Arbitrary polynomial chaos expansion and its application to power flow analysis-Fast approximation of probability distribution by arbitrary polynomial expansion
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Abstract This paper introduces an arbitrary polynomial chaos expansion method for performing probabilistic power flow analysis in power systems. The proposed method is used for uncertainty analysis, expressing the uncertainty of a system as random variables with an arbitrary output distribution based on orthogonal polynomial expansion. This method is advantageous because of its calculation speed and accuracy. This study expresses probabilistic power flow in a power system with many uncertain power sources using linear combination polynomial expansion. The orthogonal polynomial system employed is generated by moment analysis from renewable energy output data, with the polynomial coefficients derived from a collocation method. Simulation of probabilistic power flow using the proposed method is applied to a 29-bus transmission network model including three renewable energies, and the calculation speed and accuracy are evaluated by changing the expansion order of the polynomial. In addition, the influence on the polynomial coefficient is assessed when the system topology is changed due to a line fault. Therefore, since the arbitrary polynomial chaos expansion method can represent complex networks by linear combination of orthogonal polynomial sets, calculation based on it is several hundred times faster than the conventional Monte Carlo method. The results demonstrate that the proposed method is very useful for analyzing the probabilistic power distribution and that third-order expansion is practically appropriate.Keywords:
Polynomial Chaos
Polynomial expansion
Collocation (remote sensing)
Vector parabolic equation (VPE) methods have been widely applied to model wave propagation in guiding structures with high accuracy and computational efficiency. A detailed specification of the modeled environment is required. However, considerable uncertainty can exist in the description of the environment. This paper presents a non-intrusive formulation of polynomial chaos expansion (PCE) method to quantify the uncertainties in VPE-based channel models for tunnel environments. The performance is validated against Monte Carlo simulations and measurements.
Polynomial Chaos
Polynomial expansion
Uncertainty Quantification
Propagation of uncertainty
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Polynomial Chaos
Sobol sequence
Polynomial expansion
Uncertainty Quantification
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Galerkin polynomial chaos and collocation methods have been widely adopted for uncertainty quantification purpose. However, when the stiff system is involved, the computational cost can be prohibitive, since stiff numerical integration requires the solution of a nonlinear system of equations at every time step. Applying the Galerkin polynomial chaos to stiff system will cause a computational cost increase from O(n3) to O(S3n3). This paper explores uncertainty quantification techniques for stiff chemical systems using Galerkin polynomial chaos, collocation and collocation least-square approaches. We propose a modification in the implicit time stepping process. The numerical test results show that with the modified approach, the run time of the Galerkin polynomial chaos is reduced. We also explore different methods of choosing collocation points in collocation implementations and propose a collocation least-square approach. We conclude that the collocation least-square for uncertainty quantification is at least as accurate as the Galerkin approach, and is more efficient with a well-chosen set of collocation points.
Polynomial Chaos
Collocation (remote sensing)
Orthogonal collocation
Uncertainty Quantification
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Polynomial Chaos
Polynomial expansion
Uncertainty Quantification
Solver
Helmholtz free energy
Helmholtz equation
Model order reduction
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We present a new response surface based stochastic finite element method to obtain solutions for general random uncertainty problems using the polynomial chaos expansion. The approach is general but here a typical elastostatics example only with the random field of Young's modulus is presented to illustrate the stress analysis, and computational comparison with the traditional polynomial expansion approach is also performed. It shows that the results of the polynomial chaos expansion are improved compared with that of the second polynomial expansion method.
Polynomial Chaos
Polynomial expansion
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The polynomial chaos (PC) method has been used in many engineering applications to replace the traditional Monte Carlo (MC) approach for uncertainty quantification (UQ) due to its better convergence properties. Many researchers seek to further improve the efficiency of PC, especially in higher dimensional space with more uncertainties. The intrusive PC Galerkin approach requires the modification of the deterministic system, which leads to a stochastic system with a much bigger size. The non-intrusive collocation approach imposes the system to be satisfied at a set of collocation points to form and solve the linear system equations. Compared with the intrusive approach, the collocation method is easy to implement, however, choosing an optimal set of the collocation points is still an open problem. In this paper, we first propose using the low-discrepancy Hammersley/Halton dataset and Smolyak datasets as the collocation points, then propose a least-squares (LS) collocation approach to use more collocation points than the required minimum to solve for the system coefficients. We prove that the PC coefficients computed with the collocation LS approach converges to the optimal coefficients. The numerical tests on a simple 2-dimensional problem show that PC collocation LS results using the Hammersley/Halton points approach to optimal result.
Collocation (remote sensing)
Polynomial Chaos
Orthogonal collocation
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Polynomial Chaos
Collocation (remote sensing)
Orthogonal collocation
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In this paper a Two Step approach with Chaos Collocation for efficient uncertainty quantification in computational fluid-structure interactions is followed. In Step I, a Sensitivity Analysis is used to efficiently narrow the problem down from multiple uncertain parameters to one parameter which has the largest influence on the solution. In Step II, for this most important parameter the Chaos Collocation method is employed to obtain the stochastic response of the solution. The Chaos Collocation method is presented in this paper, since a previous study showed that no efficient method was available for arbitrary probability distributions. The Chaos Collocation method is compared on efficiency with Monte Carlo simulation, the Polynomial Chaos method, and the Stochastic Collocation method. The Chaos Collocation method is non-intrusive and shows exponential convergence with respect to the polynomial order for arbitrary parameter distributions. Finally, the efficiency of the Two Step approach with Chaos Collocation is demonstrated for the linear piston problem with an unsteady boundary condition. A speed-up of a factor of 100 is obtained compared to a full uncertainty analysis for all parameters.
Polynomial Chaos
Collocation (remote sensing)
Uncertainty Quantification
Orthogonal collocation
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In this paper, we present a method for finding zeros of polynomial equations in a given domain. We apply a numerical eigensolver using contour integral for a polynomial eigenvalue problem that is derived from polynomial equations. The Dixon resultant is used to derive the matrix polynomial of which eigenvalues involve roots of the polynomial equations with respect to one variable. The matrix polynomial obtained by the Dixon resultant is sometimes singular. By applying the singular value decomposition for a matrix which appears in the eigensolver, we can obtain the roots of given polynomial systems. Experimental results demonstrate the efficiency of the proposed method.
Companion matrix
Characteristic polynomial
Matrix (chemical analysis)
Wilkinson's polynomial
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Polynomial chaos expansion and Gaussian mixture models are combined in a hybrid fashion to propagate state uncertainty for spacecraft with initial Gaussian errors. Polynomial chaos expansion models uncertainty by performing an expansion using orthogonal polynomials. The accuracy of polynomial chaos expansion for a given problem can be improved by increasing the order of the orthogonal polynomial expansion. The number of terms in the orthogonal polynomial expansion increases factorially with dimensionality of the problem, thereby reducing the effectiveness of the polynomial chaos expansion approach for problems of moderately high dimensionality. This paper shows a combination of Gaussian mixture model and polynomial chaos expansion, Gaussian mixture model–polynomial chaos expansion as an alternative form of the multi-element polynomial chaos expansion. Gaussian mixture model–polynomial chaos expansion reduces the overall order required to reach a desired accuracy. The initial distribution is converted to a Gaussian mixture model, and polynomial chaos expansion is used to propagate the state uncertainty represented by each of the elements through the nonlinear dynamics. Splitting the initial distribution into a Gaussian mixture model reduces the size of the covariance associated with each new element, thereby reducing the domain of approximation and allowing for lower-order polynomials to be used. Several spacecraft uncertainty propagation examples are shown using Gaussian mixture model–polynomial chaos expansion. The resulting distributions are shown to efficiently capture the full shape of the true non-Gaussian distribution.
Polynomial Chaos
Polynomial expansion
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Citations (70)