Robust Nonlinear Control of DFIG-Based Wind Farms for Damping Inter-Area Oscillations of Power Systems

This article presents a perturbation observer–based robust nonlinear damping control (RNDC) scheme for a doubly fed induction generator (DFIG)–based wind farm to damp the inter-area oscillations of multi-area power systems. In the RNDC scheme, the perturbation term is introduced to describe the combined effects of nonlinearities, uncertainties, and disturbances of multi-area power systems with wind farms. The feedback linearization control is realized for reactive power control loops of the DFIG based on state estimation and perturbation estimation derived from a perturbation observer, thus achieving the damping control of the DFIG. The proposed RNDC scheme only needs one measurement signal, and it does not require any model parameters of the power grid and wind farm. Simulation studies are carried out on a two-area power system model connected with the wind farm to validate the control performance of the RNDC scheme under the conditions of three-phase-to-ground faults, parameter variations, and time delays.

1 Introduction

Wind energy is a cost-effective and environmental-friendly resource, and it has become one of the most promising renewable energy resources to solve the fossil energy crisis and environmental pollution problems (Li et al., 2019; Xiahou K. et al., 2020; Moness and Moustafa, 2020; Xiahou et al., 2021; Xiao et al., 2022). The wind power generation technology has developed rapidly over the past decades. Among various wind power generators, the doubly fed induction generator (DFIG)–based wind turbine (Xue et al., 2019; Xiahou et al., 2018a,b; Wei et al., 2022) has the advantages of high energy conversion efficiency, full active and reactive power control capabilities, and small power converter rating of back-to-back converter. Therefore, the DFIG-based wind turbines have been broadly installed around the world for wind power generation systems, and the DFIG-based wind farms play an important role in the control and operation of power systems.

The modern power system is interconnected by different control areas, and the size of the power system is growing. The inter-area low-frequency oscillation is a threat to the stable operation of the power system, and it has been reported by many countries (Yao et al., 2013; Liu et al., 2022, to be published). With the integration of large-scale wind power, the inter-area oscillation issue becomes more and more severe since the intermittency and volatility of wind power may aggravate the effects of inter-area oscillation and even lead to instability of the power system. Recently, researchers have investigated various damping control methods to mitigate the inter-area oscillations and ensure the stable operation of the power system. Authors have adopted the lead–lag compensator (Yao et al., 2011), networked predictive control (Yao et al., 2015), and dynamic programming control (Zhao et al., 2022, to be published) for designing wide-area damping controllers to suppress the inter-area oscillations of the power system. Nevertheless, the damping controller (Yao et al., 2011, 2015; Zhao et al., 2022, to be published) is only applied for synchronous generators and does not consider the wind farms.

Due to the merits of flexible active and reactive power control ability of the DFIG, recent researchers have used the DFIG-based wind farm to damp the inter-area oscillations of power systems. In the study by Hughes et al., (2006), a classical damping controller is designed for the DFIG to enhance the system dynamic response, and the damping control is similar to the power system stabilizer (PSS) for the synchronous generator. In the study by Miao et al., (2009); Fan et al., (2011), a damping control scheme is designed for the rotor-side converter of the DFIG-based wind farm based on the root locus analysis method so as to improve the damping performance of inter-area oscillations. In the study by Singh et al., (2014); Kunjumuhammed et al., (2017), the classical lead–lag compensator is applied to design the damping controller for the DFIG. The linear robust control method (Yogarathinam and Chaudhuri, 2018) and particle swarm optimization method (Huang and Chung, 2012) are used by researchers for the parameter design of damping controllers of the DFIG.

Nevertheless, the abovementioned linear damping controllers (Hughes et al., 2006; Miao et al., 2009; Fan et al., 2011; Huang and Chung, 2012; Singh et al., 2014; Kunjumuhammed et al.,2017; Yogarathinam and Chaudhuri, 2018) are designed based on the linearized model of the DFIG and power system with respect to one fixed operation point; thus, these damping controllers are weak to model uncertainties and operation point uncertainties. Nonlinear intelligent control methods, such as data-driven, model-free adaptive control (Shi et al., 2020) and adaptive dynamic programming control (Mir and Senroy, 2020), have been applied to design the damping controller for the DFIG, while these controllers are complex and cannot ensure the closed-loop stability of the system. Nonlinear sliding-mode control (SMC)–based damping control (Liao et al., 2016, 2017) has been proposed for the DFIG-based wind farm, and it shows better damping performance than the conventional linear controller. However, the SMC (Liao et al., 2016, 2017) requires many model parameters and measurement signals, which is difficult to be implemented in practice.

This study deals with the aforementioned problems and proposes a perturbation observer–based robust nonlinear damping control (RNDC) scheme for the DFIG-based wind farms to damp the inter-area oscillations of the power system. The nonlinear model of the multi-area power system integrated with the DFIG-based wind farm is built. A perturbation observer is designed based on the input–output linearization model of the nonlinear power system to simultaneously estimate the system state and perturbation state. The nonlinear power system model with the wind farm is regulated by the feedback linearization control based on the state and perturbation estimations provided by the perturbation observer. Simulation studies are carried out on a two-area power system to demonstrate the effectiveness of the proposed RNDC scheme.

This article is organized as follows. Section 2 depicts the nonlinear model of the two-area power system with the DFIG-based wind farm. Section 3 illustrates the proposed perturbation observer–based robust nonlinear damping control scheme. Simulation results are given in Section 4, and conclusions are drawn in Section 5.

2 Two-Area Power System Model With Wind Farm

The configuration of the studied two-area power system is shown in Figure 1, in which Area 1 and Area 2 power systems are connected via transmission lines, and a wind farm is connected to the Area 1 power system. The wind farm is represented by an equivalent aggregated DFIG-based wind turbine model, which consists of a rotor-side converter (RSC) and grid-side converter (GSC). A damp controller is installed on the DFIG-based wind farm to mitigate the inter-area oscillations between Area 1 and Area 2 power systems.

FIGURE 1

FIGURE 1. Two-area power system integrated with the DFIG-based wind farm.

2.1 Wind Farm Model

The DFIG model (Xiahou and Wu, 2018; Xiahou K. S. et al., 2020) can be expressed in the synchronous rotating d-q reference frame. The stator flux linkages of the DFIG are given as

${\psi }_{\mathrm{s}\mathrm{d}}=L_{\text{s}}{I}_{\mathrm{s}\mathrm{d}}+L_{\text{m}}{I}_{\mathrm{r}\mathrm{d}}.(1)$

${\psi }_{\mathrm{s}\mathrm{q}}=L_{\text{s}}{I}_{\mathrm{s}\mathrm{q}}+L_{\text{m}}{I}_{\mathrm{r}\mathrm{q}}.(2)$

and the rotor flux linkages of the DFIG are represented by

${\psi }_{\mathrm{r}\mathrm{q}}=L_{\text{r}}{I}_{\mathrm{r}\mathrm{q}}+L_{\text{m}}{I}_{\mathrm{s}\mathrm{q}},(3)$

${\psi }_{\mathrm{r}\mathrm{q}}=L_{\text{r}}{I}_{\mathrm{r}\mathrm{q}}+L_{\text{m}}{I}_{\mathrm{s}\mathrm{q}},(4)$

where ψ_sd and ψ_sq are the d-axis and q-axis components of the stator flux, respectively. ψ_rd and ψ_rq are the d-axis and q-axis components of rotor fluxes, respectively. I_sd and I_sq are d-axis and q-axis components of stator currents, respectively. I_rd and I_rq are the d-axis and q-axis components of rotor currents, respectively. L_ls, L_lr, and L_m are the stator leakage inductance, rotor leakage inductance, and mutual inductance, respectively. L_s = L_ls + L_m is the stator self-inductance, and L_r = L_lr + L_m is the rotor self-inductance.

The dynamics of stator flux linkages of the DFIG are expressed by

${\dot{\lambda }}_{\mathrm{s}\mathrm{d}}-{\omega }_{\text{s}}{\lambda }_{\mathrm{s}\mathrm{q}}+R_{\text{s}}{I}_{\mathrm{s}\mathrm{d}}-V_{\mathrm{s}\mathrm{d}}=0.(5)$

${\dot{\lambda }}_{\mathrm{s}\mathrm{q}}+{\omega }_{\text{s}}{\lambda }_{\mathrm{s}\mathrm{d}}+R_{\text{s}}{I}_{\mathrm{s}\mathrm{q}}-V_{\mathrm{s}\mathrm{q}}=0.(6)$

and the dynamics of rotor flux linkages are given by

${\dot{\lambda }}_{\mathrm{r}\mathrm{d}}-{\omega }_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}}{\lambda }_{\mathrm{r}\mathrm{q}}+R_{\text{r}}{I}_{\mathrm{r}\mathrm{d}}-V_{\mathrm{r}\mathrm{d}}=0.(7)$

${\dot{\lambda }}_{\mathrm{r}\mathrm{q}}+{\omega }_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}}{\lambda }_{\mathrm{r}\mathrm{d}}+R_{\text{r}}{I}_{\mathrm{r}\mathrm{q}}-V_{\mathrm{r}\mathrm{q}}=0,(8)$

where ω_s, ω_r, and ω_slip = ω_s − ω_r are the synchronous speed, rotor speed, and slip speed, respectively. V_sd and V_sq are the d-axis and q-axis components of stator voltages, respectively; V_rd and V_rq are the d-axis and q-axis components of rotor voltages, respectively; and R_s and R_r are the stator and rotor resistances, respectively.

The stator active and reactive powers of the DFIG are calculated as

$P_{\mathrm{s}}=-1.5\left(V_{\mathrm{s}\mathrm{d}}{I}_{\mathrm{s}\mathrm{d}}+V_{\mathrm{s}\mathrm{q}}{I}_{\mathrm{s}\mathrm{q}}\right),(9)$

$Q_{\mathrm{s}}=-1.5\left(V_{\mathrm{s}\mathrm{q}}{I}_{\mathrm{s}\mathrm{d}}-V_{\mathrm{s}\mathrm{d}}{I}_{\mathrm{s}\mathrm{q}}\right),(10)$

where P_s is the stator active power and Q_s is the stator reactive power. The speed motion equation of the DFIG can be represented as

$2H_{\text{w}}{\dot{\omega }}_{\mathrm{r}}=T_{\text{e}}-T_{\text{m}}-D_{\text{w}}{\omega }_{\text{r}},(11)$

where T_m is the mechanical torque, H_w is the inertia constant, D_w is the damping factor, and T_e is the electromagnetic torque given by

$T_{\text{e}}=1.5L_{\text{m}}\left(I_{\mathrm{s}\mathrm{q}}{I}_{\mathrm{r}\mathrm{d}}-I_{\mathrm{s}\mathrm{d}}{I}_{\mathrm{r}\mathrm{q}}\right).(12)$

2.2 Power System Model

The two-area power system model integrated with the wind farm illustrated in Figure 1 is considered, which can be expressed by the swing equation (Kundur, 1994; Tobergte and Curtis, 2013; Xiahou et al., 2022)

$\left\{\begin{array}{cc}\hfill {\dot{\delta }}_{12}=& {\omega }_{12}\hfill \\ \hfill {\dot{\omega }}_{12}=& \frac{P_{\text{m}1}}{H_1}-\frac{P_{\text{L}1}}{H_1}-\frac{P_{\text{m}2}}{H_2}+\frac{P_{\text{L}2}}{H_2}\hfill \\ \hfill & -\frac{V_1}{H_1{V}_2}\sin {\delta }_{12}-\frac{V_1}{H_2{V}_2}\sin {\delta }_{12}\hfill \end{array}\right.(13)$

where δ₁₂ = δ₁ − δ₂ is the rotor angle deviation between Area 1 and Area 2 and δ₁ and δ₂ are the rotor angles of Area 1 and Area 2, respectively. ω₁₂ = ω₁ − ω₂ is the generator speed deviation between Area 1 and Area 2, and ω₁ and ω₂ are the generator speed of Area 1 and Area 2, respectively. H₁ and H₂ are the equivalent inertia of Area 1 and Area 2, respectively. P_m1 and P_m2 are the mechanical power of Area 1 and Area 2, respectively. P_L1 and P_L2 are the load power of Area 1 and Area 2, respectively. V₁ and V₂ are the terminal voltage of Area 1 and Area 2, respectively.

The DFIG-based wind farm is connected to Area 1; thus, the two-area power system model with the wind farm becomes

$\begin{array}{cc}\hfill {\dot{\omega }}_{12}=& \frac{P_{\text{m}1}}{H_1}+\frac{P_{\text{w}}}{H_1}-\frac{P_{\text{L}1}}{H_1}-\frac{P_{\text{m}2}}{H_2}+\frac{P_{\text{L}2}}{H_2}\hfill \\ \hfill & -\frac{V_1}{H_1{V}_2}\sin {\delta }_{12}-\frac{V_1}{H_2{V}_2}\sin {\delta }_{12}\hfill \end{array},(14)$

where P_w is the active power of the wind farm. A supplementary reactive power controller is adopted to damp the inter-area oscillations of the power system. The reactive power Q₁ of Area 1 can be expressed as

$\left\{\begin{array}{cc}\hfill Q_1=& Q_{\text{w}}+Q_{\text{c}}\hfill \\ \hfill Q_1=& \frac{V_2^2-V_1{V}_2\cos {\delta }_{12}}{X}\hfill \end{array}\right.(15)$

where Q_w is the reactive power of the wind farm, Q_c is the reactive power of the capacitive compensator and synchronous generator, and X is the impedance of the transmission line.

3 Perturbation Observer–Based Robust Nonlinear Damping Control

3.1 Nonlinear System Model for Damping Control

Taking the power angle δ₂ = 0 as the reference value, it has δ₁₂ = δ₁. Thus, the two-area power system model becomes

$\left\{\begin{array}{cc}\hfill {\dot{\delta }}_1=& {\omega }_{12}\hfill \\ \hfill {\dot{\omega }}_{12}=& \frac{P_{\text{m}1}}{H_1}+\frac{P_{\text{w}}}{H_1}-\frac{P_{\text{L}1}}{H_1}-\frac{P_{\text{m}2}}{H_2}+\frac{P_{\text{L}2}}{H_2}\hfill \\ \hfill & -\frac{V_1}{H_1{V}_2}\sin {\delta }_1-\frac{V_1}{H_2{V}_2}\sin {\delta }_1\hfill \end{array}\right.(16)$

From (15), the terminal voltage V₁ of the Area 1 power system can be expressed as

$V_1=\frac{V_2^2-Q_{\text{w}}X-Q_{\text{c}}X}{V_2\cos {\delta }_1}.(17)$

Replacing V₁ with (17) in (16), it has

$\left\{\begin{array}{cc}\hfill {\dot{\delta }}_1=& {\omega }_{12}\hfill \\ \hfill {\dot{\omega }}_{12}=& \frac{P_{\text{m}1}}{H_1}+\frac{P_{\text{w}}}{H_1}-\frac{P_{\text{L}1}}{H_1}-\frac{P_{\text{m}2}}{H_2}+\frac{P_{\text{L}2}}{H_2}\hfill \\ \hfill & -\frac{\left(V_2^2-Q_{\text{w}}X-Q_{\text{c}}X\right)\sin {\delta }_1}{H_1{V}_2^2\cos {\delta }_1}\hfill \\ \hfill & -\frac{\left(V_2^2-Q_{\text{w}}X-Q_{\text{c}}X\right)\sin {\delta }_1}{H_2{V}_2^2\cos {\delta }_1}.\hfill \end{array}\right.(18)$

The reactive power control loop of the DFIG can be approximated by a first-order model as

$Q_{\text{w}}=\frac{Q_{\text{w}}^{*}}{T_{\text{w}}s+1},(19)$

where $Q_{\text{w}}^{*}$ is the reference value of the reactive power Q_w generated from the DFIG-based wind farm. Thus, the damping control model of reactive power loops can be expressed as

$\mathrm{\Delta }{Q}_{\text{w}}=\frac{\mathrm{\Delta }{Q}_{\text{w}}^{*}}{T_{\text{w}}s+1},(20)$

where ΔQ_w is the reactive power output caused by the damping controller and $\mathrm{\Delta }{Q}_{\text{w}}^{*}$ is the damping control input. Based on the abovementioned analysis, the whole system dynamic equation can be written as

$\left\{\begin{array}{cc}\hfill {\dot{\delta }}_1=& {\omega }_{12}\hfill \\ \hfill {\dot{\omega }}_{12}=& \frac{P_{\text{m}1}}{H_1}+\frac{P_{\text{w}}}{H_1}-\frac{P_{\text{L}1}}{H_1}-\frac{P_{\text{m}2}}{H_2}+\frac{P_{\text{L}2}}{H_2}\hfill \\ \hfill & -\frac{\left(V_2^2-\mathrm{\Delta }{Q}_{\text{w}}-Q_{\text{w}}X-Q_{\text{c}}X\right)\sin {\delta }_1}{H_1{V}_2^2\cos {\delta }_1}\hfill \\ \hfill & -\frac{\left(V_2^2-\mathrm{\Delta }{Q}_{\text{w}}-Q_{\text{w}}X-Q_{\text{c}}X\right)\sin {\delta }_1}{H_2{V}_2^2\cos {\delta }_1}\hfill \\ \hfill \mathrm{\Delta }{\dot{Q}}_{\text{w}}=& \frac{u-\mathrm{\Delta }{Q}_{\text{w}}}{T_{\text{w}}}\hfill \end{array}\right.(21)$

where $u=\mathrm{\Delta }{Q}_{\text{w}}^{*}$ is the control input of the damping controller.

3.2 Controller Design

Taking $x={[x_1,x_2,x_3]}^{\top }={[{\delta }_1,{\omega }_{12},\mathrm{\Delta }{Q}_{\text{w}}]}^{\top }$ as the state vector, the power system model (21) can be expressed as the following nonlinear system model

$\dot{x}=f\left(x\right)+g\left(x\right)u,(22)$

where $g(x)={[\mathrm{0,0,1}/T_{\text{w}}]}^{\top }$, $f(x)=\left[f_1(x),f_2(x)\right.$, ${\left.f_3(x)\right]}^{\top }$, f₁(x) = x₂, $f_3(x)=-T_{\text{w}}^{-1}{x}_3$, and

$\begin{array}{cc}\hfill f_2\left(x\right)& =\frac{P_{\text{m}1}}{H_1}+\frac{P_{\text{w}}}{H_1}-\frac{P_{\text{L}1}}{H_1}-\frac{P_{\text{m}2}}{H_2}+\frac{P_{\text{L}2}}{H_2}\hfill \\ \hfill & -\left(\frac{1}{H_1}+\frac{1}{H_2}\right)\frac{V_2^2-x_3-Q_{\text{w}}X-Q_{\text{c}}X}{V_2^2}\frac{\sin x_1}{\cos x_1}.\hfill \end{array}(23)$

Taking the control target y = x₂ = ω₁₂ as the output signal, the measurement equation is given as

$y=h\left(x\right)=x_2.(24)$

The input–output linearization method is applied for the single-input-single-output (SISO) nonlinear system model (22)–(24). Differentiating the output variable y until the control signal u explicitly appears, which can be represented by

$\ddot{y}={\mathcal{L}}_f^{2}h\left(x\right)+{\mathcal{L}}_g{\mathcal{L}}_{f}h\left(x\right)\cdot u,(25)$

where the Lie derivations ${\mathcal{L}}_f^{2}h(x)$, ${\mathcal{L}}_g{\mathcal{L}}_{f}h(x)$ are calculated as

$\begin{array}{cc}\hfill {\mathcal{L}}_f^{2}h\left(x\right)& =-\left(\frac{1}{H_1}+\frac{1}{H_2}\right)\frac{\left(V_2^2-x_3-Q_{\text{w}}X-Q_{\text{c}}X\right)x_2}{V_2^2{\cos }^2{x}_1}\hfill \\ \hfill & -\left(\frac{1}{H_1}+\frac{1}{H_2}\right)\frac{x_3\sin x_1}{V_2^2{T}_{\text{w}}\cos x_1}\hfill \end{array}.(26)$

${\mathcal{L}}_g{\mathcal{L}}_{f}h\left(x\right)=\left(\frac{1}{H_1}+\frac{1}{H_2}\right)\frac{\sin x_1}{V_2^2{T}_{\text{w}}\cos x_1}.(27)$

Since the Lie derivative ${\mathcal{L}}_g{\mathcal{L}}_{f}h(x)\ne 0$, the relative degree of y is obtained as r = 2.

The perturbation term ψ of the damping control system is introduced as

$\psi \left(x,u,t\right)={\mathcal{L}}_f^{2}h\left(x\right)+\left({\mathcal{L}}_g{\mathcal{L}}_{f}h\left(x\right)-b_0\right)u(28)$

to represent the combined effects of nonlinearities, uncertainties, and external disturbances of the system, where the constant parameter b₀ is to be tuned. Based on the definition of the perturbation term in Eq. 28, the system model (25) can be transformed as

$\ddot{y}=\psi \left(x,u,t\right)+b_{0}u.(29)$

The new state variable $\xi ={[{\xi }_1,{\xi }_2]}^{\top }$ is defined to represent the system state of Eq. 29, where ξ₁ = y and ${\xi }_2=\dot{y}$. Meanwhile, the extended state is defined as ξ₃ = ψ to represent the perturbation term. Therefore, the nonlinear system model (29) can be represented with the third-order state-space form as

$\mathrm{\Theta }:\left\{\begin{array}{c}{\dot{\xi }}_1=A\xi +B\left({\xi }_3+b_{0}u\right)\hfill \\ {\dot{\xi }}_3=\dot{\psi }\left(\cdot \right)\hfill \\ y=C\xi \hfill \end{array}\right.(30)$

with

$\begin{array}{c}A=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],B=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right],C=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \end{array}\right].\hfill \end{array}(31)$

With the feedback linearization control law

$u=\frac{-{\xi }_3-K\xi }{b_0},(32)$

the nonlinear system Θ can be linearized exactly into the following linear system

$\dot{\xi }=\left(A-BK\right)\xi ,(33)$

where K is the feedback gain matrix which is to be designed.

However, the perturbation state ξ₃ of control law (32) is unknown, which is almost impossible to be implemented. Thus, the perturbation observer is designed to estimate the system perturbation state. Based on Eq. 30, a perturbation observer $\widehat{\mathrm{\Theta }}$ is designed for the nonlinear system Θ as follows

$\widehat{\mathrm{\Theta }}:\left\{\begin{array}{c}\dot{\widehat{\xi }}=A\widehat{\xi }+B\left({\widehat{\xi }}_3+b_{0}u\right)+L\left(y-C\widehat{\xi }\right)\hfill \\ {\dot{\widehat{\xi }}}_3=l_3\left(y-C\widehat{\xi }\right)\hfill \end{array}\right.,(34)$

where $\widehat{\xi }={[{\widehat{\xi }}_1,{\widehat{\xi }}_2]}^{\top }$ is the estimated value of the system state ξ, ${\widehat{\xi }}_3$ is the estimated value of the perturbation state ξ₃, $L={[l_1,l_2]}^{\top }$, l₃ are observer gains, which are selected as l_j = α_jɛ^−j (j = 1, 2, 3), where α_j is chosen such that all the roots of s³ + α₁s² + α₂s + α₃ have negative real parts, and ɛ ∈ (0, 1] is a small gain parameter.

Based on the system state estimation $\widehat{\xi }$ and perturbation state estimation ${\widehat{\xi }}_3$ derived from the perturbation observer, the damping control law of the DFIG-based wind farm is given as

$u=\frac{-{\widehat{\xi }}_3-K\widehat{\xi }}{b_0},(35)$

where the perturbation estimation ${\widehat{\xi }}_3$ is adopted to compensate the effects of the perturbation term and the system state estimation $\widehat{\xi }$ is used for the output feedback control. The classical linear quadratic regulator (LQR) method is utilized to design the optimal feedback gain matrix K since the relative degree of y is equal to 2, which is less than the nonlinear system order of 3. Thus, the internal dynamics can be obtained as ${\dot{x}}_1=x_2=y={\xi }_1$. When ξ₁ = ξ₂ = 0, the zero dynamics of the nonlinear system is given as ${\dot{x}}_1=0$, which indicates the internal stability of the nonlinear system model.

Based on the abovementioned analysis, the block diagram of the proposed perturbation observer–based robust nonlinear damping control scheme is illustrated in Figure 2. As seen, the PI-based dual-loop vector control scheme is adopted to control the RSC of the DFIG-based wind farm. The rotor currents I_rd, I_rq of the DFIG is regulated to their reference values $I_{\text{rd}}^{*},I_{\text{rq}}^{*}$ with the application of PI current controllers with compensation terms Δ_rd, Δ_rq. Meanwhile, the active power P_s and reactive power Q_s are, respectively, regulated to their reference values $P_{\text{s}}^{*}$ and $Q_{\text{s}}^{*}$ by the PI power controllers. Taking the speed deviation as the measured signal y = ω₁₂, the perturbation observer (34) is implemented based on the input–output linearization model to simultaneously estimate the state and perturbation of the system. Based on Eq. 35, the state estimation and perturbation estimation of the perturbation observer are used for output feedback control and perturbation compensation, respectively, thus realizing the robust nonlinear damping control of the DFIG-based wind farm. It is noted that the proposed RNDC scheme only requires the measurement signal of speed deviation, and it does not need any model parameters.

FIGURE 2

FIGURE 2. Schematic of the proposed perturbation observer-based RNDC scheme.

4 Simulation Studies

Simulation studies are undertaken on the detailed two-area four-machine power system with the wind farm illustrated in Figure 3 built in Simulink/Sim Power Systems.

FIGURE 3

FIGURE 3. Configuration of the simulated power system.

The detailed parameters of the two-area four-machine power system are given in the study by Kundur, (1994). The rated power of Area 1 and Area 2 power is set as 1800 MW;, and the rated power of the DFIG-based wind farm is set as 180 MW; thus, the penetration level of the wind power of Area 1 reached 10%. The parameters of the DFIG-based wind farm are given as follows: rated voltage V_nom = 690 V, rated frequency f_nom = 60 Hz, rated power P_nom = 2 MW, rated wind speed V_wnom = 11 m/s, stator leakage inductance L_ls = 0.18 p. u., rotor leakage inductance L_lr = 0.16 p. u., mutual inductance L_m = 2.9 p. u., stator resistance R_s = 0.023 p. u., and rotor resistance R_r = 0.016 p. u. The proposed RNDC scheme is compared with the classical damping control (CDC) (Miao et al., 2009) and sliding-mode control (SMC) (Liao et al., 2016) schemes under the conditions of three-phase-to-ground faults, parameter variations, and time delays.

4.1 Three-phase-to-ground Fault

In the first case, a three-phase-to-ground short-circuit fault is caused in the middle between Bus 7 and Bus 8 at t = 0.5 s, which means the fault location is 110 km away from the Area 1 power system. The simulation results of power angle deviation δ₁₂, speed deviation ω₁₂, active power P₁ of Area 1, active power P₂ of Area 2, active power P_w of wind farm, reactive power Q_w of wind farm, rotor speed ω_r of wind farm, and DC-link voltage V_dc of wind farm are given in Figure 4.

FIGURE 4

FIGURE 4. Simulation results obtained in the case where a three-phase-to-ground fault occurs in the middle of the transmission line.

As seen, after the three-phase-to-ground fault happens, the responses δ₁₂ and ω₁₂ of the two-area power system present low-frequency oscillations. All the three damping control schemes of the CDC, SMC, and RNDC can damp the inter-area oscillations, while the proposed RNDC scheme shows better damping performance than both the CDC and SMC. Meanwhile, the RNDC scheme adjusts the reactive power Q_w of the wind farm to damp the oscillations, and the responses of active power P_w, reactive power Q_w, rotor speed ω_r, and DC-link voltage V_dc of the wind farm can quickly reach the stable states.

In addition, a three-phase-to-ground short-circuit fault is caused 20 km away from the Area 1 power system, and the simulation results are shown in Figure 5. Since the fault location is close to Bus 7, the responses of power angle deviation δ₁₂, speed deviation ω₁₂, active power P₁ of Area 1, and active power P₂ of Area 2 present much larger oscillations than the simulation results in Figure 4. According to the simulation results, the proposed RNDC scheme still shows better damping performance than both the CDC and SMC schemes, which verify the effectiveness of the RNDC scheme.

FIGURE 5

FIGURE 5. Simulation results obtained in the case where a three-phase-to-ground fault occurs close to the Area 1 power system.

4.2 Parameter Variations

The performance of the proposed RNDC scheme is further compared with that of the SMC scheme under the condition of parameter variations. In this case, the equivalent inertia H₂ of Area 2 and the impedance X of transmission line are varied with +20% and −20% errors, respectively. Meanwhile, a three-phase-to-ground short-circuit fault is caused at the middle between Bus 7 and Bus 8 at t = 0.5 s, and the simulation results of the SMC and RNDC are illustrated in Figures 6A,B, respectively. As can be seen from the responses of δ₁₂, ω₁₂, P₁, and P₂, the parameter variations deteriorate the control performance of the SMC. This is due to the fact that the implementation of the SMC requires many model parameters of Area 1 and Area 2 power systems. Nevertheless, the implementation of the RNDC does not require any model parameters. The simulation results of the RNDC obtained with and without parameter variations are almost the same, which demonstrates the stronger robustness of the RNDC against parameter uncertainties than the SMC.

FIGURE 6

FIGURE 6. Simulation results of the (A) SMC and (B) RNDC obtained under parameter variations.

4.3 Time Delays

In this case, the proposed RNDC scheme is tested under the condition of time delay. A time delay of 50 ms is added to the measurement signal derived from the Area 2 power system, a three-phase-to-ground short-circuit fault is caused at the middle between Bus 7 and Bus 8 at t = 0.5 s, and the simulation results are shown in Figure 7.

FIGURE 7

FIGURE 7. Simulation results obtained under the condition of time delay.

As observed from the responses of δ₁₂, ω₁₂, P₁, and P₂, the time delay of the measurement signal degrades the performance of the CDC, SMC, and RNDC schemes. Since the SMC scheme requires many measurement signals from Area 2 power systems, the SMC scheme presents large fluctuations, and the power systems is almost unstable. Both the CDC and RNDC can still damp the inter-area oscillations, and the RNDC shows much better control performance than the CDC, which reveals the strong adaption ability of the RNDC to time delays.

5 Conclusion

This study has proposed an RNDC scheme for DFIG-based wind farms to damp the inter-area oscillations of power systems. The RNDC scheme is implemented based on the perturbation observer and feedback linearization control method. Simulation studies are undertaken on a two-area power system model with 10% level penetration of the wind farm. According to the simulation results obtained under the conditions of three-phase-to-ground short-circuit faults, parameter variations, and time delays, conclusions can be drawn as follows.

First, the proposed RNDC scheme can damp the inter-area oscillations of power systems caused by three-phase-to-ground short-circuit faults, and it shows better damping performance than both the CDC scheme and SMC scheme. Meanwhile, the proposed RNDC scheme presents stronger robustness against parameter variations and better adaption ability to time delays. This is due to the fact that the RNDC scheme only requires one measurement signal and does not need any model parameters. In addition, the accurate perturbation estimation derived from the perturbation observer is applied to compensate the unknown dynamics of nonlinear systems in the RNDC scheme. Finally, with the perturbation compensation and output feedback control, the RNDC can realize the exact feedback linearization control of DFIG-based wind farms and effectively damp the inter-area oscillations of multi-area power systems.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author Contributions

BL: Investigation, methodology, software, and writing—original draft. SY: Supervision, data curation, and writing—review and editing. BY: Visualization and writing—review and editing. KF: Writing—review and editing.

Funding

This work was supported by the project of State Grid Jiangsu Electric Power Co., Ltd. (research and economic analysis of cluster control strategy for multi-client adjustable loads, No. J2021152).

Conflict of Interest

Authors BL, SY, BY, and KF were employed by Marketing Service Center, State Grid Jiangsu Electric Power Co., Ltd.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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