The multi time-scale phenomenon which results from the actual time effect, spatial effect, structural effect and physical effect in a real system, exists in many complex mathematical models.^[1–3] The complex mathematical model in which the state variables have very different changing rates is called the fast-slow coupled model.^[4–8] The response of a fast-slow coupled system is performed by the alternate change of large amplitude oscillation (i.e, spiking state) and micro amplitude oscillation (i.e, quiescent state). The time history and the trajectory connecting the two special behavior process are referred to as bursting oscillations.^[9–11]

Multi time-scale dynamics can reveal nonlinear essential characteristics of the system from the dynamic point of view, and it is extensively used in biology, chemistry, life sciences, physics, etc.^[12–17] For example, Gu et al.^[18] not only revealed the nonlinear dynamics of the firing patterns of neural pacemakers but also provided experimental evidence for the existence of bifurcations from bursting to spiking simulated by using the theoretical models. Song and Xu^[19] showed that the time delay must be large enough for bursting behavior to occur in a delayed system and the different bursting behaviors are related to the delay coupling and external inputs. Ji and Bi^[20] found that the multiple crossing bifurcation, which is the combination of the Hopf bifurcation and the turning point bifurcation and occurs at the non-smooth boundaries, causes the periodic buster to the quasi-periodic oscillation. Guo et al.^[21] investigated how intrinsic channel noise affects the response of stochastic Hodgkin–Huxley neuron to external fluctuating inputs with different amplitudes and correlation times. Hou et al.^[22] investigated the fractional-order Belousov–Zhabotinsky (BZ) reaction with different time scales and studied the slow-fast effect in fractional-order BZ reaction with two time scales coupled.

As is well known, few researchers concentrate on modeling and analyzing the fast-slow coupled engineering system such as the HTGS. The HTGS is a complex nonlinear, time-variant and non-minimum phase system involving hydro dynamics, mechanical dynamics, and electrical dynamics. From the point of view of dynamically coupled hydraulic-mechanical-electric-structural system for a hydropower station, the parameters in the HTGS can be divided into a fast transmission rate (electrical transient) and slow strain rate (hydraulic transient).^[23,24] In recent years, achievements about modeling and controlling of the HTGS are becoming more and more abundant. For example, Zhang et al.^[25] and Li et al.^[26–28] concentrated on the modeling and analyzing of the HTGS in the process of large fluctuation. Zeng et al.^[29] and Xu et al.^[30] focused on Hamiltonian mathematical modeling of HTGS which can be used to describe general open systems with the structure of energy dissipation and the exchange of energy with the environment. Pico et al.^[31] presented an investigation of hydro power plants with pelton-type turbine-generator sets and robust control theory was used to design a hydro-stabilizer. Salhi et al.^[32] stated an approach to modeling and controlling a micro hydro power plant considered as a non-linear system by using a Takagi–Sugeno fuzzy system. Husek^[33] presented a proportional-integral-derivative controller design for hydraulic turbine based on sensitivity margin specifications. These research results not only explain many complex phenomena in the actual operation of the hydropower station, but also provide a theoretical basis for the safe and stable operation of the hydropower station system. Although great achievements have been made in autonomous HTGS with single time scale, many time-relative non-autonomous factors existing in the real HTGS with multi time-scale have not been considered carefully. Some complex disturbances that are defined as risks constantly occur in the process of operation of the HTGS, like the load disturbance, the frequency disturbance, and the flow rates disturbance.^[34] Besides, the non-autonomous factors not only change the structure of the HTGS, but also make the system more complex. Therefore, it is necessary to study the non-autonomous HTGS with multi time-scale, which will provide a theoretical basis for studying the operation stability and control strategy for the system.

In this paper, a mathematical model of the non-autonomous HTGS with two time scales is proposed by introducing frequency disturbance as periodic excitation. Next, compared with the traditional model of HTGS, the stabilities and dynamic behaviors of the mathematic model of non-autonomous HTGS with two time scales are investigated by numerical simulation. Besides, combining the time waveforms and the phase portraits, the phenomena of the bursting oscillations are described in detail. At the same time, the bursting oscillations of the system, with different amplitudes of periodic excitation, are analyzed by numerical simulation. Finally, the conclusions of this study and discussion about future work are presented.

The rest of the paper is organized as follows. In Section 2, a mathematical model of the non-autonomous HTGS with two time scales is established. In Section 3 its dynamic behaviors are analyzed and described are the phenomena of the bursting oscillations in the above model. In Section 4, some conclusions are drawn from this study.

As shown in Fig. 1,^[35] the HTGS consists of four parts, i.e., a hydro-turbine, a hydraulic system, a generator and a controller. In view of that the four parts of the HTGS were discussed in the previous papers,^[23–28] and here we will directly utilize the four parts. Therefore, based on the four parts, in this section we directly give the traditional mathematical model of the autonomous HTGS with a single time scale. Then, a mathematical model of the non-autonomous HTGS with two time scales is established by introducing the periodic excitation.

Fig. 1.

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	Fig. 1. Diagrammatic sketch of the HTGS.

In this section, a traditional model and a fast–slow coupled model of the HTGS are used to study the dynamical characteristics of the HTGS with two time scales. Here, the numerical simulations are carried out by using the Runge–Kutta method. Specifically, the initial values of [ , , , ] in Eqs. (5) and (6) are [0.001, 0.001, 0.001, 0.001,] and the iteration step is 2000 for each simulation. The values of HTGS parameters are shown in Table 1.

Table 1.

Table 1.

Table 1. Values of HTGS parameters. . Parameters Values Units Parameters Values Units eqh 0.50 p.u. Tw 2.00 s eqx 0.01 p.u. Ty 1.00 s eqy 1.00 p.u. r 0.001 s emh 1.50 p.u. mg0 0.001 s emy 1.00 p.u. kp 2.00 p.u. e 1.00 p.u. ki 1.00 s−1 en 1.00 s a 0.001 s Tab 8.00 s

Table 1. Values of HTGS parameters. .

Bifurcation diagrams of the mechanical torque m_t with respect to the governor parameter k_d are presented in Fig. 2 to study the stability and operational feature of the HTGS. Then, the bifurcation diagram and its local amplification of a traditional model of the HTGS are obtained as shown in Figs. 2(a) and 2(b), respectively. The bifurcation diagrams of the mechanical torque m_t with respect to the governor parameter k_d are also presented in Figs. 2(c) and 2(d).

Fig. 2.

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Fig. 2. (color online) Bifurcation diagrams of the mechanical torque mt with respect to the governor parameter kd: (a) bifurcation diagram of the autonomous HTGS with single time scale at 2.5 ≤ kd ≤ 5.5; (b) local amplification of the bifurcation diagram of the autonomous HTGS with single time scale at 3.8 ≤ kd ≤ 4.2; (c) bifurcation diagram of the non-autonomous HTGS with two time scales at 2.5 ≤ kd ≤ 5.5; and (d) local amplification of the bifurcation diagram of the non-autonomous HTGS with two time scales at 3.8 ≤ kd ≤ 4.2.

Fig. 3.

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	Fig. 3. (color online) Time waveform and phase orbit of the non-autonomous HTGS with two time scales at kd = 3.5. (a) Time waveform with mt–t, (b) phase orbit with mt–x, (c) local amplification of the time waveform with 1800 < time < 2000, and (d) local amplification of the phase orbit with mt–x.

From Figs. 2(a)–2(b), when 2.5≤ k_d ≤ 3.96, the maximum of the relative deviation of the mechanical torque m_t decreases gradually and the track becomes narrow with the increase of k_d. For 3.96 ≤ k_d ≤ 5,5, the maximum of m_t first decreases and then gradually tends to be zero with increasing k_d, which indicates that the governing system operates in a steady state. Note that there is an important bifurcation point, namely k_d = 3.96, which is consistent with N₁ in Fig. 2(b). At this point, the system is in a critical state.

Figures 2(c)–2(d) show bifurcation diagrams of a fast-slow coupled model of the HTGS with two time scales. Meanwhile, figures 2(c)–2(d) show more interesting physical phenomena than the bifurcation diagrams of a traditional model of the HTGS. Firstly, the relative deviation of the mechanical torque m_t includes negative values. In addition, the maximum of m_t increases by more than five percent when m_d = 2.5. For 2.5 ≤ k_d ≤ 3, the values of m_t show a fan-shaped trend and then gradually tends to be zero with increasing k_d. Comparison of Fig. 2(d) with Fig. 2(b) which have typical Hopf bifurcation lines and critical point, shows that figure 2(b) has no obvious critical point and the value of m_t fluctuating between −0.004 and 0.007. Furthermore, the maximum of m_t is less than 0.007 when k_d = 3.8 in Fig. 2(b), but the maximum of m_t is obviously bigger than 0.007 in Fig. 2(d). All of these results show that the frequency disturbance introduced into the traditional model of the HTGS makes the physical phenomena more complex.

In order to further study the dynamical behaviors of the system with two time scales, we select three typical points in Fig. 2(c), which are k_d = 3.5, k_d = 3.96, and k_d = 4.5. Time waveforms and phase orbits are presented as shown below.

For k_d = 3.5, as shown in Fig. 3(a), the relative deviation of the mechanical torque m_t appears as periodically increasing oscillation as time goes on. The periodic boundary points emerge clearly in the wave crest and trough at the first half of the time, and they seem to disappear at the last part of the time. The phase orbit with m_t–x described in Fig. 3(b) is divergent gradually and the thickness of track line changes back and forth alternately. The corresponding magnifications shown in Figs. 3(c) and 3(d) are utilized to further analyze the dynamic behavior of the system with two time scales. As shown in Fig. 3(c), the track line of m_t has the characteristic of a periodic oscillation between the micro amplitude oscillation and the stationary state, which indicates that the spiking state and the quiescent state are alternately periodic. From Fig. 3(d), it is clear that there are transitions between the spiking state and the quiescent state in the system. These results indicate that the HTGS exhibits a bursting oscillation phenomenon over time and is in an unstable state.

Figure 4(a) shows a time waveform of m_t with k_d = 3.96. The amplitude of m_t does not change as time varies. In addition, a more obvious oscillation phenomenon and a clear demarcation point occur near the wave crest and trough. Figure 4(b) shows phase orbit with k_d = 3.96, which has two distinct intersections near m_t = 0. The dynamic law of the interaction between the spiking state and the quiescent state of the system is clearly shown in Figs. 4(c) and 4(d). Therefore, the HTGS turns a periodic vibration into a bursting oscillation.

Fig. 4.

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	Fig. 4. (color online) Time waveform and phase orbit of the non-autonomous HTGS with two time scales at kd = 3.96. (a) Time waveform with mt–t, (b) phase orbit with mt–x, (c) local amplification of the time waveform with 1800 < time < 2000, and (d) local amplification of the phase orbit with mt–x.

The responses of the HTGS with k_d = 4.5 are shown in Fig. 5. From Fig. 5(a), it is obviously observed that m_t presents a periodic decay oscillation and has the similar characteristics to those in Figs. 3(a) and 4(a) near the wave crest and trough. It can be seen that the trajectory of m_t changes significantly at the second half of the time, which indicates that the effect of periodic oscillation on system behavior is becoming more and more serious as time goes by. As shown in Fig. 5(b), the track line of the system converts between the sharp shock and micro shock, which presents a chaotic state. The states of the spiking and the quiescent are described in detail in Figs. 5(c) and 5(d). These results illustrate that the system keeps steady at k_d = 4.5 and the bursting oscillation has a great influence on the stability of the system.

Fig. 5.

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	Fig. 5. (color online) Time waveform and phase orbit of the non-autonomous HTGS with two time scales at kd = 4.5. (a) Time waveform with mt–t, (b) phase orbit with mt–x, (c) local amplification of the time waveform with 0 < time < 200, and (d) local amplification of the phase orbit with mt–x.

In order to reveal the bursting oscillation of the system with the amplitude of the periodic excitation, time waveforms and phase orbits are presented with different values of the amplitude of the periodic excitation a at k_d = 3.96, namely a = 0.00, a = 0.001, a = 0.005, and a = 0.01.

The value of the amplitude of the periodic excitation is equal to 0.00, which means that the frequency disturbance does not occur in the operation process of the hydropower station. For a = 0.00, as shown in Fig. 6, the motion regularity of the hydro-turbine mechanical torque m_t is periodical and a limit cycle appears in the phase orbit. Besides, the maximal value of m_t is approximately equal to 4.5 × 10⁻³. The results indicate that the governing system is in a critical state. For a = 0.001 and k_d = 3.95, as shown in Fig. 4, the bursting oscillations of the system behave as that described above and the maximal value of m_t is nearly equal to 5.0 × 10⁻³. Figure 7 shows the time waveform and local amplification of the time waveform with a = 0.005. It can be seen that the phenomenon is almost the same as that in Fig. 4, while the maximal value of m_t is about 8.0 × 10⁻³. As shown in Fig. 8, the hydro-turbine mechanical torque m_t with a = 0.01 tends to be more complex due to the stronger bursting oscillations effect. The maximal value of the relative deviation of the mechanical torque m_t is bigger than those in Figs. 4–6, which means that vibration of the system in this case is more intense.

Fig. 6.

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	Fig. 6. (color online) Time waveform and phase orbit of the autonomous HTGS with single time scale for a = 0.00 at kd = 3.96. (a) Time waveform with mt–t, (b) local amplification of the time waveform with 1800 < time < 2000.

Fig. 7.

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	Fig. 7. (color online) Time waveform and phase orbit of the non-autonomous HTGS with two time scales for a = 0.005 at kd = 3.96. (a) Time waveform with mt–t, and (b) local amplification of the time waveform with 1800 < time < 2000.

Fig. 8.

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	Fig. 8. (color online) Time waveform and phase orbit of the non-autonomous HTGS with two time scales for a = 0.01 at kd = 3.96. (a) Time waveform with mt–t, (b) local amplification of the time waveform with 1800 < time < 2000.

These results show that the amplitude of the periodic excitation has a great influence on the dynamic behaviors of the system. As the value of the amplitude of the periodic excitation a increases, the effect of the bursting oscillations on the system becomes stronger.

The HTGS is a complex nonlinear, time-varying, and non-minimum phase system involving hydro dynamics, mechanical dynamics, and electrical dynamics, which is accurately a multi-time-scale dynamic system. However, few papers have reported on the fast–slow dynamical analysis of a typical complex engineering system coupled hydraulic-mechanical-electric-structure.

In this paper, some interesting physical phenomena of the non-autonomous HTGS with two time scales are investigated. First, the stabilities and dynamic behaviors of the above system are investigated in detail by numerical simulation, which is carried out by using the Runge–Kutta method. Meanwhile, the bifurcation diagram of a traditional model of the HTGS is presented to further study the bifurcation characteristics for the system with two time scales. Third, three typical points are respectively selected based on the bifurcation diagram. Then, the physical phenomena of the bursting oscillations existing in the system with two time scales are described in detail with the combination of the time waveforms and the phase portraits. From an experimental point of view, the spiking state and the quiescent state alternate periodically as the time goes by. Also, stability and operational characteristics vary with system parameter k_d. Finally, the comparative analyses for the effect of the bursting oscillations on the system with different amplitudes of the periodic excitation a are carried out. We obtain that the relative deviation of the mechanical torque m_t rises with the increase of a.

The future work will involve the investigation into the bifurcation characteristics in theory of the non-autonomous HTGS with two time scales. In addition, the numerical experiments will focus far more on the behavior characteristics of the HTGS with two time scales in the transient process.