Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

The strong coupling, multivariate, and nonlinear characteristics of the mathematical model of a doubly‐fed induction generator (DFIG) make it challenging to analyze the bifurcation category of the DFIG. Therefore, in this study, we developed a method for analyzing the DFIG model via topologically equivalent dimensionality reduction in the neighborhood of bifurcation values based on the central manifold theorem and realized the discrimination of the Hopf bifurcation type of the DFIG. We established a complex frequency domain model of the DFIG and then calculated the critical open‐loop gain point using the root trajectory method to obtain the relationship between the critical open‐loop gain point and Hopf bifurcation. Thereafter, we performed topological equivalence dimensionality reduction in the neighborhood of the DFIG bifurcation values based on the central manifold theorem. Subsequently, the Hopf bifurcation type of the system was determined using the stability index (first Lyapunov exponent) of the Hopf bifurcation. Finally, a nonlinear time‐domain simulation was used to verify that the four and two‐dimensional DFIGs generate a stable limit cycle by supercritical Hopf bifurcation under the tuning parameter. The results indicated that the proposed method was feasible and effective in the neighborhood of bifurcation values in the state parameter space of DFIG, which provides a theoretical basis for the control of Hopf bifurcation of the DFIG.

dynamic characteristic analysis. The linear analysis method is divided into eigenvalue analysis, 3,4 the Routh-Hurwitz method, 5 and the oscillation energy method. 6 However, these methods face the "dimensional disaster" when analyzing high-dimensional systems, which makes the stability analysis of highdimensional systems difficult, and they cannot reflect the nonlinear dynamic characteristics of the DFIG. Therefore, it is necessary to study dimensionality reduction methods for DFIG systems to retain nonlinear terms.
Dimensionality reduction methods for nonlinear dynamical systems can be classified into two types: linear and nonlinear. As linear dimensionality reduction methods cannot preserve higher-dimensional modes and nonlinear terms, nonlinear dimensionality reduction methods must be utilized for DFIG models with strong nonlinearity. Currently, nonlinear dimensionality reduction methods are classified into four types: the central manifold method, Lyapunov-Schmidt method, the nonlinear Galerkin method, and the proper orthogonal decomposition method. 7 Among them, the central manifold decomposition method has attracted considerable attention owing to its advantages of strict mathematical theory and requiring little computational effort. The central manifold dimensionality reduction method has been used in fold bifurcation and the voltage collapse phenomenon of a simple power system caused by fold bifurcation was observed. 8 Furthermore, Yu and Gumel 9 studied the bifurcation category and stability of a coupled Brusselator model in the context of the central manifold theorem, and the bifurcation types of the model with different bifurcation values were confirmed theoretically.
Hopf bifurcation for a high-dimensional nonlinear DFIG can be classified as non-smooth or smooth bifurcation. Regarding the bifurcation characteristics of the non-smooth DFIG model containing dead zones and limits in the control link, the causes of switching-type chaotic oscillations were analyzed, and non-smooth bifurcation in the control system of a non-smooth DFIG due to the limiting and dead zones was investigated to reveal the influence of the control system on the system stability from the perspective of non-smooth bifurcation. 10,11 In contrast to non-smooth bifurcation, the smooth Hopf bifurcation phenomenon has been widely demonstrated in power systems. [12][13][14] Regarding the type of Hopf bifurcation, a nonlinear time-domain simulation method was used to determine the Hopf bifurcation category of a hydroelectric DC islanding system. 15 The stable limit ring formed by the supercritical Hopf occurred in the Lorenz system theoretically using the first Lyapunov coefficient. 16 However, few studies have been conducted on the Hopf bifurcation type of highdimensional nonlinear DFIGs, which has been proven theoretically.
In this study, a method for obtaining the reducedorder model of the DFIG based on the central manifold theory was used as an example of the fourth-order nonlinear dynamic model of the DFIG. In addition, a theoretical proof was developed for the Hopf bifurcation type of the DFIG using the first Lyapunov coefficient. First, a DFIG complex frequency domain model was established, and the relationship between the critical open-loop gain of the system and Hopf bifurcation was obtained using the root trajectory. Subsequently, the reduced-order model in the neighborhood of the bifurcation was obtained using the central manifold theorem. Finally, the results of the nonlinear time-domain and roottrajectory methods were compared. The results indicated that the DFIG system exhibits supercritical Hopf bifurcation at the critical open-loop gain and forms a stable limit loop. The central manifold theorem can well preserve the nonlinear characteristics of the system in the Hopf bifurcation neighborhood, providing the theoretical basis for the subsequent Hopf bifurcation control.
The remainder of this paper is organized as follows. In Section 1, a fourth-order nonlinear DFIG dynamic model is developed. In Section 2, the DFIG complex frequency-domain model is established. In Section 3, the reduced-dimension model for the DFIG bifurcation neighborhood is obtained, which is based on the central manifold theorem. Section 4 presents the numerical simulations, in which the DFIG model converged to the stability limit cycle under different initial values.

| DYNAMIC MATHEMATICAL MODEL OF DFIG
DFIGs are gradually becoming mainstream generators in wind power systems, contrary to direct-drive generators, 17 which enable energy exchange between the stator and rotor sides in the grid. 18 To facilitate the research, the dynamic relations of rotor-side and stator-side control links are ignored in this study, and the doubly-fed asynchronous power generation system is selected and described as follows, 19 as shown in Figure 1.
Here, u sd and u sq represent the d-and q-axis components of the stator voltage, respectively; i sd and i sq represent the d-and q-axis components of the stator current, respectively. u rd and u rq represent the d-and q-axis components of the rotor voltage, respectively; i rd and i rq represent the d-and q-axis components of the rotor current, respectively. r s and r r represent the stator and rotor resistances, respectively; and ω ω ω = − s n r ω s represent the relative speed of the d-q coordinate system with respect to the rotor.

| Magnetic chain equation
Here, ψ sd and ψ rd represent the d-axis components of the stator and rotor magnetic chains, respectively; ψ sq and ψ rq represent the q-axis components of the stator and rotor magnetic chains, respectively. L m , L s , and L r represent the equivalent mutual inductances between the stator and rotor in the coaxial phase, as well as the equivalent self-inductances of the stator and rotor in the two phases, respectively.

| Rotor equations of motion and control equations
F I G U R E 1 Schematic diagram of a simplified grid-connected system of doubly-fed induction generator.
Here, J g represents the generator rotational inertia, D g is the torque damping factor, p n represents the number of pole pairs of the asynchronous motor, and T w represents the mechanical torque. In the speed control system of a doubly-fed wind turbine, indirect vector control does not require the calculation of the amplitude and phase of the magnetic chain, and the position of the magnetic flux with respect to the stator is estimated by summing the differential frequency and the measured speed, which can overcome the shortcomings of vector control. The integral form of position regulation is converted into a differential form. This control strategy can be described simply using Equation (4).
Here, ĉ 1 is the estimated time constant of the rotor, and c 1 is the real rotor time constant of the doubly-fed asynchronous motor. For stability analysis of the doublyfed wind power system, the rotor stability plays a dominant role; thus, the tuning parameter is defined as k = ĉ 1 /c 1 . The physical meanings of the remaining variables can be found in the literature. 20 2.4 | Time-domain equations and equilibrium points of the system If the left side of Equation (5) is equal to zero, the equilibrium point of the system can be obtained as Therefore, the origin is the only equilibrium point of the DFIG system after translation. 19

| Hopf bifurcation Routh criterion for nonlinear dynamical systems
According to the literature, 13,14 it is known that for the nonlinear system, x f x k = ( , ), ∈ x R n , and ∈ k R m are the bifurcation parameters of the system. In the DFIG model, n = 4 and m = 1. Hopf bifurcation of the system at k k = 0 occurs if the following conditions are satisfied for the characteristic equations of the system after linearization near the origin.
(1) Only one pair of complex conjugate roots exists in the eigenvalues.
(2) The remaining n − 2 roots of the characteristic equation have negative real components. If these two conditions are satisfied, there exists a periodic solution for the system at k = k 0 ; that is, the system undergoes Hopf bifurcation at that point.

| Routh-Hurwitz criterion for Hopf bifurcation of DFIG
From Section 2.4, the DFIG is a smooth system of order four; thus, the Jacobian matrix of this system at the origin can be expressed as From Lei and Lightbody, 18 the actual parameters yield the Jacobi matrix given as follows: . When the linearized system has a pair of pure imaginary roots at k k iω = ± 0 0 , and the real parts of the remaining characteristic roots are negative. Let U and V be the eigenvectors of iω 0 and iω − 0 , respectively; if the equations U V = 1 are satisfied, it is proved that the DFIG model exhibits a Hopf bifurcation at k k = 0 ; the system is in a cyclic motion at this time, and the oscillation frequency is ω 0 . From Equation (8) ; thus, the stability of the DFIG system depends on the positive or negative of a 4 . From Table 1, When the value of the element present in the first column is 0, then the DFIG undergoes Hopf bifurcation. Therefore, when a = 0 4 , the system undergoes Hopf bifurcation at k = 1.325 0 . The system oscillates lower, and the oscillation frequency ω 0 is expressed as:

| Analysis of low-frequency oscillations of DFIG
The power spectrum is based on time-domain simulation data describing the distribution of the signal power in frequency. Therefore, the oscillation frequency and power magnitude of the periodic motion occurring in the DFIG can be discerned. From Equation (10), it can be observed that the DFIG undergoes Hopf bifurcation at k = 1.325 0 , and the rotor curve oscillates periodically with an oscillation angular frequency of 1.461 rad/s. As shown in Figure 2, the system oscillates at a low frequency at the Hopf bifurcation with a periodic motion frequency of 0.21 Hz. This is consistent with the oscillation angular frequency analyzed using the Hopf bifurcation Routh stability criterion.

| DFIG system complex frequencydomain stability analysis
From Section 3.3, it is clear that the Routh stability criterion for the Hopf bifurcation cannot be used to visually analyze the stability of the system at the equilibrium point for k > 1.325. Therefore, the distribution of the roots of the DFIG characteristic equation (the closed-loop poles of the system) in the complex frequency-domain planes can be visually described using the root trajectory diagram. When the closed-loop poles of the system are all located in the left half of the s-plane, the system is in a stable state. When the root trajectory crosses the imaginary axis into the right half-plane of s, the system is critically stable, and the value of k at the intersection of the root trajectory and the imaginary axis at this time is called the critical open-loop gain. As  F I G U R E 2 Power spectrum of the Hopf bifurcation moment of the doubly-fed induction generator system. shown in Figure 3, Equation (9) has the general form of a negative feedback system for analyzing the stability in the neighborhood of the equilibrium point. At this point the closed-loop transfer function of the DFIG can be expressed as For simplicity, the DFIG system is a unit negative feedback system, and the characteristic equation of the system can be expressed as where the transfer function of the DFIG system is As indicated by Equation (13), the DFIG system has four closed-loop poles, but the trajectories of two closedloop poles do not cross the imaginary axis. To visually analyze the stability of the system, root trajectories were obtained as shown in Figure 4.
As shown in Figure 4B, there are two closed-loop poles of the system such that the root trajectory crosses the imaginary axis. It can be calculated that the critical open-loop gain of the system at this time is k c = 1.325; thus, when k c < 1.325, all four poles of the DFIG system are located in the left half-plane of s, and the system is in a stable state; when k c > 1.325, there are two closed-loop poles of the system in the right half-plane of s, and the system is in an unstable state. As shown in Table 1, when the DFIG is critically stable, the critical open-loop gain is equal to the Hopf bifurcation parameter, that is, k k = c 0 .

| EQUIVALENT DIMENSIONALITY REDUCTION MODEL OF DFIG
As discussed in Section 2.4, it is cumbersome to calculate the bifurcation values of the system directly for the DFIG four-dimensional nonlinear dynamical model. The asymptotic behavior of a high-dimensional dynamical system in a specific neighborhood of its local bifurcation point is topologically equivalent to the asymptotic behavior of this system on its central manifold. Therefore, the Hopf bifurcation associated with the angular velocity instability of the DFIG can be analyzed in a reduced-order and simplified manner using the central manifold dimensionality reduction method.

| Central manifold theorem
Let the smooth autonomous system be expressed as follows 21,22 : Here, if ∈ x R 2 has two eigenvalues on the imaginary axis and A 1 is the corresponding 2 × 2 matrix and, similarly, ∈ y R 2 has a negative real part of eigenvalues and A 2 is the corresponding 2 × 2 matrix, then from the central manifold theorem, the central manifold of Equation (15), that is, W C can be expressed in the local neighborhood at the equilibrium point can be expressed as follows: Let y φ x = ( ), substituting this into Equation (15) After Taylor expansion at the origin, if y φ x = ( ) is substituted into (15), the coefficients obtained from the expansion term match the coefficients of Equation (15); thus, it can be obtained as (let Taylor expansion ignore the higher-order terms) The coefficients (a a a , , 0 1 2 ) can be obtained, and the reduced-order planar system is obtained as follows:

| DFIG topological equivalence dimensionality reduction model
An actual nonlinear dynamic system is more complex to directly analyze using the central flow descending-order theory. First, the dynamic model should be normalized to obtain the general form of the normative system type, which can reduce the complexity of the subsequent descending-order calculation.
4.2.1 | General form of canonical type of nonlinear dynamical system 22,23 Consider nonlinear dynamical systems where A represents the linear part of the system and f x ( ) represents the nonlinear part. By introducing the nonsingular linear transformation matrix Q such that x = Qy, the linear part of Equation (20), that is, x Aẋ = can be transformed into Equation (21).
Similarly, the nonlinear part of Equation (19), that is, f x ( ) can be obtained by substituting y Q x = −1 into Equation (19) to obtain Equation (21) dx dt Q AQy f y = + ( ) .
Here, ∈ x y R , n , A, and Q are n n × matrices, with the determinant  Q Δ 0. Therefore, the canonical type of the nonlinear autonomous dynamical system that can be transformed into is obtained as shown in Equation (22).

| DFIG system specification type
In Section 2.2, the nonlinear dynamic model of the DFIG system is given by Equation (5). From Equation (19), the nonsingular linear transformation x Qy = is introduced, and the matrix formed by the eigenvectors of J x ( )| x ′ e is the transformation matrix Q: (23) 4 , the Poincare canonical type of the transformed system is expressed as where the canonical type of the nonlinear term is expressed as: From Equation (15) Ignoring the higher terms, the Taylor expansion of Equation (24) at the origin gives Equation (25): , y y = = 0 1 4 , and a a = = 0 10 20 . By matching the canonical DFIG system (23) with Equation (26), the coefficients are obtained as follows: Substituting Equation (28) into Equation (23) yields the following simplified system: Because the coefficients of the higher-order terms are small, ignoring the higher-order terms U y y ( , ) 2 3 , and collating Equation (28) yields the planar descending system of the DFIG. (31)

| Hopf bifurcation phenomenon of the dimensionality reduction model
As discussed in Section 2, the DFIG undescended order has a Hopf bifurcation at k = 1.325 0 , which is shown in Figure 5A. From Equation (31), the eigenvalues of the system λ ε i = ± 1.461 2,3 , and ε are infinitesimal in the neighborhood of the bifurcation point. This is consistent with the critical dominant eigenvalues in Equation (5) within the error range. When ε < 0, the dimensionality reduction system is stable, when ε = 0, the Hopf bifurcation occurs, and when ε > 0, the system is destabilized. Therefore, when Hopf bifurcation occurs in the DFIG, its reduced-order model at the origin has the same properties as the original model, and under the action of the nonlinear term of the reduced-order model, the system generates a limit cycle at this time, which can be compared in the neighborhood of the bifurcation parameter values, as shown in Figure 5.

| ANALYSIS OF HOPF BIFURCATION TYPE OF DFIG
From the above subsection, it is known that the DFIG system undergoes a Hopf bifurcation to produce the limit ring: however, the relationship between the type of bifurcation and the stability of the limit ring cannot be determined. Therefore, in this section, the first Lyapunov exponent is used to theoretically analyze the bifurcation type of the DFIG. CHEN ET AL. | 2163 5.1 | First Lyapunov coefficient 16 We define C n as a linear area in the domain of C-complex numbers. The number product   x y , of space X, Y satisfies the following properties: n The smooth autonomous system can be expressed as follows: where And   p q , = 1. Therefore, the first Lyapunov coefficient at the origin is defined as If Hopf bifurcation occurs in the nonlinear dynamical system of Equation (32), when l (0) < 0 1 , system (32) undergoes a supercritical Hopf bifurcation in the neighborhood of the origin, that is, the limit ring formed after the bifurcation is stable. When l (0) > 0 1 , the system undergoes subcritical Hopf bifurcation in the neighborhood of the head, and an unstable limit ring is formed.

| Hopf bifurcation type of DFIG system
From Equation (5) is negative, the DFIG system undergoes nondegenerate supercritical Hopf bifurcation, which results in a stable limit loop. Therefore, the stable limit loop generated by the nonlinear smooth DFIG system is caused by the supercritical Hopf bifurcation.

| TIME-DOMAIN SIMULATION VERIFICATION
Time-domain simulation analysis was used to determine the stability of the DFIG system from the convergence and divergence of the trajectory using the time-domain evaluation of the system at different initial values. The Runge-Kutta method is widely used for the numerical solution of nonlinear ordinary differential equations, because of the high-precision single-step algorithm and measures for suppressing errors. The simulation results indicated that the trajectory of the DFIG system at different initial values is related to the value of the harmonic parameter k. There exist two states: (1) the decay of the oscillation to the equilibrium point and (2) the formation of periodic oscillation as well as the generation of limit cycles. In this section, the trajectory of the DFIG system under different initial values is used to determine whether it has a stable limit cycle.

| Convergence to a stable equilibrium point
The time-domain simulation indicated that the system converges to the equilibrium point from a wide range of initial values when  k 1.325, and eventually reaches a steady state. Considering

| 2165
The phase space trajectory and speed curves of the DFIG from the initial equilibrium point at k = 0.1 and k = 1 are presented in Figures 5 and 6, respectively. As shown in Figure 7, the time history of speed x 3 converges, which means that DFIG is stable at k = 1.

| Convergence to a stable limit cycle
For k > 1.325, the system was unstable in the neighborhood of the equilibrium point, and instead of converging to a stable equilibrium point, the system formed periodic oscillations and a limit cycle. As shown in Figure 8, the original system forms periodic oscillations from the initial state, and at k = 1.325 the system underwent a Hopf bifurcation to form a limit cycle.

| Stability of limit cycle for different initial values
For a smooth autonomous system, the class of bifurcations produced by the system is determined when the bifurcation parameters are inevitable, and it is known from Equation (39) that the DFIG system has a supercritical Hopf bifurcation at k = 1.325 0 ; therefore, the system appears as a stable limit cycle. The stability of the DFIG bifurcation type at different initial values was analyzed using the time-domain method. The simulation results are shown in Figure 9, considering the initial values of the DFIG system at In Figure 9, x 3 and x 4 are selected as the phase planes. When the initial value was x 1 , the trajectory converges from the outside to the limit cycle as observed from the phase plane. When the initial value is x 0 , the DFIG model oscillates and diverges from the inside to the limit cycle. Therefore, the DFIG system undergoes a supercritical Hopf bifurcation at k = 1.325 0 and formed a stable limit cycle.

| CONCLUSION
The effect of the turning parameter k on the stability of a DFIG was investigated by considering a DFIG nonlinear fourth-order dynamic model as an example. A method based on the central manifold theory was used to obtain a reduced-order model of the DFIG in the neighborhood of the equilibrium point. In addition, the limit cycle stability of the model at the Hopf F I G U R E 8 Limit loop formed by doubly-fed induction generator system at k = 1.33.
F I G U R E 9 Limit ring formed by doubly-fed induction generator with different initial values.
bifurcation point was analyzed. The following con-