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.

本文提出一种基于树形结构图模型和区间离散化的近似算法，使用约束规划技术和自适应边界传播算法，能够解决优化问题，适用于任意分配网络和混合整数优化问题，可在智能电网应用中使用，实现分布式消息传递方式，具有良好的可伸缩性和实际效用。

最优潮流的图模型

Graphical Models for Optimal Power Flow

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.

通过研究注入区域的几何结构及其与树网络功率流优化的关系，提出了一种在实践条件下通过凸壳有效解决凸化功率流问题的方法。