IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 6, NO. 4, DECEMBER 2019 1391

Robust Power System State Estimation FromRank-One Measurements

Gang Wang , Member, IEEE, Hao Zhu , Member, IEEE, Georgios B. Giannakis , Fellow, IEEE,and Jian Sun , Member, IEEE

Abstract—The unique features of current and upcomingenergy systems, namely, high penetration of uncertain re-newables, unpredictable customer participation, and pur-poseful manipulation of meter readings, all highlight theneed for fast and robust power system state estimation(PSSE). In the absence of noise, PSSE is equivalent tosolving a system of quadratic equations, which, also re-lated to power flow analysis, is NP-hard in general. Assum-ing the availability of all power flow and voltage magnitudemeasurements, this paper first suggests a simple algebraictechnique to transform the power flows into rank-one mea-surements, for which the �1 -based misfit is minimized. Touniquely cope with the nonconvexity and nonsmoothnessof �1 -based PSSE, a deterministic proximal-linear solver isdeveloped based on composite optimization, whose gener-alization using stochastic gradients is discussed too. Thispaper also develops conditions on the �1 -based loss func-tion such that exact recovery and quadratic convergence ofthe proposed scheme are guaranteed. Simulated tests usingseveral IEEE benchmark test systems under different set-tings corroborate our theoretical findings, as well as the fastconvergence and robustness of the proposed approaches.

Index Terms—Bad data analysis, composite optimization,least-absolute-value estimator, proximal-linear algorithm,supervisory control and data acquisition measurement.

I. INTRODUCTION

THE North American power grid is praised as the great-est engineering achievement of the 20th century [1]. To

Manuscript received September 17, 2018; revised September 18,2018, November 10, 2018, and November 30, 2018; accepted Decem-ber 5, 2018. Date of publication January 4, 2019; date of current versionDecember 17, 2019. The work of G. Wang and G. B. Giannakis wassupported by the NSF under Grant 1514056, Grant 1505970, and Grant1711471. The work of H. Zhu was supported by the NSF under Grant1802319. The work of J. Sun was supported in part by the NationalNatural Science Foundation of China (NSFC) under Grants 61621063,61522303, in part by the Projects of Major International (Regional) JointResearch Program NSFC under Grant 61720106011, and in part bythe Program for Changjiang Scholars and Innovative Research Teamin University (IRT1208). Recommended by Associate Editor Y. Hong.(Corresponding author: Jian Sun.)

G. Wang and G. B. Giannakis are with the Digital Technology Centerand the Department of Electrical and Computer Engineering, Univer-sity of Minnesota, Minneapolis, MN 55455 USA (e-mail:, gangwang@umn.edu; georgios@umn.edu).

H. Zhu is with the Department of Electrical and Computer Engineer-ing, The University of Texas at Austin, Austin, TX 78712 USA (e-mail:,haozhu@utexas.edu).

J. Sun is with the State Key Laboratory of Intelligent Control and Deci-sion of Complex Systems and the School of Automation, Beijing Instituteof Technology, Beijing 100081, China (e-mail:,sunjian@bit.edu.cn).

Digital Object Identifier 10.1109/TCNS.2019.2890954

maintain grid efficiency, reliability, and sustainability, systemoperators constantly monitor the operating conditions of elec-tricity networks [2], [3]. In the 1960s, power system engineerstried to compute voltages at critical buses based on meter read-ings manually collected from geometrically distributed currentand potential transformers. Due to timing, model mismatches,and metering errors, however, the exact (ac) power flow equa-tions were never infeasible.

With the development of supervisory control and data acqui-sition (SCADA) systems, a wealth of improved data meteredfrom across the network became available. In the seminal workof Schweppe et al. [4], the modern statistical foundation forpower system state estimation (PSSE) was laid. Given a col-lection of SCADA data along with corresponding measurementmatrices, the goal of PSSE is to compute the complex voltages(or, the voltage magnitudes and angles if polar coordinates areused) at all network buses. Since then, substantial contributionshave been devoted to PSSE. Interested readers can refer to [3]for a review of recent developments on PSSE.

Based on the weighted least-squares (WLS) estimation cri-terion, the Gauss–Newton solver is arguably the “workhorse”for PSSE, and it is also employed in practice [2]. Yet, the non-convex nature of WLS poses challenges on the Gauss–Newtonmethod, including sensitivity to initialization and outliers, aswell as no convergence guarantee [5]. To address these chal-lenges, semidefinite programming (SDP) relaxation approacheshave been pursued [6]–[8]. However, SDP incurs computationalcomplexity that does not scale well with problem dimension,discouraging its use in practical settings.

With utilities increasingly shifting toward smart grid tech-nology and other upgrades with inherent cyber vulnerabilities,correlative threats from adversarial cyberattacks on the NorthAmerican power grid continue to grow in frequency and form[9]. These introduce new yet critical challenges to PSSE, partic-ularly to the WLS-based SE solvers, concerning data integrityand uninformed model changes [10]–[14]. Such concerns moti-vate well the development of accurate and robust approaches toendow PSSE with resilience to anomalous (i.e., bad) data andmodel inaccuracies.

A. Related Work

In this context, robust PSSE has recently received renewedinterest. To cope with the malicious data, the largest normal-ized residual test was incorporated while performing PSSE

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1392 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 6, NO. 4, DECEMBER 2019

[10]. The least-median-squares and least-trimmed-squaresbased alternatives were pursued [15]. Unfortunately, the afore-mentioned robust PSSE proposals incur unfavorable computa-tional complexities and/or stringent storage requirements, whichlimit their practical uses in real-world power networks.

On the other hand, the �1-based criterion has been well knownin optimization and statistics for its robustness to outliers [16],[2, Ch. 6]. In addition to being robust, the �1-based estimator isalso statistically optimal in the maximum likelihood sense, whenthe independent additive noise follows a Laplacian distribution.In the PSSE literature, the �1-based criterion was advocated forbad data identification and rejection in [17]. Research focus hasshifted toward devising efficient and user-friendly algorithms tohandle the nonconvexity and nonsmoothness issues of �1-lossfunction; see, e.g., [18] and [19].

B. This Paper

This paper revisits the �1-based robust PSSE with a focus ondevelopment of efficient algorithms and theory on exact staterecovery in the noiseless case. Leveraging recent advances insolving rank-one quadratic equations (i.e., phase retrieval) [20],[21], we first suggest a simple algebraic procedure to transformthe power flows into rank-one measurements, namely with cor-responding transformed measurement matrices being rank one.Subsequently, we develop two efficient and easy-to-implementalgorithms to optimize the �1-loss of the obtained rank-one mea-surements. With appropriate conditions on the �1-loss function,we establish exact recovery as well as quadratic convergencefor our approach. Simulated tests using three IEEE benchmarksystems showcase the robustness and computational efficiencyof our proposed scheme relative to competing Gauss–Newtonmethod.

The rest of this paper is organized as follows. System mod-eling and problem formulation are given in Section II. Theprocedure to obtain rank-one measurements is presented inSection III, followed by two algorithms in Section IV. Exactstate recovery and convergence are established in Section V.Numerical tests are provided in Section VI, and this paper isconcluded in Section VII.

Notation: Matrices (column vectors) are denoted by upper-(lower-) case boldface letters; e.g., A (a). Sets are denotedusing calligraphic letters. Symbol T (H) represents (Hermitian)transpose, and (·) complex conjugate, whereas �(·) (�(·)) takesthe real (imaginary) part of a complex number.

II. SYSTEM MODELING AND PROBLEM FORMULATION

A. System Modeling

Consider an electric power grid modeled as a graph G =(N , L), whose nodesN := {1, 2, . . . , N} correspond to busesand whose edgesL := {(n, n′)} ⊆ N ×N correspond to lines.Throughout this paper, all analysis pertains to the per unit(p.u.) system. The complex voltage per bus n ∈ N can begiven in rectangular coordinates as vn = �(vn ) + j�(vn ). Forbrevity, all nodal voltages are stacked up to form the vectorv := [v1 · · · vN ]T ∈ CN . In the ac-based SE literature, a sub-

set of following system variables can be measured by SCADA[2, Ch. 2]:

1) |vn |: the voltage magnitude at bus n;2) Pnn ′ (Qnn ′ ): the active (reactive) power flow from buses

n to n′ at the sending terminal;3) Pn (Qn ): the active (reactive) power injection into bus n.

Compliant with the ac power flow model [2], these systemvariables can be expressed as quadratic functions of v. Thisjustifies why the voltage vector v is referred to as the systemstate. To this end, observe that the squared voltage magnitudesVn := |vn |2 = [�(vn )]2 + [�(vn )]2 can be written as

Vn = vHHVn v, with HV

n := hnhTn (1)

for all n ∈ N , where we have introduced the measurementvector hn := en with en being the nth canonical vector inRN . To express power injections as functions of v, introducethe bus admittance matrix Y = G + jB ∈ CN , where G andB ∈ RN ×N are the real and imaginary parts of Y [2], respec-tively. In rectangular coordinates, the active and reactive powersPn and Qn injected into bus n can be expressed as

Pn = �(vn )N∑

n ′=1

[�(vn ′)Gnn ′ − �(vn )Bnn ′

]

+ �(vn )N∑

n ′=1

[�(vn )Gnn ′ + �(vn )Bnn ′

](2)

Qn = �(vn )N∑

n ′=1

[�(vn )Gnn ′ − �(vn )Bnn ′

]

−�(vn )N∑

n ′=1

[�(vn )Gnn ′ + �(vn )Bnn ′

](3)

which can be compactly expressed as

Pn = vHHPn v, with HP

n :=YH

n + Yn

2(4a)

Qn = vHHQn v, with HQ

n :=YH

n − Yn

2j(4b)

with Yn := eneTn Y for all n ∈ N .With regards to power flows, Kirchhoff’s current law dictates

that the complex current over the line (n, n′) at the “sending”end is inn ′ = ys

nn ′vn + ynn ′(vn − v′n ), where ys

nn ′ is the shuntadmittance at bus n′ associated with the line (n, n′). The acpower flow model, in conjunction with Ohm’s law, further as-serts that the complex power flowing over line (n, n′) at the“sending” end can be expressed as

Snn ′ = Pnn ′ + jQnn ′ = vn inn ′

= |vn |2(ysnn ′ + ynn ′) − vnvnynn ′ . (5)

It is worth pointing out that the complex power flow over line(n, n′) ∈ L at the “receiving” end is captured by that over line(n′, n) ∈ L at the “sending” end. Upon defining the followingmatrices for all lines (n, n′) ∈ L:

Ynn ′ := (ysnn ′ + ynn ′)eneTn − ynn ′en ′eTn ′ (6)

WANG et al.: ROBUST POWER SYSTEM STATE ESTIMATION FROM RANK-ONE MEASUREMENTS 1393

the active and reactive power flows at the “sending” terminal,namely the real and imaginary parts of Snn ′ in (5) can be givenin a compact representation as

Pnn ′ = vHHPnn ′v, with HP

nn ′ :=YH

nn ′ + Ynn ′

2(7a)

Qnn ′ = vHHQnn ′v, with HQ

nn ′ :=YH

nn ′ − Ynn ′

2j. (7b)

Having expressed all SCADA measurements as functions ofv, the PSSE problem can be presented next.

B. Power System State Estimation

In practice, the SCADA system measures a subset of thesystem variables specified in (1), (4), and (7). Suppose now atotal of M such variables are measured, which are stacked upto form the following M × 1 measurement vector:

z :=[{Vn}n∈NV

, {Pn}n∈NP, {Qn}n∈NQ

{Pnn ′ }(n,n ′)∈LP, {Qnn ′ }(n,n ′)∈LQ

]T ∈ RM (8)

where the check-marked terms represent possibly noisy obser-vations of the corresponding error-free variables. The subsetsNV , NP , NQ ⊆ N , and LP , LQ ⊆ L specify the locationswhere meters are installed and the associated type of variablesare measured. Succinctly, per mth measurement in z can beequivalently rewritten as

zm := vHHmv + εm (9)

for all m ∈ {1, 2, . . . , M}, where the terms εm ∈ R capturethe metering errors and modeling inaccuracies, and the Hermi-tian measurement matrices Hm ∈ CN ×N can correspond to anysubset of the matrices defined in (1), (4), and (7). The criticalgoal of PSSE is to obtain v ∈ CN based on the available data{(zm ; Hm )}M

m=1 .Without loss of generality, adopting the LS error objective,

which coincides with the maximum likelihood criterion assum-ing additive white Gaussian noise, PSSE pursues problem1

minimizex∈CN

�(x) :=1

2M

M∑

m=1

(zm − xHHmx

)2. (10)

Because of the quadratic terms inside the squares, the quarticfunction �(x) is nonconvex, whose general instance is NP-hard[22]. Hence, it is computationally intractable to compute the LSor maximum likelihood estimate of v in general.

C. Prior Contributions

Minimizing the nonlinear LS loss in (10), the Gauss–Newtonmethod is the “workhorse” [2, Ch. 2]. Upon linearizing allquadratic terms xHHmx around a given point using Taylor’sexpansion, the Gauss–Newton subsequently approximates thenonlinear LS fit in (10) using a linear one per iteration, andrelies on its resultant minimizer to obtain the next iterate [3].

1Throughout, v is fixed for the actual system state, whereas x is used for theoptimization variable and the state estimate.

It typically converges in a few (≤ 10) iterations, very fast forsmall- or medium-size problems. However, it is known that theGauss–Newton iterations for nonconvex LS are sensitive to theinitial point, and they may diverge in certain cases; see e.g., [5,Ch. 5].

On the other hand, several numerical polynomial-time SE al-gorithms have been pursued based on convex programming [6].By means of matrix lifting, such convex approaches start ex-pressing all quadratic measurementsxHHmx as linear functionsTr(HmX) of the rank-one matrix variable X := xxH � 0.Upon discarding the nonconvex rank constraint, the nonlinearLS in (10) boils down to (or can be converted into) a convexSDP. In terms of computational efficiency, such convex schemesentail solving for an N × N positive semidefinite matrix fromM SDP constraints, whose worst case computational complex-ity is O(M 4N 1/2 log(1/ε)) for any given solution accuracyε > 0 [23, Sec. 6.6.2]. This complexity and the resultant storagerequirement evidently do not scale nicely to the increasinglyinterconnected large power networks.

III. RANK-ONE MEASUREMENT APPROACH

Drawing from advances in nonconvex optimization, this sec-tion presents a new framework for scalable, accurate, and ro-bust PSSE. Specifically, our proposed approach reformulatesthe rank-two power flows in (7) into rank-one quadratic mea-surements (namely with corresponding measurement matriceshaving rank one), followed by two efficient algorithms for min-imizing the �1-based misfit of the transformed measurements.In contrast to previous SE approaches that minimize the quar-tic polynomial in (10), a novel quadratic objective functionalis obtained and subsequently minimized. With more compli-cated algebraic manipulations, the power injections can alsobe accounted for in our proposed framework. In the presenceof additive noise, near-optimal statistical performance of thedeveloped approach is numerically demonstrated.

A. Measurement Transformation

In this paper, we focus on the following types of measure-ments: first, {|vm |}N

m=1 the voltage magnitudes at all buses,and second, {Pnn ′ }(m,n)∈LP

and/or {Qnn ′ }(m,n)∈LQthe ac-

tive and/or reactive power flows on a selected subset of lines,namely LP , LQ ⊆ L. Consider first the noise-free case, whereall available measurements can be described as

zm = vHHmv, ∀m = 1, . . . , M. (11)

Without loss of generality, let the first N measurements be thesquared voltage magnitudes at the N buses, namely zm = |vm |2for m = 1, 2, . . . , N , and the remaining ones be the (active orreactive) power flows. It is clear from (1) that the squared voltagemagnitude measurements are given by

zm =∣∣hH

mv∣∣2 , ∀m = 1, . . . , N (12)

whose corresponding measurement matrices have rank one; thatis, {Hm = hmhH

m}Nm=1 .

Now let us consider the power flow data pairs (Pnn ′ ;HPnn ′)

and (Qnn ′ ;HQnn ′) in (7). Upon substituting Ynn ′ in (6) into (7),

1394 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 6, NO. 4, DECEMBER 2019

one can rewrite for all lines (n, n′) ∈ LP

HPnn ′ =

12

(αP

nn ′eneTn − βPnn ′eneTn ′ − βP

nn ′en ′eTn)

(13)

and similarly for all lines (n, n′) ∈ LQ

HQnn ′ =

12

(αQ

nn ′eneTn − βQnn ′eneTn ′ − βQ

nn ′en ′eTn)

(14)

where the four coefficients are given by

αPnn ′ := 2�(ys

nn ′ + ynn ′), βPnn ′ := ynn ′ (15a)

αQnn ′ := −2�(ys

nn ′ + ynn ′), βQnn ′ := jynn ′ . (15b)

It is worth pointing out that αPnn ′ and αQ

nn ′ are both real,whereas βP

nn ′ and βQnn ′ are in general complex.

It is self-evident from (13) and (14) that each of the activeor reactive power flow measurement matrices {HP

nn ′ }(n,n ′)∈LP

or {HQnn ′ }(n,n ′)∈LQ

is of at most rank-two, with three nonzeromatrix entries at the (n, n)-, (n, n′)-, and (n′, n)th positionsdepending on whether the involved coefficients are nonzero ornot. Yet, these power flow matrices are generally indefinite.Upon neglecting the superscripts for notational brevity, bothHP

nn ′ and HQnn ′ are in the following form:

Hnn ′ =12

(αnn ′eneTn − βnn ′eneTn ′ − βnn ′en ′eTn

). (16)

We establish the following result for power flow data {zm}.Proposition 1 (Rank-one measurements): Suppose that

the voltage magnitudes are measured at all buses. Then, one canconstruct equivalently a new measurement zm for each powerflow zm so that its corresponding measurement matrix Hnn ′ isof rank one, and positive semidefinite.

Proof: Depending on whether the coefficient αnn ′ or βnn ′

is zero or not, we discuss separately the ensuing three cases:c1) αnn ′ = 0, βnn ′ = 0;c2) αnn ′ = 0, βnn ′ = 0;c3) αnn ′ = 0, βnn ′ = 0

where the trivial case of αnn ′ = βnn ′ = 0 is excluded becauseall SCADA measurements are assumed nonzero. The goal ofthe following section is to transform all power flow measure-ment matrices into rank-one (symmetric) positive semidefinitematrices. The three cases are individually discussed next.

Consider first the case c1). When both αnn ′ and βnn ′ arenonzero, the matrix Hnn ′ has exactly rank two with threenonzero entries. Its two nonzero eigenvalues can be obtainedby solving the quadratic equation

λ(λ − αnn ′

2

)− |βnn ′ |2

4= 0

which is derived by setting the determinant of (λIN − Hnn ′) tozero. Its closed-form solutions are given by

λ1 =αnn ′ +

√α2

nn ′ + 4|βnn ′ |24

> 0

λ2 =αnn ′ −

√α2

nn ′ + 4|βnn ′ |24

< 0.

Let u1 , u2 ∈ CN be the unit eigenvectors of Hnn ′ associ-ated with the eigenvalues λ1 , λ2 , respectively. Hence, one canwrite Hnn ′ := λ1u1uH

1 + λ2u2uH2 . To obtain a rank-one pos-

itive semidefinite matrix, the first attempt would be to com-pensate for the negative eigenvalue λ2 and make it zero. Thisis tantamount to adding −λ2u2uH

2 to Hnn ′ , and accordinglyadding −λ2vH(u2uH

2 )v to the measurement znn ′ ; that is

Hnn ′ := Hnn ′ − λ2u2uH2 (17a)

znn ′ := znn ′ − λ2vH(u2uH2 )v (17b)

in which the transformed measurement matrix Hnn ′ is rank-oneand symmetric positive semidefinite, and znn ′ is the resultanttransformed measurement. To realize this however, entails, eval-uating the term vH(u2uH

2 )v, which requires knowledge of thetrue state vector v. This procedure is thus not feasible, and onehas to develop a new twist to bypass this hurdle.

Recall from our working assumption that we have access toall squared voltage magnitudes {|vn |2}N

n=1 . Based on this fact,we show in the following that it is sufficient to add a matrix ofthe form (δnn ′/2)en ′eTn ′ to Hnn ′ such that the resulting sum,denoted by

Hnn ′ := Hnn ′ + (δnn ′/2)en ′eTn ′ (18)

can be rendered rank-one. Here, δnn ′ is an unknown coefficientto be determined next. Toward this end, setting the determinantof (λIN − Hnn ′) to zero leads to

(λ − αnn ′

2

) (λ − δnn ′

2

)− |βnn ′ |2

4= 0. (19)

To yield a rank-one matrix Hnn ′ , it is sufficient for thequadratic equation (19) to have exactly one nonzero solution. Bybasic linear algebra, this is equivalent to having a zero constantterm in (19), giving rise to

αnn ′δnn ′ − |βnn ′ |2 = 0

or alternatively

δnn ′ := |βnn ′ |2/αnn ′ .

It can be verified that the transformed measurement matrix

Hnn ′ := Hnn ′ + (|βnn ′ |2/(2αnn ′))en ′eTn ′ (20)

is rank-one. In addition, if αnn ′ > 0, then Hnn ′ is positivesemidefinite. Therefore, one can write

Hnn ′ := hnn ′hHnn ′ (21)

with the equivalent measurement vector being

hnn ′ :=√

αnn ′

2en +

βnn ′√2αnn ′

en ′ . (22)

The transformed measurement znn ′ corresponding to Hnn ′ canbe given by

znn ′ := vHHnn ′v = vHHnn ′v + vH (δnn ′en ′eTn ′/2

)v

= znn ′ + (|βnn ′ |2/2αnn ′)|vn ′ |2 (23)

WANG et al.: ROBUST POWER SYSTEM STATE ESTIMATION FROM RANK-ONE MEASUREMENTS 1395

for which the required quantity (|βnn ′ |2/2αnn ′)|vn ′ |2 is avail-able, or can be obtained as long as |vn |2 is available.

If αnn ′ < 0, one can instead define

Hnn ′ := −Hnn ′ + (|βnn ′ |2/(2αnn ′))en ′eTn ′ (24)

and write the equivalent measurement vector as

hnn ′ :=

√−αnn ′

2en +

βnn ′√−2αnn ′

en ′ (25)

for which the corresponding measurement becomes

znn ′ := vHHnn ′v = −vHHnn ′v − vH (δnn ′en ′eTn ′/2

)v

= −znn ′ − (|βnn ′ |2/2αnn ′)|vn |2 . (26)

Likewise, the quantity −(|βnn ′ |2/2αnn ′)|vn |2 is also availableunder our working assumption.

Let us now focus on the case c2). To transform Hnn ′ into arank-one positive semidefinite matrix, it is sufficient to add amatrix of the form (γnn ′/2)eneTn + (δnn ′/2)en ′eTn ′ , to yield

Hnn ′ := Hnn ′ + (γnn ′/2)eneTn + (δnn ′/2)en ′eTn ′ (27)

for some coefficients γnn ′ > 0 and δnn ′ > 0 to be determined.Similar to the discussion for case c1), to find γnn ′ and δnn ′ , onesets the determinant of (λIN − Hnn ′) to zero, leading to

(λ − γnn ′

2

)(λ − δnn ′

2

)− |βnn ′ |2

4= 0.

The fact that Hnn ′ is rank-one implies that

γnn ′δnn ′ − |βnn ′ |2 = 0.

Without loss of generality, one can take

γnn ′ := 1, and δnn ′ := |βnn ′ |2

and Hnn ′ in (27) becomes rank-one and can be written as

Hnn ′ := hnn ′hHnn ′ (28)

with

hnn ′ :=1√2en − βnn ′√

2en ′ . (29)

The transformed measurement associated with Hnn ′ can befound as follows:

znn ′ := vHHnn ′v

= vHHnn ′v + vH (βnn ′eneTn /2 + δnn ′en ′en ′/2

)v

= znn ′ + (1/2)|vn |2 + (|βnn ′ |2/2)|vn ′ |2 (30)

for which the required quantities |vn |2 and |βnn ′ |2 |vn ′ |2 can becomputed when |vn |2 and |vn ′ |2 are available.

Let us now turn to the last case c3). Since βnn ′ = βnn ′ = 0,one can write Hnn ′ = αnn ′eneTn /2, which is already rank-one.To make it positive semidefinite, it suffices to take the absolutevalue, and define

Hnn ′ := hnn ′hHnn ′ =

(√|αnn ′ |

2en

) (√|αnn ′ |

2en

)T

.

(31)

The transformed measurement is given by

znn ′ = |znn ′ |. (32)

To summarize, for any active or reactive power flow data(znn ′ ;Hnn ′), we have developed a strategy to obtain a newmeasurement pair (znn ′ ; Hnn ′), in which Hnn ′ becomes rank-one and positive semidefinite. Specifically, this is accomplishedthrough steps in (21)–(32), by depending upon the values ofcoefficients αnn ′ and βnn ′ , provided that the voltage magnitudesat all buses are available. �

The assumption on full voltage measurements can be relaxed.Indeed, it is possible to build up rank-one measurements vialinear combinations, so long as two of the following SCADAquantities {|vn |, |vn ′ |, pnn ′ , pn ′n , qnn ′ , qn ′n} are measured onevery line (n, n′) ∈ L. In a nutshell, one can readily rewrite allmeasurements as intensities of some known and deterministiclinear transforms of the state vector, namely

zm =∣∣hH

mv∣∣2 , ∀m = 1, . . . , M (33)

where the measurement vectors hm ∈ CN are given in (1), (22),(25), (29), and (31), whereas the corresponding transformedmeasurements zm > 0 are defined in (1), (23), (26), (30), and(32). Moreover, all vectors hm are highly sparse, each havingat most two nonzero entries independent of the system size N .This feature can be carefully exploited to endow the iterativePSSE solvers with computational efficiency and scalability.

IV. PROX-LINEAR SE SOLVERS

In general, given a set of (consistent) quadratic equations,there may exist multiple solutions even after excluding triv-ial ambiguities. In the context of phase retrieval, in which themeasurement matrices are rank-one positive semidefinite (i.e.,Hm = hmhH

m ), a number M ≥ 4N − 4 of random quadraticequations suffice for uniqueness of the solution [24]. In thepower systems literature though, it remains an open questionthat how many quadratic measurements as in (1), (4), and (7)are required for the uniqueness of PSSE solution. For concrete-ness, this contribution assumes that a large enough number ofmeasurements are available, and they collectively determine aunique solution, namely the underlying true system state.

Leveraging the rank-one measurement model, this sectionpresents two algorithms for scalable and exact power systemstate recovery based on nonconvex optimization. Specifically,we focus on the �1-loss to fit the intensity measurements zm

in (33) instead of zm in (11). Despite the nonconvexity andnonsmoothness of the resulting loss function, we first developa deterministic prox-linear algorithm. When the initialization issufficiently close to the underlying true voltage state vector, andthe loss function satisfies a certain local “stability condition,”we show that our first deterministic approach recovers the truevoltage vector at a quadratic convergence rate. It entails solv-ing a quadratic program per iteration, for which off-the-shelfconvex programming toolboxes are widely available, but the re-sulting complexity does not scale well with the system size. Toendow the algorithm with scalability, a stochastic generalizationis pursued, which processes a single measurement per iteration.

1396 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 6, NO. 4, DECEMBER 2019

It is well known in statistics and power systems literature that�1-based loss functions yield median-based estimators, and theycan cope with gross errors in the measurements zm in a relativelybenign way [21]. This prompts us to consider the �1-loss [i.e.,least-absolute-value (LAV)] formulation

minimizex∈CN

�(x) :=1M

M∑

m=1

∣∣zm − |hHmx|2

∣∣ . (34)

Evidently, this loss function is nonsmooth and nonconvex, whichis not even locally convex near v. This can be understood fromthe scalar case �(x) = |1 − x2 |. As such, it is unclear how toefficiently minimize such functions.

However, the function exhibits several appealing structuralproperties that we explore in the following to develop iterativealgorithms to locally solve the problem (34) efficiently [21],[25], [26]. To this end, consider first expressing the loss func-tion as the composition �(x) := c(s(x)), of the convex func-tion c(·) := ‖ · ‖1 and the smooth one s(x) := 1

M (z − |Hx|2),where z := [z1 · · · zM ]T , H := [h1 · · · hM ]H, and the mod-ulus operator | · | is understood elementwise when applied toa vector. Such a compositional structure lends itself nicely toiterative procedures that are referred to as proximal-linear (prox-linear) algorithms, which we are described in detail ahead.

Consider a real-valued x ∈ RN for now. The (deterministic)prox-linear method for minimizing �(x) = c(s(x)) is to lin-earize s only, followed by successively minimizing a sequenceof locally regularized models. Specifically, starting with someinitialization x0 ∈ RN , the prox-linear method defines a local“linearization” of � around the current iterate xt ∈ RN as

�xt(x) := c

(s(xt) + ∇T s(xt)(x − xt)

)(35)

with ∇s(xt) ∈ RN ×M representing the Jacobian matrix of sevaluated at point xt ; and subsequently, it proceeds inductivelyto obtain iterates x1 , x2 , ...by minimizing the quadratically reg-ularized models [25], [26]

xt+1 = arg minx∈RN

{�xt

(x) +1

2μt‖x − xt‖2

2

}(36)

where μt > 0 is a step size that can be fixed a priori to someconstant, or be determined “on-the-fly” through a line search[25], [26]. Furthermore, observing that the linearization �xt

(x)is convex in x, so problem (36) is convex in x as well. It hasbeen shown in [26] that when c is L-Lipschitz and ∇s is κ-Lipschitz, choosing any step size 0 < μ < 1

κL guarantees thatthe algorithm (36) is a descent method; that is, the iterates {xk}monotonically decrease the function value of �(x); and finds an(approximate) stationary point of (34).

Nevertheless, the PSSE problem (34) involves optimizationover complex-valued variables in x ∈ CN . It can be checkedthat the functions � and s do not satisfy the Cauchy–Riemann(CR) equations; see e.g., [27, Th. 7.2] for the definition of CRequations. Hence, functions � and s are not holomorphic (i.e.,complex-differentiable) in x. As such, the “linearization,” or thefirst-order Taylor’s expansion of s(x) in x ∈ CN alone [cf. (35)]does not exist. To address this challenge, we invoke Wirtinger’scalculus to generalize prox-linear algorithms to optimization

Algorithm 1: Deterministic Prox-linear SE Solver.

1: Input data {(zm , Hm )}Mm=1 , step size μ > 0,

initialization v0 ∈ CN , solution accuracy ε > 0, and sett = 0.

2: Prepare the power flow data {(zm , Hm )}Mm=N +1

according to (1), (22)–(23), (25)–(26), (29)–(30), and(31)–(32) to obtain {(zm , hm )}M

m=N +1 based on{zm = |vm |2}N

m=1 .3: Repeat4: Evaluate am,t and bm,t in (38).5: Solve (37) to yield xt+1 .6: t = t + 1.7: Until ‖xt − xt−1‖2 ≤ ε

√N .

8: Return xt .

over complex-valued arguments in the sequel. Please refer to[28] for basics of Wirtinger’s calculus.

A. Deterministic Prox-Linear SE Solver

Our first deterministic prox-linear approach to (34) is simplystated: begin with initialization x0 := 1 ∈ RN , and proceedby successively minimizing quadratically regularized functionsaround the current iterate xt ∈ CN to yield the next iterate

xt+1 = arg minx∈CN

1M

M∑

m=1

∣∣bm,t − 2�(aH

m,tx)∣∣+ 1

2μt‖x − xt‖2

2

(37)

where the term bm,t − 2�(aHm,tx) can be interpreted as the first-

order Taylor’s approximation of the nonholomophic functionzm − |hH

mx|2 at xt based upon the Wirtinger derivatives; seeAppendix A for the rigorous derivation. The coefficients am,t

and bm,t are given by

am,t :=(hH

mxt

)hm (38a)

bm,t := zm +∣∣hH

mxt

∣∣2 . (38b)

Observe that the problem (37) to be tackled per iteration ofour deterministic prox-linear SE solver is a convex quadraticprogram, which can be efficiently solved with standard con-vex programming methods. Under appropriate conditions, ourscheme converges quadratically fast to the true state vectorv, meaning that we have to solve only about log2 log2(1/ε)such quadratic programs to obtain an estimate x of v satisfyingdist(x, v) ≤ ε‖v‖2 . As will be corroborated by our numericaltests in Section VI, this boils down to 5 ∼ 8 convex quadraticprograms in practice. Moreover, our approach applies both inthe noiseless setting, and when a constant (random) portion ofthe measurements are even adversarially corrupted.

For implementation, our deterministic prox-linear solver issummarized in Algorithm 1. Regarding computational com-plexity, preparing the data in Step 2 can be performed withinO(M) operations. Exploiting the sparsity of hm ’s, evaluatingthe coefficients {(bm,t , am,t)}M

m=1 in Step 5 can also be donewith O(M) operations. The overall complexity of Algorithm 1

WANG et al.: ROBUST POWER SYSTEM STATE ESTIMATION FROM RANK-ONE MEASUREMENTS 1397

is indeed dominated by solving the quadratic program of (37) inStep 6. With standard convex programming solvers, the resultantcomplexity is often O(MN 2). Iterative procedures dependingon the alternating direction method of multipliers can reduce thisnumber to O(MN log(1/ε)) [21], [29]. The latter complexity,however, may still become unfavorable for large-size power net-works. Furthermore, even though {hm}M

m=1 have at most twononzero entries, this property cannot be fully exploited to speedup computations for solving the quadratic program of (37). Toaddress these issues, we advocate a stochastic alternative of (37)ahead for solving problem (34).

B. Stochastic Prox-Linear SE Solver

The stochastic prox-linear method deals with a single mea-surement per iteration. Initialized with x0 , our (stochastic) prox-linear SE solver operates by first sampling uniformly a lossfunction via randomly picking mt ∈ {1, 2, . . . , M}, and re-lies on minimizing its quadratically regularized “linearization”around xt to yield xt+1 [30]; that is, define inductively fort = 0, 1, 2, . . . that

xt+1 =arg minx∈CN

∣∣bmt ,t − 2�(aH

mt ,tx)∣∣ +

12μt

‖x − xt‖22

(39)where the coefficients bmt ,t and amt ,t are given in (38), withbmt ,t − 2�(aH

mt ,tx) being the first-order Taylor’s expansion ofthe mt th error function zm t

− |hHmt

x|2 around xt . Evidently,problem (39) is again a quadratic program too. Compared withthe first quadratic program in (36), fortunately, the solution to(39) can be found in simple closed form.

To that end, we invoke an earlier result in [29, Prop. 3],which is included in Appendix B for completeness. Upon defin-ing w := x − xt , a := amt ,t , and b := bmt ,t − 2�(aH

mt ,txt)

in Proposition 2, one can readily find the solution to (39) asfollows:

xt+1 = xt + projμt

(bmt ,t − 2�

(aH

mt ,txt

)

‖amt ,t‖22

)amt ,t .

Substituting amtand bmt

of (38) into the last equality leads toour stochastic prox-linear SE solver

xt+1 = xt + projμt

(zm t

−∣∣hH

mtxt

∣∣2

4∣∣hH

mtxt

∣∣2 · ‖hmt‖2

2

)· 2

(hH

mtxt

)hmt

.

(40)We summarize our second (stochastic) prox-linear SE solver

in Algorithm 2 for further reference. In terms of computationalcomplexity, we report the exact number of complex scalar op-erations (e.g., additions, multiplications) needed per stochas-tic prox-linear SE iteration of (40) next. Relying on whetherhmt

has 1 or 2 nonzero entries, the following statements holdtrue. If hmt

has 1 (2) nonzero entries, evaluating |hHmt

xt |2 re-quires 2 (4) operations, and ‖hmt

‖22 requires 1 (3) operations,

plus another 5 operations for the remaining, all summing toa total of 8 (12) operations. In other words, per iteration ofAlgorithm 2 (cf. Steps 4–6) must perform only 12 complexscalar operations or so. Interestingly, this per-iteration com-plexity of O(1) holds regardless of the power network under

Algorithm 2: Stochastic Prox-linear SE Solver.

1: Input data {(zm , Hm )}Mm=1 , step size μ > 0,

initialization v0 ∈ CN , solution accuracy ε > 0, and sett = 0.

2: Prepare the power flow data {(zm , Hm )}Mm=N +1

according to (1), (22)–(23), (25)–(26), (29)–(30), and(31)–(32) to obtain {(zm , hm )}M

m=N +1 based on{zm = |vm |2}N

m=1 .3: Repeat4: Draw mt ∈ {1, 2, . . . , M} uniformly at random.5: Evaluate Evaluate amt ,t and bmt ,t in (38).6: Update xt+1 via (40).7: t = t + 1.8: Until ‖xt − xt−1‖2 ≤ ε

√N .

9: Return xt .

investigation, or more precisely, the system size N . It is self-evident that this O(1) per-iteration complexity scales nicely tolarge- and even massive-size power networks.

V. CONVERGENCE ANALYSIS AND EXACT RECOVERY

In this section, we begin our development by providing con-vergence guarantees for the proposed prox-linear SE solvers.For concreteness, we will focus on the deterministic prox-linearAlgorithm 1, whereas convergence can be also established forAlgorithm 2 in a probabilistic sense. Interested readers are re-ferred to [30]. Under certain conditions on the loss function �,exact recovery results are also established.

Recall our loss function c(s(x)) = 1M ‖z − |Hx|2‖1 , where

c(u) = ‖u‖1 : RM → R, and s(x) = 1M (z − |Hx|2) : CN →

RM . It is easy to verify for any u, v ∈ RN that it holds

|c(u) − c(v)| ≤M∑

n=1

|um − vm | = ‖u − v‖1 ≤√

M‖u − v‖2

where the last inequality arises from the equivalence of norms.By definition, this asserts that c is

√M -Lipschitz continuous.

Focusing now on the complex Jacobian ∇xs for any x andy ∈ CN , we deduce

‖∇xs(x) −∇xs(y)‖2 =1M

∥∥HHH(x − y)∥∥

2 ≤ L‖x − y‖2

with L := (2/M)λmax(HHH), which confirms that ∇xs is L-Lipschitz continuous.

Appealing to the results in [26, Th. 5.3], one can concludethat our deterministic prox-linear SE solver with constant stepsize μ ≤ 1/(L

√M) =

√M/(2λmax(HHH)) converges to a

stationary point of �(x) in (34).We provide in the sequel conditions on the function � such

that exact recovery of power system states by our prox-linearSE solvers is guaranteed. Going beyond [21], which is limitedto optimization over real-valued variables, we introduce twocomplimentary conditions on �(x) and its linearization �x(y) ofcomplex-valued variables x, y ∈ CN .

1398 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 6, NO. 4, DECEMBER 2019

If � is defined over real-valued vectors, namely �(x) : RN →R, we can readily have the following stability condition for es-tablishing fast convergence: for any given v ∈ RN , there existsa constant ρ > 0 such that �(x) − �(v) ≥ ρ‖x − v‖2 ‖x + v||2holds for all x ∈ RN , which is included in Appendix B forcompleteness. This condition, though crucial for establishingfast convergence, does not generalize to functions of complex-valued variables. It is obvious in the complex case that ifv ∈ CN is an optimal solution to (34), then ejφv with any φ ∈[0, 2π) is also an optimal solution. This ambiguity thus promptsus to define the Euclidean distance of any estimate x ∈ CN tov ∈ CN as dist(x, v) := minφ∈[0,2π ) ‖x − ejφv‖2 . This def-inition can be thought of as enforcing zero-phase angle at thereference bus, a standard procedure in power flow problemsto eliminate the phase ambiguity. Invoking a result for stablephase retrieval in [31, Th. 3.1], we define the following stabilitycondition for functions �(x) : CN → R, which also applies topractical settings where the measurements may contain noise orbe even adversarially corrupted.

Condition 1: There exists some constant ρ > 0 such thatthe inequality holds for all x ∈ CN

�(x) − �(v) ≥ ρ

√‖x − v‖2

2 ‖x + v‖22 − 4|�(xHv)|2 . (41)

Evidently, when the measurements z = |Hv|2 are noiseless,it holds that �(v) = 0. Similar to the real-valued case stud-ied in [21], our Condition 1 is instrumental in establishingfast convergence of the prox-linear algorithm for optimizingfunctions of complex-valued variables. Besides Condition 1,we require a condition on the linearization �x(y) : CN → R of�(x) = c(s(x)) around x defined by

�x(y) := c(s(x) + 2�

(∇H

x s(x)(y − x)))

. (42)

Condition 2: There exists a constant L < +∞ such that theinequality holds for all x, y ∈ CN

|�(y) − �x(y)| ≤ L

2‖x − y‖2

2 . (43)

This condition basically requires that the locally linearizedconvex approximation �x(y) is quadratically close [cf. (43)] tothe nonconvex function �(y). Indeed, the �1-based PSSE costfunction � in (34) automatically satisfies Condition 2 globally.To show this, let us first express �x(y) according to the definitionof (42)

�x(y) =1M

M∑

m=1

∣∣|hHmx|2 − zm + 2�

(xHhmhH

m (y − x))∣∣ .

On the other hand, for any m ∈ {1, 2, . . . , M} and y ∈ CN ,the following holds true

∣∣hHmy

∣∣2 =∣∣hH

mx∣∣2 + 2�

(xHhmhH

m (y − x))

+∣∣hH

m (y − x)∣∣2 .

Subtracting zm from both sides, summing from m = 1 to M ,and leveraging the triangle inequality, we have that

�(y) =1M

M∑

m=1

∣∣|hHmy|2− zm

∣∣≤ �x(y)+1M

M∑

m=1

∣∣hHm (y− x)

∣∣2

�(y) =1M

M∑

m=1

∣∣|hHmy|2− zm

∣∣≥�x(y)− 1M

M∑

m=1

∣∣hHm (y− x)

∣∣2 .

Rewriting the last terms in matrix-vector form yields

|�(y) − �x(y)| ≤ (y − x)H( 1M

HHH)(y − x) ≤ L

2‖y − x‖2

2

(44)which proves that Condition 2 is satisfied globally by the �1-lossin (34). Moreover, one can take L = λmax(HHH/M), namelythe largest eigenvalue of matrix HHH/M .

Under Conditions 1 and 2, we now devote to exact recov-ery guarantees for the deterministic prox-linear SE solver inAlgorithm 1. The following result implies exact recovery ofpower system states at a quadratic rate by our proposed prox-linear SE solver in Algorithm 1 under suitable conditions.

Theorem 1: Let Conditions 1 and 2 hold. Assuming thatthe quadratic program (37) is solved exactly per iteration ofAlgorithm 1, the successive prox-linear SE iterates xt satisfy

dist(xt ,v)‖v‖2

≤ ρ

L

(L

ρ· dist(x0 ,v)

‖v||2

)2t

. (45)

For readability, the proof of Theorem 1 is postponed toAppendix C. Regarding Theorem 1, three observations comein order.

Remark 1 (Exact recovery): If the initialization x0 of theiterations is accurate enough, meaning that it satisfies the con-dition dist(x0 , v) < (ρ/L)‖v‖2 , the prox-linear SE solver re-covers exactly the true state vector v ∈ CN . In terms of initial-izations, there are several approaches for this desideratum, threeof which are discussed next. Since power systems are typicallyoperating close to the flat voltage profile 1, it is reasonable toinitialize the algorithm with x0 = 1. Moreover, as the voltagemagnitudes at all buses are assumed available, one can use thevoltage magnitude vector x0 = |v| as the initializer. Alterna-tively, it is also feasible to initialize with the estimate found bysolving the linearized dc power flow equations.

Remark 2 (Quadratic convergence rate): When dist(x0 ,v) < (ρ/L)‖v‖2 holds true, our prox-linear SE algorithm con-verges quadratically fast to the globally optimal solution ofthe nonconvex and nonsmooth optimization problem (34). Ex-pressed differently, to obtain a solution xt of (at most) ε-relativeerror, namely dist(xt , v)/‖v‖2 ≤ ε, we must only run Algo-rithm 1 for about log2 log2(1/ε) iterations, or equivalently, solvelog2 log2(1/ε) convex quadratic programs as in (37). This, inpractice, amounts to about 5 ∼ 10 such quadratic programs.

Remark 3: Under the condition dist(x0 , v) < (ρ/L)‖v‖2 ,it is worth pointing out that Condition 1 can be replaced by acondition requiring only the function �(x) to satisfy the inequal-ity (49) locally for all x within the neighborhood of v definedby dist(x, v) < (ρ/L)‖v‖2 .

WANG et al.: ROBUST POWER SYSTEM STATE ESTIMATION FROM RANK-ONE MEASUREMENTS 1399

Fig. 1. Exact recovery performance of Algorithm 1 for the IEEE 14-bussystem (noiseless case).

VI. NUMERICAL TESTS

In this section, we perform a number of numerical tests toevaluate our approach and compare with the “workhorse” LS-based Gauss–Newton method. Several power network bench-marks including the IEEE 14-, 118-, and 300-bus systemswere simulated, following the MATLAB-based toolbox MAT-POWER [32], [33]. The Gauss–Newton iterations were im-plemented by using the embedded SE function “doSE.m”in MATPOWER. To carefully isolate the relative perfor-mance of the iterative algorithms, rather than initializationemployed, all simulated schemes were initialized with theflat voltage vector (i.e., the all-one vector) in all reportedexperiments.

A. Tests With Zero Noise

The first experiment examines the exact recovery and conver-gence performance of Algorithm 1 from noiseless data on theIEEE 14-, 118-, and 300-bus test systems. The actual voltagemagnitude (in p.u.) and angle (in radians) of each bus were uni-formly distributed over [0.9, 1.1], and over [−0.1π, 0.1π]. Thevoltage magnitude squares at all buses as well as the active powerflows across all lines were measured. The quadratic programsin Step 5 of Algorithm 1 were solved by the standard convexprogramming solver SeDuMi [34] with a constant step size ofμt = 1, 000. Algorithm 1 terminates either when a maximumnumber 20 of iterations are simulated, or when the normal-ized distance between two consecutive iterates becomes smallerthan 10−10 , namely dist(xt , xt−1)/

√N ≤ 10−10 . A total of

100 Monte Carlo (MC) runs were carried out. Figs. 1–3 plot thenormalized estimation errors dist(xt , v)/

√N for the 100 MC

realizations on the simulated three systems, whose correspond-ing L values are 0.9980, 3.0201, and 6.3102. Furthermore, Fig. 4depicts the convergence of the normalized estimation error ofAlgorithm 1 for the 100 runs on the 14-bus system. Evidently,Algorithm 1 achieves exact recovery over the 100 runs, and en-joys quadratic convergence in this noiseless setting, validatingour theoretical findings in Theorem 1.

Fig. 2. Exact recovery performance of Algorithm 1 for the IEEE118-bus system (noiseless case).

Fig. 3. Exact recovery performance of Algorithm 1 for the IEEE300-bus system (noiseless case).

Fig. 4. Quadratic convergence of Algorithm 1 in 100 runs for the IEEE14-bus system (noiseless case).

B. Tests With Outlying Measurements

One of the claimed advantages of the �1-based loss functionin (34) is its robustness to outliers. In this test, we evaluatethe robustness of Algorithm 1 to measurements with outliers

1400 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 6, NO. 4, DECEMBER 2019

Fig. 5. Recovery performance for the IEEE 118-bus system with 1outlying measurement (noiseless case).

in terms of the exact recovery. Concretely, the IEEE 118-bussystem with its default voltage profile was simulated. The volt-age magnitudes at all buses along with the active and reactivepower flows across all lines were measured. Considering the factthat the nodal voltage magnitudes in transmission networks aremaintained close to one, we assume that only the power meterscan be compromised. A total of 100 MC runs were performed.Per run, one power flow meter was randomly compromised,whose measurement was purposefully manipulated and ampli-fied to five times its original value.

Both the Gauss–Newton and Algorithm 1 were simulated inthis experiment, whose corresponding normalized estimationerrors for the 100 MC realizations are presented in Fig. 5. It isself-evident from the plots that the Gauss–Newton method is notrobust to outlying measurements, whereas our proposed prox-linear scheme in Algorithm 1 can identify and automaticallyreject the bad data, yielding exact recovery of the true states inmost cases even under adversarial attacks.

C. Tests With Additive Noise and Outliers

The third experiment assesses the robust estimation perfor-mance of Algorithm 1 relative to Gauss–Newton, in a settingwhere both additive noises and outliers are present. The largeIEEE 300-bus benchmark system with its default voltage pro-file was simulated. All active and reactive power flows as wellas all voltage magnitudes were measured. Additive noise wasindependently generated from normal distributions having zero-mean and standard deviations 0.004 and 0.008 for the voltagemagnitude and line flow measurements, respectively [18]. In ad-dition to additive noise, 5% of the entire measured power flowswere corrupted uniformly at random with “outliers” drawn in-dependently from a Gaussian distribution with zero-mean andstandard deviation 5. The normalized estimation errors obtainedby the Gauss–Newton method and Algorithm 1 for 100 MC in-dependent realizations are reported in Fig. 6. Evidently, ourdeveloped algorithm consistently exhibits more robust estima-

Fig. 6. Estimation performance for the IEEE 300-bus system underadditive noise and 5% outlying measurements.

tion performance than LS-based Gauss–Newton against additivenoise and outlying measurements.

VII. CONCLUSION

Robust PSSE is approached by minimizing the �1-based lossfunction from the vantage point of composite optimization. Toenable efficient algorithms and exact state recovery, the powerquantities were first transformed into rank-one measurements.Building on advances in nonconvex and nonsmooth compositeoptimization, two algorithms were put forth for minimizing the�1-based loss of the transformed rank-one measurements. Ouralgorithms require no tuning of parameters, except for a stepsize. We also developed “stability conditions” on the �1-basedloss function such that exact state recovery and quadratic con-vergence are guaranteed by our approach in the noiseless case.Simulated tests using three IEEE benchmark networks underdifferent settings validate our theoretical findings, and show-case the efficacy of our approach.

APPENDIX

A. Wirtinger’s Calculus

Introducing the complex conjugate coordinates [xT xT ]T ∈C2N , one can rewrite s(x) = s(x, x) ∈ CM . It is obvious nowthat s(x, x) becomes holomorphic in x for a fixed x, and viceversa. This leads to the partial Wirtinger derivatives [28]

∂sm

∂x:=

∂sm (x,x)∂x

∣∣∣∣x=constant

=[∂sm

∂x1

∂sm

∂x2· · · ∂sm

∂xN

]

∂sm

∂x:=

∂sm (x,x)∂x

∣∣∣∣x=constant

=[∂sm

∂x1

∂sm

∂x2· · · ∂sm

∂xN

]

where the partial derivative with respect to x (x) treats x (x) as aconstant in sm . The complex gradient of sm (x, x) with respectto x or x can be defined by

∇xsm :=(

∂sm

∂x

)H, and ∇xsm :=

(∂sm

∂x

)H

WANG et al.: ROBUST POWER SYSTEM STATE ESTIMATION FROM RANK-ONE MEASUREMENTS 1401

yielding the complex gradient of sm in new coordinate system

∇csm :=[∇T

x sm ∇Tx sm

]T=

[∂sm

∂x∂sm

∂x

]H.

Upon introducing the complex Jacobian

∇cs := [∇cs1 ∇cs2 · · · ∇csM ] ∈ C2N ×M

we can define for given vectors x and Δx ∈ CN the followingfirst-order Taylor’s expansion:

s(x + Δx) ≈ s(x) + ∇Hc s(x)

[ΔxΔx

]

= s(x) + 2�(∇Hsx(x)Δx

)∈ RM ×N . (46)

B. Supporting Results

Proposition 2 ([29, Prop. 3]): Given a ∈ CN and b ∈ R,the solution of

minimizew∈CN

∣∣b −�(aHw

)∣∣ +12μ

‖w‖22 (47)

can be obtained as w := projμ(b/‖a‖22) · a, where projμ(x) :

R × R+ → R is the projection operator that returns the realnumber in interval [−τ, τ ] closest to any given x ∈ R.

Condition 3 [21, Condition 1]: For any given v ∈ RN ,there exists a parameter ρ > 0 such that function �(x) satisfiesthe following for all x ∈ Rn :

�(x) − �(v) ≥ ρ‖x − v‖2‖x + v‖2 . (48)

Concerning the lower bound in (41), we have the next result.Lemma 1: For any fixed v ∈ CN , the inequality holds for

all x, v ∈ CN

‖x − v‖22 ‖x + v‖2

2 − 4∣∣�(xHv)

∣∣2 ≥ ‖v‖22 dist2(x,v).

(49)Proof: The left-hand-side term of (49) can be rewritten as

‖x − v‖22 ‖x + v‖2

2 − 4�2(xHv)

=(‖x‖2

2 + ‖v‖22 − 2�(xHv)

) (‖x‖2

2 + ‖v‖22 + 2�(xHv)

)

− 4�2(xHv)

=(‖x‖2

2 + ‖v‖22)2 − 4

[�2(xHv) + �2(xHv)

]

=(‖x‖2

2 + ‖v‖22)2 − 4|xHv|2

=(‖x‖2

2 + ‖v‖22 + 2|xHv|

) (‖x‖2

2 + ‖v‖22 − 2|xHv|

)

≥ ‖v‖22(‖x‖2

2 + ‖v‖22 − 2|xHv|

)

= ‖v‖22 dist2(x,v).

Taking the square root from both sides of the inequality yieldsthe statement of Lemma 1. �

C. Proof of Theorem 1

The proof is based on that of [21, Th. 1], but we here gen-eralize its results to function optimization over complex do-mains. Observe that the regularized function g(x) := �xt

(x) +

L2 ‖x − xt‖2

2 is L-strongly convex in x ∈ CN , and its min-imum is attained at xt+1 [cf. (37)]. The standard optimal-ity conditions for strongly convex minimization confirms thatg(xt+1) ≤ g(ej∠xH

t vv) − L2 ‖ej∠xH

t vv − xt+1‖22 ; that is,

�xt(xt+1) +

L

2‖xt+1 − xt‖2

2 ≤ �xt(ej∠xH

t vv)

+L

2

∥∥∥ej∠xHt vv − xt

∥∥∥2

2− L

2

∥∥∥ej∠xHt vv − xt+1

∥∥∥2

2. (50)

Recalling now Condition 2, we have that

�(xt+1) ≤ �xt(xt+1) +

L

2‖xt+1 − xt‖2

2 (51a)

�xt(ej∠xH

t vv) ≤ �(ej∠xHt vv)+

L

2

∥∥∥ej∠xHt vv−xt

∥∥∥2

2. (51b)

Substituting (51a) and (51b) into (50) gives rise to

�(xt+1) ≤ �xt(ej∠xH

t vv) +L

2

∥∥∥ej∠xHt vv − xt

∥∥∥2

2

− L

2

∥∥∥ej∠xHt vv − xt+1

∥∥∥2

2

≤ �(ej∠xHt vv)+L

∥∥∥ej∠xHt vv−xt

∥∥∥2

2−L

2

∥∥∥ej∠xHt vv−xt+1

∥∥∥2

2

which, in conjunction with �(ej∠xHt vv) = �(v) in (34), yields

L∥∥∥ej∠xH

t vv − xt

∥∥∥2

2≥ �(xt+1) − �(v)

+L

2

∥∥∥ej∠xHt vv − xt+1

∥∥∥2

2. (52)

Invoking further Condition 1 in (52), we have that

L∥∥∥ej∠xH

t vv − xt

∥∥∥2

2≥ L

2

∥∥∥ej∠xHt vv − xt+1

∥∥∥2

2

+ ρ

√‖xt+1 − v‖2

2 ‖xt+1 + v‖22 − 4

∣∣�(xHt+1v)

∣∣2 (53)

in which the last term can be replaced with its lower boundρ‖v‖2 dist(xt+1 ,v) established in Lemma 1. Upon droppingthe nonnegative term L

2 ‖ej∠xHt vv − xt+1‖2

2 , and recalling thedefinition of dist(xt ,v), we immediately have

ρ‖v‖2 dist(xt+1 ,v) ≤ Ldist2(xt ,v)

dividing both sides of which by ρ‖v‖22 yields

dist(xt+1 ,v)‖v‖2

≤ L

ρ· dist2(xt ,v)

‖v‖22

=ρ

L

(L

ρ· dist(xt ,v)

‖v‖2

)2

.

Applying the above-mentioned inequality successively fromthe initialization x0 for t iterations through xt gives rise to (45),concluding the proof.

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Gang Wang (M’18) received the B.Eng. de-gree in electrical engineering and automationfrom the Beijing Institute of Technology, Beijing,China, in 2011, and the Ph.D. degree in elec-trical and computer engineering from the Uni-versity of Minnesota, Minneapolis, MN, USA, in2018.

He is currently a Postdoctoral Associate withthe Department of Electrical and Computer En-gineering, University of Minnesota. His researchinterests focus on the areas of statistical signal

processing, stochastic and (non-)convex optimization with applicationsto autonomous energy grids, and deep learning.

Dr. Wang was a recipient of a National Scholarship (2014), a Guo RuiScholarship (2017), and an Innovation Scholarship (first place in 2017),all from China, as well as a Best Student Paper Award at the 2017 Eu-ropean Signal Processing Conference.

Hao Zhu (M’12) received the B.Eng. degree inelectrical engineering from Tsinghua University,Beijing, China in 2006, and the M.Sc. (minor:Math) and Ph.D. degrees in electrical engineer-ing from the University of Minnesota, Minneapo-lis, MN, USA, in 2009 and 2012, respectively.

She is currently an Assistant Professor inelectrical and computer engineering (ECE) withThe University of Texas at Austin, Austin, TX,USA. She was a Postdoctoral Researcher, from2012 to 2013, and then an Assistant Profes-

sor in ECE from 2014–2017 with the University of Illinois at Urbana-Champaign. Her current research interests include power system moni-toring and operations considering its cyber-physical coupling, as well asenergy data analytics.

Dr. Zhu was a recipient of the NSF CAREER Award in 2017, theSiebel Energy Institute Seed Grant, and the U.S. AFRL Summer Fac-ulty Fellowship in 2016. She is the faculty advisor and/or coauthor of twobest papers at the North American Power Symposium. She is currently amember of the steering committee of the IEEE Smart Grid representingthe IEEE Signal Processing Society.

WANG et al.: ROBUST POWER SYSTEM STATE ESTIMATION FROM RANK-ONE MEASUREMENTS 1403

Georgios B. Giannakis (F’97) received theDiploma degree in electrical engineering fromthe National Technical University of Athens,Athens, Greece, in 1981, the M.Sc. degree inelectrical engineering in 1983, the M.Sc. de-gree in mathematics, and the Ph.D. degree inelectrical engineering both in 1986 from the Uni-versity of Southern California, Los Angeles, CA,USA.

He was with the University of Virginia, from1987 to 1998 and since 1999, he has been a Pro-

fessor with the University of Minnesota, Minneapolis, MN, USA, wherehe holds an Endowed Chair in Wireless Telecommunications, Univer-sity of Minnesota McKnight Presidential Chair in Electrical and Com-puter Engineering, and serves as the Director of the Digital TechnologyCenter. His general interests include the areas of communications, net-working, and statistical signal processing—subjects on which he hasauthored/coauthored more than 430 journal papers, 730 conference pa-pers, 25 book chapters, two edited books, and two research monographs(h-index 135). His current research focuses on learning from big data,wireless cognitive radios, and network science with applications to so-cial, brain, and power networks with renewables. He is the (co-)inventorof 30 patents issued.

Dr. Giannakis was the (co-)recipient of nine best paper awards fromthe IEEE SIGNAL PROCESSING (SP) and Communications Societies, in-cluding the G. Marconi Prize Paper Award in Wireless Communications.He is also a recipient of the Technical Achievement Awards from theSP Society (2000), from the EURASIP (2005), a Young Faculty Teach-ing Award, the G. W. Taylor Award for Distinguished Research from theUniversity of Minnesota, and the IEEE Fourier Technical Field Award(inaugural recipient in 2015). He is a Fellow of the EURASIP, and hasserved the IEEE in a number of posts, including that of a DistinguishedLecturer for the IEEE-SP Society.

Jian Sun (M’10) was born in Jilin Province,China, in 1978. He received the Bachelor’sdegree in electrical engineering from the JilinInstitute of Technology, Changchun, China, in2001, the Master’s degree in mechanical engi-neering from the Changchun Institute of Optics,Fine Mechanics and Physics, Chinese Academyof Sciences (CAS), Changchun, China, in 2004,and the Ph.D. degree in control theory and con-trol engineering from the Institute of Automation,CAS, Beijing, China, in 2007.

He was a Research Fellow with the Faculty of Advanced Technology,University of Glamorgan, U.K., from April 2008 to October 2009. He wasa Postdoctoral Research Fellow with the Beijing Institute of Technology,Beijing, China, from December 2007 to May 2010. In May 2010, he joinedthe School of Automation, Beijing Institute of Technology, where he hasbeen a Professor since 2013. His research interests include control,security, and learning of networked control systems and cyber-physicalsystems.

Dr. Sun is currently on the Editorial Boards of the Journal of SystemsScience and Complexity and Acta Automatica Sinica.