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# Optimal Control Theory

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Identification &amp; Tracking of Harmonic Sources in a Power System using a Kalman Filter

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### Optimal Control Theory

1. 1. Identification & Tracking of Harmonic Sources in a Power System using a Kalman Filter<br />Optimal Control Theory II (ECGR 6116)<br />Prof. Dr. Kakad<br />Nikhil Kulkarni (Student ID 800696839)<br />Adly A. Girgis, Fellow, IEEE<br />Dept. of Elect & CompEngg.<br />Clemson University<br />Clemson, SC 29634<br />Haili Ma, Student Member, IEEE<br />Illinois Institute of Technology<br />Chicago, IL 60616<br />
2. 2. Introduction<br />Objective is to solve following 2 problems using Kalman Filtering approach.<br /> 1) Optimal locations of a limited number of harmonic meters.<br />2) Optimal dynamic estimates of harmonic source locations and their injections in unbalanced three-phase power systems.<br /><ul><li> System error covariance analysis by the Kalman Filter associated with the harmonic injection estimate determines the optimal arrangement of limited harmonic meters. Based on optimally arranged harmonic metering locations, the Kalman filter then yields the optimal dyanamic estimates of harmonic injections with few noisy harmonic measurements. </li></li></ul><li>Harmonic injection<br />Its due to increasing application of non linear loads.<br />Causes serious problems such as voltage distortion, increased losses and heating, and wrong operation of protective equipment.<br />Due to cost, the number of harmonic meters in power systems is very limited as compare to frequency measurement<br />Harmonic State Estimation<br />Limitations in number of meters makes harmonic state estimations an underdetermined estimation problem.<br />The quality of estimations is a function of the number & locations of the harmonic measurements.<br />Non linear load devices may be 3 phase unbalanced. As a result, the unbalance of harmonic sources further complicates the harmonic sources identification problem.<br />
3. 3. Mathematical Formulation<br />Bus injection current expressed as<br />c(t) = I cos(wt + θ) = I coswtcos θ- I sinwt sin θ<br />Let the state variable XR be Icos θand XI be I sin θ<br />Then 2 state variable equation for injection of current may be expressed as<br />
4. 4. For powers system with n buses, all bus current injections are treated as state variable,<br />Vector W is process noise which represents the random variation of the state variable.<br />Bus voltages are related to bus injection current by Z matrix.<br />
5. 5. Hence, bus voltage expressed as<br />
6. 6. We assume that m measuring meters are available and that the measured quantities are sampled values of bus voltage and injection current waveforms at m buses. The final measurement equation at the frequency w is given as<br />
7. 7. A harmonic injection current i(t) which includes rharmonics may be represented by<br />For an n-bus power system, the state equation representation of the harmonic injection current is<br />
8. 8. The system harmonic measurement equation is<br />A 3 phase Z bus matrix is considered in the equation.<br />Also last 2 equations together gives mathematical model on which Kalman Filtering is based.<br />
9. 9. A Kalman Filter Algorithm<br />For simplicity mathematical model is rewritten in following form<br />State Equation: <br />Measurment Equation:<br />System covariance matrices are assumed for Wkand Vk are assumed as<br />Initial variable is assumed to be zero<br />Initial covariance matrix is<br />
10. 10. Sequential recursive computation steps for harmonic injection estimate by Kalman Filter are<br />
11. 11. Optimal Metering Locations<br />From recursive Kalman Filter Equations, the error covariance matrix Pk is independent of the measurements,thus Pk+1 can be explicitly expressed as a function of Pk.<br />Advantage of Kalman filter method when used for optimal metering locations is that it accounts for the Measurement noise variance Rk and system covariance matrix Qk.<br />These parameters may change the arrangements of harmonic metering locations up to some extend. Hence, method is suitable for Stochastic & Deterministic variations.<br />Optimal metering locations are based on error covariance analysis of the harmonic injection.<br />Optimal metering locations in power system are arranged so that <br />trace[Pk] = minimal<br />
12. 12. Simulation Test<br />Actual power system shown in figure is use as test case<br />
13. 13. Power system brief<br />Capacitors are considered as out of service.<br />6 pulse drive is major source of harmonic.<br />Actual 3 phase harmonic measurement data on the system is used to test the method<br />3 phase bus voltage waveforms were observed on October 15, 1993. <br />All signals were sampled at 7680 Hz i.e. 128 samples/cycle<br />
14. 14. Initial covariance estimation<br />Measurement error vi is unknown for specific measurement, but for stable measurement system, errors in number of measurement have certain statistical feature of normal distribution.<br />Standard deviation of error is assumed to know.<br />In this study, based of actual field measurement standard deviation is taken as 5%.<br />P0(-) is denoted by σ1=3.8, σ4=1.1, σ5=3, σ6=1, σ7=2.5, σ9= 1.5.<br />These initial parameters model the possibilities of occurrences of harmonic sources at these buses, and are determined by average load levels & prior information about harmonic sources.<br />For buses without loads σ is chosen as 0.1<br />Diagonal elements of Qk taken as 0.05 (p.u.)<br />
15. 15. Optimal Arrangements of Harmonic Meters<br />3 harmonic meters are assumed to be available<br />Steady state value of trace[Pk] depends on location of meters<br />
16. 16.
17. 17. Identification and Tracking of Harmonic Sources<br />Based on results in table I meters are placed at buses 4,5,7.<br />Actual voltage waveform at buses 4,5,7 are sampled.<br />No current injection were observed, i.e. load levels are low during sampling.<br />Recursive Kalman Filter equations are used to estimate harmonic injection at all buses.<br />Estimates of harmonic injections at all buses are very small except Bus no. 9<br />
18. 18.
19. 19. Phase A harmonic injection at bus 9 with harmonic meters placed at 2,3,4.<br />
20. 20. Comparison of estimated harmonic injections at bus 9 with actual values.<br />
21. 21. Conclusion<br />