Evaluations of current control in weak grid case for grid ... - IEEE Xplore

www.ietdl.org Published in IET Power Electronics Received on 28th December 2011 Revised on 12th August 2012 Accepted on 29th October 2012 doi: 10.1049/iet-pel.2012.0192

ISSN 1755-4535

Evaluations of current control in weak grid case for grid-connected LCL-filtered inverter Jinming Xu, Shaojun Xie, Ting Tang College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, No. 29, Yudao Street, Nanjing 210016, Jiangsu Province, People’s Republic of China E-mail: [email protected]

Abstract: For grid-connected inverters, switching harmonics can be effectively attenuated through an LCL-type filter. In order to suppress resonance and guarantee good performance, many strategies (e.g. active damping (AD), harmonic resonant control, repetitive control and grid feedforward) have been proposed. However, the wide variation of grid impedance value challenges system stability in practical applications. The aforementioned methods need to be investigated. This study evaluates the applicability of each part of the overall control in a weak grid case with the use of a stability criterion. It has been demonstrated that the feedback-based AD control can work well in a wide range of grid conditions. However, the resonant and repetitive control methods meet constraints. The grid feedforward method brings in an extra positive feedback path, and consequently results in high harmonics or even instability. Finally, a recommendation for system design has been presented. Simulations and experiments have been provided to verify the analysis.

1

Introduction

With the development of distributed grid-connected generations based on renewable energies, grid-connected inverters have gained more and more attention. Difficulties in the control of grid-connected inverters have been analysed in [1], which consist of dc-link voltage control, phase-locked loop (PLL), passive filters and low-harmonics current control. Among them, in order to suppress the switching-frequency harmonics caused by pulse width modulation (PWM), an L-type or LCL-type filter is required [2, 3]. Compared with the L-type filter, the LCL-type filter is a high-order low-pass one with −60 dB/decade attenuation of the frequencies higher than resonance frequency and thus guarantees better high-frequency harmonics rejection. Thus, the LCL-type filter can effectively attenuate the harmonics with less weight and volume. However, the resonance caused by the third-order filter puts forward a higher requirement for overall control. First, the system response is badly affected. Although the LCL-type filter has similar performance with the L-type filter in the low-frequency range, many researchers have tried to directly address the L-type control into the LCL-filtered system [4–9]. It has been indicated in [4] that the single-grid current control works stably when the filter resonance frequency is high; with the decrease of resonance frequency, the performance gets worse or instability occurs. In [5, 9], it is also proved that the bandwidth of the single-loop control was low to avoid resonance. Besides, some researchers have tried to feedback the inverter-side current to perform close-loop control and found that the stability was a little better than grid current control [6–8]; IET Power Electron., 2013, Vol. 6, Iss. 2, pp. 227–234 doi: 10.1049/iet-pel.2012.0192

however, this indeed is an indirect control method which cannot provide good controllability of injected grid current. In summary, with single-loop control methods, system bandwidth and resonance rejection cannot be guaranteed at the same time. Thus, there must be a trade-off between them. In order to solve this, an idea which used a passive resistor in the series to filter capacitor has been reported in [2]. It is simple and achieves good resonance rejection; however, the power losses and weakened high-frequency harmonics rejection cannot be tolerated. Alternatively, active damping (AD) based on additional feedback control has been proposed to replace the passive method. Typical applications can be sorted into the ADs based on capacitor current feedback [8, 10–13], capacitor voltage feedback [13, 14] and multi-variable feedbacks [15]. These commonly-used methods have been proven to realise good resonance rejection. Secondly, precise tracking performance is also an important issue, especially when grid distortion is quite bad. Since grid distortion is the main cause, grid feedforward can guarantee good performance [16]. Besides, a harmonic resonant controller which achieves an extremely high loop gain can maintain the same perfect tracking of the fundamental component and harmonics [6, 7, 9]. However, for the purpose of multi-harmonics elimination, multiresonant controllers are required. Thus, compared with the feedforward method resonant control is more complex. In order to improve this, Loh et al. [17] have proposed a novel repetitive controller which can realise multi-harmonics rejection with a simple structure. The novel controller has been only verified in an L-filtered inverter. 227

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www.ietdl.org The aforementioned studies have proved that the existing strategies for LCL-filtered inverters can realise satisfactory performance in a stiff grid case. That is, with AD control, resonance can be highly attenuated; with resonant, repetitive control or grid feedforward, current harmonics caused by grid distortion can be eliminated. However, in practical applications, because of a large amount of distributed power generation, long transmission wires and transformers, the grid seen from the point of common coupling (PCC) actually has large inductive impedance [6]. In this case, the weak grid situation may challenge system stability and control design. The studies proposed by Sun in [18] have indicated that the grid current would have large low-frequency harmonics when interfaced with a large inductor. It has been proved by Liserre et al. in [6] that filter natural resonance frequency and system bandwidth decreases with increase of grid impedance, and the harmonic resonant controller causes low-frequency harmonics or even instability. In order to improve these, a damping method is used. By placing an additional pair of pole-zeros around the resonance frequency, the response is improved; however, in a wide impedance variation case, the frequency varies largely and the effectiveness is hardly guaranteed. In [19], an uncertainty rejection controller has been proposed to eliminate the impact of grid parameter variation. However, its design needs the observation method and its convergence study. The design procedure is difficult. In Gabe et al. [20] use a robust design method to fix the problem in a weak grid case; however, it is only suitable for a small grid impedance. Besides, another type method is to detect grid impedance and adjust the control parameters to fulfil different grid conditions. However, it requires a precise online parameter estimation method, and may decrease the current quality if the method which, for instance, adds a specific harmonic periodically in the grid current and measures the PCC voltage is adopted [21]. In conclusion, the search for control strategies that are simple and of good grid applicability is still an arduous task. The premise is to clearly acquaint with the grid impact on system control. However, the existing knowledge is still lacking. The research in [6] only indicated the instability phenomena and provided experimental verification; however, the causes have not been clearly studied. The possibility of addressing novel repetitive control in the LCL-filtered inverter also needs to be investigated. Besides, AD in a weak grid case has not been analysed, along with the grid feedforward method when grid impedance varies largely. This paper is to fill the aforementioned blanks, provide design recommendation and promote study in a weak grid case. This paper is organised as follows. Section 2 presents a frequency-domain stability criterion and describes inverter topology. The studies of AD, resonant, repetitive control and grid feedforward in stiff and weak grid cases have been provided in Sections 3 and 4, respectively. Then in Section 5, results obtained from the above analysis are all verified. Finally, Section 6 concludes this paper and provides some recommendations.

2 2.1

Stability criterion and system description Stability criterion

The most basic criterion is the Ruth criterion. However, for a high-order LCL system, it is hard to use this criterion. Thus, a frequency-domain stability criterion is adopted. For a control 228 & The Institution of Engineering and Technology 2013

Fig. 1 Single-phase grid-connected LCL-filtered inverter

system, if its open-loop logarithm amplitude- and phasefrequency response curves are plotted, denoted as L(ω) and w(ω), respectively, the closed loop would be stable if the criterion shown in (1) has been satisfied [22]. N = N+ − N− (1) P − 2N = 0 where, P represents the number of open-loop poles with positive real part; N+ and N−, respectively, denote the times that w (ω) crosses the (2m + 1) π-line in the downward and upward directions when L(ω) > 0, where m equals to 0, ±1, ±2,... It is to be noted, for the inverters with L or LCL filters, P equals to zero. Thus, the stability criterion can be simplified N+ = N−

2.2

(2)

System description

Through frame transformation [1], a three-phase system can be equivalent to two single-phase systems. Generally, the current control for single-phase grid-connected LCL-filtered inverter is discussed. As shown in Fig. 1, the overall system consists of inverter-side inductance L1, filter capacitance C1, grid-side inductance L2, dc-link voltage udc, inverter output voltage uinv, PCC voltage ug (sampled for feedforward and PLL, consisting of grid impedance Lg and real grid us), inverter current iL1, capacitor current iC1 and grid current ig. The relation between uinv and ig is shown in (3). The grid current is easily resonated at the natural resonance frequency [ fres, ωres, expressed in (4)]. Besides, with the increase of grid impedance, the frequency decreases. i

Guginv (s) =

fres

3

1 L1 (L2 + Lg )C1 s3 + (L1 + L2 + Lg )s L1 + L2 + Lg vres 1 = = 2p 2p L1 (L2 + Lg )C1

(3)

(4)

Control in stiff grid case

Wessels et al. [10], Xu et al. [11], Liu et al. [12], Dannehl et al.[13] have indicated that proportional capacitor current feedback could effectively suppress resonance. In addition, the AD methods based on capacitor current and its voltage have some correlations [13]. Thus, AD based on capacitor current is typically chosen for analysis purpose. In a stiff grid case, the grid impedance is so small that it can be IET Power Electron., 2013, Vol. 6, Iss. 2, pp. 227–234 doi: 10.1049/iet-pel.2012.0192

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Fig. 2 Current control structure in stiff grid case

neglected and ug = us. The overall control is shown in Fig. 2, where iref is the reference current, um is the modulated wave and kpwm represents the PWM gain (for simplification, equals to 1). 3.1

been highly attenuated; however, the phase margin decreases and may result in a poor transient response. Thus, in order to maintain enough stability margins, kc is chosen to be 16 when the gain and phase margins are 10.4 dB and 57.8°.

AD feedback 3.2

In order to perform the AD design, a simple proportional-integral (PI) current controller is chosen to be the outer loop regulator Gc(s). According to Dannehl et al. [4], the optimal PI parameters are expressed as Gc (s) = kp ·

Ti s + 1 Ti s

1 kp = L1 + L2 fs 2

a2 Ti = i fs

(5) ai = 3

where, fs is the switching frequency. Then the open-loop transfer function is (subscript ‘_o’ denotes open loop) i

g Giref o (s) = kp ·

·

Ti s + 1 Ti s

1 L1 L2 C1 s3 + kc L2 C1 s2 + L1 + L2 s

(6)

Bode diagrams are shown in Fig. 3. The filter parameters for AD analysis are L1 = 0.6 mH, L2 = 0.36 mH, C1 = 7 μF and fs = 15 kHz. Clearly, when kc is rather small (e.g. equals to 1), the phase curve crosses −π-line around ωres only once; however, the amplitude curve is above the 0 dB line. According to the stability criterion, N+ = 1 and N− = 0, and instability occurs. With the increase of kc (e.g. larger than 5, and N+ = 0), the damping factor improves and the peak has

Harmonic resonant controller

The harmonic resonant controller can be expressed as Gc (s) = kp +

IET Power Electron., 2013, Vol. 6, Iss. 2, pp. 227–234 doi: 10.1049/iet-pel.2012.0192

ki s 2 + as + (nv )2 s 0 n=1,3,5,...

(7)

where, kp denotes the proportional part, n is the number of harmonics, ω0 is the fundamental angular frequency (here, 100π rad/s), α indicates the damping of the resonant controller and ki and α determine amplification of the harmonic frequency response. The filter parameters for resonant control and the following repetitive control studies are: L1 = 0.755 mH, L2 = 0.125 mH and C1 = 22 μF. Since the resonant control part has a slight impact on the performance around filter resonance frequency, the design method in the previous section can be applied to the AD and proportional part design, that is, kc = 19 and kp = 6.6. Besides, it is to be noted that grid frequency would have a small variation and it would be impossible to realise non-damped and extremely high gain-featured resonant control. In order to guarantee that the grid frequency is within the bandwidth of resonant control [9], α is chosen to be 6. Then, the open-loop transfer function is the product of (3) and (7) ig Giref o PR (s)

= kp + ·

Fig. 3 Open-loop characteristics with PI and AD control

ki s

2 s2 + as + nv0

L1 L2 C1

s3

1 + kc L2 C1 s2 + L1 + L2 s

(8)

Table 1 shows the cut-off and crossover frequencies with respect to different controllers centred at different frequencies. When ω < ωC1 and ωC2 < ω < ωC3, the condition L(ω) > 0 is satisfied; the phase crossovers occur at ωX1, ωX2 and ωX3 in the downward, upward and downward directions, respectively. With only proportional part (the first case in Table 1), the cut-off frequency is around 1.3 kHz and the crossover occurs around 3.3 kHz. The phase w (ω) crosses −π at the frequency, where L(ω) < 0 and N+ and N− are both zero. The system is stable with suitable margins. Besides, the plug-in 13th resonant controller does not cause instability, because no crossover occurs around 13 times of ω0. However, with the increase of n, the phase curve crosses −π in the downward direction around nω0 (that is, L(ω) > 0, N+ = 1). Clearly, when the 229


www.ietdl.org Table 1 Cut-off, crossover frequencies and stability criterions of system with PR control Cut-off: ωC, rad/s

Controller

Crossover: ωX, rad/s

Stability criterions

ki

n

ωC1

ωC2

ωC3

ωX1

ωX2

ωX3

N+

N−

Stable?

0 1000 1000 1000 1000 1000

0 13 25 31 37 43

7732 7735 8112 7737 7733 7732

(null) (null) (null) 9631 11 552 13 454

(null) (null) (null) 9842 11 695 13 563

(null) (null) 7861 9744 11 627 13 511

(null) (null) 7888 9790 11 697 13 614

20 587 20 492 20 480 20 469 20 453 20 425

0 0 1 1 1 1

0 0 1 1 0 0

yes yes yes yes no no

37th or 43rd resonant controller is involved, the upward crossover is at the frequency larger than ωC3. The additional phase variation causes instability when the upward crossover shifts into the range, where L(ω) < 0, or even when no upward crossover occurs if the centred frequency is too large, for example around the LCL resonance frequency. One thing to be noted, although it is stable with the 31st resonant controller, is that too close a distance between ωC2 and ωX1 or ωC3 and ωX2 means that the stability margins would be relatively small. It is recommended to design the plug-in resonant controllers centred at frequencies lower than the cut-off frequency of the P-controlled system (ωC1 in the first line of Table 1).

the cut-off frequency. As a result, the repetitive controller would cause instability of the LCL-filtered system.

3.3

As long as the feedforward function does not occur in the denominator, it does not affect system stability in the stiff grid case. In order to comprehensively compensate grid impact, the feedforward function is

Repetitive controller

The repetitive controller proposed in [17] for the stationary frame is shown in (9). The delay part of the controller can be easily implemented by analogue or digital control. Td expresses the delay time. In case that Td is 1/6 of a fundamental period, the controller realises the resonant control at (6k ± 1) times of fundamental frequency, where k is an infinite integer. Consequently, this repetitive controller can be equivalent to a bank of resonant controllers, as shown in (10). Gc (s) = kp +

1 − e−s2Td 1 + e−s2Td − e−sTd

1 6v0 s s Gc (s) = kp + +

p s2 + v20 k=1 s2 + (6k + 1)v0 2

s +

2 s2 + (6k − 1)v0

(9)

3.4

Seen from Fig. 2, with grid feedforward compensation, the transfer function from the grid voltage to grid current is expressed as i Gugs (s)

As discussed in Section 3.2, for the purposes of stability and good performance, the centred frequency is limited. However, k can be large enough to make the centred frequency exceed

Gf (s) − L1 C1 s2 + kc C1 s + 1 = (11) L1 L2 C1 s3 + kc L2 C1 s2 + L1 + L2 s + Gc (s)

Gf (s) = L1 C1 s2 + kc C1 s + 1

(12)

This result agrees with that in [16]. The proportional part is mainly in charge of the low-frequency distortion, and the first-order and second-order differential compensators mainly achieve middle- and higher-frequency grid harmonics rejection.

4

(10)

Grid feedforward method

Control in weak grid case

In the weak grid case, the grid impedance varies largely, for example, Lg would be close to or higher than the total inductance of grid-connected inverter. Meanwhile, the system sampling and control parameters remain constant, and the robustness need to be investigated. The equivalent overall control is shown in Fig. 4. As long as the PCC voltage expressed in (13) is sampled for the feedforward function, the feedforward actually introduces an additional grid current feedback loop. In the following, the parameters

Fig. 4 Equivalent current control structure in weak grid case 230 & The Institution of Engineering and Technology 2013


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Fig. 5 Open-loop response with PI and AD in weak grid case

Fig. 6 Open-loop response with proportional and 13th resonant controller in weak grid case

are the same as the above corresponding analysis section. ug = us + ig · Lg s

4.1

(13)

guaranteed; however, the cut-off frequency reduces close to or lower than 650 Hz in the weak grid case, and instability may occur, as shown in the plot, where instability occurs when β = 1 (N+ = 1, N− = 0).

AD feedback 4.3

In order to analyse AD applicability, the additional feedback loop introduced by grid feedforward has been neglected in this part. The open-loop transfer function can be expressed as (where, the subscript ‘_Lg’ represents the weak grid case) (see (14)) Denote Lg = β · (L1 + L2), the open-loop characteristics with respect to different grid conditions are shown in Fig. 5. By the use of the criterion, even when the grid impedance varies largely the stability is still guaranteed (N+ = N− = 0). Besides, in case the grid impedance equals total inductance, the crossover frequency reduces almost 30% of that in the stiff grid case; the gain margin increases, which indicates that filter resonance rejection performance becomes better. Similarly, as long as the zero of PI regulator remains constant and the cut-off frequency decreases largely, the phase margin is decreased slightly, which indicates that low-frequency rejection might become poor; however, the phase margins are still about 45°, and low-frequency rejection hardly changes actually. In addition, when Lg becomes bigger, the crossover and cut-off frequencies reduce slowly, and the stability margins remain almost unchanged. In conclusion, the AD can guarantee effective resonance rejection with wide range variation of grid impedance; and the low-frequency harmonics current is hardly increased. It is to be mentioned with different filter parameters that low-frequency performance might be different; however, the above analysis method can still be followed.

Repetitive controller

As for the repetitive controller, the same conclusion with resonant controller has been obtained because of its equivalent performance. Thus, the repetitive controller is hard to be implemented in grid current close-loop control. 4.4

Grid feedforward method

As shown in (13), the sampled voltage consists of real grid and grid inductor voltage. A positive feedback has been added into the control and would cause pollution. With only grid feedforward function, the open-loop transfer function is (the subscript ‘_f ’ represents the feedforward impact) i

g GLCL

o Lg f (s)

=

1 L1 (L2 + Lg )C1 s + (L1 + L2 + Lg )s − Gf (s)Lg s 3

(15) Considering that derivative functions are hard to be implemented and mainly for middle- and higher-frequency harmonics, in the follow-up studies commonly adopted proportional feedforward is used, and Gf(s) = 1. Consequently, the resonance frequency changes to be fres

4.2

Harmonic resonant controller

In the weak grid case, system open-loop cut-off frequency decreases, and even the resonant control designed to be stable in the stiff grid case would cause instability. Fig. 6 indicates open-loop response with proportional and 13th resonant controller. In the stiff grid case, the cut-off frequency is higher than 650 Hz so that stability has been i

g Giref

o Lg (s)

= kp ·

f

1 = 2p

L1 + L2 L1 (L2 + Lg )C1

Compared with (4), the frequency becomes much lower. Fig. 7a expresses the relation between resonance frequency and Lg with and without feedforward in the weak grid case. Obviously, with the grid feedforward method the grid impedance impact is extremely large; the resonance frequency reduces at twice or even higher speed compared

Ti s + 1 1 · Ti s L1 (L2 + Lg )C1 s3 + kc (L2 + Lg )C1 s2 + (L1 + L2 + Lg )s


(16)

(14)

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Fig. 7 Weak grid feedforward impact a Resonance frequency b Open-loop characteristics

with the control without feedforward. In this case, with the PI and AD control, the open-loop transfer function can be expressed as (see (17)) Bode plots of (17) are shown in Fig. 7b. Unlike those in Fig. 5, the crossover and cut-off frequencies vary largely. The gain margin does not change significantly; however, the phase margin reduces rapidly. This results in a large amount of low-frequency harmonics. Besides, seen from the open-loop response, the crossover frequency reduces close to the cut-off frequency. According to the stability criterion, when the crossover frequency is lower than the cut-off frequency, instability occurs (N+ = 1, N− = 0). In conclusion, the grid feedforward badly affects system performance in the weak grid case.

5 5.1

Verifications AD in the weak grid case

A 5 kW single-phase grid-connected LCL-filtered inverter with a TI TMS320F28035 digital signal processor (DSP) has been built for further verification in the laboratory. In order to reduce the leakage current, inverter topology (oH5) in [23] is used. The dc-link voltage is rectified from a three-phase voltage regulator and manually set to be 400 V. The grid side is the mains ac grid 220 V/50 Hz. The filter and control parameters are the same as the corresponding analysis sections. The overall control consists of PI, AD i

g Giref

o Lg f (s)

= kp ·

Fig. 8 AD applicability results a In rated case b Inserted with a 0.24 mH inductor c Inserted with a 1.4 mH inductor

based on capacitor current and grid feedforward. The follow-up experiments are tested in 30% of rated-power case. Fig. 8a shows the grid current and its spectrum in the default parameters case. In the following, this case is defined as the stiff grid case. It can be seen that a small amount of resonance-frequency harmonics existed in grid current, and the total current distortion (THD) is 5.6% obtained by a PM3300 power analyser. Furthermore, AD applicability has been tested. In order to avoid the feedforward impact, an equivalent method of adding inductance into L2 is adopted. The experimental waveforms are shown in Figs. 8b and c when an inductor with the

Ti s + 1 1 · Ti s L1 (L2 + Lg )C1 s3 + kc (L2 + Lg )C1 s2 + (L1 + L2 )s

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(17)


www.ietdl.org value of 0.24 or 1.4 mH is inserted. The resonance occurring in Fig. 8a does not exist anymore, and current THDs are 5.5 and 4.1%, respectively. Larger inductance results in better switching-frequency harmonics rejection and might be helpful for attenuation of lower-frequency harmonics, the result in the 1.4 mH inductance case gets better. The aforementioned results indicate that the AD method guarantees good applicability in the weak grid case. 5.2

Harmonic resonant and repetitive controllers

A grid-connected full-bridge LCL-filtered inverter is built for Saber simulations. The capacitor current feedback kc, proportional controller kp and plug-in resonant controller are used. The control parameters and filter parameters are the same as Section 3.2. Fig. 9a shows the simulation results when n equals to 13, 25, 31, 37 or 43 from top to bottom. Clearly, when the centred frequency is low, the grid current

is stable and synchronous with grid voltage; however, when the 37th or 43rd resonant controller is plugged in, current is obviously resonated. These results cooperate well with Table 1. The experiments have also been tested with the same parameters used above. The waveforms indicate the same conclusion and have not been presented. Furthermore, experiments in the weak grid case have been tested. The centred frequency of plug-in resonant controller is 650 Hz (n = 13) and a 0.8 mH (β = 0.91) inductor is inserted. The waveforms are shown in Fig. 9b. During the start-up procedure, the grid current is resonated so that the protection is tripped and the system cannot work stably. Seen from the current spectrum, the resonance frequency is only 650 Hz. This agrees well with that in Fig. 6. As for the repetitive controller, its model for simulation has been built according to (9) by the use of a delay block. Fig. 9c shows the grid current waveform with repetitive control. It is clear that the grid current is resonated and the system becomes unstable. As it is hard to be stable, no steady-state experimental result has been obtained. The protection is activated during the start-up procedure. 5.3

Grid feedforward in the weak grid case

In this section, grid feedforward impact in the weak grid case has been successfully verified based on the setup built in Section 5.1. The experiments are tested when the grid impedance is 0.24 or 1.4 mH and the PCC voltage is sampled. The corresponding results are shown in Fig. 10. When interfaced with a small value inductor, the system does not change apparently and the results in Fig. 10a with

Fig. 9 Results with a Proportional resonant (PR) controller in stiff grid case b Thirteenth resonant controller in weak grid case and c Repetitive controller IET Power Electron., 2013, Vol. 6, Iss. 2, pp. 227–234 doi: 10.1049/iet-pel.2012.0192

Fig. 10 Grid feedforward applicability results a With a 0.24 mH grid impedance b With a 1.4 mH grid impedance 233


www.ietdl.org Table 2 Current control applicability in stiff or weak grid case Performance Current control

AD control resonant control repetitive control grid feedforward

In stiff grid case

In weak grid case

effective damping unstable if centred frequency is high unstable

effective damping worse than that in stiff grid case unstable

effective harmonics rejection

ineffective, even unstable if Lg is large

a THD of 5.3% are the same as in Fig. 8b. In contrast, with a large value inductor additional positive feedback affects the system significantly as indicated in Fig. 7, and a worse current with a THD of 10.4% is produced as shown in Fig. 10b. Meanwhile, because of the distorted current the PCC voltage gets even worse. This would cause system instability and unreliability. For clarity, the results obtained from the above analysis and the verification sections are illustrated in Table 2. For detailed reasons, refer to the above sections.

6

Conclusions

This paper has focused on system stability and control applicability in the weak grid case. The researches have indicated that feedback-based AD guarantees good performance even in the wide grid impedance variation case; the centred frequencies of harmonic resonant controllers should be lower than the cut-off frequency of the proportional regulator-controlled system; the repetitive controller still faces difficulties if it is adopted in the grid current close-loop control; in the weak grid case, grid feedforward compensation introduces a positive feedback loop, and yields a large amount of harmonics or even instability. Thus, grid impedance should be taken into account during the design procedure. For resonant control, the condition that even in feasible largest impedance case, the centred frequencies be lower than the cut-off frequency of the proportional regulator-controlled system should be satisfied; for grid feedforward, a similar stability condition should be maintained with consideration of the positive feedback loop.

7

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant number 51077070), the Science and Technology Support Programme of Jiangsu Province (grant number BE2010188) and the Funding of Jiangsu Innovation Programme for Graduate Education (grant number CXZZ12_0153, the Fundamental Research Funds for the Central Universities).

8

References

1 Blaabjerg, F., Teodorescu, R., Liserre, M., et al: ‘Overview of control and grid synchronization for distributed power generation systems’, IEEE Trans. Ind. Electron., 2006, 53, (5), pp. 1398–1409

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2 Liserre, M., Blaabjerg, F., Hansen, S.: ‘Design and control of an LCL-filter-based three-phase active rectifier’, IEEE Trans. Ind. Appl., 2005, 41, (5), pp. 1281–1291 3 Jalili, K., Bernet, S.: ‘Design of LCL filters of active-front-end two-level voltage-source converters’, IEEE Trans. Ind. Electron., 2009, 56, (5), pp. 1674–1689 4 Dannehl, J., Wessels, C., Fuchs, F.: ‘Limitations of voltage-oriented pi current control of grid-connected PWM rectifiers with LCL filters’, IEEE Trans. Ind. Electron., 2009, 56, (2), pp. 380–388 5 Shen, G., Xu, D., Cao, L., Zhu, X.: ‘An improved control strategy for grid-connected voltage source inverters with an LCL filter’, IEEE Trans. Power Electron., 2008, 23, (4), pp. 1899–1906 6 Liserre, M., Teodorescu, R., Blaabjerg, F.: ‘Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values’, IEEE Trans. Power Electron., 2006, 21, (1), pp. 263–272 7 Teodorescu, R., Blaabjerg, F., Borup, U., et al: ‘A new control structure for grid-connected LCL PV inverters with zero steady-state error and selective harmonic compensation’. Proc. IEEE Applied Power Electronics Conf. and Exposition, Anaheim, California, 2004, pp. 580–586 8 Tang, Y., Loh, P.C., Wang, P., et al: ‘Exploring inherent damping characteristic of LCL-filters for three-phase grid-connected voltage source inverters’, IEEE Trans. Power Electron., 2012, 27, (3), pp. 1433–1442 9 Shen, G., Zhu, X., Zhang, J., Xu, D.: ‘A new feedback method for PR current control of LCL-filter-based grid-connected inverter’, IEEE Trans. Ind. Electron., 2010, 57, (6), pp. 2033–2041 10 Wessels, C., Dannehl, J., Fuchs, F.: ‘Active damping of LCL-filter resonance based on virtual resistor for PWM rectifiers stability analysis with different filter parameters’. Proc. IEEE Power Electronics Specialists Conf., Rhodes, August 2008, pp. 3532–3538 11 Xu, A., Xu, Z., Xie, S., Zou, M.: ‘Study on dual-loop grid current control strategy for grid-connected inverter with an LCL-filter’. Proc. IEEE Conf. Industrial Electronics and Applications, Xi’an, May 2009, pp. 3200–3203 12 Liu, F., Zhou, Y., Duan, S., Yin, J., Liu, B., Liu, F.: ‘Parameter design of a two-current-loop controller used in a grid-connected inverter system with LCL filter’, IEEE Trans. Power Electron., 2010, 56, (11), pp. 4483–4491 13 Dannehl, J., Fuchs, F., Hansen, S., Thøgersen, P.B.: ‘Investigation of active damping approaches for PI-based current control of grid-connected pulse width modulation converters with LCL filters’, IEEE Trans. Ind. Appl., 2010, 46, (4), pp. 1509–1517 14 Malinowski, M., Bernet, S.: ‘A simple voltage sensorless active damping scheme for three-phase PWM converters with an LCL filter’, IEEE Trans. Ind. Electron., 2008, 55, (4), pp. 1876–1880 15 Dannehl, J., Fuchs, F., Hansen, S., et al: ‘PI state space current control of grid-connected PWM converters with LCL filters’, IEEE Trans. Power Electron., 2010, 25, (9), pp. 2320–2330 16 Wang, X., Ruan, X., Liu, S., et al: ‘Full feedforward of grid voltage for grid-connected inverter with LCL filter to suppress current distortion due to grid voltage harmonics’, IEEE Trans. Power Electron., 2010, 25, (12), pp. 3119–3127 17 Loh, P.C., Tang, Y., Blaabjerg, F., Wang, P.: ‘Mixed-frame and stationary-frame repetitive control schemes for compensating typical load and grid harmonics’, IET Power Electron., 2011, 4, (2), pp. 218–226 18 Sun, J.: ‘Impedance-based stability criterion for grid-connected inverters’, IEEE Trans. Power Electron., 2011, 26, (11), pp. 3075–3078 19 Mohamed, Y.A.-R.I.: ‘Suppression of low- and high-frequency instabilities and grid-induced disturbances in distributed generation inverters’, IEEE Trans. Power Electron., 2011, 26, (12), pp. 3790–3803 20 Gabe, I.J., Montagner, V.F., Pinheiro, H.: ‘Design and implementation of a robust current controller for VSI connected to the grid through an LCL filter’, IEEE Trans. Power Electron., 2009, 24, (6), pp. 1444–1452 21 Shen, G., Zhang, J., Li, X., et al: ‘Current control optimization for grid-tied inverters with grid impedance estimation’. Proc. IEEE Applied Power Electronics Conf. and Exposition, Palm Springs, CA, February 2010, pp. 861–866 22 Hu, S.: ‘Principles of automatic control’ (Science Press, Beijing, 2001, 4th edn.) (in Chinese) 23 Xiao, H., Xie, S., Chen, Y., et al: ‘An optimized transformerless photovoltaic grid-connected inverter’, IEEE Trans. Ind. Electron., 2011, 58, (5), pp. 1887–1895
