POWER electronic converters contribute significantly to

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1 A Time-Domain Method for alculating Harmonics Produced by A Power onverter K.. ian Member, IEEE and P. W. ehn, Senior Member, IEEE PEDS2009 Abstract In general, power converters are nonlinear and time varying devices. Power electronic converters contribute significantly to the harmonic pollution of transmission systems. Being able to predict accurately the values of harmonics injected by the converters is essential for the power system planning and studies. Power converters can be classified into two types line-commutated and force-commutated converters. In this paper, a unified time-domain method is presented to show how the harmonics produced by a line-commutated or a forcedcommutated circuit can be easily calculated, provided that the switching times are known. All the results are validated with /EMTD. urrent or Voltage i ac v ac Index Terms Diode Bridge ectifier, Harmonics, ontinuous onduction Mode, Discontinuous onduction Mode, Steady State Analysis. Mode I Mode II I. INTODUTION POWE electronic converters contribute significantly to the harmonic pollution of transmission systems. Being able to predict accurately the values of harmonics injected by the converters is essential for the power system planning and studies. Power electronic converters are in general nonlinear and time-varying devices 1. A simple example illustrates this fact. onsider the single-phase capacitor filtered diode bridge rectifier, shown in Figure 1, whose typical (balanced) ac current waveform, i ac, is shown in Figure 2. 0 γ μ T/2 T/2+γ T Time Fig. 2. Single phase diode voltage and current waveforms. for the dc voltage harmonics, V dc. Therefore, the following equation can be written: ( ) Iac Vac = f,t V dc I sw. (1) dc Or v ac Fig. 1. i ac + vdc Single-phase capacitor filtered diode bridge rectifier - load As can be seen from Figure 2, the current waveform, i ac, is greatly dependent on the time the diodes turn on (i.e. γ), and the duration the diodes remains on (i.e. μ). The shorter the value of μ, the narrower the current pulse, and hence, the more distorted the current waveform. Thus, the ac current harmonics, I ac, are not only a function of inputs (V ac and I dc ), but also the switching times, t sw 1. Similar argument holds true K.. ian is with the Power and Energy Group of National Taiwan University of Science and Technology, Taipei, Taiwan ( ryankuolian@ieee.org). P. W. ehn is with the Energy System Group of University of Toronto, Toronto, ON M5S 3G4, anada ( lehn@ecf.utoronto.ca). 1 t sw = γ μ T H out = f (H in,t sw ), (2) where H out represents the output harmonics and H in represents the input harmonics. To determine the switching times, t sw, switching time constraint equations generally need to be solved. For the rectifer, the values of γ and μ are determined by solving the following constraints: (γ) v ac (γ) = 0 (3) i ac (γ + μ) = 0 (4) Equation (3) and (4) indicate that the switching times, γ and μ are, in turn, a function of both the stimulus vector, Vac I dc T, and the state vector Iac V dc T. Thus, one can write that Or ( ) Iac Vac t sw = g,. (5) V dc I dc ombing (2) and (6) leads to t sw = g (H in,h out ), (6) 528

2 H out = f (g (H out,h in ),H in ). (7) Although equation (7) is derived for a line-commutated circuit, it is also applicable to a forced-commutated convertor such as a voltage source converter. A forced-commutated converter is a switching circuit, so it satisfies (2). Moreover, a forced-commutated converter will also have the form of (6) due to feedback controls 1. Equation (7) is a nonlinear equation and in general does not have an analytical solution. However, if the switching times are known or the switching times are independent of the input and output harmonics, equation (7) reverts to the form of equation (2). An analytical input-output formulation can be obtained and can be written as H out = FH in. (8) Section II briefly describes how F (also called the frequency coupling matrix 2, 3)can be analytically obtained in the time domain. This is different from the state of the art, which evaluates F in the frequency domain 2-7. Section III shows some of the results evaluated by (8), and they are compared with those obtained by / EMTD. Finally, a conclusion is given in section IV. II. UNIFIED TIME-DOMAIN METHOD Assume that the converter system to be analyzed is composed of linear components and ideal switches. For a given set of switching times, t 1, t 2, t 3,, t n, a linear state-space differential equation can be written for the interval between t k 1 and t k. ẋ k 1 = A k 1 x k 1 + B k 1 u k 1, (9) where x k 1, A k 1, B k 1,andu k 1 are the state variable vector, state matrix, input matrix, and input vector between t k 1 and t k. Expressing the input vector in terms of a linear combination of the ideal oscillators 2, one can turn (9) to an augmented autonomous equation 8, 9: x k 1 = Âk 1 x k 1, () x Ak 1 F where x k 1 =,  z k 1 = k 1, F 0 Ω k 1 is the corresponding input matrix, Ω = diag(0, Ω 1,, Ω m ), 0 indicates a zero matrix, and m = integer. Equation () is easier to be solved as compared to (9) because it is an autonomous equation and does not involve convolution integral. One can further exploit the concept of augmentation for evaluating the harmonics of the system. According to 9, the Fourier coefficients of any periodic signal, g(t) can be evaluated by solving the following differential equation over one period T : dy = my + hg(t), (11) dt 2 The state space equation of an ideal oscillator is ż =Ωz, whereż = zx 0 mω, Ω=, ω is the radian frequency and m is the zy mω 0 integer. where m = jnω and h = 1 T e jnωt. Augmenting (11) to (), one will obtain the following equation: x k 1 = Ãk 1 x k 1, (12), à k 1 =. where x k 1 = x z A k 1 F k Ω 0 y H 0 M When one evaluates (12) successively for each interval over the entire period T, there are two situations which he may encounter: A. The state variables are the same for each interval This situation can be seen from an ideal voltage source converter 3 (VS). Fig. 3 shows a three-phase voltage source converter whose state variables (x = T i a i b i c ) do not change throughout the switching intervals 11. The circuit changes are reflected from the switching functions 12, which represents the status of the switches. Since the VS circuit is linear during each interval, the solution of (12) over T for the VS is given by (13). x(t )= Φ x(0), (13) where Φ =eãn(t tn) eã2(t2 t1) eã1(t1 0). Fig. 3. v sc i ta i tb i tc Sa Sb Sc Sa' Sb' Sc' Three-phase ideal VS with dc storage capacitor B. The state variables are not the same for each interval This situation can be seen from a three-phase diode bridge rectifier. Fig. 4 shows a three-phase diode bridge rectifier whose typical waveform is shown in Fig. 5. Fig. 6 shows the circuit involved during the conduction (diodes D 1 and D 6 turn on) and non-conduction (diodes turn off) intervals. During the conduction interval, the state variables x on is T id, while the state variables xoff is during the non-conduction interval. Thus, the size of state matrix of the conduction interval à on is different from that of the non-conduction interval à off. To evaluate the harmonics of such a system, one needs to use the concept of projection and injection matrices 13, 14. The projection matrix (P ) is associated with the change of basis from conduction to non-conduction interval whereas the injection matrix (Q) is associated with the change of basis at the transition instant 3 An ideal VS is the one whose switches are switched in a complementary fashion

3 from non-conduction to conduction interval. Thus, the solution of (12) over T for the three-phase rectifier is given by x(t )= Φ x(0), where Φ =eãoff (γ) PeÃon(η) Q. n Fig. 4. urrent Fig. 5. v sc i a i b i c A three-phase diode bridge rectifier with capacitive dc smoothing ξ η ξ η ξ η ξ η ξ η ξ η 0 γ T/6+ γ T/2 T Time A typical waveform of a three-phase phase diode bridge rectifier i d i d D 1 l D 6 Fig. 6. (a) ectifier model during the conduction subinterval; (b) rectifier model during the non-conduction subinterval To obtain the harmonics of the system, one needs to ensure the system is at steady state, which requires x(t )= x(0). (14) N l i a i b i c l As proved in 15, Φ has the following unique structure: Φ = A p N p 0 0 Ω p 0, (15) H p Q p M p where subscript, p represents the product term. Expanding (12) and applying (14) to the equation, one will obtain (16) 15: y(t )={H p (I A p ) 1 N p +Q p }z(0) (16) Equation (16) relates the output harmonics 4 to the input initial conditions, rather than the output harmonics to the input harmonics. A one-to-one relation exists between the initial conditions of the ideal oscillators and its Fourier coefficients. Thus, one can turn (16) to (8) by a simple linear transformation matrix 15. For example, an input voltage harmonic phasor, V s = A k θ k can be expressed as V s = A k cos(θ k )+ ja k sin(θ k ). Noting that A k cos(θ k ) and A k sin(θ k ) are also the initial conditions of the ideal oscillator, one can obtain the relation between the input harmonic and the input initial conditions, as stated in (17). e{v k s } Im{Vs k = } Ak cos(θ k ) A k sin(θ k ) = zx (0) z y (0). (17) III. SIMUATION EXAMPES A. Example 1 Fig. 7 shows a VS with the reactive current (i tq ) control loop. The reference value of the reactive current, i tdref is set to be 30A and the dc voltage is regulated at 230V.The switching frequency is set to be 1620 Hz. The other values of the system parameters and the control strategy are the same as those of 16. To obtain the harmonics of interest by means of (8), one needs to know the switching times, t sw, first. This can be achieved by using the algorithm developed by 17. Then, t sw is substituted to F of (8) and the resulting F is multiplied with the input harmonics to get the output harmonics. Fig. 9 compares the ac current harmonics (in space vector) and dc voltage harmonics obtained by /EMTD and those obtained by the proposed method when the input only contains the fundamental ac voltage harmonics. As the figures show, they are in good agreement, showing the validity of the proposed method. B. Example 2 The second example simulates the three-phase diode bridge rectifier in Fig. 4. The system parameters are as follows: = 0.001Ω, =0.1mH, = 00μF, l = 25Ω, and source voltages: = sin(377t) = sin(377t 2π 3 ) v sc = sin(377t + 2π 3 ) Similar to Example 1, one needs to first find out the values of the switching intervals: γ, η, andξ. By applying the algorithm 4 y(t ) is equal to H out. 530

4 es θ s i t η = ,andξ = Thesevalues are substituted to F of (8) and the resulting F is multiplied with the input harmonics to get the output harmonics. Fig. compares the ac current harmonics (in space vector form) and dc voltage harmonics obtained by /EMTD and those obtained by the proposed method. As the figures show, they are also in good agreement. t sw Fig A three-phase VS with reactive control i t i qref vdc ref harmonic amplitude (A) harmonic amplitude (A) Fig.. Ac current of a three-phase diode bridge rectifier Fig Ac current harmonics of a three-phase VS harmonic amplitude (V) in log scale harmonic amplitude (V) in log scale Fig. 9. Dc voltage harmonics of a three-phase VS developed by 17 and 18, one obtains γ = , 0.01 Fig Dc voltage of a three-phase diode bridge rectifier IV. ONUSIONS A time-domain based method for calculating the frequency coupling matrices of a power converter is presented. The method is applicable to both self-commutated and forcecommutated converters. In contrast to the frequency-domain based method, the proposed method offers great accuracy as it is a time-domain method and does suffer from the harmonic truncation errors. The application of the proposed method can be used in a harmonic power flow program to improve its overall accuracy. 531

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