Numerical Methods Coupled with Richardson Extrapolation for Computation of Transient Power Systems

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Whady Felipe Florez
Jorge W Gonzalez
Alan F Hill
Gabriel J Lopez
Juan D Lopez


Power system transient stability, Richardson extrapolation, dynamic equations


The numerical solution of transient stability problems is a key element for electrical power system operation. The classical model for multi-machine systems is defined as a set of non-linear differential equations for the rotor speed and the generator angle for each electrical machine, this mathematical model is usually known as the swing equations. This paper presents how to use direct Richardson extrapolation of several orders for the numerical solution of the swing equations and compares it with other commonly used implicit and explicit solvers such as Runge-Kutta, trapezoidal, Shampine and Radau methods. A numerical study on a simple three machine system is used to illustrate the performance and implementation of algebraic Richardson extrapolation coupled to several solution methods. Normally, the order of accuracy of any numerical solution can be increased when Richardson Extrapolation is used. A numerical example is provided for an electrical grid consisting of three machines and nine buses undergoing a disturbance. It is shown that in this case Richardson extrapolation effectively increases the order of accuracy of the explicit methods making them competitive with the implicit methods. 


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[1] M. Watanabe, Y. Mitani, and K. Tsuji, “A numerical method to evaluate power system global stability determined by limit cycle,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1925–1934, Nov 2004.

[2] J. H. Chow and K. W. Cheung, “A toolbox for power system dynamics and control engineering education and research,” IEEE Transactions on Power Systems, vol. 7, no. 4, pp. 1559–1564, Nov 1992.

[3] F. Milano, “Semi-implicit formulation of differential-algebraic equations for transient stability analysis,” IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–10, 2016.

[4] L. Shampine, “Solving 0 = f(t, y(t), y(t)) in matlab,” Numerical Mathematics, vol. 10, pp. 291 – 310, 2002.

[5] E. Hairer and G. Wanner, “Stiff differential equations solved by radau methods,” Journal of Computational and Applied Mathematics, vol. 111, no. 1002, pp. 93 – 111, 1999. [Online]. Available:

[6] A. Aubry and P. Chartier, “On the structure of errors for radau ia methods applied to index-2 daes,” Applied Numerical Mathematics, vol. 22, no. 1, pp. 23 – 34, 1996. [Online]. Available:

[7] K. S. Shetye, T. J. Overbye, S. Mohapatra, R. Xu, J. F. Gronquist, and T. L. Doern, “Systematic determination of discrepancies across transient stability software packages,” IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 432–441, Jan 2016.

[8] A. Singh and S. Parida, “A multiple strategic evaluation for fault detection in electrical power system,” International Journal of Electrical Power & Energy Systems, vol. 48, pp. 21 – 30, 2013. [Online]. Available:

[9] A. Paul M. Anderson, Power System Control and Stability. Wiley, 2002.

[10] W. John J. Grainger, Power system analysis. McGraw-Hill, 1994.

[11] P. K. Iyambo and R. Tzoneva, “Transient stability analysis of the ieee 14-bus electric power system,” in AFRICON 2007, Sept 2007, pp. 1–9.

[12] A. Farraj, E. Hammad, and D. Kundur, “A cyber-physical control framework for transient stability in smart grids,” IEEE Transactions on Smart Grid, vol. PP, no. 99, pp. 1–10, 2016.

[13] C. Rehtanz and X. Guillaud, “Real-time and co-simulations for the development of power system monitoring, control and protection,” in 2016 Power Systems Computation Conference (PSCC), June 2016, pp. 1–20.

[14] S. Teleke, “Modeling and comparison of synchronous condenser and svc,” Master’s thesis, Chalmers University Of Technology, 2006.

[15] Z. Zlatev, K. Georgiev, and I. Dimov, “Studying absolute stability properties of the richardson extrapolation combined with explicit runge kutta methods,” Computers & Mathematics with Applications, vol. 67, no. 12, pp. 2294 – 2307, 2014, efficient Algorithms for Large Scale Scientific Computations. [Online]. Available:

[16] Y. C and H. Dag, “Detailed model for power system transient stability analysis,” Journal of Engineering & Computer Science IJECS-IJENS, vol. 13, no. 2, pp. 1 – 5, 2012.

[17] D. XIE, “A new numerical algorithm for efficiently implementing implicit runge-kutta methods,” Department of Mathematical Sciences. University of Wisconsin, Milwaukee, Wisconsin, USA, pp. 1 – 17, 2009.

[18] R. Zarate-Minano, T. V. Cutsem, F. Milano, and A. J. Conejo, “Securing transient stability using time-domain simulations within an optimal power flow,” IEEE Transactions on Power Systems, vol. 25, no. 1, pp. 243–253, Feb 2010.

[19] S. Sharma, S. Kulkarni, A. Maksud, S. R. Wagh, and N. M. Singh, “Transient stability assessment and synchronization of multimachine power system using kuramoto model,” in North American Power Symposium (NAPS), 2013, Sept 2013, pp. 1–6.

[20] S. Ekinci and A. Demiroren, “Transient stability simulation of multi-machine power systems using simulink,” Electrical & Electronics Engineering, vol. 15, no. 2, pp. 1 – 8, 2015.

[21] Q. M. R. M. A. Salam, M. A. Rashid and M. Rizon, “Transient stability analysis of a three-machine nine bus power system network,” Engineering Letters, vol. 22, no. 1, pp. 1 – 7, 2014.

[22] F. Dorfler and F. Bullo, “Kron reduction of graphs with applications to electrical networks,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 1, pp. 150–163, Jan 2013.

[23] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed. New York, NY, USA: Cambridge University Press, 2007.

[24] MATLAB, version R2015. Natick, Massachusetts: The MathWorks Inc., 2015.

[25] J. C. Butcher, “Implicit runge-kutta processes,” Mathematics of Computation, vol. 18, no. 85, pp. 50–64, 1964. [Online]. Available:

[26] D. Süli, Endre; Mayers, An Introduction to Numerical Analysis. Cambridge University Press, 2003.

[27] Digsilent powerfactory. [Online]. Available: