Passivity-based stabilization of voltage-source ... - IEEE Xplore

Passivity-Based Stabilization of Voltage-Source Converters Equipped With LCL Input Filters Lennart Harnefors

Alejandro G. Yepes, Ana Vidal, Jes´us Doval-Gandoy

ABB, Corporate Research 72178 V¨aster˚as, Sweden E-mail: [email protected]

Dept. of Electronics Technology, Univ. of Vigo 36310 Vigo, Spain E-mail: [email protected], [email protected], [email protected]

Abstract—The time delay in the current control loop of a gridconnected voltage-source converter (VSC) may cause instability if the VSC is equipped with an inductive–capacitive–inductive (LCL) input filter. However, instability is prevented if the input admittance of the VSC can be made passive. This paper presents a systematic method for gaining passivity. Thereby, stability is obtained for any LCL-filter resonance and for any passive grid impedance. The method is equally applicable to single-phase and three-phase systems.

v ref

Ts =

1 fs

Cf

Zg(s) ig

vf

vg

i ref

F(s)

i v Hv(s)

L

Z (s) ig vf

vg

(b)

Fig. 1. Block and circuit diagram of a VSC with an LCL filter. (a) “Active damping” with capacitor-current feedback. (b) “Active damping” with capacitor-voltage feedforward. The impedances in (a) and (b) are related as Z(s) = 1/(sCf )||[sLf + Zg (s)].

(1)

of which Ts and 0.5Ts respectively are due to computation and pulsewidth modulation (PWM). In the ideal case of infinite sampling and switching frequencies, a passive input admittance, i.e., a nonnegative real part for all frequencies, is easily achievable using standard current control [18]. In practice, though, passivity is not obtained and a so-called “active damping” loop must often be added to achieve asymptotic stability. Commonly, feedback of the filter’s capacitor current is used [3], [7], [14]. The contribution of this paper is the development of a systematic method for the design of an “active damping” scheme whereby a passive input admittance is obtained for Td given by (1). This prevents destabilization regardless of the LCL filter’s inherent resonant frequency and the characteristics of the grid impedance [as long as the latter is a

k,(((

Lf

(a)

v ref

Destabilization of the resonance of an inductive– capacitive–inductive (LCL) input filter—which is used with grid-connected voltage-source converters (VSCs) to give improved harmonics rejection at the point of common coupling— is a well-studied problem [1]–[15]. Instabilities are effectively caused by the converter time delay Td . For a two-level VSC with synchronous sampling of the converter current (as well as other variables), and with sampling frequency fs , the time delay is given by [8], [16], [17]

L

H i(s)

I NTRODUCTION

Td = 1.5Ts ,

i v

Keywords—Converter control, LCL filter, passivity, resonance, stabilization.

I.

i ref

F(s)

resistive–inductive–capacitive (RLC) impedance]. The method is designed and analyzed in Section II, whereas it is evaluated by simulations and experiments in Section III. Extended results can be found in [19].

II.

C ONTROLLER D ESIGN AND A NALYSIS

For three-phase systems, current-controller implementation in stationary coordinates (the αβ frame) is assumed. Hence, the results developed are applicable also to single-phase systems or to per-phase control of multiphase systems. Scalar notation of all variables is used, but the variables should be interpreted as space vectors for αβ-frame controller implementation.



A. Control Laws

B. Inner and Outer Loops

The circuit and block diagram under consideration is depicted in Fig. 1(a), where L, Cf , and Lf are the LCL-filter parameters, ig is the grid current, and vg is the stiff grid voltage behind the RLC grid impedance Zg (s). The inner resistances of the inductors are neglected. For a negligible Zg (s), the resonant frequency of the LCL filter is given by  L + Lf . (2) ωres = LLf Cf

System stability can preferably be analyzed in an inner and an outer loop. The former comprises closed-loop control of the converter current through inductor L with the capacitor voltage regarded as a disturbance. Solving for i in (7) yields

The converter current i is forced to follow its reference iref via a linear controller F (s). The standard method for “active damping,” i.e., feedback of the capacitor current ig − i through a filter Hi (s) [3], [14], is used, giving the control law1

1 − Hv (s)e−sTd . sL + F (s)e−sTd (9) Asymptotic stability as well as good damping of Gc (s) and Y (s) is easy to achieve [18]. From Fig. 1(b), the following relation for the outer loop is obtained:

vref = −F (s)(iref − i) + Hi (s)(ig − i).

(3)

i = Gc (s)iref + Y (s)vf

where the (inner) closed-loop system and the (inner) input admittance respectively are given by Gc (s) =

An alternative is shown in Fig. 1(b), where “active damping” instead is made via feedforward of the capacitor voltage vf through a filter Hv (s) [7], giving vref = −F (s)(iref − i) + Hv (s)vf .

(4)

The relation between the capacitor voltage and current is vf =

1 (ig − i) sCf

(5)

so if the filters are related as Hv (s) = sCf Hi (s)

(6)

then (3) and (4) have identical impact. Henceforth, (6) is assumed always to hold. Since (4) uses differentiation, as is evident from (6), control law (3) is normally preferable. However, for convenience we shall use (4) in the analyses to follow, but all results are valid also for (3). Note that, in Fig. 1(b), a one-port equivalent is used where the LCLfilter elements Cf and Lf are included in the modified grid impedance Z(s) (and where the modified grid current and voltage ig and vg are introduced as well). The converter voltage v is created by PWM. As the converter is assumed always to operate within the limits set by the dc-link voltage, nonlinear effects can be disregarded. A pure time delay suffices as converter model, i.e., v = e−sTd vref [8], [16], [17]. Discrete-time controller implementation is assumed, but analysis will, in order to obtain simpler equations, be made in the continuous-time s domain. Remark 1: The term involving vf in (4) is called a feedforward rather than a feedback, since (at least structurally) it partially cancels vf in the dynamic relation i = (vf − v)/(sL) for the converter current. With (4), we get i=

e−sTd F (s)(iref − i) + [1 − e−sTd Hv (s)]vf . sL

(7)

1 Variable s shall, where appropriate, be considered either as the derivative operator s = d/dt or as the complex Laplace variable.

(8)

F (s)e−sTd sL + F (s)e−sTd

i=

Y (s) =

Gc (s) Y (s) iref + v . 1 + Y (s)Z(s) 1 + Y (s)Z(s) g

(10)

Stability of the outer loop can be analyzed by applying the Nyquist criterion to the open-loop transfer function Y (s)Z(s). However, if Z(s) and Y (s) are both passive, then stability of the outer closed-loop system is guaranteed [18]. Since Z(s) is an RLC impedance, it is obviously passive. Remaining is thus to make Y (s) passive. C. Design for Passivity A suitable “active damping” filter Hv (s) for control law (4) will now be designed. This gives, via (6), a suitable filter Hi (s) also for control law (3). To enhance clarity, this will be made successively in four steps. In the evaluations to be presented, Zg (s) ≈ 0 and different values of ωres relative to the angular sampling frequency ωs = 2πfs will be considered, all for the special case Lf = L. The current control loop will be designed to have bandwidth αc , for which the following recommendation is given [21]: αc ≤

ωs . 10

(11)

In all numerical studies, the upper limit of (11) will be considered. These are the main restrictions of the numerical studies. As will be seen, normalizations can be made, which allow general conclusions to be drawn. Due to the discrete nature of the converter control system, frequencies above the Nyquist frequency fs /2 are aliased down into the region [0, fs /2]. The criterion for passivity of Y (s) is thus ωs . (12) Re{Y (jω)} ≥ 0, 0≤ω≤ 2 In all diagrams to be shown, evaluation is made only up to the Nyquist frequency.



0.8

(a): P control

0.6

15 10

0.4

5

s

ω L Re{Y(jω)}

20

−5 0

0.1

0.2

ω/ω

0.3

0.4

Im{Y(jω)Z(jω)}

0 0.5

s

(b): PR control 15

0 −0.2 −0.4

10 −0.6

5

s

ω L Re{Y(jω)}

20

0.2

0

−0.8

−5 0

0.1

0.2

ω/ω

0.3

0.4

s

Fig. 2. Real part of the input admittance for (a) P and (b) PR control: (solid) without “active damping,” (dashed) with “active damping,” and (dasheddotted) with modified “active damping.” 0.8

−0.5 Re{Y(jω)Z(jω)}

0

Fig. 4. Nyquist curves for PR control with “active damping”: (solid) ωres = 0.1ωs , (dashed) ωres = 0.15ωs , and (dashed-dotted) ωres = 0.25ωs .

instability for ωres = 0.25ωs and even for ωres = 0.15ωs , but asymptotic stability for ωres = 0.1ωs . 2) P Control With “Active Damping”: Control law (4) is now augmented with an “active damping” filter in the form of a differentiator [7], Hv (s) = sK ⇒ Hi (s) = K/Cf , giving the input admittance

0.6 0.4 Im{Y(jω)Z(jω)}

−1

0.5

0.2

Y (s) =

0

1 − sKe−sTd . (s + αc e−sTd )L

(13)

Taking the real part yields −0.2

Re{Y (jω)} =

−0.4

−1

−0.5 Re{Y(jω)Z(jω)}

(αc − ω 2 K) cos ωTd . + ω 2 − 2αc ω sin ωTd )L

(14)

The sign change at ω = π/(2Td ) = ωs /6 caused by the cosine function in the numerator of (14) can now be lifted by selecting K such that the factor (αc − ω 2 K) changes sign exactly when the cosine function does. This is achieved by letting

−0.6 −0.8

(α2c

0

Fig. 3. Nyquist curves for P control: (solid) ωres = 0.1ωs , (dashed) ωres = 0.15ωs , and (dashed-dotted) ωres = 0.25ωs .

1) P Control Without “Active Damping”: We shall at first restrict the analysis to the simplest controller selection, i.e., F (s) = αc L [21] and Hv (s) = 0. This yields Y (s) = 1/[(s + αc e−sTd )L]. It is immediately found that Re{Y (jω)} changes sign from positive to negative at ω = π/(2Td ). With (1), this critical angular frequency equals ω = ωs /6 ≈ 0.17ωs. This is illustrated by the solid curve in Fig. 2(a), where normalization by ωs L is made to obtain dimensionless values. The input admittance is not passive and it can be conjectured that the outer loop will turn unstable, roughly, for ωres > 0.17ωs . This is confirmed by the Nyquist curves in Fig. 3, showing

4αc Td2 . (15) π2 As shown by the dashed curve in Fig. 2(a), a nonnegative real part is now obtained in the entire frequency range of interest, i.e., the input admittance is passive. K=

3) PR Control With “Active Damping”: To facilitate zero steady-state control error of the fundamental component—for a three-phase system both the positive- and negative-sequence components—an R part at the fundamental angular frequency ω1 is added to the current controller. This yields F (s) = αc L+ FR1 (s), where FR1 (s) is a special case (h = 1) of a general R part for a harmonic of order h [22], [23], [24]



FRh (s) = KIh

s cos φh − hω1 sin φh s2 + (hω1 )2

(16)

0.8

0.6

0.6

0.4

0.4

0.2

0.2

Im{Y(jω)Z(jω)}

Im{Y(jω)Z(jω)}

0.8

0 −0.2

0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−1

−0.5 Re{Y(jω)Z(jω)}

−0.8

0

Fig. 5. Nyquist curves for PR control with “active damping” and with inner resistances R = 0.32ω1 L = 0.32ω1 Lf added to the filter inductors: (solid) ωres = 0.1ωs , (dashed) ωres = 0.15ωs , and (dashed-dotted) ωres = 0.25ωs .

with gain KIh . There are several possible choices of the compensation angle φh ; φh = hω1 Td gives a direct compensation of the time delay [25]. In the numerical examples to follow, the R-part gain is selected as KIh = 0.1α2c L [18], whereas αc = 0.1ωs and ω1 = 0.01ωs are considered. The latter ratio is realistic, e.g., 50 Hz/5 kHz. The dashed curve in Fig. 2(b) is now obtained, showing a negative-real-part region about ω = ωs /6. If a resonance falls within—or even is close to— this region, the outer loop may turn unstable. This is confirmed by the dashed Nyquist curve in Fig. 4. Nonnegligible inner resistances of the filter inductors gives improved damping, but often not enough to prevent instability. In the case exemplified, inner resistances R = 0.32ω1 L = 0.32ω1Lf or larger have to be added to both filter inductors to stabilize the outer loop, as shown in Fig. 5. These inner resistances correspond to a quality factor 3.1, which is unrealistically low for a high-power converter. 4) PR Control With Modified “Active Damping”: The negative-real-part region about ω = ωs /6 can be eliminated by modifying the “active damping” with a compensation filter Hc (s) as Hv (s) = sKHc (s) ⇒ Hi (s) = KHc (s)/Cf . At the critical frequency Re{Y (jωs /6)} = −

ωs K Im{Hc (jωs /6)} . ωs − 6αc

(17)

−1

−0.5 Re{Y(jω)Z(jω)}

0

Fig. 6. Nyquist curves for PR control with modified “active damping”: (solid) ωres = 0.1ωs , (dashed) ωres = 0.15ωs , and (dashed-dotted) ωres = 0.25ωs .

and phase lead or zero phase shift at ωs /2. These requirements can be satisfied with a biquad filter Hc (s) =

s2 + 2ζz ωz s + ωz2 . s2 + 2ζp ωp s + ωp2

(19)

By selecting ωp < ωs /6 < ωz , phase lag at the critical frequency is obtained. Zero phase shift at the Nyquist frequency results from selecting ζz ωp (ωs2 − 4ωz2 ) . = ζp ωz (ωs2 − 4ωp2 )

(20)

Some iterations show that ωp = 0.1ωs

ωz = 0.2ωs

ζp = 2

(21)

are reasonable choices, for which (20) yields ζz = 0.875. This results in the following selection of the filter in (3): Hi (s) =

4αc Td2 s2 + 2ζz ωz s + ωz2 . π 2 Cf s2 + 2ζp ωp s + ωp2

(22)

As shown by the dashed-dotted curves in Fig. 2, the negativereal-part region about ω = ωs /6 is now gone. Asymptotic stability of the outer loop in all three cases being exemplified is verified in Fig. 6.

(18)

Equation (22), together with parameter selections (21) and (20), is the core result of the paper. With this filter in the feedback path for the capacitor current, an asymptotically stable outer loop, i.e., total closed-loop system, is obtained regardless of the circuit parameters and the choice of sampling and switching frequencies, provided that (1) holds and (11) is observed.

so Im{Hc (jωs /2)} ≥ 0 is required to prevent negative real part there. Thus, Hc (s) must be a filter with phase lag at ωs /6

Remark 2: Since the selections in (21) are given in relation to ωs , they are valid for any sampling frequency. Moreover,

Since from (11), ωs − 6αc > 0, Im{Hc (jωs /6)} < 0 is required if the negative-real-part region is to be eliminated. On the other hand, at the Nyquist frequency Re{Y (jωs /2)} =

ωs K Im{Hc (jωs /2)} ωs + 2αc



Re{Y(jω)}

(a) 0.06 0.04 0.02 0 0.1

0.2

ω/ω

0.3

0.4

0.5

0.4

0.5

s

Re{Y(jω)}

(b) 0.06 0.04 0.02 0 0.1

0.2

0.3 ω/ωs

Re{Y(jω)}

(c)

(a)

0.06 0.04 0.02 0 0.1

0.2

0.3 ω/ωs

0.4

0.5

Fig. 7. Real part of the input admittance [Ω−1 ]. PR control for (a) no “active damping,” (b) “active damping, and (c) modified “active damping.” Solid and dashed curves respectively show simulated and calculated results.

they are valid for any converter inductance (owing to controller gain selection proportional to L) and give a passive input admittance for any bandwidth selected according to (11) [but not necessarily if (11) is violated]. Remark 3: Although passivity arguments are not used in [14], a critical frequency region about ω = π/(3Ts ) = ωs /6 is identified, in agreement with the results found in this paper. If the LCL-filter resonant frequency (2) coincides with the critical frequency region, then adequate stability margins are in [14] found hard to obtain. Usage of modified “active damping” removes this obstacle. III.

S IMULATIONS AND E XPERIMENTS

To verify the theoretical results, some simulations and experiments have been performed. In the simulation study, a single-phase converter with dc-link voltage 1 kV, f1 = 50 Hz, switching frequency fsw = 5 kHz, fs = 10 kHz, and L = 10 mH is considered. (Note that this is a commutating-model—not an averagedmodel—simulation.) An inner resistance of 0.1 Ω is added to the converter-side inductor. PR control, with αc = ωs /10 and KIh = 0.1α2c L, is used. Controller discretization is made using the prewarped Tustin method [20]. For Cf = Lf = Zg (s) = 0, vg is set to a fundamental-frequency sinusoid with peak value 800 V. To this, a test signal with peak value 40 V and whose frequency is swept from 10 Hz to fs /2 is added. The amplitude and phase angle of the current component which results from this test signal are extracted by a phase-locked loop (PLL), allowing the real part of the input admittance to be computed. The results are shown in Fig. 7. As can be seen, the agreement between the simulated and calculated results is good, despite the fact that controller and

(b) Fig. 8. Results of short-circuiting the passive damping resistor, for ωres = 0.25ωs , with (a) “active damping” and (b) modified “active damping.”

filter discretization is not considered in the calculations. Close to ωs /2, accurate simulation results are virtually impossible to obtain. As the frequency of the test signal then approaches the switching frequency, the PLL fails to lock on the signal. For the experimental tests, a three-phase active rectifier is connected to the grid through an LCL filter. Control law (3), i.e., with capacitor-current feedback, is used. The dc-link and rms grid voltages are 750 V and 230 V, respectively. The values of f1 , fsw , fs , and αc coincide with those in the simulation. Two LCL-filter configurations are employed: ωres = 0.25ωs (L = 4.7 mH, Cf = 3 μF, and Lf = 1.85 mH) and ωres = 0.15ωs (L = 4.7 mH, Cf = 5 μF and Lf = 4.35 mH). The control algorithms are implemented on a dSPACE MABXII DS1401 platform. The behavior when a 5-Ω damping resistor in series with capacitor Cf is short-circuited is investigated. The oscilloscope trigger is synchronized with the instant at which such action is applied. Figs. 8(a) and (b) show voltage and current waveforms (where subscripts d and q indicate synchronousframe components and subscript a indicates phase) obtained for ωres = 0.25ωs with “active damping” and modified “active damping,” respectively. Both strategies give stable behavior, since the resonance is located in the positive-real-part region in both cases.



However, when the tests are repeated for ωres = 0.15ωs ,

[2]

(a)

(b) Fig. 9. Results of short-circuiting the passive damping resistor, for ωres = 0.15ωs , with (a) “active damping” and (b) modified “active damping.”

then only the system with modified “active damping” remains stable. See Figs. 9(a) and (b), which respectively show the results for “active damping” and modified “active damping.” [In Fig. 9(a), when the protection is turned on, the VSC stops switching and the dSPACE outputs become zero.] This is because the resonance in this case, for “active damping,” coincides with the negative-real-part region, and the result is in accordance with Section II-C. IV.

C ONCLUSION

We have in this paper presented a systematic method for stabilization of grid-connected VSCs equipped with LCL input filters. This is achieved by making the input admittance of the converter passive. The method is effective for any LCL-filter resonant frequency as well as for any RLC grid impedance. ACKNOWLEDGMENT This work was supported in part by ABB, in part by the Spanish Ministry of Science and Innovation, and in part by the European Commission, European Regional Development Fund (ERDF) under project DPI2012-31283. R EFERENCES [1]

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./Apr. 1997.

M. Liserre, A. DellAquilla, and F. Blaabjerg, “Genetic algorithm based design of the active damping for an LCL-filter three-phase active rectifier,” IEEE Trans. Power Electron., vol. 19, no. 1, pp. 234–240, Jan. 2003. [3] E. Twining and D. G. Holmes, “Grid current regulation of a three-phase voltage source inverter with an LCL input filter,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 888–895, May 2003. [4] M. Liserre, F. Blaabjerg and S. Hansen, “Design and control of an LCL-filter-based three-phase active rectifier,” IEEE Trans. Ind. Appl., vol. 41, no. 5, pp. 1281–1291, Sep./Oct. 2005. [5] M. Malinowski and S. Bernet, “A simple voltage sensorless active damping scheme for three-phase PWM converters with an LCL filter,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1876–1880, Apr. 2008. [6] K. Jalili and S. Bernet, “Design of LCL filters of active-front-end twolevel voltage-source converters,” IEEE Trans. Ind. Electron., vol. 56, no. 5, pp. 1674–1689, May 2009. [7] J. Dannehl, F. W. Fuchs, S. Hansen, and P. B. Thøgersen, “Investigation of active damping approaches for PI-based current control of gridconnected pulse width modulation converters with LCL filters,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 1509–1517, Jul./Aug. 2010. [8] J. L. Agorreta, M. Borrega, J. L´opez, and L. Marroyo, “Modeling and control of n-paralleled grid-connected inverters with LCL filter coupled due to grid impedance in PV plants,” IEEE Trans. Power Electron., vol. 26, no. 3, pp. 770–785, Mar. 2011. [9] J. Dannehl, M. Liserre, and F. W. Fuchs, “Filter-based active damping of voltage source converters with LCL filter,” IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3623–3633, Aug. 2011. [10] J. He and Y. W. Li, “Analysis, design, and implementation of virtual impedance for power electronics interfaced distributed generation,” IEEE Trans. Ind. Appl., vol. 47, no. 6. pp. 2525–2538, Nov./Dec. 2011. [11] Y. Tang, P. C. Loh, P. Wang, F. H. Choo, and F. Gao, “Exploring inherent damping characteristics of LCL-filters for three-phase grid-connected voltage source inverters,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1433–1443, Mar. 2012. [12] J. He and Y. Li, “Generalized closed-loop control (GCC) 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. [13] J. Yin, S. Duan, and B. Liu, “Stability analysis of grid-connected inverter with LCL filter adopting a digital single-loop controller with inherent damping characteristic,” IEEE Trans. Ind. Inform., vol. 9, no. 2, pp. 1104–1112, May 2013. [14] S. G. Parker, B. P. McGrath, and D. G. Holmes, “Regions of active damping control for LCL filters,” IEEE Trans. Ind. Appl., vol. 50, no. 1, pp. 424–432, Jan./Feb. 2014. [15] R. Pe˜ na-Azola, M. Liserre, F. Blaabjerg, R. Sebasti´an, 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. Inform., vol. 10, no. 1, pp. 43–52, Feb. 2014. [16] J. He, Y. W. Li, F. Blaabjerg, and X. Wang, “Active harmonic filtering using current-controlled, grid-connected DG units with closed-loop power control,” IEEE Trans. Power Electron., vol. 29, no. 2, pp. 642– 653, Feb. 2014. [17] A. G. Yepes, A. Vidal, J. Malvar, O. Lopez, and J. Doval-Gandoy, “Tuning method aimed at optimized settling time and overshoot for synchronous proportional-integral current control in electric machines,” IEEE Trans. Power Electron., vol. 29, no. 6, pp. 3041–3054, Jun. 2014. [18] L. Harnefors, L. Zhang, and M. Bongiorno, “Frequency-domain passivity-based current controller design,” IET Power Electron., vol. 1, no. 4, pp. 455–465, Dec. 2008. [19] L. Harnefors, A. G. Yepes, A. Vidal, and J. Doval-Gandoy, “Passivitybased controller design of grid-connected VSCs for prevention of electrical resonance instability,” IEEE Trans. Ind. Electron., 2014, [online].



[20]

L. Harnefors, “Implementation of resonant controllers and filters in fixed-point arithmetic,” IEEE Trans. Ind. Electron., vol. 56, no. 4, pp. 1273–1281, Apr. 2009. [21] L. Harnefors and H.-P. Nee, “Model-based current control of ac machines using the internal model control method,” IEEE Trans. Ind. Appl., vol. 34, no. 1, pp. 133–141, Jan./Feb. 1998. [22] R. I. Bojoi, G. Griva, V. Bostan, M. Guerriero, F. Farina, and F. Profumo, “Current control strategy for power conditioners using sinusoidal signal integrators in synchronous reference frame,” IEEE Trans. Power Electron., vol. 20, no. 6, pp. 1402–1412, Nov. 2005. [23] R. Bojoi, L. R. Limongi, F. Profumo, D. Roiu, and A. Tenconi, “Analysis of current controllers for active power filters using selective harmonic compensation schemes,” IEEJ Trans. Electr. Electron. Eng., vol. 4, no. 2, pp. 139-157, Mar. 2009. [24] M. Hanif, V. Khadkikar, W. Xiao, and J. L. Kirtley, “Two degrees of freedom active damping technique for filter-based grid connected PV systems,” IEEE Trans. Ind. Electron., vol. 61, no. 6, pp. 2795–2803, Jun. 2014. [25] L. Harnefors, A. G. Yepes, A. Vidal, and J. Doval-Gandoy, “Passivitybased stabilization of resonant current controllers with consideration of time delay,” IEEE Trans. Power Electron., [online].