사각형입니다.

https://doi.org/10.6113/JPE.2018.18.4.1165

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718



New Design Approach for Grid-Current-Based Active Damping of LCL Filter Resonance in Grid-Connected Converters


Mahmoud A. Gaafar, Gamal M. Dousoky*, Emad M. Ahmed**,***, Masahito Shoyama****, and Mohamed Orabi


†, ‡APEARC, Faculty of Engineering, Aswan University, Aswan, Egypt

*Electrical Engineering Dept., Minia University, Alminia, Egypt

**Dept. of Electrical Engineering, Jouf University, Aljouf, Saudi Arabia

***Dept. of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, Egypt

****Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan



Abstract

This paper investigates the active damping of grid-connected LCL filter resonance using high-pass filter (HPF) of the grid current. An expression for such HPF is derived in terms of the filter components. This expression facilitates a general study of the actively damped filter behavior in the discrete time domain. Limits for the HPF parameters are derived to avoid the excitation of unstable open loop poles since such excitation can reduce both the damping performance and the system robustness. Based on this study, straightforward co-design steps for the active damping loop along with the fundamental current regulator are proposed. A numerical example along with simulation and experimental results are presented to verify the theoretical analyses.


Key words: Active damping, Grid, LCL filter, Resonance


Manuscript received Aug. 29, 2017; accepted Mar. 27, 2018

Recommended for publication by Associate Editor Se-Kyo Chung.

Corresponding Author: mahmoud_gaafar@aswu.edu.eg Tel: +20-97-4661589, Fax: +20-97-4661406, Aswan University

APEARC, Faculty of Engineering, Aswan University, Egypt

*Dept. of Electrical Engineering, Minia University, Egypt

**Dept. of Electrical Engineering, Jouf University, Saudi Arabia

***Dept. of Electrical Engineering, Aswan University, Egypt

****Graduate School of Inform. Sci. & Electr. Eng., Kyushu Univ., Japan



Ⅰ. INTRODUCTION

For grid-connected converters, LCL filters are more interesting when compared to L filters due to their higher attenuation for switching harmonics along with lower weight and volume [1], [2]. From the control view point, the current control strategies developed for grid-connected converters should fulfill a number of objectives include sinusoidal grid current, fast transient response, high robustness against parameters variations, implementation simplicity and achieving zero steady-state error in the grid current [3]. LCL filters introduce two extra poles which reduce system stability. Therefore, achieving the aforementioned objectives for an LCL filter based system requires a complex control algorithm equipped with a resonance damping technique [3]. Various linear and non-linear control strategies have been proposed for LCL filter based single-phase grid-connected converters.

Hysteresis current control methods can achieve most of the aforementioned objectives such as fast transient response, high robustness to parameter variations and implementation simplicity [4]-[7]. However, keeping the measured current within a hysteresis band results in a variable switching frequency that may result in undesired current harmonics. Sliding mode control based methods offer simple implementation, fast dynamic response and high robustness [8], [9]. However, they suffer from the chattering phenomenon and variable switching frequency. Composite nonlinear feedback and Lyapunov-function based control methods have been adopted. However, their implementation is not simple [10], [11].

Other control methods, such as repetitive control [12], predictive control [13], intelligent control [14] and neural- network based control [15], have been adopted for the current loop control of inverters. However, there are some limitations associated with these methods. For example, predictive control needs precise system parameters to reach the desired performance, intelligent control has a variable switching frequency that may result in undesired current harmonics, repetitive control shows a slow dynamic response, and neural-network based control requires a complex training process.

Using a conventional PI-based controller in the stationary reference frame results in a steady state error in tracking sinusoidal references [16]. On the other hand, an imaginary control circuit is required if synchronous reference frame implementation is adopted. To overcome the problems of conventional PI controllers in the stationary reference frame, a PR controller has been proposed. PR controllers are widely used in the control of single-phase inverters [11], [17], [18].

From the stability viewpoint, a single grid current control loop can be adopted for resonant frequencies of more than one-sixth of the sampling frequency [1]. However, this technique is not always suitable especially in weak grids where the grid inductance varies significantly. Optimized loop shaping for both a current controller and an active damper (AD) has been proposed in [19] using a complicated fifth-order feedback. Passive damping configurations have been presented in [1][1]. However, they increase both the power losses and the filter size. Recently, active damping by modifying the control system has been getting a lot of attention. A cascade digital filter can be used for this purpose [20]. However, this decreases the system bandwidth and is highly dependent on the varied grid side inductance. Active damping based on state variables feedback is more desirable. In this regard, using the filter capacitor current or voltage feedback have been investigated [21]-[31]. However, these methods make additional current/voltage sensors or complicated estimation loops necessary.

Grid-current-based active damping is more desirable since there is no need for additional sensors or complicated control algorithms. Ideally, this needs an s2 term in the active damping loop [32]. However, it is not implemented practically due to noise amplification. Two approaches have been presented in the literature to overcome this issue. One of these approaches employs a second order Infinite Impulse Response (IIR) filter [33]. However, the control system is complicated and a large number of iterations are needed to meet pre-specified behavior. The second approach, which is the main focus of this paper, employs a HPF of the grid current feedback [32], [34]-[36]. In [32], a HPF is designed in the s-domain to behave as an ideal s2 term around the resonant frequency. However, these studies do not provide straightforward design steps especially for discrete implementation. In [34], an independent design for an HPF and Synchronous Rotating Frame Proportional-Integrator (SRFPI) current regulator has been proposed.

In both [16] and [18], there is no consideration of the transport delays of the digitally controlled systems and their effect on the open loop system stability which, upon violation, can decline the damping performance and the system robustness. Moreover, as indicated in [35], both the AD and the fundamental current regulator must be designed together to achieve a pre-specified performance.

Based on s-domain emulation of a digitally controlled system, a virtual impedance model for a grid-current-based actively damped filter has been derived in [35] as a shunt impedance across the grid side inductance. It was determined that unstable open loop behavior can be avoided for resonant frequencies up to 0.28 of the sampling frequency. However, for certain resonant frequencies, it cannot identify the parametric influence of the HPF on the open loop stability. Consequently, the tuning process becomes tedious and a lot of iterations are needed to design the HPF along with the fundamental current regulator without open loop stability violation. In addition, it is cost effective to design LCL filters with higher resonant frequencies, and without violation of open loop stability especially when selective harmonic compensation is of concern [25].

From the above discussion, the following challenges can be identified when handling grid-current-based active damping.

⋅ Extending the resonant frequency range over which unstable open loop behavior can be avoided.

⋅ Identifying the parametric influence of the HPF on the open loop system stability of digitally controlled systems.

⋅ Straightforward design steps for the HPF along with the fundamental current regulator.

Following this introduction, an expression of the HPF is derived. Using this expression, a detailed study for the actively damped filter is carried out in the discrete time domain to clarify the effect of HPF parameters on open loop stability. After that, straightforward design steps for the HPF along with the fundamental current regulator are presented. To verify the theoretical analyses, a numerical example along with experimental results at different resonant frequencies are presented. Finally, some conclusions are introduced.



Ⅱ. PROPOSED HPF FORM FOR ACTIVE DAMPING


A. System Description

Using an LCL filter, a grid-connected single phase inverter is shown in Fig. 1. The control system is shown in Fig. 2. A proportional resonant (PR) controller, represented in (1) as Gc(s), is used as a current regulator with 그림입니다.
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Fig. 1. Inverter connected to a grid through an LCL filter.


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Fig. 2. System block diagram with capacitor current feedback.


This system is manipulated in Fig. 3. From Fig. 3(c), the capacitor current feedback is equivalent to three feedback loops of the modulated inverter voltage (vi), the capacitor voltage (vc) and the grid current (ig). The system is further manipulated in Fig. 3(d) as follows.


Fig. 3. Manipulation of the active damping method, which uses a proportional feedback of the capacitor current, sequentially from (a) to (d).

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⋅The capacitor voltage feedback is shifted towards the grid current producing a proportional term 그림입니다.
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⋅The modulated voltage feedback is augmented as a cascaded HPF and expressed as Gh(s) in (3) where 그림입니다.
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B. Proposed Active Damper Expression

The HPF Gh(s) in Fig. 3(d) is eliminated from the main control loop and inserted in the grid current damping loop as shown in Fig. 4. The damping feedback loop is still a HPF, expressed in (4) as Gad(s). To acquire flexibility, the gain of Gad(s) is re-written in (5) as the product of a new variable (r) with a constant quantity (Li+Lg).

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Unlike the general HPF expressions used in [32],[34]-[35], expressing the HPF gain in terms of the filter inductances helps clarify the parametric influence of the HPF on open loop stability as indicated in the following sections.


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Fig. 4. System block diagram with a HPF of the grid current feedback.



Ⅲ. DISCRETE IMPLEMENTATION


A. System Discretization

Fig. 5 shows the system discrete representation where Gig(z) is the discrete  transfer function relating the modulated inverter voltage to the grid current. It is determined using a zero-order-hold (ZOH) discretization of its continuous counterpart Gig(s). Both Gig(s) and Gig(z) are expressed in (6) and (7), respectively. Gc(s) and Gad(s) are discretized using Tustin approximations as expressed in (8) and (9) where Ts denotes the sampling time. The digital signal processor (DSP) delay is modeled by one sample delay.

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Fig. 5. Discrete representation of the overall system.


where:

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To generalize the analyses, both ωres and ωh are expressed in terms of the sampling frequency (s) as in (11). Then the expressions in (10) are re-written in (12).

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Using (7), (9) and (12), the actively damped filter (Fnew(z)) is expressed in (13). Its gain depends on the specific values of the sampling frequency and the filter inductances. However, the zeros and poles of Fnew(z) do not depend on these specific values. They depend on r (the gain-multiplier of the HPF), βres (the ratio of ωres to ωs) and βh (the ratio of ωh to ωs). Since the PR controller expressed in (8) does not have any unstable poles, the open loop stability (그림입니다.
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In the next sections, the effects of r, βres and βh on the stability of Fnew(z) are investigated. From (13), Fnew(z) has one constant pole at z=1 and four poles that depend on r, βres and βh; two resonant poles, and other two are called non-resonant poles.


B. Discussing the Effects of the HPF Parameters

In Fig. 6, the pole map of Fnew(z) is plotted by sweeping βh from 0 to 0.5 (corresponding to ωh, which is equal to the Nyquist frequency) at a constant value of r=1 (corresponding to a HPF gain of (Li+Lg)) and three values of βres (βres1<βres2<βres3; corresponding to different resonant frequencies). The following remarks can be revealed from this plot.

1. In addition to the constant pole at z=1, one of the non-resonant poles is also constant at z=1. The second non-resonant pole tracks entirely inside the unit circle for all values of βres (its track direction is not shown for the sake of clarity).

2. The tracks of the resonant poles start from some point on the unit circle (corresponding to an undamped LCL filter). By increasing βh, the resonant poles can track entirely inside the unit circle (as for βres1), or they can track entirely outside the unit circle (as for βres3). In addition, they can initially track outside the unit circle before tracking inside the unit circle above a certain value of βh (as for βres2).

From the second remark, it is expected that there is a maximum limit for βres above which Fnew(z) is unstable  in  the range of βh (<0.5). This maximum limit is denoted as βres-max (corresponding to the resonant frequency of ωres-max). At βres-max, the resonant poles should track outside the unit circle and end by an intersection with the unit circle at βh=0.5. Based on this understanding, βres-max is determined by plotting a pole map of Fnew(z) while sweeping βres at a constant value of βh=0.5.


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Fig. 6. Pole maps of Fnew(z) with sweeping βh (at r=1 and different βres).


To investigate the effect of HPF gain variations, pole maps are plotted for the two regions of (0<r≤1) and (r<0) as shown in Figs. 7 (a) and (b), respectively. In these plots, βres is swept from 0.1 to 0.45 (theoretically, βres can be extended to 0.5. However, due to resonant frequency variations with discrete implementation, the resonant frequency should be adequately far from the vicinity of the Nyquist frequency [37]). From these figures, it can be revealed that:

⋅For 0<r≤1: in Fig. 7(a), the resonant poles track initially inside the unit circle before tracking outside the unit circle above a certain value of βres=βres-max. It is observed that all of the tracks of the resonant poles intersect with the unit circle at certain points corresponding to the resonant poles denoted as P1,2. Accordingly, for 0<r≤1, Fnew(z) is stable only for resonant frequencies below the value of ωres-max (=βres-maxωs).

⋅For r<0: in Fig. 7(b), the resonant poles track initially outside the unit circle before tracking inside the unite circle above a certain value of βres. This value is denoted as βres-min. Furthermore, it is observed that all of the tracks of the resonant poles intersect with the unit circle at the points corresponding to the resonant poles P1,2. For the low values of r in this region and with increasing βres above βres-min, one of the non-resonant poles tracks outside the unit circle above a certain value of βres corresponding to one of the non-resonant poles at (-1,0). This value is denoted as βres-high. Accordingly, for a certain value of r<0, Fnew(z) is stable only over the resonant frequency range between ωres-min (=βres-minωs) and ωres-high (=βres-highωs). By decreasing r in this region, the range between βres-min and βres-high shrinks untill it vanishes at a certain r corresponding to βres-minres-high. This value of r is denoted as rb with the corresponding βres denoted as βres-b.


Fig. 7. Pole maps of Fnew(z) with sweeping βres at βh=0.5 and different values of r: (a) 1≥r>0; (b) r<0.

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C. HPF Cutoff Frequency Variations at Different HPF Gains

For the two regions of r (0<r1 and r<0), the above analyses are repeated in Figs. 8 and 9 for three values of βh (0.4, 0.3 and 0.2). It is shown that by decreasing βh, the resonant poles P1,2 move to the right on the unit circle. The performance in the two ranges of r is still the same. For 0<r≤1, Fnew(z) is stable below a certain value of βres= βres-max; for r<0, Fnew(z) is stable over a certain range of βres-min <βres< βres-high.


Fig. 8. Pole maps of Fnew(z) with sweeping βres for 1≥r>0 and different values of βh: (a) βh=0.4; (b) βh=0.3; (c) βh=0.2.

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Fig. 9. Pole maps of Fnew(z) with sweeping βres for r<0 at different values of βh: (a) βh=0.4; (b) βh=0.3; (c) βh=0.2.

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Ⅳ. REGIONS FOR A STABLE OPEN LOOP SYSTEM

Two regions of r can be identified for a certain βh as follows:

1. 0<r≤1; for a certain r in this region, Fnew(z) is stable only for resonant frequencies below a certain value of ωres-max.

2. 0>r≥rb; for a certain r in this region, Fnew(z) is stable only over  the resonant frequency range of ωres-min< ωresres-high.

The values of βres-max, βres-min, βres-b and rb can be determined for a certain value of βh as follows. At βres-max or βres-min, it was shown that Fnew(z) has five poles: one pole at z=1, two resonant  poles at P1,2 and two non-resonant poles (denoted as P3 and P4). Accordingly, the denominator of Fnew(z) at βres-max or βres-min can be expressed as (14).

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By expanding (14) and equating its coefficients with the denominator coefficients of Fnew(z), expressed in (13), the expressions of (15), (16) and (17) are derived to determined r, P3 and P4, respectively.

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At a βh of 0.5, Fig. 10 plots r and the magnitudes of P3 and P4 versus βres-max/βres-min. Using this figure, it can be implied that:

a. For 0<r≤1, the minimum limit of βres-max corresponds to r=1 and is denoted as βres-a. With a decreasing r, βres-max increases untill reaches its maximum limit at r=0. On the other hand, for 0>rrb, the maximum limit of βres-min corresponds to rmin = rb. With an increasing r, βres-min decreases till it reaches its minimum limit at r=0. At r=0, both βres-max and βres-min have the same value which is denoted as βres-cr and expressed in (18) by substituting r=0 into (15). It is shown from Figs. 8 and 9 that decreasing βh causes a movement to the right for the poles P1,2 on the unit circle. This in turn increases real{P1,2}. Accordingly, from (18), βres-cr decrease as βh decreases. Then the maximum value of βres-cr corresponds to βh=0.5, where real{P1,2} is determined from Figs. 7(a) or 7(b) as -0.111. By substituting this value into (18), the maximum value of βres-cr is determined as 0.268. Therefore, for 0<r≤1, Fnew(z) can only be stable for resonant frequencies less than 0.268ωs.

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b. For 0>rrb, both rb and βres-b correspond to one pole of P3 or P4 at (-1,0). Thus, rb and βres-b can be determined from Fig. 10 by locating their values at the unity magnitudes of P3 or P4. βres-high does not correspond to the resonant poles at P1,2. It only corresponds to one pole of P3 or P4 at (-1,0). Therefore, it cannot be determined from Fig. 10. However, at high values of r in its second region, the non-resonant poles track entirely inside the unit circle over the entire range of βres as shown in Fig. 9 (e.g. the non-resonant poles track entirely inside the unit circle for r of -0.1 and -0.2). Hence, the stable range of Fnew(z) can be extended to a βres-high of 0.45 (the maximum considered limit of βres) using high values of r in its second region.


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Fig. 10. HPF gain factor (r) along with |P3| and |P4| versus βres-max/βres-min at βh=0.5.


Using Fig. 10, at a constant βh, the stable regions of Fnew(z) can be re-identified depending on βres as follows:

1. For βres ≤ βres-a, Fnew(z) can only be stable in the first region of r (0<r1).

2. For βres-aresres-cr, Fnew(z) can only be stable over a certain range in the first region of r between 0 and the value of r corresponding to βres in Fig. 10.

3. For βres-crresres-b, Fnew(z) can only be stable over a certain range in the second region of r between 0 and the value of r corresponding to βres in Fig. 10.



Ⅴ. CONTROL SYSTEM DESIGN


A. HPF Cutoff Frequency Tuning (βh tuning)

To tune βh, pole maps of Fnew(z) are plotted with sweeping βh for different values of βres in two identified regions of r:

⋅First region; 0<r≤1, pole maps are plotted in Figs. 11(a), (b) and (c) for three values of r at 0.2, 0.5 and 0.8, respectively. In each figure, three βres values of 0.17, 0.2 and 0.24 are considered (these values are less than βres-cr=0.268 since above this value, a positive r cannot be used for the stability of Fnew(z)). By increasing βh, the resonant poles may track entirely outside or inside the unit circle or they may initially track outside the unit circle before tracking inside the unit circle with an increasing βh. To ensure the stability of Fnew(z), high βh should be adopted. Theoretically, βh can be extended to 0.5. However, such a value can deteriorate the discretization process. A value of βh=0.4 is adopted. At this value, βres-cr and βres-a are determined by plotting the pole map of Fnew(z) with sweeping βres at any constant value of r (any of the pole maps in Fig. 8(a) or Fig. 9(a) can be used). From these figures, real{P1,2} is determined to be -0.0562. Using (15), (16) and (17), Fig. 13 plots r along with the magnitudes of P3 and P4 versus βres-maxres-min in the first region of r. From this figure, βres-a and βres-cr are determined to be 0.188 and 0.259, respectively. Then the first and the second regions of βres at βh=0.4 are identified since βres ≤ 0.188 and 0.188<βres<0.259, respectively.

Second region; 0>r≥rb, pole maps of Fnew(z) are plotted in Figs. 12 (a), (b) and (c) for values of r at -0.2, -0.4 and -0.6, respectively. Three βres values of 0.3, 0.34 and 0.38 are considered. By increasing βh, the resonant poles may track entirely inside the unit circle or they may initially track inside the unit circle before tracking outside the unit circle with an increasing βh. From these pole maps, using a medium value for βh (0.25) is a good tradeoff to ensure the stability of Fnew(z). At this value, βres-cr and βres-b can be determined by plotting the pole map of Fnew(z) with sweeping βres at any value of r (r=1 is used) as shown in Fig. 14(a), where real{P1,2} is determined as 0.0653. Using (15), (16) and (17), Fig. 14 (b) plots r and the magnitudes of P3 and P4 versus βres-maxres-min in the second region of r. From this figure, βres-cr and βres-b are determined as 0.239 and 0.395, respectively. In addition, rb is determined as -0.875. Then the third region of βres at βh=0.25 is identified as 0.239<βres<0.395. Note that, as indicated previously, for values of 0.395<βres<0.45, higher values of r in the second region have to be adopted to stabilize Fnew(z) (e.g. -0.1 and -0.2).

⋅If βres is between 0.239 (βres-cr at βh =0.25) and 0.259 (βres-cr at βh =0.4), a βh of either 0.25 or 0.4 with the corresponding regions of r can be used.


Fig. 11. Pole maps of Fnew(z) with sweeping βh at different βres and different values of r in the first region (0<r≤1): (a) r=0.2; (b) r=0.5; (c) r=0.8.

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Fig. 12. Pole maps of Fnew(z) with sweeping βh at different βres and different values of r in the second region (0>r≥rb): (a) r=-0.2; (b) r=-0.4; (c) r=-0.6.

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Fig. 13. HPF gain factor (r) along with |P3| and |P4| versus βres-max at βh=0.4.


Fig. 14. At βh=0.25: (a) Pole map of Fnew(z); (b) HPF gain factor (r) along with |P3| and |P4| versus βres-min.

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B. Control Parameters Design

The design objectives for the overall system are:

1. To ensure the stability of Fnew(z).

2. To meet pre-specified limits of the fundamental loop gain (Tfo) and crossover frequency (ωc).

An s-domain model, shown in Fig. 15, is used to design the control parameters. An exponential function of 그림입니다.
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Fig. 15. Equivalent s-domain model.


Since the crossover frequency should be sufficiently higher than ωo, and below both ωres and the adopted values of ωh (0.25ωs or 0.4ωs), the loop transfer function (Tloop) at ωc can be approximated using trigonometry as in (19), where Ac and θc are expressed in (20). Then the loop gain at ωc can be expressed as in (21). Then Kp can be determined as in (22).

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Similarly, the loop gain at ωo is approximated in dB in (23). Then Kr can be determined using (24).

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where:

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By substituting (22) and (24) into (8), Gc(z) is expressed in terms of r and the pre-specified quantities as in (26).

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From Fig. 5, the discrete closed loop transfer function is expressed in (27).

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Using the above-addressed expressions, Fig. 16 shows the design flow of the control parameters.


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Fig. 16. Design flow for the control parameters.



Ⅵ. NUMERICAL AND EXPERIMENTAL VERIFICATION


A. Numerical Example

Table I presents the parameters of the system shown in Fig. 1. The proposed tuning steps are applied to the four values of the resonant frequencies ωres1, ωres2, ωres3 and ωres4, corresponding to βres of 0.146 (βres1), 0.197 (βres2), 0.296 (βres3) and 0.379 (βres4), respectively. Tfo is specified as 65 dB, and c is specified as the ratio of the corresponding resonant frequency as follows: 0.3ωres1, 0.25ωres2, 0.22ωres3 and 0.18ωres4.


TABLE I SYSTEM PARAMETERS

Symbol

Quantity

Value

P

Rated power

1 kw

Vg

Grid voltage

120 V

Fo

Grid Frequency

50 Hz

Vdc

DC Voltage

220 V

Li

Inverter side inductance

2.75 mH

Lg

Grid side inductance

1.2 mH

C

Capacitance

22.2 µF, 12.2 µF, 5.4 µF, and 3.3 µF

Fsw

Switching Frequency

8 kHz

Fs

Sampling Frequency

8 kHz


At the beginning, a value of βh=0.4 is adopted for βres1 and βres2. On the other hand, a value of βh=0.25 is adopted for βres3 and βres4. Then for these values of βh, Figs. 13 and 14(b) plot r versus βres-max/βres-min. From Fig. 13, r for βres2 a determined as 0.83. From Fig. 14 (b), r for βres3 and βres4 are determined as -0.48 and -0.84, respectively. To complete the tuning process, the pole map of Tclosed(z) is plotted over the corresponding stable range of r for each value of βres as follows:

⋅0<r≤1 for βres1=0.146 (<0.188).

⋅0<r<0.83 for βres2=0.197.

⋅0>r>-0.48 for βres3=0.296.

⋅0>r>-0.84 for βres4=0.379.

These pole maps are not shown here. The values of r corresponding to the farthest closed loop poles inside the unit circle are selected as 0.24, 0.16, -0.12 and -0.18 for βres1, βres2, βres3 and βres4, respectively. Finally, Kp and Kr are determined from (22) and (24), respectively. Table II lists the control parameters along with the results of the experimental study introduced below.


TABLE II DESIGNED PARAMETERS AND EXPERIMENTAL RESULTS

C (µF)

βres

Designed Controller

Experimental Results

βh

r

Kp

Kr

Ig1

Ess

PF

22.2

0.146

0.4

0.24

6.84

1678

8

0.04

0.999

12.2

0.197

0.4

0.16

8.41

1854

8.01

0.039

0.999

5.4

0.296

0.25

-0.1

14.01

2427

7.98

0.042

0.999

3.3

0.379

0.25

-0.18

15.56

2600

8.03

0.037

0.999


B. Simulation Results

For the system shown in Fig. 1, simulation work is carried out in the PSIM environment using the parameters listed in Table I. Discrete models for the active damper and the PR controller are constructed using PSIM digital control modules. Unipolar PWM is adopted for the inverter. To verify the proposed approach, step changes in the reference current are carried out twice (from a half to the rated current and back). Figs. 17, 18 and 19 show simulation waveforms of the grid voltage and grid current using the designed parameters listed in Table II at the different resonant frequencies.


Fig. 17. Simulation results of the grid voltage (vg)andthecurrent(ig)forβres1=0.146: (a) without AD; (b) with AD.

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Fig. 18. Simulation results of the grid voltage (vg) and the current (ig) for βres2=0.197: (a) without AD; (b) with AD.

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Fig. 19. Simulation results of the grid voltage (vg) and the current (ig) with AD at: (a) βres3=0.296; (b) βres4=0.379.

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C. Experimental Results

A prototype single phase inverter was connected, using an LCL filter, to an AC source for grid emulation. The control scheme was implemented using a C6713-A DSP development board. A step change in the reference current was carried out to verify the transient characteristics. Experimental investigations have been conducted using the parameters listed in Table II at different resonant frequencies.

For ωres1 (그림입니다.
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Fig. 20. Experimental measurements of the grid voltage (vg) and the current (ig) for βres1=0.146: (a) without AD; (b) with AD.

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For resonant frequencies greater than one-sixth of the control frequency, AD is not mandatory for system stability. However, the stability can be worse with variations in grid side inductance. Moreover, oscillatory resonant currents can be generated without using resonance damping. This is confirmed in Fig. 21(a), where the current waveforms at ωres2 are shown without using AD. In this case, oscillatory resonant currents are generated at the stepping up of the reference current. Much worse oscillations can be generated if the grid voltage contains harmonic components around the resonant frequency. On the other hand, the waveforms when using AD are shown Fig. 21(b). It can be realized that the mitigation effect is introduced by the AD in this case.  Finally, Figs. 22(a) and 22(b) show the waveforms when using AD for ωres3 and ωres4, respectively.


Fig. 21. Experimental measurements of the grid voltage (vg) and the current (ig) for βres2=0.197: (a) without AD; (b) with AD.

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Fig. 22. Experimental measurements of the grid voltage (vg) and the current (ig) with AD at: (a) βres3=0.296; (b) βres4=0.379.

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At steady state conditions, Table II presents the measured fundamental current component (그림입니다.
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The total harmonic distortion of the measured grid current (THDi) has been estimated at the rated conditions as ratio (in %) of the harmonic current (of the orders 2 to 40) to the fundamental current as listed in table III. For values of the resonant frequencies, it is shown that THDi is less than 5%. For ωres2, it is shown that adding the AD loop increases the THDi value from 3.02% to 3.09%. This indicates the negligible effect of the AD loop on the THDi of the grid current.


TABLE III THD OF THE CURRENT WAVEFORMS

C (µF)

βres

Without AD

With AD

22.2

0.146

3.32%

12.2

0.197

3.02 %

3.09%

5.4

0.296

2.27%

3.3

0.379

2.7%



Ⅶ. CONCLUSIONS

This paper investigates active damping of LCL filter resonance using HPF of the grid current feedback. A new expression for this HPF, in terms of the filter components, has been derived. This expression facilitates a general stability study of the active damped filter. Through discrete time domain investigation of the active damped filter, three regions of resonant frequencies have been identified for stable open loop behavior at a certain HPF cutoff frequency. These regions cover a wide range of resonant frequencies up to 0.45 of the sampling frequency. Moreover, straightforward design steps for both the HPF and the fundamental current regulator have been proposed. A numerical example and experimental work have been introduced. The results show that good steady state and dynamic performance along with resonance damping can be obtained over a wide range of resonant frequencies using the proposed co-design steps of the control parameters.



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Mahmoud A. Gaafar received his B.Sc. and M.Sc. degrees in Electrical Engineering from Aswan University, Aswan, Egypt, in 2004 and 2010, respectively. He received his Ph.D. degree from the Department of Electrical and Electronic Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan, in 2017. He is presently working as an Assistant Professor in the Department of Electrical Engineering, Aswan University. He is a Member of the Aswan Power Electronics Application Research Center (APEARC), Aswan University. His current research interests include harmonics mitigation, grid-connected converters and digital control strategies. He is a Member of the IEEE Power Electronics Society (PELS).


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Gamal M. Dousoky was born in Minia, Egypt, in 1977. He received his B.Sc. and M.Sc. degrees from Minia University, Egypt, in 2000 and 2004, respectively, and his PhD in 2010 from Kyushu University, Japan (all in Electrical and Electronic Engineering). Since 2000 he has been associated with the Dept. of Elect.  Eng., Faculty of Eng., Minia University, in March 2017 he was promoted to the position of an Associate Professor position. He worked as Postdoctoral research fellow at Kyushu University for two years. His research interests include power electronics, particularly renewable-energy applications, energy efficiency, switching power supplies, electromagnetic interference/compatibility, and digital control. He authored and co-authored more than 70 publications, in international journals and conference proceedings of Power Electronics and Industrial Technologies. Dr. Gamal received the 2009 Excellent Student Award of the IEEE Fukuoka Section. He is an IEEE Senior member, and a reviewer in many international journals and conferences.


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Emad M. Ahmed received his B.Sc. and M.Sc. degrees from the Dept. of Electrical Engineering, Aswan University, EGYPT in 2001, 2006, respectively. In addition, he received the Ph.D. degree from the Dept. of Electrical Engineering, Kyushu University, JAPAN in 2012. Currently, he is an associate Professor with the Dept. of electrical engineering, Aswan University, Egypt. His present research interests include applied power electronics especially in renewable energy applications, Micro-grids, and Fault Tolerant control. He has incorporated in several research projects in power electronic and renewable energy applications. Dr. Emad received Baek-Hyun Award from the Korean Institute of Power Electronics (KIPE) for his academic contribution in the field of power electronics in 2012. He is a member in IEEE Power Electronics Society (PELS), IEEE Industrial Electronics Society (IES), and IEEE Industrial Application Society.


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Masahito Shoyama received his B.S. degree in Electrical Engineering and his Ph.D. degree from Kyushu University, Fukuoka, Japan, in 1981 and 1986, respectively. He joined the Department of Electronics, Kyushu University as a Research Associate, in 1986. He became an Associate Professor in 1990, and he has been a Professor since 2010. Since 2009, he has been with the Department of Electrical Engineering, Faculty of Information Science and Electrical Engineering, Kyushu University. He has been active in the field of power electronics, especially in the areas of bi-directional converters for DC/AC power systems, high-frequency switching converters for renewable energy sources, power factor correction (PFC) converters, and electromagnetic compatibility (EMC). Professor Shoyama is a Member of the IEEE, IEICE, IEEJ and SICE.


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Mohamed Orabi received his Ph.D. degree from Kyushu University, Fukuoka, Japan, in 2004. He is presently working as a Professor at Aswan University, Aswan, Egypt. He is the Founder and Director of the Aswan Power Electronics Application Research Center (APEARC), Aswan University. In addition, he was with Enpirion Inc. and Altera Corp., where was the Senior Manager of the Altera-Egypt Technology Center, from June 2011 to July 2014. He has published about 200 papers in international journals and conference proceedings. His current research interest includes dc-dc and PFC converters, integrated power management, nonlinear circuits and inverter designs for renewable energy applications. He has lead several projects related to power electronics applications for renewable energy. Professor Orabi is an Associate Editor of the IET Power Electronics Journal and a Guest Editor of the IEEE JESTPE. He is also the Chair of the PELS Egypt Chapter. Dr. Orabi was a recipient of a 2002 Excellent Student Award of the IEEE Fukuoka Section and a Best Young Research Award from the IEICE Society, Japan, in 2004. He also received a SVU Encouragement Award in 2009, a National Encouragement Award in 2010, and Aswan University Supervision Awards for 2015 and 2016.