Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Virtual Inertia Estimation Method of DFIG-based Wind Farm with Additional Frequency Control  PDF

  • Pengwei Chen (Member, IEEE)
  • Chenchen Qi
  • Xin Chen (Member, IEEE)
Jiangsu Key Laboratory of New Energy Generation and Power Conversion, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China

Updated:2021-09-27

DOI:10.35833/MPCE.2020.000908

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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.

I. Introduction

DUE to cleanliness, safety, and adequacy, wind power generation has experienced rapid development in the past decades [

1]-[3]. Doubly-fed induction generator (DFIG) is one of the most commercialized wind turbines (WTs) with good performance in integrating intermittent wind power, but its converter control makes the rotor speed decoupled with the system frequency [3]. With the continuous growth of DFIG-based wind farms, the total equivalent rotary inertia of the system is passively decreased, and the high penetration itself also generates additional disturbance because of the time-varying output. Meanwhile, these factors will challenge the system frequency stability [4], [5].

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 [

6], [7]. Against this background and demand, various methodologies have been explored in the literature or practical applications, which fall into three main categories: rotor speed control, pitch angle control, and inertia control [8], [9]. In the rotor speed control, the operating point is set to be retarded or advanced from the optimum operation conditions to achieve the active power reserve, which is usually employed at low wind speed [10]. The pitch angle control is also a de-loading technique, but its response speed participating in primary frequency regulation is still limited because of the mechanical constraints [11]. By adding an auxiliary signal to the active power reference value, the inertia control can provide temporary active power support by releasing active power reserve under a variable wind speed condition [10]-[12].

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 [

13], consisting of actual changed kinetic energy, energy loss, and captured wind power. Such a time constant of virtual inertia is decided by the integration of output active power and the control time of rotor speed, which is hard to calculate accurately when the parameters are unknown. In [14] and [15], based on the assumption about the effective inertia of rotating components, the equivalent inertia constant is acquired from the mechanical energy captured by WTs. This type of method ignores the impact of DFIG control on the inertia, and thus the accuracy of the derived equivalent time constant cannot be guaranteed. Taking the additional frequency control into consideration, a definition of virtual inertia constant based on the ratio between the changes of system angular frequency and rotor angular velocity is also defined in [16], but the ratio is difficult to determine as the dynamics of the converter has to be considered. Different from focusing on the inertia constant that relies on the conversion of the moment of inertia, rated capacity, and rated rotor angular velocity, an expression of the equivalent moment of inertia described by transfer function is derived in [17] by using the change of kinetic energy. In essence, such a moment of inertia is still a kind of dynamic process, which is significantly different from the accepted deterministic definition. If the inertia is measured in this way, the function intuitively indicating the system inertia will not exist.

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 [

18], [19]. However, each WT has its own operating point, of which operation data such as rotor velocity are usually difficult to be observed completely. Such an accumulation processing is inefficient and even inaccessible. Another approach to obtain the total virtual inertia is to utilize the established multi-machine dynamic equivalent model of wind farm [20], but this two-stage processing essentially transfers the workload of inertia accumulation to multi-machine representation as well as introduces unpredictable errors. Although the equivalent inertia of an entire system can be roughly estimated based on the power imbalance and change rate of frequency [21], or other monitoring data, the part provided by the wind farm still remains to be determined, and the estimation also relies on the integrated system with the wind farm. The direct estimation of the virtual moment of inertia is more valuable for the planning and operation of the wind farm itself, which is efficient to determine the impact on the characteristics of system frequency no matter what changes occur in the system. Regarding the estimation for the virtual inertia of devices, some techniques such as disturbance observer and recursive least squares algorithm have been explored for the grid-connected inverter with virtual synchronous generator control or droop control, the drive system based on the permanent magnet synchronous machine, and so on [22]-[24]. It is noted that the dynamics of the wind farm with additional frequency control are more complicated than that of a single inverter, and the definition of virtual inertia supported by the wind farm is not clear due to its non-synchronous nature, which also results in the inapplicability of the existing estimation techniques.

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.

II. Inertia of Power System with DFIG

A. Inertia of Synchronous Generator

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:

Jg=r2dm (1)

where r and m 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 ωmg and the torque exerting on the rotor can be expressed as:

Jgdωmgdt=Tmg-Teg-Dgωmg (2)

where Tmg, Teg, and Dg are the mechanical torque, electromagnetic torque, and damping coefficient of a generator, respectively.

Substitute Tmg=Pmg/ωmg,Teg=Peg/ωmg and ωmg=ω1/npg into (2) with Pmg, Peg, ω1, and npg 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:

Jgω10npg2dΔω1dt=ΔPmg-ΔPeg-Dgω10npg2Δω1 (3)

where ω10 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:

Δω1ΔPmg-ΔPeg=npg2Jgω10s+Dgω10 (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.

B. Inertia of DFIG

Using the motor convention, the electromechanical transient process of DFIG can be expressed as [

25]:

Jddωmddt=Ted-Tmd-DdωmdTed=-npdω1PsTmd=-ρSwCpv32ωmd (5)

where ωmd is the angular velocity of rotor; Jd and Dd are the total moment of inertia and damping coefficient converted onto the motor axis, respectively; Ted and Tmd are the electromagnetic torque and mechanical torque, respectively; Ps is the active power of the stator; and ρ, Sw, v, Cp, and npd 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.

PsPs*=1αs+1α=3KLmusd0kp1+2LsK3KLmusd0ki1 (6)

where Ps and Ps* are the active power at the stator side and its reference value, respectively; K is the gain of pulse width modulation; Lm is the mutual inductance between the coaxial equivalent stator and rotor winding; Ls is equivalent to the self-inductance of the stator winding; usd0 is the steady-state d-axis voltage of stator; and kp1 and ki1 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 Popt=kwωmd3/Ng, where Ng is the transformation ratio of the gearbox, kw=0.5ρSw(Rw/λopt)3Cp,max, and Rw, λopt, and Cp,max are the wind wheel radius, optimal tip speed ratio, and maximum value of coefficient of power, respectively [

14]. To retain the margin of active power, the reference value Ps* can be set by:

Ps*=kωmd3    k<kwNg (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:

Pw=npdωmdω1Ps (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.

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 Tmd is nonlinear, and after the linearization, we can obtain the coefficient β with respect to Δωmd, namely ΔTmd=βΔωmd, as:

β=-ρSwCv03ωmd0-2Pmd02ωmd02 (9)

where v0 is the steady-state wind speed; ωmd0 is the steady-state angular velocity of rotor; Pmd0 is the steady-state value of the mechanical power; and C is obtained by linearizing the fit expression of Cp and thus making ΔCp=CΔωmd.

As depicted in Fig. 1, if the control loop is reduced to the unity gain, namely ΔPs*=ΔPs, the relationship between Δωmd and Δω1 can be formed as:

Δωmd=npdPs0Jdω102s+3knpdω10ωmd02+βω102+Ddω102Δω1 (10)

where Ps0 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 Δωmd is much smaller than that of Δω1 under disturbance, and an assumption of Δωmd0 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:

ΔPw=npdω10ωmd0ΔPs+npdω10Ps0Δωmd-npdωmdPs0Δω1ω102=3knpdωmd03ω10Δωmd0 (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.

III. Virtual Moment of Inertia Supported by DFIG

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.

A. Frequency Response of DFIG

The additional frequency control considered in this paper consists of two loops, i.e., the virtual inertia control and the droop control [

9]-[11], [13], as shown in Fig. 2.

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:

Pid=kdfsω1Tls+1+kpf(ω1-ω1ref) (12)

where Tl is the time constant of a low-pass filter; kdf and kpf are the virtual inertia coefficient and droop coefficient, respectively; and ω1ref is the setting value of the system angular frequency.

Thus, the reference value of active power at the stator side is modified as:

Ps*=kωmd3-Pid (13)

Combined with the block diagram shown in Fig. 1, the system angular frequency dynamic considering the additional frequency control can be updated as shown in Fig. 2. The additional frequency control promotes the change of reference value of active power at the stator side when the frequency disturbance occurs, thus constituting the response to the frequency variety. Such progress can rapidly release the kinetic energy of the rotor of DFIG to compensate for the imbalance of active power.

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 Fig. 2, the relationship between Δω1 and ΔPw can be obtained as:

ΔPw=g1+g2+g3+g4g0g0Δω1 (14)
g0=1+3kωmd02npdω10(αs+1)(Jds+Dd+β)g1=3kωmd03npd2Ps0ω103(αs+1)(Jds+Dd+β)g2=-npd2Ps02(αs+1)ω103(αs+1)(Jds+Dd+β)g3=-npdωmd0(kdfs+kpf)(1+Tls)(αs+1)ω10g4=-npdωmd0Ps0ω102 (15)

As the time constant Tl in the low-pass filter is small, if the corresponding part is neglected, then (14) can be simplified as:

ΔPwΔω1=Gpω(s)=b2s2+b1s+b0a2s2+a1s+a0 (16)
a2=Jdαa1=αDd+αβ+Jda0=Dd+βb2=-kdfJdnpdωmd0Ps0ω10-npdωmd0Ps0a2ω102b1=-npdωmd0(kdfDd+kdfβ+kpfJd)ω10-         npdωmd0Ps0a1ω102-npd2Ps02αω103b0=-3kωmd02npd+npdωmd0kpfa0ω10-         npdωmd0Ps0a0ω102-3kωmd03npd2Ps0+npd2Ps02ω103 (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(a).

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.

B. Estimation for 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:

Jgω10npg2dΔω1dt=ΔPmg-ΔPL+ΔPw-Dgω10npg2Δω1 (18)

where ΔPL is the load disturbance.

Substituting (16) into (18), the transfer function between the power imbalance and the system angular frequency can be expressed as:

Δω1ΔPmg-ΔPL=Gω(s)=a2s2+a1s+a0c3s3+c2s2+c1s+c0 (19)
c0=Dgω10a0npg2-b0c1=Jgω10a0npg2+Dgω10a1npg2-b1c2=Jgω10a1npg2+Dgω10a2npg2-b2c3=Jgω10a2npg2 (20)

It can be observed that Gω(s) 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 [

26], and the reduced result is:

Ĝω(s)=1c1a0s+c0a0 (21)

If Ĝω(s) is used to replace Gω(s), the angular frequency dynamic shown in Fig. 3(a) can be further simplified, as shown in Fig. 3(b). Obviously, c1/a0 contains the total moment of inertia Jtotal from both generator and DFIG, and we can easily obtain:

Jtotal=Jg+Dga1a0-b1npg2a0ω10 (22)

More specifically, Jtotal 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 Jg and can be ignored. Therefore, the virtual moment of inertia Jeq supported by DFIG can be given by:

Jeq-b1npg2a0ω10 (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 Gpω(s). For other types of additional frequency control schemes, the expression of Jeq can also be obtained in the same way, and there are only some differences in coefficients a0 and b1. 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.

IV. Estimation Method for Virtual Moment of Inertia

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.

A. Data Processing

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:

y(k)=x(k)+σ(k)h=1Mphzhk+σ(k)    k=1,2,,N (24)

where x(k) is the real data of the grid-connected power; σ(k) is the data noise; ph=Ahejθh, zh=e(αh+j2πfh)Δt, and Ah, θh, αh, and fh are the amplitude, initial phase, damping factor, and angular frequency of component h, respectively; and Δt, M, and N 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 [

27], [28], as a signal reconstruction algorithm with good performance, is employed, of which the specific usage is as follows.

For the sampling data sequence, the Hankel matrix Y is constructed firstly in the form of

Y=y(1)y(2)y(L+1)y(2)y(3)y(L+2)y(N-L)y(N-L+1)y(N) (25)

where L is the pencil parameter, which is usually chosen from N/3 to N/2 for efficient noise filtering.

Then, the singular-value decomposition is carried out to achieve denoising. Considering the singular value σh of Y as M, thus we can obtain:

σhσmax<10-γσh+1σmax (26)

where γ is the threshold value; and σmax is the maximum singular value. The ratio smaller than 10-γ is considered as the noise-singular value.

After the number of dominant components M is determined, we can use the filtered right-unitary matrix V that contains only M dominant right-singular vectors to construct the matrix pair {V1,V2}, which satisfies

V1H+V2H-zhI=0 (27)

where the superscripts “+” and “H” represent the operations of Moore-Penrose pseudoinverse and conjugate transpose, respectively; V1 is obtained by removing the last row of V; V2 is obtained by removing the first row of V; zh is the generalized eigenvalue; and I is the unit matrix.

Once zh and M are known, the residues ph in (24) can be solved from the following least squares problem.

y(1)y(2)y(N)=111z1z2zMz1N-1z2N-1zMN-1p1p2pM (28)

Thus, the data sequence of active power can be reconstructed as:

y(t)=h=1M2Aheαhtcos(2πfht+θh)    t=0,Δt,,NΔt (29)
Ah=phθh=tan-1Im(ph)Re(ph)αh=lnzhΔtfh=12πΔttan-1Im(zh)Re(zh) (30)

B. Estimation Process of Virtual Moment of Inertia

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 s=(1-z-1)/Δt is used to realize the z-transformation. Then, we can obtain the transfer function in the z-domain as:

Gpω(z)=n2z2+n1z+n0z2+m1z+m0 (31)

where m1,m0,n2,n1,n0 are the parameters that need to be identified.

Once the Gpω(z) is obtained, Gpω(s) is automatically determined. Through the above transformation, the detailed estimation process based on the least squares algorithm can be summarized in Fig. 4, which contains 5 steps.

Fig. 4 Flow chart of estimation process based on least squares algorithm.

Step 1:   data preparation. Set the sampling interval Δt and sampling period T, then the length of all data sequences is obtained as N=T/Δt. The simulations or monitoring data are used to construct the average wind speed sequence uv, system angular frequency sequence uω, and grid-connected power sequence y, where uv is the mean result of all monitoring points at each sampling interval.

Step 2:   sequence state judgment. Let x represent the normalized result of uv or uω, if x(k),k=1,2,,N, meets (x(k)-x̂)2dmax, then x can be considered as the stationary sequence denoted by x¯; otherwise, x is the non-stationary sequence denoted by x˜. x̂ and dmax are the average of the data sequence and the threshold value of judgment, respectively.

Step 3:   sequence selection. If both uv and uω can be classified as the state group {u¯v,u˜ω}, the matrix pencil method is used to reconstruct the corresponding sequence of grid-connected power and the low-frequency component yp is obtained, and then go to Step 4; otherwise, return to Step 1.

Step 4:   frequency response identification. Let u˜pω-upω0 and yp-yp0 be the input and output sequences, where upω0 and yp0 are the initial states of u˜pω and yp, the accuracy of parameter vector θ=[m1,m0,n2,n1,n0]T can be measured by the following error sum of squares [

29].

J=k=1N(y(k)-φkθ)2 (32)

where φk=[-yp(k-1)+yp0, -yp(k-2)+yp0,u˜pω(k)-u˜pω0,u˜pω(k- 1)-u˜pω0,u˜pω(k-2)-u˜pω0].

To minimize the error, θ should satisfy

Jθ=-ΦTy+ΦTΦθ=0 (33)

where y=[y(1),y(1),,y(N)]; and ΦT=[φ1T,φ2T,,φNT]. If Φ is full rank, then θ can be solved by the least squares algorithm:

θ=(ΦTΦ)-1ΦTy (34)

Step 5:   moment of inertia estimation. When Gpω(z) is obtained, Gpω(z) is transfered to Gpω(s) using the principle as:

Gpω(s)=Gpω(z)z=11-sΔt (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 uv and uω are classified as the state group {u˜v,u˜ω}, 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.

V. Case Studies

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.

A. DFIG Grid-connected System

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 Table I.

TABLE I Parameters of DFIG Grid-connected System
ParameterValue
Rated capacity 2 MVA
Rated voltage of stator 575 V
Total moment of inertia Jd 580 kg∙m2
Resistance of stator Rs 23 mΩ
Resistance of rotor Rr 16 mΩ
Leakage inductance of stator Lls 0.18 mH
Leakage inductance of rotor Llr 0.16 mH
Mutual inductance Lm 2.9 mH
Proportional-integral (PI) coefficient of inner loop [5, 40]
PI coefficient of outer loop [0.2, 20]
Virtual inertia coefficient kdf 0.3×106
Droop coefficient kpf 0.3×106

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⋅m2 and 100 MVA, respectively, and the load is 25 MW. The frequency disturbance is set by putting into or reducing the load of the background system.

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. To illustrate the DFIG without the additional frequency control participating in the frequency regulation, the DFIG is removed from the grid-connected system and only the background system is reserved (labeled as DFIG removed). When the DFIG is not configured with the additional frequency control, its frequency response to load disturbance is a close resemblance to that of DFIG removed, which demonstrates the system angular frequency is mostly dependent on the inertia of the equivalent synchronous generator. After the additional frequency control is added, DFIG can release the kinetic energy of the DFIG rotor and even potential unused wind energy to provide the system with inertia support, and thus its angular velocity of rotor and grid-connected power are no longer stable.

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 kpf=kdf=3×105, Jeq=1143.23 kg⋅m2, and for kpf=kdf=2×105, Jeq=758.25 kg⋅m2. To demonstrate the accuracy of the derived results, Jeq is added into the moment of inertia in the equivalent synchronous generator and the additional frequency control is removed from the original DFIG grid-connected system. The modified system is referred to as the benchmark system of inertia. If the derived virtual moment of inertia is accurate enough, then the benchmark system of inertia will possess the same inertia as the original system in theory as well as the frequency response to load disturbance.

The comparison between the DFIG grid-connected system and benchmark system of inertia using the derived results is shown in Fig. 6. The error of the system angular frequency eω1 is the difference between two trajectories at the corresponding sampling time. It can be observed that the errors are all located in the range of -0.02-0.02, which proves the accuracy of the derived virtual moments of inertia.

Fig. 6 Comparison between DFIG grid-connected system and benchmark system of inertia using derived results. (a) System angular frequency with kpf=kdf=3×105 and PL=3 MW. (b) Error distribution of system angular frequency with kpf=kdf=3×105 by taking result of benchmark system of inertia as a reference. (c) System angular frequency with kpf=kdf=2×105 and PL=3 MW. (d) Error distribution of system angular frequency with kpf=kdf=2×105 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 kpf=kdf=2×105 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, where the sampling interval Δt and the sampling period T are set to be 2 ms and 10 s, respectively.

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 Jeq=787.07  kg·m2. The comparison between the DFIG grid-connected system and the benchmark system of inertia using the estimated result is shown in Fig. 8. It can be observed that the estimated Jeq is accurate enough to be applied under different conditions of frequency disturbance.

Fig. 8 Comparison between DFIG grid-connected system and benchmark system of inertia using estimated result. (a) System angular frequency with PL=4 MW. (b) Error distribution of system angular frequency with PL=4 MW. (c) System angular frequency with PL=3 MW. (b) Error distribution of system angular frequency with PL=3 MW.

Considering the effect of the parameters used in the additional frequency control, we further change the values of kdf and kpf from 2×105 to 6×105. After the estimation, the corresponding estimated results of virtual moment of inertia are summarized in Table II. In addition to the moment of inertia Jd in DFIG, the virtual moment of inertia mainly depends on the parameters of additional frequency control, and the proposed method is of high sensitivity with changes of control parameters.

TABLE II Estimated Results of Virtual Moment of Intertia with Different Parameters
kdfJeq
kpf=2×105kpf=4×105kpf=6×105
2×105 787.07 1023.04 1259.25
4×105 1303.07 1593.72 1775.92
6×105 1819.06 2055.95 2292.16

B. Modified IEEE 30-bus System with Wind Farm

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 is taken as the test system. Limited by the electromagnetic transient simulation capability of MATLAB/Simulink, the wind farm consists of three DFIGs, where the capacity of each DFIG is 10 MVA and their converters employ the average model.

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. Regarding Fig. 10(c) and (d), the load at each bus is cut in half, and the location of the wind farm is changed to Bus 13, while the generators at Buses 5 and 13 are removed.

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. 7, the sampling data sequences of wind speeds individually measured by three equivalent WTs are shown in Fig. 11(a), whereas the system angular frequency and grid-connected power are shown in Fig. 11(b) and (c). The operation mode with all WTs participating in the DFIG-based wind farm is labeled as total participation mode. After the judgment, the data segments marked by red lines shown in Fig. 11 are determined as the input of the estimation process, where the sampling interval and sampling period are the same as those in Section V-A. The virtual moment of inertia is estimated to be Jeq=3814.38 kg·m2. The benchmark system of inertia corresponding to this operation mode is constructed in a similar way to Section V-A, namely removing the additional frequency control from the wind farm and adding the estimated moment of inertia into the generator at the slack bus. The comparison between the modified IEEE 30-bus system and the corresponding benchmark system of inertia in the total participation mode is shown in Fig. 12. It can be observed that although the background system is extended and the number of WTs is increased, the proposed method is still of high accuracy.

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 ΔPL=25 MW. (b) Error distribution of system angular frequency with ΔPL=25 MW. (c) System angular frequency with ΔPL=20 MW. (d) Error distribution of system angular frequency with ΔPL=25 MW.

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. In the partial participation mode, since the accuracy performance is the same as that of the total participation mode, it is not explained in detail any longer.

Fig. 13 Comparison between modified IEEE 30-bus system and corresponding benchmark system of inertia in partial participation mode. (a) System angular frequency with ΔPL=25 MW. (b) Error distribution of system angular frequency with ΔPL=25 MW. (c) System angular frequency with ΔPL=20 MW. (d) Error distribution of system angular frequency with ΔPL=20 MW.

By setting different parameters of the additional frequency control, Table III presents the estimated results of the virtual moment of inertia provided by the DFIG-based wind farm in total participation mode, whereas Table IV summarizes the estimated results in the partial participation mode. Combined with Fig. 13, it can be observed that the proposed method is sensitive to the changes of control parameters and operation mode.

TABLE III Estimated Results of Virtual Moment of Inertia with Different Parameters in Total Participation Mode
kdfJeq
kpf=6.0×105kpf=1.2×106kpf=1.8×106
6.0×105 3814.38 4414.01 5013.59
1.2×106 7700.05 8299.74 8899.35
1.8×106 11585.89 12185.50 12785.11
TABLE IV Estimated Results of Virtual Moment of Inertia with Different Paramrters in Partial Participation Mode
kdfJeq
kpf=6.0×105kpf=1.2×106kpf=1.8×106
6.0×105 2888.41 3288.37 3688.33
1.2×106 5478.18 5878.29 6278.25
1.8×106 8068.25 8468.21 8868.17

Table V presents the average time cost of the main steps in an effective estimation process, where the estimation times is 10. The total time cost mainly depends on the process of data collection. As the sampling interval increases, the time cost of data collection will decrease significantly, whereas the time cost on inertia estimation still maintains a satisfactory level. Regarding the sampling interval Δt=2 ms adopted above, the total time cost is less than the length of the data sequence of 10 s, which means the proposed method can be used online to monitor the virtual moment of inertia in real-time. It is worth noting that the derived expression essentially completes the transformation of DFIG or DFIG-based wind farm from non-synchronous to synchronous, which determines the dominant component of the virtual moment of inertia and becomes the key point in the estimation process. In addition to the above academic contributions, the proposed method presents a framework and an implementation scheme with the satisfactory performance. From the perspective of substitutability, more techniques can be introduced to improve the efficiency and accuracy.

TABLE V Average Time Cost of Main Steps in an Effective Estimation Process
Main stepAverage time cost (s)
Δt=1 msΔt=2 msΔt=3 ms
Data collection (Steps 1-3) 21.1092 2.1523 0.6944
Inertia estimation (Steps 4, 5) 0.0633 0.0570 0.0470
Total 21.1725 2.1823 0.7414

VI. Conclusion

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

Appendix A

Taking the stator voltage-oriented vector control as example, the increment of output active power at the stator side can be expressed as:

ΔPs=1.5G1(s)Gi(s)usd0LmLs1+1.5G1(s)Gi(s)usd0LmLsΔPs* (A1)

where G1(s)=kp1+ki1/s is the PI regulator of the outer power loop, and kp1 and ki1 are the corresponding proportional and integral parameters, respectively; and Gi(s) is the transfer function of the inner current loop, i.e.,

Gi(s)=G2(s)Gpwm(s)Gc(s)1+G2(s)Gpwm(s)Gc(s) (A2)

where G2(s)=kp2+ki2/s is the PI regulator of the inner current loop, and kp2 and ki2 are the corresponding proportional and integral parameters, respectively; Gpwm(s)=K/(1+Tδs), and Tδ is the switching cycle; and Gc(s)=1/(σLrs+Rr), and Rr and Lr are the resistance and inductance at the rotor side, respectively, σ=1-Lm2/(LsLr).

Since Gpwm(s) is usually of large bandwidth, in the low-frequency band, it can be assumed as the gain of K, and then Gi(s) is approximated as:

Ĝi(s)=kp2Ks+ki2kp2sσLrs+RrσLr (A3)

Note that due to the plant pole at s=-Rr/(σLr), 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, Ĝi(s) becomes:

Ĝi'(s)=kp2KσLrs+kp2K (A4)

Replacing Gi(s) in (A1) with Ĝi'(s), we can obtain:

ΔPsΔPs*s=Gp(s)=B1s+B2A0s2+A1s+A2 (A5)
A0=2LsσLrA1=3LmKusd0kp2kp1+2kp2LsKA2=3LmKusd0kp2ki1B1=3LmKusd0kp2kp1B2=3LmKusd0kp2ki1 (A6)

By using Routh approximation, two types of the first-order transfer function close to Gp(s) in the low-frequency band are obtained.

Gp'(s)=B1A0s+A1Ĝp'(s)=B2A1s+A2 (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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