Computing Critical $k$-tuples in Power Networks

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by  Kin Cheong Sou

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Computing Critical k-tuples in Power Networks
Kin Cheong Sou, Henrik Sandberg and Karl Henrik Johansson
Abstract—Inthispapertheproblemoffindingthesparsest(i.e., I. INTRODUCTION
minimum cardinality) critical k-tuple including one arbitrarily
specifiedmeasurementisconsidered.Thesolutiontothisproblem A. Critical k-tuples
can be used to identify weak points in the measurement set, or
A modern SCADA/EMS system relies heavily on the state
aid the placement of new meters. The critical k-tuple problem
estimator, which estimates the power network states (e.g., the
is a combinatorial generalization of the critical measurement
calculation problem. Using topological network observability phase angles of bus voltages) based on measurements such
2
results,thispaperproposesanefficientandaccurateapproximate as transmission line power flows, bus power injections and
1
solutionprocedurefortheconsideredproblembasedonsolvinga busvoltages.Animportantquestionrelatedtostateestimation
0
minimum-cut(Min-Cut)problemandenumeratingallitsoptimal
2 is whether the network is observable or not; whether the
solutions.It isalso shown that thesparsest critical k-tupleprob-
states can be uniquely determined based on the available
n lem can be formulated as a mixed integer linear programming
a (MILP) problem. This MILP problem can be solved exactly measurements.Thisisacentralissueofnetworkobservability
J using available solvers such as CPLEX and Gurobi. A detailed analysis (e.g., [1]–[9] and the references therein). While the
2 numerical study is presented to evaluate the efficiency and the measurementsare typically placed so that a power network is
accuracy of the proposed Min-Cut and MILP calculations.
observable,thereexistweakpointsknownascriticalmeasure-

] ments. By definition, if a critical measurement is lost (e.g.,
E
failure of a meter), then the network becomes unobservable
C
NOTATION (i.e., the states can no longer be uniquely determined). The
.
s A A subset of transmission lines whose removal notionof critical measurementalso playsan importantrole in
c
[ partitions the network into two disjoint parts. anothervitalpowernetworkstateestimationfunction,namely,
E The set of all transmission lines. bad data detection (e.g., [3], [4], [10]–[12] and the references
1
G The graph of the power network. therein). Specifically, a bad data detection scheme based on
v
9 H The Jacobian of the measurement function. measurementresidualcannotidentifywhetherameterisfaulty
6 H(I,J) A submatrix of H, consisting of the rows and or not if the corresponding measurement is critical. A gener-
4 columns in the index sets I and J respectively. alization of the concept of critical measurement is a critical
0 H(i,:) The ith row of matrix H. k-tuple, where k is any natural number. A critical k-tuple is
.
1 I¯ The complement of an index set I. a set of k measurements such that if all measurements in the
0 j¯ The complement of a singleton set {j}. setarelostthenthenetworkbecomesunobservable.However,
2 M A large scalar constant treated as “infinity” losing any subset of p< k measurements would not result
1
in the MILP procedure. in the loss of observability. A critical 2-tuple is also referred
:
v m The number of measurements (rows of H). to as a critical set, where bad data can be detected but not
i
X n The number of buses (columns of H). identified (other terminologies include minimally dependent
r S A subset of buses (i.e., S⊂V). set or bad data group, e.g., [13]–[15]). Critical k-tuples of
a S is used to define a partition of the network. larger cardinalities are also of practical interest, as will be
v Node weights for bus i. explained later. How to compute them is the main topic of
i
w Edge weights for transmission line {i,j}. this paper.
ij
w˜ Edge weights which also account for the bus While critical k-tuple, network observability, and bad data
ij
weights connected to {i,j}. w˜ =w +v +v . detectionarecloselyrelated,themathematicaltoolstoanalyze
ij ij i j
V The set of all buses. them are different. While critical k-tuples provide the mea-
q The n×1 state vector (phase angles). surements to remove to render the network unobservable, the
d (S) Cut capacity of S. The sum of all weights of topic of network observability is the opposite. They include
the edges cut by the partition defined by S. checking whether a network is observable or not, and in
d (S) The sum of all weights of the buses connected case of an unobservable network which parts of it is still
inj
by edges which are cut by the partition of S. observable(i.e., finding observableislands). Likewise, critical
d˜(S) Modified cut capacity, where the edge weights k-tuplecomputationandbaddatadetectionareseparateissues:
w are replaced by w˜ . Essentially,themeasurementresidualbasedbaddatadetection
ij ij
theory investigates what detection can be achieved beyond
the limitations imposed by critical measurements or critical
The authors are with the ACCESS Linnaeus Center and the Automatic
Control Lab, the School of Electrical Engineering, KTH Royal Institute of k-tuples. These techniques are, consequently, not concerned
Technology, Sweden.{sou,hsan,kallej}@kth.se with finding the critical k-tuples.
This work is supported by the European Commission through the VIKING
Techniques to identify critical 1-tuples (i.e., critical mea-
project, the Swedish Foundation for Strategic Research (SSF) and the Knut
andAliceWallenberg Foundation. surements) and critical 2-tuples (i.e., critical sets) are widely
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known (e.g., [3], [11], [13]–[16]). In [17] and [18] the calcu- becomes more robust to meter failures. Other, but related,
lationofcriticalk-tuplesfork>2isconsidered.However,the meter inclusion problems are also considered in [19], [20].
procedure in [18] is efficient only for finding critical k-tuples This paper, on the other hand, solves (1) for all i, regardless
of lower cardinalities (i.e., k≤3), as will be explained and of the corresponding optimal objective value. These include
numericallydemonstratedlaterinthispaper.Thecomputation the measurements in the critical k-tuples of cardinalities less
of critical k-tuples is inherently computationally intensive, than or equal to three as in [18]. However,the informationof
because finding critical k-tuples amounts to a combinatorial thesparsestcriticalk-tuplesoflargercardinalitiescanbeused
search, as will be discussed in detail in this paper. for a measurement inclusion scheme with a more stringent
robustness requirement.
B. Problem Formulation and its Motivation 2) Planning of measurement sets: Instead of expanding a
pre-existing measurement set as in [18]–[20], it is possible to
The general setup of this paper is the standard state es-
obtainacosteffectiveyetmeterfailurerobustmeasurementset
timation problem over a linearized DC power flow network
by removingappropriate measurementsfrom the full set. The
[3], [4]. The particular problem considered is to find the
solution to (1) with H correspondingto the full measurement
sparsest (i.e., minimum cardinality) critical k-tuples involving
set can provide insight into which measurements can be re-
at least one arbitrarily specified measurement. By parameter-
moved.Thismeasurementremovalstrategy hasthe advantage
izing the sparsest critical k-tuple problem with the specified
that it does not assume any pre-existing measurement set
measurement, it is possible to examine all weak points in the
which can affect the final measurement set. In Section V-E
network and notjust the weakest pointat the boundaryof the
a numericalexampleis presentedto demonstratethe potential
network. The precise description of the considered problem
of the above measurement removal scheme.
is as follows. Let m be the number of measurements in the
power network, and n be the number of states (i.e., the phase 3) Cyber-security of power networks: Based on a result in
angles of the bus voltage phasors). It is assumed that m>n. [21] (Corollary 1 in Section II-B), problem (1) is equivalent
Let H ∈Rm×n be the Jacobian of the state-to-measurement to another cardinality minimization problem: (2) to be de-
functioninalinearizedmodel.DenoteH(I,J)asthesubmatrix scribedinSectionII-B.Problem(2)arisesfromcyber-security
of H formed by including the rows in an index set I and analysis of power networks (e.g., [21]–[25]). In particular,
the columns in an index set J. Also denote I¯ and J¯ as the [24]–[26] analyze the vulnerability of each measurement i
complements of I and J, respectively. Then according to, for using (2), where a malicious attacker inflicts “bad data” in a
instance [4] (Theorem p.165), the measurements in an index critical k-tuple. In this case, sparsest critical k-tuplesof larger
setI formacriticalk-tupleifandonlyifrank(H(I¯,j¯))<n−1 cardinalities(i.e.,>3)areofinterestbecausethe“baddata”is
forany j.Here jistheindexofanarbitraryreferencebus,and intentionalinsteadofoccurringbychance.Thesolutionof(2)
j¯denotes the set of all indices except j. The sparsest critical can be used to identify the weak points in the measurement
k-tuple problem for a specified measurement i can be written set in the cyber-security setting.
as: Finally, note that for complete safeguard against bad data
minimize card(I) or cyber-attack, the set of all critical k-tuples (in addition to
I
subject to rank(H(I¯,j¯))<n−1 (1) the sparsest ones found by solving (1)) should be computed.
However,thiswouldrequireanenumerationwhichisnotcom-
i∈I
putationallytractableforrealisticapplications.Thecalculation
where card(·) denotes either the cardinality of a set or the of the sparsestcriticalk-tuplesin (1) can identifythe network
numberofnonzeroentriesof avector,dependingontheinput vulnerabilities, subject to practical computation constraints.
argument. Notice that (1) does not explicitly impose the con-
dition that I cannot contain any strictly proper subsets whose
C. Contributions and Related Work
removal makes H rank deficient. However, this condition is
always satisfied at optimality. Problem (1) requires a combi- This paper presents two methods to solve (1). The first
natorialsearchoftherowswhoseremovalmakestheH matrix method is efficient but suboptimal. It utilizes a sufficient
rank deficient. In general, no efficient algorithm is available condition for critical k-tuples candidates in [2], [21]. This
to exactly solve (1). However, specializing (1) to the case of condition is topological. In the setting of this paper, the
power system state estimation results in a significant solution conditionstatesthatforanysetoftransmissionlineswhosecut
efficiency gain because of the special structure of H. The would separate the network into two disjoint parts, removing
demonstration of this is the main contributions of the paper. all line and injection measurements associated with these
Problem (1) is motivated from the following applications: transmission lines would make the network unobservable.
1) Identifyingmeasurementsin smallcardinality criticalk- Using this sufficient condition, a restricted version of the
tuples: While notdirectly solved as an optimizationproblem, sparsestcriticalk-tupleproblemin(1)canbestatedasfollows.
(1) is addressed in [18] (Definition 1, even though the term If the specified measurement i is a line power flow, then
“criticalset”in[18]hasadifferentmeaningthantheonehere). the corresponding transmission line must be cut. On the
For any given measurement set, [18] finds the measurements other hand, if the specified measurement is a power injection
such that optimal objective value of (1) is less than or equal at a bus, then one of the incident transmission lines must
to three. This information is used to determine the set of be cut. Then the rest of the transmission lines are cut (or
additional measurements to be included, so that the network not cut) in order to minimize the number of measurements
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removed, while dividing the network into two parts. This is less time-efficient. The method is based on the equivalence
cut problem, while being a restricted version of (1), is still between (1) and (2), to be described in Section II-B. This
combinatorial. However, if the injection measurements are means that (1) can be solved by instead solving (2). Previous
not directly counted towards the optimization objective (to be attemptsto approximatelysolve (2) include,forinstance, [22]
made precise later), then this modified problem becomes a describing an attempt to use matching pursuit (e.g., [37]),
classical minimum cut problem (Min-Cut) (e.g., [27]). Min- and [38] about the application of LASSO [39]. The MILP
Cut admits scalable solution algorithms(e.g., [28], [29]). The formulation, based on [26], does not admit any polynomial
solution to the Min-Cut problem can be used as a suboptimal timesolutionalgorithmsingeneral.However,thereexistgood
solution to (1). In fact, due to [30]–[32], it is possible to MILP solvers such as CPLEX [40] or Gurobi [41]. The
efficiently enumerate all optimal solutions to the Min-Cut major novelty of this second contribution of the paper is the
problem and pick the best available suboptimal solutions to combination of [21] and [26].
(1). This is the idea of the first method of this paper.
Two previous results are related to the first proposed D. Organization of the Paper
method. As mentioned before, [18] addresses (1). In [18], a
The rest of the paper is organized as follows. In Section II
(non-unique) set of measurements is chosen to be the basic
three known theorems from [2] and [21] are reviewed, and
measurements. Then the critical k-tuples containing exactly
a corollary is derived. These theorems form the theoretical
one basic measurement can be identified using a matrix
foundation of this paper. Section II also reviews the Min-Cut
factorization approach generalizing the one in [3] (Chapter
problem,whichisanimportantpartoftheproposedalgorithm.
4.5.4).Tofindcriticalk-tuplescontainingmorethanonebasic
Section III describes the first contribution of the paper: a
measurements,a recursive applicationof the matrix factoriza-
Min-Cut based algorithm which makes use of the topological
tion approachfor finding critical k-tuples with only one basic
characterization of network observability to find the sparsest
measurement is required. For larger k, the recursion becomes
critical k-tuples. In Section IV the second contribution, the
more expensive as there are (cid:0)n(cid:1) possible combinations of p
p exact MILP formulation, is derived with some properties
basic measurementsto be includedin the critical k-tuples, for
n discussed. In Section V some case studies are presented to
different p≤k. To solve (1) for all possible i, in total (cid:229) (cid:0)np(cid:1) evaluate the performance of the proposed algorithms. Finally
p=1
Section VI concludes the paper.
applicationsofthematrixfactorizationprocedurearerequired.
Thecomputationeffortisexponentialintermsofnetworksize
(i.e., the number of buses n). In summary, [18] is accurate II. TECHNICAL BACKGROUND
but the procedure is efficient only for a sparse measurement This section reviews some known results needed for the
set (so that critical k-tuples of high cardinalities will not be derivationofthecontributionsofthispaper.Theoremsadopted
encountered).Theproposedmethodinthispaper,ontheother from known sources are stated without proof.
hand,is efficientfor solving(1) irrespectiveof the cardinality
of the critical k-tuple, because (1) is approximately solved
A. A Topological Sufficient Condition for Critical k-tuple
via Min-Cut.However,asitwill benumericallydemonstrated
Candidates
in Section V, the accuracy of the proposed method suffers
whenthe measurementset becomessparse. Inthis sense, [18] The first statement is adopted from [2] (Theorem 5). It
and the proposed method are complementary to each other. provides a sufficient and necessary condition for network
Anothercloselyrelatedworkis[21],whichconsidersavariant observability in terms of spanning trees, which are loop-free
of(1).Inthisvariant,thesparsestcriticalk-tuplealsocontains connectedsubgraphsof the power networkretaining all buses
at least one measurement. However, instead of being user- but subsets of the transmission lines.
specified, this measurement is chosen by the optimization to Theorem 1: A power network is observable if and only
findthesparsestnon-emptycriticalk-tuple.Solving(1)forall if there exists a spanning tree with an assignment function,
i leads to the solution to the problemin [21] but the converse mappingfromthesetofthetransmissionlinesinthespanning
isnottrue.Inaddition,[21]doesnotposetheirproblemas(1) treetothesetoflinepowerflowandinjectionmeasurementsof
definedin thispaper.Mostimportantly,theproblemin [21] is the original power network. The assignment function satisfies
posed as a submodular function minimization problem [33], the following properties:
[34]. While theoreticallypolynomial-timealgorithmsexistfor 1) Twodistinctspanningtreetransmissionlinesmaptotwo
solving this problem (notably the ellipsoid method [35] and distinct measurements.
more recently[36] of which the complexityis O(m8log(m))), 2) If the line power flow of a spanning tree transmission
no practically efficient algorithms for this class of problem line is measured, then this transmission line maps to its
have been observed. On the other hand, the Min-Cut problem own line measurement under the assignment function.
encounteredby the proposedmethodcanbe solved efficiently 3) If the line power flow of a spanning tree transmission
both in theory and in practice. For example, the complexity line is not measured, then the injection measurementof
of [28] is O(mn+n2log(n)). The practical efficiency will be one of the two terminal buses of this transmission line
demonstrated by the numerical experimentlater in this paper. is the value of the assignment function.
Thesecondproposedmethod,basedonmixedintegerlinear The following theorem, which is the main theoretical basis
programming(MILP),isexactunderamildassumption,butit of this paper, is adopted from Theorem 2 of [21]. It states
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that if an appropriate choice of measurements are removed, that s and t are in different partitions, and the cut capacity
then it becomes impossible to form any spanning tree with is minimized:
an assignment function defined as in Theorem 1. Hence, the minimize d (S)
statementprovidesasufficientconditionforfindingcandidates S⊂V (4)
subject to s∈S and t∈/S
for critical k-tuples.
Theorem 2: Let A be any set of transmission lines whose For more detail regarding the Min-Cut problem in (4), see
cut would divide the power network into two disjoint parts. for example [27]. For efficient solution algorithms, see for
Then removing all line power flow measurements in A and example [28], [29]. The Min-Cut problem is a subproblem to
all power injection measurements of the buses connected by besolvedintheproposedcriticalk-tuplecalculationalgorithm
the linesin A wouldrenderthepowernetworkunobservable. to be described in the next section.
Remark 1: Any spanning tree of the network contains at
least one transmission line in A. However, under the mea-
III. APPROXIMATE CRITICAL K-TUPLECALCULATION VIA
surement removal scheme in Theorem 2, it is impossible to
MIN-CUT OPTIMIZATION
define any assignment function in Theorem 1 for this line.
A. A Graph-Oriented Optimization Problem Related to (1)
Hence the network becomes unobservable.
Remark 2: While Theorem2 providesthe sets of measure- ThesufficientconditioninTheorem2providesatopological
mentswhoseremovalwouldrenderthenetworkunobservable, characterization of a subset of the solution candidates of the
these sets are not necessarily critical k-tuples since their sparsest critical k-tuple problem in (1). This characterization
subsets might also render the network unobservable. leads to a graph-oriented optimization problem which is re-
Remark 3: TheoriginalversionofTheorem2,asin[21],is lated to, but not exactly the same as, (1). The developmentis
moregeneralinthatitallowsthesituationsinwhichA divides as follows. Denote the power network as G =(V,E), where
the network into more than two disjoint parts. However, the V is the set of all buses and E is the set of all transmission
method proposed in this paper cannot exploit the additional lines.ThenthesetA inTheorem2,whosecutwouldpartition
generality. G, can be characterized by a bus subset S⊂V such that
A =(cid:8){i,j} either (a) i∈S and j∈/S or (b) i∈/S and j∈S(cid:9)
B. Sparsest Critical k-tuple Problem as a Cardinality Mini- (5)
mization Problem To describe the number of removed measurements associated
with S (i.e., A) according to Theorem 2, the following
The following cardinality minimization problem has been
definitions are required. Let w be the number of meters
ij
studied in power network cyber-security (e.g., [22], [25]):
on a transmission line {i,j}∈E, and v be the number of
j
minimize card(Hq ) injection flow meters on a bus j∈V. Then associated with
q (2) S, the number of line power flow measurements to remove is
subject to H(i,:)q =1
d (S)definedin(3).Inaddition,thenumberofpowerinjection
measurements to remove can be defined as
The following theorem, adopted from Theorem 1 in [21],
establishes that the sparsest critical k-tuple problem in (1) is d (S), (cid:229) v such that either
inj j
equivalent to (2). j∈V
Theorem 3: An index set I is a feasible solution to (1) if (a) j∈S and ∃i∈(V \S) s.t. {i,j}∈E
and only if there exists a feasible solution q in (2) such that or (b) j∈(V \S) and ∃i∈S s.t. {i,j}∈E
H(j,:)q =0 whenever j∈/I.
Theorem 3 implies the following statement (proved in Ap- Hence, associated with S, the total number of measurements
pendix) establishing the equivalence between (1) and (2). to be removed is d (S)+d inj(S). Lastly, the constraint in (1)
Corollary 1: The optimization problems in (1) and (2) are thatonespecifiedmeasurementmustbeincludedinthecritical
equivalent in that q ⋆ is an optimal solution to (2) if and only k-tuple should be enforced. For simplicity of discussion, for
if I⋆={j H(j,:)q ⋆6=0} is an optimal solution to (1). the momentit is assumed that the specified measurementis a
line power flow. The case of power injection will be handled
in the end of this section. Now suppose the specified line
C. Min-Cut Problem on an Undirected Graph power flow meter i is on transmission line {s,t}, then the
Consider an undirected graph G =(V,E) where V and E correspondingtopologicalconstraintis thats∈S andt∈/S. In
denote the set of nodes and the set of edges respectively, and summary, the graph oriented optimization problem, set up as
let each edge {i,j}∈E be weighted with a scalar w . Let an approximation to (1), is described as:
ij
S⊂V be any subset of V. Define the cut capacity function minimize d (S)+d (S)
inj
S⊂V (6)
d (S), (cid:229) wij such that either (a) i∈S and j∈/S subject to s∈S and t∈/S
{i,j}∈E
or (b) i∈/S and j∈S. Strictly speaking an optimal solution to (6) is a set of buses,
(3) with the correspondingset of “cut” transmission lines defined
For any two distinct nodess andt, the s−t Min-Cutproblem in (5). However, it is more convenient to treat the optimal
seeks to find a partition of V into V = S∪(V \S) such solution as the corresponding set of measurements to be
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different optimization problems. Moreover, solving (7) is not
2 ((1)) 2
(1) 1 (1) 1 (3) equivalent to solving (6), as illustrated in Fig. 1. The quality
of approximately solving (6) via (7) depends on the ratio
between the number of transmission line measurements and
((22))
((11)) businjectionmeasurements.Intheextremecasewherethereis
3 3
noinjectionmeasurement,(7)isthesameas(6).Theaccuracy
meter bus and efficiency of the proposed Min-Cut procedure will be
numerically assessed in Section V. The following algorithm
summarizes the Min-Cut based approximate solution proce-
Fig. 1:Illustrationofthe modifiedcutcapacityfunction.Left: dure for (1), where the specified measurementis a line power
Metering scenario in (6). Right: Transmission line metering
flow on a transmission line.
scenario pertaining to d˜(S) in (7). The number in the paren-
Algorithm 1: Min-Cut procedure for transmission line
thesis indicates the number of times a meter is repeated. If
case:
both 1-2 and 1-3 are cut, the true cost d (S)+d (S) is 4.
inj
However, the approximate cost d˜(S) is 5. Step 1
In the power network graph G =(V,E), define arc
weights w˜ as the number of meters on a transmis-
ij
removed, as prescribed by Theorem 2. Solving (6) yields a sion line {i,j}∈E, plus the number of meters on
sparse set measurements whose removal makes the network buses i and j.
unobservable. The numerical experiment in Section V will Step 2
demonstrate the usefulness of (6). However, it should be Supposethespecified transmissionline is{s,t}∈E.
emphasized that an optimal solution to (6) is not necessarily Setup a s−t Min-Cut problem as in (7). Solve (7)
a sparsest critical k-tuple in (1). The reason is twofold. First, using algorithms such as [28], [29] for an optimal
since Theorem 2 is a sufficient condition, (6) searches only solution, which is a set of “cut” transmission lines.
through a subset of the sets of measurements whose removal The line power flows measurements and injections
would render the network unobservable. Second, as pointed measurements at the terminal buses constitute a
outinRemark2,anoptimalsolutionto(6)doesnotevenneed suboptimal solution to the sparest critical k-tuple
tobeacriticalk-tuple.Theserestrictionswillbedemonstrated problem in (1).
by the numerical experiment in Section V. Step 3
Use theresultsin[30]–[32] toenumeratealloptimal
solutionsto (7). Pick the best suboptimalsolutionto
B. Min-Cut Approximate Solution Procedure for (1)
(1) among all optimal solutions to (7).
Compared with the tractable Min-Cut problem in (4), (6)
is a combinatorial optimization problem because of the addi- Even the best available suboptimal solution to (1) might not
tional term d inj(S) in the objective function. To overcome the be a critical k-tuple in that there might be a strictly proper
computationaldifficulty,itisproposedinthispaperthatd inj(S) subset whose removal makes the network unobservable. To
is indirectly accounted for by solving the following Min-Cut make sure a critical k-tuple is obtained, an enumeration is
problem: requiredtoseewhichmeasurementsinthesuboptimalsolution
minimize d˜(S) can be eliminated. However, since the suboptimal solution
S⊂V (7) typically contains very few measurements, the enumeration
subject to s∈S and t∈/S,
is not expensive.
where d˜(S) is defined according to (3) with modified edge In the case where the specified measurement i in (1) is
weights w˜ij,wij+vi+vj for all {i,j}∈E, with wij, vi and a power injection on a bus, the following procedure can be
vj defined in Section III-A. d˜(S) corresponds to a modified applied:
meteringscenario where an injection meter of a bus is moved
Algorithm 2: Min-Cut procedure for bus injection case:
toallincidenttransmissionline(s).However,thismodification
can lead to overcounting of injection meters as opposed to Step 1
solving(6).SeeFig.1foranillustration.Since(6)and(7)have Let G =(V,E) be the power network graph and
thesameconstraint,theoptimalsolutionto(7)isasuboptimal let i∈V be the bus with the considered injection
solution to (6). In addition, by the results in [30]–[32], it is measurement. For each j∈V such that {i,j}∈E,
possible to efficiently enumerate all optimal solutions to the apply Algorithm 1 on transmission line {i,j}.
Min-Cut problemin (7). Hence, the best available suboptimal Step 2
solution to (6) can be chosen. However, it is emphasized AmongallsolutionsprovidedbyAlgorithm1applied
that the strategy of solving (7) can only provide a sparse set to {i,j}, pick the one with the minimum cost in (1)
of measurements including {s,t}, and the removal of these as the best available solution to (1).
measurements makes the network unobservable. From (1) to
(6)andthento(7)thesetwotransitionsinducetheirrespective The numerical examples in Section V illustrate the perfor-
limitations. As it was explained earlier, (1) and (6) are two mance of these algorithms.
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IV. EXACT SPARSEST CRITICAL K-TUPLEPROBLEM method is the proposed Min-Cut procedure in Algorithm 1
FORMULATION AS A MILPPROBLEM and Algorithm 2 in Section III. The third method is the
Theorem 3 and Corollary 1 state that the sparsest critical MILP procedure (with M=100) in Section IV. The solution
of the MILP procedure is used as a reference for accuracy.
k-tupleproblemin(1)canbesolvedbysolvingthecardinality
The IEEE 14-bus benchmark system is analyzed, with two
minimization problem in (2). Problem (2) can be formulated
as a MILP problem, as mentioned in [26]. The key to the differentmeasurementsets. The first measurementset is from
formulationisthecountingofthecardinalityofvectorHq .To [18] (Section IV, measurement set of Scenario 1), containing
achieve this, an additional binary decision vector y∈{0,1}m measurements from 9 out of 14 buses and 6 out of 19 trans-
mission lines. The second measurement set contains all bus
and a scalar constantM>0 are needed.If M is largeenough,
andtransmissionlinemeasurements,whichmaybeofinterest
then the constraint
for meter placement. For each i, the procedure in [18] (with
|H(j,:)q |≤My(j) three basic measurements) and Min-Cut might only provide
an overestimate of the cardinality of the sparsest critical k-
provides a cardinality counting mechanism via y. If |H(j,:
)q |>0,theny(j)=1.If|H(j,:)q |=0,theny(j)canbeeither tuple. Table I lists the percentages of i with overestimation,
theaverageoverestimation(overalli)andtheaveragerelative
0 or 1. However, since (2) seeks to minimize the cardinality
ofHq ,asitwillclearshortly,y(j)mustbezeroatoptimality. overestimation(relativetothecardinalityofthecorresponding
sparsest critical k-tuple).
The constant M must be chosen large enough so that it is
larger than max{|H(j,:)q ⋆|} for at least one optimal solution Table I confirms the statement in Section I-C that the
j procedure in [18] and Min-Cut should be used in different
q ⋆ of (2). In the special case where all line power flows are
measurement settings. [18] performs better in a sparse mea-
measured, the method in [26] can be used to compute M. In
surement set, while Min-Cut is a better choice in a dense
other cases, the generalguidelineis that M shouldbe as large
measurement set.
as possible, before the optimization solver complains about
numerical difficulties. Suppose q 0 is a typical state vector TABLE I: Comparison between the procedure in [18], Min-
under normal operation, then a max|Hq 0| with some a > 1 Cut and MILP for the IEEE 14-bus system
min|Hq 0|
can be a reasonable guess for M. The choice of M is the
measurementset1(sparsemetering) [18] Min-Cut MILP
only heuristic part of the otherwise exact sparsest critical k- solvetime(s) 0.04 0.02 3.6
tupleproblemformulation.TheMILPformulationof(2)is as percentofmeas.withoverestimation (%) 0 93 0
averageoverestimation 0 1.07 0
follows:
averagerelative overestimation (%) 0 75 0
minimize (cid:229) y(j) measurementset2(fullmetering) [18] Min-Cut MILP
q ,y j solvetime(s) 10 0.03 17
subject to Hq ≤ My percentofmeas.withoverestimation (%) 26.5 0 0
−Hq ≤ My (8) averageoverestimation 4.1 0 0
H(i,:)q = 1 averagerelative overestimation (%) 67.7 0 0
y(j) ∈ {0,1} ∀ j
Note that since the objective function is (cid:229) y(j), at optimality
j
for any j such that |H(j,:)q | = 0, the corresponding y(j) B. TheEffectofthe ProportionofLinePowerFlowMeasure-
must be zero. Hence, (cid:229) y(j)=card(Hq ). Finally, notice that ments on the Min-Cut Procedure
j
if the measurements in a certain set P are considered very As explained in Section III, the Min-Cut procedure for (1)
reliable andare immunefromfaults, then (8) can be modified achieves computation efficiency by approximately counting
accordingly by adding the constraint y(j)=0 for all j∈P. the injection measurements. Hence, the relative ratio between
the line power flow and bus injection measurements affects
the approximation quality of the Min-Cut procedure. In this
V. CASE STUDY
subsection, the relationship between approximation quality
Numerical experiment results are demonstrated in this sec-
and the proportion of transmission line measurements in the
tion. All computations are performed on a laptop with an
network is considered.
Intel Core i5 2.53GHz CPU and 4GB of memory. All Min-
The IEEE 14-bus, 57-bus and 118-bus benchmark systems
Cut problems are solved in MATLAB using [42], which calls
areconsidered.ThenetworktopologiesarefromMATPOWER
the libraries from [43]. All MILP problems are solved in
[45]. For each system, 11 different measurement sets are
MATLAB using Gurobi [41] via Gurobi Mex [44].
considered. Each measurementset contains all injection mea-
surements, but the proportions of removed line power flow
A. Comparison with the Procedure in [18]
measurementsincrease as 0%, 10%, ..., 100%. The removed
First, problem(1), for each possible specified measurement line measurements are randomly chosen. A study similar to
i, is solved using three methods. The first method is the theoneinSectionV-Aisperformed,testingonlytheproposed
recursive critical k-tuple calculation procedure in [18] im- Min-Cut and MILP procedures. The above study can be con-
plemented by the authors. As in [18], only critical k-tuples sidered as “one sample” of a random experiment involving a
containing three basic measurements are sought. The second sequenceof11measurementsetsforeachbenchmarksystem.
7
The randomness stems from choice of the removed line flow For the 2383-bus case, only 14 instances of (8) are solved.
measurements. To examine the typical phenomena, the above For all these 14 instances Min-Cut also provides the correct
random experiment is repeated five times. Table II shows the estimates.ThecomputationtimesforusingtheproposedMin-
meanvalue(over5experiments)oftheperformanceanderror Cut procedure and solving (8) are listed in Table V. This
statistics similar to those in Table I. numerical study again assures the efficiency and accuracy
WhilenotseenfromTableII,thecomputationtimeforMin- of the proposed Min-Cut procedure in the case when all
Cut remains roughly the same. The increase in solve time transmission lines and buses are metered.
ratio (up to about 0.36 when 100% of line measurements
are removed) is due to the decrease in solve time of the
E. Using Critical k-tuple Information for Meter Placement
MILP procedure. In general, Min-Cut is more efficient than
While not the main focus of this paper, a possible use of
MILP. In terms of approximation error, for up to 90% of
the critical k-tuple information in (1) is for meter placement.
line measurement removal, Min-Cut results in at most 7% of
Problem (1) is solved for all specified measurement i for the
measurements whose sparsest critical k-tuple cardinalities are
IEEE 6-bus benchmark system (see Fig. 2). It is assumed
overestimated (the number is down to 3% for up to 40% of
that the network is fully metered (with 6 injection and 11
line measurementremoval).On average,the overestimationis
line power flow measurements). The Min-Cut procedure (i.e.,
by about 1 measurement with the maximum observed in the
Algorithm1andAlgorithm2)isusedtosolve(1).Intotal,17
experiment being 3 measurements (not shown in Table II).
criticalk-tuplesarefound.Thenforeachmeasurementi,Fig.3
Finally, the average relative overestimation is at worst 35%
shows the number of critical k-tuples containing i. However,
(e.g., overestimationby 1 measurementfor a critical 3-tuple).
note that by solving (1) for all specified measurement i, all
sparsest critical k-tuples are not found. Hence, Fig. 3 only
C. TheEffectoftheProportionofInjectionMeasurementsand
showsthelowerboundsforthetruenumberofcriticalk-tuples
Arbitrary Measurements on the Min-Cut Procedure
containingi. Fig. 3 indicatesthatmeasurement12is probably
Inthissubsection,theexperimentinSectionV-Bisrepeated not important (as far as network observability is concerned),
for the 118-bus benchmark system with a difference in the because only one critical k-tuple contains it and none of the
definition of the measurementsets. Two cases are considered. other 16 critical k-tuples will be affected by the removal of
Inthe firstcase, each measurementset containsallline power measurement12.Thisisconsistentwiththenetworktopology
flow measurements and different proportions of the injection in Fig. 2, since measurement 12 is the line power flow
measurements are randomly removed. In the second case, measurement between bus 2 and bus 5 (i.e., the two buses
different proportions of arbitrary measurements (injection or with the largest degree). On the other hand, Fig. 3 suggests
line)arerandomlyremovedinawaythattheresultednetwork that measurements 2 and 5 (i.e., power injections at bus 2
is still observable. Table III lists the relevant statistics for and bus 5) are definitely important because they are involved
the first case and suggests (from the fourth row) that the in many critical k-tuples. This is again consistent with the
Min-Cut procedure is more accurate when the transmission topologyin Fig. 2 since each ofthese injectionmeasurements
lines are more densely metered. On the other hand, Table IV can substitute one of the five line measurements in case any
lists the statistics for the second case. In this case, at most oneofthemfails.Finally,notethatthesameanalysisherecan
1−118/(118+186) ≈ 41% of the measurements can be be carriedoutforlargerscale networkswhere itbecomesless
removed before the network become unobservable. However, obviousfromthetopologywhichmeasurementsareimportant
randomremovalof40%ofthemeasurementstypicallyresults or unimportant.
inaunobservablenetwork,andhencethecorrespondingresult
is not shown in Table IV. Table IV confirms again that the
1 2 3
Min-Cutprocedureisefficientandaccurateforrelativelydense
measurement sets.
D. Time Efficiency of Min-Cut and MILP for Large Networks
This numericalstudy investigatesthe possible advantageof
the proposed Min-Cut procedure for the sparsest critical k-
44 55 66
tuple analysis for larger scale power networks. The networks
considered are the IEEE 118-bus, IEEE 300-bus and the bus
Polish 2383-bus systems. The topologies of these networks
are obtained using MATPOWER [45]. On each transmission
Fig. 2: IEEE 6-bus system.
line of the benchmarksystems, there are two line power flow
meters (one from each terminal bus). In addition, all power
injections are measured. For each of the benchmark system,
problem (1) is solved using the Min-Cut procedure for all VI. CONCLUSION
possible specified measurement i. For the 118 and 300 bus In conclusion, a version of the sparsest critical k-tuple
cases, the Min-Cut procedure is experimentally found to be problem is considered. The sparsest critical k-tuple is sought
exact, compared with solving the MILP formulation in (8). for one arbitrarily specified measurement. It is possible to
8
TABLE II: Ensemble mean of the solve time and error statistics for the case study in Section V-B
linemeas.removal(relative tototallines)(%) 0 10 20 30 40 50 60 70 80 90 100
linemeas.removal (relative tototal meas.)(%) 0 6 12 18 24 29 35 41 47 53 58
14-bus solvetimeratio(Min-Cut/MILP)×10−4 90 97 110 140 151 175 262 369 557 924 3642
percentofmeas.withoverestimation (%) 0 0.63 1.0 1.4 3.1 5.4 6.0 5.5 6.1 6.9 86
averageoverestimation 0 0.1 0.2 0.3 0.5 0.7 0.96 0.97 0.85 0.8 1.0
average relative overestimation (%) 0 1.7 3.1 5.7 9.0 12 20 21 22 23 50
linemeas.removal(relative tototallines)(%) 0 10 20 30 40 50 60 70 80 90 100
linemeas.removal (relative tototal meas.)(%) 0 5.8 12 18 23 29 35 40 47 53 58
57-bus solvetimeratio(Min-Cut/MILP)×10−4 1.5 2.1 2.7 2.4 5.9 6.6 9.5 17 18 36 2740
percentofmeas.withoverestimation (%) 0 1.2 1.0 1.2 0.95 2.1 3.2 3.0 2.2 0.92 96
averageoverestimation 0 0.4 0.6 0.65 0.8 0.6 0.8 0.8 1.0 0.2 1.1
average relative overestimation (%) 0 4.7 8.9 12 14 11 14 15 23 5.0 53
linemeas.removal(relative tototallines)(%) 0 10 20 30 40 50 60 70 80 90 100
linemeas.removal (relative tototal meas.)(%) 0 6.2 12 18 24 31 37 43 49 55 61
118-bus solvetimeratio(Min-Cut/MILP)×10−4 2.1 2.1 1.7 2.0 2.1 2.4 2.7 3.9 4.6 11 1116
percentofmeas.withoverestimation (%) 0 0.77 0.90 1.4 2.7 4.2 5.4 6.7 6.7 5.7 87
averageoverestimation 0 1.0 1.2 1.1 1.2 1.2 1.2 1.2 1.2 1.2 1.1
average relative overestimation (%) 0 13 15 14 17 19 22 24 27 35 54
TABLE III: Ensemble mean of the solve time and error statistics for the case study in Section V-C with varying injection
measurement proportion
busmeas.removal(relative tototalbuses)(%) 0 10 20 30 40 50 60 70 80 90 100
busmeas.removal (relative tototal meas.)(%) 0 3.9 7.8 12 16 19 23 27 31 34 39
solvetimeratio(Min-Cut/MILP) ×10−4 2.1 3.2 6.4 9.7 18 33 61 108 185 404 1211
percent ofmeas.withoverestimation (%) 0 0.61 1.4 1.4 1.5 2.2 2.6 2.3 1.4 1.1 0
average overestimation 0 1.4 1.1 1.2 1.2 1.2 1.1 1.2 1.1 0.8 0
averagerelative overestimation (%) 0 16 14 15 16 18 19 20 22 15 0
TABLE IV: Ensemble mean of the solve time and error statistics for the case study in Section V-C with varying arbitrary
measurement proportion
meas.removal(relative tototalmeas.)(%) 0 10 20 30 40 50
solvetimeratio(Min-Cut/MILP)×10−4 2.1 3.3 8.9 21 110 345
percentofmeas.withoverestimation (%) 0 1.0 5.0 7.6 20 35
averageoverestimation 0 0.9 1.0 1.1 1.1 1.4
average relative overestimation (%) 0 12 24 28 45 69
TABLE V: CPU times for solving all instances of (1) in the
18 118, 300 and 2383 bus systems
s
uple16 Method 118-bus 300-bus 2383-bus
k−t14 MILP 763sec 6708sec (projected) 5.7days
al Min-Cut 0.30sec 1sec 31sec
c12
criti
g 10
n
ni
ai 8 paper demonstrates that the studied sparsest critical k-tuple
nt
co 6 problemcanbeformulatedasaMILPproblemsothatpower-
er of 4 fulMILP solvers such as CPLEX and Gurobican be utilized.
b Ontheotherhand,byusingtopologicalnetworkobservability
m
nu 2 results in [2], [21], a Min-Cut based approximate solution
00 2 4 6 8 10 12 14 16 18 procedure can be derived. The numerical experiment in this
measurement index
paper reveals that the Min-Cut procedure is highly accurate
andefficientwhenthereareasignificantnumberoflinepower
Fig. 3: Number of critical k-tuples containing any specific
flowmeasurementsinthepowernetwork.Consequently,Min-
measurement.
Cut should be the first method to attempt (over MILP) in this
scenario.
APPENDIX
identify the weak points in the power network by listing
all measurements which might form critical k-tuples with Proof of Corollary 1
small cardinality, even though this is short of a complete Suppose q ⋆ is an optimal solution to (2) and I⋆ is such
enumeration of all possible sparsest critical k-tuples. This that H(j,:)q ⋆ 6= 0 if and only if j ∈ I⋆. Then Theorem 3
9
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