AU Journl of Electricl Engineering AU J. Elec. Eng., 49()(27)3-38 DOI:.226/eej.26.86 Dynmic Hrmonic Modeling nd Anlysis of VSC-HVDC Systems E. Krmi*, M. Mdrigl2, G. B. Ghrehpetin 2 Deprtment of Electricl Engineering, Fculty of Electricl Engineering, Amirkbir University of echnology, ehrn, Irn Deprtment of Electricl Engineering, Instituto ecnológico de Moreli, Michocán, Mexico ABSAC: Hrmonic nlysis hs become n importnt issue in modern power systems. he widespred penetrtion of non-liner lods into the emergence of power systems hs turned power qulity nlysis into n importnt opertion issue under both stedy stte nd trnsient conditions. his pper employs Dynmic Hrmonic Domin (DHD) bsed frmework for the dynmic hrmonic nlysis of VSC-HVDC systems. hese systems re widely used in modern power systems in both distribution nd trnsmission levels in order to provide voltge profile improvement, power flow control, nd power loss reduction. In this pper, pproprite modeling of VSC-HVDC systems for hrmonic propgtion is performed by mens of switching function which provides connection between DC nd AC sides. Also in this pper, dynmics relted to DC side cpcitor re tken into ccount which cn gretly ffect the trnsient response. In order to vlidte the results, the proposed method hs been successfully tested on test system nd the obtined results re compred to those of time-domin softwre, followed by discussion on results. - Introduction Hrmonic nlysis nd power qulity ssessment hve become importnt issues in modern power systems []. Nowdys, power systems include mny power electronic devices, such s Voltge Source Converter (VSC) which is the min building block of Flexible AC rnsmission System (FACS) devices tht re widely used in modern power system in order to improve voltge profile, trnsient stbility, lodbility nd power flow control [2]. VSC uses multilevel rrys nd/or Pulse Width Modultion (PWM) to control power, voltge nd current [3]. However, these devices re one of the min hrmonic sources in the modern power systems due to their switching nture [4]. Hrmonic nlysis cn be conducted in both time nd frequency domins bsed on the vilble tools. Hrmonic nlysis in the time-domin would need n dditionl processing procedure, such s Windowed Fst Fourier rnsform (WFF) which llows the clcultion of the hrmonic content by sliding Fst Fourier rnsform (FF) window [5]. It is worth noting tht numericl errors, such s Gibbs oscilltion nd the picketfence effect re considerble especilly for fst trnsient studies [6]. A gret pproch for stedy-stte hrmonic nlysis is the hrmonic domin (HD) which models the coupling of hrmonics in the nonliner systems very ccurtely [7]. his methodology hs been effectively pplied to power electronic systems nd FACS devices [8-8]. It should be noted tht n lterntive hybrid time-frequency domin method in order to compute the stedy-stte response of n electricl system is presented in [9]. he HD pproch hs been further extended to include the dynmic nlysis of hrmonics during trnsient sttes. he eview History: eceived: 22 October 25 evised: 28 September 26 Accepted: 29 October 26 Avilble Online: 2 November 26 Keywords: VSC-HVDC Dynmic Hrmonic Domin Power Qulity Stedy Stte esponse Switching Function method is clled Extended Hrmonic Domin (EHD) or Dynmic Hrmonic Domin (DHD). As shown in [2], the DHD is powerful method which contributes to the ccurte ssessment of power qulity. his pproch provides the clcultion of hrmonic content step-by-step. One of the min slient fetures of this method is its strightforwrd initiliztion in comprison with the time domin. Combining this method with compnion circuit modeling leds to powerful nlyticl technique clled dynmic compnion circuit modeling [3]. he DHD hs been used in order to exct hrmonic nlysis of FACS devices, synchronous mchines, trnsmission lines nd trnsformers [3], [2-26]. In [26], physicl mening of trnsient hrmonics is put forwrd by using DHD methodology. A novel pproch for stedy nd dynmic sttes hrmonic nlysis of power systems is presented in [27]. It employs decomposition frmework so tht hrmonic producing devices re considered s seprte subsystems which re solved vi the extended hrmonic domin technique. Appliction of DHD pproch for investigting the effect of the source phse ngle on hrmonic content nd time domin response during both trnsient nd stedy sttes is presented in [28]. It is shown tht shifting ll the sources does not ffect the hrmonics mgnitude nd only hrmonics phse ngles re linerly shifted ccording to their hrmonic order. In this pper, the DHD methodology long with switch-ing function concept is used for the dynmic hrmonic nlysis of VSC-HVDC systems by obtining stte-spce model. Also, generl procedure for the clcultion of switching function is proposed which cn be implemented without ny complexity nd tkes into ccount different switching types. he proposed method hs been successfully tested on test system nd the obtined results re compred to those of time-domin softwre, followed by discussion on results. Corresponding uthor; Emil: ehsnkrmi@ut.c.ir 3
E. Krmi et l., AU J. Elec. Eng., 49()(27)3-38, DOI:.226/eej.26.86 2- Dynmic Hrmonic Domin he min ide behind DHD is tht periodic function x(t) cn be pproximted by time dependent Fourier series s shown in Eq. (): x(t) he complex Fourier coefficient X n (t) is time vrying. By considering only first h hrmonics, Eq. () cn be rewritten in mtrix form s follows: x(t) = G (t)x(t) (2) where jnωt = X n(t)e () n= jhωt e X h(t) ω j t e X (t) G( τ ) =,X(t) = X (t) jωt e X (t) jhωt e X h (t) Considering the following stte-spce eqution: x(t) = (t)x(t) + b(t)u(t) where, x(t), (t), b(t) nd u(t) re periodic functions. Using the generl form of (2) in (3) it is not difficult to show tht: x(t) = G ( τ)d(jh ω )X(t) + G (t)x(t) (4) where, D(jhω ) is given by: D ( ) jh j h Also (t)x(t) = G (t)ax( τ) ω = ω In Eq. (5), A hs oeplitz structure given by: A A A h A A A A = Ah A A A A h A A A Ah A A h (3) (5) where, the entries in A re the hrmonic coefficients of (t). In the sme mnner, we cn obtin tht: b(t)u(t) = G (t)bu (6) Using (4), (5) nd (6) in (3): G ( τ)d(jh ω )X(t) + G ( τ )X(t) = G (t)ax(t) + G (t)bu Dropping G (τ) from both sides of (7) nd solving for the stte vrible results in: X(t) = {A D(jh ω )}X(t) + BU (8) Eq. (8) is the trnsformtion of (3) into the DHD, where the stte vrible in (3) is x(t) nd in (8) is the hrmonics of x(t). By compring (3) nd (8), one cn observe tht DHD trnsforms liner time periodic (P) system to liner time invrint (I) system. A prticulr cse of (8) is the stedy stte which is reduced to set of lgebric equtions. Solving Eq. (8) needs numericl integrtion method. In this pper, ode45 solver in Mtlb with time step of -5 s is used. 2- - Stedy Stte esponse Stedy stte response cn be esily reched by DHD method. In the stedy stte condition, derivtive of the stte vribles in Eq. (8) is equl to zero which mens =(A D)X+BU. his eqution cn be rewritten s follows: X = {D( jh ω ) A} BU (9) Eq. (9) shows one of the min slient fetures of DHD method ginst time domin simultions. In the stedy stte ll the bove mtrices re constnt nd therefore the solution for stte vribles is strightforwrd. However, in the presence of non-liner components, n itertive process is required. It should be noted tht, Eq. (9) cn be used s initil condition for Eq. (8) for dynmic nlysis purposes. his eqution is used in following sections. 3- Voltge Source Converter Stte Spce Model he bsic VSC scheme is shown in Fig.. he opertion of VSC bsed devices depends on the conduction of semicon ductor vlves such s IGBs, nd it interconnects the AC system with the DC system. his conduction intervl of the vlves cn be described by mens of switching func tion. In this study, the switching function is used in order to provide meningful connection between AC nd DC sides. he voltge equtions tht relte the AC side with the DC side re given by: ν = sνdc, ν b = sbνdc, ν C = scνdc () where, s, s b nd s c re switching functions tht represent the opertion of VSC. If the VSC is ssumed to be lossless, instntneous power t AC side is equl to instntneous power t DC side s shown in Eq. (): i ν + i ν + i ν = i ν b b c c DC DC If the converter is not idel, losses cn be modeled through dding resistnce to the DC side [22]. Moreover, switching losses of the converter cn be included by dding current dependent resistnce to the DC side s described in [3]. Employing () in () yields n expression for i DC. (7) () 32
E. Krmi et l., AU J. Elec. Eng., 49()(27)3-38, DOI:.226/eej.26.86 v vb vc i ib ic Fig.. VSC connected to AC system si + si b b + si c c = idc (2) Keeping in mind tht DC side current my be written in terms of the dynmic eqution of the cpcitor. i DC dν = C dt DC (3) he voltge drop cross the three phse impednce of VSC circuit in Fig. is s follows: di ν ν = i + dt dib νb ν b = i b + dt dic νc ν c = i c + dt (4) Combining equtions ()-(4) yields the stte spce model for VSC s follows: di s dt di b s i v b dt i b v b (5) = + di c sc i c vc dt v dc s sb dv dc sc C C dt C his eqution cn be esily trnsformed into the DHD. According to Eq. (8), Eq. (5) in the DHD is given by: UI D S I I V U D S I I b b = Ib Vb + I (6) I c c Vc UI D Sc V dc Vdc S S b C C Sc D C where U I is identity mtrix nd is mtrix of zeroes. It should be noted tht both U I nd hve dimension of (2h+) (2h+). hese equtions re the bsics for modeling of VSC-HVDC systems. VSC-HVDC system is modeled by following the sme procedure nd considering tht DC side current nd voltge re ffected by two VSCs. v vb vc idc C + vdc - 3- - DC Side Voltge Control In the VSC, it is desired tht the voltge in the DC side be constnt ll the time. Hence, PI controller s shown in Fig. 2 cn be employed to mintin DC side voltge t the constnt vlue by pproprite shifting of switching function []. 4- Switching Function here re different switching strtegies for power electronic converters, such s hrmonic elimintion nd PWM. In some cses, there is strightforwrd mnner to clculte switching function ccording to Fourier coefficients. For instnce, in hrmonic elimintion technique switching pulses re pplied so tht predetermined hrmonics re eliminted t the AC side. It cn be shown tht in order to eliminte n hrmonics t the AC side, n pulses re needed to be pplied to semiconductor vlves which re obtined through n itertive process []. For exmple, in order to eliminte the fifth, seventh, th, 3th nd 7th hrmonics, the clculted switching ngles re.35, 7.27, 23.8, 34.88 nd 37.27, respectively. he hrmonic content of switching function is shown in Fig. 3. According to this figure, ssocited hrmonics re effectively eliminted t the AC side. Furthermore, this figure emphsizes on the significnt mgnitude of higher order hrmonics. In some cses, extr mthemticl computtions re required in order to obtin the switching function. For instnce, PWM switching is bsed on the comprison of two signls in which different prmeters like modul tion index nd switching frequency ffect the output. In order to overcome the mentioned complexities with both clcultion nd computer implementtion, generl pro cedure s shown in Fig. 4 is proposed to clculte switching function of every switching strtegy. 5- Simultion esults In order to ccess the effectiveness nd precision of DHD pproch in nlyzing VSC-HVDC, test system shown in Fig. 5 is used. Series resistnce nd inductnce re.5ω nd.27mh, respectively. DC side cpcitor hs vlue of 495μF. Both sending nd receiving end voltges re blnced with the mgnitude of p.u. nd phse shift of 2 between phses. Hrmonic elimintion technique is employed in both converters in order to eliminte fifth, seventh, th, 3 th nd 7 th t the AC sides nd ssocited switching functions re obtined by following the proposed method in Fig. 3. In order to control the power flow switching shift in sending nd receiving ends re set to nd, respectively. It should be noted tht ll results re shown in p.u. 5- - Stedy Stte esponse As mentioned in the previous sections, one of the min dvntges of DHD method is the strightforwrd solution for stedy stte response by solving Eq. (8). Stedy stte hrmonic content nd time domin responses for receiving end currents re shown in Figs 6 nd 7, respectively. Also, DC side voltge is shown in Fig. 8. By nlyzing the wveforms, it cn be seen tht the stedy stte initil condition ws exct since no trnsient t the beginning of the simultion ws identified nd simultion strts from stedy stte. 33
E. Krmi et l., AU J. Elec. Eng., 49()(27)3-38, DOI:.226/eej.26.86 δ VDC-ref + - ε K/(+S) Δδ + δ VDC-mesured Fig. 2. PI controller block digrm Fig. 6. Hrmonic content of receiving end currents Fig. 3. Hrmonic content of switching function using hrmonic elimintion technique ed the Input Dt: Switching Method nd Associted Prmeters Clcultion of Switching Function in ime Domin Fig. 7: ime domin response of receiving end currents Using FF to Extrct Hrmonic Content of Switching Function Clcultion of Switching Function Shift Yes Is Switching Function Shifted? No Clcultion of Switching Function in DHD Fig. 4. Proposed procedure to clculte switching function Fig. 8. Hrmonics nd ime domin response of sending end currents vr i2 v2 idc v i vs vrb ib2 vb2 C + vdc vb ib vsb vrc ic2 vc2 - vc ic vsc Fig. 5. Bck to Bck VSC-HVDC connected to two seprte AC systems 34
E. Krmi et l., AU J. Elec. Eng., 49()(27)3-38, DOI:.226/eej.26.86 Fig. 9. Hrmonic content of sending end current Fig. 3. ime domin response of sending end currents Fig.. ime domin response of sending end currents Fig. 4. Sending end ctive powers Fig.. Hrmonic content of receiving end current Fig. 5. eceiving end ctive powers Fig. 2. ime domin response of receiving end currents Fig. 6. Sending end rective powers 35
E. Krmi et l., AU J. Elec. Eng., 49()(27)3-38, DOI:.226/eej.26.86 Fig. 7. eceiving end rective powers Fig. 8. ime domin response of receiving end current of phse for different vlues of hrmonics under study Fig. 9. Comprison of time domin simultions nd DHD Since the system is completely blnced, otl Hrmonic Distortion (HD) of receiving end currents re the sme nd ll equl to 36%. HD for sending end currents is equl to 33.36%. It should be noted tht ccording to Eq. () nd keeping in mind tht DC side voltge cn be represented by Fourier series s Eq. () nd since switching functions nd DC side voltge contin only odd nd even hrmonics, respectively; the fifth hrmonic is not completely eliminted t AC sides. 5-2- Dynmic Anlysis In this study, it is ssumed tht disturbnce lsting ms tkes plce t time ms. During the disturbnce, the voltge v r is set to hlf of its prefult vlue. Hrmonic content of both sending nd receiving ends of phse nd c currents re depicted in Figs 9 nd, respectively. he time representtion of sending nd receiving ends currents re given in Figs nd 2, respectively. hese results show tht hrmonics rect very shrply to system disturbnces; they give very ccurte informtion of the instnt when the disturbnce strts nd finishes. DC side voltge hrmonic content long with its time domin response re depicted in Fig. 3. According to this figure, DC component strts to reduce while both second nd 4th hrmonics mgnitude increse. Figs 4-7 show ctive nd rective powers t both sending h nd receiving ends clculted by using P= VI nd, n n VI n = h Q = sin θ 2 respectively []. As these Figs imply, the vrition of phse prmeters (fulty phse) tkes plce very shrply. Since DHD method provides the exct clcultion of dynmic electricl indices, it cn be used in phsor bsed softwres in order to incorporte hrmonic nlysis during the trnsient period. In order to investigte the effects of higher order hrmonics (h vlue in Eq. ()) on the ccurcy of the results, receiving end current of phse is shown for 5 hrmonics under study in Fig. 8. his prmeter gretly ffect the simultion run time so tht simultion run time for 9 nd 5 hrmonics under study re 3.4s nd, 3.5s, respectively. According to the results of Fig. 8 nd the required simultion run time for different vlues of number of hrmonics under study, compromise between ccurcy nd required simultion time in order to use benefits of DHD method in hrmonic nlysis is essentil. Usully, in norml opertion condition like hrmonic power flow, bsed on the used switching chrcteristic in power electronic converter, nlyzing 25 hrmonics will be enough nd provides technicl requirements. However, for more ccurte purposes like resonnce nlysis, more hrmonics should be included. Fig. 9 compres the results given by time domin solution method nd the DHD (nine hrmonics re used). he ccurcy of the DHD solution increses when the number of hrmonic coefficients increses so tht both results cn be compred very well when 5 hrmonics re used. It should be noted tht the solution with reduced number of hrmonics shows to be good verging method. 6- Conclusion his pper described the dynmic hrmonic nlysis of VSC-HVDC systems by employing the Dynmic Hrmonic Domin method nd solving stte spce model. In the extended VSC-HVDC model, AC nd DC sides re connected through switching function. Moreover, generl procedure in order to clculte the switching function by using time domin nd FF is proposed. he proposed solution pproch for nlyzing VSC-HVDC systems is fully frequency dependent, which provides step-by-step procedure for following the hrmonics evolution with respect to the time. It should be mentioned tht ccording to the proposed representtion for VSC-HVDC systems, n equivlent impednce is obtined which depends on switching procedure nd electricl prmeters nd it could be used for resonnce nlysis. he proposed solution nd extended equtions were pplied to test system followed by the discussion on results. 36
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