Solving stochastic AC power flow via polynomial chaos expansion

The present contribution demonstrates the applicability of polynomial chaos expansion to stochastic (optimal) AC power flow problems that arise in the operation of power grids. For rectangular power flow, polynomial chaos expansion together with Galerkin projection yields a deterministic reformulation of the stochastic power flow problem that is solved numerically in a single run. From its solution, approximations of the true posterior probability density functions are obtained. The presented approach does not require linearization. Furthermore, the IEEE 14 bus serves as an example to demonstrate that the proposed approach yields accurate approximations to the probability density functions for low orders of polynomial bases, and that it is computationally more efficient than Monte Carlo sampling.

[1]  Gary W. Chang,et al.  Power System Analysis , 1994 .

[2]  Barbara Borkowska,et al.  Probabilistic Load Flow , 1974 .

[3]  J.H. Zhang,et al.  Probabilistic Load Flow Evaluation With Hybrid Latin Hypercube Sampling and Cholesky Decomposition , 2009, IEEE Transactions on Power Systems.

[4]  J. Morales,et al.  Point Estimate Schemes to Solve the Probabilistic Power Flow , 2007, IEEE Transactions on Power Systems.

[5]  David W. P. Thomas,et al.  Probabilistic load flow for distribution systems with wind production using Unscented Transform method , 2011, ISGT 2011.

[6]  Alberto Bemporad,et al.  Scenario-based model predictive operation control of islanded microgrids , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[7]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[8]  Z. Nagy,et al.  Distributional uncertainty analysis using power series and polynomial chaos expansions , 2007 .

[9]  Santanu S. Dey,et al.  Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow , 2014, IEEE Transactions on Power Systems.

[10]  Richard D. Braatz,et al.  Stochastic nonlinear model predictive control with probabilistic constraints , 2014, 2014 American Control Conference.

[11]  C. Cañizares,et al.  Probabilistic Optimal Power Flow in Electricity Markets Based on a Two-Point Estimate Method , 2006, IEEE Transactions on Power Systems.

[12]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[13]  Ronald N. Allan,et al.  Numerical techniques in probabilistic load flow problems , 1976 .

[14]  Nikos D. Hatziargyriou,et al.  Probabilistic load flow in distribution systems containing dispersed wind power generation , 1993 .

[15]  G. Andersson,et al.  Decentralized Optimal Power Flow Control for Overlapping Areas in Power Systems , 2009, IEEE Transactions on Power Systems.

[16]  H. Hong.  An efficient point estimate method for probabilistic analysis , 1998 .

[17]  E. Rosenblueth.  Point estimates for probability moments. , 1975, Proceedings of the National Academy of Sciences of the United States of America.

[18]  S.T. Lee,et al.  Probabilistic load flow computation using the method of combined cumulants and Gram-Charlier expansion , 2004, IEEE Transactions on Power Systems.

[19]  Enrico Carpaneto,et al.  Probabilistic characterisation of the aggregated residential load patterns , 2008 .

[20]  M. Fotuhi-Firuzabad,et al.  Probabilistic Load Flow in Correlated Uncertain Environment Using Unscented Transformation , 2012, IEEE Transactions on Power Systems.

[21]  R.N. Allan,et al.  Evaluation Methods and Accuracy in Probabilistic Load Flow Solutions , 1981, IEEE Transactions on Power Apparatus and Systems.

[22]  Rolf Findeisen,et al.  Efficient stochastic model predictive control based on polynomial chaos expansions for embedded applications , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[23]  M. Harr.  Probabilistic estimates for multivariate analyses , 1989 .

[24]  R. H. Lasseter,et al.  Stochastic optimal power flow: formulation and solution , 2000, 2000 Power Engineering Society Summer Meeting (Cat. No.00CH37134).

[25]  P. Kundur,et al.  Power system stability and control , 1994 .