Optimal power flow (OPF) is one of the most important optimization problems
in the energy industry. In its simplest form, OPF attempts to find the optimal
power that the generators within the grid have to produce to satisfy a given
demand. Optimality is measured with respect to the cost that each generator
incurs in producing this power. The OPF problem is non-convex due to the
sinusoidal nature of electrical generation and thus is difficult to solve.
Using small angle approximations leads to a convex problem known as DC OPF, but
this approximation is no longer valid when power grids are heavily loaded. Many
approximate solutions have been since put forward, but these do not scale to
large power networks. In this paper, we propose using graph neural networks
(which are localized, scalable parametrizations of network data) trained under
the imitation learning framework to approximate a given optimal solution. While
the optimal solution is costly, it is only required to be computed for network
states in the training set. During test time, the GNN adequately learns how to
compute the OPF solution. Numerical experiments are run on the IEEE-30 and
IEEE-118 test cases.

本文提出使用图神经网络在模仿学习框架下，近似求解最优潮流问题 (OPF)，并在 IEEE-30 和 IEEE-118 测试实例上进行了数值实验。

使用图神经网络进行最优潮流计算

Optimal Power Flow Using Graph Neural Networks

We consider the problem of computing a Wasserstein barycenter for a set of
discrete probability distributions with finite supports, which finds many
applications in areas such as statistics, machine learning and image
processing. When the support points of the barycenter are pre-specified, this
problem can be modeled as a linear programming (LP) problem whose size can be
extremely large. To handle this large-scale LP, we analyse the structure of its
dual problem, which is conceivably more tractable and can be reformulated as a
well-structured convex problem with 3 kinds of block variables and a coupling
linear equality constraint. We then adapt a symmetric Gauss-Seidel based
alternating direction method of multipliers (sGS-ADMM) to solve the resulting
dual problem and establish its global convergence and global linear convergence
rate. As a critical component for efficient computation, we also show how all
the subproblems involved can be solved exactly and efficiently. This makes our
method suitable for computing a Wasserstein barycenter on a large-scale data
set, without introducing an entropy regularization term as is commonly
practiced. In addition, our sGS-ADMM can be used as a subroutine in an
alternating minimization method to compute a barycenter when its support points
are not pre-specified. Numerical results on synthetic data sets and image data
sets demonstrate that our method is highly competitive for solving large-scale
Wasserstein barycenter problems, in comparison to two existing representative
methods and the commercial software Gurobi.

本文介绍了一种求解离散概率分布 Wasserstein 重心的方法，使用了对偶问题建模，并基于对称 Gauss-Seidel 方法的交替方向乘子法求解，可以高效地处理大规模线性规划问题。