ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2015. Vol. 12

The Numerical Probabilistic Analysis of Optimization Problems Hydropower

O. A. Popova

The paper considers the problem of optimizing the hydroelectric power generation in the face of uncertainty of the input data. To solve optimization problems with random hydropower input data we used numerical probability analysis. The numerical probabilistic analysis is a new section of Computational Mathematics, for applying to different tasks with random input data. The probabilistic extensions and numerical operations on the probability densities of the random variables are the base of numerical probabilistic analysis. We explore the sources of the emergence of various types of uncertainty and their methods of presentation. To demonstrate the NPA methods we present an optimization problem example of hydroelectric power generation which depends on the prediction of lateral inflow into the reservoir provided in the form of stochastic functions. It is shown that in the discrete case the problem reduces to solving a system of linear algebraic equations with random coefficients. The results of numerical simulation are presented in the graphic form of probability density histograms approximating the joint probability density function as the optimal amount of water passing through the turbines at different times.

numerical probabilistic analysis, optimization, uncertain data, hydropower

1. Gelfan A.N. Dynamic-stochastic modeling of formation of meltwater runoff. Inst waters. Problems of RAS. Moscow, Nauka, 2007. 279 p.

2. Vennikov V.A. Optimization of power plants and power systems: A Textbook for high schools. M., Energoizdat, 1981. 464 p.

3. Dobronets B.S., Popova O.A. Numerical probabilistic analysis for the study of systems under uncertainty. Bulletin of Tomsk State University. Management, Computer Science and Informatics, 2012, no 21, pp. 39–46.

4. Dobronets B.S., Popova O.A. Elements of numerical probabilistic analysis. Bulletin of the Siberian State Aerospace University, 2012, no 2, pp. 19–23.

5. Dobronets B.S., Popova O.A. Histogram approach to representation and processing of data space and data ground monitoring. Izvestiya SFedU. Engineering Sciences, 2014, № 6 (155). С. 14–22

6. Dobronets B.S., Popova O.A. The numerical probabilistic analysis of uncertain data. Krasnoyrsk, Siberian Federal University, 2014. 167 p.

7. Perepelitsa V.A., Tebueva F.B. Discrete optimization and modeling under uncertainty data, M., The Academy of Natural Sciences, 2007.

8. Popova O.A. Extraction technology and visualization of knowledge on the basis of numerical probabilistic analysis of uncertain data. Informatization and Communication, 2013, no 2, pp. 63–66.

9. Popova O.A. Linear programming problem with random input data. VSGUTU Bulletin, 2013, no2 (41), pp. 19–23.

10. Popova O.A. Numerical solution of systems of linear algebraic equations with random coefficients. VSGUTU Bulletin, 2013, no2 (41), pp. 5–11.

11. Tsvetkov E.V., Alyabysheva T.M., Parfenov L.G. Optimum modes of hydro power plants in power systems. Moscow, Energoatomizdat, 1984. 304 p.

12. Fiedler M., Nedoma J., Ram´ik J., Rohn J. and Zimmermann K. Linear Optimization Problems with Inexact Data. Springer Science+Business Media, New York, 2006.

13. Yudin D.B. Mathematical methods of control under incomplete information. Moscow, Soviet Radio, 1974. 400 p.

14. Dobronets B.S., Krantsevich A.M., Krantsevich N.M. Software implementation of numerical operations on random variables. Journal of Siberian Federal University. Mathematics & Physics,;2013, vol 6, no2, pp. 168–173.

15. Dobronets B.S., Popova O.A. Numerical Probabilistic Analysis under Aleatory and Epistemic Uncertainty. Reliable Computing. 2014, vol. 19, pp. 274–289.

16. Liu B. Theory and Practice of Uncertain Programming. 2nd ed. Springer-Verlag, Berlin, 2009.

17. Popova О.А. Optimization Problems with Random Data Journal of Siberian Federal University. Mathematics & Physics,;2013, vol. 6, no4, pp. 506–515.

18. Pr´ekopa A. On the probability distribution of the optimum of a random linear program. J. SIAM Control, 1966, vol. 4, no1, pp. 211–222.

Full text (russian)