1 Introduction

The stability of electric power systems is a well-established subject with a long history of research ^[1] . Actual power systems are forced to operate closer to their stability limits due to the increasing power demands and other factors such as environmental and economical constraints for building new power plants and transmission lines. Recently,electric power systems have become much huger and more complicated. Stability problems have become more complex as interconnections become more extensive. Therefore,the stability analysis of electric power systems is still a major issue and has been received significant attention in scientific studies.

Due to the high nonlinearity of the electric power system,its stability is closely related to a disturbance. If the disturbance is large,the system operating point varies significantly,and nonlinearities may have a considerable effect on the system performance. In this situation,the equations that describe the dynamics of the power system cannot be linearized. The tools of assessment for this type of stability belong to nonlinear system theory,which include geomet- ric methods,energy functions,bifurcation theories,normal forms of vector fields,and numerical simulations ^{[1, 2, 3]} . The stability analysis can be carried out by using nonlinear dynamic methods ^{[4, 5]} and perturbation techniques ^[6] ,which are widely used in nonlinear mechanics ^[7].

There has been great concern in the power system research field on dynamic characteris- tics relating to the stability of a power system. Some researchers focused their attention on an equivalent single-machine infinite-bus (SMIB) power system. A good example of using the perturbation method of multiple scales for stability analysis of a single machine power system was studied by Nayfeh et al. ^{[8, 9]} . Duan et al. ^[10] gave a bifurcation analysis for an SMIB power system with series capacitor compensation associated with sub-synchronous resonance. Chen et al. ^[11] showed the presence of chaos in an SMIB power system through a period-doubling bifurcation route,and they also investigated the chaotic control and chaotification problem of that system. Wang et al. ^[12] investigated dynamical behaviors and singularities in an SMIB power system formulation by using the geometric singular perturbation theory. Chen et al. ^[13] used Melnikov’s method to discuss hetero-clinic and sub-harmonic bifurcations and gave a con-dition of parameters for chaotic oscillation occurrence in an SMIB power system. Wei et al. ^[14] analyzed the effect of the Gaussian white noise on erosion of safe basin in the same model under different parameters.

The advantage of the SMIB representation is the simplicity of the model which facilitates sophisticated nonlinear analysis. However,such models are limited to study instability events involving many generators. For multi-machine power systems,many techniques have been de- veloped and applied to the stability analysis in terms of theoretical analysis ^{[15, 16]} ,numerical methods ^[17] ,and engineering applications ^{[18, 19]}.

A power system is inherently a complex nonlinear system,which exhibits complex dynamic behavior when subjected to disturbances. For two or more machines infinite-bus power sys-tem,there exist coupled power angles in the motion equations (called swing equations) which describe the dynamic characteristics of the interconnected synchronous generators. It has been pointed out that the inter-area modal phenomenon may occur as a result of a nonlinear in-teraction of the natural oscillation modes of stressed power system. The conventional linear techniques ^[20] analyze the linear modal behavior and cannot account for many complex phe-nomena that occur when the system is stressed. Therefore,for a better understanding of the underlying cause of the complex behavior of a stressed power system,many researchers have turned to seek new methods. Many scientists have made great efforts on applying the method of normal forms to quantify nonlinear modal interaction in power systems ^{[21, 22, 23, 24]} and strong modal resonance analysis ^[25] ,as similar to the most widely considered ones for studying the normal mode bifurcation in nonlinear mechanical systems^{[26, 27, 28, 29]} .

For a two-machine infinite-bus (TMIB) power system,Yuan and Sun ^[30] studied the occur- rence of chaotic phenomenon by using Melnikov’s method under some particular conditions. They proposed a two-generator electric power system model by considering that the infinite-bus maintains a voltage of fixed amplitude with a small periodic fluctuation in the phase angle. However,the internal resonance in the system subjected to disturbances has not been discussed.

The motivation of this paper is to investigate the internal resonance and its effect on the bi-furcation characteristics of a TMIB power system. Besides the fluctuation of changing phase ^[31] , the fluctuation of changing voltage is also considered to formulate the infinite-bus in our con-sidered model. The approximate solutions to normal modes of this system with or without the internal resonance are derived by using the method of multiple scales. For comparison,the bifurcation analysis for the case of no internal resonance is carried out by using the single-state-variable singularity method to show the basic bifurcation characteristics of the system. Then, the 1:3 internal resonances are proposed,and the bifurcation equations are studied successfully by employing the two-state-variable singularity method,which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on mode solution bifurcation characteristics of this system.

2 Mathematical model of TMIB power system

A simple interconnected power system model with two generators and three nodes in this paper is shown in Fig. 1,where the synchronous generators G1 and G2 are delivering power to the infinite-bus through transmission lines and main transformers X_T1 ,X_T2 . The number “3” represents the infinite-bus node. The motion equations of the two generators can be written as follows ^{[30, 31]} :

where Here,H_i(i = 1,2) is the inertia constant of the ith generator,D_i(i = 1,2) is the damping coefficient,δ_i(i = 1,2) is the rotor angle (also named power angle) measured with respect to a synchronously rotating reference frame moving with the constant angular velocity ω₀,P mi (i = 1,2) is the mechanical power input to the generator,V_i(i = 1,2) is the machine terminal voltage,δ_Bis the infinite-bus voltage phase,V_Bis the infinite-bus voltage,and B₁₂ ,B₁₃ ,B₂₃ are susceptance parameters. The voltage and phase of the infinite-bus are time varying with the frequency ω of the periodic variations. V_B1,δ_B1,φ_V,and φ_Bare assumed to be constant, and the magnitudes V_B1and δ_B1are assumed to be small. From (1),it can be seen that there exist nonlinear coupling terms in the dynamic equations of the TMIB power system,which is different from that of the single-machine power system ^{[8, 9, 10, 11, 12]} . Take the following transformation for (1a) and (1b): where θ₁₀ is the steady operating value of the phase angle δ₁around which the variation Δδ =δ₁− θ₁₀ takes place. Then,it can be obtained that

Use an identical procedure,and let

where θ₂₀ is the operating value of δ₂around which the variation Δδ = δ₂− θ₂₀ takes place. Then,one can obtain

Expanding (1) in a Taylor series around θ₁₀,θ₂₀,and retaining terms up to third order,we obtain the following modified swing equation:

where the corresponding coefficients are listed in Appendix A.

3 Perturbation and singularity analysis

In this section,we use the method of multiple scales ^[7] to obtain a set of four averaged equations that determine the amplitudes and phases of the steady-state solutions on a slow scale. It is noted that this technique is applicable to the cases with or without an internal resonance. The investigation for the case of no internal resonance is first given for comparison. Then,we study the case of internal resonance. Assume that the solutions of (7) in the neighborhood of the trivial equilibrium are represented by an expansion of the form:

where ε is a small positive parameter,T₀= t represents a fast scale,and T₁= εt is a slow scale characterizing modulation of the amplitudes and phases of two modes. In terms of T₀ and T₁, the time derivatives transform according to where

To obtain a system which is suitable for the application of the method of multiple scales, the scales transformations may be introduced as follows:

Substituting (8)-(10) into (7) and equating the coefficients of the same order of ε in both sides,one obtains the following sets of differential equations:

Order ε⁰:

Order ε¹:

From the left-hand side of (11),it can be seen that there exist coupled interaction terms of x₀ and y₀,and supposing that its solution has the following form: where “C_c ” stands for the complex conjugate of the preceding terms. The quantities A₁and A₂are unknown at this stage of the analysis,they are determined by eliminating the secular terms at the next approximation,and jω₁,and jω₂are two different pure imaginary roots of the following characteristic equation about λ:

Supposing₀< ω₁< ω₂,it is worth pointing out that ω₁,and ω₂are also called the first-order and the second-order natural frequencies of the normal modes,respectively. For the case of the parametric principal resonance,the disturbance frequency ω is assumed to be almost equal to the second-order natural frequency ω₂ according to

where σ₁is an external detuning parameter to express the nearness of Ω to ω₂

The 1:3 internal resonances may occur under the condition that two natural frequencies of the system satisfy such relationship ω₁/ω₂ = 1/3. For comparison,the bifurcation analyses of the system for both of the non-internal resonance and the 1:3 internal resonance cases are discussed,respectively.

3.1 Case of no internal resonance

In this case,the physical parameters are chosen such that the natural frequencies of the two modes are not of the ratio of 1:3,that is,

Substitute (13) into (12). In order to eliminate the terms that lead to secular terms from (12),suppose that its particular solution can be written as

Substituting (13) and (16) into (12) and equating the harmonic coefficients of the frequencies ω₁and ω₂ for both sides leads to

where

In (17),the harmonic term whose frequency is ω₁(or ω₂ ) relies on both the left and the right sides to balance each other,with the following solvability condition:

The solvability condition can be written as Introducing the polar representation for A₁(T₁) = a₁(T₁)e^jθ₁(T₁) and A₂(T₁) = a₂(T₁)e^jθ₂(T₁) into (19) and then separating real and imaginary parts give rise to where for notation purpose,four functions a₁(T₁),a₂(T₁),θ₁(T₁),and θ₂(T₁) have been ex- pressed by a₁,a₂,θ₁,and θ₂,respectively.

By letting φ= σ₁T₀+φ_V−θ₂ ,the steady solution can be obtained by finding the solutions to the four algebraic equations which can be obtained by letting D₁θ₁= D₁θ ₂ = D₁a₁= D₁a₂ = 0 in (20). Then,(20) becomes

According to the typical phenomenon of a multi-degree-of-freedom vibration system ^[32] , when the disturbance frequency is close to the second-order natural frequency,because of the presence of damping,only the second-order mode determines the long-term dynamic behavior of the system. Therefore,we can let a₁=₀in (21),which yields

(22) can be transformed into the following form: where

Considering the resonance condition,we can have 2ω₂ ≈ 2ω,by letting

(23) can be written as Eliminating the trigonometric terms by using the relations between trigonometric functions in (24) gives rise to (25) is the so-called frequency-response equation,which can reflect the bifurcation characteris- tics of the second-order mode as the disturbance frequency ω changes. Actually,some system parameters of the TMIB power system may also change,which can lead to a sudden change in the mode amplitude. This has a significant effect on the power system’s stability and secure operation. Therefore,in order to discuss the bifurcation characteristics in a wider parameter space,the engineering unfolding analysis is carried out ^[33] . Let z = a₂² . Then,(25) becomes where

Choose ω as a bifurcation parameter and V_B1and δ_B1as unfolding parameters. The tran-sition sets can be obtained by using the singularity theory with one variable as follows ^[34] :

where For a given TMIB power system,the values for the system parameters are ^[1] listed as follows: ω₀= 100π,V_B0= 1,H₁= 2.37,P_m1= 1,V₁= 1.27,H₂= 2.37,P_m2= 1,V₂= 1.27,B₁₂= 0.1,B₁₃= 1.2,B₂₃= 1.0,ω = 8.1,D₁= 0.008,D₂= 0.008. Figure 2shows the transition sets in V_B1-δ_B1symmetrical plane which is divided into four regions. The frequency-response curves in four regions are shown in Figs.3,4,5,6,respectively. It can be seen that every bifurcation dia-gram corresponding to different region possesses different topological structures. It should be noted that all of the frequency-response curves present softening nonlinearity and the jump phe-nomenon can occur at a lower frequency. As shown in Fig. 6(a),when the disturbance ratios of the infinite-bus voltage amplitude and phase angle are both less than 5%,the frequency-response curve in Region IV is nearly close to be a single-valued curve with small amplitude,and the jump and hysteresis phenomena do not appear. The frequency-response curves depicted in Fig. 6(b) also lie in Region IV,but in this case,the disturbance ratios of the infinite-bus voltage amplitude and phase angle are 5% and 25%,respectively. We can see that the jump and hys-teresis phenomena occur. Therefore,for an actual TMIB power system,it is better to choose the physical parameters lying in Region IV and guarantee the disturbance ratios of the infinite-bus voltage amplitude and phase angle are both less than 5%. For these chosen parameters,the generator rotors swing periodically with small amplitude,which can avoid losing synchronism.

3.2 Case of 1:3 internal resonance

When there exists a relationship ω₂ ≈ 3ω₁,the case of 1:3 internal resonance will occur. Introduce a detuning parameter σ₂to express the nearness of the two natural frequencies

Suppose the solution of (11) has the following form: Substituting (28) into (12) and taking into consideration (17),we can obtain where ψ₁= ω₁T₀+ β₁(T₁),ψ₂= ω₂ T₀+ β₂(T₁),and θ = ωT₀+ φθ.

According to the solvability condition expressed by (19) and separating the same harmonic term leading to secular terms yields the governing equations for the amplitudes a₁,a₂and phases γ₁,γ₂as follows:

where γ₁=3β₁−σ₂T₁−β₂,γ₂=3β₁−σ₂T₁−σ₁T₁−φ_V,and γ 3 = γ₂−γ₁,and the corresponding symbols are listed in Appendix B.

The steady-state solutions correspond to a₁,a₂=₀and ý₁,ý₂= 0. Eliminating the trigono-metric terms in (29) gives rise to

When 1:3 internal resonances take place,the singularity theory with single variable is not applicable for bifurcation characteristic analysis of (30) which has two mode amplitudes. Here, for two-machine power system,we first use the singularity method with two variables to discuss the bifurcation behaviors of the two normal mode solutions. In(30),letting x=a²₁ ,y=a₂² as two state variables,supposing λ = σ₁is the bifurcation parameter,V_B1,and δ_B1are engineer-ing unfolding parameters. According to the singularity method with two variables [35−36] ,the transition sets are obtained by the following formulae (the corresponding symbols are listed in Appendix C):

Consider a specific system with the system parameters: Rated MVA=160,Rated PF=0.85, Rated KV=15,ω₀= 100π,H₁= 2.37,P_m1= 1.0,B₁₃= 1.2,B₁₂= 1.0,V₁= 1.0,H₂= 0.6,P_m2= 0.6 B₂₃ = 1,V₂= 1.0,V_B0= 1.0,D₁= 0.008,and D₂= 0.005. Fig. 7 shows the transition sets in V_B1-δ_B1plane which is divided into seven regions. The frequency-response curves of the two modes in every region are shown in Fig. 8. As the infinite-bus voltage ampli-tude and phase fluctuate periodically,the frequency-response curves of the two modes exhibit parametric vibration characteristics as shown in Fig. 8. Owing to occurrence of 1:3 internal resonances,the first-order mode amplitude is considerably larger than that of the second-order mode. We can see that when the infinite-bus voltage amplitude or phase fluctuates,both of the two mode solutions exhibit rich bifurcation patterns leading to a change in the number of the possible normal mode solutions,for example,three to two,three to one,or two to three.

In comparison with no internal resonance,it can be found that the bifurcation behavior and stability of the system are significantly influenced by the infinite-bus voltage amplitude and phase. The coupled modes of the TMIB power system have more critical bifurcation points and quite rich bifurcation patterns,which is greatly different from that of no internal resonance case.

Such work is obviously preliminary to a closer understanding of the complex bifurcation behavior of the TMIB power system. When making parameter design for the system stability, it is better to avoid choosing these parameters corresponding to the critical points,where the state of motion may change suddenly. We can adjust the system parameters to achieve a required working pattern in practical engineering applications. Therefore,the above discussion can provide a theoretical direction for the dynamic analysis,bifurcation control,and engineering optimization design for an actual TMIB power system with different working and structural parameters.

4 Conclusions

In this paper,the bifurcation analysis of a simple electric power system involving two syn-chronous generators connected by a transmission network to an infinite-bus has been carried out. In this system,the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. By using the method of multiple scales,the bifurcation equations of this system have been obtained. Then,the singularity methods for single state variable and for two state variables have been respectively used to analyze the bifurcation characteristics of the system without and with 1:3 internal resonance. The transition sets determining different bifurcation patterns of the system are obtained and analyzed,which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. The results show that the bifurcation behavior and stability of the system are significantly influenced by the infinite-bus voltage amplitude and phase. In comparison with the non in- ternal resonance case,the 1:3 internal resonance makes the bifurcation characteristics more complicated and induces more numbers of bifurcation patterns. Owing to occurrence of 1:3 in- ternal resonance,the energy transfer makes that the first-order mode amplitude is considerably larger than that of the second-order mode. The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a TMIB power system.