1. Introduction
With the increasing number of High-Voltage Direct Current (HVDC) links integrated into the Alternating Current (AC) transmission system and converters, power electronics relying on complex control systems play a more dominant role in system stability. In particular, remote offshore wind farms are connected by high-voltage cables and rely on HVDC technology due to the high reactive power demand of AC-based systems. Additionally, modern wind farms utilize mainly fully-scaled Voltage Source Converters (VSCs) for the offshore grid connection. VSC and state-of-the-art HVDC converters, such as Modular Multilevel Converters (MMCs), have response times that are significantly faster compared to traditional power system components [
1]. Several interactions of converters with the connected AC system or other converters have been reported. For instance, in 2014, the offshore wind farm Bard Offshore 1 had to be shut down due to high oscillations in the grid current and voltage at frequencies other than the grid frequency of 50
[
2]. In addition, the 2015-commissioned HVDC INELFE system connecting Spain and France experienced resonances between the HVDC system and the AC grid [
3,
4]. Furthermore, unexpected oscillations were observed in a weak part of the China Southern Grid’s transmission system after a Static Synchronous Compensator (STATCOM) was put into operation [
5].
Consequently, electromagnetic phenomena need to be considered for assessing the stability of modern power systems. Hence, the classical power system stability classification was extended to consider the impact of power electronic interfaced technologies [
6]. Whereas this interaction phenomenon is often referred to in the literature as harmonic stability [
7,
8], the updated stability definition classifies the aforementioned interaction incidents as converter-driven stability with fast interactions [
6]. Converter-driven instabilities are mainly caused by the converters’ control system that interacts with the AC grid impedance [
6,
9]. However, disclosing manufacturer-specific converter control systems is highly improbable due to intellectual property concerns [
10], making it challenging to model the converters’ real frequency behavior. In addition, Transmission System Operators (TSOs) are unlikely to share details of their grid models with other parties.
Several methods and approaches exist to investigate converter-driven stability. For example, eigenvalue analyses have the advantage that they can assess stability globally for an entire system. Additionally, participation factors provide additional information about controllers and parameters affecting system stability [
11]. However, this requires a state-space model of the entire system, including linearized component models. Therefore, state-space modeling is not applicable when detailed information of the components is not provided. A promising method to investigate converter-driven stability problems is the Impedance-based Stability Criterion (IbSC) that models a system as two subsystems with frequency-dependent impedances and uses classical control theory to assess the stability in the frequency domain [
12,
13]. The IbSC allows for a direct and straightforward stability assessment. By providing Phase Margins (PMs), it indicates how close a system is to instability [
7,
14]. It can also be consecutively applied to assess the stability of large-scale systems by defining different interfaces and subsystems [
15]. Moreover, due to short computation times, numerous scenarios can be investigated, which would not be possible when using Electromagnetic Transient (EMT) simulations [
7,
14]. The IbSC requires frequency-dependent impedance models of the system components, which can be derived analytically using small-signal linearization techniques [
16,
17,
18] or numerically on the basis of EMT simulations or measurements using frequency sweep techniques [
19,
20].
In the literature, stability analyses often utilize analytical impedance models derived based on a white-box approach that implies that the converter structure and the control system’s block diagrams are known [
14,
21,
22,
23]. This approach offers excellent insight into the cause of interactions and can be used to investigate the impact of specific controllers and parameters on stability [
24]. However, it does not allow for stability analyses of black-boxed systems and is not applicable for industrial applications. Moreover, analytical models are often based on a simplified electrical structure of converters so that the real frequency behavior of components might not be considered [
11,
25,
26]. In addition, the models often neglect control loops, assuming that they are not relevant [
16]. To deal with the three-phase nature of the transmission system, authors advocate modeling the impedances in the dq-domain [
27,
28]. Couplings between phases and frequencies are included in the models using a 2 × 2 impedance matrix. The authors demonstrate that neglecting the coupling can result in false stability predictions [
28,
29]. However, approaches that rely on impedance representations in the dq-domain require the definition of a common reference angle
for the dq-domain transformation [
30,
31,
32]. Thus, this approach makes it challenging to use different entities’ impedance models (e.g., converter manufacturer and TSO). They would have to align their models on a reference angle which is not applicable for industrial studies.
As a result, utilized approaches are not aimed at investigating real and industrial systems. They cannot be applied to black-boxed systems such as converter controller replicas, representing the frequency behavior of industrial control systems. Thus, this paper presents an approach to assess the stability of a system, where no detailed information of a converter control system is provided and where the system’s models (e.g., converter and grid impedance) are provided by different entities. The approach builds upon a frequency-sweep approach to assess the converter-driven stability [
33,
34]. Not requiring full-system knowledge, this approach makes it possible to employ impedance models of a black-boxed system such as a converter controller replica or measurement-based models of physical laboratory converters. Additionally, models can be provided by different entities for the stability assessment. For instance, a converter manufacturer can provide the impedance model to a TSO. The TSO is then able to investigate the converter-driven stability using their grid impedance model. In this paper, the proposed impedance derivation method in [
33,
34] is extended to include additional coupling currents resulting from the frequency sweep measurements. To determine these additional coupling currents, the spectrum of physical MMC is analyzed using a laboratory system—here referred to as the MMC Test Bench (TB). The derived MMC impedances are used to investigate the stability of an MMC in grid-following control mode and the adjacent AC system, representing the onshore side of an HVDC link. The AC grid impedance is calculated analytically with respect to different Short-Circuit Ratios (SCRs). On the offshore side, the stability of a wind farm and the offshore MMC operating in grid-forming control mode is investigated utilizing the impedance measurement of an offshore Wind Turbine (WT) VSC controller replica. Therefore, this paper contributes to the evaluation of converter-driven stability under realistic conditions by evaluating the stability of an offshore HVDC system, employing laboratory converters and converter control replica systems for the analysis.
The paper is structured as follows: First, the IbSC is presented, and an overview of the impedance derivation approach is given. The IbSC is validated by comparing the stability predictions to the results of an EMT time-domain simulation. Subsequently,
Section 3 demonstrates the extended frequency-sweep approach and shows how the additional measurements are incorporated in the impedance models.
Section 4 presents the Control Hardware in the Loop (CHiL) TB setup for the WT VSC controller replica impedance measurement. Based on the derived impedance models, the following stability analysis in
Section 5 presents and discusses the results of the onshore and offshore test case. Concluding the paper, the main findings are given in
Section 6.
3. Extended MMC Impedance Derivation Method
Representing the frequency behavior of the MMCs, in this work, the corresponding impedance models are derived based on physical laboratory converters using the MMC TB laboratory at RWTH Aachen University. Thus, inaccuracies due to model simplifications can be avoided, and the impedance derivation method is demonstrated in a laboratory environment. First, the frequency spectrum of the MMC TB response is evaluated when being subjected to different perturbation signals to determine relevant frequency components. The MMC TB laboratory consists of several real-time simulators and low-voltage MMC with the parameters given in
Table 4 [
45]. For measuring the MMC impedances, two MMCs are physically connected through Pi-sections on the DC side and coupled with real-time simulators through power amplifiers on the AC side [
34]. The control system is identical to the controls of the simulated MMC model as presented in
Figure 6.
Figure 13 illustrates the MMC TB setup that is used for deriving the laboratory MMC’s impedances. The power amplifier can be operated either in current or voltage mode. In current mode, the power amplifier connected to MMC 1 feeds in a current consisting of the grid and the perturbation current. In voltage mode, the power amplifier supplies a voltage consisting of the grid voltage superimposed by the perturbation voltage. In both cases, power amplifier 2 supplies the grid voltage for MMC 2. A detailed overview of the measurement setup can be found in [
34].
The MMC TB impedance
is scaled up according to the scaling factors
and
[
34] to align with the AC grid impedance and the 1
wind turbine VSC. The scaled MMC TB impedance is
with
and
The equivalent resistance
and inductance
are calculated using the simulated full-scale MMC parameters given in
Table 1, while
and
are based on the MMC TB parameters given in
Table 4 as described in [
34].
3.1. Spectrum Analysis
The impedance derivation method introduced in
Section 2.2 is applied to measure the impedance of the MMC TB converters. A preceding analysis investigates the spectrum of the MMCs signals to determine with what frequencies the MMC TB responds to a perturbation. When operating in grid-following control mode, the MMC is perturbed by a voltage source, causing the converter to respond with a current. Therefore, the voltage injection approach shown in in
Figure 4 is used and power amplifier 2 operates in voltage mode as shown in
Figure 13. Applying a voltage perturbation at 40
, 70
, 80
and 90
, the corresponding current spectrum is analyzed.
Figure 14 shows the effect of the perturbations on the current spectrum. As expected, the converter also responds with a current at 40
, 70
, 80
and 90
. However, for positive-sequence perturbations, the spectrum depicted in
Figure 14a also contains current responses at 60
and 140
for a 40
perturbation, at 30
and 170
for a 70
perturbation, at 20
and 180
for a 80
perturbation, and at 10
and 190
for a 90
perturbation. In fact, for a 40
perturbation, the response at 60
exceeds that at 40
, indicating a high degree of coupling in that frequency region. The results indicate a pattern in the current responses. When being subjected to a voltage perturbation with
, the MMC responds with a current with
and
in addition to a current at
.
These currents are referred to as (mirror) frequency coupling [
29,
39,
46] and are subsequently defined as a negative and positive coupling current dependent on the grid frequency
fG and the perturbation frequency
with
and
Due to nonlinearities of controllers, converters respond to a perturbation frequency with harmonics at frequencies other than the perturbation frequency. These harmonics can be identified in the dq-domain when the nondiagonal elements in a dq-domain impedance matrix are not zero and cross-couplings exist [
28,
32]. This coupling effect is typical for VSC including MMCs [
39,
47]. It can be caused by the grid frequency used in the dq-transformation of the converters’ control systems [
5]. In particular, controllers that control the d- and q-axis current differently, such as the tracking controllers of the PLL, can be responsible for causing the coupling frequencies [
46]. The effect was seen in STATCOM measurements in the China Southern Grid [
5]. The PLL was considered responsible for transforming a component at the frequency
f into two simultaneous components in the stationary frame, one at
and the other at
[
5]. However, different parts of the control system, such as the PWM, can also produce the coupling currents [
48].
The spectrum depicted in
Figure 14b, shows that no significant current magnitudes at frequencies other than the perturbation frequency can be seen for a negative-sequence perturbation.
Figure 15 shows the perturbation response current
as well as the previously identified coupling currents
and
when the MMC TB is subjected to a perturbation voltage source with frequencies ranging from 1
to 10
. This means that for instance, for a perturbation frequency with
the current response at 40
, at 60
, and at 140
can be seen in the Bode plot. The current trajectories reveal that the current’s magnitude at
is significantly smaller than at
. Furthermore, the plot shows that the frequency coupling effect is only significant at frequencies close to the grid frequency. However, as already seen in the spectrum analysis, the current response at
exceeds that at the perturbation frequency between 30
and 60
. The response at the coupling frequencies converges to zero for higher frequencies, and the MMC responds with the current at
only.
When operating in grid-forming control mode, the converter is perturbed with a current source and responds with a voltage. In this case, the current injection approach shown in
Figure 4 is used and power amplifier 2 operates in current mode as shown in
Figure 13. The converter reacts only with voltages at
as it can be seen in the voltage spectrum in
Figure A2 of the
Appendix A.
To conclude, the coupling current is only significant for the given system when
the MMC operates in grid-following control mode,
the MMC is subjected to a positive-sequence perturbation,
the perturbation frequency is below 200 .
3.2. Coupling Modeling
As demonstrated in the previous section, the MMC responds to a positive-sequence voltage perturbation at
not only with a current at the same frequency but also with a current at
when operating in grid-following control mode.
Figure 16 illustrates how the additional current introduces a dependency of the grid impedance on the measured current response at
.
The voltage perturbation
causes a current response at the same frequency,
, and also one at the coupling frequency,
, as seen in the spectrum analysis. This current leads to an additional voltage
whose magnitude depends on the grid strength and the resulting grid impedance through the grid impedance. This voltage can be interpreted as an additional perturbation source that causes a current response at the same frequency, which is
. Additionally, the converter also responds to the additional perturbation
with a current
at frequency
As a result, the current response at
consists of the converter response to the voltage perturbation
and that to the additional voltage perturbation
. The latter introduces a nondesirable dependency of the converter impedance on the grid impedance, which nullifies the advantage of the IbSC that both subsystems’ impedances can be independently derived. While the measurement setup can include the grid impedance, this approach would further increase the required number of measurement series. Every grid impedance would require a new measurement; therefore, this approach is not feasible. Thus, the converter impedance is measured in ideal conditions, not including the grid impedance. The impact of the additional coupling current on the converter impedance can be subsequently included in the converter impedance model by an additional parallel impedance as presented in
Figure 17 [
39].
The additional impedance
depends on the grid impedance as well as the additional current
and is added in parallel to the converter impedance
. For simplification, it will be derived as the admittance
so that the overall converter admittance
The admittance
is derived on the basis of the model presented in [
49] which maps the current
to the perturbation voltage
by defining a admittance
Figure 16 illustrates that the current
leads to a voltage
through the parallel converter and grid and converter admittance so that
The converter responds to this additional perturbation with the current
at the same frequency and also with a second coupling current
at frequency
as derived in (
12). It can be expressed by
Replacing
in the right summand of (
13) by (
16) gives
Now
in (
17) can be represented by (
15) so that
Then
as in (
14) can be used to replace
in (
18) which results in
as in [
49]. The resulting admittance considers the influence of the grid impedance by integrating the current
into the overall impedance model
. Including the admittance
into the overall impedance model requires recording also the signals
and
as shown in (
14) and (
19). Consequently, the impedance measurement method in [
33] is extended to also record the voltage
at
and the coupling current
at
which can be seen in
Figure 15. Equation (
19) also shows that the converter admittance
and the grid admittance
need to be obtained also for the coupling frequency
for deriving
. Thus, the derived admittances are additionally shifted so that the corresponding values at
can be added to the model.
Figure 18 shows the impact of including the coupling current in the overall MMC impedance model. A low SCR (
) and consequently high grid impedance
significantly influences the MMC impedance trajectory at frequencies below 50
. A higher SCR (
) shows a significantly reduced impact with the overall impedance almost matching the impedance when no coupling is modeled and
.
Figure 18 also demonstrates that the coupling current does not affect the overall impedance at higher frequencies. Therefore, the coupling current is neglected in the impedance modeling for frequencies higher than 200
.
4. WT VSC Controller Replica System
The VSC of the wind turbine determines the frequency behavior of the offshore wind farm. The related impedance models are derived in the DNV GL Smart Grid Lab in Arnhem, The Netherlands. The goals of the laboratory are [
40,
41]:
To propose a PHiL test circuit to derive the impedance model of the power converter unit of a vendor-specific wind turbine (i.e., the power converter unit of a commercial 1
wind turbine generator). Further details for the PHiL TB can be found in [
40].
To establish a CHiL TB to derive the impedance model a VSC WT controller replica that can be used for the grid integration of offshore wind power plants.
To compare the PHiL impedance of the VSC with its equivalent CHiL and provide suggested practices for potential industrial applications.
For this work, the CHiL impedance models the frequency behavior of the wind turbine VSC since it can be obtained with a higher bandwidth than the PHiL impedance. The CHiL TB depicted in
Figure 19 is configured with the following key components:
The OPAL-RT real-time simulator consisting of OP5700.
A 1 WT VSC controller replica from Ming Yang Wind Power, Zhongshan, Guangdong, China.
The real-time simulator OPAL-RT OP5700 is the core of the system. It simulates a single wind turbine, including a wind turbine’s electrical subsystem, the generator, filters, transformer, the offshore AC-grid, and breakers. The simulations are FPGA-based. The MingYang Wind Power WT VSC controller replica connects to the OP5700 system through digital and analog channels [
40,
41]. The real-time simulator measures the grid side current and voltage signals
and
and the generator side current and voltage signals
and
. The analog output interface exports the signals. The OP5700 communicates with the WT VSC controller replica through 10
analog signals. The controller replica then measures the terminal voltage and current coming from the controller replica. Using its time-stamped digital input interface, the OP5700 can interface with the controller replica’s measured signals. The resulting dq-impedance models used for the stability analysis are transformed into the sequence domain [
50] and adjusted for outliers [
45].