Advances in Electric Power and Energy. Группа авторов

LMR 0.0018 0.0013 0.0017 0.0014

TABLE 2.17 Case study: computational performance results with bad measurements.

Method	Minimum (s)	Mean (s)	Maximum (s)	Std. dev. (s)
WLS	0.94	1.70	2.28	0.18
LAV	0.59	0.94	1.29	0.12
QC	0.22	0.31	0.45	0.05
QL	1.00	1.74	2.71	0.27
LMS	3.80	8.21	12.64	1.42
LTS	1.28	2.36	3.96	0.36
LMR	0.94	2.73	34.84	4.99

      The following observations can be made about Tables 2.16 and 2.17:

      1 As expected, the WLS approach does not provide the most accurate results. The estimates computed using the QC and QL techniques are more precise than that obtained with the conventional WLS method.

      2 The QC and LAV approaches are the most efficient ones from the computational perspective. The computational burden of the LMS technique is higher than that of any of the other procedures.

      2.5.12 Conclusions

      Considering recent advances in computational techniques, this work addresses the electric state estimation problem from a mathematical programming perspective.

      In this chapter, the most common state estimators are formulated as optimization problems and implemented, proving to be computationally efficient and numerically accurate.

From the computational point of view, QC and LAV techniques perform faster than the conventional WLS estimator, saving up to 75% CPU time (compared with the WLS method directly solved as an optimization problem). On the other hand, mathematical programming formulation of some estimators (such as LMS and LTS approaches) involves non‐convexities and a significant number of binary variables, resulting in higher computational burdens.

      With regard to estimation accuracy, numerical simulations denote that the LMR and QL techniques provide an estimation accuracy level that is similar to that obtained using WLS method. On the other hand, if the measurement set is corrupted with errors that χ² test cannot detect, WLS estimation quality deteriorates, providing the worst estimation quality (if LMS and LTS approaches are not considered). It is observed that QL and QC techniques outperform the rest of estimators, followed by LMR procedure.

      The computational analyses carried out in this chapter have led to the conclusion that alternative estimators are potential substitutes for traditional WLS method. The state of the art of current nonlinear optimization solvers and recent advances in computational equipments allow using robust estimators in real electric energy systems.

REFERENCES

      1 1. Monticelli, A. (Feb. 2000). Electric power system state estimation. Proceedings of the IEEE 88 (2): 262–282.

      2 2. Rousseeuw, P.J. (Dec. 1984). Least median of squares regression. Journal of the American Statistical Association 79 (388): 871–880.

      3 3. Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. New York: Wiley.

      4 4. Schweppe, F.C. (Jan. 1970). Power system static state estimation. Part III: implementation. IEEE Transactions on Power Apparatus and Systems 89 (1): 130–135.

      5 5. Schweppe, F.C. and Rom, D. (Jan. 1970). Power system static state estimation. Part II: approximate model. IEEE Transactions on Power Apparatus and Systems 89 (1): 125–130.

      6 6. Schweppe, F.C. and Wildes, J. (Jan. 1970). Power system static state estimation. Part I: exact model. IEEE Transactions on Power Apparatus and Systems 89 (1): 120–125.

      7 7. Abur, A. and Gómez‐Expósito, A. (2004). Power System State Estimation: Theory and Implementations. New York: Marcel Dekker.

      8 8. Chatterjee, S. and Hadi, A.S. (2006). Regression Analysis by Example, 4e. New York: Wiley.

      9 9. Conejo, A.J., de la Torre, S., and Cañas, M. (Feb. 2007). An optimization approach to multi‐area state estimation. IEEE Transactions on Power Systems 22 (1): 213–221.

      10 10. Drud, A. (2007). CONOPT. In: GAMS—The Solver Manuals. Washington, DC: GAMS Development Corporation.

      11 11. Drud, A. (2008). MINOS: A Solver for Large‐Scale Nonlinear Optimization Problems. Washington, DC: GAMS Development Corporation.

      12 12. Brooke, A., Kendrick, D., Meeraus, A., and Raman, R. (1998). GAMS: A User's Guide. Washington, DC: GAMS Development Corporation.

      13 13. Fourer, R., Gay, D.M., and Kernighan, B.W. (2002). AMPL: A Modeling Language for Mathematical Programming, 2e. New Mexico: Brooks/Cole Publishing Company.

      14 14. Bisschop, J. and Roelofs, M. (2007). AIMMS – The User's Guide. Haarlem: Paragon Decision Technology B.V.

      15 15. Caro, E., Conejo, A.J., and Mínguez, R. (2008). A mathematical programming approach to state estimation. In: Optimization Advances in Electric Power Systems (ed. E.D. Castronuovo), 1–26. New York: Nova Science Publishers Inc.

      16 16. Holten, L., Gjelsvik, A., Aam, S. et al. (Nov. 1988). Comparison of different methods for state estimation. IEEE Transactions on Power Systems 3 (4): 1798–1806.

      17 17. Larson, R., Tinney, W., Hadju, L., and Piercy, D. (Mar. 1970). State estimation in power systems. Part II: implementations and applications. IEEE Transactions on Power Apparatus and Systems 89 (3): 353–362.

      18 18. Gómez‐Expósito, A., Conejo, A.J., and Cañizares, C. (2008). Electric Energy Systems: Analysis and Operation. New York: CRC Press, Taylor & Francis Group.

      19 19. Abur, A. (Aug. 1990). A bad data identification method for linear programming state estimation. IEEE Transactions on Power Systems 5 (3): 894–901.

      20 20. Abur, A. and Çelik, M.K. (May 1993). Least absolute value state estimation with equality and in‐equality constraints. IEEE Transactions on Power Systems 8 (2): 680–686.

      21 21. Çelic, M.K. and Abur, A. (Feb. 1992). A robust WLAV state estimator using transformations. IEEE Transactions on Power Systems 7 (1): 106–113.

      22 22.