# Damping Torque Analysis of the PMSG-Based WT with Supplementary Damping Control for Mitigating Interarea Oscillations

Authors: Yanan Yang; Yujun Li; Xiaotian Yuan; Jiapeng Li; Zhengchun Du
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

This study presents a comprehensive analysis of the impact that a supplementary damping controller of the permanent magnet synchronous generator (PMSG)-based wind turbine (WT) has on low-frequency oscillation (LFO) damping. A reduced mathematical model of an interarea system is first established. Then, an auxiliary damping controller is designed using the PMSG active power control loop, enabling the WT to actively support the interarea LFO mitigation. Based on this, the damping torque analysis (DTA) method is applied to explore the contribution of the PMSG-based WT with an auxiliary damping controller to LFO damping enhancement, whose analytical expressions of the damping torque coefficients reveal the impacts of the control parameters and the operation conditions on the system damping characteristics. It is evident that the installation location of the wind power plant (WPP) plays a leading role in determining whether the damping provided by the PMSG controller is positive or negative. Also, the contribution of damping from the PMSG can be improved by tuning the droop coefficient of the PMSG controller properly. The results indicate that the proposed damping control is always helpful in improving the system damping when the wind power penetration increases. Accordingly, analytical conclusions can serve as guidelines for the control design. Case studies of a two-area test system integrated with a PMSG-based wind farm have been conducted to verify the theoretical analysis.


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## Body

## 1. Introduction

Nowadays, the large-scale integration of inverter-connected renewable energy such as wind power has radically changed the structure of the power system and posed significant challenges to the stability of the power system [1–4]. Unlike the synchronous generators (SGs), the wind turbine (WT) does not participate in electromechanical oscillations since the wind power generators are separated from the power grids by the power electronic devices. Thus, high penetration of wind power may affect the inertia [5, 6] and damping characteristics of the power system that used to be dominated by synchronous generators [7–9]. A major concern is the low-frequency oscillation (LFO) problem, especially the interarea oscillations in the interconnected systems, which long exists in the power system and will occur if adequate damping cannot be provided during disturbances [10].In order to improve the system damping with high penetration of wind power, it has been proposed to equip the WTs with an auxiliary damping control loop to mitigate LFOs [11–17]. Correspondingly, the damping control strategies of variable speed WTs realized by supplementary damping control can be divided into active power modulation [18–21] and reactive power modulation [22–25]. MokhtariAminifar and Li et al. [26, 27] commonly add the auxiliary damping controller to modulate the active power output of the rotor-side converter (RSC) so as to damp the interarea oscillations. Although active power modulation is effective in suppressing electromechanical oscillations, it is found to adversely affect the damping of turbines’ shaft control [28, 29]. Also, the reactive power control of WT is designed to enhance the damping of LFOs. In [23], satisfactory damping can be achieved by equipping the doubly fed induction generator (DFIG) with the proposed power system stabiliser (PSS), which is connected to the reactive power controller of RSC. However, it is reported that the reactive power modulation might deteriorate the voltage stability [20].Due to the complexity and high order of the power system models, the general approach to verifying the proposed control strategies is based on numerical studies. Modal analysis (MA) is one of the most conventional methods for small-signal stability analysis, which can provide numerical results by analysing the participation patterns of the different system elements. The main advantage of the MA method is that global stability of the system under specified operating conditions can be obtained [30]. The state-space model of a multimachine power system with the DFIG is reported in [10]. It is found that the impact of dynamic interactions introduced by the DFIG increases when wind power penetration increases [31]. In [32], the characteristics of the LFO modes and the dominant states have been comprehensively analysed to study the impact of virtual synchronous machine control on the LFOs. Based on the analysis of the residues of the critical eigenvalues under different conditions, the damping contribution of WTs is evaluated in [33]. To improve the control effects, the power oscillation damper parameters are optimized with the eigenvalue analysis [34]. However, the MA method relies on the numerical results of complicated modal computation, which cannot illustrate why the instability may occur from the physical interpretation. In addition, it fails to formulate a direct mathematical relationship between damping characteristics and influential factors such as the operating point and the parameters of controllers.To study the mechanism of the impacts of WT on the LFO, the damping torque analysis (DTA) method is employed in [35–43]. Through this method, the electric torque contributed by the controller is decomposed into synchronization torque and damping torque components. The damping torque component determines the damping of the LFOs. For the power system integrated with a DFIG-based wind farm, the effect of the dynamic interactions introduced by the WTs on the oscillation modes is estimated by the DTA method [38]. In [39], the authors provided a very comprehensive examination of the effect of the supplementary HVDC power control on small-signal stability by calculating the damping matrix. The analytical results of [39] were innovatively extended to multiterminal HVDC transmissions in [40]. In [41], the change of load flow and the dynamic interaction introduced by the DFIG are separated to identify the impact of the dynamic interactions of the DFIG with the SGs. Unfortunately, the damping torque coefficients obtained from the above studies are not in analytical form, making it difficult to intuitively evaluate the impacts of the parameters of the WT controller on the LFO mode. In addition, the influencing factors of the LFOs need to be further discussed.Existing studies primarily focus on the damping effects of LFOs by modulating active or reactive power but rarely care about the mechanism of the supplementary damping controller to deal with LFOs. This study develops insights into the contribution of the PMSG-based WT with an auxiliary damping controller to LFO mitigation by the DTA method. Firstly, this study developed a dynamic reduced model of an interarea system oriented to the small-signal stability analysis. Then, an auxiliary damping control strategy for the PMSG-based WT is designed by the droop control loop, which uses the rotational speed signal between generators to guide the active power modulation. Based on this, the DTA method is applied to evaluate the contribution of WT control on LFO mitigation, whose explicit expressions of the damping torque coefficients are analytically derived. Consequently, qualitative evaluations of the operation conditions and the control parameters on the system damping characteristics can be achieved, providing useful guidance for the controller design.The rest of this study is organized as follows. Section2 constructs a reduced dynamic model of an interarea system. In Section 3, an auxiliary damping control based on a droop control scheme is proposed to restrain the LFOs of the PMSG-based WT. Section 4 applies the DTA method to obtain some analytical conclusions. The analytical results are verified by simulations in Section 5. Section 6 summarizes the main conclusions of this study.


## 2. Modelling of the Interarea System

As a mainstream type of variable speed wind turbine, the PMSG-based WT is widely used due to its excellent control features. In this section, a simplified dynamic model of a two-area system integrated with a PMSG-based wind farm is first established. Then, the mathematical model and conventional control scheme of the PMSG are briefly introduced.


### 2.1. A Reduced Dynamic Model of the Interarea System

As shown in Figure1, an interarea system integrated with the PMSG-based wind power plant (WPP) is first constructed. For simplification, the AC grid of each area is modelled as the equivalent generators SG1 and SG2. Hence, the dynamics of the AC grids can be represented by two synchronous generators. The AC transmission lines of the studied system are regarded as lossless and without energy storage.Figure 1
Structure of an interarea system.As for generators, the classical machine model is adopted, with a stiff electromotive force behind a transient reactance. Under the consideration of eliminating the damping effect of the SGs, the damping windings of SG1 and SG2 (including the damping effect caused by eddy currents in the rotor core) are neglected, and the damping coefficient values of SG1 and SG2 are regarded as zero (D1 = D2 = 0).The swing equations of SG1 and SG2 can be written as follows:(1)2H1dω1dt=PM1−Pe1−PL1,2H2dω2dt=PM2−Pe2−PL2,dδ1dt=ωsω1−1,dδ2dt=ωsω2−1,where H1 and H2 are the inertia constants, PM1 and PM2 are the mechanical power inputs of generators SG1 and SG2, respectively, Pe1 and Pe2 are the active power outputs of generators SG1 and SG2, respectively, PL1 and PL2 are the local loads, ω1, ω2 and δ1, δ2 are the rotor speed and rotor angle of SG1 and SG2, respectively, and ωs is the synchronous electrical angular velocity. The mathematical model mentioned in (1) is the simplest dynamic representation of a synchronous generator. Based on this, it is possible to decouple the damping effect introduced by WT from the power system.Active power balances at Bus 3 add the relation(2)V1V3X1sinδ1−δ3+V2V3X2sinδ2−δ3+Pe3=0,where X1 and X2 are the total reactance of the transmission line in each area, as shown in Figure 1, δ3 is the voltage phase at Bus 3, and Pe3 is the active power output from the WPP. In normal operation, the voltage magnitudes of all busbars vary within a small range because of the regulation of the excitation system and the injected reactive power. For this reason, it is assumed that the voltage magnitudes of all busbars are equal, which meets the condition that V1 = V2 = V3 = V. Under this assumption, the linearization equation (2)becomes(3)V2X1cosδ10−δ30⋅Δδ1−Δδ3+V2X2cosδ20−δ30⋅Δδ2−Δδ3+ΔPe3=0,where the prefix “∆” is the incremental operator that denotes the small deviation of a variable. The subscript (0) of each variable represents its steady-state value. For simplification, the variables a1 and a2 are defined as follows:(4)a1=V2X1cosδ10−δ30,a2=V2X2cosδ20−δ30.Then, ∆δ3 can be solved based on (3) and (4):(5)Δδ3=a1a1+a2Δδ1+a2a1+a2Δδ2+ΔPe3a1+a2.Substituting (3) and (4) into the small-signal dynamic model of (1), the simplified dynamic equations of the interarea system can be obtained as follows:(6)2H1H2dΔωdt=−a1a2H1+H2a1+a2Δδ+H2a1−H1a2a1+a2ΔPe3,dΔδdt=ωsΔω,where(7)Δω=Δω1−Δω2,Δδ=Δδ1−Δδ2,where ∆ω is the rotational speed deviation between SG1 and SG2 and ∆δ is the rotor angle deviation between SG1 and SG2. If the active power output from the WPP is zero (i.e., Pe3 = 0) in (6), then Bus 3 can be considered a point to observe the voltage phase along the line since it has no impact on the system dynamics. In this case, the Laplace transform of (6) can be expressed as(8)2H1H2ωss2+a1a2H1+H2a1+a2=0.Then, the undamped natural mechanical mode frequency can be solved as follows:(9)s1,2=±jωn,ωn=ωsa1a2⋅H1+H22a1+a2⋅H1H2.As shown in (6), the dynamic behaviour of a two-area model can be similarly represented by an equivalent single-machine infinite-bus (SMIB) model. However, if the full mode of the two-area system is considered, the relative degree will be large. Thus, the design of the WT damping controller will use a high-order derivation, which will be much more difficult. Otherwise, the reduced dynamic mathematical model in (6) has many advantages. (i) Firstly, the complexity of the interarea system model has been greatly reduced since the model has been reduced from fourth order to second order. (ii) Secondly, the initial parameters of two interconnected synchronous generators are retained in (6) after some simple formula reconstructions and variable substitutions. (iii) Finally, the effect of the injected power at Bus 3 on LFOs is clearly outlined in (6), allowing the derivation of the equivalent damping torque provided by the WT controller. For these reasons, the electromechanical dynamics of the interarea system in (1) can be represented by the dynamic equations in (6).


### 2.2. PMSG-WT with the Conventional Control Scheme

As shown in Figure2, the conventional controller of PMSG is composed of a rotor-side converter (RSC) and a grid-side converter (GSC). In this study, the RSC controls the PMSG generated active power, while the GSC is utilized to maintain the DC-link voltage. Normally, the active power generated by the PMSG is controlled through the maximum power point tracking (MPPT) algorithm and the pitch angle control.Figure 2
The configuration of a PMSG connecting to a large system.The output power of the PMSG under the MPPT control can be expressed as(10)PMPPT=ρπR5k3Cpmax2λ3⋅ωt3=CM⋅ωt3,where ρ is the air density, R is the rotor blade radius, k is the gear ratio of the gearbox, λ is the tip speed ratio, Cp is the maximum power coefficient as λ takes a certain value, and ωt is the PMSG generator rotational speed.Generally, the control scheme at the back-to-back VSCs of a PMSG primarily includes the inner current control loop and the outer control loop. Due to the fast response of the double loop controller system, the converter dynamics of the PMSG can be overlooked. Therefore, the generated active powerPWT can be regarded as the same as its power reference, which is determined by the MPPT algorithm PMPPT. Under the assumption that the converter is lossless, the equation for the PMSG output power is given by(11)Pe3=PWT.The rotor motion dynamic equation of the PMSG can be described as(12)2Ht⋅ωt⋅dωtdt=Pwind−PWT,where Ht is the inertia time constant of the PMSG.


## 2.1. A Reduced Dynamic Model of the Interarea System

As shown in Figure1, an interarea system integrated with the PMSG-based wind power plant (WPP) is first constructed. For simplification, the AC grid of each area is modelled as the equivalent generators SG1 and SG2. Hence, the dynamics of the AC grids can be represented by two synchronous generators. The AC transmission lines of the studied system are regarded as lossless and without energy storage.Figure 1
Structure of an interarea system.As for generators, the classical machine model is adopted, with a stiff electromotive force behind a transient reactance. Under the consideration of eliminating the damping effect of the SGs, the damping windings of SG1 and SG2 (including the damping effect caused by eddy currents in the rotor core) are neglected, and the damping coefficient values of SG1 and SG2 are regarded as zero (D1 = D2 = 0).The swing equations of SG1 and SG2 can be written as follows:(1)2H1dω1dt=PM1−Pe1−PL1,2H2dω2dt=PM2−Pe2−PL2,dδ1dt=ωsω1−1,dδ2dt=ωsω2−1,where H1 and H2 are the inertia constants, PM1 and PM2 are the mechanical power inputs of generators SG1 and SG2, respectively, Pe1 and Pe2 are the active power outputs of generators SG1 and SG2, respectively, PL1 and PL2 are the local loads, ω1, ω2 and δ1, δ2 are the rotor speed and rotor angle of SG1 and SG2, respectively, and ωs is the synchronous electrical angular velocity. The mathematical model mentioned in (1) is the simplest dynamic representation of a synchronous generator. Based on this, it is possible to decouple the damping effect introduced by WT from the power system.Active power balances at Bus 3 add the relation(2)V1V3X1sinδ1−δ3+V2V3X2sinδ2−δ3+Pe3=0,where X1 and X2 are the total reactance of the transmission line in each area, as shown in Figure 1, δ3 is the voltage phase at Bus 3, and Pe3 is the active power output from the WPP. In normal operation, the voltage magnitudes of all busbars vary within a small range because of the regulation of the excitation system and the injected reactive power. For this reason, it is assumed that the voltage magnitudes of all busbars are equal, which meets the condition that V1 = V2 = V3 = V. Under this assumption, the linearization equation (2)becomes(3)V2X1cosδ10−δ30⋅Δδ1−Δδ3+V2X2cosδ20−δ30⋅Δδ2−Δδ3+ΔPe3=0,where the prefix “∆” is the incremental operator that denotes the small deviation of a variable. The subscript (0) of each variable represents its steady-state value. For simplification, the variables a1 and a2 are defined as follows:(4)a1=V2X1cosδ10−δ30,a2=V2X2cosδ20−δ30.Then, ∆δ3 can be solved based on (3) and (4):(5)Δδ3=a1a1+a2Δδ1+a2a1+a2Δδ2+ΔPe3a1+a2.Substituting (3) and (4) into the small-signal dynamic model of (1), the simplified dynamic equations of the interarea system can be obtained as follows:(6)2H1H2dΔωdt=−a1a2H1+H2a1+a2Δδ+H2a1−H1a2a1+a2ΔPe3,dΔδdt=ωsΔω,where(7)Δω=Δω1−Δω2,Δδ=Δδ1−Δδ2,where ∆ω is the rotational speed deviation between SG1 and SG2 and ∆δ is the rotor angle deviation between SG1 and SG2. If the active power output from the WPP is zero (i.e., Pe3 = 0) in (6), then Bus 3 can be considered a point to observe the voltage phase along the line since it has no impact on the system dynamics. In this case, the Laplace transform of (6) can be expressed as(8)2H1H2ωss2+a1a2H1+H2a1+a2=0.Then, the undamped natural mechanical mode frequency can be solved as follows:(9)s1,2=±jωn,ωn=ωsa1a2⋅H1+H22a1+a2⋅H1H2.As shown in (6), the dynamic behaviour of a two-area model can be similarly represented by an equivalent single-machine infinite-bus (SMIB) model. However, if the full mode of the two-area system is considered, the relative degree will be large. Thus, the design of the WT damping controller will use a high-order derivation, which will be much more difficult. Otherwise, the reduced dynamic mathematical model in (6) has many advantages. (i) Firstly, the complexity of the interarea system model has been greatly reduced since the model has been reduced from fourth order to second order. (ii) Secondly, the initial parameters of two interconnected synchronous generators are retained in (6) after some simple formula reconstructions and variable substitutions. (iii) Finally, the effect of the injected power at Bus 3 on LFOs is clearly outlined in (6), allowing the derivation of the equivalent damping torque provided by the WT controller. For these reasons, the electromechanical dynamics of the interarea system in (1) can be represented by the dynamic equations in (6).


## 2.2. PMSG-WT with the Conventional Control Scheme

As shown in Figure2, the conventional controller of PMSG is composed of a rotor-side converter (RSC) and a grid-side converter (GSC). In this study, the RSC controls the PMSG generated active power, while the GSC is utilized to maintain the DC-link voltage. Normally, the active power generated by the PMSG is controlled through the maximum power point tracking (MPPT) algorithm and the pitch angle control.Figure 2
The configuration of a PMSG connecting to a large system.The output power of the PMSG under the MPPT control can be expressed as(10)PMPPT=ρπR5k3Cpmax2λ3⋅ωt3=CM⋅ωt3,where ρ is the air density, R is the rotor blade radius, k is the gear ratio of the gearbox, λ is the tip speed ratio, Cp is the maximum power coefficient as λ takes a certain value, and ωt is the PMSG generator rotational speed.Generally, the control scheme at the back-to-back VSCs of a PMSG primarily includes the inner current control loop and the outer control loop. Due to the fast response of the double loop controller system, the converter dynamics of the PMSG can be overlooked. Therefore, the generated active powerPWT can be regarded as the same as its power reference, which is determined by the MPPT algorithm PMPPT. Under the assumption that the converter is lossless, the equation for the PMSG output power is given by(11)Pe3=PWT.The rotor motion dynamic equation of the PMSG can be described as(12)2Ht⋅ωt⋅dωtdt=Pwind−PWT,where Ht is the inertia time constant of the PMSG.


## 3. Proposed Damping Controller by Droop Control

This section proposes a droop control strategy for the PMSG to provide additional damping for the interarea system. For the conventional PMSG control strategy, the PMSG output power is completely handled by the fast converter control. However, this converter control strategy decoupled the rotational speed of the PMSG from the grid frequency. Consequently, the PMSG cannot provide damping after a disturbance, which might diminish the ability of damping LFOs in the power system.In order to improve the damping of the WT, most existing studies focus on implementing the interarea oscillation damping control via auxiliary droop control. The output of the droop control loop can be expressed as(13)ΔPdroop=−KWTωsys−ωnom,where ωsys is the system rotational speed in p.u., ωnom is the nominal rotational speed in p.u., and KWT is the droop coefficient. The principle of this auxiliary damping control can be explained as follows. Once the system is disturbed; for example, a sudden load increase occurs in the power system. Then, the rotational speed of the system will deviate from its nominal rotational speed, which activates the droop control loop in (13) and adds a power component related to the rotational speed to the PMSG reference power. Subsequently, the turbine slows down and releases kinetic energy from the rotor to modulate the imbalanced power in the power system.In this study, an auxiliary damping control using the signal of the rotational speed deviation between SG1 and SG2 has been particularly developed. There are several considerations for using ∆ω as the input signal. Firstly, ∆ω is in accordance with the reduced dynamic model of the interarea system derived in (6), which might simplify the design of auxiliary damping control. Additionally, the input signal ∆ω can automatically detect the speed deviation between generators, which might provide damping more effectively. Also, damping the LFOs is meant to keep the relative angle between the two areas as a constant. In other words, it is to maintain the rotational speed deviation ∆ω at a value of zero.Accordingly, the reference power of PMSG can be written as(14)PWT=PMPPT−KWTΔω.For the steady state, the relative rotor speed ∆ω and the active power for damping control ∆Pdroop are both zeros. Figure 3 displays the structure of the PMSG droop control. In Figure 3, ∆Pdroop is limited to 0.1 p.u. to ensure that the current of the IGBTs in the converter does not exceed its limits. As for the proposed droop control scheme in Figure 3, the signal of the rotational speed deviation ∆ω is sent through a deadband filter first, which is used to avoid the reaction of the controller to very small rotational speed variations during normal operation. Considering small disturbances of the rotational speed, it is necessary to set the threshold value of the rotational speed deviation ∆ω, which is the input signal of the droop control. In this study, the threshold value of the rotational speed deviation is set as ∆ω = ±0.02 p.u. in the deadband filter.Figure 3
Structure of the PMSG droop control.


## 4. Low-Frequency Oscillation Analysis of PMSG Integrated Power System

The DTA is a conventional and effective method to evaluate the ability to damp oscillations. In this section, the DTA method is applied to clearly outline the effect of the auxiliary damping control of PMSG on the LFOs. Moreover, the damping torque provided by the PMSG controller is explicitly derived for further analysis.


### 4.1. Dynamic Model of the Studied System

Considering the structure of a WPP integrated into the interarea power system in Figure1, some assumptions need to be made. (i) Firstly, the impact of the automatic voltage regulator (AVR) and the power system switch (PSS) on the power system dynamics are overlooked. (ii) Secondly, it is assumed that the deviation of voltage-angle between busbars is comparatively small in this study. (iii) Finally, it is supposed that the rotor speed signal of the synchronous generator can be transmitted to the wind turbine terminals through wide area control systems (WACS) instantaneously, i.e., the communication delay can be neglected.Suppose that the wind speed remains constant during a comparatively short period; then, the linearized model of the PMSG equipped with auxiliary damping control (cf. Section3) can be established as follows:(15)2Ht⋅ωt0⋅dΔωtdt=−ΔPe3,ΔPe3=3CMωt02Δωt−KWTΔω.Combining (15) with (6) and using the Laplace transformation yield(16)2H1H2dΔωdt=−a1a2H1+H2a1+a2Δδ−H2a1−H1a2a1+a22Htωt0KWTs3CMωt02+2Htωt0s︸FsΔω.The last term of (16), F(s), is an electrical torque, which contributes to the auxiliary damping control of the PMSG-based WT. According to the classical DTA method, the electrical torque component can be decomposed into the synchronizing and damping components, which are in phase with the rotor angle perturbation and the angle speed deviation, respectively. Since the system damping torque coefficient is comparably small to the synchronizing torque coefficient, the system oscillation frequency can be represented by the natural oscillation frequency in (9). Accordingly, (16) can be reformulated as(17)TJdΔωdt=−KSΔδ−KDΔω,where(18)KS=a1a2H1+H2a1+a2,KD=H2a1−H1a2a1+a24Ht2KWTωn23CMωt02+2Htωn2,TJ=2H1H2+H2a1−H1a2a1+a26CMHtKWTωt03CMωt02+2Htωn2,where TJ is the equivalent inertia constant provided by the synchronous generator and the PMSG, KS is the synchronizing torque coefficient, and KD is the damping torque coefficient.


### 4.2. Damping Torque Analysis

According to (18), the damping torque coefficient of the studied system is correlated with the parameters of the PMSG damping control. Due to the small variation between generators, the cosine function cos (δi(0) − δj(0)) in (18) can be approximated as 1. Then, the damping torque coefficient can be obtained as(19)KD=H2X2−H1X1X1+X24Ht2KWTωn23CMωt02+2Htωn2.In (19), the equivalent damping torque coefficient is not only related to the droop control coefficient KWT but also affected by the installed location of the WPP. The damping torque analysis will be conducted in the following section to clearly outline the effect of these factors.


#### 4.2.1. The Impact of the Location of the WPP

From (19), it can be seen that the relative location of the WPP to the two SGs affects the damping level of the studied system. The line reactance from the WPP to SG1 and SG2 are noted as Xl1 and Xl2, respectively. Assuming that the reactance of the transmission line is well distributed, then the WPP location can be expressed by the relative distance from SG1 noted as lw:(20)Xl1=lwX,Xl2=1−lwX,where X is the total reactance of the transmission line 1 and the transmission line 2.If the WPP is located at the mass-weighted electrical centre between the two areas, i.e.,H2X2 = H1X1, then the damping torque coefficient KD will reduce to zero. In this case, the PMSG damping control has no impact on the interarea LFOs, and consequently, the WT injected power has no contribution to the LFOs’ damping. The location of the mass-weighted electrical centre can be expressed by lw:(21)lw=H2XT2+X−H1XT1H1+H2X,where XT1 and XT2 are the reactance of transformers T1 and T2. Obviously, the interarea mode is uncontrollable by using the PMSG auxiliary damping control as long as the WPP is located at the mass-weighted electrical centre.Area 1: if the WPP is installed between the mass-weighted electrical centre andSG1, i.e., H1X1 < H2X2, then the WPP is relatively close to SG1 in the electrical distance. The range of this area can be expressed by lw:(22)0<lw<H2XT2+X−H1XT1H1+H2X.In this case, the damping torque coefficientKD is positive. Therefore, the damping control of PMSG has a positive contribution to LFO damping. Moreover, the closer the WPP is to SG1 (see Figure 1), the higher the damping torque coefficient KD is. Especially, the PMSG controller gives the largest contribution to the oscillation damping when the WPP terminal is located near the centre of SG1, and the maximal damping torque coefficient is(23)KD=4H2Ht2KWTωn23CMωt02+2Htωn2.Area 2: similarly, if the WPP is installed between the mass-weighted electrical centre andSG2, i.e., H1X1 > H2X2, then the WPP is relatively close to SG2 in the electrical distance. The restriction of lw can be derived as(24)H2XT2+X−H1XT1H1+H2X<lw<1.As seen in (19), if H1X1 > H2X2, then KD < 0. The PMSG damping control gives a negative contribution to the LFO damping. One may also notice that the PMSG controller placed closer to the generator SG2 (as shown in Figure 1) has a higher impact on the damping torque coefficient. The negative damping torque coefficient KD is maximized as the WPP is installed near SG2, giving(25)KD=−4H1Ht2KWTωn23CMωt02+2Htωn2.It can be concluded that the PMSG auxiliary damping control will provide a positive damping torque to the system when the WPP is installed in Area 1. Otherwise, a negative damping torque will be obtained when the WPP is installed in Area 2. One may notice that the WPP placed closer to SG1 has a better damping effect on the LFOs.


#### 4.2.2. The Impact ofKWT

The sensitivity of the damping torque coefficientKD to the droop coefficient KWT is given by(26)∂KD∂KWT=H1X1−H2X2X1+X24Ht2ωn23CMωt02+2Htωn2.Equation (26) shows the analytical relationship between the droop coefficient of the PMSG damping control and the change in damping torque coefficient. Considering the case that the WPP is located in Area 1, then the damping torque coefficient KD is positive, and its value increases with the increase in the droop coefficient KWT. In this case, the damping provision of the WT can be enhanced by increasing KWT. The higher KWT, the stronger the damping effect of the WT controller on the LFO. Moreover, the sensitivity of KD to KWT is affected by the location of the WPP. It can be seen from (26) that KD is more sensitive to KWT if the WPP is placed closer to SG1.On the contrary, if the WPP is located in Area 2, the WT controller will exhibit negative damping on system oscillation, and the value ofKD decreases as KWT increases. In this case, a higher droop coefficient will deteriorate the damping level and aggravate the LFO. To sum up, the installation location of the WPP plays a leading role in determining whether the damping provided by the WT controller is positive or negative, and even more, the damping of the interarea oscillations can be improved by turning the droop coefficient. However, the effectiveness of the WT damping control is also affected by the installation location of the WPP, with modulation being more efficient when the WPP is close to SG1.


## 4.1. Dynamic Model of the Studied System

Considering the structure of a WPP integrated into the interarea power system in Figure1, some assumptions need to be made. (i) Firstly, the impact of the automatic voltage regulator (AVR) and the power system switch (PSS) on the power system dynamics are overlooked. (ii) Secondly, it is assumed that the deviation of voltage-angle between busbars is comparatively small in this study. (iii) Finally, it is supposed that the rotor speed signal of the synchronous generator can be transmitted to the wind turbine terminals through wide area control systems (WACS) instantaneously, i.e., the communication delay can be neglected.Suppose that the wind speed remains constant during a comparatively short period; then, the linearized model of the PMSG equipped with auxiliary damping control (cf. Section3) can be established as follows:(15)2Ht⋅ωt0⋅dΔωtdt=−ΔPe3,ΔPe3=3CMωt02Δωt−KWTΔω.Combining (15) with (6) and using the Laplace transformation yield(16)2H1H2dΔωdt=−a1a2H1+H2a1+a2Δδ−H2a1−H1a2a1+a22Htωt0KWTs3CMωt02+2Htωt0s︸FsΔω.The last term of (16), F(s), is an electrical torque, which contributes to the auxiliary damping control of the PMSG-based WT. According to the classical DTA method, the electrical torque component can be decomposed into the synchronizing and damping components, which are in phase with the rotor angle perturbation and the angle speed deviation, respectively. Since the system damping torque coefficient is comparably small to the synchronizing torque coefficient, the system oscillation frequency can be represented by the natural oscillation frequency in (9). Accordingly, (16) can be reformulated as(17)TJdΔωdt=−KSΔδ−KDΔω,where(18)KS=a1a2H1+H2a1+a2,KD=H2a1−H1a2a1+a24Ht2KWTωn23CMωt02+2Htωn2,TJ=2H1H2+H2a1−H1a2a1+a26CMHtKWTωt03CMωt02+2Htωn2,where TJ is the equivalent inertia constant provided by the synchronous generator and the PMSG, KS is the synchronizing torque coefficient, and KD is the damping torque coefficient.


## 4.2. Damping Torque Analysis

According to (18), the damping torque coefficient of the studied system is correlated with the parameters of the PMSG damping control. Due to the small variation between generators, the cosine function cos (δi(0) − δj(0)) in (18) can be approximated as 1. Then, the damping torque coefficient can be obtained as(19)KD=H2X2−H1X1X1+X24Ht2KWTωn23CMωt02+2Htωn2.In (19), the equivalent damping torque coefficient is not only related to the droop control coefficient KWT but also affected by the installed location of the WPP. The damping torque analysis will be conducted in the following section to clearly outline the effect of these factors.


### 4.2.1. The Impact of the Location of the WPP

From (19), it can be seen that the relative location of the WPP to the two SGs affects the damping level of the studied system. The line reactance from the WPP to SG1 and SG2 are noted as Xl1 and Xl2, respectively. Assuming that the reactance of the transmission line is well distributed, then the WPP location can be expressed by the relative distance from SG1 noted as lw:(20)Xl1=lwX,Xl2=1−lwX,where X is the total reactance of the transmission line 1 and the transmission line 2.If the WPP is located at the mass-weighted electrical centre between the two areas, i.e.,H2X2 = H1X1, then the damping torque coefficient KD will reduce to zero. In this case, the PMSG damping control has no impact on the interarea LFOs, and consequently, the WT injected power has no contribution to the LFOs’ damping. The location of the mass-weighted electrical centre can be expressed by lw:(21)lw=H2XT2+X−H1XT1H1+H2X,where XT1 and XT2 are the reactance of transformers T1 and T2. Obviously, the interarea mode is uncontrollable by using the PMSG auxiliary damping control as long as the WPP is located at the mass-weighted electrical centre.Area 1: if the WPP is installed between the mass-weighted electrical centre andSG1, i.e., H1X1 < H2X2, then the WPP is relatively close to SG1 in the electrical distance. The range of this area can be expressed by lw:(22)0<lw<H2XT2+X−H1XT1H1+H2X.In this case, the damping torque coefficientKD is positive. Therefore, the damping control of PMSG has a positive contribution to LFO damping. Moreover, the closer the WPP is to SG1 (see Figure 1), the higher the damping torque coefficient KD is. Especially, the PMSG controller gives the largest contribution to the oscillation damping when the WPP terminal is located near the centre of SG1, and the maximal damping torque coefficient is(23)KD=4H2Ht2KWTωn23CMωt02+2Htωn2.Area 2: similarly, if the WPP is installed between the mass-weighted electrical centre andSG2, i.e., H1X1 > H2X2, then the WPP is relatively close to SG2 in the electrical distance. The restriction of lw can be derived as(24)H2XT2+X−H1XT1H1+H2X<lw<1.As seen in (19), if H1X1 > H2X2, then KD < 0. The PMSG damping control gives a negative contribution to the LFO damping. One may also notice that the PMSG controller placed closer to the generator SG2 (as shown in Figure 1) has a higher impact on the damping torque coefficient. The negative damping torque coefficient KD is maximized as the WPP is installed near SG2, giving(25)KD=−4H1Ht2KWTωn23CMωt02+2Htωn2.It can be concluded that the PMSG auxiliary damping control will provide a positive damping torque to the system when the WPP is installed in Area 1. Otherwise, a negative damping torque will be obtained when the WPP is installed in Area 2. One may notice that the WPP placed closer to SG1 has a better damping effect on the LFOs.


### 4.2.2. The Impact ofKWT

The sensitivity of the damping torque coefficientKD to the droop coefficient KWT is given by(26)∂KD∂KWT=H1X1−H2X2X1+X24Ht2ωn23CMωt02+2Htωn2.Equation (26) shows the analytical relationship between the droop coefficient of the PMSG damping control and the change in damping torque coefficient. Considering the case that the WPP is located in Area 1, then the damping torque coefficient KD is positive, and its value increases with the increase in the droop coefficient KWT. In this case, the damping provision of the WT can be enhanced by increasing KWT. The higher KWT, the stronger the damping effect of the WT controller on the LFO. Moreover, the sensitivity of KD to KWT is affected by the location of the WPP. It can be seen from (26) that KD is more sensitive to KWT if the WPP is placed closer to SG1.On the contrary, if the WPP is located in Area 2, the WT controller will exhibit negative damping on system oscillation, and the value ofKD decreases as KWT increases. In this case, a higher droop coefficient will deteriorate the damping level and aggravate the LFO. To sum up, the installation location of the WPP plays a leading role in determining whether the damping provided by the WT controller is positive or negative, and even more, the damping of the interarea oscillations can be improved by turning the droop coefficient. However, the effectiveness of the WT damping control is also affected by the installation location of the WPP, with modulation being more efficient when the WPP is close to SG1.


## 4.2.1. The Impact of the Location of the WPP

From (19), it can be seen that the relative location of the WPP to the two SGs affects the damping level of the studied system. The line reactance from the WPP to SG1 and SG2 are noted as Xl1 and Xl2, respectively. Assuming that the reactance of the transmission line is well distributed, then the WPP location can be expressed by the relative distance from SG1 noted as lw:(20)Xl1=lwX,Xl2=1−lwX,where X is the total reactance of the transmission line 1 and the transmission line 2.If the WPP is located at the mass-weighted electrical centre between the two areas, i.e.,H2X2 = H1X1, then the damping torque coefficient KD will reduce to zero. In this case, the PMSG damping control has no impact on the interarea LFOs, and consequently, the WT injected power has no contribution to the LFOs’ damping. The location of the mass-weighted electrical centre can be expressed by lw:(21)lw=H2XT2+X−H1XT1H1+H2X,where XT1 and XT2 are the reactance of transformers T1 and T2. Obviously, the interarea mode is uncontrollable by using the PMSG auxiliary damping control as long as the WPP is located at the mass-weighted electrical centre.Area 1: if the WPP is installed between the mass-weighted electrical centre andSG1, i.e., H1X1 < H2X2, then the WPP is relatively close to SG1 in the electrical distance. The range of this area can be expressed by lw:(22)0<lw<H2XT2+X−H1XT1H1+H2X.In this case, the damping torque coefficientKD is positive. Therefore, the damping control of PMSG has a positive contribution to LFO damping. Moreover, the closer the WPP is to SG1 (see Figure 1), the higher the damping torque coefficient KD is. Especially, the PMSG controller gives the largest contribution to the oscillation damping when the WPP terminal is located near the centre of SG1, and the maximal damping torque coefficient is(23)KD=4H2Ht2KWTωn23CMωt02+2Htωn2.Area 2: similarly, if the WPP is installed between the mass-weighted electrical centre andSG2, i.e., H1X1 > H2X2, then the WPP is relatively close to SG2 in the electrical distance. The restriction of lw can be derived as(24)H2XT2+X−H1XT1H1+H2X<lw<1.As seen in (19), if H1X1 > H2X2, then KD < 0. The PMSG damping control gives a negative contribution to the LFO damping. One may also notice that the PMSG controller placed closer to the generator SG2 (as shown in Figure 1) has a higher impact on the damping torque coefficient. The negative damping torque coefficient KD is maximized as the WPP is installed near SG2, giving(25)KD=−4H1Ht2KWTωn23CMωt02+2Htωn2.It can be concluded that the PMSG auxiliary damping control will provide a positive damping torque to the system when the WPP is installed in Area 1. Otherwise, a negative damping torque will be obtained when the WPP is installed in Area 2. One may notice that the WPP placed closer to SG1 has a better damping effect on the LFOs.


## 4.2.2. The Impact ofKWT

The sensitivity of the damping torque coefficientKD to the droop coefficient KWT is given by(26)∂KD∂KWT=H1X1−H2X2X1+X24Ht2ωn23CMωt02+2Htωn2.Equation (26) shows the analytical relationship between the droop coefficient of the PMSG damping control and the change in damping torque coefficient. Considering the case that the WPP is located in Area 1, then the damping torque coefficient KD is positive, and its value increases with the increase in the droop coefficient KWT. In this case, the damping provision of the WT can be enhanced by increasing KWT. The higher KWT, the stronger the damping effect of the WT controller on the LFO. Moreover, the sensitivity of KD to KWT is affected by the location of the WPP. It can be seen from (26) that KD is more sensitive to KWT if the WPP is placed closer to SG1.On the contrary, if the WPP is located in Area 2, the WT controller will exhibit negative damping on system oscillation, and the value ofKD decreases as KWT increases. In this case, a higher droop coefficient will deteriorate the damping level and aggravate the LFO. To sum up, the installation location of the WPP plays a leading role in determining whether the damping provided by the WT controller is positive or negative, and even more, the damping of the interarea oscillations can be improved by turning the droop coefficient. However, the effectiveness of the WT damping control is also affected by the installation location of the WPP, with modulation being more efficient when the WPP is close to SG1.


## 5. Simulation Studies

To demonstrate and evaluate the proposed control scheme, a simple simulation system consisting of two synchronous generators (SG1 and SG2), a PMSG-based wind plant, and two local loads (PL1 and PL2) described in Figure 1 is modelled. A WPP consisting of 196 (14 × 14) units of PMSG is modelled, as well as the rating of each WT converter is set the same as 2 MW. The rated capacity of SG is 300 MW and the wind power capacity penetration level in this study is around 30%. All variables are expressed in per-unit (p.u.) quantities for the test system. More parameters of the test system is in Tables S1 and S2 in the Supplementary Material. In this study, the simulation is developed on the MATLAB platform.


### 5.1. Impact of the WPP Installation Location

In this case, the impact of the WPP location was analysed by changing the reactance of the interconnection linesXl1 and Xl2 in Figure 1. The gain of the droop loop KWT is set to 10. Figure 4 shows the eigenvalue trajectories with lw (the relative distance from SG1) increasing from 0 to 1, as well as the WPP installation location moving along the transmission line and close to SG2. It can be seen that, with lw increasing from 0 to 1, the dominant mode λ1,2 moves from the left half of the complex plane to the right, and KD varies from 13.41 to −32.76. Furthermore, it is interesting to find that there is a critical stable point in the imaginary axis when lw = 0.29, which is indeed the mass-weighted electrical canter. Thus, it can be concluded that the WPP placed close to SG1 might increase the stability of the studied system, which also verifies the accuracy of the damping torque analysis in Section 4.2.Figure 4
Eigenvalue trajectories withlw increasing from 0 to 1.To illustrate the impact of the WPP installation location in detail, two different cases, including the curves in the right and left halves of the complex plane in Figure4, which correspond to Area 1 and Area 2, are considered, respectively. Table 1 shows the interarea oscillation mode λ1,2 when the WPP is located in Area 1. Apparently, as the WPP is placed closer to the mass-weighted electrical centre within Area 1, the damping ratio is decreasing and approaching zero. It can also be seen that mode λ1,2 dominates the LFO with a frequency of approximately 1.5 Hz. Furthermore, the confirmation of simulations corresponding to Table 1 is presented in Figure 5. Assuming that a sudden load event occurs at Bus 1, the load PL1 increases 0.1 p.u. at t = 1 s. As displayed in Figure 5, the oscillations of Location B lw=0.167 are damped better than those of Location A lw=0, confirming the correctness of the results obtained in Figure 4. In particular, the dark curve represents that the contribution to system damping is zero when the WPP is located at the mass-weighted electrical centre, which is consistent with the analysis made by (19).Table 1
Comparison of different installation locations of the WPP in area 1. (KWT = 10, wind energy ratio of 30%).
LocationLocation ALocation BThe mass-weighted electrical centreXl1 (p.u.)0.150.30.41lw00.1670.289Damping ratio0.04530.02010.0015KD13.416.120.47Oscillation frequency (Hz)1.471.511.53Eigenvalue−0.42 ± 9.25j−0.19 ± 9.51j−0.01 ± 9.64jFigure 5
Simulation results of different locations in area 1.Similarly, the oscillation mode of the example system that the WPP is installed in Area 2 is listed in Table2. One may also notice that the WPP placed closer to SG2 has a negative impact on the damping torque coefficient KD. To illustrate this, the time-domain responses for Location C lw=0.333 and Location D lw=0.388 are visualized in Figure 6. It is indicated that, with lw varying from 0.289 to 0.389, the installation location of the WPP is placed closer to SG2 and the risk of oscillatory instability is increasing. In summary, the installation location of the WPP significantly affects the system damping in the concerned oscillation mode. Based on the results, it is clear that the impact of the PMSG auxiliary controller on LFO damping can be positive or negative, depending on the WPP location. The simulation results and the eigenvalue analysis are fully consistent with the damping torque analysis.Table 2
Comparison of different installation locations of the WPP in area 2. (KWT = 10, wind energy ratio of 30%).
LocationThe mass-weighted electrical centreLocation CLocation DXl1 (p.u.)0.410.450.5lw0.2890.3330.389Damping ratio0.0015−0.0052−0.0137KD0.47−1.62−4.254Oscillation frequency (Hz)1.531.541.55Eigenvalue−0.01±9.64j0.05±9.67j0.13±9.71jFigure 6
Simulation results of different locations in area 2.


### 5.2. Impact of the Droop Coefficient

In this section, test results are presented to evaluate the impact of the droop coefficientKWT of the PMSG auxiliary damping control on LFOs. As shown in Figure 7, the eigenvalue trajectory results are depicted when the WPP is located at Location B in the stable region and Location C in the unstable region, respectively. Figure 7(a) shows the eigenvalue trajectories when the WPP is placed at Location B (in Area 1) as the droop coefficient KWT varies from 1 to 100. With the increase of KWT, the dominant mode λ1,2 moves leftward in the plane, and the damping torque coefficient increases from 2.45 to 61.16. However, if the WPP is placed at Location C (in Area 2), the dominant mode λ1,2 lies in the right half of the complex plane, as shown in Figure 7(b). In this case, a higher droop coefficient KWT will deteriorate the modal damping, and thus, the PMSG auxiliary damping control plays a negative role in damping the LFOs. The analytical results derived from equation (26)agree with the results mentioned above.Figure 7
Eigenvalue trajectories asKWT varies. (a) The WPP located at location B. (b) The WPP located at location C.
(a)(b)To better show the effect of the PMSG controller on the LFOs, assuming that a sudden load increase of 10% occurs inPL1 at t = 1 s, the simulation results with different droop coefficients KWT are shown in Figure 8. Figure 8(a) shows the time-domain responses of the power angle deviation between SG1 and SG2 with KWT = 10 and KWT = 20 when the WPP is placed at Location B. In the case of Location B, the damping controller has excellent performance in suppressing the interarea oscillations when a high droop coefficient is applied. For comparison, Figure 8(b) plots the time-domain responses under different values of KWT when the WPP is placed at Location C. As shown in Figure 8(b), the damping controller deteriorates the dynamic stability of the system as KWT increases. In addition, the damping effect of the PMSG auxiliary damping control when the WPP is located at the mass-weighted electrical centre is also fully investigated. As shown in Figure 8(c), the damping of this test basically remains the same when KWT increases from 10 to 20, i.e., the effect of the damping controller is negligible.Figure 8
Simulation results of differentKWT. (a) The WPP located at location B. (b) The WPP located at location C. (c) The WPP located at the mass-weighted electrical centre.
(a)(b)(c)The results mentioned above prove that the oscillation mode is sensitive to the droop coefficient except for the mass-weighted electrical centre. It also demonstrates that the location of the damping controller in Area 1 and Area 2 has different performances on LFO damping, which is consistent with the damping torque analysis in Section4.2.


### 5.3. Impact of the Wind Power Penetration Level

In this section, the impact of the wind penetration was analysed by gradually displacing units of SG1 and SG2 with wind turbine generators. The eigenvalue computational results of Location B are shown in Table 3. It can be seen that, with the wind energy ratio varying from 30% to 50%, the damping ratio of interarea oscillation is increasing from 0.0201 to 0.0281. In addition, the interarea oscillation mode of the example WPP is given in the last row of Table 3, which shows that the mode moves to the left when the wind penetration increases.Table 3
Comparison of different wind energy ratios at location B (KWT = 10 and lw = 0.167).
The wind energy ratio (%)304050Damping ratio0.02010.02330.0281KD6.126.877.93Oscillation frequency (Hz)1.511.471.40Eigenvalue−0.19 ± 9.51−0.21±9.22j−0.25±8.80jFigure9 plots the relationship between the wind penetration and the damping torque coefficient KD when the WPP is located at Location B. As seen in Figure 9, the damping torque coefficient KD increases with the increase of the wind penetration, and KD is more intense in high wind penetration.Figure 9
Damping torque coefficientKD with varying wind energy ratios for different KWT at location B.Table4 shows the results of the interarea oscillation mode when the WPP is installed at Location C. It is shown that the damping ratio of the interarea mode improves with the increase in the wind energy contribution. The results of Figure 10 show that the damping torque coefficient of the studied system improves more with a high droop coefficient. Moreover, as displayed in Figure 10, the damping torque coefficient becomes positive when the wind penetration exceeds 54%. It can be attributed to the change in the operating condition of the system. The results in Figures 9 and 10 also confirm that the droop coefficient plays an important role in the effect of the wind energy ratio on the system damping.Table 4
Comparison of different wind energy ratios at location C (KWT = 10 and lw = 0.333).
The wind energy ratio (%)304050Damping ratio−0.0052−0.0036−0.0012KD−1.62−1.08−0.37Oscillation frequency (Hz)1.541.511.48Eigenvalue0.05±9.67j0.03±9.51j0.01±9.28jFigure 10
Damping torque coefficientKD with varying wind energy ratios for different KWT at location C.To justify the findings in Tables3 and 4, the time‐domain simulations are presented in Figure 11. In Figure 11, the results of different wind energy ratios are compared when the WPP is installed at Location B and Location C, respectively. It can be found that high wind penetration has a beneficial impact on system damping. The simulations indicate that the PMSG auxiliary damping control is always helpful in improving the system damping as wind power penetration increases.Figure 11
Simulation results of different wind energy ratios. (a) The WPP located at location B. (b) The WPP located at location C.
(a)(b)


### 5.4. Impact of the Wind Speed

In practice, the wind speed is variable, and therefore, the PMSG output power also varies. To evaluate the effectiveness of the damping controller of PMSG, two scenarios of a sudden rise and a sudden drop in the wind speed are simulated by a nonlinear simulation. Figure12 shows the rotor angle deviation between SG1 and SG2 for a given step wind speed when the WPP is installed at Location B. The wind speed steps up from 12 to 10 m/s at 1 s as depicted in Figure 12(a). As observed, the damping control is effective in suppressing the LFO with a higher droop. In addition, Figure 12(b) shows the simulation results when the wind speed changed from 12 to 14 m/s. It can be seen that the system is stable (positive damping) in the presence of the damping controller. The results show that the proposed damping controller provides superior performance against wind speed fluctuations as long as the WPP is installed in Area 1.Figure 12
Simulation results of differentKWT at location B. (a) A sudden drop in the wind speed. (b) A sudden rise in the wind speed.
(a)(b)The effects of the proposed damping control are also investigated when the WPP is installed at Location B. As shown in Figure13, the PMSG auxiliary damping control plays a negative role in damping the LFOs. In Figure 13(a), the system becomes unstable with negative damping when the wind speed suddenly drops from 12 to 10 m/s. As observed in Figure 13(b), a higher droop coefficient KWT will deteriorate the modal damping when the WPP is installed in Area 2. Therefore, the LFO mode brings the system to instability under variable wind speed conditions as long as the WPP is installed in Area 2.Figure 13
Simulation results of differentKWT at location C. (a) A sudden drop in the wind speed. (b) A sudden rise in the wind speed.
(a)(b)


### 5.5. Results and Discussion

The damping coefficients derived by the DTA method reveal the impacts of PMSG location, the operation conditions, and the control parameters on the system damping characteristics. In conclusion, the following potentials for further investigation and practical applications of the proposed method are demonstrated.(1)
The location of WPP is an important factor that influences the damping of LFOs. The effect of the proposed damping controller on system damping can be either positive or negative, depending on the location of WPP. It is found that the interarea mode is uncontrollable through power modulation as long as the WPP is installed at the mass-weighted electrical centre between the two areas. As for the two-area system integrated with a WPP, the location area is divided into an area of positive effect and a negative one by the mass-weighted electrical centre. The modulation will be more efficient at damping the LFO when the WPP is located in the positive effect area. On the contrary, the WT controller will deteriorate the LFO damping when the WPP is located in the negative effect area.(2)
Parameters of PMSG’s control systems can be tuned to improve system damping. In this study, the explicit analytical relationships between the parameters of damping control and change in damping torque coefficient are derived. It can be concluded that the contribution of damping from the PMSG can be enhanced by increasing the droop coefficient when the WPP is installed in the area of positive effect. Conversely, a higher droop coefficient will deteriorate the damping level and aggravate the LFO when the WPP is installed in the area of negative effect. It is indicated that the parameters of the PMSG damping control should be adjusted according to the installation location of the WPP.(3)
The dynamic interactions introduced by the PMSG auxiliary damping controller are affected by the level of wind power penetration. It is found that, for the same PMSG controller parameter settings, the variation of wind power penetration will alter the damping of LFO modes in the system. It is attributed to the change in the operating condition of the system. However, the PMSG auxiliary damping control proves to be beneficial to LFO damping as the wind power penetration increases.It is shown that the damping control can provide considerable positive damping torque to effectively damp LFOs in power grids with proper control design. Particularly, using the damping torque analysis, the influences of the location of WPP, the control parameters, and the operation conditions are elaborately studied to provide design guidelines for better damping LFOs.


## 5.1. Impact of the WPP Installation Location

In this case, the impact of the WPP location was analysed by changing the reactance of the interconnection linesXl1 and Xl2 in Figure 1. The gain of the droop loop KWT is set to 10. Figure 4 shows the eigenvalue trajectories with lw (the relative distance from SG1) increasing from 0 to 1, as well as the WPP installation location moving along the transmission line and close to SG2. It can be seen that, with lw increasing from 0 to 1, the dominant mode λ1,2 moves from the left half of the complex plane to the right, and KD varies from 13.41 to −32.76. Furthermore, it is interesting to find that there is a critical stable point in the imaginary axis when lw = 0.29, which is indeed the mass-weighted electrical canter. Thus, it can be concluded that the WPP placed close to SG1 might increase the stability of the studied system, which also verifies the accuracy of the damping torque analysis in Section 4.2.Figure 4
Eigenvalue trajectories withlw increasing from 0 to 1.To illustrate the impact of the WPP installation location in detail, two different cases, including the curves in the right and left halves of the complex plane in Figure4, which correspond to Area 1 and Area 2, are considered, respectively. Table 1 shows the interarea oscillation mode λ1,2 when the WPP is located in Area 1. Apparently, as the WPP is placed closer to the mass-weighted electrical centre within Area 1, the damping ratio is decreasing and approaching zero. It can also be seen that mode λ1,2 dominates the LFO with a frequency of approximately 1.5 Hz. Furthermore, the confirmation of simulations corresponding to Table 1 is presented in Figure 5. Assuming that a sudden load event occurs at Bus 1, the load PL1 increases 0.1 p.u. at t = 1 s. As displayed in Figure 5, the oscillations of Location B lw=0.167 are damped better than those of Location A lw=0, confirming the correctness of the results obtained in Figure 4. In particular, the dark curve represents that the contribution to system damping is zero when the WPP is located at the mass-weighted electrical centre, which is consistent with the analysis made by (19).Table 1
Comparison of different installation locations of the WPP in area 1. (KWT = 10, wind energy ratio of 30%).
LocationLocation ALocation BThe mass-weighted electrical centreXl1 (p.u.)0.150.30.41lw00.1670.289Damping ratio0.04530.02010.0015KD13.416.120.47Oscillation frequency (Hz)1.471.511.53Eigenvalue−0.42 ± 9.25j−0.19 ± 9.51j−0.01 ± 9.64jFigure 5
Simulation results of different locations in area 1.Similarly, the oscillation mode of the example system that the WPP is installed in Area 2 is listed in Table2. One may also notice that the WPP placed closer to SG2 has a negative impact on the damping torque coefficient KD. To illustrate this, the time-domain responses for Location C lw=0.333 and Location D lw=0.388 are visualized in Figure 6. It is indicated that, with lw varying from 0.289 to 0.389, the installation location of the WPP is placed closer to SG2 and the risk of oscillatory instability is increasing. In summary, the installation location of the WPP significantly affects the system damping in the concerned oscillation mode. Based on the results, it is clear that the impact of the PMSG auxiliary controller on LFO damping can be positive or negative, depending on the WPP location. The simulation results and the eigenvalue analysis are fully consistent with the damping torque analysis.Table 2
Comparison of different installation locations of the WPP in area 2. (KWT = 10, wind energy ratio of 30%).
LocationThe mass-weighted electrical centreLocation CLocation DXl1 (p.u.)0.410.450.5lw0.2890.3330.389Damping ratio0.0015−0.0052−0.0137KD0.47−1.62−4.254Oscillation frequency (Hz)1.531.541.55Eigenvalue−0.01±9.64j0.05±9.67j0.13±9.71jFigure 6
Simulation results of different locations in area 2.


## 5.2. Impact of the Droop Coefficient

In this section, test results are presented to evaluate the impact of the droop coefficientKWT of the PMSG auxiliary damping control on LFOs. As shown in Figure 7, the eigenvalue trajectory results are depicted when the WPP is located at Location B in the stable region and Location C in the unstable region, respectively. Figure 7(a) shows the eigenvalue trajectories when the WPP is placed at Location B (in Area 1) as the droop coefficient KWT varies from 1 to 100. With the increase of KWT, the dominant mode λ1,2 moves leftward in the plane, and the damping torque coefficient increases from 2.45 to 61.16. However, if the WPP is placed at Location C (in Area 2), the dominant mode λ1,2 lies in the right half of the complex plane, as shown in Figure 7(b). In this case, a higher droop coefficient KWT will deteriorate the modal damping, and thus, the PMSG auxiliary damping control plays a negative role in damping the LFOs. The analytical results derived from equation (26)agree with the results mentioned above.Figure 7
Eigenvalue trajectories asKWT varies. (a) The WPP located at location B. (b) The WPP located at location C.
(a)(b)To better show the effect of the PMSG controller on the LFOs, assuming that a sudden load increase of 10% occurs inPL1 at t = 1 s, the simulation results with different droop coefficients KWT are shown in Figure 8. Figure 8(a) shows the time-domain responses of the power angle deviation between SG1 and SG2 with KWT = 10 and KWT = 20 when the WPP is placed at Location B. In the case of Location B, the damping controller has excellent performance in suppressing the interarea oscillations when a high droop coefficient is applied. For comparison, Figure 8(b) plots the time-domain responses under different values of KWT when the WPP is placed at Location C. As shown in Figure 8(b), the damping controller deteriorates the dynamic stability of the system as KWT increases. In addition, the damping effect of the PMSG auxiliary damping control when the WPP is located at the mass-weighted electrical centre is also fully investigated. As shown in Figure 8(c), the damping of this test basically remains the same when KWT increases from 10 to 20, i.e., the effect of the damping controller is negligible.Figure 8
Simulation results of differentKWT. (a) The WPP located at location B. (b) The WPP located at location C. (c) The WPP located at the mass-weighted electrical centre.
(a)(b)(c)The results mentioned above prove that the oscillation mode is sensitive to the droop coefficient except for the mass-weighted electrical centre. It also demonstrates that the location of the damping controller in Area 1 and Area 2 has different performances on LFO damping, which is consistent with the damping torque analysis in Section4.2.


## 5.3. Impact of the Wind Power Penetration Level

In this section, the impact of the wind penetration was analysed by gradually displacing units of SG1 and SG2 with wind turbine generators. The eigenvalue computational results of Location B are shown in Table 3. It can be seen that, with the wind energy ratio varying from 30% to 50%, the damping ratio of interarea oscillation is increasing from 0.0201 to 0.0281. In addition, the interarea oscillation mode of the example WPP is given in the last row of Table 3, which shows that the mode moves to the left when the wind penetration increases.Table 3
Comparison of different wind energy ratios at location B (KWT = 10 and lw = 0.167).
The wind energy ratio (%)304050Damping ratio0.02010.02330.0281KD6.126.877.93Oscillation frequency (Hz)1.511.471.40Eigenvalue−0.19 ± 9.51−0.21±9.22j−0.25±8.80jFigure9 plots the relationship between the wind penetration and the damping torque coefficient KD when the WPP is located at Location B. As seen in Figure 9, the damping torque coefficient KD increases with the increase of the wind penetration, and KD is more intense in high wind penetration.Figure 9
Damping torque coefficientKD with varying wind energy ratios for different KWT at location B.Table4 shows the results of the interarea oscillation mode when the WPP is installed at Location C. It is shown that the damping ratio of the interarea mode improves with the increase in the wind energy contribution. The results of Figure 10 show that the damping torque coefficient of the studied system improves more with a high droop coefficient. Moreover, as displayed in Figure 10, the damping torque coefficient becomes positive when the wind penetration exceeds 54%. It can be attributed to the change in the operating condition of the system. The results in Figures 9 and 10 also confirm that the droop coefficient plays an important role in the effect of the wind energy ratio on the system damping.Table 4
Comparison of different wind energy ratios at location C (KWT = 10 and lw = 0.333).
The wind energy ratio (%)304050Damping ratio−0.0052−0.0036−0.0012KD−1.62−1.08−0.37Oscillation frequency (Hz)1.541.511.48Eigenvalue0.05±9.67j0.03±9.51j0.01±9.28jFigure 10
Damping torque coefficientKD with varying wind energy ratios for different KWT at location C.To justify the findings in Tables3 and 4, the time‐domain simulations are presented in Figure 11. In Figure 11, the results of different wind energy ratios are compared when the WPP is installed at Location B and Location C, respectively. It can be found that high wind penetration has a beneficial impact on system damping. The simulations indicate that the PMSG auxiliary damping control is always helpful in improving the system damping as wind power penetration increases.Figure 11
Simulation results of different wind energy ratios. (a) The WPP located at location B. (b) The WPP located at location C.
(a)(b)


## 5.4. Impact of the Wind Speed

In practice, the wind speed is variable, and therefore, the PMSG output power also varies. To evaluate the effectiveness of the damping controller of PMSG, two scenarios of a sudden rise and a sudden drop in the wind speed are simulated by a nonlinear simulation. Figure12 shows the rotor angle deviation between SG1 and SG2 for a given step wind speed when the WPP is installed at Location B. The wind speed steps up from 12 to 10 m/s at 1 s as depicted in Figure 12(a). As observed, the damping control is effective in suppressing the LFO with a higher droop. In addition, Figure 12(b) shows the simulation results when the wind speed changed from 12 to 14 m/s. It can be seen that the system is stable (positive damping) in the presence of the damping controller. The results show that the proposed damping controller provides superior performance against wind speed fluctuations as long as the WPP is installed in Area 1.Figure 12
Simulation results of differentKWT at location B. (a) A sudden drop in the wind speed. (b) A sudden rise in the wind speed.
(a)(b)The effects of the proposed damping control are also investigated when the WPP is installed at Location B. As shown in Figure13, the PMSG auxiliary damping control plays a negative role in damping the LFOs. In Figure 13(a), the system becomes unstable with negative damping when the wind speed suddenly drops from 12 to 10 m/s. As observed in Figure 13(b), a higher droop coefficient KWT will deteriorate the modal damping when the WPP is installed in Area 2. Therefore, the LFO mode brings the system to instability under variable wind speed conditions as long as the WPP is installed in Area 2.Figure 13
Simulation results of differentKWT at location C. (a) A sudden drop in the wind speed. (b) A sudden rise in the wind speed.
(a)(b)


## 5.5. Results and Discussion

The damping coefficients derived by the DTA method reveal the impacts of PMSG location, the operation conditions, and the control parameters on the system damping characteristics. In conclusion, the following potentials for further investigation and practical applications of the proposed method are demonstrated.(1)
The location of WPP is an important factor that influences the damping of LFOs. The effect of the proposed damping controller on system damping can be either positive or negative, depending on the location of WPP. It is found that the interarea mode is uncontrollable through power modulation as long as the WPP is installed at the mass-weighted electrical centre between the two areas. As for the two-area system integrated with a WPP, the location area is divided into an area of positive effect and a negative one by the mass-weighted electrical centre. The modulation will be more efficient at damping the LFO when the WPP is located in the positive effect area. On the contrary, the WT controller will deteriorate the LFO damping when the WPP is located in the negative effect area.(2)
Parameters of PMSG’s control systems can be tuned to improve system damping. In this study, the explicit analytical relationships between the parameters of damping control and change in damping torque coefficient are derived. It can be concluded that the contribution of damping from the PMSG can be enhanced by increasing the droop coefficient when the WPP is installed in the area of positive effect. Conversely, a higher droop coefficient will deteriorate the damping level and aggravate the LFO when the WPP is installed in the area of negative effect. It is indicated that the parameters of the PMSG damping control should be adjusted according to the installation location of the WPP.(3)
The dynamic interactions introduced by the PMSG auxiliary damping controller are affected by the level of wind power penetration. It is found that, for the same PMSG controller parameter settings, the variation of wind power penetration will alter the damping of LFO modes in the system. It is attributed to the change in the operating condition of the system. However, the PMSG auxiliary damping control proves to be beneficial to LFO damping as the wind power penetration increases.It is shown that the damping control can provide considerable positive damping torque to effectively damp LFOs in power grids with proper control design. Particularly, using the damping torque analysis, the influences of the location of WPP, the control parameters, and the operation conditions are elaborately studied to provide design guidelines for better damping LFOs.


## 6. Conclusions

In this study, the DTA method is utilized to theoretically assess the impact of the WT supplementary damping controller on the LFOs. To enable the PMSG-based WT to provide damping support for interarea LFOs, an auxiliary damping controller is designed where the rotational speed signals are utilized for active power modulation. For analytical investigation, a reduced mathematical model of an interarea system oriented to the small-signal stability analysis is particularly developed in this study. Based on this, the equivalent damping torque coefficient is explicitly derived using the DTA method to examine the coupling effect between WT dynamics and synchronous generators. The effects of the WT control parameters (the droop coefficient) and the system operating condition (the WPP location) on the system damping properties are elaborately assessed, which gives guidelines on the control design for better damping enhancement. Case studies have been conducted to verify the proposed damping controller strategy and the theoretical analysis.It is found that the damping effect of the PMSG controller on the LFOs can be positive or negative depending on the location of WPP. Also, it is interesting to find that the LFO mode is uncontrollable as long as the WPP is installed at the mass-weighted electrical centre between the two areas. Moreover, the parameters of the PMSG damping control can be adjusted to improve the system damping. And it is shown that the damping control is always beneficial in improving the system damping with the increase of wind power penetration. More discussions about interactions introduced by the damping control of the WT (i.e., the reactive power modulation) on LFOs will be fully discussed in our future works.



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*Source: 2391770-2022-11-30.xml*