Passivation of Grid-Following VSCs: A Comparison Between Active Damping and Multi-Sampled PWM

—This article compares different strategies used to enhance the stability properties of grid-following voltage-source converters (VSCs). Because of digital delays, VSC admittance exhibits a nonpassive zone, which introduces negative damping and may destabilize the grid-connected operation. It is shown that typically used active damping (AD) strategies only bring positive impact up to a certain frequency, while deteriorating admittance properties around and above the Nyquist frequency. Multi-sampled pulsewidth modulation (MS-PWM) greatly extends the passive ad-mittance region, using only a single-loop current controller. Experimental admittance measurements are performed on a single-phase VSC, up to twice the switching frequency. Subsequently, different grid-connected scenarios are tested to show that MS-PWM retains stable operation, where AD methods cause instability. This article also offers analytic modeling and experimental measurements of noise propagation for compared strategies. It is shown that derivative-based AD is not highly sensitive; however, MS-PWM offers additional noise suppression.

Passivation of Grid-Following VSCs: A Comparison Between Active Damping and Multi-Sampled PWM Ivan Z. Petric , Student Member, IEEE, Paolo Mattavelli , Fellow, IEEE, and Simone Buso , Member, IEEE Abstract-This article compares different strategies used to enhance the stability properties of grid-following voltage-source converters (VSCs).Because of digital delays, VSC admittance exhibits a nonpassive zone, which introduces negative damping and may destabilize the grid-connected operation.It is shown that typically used active damping (AD) strategies only bring positive impact up to a certain frequency, while deteriorating admittance properties around and above the Nyquist frequency.Multi-sampled pulsewidth modulation (MS-PWM) greatly extends the passive admittance region, using only a single-loop current controller.Experimental admittance measurements are performed on a single-phase VSC, up to twice the switching frequency.Subsequently, different grid-connected scenarios are tested to show that MS-PWM retains stable operation, where AD methods cause instability.This article also offers analytic modeling and experimental measurements of noise propagation for compared strategies.It is shown that derivative-based AD is not highly sensitive; however, MS-PWM offers additional noise suppression.

I. INTRODUCTION
G RID-FOLLOWING voltage-source converters (VSCs)   are becoming increasingly present in electric energy distribution grids [1], [2].However, connecting an actively controlled VSC to a grid may bring stability issues [3].One of such problems may arise when poorly damped grid resonances get destabilized due to current-loop dynamics of the connecting VSC [3].This article focuses on the investigation of harmonic instability at high frequencies, which is enabled by delays of the current control system [4], [5].To prevent the harmonic instability, passivity-based control (PBC) has gained significant attention [5]- [27].Starting from impedance-based stability assessment [28]- [31], PBC provides a sufficient condition for stability by rendering the VSC input admittance passive, i.e., making its real part nonnegative.This concept has proved to be powerful for the analysis and improvement of the stability of a system that consists of many interconnected power converters [3], [7], [32]- [34].Initially, the authors focused on passivizing the admittance up to the Nyquist frequency (NF).However, more recently, it has been pointed out that passivity should be investigated also above the NF, where coupling between sidebands, generated by sampling and modulation, introduces a destabilizing mechanism [18], [27], [35], [36].
The focus of this article is placed on the most widely used VSCs that operate using pulsewidth modulation (PWM).For uniformly sampled PWM (US-PWM) [4] with one step computation delay, VSC admittance exhibits negative conductance above one-sixth of the sampling frequency f s [16].For enhancing the passivity, many different strategies of active damping (AD) have been developed.Some improvement is obtained by adding damping filters to the current controller (CC) [6]- [9]; however, most methods employ additional feedforward or feedback actions [10]- [26].Although certain multiloop methods are developed specifically for VSCs with LCL filters [10]- [15], this article compares and refers to the most generally applicable ones that are relevant also for VSCs with L filters [16]- [19].These methods rely on the derivative-based feedforward of the voltage at the point of common coupling (PCC).In terms of effectiveness, for full passivity up to the NF, analog or oversampled derivative action must be applied.This kind of AD, first analyzed in [16] and generalized in [19], will be referred to as the derivative active damping (D-AD).The passive region can also be extended without analog or oversampled implementation, which is referred to as discretized derivative active damping (DD-AD) [17], [18].DD-AD effectively increases the passive region up to approximately f s /3.Another important fact is that, for any AD implementation, passivity enhancement is affected by parameter mismatch.An alternative approach to US-PWM is the multi-sampled PWM (MS-PWM), where the control action is executed at a rate higher than double the switching frequency [37]- [54].MS-PWM reduces control update and modulation delays and, in this way, approaches the naturally sampled PWM [55].In [54], it is shown that MS-PWM inherently renders the admittance passive in a wide frequency range.
This article first compares the effectiveness of D-AD, DD-AD, and MS-PWM in passivizing the admittance near and above the switching frequency.Even though in [27], it is shown that this frequency range is important for stability, there is a lack of detailed investigation of corresponding AD effectiveness, which is addressed in this article.Furthermore, a potential practical challenge with D-AD and DD-AD concerns noise sensitivity due to the implementation of the derivative feedforward action.Accordingly, this article provides an investigation of noise propagation, which has not been analyzed in the literature yet.
Main novelties and contributions of this article are summarized as follows.
1) A comparative analysis of passivizing properties of D-AD, DD-AD, and MS-PWM: It is experimentally validated that AD does not provide the robust stability of grid-tied VSCs for harmonic resonances around and above the NF.This property is predictable by a linear single-frequency model of the VSC admittance.Experimental results are given to show the destabilization of grid antiresonances found slightly below the NF for DD-AD and above NF for D-AD.MS-PWM enables stable operation for all tested cases by passivizing the VSC in a wider frequency range.

2) Analytic modeling, experimental measurements, and comparison of noise sensitivity of D-AD, DD-AD, and MS-PWM:
The proposed analytics predict well the experimental measurements in case of both single-and double-loop control systems.Another step forward is taken by modeling the noise propagation for the case of oversampled (or analog) derivative being followed by a resampler, i.e., the case of D-AD.An important conclusion is that D-AD and DD-AD strategies are not highly sensitive to measurement noise, thanks to a relatively low required value of the derivative gain.For DD-AD, it is shown that, for all practical purposes, noise sensitivity is the same as without AD.Comparatively, MS-PWM is found to offer stronger noise suppression.The rest of this article is organized as follows.Section II describes the current control system of a grid-following VSC and explains the impact of admittance on stability.Section III analyzes strategies for passivity enhancement and shows corresponding experimental admittance measurements.Section IV analyzes the implications of connecting a VSC with a nonpassive admittance to a grid.Experimental tests are provided to show that AD strategies cause instability for certain grid impedances.Section V analyzes noise propagation and shows corresponding measurements.Finally, Section VI concludes this article.

II. CONTROL OF GRID-FOLLOWING VSCS
Grid-following VSCs are employed in power systems as current-controlled converters.As the most general case, this article considers a single-phase VSC with an inductive output filter, as shown in Fig. 1(a).However, the analysis is directly applicable to three-phase VSCs in stationary coordinates as well as to any output filter.The current reference i L,r is provided by outer loops that control the required power flow by changing the reference current magnitude |i L,r | and phase angle Φ, with respect to the voltage at the PCC, v pcc .A phase-locked loop (PLL) is used to track the fundamental component of v pcc .The CC processes the error between i L,r and the measured inductor current i L,f .It may also include an action based on v pcc , which will be referred to as AD.The controller output, i.e., the reference voltage v i,r , is processed by the digital pulsewidth modulator (DPWM), which outputs the transistor switching signal x with duty cycle D.The switching frequency is f pwm = 1/T pwm .Input dc bus voltage V in is used for v i,r scaling [4].
A closed-loop block diagram of the current control system is shown in Fig. 1(b), where v pcc is shown as a disturbance.The difference between the VSC output voltage v i and v pcc is applied to the output inductive filter where s is the complex variable of the Laplace transform and L is the inductance value.The resulting current i L is sampled by an analog-to-digital converter (ADC) with a rate f s = Nf pwm , where N is the multisampling (oversampling) factor.The frequency f s will be referred to as the sampling and control frequency, and its inverse T s = 1/f s as the control period.Values N = {1, 2} correspond to uniformly single-sampled (SS-PWM) and double-sampled (DS-PWM) control [4], respectively.For N > 2, MS-PWM control loop is formed [38].The control frequency determines the execution rate of the controller G c (z), where z is the Z-domain variable corresponding to T s .Owing to finite algorithm computation time, the controller output is often delayed by one control period [4].The control system in Fig. 1(b) also includes the AD block H ad (z 1 ) [16].The execution rate of the AD block is independently chosen as f s,1 = N ad f pwm , where N ad will be referred to as the AD oversampling factor.The Z-domain variable z 1 corresponds to T s,1 = 1/f s,1 .In case f s,1 = f s , a rate conversion needs to be included after H ad .

A. S-Domain Representation of the Control System
In this article, modeling is performed in the s-domain to allow the investigation of VSC admittance properties even above the NF [27].The impact of sidebands [18], [27], [34]- [36], [56] is not considered as it introduces additional complexity, without being crucial for the conclusions drawn in this article.The sdomain representation of the system is shown in Fig. 2(a).All time delays are represented by a unique block G d (s).
The triangular carrier is chosen for the DPWM [4].Its small-signal model is obtained using the describing function approach [57] where m is the modulating waveform, equal to v i,r after one-step delay, and d is the continuous-time duty cycle of x, used in averaged modeling [4].For D = 0.5, the DPWM model reduces to a pure time delay e −s Ts 2 .As this D is the mean value of duty cycles for sinusoidal ac operation, it will be used for the subsequent analysis.Together with one-step delay added due to algorithm execution time [4], the total time delay in the digital current control system is equal to As a standard approach for sinusoidal reference tracking, the proportional-resonant (PR) controller is used in this article [4] where k p and k r are the proportional and resonant gains, respectively, and ω 1 is the angular frequency of the current reference.As an example considered in this article, the proportional gain is chosen as k p = ω c L, where ω c is the sought angular crossover frequency of the current loop [4].The resonant gain is chosen as k r = 110 ω c k p , which limits its impact to lower frequencies [4].The value of ω c relative to the angular switching frequency ω pwm , labeled as α = ω c /ω pwm , will be referred to as the relative bandwidth.The value of α is limited by system delays and PWM carrier frequency [4], [58].

B. Impedance-Based Stability
The impedance-based stability assessment requires representing the VSC with its Norton equivalent circuit, as in Fig. 2(b).The Norton current source determines the short-circuit current of the VSC, i.e., the value of i L when v pcc = 0 where is the open-loop gain of the corresponding system.The input admittance Y i is found as the inductor current response to the voltage v pcc when current reference is set to 0 From ( 6), it can be seen that the VSC input admittance is shaped by the output filter and the current control system, which includes digital delays and, if implemented, AD.The impact of outer loops on admittance is not modeled, as this article focuses on properties at frequencies significantly above their bandwidth [5].
Considering the VSC connection to a nonideal grid, i.e., Z g = 0, the inductor current can be calculated using the equivalent circuit parameters as v g (s).
(7) Assuming that W cl is stable, the stability of the grid-connected VSC depends on the product Y i (s)Z g (s), which is referred to as the minor-loop gain [29].Stability analysis can be performed by applying the Nyquist stability criterion to the minor-loop gain. 1  The input admittance Y i is said to be passive if its phase is in the range (− π 2 , π 2 ), i.e., if its real part is greater than or equal to 0 [5].Imposing the requirement for passive Y i , such as in [59], arises from the fact that if both Y i and Z g are passive, the phase of their product never crosses ±π.Therefore, minor-loop gain always satisfies the Nyquist criterion.
Instead, a negative real part of Y i determines nonpassive regions.For a control system described by ( 3) and ( 4), without AD, Re{Y i (jω)} first turns negative above ω ≈ 2π f s 6 and its peak negative value is increased with higher ω c and τ d , and lower L [54].

A. Experimental Setup and Admittance Measurements for DS-PWM Without AD
For experimental admittance measurements and subsequent grid-connected tests, an industrial full-bridge VSC, described in Table I, is used.The VSC is operated using the bipolar modulation.Input voltage is provided by the regenerative power system Keysight RP7962A.
The control system is implemented on an NI sbRIO-9606, which is based on a Xilinix Zynq 7020 all programmable system on chip (AP-SoC).The controller is discretized using the impulse-invariant approach [4].For all subsequent admittance measurements, L = 1.5 mH and α = 0.2.For strategies other than MS-PWM, current is sampled and the control algorithm is executed twice per switching period, i.e., DS-PWM is used.Sampling instants coincide with peaks and valleys of the carrier [4].
Input admittance Y i is measured by injecting voltage perturbation components at the PCC and measuring the current response, as in [54].The perturbation voltage is generated using a power operational amplifier MP118 from APEX.Measurements are performed for 23 frequencies, one at a time, from 5 to 41 kHz, which is just above f s for N = 2. Perturbation voltage is calculated to obtain, at least, 100 mA of perturbation component of i L , to provide a good measurement resolution.A Tektronix 5 series oscilloscope is used to acquire i L and v pcc with data length equal to 40 ms and sampling rate of 250 MS/s.Data are postprocessed in MATLAB by calculating the fast Fourier transform of −i L and v pcc and determining the admittance, based on their ratio at the perturbation frequency.The same measurement approach, described in more detail, is used in [54].All the admittance measurements shown in this article were conducted for v g = 0, leaving only the perturbation voltage component at the PCC.This sets the steady-state operating point to D = 0.5, which is compliant with the previous analysis.It is interesting to note that measurements were also conducted for other steady-state values of D. It is found that, as D approaches extreme values, the nonpassive zone peaks become less emphasized, which can be predicted by including the full DPWM model (2).Therefore, for sinusoidal ac operation, the highest introduced negative damping is expected around zero crossings of v g .An analysis of  the impact of DPWM operating point dependence on admittance measurements can be found in [56].
As a benchmark, admittance measurements are given for the case of DS-PWM without AD.These results are shown in Fig. 3, where it can be seen that a very good match with analytics is obtained, even above the NF.At frequencies above 1.5f pwm , the single-frequency model starts to show a more significant mismatch, which is addressed in [56].Note the existence of two nonpassive zones, where phase drops below −90 • .The first one starts at approximately 6.5 kHz, which is consistent with the often mentioned critical frequency f s /6 [16].Another nonpassive zone appears above the NF.The goal of the passivity-based design, which is analyzed in the following subsections, is to suppress these nonpassive zones.As an additional illustration, the real part of measured Y i , relative to the nominal VSC admittance Y n = S n U 2 n = 56.7 mS, is shown in Fig. 4.

B. Impact of AD
The first method considered for comparison is D-AD [16], which extends passivity up to the NF by adding a derivative feedforward of v pcc to the controller output, H ad (s) = sk ad .AD gain is chosen to compensate for the sign change of the real part of the frequency response of (6) [16]  In [16], it is shown that the derivative action must be implemented in the analog domain or using oversampling.The latter approach is chosen in this article in order to keep the implementation fully digital.Backward Euler discretization is chosen, as advised in [16].Referring to Fig. 1 ).The AD oversampling factor was chosen to be as high as allowed by the control platform.The relative value of the measured conductance for D-AD is shown in Fig. 5.A very good match with s-domain analytics is obtained, apart for frequencies very close to f s .It is assumed that this mismatch comes from not including the sideband effects [56].Results in Fig. 5 confirm that D-AD is capable of improving passivity up to NF2 .It can be seen that, above the NF, a nonpassive zone is introduced with a peak of negative conductance being slightly higher than for the case of DS-PWM without AD.In [21], this effect is mentioned; however, it is not thoroughly investigated in the literature, even if, from [27], it is clear that admittance properties above the NF need to be examined.D-AD was also tested for other values of k ad ; however, these results are not shown due to article length limitations.The trend is that higher values of k ad add more positive damping in the frequency region from f s /6 to the NF and more negative damping in the region above the NF, where the peak negative conductance is found.Lower values of k ad decrease the peak negative conductance above the NF; however, passivity is not achieved around f s /6.
The second AD method compared, DD-AD, uses the same derivative action as D-AD, however, without oversampling: N = 2, N ad = 2, and H ad (z) = k ad f s (1 − z −1 ).DD-AD is applied in [17], with a specific parameter design procedure.However, it was verified that implementing derivative gain as (8) brings practically the same results, without needing to use one fixed value of bandwidth as in [17].In [17], Z-domain modeling was used to predict DD-AD impact on admittance, which yielded a significant mismatch as frequencies approached the NF.Hence, in this article, H ad is directly transformed to s-domain, using z = e sT s .Results are reported in Fig. 6.It can be seen that a good match between experiments and analytics is obtained, in the same frequency range as in Figs. 4 and 5. From Fig. 6, it can be seen that DD-AD improves passivity at medium frequencies; however, above approximately f s /3, passivity is lost.The same frequency limit is reported in papers that use single-loop AD, by adding derivative gain to the controller structure [7].Note that the peak of negative conductance for DD-AD is slightly reduced compared to the case of DS-PWM without AD.

C. Impact of MS-PWM
It has been recently shown that a suitable candidate for gridconnected converters is the MS-PWM control, where N > 2 and H ad = 0 [54].Instead of compensating digital delays, MS-PWM relies on reducing them by increasing the update rate of the controller output.The main advantage of MS-PWM, compared to other strategies for US-PWM delay reduction [60], [61], is that both delays, due to controller output update and digital modulation, are reduced [38].With MS-PWM, switching ripple is introduced in the feedback.While removing the switching ripple is beneficial regarding modulator linearity [47], it strongly deteriorates dynamic improvements.The analysis of MS-PWM operation without ripple filtering can be found in [38], [46], [53], [54], and [58], and such an approach is considered in this article as well.Particularly, in [54], it is shown that the presence of switching ripple does not deteriorate the passivation effectiveness of MS-PWM.For conciseness of this article, important practical aspects of MS-PWM implementation are left out, but interested readers can find detailed information in the papers cited above.
In [54], a thorough investigation of MS-PWM impact on admittance properties is given, for many values of N .There, it is shown that MS-PWM expands the passive region of Y i , without needing to resort to AD.For lower values of N , such as 4, a small nonpassive zone appears close to f pwm ; however, negative damping is greatly reduced compared to N = 2. Further increase in N strongly reduces the nonpassive zone so  that fully passive admittance is measured up to 1.5f pwm for N > 8 in [54].As an example used for this article, admittance is measured for N = 16.The oversampling factor N is limited by the available processing power with respect to the switching frequency.Therefore, in the following section, N = 4 is also implemented to show that it is sufficient to retain the stability in the tested grid-connected operation.For the case of MS-PWM, the measured conductance is shown in Fig. 7.It can be seen that Y i is almost fully passive, except for a very narrow range, where the conductance is just below 0.

D. Comparison of Effectiveness
In Fig. 8, conductances for all the tested cases are shown in an extended frequency range.It can be seen that both D-AD and DD-AD are able to extend the passive zone; however, for both the methods, another emphasized nonpassive zone is introduced, which is not present for the case of DS-PWM without AD.This points to the fact that the addition of AD only shifts the nonpassive zone to higher frequencies and can, therefore, compromise stability, instead of guaranteeing it.It is important to note that these nonpassive zones are found in the frequency range where single-frequency admittance models predict well the experimentally measured VSC responses.This is important as it can be assumed that assessing stability using the Nyquist criterion most likely yields correct conclusions, which is validated in the following section.MS-PWM shows a natural drop toward zero conductance, where Y i starts to behave as an inductor.The summary of measured conductances is shown in Table II, by providing peaks of negative values and frequencies for which they are found.Results are also given for a second inductor with L = 2.5 mH.For MS-PWM, analytics predict the nonpassive zone to fall outside of the measurement range.

IV. PASSIVITY IMPLICATIONS IN GRID-CONNECTED SCENARIOS
Impedance-based analysis from Section II-B predicts that a VSC can be destabilized if connected to a grid with a certain Z g .For grid-following VSCs with inductive filters, a parallel antiresonance that coincides with the nonpassive zone of Y i can cause instability [16].In this section, operation stability is tested for all the strategies compared in Section III, by connecting the VSC to a grid with a specific Z g .For all the results in this section, L = 1.5 mH and α = 0.2.The results presented in this section hold, in relative terms, also for VSC designs with lower switching frequencies.Therefore, irrespective of the specific design parameters, reliable conclusions can be drawn on the robustness ensured by each controller organization.

A. Analysis of the Equivalent Impedance Network
Suppose that the analyzed VSC is connected to a grid with an antiresonance formed by L g and C g . 3Grid passive damping can be modeled as a parallel conductance G g .Such a grid impedance can, therefore, be described using its antiresonant frequency and quality factor Q = 1 Z c G g .The analysis of passive damping is important, as a nonpassive Y i does not necessarily destabilize the grid antiresonance with realistic values of Q [54].
In this section, VSC connection to three different grid antiresonances is tested.Antiresonant networks that coincide with nonpassive zones are formed using an inductor L g = 72 μH and different parallel capacitors C g .Passive damping of the antiresonant network was estimated from the exponential decay of oscillations triggered by imposing a transient while the VSC was disconnected.First antiresonance, Z g,1 , is obtained with C g = 5.6 μF.It is very close to the peak of negative conductance for N = 2 without AD, and its parameters are f res = 7.9 kHz, Fig. 9. Impedance network at the frequency of grid antiresonance.VSC admittance is represented as Y i (jω res ) = G i (jω res ) + jB i (jω res ).Grid impedance features passive damping, represented by G g .Q = 28, Z c = 3.6 Ω, and G % g = 17.6%.Second antiresonance, Z g,2 , is obtained with C g = 1 μF.It is very close to the peak of negative conductance for DD-AD, and its parameters are f res = 18.4 kHz, Q = 38, Z c = 8.3 Ω, and G % g = 5.6%.Third antiresonance, Z g,3 , is obtained with C g = 440 nF.It is very close to the peak of negative conductance for D-AD, and its parameters are f res = 28.2kHz, Q = 30, Z c = 12.7 Ω, and G % g = 4.6%.
The equivalent impedance network at the frequency of grid antiresonance is illustrated in Fig. 9.At ω res , VSC input admittance is represented by its conductance G i (jω res ) and susceptance B i (jω res ).Intuitively, from Fig. 9, it can be concluded that overall system damping at ω res is impacted by the parallel connection of G i (jω res ) and G g .Hence, the value of G i (jω res ) + G g indicates the damping of the resonant network.This intuitive approach is valuable as it helps to predict the stability based solely on admittance measurements, without knowing the analytic expression for Y i .A formal approach, in case Y i is analytically known, is to analyze stability using Nyquist plots of the minor-loop gain.Examples of Nyquist plots, with analytic and measured data points, are shown in Fig. 10, for all the tested control strategies and grid impedances.From Fig. 10(a), it is clear that, although the antiresonance Z g,1 coincides with the nonpassive zone of Y i for DS-PWM without AD, grid passive damping is high enough to prevent instability.This can also be concluded from the fact that, for that case, G i (jω res ) + G g > 0.
Nyquist plots in Fig. 10(b) and (c) predict instability for DD-AD and D-AD, respectively.

B. Experimental Grid-Connected Tests
The experimental prototype is the same used for admittance measurements.The grid is formed using Chroma 6460 programmable ac power supply and passive filters to form specific antiresonances.It was noticed that our Chroma was adding some damping at high frequencies; hence, it was decoupled with an additional LC filter with an antiresonant frequency equal to 650 Hz.This filter does not have direct implications on subsequent tests, and its purpose is to form an almost ideal voltage source at frequencies of interest.Voltage levels are reduced to V in = 250 V and v pcc = 110 V RMS to reduce any impact of unstable operation.The VSC operates as a rectifier with unity power factor, as our Chroma does not support regenerative operation.
Results of all grid-connected tests are shown in Fig. 11.The oscilloscope screen shows i L (orange) and v pcc (red).Each column of subfigures shows results for one grid antiresonance.In first two rows, results are shown for the case of MS-PWM with N = 4 and N = 16, which enable stable operation for all the tested antiresonances.Transient is tested by imposing a step reference change from 4 A to 15 A, at the peak of the fundamental component.Results for N = 4 are given to show that it is not always necessary to raise N to very high values.Note that, based on results from [54], MS-PWM with N = 4 results in a small nonpassive zone around f pwm , which coincides with the antiresonance Z g,2 .However, introduced negative damping is strongly suppressed; hence, passive grid damping is enough to compensate it.
In the third row of Fig. 11, results are shown for DS-PWM strategies that feature nonpassive zones, which coincide with the correspondingly tested antiresonances.In Fig. 11(g), strong resonant oscillations are triggered; however, as predicted by Fig. 10(a), due to sufficiently high passive damping of Z g,1 , instability is prevented.In Fig. 11(h) and (i), instability occurs for AD strategies, as predicted by Fig. 10(b) and (c).For these antiresonances, DS-PWM without AD is stable; hence, results show transients when the respective AD strategy is activated.In Fig. 11(h), overvoltage protection is triggered when DD-AD is activated, and the VSC is turned OFF.In Fig. 11(i), when D-AD is activated, protection is not triggered; however, instability clearly occurs and the controller output jitters from positive to negative saturation limits.From Fig. 11(i), it is clear that oscillations are much more emphasized around zero crossings of v pcc , where D = 0.5.This is in agreement with comments made in Section III-A regarding the impact of the operating point on the negative peak of Y i .Note that this case corresponds to the instability caused by antiresonance above the NF.Its enabling mechanism is the coupling between sampling and modulation sidebands, which allows the controller to react to a disturbance above the NF [27].In the fourth row of Fig. 11, zoomed waveforms corresponding to the third row are given to show that oscillations are triggered at the antiresonant frequency.
It can be concluded that the implementation of DS-PWM with AD may destabilize operation for certain grid antiresonances.Hence, AD does not guarantee the stability of a grid-connected VSC for any passive Z g .On the other hand, MS-PWM retains stability for all the tested cases.

V. ANALYSIS OF NOISE SENSITIVITY
A question that arises for AD strategies is their noise sensitivity [13], [54], as derivative action is applied to v pcc .The goal of this section is to quantify the noise propagation from sensing to the output current i L , for passivizing control strategies tested beforehand.As seen in Fig. 1(b), noise due to current measurement, n i , is present for all the strategies, while AD brings an additional noise path to the system from n v to i L .A detailed analysis of noise propagation for dc-dc converters is given in [52], and the same approach and assumptions are taken in this article.It is assumed that n i and n v are wide-sense stationary Gaussian white noise signals [62], with corresponding variances labeled as σ 2 i and σ 2 v .In this way, an analysis of the resulting output current variance σ 2 i L can be performed applying the theory of linear time-invariant discrete-time filtering of stochastic signals [42], [62].For all the compared strategies, noise propagation due to n i is determined by [62] where f x is the band of interest, is the squared magnitude response from n i to i L in Fig. 1(b), and S ni (f ) is the noise power spectral density of the inductor current, caused by n i .It can be seen that the spectral power density is decreased by oversampling the feedback, as determined by f s .In [52], it is shown that analyzing noise using (9) provides a good prediction for SS-PWM and DS-PWM single-loop controllers.For the case of MS-PWM, it is shown that the limiting aspect is the fact that the triangular DPWM behaves as a resampler with frequency 2f pwm , thus causing aliasing of noise power above f pwm .For this reason, digital antialiasing filters are needed to suppress the resulting noise floor below the level determined by DS-PWM [52].
For AD, noise propagation due to n v is determined by where |H nv (f )| 2 is the squared magnitude response from n v to i L and S nv (f ) is the noise power spectral density of the inductor current, caused by n v .Assuming that the two noise signals are uncorrelated, total in-band variance of i L can be obtained as an algebraic sum of ( 9) and ( 10) [62].
An interesting fact is that, for the case of D-AD, white noise n v is shaped by the AD filter before being sampled with f s .In the case of digital implementation, this corresponds to decimation from f s,1 to f s .Let us, for this reason, define |H nv (f )| 2 as a product of squared magnitude responses from n v to the decimator output and from the decimator output to i L where f s is the decimation factor and |H M :1 ad (f )| 2 is the modified squared magnitude response of the AD filter that includes the decimation impact.Given that n v is a stationary white stochastic signal, it exhibits no spectral correlation and its power spectral density is constant [62], [63].For this reason, to obtain the equivalent model valid for power spectral density analysis, it is possible to directly apply the expression for frequency-domain relation between the input and the output of a downsampler, assuming integer values of M [62], [63] For the case of DD-AD, where Individual impacts of ( 9) and ( 10) on the total variance of i L for DS-PWM are illustrated in Fig. 12, by showing values of S n ∈ {S ni , S nv }.Values of input noise powers, σ 2 i = 1.7 × 10 −3 A 2 and σ 2 v = 154 × 10 −3 V 2 , are estimated from sampled i L and v pcc , obtained while the VSC was running in open-loop condition.It can be seen that, for DD-AD, spectral power density due to voltage noise sensing is well below the one due to current noise sensing; hence, voltage noise propagation is completely masked and σ 2 i L is expected to be the same as for DS-PWM without AD.This outcome is a result of a relatively low derivative gain k ad (8).For D-AD, however, the derivative in H ad first causes the amplification of the high-frequency noise content, which then gets decimated and folded back to frequencies below the NF.This causes the in-band noise power to significantly increase.From Fig. 12, it can be seen that for N ad = 16, which is used in this article, spectral power densities due to n i and n v almost overlap.For N ad = 32, n v has a higher impact than n i .Note that for the analog implementation of D-AD, M → ∞ and the impact of n v is even higher, with the exact amount being dependent on the actual bandwidth of the noise signal.These results may seem counterintuitive, stating that oversampling causes a higher resulting noise floor.However, they come from the fact that for D-AD, H ad needed to render Y i fully passive up to the NF does not include any antialiasing filters [16].In cases where full passivity up to the NF is not crucial, an antialiasing filter can be placed between H ad and the decimator, which would result in a strong noise suppression compared to DD-AD, while offering similar passivizing effectiveness.This requires additional investigation and is a potential topic of further research.Note that this effect is closely related to conclusions brought in [52], regarding DPWM decimation for MS-PWM with PID controllers.The impact of DPWM decimation is, however, more complex to analyze as it occurs inside the closed loop; hence, it is left for future research.It should be noted that results in Fig. 12 are directly proportional to given σ 2 i and σ 2 v .Hence, for a higher ratio σ 2 v /σ 2 i , D-AD can show an increased sensitivity even for lower values of N ad .However, regarding DD-AD, spectral power density line for n v stays more than 20 dBA 2 /Hz below the one for n i , meaning that σ 2 v should increase more than 100 times to result in a comparable impact, which is not a realistic case.Hence, it can be concluded that DD-AD is not likely to be prone to noise sensitivity.
For experimental verification, reference current is set to 5 A RMS and L = 2.5 mH is used in order to capture the current signal using oscilloscope with 2 A/div scale.Larger scale was resulting in noise floor of the oscilloscope compromising i L noise measurements.Output variance, up to f pwm /2, is calculated in MATLAB using 200 ms of oscilloscope data sampled with 250 MS/s.Fundamental component and its harmonics are removed, so that distortion does not impact variance calculations.As a benchmark, simulations are implemented in Simulink using PLECS Blockset, as in [52].White noise signals are injected, with variances corresponding to measured σ 2 i and σ 2 v .Results that show variance σ 2 i L for all the tested strategies are given in Fig. 13.Experimental measurements are compared with simulations and analytics.Results in Fig. 13(a) are given for relative bandwidth α = 0.1.First, it can be seen that an excellent analytic prediction is obtained for DS-PWM without AD and DD-AD.Results show almost the same output noise power for these two strategies.As both the cases feature the same magnitude response |H ni |, it is concluded that the voltage noise propagation for DD-AD is completely masked by the stronger current noise propagation, which is consistent with Fig. 12.For D-AD, analytics predict 2.6 dBA 2 noise power increase, while simulations and experiments show approximately 2.4 dBA 2 and 2.5 dBA 2 increase, respectively, which supports the validity of the model (12).As discussed above, for setups that feature a higher ratio σ 2 v /σ 2 i , higher noise sensitivity of D-AD is expected.Equation ( 9) predicts that, compared to DS-PWM, implementing MS-PWM with N = 16 would decrease the output noise power by, at least, a factor 8. The exact amount depends on the impact of delay reduction on |H ni |.However, as analyzed in [52], for N > 2 and controllers with a nearly constant highfrequency gain, the power spectral density remains almost unaffected due to aliasing caused by DPWM resampling effects; hence, additional high-frequency digital filtering is needed to enable a strong in-band noise suppression.For this article, three digital low-pass filters (DLPFs) are tested to observe their impact on noise attenuation.DLPFs are implemented as first-order filters with cutoff frequencies ω f = {2, 1.5, 1} • ω pwm and are discretized using the bilinear transform.It is important to notice that these DLPFs do not directly attenuate noise power in the observed frequency window.The design goal is to suppress noise power above f pwm with as little impact on dynamic performance. 4DLPFs with lower cutoff frequencies bring more filtering at f > f pwm ; hence, corresponding results are expected to approach the value determined by (9).This trend is confirmed by results in Fig. 13(a), where the analytic prediction is only outlined as it does not model the impact of DPWM decimation.Again, simulations are in a good match with experiments.Note that, even without DLPF, noise power is reduced compared to DS-PWM without AD by 2.2 dBA 2 , due to a higher phase margin, which brings additional filtering by |H ni |.
Same conclusions are brought based on results shown in Fig. 13(b), given for α = 0.2, except that all variances are increased due to the higher bandwidth setting.It can also be seen that a higher noise suppression of MS-PWM without DLPF relative to DS-PWM without AD is obtained, equal to approximately 5 dBA 2 , due to a higher phase margin difference between the two.Finally, compared to α = 0.1, for MS-PWM, there is a bigger mismatch between measured noise power and the analytic prediction that does not consider DPWM resampling effects.This is explained by the fact that the increased proportional gain results in a higher amplification of noise power above f pwm , which gets decimated by the DPWM.Hence, same digital filters become less effective in suppressing the aliasing impact.
The main conclusion of this section is that implementing AD does not necessarily result in excessive noise sensitivity.Still, MS-PWM is able to offer additional noise suppression without compromising the passivation effectiveness.

VI. CONCLUSION
This article has provided a comparison in the effectiveness and noise sensitivity of control strategies used to render the input admittance of grid-connected VSCs passive.MS-PWM was compared against typically used derivative-based AD methods.It was shown that MS-PWM outperforms AD in terms of effectiveness, as AD brings a negative impact around and above the NF.Grid-connected operation was tested for different antiresonances, and it was shown that only MS-PWM enables robust stability.Analytic modeling of noise propagation was developed, and it was shown that AD strategies are not highly sensitive.Still, MS-PWM outperforms AD in this respect as well.The study shown in this article further pointed out that MS-PWM is a very viable solution for achieving high bandwidths, while guaranteeing stability of current-controlled grid-connected VSCs.

Fig. 1 .
Fig. 1.Illustration of a grid-following VSC.(a) Schematic representing connection to a grid.(b) Multirate current control system with a feedforward action.Black color depicts blocks executed at current sampling rate.Red color depicts feedforward execution rate.Continuous-time domain is shown in blue.Signals n i and n v represent measurement noise for current and voltage sensing, respectively.

Fig. 2 .
Fig. 2. S-domain representation of a grid-following VSC.(a) Current control system and grid connection.(b) Norton equivalent circuit used for impedancebased stability assessment.

Fig. 5 .
Fig. 5. Relative conductance for the case of D-AD: N = 2, N ad = 16, H ad = k ad T s,1 (1 − z −1 1 ), L = 1.5 mH, and α = 0.2.The vertical line marks the switching (and the Nyquist) frequency.Orange-dashed line shows the impact of including an additional filter H bq in cascade with the derivative gain [16].

TABLE I VSC
AND CONTROL PARAMETERS

TABLE II COMPARISON
OF CONDUCTANCE MINIMUMS AND FREQUENCIES FOR WHICH THEY ARE FOUND: min{R e {Y i }} [%] @ f min [kHz].