Hamiltonian model and dynamic analyses for a hydro-turbine governing system with fractional item and time-lag. (English) Zbl 1456.74038
Summary: This paper focus on a Hamiltonian mathematical modeling for a hydro-turbine governing system including fractional item and time-lag. With regards to hydraulic pressure servo system, a universal dynamical model is proposed, taking into account the viscoelastic properties and low-temperature impact toughness of constitutive materials as well as the occurrence of time-lag in the signal transmissions. The Hamiltonian model of the hydro-turbine governing system is presented using the method of orthogonal decomposition. Furthermore, a novel Hamiltonian function that provides more detailed energy information is presented, since the choice of the Hamiltonian function is the key issue by putting the whole dynamical system to the theory framework of the generalized Hamiltonian system. From the numerical experiments based on a real large hydropower station, we prove that the Hamiltonian function can describe the energy variation of the hydro-turbine suitably during operation. Moreover, the effect of the ractional \(\alpha\) and the time-lag \(\tau\) on the dynamic variables of the hydro-turbine governing system are explored and their change laws identified, respectively. The physical meaning between fractional calculus and time-lag are also discussed in nature. All of the above theories and numerical results are expected to provide a robust background for the safe operation and control of large hydropower stations.
MSC:
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74D05 | Linear constitutive equations for materials with memory |
| 74F05 | Thermal effects in solid mechanics |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Alidai, A.; Pothof, I. W.M., Hydraulic performance of siphonic turbine in low head sites, Renew Energy, 75, 505-511 (2015) |
| [2] | Duquesne, P.; Ciocan, G. D.; Aeschlimann, V.; Bombenger, A.; Deschenes, C., Pressure probe with five embedded flush-mounted sensors: unsteady pressure and velocity measurements in hydraulic turbine model, Exp Fluids, 54 (2013) |
| [3] | Strah, B.; Kuljaca, O.; Vukic, Z., Speed and active power control of hydro turbine unit, IEEE Trans Energy Conver, 20, 424-434 (2005) |
| [4] | Nagode, K.; Skrjanc, I., Modelling and internal fuzzy model power control of a Francis water turbine, Energies, 7, 874-889 (2014) |
| [5] | Li, J. Y.; Chen, Q. J., Nonlinear dynamical analysis of hydraulic turbine governing systems with nonelastic water hammer effect, J Appl Math (2014) · Zbl 1463.93106 |
| [6] | Liu, X. L.; Liu, C., Eigenanalysis of oscillatory instability of a hydropower plant including water conduit dynamics, IEEE Trans Power Syst, 22, 675-681 (2007) |
| [7] | Zeng, Y.; Zhang, L. X.; Guo, Y. K.; Dong, K. H., Hydraulic decoupling and nonlinear hydro turbine model with sharing common conduit, Proc CSEE, 32, 103-108 (2012), (in Chinese) |
| [8] | Shen, Z. Y., Hydraulic turbine regulation (1998), China Water Power Press: China Water Power Press Beijing, [in Chinese] |
| [9] | Zeng, Y. H.; Huai, W. X., Application of artificial neural network to predict the friction factor of open channel flow, Commun Nonlinear Sci, 14 (2009) |
| [10] | Xu, B. B.; Chen, D. Y.; Zhang, H.; Wang, F. F., Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system, Chaos Solitons Fract, 75, 50-61 (2015) · Zbl 1352.93019 |
| [11] | Chen, D. Y.; Ding, C.; Ma, X. Y.; Yuan, P.; Ba, D. D., Nonlinear dynamical analysis of hydro-turbine governing system with a surge tank, Appl Math Model, 37, 14-15 (2013) |
| [12] | Zhang, H.; Chen, D. Y.; Xu, B. B.; Wang, F. F., Nonlinear modeling and dynamic analysis of hydro-turbine governing system in the process of load rejection transient, Energ Convers Manage, 90, 128-137 (2015) |
| [13] | Ortigueira, M. D.; Machado, J. A.T., What is a fractional derivative, J Comput Phys, 293, 4-13 (2015) · Zbl 1349.26016 |
| [14] | Matouk, A. E.; Elsadany, A. A.; Ahmed, E.; Agiza, H. N., Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization, Commun Nonlinear Sci, 27, 153-167 (2015) · Zbl 1457.92189 |
| [15] | Baleanu, D.; Magin, R. L.; Bhalekar, S.; Daftardar-Gejji, V., Chaos in the fractional order Bloch equation with delay, Commun Nonlinear Sci, 25, 41-49 (2015) · Zbl 1440.78002 |
| [16] | Wagoner, R. H.; Li, M., Simulation of springback: through-thickness integration, Int J Plast, 23, 345-360 (2007) · Zbl 1349.74378 |
| [17] | Metzler, R.; Nonnenmacher, T. F., Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials, Int J Plast, 19, 941-959 (2003) · Zbl 1090.74673 |
| [18] | Kwuimy, C. A.K.; Litak, G.; Nataraj, C., Nonlinear analysis of energy harvesting systems with fractional order physical properties, Nonlinear dynam, 80, 491-501 (2015) |
| [19] | Sardar, T.; Rana, S.; Chattopadhyay, J., A mathematical model of dengue transmission with memory, Commun Nonlinear Sci, 22, 511-525 (2015) · Zbl 1329.92140 |
| [20] | Hu, J. B.; Lu, G. P.; Zhang, S. B.; Zhao, L. D., Lyapunov stability theorem about fractional system without and with delay, Commun Nonlinear Sci, 20 (2015) · Zbl 1311.34125 |
| [21] | Armero, F.; Simo, J. C., Apriori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity, Int J Plast, 9, 749-782 (1993) · Zbl 0791.73026 |
| [22] | Machado, J. A.T.; Galhano, A., Fractional dynamics: a statistical perspective, J Comput Nonlinear Dyn, 3 (2008) |
| [23] | Zeng, Y.; Zhang, L. X.; Xu, T. M.; Guo, Y. K., Hamiltonian model of nonlinear hydraulic turbine with elastic water column, Proc of the CSEE, 28 (2010), 525-520. (in Chinese) |
| [24] | Cheng, D. Z.; Xi, Z. R.; Lu, Q.; Mei, S. W., Geometric structure of generalized controlled Hamiltonian systems and its application, Sci China Ser E-Technol Sci, 43, 365-379 (2000) · Zbl 0981.53077 |
| [25] | Escobar, G.; Vander Schaft, A. J.; Ortega, R., A Hamiltonian viewpoint in the modeling of switching power converters, Automatica, 35, 445-452 (1999) · Zbl 0930.94052 |
| [26] | Yu, S. M.; Yao, Y. Q.; Shen, S. F.; Ma, W. X., Bi-integrable couplings of a Kaup-Newell type soliton hierarchy and their bi-Hamiltonian, Commun Nonlinear Sci, 23, 366-377 (2015) · Zbl 1351.37253 |
| [27] | Sugino, F.; Yoneya, T., Stochastic Hamiltonians for noncritical string field theories from double-scaled matrix models, Phys Rev D, 53, 4448-4488 (1996) |
| [28] | Geng, X. G.; Li, R. M.; Xue, B., A new hierarchy of integrable equation with peakons and cuspons and its bi-Hamiltonian structure, Appl Math Lett, 46, 64-69 (2015) · Zbl 1326.35066 |
| [29] | Wang, Y. Z.; Cheng, D. Z.; Ge, S. S., Approximate dissipative Hamiltonian realization and construction of local Lypaunov functions, Syst Control Lett, 56, 141-149 (2007) · Zbl 1108.93065 |
| [30] | Klemen, N.; Igor, S., Modelling and internal fuzzy model power control of a Francis water turbine, Energies, 7, 874-889 (2014) |
| [31] | Hydraulic-turbine and turbine control-models for system dynamic studies, IEEE Trans Power Syst, 7, 167-179 (1992) |
| [32] | Wang, Y. Z.; Cheng, D. Z.; Hu, X. M., Problems on time-varying port-controlled Hamiltonian system: geometric structure and dissipative realization - brief paper, Automatica, 41, 717-723 (2005) · Zbl 1062.93013 |
| [33] | Zeng, Y.; Zhang, L. X.; Guo, Y. K.; Qian, J.; Zhang, C. L., The generalized Hamiltonian model for the shafting transient analysis of the hydro turbine generating sets, Nonlinear Dynam, 76, 1921-1933 (2014) · Zbl 1314.37057 |
| [34] | Zeng, Y.; Zhang, L. X.; Qian, J.; Xu, T. M.; Guo, Y. K., Differential algebra model simulation for hydro turbine with elastic water column, J Drain Irrig Mach Eng, 32, 8, 691-697 (2014), (in Chinese) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
