Graphical models for optimal power flow

  title={Graphical models for optimal power flow},
  author={Krishnamurthy Dvijotham and Michael Chertkov and Pascal Van Hentenryck and Marc Vuffray and Sidhant Misra},
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 NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference… 
Scalable Unit Commitment with AC Power Flow via Semidefinite Programming Relaxation
This paper designs a polynomial-time solvable third-order semidefinite programming (TSDP) relaxation, with the aim of finding a near globally optimal solution for the unit commitment problem with AC power flow constraints.
A Scalable Semidefinite Relaxation Approach to Grid Scheduling
This paper proposes for the first time a computational method, capable of solving large-scale power system scheduling problems with thousands of generating units, while accurately incorporating the nonlinear equations that govern the flow of electricity on the grid.
A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems
This paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems.
Belief propagation for supply networks: Efficient clustering of their factor graphs
A systematic way to cluster loops of naively assigned factor graphs such that the resulting transformed factor graphs have no additional loops as compared to the original network is proposed.
State Estimation of Power Flows for Smart Grids via Belief Propagation
This work gives a criterion for how to assess whether other algorithms, using only local information, are adequate for state estimation for a given grid and demonstrates how belief propagation can be utilized for coarse-graining power grids toward representations that reduce the computational effort when the coarse- grained version is integrated into a larger grid.
A Survey of Relaxations and Approximations of the Power Flow Equations
This monograph provides the first comprehensive survey of representations in the context of optimization of the power flow equations, categorized as either relaxations or approximations.
Graphical Models and Belief Propagation-hierarchy for Optimal Physics-Constrained Network Flows
It is shown that the network flow optimization problem can be stated naturally in terms of the so-called Graphical Models (GM), which are wide spread in statistical disciplines such as Applied Probability, Machine Learning and Artificial Intelligence.
Tractable Algorithms for Approximate Nash Equilibria in Generalized Graphical Games with Tree Structure
This work provides the first fully polynomial time approximation scheme (FPTAS) for computing an approximate mixed-strategy Nash equilibrium in graphical multi-hypermatrix games (GMhGs), and is the first to establish an FPTAS for tree polymatrixGames as well as tree graphical games when the number of actions is bounded by a constant.
Grid-Graph Signal Processing (Grid-GSP): A Graph Signal Processing Framework for the Power Grid
Grid-GSP provides an interpretation for the spatio-temporal properties of voltage phasor measurements, by showing how the well-known power systems modeling supports a generative low-pass graph filter model for the state variables, namely the Voltage phasors.
FPTAS for Mixed-Strategy Nash Equilibria in Tree Graphical Games and Their Generalizations
This work provides the first fully polynomial time approximation scheme (FPTAS) for computing an approximate mixed-strategy Nash equilibrium in tree-structured graphical multi-hypermatrix games (GMhGs), and is the first to establish an FPTAS for tree polymatrixGames as well as tree graphical games when the number of actions is bounded by a constant.


Exact Convex Relaxation of Optimal Power Flow in Radial Networks
It is proved that a global optimum of OPF can be obtained by solving a second-order cone program, under a mild condition after shrinking the OPF feasible set slightly, for radial power networks.
Geometry of Power Flows and Optimization in Distribution Networks
It is shown that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull and the resulting convexified optimal power flow problem can be efficiently solved via semi-definite programming or second-order cone relaxations.
Geometry of power flows in tree networks
The optimal power flow problem can be convexified and efficiently solved, and this result improves upon earlier works since it does not make any assumptions about the active bus power constraints.
Approximate inference in graphical models using lp relaxations
The authors' algorithms optimize over the cycle relaxation of the marginal polytope, which is shown to be closely related to the first lifting of the Sherali-Adams hierarchy, and is significantly tighter than the pairwise LP relaxation.
Convex relaxation methods for graphical models: Lagrangian and maximum entropy approaches
A distributed, iterative algorithm that minimizes the Lagrangian dual function by block coordinate descent and results in an iterative marginal-matching procedure that enforces consistency among the subgraphs using an adaptation of the well-known iterative scaling algorithm.
Strengthening Convex Relaxations with Bound Tightening for Power Network Optimization
It is shown that the Quadratic Convex relaxation of power flows, enhanced by bound tightening, almost always outperforms the state-of-the-art Semi-Definite Programming relaxation on the optimal power flow problem.
SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework
These extensions that were added to the constraint integer programming framework SCIP to enable it to solve convex and nonconvex mixed-integer nonlinear programs (MINLPs) to global optimality are described and insights into the performance impact of individual MINLP solver components are provided.
LP approximations to mixed-integer polynomial optimization problems
An approximation scheme for the "AC-OPF" problem on graphs with bounded tree-width is obtained and a more general construction for pure binary optimization problems where individual constraints are available through a membership oracle is described.
Exactness of Semidefinite Relaxations for Nonlinear Optimization Problems with Underlying Graph Structure
The main objective of this paper is to investigate how the (hidden) structure of a given real/complex-valued optimization problem makes it easy to solve, and to this end, three conic relaxations are proposed.
Convex Relaxation of Optimal Power Flow—Part II: Exactness
  • S. Low
  • Engineering
    IEEE Transactions on Control of Network Systems
  • 2014
This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow