Symmetrical Component Analysis of Multi-Pulse Converter Systems

Transcription

1 Electric Power Components and Systems ISSN: (Print) (Online) Journal homepage: Symmetrical Component Analysis of Multi-Pulse Converter Systems M. Abdel-Salam, S. Abdel-Sattar, A. S. Abdallah & H. Ali To cite this article: M. Abdel-Salam, S. Abdel-Sattar, A. S. Abdallah & H. Ali (2006) Symmetrical Component Analysis of Multi-Pulse Converter Systems, Electric Power Components and Systems, 34:8, , DOI: / To link to this article: Published online: 23 Feb Submit your article to this journal Article views: 63 Citing articles: 3 View citing articles Full Terms & Conditions of access and use can be found at

2 Electric Power Components and Systems, 34: , 2006 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / Symmetrical Component Analysis of Multi-Pulse Converter Systems M. ABDEL-SALAM S. ABDEL-SATTAR A. S. ABDALLAH H. ALI Electrical Engineering Department Assiut University Assiut, Egypt Introduction This article describes a new method for dynamic simulation of multi-converter systems. This simulation is based on symmetrical components in time domain analysis and general representation of converter transformers to meet Y/, Y/Y, and Y/Z connections. The simulation is suitable for harmonic analysis of balanced and unbalanced AC voltages. The computed currents and voltages agreed reasonably with those measured and reported in the literature for the characteristic and non-characteristic harmonics. Keywords symmetrical components, multi-pulse converters, harmonic analysis, phase shift transformers Static power converters have many fields of applications in modern life. These applications are extended from large power converters in electrical and industrial utilities up to small power converters in battery chargers and small power electrical devices such as personal computers and TV set receivers [1, 2]. Normally, the static power converters use electronic switching devices such as thyristors, diodes, power transistors and gate-turn off thyristors to convert bi-directional currents and voltages into unidirectional currents and voltages or other inverter system [3]. The increasing use of these electronic switching devices is due to ease of maintenance, reducing their volt-ampere absorption and their high reliability in controlling process [4]. However, they have many problems in power systems feeding due to their non-linearity and harmonic content. Such excessive use of these devices can reach 50% 70% of power load in the nearest future, especially in industrial countries [5 8]. Therefore, study of converter system simulation is an effective tool to evaluate harmonic content with the aim to minimize it to acceptable levels according to standard limits-ieee Modeling of phase shift transformer represents difficulties in both AC and converter power systems [9, 10]. Some mathematical models for dynamic simulation of multi-pulse converters are reported [9, 11] in the literature. A technique was Manuscript received in final form on 28 November Address correspondence to Prof. Mazen Abdel-Salam, Electrical Engineering Dept., Faculty of Engineering, Assiut, 71516, Egypt. mazen2000as@yahoo.com 867

3 868 M. Abdel-Salam et al. described to model phase-shifting converter transformers in the harmonic domain [9]. The results show that the characteristic DC harmonics for an individual converter in a multipulse parallel connection installation increase with pulse number rather than decrease as with a series connection. Most of these techniques used iterative harmonic analysis method that have many disadavantages such as uncovergence with strongly interaction between AC and DC systems, being modeled separately sepentensery harmonic results on previous knowledge of injection current harmonics on the AC bus-bar, and also difficulties for representation of phase-shifting transformer, which require many matrices. Other models, such as time domain simulations, were used with accurate results. However, such models have suffered from long simulation time required to reach steady-state solutions [12, 13]. The symmetrical component analysis was applied before [14, 15] for power network calculations. In this article, a dynamic simulation of multi-converters is proposed based on symmetrical components, where mathematical equations are predicted for the conduction and commutation system states. This model has the following advantageous: 1. The use of symmetrical component AC analysis has several advantageous such as easier representations of unbalanced AC voltage and more power for protections and network analysis [14, 15]. 2. Simulations of static power converters as a whole part as AC power systems during unsymmetrical instantenous faults. 3. The model has a simplified representation of phase-shift transformer which suits converter representations at steady-state and during short circuit. Simplified Multi-Pulse Converter System Converter System Description A typical arrangement of a multi-pulse converter system is shown in Figure 1. It consists of 6-pulse converters connected in series or parallel through phase-shift transformers. All are connected to a common AC bus-bar, which feeds the converter system by balanced AC voltages [9, 11]. The output of the converter system is connected to a DC load. Normally, such multi-parallel converters system have the following characteristics: 1. Each branch has the same circuit elements of resistance, inductance and capacitance. 2. All converter branches have the same rating. 3. The number of parallel converter branches N and associated phase-shift angle φ between converter branches are simply expressed by: N = P/6.0 φ = 360/P where P is the pulse number of the multi-pulse converter system. Thus, the number of pulses P of the converter system is determined by the required smoothness of the system DC output voltage. Subsequently, the number of branches N and phase shift φ are determined. 4. The converter thyristors are ideally represented by short circuit during conduction and commutation periods and by open circuits during non-conduction periods.

4 Symmetrical Component Analysis 869 Figure 1. Simplified multi-pulse static power converter system. 5. Accordingly, linear circuit theories can be applied during the conduction and commutation periods of the system. General Representation of Converter System Components The three-phase equivalent circuit for each converter branch of the simplified multi-pulse converter system is shown in Figure 2. The main elements of the converter branch, Figure 2. Equivalent circuit of static power converter branch.

5 870 M. Abdel-Salam et al. as in Figure 2, are: 1. Three-phase balanced AC power supply with emfs e A, e B, and e C and impedance elements r a1 and L a1. 2. Phase-shift transformer with phase-shift angle ψ and impedance elements r b1 and L b1. 3. Three-phase converter bridge and its feeder impedance elements r c1 and L c1. 4. DC load of impedance Z L. a. Representation of AC Power Supply The voltage of the supply being assumed unbalanced purely sinusoidal, are: e A = E a sin(θ) e B = E b sin(θ 2π/3) e C = E c sin(θ + 2π/3) (1) θ = ωt + α 0 where E a, E b, E c are the voltage amplitudes in p.u., ω is the angular frequency in radians per second, and α 0 is the phase angle which determines the voltage magnitude of phase A at t = 0. b. Representation of Phase-Shift Transformer The transformer windings are Y -connected in the primary side and zigzag connected in the secondary side with a phase-shift angle ψ of the converter branch, as shown in Figure 2. The primary winding of the phase-shift transformer is assumed to have one turn while the secondary is considered as having two parts with turns N 2 and N 3, which are less than unity. N 2 and N 3 of the secondary windings are expressed [11] in terms of the phase-shift angle as follows: N 2 = cos(ψ) sin(ψ) 3 N 3 = 2 sin(ψ) 3 (2) (N 2 ) 2 + N 2 N 3 + (N 3 ) 2 = 1 c. Representation of Three-Phase Converter Bridge The three-phase converter bridge, shown in Figure 2, normally has 6-modes of conduction and commutation states during positive or negative voltage waveforms in sequence 12, 123,...,345. The analysis of conduction and commutation states remain the same irrespective of the network topology and pulse number of converter (6, 12, 18,...). Normally, the network topology changes from mode to mode.

6 Symmetrical Component Analysis 871 Symmetrical Component Analysis of Multi-Pulse Converter System The symmetrical analysis of static power converter elements are evaluated as follows. a. Symmetrical Component Analysis of AC Power Supply The positive e A1, negative e A2, and zero e A0 sequence induced emfs of the power supply are: e A1 = E m [e jθ U n e j(θ+θυ) ]/j 2.0 e A2 = E m [e jθ U n e j(θ+θυ) ]j2.0 e A0 = U n E m sin(θ θu) U n = (2Ea E b E c ) 2 + 3(E b E c ) 2 6E m (3) E m = (E a + E b + E c )/3.0 θ u = tan 1 3(Eb E c ) (2E a E b E c ) where U n, E m, and θ u are the unbalanced voltage factor, average voltage amplitude and unbalanced voltage angle respctively. b. Symmetrical Component Analysis of Phase Shift-Transformer Normally, phase-shift transformer represents difficulties in short circuit and load flow studies of AC conventional power system [10]. The following symmetrical component analysis simplifies the phase-shift transformer representations in power systems. Y-Zigzag Transformer. The output voltage (v a, v b, v c ) and input current (i A, i B, i C ) equations of the phase-shift transformer, shown in Figure 2, are expressed as: v a = N 2 v A N 3 v C v b = N 2 v B N 3 v A v c = N 2 v C N 3 v B i A = N 2 i a N 3 i b (4) i B = N 2 i b N 3 i c i C = N 2 i c N 3 i a

7 872 M. Abdel-Salam et al. where v A, v B, v C are the input voltages and i a, i b, i c are the output currents. The symmetrical components of these voltages and currents are expressed as: v a1 = (N 2 λn 3 )v A1 v a2 = (N 2 λ 2 N 3 )v A2 v a0 = (N 2 N 3 )v A0 i A1 = (N 2 λ 2 N 3 )i a1 i A2 = (N 2 λn 3 )i a2 (5) i A0 = (N 2 N 3 )i a0 (N 2 λn 3 )(N 2 λ 2 N 3 ) = 1.0 λ = e j2π/3.0 λ 2 = e j2π/3.0 where v A1, v A2, v A0 are the positive, negative and zero sequence components of input voltages; i A1, i A2, i A0 are the positive, negative and zero sequence components of input currents; v a1, v a2, v a0 are the positive, negative and zero sequence components of output voltages; and i a1, i a2, i a0 are the positive, negative and zero sequence components of output currents. Zigzag-Y Transformer. The input voltage and output current equations of the phase-shift transformer can be written in the same way as the (Y/Z) transformer. c. Symmetrical Component Analysis of Converter during Conduction State The conduction state equations of Figure 3 are: v ar = v a v d = v br v cr v br v cr = Z L i b i b = i c (6) i a = 0.0 i d = Magnitude of (i b ) where v ar, v br, v cr, and v d are the input and DC voltages of three-phase rectifier bridge. The symmetrical equations are: v a1r v a2r = Z L i a1 i a1 = i a2 (7) i a0 = 0.0

8 Symmetrical Component Analysis 873 Figure 3. Conduction and commutation states of static power converter branch. where v a1r and v a2r are the positive and negative sequence input votage of three phase rectifier bridge. The sequence network connection during conduction period, Figure 4, gives: v A1 = e A1 (L a1 + L b1 )(di A1 /dt) (r a1 + r b1 )i A1 v A2 = e A2 (L a1 + L b1 )(di A2 /dt) (r a1 + r b1 )i A2 (8) From Eqs. (5), (7), and (8), a set of differential equations is obtained whose solution determines the symmetrical components: i a1 = N 2E m jz T [cos(θ θ ZT ) cos(α θ ZT )e (R T /L T )t ] N 3E m jz T [cos(θ ± 2π/3 θ ZT ) cos(α ± 2π/3 θ ZT )e (R T /L T )t ] N 2U n E m jz T [cos(θ + θ u θ ZT ) cos(α + θ u θ ZT )e (R T /L T )t ] + N 3U n E m [cos(θ + θ u ± 2π/3 θ ZT ) cos(α + θ u ± 2π/3 θ ZT )e (R T /L T )t] jz T + j( 3/3)I 1 e (R T /L T )t i a2 = i a1 (9) Figure 4. Sequence network connection of converter branch during conduction period.

9 874 M. Abdel-Salam et al. where +2π/3.0 in a positive phase shift angle of Y/Z or Z/Y transformer; 2π/3.0 in a negative phase shift angle of Y/Z or Z/Y transformer and I 1 is the initial current at the beginning of the conduction period. R T = 2(r a1 + r b1 + r c1 ) + R L L T = 2(L a1 + L b1 + L c1 ) + L L Z T = (R T ) 2 + (X T ) 2 X T = ωl T θ ZT = tan 1 (X T /R T ) Thus, the conduction current equations are expressed as: i a = 0.0 i b = λ 2 i a1 + λi a2 = (λ 2 λ)i a1 = j 3i a1 (10) i c = i b d. Symmetrical Component Analysis of Converter during Commutation State The commutation state equations of Figure 3 are: v ar v br = v d v ar v br = Z L i a v br = v cr (11) i a + i b + i c = 0.0 i d = Magnitude of (i a ) The symmetrical voltages and currents equations are: v a1r = v a2r v a1r = Z L (i a1 + i a2 )/3.0 (12) i a0 = 0.0 The sequence network connection during commutation period, shown in Figure 5, gives: v a1r = v a2r = (L L /3.0)(di a1 /dt + di a2 /dt) + (R L /3.0)(i a1 + i a2 ) (13)

10 Symmetrical Component Analysis 875 Figure 5. Sequence network connection of converter branch during commutation period. Solution of this differential Eq. (13) gives the symmetrical components: i a11 = N 2E m 2Z 3 [sin(θ θ Z3 ) sin(α θ Z3 )e (R 3/L 3 )t ] N 3E m 2Z 3 [sin(θ ± 2π/3 θ Z3 ) sin(α ± 2π/3 θ Z3 )e (R 3/L 3 )t ] + N 2U n E m 2Z 3 [sin(θ + θ u θ Z3 ) sin(α + θ u θ Z3 )e (R 3/L 3 )t ] N 3U n E m 2Z 3 [sin(θ + θ u ± 2π/3 θ Z3 ) sin(α + θ u ± 2π/3 θ Z3 )e (R 3/L 3 )t ] i a12 = N 2E m j2z 1 [cos(θ θ Z1 ) cos(α θ Z1 )e (R 1/L 1 )t ] N 3E m j2z 1 [cos(θ ± 2π/3 θ Z1 ) cos(α ± 2π/3 θ Z1 )e (R 1/L 1 )t ] N 2U n E m j2z 1 [cos(θ + θ u θ Z1 ) cos(α + θ u θ Z1 )e (R 1/L 1 )t ] + N 3U n E m j2z 1 [cos(θ + θ u ± 2π/3 θ Z1 ) cos(α + θ u ± 2π/3 θ Z1 )e (R 1/L 1 )t ] i a1 = i a11 + i a I 2 e (R 3/L 3 )t j( 3/6)I 2 e (R 1/L 1 )t i a2 = i a11 i a I 2 e (R 3/L 3 )t + j( 3/6)I 2 e (R 1/L 1 )t (14)

11 876 M. Abdel-Salam et al. where I 2 is the initial current value at the beginning of commutation state, R 1 = r a1 + r b1 + r c1 L 1 = L a1 + L b1 + L c1 R 3 = R 1 + (2/3)R L L 3 = L 1 + (2/3)L L Z 1 = (R 1 ) 2 + (X 1 ) 2 Z 3 = (R 3 ) 2 + (X 3 ) 2 X 1 = ωl 1 X 3 = ωl 3 θ Z1 = tan 1 (x 1 /R 1 ) θ Z3 = tan 1 (x 3 /R 3 ) Thus, the commutation current equations are expressed as: i a = i a1 + ia2 i a = 2i a11 + I 2 e (R 3/L 3 )t i b = λ 2 i a1 + λi a2 i b = i a11 j 3i a12 0.5I 2 e (R 1/L 1 )t 0.5I 2 e (R 3/L 3 )t (15) i c = λi a1 + λ 2 i a2 i c = i a11 + j 3i a I 2 e (R 1/L 1 )t 0.5I 2 e (R 3/L 3 )t e. Initial and Boundary Conditions of Converter Branch The initial and boundary conditions between subsequent states (either conduction or commutation state) of converter branch are taken from the final values of the preceding state in a manner as to ensure magnetic flux and electric charge continuity [16]. This means continuity of currents through inductors and charges across capacitors during change of converter states from conduction to commutation and vise versa. f. Bus-Bar Current and Voltage Equations of the Multi-Pulse Converter System The temporal variation of currents and voltages of the multi-pulse converter system, shown in Figure 1, are obtained as follows: 1. Determine the currents and voltages of each converter branch at a given instant according to its states (either conduction or commutation states).

12 Symmetrical Component Analysis Determine the bus-bar currents (i AS, i BS, and i CS ) of the multi-pulse converter power supply by adding the currents of all the converter branches at the same instant. 3. Determine the bus-bar current derivatives (di AS /dt, di BS /dt, and di CS /dt) of multi-pulse converter power supply by adding the current derivatives of each converter branch at the same instant. 4. Determine the bus-bar voltages of the multi-pulse converter system as: v AB = e A L a1 (di AS /dt) r a1 i AS v BC = e B L a1 (di BS /dt) r a1 i BS (16) v CB = e C L a1 (di CS /dt) r a1 i CS 5. Subsequently, the input voltages of each phase shift transformer as: v A = v AB L b1 (di A /dt) r b1 i A v B = v BB L b1 (di B /dt) r b1 i B (17) v C = v CB L b1 (di C /dt) r b1 i C g. DC Side Calculations i. DC Connected Bridges in Series. i DC = i d v DC = ii. DC Connected Bridges in Parallel. N (18) (v d ) i i=1 v DC = v d N (19) i DC = (i d ) i i=1 Computer Algorithm The flow chart of computer program is shown in Figure 6. The following steps describe the procedure followed in building a computer program for solving the above describing differential equations. At the instant of switching on, the converter bus-bar voltages are equal to the supply emfs, and the following values are computed after a time increment: 1. The rectifier currents of each converter branch and their derivatives.

13 878 M. Abdel-Salam et al. Figure 6. Flow chart of computation steps for multi-pulse power converters. 2. The input currents of each converter branch and their derivatives. 3. The total currents and their derivatives of all converter branches. 4. The bus-bar voltage of the multi-pulse converter system. 5. The input voltage of each phase-shift transformer. 6. The output voltage of each phase-shift transformer. With further time increments, steps 1 6 are repeated until a predetermined period of time is elapsed. The harmonic content of such a system is calculated according to digital fourier analysis.

14 Symmetrical Component Analysis 879 Results and Discussion 6-Pulse Converter To indicate the capability of the proposed simulation method, the computed results are compared with those measured before [8] for 6-pulse converter system with /Y phaseshift transformer. The system parameter calculations are given in Appendix A and their estimated values in per-unit are: E m = 1, r a1 = 0.02, r b1 = 0.05, r c1 = 0.02, R L = 2.03, L a1 = 8.750e 05, L b1 = 2.600e 5, L c1 = 1.590e 04, L L = 1.00e 03, and conducting angle = 15 degrees. The base values of the system are: 1. Base values at point of common coupling side are E m = 460 2, kva Base = 1.4, and base current = amp. 2. Base values at load bus side are E m = 190 2, kva Base = 1.4, and base current = amp. The current and voltage waveforms of 6-pulse converter at the input and output of /Y transformer are compared with those measured experimentally [8] (Figures 7 and 8). Tables 1 and 2 give the computed and measured values of the total harmonic distortion voltage THD V and current THD I distortion. The computed values of the characteristic harmonics agree reasonably with those measured experimentally. Also, the computed Figure 7. Current and voltage waveforms of 6-pulse converter at the input of /Y transformer (a) computed results, (b) measured results of reference [8].

15 880 M. Abdel-Salam et al. Figure 8. Current and voltage waveforms of 6-pulse converter at the output of /Y transformer (a) computed results, (b) measured results of reference [8]. Table 1 Harmonic content of 6-pulse converter at the input of /Y transformer Computed values Measured values in % Harmonic in % a of reference [8] b order Current Voltage Current Voltage a THD I = 25.68%, THD V = 4.94%. b THD I = 25.00%, THD V = 5.2%.

16 Symmetrical Component Analysis 881 Table 2 Harmonic content of 6-pulse converter at the output of /Y transformer Computed values Measured values in % Harmonic in % a of reference [8] b order Current Voltage Current Voltage a THD I = 26.04%, THD V = 6.9%. b THD I = 27.00%, THD V = 5.7%. total harmonic distortion values of the current and voltage also agree resonably with those measured experimentally. Morever, the computed results are compared with Kimbark equation [6] during commutation periods. I d = (V / 2X c )[cos(α) cos(α + µ)] (20) where V is the rms line-line voltage, α is the conduction angle, µ is the overlap angle, and X c is the commutation reactance. According to Eq. (20), I d = amp for α = 15 and µ = and the computed current of proposed simulation method = amp. The results are fairly acceptable and the difference is small due to the fact that Eq. (20) assumes that the DC current I d is ripple free and neglects circuit resistance. The difference between computed and measured results may be attributed to neglecting no-load current which contains large harmonic content, using estimated system parameters, little firing error or little unbalance in voltage and system parameters, and limited accuracy measuring instruments. 12-Pulse Converter For 12-pulse converter with Y/Y and Y/ converter transformers of phase-shift angle ψ of 0 and 30 degrees, the phase shift φ between converter branches is 30 degrees. The power supply of the system has unbalaced factor 5.4% and their unbalanced emfs in % are: E a = 88 0, E b = , and E c = The estimated impedance parameters of the system are L a1 = 5e 05, L b1 = 1.00e 05, L L = 8.50e 02, r a1 = 0.001, r b1 = 0.001, and R L = The AC input currents of 12-pulse converter and their harmonic contents for unbalanced power supply are compared with those obtaind by EMTP computer simulation

17 882 M. Abdel-Salam et al. Figure 9. AC input currents of 12-pulse converter feeding from unbalanced power supply (a) computed results, (b) previous results of reference [17]. package, as shown in Figures 9 and 10 [17]. The results are fairly acceptable and the little differences are due to the use of estimated system parameters. Also, the DC output votage of the system is comparing by EMTP results of a previous reference [17] (Figure 11). The results are fairly reconizably acceptable. Unbalanced AC Voltages on Multi-Pulse Converter Percentage of unbalance in AC voltage U n was varied from 5% to 40%. The results of zero sequence harmonics of order 3, 6, 9, 15 for both 6-pulse and 12-pulse converters are shown in Figure 12. The values of the 3rd and 9th harmonics increase sharply with the percentage of unbalance voltage when compared with the 6th and 15th harmonics (Figure 12). Also, the values of zero sequence harmonics are reduced with increasing pulse-number at the same level of unbalance in AC voltage. The DC output voltage harmonic values are plotted in Figure 13 against the percentage unbalanced in AC voltage for both 6-pulse and 12-pulse converters. The unbalance in the AC voltages of multi-pulse converters increases the even harmonics of the DC output voltage, such as 2nd, 4th, and 6th. The values of 2nd harmonic are increased sharply as compared to other even harmonics, and also, the even harmonic become less with increasing converter pulse number. Conclusions A generalized modeling technique to represent multi-pulse converter systems is presented in this article. This technique is based on symmetrical network analysis during

18 Symmetrical Component Analysis 883 Figure 10. Harmonic currents of 12-pulse converter feeding from unbalanced power supply (a) computed results, (b) results of reference [17]. conduction and commutation states by using general equations of multi-pulse converters system. The computed results are fairly acceptable with respect to previous experimental and theoretical investigations for 6-, 12-pulse converters. This system simulation is simpler and has a capability to represent in time-domain multi-pulse converters, taking into account the unbalanced power supply, source impedance and the smoothing DC reactor. The main merit of such systems is the reduction of harmonic currents and voltages with the increase of pulse number. The 3rd and 9th harmonics of the AC input currents and 2nd harmonic of the DC output voltage are very sensitive for increasing of the unbalanced AC power voltages system. Appendix A Estimated Parameters of Six-Pulse Converter The parameters calculation are based on the following equations [5]: P = 2VIcos(φ) = V d I d I d = 1.22I (A.1) (A.2)

19 884 M. Abdel-Salam et al. Figure 11. DC output voltage of 12-pulse converter with unbalanced power supply (a) computed results, (b) previous results of reference [17]. where V is the RMS line-to-line voltage, I is the RMS line current, cos(φ) is the power factor, V d is the DC voltage, and I d is the DC current. Load Bus Parameters (V d, I d, R L ) From reference [8], one obtains: Base VA = 1400 VA Base voltage = 190 volts Base impedance = Z B = (190) 2 /1400 = ohm From reference [8], the following measuring data are given: S = 1302 VA at load bus I L = = amp 3 Power factor = at load bus I d = = amp V d = S Power factor/i d = /4.826 = volts R L = V d /I d = /4.826 = ohm R L = R L /Z B = / = 2.03 p.u.

20 Symmetrical Component Analysis 885 Figure 12. Zero sequence harmonics against unbalanced AC voltage (a) 6-pulse converter, (b) 12- pulse converter. Feeder-to-Bridge Parameters (L c1, r c1 ) From reference [5]: Power Factor = 0.5[cos(α) + cos(α + µ)] (A.3) where α is the conduction angle (= 15 degrees), µ is the overlap angle. cos(α + µ) = 2 cos(φ) cos(α) I d = (V / 2X c )[cos(α) cos(α + µ)] (A.4) (A.5)

21 886 M. Abdel-Salam et al. Figure 13. DC output voltage harmonics against unbalanced AC voltage power supply (a) 6-pulse converter, (b) 12-pulse converter. where cos(α + µ) = = X c1 = V 2Id [cos(α) cos(α + µ)] (A.6) 190 X c1 = [ ] =1.67 ohm X c1 = X c1 Z B = = 0.06 p.u. L c1 = X c1 /ω, ω = 2πf L c1 = 1.59e 04 for f = 60 HZ r c1 = 0.02

22 /Y Transformer Parameters (L b1, r b1 ) From reference [8], one obtains: Symmetrical Component Analysis 887 S = 1375 VA Power factor = Transformer losses = 3I 2 r b1 = = 61 watts r b1 = 61/3(3.956) 2 = 1.3 ohm r b1 = 1.3/ = 0.05 p.u. cos(α + µ) = = Z B = X CT = [ ] =1.949 ohm X CT = X CT = 0.07 p.u. X b1 = X CT X c1 = = 0.01 p.u. L b1 = X b1 /ω, ω = 2πf L b1 = p.u. Source Impedance (L a1, r a1 ) From reference [8], one obtains: X a1 = 2.0/60 = p.u. L a1 = r a1 = 0.02 References 1. J. Arrilaga, D. A. Bradely, and B. S. Bodger, Power System Harmonics, NewYork: John Wiley & Sons, S. Hansen, P. Nielsen, and F. Blaabjerg, Harmonic cancellation by mixing nonlinear singlephase and three-phase loads, IEEE Trans. on Industry Applications, vol. 36. pp , D. A. Bradely, Power Electronics, UK: Book Society/Van Nostrand Rienhold, D. O. Kelly, Differential firing-angle control of series-connected HVDC bridges, Electric Power System Research, vol. 20, pp , J. Arrilaga, High Voltage Direct Current Transmission, London, UK: Peter Peregrinus Ltd., E. W. Kimbark, Direct Current Transmission, New York: John Wiley & Sons, D. A. Paice, Power Electronic-Multipulse Methods for Clean Power, IEEE Press, New York, C. J. Wu and W. N. Chang, Developing a harmonics education facility in a power system simulator for power education, IEEE Trans. on Power Systems, vol. 12, pp , G. N. Bathurst, B. C. Smith, N. R. Watson, and J. Arrillaga, Harmonic domain modeling of high pulse converters, IEE Proc., Electric Power Appl., vol. 146, pp , R. D. Youssef, Phase-shifting transformers in load flow and short-circuit analysis: Modelling and control, IEE Proceedings-C, vol. 140, pp , R. Yacamini and J. C. de Oliveira, Harmonics in multiple convertor systems a generalised approach, IEE Proceedings, vol. 127, pt. B, pp , G. Carpunell, F. Gagliardi, M. Russo, and D. Villaco, Generalised converter models for iterative harmonic analysis in power systems, IEE Proc. Gener. Transn. & Distr., vol. 141, pp , B. K. Parkins and M. R. Iravani, Novel calculation of HVDC converter harmonics by linearization in the time-domain, IEEE Trans. on Power Delivery, vol. 12, pp , 1997.

23 888 M. Abdel-Salam et al. 14. G. C. Paap, Symmetrical components in time domain and their application to power network calculations, IEEE Trans. on Power Systems, vol. 15, pp , W. D. Stevenson, Elements of Power System Analysis, McGraw-Hill Book Company, Inc., B. T. Ooi, N. Menemenlis, and H. L. Nakra, Fast steady-state solution for HVDC analysis, IEEE Trans. on Power Apparatus and Systems, vol. PAS-99, pp , G. Paullilo, C. A. M. Guimaraes, J. Policarpo, G. Abreu, and R. A. Olivera, Reducing harmonics in multi-converters systems under unbalanced voltage supply A novel transformer topology, IEEE Power Engineering Society Summer Meeting, IEEE CD-ROM-00H DATA/34_4. pdf, 2000.