1

Abstract— Smart grid (SG) technology transforms the traditional

power grid from a single layer physical system to a cyber-

physical network that includes a second layer of information.

Collecting, transferring, and analyzing the huge amount of data

that can be captured from different parameters in the network,

together with the uncertainty that is caused by the distributed

power generators, challenge the standard methods for security

and monitoring in future SGs. Other important issues are the

cost and power efficiency of data collection and analysis, which

are highlighted in emergency situations such as blackouts. This

paper presents an efficient dynamic solution for online SG

topology identification and monitoring by combining concepts

from compressive sensing and graph theory. In particular, the

SG is modeled as a huge interconnected graph, and then using a

DC power-flow model under the probabilistic optimal power

flow, topology identification is mathematically reformulated as a

sparse recovery problem. This problem and challenges therein

are efficiently solved using modified sparse recovery algorithms.

Network models are generated using the MATPOWER toolbox.

Simulation results show that the proposed method represents a

promising alternative for real time monitoring in SGs.

Index Terms— Smart grids, topology identification and fault

detection, cyber-physical systems, compressive sensing (CS),

sparse recovery.

I. INTRODUCTION

N intelligence revolution is happening in the

technology of power grids, supporting the development

of so-called smart grids (SGs). Roughly speaking, “the

term SG refers to electricity networks that can intelligently

integrate the behavior and actions of all parameters and users

connected to them” [1]. This revolution transforms a

traditional power grid from a single-layer physical system into

a huge cyber-physical network, using a layer of information

that flows through the system. This information includes the

status of several parameters in the network such as bus

voltages, active and reactive powers, phase angles, branch

currents, and load consumption, in addition to the actuating or

controlling commands fed back to the network from

controllers and decision making units [1]-[4], [22]-[23].

Another important feature of next generation SGs is the

aggregation of renewable energy micro-sources in distributed

generation technology [5]. The nature of the injected power

from renewables as wind turbines and solar panels is

inherently random. This randomness, along with the

uncertainty in the load, lead to probabilistic behavior in the

load-power flow at buses. The probabilistic optimal power

flow (P-OPF) problem has been fully discussed in [6]-[8] and

references therein.

Collecting, transferring, and analyzing the huge amount of

data flowing through the information layer of the SG, together

with the uncertainty caused by distributed power generators

and load characteristics, challenge the standard methods for

security, monitoring, and control. Identifying changes of lines

is particularly critical for a number of tasks, including state

estimation, optimal power flow, real-time contingency

analysis, and thus, security assessment of power systems [7].

Other important issues are the cost and power efficiency of

real-time data collection and analysis [10]. The Supervisory

Control and Data Acquisition (SCADA) system and Wide

Area Mentoring System (WAMS) provide voltage and power

data for each local system in near real time [2]-[4], [22]-[23].

The most popular sensing technology used widely in SG data

collection systems is the synchronous Phasor Measurement

Unit (PMU) [4]. The system data exchange (SDX) module of

the North American Electric Reliability Corporation (NERC)

can provide grid-wide topology information on an hourly basis

[10]; however, near real-time monitoring of transmission lines

is mandatory in order to change the grid into a smart system.

The power network (PN) topology identification (TI)

problem has been addressed in [9] based on a consuming or

economic approach. Some related work has addressed the

problem of fault or power line outage identification (POI)

using PMU data in a power grid; see [7], [10]-[14] and

references therein. In this work, we introduce an efficient and

low cost solution for TI (that can also be used to solve the POI

problem) using a few measurements of the system parameters.

Our work combines sparse sampling theory from the field of

Compressive Sensing [18] with concepts borrowed from graph

theory [15]. Most of the previous approaches (which are

mainly focused on POI) rely on a DC linear power flow model

[7], [10]-[14]. Using this approximated model, the PN is

modeled as a graph and the POI problem is expressed in terms

of certain matrices defined by this graph. In some recent

works, POI is formulated as a combinatorially complex

(integer programming) problem, which is computationally

tractable only for single or, at most, double line changes (refer

to [10]). Due to the discrete formulation of the problem, the

optimal solution can be found through an exhaustive search

(ES). However, the computational complexity of such an ES is

high, and thus it is inappropriate from a time cost perspective.

Moreover, coping with multiple line changes is critical in the

face of cascading failures in recent blackouts, especially in

case of large networks. This has motivated several existing

works [7], [10] and [13], which consider how to handle

multiple line outages. A recent alternative approach to line-

change identification adopts a Gauss–Markov graphical model

of the PN and can deal with multiple changes at affordable

complexity; the Gauss-Markov approach is based on a

stochastic characterization of power flows that models the

buses’ phase angles as Gaussian random variables and

comprehensively addresses their dependencies [7]. An

ambiguity group-based location recognition algorithm has

been also proposed for recognizing the most likely outage line

locations in multiple outages. Another new method for solving

the POI problem has been developed based on the theory of

quickest change detection. Also recently a distributed

framework to perform the identification locally at each phasor

Compressive Sensing-Based Topology

Identification for Smart Grids

A

2

data concentrator has been presented. Moreover, another

approach considered the same problem under scenarios with a

limited number of PMUs. Most recently, a sparsity-based

method has been developed in order to address the bad PMU

data records and their imperfections in power outage detection

and localization. For more information on these approaches,

please refer to [10]-[14] and references therein.

In [10] and [21], the authors formulate POI as a sparse

recovery problem (SRP). Their method leverages the fact that

the number of outage lines after a failure represents a small

fraction of the total number of lines. In [11], a global

stochastic optimization technique based on cross-entropy

optimization (CEO) is presented. The initial idea of this

method is based on the formulation presented in [10]. It has

been shown that CEO has better performance compared to

[10] in general. However, both of these methods require

hourly basecase grid topology information in addition to the

system parameters to be able to find the outage lines in the

grid. In [10], the measurement or sensing matrix of the SRP

includes the weighted Laplacian matrix 𝐵 (also called the

topology, structure, or nodal-admittance matrix) of the

corresponding graph of the power network, which is

dependent on the previous structure of the network and needs

to have the topology information of the graph delivered in an

hourly manner. In [11], the measurement vector is calculated

from multiplication of the same Laplacian matrix with the

phase angle vector, so again hourly basecase topology

information from the grid is needed. Moreover, the

performance of most of the aforementioned methods is highly

related to the level of noise. It has been shown in the literature

([10] and [11]) that the recovery performance of these

methods significantly decreases as the level of noise increases,

so in both of the approaches in [10] and [11], a pre-whitening

procedure is applied before further analysis. Another

important issue to be considered is the rank deficiency of the

matrix 𝐵. In order to avoid the problems arising from this rank

deficiency, one bus is normally considered as the reference

bus and its corresponding row and column are removed from

the matrix 𝐵, so that the matrix becomes full rank. However,

due to cyber-attacks and other precision deficiency issues,

whenever the records from the reference bus are affected by

bad data, the overall procedure can be corrupted. Another

important problem is with the switching events that are

mistakenly unreported. These kind of errors can change the

structure of the nodal-admittance matrix 𝐵. Thus, if the last

status of the topology of the network has not been reported

within a suitable time interval the final results can be affected.

On the other hand, to the best of our knowledge, most of the

methods mentioned above do not account for the random

behavior applied by renewable energy sources and the

uncertainty in loads. Finally, the mathematical formulation of

some aforementioned methods is dependent on the

connectivity of the power grid graph, where they assume that

the power outages may not be allowed to result in islanding in

any part of the network [10].

Our work is based on the probabilistic power flow model.

Our approach is completely different from previous

formulations, in that the presented reformulation relies only on

the measurements from system parameters and does not need

any a priori information about the topology of the network. In

fact, we reformulate the TI problem in such a way that the

output of the optimization problem is the structure or topology

matrix 𝐵 of the network itself. As a result, this framework is

capable of dealing with the aforementioned imperfections.

Case studies using standard IEEE test-beds [16] show that the

proposed method represents a promising new strategy for

topology identification, line change, fault detection, and

monitoring issues in SGs.

The rest of this paper is organized as follows. Section II

discusses the model of a PN as an interconnected graph and its

corresponding mathematical representation using a DC power

flow model. Section III presents the proposed sparse topology

identification technique. Moreover, it contains a short review

on CS and a short description of the SRP solvers to be used in

this paper. Simulation results of a study of the sensing matrix

coherence (Section III.A), the effectiveness of the algorithm,

and the results from applying a model of structured sparsity

are discussed in Section IV. Finally, Section V summarizes the

final conclusions of this paper.

II. POWER NETWORK MODEL AND CORRESPONDING GRAPH

In this section, the graph representation of a PN, its

corresponding matrices, and its connection with the DC power

flow model are described. Moreover, we discuss a random

Gaussian distribution assumption for the system (bus power

and voltage) parameters in a large SG with a considerable

amount of renewable energy sources.

A. Graph of the Power Network

We represent the PN as a graph 𝐺(𝑆𝑁 , 𝑆𝐸), consisting of a

set of 𝐾 nodes 𝑆𝑁 = {1, . . . , 𝐾}, where each node 𝑖 represents

an arbitrary bus of the PN, and a set of 𝐿 edges or lines 𝑆𝐸 =

{𝑙𝑖,𝑗: 𝑖, 𝑗𝜖𝑆𝑁}, which connect certain nodes and form the

general structure or topology of the grid. Following the

approaches in [7], and almost all of examples from [10], we

assume that the data from the nodes’ parameters (such as

power and phase angles) are obtained from a network of

sensors. The main goal of this paper is to identify the structure

or topology of the PN in near real time using a small set of

data, without any a priori knowledge about the structure of the

network. Nowadays, a vast amount of work is focusing on

power sensor technology especially on software-based PMUs

with much lower costs per unit; [25] and the references

therein.

B. DC Power Flow Model and Corresponding Graph of PN

The AC power-flow model, also known as the AC load-

flow model, is a numerical technique in power engineering,

applied to a power system in order to analyze the behavior of

different forms of AC power under normal steady-state

operation. In such a model two distinct sets of non-linear

equations (called power balance equations) indicate the

relationship between active and reactive power and the voltage

magnitude and phase angle as follows [17]:

𝑃𝑖𝑗 = 𝑔𝑖,𝑗𝑉𝑖2 − 𝑔𝑖,𝑗𝑉𝑖𝑉𝑗𝑐𝑜𝑠(𝜃𝑖 − 𝜃𝑗) + 𝑏𝑖,𝑗𝑉𝑖𝑉𝑗𝑠𝑖𝑛(𝜃𝑖 − 𝜃𝑗) (1)

𝑄𝑖𝑗 = 𝑏𝑖,𝑗𝑉𝑖2 − 𝑏𝑖,𝑗𝑉𝑖𝑉𝑗𝑐𝑜𝑠(𝜃𝑖 − 𝜃𝑗) + 𝑔𝑖,𝑗𝑉𝑖𝑉𝑗𝑠𝑖𝑛(𝜃𝑖 − 𝜃𝑗). (2)

Here 𝑃𝑖𝑗 and 𝑄𝑖𝑗 are the active and reactive powers injected

from node 𝑗 to node 𝑖, respectively; 𝑉𝑖 and 𝜃𝑖 represent the

amplitude and phase of the voltage on node 𝑖, respectively;

3

𝑔𝑖,𝑗 is the real part of the admittance of the line 𝑙𝑖,𝑗 (or the

conductance); and 𝑏𝑖,𝑗 is the imaginary part of the admittance

of line 𝑙𝑖,𝑗 (or the susceptance). It has been shown that when

the system is stable, under a normal steady-state condition, the

phase angle differences are small, meaning that 𝑠𝑖𝑛(𝜃𝑖 − 𝜃𝑗 ) ≈ 𝜃𝑖 − 𝜃𝑗. Using this simplification and the DC power

flow method discussed in [17], the power injected to a distinct

bus 𝑖 follows the superposition law (in practical cases equation

(4) cannot be exactly validated):

𝑃𝑖 = ∑ 𝑃𝑖𝑗𝑗 = ∑ 𝑏𝑖,𝑗𝑗𝜖𝛮𝑖

(𝜃𝑖 − 𝜃𝑗) (3)

𝑄𝑖 = ∑ 𝑄𝑖𝑗𝑗 = ∑ 𝑏𝑖,𝑗𝑗𝜖𝛮𝑖

(𝑉𝑖 − 𝑉𝑗) . (4)

Here 𝛮𝑖 is the set of neighbor buses connected directly to bus

𝑖. It is useful to rewrite these summations in a matrix-vector

format, where we have:

𝑝 = 𝐵𝜃 𝑝𝜖𝑅𝐾 (5)

𝑞 = 𝐵𝑣 𝑞𝜖𝑅𝐾 . (6)

In this notation the voltage amplitude and voltage phasor angle

values from all of the nodes are collected in two vectors

𝑣, 𝜃 𝜖𝑅𝐾 , respectively. Also, the active and reactive power

values of all of the nodes are stored in the vectors 𝑝 and 𝑞,

respectively. The matrix 𝐵 ∈ 𝑅𝐾×𝐾 is called the nodal-

admittance matrix, describing a power network of 𝐾 buses,

and its elements can be represented in the following format:

𝐵𝑖𝑗 = {

−𝑏𝑖,𝑗 , 𝑖𝑓 𝑙𝑖,𝑗𝜖𝑆𝐸

∑ 𝑏𝑖,𝑗 , 𝑗𝜖𝛮𝑖𝑖𝑓 𝑖 = 𝑗

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

} (7)

The nodal-admittance matrix of the IEEE Standard-118 Bus

is shown in Fig. 1. We see that this matrix has a sparse

structure. It has been mentioned in the literature that this

matrix can be viewed as a weighted version of the Laplacian

matrix of the graph 𝐺(𝑆𝑁 , 𝑆𝐸) [17] and [15]. Moreover, since

the Laplacian matrix has a direct relationship with other

important structural matrices of a graph, the nodal-admittance

matrix 𝐵 can give a full description of the power network

structure; see [17] for more information on graphs and their

corresponding matrices. As a result, our goal is to design an

appropriate and fast method to determine the structure of the

matrix 𝐵 from measurements of the PN parameters.

C. Gaussian Distribution of Parameters

In [6]-[8] it has been thoroughly discussed that due to the

load uncertainty in large-scale transmission networks and the

increasing contribution of distributed renewable sources in

smaller scale PNs, injected active and reactive powers can be

modeled as random variables. Moreover, since “the injected

power models the superposition of many independent factors

(e.g., loads) it can be modeled as a Gaussian random variable

[6]”. The linear relationship in (3) suggests that the difference

of phasor angles (𝜃𝑖 − 𝜃𝑗) for each sample of time across a bus

can be approximated by a Gaussian random variable as well

[7]. Moreover, allowing for the fixed phasor at the slack bus,

under steady-state conditions individual phasor angle

measurements 𝜃𝑖 can be modeled as Gaussian random

variables. Consequently, we will assume that during the

observation window of a TI problem, the measurements of the

phase angle of node 𝑖 can be well approximated by Gaussian

random variables. As we will discuss in Section III, this

random behavior has an impact on the structure of the sensing

matrix in the TI problem, and it is beneficial to the

performance of sparse recovery techniques in Section IV.

Fig.1. Corresponding normalized Laplacian matrix of the standard IEEE

118 Bus.

III. SPARSE TOPOLOGY IDENTIFICATION

We now focus on the problem of recovering the topology

matrix 𝐵 from vectors of measurements 𝑝 and 𝜃 (see (5)), or

equivalently, 𝑞 and 𝑣 (see (6)). Our key assumption here is

that the PN is a sparse interconnected system. This assumption

is based on a survey of articles and standard power network

models found in software and toolboxes such as

MATPOWER. We observe that the regular maximum

connectivity level of a bus in a network is typically less than

5-10%, especially in very large power transmission systems.

For example, the highest connectivity level of a bus in IEEE

30 bus is 6 (10%), in IEEE 118 bus is 9 (7%) and in IEEE

2383 is 9 (0.3%). This means that each column in the nodal-

admittance matrix satisfies the definition of a sparse signal

(Section III-B). This sparsity helps us formulate TI as a sparse

recovery problem, which as supported by the theory of

Compressive Sensing, can be solved with a small set of

measurements in a fast and accurate way using SRP

algorithms. If the sparsity assumption is violated, however, the

performance of the algorithm will suffer. The formulation of

TI as a SRP is presented in Section III-A. Section III-B gives a

short background on CS. Finally, in Section III-C we

introduce the types of the SRP solvers used in this work.

A. Sparse Setup of Topology Identification Problem

Given an interconnected power network of 𝐾 nodes or

buses, let the measurements of the active power and phase

angle of node 𝑖 be associated with the following time series of

𝑀 sample times:

𝑝𝑖(𝑡) 𝑓𝑜𝑟 𝑡 = 1,2, . . . , 𝑀 (8)

𝜃𝑖(𝑡) 𝑓𝑜𝑟 𝑡 = 1,2, . . . , 𝑀 . (9)

As mentioned before, in the DC model, the (active or

reactive) power injected into a distinct bus 𝑖 follows the

superposition law (3) and (4). Since (4) cannot be validated

properly in some practical situations, we prefer to use (3) to

formulate the sparse TI problem. This means that, for each

node 𝑖 in the network and at each sample time 𝑡, we have

4

𝑝𝑖(𝑡) = ∑ 𝑝𝑖𝑗𝑗 (𝑡) = ∑ 𝑏𝑖,𝑗𝑗𝜖𝛮𝑖

(𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) . (10)

As mentioned before, 𝑏𝑖,𝑗 is the susceptance along the line

𝑙𝑖,𝑗 under the DC model, and 𝛮𝑖 is the set of neighbor buses

connected directly to bus 𝑖. If we write 𝑏𝑖,𝑗 = 0 for 𝑗 ∉ 𝛮𝑖, we

can extend (10) as follows:

𝑝𝑖(𝑡) = ∑ 𝑝𝑖𝑗𝑗 (𝑡) = ∑ 𝑏𝑖,𝑗𝑗𝜖𝑆𝐾

𝑖 (𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) + 𝑢𝑖(𝑡) + 𝑒𝑖(𝑡), (11)

where 𝑆𝐾𝑖 is the set of all nodes in the network except node 𝑖

(so the set 𝑆𝐾𝑖 contains 𝐾 − 1 nodes), 𝑢𝑖 is a possible leakage

of active power in node 𝑖 itself, and 𝑒𝑖 is the measurement

noise. Since we assume data is collected at 𝑀 sample times,

we can use matrix-vector notation to write

𝑝𝑖 = 𝐴𝑖𝑦𝑖 + 𝑢𝑖 + 𝑒𝑖, (12)

where, 𝑝𝑖 , 𝑢𝑖 , 𝑒𝑖𝜖𝑅𝑀. In equation (14) we have

𝐴𝑖 = [𝑎1,𝑖𝑇 , … , 𝑎𝑖−1,𝑖

𝑇 , 𝑎𝑖+1,𝑖𝑇 , . . . , 𝑎𝐾,𝑖

𝑇 ] 𝜖 𝑅𝑀×𝐾−1 (13)

𝑎𝑗,𝑖𝑇 = (𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) 𝜖𝑅𝑀 𝑓𝑜𝑟 𝑡 = 1,2, . . . , 𝑀 (14)

𝑦𝑖 = [𝑏𝑖,1, … , 𝑏𝑖,𝑖−1, 𝑏𝑖,𝑖+1, . . . , 𝑏𝑖,𝐾]𝑇

𝜖𝑅𝐾−1. (15)

Each column of the matrix 𝐴𝑖 represents the difference of

phase angles between node 𝑖 and one node 𝑗𝜖𝑆𝐾𝑖 in the network

across 𝑀 samples of time. As discussed in Section II-C, this

difference can be modeled as a Gaussian random variable. The

vector 𝑦𝑖 is a sparse vector, with all of its elements equal to

zero except a small portion, which are located in those

positions corresponding the (few) neighbors of node 𝑖. Summing 𝑢𝑖 and 𝑒𝑖 (which can also be modeled as a vector of

white Gaussian noise), we end up with the following equation

for each node:

𝑝𝑖 = 𝐴𝑖𝑦𝑖 + 𝜂𝑖 . (16)

The TI problem can be viewed as the estimation of all

vectors {𝑦𝑖}𝑖=1𝐾 that best match the observed measurements

{𝑝𝑖}𝑖=1𝐾 . In order to solve such a problem, we can define the

following optimization problem:

𝑚𝑖𝑛{�̂�𝑖}𝑖=1

𝐾∑ ‖𝐴𝑖�̂�𝑖 − 𝑝𝑖‖2

2𝐾𝑖=1 . (17)

Problem (17) can be solved individually for each node 𝑖. If 𝐴𝑖 is full rank, the number of measurements 𝑀 exceeds the

number of unknowns 𝐾 − 1, and the variance of the noise

vector, 𝜂𝑖, is small enough, this optimization problem can be

solved using Least Square (LS) techniques. Since the value of

𝐾 depends on the number of buses in the grid, a large number

of measurements 𝑀 will be needed to solve this problem in the

case of large power grids. In order to avoid this problem, we

suggest using sparse recovery techniques [18] to solve for the

vectors {𝑦𝑖}𝑖=1𝐾 . Since each vector 𝑦𝑖 is sparse, we can view

the goal in (16) as recovering an S-sparse vector 𝑦𝑖𝜖𝑅𝐾−1

from a set of observations 𝑝𝑖𝜖𝑅𝑀, where 𝐴𝑖𝜖𝑅𝑀×𝐾−1 is the

sensing matrix, 𝜂𝑖𝜖𝑅𝑀 is a vector of white Gaussian noise, and

𝑆 is the number of nodes which are directly connected to the

one individual node 𝑖 for which we are solving the problem. In

general we want 𝑀 ≪ 𝐾 so the problem can be solved using a

reasonable set of observations in huge power grids. After

solving this problem for each sparse vector 𝑦𝑖 corresponding

to each node 𝑖, we can concatenate all of the sparse vectors

together, form the weighted Laplacian matrix 𝐵, and the

process is completed. Because 𝜃𝑖(𝑡) − 𝜃𝑗(𝑡) = 0 for 𝑖 = 𝑗, 𝑆𝐾𝑖

should include 𝐾 − 1 nodes; in other words, we should keep

the node 𝑖 out of 𝑆𝐾𝑖 since it produces a vector of zeros in the

corresponding column of the matrix 𝐴𝑖. This means that we

are not able to find the value of the parameter 𝐵𝑖𝑖 from the

recovered vector 𝑦𝑖; however, according to the definition of

the nodal-admittance matrix 𝐵,

𝐵𝑖𝑖 = ∑ 𝑏𝑖,𝑗𝑗𝜖𝛮𝑖 . (18)

Thus, after recovering the vector 𝑦𝑖 , 𝐵𝑖𝑖 can be easily

calculated. In the next section, we briefly give a summary of

the required CS and SRP background and concepts used in

interpreting and solving (16) as an SRP.

B. CS Background and Related Concepts

Definition 1 (Sparse Signal): An S-sparse signal 𝑦𝜖𝑅𝐾 is a

signal of length 𝐾 with 𝑆 non-zero entries where 𝑆 < 𝐾 (in

many cases (𝑆 ≪ 𝐾).1

Intuitively, the SRP is an optimization problem in which the

goal is to recover an S-sparse signal 𝑦𝜖𝑅𝐾 from a set of linear

observations (measurements) 𝑝 = 𝐴𝑦 𝜖𝑅𝑀 where 𝐴𝜖𝑅𝑀×𝐾 is

the sensing matrix with 𝑀 < 𝐾 (in many cases 𝑀 << 𝐾).

Typically in CS, the number of measurements 𝑀 will be

proportional to the sparsity level 𝑆 of the signal 𝑦 to be

recovered. For example, it is known that when the sensing

matrix 𝐴 is drawn randomly from a suitable distribution (i.e.,

the measurement protocol is random), with high probability

any 𝑆-sparse signal 𝑦 can be recovered from the measurements

𝑝 = 𝐴𝑦 if the number of measurements 𝑀 is merely

proportional 𝑆 𝑙𝑜𝑔 (𝐾

𝑆). This number of measurements can be

significantly smaller than the overall signal length 𝐾, and is

only greater than the fundamental limit 𝑆 by a logarithmic

factor; one can interpret this logarithmic factor as the price

that should be paid for not knowing the locations of the sparse

coefficients in advance. Remarkably, assuming the signal 𝑦 to

be exactly sparse and the measurements 𝑝 = 𝐴𝑦 to be noise-

free, this recovery is exact. If the signal to be recovered is

nearly sparse (compressible2) or if the measurements are

noisy, this recovery is provably robust. Compressible signals

appear in variety of applications such as when CS-based

techniques are used for MRI; in many such applications,

approximate signal recovery is sufficient. However, in our

work each column of the matrix 𝐵 is exactly sparse.

Due to the underdetermined nature of this recovery

problem (i.e., since 𝑀 < 𝐾), the null space of the sensing

matrix 𝐴 is non-trivial; therefore, we have infinitely many

candidate solutions to the system of equations 𝑝 = 𝐴𝑦.

However, under certain conditions on the sensing matrix 𝐴,

CS-based recovery methods can be guaranteed to efficiently

1 The notation in this Section matches that in Section III-A, except for

convenience we suppose the unknown vector 𝑦 has length 𝐾 rather than 𝐾 −1.

2 The nearest 𝑆-sparse signal to 𝑥, 𝑥𝑠, can be obtained simply by keeping

the 𝑆 largest entries of 𝑥 and setting all remaining entries to 0. If 𝑥 is exactly

𝑆-sparse, then 𝑥 = 𝑥𝑠. If the distance from 𝑥 to 𝑥𝑠 is small (but not

necessarily zero), 𝑥 is said to be compressible.

5

find the candidate solution that is sparse enough. The

following definitions and theorems provide examples of such

conditions; for more discussion and technical proofs and

details please refer to [18]-[19] and references therein.

Definition 2 (Coherence): The coherence of an 𝑀 × 𝐾

matrix 𝐴 with normalized columns 𝑎1, 𝑎2, … , 𝑎𝐾 is defined as

follows:

𝜇𝐴 = max1≤𝑚,𝑛≤𝐾,𝑚≠𝑛

|⟨𝑎𝑚, 𝑎𝑛⟩| . (19)

The coherence of an 𝑀 by 𝐾 matrix 𝐴 is a value in the interval

[1

√𝐾, 1].

Definition 3 (Restricted Isometry Property): An 𝑀 by 𝐾

sensing matrix 𝐴 satisfies the Restricted Isometry Property

(RIP) of order 𝑆 if

(1 − 𝛿𝑆)‖𝑦‖22 ≤ ‖𝐴𝑦‖2

2 ≤ (1 + 𝛿𝑆)‖𝑦‖22 (20)

holds for all coefficient vectors 𝑦 with ‖𝑦‖0 ≤ 𝑆 where

𝛿𝑆𝜖(0,1) is known as the isometry constant of order 𝑆.

Theorem 1 (Coherence-based recovery guarantee): Suppose

𝑦 is S-sparse and let 𝑝 = 𝐴𝑦. If

𝜇𝐴 <1

2𝑆−1 , (21)

then �̂� = 𝑦 is the unique solution to 𝑝 = 𝐴�̂� having sparsity 𝑆

or less. The smaller the coherence of 𝐴, the larger the

permitted value of 𝑆, and the broader the class of signals that

can be recovered. Roughly, what these conditions require is

that any two subsets of columns of the sensing matrix 𝐴 must

be almost orthogonal to each other.

Theorem 2 (RIP-based recovery guarantees): Suppose 𝐴

satisfies the RIP of order 2𝑆 with 𝛿2𝑠 < 0.4651. Let 𝑝 =𝐴𝑦 + 𝜂 be noisy measurements of a vector 𝑦. Then for any S-

sparse vector 𝑦, (22) correctly returns �̂� = 𝑦 given 𝑝:

�̂� = 𝑎𝑟𝑔𝑚𝑖𝑛�́�

‖�́�‖1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ‖𝑝 − 𝐴�́�‖2 < 휀. (22)

Moreover, for any vector 𝑦, if 휀 ≥ ‖𝜂‖2, then the solution �̂� to

(22) obeys:

‖𝑦 − �̂�‖2 ≤ 𝐶1‖𝑦−𝑦𝑆‖1

√𝑆+ 𝐶2휀 , (23)

where 𝐶1 and 𝐶2 depend only on the matrix isometry constant.

The vector 𝑦𝑆 is the closest 𝑆-sparse approximation to 𝑦. The

optimization (22) is widely known as Basic Pursuit Denoising

(BPDN) and is an SRP. In the next section, we briefly

introduce several SRP solvers which are used in this work. By

generalizing these methods, we will introduce new adaptable

strategies in order to address the practical challenges involved

in solving the sparse TI problem.

C. Sparse TI Structure and Data Correlation Issues

In general, the standard techniques for solving SRPs can be

categorized into two major groups: greedy algorithms and

algorithms based on convex optimization. In this work we will

use the popular greedy Orthogonal Matching Pursuit (OMP)

algorithm (see Algorithm 1) and the convex optimization

algorithm known as LASSO. The LASSO estimator is defined

as follows:

�̂� = 𝑎𝑟𝑔𝑚𝑖𝑛�́�‖𝑝 − 𝐴�́�‖22 + 𝜆‖�́�‖1 . (24)

This can be viewed as the Lagrangian form of the BPDN

problem (24).

Algorithm 1 Orthogonal Matching Pursuit - OMP require: matrix 𝑨, measurements 𝒑 , stopping criterion

initialize: 𝒓𝟎 = 𝒑, 𝒚𝟎 = 𝟎, 𝒍 = 𝟎, 𝑺𝑼𝑷𝟎

= ∅ repeat

1.match: 𝒉𝒍 = 𝑨𝑻𝒓𝒍 2.identify support indicator:

𝒔𝒖𝒑𝒍

= {𝒂𝒓𝒈𝒎𝒂𝒙𝒋|𝒉𝒍(𝒋)|}

3.update the support:

𝑺𝑼𝑷𝒍+𝟏 = 𝑺𝑼𝑷𝒍 ∪ 𝒔𝒖𝒑𝒍 4.update signal estimate:

𝒚𝒍+𝟏

= 𝒂𝒓𝒈𝒎𝒊𝒏𝒛: 𝒔𝒖𝒑𝒑(𝒛)⊆𝑺𝑼𝑷𝒍+𝟏‖𝒑 − 𝑨𝒛‖𝟐

𝒓𝒍+𝟏 = 𝒑 − 𝑨𝒚𝒍+𝟏

𝒍 = 𝒍 + 𝟏 Until stopping criterion met

output: �̂� = 𝒚𝒍

a. Structured Sparsity

By inspection of the columns of the Laplacian matrices of

standard IEEE test-beds (Fig.1 and Fig.2) we see that the

columns tend to exhibit a clustered sparse structure: the

column vectors not only contain few nonzero elements (i.e.,

they are sparse), but the nonzero elements tend to occur in

nearly adjacent positions. In the case of the IEEE Standard-14

Bus or IEEE Standard-30 Bus, columns 2, 5 and 6 or columns

6 and 10 can be taken as clear examples, respectively. Since in

a real PN, nodes and lines are ordered based on their position

in the network, and commonly the neighbor nodes are

connected through neighbor lines (except in special cases), it

is realistic in practice to assume that nodes have an ordering

that lends itself to clustered sparsity.

Fig.2. Corresponding Laplacian matrices of two of the standard IEEE test-

beds (Scaled between [-1, 1]).

Definition 4 (Clustered sparsity): We refer to a signal 𝑦 ∈𝑅𝐾

as (S;C)-clustered sparse if the total number of nonzero

coefficients is 𝑆 and all nonzero coefficients are distributed

within 𝐶 disjoint clusters (the locations and sizes of the

clusters are arbitrary).

Figure 3 shows the difference between the general structure

of an 𝑆-sparse signal (Fig. 3a) and a clustered-sparse signal

(Fig. 3b).

Modified Clustered OMP

Clustered OMP is a generalized version of the traditional

greedy method OMP. COMP modifies the support

identification step of OMP in order to improve the SRP results

whenever we expect clustered sparsity in the signal to be

recovered [20]. In Section IV, we show how COMP offers

improved recovery of clustered sparse signals. The step-wise

description of COMP (from [20]) is given in Algorithm 2.

Laplacian Matrix of Graph # 14

2 4 6 8 1012 14

2

4

6

8

10

12

14 -1

-0.5

0

0.5

1Laplacian Matrix of Graph # 30

10 20 30

10

20

30 -1

-0.5

0

0.5

1

Laplacian Matrix of Graph # 57

10 20 30 40 50

10

20

30

40

50

-1

-0.5

0

0.5

1Laplacian Matrix of Graph # 118

20 40 60 80 100

20

40

60

80

100

-1

-0.5

0

0.5

1

6

Algorithm 2 Clustered Orthogonal Matching Pursuit- COMP require: matrix A, measurements 𝒑 , maximum cluster size m, stopping criterion

initialize: 𝒓𝟎 = 𝒑, 𝒚𝟎 = 𝟎, 𝒍 = 𝟎, 𝑺𝑼𝑷𝟎

= ∅ repeat

1.match: 𝒉𝒍 = 𝑨𝑻𝒓𝒍 2.identify support indicator:

𝒔𝒖𝒑𝒍

= {𝒂𝒓𝒈𝒎𝒂𝒙𝒋|𝒉𝒍(𝒋)|}

3.extend support

𝐬𝐮𝐩 ̂ 𝒍 = {𝒔𝒖𝒑𝒍 − 𝒎 + 𝟏, … , 𝒔𝒖𝒑𝒍, … , 𝒔𝒖𝒑𝒍 + 𝒎 − 𝟏} 4.update the support:

𝑺𝑼𝑷𝒍+𝟏 = 𝑺𝑼𝑷𝒍 ∪ 𝐬𝐮𝐩 ̂ 𝒍 5.update signal estimate:

𝒚𝒍+𝟏

= 𝒂𝒓𝒈𝒎𝒊𝒏𝒛: 𝒔𝒖𝒑𝒑(𝒛)⊆𝑺𝑼𝑷𝒍+𝟏‖𝒑 − 𝑨𝒛‖𝟐

𝒓𝒍+𝟏 = 𝒑 − 𝑨𝒚𝒍+𝟏

𝒍 = 𝒍 + 𝟏 Until stopping criterion met

output: �̂� = 𝒚𝒍

In order to improve the handling of border effects, we make a

slight modification to Step 3 of COMP as follows:

3.1. finding the boundaries of the support

Upper Bound = 𝒎𝒊𝒏((𝒔𝒖𝒑𝒍 + 𝒎 − 𝟏), 𝟏))

Lower bound = 𝒎𝒂𝒙(𝒔𝒖𝒑𝒍 − 𝒎 + 𝟏), 𝑵)

3.2. modified extend support:

𝒔𝒖𝒑𝒍 = {𝑳𝒐𝒘𝒆𝒓 𝒃𝒐𝒖𝒏𝒅 , … , 𝒔𝒖𝒑𝒍, … , 𝑼𝒑𝒑𝒆𝒓 𝑩𝒐𝒖𝒏𝒅 }

We call the resulting algorithm Modified COMP (MCOMP).

b. Data Correlation Issue

Even though graph data analytics are essential for gaining

insight into big networks, large-scale interconnected network

processing is complex because of its graph-specific

challenges, including complicated correlations among data

entities, highly skewed distributions, and so on. Considering

large-scale interconnected power networks, the possible

correlation among the recorded data from the nodes

(especially from neighboring nodes) is an issue in sparse TI

since it may result in highly coherent sensing matrices 𝐴. As

noted in Theorem 1, high coherence can have an undesirable

effect on the performance of SRP solvers. Recently, two new

techniques named Band–exclusion (B) and Local

Optimization (LO) have been developed in [22] in order to

help address the high coherence issue in sparse signal

recovery. The BLO technique known as BLOOMP can be

interpreted as adding an additional constraint on the support

selection step in OMP to prevent the algorithm from picking

highly correlated indexes in the estimated support of the

sparse signal [22]. Within this algorithm a new concept termed

the coherence band is defined. The coherence band (𝑐𝑏) and

related concepts are defined as follows: for some tolerance

𝛼 ≥ 0,

𝐵𝛼(𝜆) = {𝑖| |⟨𝑎𝑖, 𝑎𝜆⟩| > 𝛼} (25)

𝐵𝛼(𝛬) =∪𝜆𝜖𝛬 𝐵𝛼(𝜆) (26)

𝐵𝛼(2)(𝜆) = 𝐵𝛼(𝐵𝛼(𝜆)) =∪𝑗𝜖𝐵𝛼(𝜆) 𝐵𝛼(𝑗) (27)

𝐵𝛼(2)(𝛬) = 𝐵𝛼(𝐵𝛼(𝛬)) =∪𝜆𝜖𝛬 𝐵𝛼

(2)(𝜆). (28)

However, there is an issue with applying such a method in the

sparse TI problem. Due to nature of this method, whenever the

correlation between two columns of the sensing matrix is high

one of the corresponding positions will not be selected within

the true support. On the other side, as it has been mentioned

before, in TI the positions of the nonzero elements of a signal

vector are typically clustered. Since correlation in a sensing

matrix tends to be highest for columns corresponding to

neighboring nodes, this means that applying the BLOOMP

method may cause us to miss some nonzero elements of the

signal vector. In order to deal with this problem, we suggest an

algorithm composed of BLOOMP and MCOMP that we refer

to as BLOMCOMP (Algorithm 3); this algorithm adds the pre-

knowledge of clustered sparsity to BLOOMP. This

modification prevents us from losing nonzero neighbor

elements within a cluster. In addition to band-exclusion, LO is

used in BLOMCOMP as a residual-reduction technique (for

more information please refer to [22]).

Algorithm 3 Band-excluded Locally Optimized MCOMP

(BLOMCOMP) require: matrix 𝑨, measurements 𝒑 , Coherence band 𝜶 > 𝟎, maximum

cluster size 𝒎, stopping criterion

initialize: 𝒓𝟎 = 𝒑, 𝒚𝟎 = 𝟎, 𝒍 = 𝟎, 𝑺𝑼𝑷𝟎 = ∅

repeat

1.match: 𝒉𝒍 = 𝑨𝑻𝒓𝒍 2.identify support indicator:

𝒔𝒖𝒑𝒍

= {𝒂𝒓𝒈𝒎𝒂𝒙𝒋|𝒉𝒍(𝒋)|} , 𝒋 ∉ 𝑩𝜶(𝟐)

(𝑺𝑼𝑷𝒍−𝟏)

3.update the support:

Upper Bound = 𝒎𝒊𝒏((𝒔𝒖𝒑𝒍 + 𝒎 − 𝟏), 𝟏))

Lower bound = 𝒎𝒂𝒙(𝒔𝒖𝒑𝒍 − 𝒎 + 𝟏), 𝑲)

𝒔𝒖𝒑 ̂ 𝒍 = {𝑳𝒐𝒘𝒆𝒓 𝒃𝒐𝒖𝒏𝒅 , … , 𝒔𝒖𝒑𝒍, … , 𝑼𝒑𝒑𝒆𝒓 𝑩𝒐𝒖𝒏𝒅 } 𝑺𝑼𝑷𝒍+𝟏 = 𝑳𝑶(𝑺𝑼𝑷𝒍 ∪ {𝒔𝒖𝒑 ̂ 𝒍}) 4.update signal estimate:

𝒚𝒍+𝟏

= 𝒂𝒓𝒈𝒎𝒊𝒏𝒛: 𝒔𝒖𝒑𝒑(𝒛)⊆𝑺𝑼𝑷𝒍+𝟏‖𝒑 − 𝑨𝒛‖𝟐

𝒓𝒍+𝟏 = 𝒑 − 𝑨𝒚𝒍+𝟏

𝒍 = 𝒍 + 𝟏 Until stopping criterion met

output: �̂� = 𝒚𝒍

Local Optimization Procedure require: 𝑨, 𝒚 , Coherence band 𝜶 > 𝟎, 𝑺𝑼𝑷𝟎 = ∪ {𝒔𝒖𝒑 ̂ 𝒊} 𝒇𝒐𝒓 𝒊 = 𝟏: 𝒌+1

repeat: for 𝒊 = 𝟏: 𝒌+1

1. 𝒚𝒊

= 𝒂𝒓𝒈𝒎𝒊𝒏𝒛:𝒔𝒖𝒑𝒑(𝒛)=(𝑺𝑼𝑷𝒊−𝟏 \{𝒔𝒖𝒑 ̂ 𝒊}) ∪ {𝝀𝒊},𝝀𝒊𝝐𝑩𝜶({𝒔𝒖𝒑 ̂ 𝒊})‖𝒑 − 𝑨𝒛‖𝟐

2. 𝑺𝑼𝑷𝒊 = 𝒔𝒖𝒑𝒑(𝒚𝒊)

output: 𝑺𝑼𝑷𝒌+𝟏

Fig.3 (a) Structure of a 6-sparse signal. (b) A (25;5)-clustered-sparse signal.

(c) The average value of Coherence 𝜇𝐴 vs # of measurements 𝑀 over all 118

corresponding sensing matrices '𝐴' for all of the nodes in the IEEE 118-Bus.

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mag

nitu

de

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Index

Mag

nitu

de

a) b)

c)

7

IV. SIMULATION RESULTS AND DISCUSSION

A. Setup

In this section, we test the proposed method for recovering

the topology of an SG using compressive observations (𝑀 <𝐾) collected from the system parameters. We see that the

probability of successful recovery varies for different nodes in

the network based on the local pattern of sparsity. In these

simulations, we use the IEEE Standard-30 Bus, 118 Bus and,

2383 as case studies. These three PNs include 30 Buses and 47

power transmission lines, 118 Buses and 186 lines, and 2383

Buses and 2896 lines, respectively, and their detailed

specifications have been fully described in the MATPOWER

toolbox [16]. To generate the data, we first feed the system

with Gaussian demand and simulate the PN. The

MATPOWER [16] toolbox is used for solving the power flow

equations in various demands and the resulting phase angle

and active power measurements are applied as the input to the

SRP. Finally, white noise has been added to measurement

vectors to set the SNR in the range of 10-20 dB randomly.

Each column of the matrix 𝐵 represents one of the sparse

vectors 𝑦𝑖 which we attempt to recover by solving an SRP. For

the IEEE Standard-118 Bus, Fig. 3c shows how the coherence

(19) of the corresponding sensing matrix 𝐴 changes as the

number of measurements 𝑀 increases; curves are averaged

over 1000 realizations of the network and over all 118 nodes.

This coherence measure appears to approach a non-zero

asymptote as the number of measurements increases. As has

been mentioned before, the smaller the coherence metric, the

larger the sparsity 𝑆 of signals that can be recovered. One

class of matrices well known to have low coherence are

random Gaussian matrices. As mentioned in Section II-C and

in the literature such as [6]-[8], due to the load uncertainty and

the aggregation of the renewables in SGs, the difference of the

phase angles across a bus can be approximated by a Gaussian

random variable. Moreover, in the SRP formulation in Section

III-A, the sensing matrix is formed by the variation in bus

phase angles. Thus, the closer the behavior of the phase angles

is to being Gaussian, the better one might expect the sparse

recovery algorithms to perform.

B. Results

Within the network graph, SRP solvers exhibit different

recovery performance on different nodes for a given number

of measurements, mainly because of the following two factors.

The first factor is the sparsity level of the signal 𝑦𝑖 (or in our

PN, the in-degree, or the number of connected links to a bus),

and the second factor is the presence of a possible structure in

the position of the non-zero components. Figs.4-7 show the

recovery performance of the OMP (green), LASSO (blue) and

MCOMP (red) algorithms over 1000 realizations of three

IEEE networks used as case studies. In the first case, the IEEE

standard-30 Bus system has been used. In Figs.4a and 4b,

recovery results are illustrated for nodes 3 and 10

(corresponding to the easiest and the hardest cases in the IEEE

30 bus network), respectively. At each specific number of

measurements 𝑀 (𝑀 𝜖 {4, … ,60}), the vertical axis shows the

percentage of trials in which perfect recovery is achieved, over

1000 realizations of the network. Nodes 3 and 10 are

distinguished from each other by their in-degrees. We see that

node 3, with in-degree 2, is more frequency recovered

correctly than is node 10, which has in-degree 6. This is to be

expected: in the corresponding SRPs, the signal 𝑦3 has a

sparsity level of 2, while the signal 𝑦10 has sparsity 6; signals

with higher sparsity levels are, in general, more difficult to

recover.

Fig.4. Recovery rate comparison of nodes 3 (a) and 10 (b) in the IEEE 30 bus system. These nodes have in-degree 2 and in-degree 6, respectively.

Another important factor, which affects the SRP solver’s

performance, is the existence of a possible clustered structure

in the sparse signals to be recovered. As we have mentioned,

MCOMP is an extended version of OMP, which has been

adapted to exploit the knowledge that the non-zero entries of

the signal appear in clusters. The structured sparsity in the

columns of the matrix 𝐵 can be used to improve the recovery

performance. Figs.4a and 4b show how the presence of

clustered sparsity helps the MCOMP algorithm to outperform

the LASSO and OMP, especially in the case of node 10, where

𝑦10 is a (6, 4)-clustered sparse signal and the in-degree of the

node (or the number of nonzero elements of the signal to be

recovered) is larger.

In order to compare the performance over more complicated

cases, the IEEE Standard-118 Bus and 2383 Bus systems have

also been used for modeling the PN. In the IEEE 118 bus, we

focus on recovering the connections to nodes 74 (which has

in-degree 2), 49 (in-degree 9), and 59 (in-degree 6). In the

IEEE 2383 bus, we focus on nodes 3 (in-degree 3) and 1920

(in-degree 9). From the sparsity level viewpoint, 𝑦49 and 𝑦1920

correspond to the most complicated signals 𝑦𝑖 to be recovered

in the two networks, respectively. Recovery results are again

shown over 1000 realizations of the corresponding network.

Figs.5a-c show the recovery results for the IEEE 118 Bus

network. As might be expected, we see that node 74 (with in-

degree 2) is more likely to be recovered from a given number

of measurements than is node 59 (in-degree 6). Node 49 (with

in-degree 9) is the most difficult to recover. These plots also

show how MCOMP outperforms LASSO and OMP in the case

a)

b)

8

of node 49, where 𝑦49 is a (9, 5)-clustered sparse signal and

the in-degree of the node (or the number of nonzero elements

of the signal to be recovered) is larger. On node 59 (where the

corresponding signal 𝑦59 is a (6, 3)-clustered sparse signal)

MCOMP again outperforms OMP and LASSO. On node 74

(where 𝑦74 has no clustered sparsity), LASSO is the best

performing algorithm.

Fig.5. Recovery rate comparison of nodes 74, 59, and 49 in the IEEE 118-Bus,

with in-degree 2, 5, and 9 respectively.

Moreover, Fig.6.a and b are representing the recovery

performance of each of 3 SRP solvers for 3-sparse node 3 and

9-sparse node 1920 in IEEE 2383 Bus network. Due to the

larger scale of this network, we generally need more

measurements to achieve the perfect recovery for any

individual node compared to smaller networks like the IEEE

118 Bus. As it has been mentioned before, the number of

measurements needed for full recovery depends on both the

sparsity level and the original dimension of the signal 𝑦 (here,

the dimension 𝐾 equals the number of buses in our sparse TI

problem); specifically, 𝑀 must be at least proportional to

𝑆 𝑙𝑜𝑔(𝐾/𝑆). As a rough rule of thumb, in many applications,

when the sensing matrix has low coherence, one observes

perfect recovery when 𝑀 ≈ 4𝑆. However, as the dimension 𝐾

grows, it becomes necessary to take more measurements. For

example, in the case of the IEEE 118 Bus, node 49 has in-

degree 9, and perfect recovery is possible when 𝑀 ≈ 50. In

the case of the IEEE 2383 network, node 1920 again has in-

degree 9, but perfect recovery is not possible until 𝑀 ≈ 80.

Fig.6. Recovery rate comparison of nodes 3 and 1920 in the IEEE standard 2383 network. Nodes 3 and 1920 have in-degree 3 and 9, respectively.

Comparing the results from different size networks it is

clear that number of measurements required for perfect

recovery is affected less by the network size than by the

sparsity level of the corresponding signal 𝑦𝑖 to be recovered.

Taking fewer measurements reduces the recovery performance

of any SRP solver; however, the LASSO and MCOMP

generally show better recovery performance than OMP.

Moreover, we cannot recover the nodes with higher in-degrees

(such as node 49, where 𝑦49 is a (9, 5)-clustered sparse signal)

until 𝑀 is large enough that the PN coherence metric reaches a

suitably small level, and if such a level were never to be

reached, we might be limited in the degree of nodes that we

could recover with these techniques.

Finally, in order to highlight the effect of the data

correlation issue in addition to the impact of clustered sparsity,

we have examined the recovery results for two other clustered

sparse signals in the IEEE 118 bus: signal 𝑦37 which is a (6,

2)-clustered sparse signal, and signal 𝑦80 which is an (8, 4)-

clustered sparse signal. In general, as one should expect 𝑦37 to

be easier to recover compared to 𝑦80. In order to evaluate the

results we define the interconnection order (𝐼𝑂) for node 𝑖 as

follows:

𝐼𝑂(𝑖) = ∑ 𝑖𝑛 − 𝑑𝑒𝑔𝑟𝑒𝑒(𝑗)𝑗𝜖𝛮𝑖 (29)

Node 37 has 6 direct neighbors and 𝐼𝑂(37) = 17; node 80

has 8 direct neighbors and 𝐼𝑂(80) = 22. The interconnection

order can be interpreted as a factor that can reflect the level of

data correlation within a specific bus versus the rest of the

network; however, it will not be the only factor. Fig.7

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Recovery Performance of 3 SRP solvers over 1000 realization of the network for node 74

Number of Measurments "M"

Reco

very

Perf

orm

an

ce (

%)

LASSO

MCOMP

OMP

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Recovery Performance of 3 SRP solvers over 1000 realization of the network for node 59

Number of Measurments "M"

Rec

ov

ery

Per

form

ance

(%

)

LASSO

MCOMP

OMP

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Number of Measurments "M"

Rec

ov

ery

Per

form

ance

(%

)

Recovery Performance of 3 SRP solvers over 1000 realization of the network for node 49

LASSO

MCOMP

OMP

a)

b)

c)

b)

a)

OMP

9

illustrates how BLOMCOMP outperforms MCOMP, OMP

and LASSO in the case of node 80, where we face a more

complicated and interconnected local topology and a higher

coherence in the corresponding sensing matrix.

Fig.7. Recovery rate comparison of nodes 37 and 80 in the IEEE 118-Bus.

An initial value of m = 4 is chosen in MCOMP and BLOMCOMP.

Figs.8a-c illustrate the recovery performance for 3 SRP

solvers bus-by-bus (i.e., for each of the columns of the nodal-

admittance matrix 𝐵) over 100 realizations of the IEEE 118-

Bus network. The vertical axis represents the number of

measurements 𝑀 taken for sparse recovery of each sparse

signal from the following set: {10, 20, 30, 40, 50}. The

horizontal axis shows the bus number from 1 to 118, or

equivalently, the column number from the nodal-admittance

matrix. The recovery performance is illustrated by a color

spectrum ranging from dark blue (corresponding to 0%

recovery) up to dark red (corresponding to 100% recovery).

As can be seen, almost all of the algorithms can reach 100%

recovery performance with 50 measurements. However,

LASSO and MCOMP generally exhibit better performance

when the number of measurements is lower, and MCOMP

generally has better performance than LASSO, especially in

the case of clustered sparse signals. Table I compares the

recovery performance of 3 SRP solvers using 30

measurements for some of the most significant structured

columns in the IEEE 118-Bus system.

Finally, Fig.9 illustrates the whole network topology

recovery performance for 3 SRP solvers for the standard IEEE

118 Bus system, over 1000 realizations of the network. The

vertical axis shows the percentage of trials in which all 118

columns of the nodal-admittance matrix 𝐵 (and, as a result, the

whole network topology) are perfectly recovered. This figure

also shows that the whole network topology can typically be

recovered from 50 or fewer measurements per bus.

Fig.8. Recovery performance for 3 SRP solvers over all of the buses of the

standard IEEE 118 Bus system: (a) OMP, (b) LASSO, (c) MCOMP.

Table. I. Recovery performance of 3 SRP solvers using 30 measurements for

the most significant clustered sparse buses in the IEEE 118-Bus system.

Bus (Column) # OMP (%) LASSO (%) MCOMP (%)

5 90 91 100

11 92 100 98

12 52 54 51

15 85 93 100

17 74 71 88

56 89 92 100

62 92 99 99

85 74 92 99

100 31 32 78

105 90 90 100

Fig.9. Whole network topology recovery performance for 3 SRP solvers

for the standard IEEE 118 Bus system over 1000 realizations of the network.

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Recovery Performance of 3 SRP solvers over 1000 realization of the network for node 37

Number of Measurments "M"

Rec

ov

ery

Per

form

ance

(%)

LASSO

MCOMP

OMP

BLOMCOMP

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Recovery performance of 3 SRP solvers over 1000 realization of the network for node 80

Number of Measurments "M"

Rec

ov

ery

Per

form

ance

(%

)

LASSO

MCOMP

BLOMCOMP

OMP

a)

b) OMP

LASSO MCOMP

BLOMCOMP

b)

c)

a)

10

V. CONCLUSIONS

In this study, we explore the topology identification

problem for smart grids. The DC power flow model, graph

theory, and sparse recovery techniques from compressive

sensing are composed in order to address the following

challenges in topology identification and line outage

localization of transmission lines: 1) the computational

complexity of data collection and analysis procedures in large

scale SGs; 2) the effect of distributed generation systems such

as wind turbines or PV cells; and 3) the uncertainty of system

states and parameters caused by the nonlinear and stochastic

behavior of loads. The proposed method models the SG as an

interconnected graph. Next, using an approximated version of

the AC power-flow model (a DC model) under the

probabilistic power flow regime, the TI problem is

mathematically formulated as an optimization problem.

Finally, due to the sparse nature of this optimization

problem the TI can be interpreted as an SRP, which can

efficiently be solved using SRP solvers such as greedy (OMP,

MCOMP) or optimization based (LASSO) algorithms. We

have discussed important concepts from CS such as coherence

and its behavior in a sparse representation of the TI problem

for IEEE standard networks. Network models have been

generated using the MATPOWER toolbox and standard IEEE

test-beds. The main advantage of the presented method is that

this sparse reformulation of the TI problem relies only on the

measurements from the system parameters and does not need

any a priori information about the topology of the network. In

fact, in our technique, the output of the SRP solver methods is

the structure or topology matrix of the network itself. Case

studies using standard IEEE standard test-beds have shown

that the proposed method represents a promising new strategy

for topology identification, fault detection, and monitoring

using only a small set of observations from some of the bus

parameters. This paper shows that the recovery performance

of the SRP solvers is mainly dependent on the in-degree

(number of lines connected) of each bus in the network.

Moreover, the sparse signals to be recovered often exhibit a

clustered-sparse structure, so it is possible to improve the

recovery results using SRP solvers that encourage structured

sparsity. We have also addressed the problem of correlation in

the sensing matrix among columns that are likely to belong to

the signal support.

REFERENCES

[1] X.Y.Yu, C. Cecati, T. Dillon, and M. G. Simões, “The new frontier of

smart grids,” IEEE Ind. Electron. Mag., vol. 5, no. 3, pp. 49–63, Sep. 2011.

[2] V. Gungor, D. Sahin, T. Kocak, S. Ergut, C. Buccella, C. Cecati, and G.

Hancke, Smart Grid Technologies Communications Technologies and Standards” IEEE Transactions on Industrial Informatics, vol.7, no. 4,

pp. 529-539, Sept. 2011.

[3] Weilin Li, Mohsen Ferdowsi, Marija Stevic, Antonello Monti, and Ferdinanda Ponci, “Cosimulation for Smart Grid Communications”,

IEEE Transactions on Industrial Informatics, vol. 10, no. 4, Nov 2014.

[4] X. Fang, S. Misra, G. Xue, and D. Yang, “Smart grid-The new and improved power grid: A survey,” IEEE Commun. Surveys Tutorials, vol.

14, no. 4, pp. 944–980, Fourth Quarter 2012.

[5] Harirchi, Farnaz; Simoes, M.Godoy, Al Durra, Ahmed and Muyeen, S.M. "Short transient recovery of low voltage-grid-tied DC distributed

generation". ECCE, 2015 IEEE, 20-24 Sept. 2015, pp: 1149 - 1155,

Montreal, QC, Canada

[6] H. Sedghi, E Jonckheere. Information and Control in Networks. In

Giacomo Como, Bo Bernhardsson, Anders Rantzer, editors, Lecture Notes in Control and Information Sciences, vol. 450, 2014, Springer

Cham Heidelberg New York Dordrecht London, Ch. 9, pp 277-297.

[7] M. He and J. Zhang. “A dependency graph approach for fault detection and localization towards secure smart grid”. IEEE Transactions on

Smart Grid, no. 2, pp. 342–351, June 2011.

[8] A. Schellenberg, W. Rosehart, and J. Aguado, “Cumulant-based probabilistic optimal power flow (P-OPF) with Gaussian and Gamma

distributions,”IEEE Trans. on Power Systems, vol. 20, pp. 773–781,

May 2005. [9] V. Kekatos, G. B. Giannakis, and R. Baldick, “Grid topology

identification using electricity prices,” in Proc. IEEE Power & Energy

Society General Meeting, National Harbor, MD, Jul. 2014. [10] H. Zhu and G. B. Giannakis, “Sparse Overcomplete Representations for

Efficient Identification of Power Line Outages” IEEE Trans. on Power

Systems, vol. 23, pp. 1644-1652, Oct. 2012. [11] J.C. Chen, W.T Li, C.K. Wen, J.H. Teng, P. Ting. “Efficient

identification method for power line outages in the smart power grid”.

IEEE Trans. Power Systems, vol. 29, pp. 1788–1800, 2014. [12] Zhao and W.-Z. Song, “Distributed power-line outage detection based

on wide area measurement system,” Sensors, vol. 14, no. 7, pp. 13 114–

13 133, Jan. 2014. [13] J. Wu, J. Xiong, and Y. Shi, “Efficient location identification of multiple

line outages with limited PMUs in smart grids,” IEEE Trans. Power

Syst., vol. 30, no. 4, pp. 1659-1668, Jul 2015. [14] Wen-Tai Li, Chao-Kai Wen, Jung-Chieh Chen, Kai-Kit Wong, Jen-Hao

Teng, and Chau Yuen. “Location Identification of Power Line Outages Using PMU Measurements with Bad Data”. arXiv:1502.05789v1

[cs.SY] 20 Feb 2015

[15] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, 2001. [16] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “Steady-

state operations, planning and analysis tools for power systems research

and education,” IEEE Trans. on Power Systems, vol. 26, no. 1, pp. 12–19, Feb. 2011.

[17] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and

Control, 2nd ed. New York: Wiley, 1996. [18] E.J. Candes and M.B. Wakin, “An introduction to compressive sam-

`pling”, IEEE Signal Processing Magazine, vol. 24, pp. 21–30, Mar

2008.

[19] M. B. Wakin. Compressive Sensing Fundamentals. In M. Amin, editor,

Compressive Sensing for Urban Radar, pp. 1-47. CRC Press, Boca

Raton, Florida, 2014. [20] B.M. Sanandaji, T.L. Vincent, and M.B. Wakin. “Compressive

Topology Identification of Interconnected Dynamic Systems via

Clustered Orthogonal Matching Pursuit”, in Proceedings of the 50th IEEE CDC-ECC 2011, Orlando, Florida, USA, Dec. 2011, pp. 174–180.

[21] M. Babakmehr, M.G. Simões, A. Al Durra, F. Harirchi and Qi. Han.

“Application of Compressive Sensing for Distributed and Structured Power Line Outage Detection in Smart Grids”, IEEE, ACC 2015,

Chicago, Illinois, USA, July 2015. p. 3682-3689.

[22] A. Fannjiang, W. Liao. "Coherence Pattern–Guided Compressive Sensing with Unresolved Grids". SIAM J. IMAGING SCIENCES, vol. 5,

no. 1, pp. 179–202, 2012.

[23] Debomita Ghosh, T. Ghose, and D. K. Mohanta. “Communication Feasibility Analysis for Smart Grid with Phasor Measurement Units”.

IEEE Transactions on Industrial Informatics, vol. 9, no. 3, pp 1486-

1496, Aug 2013.

[24] Ayman I. Sabbah, Student Member, IEEE, Amr El-Mougy, and

Mohamed Ibnkahla, A Survey of Networking Challenges and Routing

Protocols in Smart Grids, IEEE Transactions on Industrial Informatics, vol. 10, no. 1, pp 210-221, Feb 2014.

[25] David M. Laverty, Robert J. Best, Paul Brogan, Iyad Al Khatib, Luigi

Vanfretti, and D. John Morrow, “The OpenPMU Platform for Open-Source Phasor Measurements” IEEE Transactions on Instrumentation

and measurements, vol. 62, no. 4, Apr 2013.