Abstract
With the increasing penetration of wind power, using wind turbines to participate in the frequency regulation to support power system has become a clear consensus. To accurately quantify the inertia provided by the doubly-fed induction generator (DFIG) based wind farm, the frequency response model of DFIG with additional frequency control is established, and then by using Routh approximation, the explicit expression of the virtual moment of inertia is derived for the DFIG grid-connected system. To further enhance the availability of the expression, an estimation method is proposed based on the matrix pencil method and the least squares algorithm for estimating the virtual moment of inertia provided by the wind farm. Finally, numerical results tested by a DFIG grid-connected system and a modified IEEE 30-bus system verify the derived expression of the virtual moment of inertia and the proposed estimation method.
DUE to cleanliness, safety, and adequacy, wind power generation has experienced rapid development in the past decades [
To reduce the negative impact of high penetration of wind power, using WTs to participate in the frequency regulation has become a clear consensus among many countries [
The conventional inertia in the power system is generally defined as the total amount of kinetic energy stored in all rotating grid-connected generators and motors. Although the virtual inertia of DFIG is also provided by releasing the reserved active power, the inertia is time-varying and hidden in the complex control process. From the perspective of system planning and operation, the quantification of virtual inertia is valuable.
In terms of energy, an equivalent time constant of virtual inertia is defined for DFIG in [
As for a wind farm, its size may vary from a small number to several hundred WTs covering an extensive area. No matter what description or definition is adopted, the total virtual inertia can be accumulated by that of each WT in theory [
To efficiently analyze and reveal the impact of high penetration of wind power on the characteristics of system frequency, a method of estimating the virtual inertia supported by the additional frequency control of the DFIG-based wind farm is investigated in this paper. The main contributions are as follows.
1) Taking the inertia of power system as a reference, the response characteristic of DFIG with conventional vector control is investigated, and the principle of approximate decoupling between DFIG and system frequency is further discussed.
2) To demonstrate the origin of the virtual moment of inertia directly, the frequency response model of DFIG with additional frequency control is established, and then by using Routh approximation, the expression of the virtual moment of inertia is derived. The derivation process is based on the DFIG grid-connected system with an equivalent generator, but the final derived expression is independent of the background system, which essentially completes the transformation of DFIG from non-synchronous to synchronous and determines the basic idea of inertia estimation for a wind farm.
3) From the perspective of the application, an estimation method based on the matrix pencil method and the least squares algorithm is proposed for identifying the frequency response characteristics of the wind farm with additional frequency control. Combining the derived expression and the frequency response, the basic framework to efficiently estimate the virtual moment of inertia is obtained, and thus the complex processing of the parametric derivation and response integration for the wind farm is avoided, which makes the real-time application of virtual inertia estimation possible.
The remainder of this paper is organized as follows. Section II introduces the inertia of the power system with DFIG and the principle of approximate decoupling between DFIG and system angular frequency. Section III summarizes the frequency response modeling process of DFIG with additional frequency control as well as the basic expression of the virtual moment of inertia. Section IV presents the identification method of the frequency response characteristic on a wind farm and the main process of calculating its virtual moment of inertia. The case studies are discussed in Section V, followed by Section VI, which concludes the paper.
Inertia is an inherent property of any physical object. The inertia of a synchronous generator usually refers to the rotational inertia, which reflects the degree of resistance to the change of rotor motion state. To describe the strength of resistance, the moment of inertia that determines the torque needed for a desired angular acceleration about a rotational axis is frequently employed, which is defined as:
(1) |
where and are the rotation radius and the mass of the rotor shaft, respectively.
The synchronous generator can be considered as a rotating rigid body. According to the electric machine theory, the relationship between the mechanical angular velocity and the torque exerting on the rotor can be expressed as:
(2) |
where , , and are the mechanical torque, electromagnetic torque, and damping coefficient of a generator, respectively.
Substitute and into (2) with , , , and being the mechanical power, electromagnetic power, system angular frequency, and pole pairs, respectively. By linearization, the motion equation concerning the system angular frequency can be described as:
(3) |
where is the steady-state value of system angular frequency.
Thus, the transfer function between the system angular frequency and active power can be given by:
(4) |
It can be observed that the change of the angular frequency caused by the active power imbalance is a first-order process, where the moment of inertia and the damping coefficient jointly determine the dynamic. In particular, the higher the moment of inertia is, the slower the change rate of system angular frequency will be.
Remark 1: in term of a multi-machine system, the rotor speed of each synchronous generator is coupled to the system angular frequency, and thus the total moment of inertia in such a system can be accumulated by all participating machines, where it needs to transform all the moments of inertia to the same pole pairs. From this perspective, the estimate to the moment of inertia of each machine that actively or passively participates in frequency regulation is the basis of evaluating system inertia.
Using the motor convention, the electromechanical transient process of DFIG can be expressed as [
(5) |
where is the angular velocity of rotor; and are the total moment of inertia and damping coefficient converted onto the motor axis, respectively; and are the electromagnetic torque and mechanical torque, respectively; is the active power of the stator; and , , , , and are the air density, swept area of WT blades, wind speed, coefficient of power, and pole pairs of DFIG, respectively.
In the vector control strategy of DFIG, the control of the rotor-side converter can be simplified as a first-order dynamic process by Routh approximation. The time constant is shown in (6), but it is worth noting that we can also obtain different expressions of the time constant by using other order-reduction methods. To make the statement easier to follow and more general, the detailed derivation on this part is presented in Appendix A.
(6) |
where and are the active power at the stator side and its reference value, respectively; is the gain of pulse width modulation; is the mutual inductance between the coaxial equivalent stator and rotor winding; is equivalent to the self-inductance of the stator winding; is the steady-state d-axis voltage of stator; and and are the proportional and integral parameters of the outer control loop, respectively.
Given that DFIG is usually controlled by the scheme of maximum power point tracking, the maximum power captured by the WT can be represented as , where is the transformation ratio of the gearbox, , and , , and are the wind wheel radius, optimal tip speed ratio, and maximum value of coefficient of power, respectively [
(7) |
When the copper loss in both stator and rotor is neglected, we can obtain the approximation of the grid-connected power of DFIG as:
(8) |
Assume that the background system can be represented by an equivalent synchronous generator and the DFIG is connected to the generator. Combining the model of synchronous generator, the angular frequency dynamic of such a DFIG grid-connected system described by small-signal equations is shown in

Fig. 1 Block diagram of angular frequency dynamic in DFIG grid-connected system.
Since the above DFIG model only involves the electromechanical transient process, its small-signal equations can be easily derived by linearizing (5)-(8), which will not be listed in this paper due to the limited space. It is worth noting that the expression of is nonlinear, and after the linearization, we can obtain the coefficient with respect to , namely , as:
(9) |
where is the steady-state wind speed; is the steady-state angular velocity of rotor; is the steady-state value of the mechanical power; and is obtained by linearizing the fit expression of and thus making .
As depicted in
(10) |
where is the steady-state active power at the stator side.
We can observe that the gain of (10) is smaller than 1 so that the changing range of is much smaller than that of under disturbance, and an assumption of can be easily obtained, which means the rotor speed is decoupled from the system angular frequency under the control of rotor-side converter. Considering the above inference, the linearization of (8) can be expressed as:
(11) |
Remark 2: under the vector control strategy, the DFIG grid-connected system is equivalent to a separate constant power source. Although such a source is usually affected by system voltage and wind speed, the kinetic energy of the rotor is insensible when the frequency disturbance occurs, which means the system inertia is mainly determined by the synchronous generators. DFIG itself provides little inertia support, and such a characteristic will not change significantly with the increase of DFIG penetration.
When DFIG is configured with additional frequency control, the grid-connected power is directly affected by the system frequency, which makes DFIG actively participating in the primary frequency regulation possible. To determine the virtual moment of inertia supported by DFIG, the frequency response model of DFIG with additional frequency control is established below, and the method to estimate the virtual moment of inertia is also proposed.
The additional frequency control considered in this paper consists of two loops, i.e., the virtual inertia control and the droop control [

Fig. 2 Block diagram of system angular frequency dynamic in DFIG grid-connected system with additional frequency control.
When the system frequency disturbance occurs, the reference value of additional active power generated by the additional frequency controller can be given by:
(12) |
where is the time constant of a low-pass filter; and are the virtual inertia coefficient and droop coefficient, respectively; and is the setting value of the system angular frequency.
Thus, the reference value of active power at the stator side is modified as:
(13) |
Combined with the block diagram shown in
To determine the virtual moment of inertia provided by the above process, the frequency response is necessary. According to the simplified block diagram shown in
(14) |
(15) |
As the time constant in the low-pass filter is small, if the corresponding part is neglected, then (14) can be simplified as:
(16) |
(17) |
Through the above simplification, the frequency response of DFIG with additional frequency control is reduced to a second-order transfer function, as shown in

Fig. 3 Simplified block diagram of angular frequency dynamic in DFIG grid-connected system with additional frequency control. (a) Original block diagram. (b) Simplified block diagram with effect of DFIG represented by virtual moment of inertia.
Combining the grid-connected power of DFIG and the motion equation of synchronous generator, the response of the angular frequency dynamic to the load variation can be obtained as:
(18) |
where is the load disturbance.
Substituting (16) into (18), the transfer function between the power imbalance and the system angular frequency can be expressed as:
(19) |
(20) |
It can be observed that is a high-order transfer function, which contains the electromechanical transient process of synchronous generator and the active regulation process of DFIG. The virtual moment of inertia provided by the additional frequency control of DFIG is hidden in such a complex transfer function.
To directly demonstrate the virtual moment of inertia just like (4), the model order reduction is employed to obtain a first-order dynamic process. To ensure the approximate accuracy in the low-frequency band, we choose the Routh approximation to implement model order reduction [
(21) |
If is used to replace , the angular frequency dynamic shown in
(22) |
More specifically, is made up of three terms, where the first term is the inherent moment of inertia provided by the synchronous generator. The second term is related to the damping coefficient in both generator and DFIG, which is much smaller than and can be ignored. Therefore, the virtual moment of inertia supported by DFIG can be given by:
(23) |
Remark 3: the virtual moment of inertia obtained by the model order reduction is almost independent of the background system, and it mainly relies on the modeling of the frequency response . For other types of additional frequency control schemes, the expression of can also be obtained in the same way, and there are only some differences in coefficients and . Although we can rigorously derive the exact expression of a single DFIG with all parameters known, such a modeling process is inefficient and even inaccessible for a wind farm due to its size. Therefore, the efficient estimate to the frequency response of the wind farm with additional frequency control becomes the key point to the practical application of the proposed estimation method.
For a wind farm, its complete frequency response is made up of the characteristics of all WTs. To avoid the difficult integration of theoretical expressions, an estimation method using the simulation data or actual monitoring data as an alternative is introduced below.
The wind farm is a typical system based on power electronics, which determines the presence of both broadband harmonics and disturbances in the sampling data of active power. Different from the low-frequency oscillation related to system inertia, harmonics can be regarded as a kind of data noise, and thus the sampling data sequence of grid-connected power can be formulated as:
(24) |
where is the real data of the grid-connected power; is the data noise; , , and , , , and are the amplitude, initial phase, damping factor, and angular frequency of component , respectively; and , , and are the sample interval, number of dominant low-frequency components, and number of sample points, respectively.
To ensure the identification accuracy of the moment of inertia, the data noise should be separated from the dominant low-frequency components. Therefore, the matrix pencil method [
For the sampling data sequence, the Hankel matrix is constructed firstly in the form of
(25) |
where is the pencil parameter, which is usually chosen from to for efficient noise filtering.
Then, the singular-value decomposition is carried out to achieve denoising. Considering the singular value of as , thus we can obtain:
(26) |
where is the threshold value; and is the maximum singular value. The ratio smaller than is considered as the noise-singular value.
After the number of dominant components is determined, we can use the filtered right-unitary matrix that contains only dominant right-singular vectors to construct the matrix pair , which satisfies
(27) |
where the superscripts “+” and “H” represent the operations of Moore-Penrose pseudoinverse and conjugate transpose, respectively; is obtained by removing the last row of ; is obtained by removing the first row of ; is the generalized eigenvalue; and is the unit matrix.
Once and are known, the residues in (24) can be solved from the following least squares problem.
(28) |
Thus, the data sequence of active power can be reconstructed as:
(29) |
(30) |
The sampling data sequence is discrete so that there is a gap between it and the s-domain model shown in (16). Thus, the backward difference formula is used to realize the z-transformation. Then, we can obtain the transfer function in the z-domain as:
(31) |
where are the parameters that need to be identified.
Once the is obtained, is automatically determined. Through the above transformation, the detailed estimation process based on the least squares algorithm can be summarized in

Fig. 4 Flow chart of estimation process based on least squares algorithm.
Step 1: data preparation. Set the sampling interval and sampling period , then the length of all data sequences is obtained as . The simulations or monitoring data are used to construct the average wind speed sequence , system angular frequency sequence , and grid-connected power sequence , where is the mean result of all monitoring points at each sampling interval.
Step 2: sequence state judgment. Let represent the normalized result of or , if , meets , then can be considered as the stationary sequence denoted by ; otherwise, is the non-stationary sequence denoted by . and are the average of the data sequence and the threshold value of judgment, respectively.
Step 3: sequence selection. If both and can be classified as the state group , the matrix pencil method is used to reconstruct the corresponding sequence of grid-connected power and the low-frequency component is obtained, and then go to Step 4; otherwise, return to Step 1.
Step 4: frequency response identification. Let and be the input and output sequences, where and are the initial states of and , the accuracy of parameter vector can be measured by the following error sum of squares [
(32) |
where .
To minimize the error, should satisfy
(33) |
where ; and . If is full rank, then can be solved by the least squares algorithm:
(34) |
Step 5: moment of inertia estimation. When is obtained, is transfered to using the principle as:
(35) |
Then, according to the expression in (23), the moment of inertia supported by a wind farm can be finally estimated.
Remark 4: under the extreme condition, if no sequence that meets the selection criteria, i.e., all and are classified as the state group , the component of grid-connected power corresponding to the wind speed disturbance can be removed, and then the proposed method can be used to estimate the virtual moment of inertia for a DFIG-based wind farm.
To illustrate the performance of the proposed method, a DFIG grid-connected system and a modified IEEE 30-bus system are both used. The sampling data used to estimate the virtual moment of inertia are captured by the simulation using MATLAB/Simulink on an Intel Core i7-9700 3.00 GHz machine.
As the virtual moment of inertia provided by a single DFIG can be deduced directly, the DFIG grid-connected system is employed to facilitate the accuracy comparison and a clear demonstration, of which simulation parameters are given in
The background system is made up of an equivalent synchronous generator, a load, and a 3-bus network. The moment of inertia and the rated capacity of the generator are set to be 6500 kg⋅
Setting the frequency disturbance by changing the load (4 MW) in the background system, the dynamic responses of angular frequency in the DFIG grid-connected system with or without the additional frequency control are shown in

Fig. 5 Dynamic responses of DFIG grid-connected system with or without additional frequency control. (a) System angular frequency with DFIG removed. (b) System angular frequency. (c) Angular velocity of rotor. (d) Grid-connected power.
Given that all parameters of DFIG are known, the value of the virtual moment of inertia supported by the DFIG with additional frequency control can be directly obtained by (23). For , =1143.23 kg⋅
The comparison between the DFIG grid-connected system and benchmark system of inertia using the derived results is shown in

Fig. 6 Comparison between DFIG grid-connected system and benchmark system of inertia using derived results. (a) System angular frequency with and MW. (b) Error distribution of system angular frequency with by taking result of benchmark system of inertia as a reference. (c) System angular frequency with and MW. (d) Error distribution of system angular frequency with by taking result of benchmark system of inertia as a reference.
To further illustrate the performance of the proposed method, we take the DFIG grid-connected system with as an application object. The sampling data sequences of the DFIG grid-connected system obtained by the electromagnetic transient simulation are shown in

Fig. 7 Sampling data sequences of DFIG grid-connected system. (a) Wind speed. (b) System angular frequency. (c) Grid-connected power.
The data marked by red lines are selected as the input, and eventually, the virtual moment of inertia is estimated to be . The comparison between the DFIG grid-connected system and the benchmark system of inertia using the estimated result is shown in

Fig. 8 Comparison between DFIG grid-connected system and benchmark system of inertia using estimated result. (a) System angular frequency with . (b) Error distribution of system angular frequency with . (c) System angular frequency with . (b) Error distribution of system angular frequency with .
Considering the effect of the parameters used in the additional frequency control, we further change the values of and from 2×1
To illustrate the performance of the proposed method applied to a wind farm, a modified IEEE 30-bus system with a wind farm shown in

Fig. 9 Modified IEEE 30-bus system with a wind farm.
Setting the frequency disturbance by changing the load (25 MW) in the modified IEEE 30-bus system, the dynamic responses of the system with or without the additional frequency control are shown in

Fig. 10 Dynamic responses of modified IEEE 30-bus system with or without additional frequency control. (a) System angular frequency. (b) Grid-connected power of wind farm. (c) System angular frequency with load at each bus cut in half. (d) Grid-connected power of wind farm with load at each bus cut in half.
Apply the proposed method to estimate the virtual moment of inertia provided by these three equivalent WTs at different wind speeds. Similar to

Fig. 11 Sampling data sequences of DFIG-based wind farm in total participation mode. (a) Wind speeds of three WTs. (b) System angular frequency. (c) Grid-connected power.

Fig. 12 Comparison between modified IEEE 30-bus system and corresponding benchmark system of inertia in total participation mode. (a) System angular frequency with . (b) Error distribution of system angular frequency with . (c) System angular frequency with . (d) Error distribution of system angular frequency with .
To verify the estimation method in different operation modes of the wind farm, two equivalent WTs are set to participate in the frequency regulation and the other one is not. Such an operation mode is labeled as partial participation mode. The verification of the estimated virtual moment of inertia by using its benchmark system of inertia is shown in

Fig. 13 Comparison between modified IEEE 30-bus system and corresponding benchmark system of inertia in partial participation mode. (a) System angular frequency with . (b) Error distribution of system angular frequency with . (c) System angular frequency with . (d) Error distribution of system angular frequency with .
By setting different parameters of the additional frequency control,
Starting from the derivation of the frequency response model of DFIG, this paper proposes an estimation method to calculate the virtual moment of inertia provided by the DFIG-based wind farm with additional frequency control. The case studies tested by the DFIG grid-connected system and modified IEEE 30-bus system demonstrate the effectiveness of the derived expression and the proposed method. Some conclusions are derived below.
1) The derived expression of the virtual moment of inertia can directly represents the inertial characteristic of WT with additional frequency control, which is easy to combine with the existing moment of inertia in the power system and thus determine the overall frequency response of the power system.
2) The estimation method can efficiently calculate the virtual moment of inertia provided by the wind farm, whether the WTs are fully involved in the frequency regulation or not. Such an identification way avoids the complex processing of the parametric derivation and frequency response integration, which is very suitable for the power system with high penetration of wind power.
Appendix
Taking the stator voltage-oriented vector control as example, the increment of output active power at the stator side can be expressed as:
(A1) |
where is the PI regulator of the outer power loop, and and are the corresponding proportional and integral parameters, respectively; and is the transfer function of the inner current loop, i.e.,
(A2) |
where is the PI regulator of the inner current loop, and and are the corresponding proportional and integral parameters, respectively; , and is the switching cycle; and , and and are the resistance and inductance at the rotor side, respectively, .
Since is usually of large bandwidth, in the low-frequency band, it can be assumed as the gain of , and then is approximated as:
(A3) |
Note that due to the plant pole at , which is fairly close to the origin, the magnitude and phase of the loop gain start to drop from a relatively low frequency. When the plant pole is first canceled by the compensator zero, becomes:
(A4) |
Replacing in (A1) with , we can obtain:
(A5) |
(A6) |
By using Routh approximation, two types of the first-order transfer function close to in the low-frequency band are obtained.
(A7) |
After comparing their pole positions and selecting the one close to the imaginary axis, the expression of the time constant is finally determined.
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