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As a renewable energy resource, the pumped storage power station has great prospects for better balancing supply and demand, particularly with further development of intermittent power sources and the growing need for an intelligent electric grid. However, pump-turbines often involve problematic S-shaped regions in their machine characteristics and thus, while pumped storage may solve some problems in the grid, pump-turbine operation and control can lead to other problems including severe self-excited oscillation in the hydro-mechanical system. These operational challenges have been so severe that they have even led to some reported accidents. Based on two typical and complex water conveyance systems, including two pump-turbine system sharing a common tail tunnel and two pump-turbine systems sharing an upstream penstock, the mathematical equations for self-excited oscillation are deduced from the basic equations of pressurized pipe flow coupled to the pump-turbine’s characteristics; moreover, the behaviour of a single pump-turbine system is obtained through simplification of the first analysis. An analytical study is performed and the amplitude–frequency characteristic is investigated by means of both non-linear vibration theory and its corresponding analytical solution algorithm through a multi-scale method. With a given case study in detail, the results show that, for the turbines staying in the S-shaped regions for a relatively long time, self-excited oscillation inevitably occurs with severe oscillation superposed by several oscillation modes by the use of power spectral analysis module in MATLAB, and numerical results are shown to agree well with the theoretical analysis.

SUMMARYLet M T be the mean ÿrst passage matrix for an n-state ergodic Markov chain with a transition matrix T . We partition T as a 2 × 2 block matrix and show how to reconstruct M T e ciently by using the blocks of T and the mean ÿrst passage matrices associated with the non-overlapping Perron complements of T . We present a schematic diagram showing how this method for computing M T can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute M T by our method and show that, for large size problems, the number of multiplications is reduced by about 1=8, even if the algorithm is implemented in serial. We present ÿve examples of moderate sizes (of orders 20 -200) and give the reduction in the total number of ops (as opposed to multiplications) in the computation of M T . The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of ops can be much better than 1=8.