The Comparison of Approximations of Nonlinear Functions Combined with Harmonic Balance

Method for Power System Oscillation Frequency Estimation

Abigail C. Teron Allan Bartlett CURENT REU

Final PresentationJuly 22, 2015

Min H. Kao Building

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

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

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

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

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

• Sine estimation

• Cosine estimation

Sine and cosine estimation

Time (sec)-pi/2 -pi/4 0 pi/4 pi/2

y =

sin(

t)-1

-0.5

0

0.5

1

sin(t)

TECHEB-POL

PADE

CONFRAC

Time (sec)

-pi/2 -pi/4 0 pi/4 pi/2

y =

cos(

t)

-0.5

0

0.5

1

cos(t)TE

CHEB-POL

PADECONFRAC

Function approximation techniques applied to SMIB

5-8

• TE

• CHEB-POL

• PADE

• CONFRAC

HBM ASSUMPTION

• 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

Numerical Integration

5-10

Time (sec)0 1 2 3 4

δ (r

ad)

-2

-1

0

1

2

3T is period between peaks ω = 2*π / T

Time (sec)0.5 1 1.5 2

δ (

rad

)

-1

0

1

2

3T is period between first two peaks ω = 2*π / T

Variation of oscillation frequencies under different operating conditions

Results: Single Machine system

Chebyshev Polynomials approximation is superior

1 5 9 13 17 21 25 296

7

8

9

10

11

12

13

Single-machine oscillation frequencies comparison between different function approxima-tion approaches

Numerical Integration

TE

CHEB-POL

PADE

CONFRAC

Fault Duration (cycles)

ω(r

ad/s

)

Results: 2-machine system

Chebyshev Polynomials approximation is superior

1 5 9 13 17 21 25 293

3.5

4

4.5

5

5.5

6

2-machine oscillation frequencies comparison between different function approximation approaches

Numerical Integration

TE

CHEB-POL

PADE

CONFRAC

Fault Duration (cycles)

ω(r

ad/s

)

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.

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

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.

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Questions

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