Lyapunov Energy Function for an SMES

Transcription

1 roceedngs of the 6th WSEAS/ASME nt. Conf. on Electrc ower Systems, Hgh Voltages, Electrc Machnes, enerfe, Span, December 16-18, Lyapunov Energy Functon for an SMES VALENN AZBE, RAFAEL MHALC Faculty of Electrcal Engneerng nversty of Lublana rzaska 25, 1000 Lublana SLOVENA Abstract: - A superconductng magnetc energy storage (SMES can provde dynamc power-flow control, and n ths way mprove electrc-power-system stablty. n order to assess ts nfluence on the system s dynamc behavor or to determnate the devce s control strategy usng drect methods, proper energy functons for ths devce are needed. n ths paper the energy functons for a seres connecton and for a parallel connecton of SMES nto the electrc-power system have been developed. he energy functons were constructed as addtonal terms that can be added to any exstng structure-preservng energy functon. ests wthn a sngle-machne nfnte-bus system proved the correctness of the proposed energy functons. he applcaton of new energy functons was demonstrated on the problem of transent-stablty assessment. Key-Words: - FACS devces, Lyapunov energy functons, Lyapunov drect methods, power-system control, power-system transent stablty, superconductng magnetc energy storage 1 ntroducton Superconductng Magnetc Energy Storage (SMES systems store energy n the magnetc feld created by the flow of drect current n a superconductng col whch has been cryogencally cooled to a temperature below ts superconductng crtcal temperature. A typcal SMES system ncludes three parts: superconductng col, power condtonng system and cryogencally cooled refrgerator. SMES loses the least amount of electrcty n the energy storage process compared to other methods of storng energy. Due to the energy requrements of refrgeraton and the hgh cost of superconductng wre, SMES s currently used for short duraton energy storage. herefore, SMES s devoted to load levellng, dampng system osclatons or mprovng power qualty. t could also perform a varety of functons such as automatc generaton control, fast spnnng reserve, black start, mproved transmsson effcency, voltage control and transent-stablty mprovement. ts applcaton can be smlar to an applcaton of FACS devces wth addtonal possblty of nectng actve power nto (or out of the electrc-power system (ES. n ths paper we have attempted to develop the energy functons for an SMES. Our dervatons were demonstrated on the phenomenon of transent stablty as a typcal task where energy functons are tradtonally appled n drect methods for transent-stablty assessment. Wth the ncreased mportance of onlne dynamc securty assessment, drect methods mght be appled to avod the tme-consumng repetton of solvng a system s nonlnear dfferental equatons. Regardless of how SMES acts durng and after a fault, ts nfluence should be consdered n drect methods that apply proper energy functons. Of course, the applcaton of energy functons s much wder, e.g., for the control strateges of SMES [1]. We tred to construct the energy functons for both of the possble connecton of an SMES nto the ES.e., for a seres connecton and for a parallel connecton. he energy functons were constructed for the model of the ES wth structure-preservng topology, and therefore they are n the form of an addtonal term that can be added to any exstng structure-preservng energy functon. he same prncples were appled for constructon of energy functons for FACS devces [2]- [5]. he paper s organzed as follows: Secton 2 descrbes an SMES s characterstcs and the dervaton of ts energy functons. Secton 3 presents numercal examples of the transent-stablty assessment of a sngle-machne nfnte-bus test system that prove the correctness of the proposed energy functons. Secton 4 draws conclusons. 2 Characterstcs of an SMES A superconductng col of an SMES s normally connected to the electrc-power system va a power condtonng system,.e., semconductor converter. Generally, t can be connected ether n seres or n parallel to the system. Because the effect on power system dynamcs depends essentally on the way of ts connecton, the necton model and constructon

2 roceedngs of the 6th WSEAS/ASME nt. Conf. on Electrc ower Systems, Hgh Voltages, Electrc Machnes, enerfe, Span, December 16-18, procedure for the energy functon are presented separately for the seres and for the parallel connecton of an SMES. n order to be able to derve the energy functons for the SMES, t has to be represented by the necton model that represents ts behavor n an ES wth ts basc characterstcs. 2.1 Seres connecton of SMES f SMES s connected n seres to the system, t s connected va a voltage-source converter (VSC and a seres transformer, smlarly to a statc synchronous seres compensator (SSSC. herefore, the necton model for the SMES n seres connecton s the same as a general SSSC necton model presented n [6] necton model for a seres connecton Fg. 1 presents an SMES scheme, ts phasor dagram and the necton model. Lke n an SSSC, the seres-nected voltage magntude of a VSC,, does not depend on the current through the lne or on the bus voltage. herefore, s lmted only by the constructon of a VSC and a seres transformer and can be treated as an SMES s control parameter. n contrast to an SSSC, the angle of the seres-nected voltage, ϕ, can be set to any value because an SMES can nect actve and/or reactve power. he actve and reactve powers nected accordng to Fg. 1 (c can be denoted as: θ θ +ϕ Τ θ' θ ' Q s sn ( ϕ (1 ( θ ϕ s sn + (2 s Q s Χ cos ( ϕ (3 Χ cos ( θ + ϕ (4 where θ θ θ, accordng to Fg. 1. hese power nectons are the bass for the constructon of the energy functon Constructon of an Energy Functon he followng step s the constructon of an energy functon for an SMES that can be added to the structurepreservng energy functon (SEF presented n [7]. n [7] the SEF s constructed for the ES wthout SMES and we denote t as V wthout SMES. hs energy functon can be treated as the sum of the knetc energy V k and the potental energy V p : Vwthout SMES Vk + Vp (5 sng the above energy functon formulaton the system can be llustrated wth a mechancal analogy as a ball rollng on a potental energy surface, as n [8]. hs vsualzaton s presented n Fg. 2. a ϕ Τ ' θ b c θ θ s + Q s s + Q s Fg. 1. Seres connecton of an SMES; a scheme; b phasor dagram; c necton model. Fg. 2. A ball on a potental-energy surface. he potental energy V p of a gven post-fault system depends on the machne angles. n the case of a threemachne system the potental energy V p can be presented as a surface on a three-dmensonal chart, where the horzontal axes represent the angle of two machnes accordng to the thrd machne (.e., δ 1 and δ 2 accordng

3 roceedngs of the 6th WSEAS/ASME nt. Conf. on Electrc ower Systems, Hgh Voltages, Electrc Machnes, enerfe, Span, December 16-18, to Fg. 2. he potental-energy surface has a local mnmum at the stable equlbrum pont, whch corresponds to the machne angles durng the post-fault steady-state operaton. Around ths stable equlbrum pont the potental-energy surface forms a bowl-shaped area, whch s the area of stable system operaton. he knetc energy of the system s equated to the knetc energy of a ball that rolls along the potentalenergy surface accordng to the generator swng traectory. n steady-state operaton the ball stands stll at the stable equlbrum pont. However, when the fault occurs, the ball s pushed toward the edge of the bowlshaped area of the potental-energy surface untl the fault s cleared. Dependng on the total of the knetc and potental energes of the ball at the tme of fault clearng, the ball can ether escape from the bowl over the saddle (.e., an unstable case or t can contnue to oscllate wthn the bowl (.e., a stable case. o assess the stablty of the system the knetc energy s compared wth the potental energy at the border of the stable area. o construct an energy functon for an SMES that can be added as an addtonal term to the V wthout SMES we follow the constructon procedure presented n [7]. Accordng to ths procedure, the actve-power nectons s and s are multpled by the tme dervatve of the voltage angle and the reactve-power nectons Q s and Q s are dvded by the voltage magntude and multpled by the tme dervatve of the voltage magntude: θ sn ( ϕ Χ θ (6 s θ sn θ + ϕ Χ θ (7 ( s Qs ( ϕ cos (8 Qs ( θ ϕ (9 cos + hen, (6 (9 are summed and we obtan the followng expresson: sn ϕ θ θ ϕ Χθ ( Χ sn ( + ( ϕ cos ( θ ϕ + cos + (10 Equaton (10 s the result of the constructon procedure presented n [7]. Now we have to fnd the frst ntegral of (10, whch represents the Lyapunov energy functon for an SMES. t s well known that there s no procedure for fndng the Lyapunov energy functon and that t always has to be found ntutvely. Consderng the smlarty of an SMES to an SSSC and that the already-known energy functons for some FACS devces are equal to half or to the total sum of the reactve power nected nto the system, we can search for the Lyapunov energy functon for an SMES n ths drecton. Let us denote the sum of the SMES s reactvepower nectons (3 and (4 as Q n and make ts dervatve: dqn dt Χ Χ Χ cos ( ϕ + cos ( ϕ sn ( ϕ Χϕ Χ Χ Χcos ( θ + ϕ Χcos ( θ + ϕ Χ + Χsn ( θ + ϕ Χι θ + ϕ ω λ ϋ (11 Comparng (10 and (11 t s clear that some parts are equal. Now we can reproduce the frst ntegral of (10, whch represents the energy functon for an SMES n seres connecton: V SMES-seres ς ι ι Χ κκ κλ λ n ( ϕ sn ( θ ϕ Χ( θ ϊ ϕ + sn + + ι κ Q cos Χ ω ω + Χ dt ϊ (12 ( ϕ cos ( θ ϕ ( ϊ λ ϋ ϋ he frst square bracket n (12 s equal to the actve power nected, and we denote t as n. he second square bracket represents the quotent Q n /. herefore (12 can be rewrtten as: Qn V Q + ι Χ( θ ς + ϕ + Χ ω ϋ dt λ (13 SMES-seres n n Equaton (13 ncludes the terms that depend on the tme dervatve of the SMES s control parameters. t can be expected that voltage magntude should be set to ts maxmum possble (.e. constant value n order to acheve maxmum mprovement n power system s dynamcs (e.g. transent-stablty mprovement. Consequently can be consdered as a constant. he second controlled parameter can be as SMES s most basc control parameter actve power n. Consderng constant n and constant, the ntegral n (13 can be ω ϋ

4 roceedngs of the 6th WSEAS/ASME nt. Conf. on Electrc ower Systems, Hgh Voltages, Electrc Machnes, enerfe, Span, December 16-18, easly solved and the energy functon for an SMES s: V Q + θ + ϕ SMES-seres n n Χ ( (14 he energy functon (14 can be appled not only when n and are constant, but even when they are sectonal-constant, n a way that s presented n [2]. Wth sectonal-constant parameters any control strategy can be approxmated. he SEF for the system wthout SMES V wthout SMES (5 can now be upgraded to represent the SEF for the system wth an SMES n seres connecton: Vwth SMES-seres Vwthout SMES + VSMES-seres ( arallel connecton of SMES Lke a seres-connected SMES, a parallel connected SMES s also coupled wth an ES va a converter. hs converter s smlar to a SACOM, except that t can nect to the system not only reactve power but also an actve power. he angle between the nected current and the bus-voltage can take any value and s not lmted to the angles 90 o and 90 o, as t s n the case of a SACOM necton Model of a parallel connecton. Fg. 3 presents the scheme, the phasor dagram and the necton model of an SMES n parallel connecton. ower nectons are the product of the bus voltage and the nected current: 1 θ 2 Fg. 3. arallel connecton of an SMES; a scheme; b phasor dagram; c necton model. s Χ Χ cos ( β (16 Q β s Χ Χ sn ( (17 hese equatons are the bass for the constructon of an energy functon for an SMES n parallel connecton Constructon of an Energy Functon n a smlar way to the SMES n seres connecton we construct ts energy functon for a parallel connecton to be added as an addtonal term to the SEF V wthout SMES (5. he frst steps are the same: the actve power necton, s, s multpled by the tme dervatve of the voltage angle, and the reactve-power necton Q s s dvded by the voltage magntude and multpled by the tme dervatve of the voltage magntude. Next, both terms are summed together and we get the followng expresson: ( β θ + sn ( β (18 Χ Χcos Χ Χ Χ he frst ntegral of (18 can be rewrtten as: ς Χcos ( β Χ dθ + ς Χsn ( β d (19 he frst part of (19 s the ntegral of the actve-power necton over the angle θ. Assumng that should be set to ts maxmum possble (.e. constant value n order to acheve maxmum mprovement of power-system dynamc and that the second controlled parameter of an SMES ts most basc parameter s s constant, (19 can be rewrtten as: ( β sn s Χθ + ς Χ d (20 a β Because the angle β n the ntegral part of (20 s not constant, we transform ths ntegral as: b β ς Χsn ( β d ( cos ( ι Χsn β Χ + Χ Χ β Χβ ςλ ωdt ( β Χ Χcos Χβ ϋ (21 c θ s + Q s he frst two terms n square brackets can be denoted as the tme dervatve of the nected reactve power Q s, whle the thrd part of (21 nherts a term for nected actve power that s constant. herefore, we can rewrte (21 as:

5 roceedngs of the 6th WSEAS/ASME nt. Conf. on Electrc ower Systems, Hgh Voltages, Electrc Machnes, enerfe, Span, December 16-18, ι d ω ςκ ι sn ( β dt s d β dt λ Χ Χ ωϋ ϊ ς λ ϋ Χsn Χ Χβ s ( β Q Χβ s s (22 30 o. he dsturbance s a three-phase short-crcut near BS1, accordng to Fg. 4 or Fg. 5, and t s assumed to be eventually cleared,.e., the system s post-fault confguraton s dentcal to the pre-fault one. Now we can nsert (22 nto (20 and we get the energy functon for an SMES n parallel connecton wth a constant current magntude,, and constant nected actve power, s : SMES-parallel s s ( V Q + Χ θ β (23 Agan, lke n seres connecton, the energy functon (23 s constructed for constant control parameters. Applyng the procedure presented n [2], ths energy functon can be appled even when the control parameters are sectonal-constant. Assumng that wth sectonal-constant parameters any control strategy can be approxmated, (23 can be appled for any control strategy. he SEF for the system wthout SMES V wthout SMES (5 can now be upgraded to represent the SEF for the system wth an SMES n parallel connecton: Vwth SMES-parallel Vwthout SMES + VSMES-parallel (24 3 Numercal examples of transentstablty assessment usng the Lyapunov drect method he proof of correctness and a demonstraton of the applcaton of the newly constructed energy functons for an ES comprsng SMES.e., (15 for SMES n seres connecton and (24 for SMES n parallel connecton were carred out on an example of transent-stablty assessment. n order to obtan the crtcal energy of the system a potental-energy boundary-surface (EBS method [9] was used, n whch the crtcal clearng tme (CC s the tme nstant when the total energy of the system along the fault-on traectory equals the maxmum of the potental energy along the same fault-on traectory. n order to prove the correctness of the proposed energy functons we compared the results.e., the CCs obtaned by the drect method wth the CCs obtaned by the smulaton method,.e., by the tmedoman step-by-step smulaton. n a sngle-machne nfnte-bus (SMB test system the traectory of the system s unformly gven and therefore the CCs obtaned should be equal, regardless of the method appled [9]. he data for the SMB test system can be found n [2]. he generator s presented as a classcal model wth the ntal voltage at BS1 set to 1 p.u. at 3.1 Seres connecton of SMES he SMB test system wth an SMES s presented n Fg. 4. he SMES s connected to the system va a seres transformer wth a short-crcut voltage u k 3.75 % and s rated at 265 MVA. he pre-fault and fault-on values of the SMES s controllable parameters are set to 0. he CCs were obtaned usng tme-doman smulatons, and drectly wth the use of the energy functon (15 that ncludes newly proposed energy functon for an SMES n seres connecton (14. GEN BS1 BS2 BS km 400 km 150 MW 15 MVAr SMES Fg. 4. Seres connecton of an SMES BS3 Grd he results for varous lmts and varous constant nected actve power n are presented n able 1. he nected actve power n consdered n ths example s the maxmum possble power, as t depends on the current through the lne n whch the SMES s nserted. he results are equal usng both methods and n ths way they valdate the proposed SMES energy functon (14. able 1. CCs obtaned n a SMB test system ncludng seres-connected SMES. Smulaton method Drect method [pu] n [pu] CC [ms] CC [ms] 0 ~ Comparng the CCs from able 1 consderng the same lmts t can be seen that accordng to the negatve sgn of n n ths example the actve power should be nected to the SMES n order to mprove the transent-stablty.

6 roceedngs of the 6th WSEAS/ASME nt. Conf. on Electrc ower Systems, Hgh Voltages, Electrc Machnes, enerfe, Span, December 16-18, arallel connecton of SMES he SMB test system wth an SMES n parallel connecton s presented n Fg. 5. he pre-fault and faulton values of the SMES s controllable parameters are set to 0. Here also, the CCs were obtaned usng a tmedoman smulaton, and drectly wth the use of the newly proposed energy functon (24. GEN BS1 BS2 200 km 400 km 150 MW 15 MVAr SMES Fg. 5. arallel connecton of an SMES BS3 Grd he energy functon (23 was developed for a parallel connecton of an SMES to be ncluded n the energy functon for the ES (24. he resultng CCs are presented n able 2 for varous lmts and varous actve-power nectons s. he voltage magntude at BS2 durng the frst-swng angles propagaton consderably decreases, and consequently the constant actve power s nected wthn ths perod s lmted to small values. he s presented n able 2 are the maxmum possble that at the same tme gve maxmum CCs. he negatve sgn of s means that the actve power flows from the system to the SMES. able 2. CCs obtaned n a SMB test system ncludng SMES n parallel connecton. Smulaton method Drect method [pu] s [pu] CC [ms] CC [ms] Concluson SMES can mprove the power system s stablty. n order to assess ts nfluence on the system s dynamc behavor or to determnate the devce s control strategy usng drect methods, proper energy functons for an SMES s needed. n ths paper two energy functons have been developed.e. for a seres connecton and for a parallel connecton of an SMES. Both energy functons are constructed as an addtonal term that can be added to any exstng structure-preservng energy functon. he correctness of the proposed energy functons was tested wth numercal examples on a sngle-machne nfnte-bus test system. he resultng CCs were compared to the reference CCs obtaned by the smulaton method. he equalty of the CCs proves the correctness of the proposed energy functons. Future work wll be focused on determnng the control strategy of an SMES, based on a maxmzaton of the tme dervatve of the power system s energy functon along the system traectory. References: [1] V. Azbe, R. Mhalc, Applcaton of the drect Lyapunov method for optmum control of a FC, roceedngs of 2005 EEE owerech, St. etersburg, [2] V. Azbe,. Gabrel, R. Mhalc and D. ovh, he Energy Functon of a General Multmachne System wth a nfed ower Flow Controller, EEE rans. ower Systems 2005, vol. 20, pp [3]R. Mhalc,. Gabrel, A structure-preservng energy functon for a Statc Seres Synchronous Compensator, EEE rans. ower Systems, vol. 19, no. 3, Aug. 2004, pp [4] R. Mhalc,. Gabrel, ransent Stablty Assessment of Systems Comprsng hase-shftng FACS Devces by Drect Methods, nternatonal Journal of Electrcal ower Energy Systems, vol. 26, no. 6, July 2004, pp [5]. Gabrel, R. Mhalc, Drect Methods for ransent Stablty Assessment n ower Systems Comprsng Controllable Seres Devces, EEE rans. ower Systems, pp , Sept [6] M. Noroozan, L. Angqust, G. ngestrom, Seres compensaton, Flexble ac transmsson systems (FACS, EE: London, 1999, pp [7]. Van Cutsem, M. Rbbens-avella, Structure preservng drect methods for transent stablty analyss of power systems, roceedngs of 24th Conference on Decson and Control, 1985, pp [8]. Athay, R. odmore,s. Vrman, A practcal method for the drect analyss of transent stablty, EEE rans. ower Apparatus and Systems, 1979, vol 98, pp [9] M.A. a, Energy Functon Analyss for ower System Stablty. Kluwer Academc ublshers, Boston, 1989.