Search

We consider a system of two sets of partial differential equations describing the water hammer in a hydroelectric power plant containing the dynamics of the tunnel, turbine penstock, surge tank and hydraulic turbine. Under standard simplifying assumptions (negligible Darcy-Weisbach losses and dynamic head variations), a system of functional differential equations of neutral type, with two delays, can be associated to the aforementioned partial differential equations and existence, uniqueness and continuous data dependence can be established. Stability is then discussed using a Lyapunov functional deduced from the energy identity. The Lyapunov functional is "weak" i.e. its derivative function is only non-positive definite. Therefore only Lyapunov stability is obtained while for asymptotic stability application of the Barbashin-Krasovskii-LaSalle invariance principle is required. A necessary condition for its validity is the asymptotic stability of the difference operator associated to the neutral system. However, its properties in the given case make the asymptotic stability non-robust (fragile) in function of some arithmetic properties of the delay ratio. [ABSTRACT FROM AUTHOR]

The present paper starts from a model with distributed parameters (i.e. described by hyperbolic partial differential equations with non-standard (derivative) boundary conditions) of a hydroelectric power plant with tunnel, surge tank and penstock. The association of a system of functional differential equations of neutral type and the one-to-one correspondence between the solutions of the two mathematical objects is given. Further, it is given the deduction - via singular pertur-bations - of the nonlinear ordinary differential equations for modeling the surge tank in order to discuss its stability under constant power delivery of the hydraulic turbine. Some other unsolved problems are pointed out. [ABSTRACT FROM AUTHOR]