Modeling and Stability Analysis of AC-DC Power System with Controlled Rectifier and Constant Power Loads

Transcription

1 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak Mdeling and Stability Analysis f AC-DC Pwer System with Cntrlled Rectifier and Cnstant Pwer La K. CHAJARURNUDOMRUNG, K-N. AREERAK *, and K-L. AREERAK Schl f Electrical Engineering nstitute f Engineering, Suranaree University f Technlgy Nakhn Ratchasima, 3 THALAND *crrespnding authr: kngpan@sut.ac.th Abstract: -Pwer cnverters with their cntrls nrmally behave as a cnstant pwer lad. This lad can significantly degrade the system stability. Therefre, the system dynamic mdel is very imprtance because it can be used fr stability analysis. t is well knwn that pwer cnverter mdels are time-varying because f their switching behaviurs. This paper presents the DQ mdeling methd t eliminate the switching actin t achieve time-invariant mdel fr the stability analysis. The pwer system studied is the AC distributin system. The small-signal mdel f the pwer system is btained by using a linearizatin technique. The small-signal simulatins are used t validate the DQ linearized mdel. The reprted mdel is then used fr the stability analysis via the classical cntrl thery. The results shw that an excellent agreement between the mathematical mdel and the three-phase benchmark mdel is achieved. Key-Wr: - Cnstant pwer la, DQ methd, simulatin, mdeling, cntrlled rectifier, stability analysis ntrductin Electrical la based n pwer electrnic cnverters are widely used in many applicatins. Unfrtunately, pwer electrnic driven la ften behave as a cnstant pwer lad (CPL). The CPL can significantly degrade the system stability []- [4]. Therefre, the dynamic mdel f the pwer system with CPLs is very imprtance. t is well knwn that the pwer cnverter mdel is time-varying because f the switching behaviur. Several appraches are cmmnly used fr eliminating the switching actins t achieve timeinvariant 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 generalized state-space averaging (SSA) mdeling methd. This methd has been used t analyze many pwer cnverters in DC distributin systems [5]-[7], as well as uncntrlled and cntrlled rectifiers in single-phase AC distributin systems [8],[9] and 6- and - pulse dide rectifiers in three phase systems []. The secnd is an average-value mdeling methd, which has been used fr 6- and - pulse dide rectifiers in many publicatins []- [3], as well as generatrs with line-cmmutated rectifiers [4]-[8]. These rectifiers can be mdeled with gd accuracy as a cnstant DC vltage surce. Hwever, this methd is nt easily applicable t the analysis f the stability f the general AC pwer system with multi-cnverter pwer electrnic systems. Anther technique widely used fr AC system analysis is that f DQ-transfrmatin thery [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 vectr-cntrlled cnverters where the SSA mdel and the average-value mdel are nt easily applicable. Mrever, the resulting cnverter mdels can be easily cmbined with mdels f ther pwer elements expressed in terms f synchrnusly rtating frames such as generatrs, frnt-end cnverters, and vectr-cntrlled drives. The DQ mdels f three-phase AC-DC pwer systems have been reprted in the previus wrks [9]-[]. But these d nt include a cnstant pwer lad (CPL). Applying the DQ mdeling apprach fr stability studies f the pwer system including a CPL has been addressed in []-[4]. The DQ methd fr mdeling the three-phase uncntrlled and cntrlled rectifier has been reprted in [] and [5], respectively. This paper exten the wrk in [5] with the eigenvalue therem t analyze the stability f the system due t a CPL. The stability results frm the thery will be supprted by using the intensive time dmain simulatin. The paper is structured as fllws. n Sectin, the pwer system definitin and assumptins are explained SSN: ssue, Vlume 6, April

2 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak Fig. The pwer system studied The DQ dynamic mdel f the system frm [5] is explained again in Sectin 3. The steady-state value calculatin fr the small-signal mdel derived frm the prpsed methd is presented in Sectin 4. n Sectin 5, the mdel validatin using a small-signal simulatin and stability analysis fr each firing angle f cntrlled rectifier are shwn. Finally, Sectin 6 cncludes and discusses the advantages f the DQ methd t mdel the pwer cnverter fr stability analysis. The effect f L n the AC side causes an verlap angle µ in the utput wavefrms that causes as a cmmutatin vltage drp. This drp can be represented as a variable resistance r µ that is lcated n the DC side [],[6] as shwn in Fig.. The r µ can be calculated by: 3ωL rµ = () π where ω is the surce fruency. Pwer System Definitin and Assumptins The pwer system studied in this paper is depicted in Fig.. t cnsists f a three-phase vltage surce, transmissin line, 6-pulse cntrlled rectifier, DClink filters, and an ideal CPL cnnected t the DC. The ideal CPL is used t represent actuatr drive systems by assuming an infinitely fast cntrller actin f the drive system. Hence, the ideal CPL can be cnsidered as a vltage-dependent current surce given by: P CPL CPL = () Vut where V ut is the vltage acrss the CPL and P CPL is the pwer level f CPL. t is assumed that the three-phase vltage surce is balanced. The uivalent parameters f a transmissin line are represented by R, L, and C. The DC-link filters are shwn by elements r F, L F, and C F. E dc and V ut are the utput terminal vltage f a cntrlled rectifier and the vltage acrss the DC-link capacitr C F, respectively. A phase shift between the surce and the AC is λ as shwn in Fig.. Fig. Three-phase cntrlled rectifier with verlap angle resistance t can be seen frm Fig. that E dc represents the utput vltage frm the switching signal withut an verlap angle effect, while E dc represents the vltage at the rectifier utput terminal taking nt accunt the vltage drp effect. Since the cmmutatin effect has been mved n t the DC side, the switching signals fr 3-phase cntrlled rectifier can be applied withut cnsidering the effect f verlap angle. This is shw in Fig. 3 in which α is the firing angle f thyristrs. The switching functin f S a in Fig.3 can be expressed by a Furier series. SSN: ssue, Vlume 6, April

3 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak 3. DQ Dynamic Mdel f the System n this sectin, the DQ mdeling methd is applied t derive a mathematical mdel f the system as depicted in Fig.. Firstly, the cntrlled rectifier is transfrmed int a tw axis frame (DQ frame) rtating at the system fruency ω by means f: [ ] Tθ = cs( θ) 3 sin( θ) π cs( θ ) 3 π sin( θ ) 3 π cs( θ+ ) 3 π sin( θ+ ) 3 (6) Fig.3 The cntrller rectifier switching signal n this paper, neglecting the harmnics f the pwer system, the switching functins can be written fr three phases as: S abc sin( ωt+ φα) 3 π = sin( ωt + φα) π 3 π sin( ωt+ + φα) 3 (3) where φ is a phase angle f the AC vltage and α is the firing angle. The relatinship between input and utput terminal f cntrlled rectifier is given by:, = S (4) in abc abc dc T E dc = S abcv (5), abc t can be seen frm (4) that the fundamental input current is in phase with the switching signals. n additin, fr a cntrlled rectifier, the fundamental input current lags the fundamental input vltage by α [6]. Equatins (3)-(5) will be used t derive the mdel f cntrlled rectifier by using DQ mdeling methd in Sectin 3. The mdel assumptins in this paper are as fllws: - The rectifier is perated under a cntinuus cnductin mde (CCM). - The utput DC current f the rectifier is cnstant. - The amplitude f the three-phase surce is cnstant and balanced. - Only ne cmmutatin ccurs at a time. - All harmnics in the system are neglected. where π θ = ωt + φ. Cmbining uatins (4)-(6) results in: = S in, dq dq dc (7) T E dc = S dqv, dq (8) Secndly, the switching functins in (3) can be transfrmed int a DQ frame by means f (6) t give: 3 3 cs( φφ+ α) S = (9) dq π sin( φφ+ α) The vectr diagram fr the DQ transfrmatin is as shwn in Fig. 4 where V s is the peak amplitude phase vltage, in is the peak amplitude current, V is the peak amplitude AC vltage, and S is peak amplitude f the switching signal, here ual t 3 / π as shwn in (3). Fig.4 The vectr diagram fr DQ transfrmatin Frm (7)-(9), the cntrlled rectifier can be easily represented as a transfrmer having d and q- axis transfrmer rati S d, S q that depend n the SSN: ssue, Vlume 6, April

4 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak phase f the DQ frame (φ ), the phase f V (φ), and the firing angle f thyristrs (α). As a result, the uivalent circuit f the cntrlled rectifier in the DQ frame derived by using DQ mdeling methd is shwn in Fig.5. R = L + ω qs + L V, d V sd L R qs = ω qs V, q + V sq L L L V, d = C + ωv, q 3 3 πc dc V, q = ωv, d + C qs dc = V ut = 3 3 V, d πl F C F dc C F r r F µ + L F L F P CPL V ut dc L F V ut () Fig.5 The cntrlled rectifier uivalent circuit in the DQ frame Finally, using (6), the cable sectin can be transfrmed int DQ frame [7]. The DQ representatin f the cable is then cmbined with the cntrlled rectifier as shwn in Fig.5. As a result, the uivalent circuit f the pwer system in Fig. can be represented in the DQ frame as depicted in Fig.6. The uivalent circuit in Fig.6 can be simplified by fixing the rtating frame n the phase f the switching functin ( φ = φα ). This results in the circuit as shwn in Fig.7. Applying the Kirchhff s vltage law (KVL) and the Kirchhff s current law (KCL) t the circuit in Fig.7 btains the set f nnlinear differential uatins. We define: State variables: x = V [ V V ] T nput: u = V Output: y = qs [ ] T m P CPL [ ] V ut, d, q The set f nnlinear differential uatins is given as fllws: dc ut The uatin () is a nnlinear uatin. t is well knwn that the linearized mdel can be used fr a cntrller system design via a linear cntrl thery. n additin, the linearized mdel can be als used t analyze the small-signal stability f the pwer system including a CPL []-[4]. Therefre, () is 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 DQ linearized mdel f () is then f the fllwing frm: where δ x= δ u= δ y = δ x= A(x,u ) δx+ B(x,u ) δu δy= C(x,u ) δx+ D(x,u ) δu [ δ δ δv δv δ δv ] T qs [ δv ] T m δp CPL [ ] δv ut B(x,u ) =, d 3 cs( λ + α) L 3 sin( λ + α ) L, q C V F ut, dc 6 ut () SSN: ssue, Vlume 6, April

5 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak Fig.6 The uivalent circuit f the system in Fig. n DQ frame : 3 3 π Fig.7 The simplified uivalent circuit f the pwer system R L ω C A(x,u ) = ω R L C L ω 3 3 πl L ω 3 3 πc r r F µ + LF LF C F LF P C CPL FVut, Calculatin the Steady-State Value The DQ linearized mdel in () nee t define V ut, and λ. The pwer flw uatin can be applied t determine the steady state value at the AC side f the pwer system in Fig.. This lea t a system f nnlinear uatins: VsV Z VsV Z V cs( γ λ) Z V sin( γ λ) Z cs( γ ) = P sin( γ ) = Q () C(x,u ) = [ ] 6 where the fllwing steady-state values are: V, D(x,u ) = [ ] vltage at AC (rms), λ - phase shift between V s and V as mentined abve. Nte that Z γ is the transmissin line impedance, while the active and SSN: ssue, Vlume 6, April

6 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak reactive pwer (per phase) at the AC is given by: prvide a high accuracy. Therefre, this mdel can then be used fr stability analysis. P Q = V = V csα = ( P sinα CPL / 3) (3) t can be seen frm (3) that the P and Q depend n the firing angle f thyristrs (α). Equatin () can be slved by using a numerical methd such as Newtn Raphsn t achieve V, and λ at the steady-state cnditins. Cnsuently, V ut, fr DQ linearized mdel in () can then be calculated by: 3 3 V = ut, π where 3L ω V cs( α) r, π dc, F dc, (4) Fig.8 Mdel verificatin fr changing P CPL frm 7 t 9kW (α = degree) j jλ V s e V, e 3 jγ Ze z dc, = 3 3 π ωl = + = Z R ( ωl ), γ z tan R 5. Small-signal Simulatin and Stability Analysis The DQ linearized mdel in () is simulated fr small-signal transients against a crrespnding three-phase benchmark circuit mdel in SimPwerSystems TM f SMULNK. The set f parameters fr the example system accrding t Fig. is given as fllws: V s =3 V rms/phase, f=5 Hz, R =.5Ω, L =3µH, C = nf, C F =µf, r F =.3Ω, and L F =6.5mH. Fig.8 shws the V ut respnse f the system in Fig. t a step change f P CPL frm 7 t 9kW that ccurs at t =.4s. (α = degrees). Similarly, Fig.9-Fig.3 are the respnses t a step change f P CPL frm 7 t 9kW fr α ual t,, 3, 4, and 5 degrees, respectively. Nte that the firing angle cannt be allwed t exceed 6 degrees t btain the psitive utput vltage f the rectifier n the DC side [6]. Frm the results in Fig.8-Fig.3, an excellent agreement between bth mdels is achieved under small-signal simulatin. t cnfirms that the mathematical mdel f the pwer system with a cntrlled rectifier derived frm the DQ methd Fig.9 Mdel verificatin fr changing P CPL frm 7 t 9kW (α = degrees) Fig. Mdel verificatin fr changing P CPL frm 7 t 9kW (α = degrees) SSN: ssue, Vlume 6, April

7 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak simulated by using the three-phase benchmark mdel, while the respnses frm the DQ linearized mdel prvide nly the fundamental cmpnent. Hwever, it will be shwn in the stability results that the mathematical mdel derived by cnsidering nly the fundamental cmpnent can crrectly predict the unstable pint f the system. Fr stability analysis, the DQ linearized mdel in () is used with the eigenvalue therem. The eigenvalue can be calculated frm the Jacbian matrix A x, u ) in () by: ( [ λ A] (5) det = Fig. Mdel verificatin fr changing P CPL frm 7 t 9kW (α = 3 degrees) and the system is stable if { } Re λ < (6) i where i =,,3,., n (n = the number f state variables). Fig. Mdel verificatin fr changing P CPL frm 7 t 9kW (α = 4 degrees) T investigate the instability cnditin f the pwer system in Fig. due t a CPL, the eigenvalues f the system with the given parameters are calculated frm the Jacbian matrix when the P CPL varies frm kw t 5 kw. The dminant rt lcus fr α ual t degrees is shwn in Fig.4. Accrding t (6), it can be seen that the system becmes unstable when the P CPL excee ~ kw. Nte that the results depend n the system parameters such as system fruency, DC-link filters etc. f the parameters are changed, the stability results will be changed. Fig.5 shws the time-dmain simulatins that supprt the theretical results with instability ccurring at P CPL ual t 4 kw. This is greater than kw fr the unstable cnditin predicted frm the thery. Fig.3 Mdel verificatin fr changing P CPL frm 7 t 9kW (α = 5 degrees) n additin, it can be seen frm Fig.8-Fig.3 that the harmnic cmpnents due t the switching actins are included in the respnses that are Fig.4 Eigenvalue plt f the system (α= degrees) SSN: ssue, Vlume 6, April

8 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak Fig.5 The simulatin validatin f Fig.4 Fig.8 Eigenvalue plt f the system (α=4 degrees) Similarly, Fig.6-Fig.9 shw the eigenvalue plt fr α ual t, 3, 4, and 5 degrees, respectively. Fig.-Fig.3 shw the time-dmain simulatins that cnfirm the theretical results f Fig.6-Fig.9, respectively. t can be seen that the mathematical mdel derived frm the DQ methd can be used t predict the unstable cnditin f the pwer system. Fig.9 Eigenvalue plt f the system (α=5 degrees) Fig.6 Eigenvalue plt f the system (α= degrees) Fig. The simulatin validatin f Fig.6 Fig.7 Eigenvalue plt f the system (α=3 degrees) n the future, the dynamic mdel will be used t predict the instability pint fr variatins in system parameters such as system fruency, DC-link parameters etc. Mrever, the dynamic CPL explained in [4] can be als used with the DQ mdel f three-phase cntrlled rectifier frm this paper instead f the ideal CPL. This is t investigate the effect f the dynamic CPL t the stability margin. SSN: ssue, Vlume 6, April

9 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak Recently, the artificial intelligence (A) techniques are widely used in the pwer system applicatin [8], [9]. Therefre, the mathematical mdel derived by using the DQ methd frm this paper can be als used as the bjective functin f the A algrithm fr the system design. Fig. The simulatin validatin f Fig.7 Fig. The simulatin validatin f Fig.8 Fig.3 The simulatin validatin f Fig.9 6. Cnclusin n this paper, the DQ mdeling methd is presented fr mdeling a three-phase AC distributin system with a three-phase cntrlled rectifier, DC-link filters, and an ideal CPL cnnected t the DC. The prpsed apprach is very useful fr mdeling the AC distributin system and als cncerning a phase shift between surce and AC. Mrever, the resulting cnverter mdels can be easily cmbined with mdels f ther pwer elements expressed in terms f synchrnusly rtating frames such as generatrs, frnt-end cnverters, and vectr-cntrlled drives. This paper als present the DQ linearized mdel that is used t analyse the system stability due t the CPL. The three-phase benchmark mdel is used t verify the stability results in the paper. The results shw that the mathematical mdel derived frm the DQ methd can predict the instability pint with a high accuracy. Therefre, electrical engineers can use the mathematical mdel t study the pwer system behaviur and t avid the unstable cnditin. n the future, the dynamic mdel will be used t predict the instability pint fr variatins in system parameters. Mrever, the mathematical mdel frm this paper can be als used fr the A applicatin t the pwer system design t achieve the best perfrmance. References: [] R.D. Middlebrk, nput Filter Cnsideratin in Design and Applicatin f Switching Regulatrs, EEE ndustry Applicatin Sciety Annual Meeting, Chicag, llinis, Octber 976, pp [] A. Emadi, B. Fahimi, and M. Ehsani, On the Cncept f Negative mpedance nstability in the Mre Electric Aircraft Pwer Systems with Cnstant Pwer La, Sc. Autmtive Eng. Jutnal, 999, pp [3] C. Rivetta, G.A. Williamsn, and A. Emadi, Cnstant Pwer La and Negative mpedance nstability in Sea and Undersea Vehicles: Statement f the Prblem and Cmprehensive Large-Signal Slutin, EEE Electric Ship Tech. Sympsium., Philadelphia, PA USA, July 5, pp [4] A. Emadi, A. Khaligh, C.H. Rivetta, and G.A. Williamsn, Cnstant Pwer La and Negative mpedance nstability in Autmtive Systems: Definitin, Mdeling, Stability, and Cntrl f Pwer Electrnic Cnverters and Mtr Drives, EEE Trans. n Vehicular Tech., Vl. 55, N. 4, July 6, pp.-5. SSN: ssue, Vlume 6, April

10 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak [5] J. Mahdavi, A. Emadi, M.D. Bellar, and M. Ehsani, Analysis f Pwer Electrnic Cnverters Using the Generalized State-Space Averaging Apprach, EEE Trans. n Circuit and Systems., Vl. 44, August 997, pp [6] A. Emadi, Mdeling and Analysis f Multicnverter DC Pwer Electrnic Systems Using the Generalized State-Space Averaging Methd, EEE Trans. n ndus. Elect., Vl. 5, N. 3, June 4, pp [7] A. Emadi, M. Ehsani, and J.M. Miller, Vehicular Electric Pwer Systems: Land, Sea, Air, and Space Vehicles, Marcel Dekker, nc, 4. [8] A. Emadi, Mdeling f Pwer Electrnic La in AC Distributin Systems Using the Genearlized State-Space Averaging Methd, EEE Trans. n ndus. Elect., Vl. 5, N. 5, Octber 4, pp [9] K-H. Cha, Dynamic Mdeling and Rt Cntrl f Multi-Mdule Parallel Sft- Switching-Mde Rectifiers, WSEA Transactins n Systems, Vl.8, ssue 5, 9, pp [] L. Han, J. Wang, and D. Hwe, State-space average mdelling f 6- and -pulse dide rectifiers, The th Eurpean Cnf. n Pwer Elect. and Appl., Aalbrg, Denmark, Sep. 7. [] S.F. Glver, Mdeling and stability analysis f pwer electrnics based systems, Ph.D. dissertatin, Purdue University, May 3. [] A. Baghramian, and A.J. Frsyth, Averaged- Value Mdels f Twelve-Pulse Rectifiers fr Aerspace Applicatins, Pwer Electrnics, Machines, and Drives (PEMD 4), University f Edinburgh, UK, March-April 4, pp.-5. [3] A. Uan-Z-li, R.P. Burgs, F. Lacaux, F. Wang, and D. Bryevich, Assessment f Multi-Pulse Cnverter Average Mdels fr Stability Studies Using a Quasi-Statinary Small-Signal Technique, Pwer Electrnics and Mtin Cntrl Cnference 4, 4, pp [4] S.D. Sudhff, and O. Wasynczuk, Analysis and Average-Value Mdeling f Line-Cmmutated Cnverter-Synchrnus Machine Systems, EEE Trans. n Energy Cnversin., Vl. 8, N., March 993, pp [5] S.D. Sudhff, Wavefrm Recnstructin frm the Average-Value Mdel f Line-Cmmutated Cnverter-Synchrnus Machine Systems, EEE Trans. n Energy Cnversin., Vl. 8, N. 3, September 993, pp [6] S.D. Sudhff, Analysis and Average-Value Mdeling f Dual Line-Cmmutated Cnverter-6-Phase Synchrnus Machine Systems, EEE Trans. n Energy Cnversin., Vl.8, N. 3, September 993, pp [7] S.D. Sudhff, K.A. Crzine, H.J. Hegner, and D.E. Delisle, Transient and Dynamic Average- Value Mdeling f Synchrnus Machine Fed Lad-Cmmutated Cnverters, EEE Trans. n Energy Cnversin, September 996, pp [8]. Jadric, D. Brjevic, and M. Jadric, Mdeling and Cntrl f a Synchrnus Generatr with an Active DC Lad, EEE Trans. n Pwer Electrnics, Vl. 5, N., March, pp [9] C.T. Rim, D.Y. Hu, and G.H. Ch, Transfrmers as Equivalent Circuits fr Switches: General Prfs and D-Q Transfrmatin-Based Analyses, EEE Trans. n ndus. Appl., Vl. 6, N. 4, July/August 99, pp [] C.T. Rim, N.S. Chi, G.C. Ch, and G.H. Ch, A Cmplete DC and AC Analysis f Three- Phase Cntrlled-Current PWM Rectifier Using CircuitD-Q Transfrmatin, EEE Trans. n Pwer Electrnics, Vl. 9, N. 4, July 994, pp [] S.B. Han, N.S. Chi, C.T. Rim, and G.H. Ch, Mdeling and Analysis f Static and Dynamic Characteristics fr Buck-Type Three-Phase PWM Rectifier by Circuit DQ Transfrmatin, EEE Trans. n Pwer Electrnics, Vl. 3, N., March 998, pp [] K-N. Areerak, S.V. Bzhk, G.M. Asher, and D.W.P. Thmas, Stability Analysis and Mdelling f AC-DC System with Mixed Lad Using DQ-Transfrmatin Methd, EEE nternatinal Sympsium n ndustrial Electrnics (SE8), Cambridge, UK, 9 June- July 8, pp [3] K-N. Areerak, S.V. Bzhk, G.M. Asher, and D.W.P. Thmas, DQ-Transfrmatin Apprach fr Mdelling and Stability Analysis f AC-DC Pwer System with Cntrlled PWM Rectifier and Cnstant Pwer La, 3 th nternatinal Pwer Electrnics and Mtin Cntrl Cnference (EPE-PEMC 8), Pznan, Pland, -3 September 8. [4] K-N. Areerak, S. Bzhk, G. Asher, L.de Lill, A. Watsn, T. Wu, and D.W.P. Thmas, The Stability Analysis f AC-DC Systems including Actuatr Dynamics fr Aircraft Pwer Systems, 3 th Eurpean Cnference n SSN: ssue, Vlume 6, April

11 WSEAS TRANSACTONS n POWER SYSTEMS K. Chaijarurnudmrung, K.-N. Areerak, K.-L. Areerak Pwer Electrnics and Applicatins (EPE 9), Barcelna, Spain, 8- September 9. [5] K. Chaijarurnudmrung, K-N. Areerak, and K- L. Areerak, Mdeling f Three-phase Cntrlled Rectifier using a DQ methd, nternatinal Cnference n Advances in Energy Engineering (CAEE ), Beijing, China: June 9-,, pp [6] N. Mhan, T.M. Underland, and W.P. Rbbins, Pwer Electrnics: Cnverters, Applicatins, and Design, Jhn Wiley & Sn, USA, 3. [7] C-M Ong, Dynamic Simulatin f Electric Machinery using MATLAB/Simulink, Prentice Hall, 998. [8] K-L. Areerak, and T. Narngrit, Shunt Active Pwer Filter Design using Genetic Algrithm Methd, WSEA Transactins n Systems, Vl.9, ssue 4,, pp [9] L-Y. Chang, and H-C. Chen, Tuning f Fractinal PD Cntrllers using Adaptive Genetic Algrithm fr Active Magnetic Bearing System, WSEA Transactins n Systems, Vl.8, ssue, 9, pp SSN: ssue, Vlume 6, April