1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6-pulse thyristor brige connecte to an ac network. Balance firing of the converter an a balance interface transformer inuctance are assume. For the special case when the ac network is linear, a close form solution for the harmonic inection of the converter is evelope. For the more general case of a nonlinear ac network, a moular converter moel is evelope that relies on iterative methos. The converter moel moule takes as input the ac voltage harmonics at the point of common coupling an outputs the corresponing harmonic current inection into the network. The moels are first valiate against time omain simulation results. The results from the evelope moels are then compare with those obtaine from approximate analytical techniques which assume zero c ripple current. For a system with typical parameter values, it is shown that the zero ripple assumption may yiel acceptable results for some operating points, but highly erroneous results for others. Inex Terms Thyristor brige, HVDC, converter, harmonics, steay state analysis I. INTRODUCTION Power electronics converters contribute significantly to the harmonic pollution of istribution an transmission networks. In particular, converters that switch at line frequency, such as thyristor briges, inect large harmonic currents into the network. The amplitue an phase of these harmonic current inections may be significantly influence by the presence of resonance conitions on either the ac or c sie of the converter. Accurate preiction of the harmonic current inection of a converter into the ac network must therefore take into account ynamics of both the ac an c sie networks. Harmonic interactions in thyristor briges may be calculate either in the frequency omain 1, 2 or in the time omain 3, 4, 5. The focus of this paper will be on the evelopment of simplifie, computationally efficient time omain methos. In orer to solve for the steay state of a thyrsitor brige Grötzbach employe a state space formulation of the converter 6. For a simplifie ac system, he analytically etermine initial conitions of the states that woul lea to a steay state solution. Base on this founation he later etermine the influence of ac an c sie reactances on the ac sie current harmonics 3. For the special case of a linear, balance ac network Herol an Weinl emonstrate the existence of a completely analytical solution for the steay state of the thyristor brige 4. Pivotal to this analytical solution was the nee to moel the P.W. Lehn is with the Dept. of Electrical an Computer Engineering, University of Toronto, Toronto, Canaa (e-mail: lehn@ecf.utoronto.ca). G. Ebner is with the Institute of Electrical Power Systems, University of Erlangen-Nuremberg, Erlangen, Germany, (e-mail: ebner@eev.e-technik.unierlangen.e). ac system all the way back to an ieal, harmonic free, infinite bus. After fining the initial conition associate with the steay state, a 1-perio time omain simulation was employe, followe by a Fourier Analysis, to etermine converter current harmonics. Perkins 5 employe a more general iterative metho, similar to the one propose by Dobson for ioe circuits, that allows inclusion of arbitrary linear ac networks. Base again on time omain concepts, iteration is employe to fin (i) the commutation angle of the converter an (ii) the initial conition associate with the steay state. Base on the steay state initial conition a 1-perio time omain simulation coul be employe, followe by a Fourier Analysis, to etermine converter current harmonics. All the above works employ ieal switching evices (zero or infinite impeance) to avoi singularity an/or convergence problems associate with high/low impeance switch moels. This leas to a converter representation with 2 state equations uring the commutation interval an only one state equation uring the non-commutation interval (see commutation an non-commutation circuit iagrams of Fig. 3). This time varying imension of the system significantly complicates the analysis. In this paper it is shown that an assumption of ieal switching evices oes not preclue the evelopment of a fixe imension formulation of the thyristor brige. It is then emonstrate that this fixe imension moel of the converter may be easily augmente to the state matrices of an arbitrary linear time invariant, balance ac network. For analysis of ac networks that contain multiple converters, a moular converter moel is also evelope. While solution of the moular moel requires Newton iteration, convergence is extremely fast since: (i) imension of the system Jacobian is 1x1, (ii) an accurate initial estimate for the solution is available, (iii) the equation being iterate is nearly linear in the vicinity of the solution. II. CONVERTER WITH IDEAL SOURCE Steay state analysis of a 6-pulse or 12-pulse converter is most easily accomplishe if: converter interface inuctance an gating are balance the ac network parameters are balance the ac network is linear the infinite bus is balance an free of harmonics. Uner these conitions a complete analytic moel of the unregulate converter may be evelope. Key to this evelopment is the observation that the state transition matrices of the system epen only on the commutation interval length,
2 Y D Y E L D L E 5 / / G 5 G Y F Y GF D Y D Y E L D L E 5 / / G 5 G Y F Y GF E Fig. 3. The equivalent circuits of the converter: (a) uring commutation an (b) after commutation. Fig. 1. Fig. 2. Flow chart for fining the steay state solution of a thyristor brige. Y D Y E Y F L D L E L F 5 / The simplifie schematic iagram a thyristor brige. L GF / G 5 G Y GF β 8. Consequently, taking β as an input variable allows a solution algorithm to procee, as per Fig. 1, that outputs the associate firing angle, α, an the associate initial conition x(0) that leas to a steay state solution. The following subsections etail the solution algorithm. For simplicity, the ac an c networks are first replace by ieal voltage sources, as shown in Fig. 2. It will later be shown how this moel may be extene to consier more general ac network configurations. A. State Transition Map Uner balance operation, the 6-pulse converter isplays 6 th perio symmetry that may be exploite to simplify harmonic analysis. Assuming continuous conuction, the two circuits shown in Fig. 3 characterize the behavior of the converter. The circuit of Fig. 3(a) hols uring the commutation interval, while the circuit of Fig. 3(b) hols for the remainer of the 6 th perio. In general, the state transition map will epen on the choice of state variables. In this work, a new state assignment is selecte to simplify analysis. States are selecte as phase currents i a an i b subect to the constraint that i a = 0 for the entire interval after commutation. The propose selection of state is in contrast to the common approach associating two state variables with the circuit of Fig. 3(a), an only one state variable with the circuit of Fig. 3(b). By maintaining the same number of state variables uring commutation an non-commutation intervals, the propose approach avois the nee for proection an inection matrices (as use in 5, ), or the nee for partial matrix inverses (as use in 4), thereby simplifying implementation. During commutation the state moel of the system is given by: ia ia va L 1 = R t i 1 + B b i 1 + D b v 1 v c (1) b 2L L L L L 1 = L L 2L L 2R + R R + R R 1 = R + R 2R + R 1 D 1 = 1 2 1 B 1 = 1 2 After commutation phase a becomes open circuite an i a = 0 for the remainer of the sixth perio. To avoi elimination of state variable i a we employ an auxiliary ifferential equation of the form: i a t = 0 t ɛ β, π/3. Given that i a (β) = 0 by efinition, this simple state moel yiels the esire solution: i a (t) = 0 t ɛ β, π/3. The state equations after commutation are therefore given by: ia ia va L 2 = R t i 2 + B b i 2 + D b v 2 v c, (2) b
3 1 0 L 2 = 0 2L L 0 0 R 2 = 0 2R + R 0 D 2 = 1 0 0 B 2 = 1 2 Assuming the converter connection allows no zero sequence currents to flow, the αβ frame equations of the system uring an after commutation are obtaine by transformation of (1) an (2): iα t i β = CL 1 R C 1 iα i β. + CL 1 B C 1 vα v β + CL 1 D v c (3) setting subscript = 1 gives the equation uring commutation an = 2 gives the equations after the commutation. The matrix C is a 2 2 variant of the Clarke Transform, as given in the Appenix. The ieal source voltage vector T v α v β is represente by a set of ieal oscillator equations, while the c source is represente by the equation z c /t = 0 as per 9. This allows formation of an equivalent autonomous system equation: with A = t CL 1 x z ac z c = A x z ac z c (4) R C 1 CL 1 B C 1 CL 1 0 Ω ac 0 0 0 Ω c Ω ac = 0 1 1 0 Ω c = 0 an the frequency of the ac source has been normalize. Amplitue an phase information of the ac voltage source, as well as amplitue information of the c voltage source is containe only in the initial conitions z(0) = T. Over a sixth of a perio the state transitions may be foun accoring to: x(π/3) z ac (π/3) z c (π/3) = Φ x(0) z ac (0) D (5) Φ is the state transition matrix of the system over a sixth of a perio given by: Φ = e A2(π/3 β) e A1β. (6) It is important to note that the above state transition matrix is only a function of the commutation angle β an oes not explicitly epen on the the firing angle of the converter. B. Perioicity Constraint on State Traectories For a time omain formulation, the conitions that must be satisfie in the steay state are 10: (i) perio excitation, (ii) perioicity of the state traectories (iii) perioicity of the switching events. Perioicity of the excitation is assume for any harmonic analysis an will not be iscusse further. Perioicity of state variables will be aresse in this section, while perioicity of the switching times will be aresse in the subsequent section. Perioicity of the state traectories requires that x(t+2π) = x(t). For a balance system this constraint may be converte to an equivalent constraint on the state traectories over a sixth of a perio 10. Over a sixth of a perio ac current an source space vectors unergo a rotation of π/3 raians: x(π/3) = Θ ac x(0) () Θ ac is the π/3 rotation matrix: cos π/3 sin π/3 Θ ac = sin π/3 cos π/3. (8) Dc quantities repeat ever sixth perio, thus their state rotation matrix is simply the unity matrix: Θ c = 1. (9) To apply the state perioicity constraint () to the state traectory equation (5), the state transition matrix (6) is first evaluate for the specifie commutation angle. It has the form 1 : x(π/3) z ac (π/3) z c (π/3) = Ap N p ac N p c 0 Ω p ac 0 0 0 Ω p c x(0) z ac (0) (10) Applying the state perioicity constraint to (10) yiels a constraints on the initial conitions associate with the steay state solution: x(0) = (Θ ac A p ) 1 N p ac N p c z ac (0).. (11) Contrary to the analysis of VSC circuits 9, in analysis of the thyristor brige (11) provies only one of two constraints necessary to solve for the steay state. A secon constraint must be impose to ensure perioicity of the switching times. C. Perioicity Constraint on Switching Times Analysis leaing up to (11) is base on the assumption that the commutation interval length β is known a priori. In fact, the commutation interval length is equal to β if an only if the current i a in Fig. 3(a) reaches zero precisely at time t = β. Mapping this constraint into the αβ-frame yiels: i α (β) = 0. (12) It is assume that a unique firing angle α is associate with each commutation interval length β. Thus we may aust the phase of the ac voltage vector until constraint (12) is satisfie. 1 It may be easily proven that Ω p ac = Θ ac an that Ω p c = Θ c.
4 Solution procees as follows. First x(β) is expresse in a from similar to (10) by evaluating the state transition matrix e A1β : x(β) A β N β ac N β c z ac (β) = 0 Ω β ac 0 x(0) z ac (0). (13) z c (β) 0 0 Ω β z c c (0) Introucing the constraint (11) an solving for x(β) yiels: x(β) = F (14) F = A β (Θ ac A p ) 1 N p ac N p c + N β ac N β c. (15) Both the c an ac voltage information as well as the firing angle information is containe in the initial conition vector z(0). Assuming a c voltage of V c, an ac voltage of V an a firing angle of α, the associate initial conition vector is given by (16): z(0) = = V cos(α + π/3) V sin(α + π/3) V c An expression for i α (β) is extracte from (14):. (16) i α (β) = F 1,1 F 1,2 F 1,3 z(0). (1) Finally the constraint (12) is applie to etermine the firing angle α. α = cos 1 F 1,3 F 2 1,1 + F 2 1,2 V c V + tan 1 { F1,2 F 1,1 } π 3. (18) In other wors, a converter operate at the above calculate firing angle will settle into a steay state with the stipulate commutation interval length β. The initial conitions of the states, x(0), associate with this solution may be etermine by applying initial conition (16) to (11). This yiels a fully analytic solution of the converter. III. CONVERTER WITH ARBITRARY AC NETWORK It is often necessary to stuy the harmonic interaction of a converter with its filters an an existing AC network. In this case, ac network equations must be ae to the basic converter equations. This may be one either by augmenting the abc-frame equations of of (1) an (2) augmenting the αβ-frame equations of of (3). Generally, it is easier to employ the latter metho, as it leas to a more moular moel of the system. Network equations are therefore expresse in the form: x ac = A ac x ac + B ac u ac (19) t y ac = C ac x ac (20) input excitation u ac comes from an ieal oscillator then is use to represent the infinite bus within the ac network: u ac = z ac, an output of the network equations gives the ac input voltage neee by the converter moel of (3): vα = y ac. (21) v β This approach allows the influence of network parameters on the operation of the converter to be etermine uner balance operation. IV. CONVERTER WITH AC SOURCE DISTORTION In the previous section, a set of linear network equations were simply augmente to the converter moel. A maor limitation of this classical approach is that no other switching circuits may exist within the moele ac network. Possible interactions between neighboring converters cannot be stuie. To overcome this limitation, a fully moular harmonic moel of the converter can be evelope subect to the following reuce set of limitations: converter interface inuctance an gating are balance only characteristic harmonics exist in the ac network, i.e. negative sequence 5, 11, 1, etc. an positive sequence 1,, 13, etc. The moel buils irectly on the results of Section II. Ac excitation is now provie not by merely a funamental frequency ac source, but by a set of harmonic sources all summe together. The harmonic oscillator matrix Ω ac is now given by: Ω ac = iag(ω 1, 5Ω 1, +Ω 1, 11Ω 1, +13Ω 1,...) (22) Ω 1 = 0 1 1 0 The phase angles of the bus voltage harmonics are efine with respect to the phase angle of the funamental: v α + v β = V +1 0e t + V 5 φ 5 e 5t + V + φ + e t +.. Assuming the funamental of the bus voltage to have zero phase, the necessary initial conitions for the oscillator states are: V +1 cos(0) V +1 sin(0) V 5 cos(φ 5 ) z ac (0) = V 5 sin(φ 5 ). (23) V + cos(φ + ) V + sin(φ + ) : The solution algorithm time shifts the ac excitation voltage to meet the commutation constraint i α (β) = 0. Introucing harmonics on the bus voltage has one critical implication on this solution algorithm. Since harmonics have their phase angles efine relative to the funamental, time shifting of the funamental results in an associate phase shift of all.
5 Fig. 4. Test system for valiation of the analytical moel. harmonics. The require initial conition vector therefore has the form: V +1 cos(α + π/3) V +1 sin(α + π/3) V 5 cos(φ 5 5(α + π/3)) z(0)= = V 5 sin(φ 5 5(α + π/3)) V + cos(φ + + (α + π/3)). (24) V + sin(φ + + (α + π/3)) : V c Matrix F may be evaluate ust as in (15), albeit with larger matrix imensions for A β, N p ac an N β ac. Assuming a total of n ac sie harmonics (incluing the funamental) (1) becomes: i α (β) = 0 = F 1,1 F 1,2 : F 1,2n 1 F 1,2n F 1,2n+1 T. (25) All terms in the vector z ac (0) have trigonometric epenance on firing angle α, hence a close form solution to constraint equation (25) oes not exist. Instea the firing angle is solve through iteration. V. MODEL VALIDATION A test system, as epicte in Fig. 4, is employe to valiate the propose moel against time omain simulation results. Parameters for the test system are given in Table I. A complete analytical representation of the system is evelope, as per Sections II an III, an converter current harmonic inections an bus voltage harmonics at the point of common coupling are valiate. The operating point is specifie by the selection of β = 18.00 o. The resulting firing angle of the converter is solve using the analytical moel to be 45.98 o. This firing angle is use in the time omain simulation. Table II compares the α an β values from simulation with those from the analytical moel. Table III compares the resulting voltage harmonics at the point of common coupling (PCC) an the converter current harmonics. The results of Tables II an III clearly valiate the accuracy of the analytical moel propose in Sections II an III. TABLE I TEST SYSTEM PARAMETERS. Ac System Converter System Quantity Value Quantity Value V s 10.00 kv ln, peak V c 9.00 kv R s 0.040 Ω R 0.200 Ω X s 0.400 Ω X 3.600 Ω R l 0.128 Ω R 0.100 Ω X l 0.10 Ω X 1.400 Ω X Cl 3858 Ω X Cf1 14.0 Ω R f1 0.026 Ω X f1 0.400 Ω X Cf2 40.0 Ω R f2 0.0186 Ω X f2 0.280 Ω TABLE II VALIDATION OF ANALYTICAL MODEL - ANALYTICALLY OBTAINED FIRING AND COMMUTATION ANGLES COMPARED WITH SIMULATION Quantity Analysis Simulation β 18.00 o 18.12 o α 45.98 o 45.98 o TABLE III VALIDATION OF THE ANALYTICAL MODEL - ANALYTICALLY OBTAINED PCC VOLTAGE AND CURRENT HARMONICS COMPARED WITH SIMULATION Harmonic Analysis Simulation Analysis Simulation Number Vpcc h Vpcc h I h I h h (V ln, peak ) (V ln, peak ) (A peak ) (A peak ) +1 9223.9 922.8 1621.9 1621.6-5 282.28 282.29 334.99 334.86 + 124.90 124.9 141.8 141.8-11 53.01 53.11 2.84 2.1 +13 24.2 24.68 52.05 52.01-1 15.52 15.39 9.91 9.82 +19 20.55 20.51 10.35 10.34 Next the moular converter moel of Section IV is valiate. The moular moel takes as input voltage harmonics at the PCC. For a given β value, it outputs the resulting converter current harmonics. Again we assume β = 18.0 o. Table IV shows the PCC voltage harmonics applie to the converter an the resulting converter current harmonics obtaine from simulation an from the moular converter moel. In steay state, the firing angle associate with these PCC harmonics an the specifie β value is foun to be α = 44.63 o. Excellent agreement is again seen between the results from time simulation an those obtaine from the propose metho.
6 TABLE IV VALIDATION OF THE MODULAR MODEL FOR AN ARBITRARY SET OF V pcc 350 HARMONICS Output Output Harmonic Input Input I h I h Number Vpcc h Vpcc h (A peak ) (A peak ) h (V ln, peak ) (egrees) Analysis Simulation +1 10000 0 263.6 256.26-5 4000 0 612.18 611.00 + 2000 0 20.42 206.29-11 0 0 124.0 124.34 +13 0 0 85.4 85.59-1 0 0 22.04 22.00 +19 0 0 20.00 20.12 I 5, I (Amps) 300 250 200 150 I 5 with c ripple no c ripple I 100 In contrast to the previous moel, no assumption on the linearity of the ac network is mae in the evelopment of the moular moel. Thus the moular moel may be interface to an arbitrary network, provie the network is balance. VI. COMPARISON WITH APPROXIMATE MODELS Many classical texts analyze the thyristor brige base on an assumption of zero c ripple current 11. In other wors, they assume the c smoothing reactor (X in Fig. 4) is infinitely large. In this section, the accuracy of this zero ripple assumption is investigate. The system of Fig. 4 is first simulate with the nominal smoothing reactance of the test system: X = 3.982Ω. It is then simulate with a near infinite smoothing reactance. Fig. 5 shows the resulting ac sie 5 th an th harmonic currents as a function of the commutation angle β. As may be seen from Fig. 5, the zero ripple assumption yiels fairly accurate results for some operating points (e.g. for β > 30 o ), however for other operating points (e.g. for β < 30 o ) large errors result. While the zero ripple assumption sometimes yiels sufficiently accurate results, there is no obvious means of etermining whether its results shoul be truste. VII. CONCLUSION A time omain metho for harmonic analysis of thyristor briges was presente. The complexity of the time omain formulation was reuce through the introuction of an auxiliary ifferential equation. The auxiliary ifferential equation makes the equations uring commutation equal to the orer of the equations after commutation. This eliminates the nee for inection/proection matrices or the nee for non-unique partial matrix inverses, significantly simplifying the solution. The propose metho leas to a fully analytic solution of the converter equations provie the ac system is linear, balance an contains no other harmonic sources. A more general moular converter moel is also evelope. The moular moel may be interface to ac networks containing nonlinearities, an other harmonic sources, however, the solution of the moular moel relies on iterative methos. The propose moel is employe for a simple emonstrative stuy investigating the accuracy of the common zero c ripple assumption. The stuy shows that significant errors 50 0 0 10 20 30 40 50 60 BETA (eg) Fig. 5. Errors resulting from a zero c current ripple assumption. 5 th an th ac current harmonics with an without inclusion of c ripple. may occur when using simplifie analytical moels base on a zero c ripple assumption. VIII. ACKNOWLEDGEMENTS The authors woul like to thank Prof. Dr. G. Herol of the University of Erlangen-Nuremberg for hosting this research proect. Without his support this collaborative work woul not have been possible. REFERENCES 1 J. Rittiger an B. Kulicke, Calculation of HVDC converter harmonics in frequency omain with regar to asymmetries an comparison with time omain simulations, IEEE Transactions on Power Delivery, Vol. 10, No. 4, Oct. 1995, pp. 1944-1949. 2 B.C. Smith, N.R. Watson, A.R. Woo an J. Arrillaga, Steay state moel of AC/DC converter in the harmonic omain, IEE Proc.- Generation, Transmission an Distribution, Vol. 142, No. 2, March 1995, pp. 109-118. 3 M. Grötzbach, an W. Frankenberg, Inecte currents of controlle AC/DC converters for harmonic analysis in iustrial power plants, IEEE Transactions on Power Delivery, Vol. 8, No. 2, April 1993, pp. 511-51. 4 G. Herol an C. Weinl, Calculation of 6-pulse current converters in steay state an optimise esign of AC series wave traps, PEMC Proceeings, 1996, vol. 3, pp. 510-516. 5 B. K. Perkins, an M. R. Iravani, Novel calculation of HVDC converter harmonics by linearization in the time-omain, IEEE Transactions on Power Delivery, Vol. 12, No. 2, April 199, pp. 86-83. 6 M. Grötzbach, an R. von Lutz, Unifie moelling of rectifer-controlle DC-power supplies, IEEE Transactions on Power Electronics, Vol. PE-1, No. 2, April 1986, pp. 90-100. I. Dobson, an S. G. Jalali, Surprising simplification of the acobian of a ioe switching circuit, IEEE Intl. Symposium on Circuits an Systems, May 1993, pp. 2652-2655. 8 G. Herol, Resonanzbeingungen sechspulsiger Stromrichtersysteme, Elektrie, vol. 40, 1986, pp. 255-258. 9 P. W. Lehn, an K. L. Lian Frequency coupling matrix of a voltage source converter erive from piecewise linear ifferential equations, submitte to the IEEE Transactions on Power Delivery, Jan. 2006. 10 P. W. Lehn, Exact moeling of the voltage source converter, IEEE Transactions on Power Delivery, Vol. 1, No. 1, January 2002, pp. 21-222. 11 S.B. Dewan, A. Straughen, Power semiconuctor circuits, Wiley, New York, 195.