Page - 175 - in Book of Full Papers - Symposium Hydro Engineering

Fig. 2 The First ten Eigen Values of the ESP ESP Les valeurs de Eigen dix premiers de l’ESP It is observed from Fig.2 that the first three eigen-value are much larger than the rest, accounting for a large proportion in the sum of eigen-value. According to the KL expansion theory, this result means that significant portion of the random process can be approximately represented by the first three terms (Namely M=3 in this case), and is written in the following: 𝑄 (𝑡 ) = 𝑄 (𝑡 )̅̅ ̅̅ ̅̅ +∑ √λ𝑘 𝜑 𝑘 (𝑡 ) 3 𝑘 =1 𝜉 𝑘 [5] In the Eq.[5], the average of Q(t) can be easily calculated. The eigen-value and eigen-function for the first three terms are calculated and are deterministic parameters in the equation. Therefore, the uncertainty of the ESP is only determined by three uncorrelated random variables (coefficients) with mean of 0 and variance of 1. In other words, we could use only three variables to approximate the uncertainty that are associated with the ESP of the inflow to the Qinshitan Reservoir. 4. DISCUSSIONS The KL expansion form of the inflow ESP to the Qinshitan Reservoir (Eq.[5]) results in a great reduction for approximating the ESP uncertainty. This is particular useful for optimization the reservoir operation under uncertainty. In the conventional modeling, Monte Carlo method is used for modeling the uncertainty, which lead to a very high computation cost as thousands or even millions of simulations are carried in each time step. As a comparison, the KL expansion form of the inflow ESP has only three variables and is invariant to the time step. A 175