An official website of the United States government
Abstract We analyze and test a simple-to-implement two-step iteration for the incompressible Navier-Stokes equations that consists of first applying the Picard iteration and then applying the Newton iteration to the Picard output. We prove that this composition of Picard and Newton converges quadratically, and our analysis (which covers both the unique solution and non-unique solution cases) also suggests that this solver has a larger convergence basin than usual Newton because of the improved stability properties of Picard-Newton over Newton. Numerical tests show that Picard-Newton converges more reliably for higher Reynolds numbers and worse initial conditions than Picard and Newton iterations. We also consider enhancing the Picard step with Anderson acceleration (AA), and find that the AAPicard-Newton iteration has even better convergence properties on several benchmark test problems.
more »
« less
A Machine Learning Initializer for Newton-Raphson AC Power Flow Convergence
Power flow computations are fundamental to many power system studies. Obtaining a converged power flow case is not a trivial task especially in large power grids due to the non-linear nature of the power flow equations. One key challenge is that the widely used Newton based power flow methods are sensitive to the initial voltage magnitude and angle estimates, and a bad initial estimate would lead to non-convergence. This paper addresses this challenge by developing a random-forest (RF) machine learning model to provide better initial voltage magnitude and angle estimates towards achieving power flow convergence. This method was implemented on a real ERCOT 6102 bus system under various operating conditions. By providing better Newton-Raphson initialization, the RF model precipitated the solution of 2,106 cases out of 3,899 non-converging dispatches. These cases could not be solved from flat start or by initialization with the voltage solution of a reference case. Results obtained from the RF initializer performed better when compared with DC power flow initialization, Linear regression, and Decision Trees.
more »
« less
- Award ID(s):
- 2243204
- PAR ID:
- 10496868
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- 2024 IEEE Texas Power and Energy Conference (TPEC)
- ISBN:
- 979-8-3503-3120-2
- Page Range / eLocation ID:
- 1 to 6
- Subject(s) / Keyword(s):
- Machine Learning Newton Raphson Power flow convergence Power flow initialization Random Forest
- Format(s):
- Medium: X
- Location:
- College Station, TX, USA
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)This article establishes sufficient conditions for the uniqueness of AC power flow solutions via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that the P-Θ power flow problem has at most one solution for any acyclic or GSP graph. In addition, for arbitrary cyclic power networks, we show that multiple distinct solutions cannot exist under the assumption that angle differences across the lines are bounded by some limit related to the maximal girth of the network. In these cases, a vector of voltage phase angles can be uniquely determined (up to an absolute phase shift) given a vector of real power injections within the realizable range. The implication of this result for the classical power flow analysis is that under the conditions specified above, the problem has a unique physically realizable solution if the phasor voltage magnitudes are fixed. We also introduce a series-parallel operator and show that this operator obtains a reduced and easier-to-analyze model for the power system without changing the uniqueness of power flow solutions.more » « less
-
We study the convergence properties of the Expectation-Maximization algorithm in the Naive Bayes model. We show that EM can get stuck in regions of slow convergence, even when the features are binary and i.i.d. conditioning on the class label, and even under random (i.e. non worst-case) initialization. In turn, we show that EM can be bootstrapped in a pre-training step that computes a good initialization. From this initialization we show theoretically and experimentally that EM converges exponentially fast to the true model parameters. Our bootstrapping method amounts to running the EM algorithm on appropriately centered iterates of small magnitude, which as we show corresponds to effectively performing power iteration on the covariance matrix of the mixture model, although power iteration is performed under the hood by EM itself. As such, we call our bootstrapping approach “power EM.” Specifically for the case of two binary features, we show global exponentially fast convergence of EM, even without bootstrapping. Finally, as the Naive Bayes model is quite expressive, we show as corollaries of our convergence results that the EM algorithm globally converges to the true model parameters for mixtures of two Gaussians, recovering recent results of Xu et al.’2016 and Daskalakis et al. 2017.more » « less
-
For fast timescales or long prediction horizons, the AC optimal power flow (OPF) problem becomes a computational challenge for large-scale, realistic AC networks. To overcome this challenge, this paper presents a novel network reduction methodology that leverages an efficient mixed-integer linear programming (MILP) formulation of a Kron-based reduction that is optimal in the sense that it balances the degree of the reduction with resulting modeling errors in the reduced network. The method takes as inputs the full AC network and a pre-computed library of AC load flow data and uses the graph Laplacian to constraint nodal reductions to only be feasible for neighbors of non-reduced nodes. This results in a highly effective MILP formulation which is embedded within an iterative scheme to successively improve the Kron-based network reduction until convergence. The resulting optimal network reduction is, thus, grounded in the physics of the full network. The accuracy of the network reduction methodology is then explored for a 100+ node medium-voltage radial distribution feeder example across a wide range of operating conditions. It is finally shown that a network reduction of 25-85% can be achieved within seconds and with worst-case voltage magnitude deviation errors within any super node cluster of less than 0.01pu. These results illustrate that the proposed optimization-based approach to Kron reduction of networks is viable for larger networks and suitable for use within various power system applications.more » « less
-
null (Ed.)Power system state estimation (PSSE) aims at finding the voltage magnitudes and angles at all generation and load buses, using meter readings and other available information. PSSE is often formulated as a nonconvex and nonlinear least-squares (NLS) cost function, which is traditionally solved by the Gauss-Newton method. However, Gauss-Newton iterations for minimizing nonconvex problems are sensitive to the initialization, and they can diverge. In this context, we advocate a deep neural network (DNN) based “trainable regularizer” to incorporate prior information for accurate and reliable state estimation. The resulting regularized NLS does not admit a neat closed form solution. To handle this, a novel end-to-end DNN is constructed subsequently by unrolling a Gauss-Newton-type solver which alternates between least-squares loss and the regularization term. Our DNN architecture can further offer a suite of advantages, e.g., accommodating network topology via graph neural networks based prior. Numerical tests using real load data on the IEEE 118-bus benchmark system showcase the improved estimation performance of the proposed scheme compared with state-of-the-art alternatives. Interestingly, our results suggest that a simple feed forward network based prior implicitly exploits the topology information hidden in data.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/TPEC60005.2024.10472261
