AC-Side Impedance-Based Stability Assessment in Grid-Forming Modular Multilevel Converters

Grid-forming converters can emulate the behavior of a synchronous generator through frequency droop control. The stability of grid-forming modular multilevel converters can be studied via the impedance-based stability criterion. This paper presents an ac-side impedance model of a grid-forming modular multilevel converter which includes a complete grid-forming control structure. The impact of different control schemes and parameters on the closed-loop output impedance of the converter is thoroughly analyzed and the learnings have been used in mitigating undesired control interactions with the grid. The results are verified through simulations in time- and frequency-domains along with experiments on a down-scaled laboratory prototype.


I. INTRODUCTION
Voltage source converter (VSC)-based high voltage direct current (HVDC) transmission systems enable large-scale integration of renewable energy sources and efficient transfer of bulk power over long distances [1], [2].The modular multilevel converter (MMC) is the preferred topology in HVDC transmission systems due to its modularity, scalability, and low distortions [3].
VSCs are conventionally controlled as current sources and in a grid-following manner where the synchronization is realized by a phase-locked loop (PLL).This makes them prone to stability issues, especially in weak grid conditions due to the inadequacy of PLLs in tracking disturbed voltage angles [4].Considering a shift toward a cleaner energy matrix, the grid stiffness may substantially decrease in the future as synchronous generators are replaced by inverterbased resources.Grid-forming control schemes primarily control the ac-side voltage and operate the converter as a The associate editor coordinating the review of this manuscript and approving it for publication was Xueguang Zhang .voltage source.By using a power-frequency droop function, they also emulate the behavior of a synchronous generator and add virtual inertia to the power system, essentially enabling operation in weak grid conditions [5], [6].
VSC-based HVDC transmission systems are prone to system instabilities that need to be predicted, analyzed, and mitigated [7], [8], [9], [10].The stability of power electronics-based systems, including VSC-HVDC systems, can be evaluated using several methods such as passivitybased analysis, eigenvalue-based analysis, the net-damping criterion, and the impedance-based stability criterion [11].In the latter, the grid-connected converter system can be partitioned into a source and load subsystem where the converter is modeled as an impedance behind a voltage source (Thévenin equivalent) or an admittance in parallel to a current source (Norton equivalent).The small-signal stability of the system is then assessed via the ratio of the source and load impedance, known as the minor-loop gain.Accurate stability assessment requires either precise measurement of the converter's impedance at its terminals or a well-constructed model of the same.Nonetheless, owing to the nonlinearities introduced by the converter dynamics, impedance modeling of MMCs is challenging and intricate.
Research on impedance modeling of grid-following VSCbased HVDC systems and MMCs is abundant [12], [13], [14], [15], [16], [17], [18], [19], [20].Several new studies analyze converter-driven stability with regards to grid-forming converters.For instance, stability assessment of grid-forming control applied to a two-level VSC is studied in [21].An ac-side impedance model in the αβ-domain is developed, based on which a subsynchronous control interaction with a strong grid is identified.The study includes experimental time-domain verification of the control interaction.However, it lacks experimental frequency-domain verification of the developed impedance model.Later in [22], the authors complement the prior study by a control decomposition methodology based on tuning of the controllers to improve the stability margin.It has been shown both in [21] and [23] that the root cause of impedance interaction and instability is the non-passivity of the converter around the fundamental frequency introduced by the resonant/integral action of the voltage control loop.
A few studies have also analyzed the impact of grid-forming control on the closed-loop output impedance of MMCs.For instance, Reference [24] has presented an ac-side impedance model of a voltage-controlled MMC regulated via single-loop and dual-loop structures.Nevertheless, the power droop functions emulating a synchronous machine have been neglected.Moreover, the regulators have been limited to proportional-resonant types used in the abc or αβ stationary reference frame.Reference [25] has analyzed the stability of a voltage-controlled MMC connected to a wind farm.The droop functions and dual-loop voltage control are however not included.The authors of [26] have analyzed the ac-and dc-side impedances of an MMC with grid-forming control.In this study, the inner-loop voltage and current control loops have been ignored and the analysis has been limited to outer-loop droop functions.Harmonic instabilities have been identified and virtual impedances have been proposed to damp the unstable oscillations.Reference [27] has conducted an impedance based small-signal stability analysis for MMCs with power synchronization control and proposed a virtual impedance loop on the computed insertion indices that handles the negative resistance below the fundamental frequency.In [25], [26], and [27], the cause of instability has also been determined as the non-passivity of the converter in sub-synchronous frequencies, similar to that present in generic two-level converters.High frequency unstable oscillations may also happen in grid-forming MMCs connected to wind farms [28].As described above, the investigation into the influence of grid-forming control on the small-signal stability of VSCs represents an ongoing area of research.None of the studies introduced above, however, have included an experimental verification of the developed impedance models, time-domain verification of instability scenarios, and the workings of their proposed stabilizing loops.
This paper presents an accurate impedance model of a grid-forming MMC which takes into account the entire grid-forming control structure not excluding the inner voltage and current control loops, or the outer droop functions.The effect of using both single-loop voltage control and dual-loop voltage control on the ac-side impedance of the converter has been investigated and the small-signal stability of the system is analyzed.It has been shown that instabilities can occur under certain control and system parameters due to the interaction of the grid impedance and the negative resistance behavior of the converter's output impedance that is introduced by the voltage control loop.The instability and the frequency of the unstable oscillations can be predicted by impedance analysis.
Compared to the state-of-the-art in the literature, the contributions of this work are the proposal of three novel instability mitigation strategies based on 1) cross-coupled virtual resistance on ac-side currents, 2) low-pass filters on the ac-side currents, and 3) lead compensators on the PCC voltage that improve the stability margin of the system.Moreover, not only the developed frequency-domain impedance models are verified on a down-scaled laboratory prototype, but also the prediction of the instability and the efficacy of the mitigation strategies have been verified on power electronics hardware.
The paper is organized as follows: Section II describes the converter dynamics and discusses the considered control solutions for grid-forming MMCs.Section III describes the derivation procedure of the ac-side impedance for all control solutions.Section IV presents and discusses the results of the study, analyzes the effect of different control schemes and parameters on converter's impedance, assesses small-signal system stability, and proposes methods to improve stability margin.Finally, Section V draws conclusions based on the results.

II. SYSTEM MODEL
The MMC for ac/dc conversion comprises three phase-legs with two arms within each leg.The converter arms consist of a number of cascaded submodules (SMs) and an arm inductor designed to reduce the harmonic distortion in arm currents and prevent high transient currents.Fig. 1 shows the single-line diagram of a grid-connected MMC in which L and L g denote the arm and grid inductances, respectively, and R and R g denote the arm and grid resistances, respectively.A phase-domain impedance model is derived based on dynamics of the converter and the considered control scheme.For the sake of brevity, throughout the text, a per-phase modeling approach is taken where the subscript indicating specific phases is dropped.

A. CONVERTER DYNAMICS
A dynamic model of the MMC is adopted in which the switching operations are neglected and balanced submodule capacitor voltages are assumed within an arm [29].The dynamics governing the MMC can then be obtained as where i u (i l ) is the upper (lower) arm current, v du (v dl ) is the upper (lower) dc-side voltage, and e is the voltage of the point of common coupling (PCC).Arm voltages are denoted by v u,l , and v Cu,l are the sum-capacitor voltages.C SM is the submodule capacitance, N is the number of submodules and v C0 is the initial sum capacitor voltage.Moreover, the (upper and lower arm) insertion indices, indicated by n u,l , are the outputs of the control system computed in a closed-loop fashion.

B. CONVERTER CONTROL
VSC control is often implemented in a cascaded manner, with outer control loops setting the reference for the inner loops [1].The grid-forming control scheme for MMCs is shown in Fig. 2. In grid-forming control, the primary control objectives are controlling the converter output voltage while transferring active and reactive power.The powers are controlled via droop functions which set the reference for the faster inner-loop voltage control.Voltage control can be realized either with a single-loop, or can also nest a current control loop whose primary objective is to control transient currents, referred to commonly in the literature as dual-loop control [23].The outputs of the control system is then modulated (by e.g., phase-shifted carrier pulsewidth modulation) to generate the switching commands which are applied to the submodules of the converter.
In addition, a circulating current control loop, suppresses the second-order harmonic currents that appear in the arm currents.
Assuming v du = v dl = v dc /2, adding and subtracting upper/lower arm current equations yields where v s is the voltage driving the ac-side current i s and v c is the voltage driving the circulating current i c .The reference voltages v ⋆ s and v ⋆ c can be used to compute the insertion indices of the arms.Assuming a direct voltage control scheme (also known as direct modulation), we define which provides an asymptotically stable system [30].
Ideal insertion indices can also be considered where the arm reference voltages are normalized by the measured sum capacitor voltages.Referred to as closed-loop voltage control (or compensated modulation), the insertion indices in this case can then be calculated by With closed-loop voltage control, the internal dynamics of the MMC are only marginally stable and arm-balancing control is required to stabilize the system [31].In this work, a direct voltage control scheme is considered.This choice is motivated by the fact that the studied stability problem is related to the non-passivity of the converter introduced by the control structure shown in Fig. 2, which is present irrespective of the choice of the modulation scheme.
In a three-phase system, the powers are calculated in the dq-frame as where e dq is the PCC voltage in the synchronous (dq) reference frame and i s,dq is the output current in the dq-frame.

III. AC-SIDE IMPEDANCE CALCULATION
To calculate the ac-side impedance, a fixed-frequency perturbation is superimposed on the terminal voltage or current.Then, the response of converter and control variables is calculated for the applied perturbation under frequency sweeps of the perturbation source [32].In this work, a three-phase balanced small-signal perturbation voltage is superimposed on the grid voltage, i.e., such that |V g (jω p )| ≪ |V g (jω 1 )|, i.e., the perturbation is small-signal.In (10), V g denotes the Fourier coefficient of v g , ω p is the angular perturbation frequency, ω 1 is the fundamental frequency, and φ 1 is the initial phase angle of the grid voltage.φ p is the initial angle of the perturbation signal.
The harmonic responses of the converter and the control variables to the perturbation are calculated in the frequencydomain.Multiplication of the converter variables in the time-domain in (3)-( 4) leads to a convolution of their spectra in the frequency-domain [18].The most significant of these harmonic responses appear at a combination of the perturbation frequency f p and integer multiples of the fundamental frequency f 1 , i.e., f = {f p ± nf 1 }, n ∈ Z.To keep the complexity of the resulting impedance model at bay, n ≤ 3 is assumed in this work.
In the following, the harmonic responses of the converter and control variables to the applied perturbation are evaluated and calculated.The capitalized letters indicate the Fourier coefficient of the variables.Due to the symmetry of the threephase topology, frequency-domain analysis of a single-armin this case upper-arm of phase a-suffices.

A. RESPONSE OF THE CONVERTER VARIABLES
The harmonic responses of the converter variables, i u , v u , and v Cu , to the applied perturbation are calculated at jω p .The rest of the expressions for other frequencies (including the steady-state frequencies) are derived in a similar fashion and are given in [19].

1) ARM AND SUM-CAPACITOR VOLTAGES
The multiplication in the time-domain in (3) and ( 4) translates to the convolution of the elements in the frequency-domain.From (3), the perturbation frequency component of the arm voltage at jω p can be derived as In a similar fashion, the perturbation frequency component of V Cu at jω p is calculated using (4) as The rest of the perturbation frequency components and the steady-state solution are given in [19].The bar denotes the complex conjugate operation.

2) ARM CURRENTS
The response of i u to the applied perturbation can be calculated from (1) as where ω ∈ {ω p ± nω 1 }, n ∈ N, n ≤ 3.

3) AC-SIDE VECTOR-CURRENTS
The dq-components of the ac-side current can be obtained by Park transformation as where and i sα and i sβ are the ac-side currents in the αβ-reference frame, and ϑ is obtained from the active power droop characteristics.
After mathematical manipulation, the responses of the ac-side current in the dq-frame to the applied perturbation can be obtained as where I s (jω p ) = 2I u (jω p ).The steady-state frequency components A(jω 1 ) and B(jω 1 ) are given in the Appendix.

4) CIRCULATING VECTOR-CURRENTS
The circulating currents have a dominant negative sequence double line frequency component.We define used in the Park transformation of the circulating currents.
The responses of the circulating currents in the synchronous reference frame to the applied ac-side perturbation are also calculated similarly as Via Kirchhoff's voltage law applied to the circuit in Fig. 1, the response of the PCC voltage to the applied perturbation can be calculated as where ω ∈ {ω p , ω p − 2ω 1 } due to the mirror frequency effect [33], and V g (jω) = e p /2, with e p as the amplitude of the perturbation.
Vector-voltage control has been applied in this work.Therefore, the dq-components of the PCC voltage can be obtained by the Park transformation as where e α and e β are the PCC voltages in the αβ-reference frame.
After mathematical manipulation, the Fourier coefficients E d and E q can be obtained as

B. RESPONSE OF THE CONTROL VARIABLES
The response of the control system comprising the outer power synchronization loop, the inner ac-side current and PCC voltage loops, and the circulating current control loop is analyzed in the frequency-domain.

1) ACTIVE AND REACTIVE POWER
The active and reactive power droops set the PCC voltage amplitude and angle, respectively.Linearizing ( 8) and ( 9) around the steady-state trajectory yields where ω = ω p − ω 1 .

2) VOLTAGE PROFILE MANAGEMENT
The harmonic response of the desired PCC voltage magnitude e m can be calculated by evaluating the reactive power control loop in Fig. 2 as where ω = ω p − ω 1 , due to the Park transformation.The transfer function F LPF is a first-order low-pass filter of the form and n Q is the reactive power droop gain.The PCC voltage reference phasor is generated by 23518 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Similar to the actual PCC voltage, the references have harmonic responses at ω = {ω p , ω p ± 2ω 1 }.The harmonic responses of this reference voltage are calculated as Subsequently, the harmonic response of the PCC reference voltages in the dq-frame can be obtained as 3) PCC VOLTAGE CONTROL The perturbation frequency response of i ⋆ s,dq can be calculated from (28) as where ω = ω p − ω 1 , and F v (s) is a proportional-integral (PI) controller given by

4) AC-SIDE VECTOR-CURRENT CONTROL
Similarly, the response to the applied perturbation can be calculated as where ω = ω p − ω 1 , and F s,dq (s) is a PI controller given by The Fourier coefficients of the converter output voltage references in the phase-domain can be obtained through dq/abc transformation as

5) CIRCULATING VECTOR-CURRENT CONTROL
The small-signal response of the voltages driving the circulating currents can be calculated as where ω = ω p − ω 1 , and F c,dq (s) is a PI controller given by With the dq/abc transformation, the harmonic responses of the reference voltages driving circulating currents in the phase-domain can be obtained as The small-signal perturbation of insertion indices follows (6) and thus obtained as where ω is the available frequency components in the insertion indices according to the chosen control solution.

C. DERIVATION OF THE AC-SIDE IMPEDANCE
Based on the expressions describing the frequency responses of the converter and control variables to the applied perturbation, Eqs. ( 11)-( 37), a linear system of equations with the unknown variables of the form is formulated where U p comprises the coefficients of the linear system, T p consists of the constant terms in the equations, and x p contains the system variables The elements of U p and T p for I u , V u , and [19], while those of the rest of the variables are obtained in ( 16)- (37).Solving the linear system of equations (39) yields the harmonic responses of all considered converter and control variables at the considered frequencies.It is worth mentioning that the developed impedance model is valid for a specified steady-state point at which the mathematical problem is solved.
The injected perturbation frequency f p yields harmonic responses through the grid impedance in the PCC voltage E and the converter current I s , at f p and f p − 2f 1 , respectively.
The assumption of small-signal study is valid here; the responses to the applied perturbation signal must be small signal, i.e., they should not diverge the system from its steady-state point.While the impact of the cross-coupling impedance terms are smaller than that of the direct terms, for accurate stability assessment, a multiple-input multipleoutput (MIMO) stability problem needs to be analyzed.Accordingly, the ac-side impedance of the converter is a 2×2 MIMO matrix of the form which indicates that four distinct equations are needed to calculate the four unknown elements of the impedance matrix.Therefore, two independent perturbation signals need to be injected from which the responses of the PCC voltage and current need to be captured at f p and f p − 2f 1 .The perturbations must be chosen to be linearly independent.Thus, firstly a positive sequence perturbation signal of the form is applied, followed by a negative sequence perturbation signal of the form After applying the two perturbations, the ac-side impedance matrix is calculated based on the method described in [34] as where subscripts 1 and 2 indicate the respective perturbation sets.Since the considered system is linear time-invariant, the resulting impedance is not affected by the initial phase angle of the perturbations.Thus, φ p can be assumed to be equal to zero [15].

1) AC-SIDE IMPEDANCE WITH CLOSED-LOOP VOLTAGE CONTROL
A closed-loop voltage control scheme can be adopted for MMCs that computes the ideal insertion indices based on measured sum-capacitor voltages, i.e., However, this approach yields an asymptotically unstable system which requires additional sum/imbalance arm energy (or alternatively voltage) control to achieve asymptotic stability [31].Assuming that perfect arm-balancing control 3. Small-signal frequency-domain representation of the grid-connected converter by its Thévenin equivalent circuit.
is achieved, the response of v ⋆ c to the applied perturbation is infinitesimal and/or can be ignored.Thus, the closed-loop output impedance becomes independent of the internal dynamics of the MMC and can be obtained based on the acside dynamics, i.e., (5a).The response of the PCC voltage to the applied perturbation is given in (19).Ignoring delays, substituting in (44) yields v s = v ⋆ s .Using equations ( 16), ( 21)-( 29), (31), and (33a) (which are all equations not involving internal dynamics of the MMC), the response of E(jω p ) and I s (jω p ) to the applied perturbation can be derived analytically by evaluating (5a) and thus, the closed-loop output impedance can be obtained.With compensated modulation, mirror frequency effect is negligible and the output impedance is single-input single-output (SISO), i.

D. STABILITY ANALYSIS: THE IMPEDANCE-BASED CRITERION
Fig. 3 shows the small-signal representation of the system consisting of the grid-forming converter which is modeled by a voltage source in series with an impedance and a grid represented by a voltage source behind the grid impedance.Grid-forming converters are prone to instability and control interactions in strong grid conditions or when the voltage control bandwidth is low [6].In this work, the interactions arising from low bandwidth of the voltage control loop are investigated.
The frequency response of the converter current can be obtained as In order to assess the small-signal stability of the system, two prerequisite conditions apply, namely that, • E must be stable, i.e., the grid-forming converter must be stable when unloaded; • Z g must be stable, i.e., the grid current must be stable when supplied by an ideal voltage-source.

VOLUME 12, 2024
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The characteristic equation of the system in (47) is a MIMO system.The right-half-plane zeros of the characteristic equation, i.e., Z ac + Z g can indicate unstable closed-loop poles.It should be noted that no cross-couplings exist in the grid impedance as it has been modeled in the phase-domain, not the synchronous reference frame.Thus, the determinant of the characteristic equation can be obtained as where After mathematical manipulations, the determinant can be reformulated as where Z eq,ac is the equivalent single-input single-output acside impedance of the MMC taking into account its interaction with the grid impedance [21].If the two prerequisites defined above are satisfied, the minor-loop gain defined as Z eq,ac /Z g,11 can be used to assess the small-signal stability of the system via the Nyquist stability criterion.

IV. RESULTS AND DISCUSSIONS
The impedance obtained via the analytical model developed above is validated against that obtained from post-processing the PCC voltage and current waveforms in simulations and in a down-scaled experimental setup.The analytical impedance model is then used to analyze and showcase the system stability.A logarithmically spaced set of perturbation frequencies between 1 Hz and 200 Hz is considered.Fig. 4 shows the ac-side impedances of an MMC with direct voltage control, with closed-loop voltage control, and for comparison that of a two-level converter.For the two level converter, a phase inductance equal to half the arm inductances of the equivalent MMC is opted.It can be seen that with closed-loop voltage control, the ac-side impedance of the MMC is similar to that of the two-level converter, implying that they exhibit the same behavior on the ac-side.The closed-loop output impedance of the MMC with direct voltage control, however, is different due to the influence of the internal dynamics of the MMC on the impedance.A nonpassive frequency region (φ Zac < −90 • ) exists around the fundamental frequency in all cases which is a byproduct of the applied grid-forming control shown in Fig. 2. For the ease of hardware implementation, the direct voltage control scheme is assumed in the remainder of the study to demonstrate the workings of the proposed active damping methods.

A. SIMULATION RESULTS
A simulation model of an MMC-based HVDC system is built in MATLAB/Simulink.The ac-side of an MMCbased HVDC system is considered, neglecting the dc-side  dynamics, i.e., a stiff voltage source on the dc-bus is assumed.The parameters of the simulated system are summarized in Tables 1 and 2. Fig. 5 shows the Bode diagram of the ac-side impedance of the HVDC converter Z ac , indicating a close match between the analytical impedance model and the impedance measured via simulations.Fig. 6 shows the equivalent single-input single-output closed-loop output (ac-side) impedance of the HVDC converter when the integral gain of the voltage controller is reduced, and the grid impedance.With a low integral gain, a narrower band with negative resistance (φ Zac,eq < −90 • ) exists below the fundamental frequency and at the magnitude intersection point of 46.7 Hz, the phase margin of the minor-loop gain is negative.Fig. 7 shows the Nyquist diagram of the minor-loop

6.
Bode diagram of the equivalent ac-side impedance of the converter Z eq,ac for different values of the integral gain of the voltage controller and the grid impedance Z g,11 .The asterisks indicate simulation results.The close-up box shows the phase margin of Z eq,ac /Z g,11 for the two cases.gain of the system for the two cases studied in Fig. 6.It can be observed that with a low integral gain, the contour encircles the (−1, 0) critical point, implying the instability of the system.

B. THE EXPERIMENTAL SETUP
Fig. 8 shows the experimental setup used in this work.An MMC with one half-bridge submodule per arm (based on SiC MOSFETs) is assembled.The control system is implemented on a rapid control prototyping platform by Imperix which utilizes fully programmable digital signal processing (DSP) and field-programmable gate array (FPGA) units.External current and voltage sensors placed on the back plane of the rack measure the arm currents and the PCC  voltages that are used to control the system quantities, see Fig. 2. A phase-shifted pulse-width modulation (PSC-PWM) scheme has been used that ensures individual capacitor voltage balancing provided that the carrier frequency is a non-integer multiple of the fundamental frequency [35].The system and control parameters of the experimental setup are given in Tables 3 and 4, respectively.
A Chroma regenerative grid simulator is used as the three-phase ac grid and also as the perturbation source.The perturbation signals are superimposed on the grid voltage by overlaying desired perturbation frequencies on the reference voltage of the grid simulator.An inductive-resistive impedance models the grid weakness.
The ac-side impedance is measured by perturbing the grid voltage with two independent perturbation sets through externally amplifying the voltage of the regenerative grid simulator.The measured voltages and currents are then post-processed to obtain the measured ac-side impedance matrix of the converter.

C. IMPACT OF CONTROL ON CLOSED-LOOP OUTPUT IMPEDANCE
The choice of control solutions and selected control gains can impact the equivalent ac-side impedance.A singleloop voltage control is the simplest form that the control of the PCC voltage can be realized in grid-forming applications [23].In this control scheme, there is no explicit control of ac-side currents.The ac-side impedance of the MMC with single-loop voltage control is shown in Fig. 9.As seen in the figure, non-passive frequency regions exist around the fundamental frequency in Z 11 .It is also evident from the magnitude plots that the impact of the direct term impedances are more pronounced than the crosscoupling terms, indicating that the equivalent impedance is impacted more by Z 11 and Z 22 , rather than the cross-coupling impedances.Fig. 10 shows the ac-side impedance of the MMC with dual-loop voltage control, i.e., the default form in this work as shown in Fig. 2. Differences exist between the impedance with dual-loop versus the single-loop case, especially around the fundamental frequency where the impedance of the dual-loop control case has a more pronounced valley in magnitude of Z 11 and non-passive behavior in Z 22 .In addition, the impedance magnitudes are greater in value with dual-loop control at frequencies other than the fundamental frequency.
Fig. 11 shows the equivalent ac-side impedance of the converter Z eq,ac and the grid impedance Z g,11 , when the integral gain of the voltage controller is altered.When the integral gain is high, the resonance at the fundamental frequency has a wide bandwidth as observed in the impedance magnitude plot.In this case, there are intersection points between the converter equivalent impedance and the grid impedance at 43.4 Hz, 57.4 Hz and 71.8 Hz.At all of these points, the phase margin is positive, indicating that the system is stable.When k iv is reduced, the bandwidth of the resonance at the fundamental frequency gets narrower, and the intersection points move to 47.9, 51.9 and 67.4 Hz, respectively.The phase margin of the two systems with different integral gains of the voltage controller are shown in the zoomed-in phase plot of the same figure.As highlighted in the figure, at f = 47.9Hz, the phase margin of the minor-loop gain of the case with a low integral gain of the controller is below zero.This indicates a risk of smallsignal instability.Nevertheless, the stability has to be verified with the Nyquist stability criterion which is a necessary and sufficient condition for stability.
The Nyquist diagram of the single-input single-output minor loop gain of the system, i.e., Z eq,ac /Z g,11 , shown in Fig. 12 indicates that the system does go unstable when the integral gain of the voltage controller is decreased.As expected, the contours intersect with the unit circle three times corresponding to the three intersections in the Bode diagram of the impedance magnitudes.Unstable oscillations in the phase variables, therefore, should occur at the predicted frequency of f osc = 47.9Hz.Due to Park transformation, the frequency of the oscillations in direct (non-alternating) system variables is at f osc,dq = 50 − 47.9 = 2.1 Hz.

D. ACTIVE DAMPING TECHNIQUES
Filter-based active damping to handle stability issues in grid-following systems has been established in [36] and [37].Three active damping methods for grid-forming converters are presented in this work that can add phase lead to the system around the fundamental frequency in order to stabilize the system.In this way they indirectly modify also the ac-side impedance at these frequencies.Fig. 13 summarizes the presented modifications that improve the phase margin and therefore, the small-signal stability.
A virtual resistance on the path of the ac-side currents damps the ac-side impedance around the fundamental frequency.Virtual resistances R v can be added in a cross-coupling fashion to the input reference of the current controller similar to that presented in [26].The value of the R v gain can be selected based on trial and error.To this end, the Nyquist diagram of the minor loop-gain of the closed-loop system can drawn and a suitable value that ensures closed-loop stability and phase margin can be selected.In the case studies of this work, a gain of R v = 0.2 for simulations and R v = 0.9 in the experimental setup is chosen.In the impedance model, the responses of the voltage in the dq-frame to the applied perturbation are modified where ω = ω p − ω 1 .
Alternatively, the dq-frame currents can be filtered at a lower frequency than the predicted oscillation.A first-order low-pass filter of the form is used in this work with a cut-off frequency of 2π3 rad/s in the HVDC system simulation and 2π1.5 rad/s in the experimental setup.The cut-off frequencies are chosen to be lower than the predicted oscillation frequencies which In this way, extra damping is provided by diminishing the oscillations that enter the current control loop.In the frequency-domain, the harmonic responses of the ac-side voltage references are modified to where ω = ω p − ω 1 .
A third option is to cascade a lead compensator on the path of the synchronous reference frame PCC voltage measurements.Lead compensators add phase to the control loop, affecting the converter impedance and improving the stability.In this work, a first-order lead compensator of the form has been designed that adds phase around 2 Hz to the PCC voltages in the dq-frame, and indirectly improves the impedance phase.To this end, the time constants of the compensator are selected as T 1 = 0.1 and T 2 = 0.05.Fig. 14 shows the Bode diagram of the designed lead compensator in this case.The selected compensator is shown to be able to sufficiently dampen the predicted oscillation both in the simulation case study and in the experimental work.Adding phase lead to the measured ac-side voltages translates in the frequency domain to modifying the ac-side current references to where ω = ω p − ω 1 .The tuning of appropriate compensator poles and zeros can be done by quantifying the phase and gain margins off the Nyquist diagram of the minor loop-gain that is obtained by using the developed ac-side impedance model.Using each one of the three proposed active damping methods provides enough phase lead at non-passive frequencies around the fundamental frequency such that the system stabilizes with a positive phase margin.Fig. 15 shows the simulation results of the HVDC system.It is observed that in t < 2 s and t > 8 s with a low integral gain the system is unstable.When the active damping loops are enabled, the system stabilizes, albeit with different dynamic responses.The transient behavior is largely dependent on the design of the filters and the choice of the gains.
Fig. 16 shows the waveforms of the converter variables during experiments with a laboratory setup to verify the unstable case studies and the performance of the stabilizing active damping techniques.The integral gain of the voltage controller is manipulated to cause system instability.The stabilizing loops are then enabled for a short duration starting at t = 2 s and disabled at t = 8 s.As seen in the figure, unstable oscillations are present in the system before t = 2 s when the stabilizing loops are bypassed.The frequency of the oscillations is computed using a fast Fourier transform of the signals during the instability, and corresponds to the predicted f osc,dq = 2.1 Hz for the variables in the direct reference frame, as seen in Fig. 17.The system is stabilized upon enabling of the damping compensators in all cases.As an example, Fig. 18 shows the harmonic responses of the ac-side current and the PCC voltage to the applied perturbation with and without the virtual resistance loop for the experimental case study.As seen in the figure, as a result of the virtual resistance loop, the predicted oscillations at 47.9 Hz in the system are dampened and no longer appear.Fig. 19 shows the Nyquist diagram of the minor loop gain Z eq,ac /Z g,11 with and without the active damping compensators, for the simulations of the HVDC system and the   down-scaled laboratory prototype.It can be concluded from the time-domain results and frequency-domain verification that the proposed loops successfully stabilize the system as the contours no longer encircle the (−1, 0) point.Future research on the topic is required to unravel the small-signal stability conditions for grid-forming converters connected in   parallel both in grid-connected and off-grid conditions, the effect of modulation schemes on the ac-side impedance of the MMCs in grid-forming control mode, and the impact of discretization of the control system on the terminal impedance and the system stability.

V. CONCLUSION
This paper presents an analytical ac-side impedance model of a droop-controlled grid-forming modular multilevel converter.The effect of single-loop and dual-loop voltage grid-forming control schemes on the closed-loop output impedance of the converter is analyzed, followed by showcasing a possible subsynchronous control interactions of the converter with the grid.It is shown that the parameters of the voltage controller can impact the small-signal stability of the grid-connected converter.Three stabilizing loops based on 1) virtual resistance added to ac-side currents, 2) low-pass filters in the path of ac-side currents, and 3) lead compensators in the path of PCC voltage have been proposed in order to omit the small-signal instability and improve stability margins.Experimental results verified on a down-scaled laboratory prototype validate the accuracy of the developed analytical model and the workings of the proposed stabilizing compensators.It is shown that the unstable oscillations and their frequencies can be accurately predicted, and the proposed active damping techniques can bring the system back to stability.

FIGURE 1 .
FIGURE 1. Single-line diagram of a modular multilevel converter (MMC).A half-bridge submodule configuration is shown as an example.

FIGURE 2 .
FIGURE 2. Droop-controlled dual-loop grid-forming control structure for an MMC.

FIGURE 4 .
FIGURE 4. Bode diagram of the ac-side impedance Z eq,ac of an MMC with direct voltage control (blue), an MMC with closed-loop voltage control (red), and a two-level converter (black).

FIGURE 5 .
FIGURE 5. Bode diagram of the ac-side impedance of a droop-controlled grid-forming MMC-HVDC with dual-loop voltage control.The solid lines show the analytical impedance, and the circles demonstrate the measured impedance in simulations.

FIGURE 7 .
FIGURE 7. Nyquist diagram of Z eq,ac /Z g,11 for droop-controlled grid-forming MMC with dual-loop voltage control in the simulation model of the HVDC system.

FIGURE 9 .
FIGURE 9. Bode diagram of the ac-side impedance of a droop-controlled grid-forming MMC with single-loop voltage control.The solid lines show the analytical impedance, and the asterisks demonstrate the measured impedance from the experimental prototype.

FIGURE 10 .
FIGURE 10.Bode diagram of the ac-side impedance of a droopcontrolled grid-forming MMC with dual-loop voltage control.The solid lines show the analytical impedance, and the asterisks demonstrate the measured impedance from the experimental prototype.

FIGURE 11 .
FIGURE 11.Bode diagram of the equivalent ac-side impedance of the converter Z eq,ac for different values of the integral gain of the voltage controller and the grid impedance Z g,11 .The asterisks indicate experimental results.The close-up box shows the phase margin of Z eq,ac /Z g,11 for the two cases.

FIGURE 12 .
FIGURE 12.Nyquist diagram of Z eq,ac /Z g,11 for droop-controlled grid-forming MMC with dual-loop voltage control for the experimental setup.The contour encircles (−1, 0) with low integral gain indicating an unstable system.A close-up visualizing the critical point is also shown.

FIGURE 13 .
FIGURE 13.Proposed active damping techniques to increase system phase margin.

FIGURE 14 .
FIGURE 14. Bode diagram of the lead compensator F Lead (s).

FIGURE 15 .
FIGURE15.Time-domain verification of the instability scenarios and the impact of proposed modifications in simulations of the HVDC system.The stabilizing loops are enabled at t = 2 s and disabled at t = 10 s.

FIGURE 16 .
FIGURE 16.Experimental time-domain verification of the instability scenarios and the impact of proposed modifications.The stabilizing loops are enabled at t = 2 s and disabled at t = 10 s.

FIGURE 17 .
FIGURE 17. Fast Fourier transform of e q during instability based on recording from the experiments.

FIGURE 18 .
FIGURE 18. Fourier coefficient magnitudes of i s (top) and e (bottom) to the applied perturbation with and without the cross-coupled virtual resistance loop.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

FIGURE 19 .
FIGURE 19.Nyquist diagram of Z eq,ac /Z g,11 .The base unstable case's contour encircles (−1, 0) indicating an unstable system.For all other cases the Nyquist diagram predicts a stable system.

TABLE 2 .
Controller settings of the simulation model of the HVDC system.

TABLE 4 .
Controller settings of the experimental setup.