1. Least Squares Estimate Algorithm Bei Zhang 2014-3-5

2. Basics of Least Squares Estimate Algorithm  Estimating the parameter  Minimizing the squared discrepancies between the observed data and the modeled data  Redundant measurements

3. Implementation of Least Squares Estimate on Distance Relaying Least Squares Estimate algorithm can be applied on:  Line model: based on differential equation  Signal model: based on modeling the voltage and current signal

4. Implementation of Least Squares Estimate on Distance Relaying – signal model  Taylor expansion and reserve the first three items  No even harmonics :Output measurement; :Parameters to be estimated; The rest: Inputs  Filtering out the high frequency harmonics Decaying DC component Harmonic components

5.  Harmonics (non-sensitive if under assumption)  Transient and DC offset (non-sensitive if under assumption)  Noise (sensitivity will be reduced)  Frequency variation (sensitive) Implementation of Least Squares Estimate on Distance Relaying – signal model  Sampling rate (low)  Data window (long)  Time reference

6.  Improved Method to Reduce the Impact of Coupling Capacitor Voltage Transformer (CCVT) Transients  Adaptive Least Squares Estimate Technique: reduce the algorithm inertia to increase the speed of making decision  Forgetting factor  Adjustable data window  Recursive Least Squares Estimate Technique: decrease the calculation burden Implementation of Least Squares Estimate on Distance Relaying – signal model

7.  Similarly, the adaptive way and recursive form of least squares estimate algorithm are also developed in the line model application. Implementation of Least Squares Estimate on Distance Relaying – line model Numerical approximation √ Other forms √ √ Integration √ :Output measurement; :Parameters to be estimated; : Inputs

8.  Harmonics (theoretically not affected, actually might be sensitive, therefore an input filter with a very low cut off frequency will be needed)  Transient and DC offset (not sensitive)  Noise (sensitivity will be reduced)  Frequency variation (not affected) Implementation of Least Squares Estimate on Distance Relaying – line model  Sampling rate (high)  Data window (long) The error of estimation result

9. Advantages (least square estimate algorithm)  Linear least-squares solutions can be explicitly evaluated in closed forms  They can be recursively updated as more input data is available  Statistical advantages: the best linear unbiased estimator in the case when the errors are uncorrelated to each other

10. Signal ModelLine Model Computation burden (inversion involved) Easily get impacted by the transients Mainly come from CCVT saturation, a lot of unexpected components: inter-harmonic components, and etc. Easily get impacted by the noise The noise in the measurement may result in overreaching tendency Time delay Usually one cycle or even more time will be needed when CCVT saturation happens. Time delay The response time is not that fast compared with other algorithm (e.g. Kalman) Disadvantages (least square estimate algorithm)

11. References [1] M. S. Sachdev and M. A. Baribeau, "A New Algorithm for Digital Impedance Relays," Power Apparatus and Systems, IEEE Transactions on, vol. PAS- 98, pp. 2232-2240, 1979. [2] E. Pajuelo, G. Ramakrishna, and M. S. Sachdev, "An improved voltage phasor estimation technique to minimize the impact of CCVT transients in distance protection," in Electrical and Computer Engineering, 2005. Canadian Conference on, 2005, pp. 454-457. [3] E. Pajuelo, G. Ramakrishna, and M. S. Sachdev, "Phasor estimation technique to reduce the impact of coupling capacitor voltage transformer transients," Generation, Transmission & Distribution, IET, vol. 2, pp. 588-599, 2008. [4] P. Jafarian and M. Sanaye-Pasand, "Weighted least error squares based variable window phasor estimator for distance relaying application," Generation, Transmission & Distribution, IET, vol. 5, pp. 298-306, 2011. [5] P. Jafarian and M. Sanaye-Pasand, "An adaptive phasor estimation technique based on LES method using forgetting factor," in Power & Energy Society General Meeting, 2009. PES '09. IEEE, 2009, pp. 1-8. [6] M. Sanaye-Pasand and M. Davarpanah, "Performance evaluation of an extended adaptive distance relaying algorithm," in Developments in Power System Protection, 2004. Eighth IEE International Conference on, 2004, pp. 449-452 Vol.2. [7] T. S. Sidhu, D. S. Ghotra, and M. S. Sachdev, "An adaptive distance relay and its performance comparison with a fixed data window distance relay," Power Delivery, IEEE Transactions on, vol. 17, pp. 691-697, 2002. [8] M. S. Sachdev and M. Nagpal, "A recursive least error squares algorithm for power system relaying and measurement applications," Power Delivery, IEEE Transactions on, vol. 6, pp. 1008-1015, 1991. [9] Y. Huang, Y. Cai, J. Xiao, L. Tang, and J. Li, "Research of implementing least squares in digital distance relaying for AC electrified railway," in Developments in Power System Protection, 2004. Eighth IEE International Conference on, 2004, pp. 116-118 Vol.1. [10] A. S. AlFuhaid and M. A. El-Sayed, "A recursive least-squares digital distance relaying algorithm," Power Delivery, IEEE Transactions on, vol. 14, pp. 1257-1262, 1999. [11] T. Segui, P. Bertrand, M. Guillot, P. Hanchin, and P. Bastard, "Fundamental basis for distance relaying with parametrical estimation," Power Delivery, IEEE Transactions on, vol. 15, pp. 659-664, 2000. [12] C. Fernandez and F. L. Pagola, "Total least squares and discrete-time line models in HV distance protection," Power Delivery, IEEE Transactions on, vol. 14, pp. 74-79, 1999. [13] A. T. Ghainani, "Digital protection scheme for distance relaying of a six-phase power system," Universiti Teknologi Malaysia, Faculty of Electrical Engineering, 2010. [14] A. P. Morais, G. Cardoso Jr, L. Mariotto, and G. D. Ferreira, "Numerical distance relaying algorithm based on Mathematical Morphology and Least- Squares Curve Fitting method," Electric Power Systems Research, vol. 81, pp. 1144-1150, 5// 2011.

12. Thank you!