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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < overall converter. References [14] and [16] have therefore proposed the lead-lag network, in place of the HPF function, for resonance damping. It was particularly proven in these references that the lead-lag network behaves like a differentiator around the resonance frequency. It is therefore suitable for resonance damping, but sensitive to resonance frequency variation, which in practice, cannot be tracked if the grid inductance is unknown. Because of that, the lead-lag differentiator is probably only suitable for strong grids with nearly constant grid impedances. Even for strong grids, the lead-lag differentiator is not an optimal choice, which this paper will prove by proposing a derivative based on the Quadrature-Second-Order Generalized Integrator (Q-SOGI), before discretization. In-phase SOGI has earlier been used as a phase-locked loop [26]. Its quadrature version is now studied to identify its selective derivative characteristics around the resonance frequency, which theoretically, are more accurate than the lead-lag derivative. Its unique phase lead above 90 introduced by differentiation can also help to nullify delay found in a digital system. It is thus emphasized during the design for improving robustness even with no algorithm included for estimating the resonance frequency. A second new derivative based on the non-ideal Generalized Integrator (GI) before discretization has also been presented. The GI was first mentioned in [27], and has since been used with Proportional-Resonant (PR) controllers and filters. Its derivative characteristics have been studied in our recent work [28]. This method is now expanded in this paper with more comprehensive evaluation about its performance for AD, including the discrete root locus analysis, parameter design and more experimental verifications. To do that systematically, Section II begins by succinctly reviewing active damping of a LCL-filtered grid converter with either capacitor voltage or current feedback. Those derivatives mentioned earlier are then compared more deeply in Section III, to better identify their features and constraints. This is followed by Section IV, which explains the proposed Q-SOGI and GI derivatives, their parameter design and discretization. Effectiveness of the proposed derivatives is finally verified through experiments in Section V, before concluding the findings in Section VI. It should, in addition, be emphasized that the proposed derivatives can be used with other applications, even though they have only been analyzed for resonance damping in the paper. II. SYSTEM MODELING AND DESCRIPTION Fig. 1 shows a three-phase grid-connected voltage-source converter with an output LCL-filter. The filter includes a converter-side inductor L1, a middle capacitor Cf, and a grid-side inductor L2, whose equivalent series resistances are usually ignored for emulating the worst case without any physical damping. Also shown in the figure is an equivalent grid inductance Lg, whose value is usually non-zero. The grid current i2 is then controlled to regulate the amount of power injected to the grid. For synchronization, the middle capacitor voltage vc or voltage at the point-of-common-coupling (PCC) is also sensed. In case the capacitor voltage is sensed for synchronization, it can additionally be used for resonance

2

Fig. 1. Three-phase grid converter with an LCL-filter.

damping. Otherwise, the capacitor current ic should be sensed and fed back for damping. Ideally, feeding back the capacitor voltage or current will produce the same results, if they are related precisely by an ideal “s” derivative. In other words, the damping function used with capacitor voltage feedback should be kaCfs like in Fig. 2(b), if a simple damping gain ka is used with capacitor current feedback like in Fig. 2(a) [19]. This requirement is however not realizable in practice, which means there will be differences between the two feedback methods. These differences are better clarified by deriving transfer functions i2 (s)⁄ic (s) and i2 (s)⁄vc (s) used in Fig. 2(a) and (b), beginning with the function in (1) for relating the converter output voltage v to the grid current i2 .

Gi 2 s

i2 s 1 1 v s sL1 L2 Lg C f s 2 r2

where ωr is the filter resonance frequency expressed as: 1 r L1 L2 Lg C f

(1)

(2)

Similarly, the function for relating the converter output voltage v to the capacitor current ic can be derived as: i s 1 s Gic s c (3) v s L1 s 2 r2 Further noting that vc (s) = ic (s)⁄(sCf ) , which substituted to (3), gives rise to: v s 1 1 Gvc s c v s L1C f s 2 r2

upon (4)

Transfer functions i2 (s)⁄ic (s) and i2 (s)⁄vc (s) in Fig. 2(a) and (b) can eventually be derived by dividing (3) from (1) and (4) from (1). The resulting transfer functions are given as follows. i2 s Gi 2 s 1 2 (5) ic s Gic s s L2 Lg C f

i2 s Gi 2 s 1 . vc s Gvc s s L2 Lg

(6)

With (3) to (6) now defined, transfer functions for relating the grid current i2 to the controller output voltage u in Fig. 2(a) and (b) can be proven the same, which certainly is expected. The common transfer function is given as follows. 1 1 Gad s . (7) 2 sL1 L2 Lg C f s s ka L1 r2 damping

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < TABLE II ADDITIONAL DISCRETIZATION METHODS (

1⁄ AND

Method Tustin with pre-warping

5

)

RULE

s

o z 1 oTs z 1 tan 2

Zero-order hold

X s X z 1 z 1 Z s

First-order hold

X z

z 1 zTs

2

X s Z 2 s

Zero-pole matching

z e sTs

Impulse invariant

X z Z X s

3) Direct Backward Euler Derivative Backward Euler discretization is observed from Fig. 5 to have the same magnitude response as forward Euler discretization. It therefore faces no amplification problem, unlike Tustin discretization. Its phase characteristic is however significantly different, which instead of a phase-lead, is now a phase-lag measured from the 90-phase of the continuous “s” function. This phase-lag increases with frequency until it reaches 90 at the Nyquist frequency. The actual phase of the backward Euler derivative is then zero, which strictly, is a gain and not derivative. To illustrate the phase-lag more distinctly, time-domain inputs at 50 Hz and 800 Hz are defined at the top of Fig. 8, which when fed to the continuous “s” and backward Euler derivatives, yield those results shown at the bottom of Fig. 8. Like in Fig. 5, the introduced phase-lag in Fig. 8 will only be significant at 800 Hz and above. Capacitor voltage feedback based on backward Euler derivative will therefore not be as well-damped as capacitor current feedback if the resonance frequency is located at 800 Hz or higher. Despite that, backward Euler derivative is still generally stable and noiseless, which has led to its usages in [19] for active damping based on capacitor voltage feedback and in [24] for improving system passivity. Nonetheless, it was pointed out in [19] that capacitor voltage feedback with backward Euler derivative cannot provide sufficient damping at a higher resonance frequency, which according to Fig. 5, is caused by the introduced phase-lag. Effects of the phase-lag on the system poles can also be viewed by plotting root loci of (13) after substituting Ga(z) with the backward Euler function listed in

(a) (b) Fig. 8. Waveform comparison between backward Euler and ideal “s” derivatives, when the frequency of input signal is (a) 50 Hz, and (b) 800 Hz.

Table I. The loci obtained by varying damping gain ka at different resonance frequencies ωr are given in Fig. 7(c), which noticeably, are always out of the unit circle. In theory, the loci will move into the unit circle if the filter is designed to have a sufficiently small ωr . This is however at the expense of size and cost. It is thus not encouraged, which indirectly, also means capacitor voltage feedback with backward Euler derivative is very hard to stabilize, even though possible. B. Indirect Discretization Instead of the “s” function, discretization can be applied to a filter function, whose purpose is to trim off the high gain of the “s” function at high frequency. The same approach has been applied to the pure integrator for resolving dc drift problems rather than noise amplification [30]–[32]. The eventual filter function chosen must still be discretized for digital implementation, which in case of derivative, leads to the following observations. 1) Indirect HPF Derivative Transfer function of the HPF is given in (16), where ωHP is its cutoff frequency. s GHP s HP (16) s HP Below ωHP , characteristics of the HPF mimic an ideal differentiator well, but above it, the HPF becomes a constant gain with much lesser noise problems. The HPF can thus be used for approximating the “s” function, before discretization. Among the three methods listed in Table I and four additional

(a) (b) (c) (d) Fig. 7. Root loci of (a) (14) based on Capacitor Current (CC) feedback, (b) (13) based on capacitor voltage feedback with direct forward Euler derivative, (c) (13) based on capacitor voltage feedback with direct backward Euler derivative, and (d) (13) based on capacitor voltage feedback with indirect non-ideal GI derivative.

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < those discretized Bode diagrams drawn in Fig. 15 and Fig. 16. In common, both figures have their ω'' tuned to the Nyquist frequency, which for 10-kHz sampling, is equal to 10000×π rad/s. Their common effective derivative range is thus extended to the Nyquist frequency, which for a digital controller, is its full controllable range. Such broadness has undeniably distinguished the non-ideal GI derivative from the more selective Q-SOGI derivative. Regardless of that, both derivatives, and in fact all derivatives studied, must have an additional filter for removing switching component if both resonance and switching frequencies are close especially for high power conversion [25]. The same filtering will also be required when the capacitor current is measured directly. The proposed derivatives are therefore not inferior even when used for high power conversion with capacitor voltage fed back. Returning to Fig. 15, an immediate observation noted is the best magnitude match yielded by the backward and forward Euler discretization methods. They however introduce either a large phase-lag or unstable poles, like explained in Section III (A). They are hence not recommended. On the other hand, Tustin method has shifted its magnitude peak towards the left of the Nyquist frequency. Its effective derivative range is thus narrowed, which technically, can be resolved by pre-warping close to the Nyquist frequency. The pre-warped magnitude response is however significantly attenuated, and has an unacceptably high gain at the Nyquist frequency, which can cause noise amplification. It is therefore not recommended too. Proceeding to Fig. 16 where four more discretized responses are shown, the Impulse Invariant (IMP) method is immediately recognized as unsuitable because of its approximately zero phase over a wide range of frequency. Zero-Order Hold (ZOH) and Zero-Pole Matching (ZPM) methods are also not acceptable because of their increasing phase-lag below 90, as frequency increases. The last First-Order Hold (FOH) method is, in contrast, having interestingly close phase characteristic when compared with the continuous “s” derivative. A simple example for demonstrating it can be extracted from Fig. 5, Fig. 9 and Fig. 16, where at 2 kHz, phase errors of backward Euler, HPF and non-ideal GI derivatives are respectively read as 30, 20 and 5. Non-ideal GI derivative discretized by FOH is thus the most accurate with only a slightly elevated magnitude response detected at its Nyquist frequency. The discretized expression for the non-ideal GI with a specified ωc is not easy to be expressed, but it is easy to be obtained by using the matlab command “c2d” to (24). This can clearly be seen in Fig. 17, where time-domain input and output waveforms of the non-ideal GI derivative are compared with those of the ideal “s” derivative. As expected, responses from both derivatives match each other closely, unlike the direct Tustin and direct backward Euler derivatives represented by Fig. 6 and Fig. 8, respectively. To next verify its impact on stability when used for active damping with capacitor voltage feedback, root loci of the non-ideal GI derivative are plotted by varying damping gain ka in (13). The obtained loci are shown in Fig. 7(d), which evidently, is similar to Fig. 7(a) for representing capacitor current feedback. Such similarities can only be achieved by the proposed non-ideal GI derivative, and not other derivatives reviewed in the paper.

9

Fig. 15. Frequency responses of different discretized non-ideal GI derivatives.

Fig. 16. Frequency responses of different discretized non-ideal GI derivatives.

(a) (b) Fig. 17. Waveform comparison between non-ideal GI and ideal “s” derivatives, when the frequency of input signal is (a) 50 Hz, and (b) 800 Hz.

2) Parameter Design The non-ideal GI derivative in (24) uses ω'' and ωc for notating the frequency with maximum gain and cutoff frequency, respectively. The former should be set to the Nyquist frequency, as explained earlier. Its tuning is therefore relatively straightforward. The same however does not apply to ωc , where it is shown in Fig. 18 to have countering influences on the phase and gain of the non-ideal GI derivative (trimming infinite gain to prevent noise amplification results in visible

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < [9]

[10] [11]

[12]

[13] [14] [15]

[16]

[17]

[18] [19]

[20]

[21]

[22]

[23]

Y. Tang, P. C. Loh, P. Wang, F. H. Choo, F. Gao, and F. Blaabjerg, “Genrealized design of high performance shunt active power filter with output LCL filter,” IEEE Trans. Ind. Electron., vol. 59, no. 3, pp. 1443–1452, Mar. 2012. M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCLfilter based three-phase active rectifier,” IEEE Trans. Ind. Appl., vol. 41, no. 5, pp. 1281–1291, Sep./Oct. 2005. R. N. Beres, X. Wang, F. Blaabjerg, M. Liserre, and C. L. Bak, “Optimal design of high-order passive-damped filters for grid-connected applications,” IEEE Trans. Power Electron., 2015, 10.1109/TPEL.2015.2441299. R. Pena-Alzola, M. Liserre, F. Blaabjerg, R. Sebastian, J. Dannehl, and F. W. Fuchs, “Analysis of the passive damping losses in LCL-filter-based grid converters,” IEEE Trans. Power Electron., vol. 28, no. 6, pp. 2642– 2646, Jun. 2013. P. Channegowda and V. John, “Filter optimization for grid interactive voltage source inverters,” IEEE Trans. Ind. Electron., vol. 57, no. 12, pp. 4106–4144, Dec. 2010. V. Blasko and V. Kaura, “A novel control to actively damp resonance in input LC filter of a three-phase voltage source converter,” IEEE Trans. Ind. Appl., vol. 33, no. 2, pp. 542–550, Mar. 1997. D. Pan, X. Ruan, C. Bao, W. Li, and X. Wang, “Capacitor-current-feedback active damping with reduced computation delay for improving robustness of LCL-type grid-connected inverter,” IEEE Trans. Power Electron., vol. 29, no. 7, pp. 3414–3427, Jul. 2014. R. Pena-Alzola, M. Liserre, F. Blaabjerg, R. Sebastian, J. Dannehl, and F. W. Fuchs, “Systematic design of the lead-lag network method for active damping in LCL-filter based three phase converters,” IEEE Trans. Ind. Inf., vol. 10, no. 1, pp. 43–52, Feb. 2014. X. Li, X. W, Y. Geng, X. Yuan, C. Xia, and X. Zhang, “Wide damping region for LCL-type grid-connected inverter with an improved capacitor-current-feedback method,” IEEE Trans. Power Electron., [online]. X. Wang, F. Blaabjerg, and P. C. Loh, “Virtual RC damping of LCL-filtered voltage source converters with extended selective harmonic compensation,” IEEE Trans. Power Electron., [online]. J. Dannehl, F. W. Fuchs, S. Hansen, and P. B. Thogersen, “Investigation of active damping approaches for PI-based current control of grid-connected pulse width modulation converters with LCL filters,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 1509–1517, Jul. –Aug. 2010. J. Xu, S. Xie, and T. Tang, “Active damping-based control for grid-connected LCL-filtered inverter with injected grid current feedback only,” IEEE Trans. Ind. Electron., vol. 61, no. 9, pp. 4746–4758, Sep. 2014. X. Wang, F. Blaabjerg, and P. C. Loh, “Analysis and design of grid-current-feedback active damping for LCL resonance in grid-connected voltage source converters,” in Proc. of IEEE ECCE, 2014, pp. 373–380. J. He and Y. W. Li, “Generalized closed-loop control schemes with embedded virtual impedances for voltage source converters with LC or LCL filters,” IEEE Trans. Power Electron., vol. 27, no. 4, pp. 1850–1861, Apr. 2012. L. Harnefors, A. G. Yepes, A. Vidal, and J. Doval-Gandoy, “Passivity-based controller design of grid-connected VSCs for prevention of electrical resonance instability,” IEEE Trans. Ind. Electron., vol. 62, no. 2, pp. 702–710, Feb. 2015.

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[24] X. Wang, F. Blaabjerg, and P. C. Loh, “Proportional derivative based stabilizing control of paralleled grid converters with cables in renewable power plants,” in Proc. IEEE of ECCE, 2014, pp. 4917–4924. [25] K. Hatua, A. K. Jain, D. Banerjee, and V. T. Ranganathan, “Active damping of output LC filter resonance for vector-controlled VSI-fed AC motor drives,” IEEE Trans. Ind. Electron., vol. 59, no. 1, pp. 334–342, Jan. 2012. [26] P. Rodriguez, R. Teodorescu, I. Candela, A. V. Timbus, M. Liserre, and F. Blaabjerg, “New positive-sequence voltage detector for grid synchronization of power converters under faulty grid conditions,” in Proc. IEEE PESC’06, Jun. 2006, pp. 1–7. [27] X. Yuan, W. Merk, H. Stemmler, and J. Allmeling, “Stationary-frame generalized integrators for current control of active power filters with zero steady-state error for current harmonics of concern under unbalanced and distorted operating conditions,” IEEE Trans. Ind. Appl., vol. 38, no. 2, pp. 523–532, Mar./Apr. 2002. [28] Z. Xin, X. Wang, P. C. Loh, and F. Blaabjerg, “Realization of Digital Differentiator Using Generalized Integrator for Power Converters,” IEEE Trans. Power Electron., 2015, 10.1109/TPEL.2015.2442414. [29] D. G. Holmes, T. A. Lipo, B. P. McGrath, and W. Y. Kong, “Optimized design of stationary frame three phaseACcurrent regulators,” IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2417–2426, 2009. [30] M. H. Shin, D. S. Hyun, S. B. Cho, and S. Y. Choe, “An improved stator flux estimation for speed sensorless stator flux orientation control of induction motors,” IEEE Trans. Power Electron., vol. 15, no. 2, pp. 312– 318, Mar. 2000. [31] Y. Wang and Z. Deng, “An integration algorithm for stator flux estimation of a direct-torque-controlled electrical excitation flux-switching generator,” IEEE Trans. Energy Convers., vol. 27, no. 2, pp. 411–420, Jun. 2012. [32] A. Veltman and J. L. Duarte, “Fish method based on-line optimal control for PWM rectifiers,” in Proc. of IEEE PESC’06, 1995, pp. 549–555 [33] M. Liserre, A. Dell’ Aquila, and F. Blaabjerg, “Stability improvements of an LCL-filter based three-phase active rectifier,” in Proc. of IEEE PESC’02, 2002, pp. 1195–1201. [34] A. Ghoshal and V. John, “Active damping of LCL filter at low switching to resonance frequency ratio,” IET Power Electron., vol. 8, pp. 574–582, 2015. [35] J. Wang, J. D. Yan, L. Jiang and J. Zou, “Delay-dependent stability of single-loop controlled grid-connected inverters with LCL filters,” IEEE Trans. Power Electron. Early Access Articles. [36] P. Rodriguez, A. Luna, I. Candela, R. Teodorescu, and F. Blaabjerg, “Grid synchronization of power converters using multiple second order generalized integrators,” in Proc. 34th Annu. Conf. IEEE Ind. Electron., Nov. 2008, pp. 755–760. [37] R. Teodorescu, F. Blaabjerg, M. Liserre, and P. C. Loh “Proportionalresonant controllers and filters for grid connected voltage-source converters,” in IEE Proc. Electr. Power Appl., vol. 153, no. 5, pp. 750– 762, Sep. 2006. [38] C. Zou, B. Liu, S. Duan, and R. Li, “Influence of delay on system stability and delay optimization of grid-connected inverters with LCL filter,” IEEE Trans. Ind. Inform., vol. 10, no. 3, pp. 1775-1784, Aug. 2014. [39] S. G. Parker, B. P. McGrath, and D. G. Holmes, “Regions of Active Damping Control for LCL Filters,” IEEE Trans. Ind. Appl., vol. 50, pp. 424-432, Jan/Feb. 2014.

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < overall converter. References [14] and [16] have therefore proposed the lead-lag network, in place of the HPF function, for resonance damping. It was particularly proven in these references that the lead-lag network behaves like a differentiator around the resonance frequency. It is therefore suitable for resonance damping, but sensitive to resonance frequency variation, which in practice, cannot be tracked if the grid inductance is unknown. Because of that, the lead-lag differentiator is probably only suitable for strong grids with nearly constant grid impedances. Even for strong grids, the lead-lag differentiator is not an optimal choice, which this paper will prove by proposing a derivative based on the Quadrature-Second-Order Generalized Integrator (Q-SOGI), before discretization. In-phase SOGI has earlier been used as a phase-locked loop [26]. Its quadrature version is now studied to identify its selective derivative characteristics around the resonance frequency, which theoretically, are more accurate than the lead-lag derivative. Its unique phase lead above 90 introduced by differentiation can also help to nullify delay found in a digital system. It is thus emphasized during the design for improving robustness even with no algorithm included for estimating the resonance frequency. A second new derivative based on the non-ideal Generalized Integrator (GI) before discretization has also been presented. The GI was first mentioned in [27], and has since been used with Proportional-Resonant (PR) controllers and filters. Its derivative characteristics have been studied in our recent work [28]. This method is now expanded in this paper with more comprehensive evaluation about its performance for AD, including the discrete root locus analysis, parameter design and more experimental verifications. To do that systematically, Section II begins by succinctly reviewing active damping of a LCL-filtered grid converter with either capacitor voltage or current feedback. Those derivatives mentioned earlier are then compared more deeply in Section III, to better identify their features and constraints. This is followed by Section IV, which explains the proposed Q-SOGI and GI derivatives, their parameter design and discretization. Effectiveness of the proposed derivatives is finally verified through experiments in Section V, before concluding the findings in Section VI. It should, in addition, be emphasized that the proposed derivatives can be used with other applications, even though they have only been analyzed for resonance damping in the paper. II. SYSTEM MODELING AND DESCRIPTION Fig. 1 shows a three-phase grid-connected voltage-source converter with an output LCL-filter. The filter includes a converter-side inductor L1, a middle capacitor Cf, and a grid-side inductor L2, whose equivalent series resistances are usually ignored for emulating the worst case without any physical damping. Also shown in the figure is an equivalent grid inductance Lg, whose value is usually non-zero. The grid current i2 is then controlled to regulate the amount of power injected to the grid. For synchronization, the middle capacitor voltage vc or voltage at the point-of-common-coupling (PCC) is also sensed. In case the capacitor voltage is sensed for synchronization, it can additionally be used for resonance

2

Fig. 1. Three-phase grid converter with an LCL-filter.

damping. Otherwise, the capacitor current ic should be sensed and fed back for damping. Ideally, feeding back the capacitor voltage or current will produce the same results, if they are related precisely by an ideal “s” derivative. In other words, the damping function used with capacitor voltage feedback should be kaCfs like in Fig. 2(b), if a simple damping gain ka is used with capacitor current feedback like in Fig. 2(a) [19]. This requirement is however not realizable in practice, which means there will be differences between the two feedback methods. These differences are better clarified by deriving transfer functions i2 (s)⁄ic (s) and i2 (s)⁄vc (s) used in Fig. 2(a) and (b), beginning with the function in (1) for relating the converter output voltage v to the grid current i2 .

Gi 2 s

i2 s 1 1 v s sL1 L2 Lg C f s 2 r2

where ωr is the filter resonance frequency expressed as: 1 r L1 L2 Lg C f

(1)

(2)

Similarly, the function for relating the converter output voltage v to the capacitor current ic can be derived as: i s 1 s Gic s c (3) v s L1 s 2 r2 Further noting that vc (s) = ic (s)⁄(sCf ) , which substituted to (3), gives rise to: v s 1 1 Gvc s c v s L1C f s 2 r2

upon (4)

Transfer functions i2 (s)⁄ic (s) and i2 (s)⁄vc (s) in Fig. 2(a) and (b) can eventually be derived by dividing (3) from (1) and (4) from (1). The resulting transfer functions are given as follows. i2 s Gi 2 s 1 2 (5) ic s Gic s s L2 Lg C f

i2 s Gi 2 s 1 . vc s Gvc s s L2 Lg

(6)

With (3) to (6) now defined, transfer functions for relating the grid current i2 to the controller output voltage u in Fig. 2(a) and (b) can be proven the same, which certainly is expected. The common transfer function is given as follows. 1 1 Gad s . (7) 2 sL1 L2 Lg C f s s ka L1 r2 damping

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < TABLE II ADDITIONAL DISCRETIZATION METHODS (

1⁄ AND

Method Tustin with pre-warping

5

)

RULE

s

o z 1 oTs z 1 tan 2

Zero-order hold

X s X z 1 z 1 Z s

First-order hold

X z

z 1 zTs

2

X s Z 2 s

Zero-pole matching

z e sTs

Impulse invariant

X z Z X s

3) Direct Backward Euler Derivative Backward Euler discretization is observed from Fig. 5 to have the same magnitude response as forward Euler discretization. It therefore faces no amplification problem, unlike Tustin discretization. Its phase characteristic is however significantly different, which instead of a phase-lead, is now a phase-lag measured from the 90-phase of the continuous “s” function. This phase-lag increases with frequency until it reaches 90 at the Nyquist frequency. The actual phase of the backward Euler derivative is then zero, which strictly, is a gain and not derivative. To illustrate the phase-lag more distinctly, time-domain inputs at 50 Hz and 800 Hz are defined at the top of Fig. 8, which when fed to the continuous “s” and backward Euler derivatives, yield those results shown at the bottom of Fig. 8. Like in Fig. 5, the introduced phase-lag in Fig. 8 will only be significant at 800 Hz and above. Capacitor voltage feedback based on backward Euler derivative will therefore not be as well-damped as capacitor current feedback if the resonance frequency is located at 800 Hz or higher. Despite that, backward Euler derivative is still generally stable and noiseless, which has led to its usages in [19] for active damping based on capacitor voltage feedback and in [24] for improving system passivity. Nonetheless, it was pointed out in [19] that capacitor voltage feedback with backward Euler derivative cannot provide sufficient damping at a higher resonance frequency, which according to Fig. 5, is caused by the introduced phase-lag. Effects of the phase-lag on the system poles can also be viewed by plotting root loci of (13) after substituting Ga(z) with the backward Euler function listed in

(a) (b) Fig. 8. Waveform comparison between backward Euler and ideal “s” derivatives, when the frequency of input signal is (a) 50 Hz, and (b) 800 Hz.

Table I. The loci obtained by varying damping gain ka at different resonance frequencies ωr are given in Fig. 7(c), which noticeably, are always out of the unit circle. In theory, the loci will move into the unit circle if the filter is designed to have a sufficiently small ωr . This is however at the expense of size and cost. It is thus not encouraged, which indirectly, also means capacitor voltage feedback with backward Euler derivative is very hard to stabilize, even though possible. B. Indirect Discretization Instead of the “s” function, discretization can be applied to a filter function, whose purpose is to trim off the high gain of the “s” function at high frequency. The same approach has been applied to the pure integrator for resolving dc drift problems rather than noise amplification [30]–[32]. The eventual filter function chosen must still be discretized for digital implementation, which in case of derivative, leads to the following observations. 1) Indirect HPF Derivative Transfer function of the HPF is given in (16), where ωHP is its cutoff frequency. s GHP s HP (16) s HP Below ωHP , characteristics of the HPF mimic an ideal differentiator well, but above it, the HPF becomes a constant gain with much lesser noise problems. The HPF can thus be used for approximating the “s” function, before discretization. Among the three methods listed in Table I and four additional

(a) (b) (c) (d) Fig. 7. Root loci of (a) (14) based on Capacitor Current (CC) feedback, (b) (13) based on capacitor voltage feedback with direct forward Euler derivative, (c) (13) based on capacitor voltage feedback with direct backward Euler derivative, and (d) (13) based on capacitor voltage feedback with indirect non-ideal GI derivative.

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < those discretized Bode diagrams drawn in Fig. 15 and Fig. 16. In common, both figures have their ω'' tuned to the Nyquist frequency, which for 10-kHz sampling, is equal to 10000×π rad/s. Their common effective derivative range is thus extended to the Nyquist frequency, which for a digital controller, is its full controllable range. Such broadness has undeniably distinguished the non-ideal GI derivative from the more selective Q-SOGI derivative. Regardless of that, both derivatives, and in fact all derivatives studied, must have an additional filter for removing switching component if both resonance and switching frequencies are close especially for high power conversion [25]. The same filtering will also be required when the capacitor current is measured directly. The proposed derivatives are therefore not inferior even when used for high power conversion with capacitor voltage fed back. Returning to Fig. 15, an immediate observation noted is the best magnitude match yielded by the backward and forward Euler discretization methods. They however introduce either a large phase-lag or unstable poles, like explained in Section III (A). They are hence not recommended. On the other hand, Tustin method has shifted its magnitude peak towards the left of the Nyquist frequency. Its effective derivative range is thus narrowed, which technically, can be resolved by pre-warping close to the Nyquist frequency. The pre-warped magnitude response is however significantly attenuated, and has an unacceptably high gain at the Nyquist frequency, which can cause noise amplification. It is therefore not recommended too. Proceeding to Fig. 16 where four more discretized responses are shown, the Impulse Invariant (IMP) method is immediately recognized as unsuitable because of its approximately zero phase over a wide range of frequency. Zero-Order Hold (ZOH) and Zero-Pole Matching (ZPM) methods are also not acceptable because of their increasing phase-lag below 90, as frequency increases. The last First-Order Hold (FOH) method is, in contrast, having interestingly close phase characteristic when compared with the continuous “s” derivative. A simple example for demonstrating it can be extracted from Fig. 5, Fig. 9 and Fig. 16, where at 2 kHz, phase errors of backward Euler, HPF and non-ideal GI derivatives are respectively read as 30, 20 and 5. Non-ideal GI derivative discretized by FOH is thus the most accurate with only a slightly elevated magnitude response detected at its Nyquist frequency. The discretized expression for the non-ideal GI with a specified ωc is not easy to be expressed, but it is easy to be obtained by using the matlab command “c2d” to (24). This can clearly be seen in Fig. 17, where time-domain input and output waveforms of the non-ideal GI derivative are compared with those of the ideal “s” derivative. As expected, responses from both derivatives match each other closely, unlike the direct Tustin and direct backward Euler derivatives represented by Fig. 6 and Fig. 8, respectively. To next verify its impact on stability when used for active damping with capacitor voltage feedback, root loci of the non-ideal GI derivative are plotted by varying damping gain ka in (13). The obtained loci are shown in Fig. 7(d), which evidently, is similar to Fig. 7(a) for representing capacitor current feedback. Such similarities can only be achieved by the proposed non-ideal GI derivative, and not other derivatives reviewed in the paper.

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Fig. 15. Frequency responses of different discretized non-ideal GI derivatives.

Fig. 16. Frequency responses of different discretized non-ideal GI derivatives.

(a) (b) Fig. 17. Waveform comparison between non-ideal GI and ideal “s” derivatives, when the frequency of input signal is (a) 50 Hz, and (b) 800 Hz.

2) Parameter Design The non-ideal GI derivative in (24) uses ω'' and ωc for notating the frequency with maximum gain and cutoff frequency, respectively. The former should be set to the Nyquist frequency, as explained earlier. Its tuning is therefore relatively straightforward. The same however does not apply to ωc , where it is shown in Fig. 18 to have countering influences on the phase and gain of the non-ideal GI derivative (trimming infinite gain to prevent noise amplification results in visible

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2015.2467313, IEEE Transactions on Power Electronics

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