Almost global asymptotic stability of a grid-connected synchronous generator. (English) Zbl 1396.93104
Summary: We study the global asymptotic behavior of a grid-connected constant field current Synchronous Generator (SG). The grid is regarded as an “infinite bus”, i.e., a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo \(2\pi \)). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the exact swing equation, which resembles a forced pendulum equation and is equivalent to our fourth-order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500-kW synchronverter.
MSC:
| 93D20 | Asymptotic stability in control theory |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C95 | Application models in control theory |
Keywords:
synchronous machine; infinite bus; almost global asymptotic stability; forced pendulum equation; synchronverterReferences:
| [1] | Angeli, D; Efimov, D, Characterizations of input-to-state s tability for systems with multiple invariant sets, IEEE Trans Autom Control, 60, 3242-3256, (2015) · Zbl 1360.93627 · doi:10.1109/TAC.2015.2418676 |
| [2] | Barabanov N, Schiffer J, Ortega R, Efimov D (2016) Almost global attractivity of a synchronous generator connected to an infinite bus. In: Proceedings of the 55th IEEE conference on decision and control, Las Vegas, December 2016 · Zbl 1390.93597 |
| [3] | Barabanov, N; Schiffer, J; Ortega, R; Efimov, D, Conditions for almost global attractivity of a synchronous generator connected to an infinite bus, IEEE Trans Autom Control, 62, 4905-4916, (2017) · Zbl 1390.93597 · doi:10.1109/TAC.2017.2671026 |
| [4] | Beck HP, Hesse R (2007) Virtual synchronous machine. In: Proceedings of 9th international conference on electrical power quality and utilisation (EPQU), Barcelona, Spain, pp 1-6 |
| [5] | Brown E (2015) A study of the use of synchronverters for grid stabilization using simulations in SimPower. M.Sc. thesis, Tel Aviv University |
| [6] | Caliskan, SY; Tabuada, P, Compositional transient stability analysis of multimachine power networks, IEEE Trans Control Netw Syst, 1, 4-14, (2014) · Zbl 1370.90036 · doi:10.1109/TCNS.2014.2304868 |
| [7] | Demello, FP; Concordia, C, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans Power Appar Syst, 88, 316-327, (1969) · doi:10.1109/TPAS.1969.292452 |
| [8] | Dong, S; Chi, YN; Li, Y, Active voltage feedback control for hybrid multi-terminal HVDC system adopting improved synchronverters, IEEE Trans Power Deliv, 31, 445-455, (2016) · doi:10.1109/TPWRD.2015.2420657 |
| [9] | Dörfler, F; Bullo, F, Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators, SIAM J Control Optim, 50, 1616-1642, (2012) · Zbl 1264.34105 · doi:10.1137/110851584 |
| [10] | Dörfler, F; Bullo, F, Synchronization in complex networks of phase oscillators: a survey, Automatica, 50, 1539-1564, (2014) · Zbl 1296.93005 · doi:10.1016/j.automatica.2014.04.012 |
| [11] | Driesen J, Visscher K (2008) Virtual synchronous generators. In: IEEE Power and energy society general meeting—conversion and delivery of electrical energy in the 21st century, Pittsburg, PA, July 2008, pp 1-3 · Zbl 0063.06720 |
| [12] | Efimov, D; Schiffer, J; Ortega, R, Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids, Int J Control, 89, 909-918, (2016) · Zbl 1338.93333 · doi:10.1080/00207179.2015.1104555 |
| [13] | Efimov, D; Schiffer, J; Barabanov, N; Ortega, R, A relaxed characterization of ISS for periodic systems with multiple invariant sets, Eur J Control, 37, 1-7, (2017) · Zbl 1373.93295 · doi:10.1016/j.ejcon.2017.04.002 |
| [14] | Farkas, B; Wegner, SA, Variations on barbălat’s lemma, Am Math Monthly, 123, 825-830, (2016) · Zbl 1391.26005 · doi:10.4169/amer.math.monthly.123.8.825 |
| [15] | Fiaz, S; Zonetti, D; Ortega, R; Scherpen, JMA; Schaft, AJ, A port-Hamiltonian approach to power network modeling and analysis, Eur J Control, 19, 477-485, (2013) · Zbl 1293.90007 · doi:10.1016/j.ejcon.2013.09.002 |
| [16] | Fitzgerald AE, Kingsley C, Umans SD (2003) Electric machinery, 6th edn. McGraw-Hill, New York |
| [17] | Galaz, M; Ortega, R; Bazanella, AS; Stankovic, AM, An energy-shaping approach to the design of excitation control of synchronous generators, Automatica, 39, 111-119, (2003) · Zbl 1006.93560 · doi:10.1016/S0005-1098(02)00177-2 |
| [18] | Grainger JJ, Stevenson WD (1994) Power systems analysis. McGraw-Hill, New York |
| [19] | Halanay, A; Leonov, GA; Răsvan, V, From pendulum equation to an extended analysis of synchronous machines, Rend Seminario Matematico Univers e Politecnico di Torino, 45, 91-106, (1987) · Zbl 0668.34054 |
| [20] | Hayes, WD, On the equation for a damped pendulum under a constant torque, Zeitschrift für angewandte Mathematik und Physik, 4, 398-401, (1953) · Zbl 0053.24503 · doi:10.1007/BF02074983 |
| [21] | Jayawardhana, B; Weiss, G, State convergence of passive nonlinear systems with an \(L^2\) input, IEEE Trans Autom Control, 54, 1723-1727, (2009) · Zbl 1367.93435 · doi:10.1109/TAC.2009.2020661 |
| [22] | Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River · Zbl 1003.34002 |
| [23] | Kothari DP, Nagrath IJ (2004) Electric machines, 3rd edn. Tata McGraw-Hill, New Delhi |
| [24] | Kundur P (1994) Power system stability and control. McGraw-Hill, New York |
| [25] | Leonov GA, Ponomarenko DV, Smirnova VB (1996) Frequency-domain methods for nonlinear analysis: theory and applications. World Scientific Publishing Company, Singapore · Zbl 0954.65091 · doi:10.1142/2638 |
| [26] | Logemann, H; Ryan, EP, Asymptotic behaviour of nonlinear systems, Am Math Monthly, 111, 864-889, (2004) · Zbl 1187.34068 · doi:10.1080/00029890.2004.11920152 |
| [27] | Mandel, Y; Weiss, G, Adaptive internal model based suppression of torque ripple in brushless DC motor drives, Syst Sci Control Eng Open Access J, 3, 162-176, (2015) · doi:10.1080/21642583.2014.999387 |
| [28] | Natarajan V, Weiss G (2014) Almost global asymptotic stability of a constant field current synchronous machine connected to an infinite bus. In: Proceedings of the 53rd IEEE conference on decision and control, Los Angeles, pp 3272-3279 · Zbl 1006.93560 |
| [29] | Natarajan V, Weiss G (2014) A method for proving the global stability of a synchronous generator connected to an infinite bus. In: Proceedings of the IEEE 28th Convention of Electrical and Electronics Engineers in Israel, Eilat |
| [30] | Natarajan, V; Weiss, G, Synchronverters with better stability due to virtual inductors, virtual capacitors and anti-windup, IEEE Trans Ind Electron, 64, 5994-6004, (2017) · doi:10.1109/TIE.2017.2674611 |
| [31] | Park, RH, Two-reaction theory of synchronous machines. generalized method of analysis-part I, Trans Am Inst Electr Eng, 48, 716-727, (1929) · doi:10.1109/T-AIEE.1929.5055275 |
| [32] | Sauer PW, Pai MA (1997) Power systems dynamics and stability. Stipes Publishing, Champaign |
| [33] | Sauer, PW; Pai, MA; Savulescu, SC (ed.), Power system dynamic equilibrium, power flow and steady-state stability, 1-26, (2014), Cham |
| [34] | Sard, A, The measure of the critical values of differentiable maps, Bull Am Math Soc, 48, 883-890, (1942) · Zbl 0063.06720 · doi:10.1090/S0002-9904-1942-07811-6 |
| [35] | Schiffer, J; Efimov, D; Ortega, R; Barabanov, N, An input-to-state stability approach to verify almost global stability of a synchronous-machine-infinite-bus system, Philos Trans R Soc A, (2017) · Zbl 1390.93597 · doi:10.1098/rsta.2016.0304 |
| [36] | Szlenk W (1984) An introduction to the theory of smooth dynamical systems. John Wiley & Sons, Chichester · Zbl 0566.58015 |
| [37] | Venezian E, Weiss G (2016) A warning about the use of reduced models of synchronous generators. In: Proceedings of international conference on the science of electrical engineering (ICSEE) Eilat, Israel |
| [38] | Walker JH (1981) Large synchronous machines: design, manufacture and operation. Oxford University Press, Oxford |
| [39] | Yu, YN; Vongsuriya, K, Nonlinear power system stability study by Liapunov function and zubov’s method, IEEE Trans Power Appar Syst, 86, 1480-1485, (1967) · doi:10.1109/TPAS.1967.291911 |
| [40] | Zhong, QC; Nguyen, PL; Ma, Z; Sheng, W, Self-synchronized synchronverters: inverters without a dedicated synchronization unit, IEEE Trans Power Electron, 29, 617-630, (2014) · doi:10.1109/TPEL.2013.2258684 |
| [41] | Zhong QC, Weiss G (2009) Static synchronous generators for distributed generation and renewable energy. In: proceedings of the IEEE PES power systems conference & exhibition, Washington, USA · Zbl 1390.93597 |
| [42] | Zhong, QC; Weiss, G, Synchronverters: inverters that mimic synchronous generators, IEEE Trans Ind Electron, 58, 1259-1267, (2011) · doi:10.1109/TIE.2010.2048839 |
| [43] | Zhou, J; Ohsawa, Y, Improved swing equation and its properties in synchronous generators, IEEE Trans Circuit Syst, 56, 200-209, (2009) · Zbl 1468.94889 · doi:10.1109/TCSI.2008.924895 |
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.
