1. The Comparison of Approximations of Nonlinear Functions Combined with Harmonic Balance Method for Power System Oscillation Frequency Estimation Abigail C. TeronAllan Bartlett CURENT REU Final Presentation July 22, 2015 Min H. Kao Building

2. Background Our project: analyze power system oscillation frequencies Oscillations are natural within a grid, but their stability is required for reliable and secure system operation When there is a disturbance, such as a tree falls on a line, the oscillations are affected If the oscillations are fast (high frequency), the system can naturally dampen the oscillations If the oscillations are slow (low frequency), there can be problems  Voltage instability  Outages  Loss of synchronism We want to find these oscillations If a system operator knows that the oscillation frequency is too low, he/she can utilize preventative measures, such as adjusting the PSS settings, in order to maintain grid stability 5-2

3. Problem: Numerical Integration Currently the most accurate technique for studying the electromechanical oscillations is by using numerical integration (NUMINT). NUMINT can be computationally expensive, with extremely long run times for a large system. A quicker, more efficient approach for estimating oscillations is needed. The function approximations can be analyzed and performed much faster than NUMINT, though not as precisely. 5-3

4. Harmonic Balance Method The HBM can be utilized to obtain an explicit expression in the time-domain to describe oscillatory motion. Previous HBM applications: -Aeronautics -Wireless applications -Analyzing atomic forces The HBM may also be applied to study frequency oscillations of power systems. 5-4

5. Swing Equation Describes how the rotor of synchronous machine will move when there is an unbalance between mechanical power fed into the machine and the electrical power extracted from it. Single machine system 2 machine system 5-5

6. Function approximation techniques Use four nonlinear function approximation techniques…  Taylor Expansion  Chebyshev Polynomials  Padé Approximant  Continuous Fraction Representation …to estimate or in the swing equations

7. Sine estimation Cosine estimation Sine and cosine estimation

8. Function approximation techniques applied to SMIB 5-8 TE CHEB-POL PADE CONFRAC HBM ASSUMPTION

9. Steps: - Substitute in the assumptions (previous slide) - Obtain equations to be of the form -Set -Use “solve” method in Maple to solve for A, B, C, - is oscillation frequency Harmonic Balance Method

10. Numerical Integration 5-10 Variation of oscillation frequencies under different operating conditions

11. Results: Single Machine system Chebyshev Polynomials approximation is superior

12. Results: 2-machine system Chebyshev Polynomials approximation is superior

13. Conclusions Chebyshev Polynomials is the best technique for both SMIB and 2 machine systems. Advantages: -They are not dependent upon an operating point -System operators can use the method offline to deduce the analytic frequency expressions. -If the system operator finds that the frequency of a system is unacceptable, then the utility will be able to enact anticipatory preventative measures to avoid the outages.

14. Future Work Scalability of Chebyshev Polynomials to more complex systems Explore Chebyshev Polynomials of second-,third-,fourth- kind Utilize higher order of Chebyshev Polynomial’s for better approximation

15. Acknowledgements This work was supported primarily by the ERC Program of the National Science Foundation and DOE under NSF Award Number EEC-1041877. Other US government and industrial sponsors of CURENT research are also gratefully acknowledged. 15

16. Questions 16