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Gao Xiang, Chen Diyi, Zhang Hao, Xu Beibei, Wang Xiangyu. Nonlinear fast–slow dynamics of a coupled fractional order hydropower generation system. Chinese Physics B, 2018, 27(12): 128202  
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Nonlinear fast–slow dynamics of a coupled fractional order hydropower generation system
Gao Xiang1, 2, Chen Diyi1, 2, 3, †, Zhang Hao1, 2, Xu Beibei1, 2, Wang Xiangyu3
Key Laboratory of Agricultural Soil and Water Engineering in Arid and Semiarid Areas, Ministry of Education, Northwest A & F University, Yangling 712100, China
Institute of Water Resources and Hydropower Research, Northwest A & F University, Yangling 712100, China
Australasian Joint Research Centre for Building Information Modelling, School of Built Environment, Curtin University, WA 6102, Australia

 

† Corresponding author. E-mail: diyichen@nwsuaf.edu.cn

Project supported by the National Natural Science Foundation of China for Outstanding Youth (Grant No. 51622906), the National Natural Science Foundation of China (Grant No. 51479173), the Fundamental Research Funds for the Central Universities (Grant No. 201304030577), the Scientific Research Funds of Northwest A & F University (Grant No. 2013BSJJ095), and the Science Fund for Excellent Young Scholars from Northwest A & F University and Shaanxi Nova Program, China (Grant No. 2016KJXX-55).

Abstract

Internal effects of the dynamic behaviors and nonlinear characteristics of a coupled fractional order hydropower generation system (HGS) are analyzed. A mathematical model of hydro-turbine governing system (HTGS) with rigid water hammer and hydro-turbine generator unit (HTGU) with fractional order damping forces are proposed. Based on Lagrange equations, a coupled fractional order HGS is established. Considering the dynamic transfer coefficient eis variational during the operation, introduced e as a periodic excitation into the HGS. The internal relationship of the dynamic behaviors between HTGS and HTGU is analyzed under different parameter values and fractional order. The results show obvious fast–slow dynamic behaviors in the HGS, causing corresponding vibration of the system, and some remarkable evolution phenomena take place with the changing of the periodic excitation parameter values.

PACS: 82.40.Bj;88.60.K-;88.40.fc
Keyword:fast-slow dynamics;fractional order;nonlinear dynamics;hydropower generation system
1. Introduction

Due to the hydro-mechanical-electric coupling characteristics of the hydropower station system, there are a wide variety of complex nonlinear dynamic behaviors in the operation.[1] As two core parts of the hydropower station, the nonlinear analysis of the hydro-turbine governing system (HTGS) and hydro-turbine generator unit (HTGU) has attracted wide interest from scholars.[2–9] The development of the nonlinear theory and the introduction of the fractional order makes analysis on the nonlinear behaviors more diverse.[10–20] Related researches gradually evolved from the single subsystem to the internal connection of overall system.

Former correlational researches on the hydropower station system mainly focus on the model optimization and the stability control.[21] Researches on HTGS investigated the dynamics of the hydropower system by an accurate model that is based on the practical hydropower station and introduction of other physics phenomena such as time-delay.[5,22–24] In addition, developments of various control methods, such as sliding mode control and fuzzy logic control, are applied to the stability control of hydro-turbine governing system.[25] For the HTGU, shaft vibration and the description of multiple complex forces are emphasized in the majority studies. The vibration problem directly affects the stable operation and efficiency of the hydropower station.[26]

Although a lot of results have been made in the HTGS and HTGU, few studies have examined the internal effects of the dynamic phenomena in HGS. As a hydro-mechanical–electric coupled system, there is a tight link between each component. Therefore, an overall analysis of the system dynamics can give a more complete picture of the interaction in HGS, as well as the mechanism of various dynamic behaviors.

In light of this, this paper proposed a fractional order HGS model constituted with the HTGS and HTGU to analyze the dynamic characteristics in detail. First, mathematical models of HTGS with rigid water hammer and HTGU with fractional order damping forces are established. According to the Lagrange theory, coupling the HTGS and HTGU to get the coupled fractional order HGS. Because the transfer coefficient e changes during the operation, e is introduced into the HGS model as a periodic excitation. The internal effect in HGS is obtained by comparing the dynamic behaviors of rotational speed and shaft vibration. Meanwhile, effects of fractional order on the changes of the HGS behaviors are analyzed. Finally, the bifurcation diagram of the HGS shows the evolution characteristics of the dynamic behaviors of the HGS under different excitation parameter values.

The rest of this paper is organized as follows. In Section 2, the coupled fractional order HGS model is derived. In Section 3, the results of numerical simulations are discussed from two aspects. The dynamic behaviors under different periodic parameter values are discussed, and the HGS stability under different dynamics are analyzed. Our concluding remarks and the future work are provided in Section 4.

2. Modeling
2.1. HTGS

The hydro-turbine used in this paper is a Francis turbine. The relationship between the hydro-turbine torque and the output power in the dynamic model of the hydro-turbine and water diversion system is

where Pm, mt, and n are the relative division of the output power, hydro-turbine torque, and rotational speed.

The transfer function is

where Gh(s) is the transfer function of the conduit system, e is the intermediate variable, ey and eqh are the transfer coefficient of the servomotor and water head.

From Fig. 1, the output torque of the hydro-turbine with rigid water hammer can be written as

where mt and yg are the relative values of the output torque and the guide vane opening, Ty and Tw are the inertial time constants of the servomotor and the water in the diversion system, u is the regulator output.

Fig. 1. (color online) Structure chart of the hydro-turbine governing system (HTGS).

The nonlinear model of a second order generator is

where D and Tab are the hydro-turbine damping coefficient and inertial time constants, me is the electromagnetic torque.

The electromagnetic power is

where is the electric potential, Vs is the infinite busbar voltage.

2.2. HTGU

The changeability of the differential and integral order in the fractional calculus extends the dynamic descriptive ability of the commonly used integer order calculus. Especially for complex systems, since the dynamic process of the actual system is fractional, the fractional order can accurately describe the essential characteristics of the system and its dynamic behaviors. The fractional order damping forces acting on the HTG shaft in the stable operation can be written as

where Fx and Fy are the damping forces on the x axis and y axis and c is the damping coefficient.

The fractional order model of the oil film forces can be described as

where Fx0-off and Fy0-off are the oil-film forces at the quiescent operation point and the symbols kxx, kxy, kyx, kyy, cxx, cxy, cyx, and cyy are the linearized stiffness and damping coefficients.

The generator included in the HTG is composed by the rotor and the stator. When the rotor is moved rapidly relative to the stator, the uneven air gap causes an asymmetric magnetic pull. When the generator has more than three pole pairs, the asymmetric magnetic pull can be described as

where d is the radius of the rotor, L is the length of the generator rotor, μ0 denotes the magnetic permeability of the air, ki is the coefficient of magneto-motive force fundamental wave of the air gap, Ii represents the exciting current of the generator and v0, v1, v2, and v3 are intermediate variables.

From Fig. 2, we can obtain the relationship of the axis of the generator rotor and turbine runner

where x1, y1 and x2, y2 are the coordinates of the axis of the generator rotor and turbine runner, respectively, r is the eccentric distance, ω is the angular speed and θ0 is the phase shift.

Fig. 2. (color online) Schematic diagrams of the shaft misalignment and generator air-gap eccentricity.

We have

where xm1, ym1, xm2, and ym2 are the mass center of the generator rotor and the turbine runner, respectively, φ0 is the initial phase of the generator rotor, and e1 and e2 denote the mass eccentricity of the generator and turbine.

2.3. Coupled HGS modeling

Assuming that the static balance is the point of potential energy, the potential U and kinetic T energies of the shaft system can be described as

where J1 and J2 are the rotary inertia of the generator rotor and turbine runner, m1 and m2 is the quality of the generator rotor and turbine runner, respectively, k1 and k2 are the stiffness of the rotor bearing and the turbine runner, respectively.

The Lagrange equations of the hydraulic turbine generator shaft system are

where Mt is the hydro-turbine torque, Mg is the generator magnetic torque and
Then, we obtain
where
and

Besides, the relationship between the relative deviation of the angular speed and the angular speed is

where ωB is the standard value of the angular speed.

From the above equations, the coupled fractional model of the HGS can be written as

3. Numerical simulation

The intermediate variable e, as a transfer coefficient in the HTGS, is always changing with the operating condition. To study the effect of this variability of e on the dynamic characteristics of the HGS, introducing e as a periodic excitation into the HGS model, and the equation of e is

where A and Ω are the amplitude and frequency of the periodic excitation.

The parameters values and initial values of the HGS variables are shown in Table 1 and Table 2.

Table 1.

Parameters values.

.
Table 2.

Initial values of the HGS variables for the simulation.

.
3.1. Dynamic behaviors

In this section, the effect of the periodic excitation e on the dynamic characteristics of the HGS is investigated. Considering the influence of different frequency of e, taking four sets of different frequency e to simulate the HGS dynamic behaviors. The basic value of e, i.e., the amplitude of the excitation A = 1.1 in this paper and PID controller parameter values are kp = 2.7, ki = 1, and kd = 1.5, and four sets e frequency are 1, 0.1, 0.01, and 0.001. Four sets dynamic behaviors of the relative value of rotational speed n and displacement of x axis are shown in Fig. 3.

Fig. 3. (color online) The time waveforms of the rotational speed n and displacement of the x axis (a1) and (a2) Ω = 0.001; (b1) and (b2) Ω = 0.01; (c1) and (c2) Ω = 0.1; (d1) and (d2) Ω = 1.

From Fig. 3, the periodic excitation has significant effect on the dynamic behaviors of the HGS, and the dynamics phenomena have obvious difference under different e frequency. Periodic fast–slow dynamics (PFS) appears as an alternating transition from the quiescent state to the spiking state.[27,28] As shown in Figs. 3(a1) and 3(a2) and magnification Fig. 3(e), n shows obvious periodic fast–slow dynamics when Ω = 0.001. Meanwhile, the displacement of the x axis shows disordered vibration when n is under PFS effects. The fast–slow effect accused by the different scale between the natural frequency and the excitation frequency. Spiking state is in the fast time-scale, which refers to the concentrative field of the fast changing trajectory in the time history, and the trajectory connecting the spiking states corresponds to the quiescent state.

When Ω = 0.01, the dynamic behaviors of n transitions from the non-periodic fast–slow state (NPFS) to the non-periodic oscillation state (NPO) and finally to the stable state. From magnification 2 (Fig. 3(f)), the spiking state of the fast–slow phenomenon gradually weakens until it enters the NPO. When n at the NPO and NPFS, x axis shows disordered vibration, and enters the stable periodic vibration state when n enters stable state.

When Ω = 0.1, n shows periodic motion from ENPO (gradually enhanced NPO) to PFS to NPO to stable state. From magnification 3 (Fig. 3(g)), the dynamic behaviors of n directly transition from the PFS to the NPO instead of the NPFS weakening to the NPO. Under this kind of effect, the displacement of the x axis shows periodic alternating transition from the disordered vibration state to the stable periodic vibration state.

When Ω = 1, a new fast–slow effect appears that n transition from the quiescent state 1 to the spiking state 1 to the quiescent state 2 to an oscillation state and then to the spiking state 2, as shown in magnification 4 (Fig. 3(h)). The displacement of x axis shows the same disordered vibration as Ω = 0.001.

From this analysis, it can be seen that the periodic excitation has an obvious fast–slow effect on the HGS dynamics, and there is significant difference of the dynamic behaviors under different frequency of the periodic excitation. The PFS, NPFS, NPO, and stable state alternately transformed each other with the change of the e. Meanwhile, the dynamic behaviors of the displacement of x axis correspondingly changes with n. For the further analysis of the fast–slow effect, the phase diagrams of n versus rotor angle δ are shown in Fig. 4.

Fig. 4. (color online) The phase diagrams of the rotational speed n versus rotor angle δ. (a) Ω = 0.001; (b) Ω = 0.01; (c) Ω = 0.1; (d) Ω = 1.

From Fig. 4, the phase trajectories show obvious fast–slow dynamics phenomena, there is big difference under different excitation frequency. Figure 4(a) shows standard fast–slow dynamics phase trajectory that is an alternating transition process from quiescent state 1 to spiking state 1 to quiescent state 2 to spiking state 2. When Ω = 0.01, consistent with the time waveforms, the phase trajectory shows different scale fast–slow phenomena and gradually enters the NPO state. Due to the small scale PFS when Ω = 0.1, the major part of the phase trajectory shows the NPO and stable state. When Ω = 1, the phase trajectory shows the cross transitions of the spiking state, quiescent state and the oscillation state.

3.2. Fractional order effect

The fractional order damping forces are more accurate than the integer order in the description of the dynamic behaviors of the vibration system. To analyze the effect of the fractional order on the dynamic characteristics of the HGS, the x axis responses under different fractional order are shown in Fig. 5.

Fig. 5. (color online) Dynamic responses of the x-axis under different fractional orders. (a1)–(a3) Ω = 0.001; (b1)–(b3) Ω = 0.01; (c1)–(c3) Ω = 0.1; (d1)–(d3) Ω = 1.

From Fig. 5, when Ω = 0.001 and 1, the dynamic responses of x axis are in the disordered vibration state affected by PFS, ENPO, and NPO (Ω = 1). As the increase of the fractional order, the amplitude of the disordered vibration gradually enhanced, and this phenomenon is more noticeable when the fractional order increases to 1. When Ω = 0.01 and 0.1, the x-axis state is alternating transforming from disordered vibration to stable periodic vibration. The stable periodic vibration states experience almost no change with the increase of the fractional order, while the amplitude of the disordered vibration states are enhanced. It is worth noting that the degree of amplitude enhancement at the two state transitions are more pronounced.

3.3. Bifurcation characteristics

The introduction of the periodic excitation into the intermediate variable e has a significant effect on the dynamic behaviors of the HGS. Meanwhile, the difference of excitation parameter values caused diverse dynamic phenomena. To further analyze the effect of the periodic excitation on the stability of the HGS, the bifurcation diagrams of the rotational speed n with increase of the periodic excitation frequency Ω are shown in Fig. 6.

Fig. 6. (color online) The bifurcation diagrams of the rotational speed n with increase of the periodic excitation frequency Ω. (a) Ω from 0.01 to 1; (b) Ω from 0.01 to 0.1; (c) Ω from 0.05 to 0.2; (d) Ω from 0.2 to 0.5; (e) Ω from 0.5 to 1; (f) Ω from 1 to 5.

As shown in Fig. 6(a), the bifurcation of n has significantly evolution characteristic with the increase of the frequency Ω, and there are three key parts in the evolution process. First, as shown in Figs. 6(b)–6(c), n is converted from two clusters of symmetrical bifurcations into a cluster of convergent bifurcations that finally enter stable state. When Ω> 0.16, the bifurcation behaviors of n enter an attenuation bifurcation cluster with obvious fluctuation rules. It worth noting that when Ω is around 0.34, the wave trough of the fluctuating bifurcation clusters overlapped makes this point close to the stable state. As shown in Fig. 6(f), when the Ω > 1, the bifurcation behaviors enter the chaos state, and the degree of the chaos gradually decrease. The slow-fast dynamics has significant effects on the system stability, meanwhile, the effect degree is related to the state of the slow-fast behaviors caused by the different frequency of e.

4. Conclusions

In this work, a fractional order model of HGS is presented, coupled by the HTGS with rigid water hammer and HTGU with fractional order damping forces. To describe the effect of the dynamic transfer coefficients e on the HGS accurately, the periodic excitation is introduced in to the HGS. Dynamic behaviors, effects of the fractional order and the evolution characteristics are analyzed in detail.

Obvious fast–slow dynamic phenomena are found in the process and multiple dynamic behaviors alternately transform each other as the periodic excitation frequency changes. Fast–slow dynamic behaviors caused corresponding disordered vibration on the shaft. Fractional order changes have little effect on the stable dynamic behaviors of the x axis, while they have the greatest impact on the junction points of unstable and stable states, causing larger vibration amplitude. Moreover, the HGS bifurcation behaviors shows an evolution trend from bifurcation to chaos with the increase of the excitation frequency and some interesting phenomena take place in the process.

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