Inertia Support Capability Evaluation for Wind Turbine Generators Based on Symmetrical Operation

Abstract

With the increasing integration of new energy into the grid, the level of system inertia has been significantly reduced, posing a severe challenge to frequency stability. Consequently, there is an urgent need for wind turbine generators (WTGs) to actively provide inertia support through virtual inertia control. Assessing the inertia support capability of WTGs reasonably and setting appropriate controller parameters based on this assessment is a topic worthy of discussion. As WTGs’ characteristics are mostly ignored in the evaluation of inertia support capability for WTGs, an evaluation method based on symmetrical operation is proposed. The proposed method considers the impact of real inertia and aerodynamic characteristics, thereby helping to determine reasonable virtual inertia coefficients and de-loading reserve capacity for WTGs. With the proposed method, it can be determined that large WTGs can provide inertia support capabilities close to those of synchronous generators to the grid without exceeding a 0.1% reduction in reserve capacity during de-loading operation.

1. Introduction

In recent years, with the continuous maturation of new energy generation technologies, the proportion of new energy in the grid has been increasing. New energy sources operate dynamically decoupled from the grid frequency and no longer exhibit the inertia characteristics that hinder the change of rotor speed (i.e., grid frequency) of traditional synchronous generators (SGs), which are predominantly interfaced to the grid through power electronic converters [1,2]. Consequently, the grid with a high penetration of new energy increasingly exhibits “low-inertia” characteristics, with a diminishing capacity to withstand disturbances, thereby threatening system frequency stability [3,4].

A feasible solution to mitigate the issue of reduced inertia in the grid is to employ virtual inertia control, effectively simulating the inertia characteristics of synchronous generators [5,6]. Virtual inertia control integrates the rate of change of the grid frequency into the power control of new energy sources, thereby endowing new energy sources with a supporting effect that hinders the frequency drop of the grid (i.e., inertia support). When frequency oscillations occur in the grid due to an imbalance between generation and load, these sources adjust their output power to suppress the frequency variations [7].

The provision of inertia support to the grid by new energy sources is primarily determined by the gain coefficient (referred to as the virtual inertia coefficient hereinafter) of the virtual inertia control [8,9,10]. Concurrently, new energy sources must address the energy source for inertia support through measures such as de-loading [11] or the integration of energy storage systems [12]. In this context, while a higher virtual inertia coefficient can enhance the equivalent inertia provided by renewable energy sources to the grid, it also necessitates the expenditure of greater energy for inertia support.

Distinct from photovoltaic generation, wind power generation retains a physical rotor that can mitigate the imbalanced power between the prime mover (i.e., wind rotor) and the generator through the absorption and release of rotor kinetic energy [11]. With the increasing capacity of single wind turbine generators (WTGs), the dimensions of the blades and the inertia of the wind rotor are also progressively expanding [13], allowing the significant kinetic energy contained in the wind rotor to be utilized as an energy source for virtual inertia control in WTGs.

On this basis, research results have suggested using wind rotor inertia to offer grid inertia support without de-loading [14], but improper virtual inertia coefficient settings can excessively decrease rotor speed, leading to instability or power drop in WTGs [15,16]. To address this, References [17,18,19] set the virtual inertia coefficient based on the impact of control parameters on frequency stability, guiding practical parameter tuning. References [20,21] adjust the virtual inertia coefficient in real time according to kinetic energy release, improving rotor stability for WTGs. Reference [15] enhanced traditional virtual inertia control with a nonlinear controller, dynamically regulating support power based on wind and rotor speed variations, stabilizing rotor stability for WTGs, and preventing secondary frequency drop in the grid due to power drop. In summary, the above research has focused on tuning the virtual inertia coefficient, variable coefficient schemes, and enhanced virtual inertia control to maintain stability for WTGs.

Therefore, the inertia support provided by WTGs is primarily determined by the virtual inertia coefficient, as their rotor speed is decoupled from grid frequency, rather than being constrained by actual rotor inertia. However, the magnitude of the real rotor inertia influences the rate of speed change under virtual inertia control. Additionally, the unique aerodynamic characteristics of the wind rotor couple the rotor speed with the mechanical power output, indirectly affecting the inertia support capability. Current research on virtual inertia control mainly focuses on the rational setting of the virtual inertia coefficient to maximize inertia support while ensuring stable operation for WTGs. However, most research evaluates the appropriateness of the virtual inertia coefficient based on practical outcomes, with limited attention given to the effects of real rotor inertia and aerodynamic properties on virtual inertia control. Therefore, it is difficult to evaluate the ability of WTGs to support inertia based on their own inertia.

To address the aforementioned issues, this paper investigates the impact of prime mover characteristics on the traditional virtual inertia control of WTGs, aiming to explore whether the kinetic energy contained in the rotor of the WTG has the potential to serve as an energy source for providing inertia support to the grid. The contributions of this paper are as follows:

(1)

Considering the coupling effects among power, inertia, rotor speed, and aerodynamics, this paper analyzes the energy transfer process during the active inertia support of WTGs under virtual inertia control, identifying two key factors that influence the inertia support capability, which are real inertia and aerodynamic characteristics.

(2)

Furthermore, a symmetrical operation mode for WTGs is developed to assess the inertia support capability while accounting for prime mover characteristics, which is defined as an operation mode with a fixed ratio of WTGs’ rotor speed to SGs’ rotor speed.

(3)

Analysis results based on a standard IEEE 10-machine 39-bus system and an NREL 5MW WTG simulation model indicate that the inertia support provided by large WTGs has a minimal impact on their aerodynamic efficiency under non-fault conditions, allowing them to deliver inertia support equivalent to that of SGs with a power reduction of no more than 0.1%.

2. Mathematical Model and Virtual Inertia Control of WTGs

This section mainly focuses on the mathematical model and virtual inertia control of WTGs, which constitutes the basis for the analysis in the later sections.

2.1. Mathematical Model of the WTG

A block diagram of the WTG structure based on a permanent magnet synchronous generator [22] is shown in Figure 1. The WTG utilizes the kinetic energy inherent in the moving air, which is converted into rotor kinetic energy by the lift force exerted on the blades. Thereafter, the rotor, via a gearbox, drives the rotation of the generator, and the electromagnetic power output from the generator is fed into the power grid through the power electronic converters.

The aerodynamic power $P_{\mathrm{m}}^{\mathrm{WT}}$ can be expressed as

$$P_{\mathrm{m}}^{\mathrm{WT}}=0.5\rho \mathsf{\pi }{R}^2{v}^3{C}_{\mathrm{P}}\left(\lambda , \beta \right)$$

(1)

where $\rho$ is the air density, $R$ is the wind rotor radius, $v$ is the wind speed, and $C_{\mathrm{P}}\left(\lambda , \beta \right)$ is the wind energy utilization coefficient, which is a function of the tip speed ratio $\lambda$ and the pitch angle $\beta$. The term $\lambda$ is defined as the ratio of the linear velocity of the wind turbine blade tip to the wind speed

$$\lambda ={\displaystyle \frac{{\omega }_{\mathrm{r}}^{\mathrm{WT}}R}{v}}$$

(2)

where ${\omega }_{\mathrm{r}}^{\mathrm{WT}}$ is the rotor speed of the WTG. When the wind speed is lower than the rated wind speed, the pitch angle is generally fixed at 0° and $C_{\mathrm{P}}$ can be regarded as a function of $\lambda$. At this time, the $C_{\mathrm{P}}\left(\lambda \right)$ has a single-peak characteristic, that is, there is an optimal tip speed ratio ${\lambda }_{\mathrm{opt}}$ so that the wind energy utilization coefficient obtains the maximum $C_{\mathrm{P}\max }$, and the WTG can capture the maximum wind energy.

The drive chain consists of a low-speed shaft on the wind rotor side and a high-speed shaft on the generator side which are connected via a gearbox. If the drive chain is approximated as a rigid shaft, the drive chain model based on a simplified single mass block [23] is obtained as follows:

$$\left\{\begin{array}{l}{J}_{\mathrm{h}}^{\mathrm{WT}}{\omega }_{\mathrm{g}}^{\mathrm{WT}}{\displaystyle \frac{d{\omega }_{\mathrm{g}}^{\mathrm{WT}}}{dt}}=P_{\mathrm{m}}^{\mathrm{WT}}-P_{\mathrm{e}}^{\mathrm{WT}}\\ J_{\mathrm{h}}^{\mathrm{WT}}={\displaystyle \frac{J_{\mathrm{r}}}{N_{\mathrm{g}}^2}}+J_{\mathrm{g}}\end{array}\right.$$

(3)

where $J_{\mathrm{r}}$ and $J_{\mathrm{g}}$ are the rotor inertia of the wind rotor and the generator, respectively, $N_{\mathrm{g}}$ is the gear ratio, and $J_{\mathrm{h}}^{\mathrm{WT}}$ is the overall rotor inertia of the wind turbine converted to the high-speed side. The above parameters can be obtained according to the hardware parameters of the WTG when it leaves the factory. $P_{\mathrm{e}}^{\mathrm{WT}}$ is the generator electromagnetic power, and ${\omega }_{\mathrm{g}}^{\mathrm{WT}}$ is generator speed, which has

$${\omega }_{\mathrm{g}}^{\mathrm{WT}}={\omega }_{\mathrm{r}}^{\mathrm{WT}}\cdot N_{\mathrm{g}}$$

(4)

The comprehensive dynamic response process of the WTG in Figure 1 includes both electromechanical dynamics and electromagnetic dynamics. The former concentrates on the energy conversion process from wind to electrical energy within the wind turbine system, while the latter addresses the variations in the states of the electrical components during power generation. Given that the regulation process and response time of the electromagnetic dynamics are on the millisecond timescale, which is significantly lower than the second-scale timescale associated with virtual inertia control for WTGs, it is feasible within the scope of this study to decouple the fast and slow subsystems, thereby neglecting the electromagnetic dynamics and assuming that the generator can instantaneously output the reference power command $P_{\mathrm{e}.\mathrm{ref}}^{\mathrm{WT}}$ [14], which has $P_{\mathrm{e}}^{\mathrm{WT}}=P_{\mathrm{e}.\mathrm{ref}}^{\mathrm{WT}}$. Therefore, for the rotor-side converter in Figure 1, the virtual inertia control power command of the WTG through the generator control response is also applicable to the grid-side converter, which can be referred to [24,25].

2.2. Virtual Inertia Control

Virtual inertia control of the WTG is used to adjust the active power output according to the grid frequency change, thus emulating the inertia response characteristics of the SG and providing inertia support to the grid. As shown in Figure 2, the reference power $P_{\mathrm{e}.\mathrm{ref}}^{\mathrm{WT}}$ of the WTG under virtual inertia control can be expressed as

$$\left\{\begin{array}{l}{P}_{\mathrm{e}.\mathrm{ref}}^{\mathrm{WT}}=P_0^{\mathrm{WT}}+\Delta P_{\mathrm{VIC}}\left(t\right)\\ \Delta P_{\mathrm{VIC}}\left(t\right)=-K_{\mathrm{pf}}\cdot {\displaystyle \frac{df\left(t\right)}{dt}}\cdot {\displaystyle \frac{S_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\end{array}\right.$$

(5)

where $P_0^{\mathrm{WT}}$ is the output power in the initial equilibrium state, which is related to the operating scenario of the WTG. In the mode of maximal wind energy capture, the WTG is operated under maximum power point tracking (MPPT), where $P_0^{\mathrm{WT}}$ is the output power when the WTG is operated at the MPPT (defined as the MPPT operation hereafter). In the mode of de-loading control, the speed regulation-based active power control is adopted for the WTG, where $P_0^{\mathrm{WT}}$ is the output power when the WTG is operated at the balance point of de-loading (defined as the de-loading operation hereafter). $\Delta P_{\mathrm{VIC}}$ is the inertia support power calculated by Equation (5), $K_{\mathrm{pf}}$ is the virtual inertia coefficient to be set, $S_{\mathrm{N}}^{\mathrm{WT}}$ is the rated power of the WTG determined when the WTG leaves the factory, $f$ is the real-time grid frequency obtained by measurement, and $f_{\mathrm{N}}$ is the nominal frequency of the grid determined by the power grid dispatching. In either scenario, the method relies on rapid regulation of the output power of the WTG, changing its speed to release/absorb rotor kinetic energy and buffering the unbalanced power between aerodynamic and electromagnetic power, so as to provide inertial support to the grid.

It is pointed out that according to the principle of frequency regulation in the grid, the frequency change is negatively correlated with the regulation of the power supply, that is, when the power shortage of power grid leads to a decrease in the grid frequency, an increase in the wind power output hinders the decrease in frequency. Therefore, there is a negative sign for calculating $\Delta P_{\mathrm{VIC}}$ in Equation (5) and Figure 2 [11].

3. Analysis of Influencing Factors for the Inertia Support Capacity of WTGs

The WTGs actively adjust their output power in response to grid frequency changes through virtual inertia control, and thereby provide inertia support. This section will further analyze the energy transfer in the active support process, and discuss the difference in energy utilization requirements between inertia support based on virtual inertia control and primary frequency regulation based on droop control. The impact of the real inertia and the inertia support capability of WTGs considering aerodynamic characteristics are also analyzed.

3.1. Energy Transfer in Inertia Support Processes

As can be seen from Section 2.1, the WTG converts the captured wind energy into rotor kinetic energy. Furthermore, the kinetic energy is converted into electrical energy using the generator and transmitted to the electrical grid. However, during the process of providing inertial support to the grid, the aerodynamic power of the WTG changes with the wind speed and the rotor speed. The electromagnetic power has to respond to the grid frequency changes, and thus the aerodynamic power and electromagnetic power cannot be guaranteed to be equal in real time, which makes the WTG accelerate/decelerate to store/release the rotor kinetic energy to achieve power balance.

For the mathematical model introduced in Section 2.1, the rotor kinetic energy $E_{\mathrm{k}}$ of the WTG at time $t$ is

$$E_{\mathrm{k}}\left(t\right)={\displaystyle \frac{1}{2}}{J}_{\mathrm{h}}^{\mathrm{WT}}{\left({\omega }_{\mathrm{g}}^{\mathrm{WT}}\right)}^2\left(t\right)$$

(6)

Neglecting energy losses, the rate of change of the kinetic energy ${\dot{E}}_{\mathrm{k}}$ can be expressed as

$${\dot{E}}_{\mathrm{k}}\left(t\right)=P_{\mathrm{m}}^{\mathrm{WT}}\left(t\right)-P_{\mathrm{e}}^{\mathrm{WT}}\left(t\right)$$

(7)

When the electromagnetic power is greater than the aerodynamic power, the kinetic energy of the WTG decreases, and vice versa. Combining Equations (1) and (5), the following equation can be obtained by substituting the aerodynamic power expression of the WTG and the virtual inertia control expression into Equation (7).

$${\dot{E}}_{\mathrm{k}}\left(t\right)={\displaystyle \frac{1}{2}}\rho \pi R^5{\omega }_{\mathrm{r}}^3\left(t\right){\displaystyle \frac{C_{\mathrm{P}}\left(\lambda \right)}{{\lambda }^3}}-\left(P_0^{\mathrm{WT}}+\Delta P_{\mathrm{VIC}}\left(t\right)\right)$$

(8)

A schematic diagram of a typical inertia support dynamic process for WTGs that deals with frequency dip in the grid under the MPPT operating scenario is shown in Figure 3. For virtual inertia control, the WTGs operate at a stable equilibrium state at the beginning of the frequency disturbance. At this time, ${\dot{E}}_{\mathrm{k}}$ is mainly determined by $\Delta P_{\mathrm{VIC}}$. As the rotor kinetic energy is released, the rotor speed of WTGs will reach a minimum point, followed by a gradual increase in aerodynamic and electromagnetic power. When the grid frequency reaches the steady state again, the rate of change of frequency will become 0. At this time, the $\Delta P_{\mathrm{VIC}}$ becomes 0 and the WTGs return to the equilibrium state before the disturbance occurs. Therefore, the WTGs not only provide a certain degree of inertial support in the early stage but also reduce their power output for speed recovery in the later stage. Compared with the case where no inertial support is provided, the process of recovering the rotor speed of WTGs is equivalent to “absorbing” some energy from the grid.

3.2. Difference in Energy Utilization Requirements Between Virtual Inertia Control and Primary Frequency Regulation

Section 3.1 analyzes the energy transfer in the inertial support process for WTGs. This section will analyze the difference in kinetic energy release requirements between inertia support and primary frequency regulation, to reveal the inertia support characteristics under inertia support control and to provide the basis for a summary of the factors influencing inertia support capacity in Section 3.3.

In analogy to the primary frequency regulation of SGs, the WTGs adjust their output power by introducing deviations from the grid frequency, that is, droop control [11]. As shown in Figure 4, the output power of the WTG under this control can be expressed as

$$\left\{\begin{array}{l}{P}_{\mathrm{e}.\mathrm{ref}}^{\mathrm{WT}}=P_0^{\mathrm{WT}}+\Delta P_{\mathrm{DC}}\left(t\right)\\ \Delta P_{\mathrm{DC}}\left(t\right)=-K_{\mathrm{df}}\cdot \Delta f\left(t\right)\cdot {\displaystyle \frac{S_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\end{array}\right.$$

(9)

where $\Delta P_{\mathrm{DC}}$ is the support power for primary frequency regulation, $K_{\mathrm{df}}$ is the gain coefficient of droop control, and $\Delta f$ is the deviation of $f$ from $f_{\mathrm{N}}$, that is $\Delta f\left(t\right)=f\left(t\right)-f_{\mathrm{N}}$.

Comparing (5) with (9), it can be found that both controls adjust the output power according to the grid frequency, and therefore, both are often regarded as the same type of control, called integrated inertia control [26]. However, in terms of control objectives, the former aims to reduce the frequency variation caused by power imbalance under load disturbance, while the latter seeks to compensate for the power deficit after the frequency exceeds the dead zone.

The difference in energy utilization requirements between the two types of control described above is analyzed below. It is assumed that the grid initially operates steadily at $f_{\mathrm{N}}$, that a frequency event occurs at time 0, and that $\Delta f\left(t\right)$ is bounded stable. For virtual inertia control,

$$\Delta E_{\mathrm{VIC}}={\displaystyle {\int }_0^{\infty }\Delta P_{\mathrm{VIC}}\left(t\right) dt}=-{\displaystyle {\int }_0^{\infty }{K}_{\mathrm{pf}}\cdot \frac{df\left(t\right)}{dt}\cdot \frac{S_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}} dt}=-{\displaystyle \frac{K_{\mathrm{pf}}{S}_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\cdot \underset{t\to \infty }{\lim }\Delta f\left(t\right)$$

(10)

At this point, the $\Delta E_{\mathrm{VIC}}$ is bounded. Assuming that the grid frequency is out of the dead zone at $t_0$, for droop control, $\exists t_1\in \left(t_0, \infty \right)$ that satisfies $\left|\Delta f\left(t\right)-\underset{t\to \infty }{\lim }\Delta f\left(t\right)\right|\le \epsilon$ when $t\ge t_1$ ($\epsilon$ is a very small positive number), so $\Delta f\left(t\right)\le \epsilon +\underset{t\to \infty }{\lim }\Delta f\left(t\right)$. Thus,

$$\begin{array}{l}\Delta E_{\mathrm{DC}}={\displaystyle {\int }_{t_0}^{\infty }\Delta P_{\mathrm{DC}}\left(t\right) dt}=-{\displaystyle {\int }_{t_0}^{\infty }{K}_{\mathrm{df}}\cdot \Delta f\left(t\right)\cdot \frac{S_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}dt}\\ =-{\displaystyle \frac{K_{\mathrm{df}}{S}_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\cdot {\displaystyle {\int }_{t_0}^{t_1}\Delta f\left(t\right) dt}-{\displaystyle \frac{K_{\mathrm{df}}{S}_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\cdot {\displaystyle {\int }_{t_1}^{\infty }\Delta f\left(t\right) dt}\\ \ge \alpha -{\displaystyle \frac{K_{\mathrm{df}}{S}_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\cdot {\displaystyle {\int }_{t_1}^{\infty }\left(\epsilon +\underset{t\to \infty }{\lim }\Delta f\left(t\right)\right) dt}\\ =\alpha -{\displaystyle \frac{K_{\mathrm{df}}{S}_{\mathrm{N}}^{\mathrm{WT}}}{f_{\mathrm{N}}}}\cdot {\left.\left(\epsilon +\underset{t\to \infty }{\lim }\Delta f\left(t\right)\right)\cdot t\right|}_{t_1}^{\infty }\end{array}$$

(11)

where $\alpha$ is constant and $\alpha =-K_{\mathrm{df}}{S}_{\mathrm{N}}^{\mathrm{WT}}/f_{\mathrm{N}}\cdot {\displaystyle {\int }_{t_0}^{t_1}\Delta f\left(t\right) dt}$. At this point, $\Delta E_{\mathrm{DC}}$ has no lower boundary. This shows that the output energy of WTGs under inertial support is a finite value, but the output energy under primary frequency regulation will be monotonically increased or decreased, and a continuous energy supply is needed.

Furthermore, a typical grid frequency dip event is used as an example to analyze the difference in energy utilization requirements between the two types of control under the MPPT operation scenario. For the inertia support dynamics of WTGs shown in Figure 3, the sum of the energy “released” and “absorbed” by WTGs is limited, and $\Delta E_{\mathrm{VIC}}$ is bounded. For the schematic diagram of primary frequency regulation dynamics of WTGs shown in Figure 5, the energy released continues to increase, but the aerodynamic power under the MPPT operation scenario gradually decreases with the decreasing rotor speed, which results in severe kinetic energy overdraft and causes rotor speed instability. Therefore, it is necessary to exit the frequency regulation process and reduce the output power in time, but this also causes a secondary drop in the grid frequency.

The true inertia of WTGs is therefore more suitable for providing inertial support to the grid. To avoid a secondary drop in the frequency or instability of WTGs, a safe way for the WTGs to participate in primary frequency regulation is to reserve a certain capacity through de-loading control so that the output energy can be supplemented in time by the increase in aerodynamic power.

3.3. Factors That Influence Inertia Support Capacity

Based on the analysis of energy transfer in virtual inertia control and its differences from primary frequency regulation in Section 3.1 and Section 3.2, this section will further discuss two important factors affecting the inertia support capability for WTGs—real inertia and aerodynamic characteristics.

3.3.1. Real Inertia

Equation (5) shows that the WTG with larger rotor inertia can store more kinetic energy, while the rotor kinetic energy change will slow down with the increase in rotor inertia. Therefore, the magnitude of rotor inertia is one of the key factors that determine the inertia support capability of WTGs, and larger rotor inertia allows the WTG to provide more inertia support within the same speed variation range. With an increase in the capacity, the size and real inertia of the WTG gradually increase. The following example calculates and compares the real inertia of ten SGs in a standard IEEE 10-machine 39-bus system, an NREL 5 MW WTG [27], and an NREL 1.5 MW WTG [28]. The details can be seen in Appendix A.

Assuming the standard IEEE 10-machine 39-bus system $f_{\mathrm{N}}$ is 50 Hz (that is, the rated speed of all SGs ${\omega }_{\mathrm{fN}}^{\mathrm{SG}}$ is 3000 rpm), checking reference [29], the rated capacity of the G2 $S_{\mathrm{N}}^{\mathrm{SG}2}$ is 700 MVA, the time constant of inertia $H^{\mathrm{SG}2}$ is 4.329 s and the number of pole pairs $p$ is 1, and the inertia of this SG can be calculated as

$$J^{\mathrm{SG}2}={\displaystyle \frac{2H^{\mathrm{SG}2}{S}_{\mathrm{N}}^{\mathrm{SG}2}{p}^2}{{\left({\omega }_{\mathrm{fN}}^{\mathrm{SG}}\right)}^2}}=4.27\times {10}^4{ \mathrm{kgm}}^2$$

(12)

It is known that $J_{\mathrm{r}}$ of NREL 5 MW WTG is $3.5444067\times {10}^7{ \mathrm{kgm}}^2$, $J_g$ is $534.116{ \mathrm{kgm}}^2$, $N_{\mathrm{g}}$ is 97 and ${\omega }_{\mathrm{gN}}^{\mathrm{WT}}$ is 1173.7 rpm. The following calculation of the true inertia of the WTG is based on the rated speed of the SG because the rated speed of the WTG is different from that of the SG. Therefore, from (3), the true inertia of a 5 MW WTG converted to the high-speed side is calculated as

$$J_{\mathrm{ac}}^{\mathrm{WT}}=J_{\mathrm{h}}^{\mathrm{WT}}\cdot {\left({\displaystyle \frac{{\omega }_{\mathrm{gN}}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}\right)}^2=658.35{ \mathrm{kgm}}^2$$

(13)

As the rated capacity of the SG is 140 times larger than that of the WTG, the true inertia of the WTG with the same capacity size as the SG is $6.4\times {10}^4{ \mathrm{kgm}}^2$. A comparison of the data of the true inertia of the WTGs and all SGs in the system is shown in

Table 1. Comparison of real inertia of WTGs and SGs.

SGs in the Standard IEEE 39-Bus System	Real Inertia of Each SG/kgm2	Real Inertia of the NREL 5 MW WTG Under the Same Capacity/kgm2	Real Inertia of the NREL 1.5 MW WTG Under the Same Capacity/kgm2
G1	7.04 × 105	1.32 × 106	1.05 × 106
G2	4.27 × 104	9.22 × 104	7.32 × 104
G3	5.04 × 104	1.05 × 105	8.36 × 104
G4	4.03 × 104	1.05 × 105	8.36 × 104
G5	3.66 × 104	3.95 × 104	3.14 × 104
G6	4.90 × 104	1.05 × 105	8.36 × 104
G7	3.72 × 104	9.22 × 104	7.32 × 104
G8	3.42 × 104	9.22 × 104	7.32 × 104
G9	4.86 × 104	1.32 × 105	1.05 × 105
G10	5.91 × 104	1.32 × 105	1.05 × 105

. Except for G5, the true inertia of the WTGs with the same capacity is greater than the true inertia of the SGs.

3.3.2. Aerodynamic Characteristics

For the mathematical model in Section 2.1, the aerodynamic characteristics of different WTGs can be expressed with different functions of $C_{\mathrm{P}}\left(\lambda \right)$. The function of $C_{\mathrm{P}}\left(\lambda \right)$ has a single-peaked characteristic and $\lambda ={\omega }_{\mathrm{r}}^{\mathrm{WT}}R/v$, so $P_{\mathrm{m}}^{\mathrm{WT}}$ is a function of ${\omega }_{\mathrm{r}}^{\mathrm{WT}}$ and $P_{\mathrm{m}}^{\mathrm{WT}}\left({\omega }_{\mathrm{r}}^{\mathrm{WT}}\right)$ still has a single-peaked characteristic. This shows that, unlike the mechanical characteristics of the SG, the aerodynamic power of the WTG is influenced by its rotor speed and the aerodynamic power will further influence its rotor speed. This coupling relationship between aerodynamic power and rotor speed makes the operation dynamics of the WTG more complex. In addition, different WTGs have specific $C_{\mathrm{P}}\left(\lambda \right)$, so the variations in their aerodynamic power influenced by fluctuations in rotor speed are also related to their own $C_{\mathrm{P}}\left(\lambda \right)$. The common $C_{\mathrm{P}}\left(\lambda \right)$ curves [27,28,30] are shown in Figure 6.

In summary, the rotor speed dynamics of WTGs under virtual inertia control are influenced by the two factors mentioned above, specifically by the real inertia preventing large fluctuations in the rotor speed, and by the aerodynamic characteristics coupling the aerodynamic power with the rotor speed, which indirectly influences the inertia support capacity of WTGs to the grid.

4. Inertia Support Capability Evaluation for WTGs Based on Symmetrical Operation

To evaluate the inertia support capabilities of WTGs, this section constructs a symmetric operation mode that mirrors the rotor speed dynamic changes of conventional SGs, based on the principle of inertia support for SGs. In this mode, the inertia provided by WTGs without passing through a converter can be directly calculated. For WTGs that are more commonly grid-connected through converters, the symmetric operation serves as an approximation of the dynamic behavior for WTGs under virtual inertia control. However, this mode bridges the real and virtual inertia of WTGs, facilitating the analysis of the impact on aerodynamic efficiency when providing varying levels of inertia support.

4.1. Symmetrical Operation Mode of WTGs

In the absence of converters, the frequency of electricity generated by WTGs matches the grid frequency. Consequently, the speed variations of WTGs are in a fixed ratio to those of the SGs, governed by the generator’s pole pairs and gearbox ratio. This paper denotes this operational mode as the symmetric operation mode, that is,

$${\displaystyle \frac{{\omega }_{\mathrm{g}}^{\mathrm{WT}}-{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}}=k\cdot {\displaystyle \frac{{\omega }_{\mathrm{f}}^{\mathrm{SG}}-{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}$$

(14)

where $k$ is the proportional coefficient of rotor speed change, $k>0$, ${\omega }_{\mathrm{g}0}^{\mathrm{WT}}$ is the reference generator speed of WTGs, and ${\omega }_{\mathrm{f}}^{\mathrm{SG}}$ is the actual speed of SGs.

Under symmetric operation, the speed variation of WTGs is consistent with that of SGs, and WTGs interfaced through converters under virtual inertia control exhibit similar dynamic characteristics. Thus, the concept of symmetric operation can also be applied to converter-interfaced WTGs, where control is employed to align the speed of WTGs with that of SGs as described in Equation (14), termed as symmetric operation for WTGs. Due to the presence of converters, the rotor speed of WTGs is decoupled from grid frequency. Hence, the symmetric operation is considered an approximation of the dynamic behavior under virtual inertia control. The primary distinction lies in that the aerodynamic power variations are reflected in the output power for the former, while for the latter, they are manifested in rotor speed changes.

4.2. Equivalent Inertia Calculation of WTGs Under Symmetrical Operation

It can be seen from Equation (14) that the rotor speed for WTGs under symmetrical operation can be expressed as

$${\omega }_{\mathrm{g}}^{\mathrm{WT}}=k\cdot {\omega }_{\mathrm{f}}^{\mathrm{SG}}{\displaystyle \frac{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}-(k-1){\omega }_{\mathrm{g}0}^{\mathrm{WT}}$$

(15)

If the SG and the WTG are initially operated in a stable equilibrium state, the dynamic expression of the SG is

$$\Delta P_{\mathrm{m}}^{\mathrm{SG}}-\Delta P_{\mathrm{e}}^{\mathrm{SG}}=J^{\mathrm{SG}}{\omega }_{\mathrm{f}}^{\mathrm{SG}}{\displaystyle \frac{d{\omega }_{\mathrm{f}}^{\mathrm{SG}}}{dt}}$$

(16)

where $\Delta P_{\mathrm{m}}^{\mathrm{SG}}$ is the mechanical power deviation of SGs and $\Delta P_{\mathrm{e}}^{\mathrm{SG}}$ is the electromagnetic power deviation of SGs. The deviation of the speed fluctuation relative to ${\omega }_{\mathrm{fN}}^{\mathrm{SG}}$ can be ignored during the normal operation of the SGs [31], so it can be regarded as ${\omega }_{\mathrm{f}}^{\mathrm{SG}}={\omega }_{\mathrm{fN}}^{\mathrm{SG}}$. Combined with Equations (3) and (15), the dynamic expression of the WTG is

$$\begin{array}{l}\Delta P_{\mathrm{m}}^{\mathrm{WT}}-\Delta P_{\mathrm{e}}^{\mathrm{WT}}=J_{\mathrm{h}}^{\mathrm{WT}}{\omega }_{\mathrm{g}}^{\mathrm{WT}}{\displaystyle \frac{d{\omega }_{\mathrm{g}}^{\mathrm{WT}}}{dt}}\\ =J_{\mathrm{h}}^{\mathrm{WT}}\cdot \left(k\cdot {\omega }_{\mathrm{f}}^{\mathrm{SG}}{\displaystyle \frac{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}-\left(k-1\right){\omega }_{\mathrm{g}0}^{\mathrm{WT}}\right)\cdot \left(k\cdot {\displaystyle \frac{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}\cdot {\displaystyle \frac{d{\omega }_{\mathrm{f}}^{\mathrm{SG}}}{dt}}\right)\\ =k\cdot J_{\mathrm{h}}^{\mathrm{WT}}\cdot {\left({\displaystyle \frac{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}\right)}^2\cdot {\omega }_{\mathrm{fN}}^{\mathrm{SG}}{\displaystyle \frac{d{\omega }_{\mathrm{f}}^{\mathrm{SG}}}{dt}}\end{array}$$

(17)

where $\Delta P_{\mathrm{m}}^{\mathrm{WT}}$ is the aerodynamic power deviation of WTGs and $\Delta P_{\mathrm{e}}^{\mathrm{WT}}$ is the electromagnetic power deviation of WTGs. The expression of equivalent inertia for WTGs is obtained by comparing Equations (16) and (17); it can be obtained that

$$J_{\mathrm{eq}}^{\mathrm{WT}}=k\cdot J_{\mathrm{h}}^{\mathrm{WT}}{\left({\displaystyle \frac{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}\right)}^2$$

(18)

It should be noted that the symmetric operation can be achieved through the closed-loop speed control of WTGs, with the specific implementation process detailed in reference [32], and thus will not be reiterated here. The simulation results are presented in Section 4.3. Consequently, Equation (18) calculates the equivalent inertia under ideal conditions where there is no deviation in closed-loop speed control.

4.3. Equivalent Inertia of WTGs Under Symmetrical Operation and Its Influence on Aerodynamic Efficiency

As indicated in Section 4.2, the equivalent inertia of WTGs under symmetric operation is related to their initial operating conditions. Therefore, two constant wind speed scenarios of 10 m/s and 7 m/s are set, with the grid frequency fluctuating within the range of 49.5 Hz to 50.5 Hz and $k=1$. This case evaluates and analyzes the equivalent inertia of WTGs operating under MPPT operation and its impact on aerodynamic efficiency. The deviation of ±0.5 Hz is the frequency deviation limit of power system under normal operating conditions specified in the Chinese standard [33], which can be used on demand.

4.3.1. Equivalent Inertia Evaluation

Building upon the equivalent inertia calculation method for the symmetric operation detailed in Section 4.2, this section compares the equivalent inertia of WTGs with the real inertia of SGs within the standard IEEE 10-machine 39-bus system, as mentioned in Section 3.3 (refer to Appendix A). The real inertia of the SGs has been calculated in Section 3.3, and thus will not be reiterated here.

By combining Equations (2) and (3) and substituting the relevant data, the calculation result for the equivalent inertia on the high-speed side of a single 5 MW WTG at a constant wind speed of 10 m/s is obtained as

$$J_{\mathrm{eq}1}^{\mathrm{WT}}=J_{\mathrm{h}}^{\mathrm{WT}}{\left({\displaystyle \frac{{\omega }_{\mathrm{g}0}^{\mathrm{WT}}}{{\omega }_{\mathrm{fN}}^{\mathrm{SG}}}}\right)}^2=596.81{ \mathrm{kgm}}^2$$

(19)

Given that the rated capacity of G2 is 140 times that of the WTG, the equivalent inertia of the WTG with the same capacity as G2 should be $8.36\times {10}^4{ \mathrm{kgm}}^2$. Furthermore, the equivalent inertia of both WTGs at the same capacity is calculated and compared with the real inertia of all SGs in the system, as shown in

Table 2. Comparison of real inertia of WTGs and SGs under 10 m/s wind speed scenario.

SGs in the Standard IEEE 39-Bus System	Real Inertia of Each SG/kgm2	Real Inertia of the NREL 5 MW WTG Under the Same Capacity/kgm2	Real Inertia of the NREL 1.5 MW WTG Under the Same Capacity/kgm2
G1	7.04 × 105	1.19 × 106	7.42 × 105
G2	4.27 × 104	8.36 × 104	5.20 × 104
G3	5.04 × 104	9.55 × 104	5.94 × 104
G4	4.03 × 104	9.55 × 104	5.94 × 104
G5	3.66 × 104	3.58 × 104	2.23 × 104
G6	4.90 × 104	9.55 × 104	5.94 × 104
G7	3.72 × 104	8.36 × 104	5.20 × 104
G8	3.42 × 104	8.36 × 104	5.20 × 104
G9	4.86 × 104	1.19 × 105	7.42 × 104
G10	5.91 × 104	1.19 × 105	7.42 × 104

. Consistent with the results from Section 3.3 regarding real inertia, the equivalent inertia of WTGs at the same capacity exceeds that of the real inertia of SGs, with the exception of G5.

Similarly, the equivalent inertia provided by both WTGs under a constant wind speed scenario of 7 m/s is calculated and compared with the real inertia of all SGs in the system, with the results presented in

Table 3. Comparison of real inertia of WTGs and SGs under 7 m/s wind speed scenario.

SGs in the Standard IEEE 39-Bus System	Real Inertia of Each SG/kgm2	Real Inertia of the NREL 5 MW WTG Under the Same Capacity/kgm2	Real Inertia of the NREL 1.5 MW WTG Under the Same Capacity/kgm2
G1	7.04 × 105	5.85 × 105	3.64 × 105
G2	4.27 × 104	4.09 × 104	2.55 × 104
G3	5.04 × 104	4.68 × 104	2.91 × 104
G4	4.03 × 104	4.68 × 104	2.91 × 104
G5	3.66 × 104	1.75 × 104	1.09 × 104
G6	4.90 × 104	4.68 × 104	2.91 × 104
G7	3.72 × 104	4.09 × 104	2.55 × 104
G8	3.42 × 104	4.09 × 104	2.55 × 104
G9	4.86 × 104	5.85 × 104	3.64 × 104
G10	5.91 × 104	5.85 × 104	3.64 × 104

. Although the equivalent inertia of the NREL 5 MW WTG is less than the real inertia of the SGs, it remains within the same order of magnitude. The equivalent inertia of the NREL 1.5 MW WTG is close to that of the SGs. In summary, under different scenarios, WTGs operating under the symmetric operation mode can provide inertia support to the grid that is comparable to, or even greater than, that of SGs.

4.3.2. The Influence of Inertia Support on Aerodynamic Efficiency

Furthermore, under 10 m/s and 7 m/s wind speed scenarios, the WTGs achieve symmetric operation with the grid frequency through closed-loop speed control, with simulation results depicted in Figure 7 and Figure 8. As shown in the figures, the rotor speed of the WTG varies in near consistency with the grid frequency (that is, the rotor speed of SGs) under both wind speed scenarios. Concurrently, the fluctuation in the C_p of the WTGs is minimal, with the maximum aerodynamic power loss being only 0.022% at a wind speed of 10 m/s and 0.023% at a wind speed of 7 m/s. Therefore, the impact of inertia support on the aerodynamic efficiency of WTGs under symmetric operation is negligible, implying that changes in the operating state for WTGs during the inertia support process have a minimal effect on aerodynamic power.

4.4. The Influence of WTG Speed Variation Range on Equivalent Inertia and Aerodynamic Efficiency

Furthermore, the relationship between equivalent inertia under symmetric operation mode and virtual inertia control is derived. As indicated by Equation (5), the electromagnetic power deviation of WTGs under virtual inertia control is equal to the inertia support power, that is,

$$\Delta P_{\mathrm{e}}^{\mathrm{WT}}=\Delta P_{\mathrm{VIC}}\left(t\right)=-K_{\mathrm{pf}}{\displaystyle \frac{S_{\mathrm{N}}^{\mathrm{WT}}}{{\left({\omega }_{\mathrm{fN}}^{\mathrm{SG}}\right)}^2}}{\omega }_{\mathrm{fN}}^{\mathrm{SG}}{\displaystyle \frac{d{\omega }_{\mathrm{f}}^{\mathrm{SG}}}{dt}}$$

(20)

Based on the analysis in Section 4.3, by neglecting the impact of inertia support in the symmetric operation on $\Delta P_{\mathrm{m}}^{\mathrm{WT}}$, and combining Equations (17), (18), and (20), we can obtain

$$K_{\mathrm{pf}}={\displaystyle \frac{J_{\mathrm{eq}}^{\mathrm{WT}}{\left({\omega }_{\mathrm{fN}}^{\mathrm{SG}}\right)}^2}{S_{\mathrm{N}}^{\mathrm{WT}}}}$$

(21)

It is evident that under the symmetric operation, the equivalent inertia of WTGs corresponds directly to the virtual inertia coefficient. Furthermore, by combining Equation (18) with Equation (21), we can derive

$$k={\displaystyle \frac{K_{\mathrm{pf}}{S}_{\mathrm{N}}^{\mathrm{WT}}}{J_{\mathrm{h}}^{\mathrm{WT}}{\left({\omega }_{\mathrm{g}0}^{\mathrm{WT}}\right)}^2}}$$

(22)

In other words, the virtual inertia coefficient determines the $k$. When the grid frequency (i.e., the rotor speed of SGs) is within the normal operating range, the virtual inertia coefficient can be used to establish the range of speed variation for WTGs. Concurrently, increasing the virtual inertia coefficient not only amplifies the equivalent inertia of the WTGs but also broadens the range of their speed variation, which in turn increases the impact on aerodynamic efficiency. Therefore, it is necessary to set aside a reserve capacity that exceeds the loss in aerodynamic efficiency to enable WTGs to provide inertia support under de-loading operation, ensuring the stability of the WTGs.

4.5. Verification of Virtual Inertia Control for WTGs Under De-Loading Operation

Section 4.4 establishes the relationship between the virtual inertia control coefficient of WTGs, the equivalent inertia under symmetric operation, and the range of speed variation. In this section, based on the simulation model provided in Appendix B, two case studies will be set up for WTGs in de-loading operation. The relationships derived in Section 4.4 will be utilized to calculate the virtual inertia coefficient, thereby validating that the virtual inertia control of WTGs can provide adequate inertia support to the grid.

4.5.1. Case 1: Operating in the Same Rotor Speed Range

In this case, the grid frequency is set to operate normally between 49.5 Hz and 50.5 Hz, with the speed variation range of WTGs being the same as that of the grid frequency (i.e., $k=1$). If the WTGs initially operate at a constant wind speed of 10 m/s under MPPT, Section 4.3 indicates a loss of 0.022% in aerodynamic power, thus allowing for the determination to reserve a 0.1% reserve capacity. Furthermore, Section 4.3 calculates the equivalent inertia provided by the WTGs as $J_{\mathrm{eq}}^{\mathrm{WT}}=596.81{ \mathrm{kgm}}^2$, and $K_{\mathrm{pf}}=11.78$ can be derived from Equation (21). Simulations based on these calculations are conducted, with the system operation curves under virtual inertia control shown in Figure 9. Consequently, the WTGs effectively reduce the frequency fluctuations during normal grid operation while maintaining stable operation, demonstrating the effectiveness of the virtual inertia coefficient setting under the same rotor speed variation range.

4.5.2. Case 2: Provide the Same Equivalent Inertia as G1

In this case, the grid frequency is set to operate normally between 49.5 Hz and 50.5 Hz, and the WTGs are required to provide an equivalent inertia comparable to that of G1, hence calculating the equivalent inertia provided by the WTGs as $J_{\mathrm{eq}}^{\mathrm{WT}}={352 \mathrm{kgm}}^2$. If the WTGs initially operate at a constant wind speed of 7 m/s under MPPT control, then according to Equation (18), $k=1.2$. The time-domain curves of the grid frequency, rotor speed, and C_p curve for WTGs under symmetrical operation are shown in Figure 10. The aerodynamic power loss is statistically 0.03%, thus allowing for the determination to reserve a 0.1% reserve capacity.

Furthermore, by substituting $J_{\mathrm{eq}}^{\mathrm{WT}}$ into Equation (21), $K_{\mathrm{pf}}=6.94$ is calculated. Based on the aforementioned computational results, simulations are conducted, and the system operation curves under virtual inertia control are depicted in Figure 11. Similarly, the virtual inertia control of WTGs mitigates frequency fluctuations in the grid during normal operation while ensuring stable running, indicating that the virtual inertia coefficient is effective when tuned to provide an equivalent inertia comparable to that of G1.

Finally, it can be seen from the above simulation that in the large-scale power systems dominated by SGs, the fluctuation range of output power and rotor speed for virtual inertia control of the WTGs under symmetrical operation mode is acceptable. Since the virtual inertia control provides inertia support for the grid, the WTGs will no longer be able to achieve maximum power tracking with MPPT control, which will affect the tip speed ratio and aerodynamic power extraction. However, the inertia support provided by large WTGs has a minimal impact on their aerodynamic efficiency under non-fault conditions, allowing them to deliver inertia support equivalent to that of SGs with power reduction of no more than 0.1%. When facing the actual turbulent wind speed, the power change and speed fluctuation of the WTG only under MPPT control are much larger than those in the above simulation [34,35].

5. Conclusions

In this paper, following an analysis of the energy utilization process and influencing factors of inertia support in WTGs, a symmetric operation mode is constructed to calculate and analyze the equivalent inertia under various scenarios and control parameters, as well as its impact on aerodynamic efficiency. The results indicate that, unlike primary frequency regulation which relies on continuous kinetic energy release, large WTGs can provide inertia support to the grid approaching that of SGs under de-loading operation with a reserve capacity not exceeding 0.1%. Currently, the design dimensions of WTGs are continually increasing with rotor diameters now exceeding 250 m [36], particularly offshore WTGs, resulting in a more substantial equivalent inertia and thus a stronger inertia support capability. Finally, considering the distinct energy requirements of primary frequency regulation and inertia support, it is recommended that WTGs utilize the reduced reserve capacity reserved for overspeed or pitch control to participate in the grid’s primary frequency regulation while reserving their inherent inertia solely for virtual inertia support.