Inter-area oscillation damping in large scale power systems with unified power flow controllers

Transcription

1 Scholars' Mne Doctoral Dssertatons Student Research & Creatve Works Fall 2008 Inter-area oscllaton dampng n large scale power systems wth unfed power flow controllers Mahyar Zargham Follow ths and addtonal works at: Part of the Electrcal and Computer Engneerng Commons Department: Electrcal and Computer Engneerng Recommended Ctaton Zargham, Mahyar, "Inter-area oscllaton dampng n large scale power systems wth unfed power flow controllers" (2008) Doctoral Dssertatons Ths Dssertaton - Open Access s brought to you for free and open access by Scholars' Mne It has been accepted for ncluson n Doctoral Dssertatons by an authorzed admnstrator of Scholars' Mne Ths work s protected by U S Copyrght Law Unauthorzed use ncludng reproducton for redstrbuton requres the permsson of the copyrght holder For more nformaton, please contact scholarsmne@mstedu

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3 INTER-AREA OSCILLATION DAMPING IN LARGE SCALE POWER SYSTEMS WITH UNIFIED POWER FLOW CONTROLLERS by MAHYAR ZARGHAMI A DISSERTATION Presented to the Faculty of the Graduate School of the MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY In Partal Fulfllment of the Requrements for the Degree DOCTOR OF PHILOSOPHY n ELECTRICAL ENGINEERING 2008 Approved by Maresa L Crow, Advsor Jagannathan Sarangapan Badrul H Chowdhury Mehd Ferdows Bruce M McMlln

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5 PUBLICATION DISSERTATION OPTION Ths dssertaton has been prepared n the form of ten papers for publcaton The frst paper consstng from pages 3 to 2 has been publshed n the conference proceedngs of the 38 th North Amercan Power Symposum The second paper consstng from pages 22 to 28 has been publshed n IEEE Transactons on Power Systems, volume 22, ssue 4, Nov 2007 The thrd paper consstng from pages 29 to 46 has been publshed n the conference proceedngs of the 39 th North Amercan Power Symposum The fourth paper consstng from pages 47 to 65 s under revew for publcaton n the IEEE Transactons on Power Delvery The ffth paper consstng from pages 66 to 84 has been submtted for revew for publcaton n the IEEE Transactons on Power Systems The sxth paper consstng from pages 85 to 99 has been publshed n the conference proceedngs of the IEEE Power and Energy Socety 2008 Summer Meetng The seventh paper consstng from pages 00 to has been publshed n the conference proceedngs of the 40 th North Amercan Power Symposum The eghth paper consstng from pages 2 to 30 s n preparaton for submsson to the IEEE Transactons on Power Systems The nnth paper consstng from pages 3 to 46 has been accepted to be publshed n the conference proceedngs of the Power Systems Conference and Exposton 2009 The tenth paper consstng from pages 47 to 63 s n preparaton for submsson to the IEEE Transactons on Power Systems

6 v ABSTRACT Power system oscllatons occur n power networks as a result of contngences such as faults or sudden changes n load or generaton They are detrmental to the operaton of the system snce they affect system stablty and the optmal power flow through t These oscllatons do not usually damp out n te-lnes unless certan controls are appled to the system Local and nter-area oscllatons have tradtonally been controlled by Power System Stablzers (PSS) However, Flexble Alternatng Current Transmsson Controllers (FACTS) have sgnfcant potental as alternatves to PSS The man goal of ths research s to damp nter-area oscllatons by Unfed Power Flow Controllers (UPFC) UPFC s a seres-shunt FACTS devce whch s used for purposes such as the control of actve and reactve power flow through the corrdors of the system However, usng supplementary controls and proper coordnaton of UPFCs, they can be used for fast dampng of nter-area oscllatons n mult-area power systems The research conssts of ten papers There are several ssues assocated wth dynamc control of FACTS devces whch need to be taken nto consderaton In the frst two papers the role of pre-fault UPFC operatng ponts on the stablty and dynamc behavor of power systems s dscussed Lnear approaches for the control of nter-area oscllatons have been dscussed n the thrd and fourth papers Snce the dscussed algorthms for dampng oscllatons need global feedback data for control mplementaton, decentralzed and wde-area methods for dynamc state estmaton have been presented n the ffth and sxth papers Seventh paper shows that usng smlar methodologes to UPFCs, multple coordnated Statc Synchronous Compensators (STATCOM) can also be used for controllng power system oscllatons A nonlnear method for controllng oscllatons has been presented n the eghth paper Fnally, snce FACTS placement plays an mportant role n the dynamc behavor of the system, the last two papers propose two dfferent methods for optmal dynamc placement of UPFCs

7 v ACKNOWLEDGMENTS The author would lke to express hs deepest apprecaton to hs advsor, Dr Maresa L Crow for her contnuous gudance, advce and help durng ths research My apprecaton s extended to the members of advsory commttee, Dr Jagannathan Sarangapan, Dr Badrul H Chowdhury, Dr Mehd Ferdows and Dr Bruce M McMlln for ther tme and effort for revewng ths dssertaton Specal thanks to Dr Sarangapan for hs advces durng my work Fnancal support of the Natonal Scence Foundaton s gratefully acknowledged Last but not least, my work would have not been fulflled wthout blessngs of my dear parents, brothers and my lovely wfe Atousa

8 v TABLE OF CONTENTS PUBLICATION DISSERTATION OPTION ABSTRACT ACKNOWLEDGEMENTS LIST OF ILLUSTRATIONS LIST OF TABLES SECTION PAPER INTRODUCTION The Effect of Varous UPFC Operatng Ponts on Transent Stablty ABSTRACT I INTRODUCTION II THE POWER INJECTION MODEL OF THE UPFC III MULTIPLE OPERATING POINTS OF THE UPFC IV EFFECT OF PRE-FAULT OPERATING CONDITIONS ON TRANSIENT STABILITY V SIMULATIONS AND RESULTS VI CONCLUSIONS REFERENCES The Exstence of Multple Equlbra n the UPFC Power Inecton Model ABSTRACT I INTRODUCTION II THE UPFC STATE MODEL III ILLUSTRATIVE EXAMPLE IV SUMMARY AND CONCLUSIONS REFERENCES Dscusson on Effectve Control of Inter-Area Oscllatons by UPFCs ABSTRACT I INTRODUCTION II SYSTEM MODELING FOR CONTROLLER DESIGN III ONE STAGE CONTROLLER DESIGN Page v v x xv

9 v IV TWO STAGE CONTROLLER DESIGN V EXAMPLE AND DISCUSSION VI CONCLUSION REFERENCES A Novel Approach to Inter-Area Oscllaton Dampng by UPFC Voltage Control ABSTRACT I INTRODUCTION II THE UPFC MODEL III VOLTAGE CONTROL CONTROLLER DESIGN IV POWER CONTROL CONTROLLER DESIGN V THE TEST SYSTEM VI CONTROLLER RESULTS AND COMPARISONS VII CONCLUSIONS AND FUTURE WORK ACKNOWLEDGEMENTS APPENDIX REFERENCES Decentralzed Control and Placement of Multple UPFCs for Dampng Interarea Oscllatons: An LMI Approach ABSTRACT I INTRODUCTION II UPFC INTERACTIONS III CONTROL DEVELOPMENT IV CONTROL VALIDATION V UPFC PLACEMENT VI CONCLUSIONS APPENDIX REFERENCES

10 v 6 Dampng Inter-Area Oscllatons by UPFCs Based on Selected Global Measurements ABSTRACT I INTRODUCTION II TWO STAGE CONTROLLER III REDUCED ORDER ESTIMATOR DESIGN IV EXAMPLE AND DISCUSSION V CONCLUSIONS AND FURTHER WORK REFERENCES Dampng Inter-Area Oscllatons n Power System by STATCOMs ABSTRACT I INTRODUCTION II STATCOM MODEL III CONTROLLER DESIGN IV DECENTRALIZED CONTROLLER V THE TEST SYSTEM VI EXAMPLES AND RESULTS VII CONCLUSIONS AND FURTHER WORK APPENDIX REFERENCES Nonlnear Control of FACTS Devces for Dampng Inter-Area Oscllatons n Wde-Area Power Systems ABSTRACT I INTRODUCTION II THE UPFC MODEL III SYSTEM MODEL IV CONTROLLER DESIGN V SELECTIVE FEEDBACK MEASUREMENTS BASED ON DOMINANT MACHINES VI EXAMPLE AND RESULTS

11 x VII CONCLUSIONS APPENDIX REFERENCES Optmal Placement and Sgnal Selecton for Wde-Area Controlled UPFCs for Dampng Power System Oscllatons ABSTRACT I INTRODUCTION II UPFC MODEL III CONTROLLER DESIGN IV PLACEMENT FOR STABILITY IMPROVEMENT V OBSERVER DESIGN VI EXAMPLES AND RESULTS VII CONCLUSIONS AND FURTHER WORK APPENDIX REFERENCES Dynamc Placement and Sgnal Selecton for UPFCs n Wde-Area Controlled Power Systems ABSTRACT I INTRODUCTION II DETERMINING THE TABLE OF MOST DOMINANT BRANCHES (MDB) SECTION III ALGORITHM FOR DYNAMIC FACTS PLACEMENT- AND SIGNAL SELECTION IV CONTROLLER DESIGN V EXAMPLES AND RESULTS VI CONCLUSIONS AND FURTHER WORK APPENDIX REFERENCES CONCLUSIONS VITA

12 x LIST OF ILLUSTRATIONS Fgure Page PAPER Power Inecton Model of the UPFC Embedded nto the Lne from Bus to Bus Actve Power Balance n Steady-State Condtons Operatng Plots of the UPFC UPFC Inected Seres Actve Power n Example a UPFC Inected Seres Reactve Power n Example a ac voltage of the shunt converter of the UPFC n Example a ac bus voltage of the UPFC n Example a UPFC Inected Seres Actve Power n Example b UPFC Inected Seres Reactve Power n Example b ac voltage of the shunt converter of the UPFC n Example b ac bus voltage of the UPFC n Example b UPFC Inected Seres Actve Power n Example 2a UPFC Inected Seres Reactve Power n Example 2a ac voltage of the shunt converter of the UPFC n Example 2a ac bus voltage of the UPFC n Example 2a UPFC Inected Seres Reactve Power n Example 2b UPFC Inected Seres Reactve Power n Example 2b ac voltage of the shunt converter of the UPFC n Example 2b ac bus voltage of the UPFC n Example 2b PAPER 2 Unfed Power Flow Controller Dagram UPFC Equvalent Model UPFC parameters for varatons n P Dynamc response of UPFC seres actve power

13 x PAPER 3 Unfed Power Flow Controller Dagram Equvalent Power System from the Controller s Vew Power Inecton Model for UPFC Two Stage Control Desgn IEEE 8 Bus Test System Speed Devatons Modulaton Ampltudes and Angles UPFC Currents UPFC ac & dc Voltages PAPER 4 Unfed Power Flow Controller Dagram Equvalent power system from the voltage control vew Two stage control desgn IEEE 8 bus test system Generator frequency UPFC voltages UPFC recevng end power flows UPFC nected voltage UPFC nected actve power UPFC nected reactve power UPFC shunt actve power UPFC dc lnk capactor voltages UPFC dc lnk capactor voltages under power control longer duraton PAPER 5 UPFC nteracton n the 8 bus system generator frequences UPFC nteracton n the 8 bus system actve power flow lnes Ae matrx

14 x 4 IEEE 8 bus test system Proposed control response n the IEEE 8 bus test system Actve power flow on te lnes wth proposed control UPFC seres actve power flow Comparson of dfferent placements PAPER 6 Unfed Power Flow Controller Dagram Power Inecton Model for UPFC Equvalent Power System from the Controller s Vew Two Stage Control Desgn IEEE 4 Bus Test System Machne speeds 2,4, Machne speeds 2,4, Machne speeds 2,4, Machne speeds,2, PAPER 7 STATCOM Dagram Equvalent Power System from the Controller s Vew Two Stage Control Desgn IEEE 8 Bus Test System Speed devatons STATCOM bus voltage magntudes STATCOM bus voltage angles STATCOM dc voltages STATCOM nected actve powers

15 x PAPER 8 Unfed Power Flow Controller Dagram Equvalent power system from the controller vewpont The three stage control process bus, 6 generator test system Generator speeds for no FACTS devces (bold) and Case I (thn) Generator speeds for Case I (bold), Case II (thn), and Case III (dashed) UPFC nected actve power STATCOM actve power necton FACTS Vdc PAPER 9 Unfed Power Flow Controller Dagram Power Inecton Model for UPFC Equvalent Power System from the Controller s Vew Machne rotor speeds Generator rotor speeds PAPER 0 Unfed Power Flow Controller Dagram Equvalent Power System from the Controller s Vew IEEE 8 Bus Test System Comparson of placements for rotor speeds Comparson of rotor speeds Comparson of rotor speeds for controlled and uncontrolled

16 xv LIST OF TABLES Table Page PAPER I PI Controlled Parameters for Examples and PAPER 4 I UPFC Parameters II Controller Comparsons III Domnant Modal Content PAPER 5 I Assessment of Control Performance and Control Effort of UPFC Placements PAPER 6 I Smulaton Cases Accordng to the Feedback and Estmated States PAPER 7 I STATCOM Parameters

17 xv PAPER 8 I FACTS Parameters PAPER 9 I UPFC Parameters II UPFC Placements Sorted by Placement Indces (PIs) III Comparson of the Placements Based on Speed Profle Index IV Measurement Canddates Sorted by Observaton Indces (OIs) PAPER 0 I UPFC Parameters II Most Domnant Branches and Ther Influence on Inter-Area Modes III Comparson of the Placements Based on Speed Profle Index IV Output Measurements for Observer Desgn

18 INTRODUCTION Today, FACTS controllers have become less expensve and because of ther fast response to system dsturbances they wll be used even more extensvely n the future However, there are stll problems assocated wth ther applcaton especally n the dynamc control area Phenomena such as nter-area oscllatons mpact power systems n wde area dstances For the same reason, controllng such phenomena needs proper coordnaton between the controllng devces The man goal of ths research s to damp nter-area oscllatons n mult-area power systems usng multple FACTS devces The UPFC s the most versatle FACTS devce and the focus of ths research has been based on the applcaton of multple UPFCs for dampng oscllatons In the lterature, there have been several methods proposed for the control of UPFCs based on lnear and nonlnear methods However, the applcaton of most of these methods s lmted to the control of an ndvdual UPFC wthout ts coordnaton wth other devces Consderng that mult-area power systems have several oscllaton modes, more than one FACTS devce s needed n order to affect all those modes Thus, coordnaton of the devces becomes an mportant ssue In ths research whch conssts of ten papers, several contrbutons have been made n the area of dynamc control of the UPFC for the purpose of dampng nter-area oscllatons In the ntal work, t has been shown that the pre-fault operatng status of UPFC plays an mportant role n ts dynamc behavor Ths has been dscussed n the frst two papers Then, a novel method has been proposed for the control of nter-area oscllatons usng UPFCs based on controllng ther bus voltages Papers three and four descrbe ths method and show ts comparson wth other methods The proposed methods for oscllaton dampng generally need global feedback data for mplementaton However n practce, global feedback s not avalable and dynamc feedback estmaton s usually needed A decentralzed method based on Lnear Matrx Inequalty (LMI) approach has been dscussed n the ffth paper Decentralzed methods usually use local measurements for data estmaton As these local measurements mght not always be adequate for proper feedback estmaton, a centralzed wde-area method for feedback estmaton based on selected global measurements usng reduced order observers has been dscussed n the sxth paper Although the focus of the research s on the UPFC, n the seventh paper t has been shown that usng smlar methodologes for system modelng and control, t s also possble to use multple STATCOMs for dampng power system oscllatons The next contrbuton of ths research s to propose a new nonlnear method for oscllaton dampng whch has been descrbed n the eghth paper Snce dynamc placement of FACTS controllers plays an mportant role on

19 2 the dynamc behavor of the power system, the other contrbuton of ths research has been devoted to two dfferent methods for dynamc placement of UPFCs whch have been dscussed n the nnth and tenth papers

20 3 The Effect of Varous UPFC Operatng Ponts on Transent Stablty Mahyar Zargham Maresa L Crow Department of Electrcal and Computer Engneerng Mssour Unversty of Scence and Technology, MO, 65409, USA ABSTRACT: In ths paper, t s shown that usng the power necton model for a UPFC, there exst sets of operatng ponts for whch the same amount of ac bus voltage and actve/reactve seres power nectons can be evaluated These varous operatng ponts are desgnated by dfferent dc bus voltages ( ) and modulaton ampltudes ( k, k ) and phases Vdc 2 ( α α ) Smulatons show that the dynamc behavor of a UPFC could be related to ts pre-fault, 2 operatng stuatons The queston to be answered s to verfy whch operatng pont would be the best canddate from a network stablty pont of vew A lnear approach based on the egenvalue problem has been used to answer the queston and satsfactory results have been outlned Index Terms: UPFC, Transent Stablty I INTRODUCTION The power necton model has so far been wdely used for power system smulatons where FACTS devces such as UPFCs exst n a network In ths approach, UPFCs are modeled by convertng ther phasor varables nto the dq0 doman, whch has the advantage that n steadystate, the UPFC varables are constant In addton, t s possble to drectly ncorporate the dfferental-algebrac equatons of the UPFC nto the power network Usng the power necton model, the acton of a UPFC n the network s represented by ts seres and shunt current nectons where these currents are controlled by the varatons of the dc bus ( V dc ) and modulaton ampltudes ( k, k2) and phases ( α, α 2) [] In ths paper, t wll be shown that for the same ac bus voltage and actve/reactve seres power necton n a UPFC, there exsts a set of operatng ponts desgnated by dfferent sets of V, dc k, α,, k2 α 2 values Although each of these sets of control parameters result n the same power necton model,

21 4 the dfferent control parameter sets have dfferent mpacts on transent stablty Therefore, the choce of a proper operatng pont s of vtal mportance Some work already exsts n the lterature related to the choce of optmal set of operatng ponts for the FACTS devces n steadystate condtons These approaches mostly focus on the operaton of the system from aspects such as optmal power flow and UPFC nternal constrants [2, 3] Despte the work done so far, there s stll consderable work to be done n the area of the system dynamcs to analyze the effect of varous operatng ponts on the stablty and control of the power system In the followng sectons, the power necton model wll be descrbed along wth an algorthm for fndng sets of operatng ponts of UPFCs These operatng ponts are vsualzed usng correspondng operatng plots In the plots, the stable/unstable regons are desgnated based on determnng the egenvalues of the lnearzed power system state space matrx evaluated at steady-state Power system smulatons have been accomplshed to fnd a probable relaton between the stable/unstable pre-fault operatng areas and the lkelhood of the power system to go unstable after a fault Observatons are reported and further work s proposed II THE POWER INJECTION MODEL OF THE UPFC The equatons of the power necton model for the UPFC are taken from [4] It s assumed that the power system s balanced and therefore no zero sequence voltages and currents exst n the network There are bascally nne equatons governng a UPFC and ts nterface to the power network Each of the seres and shunt parts has two dfferental equatons and there s one dfferental equaton relatng the seres and shunt parts through the dc lnk The four remanng algebrac equatons nterface the UPFC wth the power network through the two buses of the UPFC The two buses of the UPFC are denoted as Bus and Bus2, where Bus s the bus connected to the shunt transformer of the UPFC The equatons that descrbe the shunt part of the UPFC are: R k V = + + cos( + ) V cosθ () ω B d d q θ α dc X X X R k V = + sn( + ) V snθ (2) ω B q q d θ α dc X X X Smlarly, the seres equatons are:

22 5 R k V V = + + cos( + ) V cos + cosθ (3) ω B d2 d2 q θ 2 α2 dc θ2 X2 X2 X2 X2 R k V V = + sn( + ) V sn + snθ (4) ω B q2 q2 d θ 2 α2 dc θ2 X2 X2 X2 X2 The equaton for the dc capactor that lnks the shunt and seres parts s: C V dc = k cos( θ+ α) d k sn( θ+ α) q B ω V k2cos( θ+ α2) d k 2 2sn( θ+ α2) q 2 R dc p (5) The power balance equatons at Bus are: V (cos( θ )( ) + sn( θ )( )) V V Y cos( θ θ Φ ) = 0 (6) d d2 q q2 = V (sn( θ )( ) cos( θ )( )) V VY sn( θ θ Φ ) = 0 (7) d d2 q q2 = n n Fnally the power balance equatons at Bus2 are: n V2(cos( θ2) d + sn( θ = 2 2) q ) V 2 2 VY 2cos( θ2 θ Φ2) 0 (8) = n V2(sn( θ2) d cos( θ = 2 2) q ) V 2 2 VY2 cos( θ2 θ Φ2 ) 0 (9) = where the followng varables are defned: ω B : Base frequency of the network (rad/s) R, X R 2, X 2 : Equvalent Resstance & Reactance of the shunt transformer (pu) : Equvalent Resstance & Reactance of the seres transformer (pu)

23 6 d, q : The shunt currents of the UPFC n the drect and quadrature axes, respectvely (pu) d, 2 q2 : The seres currents of the UPFC n the drect and quadrature axes, respectvely (pu) k, k 2 : Ampltude modulaton ndces of the shunt and seres parts, respectvely α, α 2 : Phase modulaton ndces of the shunt and seres parts, respectvely Vdc : dc lnk capactor voltage (pu) C : Capactance of the dc lnk (pu) R p losses (pu) : Equvalent resstance parallel wth the capactor of the dc lnk, representng the dc III MULTIPLE OPERATING POINTS OF THE UPFC In ths secton, we verfy the exstence of multple operatng ponts of a UPFC n steadystate condtons Fg shows a UPFC embedded nto a lne between buses and 2 We mean to regulate the actve/reactve power flow through the lne as V sc the ac bus of the UPFC as P sc / Q sc and the voltage magntude at V sc + Bus d 2 q 2 Bus 2 + d q P sc + Q sc Fg Power necton model of the UPFC embedded nto the lne from Bus to Bus2 In a conventonal power system wth no FACTS devce, two actve/reactve power flow equatons can be wrtten at every PQ bus Wth the ntroducton of a UPFC nto the power system, the four power flow equatons at buses and 2 cannot be wrtten n ther conventonal form anymore These four equatons are vtal for the evaluaton of two voltage magntudes and two voltage phases Instead of the four powerflow equatons to determne the bus voltages magntude and angle, equatons (6)-(9) are used But equatons (6)-(9) ntroduce four addtonal

24 7 unknowns of nto the system However, because V = Vsc s known, there are actually, d q, d2 q 2 only three addtonal unknowns Usng equatons ()-(5) also adds the addtonal unknowns ofv, dc k, α,, k2 2 α So up to ths pont, the UPFC has 9 addtonal equatons and 2 addtonal unknowns In order for the system to be solvable, 3 more equatons are needed for every UPFC From the power necton model, two of the equatons can be wrtten as: V (cos( θ ) + sn( θ ) ) = P UPFC (0) 2 2 d2 2 q2 V (sn( θ ) cos( θ ) ) = Q () 2 2 d2 2 q2 UPFC To develop the last equaton, consder Fg 2 Here, actve power balance at the dc lnk n steady-state can be wrtten as: P shunt 2 dc V + Pseres = (2) R p where P and P are the actve powers nected from the shunt and seres parts of the shunt seres UPFC, respectvely More specfcally, these two powers can be wrtten as: P = kv + + kv + (3) cos( θ α ) sn( θ α ) shunt dc d dc q P cos( ) sn( ) seres =+ kv 2 dc θ + α2 d + kv 2 2 dc θ +α 2 q (4) 2 P shunt P seres Shunt Converter V R 2 dc p Seres Converter Fg 2 Actve Power Balance n Steady-State Condtons Consderng P shunt to be specfed, equaton (3) s the last equaton needed to determne the UPFC parameters Snce P shunt s a specfed value, each value of P shunt determnes a possble UPFC operatng pont Fgs 3 show sets of operatng k, k and V plots for a UPFC nstalled between buses and 2 dc

25 8 2 n the IEEE 8 bus test system The parameters of the UPFC have been gven n Secton V As can be seen n Fg 3a, there s a large varaton n the value of However, the value of UPFC to be equal to V sc kv dc k wth changes of P remans around pu to regulate the voltage of the ac bus of the The dotted lnes n Fg 3 represent the unstable regons determned by the egenvalues of the lnearzed system explaned n secton IV shunt IV EFFECT OF PRE-FAULT OPERATING CONDITIONS ON TRANSIENT STABILITY In ths secton, the effect of the ntal operatng pont on the transent stablty of the system wll be consdered One basc approach to determnng stablty s to lnearze the system equatons around ts equlbrum pont Although power systems are nonlnear, ths approach s able to roughly predct the relatve lkelhood of the system to go unstable Consder the followng nonlnear dfferental/algebrac equatons of the system [5]: X & = f( X, Y) (5) 0 = g( X, Y) (6) system where X s the state vector and Y s the vector of voltage magntude/angles of the power The lnearzed equatons are of the form: Δ X& = AX + BY (7) 0 = CX + DY (8) where Δ shows the lnearzed varable around ts equlbrum pont system as: Usng equatons (7) and (8), t s possble to get the state space matrx of the lnearzed Asys A BD C = (9) The egenvalues of A sys determne the relatve stablty of the system after a fault If the egenvalues have postve real parts, then the relatve stablty of the system s poor

26 k Pshunt (pu) (a) k vs P shunt x k Pshunt (pu) (b) k 2 vs P shunt Vdc (pu) Pshunt (pu) (c) V dc vs P shunt Fg 3 Operatng Plots of the UPFC

27 0 V SIMULATIONS AND RESULTS To verfy the assumpton made n the prevous secton, we show two examples usng the IEEE 8 bus test system [6] Ths system has 20 machnes, where the order of each machne s 0, contanng the two-axs generator model, Type I Excter/AVR model and turbne and governor models As we dscussed n the above sectons, every UPFC would add 5 state varables nto the system So the order of the lnearzed power system wth one UPFC would be 205 Two dfferent UPFC placements have been consdered n the followng examples and ther results are explaned The parameters of the UPFC are: R = 00pu, X = 05pu R2 = 000pu, X 2 = 005pu Rp = 00 pu, C = 364 pu Example As the frst example, a UPFC placement between buses and 2 has been consdered wth P = 078pu, = 0353pu andv = 09528pu These are bascally the lne sc Q sc power and voltage values before the UPFC nstallaton sc (a) When P = 00pu, we get an operatng pont where: shunt k = 09626, k 2 = α = 6285rad, α 2 = 34029rad kv dc = 09533pu d = 000 pu, q d = pu 2, q2 = 0005pu = 0827 pu Wth the above condtons, all egenvalues of Asys have negatve real values To observe the transent stablty of the system, a fault s appled to bus 30 at s and s cleared after 0 s Fgs 4-7 show the behavor of the system wth and wthout PI controllers Four smple PI controllers have been appled to control the seres actve power, seres reactve power and voltages of the dc and ac buses of the UPFC [4] The controller parameters are shown n Table I

28 TABLE I PI Controller Parameters for Examples and 2 Seres Controller k P k I Actve Power e-3 e-3 Reactve Power e-3 e-3 Shunt Controller k P k I dc voltage 5e-2 5e-2 ac voltage 5e-3 5e-3 Fgs 4a and 4b show the seres nected power ( P seres ) of the UPFC before and after the fault for the uncontrolled and controlled cases, respectvely Fgs 5, 6 and 7 depct the varatons of the seres nected reactve power ( Q ), kv and ac voltage of the UPFC ( V ) for the uncontrolled and controlled cases In ths example, t s obvous that the system s stable and easly controlled seres dc (a) Pseres (pu) Tme (S) (b) Pseres (pu) Tme (S) Fg 4 UPFC Inected Seres Actve Power n Example a (wthout control top, wth control bottom)

29 2 0 (a) -0 Qseres (pu) Tme (S) 0 (b) -0 Qseres (pu) Tme (S) Fg 5 UPFC Inected Seres Reactve Power n Example a (wthout control top, wth control bottom) 05 (a) kvdc (pu) Tme (S) 05 (b) kvdc (pu) Tme (S) Fg 6 ac voltage of the shunt converter of the UPFC n Example a (wthout control top, wth control bottom) (a) V (pu) Tme (S) 098 (b) 096 V (pu) Tme (S) Fg 7 ac bus voltage of the UPFC n Example a (wthout control top, wth control bottom)

30 3 (b) Next, an alternate operatng pont s chosen wth the same values of Psc, QscandV sc, but where P = 06 pu Note that for the power necton model, these two operatng ponts are shunt dentcal because they have the same seres actve and reactve power and bus voltage For ths stuaton, two complex conugate egenvalues have postve real parts, thus ths operatng pont s locally unstable From the partcpaton factors, t s determned that these unstable egenvalues are assocated mostly wth the angular frequency of the generators connected to buses 00 and 2 Ths means that the UPFC tself s not contrbutng drectly to the nstablty The control parameters of the UPFC at ths pont are: k = 0287 k 2 = α = 6842rad α 2 = 6396rad kv dc = 0989pu d = 0664pu q = 0577 pu d2 = pu q2 = 0853pu The same fault s appled to the system as n the prevous example The same PI controller s used to stablze the system As can be seen n Fgs 8-, the controlled network becomes unstable rapdly Ths shows how mportant the choce of ntal operatng condton s to the controllablty and stablty of the system, regardless of the choce of ntal seres actve and reactve power flows and system bus voltage 0 (a) Pseres (pu) Tme (S) (b) 4 Pseres (pu) Tme (S) Fg 8 UPFC Inected Seres Actve Power n Example b (wthout control top, wth control bottom)

31 4-005 (a) Qseres (pu) Tme (S) (b) 4 Qseres (pu) Tme (S) Fg 9 UPFC Inected Seres Reactve Power n Example b (wthout control top, wth control bottom) 05 (a) kvdc (pu) Tme (S) (b) 2 kvdc (pu) Tme (S) Fg 0 ac voltage of the shunt converter of the UPFC n Example b (wthout control top, wth control bottom)

32 5 (a) V (pu) Tme (S) (b) 5 V (pu) Tme (S) Fg ac bus voltage of the UPFC n Example b (wthout control top, wth control bottom) Example 2 In the second example, a UPFC placement s changed and t s now placed between buses 02 and 0 n the 8 bus network The ntal necton model settngs are P = pu, Qsc = 0063pu, and V = pu These are bascally the lne power and voltage values before UPFC nstallaton sc sc (a) When P = 00pu, the ntal parameters are: shunt k = 008 k 2 = α = 6286rad α 2 = 737rad kv dc = pu d = 0003pu = 8934e 4 pu q d2 = 043pu q2 = pu Wth the above condtons, all egenvalues of A sys have negatve real parts Smlar to example, a fault s appled to bus 30 of the network at s and cleared after 0 s Fgs 2-5 show the behavor of the system wth and wthout PI controllers In fgures denoted by (a), no control has been appled untl 425 s On the other hand, n the fgures dstngushed by (b), the PI controller has been appled from the very begnnng The same control parameters as n Table I have been

33 6 used n ths example As t mght be vewed n Fgs 4a and 5a, after about 425 s, the PI controllers do a good ob for stablzng the system 08 (a) Pseres (pu) Pseres (pu) Tme (S) (b) Tme (S) Fg 2 UPFC Inected Seres Actve Power n Example 2a (wthout control untl 425s top, wth control bottom) 0 (a) Qseres (pu) Qseres (pu) Tme (S) (b) Tme (S) Fg 3 UPFC Inected Seres Reactve Power n Example 2a (wthout control untl 425 top, wth control bottom)

34 7 (a) kvdc (pu) Tme (S) (b) kvdc (pu) Tme (S) Fg 4 ac voltage of the shunt converter of the UPFC n Example 2a (wthout control untl 425 top, wth control bottom) 04 (a) V02 (pu) V02 (pu) Tme (S) (b) Tme (S) Fg 5 ac bus voltage of the UPFC n Example 2a (wthout control untl 425 top, wth control bottom) (b) Next, an alternate operatng pont s chosen wth the same values of Psc, Qscand V sc where P = 07 pu For ths stuaton, two complex conugate egenvalues have postve real shunt parts, thus ths operatng pont s locally unstable From the partcpaton factors, t s determned that these unstable egenvalues are assocated mostly wth the angular frequency of the generators connected to buses 40 and 2 Here agan we conclude that the power system nter-area modes

35 8 are correspondng to unstable equlbrum ponts and not the UPFC modes The operatng condtons of the UPFC n these stuatons are: k = 026 k 2 = 0005 α = 6752rad α 2 = 6288rad kv dc = 00 pu d = pu q = 039 pu d2 = 0458pu q2 = 0068pu The control s appled at 28s As t s seen n Fgs 6a, 7a, 8a and 9a, the controller s not able to stablze the system Actually the control acton makes the stuaton worse However, n Fgs 6b, 7b, 8b and 9b t s seen that when the controllers come nto acton from the very begnnng, they could stablze the system Ths agan shows that pckng an napproprate operatng pont could lead the power system nto undesrable behavor after a fault 08 (a) Pseres (pu) Tme (S) (b) 08 Pseres (pu) Tme (S) Fg 6 UPFC Inected Seres Reactve Power n Example 2b (wthout control untl 28s top, wth control bottom)

36 9 (a) 0 Qseres (pu) Tme (S) (b) 0 Qseres (pu) Tme (S) Fg 7 UPFC Inected Seres Reactve Power n Example 2b (wthout control untl 28s top, wth control bottom) 06 (a) kvdc (pu) Tme (S) (b) 06 kvdc (pu) Tme (S) Fg 8 ac voltage of the shunt converter of the UPFC n Example 2b (wthout control untl 28s top, wth control bottom)

37 20 (a) V02 (pu) Tme (S) (b) V02 (pu) Tme (S) Fg 9 ac bus voltage of the UPFC n Example 2b (wthout control untl 28s top, wth control bottom) VI CONCLUSIONS Ths work shows that the steady-state power necton model of the UPFC s nsuffcent for transent stablty analyss In the steady-state power necton model, only the seres reactve and actve powers and the shunt bus voltage magntude are specfed Even wth these values specfed, there are many operatng states that can be acheved from the dynamc equatons Ths may lead to unstable ntal operatng condtons In addton, even f the ntal operatng condtons are stable, the stablty margn may be suffcently decreased so as to adversely affect the controllablty of the system durng transents Future work would be to apply nonlnear control theory for the evaluaton of an optmal operatng condton from the system s stablty pont of vew Other approaches such as applcaton of the energy functon methods mght also be able to explan and predct system behavor after a contngency happens to the power system REFERENCES [] Tmothy L Skvarenna, The Power Electroncs Handbook CRC Press 2002 [2] Yng Xao; Song, YH; Sun, YZ, Power Flow Control Approach to Power Systems wth Embedded FACTS devces, IEEE Transactons on Power Systems, Vol 7, No 4, Nov 2002, pp

38 2 [3] Jun-Yong Lu; Yong-Hua Song; Mehta, PA, Strateges for Handlng UPFC Constrants n Steady-State Power Flow and Voltage Control, IEEE Transactons on Power Systems, Vol 5, No 2, May 2000, pp [4] L Dong, ML Crow, Z Yang, C Shen, L Zhang, S Atctty, A Reconfgurable FACTS System for Unversty Laboratores, IEEE Transactons on Power Systems, Vol 9, No, Feb 2004, pp [5] Peter W Sauer, MA Pa, Power System Dynamcs and Stablty, Prentce Hall 998 [6]

39 22 2 The Exstence of Multple Equlbra n the UPFC Power Inecton Model M Zargham, Student Member, IEEE, M L Crow, Senor Member, IEEE ABSTRACT: Ths letter shows the exstence of multple equlbra that arse from the use of the state model of the UPFC These multple equlbra can arse from a common power necton model for the same termnal condtons of shunt bus voltage and seres actve and reactve power nectons The multple equlbra result n two or more sets of egenvalues, some of whch may ndcate an unstable operatng condton Therefore, the use of the UPFC power necton model must be used wth cauton to ensure stable operaton of the UPFC Index Terms UPFC, oscllaton dampng, power system stablty I INTRODUCTION The UPFC power necton model s wdely used for power system smulatons (recent examples nclude []-[3]) In the power necton model, the mpact of the UPFC n the network s represented by ts seres and shunt current nectons, or smlarly, ts seres and shunt actve and reactve power nectons A common approach to ncorporatng the UPFC power necton model nto the system s to represent the UPFC as two buses: a PQ bus at the recevng end n whch both actve and reactve power are specfed, and a PV bus at the sendng end n whch voltage and actve power are specfed [4] In ths letter, t wll be shown that f the power necton model s used nstead of the dynamc model for the same operatng condtons, then multple equlbra (wth possbly dfferent stablty propertes) can exst

40 Transformer 23 2 Transformer To system To system P +Q P dc P sc +Q sc va, vb, vc, Ls, Rs, e a, ea,2 dc a, e b, b, e c, c, Converter R dc C + v dc Converter e a,2 b,2 e b,2 c,2 c,2 R s,2 L s,2 v a,2 v b,2 vc,2 Fg Unfed Power Flow Controller Dagram II THE UPFC STATE MODEL The UPFC s a combnaton of the STATCOM (statc synchronous compensator) and SSSC (statc seres synchronous compensator) as shown n Fgure The seres connected nverter nects a voltage wth controllable magntude and phase angle n seres wth the transmsson lne, thereby provdng actve and reactve power to the transmsson lne The shunt-connected nverter provdes the actve power drawn by the seres branch and the losses and can ndependently provde reactve compensaton to the system The UPFC state model s: d ω s dt d = k V dc cos (α + θ )+ ω q R s d V cos θ L s ω s L s L s () d ω s dt q = k V dc sn (α + θ ) R s q ω d V sn θ L s L s ω s L s (2) d ω s dt d 2 = R s 2 d2 + ω q2 + k 2 cos (α 2 + θ ) V dc L s2 ω s L s2 (V 2 cos θ 2 V cos θ ) (3) L s2 d ω s dt q 2 = R s 2 q2 ω d2 + k 2 sn (α 2 + θ ) V dc L s2 ω s L s2 (V 2 sn θ 2 V sn θ 2 ) (4) L s2 C d ω s dt V dc = k cos (α + θ ) d k sn (α + θ ) q k 2 cos (α 2 + θ ) d2 k 2 sn (α 2 + θ ) q2 V dc R dc (5)