Smart Grid Topology Identification Using Sparse Recoverymsimoes/documents/papers2016/MGS_pape… ·...

0093-9994 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIA.2016.2574767, IEEETransactions on Industry Applications

Abstract -- Smart grid (SG) technology reshapes the traditional

power grid into a dynamical network with a layer of

information that flows along the energy system. Recorded data

from a variety of parameters in SGs can improve analysis of

different supervisory problems, but an important issue is their

cost and power efficiency in data analysis procedures. This

paper develops an efficient solution for power network topology

identification (PNTI) and monitoring activities in SG. The basic

idea combines optimization-based sparse recovery techniques

with a graph theory foundation. The power network (PN) is

modeled as a large interconnected graph, which can be

evaluated with the DC power-flow model. It has been shown that

topology identification for such a system can mathematically be

reformulated as a sparse recovery problem (SRP), and the

corresponding SRP can efficiently be solved using SRP solvers.

In this work we especially exploit the concentration of nonzero

elements in the corresponding sparse vectors around the main

diagonal in the nodal-admittance or structure matrix of the PN

to improve the results. The network models have been generated

with the MATPOWER toolbox, and Matlab-based simulation

results have indicated the promising performance of the

proposed method for real time TI in SGs.

Index Terms—Smart Grid, Sparse Recovery, Topology

Identification, Reweighted 𝒍𝟏-minimization.

I. INTRODUCTION

Smart grid (SG) technology reshapes the traditional power

grid from a single layer physical system to a huge dynamical

network that includes a second layer of information that

flows through the system [1]. This information layer is

formed from two major classes of data: the first class contains

the records collected from the status of different parameters

in the network such as bus voltages, powers, currents, and so

on [2], while the second class comprises the controlling

commands which are fed back to the network from decision

making units [13]. The supervisory control and data

acquisition (SCADA) system in addition to the technology of

the wide area monitoring system (WAMS) can provide such

massive voltage and power data in near real time [2]-[3].

Based on this vast amount of data, it is possible to define new

appropriate analysis tools which can transfer the standard

security and monitoring methods from static frameworks into

dynamical frameworks [12].

In addition to normal monitoring issues, PNTI and power

line status monitoring is particularly critical for a number of

1 M. Babakmehr, M.G. Simões, M.B. Wakin and F. Harirchi are with the

Division of Electrical Engineering, Department of Electrical Engineering and

Computer Science, Colorado School of Mines, Golden, CO 80401 USA, (e-mails: {mbabakme, msimoes, mwakin, harirchi}@ mines.edu). A. Al Durra

is with the Electrical Engineering Department, Petroleum Institute, Abu

Dhabi, (email: [email protected]). This work was partially supported by Petroleum Institute (PI) grant

470039 and NSF CAREER grant CCF-1149225.

tasks, including real-time contingency analysis, power-flow

analysis, power outage identification (POI), state estimation,

resiliency against natural disasters, and security assessment of

power systems [4]-[6]. Both network structural change and

fault identification also play a crucial role in power quality

improvement, system operation, and Microgrid technology

[7]. Smart Microgrid units will form the basis of the future

generation of power systems [8]. Numerous advanced

Microgrid design schemes have been developed to date [10],

where power network situational awareness is the most

critical point for islanding detection, system planning, and

protection [9], [11]. The system data exchange (SDX) module

of the North American Electric Reliability Corporation

(NERC) can provide grid-wide topology information on an

hourly basis. However, near real-time monitoring of

transmission lines is mandatory in order to make the PN act

as a smart system.

The PNTI problem has been addressed using economics in

[14]. Most other related works are focused on the problem of

fault or outage line detection in a power grid [4]-[5], [14]-

[21], [35]. In general, identifying the position of possible

outage lines can be interpreted as power network topology

identification; we call this the Power Outage Identification

Problem (POIP). Most of these works rely on the DC linear

power flow model (an approximation of the AC model) and

data measured from phasor measurement units (PMUs). The

POIP has been formulated as a combinatorial problem that is

computationally tractable only for single or double line

changes [4]. However, one must be able to cope with multiple

line changes in the face of cascading failures as in recent

blackouts. This has motivated several existing works [5],

[14]-[21]. A recent alternative approach for line-change

identification adopts a Gauss-Markov graphical model of the

power network and can deal with multiple changes [5];

Banerjee et al. [15] developed a new method for solving the

POIP based on the theory of quickest change detection, and

an ambiguity group-based location recognition algorithm has

been proposed in [16]. Moreover, a non-iterative method for

wide-area fault location has been presented in [17], which

applies the substitution theorem. In [18]-[20], the authors

used the sparse nature of the POIP to reformulate this

problem as a sparse recovery problem (SRP). In [21], a global

stochastic optimization technique based on cross-entropy

optimization (CEO) was presented. Most of these methods

require hourly base case grid topology information

(corresponding to the nodal-admittance matrix 𝐵, which we

will discuss) in addition to the system parameter data to be

able to find the outage lines. Also, a pre-whitening procedure

is mandatory for most of methods discussed above. There is a

rank deficiency issue when using such a grid topology matrix

[19]. In order to deal with this rank deficiency, one bus is

Smart Grid Topology Identification Using

Sparse Recovery Mohammad Babakmehr, Student Member, IEEE, Marcelo G. Simões, Fellow, IEEE,

Michael B. Wakin, Senior Member, IEEE, Ahmed Al Durra, Senior Member, IEEE,

Farnaz Harirchi, Student Member, IEEE1



normally considered as the reference bus, and its

corresponding row and column are removed from the matrix

𝐵 in order to obtain a full rank matrix [18]. However, due to

cyber-attacks and other precision deficiency issues, whenever

records from the reference bus are affected by bad data the

overall procedure might be completely corrupted. Another

important problem is with the switching events that are

mistakenly unreported. These kinds 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. In addition, to the best of our knowledge, none of

the methods mentioned above account for the random

behavior applied by renewable energy sources and the

uncertainty in loads [22].

Recently, we have addressed some of these issues in [23],

which employs a compressive sensing-based approach

involving some modifications to greedy-based sparse

recovery solvers. In this work we further investigate the

optimization-based formulations of the sparse PNTI, and we

aim to exploit additional structure within the sparse vectors in

order to improve the performance of the optimization-based

sparse recovery techniques for PNTI using a few

measurements of the system parameters (that can also be used

to solve the POIP). We particularly exploit and emphasize the

concentration of the nonzero elements in the corresponding

sparse vectors around the main diagonal of the nodal-

admittance or structure matrix of the PN, and we improve the

final identification performance using reweighted 𝑙1-

minimization [26].

Recently, sparse recovery has found interest in fault-type

identification and localization in distribution networks as well

[24], [25]. The approaches utilized here and in [23] are

completely different from the literature in the formulation.

Our methods rely only on the measurements from system

parameters and do not require any a priori information about

the topology of the network. In fact, we reformulate the PNTI

problem in such a way that the output of the optimization

problem is the structure or topology matrix 𝐵 of the network

itself. Case studies using IEEE standard test-beds [27] show

that the proposed method represents a promising fast and

accurate strategy for line change, fault detection, and

monitoring issues in SGs. This paper is the journal version of

our presented work in 2015 IAS annual meeting [36].

II. POWER NETWORK MODELING

This section describes the graph representation of power

networks, which are modeled with DC power flow equations.

In addition, we relate the Laplacian matrix of the graph to the

nodal-admittance matrix of the PN.

A. Network Model

In this work, the PN is modeled as a graph 𝐺(𝑆𝑁 , 𝑆𝐿),

consisting of a set of 𝑁 nodes 𝑆𝑁 = {1, . . . , 𝑁}, where each

bus of the PN is associated with a node 𝑖 ∈ 𝑆𝑁, and a set of 𝐿

edges or links 𝑆𝐿 ⊆ {𝑙𝑖,𝑗: 𝑖, 𝑗𝜖𝑆𝑁}, where each edge models a

transmission line.2 Following the same approach as [5] and

[18], the data from the buses’ electrical parameters such as

powers and voltages are assumed to be obtained using a

network of sensors. As an example, Fig.1 shows the structure

of the corresponding graph of the IEEE Standard-30 Bus.

Fig.1. Graph of IEEE Standard-30 Bus

B. DC Power Flow Model and its Graph

A simplified linearized approximation of the AC power

flow model has been introduced in [28] and [29]. This is

widely known as a DC load-flow (power-flow) model. Under

the DC load-flow regime, the active power injected to a

particular bus 𝑖 follows the superposition law

𝑃𝑖 = ∑ 𝑃𝑖𝑗𝑗 = ∑ 𝑏𝑖𝑗𝑗𝜖ℵ𝑖

(𝜃𝑖 − 𝜃𝑗) , (1)

where 𝑃𝑖𝑗 is the active power injected from node 𝑗 to node 𝑖,

𝜃𝑖 represents the phase of the voltage on each node, and 𝑏𝑖𝑗 is

the imaginary part of the admittance of line 𝑙𝑖,𝑗 or the

susceptance. Susceptance (B) is the imaginary part of

admittance (the inverse of impedance), in SI units,

susceptance is measured in Siemens. Considering, ℵ𝑖 as 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 𝒑, 𝜽 𝜖𝑅𝑁. (2)

In this equation the voltage phasor angle values of the nodes

are collected in the vector 𝜽𝜖𝑅𝑁, and the active power values

of the nodes are stored in the vector 𝒑𝜖𝑅𝑁. The matrix 𝐵 ∈𝑅𝑁×𝑁 is called the nodal-admittance matrix describing a

2 We will use bus/node and edge/(transmission) line interchangeably

within the rest of the manuscript.

Node 1

Node 2Node 3

Node 4 Node 5

Node 6

Node 7Node 8Node 9

Node 10 Node 11

Node 12

Node 13 Node 14

Node 15 Node 16

Node 17Node 18

Node 19

Node 20

Node 21

Node 22Node 23

Node 24

Node 25

Node 26 Node 27

Node 28

Node 29

Node 30



power network of 𝑁 buses, and it can be represented in the

following format:

𝐵𝑖𝑗 = {

−𝑏𝑖𝑗 , 𝑖𝑓 𝑙(𝑖,𝑗)𝜖𝑆𝐿

∑ 𝑏𝑖𝑗 ,𝑗𝜖𝛮𝑖 𝑖𝑓 𝑖 = 𝑗

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

} . (3)

It has been shown that this matrix can be viewed as a

weighted version of the Laplacian matrix of the graph

𝐺(𝑆𝑁 , 𝑆𝐿) [28], [18]. Since the nodal-admittance matrix gives

a full description of the structure of the networks’ graph, we

can represent the PNTI problem as that of determining the

structure of the nodal-admittance matrix 𝐵 [23]. As has been

discussed before [5], [22], and [30], power flow injection

originates from the aggregated load requests of a large

number of users and can be well approximated using

Gaussian random variables; in addition, there is uncertainty

caused by the utilization of renewable resources. In light of

this, the difference of phasor angles (𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) across a

bus in equation (2) can be approximated by a Gaussian

random variable at each sample of time [5], [22]. This

random behavior affects the structure of the sensing matrix

that is introduced in the next section; the randomness is

beneficial to the performance of sparse recovery techniques.

III. SMART GRID SPARSE TOPOLOGY IDENTIFICATION

We frame the PNTI problem as that of recovering the

topology matrix 𝐵 from the set of measurements of 𝑝 and 𝜃.

Similar to the CS-based approach that we have developed in

[23], our key assumption is that the PN is a sparse

interconnected system. This sparse structure helps us to

reformulate the PNTI problem as a sparse recovery problem

that can be solved using a small set of measurements in a fast

and accurate way using SRP solvers. This assumption has

been made based on our observations on a survey of standard

power grids models found on databases and toolboxes such as

MATPOWER. Considering a variety of standard PN models,

it can be observed that the regular maximum connectivity

level of an electrical bus in a grid is typically less than 5-

10%, especially in case of large-scale PNs. When the sparsity

assumption is not valid, however, the effectiveness of

sparsity-based PNTI will be limited.

A. PNTI Sparse Optimization-based Formulation

Given an interconnected power network of 𝑁 nodes or

buses, let the measurements of the active power and phase

angle of node 𝑖 be associated with the two following time

series, each of 𝑀 sample times:

𝑃𝑖(𝑡) , 𝜃𝑖(𝑡) 𝑓𝑜𝑟 𝑡 = 1,2, . . . , 𝑀 . (4)

Under the DC power flow regime, for each node in the

network and at each sample time 𝑡, we have the following

superposition law:

𝑃𝑖(𝑡) = ∑ 𝑃𝑖𝑗𝑗 (𝑡) = ∑ 𝑏𝑖,𝑗𝑗𝜖ℵ𝑖

(𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) . (5)

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

𝑖 and 𝑏𝑖,𝑗 is the susceptance along the line 𝑙𝑖,𝑗 under the DC

model. Since 𝑏𝑖,𝑗 = 0 for 𝑗 ∉ ℵ𝑖, we can extend (5) as

follows:

𝑃𝑖(𝑡) = ∑ 𝑃𝑖𝑗

𝑗

(𝑡) =

∑ 𝑏𝑖,𝑗𝑗𝜖𝑆𝑁𝑖 (𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) + 𝑢𝑖(𝑡) + 𝑒𝑖(𝑡), (6)

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

𝑢𝑖 is the possible leakage active power in node 𝑖 itself, and 𝑒𝑖

is the measurement noise. Representing the vector of active

power values as 𝒚𝒊, summing 𝒖𝒊 and 𝒆𝒊𝜖𝑅𝑀 (which we

assume to be modelled as a vectors of white Gaussian noise),

and dropping the time-sample notation, we end up with the

following equation for each node:

𝑦𝑖 = 𝐴𝑖𝑥𝑖 + 𝜂𝑖 (7)

𝐴𝑖 = [𝑎1,𝑖 , … , 𝑎𝑖−1,𝑖 , 𝑎𝑖+1,𝑖, . . . , 𝑎𝑁,𝑖] 𝜖 𝑅𝑀×𝑁−1 (8)

𝑎𝑗,𝑖(𝑡) = (𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)) 𝑓𝑜𝑟 𝑡 = 1,2, . . . , 𝑀 (9)

𝑥𝑖 = [𝑏𝑖,1, … , 𝑏𝑖,𝑖−1, 𝑏𝑖,𝑖+1, . . . , 𝑏𝑖,𝑁]𝑇

𝜖𝑅𝑁−1, (10)

where 𝒚𝒊𝜖𝑅𝑀, and where 𝜼𝒊𝜖𝑅𝑀 is a vector of white

Gaussian noise. In our formulation, each column of the

matrix 𝐴𝑖 represents the difference between phase angles of

node 𝑖 vs. each node 𝑗𝜖𝑆𝑁𝑖 in the network for 𝑀 samples of

time. Since 𝑏𝑖,𝑗 = 0 for 𝑗 ∉ ℵ𝑖, the vector 𝒙𝒊 is a 𝐾-sparse

vector (see Definition.1) where 𝐾 is the number of nodes

which are directly connected to node 𝑖.3

Given 𝒚𝒊, solving for 𝒙𝒊 can be approached using a Least

Squares formulation. However, a large number of

measurements 𝑀 will be needed to solve such a problem in

the case of large power grids. Since all vectors 𝒙𝑖 are sparse

vectors, in order to reduce the measurement requirement, we

suggest using sparse recovery techniques [31]. Solving this

problem for each sparse vector 𝒙𝒊, we can concatenate all of

the sparse vectors together, form the nodal-admittance matrix

𝐵, and the process is completed4 (PNTI-SRP).

For each step the goal is to recover one individual column 𝑖 (termed 𝒙𝒊 in our formulation) of the nodal-admittance

matrix5 𝐵. Once this column is recognized, the connectivity

structure between bus 𝑖 and all other buses in the network can

be determined by considering the location of nonzero

elements in that specific column. In particular, if 𝑥𝑖(𝑗) has a

nonzero value we consider bus 𝑖 to be connected to bus 𝑗

through a transmission line with susceptance 𝑥𝑖(𝑗). If the

3 Therefore, the topology identification problem can be viewed as the

estimation of all sparse vectors {𝑥𝑖}𝑖=1𝑁 that best match the observed

measurements {𝑦𝑖 , 𝐴𝑖}𝑖=1𝑁 .

4 Since 𝜃𝑖(𝑡) − 𝜃𝑗(𝑡) = 0 for 𝑖 = 𝑗 & 𝑓𝑜𝑟 𝑡 = 1,2, . . . , 𝑀, we should

keep the node 𝑖 out of 𝑆𝑁𝑖 to avoid producing 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 directly from recovered vector 𝑥𝑖;

however, according to the definition of the nodal-admittance matrix 𝐵, 𝐵𝑖𝑖 =∑ 𝑏𝑖,𝑗𝑗𝜖𝛮𝑖

. After recovering the vector 𝑥𝑖, 𝐵𝑖𝑖 can be easily calculated.

5 Regarding the mathematical formulation, the recovery step can be done individually and in parallel over all columns.



estimated 𝑥𝑖(𝑗) does not equal the estimated 𝑥𝑗(𝑖) for some

pair 𝑖 and 𝑗, this could be a sign of an outage or an anomaly

warranting further investigation.

In general, standard techniques for solving SRPs can be

categorized into two major groups: (𝑖) greedy algorithms, and

(𝑖𝑖) convex optimization based algorithms [16]. In [23] we

have applied the OMP greedy algorithm as a basic SRP

solver. We also presented modifications to the OMP

algorithm to handle certain practical problems such as a data

correlation issue. In this work we aim to discuss the

optimization based-rather than greedy-formulations of the

sparse PNTI, and in particular we exploit and emphasize the

concentration of nonzero elements in the corresponding

sparse vectors around the main diagonal of the nodal-

admittance matrix 𝐵. To exploit this structure, we present a

new PNTI formulation using reweighted 𝑙1-minimization.

B. 𝑆𝑝𝑎𝑟𝑠𝑒 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝐹𝑢𝑛𝑑𝑎𝑚𝑒𝑛𝑡𝑎𝑙𝑠

The sparse recovery problem (SRP) can be interpreted as a

paradigm for recovering an unknown signal from a set of

underdetermined linear measurements. The ability to solve an

SRP requires the assumption of sparsity for the signal to be

recovered, and it requires the measurement or sensing matrix

to satisfy certain conditions.6

Definition 1: A 𝐾-sparse signal 𝒙𝜖𝑅𝑁 is a signal of length

𝑁 with 𝐾 nonzero entries where 𝐾 < 𝑁 (in many cases

(𝐾 << 𝑁)). The sparsity level of a signal 𝑥 is denoted by the

𝑙0 norm ‖𝑥‖0.

Intuitively, an SRP is an optimization problem in which the

goal is to recover a 𝐾-sparse signal 𝒙𝜖𝑅𝑁 from a set of

observations 𝒚 = 𝐴𝒙 𝜖𝑅𝑀 where 𝐴𝜖𝑅𝑀×𝑁 is the sensing

matrix with 𝑀 < 𝑁 (in many cases 𝑀 << 𝑁) [31]-[32]. Due

to the underdetermined nature of this recovery problem (since

𝑀 < 𝑁), the null space of the matrix 𝐴 is non-trivial;

therefore, there exist infinitely many candidate solutions for

this problem. However, under certain conditions on the

sensing matrix 𝐴, various sparsity based recovery methods

can be guaranteed to efficiently find the candidate solution

that is sufficiently sparse. Examples of such conditions are

the restricted isometry property (RIP), the exact recovery

condition, and low coherence. In essence, what these

conditions require is that any two small subsets of columns of

the sensing matrix 𝐴 must be almost orthogonal to each other.

Definition 2: An 𝑀 by 𝑁 sensing matrix 𝐴 is said to satisfy

the Restricted Isometry Property (RIP) of order 𝐾 if there

exists a constant 𝛿𝐾𝜖(0,1) such that:

(1 − 𝛿𝑘)‖𝑥‖22 ≤ ‖𝐴𝑥‖2

2 ≤ (1 + 𝛿𝑘)‖𝑥‖22 (11)

holds for all 𝐾-sparse vectors 𝒙. The parameter 𝛿𝐾 is known

as the isometry constant of order 𝐾.

Definition.3: The coherence of an 𝑀 × 𝑁 matrix 𝐴 is the

maximum normalized inner product (correlation) between

any two distinct columns of 𝐴:

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

convenience we suppose the unknown vector 𝑥 has length 𝑁 rather than 𝑁 − 1.

𝜇A = max1≤𝑚,𝑛≤𝑁,𝑚≠𝑛

|⟨𝑎𝑚,𝑎𝑛⟩|

‖𝑎𝑚‖2‖𝑎𝑛‖2. (12)

The following sparse PNTI formulations have been

developed based on sparse recovery theorems [31]-[32].

C. 𝑙1-Minimization Based PNTI

If the 𝑀 by 𝑁 sensing matrix 𝐴 satisfies the RIP of order

2𝐾 with any isometry constant less than 1, then the following

optimization problem can recover the original 𝐾-sparse signal

𝒙 from the set of measurements 𝒚 = 𝐴𝒙:

𝑃0: �̂� = 𝑎𝑟𝑔𝑚𝑖𝑛�́� ‖�́�‖0 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑦 = 𝐴𝑥 ́ . (13)

In general, this 𝑙0-minimization problem is known to be

NP-hard. Fortunately, there is a relaxed version of this 𝑙0-

minimization problem, known as 𝑙1-minimization, that can

still guarantee the recovery of the sparse signal

𝑃1: �̂� = 𝑎𝑟𝑔𝑚𝑖𝑛�́�

‖�́�‖1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑦 = 𝐴𝑥 ́ . (14)

Here, ‖�́�‖1 = ∑ |�́�(𝑛)|𝑁𝑛=1 . Since the 𝑙1-norm is convex, this

results in a tractable convex optimization problem, widely

known as Basis Pursuit (BP).

D. Noisy 𝑙1-Minimization Based PNTI

In the case of noisy measurements, one can solve the

following problem instead:

𝑁𝑃1: �̂� = 𝑎𝑟𝑔𝑚𝑖𝑛�́�

‖�́�‖1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ‖𝑦 − 𝐴�́�‖2 < 𝜂. (15)

This problem is widely known as Basis Pursuit De-Noising

(BPDN). Suppose 𝐴 satisfies the RIP condition of order 2𝐾

with isometry constant 𝛿2𝐾 < 0.4651. Let 𝒚 = 𝐴𝒙 + 𝒏 be

noisy measurements of any vector x. If 𝜂 ≥ ‖𝑛‖2, then the

solution 𝒙 to (15) obeys:

‖𝑥 − �̂�‖2 ≤ 𝐶1‖𝑥−𝑥𝑆‖1

√𝑆+ 𝐶2𝜂, (16)

where 𝐶1 and 𝐶2 depend only on 𝛿2𝐾.7

In general, it is difficult to check whether a given sensing

matrix 𝐴 satisfies the RIP condition (11). An alternative, but

weaker property known as coherence (12) is easier to check

in practice. For a 𝐾-sparse vector 𝒙 and a vector of

measurements 𝒚 = 𝐴𝒙, if 𝜇𝐴 <1

2𝐾−1 , then, 𝑃1 can recover

the original 𝐾-sparse vector 𝒙 from the set of measurements

𝒚. Fig.2 shows how the average value of the coherence of the

sensing matrices 𝐴𝑖 (for 𝑖 = 1: 30) of IEEE Standard-30 Bus

changes as the number of measurements 𝑀 increases. The

smaller the coherence, the larger the permitted value of 𝐾,

and the broader the class of nodal-admittance matrix columns

(sparse vectors) 𝒙𝒊 that can be recovered. The minimum

number of measurements needed for perfect recovery is

related to both the original dimension 𝑁 of the signal 𝒙

(which in sparse PNTI equals the number of buses of the PN)

and the sparsity level 𝐾 (which equals the maximum in-

degree of buses); specifically, 𝑀 must be at least proportional

7 The vector 𝑥𝐾 is the closest 𝐾-sparse approximation to 𝑦



to 𝐾 𝑙𝑜𝑔(𝑁/𝐾). Successful recovery with 𝐾 𝑙𝑜𝑔(𝑁/𝐾) is

generally possible (with high probability) when the sensing

matrix 𝐴 is generated randomly, such as with independent

and identically distributed (i.i.d.) Gaussian entries.

E. Reweighted 𝑙1-Minimization Based PNTI

Some specialized methods have also been introduced

which are able to exploit extra prior information to solve the

SRP more efficiently and more accurately from even fewer

measurements than 𝑃1 or 𝑁𝑃1. Examining the structure of the

columns of the Laplacian (nodal-admittance) and adjacency

matrices for some standard IEEE test-beds (Fig.3) we can see

that due to the roughly sequential numbering of neighbor

buses within a PN, the positions of the nonzero elements in

both of these matrices exhibit a certain structure. In

particular, Fig.3 shows that most of the nonzero entries

concentrate close to the main diagonal. This means that in

each sparse vector 𝒙𝒊, certain entries are more likely than

others to be nonzero.

Fig.2. Average coherence versus number of measurements over all

corresponding sensing matrices 𝐴𝑖 of all of the nodes in IEEE Standard-30 Bus system, where the curve is averaged over 100 realizations of the network

Fig.3. Corresponding normalized Laplacian matrices of (a) IEEE Standard-57 Bus and (b) IEEE Standard-118 Bus, and the adjacency matrices of

(c) IEEE Standard-57 Bus and (d) IEEE Standard-118 Bus.

To exploit this anticipated behavior, we can replace 𝑃1

with the following weighted optimization problem instead:

𝑊𝑃1: �̂� = 𝑚𝑖𝑛�́�

∑ 𝜔(𝑛)|�́�(𝑛)|𝑁𝑛=1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑦 = 𝐴𝑥 ́, (17)

where 𝜔(1), 𝜔(2), … , 𝜔(𝑁) are non-negative weights. Similar

to 𝑃1, 𝑊𝑃1 is a convex optimization problem that can be

solved as a linear program (LP). By assigning smaller

weights near the diagonal elements, we can encourage those

elements to be selected in the recovered sparse signal 𝒙𝑖. 𝑊𝑃1

can be solved within a single step using our prior knowledge

for setting the weights. However, in order to increase the

accuracy of the final results we have extended this

formulation through an iterative procedure called Reweighted

𝑙1 minimization. This iterative algorithm (𝑅𝑤𝑙1) updates the

weights in each step based on the estimated sparse vector

magnitudes from the previous step (see Algorithm.1). In

Algorithm.1, 휀 is defined as a stabilizer parameter that is

used in order to obviate the effect of zero-valued components

in 𝒙𝑙. It has been shown that in general the 𝑅𝑤𝑙1 recovery

process tends to be reasonably robust to the choice of this

parameter (휀 > 10−3 is suggested for practical situations).

The diagonal matrix 𝑊𝑙 is defined as follows:

𝑊𝑙 = [

𝜔(1)𝑙 ⋯ 0

0 ⋱ 0

0 ⋯ 𝜔(𝑁)𝑙

]. (18)

Algorithm.1 Reweighted 𝒍𝟏-minimization (𝑹𝒘𝒍𝟏) PNTI

require: stopping criterion, phase angle measurements,

active power measurements, each of 𝑴 sample times

𝒑𝒊(𝒕) , 𝜽𝒊(𝒕) 𝒇𝒐𝒓 𝒕 = 𝟏, 𝟐, . . . , 𝑴.

1. form: measurement matrix 𝑨 ∈ 𝑹𝑵−𝟏×𝑴, and

measurement vector 𝒚

2. Set counter 𝒍 to zero and initialize 𝝎(𝒏)𝟎 𝒇𝒐𝒓 𝒏 = 𝟏: 𝑵 − 𝟏

3. solve:

𝒙𝒍 = 𝒂𝒓𝒈𝒎𝒊𝒏‖𝑾(𝒍)𝒙‖𝟏

𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐 𝒚 = 𝑨𝒙

4. update weights:

𝝎(𝒏)𝒍+𝟏 =

𝟏

|𝒙(𝒏)𝒍 | + 𝜺

𝒇𝒐𝒓 𝒏 = 𝟏: 𝑵 − 𝟏

5. Go back to step.3. Until stopping criterion met.

6. output: 𝒙 = 𝒙𝒍

F. Noisy Reweighted 𝑙1-Minimization Based PNTI

In short, to adapt the 𝑅𝑤𝑙1 algorithm in the case of noisy

measurements the following change should be made in the 3rd

step:

𝑥𝑙 = 𝑎𝑟𝑔𝑚𝑖𝑛‖𝑊(𝑙)𝑥‖1

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ‖𝑦 − 𝐴𝑥‖𝑙2< 𝜂. (19)

Additional discussion of 𝑊𝑃1 and 𝑅𝑤𝑙1 is contained in [26].

There are a variety of optimization packages that can be used

to solve 𝑃1, 𝑁𝑃1, 𝑊𝑃1 and 𝑅𝑤𝑙1; examples include CVX [33]

and NESTA [34].

(b)

(c)

(d)

(a)



Our justification for this modeling assumption of quasi-

sequential node numbering was based on the structure

observed in IEEE standard test beds as well as some real-

world power networks. We believe that this modeling

assumption will be valid in a considerable number (but not

all) networks. Moreover, whenever a new area is constructed

and is added to a power network the new buses will usually

be numbered similarly to their neighbors, so some sequential

numbering may be preserved. In case such a sequential

numbering is not present within a network then one may see

only limited improvement from weighted 𝑙1 minimization.

Even in such a situation, one may still use ordinary sparsity-

based optimization approaches (including the iterative re-

weighted 𝑙1-minimization algorithm, but with weights

initialized to 1) to solve the problem, although with a slightly

greater number of measurements.

IV. SIMULATION RESULTS AND DISCUSSION

The authors tested the proposed method for recovering the

topology of a SG using compressive observations, collected

from the system parameters. In these simulations, we used the

IEEE Standard-30 Bus and IEEE Standard-300 Bus as case

studies. These power networks include 30, 300, and 2383

buses and 47, 411, and 2896 power transmission lines, and

their detailed specifications have been fully described in

MATPOWER toolbox [27]. The MATPOWER 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 sparse solver. In

order to generate the data, first, we fed the system with

Gaussian demands and simulated the PN. Based on new PMU

technology standards, measurement SNR has been randomly

set to 20-100 dB.

Fig.4. Recovery rate comparison of nodes a) 3 (in-degree 2), b) 10 (in-degree

6) in the network of Fig. 1, respectively. Success rate is calculated over 100

realizations of the network for a given number of measurements.

Within the network graphs, SRP solvers have different

recovery performance for different nodes, mainly because of

the sparsity level of the signal (or in our PN, in-degree or the

number of incoming transmission lines to an individual bus).

Fig.4.a, and b show the recovery performance of the 𝐵𝑃 and

𝑅𝑤𝑙1 solutions for nodes 3 and 10 of the IEEE Standard-30

Bus, respectively. These buses are distinguished from each

other by their in-degrees. The success rate8 is calculated over

100 realizations of the network for a given number of

measurements. From the sparsity level viewpoint, the 6-

sparse signal 𝑥10 corresponds to one of the most complicated

signals 𝑥𝑖 to be recovered in the IEEE Standard-30 Bus. The

2-sparse signal 𝑥3 (corresponding to the double in-connection

node 3) is more likely to be recovered using a smaller number

of measurements than the 6-sparse signal 𝑥10 (corresponding

to the 6 in-connection node 10). In Fig.5, the same trends can

be observed by looking at the recovery rate over the 1st, 2nd,

15th, 3rd and 130th buses, with sparsity levels (in-degrees) 3, 4,

6, 7, and 9, respectively, in the IEEE Standard-300 Bus.

Fig.5. Recovery rate comparison of nodes (downward): 1 (in-degree 3), 2 (in-

degree 4), 15 (in-degree 6), 3 (in-degree 7), 130 (in-degree 9) respectively.

Success rate is calculated over 100 realizations of the IEEE Standard-300

Bus for a given number of measurements.

We occasionally observe that nodes with similar in-degree

actually have a different recovery performance. For example,

8 For our success criterion we first check to see whether the support (the

positions of the non-zeros) of the true signal is correctly identified, and if so,

we then check to see whether the recovery error ‖𝑥𝑖 − 𝑥�̂�‖2/‖𝑥𝑖‖2 is within a

certain bound 휀. Both conditions must be satisfied for recovery to be

considered successful. We picked 휀 = 0.05 to illustrate the acceptable recovered signals.



nodes 3 and 15 (corresponding to the 7 and 6-sparse signals

𝑥3 and 𝑥15, respectively) have close in-degrees, but Fig.5

shows that the recovery rate curve of 𝑥3 is closer to that of

the 9-sparse signal 𝑥130, i.e., it requires a larger number of

measurements for correct recovery. This difference can be

caused by the network-wide location of the bus and also the

structure of the incoming transmission lines to each bus.

Moreover, due to the close geographical position and

similarity in load pattern within an interconnected network,

the parameters of different nodes may share a level of

correlation. This will affect the coherence of the resulting

sensing matrix and may result in different recovery

performance over the network nodes. Figs.4-5 also show how

the presence of structured sparsity helps the 𝑅𝑤𝑙1 algorithm

to outperform 𝐵𝑃, especially in case of the IEEE Standard-

300 Bus where the in-degree of the nodes is larger. Figs.6a-c

provide a node-by-node9 comparison of the recovery

performance for the 2 aforementioned SRP approaches over

100 realizations of the IEEE Standard-30 Bus. The vertical

axis indicates the number of measurements 𝑀, while the

horizontal axis represents the bus number from 1 to 30.10 A

color spectrum ranging from dark blue (corresponding to 0%

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

has been used to illustrate the recovery performance

percentage. Results indicate that, for almost all nodes, both of

the algorithms can arrive at full recovery performance using

20-25 measurements. In general, however, the 𝑅𝑤𝑙1

algorithm demonstrates better performance with lower

numbers of measurements, especially for columns with high

congestion around the diagonal elements. Finally, Fig.7a-c

demonstrate the network-wide topology recovery

performance for 𝐵𝑃 and 𝑅𝑤𝑙1 on the IEEE Standard-30, 300,

and 2383 Bus networks, over 100 realizations of each

network, respectively. For each curve, the vertical axis

represents the percentage of trials in which all 30, 300, or

2383 columns of the corresponding nodal-admittance matrix

𝐵 (and, as a result, the network-wide topology) are

successfully recovered.

As has been discussed, theoretically, the number of

measurements required for full recovery 𝑀 should be at least

proportional to 𝐾 𝑙𝑜𝑔(𝑁/𝐾). In the case of the IEEE

Standard-30 Bus, node 6 has the highest in-degree of 7; as a

result the whole network-wide recovery is expected at

roughly 𝑀 = 7 𝑙𝑜𝑔(30/7) ≈ 10 or more. Moreover, based

on the discussion in Section III-B, as the number of buses of

a particular network, 𝑁, grows, more measurements are

needed. For example, in case of the IEEE Standard-300 Bus,

node 276 has in-degree 12, and full recovery is theoretically

possible when roughly 𝑀 = 12 𝑙𝑜𝑔 (300

12) ≈ 39. In the case

of the IEEE Standard-2383 network, node 1920 has in-degree

9; as a result, full recovery should be expected at 𝑀 =

9 𝑙𝑜𝑔 (2383

9) ≈ 50 measurements or more. However, Figs.6-7

indicate that in practice the whole network topology for these

9 That is, for each of the columns of the nodal-admittance matrix 𝐵. 10 Or equivalently, the column number from the nodal-admittance matrix.

two systems can typically be recovered from 𝑀 ≈ 20, 60 or

90 measurements per bus, respectively. This is likely due to

the coherence of the sensing matrices (because the elements

are not perfectly independent Gaussian random variables) and

the additional noise.

An important factor is that, although the number of

measurements needed for full recovery increases with the

network size 𝑁, this is not a linear relationship. As can be

seen, although the IEEE Standard-2383 and 300 Bus are

almost 80 and 10 times larger than IEEE Standard-30 Bus in

scale, respectively, they require less than 3 and 5 times more

measurements for full recovery, respectively. This fact

highlights the suitability of the sparse PNTI setup especially

for large scale power networks.

Fig.6. Recovery performance for 2 SRP solvers over all of the buses of the

standard IEEE Standard-30 Bus: (a) 𝐵𝑃, (b) 𝑅𝑤𝑙1, (c) 𝑅𝑤𝑙1 − 𝐵𝑃 .

(a)

(b)

(c)



In general, a lack of measurements reduces the recovery

performance for any SRP solver; however, the reweighted 𝑙1

minimization algorithm (𝑅𝑤𝑙1), which uses additional

structural knowledge on the sparse vectors, is less affected

and exhibits a better recovery performance when compared

with 𝐵𝑃. Finally, we do note that it is not possible to recover

the nodes with higher in-degrees until 𝑀 is large enough that

the PN coherence metric reaches a suitably small level. If

such a level were never to be reached, we might be limited in

the degree of nodes that we could recover with this technique.

Fig.7. Whole network topology recovery performance for 2 SRP solvers

for a) the IEEE Standard-30 Bus, b) the IEEE Standard-300 Bus over 100

realizations of the network, and c) the standard IEEE Standard-2383 Bus

over 100 realizations of the network.

V. CONCLUSIONS

In this paper we described a novel approach to address

topology identification and line outage localization in smart

power grids, using graph and sparse recovery theories. We

have discussed the computational complexity and expensive

data analysis required for large scale PNs and the uncertainty

of system states and records from parameters that may have

caused by uncertain behavior of the load. In addition, cyber-

attacks may cause unreported status changes, and there may

be random behavior of distributed generation systems (such

as wind turbines or PV cells). We presented an efficient and

low-cost solution for topology identification of line changes

and/or fault detection, plus possible monitoring tasks. This

method can efficiently overcome the inherent challenges in

the analysis of big data (that can capture different parameters

in the network). The presented approach formalizes the use of

the theory of sparse recovery for enhancing the smart-grid

analysis. The PN has been modeled as a sparse

interconnected graph. Therefore, the topology identification

problem was mathematically reformulated as a sparse

recovery problem (PNTI-SRP) based on the application of a

DC power-flow model. The conclusion is the appropriate use

of SRP solvers can solve the TI problem for IEEE standard

networks. The network model used in this study was

generated with the MATPOWER toolbox through standard

IEEE 30, 300, and 2383 bus test-beds. Such a sparse

reformulation for the TI problem depends only on

measurements from the system parameters; there is no need

for previous information from the network topology. In the

technique presented in this paper, SRP solver methods

supported finding the structure or topology matrix of the

network. Several case studies demonstrated that our proposed

method represents a promising alternative strategy for

topology identification, fault detection, and monitoring issues

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

parameters. The recovery performance of the SRP solvers is

mainly dependent on the in-degree (number of lines

connected to each bus) of each bus in the network. Moreover,

it has been shown that columns of the matrix 𝐵 of a sample

PN (each corresponds to one of the sparse vectors 𝒙𝒊 to be

recovered in the 𝐵𝑃 or 𝑅𝑤𝑙1 solutions), can be assumed as

structured-sparse signal. This structural assumption can be

used to improve the recovery performance, using an even

smaller number of measurements, which makes the retrofit

and real-time application very easy to be done with

inexpensive hardware.

Finally, we note that within the state of the art, most of the

proposed power line status identification (essentially, PNTI)

algorithms rely on the voltage phasor analysis within the DC

power flow model and try to define a mathematical

formulation that can directly extract the structure of the

nodal-admittance or another structural matrix or criterion of

the PN. Our algorithm also fits into this category; such data

has traditionally been the readily available. However, PMU

technology today also provides real time measurements from

electrical current phasors. This functionality could be utilized

in order to develop new current phasor-based frameworks

which directly identify the status of power lines from the

current data. However, due to the uncertainty in loads and

within the aggregation of renewables, in addition to

nonlinearity in demand within various scales of power

networks, the general behavior of electrical currents may vary

over a wide range. Accounting for this uncertainty may

require a deep and comprehensive study before current

measurements can be used as a trustable criterion for

topology identification and power outage detection.

(a)

(b)

(c)



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] 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. [3] J. De La Ree, V. Centeno, J. S. Thorp, and A. G. Phadke. Synchronized

phasor measurement applications in power systems. IEEE Trans. Smart

Grid, 1(1):20–27, 2010. [4] J. E. Tate and T. J. Overbye, “Line outage detection using phasor angle

measurements,” IEEE Trans on Power Syst, vol. 23, pp: 164-165, Nov. 2008.

[5] M. He and J. Zhang. A dependency graph approach for fault detection and localization towards secure smart grid. IEEE Trans on Smart Grid, vol:2,

pp:342–351, June 2011.

[6] B. Ansari and S. Mohagheghi. "Optimal energy dispatch of the power distribution network during the course of a progressing wildfire". Int. Trans.

Electr. Energ. Syst. 2015; 25:3422–3438.

[7] A.S. Bubshait, M.G. Simoes, A. Mortezaei, and T.D.C. Busarello. "Power quality achievement using grid connected converter of wind turbine

system." In Proc. IEEE IAS Annu. Meeting, Oct. 2015, pp.1–8., Dallas, USA.

[8] S. Thale, R. Wandhare, V Agarwal, "A Novel Reconfigurable Microgrid Architecture with Renewable Energy Sources and Storage", IEEE

Transactions on Industry Appl. Vol. 51, No. 2, pp: 1805:1816, March 2015.

[9] F. Harirchi, M.G. Simoes, A. Al-Durra, S.M.Muyeen and M. Babakmehr. “Designing Smart Inverter with Unified Controller and Smooth Transition

between Grid Connected and Islanding Modes for Microgrid Application”, In

Proc. IEEE IAS Annu. Meeting, Oct. 2015, pp. 1–8, Dallas, USA. [10] M.G. Simoes, T.D.C. Busarello, A.S. Bubshait, F. Harichi, J.A. Pomilio,

and F. Blaabjerg, “Interactive smart battery storage for a PV and wind hybrid

energy management control based on conservative power theory,” International Journal of Control, in press, DOI: 10.1080/00207179.

2015.1102971, 2016.

[11] A. Luna, J. Rocabert, J. Candela, J. Hermoso, R. Teodorescu, F. Blaabjerg, and P. Rodríguez. "Grid Voltage Synchronization for Distributed

Generation Systems Under Grid Fault Conditions", IEEE Transactions on

Industry Applications. Vol. 51, No. 4, pp: 3414:3425, Jul 2015. [12] D.C. Mazur, R.A. Entzminger, and J.A. Kay. "Enhancing Traditional

Process SCADA and Historians for Industrial and Commercial Power

Systems with Energy (Via IEC 61850)", IEEE Transactions on Industry Applications. Vol. 52, No. 1, pp: 76:82, Jan 2016.

[13] R. Yousefian, and S. Kamalasadan. "Design and Real-Time

Implementation of Optimal Power System Wide-Area System-Centric Controller Based on Temporal Difference Learning", IEEE Transactions on

Industry Applications. vol. 52, no. 1, pp: 395:406, Jan 2016.

[14] 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.

[15] Y.C. Chen, T. Banerjee, A.D. Domínguez-García, and V.V. Veeravalli. "Quickest Line Outage Detection and Identification". IEEE Trans on Power

Syst, vol. 31, no. 1, pp 749-758, Jan 2016.

[16] Wu, J. Xiong, J. Shi, Y. “Ambiguity group based location recognition for multiple power line outages in smart grids”. In Proceedings of the 2014

IEEE PES on ISGT, Washington, DC, USA. [17] S. Azizi and M. Sanaye-Pasand, “A straightforward method for wide-

area fault location on transmission networks,” IEEE Trans. Power Del., vol.

30, no. 1, pp 264–272, Feb. 2015. [18] H. Zhu and G. B. Giannakis, “Sparse Overcomplete Representations for

Efficient Identification of Power Line Outages,” IEEE Tran. on Power Syst,

vol. 23, no. 4, pp. 1644–1652, Oct. 2012. [19] 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”, in Proceeding with 2015 American Control Conference-ACC 2015, Chicago, Illinois, USA, July 2015.

[20] A.Arabali, M.Majidi, M.S.Fadali, and M.Etezadi-Amoli, “Line Outage

Identification-Based State Estimation in a Power System with Multiple Line Outages,” Electric Power Systems Research, Vol.133, pp.79–86, April 2016.

[21] Chen, J.C.; Li, W.T.; Wen, C.K.; Teng, J.H.; Ting, P. “Efficient

identification method for power line outages in the smart power grid”. IEEE Trans. Power Syst. 2014, vol: 29, pp: 1788–1800.

[22] 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 ISBN: 978-3-319-02149-2, Ch. 9, pp 277-297. [23] M.Babakmehr, M.G.Simoes, M.B.Wakin, and F.Harirchi. “Compressive

Sensing-Based Topology Identification for Smart Grids”.

DOI:10.1109/TII.2016.2520396, IEEE Trans on Industrial Informatics. [24] M.Majidi, M.Etezadi-Amoli, and M.S.Fadali, “A Novel Method for

Single and Simultaneous Faults Location in Distribution Networks,” IEEE

Transactions on Power System. Vol. 30, Issue 6, pp. 3368-3376, Nov. 2015. [25] M.Majidi, A.Arabali, and M.Etezadi-Amoli, “Fault Location in

Distribution Networks by Compressive Sensing,” IEEE Transactions on

Power Delivery. Vol. 30, Issue 4, pp. 1761-1769, Aug. 2015. [26] E. J. Candes, M. B. Wakin, and S. P. Boyd, Enhancing sparsity by

reweighted 𝑙1-minimization, J Fourier Anal Appl (2008) 14: 877–905. [27] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas,

“Matpower: 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.

[28] A. Abur and A.G Exposito. Power System State Estimation, Theory and Implementation. Marcel Dekker, 2004.

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

Control, 2nd ed. New York: Wiley, 1996. [30] A. Schellenberg, W. Rosehart, and J. Aguado, “Cumulant-based

probabilistic optimal power flow (P-OPF) with Gaussian and Gamma

distributions”, IEEE Trans. on Power Syst, vol. 20, pp. 773–781, May 2005. [31] E.J. Candes and M.B. Wakin, “An introduction to compressive

sampling”, IEEE Sig Proc Magazine, vol. 24, pp. 21–30, March 2008.

[32] M. B. Wakin. Compressive Sensing Fundamentals. In M. Amin, editor, Compressive Sensing for Urban Radar, pp. 1-47. CRC Press, Boca Raton,

Florida, 2014.

[33] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version 1.1. September 2007.

[34] S. Becker, J. Bobin, and E. J. Candes, NESTA: A fast and accurate first-

order method for sparse recovery, SIAM J. Imaging Sci., 4 (2011), pp. 1–39. [35] Yang, C., Guan, Z. H., Liu, Z. W., Chen, J., Chi, M., & Zheng, G. L.

“Wide-area multiple line-outages detection in power complex networks”.

International Journal of Electrical Power & Energy Systems, 79, 132-141, 2016.

[36] M. Babakmehr, M. G. Simoes, M. B. Wakin, A. Durra, and F. Harirchi,

827 “Smart grid topology identification using sparse recovery,” in Proc.

IEEE IAS Annu. Meeting, Oct. 2015, pp. 1–8.

Mohammad Babakmehr (S’14) received the

B.S. degree in electrical engineering in 2008 from Central Tehran University and the M.Sc. degree in

Biomedical-Bioelectric engineering in 2011 from

the Amirkabir University of Technology, Tehran, Iran. He is currently a Ph.D. degree candidate in

the Department of Electrical Engineering and

Computer Science, Colorado School of Mines (CSM), Golden. He has been with the Center for

the Advanced Control of Energy and Power

Systems since 2013. His research interests include smart grid technologies, compressive sensing, advance signal processing and control theory.

Marcelo G. Simões (S’89–M’95–SM’98-F’15)

Marcelo Godoy Simões received a B.Sc. degree from the University of São Paulo, Brazil, an M.Sc.

degree from the University of São Paulo, Brazil,

and a Ph.D. degree from The University of Tennessee, USA in 1985, 1990 and 1995

respectively. He received his D.Sc. degree (Livre-

Docência) from the University of São Paulo in 1998. Dr. Simões was an US Fulbright Fellow for

AY 2014-15, working for Aalborg University,

Institute of Energy Technology (Denmark). He is currently with Colorado School of Mines. He has been elevated to the grade

of IEEE Fellow, Class of 2016, with the citation: "for applications of artificial intelligence in control of power electronics systems."



Miachael B. Wakin (S’01–M’06–SM’13) Michael B. Wakin is the Ben L. Fryrear Associate Professor

in the Department of Electrical Engineering and

Computer Science at the Colorado School of Mines (CSM). Dr. Wakin received a Ph.D. in Electrical

Engineering from Rice University. He was an NSF

Mathematical Sciences Postdoctoral Research Fellow at Caltech from 2006-2007 and an Assistant

Professor at the University of Michigan from 2007-

2008. His research interests include sparse, geometric, and manifold-based models for signal

processing and compressive sensing. Dr. Wakin received the NSF CAREER

Award and also received the CSM Excellence in Research Award for his research as a junior faculty member.

Ahmed Al-Durra (S'07-M'10-SM'14) received the

B.S., M.S., and PhD in Electrical and Computer

Engineering from the Ohio State University in 2005, 2007, and 2010, respectively. He is an

Associate Professor in the Electrical Engineering

Department at the Petroleum Institute, Abu Dhabi, UAE. His research interests are application of

estimation and control theory in power system

stability, Micro and Smart Grids, renewable energy, and process control. He has published over

70 scientific articles in Journals and International Conferences. Dr. Ahmed has successfully

accomplished several research projects at international and national levels.

He is the co-founder of Renewable Energy Laboratory at the Petroleum Institute.

Farnaz Harirchi (S’14) received the B.S. degree in

electrical engineering in 2008 from Central Tehran

University and the M.Sc. degree in Electrial and

Electronic engineering in 2011 from the Iran

University of Science and Technology, Tehran,

Iran. She is currently a Ph.D. degree candidate in the Department of Electrical Engineering and

Computer Science, Colorado School of Mines

(CSM), Golden. She has been with the Center for the Advanced Control of Energy and Power

Systems since 2012. Her research interests include renewable energy

aggregation, smart Microgrids, power electronics and, intelligent control for high-power electronics applications.

Smart Grid Topology Identification Using Sparse Recoverymsimoes/documents/papers2016/MGS_pape… ·...

Documents

Transcript of Smart Grid Topology Identification Using Sparse Recoverymsimoes/documents/papers2016/MGS_pape… ·...