Design of LCL Filters With LCL Resonance Frequencies ... - IEEE Xplore

IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 4, NO. 1, MARCH 2016

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Design of LCL Filters With LCL Resonance Frequencies Beyond the Nyquist Frequency for Grid-Connected Converters Yi Tang, Member, IEEE, Wenli Yao, Student Member, IEEE, Poh Chiang Loh, and Frede Blaabjerg, Fellow, IEEE

Abstract— This paper proposes a novel LCL filter design method and its current control for grid-connected converters. With the proposed design method, it is possible to set the resonance frequency of the LCL filter to be higher than the Nyquist frequency, i.e., half of the system sampling frequency, and this observation is so far not discussed in the literature. In this case, a very cost-effective LCL filter design can be achieved for the grid-connected converters, whose dominant switching harmonics may appear at double the switching frequency, e.g., in unipolar-modulated three-level full-bridge converters and 12-switch-based three-phase pulsewidth-modulated converters. Moreover, a single-loop current control strategy is proposed for the designed LCL filter, and the control system is inherently stable without introducing any passive or active damping. Based on the new stability region, two LCL filter design examples are given, with one of them optimizing the utilization of passive filter inductors, and another one being robust against grid impedance variation. Comprehensive experimental results, showing the highquality output current and excellent resonance attenuation, are presented in this paper, which are also in very good agreement with those of the simulated ones. These results successfully verify the feasibility of the proposed LCL filter design and its current control. Index Terms— Current control, grid-connected converter, LCL filter, Nyquist frequency.

I. I NTRODUCTION

W

ITH the ever-increasing developments in power semiconductors and power electronics technologies, grid-connected converters may now find extremely wide and important applications in modern electric power systems, e.g., photovoltaic inverters for distributed generation, active power filters for power quality enhancement, and active front-end rectifiers for variable-speed motor drives. Manuscript received February 23, 2015; revised June 11, 2015; accepted June 18, 2015. Date of publication July 17, 2015; date of current version January 29, 2016. Recommended for publication by Associate Editor P. Mattavelli. Y. Tang is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). W. Yao is with the School of Automation, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]). P. C. Loh and F. Blaabjerg are with the Department of Energy Technology, Aalborg University, Aalborg 9100, Denmark (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JESTPE.2015.2455042

For voltage-sourced grid-connected converters, an L filter or an LCL filter has to be used at the output to filter the pulsed voltages and limit the harmonic contents. The LCL filter, being a third-order low-pass filter (LPF), can realize very effective switching harmonic attenuation with much reduced inductance requirement [1]. Therefore, it is usually preferred to the conventional L filter in order to achieve the high-quality sinusoidal grid current to comply with the stringent grid code. Nonetheless, LCL filters may also introduce the wellknown resonance issue into the system, making the closedloop current controller design challenging. A great number of research papers have been published to propose solutions to cope with the LCL resonance, among which the most straightforward way is to put a damping resistor in series with the capacitor of the LCL filter. In this case, the LCL resonance peaks can be attenuated, and a stable current control can be easily achieved in a wide frequency range. Despite the simplicity and the effectiveness of passive damping, the damping resistor may inevitably cause power losses to the system and decrease the conversion efficiency and the reliability of passive components [2]–[4]. Moreover, the passively damped LCL filter essentially becomes a second-order LPF rather than a third-order one at the high-frequency range, and therefore, its harmonic attenuation capability will be much reduced, and the original advantage of the LCL filters is lost. To obtain a more efficient and effective solution, active damping methods are also extensively researched in the literature, which could be good alternatives for the LCL resonance mitigation [5]–[20]. Through the proportional feedback of filter capacitor current, it is possible to emulate a virtual resistor in the control loop, and the resonance peaks can be attenuated without sacrificing the system efficiency and the harmonic reduction at high frequencies [5], [6], [12]. In order to achieve satisfactory current regulation, a step-by-step controller design for capacitor current feedback active damping is given in [7], where the system steady-state and dynamic responses can be optimized through careful design of the active damping and the current controllers. In addition to simply convert the active power, grid-interfaced converters with the LCL-filters can also be used for the power quality improvement, and the control and the implementation of such systems are discussed in [8] and [10]. Again, the inner capacitor current loop has to be employed to mitigate the LCL resonance. In order to

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further reduce the number of sensors and the implementation cost of active damping, Xu et al. [11] derived the equivalence of the capacitor current active damping and showed that the active damping can also be accomplished through the injected grid current. Similarly, in [13], it is reported that the capacitor current can be obtained by an observer instead of direct measurement, and in this case, one set of current transducers can be saved. Alternatively, the filter capacitor voltage can be measured for active damping, because it can be used for both the synchronization of the grid current and the derivation of the filter capacitor current for active damping [15]. Since the capacitor current is derived from the differentiation of the capacitor voltage, which could be noise sensitive, its digital implementation and lead–lag compensatorbased design have been discussed in [18]–[20] in order to improve the stability of the active damping. Due to the system delay caused by the digital computation and the pulsewidth modulation (PWM), the proportional feedback of the capacitor current may not always guarantee a stable active damping to the LCL resonance [9], [17], and the stability regions of active damping control have been identified in [21], where it is claimed that the active damping is necessary only at a low resonance frequency region. Similar conclusions are derived in [9], and the virtual impedance emulated by the proportional feedback of the capacitor current is found to contain a negative resistance part if the LCL resonance frequency is higher than 1/6 of the sampling frequency [14], [16]. Implementing active damping, in this case, may worsen the system stability. To cope with this issue, it is proposed in [9] that the sampling instant of the capacitor current can be shifted in order to reduce the signal delay time and widen the region of active damping. However, careful design must be devoted to avoid signal aliasing and switching noise issues with this sampling technique. In order to reduce the implementation complexity, Li et al. [22] proposed to introduce a repetitive control element in the capacitor current feedback loop, and in this way, the available active damping region can be widened up to 1/4 of the sampling frequency without complicated software and hardware design. In addition to the extensive research works revolved with the active damping, another research direction in LCL filters is to design a stable current control using the so-called system inherent damping characteristics [23]–[28]. Tang et al. [23] found that the converter current control is inherently stable as long as the system delay time can be minimized, e.g., using oversampling techniques and designing a low LCL resonance frequency. However, being similar to the active damping case, the system delay time is unavoidable in digitally controlled power converters, especially in high-power conversion system, where the switching or sampling frequency cannot be high. In this case, the stability of the current control may greatly be affected by the delay, and again, 1/6 of the sampling frequency is identified as the critical value for the LCL resonance frequency, above which the grid current control will inherently be stable, and otherwise, the converter current control should be adopted [26]. Wang et al. [27] comprehensively analyzed the influence of the delay time to the current control stability, the stable control regions, and the optimal control regions

Fig. 1. Circuit diagram of a full-bridge converter connected to the grid through an LCL filter.

for both current control schemes are identified, based on two different sampling techniques with different delay times. Regardless of the damping methods (passive, active, or damping less) adopted, a common principle of designing an LCL filter is to set its resonance frequency to be below the Nyquist frequency, i.e., half of the system sampling frequency, and in this case, the system controllability can be ensured, as claimed in [9] and [29]. However, in this paper, it is found that this conclusion is incomplete, and it is possible to exceed the Nyquist frequency limit by properly setting the resonance frequency of the LCL filters. Designing the LCL resonance frequency to be above the Nyquist frequency may find excellent applications in unipolar-modulated full-bridge converters or 12-switch-based three-phase PWM converters, where the dominant switching harmonics may appear at double the switching frequency, because very cost-effective design of the LCL filters can be achieved by setting a relatively higher LCL resonance frequency in these applications. Meanwhile, the quality of the output current is not compromised, and the current injected to the grid is still a clean sinusoidal with very low total harmonic distortion (THD). Furthermore, a new inherent stability region is identified, and a single-loop current control can be designed to stabilize the system without introducing any passive or active damping. Based on the new stability region, the two LCL filter design examples are given, with one of them optimizing the utilization of passive filter inductors, and another one being robust against grid impedance variation. Comprehensive experimental results, showing the high-quality output current and excellent resonance attenuation, are presented in this paper, which are also in good agreement with those of simulated ones. These results successfully verify the feasibility of the proposed LCL filter design and its current control. II. S YSTEM M ODELING AND S TABILITY A SSESSMENT The typical circuit diagram of a grid-connected full-bridge converter is shown in Fig. 1, where it is used for ac/dc rectifier applications. The direction of the converter current i 1 and the grid current i 2 is labeled according to the inverter case for ease of explanation. It should be noted that doing this will not affect the modeling and the stability analysis presented later on. The dc-link voltage control is not discussed in this paper, because it is usually designed to be a slow control loop, and will not interact with the inner current control as long

TANG et al.: DESIGN OF LCL FILTERS WITH LCL RESONANCE FREQUENCIES BEYOND THE NYQUIST FREQUENCY

Fig. 2.

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Equivalent block diagram of the full-bridge converter with digital converter current control.

as a proper control bandwidth is chosen. By neglecting the equivalent series resistances (ESRs) of passive components, the plant-equivalent block diagram is shown in Fig. 2. Based on this equivalent model, it can be readily derived that the converter current i 1 will be related to the converter pole voltage vo and the grid voltage vg with the following transfer function in the s-domain: (L 2 + L g )C f s 2 + 1 i 1 (s) = vo (s) L 1 (L 2 + L g )C f s 3 + (L 1 + L 2 + L g )s 1 vg (s) (1) − L 1 (L 2 + L g )C f s 3 + (L 1 + L 2 + L g )s

TABLE I S YSTEM PARAMETERS U SED FOR THE A NALYSIS , S IMULATIONS , AND E XPERIMENTS

where L 1 and L 2 are the converter-side inductance and the grid-side inductance, respectively. L g is the grid inductance, and C f is the filter capacitance. Further assuming a stiff grid voltage will allow the omission of the second term on the right-hand side of (1), and result in the following output voltage to converter’s current transfer function G i1 (s) for the system stability assessment: G i1 (s) = where γ =

1 s 2 + γ2 i 1 (s) = · 2 2 vo (s) L 1 s s + ωres

1 , ωres = (L 2 + L g )C f

(2)

L1 + L2 + Lg L 1 (L 2 + L g )C f

is the resonance frequency of the LCL filter. Since the current control is normally implemented in a digital signal processorbased control platform, the plant transfer function should be discretized before being used for the stability analysis, and this can be done by finding the zero-order hold (ZOH) equivalent of G i1 (s) as follows: G i1 (z)

G i1 (s) = (1 − z )Z L s 1 s 2 + γ2 = (1 − z −1 )Z L −1 · 2 L s 2 s 2 + ωres 1 Lt − L1 ωres 1 1 = (1 − z −1 )Z L −1 · 2 + · 2 2 Lt s L 1 L g ωres s + ωres t (L t − L 1 ) sin(ωres t) + = (1 − z −1 )Z Lt L 1 L t ωres z−1 Lt − L1 sin(ωres Ts )z Ts z = + · z L t (z − 1)2 L 1 L t ωres z 2 − 2z cos(ωres Ts ) + 1 (L t − L 1 ) sin(ωres Ts ) Ts + = L t (z − 1) L 1 L t ωres z−1 (3) · 2 z − 2z cos(ωres Ts ) + 1 −1

−1

where L t = L 1 + L 2 + L g is the total inductance formed by L 1 , L 2 , and L g . Ts is the system sampling period, which is normally chosen to be the reciprocal of the switching frequency f s if the symmetrical sampling technique is adopted [31]. In order to simplify the analysis, it is assumed that the gains of the integrator and the generalized integrator (or resonant controllers) are properly designed, so that the phase lag caused by them is negligible. In this case, a simple proportional gain K p can be considered. Considering the delay from the digital computation of duty cycles, the final forward path open-loop transfer function G ol_i1 (z) in the discrete z-domain can be written as G ol_i1 (z) = z −1 K p Vdc G i1 (z) K p Vdc (L t − L 1 ) sin(ωres Ts )(z − 1) K p Vdc Ts + = L t z(z − 1) L 1 L t ωres z[z 2 − 2z cos(ωres Ts ) + 1] (4) where Vdc is the dc bus voltage. Equation (4) can then be used to assess the stability of the converter current control. In order to investigate the new stability region, the LCL resonance frequency is purposely designed to be higher than the Nyquist frequency using the parameters listed in Table I (Case I). The Bode diagram of (4) is then plotted in Fig. 3, where it is interesting to note that the resof nance frequency ωres in the z-domain (or seen by the digital controller) is no longer equal to the one in the original plant. This is because the resonance frequency of the designed LCL filter already exceeds the Nyquist frequency, i.e., half of

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It is also noted from Fig. 3 that the resonance peaks may still exist in the discrete system, and its phase has a sharp f −180° change at the new resonance frequency fres . This step phase change may greatly affect the closed-loop system stability, and therefore, it is important to investigate the phase response of (4). However, a direct examination of (4) is not so straightforward if simply substituting z = e j ωTs and calculating its phase angle G ol_i1 (e j ωTs ). Fortunately, it is observed from Fig. 3 that the LCL resonance basically makes no phase contribution until the resonance frequency is actually reached [21]. In this case, the phase of (4) can be approximated by an ideal L filter-based system, whose transfer function G ol_it (z) can be written as G ol_it (z) = z −1 K p Vdc G it (z) = Fig. 3. Bode diagram of the system open-loop transfer function, showing the folded resonance frequency.

K p Vdc Ts L t z(z − 1)

(7)

which is essentially the first term on the right-hand side of (4). f f In this case, two critical phases ϕa and ϕb can be identified from Fig. 3 and can be calculated as f

ϕa = G ol_it (z)|z= j ω f

res Ts

= G ol_it (z)|z= j ω f

res Ts

f

ωres Ts π f − ωres =− − Ts 2 2 f ωres Ts 3π f f f − − ωres ϕb = ϕa − π = − Ts . 2 2 f

Fig. 4. Original resonance frequency folded into the digital system with respect to the Nyquist frequency.

the sampling frequency f s /2, and signal aliasing will occur in the digital sampling process. The system resonance frequency in the digital control system is mainly determined by the complex conjugate poles of (4). Since cos(ωres Ts ) is an even function with respect to ωres Ts = π if ωres Ts is greater than π, the pole location will be exactly the same as the case of (2π − ωres Ts ). In other words, the frequency of sampled resonance signals will be folded back with respect to the Nyquist frequency, as shown in Fig. 4, and the new resonance frequency in the z-domain will be related to the original one as follows: f f Ts + ωres Ts = 2π, or namely f res + f res = f s . ωres

(5)

It should be noted that this kind of frequency folding is, however, not applicable to the system zeros, because they are mainly determined by the numerator of (4), which consists of both sin(ωres Ts ) and cos(ωres Ts ) terms. Since sin(ωres Ts ) is an odd function with respect to ωres Ts = π, the zero location will be changed but is not simply folded with respect to the Nyquist frequency, as shown in Fig. 3. It order to clearly show the aliasing effect of the digital sampling, the Bode diagram of (4) is also shown in comparison with the following s-domain counterpart in Fig. 3, and the two resonance frequencies strictly comply with the relationship defined by (5) G ol_i1 (s) = K p Vdc e−1.5

Ts

G i1 (s).

(6)

f

(8) (9)

f

For ϕa , in (8), −π/2, −ωres Ts /2, and −ωres Ts are, respectively, caused by the filter inductor, the PWM, and the digital computation. In order to avoid the Nyquist encirclement of −1 f f (or the line of ϕa and ϕb crossing −180°), the following condition should be satisfied: f

ϕa < −π

(10)

f ϕb

(11)

> −3π.

Combining (5) and (8) to (11), it is possible to derive the new stability region for the converter current control, when the LCL resonance frequency is higher than the Nyquist frequency as follows: 5π fs 5 fs π < f res < . (12) < ωres < or namely, Ts 3Ts 2 6 By closing the open-loop transfer function of (4) using the parameters listed in Table I (Case I), the system closed-loop response can be obtained as G ol_i1 (z)/[1 + G ol_i1 (z)], and the pole–zero maps of this system can be plotted in Fig. 5, where the LCL resonance frequency is swept from 5 to 11 kHz by changing only the filter capacitance. The sampling frequency is fixed at 12 kHz. As it can be seen, there is always a pair of conjugate poles located inside of the unit circle with very slight changes. These two poles are mainly determined by the total inductance L t and the digital computation delay. Since they are at low-frequency bands, they may dominate the dynamic response of the system. It is also noted from Fig. 5 that the two LCL resonance poles are moving from the outside to the inside of the unit circle as the resonance frequency increases. Two critical resonance frequencies can be identified, i.e., 6 and 10 kHz, and they correspond to 1/2 and 5/6 of


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Fig. 5. Trajectories of the closed-loop poles with changing resonance frequencies, showing the stability region of the converter current control.

Fig. 7. Root loci of the converter current control with f res = 5 kHz, i.e., 5/12 of the sampling frequency.



the sampling frequency, respectively. Further increase of the resonance frequency may push the two resonance poles to the outside of the unit circle, making the system unstable again. This observation matches well with the stability condition dictated by (12), and it is not discussed in the literature yet. The controllability of the system under different stability regions can be further investigated with the root loci results, as shown in Figs. 6–8. Fig. 6 shows that, when the LCL resonance frequency is designed to be 8 kHz, which falls within the stability region specified by (12), the system is inherently stable as long as the proportional gain of the current controller is properly chosen. On the contrary, if the resonance frequency is placed to be out of the stability region, as shown in Figs. 7 and 8, the system will always be unstable, even though a very small proportional gain is adopted. In order to stabilize the system under these circumstances, the similar active damping concept can be utilized, and the capacitor current can be fed back into the control loop, as shown in Fig. 9. From Fig. 9, the capacitor current i c will be related to the converter output voltage vo with the following transfer

Fig. 9. Block diagram of the capacitor current active damping control with an ideal grid voltage assumed.

function G ic (s): G ic (s) =

1 s2 i c (s) = · 2 . 2 vo (s) L 1 s s + ωres

(13)

For the digital implementation, its ZOH equivalent will be G ic (z) =

sin(ωres Ts ) i c (z) z−1 = . · 2 vo (z) L 1 ωres z − 2z cos(ωres Ts ) + 1 (14)

Considering the proportional feedback of the capacitor current, the new open-loop transfer function of the

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Fig. 10. Harmonic spectrum of the unipolar-modulated full-bridge converter, as shown in Fig. 1.

system G ol_i1ic (z) will be G ol_i1ic (z) =

z −1 K p Vdc G i1 (z) 1 + z −1 K d Vdc G ic (z)

(15)

where K d is the active damping gain, and similarly, only a simple proportional gain K p is considered for G ci (z). With this active damping loop introduced to the feedback control loop, the LCL resonance poles can be attracted into the unit circle, as shown in Figs. 7 and 8, and the system can still be stabilized even if the LCL resonance frequency is out of the new stability region dictated by (12). III. LCL F ILTER D ESIGN E XAMPLES AND C URRENT C ONTROLLER D ESIGN As mentioned, for symmetrical sampled systems, where the duty cycles are updated once in each switching period, the system switching frequency f sw simply equals to the sampling frequency f s . In this case, designing the LCL resonance frequency to be above the Nyquist frequency is then not a wise choice for the conventional half-bridge-based topologies or bipolar-modulated full-bridge converters, because the resonance frequency is very close to the switching frequency, and the switching harmonic attenuation will be quite limited. Therefore, the LCL resonance frequency in such systems is typically designed to be lower than half of the switching frequency or even lower in order to obtain sufficient harmonic attenuation. However, the proposed LCL filter design will be very suited for some applications, e.g., unipolar-modulated full-bridge converters and 12-switch-based three-phase converters, where the first switching harmonic is almost zero because of the cancelation effects between the two switching legs in one phase [30]. Fig. 10 shows the spectrum of the output voltage of a unipolar-modulated full-bridge converter, where it is clear that the switching harmonics of interest are significant only at double the switching frequency. Therefore, designing the LCL resonance frequency in the range of 1/6 f s –1/2 f s becomes improper, because the passive components will be unnecessarily large. The proposed LCL filter design is, therefore, more reasonable and may provide a more cost-effective solution over the conventional case. A. LCL Filter Design With Optimized Inductance Utilization For the first design case, a stiff ac power grid is assumed, and the grid inductance L g is zero. The proposed LCL filter design method can then starts with the converter-side filter inductor design. For unipolar-modulated full-bridge

Fig. 11.

Design flowchart of the proposed LCL filter.

converters, the switching ripple current on L 1 can be approximated by the following equation: L 1 I L1 =

0.5 × 0.5 × Vdc 2 fs

(16)

where I L1 is the maximum peak-to-peak ripple current of the converter-side inductor. In order to limit the amplitude of the switching current as well as the core loss of L 1 , I L1 is designed to be around 40% of the nominal current. Referring to Table I, the required inductance can be calculated to be L1 =

Vg 0.5 × 0.5 × Vdc Vdc = × ≈ 0.44 mH. √ 2 f s I L1 8 fs 0.4 × 2Pn (17)

L 1 is finally designed to be 0.45 mH. The grid-side inductor L 2 is chosen to be identical to L 1 in order to maximize the utilization of inductance, because the lowest resonance frequency can be achieved with such a design [10]. The filter capacitance C f should be designed in such a way that the LCL resonance frequency is in the range of 1/2 f s to 5/6 f s . In order to have more stability margins and account for the inductance reduction at high current, the midpoint resonance frequency 2/3 f s , i.e., 8 kHz is chosen, and therefore L1 + L2 ≈ 1.76 μF. (18) Cf = 2 L 1 L 2 ωres Before completing the LCL-filter design, the value of C f should be checked. If it is greater than 0.5 p.u., it will cause too much reactive current [1]. In this case, the filter inductor should be increased until C f = 0.05 p.u. can be satisfied. A detailed design flowchart is shown in Fig. 11 to


facilitate the design process of the proposed LCL-filter. With C f determined, the resonance frequency will be well within the stability region, as shown in Fig. 5, and it is also far away from the dominant switching harmonics, indicating that sufficient harmonic attenuation can successfully be achieved. Its effectiveness will be evidenced by the results shown in a later Section IV.

dynamics will be dominated by the total inductance L t and the system time delay. In order to have a fast dynamic response and meanwhile ensure enough phase margins, the crossover frequency is designed to be 1/15 of the sampling frequency, i.e., ωc = 2π f s /15. At the crossover frequency

For the studied system, f res_ min and f res_ max are finally designed to be 7 and 9 kHz, respectively, in order to have some stability margins. L 1 can still be determined by (17) for the same reasons, i.e., limiting the amplitude of switching current and the core loss of L 1 . In this case, C f can be calculated according to (19) Cf =

1 ≈ 1.15 μF. L 1 (2π fres_ min )2

(21)

C f is finally chosen to be 1.15 μF. Equation (20) can then be used to determine L 2 , which is L2 =

L1 ≈ 0.69 mH. L 1 C f (2π fres_ max )2 − 1

G ol_it (e j ωc Ts ) = =

B. LCL Filter Design With Robustness Against Grid Inductance Variation The grid inductance may be uncertain in a weak power system, and this may lead to varying LCL resonance frequencies for the power converter. According to (2), the minimum LCL resonance frequency f res_ min and the maximum LCL resonance frequency f res_ max can be, respectively, calculated to be 1 1 (19) f res_ min = f res | L g →∞ = 2π L 1 C f 1 L1 + L2 f res_ max = f res | L g =0 = . (20) 2π L 1 L 2 C f

(22)

With such an LCL-filter design, the LCL resonance frequency is always within the stability region regardless of the change of the grid inductance.

= = |G ol_it (e j ωc Ts )| = =

The current controller design is applied to the first LCL filter design case, where the inductance utilization is maximized. The same design procedure can also be repeated to the second case by assuming L g = 0. Increasing of L g may push the resonance frequency to be lower and give more damping effects to the LCL resonance poles, as long as the proportional gain of the current controller is fixed. However, the system dynamic performance may inevitably be compromised in this case, because the crossover frequency ωc also becomes lower. An adaptive gain scheduling scheme can be adopted in order to keep a relatively fixed ωc . This control scheme will not be further discussed here, because it is out of the scope of this paper. Referring to Fig. 3, it is clear that the magnitude and the phase responses of (4) can be approximated by (7) under the low-frequency range. In this case, the system stability and

K p Vdc G it (e j ωc Ts ) e j ωc Ts K p Vdc Ts 1 j ω T j c s Lt e (e ωc Ts − 1) π π ωc Ts π − − − ωc Ts = − − 2 2 2 5 7π = −126° (23) − 10 | e− j ωc Ts K p Vdc G it (e j ωc Ts )| K p Vdc Ts (24) L e j ωc Ts (e j ωc Ts − 1) = 1.

t

Equation (23) indicates that there is still a −126° + 180° = 54° phase margin left, which can be used by the integral or generalized integral terms. Substituting ωc Ts = 2π/15 and the parameters listed in Table I (Case I) into (24), the desired proportional gain will be L t |cos ωc Ts − 1 + j sin ωc Ts | ≈ 0.022. (25) Kp = Vdc Ts The current controller is implemented in the stationary reference frame using the well-known proportional-resonant controller [32]. Another resonant controller tuned at the triple of the fundamental frequency ωn is also implemented in order to reject the second-order harmonic disturbance from the dc-link. The final current controller can be written as 2K i s 2K i s + 2 (26) G ci (s) = K p + 2 2 s + s + ωn s + s + 9ωn2 where K i1 is the gain of the sinusoidal integrators. The above current controller is discretized by the Tustin method to obtain G ci (z) for its digital implementation. Since the resonance frequencies of (26) are much lower than the sampling frequency, the phase lag introduced by the current controller at ωc will be

C. Current Controller Design

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G ci (z) ≈ G ci (s) 2K i j ωc 2K i j ωc = + Kp + −ωc2 + ωn2 + j ωc −ωc2 + 9ωn2 + j ωc 1 1 K p + j 2K i ωc . + ≈ −ωc2 + ωn2 −ωc2 + 9ωn2 (27)

If the required phase margin is 45°, the integral gain can be calculated to be K p tan(45°–54°)

≈ 4.25. Ki = (28) 2ωc −ω21+ω2 + −ω2 1+9ω2 c

n

c

n

The final Bode diagram of the open-loop transfer function, considering G ci (z) and G i1 (z) is shown in Fig. 12. As it can be seen, both the crossover frequency and the phase margin of the system are very close to the predefined values, and the highquality output current can be expected because of the high compensation gains introduced by the resonant controllers.

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Fig. 12. Bode diagram of the system open-loop transfer function considering the resonant current controllers. Fig. 14. Simulated dynamic results with 40 V step-up change on the dc bus voltage using the parameters listed in Table I (Case I) (note that the currents are inverted).

Fig. 13. Simulated steady-state results with an 840-W load operation using the parameters listed in Table I (Case I) (note that the currents are inverted).

IV. S IMULATION AND E XPERIMENTAL R ESULTS A. Simulation Results The system shown in Fig. 1 was simulated with piecewise linear electrical circuit simulation (PLECS) in order to show the effectiveness of the proposed LCL filter design method and its current control. The circuit parameters are listed in Table I. Only the first LCL filter design case was simulated, and the second case will be proved by the experimental results shown in a later Section IV B. Fig. 13 shows the simulated steady-state results with 840-W load power on the dc side. As it can be seen, the grid current can be well regulated in the steady state. It is highly sinusoidal, and its THD is as low as 0.76%. This is mainly because the proposed LCL-filter can provide enough attenuation (−85.6 dB) at the dominant switching harmonic frequency, i.e., twice of the switching frequency. For a conventional LCL-filter design, e.g., f res = f sw /3 = 4 kHz, the resonance frequency is halved, and therefore, either the filter inductors or the filter capacitor have to be four times of the ones in the proposed design. However, the improvement

on the harmonic performance will be very limited, because the switching harmonics are already well suppressed by the proposed LCL-filter, as shown in Fig. 13. This confirms that setting the LCL resonance frequency to be above the Nyquist frequency may give a more cost-effective design for unipolarmodulated full-bridge converters, as the passive components for the output filter can be reduced. The ripple current on the converter-side inductor is also in an acceptable range, i.e., its peak-to-peak value is smaller than 40% of the nominal current. In this case, the switching losses of semiconductors and the core loss of L 1 can be limited. In order to test the dynamic response, at 0.3 s, the reference voltage of the dc-link is suddenly changed from 1 pu (200 V) to 1.2 pu (240 V), giving an 8 A step-up change on the current reference as the proportional gain adopted in the voltage control loop is 0.2 (0.2 × Vdc = 8). Fig. 14 shows the simulated results, and it is clear that the system has excellent dynamic response, because the converter current can track its reference in a very short time without obvious resonance. Fig. 15 shows the zoomed-in view of the transient period. It is noted that the grid current is slightly ringing with a frequency of 8 kHz, which is clearly caused by the LCL resonance poles. However, this ringing is basically invisible, because they are far away from the dominant poles, as shown in Fig. 6. B. Experimental Results A 1 kW experimental prototype was built and tested in the laboratory, and the circuit parameters are basically the same as those used in the simulation. The filter inductance was purposely made to be slightly larger than the nominal value to account for the inductance reduction at high currents. Litz wires were used for inductor windings in order to minimize the damping effects of ESRs. The ac power grid was emulated by a Chroma 61502 programmable ac power supply, and the maximum dc load power tested in experiments was 840 W due to the limitation of the ac power supply. The current control algorithm was executed on a Dspace 1006 digital control platform. Again, the switching deadtime was set to be only


Fig. 15. Zoomed-in view of the transient period, as shown in Fig. 14 (note that the currents are inverted).

Fig. 17. Experimental steady-state results with an 840-W load operation using the parameters listed in Table I (Case I). Top to bottom: vdc , 200 V/div, vpcc , 200 V/div, i2 , 10 A/div (inverted), and i1 , 10 A/div (inverted). Time axis: 5 ms/div.

Fig. 18.

Fig. 16.

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Spectrum of the grid current, as shown in Fig. 17.

Experimental setup in the lab.

200 ns in order to minimize its damping effects, and the SiC MOSFET C2M0080120D from Cree was chosen as the active switch because of its very fast switching capability and low power losses. All the experimental waveforms were captured from a Lecory Wavesurfer 3024 digital oscilloscope, and the implemented experimental setup is shown in Fig. 16. The steady-state experimental results with 840-W dc load power operation is shown in Fig. 17, which clearly match well with those of the simulated ones in Fig. 13. The data of the grid current were also saved from the oscilloscope and then replotted in MATLAB for the fast Fourier transform analysis, and the result is shown in Fig. 18. As expected, the dominant switching harmonics appear at 2 f s , and the harmonic amplitude is as low as 43 mA, which is less than 0.5% of the fundamental component. The low-order harmonics are also well regulated because of the inclusion of the third-order resonant controller. The resulting THD of the grid current is only 1.62%, which can well comply with the grid code. The resonance component at 8 kHz is basically zero mainly because of the stabilization of the designed current controller as well as the damping effects provided by the ESRs of passive components.

Fig. 19. Experimental dynamic results with 40 V step-up change on the dc bus voltage using the parameters listed in Table I (Case I). Top to bottom: vdc , 200 V/div, vpcc , 200 V/div, i2 , 10 A/div (inverted), and i1 , 10 A/div (inverted). Time axis: 10 ms/div.

Next, the similar dynamic test was performed, and the dc reference voltage was commanded to have a 40 V step-up change, and the captured experimental results are shown in Figs. 19 and 20. Again, they are in very good agreement

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Fig. 20. Zoomed-in view of the transient period, as shown in Fig. 19. Time axis: 0.5 ms/div.

Fig. 21. Experimental dynamic results with 40 V step-up change on the dc bus voltage using the parameters listed in Table I (Case II) (L g = 0). Top to bottom: vdc , 200 V/div, vpcc , 200 V/div, i2 , 10 A/div (inverted), and i1 , 10 A/div (inverted). Time axis: 10 ms/div.

with those of the simulated ones in Figs. 14 and 15. It is noted in Fig. 20 that the ringing of the grid current becomes shorter and basically diminishes to zero after the fourth period. This is mainly because of the passive damping effects of the real converter, which is not fully considered in the simulation. This dynamic test was also performed with the second LCL filter design case, where the system is made to be robust against the grid inductance variation. Using the parameters listed in Table I (Case II), the first set of dynamic experimental results is shown in Figs. 21 and 22, where L g = 0 is assumed. As it can be seen, the dynamic response is very similar to the one shown in Fig. 19, and this is because the current controller was designed by the same method. The zoomed-in view


Fig. 23. Experimental dynamic results with 40 V step-up change on the dc bus voltage using the parameters listed in Table I (Case II) (L g = 3 mH). Top to bottom: vdc , 200 V/div, vpcc , 200 V/div, i2 , 10 A/div (inverted), and i1 , 10 A/div (inverted). Time axis: 10 ms/div.

shown in Fig. 22 proves that the maximum resonance frequency is ∼9 kHz, and it is always within the stability region. The second set of dynamic experimental results is shown in Figs. 23 and 24, where L g = 3 mH was used in experiments to emulate a weak power grid. The gains in the current controller remain the same as the previous case. As discussed, the increasing of L g may push the crossover frequency to be lower, and thus provide more damping effects to the LCL resonance poles. Therefore, there is basically no ringing in the grid current, as shown in Fig. 24. However, the system dynamic response is deteriorated and it takes longer time for the current to track its reference, as shown in Fig. 23. Again, the stability of the current control can be ensured, because the LCL resonance frequency will



always be higher than the stability margin, i.e., 6 kHz in this case. V. C ONCLUSION This paper has proposed a novel design method for gridconnected converters with LCL filters, where the resonance frequency of the LCL filter can be placed to be higher than the system Nyquist frequency. The signal aliasing and frequency folding effects under such design have been identified. Based on the folded resonance frequency, the converter current control is found to be inherently stable when used as the feedback control variable, and it may greatly facilitate the current controller design. The proposed LCL filter design may find suitable applications in the unipolar-modulated full-bridge converters and the 12-switch-based three-phase PWM converters, where the dominant switching harmonics appear at double the switching frequency. The required passive components for the LCL filter can be much reduced, and more cost-effective filter design can be achieved over the conventional method. The two LCL filter design examples have been presented, with one of them optimizing the utilization of inductance, and another one being robust against grid inductance variation. The injected grid current can be well regulated under both the cases without any resonance. The simulation and the experimental results are finally presented to verify the feasibility of the proposed LCL filter and the current control design. R EFERENCES [1] M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCLfilter-based three-phase active rectifier,” IEEE Trans. Ind. Appl., vol. 41, no. 5, pp. 1281–1291, Sep./Oct. 2005. [2] R. Peña-Alzola, M. Liserre, F. Blaabjerg, R. Sebastián, J. Dannehl, and F. W. Fuchs, “Analysis of the passive damping losses in LCL-filterbased grid converters,” IEEE Trans. Power Electron., vol. 28, no. 6, pp. 2642–2646, Jun. 2013. [3] A. A. Rockhill, M. Liserre, R. Teodorescu, and P. Rodriguez, “Grid-filter design for a multimegawatt medium-voltage voltage-source inverter,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1205–1217, Apr. 2011.

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Yi Tang (S’10–M’14) received the B.Eng. degree in electrical engineering from Wuhan University, Wuhan, China, in 2007, and the M.Sc. and Ph.D. degrees from the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2008 and 2011, respectively. He was a Senior Application Engineer with Infineon Technologies Asia Pacific, Singapore, from 2011 to 2013. From 2013 to 2015, he was a Post-Doctoral Research Fellow with Aalborg University, Aalborg, Denmark. He is currently with Nanyang Technological University, Singapore, as an Assistant Professor. Dr. Tang received the Infineon Top Inventor Award in 2012.

Wenli Yao (S’14) received the B.S. and M.S. degrees in electrical engineering from the School of Automation, Northwestern Polytechnical University, Xi’an, China, in 2009 and 2012, respectively, where he is currently pursuing the Ph.D. degree in power electronics. His current research interests include current control, grid connected inverter, multipulse converter, and power decoupling.

Poh Chiang Loh received the B.Eng. (Hons.) and M.Eng. degrees from the National University of Singapore, Singapore, in 1998 and 2000, respectively, and the Ph.D. degree from Monash University, Melbourne, VIC, Australia, in 2002, all in electrical engineering. He has been with Aalborg University, Aalborg, Denmark, since 2013.

Frede Blaabjerg (S’86–M’88–SM’97–F’03) received the Ph.D. degree from Aalborg University, Aalborg, Denmark, in 1992. He was with ABB-Scandia, Randers, Denmark, from 1987 to 1988. He became an Assistant Professor in 1992, an Associate Professor in 1996, and a Full Professor of Power Electronics and Drives with Aalborg University in 1998. His current research interests include power electronics and its applications, such as wind turbines, PV systems, reliability, harmonics, and adjustable speed drives. Dr. Blaabjerg received 15 IEEE Prize Paper Awards, the IEEE Power Electronics Society (PELS) Distinguished Service Award in 2009, the EPE-PEMC Council Award in 2010, the IEEE William E. Newell Power Electronics Award in 2014, and the Villum Kann Rasmussen Research Award in 2014. He was the Editor-in-Chief of the IEEE T RANSACTIONS ON P OWER E LECTRONICS from 2006 to 2012. He was a Distinguished Lecturer of the IEEE PELS from 2005 to 2007, and the IEEE Industry Applications Society from 2010 to 2011.