Hopf Bifurcation Control of Subsynchronous Resonance Utilizing UPFC

Transcription

1 Engneerng, Technology & Appled Scence Rearch ol. 7, No. 3, 07, Hopf Bfurcaton Control of Subsynchronous Resonance Utlzng UPFC Majd M. Alomar Mohammad S. Wdyan Mohammed Abdul-Nby Alreza Ghetas The Australan College of Kuwat, Kuwat The Haemte Unversty Jordan The Australan College of Kuwat, Kuwat The Australan College of Kuwat, Kuwat Abstract The u of a unfed power flow controller (UPFC to control the bfurcatons of a subsynchronous resonance (SSR n a mult-machne power system s ntroduced n ths study. UPFC s one of the flexble AC transmsson systems (FACTS where a voltage source converter (SC s ud bad on gate-turn-off (GTO thyrstor valve technology. Furthermore, UPFC can be ud as a stablzer by means of a power system stablzer (PSS. The consdered system s a modfed verson of the cond system of the EEE cond benchmark model of subsynchronous resonance where the UPFC s added to ts transmsson lne. The dynamc effects of the machne components on SSR are consdered. Tme doman smulatons bad on the complete nonlnear dynamcal mathematcal model are ud for numercal smulatons. The results n ca of ncludng UPFC are compared to the ca where the transmsson lne s conventonally compensated (wthout UPFC where two Hopf bfurcatons are predcted wth unstable operatng pont at wde range of compensaton levels. For UPFC systems, t s worth to menton that the operatng pont of the system never los stablty at all realstc compensaton degrees and therefore all power system bfurcatons have been elmnated. Keywords-Hopf Bfurcatons; Subsynchronous Resonance; UPFC; GTO. NTRODUCTON Seres compensaton s ud n power systems as a valuable technque to effectvely ncrea power transfer capablty and mprove stablty. SubSynchronous Resonance (SSR s consdered a major drawback of res compensaton and can be defned as the electromechancal nteracton between electrcal resonant crcuts of the transmsson system and the torsonal natural frequences of the turbne-generator rotor. Bascally, SSR can cau aft fatgue as well as damage or falure. The subsynchronous torques on the rotor s an ssue of great concern becau the turbne-generator aft has natural modes of oscllaton that are typcal of any sprng-damper-mass system. SSR has been consdered ntensvely snce 970, when a aft falure took place to ts turbne-generator unt n the major transmsson network wth conventonal res compensaton n southern Calforna. The bfurcaton phenomenon also occurs n power systems. Rearchers study power system dynamcs from the nonlnear dynamcs pont of vew utlzng the modern nonlnear theory (bfurcaton theory. Bfurcaton s defned as a qualtatve change n the features of a system, such as the types and number of solutons when a small varaton occurs n the parameters of the system. Bfurcaton theory has been ud to ow the prence of the dynamc bfurcaton (Hopf bfurcaton n a sngle machne nfnte bus (SMB power system n the ca of neglectng the dynamcs of the damper wndngs and the automatc voltage regulator (AR of the synchronous generator []. The results ow that the system operatng pont loss stablty through supercrtcal Hopf bfurcaton. Bfurcaton theory has also been appled on modelng the characterstcs of the BOARDMAN generator wth respect to the North-Western Amercan power system and the CHOLLA generator wth respect to the SOWARA staton n the USA. The effect of the lnear and nonlnear controllers on the bfurcatons of SSR of CHOLLA generator s studed n []. t s found that the lnear controllers ncrea the compensaton level at whch SSR occurs whle the nonlnear controller does not affect the locaton and type of the Hopf bfurcaton. t s also own that the larger the nonlnear controller gan s, the smaller the ampltude of the lmt cycle tends to be. Addtonally, the bfurcaton theory has been appled on the frst system of the EEE cond benchmark model of SSR where supercrtcal Hopf bfurcaton s 3 3 experenced [3]. A nonlnear controller of the K ( g form has been desgned where all bfurcatons of the system are elmnated at all realstc compensaton factors despte the successve nteractons of the subsynchronous electrcal mode wth the three torsonal mechancal modes of the frst system of the EEE cond benchmark model of SSR [4]. The recent mprovements of power electronc devces have launched the u of flexble AC transmsson system (FACTS controllers n power systems. n fact, FACTS controllers can control the network condton va a very fast approach whch n turn can be ud to mprove the dynamc and steady-state stablty of power systems [5]. Utlzng FACTS controllers n power transmsson system have led to many applcatons of the controllers for mprovng the stablty of the exstng power network resources as well as provdng some operatng flexbltes. Bascally, FACTS devces have been defned as

2 Engneerng, Technology & Appled Scence Rearch ol. 7, No. 3, 07, alternatng current transmsson system ncorporatng power electronc-bad and other statc controllers to enhance controllablty and ncrea power transfer capablty [6]. The most famlar and common FACTS devces are thyrstorcontrolled res capactor (TCSC, statc synchronous compensator (STATCOM, statc synchronous res compensator (SSSC, statc var compensator (SC and unfed power flow controller (UPFC [6 7]. Techncally, each of the devces has specfc lmtatons, characterstcs and features. n power systems, UPFC s consdered as a powerful technque bad on economc and techncal consderatons for controllng actve and reactve power flows n a transmsson lne as well as the bus voltage. n addton, UPFC can be ud as a stablzer wth the help of the power system stablzer (PSS. Controllng the bfurcatons of the frst system of the EEE cond benchmark model usng TCSC has been nvestgated [8]. t s found that the operatng pont of the system never los stablty at any realstc frng angle and therefore all bfurcatons of the system are elmnated at all practcal values of res compensaton. Tme doman smulatons bad on the nonlnear dynamcal mathematcal model after step reducton n the nfnte bus voltage and nput power concde wth the results of the bfurcaton analyss. The effect of SSSC on the bfurcatons of the SSR n power system has also been nvestgated where the effect of replacng the conventonal compensaton wth SSSC s hghlghted [9]. aryng the SSSC controller reference voltage changes the compensaton degree. The results ow that the operatng pont of the system never los stablty at any realstc compensaton degree n ca of SSSC.e. all bfurcatons of the system have been elmnated. n ths paper, the effect of ncorporatng the cond system of the EEE cond benchmark model of SSR wth UPFC on the Hopf bfurcaton of the system s nvestgated whch s consdered as the man contrbuton of the paper where unt current and res voltage controllers are consdered as controllers for UPFC. The paper s constructed n the followng manner: Secton descrbes the system under study and ts nonlnear dynamcal mathematcal model ncludng the system controllers. n cton 3, the model of the UPFC wth three-level SC s outlned. Numercal smulaton results n ca of ncludng UPFC as compared wth the ca of conventonal compensaton wthout UPFC are prented n cton 4. Fnally conclusons are drawn n cton 5.. SYSTEM DESCRPTON AND MATHEMATCAL MODEL The system consdered n ths study s the cond system of the EEE cond benchmark model of subsynchronous resonance (Fgure [0]. t s a two-machne nfnte-bus (TMB power system connected to a sngle rescompensated transmsson lne where the termnals of the two generators are connected at the same bus. The capactve reactance vares n the range of 0 to 00% of the transmsson lne nductve reactance. UPFC s ncorporated at the end of the transmsson lne njectng certan amounts of voltage after the capactve reactance. Fg.. Electrcal system under study (modfed verson of cond system of the EEE cond benchmark model of SSR The electro-mechancal systems of the two generators and the data of the electrcal and mechancal systems are provded n [0]. Each generator s reprented n ts own rotor frame of reference wth the proper transformatons as the substtuton of the two generators n one equvalent generator wll change the resonance characterstcs. The electrcal system conssts of the nonlnear dynamcal mathematcal model of the two synchronous generators and that of the transmsson lne n dq statonary reference frame related to the d-axs stator wndng, q-axs stator wndng, d-axs rotor feld wndng, q-axs rotor damper wndng and d-axs rotor damper wndng utlzng Park's transformaton. Each cton of the mechancal system of the two generators s reprented by cond order ordnary dfferental equaton (swng equaton whch s prented n state space model as two frst order ordnary dfferental equatons. The mathematcal model of the electrcal and mechancal systems s gven n []. As a result, ths systems can be mathematcally reprented as a t of frst order nonlnear ordnary dfferental equatons wth the compensaton factor ( X c / X L as a bfurcaton parameter. Hence, bfurcaton theory can be appled to ths nonlnear dynamcal system whch can be wrtten n the followng form: dx F( x; ( dt

3 Engneerng, Technology & Appled Scence Rearch ol. 7, No. 3, 07, x x,..., x, where x s the vector of state varables n, n s an nteger number reprentng the number of state varables, F s the feld vector and s the control parameter of the system whch s n ths study s the transmsson lne compensaton degree. Unfed power flow controller (UPFC s a FACTS devce utlzng a voltage source converter (SC. t depends on gateturn-off (GTO thyrstor valve technology and can be desgned as a coordnated combnaton of a statc synchronous res compensator (SSSC and a statc synchronous compensator (STATCOM, whch us the same technology, coupled through a common DC voltage lnk [-]. UPFC s a multfunctonal FACTS controller where ts prmary duty s to control the power flow [3]. The condary functons of the UPFC are voltage control, trannt and steady-state stablty mprovement and oscllaton dampng. Recently, there s a rsng attenton n studyng UPFC whch ncludes ts modelng [4], ts effect on controllng the power flow [5] and ts capablty to mprove system trannt stablty [6]. However, so far, lttle work has been done to nvestgate the effect of UPFC on SSR problems. Dampng the oscllatons of The EEE frst benchmark system of SSR usng UPFC has been examned [7]. A compensator utlzng specfc technque bad on model control theory wth two feedback sgnals s employed to generate two supplementary dampng sgnals. The two port voltages of UPFC are regulated through control of unt current and res real voltage as own n Fgure. Moreover, t has varous operatng control modes such as lne mpedance compensaton, voltage and power regulaton. The UPFC s realzed by two three-level -pul voltage source converters (SC. t conssts of a unt connected voltage source converter (SC and a res connected voltage source converter (SC. SC njects a res voltage whle SC s controlled to nject reactve current. The res and unt branches of UPFC can generate/absorb reactve power ndependently and the two branches can exchange actve power. The njecton of res reactve voltage provdes actve res compensaton whle the njecton of the unt reactve current can be controlled to regulate the voltage at the bus where SC s connected. The njecton of res real voltage (n-pha wth the lne current can be controlled to regulate the reactve power n the lne or the voltage at the output port of the UPFC. UPFC can be vewed as a two port devce on a sngle pha bass [8].. MATHEMATCAL MODEL OF UPFC AND TS CONTROLLERS A. Mathematcal Model of UPFC Typcally, the converter n the power crcut of a UPFC s ether a mult-pul or a mult-level confguraton. n UPFC, t s consdered to change the magntude of AC output voltage of the converter wthout varyng the magntude of the dc voltage. Ths can be accompled by utlzng pul wdth modulaton (PWM wth two-level topology whch demands hgher swtchng frequency and leads to ncread loss. The objectve can be accompled by varyng dead angle β wth fundamental swtchng frequency of the three-level converter topology [9]. The converters that allow the varaton of both magntude and the pha angle of the converter output voltage are classfed as type- converters [30]. n ths study, a combnaton of mult-pul and three-level confguraton s ud [3]. The three-level converter topology s consdered to reduce the harmonc dstorton on the asde [5, 9, 3-33]. Both the unt and res branches of the UPFC contan -pul converter wth three-level poles. A specal desgn of a converter operaton by swtchng functons to mprove a complete three-pha model of UPFC s consdered [3]. UPFC can be desgned by transformng the three-pha voltages and currents to D-Q varables usng Park s transformaton. Ths s can be done when swtchng functons are approxmated by ther fundamental frequency components neglectng [34]. As own n Fgure, the resstance and reactance of the nterfacng transformer of unt and res SC are symbolzed as R, X, R and X, respectvely. The magntude control of unt and res converter output voltages and are accompled by modulatng the conducton perod affected by dead angles β and β of ndvdual converters whle DC voltage s mantaned constant. The unt converter output voltage can be wrtten as: ( D ( k m dc sn( (3 Q ( k m dc cos( (4 Where km ( 6 cos( for a -pul converter, ρ s the transformaton rato of the nterfacng transformer T and α s the angle by whch the fundamental component of unt converter output voltage leads the port- voltage. Fg.. UPFC as a two port FACTS controller On the other hand, the res converter output voltage can also be wrtten as: (5 D ( km dc sn( (6 Q ( km dc cos( (7

4 Engneerng, Technology & Appled Scence Rearch ol. 7, No. 3, 07, where km = ( 6 p r cos( b, s the transformaton rato of the nterfacng transformer T and γ s the angle by whch the fundamental component of res converter output voltage leads the port- current Ι.The DC sde capactor can be mathematcally descrbed as: d dt g dc c B B B dc dc (8 bc bc bc =- [ k sn( q+ a + k cos( q+ a ],,D where dc m, D m, Q and,q are D-Q components of the unt converter current, respectvely and: [ km sn(, D km cos(, dc Q B. Shunt Current Controller The real current drawn by the unt SC s controlled by pha angle and the reactve current by modulatng the converter output voltage magntude whch depends on β. The dynamcal equatons of the unt current control are prented n detals n [8]. The schematc reprentaton of type- controller for unt current control s own n Fgure 3. n order to keep the port- voltage magntude at the specfed value, the reactve current reference of unt converter can be mantaned constant or regulated. ]. Fg. 3. Type- unt current controller n Fgure 3, real and reactve currents can be reprented as: P( sn( cos( (9 R( cos( sn( (0 Where and are calculated as: R(, ord tan ( P(, ord P ord (, R(, ord cos ( kdc n the ca the unt SC s drawng real current and nductve reactve current, (9 and (0 ould be postve values. C. Seres oltage Controller Fgure 4 ows the type- controller structure for res converter. The ndependent njecton of real and reactve voltage caus two operatng combnatons of res converter. The frst one s about constant reactve voltage and port- voltage control whle the cond s about constant reactve voltage and constant resstance emulaton. The actve power can be regulated and/or modulated by controllng res reactve voltage reference [35-37]. On the other hand, the reactve power n the lne can be regulated by controllng res real voltage njecton and ths s equvalent to port- voltage control [8]. The voltage at port- of the UPFC s related to that at port- and the voltage njected by res SC. For smplcty, the res transformer reactance s clubbed wth the lne mpedance. The voltage relaton s gven by: D ( P Q P( ( D ( R ( R( D (3

5 Engneerng, Technology & Appled Scence Rearch ol. 7, No. 3, 07, where P and R are the n-pha and quadrature components of port- voltage wth respect to port- current and P( and R( are the n-pha and quadrature components of wth respect to port- current and can be wrtten as: system has a stable operatng pont to the left of H» 4 and to the rght ofh» 0.84 and an unstable operatng pont between H and H. sn cos (4 cos sn (5 P( R( Rotor Speed of the Frst Generator (pu Frst Unt Tme (c.06 Second Unt Rotor Speed of the Second Generator (pu Fg. 4. Type- Seres voltage controller P and R can be computed from D, Q and by assumng R( = R(,ref. Real voltage P( can be calculated to be njected to obtan desred magntude of as own n Fgure 4. n ths ca, there are two solutons of P(. The soluton whch has the lower magntude s chon. n Fgure 4, γ and β are expresd as: R(, ord tan (6 P(, ord P ref (, R(, ref cos (7 k dc. NUMERCAL SMULATONS A. System Respon wthout UPFC Ths subcton prents the results of the numercal smulatons when the power system has no UPFC. n ths ca, the mathematcal model of the system s reprented by 7 ordnary nonlnear coupled dfferental equatons []. The synchronous generators are heavly loaded wth an output actve power of 0.9 pu and output reactve power of 0.43 pu. The respon of the system wth % ntal dsturbance on the speed of the generator at μ=0.5 wthout UPFC s own n Fgure 5. t can be concluded that the operatng pont of the system s unstable. Fgure 6 ows the bfurcaton dagram of the power system n ths ca, whch s the steady-state value of the frst generator delta angle as functon of the compensaton degree. Two Hopf bfurcaton ponts are predcted. The power Fg Tme (c Rotor speed of the generators at μ=0.5 wth % ntal dsturbance n rotor speed of the generators (wthout UPFC. The operatng pont los stablty through subcrtcal Hopf bfurcaton at H. t returns to stablty ca n a rever Hopf bfurcaton at H. n Hopf bfurcaton, a par of complex conjugate egenvalues of the lnearzed model around the operatng pont transversally cross from left to rght sde of the complex plane or vce versa. The type of the frst Hopf bfurcaton at H whch decdes whether the lmt cycles created due to the Hopf bfurcaton are stable or unstable has been determned numercally by obrvng the tme respon of the system after small dsturbance slghtly before H. Fg. 6. Bfurcaton dagram owng varaton of the frst generator rotor angle wth the compensaton factor (wthout UPFC. Fgure 7 ows the respon of the system wth % ntal dsturbance on the speed of the frst generator at μ=0.975,

6 Engneerng, Technology & Appled Scence Rearch ol. 7, No. 3, 07, whch s slghtly lower than H. t can be concluded that the system operatng pont has an unstable behavor. Conquently, the type of ths Hopf bfurcaton s subcrtcal and the perodc soluton emergng at the bfurcaton pont s unstable. predct any bfurcatons at all realstc compensaton degrees and therefore one can conclude that the UPFC and ts controllers have succusd n elmnatng all bfurcatons of the power system at all practcal compensaton degrees..4 Frst Unt.05 Frst Unt Rotor Speed of the Frst Generator (pu Rotor Speed of the Second Generator (pu Tme (c Second Unt Tme (c Fg. 7. Rotor speed of the generators at μ=0.975 (slghtly lower than H wth % ntal dsturbance n rotor speed of generator (wthout UPFC. B. System Respon wth UPFC The system respon n ca of addng UPFC after % ntal dsturbance n generator rotor speed at μ=0.6 s own n Fgure 8. n fact, testng of the system at all values of μ ows that the system operatng pont does not lo stablty for all realstc values of compensaton factor. Hence, the system never experences any bfurcatons for ths ca. As a result, t can be concluded that UPFC has succusd n stablzng the system operatng pont at all realstc bfurcaton parameters and therefore the Hopf bfurcaton wll never occur for all values of the compensaton factor.. CONCLUSONS n ths study, a controller s desgned and consdered for UPFC to control bfurcatons of subsynchronous resonance n mult-machne power system. The system consdered s a modfed verson of the cond system of the EEE cond benchmark model of subsynchronous resonance where the compensaton degree μ=χ c /X L s the bfurcaton parameter. UPFC s ud as a powerful technque bad on economc and techncal consderatons for controllng actve and reactve power flows n ac transmsson lnes as well as controllng the bus voltage. t can also be ud to mprove the power system stablty wth the help of the Power System Stablzer (PSS. When UPFC s not ncluded, t s found that as the compensaton factor ncreas the system operatng pont los stablty through subcrtcal Hopf bfurcaton pont at relatvely early compensaton degree before reganng the stablty at a very late compensaton level n a rever Hopf bfurcaton. When UPFC s added to the system, the analyss does not Rotor Speed of the Frst Generator (pu Rotor Speed of the Second Generator (pu Tme (c Second Unt Tme (c Fg. 8. Rotor speed of the frst and cond generator at μ= 0.60 wth % ntal dsturbance n rotor speed of generator (wth UPFC controller. REFERENCES [] W. Zhu, R. R. Mohler, R. Spce, W. A. Mttelstadt, D. Maratukulman, Hopf Bfurcaton n a SMB Power System wth SSR, EEE Trans. on Power Systems, ol., No. 3, pp , 996 [] A. M. Harb, Applcaton of bfurcaton theory to subsynchronous resonance n power systems, Doctoral Dsrtaton, rgna Polytechnc nsttute and State Unversty, USA, 996 [3] A. M. Harb, M. S. Wdyan, Modern nonlnear theory as appled to SSR of the EEE cond benchmark model, EEE Bologna Power Tech. 003 Conf., Bologna, taly, June 3 6, 003 [4] A. M. Harb, M. S. Wdyan, Chaos and bfurcaton control of SSR n the EEE cond benchmark model, Chaos, Soltons and Fractals Journal, ol., pp , 004 [5] N.G. Hngoran, L. Gyugy, Understandng FACTS, EEE Press, New York, 000. [6] Cgre/EEE FACTS Workng Group, FACTS Overvew, EEE Power Engneerng Socety, No. 95 TP08, 995 [7] C. A. Canzares, Power Flow and Trannt Stablty Models of FACTS Controllers for oltage and Angle Stablty Studes, Proceedngs of the 000 EEE/PES Wnter Meetng, Sngapore, January, 000 [8] M. S. Wdyan, Controllng Chaos and Bfurcatons of SSR Usng TCSC, nternatonal Journal of Modellng and Smulaton, ol. 30, No. 3, pp , 00 [9] M. S. Wdyan, Controllng chaos and bfurcatons of SMB power system experencng SSR phenomenon usng SSSC, nternatonal Journal of Electrcal Power and Energy Systems, ol. 49, pp , 03 [0] EEE SSR Workng Group, Second Benchmark Model for Computer Smulaton of Subsynchronous Resonance, EEE Trans. on Power Apparatus and Systems. ol. PAS- 04, No. 5, , 985

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Stacey, L. Kovalsky, L. Gyugy, A. Edrs, AEP Unfed Power Flow Controller Performance, EEE Trans. Power Delvery, ol. 4, No. 4, pp , 999 [37] K. Esakk Shenbaga Loga, S. Umayal, T. C. Aa Raghavan, Mtgaton of Subsynchronous Resonance n Seres Compensated Wnd Farm Usng UPFC and ANN wth Back Propagaton Algorthm, nternatonal Journal of Engneerng Rearch and General Scence, ol. 3, No., 05. AUTHORS PROFLE Majd M. Alomar (BSc 00, MSc 004, PhD 0 s an Assstant Professor n Electrcal Engneerng Department at the Australan College of Kuwat. Hs rearch nterests nclude: Power System Analyss, Power System Stablty and Control, Power System Plannng, Power System Protecton, Power Electroncs, Electrcal Machnes and Drves, Electrc Machne Desgn, Renewable Energy, Smart Grd, Energy Plannng and Polcy, Control Systems, Optmal Control, Nonlnear Dynamcs, Lnear System Analyss and other related topcs. Mohammad S. Wdyan (BSc 000, MSc 00, PhD 006 s an Assocate Professor n Electrcal Engneerng Department at the Haemte Unversty. Hs rearch nterests nclude: Conventonal and Permanent-Magnet Electrcal Machnes Desgn, Fnte Element Technque, Bfurcaton Theory, Nonlnear Dynamcs and Control, Subsynchronous Resonance n Power Systems (Analyss & Control, Power System and Electrcal Machne Dynamcs, Renewable and Hybrd Energy Systems (Wnd and Photovoltac, Dynamcal Analyss of arous P-Powered and Hybrd Electrcal Systems, Smulatons of Power Electronc Systems n MATLAB. Mohammed Abdul-Nby receved the B.Sc. degree n electrcal engneerng and MSc n Electroncs and communcatons from the college of engneerng, Unversty of Basrah, Basrah, raq and the Ph.D degree from the Unversty of Surrey, Unted Kngdom n 998. He s currently an Assstant Professor at the Electrcal Engneerng Department, Australan College of Kuwat, Kuwat. He has publed n areas of sgnal processng and electronc crcuts. Hs current rearch nterest nclude smulaton modelng of mconductor devces, characterzaton of mplanted slcon, and renewable energes. Alreza Ghetas (BSc 00, MEngSc 007, Ph.D 0 has been nvolved n a number of ndustral and rearch projects snce 008 n New Zealand. He s currently workng for the Australan College of Kuwat as a lecturer. Hs man rearch nterests are power system protecton, fault detecton and dagnoss and smart grds.