# Active Damping Strategies for Mitigating Resonances in Offshore Wind Farms: A Comparative Study of PWM and SHEPWM

Iñigo Pascual, Naroa Ramos, Eugenio Gubia, Jesús López, Javier Samanes

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Iñigo Pascual, Naroa Ramos, Eugenio Gubia, Jesús López, Javier Samanes. Active Damping Strategies for Mitigating Resonances in Offshore Wind Farms: A Comparative Study of PWM and SHEPWM. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0294 . hal-05082045

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# Active Damping Strategies for Mitigating Resonances in Offshore Wind Farms: A Comparative Study of PWM and SHEPWM

Iñigo Pascual, Naroa Ramos, Eugenio Gubia, Jesús López and Javier Samanes

Institute of Smart Cities (ISC), Public University of Navarre (UPNA)

Campus of Arrosadia, 31006

Pamplona, Spain

inigo.pascual@unavarra.es

## ACKNOWLEDGMENT

This work is part of the project PID2022-142791OB-I00 and TED2021-132604B-I00, funded by MCIN/AEI/10.13039/501100011033, by the FSE+, by ERDF/UE and by the European Union NextGenerationEU/PRTR. The authors want to acknowledge Ingeteam Power Technology for its financial and ongoing support.

**Index Terms**—Active Damping, Converter control, Stability Analysis, Grid-Connected Converter, Selective Harmonic Elimination PWM.

**Abstract**—This paper presents an active damping strategy to mitigate resonances in offshore wind farms caused by the parasitic capacitance of submarine cables, which challenge system stability and grid code compliance. The proposed method emulates a virtual impedance damping the wide range of resonances that appear in offshore wind farms, depending on its configuration and number of turbines connected. Its effectiveness is evaluated for two modulation techniques: PWM and selective harmonic elimination PWM. PWM generates predictable harmonic content outside the resonance range, enabling robust damping of resonances. In contrast, selective harmonic elimination PWM—though optimized to eliminate the harmonics in the resonance frequency range—introduces distortion in this range due to modulation-control interactions in closed-loop operation. This distortion demands excessive corrective action from the AD, limiting its practicality. The strategy and its results are validated through simulation.

## I. INTRODUCTION

Offshore wind energy has experienced exponential growth in recent years [1]. Offshore wind power plants (WPPs) harness the more consistent and stronger winds found at sea, obtaining a higher capacity factor than in onshore WPPs. However, technical challenges emerge

that must be addressed to ensure reliable and stable operation [2].

Offshore WPPs are connected to the onshore grid through long submarine cables due to their remote location. The predominant capacitive behavior of these cables resonates with the equivalent inductance of the park, formed by transformers and filters [3]. This resonances are detrimental to system stability due to their interaction with the control, as well as to grid code compliance, as they amplify the content at the frequency at which the resonance occurs.

The resonance frequency can vary depending on the connection or disconnection of wind turbines (WTs), as the equivalent impedance of the system changes [3]. Additionally, the equivalent grid impedance also changes based on operating conditions, such as the power output of the turbines, further impacting the possible range of resonance frequencies [4].

Several authors have studied and modeled this resonances in offshore WPPs. In [4], the output impedance of the WT is considered purely inductive and approximately equal to the transformer reactance. The resonance occurs within a frequency range from a few hundred up to 800 Hz, depending on the operational state of the WPP [3]–[6]. Due to the high number of WTs in a WPP, several medium-voltage buses connect them. Under favorable conditions, with balanced buses, an amplification factor of 10:1, between the voltage of the converter and the medium-voltage buses, can be reached at the resonance frequency. Conversely, in the most critical unbalanced scenario between two buses, an 80:1 amplification factor would be achieved at the resonance frequency [4]. The distortion generated by each high-power converter must be significantly minimized to ensure compliance with stringent grid codes that impose emissions limits in the point of common coupling (PCC) [7], [8].
In offshore applications with high-power electronic
converters, optimal pulse-width modulation can be em-
ployed to minimize switching losses by reducing the
minimum number of switching transitions [9], [10]
to meet the grid harmonic emission limits [7]. While
selective harmonic elimination pulse-width modulation
(SHEPWM) can eliminate the harmonics in the range of
resonance frequencies, if the resonances are not damped,
any harmonic in the grid can be amplified in the wind
farm grid.

Additionally, instability problems can still appear by
the interaction between the inherent delays introduced
by the closed-loop control of the converter and the wind
farm resonances [11]. To avoid these problems, passive
damping (PD) and active damping (AD) techniques can
be applied [12], [13]. PD strategies involve installing
passive filters [14], [15], which increase the system cost
and produce additional power losses to damp resonances.

This article focuses on damping grid resonances
through converter control, using AD techniques [16],
[17]. These techniques are based either on the modifica-
tion of both the magnitude and phase of the forward-path
[18], or on the emulation of impedances [19]–[22].

Among the techniques based on modifying the mag-
nitude of the forward-path, the simplest solution would
be to drastically reduce the proportional gain of the
controller. However, in a WPP this measure would
excessively reduce bandwidth and disturbance rejection
capability. If the resonance frequency were known, an-
other option would be to introduce a notch filter, which
would greatly reduce the impact of the resonance on the
magnitude of the forward-path [18].

The phase of the forward-path can be modified by
AD techniques based on filtering the converter voltage
reference computed by the controller. With a low-pass
filter, a phase reduction is achieved, while a lead-lag

filter would result in a phase increase [18]. The aim of
modifying the control loop phase is to ensure system
stability. However, these filters negatively impact phase
margins, thereby limiting the achievable bandwidth.

Other AD strategies are based on emulating a virtual damping resistor, which becomes a virtual impedance due to control delays [19], [20]. These strategies rely on current or voltage feedback from the capacitor. In an offshore WPPs, access to capacitor current is not possible, as it is a parasitic capacitance of the power cable. Therefore, the voltage at the PCC must be fed back.

Among the AD strategies based on the voltage of
the PCC, this paper presents the study and adaptation
of the AD technique previously developed in [22]. This
AD strategy adapts the stability range to the range of
resonance frequencies present in the system, a suitable
approach for offshore WPP given the wide variability of
the resonance frequency.

Therefore, this paper analyzes the performance of
this AD technique in offshore wind farm applications,
addressing the aforementioned resonance issues. The
objective is to compare the behavior of the AD when
modulating with PWM and SHEPWM, evaluating how
the modulation affects the robustness and effectiveness
of the strategy.

II. SYSTEM DESCRIPTION AND STABILITY ANALYSIS

A. System Description and Modeling

To conduct the analysis of resonances in offshore
WPPs, which mainly depend on the number of intercon-
nected turbines, their output power, and the impedance
of the grid, an equivalent WT and a simplified model
of the WPP electrical system is used, as illustrated in
Fig. 1. The grid inductance, $L_g$, and capacitor, $C$, are
adjusted to reproduce the different resonance frequencies

<figure><img src="image_2.png" /><figcaption>Fig. 1. Grid-connected NPC Power converter.</figcaption></figure>
and amplification phenomena observed in the grid of an offshore WPP.

Offshore wind turbines based on permanent magnet generators are the dominant technology, so a full converter architecture is considered. Therefore, the analysis focuses on the grid-side converter represented in Fig. 1 (red), which is a three-level neutral point clamped (NPC) voltage source converter (VSC) with a DC bus voltage, $v_{dc}$, assumed to be constant and balanced. This VSC is connected to the PCC through an inductance $L$, representing both the converter's inductance and the step-up transformer's leakage inductance used to connect the WT to the medium-voltage wind farm grid. Additionally, in Fig. 1 the converter's control scheme (blue), which is executed on a digital signal processor (DSP), is represented. The modulator (orange) determines the switching state of the IGBTs.

Different factors such as the connection or disconnection of WTs, their output power or the balance between feeders, influence the WPP configuration. These variables, as well as the variation of the unknown $L_g$, modify both the amplitude and the frequency of the resonance, as the equivalent impedance of the system changes. The range of resonance frequencies reported in the literature varies between the 8th and 17th harmonic orders, being $f_r^l$ the lowest frequency and $f_r^h$ the highest frequency in the range [4]–[6]. According to [4], [5] in certain offshore WPP configurations, the most critical resonance occurs at the 11th harmonic order, at $f_r^c$ — critical frequency resonance—, where amplifications of up to 80 V/V are reached at the medium-voltage buses. In this paper, three different grid configuration to reproduce the grid resonances at $f_r^l$, $f_r^c$ and $f_r^h$ are used for the stability analysis. To ensure robust operation of the AD strategy, the maximum amplitude resonance, 80 V/V, is modeled between the converter and the PCC voltage for the entire frequency range as the worst-case scenario.

To evaluate the system's stability, all components must be modeled in the same reference frame. In this case, the $\alpha\beta$ reference frame is used. Since the control implemented is the same in both axes, a generic nomenclature without subscripts is used in block diagram figures and equations, which is valid for both $\alpha$ and $\beta$. The converter current is regulated using a proportional resonant (PR) controller, to ensure zero-error current tracking at the fundamental frequency $f_0$, defined as

$$PR(z) = K_p + \frac{a_0^R + a_2^R z^{-2}}{1 + b_1^R z^{-1} + b_2^R z^{-2}}, \quad (1)$$

where

$$a_0^R = \frac{2K_pT_s}{T_n(4+T_s^2\omega_0^2)}, \quad a_2^R = -a_0^R,$$

$$b_1^R = \frac{2T_s^2\omega_0^2-8}{4+T_s^2\omega_0^2} \quad \text{and} \quad b_2^R = \frac{4+T_s^2\omega_0^2}{4+T_s^2\omega_0^2},$$

$T_s$ being the sampling period. The parameter $\omega_0 = 2\pi f_0$, while $K_p$ and $T_n$ are the PR's parameters adjusted to provide the desired dynamics in the NPC converter. The current error, $\epsilon(z)$, is obtained from the difference between the reference current setpoint, $i_{ref}(z)$, and the measured and filtered current at the converter output, $i_{conv,f}(z)$. Both currents and voltages measurements are filtered by an analog low-pass filter, $LPF(s) = 1/(\tau s + 1)$ with a cutoff frequency of $2f_s/5$ to eliminate noise and switching harmonics. The plant model, $G(s)$, is composed of the converter and the simplified grid, and it relates the converter voltage to its current.

$$G(s) = \frac{i_{conv}(s)}{v_{conv}(s)} = \frac{L_g C s^2 + 1}{L L_g C s^3 + (L + L_g)s} \quad (2)$$

All control operations are executed in a DSP as shown in Fig. 1 (blue), sampling at each sampling period. To analyze stability at different resonance frequencies, especially as the Nyquist frequency limit is approached, it is crucial to have an accurate model in that frequency range. For this reason, a discrete-time domain model of the system is developed.

In Fig. 2, the block diagram of the converter current control loop is presented, which illustrates the detailed structure of the control system. The digital implementation of the control system introduces a sampling time delay caused by computation, modeled as $z^{-1}$. Besides that, the delay introduced by keeping the converter voltage reference constant during a $T_s$ is modeled by discretizing the system plant model, $G(s)$, and the $LPF(s)$, using the Zero-Order Hold (ZOH) method, resulting in $G_{cc}(z)$.

## B. Stability Analysis

The stability of the system is evaluated by observing the frequency response of the open-loop transfer function, from $\epsilon(z)$ to $i_{conv,f}(z)$, in a Bode diagram. The system and controller parameters are reported in Table I.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Block diagram of the NPC current control loop.</figcaption></figure>
From the Bode diagram in Fig. 3, it can be seen that instability occurs when the resonance frequency occurs for values higher than $f_r^c$ (blue). The system is not stable within the required frequency range due to the presence of a -180° phase crossing when the gain is over 0 dB. As stated by the Bode Generalized Criterion [23], the closed-loop system will have 2 unstable poles.

System instability within the possible resonance frequency range highlights the need for an active damping strategy. The aim of this AD strategy is to damp the closed-loop poles so that the system becomes stable.

<table><caption>TABLE I
SYSTEM PARAMETERS</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td colspan="3">Grid</td></tr><tr><td>Grid fundamental frequency</td><td>f<sub>0</sub></td><td>50 Hz</td></tr><tr><td>Rated grid voltage</td><td>U<sub>R</sub></td><td>3100 V</td></tr><tr><td colspan="3">Power converter</td></tr><tr><td>Rated power</td><td>P<sub>R</sub></td><td>12 MW</td></tr><tr><td>Rated current</td><td>I<sub>R</sub></td><td>2483 A</td></tr><tr><td>Converter &amp; Transformer inductance</td><td>L</td><td>0.196 p.u.</td></tr><tr><td>Grid inductance</td><td>L<sub>g</sub></td><td>0.100 p.u.</td></tr><tr><td>Commutation angles per quarter wave</td><td>N</td><td>11</td></tr><tr><td colspan="3">Control strategy</td></tr><tr><td>Sampling frequency</td><td>f<sub>s</sub></td><td>4000 Hz</td></tr><tr><td>Low-pass current filter time constant</td><td>τ</td><td>99.5 μs</td></tr><tr><td>PR controller proportional gain</td><td>K<sub>p</sub></td><td>0.25</td></tr><tr><td>PR controller integral time constant</td><td>T<sub>n</sub></td><td>0.02 s</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Bode diagram of the open-loop transfer function.</figcaption></figure>

### III. AD STRATEGY

#### A. Fundamentals of the AD Strategy

The AD strategy is based on emulating a resistance in parallel with the grid impedance to damp the resonances. Since the capacitor current measurement is not available, the PCC voltage is used and the derivative is applied to estimate the current through the capacitor [22]. For this reason the block diagram in Fig. 2 is modified as shown in Fig. 4. An additional block, $G_{vc}(z)$, is included, which relates the converter voltage, $v_{conv}(z)$, to the filtered PCC voltage, $v_{pcc,f}(z)$. As for $G_{cc}(z)$, the ZOH discretization method is used to model $G_{vc}(z)$. $v_{pcc,f}(z)$ is derivated using $Der(z)$ as indicated in Fig. 2 to estimate the capacitor current.

Despite the AD strategy aiming to implement a resistance, the effect of delays actually results in emulating an impedance. The resistive component of the emulated impedance, $Re(Z_{AD})$, changes its sign depending on the frequency. When the sign change occurs in the frequency range where system resonance can occur, the AD strategy, instead of stabilizing, tends to destabilize at certain resonance frequencies [19], [21]. Therefore, it is crucial to adjust the delay in the emulation of the impedance to achieve robust damping. As a first step in determining the required delay, it is necessary to analyze the factors that influence the real part of the emulated impedance.

As illustrated in Fig. 4, the emulated virtual impedance depends on the feedback elements and the delays present in the control loop. Consequently, the sign change of the real part of the impedance depends on the phase contributions from the delay $z^{-1}$, $LPF(z)$ and the ZOH. To prevent the sign change of the real part of the emulated impedance within the range of possible resonance frequencies, it is necessary to adjust the phase. This is accomplished through $H_{AD}(z)$, which comprises two components as expressed Eq. (3).

$$H_{AD}(z) = D_{AD}(z) \cdot DF(z) \quad (3)$$

The first term, $D_{AD}(z)$, introduces an additional delay modeled for y sample periods by means of linear

<figure><img src="image_5.png" /><figcaption>Fig. 4. Block diagram of the proposed AD control strategy.</figcaption></figure>
interpolation. As shown in Eq. (4), $y_i$ denotes the integer delay component, while $y_f$ represents its fractional part [24].

$$D_{AD}(z) = ((1 - y_f) + y_f z^{-1}) z^{-y_i} \quad (4)$$

The second term, $DF(z)$, incorporates a series of digital filters designed to constrain the AD action to the targeted frequency band, thereby suppressing undesired amplification outside this range.

Therefore, to adjust the AD strategy and enhance its robustness, the following procedure is implemented:

1) Define the operational frequency range and adjust the filters to prevent amplification of undesired content.

2) Tune the delay $y$ via Eq. (4) to maintain a constant sign of $Re(Z_{AD})$ within the resonance frequency range.

3) Select the damping gain $k_{AD}$ to ensure stability over all potential resonance frequencies.

## B. AD Strategy Adjustment

To begin the adjustment of the strategy, it is necessary to determine the frequency range in which the AD has to damp the wind farm resonances. Between $f_r^l$ and $f_r^h$ the filters should maintain a magnitude as close as possible to unity to allow the resonances within this range to pass through and be actively damped.

Thus, a band-pass filter, $BPF(z)$, is adjusted to define the operational range, with its lower cutoff frequency set at $f_r^l/3$ to ensure the proper attenuation outside the resonance range. Its higher cutoff frequency is set at $2f_r^h$, slightly below the Nyquist frequency, to minimize noise at higher frequencies. However, at $f_0$, the filter exhibits an attenuation of -9 dB, which may require significant AD action.

To address this, a notch filter, $NF(z)$, is introduced at $f_0$ to suppress the fundamental component [25]. This reduces the demand on the AD, preventing it from acting unnecessarily on the fundamental frequency while maintaining its focus on damping resonances within the desired range. The expression for the notch filter is

$$NF(z) = \frac{a_0 + a_1 z^{-1} + a_2 z^{-2}}{b_0 + b_1 z^{-1} + b_2 z^{-2}}, \quad (5)$$

where $a_0 = 1+2d_1\lambda+\lambda^2$, $a_1 = 2\lambda^2-2$, $a_2 = 1-2d_1\lambda+\lambda^2$, $b_0 = 1+2d_2\lambda+\lambda^2$, $b_1 = a_1$, $b_2 = 1-2d_2\lambda+\lambda^2$ and $\lambda = \omega_{notch}T_s/2$. The parameter $\omega_{notch} = 2\pi f_0$ represents the central angular frequency that is filtered, while $d_1 = 0.004$ and $d_2 = 0.02$ define the frequency range attenuated.

The $BPF(z)$ and $NF(z)$ together form the digital filter $DF(z)$ defined in Eq. (6). As previously mentioned, it is one of the components of $H_{AD}(z)$. Once $DF(z)$ is adjusted, the next step involves tuning the delay $D_{AD}(z)$ to ensure robust damping across the resonance frequency range.

$$DF(z) = BPF(z) \cdot NF(z) \quad (6)$$

Although in the original strategy the delay is calculated so that the virtual impedance exhibits a purely resistive behavior at the center of the resonance frequencies, $f_r^R = (f_r^l + f_r^h)/2$, the reduced damping of the system's resonant poles for the cases of higher resonance frequencies requires a slight shift in that direction. Thus, this shift is 10% of the amplitude of the frequency range of resonances determined by $f_r^l$ and $f_r^h$.

As previously mentioned, the virtual impedance depends on the delay $z^{-1}$, $LPF(z)$, the ZOH and $H_{AD}(z)$. Additionally, since the derivative is implemented in the DSP, it is not an ideal derivative, and its phase differs from the expected 90° at all frequencies. As the Tustin discretization is used, this phase difference must be accounted for when adjusting $D_{AD}(z)$. The extra delay to be introduced in $D_{AD}(z)$ can be calculated by ensuring that Eq. (7) equals $\pi$ when $\omega = 1.1 \cdot 2\pi f_r^R$, being $z = e^{j\omega T_s}$.

$$\varphi[Z_{AD}(z)] = -\varphi[z^{-1}] - (\varphi[Der(z)] - 90^{\circ}) - \varphi[LPF(z)] - \varphi[H_{AD}(z)]. \quad (7)$$

Once the $H_{AD}(z)$ term has been fully adjusted, a negative emulated resistance is achieved over the whole range of possible resonance frequencies. To obtain the desired damping action, the sign of the measured capacitor voltage feedback has to be changed to positive.

The last adjustment is made on the damping gain $k_{AD}$. The goal is to achieve maximum damping across the entire range of resonance frequencies. Each resonance frequency has an optimal $k_{AD}$ value. However, $k_{AD}$ is selected to provide the greatest damping in the most unstable case, which is for the highest resonance frequency, i.e. $k_{AD} = k_{AD}^{opt}(f_r^h)$, as long as stability is guaranteed for all the possible resonances. In Fig. 5, the resonant poles in the complex z-plane for different resonance frequencies and the different $k_{AD}$ values shown in the legend are observed. The asterisk markers correspond to the case of $f_r^l$, the squares to $f_r^c$ and the 'x' to $f_r^h$. A value for $k_{AD}$ of $14.10^{-5}$ (yellow marker) is selected, guaranteeing stability in the range of resonance frequencies and maximum damping for $f_r^h$. It should be noted that for each case studied, the amplification gain
at the resonance frequency corresponds to the worst-case scenario, 80:1, thus demonstrating the robustness of the strategy's adjustment. A summary of the AD parameters established for each resonant frequency, along with the damping of the system's resonant poles, is presented in Table II.

## IV. INFLUENCE OF THE MODULATION TECHNIQUE

### A. PWM and SHEPWM Implementation

In this section, the influence of the modulator on the adjusted damping strategy is analyzed. With this purpose, the simulation results for PWM and SHEPWM are performed and the results obtained compared.

The PWM technique used in this work is Level-Shifted Sinusoidal PWM (LS-SPWM). This method employs two in-phase triangular carriers for each phase of the converter. The PWM technique allows obtaining a control average voltage during a sampling period by controlling the pulse width. With PWM, the resulting harmonic content is known and appears grouped around the switching frequency and its multiples. Therefore, this modulation does not generate harmonics in the resonance frequency range if the switching frequency is greater than $f_r^h$. In Fig. 6 the implementation of the PWM is

<figure><img src="image_6.png" /><figcaption>Fig. 5. Closed-loop resonant poles of the system for the three offshore grid configurations analyzed.</figcaption></figure>

<table><caption>TABLE II
AD PARAMETERS AND DAMPING IN THE POLES AT THE RESONANCE FREQUENCY</caption><thead><tr><th>f<sub>r</sub></th><th>k<sub>AD</sub><sup>opt</sup></th><th>ξ<sup>opt</sup></th><th>k<sub>AD</sub></th><th>ξ</th><th>Delay y</th></tr></thead><tbody><tr><td>f<sub>r</sub><sup>l</sup></td><td>1·10<sup>-5</sup></td><td>0.0177</td><td rowspan="3">14·10<sup>-5</sup></td><td>0.0088</td><td rowspan="3">0.0999</td></tr><tr><td>f<sub>r</sub><sup>c</sup></td><td>20.5·10<sup>-5</sup></td><td>0.0715</td><td>0.0536</td></tr><tr><td>f<sub>r</sub><sup>h</sup></td><td>14·10<sup>-5</sup></td><td>0.0324</td><td>0.0324</td></tr></tbody></table>

illustrated, where the voltage to be imposed, coming from the controller and the required AD action, is scaled and then, after adding the third harmonic, compared with the carrier waves. In this way, the comparison of the carrier waves with the modulating signals from the DSP determines the switching commands, which are sent to the converter.

Regarding SHEPWM, it is a modulation technique used to eliminate specific harmonics by appropriately selecting the switching angles. The maximum harmonic order that can be eliminated depends on the number of angles per quarter period of the fundamental, $N$. For the case study presented $N = 11$. Given that even harmonics are eliminated due to symmetry, and triplen harmonics do not generate currents because neutral is not connected, harmonics up to the 31st order are eliminated. This means that harmonics at the system's resonance frequency are also not expected. In Fig. 7 is shown a schematic of the SHEPWM modulation implementation. Every sampling period, the magnitude and phase of the converter's reference voltage in $α$ and $β$ are computed. With these values, the operating point or modulation index (MI) can be calculated, and the lookup table (LUT) information can be accessed. The LUT contains the pre-calculated switching angles for different MIs. These angles obtained in the DSP are sent to the modulator, which, when compared with a carrier, in this case, a sawtooth wave, provides the switching commands.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Implementation schematic for PWM.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Implementation schematic for SHEPWM.</figcaption></figure>
## B. Simulations results for PWM

Once the strategy is adjusted, it is validated using Matlab/Simulink creating a model using the Simscape Power Systems library. The model includes a S-Function where the DSP control is implemented in C language. In this simulation environment, the adjusted control strategy is tested for the three resonance frequencies studied so far. The system has been adjusted to ensure its stability across the entire frequency range, as well as to damp resonances. Therefore, the following test is conducted to demonstrate the effectiveness of the strategy for both PWM and SHEPWM.

The test begins with the system operating in closed-loop mode, with the AD strategy activated and an active reference setpoint at 80% of the nominal value. At time $t = 0$ s, a power step from 80% to 100% is introduced to evaluate the control dynamics and robustness against abrupt changes in the power setpoint. Once the system stabilizes, at time $t = 0.3$ s, the AD strategy is deactivated, making it possible to observe the system's evolution. In scenarios where the system was stable without the need for AD, it is expected to continue providing adequate current tracking; whereas, in cases where the system was unstable, it is expected to evolve towards instability.

The results obtained from this test with PWM are presented in Fig. 8. For this purpose, the time evolution of the converter current (blue) along with its reference (pink) for the $\alpha$ axis has been plotted for each case study (Case 1: $f_r^l$; Case 2: $f_r^c$; and Case 3: $f_r^h$), allowing the observation of the reference current tracking. Additionally, the voltage demanded by the AD branch in the $\alpha\beta$ axes is shown to clearly observe the moment of strategy disconnection.

The system for Case 1: $f_r^l$ was stable without the need to implement the AD strategy. This is observed when the AD is disconnected at $t = 0.3$ s, maintaining the reference current tracking. In this case, once the steady-state regime at rated power is achieved ($t \approx 0.25$ s), the voltage demand of the AD branch is slightly less than 20 V peak. The system without active damping was also stable, so, in this case the AD only increases the damping of the resonance.

On the other hand, in the cases for $f_r^c$ and $f_r^h$, the systems without AD were unstables. In Case 2: $f_r^c$ and more clearly in Case 3: $f_r^h$ in Fig. 8, it can be observed that the system is stable with AD, thus achieving the strategy's objective. However, it can be seen after the AD disconnection that the system becomes unstable, ceasing to follow the reference setpoint.

<figure><img src="image_9.png" /><figcaption>Fig. 8. Current tracking and AD voltage demand for 3 simulation cases modulating with PWM.</figcaption></figure>

This allows for the validation of the model in the discrete domain with which the strategy adjustment has been carried out and ensures the correct functioning of the strategy for PWM modulation.

## C. Simulations results for SHEPWM

The results of the same test for the converter with SHEPWM are shown in Fig. 9. As previously mentioned, this modulation is based on loading a switching pattern in each $T_s$ according to the MI. Due to the substantial fluctuation of the MI within a period, highly incremented by the AD strategy, different patterns are loaded, causing the harmonic content to vary significantly from the theoretical one. This introduces harmonic content in the range of possible resonance frequencies of the offshore grid. Consequently, the AD requires significantly more action, which in turn leads to even greater oscillations in the MI.
<figure><img src="image_10.png" /><figcaption>Fig. 9. Current tracking, AD voltage demand and modulation index for the simulation case modulating with SHEPWM.</figcaption></figure>

These fluctuations in the MI values introduce current and voltage distortion, which is fed back into the system, causing further perturbations in the PCC voltage. The AD strategy, which derives the voltage harmonic content, amplifies its effect, resulting in an increased oscillation of the MI. This feedback loop intensifies the distortion and ultimately causes converter saturation leading to converter disconnection.

## V. CONCLUSION

This article implements and adjusts a robust AD strategy previously developed in the literature to address a critical challenge in high-power offshore wind farms: the high-magnitude resonances caused by the parasitic capacitance of submarine cables. These resonances, which appear across a wide frequency range depending on the configuration and number of turbines connected, can compromise system stability and compliance with grid codes, highlighting the need to implement AD techniques.

The proposed AD strategy is effective when using PWM, as it generates predictable harmonic content outside the frequency resonance range, ensuring stable operation. However, with SHEPWM, an interaction between the modulation and control distorts the harmonic content and creates a feedback loop that amplifies this distortion, leading to converter disconnection.

The results highlight that while the AD strategy is effective in damping resonances for a small-signal perspective, its performance under real operating conditions depends heavily on the modulation technique used. PWM ensures the robustness of the AD strategy, whereas SHEPWM, due to its inherent feedback issues, limits the applicability of the strategy.

## REFERENCES

[1] R. Williams and F. Zhao, *Global offshore wind report 2024*. Global Wind Energy Council, 2024.

[2] S. W. Ali, M. Sadiq, Y. Terriche, S. A. R. Naqvi, L. Q. N. Hoang, M. U. Mutarraf, M. A. Hassan, G. Yang, C.-L. Su, and J. M. Guerrero, "Offshore wind farm-grid integration: A review on infrastructure, challenges, and grid solutions," *IEEE Access*, vol. 9, pp. 102811–102827, 2021.

[3] E. Ebrahimzadeh, F. Blaabjerg, X. Wang, and C. L. Bak, "Harmonic stability and resonance analysis in large pmsg-based wind power plants," *IEEE Transactions on Sustainable Energy*, vol. 9, no. 1, pp. 12–23, 2018.

[4] E. V. Larsen and J. Sun, "Managing resonances in windfarm collector systems," *20th International Workshop on Large-Scale Integration of Wind Power into Power Systems as well as on Transmission Networks for Offshore Wind Power Plants (WIW 2021)*, vol. 2021, pp. 335–339, 2021.

[5] M. Bradt, B. Badrzadeh, E. Camm, D. Mueller, J. Schoene, T. Siebert, T. Smith, M. Starke, and R. Walling, "Harmonics and resonance issues in wind power plants," *2011 IEEE Power and Energy Society General Meeting*, pp. 1–8, 2011.

[6] M. Cheah-Mane, L. Sainz, J. Liang, N. Jenkins, and C. E. Ugalde-Loo, "Criterion for the electrical resonance stability of offshore wind power plants connected through hvdc links," *IEEE Transactions on Power Systems*, vol. 32, no. 6, pp. 4579–4589, 2017.

[7] I. E. Kommission and IEC, *Electromagnetic compatibility (EMC). 3-6, Limits - Assessment of emission limits for the connection of disorting installations to MV, HV and EHV power systems : Edition 2.0 2008-02*. IEC, 2008.

[8] M. H. J. Bollen, L. Yao, S. K. Rönnberg, and M. Wahlberg, "Harmonic and interharmonic distortion due to a windpark," in *IEEE PES General Meeting*, pp. 1–6, 2010.

[9] H. S. Patel and R. G. Hoft, "Generalized techniques of harmonic elimination and voltage control in thyristor inverters: Part i - harmonic elimination," *IEEE Transactions on Industry Applications*, vol. IA-9, pp. 310–317, 1973.

[10] R. G. Hoft and H. S. Patel, "Generalized techniques of harmonic elimination and voltage control in thyristor inverters: Part ii - voltage control techniques," *IEEE Transactions on Industry Applications*, vol. IA-10, pp. 666–673, 1974.

[11] J. Wang, J. D. Yan, L. Jiang, and J. Zou, "Delay-dependent stability of single-loop controlled grid-connected inverters with lcl filters," *IEEE Transactions on Power Electronics*, vol. 31, no. 1, pp. 743–757, 2016.

[12] M. Büyük, A. Tan, M. Tümay, and K. Çağatay Bayındır, "Topologies, generalized designs, passive and active damping methods of switching ripple filters for voltage source inverter: A comprehensive review," *Renewable and Sustainable Energy Reviews*, vol. 62, pp. 46–69, 2016.
[13] C. C. Gomes, A. F. Cupertino, and H. A. Pereira, "Damping techniques for grid-connected voltage source converters based on lcl filter: An overview," *Renewable and Sustainable Energy Reviews*, vol. 81, pp. 116-135, 2018.

[14] A. B. Nassif, W. Xu, and W. Freitas, "An investigation on the selection of filter topologies for passive filter applications," *IEEE Transactions on Power Delivery*, vol. 24, no. 3, pp. 1710-1718, 2009.

[15] W. Xu, T. Ding, X. Li, and H. Liang, "Resonance-free shunt capacitors—configurations, design methods and comparative analysis," *IEEE Transactions on Power Delivery*, vol. 31, no. 5, pp. 2287-2295, 2016.

[16] M. Liserre, R. Teodorescu, and F. Blaabjerg, "Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values," *IEEE Transactions on Power Electronics*, vol. 21, no. 1, pp. 263-272, 2006.

[17] J. Xu, S. Xie, and T. Tang, "Active damping-based control for grid-connected lcl-filtered inverter with injected grid current feedback only," *IEEE Transactions on Industrial Electronics*, vol. 61, no. 9, pp. 4746-4758, 2014.

[18] J. Dannehl, M. Liserre, and F. W. Fuchs, "Filter-based active damping of voltage source converters with lcl filter," *IEEE Transactions on Industrial Electronics*, vol. 58, no. 8, pp. 3623-3633, 2011.

[19] 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 Transactions on Power Electronics*, vol. 29, no. 7, pp. 3414-3427, 2014.

[20] X. Wang, F. Blaabjerg, and P. C. Loh, "Virtual rc damping of lcl-filtered voltage source converters with extended selective harmonic compensation," *IEEE Transactions on Power Electronics*, vol. 30, no. 9, pp. 4726-4737, 2015.

[21] X. Wang, F. Blaabjerg, and P. C. Loh, "Virtual rc damping of lcl-filtered voltage source converters with extended selective harmonic compensation," *IEEE Transactions on Power Electronics*, vol. 30, no. 9, pp. 4726-4737, 2015.

[22] J. Samanes, A. Urtasun, E. Gubia, and A. Petri, "Robust multisampled capacitor voltage active damping for grid-connected power converters," *International Journal of Electrical Power And Energy Systems*, vol. 105, pp. 741-752, 2019.

[23] A. Urtasun, J. Samanes, E. L. Barrios, and P. Sanchis, "Control design and stability analysis of power converters: The discrete generalized bode criterion," *IEEE Access*, vol. 9, pp. 37840-37854, 2021.

[24] T. Laakso, V. Valimaki, M. Karjalainen, and U. Laine, "Splitting the unit delay [fir/all pass filters design]," *IEEE Signal Processing Magazine*, vol. 13, no. 1, pp. 30-60, 1996.

[25] S. Nishimura, K. Hirano, and S. K. Mitra, "Design of digital notch filters," *IEEE Transactions on Communications*, vol. 22, pp. 964-970, 1974.