An Information-Theoretic Approach to PMU Placement in Electric Power Systems

0.2 MB·English

by  Qiao Li

1
An Information-Theoretic Approach to PMU
Placement in Electric Power Systems
Qiao Li, Student Member, IEEE, Tao Cui, Student Member, IEEE, Yang Weng, Student Member, IEEE,
Rohit Negi, Member, IEEE, Franz Franchetti Member, IEEE, and Marija D. Ilic´, Fellow, IEEE
2 Abstract—This paper presents an information-theoretic ap- Currently, there is a significant performance gap between the
1
proach to address the phasor measurement unit (PMU) place- existing research and the desired ‘informative’PMU configu-
0
ment problem in electric power systems. Different from the
ration [6], [7]. One particular reason is that most researches
2
conventional ‘topological observability’ based approaches, this
center around the topological observability criterion, which
n paper advocates a much more refined, information-theoretic
a criterion,namelythemutualinformation(MI)betweenthePMU only specifies that power system states should be uniquely
J measurements and the power system states. The proposed MI estimatedusingminimumnumberofPMU measurements[8].
3 criterion can not only include the full system observability as Based on such criterion, many solutions were proposed, such
a special case, but also can rigorously model the remaining
1 as the ones based on mixed integer programming [9], [10],
uncertainties in the power system states with PMU measure-
binary search [11], and metaheuristics [12], [13]. While it
ments,so astogenerate highlyinformative PMUconfigurations.
]
C Further, the MI criterion can facilitate robust PMU placement is true that all PMU configurations can monitor the power
byexplicitly modelingprobabilisticPMU outages. Wepropose a system states with similar accuracy once the system becomes
O
greedy PMU placement algorithm, and show that it achieves an fully observable, these PMU placement approaches can yield
h. approximationratioof(1−1/e)foranyPMUplacementbudget. quite suboptimal results for the important and current sit-
We further show that the performance is the best that one can
t uation, where the number of installed PMUs is far from
a achieve in practice, in the sense that it is NP-hard to achieve
m any approximation ratio beyond (1−1/e). Such performance sufficient to achieve full system observability. The reason is
guarantee makes the greedy algorithm very attractive in the as follows. Firstly, the ‘observability’criterion is very coarse,
[
practical scenario of multi-stage installations for utilities with whichspecifiestheinformationgainedonthesystemstatesas

1 limited budgets. Finally, simulation results demonstrate near-
binary, i.e., either observable or non-observable. Such crude
v optimalperformanceoftheproposedPMUplacementalgorithm.
approximation essentially assumes that the states at different
4
3 buses are completely independent (with exceptions for buses
Index Terms—Phasor measurement unit, electric power sys-
9 with zero injection), in that the knowledge of the state of a
tems, submodular functions, mutual information, greedy algo-
2 bus has zero information gain on the state of any other bus,
rithm.
1. as long as that bus is not ‘observable’. This is clearly not
0 the case for power systems, where the system states exhibit
2 I. INTRODUCTION high correlations, due to the fundamental physical laws, such
1 SYNCHRONIZED measurement technology (SMT) has as KVL and KCL. Secondly, the observability approaches
:
v beenwidelyrecognizedasanenableroftheemergingreal- neglect important parameters of the power system, such as
i timewideareamonitoring,protectionandcontrol(WAMPAC) transmission line admittances, by focusing only on the binary
X
systems [1], [2]. Phasor measurement unit (PMU), being the connectivity graph. In this sense, if zero injection are not
r
a most advanced and accurate instrument of SMT, plays a considered, the current researches is essentially the classic
critical role in achieving key WAMPAC functionalities [3]. ‘dominating set’ problem [14], where a subset of buses in
With better than one microsecond global positioning system the system are selected, so that every bus is either in the
(GPS)synchronizationaccuracy,thePMUscanprovidehighly subset, or a neighbor of the subset. Such over-simplification
synchronized, real-time, and direct measurements of voltage of the power system is very likely to result in suboptimal
phasors at the installed buses, as well as current phasors of design and significant performance loss. For example, it has
adjacent power branches. Such measurements are vital for beenshownin[6],[7]thatPMUconfigurationscanhavelarge
the efficient and reliable operations of the power systems by influenceon the accuracyof the state estimation, eventhough
both improving the Situational Awareness (SA) of the grid the observability result stays the same.
operators, and facilitating synchronized and just-in-time (JIT) To overcome the performance limitation of current ap-
automated control actions [4], [5] . proaches, in this paper, we advocate a much more refined,
Given the critical role of PMUs for the power system, it information-theoretic criterion to generate highly informative
is important that these instruments are installed at carefully PMU placement configurations. Specifically, we rigorously
chosenbuses, soastomaximizethe ‘informationgain’onthe model the ‘information gain’ achieved by the PMUs states
system states, and achieve desired functionalities efficiently. as the Shannon mutual information (MI) [15] between the
PMU measurements and the power system states. The MI
All authors are affiliated withthe Department ofElectrical andComputer
criterionis verypopularin the statistics andmachinelearning
Engineering,CarnegieMellonUniversity,Pittsburgh,PA,15213USA(email:
[email protected], {tcui, yweng,negi,franzf,milic}@ece.cmu.edu). literature [16], [17], which has found many applications in
2
sensor placementproblems.For powersystems, we will show
that it can include the ‘topological observability’ by current
researchesasaspecialcase.Notonlythis,theMIcriterioncan
alsoincorporateprobabilisticPMUfailures,tofacilitaterobust
PMU placementconfigurations.A related work is the entropy
based approach [18], where the PMU buses were selected, so
that the ‘information content’ of the PMU response signals
Fig. 1. Theone line diagram ofa power system with 5buses. Thesquare
from certain transient-stability program can be maximized. noderepresents themeasurementofactivepowerflowfrombus1to2.Two
In [18], the ‘information’ is represented by the norm of PMUsareinstalled atbus1and4.
the entropy matrix of PMU response signals. Compared to
their method, the MI criterion in this paper directly models
bythenon-referencebusangles,duetothelawofconservation
the uncertainties in the power system states. Further, the MI
of energy. Write the non-reference bus angles in vector form
criterion is rigorous, based on the analyticalDC model of the
as θ = (θ ,θ ,...,θ )T. Note that the system states θ are
power system. 1 2 N
highly correlated statistically, due to the DC model in (1).
As a secondcontributionof thispaper,we presenta greedy
Formally, the dependency of these variables are described by
PMU placement algorithm, and show that it can achieve
the following theorem:
(1−1/e)oftheoptimalinformationgainforanyPMUbudget
Theorem 1: Assume the power system is fully connected.
K. We further prove that the approximation ratio is the best
UndertheDCmodel,θ formsaGMRFwithmeanB−1µand
that one can achieve in practice, by showing that it is NP-
covariance matrix B−1ΣB−1.
hard to approximate with any factor larger than (1 − 1/e).
Proof: Since the power system is fully connected, the
Compared to existing approaches, the greedy algorithm can
matrix B is invertible [24]. Thus, the states can be calculated
not only achieve the best performance guarantee, but also
as θ =B−1Pinj, from which the theorem follows.
can be easily extended to large-scale power systems. Further,
Fig. 1 illustrates a 5-bus power system, with its GRMF
the greedy PMU placement is very attractive in the practical
model shown as the shaded region in Fig. 2. In this case,
scenarioofmulti-stagePMUinstallation,whereutilitiesprefer
the GRMF is formedby connectingtwo-hop neighborsof the
to installthe PMUs overa horizonofmultiple periods,due to
buses in the original power system, as the power injections
limited budgets [19]. In such cases, utilities can simply adopt
are assumed to be independent. We next describe the PMU
the greedyplacementstrategy,as the (1−1/e)approximation
measurement model.
ratio holdsfor anyK. On the otherhand,existing multi-stage
methods [19], [20] may incur significant performance loss if
the multi-period budget changes unexpectedly. B. Measurement Model
Theremainingofthispaperisorganizedasfollows.Section 1) Conventional Measurements: As the DC model is as-
II describes the power system and measurement models, and sumed in this paper, the conventional measurements only
Section III formulates the optimal PMU placement problem. include the real power injection {zconv} and real power flow
i
Section IV proposes the greedy PMU placement algorithm {zconv}. Under the DC model, these measurements can be
ij
andanalyzesitsperformance,andSectionV demonstratesthe described as follows:
numerical results. Finally, Section VI concludes this paper.
ziconv = Biiθi+ X Bijθj +ecionv (2)
II. SYSTEMMODEL j∈Ni
zconv = B (θ −θ )+econv (3)
In this section, we formulate a Gaussian Markov random ij ij j i ij
field (GMRF) model [21] for the system states, and describe where econv and econv are measurement noises, which are
i ij
the measurement models. distributed as N(0,κconv) and N(0,κconv), respectively.
i ij
2) PMU Measurements: A PMU placed at bus i can
A. GMRF Model for Phasor Angles measure both the voltage at bus i and the currentsof selected
A DC power flow model [22] is assumed in this paper, incident branches. This implies that the phasor angles of
where the power injection Pinj at bus i can be expressed as corresponding adjacent buses can also be directly calculated.
i
follows: Thus,we assume the followingequivalentPMU measurement
Piinj =Biiθi+ X Bijθj (1) model. We associate a PMU placed at bus i with a vector
zPMU(i), such that
j∈Ni
In above, Ni is the set of neighboring buses of i, Bij is ziPMU(i) = θi+ePiMU (4)
imaginarypartofthenodaladmittancematrixY,andθi isthe zPMU(i) = (θ −θ )+ePMU,∀j ∈P (5)
ij i j ij i
voltage phasor angle at bus i. The uncertainties in the power
injectionvectorPinjcanoftenbeapproximatedasGaussianby where ePMU and ePMU are measurement noises with distribu-
i ij
existingstochasticpowerflow methods[23]. Inthispaper,we tion N(0,κPMU) and N(0,κPMU), respectively. P ⊆ N is
i ij i i
assumethatPinjisdistributedasN(µ,Σ).Denotebus0asthe a subset of neighbors of bus i. This is because of the PMU
slackbus.Weareonlyinterestedinthestatesatnon-reference channel limits, which implies that only a subset of adjacent
buses, as the angle of the slack bus can be uniquely specified branches can be monitored. The variances κPMU and κPMU
i ij
3
reduction corresponds to the conditional MI:
I(θ;zPMU(S)|zconv)=H(θ|zconv)−H(θ|zPMU(S),zconv)
The MI criterion is widely adopted in the machine learning
literature to generate highly informative sensor placement
configurations[16].ForthePMUplacementprobleminpower
systems, we claim that the MI criterion is also intimately
related to the power system observabilityand state estimation
Fig. 2. The probabilistic graphical model forthe non-reference bus angles
accuracy, which we elaborate as follows:
in the power system in Fig. 1. The shaded region illustrates the GMRF for
systemstates.Thepowerinjections areassumedtobeindependent. 1) (Observability): The proposed MI criterion can include
theobservabilitycriterionasaspecialcase.Toseethis,assume
no PMU failure and PMU measurement noise. We claim that
depend on various sources of uncertainties, such as the GPS the maximum information gain is achieved if and only if the
synchronization,instrument transformers, A/D convertersand power system is completely observable from PMU measure-
cable parameters, which can be estimated appropriately [25]. ments. This can be clearly observed from (7), where the MI
Further, PMU failures can be modeled by assuming that each function is maximized if and only if H(θ|zPMU(S)) = 0, in
current measurement ziPjMU(i) outputs a failure message with which case the system states are deterministic giventhe PMU
probability1−aPijMU,andsimilarly,eachvoltagemeasurement measurements zPMU(S).
fails with probability 1 − aPMU, where aPMU and aPMU are 2) (State Estimation Error): The proposed MI function
i ij i
the availability of the current and voltage measurements, is also intimately related to the minimization of the state
respectively. estimation error. In fact, the conditional entropy H(θ|zPMU)
As an illustration, Fig. 2 represents the probabilistic graph can then be expressed as follows:
model for the measurement configuration in Fig. 1, where
the gray nodes represents the measurement variables. From H(θ|zPMU)≅logdetCov(e)+C−Nlog∆ (8)
the figure, it is clear that the information gain of PMU where C is a constant, and e = θ − θˆ (zPMU) is the
MMSE
measurements depends heavily on the placement buses. This
estimationerroroftheMinimumMeanSquareError(MMSE)
will be formalized in the next section by the optimal PMU
estimation of θ given zPMU, and Cov(·) is the covariance
placement problem.
matrix. The error in the above approximation is due to the
quantization, and goes to zero as the quantization step size
III. OPTIMAL PMU PLACEMENT ∆→0. Thus,it isclear thatthe maximizationof MI canalso
lead to minimizing the MMSE error e of the power system
The placement configuration of PMUs should be highly
states. Intuitively, H(θ|zPMU) specifies how ‘peaked’ the
‘informative’ to effectively monitor power system states. In
distribution of estimated power system states behaves around
this paper, we advocate an information-theoretic criterion
the mean value. In the statistics literature, such criterion is
to quantitatively assess the ‘information gain’ that can be
referred to as the ‘D-optimality’ [16].
obtainedfromthePMUmeasurements.Specifically,wemodel
WearenowreadytoformulatetheoptimalPMUplacement
the uncertainties in the system states as the Shannon entropy
problem. Assume that there are a total of K PMUs to be
[15]:
installed in the power system. The goal is to choose a subset
H(θ)=X−p(θ)logp(θ) (6) of PMU configurations S⋆ from a set of candidate PMU
θ
configurations, such that
wherep(θ)istheprobabilitymassfunctionofθ.Inthispaper,
S⋆ ∈arg max F (S), i=1,2. (9)
we assume that the entropies of all variables are calculated i
|S|≤K
afterquantizationwithasufficientlysmallstepsize∆.Thisis
The objective functions are illustrated as follows:
motivatedby the fact thatthe phasor anglesin power systems
are observed by meters with finite accuracy. Denote S as the 1) (PMU Measurements Only) In this case, the objective
set of PMU configurations, where each element s = {i,P } function associated with a PMU placement set S is
i
in the set S correspondsto a candidatePMU configurationas T
1
in (4) and(5).Note thatourmodelis verygeneral,whichcan F1(S) = XIt(θ;zPMU(S)) (10)
T
be used to model PMU channel limits. The information gain
t=1
of the PMU configuration S can be assessed by the entropy
where the dependence on time index t is because the
reduction due to PMU measurements zPMU(S):
power system states θ have time-dependent distribution
I(θ;zPMU(S))=H(θ)−H(θ|zPMU(S)) (7) N(B−1µt,B−1ΣtB−1), due to the changes in real power
injections over time. Thus, the objective function in (10)
whereI(θ;zPMU(S))istheShannonmutualinformation(MI) describesthe‘timeaveragedinformationgain’aboutthepower
betweenthePMU measurementsandthepowersystemstates, system state over a time period of interest (such as one day).
and H(θ|zPMU(S)) is the conditional entropy. Finally, when 2) (With Conventional Measurements) When conventional
conventional measurements are considered, the uncertainty measurements are considered, the objective function should
4
TABLEI
be replaced with the conditional MI function, as follows:
COMPUTATIONTIME
T
F2(S) = 1 XIt(θ;zPMU(S)|zconv) (11) Test Systems Time (Seconds)
T
IEEE14-bus(PMUOnly) 0.7093
t=1
IEEE14-bus(Conventional) 2.5543
Note that it is possible that the time scales can be different
IEEE57-bus(PMUOnly) 2.9141×103
in both cases, as the conventional measurements can have
IEEE57-bus(Conventional) 3.4556×104
much slower sampling rate (on the order of minutes) than
PMU measurements. Having formulated the optimal PMU
placement problem, we will discuss the solutions in the next
Theorem 3: ThegreedyPMU placementin (1) canachieve
section.
anapproximationratioof(1−1/e)forbothobjectivefunctions
F (·) and F (·).
1 2
IV. GREEDY PMU PLACEMENT Proof: The proof is obtained by identifying a key prop-
It is highly desired that the PMUs are optimally placed erty,submodularity,of thePMU placementproblem.Detailed
in the power system. However, the optimal solution is very proof is in Appendix B.
hard to obtain, as the optimal PMU placement problem is We have the following remarks:
NP-complete [26]. In this section, we propose a greedy PMU 1) Optimality: Based on Theorem 2 and 3, we claim
placementalgorithm,andshowthatitcanachievetheoptimal that the greedy algorithm can achieve the best performance
performance guarantee among the class of polynomial time guarantee that is possible. Further, compared to methods
algorithms. such as mixed integer programming [9], [10], binary search
[11], or metaheuristics [12], [13], the greedy algorithm is
not only the best in performance guarantees, but also can
A. Hardness Result
be easily implemented in large-scale systems, due to the low
Beforepresentingthegreedyalgorithm,wefirstdemonstrate computation complexity.
the hardness result. We extend the hardness result in [26], by 2) Multi-stage Installation: The greedy algorithm is very
showingthatthe optimalPMU placementproblemisnotonly attractive in the case of multi-stage installations, where the
NP-hardtosolve,butalsoNP-hardtoapproximatebeyondthe utilities plan to install the PMUs over a horizon of multiple
approximation ratio of (1−1/e): years, due to the limited (and possibly uncertain) budgets. In
Theorem 2: Unless P =NP, there is no polynomial time suchscenarios,thegreedyalgorithmcanalwaysachieveanap-
algorithmfortheoptimalPMUplacementproblemin(9)with proximationratioof(1−1/e)foranygivenK,whereasfixed
better approximation ratio than (1−1/e). multi-stage planning algorithms may suffer from substantial
Proof: See in Appendix A. performance loss when the budget K changes unexpectedly.
We next propose a greedy PMU placement algorithm, which Having formulated the greedy PMU placement algorithm
can achieve the (1−1/e) approximation ratio. and proved its optimality results, we will test it against other
methods in standard IEEE test systems in the next section.
B. Greedy PMU Placement
V. NUMERICAL RESULTS
The greedy PMU placement algorithm is shown in Al-
This section demonstratesthe performanceof the proposed
gorithm 1. Compared to the optimal placement, the greedy
greedy PMU placement algorithm, and compare it with the
algorithm has low complexity, and is easy to implement in
PMU placement results in the literature.
large-scale systems. In each step, the algorithm chooses the
nextcandidatePMU configurationthatcanachievethelargest
A. System Description
‘marginalinformationgain’,wheretheobjectivefunctionF(·)
can be chosen as either F or F , depending on whether In the simulation, real power injections are assumed to be
1 2
conventionalmeasurements are included. normally distributed, and independent across different buses
[23]. For each bus, the standard deviation of the real power
Algorithm 1 Greedy PMU Placement injection is assumed to be 10% of the mean value. For
1: Initialize: S ←∅; each time slot, the mean real power injections at all buses
2: for k =1 to K do are obtained by properly scaling the case profile description
3: S ←S∪{s⋆}, where s⋆ solves the following: of the standard test system [27]. Thus, the MI function at
differenttimeslotsareonlydifferentbyamultiplicativefactor.
s⋆ =argmaxF(S∪{s}); (12)
Each PMU measurement is assumed to fail independently
s6∈S
with probability 0.03 [20]. The standard error of all PMU
4: end for measurements are assumed to be 0.02◦, whereas the standard
5: return S error of conventionalmeasurements are assumed to be 0.57◦.
Finally,forcomparisonpurposes,weconsiderthe‘topological
The next theorem shows that the greedy algorithm can observability’ based PMU placement configurations in [28],
achieve the largest approximation ratio of (1−1/e). which are obtained based on mixed integer programming.All
5
IEEE 14−Bus System, PMU Only IEEE 14−Bus System, With Conventional Measurements
1 1
0.9 0.9
0.8 0.8
Gain0.7 Gain0.7
mation 0.6 mation 0.6
malized Infor00..45 malized Infor00..45
Nor0.3 Nor0.3
0.2 0.2
Upper Bound Upper Bound
0.1 Greedy 0.1 Greedy
Observable Observable
0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Number of PMU Number of PMU
(a) (b)
Fig. 3. Normalized information gain of different PMU placement schemes in the IEEE 14-bus system. (a) with PMU measurements only and (b) with
conventional measurements.
TABLEII Standard Deviation in IEEE 14−Bus System
PMULOCATIONSFORIEEE14-BUSSYSTEM 1
K (POMpUtimOanlly) (PMGUreeOdynly) (CoOnpvteinmtaiolnal) (CoGnrveeendtiyonal) Degree)0.8
1234 4,464,,,4691,,3913 4,4,46,,94,193,1313 4,4,46,,66,1931,413 4,46,6,,669,9,913 Standard Devation (000...246
0
2 4 6 8 10 12 14
Bus Index
simulation results are obtained with MATLAB on an Intel
Xeon E2540 CPU with 8GB RAM. The computation time Fig.4. Standarddeviation ofvoltageangles intheIEEE14-bussystem.
is shown in Table I.
B. IEEE 14-Bus System greedy PMU placement strategy, in that the ‘Greedy’ curve
is very close to the ‘Upper Bound’. Further, the greedy
1) PMU Measurements Only: For this case, the optimal
algorithm has a significant improvement on information gain
PMU locations are calculated by an exhaustive search among
compared to the conventional‘observability’ based approach.
all possible configurations. The PMU locations for both opti-
For example, for K = 3, the improvement is around 20%
malplacementandgreedyplacementfor K ≤4 are shown in
as compared to the observability based placement [28]. This
TableII.Fromthetable,onecanobservethattheoptimalPMU
clearly demonstratesthe performanceloss associated with the
configurationcanchangesignificantlyoverdifferentplacement
coarse observability based criterion. Finally, one can observe
budget K. For example, when K =2, the optimal placement
fromthe ‘UpperBound’curvethat the maximuminformation
is {4,13}, whereas for K = 3, the optimal placement is
gain has a ‘diminishing marginal return’ property, in that the
{4,6,9}. On the other hand, the greedy placement configu-
marginalinformationgain tends to decrease as the number of
ration S (K) always satisfy S (K −1) ⊂ S (K). Thus, the
g g g
installedPMUsinthepowersystemgrows.Thisalsoconfirms
greedy placement strategy is robust against the uncertainties
the submodularity of the MI objective function.
in the placement budget K. To verify the placement results,
weplotthestandarddeviationsofthephasoranglesatallnon- 2) With Conventional Measurements: In this case, real
referencebusesinFig.4.Fromthefigure,onecanobservethat power flow measurements are assumed to be configured at
thestateatbus3hasthelargestvariance.However,bus3isnot all branchesand buses. The resulting PMU configurationsare
chosen as the first PMU bus, since, intuitively,it is connected shown in Table II. From the table, one can conclude that
toonlytwoneighborsinthesystem(seethetopologyin[27]). the optimal PMU placement is even more vulnerable to the
Instead, bus 4 is chosen, since it has five neighbors. changes in the PMU placement budget K than the PMU
Fig. 3 (a) shows the normalized information gain for the only case, as the configurations changes significantly as K
IEEE 14-bus system with only PMU measurements. In the increases. On the other hand, the greedy algorithm is robust,
figure, the ‘Upper Bound’ curve is computed by the op- as S (K − 1) ⊂ S (K) for any K ≥ 1. The normalized
g g
timal PMU configuration assuming no PMU failure. Thus, informationgain is shown in Fig. 3 (b). As the power system
it overestimates the information gain on the system states. is already observable with conventional measurements, we
One can easily observe the near optimal performance of the use the same configuration for the ‘Observable’ curve as the
6
IEEE 57−Bus System, PMU Only IEEE 57−Bus System, With Conventional Measurements
1 1
0.9 0.9
0.8 0.8
Gain0.7 Gain0.7
mation 0.6 mation 0.6
malized Infor00..45 malized Infor00..45
Nor0.3 Nor0.3
0.2 0.2
0.1 Greedy 0.1 Greedy
Observable Observable
0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30
Number of PMU Number of PMU
(a) (b)
Fig.5. Normalizedinformationgainofdifferent PMUplacement schemesintheIEEE57-bussystem.(a)withoutconventional measurementsand(b)with
conventional measurements.
TABLEIII
are shown in Table III. The normalized information gain is
PMULOCATIONSFOR57-BUSSYSTEMBYGREEDYALGORITHM
shown in Fig. 5 (b), where the greedy algorithm is compared
TestScenarios PMULocations againstthesameconfigurationinthepreviouscase,Similarto
IEEE57-Bus(PMUOnly) 9,56,18,31,12,49,29,6, the case withoutconventionalmeasurements,one can observe
25, 54, 20, 41, 38, 51, 32, the significant performance gain of the greedy algorithm.
13,27,53,57,15,19,8,30,
Note that, in this case, the power system is observable with
50,17,5,16,42,52,48,55,
44,24,34 conventionalmeasurements.Therefore,theobservabilitybased
IEEE57-Bus(Conventional) 56, 31, 19, 12, 54, 49, 25, methodsfailto computeeffectivePMU configurations,dueto
41,32,9,29,18,6,50,20, the coarse modeling of the uncertainties in the power system
57, 27, 53, 38, 30, 13, 42,
states.
51,17,55,52,5,24,34,43,
16,44,8,10
VI. CONCLUSION
This paper proposed an information-theoretic approach to
previous case. One can observe that the performance gain is
address the phasor measurement units (PMUs) placement
larger compared to the case with only PMU measurements.
for power system. Different from the topological observabil-
This, again, confirms the conclusion that the pure topology
ity based criterion in the literature, this paper proposed a
based observability criterion can not efficiently model the
muchmorerefined,information-theoreticcriterion,namelythe
uncertainties in the power system states.
mutual information (MI), as the PMU placement objective
function. The proposed MI criterion can not only include
C. IEEE 57-Bus System the complete observability criterion as a special case, but
also can accurately model the uncertainties in the system
1) PMUMeasurementsOnly: FortheIEEE57-bussystem,
states. We further proposed a greedy PMU placement algo-
it is computationally infeasible to obtain the optimal PMU
rithm, and showed that it achieves an approximation ratio of
configuration for large K. In such case, we only demonstrate
(1−1/e) for any PMU budget K, which is the best guar-
the performance of the greedy PMU placement, and compare
antee among polynomial-time algorithms. Such performance
it against the observability based results in [28]. For the case
guaranteemakesthe greedyalgorithmattractive in the typical
with only PMU measurements, the resulting PMU configura-
scenario of phased installations, as the performance is robust
tions are shown in Table III, and the normalized information
to the changes in the PMU budget. Finally, the performance
gainisshowninFig.5(a).SimilartotheIEEE14-bussystem,
of the proposed PMU placement algorithm was demonstrated
one can conclude that the greedy algorithm can achieve a
by simulation results.
significant information gain compared to the observability
based criteria. This is because of the much more refined
modeling of the MI function, which can effectively capture APPENDIX A
the remaining uncertainties in the states of the power system. PROOF OF THEOREM2
Further, the information gain of the greedy placement curve Proof: We prove the hardness result by constructing
also demonstrates the ‘diminishing marginal return’ property. polynomial-time reduction of an arbitrary instance of the
2) With Conventional Measurements: In this case, real max k-cover problem [29] to a PMU placement problem.
power flow measurementsare assumed to be configuredat all Thus, the hardness result easily follows from the (1−1/e)
branchesandbuses. TheresultinggreedyPMU configurations inapproximability of the max k-cover problem [29]. The
7
max k-cover problem is as follows. We are given a set For nondecreasing submodular functions, the following guar-
of elements U = {1,2,...,N}, and a collection of sets antee always holds for the greedy algorithm [30]:
P = {P ,P ,...,P }, where each set P is a subset of Lemma 1: Let a set function F(·) be submodular, nonde-
1 2 M i
U. The task is to compute a subcollection of K subsets creasing and F(∅) = 0. For any K ≥ 1, denote S⋆(K) and
P ,P ,...,P , such that the cardinality | ∪K P | is S (K) as the optimal solution to the problem (18) and the
l1 l2 lK k=1 lk g
maximized. Now, the reduction is as follows. Given a max solution obtained by Algorithm 1, respectively. Then,
k-cover problem instance, we construct a power system with
F(S (K))≥(1−1/e)F(S⋆(K)) (19)
N + 1 buses, so that two buses (i,j) are connected by g
a transmission line if and only if they both appear in a
always holds. Thus, Algorithm 1 can achieve at least an
certain subset in P. Further, we associate each subset P
i approximation ratio of (1−1/e).
in P with a PMU, which is installed at a fixed (but can
be arbitrarily chosen) bus in P . Assume that the covariance
i
matrix of the power injectionsPinj is Σ=γB2, where γ is a B. Proof of Theorem 3
proper scalar that will be discussed later. Thus, according to Proof: We now prove the theorem by showing that the
Theorem 1, the resulting GMRF of θ has covariance matrix objective functions F (·) and F (·) for the optimal PMU
1 2
γI. Finally, chooseγ so thateach discrete randomvariable θi placement problem satisfy all assumptions in Lemma 1. We
hasentropy1.AssumethereisnoPMUfailure,andzeroPMU first fix time index t, and consider the case without conven-
measurementnoise,wecanwritetheMIobjectivefunctionas tionalmeasurements.ItiseasytoseethatI (θ;zPMU(∅))=0.
t
follows: We next verify (17) as follows:
I(θ;zPMU(S)) = H(θ)−H(θ|zPMU(S)) (13) I(θ;zPMU(A∪s))−I(θ;zPMU(A)) (20)
(=a) N −H(θ|zPMU(S)) (14) (=a) I(θ;zPMU(s)|zPMU(A)) (21)
(b)
= N −|∪k∈SPk| (15) (=b) H(zPMU(s)|zPMU(A))
= |∪k∈SPk| (16) −H(zPMU(s)|θ,zPMU(A)) (22)
where (a) is because the phasor angles are independent, and (=c) H(zPMU(s)|zPMU(A))−H(zPMU(s)|θ) (23)
each has entropy 1, by construction. (b) is because given
the PMU measurements, the uncertainties only remain at the In above, time index is omitted for notation simplicity. (a)
phasor angles of the ‘unobservable buses’ ∪ P . Denote follows from the chain rule of MI [15]. (b) follows from
k∈S k
P as the set of all possible PMU placement configurations. the definition of conditional MI. (c) is because zPMU(s) is
Thus, if we can solve the optimal PMU placement problem conditionally independent of zPMU(A) given θ, due to the
by maximizing I(θ;zPMU(S)) subject to |S| ≤ k beyond measurement model in (4) and (5). This can also be clearly
(1 − 1/e) in polynomial time, we can also achieve the observed from the probabilistic graphical model in Fig. 1(b),
same performance guarantee for the max k-cover problem in where state variables θ serve as the parent nodes of the
polynomial time, from which the theorem follows. PMU measurementsin the ‘Bayesian part’ of the graph. Note
that H(zPMU(s)|zPMU(A)) in (23) is decreasing in A, as
APPENDIX B conditioning always reduces entropy [15]. Since the second
PROOF OF THEOREM3 term in (23) is independentof A, (17) follows easily. Finally,
asconditionalMIisalwaysnonnegative,onecandeducefrom
We now prove the (1 − 1/e) performance guarantee of
Algorithm 1. The key lies in identifying the submodular (21)thattheMIfunctionI(θ;zPMU(A))isalsonondecreasing.
Now, the above analysis can be immediately extended to
property of the PMU placement problem, which is a widely
used concept in combinatorial optimizations. a time period of length T, where (17) holds for F1(·) by
summing up the inequalities corresponding to each time slot
t, and then dividing both sides by T. Thus, we conclude that
A. Introduction to Submodular Functions
the claim holds for F (·). Similarly, an identical analysis can
1
A set function F is called submodular [30] if
be carried out in the case with conventional measurements,
F(A∪{k})−F(A)≥F(B∪{k})−F(B) (17) where all functions in (21)-(23) hold when conditioned on
conventionalmeasurements.
for any sets A and B such that A⊆B. Essentially, this is the
‘diminishing marginal return’ property, which, in the context
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