First characteristic harmonic component of output ripple on dc railway rectifiers

Transcription

1 First characteristic harmonic component of output ripple on dc railway rectifiers J. Allan, J. H. Jin & K. Payne ^ The University ofbirmingham, UK. ^London Underground Limited, UK. Abstract A research project is being carried out by the University of Birmingham, sponsored by London Underground Ltd, to research into a new earth fault detection system on a fourth rail power supply system. It is proposed that the fault detection system will make use of the first characteristic harmonic component of ripple on the output side of the substation rectifiers. It was therefore necessary to have knowledge of the variation in the voltage of the first harmonic component of the ripple with load current on the output side of the substation rectifier. Analytical and simulation studies were carried out, which revealed that the variation in first characteristic harmonic component of substation ripple with load current did not appear to match classic theory where load inductance is assumed to be infinite. The analysis presented in this paper takes account of variable load inductance in order to explain the mismatch. Both the simulation and calculated results have shown that the first harmonic of load voltage with finite load inductance behaves differently from the theoretical calculation where it is assumed that the load current is smooth. The analysis technique provides a sufficiently accurate prediction, which simpler analysis with an assumption of infinite inductance fails to do. List of symbols Instantaneous values H» % Vc Phase voltages /*, 4, ic Phase currents

2 Vd Direct voltage id Direct current id" Direct current during commutation ic Commutation current Vm Amplitude of Voltage Mean values Vd Direct voltage Id Direct current R. M. S. values E R.m.s. value of phase voltages Vdn The r.m.s value of nth harmonics of vd Parameters R Resistance L Inductance Lc Commutation inductance Xc Commutation reactance Rd Load Resistance Ld Load inductance n Harmonic order Angles (o Angular frequency of alternating voltage ji Commutation angle t2-tl Commutation time cot Angular interval at supply frequency a Delay angle a CC+ILI c )l,<()2 Phase angle 1 Introduction Computers in Railways VII The results presented here have been determined as a stage in development of a fault detection system for a DC rapid transit railway application. It is proposed that this system will make use of the first harmonic component of ripple on the output side of the sub-station rectifiers. It was therefore necessary to have a knowledge of the variation in the voltage of thefirstharmonic component of the ripple with load current. A detailed model of a 12-pulse series bridge was developed in SABER [1] and a Fourier transform was applied to the resulting output. The variation in thefirstcharacteristic harmonic voltage with load current for a moving train seen in Figure 1 did not seem to match classical theory in which it is assumed that the load inductance is infinite [2]. This assumption is also present in a recent paper on harmonic analysis of a parallel-connected 12- pulse uncontrolled rectifier [3]. Experimentation with the simulator showed that the variation was particularly dependent on the load inductance. In practise, the load inductance could be small at high train speed and does vary with train position. An analysis was carried out by deriving the waveform as a function of

3 Computers in Railways c a I c u LatedTU=X). 2 m H) <- calculated (L=oo ld(a) Figure 1: 600Hz Component of Output Voltage Versus Load Current time, taking into account thefirstmode of operation of the rectifier [4], overlap and the inductance (and resistance) of the load. The 600 Hz component of the time-based waveform was derived. 2 DC Side Harmonics with Infinite Load Inductance Consider an uncontrolled 12-pulse rectifier which consists of two 6-pulse bridges connected in series as shown in Figure 2, voltage Vd is the sum of Vdj and Vd2. In this case, the analysis of DC side harmonics of the 12-pulse rectifier can be derived from that of a 6-pulse bridge. Figure 3 shows the 6-pulse rectifier circuit. Figure 4 shows the equivalent circuit before and during commutation. Figure 4 (a) shows the reduced circuit when only two diodes Dl and D2 are conducting. During this time there are no voltage drops across the commutation reactances, The load voltage Vd is the difference between the terminal voltages, that is, Vd = Va - Vc = Vac. The switch of current from Dl to D3 cannot occur instantaneously because of the inductors in the circuit. There are voltage drops across the inductors due to the current variation, i.e., the inductances restrict the rate of change of the current. Therefore, afinitetimefe-ti),called the overlap or

4 530 Computers in Railways VII Figure 2: Two 6-pulse Circuits Connected in Series % Vd Figure 3 three-phase six-pulse rectifier circuit ±

5 Computers in Railways "S Figure 4: Equivalent Circuits for Commutation (a) Before (b) During Commutation , Figure 5 Load Voltage of First Mode Operation with Infinite Load Inductance and Commutation Angle p = 71/6, Vm=331V commutation time (Figure 5), is required for the current to transfer from one phase to another. The equivalent circuit during commutation is shown in Figure 4 (b) with three diodes conducting. The angle co(t2-ti) is called the commutation angle, which is usually denoted as u. For a 6-pulse rectifier in mode I, the r.m.s. magnitudes of the harmonics voltages of the DC voltage can be obtained from the following expression, assuming ripple free DC load current, i.e. infinite load inductance:

6 Computers in Railways VII In which, n is the order of the harmonics, n = 6k, k=l, 2, 3,..., ji is the commutation angle (0<p<%/3), Vm is the maximum value of the source voltage, and Vdn represents the nth harmonic of Vd. Hence, the 600 Hz harmonic of the load voltage of a 12-pulse rectifier Vd600 should be 2*Vd,2, from which the variation of Vd600 versus Id is plotted in Figure 1. It can be seen that Vd600 increases with load current Id. However, in practice, the load inductance is finite in a power supply system for railway traction [5]. Furthermore the load inductance will vary with train movement. Therefore, the variation in the harmonic voltage with load current has been researched for the condition of finite load inductance (and load resistance). 3 DC Side Harmonics with Finite Load Inductance Guided by Figure 4, by using Maxwell's loop methods, three equations can be obtained: ^ ((2Lc + Ld)D + R)id = Va-Vc = V6Esin(W + -;r) (2) /\ (2Lc + Ld)D + R)id"-LcDic = Va-Vc = y/6esm(cot + -ar) (3) - LcDid"+2LcDic = Vb-Va = 46E sin a* (4) Where E is the r.m.s. value of source voltage, D is the operator d/dt. The solutions for id, id and ic take the form 2. \. -fhr"), a - re/3 < cot < a (5) -- Ae ^ ' ' ^ ' 2Z, *"Y*~*"2 *J 1 fflj? ic= id COS6Y 4-C, a < cot < a (^) 2 2A", in which a is the delay angle and JLI is the commutation angle. While the resistance is the major part of the load, a delay angle a occurs even in the first mode, a = a + a is used for convenience to denote the start of thefirstsingle conduction period. Xi = (2Lc+Ld)co tan ^ = X,/R (8) It is necessary to determine a the point at which overlap begins. This can occur only where the terminal voltage of the rectifier intersects with the terminal voltage of the next phase to conduct current: Va - LcDid = Vb, that is 6 which is satisfied at cot = a.

7 Computers in Railways VII 533 Ztrack(DC) Rectifier Rbp Ztrain 0 Q RBN 100 Ztrack(DC) Figure 6 Simplified Circuit with Moving Train Load After making necessary substitutions, the following equation can be obtained: cole. (n.\ ARL^co (T»] sma = sin + a-^ +,- e ^ Z, 16 ^J V6&Y, (11) To determine the constant A, B and C and the angle a and ju, the necessary conditions is used that id at the beginning of single conduction is equal to id at the end of overlap, and id at the end of single conduction is equal to id at the beginning of overlap. A numeric method was applied, which used MATLAB programming, to solve the equations and obtain the instantaneous waveform. A discrete Fourier transformation is then used to obtain the harmonics of both Vd and Id 4 Simulations Simulations were carried out using the 12 pulse rectifier model shown in Figure 2, which composed of a series-connected uncontrolled 12-pulse rectifier, a resistor (R) and an inductance (L), with commutation inductance Lc= 47uH and a deliberately small load inductance L=5pH. Also a realistically dimensioned inductance for a train and a variable inductance representing the load under the fault conditions being studied. Figure 6 shows a simplified circuit with a moving train load, with the models for the track and train as following: Ztrack = ( mh) per kilometre, Ztrain = (0.2 Q mh) per kilometre. It can be calculated from Figure 6 that, when the train is moving from substation to 1 km from the substation, the range for equivalent load resistance Rd will be

8 534 Computers in Railways VII approximately from 0.2 to Q, whilst for the load inductance Ld, the range is from 0.2m to 2.3m H. These values are used in the calculation and simulation for the case with train moving. The first characteristic harmonic components Vd600 and Id600 can be obtained by applying an FFT transform to the simulated waveforms Vd and Id from SABER. 5 Results The calculated and simulated results with two finite load inductances are shown in Figure 1. The graph with smaller inductance shows that the 600 Hz component of voltage can decrease with increasing load current. The analysis has also been applied to the prediction of the case of variable load due to train movement. The 600 Hz component of voltage also decreases with increasing load current. The match between the analysis and the simulation is good. 6 Conclusion The simulation and calculated results have shown that thefirstharmonic of load voltage with finite load inductance behaves differently from the theoretical calculation where it is assumed that the load current is smooth. The analysis technique described provides a sufficiently accurate prediction, which simpler analysis with an assumption of infinite inductance fails to do. 7 Acknowledgements The authors would like to thank London Underground Ltd. for its support in this work. References: [1] Goodman C J and Zhang Z, "Simulation of traction rectifier using proprietary simulators", IEE Conference Publication, 405, pp , [2] Arrillaga, J, Bradley D A and Bodger P S, "Power system harmonics", JOHN WILEY & SONS, (location), 1985 [3] Tzeng Y S, "Harmonic analysis of paralled-connected 12-pulse uncontrolled rectifier without an interphase transfromer", DEE Proc. Electr. Power Appl. Vol. 145, No.3, pp , [4] Kassakian J G, Schlecht M F and Verghese G C, "Principles of power electronics", Addison-Wesley Publishing Company, [5] Brown J C, Allan J and Mellitt B, "Calculation and measurement of rail impedances applicable to remote short circuit currents", IEE Proceedings-B, Vol. 139, No.4, pp , July 1992.