I. INTRODUCTION

Due to their higher attenuation for switching harmonics with a lower size and weight, LCL filters are widely used with grid-connected converters to limit the harmonic contents of the injected grid current to comply with the grid codes; i.e. IEEE 519-1992 [1]. However, the inherent resonance of the LCL filters represents a challenge for control system designers. Damping techniques have to be adopted to cope with this challenge. With discrete implementation, closed loop system stability can be maintained by the inherent damping characteristic of a single grid current control loop for resonant frequencies greater than one-sixth of the control frequency [2]. However, this strategy gives rise instability due to resonant frequency variations which are likely to occur particularly in weak grids where the grid inductance changes significantly [3]. Passive damping, by using a resistor, was used to cope with this issue [4]. However, it causes power losses. Thus active damping (AD) by modifying the control algorithm is preferred [5].

Number of active damping techniques have been discussed in the literature [6]-[20]. A cascaded filter in the current control loop was used in [6] and [7]. However, this method is highly sensitive to resonant frequency variations. In addition, it causes a reduction in the system bandwidth. To overcome these issues, an inner feedback loop of one of the filter states has been employed to produce a damping effect [8]-[20]. A proportional feedback of the filter capacitor current was employed in [8]-[10]. To stabilize the closed loop system, it was proved that excitation of unstable open loop poles is mandatory for resonant frequencies greater than one-sixth of the control frequency [11]- [12]. This non-minimum phase behavior can decline the system performance especially when selective harmonic mitigation is of concern [11]. Modified feedback loops of the capacitor current have recently been proposed to avoid this behavior over wider range of resonant frequencies [11]-[15]. However, a high precision current sensor or a complicated observer loop is needed [16]. The capacitor voltage differentiation can be used to produce a damping effect. However, this technique results in noise amplification. To cope with this issue, a lead-lag network has been adopted to behave as a differentiator around the resonant frequency [17]- [18]. However, as shown in [17] , this method can be used effectively over limited range of resonant frequencies between 1/3.2 and 1/3.4 of the control frequency.

Grid-current-based active damping has recently been discussed [19]-[20]. Ideally, this needs an s2 term which cannot be implemented practically due to noise amplification. A high pass filter (HPF) is employed in [19] instead of the s2 term. This technique is further discussed in [20] where it was shown that non-minimum phase behavior can be avoided for resonant frequencies up to a theoretical limit of 0.27 of the sampling frequency. However, the co-design steps of this HPF along with the fundamental current regulator are very complicated since many iterations are needed. Moreover, avoiding non-minimum behavior at high resonant frequencies requires increasing the cutoff frequency of the HPF to its maximum practical limit of 0.5 of the sampling frequency (Nyquist frequency). However, using such high value can deteriorate the HPF performance. This in turn, makes it unsuitable for practical implementation. Due to this practical limitation, and during the performance verification presented in [20], non-minimum phase behavior has been avoided up to a maximum resonant frequency of about 0.24 of the sampling frequency.

In this paper, two feedback loops of the grid current and the capacitor voltage are proposed as a new active damping strategy. By using the proposed strategy, the non-minimum phase characteristics can be avoided over a wide range of resonant frequencies. Moreover, straightforward co-design procedures for both the fundamental current regulator and the active damping loops are proposed. For reduced number of sensors, virtual flux technique can be employed to estimate the capacitor voltage [18]. However, this is not adopted here.

Following this introduction, section II presents   the continuous time domain derivation of the proposed active damping strategy. In section III, discrete implementation of the proposed strategy is presented along with the co-design steps of the fundamental current regulator and the active damping loops. In section IV, a numerical example along with experimental work are introduced to verify the proposed strategy performance at different resonant frequencies. Finally, section V presents some conclusions.

 

II. PROPOSED ACTIVE DAMPING STRATEGY

A. System Description

Fig. 1 shows a single phase inverter connected to the grid through an LCL filter. The block diagram of the capacitor-current-based active damping system is shown in Fig. 2, where a proportional feedback (Hd) of the capacitor current is used to actively damp the filter resonance. The un-damped filter transfer function is denoted as Gig(s) and is expressed in (1), where ωres - expressed in (2) - is the LCL filter resonant frequency. A proportional resonant (PR) controller with a transfer function of Gc(s) – expressed in (3) – is employed for fundamental current regulation.

Fig. 1.Grid-connected Single phase inverter through an LCL filter.

Fig. 2.Block diagram of capacitor-current-based active damping.

where ωo and ωi are the fundamental frequency and the bandwidth of the resonant part of the PR regulator, respectively.

According to Fig. 2, the transfer function of the actively damped filter is expressed in (4).

In Figs. 3(a) till 3(d), the capacitor-current-based active damping system is manipulated using signal flow graph manipulation. In Fig. 3(a), the capacitor current is replaced by the difference between the inverter output current and the grid injected current. With further manipulation, it is shown in Fig. 3(c) that the capacitor current feedback is equivalent to using three feedback loops of the grid current (ig), the capacitor voltage (vc), and the modulated inverter voltage (vi). By further manipulation, the modulated voltage feedback is augmented in the main loop as a HPF, which is denoted as Gh(s) and expressed in (5), with a cut off frequency of ωh = Hd/Li; this system is shown in Fig. 3(d). The typical rang for ωh can be calculated by expressing the transfer function of the actively damped filter (Fad) in terms of ωh and writing it in a standard form as in (6). The damping ratio ζ is typically around 0.7 [8], [21]. Therefore, the typical range for ωh can be determined as ωres < ωh < 2ωres.

Fig. 3.Manipulation of the capacitor-current-based active damping technique.

Note that the capacitor voltage feedback is an integrator, denoted as Gi(s) in Fig. 3(d), with a time constant of Li.

The presence of the HPF (Gh) in cascade with the main control loop can deteriorate the system disturbance rejection capability. This deterioration can be determined by comparing the system transfer function to the grid voltage (vg) for both the original system (shown in Fig. (2)) and the final manipulated system (shown in Fig. 3 (d)) considering the above-determined typical range of ωh [21].

B. Proposed Active Damping System

The proposed system is derived in two steps as follow:

Fig. 4.Block diagram of the proposed active damping strategy.

Substituting (9) and (10) into (8), Fnew(s) is re-written as:

Using Routh’s criteria, Kd has to follow the constraint in (12) to guarantee the open loop system stability and hence minimum phase behavior.

To generalize the following analysis, Kd is expressed in terms of the above maximum limit (Li+Lg) as in (13), where 0 < βd < 1 for a stable open loop system.

Substituting (13) into (11), the actively damped filter of the proposed system is finally expressed in (14).

 

III. DISCRETE IMPLEMENTATION

A. System Discretization

The discrete system representation of the proposed active damping strategy is shown in Fig. 5 where the DSP delay is represented by one sample delay. Using Tustin approximation with pre-warping at the fundamental frequency, the discrete PR regulator is determined in (15) where Ts is the sampling period.

Fig. 5.Discrete representation of the proposed system.

In addition to Gig(s), expressed in (1), two other transfer functions should be defined for system discretization:

Using Zero-Order-Hold (ZOH) discretization, Gig(z) and Giv(z) are expressed as (18) and (19), respectively. Gvg(z) is determined as Gig(z)/Giv(z).

where δ = ωresTs and

For the active damping loops, Gi(z) and Gad(z) are determined using Tustin approximation and expressed in (20) and (21), respectively.

where and

Finally, the discrete actively damped filter and the loop transfer function are expressed in (22) and (23), respectively.

B. Control Parameters Design

For tuning purpose, the equivalent s-domain representation, shown in Fig. 6, is used. The DSP delay is modelled by an exponential transfer function of Gd(s)=e–15sTs [9]. According to this representation, both the actively damped filter transfer function (Fnew-d) and the loop transfer function (Tloop-d) are expressed in (24) and (25), respectively.

Fig. 6.System equivalent discrete implementation.

It was shown in [12] that the resonant frequency changes with discrete implementation. The new resonant frequency will be denoted as ωres–ad. At this resonant frequency the gain of Fnew-d can be approximately expressed in (26).

According to (26), higher values of ωh should be used to acquire better damping effect. Theoretically, for discrete implementation, ωh can be extended up to 0.5ωs (Nyquist sampling theory, where ωs is the control frequency in rad/sec). However, such high value can deteriorate the discretization process. A value of ωh =0.4ωs is adopted here.

Since the resonant gain of the PR regulator is mainly effective at the fundamental frequency, the PR controller can be approximated as (27)

At the crossover frequency ( ωc ), which should be sufficiently higher than ωo and below both ωres and the adopted ωh (0.4ωs), the loop gain can be approximated as (28).

Using Trigonometry, this gain is reduced to (29).

where

Hence, for certain value of βd, Kp should be calculated as in (31) to obtain certain crossover frequency.

Substituting (31) into (23), the loop transfer function is expressed in (32)

At the fundamental frequency, the loop gain can be approximated as in (33).

where

This is expressed in dB in (34) from which Kr can be determined from (35) for certain fundamental loop gain (Tfo).

Using the above-derived expressions, the following steps are proposed to co-design the control system parameters.

 

IV. VERIFICATION

A. Numerical Example

Table I lists the parameter values of the grid-connected inverter shown in Fig. 1. Four capacitance values, corresponding to resonant frequencies of 0.143ωs, 0.179ωs, 0.209ωs and 0.241ωs are used to verify the performance of the proposed system over a wide range of resonant frequencies with respect to the control frequency. These resonant frequencies are denoted as ωres1, ωres2, ωres3 and ωres4, respectively. The HPF cut off frequency (ωh) value is taken as 0.4ωs to mitigate the resonant peak as much as possible. In addition, a value of 60 dB is adopted for the fundamental loop gain (Tfo). Finally, an initial value for the crossover frequency of 0.3 of each corresponding resonant frequency is adopted. Using the tuning steps presented in the last section, a pole-map of Fnew(z) is plotted with variation of βd. These pole maps are plotted in Figs. 7(a), 8(a), 9(a) and 10(a) for the resonant frequencies ωres1, ωres2, ωres3 and ωres4, respectively. To achieve the best damping effect, the values of βd corresponding to the farthest resonant poles inside the unit circle are selected. These values are determined as 0.55, 0.45, 0.3 and 0.15 for ωres1, ωres2, ωres3 and ωres4, respectively. Using the selected values of βd along with the pre-specified values of ωc and Tfo, the corresponding values of Kp and Kr are determined from (31) and (35), respectively.

TABLE ISYSTEM PARAMETERS

Fig. 7.(a) pole-map of Fnew(z) for ωres1=0.143ωs with sweeping βd (b) corresponding bode plot for Tloop at βd =0.55.

Fig. 8.(a) pole-map of Fnew(z) for ωre2 =0.179ωs with sweeping βd, (b) corresponding bode plot for Tloop at βd =0.45.

Fig. 9.(a) pole-map of Fnew(z) for ωre3 =0.209ωs with sweeping βd, (b) corresponding bode plot for Tloop at βd =0.3 and different crossover frequencies.

Fig. 10.(a) pole-map of Fnew(z) for ωre4=0.241ωs with sweeping βd, (b) corresponding bode plot for Tloop at βd =0.15 and different crossover frequencies.

For ωres1 and ωres2, Figs. 7(b) and 8(b) show bode plots of the loop transfer function, expressed in (23), respectively. It is shown that the resonance peak is less than 0 dB. For ωres3 and ωres4, it is found that the frequency response exhibits a resonant peak of more than of 0 dB. To overcome this issue, a reduction in the crossover frequency has to be adopted. For ωres3, it is found that a reduction of the crossover frequency of 0.12ωres3 can reduce the resonant peak to less than 0 dB However, for ωres4, a large crossover frequency reduction is required to obtain a resonant peak of less than 0 dB. Such a reduction can deteriorate the system dynamic performance.

Moreover, the phase lag introduced by the PR controller at low frequencies dramatically reduces the phase margin. Therefore, only a reduction of the crossover frequency to 0.1ωres4 is adopted. Figs. 9(b) and 10(b) show the frequency responses for ωres3 and ωres4, respectively.

TABLE II summaries the designed control parameters and the achieved performance of the phase margin (PM), ωc and Tfo. These results indicate the well damped performance of the proposed method over a wide range of resonant frequencies while meeting the pre-specified values of ωc and Tfo.

TABLE IIDESIGNED CONTROL PARAMETERS & FREQUENCY RESPONSE RESULTS

B. Robustness against Grid Inductance Variations

In real operation, the grid side inductance (Lg) may vary significantly. To investigate system robustness against such variations, the pole maps of the closed loop system Tclosed, expressed in (36), are plotted in Fig. 11 while sweeping Lg between 100-300% of its original value.

For the considered resonant frequencies, it is shown that the closed loop poles move inside the unit circle with an increasing Lg. These plots reflect the system robustness against grid inductance variations.

Fig. 11.Closed loop pole maps with grid inductance (Lg) variation for (a) ωres1=0.143ωs, (b) ωres2=0.179ωs, (c) ωres3=0.201ωs, (d) ωres4=0.241ωs.

C. Comparative Study

To show the superiority of the proposed method compared to the existing capacitor voltage/current based AD methods, the limitations of these methods are clarified under the same parameters used in the aforementioned numerical example.

Fig. 12.Block representation of capacitor-voltage-based AD method.

Fig. 13.Closed loop pole map using capacitor-voltage-based AD method with sweeping Kd, a) for ωres1=0.143ωs (Kp=8.47), b) for ωres2=0.179ωs (Kp=10.6).

It is shown in these plots that the system cannot be stable for any values of Kd. This ensures the difficulty of using this method for resonant frequencies outside specific limits. On the other hand, the proposed method behaves effectively over a wide range of resonant frequencies as verified in the above numerical example.

Fig. 14.Closed loop pole map of capacitor-current-based AD method with sweeping Lg, a) for ωres2 (Kp=10.6, Hd=5) , b) for ωres3 (Kp=4.96, Hd=1).

D. Experimental Work

Using the system parameters listed in Table I, a single phase inverter prototype has been built and connected through an LCL filter to an AC power supply to emulate a grid. The control algorithm has been implemented using the PE-Expert3 platform, which consists of a C6713-A DSP development board along with a high-speed PEV board for analog-to-digital conversion and PWM signal generation. To verify the dynamic response, the reference current is stepped up from 2 A (0.5Irated) to 4 A (Irated). Using the designed parameters listed in Table II, some tests are carried out with and without the proposed active damping method.

For ωres1, which is lower than one-sixth of the sampling frequency, the system cannot be stabilized without active damping (AD). Thus, removing the active damping loop for this case causes a high oscillatory current as shown in Fig. 15(a). On the other hand, Fig. 15(b) shows the waveforms when using active damping loops.

Fig. 15.Experimental waveforms of grid current (ig) and grid voltage (vg) for ωres1 =0.143ωs (a) without AD, (b) with AD.

For ωres2, ωres3 and ωres4, the system can be stabilized without active damping as shown in Figs. 16(a), 17(a) and 18(a). However, it can recognize the dynamic oscillations which are caused by weak damping (there is some damping introduced by the small resistance of the coils). Figs. 16(b), 17(b) and 18(b) show the waveforms when using the proposed active damping loops. It can recognize the mitigation effect of the dynamic oscillations when using the proposed active damping method. This mitigation effect can be further clarified in Figs. 19 and 20. These figures show the spectrum of the grid current for each resonant frequency with and without the proposed active damping method.

Fig. 16.Experimental waveforms of grid current (ig) and grid voltage (vg) for ωres2 =0.179ωs (a) without AD, (b) with AD.

Fig. 17.Experimental waveforms of grid current (ig) and grid voltage (vg) for ωres3 =0.209ωs (a) without AD, (b) with AD.

Fig. 18.Experimental waveforms of grid current (ig) and grid voltage (vg) for ωres4 =0.241ωs (a) without AD, (b) with AD.

Fig. 19.Spectrum of the grid current (ig) without active damping loops for (a) ωres2=0.179ωs (b) ωres3 =0.209ωs and (c) ωres4=0.241ωs.

Fig. 20.Spectrum of the grid current (ig) with active damping loops for (a) ωres2=0.179ωs (b) ωres3 =0.2ωs and (c) ωres4=0.241ωs.

For experimental verification of its ineffective damping for resonant frequencies of more than one-sixth of the sampling frequency, the capacitor current based AD method has been used for the resonant frequency ωres2 (=0.179ωs), and Fig. 21 shows the corresponding experimental waveforms. It can be seen that the resonant current oscillations are still present in this case. On the other hand, the damping of the proposed AD method at the same resonant frequency has been clarified in Fig. 16(b).

Fig. 21.Experimental waveforms of grid voltage (vg) and grid current (ig) using capacitor-current-based AD at ωres2 =0.179ωs.

These results, along with the frequency response analysis introduced in the above numerical example, reflect satisfactory steady state and transient performances along with resonance damping over a wide range of resonant frequencies using the proposed active damping method and the control parameters tuning steps.

 

V. CONCLUSION

A novel active damping strategy using two feedback loops of the grid current and filter capacitor voltage is proposed in this paper. Compared to the previous active damping methods, the proposed one can offer the following merits.

A numerical example has been introduced to verify the performance of the proposed method over a wide range of resonant frequencies. To show the superiority of the proposed method, the drawbacks of the capacitor voltage/current based methods have been clarified. This example along and experimental results reflect the satisfactory performance of the proposed method.