gs N N N r3 0 0 r gs3 N N L LPS LPT gs LPS L L ST g s LPT LST L gs3 gn c N N N N 0 N N N i 0 50

68 Y A n4 g, i g, g,3 g,4 g,5 g,6 g,7 g,8 g,9 g,0 g, g, g,3 g,4 i n4 g, g, i g,3 g,4 g,5 g,6 g,7 g,8 g,9 g,0 g, g, g,3 g, i n4 g3, g3, g3, i g3,4 g3,5 g3,6 g3,7 g3,8 g3,9 g3,0 g3, g3, g3,3 g3, i3 g g g g n4 g g g g g g g g g g n4 g g g g g 4, 4, 4,3 4, i 4,5 4,6 4,7 4,8 4,9 4,0 4, 4, 4,3 4,4 i4 5, 5, 5,3 5,4 5, i i5 5,6 5,7 5,8 5,9 5,0 5, 5, 5,3 5,4 n4 g6, g6, g6,3 g6,4 g6,5 g6, i g6,7 g6,8 g6,9 g6,0 g6, g6, g6,3 g6, i6 n4 g7, g7, g7,3 g7,4 g7,5 g7,6 g7, i g7,8 g7,9 g7,0 g7, g7, g7,3 g7, i7 n4 g8, g8, g8,3 g8,4 g8,5 g8,6 g8,7 g8, i g8,9 g8,0 g8, g8, g8,3 g8, i8 g g g g g g g g g g g g g g n4 9, 9, 9,3 9,4 9,5 9,6 9,7 9,8 9, i 9,0 9, 9, 9,3 9,4 i9 n4 g g g g g g g g g g g g g g , 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0, i 0, 0, 0,3 0,4 i0 g g g g g g g g g g g g g g g g g g g ,,,3,4,5,6,7,8,9,0 n4 g g g g, i,,3,4 i n4 g, i g,3 g, g, g,3 g,4 g,5 g,6 g,7 g,8 g,9 g,0 g, i g,4 n4 g3, g3, g3,3 g3,4 g3,5 g3,6 g3,7 g3,8 g3,9 g3,0 g3, g3, g, i g3,4 i n4 g4, g4, g4,3 g4,4 g4,5 g4,6 g4,7 g4,8 g4,9 g4,0 g4, g4, g4,3 g i, i YA 0 YA 0 D eqx D D 33 D 3 A A 4 3 D 4 A A 5

69 D A DA 0 DA 0 5

70 D A C C C C C C C C C C C C C C C g T g T eqc ambien ambien Yeqx 3 Y B YB Y B T 53

71 YB F eqxx 3 0,(6,6) 0 F,(8,8) eqxx3 F,(9,9) eqxx3 F 3,(8,0) eqxx3 F eqxx 3 F F F F F F F eqxx3 eqxx3 4,(6,) 5,(6,6) 6,(6,6) 7,(5,5) r 8,(5,5) r 9,(5,5) r 4 9,(6,6) r 4 0,(6,6) r eqxx3 eqxx3 eqxx3 eqxx3 eqxx3 0 e h e h 54

72 F F F F,(6,6) r,(7,7) r 3,(7,7) r 4,(7,7) r eqxx3 eqxx3 3 eqxx3 3 eqxx3 3 Ceqc3 0 There are 45 sae variables and 45 equaion in he model. The definiion of he exernal sae, inernal sae, and hrough variables of he auo-ransformer are lised in Table 4-4 o Table 4-6, respecively. Table 4-4. Exernal saes of he auo-ransformer. Index Variable Descripion 0 v ( ) erminal volage of ransformer primary side (V) v ( ) erminal volage of ransformer secondary side (V) v ( ) erminal volage of ransformer eriary side (V) 3 3 v () 4 4 v () 5 erminal volage of ransformer eriary neural (V) erminal volage of ransformer primary neural (V) Table 4-5. Inernal saes of he auo-ransformer. Index Variable Descripion 5 i () L curren hrough he primary side inducance (A) 6 i () L curren hrough he secondary side inducance (A) 7 i () L3 curren hrough he eriary side inducance (A) 8 i () gs curren hrough he primary side sabilizer (A) 55

73 Table 4 5 coninued Index Variable Descripion 9 i () gs curren hrough he secondary side sabilizer (A) 0 i () gs3 curren hrough he eriary side sabilizer (A) z () z () 3 z () 3 4 ec () inroduced sae inroduced sae inroduced sae volage generaed by he flux (A) 5 i c () curren hrough eriary side windings (A) 6 () flux linkage (Web) 7 i m () magneizing curren (A) 8 y ( ) inroduced sae (p.u.) 9 y () 0 y () 3 y () 4 y () 5 3 ET () inroduced sae (p.u.) inroduced sae (p.u.) inroduced sae (p.u.) inroduced sae (p.u.) emperaure a core up poin (Celsius) 4 ET ( ) emperaure a core down poin (Celsius) 5 ET () 3 6 ET () 4 7 ET () 5 emperaure a primary coil up poin (Celsius) emperaure a primary coil mid poin (Celsius) emperaure a primary/secondary coil connecion 56

74 Table 4 5 coninued Index Variable Descripion 8 ET () 6 emperaure a secondary coil mid poin (Celsius) 9 ET ( ) emperaure a secondary coil down poin (Celsius) 7 30 ET () 8 3 ET () 9 emperaure a eriary coil up poin (Celsius) emperaure a eriary coil midpoin (Celsius) 3 ET () 0 emperaure a eriary coil down poin (Celsius) 33 ET () emperaure a inernal oil up poin (Celsius) 34 ET () emperaure a inernal oil down poin (Celsius) 35 ET () 3 emperaure a ransformer ank fron poin (Celsius) 36 ET () 4 emperaure a ransformer ank rear poin (Celsius) 37 Q () core, hea generaed a core up poin (W) 38 Q () core, hea generaed a core down poin (W) 39 Q () _, hea generaed a primary coil up poin (W) coil pri 40 Q () _, hea generaed a primary coil down poin (W) coil pri 4 Q () _, hea generaed a secondary coil up poin (W) coil sec 4 Q () _, hea generaed a secondary coil down poin (W) coil sec 43 Q () _, hea generaed a eriary coil up poin (W) coil er 44 Q () _, hea generaed a eriary coil down poin (W) coil er 57

75 Table 4-6. Through variables of he auo-ransformer. Index Variable Descripion 0 i ( ) curren hrough ransformer primary side (A) i () i () 3 3 i () 4 4 i () 5 curren hrough ransformer secondary side (A) curren hrough ransformer eriary side (A) curren hrough ransformer eriary neural (A) curren hrough ransformer primary neural (A) 4.3 Transformer AQCF Device Model In his secion, he ransformer AQCF device model is presened. The device AQCF model is a mahemaical model derived from he physical elecro-hermal circui of he ransformer direcly, using he quadraic inegraion mehod, seen in Figure 4-3. Secion. Deails of he quadraic inegraion mehod is presened in Appendix A. Quadraized Transformer Model Se of Linear & Quadraic Equaions Quadraic Inegraion AQCF Transformer Model Algebraic Quadraic Companion Form Figure 4-3. Derive he AQCF forma wih quadraic inegraion mehod Because he AQCF forma auomaically derived in an objec-oriened manner, here are a sandard procedure and sandard synax for he ransformer AQCF model, 58

76 no maer wha kind of he ransformer is. Therefore, wih he quadraic device models in secion 4., he AQCF models of he single-phase saurable ransformer and he single-phase auo-ransformer are auomaically generaed in he following forma: i () 0 0 T i Y x x F x B i ( m) 0 0 eqx eqx eq B N x ( h) M i( h) K eq eqx eq eq T i h( x) Y x x F x C feqx feqxxx feqc Conneciviy: TerminalNodeName subjec o : h h( x) h, x x x min max min max where i( ) and i( m) are he hrough variables of he device model; x is he exernal and inernal sae variables of he device model; Yeqx is he marix defining he linear par of sae variables, F is he marix defining he eqx quadraic par of sae variables; B is he hisory dependen vecor of he device model; eq Neqx is he marix defining he las inegraion sep sae variables par; M eq is he marix defining he las inegraion sep hrough variables par; Keq is he consan vecor of he device mode; TerminalNodeName is he erminal names defining he conneciviy of he device 59

77 model; Y is he consrain marix defining he linear par of sae variables; feqx F is he consrain marix defining he quadraic par of sae variables; feqx C feqc is he consrain hisory dependen vecor of he device model h h( x, u) h is he funcional consrain; min max x, x are he lower and upper bounds of he sae variables. min max The ransformer AQCF model is auomaically generaed wih he QDM model wih he ime sep h as: Y F eqx eqx 4 8 D Y D h h h h Deqxd Yeqx Yeqx 6 3 Yeqx3 0 D D Y h h h h Y D Y Yeqx3 eqxd eqx eqxd Feqxx F eqxd eqxd eqx eqx eqxd eqx eqxx3 60

78 N M eqx eq 4 Y D h h Y D Y D h 5h Y D 4 0 I 0 0 I 0 0 eqx eqxd eqx eqxd eqx eqxd eqx eqxd size( i( )) size( i( )) K eq 0 hc C 3 C hc C eqc eqc3 eqc eqc eqc3 4.4 Transformer AQCF Measuremen Model The AQCF measuremen model is obained by expressing measuremens using he 6

79 saes or equaions in he ransformer model. The ransformer AQCF measuremens are classified ino four ypes: () acual measuremens; () virual measuremens; (3) derived measuremens and (4) pseudo measuremens. Acual measuremens are he real measuremens obained by ypical measuring equipmen. For example, he erminal volages and currens of ransformer are he acual measuremens. The acual measuremens conain noises due o he daa acquisiion sysem. Virual measuremens presen he zeros on he lef sides of he inernal equaions. These measuremens are physical laws ha ransformer mus obey. These physical laws are expressed wih specific equaions, which mus be exacly saisfied and herefore he virual measuremens are exac measuremens. Since hose inernal equaions are derived from he physical laws, he virual measuremens are noiseless. Derived measuremens are he measuremens derived from he acual available measuremens. For example, hree erminals are conneced ogeher and wo erminal currens of hem are measured as acual measuremens, so he hird curren can be derived based on he Kirchhoff's curren law (KCL). The derived measuremens have same noise (error) as he acual ones. Pseudo measuremens are he measuremens normally no measured, like he volage a he neural erminal. I represens a quaniy for which one can expec o be a a cerain level bu do no have an acual measuremen. Since he value of he pseudo measuremen is no precise, i is considered ha pseudo measuremens conain larger 6

80 error (noise) han hose acual, derived and virual measuremens. The measuremen definiion and device model is sufficien o auomaically derive he measuremen model. The AQCF measuremen model can be wrien as T i z Y, x x F, x N, x ( h) M i( h) C m x m x m x m m Measuremen noise error: dmeerscale, dmeersigmapu Noe: All he above variables are in per uni sysem. where: z : measuremen variables a boh ime and ime m, z [ z( ), z ( m )] x : exernal and inernal sae variables of he measuremen model, x [ x( ), x ( m )] Y mx, : marix defining he linear par for sae variables, F mx, : marices defining he quadraic par for sae variables, C m : hisory dependen vecor of he measuremen model, N mx, : marix defining he las inegraion sep sae variables par, M m : marix defining he las inegraion sep hrough variables par, K m : consan vecor of he measuremen model, dmeerscale: he scale ha meers use (in meric unis), dmeersigmapu : he sandard deviaion for he measuremens (in per. uni), Deails of he creaion of AQCF measuremen model are described in [94]. 4.5 Consrucing a Three-Phase Transformer In his secion, he mehod of consrucing a hree-phase ransformer is described. 63

81 No maer i is a regular saurable-core ransformer or an auo-ransformer, he AQCF model of a hree-phase ransformer can be consruced by connecing hree single-phase models ino one composie model. In his secion, he regular saurable-core ransformer is aken as an example. To ge he overall model, each phase are inerconneced. The hree-phase ransformer AQCF model is shown below. i3 () 0 0 Y x x F x B T i eqx3 3 3 eqx3 3 eq3 i3 ( m) 0 0 B N x ( h) M i ( h) K eq3 eqx3 3 eq3 3 eq3 h ( x) Y x x F x C T i 3 feqx3 3 3 feqxxx3 3 feqc3 Conneciviy: TerminalNodeName 3 subjec o : hmin3 h3 ( x) hmax3, xmin3 x3 x max3 where x3 are he saes for hree-phase ransformer and he marix eqx3 Y, Feqx 3, Neqx3, eq3 M and Keq3 are decided by he combinaions of connecing he erminals in wye or dela configuraion. The oal sae vecor of he hree-phase ransformer x3 consiss of 3n in ph n ex saes. The number of exernal saes is deermined by he configuraion ha he phases of he ransformer are conneced. In general here are four configuraions, namely he 64

82 wye-wye, wye-dela, dela-wye and dela-dela configuraions. In subsequen paragraphs each one of hese four cases are documened. The inernal saes are commonly defined for all hese cases by direcly appending o he sae vecor he inernal saes of each phase Wye-Wye Conneced Transformer The configuraion of wye-wye conneced ransformer is illusraed in Figure 4-4. In his case, he number of exernal saes is eigh. v 0 () v () v () v 3 () i 0 () i () i () i 3 () Phase A Module Phase B Module Phase C Module i 4 () i 5 () i 6 () i 7 () v 4 () v 5 () v 6 () v 7 () Figure 4-4. Three-phase wye-wye conneced ransformer In order o inegrae he hree ses of he AQCF of he single-phase ransformer, he sae poiners of he AQCF of he single-phase ransformer need o be re-assigned o hose of he AQCF of he hree-phase ransformer. The indices relaionship beween he AQCFs of he single-phase ransformer and he hree-phase wye-wye conneced ransformer is shown in Figure 4-5. The oal sae number of his ransformer configuraion is 3n in 8. ph 65

83 0 in n Single Phase H,A 0 Three Phase wye-wye in n A L,A 4 3 H,B L,B in n B 5 H,C L,C H,N in n C 6 L,N 3 7 Figure 4-5. The indices relaionship of he hree-phase wye-wye conneced ransformer The exernal saes in his case are lised in Table 4-7: Table 4-7. Exernal Saes of wye-wye conneced ransformer Exernal Saes Sae Index Sae Name 0 v ( ) H, A ( ) v H, B ( ) v H, C 3 ( ) v H, N 4 ( ) v L, A 5 ( ) v L, B 6 ( ) v L, C 7 ( ) v L, N 66

84 The correspondence beween each exernal phase saes (and equaions) and he hree phase ransformer exernal saes (and equaions) for each phase is defined in Table 4-8. Table 4-8. Correspondence beween he exernal phase and bank saes index Phase A Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 v A ( ) 0 v ( ) 0, H, A ( ) v, A ( ) v, A 3 ( ) v3, A 3 ( ) v H, N 4 ( ) v L, A 7 ( ) v L, N Phase B 0 ( ) v0, B ( ) v, B ( ) v, B 3 ( ) v3, B ( ) v H, B 3 ( ) v H, N 5 ( ) v L, B 7 ( ) v L, N Phase C 0 ( ) v0, C ( ) v, C ( ) v, C 3 ( ) v3, C ( ) v H, C 3 ( ) v H, N 6 ( ) v L, C 7 ( ) v L, N Because each single-phase ransformer is modeled in he sandard AQCF synax, i is 67

85 very easy o combine he hree AQCF models ino a (composie) hree-phase ransformer AQCF model. The erminals of hree single-phase ransformers are inerconneced, by applying he Kirchhoff's Curren Law (KCL) he erminal equaions can be easily combined. The inernal equaions of each single-phase ransformer will keep unchanged. The hree-phase ransformer AQCF model is shown below. i3 () 0 0 Y x x F x B T i eqx3 3 3 eqx3 3 eq3 i3 ( m) 0 0 B N x ( h) M i ( h) K eq3 eqx3 3 eq3 3 eq3 h ( x) Y x x F x C T i 3 feqx3 3 3 feqxxx3 3 feqc3 Conneciviy: TerminalNodeName 3 subjec o : hmin3 h3 ( x) hmax3, xmin3 x3 x max3 where x3 are he saes for hree-phase ransformer and he marix eqx3 Y, Feqx 3, Neqx3, eq3 M and Keq3 are decided by he combinaions of connecing he erminals in wye or dela configuraion Wye-Dela Conneced Transformer The configuraion of wye-dela conneced ransformer is illusraed in Figure 4-6. In his case, he number of exernal saes is seven. 68

86 v 0 () v () v () v 3 () i 0 () i () i () i 3 () Module A Module B Module C i 4 () i 5 () i 6 () v 4 () v 5 () v 6 () Figure 4-6. Three-phase wye-dela conneced ransformer In order o inegrae he hree ses of he AQCF of he single-phase ransformer, he sae poiners of he AQCF of he single-phase ransformer need o be re-assigned o hose of he AQCF of he hree-phase ransformer. The indices relaionship beween he AQCFs of he single-phase ransformer and he hree-phase wye-dela conneced ransformer is shown in Figure 4-7. The oal sae number of his ransformer configuraion is 3n in 7. ph 69

87 0 in n Single Phase H,A 0 Three Phase wye-dela in n A L,A 4 3 H,B L,B in n B 5 H,C L,C H,N in n C 6 3 Figure 4-7. The indices relaionship of he hree-phase wye-dela conneced ransformer The exernal saes in his case are lised in Table 4-9: Table 4-9. Exernal Saes of wye-dela conneced ransformer Exernal Saes Sae Index Sae Name 0 v ( ) H, A ( ) v H, B ( ) v H, C 3 ( ) v H, N 4 ( ) v L, A 5 ( ) v L, B 6 ( ) v L, C 70

88 The correspondence beween he exernal phase saes (and equaions) and he hree phase ransformer exernal saes (and equaions) for each phase is defined as: Table 4-0. Correspondence beween he exernal phase and bank saes index Phase A Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, A ( ) v, A ( ) v, A 3 ( ) v3, A 0 v ( ) H, A 3 ( ) v H, N 4 ( ) v L, A 6 ( ) v L, C Phase B Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, B ( ) v, B ( ) v, B 3 ( ) v3, B ( ) v H, B 3 ( ) v H, N 5 ( ) v L, B 4 ( ) v L, A Phase C Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, C ( ) v, C ( ) v, C 3 ( ) v3, C ( ) v H, C 3 ( ) v H, N 6 ( ) v L, C 5 ( ) v L, B 7

89 Because each single-phase ransformer is modeled in he sandard AQCF synax, i is very easy o combine he hree AQCF models ino a (composie) hree-phase ransformer AQCF model. The erminals of hree single-phase ransformers are inerconneced, by applying he Kirchhoff's Curren Law (KCL) he erminal equaions can be easily combined. The inernal equaions of each single-phase ransformer will keep unchanged. The hree-phase ransformer AQCF model is shown below. i3 () 0 0 Y x x F x B T i eqx3 3 3 eqx3 3 eq3 i3 ( m) 0 0 B N x ( h) M i ( h) K eq3 eqx3 3 eq3 3 eq3 h ( x) Y x x F x C T i 3 feqx3 3 3 feqxxx3 3 feqc3 Conneciviy: TerminalNodeName 3 subjec o : hmin3 h3 ( x) hmax3, xmin3 x3 x max3 where x3 are he saes for hree-phase ransformer and he marix eqx3 Y, Feqx 3, Neqx3, eq3 M and Keq3 are decided by he combinaions of connecing he erminals in wye or dela configuraion Dela-Wye Conneced Transformer The configuraion of dela-wye conneced ransformer is illusraed in Figure

90 In his case, he number of exernal saes is seven. v 0 () v () v () i 0 () i () i () Module A Module B Module C i 3 () i 4 () i 5 () i 6 () v 3 () v 4 () v 5 () v 6 () Figure 4-8. Three-phase dela-wye conneced ransformer In order o inegrae he hree ses of he AQCF of he single-phase ransformer, he sae poiners of he AQCF of he single-phase ransformer need o be re-assigned o hose of he AQCF of he hree-phase ransformer. The indices relaionship beween he AQCFs of he single-phase ransformer and he hree-phase dela-wye conneced ransformer is shown in Figure 4-9. The oal sae number of his ransformer configuraion is 3n in 7. ph 73

91 0 in n Single Phase H,A 0 Three Phase dela-wye in n A L,A 3 3 H,B L,B in n B 4 H,C L,C in n C 5 L,N 6 Figure 4-9. The indices relaionship of he hree-phase dela-wye conneced ransformer The exernal saes in his case are lised in Table 4-: Table 4 -. Exernal Saes of dela-wye conneced ransformer Exernal Saes Sae Index Sae Name 0 v ( ) H, A ( ) v H, B ( ) v H, C 3 ( ) v L, A 4 ( ) v L, B 5 ( ) v L, C 6 ( ) v L, N The correspondence beween he exernal phase saes (and equaions) and he hree 74

92 phase ransformer exernal saes (and equaions) for each phase is defined as: Table 4 -. Correspondence beween he exernal phase and bank saes Index Phase A Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, A ( ) v, A ( ) v, A 3 ( ) v3, A 0 v ( ) H, A ( ) v H, C 3 ( ) v L, A 6 ( ) v L, N Phase B Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, B ( ) v, B ( ) v, B 3 ( ) v3, B ( ) v H, B 0 v ( ) H, A 4 ( ) v L, B 6 ( ) v L, N Phase C Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, C ( ) v, C ( ) v, C 3 ( ) v3, C ( ) v H, C ( ) v H, B 5 ( ) v L, C 6 ( ) v L, N Because each single-phase ransformer is modeled in he sandard AQCF synax, i is 75

93 very easy o combine he hree AQCF models ino a (composie) hree-phase ransformer AQCF model. The erminals of hree single-phase ransformers are inerconneced, by applying he Kirchhoff's Curren Law (KCL) he erminal equaions can be easily combined. The inernal equaions of each single-phase ransformer will keep unchanged. The hree-phase ransformer AQCF model is shown below. i3 () 0 0 Y x x F x B T i eqx3 3 3 eqx3 3 eq3 i3 ( m) 0 0 B N x ( h) M i ( h) K eq3 eqx3 3 eq3 3 eq3 h ( x) Y x x F x C T i 3 feqx3 3 3 feqxxx3 3 feqc3 Conneciviy: TerminalNodeName 3 subjec o : hmin3 h3 ( x) hmax3, xmin3 x3 x max3 where x3 are he saes for hree-phase ransformer and he marix eqx3 Y, Feqx 3, Neqx3, eq3 M and Keq3 are decided by he combinaions of connecing he erminals in wye or dela configuraion Dela-Dela Conneced Transformer The configuraion of dela-dela conneced ransformer is illusraed in Figure 4-0. In his case, he number of exernal saes is six. 76

94 v 0 () v () v () i 0 () i () i () Module A Module B Module C i 3 () i 4 () i 5 () v 3 () v 4 () v 5 () Figure 4-0. Three-phase dela-dela conneced ransformer In order o inegrae he hree ses of he AQCF of he single-phase ransformer, he sae poiners of he AQCF of he single-phase ransformer need o be re-assigned o hose of he AQCF of he hree-phase ransformer. The indices relaionship beween he AQCFs of he single-phase ransformer and he hree-phase dela-dela conneced ransformer is shown in Figure 4-. The oal sae number of his ransformer configuraion is 3n in 6. ph 77

95 0 in n Single Phase H,A 0 Three Phase dela-dela in n A L,A 3 3 H,B L,B in n B 4 H,C L,C in n C 5 L,N 6 Figure 4 -. The indices relaionship of he hree-phase dela-wye conneced ransformer The exernal saes in his case are lised in Table 4-3: Table 4-3. Exernal Saes of dela-dela conneced ransformer Exernal Saes Sae Index Sae Name 0 v ( ) H, A ( ) v H, B ( ) v H, C 3 ( ) v L, A 4 ( ) v L, B 5 ( ) v L, C The correspondence beween he exernal phase saes (and equaions) and he hree 78

96 phase ransformer exernal saes (and equaions) for each phase is defined as: Table 4-4. Correspondence beween he exernal phase and bank saes Index Phase A Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, A ( ) v, A ( ) v, A 3 ( ) v3, A 0 v ( ) H, A ( ) v H, C 3 ( ) v L, A 5 ( ) v L, C Phase B Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, B ( ) v, B ( ) v, B 3 ( ) v3, B ( ) v H, B 0 v ( ) H, A 4 ( ) v L, B 3 ( ) v L, A Phase C Phase Sae Index Sae Name Bank Sae Index Bank Sae Name 0 ( ) v0, C ( ) v, C ( ) v, C 3 ( ) v3, C ( ) v H, C ( ) v H, B 5 ( ) v L, C 4 ( ) v L, B Because each single-phase ransformer is modeled in he sandard AQCF synax, i is 79

97 very easy o combine he hree AQCF models ino a (composie) hree-phase ransformer AQCF model. The erminals of hree single-phase ransformers are inerconneced, by applying he Kirchhoff's Curren Law (KCL) he erminal equaions can be easily combined. The inernal equaions of each single-phase ransformer will keep unchanged. The hree-phase ransformer AQCF model is shown below. i3 () 0 0 Y x x F x B T i eqx3 3 3 eqx3 3 eq3 i3 ( m) 0 0 B N x ( h) M i ( h) K eq3 eqx3 3 eq3 3 eq3 h ( x) Y x x F x C T i 3 feqx3 3 3 feqxxx3 3 feqc3 Conneciviy: TerminalNodeName 3 subjec o : hmin3 h3 ( x) hmax3, xmin3 x3 x max3 where x3 are he saes for hree-phase ransformer and he marix eqx3 Y, Feqx 3, Neqx3, eq3 M and Keq3 are decided by he combinaions of connecing he erminals in wye or dela configuraion. 4.5 Summary This chaper has presened he elecro-hermal model of ransformers. A firs he quadraized device model (QDM) is presened, hen i is inegraed ino he AQCF 80

98 device model by he quadraic inegraion mehod, and hen nex is he AQCF measuremen model. Finally, he way of consrucing a hree-phase ransformer wih hree single-phase ransformers is presened along wih he wye-wye, wye-dela, delawye and dela-dela four configuraions. The DSE-based proecion scheme direcly works on he ransformer AQCF objecs so ha his mehod is objec-oriened. This mehod can be applied o oher proecion zone wih he device and measuremen models also in AQCF synax. 8

99 CHAPTER 5 TRANSFRMER PRTECTIN BASED N 5. verview DYNAMIC STATE ESTIMATIN Dynamic sae esimaion has been widely sudied in power sysem analysis. Wih redundancy in measuremens, dynamic sae esimaor can provide more accurae esimaed saes and measuremens based on mahemaic models. In his chaper, he dynamic sae esimaion mehod is applied o compue he bes esimaes of he operaing saes of ransformers based on he ransformer elecro-hermal model in Chaper 4. Three approaches have been developed o solve he dynamic sae esimaion problem, namely he unconsrained weighed leas square (UCWLS) mehod, he consrained weighed leas square (CWLS) mehod and he exended Kalman filer (EKF) mehod. The hree mehods are discussed in his chaper. Wih he soluion of dynamic sae esimaion, a Chi-square es is performed o calculae he probabiliy ha he ransformer measuremens are consisen wih is dynamic model. Based on his probabiliy, appropriae proecion decision can be made. The DSE-based proecion mehod algorihm is also inroduced in his chaper. 5. Three Mehods for he DSE Problem 5.. Approach ne: UCWLS Mehod Unconsrained Weighed Leas Square (UCWLS) mehod has been widely used for sae esimaion [03]-[05]. WLS mehod provides a soluion ha minimizes he sum 8

100 of he squares of he errors (or residuals) of every single measuremen equaion. If he measuremen model is linear, he WLS mehod provides a closed-form soluion in a sraighforward manner. Regarding he nonlinear measuremen model, a local opimal soluion can usually be reached using he Newon s mehod. Specifically, in he UCWLS mehod for ransformer dynamic sae esimaion, any measuremen (acual, derived, pseudo and virual measuremen) can be expressed in erms of he ransformer saes wih he aid of he ransformer dynamic model in AQCF form: k k,,, z h x a x b x x c k k k i x i i j x i j k k i i, j where z is he combinaion of he acual measured measuremen, derived measuremen, pseudo measuremen and virual measuremen, x is he sae variables, a is he coefficien of linear erms, b is he coefficiens of nonlinear erms, c is he consan erm [0]. The objec of UCWLS mehod is o minimize he weighed square of he measuremen residuals, mahemaically, n n hi( x) z i T Minimize J si W i i i i where h( x) z, si and W diag,, i i and i is he sandard deviaion of he meer by which he corresponding measuremen z is measured; W is he diagonal weigh marix whose non-zero enries are he inverse of he variance of he measuremen errors. 83

101 The bes esimae of he sysem sae is obained from he Gauss-Newon ieraive algorihm: xˆ xˆ ( H T WH) T H W( h( xˆ ) z) where ˆ x refers o he bes esimae of he sae vecor x, and H is he Jacobian marix of he measuremen equaions. H h( x ) x The covariance marix of he sae is defined as C x E xˆ xxˆ x where x denoes he rue sae value, and he covariance marix compued as C T H WH x. T 5.. Approach Two: CWLS Mehod The UCWLS mehod works well o esimae he ransformer saes wih a measuremen se ha represens acual measuremens wih usual measuremen errors. However, when i is used o handle virual measuremens wih very small uncerainy, i may generae numerical insabiliies due o he large separaion beween he variances of he acual measuremens and he virual measuremens [06]. To avoid hese numerical insabiliies, he consrained weighed leas square (CWLS) mehod has been used [07]-[09]. The CWLS mehod is very similar o he unconsrained one, excep for he virual measuremens ha are reaed as consrains here. Noe ha if virual measuremens do 84

102 no exis, hen he mehod revers o an unconsrained opimal problem. Given he sandard measuremens model in AQCF form, hey are separaed ino acual, derived and pseudo measuremens as:,,, k k z h x a x b x x c, k k k i x i i j x i j k k i i, j and virual measuremens expressed as: 0 g ( x) Y x F x x b k k k k eqx, i i eqx, ij i j eq, i i i, j i I is noed ha he acual, derived and pseudo measuremens conain some errors, while he virual measuremens do no have any errors. Tha is because he virual measuremens sand for he physical laws ha he ransformer model mus obey, hey are noiseless. The objec of CWLS mehod is o minimize he weighed square of he residuals of acual, pseudo and derived measuremens subjec o he virual measuremen consrains. Mahemaically, Minimize J m hi ( x) z i i i i m s i T W Subjec o : 0 g ( x) k,... n k i where h( x) z, si and W diag,, i i and i is he sandard deviaion of he meer by which he corresponding measuremen z is measured; W is he diagonal weigh marix whose non-zero enries are he inverse of he variance of he measuremen errors. The mehod of Lagrange mulipliers is applied here. A new variable λ called a Lagrange muliplier is inroduced and he Lagrange funcion is defined as follows: 85

103 T L( x, λ) J λ g( x ) The necessary condiions for he Lagrange funcion are: L( x, λ) T hnonvirual ( x) T g( x) hnonvirual ( x) z W λ 0 x x x L( x, λ) gx ( ) 0 λ The soluion of above equaions could be obained ieraively by Newon s ieraive mehod wih an iniial guess x0 and λ0. The updae is given by: v v x x x v v λ λ λ Use he Taylor expansion a v+ ieraion and ignore he higher order erms. v v hnonvirual ( x x) hnonvirual ( x ) Hx v v g( x x) g( x ) Gx where H and G are he Jacobian marices H hnonvirual ( x) gx ( ), G. x x Therefore, he necessary condiions for mehod of Lagrange mulipliers are: v v hnonvirual ( x ) Hx z WH ( λ λ ) G 0 v g( x ) Gx 0 The above equaions give: T T v T v T T x H WH G H W h( x ) z G v G 0 gx ( ) The bes esimae of he sysem sae is also obained from he Gauss-Newon ieraive algorihm: 86

104 v v T T T v T v x x H WH G nonvirual v v v λ λ G 0 g( x ) H W h ( x ) z G λ The covariance of he sae for his CWLS mehod is: T T T T T T m T T T x, H WH G H W G H WH G Cov E W H G 0, m G 0 0 G 0 T Wih some addiional calculaion, he covariance is: C T T T T T T H WH H WH G G H WH G G H WH 0 0 T T T GH WH G 5..3 Approach Three: Exended Kalman Filer Mehod Anoher widely used esimaor is he Exended Kalman Filer (EKF) [0]-[]. The EKF linearizes he nonlinear sysem o is firs-order so ha he radiional Kalman filer equaions can be applied. Given device quadraized model quadraized model: i( ) Y x ( ) D dx() C d 0 Y x ( ) D dx() C d eqx eqxd eqc eqx eqxd eqc 0 Y x( ) x( ) F x ( ) C T i eqx3 eqxx3 eqc3 The above equaions are discreized wih rapezoidal inegraion mehod: i i Y x x D ( x x ) C h 0 Yeqx xk xk Deqxd ( xk xk ) Ceqc h k k eqx k k eqxd k k eqc 87

105 0 Y x x F x C T i eqx3 k k eqxx3 k eqc3 ne furher sep, he above equaions can be re-wrien as: D ( x x ) h i i Y x x C Deqxd ( xk x ) h k Yeqx xk xk Ceqc eqxd k k k k eqx k k eqc 0 Y x x F x C T i eqx3 k k eqxx3 k eqc3 The above equaions are finally convered ino sandard EKF forma: x Ax B( i, i ) C k k mk mk cons where z h( x ) E x F x x F x C T i k k m k m k k m k m h h Deqxd Y eqx Deqxd Yeqx A h h Deqxd Y eqx Deqxd Y eqx 0 0 Yeqx 3 xk Feqxx 3 Feqxx 3x k 0 h Deqxd Yeqx h B h Deqxd Y eqx Yeqx 3 xk Feqxx 3 Feqxx 3x k C cons h Deqxd Yeqx hceqc h hc C eqc3 0 0 Yeqx 3 xk Feqxx 3 Feqxx 3x k eqc Deqxd Y eqx 88

106 The EKF mehod is implemened as a wo-sep predicion-correcion process. The equaions are given below: Predicion: Correcion: where Hk is he Jacobian marix: x Ax B( i, i ) C k k mk mk cons P A P A Q T k k k k k K P H ( H P H R ) xk xk Kk zk hxk Pk I KkHk Pk T T k k k k k k k H k hx ( k ) x and Qk and Rk are he process and measuremen noises respecively. The covariance marix of EKF mehod is conribued by boh he marix Pk and Q k. 89

107 5.3 Proposed DSE-Based Transformer Proecion Logic The logic of proposed DSE-based ransformer proecion scheme is illusraed in Figure 5-. According o his figure, he firs sep is o perform he dynamic sae esimaion wih he dynamic model and measuremens of ransformer. The dynamic sae esimaor gives he bes esimaes of ransformer saes wih aforemenioned hree mehods. Figure 5 -. Proposed DSE-based proecion logic Wih he bes esimaes of ransformer saes from he DSE soluions, a Chi-square es is performed o calculae he probabiliy ha measuremen daa are consisen wih he ransformer model. The normalized residual (or error) for each measuremen i is defined as: s i i i 90

108 and hus he vecor of normalized residuals is s W The value of Chi-square es is defined as n h( ˆ i ) z i x i i The Chi-square es quanifies he preciseness of fi beween he model and measuremens, i.e., he confidence level. The confidence level is expressed as he probabiliy ha he measuremen errors are disribued wihin heir expeced range. Specifically, he preciseness of fi (confidence level) can be obained as Pr[ ] Pr[ ] Pr(, v). where ζ is he Chi-square criical value, and v is he degree of freedom. The degree of freedom is defined as difference beween he number of measuremens m and number of saes n. The degrees of freedom are always posiive because ha m is always greaer han n. v m n The chi-square es is uilized o provide he probabiliy ha he expeced error of he esimaed sae values will be wihin a specific range. Because here are many daa acquisiion devices wih differen accuracy, a normalizaion consan k has been inroduced. The variable k is defined as follows: if i is.0 hen he sandard deviaion of each measuremen is equal o he accuracy of he meer wih which his measuremen was obained. If differen han.0 hen he sandard deviaion of he measuremen error equal he accuracy of he meer imes k. The inroducion of he variable k allows us o 9

109 characerize he accuracy of he esimaed sae wih only one variable. This is equivalen of providing he expeced error (which equals he variable k imes he sandard deviaion of he measuremen error) versus probabiliy (confidence level). Figure 5- illusraes he graph of he parameer k versus confidence level. Figure 5 -. K-facor curve for chi-square es The proposed mehod uses he confidence level as he healh index of ransformer. I is obvious ha confidence level around.0 (small Chi-square value) infers he measuremens are highly consisen wih he ransformer dynamic mode, which means here is no inernal abnormaliy. n he oher side, confidence level around 0.0 (large Chi-square value) infers he measuremens do no fi wih he ransformer dynamic model. The proposed DSE-based scheme accuraely makes he proecion decision only based on he operaing condiion of he ransformer. I akes wo consecuive samples o perform he dynamic sae esimaion. Theoreically, he proposed DSE-based mehod only requires wo samples ime (a 9

110 fracion of one ms) o deermine wheher an inernal faul happens in he ransformer. This means he proposed DSE-based scheme is faser han any exising ransformer proecion mehod for deecing he fauls. I is also noed ha here is a possibiliy ha he confidence level may drop (bu reurn o.0) for a few samples when ransiens suddenly happen. To avoid false ripping caused by ransiens, he values of he chi-square es and confidence level are inegraed over a user-seleced inerval (ypically half or one cycles) before a rip command is issued. 5.3 Summary This chaper has discussed he algorihm of proecing ransformer using he dynamic sae esimaion mehod. Three approaches are used o solve he dynamic sae esimaion problem, namely he unconsrained weighed leas square (UCWLS) mehod, he consrained weighed leas square (CWLS) mehod and he exended Kalman filer (EKF) mehod. In he end of his chaper, he proecion logic of he proposed DSEbased mehod is presened. A Chi-square es is performed o calculae he probabiliy (confidence level) ha measuremen daa are consisen wih he ransformer model. If he confidence level is around.0, he ransformer is healh; oherwise if he confidence level is around 0.0, somehing is wrong inside he ransformer and he proposed mehod will send a signal o rip and proec he ransformer. 93

111 CHAPTER 6 TRANSFRMER MDEL PARAMETER 6. verview CALIBRATIN Dynamic sae esimaion-based ransformer proecion mehod is a model-based mehod. Modeling accuracy and fideliy are fundamenal o guaranee he feasibiliy of he proposed mehod. The ransformer parameers used for sae esimaion come from he nameplae provided by he manufacurers. However, he acual parameers are always quie differen from he nameplae raings because of he ransformer ageing [3]-[4]. If he ransformer parameers are no correc, he resuls of he proposed DSE-based proecion mehod mus include sizeable errors. To solve his problem, he proposed DSE-based proecion mehod is also uilized o fine une he ransformer models and/or deermine he parameers of he model wih greaer accuracy. The basic approach of ransformer parameers calibraion is o expand he dynamic sae esimaion o include some independen parameers as sae variables. Through his way he ransformer model can be validaed. Therefore, he proposed overall approach can also provide beer models wih validaed parameers. In his chaper, a demonsraing example of auo-ransformer parameer calibraion is presened. 94

112 6. Parameers Calibraion This secion describes he parameer calibraion of single-phase auoransformer wih eriary winding. The auo-ransformer hermal conducance physical configuraion is show in Figure 6-. ET ET ET 3 ET4 ET 3 ET 4 ET 8 ET 5 ET 9 ET 6 ET 0 ET 7 gy gx gz ET ET ET amb Figure 6 -. Auo-ransformer physical parameer configuraion Because of he auoransformer physicals, he conducance parameers depend on few parameers, and hey can be classified ino hree groups ha are represened by green, brown and blue dash-lines respecively. In he firs (green) group, all he hermal conducance can be represened by he value of core (ET) o coil (ET3) conducance muliple cerain coefficiens; in he second (brown) group, all he hermal conducance can be represened by he value of coil (ET3) o oil (ET) conducance muliple cerain coefficiens; in he hird (blue) group, all he hermal conducance can be represened by he value of ank (ET3) o ambien (ETamb) conducance muliple cerain coefficiens. 95

113 The independen parameers of he auo-ransformer as reaed as saes in he dynamic sae esimaion. These independen parameers ha are summarized in Table 6-. Table 6 -. Parameers o be calibraed No. Variables Descripion x g Thermal conducance beween Core Coil ( ET ET3) y g Thermal conducance beween Coil il ( ET3 ET ) 3 z g Thermal conducance beween Tank Amb ( ET3 ET amb ) The conducance beween each emperaure hermal poin depends on he above hree variables and he relaionship is show in Table 6-: Table 6 -. Auoransformer hermal conducance The quadraized device model of single-phase auoransformer wih eriary winding for he purpose of parameer calibraion is wrien as: 96

114 i ( ) i ( ) i ( ) L gs i ( ) i ( ) i ( ) i ( ) i ( ) L gs L gs i ( ) i ( ) i ( ) 3 L3 gs3 i ( ) i ( ) i ( ) 4 L3 gs3 i ( ) i ( ) i ( ) 5 L gs dil () 0 z( ) d dil () 0 z( ) d dil3() 0 z3( ) d d() 0 ec ( ) d 0 v ( ) v ( ) r i ( ) i ( ) i ( ) / g N e ( ) L gs gs s c N N N 0 v ( ) v ( ) r i ( ) i ( ) i ( ) / g e ( ) 5 L gs gs s c N N N 0 v ( ) v ( ) r i ( ) i ( ) i ( ) / g e ( ) L3 gs3 gs3 s3 c N N 0 i ( ) / g L z ( ) L z ( ) L z ( ) gs s PS PT 3 0 i ( ) / g L z ( ) L z ( ) L z ( ) gs s PS ST 3 0 i ( ) / g L z ( ) L z ( ) L z ( ) gs3 s3 3 3 ST PT 3 0 il3( ) igs3( ) ic ( ) im( ) gc ec ( ) N N 0 N i ( ) i ( ) N i ( ) i ( ) N i ( ) L gs L gs 3 c 0 0 i ( ) i y ( ) sign ( ) n m m det () 0 C 6.7 g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 x y, j j core, d j det () 0 C 7.3 g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 x y, j j core, d j N 97

115 det () 0 C 7.4 g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 3 3 x y 3 3, j j coil _ pri, d j3 det () 0 C 8 g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 4 4 x y 4 4, j j coil _ pri, d j4 det () 0 C 8. g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) Q ( ) n4 x y j j coil pri coil , _, _sec, d j5 det () 0 C 8. g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 6 6 x y 6 6, j j coil _sec, d j6 det () 0 C 7.7 g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 7 7 x y 7 7, j j coil _ sec, d j7 det () 0 C 8. g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 8 8 x y 8 8, j j coil _ er, d j8 det () 0 C 8. g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) Q ( ) n4 x y j j coil er coil er , _, _, d j9 det () 0 C 7.7 g ( ).6 g ( ) ET ( ) g ET ( ) Q ( ) n4 0 0 x y 0 0, j j coil _ er, d j0 det () 0 C 0.6 g ( ) ET ( ) g ET ( ) n4 y, j j d j det () 0 C 0.6 g ( ) ET ( ) g ET ( ) n4 y, j j d j det () 0 C 0.9 g ( ) 3 g ( ) ET ( ) g ET ( ) g ET n4et amb 3 3 y z 3 3, j j 3, T amb d j3 det () 0 C 0.9 g ( ) 3 g ( ) ET ( ) g ET ( ) g ET n4et amb 4 4 y z 4 4, j j 4, T amb d j4 () 0 y ( ) 0 0 y ( ) y ( )... 98

116 0 y ( ) y ( ) m m 0 y ( ) y ( ) y ( ) m i j 0 y ( ) y ( ) y ( ) m m j... 0 ym( ) ym ( ) y jm( ), if n even () 0 ym( ) ym ( ), if n odd 0 0 Qcore,( ) eh( ) 0 Qcore,( ) eh( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ pri,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ sec,( ) ri L( ) 0 Qcoil _ er,( ) r3i L3( ) 0 Qcoil _ er,( ) r3i L3( ) Afer he quadraizaion, he quadraic inegraion mehod is applied o above he auo-ransformer quadraized model o conver he model o he sandard AQCF synax. This procedure is he same as he procedure discussed in Chaper 4, and he deails are no lised here. The exernal saes, inernal saes and he acual measuremens of he auoransformer physical parameers idenificaion problem are lised by Table 6-3 o Table 6-5, respecively. 99

117 Table 6-3. Exernal sae variables of auo-ransformer Index Variable Descripion 0 v ( ) v ( ) v ( ) 3 3 v () 4 4 v () 5 erminal volage of ransformer primary side (V) erminal volage of ransformer secondary side (V) erminal volage of ransformer eriary side (V) erminal volage of ransformer eriary neural (V) erminal volage of ransformer primary neural (V) Table 6-4. Inernal sae variables of auo-ransformer Index Variable Descripion 5 i () L curren hrough he primary side inducance (A) 6 i () L curren hrough he secondary side inducance (A) 7 i () L3 curren hrough he eriary side inducance (A) 8 i () gs curren hrough he primary side sabilizer (A) 9 i () gs curren hrough he secondary side sabilizer (A) 0 i () gs3 curren hrough he eriary side sabilizer (A) z () z () 3 z () 3 4 ec () inroduced sae inroduced sae inroduced sae volage generaed by he flux (A) 5 i c () curren hrough eriary side windings (A) 6 () flux linkage (Web) 00

118 Table 6 4 coninued Index Variable Descripion 7 i m () magneizing curren (A) 8 y ( ) 9 y () inroduced sae (p.u.) inroduced sae (p.u.) inroduced sae (p.u.) inroduced sae (p.u.) 7+m ym() inroduced sae (p.u.) 8+m ET () emperaure a core up poin (Celsius) 9+m ET ( ) emperaure a core down poin (Celsius) 0+m ET () 3 +m ET () 4 +m ET () 5 3+m ET () 6 emperaure a primary coil up poin (Celsius) emperaure a primary coil mid poin (Celsius) emperaure a primary/secondary coil connecion emperaure a secondary coil mid poin (Celsius) 4+m ET ( ) emperaure a secondary coil down poin (Celsius) 7 5+m ET () 8 6+m ET () 9 emperaure a eriary coil up poin (Celsius) emperaure a eriary coil midpoin (Celsius) 7+m ET () 0 emperaure a eriary coil down poin (Celsius) 8+m ET () emperaure a inernal oil up poin (Celsius) 9+m ET () emperaure a inernal oil down poin (Celsius) 30+m ET () 3 emperaure a ransformer ank fron poin (Celsius) 0

119 Table 6 4 coninued Index Variable Descripion 3+m ET () 4 emperaure a ransformer ank rear poin (Celsius) 3+m Q () core, hea generaed a core up poin (W) 33+m Q () core, hea generaed a core down poin (W) 34+m Q () _, hea generaed a primary coil up poin (W) coil pri 35+m Q () _, hea generaed a primary coil down poin (W) coil pri 36+m Q () _, hea generaed a secondary coil up poin (W) coil sec 37+m Q () _, hea generaed a secondary coil down poin (W) coil sec 38+m Q () _, hea generaed a eriary coil up poin (W) coil er 39+m Q () _, hea generaed a eriary coil down poin (W) coil er 40+m g x 4+m g y 4+m g z Thermal conducance beween Core Coil ( ET ET ) 3 Thermal conducance beween Coil il ( ET ET ) 3 Thermal conducance beween Tank Amb ET3 ET amb ( ) 0

120 Table 6-5. Through variables of auo-ransformer Index Variable Descripion 0 i ( ) i () i () 3 3 i () 4 4 i () 5 curren hrough ransformer primary side (A) curren hrough ransformer secondary side (A) curren hrough ransformer eriary side (A) curren hrough ransformer eriary neural (A) curren hrough ransformer primary neural (A) These acual measuremens ogeher wih he ransformer model which are reaed as virual measuremens, conribue 46+m measuremens for he sae esimaor, while he sae esimaor has 48+m saes. Besides, wo addiional pseudo measuremens (Phase N volages) and hree emperaure measuremens (emperaure a he ank, oil and primary coil) are available, which giving he parameer calibraion problem a degree of freedom equals hree. Therefore, we have enough observabiliy o realize he parameer calibraion of auo-ransformer. By applying a muli-sep dynamic sae esimaion, he degree of freedom would increase again. The reason o implemen his approach is ha he dynamic sae esimaor runs a high raes and he physical parameers of he auo-ransformer can be assumed o be consan wihin -3 consecuive ime seps. For example, consider a wo ime sep case where we will have (48+m) + (45+m) = 93+m saes (for he second ime sep he 3 physical parameers are considered he same as in he firs ime sep). In conras, we have (49+m) * =98 + m measuremens, which will make he sae esimaor more observable. 03

121 6.3 Auoransformer Parameers Idenificaion Numerical Resuls In his secion, a 300MVA, 765/345/3.8 kv single-phase auo-ransformer wih eriary winding is seleced o validae he performance of he proposed mehod. The es sysem consiss of a generaor source, several ransformers and ransmission lines, as shown in Figure 6-. A ime 9.9 sec, a single-phase line o ground faul is generaed a he bus LINE345 o generae some ransiens for he purpose of beer calibraion. Figure 6 -. Auo-ransformer parameer calibraion es sysem The waveforms of he measuremens are illusraed in Figure

122 048.7 kv Volage_Primary_AN (V) kv 45.6 kv Volage_Secondary_AN (V) -9. kv kv Volage_Teriary_AN (V) kv 637. A Curren_Primary_Phase_A (A) A 545. A Curren_Secondary_Phase_A (A) A 50.0 ma Curren_Teriary_Phase_A (A) ma 54.3 Celsius ET3_Coil (Celsius) 5.65 Celsius Celsius ET_il (Celsius) Celsius Celsius ET3_Tank (Celsius) 45.8 Celsius 9.55 s s Figure 6-3. Auo-ransformer measuremens There are six volage and curren measuremens for he primary, secondary and eriary erminals, as well as hree emperaure measuremens: primary coil (ET3), oil (ET) and ank (ET3). The above measuremens are used for auo-ransformer parameer calibraions. The real and esimaed parameers are presened in Table 6-6. Parameers Table 6-6. Parameers calibraion resuls Acual Value (kj/celsius) Esimaed Value (kj/celsius) Error g x % g y % g z % From he above Table, i is known ha he esimaion errors for hese auoransformer physical parameers are below %, which verifies he effeciveness of he 05

123 proposed parameer calibraion mehod. 6.4 Summary In his chaper, he proposed DSE-based mehod is used o calibrae ransformer physical parameers. The independen physical parameers are modeled as sae variables in he esimaion process. The mahemaical formulaion of a single-phase auoransformer physical parameers idenificaion problem is presened o illusrae he procedure. Numerical resuls validae he feasibiliy of using DSE mehod o calibrae ransformer physical parameers. 06

124 CHAPTER 7 DEMNSTRATING EXAMPLES: DSE-BASED 7. verview TRANSFRMER PRTECTIN In his chaper, proposed DSE-based ransformer proecion scheme is compared wih he legacy proecion schemes o es he securiy and dependabiliy. The example es sysem comprises a 35MVA, 5/35kV hree-phase dela-wye saurable-core ransformer wih impedance Z = j0.0 pu, designaed as T in Figure 7-. The ambien emperaure is 5 C, he ransformer is 50% loaded and i is proeced wih six aforemenioned legacy schemes: (a) percenage differenial proecion; (b) harmonic-resrain differenial proecion; (c) negaive-sequence differenial proecion; (d) overcurren proecion; (e) vols-over-herz proecion and (f) hermal proecion. The performance of he legacy proecion funcions is compared wih he performance of proposed DSE-based proecion relay. B T B3 G B Load Load Figure 7 -. Transformer esing sysem. The legacy proecion funcions have he following seings: (a) percenage differenial proecion: he percen differenial hreshold seing is 0%, he minimum 07

125 pickup operaing curren is 0A (referred o primary side); (b) harmonic-resrain differenial proecion: he percen differenial hreshold seing is 0%, he minimum pickup operaing curren is 0A (referred o primary side) and he nd harmonic blocking level I / I is 0%; (c) negaive-sequence differenial proecion: he op nd op percen differenial hreshold seing is 0%, he minimum pickup operaing curren is.0a (referred o primary side); (d) ime-overcurren proecion: he pickup curren referred o primary side is 900A and he ime dial is 0. and very inverse; (e) vols-overherz proecion: he ime characerisic relaed o he raio of vols-over-herz is shown in Figure 7- ; (f) hermal proecion: he emperaure limi a he ho-spos is 05. Figure 7 -. Transformer ime characerisic relaed o vols-over-herz. 7. Even ne: Transformer Energizaion In his even, ransformer energizaion is presened. Breakers B and B are 08

126 iniially open while he breaker B3 is closed. A ime = 5.s he breaker B is suddenly closed and he ransformer is energized, as shown in Figure 7-3. Figure 7-3. Transformer energizaion siuaion The erminal volages and currens in his even are shown in Figure 7-4 for he ime period [ ] seconds. Noe ha he firs se of races show he volages a he wo ends of he ransformer and he second se of races show he currens a he wo sides of he ransformer. Noe obvious inrush currens occur on he primary side erminal currens when he breaker B is suddenly closed. The resuls of legacy proecion funcions as well as he proposed DSE-based mehod are presened nex: 09

127 66.5 kv Volage XFMR-High Phase_BC (V) Volage XFMR-High Phase_CA (V) Volage XFMR-High Phase_AB (V) kv 5.66 kv Volage XFMR-Low Phase_AB (V) Volage XFMR-Low Phase_BC (V) Volage XFMR-Low Phase_CA (V) kv 46.5 A Line_Curren_Meer_TXMR_high_A (A) Line_Curren_Meer_TXMR_high_B (A) Line_Curren_Meer_TXMR_high_C (A) A 55.9 A Line_Curren_Meer_TXMR Low_A (A) Line_Curren_Meer_TXMR_Low_B (A) Line_Curren_Meer_TXMR_Low_C (A) A 5.00 s 5.30 s Figure 7-4. Terminal volages and currens for ransformer energizaion (a) Percenage Differenial Proecion The resuls of he percenage differenial proecion are shown in Figure 7-5. When he energizaion happens, because of he inrush currens, he operaing curren referring o he primary side is abou 40 A, which is larger han he 0A seing. The resraining curren is abou 85A, and he differenial percen is 4%, and i is also more han he 0% seing. Because boh seings are exceeded, a ripping command is issued. As a consequence, he percenage differenial funcion would falsely rip he ransformer in his energizaion siuaions. 0

128 43.9 A peraing_i_op (A) 3.3 ma 85.7 A Resraining_I_res (A).93 ma 4.0 % Diff_Percen_Iop/Ires (%) 0.65 % 5.00 s 5.30 s Figure 7-5. Percenage differenial proecion resuls for energizaion (b) Harmonic-resrain differenial proecion Excep for he operaing curren, resraining curren and differenial percen, he harmonic-resrain differenial funcion also moniors he second-harmonic of he operaing-curren, as shown in Figure 7-6. The values of operaing curren, resraining curren and he differenial percen are he same as hose in Figure 7-5. The secondharmonic curren is abou 69A, and he measured second-harmonic level is 30%. This value is larger han he 0% seing. Therefore, he harmonic-resrain differenial funcion would block rip signals, and he ransformer would no be falsely ripped in his energizaion siuaion.

129 43.9 A peraing_i_op (A) 3.3 ma 85.7 A Resraining_I_res (A).93 ma 4.0 % Diff_Percen_Iop/Ires (%) 0.65 % A Second_Harmonic_Curren (A) 0.3 na 9.93 % nd_harmonic_level (%) 3.76 u% 5.00 s 5.30 s Figure 7-6. nd harmonic level for ransformer energizaion (c) Negaive-sequence differenial proecion The resuls of negaive-sequence differenial proecion are shown in Figure 7-7. When he energizaion happens, he negaive-sequence of operaing curren referring o he primary side is 95 A, which is higher han he 5 A seing. The negaive-sequence resraining curren is 04 A. The negaive-sequence differenial percen is 89%, which is also larger han he 0% seing. Because boh seings are exceeded, a ripping command is issued. As a consequence, he negaive-sequence differenial funcion would falsely rip he ransformer in his energizaion siuaions.

130 95.7 A Neg_Seq_peraing_I_op (A).457 ma 04. A Neg_Seq_Resraining_I_res (A).457 ma 89.8 % Neg_Seq_Diff_Percen_Iop/Ires (%) 0.3 % 5.00 s 5.96 s Figure 7-7. Negaive-sequence differenial proecion resuls (d) Time-overcurren proecion The resul of ime-overcurren proecion is shown in Figure 7-8. The RMS value of ransformer primary-side curren is abou 93 A and i is less han he seing (900A). Therefore, he ransformer would no be falsely ripped by he ime-overcurren proecion funcion in his energizaion siuaion. 93. A Primary Side Curren RMS (A) ma 5.0 s 5.97 s Figure 7-8. Time-overcurren proecion for ransformer energizaion (e) Vols-over-herz proecion and hermal proecion The vols-over-herz proecion funcion and hermal proecion funcion are no for his siuaion. 3

131 (f) Proposed DSE-based proecion The resuls of proposed DSE-based proecion scheme are shown in Figure 7-9. When he energizaion happens, he currens a boh ends of ransformer change from zero o large values. Though large inrush currens occur o he primary side erminal currens, he residuals of erminal currens remain a very small values. The chi-square value is also very small, while he confidence level says around 00%. This high confidence level means here is nohing wrong inside he ransformer so ha he DSEbased proecion scheme will no falsely rip he ransformer, hus avoiding he misoperaion under energizaion siuaions. 48. A Acual_Measuremen_Curren_Highside (A) A 5.7 A Acual_Measuremen_Curren_Lowside (A) -56. A.330 ma Residual_Curren_Highside (A) ua 5.36 ma Residual_Curren_Lowside (A) ma 0.9 Chi_Square u Confidence_Level u Trip_Decision u 5.00 s 5.30 s Figure 7-9. Proposed DSE-based proecion for ransformer energizaion (h) Summary of Even ne 4

132 A summary of he proecion mehod resuls for Even ne is shown in Table 7-. In his even wih ransformer energizaion, some legacy funcions such as he percenage differenial and negaive-sequence differenial proecion mehods end o falsely rip he ransformer, which means hey are insecure. In conras, proposed DSEbased mehod guaranees 00% securiy when ransformer is energized in his even. Table 7 -. Summary of Even ne: Energizaion Proecion Mehods Falsely Trip Percenage differenial Yes Insecure Harmonic-resrain differenial No Secure Negaive-sequence differenial Yes Insecure Time-overcurren No Secure Vols-over-herz NA NA Thermal NA NA Proposed DSE-based No Secure 7.3 Even Two: Secondary Side Coil Faul, 5% from Neural In his even an inernal faul happens o he ransformer. Breakers B and B3 are iniially closed while he breaker B is open. A ime = 0.0s he breaker B is suddenly closed and a 5% coil-neural faul happens o he phase A of secondary windings, as shown in Figure

133 Figure 7-0. Transformer 5% faul near neural siuaion The erminal measuremens are shown in Figure 7- for he ime period [ ] seconds. Noe he firs se of races show he volages a he wo ends of he ransformer and he second se of races show he currens a he wo sides of he ransformer. Noe ha very lile change occurs o he erminal volages and currens due o his inernal faul. The resuls of legacy proecion funcions as well as he proposed mehod are presened nex: 6

134 56.5 kv Volage_High_BC (V) Volage_High_CA (V) Volage_High_AB (V) kv kv Volage_Low_AB (V) Volage_Low_BC (V) Volage_Low_CA (V) kv 78.0 A Curren_High_A (A) Curren_High_B (A) Curren_High_C (A) A 59.6 A Curren_Low_A (A) Curren_Low_B (A) Curren_Low_C (A) -59. A.08 s.3 s Figure 7 -. Terminal volages and currens for ransformer inernal fauls (a) Percenage Differenial Proecion The resuls for percenage differenial proecion are shown in Figure 7-. When he inernal faul happens, he operaing-curren referring o he primary side is abou 6.5 A, which is smaller han he 0 A seing. The resraining curren is abou A. The differenial percen is 5.4%, and i is also less han he 0% seing. Because neiher seing is exceeded, he percenage differenial funcion would no send a rip signal, so he ransformer is no proeced by his funcion regarding his inernal faul. 7

135 6.5 A peraing_i_op (A).39 A.3 A Resraining_I_res (A) 7. A 5.36 % Diff_Percen_Iop/Ires (%).87 %.08 s.3 s Figure 7 -. Percenage differenial proecion resuls (b) Harmonic-resrain differenial proecion The resuls of he harmonic-resrain differenial proecion are shown in Figure 7-3. The values of operaing curren, resraining curren and he differenial percen are he same as hose in Figure 7-. The second-harmonic curren is abou.36a. The measured second-harmonic level is around %, which is almos he same as he seing. However, because he oher seings are no exceed, no rip signal will be sen. Therefore, he ransformer is no proeced by he harmonic-resrain differenial funcion regarding his inernal faul. 8

136 6.5 A peraing_i_op (A).39 A.3 A Resraining_I_res (A) 7. A 5.36 % Diff_Percen_Iop/Ires (%).87 %.36 A Second_Harmonic_Curren (A) 0.46 A.48 % nd_harmonic_level (%).8 %.08 s.3 s Figure 7-3. nd harmonic level for ransformer inernal fauls (c) Negaive-sequence differenial proecion The resuls for negaive-sequence differenial proecion are shown in Figure 7-4. When he faul happens, he negaive-sequence operaing-curren is 5.4 A (larger han seing). The negaive-sequence resraing curren is 6.84A and he negaive-sequence differenial percen is 66% (larger han seing). Because boh seings are exceeded, a rip signal will be sen. Therefore, he negaive-sequence differenial funcion is able o proec he ransformer agains his inernal faul. This funcion deecs he faul a.06 s and i will rip he ransformer a.6 s. 9

137 5.430 A Neg_Seq_peraing_I_op (A) 0.7 A A Neg_Seq_Resraining_I_res (A) 0.88 A 65.8 % Neg_Seq_Diff_Percen_Iop/Ires (%) %.07 s.30 s Figure 7-4. Negaive-sequence differenial proecion resuls (d) Time-overcurren proecion The resul of ime-overcurren proecion is shown in Figure 7-5. The RMS value of ransformer primary-side curren is abou 6 A when he faul happens. This value is less han he seing (900A) so no rip signal is sen. Therefore, he ime-overcurren scheme would fail o proec he ransformer. 5.8 A Primary Side Curren RMS (A) 7.5 A.08 s.3 s Figure 7-5. Time-overcurren proecion for ransformer inernal fauls (e) Vols-over-herz proecion and hermal proecion The vols-over-herz proecion funcion and hermal proecion funcion are no for his siuaion. 0

138 (f) Proposed DSE-based proecion The resuls of proposed DSE-based proecion are shown in Figure 7-6. When he inernal faul happens, here are no obvious changes in he erminal currens. However, he residuals of erminal currens increase from zero o considerable values. The chisquare value also increase from zero o 5, while he confidence level drops from 00% o zero immediaely. This zero confidence level indicaes abnormaliies inside he ransformer and proecion acions would be aken. I is noiced ha confidence level is oscillaing during he faul period since he inernal faul is oo small. An inegral funcion is applied o accumulae he confidence level values and a rip decision is aken o proec he ransformer, as shown in he figure. The DSE-based proecion scheme deecs he fauls a.00 s and i rips he ransformer a.08 s A Acual_Measuremen_Curren_HighSide (A) A 58.7 A Acual_Measuremen_Curren_LowSide (A) A.64 ma Residual_Curren_HighSide (A) -.64 ma 6.0 ma Residual_Curren_Lowside (A) -6.0 ma 5. Chi_Square 5.88 u Confidence_Level Trip_Decision s.30 s Figure 7-6. Proposed DSE-based proecion for ransformer inernal fauls

139 (h) Summary of Even Two A summary of he proecion mehod resuls for Even Two is shown in Table 7-. In his even where a faul near neural happens, none of he legacy funcion could successfully deec he exisence of faul excep for he negaive-sequence differenial funcion, which means mos of hem are undependable. In conras, he proposed DSEbased scheme is able o deec i dependably. Moreover, he speed of he proposed scheme is much faser han ha of negaive-sequence differenial funcion. Table 7 -. Summary of Even Two: Inernal Faul Proecion Mehods Trip Percenage differenial No Undependable Harmonic-resrain differenial No Undependable Negaive-sequence differenial Yes Dependable Time-overcurren No Undependable Vols-over-herz NA NA Thermal NA NA Proposed DSE-based Yes Dependable 7.4 Even Three: 5% Secondary Side Coil Faul during Energizaion In his even he ransformer energizaion happens firs, following an inernal faul. Breakers B and B are iniially open while he breaker B is closed. A ime =.3s he B is suddenly closed and he ransformer is energized. ne cycle laer a ime =.36s, he breaker B is also closed and a 5% coil-neural faul happens o he phase

140 A of secondary windings, as shown in Figure 7-7. Figure 7-7. Transformer 5% faul during ransformer energizaion The erminal measuremens are shown in Figure 7-8 for he ime period [.3-.38] seconds. Noe ha he firs se of races show he volages a he wo ends of he ransformer and he second se of races show he currens a he wo sides of he ransformer. Noe obvious inrush currens occur on he primary side erminal currens when he energizaion happens. The resuls of legacy proecion funcions as well as he proposed mehod are presened nex: 3

141 55.5 kv Volage XFMR-High Phase_BC (V) Volage XFMR-High Phase_CA (V) Volage XFMR-High Phase_AB (V) kv kv Volage XFMR-Low Phase_AB (V) Volage XFMR-Low Phase_BC (V) Volage XFMR-Low Phase_CA (V) kv 478. A Curren_TXMR_high_A (A) Curren_TXMR_high_B (A) Curren_TXMR_high_C (A) -66. A 55.5 A Curren_TXMR_Low_A (A) Curren_TXMR_Low_B (A) Curren_TXMR_Low_C (A) A.30 s.38 s Figure 7-8. Terminal volages and currens for ransformer inernal fauls under energizaion (a) Percenage Differenial Proecion The resuls of he percenage differenial proecion are shown in Figure 7-9. When he energizaion happens, he operaing curren is abou 78 A, which is larger han he 0A seing. The resraining curren is abou 90A. The differenial percen is 54%, and i is also more han he 0% seing. Because boh seings are exceeded, a ripping command is issued and ransformer will be falsely ripped because of he energizaion. ne cycle laer when he inernal faul happens, here are no much change o he above values, hus he ripping decision is sill effecive. Though he percenage differenial funcion would send a rip signal, i does no necessary mean his funcion is able o deec he inernal faul. The large operaing curren and high differen percen 4

142 are acually caused by he inrush currens. The percenage differenial funcion dose no perform he correc reacion regarding he ransformer energizaion in his even A peraing_i_op (A) ma 90.7 A Resraining_I_res (A).69 ma 53.9 % Diff_Percen_Iop/Ires (%) %.30 s.38 s Figure 7-9. Percenage differenial proecion resuls for inernal fauls under energizaion (b) Harmonic-resrain differenial proecion Excep for he operaing-curren, resraining-curren and differenial percen, he harmonic-resrained differenial funcion also moniors he second-harmonic of he operaing-curren, as shown in Figure 7-0. The values of operaing curren, resraining curren and he differenial percen are he same as hose in Figure 7-9. The secondharmonic curren is abou 7 A, and he measured second-harmonic level is 7% (larger han seing). Therefore, he harmonic-resrained differenial funcion will block any false rip during ransformer energizaion in his even. 5

143 78.4 A peraing_i_op (A) ma 90.7 A Resraining_I_res (A).69 ma 53.9 % Diff_Percen_Iop/Ires (%) % 7.38 A Second_Harmonic_Curren (A) 0.68 na 7.4 % nd_harmonic_level (%) 33.7 u%.30 s.38 s Figure 7-0. nd harmonic level for inernal fauls under energizaion ne cycle laer when he inernal faul happens, here are no much change o he above values. Therefore, any rip signal will be sill blocked, even when he inernal faul happens. As a consequence, he ransformer is no proeced when he inernal faul happens in his siuaion. (c) Negaive-sequence differenial proecion The resuls of negaive-sequence differenial proecion are shown in Figure 7-. When he energizaion happens, he negaive-sequence operaing-curren is 00 A, which is higher han he 5 A seing. The negaive-sequence resraining curren is 08 A. The negaive-sequence differenial percen is 9%, which is also larger han he 0% seing. Therefore, he negaive-sequence differenial proecion would falsely rip he ransformer when he energizaion happens in his even. 6

144 00.6 A Neg_Seq_peraing_I_op (A) 3.09 ma 08.8 A Neg_Seq_Resraining_I_res (A) 4.58 ma 90.8 % Neg_Seq_Diff_Percen_Iop/Ires (%) 0.55 %.30 s.38 s Figure 7 -. Negaive-sequence differenial proecion resuls ne cycle laer when he inernal faul happens, here are no much change o he above values. So his funcion would rip he ransformer when inernal faul happens. However, his funcion dose no perform he correc reacion for ransformer energizaion in his even. Moreover, if he harmonic-resrain differenial proecion funcion is also implemened, he rip signal would be blocked (he reason has been explained in he former proecion funcion) and he ransformer would no be proeced. (d) Time-overcurren proecion The resul of ime-overcurren proecion is shown in Figure 7-. When he energizaion happens, he RMS value of ransformer primary-side curren is abou 37 A and i is less han he seing (900A). Therefore, he ransformer would no be falsely proeced during energizaion. When he inernal faul happens, he RMS value is sill below he seing. Therefore i canno proec he ransformer for he inernal faul in his even. 7

145 37. A Primary_Side_Curren_RMS (A) 3.94 ma.38 s.387 s Figure 7 -. Time-overcurren proecion for inernal fauls under energizaion (e) Vols-over-herz proecion and hermal proecion The vols-over-herz proecion funcion and hermal proecion funcion are no for his siuaion. (f) Proposed DSE-based proecion The resuls of proposed DSE-based proecion scheme are shown in Figure 7-3. A firs, when he energizaion happens, he currens a boh ends of ransformer change from zero o large values. Though a large inrush curren occurs o he primary side erminal curren, he residuals of erminal currens remains a very small values. The chi-square value is small and he confidence level says around 00%. This high confidence level means here is nohing wrong inside he ransformer during he firs cycle. The DSE-based proecion scheme will no falsely rip he ransformer, hus avoiding he mis-operaion under energizaion siuaions. ne cycle laer when he inernal faul happens, here are no much change a he erminal currens and he inrush curren sill exiss. However, he residuals of erminal currens change from zero o considerable values. The chi-square value also increase 8

146 from zero o 39, while he confidence level drops from 00% o zero immediaely. This zero confidence level indicaes abnormaliies inside he ransformer and proecion acions would be aken. I is noiced ha confidence level is oscillaing during he faul period since he inernal faul is oo small. An inegral funcion is applied o accumulae he confidence level values and a rip decision is aken o proec he ransformer, as shown in he figure. The DSE-based proecion scheme makes he correc response for boh he energizaion period and inernal faul period in his even. I deecs he faul a.36 s and rips he ransformer a.334 s A Acual_Measuremen_Curren_Highside (A) A A Acual_Measuremen_Curren_Lowside (A) A.00 ma Residual_Curren_Highside (A) -.0 ma ma Residual_Curren_Lowside (A) ma 39. Chi_Square u Confidence_Level Trip_Decision s.38 s Figure 7-3. Proposed DSE-based proecion for inernal fauls under energizaion (h) Summary of Even Three 9

147 A summary of he proecion mehod resuls for Even Three is shown in Table 7-3. In his even where boh inernal faul and ransformer energizaion happen in a shor ime inerval, none of he legacy funcion could successfully make he righ response for boh siuaion. In conras, he proposed scheme is able o say inacive during ransformer energizaion, while making he righ rip decision when i deec he exisence of inernal faul wih high dependabiliy and sensiiviy. Moreover, he speed of he proposed scheme is so fas ha i can rip he ransformer before he faul escalae o more server siuaion. Table 7-3. Summary of Even Three: Energizaion and Inernal Faul Proecion Mehods Falsely Trip a Energizaion Correc Trip a Inernal Faul Percenage differenial Yes Yes Insecure Harmonic-resrain differenial Negaive-sequence differenial No No Undependable Yes Yes Insecure Time-overcurren No No Undependable Vols-over-herz NA NA NA Thermal NA NA NA Proposed DSE-based No Yes Secure and Dependable 7.5 Even Four: Transformer % Iner-urn Faul In his even a ransformer iner-urn faul is presened. Breakers B and B3 are 30

148 iniially closed while he breaker B is open. A ime = 30.0s he breaker B is suddenly closed and a % iner-urn faul happens o he phase A of ransformer secondary side windings, as shown in Figure 7-4. Figure 7-4. Transformer % iner-urn faul siuaion The erminal measuremens are shown in Figure 7-5 for he ime period [ ] seconds. Noe ha he firs se of races show he volages a he wo ends of he ransformer and he second se of races show he currens a he wo sides of he ransformer. Noe ha very lile change occurs o he erminal volages and currens due o his iner-urn faul. The resuls of legacy proecion funcions as well as he proposed mehod are presened nex. 3

149 56.4 kv Volage_High_BC (V) Volage_High_CA (V) Volage_High_AB (V) kv kv Volage_Low_AB (V) Volage_Low_BC (V) Volage_Low_CA (V) kv 66.4 A Curren_High_A (A) Curren_High_B (A) Curren_High_C (A) A 58.5 A Curren_Low_A (A) Curren_Low_B (A) Curren_Low_C (A) A 30.0 s 30.7 s Figure 7-5. Terminal volages and currens for ransformer inernal fauls (a) Percenage Differenial Proecion The resuls for percenage differenial proecion are shown in Figure 7-6. When he inernal faul happens, he operaing-curren is abou 0.5 A, which is smaller han he 0 A seing. The resraining curren is abou 3A. The differenial percen is 0.43%, and i is also less han he 0% seing. Because neiher seing is exceeded, he percenage differenial funcion would no send a rip signal, so he ransformer is no proeced by his funcion regarding his iner-urn faul. 3

150 0.53 A operaing_i_op (A) 0. A.8 A Resraining_I_res (A).5 A 0.46 % Diff_Percen (%) 0.8 % 30.0 s 30.7 s Figure 7-6. Percenage differenial proecion resuls (b) Harmonic-resrained differenial proecion Excep for he operaing-curren, resraining-curren and differenial percen, he harmonic-resrained differenial funcion also moniors he second-harmonic of he operaing-curren, as shown in Figure 7-7. The values of operaing curren, resraining curren and he differenial percen are he same as hose in Figure 7-6. The secondharmonic of operaing curren is abou 0.07A, and he measured second-harmonic level is around 6%, which is smaller han he seing. Because neiher seing is exceeded, no rip signal will be sen when he iner-urn faul happens. Therefore, he ransformer is no proeced by he harmonic-resrained differenial funcion regarding he iner-urn faul in his even. 33

151 0.53 A operaing_i_op (A) 0. A.8 A Resraining_I_res (A).5 A 0.46 % Diff_Percen (%) 0.8 % ma Second_Harmonic_Curren (A) ma 6.0 % nd_harmonic_level (%) 6.7 % 30.0 s 30.7 s Figure 7-7. nd harmonic level for ransformer inernal fauls (c) Negaive-sequence differenial proecion The resuls for negaive-sequence differenial proecion are shown in Figure 7-8. When he iner-urn faul happens, he negaive operaing-curren is abou 0.43 A, which is smaller han he 0.75 A seing. The negaive-sequence resraining curren is.8 A. The negaive sequence differenial percen is 4.68%, and i is also less han he 0% seing. Because neiher seing is exceeeded, no rip signal would be sen. Therefore, he ransformer is no proeced by he negaive-sequence differenial funcion regarding he iner-urn faul in his even. 34

152 0.43 A Negive_I_oper (A) 9.0 ma.83 A Negaive_I_Res (A).08 A 4.68 % Neg_Seq_Diff_Percen (%).786 % s 30.6 s Figure 7-8. Negaive-sequence differenial proecion resuls (d) Time-overcurren proecion The resul of ime-overcurren proecion is shown in Figure 7-9. The RMS value of ransformer primary-side curren is abou A when he faul happens. This value is less han he seing (900A) so no rip signal is sen. Therefore, he ime-overcurren scheme would fail o proec he ransformer regarding he iner-urn faul in his even..4 A Primary_Side_Curren_RMS (A). A 30.0 s 30.8 s Figure 7-9. Time-overcurren proecion for ransformer inernal fauls (e) Vols-over-herz proecion and hermal proecion The vols-over-herz proecion funcion and hermal proecion funcion are no for 35

153 his siuaion. (f) Proposed DSE-based proecion The resuls of proposed DSE-based proecion are shown in Figure A Acual_Measuremen_Curren_Highside (A) -0. A 966. A Acual_Measuremen_Curren_Lowside (A) A 7.35 ma Residual_Curren_Highside (A) ma ma Residual_Curren_Lowside (A) ma 08.4 Chi_Square 9.99 u Confidence_Level Trip_Decision s 30.7 s Figure Proposed DSE-based proecion for ransformer inernal fauls When he iner-urn faul happens, here are no obvious changes in he erminal currens. However, he residuals of erminal currens increase from zero o considerable values. The chi-square goes from a very small value o 08, while he confidence level drops from 00% o zero immediaely. This zero value indicaes abnormaliies inside he ransformer and proecion acions would be aken as soon as possible. I is noiced ha confidence level is oscillaing during he faul period because he faul is oo small. An inegral funcion is applied o accumulae he confidence level values, and a rip 36

154 decision is aken o proec he ransformer. I deecs he faul a s and rips he ransformer a s. (h) Summary of Even Four A summary of he proecion mehod resuls for Even Four is shown in Table 7-4. In his even where a % iner-urn faul happens, none of he legacy funcion could successfully deec he exisence of faul. In conras, he proposed scheme is able o deec i dependably wih high sensiiviy. Moreover, he speed of he proposed scheme is very fas and i only ake 0.ms o deec he exisence of faul. Table 7-4. Summary of Even Four: Inernal Faul Proecion Mehods Trip Percenage differenial No Undependable Harmonic-resrain differenial No Undependable Negaive-sequence differenial No Undependable Time-overcurren No Undependable Vols-over-herz NA NA Thermal NA NA Proposed DSE-based Yes Dependable 7.6 Even Five: Auo-Transformer % Faul near Neural In his even, he regular ransformer T is replaced wih an auo-ransformer T 37

155 (he capaciy and volage raings are unchanged). Breakers B and B3 are iniially closed while he breaker B is open. A ime = 38.0s he breaker B is suddenly closed and a % faul near neural happens o he phase B of ransformer secondary side windings, as shown in Figure 7-3. % Faul Figure 7-3. Auoransformer % iner-urn faul siuaion The erminal measuremens are shown in Figure 7-3 for he ime period [ ] seconds. Noe ha he firs se of races show he volages a he wo ends of he ransformer and he second se of races show he currens a he wo sides of he ransformer. Noe ha very lile change occurs o he erminal volages and currens due o his inernal faul. The resuls of legacy proecion funcions as well as he proposed mehod are presened nex. 38

156 60.0 kv Volage_High_AB (V) Volage_High_BC (V) Volage_High_CA (V) kv kv Volage_Low_AB (V) Volage_Low_BC (V) Volage_Low_CA (V) kv 57.4 A Curren_High_A (A) Curren_High_B (A) Curren_High_C (A) A 84.9 A Curren_Low_A (A) Curren_Low_B (A) Curren_Low_C (A) A s s Figure 7-3. Terminal volages and currens for auo-ransformer inernal fauls (a) Percenage Differenial Proecion The resuls for percenage differenial proecion are shown in Figure When he inernal faul happens, he operaing-curren is abou 0.75 A, which is smaller han he 0 A seing. The resraining curren is abou 80A. The differenial percen is 0.4%, and i is also less han he 0% seing. Because neiher seing is exceeded, he percenage differenial funcion would no send a rip signal, so he ransformer is no proeced by his funcion regarding he inernal faul in his even. 39

157 0.750 A peraing_iop (A) 0.78 A 8.0 A Resraining_Ires (A) 8.3 A 0.44 % Diff_Percen (%) % s s Figure Percenage differenial proecion resuls (b) Harmonic-resrained differenial proecion Excep for he operaing-curren, resraining-curren and differenial percen, he harmonic-resrained differenial funcion also moniors he second-harmonic of he operaing-curren, as shown in Figure The values of operaing curren, resraining curren and he differenial percen are he same as hose in Figure The secondharmonic curren is abou 3.8mA, and he measured second-harmonic level is around 0.5%, which is smaller han he seing. Because neiher seing is exceeded, no rip signal will be sen. Therefore, he ransformer is no proeced by Harmonic-resrained differenial funcion regarding he inernal faul in his even. 40

158 0.750 A peraing_iop (A) 0.78 A 8.0 A Resraining_Ires (A) 8.3 A 0.44 % Diff_Percen (%) % ma Second_Harmonic_Curren (A) 0.38 ma 0.5 % nd_harmonic_level (%) 8.95 m% s s Figure nd harmonic level for ransformer inernal fauls (c) Negaive-sequence differenial proecion The resuls for negaive-sequence differenial proecion are shown in Figure When he inernal faul happens, he negaive operaing-curren is abou 0.54 A, which is a lile smaller han he 0.75 A seing. The negaive-sequence resraining curren is.77 A. The negaive sequence differenial percen is 9.56%, and i is more or less he 0% seing. Because he minimun pickup curren seing is no exceeeded, no rip signal would be sen. Therefore, he ransformer is no proeced by he negaivesequence differenial funcion regarding he inernal faul in his even. 4

159 0.538 A Negive_I_oper (A).576 ma.77 A Negaive_I_Res (A) 0.90 A 9.56 % Neg_Seq_Diff_Percen (%) 0.33 % s s Figure Negaive-sequence differenial proecion resuls (d) Time-overcurren proecion The resul of ime-overcurren proecion is shown in Figure The RMS value of ransformer primary-side curren is abou 8 A when he faul happens. This value is less han he seing (900A) so no rip signal is sen. Therefore, he ime-overcurren scheme would fail o proec he ransformer in his even. 8.0 A Primary_Curren_RMS (A) 8.3 A s s Figure Time-overcurren proecion for ransformer inernal fauls (e) Vols-over-herz proecion and hermal proecion The vols-over-herz proecion funcion and hermal proecion funcion are no for his siuaion. 4

160 (f) Proposed DSE-based proecion The resuls of proposed DSE-based proecion are shown in Figure When he iner-urn faul happens, here are no obvious changes in he erminal currens. However, he residuals of erminal currens increase from zero o considerable values. The chi-square goes from a very small value o 36, while he confidence level drops from 00% o zero immediaely. This zero value indicaes abnormaliies inside he auoransformer and proecion acions would be aken as soon as possible. I is noiced ha confidence level is oscillaing during he faul period because he faul is oo small. An inegral funcion is applied o accumulae he confidence level values, and a rip decision is aken o proec he auo-ransformer. I deecs he faul a s and rips he ransformer a s. 43

161 56.4 A Acual_Measuremen_Curren_XFMRH_B (A) A 85.4 A Acual_Measuremen_Curren_XFMRL_B (A) A ma Residual_Curren_XFMRH_B (A) -.37 ma 4.4 ma Residual_Curren_XFMRL_B (A) ma Chi_Square 0.4 m Confidence_Level u.000 Trip_Decision s s Figure Proposed DSE-based proecion for ransformer inernal fauls (h) Summary of Even Five A summary of he proecion mehod resuls for Even Five is shown in Table 7-5. In his even where a % inernal faul happens o he auo-ransformer, none of he legacy funcion could successfully deec he exisence of iner-urn faul. In conras, he proposed DSE-based scheme is able o deec i dependably wih high sensiiviy. Moreover, he speed of he proposed scheme is very fas and i only ake 0.ms o deec he exisence of faul. 44

162 Table 7-5. Summary of Even Five: Inernal Faul Proecion Mehods Falsely Trip Percenage differenial No Undependable Harmonic-resrain differenial No Undependable Negaive-sequence differenial No Undependable Time-overcurren No Undependable Vols-over-herz NA NA Thermal NA NA Proposed DSE-based Yes Dependable 7.7 Even Six: Auo-Transformer ver-exciaion In his even, he auo-ransformer T is over-excied during he simulaion, as shown in Figure The auoransformer is 35MVA 5kV/35kV/3.8kV and i is around % over-excied. The simulaion ime is min 30 sec. 45

163 Figure Auo-ransformer over-exciaion The phase A erminal measuremens of he auoransformer are shown in Figure Noe ha he firs hree races show he volages a he primary, secondary and eriary of he ransformer and he second hree races show he currens. The resuls of legacy proecion funcions as well as he proposed mehod are presened nex kv Volage_Primary_AN (V) kv 3.8 kv Volage_Secondary_AN (V) kv V Volage_Teriary_AN (V) V 9.3 A Curren_Primary_A (A) -9.3 A 58. A Curren_Secondary_A (A) -58. A 94.0 A Curren_Teriary_A (A) A s 50.7 s Figure Auo-ransformer phase A erminal measuremens (a) Percenage differenial proecion, harmonic-resrained differenial proecion, 46

164 negaive-sequence differenial proecion and ime-overcurren proecion The percenage differenial proecion funcion, harmonic-resrained differenial proecion funcion, negaive-sequence differenial proecion funcion and imeovercurren proecion funcion are no for his siuaion. (b) Vols-over-herz proecion The resul of vols-over-herz proecion funcion is shown in Figure The vols-over-herz value is abou.9%. According o he ime characerisic of he auoransformer, his funcion would rip he ransformer a ime = 0.0 sec..9 % Vols_over_Hz (%) 0.86 m% s 5.0 s Figure Vols-over-herz proecion for auoransformer over-exciaion (c) Thermal proecion The resul of hermal proecion funcion is shown in Figure 7-4. The hospo emperaure increases o 05.0 Celsius (he hreshold) a ime =. sec, and i coninues o increase. Therefore, he hermal proecion funcion would rip he ransformer a ime =. sec. 47

165 Figure 7-4. Thermal proecion for auoransformer over-exciaion (e) Proposed DSE-based proecion The resul of proposed DSE-based proecion is shown in Figure 7-4. Based on he elecro-hermal model, he DSE can esimae he emperaure inside he auoransformer. The esimaed hoes emperaure is he ET a he core, an i reaches 05.0 Celsius (he hreshold) a ime = 00.7 sec. Therefore, he DSE-based proecion scheme will rip he ransformer a ime = 00.7 sec. Figure 7-4. Proposed DSE-based proecion for auoransformer over-exciaion (h) Summary of Even Five A summary of he proecion mehod resuls for Even Five is shown in Table 7-6. In his even where he auoransformer is % overexcied, he vols-over-herz 48