High Dimensional Model Representation With Principal Component Analysis

In engineering design, spending excessive amount of time on physical experiments or expensive simulations makes the design costly and lengthy. This issue exacerbates when the design problem has a large number of inputs, or of high dimension. High dimensional model representation (HDMR) is one powerful method in approximating high dimensional, expensive, black-box (HEB) problems. One existing HDMR implementation, random sampling HDMR (RS-HDMR), can build an HDMR model from random sample points with a linear combination of basis functions. The most critical issue in RS-HDMR is that calculating the coefficients for the basis functions includes integrals that are approximated by Monte Carlo summations, which are error prone with limited samples and especially with nonuniform sampling. In this paper, a new approach based on principal component analysis (PCA), called PCA-HDMR, is proposed for finding the coefficients that provide the best linear combination of the bases with minimum error and without using any integral. Several benchmark problems of different dimensionalities and one engineering problem are modeled using the method and the results are compared with RS-HDMR results. In all problems with both uniform and nonuniform sampling, PCA-HDMR built more accurate models than RS-HDMR for a given set of sample points.

Download Full-text

A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques

Energies ◽

10.3390/en13143520 ◽

2020 ◽

Vol 13 (14) ◽

pp. 3520 ◽

Cited By ~ 1

Author(s):

Hang Li ◽

Zhe Zhang ◽

Xianggen Yin

Keyword(s):

Principal Component Analysis ◽

Power System ◽

Power Flow ◽

Random Variables ◽

Principal Component ◽

Component Analysis ◽

High Order ◽

High Dimensional ◽

High Dimensional Model Representation ◽

Dimensional Model

Because the penetration level of renewable energy sources has increased rapidly in recent years, uncertainty in power system operation is gradually increasing. As an efficient tool for power system analysis under uncertainty, probabilistic power flow (PPF) is becoming increasingly important. The point-estimate method (PEM) is a well-known PPF algorithm. However, two significant defects limit the practical use of this method. One is that the PEM struggles to estimate high-order moments accurately; this defect makes it difficult for the PEM to describe the distribution of non-Gaussian output random variables (ORVs). The other is that the calculation burden is strongly related to the scale of input random variables (IRVs), which makes the PEM difficult to use in large-scale power systems. A novel approach based on principal component analysis (PCA) and high-dimensional model representation (HDMR) is proposed here to overcome the defects of the traditional PEM. PCA is applied to decrease the dimension scale of IRVs and eliminate correlations. HDMR is applied to estimate the moments of ORVs. Because HDMR considers the cooperative effects of IRVs, it has a significantly smaller estimation error for high-order moments in particular. Case studies show that the proposed method can achieve a better performance in terms of accuracy and efficiency than traditional PEM.

Download Full-text

A Global High Dimensional Metamodeling Approach With the Ability of Using Non-Uniform Sampling

Volume 2B: 40th Design Automation Conference ◽

10.1115/detc2014-35316 ◽

2014 ◽

Author(s):

Kambiz Haji Hajikolaei ◽

G. Gary Wang

Keyword(s):

Linear Combination ◽

Principal Component ◽

Critical Issue ◽

Basis Functions ◽

High Dimensional ◽

Benchmark Problems ◽

High Dimensional Model Representation ◽

Uniform Sampling ◽

Non Uniform Sampling ◽

Sample Points

In engineering design, spending excessive amount of time on physical experiments or expensive simulations makes the design costly and lengthy. This issue exacerbates when the design problem has a large number of inputs, or of high dimension. High Dimensional Model Representation (HDMR) is one powerful method in approximating high dimensional, expensive, black-box (HEB) problems. One existing HDMR implementation, Random Sampling HDMR (RS-HDMR), can build a HDMR model from random sample points with a linear combination of basis functions. The most critical issue in RS-HDMR is that calculating the coefficients for the basis functions includes integrals that are approximated by Monte Carlo summations, which are error prone with limited samples and especially with non-uniform sampling. In this paper, a new approach based on Principal Component Analysis (PCA), called PCA-HDMR, is proposed for finding the coefficients that provide the best linear combination of the bases with minimum error and without using any integral. Benchmark problems are modeled using the method and the results are compared with RS-HDMR results. With both uniform and non-uniform sampling, PCA-HDMR built more accurate models than RS-HDMR for a given set of sample points.

Download Full-text