Recent Researches in Pwer Systems and Systems Science The Mathematical Mdel f a Three-Phase Dide Rectifier with Multi-nverter Pwer Electrnic ads T. Spapirm, -N. Areerak, -. Areerak Pwer electrnics, Machines and ntrl (PeM) Research Grup Schl f Electrical Engineering, nstitute f Engineering, Suranaree University f Technlgy Nakhn Ratchasima, 0000 THAAND crrespnding authr: kngpan@sut.ac.th Abstract: - Dynamic mdels f pwer cnverters are nrmally time-varying because f their switching actins. Averaging methds are widely used t eliminate the switching behavir t achieve the time-invariant mdels. This paper presents hw t derive the mathematical mdel f a three-phase dide rectifier feeding bth resistive lad and cntrlled buck cnverters behaving as cnstant pwer lads. The DQ mdeling methd is used t analyze the dynamic mdel f a three-phase rectifier including the transmissin line n A side, while the generalized state-space averaging (GSSA) mdeling methd is applied t derive the dynamic mdel f a buck cnverter. ntensive time-dmain simulatins via the well-knwn sftware packages with the exact tplgy mdels are used t validate the prpsed mdels. The simulatin results shw that the prpsed mathematical mdels prvide high accuracies in bth transient and steady-state respnses. The reprted mdels are suitable fr the system stability analysis and design. ey-wrds: - Three-phase dide rectifier; ntrlled buck cnverter; DQ mdeling methd; Generalized statespace averaging methd; Mdeling and simulatin. ntrductin The pwer cnverter mdels are time-varying in nature due t their switching behavirs. t is very cmplicated t use the time-varying mdel fr the system analysis and design. Therefre, there are several appraches cmmnly used fr eliminating the switching actins t achieve a time-invariant mdel. Then, the classical linear cntrl thery can be easily applied t the mdel fr a system analysis and design. The first methd is the GSSA mdeling methd. This methd has been used t analyze many pwer cnverters in D distributin systems []-[], as well as uncntrlled and cntrlled rectifiers in single-phase A distributin systems [4],[5] and 6- and - pulse dide rectifiers in three phase systems [6]. The secnd widely used fr A system analysis is that f DQ-transfrmatin thery [7]-[9], in which pwer cnverters can be treated as transfrmers. The DQ mdeling methd can als be easily applied fr mdeling a pwer system cmprising vectrcntrlled cnverters where the GSSA and A mdels are nt easily applicable. The DQ mdels f three-phase A-D pwer systems have been reprted in the previus wrks fr stability studies f the pwer system including a cnstant pwer lad (P) [0]-[]. The DQ methd fr mdeling the three-phase uncntrlled and cntrlled rectifier has been reprted in [0] and [], respectively. Frm the literature reviews, this paper presents the cmbinatin between the DQ mdeling apprach and the GSSA mdeling methd t derive the mathematical mdel f a three-phase rectifier feeding bth resistive lad and paralleled buck cnverters in which it has nt been reprted in the previus publicatins. Accrding t the advantages f DQ and GSSA methds, the DQ methd is selected t analyze the three-phase dide rectifier including the transmissin line cmpnents n A side, while the GSSA methd is used t analyze the buck cnverters with their cntrls. The prpsed mdel derived frm bth DQ and GSSA methds is validated by the intensive time-dmain simulatin via the exact tplgy mdel. The results shw that the prpsed mathematical mdels prvide high accuracies in bth transient and steady-state respnses. n the future wrk, the reprted mdels will be used fr stability studies f the system due t the effect f a P. The paper is structured as fllws. n Sectin, the cnsidered system is described. Deriving the dynamic mdel f the cnsidered pwer system is fully explained in Sectin. The mdel in Sectin is a nnlinear mdel derived frm bth DQ and SBN: 978--6804-04- 00
Recent Researches in Pwer Systems and Systems Science GSSA methds called DQGSSA mdel. Therefre, the linearizatin technique using the first rder term f Taylr s series expansin including the steadystate value calculatin is fully explained in Sectin 4. n Sectin 5, the mdel validatin using the small-signal simulatin is illustrated. Finally, Sectin 6 cncludes and discusses the advantages f the DQ and GSSA mdeling methds t derive the mdel f the A-D pwer system with multicnverter pwer electrnic lads. nsidered Pwer System The cnsidered system is depicted in Fig.. t cnsists f a balanced three-phase vltage surce, transmissin line, three-phase dide rectifier, and D-link filters feeding a resistive lad (R ) and cntrlled buck cnverters. t is assumed that the dide rectifier and the buck cnverter are perated under a cntinuus cnductin mde (M) and the higher harmnics f the fundamental are neglected. Deriving the Dynamic Mdel n this paper, the DQ mdeling methd is firstly selected t derive the dynamic mdel f a threephase dide rectifier in which such rectifier can be treated as a transfrmer [0]. As a result, the uivalent circuit f the pwer system in Fig. can be represented in the DQ frame as depicted in Fig.. Nte that the uivalent circuit in Fig. is simplified by fixing the rtating frame n the phase f the dide rectifier switching functin (φ =φ). n Fig., the three-phase dide rectifier including the transmissin line n A side is transfrmed int the DQ frame via the DQ mdeling methd. The GSSA mdeling methd is then used t eliminate the switching actin f the cntrlled buck cnverter. The P cntrllers f the current lp (inner lp) and the vltage lp (uter lp) fr each buck cnverter are represented by pv, iv, pi, ii, pv, iv, pi, and ii respectively. Frm Fig., d and d can be derived and given in (). d =pi, pv, pi, iv, pi, v ii, i pv, pi, d = ii, i pv, pi, pi, pv, pi, iv, pi, v () The dynamic mdel f the system in Fig. using GSSA mdeling methd with () can be expressed as: Fig.. nsidered pwer system SBN: 978--6804-04- 0
Recent Researches in Pwer Systems and Systems Science : π Fig.. The uivalent circuit f the cnsidered pwer system n DQ frame R sd = sd ω sq bus, d m csλ R sq = ω sd sq bus, q m sinλ bus, d = sd ωbus, q π bus, q = ωbus, d sq rµ r r R r r c c pi, = bus, d π R rc pv, pi, rc iv, pi, rc ii, v i rc pv, pi, r c pi, r c pv, pi, rc iv, pi, rc ii, rc pv, pi, v i pi, pv, pi, iv, pi, = R ii, pv, pi, pi, pv, pi, i iv, pi, ii, pv, pi, v i, (,, pi pv pi ) iv, pi, = v ii, pv, pi, i O = R v = i = pv, iv, v pv, pi, ( pv, ) pi, iv, pi, = v ii, pv, pi, i O = R v = i = pv, iv, v pv, v () t can be seen in () that the state variables v, i, v, and i are als included. Eq.() is the nnlinear differential uatins. Therefre, () can be linearized using the first rder terms f the Taylr expansin s as t achieve a set f linear differential uatins arund an uilibrium pint. The details f the DQGSSA linearized mdel f () are given in Sectin 4 4 DQGSSA inearized Mdel and Steady- State alue alculatin As mentined in Sectin, () can be linearized using the first rder terms f the Taylr expansin s as t achieve a set f linear differential uatins arund an uilibrium pint. The DQGSSA linearized mdel f () is then f the frm in (). δ x= A(x,u ) δ x B(x,u ) δ u δ y= (x,u ) δ x D(x,u ) δ u where δ x= δ ds δ qs δbus d δbus q δ δ δ δ δ δ δ δ δ δ,, v i v i T = m T δu δ δ δ [ ] δy = δ δ δ ] () SBN: 978--6804-04- 0
Recent Researches in Pwer Systems and Systems Science R ω 0 0 0 0 0 0 0 0 0 0 0 R ω 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0 0 0 0 0 0 0 0 0 π 0 ω 0 0 0 0 0 0 0 0 0 0 0 rµ r rc R r r c c pv, pi, rc iv, pi, rc ii, rc pv, pi, rc iv, pi, rc ii, 0 0 0 a(5, 7) a(5,) π R pv, pi, iv, pi, ii, pv, pi, iv, pi, ii, 0 0 0 0 0 a(6, 7) a(6,) A(x,u ) = pi,,0 pv, pi,,0 iv, pi,,0 ii,,0 0 0 0 0 0 a(7,6) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 pv, iv 0 0 0 0 0, pi,,0 pv, pi,,0 iv, pi,,0 ii,,0 0 0 0 0 0 a(,6) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 pv iv 0,, 4 4 r r r r r r r r r r a(5,7) =, a(5,) = c pi,,0 c pv, pi,,0 c iv, pi, v,0 c ii, i,0 c pv, pi,,0 c pi,,0 c pv, pi,,0 c iv, pi, v,0 c ii, i,0 c pv, pi,,0 pi,,0 pv, pi,,0 iv, pi, v,0 ii, i,0 pv, pi,,0 pi,,0 pv, pi,,0 iv, pi, v,0 ii, i,0 pv, pi,,0 a(6, 7) =, a(6,) = pi,,0 pv pi,,0 iv, pi, v,0 ii, i,0 pv, pi,,0 pi,,0 pv pi,,0 iv, pi, v,0 ii,i,0 pv, pi,,0 i i a(7,6) =, a(7,) = cs( λ ) 0 0 sin( λ ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B(x,u ) 0 0 0 0 0 0 pv, 0 0 0 0 0 0 0 0 0 0 pv, pv, pi,,0 = 0 0 (x,u ) = D(x,u ) = pv, pi,,0 4 [ 0 0 0 0 0 0 0 0 0 0 0 ] 4 [ 0 0 0 ] Accrding t DQGSSA linearized mdel in (), the mdel needs t define,, λ,,,,,,,,, v,, v,, i, and i,. n the paper, the pwer uatin can be first applied t determine the steady state values at the A side, here are bus, and λ. Other steady-state values can be calculated by: where ω = ( ) r π π, =,, =,,, =,, = R R,,, =,, = iv, iv, i, =, i, = ii,, ii,,, bus,,, j 0 jλ, s e b u s e Z e jγ d c, = π ( ), ta n ω e q Z = R e q ω e q γ = R e q (4) 5 Small-Signal Simulatin The DQGSSA linearized mdel in () is simulated fr small-signal transients against a crrespnding exact tplgy mdel as shwn in Fig.. The set f system parameters is given in Table with the vltage lp cntrllers pv = pv = 0.05 and iv = SBN: 978--6804-04- 0
Recent Researches in Pwer Systems and Systems Science Fig.. The exact tplgy mdel (SimPwerSystem TM f SMUN) () 0 8 6 exact tplgy mdel DQGSSA linearized mdel () 4 0.4 0.6 0.8..4.6.8. 0 5 0 5 exact tplgy mdel DQGSSA linearized mdel () 0 0.4 0.6 0.8..4.6.8..4 0 exact tplgy mdel DQGSSA linearized mdel 0 0 0.4 0.6 0.8..4.6.8. time(sec.) Fig.4., and respnses f the system in Fig. t a step change f and iv = 50 ( ω n, vltage = 64 Hz, ζ v =.0), and the current lp cntrllers iv = iv = 0.778 and ii = ii = 040 ( ω n, current = 00 Hz, ζ i = 0.7). Table : Parameters f the Pwer System in Fig. Parameter alue s 0 rms/phase πx50 rad/s R 0. Ω 4 µh nf r 0.0 Ω r c 0.4 Ω (.5 A) 50 mh ( 50 ) 500 µf = ( 0.5 A) 4.68 mh = ( 50 m) 5 µf R = R 0 Ω Fig. 4 shws the, and respnses f the system in Fig. t a step change f frm 5 t 5 that ccurs at t = 0.6 sec. and frm 5 t 5 at t =.5 sec.. Frm the result in Fig. 4, an excellent agreement between bth mdels is achieved under the smallsignal simulatin. t cnfirms that the mathematical mdel f the pwer system with paralleled cntrlled buck cnverters and a resistive lad derived frm bth DQ and GSSA methds prvide a gd accuracy. The DQGSSA linearized mdel in the paper will be als used fr the stability analysis due t the effect f P in the future wrk. n additin, it is well knwn that simulatins f pwer electrnic system using sftware packages (such as MATAB, PSM, and etc.) via the exact tplgy mdels cnsume a huge simulatin time due t a switching behavir. t is nt easily applicable fr simulatin f cmplex systems. Hence, the averaging mdel f the pwer electrnic based system derived by the prpsed mdeling methd in the paper can be used t reduce the simulatin time. SBN: 978--6804-04- 04
Recent Researches in Pwer Systems and Systems Science 6 nclusin This paper presents hw t derive the dynamic mdel f the three-phase dide rectifier feeding multi-cnverter pwer electrnic lads with their cntrls. The DQ and GSSA mdeling methds are used t eliminate the switching behaviur f the pwer cnverter in which the DQ methd is used t analyze the three-phase rectifier and the GSSA methd is als applied t the buck cnverter. The prpsed mdels are suitable fr the system design and simulatin. Mrever, it is well knwn that when the pwer cnverters are regulated, they behave as a P. This P can significantly degrade pwer system stability margins. Therefre, the dynamic mdel f the pwer system is very imprtant. The DQGSSA linearized mdel in the paper can be als used fr stability studies in the future wrk. Acknwledgements This wrk was supprted by Suranaree University f Technlgy (SUT) and by the ffice f the Higher Educatin mmissin under NRU prject f Thailand. References: [] J. Mahdavi, A. Emadi, M.D. Bellar, and M. Ehsani, Analysis f Pwer Electrnic nverters Using the Generalized State-Space Averaging Apprach, EEE Trans. n ircuit and Systems., l. 44, August 997, pp.767-770. [] A. Emadi, Mdeling and Analysis f Multicnverter D Pwer Electrnic Systems Using the Generalized State-Space Averaging Methd, EEE Trans. n ndus. Elect., l. 5, n., June 004, pp. 66-668. [] A. Emadi, M. Ehsani, and J.M. Miller, ehicular Electric Pwer Systems: and, Sea, Air, and Space ehicles, (Marcel Dekker, nc, 004). [4] A. Emadi, Mdeling f Pwer Electrnic ads in A Distributin Systems Using the Genearlized State-Space Averaging Methd, EEE Trans. n ndus. Elect., l. 5, n. 5, Octber 004, pp. 99-000. [5] -H. ha, Dynamic Mdeling and Rbust ntrl f Multi-Mdule Parallel Sft- Switching-Mde Rectifiers, WSEA Transactins n Systems, l.8, 009, pp.659-67. [6]. Han, J. Wang, and D. Hwe, State-space average mdelling f 6- and -pulse dide rectifiers, The th Eurpean nf. n Pwer Elect. and Appl., September, 007, Aalbrg, Denmark. [7].T. Rim, D.Y. Hu, and G.H. h, Transfrmers as Equivalent ircuits fr Switches: General Prfs and D-Q Transfrmatin-Based Analyses, EEE Trans. n ndus. Appl., l. 6, n. 4, July/August 990, pp. 777-785. [8].T. Rim, N.S. hi, G.. h, and G.H. h, A mplete D and A Analysis f Three- Phase ntrlled-urrent PWM Rectifier Using ircuitd-q Transfrmatin, EEE Trans. n Pwer Electrnics, l. 9, n. 4, July 994, pp. 90-96. [9] S.B. Han, N.S. hi,.t. Rim, and G.H. h, Mdeling and Analysis f Static and Dynamic haracteristics fr Buck-Type Three-Phase PWM Rectifier by ircuit DQ Transfrmatin, EEE Trans. n Pwer Electrnics, l., n., March 998, pp.-6. [0] -N. Areerak, S.. Bzhk, G.M. Asher, and D.W.P. Thmas, Stability Analysis and Mdelling f A-D System with Mixed ad Using DQ-Transfrmatin Methd, EEE nternatinal Sympsium n ndustrial Electrnics (SE08), ambridge, U, 9 June- July 008, pp. 9-4. [] -N. Areerak, S.. Bzhk, G.M. Asher, and D.W.P. Thmas, DQ-Transfrmatin Apprach fr Mdelling and Stability Analysis f A-D Pwer System with ntrlled PWM Rectifier and nstant Pwer ads, th nternatinal Pwer Electrnics and Mtin ntrl nference (EPE-PEM 008), Pznan, Pland, - September 008. [] -N. Areerak, S. Bzhk, G. Asher,.de ill, A. Watsn, T. Wu, and D.W.P. Thmas, The Stability Analysis f A-D Systems including Actuatr Dynamics fr Aircraft Pwer Systems, th Eurpean nference n Pwer Electrnics and Applicatins (EPE 009), Barcelna, Spain, 8-0 September 009. []. haijarurnudmrung, -N. Areerak, and -. Areerak, Mdeling f Three-phase ntrlled Rectifier using a DQ methd, 00 nternatinal nference n Advances in Energy Engineering (AEE 00), Beijing, hina: June 9-0, 00, pp.56-59. SBN: 978--6804-04- 05