Bursting oscillation process and formation mechanism of doubly fed induction generator

This paper proposes an analytical method based on bifurcation theory and frequency domain fast-slow scale coupling to explain the oscillation process of a doubly fed induction generator (DFIG) and its formation mechanism, in view of the bursting oscillation behavior of DFIG when the system parameters vary under external disturbance. First, a mathematical model of the fourth-order generalized autonomous system of DFIG is established. Then, a two-parameter bifurcation diagram is introduced to study the bifurcation mechanism of the system when the internal parameters of the DFIG vary under external disturbance. To address the codimension 2-bifurcation under the two-parameter variation, a three-dimensional overlay diagram is used to analyze the bursting oscillation process caused by the codimension 2-bifurcation. In addition, the system operation is classified into three categories according to the bifurcation type of the two parameters. The oscillation state of the system when the DFIG is in different parameter ranges and the bifurcation type and formation mechanism causing bursting oscillation are analyzed using the equilibrium point distribution, transition phase, and timing diagrams. The research enriches the dynamic characteristics of the DFIG system and provides a theoretical basis for suppressing the bursting oscillation of the DFIG system.


Introduction
Access to a high percentage of power electronics and a high percentage of new energy sources has become an important trend and a key feature in the development of power systems [1,2].The development of wind power generation technology is particularly rapid, whereby a large number of doubly fed induction generators (DFIGs) have been connected to the grid.However, the strong nonlinear and strong coupling characteristics of DFIGs would produce complex dynamic behaviors.These, in turn, cause significant challenges to the safe and stable operation of the power system [3,4].There are relevant reports on broadband oscillation caused by the DFIG grid connection.Linearization analysis methods represented by impedance analysis [5] and eigenvalue analysis [6] are incapable of analyzing and characterizing the characteristics and processes of nonlinear oscillations in the system, which has certain limitations.
The DFIG system involves multiple nonlinear factors such as limiting, dead zone, and multiple timescale coupling.These render it highly susceptible to switching-type oscillations or chaotic oscillations.Ref [7] presents a mechanistic analysis of chaotic oscillations caused by limited saturation inside DFIG systems from the perspective of nonlinear switching oscillations.In Ref [8], a mathematical model of a DFIG was developed to demonstrate the chaotic motion of the system under the variation in the degree of tuning k or diffusion coefficients r, and to design a relevant controller to suppress the nonlinear oscillations generated by the system.Ref [9] developed a mathematical model of the DFIG closed-loop system; it was used an internal controller as the slow scale to study the nonlinear oscillation phenomenon caused by Hopf bifurcation in the system and determine the normal operation boundary of the system.The above nonlinear study methods are used to address the switching-type oscillations and chaotic oscillations caused by a DFIG.However, in reality, wind turbines are typically installed in deserts, grasslands, and near the sea.As a type of wind turbine, DFIG operates in harsh environments and is more susceptible to disturbances at different scales, which can cause burst oscillations.Two frequency couplings occur between the frequency of the external disturbance and that of the system when the system is subjected to a slowperiod external disturbance.This forms a two-scale system in the frequency domain, which, in turn, results in a significant fast-slow effect between the state variables [10].The system shows an oscillation state in which the spiking state (SP) with a relatively largeamplitude oscillation and the quiet state (QS) with a relatively gentle amplitude oscillation vary alternately [11].Bursting oscillation occurs when a variable alternates between two states.The system would produce different bifurcations when bursting oscillation occurs.This would, in turn, cause a switch between the QS and SP [12].Bursting oscillations have gradually attracted the attention of scholars since the reproduction of the bursting behavior of neurons by Hodgkin and Huxley [13].However, owing to the deficiency of effective analysis methods, related research remains at the stage of phenomenon description.The problem of coupling at different scales was brought to the level of mechanism research only after Izhikevich explained and classified the formation process of bursting oscillations [12] based on the fast-slow dynamics method proposed by Rinzel [14].In recent years, Han [15] proposed the concept of transformation phase diagram.It provides an important method for analyzing the bursting mechanism.Bi et al. [16] revealed the bursting mechanism of nonautonomous systems by studying controlled Lorenz systems to transform nonautonomous systems with periodic excitations into generalized autonomous systems.Kovacic et al. [17] studied the burst synchronization of low-frequency pure nonlinear oscillators under frequency domain coupling, where this oscillation mode is composed of fast oscillations around the slow manifold.In a ''double-high'' power system, a DFIG and power electronic device would also produce complex bursting oscillation phenomena at the multi-time domain scale or frequency domain scale with amplitude limiting and disturbance.Ref [18] observed the bursting oscillation phenomenon of the improved memristive Wien bridge and explained its bifurcation mechanism.References [19,20] studied the bursting oscillations in piecewise nonsmooth circuit structures and performed a nonsmooth bifurcation analysis.Reference [21] studied the bursting oscillation and bifurcation mechanism of a permanentmagnet synchronous motor system under external load disturbance.Reference [22] studied the bursting oscillation of a single-machine direct-drive permanent magnet synchronous generator.It is indicated that the bursting oscillation of a direct-drive permanent magnet synchronous generator may result in the chaotic oscillation of each node of the entire system in a complex network.
It should be indicated that most studies have focused on the switching oscillation and chaotic oscillation caused by the DFIG system.It is more susceptible to interference from various external factors owing to the harsh working environment of a DFIG.When the external disturbance acts directly on the rotating shaft and blades of the motor in the form of load, the bursting oscillation affects the normal operation of the blades and rotating shaft as well as the controller and converter in the system.This adversely affects the safe and stable operation of the power system.Therefore, it is necessary to study the bursting oscillation process of DFIG and its formation mechanism.
Based on the above analysis and the simplified fourth-order DFIG model [8,23], this study discusses the bursting oscillation process when the internal parameters of a DFIG are transformed under external disturbance.First, based on the simplified DFIG model, a sine function is introduced as an external disturbance term.Furthermore, a mathematical model under the fast-slow coupling in the frequency domain with external disturbances is established.A method is proposed to analyze the bursting behavior of DFIG systems using two-parameter bifurcation diagrams and overlay diagrams of three-dimensional equilibrium point distribution and bifurcation diagrams, three-dimensional transition phase diagrams, and three-dimensional two-parameter bifurcation diagrams.The bursting oscillation process and its bifurcation mechanism of the system with external disturbance and internal parameter variations are revealed.Finally, the oscillation states of the DFIG system in different parameter ranges are obtained.This enriches the dynamic characteristics of the DFIG system and provides a theoretical basis for controlling its bursting oscillation.

Mathematical model with external disturbance in time-frequency domain scale fast-slow coupling
The mathematical model with disturbance is established as follows based on the mathematical model of a DFIG under a synchronously rotating axis [8,23]: here x s ¼ x n À x r .u sd , u sq , i sd , and i sq are the d-axis and q-axis components of the stator voltage and current, respectively; u rd , u rq , i rd , and i rq are the d-axis and q-axis components of the rotor voltage and current, respectively; w sd , w sq , w rd , and w rq are the daxis and q-axis components of the stator and rotor flux linkages; and x n , x r , and x s are the synchronous speed, rotor speed, and angular velocity, respectively, of the coordinate system relative to the rotor.

Flux equation
here L s , L r are the self-inductance between two stator and rotor windings, respectively.L m is the equivalent mutual inductance between the coaxial stator and rotor windings.

Rotor equation of motion
here J g is the moment of inertia of the generator, D g is the torque damping coefficient, p n is the number of rotor poles, and T L is the torque load.The external periodic disturbance T w is added to Eq. ( 3) to study the bursting oscillation process of the DFIG system under external disturbance.Here, T w is defined as where A and X are the amplitude and frequency, respectively, of external disturbance T w .
Let and c 5 ¼ p n .Integrate (1-4) as follows: At present, indirect field-oriented control (IFOC) is the most widely used type of field-oriented control technology for motor control.It uses the slip angle frequency as the control amount.In the M-T coordinate system, the slip angle frequency is proportional to the inverse of the rotor time constant c 1 .Hence, the key to accurate control of the system is the accurate determination of the rotor time constant.The indirect field-oriented control strategy for model (5) of the above closed-loop DFIG system can be described as [22] where ĉ1 is the estimated value of c 1 ; k c ¼ k i À k p c 3 ; k p and k i are PI parameters; and x ref is the reference speed.d and d ref are the rotor angle and rotor reference angle, respectively.k ¼ ĉ1 =c k[ 0 ð Þis introduced.k is defined as the degree of tuning.The control effect of IFOC depends entirely on the selection of the k value.
The fourth-order system model of a DFIG when disturbed by external loads is obtained as follows by combining Eqs. ( 5) and ( 6): System ( 7) is an autonomous system when the amplitude of periodic external disturbance T w of the system A ¼ 0. Since the state variables oscillate mainly at the intrinsic frequency, i.e., oðd _ x 1 /dt; d _ x 2 /dt; d _ x 3 /dt; d _ x 4 /dtÞ ¼ oð1:0Þ X 0 , and the external perturbation frequency, oðdT w /dtÞ ¼ oð0:04Þ X, it is clear that the X ( X 0 , DFIG system has both T A % T B external perturbation terms This implies that the perturbation T w remains almost constant for any period.Thus system Eq.( 7) can be identified as a generalized autonomous system.

Bifurcation conditions and bifurcation sets of DFIG systems
As the degree of tuning k varies within the DFIG system, generalized autonomous DFIG system (7) generates different bifurcation behaviors.In addition, the bifurcation results in different dynamical states of the system.Hence, a bifurcation analysis is required for the system to determine the bifurcation conditions and the set of bifurcations constituting the destabilization of the equilibrium point in the system.The unique equilibrium point E 0 ½c 2 u 0 2 =c 1 ; À T w =c 5 u 0 2 ; 0; 0 of system ( 7) can be obtained.Then, the Jacobi matrix of the system at the equilibrium point can be obtained as: Thus, it can be concluded that the characteristic equation of system (7) at the equilibrium point is where when the parameters satisfy Routh Criterion (11), all the corresponding eigenvalues have negative real parts, i.e., the equilibrium point is stable.
The equilibrium point of system ( 7) is unstable when it does not satisfy Eq. (11).Thereby, bifurcations would occur.These would, in turn, cause bursting oscillation.It can be established from characteristic polynomial (9) that there are two instability modes in the equilibrium point E 0 .Furthermore, two codimension 1 bifurcations [24] (namely, Fold bifurcation and Hopf bifurcation) would be generated in the two instability modes.
According to the conditions of Fold bifurcation, the system has Fold bifurcation when the eigenvalue of the Jacobi matrix at the equilibrium point of the system is zero.Therefore, when the parameters satisfy the conditions the point set corresponding to the bifurcation parameter T w in the system is the Fold bifurcation set.
It can be established from the condition of Hopf bifurcation that the system would have Hopf bifurcation when the entire row of the Routh table is zero (i.e., the Jacobi matrix of the equilibrium point of the system has a pair of conjugate imaginary roots) and the real parts of other eigenvalues in the characteristic equation are nonzero.Therefore, when the parameters satisfy the condition the point set corresponding to the bifurcation parameter T w in the system is the Hopf bifurcation set.
4 Bursting oscillation behavior of DFIG system and its formation mechanism The external disturbance would vary, while the DFIG system is operating.The degree of tuning k of its internal parameter would also vary.Because its internal parameter k is defined by the rotor time constant, the rotor time constant is affected by the slip angle frequency and by various external environmental factors such as temperature.In addition, the control effect of IFOC is closely related to k.Therefore, this study investigates the dynamic evolution of the system when subjected to external perturbations and variations in the degree of tuning.

Conversion phase diagram and definition of bifurcation point
Entire system (7) would have a bifurcation behavior with the slow-scale variable T w when there is a twoscale coupling in the frequency domain in the system.The bifurcation would cause the bursting oscillation phenomenon.The concept of the conversion phase diagram [25] and the definition of relevant bifurcation points [26] are introduced to better explain the relationship between bifurcation and the actual oscillation.
Conversion phase diagram: For system (7), the conventional phase diagram can be represented as However, for the transformed phase diagram, its trajectory can be defined as That is, T w ¼ A sinðXtÞ is regarded as a generalized state variable.Therefore, the conversion phase diagram of periodic excitation system (7) with respect to the slow variable T w is the projection of C Z on any subspace with respect to the slow variable.
T w .
Fold bifurcation: Consider the autonomous systems where f is a smooth function.If the following conditions are satisfied, (1) There is an equilibrium point a ¼ 0 at x ¼ 0; then, system (15) has a Fold bifurcation at a¼ 0. The Fold bifurcation is the inflection point of the equilibrium point distribution curve.When the trajectory of the fast subsystem reaches to this bifurcation point, there would be an immediate jump, and the bursting phenomenon would occur without delay.The Fold bifurcation is a codimension 1 bifurcation.(2) k ¼ f x 0; 0 ð Þ ¼ 0; then, system (15) has a Fold bifurcation at a¼ 0. The Fold bifurcation is the inflection point of the equilibrium point distribution curve.When the trajectory of the fast subsystem reaches to this bifurcation point, there would be an immediate jump, and the bursting phenomenon would occur without delay.The Fold bifurcation is a codimension 1 bifurcation.
(3) f xx 0; 0 ð Þ 6 ¼ 0; and then system (15) has a Fold bifurcation at a¼ 0. The Fold bifurcation is the inflection point of the equilibrium point distribution curve.When the trajectory of the fast subsystem reaches to this bifurcation point, there would be an immediate jump, and the bursting phenomenon would occur without delay.The Fold bifurcation is a codimension 1 bifurcation.(4) f a 0; 0 ð Þ 6 ¼ 0; then, system (15) has a Fold bifurcation at a ¼ 0 .The Fold bifurcation is the inflection point of the equilibrium point distribution curve.When the trajectory of the fast subsystem reaches to this bifurcation point, there would be an immediate jump, and the bursting phenomenon would occur without delay.The Fold bifurcation is a codimension 1 bifurcation.
Cusp bifurcation: The Cusp bifurcation belongs to the codimension 2 bifurcation.The difference from the above codimension 1 bifurcation is that the system should adjust two parameters simultaneously to achieve this bifurcation.
Consider a one-dimensional system where f is a smooth function.If the following conditions are satisfied, (1) There is an equilibrium point x ¼ 0 at a¼ 0; (2) then, system ( 17) is at Cusp, and there is a a ¼ 0 bifurcation point.Two parameters should be adjusted simultaneously to realize Cusp bifurcation.In threedimensional space, a plane containing cusp mutations can be obtained by the relationship between the two parameters, when these vary, and the state variables.The plane folds in specific regions.Furthermore, the projection of these folds on a plane composed of two parameters yields a two-dimensional bifurcation curve (see Fig. 1).The system state variables would vary abruptly at the cusp point.This may cause the system state to descend abruptly from the edge of the upper half plane to the lower half plane.This type of mutation would severely affect the safe and stable operation of the system, and the resulting mutation is generally catastrophic.

Two-parameter bifurcation analysis of DFIG systems
To study the burst oscillation phenomenon of system (7), the parameters of a 2 MW mainstream doubly fed machine are selected as shown in Ref [8,23].Let the external disturbance frequency be X ¼ 0:04 and T w transform in the interval ½À10; 10. Figure 1 shows the two-parameter bifurcation diagram of T w and parameter k.It reveals the two-parameter bifurcation mechanism of the DFIG.In Fig. 1, the blue solid line represents the Fold bifurcation and the red solid line represents the Hopf bifurcation curve.To facilitate the analysis of whether the burst oscillation caused by the Hopf bifurcation will occur in the system, the parameter k ¼ 2 is selected for analysis and interpretation, i.e., Case 1 k ¼ 2. When k [ 3, the system goes through both the Fold bifurcation curve and Hopf curve.To study the burst oscillations caused by Fold bifurcation and explore the difference between Fold bifurcation and Hopf bifurcation, k ¼ 5 is chosen for analysis and interpretation, i.e., Case 3, k ¼ 5.As the parameter k changes, the perturbation causes the two branches of the Fold bifurcation set to a tangent to produce a tip bifurcation point, and to analyze the effect of tip bifurcation and Hopf bifurcation on the burst oscillations of the system, k ¼ 3 is chosen as case 2 for analysis.

Case 1 k ¼ 2
The equilibrium point distribution and bifurcation diagram of the system under k ¼ 2 are shown in Fig. 2.Under this condition, the system has two Hopf bifurcation points distributed on either side of T w ¼ 0. These are HB1 and HB2.Here, HB1 : ðT w ; x 1 Þ ¼ ð5:127 84; 0:101 99Þ and HB2 : ðT w ; x 1 Þ ¼ ðÀ5:127 84; 0:101 99Þ.The two Hopf bifurcation points are supercritical Hopf bifurcation points because their first Lyapunov coefficient l 1 ¼À7:607 98 Â 10 À4 \0.In the figure , E þ T w , E À T w , and E 0 T w are the positive and negative stable branches and the unstable branches, respectively, of the system equilibrium point.
Next, it is discussed whether the system would oscillate in case 1.It is evident from Fig. 3 whether the system would oscillate when the gradually varying parameter T w starts to vary from -10.It is evident from the overlay of the system transition phase diagram and the equilibrium point diagram in Fig. 3c that when x 1 starts at point p 1 , the system operates along a stable equilibrium curve E À T w as T w increases from -10.However, while operating to bifurcation point HB2, x 1 enters the unstable equilibrium curve E 0 T w and operates along the unstable equilibrium curve E 0 T w to the stable equilibrium curve E þ T w to reach point p 2 .At this time, the slow variable T w of the system is 10.Then, T w decreases gradually as time increases.x 1 starts to operate from point p 2 along E þ T w in the direction of decreasing T w .Then, it operates to the bifurcation point HB1, enters the unstable equilibrium curve E 0 T w again, continues to operate along the unstable equilibrium curve E 0 T w to the stable equilibrium curve E À T w , and finally returns to point p 1 to complete one cycle of motion.
In the motion period mentioned above, T w values corresponding to the two Hopf bifurcation points HB1 and HB2 are relatively close when k ¼ 2. Furthermore, the oscillation hysteresis effect exists in the supercritical Hopf bifurcation, which results in the absence of oscillation.This is evident from the phase diagram in Fig. 3a and the timing diagram in Fig. 3b.Although the system contains bifurcation points, bursting oscillation that jeopardizes the security of the system does not occur.Therefore, in case 1, system (7) is in safe and stable operation.

Case 1:k ¼ 3
The equilibrium point distribution and bifurcation diagram of the system under k ¼ 3 are shown in Fig. 4. In the case of k ¼ 3, the system has two Hopf bifurcation points distributed on either side of T w ¼ 0. These are named HB1 and HB2.Here, HB1:ðT w ; x 1 Þ ¼ ð6:034 87; 0:096 99Þ and HB2:ðT w ; x 1 Þ ¼ ðÀ6:034 87; 0:096 99Þ.The first Lyapunov coefficient l 1 ¼À1:158 76 Â 10 À3 \0 of these two Hopf bifurcation points is supercritical bifurcation.LC1 is the stable limit cycle generated when the stable equilibrium curve enters the unstable equilibrium curve through supercritical Hopf bifurcation.However, unlike case 1, in case 2, the T w value corresponding to the Hopf bifurcation point is at a distance.This would cause a delayed oscillation phenomenon.In addition, when k ¼ 3, the system passes through the Cusp bifurcation point under the codimension 2 bifurcation, i.e., the point CP:ðT w ; x 1 Þ ¼ ðAE9:814 695; 0:053 549Þ.However, Cusp bifurcation points cannot be shown in the equilibrium point distribution and bifurcation diagrams of single bifurcation parameters or the two-dimensional conversion phase diagrams.Therefore, the three-dimensional overlay diagrams are introduced in the following analysis.In the figure , E þ T w , E À T w , and E 0 T w are the positive and negative stable branches and the unstable branches, respectively, of the equilibrium point of the system.
When k ¼ 3, under the external disturbance T w , the system trajectory would pass through the Hopf bifurcation set and the Cusp bifurcation point caused by the tangency of the two branches of the Fold bifurcation set.Because Cusp bifurcation belongs to  5c are introduced for analysis.Here, CP1 and CP2 are the projections of the Cusp bifurcation point CP on the x 1 -axis.To further illustrate the oscillation phenomenon, the bursting oscillation behavior of the system is revealed below in conjunction with Fig. 5. Assuming that the trajectory of the system state variable x 1 still operates from T w ¼ À10, corresponding to Fig. 5b, c, x 1 shifts from point p 1 in the direction of increasing T w .Owing to the existence of the stable equilibrium curve E À T w , the x 1 trajectory moves almost stringently along E À T w until it passes the Hopf bifurcation point HB2.Furthermore, the system enters the unstable equilibrium curve E 0 T w .However, owing to the existence of the hysteresis effect of Hopf bifurcation, the system continues to operate stably along the unstable equilibrium curve E 0 T w , and it is in a QS.While shifting to point P2 in Fig. 5b, c, it enters the SP, and the system starts to oscillate owing to supercritical Hopf bifurcation.The system operates on the unstable equilibrium curve E 0 T w , as shown in Fig. 5c.Thereby, it would also be affected by the Cusp bifurcation point CP.This, in turn, results in oscillation.On the unstable equilibrium curve E 0 T w , the closer the value of the state quantity x 1 corresponding to the Cusp bifurcation point, the larger the oscillation amplitude of the system.As shown in Fig. 5b I, the system undergoes a complex oscillation pattern combining large oscillations of gradually increasing amplitude affected by the Cusp bifurcation and oscillations of relatively small amplitude caused by the Hopf bifurcation.When the system enters the stable equilibrium curve E þ T w from the unstable equilibrium curve E 0 T w , the delayed oscillation phenomenon caused by the supercritical Hopf bifurcation of the system disappears under the influence of the stable equilibrium curve E þ T w .However, owing to the influence of the Cusp bifurcation point CP, oscillation occurs although the system is in a stable equilibrium curve E þ T w .Furthermore, as the value of the state variable x 1 gradually approaches the Cusp bifurcation point, the oscillation amplitude continues to increase until it attains x 1 ¼ 0:053 55 (CP1 point).The oscillation amplitude attains the maximum.Figure 5b shows an enlarged plot of the system exhibiting a large oscillatory state curve caused by the Cusp bifurcation on the stable equilibrium curve E þ T w .Subsequently, the oscillations converge rapidly to the stable equilibrium curve (E þ T w ) under the influence of the stable equilibrium curve.The above-excited state can also be verified from the timing diagram of state variable x 1 in Fig. 5a.Then, the system state x 1 continues to run steadily along E þ T w to the point p 3 , when T w attains the maximum value of 10.As time increases, T w decreases gradually, and x 1 starts to move from point p 3 in the direction of decreasing T w along E þ T w .The track of x 1 runs stably because of the stable balance curve E þ T w .It enters the unstable equilibrium curve E 0 T w after passing through the Hopf bifurcation point HB1.The system is in the QS until it reaches point p 4 .From point p 4 , the system enters the SP.The trajectory of the system oscillates owing to the supercritical Hopf bifurcation.Because the system runs on the unstable equilibrium curve E 0 T w , it would be affected by the Cusp bifurcation point to produce an oscillation with a gradually increasing amplitude (the closer one is to the value of the state quantity x 1 corresponding to the CP point, the larger is the oscillation amplitude).At this time, the system is in a complex oscillation mode combining two types of oscillations.With the gradual decrease in T w , the system enters the stable equilibrium curve E À T w , and the complex oscillation mode with the combination of the two oscillations is transformed into a large-amplitude oscillation mode with only Cusp bifurcation.Although the system is on a stable equilibrium curve E À T w , it would continue to be in a mode of large oscillations until it runs to x 1 ¼ 0:053 55 (CP2 point, where the amplitude of oscillations attains its maximum) owing to the increasing proximity to the CP point.Subsequently, the system oscillation converges rapidly to the stable equilibrium curve E À T w , and the trajectory runs stably to the point p 1 .This completes a cycle of oscillation.The above oscillations constitute the bursting oscillation process of the system under k ¼ 3.This type of nonlinear oscillation would cause the complex alternating phenomenon of large-amplitude oscillation and small-amplitude oscillation.Furthermore, the oscillation is periodic and symmetrical.Therefore, when the system works in case 2, it would cause severe damage to the motor as well as its connected controller and converter.
Next, the mechanism of bursting oscillation is analyzed in case 3 with reference to Fig. 7.It is evident from Fig. 7c that when k ¼ 5, the trajectory of x 1 shifts from the p 1 point T w ¼ À10 to the direction in which T w increases.The trajectory of the system x 1 moves almost stringently along E À T w 1 owing to the existence of the stable equilibrium curve E À T w 1 .At this point, the system is in a QS until the trajectory operates the FB2 point and generates a Fold Because the Fold bifurcation point FB2 is the inflection point of the equilibrium distribution curve and connects the stable equilibrium curve E À T w 1 and unstable equilibrium curve E 0 T w 2 , the running track of the system would jump immediately, and the motor would soon lose its stability.The trajectory of x 1 jumps from the stable equilibrium curve E À T w 1 to the Fig. 6 Equilibrium point distribution and bifurcation diagrams for k ¼ 5 stable limit cycle LC1 through the Fold bifurcation point FB2.This results in a large-amplitude oscillation, and the system appears as the SP.While running to the next bifurcation point (namely, the Fold bifurcation point FB1), the system trajectory jumps from the stable limit cycle LC1 to the stable equilibrium curve E þ T w 4 via FB1 and runs along the stable equilibrium curve.The large oscillation converges rapidly to no-oscillation, the system trajectory also transitions from unstable to stable, and the system state transitions from the SP to QS, until the trajectory runs to point p 2 (i.e., T w attains the maximum value of 10).Then, with the increase in time, T w decreases gradually, and the trajectory runs stably along the stable equilibrium curve E þ T w 4 .Subsequently, the trajectory jumps into the stable limit cycle LC1 again through the Fold bifurcation point FB2.This generates a large-amplitude oscillation.The trajectory reaches the FB2 bifurcation point and, then, enters the stable equilibrium curve E À T w 1 from the FB2 bifurcation point.The oscillation convergence of the system disappears and runs steadily along E À T w 1 back to point p 1 .This completes a period of oscillation.The above oscillation constitutes the periodic bursting oscillation phenomenon of the system under k ¼ 5.It can also be concluded from the timing diagram of Fig. 7b that the bursting oscillation is periodic.The analysis shows that the oscillation is a bursting oscillation with a Similarly, the bursting oscillation that occurs in this situation would also affect the normal and stable operation of the DFIG and even damage the motor.
Considering the above situation, the system would have different bifurcation modes and operating states when k differs.To prevent bursting oscillation of the DFIG system, the value of k of the parameter harmonic scheduling is made to increase in the ð0; 12 interval in steps of 0.1.The operating states of DFIG system (7) and the corresponding range of values are obtained by numerical simulation as shown in Table 1.The system would have two types of codimension 1 bifurcation under the condition of external disturbance and internal parameter variation: Fold bifurcation and Hopf bifurcation.In addition, a codimension 2 bifurcation point would be generated by the tangency of two branches of the Fold bifurcation set.When k 2 ð0; 1:37Þ, the motor is in a stable operation state under external disturbance, and there is no oscillation phenomenon.Hopf bifurcation occurs when k 2 ½1:37; 3Þ.However, the system does not oscillate.When k ¼ 3, the system would produce complex bursting oscillations caused by Hopf bifurcation and Cusp bifurcation.When k 2 ð3; 12, Hopf bifurcation and Fold bifurcation would occur, and the system would display bursting oscillation dominated by Fold bifurcation and the limit cycle LC1 generated by Hopf bifurcation.The appropriate parameter values can be selected according to the data in Table 1.

Conclusion
A system would display bursting oscillation when a magnitude difference exists between the external disturbance and its natural frequency (i.e., a different scale coupling exists in the frequency domain).When the bursting oscillation occurs, the variation in certain internal parameters of the system would also cause different bursting oscillation phenomena and bifurcation behaviors.For this problem, this study investigated the bursting oscillation of a DFIG system when the external disturbance T w and internal parameter degree of tuning k vary.Using the generalized autonomous fourth-order DFIG system model, the external disturbance was regarded as a slow variable, the bifurcation mechanism of the system was analyzed, and the corresponding bifurcation set was obtained.Then, a two-parameter bifurcation diagram was introduced to study the bifurcation mechanism of the DFIG system when the internal parameter k varies under external perturbations.The system was divided into three different bifurcation modes according to the different bifurcation mechanisms of k in different areas of the two-parameter bifurcation diagram.Finally, the occurrence of bursting oscillation in the three bifurcation modes and the different bifurcation mechanisms after bursting oscillation were studied in detail.By studying the mechanism of the influence of the control strategy parameters of DFIG on the burst oscillation, the research work can ensure that the DFIG system can avoid the burst oscillation phenomenon caused by the influence of external perturbation under the actual operation situation and provide theoretical support for the stable operation of the DFIG system.At the same time for the wind power grid-connected system burst oscillation inhibition strategy development provides a theoretical basis, so as to ensure the stable operation of the system.The research indicated the following: (1) When the internal parameter k of the system varies in different bifurcation modes, the system would display a state of oscillation and nonoscillation.Two main bifurcation modes exist when oscillation occurs: the bursting oscillation dominated by the cusp bifurcation in codimension 2 and the bursting oscillation state dominated by the Fold bifurcation in codimension 1. (2) The system is subject to external disturbances.
In addition, the range of the degree of tuning k has a substantial influence on whether the system produces bursting oscillations.No oscillation occurs in the DFIG system when k 2 ð0; 3Þ.When k 2 ½3; 12, the bursting oscillation process caused by different bifurcations occurs in the system.

Appendix
See Table 2.

Fig. 2
Fig. 2 Equilibrium point distribution and bifurcation diagrams of k ¼ 2

Fig. 3 Fig. 4
Fig. 3 Oscillatory state of system at k ¼ 2. a Phase portrait on ðx 1 ; x 3 Þ plane, b time series of x 1 , c overlay of the transformed phase portrait and equilibrium distribution diagram on the ðT w ; x 1 Þ plane

Fig. 5
Fig. 5 Oscillatory state of system at k ¼ 3. a Time series of x 1 , b overlay of transformed phase portrait and equilibrium distribution diagram on ðT w ; x 1 Þ plane, c overlay of 3D transformation phase diagram, 3D two-parameter bifurcation diagram and 3D equilibrium point diagram in ðx 1 ; k; T w Þ space

Fig. 7
Fig. 7 Oscillatory state of system at k ¼ 5. a Phase portrait on ðx 1 ; x 3 Þ plane, b time series of x 1 , c overlay of transformed phase portrait and equilibrium distribution diagram on ðT w ; x 1 Þ plane

Table 1
Table of DFIG operating states for different values of parameters