- Email: [email protected]
Mathematical study of two controlled converter-type utility interface for harmonic attenuation
Electric Power Systems Research 75 (2005) 124–133
Mathematical study of two controlled converter-type utility interface for harmonic attenuation Sangshin Kwak ∗,1 , Hamid A. Toliyat Advanced Electric Machines and Power Electronics Lab, Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128, USA Received 29 November 2004; accepted 24 January 2005 Available online 8 June 2005
Abstract Diode rectifiers are very popular in industry from a few kW to a few hundreds of kW. Two methods based on controlled converter-type utility interface are possible for harmonic-pollution problems caused by the diode rectifiers: one replacing the diode rectifier with a PWM rectifier and the other connecting an active power filter to the diode rectifier. Despite numerous publications for the two converters, the features and evaluations between them have not been clearly explained. This paper, in detail, presents theoretical analyses and systematic comparisons of the two converters, from the point of view of converter kVA ratings, dc bus voltage requirements, switch ratings, and reactive component designs. © 2005 Elsevier B.V. All rights reserved. Keywords: Power quality; PWM-voltage source rectifier; Active power filter
1. Introduction In recent years, the power quality issues in the utility grids have received considerable attention to suppress harmonicsrelated problems resulted from a proliferation of nonlinear loads. This has led to restricted norms regarding utility power quality; such as IEEE-519 standards. As a result, a number of research works are directed to fulfill these requirements by eliminating the power quality degradation problems. The diode rectifier, which is an uncontrolled utility interface, is the most common harmonic-polluting load in the distributing networks from a few kW small power consumer electronics to high power industrial variable-speed drives with a few hundreds of kW. The two methods based on controlled converter-type utility interface have been considered to solve utility contamination problems. One is to replace the diode rectifier to the PWM-Voltage Source Rectifier (PWM-VSR). The PWM-VSR performs the dc power conversion for its dc ∗
Corresponding author. Tel.: +1 979 845 1171; fax: +1 979 845 6259. E-mail addresses: [email protected] (S. Kwak), [email protected] (H.A. Toliyat). 1 Tel.: +1 979 862 3034; fax: +1 979 845 6259. 0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.01.004
load as well as draws the sinusoidal current waveforms from the utility. The other one is connecting the active power filter (APF) to the diode rectifier. The APF is used to inject cancellation current to compensate for the distortion produced by the diode rectifier [1–3]. As a result, the utility can provide real power with sinusoidal supply current. The proposition of using the APF is that a relatively small inverter can be employed for the purpose [4]. Although both converters have been, together, recognized as solutions to power quality problems in the distribution network, the operational principles are quite different. The PWM-VSR is based on direct sinusoidal current generation, whereas the APF works on the principle of load harmonic compensation. As a consequence, the PWM-VSR deals with the real power, while the reactive and harmonic power of the diode rectifier are supplied by the APF. The objective of this paper is to clearly analyze and compare the features and requirements of the two converters, to offer deciding factors when choosing between them. For this comparison, detail and theoretical analyses are presented for the PWM-VSR and the APF with a typical diode rectifier load. Intuition suggests that the APF is likely to have lower ratings because of the no real power the APF theoret-
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
ically needs to carry. This paper first examines the question of the converter rating for the two converters, by deriving closed-form expressions for their ratings. Having compared the converter ratings, the paper tackles the issues of comparing the dc-link voltage requirements, switch ratings, and the filter components. The physical features are quantified, and then, closed-form analytical expressions to reflect the converter characteristics are established through mathematical derivations. The comparisons and assessments based on the theoretical analyses are presented for the two converters. 2. Basis of operations and specifications The PWM-VSR, which is subsequently referred to as the VSR, and the APF, are depicted in Figs. 1 and 2, respectively. Both the VSR and the APF have basically the same circuit topology, where a dc capacitor is used at the dc-side as an energy storage element. The ac-sides of the converters are connected to the utility via the input reactors, which serve as low-pass filters so that the input currents on the converter ac sides can be actively waveshaped. It is worth noting that the input inductors are essential for proper operations of both the VSR and the APF [5]. Notwithstanding the topology similarity of the two converters, their operational principles and features are very different due to different types of power delivered by them. The VSR delivers real power to its dc load by serving as a rectifier. On the other hand, the APF compensates the reactive and harmonic power caused by the diode rectifier, while the diode rectifier carries the real power to its load. The reactive and harmonic power levels of the APF are
125
also dependent on the real power level of the diode rectifier. Thus, it is felt that the real output power carried by the rectifiers is a proper criterion to compare the two configurations. For fair comparisons and evaluations of the two converters, the analyses are done based on only utility and load terminal constraints. The utility terminal constraints are that the balanced, three-phase supply voltages with line-to-line rms voltage VLL = 460 V and frequency fg = 60 Hz are defined for both configurations. Further, the supply currents have sinusoidal waveforms in phase with the utility voltages. In the load terminals, both the VSR and the diode rectifier deliver equal output power to their dc loads, denoted by Pout . The losses of the converters and the reactive components are assumed to be negligible. As a result, the supply power is equal to the load power, and no real power is consumed by the APF. Compared with the VSR, the analysis of the APF with the diode rectifier in Fig. 2 is quite complicated because of its highly distorted input current/voltage waveforms. In analyzing the APF features, the followings are assumed to simplify the diode rectifier operation: 1. The dc-side filter of the diode rectifier is assumed to be ideal. This implies that the rectifier operates an ideal sixpulse rectifier load with a dc-side current id (ωt) = Idc [6]. 2. Current commutations of the diode rectifier are instantaneous. 3. The displacement factor angle (ϕ1 ) of the diode rectifier input current ranges from 0◦ to 30◦ . The APF compensates the reactive power proportional to the angle. Fig. 3 shows the supply current, the diode rectifier input current, and the APF input current under the aforementioned assumptions with ϕ1 = 10◦ and Pout = 7.5 kW. 3. Analysis of converters ratings 3.1. VSR rating The VSR kVA rating SVSR is given by rms SVSR = 3urms c(VSR) is
(1)
Fig. 1. The PWM-VSR.
Fig. 2. The APF and the diode rectifier.
Fig. 3. Supply current, diode rectifier input current, and APF input current with ϕ1 = 10◦ and Pout = 7.5 kW.
126
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
power factor. This implies that the VSR operating with leading power factor should cancel the lagging VAR of the input reactor by delivering the reactive power, to operate with unity displacement factor on the supply terminal. 3.2. APF rating
Fig. 4. VSR kVA rating as a function of Pout and Ls (VLL = 460 V). rms are the rms value of the voltage where, urms c(VSR) and is uc(VSR) and current is of the VSR ac side, respectively. In order to find a closed-form expression for the kVA rating, only fundamental components of VSR ac side voltage and current are considered. The supply phase voltage and current are defined as √ 2 vs (ωt) = is (ωt) = 2Is cos ωt (2) VLL cos ωt, 3
where VLL and Is denotes the rms value of the line-to-line supply voltage and the supply phase current, respectively. Using the assumption for no resistance in the filter inductor, then the VSR kVA rating, SVSR , is given by VLL 2 √ (3) SVSR = 3Is + (XL Is )2 3 where XL denotes the input filter reactance (XL = 2πfg Ls ). With constraints of unity displacement factor and no converter loss, the rms value of the supply current is given by Pout Is = √ 3VLL Reflecting (4) to (3), the VSR kVA rating is XL Pout 2 SVSR = Pout 1 + 2 VLL
(4)
(5)
Eq. (5) clearly describes that the VSR rating is determined by the output power level and the input reactor size due to the fact that the VSR delivers the real power to its dc output load and the reactive power to the input inductor. The rating is increased with the higher output power and the bigger reactor, as illustrated in Fig. 4. The input filter effect on the VSR rating is apparent from the phasor diagram for the VSR operating with unity displacement factor shown in Fig. 5. With the current in phase with the voltage at the supply terminal, the converter voltage Uc(VSR) and current Is on the ac side of the VSR have a γ degree phase shift, which shows a leading
The kVA rating of the APF can be, likewise, derived with the rms voltage and current on its ac side. However, deriving the closed-form expression is not straightforward due to its highly distorted waveforms of the ac side voltage uc(APF) and current ic . Based on its operational principle, the APF kVA rating is likely to be influenced by the reactive and harmonic power of the diode rectifier, which can be quantified by the displacement factor angle (ϕ1 ) and the total harmonic distortion (THDi ) in conjunction with the output power delivered by the diode rectifier [6]. In addition, the APF rating is also dependent on its input reactor size like the VSR rating, since the APF provides the reactive power to the input filter. As before, assume that the switching frequency components of the APF ac side current and voltage are negligible. The diode rectifier input current can be described in terms of the fundamental and harmonic components [6]: iL (ωt) =
√ 2 3 1 Idc cos(ωt − ϕ1 ) − cos 5(ωt − ϕ1 ) π 5 1 + cos 7(ωt − ϕ1 ) · · · 7
(6)
Since the utility delivers the only real power to the diode rectifier, the supply current is given by √ 2 3Idc is (ωt) = (7) cos ϕ1 cos(ωt) π The APF input current is then written by ic (ωt) = is (ωt) − iL (ωt) √ √ 2 3Idc 2 3Idc = − sin ϕ sin ωt + π π 1 1 × (8) cos 5(ωt − ϕ1 )− cos 7(ωt − ϕ1 ) · · · 5 7
Fig. 5. Phasor diagram of the VSR operating with unity displacement factor.
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
The dc-side current of the diode rectifier, Idc can be related to the output power supplied through the diode rectifier: Pout π Idc = √ 3 2 cos ϕ1 VLL
(9)
Then, the following expression for the rms value of the APF input current is derived: 2 π P out rms ic = √ −1 (10) 3 cos ϕ1 3VLL
127
the output power and the input filter. The rating is increased with the increasing Pout and Ls , like VSR rating. The effect of n and ϕ1 on the APF rating is illustrated in Fig. 7. Increasing n and ϕ1 implies that the APF compensates increasing harmonic and reactive power to the diode rectifier, respectively. Therefore, this results in the increasing APF rating. Several n values (10, 16, 22, 30, and 42) used in Fig. 7 yields THDi of 10, 8, 7, 6, and 5% in the supply current is , respectively. The rating comparisons of the VSR and the APF are depicted in Figs. 8–10. Fig. 8 shows the VSR and APF ratings as a function of the size of input filter inductor. The APF rat-
The rms value of the APF ac side voltage, uc(APF) is likewise derived by
2 2 2 cos ϕ
V XL Pout 2VLL 1 rms LL 2 uc(APF) = sin ϕ1 + + √ sin ϕ1 + n 3 XL Pout 3VLL cos ϕ1 where n denotes the number of harmonic components to be canceled by the APF. Therefore, the kVA rating of APF, SAPF , can now be written as
2 2 2 cos ϕ
π XL Pout 2VLL 1 SAPF = Pout n + sin ϕ1 + sin ϕ1 −1 1+ 2 cos ϕ 3 cos ϕ1 XL Pout VLL 1
(11)
(12)
As anticipated, the APF rating is determined by the input reactance XL , output power delivered by the diode rectifier Pout , displacement factor angle of the rectifier current ϕ1 , and the number of harmonics the APF compensates n. The specification for the number of harmonic components n to be cancelled by the APF are set from a supply current THDi limitation to meet IEEE 519 harmonic standards. IEEE 519 harmonic current limits are to be met up to the 50th harmonic in the supply current [7]. This results in n = 16, the total harmonic distortion of the supply current THDi = 0.08, and the supply power factor pf = 0.997. The lower THDi of the supply current and the better utility interface can be realized with the increasing n, for special applications such as for hospital environments. Fig. 6 shows a plot of the APF rating versus Fig. 7. APF kVA rating vs. n and ϕ1 (VLL = 460 V, Pout = 100 kW, and Ls = 2 mH).
Fig. 6. APF kVA rating as a function of output power and input filter (VLL = 460 V, n = 16, and ϕ1 = 10◦ ).
Fig. 8. Rating comparisons vs. input filter inductance (VLL = 460 V, n = 16, and Pout = 50 kW).
128
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
Fig. 9. Rating comparisons vs. output power (VLL = 460 V, n = 16 and Ls = 1 mH).
ing is increased with increasing ϕ1 due to increasing reactive power compensation. It is important to note that the dependency of the APF rating on the input filter size is higher than the VSR rating. This is due to the fact that the input filter of the VSR is subject to the supply current with the fundamental frequency 60 Hz, while the harmonic currents with higher frequency flow through the filter of the APF. As a result, the APF filter has the higher reactance than the VSR filter with the same inductance. It is shown that the APF with small ϕ1 (ϕ1 < 10◦ ) has the lower rating than the VSR. On the other hand, the APF rating subjected to big ϕ1 and Ls becomes higher than the VSR rating. The ratings of the VSR and APF are illustrated with the output power in Fig. 9. Here again, the APF ratings are substantially increased with the increase of ϕ1 . The APF ratings with ϕ1 less than 10◦ are lower than the VSR rating from the low to high power applications. However, high reactive power cancellation by the APF (ϕ1 > 10◦ ) leads to the higher APF rating than the VSR rating in high power applications. Considering that most of practical diode rectifiers have ϕ1 less than 15◦ , the APF rating in practical cases is lower than the VSR rating in low and medium power applications. It should be noticed that irrespective of ϕ1 values, the APF ratings are more sharply increased with the increase of the output power than the VSR rating due to
its higher dependency on the input filter. Thus, economical aspect regarding the converter size, which is in favor of the APF in low and medium power ranges, changes in high power levels. Fig. 10 depicts the VSR and APF ratings versus total harmonic distortion in the supply current. Note that because the APF operation is based on the principle of load harmonic compensation, the supply current has harmonics. Notice that since the THDi value of the supply current is derived from n in (12), the values are not linearly distributed in Fig. 10. Since the APF must deliver more harmonic power to the diode rectifier to achieve the lower supply current THDi , the APF ratings increase with the decrease in supply current THDi . The above analyses clearly show that the APF with high reactive power compensation due to large Ls and ϕ1 at the high-power distribution level is not regarded as viable in terms of economical considerations.
4. Analysis of dc-link voltage levels Because both the VSR and the APF should actively shape their input currents on their ac sides, the dc-link voltage levels for both converters are required to have a higher value than the peak value of the ac side voltages (uc ) for the required input currents. The normal dc-link voltage level for the VSR and the APF is set at 5% higher than the peak line-to-line voltage across the converter ac side, assuming the maximum modulation indices for the converters is 0.95 [8]. Thus, the required dc-link voltage level for the VSR is √ peak Vdc(VSR) = 1.05 3uc(VSR)
The peak converter voltage is the peak value of the converter voltage on the ac side, calculated by uc(VSR) (ωt) = vs (ωt) − XL
dis (ωt) d(ωt)
(14)
The dc-link voltage level of the VSR is derived, by reflecting (2), (4), and (14) into (13), as √
Vdc(VSR) = 1.05 2VLL
Fig. 10. Rating comparisons as a function of total harmonic distortion in supply current (VLL = 460 V, Pout = 50 kW and Ls = 3 mH).
(13)
1+
XL Pout 2 VLL
2 (15)
Fig. 11 illustrates the plot of the VSR dc-link voltage level versus the output power level and the input filter size. It is shown that the required voltage level is slightly increased with the increase of output power and filter size. In the case of APF (13) is also effective to design the dclink voltage requirement. The APF ac side voltage uc(APF) can be derived similar to (14), though it has a complicated form because of its distorted current waveform. Using the parameters used in the previous analysis, the voltage is
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
Fig. 11. The dc-link voltage level of the VSR with Pout and Ls (VLL = 460 V).
129
Fig. 13. The dc-link voltage level of the APF with the output power and the input filter (VLL = 460 V and n = 16).
derived by uc(APF) (ωt) 2 VLL (1 + M sin ϕ1 )cos ωt − 2M sin(ωt − ϕ1 ) = 3 ×
n/2
m=1
cos 6m(ωt − ϕ1 )
M=
XL Pout 2 cos ϕ VLL 1
(16)
Fig. 12 shows the diode rectifier current and the corresponding APF ac side voltage waveform of (16). The APF ac side voltage uc(APF) has sudden voltage spikes at the every commutation instant of the diode rectifier current. Notice that ideal current commutations for the diode rectifier were assumed with infinite diL /dt in this analysis. In practice, the diode rectifier current will be trapezoidal due to parasitic line inductance and the voltage spike level at the commutation will be lower than that of Fig. 12. Moreover, additional input inductors might be installed on the diode rectifier ac side to reduce the voltage spike level by limiting the current
Fig. 12. Diode rectifier input current and APF ac side voltage (VLL = 460 V, Ls = 0.5 mH, Pout = 80 kW, ϕ1 = 10◦ and n = 16).
slope during diode commutations. The voltage spike at the commutation instant is exacerbated with the increase of the inductance Ls and the output power Pout . Therefore, it can be deducted that the peak value of uc(APF) occurs at the moment of either ωt = 0 or the commutation instant of the diode rectifier current according to constraints of Pout , Ls and n. The condition and the corresponding APF peak ac side voltage are derived by peak
uc(APF) 2 2 VLL √ V , if X P ≤ LL L out 3 3n √ = 2 1 2 3 nXL Pout VLL √ VLL P , if X + > L out 2 3 2 2 VLL 3n (17) As a result, the required dc-link voltage level of the APF is found by (13) and (17). Vdc(APF)
√ if XL Pout ≤ 1.05 2VLL , = 1 3 nXL Pout , if XL Pout > 2 1.05 · VLL √2 + 2 VLL
V2 √LL 3n V2 √LL 3n
(18)
It should be √ noted that the APF dc-link voltage level can be set to 1.05 2VLL in all cases if the commutation effects are disregarded. The dc-link voltage level of the APF is depicted versus the output power and the input inductance in Fig. 13. The voltage level is comparable with that of the VSR for low power applications with small inductance, as seen in (15) and (18), because commutation voltage spikes have no effect on the dc-link voltage level of the APF. However, the APF voltage level is almost linearly increased with high power and reactance due to the commutation effect. Comparing Figs. 11 and 13, it is shown that the APF dc-link voltage level is much higher than the VSR level, in cases of high power applications with high input reactance. Theoretically,
130
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
will increase with the increase of Ls . Here again, the APF is not suitable for high power applications with large Ls due to its high dc-link voltage requirement. 5. Analysis of switch ratings
Fig. 14. The dc-link voltage levels of the VSR and the APF vs. the output power (VLL = 460 V, n = 16 and Ls = 0.1 mH).
the APF dc-link voltage level should be set to a high value enough to consider the voltage spikes of the commutation instants. However, in practical case, the nominal reference dc-link voltage for an APF is designed by neglecting the commutation effect, since the high voltage spikes at commutation points leads to high dc-link voltage requirement, which yields high switch losses and cost. For instance, given VLL = 460 V, the typical dc-link voltage level of the APF is set to 683 V
Since the VSR has sinusoidal input current, it is straightforward to determine the switch currents. The mean, rms, and peak switch current of the VSR are related with the supply voltage and the output power as √ 2 Pout 1 Pout mean rms , Isw(VSR) =√ , Isw(VSR) = √ 3π VLL 6 VLL √ 2 Pout peak Isw(VSR) = √ (19) 3 VLL The APF switch currents have more complex forms because of its highly distorted APF input current waveform shown in Fig. 3. The mean switch current of the APF can be calculated by integrating the absolute value of the APF input current ic over an entire cycle and dividing by 4π [9]. Notice that the shape of the APF input current varies by the displacement factor angle of the diode rectifier current. Therefore, the mean switch current of the APF can be categorized by ϕ1 . The mean switch current is derived as
√ π π √ 2Pout π 1 − 3 cos ϕ1 + 2 sin γ + √ , if 0 ≤ ϕ1 ≤ cos−1 −γ √ √3πV 2 3 3 cos ϕ1 6 LL mean Isw(APF) = √ π √ 2P π2 π √ out 1 − 3 cos ϕ1 + √ , if cos−1 < ϕ1 ≤ √ 6 3 cos ϕ1
3πVLL
from the first value of (18), regardless of the output power and reactance levels [7]. If this dc-link voltage is lower than the required voltage level at commutation instants, the APF will lose its harmonic current controllability at every commuta-
peak
Isw(APF)
2 3
γ = cos−1
√
π
2 3 cos ϕ1
(20)
6
The rms switch current is given by 2 π Pout rms Isw(APF) = √ −1 3 cos ϕ1 6VLL
(21)
The peak switch current is also dependent on the displacement factor angle, derived by √ 2Pout π π √ √ cos ϕ1 − , if 0 ≤ ϕ1 ≤ cos−1 3 2 3 3VLL = √ (22) 2Pout π 2π π π −1 √ √ < ϕ1 ≤ cos ϕ1 − , if cos + √ 3 6 3VLL 2 3 cos ϕ1 2 3
tion points. This will result in a notch in the supply current at commutation points of the diode rectifier currents. This will yield the deterioration in supply current quality, though it can be permissible in low and medium power applications. Fig. 14 shows the dc-link voltage levels of the VSR and the APF as a function of the output power. The voltage levels for the two converters are roughly the same in the low and medium power ranges. However, in high power levels, the APF dclink voltage level is higher than the VSR voltage level since the voltage spikes at the commutation instants determine the dc-link voltage. Note that the slopes for converters in Fig. 14
The mean, rms, and peak switch currents of the APF are depicted as a function of Pout and ϕ1 in Figs. 15–17, respectively. The switch currents of the VSR are also illustrated for the purpose of the comparisons. The mean and rms switch currents of the VSR are linearly increased with increasing Pout since the power directly flows through the converter. The APF mean and rms switch currents are also increased with increasing Pout and ϕ1 . This is because more reactive and harmonic power are required to flow from the APF to the diode rectifier, resulting in increase of the mean and the rms APF input currents. However, the mean and rms switch currents of the APF are smaller than those of the VSR due to
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
131
Fig. 18. Switch current ratios of the APF and the VSR. Fig. 15. The mean switch currents vs. the output power and the displacement angle (VLL = 460 V).
no real power flown through the APF. For clear comparison of the switch current ratings, a switch current ratio index is introduced by Isw =
Isw(APF) × 100% Isw(VSR)
(23)
Notice that the switch current ratio depends only on ϕ1 of the diode rectifier input current. Fig. 18 shows the switch current ratios of the APF and the VSR as a function of ϕ1 . It is shown the APF has a lower switch rating than the VSR.
6. Analysis of reactive components ratings 6.1. Input filter inductor
Fig. 16. The rms switch currents vs. the output power and the displacement angle (VLL = 460 V).
Peak ripple current is chosen as a criterion to design an input filter inductor. In order to calculate the ripple current, no-load condition is considered and the effect of inductor resistance is assumed to be negligible. Under these conditions, the inductance value is given by [10]: VLL Lf = √ 6 2fs iripple,peak
(24)
where fs denotes a switching frequency. Under the same supply voltage and switching frequency of the VSR and the APF, it is expected that the two converters require an equal input inductance with a given identical peak ripple current constraint. 6.2. dc capacitor The peak ripple voltage of the dc capacitor is, likewise, adopted as the design criterion for the dc capacitor size. 6.2.1. VSR With the assumption of a balanced three-phase utility and neglecting the losses of the power switches, the VSR in the dc-link in Fig. 1 is modeled as Fig. 17. The peak switch currents vs. the output power and the displacement angle (VLL = 460 V).
icap = iconv − idc
(25)
132
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
It can be written as dvdc(VSR) Pout = isδ dδ − dt vdc(VSR) 3
Cdc(VSR)
(26)
δ=1
where isδ and dδ (δ = 1,2,3) is the input current and the switch3 ing function of the VSR. By noting that isδ dδ is equal to δ=1
one of the VSR input currents at all times, the possible maximum ripple value of vdc(VSR) is approximately obtained with the negative peak value of the VSR input current and 50% duty ratio [11]. The ripple factor rv is defined with the allowable ripple voltage amplitude and the dc-link voltage level. rv =
vdc(VSR) Vdc(VSR)
Cdc(VSR)
Pout = VLL
(27) √
√ 2 + 3VLL /Vdc(VSR) √ 2 3rv Vdc(VSR) fs
(28)
The capacitor size increases with increasing Pout and decreasing rv , as shown in Fig. 19. 6.2.2. APF The capacitor size of the APF is likewise designed based on the peak APF input current in (22) as Cdc(APF)
=
Pout cos(ϕ1 − π/3) , VLL √6r V f
if 0 ≤ ϕ1 ≤ cos−1
Pout
if cos−1
v dc(APF) s
VLL
Fig. 20. APF capacitor size vs. output power and displacement factor angle (VLL = 460 V, Vdc(APF) = 683 V, fs = 10 kHz, and rv = 5%).
√ cos(ϕ1 − 2π/3) + π/(2 3cosϕ1 ) , √ 6rv Vdc(APF) fs
π
π
√ 2 3
π √ < ϕ1 ≤ 6 2 3
The APF capacitor size versus Pout and rv is depicted in Fig. 21 with ϕ1 = 30◦ . It is important to note that the APF capacitance is substantially smaller than the VSR, compared with Fig. 19. With the smaller ϕ1 , the APF capacitor size will become smaller, compared with the VSR capacitance. This is due to the different power components dealt by the two converters. The real power with the dc components are supplied through the VSR capacitor, while the reactive and harmonic power with high frequency components are carried through the APF capacitor. As a result, the voltage variation of the APF dc-link is expected to be smaller than that of the VSR. Thus the APF capacitor size can be substantially smaller than the VSR capacitor size. With the capacitance ratio defined by (30), the ratio is only dependent on ϕ1 , assuming that the same dc-link voltage level is used for the converters:
(29) Cdc =
Cdc(APF) × 100% Cdc(VSR)
(30)
Increasing Pout and ϕ1 implies that more harmonic and reactive power flow in and out of the capacitor. Thus, the capacitor size should be increased to regulate the dc voltage within the specified ripple factor. This is illustrated in Fig. 20.
In Fig. 22, it is shown that the APF capacitive filter size is smaller than the VSR capacitor.
Fig. 19. VSR capacitor size vs. output power and ripple factor (VLL = 460 V, Vdc(VSR) = 683 V, and fs = 10 kHz).
Fig. 21. APF capacitor size vs. output power and ripple factor (VLL = 460 V, Vdc(APF) = 683 V, fs = 10 kHz, and ϕ1 = 30◦ ).
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
133
References
Fig. 22. Capacitance ratio vs. displacement factor angle (VLL = 460 V and Vdc(VSR) = Vdc(APF) = 683 V).
7. Conclusions In solving harmonic-polluting problems by diode rectifiers, two approaches based on controlled converter-type utility interface are possible; one replacing the diode rectifier with a VSR, and the other connecting an APF. The paper presents theoretical and comprehensive analysis for the VSR and the APF with only terminal constraints of unity power factor and identical output power delivery. The closed-form analytical expressions have been derived based on the quantification of the converter factors. The distinctive features of two converters have been compared and assessed based on the theoretical analyses. As expected, the APF requires a lower rating than the VSR in most low and medium power areas. However, the reactive power compensation for the diode rectifier substantially increases the APF rating. This feature, sometimes, leads to the APF applications to compensate either the reactive or the harmonic power in industry. In addition, the APF shows higher influence of the input filter on the ratings. The APF rating is more dependent on the output power increase than the VSR. Therefore, the APF implementation for low and medium power applications with small input filter is cost-effective. The dc-link voltage level is comparable for both converters in low and medium power areas. In high power applications, the APF dc-link voltage level, theoretically, should be set high enough to overcome the voltage spikes due to the diode current commutation, which leads to high switch losses and cost. However, the practical voltage level is determined disregard of the commutation effect. The lower voltage level than the required voltage results in the loss of current controllability at every instant of diode rectifier current commutation, and therefore the deterioration of supply current quality. The APF results in lower mean and rms switch ratings than the VSR because of no real power delivery. Equal input inductance value can be used for both converters. The APF requires significantly smaller capacitor size than the VSR because it is subject to harmonic and reactive power with higher frequencies.
[1] T. Thomas, K. Haddad, G. Joos, A. Jaafari, Design and performance of active power filters, IEEE Ind. Appl. Mag. 4 (5) (1998) 38–46. [2] W. Mgrady, M.J. Samotyj, A.H. Noyola, Survey of active power line conditioning methodologies, IEEE Trans. Power Del. 5 (1990) 1536–1542. [3] J.W. Dixon, J.M. Contardo, L.A. Moran, A fuzzy-controlled active front-end rectifier with current harmonic filtering characteristics and minimum sensing variables, IEEE Trans. Power Electron. 14 (4) (1999) 724–729. [4] T.C. Green, J.H. Marks, Ratings of active power filters, IEE Proc. Electron. Power Appl. 150 (5) (2003) 607–614. [5] N.R. Zargari, G. Joos, P.D. Ziogas, A performance comparison of PWM rectifiers and synchronous link converters, IEEE Trans. Ind. Electron. 41 (5) (1994) 560–562. [6] N. Mohan, T.M. Undeland, W.P. Robbins, Power Electronics, 2nd Ed., New York, Wiley. [7] S. Bhattacharya, T.M. Frank, D.M. Divan, B. Banerjee, Active filter system implementation, IEEE Ind. Appl. Mag. 4 (5) (1998) 47–63. [8] S. Ponnaluri, V. Krishnamurthy, V. Kanetkar, Generalized system design & analysis of PWM based power electronic converters, IEEE Ind. Appl. Conf. 3 (2000) 1972–1979. [9] J.H. Marks, T.C. Green, Rating analysis of active power filters, Proc. IEEE PESC’01 3 (2001) 1420–1425. [10] S. Bernet, S. Ponnaluri, R. Teichmann, Design and loss comparison of matrix converters and voltage-source converters for modern AC drives, IEEE Trans. Ind. Electron. 49 (2) (2002) 304–314. [11] Y. Yang, Advances in modeling and applications of three-phase power converters, PhD Dissertation, Department of Electronics and Computer Engineering, University Waterloo, Ont., Canada, 2001. Sangshin Kwak received the B.S. and M.S. degree in electronics engineering from Kyungpook National University, Daegu, Korea, in 1997 and 1999, respectively. He is currently working toward the Ph.D. degree in electrical engineering at Texas A&M University, College Station, TX. His main research interests are power converters topologies and controls, adjustable speed drives, and DSP-based power electronics control. Hamid A. Toliyat received the B.S., degree from Sharif University of Technology, Tehran, Iran in 1982, the M.S. degree from West Virginia University, Morgantown, WV in 1986, and the Ph.D. degree from University of Wisconsin-Madison, Madison, WI in 1991, all in electrical engineering. Following receipt of the Ph.D. degree, he joined the faculty of Ferdowsi University of Mashhad, Mashhad, Iran as an Assistant Professor of Electrical Engineering. In March 1994 he joined the Department of Electrical Engineering, Texas A&M University where he is currently E.D. Brockett professor of electrical engineering. He has received the prestigious Cyrill Veinott Award in Electromechanical Energy Conversion from the IEEE Power Engineering Society in 2004, Distinguished Teaching Award in 2003, E.D. Brockett Professorship Award in 2002, Texas A&M Select Young Investigator Award in 1999, and Eugene Webb Faculty Fellow Award in 2000. He has also received the Space Act Award from NASA in 1999, and the Schlumberger Foundation Technical Awards in 2000 and 2001. He is an Editor of IEEE Transactions on Energy Conversion, and was an associate editor of IEEE Transactions on Power Electronics. He is also Chairman of IEEE-IAS Electric Machines Committee, and is a member of Sigma Xi. He is a senior member of the Power Engineering, Industrial Applications, Industrial Electronics, Power Electronics Societies of the IEEE, and the recipient of the 1996 IEEE Power Engineering Society Prize Paper Award for his paper on the Analysis of Concentrated Winding Induction Machines for Adjustable Speed Drive Applications-Experimental Results. His main research interests and experience include analysis and design of electrical machines, variable speed drives for traction and propulsion applications, fault diagnosis of electric machinery, and sensorless variable speed drives. He has published over 220 technical papers in these fields and has 10 issued or pending US patents.
Mathematical study of two controlled converter-type utility interface for harmonic attenuation Sangshin Kwak ∗,1 , Hamid A. Toliyat Advanced Electric Machines and Power Electronics Lab, Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128, USA Received 29 November 2004; accepted 24 January 2005 Available online 8 June 2005
Abstract Diode rectifiers are very popular in industry from a few kW to a few hundreds of kW. Two methods based on controlled converter-type utility interface are possible for harmonic-pollution problems caused by the diode rectifiers: one replacing the diode rectifier with a PWM rectifier and the other connecting an active power filter to the diode rectifier. Despite numerous publications for the two converters, the features and evaluations between them have not been clearly explained. This paper, in detail, presents theoretical analyses and systematic comparisons of the two converters, from the point of view of converter kVA ratings, dc bus voltage requirements, switch ratings, and reactive component designs. © 2005 Elsevier B.V. All rights reserved. Keywords: Power quality; PWM-voltage source rectifier; Active power filter
1. Introduction In recent years, the power quality issues in the utility grids have received considerable attention to suppress harmonicsrelated problems resulted from a proliferation of nonlinear loads. This has led to restricted norms regarding utility power quality; such as IEEE-519 standards. As a result, a number of research works are directed to fulfill these requirements by eliminating the power quality degradation problems. The diode rectifier, which is an uncontrolled utility interface, is the most common harmonic-polluting load in the distributing networks from a few kW small power consumer electronics to high power industrial variable-speed drives with a few hundreds of kW. The two methods based on controlled converter-type utility interface have been considered to solve utility contamination problems. One is to replace the diode rectifier to the PWM-Voltage Source Rectifier (PWM-VSR). The PWM-VSR performs the dc power conversion for its dc ∗
Corresponding author. Tel.: +1 979 845 1171; fax: +1 979 845 6259. E-mail addresses: [email protected] (S. Kwak), [email protected] (H.A. Toliyat). 1 Tel.: +1 979 862 3034; fax: +1 979 845 6259. 0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.01.004
load as well as draws the sinusoidal current waveforms from the utility. The other one is connecting the active power filter (APF) to the diode rectifier. The APF is used to inject cancellation current to compensate for the distortion produced by the diode rectifier [1–3]. As a result, the utility can provide real power with sinusoidal supply current. The proposition of using the APF is that a relatively small inverter can be employed for the purpose [4]. Although both converters have been, together, recognized as solutions to power quality problems in the distribution network, the operational principles are quite different. The PWM-VSR is based on direct sinusoidal current generation, whereas the APF works on the principle of load harmonic compensation. As a consequence, the PWM-VSR deals with the real power, while the reactive and harmonic power of the diode rectifier are supplied by the APF. The objective of this paper is to clearly analyze and compare the features and requirements of the two converters, to offer deciding factors when choosing between them. For this comparison, detail and theoretical analyses are presented for the PWM-VSR and the APF with a typical diode rectifier load. Intuition suggests that the APF is likely to have lower ratings because of the no real power the APF theoret-
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
ically needs to carry. This paper first examines the question of the converter rating for the two converters, by deriving closed-form expressions for their ratings. Having compared the converter ratings, the paper tackles the issues of comparing the dc-link voltage requirements, switch ratings, and the filter components. The physical features are quantified, and then, closed-form analytical expressions to reflect the converter characteristics are established through mathematical derivations. The comparisons and assessments based on the theoretical analyses are presented for the two converters. 2. Basis of operations and specifications The PWM-VSR, which is subsequently referred to as the VSR, and the APF, are depicted in Figs. 1 and 2, respectively. Both the VSR and the APF have basically the same circuit topology, where a dc capacitor is used at the dc-side as an energy storage element. The ac-sides of the converters are connected to the utility via the input reactors, which serve as low-pass filters so that the input currents on the converter ac sides can be actively waveshaped. It is worth noting that the input inductors are essential for proper operations of both the VSR and the APF [5]. Notwithstanding the topology similarity of the two converters, their operational principles and features are very different due to different types of power delivered by them. The VSR delivers real power to its dc load by serving as a rectifier. On the other hand, the APF compensates the reactive and harmonic power caused by the diode rectifier, while the diode rectifier carries the real power to its load. The reactive and harmonic power levels of the APF are
125
also dependent on the real power level of the diode rectifier. Thus, it is felt that the real output power carried by the rectifiers is a proper criterion to compare the two configurations. For fair comparisons and evaluations of the two converters, the analyses are done based on only utility and load terminal constraints. The utility terminal constraints are that the balanced, three-phase supply voltages with line-to-line rms voltage VLL = 460 V and frequency fg = 60 Hz are defined for both configurations. Further, the supply currents have sinusoidal waveforms in phase with the utility voltages. In the load terminals, both the VSR and the diode rectifier deliver equal output power to their dc loads, denoted by Pout . The losses of the converters and the reactive components are assumed to be negligible. As a result, the supply power is equal to the load power, and no real power is consumed by the APF. Compared with the VSR, the analysis of the APF with the diode rectifier in Fig. 2 is quite complicated because of its highly distorted input current/voltage waveforms. In analyzing the APF features, the followings are assumed to simplify the diode rectifier operation: 1. The dc-side filter of the diode rectifier is assumed to be ideal. This implies that the rectifier operates an ideal sixpulse rectifier load with a dc-side current id (ωt) = Idc [6]. 2. Current commutations of the diode rectifier are instantaneous. 3. The displacement factor angle (ϕ1 ) of the diode rectifier input current ranges from 0◦ to 30◦ . The APF compensates the reactive power proportional to the angle. Fig. 3 shows the supply current, the diode rectifier input current, and the APF input current under the aforementioned assumptions with ϕ1 = 10◦ and Pout = 7.5 kW. 3. Analysis of converters ratings 3.1. VSR rating The VSR kVA rating SVSR is given by rms SVSR = 3urms c(VSR) is
(1)
Fig. 1. The PWM-VSR.
Fig. 2. The APF and the diode rectifier.
Fig. 3. Supply current, diode rectifier input current, and APF input current with ϕ1 = 10◦ and Pout = 7.5 kW.
126
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
power factor. This implies that the VSR operating with leading power factor should cancel the lagging VAR of the input reactor by delivering the reactive power, to operate with unity displacement factor on the supply terminal. 3.2. APF rating
Fig. 4. VSR kVA rating as a function of Pout and Ls (VLL = 460 V). rms are the rms value of the voltage where, urms c(VSR) and is uc(VSR) and current is of the VSR ac side, respectively. In order to find a closed-form expression for the kVA rating, only fundamental components of VSR ac side voltage and current are considered. The supply phase voltage and current are defined as √ 2 vs (ωt) = is (ωt) = 2Is cos ωt (2) VLL cos ωt, 3
where VLL and Is denotes the rms value of the line-to-line supply voltage and the supply phase current, respectively. Using the assumption for no resistance in the filter inductor, then the VSR kVA rating, SVSR , is given by VLL 2 √ (3) SVSR = 3Is + (XL Is )2 3 where XL denotes the input filter reactance (XL = 2πfg Ls ). With constraints of unity displacement factor and no converter loss, the rms value of the supply current is given by Pout Is = √ 3VLL Reflecting (4) to (3), the VSR kVA rating is XL Pout 2 SVSR = Pout 1 + 2 VLL
(4)
(5)
Eq. (5) clearly describes that the VSR rating is determined by the output power level and the input reactor size due to the fact that the VSR delivers the real power to its dc output load and the reactive power to the input inductor. The rating is increased with the higher output power and the bigger reactor, as illustrated in Fig. 4. The input filter effect on the VSR rating is apparent from the phasor diagram for the VSR operating with unity displacement factor shown in Fig. 5. With the current in phase with the voltage at the supply terminal, the converter voltage Uc(VSR) and current Is on the ac side of the VSR have a γ degree phase shift, which shows a leading
The kVA rating of the APF can be, likewise, derived with the rms voltage and current on its ac side. However, deriving the closed-form expression is not straightforward due to its highly distorted waveforms of the ac side voltage uc(APF) and current ic . Based on its operational principle, the APF kVA rating is likely to be influenced by the reactive and harmonic power of the diode rectifier, which can be quantified by the displacement factor angle (ϕ1 ) and the total harmonic distortion (THDi ) in conjunction with the output power delivered by the diode rectifier [6]. In addition, the APF rating is also dependent on its input reactor size like the VSR rating, since the APF provides the reactive power to the input filter. As before, assume that the switching frequency components of the APF ac side current and voltage are negligible. The diode rectifier input current can be described in terms of the fundamental and harmonic components [6]: iL (ωt) =
√ 2 3 1 Idc cos(ωt − ϕ1 ) − cos 5(ωt − ϕ1 ) π 5 1 + cos 7(ωt − ϕ1 ) · · · 7
(6)
Since the utility delivers the only real power to the diode rectifier, the supply current is given by √ 2 3Idc is (ωt) = (7) cos ϕ1 cos(ωt) π The APF input current is then written by ic (ωt) = is (ωt) − iL (ωt) √ √ 2 3Idc 2 3Idc = − sin ϕ sin ωt + π π 1 1 × (8) cos 5(ωt − ϕ1 )− cos 7(ωt − ϕ1 ) · · · 5 7
Fig. 5. Phasor diagram of the VSR operating with unity displacement factor.
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
The dc-side current of the diode rectifier, Idc can be related to the output power supplied through the diode rectifier: Pout π Idc = √ 3 2 cos ϕ1 VLL
(9)
Then, the following expression for the rms value of the APF input current is derived: 2 π P out rms ic = √ −1 (10) 3 cos ϕ1 3VLL
127
the output power and the input filter. The rating is increased with the increasing Pout and Ls , like VSR rating. The effect of n and ϕ1 on the APF rating is illustrated in Fig. 7. Increasing n and ϕ1 implies that the APF compensates increasing harmonic and reactive power to the diode rectifier, respectively. Therefore, this results in the increasing APF rating. Several n values (10, 16, 22, 30, and 42) used in Fig. 7 yields THDi of 10, 8, 7, 6, and 5% in the supply current is , respectively. The rating comparisons of the VSR and the APF are depicted in Figs. 8–10. Fig. 8 shows the VSR and APF ratings as a function of the size of input filter inductor. The APF rat-
The rms value of the APF ac side voltage, uc(APF) is likewise derived by
2 2 2 cos ϕ
V XL Pout 2VLL 1 rms LL 2 uc(APF) = sin ϕ1 + + √ sin ϕ1 + n 3 XL Pout 3VLL cos ϕ1 where n denotes the number of harmonic components to be canceled by the APF. Therefore, the kVA rating of APF, SAPF , can now be written as
2 2 2 cos ϕ
π XL Pout 2VLL 1 SAPF = Pout n + sin ϕ1 + sin ϕ1 −1 1+ 2 cos ϕ 3 cos ϕ1 XL Pout VLL 1
(11)
(12)
As anticipated, the APF rating is determined by the input reactance XL , output power delivered by the diode rectifier Pout , displacement factor angle of the rectifier current ϕ1 , and the number of harmonics the APF compensates n. The specification for the number of harmonic components n to be cancelled by the APF are set from a supply current THDi limitation to meet IEEE 519 harmonic standards. IEEE 519 harmonic current limits are to be met up to the 50th harmonic in the supply current [7]. This results in n = 16, the total harmonic distortion of the supply current THDi = 0.08, and the supply power factor pf = 0.997. The lower THDi of the supply current and the better utility interface can be realized with the increasing n, for special applications such as for hospital environments. Fig. 6 shows a plot of the APF rating versus Fig. 7. APF kVA rating vs. n and ϕ1 (VLL = 460 V, Pout = 100 kW, and Ls = 2 mH).
Fig. 6. APF kVA rating as a function of output power and input filter (VLL = 460 V, n = 16, and ϕ1 = 10◦ ).
Fig. 8. Rating comparisons vs. input filter inductance (VLL = 460 V, n = 16, and Pout = 50 kW).
128
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
Fig. 9. Rating comparisons vs. output power (VLL = 460 V, n = 16 and Ls = 1 mH).
ing is increased with increasing ϕ1 due to increasing reactive power compensation. It is important to note that the dependency of the APF rating on the input filter size is higher than the VSR rating. This is due to the fact that the input filter of the VSR is subject to the supply current with the fundamental frequency 60 Hz, while the harmonic currents with higher frequency flow through the filter of the APF. As a result, the APF filter has the higher reactance than the VSR filter with the same inductance. It is shown that the APF with small ϕ1 (ϕ1 < 10◦ ) has the lower rating than the VSR. On the other hand, the APF rating subjected to big ϕ1 and Ls becomes higher than the VSR rating. The ratings of the VSR and APF are illustrated with the output power in Fig. 9. Here again, the APF ratings are substantially increased with the increase of ϕ1 . The APF ratings with ϕ1 less than 10◦ are lower than the VSR rating from the low to high power applications. However, high reactive power cancellation by the APF (ϕ1 > 10◦ ) leads to the higher APF rating than the VSR rating in high power applications. Considering that most of practical diode rectifiers have ϕ1 less than 15◦ , the APF rating in practical cases is lower than the VSR rating in low and medium power applications. It should be noticed that irrespective of ϕ1 values, the APF ratings are more sharply increased with the increase of the output power than the VSR rating due to
its higher dependency on the input filter. Thus, economical aspect regarding the converter size, which is in favor of the APF in low and medium power ranges, changes in high power levels. Fig. 10 depicts the VSR and APF ratings versus total harmonic distortion in the supply current. Note that because the APF operation is based on the principle of load harmonic compensation, the supply current has harmonics. Notice that since the THDi value of the supply current is derived from n in (12), the values are not linearly distributed in Fig. 10. Since the APF must deliver more harmonic power to the diode rectifier to achieve the lower supply current THDi , the APF ratings increase with the decrease in supply current THDi . The above analyses clearly show that the APF with high reactive power compensation due to large Ls and ϕ1 at the high-power distribution level is not regarded as viable in terms of economical considerations.
4. Analysis of dc-link voltage levels Because both the VSR and the APF should actively shape their input currents on their ac sides, the dc-link voltage levels for both converters are required to have a higher value than the peak value of the ac side voltages (uc ) for the required input currents. The normal dc-link voltage level for the VSR and the APF is set at 5% higher than the peak line-to-line voltage across the converter ac side, assuming the maximum modulation indices for the converters is 0.95 [8]. Thus, the required dc-link voltage level for the VSR is √ peak Vdc(VSR) = 1.05 3uc(VSR)
The peak converter voltage is the peak value of the converter voltage on the ac side, calculated by uc(VSR) (ωt) = vs (ωt) − XL
dis (ωt) d(ωt)
(14)
The dc-link voltage level of the VSR is derived, by reflecting (2), (4), and (14) into (13), as √
Vdc(VSR) = 1.05 2VLL
Fig. 10. Rating comparisons as a function of total harmonic distortion in supply current (VLL = 460 V, Pout = 50 kW and Ls = 3 mH).
(13)
1+
XL Pout 2 VLL
2 (15)
Fig. 11 illustrates the plot of the VSR dc-link voltage level versus the output power level and the input filter size. It is shown that the required voltage level is slightly increased with the increase of output power and filter size. In the case of APF (13) is also effective to design the dclink voltage requirement. The APF ac side voltage uc(APF) can be derived similar to (14), though it has a complicated form because of its distorted current waveform. Using the parameters used in the previous analysis, the voltage is
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
Fig. 11. The dc-link voltage level of the VSR with Pout and Ls (VLL = 460 V).
129
Fig. 13. The dc-link voltage level of the APF with the output power and the input filter (VLL = 460 V and n = 16).
derived by uc(APF) (ωt) 2 VLL (1 + M sin ϕ1 )cos ωt − 2M sin(ωt − ϕ1 ) = 3 ×
n/2
m=1
cos 6m(ωt − ϕ1 )
M=
XL Pout 2 cos ϕ VLL 1
(16)
Fig. 12 shows the diode rectifier current and the corresponding APF ac side voltage waveform of (16). The APF ac side voltage uc(APF) has sudden voltage spikes at the every commutation instant of the diode rectifier current. Notice that ideal current commutations for the diode rectifier were assumed with infinite diL /dt in this analysis. In practice, the diode rectifier current will be trapezoidal due to parasitic line inductance and the voltage spike level at the commutation will be lower than that of Fig. 12. Moreover, additional input inductors might be installed on the diode rectifier ac side to reduce the voltage spike level by limiting the current
Fig. 12. Diode rectifier input current and APF ac side voltage (VLL = 460 V, Ls = 0.5 mH, Pout = 80 kW, ϕ1 = 10◦ and n = 16).
slope during diode commutations. The voltage spike at the commutation instant is exacerbated with the increase of the inductance Ls and the output power Pout . Therefore, it can be deducted that the peak value of uc(APF) occurs at the moment of either ωt = 0 or the commutation instant of the diode rectifier current according to constraints of Pout , Ls and n. The condition and the corresponding APF peak ac side voltage are derived by peak
uc(APF) 2 2 VLL √ V , if X P ≤ LL L out 3 3n √ = 2 1 2 3 nXL Pout VLL √ VLL P , if X + > L out 2 3 2 2 VLL 3n (17) As a result, the required dc-link voltage level of the APF is found by (13) and (17). Vdc(APF)
√ if XL Pout ≤ 1.05 2VLL , = 1 3 nXL Pout , if XL Pout > 2 1.05 · VLL √2 + 2 VLL
V2 √LL 3n V2 √LL 3n
(18)
It should be √ noted that the APF dc-link voltage level can be set to 1.05 2VLL in all cases if the commutation effects are disregarded. The dc-link voltage level of the APF is depicted versus the output power and the input inductance in Fig. 13. The voltage level is comparable with that of the VSR for low power applications with small inductance, as seen in (15) and (18), because commutation voltage spikes have no effect on the dc-link voltage level of the APF. However, the APF voltage level is almost linearly increased with high power and reactance due to the commutation effect. Comparing Figs. 11 and 13, it is shown that the APF dc-link voltage level is much higher than the VSR level, in cases of high power applications with high input reactance. Theoretically,
130
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
will increase with the increase of Ls . Here again, the APF is not suitable for high power applications with large Ls due to its high dc-link voltage requirement. 5. Analysis of switch ratings
Fig. 14. The dc-link voltage levels of the VSR and the APF vs. the output power (VLL = 460 V, n = 16 and Ls = 0.1 mH).
the APF dc-link voltage level should be set to a high value enough to consider the voltage spikes of the commutation instants. However, in practical case, the nominal reference dc-link voltage for an APF is designed by neglecting the commutation effect, since the high voltage spikes at commutation points leads to high dc-link voltage requirement, which yields high switch losses and cost. For instance, given VLL = 460 V, the typical dc-link voltage level of the APF is set to 683 V
Since the VSR has sinusoidal input current, it is straightforward to determine the switch currents. The mean, rms, and peak switch current of the VSR are related with the supply voltage and the output power as √ 2 Pout 1 Pout mean rms , Isw(VSR) =√ , Isw(VSR) = √ 3π VLL 6 VLL √ 2 Pout peak Isw(VSR) = √ (19) 3 VLL The APF switch currents have more complex forms because of its highly distorted APF input current waveform shown in Fig. 3. The mean switch current of the APF can be calculated by integrating the absolute value of the APF input current ic over an entire cycle and dividing by 4π [9]. Notice that the shape of the APF input current varies by the displacement factor angle of the diode rectifier current. Therefore, the mean switch current of the APF can be categorized by ϕ1 . The mean switch current is derived as
√ π π √ 2Pout π 1 − 3 cos ϕ1 + 2 sin γ + √ , if 0 ≤ ϕ1 ≤ cos−1 −γ √ √3πV 2 3 3 cos ϕ1 6 LL mean Isw(APF) = √ π √ 2P π2 π √ out 1 − 3 cos ϕ1 + √ , if cos−1 < ϕ1 ≤ √ 6 3 cos ϕ1
3πVLL
from the first value of (18), regardless of the output power and reactance levels [7]. If this dc-link voltage is lower than the required voltage level at commutation instants, the APF will lose its harmonic current controllability at every commuta-
peak
Isw(APF)
2 3
γ = cos−1
√
π
2 3 cos ϕ1
(20)
6
The rms switch current is given by 2 π Pout rms Isw(APF) = √ −1 3 cos ϕ1 6VLL
(21)
The peak switch current is also dependent on the displacement factor angle, derived by √ 2Pout π π √ √ cos ϕ1 − , if 0 ≤ ϕ1 ≤ cos−1 3 2 3 3VLL = √ (22) 2Pout π 2π π π −1 √ √ < ϕ1 ≤ cos ϕ1 − , if cos + √ 3 6 3VLL 2 3 cos ϕ1 2 3
tion points. This will result in a notch in the supply current at commutation points of the diode rectifier currents. This will yield the deterioration in supply current quality, though it can be permissible in low and medium power applications. Fig. 14 shows the dc-link voltage levels of the VSR and the APF as a function of the output power. The voltage levels for the two converters are roughly the same in the low and medium power ranges. However, in high power levels, the APF dclink voltage level is higher than the VSR voltage level since the voltage spikes at the commutation instants determine the dc-link voltage. Note that the slopes for converters in Fig. 14
The mean, rms, and peak switch currents of the APF are depicted as a function of Pout and ϕ1 in Figs. 15–17, respectively. The switch currents of the VSR are also illustrated for the purpose of the comparisons. The mean and rms switch currents of the VSR are linearly increased with increasing Pout since the power directly flows through the converter. The APF mean and rms switch currents are also increased with increasing Pout and ϕ1 . This is because more reactive and harmonic power are required to flow from the APF to the diode rectifier, resulting in increase of the mean and the rms APF input currents. However, the mean and rms switch currents of the APF are smaller than those of the VSR due to
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
131
Fig. 18. Switch current ratios of the APF and the VSR. Fig. 15. The mean switch currents vs. the output power and the displacement angle (VLL = 460 V).
no real power flown through the APF. For clear comparison of the switch current ratings, a switch current ratio index is introduced by Isw =
Isw(APF) × 100% Isw(VSR)
(23)
Notice that the switch current ratio depends only on ϕ1 of the diode rectifier input current. Fig. 18 shows the switch current ratios of the APF and the VSR as a function of ϕ1 . It is shown the APF has a lower switch rating than the VSR.
6. Analysis of reactive components ratings 6.1. Input filter inductor
Fig. 16. The rms switch currents vs. the output power and the displacement angle (VLL = 460 V).
Peak ripple current is chosen as a criterion to design an input filter inductor. In order to calculate the ripple current, no-load condition is considered and the effect of inductor resistance is assumed to be negligible. Under these conditions, the inductance value is given by [10]: VLL Lf = √ 6 2fs iripple,peak
(24)
where fs denotes a switching frequency. Under the same supply voltage and switching frequency of the VSR and the APF, it is expected that the two converters require an equal input inductance with a given identical peak ripple current constraint. 6.2. dc capacitor The peak ripple voltage of the dc capacitor is, likewise, adopted as the design criterion for the dc capacitor size. 6.2.1. VSR With the assumption of a balanced three-phase utility and neglecting the losses of the power switches, the VSR in the dc-link in Fig. 1 is modeled as Fig. 17. The peak switch currents vs. the output power and the displacement angle (VLL = 460 V).
icap = iconv − idc
(25)
132
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
It can be written as dvdc(VSR) Pout = isδ dδ − dt vdc(VSR) 3
Cdc(VSR)
(26)
δ=1
where isδ and dδ (δ = 1,2,3) is the input current and the switch3 ing function of the VSR. By noting that isδ dδ is equal to δ=1
one of the VSR input currents at all times, the possible maximum ripple value of vdc(VSR) is approximately obtained with the negative peak value of the VSR input current and 50% duty ratio [11]. The ripple factor rv is defined with the allowable ripple voltage amplitude and the dc-link voltage level. rv =
vdc(VSR) Vdc(VSR)
Cdc(VSR)
Pout = VLL
(27) √
√ 2 + 3VLL /Vdc(VSR) √ 2 3rv Vdc(VSR) fs
(28)
The capacitor size increases with increasing Pout and decreasing rv , as shown in Fig. 19. 6.2.2. APF The capacitor size of the APF is likewise designed based on the peak APF input current in (22) as Cdc(APF)
=
Pout cos(ϕ1 − π/3) , VLL √6r V f
if 0 ≤ ϕ1 ≤ cos−1
Pout
if cos−1
v dc(APF) s
VLL
Fig. 20. APF capacitor size vs. output power and displacement factor angle (VLL = 460 V, Vdc(APF) = 683 V, fs = 10 kHz, and rv = 5%).
√ cos(ϕ1 − 2π/3) + π/(2 3cosϕ1 ) , √ 6rv Vdc(APF) fs
π
π
√ 2 3
π √ < ϕ1 ≤ 6 2 3
The APF capacitor size versus Pout and rv is depicted in Fig. 21 with ϕ1 = 30◦ . It is important to note that the APF capacitance is substantially smaller than the VSR, compared with Fig. 19. With the smaller ϕ1 , the APF capacitor size will become smaller, compared with the VSR capacitance. This is due to the different power components dealt by the two converters. The real power with the dc components are supplied through the VSR capacitor, while the reactive and harmonic power with high frequency components are carried through the APF capacitor. As a result, the voltage variation of the APF dc-link is expected to be smaller than that of the VSR. Thus the APF capacitor size can be substantially smaller than the VSR capacitor size. With the capacitance ratio defined by (30), the ratio is only dependent on ϕ1 , assuming that the same dc-link voltage level is used for the converters:
(29) Cdc =
Cdc(APF) × 100% Cdc(VSR)
(30)
Increasing Pout and ϕ1 implies that more harmonic and reactive power flow in and out of the capacitor. Thus, the capacitor size should be increased to regulate the dc voltage within the specified ripple factor. This is illustrated in Fig. 20.
In Fig. 22, it is shown that the APF capacitive filter size is smaller than the VSR capacitor.
Fig. 19. VSR capacitor size vs. output power and ripple factor (VLL = 460 V, Vdc(VSR) = 683 V, and fs = 10 kHz).
Fig. 21. APF capacitor size vs. output power and ripple factor (VLL = 460 V, Vdc(APF) = 683 V, fs = 10 kHz, and ϕ1 = 30◦ ).
S. Kwak, H.A. Toliyat / Electric Power Systems Research 75 (2005) 124–133
133
References
Fig. 22. Capacitance ratio vs. displacement factor angle (VLL = 460 V and Vdc(VSR) = Vdc(APF) = 683 V).
7. Conclusions In solving harmonic-polluting problems by diode rectifiers, two approaches based on controlled converter-type utility interface are possible; one replacing the diode rectifier with a VSR, and the other connecting an APF. The paper presents theoretical and comprehensive analysis for the VSR and the APF with only terminal constraints of unity power factor and identical output power delivery. The closed-form analytical expressions have been derived based on the quantification of the converter factors. The distinctive features of two converters have been compared and assessed based on the theoretical analyses. As expected, the APF requires a lower rating than the VSR in most low and medium power areas. However, the reactive power compensation for the diode rectifier substantially increases the APF rating. This feature, sometimes, leads to the APF applications to compensate either the reactive or the harmonic power in industry. In addition, the APF shows higher influence of the input filter on the ratings. The APF rating is more dependent on the output power increase than the VSR. Therefore, the APF implementation for low and medium power applications with small input filter is cost-effective. The dc-link voltage level is comparable for both converters in low and medium power areas. In high power applications, the APF dc-link voltage level, theoretically, should be set high enough to overcome the voltage spikes due to the diode current commutation, which leads to high switch losses and cost. However, the practical voltage level is determined disregard of the commutation effect. The lower voltage level than the required voltage results in the loss of current controllability at every instant of diode rectifier current commutation, and therefore the deterioration of supply current quality. The APF results in lower mean and rms switch ratings than the VSR because of no real power delivery. Equal input inductance value can be used for both converters. The APF requires significantly smaller capacitor size than the VSR because it is subject to harmonic and reactive power with higher frequencies.
[1] T. Thomas, K. Haddad, G. Joos, A. Jaafari, Design and performance of active power filters, IEEE Ind. Appl. Mag. 4 (5) (1998) 38–46. [2] W. Mgrady, M.J. Samotyj, A.H. Noyola, Survey of active power line conditioning methodologies, IEEE Trans. Power Del. 5 (1990) 1536–1542. [3] J.W. Dixon, J.M. Contardo, L.A. Moran, A fuzzy-controlled active front-end rectifier with current harmonic filtering characteristics and minimum sensing variables, IEEE Trans. Power Electron. 14 (4) (1999) 724–729. [4] T.C. Green, J.H. Marks, Ratings of active power filters, IEE Proc. Electron. Power Appl. 150 (5) (2003) 607–614. [5] N.R. Zargari, G. Joos, P.D. Ziogas, A performance comparison of PWM rectifiers and synchronous link converters, IEEE Trans. Ind. Electron. 41 (5) (1994) 560–562. [6] N. Mohan, T.M. Undeland, W.P. Robbins, Power Electronics, 2nd Ed., New York, Wiley. [7] S. Bhattacharya, T.M. Frank, D.M. Divan, B. Banerjee, Active filter system implementation, IEEE Ind. Appl. Mag. 4 (5) (1998) 47–63. [8] S. Ponnaluri, V. Krishnamurthy, V. Kanetkar, Generalized system design & analysis of PWM based power electronic converters, IEEE Ind. Appl. Conf. 3 (2000) 1972–1979. [9] J.H. Marks, T.C. Green, Rating analysis of active power filters, Proc. IEEE PESC’01 3 (2001) 1420–1425. [10] S. Bernet, S. Ponnaluri, R. Teichmann, Design and loss comparison of matrix converters and voltage-source converters for modern AC drives, IEEE Trans. Ind. Electron. 49 (2) (2002) 304–314. [11] Y. Yang, Advances in modeling and applications of three-phase power converters, PhD Dissertation, Department of Electronics and Computer Engineering, University Waterloo, Ont., Canada, 2001. Sangshin Kwak received the B.S. and M.S. degree in electronics engineering from Kyungpook National University, Daegu, Korea, in 1997 and 1999, respectively. He is currently working toward the Ph.D. degree in electrical engineering at Texas A&M University, College Station, TX. His main research interests are power converters topologies and controls, adjustable speed drives, and DSP-based power electronics control. Hamid A. Toliyat received the B.S., degree from Sharif University of Technology, Tehran, Iran in 1982, the M.S. degree from West Virginia University, Morgantown, WV in 1986, and the Ph.D. degree from University of Wisconsin-Madison, Madison, WI in 1991, all in electrical engineering. Following receipt of the Ph.D. degree, he joined the faculty of Ferdowsi University of Mashhad, Mashhad, Iran as an Assistant Professor of Electrical Engineering. In March 1994 he joined the Department of Electrical Engineering, Texas A&M University where he is currently E.D. Brockett professor of electrical engineering. He has received the prestigious Cyrill Veinott Award in Electromechanical Energy Conversion from the IEEE Power Engineering Society in 2004, Distinguished Teaching Award in 2003, E.D. Brockett Professorship Award in 2002, Texas A&M Select Young Investigator Award in 1999, and Eugene Webb Faculty Fellow Award in 2000. He has also received the Space Act Award from NASA in 1999, and the Schlumberger Foundation Technical Awards in 2000 and 2001. He is an Editor of IEEE Transactions on Energy Conversion, and was an associate editor of IEEE Transactions on Power Electronics. He is also Chairman of IEEE-IAS Electric Machines Committee, and is a member of Sigma Xi. He is a senior member of the Power Engineering, Industrial Applications, Industrial Electronics, Power Electronics Societies of the IEEE, and the recipient of the 1996 IEEE Power Engineering Society Prize Paper Award for his paper on the Analysis of Concentrated Winding Induction Machines for Adjustable Speed Drive Applications-Experimental Results. His main research interests and experience include analysis and design of electrical machines, variable speed drives for traction and propulsion applications, fault diagnosis of electric machinery, and sensorless variable speed drives. He has published over 220 technical papers in these fields and has 10 issued or pending US patents.