Matrix Completion for Low-Observability Voltage Estimation
Priya L. Donti, Yajing Liu, Andreas J. Schmitt, Andrey Bernstein, Rui Yang, Yingchen Zhang
11 Matrix Completion for Low-ObservabilityVoltage Estimation
Priya L. Donti,
Student Member, IEEE,
Yajing Liu,
Member, IEEE,
Andreas J. Schmitt,
Student Member, IEEE,
Andrey Bernstein,
Member, IEEE,
Rui Yang,
Member, IEEE, and Yingchen Zhang,
Senior Member, IEEE
Abstract —With the rising penetration of distributed energyresources, distribution system control and enabling techniquessuch as state estimation have become essential to distribu-tion system operation. However, traditional state estimationtechniques have difficulty coping with the low-observability conditions often present on the distribution system due tothe paucity of sensors and heterogeneity of measurements. Toaddress these limitations, we propose a distribution systemstate estimation algorithm that employs matrix completion(a tool for estimating missing values in low-rank matrices)augmented with noise-resilient power flow constraints. Thismethod operates under low-observability conditions wherestandard least-squares-based methods cannot operate, andflexibly incorporates any network quantities measured in thefield. We empirically evaluate our method on the IEEE 33- and123-bus test systems, and find that it provides near-perfect stateestimation performance (within 1% mean absolute percenterror) across many low-observability data availability regimes.
I. I
NTRODUCTION S TATE estimation is one of the most critical inferencetasks in power systems. Classically, it entails estimatingvoltage phasors at all buses in a network given some noisyand/or bad data from the network [1]. Estimates are obtainedvia the (generally non-linear) measurement model: z = h ( x ) + (cid:15) , (1)where z ∈ C m is a vector of measurements, x ∈ C n is avector of quantities to estimate (typically, voltage phasors), h ( · ) is a vector of functions representing the system physics(i.e., power-flow equations), and (cid:15) is a vector of measure-ment noise. The state-estimation task is then to estimate x given z and some knowledge of h ( · ) (e.g., its Jacobian Financial support from the Department of Energy Computational ScienceGraduate Fellowship under Grant No. DE-FG02-97ER25308 is gratefullyacknowledged. This work was authored in part by the National RenewableEnergy Laboratory, managed and operated by Alliance for SustainableEnergy, LLC, for the U.S. Department of Energy (DOE) under ContractNo. DE-AC36-08GO28308. Funding provided by the U.S. Department ofEnergy Office of Energy Efficiency and Renewable Energy Solar EnergyTechnologies Office. The views expressed in the article do not necessarilyrepresent the views of the DOE or the U.S. Government. The U.S. Gov-ernment 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 formof this work, or allow others to do so, for U.S. Government purposes. matrix). State estimation has been thoroughly addressedin transmission networks, wherein system (1) is typically overdetermined and fully observable : that is, (i) the numberof measurements m is at least the number of unknowns n , and (ii) the Jacobian J ∈ C m × n of h ( · ) is (pseudo)invertible in the sense that ( J T J ) − exists. As transmissionsystems conventionally have redundant measurements thatsatisfy the observability requirement, classical least-squaresestimators are applicable and can operate efficiently [2].In contrast, the use of state estimation has historicallybeen limited in distribution networks [3]. Due to limitedavailability of real-time measurements from SupervisoryControl and Data Acquisition (SCADA) systems, equa-tion (1) is typically underdetermined ( m < n ), renderingstandard least-squares methods inapplicable. Accurate dis-tribution system state estimation was also previously un-necessary since distribution networks only delivered powerin one direction towards the customer, requiring minimaldistribution system control. This led industry to in practiceuse only simple heuristics (e.g. based on simple load-allocation rules [4], [5]) to roughly calculate power flow.However, due to the increasing adoption of distributedenergy resources (DERs) at the edge of the network [6],distribution system state estimation has become increasinglyimportant [7]. There is thus a large focus in the literature onlow-observability state estimation techniques. Many existingmethods attempt to improve system observability, e.g., byoptimizing the placement of additional system sensors [8]–[10] or by deriving pseudo-measurements from existingsensor data [11], [12]. Unfortunately, installation of ad-ditional sensors may be expensive or slow, and pseudo-measurements can introduce estimation errors [13] or beextremely data-intensive to obtain [10], [11]. Other methodsseek to perform state estimation using neural networks,without constructing an underlying system model [14].While such machine learning methods can obtain accurateestimation results, training these methods requires a signifi-cant amount of historical data, which may not be available.As such, there is a need for state estimation methods thatcan exploit problem structure to perform state estimation atcurrent levels of data availability and observability.In this paper, we propose a low-observability state esti-mation algorithm based on matrix completion [15], a tool a r X i v : . [ m a t h . O C ] A p r for estimating missing values in low-rank matrices. Weapply this tool to state estimation for a given time step byforming a structured data matrix whose rows correspond tomeasurement locations, and whose columns correspond tomeasurement types (e.g. voltage or power). While methodssuch as [9] require collecting data over large time windows,our approach enables “single shot” state estimation thatemploys only data from a single time instance. Our approachis closely related to recent works [16], [17] that use matrixcompletion to estimate lost PMU data over a time series, butwhile these works estimate missing quantities exclusively atmeasurement points, we consider the problem of estimatingquantities even at non-measurement points where the quan-tities to be estimated may have never been measured.The main contributions of our paper are: • A novel distribution system state estimation methodbased on constrained matrix completion. By augment-ing matrix completion with noise-resilient power flowconstraints, the proposed method can accurately esti-mate voltage phasors under low-observability condi-tions where standard (least-squares) methods cannot. • A flexible framework for employing various types ofdistribution system measurements into state estimation.Whereas many works (e.g. [9], [12], [18]) requirespecific measurements for estimation, our approach canaccommodate any quantities measured in the field. • An empirical demonstration of the robustness of ourmethod to data availability and measurement loss.The rest of the paper is organized as follows: Section IIintroduces the concept of constrained matrix completion.The proposed distribution system state estimation algorithmis presented in Section III. Simulation results for the IEEE33- and 123-bus test systems are presented in Section IV.Section V concludes the paper.II. M
ATRIX C OMPLETION M ETHODS
We start by introducing constrained matrix completion , amethod that is central to our proposed approach.
A. Matrix Completion
Given an incomplete matrix that is assumed to be low-rank, the matrix completion problem aims to determinethe unknown elements in this matrix. Formally, let M ∈ R n × n be a real-valued data matrix, Ψ ⊆ { , . . . , n } ×{ , . . . , n } describe the known elements in M , and M Ψ ∈ R n × n denote the observation matrix, where ( M Ψ ) j,k = M j,k for ( j, k ) ∈ Ψ and otherwise. Matrix completion canthen be formulated as a rank-minimization problem [15]: minimize X ∈ R n × n rank( X )subject to X Ψ = M Ψ , (2)where the decision variable X estimates M . As the opti-mization problem (2) is NP-hard due to the non-convexity of the rank function, it is common to use a heuristic approachthat instead minimizes the nuclear norm of the matrix [15]: minimize X ∈ R n × n (cid:107) X (cid:107) ∗ subject to X Ψ = M Ψ , (3)where (cid:107) X (cid:107) ∗ sums the singular values of X . Given asufficient number of randomly-sampled entries in M Ψ (depending on the matrix size and rank), problem (3) oftenhas a unique minimizer X that equals M [15]. Additionally,this problem can be solved efficiently [19]–[21].Due to the nature of the equality constraint, formulation(3) is highly susceptible to noise. To alleviate this problem,[22] proposed an algorithm to handle noisy measurements.The algorithm modifies the equality constraint in (3) to (cid:107) X Ψ − M Ψ (cid:107) F ≤ δ, (4)where (cid:107)·(cid:107) F is the Frobenius norm and δ ≥ is a parameterthat can be tuned based on the extent of measurement noise. B. Constrained Matrix Completion
Now suppose that the values in the matrix M come fromsome physical system (e.g., a power system). It is thennatural to extend formulation (3)/(4) to incorporate systemphysics via the following constrained optimization problem: minimize X ∈ R n × n (cid:107) X (cid:107) ∗ subject to (cid:107) X Ψ − M Ψ (cid:107) F ≤ δ, (cid:107) g ( X ) (cid:107) ≤ β, (5)for δ, β ≥ , where g ( · ) is a vector of functions representingsystem physics (e.g., power-flow equations). We note that: • The additional constraint (cid:107) g ( X ) (cid:107) ≤ β incentivizeslow-rank solutions that respect the system physics. • The choice of δ and β is problem dependent. Theseparameters can be chosen based on the extent ofmeasurement noise, or the objective function can beaugmented with terms that try to minimize their values. • If g ( · ) is nonlinear, (5) is non-convex and NP-hard.III. L OW -O BSERVABILITY S TATE E STIMATION
We now present our low-observability distribution systemstate estimation algorithm, which employs the constrainedmatrix completion model (5). We describe our power systemmodel, possible formulations of M , and possible powersystem constraints g ( · ) before showing our full formulation. A. Power System Model
Let B denote the set of buses, where bus 1 is the slack busand the remaining |B| − buses are P Q buses. Further, let
L ⊆ B × B denote the set of distribution lines. We describethe nodal admittance matrix Y ∈ C |B|×|B| in block form as Y = (cid:20) Y ∈ C Y L ∈ C × ( |B|− Y L ∈ C ( |B|− × Y LL ∈ C ( |B|− × ( |B|− (cid:21) . Let v ∈ C |B| and s ∈ C |B| be the vectors of (partially un-known) voltage phasors and net complex power injections,respectively, at each bus. We denote the slack bus voltagephasor and power injection as v and s , respectively, andsimilarly denote the vectors of non-slack bus voltages andpower injections as v − and s − . Finally, let i ∈ C |L| bethe vector of (partially unknown) complex currents in eachbranch, where i ft is the current in line ( f, t ) ∈ L . B. Data Matrix Formulation
The formulation of the data matrix M (and thus theoptimization variable X ) can vary based on the particularattributes of the problem setting, e.g. the kinds of measure-ments available and problem scale. We present two possibleformulations here, one indexed by branches and one indexedby buses. However, we emphasize that the proposed methodis not limited to using these matrix structures . The matrix M can be flexibly structured to accommodate availablemeasurements, as long as these measurements are correlatedso that M is (approximately) low rank.
1) Branch Formulation: M can be structured such thateach row represents a power system branch and each columnrepresents a quantity relevant to that branch. This structureallows us to take advantage of both bus- and branch-relatedmeasurements. Specifically, for every line ( f, t ) ∈ L , thecorresponding row in the matrix M ∈ R n × n contains: [ (cid:60) ( v f ) , (cid:61) ( v f ) , | v f | , (cid:60) ( s f ) , (cid:61) ( s f ) , (cid:60) ( v t ) , (cid:61) ( v t ) , | v t | , (cid:60) ( s t ) , (cid:61) ( s t ) , (cid:60) ( i ft ) , (cid:61) ( i ft ) ] , where n = |L| and we employ n = 12 quantities per row.
2) Bus Formulation: M can also be structured such thateach row represents a bus and each column represents aquantity relevant to that bus. That is, for every bus b ∈ B ,the corresponding row in the matrix M ∈ R n × n contains: [ (cid:60) ( v b ) , (cid:61) ( v b ) , | v b | , (cid:60) ( s b ) , (cid:61) ( s b )] , where n = |B| and we employ n = 5 quantities perrow. While this structure only employs bus-related measure-ments, its advantage is that it yields small matrices that canbe used for efficient estimation on large-scale problems.
3) On the Low-rank Assumption:
We note that thecomplex power system quantities in both the bus andbranch formulations are approximately linearly correlated,as implied by the fact that the power system equationsemploying them can be expressed in (approximate) linearform (see Section III-C). In other words, these data matrixformulations should be approximately low-rank.We observe empirically that this low-rank assumptionholds in practice. Fig. 1 shows the cumulative percentagedistribution of singular values for the IEEE 33-bus feeder(using a branch formulation) and the IEEE 123-bus feeder(using a bus formulation). In both cases, we see that a fewsingular values comprise much of the singular value sum,implying that these matrices are (approximately) low-rank.
C. Physical Power Flow Constraints
As described in Section II-B, we augment matrix comple-tion with power system constraints to encourage physicallymeaningful solutions. These constraints are linear to ensurethat problem (5) is convex. We describe the constraints weuse below, but note that constraints can be added, removed,or modified depending on the types of measurements in M .
1) Duplication Constraint:
Depending on the formula-tion, some quantities may appear in more than one locationin M . For example, in our branch formulation, quantitiesrelated to a given bus appear in multiple rows if the bus is inmultiple branches. We thus constrain equivalent quantitiesin the matrix to be equal. Formally, let Λ contain all pairsof indices of duplicated quantities in M . We require that X λ = X λ , ∀ ( λ , λ ) ∈ Λ . (6)
2) Ohm’s Law Constraint:
When M contains both bus-and branch-related quantities (as in the branch formulation),we can apply Ohm’s Law, defined as ( v f − v t ) y ft = i ft , ∀ ( f, t ) ∈ L , (7)where y ft is the line admittance. However, using an exactequality constraint may cause the matrix completion prob-lem to become infeasible, e.g. due to measurement noise.We thus employ a noise-resilient version of Ohm’s Law, i.e., (cid:20) − ξ r,ft − ξ c,ft (cid:21) ≤ (cid:20) (cid:60) (( v f − v t ) y ft − i ft ) (cid:61) (( v f − v t ) y ft − i ft ) (cid:21) ≤ (cid:20) ξ r,ft ξ c,ft (cid:21) , (8)where ξ r,ft , ξ c,ft ∈ R + are respective error tolerances forthe real and complex parts of Ohm’s Law on line ( f, t ) ∈ L .
3) Linearized Power Flow Constraints:
As the exact ACpower flow equations are non-linear, we employ Cartesianlinearizations of these equations. For non-slack voltages andpower injections, we employ approximations of the form v − ≈ A (cid:20) (cid:60) ( s − ) (cid:61) ( s − ) (cid:21) + w , (9a) | v − | ≈ C (cid:20) (cid:60) ( s − ) (cid:61) ( s − ) (cid:21) + | w | . (9b)For example, using the method proposed in [23], we canlet w = − v Y − LL Y L ∈ C |B|− be the vector of non-slack zero-load voltages and A , C ∈ C ( |B|− × |B|− bedefined for some non-slack voltage estimates ˆ v − as A = (cid:2) Y − LL diag (ˆ v − ) − − j Y − LL diag (ˆ v − ) − (cid:3) , (10a) C = diag ( | ˆ v − | ) − (cid:60) (cid:0) diag ( | ˆ v − | ) A (cid:1) . (10b)For our case, we let ˆ v − = w . We note, however, thatother methods to obtain the linear approximations (9) (e.g.,data-driven regression methods [24]) can also be leveraged.To relate voltages with the power injection at the slackbus, we employ the exact power flow equation s = v ( Y v + Y L v − ) . (11) C u m u l a t i v e P e r c e n t a g e o f S i n g u l a r V a l u e S u m (a) IEEE 33-bus feeder (branch formulation). The largest singular value(out of 12) comprises over 98% of the singular value sum. C u m u l a t i v e P e r c e n t a g e o f S i n g u l a r V a l u e S u m (b) IEEE 123-bus feeder (multi-phase bus formulation). The first 3 (outof 5) largest singular values comprise 95% of the singular value sum. Fig. 1: Cumulative percentage of matrix singular value sum for two test cases. Both matrices are approximately low-rank.This equation is linear in voltages since v is known.As in Section III-C2, we relax these constraints intonoise-resilient versions as − τ r − τ c ≤ (cid:60) (cid:16) v − − (cid:16) A (cid:20) (cid:60) ( s − ) (cid:61) ( s − ) (cid:21) + w (cid:17)(cid:17) (cid:61) (cid:16) v − − (cid:16) A (cid:20) (cid:60) ( s − ) (cid:61) ( s − ) (cid:21) + w (cid:17)(cid:17) ≤ τ r τ c , (12a) − γ ≤ | v − | − (cid:18) C (cid:20) (cid:60) ( s − ) (cid:61) ( s − ) (cid:21) + | w | (cid:19) ≤ γ , (12b) (cid:20) − α r − α c (cid:21) ≤ (cid:34) (cid:60) (cid:0) s − (cid:0) v ( Y v + Y L v − ) (cid:1) (cid:1) (cid:61) (cid:0) s − (cid:0) v ( Y v + Y L v − ) (cid:1)(cid:1)(cid:35) ≤ (cid:20) α r α c (cid:21) , (12c) where τ r , τ c , γ ∈ R |B|− , α r , α c ∈ R + are error toler-ances, and inequalities are evaluated elementwise. D. Full Problem Formulation
Given these power flow constraints, we collect our errortolerances into the set T = { ξ r , ξ c , τ r , τ c , γ , α r , α c } andform our constrained matrix completion problem (5) as minimize X ∈ R n × n , T (cid:107) X (cid:107) ∗ + (cid:88) t ∈T w t (cid:107) t (cid:107) (13a) subject to (cid:107) X Ψ − M Ψ (cid:107) F ≤ δ, (13b)(6) , (8) , (12a) , (12b) , (12c) , (13c) t ≥ , ∀ t ∈ T , (13d)where in this case we add each constraint tolerance t ∈ T tothe objective with an associated weight w t . Each weight ischosen to reflect the relative importance of its constraint. Fora branch-formulated M , the above formulation can be usedas-is. For a bus-formulated M , equations (6) and (8) areremoved from the constraints (and their associated parame-ters ξ r , ξ c are removed from T ) since M does not containduplicated quantities or branch current measurements.Formulation (13) allows the matrix completion optimiza-tion to explicitly trade off between the low-rank assumptionand fidelity to the power flow constraints, without requiring tuning of each entry of each constraint tolerance vector. Wefurther observe that, since the objective is convex and allconstraints are linear in the entries of X , this formulation isa convex optimization problem and can be solved efficiently. E. Extension to the Multi-phase Setting
While for brevity we formally present only a single-phase, balanced formulation, our approach can easily beextended to the general multi-phase setting. Specifically, M can be structured to include phase-wise quantities, and theconstraints presented in Section III-C can be replaced withmulti-phase versions (e.g. see [23], [25]). We empiricallyillustrate the application of our method to a multi-phase -bus test case in Section IV-B.IV. S IMULATION AND R ESULTS
We demonstrate the performance of our matrix comple-tion method on the IEEE 33-bus and 123-bus test cases.We employ the branch-formulation of our algorithm onthe 33-bus feeder, and show that it performs well in low-observability settings (where traditional state estimationtechniques cannot operate) as well as in full-observabilitysettings. We also evaluate our method on the multi-phase123-bus feeder using a bus formulation matrix, demonstrat-ing that our method scales robustly to larger systems.
A. 33-Bus System
We test our branch-formulated algorithm on a modifiedversion of the IEEE 33-bus test case with solar panelsadded at buses 16, 23, and 31. On this system, voltagemagnitudes range from approximately . - . p.u., andall angles (relative to the substation voltage angle) are closeto 0. We assume that voltage phasors are known only at theslack bus, and must be estimated elsewhere. Non-voltagephasor quantities (i.e. voltage magnitude, power injections,and current flows) are assumed to be known exactly at thefeeder head, and are “potentially known” at other buses. M A P E o f V o l t a g e M a g n i t u d e F u ll O b s e r v a b ili t y M A E o f V o l t a g e A n g l e ( d e g r ee s ) MC WLS
Fig. 2: Performance on the 33-bus test case with 1% noiseand random sampling of data. (Each point represents 50runs.) We achieve less than voltage magnitude MAPEand less than . degrees voltage angle MAE when or more of “potentially known” quantities are measured.
1) Randomly-sampled data:
In one set of experiments,we model data unavailability among “potentially known”quantities via random sampling. That is, we randomlychoose sets of buses (ranging between - of buses)at which all “potentially known” quantities are known, andset these quantities to be unknown at all other buses. Resultsunder Gaussian measurement noise are shown in Fig. 2.In both cases, the mean absolute percent error (MAPE)of our voltage magnitude estimates drops to below and the mean absolute error (MAE) of our voltage angleestimates drops to below . degrees when or moremeasurements are known. (We report MAEs rather thanMAPEs for voltage angles since all angles are close to 0.)When or more of measurements are known, voltagemagnitude MAPEs are much less than , and voltageangle MAEs are less than . degrees. This result calibrateswell with guarantees for matrix completion performanceunder random data removal [22]. In contrast, traditionalfull-observability state estimation techniques cannot operateon this system until of “potentially known” quantitiesare measured. At full observability, our method’s estimatesare competitive with those of a state-of-the-art weightedleast-squares (WLS) state estimation algorithm, with voltagemagnitude MAPEs and angle MAEs within . and . degrees, respectively, for our matrix completion algorithmand within . and . degrees, respectively, for WLS.
2) Data-driven assumptions:
In practice, data unavail-ability is not uniformly random, but instead systematic andcorrelated. For instance, a utility may only have certaintypes of sensors at certain types of buses. We thus runa second set of experiments where we classify the buses into four categories: slack bus (1 bus), solar generators (3buses), large loads (6 buses), and small loads (17 buses).As before, all quantities are known exactly at the slackbus. At solar PV generators, real power injections, reactivepower injections, and voltage magnitudes are potentiallyknown. At loads (large or small), real power injections arepotentially known. Since actual sensor availability may varybetween utilities, we model different scenarios in whichthese groups of non-slack buses have real, pseudo-, or nomeasurements. In the best case (when all three groupsof buses have some measurements), of “potentiallyknown” quantities are measured. Thus, all our data-drivenscenarios are at low observability , and traditional full-observability state estimation methods cannot be used.The performance of our method on of these scenarios isshown in Fig. 3, assuming measurements have Gaussiansensor noise and pseudomeasurements have
Gaussianerror. Our algorithm achieves less than MAPE inits magnitude estimates and less than . degrees MAEin its angle estimates when accurate measurements areavailable for solar generators and either measurements orpseudomeasurements are available for loads. Results for arepresentative run in this scenario are shown in Fig. 4. Moregenerally, our estimates (averaged across all runs) have atmost . ± . % voltage magnitude MAPE and . ± . degrees voltage angle MAE in any scenario where solargenerators are measured, and at most . ± . % magnitudeMAPE and . ± . degrees angle MAE in any scenariowhere solar generators have pseudomeasurements (whereangle errors are high in this latter case due to solar generatormeasurement noise). If solar generator measurements areunknown, magnitude MAPEs range from - , and angleMAEs range from . - degrees.A potential alternative to using low-observability stateestimation techniques is to enable full-observability tech-niques by deploying additional sensors. To model thisalternative, we randomly add AMI sensors (which collectcoarse-granularity load data, modeled as pseudomeasure-ments) and “magnitude sensors” (which collect voltage andcurrent magnitudes) to our system until full observabilityis achieved. We then compare our method to WLS on thisaugmented system. Voltage phasor estimates for a represen-tative run are shown in Fig. 5. In this case, both our matrixcompletion algorithm and WLS quite accurately estimatevoltages, with voltage magnitude MAPEs and angle MAEswithin . and . degrees, respectively, for our matrixcompletion algorithm and within . and . degrees,respectively, for WLS. However, achieving full observabilityrequired adding 28-63 sensors to the system (depending onthe baseline data availability scenario), which represents apotentially high cost to the distribution utility.Overall, our results on the 33-bus system demonstrate thatour matrix completion algorithm can provide accurate stateestimation performance in the low-observability case (where M A P E o f V o l t a g e M a g n i t u d e Solar: 0 Solar: P Solar: M0 P MLarge Loads051015202530 M A E o f V o l t a g e A n g l e ( d e g r ee s ) Fig. 3: Performance on the 33-bus test case with noise over different data availability for solar, large load, and smallload buses (0 = not measured, P = some pseudomeasurements, M = some measurements; each point represents 50 runs.) Ourmethod achieves good estimation error in many data availability scenarios, even though all scenarios exhibit low observability. V o l t a g e M a g n i t u d e ( p . u . ) V o l t a g e A n g l e ( d e g r ee s ) True Estimated
Fig. 4: Representative voltage phasor estimates for the 33-bus test case with noise in the low-observability scenariowith solar, large loads, and small loads partially measured.traditional state estimation techniques cannot operate), aswell as in the full-observability case. B. 123-Bus Feeder
We next demonstrate that our matrix completion methodeffectively scales to larger systems via experiments on theIEEE 123-bus feeder. The 123-bus feeder is a multi-phaseunbalanced radial distribution system, in which buses aresingle-, double-, or three-phase (with 263 phases in total.)On this system, voltage magnitudes range from . - p.u.,and voltage angles are around or ± degrees. Weemploy a bus-formulation matrix for this test case, where V o l t a g e M a g n i t u d e ( p . u . ) V o l t a g e A n g l e ( d e g r ee s ) True Estimated (MC) Estimated (WLS)
Fig. 5: Representative estimates for the 33-bus test case with noise and data-driven sampling at full observability.each matrix row represents a phase at one bus. To validateour approach, we use two hours of system voltage andpower injection data. This data was simulated at one-minuteresolution using power flow analysis with diversified loadand solar profiles created for each bus.As in the previous section, we assume that voltagephasors are known only at the slack bus (which herehas three phases) and must be estimated elsewhere. Allother measurements (i.e. voltage magnitudes and powerinjections) are known at the slack bus and are “potentiallyknown” for non-slack system phases.We first employ our algorithm at one point in the timeseries during which solar injections are nonzero. In ourexperiments, we vary the percentage of “potentially known” M A P E o f V o l t a g e M a g n i t u d e F u ll O b s e r v a b ili t y M A E o f V o l t a g e A n g l e ( d e g r ee s )
1% noise 0.2% noise
Fig. 6: Performance on the 123-bus test case for one timestep. (Each point represents 50 runs.) We achieve lessthan 1% voltage magnitude MAPE when 10% or more of“potentially known” quantities are measured. The voltageangle MAE is always below . degrees at measurementnoise, and below degree at . measurement noise.
50 100 150 200 2500.0000.0050.0100.0150.020 A E o f V o l t a g e M a g n i t u d e ( p . u . )
50 100 150 200 250Bus Number0.00.51.01.52.0 A E o f V o l t a g e A n g l e ( D e g r ee s ) Fig. 7: Performance on the 123-bus test case for a repre-sentative run with data availability and noise.quantities that are measured, and note that this systemexhibits low-observability if less than two-thirds of thesequantities are measured. Results for the cases of . and measurement noise are shown in Fig. 6. We also showrepresentative results for one run under measurementnoise and measurement availability (which is in thelow-observability realm) in Fig. 7.These results show that our algorithm estimates voltagephasors with relatively high accuracy across all levels ofdata availability. The MAPE of our voltage magnitudeestimates is less than . even when no “potentially M A P E o f V o l t a g e M a g n i t u d e Percentage of "Potentially Measured" Quantities Available30% 50% 70%
Fig. 8: Time-series performance for the 123-bus test casewith noise. Voltage magnitude MAPEs are at most . with of data, and at most with or of data.known” quantities are measured, and falls to less than once or more of these quantities are available.Unsurprisingly, our voltage magnitude estimates are betterwhen measurement noise is lower. We do note, however,that the MAPE is not equal to zero even when ofmeasurements are available, since we use approximate linearpower flow equations as constraints in our formulation (13).For voltage angle estimates, we again report MAEs ratherthan MAPEs since the angles at some phases are close tozero. For measurement noise, we see that the MAEis at most . degrees across all data availability levels,which is small given that most voltage angles are around ± degrees. However, the accuracy of our angle estimatesdoes not decrease monotonically as more measurementsare added. This is because our algorithm directly estimatesthe real and imaginary parts of voltage phasors; whilewe estimate these quantities accurately, their errors arenot correlated (especially under large measurement noise),leading to inaccuracies in their implied angle estimates.At . measurement noise, the MAE for voltage angledecreases as more measurements are available, droppingbelow 1 degree even when no measurements are available.To demonstrate the scalability of our method, we nextimplement our matrix completion algorithm on the entiretwo-hour time series. We model data availability by placingsensors at fixed sets of buses comprising , , or of all buses (with larger sets of buses inclusive of smallersets). The MAPEs of our voltage magnitude estimates under1% measurement noise are shown in Fig. 8. We find thatunder % data availability, the MAPEs of these estimatesare within for of time steps, and within . for of time steps, with a maximum MAPE of . . Under and data availability, the MAPEs of our voltagemagnitude estimates are within across all points in thetime series. The MAEs of our voltage angle estimates ateach time step (not pictured) are similar to those in Fig. 6.Finally, we demonstrate the robustness of our algorithm M A P E o f V o l t a g e M a g n i t u d e Baseline (50% Data Available) 20% Measurement Loss
Fig. 9: Time-series performance for the 123-bus test casewith noise when measurements are randomly lost. Ourmagnitude estimates are relatively robust to data loss.to dynamic measurement loss, which commonly occurs onthe distribution system due to failures such as packet dropsand equipment malfunctions. We model the baseline dataavailability as being (low-observability), and randomlyremove of the available measurements at each point inthe two-hour time series. The MAPEs of our voltage mag-nitude estimates under measurement noise are shownin Fig. 9. We find that in most cases, our estimates underdata loss are similar to the estimates without data loss, withlarger deviations in some cases when critical measurementsare lost. In all cases, the MAPEs of our voltage magnitudeestimates are less than . , demonstrating the robustnessof our algorithm to real-world operating conditions.V. C ONCLUSION
We present an algorithm for low-observability distributionsystem voltage estimation based on constrained low-rankmatrix completion. This method can accurately estimatevoltage phasors under low-observability conditions wherestandard state estimation methods cannot operate, and canflexibly accommodate any distribution network measure-ments available in the field. Our empirical evaluations ofthis method demonstrate that it produces accurate and robustvoltage phasor estimates on the IEEE 33- and 123-bus testsystems under a variety of data availability conditions. Assuch, we believe that our algorithm is a useful mechanismfor voltage estimation on modern distribution systems.R
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