Graphical models for optimal power flow
Abstract
Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NPhard in general, and weakly NPhard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a treestructured graphical model where the nodal variables are lowdimensional vectors. We adapt the standard dynamic programming algorithm for inference over a treestructured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary treestructured distribution networks and handle mixedinteger optimization problems. Further, it can be implemented in a distributed messagepassing fashion that is scalable and is suitable for “smart grid” applications like control of distributed energy resources. In conclusion, numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach.
 Authors:

 California Inst. of Technology (CalTech), Pasadena, CA (United States)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Univ. of Michigan, Ann Arbor, MI (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Office of Electricity Delivery and Energy Reliability (OE); USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1345156
 Report Number(s):
 LAUR1624896
Journal ID: ISSN 13837133
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Constraints
 Additional Journal Information:
 Journal Volume: 22; Journal Issue: 1; Journal ID: ISSN 13837133
 Publisher:
 Springer
 Country of Publication:
 United States
 Language:
 English
 Subject:
 24 POWER TRANSMISSION AND DISTRIBUTION; 97 MATHEMATICS AND COMPUTING; Computer Science; Energy Sciences; Information Science
Citation Formats
Dvijotham, Krishnamurthy, Chertkov, Michael, Van Hentenryck, Pascal, Vuffray, Marc, and Misra, Sidhant. Graphical models for optimal power flow. United States: N. p., 2016.
Web. doi:10.1007/s106010169253y.
Dvijotham, Krishnamurthy, Chertkov, Michael, Van Hentenryck, Pascal, Vuffray, Marc, & Misra, Sidhant. Graphical models for optimal power flow. United States. doi:10.1007/s106010169253y.
Dvijotham, Krishnamurthy, Chertkov, Michael, Van Hentenryck, Pascal, Vuffray, Marc, and Misra, Sidhant. Tue .
"Graphical models for optimal power flow". United States. doi:10.1007/s106010169253y. https://www.osti.gov/servlets/purl/1345156.
@article{osti_1345156,
title = {Graphical models for optimal power flow},
author = {Dvijotham, Krishnamurthy and Chertkov, Michael and Van Hentenryck, Pascal and Vuffray, Marc and Misra, Sidhant},
abstractNote = {Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NPhard in general, and weakly NPhard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a treestructured graphical model where the nodal variables are lowdimensional vectors. We adapt the standard dynamic programming algorithm for inference over a treestructured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary treestructured distribution networks and handle mixedinteger optimization problems. Further, it can be implemented in a distributed messagepassing fashion that is scalable and is suitable for “smart grid” applications like control of distributed energy resources. In conclusion, numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach.},
doi = {10.1007/s106010169253y},
journal = {Constraints},
number = 1,
volume = 22,
place = {United States},
year = {2016},
month = {9}
}
Web of Science
Works referenced in this record:
Convex Relaxation of Optimal Power Flow—Part II: Exactness
journal, June 2014
 Low, Steven H.
 IEEE Transactions on Control of Network Systems, Vol. 1, Issue 2, p. 177189
Optimal capacitor placement on radial distribution systems
journal, January 1989
 Baran, M. E.; Wu, F. F.
 IEEE Transactions on Power Delivery, Vol. 4, Issue 1
An algorithmic framework for convex mixed integer nonlinear programs
journal, May 2008
 Bonami, Pierre; Biegler, Lorenz T.; Conn, Andrew R.
 Discrete Optimization, Vol. 5, Issue 2
Exact Convex Relaxation of Optimal Power Flow in Radial Networks
journal, January 2015
 Gan, Lingwen; Li, Na; Topcu, Ufuk
 IEEE Transactions on Automatic Control, Vol. 60, Issue 1, p. 7287
SCIP: solving constraint integer programs
journal, January 2009
 Achterberg, Tobias
 Mathematical Programming Computation, Vol. 1, Issue 1
ACFeasibility on Tree Networks is NPHard
journal, January 2016
 Lehmann, Karsten; Grastien, Alban; Van Hentenryck, Pascal
 IEEE Transactions on Power Systems, Vol. 31, Issue 1