R. Erksson an V. Knazkns / GMSARN nternatonal Journal 2 (2008) 15-20 On the Coornate Control of Multple HVDC Lnks: Moal Analyss Approach Robert Erksson an Valerjs Knazkns Abstract There are several possbltes to mprove the small-sgnal stablty n a power system. One aequate opton s to make use of avalable power system components that possess hgh controllablty propertes such as, for nstance, hgh voltage rect current (HVDC) systems. Ths paper presents results from a stuy ame at the nvestgaton of small-sgnal stablty enhancement acheve by proper coornate control of multple HVDC lnks. Moal analyss was use as the man tool for the theoretcal nvestgaton. The obtane results ncate that the coornate control of several HVDC lnks n a power system may assst achevng n an essental ncrease of ampng n the power system. Another mportant concluson from the paper s that the possbltes of the coornaton of the HVDCs to a certan extent epen on the structure of the gr, whch can be nvestgate by examnng the controllable subspace of the lnearze moel of the power system. Keywors Coornate control, HVDC, Moal analyss, Power System Stablty. 1. NTRODUCTON Moern nterconnecte electrc power systems are characterze by large mensons an hgh complexty of the structure an the ynamc phenomena assocate wth the power system operaton an control. Power system eregulaton that took place n many countres worlwe was one of the rvng forces that stmulate a fuller utlzaton of power systems, whch n some cases lea to a reuce stablty margn, as the power systems became more stresse. Uner these crcumstances t becomes qute mportant to seek new possbltes of enhancement of both transent an small-sgnal stablty of the power systems. There are several obvous ways of mprovng power system stablty, namely, (1) bulng new transmsson lnes, (2) nstallng new generaton capactes, (3) better utlzaton of the exstng equpment n the power system, or (4) a combnaton of the above. Ths paper s prmarly concerne wth thr opton, snce compare to the other optons t s less costly an can be easly mplemente n a real power system. The central ea of the stuy presente n ths paper s the utlzaton of several HVDC lnks for small-sgnal stablty enhancement. The central purpose of conventonal HVDC transmsson s to transfer a certan amount of electrcal power from one noe to another an to prove the fast controllablty of real power transfer. f the HVDC lnk s operate n parallel wth a crtcal ac lne the loa-flow of the ac lne can be controlle rectly. The presence of an HVDC can assst n mprovng the stablty margn n the power system [1]. n case there are several HVDC R. Erksson (corresponng author) s wth the Dvson of Electrc Power Systems at Royal nsttute of Technology, (KTH), Teknkrngen 33, 100 44 Stockholm, Sween. Emal: robert.erksson@ee.kth.se. V. Knazkns s wth the Dvson of Electrc Power Systems at Royal nsttute of Technology, (KTH), Teknkrngen 33, 100 44 Stockholm, Sween. Emal: valerjs.knazkns@ee.kth.se. lnks n the system, there s also a possblty of coornatng the HVDC lnks to enhance the operaton of the system [by, for example, alterng the loa-flow patterns] an to mprove the system stablty evermore. The power systems are known to be operate most of the tme n the so-calle `quas-steay state'. That s to say, the power systems are always subject to varous often small sturbances [2]. A change n the loang level or capactor swtchng s typcal examples of such small sturbances that sometmes gve rse to oscllatons n the power system. The oscllatons are often postvely ampe an ther magntues ecrease after a whle thus the system remans stable. n case of negatve ampng of the oscllatons the stuaton s opposte an may result n loss of synchronsm unless preventve measures are taken. 2. CASE STUDY The am of ths paper s to perform moal analyss usng coornate control of two conventonal HVDC lnks n a benchmark power system. Ths paper also explores the possbltes brought by the controllablty an coornaton of the HVDC lnks to enhance the rotor angle stablty upset by a sturbance. n the benchmark power system, as s n all realstc cases, the turbne acton s very slow compare to the fast controllablty of the HVDC. The theory n ths paper s base on small-sgnal analyss by lnearzng the system aroun the stable or unstable equlbrum pont. The moal analyss proves valuable nformaton about the nherent ynamc characterstcs of the system. By controllng the current through the HVDCs an usng state feeback t s possble to move the egenvalues to pre-specfe locatons n the complex plane an thereby ncrease the ampng. 3. TEST POWER SYSTEM The system n ths case stuy conssts of three generators connecte to noes A, B, an C. The HVDC lnks are 15
R. Erksson an V. Knazkns / GMSARN nternatonal Journal 2 (2008) 15-20 connecte between noes A an C an between B an C. The power proucton s large n noe A an the loa center s assume to be n noe C,.e., there s a sgnfcant power flow from noes A to C. The power can go ether through the ac lnes or through the HVDC lnks. An overvew of the test power systems s shown n Fg. 1. 4. SYSTEM MODEL Fg. 1. Test Power system. The HVDC lnk s of a classcal type, whch mples that t consumes reactve power an that the actve power can be controlle. The HVDC lnk conssts of an eal rectfer, nverter, an seres reactor whch moels the c-lne an creates a smooth c current. The power through the HVDC s controlle by controllng the frng angle, α. The frng angle s controlle by a basc Pcontroller an the nput s the set pont current. The generators are moelle by the one axs moel, escrbe by equatons (1)-(3) [3], thus three states per generator are use. The ynamcs of the HVDC [4] are escrbe n equatons (4)-(16), the subscrpts r an refer to rectfer an nverter, respectvely. x x ' E f s the -axs synchronous reactance s the -axs transent reactance s the constant generator fel voltage 1 R ( r ) & = V V (4) L L x& = K ( ) (5) r setp x& = K ( ) (6) setp cos( α ) = x + K ( ) (7) r r P setp 3 2 3 V = V cos( ) r r α pu r X C r π π (8) 3 2 Vb c bc Sr = Vr pu π S (9) bc V P = (10) bc bc r V r Sb c 2 2 r r r Q = S P (11) cos( γ ) = x + K ( ) (12) P setp 3 2 3 V = V cos( ) γ pu X C π π (13) 3 2 Vb c bc S = V pu (14) π S bc V P = (15) bc bc V Sb c 2 2 Q = S P (16) & δ = ω (1) 1 & ω = ( Pm Pe Dω ) (2) M x ' x x ' E& ' q = E f Eq + V cos( δ θ ) (3) x ' x ' δ s the rotor angle ω s the rotor synchronous spee ' E δ, V θ are the voltage phasors at the nternal q an termnal buses T o ' s the -axs transent open-crcut tme constant P m D M s the mechancal power apple to the generator shaft s the generator s shaft ampng constant s the machne nerta V r, V are the per unt c termnal voltages at the rectfer an nverter V,, S are the base quanttes at the c bc bc bc X cr, X c se for the voltage, current an power, respectvely are the unt commutaton reactances R, L are the per unt c lne parameters V, V are the per unt ac bus voltages rpu pu setp s the current set pont through the HVDC 5. MODAL ANALYSS To stuy the power system small-sgnal stablty problem, an approprate moel for the machnes, loas 16
R. Erksson an V. Knazkns / GMSARN nternatonal Journal 2 (2008) 15-20 an HVDC ynamcs s requre. The behavor of a power can be escrbe by a set of frst orer nonlnear ornary fferental equatons an a set of nonlnear algebrac equatons [3]. T T ' T T T T T q r x& = f ( x, y, u) (17) 0 = g( x, y, u) (18) x = ( δ ω E x x ) s the vector contanng the T T T state varables, y = ( θ V ) s the vector esgnatng algebrac varables, an fnally u = setp the efne as the vector of control varables. n the small sgnal stablty analyss, the equatons (17) an (18) are lnearze at an equlbrum pont an the hgher orer terms are neglecte. The lnearzaton gves the structure as follows: x& = fx x + f y y + fu u => 0 = gx x + g y y + gu u x& = ( f f g g ) x + ( f f g g ) u 1 1 x y y x u y y u = J x + J u. x u (19) (20) The prefx means a small ncrement n corresponng varables. The matrces n equaton (19) an (20) can be foun n appenx. The Lyapunov s frst stablty metho s the funamental analytcal bass for power system smallsgnal stablty assessment. t s base on egenvalue analyss an proves valuable nformaton of the behavor of the system,.e. the tme oman characterstcs of a system moe. t s usual to assocate each egenvalue λ wth a moe of the system. Real egenvalues represent non-oscllatory moes, a negatve one correspons to ecayng moe, whle a postve one relates to aperoc nstablty. Complex egenvalues are assocate wth system oscllatory moes, the par of complex egenvalues wth negatve real parts ncate a ecreasng oscllatory behavor, an those wth postve real parts result n an ncreasng oscllatory behavor. The ampng of the :th moe s efne as follows: σ =Re(λ ) ω =m(λ ) ξ = σ σ 2 2 + ω (21) By controllng the HVDCs an thereby controllng the moes t s thereby possble to change the behavor of the system. The controllable subspace proves nformaton how the egenvalues can be move.e. how the system can be controlle. The Kalman ecomposton transforms the state space moel nto controllable, uncontrollable, observable an unobservable subspaces [5],[6]. Where the controllable an uncontrollable subspaces are of nterest, snce they prove nformaton about how the egenvalues can be move. The matrces J x an J y span the controllablty matrx an the mage of the corresponng map s the controllable space. The propose metho can thereby be use for practcal an large scale power systems. By followng the above escrbe proceure the controllablty graman proves the nformaton how the poles can be move,.e., how the ampng n the system can be ncrease. The egenvalues can nor be move arbtrary n the controllable subspace ue to lmtatons n the current through the HVDCs. 6. SMULATON STUDY n the test power system the egenvalues of J x n equaton (20) are etermne. t results n two oscllatory moes whch are referrng to the system moel. The egenvalues can be foun n Table 1. The ampng n the system s about 5%. By usng state feeback t s possble to ncrease the ampng. The control s performe as showe n Fg. 2. Ths s full state feeback whch means all states are use n the feeback, f not all the states are avalable, t possble to perform state estmaton. The unavalable sgnals can be estmate by an observer, but ths stuaton s left outse of the scope of ths paper. Table 1. Egenvalues of the lnearze system λ no feeback ξ no feeback λ feeback ξ feeback -0.552 + 9.780 0.0564-2.887 1-0.552-9.780 0.0564-2.887 1-1.179 + 13.258 0.0886-3.464 1-1.179-13.258 0.0886-3.464 1-0.102 1-2 1-0.496 1-2.25 1-0.716 1-2.5 1-0.849 1-2.75 1-1.138 1-3 1-1.138 1-3.25 1-2.373 1-3.5 1-2.373 1-3.75 1 0 - -4 1 0-0 - 0-0 - 17
R. Erksson an V. Knazkns / GMSARN nternatonal Journal 2 (2008) 15-20 The output may be chosen arbtrarly from the states or a lnear combnaton of the states. However, n ths paper generator A s taken as the reference an the outputs are the evaton n spee of generator B an generator C,.e. output one = ω C -ω A an output two = ω B -ω A. The output s relate to the states by a matrx whch s lnearly combnng the state to the output. Ths s shown n Fg. 2 an the relatng matrx s enote as J y. Fg. 4. Dsturbance one.e. ncrease of loa 3 an ecrease of loa 1. Output two.e. spee fference of generator B an A. Fg. 2. State feeback. n the case stuy n ths paper all the egenvalues can be move arbtrarly. One possblty s to ncrease the ampng or remove the oscllatory moes completely. n the test power system the close loop system has the egenvalues gven n Table 1. Two sturbances are apple to the system, the sturbances are ue to loa changes. Case one s ue to a ecrease of loa 1 an ncrease of loa 3. Case two s ue to ecrease of loa 2 an ncrease of loa 3. n both the cases the ncrease an the ecrease are equal, whch s one to preserve the equlbrum pont. Fg. 3 an Fg. 4 show the output sgnals when sturbance one s apple. t s obvous that nspecton of the fgures reveals that the system behaves more ncely when usng the state feeback an the oscllatory behavor shoul be totally elmnate, snce all ω.=0 Fg. 5 an Fg. 6 show the system behavor uner sturbance two, an also n ths case, the response s better n the controlle system compare to the uncontrolle. Fg. 5. Dsturbance two.e. ncrease of loa 3 an ecrease of loa 2. Output one.e. spee fference of generator C an A. Fg. 6. Dsturbance two.e. ncrease of loa 3 an ecrease of loa 2. Output two.e. spee fference of generator B an A. Fg. 3. Dsturbance one.e. ncrease of loa 3 an ecrease of loa 1. Output one.e. spee fference of generator C an A. 7. CONCLUSONS Ths paper presents moal analyss of a power system 18
R. Erksson an V. Knazkns / GMSARN nternatonal Journal 2 (2008) 15-20 wth several HVDC lnks an nvestgates the possblty to ncrease the ampng n the system by changng the moes n the system. The obtane results strongly ncate that essental enhancement of small-sgnal stablty can be acheve by proper coornaton of the control of the HVDC lnks. The paper also escrbes the eas about the controllablty of the moes, whch are epenng of the structure of the gr an may be etermne as the controllable subspace. REFERENCES [1] Aapa, R. 2002. Facts system stues. EEE Power Engneerng Revew. [2] Hanschn, E. Et al. 1972 Real-Tme Control of Electrc Power Systems. Elsever Verlag Amsteram, 1972. [3] Kunur, P. 1993. Power System Stablty an Control. McGraw-Hll nc. [4] Canzares, C.A. 1991. Voltage Collapse an Transent Energy Functon Analyss of AC/DC Systems. PhD thess, Unversty of Wsconsn- Mason. [5] Sontag, E.D. 1998. Mathematcal Control Theory: Determnstc Fnte Dmensonal Systems. Sprnger. [6] Sgur Skogesta.P. 2001. Multvarable Feeback Control, Analyss an Desgn. Wley. APPENDX 19
20 R. Erksson an V. Knazkns / GMSARN nternatonal Journal 2 (2008) 15-20