# Passivity Analysis in Power Electronic Converters with SHE Modulation

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

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Naroa Ramos, Iñigo Pascual, Eugenio Gubía, Jesús López, Javier Samanes. Passivity Analysis in Power Electronic Converters with SHE Modulation. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0301. hal-05079965

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# Passivity Analysis in Power Electronic Converters with SHE Modulation

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

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

Campus Arrosadia, 31006

Pamplona, Spain

naroa.ramos@unavarra.es

## ACKNOWLEDGMENT

This work is part of the Projects PID2022-142791OB-I00 and TED2021-132604B-I00, funded by MICIU/AEI /10.13039/501100011033, by ERDF/UE and by the European Union NextGenerationEU/PRTR. It has also been partially supported by Ingeteam Power Technology and the Public University of Navarre.

*Index Terms*—Passivity, Selective Harmonic Elimination (SHE), Stability Analysis, Frequency Domain Analysis.

*Abstract*—To ensure the stability of modern power grids, particularly those dominated by power electronics, it is convenient for grid-connected converters to exhibit passivity across the entire frequency spectrum. Furthermore, it is interesting not only for the system to be passive, that is, for the converter system's resistance to be positive, but also for this resistance to have a high value, as it helps dampen potential grid resonances. This article presents a study of a grid-connected converter, achieving passivity and increasing the converter's resistance through control strategies. Given the growing use of Selective Harmonic Elimination (SHE) modulation techniques in high-power converters, the analysis will also focus on how to assess the passivity of a system employing these modulation techniques.

## I. INTRODUCTION

In recent years, there has been a significant increase in the use of renewable energy sources, with solar and wind energies being the most employed. The use of converters to control these technologies and the growing presence of electronic loads have led to an increase in power electronics on the grid, making the system more prone to instability [1].

It is common to analyze the stability of a system considering a simplified grid model, which is represented

by an impedance. In this way, a complete model of the system can be derived to analyze stability and design the control strategy. However, in grids dominated by inverters, due to the complexity of the grid, this no longer can be done, so local stability criteria have gained attention, particularly passivity [2]. Passivity criterion establishes that if every subsystem of a grid exhibits passive behaviour, the entire system will be stable. Thus, if the impedance of the converter is made passive, assuming the grid is also passive, the entire system will remain stable [3]. Passivity provides a sufficient, although not necessary, stability condition for interconnected dynamic systems [4].

Converters, due to the delay inherent to their control systems, typically function as non-passive systems [5]; therefore, measures must be implemented to make them passive. This can be achieved through control techniques or through passive filters. While adding passive filters would be the most straightforward approach, the cost and power losses of the system will increase [6]. Consequently, this article will focus on control techniques to improve passivity.

Control techniques to enhance passivity have been extensively documented in the literature. The three widely employed methods are (1) Point of Common Coupling (PCC) voltage feedforward, (2) feedback of the converter current, and (3) feedback of the capacitor current in the case of having an LCL filter [7]. For instance, in [7], the authors used the PCC voltage feedforward to achieve passivity, in [8], they used PCC voltage feedforward and converter current feedback and in [3], [9] and [10], both voltage feedforward and capacitor current feedback were implemented. The limitations of these approaches is that they focus solely on achieving passivity, typically in regions where grid resonances may occur or up to the Nyquist frequency, without considering the need for a minimum system resistance
to improve damping.

A Single-Input Single-Output (SISO) system is passive when the real part of the converter output impedance is non-negative [3]. So passivity can be analyzed by examining the real part of the system impedance. Previous research has concentrated on passivity up to the Nyquist frequency, which is $f_s/2$, where $f_s$ represents the sampling frequency. Assuming that the total delay of the control and modulation is $1.5 T_s$, where $T_s$ is the sampling period, the real part of the output impedance of a converter becomes negative at $f_s/6$, also known as the critical frequency [9]. In high-power applications, where the sampling frequency is typically low, grid resonances can exceed this critical frequency. Additionally, the necessity for passivity above the Nyquist frequency is also noted in [11].

The application of passivity has gained significant interest in the context of offshore wind power. Due to the presence of submarine cables, the grid presents a capacitive behavior that is prone to cause instabilities when interacting with the converter's control. This issue has been previously reported in [12], where resonance frequency appears in the range of 300 Hz to 1 kHz. Consequently, there is a growing interest in ensuring that the system remains passive, because if all converters exhibit passive behavior, system stability can be guaranteed. Additionally, [12] also introduces the idea of increasing the system's resistance through control to help dampen resonances.

In high-power converters, there has been a transition from conventional modulation techniques, such as Pulse Width Modulation (PWM), to Selective Harmonic Mitigation (SHM) and Selective Harmonic Elimination (SHE) modulations. This shift aims to ensure compliance with harmonic emission grid codes and additionally allows the use of inductive filters [13].

However, these modulation techniques present certain challenges. In [14], it is proposed modeling SHM modulation as a Zero-Order-Hold (ZOH) for the purpose of stability analysis. The study further illustrates how the real frequency response varies from that of the ZOH. The difference occurs because a modulation index is used to load a precomputed pattern designed to provide the necessary voltage over one fundamental period, if this modulation index oscillates during this period, it affects the frequency response. As a result, conclusions drawn from stability analysis using a ZOH model may differ from those obtained by examining passivity through the frequency response of an electromagnetic model.

This article has a dual objective: to derive a model for a wind power system that ensures passivity while

maximizing system resistance across all frequencies, that is done in Section II, and to develop an understanding of how to study passivity in the presence of SHE modulation, done in Section III and Section IV. The study aims to determine whether it is more appropriate to analyze the system using a ZOH model or the real frequency response.

## II. CONTROL DESIGN TO IMPROVE PASSIVITY

### A. *Description and Modeling of the System*

The passivity analysis is performed on the wind power system illustrated in Figure 1 (a), whose passivity can be studied by examining the impedance seen from the PCC. Therefore, this article focuses on the analysis of the Grid Side Converter (GSC), which includes the elements highlighted in Figure 1 (a). This converter is a three-phase three-level Neutral-Point Clamped (NPC) medium-voltage back-to-back converter which is connected to the grid through an output inductive filter and a step-up transformer. As it was mentioned in the introduction, an SHE modulation strategy is employed.

The converter is digitally controlled by a current control loop. A Proportional-Resonant (PR) controller is used and additional control techniques are added in order to improve passivity. Since the control is digitally implemented, there is an inherent delay of one sampling period. All parameters that define the system are summarized in Table I.

In order to study the system, a model in the Laplace domain is derived. Everything is modeled in the stationary reference frame, known as $\alpha\beta$. Assuming a symmetric system, the two axes remain decoupled, allowing the system to be considered as a SISO system. The block diagram of the current control loop for a single axis, in this case the $\alpha$ axis, is represented in Figure 1 (b), where all the elements are explained in the subsequent paragraphs.

$Y_{f+tr}(s)$ represents the admittance of the output inductive filter and the transformer, with the electrical circuit diagram for both components shown in Figure 2. The L filter is represented by an inductance, $L_f$, in series with its corresponding resistance, $R_f$. The transformer model is chosen to account for its high-frequency response, as inductor losses exhibit a strong frequency dependence. In fact, these losses can increase the transformer resistance by up to 100 times as the frequency increases [15]. To address this, the high-frequency transformer model, presented in [16], is used and represented in Figure 2. Accurately modeling this response is essential, as the literature suggest that achieving passivity at all frequencies through control techniques is a challenging
<figure><img src="image_2.png" /><figcaption>Fig. 1. Electric schematic of a wind power back-to-back converter (a) and its current control loop in the axis α of the stationary reference frame (b).</figcaption></figure>

task, even non-feasible according to some authors [17]. However, few studies consider the transformer positioned between the inverter and the grid, which provides a positive resistance at high frequencies.

In conclusion, the admittance $Y_{f+tr}(s)$, which includes both the converter output filter and the transformer, is modeled by the transfer function

$$Y_{f+tr}(s) = \frac{1}{Z_f(s) + Z_{tr}(s)}, \quad (1)$$

where

$$Z_f(s) = sL_f + R_f \quad (2)$$

$$Z_{tr}(s) = sL_{tr1} + R_{tr1} + \frac{sL_{tr2}R_{tr2}}{sL_{tr2} + R_{tr2}}. \quad (3)$$

The inductances $L_{tr1}$ and $L_{tr2}$, along with the resistances $R_{tr1}$ and $R_{tr2}$ are chosen to replicate the increase in resistance across all frequencies. The real part of the transformer impedance, $\Re\{Z_{tr}(j\omega)\}$, is represented in Figure 3, where it is shown to be $R_{tr1}$ (0.003 p.u.) at low frequencies and increases due to losses, reaching $R_{tr2}$ (0.694 p.u.) at 10 kHz.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Converter and transformer impedance electric schematic.</figcaption></figure>

The control system is implemented digitally, introducing a computational delay equivalent to one sample period, $T_s$, modeled by

$$D_{conv}(s) = e^{-sT_s}. \quad (4)$$

The SHE modulation is modeled using a ZOH, with its transfer function being

$$ZOH(s) = \frac{1 - e^{-sT_s}}{sTs}. \quad (5)$$

For the current control, two PR controllers are used, one for each axis. The transfer function of these con-
<figure><img src="image_4.png" /><figcaption>Fig. 3. Real part of the impedance of the high frequency transformer model.</figcaption></figure>

trollers is

$$PR(s) = K_p + K_i \frac{s}{s^2 + \xi_r \omega_0 s + \omega_0^2}, \quad (6)$$

where $K_p$ is the proportional gain, and $K_i$ the integral gain. $\omega_0$ is the fundamental angular frequency, indicating the frequency at which infinite gain is targeted, while $\xi_r$ is the damping factor of the resonant. Ideally, since infinite gain at $\omega_0$ is desired, $\xi_r$ would be set to 0. Nevertheless, this negatively affects passivity, so it is adjusted in a subsequent subsection.

The literature presents some control techniques aimed at achieving a passive response, such as PCC voltage feedforward and converter current feedback, as discussed in the introduction and illustrated in Figure 1 (b). Initially, these strategies are not implemented ($H_{ff}(s) = H_{vr}(s) = 0$), and passivity is studied using only the PR controller.

## B. Passivity Analysis of Power Converters

In this subsection, the passivity analysis of the described system is performed. A SISO system, like the one under study, can be classified as passive when the real part of the output converter impedance is positive at all frequencies. The converter output impedance is the impedance seen from the PCC, located between the transformer and the grid. The transfer function of the impedance is

$$Z_{conv}(s) = \frac{1 + PR(s) D_{conv}(s) ZOH(s) Y_{f+tr}(s)}{Y_{f+tr}(s)}, \quad (7)$$

where the controller is only a PR. Other additional control techniques, such as PCC voltage feedforward and converter current feedback, are set to 0.

Next, the real part of the $Z_{conv}(s)$ impedance is analytically studied. For this purpose, several simplifications have been made. Firstly, $Z_{tr}(s)$ has been replaced with a model that does not account for the increase of the transformer power losses at high frequencies,

$$Z'_{tr}(s) = sL_{tr} + R_{tr}, \quad (8)$$

<table><caption>TABLE I
SYSTEM PARAMETERS</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Magnitude</th></tr></thead><tbody><tr><td colspan="3"><strong>Converter parameters</strong></td></tr><tr><td>Converter output filter inductance</td><td>L<sub>f</sub></td><td>0.087 p.u.</td></tr><tr><td>Converter output filter inductance series resistance</td><td>R<sub>f</sub></td><td>0.003 p.u.</td></tr><tr><td colspan="3"><strong>Transformer parameters, simple model</strong></td></tr><tr><td>Transformer inductance</td><td>L<sub>tr</sub></td><td>0.163 p.u.</td></tr><tr><td>Transformer inductance series resistance</td><td>R<sub>tr</sub></td><td>0.006 p.u.</td></tr><tr><td colspan="3"><strong>Transformer parameters, high frequency model</strong></td></tr><tr><td>Transformer inductance 1</td><td>L<sub>tr1</sub></td><td>0.131 p.u.</td></tr><tr><td>Transformer inductance 2</td><td>L<sub>tr2</sub></td><td>0.033 p.u.</td></tr><tr><td>Transformer resistance 1</td><td>R<sub>tr1</sub></td><td>0.003 p.u.</td></tr><tr><td>Transformer resistance 2</td><td>R<sub>tr2</sub></td><td>0.694 p.u.</td></tr><tr><td colspan="3"><strong>Control parameters</strong></td></tr><tr><td>Sample time</td><td>T<sub>s</sub></td><td>250 µs</td></tr><tr><td>Sample frequency</td><td>f<sub>s</sub></td><td>4 kHz</td></tr><tr><td>Critical frequency</td><td>f<sub>crit</sub></td><td>666.67 Hz</td></tr><tr><td>Proportional gain</td><td>K<sub>p</sub></td><td>0.34 V/A</td></tr><tr><td>Integral gain</td><td>K<sub>i</sub></td><td>17 V/(A s)</td></tr><tr><td>Resonant damping factor</td><td>ξ<sub>r</sub></td><td>0.001</td></tr><tr><td>Feedforward gain (Passive system)</td><td>K<sub>ff</sub></td><td>0.05 V/V</td></tr><tr><td>Feedforward gain (Non-passive system)</td><td>K<sub>ff</sub></td><td>0.3 V/V</td></tr><tr><td>Virtual resistance gain</td><td>K<sub>vr</sub></td><td>0.16 V/A</td></tr><tr><td>Cutoff frequency of the high-pass filter</td><td>f<sub>c<sub>hpf</sub></sub></td><td>666 Hz</td></tr><tr><td>Time constant of the high-pass filter</td><td>τ<sub>hpf</sub></td><td>239 µs</td></tr><tr><td colspan="3"><strong>Grid parameters</strong></td></tr><tr><td>Fundamental frequency</td><td>f<sub>0</sub></td><td>50 Hz</td></tr><tr><td>Fundamental angular frequency</td><td>ω<sub>0</sub></td><td>314 rad/s</td></tr><tr><td>Grid inductance</td><td>L<sub>g</sub></td><td>0.1 p.u.</td></tr><tr><td>Grid capacitance</td><td>C<sub>g</sub></td><td>0.34 p.u. - 0.009 p.u.</td></tr></tbody></table>

where $L_{tr}$ is the transformer inductance and $R_{tr}$ its corresponding series resistance. The resonant part of the PR is neglected ($PR(s) = K_p$), since the resonant effects become negligible at frequencies far from $\omega_0$. Finally, the ZOH is represented by a delay of half a sample time ($ZOH(s) = e^{-s0.5T_s}$). With all this changes, and assuming that the contribution of $R_f$ and $R_{tr}$ to passivity is small, the expression of the real part of $Z_{conv}(s)$ is as follows:

$$\Re\{Z_{conv}(j\omega)\} = K_p \cos(1.5 T_s \omega). \quad (9)$$

From (9) it can be concluded that passivity depends of the value of $K_p$ and the sampling time. At low frequencies, the real part of the impedance has the same
<figure><img src="image_5.png" /><figcaption>Fig. 4. Real part of the converter output impedance obtained with different transformer models and different values of $K_p$.</figcaption></figure>

value as $K_p$. However, when $cos(1.5 T_s ω)$ reaches a value of $π/2$, which occurs at $ω = π/(3T_s)$ or equivalently at $f = f_s/6$, previously defined as the critical frequency, it transitions to exhibit a negative resistance. In summary, passivity is significantly influenced by the system delay.

### C. PR Adjustment

The PR controller, described by its transfer function in (6), can be adjusted based on the passivity curves. To enhance system dynamics, it is preferable to maximize the value of $K_p$. However, the primary objective is to maintain passivity, so the highest possible $K_p$ that does not compromise passivity is selected.

Following the expression (9), the resistance will become negative at the critical frequency ($f = f_s/6$). This is illustrated for different values of $K_p$ with dashed lines in Figure 4. It is important to note that this analysis is based on certain simplifications, such as simplifying the transformer model and the ZOH and neglecting the resonant part of the PR. Considering the whole model described in Section II A, with the transformer model in (3), the ZOH in (5) and the PR in (6), the real part of the converter impedance for the same values of $K_p$ is represented in continuous lines in Figure 4. It can be observed how the transformer introduces resistance at high frequencies. From this, it can be concluded that although a lower $K_p$ results in a slower control loop, it improves passivity at high frequencies.

By analyzing these curves, the highest $K_p$ that does not present a negative resistance is selected; in this case $K_p = 0.34$. Maximizing resistance across all frequencies has not been prioritized, as the next subsection will introduce two control techniques specifically aimed at increasing resistance where its value is close to zero. Also, the non-passive region at 50 Hz is disregarded,

<figure><img src="image_6.png" /><figcaption>Fig. 5. Real part of the converter output impedance with different $ξ_r$ values.</figcaption></figure>

as it can be eliminated by adjusting $ξ_r$, which has been assumed to be $ξ_r = 0$ up to this point.

A lower value of $ξ_r$ is generally favorable as it allows for higher gain at the fundamental frequency and reduced current tracking error. Nevertheless, the damping factor also affects passivity, as illustrated in Figure 5. While a lower $ξ_r$ would improve current tracking, it can also result in more negative resistance. Therefore, $ξ_r = 0.001$ is chosen, providing a satisfactory balance between effective current tracking and maintaining a non-negative resistance.

### D. Control Techniques to Improve Passivity

The system under study is already passive, but increasing $K_p$ also enhances the system dynamics. Nevertheless, this will result in a non-passive system. To achieve this objective while preserving passivity and further increasing the system's resistance at high frequencies, maximizing its value across all frequencies, some control techniques are integrated into the control loop. As outlined in the introduction, the literature identifies
<figure><img src="image_7.png" /><figcaption>Fig. 6. Real part of the converter output impedance with different $K_{ff}$ values.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Real part of the converter output impedance with $PR = H_{ff} = 0$.</figcaption></figure>

three primary methods for accomplishing this. Since this system does not include an LCL output filter, only two of these methods are applicable: PCC voltage feedforward and converter current feedback, which are shown in Figure 1 (b) in purple and orange, respectively. These techniques are sequentially implemented and analyzed.

The design of these control strategies is done to distance the real part of the impedance from the non-passive region. For this analysis, $K_p = 0.34$ and $\xi_r = 0.001$ are considered.

1) PCC Voltage Feedforward: A constant value for the PCC voltage feedforward has been proposed,

$$H_{ff}(s) = K_{ff}, \quad (10)$$

where $K_{ff}$ is the feedforward gain. To adjust $K_{ff}$, a sweep of the real part of the converter impedance is conducted by varying the value of $K_{ff}$, Figure 6. The value of $K_{ff}$ that maximizes the resistance across all frequencies is selected, with $K_{ff} = 0.05$. This is equivalent to implementing a feedforward of 5% of the PCC voltage.

As illustrated in Figure 6, the incorporation of this feedforward element shifts the decline in resistance to higher frequencies, but does not completely prevent it. Therefore, additional measures must be implemented.

2) Converter Current Feedback: The next step involves incorporating converter current feedback, which acts as a virtual resistance in series with the $Z_f(s)$,

$$H_{vr}(s) = K_{vr}, \quad (11)$$

where $K_{vr}$ is the virtual resistance gain. To better understand the effects of this element, an analysis similar to that in (9) is conducted. For this, it is assumed that $PR(s) = H_{ff}(s) = 0$. Additionally, as in the previous analysis, $Z'_{tr}(s)$ is considered instead of $Z_{tr}(s)$, $R_f$ and $R_{tr}$ are neglected and the ZOH is simplified to a delay of 0.5 s. The real part of the converter is then given by

$$\Re\{Z_{conv}(j\omega)\} = -K_{vr} \cos(1.5 T_s \omega). \quad (12)$$

This behaves similarly to the PR controller, but with the opposite sign, as it is shown in Figure 7 in light orange. Therefore, it can be used to compensate for the loss of resistance that previously occurred above the critical frequency ($f_s/6$).

For the effect of this virtual resistance to take place only at high frequencies, it is combined with a high-pass filter. The transfer function is thus given by

$$H_{vr}(s) = K_{vr} \frac{\tau_{hpf} s}{1 + \tau_{hpf} s}, \quad (13)$$

where

$$\tau_{hpf} = \frac{1}{2\pi f_{c_{hpf}}}, \quad (14)$$

$\tau_{hpf}$ is the time constant of the high-pass filter and $f_{c_{hpf}}$ is the cutoff frequency. A value of $f_{c_{hpf}} = f_s/6$ has been selected. In Figure 7, the real part of the converter output impedance with $PR(s) = H_{ff}(s) = 0$ is represented. In light orange the virtual resistance without a high-pass filter, (11), is displayed, while in dark orange the virtual resistance with the filter, (13), is shown.

To maximize the real part of the converter output impedance, both PCC voltage feedforward and converter current feedback are combined. To flatten the curve indicating consistent resistance across all frequencies, a value of $K_{vr} = 0.16$ is selected. The final passivity curve for the entire system is shown in green in Figure 8, where by combining these two control techniques with the PR, a minimum resistance of 0.18 p.u. is achieved. This result is compared with the passivity curve without any additional control techniques ($H_{ff}(s) = H_{vr}(s) = 0$) and only with the PR controller in blue and with the passivity curve with the PR and the PCC voltage feedforward ($H_{vr}(s) = 0$) in purple. Section III and IV will provide simulation validations for this configuration.
<figure><img src="image_9.png" /><figcaption>Fig. 8. Real part of the converter output impedance with different control techniques.</figcaption></figure>

### III. INFLUENCE OF SHE MODULATION ON PASSIVITY

The frequency response of the SHE modulation, in comparison with the ZOH model, presents variations in both gain and phase. This section and the next focus on analyzing these effects through an electromagnetic simulation model and studying their impact on passivity and the differences compared to the small-signal model in Section II. The model is derived in two different ways. The first approach, referred to as the small-signal model, is obtained by using the transfer functions from Section II in MATLAB. The second approach, designated as the EMT model, employs the Simulink environment from MATLAB, specifically the Electrical Simscape library, to create a more realistic electromagnetic model. In this second model, the control system is implemented digitally through the use of S-Function. The discretization of the control transfer functions (6), (10) and (13) is carried out using the Tustin method.

The elements of the EMT model are the same as those previously discussed. It includes an NPC converter, along with the output inductive filter and transformer model from Figure 2. The control configuration is the one established in Section II, with the PR and both PCC voltage feedforward and converter current feedback, and the tests are conducted both with and without SHE modulation.

As in Section II, passivity is analyzed by examining the real part of the converter output impedance. This impedance is measured as follows: a voltage is applied at the PCC, sweeping across a range of frequencies. For each frequency, the corresponding converter current is measured. The impedance is then calculated by dividing

one phase of the applied voltage in that disturbance frequency by the corresponding converter current at that same frequency. Finally, the real part of the impedance is obtained.

#### A. Simulation Results Without Modulation

First, passivity is evaluated through the EMT model without the inclusion of any modulation. In this scenario, the NPC converter has been removed and replaced with ideal equivalent voltage sources that directly provide the control voltage. Figure 9 (a) presents the resulting resistance. The close alignment between the small-signal model and the EMT model without modulation validates the model and its adjustment developed in Section II.

#### B. Simulation Results with SHE Modulation

Once verified that the small-signal and EMT model match, the SHE modulation is incorporated. This modulation exhibits quarter-wave symmetry, which results in the absence of even harmonics and harmonics that are multiples of three. The SHE used consists of 11 angles, thus switching 44 times each period of the fundamental, and eliminating up to the 31st harmonic. A pattern is calculated for each modulation index. This index is determined at each sampling period, meaning that the pattern can vary over the course of the fundamental period.

The corresponding resistance obtained with SHE modulation is shown in Figure 9 (b). In contrast to the previous results, the resistance obtained with the EMT model that includes modulation exhibits variations compared to the small-signal model. These variations arise from fluctuations in the modulation index, which lead to calculation errors.
<figure><img src="image_10.png" /><figcaption>Fig. 9. Real part of the converter output impedance obtained through simulation: (a) without modulation and (b) with SHE modulation.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Real part of the converter output impedance obtained through simulation: (a) system A and (b) system B.</figcaption></figure>

In Figure 9 (b), the light green curve represents a passive system derived from the small-signal model, while the dark green curve displays a non-passive system from the EMT model. The passive system is guaranteed to be stable; however, the non-passive system may become unstable. In Section IV, tests are conducted to determine which of the two models should be taken into account.

## IV. PASSIVITY ANALYSIS VALIDATION BY SIMULATION

The study conducted so far is complemented by a stability analysis. A passive system will always remain stable if connected to another passive system, whereas a non-passive may become unstable if a resonance falls at a frequency where the resistance is negative [9]. As indicated by the results in Figure 9 (b), the EMT model is non-passive, which can lead to instability, while the small-signal model results might conclude the opposite.

To validate the results from Section III, two systems are defined: one that is passive in the small-signal model (the one used in previous analysis), referred to as system A, and another that is non-passive in the small-signal model, designated as system B, with the only change being an increase in the PCC voltage feedforward gain ($K_{ff} = 0.3$). The real part of the impedance of these

systems are presented in Figure 10, with the system A in (a) and the system B in (b).

In the small-signal models, system A, in Figure 10 (a), is passive, while system B, in Figure 10 (b), is non-passive. This implies that, according to the small-signal models, system A cannot be destabilized when connected to other passive systems, whereas system B can potentially become destabilized. Nonetheless, in the EMT models, both systems are non-passive, which introduces the possibility of instability.

To determine which of the models has to be considered, a grid with a varying resonance is introduced. This is accomplished using an LC filter, shown in Figure 11. The grid inductance, $L_g$, remains constant, while the grid capacitance, $C_g$, is varied to alter the grid resonance from 300 Hz to 2000 Hz. Simulations using the Simulink Simscape EMT model connected to this variable grid are performed on both systems, with a current reference of $I_{nom}$. The results are illustrated in Figure 12.

In Figure 12 (a), the evolution of $C_g$ to achieve grid resonance from 300 Hz to 2000 Hz is presented, Figure 12 (b) shows the current response of one phase of system A, and Figure 12 (c) the current response of one phase of system B. System A, shown in Figure 12 (b), that according to the small-signal model is passive, remains stable throughout all the simulation. On the other hand,
<figure><img src="image_12.png" /><figcaption>Fig. 11. Model of the grid.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Evolution of $C_g$ (a), converter current response of one phase of system A (b) and converter current response of one phase of system B (c).</figcaption></figure>

system B, illustrated in Figure 12 (c), that exhibits non-passive behavior in the small-signal model, becomes destabilized for a $C = 0.019$ p.u., corresponding to a resonance frequency of 1420 Hz. This frequency aligns with the point at which the small-signal model begins to exhibit negative resistance, as indicated in Figure 10 (b).

In summary, the simulation results are consistent with the predictions of the small-signal model. Therefore, when the resonance frequency coincides with the range

where the real part of the converter output impedance derived from the small-signal model becomes negative, the system, as shown in Figure 12 (c), becomes unstable. It is concluded that the modulator introduces harmonics that affect the calculation of passivity, resulting in a model that is not entirely reliable for studying system passivity.

## V. CONCLUSION

This article presents the development of a passive system that ensures passivity across all frequencies. To achieve this, control techniques to improve passivity, such as PCC voltage feedforward and converter current feedback, are implemented. Additionally, the high-frequency model of the transformer is incorporated into the design. The impact of SHE modulation on the system passivity is also examined. Two key conclusions are drawn:

(1) In a system with SHE modulation, the modulator introduces harmonics in the calculation of passivity. Therefore, to reliably predict the current response and determine system stability, it is sufficient to analyze the small-signal model.

(2) While control techniques to improve passivity are crucial for ensuring passivity, it is essential that all passive elements are accurately represented across the entire frequency spectrum.

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