Graphical Models for Optimal Power Flow

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 over a tree-structured 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 distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for "smart grid" applications like control of distributed energy resources. We evaluate our technique numerically on several benchmark networks and show that practical OPF problems can be solved effectively using this approach.Comment: To appear in Proceedings of the 22nd International Conference on Principles and Practice of Constraint Programming (CP 2016

Quadratically Constrained Quadratic Programs on Acyclic Graphs With Application to Power Flow

This paper proves that nonconvex quadratically constrained quadratic programs can be solved in polynomial time when their underlying graph is acyclic, provided the constraints satisfy a certain technical condition. We demonstrate this theory on optimal power-flow problems over tree networks

Graphical models for optimal power flow

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 over a tree-structured 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 tree-structured distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for “smart grid” applications like control of distributed energy resources. Numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach

Distributed algorithms for optimal power flow problem

Optimal power flow (OPF) is an important problem for power generation and it is in general non-convex. With the employment of renewable energy, it will be desirable if OPF can be solved very efficiently so that its solution can be used in real time. With some special network structure, e.g. trees, the problem has been shown to have a zero duality gap and the convex dual problem yields the optimal solution. In this paper, we propose a primal and a dual algorithm to coordinate the smaller subproblems decomposed from the convexified OPF. We can arrange the subproblems to be solved sequentially and cumulatively in a central node or solved in parallel in distributed nodes. We test the algorithms on IEEE radial distribution test feeders, some random tree-structured networks, and the IEEE transmission system benchmarks. Simulation results show that the computation time can be improved dramatically with our algorithms over the centralized approach of solving the problem without decomposition, especially in tree-structured problems. The computation time grows linearly with the problem size with the cumulative approach while the distributed one can have size-independent computation time.postprin

Outage detection in power distribution networks with optimally-deployed power flow sensors

§These authors contributed equally to this work. Abstract—An outage detection framework for power distri-bution networks is proposed. The framework combines the use of optimally deployed real-time power flow sensors and that of load estimates via Advanced Metering Infrastructure (AMI) or load forecasting mechanisms. The distribution network is modeled as a tree network. It is shown that the outage detection problem over the entire network can be decoupled into detection within subtrees, where within each subtree only the sensors at its root and on its boundary are used. Outage detection is then formulated as a hypothesis testing problem, for which a maximum a-posteriori probability (MAP) detector is applied. Employing the maximum misdetection probability Pmaxe as the detection performance metric, the problem of finding a set of a minimum number of sensors that keeps Pmaxe below any given probability target is formulated as a combinatorial optimization. Efficient algorithms are proposed that find the globally optimal solutions for this problem, first for line networks, and then for tree networks. Using these algorithms, optimal three-way trade-offs between the number of sensors, the load estimate accuracy, and the outage detection performance are characterized for line and tree networks using the IEEE 123 node test feeder system. I

Geometry of Power Flows and Optimization in Distribution Networks

We investigate the geometry of injection regions and its relationship to optimization of power flows in tree networks. The injection region is the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region. If the voltage magnitudes are fixed, the injection region of a tree network can be written as a linear transformation of the product of two-bus injection regions, one for each line in the network. Using this decomposition, we show 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. Moreover, the resulting convexified optimal power flow problem can be efficiently solved via }{ semi-definite programming or second order cone relaxations. These results improve upon earlier works by removing the assumptions on active power lower bounds. It is also shown that our practical angle assumption guarantees two other properties: (i) the uniqueness of the solution of the power flow problem, and (ii) the non-negativity of the locational marginal prices. Partial results are presented for the case when the voltage magnitudes are not fixed but can lie within certain bounds.Comment: To Appear in IEEE Transaction on Power System

Convex Relaxation of Optimal Power Flow, Part II: Exactness

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 models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems, June 2014. This is an extended version with Appendex VI that proves the main results in this tutoria

Power Aware Routing for Sensor Databases

Wireless sensor networks offer the potential to span and monitor large geographical areas inexpensively. Sensor network databases like TinyDB are the dominant architectures to extract and manage data in such networks. Since sensors have significant power constraints (battery life), and high communication costs, design of energy efficient communication algorithms is of great importance. The data flow in a sensor database is very different from data flow in an ordinary network and poses novel challenges in designing efficient routing algorithms. In this work we explore the problem of energy efficient routing for various different types of database queries and show that in general, this problem is NP-complete. We give a constant factor approximation algorithm for one class of query, and for other queries give heuristic algorithms. We evaluate the efficiency of the proposed algorithms by simulation and demonstrate their near optimal performance for various network sizes