Page - 173 - in Book of Full Papers - Symposium Hydro Engineering

expansion to model the uncertainty associated with the ESP and demonstrate its high efficiency with a case study. 2. METHODOLOGY The Karhunen-Loeve (KL) expansion[2] [3] [4]is a representation of a random process as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Consider a random process of 𝑄 (𝑡 )and let 𝑄 (𝑡 )̅̅ ̅̅ ̅̅ be its mean and C(s,t)=cov(𝑄 (𝑠 ), 𝑄 (𝑡 )) be its covariance function. The Q(s) and Q(t) are variables at different time step. Then, the KL expansion of 𝑄 (𝑡 )can be represented by the following function: 𝑄 (𝑡 ) = 𝑄 (𝑡 )̅̅ ̅̅ ̅̅ + ∑ √λ𝑘 ψ𝑘 (𝑡 ) ∞ 𝑘 =1 𝜉 𝑘 [1] where {ψ𝑘 ,λ𝑘 }𝑘 =1 =∞ are the orthogonal eigen-functions and the corresponding eigen-values, obtained as solutions of the equation: λψ(𝑡 ) = ∫𝐶 (𝑠 ,𝑡 )ψ(𝑠 )𝑑 𝑠 [2] Equation (3) is a Fredholm integral equation of the second kind. When applied to a discrete and finite process, this equation takes a much simpler form that can be easily solved. In its discrete form, the covariance matrix C(s,t) is represented as an N×N matrix, where N is the time steps of the random process. Then the above integral form can be rewritten as ∑ C(s,t)Ψ(s)Ns,t=1 to suit the discrete case. In Eq. [1],{𝜉 𝑘 }𝑘 =1 =∞ is a sequence of uncorrelated random variables (coefficients) with mean of 0 and variance of 1 and are defined as: 𝜉 𝑘 = 1 √λ𝑘 ∫[𝑄 (𝑡 )− 𝑄 (𝑡 )̅̅ ̅̅ ̅̅ ]ψ𝑘 (𝑡 )𝑑 𝑡 [3] The form of the KL expansion in Eq.[1] is often approximated by a finite number of discrete terms (e.g., M), for practical implementation. The truncated KL expansion is then written as: 𝑄 (𝑡 ) ≈ 𝑄 (𝑡 )̅̅ ̅̅ ̅̅ + ∑ √λ𝑘 ψ𝑘 (𝑡 ) 𝑀 𝑘 =1 𝜉 𝑘 [4] The number of terms M is determined by the desired accuracy of approximation and strongly depends on the correlation of the random process. The higher the correlation of the random process, the fewer the terms that are required for the approximation[5]. One approach to roughly determine M is to compare the magnitude of the eigen-values (in descending order) with respect to the first eigen-value and consider the terms with the most significant eigen- values. With the truncated KL expansion, the large number of variables in time- domain is reduced to fewer coefficients in the transformed space (i.e., frequency- domain). The KL expansion has found many applications in science and 173