1 Unsupervised Optimal Power Flow Using Graph Neural Networks

2025-04-30 0 0 1.12MB 12 页 10玖币

侵权投诉

Unsupervised Optimal Power Flow

Using Graph Neural Networks

Damian Owerko, Fernando Gama, and Alejandro Ribeiro

Abstract—Optimal power ﬂow (OPF) is a critical optimization

problem that allocates power to the generators in order to satisfy

the demand at a minimum cost. Solving this problem exactly

is computationally infeasible in the general case. In this work,

we propose to leverage graph signal processing and machine

learning. More speciﬁcally, we use a graph neural network to

learn a nonlinear parametrization between the power demanded

and the corresponding allocation. We learn the solution in an

unsupervised manner, minimizing the cost directly. In order

to take into account the electrical constraints of the grid,

we propose a novel barrier method that is differentiable and

works on initially infeasible points. We show through simulations

that the use of GNNs in this unsupervised learning context

leads to solutions comparable to standard solvers while being

computationally efﬁcient and avoiding constraint violations most

of the time.

Index Terms—optimal power ﬂow, unsupervised learning,

graph neural networks, graph signal processing

I. INTRODUCTION

Optimal power ﬂow (OPF) is a critical optimization problem

for the energy industry. It consists in allocating power to each

generator in the grid, so that the energy demand is satisﬁed

with minimum cost (optimally). OPF is used to allocate

electricity generation throughout the day, establish day-ahead

market prices, and plan for grid infrastructure [1].

The objective of the OPF problem is to minimize the cost of

generating electrical power, subject to the constraints imposed

by the grid infrastructure, the physical laws of electromag-

netism, and the demand patterns. This problem is nonconvex

due to the sinusoidal nature of the alternating current and volt-

age and the constraints imposed by electrical interconnections

of the grid. Therefore, solving the OPF problem exactly is

computationally infeasible in the general case [1], [2]. In fact,

it has been show to be NP-hard [3].

One of the most common ways used to address the non-

tractability of the OPF problem is to solve a linear surrogate

based on small-angle approximations. [4]. In practical cases,

however, power grids are typically highly loaded thus violating

the small-angle approximation [5]. A more accurate, but com-

putationally intensive solution, is to rely on solvers for interior

point methods [6]. The OPF problem can also be approximated

by nonlinear, convex optimization problems. Examples include

quadratic [7], [8], second order conic [9] and semi-deﬁnite

Supported by NSF CCF 1717120, ARO W911NF1710438, ARL DCIST

CRA W911NF-17-2-0181, ISTC-WAS and Intel DevCloud. D. Owerko and

A. Ribeiro are with the Dept. of Electrical and Systems Eng., Univ. of Pennsyl-

vania., F. Gama is with the Electrical and Computer Engineering Department,

Rice University, Houston, TX. Email: {owerko,aribeiro}@seas.upenn.edu,

and fgama@rice.edu.

programming (SDP) [10], [11] relaxations. There exists a

small class of topologies for which sufﬁcient conditions exist

for convex relaxation optimality [12]. In particular, these

relaxations work well for radial networks, but often lead to

infeasible solutions or sub-optimal for meshed topologies [12].

Nevertheless, SDP relaxations were shown to perform well

on a variety of topologies [13], promting research into more

advanced relaxations with more general optimality guarantees

[14]. An alternative approach is to use successive linear

approximations that converge to the exact formulation [15].

This allows to balance trade-offs between accuracy and com-

putation time and integrates well into existing infrastructure

that relies on LP solvers. The implementation is slower than

interior point methods. However, unlike interior point methods,

successive linear approximations can be implemented on LP

solvers already used in industry.

Machine learning has arisen as a promising approach to

overcome computational tractability. Inference of machine

learning models is typically much faster than traditional

solvers and feasibility of the solution can be quickly veriﬁed

or used to hot-start a computationally expensive solver. Histor-

ically, there were many attempts to apply machine learning to

this problem, with the work in [16] providing a comprehensive

review until 2009. Many of these approaches such as genetic

algorithms and particle swarms, do scale well and therefore

have not between shown to be effective for networks with

more than 30 buses [17]–[20]. Early approaches used imitation

learning to learn to replicate solutions obtained using interior

point methods [21], [22], but often violated constraints and

could not perform that the method used to train them. Newer

approaches such as DeepOPF [23], use constrained learning

and are able to provide more robust solutions.

Graph signal processing (GSP) has emerged as a conve-

nient mathematical framework to describe problems involving

network data [24], [25]. By extending concepts of traditional

signal processing, such as ﬁltering and frequency analysis, to

graph-based data, GSP provides novel tools for the analysis

and design of distributed solutions [26], [27]. Of particu-

lar interest are graph neural networks (GNNs) [28], which

have been shown to be successful in both signal processing

and machine learning problems involving graph data [29]–

[31]. GNNs are built as extensions of graph convolutional

ﬁlters, followed by (typically pointwise, typically nonlinear)

activation functions. This allows GNNs to learn nonlinear

behaviors while retaining a decentralized nature and exploiting

the underlying graph structure. Thus, GNNs are promising

candidates for learning optimal power ﬂow allocations from

state measurements, while respecting the topology of the grid.

arXiv:2210.09277v1 [eess.SY] 17 Oct 2022

In this paper we propose a novel, unsupervised, constrained

learning approach. We make the following contributions.

1) We use GNNs to parametrize a mapping between the state

of the buses in the grid and the target generated power.

2) We learn the resulting parametrization by means of opti-

mizing the constrained OPF problem.

3) We introduce a differentiable, piece-wise penalty function

based on the log-barrier in order to enforce constraints.

This allows for the use of gradient-based methods.

4) We propose new methods of evaluating machine learning

OPF models. Since we are dealing with a physical system,

it is important to evaluate not only the rate at which a

model violates constraints, but also the severity of those

violations.

5) We show that graph neural networks are ideally suited to

optimal power ﬂow as they scale well for sparse graphs

and can be implemented in a distributed manner.

Section II introduces the OPF problem. Section III leverages

graph signal processing to present an appropriate descrip-

tion of the OPF problem and introduces the GNN-based

parametrization of the solution. In section IV we propose a

novel approach to unsupervised learning of OPF, where piece-

wise penalty functions IV-D are used as a method of enforcing

constraints IV-D. Finally, we provide experimental results V

on the efﬁcacy of our architecture on the IEEE 30 and 118

bus power system test cases.

II. OPTIMAL POWER FLOW

An electrical grid is an interconnected system that generates

electricity and delivers it to consumers. Its two most important

electrical components are generators that produce electricity

and loads that demand electric power [32]. Denote by BGbe

the set of all generators such that generator i∈ BGproduces

i∈Cunits of power. Similarly, load i∈ BDdemands

i∈Cunits of power. Note that the power is described by

means of a complex number to indicate its active (real) and

reactive (imaginary) power. The cost cito operate a generator

is a function ci:C7→ Rof the power produced.

The problem of OPF is concerned with minimizing the

generation costs, while meeting the demand and satisfying

the electrical constraints of the grid [5]. Speciﬁcally, in OPF

we are trying to ﬁnd the generator output power Sg

iwhich

minimize the total production cost

min

{Sg

i}i∈BGX

ci(Sg

i).(1)

The quantity of power produced by each generator is con-

strained within a certain range by two complex limits

i,min Sg

iSg

i,max (2)

for all i∈ BG, and where Sg

i,min, Sg

i,max ∈Cand is a gener-

alized inequality over the complex plane. The lower bound for

the generator output is often zero, but not necessarily. Some

generators, like nuclear power plants, cannot be turned off

within the time-frame of the optimization problem [33]. Other

generators might be able to store power leading to negative

lower bounds.

Tij : 1

Yij Vj

ij Yc

Figure 1. Circuit diagram of π-section branch model with transformer.

Abus is a grid element to which all other electrical

components connect. Consider the set of all buses B. Each

generator and load is connected to exactly one bus. Denote

the sets of generators and loads connecting to bus i∈ B as

i⊆ BGand BD

i⊆ BD, respectively. There may be multiple

generators and loads connected to one bus.

Additionally, the bus i∈ B is characterized by its voltage

Vi∈C, the net power injected at the bus Si∈C, and its

shunt admittance Ys

i. The shunt admittance is the combined

admittance between a bus’ connected devices and ground.

Therefore, the net power injected is the total power injected

by generators on the bus minus the power consumed by the

load on the bus, which can be conveniently written as follows.

[34]

Si=X

j∈BG

j−X

j∈BD

j(3)

The power lost due to the shunt admittance is given by Ohm’s

law Ys

i|Vi|2[34].

The physical limitations of the components connected to a

bus constrain the voltage magnitude between Vi,min, Vi,max ∈

R[34]. That is,

Vi,min ≤ |Vi| ≤ Vi,max.(4)

While the components may be rated to operate at voltages out-

side this range, eq. (4) deﬁnes the stable operating conditions

[34].

Branches connect different buses with each other. A branch

(i, j)∈ E ⊆ B × B connects bus ito bus j. We model

branches using an ideal transformer and a line in series, a

model known as a π-section branch model with a transformer,

see ﬁgure 1 for a circuit diagram and [34]–[36] for further

details. An ideal transformer is characterized by the complex

transformation ratio Tij ∈Cwhich describes the ratio of the

input voltage to the output voltage [34], [35], i.e. Vj=Vi/Tij ,

and assumes no internal losses. This transformer is in series

with a π-section line, which is described by three parameters:

the line admittance Yij ∈Cdescribing the ﬂow of current from

one end to the other, the forward charging admittance Yc

ij ∈C

and backward charging admittance Yc

ji ∈C. Note that the air

is a dielectric and therefore we model its effect on the line

by adding a capacitor connected to the ground. Therefore, the

total power ﬂowing forward Sij ∈Cand backward Sji ∈C

along a branch is [34]

Sij = (Yij +Yc

ij )∗|Vi|2

|Tij |2−Y∗

ViV∗

Tij

(5)

Sji = (Yij +Yc

ji)∗|Vj|2−Y∗

V∗

iVj

T∗

.(6)

There is a physical limit to the amount power a branch can

handle for extended periods of time [34], [35]. This can be

represented by Sij,max ∈Rsuch that

|Sij | ≤ Sij,max.(7)

Additionally a branch may have limits, ∆θij,min and ∆θij,max

in R, on the difference in voltage phase angle between the

buses it connects [34]. Speciﬁcally,

∆θij,min ≤∠(ViV∗

j)≤∆θij,max (8)

where ∠(ViV∗

j)is the angle difference between the voltage

phase at bus iand the voltage phase at bus j. It is common

to deﬁne reference buses to eliminate ambiguity in terms of

voltage angles [34], [35], which are deﬁned to have zero

voltage angle.

Buses and branches are related by the power-ﬂow equation

as a direct consequence of Kirchhoff’s current law [34]. The

net power injected at the bus (3) must be equal to the power

ﬂowing out of the bus, so it holds that

Si=Ys

i|Vi|2−X

(i,j)∈E∩ET

Sij .(9)

The distinction between the power ﬂow and the optimal

power ﬂow problems is important [5], [35]. In the power

ﬂow problem the goal is to solve for the voltage Vi, given

the net power injected Siat each node and the topologi-

cal characteristics of the grid, as described by the values

i, Tij , Yij , Y c

ij . The power generated and demanded at each

node are exogenous variables. Optimal power ﬂow (OPF), on

the other hand, is a constrained optimization problem where

the power generated is endogenous. The goal is to determine

the power output of each generator Sg

ithat minimizes the

total generation cost Pi∈B ci(Sg

i)while satisfying power

ﬂow equation (9) and the aforementioned constraints. In this

problem, only the power demanded at each node Sd

iis an

exogenous variable. Table I summarizes the optimal power-

ﬂow problem.

A. Solutions

There is extensive research into solving the optimal power

ﬂow problem [1]. The branch equations, (5) and (6), make

it non-convex [2], as the complex multiplications involve

trigonometric functions. Finding the exact solution is strongly

NP-hard [3]. Consequently, many research efforts focus on

ﬁnding approximations, relaxations, and solutions for OPF on

special families of graphs.

One common approach, named DC-OPF, is a linear approx-

imation to the exact OPF problem (sometimes referred to as

AC-OPF). DC-OPF is a ﬁrst order approximation around the

point where voltage angles are close to zero. Since the voltage

angle differences are small, complex multiplications in (5) and

(6) are approximated using small-angle approximations. The

problem becomes linear if we normalize voltage magnitude.

Additionally, if the cost function is convex, so too is the

DC-OPF problem. Most commonly, the cost function is a

second-order polynomial. Nevertheless, DC-OPF solutions are

not guaranteed to be feasible in the exact OPF case [37].

Table I

THE COMPLETE OPTIMAL POWER FLOW EQUATIONS FOR THE

FORMULATION USED IN THIS PAPER. REFER TO SECTION II FOR

EXPLANATIONS OF INDIVIDUAL EQUATIONS.

minimize X

i∈BG

ci(Sg

i)(1)

subject to

i,min ≤Sg

i≤Sg

i,max,for all i∈ BG(2)

Si=X

j∈BG

i−X

j∈BD

i.(3)

Vi,min ≤ |Vi| ≤ Vi,max,for all i∈ B (4)

Sij = (Yij +Yc

ij )∗|Vi|2

|Tij |2−Y∗

ViV∗

Tij

,for all (i, j)∈ E

(5)

Sji = (Yij +Yc

ji)∗|Vj|2−Y∗

V∗

iVj

T∗

,for all (i, j)∈ E (6)

|Sij | ≤ Sij,max,for all (i, j)∈ E (7)

∆θij,min ≤∠(ViV∗

j)≤∆θij,max,for all (i, j)∈ E (8)

Si=Ys

i|Vi|2+X

(i,j)∈E∪E T

Sij ,for all i∈ B (9)

Particularly, the assumptions of the DC-OPF problem are

violated when demand is high, which coincidentally is the

most critical case of the grid operation [5], [37].

In some situations, the OPF problem can be solved exactly

by using using one of several off-the-shelf optimization prob-

lem solvers, such as CONOPT, IPOPT, KNITRO, MINOS or

SNOPT [38]. However, in general, they are slow to converge

for large networks [6] or have no guarantee of convergence. In

particular, the IPOPT (Interior Point OPTimizer) [39] solver

has found widespread use due to its robustness, but it is

computationally costly [6].

III. GRAPH NEURAL NETWORKS

The objective of this work is to approximate the OPF

solution by taking the power demanded by each load Sd

ifor

all i∈ BDas an input and outputting the optimal generation

scheme, Sg

ifor all i∈ BG, in a scalable and distributed way.

To do so, we parametrize the solution by means of a graph

neural network.

A. Graph Signal Processing

Graph signal processing (GSP) has emerged as a convenient

framework to describe, analyze and solve problems that are

distributed in nature. Let G= (B,E,W)be a graph where

B={1, ..., N}is the set of Nnodes, E ⊆ B × B is the set

of edges, and W:E → Ris a weight function that assigns

a (positive) scalar to each edge. Data is described as a signal

z:B → RFdeﬁned on top of the nodes of the graph. The

signal at a node is a vector of Fmeasurements or features.

It is often convenient to represent a graph signal as a matrix

Z∈RN×F. The ith row of Z,zi∈RF, collects the Ffeatures

at node i,zi=z(i)[24], [25], [27]. We represent complex

quantities by a pair of features.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1UnsupervisedOptimalPowerFlowUsingGraphNeuralNetworksDamianOwerko,FernandoGama,andAlejandroRibeiroAbstractOptimalpowerow(OPF)isacriticaloptimizationproblemthatallocatespowertothegeneratorsinordertosatisfythedemandataminimumcost.Solvingthisproblemexactlyiscomputationallyinfeasibleinthegeneralcase.I...

收起<<

1 Unsupervised Optimal Power Flow Using Graph Neural Networks.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 Unsupervised Optimal Power Flow Using Graph Neural Networks

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: