Structure Learning and Statistical Estimation in Distribution Networks - Part I

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by  Deepjyoti Deka

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Structure Learning and Statistical Estimation in
Distribution Networks - Part I
Deepjyoti Deka*, Scott Backhaus†, and Michael Chertkov‡
*Corresponding Author. Electrical & Computer Engineering, University of Texas at Austin
†MPA Division, Los Alamos National Lab
‡Theory Division and the Center for Nonlinear Systems, Los Alamos National Lab
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Abstract—Traditionallypowerdistributionnetworksareeither substation at the root and customers positioned at the other
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not observable or only partially observable. This complicates nodes. Switching from one tree-like operational configuration
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developmentandimplementationofnewsmartgridtechnologies,
toanotheristypicallycausedbysystemupsets,e.g.faultsand
b such as those related to demand response, outage detection and
outages, and may occur few times a day or even an hour.
e management, and improved load-monitoring. In this two part
F paper, inspired by proliferation of metering technology, we dis- The radial configuration distinguishes distribution networks
cuss estimation problems in structurally loopy but operationally fromtransmissionnetworksthatgenerallyhavemultipleloops
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radial distribution grids from measurements, e.g. voltage data, energized all the time to guarantee continuous delivery of
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whichareeitheralreadyavailableorcanbemadeavailablewith
power to every node, even in case of occasional line faults
a relatively minor investment. In Part I, the objective is to learn
] andoutages.Radialconfigurationsandone-wayflowofpower
C the operational layout of the grid. Part II of this paper presents
algorithms that estimate load statistics or line parameters in have led to much less monitoring, observability, and state
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additiontolearningthegridstructure.Further,PartIIdiscusses estimation in distribution as compared to meshed transmis-
h. the problem of structure estimation for systems with incomplete sion networks [1]. The recent proliferation of smart grid
measurement sets. Our newly suggested algorithms apply to
t technology, including smart meters that measure electricity
a a wide range of realistic scenarios. The algorithms are also
consumption at the node level, is creating a new opportunities
m computationallyefficient–polynomialintime–whichisproven
theoreticallyandillustratedcomputationallyonanumberoftest toextractinformationimportanttogridoperatorsandplanners.
[
cases. The technique developed can be applied to detect line Such efforts are also getting additional attention in view of

2 failuresinrealtimeaswellastounderstandthescopeofpossible mounting concerns over data security and protection of user
v adversarial attacks on the grid. privacy [2].
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Index Terms—Power Distribution Networks, Power Flows, Inthispaper(PartI),weseektodeveloplow-complexityal-
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Struture/graph Learning, Voltage measurements, Transmission gorithmstolearnthecurrentoperationalstructurein‘radial’
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Lines.
4 distributionnetworksusingonlynodalmeasurements.Nodal
0 measurements may include voltage magnitudes, voltage phase
1. I. INTRODUCTION (potentially), and power injections and are typically available
0 at smart meters, pole-mount or pad-mount transformers, and
5 The power grid is composed of a network of transmission distributionphasormeasurementunits(PMU).Accuratestruc-
1 and distribution lines that enable the transfer of electrical
turalestimationimpactsmanyimportantapplicationsincluding
: power from generators to loads. The design, operation and
v failure identification [3], outage management, and recovery
i controlofthesenetworksistypicallyhierarchicalwithamajor following major and minor disruptions (e.g. hurricanes to
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division occurring between the transmission network of high
individuallightningstrikes),gridreconfiguration[4]forpower
r voltage lines connecting sub-stations and power plants, and
a flow optimization and generation scheduling [1], [4]–[6], and
the distribution network of medium and low voltage lines that
quantifying the need for additional meter placement. From
connect the transmission sub-stations to the end-users. Here,
an adversarial viewpoint, our work can be viewed as low-
we focus on distribution networks.
intrusionlearningbyarogueagentinterestedinestimatingthe
Thedesignofdistributionnetworksmayappeartobeloopy
grid structure for a data attack [7], [8]. In the subsequent part
or meshed, however for practical engineering concerns, the
(PartII),wewilllookatdevelopingalgorithmsthatareableto
vast majority of distribution grids are operated as ”radial”
estimate the statistics of power consumption at the grid nodes
networks, i.e. as a set of non-overlapping trees. Switches
or estimate the parameters of operational lines in addition to
in the network are used to achieve one radial configuration
determiningthegrid’sradialstructure.Further,wewillanalyze
out of many possibilities. Each tree in the network has a
learningtheoperationalgridstructurewithmissingdata,where
observations from a subset of nodes are not available.
D. Deka is with the Department of Electrical and Computer Engineering,
The University of Texas at Austin, Austin, TX 78712. Email: deepjy-
[email protected]
A. Related Work
S.BackhausiswiththeMPADivisionofLANL,LosAlamos,NM87544.
Email:*[email protected]
Our work falls in the broad category of ‘graph learning’
M. Chertkov is with the Theory Division and the Center for Nonlinear
SystemsofLANL,LosAlamos,NM87544.Email:*[email protected] problemsthathavebeenapproachedfromdifferentdirections.
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For general graphs and graphical models [9], maximum-
likelihood structure estimation has been researched in several
papers by utilizing prior information such as sparsity of the
parameter space [10], [11], size of the graph neighborhood
[12],etc.Techniquesemployedincludebothtraditionalconvex
optimization [10], [13] as well as greedy learning [12]. For
power grids, structure estimation techniques discussed in the
literature can be classified based on the type of measurements
availableaswellasassumptionsmaderegardinggridstructure
and user behavior. In [14], a maximum likelihood estimator
(MLE) with regularizers for low-rank and sparsity is used
to recover the grid structure using locational marginal prices
(LMPs). In [15], a model using bus phase angles as a Markov
randomfieldfortheDCpowerflowbuildsadependencygraph
Fig. 1. Schematic layout of an example of a distribution grid fed from 4
based approach to detect faults in grids. In work specific to
substations,withsubstationsrepresentedbylargerednodes.Theoperational
radialdistributiongrids,[16]providesastructureidentification gridhasabase-4spanningforestconfigurationthatisformedbysolidlines
algorithm that uses signs within the inverse covariance matrix (black).Dottedgreylinesrepresentopenswitches.Eachtreeintheforesthas
onesubstationattheroot.Theloadnodeswithineachtreearemarkedwith
(orconcentrationmatrix)ofvoltagemeasurementstogenerate
thesamecolor.
a minimum spanning tree. In [3], topology identification with
limited measurements in a distribution grid with Gaussian
loads is used to design a machine learning (ML) estimate spanning forest, by configuring switches as shown in Fig. 1.
with approximate schemes. Our work uses ordering of second There are exponentially many (in the number of switches)
moments, not a ML approach, to reconstruct a radial grid possible configurations of spanning forests. The grid-graph
sequentially from the leaves to the root, making it distinct with all the switches closed is denoted G=(V,E), where V
frompreviouswork.Ouralgorithmdesignisbasedonalinear is the set of nodes of the graph and E is the set of undirected
coupled approximation for lossless AC power flow that is edgesofthegraph.WedenotenodeswithsingleRomanletter
idealized but practical [17], [18] for analyzing distribution subscripts a and undirected edges with pairs of Roman letter
grids where the line and voltage characteristics limit the subscripts (ab). The operational grid is a forest denoted by F
accuracy of traditional approximations. Unlike related work, which spans all the nodes in V. Specifically, F is a special
our topology learning algorithm is agnostic to the load profile subgraph of G (F⊂G) such that
distributionsorvariabilityinlineimpedancesandrequiresonly • F is a union of K non-overlapping trees covering all the
alessrestrictiveassumptiononthecorrelationofloadprofiles. nodes of the graph
The rest of the paper is organized as follows. We start the • Each tree contains exactly one of the K bases (substa-
next section with a description of the distribution grid and tions), F=∪ T .
k=1,···,K k
summaryofthelearningproblemsdiscussedinthemanuscript. The distribution system F is a ‘base-constrained spanning
Section III and Appendix A describe the linear coupled (LC) forest’ with operational edges EF where EF ⊂ E. Table I
powerflowmodelanditsspecialcase,theDC-resistivepower
provides other relevant notations (nomenclature) used through
flowmodel.Statisticaltrendsinobservednodalmeasurements
out this manuscript to denote various nodal and edge features
arediscussedinSectionIII-C.Next,weusethederivedresults of the grid G and the operational forest F.
todesignalgorithmsforlearningthedistributiongridstructure
using the power flow models in Section V. Simulation results
B. Problem Formulation and Contribution
elucidating the performance of our algorithms on test distri-
bution grids are presented in Section VI. Finally, Section VII We consider large distribution grids where the utility (ob-
concludes and suggests future directions that will be explored server) is unsure of the grid configuration because of insuffi-
in a subsequent work. cient or inaccurate switching data, perhaps caused by a recent
systemupset.Alternatively,wecouldtakethepointofviewof
II. TECHNICALPRELIMINARIES athirdpartyobserver,whomaybeanaggregatororadversary,
tryingtoextractthecurrentforestconfigurationfromavailable
The structure of a radial distribution network has important
nodal measurements. We assume the current spanning forest
featuresthatmotivatesouralgorithmdevelopment.Wediscuss
configuration is kept intact sufficiently long for load profiles
the radial structure in detail here and introduce the notation
at grid nodes to attain a steady distribution (longer than the
used in this paper. We then formulate the learning problem
fluctuations but shorter than changes in the mean load).
tackledinthispaperintermsofitsinputdataanddeliverables
We assume that the observer has access to nodal measure-
and discuss the underlying motivation.
ments, but not edge measurements – an assumption consistent
with the recent expansion of smart grid monitoring devices.
A. Structure of Radial Distribution Network
Smart meters generally provide nodal voltages and power
We consider a meshed distribution network which is op- injections at fine spatial resolution, i.e. at the individual
erated as a union of non-intersecting ‘radial’ trees, i.e. a customer level, but they do not provide any edge flow data.
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TABLEI
NOTATIONTABLE III. POWERFLOWMODELSANDSTATISTICAL
CORRELATIONS
Our approach to the structure learning problem relies on
F∈G aofptahreticpuhlyasricfoalredsitstcroibnufitgiounrantieotnworkG linearized PF models on radial spanning forests that enable
V vertexsetofG efficient reconstruction of the grid structure via a second-
N #ofnodesotherthansub-stationsinV
moment analysis. The most general of the two, termed the
K numberofsub-stationsinthenetwork
E edgesetofG Linear Coupling (LC) model, ignores losses of active and
EF setofedgesoperationalwithinF reactive powers and consistently assumes small voltage mag-
treewithinF containing
Tk∈F thekth sub-station nitude and phase drops between connected nodes. For tree-
Mk reducedincidencematrixofthetreeTk like distribution grids, the LC-PF model becomes equivalent
ETk setofedgesinTk to the LinDistFlow PF model in [17], [18]. The second
VTk setofnodesinTk modelconsideredinthepaper,coinedtheDC-resistivemodel,
subsetofedgesfromETk s.t.each
pathbetweena,b∈VTk nodewithedgeinthesubset,except corresponds to the special resistance dominating case of the
aandb,contributesexactlytwoedges LC-PF model. These PF models are described in more detail
EaTk pathfromatoslackbusinTk in Appendix A.
bisadescendantofa acontributesETk
b
bistheparentofa (ab)∈ETak andbcontributesETak A. Linear Coupled Power Flow (LC-PF) model
θa,va voltagephaseandmagnituderesp.atbusa
θ,v Vectorofnon-substationvoltage As noted in Eqs. (25,26) in Appendix A, the LC-PF model
phasesandmagnitudesresp.
isderivedfromthegeneralACpowerflowmodelbyassuming
εa =1−va,voltagedeviationatnodea
ε =1−v,voltagedeviation small voltage magnitude deviations and phase differences
pa,qa rinesjepc.tiaocnti/vceonasnudmrpetaioctniv(e+p/o−w)eartbusa between neighboring buses in the grid. It is convenient to
restate the linear equations in LC-PF model in matrix form
Vectorofnon-substation
p,q
powerinjections/consumptions as:
susceptance,conductance,resistance,
βab,gab,rab,xab reactanceresp.ofedge(ab) p=H ε+H θ, q=H ε−H θ (1)
g β β g
diagonalmatrixoflinesusceptances,
β,g,r,x
conductances,resistances,reactancesresp. where p,q,ε and θ are defined in Table I. H and H are the
Σy matrixofsecondmomentsforvariabley weighted graph Laplacian matrices associategd withβforest F
Ωy matrixofcovariancesforvariabley
µy vectorofmeansforvariabley such that
HDMyTak Dseerwtesoidctfeuhncuedenddaognbwetsseew(irgivenhiecgtdlehudtndsoiLnidnagepsyliatsceilafn)omfaatriixnTk Hg(a,b) =0−∑cg:(aab,c)∈EFgac oiiffthb(ea=rbw)ai∈seEF (2)
H has a similar structure with g-weights replaced by β-
β
weights.TheweightedgraphLaplacianscanbestatedinterms
Additionalinstrumentationisemergingforpole-mountorpad-
of the directed incidence matrix M as
mount transformers [19], however, these new devices still
only provide nodal voltages and aggregated customer power H =MTgFM, H =MTβFM (3)
g β
injections.1 Some edge flow data is available to utilities,
however, this is generally at a few select locations in the Here, gF and βF are diagonal matrices representing, respec-
tively, line conductances and susceptances for edges within
distribution grid, e.g. at the substation/root node, voltage
F. M is the edge to node directed incidence matrix of F.
regulators, reclosers, or other major utility equipment. These
See Fig. 2 for an example. Every row m in M is equal to
select locations may also have nodal and edge data from ab
±(eT−eT) and represents the directed edge (ab), where the
anotheremergingtechnology,i.e.distributiongridPMUs[20]. a b
direction of an edge is chosen arbitrarily. e ∈RN+k is the
However,wecontinuetorestrictourinputdatatonodalvalues, a
standard basis vector associated with the vertex a, with 1 at
which is consistent with the new, ubiquitous sensing provided
the ath position and zero everywhere else. We can combine
by smart meters.
Eqs. (1) and express the complex power flows as:
The nodal devices provide the observer with temporal
samples of the nodal voltage magnitudes. The observer seeks p+iˆq=MT(gF+iˆβF)M(ε−iˆθ) (4)
to use these samples to learn the current configuration of
switchesthatdeterminethe‘base-constrainedspanningforest’. Both Hβ and Hg are weighted graph Laplacians and are
To supplement the voltage magnitude samples, the observer degenerate—showingK zero-eigenvaluesassociatedwiththe
has historical information about statistics of the nodal con- freedom in fixing phase and voltage deviation (from nominal)
sumption. at any node within each tree of the forest. It is natural to fix
phases and voltages at the sub-stations making these ‘slack
buses’ a for trees T of the (operational) forest, F such that
1We use the term ‘power injection’, ‘power consumption’ and ‘load’ k k
θ =ε =0, for any k, 1≤k ≤K. Formally, elimination
interchangeablytodenotethepowerprofileateachinterior(non-substation) ak ak
nodeofthedistributionsystem. of the set of K sub-stations corresponds to elimination of K
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a
components from all the vectors contributing Eqs. (1), and re-
2 1
ductionofKrowsandKcolumnsfromtheweightedLaplacian
matrices. All the eigenvalues of the resulting reduced graph c b nodes: a b c d e dges
Laplacian matrices are thus strictly positive. 1 -1 0 0 1
3
Without loss of generality, we will use the same notation 1 0 -1 0 2
for the original and reduced dimension variables θ,ε,p and q Slack Bus d 0 1 0 -1 3
andalsorefertoEqs.(1,4)asappliedtothereducedvectorsof
(a) (b)
dimension N×1. We will also keep notations, H and H for
β g
the reduced graph Laplacian matrices, and M for the reduced nodes: a b c edges edges: 1 2 3 nodes
incidence matrix respectively. The reduced M has a block 1 -1 0 1 1 0 1 a
diagonal structure: M =diag(M1,M2,···,MK), where, Mk is Mk = 1 0 -1 2 Mk-1 = 0 0 1 b
the invertible reduced incidence matrix of tree Tk in F. Thus, 0 1 0 3 1 -1 1 c
M andcorrespondinglyH andH arefullrank,invertibleand
β g
(c) (d)
block-diagonal matrices. Inverting the linear non-degenerate
Fig. 2. Illustration of the reduced incidence matrix construction on the
Eqs. (1) we arrive at exampleofatreewith4nodes.(a)Treegraphwithfournodes(a,b,c,d)and
three edges (1,2,3). (b) Complete directed incidence matrix. The “directed”
θ= M−1xFM−1Tp−M−1rFM−1Tq=H−1p−H−1q (5) freedominchoosingedgeorientationsisfixedasfollows,1=(ab),2=(ac),
1/x 1/r 3=(bd).(c)Thecolumncorrespondingtotheslackbus(busd)isremovedin
ε= M−1rFM−1Tp+M−1xFM−1Tq=H−1p+H−1q (6) thereducedincidencematrixMk.(d)Inverseofthereducedincidencematrix
1/r 1/x Mk.
whereH =. MTrF−1MandH =. MTxF−1M.rF andxF are
1/r 1/x IV. TRENDSINSECONDMOMENTSOVERTREE
diagonal matrices representing, respectively, line resistances
and reactances within the forest F. Their relation to gF and NETWORKS
F
β are expressed in Eqs. (28). We now derive key results related to the second moments
in voltage magnitudes that arise from the properties of the
forest F. We denote the unique path from node a to the slack
B. Relations between second moments bus in tree T by ETk. From [21], the inverse of the reduced
k a
The real and reactive nodal power injections p and q in incidence matrix of a tree has the following special structure:
Eqs. (5,6) fluctuate because of exogenous processes, and their
second moEm[eθnθtTs]a=reHr1−e/l1xaEte[dppbTy]:H1−/1x+H1−/1rE[qqT]H1−/1r 1 aifloendggeparth∈fEroTamk isadtoiresclatecdk bus,
⇒ Σθ=H−1−/1HxΣ1−p/1xHE1−[/p1xq+TH]H1−1/−1/r1rΣ−qHH1−1/−1/r1rE[qpT]H1−/1x Mk−1(a,r)=−01 iaiffgaeeiddnggseet prra∈(cid:54)∈thEEfTaTrkkomisadirtoecsteladck bus, (11)
(cid:104) (cid:105)T a
−H−1Σ H−1− H−1Σ H−1 (7)
1/x pq 1/r 1/x pq 1/r
Here, the direction of edge r = (cd) is specified by its
Similarly, Σ =H−1Σ H−1+H−1Σ H−1
ε 1/r p 1/r 1/x q 1/x representative row mcd in the directed incidence matrix. For
(cid:104) (cid:105)T example, if m =eT−eT, the edge is directed from node c
+H−1Σ H−1+ H−1Σ H−1 (8) cd c d
1/r pq 1/x 1/r pq 1/x to node d, whereas for m =eT −eT, the direction is from
cd d c
Σ =H−1Σ H−1−H−1Σ H−1 node d to node c.
θε 1/x p 1/r 1/r q 1/x
+H−1Σ H−1−H−1Σ H−1 (9) Animmediatecorollaryof(11)isthatM−1(a,r)=0ifedge
1/x pq 1/x 1/r qp 1/r r and node a lie on separate trees within the forest F, a fact
consistent with the block diagonal structure of M. Using (11)
These formulas are the basis for reconstruction/learning anal-
ysis in the rest of the paper. in Hg−1=M−1gF−1M−1T, we derive for forest F
H−1(a,b)=0 if a,b are on different trees T and (12)
g k
C. DC-resistive model H−1(a,b)= ∑ M−1(a,r)gF−1(r,r)M−1(b,r) if a,b∈T
g k
TheDC-resistivePFmodel(seeAppendixA)isanextremal r∈ETk
case of the LC-PF model realized when line reactance can be 1
= ∑ if a,b∈T (13)
ignored in comparison with resistance (x/r→0). The relation g k
between the statistics of active powers and voltage second- (cd)∈ETak(cid:84)ETbk cd
order moments deviations reduces to
Thus, H−1(a,b) is equal to the sum of the inverse conduc-
g
E[ppT]=H E[εεT]H ⇒ Σ =H−1Σ H−1 (10) tances of lines that are common to the paths from both nodes
g g ε g p g
totheslackbus.Ifnosuchlineexists,thecorrespondingentry
in H−1 is 0. See Fig. 3(a) for illustration. Similar results hold
g
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consumes strictly more than it produces in active and reactive
0 powers,i.e. p <0,q <0.Thisassumptioniscertainlytruein
a a
anydistributiongridwithsmalland/ormoderatepenetrationof
(𝑒0)
renewables [22]. However, the assumption is also reasonable
𝑒
for a system with significant penetration of generation which
(𝑏𝑒)
is still dominated in average by the consumption.
𝑏 𝑏
(𝑎𝑏) (𝑑𝑏) (𝑎𝑏) Theorem 1. If node a(cid:54)=b is a descendant of node b within
tree T , then for the DC-resistive model, Σ (a,a)>Σ (b,b).
𝑎 𝑎 k ε ε
𝑑 𝑑
𝐷𝑎𝑇𝑘 Proof: We first show that for any node a and its par-
𝑐 𝑐 ent b, Σε(a,a)>Σε(b,b). Consider Σε(a,a)−Σε(a,b). From
Eq. (10), we derive
(a) (b) Σ (a,a)−Σ (a,b)=∑H−1(a,c)Σ (c,d)
ε ε g p
c,d
rFeipgr.e3s.entSedchbeymlaatricgelaryeoduntoodfeaidsisthtreibsulaticoknbgursid.(tare)eDTokt.teTdhleinseusb-rsetparteiosennntotdhee (cid:0)Hg−1(a,d)−Hg−1(b,d)(cid:1) (16)
paths from nodes a and d to the slack bus. The common edges on those
1
paths are edges (be) and (e0). ThTus, Hg−1(a,d)=1/gbe+1/ge0. (b) the set (using Lemma 1)=∑Hg−1(a,c)Σp(c,d)g 1(d∈DTak)
ofdescendantsofa,denotedbyDak.Here,nodesaandcaredescendantsof c,d ab
bothnodesaandb,whilenoded isnotadescendantofabutonlyofnode
⇒Σ (a,a)−Σ (a,b)>0 (using Assumption 1) (17)
b. ε ε
1
Also,Σ (a,b)−Σ (b,b)=∑H−1(b,d)Σ (c,d) >0 (18)
for other measurement matrices like ε ε g p g
T ab
H−1(a,b) =(cid:40)∑(cd)∈ETak(cid:84)ETbkrcd if nodes a,b∈Tk(14) Combining Eqs. (17)c∈aDnadk,d(18) we derive Σε(a,a) >
1/r 0 otherwise, Σε(a,b)>Σε(b,b). Since node a is a descendant of node b,
there is a path a,c ,...c ,b, such that each node in the path
T 1 r
Let Dak denote the set of descendants of node a within the is a parent of its predecessor. Then, we derive Σε(a,a) >
tree Tk. We call b a descendent of a, if a lies on the (unique) Σε(c1,c1)>...>Σε(cr,cr)>Σε(b,b).
path from b to the slack bus of Tk, also including a itself in The following theorem is the LC-PF version of Theorem 1.
the set of its descendants. We call b the parent of a within T
k
Theorem 2. If node a(cid:54)=b is a descendant of node b on tree
(there can only be one) if (ab)∈T and a is an immediate
k T in forest F, then Σ (a,a)>Σ (b,b) in the LC-PF model.
descendant of b as illustrated in Fig. 3(b). k ε ε
The following statement holds. Proof: Consider Eq. (8). Notice that the right hand
side has four constituent terms (H−1Σ H−1, H−1Σ H−1,
Lemma 1. For two nodes, a and its parent b, in tree T 1/r p 1/r 1/x q 1/x
k
H−1Σ H−1 andH−1Σ H−1).Wedenoteeachoftheseterms
Hg−1(a,c)−Hg−1(b,c) =(cid:40)g1ab if node c∈DTak (15) by1/Σrεjpwqhe1r/ex j∈{1,12/x,3q,4p}.1F/roreachindividualterm,applying
0 otherwise, Assumption 1 and the analysis in Theorem 1, we find that
Σj(a,a)>Σj(b,b) if node a is a descendant of node b, other
Proof: For any node c which belongs to a tree not ε ε
than itself. Thus, the statement also holds for the sum.
containingnodesaandb,H−1(a,c)−H−1(b,c)=0according
g g We now focus on evaluating the term E[(ε −ε )2], which
to (12). Now, focus on nodes contained, together with a and a b
b,withinT .Sincebisa’sparent,ETk =ETk(cid:83){(ab)}andwe is the expected value of the squared difference between two
k a b node voltage deviations (ε). For any two nodes a and b, the
derive(alsovalidatingontheillustrativeexampleinFig.3(b))
DC-resistive model yields:
that for any node c in tree T ,
k
ETk(cid:92)ETk =ETk(cid:92)ETk if node c(cid:54)∈DTk E[(εa−εb)2]=Σε(a,a)−Σε(a,b)−Σε(b,a)+Σε(b,b)
a c b c a (using (16))=∑H−1(a,c)Σ (c,d)(cid:0)H−1(a,d)−H−1(b,d)(cid:1)
ETak(cid:92)ETck =[ETbk(cid:92)ETck](cid:91){(ab)} if node c∈DTak c,d g p g g
−∑H−1(b,c)Σ (c,d)(cid:0)H−1(a,d)−H−1(b,d)(cid:1)
resulting in Eq. (15). g p g g
c,d
We now prove our main results regarding trends in second
=∑(cid:0)H−1(a,c)−H−1(b,c)(cid:1)Σ (c,d)
moments of deviations in voltage magnitudes (ε) along any g g p
c,d
treeinthenetwork.Theresultsareconditionedonthefollow-
ing assumption regarding correlations in power injections at (cid:0)Hg−1(a,d)−Hg−1(b,d)(cid:1) (19)
the non-substation buses. Our next Lemma follows directly by applying Lemma 1 to
Assumption 1: For any two buses a and b drawing power Eq. (19).
fromthesamedistributionsub-station,Σ (a,b)>0,Σ (a,b)>
p q
Lemma 2. For two nodes, a and its parent b belonging to
0,Σ (a,b)>0.
pq
tree T , in the DC-resistive model, we derive E[(ε −ε )2]=
Note that this assumption holds, in particular, if the overall k a b
node balance is such that each non-substation node a always ∑c,d∈DTak g12abΣp(c,d)
6
For the LC-PF model, we evaluate the expression E[(ε − Algorithm1BaseConstrainedSpanningForestLearning:LC-
a
ε )2] as well. In this case, for two nodes, a and its parent b, PF Model
b
that lie on tree T , we arrive at Input: True Σp,Σq and Σpq, m voltage deviation observations
k
E[(ε −ε )2] =Σ (a,a)−Σ (a,b)−Σ (b,a)+Σ (b,b) εj,1≤ j≤m, all line resistances r and reactances x
a b ε ε ε ε
wsyhmermeetΣrεicistergmivsenH−by1ΣEHq.−(18)a.nLdeHt −Σ11εΣanHd−1Σ2εrerseppercetsievneltythine 1: Compute Σε(a,a)=∑mj=1εajεaj/m for all nodes a.
1/r p 1/r 1/x q 1/x 2: Undiscovered Set U ←{1,2,...,N+K}, Leaf Set L←φ,
Σ . Extending the result of Lemma 2, we derive
ε Descendant Sets D ←{a}∀ nodes a.
a
Σ1(a,a)−Σ1(a,b)−Σ1(b,a)+Σ1(b,b)=∑r2 Σ (c,d) 3: while (U (cid:54)=φ) do
ε ε ε ε c,d∈DTaakb p 54:: fbo∗r←alml aax∈b∈LUdΣoε(b,b)
(20)
Σ2ε(a,a)−Σ2ε(a,b)−Σ2ε(b,a)+Σ2ε(b,b)=c,d∑∈DTaxka2bΣq(c,d) 67:: xa2bΣq(ci,fd)D+∑rm2ja=rw1a(beεxdaajbg−Σepbqεe(bjct∗w,)d2e/)emnthne=onde∑sca,d∈aDnadrba2b∗Σp(c,d) +
(21) 8: Db∗ ←Db∗(cid:83)Da
Similarly, let Σ3 = H−1Σ H−1 and Σ4 = Σ3T = 9: L←L−{a}
ε 1/r pq 1/x ε ε 10: end if
(cid:104) (cid:105)T
H−1Σ H−1 , the non-symmetric terms in Σ . Using 11: end for
Lem1/mr apq1 w1/ixth Eq. (19), we get ε 12: L←L(cid:83){b∗}
13: end while
Σ3(a,a)−Σ3(a,b)−Σ3(b,a)+Σ3(b,b)= ∑r Σ (c,d)x
ε ε ε ε ab pq ab
T
c,d∈Dak
=Σ4(a,a)−Σ4(a,b)−Σ4(b,a)+Σ4(b,b) (22) Step 6 adds edges between node b∗ and nodes in set L of the
ε ε ε ε
growingtreeusingLemma3.Intheidealcasewheninfinitely
CombiningEqs.(20,21,22)wearriveatthefollowingLemma.
many voltage magnitude samples are collected, second-order
Lemma 3. In the LC-PF model, E[(ε − ε )2] = momentsofthepowerinjectionssatisfytherelationinLemma
a b
∑c,d∈DTakra2bΣp(c,d)+xa2bΣq(c,d)+2rabxabΣpq(c,d), holds for 3p.reHseonwceevoefr,anweedgheaviesdaetfienrmiteinenduminbAerlgoofristhammp1lebsy. Tchheucskitnhge
a node a and its parent b belonging to the (operational) tree
T . if the relative difference between the reals on the left and
k
right sides of the condition in Step 6 is less than a predefined
tolerance, τ:
V. LEARNINGSTRUCTUREOFBASE-CONSTRAINED
SPANNINGFOREST 1−(cid:12)(cid:12)(cid:12) ∑mj=1(εaj−εbj∗)2/m (cid:12)(cid:12)(cid:12)<τ
Here, we propose Algorithm 1 to learn the structure of (cid:12)(cid:12)∑c,d∈Dara2bΣp(c,d)+xa2bΣq(c,d)+2rabxabΣpq(c,d)(cid:12)(cid:12)
thedistribution networkusing propertiesof voltagedeviations (23)
for the LC-PF model. The polynomial time algorithm, based
Steps 9 and 12 update the set of current leaves L before
on the Theorems proved in the previous section, requires
repeating the reconstruction steps with a new undiscovered
positivity of the correlation between nodal power injections
node.
(Assumption 1). The Algorithm is also agnostic to the prob-
ability distribution of active and reactive power injections.
AlgorithmComplexity:Ignoringcomplexityofcomputing
To reconstruct the Base-Constrained Spanning Forest (F =
the second moments in Steps 4 and 6 (part of the data
∪ T ), Algorithm 1 takes as input ‘m’ measurements
k=1,···,K k pre-processing), there are N+K steps (N load nodes and
of nodal voltage deviations. These measurements are used to
K substation nodes) in the ‘while’ loop, and at most N
create the empirical voltage deviation second moment matrix
j comparisons in the ‘for’ loop for each node. Therefore, the
Σ . The voltage deviations ε at the substation nodes are
ε a worst-case complexity of this algorithm is O(N2+NK).
assumed to be zero which implies the elements of the row
Note that we use LC-PF model in Algorithm 1. To design
Σ (a,:)=0 for each of the K substation nodes. The observer
ε the DC-resistive version of Algorithm 1, the condition in Step
has prior information (or estimates using power injection
6 should be replaced with the result in Lemma 2. (Required
measurements) for the true second moment matrix of power
modifications are straightforward and thus their description is
injections Σ for non-substation nodes.
p omitted.)
Algorithm1Overview:Wereconstructeachtreewithinthe
distribution grid forest sequentially moving from the leaves to
the root nodes. At every stage,U represents the set of undis- VI. EXPERIMENTS
covered nodes that are not part of the current reconstructed We perform a set of numerical experiments to test and
tree while L represents the set of ‘current leaves’ (nodes that demonstrate the performance of Algorithm 1 in extracting
are in the current reconstructed tree but with undiscovered the operational radial forest F from meshed “as-designed”
parents).Ateachiteration,Step4selectsthenodeb∗ fromset distributionnetworksG.Weremindthereaderthattheobserver
U with the largest second moment of voltage deviation. Next in Algorithm 1 has information of the full graph G, the
7
impedance (resistance and reactance) of all lines (operational
or open) as well as the number of connected substation buses. SS SS SS
Further, true second moments of active and reactive power
injections at each non-substation node are assumed to be
known. The set of measurements available as input with the 11 55
10
observer comprises of deviations in voltage magnitudes (ε) at
the grid nodes. 22 88 66 77 1111
Table II summarizes the distribution grid test systems by
99
the number of load busses, number of substation busses, and
the number of tie switches. Additional information on these 33 44 1133 1122
test systems [23]–[25] can be found online at [26]. In normal
operation, each test grid consists of nodes in a forest- or tree- (a)
like configuration F with open tie-switches. We construct the
complete meshed network G by closing all the tie-switches.
12 13 14 15 27 28 29
S
To test the scalability of our algorithms, we increase the
numberofpossibleforestconfigurationbyintroducingseveral 1 2 3 4 5 6 7 8 9 10 11
additional non-operational lines into each system as noted in
16 17 18 19 20
TableII.AlthoughthesecontributetoG,theyarekeptopenand
donotcontributetopowerflowsintheoperationforestF.The 21
impedances(reactancesandresistances)oftheadditionallines 22
are generated by assigning random values uniformly between
23
the minimum and maximum impedances of the operational
lines. Fig. 4 displays the test networks G (solid and dashed 26 25 24
lines) and the respective operational forests F (solid lines
(b)
only).
TABLEII S 8 S
SUMMARYOFTHETESTEDDISTRIBUTIONGRIDS 1 2 3 4 5 6 7 9 55 54 53 52 51 50 49 48 47
S 10 S
Test Numberofbuses/substa- Non-operational Source
11 12 13 64 63 62 61 60 59 58 57 56
Case tions/tie-switches linesadded
S 14 22 S
bus 13 3 13/3/3 10 [23]
15 16 17 18 19 20 21 23 24 72 71 70 69 68 67 66 65
bus 29 1 29/1/1 20 [24]
bus 83 11 83/11/13 30 [25] S S
25 26 27 28 29 76 75 74 73
S 39 40 S
30 31 32 33 34 35 36 37 38 41 42 83 82 81 80 79 78 77
For each numerical experiment on a grid from Table II, we
S
pickanoperationalspanningforestF byopeningtieswitches. 43 44 45 46
We also choose the statistics of the injections at each node
(c)
usingGaussiandistributions,unlessotherwisespecified.These
Fig.4. Layoutsofdistributiongrids,alsoshowingoperationalcasestested,
distributions are used to generate multiple power injection inaccordancewithdescriptioninthetextandsummaryinTableII.Thered
samples,andfromeachvector-valuedsample,wesolveaPFto circles represent substations (marked as S). The blue circles represent load
nodesthatarenumbered.Blacklinesrepresentoperationaledges,whiledotted
computevoltagesandphasesateverynodeinthenetworkwith
bluelinesrepresentopentie-switches.Theadditionallinesarerepresentedby
thevoltagesatthesubstationsfixed(i.e.theyareslackbusses). dottedgreenlines.(a)bus 13 3testcase(b)bus 29 1testcase(c)bus 83 11
Averaging over all PF solutions, we compute empirical cor- testcase
relations of voltage magnitudes and phases. Using only these
correlations, we run our algorithms and compare the resulting ponentially with the number of samples used by the observer
reconstruction with the actual operational configuration. In in the learning algorithm. For the tolerance values considered
the reconstruction, we assume the observer has access to the here, the majority of structural errors arise due to connected
resistance and reactance of all the lines in G. All powers and nodes not satisfying condition (23) and hence being labelled
voltages are presented in per unit (p.u.) values. as open. The decay is then intuitive as an increase in the
Figs. 5(a)-5(c) display the accuracy of Algorithm 1 for sample size makes empirical moments in voltage magnitudes
the different test grids from Table II. The relative error is approximate their true values better which in turn leads to an
defined to be the number of mislabeled lines (connected increaseinthenumberofoperationallinessatisfying(23)and
when actually open and vice versa) divided by the size of being correctly identified. On the other hand, if a sufficiently
the operational edge set. The relative error is averaged by large value of τ is used, condition (23) will be relaxed and
computing many reconstructions using the same nodal power possibly be satisfied even by unconnected nodes. In such a
injection distributions. Different curves (colors) in Fig. 5(a) case, a majority of errors will be recorded due to open lines
show the effect of changing the tolerance τ in Eq. 23. beingincorrectlylabelledasoperational.Aserrorsofthistype
The average fractional error in Figs. 5(a)-5(c) decays ex- (open edges classified as operational) does not improve with
8
the number of measurement samples, the average fractional
0.7
errorswillnotdecaywiththesamplesize.Thisiselucidatedin tolerance 10−1.5
Fig. 6, where the bus 13 3 structure is learnt, in the presence 0.6 tolerance 10−1.7
of 50 non-operational lines. Note that for larger values of 3 tolerance 10−2
_
τ in Fig. 6, the errors do not decay with the sample size 130.5
wdinihsdAceeurecesdcasusiproafrnoete.VfreFsIsrmaIotr.bruallcCellateuOarrgrsNvaealCorlbLuessUseaetsSmir,mvIptOehaldNeetiySoisnnd&izoFeoPiasfgs,A.dTja6uiHs.ssttrFmiifibOaeuRldltWeioirnAnvRthagDelruipderseocifsedτiimnigs- average relative error in arning grid structure in bus_000...234
e
portant to many applications including failure identification, l
0.1
power flow optimization and estimation of state variables. In
this manuscript, we have developed algorithms to learn the
0
structureofaradialdistributiongridusingobservednodalvolt- 20 40 60 80 100 120 140 160 180 200
number of measurement samples
agemagnitudemeasurements.WehaveusedaLinearCoupled
(a)
(LC)approximationthatrelatescomplexnodalvoltagestothe
complex power consumptions to prove that second moments
0.8
ofnodalvoltagemagnitudesinradialdistributiongridsfollow
tolerance 10−1.5
certain statistical structure/ordering. Our algorithm relies on 0.7 tolerance 10−1.7
these results to reconstruct the operational tree in a bottom up tolerance 10−2
fashion – starting from the leaves and progressing to the root 9_10.6
2
omaothtfnoelTdtdyhpiheseoealtrssigpiissbtrruiiuipvdmmtri.iatoapycrnttyioiocgfbnarenlidnuoaesnsnefi-vodctosderlneotestafgrgaoaaelrnudcmdrionaeirpgnarpedstlurihavorteieaidomcsuhnteaaanlotritfdesstenaitvcworisedcoaee-olfsafo.lsolitSdlhaye.decFaoplviornrasoaditdlfi,,aslobtehuliseesr, average relative error in earning grid structure in bus_0000....2345
which is natural for most distribution grids and more general l
than other assumptions discussed in the literature. We tested 0.1
our algorithms on sample distribution grids and observed an
0
exponential decay of average errors with increasing number 20 40 60 80 100 120 140 160 180 200
number of measurement samples
of measurement samples. In the next part, we extend the
(b)
work in this paper along the following directions: coupling
structure learning with estimation of load statistics in the grid
0.8
or estimation of line parameters in the grid, and learning grid
tolerance 10−1.5
structureevenwhenmeasurementdataismissing.Developing 0.7 tolerance 10−1.7
general algorithms for learning distribution grid operational 1 tolerance 10−2
forests that are not restricted to linearized power flow models 3_10.6
remains a potential future direction of research. e error in e in bus_80.5
A.NBoatseicthPatowtheeroFpleorwatsionaAlP‘PbEasNeD-cIoXnstrainedspanningforest’ average relativng grid structur00..34
ni
Fcanbethoughtofasaspanningtreeoveranextendedgraph, ear0.2
l
where an (artificial) super node is introduced and connected
0.1
with (artificial) lines to all the sub-station nodes. This trick
allows, without a loss of generality, to limit our discussion in 0
20 40 60 80 100 120 140 160 180 200
the following to spanning trees, thus replacing F by T. number of measurement samples
Let z=(zab=rab+ixab|(ab)∈E) as the vector of complex (c)
line impedances in the grid. (i2=−1.) Expressed in terms of Fig.5. Averagefractionalerrorsplottedagainstthenumberofmeasurement
the complex powers and potentials (voltages and phases) the samples for the LC-PF model with different tolerance thresholds - corre-
spondent to the Algorithm 2.(a), (b) and (c) show results for the bus 13 3,
Kirchoff laws over the operational (spanning tree) configura-
bus 29 1andbus 83 11modelsrespectively.
tion T become
v2−v v exp(iθ −iθ )
∀a∈V:Pa=pa+iqa= ∑ a a b z∗ a b where the real valued scalars, va and θa, characterize voltage
b:(ab)∈ET ab magnitude and phase respectively at node a. We assume that
(24)
the power within the system considered is balanced through
9
0.08 in comparison with the resistance, ∀{a,b} : βab (cid:28) gab,
we arrive at the following resistance-dominating version of
0.07 Eqs. (25,26)
average relative error in ning grid structure in bus_13_30000....00003456 ttttoooolllleeeerrrraaaannnncccceeee 11110000−−−−..1102.42 WlaD∀acatCetii∈ovmwneVioiplsdl:oepwsclioa,meiu≈rnislfleabtohdr:(wiatsp∑bos)r∈aiatmpEhrpeTearrgrotieralxyalbiad(mtfεieotaadirot−itntoroaεnablnp)DsDh,maqCCsai-esr≈flseaiosonbwing:s(tlainem∑bves)e∈o.twEdTPTeohFlrge.k[aIs2tbtr,(7saθr]defbioqwt−iruohmiθnerueraaes-)l
ear0.02 line inductances to dominate resistances, which is seldom
l
observed in distribution grids.
0.01
0
10 20 30 40 50 60 70 80 90 100 D. From Power Flows to DistFlow and LinDistFlow
number of measurement samples
The DistFlow Eqs. introduced by Baran and Wu in [17],
Fig. 6. Effect of threshold on the average fractional errors in Algorithm 1 [18] are derived from the PF Eq. (24)
usingtheLC-PFmodelforthebus 13 3(with50additionalnon-operational
edges). p2 +q2
p −r a→b a→b =p + ∑ p , (29)
a→b ab v2 b b→c
a slack bus, a=0. The set of Eqs. (24), expressing potentials a (bc)∈ET;c(cid:54)=a
via complex powers injected at the nodes of the power graph, p2 +q2
q −x a→b a→b =q + ∑ q , (30)
are called Power Flow (PF) equations. a→b ab v2 b b→c
a (bc)∈ET;c(cid:54)=a
B. Linear Coupled (LC) Approximation of Power Flows v2=v2−2(r p +x q )+(cid:0)r2 +x2 (cid:1) p2a→b+q2a→b
b a ab a→b ab a→b ab ab v2
a
We linearize the PF Eqs. (24) in the first order jointly (31)
over phase difference and voltage deviations (v −1 = ε )
a a
whereeachofthethreeequationsabovearestatedintermsof
fromnominal,i.e.thetwocorrectionsareconsideredonequal
active, p , and reactive, q , powers over directed lines,
footing. We arrive at the following set of equations: a→b a→b
a→b, and voltages over nodes, a∈V. If power losses at any
pa= ∑ (βab(θa−θb)+gab(εa−εb)), (25) line segment is negligible, Eqs. (29,30,31) reduce to [4].
b:(ab)∈ET
p ≈p + ∑ p ,q ≈q + ∑ q , (32)
q = ∑ (−g (θ −θ )+β (ε −ε )) (26) a→b b b→c a→b b b→c
a b:(ab)∈ET ab a b ab a b (bc)∈ET (bc)∈ET
c(cid:54)=a c(cid:54)=a
where ∀a∈V,∀(ab)∈ET : |εa|(cid:28)1,|θa−θb|(cid:28)1, (27) ϕb≈ϕa−2(rabpa→b+xabqa→b), ϕa≡v2a (33)
. rab . xab
g = ,β = (28)
ab x2 +r2 ab x2 +r2 On a general graph with loops, the number of these (directed)
ab ab ab ab
edge-related variables in Eq. (24) is larger then the number of
Eqs. (25,26) show coupling between phases and voltages,
phases and voltages. However, the former becomes equal to
thus calling it the Linear-Coupled (LC) approximation, is
the later if the grid is a tree, where the number of edges is
proper.ThelinearityofLC-PFisusedinthepaperinderiving
equal to the number of nodes minus one. Therefore when the
results to learn the grid structure. Two comments are in order.
phase and voltage at one special node (each substation node
First of all, notice that the LC-PF approximation does not
per tree in our case) is fixed at v (say), phases and voltages
0
makeanyassumptionabouttherelativestrengthofinductance
at all other nodes can be reconstructed from the directional
andresistance,thusmakingitapplicabletopowerdistribution
line flows, and vice versa.
systems where the two line characteristics are typically of the
Note that additional boundary conditions arise in a tree
same order. Second, expressions under the sum on the rhs of
due to the requirement of active and reactive power flowing
Eqs. (25,26) represent active and reactive power flows which
from/into any leaf being equal to its nodal injection. Thus,
are antisymmetric (p = −p , q = −q ). This
a→b b→a a→b b→a
emphasizesanimportantconsequenceoflinearapproximation ∀a∈V : v =v , (34)
0 a 0
in the LC-PF model – both active and reactive losses in lines ∀b∈VT, & (a→b)∈ET :p =p , q =q , (35)
l a→b b a→b b
are ignored as such losses occur at second order.
whereV ⊂VisthesetofK substationsnodes(coloredinred
0
in Fig. 1), andVT is the set of leaf-nodes in tree T. Based on
C. DC-resistive approximation l
the assumption that voltage drop across any line segment is
In low-voltage distribution grids line inductances may be
sufficiently small, we linearize Eq. (33) by substituting v =
a
much smaller in magnitude than line resistances. Then the
1+ε , with |ε |(cid:28)1 and have
a a
LC-PF model can be simplified even further. Indeed, taking
this case to the extreme where inductance can be ignored ∀(ab)∈ET : ε −ε ≈(r p +x q ). (36)
a b ab a→b ab a→b
10
Notice that the LinDistFlow Eqs. (32,36), derived assum- [19] http://www.powerstandards.com/PQube.php.
ing that the grid graph is a tree, are exactly equivalent to [20] A. Phadke, “Synchronized phasor measurements in power systems,”
Computer,1993.
Eqs. (25,26) of the Linear Coupled (LC) Approximation.
[21] J.Resh,“Theinverseofanonsingularsubmatrixofanincidentmatrix,”
IEEETransactionsonCircuitTheory,vol.10,p.131.
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andelectricitydemandinbuildingsforthepurposeofplanningformixed
The work at LANL was funded by the Advanced Grid energydistributionsystems,”EnergyandBuildings,vol.40,no.7,pp.
Modeling Program in the Office of Electricity in the US 1124–1134,2008.
[23] S. Civanlar, J. Grainger, H. Yin, and S. Lee, “Distribution feeder
Department of Energy and was carried out under the auspices
reconfigurationforlossreduction,”PowerDelivery,IEEETransactions
of the National Nuclear Security Administration of the U.S. on,vol.3,no.3,pp.1217–1223,Jul1988.
Department of Energy at Los Alamos National Laboratory [24] U.EminogluandM.H.Hocaoglu,“Anewpowerflowmethodforradial
distributionsystemsincludingvoltagedependentloadmodels,”Electric
under Contract No. DE-AC52-06NA25396.
PowerSystemsResearch,vol.76,no.13,pp.106–114,2005.
[25] C.-T.Su,C.-F.Chang,andJ.-P.Chiou,“Distributionnetworkreconfigu-
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