NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

Conference Paper

NREL/CP-5D00-73540

May 2019

State Estimation in Low-Observable 

Distribution Systems Using Matrix 

Completion 

Preprint

Yingchen Zhang,

1 Andrey Bernstein,1 Andreas Schmitt,2

and Rui Yang1

1 National Renewable Energy Laboratory 

2 Virginia Polytechnic Institute and State University 

Presented at the Hawaii International Conference on System Sciences 

Maui, Hawaii

January 8–11, 2019
NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

National Renewable Energy Laboratory

15013 Denver West Parkway

Golden, CO 80401

303-275-3000 • www.nrel.gov

Conference Paper

NREL/CP-5D00-73540

May 2019

State Estimation in Low-Observable 

Distribution Systems Using Matrix 

Completion

Preprint

Yingchen Zhang,

1 Andrey Bernstein,1 Andreas Schmitt,2

and Rui Yang

1 National Renewable Energy Laboratory 

2 Virginia Polytechnic Institute and State University 

Suggested Citation 

Zhang, Yingchen, Andrey Bernstein, Andreas Schmitt, and Rui Yang. 2019. State 

Estimation in Low-Observable Distribution Systems Using Matrix Completion: Preprint. 

Golden, CO: National Renewable Energy Laboratory. NREL/CP-5D00-73540. 

https://www.nrel.gov/docs/fy19osti/73540.pdf. 
NOTICE

This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable 

Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding 

provided by U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Solar Energy 

Technologies Office. The views expressed herein do not necessarily represent the views of the DOE or the U.S. 

Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges

that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce 

the published form of this work, or allow others to do so, for U.S. Government purposes.

This report is available at no cost from the National Renewable 

Energy Laboratory (NREL) at www.nrel.gov/publications. 

U.S. Department of Energy (DOE) reports produced after 1991 

and a growing number of pre-1991 documents are available 

free via www.OSTI.gov. 

Cover Photos by Dennis Schroeder: (clockwise, left to right) NREL 51934, NREL 45897, NREL 42160, NREL 45891, NREL 48097, 

NREL 46526.

NREL prints on paper that contains recycled content. 
State Estimation in Low-Observable Distribution Systems Using Matrix

Completion

Yingchen Zhang

NREL

yingchen.zhang@nrel.gov

Andrey Bernstein

NREL

andrey.bernstein@nrel.gov

Andreas Schmitt

Virginia Tech

andreas9@vt.edu

Rui Yang

NREL

rui.yang@nrel.gov

Abstract

The need for distribution system state estimation is on

the rise because of the increased penetration of distributed

energy resources and flexible load. To manage the

distribution systems in real time, operators need to firstly

overcome the challenge of low observability in distribution

systems. Also, because of the amount of data present

from smart meters, distributed generation measurements,

switches, etc., the ideal distribution state estimation methods

need to be able to process heterogeneous data.

In this paper, an algorithm is developed for voltage

phasor estimation in low-observability distribution systems.

The algorithm is based on the matrix completion approach

from signal processing. The traditional matrix completion

formulation is augmented with power-flow constraints to

improve results while requiring less data. This method

can also use all types of measurements (voltage magnitude,

voltage angle, real power, reactive power) to complete the

state matrix.

1. Introduction

Power systems state estimation (SE) is a long-established

tool for bulk grid operation. It provides full visibility

and the best estimate of the entire network status. It

forms the foundational block for power system energy

management applications, such as real-time power flow,

real-time dispatch, and stability applications [1]. SE has not

been broadly used in distribution system operation largely

because traditionally distribution management systems are

a collection of individual applications that do not require

visibility of the entire system. Traditionally, power has

flowed from the transmission grid to distribution systems

and there were nearly no resources or controllable devices

to manage in real time.

However, distribution systems are undergoing dramatic

changes because of the fast adoption of the distributed energy

resources (DERs), especially distributed photovoltaic (PV)

systems. According to the SEIA report [2], the U.S. market

recently installed 10.6 GW of solar PV and it reached 55.9

GW of total installed capacity (at the end of Q1 2018). For

the new installations, 59% of the installed capacity came

from the utility PV, and 41% from distributed solar. For the

residential solar market, even though California and all major

Northeast markets experienced annual contractions, 25 of 44

states saw growth in annual residential PV installations [3].

The other major change happening in the distribution grid

is the implementation of demand response, which allows

loads to adjust their consumption according to either direct

control signals from the utility [4] or price signals [5].

Further, with the rapid development of smart appliances

and home automation, in-house home energy management

systems are developed that operate appliances based on

customers own control objective [6] [7].

To better manage distribution-level resources while

maintaining system reliability and stability, most advanced

distribution energy management systems propose to

implement distribution system state estimation (DSSE)

for real-time applications. Weighted least-square (WLS)

estimators are the most popular for DSSE [8]. There

are voltage-based WLS and branch current-based WLS

according to the selection of state variables. The algorithm

in [9] has a better computation efficiency because it has a

constant measurement Jacobian matrix at each iteration.

Dynamic DSSE methods [10] [11] generally apply a Kalman

filter to obtain a priori estimate, which is combined with

latest measurements to forecast the system state. This

method is based on several measurement snapshots in a time

sequence.

The biggest challenge for DSSE comes from the lack

of sensors [12]. Unlike the supervisory control and

data acquisition (SCADA) systems in transmission grids,

distribution systems normally do not have full observability

from the SCADA systems, which roughly speaking means

that the number of measurements is strictly less than that

of the quantities being estimated. Also, measurements in

distribution systems are truly heterogeneous. Bus voltage,

branch current, power flows and switch status are measured

from SCADAs, smart meters, autonomous switches,

distribution synchrophasor units and intelligent electronics

devices (e.g., inverters) in different time resolutions.

Several approaches in the literature tackle the issue of

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 1
low observability. The works [13–16] proposed to improve

observability by optimizing the minimal number of locations

needed to perform the state estimation task. In [17], smart

meter data was used to enhance observability of the system

by solving the power flow over several consecutive time

instances. Pseudo measurements are normally used to

solve the mathematically underdetermined problem in DSSE

because of the limited number of measurements [18–20].

The advances in machine learning have allowed for

techniques such as neural networks to be used for state

estimation [19, 21, 22]. Although machine learning

techniques obtain accurate estimation results, they require a

significant amount of historical data to train the algorithm,

and this data might not be available.

Matrix completion is a well-known method in signal

processing for the estimation of missing values within

matrices [23]. Prior attempts to apply the algorithm to power

systems include [24, 25], which perform load forecasting

while accounting for missing data. Both of these approaches

transform the data in a way that it is suitable for use with the

matrix completion algorithm.

In this paper, we propose applying the matrix completion

algorithm to estimate voltage phasors in low-observability

distribution networks. To improve upon the algorithm’s

application to power systems, power flow equations are

added as constraints to the problem formulation.

This approach is advantageous compared to previous

works due to the large variety of situations in which it

can be applied. While many works, such as [17, 20, 26],

require voltage and power measurements to perform their

estimations, this approach can make use of any quantity

measured in the field. In this way, the algorithm can be used

for a larger number of systems. Additionally, while methods

such as [17] require collecting data over large time windows,

this matrix completion approach is capable of estimating the

state with a single time instance.

The paper is structured as follows. Section 2 overviews

matrix completion and its implementation. Section 3

discusses how to apply this algorithm to solve DSSE and

the modifications that were made. Section 4 demonstrates

illustrative numerical results of the algorithm and Section 5

provides conclusion of the work.

2. Matrix Completion

Matrix completion method has been developed to

estimate missing values of partially observed matrix [23].

It has been broadly used in signal processing community,

e.g., to complete missing pixels in images [27] and to detect

sensor location with partial distance information [28]. The

concept of matrix completion is, for a given low rank matrix

with a set of known elements Ω, to make use of its low

rank nature to estimate the missing values. When a matrix

is low rank, there is a direct connection between the row and

column elements within the matrix. This connection can then

be used to determine elements that are unknown. This does,

however, rely on the assumption that the matrix actually is in

fact low rank, and that there are enough known elements to

find the connection.

The matrix completion algorithm is first defined formally.

Consider the space of real-valued n1 × n2 matrices R

n1×n2

.

Let I := {1, . . . , n1} × {1, . . . , n2} denote the index set, so

that (i, j) ∈ I represents an element’s index. For any matrix

W ∈ R

n1×n2 and a subset Ω ⊆ I, let WΩ denote

WΩ =

(

Wi,j , (i, j) ∈ Ω

0, otherwise.

(1)

Define M as the matrix containing all measurements

and MΩ as the observation matrix, where Ω is the set

of known elements within the matrix. From [23], the

matrix completion problem can then be formulated as a rank

minimization problem:

min

X∈Rn1×n2

rank(X)

s.t. XΩ = MΩ,

(2)

However, it can be seen that the formulation of (2) is

a non-convex, NP-Hard optimization problem to solve,

therefore making it unsuitable for practical use. Thus, the

original problem is relaxed to that of minimizing the nuclear

norm:

min

X∈Rn1×n2

kXk∗

s.t. XΩ = MΩ,

(3)

where

kXk∗ =

Xn

i=1

σi(X), (4)

σi(X) is the i

th singular value, and n := min{n1, n2}.

Due to the nature of the equality constraint, formulation

(3) is highly susceptible to noise. To alleviate this,

[29] proposed a robust algorithm which allows for the

matrix completion algorithm to account for noise in the

observations. The algorithm modifies the equality constraint

in (3) to

kXΩ − MΩkF ≤ δ (5)

where kXkF is the Frobenius norm of X which is defined as

kXkF =

vuut

Xm

i=1

Xn

j=1

|Xi,j |

2. (6)

The value of δ has a significant impact on the accuracy of the

estimations. Since the parameter δ corresponds to the amount

of noise on the data, the estimation accuracy will improve

as the value decreases as this is equivalent to the amount of

noise in the signal decreasing.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 2
3. Low-Observability DSSE

In this section, we first define the notion of low

observability. Then, the matrix completion algorithm is

proposed and applied to DSSE.

3.1. Power System Model

For brevity, we consider a balanced distribution network

with a single slack bus and N P Q buses. Let N =

{1, . . . , N} denote the set of the P Q buses, and L ⊆ N ×

N denote the set of distribution lines. We note that the

formulation proposed in this paper can be easily extended

to the general multiphase setting with both wye and delta

connections, as proposed, e.g., in [30].

3.2. Low observability

Consider a linear (or linearized) measurement model of

the form

y = Hx + w (7)

where y denotes the measurement vector, x denotes the

state vector and w denotes the measurements error. Then

observability can be defined as invertibility of HT H, namely

as the existence of the explicit least-square solution

x = (HT H)

−1HT

y. (8)

Therefore, we say that the system is under low-observability

conditions whenever HT H is not full rank, at which point

that conventional methods such as weighted least square are

not directly applicable.

3.3. Matrix Set-Up

Since matrix completion relies on the assumption of low

rank matrices, the choice of quantities with which the matrix

is created has a significant impact on the accuracy of the

results. The higher the link between the quantities and

voltage, the more accurate the resulting estimation will be.

However, as long as there is even a slight correlation between

the quantity and voltage, the quantity is able to be used in the

estimation algorithm.

In order to allow for the use of the largest number and

variety of variables, we set up the matrix in terms of system

connection lines rather than buses. In this way, each row of

the matrix represents one line in the system, and each column

represents one variable. The resultant matrix columns then

are the real and reactive voltage from the source bus of

the line; the total real and reactive power entering the line

source bus from all sources (generators and lines); the real

and reactive current flowing through the line; the load at the

source bus; and the voltage magnitude at the source bus.

Formally, for every line (f, t) ∈ L, the corresponding row

in the matrix M is given by

[Re(Vf ), Im(Vf ), Pinf, Qinf, Re(If,t), Im(If,t), Pf lowf,t ,

Qf lowf,t , Ploadf, Qloadf, |Vf |].

We note that the proposed algorithm is not limited to

the variables chosen. In fact, any quantity which has a

correlation to voltage can be used to supplement and improve

the results. In this paper, the elements that will be included

in the matrix and their correlation with state variables will be

described in the follow section.

3.4. Power Flow Constraints

The standard matrix completion algorithm may be able

to return estimates of the voltage under specific conditions,

but the formulation can be significantly improved upon

through the inclusion of power flow constraints. We next

formulate several corresponding constraints to be included

to the original formulation detailed in Section 2.

First, since the objective matrix contains both the voltages

at each bus and the current flowing between buses, power

flow constraints can be introduced to the optimization in a

form of the following linear equality constraint:

(Vf − Vt)Yf t = If t, ∀(f, t) ∈ L. (9)

One potential issue when including new constraints is

ensuring the problem remains feasible. To avoid these

potential issues and increase the robustness of the algorithm,

the new constraints are relaxed and bounded by a tolerance

thus becoming

− f,t ≤ (Vf − Vt)Yf t − If t ≤ f,t, ∀(f, t) ∈ L, (10)

where f,t is the error tolerance for line (f, t) ∈ L.

Next, it is a natural requirement for there to be net zero

power at each bus. Thus, the power flowing into the bus must

be equal to the power consumed at the bus in addition to the

power leaving the bus. This can be formulated as

Pinf −

X

t∈N

Pf lowf,t − Ploadf = 0, ∀f ∈ N ,

Qinf −

X

t∈N

Qf lowf,t − Qloadf = 0, ∀f ∈ N .

(11)

Similarly to constraint (10), we relax these equality

constraints as follows:

−τf ≤ Pinf −

X

t∈N

Pf lowf,t − Ploadf ≤ τf , ∀f ∈ N

−τf ≤ Qinf −

X

t∈N

Qf lowf,t − Qloadf ≤ τf , ∀f ∈ N

(12)

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 3
where τf is the error tolerance for bus f ∈ N . Feasibility

can always be ensured by selecting tolerance values which

create a feasible problem. However, since the accuracy

of the resultant estimation is dependent on the tolerance

being minimal, the values used need to be included in

the optimization objective. Therefore, augmenting the

original (robust) matrix completion problem (3), (5) with the

power flow constraints, results in the following optimization

problem:

min

X∈R

n1×n2 ,{f,t},{τf }

kXk∗ + w1

X

(f,t)∈L

f,t + w2

X

f∈N

τf

(13a)

s.t. kXΩ − MΩkF ≤ δ (13b)

(10), (12) (13c)

f,t ≥ 0, ∀(f, t) ∈ L (13d)

τf ≥ 0, ∀f ∈ N , (13e)

where w1, w2 > 0 are weighting parameters; n1 := |L| is

the number of lines in the network; and n2 is the number

of variables used for estimation. Observe that, since the

measurements are in rectangular coordinates, (13) is a convex

optimization problem and hence can be solved efficiently.

We note that, when the measurements of voltage

magnitudes and/or power injections are available,

formulation (13) can be augmented with additional

constraints to capture the dependence between these

variables.

The LinDistFlow approximation for power-flow

equations [31] can be used under the assumption that the

network is radial and of low power, as found in distribution

networks. The LinDistFlow voltage approximation is given

by

|Vt| = |Vf | − (rf,tPf lowf,t + xf,tQf lowf,t )/|V0| (14)

where V0 is the voltage magnitude at the feeder head, and

rf,t and xf,t are the resistance and reactance of line (f, t),

respectively. This constraint is once again relaxed and added

to the optimization problem, resulting in:

min

X∈R

n1×n2 ,{f,t},{τf },{γf,t}

kXk∗

+ w1

X

(f,t)∈L

f,t + w2

X

f∈N

τf + w3

X

(f,t)∈L

γf,t

(15a)

s.t. (13b) − (13e) (15b)





|Vt| − |Vf | + (rf,tPf lowf,t

+ xf,tQf lowf,t )/|V0|





 ≤ γf,t, ∀(f, t) ∈ L (15c)

γf,t ≥ 0, ∀(f, t) ∈ L (15d)

All the tunning parameters are decided through trial and

error.

4. Simulation and Results

The test case for this work is the IEEE standard 33

bus system as shown in Figure 1. The system parameters

are chosen specifically to demonstrate the accuracy of the

proposed DSSE method over a large range of voltage

values, therefore the voltage profile range (1 p.u. - 0.82

p.u.) in the system does not reflect a normal distribution

voltage profile (see Figure 2). Unless otherwise stated, all

simulations are run under the assumption of a single voltage

phasor measurement and 50% of the remaining quantities.We

acknowledge that the test system is a three-phase balanced

distribution system. However the developed method can be

easily scaled to solve unbalanced system by utilizing the

linear power multiphase model in [30]

Figure 1. IEEE Standard 33 Bus System

Figure 2 shows the results of applying the matrix

completion formulation (15) to the entire IEEE 33 bus radial

system.

Figure 2. Voltage Estimation of Radial Network

It can be seen that the results of the estimation are

accurate, following the trend of the system properly. There

are occasional large deviations from the true value due to

the high dependence on which data was available to make

the estimation. These results were produced under the

assumption that the data available was uniformly randomly

sampled from the complete data set, which is not necessarily

the case for real systems.

Figure 3 shows the impact of observability on the mean

absolute percentage error(MAPE) of the estimations.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 4
Figure 3. MAPE as Observability Increases

The results in Figure 4 show that the approach is possible

for angle estimations as well, with a slight increase on the

noise of the estimations. Additionally, the amount of voltage

measurements required to obtain accurate results is increased

when estimating angle values. The plot in Figure 4 was

obtained with a voltage phasor measurement at Bus 1 and

the ending Bus of all branches(18,22,25,33) in the IEEE 33

Bus system.

Figure 4. Voltage Angle Estimation of Radial Network

While this approach results in accurate estimations, no

algorithm would be acceptable without being able to handle

error on the measurements. To show that this procedure

is robust against system errors, varying amounts of noise

were added to the data. All plots were run using 50 percent

data availability, while only a single voltage measurement

was used at bus 1. Additionally, the missing elements were

the same for each case such that the only difference in

estimations was the error.

From Figure 5 it can be seen that while measurement

errors do have some impact on the resultant estimation, the

availability of data has a much more significant impact. The

presence of data or lack thereof will determine the trend of

the estimations, while the differences in error only cause an

offset in the final estimations. Since the estimations are done

based upon the value of the measurement, rather than the type

of measurement, errors from one parameter, such as voltage

angle, will effect all other parameters as well.

Figure 5. Impact of Measurement Errors

5. Conclusion

Due to the rapid growth in distributed generation and

active load participation in system operation, the need for

state estimation in distribution networks can no longer be

ignored. However, the issue of scale and observability

has not gone away; thus it has become necessary to be

able to perform state estimation in systems that have low

observability using heterogeneous measurements input.

In this paper, a matrix completion approach is detailed

which makes use of supplemental measurements from smart

meters, distributed resources, and any data which are

available to perform state estimation. The traditional matrix

completion algorithm is improved through the inclusion of

power flow constraints which allows for accurate results

in the estimation while required less data than previously.

Through the estimation of voltage phasors, it is shown that

this approach obtains relatively accurate results under various

scenarios of low observability. The method theoretically

should work for a low rank matrix. From the numerical

tests, the data matrix is verified to be approximately low-rank

(there are few singular values that are close to zero). This

depends on the accuracy provided by the linear power flow

model that presented in [31].

As described at the beginning of the paper, the presented

method will work with any type of measurements. Currently,

the AMI data are assumed to be aggregated at the primary

side of the distribution transformer because the secondary

system topology and parameters are not always available.

However, some proposed work [32] uses AMI data to

estimate secondary systems should work nicely with the

matrix completion method for state estimation. The authors

will pursue such knowledge in the following works.

Acknowledgement

This work was authored in part by Alliance for

Sustainable Energy, LLC, the Manager and Operator

of the National Renewable Energy Laboratory for the

U.S. Department of Energy (DOE) under Contract No.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 5
DE-AC36-08GO28308. Funding provided by U.S.

Department of Energy Office of Energy Efficiency and

Renewable Energy Solar Energy Technologies Office. The

views expressed in the article do not necessarily represent

the views of the DOE or the U.S. Government. The

U.S. Government retains and the publisher, by accepting

the article for publication, acknowledges that the U.S.

Government retains a nonexclusive, paid-up, irrevocable,

worldwide license to publish or reproduce the published

form of this work, or allow others to do so, for U.S.

Government purposes.

References

[1] D. Salem-Natarajan, L. Zhao, W. Shao, M. Varghese,

S. Ghosh, M. Subramanian, G. Lin, H.-D. Chiang, and

H. Li, “State estimator for CA ISO market and security

applications-relevance and readiness,” in IEEE Power and

Energy Society General Meeting-Conversion and Delivery of

Electrical Energy in the 21st Century, 2008.

[2] Solar Energy Industries Association and others, “US solar

market insight,” 2016.

[3] Solar Energy Industries Association and others, “Solar market

insight report 2017,” SEIA. org, 2018.

[4] H. Jorge, C. H. Antunes, and A. G. Martins, “A multiple

objective decision support model for the selection of remote

load control strategies,” IEEE Transactions on Power Systems,

vol. 15, no. 2, pp. 865–872, 2000.

[5] L. Xie, J.-Y. Joo, and M. D. Ilic, “Integration of intermittent  ́

resources with price-responsive loads,” in North American

Power Symposium (NAPS), 2009.

[6] Z. Yu, L. Jia, M. C. Murphy-Hoye, A. Pratt, and

L. Tong, “Modeling and stochastic control for home energy

management,” IEEE Transactions on Smart Grid, vol. 4, no. 4,

pp. 2244–2255, 2013.

[7] R. Yang and Y. Zhang, “Coordinated optimization of

distributed energy resources and smart loads in distribution

systems,” in Power and Energy Society General Meeting

(PESGM), July 2016.

[8] A. Primadianto and C.-N. Lu, “A review on distribution

system state estimation,” IEEE Transactions on Power

Systems, vol. 32, no. 5, pp. 3875–3883, 2017.

[9] M. E. Baran and A. W. Kelley, “State estimation for real-time

monitoring of distribution systems,” IEEE Transactions on

Power Systems, vol. 9, no. 3, pp. 1601–1609, 1994.

[10] C. Carquex, “State estimation in power distribution systems,”

Master’s thesis, University of Waterloo, 2017.

[11] S. Sarri, M. Paolone, R. Cherkaoui, A. Borghetti,

F. Napolitano, and C. A. Nucci, “State estimation of active

distribution networks: comparison between wls and iterated

kalman-filter algorithm integrating pmus,” in 3rd IEEE PES

International Conference and Exhibition on Innovative Smart

Grid Technologies (ISGT Europe), 2012.

[12] M. E. Baran, “Challenges in state estimation on distribution

systems,” in 2001 Power Engineering Society Summer

Meeting. Conference Proceedings (Cat. No.01CH37262),

vol. 1, pp. 429–433 vol.1, July 2001.

[13] M. E. Baran, J. Zhu, and A. W. Kelley, “Meter placement for

real-time monitoring of distribution feeders,” in Proceedings

of Power Industry Computer Applications Conference,

pp. 228–233, May 1995.

[14] A. Shafiu, N. Jenkins, and G. Strbac, “Measurement

location for state estimation of distribution networks with

generation,” IEE Proceedings - Generation, Transmission and

Distribution, vol. 152, pp. 240–246, March 2005.

[15] R. Singh, B. C. Pal, and R. B. Vinter, “Measurement

placement in distribution system state estimation,” IEEE

Transactions on Power Systems, vol. 24, pp. 668–675, May

2009.

[16] M. Barukciˇ c, M. Vukobratovi  ́ c, D. Masle, D. Bulji  ́ c, and  ́

Z. Herderi  ̆ c, “The evolutionary optimization approach for  ́

voltage profile estimation in a radial distribution network with

a decreased number of measurements,” in 15th International

Conference on Electrical Machines, Drives and Power

Systems (ELMA), pp. 26–31, June 2017.

[17] S. Bhela, V. Kekatos, and S. Veeramachaneni, “Enhancing

observability in distribution grids using smart meter data,”

IEEE Transactions on Smart Grid, vol. PP, no. 99, 2017.

[18] K. A. Clements, “The impact of pseudo-measurements on

state estimator accuracy,” in IEEE Power and Energy Society

General Meeting, July 2011.

[19] E. Manitsas, R. Singh, B. C. Pal, and G. Strbac,

“Distribution system state estimation using an artificial neural

network approach for pseudo measurement modeling,” IEEE

Transactions on Power Systems, vol. 27, pp. 1888–1896, Nov.

2012.

[20] J. Wu, Y. He, and N. Jenkins, “A robust state estimator for

medium voltage distribution networks,” IEEE Transactions on

Power Systems, vol. 28, pp. 1008–1016, May 2013.

[21] M. Pertl, K. Heussen, O. Gehrke, and M. Rezkalla,

“Voltage estimation in active distribution grids using neural

networks,” in IEEE Power and Energy Society General

Meeting (PESGM), July 2016.

[22] H. Jiang and Y. Zhang, “Short-term distribution system state

forecast based on optimal synchrophasor sensor placement

and extreme learning machine,” in IEEE Power and Energy

Society General Meeting (PESGM), July 2016.

[23] E. J. Candes and B. Recht, “Exact matrix completion `

via convex optimization,” Foundations of Computational

Mathematics, vol. 9, pp. 717–772, Apr. 2009.

[24] S. J. Kim and G. B. Giannakis, “Load forecasting via low rank

plus sparse matrix factorization,” in Asilomar Conference on

Signals, Systems and Computers, pp. 1682–1686, Nov. 2013.

[25] J. A. Bazerque and G. B. Giannakis, “Nonparametric basis

pursuit via sparse kernel-based learning: A unifying view

with advances in blind methods,” IEEE Signal Processing

Magazine, vol. 30, pp. 112–125, July 2013.

[26] C. Klauber and H. Zhu, “Distribution system state estimation

using semidefinite programming,” in North American Power

Symposium (NAPS), Oct. 2015.

[27] H. Ji, C. Liu, Z. Shen, and Y. Xu, “Robust video denoising

using low rank matrix completion,” in IEEE Conference

on Computer Vision and Pattern Recognition (CVPR),

pp. 1791–1798, 2010.

[28] Y. Ding, N. Krislock, J. Qian, and H. Wolkowicz, “Sensor

network localization, euclidean distance matrix completions,

and graph realization,” Optimization and Engineering, vol. 11,

no. 1, pp. 45–66, 2010.

[29] E. J. Candes and Y. Plan, “Matrix completion with noise,”

Proceedings of the IEEE, vol. 98, pp. 925–936, June 2010.

[30] A. Bernstein, C. Wang, E. Dall’Anese, J.-Y. Le Boudec,

and C. Zhao, “Load-flow in multiphase distribution networks:

Existence, uniqueness, and linear models,” 2017. [Online]

Available at: http://arxiv.org/abs/1702.03310.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 6
[31] K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov,

“Local control of reactive power by distributed photovoltaic

generators,” in First IEEE International Conference on Smart

Grid Communications, pp. 79–84, Oct. 2010.

[32] J. Peppanen, C. Rocha, J. A. Taylor, and R. C. Dugan,

“Secondary low-voltage circuit modelshow good is good

enough?,” IEEE Transactions on Industry Applications,

vol. 54, no. 1, pp. 150–159, 2018.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 7