Analytical solution to swing equations in power grids
Autoři:
HyungSeon Oh aff001
Působiště autorů:
Department of Electrical and Computer Engineering, United States Naval Academy, Annapolis, Maryland, United States of America
aff001
Vyšlo v časopise:
PLoS ONE 14(11)
Kategorie:
Research Article
doi:
https://doi.org/10.1371/journal.pone.0225097
Souhrn
Objective
To derive a closed-form analytical solution to the swing equation describing the power system dynamics, which is a nonlinear second order differential equation.
Existing challenges
No analytical solution to the swing equation has been identified, due to the complex nature of power systems. Two major approaches are pursued for stability assessments on systems: (1) computationally simple models based on physically unacceptable assumptions, and (2) digital simulations with high computational costs.
Motivation
The motion of the rotor angle that the swing equation describes is a vector function. Often, a simple form of the physical laws is revealed by coordinate transformation.
Methods
The study included the formulation of the swing equation in the Cartesian coordinate system, which is different from conventional approaches that describe the equation in the polar coordinate system. Based on the properties and operational conditions of electric power grids referred to in the literature, we identified the swing equation in the Cartesian coordinate system and derived an analytical solution within a validity region.
Results
The estimated results from the analytical solution derived in this study agree with the results using conventional methods, which indicates the derived analytical solution is correct.
Conclusion
An analytical solution to the swing equation is derived without unphysical assumptions, and the closed-form solution correctly estimates the dynamics after a fault occurs.
Klíčová slova:
Differential equations – Eigenvalues – Electrical faults – Inertia – Rotors – Simulation and modeling – System instability – Mechanical energy
Zdroje
1. U. S. Department of Energy. United States Electricity Industry Primer. July 2015; [Online] Available at https://www.energy.gov/sites/prod/files/2015/12/f28/united-states-electricity-industry-primer.pdf.
2. A Harris Williams & Co. Transmission & Distribution Infrastructure White Paper. Summer 2010, [Online] Available at https://www.harriswilliams.com/sites/default/files/industry_reports/final%20TD.pdf.
3. U.S. Department of Energy. Large Power Transformers and the U.S. Electric Grid. June 2012; [Online] Available at https://www.energy.gov/sites/prod/files/Large%20Power%20Transformer%20Study%20-%20June%202012_0.pdf
4. Pfeifenberger J., Spees K., and Carden K, Resource adequacy requirements: reliability and economic implications., September 2013; [Online] Available at https://www.ferc.gov/legal/staff-reports/2014/02-07-14-consultant-report.pdf.
5. Kundur P. Power system stability and control. New York, NY, USA: McGraw-Hill, 1994.
6. Winfree A. Biological rhythms and the behavior of populations of coupled oscillators. Journal of Theoretical Biology. 1967; 16: 15–42. doi: 10.1016/0022-5193(67)90051-3 6035757
7. Rodrigues F., Peron T., Ji P., and Kurths J. The Kuramoto model in complex networks. Physics Reports. 2016; 610: 1–98.
8. Kuramoto Y, Araki H. Lecture notes in Physics: International Symposium on Mathematical Problems in Theoretical Physics. 1975. 39 Springer-Verlag, New York, 420.
9. Dorfler F., Chertkov M., and Bullo F. Synchronization in complex oscillator networks and smart grids., PNAS. 2013; 110: 2005–2010. doi: 10.1073/pnas.1212134110 23319658
10. Willems J, Willems J. The application of Lyapunov methods to the computation of transient stability regions for multimachine power systems. IEEE Pow. App. Syst. 1970; PAS-89: 795–801.
11. Pai M., Padiyar K., and Radihakrishna C. Transient stability analysis of multimachine ac-dc power systems via energy function method. IEEE Pow. Eng. Rev. 1981; PAS-100: 49–50.
12. Chiang H. Direct methods for stability analysis of electric power systems. Hoboken, NJ, USA: Wiley, 2011.
13. Vu T, Turitsyn K. A framework for robust assessment of power grid stability and resiliency. IEEE T. Auto. Cont. 2017; 62
14. Chiang H. Study of the existence of energy functions for power systems with losses. IEEE T. Circ. Syst. 1989; 39: 1423–1429.
15. Vu T, Turitsyn K. Synchronization stability of lossy and uncertain power grids. 2015 Amer. Cont. Conf. Chicago, IL, USA, July, 2015.
16. Hill D, Chong C. Lyapunov functions of Lur’e-Postnikov form for structure preserving models of power systems. Automatica. 1989; 25: 453–460.
17. Filatrella G., Nielsen A., and Pedersen N. Analysis of a power grid using a Kuramoto-like model. Europ. Phys. J. 2008; 61: 485–491.
18. Huang Z. Jin S., amd Diao R. Predictive dynamic simulation for large-scale power systems through high-performance computing. High Perf. Comp. Net. Stor. Anal. (SCC). 2012; 347–354.
19. Nagel I., Fabre L, Pastre M., Krummenacher F., Cherkaoui R., and Kayal M. High-speed power system transient stability simulation using highly dedicated hardware. IEEE T. Pow. Syst. 2013; 28: 4218–4227.
20. Butcher J. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, New York, USA, 2003.
21. Ascher U, Petzold L. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia, Society for Industrial and Applied Mathematics, 1998.
22. Runge C. Über die numerische Auflösung von Differentialgleichungen. Mathematische Annalen, 1895; 46: 167–178.
23. Kosterev D., Taylor C., and Mittelstadt W. Model validation for the August 10, 1996 WSCC system outage. IEEE T. Pow. Syst. 1999; 14: 967–979.
24. Anderson P, Fouad A. Power system control and stability. Piscataway, NJ, USA: Wiley, 2003.
25. Karamata J. Sur un mode de croissance r´eguli`ere. Th´eor`emes fondamentaux. Bull. Soc. Math. France. 1933; 61: 55–62.
26. Grainger J, Stevenson W. Power System Analysis, New York, NY USA: McGraw-Hill. 1994.
27. Karlsson D, Hill D. Modeling and identification of nonlinear dynamic loads in power systems. IEEE Transaction on Power Systems. 1994; 9: 157–166.
28. Jereminov M, Pandey A, Song HA, Hooi B, Faloutsos C, Pileggi L. Linear load model for robust power system analysis. IEEE PES Innovative Smart Grid Technologies. Torino Italy, September 2017.
29. Song HA, Hooi B, Jereminov M, Pandey A, Pileggi L, Faloutsos C. PowerCast: Mining and forecasting power grid sequences. Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Springer 2017.
30. Arif A., Wang Z., Wang J., Mather B., Bashualdo H., and Zhao D. Load modeling–a review. IEEE Transaction of Smart Grid. 2017; 9: 5986–5999.
31. Golub G, Van Loan C. Matrix computation. Baltimore, MD, USA: The Johns Hopkins Univ. Press, 2013.
32. Overbye T. Power system analysis lecture 14. [Online] Available at https://slideplayer.com/slide/9086637/
33. Sauer P, Pai M, Chow J. Power System Dynamics and Stability. Hoboken, NJ, USA: John Wiley and Sons, 2017.
34. Hairer E, Wanner G. Solving ordinary differential equations II–stiff and differential-algebraic problems. Springer series in computational mathematics, 2nd ed., Geneva, Switzerland, 2000.
35. Berzins M., Dew P., and Furzeland R. Developing software for time dependent problems using the method of lines and differential algebraic integrators. Applied Numerical Mathematics. 1988; 5: 375–397.
36. Kröner A., Marquardt W., and Gilles E. Computing consistent initial conditions for differential algebraic equations. Computers and Chemical Engineering. 1992; 16: 131–138 (suuplement).
37. Campbell S. Consistent initial conditions for linear time varying singular systems. Frequency Domain and State Space Methods for Linear Systems. 1986; 313–318.
38. Campbell S. A computational method for general higher index nonlinear singular systems of differential equations. IMACS Transactions on Scientific Computing. 1989; 12: 555–560.
39. Pantelides C. The consistent initialization of differential algebraic systems. SIAM Journal of Scientific and Statistical Computing. 1988; 9: 213–231.
40. Pantelides C. Speedup–recent advances in process simulation. Computers and Chemical Engineering. 1988; 12: 745–755.
41. Barlow J. Error analysis of update methods for the symmetric eigenvalue problem. SIAM J. Matrix Anal. Appl. 1993; 14: 598–618.
42. Oh H, Hu Z. Multiple-rank modification of symmetric eigenvalue problem. MethodsX. 2018; 5: 103–117. doi: 10.1016/j.mex.2018.01.001 30619724
43. Wilkinson J. The algebraic eigenvalue problem. Oxford Univ. Press, London, 1965.
44. Oh H. A unified and efficient approach to power flow analysis. Energies, Multidisciplinary Digital Publishing Institute. 2019; 12: 2425.
45. Weber J. Description of machine models GENROU, GENSAL, GENTPF and GENTPJ. December 3, 2015. [Online] Available at https://www.powerworld.com/files/GENROU-GENSAL-GENTPF-GENTPJ.pdf.
46. Runge C. Über die numerische Auflösung von Differentialgleichungen. Mathematische Annalen, Springer. 1895; 46: 167–178.
Článek vyšel v časopise
PLOS One
2019 Číslo 11
- S diagnostikou Parkinsonovy nemoci může nově pomoci AI nástroj pro hodnocení mrkacího reflexu
- Proč při poslechu některé muziky prostě musíme tančit?
- Je libo čepici místo mozkového implantátu?
- Chůze do schodů pomáhá prodloužit život a vyhnout se srdečním chorobám
- Pomůže v budoucnu s triáží na pohotovostech umělá inteligence?
Nejčtenější v tomto čísle
- A daily diary study on maladaptive daydreaming, mind wandering, and sleep disturbances: Examining within-person and between-persons relations
- A 3’ UTR SNP rs885863, a cis-eQTL for the circadian gene VIPR2 and lincRNA 689, is associated with opioid addiction
- A substitution mutation in a conserved domain of mammalian acetate-dependent acetyl CoA synthetase 2 results in destabilized protein and impaired HIF-2 signaling
- Molecular validation of clinical Pantoea isolates identified by MALDI-TOF
Zvyšte si kvalifikaci online z pohodlí domova
Všechny kurzy