# Introducing MRAC-PSS-VI to Increase Small-Signal Stability of the Power System after Wind Power Integration

Authors: Ping He; Mingming Zheng; Haoran Jin; Zhijie Gong; Jie Dong
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

With the gradual increase of wind power penetration in the power system, the impact of wind farm on system stability is becoming more significant. This study designs the power system stabilizer (PSS) based on the model reference adaptive control (MRAC) method and virtual impedance (VI) control strategy. Then, the active power difference of the doubly fed induction generator (DFIG) is selected as input signal of MRAC-PSS-VI. The small-signal stability of the power system with the DFIG is enhanced by installing MRAC-PSS-VI on the rotor side control (RSC) control link. The controller model is built in DigSILENT/PowerFactory. The improvement effect of the controller on the low-frequency oscillation (LFO) characteristics of the power system is verified by using eigenvalue analysis and the time domain simulation method under different transmission powers of the tie line and different installed positions of the DFIG.


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## Body

## 1. Introduction

While wind power is used as potential source of electricity, the development of wind energy is increasing rapidly around the world. During wind power conversion, the doubly fed induction generator (DFIG) has the most important position in electrical machine, thanks to the DFIG that has an advantage in low power rating of the rotor side converter (RSC), high power capacity in both subsynchronous and supersynchronous operating conditions, and low power rotor side easy to control. In addition, the DFIG can realize constant frequency power generation under different wind speed conditions. For good measure, decoupled power injection capability and harmonic suppression capability of the DFIG are similar to the conventional synchronous generator.A variety of control strategies of the DFIG are presented in published studies. Pena et al. [1] illustrated a stator and rotor field-oriented vector control strategy, while the direct torque control scheme (DTC), direct power control strategy (DPC), and direct flux control strategy are illustrated in [2–5]. All of these control schemes require coordinate axis transformation of rotor position and control of velocity information. In order to stabilize the oscillation of the power system, Surinkaew and Ngamroo [6] proposed a coordinated control method between the DFIG and synchronous generator while installed the power oscillation damper (POD) and power system stabilizer (PSS), respectively. In the study by Rahim and Habiballah [7], the DFIG rotor voltage control for system dynamic performance enhancement is proposed. The POD equipped with the DFIG based on a phase lead compensation is able to improve the network damping. In recent years, many literatures are devoted to researching the additional damping control strategy of the DFIG. In the study by Hughes et al. [8], the PSS of wind turbine with the DFIG is proposed, which proves that the PSS can significantly affect the contribution of wind farm based on the DFIG to power grid damping. The feasibility of the DFIG providing additional damping is investigated, and the DFIG wind generators installing the PSS are synchronously applied to increase damping levels of a multimachine power system [9]. Mohamed et al. [10] pointed out the influence of the DFIG operation and control on rotor angle stability and proposed a PSS with the DFIG reactive power as input signal to improve rotor angle stability of the system. Surinkaew and Ngamroo [11] proposed a two-level stratified strategy consisting of wide area centralized and local controls of POD installed with the DFIG wind power turbine, and the PSS has been proposed for robust power oscillation damping. But all these control schemes do not take the output impedance of the PSS into account. Virtual impedance (VI) control method is very common in microgrid, but it has not been applied in the PSS controller. In order to adjust the output impedance of the PSS, this work introduces the virtual impedance method. The positive sequence virtual impedance can be introduced to improve the power distribution effect of the distributed generators (DGs), and the negative sequence VI is introduced to make up for the voltage drop because of the negative sequence current on the line impedance and to reduce the negative sequence circulating current [12]. Pham and Lee [13] proposed a heightened DG-VI controller to supply harmonic voltage compensation and accurate power harmonic sharing of microgrid. Qian et al. [14] proposed an adaptive VI control strategy considering both power quality and stability constraints based on the small-signal model of the global positioning system (GPS) microgrid. It is well known that the PSS is an additional damping controller used to suppress LFO in the power system. In order to facilitate the application of the PSS, the MRAC method is introduced in this work. Coman and Boldisor [15] studied the adjustment mechanism of MRAC by using a gradient method (Massachusetts Institute of Technology (MIT) rule) or using a stability theory (Lyapunov method). Sharma and Kumar Palwalia [16] proposed a modified proportion integration differentiation (PID) control using an adaptive controller. Hung et al. [17] presented MRAC compared to cascade PID control for the purpose of evaluating their performance. In the study by Mohamed et al. [18], the applicability of sensorless speed estimation technology based on different model reference adaptive systems (MRAS) in the brushless doubly fed reluctance generator is discussed.Although the oscillation damping characteristics have been investigated, there are still several important problems with regard to improving LFO characteristics in a power system with wind power integrated to be solved. With the gradual increase of wind power penetration in the power system, the impact of wind farm on system stability gets more significant. The objective of this work is to ensure the small-signal stability of a power system effectively with plentiful integration of wind farms generation. The main contributions of this work are summarized as follows:(a)
The PSS controller is improved based on the VI control method(b)
MRAC method is used to optimize the control of the designed PSS-VI(c)
The reference model of MRAC is designed by the TLS-ESPRIT method and regional pole assignment method(d)
Eigenvalue analysis method and time domain simulations method are employed to estimate the effects of heightening the stability after MRAC-PSS-VI is installed in the DFIGThis study is structured as follows. The mathematical model of MRAC and PSS-VI is expounded in Section2. The mathematical model of the DFIG, the DFIG-MRAC-PSS-VI installation location, and the small-signal stability analysis model are given in Section 3. Finally, the effectiveness of the designed controller is verified by using the eigenvalue analysis method and time domain simulation method through a four-machine system, and some conclusions are given at the end of the study.


## 2. MRAC-PSS-VI Modeling

### 2.1. PSS-VI Model

The PSS is an auxiliary controller widely used in excitation control. The PSS can provide a control signal which includes a phase compensation through AVR to the excitation system in order to increase the system damping and obtain the damping effect on power oscillations. Figure1 shows the block diagram of the single-input PSS used in the simulations of this study.Figure 1
Block diagram of the applied PSS.Tw is the time constant of the straight block, T1, T2, T3, and T4 are the time constants of the lead and lag blocks, and KP is the gain of the PSS. The block diagram consists of a wash-out section, a gain section, two phase compensation sections, and a limit section for the output signal.In the inverter, in order to decrease the cost of network construction and line loss, the virtual impedance can be introduced to replace the series inductance to make the inverter line impedance inductive. Based on this background, this study proposes that virtual impedance can be introduced to the power system stabilizer to adjust its output impedance without affecting its regulation effect.The controller parameters not only directly affect the output impedance but also determine the rapidity and stability of the system. Therefore, the design of virtual impedance based on the controller parameters should take both factors into account, which will lead to a smaller design range of the controller parameters. Thus, this study proposes the control strategy of adding an impedance loop to the PSS control loop, which is used to adjust the PSS output impedance without affecting the regulation effect of the PSS.PSS-VI output impedance is synthesized by selecting the reasonable virtual impedance and PSS output impedance, and then, the properties of PSS impedance are changed to improve the stability of the system. Figure2 shows the vector diagram representation of virtual impedance. It can be seen from Figure 2 that the phase angle of PSS-VI output impedance will change when the virtual impedance is changed, so the PSS-VI output impedance can be regulated by changing the virtual impedance modulus value, while ensuring that the controller has a good unit step response.Figure 2
Vector diagram representation of virtual impedance.Zvi is the virtual impedance, Zoi is the PSS output impedance, and Zi is the PSS-VI output impedance.The block diagram of the PSS-VI is shown in Figure3.Figure 3
PSS-VI model.PSS-VI input signal which can be selected rotor speed, voltage, current, or frequency has a great impact on the performance of the controller. Residual analysis is the most common method of signal selection. The input signal of the virtual impedance link adapts the filtering signal to ensure this link is not affected by the clutter of the PSS input signal.


### 2.2. Mathematical Model of MRAC

MRAC is designed based on Lyapunov theory and Barbalet’s lemma to ensure stability and convergence of the system. The second order system can be selected as the reference model to meet requirements of settling time, rise time, peak time, and overshoot. Based on the reference model, general adaptive control laws are established. MRAC contributes greatly to the elimination of modeling errors and dynamic uncertainties. The general block diagram of MRAC is shown in Figure4.Figure 4
General block diagram of MRAC.In this study, based on the general block diagram of MRAC, the reference model is designed by total least squares-estimation of signal parameters via rotational invariance techniques (TLS-ESPRIT) and the regional pole assignment method. Because the proportional integral derivative (PID) controller has a good effect on the elimination of trajectory tracking error, the PID controller is used to eliminate the error betweenup and uref. The adaptive law is used to adjust the paraments of the PID controller automatically.


#### 2.2.1. Reference Model Design

TLS-ESPRIT is an improved estimation of signal parameters via the rotational invariance techniques (ESPRIT) algorithm proposed by Roy et al. It is a valid identification algorithm to estimate vibration attenuation and sinusoidal signal. In this work, the TLS-ESPRIT algorithm is used for estimating the parameters of signal from the metrical data, which is the angle of the synchronous generator [18, 19].The process of the TLS-ESPRIT algorithm is as follows:(1)
First, LFO signal is sampled, and then, the sampled signal is sorted into a combination of attenuated sinusoidal signal and white noise whose characteristic is the value of amplitude varying exponentially. The mathematical model of the low-frequency oscillation signal is described as follows:(1)xn=∑k=1Pakejθke−σk+jωknTs+ωn,whereP is the equivalent to twice of actual component sine wave, while it also represents the mode order. ak, θk, σk, and ωk, respectively, represent the amplitude, initial phase, attenuation coefficient, and angular frequency of the kth damped sinusoidal quantity. Ts is the sampling period, and ωn is the white Gaussian noise whose average value equals 0.(2)
Then, an appropriate parameterL should be selected. The Hankel matrix H whose order is L×M can be constructed with preprocessing signals xn as follows:(2)H=x0x1⋯xM−1x1x2⋯xM⋮⋮⋮⋮xL−1xN−L⋯xN−1,whereL is the number of data records, M is the time-window length, and N is the number of the sampling points of low-frequency oscillation signal. Additionally, L > P, M > P, and N = L + M − 1.(3)
Singular value decomposition (SVD) is performed onH. It is divided into a signal subspace and a noise subspace as follows:(3)X=UΛVH=USUN∑S00∑NVSHVNH,where the signal space is classified byU and V according to the magnitude of singular values, and the subscripts S and N correspond to the signal subspace and the noise subspace, respectively. The superscript H stands for conjugate transpose, that is to say, UHU = I and VHV = I. Λ is a diagonal matrix whose diagonal elements are the singular values of matrix X, U is an L × L unitary matrix, V is an M × M unitary matrix, and I is the unit diagonal matrix.(4)
Since the signal subspace after SVD has rotation invariance, the signal space is divided into two interwoven subspaces, while there must be reversible diagonal matrix Ψ to make the equation be established.(4)Vs=V1⋮=⋮V2V2=V1Ψ,whereV1 is the matrix obtained by removing the last row of VS and V2 is the matrix obtained by removing the first row of VS. Then, matrix V′ is structured by V1 and V2, and matrix VT is obtained after SVD performed on V′:(5)V′=V1V2Vt=V11V12V21V22,whereVt is divided into four matrix blocks P × P and ΨTLS=−V12V22−1.(5)
By calculating the eigenvalueλk (k = 1, 2, 3, ..., P) of the matrix ΨTLS, the signal frequency fk, damping factor σk, and damping ratio ξk can be estimated.(6)fk=arctanImλk/Reλk2πTs,σk=−lnλk/Ts,ξk=−σkσk2+2πfk2.(6)
The amplitude and initial phase angle are acquired through the least squares (LS) method. According ton points,(7)H=λb×Y,where(8)H=x0x1LxN−1T,Y=Y0Y1LYN−1T,λb=11⋯1λ1λ2⋯λk⋮⋮⋱⋮λ1N−1λ2N−1⋯λkN−1,and Y can be obtained by the LS method:(9)Y=λbHλb−1λbHH.Then, the original amplitudeak and the initial phase θk can be obtained as follows:(10)ak=2Yk,θk=arctanImYkReYk.According to the oscillation frequency band identified by TLS-ESPRIT, a Butterworth bandpass filter is added, with a frequency range of 0.2–2.5 Hz. In the process of identification, a 2% disturbance is added to simulate the small disturbance of the system, and the corresponding eigenvalues are obtained by the algorithm. The system mainly participates in the local oscillation modes of approximately 1.1 Hz and 1.6 Hz, and the damping ratios are low. They all belong to the poorly damped oscillation modes to be suppressed, which are called the main oscillation modes. The system also participates in an interarea mode of approximately 0.5 Hz, where the damping ratio is slightly higher than the local mode. A Butterworth bandpass filter with a frequency range of 0.2–1.7 Hz is added, and these modes are taken as the identification target. Using TLS-ESPRIT for identification again, the 4th-order transfer functionG(s) of the system is given in the following equation:(11)Gs=−7.32×10−5s4−1.1972s3−24.9377s2−29.058s−11.5661s4+173379.3s3+6228.2s2+7814.1s+117.4866.In order to verify the accuracy of the system transfer function after the order reduction, frequency domain analysis is carried out in MATLAB. As shown in Figure5, the reduced system has similar magnitude and phase response to the original system.Figure 5
The Bode diagram of system transfer function.Then, the controller is designed by using the regional pole assignment method to enhance the stability of the system. The transfer function of the designed controller is as follows:(12)Ks=−1.16s5−165.31s4−5.44×103s3−4.99×104s2−5.48×104s−2.11×104s6+286.29s5+2.85×104s4+1.11×106s3+1.26×107s2+1.73×107s+2.15×105.It can be seen from equation (12) that the order of the controller obtained is high, which is not conducive to practical application. In this study, the order reduction method of the balanced truncation model based on Hankel SVD is used to reduce the order of the controller. The reduced controller function is as follows:(13)K1s=−3.26×10−9s−2.32×10−10s2+0.50s+10.89.In this study, the obtained controllerK1s is designed as the reference model, so that the PSS-VI can output ideal signal. In order to demonstrate the effect of the controller, step response experiment of the system is done in MATLAB. The simulation results are shown in Figure 6, which show that the step response of the system is significantly improved after the design controller is added to the system.Figure 6
Step response after the controller is connected to the system.


#### 2.2.2. Conventional PID Controller

In the control process, the PID controller is widely used as the reason of simple structure and excellent robustness. Tuning methods are used to improve the performance of PID controllers. Then, the application of PID controllers has grown tremendous. The adopted PID controller in this study is shown in Figure7.Figure 7
Block of the PID controller.Kp1, Ki1, and Kd1 are the parameters of the PID controller.The consequence of variation of the PID controller is given in Table1. So, parameter tuning of the PID controller is of great significance to improve its performance. Because the derivative link is too sensitive, the parameters of the differential link are rounded to a very small value to reduce the influence of the differential link on the system.Table 1
The consequence of parameter variation of the PID controller.
Parameter increaseRise timeOvershootSettling timeSteady state errorKpDecreasesIncreaseSmall changeDecreaseKiDecreasesIncreaseIncreaseHighly reducedKdSmall changeDecreaseDecreaseSmall change


#### 2.2.3. Adaptive Ratio Controller

Assuming that the system of the controlled object is a linear time invariant system, the default state variable is equal to the output quantity, and the state and output equation can be expressed as(14)x˙t=A1xt+B1ut,yt=xt.The state and output equation of the reference model is expressed as(15)x˙mt=Amxmt+Bmcmt,ymt=xmt,where xm(t) is the reference state equation of the reference model, and c(t) is the input signal of the reference model.First, this work makes a difference between equations (15) and (14):(16)e˙t=Amxmt+Bmct−A1xt−B1ut,and then, further simplify the following equation:(17)e˙t=Amet+Amxt−A1xt+Bmct−B1ut,where et=xmt−xt. If set the error e(t) is equal to 0, the system needs to satisfy that the eigenvalue of the matrix Am is negative and the output u(t) of the controller of the following equation is as follows:(18)ut=act+bxt.Therefore, the equation of the controlleru(t) can be expressed as equation (18), while parameters a and b are unknown. In a closed-loop system, it should be considered that the controller has only one adjustable parameter θ. The parameter e is defined as the deviation between the output signal (y) of the plant and the output signal (ym) of the reference model. So, this work adopts the stability theory to calculate a and b.First of all, the gradient of the state variable deviation of the reference model and the controlled object can be expressed as follows:(19)e˙=x˙mt−x˙t.According to equations (14) and (15), equations (18) and (19) can be simplified as(20)e˙t=Amet+Am−A1−B1bxt+Bm−B1act.When the eigenvalue of the matrixAm is negative and the following equation sets up, the error will converge to zero.(21)e1=Am−A1−B1b=0,e2=Bm−B1a=0.Whene, e1, and e2 converge to zero, the system will be stable. According to the second method of Lyapunov, this work constructs the following scalar function (V (e, e1, e2)) containing the introduced adaptation gain γ to calculate equation (21).(22)V=12e2+1B1γAm−A1−B1b+1B1γBm−B1a2=12e2+1B1γe12+1B1γe22.Take the time derivative of this scalar function,(23)V˙e,e1,e2=dVe,e1,e2dt=ededt+1B1γe1de1dt+1B1γe2de2dt.Because equation (23) is regarded as a first-order full state equation under the action of zero input, the stabilization point ex of the system state is(24)e=0e1=0e2=0.The scalar functionV based on equation (24) satisfies that it has a first-order continuous partial derivative with respect to all variables, and it satisfies as follows:(25)Ve,e1,e2>0e≠0,e1≠0,e2≠0,V0=0,V˙e,e1,e2=dVe,e1,e2dt<0.When the state trajectories of these three state variables all converge at the stabilization pointex, it means that e, e1, and e2 converge to zero. So,(26)V˙e,e1,e2=dVe,e1,e2dt<0⇒e=e1=e2=0.According to equations (20) and (23), the equation can be simplified as(27)dVe,e1,e2dt=ededt+1B1γe1dedt+1B1γe2dedt,=Ame2t+Am−A−B1btetxt+Bm−B1atetct−e1γdbtdt−e2γdatdt,=Ame2t+e1γγetxt−dbtdt+e2γγetct−datdt.Because ofAme2t<0, while(28)dbtdt=γetxt,datdt=γetct,V˙e,e1,e2<0, and e, e1, and e2 converge to zero.The result is(29)dVe,e1,e2dt=Ame2t.The adjustment rules obtained by the Lyapunov theory are simple and do not require to filter the signal.So, MRAC based on the second method of Lyapunov could use equation (28) calculating parameters a and b.The equation (13) can be expressed in the form of state space:(30)x˙mt=01−10.89−0.50xmt+01ct,ymt=−2.32×10−10−3.26×10−9xmt.The transfer function of the PSS-VI is(31)G1s=KPsTW1+sTW1+sT11+sT31+sT21+sT4+Kr+Kls.In this work, the author selectedKP = −0.5, Tw = 0.02, Kr = −1.2, Kl = −36, T1 = 0.3, T2 = 0.08, T3 = 0.06, and T4 = 0.3 to meet the systems requirements for dynamic performance and stability. So, G1(s) can be rewritten as follows.(32)G1s=−0.06s2+19.88s+0.054s2+49.16s+150.8.The equation (32) can be rewritten in the form of state space:(33)x˙t=01−150.8−49.16xt+01ut,yt=0.05419.88xt.According to equations (14)–(28), the paraments of the PID controller can be calculated.


## 2.1. PSS-VI Model

The PSS is an auxiliary controller widely used in excitation control. The PSS can provide a control signal which includes a phase compensation through AVR to the excitation system in order to increase the system damping and obtain the damping effect on power oscillations. Figure1 shows the block diagram of the single-input PSS used in the simulations of this study.Figure 1
Block diagram of the applied PSS.Tw is the time constant of the straight block, T1, T2, T3, and T4 are the time constants of the lead and lag blocks, and KP is the gain of the PSS. The block diagram consists of a wash-out section, a gain section, two phase compensation sections, and a limit section for the output signal.In the inverter, in order to decrease the cost of network construction and line loss, the virtual impedance can be introduced to replace the series inductance to make the inverter line impedance inductive. Based on this background, this study proposes that virtual impedance can be introduced to the power system stabilizer to adjust its output impedance without affecting its regulation effect.The controller parameters not only directly affect the output impedance but also determine the rapidity and stability of the system. Therefore, the design of virtual impedance based on the controller parameters should take both factors into account, which will lead to a smaller design range of the controller parameters. Thus, this study proposes the control strategy of adding an impedance loop to the PSS control loop, which is used to adjust the PSS output impedance without affecting the regulation effect of the PSS.PSS-VI output impedance is synthesized by selecting the reasonable virtual impedance and PSS output impedance, and then, the properties of PSS impedance are changed to improve the stability of the system. Figure2 shows the vector diagram representation of virtual impedance. It can be seen from Figure 2 that the phase angle of PSS-VI output impedance will change when the virtual impedance is changed, so the PSS-VI output impedance can be regulated by changing the virtual impedance modulus value, while ensuring that the controller has a good unit step response.Figure 2
Vector diagram representation of virtual impedance.Zvi is the virtual impedance, Zoi is the PSS output impedance, and Zi is the PSS-VI output impedance.The block diagram of the PSS-VI is shown in Figure3.Figure 3
PSS-VI model.PSS-VI input signal which can be selected rotor speed, voltage, current, or frequency has a great impact on the performance of the controller. Residual analysis is the most common method of signal selection. The input signal of the virtual impedance link adapts the filtering signal to ensure this link is not affected by the clutter of the PSS input signal.


## 2.2. Mathematical Model of MRAC

MRAC is designed based on Lyapunov theory and Barbalet’s lemma to ensure stability and convergence of the system. The second order system can be selected as the reference model to meet requirements of settling time, rise time, peak time, and overshoot. Based on the reference model, general adaptive control laws are established. MRAC contributes greatly to the elimination of modeling errors and dynamic uncertainties. The general block diagram of MRAC is shown in Figure4.Figure 4
General block diagram of MRAC.In this study, based on the general block diagram of MRAC, the reference model is designed by total least squares-estimation of signal parameters via rotational invariance techniques (TLS-ESPRIT) and the regional pole assignment method. Because the proportional integral derivative (PID) controller has a good effect on the elimination of trajectory tracking error, the PID controller is used to eliminate the error betweenup and uref. The adaptive law is used to adjust the paraments of the PID controller automatically.


### 2.2.1. Reference Model Design

TLS-ESPRIT is an improved estimation of signal parameters via the rotational invariance techniques (ESPRIT) algorithm proposed by Roy et al. It is a valid identification algorithm to estimate vibration attenuation and sinusoidal signal. In this work, the TLS-ESPRIT algorithm is used for estimating the parameters of signal from the metrical data, which is the angle of the synchronous generator [18, 19].The process of the TLS-ESPRIT algorithm is as follows:(1)
First, LFO signal is sampled, and then, the sampled signal is sorted into a combination of attenuated sinusoidal signal and white noise whose characteristic is the value of amplitude varying exponentially. The mathematical model of the low-frequency oscillation signal is described as follows:(1)xn=∑k=1Pakejθke−σk+jωknTs+ωn,whereP is the equivalent to twice of actual component sine wave, while it also represents the mode order. ak, θk, σk, and ωk, respectively, represent the amplitude, initial phase, attenuation coefficient, and angular frequency of the kth damped sinusoidal quantity. Ts is the sampling period, and ωn is the white Gaussian noise whose average value equals 0.(2)
Then, an appropriate parameterL should be selected. The Hankel matrix H whose order is L×M can be constructed with preprocessing signals xn as follows:(2)H=x0x1⋯xM−1x1x2⋯xM⋮⋮⋮⋮xL−1xN−L⋯xN−1,whereL is the number of data records, M is the time-window length, and N is the number of the sampling points of low-frequency oscillation signal. Additionally, L > P, M > P, and N = L + M − 1.(3)
Singular value decomposition (SVD) is performed onH. It is divided into a signal subspace and a noise subspace as follows:(3)X=UΛVH=USUN∑S00∑NVSHVNH,where the signal space is classified byU and V according to the magnitude of singular values, and the subscripts S and N correspond to the signal subspace and the noise subspace, respectively. The superscript H stands for conjugate transpose, that is to say, UHU = I and VHV = I. Λ is a diagonal matrix whose diagonal elements are the singular values of matrix X, U is an L × L unitary matrix, V is an M × M unitary matrix, and I is the unit diagonal matrix.(4)
Since the signal subspace after SVD has rotation invariance, the signal space is divided into two interwoven subspaces, while there must be reversible diagonal matrix Ψ to make the equation be established.(4)Vs=V1⋮=⋮V2V2=V1Ψ,whereV1 is the matrix obtained by removing the last row of VS and V2 is the matrix obtained by removing the first row of VS. Then, matrix V′ is structured by V1 and V2, and matrix VT is obtained after SVD performed on V′:(5)V′=V1V2Vt=V11V12V21V22,whereVt is divided into four matrix blocks P × P and ΨTLS=−V12V22−1.(5)
By calculating the eigenvalueλk (k = 1, 2, 3, ..., P) of the matrix ΨTLS, the signal frequency fk, damping factor σk, and damping ratio ξk can be estimated.(6)fk=arctanImλk/Reλk2πTs,σk=−lnλk/Ts,ξk=−σkσk2+2πfk2.(6)
The amplitude and initial phase angle are acquired through the least squares (LS) method. According ton points,(7)H=λb×Y,where(8)H=x0x1LxN−1T,Y=Y0Y1LYN−1T,λb=11⋯1λ1λ2⋯λk⋮⋮⋱⋮λ1N−1λ2N−1⋯λkN−1,and Y can be obtained by the LS method:(9)Y=λbHλb−1λbHH.Then, the original amplitudeak and the initial phase θk can be obtained as follows:(10)ak=2Yk,θk=arctanImYkReYk.According to the oscillation frequency band identified by TLS-ESPRIT, a Butterworth bandpass filter is added, with a frequency range of 0.2–2.5 Hz. In the process of identification, a 2% disturbance is added to simulate the small disturbance of the system, and the corresponding eigenvalues are obtained by the algorithm. The system mainly participates in the local oscillation modes of approximately 1.1 Hz and 1.6 Hz, and the damping ratios are low. They all belong to the poorly damped oscillation modes to be suppressed, which are called the main oscillation modes. The system also participates in an interarea mode of approximately 0.5 Hz, where the damping ratio is slightly higher than the local mode. A Butterworth bandpass filter with a frequency range of 0.2–1.7 Hz is added, and these modes are taken as the identification target. Using TLS-ESPRIT for identification again, the 4th-order transfer functionG(s) of the system is given in the following equation:(11)Gs=−7.32×10−5s4−1.1972s3−24.9377s2−29.058s−11.5661s4+173379.3s3+6228.2s2+7814.1s+117.4866.In order to verify the accuracy of the system transfer function after the order reduction, frequency domain analysis is carried out in MATLAB. As shown in Figure5, the reduced system has similar magnitude and phase response to the original system.Figure 5
The Bode diagram of system transfer function.Then, the controller is designed by using the regional pole assignment method to enhance the stability of the system. The transfer function of the designed controller is as follows:(12)Ks=−1.16s5−165.31s4−5.44×103s3−4.99×104s2−5.48×104s−2.11×104s6+286.29s5+2.85×104s4+1.11×106s3+1.26×107s2+1.73×107s+2.15×105.It can be seen from equation (12) that the order of the controller obtained is high, which is not conducive to practical application. In this study, the order reduction method of the balanced truncation model based on Hankel SVD is used to reduce the order of the controller. The reduced controller function is as follows:(13)K1s=−3.26×10−9s−2.32×10−10s2+0.50s+10.89.In this study, the obtained controllerK1s is designed as the reference model, so that the PSS-VI can output ideal signal. In order to demonstrate the effect of the controller, step response experiment of the system is done in MATLAB. The simulation results are shown in Figure 6, which show that the step response of the system is significantly improved after the design controller is added to the system.Figure 6
Step response after the controller is connected to the system.


### 2.2.2. Conventional PID Controller

In the control process, the PID controller is widely used as the reason of simple structure and excellent robustness. Tuning methods are used to improve the performance of PID controllers. Then, the application of PID controllers has grown tremendous. The adopted PID controller in this study is shown in Figure7.Figure 7
Block of the PID controller.Kp1, Ki1, and Kd1 are the parameters of the PID controller.The consequence of variation of the PID controller is given in Table1. So, parameter tuning of the PID controller is of great significance to improve its performance. Because the derivative link is too sensitive, the parameters of the differential link are rounded to a very small value to reduce the influence of the differential link on the system.Table 1
The consequence of parameter variation of the PID controller.
Parameter increaseRise timeOvershootSettling timeSteady state errorKpDecreasesIncreaseSmall changeDecreaseKiDecreasesIncreaseIncreaseHighly reducedKdSmall changeDecreaseDecreaseSmall change


### 2.2.3. Adaptive Ratio Controller

Assuming that the system of the controlled object is a linear time invariant system, the default state variable is equal to the output quantity, and the state and output equation can be expressed as(14)x˙t=A1xt+B1ut,yt=xt.The state and output equation of the reference model is expressed as(15)x˙mt=Amxmt+Bmcmt,ymt=xmt,where xm(t) is the reference state equation of the reference model, and c(t) is the input signal of the reference model.First, this work makes a difference between equations (15) and (14):(16)e˙t=Amxmt+Bmct−A1xt−B1ut,and then, further simplify the following equation:(17)e˙t=Amet+Amxt−A1xt+Bmct−B1ut,where et=xmt−xt. If set the error e(t) is equal to 0, the system needs to satisfy that the eigenvalue of the matrix Am is negative and the output u(t) of the controller of the following equation is as follows:(18)ut=act+bxt.Therefore, the equation of the controlleru(t) can be expressed as equation (18), while parameters a and b are unknown. In a closed-loop system, it should be considered that the controller has only one adjustable parameter θ. The parameter e is defined as the deviation between the output signal (y) of the plant and the output signal (ym) of the reference model. So, this work adopts the stability theory to calculate a and b.First of all, the gradient of the state variable deviation of the reference model and the controlled object can be expressed as follows:(19)e˙=x˙mt−x˙t.According to equations (14) and (15), equations (18) and (19) can be simplified as(20)e˙t=Amet+Am−A1−B1bxt+Bm−B1act.When the eigenvalue of the matrixAm is negative and the following equation sets up, the error will converge to zero.(21)e1=Am−A1−B1b=0,e2=Bm−B1a=0.Whene, e1, and e2 converge to zero, the system will be stable. According to the second method of Lyapunov, this work constructs the following scalar function (V (e, e1, e2)) containing the introduced adaptation gain γ to calculate equation (21).(22)V=12e2+1B1γAm−A1−B1b+1B1γBm−B1a2=12e2+1B1γe12+1B1γe22.Take the time derivative of this scalar function,(23)V˙e,e1,e2=dVe,e1,e2dt=ededt+1B1γe1de1dt+1B1γe2de2dt.Because equation (23) is regarded as a first-order full state equation under the action of zero input, the stabilization point ex of the system state is(24)e=0e1=0e2=0.The scalar functionV based on equation (24) satisfies that it has a first-order continuous partial derivative with respect to all variables, and it satisfies as follows:(25)Ve,e1,e2>0e≠0,e1≠0,e2≠0,V0=0,V˙e,e1,e2=dVe,e1,e2dt<0.When the state trajectories of these three state variables all converge at the stabilization pointex, it means that e, e1, and e2 converge to zero. So,(26)V˙e,e1,e2=dVe,e1,e2dt<0⇒e=e1=e2=0.According to equations (20) and (23), the equation can be simplified as(27)dVe,e1,e2dt=ededt+1B1γe1dedt+1B1γe2dedt,=Ame2t+Am−A−B1btetxt+Bm−B1atetct−e1γdbtdt−e2γdatdt,=Ame2t+e1γγetxt−dbtdt+e2γγetct−datdt.Because ofAme2t<0, while(28)dbtdt=γetxt,datdt=γetct,V˙e,e1,e2<0, and e, e1, and e2 converge to zero.The result is(29)dVe,e1,e2dt=Ame2t.The adjustment rules obtained by the Lyapunov theory are simple and do not require to filter the signal.So, MRAC based on the second method of Lyapunov could use equation (28) calculating parameters a and b.The equation (13) can be expressed in the form of state space:(30)x˙mt=01−10.89−0.50xmt+01ct,ymt=−2.32×10−10−3.26×10−9xmt.The transfer function of the PSS-VI is(31)G1s=KPsTW1+sTW1+sT11+sT31+sT21+sT4+Kr+Kls.In this work, the author selectedKP = −0.5, Tw = 0.02, Kr = −1.2, Kl = −36, T1 = 0.3, T2 = 0.08, T3 = 0.06, and T4 = 0.3 to meet the systems requirements for dynamic performance and stability. So, G1(s) can be rewritten as follows.(32)G1s=−0.06s2+19.88s+0.054s2+49.16s+150.8.The equation (32) can be rewritten in the form of state space:(33)x˙t=01−150.8−49.16xt+01ut,yt=0.05419.88xt.According to equations (14)–(28), the paraments of the PID controller can be calculated.


## 2.2.1. Reference Model Design

TLS-ESPRIT is an improved estimation of signal parameters via the rotational invariance techniques (ESPRIT) algorithm proposed by Roy et al. It is a valid identification algorithm to estimate vibration attenuation and sinusoidal signal. In this work, the TLS-ESPRIT algorithm is used for estimating the parameters of signal from the metrical data, which is the angle of the synchronous generator [18, 19].The process of the TLS-ESPRIT algorithm is as follows:(1)
First, LFO signal is sampled, and then, the sampled signal is sorted into a combination of attenuated sinusoidal signal and white noise whose characteristic is the value of amplitude varying exponentially. The mathematical model of the low-frequency oscillation signal is described as follows:(1)xn=∑k=1Pakejθke−σk+jωknTs+ωn,whereP is the equivalent to twice of actual component sine wave, while it also represents the mode order. ak, θk, σk, and ωk, respectively, represent the amplitude, initial phase, attenuation coefficient, and angular frequency of the kth damped sinusoidal quantity. Ts is the sampling period, and ωn is the white Gaussian noise whose average value equals 0.(2)
Then, an appropriate parameterL should be selected. The Hankel matrix H whose order is L×M can be constructed with preprocessing signals xn as follows:(2)H=x0x1⋯xM−1x1x2⋯xM⋮⋮⋮⋮xL−1xN−L⋯xN−1,whereL is the number of data records, M is the time-window length, and N is the number of the sampling points of low-frequency oscillation signal. Additionally, L > P, M > P, and N = L + M − 1.(3)
Singular value decomposition (SVD) is performed onH. It is divided into a signal subspace and a noise subspace as follows:(3)X=UΛVH=USUN∑S00∑NVSHVNH,where the signal space is classified byU and V according to the magnitude of singular values, and the subscripts S and N correspond to the signal subspace and the noise subspace, respectively. The superscript H stands for conjugate transpose, that is to say, UHU = I and VHV = I. Λ is a diagonal matrix whose diagonal elements are the singular values of matrix X, U is an L × L unitary matrix, V is an M × M unitary matrix, and I is the unit diagonal matrix.(4)
Since the signal subspace after SVD has rotation invariance, the signal space is divided into two interwoven subspaces, while there must be reversible diagonal matrix Ψ to make the equation be established.(4)Vs=V1⋮=⋮V2V2=V1Ψ,whereV1 is the matrix obtained by removing the last row of VS and V2 is the matrix obtained by removing the first row of VS. Then, matrix V′ is structured by V1 and V2, and matrix VT is obtained after SVD performed on V′:(5)V′=V1V2Vt=V11V12V21V22,whereVt is divided into four matrix blocks P × P and ΨTLS=−V12V22−1.(5)
By calculating the eigenvalueλk (k = 1, 2, 3, ..., P) of the matrix ΨTLS, the signal frequency fk, damping factor σk, and damping ratio ξk can be estimated.(6)fk=arctanImλk/Reλk2πTs,σk=−lnλk/Ts,ξk=−σkσk2+2πfk2.(6)
The amplitude and initial phase angle are acquired through the least squares (LS) method. According ton points,(7)H=λb×Y,where(8)H=x0x1LxN−1T,Y=Y0Y1LYN−1T,λb=11⋯1λ1λ2⋯λk⋮⋮⋱⋮λ1N−1λ2N−1⋯λkN−1,and Y can be obtained by the LS method:(9)Y=λbHλb−1λbHH.Then, the original amplitudeak and the initial phase θk can be obtained as follows:(10)ak=2Yk,θk=arctanImYkReYk.According to the oscillation frequency band identified by TLS-ESPRIT, a Butterworth bandpass filter is added, with a frequency range of 0.2–2.5 Hz. In the process of identification, a 2% disturbance is added to simulate the small disturbance of the system, and the corresponding eigenvalues are obtained by the algorithm. The system mainly participates in the local oscillation modes of approximately 1.1 Hz and 1.6 Hz, and the damping ratios are low. They all belong to the poorly damped oscillation modes to be suppressed, which are called the main oscillation modes. The system also participates in an interarea mode of approximately 0.5 Hz, where the damping ratio is slightly higher than the local mode. A Butterworth bandpass filter with a frequency range of 0.2–1.7 Hz is added, and these modes are taken as the identification target. Using TLS-ESPRIT for identification again, the 4th-order transfer functionG(s) of the system is given in the following equation:(11)Gs=−7.32×10−5s4−1.1972s3−24.9377s2−29.058s−11.5661s4+173379.3s3+6228.2s2+7814.1s+117.4866.In order to verify the accuracy of the system transfer function after the order reduction, frequency domain analysis is carried out in MATLAB. As shown in Figure5, the reduced system has similar magnitude and phase response to the original system.Figure 5
The Bode diagram of system transfer function.Then, the controller is designed by using the regional pole assignment method to enhance the stability of the system. The transfer function of the designed controller is as follows:(12)Ks=−1.16s5−165.31s4−5.44×103s3−4.99×104s2−5.48×104s−2.11×104s6+286.29s5+2.85×104s4+1.11×106s3+1.26×107s2+1.73×107s+2.15×105.It can be seen from equation (12) that the order of the controller obtained is high, which is not conducive to practical application. In this study, the order reduction method of the balanced truncation model based on Hankel SVD is used to reduce the order of the controller. The reduced controller function is as follows:(13)K1s=−3.26×10−9s−2.32×10−10s2+0.50s+10.89.In this study, the obtained controllerK1s is designed as the reference model, so that the PSS-VI can output ideal signal. In order to demonstrate the effect of the controller, step response experiment of the system is done in MATLAB. The simulation results are shown in Figure 6, which show that the step response of the system is significantly improved after the design controller is added to the system.Figure 6
Step response after the controller is connected to the system.


## 2.2.2. Conventional PID Controller

In the control process, the PID controller is widely used as the reason of simple structure and excellent robustness. Tuning methods are used to improve the performance of PID controllers. Then, the application of PID controllers has grown tremendous. The adopted PID controller in this study is shown in Figure7.Figure 7
Block of the PID controller.Kp1, Ki1, and Kd1 are the parameters of the PID controller.The consequence of variation of the PID controller is given in Table1. So, parameter tuning of the PID controller is of great significance to improve its performance. Because the derivative link is too sensitive, the parameters of the differential link are rounded to a very small value to reduce the influence of the differential link on the system.Table 1
The consequence of parameter variation of the PID controller.
Parameter increaseRise timeOvershootSettling timeSteady state errorKpDecreasesIncreaseSmall changeDecreaseKiDecreasesIncreaseIncreaseHighly reducedKdSmall changeDecreaseDecreaseSmall change


## 2.2.3. Adaptive Ratio Controller

Assuming that the system of the controlled object is a linear time invariant system, the default state variable is equal to the output quantity, and the state and output equation can be expressed as(14)x˙t=A1xt+B1ut,yt=xt.The state and output equation of the reference model is expressed as(15)x˙mt=Amxmt+Bmcmt,ymt=xmt,where xm(t) is the reference state equation of the reference model, and c(t) is the input signal of the reference model.First, this work makes a difference between equations (15) and (14):(16)e˙t=Amxmt+Bmct−A1xt−B1ut,and then, further simplify the following equation:(17)e˙t=Amet+Amxt−A1xt+Bmct−B1ut,where et=xmt−xt. If set the error e(t) is equal to 0, the system needs to satisfy that the eigenvalue of the matrix Am is negative and the output u(t) of the controller of the following equation is as follows:(18)ut=act+bxt.Therefore, the equation of the controlleru(t) can be expressed as equation (18), while parameters a and b are unknown. In a closed-loop system, it should be considered that the controller has only one adjustable parameter θ. The parameter e is defined as the deviation between the output signal (y) of the plant and the output signal (ym) of the reference model. So, this work adopts the stability theory to calculate a and b.First of all, the gradient of the state variable deviation of the reference model and the controlled object can be expressed as follows:(19)e˙=x˙mt−x˙t.According to equations (14) and (15), equations (18) and (19) can be simplified as(20)e˙t=Amet+Am−A1−B1bxt+Bm−B1act.When the eigenvalue of the matrixAm is negative and the following equation sets up, the error will converge to zero.(21)e1=Am−A1−B1b=0,e2=Bm−B1a=0.Whene, e1, and e2 converge to zero, the system will be stable. According to the second method of Lyapunov, this work constructs the following scalar function (V (e, e1, e2)) containing the introduced adaptation gain γ to calculate equation (21).(22)V=12e2+1B1γAm−A1−B1b+1B1γBm−B1a2=12e2+1B1γe12+1B1γe22.Take the time derivative of this scalar function,(23)V˙e,e1,e2=dVe,e1,e2dt=ededt+1B1γe1de1dt+1B1γe2de2dt.Because equation (23) is regarded as a first-order full state equation under the action of zero input, the stabilization point ex of the system state is(24)e=0e1=0e2=0.The scalar functionV based on equation (24) satisfies that it has a first-order continuous partial derivative with respect to all variables, and it satisfies as follows:(25)Ve,e1,e2>0e≠0,e1≠0,e2≠0,V0=0,V˙e,e1,e2=dVe,e1,e2dt<0.When the state trajectories of these three state variables all converge at the stabilization pointex, it means that e, e1, and e2 converge to zero. So,(26)V˙e,e1,e2=dVe,e1,e2dt<0⇒e=e1=e2=0.According to equations (20) and (23), the equation can be simplified as(27)dVe,e1,e2dt=ededt+1B1γe1dedt+1B1γe2dedt,=Ame2t+Am−A−B1btetxt+Bm−B1atetct−e1γdbtdt−e2γdatdt,=Ame2t+e1γγetxt−dbtdt+e2γγetct−datdt.Because ofAme2t<0, while(28)dbtdt=γetxt,datdt=γetct,V˙e,e1,e2<0, and e, e1, and e2 converge to zero.The result is(29)dVe,e1,e2dt=Ame2t.The adjustment rules obtained by the Lyapunov theory are simple and do not require to filter the signal.So, MRAC based on the second method of Lyapunov could use equation (28) calculating parameters a and b.The equation (13) can be expressed in the form of state space:(30)x˙mt=01−10.89−0.50xmt+01ct,ymt=−2.32×10−10−3.26×10−9xmt.The transfer function of the PSS-VI is(31)G1s=KPsTW1+sTW1+sT11+sT31+sT21+sT4+Kr+Kls.In this work, the author selectedKP = −0.5, Tw = 0.02, Kr = −1.2, Kl = −36, T1 = 0.3, T2 = 0.08, T3 = 0.06, and T4 = 0.3 to meet the systems requirements for dynamic performance and stability. So, G1(s) can be rewritten as follows.(32)G1s=−0.06s2+19.88s+0.054s2+49.16s+150.8.The equation (32) can be rewritten in the form of state space:(33)x˙t=01−150.8−49.16xt+01ut,yt=0.05419.88xt.According to equations (14)–(28), the paraments of the PID controller can be calculated.


## 3. Power Network Modeling

The power system modeling is the primary task in studying the power system. The dynamic behavior of a power system can be expressed as a sequence of nonlinear differential-algebraic equations (DAEs) involvingNs state variables, xi, and Nt input variables, yi. This behavior may be described by the following equations:(34)X˙=fX,U,Y=gX,U,whereX=x1,x2,…,xNsT,U=u1,u2,…,uNtT,andY=y1,y2,…,yNoT.In this section, we provide the model of a simple interconnected system with the DFIG.


### 3.1. Mathematical Model of the DFIG

The controller model in the WTG system is shown in Figure6. The mechanical controller consists of an aerodynamic model, mechanical drive model, and a pitch angle controller. The electrical controller consists of the DFIG and converter model.


#### 3.1.1. Aerodynamic Model

The aerodynamic turbines model captures the wind speed and converts the wind energy into mechanical energy. It follows the energy conservation principle. The electrical power available in the wind is as follows:(35)Pwt=0.5πρR2vω2cpλ,Tm=Pwtωm,where ρ is the air density (m3), vω is the wind speed, R is the radius of the turbine (m/sec), and cpλ is the coefficient of the power that is the function of the tip speed ratio and pitch angle.


#### 3.1.2. Mechanical Drive Model

The mechanical drive consists of the gearbox which synchronizes the wind turbine and generator. This model is given as(36)dωrdt=Tr−KsθrMr−1,dωwrdt=Ksθr−TeMwr−1,dθrdt=ωbωr−ωwt,Mwt=2Hwt,Mr=2Hr,where Hwt and Hr are the wind turbine and generator inertia constant, ωwt and ωr are the wind turbine and generator speed, respectively, θr is the shaft twist angle, ωb is the reference wind speed, Ks is the shaft stiffness, and Tr is the torque of the generator.


#### 3.1.3. Pitch Angle Controller

To limit the generator’ speed up to a maximum value, the pitch angle controller is employed and is expressed as(37)dβdx=βref−βTβ−1,where β is the blade pitch angle, βref is the pitch angle reference, and Tβ is the controller time constant for the pitch control.


#### 3.1.4. DFIG Model

The DFIG scheme is shown in Figure8 [20]. The stator terminal is directly connected to the local AC power grid, while rotor slip-ring terminal integrated into the same grid via a dual power converter and transformer. The two converters are named rotor side converter (RSC) and grid side converter (GSC), respectively. The wind generator model contains the wind speed model, wind turbine model, mechanical drive model, and DFIG model with its control system.Figure 8
Schematic diagram of the DFIG-based WTG with control scheme and integrated to the grid.According to the block diagram of RSC shown in Figure9, the equation can be written as(38)dx1dt=Ps_ref−Ps,iqr_ref=Kp1Ps_ref−Ps+Ki1x1,dx2dt=iqr_ref−iqr=Kp1Ps_ref−Ps+Ki1x1−iqr,dx3dt=vs_ref−vs,idr_ref=Kp3vs_ref−vs+Ki3x3,dx4dt=idr_ref−idr=Kp3vs_ref−vs+Ki3x3−idr,vqr=Kp2Kp1ΔP+Ki1x1−iqr+Ki2x2+srωsLmids+srωsLrriqr,vdr=Kp2Kp3ΔP+Ki3x1−idr+Ki2x4−srωsLmiqs−srωsLrriqr,where Kp1 and Ki1 are the proportional and integral coefficients of active power control, respectively, Kp2 and Ki2 are the proportional and integral coefficients of current control of RSC, respectively, Kp3 and Ki3 are the proportional and integral coefficients of voltage control, respectively, and x1, x2, x3, and x4 are the intermediate variable of the hypothesis.Figure 9
Block diagram of RSC with additional MRAC-PSS-VI.It is assumed that the stator and rotor winding are three-phase sinusoidal and symmetrical. The stator and rotor voltage in an arbitrarydq-axis rotating reference frame at a speed of ωs is expressed as(39)vdsvqsvdrvqr=−rsxs+xm0xmxs+xm−rsxm001−ωmxm−rr1−ωmxr+xm−1−ωmxm0−1−ωmxr+xm−rridsiqsidriqr,where ids, iqs, and idr, iqr are the stator and rotor currents in dq-axis; vds, vqs and vdr, vqr are stator and rotor voltages in dq-axis; xm is the magnetizing reactance; rs and xs are the stator resistance and reactance; rr and xr of the rotor resistance and reactance; and ωm is speed of the rotor.The stator output active and reactive power delivered to the grid is expressed as(40)PQT=Ps+PcQs+QcT.The stator side output power is(41)PsQs=vdsvqsvqs−vdsidsiqs.On the rotor side, power is given as(42)PrQr=vdrvqrvqr−vdridriqr.The grid side converter output power is(43)PcQc=vdcvqcvqc−vdcidciqc.By neglecting the power losses on GSC,(44)PcQcT=Pr0T.Hence, the power delivered to the grid is given as(45)PQT=Ps+PrQsT.The generator is modeled as a single shaft as(46)ωm=Tm−Te2Hm−1,where ωm is the speed of the rotor, Te and Tm are the electrical and mechanical torque, respectively, and Hm is inertia of the rotor.After MRAC-PSS-VI is installed in the RSC loop, the rotor side power is given as(47)Pr′Qr′=vdrvqr+upvqr+up−vdridriqr.Power delivered to the grid is given again as(48)P′QT=Ps+Pr′QsT.


### 3.2. Small-Signal Stability Analysis Model

The Lyapunov linearization method is usually used in the stability analysis of small signals. It is proved that the nonlinear system should have characteristics similar to its linearization in sufficiently small motion range [21]. On basis of this, if the real part of all the eigenvalues of A is negative, the system is stable. The dynamic process of the power system can be described by DAEs and can be linearized as(49)x˙=fx,y,0=gx,y,where x is the state variable which is used to describe the dynamic characteristics of the system and y is the input vector of the system. In the light of Lyapunov’s first law, equation (49) can be linearized as follows at stable operating point x0,y0.(50)Δx˙0=∇xf∇yf∇xg∇ygΔxΔy=ACΔxΔy,where ∇xf=∂fx,y/∂x is the gradient of the function fx,y with respect to x, and other symbols have similar meanings.Assuming that∇xf is nonsingular, it can be obtained by equation (50) that(51)Δx˙=∇xf−∇yf∇yg−1∇xgΔx=AΔx,where A is the state matrix of the system.The small-signal stability analysis is mainly to calculate the eigenvalue ofA. For the complex eigenvalue λ=σ+jΩ, the corresponding oscillation frequency is f=ω/2π, and the corresponding damping ratio is defined as(52)ξ=−σσ2+ω2.The participating factorpij describing the degree of correlation between state variables and modes of the ith state variable for the jth eigenvalue can be calculated by the corresponding left and right eigenvectors v and w, where pij=wijvji/wjTvj. For any eigenvalue λi, the n-dimensional column vectors wi satisfying Awi=λiwii=1,2,…,n is defined the right eigenvector of λi; the n-dimensional row vectors vi satisfying viA=viλii=1,2,…,n is defined the left eigenvector of λi.In the small-signal stability analysis of the power system, it is of great significance to study the distribution of eigenvalues on the complex axis. If the real parts of all eigenvalues are negative, the system is stable under small disturbance; if at least one of the real parts of all eigenvalues are positive, the system is unstable under small disturbance; if all eigenvalues have no real part that is positive, but at least one real part is zero, then the system is critically stable.


## 3.1. Mathematical Model of the DFIG

The controller model in the WTG system is shown in Figure6. The mechanical controller consists of an aerodynamic model, mechanical drive model, and a pitch angle controller. The electrical controller consists of the DFIG and converter model.


### 3.1.1. Aerodynamic Model

The aerodynamic turbines model captures the wind speed and converts the wind energy into mechanical energy. It follows the energy conservation principle. The electrical power available in the wind is as follows:(35)Pwt=0.5πρR2vω2cpλ,Tm=Pwtωm,where ρ is the air density (m3), vω is the wind speed, R is the radius of the turbine (m/sec), and cpλ is the coefficient of the power that is the function of the tip speed ratio and pitch angle.


### 3.1.2. Mechanical Drive Model

The mechanical drive consists of the gearbox which synchronizes the wind turbine and generator. This model is given as(36)dωrdt=Tr−KsθrMr−1,dωwrdt=Ksθr−TeMwr−1,dθrdt=ωbωr−ωwt,Mwt=2Hwt,Mr=2Hr,where Hwt and Hr are the wind turbine and generator inertia constant, ωwt and ωr are the wind turbine and generator speed, respectively, θr is the shaft twist angle, ωb is the reference wind speed, Ks is the shaft stiffness, and Tr is the torque of the generator.


### 3.1.3. Pitch Angle Controller

To limit the generator’ speed up to a maximum value, the pitch angle controller is employed and is expressed as(37)dβdx=βref−βTβ−1,where β is the blade pitch angle, βref is the pitch angle reference, and Tβ is the controller time constant for the pitch control.


### 3.1.4. DFIG Model

The DFIG scheme is shown in Figure8 [20]. The stator terminal is directly connected to the local AC power grid, while rotor slip-ring terminal integrated into the same grid via a dual power converter and transformer. The two converters are named rotor side converter (RSC) and grid side converter (GSC), respectively. The wind generator model contains the wind speed model, wind turbine model, mechanical drive model, and DFIG model with its control system.Figure 8
Schematic diagram of the DFIG-based WTG with control scheme and integrated to the grid.According to the block diagram of RSC shown in Figure9, the equation can be written as(38)dx1dt=Ps_ref−Ps,iqr_ref=Kp1Ps_ref−Ps+Ki1x1,dx2dt=iqr_ref−iqr=Kp1Ps_ref−Ps+Ki1x1−iqr,dx3dt=vs_ref−vs,idr_ref=Kp3vs_ref−vs+Ki3x3,dx4dt=idr_ref−idr=Kp3vs_ref−vs+Ki3x3−idr,vqr=Kp2Kp1ΔP+Ki1x1−iqr+Ki2x2+srωsLmids+srωsLrriqr,vdr=Kp2Kp3ΔP+Ki3x1−idr+Ki2x4−srωsLmiqs−srωsLrriqr,where Kp1 and Ki1 are the proportional and integral coefficients of active power control, respectively, Kp2 and Ki2 are the proportional and integral coefficients of current control of RSC, respectively, Kp3 and Ki3 are the proportional and integral coefficients of voltage control, respectively, and x1, x2, x3, and x4 are the intermediate variable of the hypothesis.Figure 9
Block diagram of RSC with additional MRAC-PSS-VI.It is assumed that the stator and rotor winding are three-phase sinusoidal and symmetrical. The stator and rotor voltage in an arbitrarydq-axis rotating reference frame at a speed of ωs is expressed as(39)vdsvqsvdrvqr=−rsxs+xm0xmxs+xm−rsxm001−ωmxm−rr1−ωmxr+xm−1−ωmxm0−1−ωmxr+xm−rridsiqsidriqr,where ids, iqs, and idr, iqr are the stator and rotor currents in dq-axis; vds, vqs and vdr, vqr are stator and rotor voltages in dq-axis; xm is the magnetizing reactance; rs and xs are the stator resistance and reactance; rr and xr of the rotor resistance and reactance; and ωm is speed of the rotor.The stator output active and reactive power delivered to the grid is expressed as(40)PQT=Ps+PcQs+QcT.The stator side output power is(41)PsQs=vdsvqsvqs−vdsidsiqs.On the rotor side, power is given as(42)PrQr=vdrvqrvqr−vdridriqr.The grid side converter output power is(43)PcQc=vdcvqcvqc−vdcidciqc.By neglecting the power losses on GSC,(44)PcQcT=Pr0T.Hence, the power delivered to the grid is given as(45)PQT=Ps+PrQsT.The generator is modeled as a single shaft as(46)ωm=Tm−Te2Hm−1,where ωm is the speed of the rotor, Te and Tm are the electrical and mechanical torque, respectively, and Hm is inertia of the rotor.After MRAC-PSS-VI is installed in the RSC loop, the rotor side power is given as(47)Pr′Qr′=vdrvqr+upvqr+up−vdridriqr.Power delivered to the grid is given again as(48)P′QT=Ps+Pr′QsT.


## 3.1.1. Aerodynamic Model

The aerodynamic turbines model captures the wind speed and converts the wind energy into mechanical energy. It follows the energy conservation principle. The electrical power available in the wind is as follows:(35)Pwt=0.5πρR2vω2cpλ,Tm=Pwtωm,where ρ is the air density (m3), vω is the wind speed, R is the radius of the turbine (m/sec), and cpλ is the coefficient of the power that is the function of the tip speed ratio and pitch angle.


## 3.1.2. Mechanical Drive Model

The mechanical drive consists of the gearbox which synchronizes the wind turbine and generator. This model is given as(36)dωrdt=Tr−KsθrMr−1,dωwrdt=Ksθr−TeMwr−1,dθrdt=ωbωr−ωwt,Mwt=2Hwt,Mr=2Hr,where Hwt and Hr are the wind turbine and generator inertia constant, ωwt and ωr are the wind turbine and generator speed, respectively, θr is the shaft twist angle, ωb is the reference wind speed, Ks is the shaft stiffness, and Tr is the torque of the generator.


## 3.1.3. Pitch Angle Controller

To limit the generator’ speed up to a maximum value, the pitch angle controller is employed and is expressed as(37)dβdx=βref−βTβ−1,where β is the blade pitch angle, βref is the pitch angle reference, and Tβ is the controller time constant for the pitch control.


## 3.1.4. DFIG Model

The DFIG scheme is shown in Figure8 [20]. The stator terminal is directly connected to the local AC power grid, while rotor slip-ring terminal integrated into the same grid via a dual power converter and transformer. The two converters are named rotor side converter (RSC) and grid side converter (GSC), respectively. The wind generator model contains the wind speed model, wind turbine model, mechanical drive model, and DFIG model with its control system.Figure 8
Schematic diagram of the DFIG-based WTG with control scheme and integrated to the grid.According to the block diagram of RSC shown in Figure9, the equation can be written as(38)dx1dt=Ps_ref−Ps,iqr_ref=Kp1Ps_ref−Ps+Ki1x1,dx2dt=iqr_ref−iqr=Kp1Ps_ref−Ps+Ki1x1−iqr,dx3dt=vs_ref−vs,idr_ref=Kp3vs_ref−vs+Ki3x3,dx4dt=idr_ref−idr=Kp3vs_ref−vs+Ki3x3−idr,vqr=Kp2Kp1ΔP+Ki1x1−iqr+Ki2x2+srωsLmids+srωsLrriqr,vdr=Kp2Kp3ΔP+Ki3x1−idr+Ki2x4−srωsLmiqs−srωsLrriqr,where Kp1 and Ki1 are the proportional and integral coefficients of active power control, respectively, Kp2 and Ki2 are the proportional and integral coefficients of current control of RSC, respectively, Kp3 and Ki3 are the proportional and integral coefficients of voltage control, respectively, and x1, x2, x3, and x4 are the intermediate variable of the hypothesis.Figure 9
Block diagram of RSC with additional MRAC-PSS-VI.It is assumed that the stator and rotor winding are three-phase sinusoidal and symmetrical. The stator and rotor voltage in an arbitrarydq-axis rotating reference frame at a speed of ωs is expressed as(39)vdsvqsvdrvqr=−rsxs+xm0xmxs+xm−rsxm001−ωmxm−rr1−ωmxr+xm−1−ωmxm0−1−ωmxr+xm−rridsiqsidriqr,where ids, iqs, and idr, iqr are the stator and rotor currents in dq-axis; vds, vqs and vdr, vqr are stator and rotor voltages in dq-axis; xm is the magnetizing reactance; rs and xs are the stator resistance and reactance; rr and xr of the rotor resistance and reactance; and ωm is speed of the rotor.The stator output active and reactive power delivered to the grid is expressed as(40)PQT=Ps+PcQs+QcT.The stator side output power is(41)PsQs=vdsvqsvqs−vdsidsiqs.On the rotor side, power is given as(42)PrQr=vdrvqrvqr−vdridriqr.The grid side converter output power is(43)PcQc=vdcvqcvqc−vdcidciqc.By neglecting the power losses on GSC,(44)PcQcT=Pr0T.Hence, the power delivered to the grid is given as(45)PQT=Ps+PrQsT.The generator is modeled as a single shaft as(46)ωm=Tm−Te2Hm−1,where ωm is the speed of the rotor, Te and Tm are the electrical and mechanical torque, respectively, and Hm is inertia of the rotor.After MRAC-PSS-VI is installed in the RSC loop, the rotor side power is given as(47)Pr′Qr′=vdrvqr+upvqr+up−vdridriqr.Power delivered to the grid is given again as(48)P′QT=Ps+Pr′QsT.


## 3.2. Small-Signal Stability Analysis Model

The Lyapunov linearization method is usually used in the stability analysis of small signals. It is proved that the nonlinear system should have characteristics similar to its linearization in sufficiently small motion range [21]. On basis of this, if the real part of all the eigenvalues of A is negative, the system is stable. The dynamic process of the power system can be described by DAEs and can be linearized as(49)x˙=fx,y,0=gx,y,where x is the state variable which is used to describe the dynamic characteristics of the system and y is the input vector of the system. In the light of Lyapunov’s first law, equation (49) can be linearized as follows at stable operating point x0,y0.(50)Δx˙0=∇xf∇yf∇xg∇ygΔxΔy=ACΔxΔy,where ∇xf=∂fx,y/∂x is the gradient of the function fx,y with respect to x, and other symbols have similar meanings.Assuming that∇xf is nonsingular, it can be obtained by equation (50) that(51)Δx˙=∇xf−∇yf∇yg−1∇xgΔx=AΔx,where A is the state matrix of the system.The small-signal stability analysis is mainly to calculate the eigenvalue ofA. For the complex eigenvalue λ=σ+jΩ, the corresponding oscillation frequency is f=ω/2π, and the corresponding damping ratio is defined as(52)ξ=−σσ2+ω2.The participating factorpij describing the degree of correlation between state variables and modes of the ith state variable for the jth eigenvalue can be calculated by the corresponding left and right eigenvectors v and w, where pij=wijvji/wjTvj. For any eigenvalue λi, the n-dimensional column vectors wi satisfying Awi=λiwii=1,2,…,n is defined the right eigenvector of λi; the n-dimensional row vectors vi satisfying viA=viλii=1,2,…,n is defined the left eigenvector of λi.In the small-signal stability analysis of the power system, it is of great significance to study the distribution of eigenvalues on the complex axis. If the real parts of all eigenvalues are negative, the system is stable under small disturbance; if at least one of the real parts of all eigenvalues are positive, the system is unstable under small disturbance; if all eigenvalues have no real part that is positive, but at least one real part is zero, then the system is critically stable.


## 4. Simulation Results and Analysis

### 4.1. Test Case Power System Description

In this study, the author adapts the two-area system as the test system. The proverbial two-area system [22] is used for stability analysis to verify the effectiveness of MRAC-PSS-VI on improving power system small-signal stability and transient performance, as shown in Figure 10. In the system, G1–G4 present a group of synchronous generators that are strongly coupled, and both local and interarea oscillation modes are researched. G1 and G2 are equipped with quick excitation, while G3 and G4 are equipped with slow excitation. All the synchronous are equipped with an IEEE (Institute of Electrical and Electronic Engineer) type 1 regulator. The base capacity of the slow excitation system is 100 MVA, the frequency is 50 Hz, and the transmission power of tie line is 400 MW. Regions 1 and 2 of the system are connected by double loop tie lines. G1–G4 are four thermal power generator units whose rated capacity is 900 MVA and rated voltage is 20 kV. The node 3 of the system is the reference node, and the active power output of each generator is 700 MW. The wind farm node is on the area 2 bus 10 (Figure 10). The DFIG was connected to the system via bus 10, and the wind power output is 5 MW.Figure 10
Two-area system.


### 4.2. Influence of DFIG-MRAC-PSS-VI on Damping Characteristics of the System

It is an appropriate method to evaluate the dynamic stability of the power system by analyzing the damping characteristics of the system, such as damping ratio. In the small-signal stability analysis of the power system, it is significant to study the distribution of eigenvalues on the complex axis. If the real parts of all eigenvalues are negative, the system is stable under small disturbance; if at least one of the real parts of all eigenvalues is positive, the system is unstable under small disturbance; if all eigenvalues have no real part that is positive, but at least one real part is zero, then the system is critically stable.When the DFIG is connected, the DFIG output is 5 MW, and the original system generator output is not changed. Some eigenvalues of the system are given in Table2. As given in Table 2, the system has three original oscillation modes: mode 1 and mode 2 represent the local oscillation of the generator sets in region 1 and region 2, respectively, and mode 3 represents the regional oscillation of the generator sets between region 1 and region 2. When the DFIG is connected, an interregional oscillation mode 4 related to the DFIG is added to the system.Table 2
The eigenvalues of the system.
ModeWithout the DFIGWith the DFIGDominant statesλf/Hzξ (%)λf/Hzξ (%)1−0.651 ±j6.5471.0429.898−0.651 ±j6.5471.0429.897G1, G22−0.619 ±j6.3921.0179.645−0.620 ±j6.3931.0179.658G3, G43−0.299 ±j3.1350.4999.526−0.300 ±j3.1350.4999.532G1–G44———−0.513 ±j3.0590.48716.539G1–G4, DFIGAssuming that the load on bus 9 fluctuates upward by 5% during the period of 1.0–1.5 s, the active power emitted by generators G1, G2, and G4 is 700 MW, the DFIG output is 5 MW, and the transmission power of the tie line is 400 MW. Figure11 shows the voltage curve (unit value) of bus 6 and the angular response curve of generator G1 relative balance node. It can be seen that the response curve without the controller oscillates with a larger amplitude and tends to be stable slowly when the system oscillates at low frequency. When the PSS is installed in the DFIG, the improvement of the response curve is minimal. After the PSS-VI is installed in the DFIG, the power angle curve of generator G1 and the voltage curve of bus 6 decreased, but the effect was not particularly ideal, as shown in Figure 8. However, the response curve with MRAC-PSS-VI has a small amplitude of oscillation and tends to be stable in a short time.Figure 11
Effects of the MRAC-PSS-VI controller on system damping. (a) Power angle of G1. (b) Voltage of bus 6.
(a)(b)In order to testify the effectiveness of the designed MRAC-PSS-VI controller, this study analyzes the system eigenvalues. It is shown that the power of the system tie line is 400 MW as part of the eigenvalues of the system (Table3).Table 3
The eigenvalues of the system with different controllers.
ModeWithout the controllerWith the PSSDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1−0.651 ±j6.5471.0429.897−0.651 ±j6.5471.0429.897G1, G22−0.620 ±j6.3931.0179.658−0.615 ±j6.3901.0179.580G3, G43−0.300 ±j3.1350.4999.532−0.368 ±j3.1800.50611.521G1–G44−0.513 ±j3.0590.48716.539−0.396 ±j3.3590.53511.710G1–G4, DFIGModeWith the PSS-VIWith MRAC-PSS-VIDominant statesΛf/Hzξ (%)λf/Hzξ (%)1−0.651 ±j6.5471.0429.897−0.670 ±j6.3591.01210.479G1, G22−0.620 ±j6.3931.0179.658−1.778 ±j6.1060.97227.954G3, G43−0.298 ±j3.1370.4999.464−0.304 ±j3.0330.4839.968G1–G44−0.689 ±j3.4820.55419.417−0.967 ±j3.5460.56426.305G1–G4, DFIGAs shown in Figure12, for intraregional oscillation mode 2 and intraregional oscillation mode 4, the eigenvalues of the DFIG with the PSS are slightly shifted to the right, which prove that system damping is reduced. After the addition of the PSS-VI in the DFIG, the eigenvalues for intraregional oscillation mode 2 almost remain unchanged compared with the previous ones, while for interregional oscillation mode 4, the eigenvalues exist a little left shift trend, and the system stability is improved. When the MRAC-PSS-VI controller is installed in the DFIG, the eigenvalues of intraregional oscillation mode 2 and interregional oscillation mode 4 show a large tendency to shift to the left, and the system robustness is significantly improved.Figure 12
The eigenvalues of the system installed different controllers. (a) Mode 2. (b) Mode 4.
(a)(b)In order to prove the effectiveness of the controller when the system is subjected to different types of perturbations, in this section, it is assumed that a three-phase short circuit fault occurs in the transmission line between bus 6 and bus 7 of the system, and the fault occurs at 1 s, with the duration of 0.01 s and a simulation time of 20 s. Figure13 shows the power angles of generators G1 and G4, as well as the frequency and voltage curves of bus 9 when the system has a three-phase short circuit fault.Figure 13
Three-phase short circuit response curve. (a) Power angle of G1. (b) Power angle of G4. (c) Frequency of bus 9. (d) Voltage of bus 9.
(a)(b)(c)(d)As shown in Figure13, when faults occur, the power angles of generators G1 and G4 are significantly improved by the addition of MRAC-PSS-VI to the DFIG, while the effects of the addition of the PSS-VI and PSS are not ideal. As shown in Figure 13(c), when low-frequency oscillation occurs in the system, the frequency response curve of bus 9 equipped with the PSS controller oscillates with a larger amplitude and tends to be stable slowly. However, the oscillation amplitude of the response curve is obviously improved and tends to be stable in a short time after the addition of the MRAC-PSS-VI controller. As shown in Figure 13(d), the voltage fluctuation of bus 9 is significantly suppressed after the installation of MRAC-PSS-VI, and the system stability is improved, which is consistent with the results obtained from the above eigenvalue analysis.


### 4.3. System Simulation of Tie Line Transmission Power Variation

Generator output or load power changes will make the transmission power of the tie line change. Area 1 sends power to area 2. By changing the output of the generators in area 1, the transmission power of tie line is adjusted. In this work, eigenvalues are analyzed for the system in three cases. These cases are as follows:Case a: the power transmitted on the tie line from region 1 to region 2 is 300 MWCase b: the power transmitted on the tie line from region 1 to region 2 is 450 MWCase c: the power transmitted on the tie line from region 1 to region 2 is 600 MWAs given in Table4, compared with the case of the PSS, the damping characteristics of the system are slightly improved with the addition of the PSS-VI. With the installation of the PSS, the damping ratio of the interregional oscillation mode 4 decreases while the interregional oscillation mode 3 increases slightly. After the addition of the PSS-VI to the system, the damping ratio of the interregional oscillation mode 4 is improved, but the damping ratio of the interregional oscillation mode 3 decreases slightly compared to addition of the PSS to the system. After MRAC-PSS-VI is installed in the DFIG, the damping ratio of interregion oscillation mode 4 and intraregion oscillation mode 2 of the system is greatly improved. For example, in cases a, b, and c, the damping ratio of intraregion oscillation mode 2 increases from 9.154%, 9.921%, and 11.021% to 19.204%, 16.713%, and 17.846%, respectively. The damping ratio of interregion oscillation mode 4 increases from 16.518%, 16.524%, and 16.387% to 28.783%, 29.563%, and 27.713%, respectively, in different cases. Adoption of DFIG-MRAC-PSS-VI further improved the damping characteristics of the system, and the damping ratio of the regional mode was greatly improved, especially the regional mode 4, whose damping ratio was improved by more than 70%. The small disturbance stability of the system is improved, and the system robustness is improved.Table 4
The eigenvalues of the system in different tie line power conditions.
ModeCaseWithout the controllerPSSDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1a−0.671 ±j6.5501.04210.195−0.671 ±j6.5501.04310.196G1, G2b−0.643 ±j6.5421.0419.790−0.644 ±j6.5421.0419.789c−0.618 ±j6.5021.0359.465−0.618 ±j6.5021.0349.4662a−0.589 ±j6.4041.0199.154−0.583 ±j6.4011.0199.079G3, G4b−0.637 ±j6.3861.0169.921−0.631 ±j6.3831.0169.839c−0.704  ±j6.3501.01111.021−0.697 ±j6.3481.01010.9203a−0.295 ±j3.2890.5248.919−0.259 ±j3.2930.5247.827G1−G4b−0.305 ±j3.0310.48210.039−0.386 ±j3.0550.48612.539c−0.343 ±j2.3140.36814.672−0.432 ±j2.3540.37518.0384a−0.513 ±j3.0630.48816.518−0.513 ±j3.4070.54214.877G1−G4, DFIGb−0.512 ±j3.0560.48616.524−0.379 ±j3.3790.53811.159c−0.503 ±j3.0280.48216.387−0.389 ±j3.3900.54011.412ModeCasePSS-VIMRAC-PSS-VIDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1a−0.671 ±j6.5501.04310.195−0.628 ±j6.3551.0119.827G1, G2b−0.644 ±j6.5421.0419.789−0.679 ±j6.3351.00810.669c−0.618 ±j6.5021.0359.464−0.746 ±j6.3121.00411.7302a−0.589 ±j6.4041.0199.154−1.059 ±j5.4140.86219.204G3, G4b−0.637 ±j6.3861.0169.920−0.959 ±j5.6600.90116.713c−0.704 ±j6.3501.01111.021−1.153 ±j6.3561.01117.8463a−0.293 ±j3.2900.5248.879−0.297 ±j3.1910.5089.278G1−G4b−0.304 ±j3.0340.4829.970−0.305 ±j2.9280.46610.345c−0.342 ±j2.3210.36914.589−0.332 ±j2.1940.34914.9784a−0.687 ±j3.4890.55519.319−1.050 ±j3.4950.55628.783G1−G4, DFIGb−0.689 ±j3.4760.55319.467−1.081 ±j3.4960.55629.563c−0.689 ±j3.4310.54619.694−0.946 ±j3.2810.52227.713In this work, the author gives the changes of eigenvalues corresponding to oscillation mode 2 and oscillation mode 4 in case a, as shown in Figure14.Figure 14
The eigenvalues of the system installed different controllers in case a. (a) Mode 2. (b) Mode 4.
(a)(b)As shown in Figure14, after the addition of the PSS in the DFIG, the eigenvalues for intraregional oscillation mode 2 almost remains unchanged compared with the DFIG with the PSS-VI. While for interregional oscillation mode 4 when the PSS-VI is installed in the DFIG, the eigenvalues exist a little left shift trend, and the system stability is improved. While the PSS is installed in the DFIG, the eigenvalues of the system change very little. When the MRAC-PSS-VI controller is installed in the DFIG, the eigenvalues of intraregional oscillation mode 2 and interregional oscillation mode 4 show a large tendency shifting to the left, and the system robustness is significantly improved.Under the same fault conditions as described in Section3.1, the system response curves under different tie line powers are shown in Figure 15. It can be seen that with the increase of the power of the tie line, the relative power angle of G1 and G3 gradually increases and the voltage at bus 6 gradually decreases. After the failure, the oscillation amplitude of the curves increased with the increase of the power of the tie line, and the time for the system to stabilize also increased slightly. Under different tie line power conditions, the damping improvement of the DFIG installed the PSS is not obvious. The addition of virtual impedance link to the DFIG-PSS has a certain effect on the stability of the system with small disturbance. Under different tie line power conditions, the power angle curve of G1 and the voltage response curve of bus 6 are improved, but the improvement effect is not ideal in the first cycle. After the MRAC control method is introduced into the DFIG-PSS-VI control, as shown in Figure 12, the power angle curve of G1 and voltage curve of bus 6 are significantly reduced in the case of small interference. The improvement effect of the DFIG with MRAC-PSS-VI is ideal under different tie line power conditions. It can be concluded that after the introduction of the DFIG-MRAC-PSS-VI, the damping of the system is improved and the robustness of the system is significantly improved.Figure 15
Response curves with different cases under a small disturbance. (a) Power angle of G1 in case a. (b) Voltage of bus 6 in case a. (c) Power angle of G1 in case b. (d) Voltage of bus 6 in case b. (e) Power angle of G1 in case c. (f) Voltage of bus 6 in case c.
(a)(b)(c)(d)(e)(f)


### 4.4. System Simulation of DFIG Access Point Changes

The change of the DFIG access position has an impact on the stability of the system. In this section, the controller effect is studied when the DFIG access position is changed. The effectiveness of the designed controller is verified by the following calculation cases:Case d: DFIG accesses bus 6 in area 1Case e: DFIG accesses transmission lines 7-8 in area 1Case f: DFIG accesses bus 9 in area 2Some oscillation modes of the system after changing the DFIG access position is given in Table5.Table 5
The eigenvalues of the system in different DFIG access positions.
ModeCaseWithout the controllerPSSDominant statesλf (Hz)ξ (%)Λf (Hz)ξ (%)1d−0.652 ±j6.5461.0429.908−0.649 ±j6.5461.0429.879G1, G2e−0.651 ±j6.5471.0429.901−0.651 ±j6.5461.0429.899f−0.651 ±j6.5471.0429.896−0.651 ±j6.5471.0429.8962d−0.620 ±j6.3931.0179.662−0.620 ±j6.3931.0179.656G3, G4e−0.621 ±j6.3931.0179.661−0.619 ±j6.3931.0179.637f−0.620 ±j6.3931.0179.654−0.616 ±j6.3921.0179.5973d−0.303 ±j3.1280.4989.639−0.327 ±j2.9940.47610.864G1−G4e−0.302 ±j3.1330.4999.591−0.257 ±j3.0790.4908.306f−0.299 ±j3.1370.4999.500−0.290 ±j3.1720.5059.1214d−0.807 ±j3.0510.48625.571−0.919 ±j3.0630.48728.742G1−G4, DFIGe−0.791 ±j3.0100.47925.419−0.894 ±j3.1470.50127.339f−0.807 ±j3.0420.48425.648−0.892 ±j2.9360.46729.075ModeCasePSS-VIMRAC-PSS-VIDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1d−0.651 ±j6.5451.0429.892−0.760 ±j6.2670.99712.030G1, G2e−0.651 ±j6.5471.0429.902−0.668 ±j6.3621.01210.441f−0.651 ±j6.5471.0429.896−0.675 ±j6.3511.01110.5632d−0.620 ±j6.3931.0179.658−1.057 ±j6.6321.05515.745G3, G4e−0.619 ±j6.3921.0179.645−1.426 ±j5.5500.88324.891f−0.615 ±j6.3901.0179.589−2.198 ±j7.2051.14629.1873d−0.366 ±j3.1870.50711.405−0.307 ±j3.0480.48510.011G1−G4e−0.360 ±j3.1400.50011.385−0.303 ±j3.0340.4839.926f−0.371 ±j3.0590.48712.049−0.300 ±j3.0360.4839.8294d−1.120 ±j2.9270.46635.749−1.421 ±j3.7200.59235.692G1−G4, DFIGe−1.033 ±j2.8440.45334.144−1.187 ±j3.1740.50535.035f−1.375 ±j2.8140.44843.911−1.225 ±j3.5650.56732.486As given in Table5, the change of the DFIG access point position causes the change of system eigenvalues. For the intraregion oscillation mode 2, the change of system eigenvalue is not obvious after the installation of the PSS and PSS-VI in the DFIG. When the DFIG is in different access positions, the DFIG installed the PSS and PSS-VI has both positive and negative effects on the damping ratio of the system, while the eigenvalue of the MRAC-PSS-VI system shifted significantly to the left, and the damping ratio of the system was greatly improved. For the interregion oscillation mode 4, the damping ratio improvement effect of adding the PSS to the DFIG is limited, while the effect of adding the PSS-VI and MRAC-PSS-VI is more obvious. Although the damping ratio improvement effect of MRAC-PSS-VI is not as good as that of the PSS-VI, the left shift trend of the system characteristic value is obviously better than that of the PSS-VI. In order to further illustrate the effect of the controller, the system is simulated in time domain, and the results are shown in Figure 16.Figure 16
The eigenvalues of system installed different controllers in case d. (a) Mode 2 in case d. (b) Mode 4 in case d.
(a)(b)In this work, the author gives the changes of eigenvalues corresponding to oscillation mode 2 and oscillation mode 4 in case d, as shown in Figure16.As shown in Figure16, after the addition of the PSS and PSS-VI in the DFIG, the eigenvalues for intraregional oscillation mode 2 almost remains unchanged compared with the DFIG without any controller. However, for interregional oscillation mode 4 when the PSS is installed in the DFIG, the eigenvalues exist a little left shift trend, and the system stability is improved. While the PSS-VI is installed in the DFIG, the left shift trend of eigenvalues is bigger than that of the DFIG-PSS. When the MRAC-PSS-VI controller is installed in the DFIG, the eigenvalues of intraregional oscillation mode 2 and interregional oscillation mode 4 show a large tendency shifting to the left, and the system robustness is significantly improved.In order to verify the effectiveness of the control at different access positions of the DFIG, time domain simulation was carried out. Figure17 shows the response curves of power angle of G1 and voltage of bus 9.Figure 17
Response curves with different cases. (a) Power angle of G1 in case d. (b) Voltage of bus 9 in case d. (c) Power angle of G1 in case e. (d) Voltage of bus 9 in case e. (e) Power angle of G1 in case f. (f) Voltage of bus 9 in case f.
(a)(b)(c)(d)(e)(f)As shown in Figure17, the installation of the MRAC-PSS-VI at different access positions of the DFIG has a good effect on system stability. As shown in Figure 17, when the DFIG is equipped with the PSS, the power angle curve oscillation of G1 has no obvious inhibition effect, and the inhibition effect of voltage fluctuation of bus 9 is not obvious either. When the PSS-VI is added, the oscillation is suppressed to some extent, but the effect is not ideal. While the MRAC-PSS-VI is installed in the DFIG, the oscillation of the system is obviously suppressed, and the rotor angle stability and voltage stability of the system are improved.


## 4.1. Test Case Power System Description

In this study, the author adapts the two-area system as the test system. The proverbial two-area system [22] is used for stability analysis to verify the effectiveness of MRAC-PSS-VI on improving power system small-signal stability and transient performance, as shown in Figure 10. In the system, G1–G4 present a group of synchronous generators that are strongly coupled, and both local and interarea oscillation modes are researched. G1 and G2 are equipped with quick excitation, while G3 and G4 are equipped with slow excitation. All the synchronous are equipped with an IEEE (Institute of Electrical and Electronic Engineer) type 1 regulator. The base capacity of the slow excitation system is 100 MVA, the frequency is 50 Hz, and the transmission power of tie line is 400 MW. Regions 1 and 2 of the system are connected by double loop tie lines. G1–G4 are four thermal power generator units whose rated capacity is 900 MVA and rated voltage is 20 kV. The node 3 of the system is the reference node, and the active power output of each generator is 700 MW. The wind farm node is on the area 2 bus 10 (Figure 10). The DFIG was connected to the system via bus 10, and the wind power output is 5 MW.Figure 10
Two-area system.


## 4.2. Influence of DFIG-MRAC-PSS-VI on Damping Characteristics of the System

It is an appropriate method to evaluate the dynamic stability of the power system by analyzing the damping characteristics of the system, such as damping ratio. In the small-signal stability analysis of the power system, it is significant to study the distribution of eigenvalues on the complex axis. If the real parts of all eigenvalues are negative, the system is stable under small disturbance; if at least one of the real parts of all eigenvalues is positive, the system is unstable under small disturbance; if all eigenvalues have no real part that is positive, but at least one real part is zero, then the system is critically stable.When the DFIG is connected, the DFIG output is 5 MW, and the original system generator output is not changed. Some eigenvalues of the system are given in Table2. As given in Table 2, the system has three original oscillation modes: mode 1 and mode 2 represent the local oscillation of the generator sets in region 1 and region 2, respectively, and mode 3 represents the regional oscillation of the generator sets between region 1 and region 2. When the DFIG is connected, an interregional oscillation mode 4 related to the DFIG is added to the system.Table 2
The eigenvalues of the system.
ModeWithout the DFIGWith the DFIGDominant statesλf/Hzξ (%)λf/Hzξ (%)1−0.651 ±j6.5471.0429.898−0.651 ±j6.5471.0429.897G1, G22−0.619 ±j6.3921.0179.645−0.620 ±j6.3931.0179.658G3, G43−0.299 ±j3.1350.4999.526−0.300 ±j3.1350.4999.532G1–G44———−0.513 ±j3.0590.48716.539G1–G4, DFIGAssuming that the load on bus 9 fluctuates upward by 5% during the period of 1.0–1.5 s, the active power emitted by generators G1, G2, and G4 is 700 MW, the DFIG output is 5 MW, and the transmission power of the tie line is 400 MW. Figure11 shows the voltage curve (unit value) of bus 6 and the angular response curve of generator G1 relative balance node. It can be seen that the response curve without the controller oscillates with a larger amplitude and tends to be stable slowly when the system oscillates at low frequency. When the PSS is installed in the DFIG, the improvement of the response curve is minimal. After the PSS-VI is installed in the DFIG, the power angle curve of generator G1 and the voltage curve of bus 6 decreased, but the effect was not particularly ideal, as shown in Figure 8. However, the response curve with MRAC-PSS-VI has a small amplitude of oscillation and tends to be stable in a short time.Figure 11
Effects of the MRAC-PSS-VI controller on system damping. (a) Power angle of G1. (b) Voltage of bus 6.
(a)(b)In order to testify the effectiveness of the designed MRAC-PSS-VI controller, this study analyzes the system eigenvalues. It is shown that the power of the system tie line is 400 MW as part of the eigenvalues of the system (Table3).Table 3
The eigenvalues of the system with different controllers.
ModeWithout the controllerWith the PSSDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1−0.651 ±j6.5471.0429.897−0.651 ±j6.5471.0429.897G1, G22−0.620 ±j6.3931.0179.658−0.615 ±j6.3901.0179.580G3, G43−0.300 ±j3.1350.4999.532−0.368 ±j3.1800.50611.521G1–G44−0.513 ±j3.0590.48716.539−0.396 ±j3.3590.53511.710G1–G4, DFIGModeWith the PSS-VIWith MRAC-PSS-VIDominant statesΛf/Hzξ (%)λf/Hzξ (%)1−0.651 ±j6.5471.0429.897−0.670 ±j6.3591.01210.479G1, G22−0.620 ±j6.3931.0179.658−1.778 ±j6.1060.97227.954G3, G43−0.298 ±j3.1370.4999.464−0.304 ±j3.0330.4839.968G1–G44−0.689 ±j3.4820.55419.417−0.967 ±j3.5460.56426.305G1–G4, DFIGAs shown in Figure12, for intraregional oscillation mode 2 and intraregional oscillation mode 4, the eigenvalues of the DFIG with the PSS are slightly shifted to the right, which prove that system damping is reduced. After the addition of the PSS-VI in the DFIG, the eigenvalues for intraregional oscillation mode 2 almost remain unchanged compared with the previous ones, while for interregional oscillation mode 4, the eigenvalues exist a little left shift trend, and the system stability is improved. When the MRAC-PSS-VI controller is installed in the DFIG, the eigenvalues of intraregional oscillation mode 2 and interregional oscillation mode 4 show a large tendency to shift to the left, and the system robustness is significantly improved.Figure 12
The eigenvalues of the system installed different controllers. (a) Mode 2. (b) Mode 4.
(a)(b)In order to prove the effectiveness of the controller when the system is subjected to different types of perturbations, in this section, it is assumed that a three-phase short circuit fault occurs in the transmission line between bus 6 and bus 7 of the system, and the fault occurs at 1 s, with the duration of 0.01 s and a simulation time of 20 s. Figure13 shows the power angles of generators G1 and G4, as well as the frequency and voltage curves of bus 9 when the system has a three-phase short circuit fault.Figure 13
Three-phase short circuit response curve. (a) Power angle of G1. (b) Power angle of G4. (c) Frequency of bus 9. (d) Voltage of bus 9.
(a)(b)(c)(d)As shown in Figure13, when faults occur, the power angles of generators G1 and G4 are significantly improved by the addition of MRAC-PSS-VI to the DFIG, while the effects of the addition of the PSS-VI and PSS are not ideal. As shown in Figure 13(c), when low-frequency oscillation occurs in the system, the frequency response curve of bus 9 equipped with the PSS controller oscillates with a larger amplitude and tends to be stable slowly. However, the oscillation amplitude of the response curve is obviously improved and tends to be stable in a short time after the addition of the MRAC-PSS-VI controller. As shown in Figure 13(d), the voltage fluctuation of bus 9 is significantly suppressed after the installation of MRAC-PSS-VI, and the system stability is improved, which is consistent with the results obtained from the above eigenvalue analysis.


## 4.3. System Simulation of Tie Line Transmission Power Variation

Generator output or load power changes will make the transmission power of the tie line change. Area 1 sends power to area 2. By changing the output of the generators in area 1, the transmission power of tie line is adjusted. In this work, eigenvalues are analyzed for the system in three cases. These cases are as follows:Case a: the power transmitted on the tie line from region 1 to region 2 is 300 MWCase b: the power transmitted on the tie line from region 1 to region 2 is 450 MWCase c: the power transmitted on the tie line from region 1 to region 2 is 600 MWAs given in Table4, compared with the case of the PSS, the damping characteristics of the system are slightly improved with the addition of the PSS-VI. With the installation of the PSS, the damping ratio of the interregional oscillation mode 4 decreases while the interregional oscillation mode 3 increases slightly. After the addition of the PSS-VI to the system, the damping ratio of the interregional oscillation mode 4 is improved, but the damping ratio of the interregional oscillation mode 3 decreases slightly compared to addition of the PSS to the system. After MRAC-PSS-VI is installed in the DFIG, the damping ratio of interregion oscillation mode 4 and intraregion oscillation mode 2 of the system is greatly improved. For example, in cases a, b, and c, the damping ratio of intraregion oscillation mode 2 increases from 9.154%, 9.921%, and 11.021% to 19.204%, 16.713%, and 17.846%, respectively. The damping ratio of interregion oscillation mode 4 increases from 16.518%, 16.524%, and 16.387% to 28.783%, 29.563%, and 27.713%, respectively, in different cases. Adoption of DFIG-MRAC-PSS-VI further improved the damping characteristics of the system, and the damping ratio of the regional mode was greatly improved, especially the regional mode 4, whose damping ratio was improved by more than 70%. The small disturbance stability of the system is improved, and the system robustness is improved.Table 4
The eigenvalues of the system in different tie line power conditions.
ModeCaseWithout the controllerPSSDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1a−0.671 ±j6.5501.04210.195−0.671 ±j6.5501.04310.196G1, G2b−0.643 ±j6.5421.0419.790−0.644 ±j6.5421.0419.789c−0.618 ±j6.5021.0359.465−0.618 ±j6.5021.0349.4662a−0.589 ±j6.4041.0199.154−0.583 ±j6.4011.0199.079G3, G4b−0.637 ±j6.3861.0169.921−0.631 ±j6.3831.0169.839c−0.704  ±j6.3501.01111.021−0.697 ±j6.3481.01010.9203a−0.295 ±j3.2890.5248.919−0.259 ±j3.2930.5247.827G1−G4b−0.305 ±j3.0310.48210.039−0.386 ±j3.0550.48612.539c−0.343 ±j2.3140.36814.672−0.432 ±j2.3540.37518.0384a−0.513 ±j3.0630.48816.518−0.513 ±j3.4070.54214.877G1−G4, DFIGb−0.512 ±j3.0560.48616.524−0.379 ±j3.3790.53811.159c−0.503 ±j3.0280.48216.387−0.389 ±j3.3900.54011.412ModeCasePSS-VIMRAC-PSS-VIDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1a−0.671 ±j6.5501.04310.195−0.628 ±j6.3551.0119.827G1, G2b−0.644 ±j6.5421.0419.789−0.679 ±j6.3351.00810.669c−0.618 ±j6.5021.0359.464−0.746 ±j6.3121.00411.7302a−0.589 ±j6.4041.0199.154−1.059 ±j5.4140.86219.204G3, G4b−0.637 ±j6.3861.0169.920−0.959 ±j5.6600.90116.713c−0.704 ±j6.3501.01111.021−1.153 ±j6.3561.01117.8463a−0.293 ±j3.2900.5248.879−0.297 ±j3.1910.5089.278G1−G4b−0.304 ±j3.0340.4829.970−0.305 ±j2.9280.46610.345c−0.342 ±j2.3210.36914.589−0.332 ±j2.1940.34914.9784a−0.687 ±j3.4890.55519.319−1.050 ±j3.4950.55628.783G1−G4, DFIGb−0.689 ±j3.4760.55319.467−1.081 ±j3.4960.55629.563c−0.689 ±j3.4310.54619.694−0.946 ±j3.2810.52227.713In this work, the author gives the changes of eigenvalues corresponding to oscillation mode 2 and oscillation mode 4 in case a, as shown in Figure14.Figure 14
The eigenvalues of the system installed different controllers in case a. (a) Mode 2. (b) Mode 4.
(a)(b)As shown in Figure14, after the addition of the PSS in the DFIG, the eigenvalues for intraregional oscillation mode 2 almost remains unchanged compared with the DFIG with the PSS-VI. While for interregional oscillation mode 4 when the PSS-VI is installed in the DFIG, the eigenvalues exist a little left shift trend, and the system stability is improved. While the PSS is installed in the DFIG, the eigenvalues of the system change very little. When the MRAC-PSS-VI controller is installed in the DFIG, the eigenvalues of intraregional oscillation mode 2 and interregional oscillation mode 4 show a large tendency shifting to the left, and the system robustness is significantly improved.Under the same fault conditions as described in Section3.1, the system response curves under different tie line powers are shown in Figure 15. It can be seen that with the increase of the power of the tie line, the relative power angle of G1 and G3 gradually increases and the voltage at bus 6 gradually decreases. After the failure, the oscillation amplitude of the curves increased with the increase of the power of the tie line, and the time for the system to stabilize also increased slightly. Under different tie line power conditions, the damping improvement of the DFIG installed the PSS is not obvious. The addition of virtual impedance link to the DFIG-PSS has a certain effect on the stability of the system with small disturbance. Under different tie line power conditions, the power angle curve of G1 and the voltage response curve of bus 6 are improved, but the improvement effect is not ideal in the first cycle. After the MRAC control method is introduced into the DFIG-PSS-VI control, as shown in Figure 12, the power angle curve of G1 and voltage curve of bus 6 are significantly reduced in the case of small interference. The improvement effect of the DFIG with MRAC-PSS-VI is ideal under different tie line power conditions. It can be concluded that after the introduction of the DFIG-MRAC-PSS-VI, the damping of the system is improved and the robustness of the system is significantly improved.Figure 15
Response curves with different cases under a small disturbance. (a) Power angle of G1 in case a. (b) Voltage of bus 6 in case a. (c) Power angle of G1 in case b. (d) Voltage of bus 6 in case b. (e) Power angle of G1 in case c. (f) Voltage of bus 6 in case c.
(a)(b)(c)(d)(e)(f)


## 4.4. System Simulation of DFIG Access Point Changes

The change of the DFIG access position has an impact on the stability of the system. In this section, the controller effect is studied when the DFIG access position is changed. The effectiveness of the designed controller is verified by the following calculation cases:Case d: DFIG accesses bus 6 in area 1Case e: DFIG accesses transmission lines 7-8 in area 1Case f: DFIG accesses bus 9 in area 2Some oscillation modes of the system after changing the DFIG access position is given in Table5.Table 5
The eigenvalues of the system in different DFIG access positions.
ModeCaseWithout the controllerPSSDominant statesλf (Hz)ξ (%)Λf (Hz)ξ (%)1d−0.652 ±j6.5461.0429.908−0.649 ±j6.5461.0429.879G1, G2e−0.651 ±j6.5471.0429.901−0.651 ±j6.5461.0429.899f−0.651 ±j6.5471.0429.896−0.651 ±j6.5471.0429.8962d−0.620 ±j6.3931.0179.662−0.620 ±j6.3931.0179.656G3, G4e−0.621 ±j6.3931.0179.661−0.619 ±j6.3931.0179.637f−0.620 ±j6.3931.0179.654−0.616 ±j6.3921.0179.5973d−0.303 ±j3.1280.4989.639−0.327 ±j2.9940.47610.864G1−G4e−0.302 ±j3.1330.4999.591−0.257 ±j3.0790.4908.306f−0.299 ±j3.1370.4999.500−0.290 ±j3.1720.5059.1214d−0.807 ±j3.0510.48625.571−0.919 ±j3.0630.48728.742G1−G4, DFIGe−0.791 ±j3.0100.47925.419−0.894 ±j3.1470.50127.339f−0.807 ±j3.0420.48425.648−0.892 ±j2.9360.46729.075ModeCasePSS-VIMRAC-PSS-VIDominant statesλf (Hz)ξ (%)λf (Hz)ξ (%)1d−0.651 ±j6.5451.0429.892−0.760 ±j6.2670.99712.030G1, G2e−0.651 ±j6.5471.0429.902−0.668 ±j6.3621.01210.441f−0.651 ±j6.5471.0429.896−0.675 ±j6.3511.01110.5632d−0.620 ±j6.3931.0179.658−1.057 ±j6.6321.05515.745G3, G4e−0.619 ±j6.3921.0179.645−1.426 ±j5.5500.88324.891f−0.615 ±j6.3901.0179.589−2.198 ±j7.2051.14629.1873d−0.366 ±j3.1870.50711.405−0.307 ±j3.0480.48510.011G1−G4e−0.360 ±j3.1400.50011.385−0.303 ±j3.0340.4839.926f−0.371 ±j3.0590.48712.049−0.300 ±j3.0360.4839.8294d−1.120 ±j2.9270.46635.749−1.421 ±j3.7200.59235.692G1−G4, DFIGe−1.033 ±j2.8440.45334.144−1.187 ±j3.1740.50535.035f−1.375 ±j2.8140.44843.911−1.225 ±j3.5650.56732.486As given in Table5, the change of the DFIG access point position causes the change of system eigenvalues. For the intraregion oscillation mode 2, the change of system eigenvalue is not obvious after the installation of the PSS and PSS-VI in the DFIG. When the DFIG is in different access positions, the DFIG installed the PSS and PSS-VI has both positive and negative effects on the damping ratio of the system, while the eigenvalue of the MRAC-PSS-VI system shifted significantly to the left, and the damping ratio of the system was greatly improved. For the interregion oscillation mode 4, the damping ratio improvement effect of adding the PSS to the DFIG is limited, while the effect of adding the PSS-VI and MRAC-PSS-VI is more obvious. Although the damping ratio improvement effect of MRAC-PSS-VI is not as good as that of the PSS-VI, the left shift trend of the system characteristic value is obviously better than that of the PSS-VI. In order to further illustrate the effect of the controller, the system is simulated in time domain, and the results are shown in Figure 16.Figure 16
The eigenvalues of system installed different controllers in case d. (a) Mode 2 in case d. (b) Mode 4 in case d.
(a)(b)In this work, the author gives the changes of eigenvalues corresponding to oscillation mode 2 and oscillation mode 4 in case d, as shown in Figure16.As shown in Figure16, after the addition of the PSS and PSS-VI in the DFIG, the eigenvalues for intraregional oscillation mode 2 almost remains unchanged compared with the DFIG without any controller. However, for interregional oscillation mode 4 when the PSS is installed in the DFIG, the eigenvalues exist a little left shift trend, and the system stability is improved. While the PSS-VI is installed in the DFIG, the left shift trend of eigenvalues is bigger than that of the DFIG-PSS. When the MRAC-PSS-VI controller is installed in the DFIG, the eigenvalues of intraregional oscillation mode 2 and interregional oscillation mode 4 show a large tendency shifting to the left, and the system robustness is significantly improved.In order to verify the effectiveness of the control at different access positions of the DFIG, time domain simulation was carried out. Figure17 shows the response curves of power angle of G1 and voltage of bus 9.Figure 17
Response curves with different cases. (a) Power angle of G1 in case d. (b) Voltage of bus 9 in case d. (c) Power angle of G1 in case e. (d) Voltage of bus 9 in case e. (e) Power angle of G1 in case f. (f) Voltage of bus 9 in case f.
(a)(b)(c)(d)(e)(f)As shown in Figure17, the installation of the MRAC-PSS-VI at different access positions of the DFIG has a good effect on system stability. As shown in Figure 17, when the DFIG is equipped with the PSS, the power angle curve oscillation of G1 has no obvious inhibition effect, and the inhibition effect of voltage fluctuation of bus 9 is not obvious either. When the PSS-VI is added, the oscillation is suppressed to some extent, but the effect is not ideal. While the MRAC-PSS-VI is installed in the DFIG, the oscillation of the system is obviously suppressed, and the rotor angle stability and voltage stability of the system are improved.


## 5. Conclusion

To solve the low-frequency oscillation problem of the wind power system, this study constructed a MRAC-DFIG-PSS controller based on virtual impedance which was built in DigSILENT/PowerFactory simulation software. This study adopts the 4-machine 2-region system as an example, while the constructed controller was installed into the reactive power control loop of the rotor side controller of the DFIG, and the improvement effect of the designed controller on the low-frequency oscillation characteristics of the system was verified by time domain simulation. The main conclusions are as follows:(1)
The PSS-VI controller whose step response characteristics is better than the traditional controller can improve the stability of the system more obviously(2)
MRAC-PSS-VI installed in the DFIG rotor side reactive power control loop can improve the low-frequency oscillation of the wind power system, and it also has a certain effect when the power of the tie line changes.(3)
The DFIG-MRAC-PSS-VI has a nice influence on system damping whose dominant states contain the DFIG; however, this method only has a little influence on system damping whose dominant states exclude the DFIG.



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*Source: 3525601-2022-01-31.xml*