Modal Analyss of a PMSG-Based D Electrcal Power System n the More Electrc Arcraft Usng Egenvalues Senstvty Fe Gao, Xancheng Zheng, Member, IEEE, Serhy Bozhko, Member, IEEE, hrstopher Ian Hll, Member, IEEE, and Greg Asher, Fellow, IEEE Abstract Ths paper deals wth the modelng and small sgnal stablty analyss of a D-dstrbuton electrcal power system (EPS) sourced by a permanent magnet synchronous generator (PMSG). The topology employed here s one of the man canddates for future more-electrc arcraft (MEA). A detaled mathematcal model s developed and comprehensve EPS modal analyss s performed. Egenvalue senstvty and partcpaton factor are utlzed n order to assess the effect of machne and control parameters, as well as system operatng condtons, on EPS stablty. Furthermore, ths paper also presents comparatve analyss of system models wth and wthout the ncluson of system cablng. Ths crucal analyss shows that the tendences n stablty behavor can be sgnfcantly dfferent wth and wthout cablng. It s therefore shown that system smplfcaton, by removal of cablng, can delver remarkably msleadng results. Tme doman smulatons are carred out to support the theoretcal analyss. The comprehensve analyss presented n ths paper provdes EPS desgners wth an extremely useful methodology for the selecton of approprate EPS parameters at the early desgn stages. Index Terms Modal analyss, stablty, egenvalue senstvty, more electrc arcraft, D power dstrbuton. I. INTRODUTION Wth ncreasng concern about envronmental protecton, the development of a more electrc system to replace the conventonal mechancal and hydraulc power for vehcles has become popular n recent years. Ths concept has been wdely accepted n transportaton ncludng automotve, shp and arcraft power systems [], [] The electrcal power system (EPS) s a hot topc for researchers n the more electrc arcraft (MEA). Varous alternatve archtectures are nvestgated n [3]. Among all the canddates, D EPS s the most promsng for the future MEA applcaton due to several advantages such as hgh effcency, reduced weght and the absence of reactve power compensaton devces [4], [5]. The modal analyss method [6]-[] ncludng partcpaton factor, dampng rato and oscllaton frequency s an effcent tool to dentfy the physcal nature of modes, extract the crtcal modes and analyse the dynamc response. It has been Ths work was supported by the leansky JTI Project, a FP7 European Integrated Project-http: //www.cleansky.eu. F. Gao s wth the Department of Electrcal and Electronc Engneerng, Unversty of Nottngham, Nottngham, NG7 RD, U.K. (e-mal: eexfg5@nottngham.ac.uk). X. Zheng s wth the School of Automaton, Northwestern Polytechncal Unversty, X an, hna. (e-mal: zxcer@sna.com). S. Bozhko,. Hll, and G. Asher are wth the Department of Electrcal and Electronc Engneerng, Unversty of Nottngham, Nottngham, NG7 RD, U.K. (e-mal: serhy.bozhko@nottngham.ac.uk,.hll@nottngham.ac.uk, greg.asher@nottngham.ac.uk). wdely appled to power grds [6] and doubly-fed nducton generator (DFIG) based wnd power generaton systems [7]- []. Partcpaton factor analyss s benefcal for dentfyng the mode type and analysng the partcpaton degree of state varables on the gven egenvalues [8], []. Thus, t provdes an effcent tool for power system stablzer (PSS) desgn [6] and controller tunng [7]. Moreover, senstvty analyss has wde applcatons n power system modelng and analyss []-[]. The egensenstvty theory was ntroduced n [] and was used as a small sgnal stablty assessment tool n a turbne/governor system and PSS desgn. Repeat computaton for the egenvalues wth varyng parameters s avoded whch not only predcts the movement of the domnant modes of the system but also reduces the computaton burden, especally n a largescale power system. In [5], comprehensve modal analyss ncludng partcpaton factor and egenvalue senstvty analyss wth respect to machne and control parameters s performed for a DFIG-based wnd turbne system. In [6], the small sgnal stablty of an EPS wth ncreased photovoltac (PV) generaton s examned usng egenvalue senstvty wth respect to system nerta. It shows that ncreased PV and DFIG-based penetraton generaton may render the system unstable as the dampng of the crtcal modes s reduced. The study of senstvty provdes useful gudance for analyss, plannng, and operaton of power systems. It was successfully appled for optmal tunng of control parameters [7], [8], determnng locatons of compensatng devces, for dampng mprovement and stablty enhancement such as capactor compensaton and FATS devces [4]. Permanent magnet machnes have been wdely used n arcrafts and hybrd electrc vehcles due to ther hgh power densty, hgh effcency, fast dynamc response and mproved relablty. ompared to a DFIG-based generaton system, permanent magnet synchronous generators (PMSGs) have some unque propertes such as havng a constant flux lnkage establshed by the permanent magnets. ontrol strateges for hgh speed PMSG-nterfaced converters are also dfferent. Varyng machne speeds, whch gve rse to dfferent generaton frequences, may mpact the system stablty. Transmsson lnes are not neglgble n such mcrogrd structures and may nfluence the system dynamc and steadystate performance. Thus, ther effect on stablty s uncertan wthout systematc analyss. One cannot smply apply the results obtaned from a DFIG system to the targeted PMSGbased MEA EPS. Several publcatons have studed the stablty of the EPS n the MEA usng small-sgnal analyss []-[4]. In [3], small sgnal stablty analyss of a hybrd EPS for the MEA s presented. The effect of control
{ GEN A/D cable 7 V D Bus apactor Bank D/A Ordnary Load Fg.. onfguraton of the studed system n the MEA. Motor Tghtly-controlled motor drve (PL) bandwdth, crcut component parameters and operatng condtons on stablty s dscussed usng the egenvalue loc. However, the egenvalues need to be calculated repettvely wth varyng parameters. So far, there are few publshed reports dealng wth comprehensve modal analyss usng egenvalue senstvty and partcpaton factor for the MEA EPS, especally for the potental D EPS. A systematc analyss and desgn framework are requred to ensure EPS stablty, desred performance and robustness. The man contrbutons of ths paper can be summarzed as follows: () A nonlnear mathematcal model and the correspondng lnearzed model for a generalzed PMSG-based MEA EPS are developed. () Analyss of the mpact on EPS stablty of the key parameters (machne parameters, cable lengths, fluxweakenng and D current controllers, as well as operatng parameters such as the generator speed and the load power) has been carred out through nvestgaton nto the system egenvalues senstvty and correspondng partcpaton factors. The paper s organzed as follows. Secton II defnes the EPS confguraton and develops the correspondng mathematcal model. In Secton III, based on the developed model, analyss s performed whch ncludes computaton of egenvalue senstvtes wth respect to dfferent system parameters. In order to mmc practcal operaton, the D cable mpedance s taken nto account n ths secton. Secton IV reassesses the nfluence of parameters uncertanty by gnorng the cables. omparatve egenvalue senstvtes studes wth regard to some key EPS parameters are presented. Nonlnear model tme-doman smulatons are performed n Secton V n support of the major fndngs of ths study. Secton VI draws together the conclusons of the paper. II. SYSTEM ONFIGURATION AND MATHEMATIAL MODEL The representatve D-dstrbuton MEA EPS archtecture ncludes two man PMSGs takng power from the man engne and another PMSG drven by an auxlary power unt (APU) [6]. Dependng on the flght scenaro and EPS operatonal regme, the system can be operated wth only one actve source, or wth multple sources feedng the same D bus. A mult-generator EPS s a very complex task for egenvalue senstvty analyss and not possble to nclude n a sngle paper. Hence, a sngle PMSG-based generaton EPS s nvestgated n ths paper. The generalzed EPS representaton utlzed n ths paper s shown n Fg.. The PMSG provdes power through an actve front-end (AFE) rectfer to the man bus and loads. The on-board loads are represented by conventonal resstve loads and by constant power loads Imax θ V dc I o -Imax 3 ω m V max SM PMM ω e p/ φ m I o Ld,Lq Flux Weakenng ontroller (PL) assocated wth tghtly controlled power-electronc converters. The capactor bank s nstalled on the man 7 V bus. The PMSG-controlled AFE utlzes the engne mechancal power and converts t to the electrc power to supply the EPS D lnk. lasscal vector control s used for the generator and converter to acheve decoupled control of actve and reactve power [7], [8]. Fg. llustrates the control scheme of the system consstng of an nner loop for current control n the dq reference frame, a flux weakenng controller for hgh speed operaton of the PMSG and a D current controller for actve power regulaton. To ensure approprate current sharng between parallel modules, a droop characterstc that operates on ndvdual modules s used. The control desgn process s dscussed n detal n [5]. A full-order model based on ths structure s developed below. Droop control s employed n lne wth the I-V characterstc shown n Fg. 3. The slope of the curve n the lnear secton s k D and the reference current (I o ) s calculated correspondng to the bus voltage (nput). In steady state, one can obtan the D current reference as follows, I ( V V ) k () o I abc abc dq where V dc s an actual D-lnk voltage and V o s the nomnal voltage for the man bus (7 V n ths study). A comprehensve model for the representatve droop-controlled PMSG-based EPS s developed below. A. PMSG and onverter Model PI PI I d ref ref Iq max Imax I d urrent Sharng ontroller I q The synchronously rotatng reference frame has been wdely used to model the PMSG [7], [8]. The converter s also o V PI PI m abc Modulaton dc abc dq D V d ref V q ref Fg.. ontrol scheme utlzed n the studed system [5]. I O I max -I max Fg. 3. Droop characterstc. k D Vo V dc. I dc I o 7 V D Bus V dc
3 modelled based on ths frame n ths paper. The dynamc equatons for PMSG n the dq frame are as follows [9]: dd ( vd Rsd elqq ) dt Ld dq ( vq Rsq eldd em ) dt Lq where v d,v q : d-axs, q-axs component of stator voltage; d, q : d-axs, q-axs component of stator current; L d,l q : d-axs, q- axs nductance; R s : stator resstance; φ m : flux lnkage of permanent magnet; ω e : electrcal rotor angular velocty. In ths study, a surface-mounted PMSG s used, thus the machne nductances n the d-axs and q-axs are dentcal (L d =L q = L S ) [3]. The machne speed and torque (.e. q-axs current) wll defne the d-axs current n order to lmt the machne voltage by approprate defluxng. The load power has a sgnfcant smultaneous effect on both the q-axs and the d- axs current whch can be seen from (3): vc vd vq 3 dc ( ddd dqq ) 4 v v d q dd, dq.5vdc.5v where d d, d q are the modulaton ndexes n correspondng drectons; v c s the ac sde voltage of the converter; dc s the output D current flowng nto the capactor. Maxmum allowable phase currents are determned by the nverter and machne ratngs. Maxmum voltage s dependent on the avalable D-lnk voltage and modulaton method. The voltage and current lmtatons can be wrtten as follows by neglectng stator resstance and the transent terms [3]. max d q c dc () (3) max e ( Lsq ) ( Ls d m) Vc (4) I I I where V c max and I c max are the maxmal phase voltage ampltude at the fundamental frequency and maxmal phase current, respectvely. Accordng to the control block dagram shown n Fg., t s possble to derve equatons (5) where X vc, X o, X d, X q are the states assocated wth the PI controllers for the flux weakenng, D current, and dq-axes current controllers, respectvely: X X X X max vc c c o V o v o K X K v K v max d vc vc vcp c vcp c d K X K K q o o op o op o q (5) Here o s the load current (out of the bus capactor); o s the reference of the D current calculated by droop; K vc, K vcp, K o and K op are the ntegral gans and proportonal gans for the flux weakenng controller, and D current controller, respectvely (see Appendx A). B. D-Lnk Model The dynamcs on the D-lnk can be expressed as: dv dt dc ( dc o ) (6) If we gnore the mpedance of the D cable between the converter and the man D bus, v dc s equal to the man bus voltage (v b ). The nomnal voltage of the man bus s 7 V, but the acceptable steady state range s between 5 V and 8 V as defned n standard MIL-STD-74F [3]:. Load Model 5 V V b 8 V (7) As mentoned above, the load s represented by a combnaton of resstve load and constant power load: V P P P P (8) b L res cpl cpl Rres where P cpl and P res are the total power of the PL and resstve load, respectvely. Hence, one can derve: v P b cpl o (9) Rres vb From ()-(9), a set of state equatons to represent the entre system can be establshed as: X f ( X, U) () where X and U are the vectors wth respect to the state and nput varables whch are defned as X=[v dc, d, q, X vc, X o, X d, X q ] T. The nonlnear system equatons are lnearzed around the equlbrum pont to obtan a set of lnearzed dynamc equatons. They can be formulated n state-space as follows: ^ ^ X A X () The derved system matrx A s too large and awkward to present here. In many practcal cases the D cable mpedances cannot be neglected, especally n low voltage D power networks. The smplfed schematc for the system wth cable ncluded s depcted n Fg. 4. Snce the parastc capactance s much smaller than the bus capactance ( b ) and the local capactance ( ), the cablng s represented by seres R -L branch n ths secton. Two more state varables (the lne current Lc and bus voltage v b ) are added to the entre system; the correspondng equatons can be derved as follows:
4 G AR dc dlc ( vdc Rc dt Lc dvb ( Lc o ) dt b Lc v ) b () where v dc and v b are the voltage across the local capactor and bus capactor, respectvely. The whole system becomes 9 th -order and t can be formulated as n (3); ^ X A ^ X (3) where the entre system matrx A s shown n Appendx B. X =[v dc, v b, Lc, d, q, X vc, X o, X d, X q ] T. III. MODAL ANALYSIS-SINGLE MAHINE SYSTEM In practce, the system parameters such as generator nductance and resstance, control system parameters and operatng condtons (machne speed, load power, etc) vary durng operaton. Any such change affects the system egenvalues. In order to estmate the egenvalue change tendency wthout calculatng the egenvalues repettvely, a senstvty analyss should be undertaken. Ths secton deals wth a comprehensve modal analyss ncludng partcpaton factor and egenvalue senstvty. Egenvalues calculated from the state matrx may be real or complex. The complex egenvalues can be expressed n conjugate pars as follows: (4) j The dampng rato and oscllaton frequency correspondng to mode λ can be defned as:, fosc (5) Partcpaton factor analyss ads n the dentfcaton of how each state varable affects a gven mode. It s a measure of the relatve partcpaton of the k th state varable n th mode. It can be computed from the egenvector matrces as follows: p k V dc w k n k able v w k k v k (6) where w k, v k are the k th element n the left and rght egenvector correspondng to the th egenvalue. In general, L 7 V D Bus b Load Fg. 4. onfguraton of the studed system wth cablng ncluded. R L V b O the partcpaton degree of the k th varable n the th mode s measured by p k. As dscussed n [5], p k can also be regarded as the senstvty of λ wth respect to the k th dagonal element of state matrx [6]. Egenvalue senstvty s used to measure the rate and drecton of the mode movement due to varatons n system parameters. The egenvalue senstvty of a mode (λ ) wth respect to an uncertan parameter µ can be calculated as follows: T w ( A/ ) v w v T (7) where w and v are the left and rght egenvector correspondng to the egenvalue λ respectvely. The nomnal system parameters used n ths analyss are shown n Appendx A. Table I lsts the egenvalues of the studed system and the correspondng dampng rato and oscllaton frequency for each mode. The modes λ, are poorly damped and regarded as the domnant modes. Table II llustrates the partcpaton factor of the state varables. It s seen that the D state varables (v dc, Lc, v b ) exhbt substantal partcpatons n the domnant modes, whereas the other states partcpate n λ, weakly. able parameters wll have a drect nfluence on the D state varables. Ths explans why the cable mpedance has strong nfluence on the modes λ,. The effect of cable mpedance s assessed n ths secton together wth the other aforementoned parameters. A. Effect of able Parameters It s assumed that the cable length s 5 m. Nomnal values of cable resstance and nductance are defned as.6 mω/m and. µh/m respectvely. Table III lsts the egenvalue senstvtes wth respect to the cable nductance and cable resstance (calculated by (7)). The nd column n the table shows the standardzed unt for the egenvalue senstvtes calculaton correspondng to the varable n the st column. For nstance, the egenvalue senstvtes of modes λ, wth regard to a µh ncrement of L are 569 ± 644. Ths value means that an ncrease of nductance wll shft the real part of the egenvalue n a postve drecton. Therefore, cable nductance has a sgnfcant adverse mpact on stablty as the egenvalues λ, wll move towards to the rght half plane (RHP) at a fast rate wth ncreased L. In contrast, an ncrement of cable resstance s helpful for stablzng the system as better dampng s obtaned and the domnant egenvalues move further towards left. Fgs. 5 and 6 llustrate the movement of the crtcal egenvalues wth respect to varyng cable nductance and resstance. The domnant modes (λ, ) change sgnfcantly compared to the other modes. Increasng cable nductance (L ) results n the system beng less damped (the poles move towards the RHP). It can be clearly seen that the egenvalue plots n Fgs. 5 and 6 match the aforementoned analyss (Table III). B. Effect of Generator Parameters The key parameters of the PMSG consdered are the synchronous nductance (L S ), stator resstance (R S ) and flux lnkage of permanent magnet (φ m ). () Synchronous Inductance
5 TABLE I EIGENVALUES, DAMPING RATIO AND OSILLATION FREQUENY OF THE STUDIED SYSTEM WITH ABLES Mode Egenvalue Dampng rato Oscllaton frequency(hz) λ, -93.7±5355 8333 λ 3,4-549±779.58 9 λ 5,6-9±4838.5 77 λ 7,8-797±57.77 4.3 λ 9-87 TABLE II PARTIIPATION FATORS OF THE STUDIED SYSTEM WITH ABLES Mode vˆdc vˆb ˆ ˆLc ˆd ˆq X ˆ vc X ˆ o X d λ,.3.34.48.3. Xˆ q λ 3,4.7.9.7....5.4 λ 5,6.7.6..3.8.3.4.9 λ 7,8..5.7.4.8.3.. λ 9.8.5..33.5.8. TABLE III EIGENVALUE SENSITIVITIES FOR SINGLE GENERATOR SYSTEM WITH ABLES Varable Standardzed unt λ, λ 3,4 λ 5,6 λ 7,8 λ 9 L µh 569±644-6.9±3.75-7.4±.4.45±.4.55 R mω -56.4±7.93 5.±3.7.6±..±.6 -.46 L S µh -.99±.4 68.56±6.94.57±3. 4.3±.5 9.5 R S mω -7.55±.9.7 ±4.86-7.8±.3.47±4.64-5.4 φ m mvs/rad 6.±5.7 -±88.6 -.6±5.9-5.37±.974 9.9 K vcp -38.±88.6 84.54±447. -33.99±65.5 379.84±9. -64. K vc -.5±. -.3±.56 -.4±.9.±.4.4 K op.3±54.5 -.83±949.5 549.36±335.6-487.54±943.8 5637.96 K o -.8±..5±.3.4±..38±.56 -.45 b mf 5.44±8793-444.55±887.96-974.8±94.77 34.5±494. -44 P cpl W.±.6 -.±..±. -.±.. P res W -.3±.6 -.±..±.3 -.±.. ω e rad/s.±.6.6±. -.3±. -.±.7.8 k -44.3±5.36 8.±7.4 74.4±6.88-53.5±79.64 8.57 D As shown n Table III, the egenvalue senstvtes of modes λ, wth regard to a µh ncrement of L S are -.99 ±.4. In other words, an ncrease of nductance wll shft the real part of the egenvalue n a negatve drecton. Hence, the EPS wll be better damped and ts stablty margn wll ncrease. ()Stator Resstance From the results shown n Table III, t can be concluded that the effect of PMSG resstance s not sgnfcant. Ths s because the senstvtes of all egenvalues, wth respect to R S, are relatvely small. In other words, the egenvalues are almost dentcal for systems wth dfferng R S values. Ths s due to the fact that the stator resstance s qute small and as a result the voltage drop on the stator resstor s much smaller than the couplng terms n (). Thus, ts contrbuton to the operatng pont s nsgnfcant compared to the machne nductance. (3)Flux Lnkage of Permanent Magnet A small ncrement of flux lnkage wll sgnfcantly compromse the dampng snce the domnant modes (λ, ) wll move towards left half plane (LHP). It can be nferred from (4) that a more negatve d s needed to keep ac sde voltage (V c ) constant, n the hgh speed regon, f φ m s ncreased. It s obvous that φ m cannot be decreased arbtrarly for a manufactured machne; however ths result may be useful for generator and EPS desgners. Imagnary Axs (seconds - ) 5-5 x 4.5 p.u. p.u. p.u. 3 p.u. Pole-Zero Map -6-4 - Real Axs (seconds - ) Fg. 5. Egenvalue movement wth respect to cable nductance (L ) varaton. Imagnary Axs (seconds - ) 6 x 4 Pole-Zero Map.5 p.u. 4 - -4 p.u. p.u. p.u. -6-6 -4 - Real Axs (seconds - ) Fg. 6. Egenvalue movement wth respect to cable resstance (R ) varaton.
6-6 -4 - Fg. 7. Egenvalue loc wth respect to varyng synchronous nductance. Fg. 8. Egenvalue loc wth respect to varyng flux lnkage of permanent magnet. In summary, from the results n Table III, t can be concluded that the machne parametrc uncertantes, or parameter varatons, have a sgnfcant nfluence on EPS stablty (except for stator resstance). The system stablty margn s mproved wth ncreased machne nductance. On the other hand, a small decrease n the flux lnkage of the permanent magnet s helpful for stablzng the entre system. The prevous numerc egenvalue senstvty analyss s valdated by the pole movement plots above. Fgs. 7 and 8 show the egenvalue plots wth regard to dfferent values of synchronous nductance and permanent magnet flux lnkage respectvely. It can be seen n Fg. 7 that domnant modes move to the left wth ncreased L S. On the contrary, the crtcal egenvalues move nto the RHP f the flux lnkage of permanent magnet s ncreased. The egenvalue plots therefore match the prevously detaled analyss (Table III). Ths demonstrates the feasblty and effcacy of egenvalue senstvty analyss. The pole movement wth respect to stator resstance varaton s shown n Fg. 9. The movement rate s much slower compared to the pole movement shown n Fgs. 7 and 8. Agan, ths therefore agrees wth the values lsted n Table III.. Effect of ontrol Parameters Imagnary Axs (seconds - ) 6 x 4 Pole-Zero Map.9 p.u. 4 - -4-6 Imagnary Axs (seconds - ) 4 - -4 p.u.. p.u. Real Axs (seconds - ) 6 x 4 Pole-Zero Map.9 p.u. p.u.. p.u. -6-6 -4 - Real Axs (seconds - ) Due to the requred fast dynamc response of the entre system, the nner current loops ( d, q loops) are desgned wth a khz bandwdth and a.77 dampng factor. Thus, the analyss n ths paper s focused on flux-weakenng control and D current control. The correspondng control parameters are Imagnary Axs (seconds - ) -6-6 -4 - Imagnary Axs (seconds - ) 6 x 4 Pole-Zero Map.9 p.u. Fg. 9. Egenvalue loc wth respect to varyng stator resstances. Overvew. Zoomed area of domnant poles. gven n Appendx A. Table III dsplays the egenvalue senstvtes wth regard to the proportonal gan and ntegral gan of the feld weakenng controller (K vcp, K vc ) and the D current controller (K op, K o ). A small ncrement n the proportonal gan of the flux weakenng controller K vcp (wthn a lmted range) leads to λ, shftng to the left. However, the ncrease of K vcp s lmted by λ 3,4 movng towards rght semplane whch wll lead to nstablty. In the case of the D current controller, the domnant modes wll move towards RHP wth ncreased K op. Overall, t can be concluded that the proportonal gans of the flux weakenng and current sharng controllers have a sgnfcant mpact on system stablty. D. Effect of Output Bus apactance 4 - -4 4 - -4 p.u.. p.u. Real Axs (seconds - ) In lne wth practcal MEA EPS, the local capactor ( ) s fxed to mf. The egenvalue senstvty wth regard to bus capactance ( b ) can be seen from Table IV. rtcal egenvalues wll move to the left when b s ncreased from. mf to.4 mf and then move to the rght (towards the RHP) when b s more than.4 mf. In other words, beyond a certan bus capactance, an ncreased b wll decrease the dampng due to the domnant modes. Thus, the system stablty cannot be mproved by smply ncreasng b. As the egenvalue senstvtes, wth respect to mf or 3 mf b, are reduced compared to the case of mf b, t can be nferred that all the modes ncludng the crtcal modes wll gradually converge to a sngle pont n the s-plane. 6 x 4 Pole-Zero Map.9 p.u. p.u.. p.u. -94.6-94.4-94. -94-93.8 Real Axs (seconds - )
7 TABLE IV FIRST-ORDER EIGENVALUE SENSITIVITIES WITH DIFFERENT BUS APAITANES b λ, λ 3,4 λ 5,6 λ 7,8 λ 9.mF -5358±645-374±58-545±4 585±63.8-39..3mF -736.5±8653-3±468-5±94.5 574±38.6-96.7.4mF 657.7±5434-83±6-47±498.3 488±53-7.8 mf 48±39-567±69-564.8±38.9 688±94.5-35.5 mf 548.5±355-84.8±95.6-5.±98 4.8±5.4-5.7 3mF 99.7±443-447.9±596.4±85.7.±37.6-5.4 E. Effect of Operatng ondtons Load power and generator speed are the man concern when consderng EPS operatng condtons. It can be seen n Table III that hgher speed operaton deterorates the system stablty as λ, and λ 3,4 move towards the RHP wth ncrease of machne speed. The less damped modes, λ,, wll also approach the RHP f load power s ncreased. Ths s therefore a good match wth the well-known PL property that negatve ncremental mpedance compromses system stablty [33], [34]. IV. MODAL ANALYSIS WITHOUT ABLES In many cases, cables are neglected n the modellng process n order to smplfy the analytcal result. However, ths secton wll show that ths may lead to naccurate, and sometmes even ncorrect, outcomes and conclusons. Here the system model wthout cablng s reassessed usng the same procedure detaled prevously. Based on the matrx n (3), egenvalues, the relevant dampng ratos and the oscllaton frequences for ths case are shown n Table V. In addton, Table VI llustrates the new partcpaton factors. It can be seen that modes λ, are the least damped modes (see Table V) and they are manly assocated wth the D lnk voltage and q- axs current. λ 3,4 are more assocated wth the d-axs current and λ 7 s manly related to flux-weakenng controller and D current controller. Table VII lsts the egenvalues senstvtes for the system wthout cables. It can be observed that the nfluences of some parameters on stablty are the opposte of those found prevously wth cablng ncluded, as wll be dscussed below. A. Effect of Generator Parameters In contrast to the system wth cables ncluded, the real part of the egenvalue senstvty of mode λ,, wth regard to the machne nductance (L S ), s now postve. Hence the system stablty degrades wth an ncrease n stator nductance. As for the flux lnkage, an ncrement of φ m wll now shft the egenvalues to the left. Ths s agan opposte to the concluson drawn from Table III. B. Effect of ontrol Parameters It shows n Table VII that the crtcal modes λ, wll move dramatcally to the magnary axs wth a small ncrement of K vcp. In other words, a small ncrement n proportonal gan for flux-weakenng controller wll lead to nstablty. Ths s opposte to the cabled case n whch a small K vcp can stablze the system (shown n Table III).. Effect of Operatng ondtons As the machne speed manly affects the d-axs current when the generator s operatng n the flux-weakenng regon, λ 3,4 are the domnant modes related to the machne speed. It can be seen that the partcpaton factor of d n modes λ 3,4 s the largest one among all the modes. Although ncreased machne speed renders less dampng for λ 3,4, the dampng of the crtcal modes λ, s ncreased. Overall, t can be found that the ncrement of the speed s benefcal for renforcng the dampng of the entre system. Agan ths s opposte to the prevous fndngs n Table III. V. SIMULATION STUDIES As dscussed n [], functonal models are well suted to tasks such as the nvestgaton of EPS stablty. Therefore, n support of the analyss n the prevous sectons, tme doman smulatons are presented n ths secton whch utlze nonswtchng functonal models of the EPS shown n Fg.. The smulaton scenaro assumes that the load power (PL) ncreases step-wse every. s as detaled n Appendx. Smulaton studes have been carred out for sngle generator systems wth cables and wthout cables. A. Sngle Generator System wthout ables Frst, smulatons are carred out to test the system wthout cablng ncluded. Results wth respect to the effect of the stator nductance (L S ) and permanent magnet flux lnkage (φ m ) on stablty are shown n Fgs. and. L S s 99 µh p.u. and φ m s.3644 Vs/rad p.u. (See Appendx A). It can be seen from Fg. that f L S s ncreased to. p.u. nstablty occurs when the load power exceeds 4 kw. However, under the same test, the system s stable wth L S set at.9 p.u. Fg. shows the results for varaton of the permanent magnet flux lnkage. It can be seen that at.9 p.u. the system becomes unstable under hgh load power (3 kw) whlst at. p.u. the system remans stable. Fg. and show the dynamc responses obtaned when varyng the proportonal gans of the flux-weakenng and D current controllers. As expected from the prevously detaled analytcal analyss, t shows that the EPS s very senstve to varatons n the proportonal gan of the fluxweakenng controller, K vcp. After t =. s, nstablty s observed f K vcp s ncreased from to 3. From Fg. t can be seen that the system becomes unstable when K op s changed from.4 to.8 at t =.4 s.
8 TABLE V EIGENVALUES OF THE STUDIED SYSTEM WITHOUT ABLES Dampng Oscllaton frequency Mode Egenvalue rato (Hz) λ, -765±836.6 74 λ 3,4-83±634.4 96 λ 5,6-334±66.98 5 λ 7-556 TABLE VI PARTIIPATION FATORS OF THE STUDIED SYSTEM WITHOUT ABLES Mode v ˆdc ˆd ˆq X ˆ ˆ vc X ˆ o X ˆ d X q λ,.7.9.3..4.5.3 λ 3,4.5.38.6.8..4.8 λ 5,6.7..7.3.8.7.5 λ 7.8...7. TABLE VII FIRST-ORDER EIGENVALUE SENSITIVITIES FOR SINGLE GENERATOR SYSTEM WITHOUT ABLES Varable Standardzed unt λ, λ 3,4 λ 5,6 λ 7 L S µh 7.4±64.85-6.84±.66±4.66.6 R S mω 4.73±9.85 -.±.4.65±5.5-3.67 φ m mvs/rad -33.68±.73 7.3±67.7 -.67±9..84 K vcp 7.87±85.7-78.46±4. 988.34±543.4-79. K vc.7±.58 -.46±..±.5. K op 576.5±6544.3-577.36±3. -97.7±48.3 5537 K o.4±.67.3±.8 -.7±.7 -.9 mf -3.4±376. -3589.3±77.45 75±543.7-658 P cpl W.±.4 -.±.3.±. -. P res W.±. -.±.3 -.±.. ω e rad/s -.7±.75.8±.9 -.±.8 -.3 k 35±66-89.4±. -4.4±39.8 D..4.6.8....4.6.8.. Vdc(V) 5 Vdc(V) 5 Vdc(V)..4.6.8....4.6.8.. 5..4.6.8.. Fg.. Smulaton results for dfferent machne nductances. L S =.9 p.u. L S =. p.u. Vdc(V)..4.6.8....4.6.8.. 5..4.6.8.. Fg.. Smulaton results for dfferent permanent magnet flux lnkages. φ m =.9 p.u. φ m =. p.u.
9 Vdc(V) 4.5..5.3.35.4 5 Kvcp= Kvcp=3.5..5.3.35.4 Kop=.4 Kop=.8.36.38.4.4.44.46 Id (A) Iq(A) Speed (krpm) 3 - - - - -4 5krpm 3krpm krpm 5krpm krpm.6.8...4.6.8...4.6.8...4 Vdc(V) 8 6 4.36.38.4.4.44.46 Fg.. Smulaton results for control parameter varaton. K vcp s changed from to 3. K op s changed from.4 to.8. Fg. 3 shows the effect of generaton speed on stablty. In ths case.5 mf s used for the local capactor ( ). Fg. 3 shows the machne speed and the stator current n both d and q axes. Fg. 3 presents the D lnk current and voltage waveforms. Onset of nstablty can be observed n Fg. 3 at low generaton speed (< 5 krpm), whch confrms the fndngs n the prevously detaled analytcal analyss (n Secton IV). B. Sngle Generator System wth ables Smulatons were also performed usng the sngle generator system shown n Fg. 4 wth cablng ncluded. Fgs. 4 and 5 show the effect on the system wth cablng ncluded when generator parameters are vared. The local capactor (at the output of the rectfer) s set to mf. A small bus capactance (.3 mf) s used wthn the system model n order to demonstrate the effect of capactance on system behavor. ompared to the results n Fg., t can be seen n Fg. 4 that n ths case, wth L S set at. p.u, the system does not exhbt nstablty. On the contrary, the system wth.9 p.u. L S exhbts nstablty under hgh loads. Ths s opposte to the case wthout cablng ncluded, as expected from the prevous analytcal analyss. Fg. 5 shows that the system wth.9 p.u. φ m now remans stable under a hgh power load. However, the system wth. p.u. φ m shows oscllatons when the load s approachng full power after t =.8 s. Agan, as expected, ths s opposte to the case wthout cablng ncluded. Fg. 6 shows the results for generator speed varaton from krpm to 3 krpm. It can be seen that n the hgh speed regon the whole system destablzes. Ths agan confrms the fndngs presented n Table III. Fg. 7 confrms that the system can be stablzed at hgh speed by a bus capactance b of.5 mf. Ths emphaszes that Vdc(V) Vdc(V) Vdc(V)..4.6.8.. 5 5..4.6.8.. Tme (s)..4.6.8.. 5 5 5.6.8...4.6.8...4 5..4.6.8.. Tme (s) Fg. 4. Smulaton results for dfferent machne nductances. L S =.9 p.u. L S =. p.u. Fg. 3. Smulaton results for varyng machne speeds ( b =.5 mf). Machne speed and stator current n dq axes. D current and voltage. an approprate bus capactance should be selected n order to renforce the dampng of the entre system. Ths s consstent wth the fndngs n Table IV.
Vdc(V) Vdc(V) 5..4.6.8.. Tme (s) Fg. 5. Smulaton results for dfferent permanent magnet flux lnkages. φ m =.9 p.u. φ m =. p.u. Speed (krpm) Vdc(V)..4.6.8.. 5 5..4.6.8.. Tme (s)..4.6.8.. 5 Fg. 6. Smulaton results for varyng machne speeds ( b =.3 mf). Speed (krpm) Vdc(V) 3 5 5 krpm 5krpm 5krpm 3krpm 3krpm.6.8...4.6.8...4.6.8...4 3 krpm 5krpm 5krpm 3krpm 3krpm 5 5.6.8...4.6.8...4.6.8...4 Fg. 7. Smulaton results for varyng machne speeds ( b =.5 mf). has been performed. Analyss of partcpaton factors, dampng ratos and oscllaton frequences helps to understand the system characterstc. Wthn the analyss cable mpedance was taken nto consderaton and the correspondng modal analyss shows that some parameters have opposte mpacts on stablty when cablng s gnored. The mpact of parameter varaton on system stablty was llustrated usng egenvalue senstvty. The appled method sgnfcantly reduces the computaton burden as repettve calculaton of egenvalues s not needed. By utlzaton of ths technque, the effect of parameter varaton on EPS behavour, tendences and drecton of modal shft can be presented n a convenent and llustratve form. Generator parameters, control parameters and operatng condtons were vared to demonstrate ther effects on small sgnal stablty. Tme-doman smulatons usng a non-lnear functonal model of the example EPS have been carred out to verfy analytcal results. It can be concluded that n order to obtan accurate system characterstcs, cable nfluence cannot be neglected. Overall, the comprehensve analyss conducted n ths paper offers nsghtful gudance for parameter optmzaton and system stablty predcton n practcal applcatons. APPENDIX A THE EPS PARAMETERS ategory Parameter Symbol Value PMSG Machne resstance R S.58 mω ( p.u.) Machne nductance L S 99 µh ( p.u.) Permanent magnet flux lnkage m.3644 Vs/rad ( p.u.) Number of poles p 6 Nomnal power P N 45 kw able Local capactor mf able resstor R 3 mω (.6 mω/m) able nductor L µh (. µh /m) Man bus Bus apactor b.5 mf Droop characterstc Stator current loop Flux weakenng control D current control Nomnal voltage V o 7 V Droop slope k D 8.5 Proportonal gan K dp, K qp.8785 Integral gan K d, K q 398 Proportonal gan K vcp Integral gan K vc 5 Proportonal gan K op.4 Integral gan K o 6 APPENDIX B STATE MATRIX A OF SINGLE GENERATOR SYSTEM VI. ONLUSION Ths paper employed a D EPS as a research objectve whch s a promsng confguraton for future MEA. A comprehensve modal analyss of a PMSG based MEA EPS
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