Local minima of quadratic functionals and control of hydro-electric power stations. (English) Zbl 1331.49046
Summary: We consider a control problem for a cascade of hydro-electric power stations, where some of the stations have reversible turbines. Our aim was to optimize the profit of power production satisfying restrictions on the water level in the reservoirs. From the mathematical point of view, this consists in minimizing an infinite-dimensional quadratic functional subject to cone constraints. Sufficient conditions of optimality for the abstract problem are derived and are then specialized for our problem. Noteworthy, the restrictions imposed on the power stations problem are in the form of control constraints and pure state constraints. The particular case of one power station is analyzed in detail, showing that reversible turbines always improve the profit.
MSC:
| 49N10 | Linear-quadratic optimal control problems |
| 49N15 | Duality theory (optimization) |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 49K27 | Optimality conditions for problems in abstract spaces |
| 49N90 | Applications of optimal control and differential games |
Keywords:
optimal control; pure state constraints; quadratic functionals; maximum principle; hydro-electric power stationsReferences:
| [1] | Labadie, J.W.: Optimal Operation of multireservoir systems: state-of-the-art review. J. Water Resour. Plan. Manag. 130(2), 93-111 (2004) · doi:10.1061/(ASCE)0733-9496(2004)130:2(93) |
| [2] | Korobeinikov, A., Kovacec, A., McGuinness, M., Pascoal, M., Pereira, A., Vilela, S.: Optimizing the profit from a complex cascade of hydroelectric stations with recirculating water. Math. Ind. Case Studies J. 2, 111-133 (2010) |
| [3] | International Atomic Energy Agency: Valoragua: a model for the optimal operating strategy of mixed hydrothermal generating systems, Users’ manual for the mainframe computer version. IAEA computer manual series no. 4, Vienna (1992) |
| [4] | Liang, R., Hsu, Y.: Hydroelectric generation scheduling using self-organizing feature maps. Electr. Power Sys. Res. 30(1), 1-8 (1994) · doi:10.1016/0378-7796(94)90053-1 |
| [5] | Ribeiro, A.F., Guedes, M.C.M., Smirnov, G.V., Vilela, S.: On the optimal control of a cascade of hydro-electric power stations. Electr. Power Sys. Res. 88, 121-129 (2012) · doi:10.1016/j.epsr.2012.02.010 |
| [6] | Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979) · Zbl 0407.90051 |
| [7] | Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer Academic Publishers, Norwell, MA (2000) · Zbl 0987.49001 · doi:10.1007/978-94-015-9438-7 |
| [8] | Malanowski, K., Maurer, H., Pickenhain, S.: Second-order sufficient conditions for state-constrained optimal control problems. J. Optim. Theory Appl. 123(3), 595-617 (2004) · Zbl 1059.49027 · doi:10.1007/s10957-004-5725-0 |
| [9] | Osmolovskii, N., Maurer, H.: Applications to regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM, Philadelphia (2012) · Zbl 1263.49002 · doi:10.1137/1.9781611972368 |
| [10] | Hoehener, D.: Variational approach to second-order optimality conditions for control problema with pure state constrains. SIAM J. Control Optim. 50(3), 1139-1173 (2012) · Zbl 1246.49016 · doi:10.1137/110828320 |
| [11] | Mariano, S.J.P., Catalao, J.P.S., Mendes, V.M.F., Ferreira, L.A.F.M.: Profit-Based Short-Term Hydro Scheduling considering Head-Dependent Power Generation.In: Proceedings of the IEEE Power Tech, pp. 1362-1367, Lausanne (July 2007) · Zbl 0166.15601 |
| [12] | Bushenkov, V.A., Ferreira, M.M.A., Ribeiro, A.F., Smirnov, G.V.: Numerical approach to a problem of hydroelectric resources management. In: AIP Conference Proceedings on International Conference of Numerical Analysis and Applied Mathematics, vol. 1558, pp. 630-633 (2013) |
| [13] | McCormick, G.: Second order conditions for constrained minima. SIAM J. Appl. Math. 15(3), 641-652 (1967) · Zbl 0166.15601 · doi:10.1137/0115056 |
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