Frequency and Voltage Compliance Capabilities of Grid-Forming Wind Turbines in Offshore Wind Farms in Weak AC Grids ^†

Abstract

Weak grid conditions challenge the grid integration of offshore wind farms. Especially grids with low inertia and large grid impedance questions frequency and voltage compliance capabilities. Grid-forming wind turbines are a promising technology for weak grids due to the nature of their control strategy. This paper explains the difference in how weak grid conditions are described in the literature and shows how the voltage stability margin changes with the short-circuit ratio and X/R ratio. With that knowledge, the frequency and voltage compliance capabilities of three grid-forming controls in an offshore wind farm are investigated and benchmarked. These three controls are a droop control, a virtual synchronous machine, and a synchronverter. This was done by quantifying their performance during a frequency disturbance with sensitivity to the short-circuit ratio, X/R ratio, and the inertia constant, H. It is concluded that the virtual synchronous machine is the most compliant grid-forming control and that DC-link modeling is of great importance when testing compliance during frequency disturbances.

1. Introduction

The penetration rate of wind energy has increased in the last two decades [1]. However, the introduction of more wind power from potent wind energy sites (PWESs) into power systems is challenged by several factors: (I) PWESs are often far away from the site of consumption, and the large impedance at the point of interconnection (POI) due to the long transmission lines is known as a weak AC grid connection point. Such connection points are related to volatile voltage transient responses [2]. (II) By replacing the power generation from synchronous generators (SGs) with offshore wind farms (OWFs), the inherent stability properties of SGs in terms of damping torque and inertia are somewhat lost or reduced [3,4]. A grid-following (GFL) control strategy for wind turbines (WTs) has been dominating the industry, but the grid-forming (GFM) control strategy is now gaining attention because it introduces some of the stability properties of an SG into converter-based power generation. The main motivation of this paper is, thus, to investigate how an OWF implemented with GFM control performs in terms of frequency and voltage compliance.. Note that this paper is an extension of a conference paper presented at the Universities Power Engineering Conference (UPEC) 2022 [5].

The definition of a weak grid is ambiguous and requires some elaboration. When the first OWFs were built in the 1990s, a weak grid referred to a volatile grid voltage [6]. Today, grids are classified as weak if the voltage and angle are sensitive to changes in the current injection [7]. This sensitivity is usually classified according to the short-circuit ratio (SCR) [2,8]. The SCR is the ratio between the initial symmetric short-circuit power of the grid and the short-circuit power of the infeed [9]. However, because OWFs are often far away from synchronous generators (SGs), the initial symmetric short-circuit contribution from the SGs is neglected, as indicated by IEC 60909 [10]. As the sub-transient reactance of SGs is often around 0.2 pu [11], the initial short-circuit contribution is five times larger than the steady-state short-circuit current [12]. The difference in the resulting short-circuit power is huge and the effect has not been elaborated upon in the literature considered in this study. Many papers proposed that the short-circuit ratio (SCR) is fundamental for system strength classification [3,6,13,14,15,16,17] (these are elaborated upon later in Section 2.1), but there is disagreement about the SCR values distinguishing weak grids from strong ones. This might be due to the unclear short-circuit contributions from SGs, but it is not evident. That being said, there are other formulations of the SCR as a grid strength classification parameter accounting for neighboring inverter-based resources in various ways, which will not be covered in this paper [3,18].

Furthermore, the SCR is a fictitious value, as it is determined by the product of the initial symmetrical short-circuit current $i_{sc}^{″}$, the nominal system voltage $V_{nom}$, and the factor $\sqrt{3}$ [10]. This implies that the impedance specifications, such as the reactance–resistance ratio $X/R$, are neglected in most cases—even though they impact the transient current limitation, the damping [19], and the active power transfer capabilities [20]. A few studies have included the $X/R$ ratio [21,22] in their investigations. Parts of the WT manufacturing industry acknowledge that the SCR cannot reflect all grid parameters as well, and they have proposed a complete WT control solution based on the analysis of the relations of $\delta P/\delta \theta$ and $\delta Q/\delta V$ [23] The $X/R$ is fundamental for this analysis.

Low-inertial-power systems have also been addressed as a problem in weak grids [3]. This is due to the low inertia and weak damping, which easily cause undesired oscillations in the system through grid dynamics [24]. There is a clear correlation between more converter-based power generation and lower system inertia [25]. This means that the frequency control from the converter-based generation must be sufficient for a successful implementation of renewable energy sources in a power system. The operational system frequency is required to be to be within a small band from the nominal frequency by the transmission system operators (TSOs), as some components are sensitive to frequency deviations and are linked to protective requirements. The following examples serve to provide an idea of the numbers and variety of grid code requirements. The Danish TSO, Energinet, requires the protection of wind power plants to withstand a rate-of-change-of-frequency (ROCOF) of ±2.5 Hz/s and to stay online in system frequencies between 47 and 52 Hz [26]. The TSO in Great Britain, the National Grid, requires an operation frequency on a system level within ±0.2 Hz and an ROCOF protection to withstand ±0.125 Hz/s [27,28]. Many events can cause a frequency disturbance directly or indirectly. Examples are a sudden change in demand, generators failing, critical lines tripping, or area separations (islanding). Cases of such incidents cover, among other things, sub-synchronous small-signal stability issues that cause converters to malfunction and black-outs to emerge through a cascade of events ($f_{min}=48.8$ Hz, ${\left|\mathit{ROCOF}\right|}_{\mathit{max}}>0.125$ Hz/s) [28], as well as area separation in the continental power system ($f_{max}=50.6$ Hz, ${\left|\mathit{ROCOF}\right|}_{\mathit{max}}=0.3$ Hz/s) [29,30].

The main problem with weak grid classification is that it depends on the impedance magnitude, impedance angle, and the inertia, but no clear methodology has been devised for a mix of these three parameters. CIGRE is the only source found [3] that defined weak grid thresholds for the SCR, the $X/R$ ratio, and the inertia in the same document, but without relating them for classification purposes. It is the motivation of this paper to investigate frequency and voltage compliance capabilities in terms of the SCR, $X/R$, and inertia.

The traditional control of WTs is based on a GFL technique, where a phase-locked loop (PLL) is absolutely necessary. The presence of a PLL reduces the stability margin [31,32] and might lead to synchronization instabilities [33]. It is, thus, obvious that the stability margin of GFL wind turbines operating in weak grid conditions is further minimized. In [34], a relatively slow PLL was identified to be more prone to instability issues than a faster one. In [35], the critical point for inverter–grid voltage stability was described with respect to a minimum SCR regarding small-signal stability.

GFM WTs are a promising new solution for OWF applications, as they do not depend on a reference voltage for the PLL [36]. Papers emerged in 2009 and 2011 with options for the current loop of the converter to mimic a synchronous machine by using a virtual impedance. The pioneering formulations of the GFM control are known as the virtual synchronous machine (VSM) [37,38] and the synchronverter (SV) [39]. Other GFM control strategies, such as droop control, have emerged since then, and elaborate reviews on these GFM control strategies are available [31,40]. This paper will not go into detail with other control strategies, as they are outside the scope of the study. Only the control strategies listed in

Table 1. GFM control strategies considered in this study.

Control Strategy		Sources
Droop control	(D)	[41]
Virtual synchronous machine	(VSM)	[42,43]
Synchronverter	(SV)	[39]

will be investigated.

The choice of these is relevant due to their wide acceptance and applicability. Droop control is simple and well known. The VSM has been used in some pilot projects [44] and has been the framework for grid codes and grid-forming capabilities for the National Grid [45]. The SV was studied for comparative purposes with respect to another “virtual synchronous machine” regarding inertia emulation.

The choice of these GFM control strategies was further backed by the availability of templates in DIgSILENT’s PowerFactory (PF, version 2021 SP4), i.e. pre-designed implementations of the GFM control strategies. Because the focus of this paper is grid-converter dynamics in weak grid conditions and not the optimization of tuning parameters, no further changes to the templates are made. The tuning parameters are found in

Table A1. The tuning constants implemented in the grid-forming control in the DSL modules from the PowerFactory templates.

Block	Description	Variable	Value	Unit	D	SV	VSM
Droop	Active power droop coefficient	mp	0.01	[-]	√
	Reactive power droop coefficient	mq	0.05	[-]	√
	Low-pass filter cut-off frequency	w_c	60	[Hz]	√
	initial speed setting	f_setpoint	1	[p.u.]	√
Synchronverter	Acceleration time const.	Ta	3	[s]		√
	Damping coefficient	Dp	100	[-]		√
	Voltage gain	Kq	1000	[-]		√
	Reactive power droop coefficient	Dq	20	[-]		√
	Damping filter cut-off freq.	w_c	0	[rad/s]		√
	Initial speed setting	f_setpoint	1	[p.u.]		√
VSM	Acceleration time constant	Ta	3	[s]			√
	Damping coefficient	Dp	100	[-]			√
	Damping filter cut-off frequency	w_c	0	[rad/s]			√
	Voltage setpoint low-pass filter time constant	T_LPF_u	0.003	[s]			√
	Initial speed setting	f_setpoint	1	[p.u.]			√
Virtual impedance	Basic virtual resistance	r	0.006	[p.u.]	√	√	√
	Basic virtual reactance	x	0.006	[p.u.]	√	√	√
	Control mode: 0 = const. Z; 1 = Proportional over-current limitation	Mode	1		√	√	√
	Over-current threshold	i_lim	1.01	[p.u.]	√	√	√
	Proportional factor for additional r	kpr	8	[-]	√	√	√
	Proportional factor for additional x	kpx	8	[-]	√	√	√
	Time constant of low-pass filter	T_lpf	0.0001	[s]	√	√	√
Output voltage	Current limitation: 0 = disabled; 1 = enabled	Mode	1		√	√	√
	Maximum current	i_max	1.2	[p.u.]	√	√	√
	Series resistance	Rseries	0	[%]	√	√	√
	Series reactance	Xseries	10	[%]	√	√	√
Voltage control	Measurement filter time constant	Tr	0.02	[s]			√
	Controller gain	K	0.5	[-]			√
	Minimum AVR output voltage	u_set_min	-1.5	[p.u.]			√
	Maximum AVR output voltage	u_set_max	1.5	[p.u.]			√

.

The main contributions of this paper are, thus, (a) an extensive sensitivity analysis of the SCR, $X/R$ ratio, and inertia constant H, (b) a proof of concept of an OWF with GFM control serving as the main frequency support in some weak grids, and (c) the highlighting of DC-link modeling as inevitable for studies of GFM controls subjected to frequency disturbances.

This paper is organized as follows: Section 2 sketches the discrepancies in the common definition of a weak grid in the literature. Section 3 explains the dependence of the voltage on the power flow across a line and the impacts of the impedance magnitude and phase. This is the context in which the GFM controls are explained in Section 4. Section 5 elaborates on the case study’s configuration and the model used. Section 6 presents the results. Section 7 and Section 8 evaluate and conclude on the benchmarking results, respectively.

2. Weak Grid Definition

Weak grids are considered as high-impedance connection points and have a volatile voltage. Thus, the short-circuit ratio (SCR) and the $X/R$ ratio are central concepts for grid strength classification. The inertia of the system also plays a role in voltage stability, as low-inertial systems require large changes in the power flow during power imbalances.

2.1. Short-Circuit Ratio (SCR)

The SCR is defined as the short-circuit power level from the grid divided by the nominal infeed capacity, which is the OWF in this application:

$$\mathit{SCR}=\frac{s_{sc}^{″}}{S_{\mathit{nom},\mathit{infeed}}}$$

(1)

It is, thus, noted that the SCR is a local parameter and is considered at the point of interconnection (POI). There is a large discrepancy in the grid strength for defining SCR levels in the literature, as summarized in

Table 2. Grid strength for defining SCR levels.

Source	Very Weak	Weak	Strong
[6]	N.A.	< 10	>20
[3]	<3	3 < SCR < 5	N.A.
[13]	<2	2 < SCR < 3	>3
[14]	N.A.	< 6–10	>20–25
[15]	1	N.A.	N.A.
[16]	N.A.	6	N.A.

.

The initial short-circuit power, $s_{sc}^{″}$, is a fictitious value [10] and does not consider the $X/R$ ratio of the impedance, but only the magnitude. The initial short-circuit current, $I_{sc}^{″}\propto s_{sc}^{″}$, differs from the short-circuit peak current, which considers the $X/R$ ratio, as seen in Figure 1. The SCR is, thus, an ambiguous grid-strength-defining parameter alone, as lower $X/R$ ratio values limit the current flow.

2.2. X/R Ratio

Ignoring sub-transient reactance from the grid, which is fair for far-from-generator locations, the grid impedance is defined as:

$$Z_{\mathit{g}}=\frac{V_{\mathit{g}}^2}{S_{\mathit{g}}}=\frac{V_g^2}{S_{\mathit{nom},\mathit{OWF}}·\mathit{SCR}}$$

(2)

The resistance and reactance are found by noting that the impedance angle is given by $\theta =arctan\left(X/R\right)$:

$$\begin{array}{cc}\hfill R_g& =Z_g·cos\theta \hfill \\ \hfill X_g& =Z_{\mathit{g}}·sin\theta \hfill \end{array}$$

(3)

A trigonometric relation of the $X/R$ ratio is noted, and this explains the development of the limited short-circuit peak current shown in Figure 1. The $X/R$ ratio is a less-debated grid strength parameter in academia, but it plays a subtle role in terms of damping. In [3], an $X/R$ ratio below 3 was considered to indicate weak grid conditions, whereas an $X/R$ ratio of between 1 and 5 was defined as a low $X/R$ ratio. The $X/R$ ratio is also a local parameter, as it depends on the Thevenin equivalent of the grid impedance.

2.3. Inertia Constant H

The inertia constant expresses the acceleration time constant of the system, $H={\tau }_a/2$, and is a global parameter, unlike the SCR and the $X/R$ ratio. CIGRE provided indirectly grid-strength-defining inertia levels [3]. The relative inertia constant $H_i$ of the ith machine indicates the total kinetic energy $E_{kin,tot}$ of the system with respect to the N online generators as a linear combination. This further relates to the absolute moment of inertia J and the rotor frequency ${\omega }_r$:

$$E_{kin,tot}=\sum _i^N{H}_i{S}_{i,rat}=\sum _i^N\frac{1}{2}{J}_i{\omega }_r^2$$

(4)

where $S_{i,rat}$ is the ith machine’s rating. It is known that the ROCOF was derived from the swing equation [46]:

$$\frac{\mathrm{d}f}{\mathrm{d}t}=f_{\mathit{nom}}\frac{\Delta P}{2H}$$

(5)

The greater the inertia is, the lower the ROCOF realized from a power imbalance of a magnitude $\Delta P$ will be. Because the ROCOF depends on the power imbalance, as shown in (5), the grid strength from the inertia depends on the event parameter $\Delta P$. According to [3], the grid is considered weak if the ROCOF is greater than 1 and very weak if it is greater than 2. The event parameter $\Delta P$ is fixed in this study to 5% of the nominal OWF capacity. It is acknowledged that frequency compliance has many aspects in terms of duration. This study will only consider the immediate dynamic frequency response period (primary frequency response). In this context, it is worth noting that the frequency regulation services of OWFs must be aligned with dynamic wind forecasts.

3. Power Flow Changes and Voltage Stability Margin

The grid impedance magnitude and impedance angle impact the maximum power transfer capabilities of a line [20,47]. Any frequency recovery from a power imbalance in a grid caused by OWFs thus heavily relies on their voltage compliance capabilities and voltage stability margin. To elaborate on this sensitivity with respect to the SCR and the $X/R$ ratio, PV curves and PQV surface plots will be presented in this section. Note that the inertia constant parameter is omitted from this analysis, as it only affects the transient behavior.

The formal analysis is based on a simplified transmission line that depends on the voltage magnitude and angle of the sending and receiving ends, as illustrated in Figure 2.

This network is comparable to an OWF generating power for the grid, which is seen as a load, as illustrated in Figure 2b. The total complex power is considered:

$$I=\frac{V_s\angle (\delta -\theta )}{Z_{LN}\angle \varphi +Z_{LD}\angle \theta }$$

(6)

It is possible to eliminate the dependency of the angles in the above expression via a network analysis [48,49]. The rewritten equation only depends on the magnitudes of the sending and receiving ends’ voltages, the active and reactive power flow, and the resistance and reactance of the line ($V_s,V_r,P,Q,R,X$, respectively):

$$V_r^4+V_r^2\left(2(RP+XQ)-V_s^2\right)+\left(R^2+X^2\right)\left(P^2+Q^2\right)=0$$

(7)

This equation yields two solutions for the active and reactive power and four solutions for voltage. It is noticed that the feasible solution space $\mathcal{F}$ only obtains voltages, resistances, and reactances equal to or greater than zero for physical consistency, i.e., $V_s,V_r,R,X\in {\mathbb{R}}^{+}$. It is noted that the feasible solution space $\mathcal{F}$ is a subset of the total solution space $\mathcal{S}$, such that $\mathcal{F}\subset \mathcal{S}$.

3.1. PV Curves, Power Factor, and Weak Grid Sensitivity

PV curves are established by solving for the voltage in (7) with respect to a range of active power set points. The reactive power is evaluated at $Q=Ptan\left(\theta \right)$. The rest of the system values are set to somewhat arbitrary values for illustrative purposes at this point (i.e., $V_s=132$ kV, $X=100\mathsf{\Omega }$, and $R=0\mathsf{\Omega }$). In Figure 3, various PV curves are plotted for different power factors.

The locus of critical points shows the respective point of voltage instability [50]. Any equilibrium point achieved below this point would affect the passive instability through other factors in the power system, e.g., tap-changers [51] and, thus, result in a voltage collapse. The effect of injecting reactive power into the system shows that voltage support also increases the voltage stability margin. The voltage stability margin refers to two relations in this context: (1) the margin from the point of operation to the tip of the PV curve and (2) the voltage deviation due to the change in active power. It is clear from the figure that $\theta =-{10}^{\circ }$ is more robust to changes in the active power flow than $\theta =0^{\circ }$, and that the former yields the largest margin with respect to the critical point.

When considering the influence of weak grid parameters on the means of a large and fairly resistive Thevenin equivalent, the PV curves are drastically altered. In Figure 4, the PV curves are plotted for values of $SCR,X/R\in \{1,5,10\}$.

The colors distinguish the $X/R$ ratio; orange is the highest and blue the lowest. For each $X/R$ ratio plotted, there is a dotted, dashed, and solid line. The solidity of the line indicates the corresponding relative grid strength. In this manner, the orange solid line represents the PV curve of a strong grid, and the blue dotted line represents the PV curve of a weak grid ($SCR=X/R=10$ and $SCR=X/R=1$, respectively).

It is acknowledged that the SCR has a larger impact on the voltage stability margin seen at an operation point at $P=1$ pu, and it is emphasized that a combination of weak grid conditions raises compliance concerns without any other grid-stabilizing equipment being considered. Acquiring flexible AC transmission system (FACTS) devices is costly [52] and would eventually challenge the techno-economic feasibility of an OWF project.

3.2. PQV Surface Plots and Voltage Stability Margin

To better illustrate the impacts of the SCR and the $X/R$ ratio on the steady state from the feasible solution space $\mathcal{F}$ provided by (7) and the increased voltage stability margin obtained by injecting reactive power, the solution of the voltage from the equation was mapped into a PQ space. This resulted in the PQV surface plots in Figure 5, where $SCR,X/R\in \{3,10\}$, and the PV curves are for $\theta \in \{0^{\circ },-{10}^{\circ },-{20}^{\circ }\}$.

Apart from the large volume into which the feasible solution space $\mathcal{F}$ expands, not much of the space is compliant with respect to the general grid code requirements and the PQ capabilities of an OWF by design. It is acknowledged that the compliant solution space $\mathcal{C}\subset \mathcal{F}$ is delimited by $V_{min},V_{max},Q_{min},Q_{max},P_{min},P_{max}$.

The plots in Figure 5 illustrate that the compliant solution space is rather large for strong grids and quite small for weak grid conditions. In particular, the impedance magnitude limits the active power transfer capabilities, which, in strong grid conditions, exceeds the OWF’s capacity by far. This provides good stability margins for the operation of the OWF.

The plots show that, for achieving compliant voltage levels, reactive power compensation is crucial for weak grids with low SCRs. However, if the $X/R$ ratio is also low, reactive power compensation loses its effect. This is explained by the increasing impact of coupling effects between resistance and reactance with respect to active power and reactive power (see [53]). This corresponds to the findings from Figure 4.

In Figure 6, the compliant solution space $\mathcal{C}$ of the voltage is shown by the blue–white–red gradient in the range of $[0.95;1.05]$ pu. This is plotted on top of the feasible solution space $\mathcal{F}$, which is plotted with a gray-scale gradient.

The present PQV analysis is only valid for the steady state and ignores all dynamics of the system.

4. Grid-Forming Topologies

A well-established formulation of the concept of GFM converters is still under discussion [31]. However, a key difference between a GFL and a GFM converter is that they behave as a current source (CS) and a voltage source (VS), respectively. Even though both are voltage source inverters, they emulate this behavior with the control. Consider these converter topologies connected to the grid, as shown in Figure 7.

By applying the KVL, the following can be derived:

$$\begin{array}{cc}\hfill \mathrm{CS}:V_c& =V_g+I_c·\left(Z_g\right)\hfill \\ \hfill \mathrm{VS}:V_c& =E-I_c·\left(Z_c\right)\hfill \end{array}$$

(8)

Thus, the GFM control adjusts the current to keep the virtual internal voltage, E, constant despite voltage perturbations. This is illustrated in Figure 8.

In this paper, three different control methods are investigated: droop control, the virtual synchronous machine (VSM), and the synchronverter.

4.1. Droop Control

GFM droop control is achieved by implementing the droop equations:

$$\begin{array}{cc}\hfill f-f_{\mathit{ref}}& =-m_p\left(P-P_{\mathit{ref}}\right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill V-V_{\mathit{ref}}& =-m_q\left(Q-Q_{\mathit{ref}}\right)\hfill \end{array}$$

(10)

where f is the measured frequency, $f_{\mathit{ref}}$ is the frequency reference, $m_p$ is the active power droop gain, P is the measured active power, $P_{\mathit{ref}}$ is the active power reference, V is the measured voltage, $V_{\mathit{ref}}$ is the voltage reference, $m_q$ is the reactive power droop gain, Q is the measured reactive power, and $Q_{\mathit{ref}}$ is the reactive power reference. These were implemented in PowerFactory, as seen in Figure A1 in Appendix A.

4.2. Virtual Synchronous Machine (VSM)

The power synchronization loop of the VSM is created by implementing the swing equation while including damping. By rewriting the swing equation from (5) and recalling that ${\tau }_a=2H$, the following can be derived:

$$\frac{\mathrm{d}}{\mathrm{d}t}{f}_{\mathit{pu}}=\frac{\Delta P}{{\tau }_a}=\frac{P_m-P_e-P_{\mathit{friction}}}{{\tau }_a}$$

(11)

where $\frac{\mathrm{d}f}{\mathrm{d}t}$ is the rate of change of frequency, $\Delta P$ is the power imbalance, ${\tau }_a$ is the acceleration constant, $P_m$ is the mechanical power, $P_e$ is the electrical power, and $P_{\mathit{friction}}$ is the damping loss. This was implemented as shown in Figure A2 in Appendix A.

4.3. Synchronverter (SV)

The SV calculates the frequency and angle by considering the rotor motion (including the damping part). Power synchronization is performed by considering Newton’s second law for an SG:

$$J\alpha \left(t\right)=T_m\left(t\right)-T_e\left(t\right)-D_p{\omega }_m\left(t\right)=T_a\left(t\right)$$

(12)

where J is the inertia, $\alpha$ is the angular acceleration, $T_m$ is the mechanical torque, $T_e$ is the electrical torque, $D_p$ is the damping coefficient, ${\omega }_m$ is the mechanical angular velocity, and $T_a$ is the accelerating torque.

The model of the synchronverter was given in [39] and was implemented in PowerFactory, as shown in Figure A3 in Appendix A. The Zhong equations were implemented in PowerFactory (see also [54]) as follows:

$$\begin{array}{ccccc}\hfill \mathrm{Equation}& \left(7\right)\hfill & \hfill & T_e=M_f{i}_f\widehat{i}\mathbf{cos}(\mathsf{\theta }-\mathsf{\phi })\hfill & \hfill \\ \hfill \mathrm{Equation}& \left(8\right)\hfill & \hfill & e=\omega M_f{i}_f\mathbf{sin}\left(\mathsf{\theta }\right)\hfill & \hfill \\ \hfill \mathrm{Equation}& \left(9\right)\hfill & \hfill & P=\omega M_f{i}_f\widehat{i}\mathbf{cos}(\mathsf{\theta }-\mathsf{\phi })\hfill & \hfill \\ \hfill & \hfill & \hfill & Q=\omega M_f{i}_f\widehat{i}\mathbf{sin}(\mathsf{\theta }-\mathsf{\phi })\hfill & \hfill \end{array}$$

(13)

where $M_f$ is the mutual virtual inductance, $i_f$ is the excitation current, $\widehat{i}$ is the current amplitude, $\omega$ is the frequency, $\mathsf{\theta }$ is the rotor angle, and $\mathsf{\phi }$ is the phase angle. The expressions in bold are vectors.

5. The Model of Offshore Wind Farm

A generic OWF design with 8 MW WTs are introduced of one cluster with six strings in the collector system as shown in Figure 9. Balanced RMS-simulations are performed to study frequency and voltage responses from a frequency disturbance from a power imbalance in the grid scaled to 5% of the installed OWF capacity. The grid voltage is modelled as a synchronous generator (SG) [12].

The power flow in the system of the base case while using the initial conditions for the frequency disturbance was considered, as shown in Figure 10.

To investigate the upward frequency regulation of the OWF, curtailed operation scenarios were also considered. The National Grid requires a minimum of 10% curtailment to enter balancing markets [55]. The test cases are defined in

Table 3. Test case definitions.

Test		Generation and Demand ${}^{\left(\mathit{b}\right)}$			Disturbance ${}^{\left(\mathit{b}\right)}$
#	Op. ${}^{\left(\mathit{a}\right)}$	Grid	OWF	Load	$\mathbf{\Delta }\mathit{P}$
1	C	$1.1$	$0.9$	$2+\Delta P·u\left(t\right)$	$0.05$
2	C	$1.1$	$0.9$	$2+\Delta P·u\left(t\right)$	$-0.05$
3	N	$1.0$	$1.0$	$2+\Delta P·u\left(t\right)$	$-0.05$

^(a) Op. = Operation, C = Curtailed, N = Nominal. ^(b) All quantities are scaled to nominal capacity, P_OWF.

.

All simulations were performed in DIgSILENT’s PowerFactory (version 2021, service package 4) while using the available GFM templates.

6. Results

6.1. Types of Responses

The types of responses of the different GFM controls implemented in the OWF to a frequency disturbance under weak grid conditions are found in Figure 11.

The droop control showed an aggressive and tightly tuned response. It was noticed that the droop coefficient was very small, as the frequency oscillated around the nominal value. Because the droop control compensated for the entire power imbalance, including accounting for the lost kinetic energy by accelerating the system again, it was noted that the active current was saturated by the converter’s current limit at 1.2 pu. The imbalance was only equal to 0.05 pu, and the additional current losses were associated with the frequency and voltage.

The voltage angle $\delta$ did not follow a typical second-order system response, as it did for the VSM and SV. However, this was in accordance with the theory, since the latter two were derived from the swing equation, and the droop control was not.

The VSM and SV had a frequency disturbance response that would have been typical for a synchronous generator. This proved that the implementation and setup were correct. It was noticed that the voltage angle $\delta$ of the VSM suffered from an unstable limit cycle (as seen in Figure 11b). This was caused by various limiters in the control systems [56]. Furthermore, in [43], numerical issues with the VSM were also addressed. The responses of the VSM and SV looked very much alike, but the SV managed to restore the voltage because of the coupling between the inertial emulation and the voltage regulation via the Zhong equations, as seen in Figure 11c.

6.2. Trends of the SCR, $X/R$ Ratio, and Inertia Constant H

An automation script was implemented in Python, which allowed the parameters in PowerFactory to be swept. The SCR, $X/R$ ratio, and inertia constant H were studied in the range of 1 to 10 in steps of 1 for the three GFM control topologies, and the three test cases are given in Table 3. This gave a total of 9000 dynamic simulations. To document the frequency and voltage compliance capabilities, it was desired to extract certain parameters. The benchmark parameters are given in

Table 4. Impacts of the compliance parameters.

Benchmark Parameter			Impact Level ${}^{\left(\mathit{b}\right)}$			Data Manipulation
Symbol	Unit	Type ${}^{\left(\mathbf{a}\right)}$	SCR	X/R	H	Dim. ${}^{\left(\mathbf{c}\right)}$	P < W < F ${}^{\left(\mathbf{d}\right)}$
$\frac{\mathrm{d}f}{\mathrm{d}t}$	Hz/s	P	H	L	H	X/R	0 < 1 < 2
$\Delta f$	Hz	C	H	L	H	X/R	0 < 0.5 < 2
$f_{post,ss}$	Hz	C	L	M	-	H	0 < 0.5 < 2
$\frac{\mathrm{d}v}{\mathrm{d}t}$	p.u./s	P	H	L	H	X/R	0 < 0.2 < 0.4
$\Delta v$	p.u.	C	H	L	H	X/R	0 < 0.02 < 0.05
$v_{post,ss}$	p.u.	C	H	L	-	X/R	0 < 0.02 < 0.05
$\Delta E_{kin}$	kWh	P	L	M	-	H	0 < 10 < 20

^(a) P = Performance, C = Compliance, ^(b) H = High, M = Medium, L = Low, ^(c) Reduced dimension, ^(d) P = Pass, W = Warn, F = Fail.

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Some benchmark parameters were directly related to grid codes and provided qualification thresholds. Other parameters could be considered as performance indicators, and pass/fail thresholds were tentatively derived. Each benchmark parameter in the simulation results could be stored in a three-dimensional array for each test case and GFM scheme, as shown in Figure 12.

To evaluate the performance, the arrays were reduced to two-dimensional heatmaps. The values were given by the maximum value in the dimension of the lowest-impact sweeping parameter (i.e., the SCR, $X/R$ ratio, or H). The impact level was based on the analysis of the sensitivity for test case #1 given in Figure 13. The performance evaluation is found in Figure 14.

The analysis of the voltage and frequency trends with respect to the SCR, $X/R$ ratio, and H was carried out by investigating the responses in Figure 13 for a range of fixed values, which are listed in

Table 5. Impacts of the compliance parameters.

Benchmark Parameter			Impact Level ${}^{\left(\mathit{b}\right)}$			Data Manipulation
Symbol	Unit	Type ${}^{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$	SCR	X/R	H	Dim. ${}^{\mathbf{\left(}\mathit{c}\mathbf{\right)}}$	P < W < F ${}^{\mathbf{\left(}\mathit{d}\mathbf{\right)}}$
$\frac{{\mathrm{d}}_f}{{\mathrm{d}}_t}$	Hz/s	P	H	L	H	X/R	0 < 1 < 2
$\Delta f$	Hz	C	H	L	H	X/R	0 < 0.5 < 2
$f_{post,ss}$	Hz	C	L	M	-	H	0 < 0.5 < 2
$\frac{\mathrm{d}v}{\mathrm{d}t}$	p.u./s	P	H	L	H	X/R	0 < 0.2 < 0.4
$\Delta v$	p.u.	C	H	L	H	X/R	0 < 0.02 < 0.05
$v_{post,ss}$	p.u.	C	H	L	-	X/R	0 < 0.02 < 0.05
$\Delta E_{kin}$	kWh	P	L	M	-	H	0 < 10 < 20

^(a) P = Performance, C = Compliance, ^(b) H = High, M = Medium, L = Low, ^(c) Reduced dimension, ^(d) P = Pass, W = Warn, F = Fail.

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General Remarks

The frequency compliance from the disturbances were very small and far from being uncompliant. However, these frequency disturbances challenged voltage compliance.

Lower $X/R$ ratios were shown to support transient voltage stability, but not with a great impact, and this is far from desired when considering the compliant solution space in Section 3. The note on the limited power transfer capabilities with low $X/R$ ratios in [20] and the increased resistance with increased losses did not further aid the imperative to strengthen the grid with lower $X/R$ ratios. Lower $X/R$ ratios also increased the frequency of the response, $\omega$, in the frequency stability plot (Figure 13c,e).

6.3. Compliance Capabilities

The compliance results are found in Figure 14 and are commented on in the following subsections.

6.3.1. Frequency Parameters

Only the performance parameter for the ROCOF with thresholds defined by CIGRE’s weak grid definition [3] “failed” for SCR = 1 and X/R = 1. This meant that the ROCOF could be classified as a very weak grid for disturbances of ±5% of the nominal power of the OWF.

This result was not different across the GFM controls and was almost completely the same across the tests. Only the droop control differed by not converging with any simulations in this grid condition. However, the droop control would fail every response in test case #1 anyway if a DC-link were implemented in the model. It is acknowledged in this study that it is essential to model the DC-link for frequency disturbances, as the stored energy in the DC-link is exploited during the emulation of inertia [57].

The compliance parameters regarding frequency passed every successive RMS simulation, regardless of the grid strength. This indicated that the GFM controls were compliant with the frequency requirements during frequency disturbances with power imbalances up to 5% of the OWF’s available capacity.

It must be noted that the lack of DC-link modeling challenges this standpoint. It is, however, expected that the frequency will adjust accordingly for the VSM and SV until a sufficient current capacity is gained.

6.3.2. Voltage Parameters

It is obvious from the figures of the pass/fail criteria that the biggest issue resulting from frequency disturbances is the voltage stability in weak grids. The SCR parameters are the dominant deciding factors for compliance issues. In particular, when SCR = 1, it is almost impossible for an OWF to counter the power imbalance and avoid classifications of system incidents, i.e., when the voltage diverges more than 5% from the nominal voltage.

The performance parameter was almost identical in its results across the tests and GFM controls. The more important compliance parameters, $\Delta V$ and $V_{post,ss}$, had some small differences. The VSM is the best-performing GFM control in weak grids, followed closely by the SV. The droop control consequently achieved warning flags from SCR ≤ 4.

6.3.3. Energy Parameters

The kinetic energy lost due to the frequency deviations after a disturbance increased for a larger SCR and a lower X/R, and the SCR was the most dominant factor. The lost kinetic energy parameter indicated how much energy the grid had contributed in order to support the frequency stability.

7. Evaluation of Results

Frequency and Voltage Stability (Benchmark Parameters)

Regarding the frequency stability, no compliance issues were discovered across the various test cases, GFM controls, and SCR, X/R, and H values. The grid was only classified as weak or very weak when $SCR\le 2$ and $X/R\le 2$ while being subjected to a frequency disturbance equivalent to a power imbalance of $\pm 5\%P_{\mathit{OWF}}$.

The following table summarizes the best GFM controls in the individual tests. The conclusions were drawn by counting the numbers of passed tests. Cells noted with a “-” indicate that the performance was identical and, thus, non-comparable. The GFM controls has the following abbreviations: droop control (D), virtual synchronous machine (VSM), and synchronverter (SV).

It is seen in

Table 6. Summary of the best-performing GFM controls with respect to the benchmark parameters.

Test	$\frac{\mathbf{d}\mathit{f}}{\mathbf{d}\mathit{t}}$	$\mathbf{\Delta }\mathit{f}$	${\mathit{f}}_{\mathit{post},\mathit{ss}}$	$\frac{\mathbf{d}\mathit{v}}{\mathbf{d}\mathit{t}}$	$\mathbf{\Delta }\mathit{v}$	${\mathit{v}}_{\mathit{post},\mathit{ss}}$	${\mathit{E}}_{\mathit{kin},\mathit{loss}}$
#1	SV/VSM	-	SV/VSM	D	VSM	SV/VSM	D
#2	-	-	SV/VSM	-	VSM	-	D
#3	SV/VSM	D/VSM	SV/VSM	-	VSM	SV/VSM	D

that the droop control is represented in five cells, the SV is represented in seven cells, and the VSM is represented in 11 cells. The VSM is, thus, the most compliant and reliable GFM control for weak grid conditions. The VSM is dominant in almost all benchmark parameters and shows supremacy in terms of the voltage nadir. This is a significant feature, as the voltage stability is a general issue for weak grids.

All GFM controls failed the performance test when SCR = 1. Recalling that the GFM control responds to the SCR at the terminals, it is acknowledged that the SCR demarcated for failing GFM controls might be even lower if the SCR is considered at the WT terminals and not at the POI.

8. Conclusions and Future Work

There is a need for a standardization of grid-strength classification that considers the inertia, and impedance magnitude, and angle at the connection point.

This was made clear in a sensitivity study of 9000 RMS simulations, where seven performance parameters were used to quantify the frequency and voltage compliance capabilities of GFM control strategies. The quantification resulted in VSM being the best GFM control and droop control being the worst with respect to frequency disturbances.

Future studies will concern the following: (a) the inclusion of DC-link modeling in GFM-controlled converters; (b) a continuation of the studies of GFM controls in weak grid conditions with EMT simulations for more precise and detailed results;(c) eigenvalue analyses of weak grid conditions with respect to the SCR, $X/R$, and inertia constant H; (d) studies of the role of GFM control in converter-based power generation at the distribution level in weak grid conditions.