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Electric Power Systems Research 106 (2014) 51– 61

Contents lists available at ScienceDirect

Electric Power Systems Research

jou rn al hom e page: www.elsev ier .com/ locate /epsr

MU placement for dynamic equivalencing of power systems underow observability constraints�

oel E. Anderson, Aranya Chakrabortty ∗

lectrical & Computer Engineering Department, North Carolina State University, Raleigh, NC 27695, USA

r t i c l e i n f o

rticle history:eceived 10 May 2013eceived in revised form 5 August 2013ccepted 7 August 2013

eywords:MU placementynamic equivalencing

a b s t r a c t

In this paper we develop two graph-theoretic algorithms for placing Phasor Measurement Units (PMUs)in a multi-area power system network with the objective of identifying its dynamic equivalent model.The system is considered to be divided into clusters of synchronous generators and loads, with each areaconnected to other sets of areas through designated transmission networks. We first show that in orderto derive the equivalent line parameters connecting the different areas we must have PMUs placed atthe minimum vertex cover of the bipartite graphs formed between every pair of node-sets arising fromthe boundary buses of these areas. Considering further that the number of tie-lines observable from any

bservabilitydentificationlgorithms

given PMU is constrained by an upper limit, we derive two sets of algorithms to compute the sub-optimalminimum cover, first for a bipartite graph and then for any general topology. The respective algorithmsare referred to as CONPLAC and CONITPLAC. Results are illustrated using a IEEE 34-bus system pointingto the robustness of the proposed algorithms against time-varying network topology. Finally, we presentstatistical analyses to describe how the final set of chosen PMU locations and the computational time of

on ne

these algorithms depend

. Introduction

The problem of sensor placement in electric power systems hasmerged with renewed interest over the past few years owingo the tremendous advancement of instrumentation technologiesuch as Wide-area Measurement Systems (WAMS) [1]. Triggeredy the smart grid investment initiatives of the US Department ofnergy, GPS-synchronized Phasor Measurement Units (PMU) areurrently being deployed at different nodes in the US power sys-em for continuous real-time monitoring of the system dynamicsollowed by rigorous postmortem data analysis to enhance grid reli-bility and stability. One primary concern of current interest in theynchrophasor research community is to determine the optimalsage of PMU data as thousands of PMUs get deployed in the Northmerican grid over the next few years yet minimizing the computa-

ional cost associated with such usage. Majority of the optimal PMUelection methods so far has focussed on observability analysis,tarting from early papers such as [2,3] to more recent algorithms

n [4,5] using combinatorial optimization-based techniques. Sev-ral of these approaches rely on meta-heuristic methods such asimulated annealing [6] and particle swarm optimization [7], and

� This research is partially supported by NSF grant ECCS 1062811.∗ Corresponding author. Tel.: +1 9195133529.

E-mail addresses: jeander4@ncsu.edu (J.E. Anderson), achakra2@ncsu.edu,ranya.chakrabortty@ncsu.edu (A. Chakrabortty).

378-7796/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2013.08.002

twork size, complexity and measurement constraints.© 2013 Elsevier B.V. All rights reserved.

generally lead to a NP-hard solution, which become numericallyintractable as the size of the power grid increases. An alternativeapproach for solving the observability problem has recently beenpresented in [8] by using graph-theoretic optimization methodsthat maximize the number of observed buses while minimizingthe number of PMUs used. With increase in the number of PMUsin the US grid over the past decade, placement has also been dic-tated by several other driving purposes such as state estimation [9],and voltage monitoring near wind power sources [10]. An illustra-tive case study of this problem for the New York grid has recentlybeen presented in [11]. An excellent taxonomy of various numer-ical optimization methods used for solving these PMU placementproblems can be found in [12].

In this paper we address the problem of optimal selection ofPMU locations from the standpoint of a new application – namely,identifying the dynamic equivalent electromechanical models ofthe coherent areas in a power system interconnected via networksof transmission lines. In a recent paper [13] we addressed theproblem of reducing a two-area power system to its two-machinedynamic equivalent using transient phasor measurements tappedfrom the terminal buses of the radial transmission line intercon-necting the two areas. We considered the structure of the two-areasystem as in Fig. 1(a), where all the generators inside each area,

irrespective of their internal connection topologies, are assumedto be connected to a single terminal bus, namely Bus 1 and Bus 2,respectively for Area 1 and Area 2. The transfer path is assumed tobe radial, i.e., Bus 1 and Bus 2 are connected by a single transmission

52 J.E. Anderson, A. Chakrabortty / Electric Power Systems Research 106 (2014) 51– 61

stem

le1mtetw

ebrpw

Fig. 1. Two-area power sy

ine, which may be the equivalent of several parallel lines. Consid-ring that Phasor Measurement Units (PMUs) are installed at buses

and 2, [13] derived algorithms by which the dynamic measure-ents of the voltages, currents and frequencies available from

hese PMUs can be utilized to identify a two-machine dynamicquivalent of the two-area system, as shown in Fig. 1(b), in ordero characterize its inter-area dynamics. This identification problemas referred to as Interarea Model Estimation (IME).

However, in an actual power system it is highly impractical toxpect all generators in an area to be connected to a single ‘ideal’us. The transmission network cannot be expected to be strictly

adial either. Instead a more realistic representation of the idealower system of Fig. 1(a) will be the system shown in Fig. 2(a),here generators in Area 1 and Area 2 are connected arbitrarily to

(a) Two-area system w

(b) Equivalen

Fig. 2. Radial equivalencing of two-

with radial transfer path.

respective sets of multiple buses, which, in turn, are connected arbi-trarily to a transmission network (TN) serving as a bridge betweenthe two areas. Four sets of nodes or buses, referred to as the bor-dering buses, are identified in this figure. A bordering bus set of anyarea is defined as the intersection of the node set of that area andof the TN. The bordering bus set of the TN, one each correspond-ing to the two areas, is defined as its nodes that share a connectionwith the bordering buses of the respective areas. In the recent paper[14] these bordering cutsets have been referred to as nodal cutsets.Throughout our discussion, we assume that the buses belonging toArea 1, 2 and the TN are known distinctly.

Our objective is to facilitate the IME problem for this compli-cated system by reducing it to an equivalent radial topology shownin Fig. 2(b). In other words, assuming that PMUs are located at some

ith meshed TN

t radial TN

area system with meshed TN.

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elected boundary buses in Area 1, Area 2 and the TN, we want to usehe voltage, current and frequency measurements available fromhese PMUs to derive the following six quantities:

. Equivalent voltage phasors V1 and V2 at the two respective ter-minal buses 1 and 2

. Line current I flowing across the radial path connecting the twoareas

. Equivalent impedances z1, ze and z2 as indicated in Fig. 2(b).

In the following we discuss how this network reduction prob-em can be handled using fundamental graph-theoretic circuit laws14], and related network cutset problems some preliminary resultsn which have been presented in [15]. Thereafter, we investigatehe problem of placing a minimum number of PMUs at the bound-ry buses so that their measurements can be used for completeerivation of the equivalent voltages and currents. We first presenthe results for a two-area power system, and then extend themo systems with multiple interconnected areas. Considering fur-her that the number of tie-lines observable from any given PMU isonstrained by an upper limit, we derive an algorithm to computehe sub-optimal minimum cover of such multi-area systems with

eshed network structure.Since both algorithms proposed in this work are inherently

raph-theoretic, it is important to compare their structural prop-rties with [8]. The main difference between our approach and [8]s that in [8] the authors do not consider any constraint on the

easurement capacity of the PMUs. As a result, their algorithmsre all NP-complete, similar to the graph-theoretic algorithms foraximum matching in bipartite graphs. Our problem, on the other

and, is severely restrained by an upper bound on the number ofariables that can be measured by a PMU, and, therefore, yieldsnly sub-optimal results at the cost of this practical consideration.urthermore, the simulation results in [8] showed a 97% optimalMU placement without constraints, while our algorithm achieves

comparable 96% optimal result under the channel observabilityonstraint (please see example 5.2). The remainder of the papers organized as follows. Section 2 formulates the PMU placementroblem for reduction of two-area systems. Section 3 extends ito multi-area systems. Sections 4 and 5 present two sub-optimallgorithms for placement under channel observability constraints.ection 6 discusses statistical properties of the algorithms and theirensitivity with respect to network size, meshness and observabil-ty constraint. Section 7 concludes the paper.

. Network reduction and PMU placement

We first consider the two-area power system shown in Fig. 2(a).t should be noted that the boundary buses in the areas as well asn the TN may share edges with buses internal to the respectiveegions, including other boundary buses in that region. However,e are interested in the power flow and current transfer from one

egion to another, and, therefore, we will focus only on the edgeshat are shared between the boundary buses of one region withhose of another. Let D1T denote the adjacency matrix of the cutsetraph joining Area 1 and the TN, while D2T denote that for the cutsetoining the TN with Area 2. Clearly both of these graphs are bipartiten nature. Denoting

B1 as the set of boundary nodes in Area 1BT1 as the set of boundary buses of TN that share edges with Area

1,BT2 as the set of the boundary buses of TN that share edges withArea 2, andB2 as the set of boundary buses of Area 2,

er Systems Research 106 (2014) 51– 61 53

the adjacency matrices for these two bipartite graphs are

D1T (s, k) ={

1 if (s ∈ B1)∼(k ∈ BT1 )

0 otherwise(1)

D2T (s, k) ={

1 if (s ∈ BT2)∼(k ∈ B2)

0 otherwise(2)

where ∼ means that the two nodes are connected. The elements ofeach of these boundary sets are indexed in the fixed order {1, 2, 3,4, . . . }.

In order to reduce the boundary nodes to respective equivalentnodes, we recall the following three simple results from power sys-tem modeling. For simplicity of notation, we refer to the two-areameshed system of Fig. 2(a) as �1, and the reduced two-machineradial system of Fig. 2(b) as �2.

1. Result 1: The phasor current flowing from node 1 to node T1 in�2 is equal to the sum of the phasor currents flowing throughthe edges of D1T in �1. This is also equal to sum of the phasorcurrents flowing through the edges of D2T .

2. Result 2: The net effective power flowing out of Area 1 in �1 isequal to the net effective power flowing out of node 1 in �2. Also,the net effective power flowing into the TN through the graphD1T in �1 is equal to that entering node T1 in �2. Similarly, thetotal power out of the TN in �1 is equal to that out of node T2 in�2, and the total power entering Area 2 through the graph DT2in �1 is equal to that entering node 2 in �2.

3. Result 3: The impedance of each edge in D1T and D2T is known.

From Result 1, it follows that

I =∑

i, j

D1T(i, j) = 1

Iij =∑

i, j

D2T(i, j) = 1

Iij (3)

where Iij � Iij∠�Iij (or, equivalently Iijes�Iij where s = √−1) denotes

the phasor current through any edge connecting node i of one graphto node j of another. Similarly, from Result 2 we have:

V1 � V1∠�1 =

∑i∈B1

{Vi(∑

j

D1T (i, j) = 1I∗ij)}

I∗(4)

VT1 � VT1∠�T1 =

∑i∈BT1

{Vi(∑

j

D1T (j, i) = 1I∗ji)}

I∗(5)

VT2 � VT2∠�T2 =

∑i∈BT2

{Vi(∑

j

D2T (i, j) = 1I∗ij)}

I∗(6)

V2 � V2∠�2 =

∑i∈B2

{Vi(∑

j

D2T (j, i) = 1I∗ji)}

I∗(7)

where ‘∗’ denotes complex conjugate. In the above derivations wehave assumed that there is no current loss in the TN. If the systemis serving intermediate loads at selected nodes inside the TN, thenthe currents will vary accordingly. It should also be noted that since

Result 1 implies the equivalence of active power at the respectivenodes of �1 and �2, for a lossless system we must have

R(V1 I∗) = R(VT1 I∗) = R(VT2 I∗) = R(V2 I∗) (8)

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4 J.E. Anderson, A. Chakrabortty / Electri

here R( · ) denotes the real part of a complex number. Once theoltage phasors (4)–(7) are computed, the line impedances follows

1 = V1 − VT1

I, ze = VT1 − VT2

I, z2 = VT2 − V2

I. (9)

If the transmission line is lossless, the line reactances can belso be written in terms of active power balance resulting in theollowing expressions:

x1 = V1VT1 sin(�1 − �T1)

R(V1 I∗), x2 = VT1V2 sin(�T1 − �2)

R(VT2 I∗),

xe = VT1VT2 sin(�T1 − �T2)

R(VT1 I∗)(10)

ll quantities on the RHS of the three equations above are avail-ble from (3)–(7). However, in order to compute the five quantitiesn (3)–(7) we must know the voltage phasors at each and everyoundary nodes contained in B1, BT1, BT2 and B2 as well as the lineurrents through each and every edges of D1T and D2T . Ideally thatndicates that PMUs should be installed at each and every bound-ry node. In reality, however, the number of PMUs is intendedo be minimal owing to constraints such as cost and installation.o solve this problem, we consider Assumption 3 and recall that

PMU installed at any boundary node can measure the individ-al edge currents entering or leaving it. Then it naturally followshat to compute all boundary voltages and all cutset currents it isufficient to place the PMUs in the minimum vertex cover of theipartite graphs D1T and D2T . For any general graph, its maximumatching can be identified by using the augmenting path algorithm,opcroft–Karp algorithm, etc. [16], and the minimum vertex coveran be selected from iterative permutation of these matchings.lso, since the branch impedances are known following Assump-

ion 3, one can easily compute the voltages at all nodes in the cutsetsing these measured currents and the voltages from the PMUs. Forxample, in this case

˜1 = Va + z1aI1a, Vb = V2 − z2bI2b, (11)

˜3 = Vc + z3c I3c, V4 = Vc + z4c I4c. (12)

Once all the bus voltages and line currents in the cutset arenown, the equivalent radial model can be derived by using (3)–(7).

. Network reduction for multi-area power systems

The results from Section 2 can be easily extended to power sys-ems with more than two areas. We consider a n-area power systemith n ≥ 2, where each area is connected arbitrarily to several sets

f transmission networks (TN) or other areas. An example is shownn Fig. 3(a), where the boundary buses of Area 1 are connected tohree sets of transmission network, namely TN-a, TN-b and TN-c,ach of which, in turn, may share boundary buses with other areasnd TN’s. The internal topologies of these three TN’s are describedy the graphs GTA, GTB and GTC , respectively, as indicated in the fig-re. Referring to an area or a TN simply as a ‘region’, our strategy oferiving equivalent boundary nodes under such a condition followstraightforward from network reduction:

If the boundary buses of a region are connected to the boundaryuses of m other regions, then it will have m equivalent nodes corre-ponding to an one-to-one connection with its neighboring regions.

For example, in Fig. 3(b), we consider the equivalent generator

n Area 1 to be connected to three equivalent buses, namely Bus, Bus 2 and Bus 3, corresponding to its connection to TN-a, TN-

and TN-c, respectively. Let the set of boundary buses of Area 1e B1, of TN-a be Ba, of TN-b be Bb and of TN-c be Bc . Let B1a ⊂

er Systems Research 106 (2014) 51– 61

B1 be the set of boundary buses of Area 1 which share a directconnection with elements of B1, B1b ⊂ B1 share a direct connectionwith elements of Bb, and B1c ⊂ B1 share a direct connection withelements of Bc . We then define D1a as the adjacency matrix forthe bipartite cutset between B1a and Ba, D1b as that between B1b

and Bb, and D1c as that between B1c and Bc . In order to determinethe equivalent voltage phasors at these buses, we repeat the threeresults listed in Section 2. For simplicity of notation, we refer to thefull-order meshed system of Fig. 3(a) as �1, and the reduced-orderradial system of Fig. 3(b) as �2.

1. Result 1: The phasor current flowing from node 1 to node T1i,i = a, b, c, in �2 is equal to the sum of the phasor currents flowingthrough the edges of D1i in �1.

2. Result 2: The net effective power flowing out of Area 1 from Bj

in �1 is equal to the net effective power flowing out of nodej in �2, j = 1, 2, 3. The net effective power flowing into any ofthe TN’s through the respective graphs B1a, B1b and B1c in �1 isequal to that entering node T1a, T1b and T1c in �2, respectively.Similarly, the total power out of the TN’s in �1 is equal to thatout of equivalent boundary nodes in �2.

2. Result 3: The impedance of each edge in D1a, D1b and Dic isknown.

The equivalent bus voltages and currents can then be calculatedas

Vk � Vk∠�k =

∑i∈Bka

{Vi(∑

j

Dka(i, j) = 1I∗ij)}

I∗1,

I1 =∑r, s

D1a(r, s) = 1

Irs, I2 =∑r, s

D1b(r, s) = 1

Irs, I3 =∑r, s

D1c(r, s) = 1

Irs

(13)

where k = 1, 2, 3. It should also be noted that in the example ofFig. 3(a), TN-b shares a connection with both Area 2 and TN-c.Therefore, the equivalent bordering network of TN-b, as shown inFig. 3(b), consists of two equivalent nodes, one corresponding toeach region. Following (13), the equivalent voltage phasor at anybordering bus connecting Region x to Region y, in general, can bewritten as

Vxy � Vxy∠�xy =

∑i∈Bxy

{Vi(∑

j

Dxy(i, j) = 1I∗ij)}

I∗xy

,

Ixy =∑r, s

Dxy(r, s) = 1

Irs(14)

where Bxy is the set of nodes of Region x that share a connectionwith Region y, and Dxy is the adjacency matrix of the bipartite cutsetconnecting the two regions. Furthermore, since B1a ∩ B1b ∩ B1c /= 0necessarily, PMUs must be placed at the minimum cover of thebipartite graph existing between B1 and the total border set (Ba ∪Bb ∪ Bc) instead of the unions of the minimum covers of (B1a, Ba),(B1b, Bb) and (B1c, Bc). Consider the three node sets B1 = {1, 2, 3},B2 = {a, b} and B3 = {c, d} with respective bipartite graphs as shownin Fig. 4. Denoting the minimum cover as M( · ), we have

M1 � M(B1, B2) = M(1, 2, 3, a, b) = {a, b}

M2 � M(B1, B3) = M(1, 2, 3, c, d) = {c, d}

implying M1 ∪ M2 = {a, b, c, d}. However, from Fig. 4 it is obviousthat M12 � M(B1, B2 ∪ B3) = {1, 2, 3}, i.e., cardinality of M12 is less

J.E. Anderson, A. Chakrabortty / Electric Power Systems Research 106 (2014) 51– 61 55

(a) Multi-area power system

(b) Network equivalen t

em w

tB

4c

w

Fig. 3. Multi-area power syst

han that of (M1 ∪ M2), the underlying reason being the fact that12 ∩ B13 /= 0.

. PMU placement under restrictions in the number of

hannels

So far we have showed how the necessary criterion for ‘net-ork equivalencing’ reduces to finding the minimum cover of the

Fig. 4. PMU placement on min cover of multiple bipartites.

ith equivalent transfer paths.

bipartite graphs formed between the boundary node sets of areasand their adjoining TNs. Since there are no constraints placed on thenumber of measurable edges from any node, we refer to these algo-rithms as the unconstrained algorithms. One significant assumptionmade, however, is that all the PMUs are capable of measuringeach individual current phasors (both magnitude and phase angle)entering or leaving the respective node where they are located. Inpractice, there may be a restriction on the number of measure-ment channels for every PMU, which means that if the numberof edges incident upon any node in the cutset matrix exceeds thechannel threshold then the PMU located at that node will not beable to measure each individual current. It will rather measure onlym designated channel currents, m > 0 being the threshold number.Such constraints in flow observability has been reported for sev-eral PMUs operating in the Eastern Interconnect [11]. This typeof constrained observability is also relevant when one or moreanalog channels of a PMU start malfunctioning, or are manipu-lated by extraneous malicious attacks, under which conditions themeasurements from those channels cannot be relied on any more.Obviously under such a situation, the number of PMUs need to bemore than the minimum vertex cover in order to solve for thesemissing currents. To solve this problem, while still maintaining sub-minimality of the number of PMUs, we consider a bipartite graphG between two independent node sets P and Q, and present thefollowing algorithm, which finds the minimum vertex cover of G

given the constraint that no chosen vertex can cover more than kedges. We illustrate the algorithm through the examples shown inFigs. 8 and 9, with the constraint k = 2. For simplicity, we refer tothe algorithm as CONPLAC, implying ‘constrained placement’.

56 J.E. Anderson, A. Chakrabortty / Electric Pow

4

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Fig. 5. PMU placement for IEEE 34-bus system.

.1. Step 1: Fig. 8(a)

The first step of the algorithm is to take all nodes of degreereater than k (referred to as supernodes) and replace them withodes of degree 1 (called subnodes) that each correspond to aingle edge of the original node. These nodes are kept in groups

ccording to the supernode that they came from. All other nodesill be referred to as normal nodes. This newly created graph will

e called G′. For this example, note that there are 6 supernodesn the original graph, creating 6 supernode groups in the expanded

ig. 6. Effect of network size on: left – number of marked nodes (unconstrained), middlor Graph 1, N2=node index for Graph 2.

er Systems Research 106 (2014) 51– 61

version – nodes 1, 2, 3, 4, C, and E. If a connection exists between twosupernodes, it is denoted by a connection between two subnodes,say 2 and C.

4.2. Step 2: Fig. 8(b)

Next, an unconstrained minimum cover algorithm using Konig’stheorem [17] is used to find the minimum cover C for G′. Any node inC is considered a marked node, shown by red squares. Recalling thatC may not be unique, the cover set with most marked normal nodescan be chosen. This will give the algorithm a better chance of findinga more optimal solution. Following the preference described above,normal nodes are marked above subnodes.

4.3. Step 3: Figs. Figs. 8(c)–9(f)

Next, each supernode group is categorized into three possibili-ties depending on the number of marked nodes n∗

iin the ith group.

Each group can be classified as one of three cases:

4.3.1. n = kIn this case, the group is re-united into its original node, and

the edges corresponding to marked subnodes are marked. Thesemarked edges are the ones that the supernode covers. All supernodegroups of this characteristic are grouped together before continuingto the next case. In Fig. 8(b), the only group fulfilling n = k is thegroup for supernode E. In Fig. 8(c), this group is recombined intonode E, and the two edges corresponding to the marked nodes aremarked with stars to signify that they are covered by the supernode.

4.3.2. n > kOnce n = k cases have been handled as above, the nodes with n > k

in the marked graph are considered next, starting with the groupwith the largest n. First, the partner node (when two nodes sharean edge, they will be called partner nodes) of each marked noderesiding in a supernode group where n < k will be considered. Thispartner node will be marked, and it’s partner node in the originalgroup will be unmarked. This will decrease n by 1 in the originalgroup, and increase n by 1 in the partner group. If the partner groupor the original group has been increase to the n = k case, then it is

combined as described in step 1. If n is still greater than k, then thisstep will be repeated. If all marked nodes with partner nodes insupernode groups of n < k have been marked and n is still greaterthan k, then marked nodes with unmarked normal partner nodes

e – number of marked nodes (constrained), right – execution time, N1=node index

J.E. Anderson, A. Chakrabortty / Electric Power Systems Research 106 (2014) 51– 61 57

unco

atmlanFCFgnP

4

nteagntitbacg1oNgtii

5

recSbnlb

Fig. 7. Effect of meshness on number of marked nodes (left –

re evaluated instead. Marked nodes are exported in this fashiono groups with the smallest n value available. This normal node is

arked, and the subnode is left unmarked, as before. This repeats asong as necessary until n = k, at which point the group is recombineds in Case 1. If all possible partner nodes have been marked and

is still greater than k, then the graph is declared unsolveable.ig. 8(d) shows the first step of this case. The supernode group for

has exported it’s first marked node to it’s partner in group 1. Inig. 8(e), the next marked node exported is to group 3. Note thatroup 2 was not chosen because it has n = 1, which is greater than

= 0 corresponding to groups 1, 3, and 4. The final assignment ofMUs is shown in Fig. 8(f).

.3.3. n < kThe final step is to consider the remaining (if any) groups with

< k. These groups are considered in increasing order of n andhe number of subnodes in the group. First, the partner node ofach subnode is evaluated for each of it’s edges. If all of it’s edgesre either marked or correspond to members of other supernoderoups of case n < k, then it is unmarked and it’s unmarked part-er nodes in groups of case n < k are marked instead. If any ofhese groups is pushed to the n = k case, then it is recombined asn Case 1. If there are no such partner nodes for a supernode group,hen it is simply recombined as in Case 1. When no more uncom-ined supernode groups remain, then the algorithm is complete,nd all marked nodes represent the minimum vertex cover of theonstrained case. This progression is shown in Fig. 9(a)–(f). First,roup 1 pulls in the marked node A, and is recombined into node. Next, group 3 is evaluated (because it has a smaller numberf subnodes than group 2), and pulls in the mark from node D.ote that node D also exports it’s mark to it’s partner subnode inroup 2. Groups 2 and 3 are both recombined as they have reachedhe n = k case. Finally, group 4 pulls in the mark from node F, ands then recombined as there are no more possible marks to pulln.

. An alternative iterative approach

The algorithm described in Section 4 has two shortcomingsegarding performance. First, it is completely dependent on thexistence of an algorithm to calculate the unconstrained vertexover, which for a non-bipartite graph is a NP-hard problem.econd, this algorithm is completely intolerant of any dynamic

ehavior. If a single edge is removed or added, the entire algorithmeeds to be re-run from the beginning. To circumvent this prob-

em, we present another algorithm in this section with the specificenefit of allowing the cover of the graph to be changed as needed

nstrained, middle – constrained), and execution time (right).

whenever small modifications are made to the graph by iterativelyadding the remaining nodes and edges. This iterative method iscapable of offering close to optimal solutions in a quick and effi-cient manner. We refer to this as CONITPLAC, implying ‘constrainediterative placement’. If necessary, CONITPLAC and CONPLAC can becombined to provide fast solutions to the minimum vertex problemthat are very close to the best possible cover within the imposedchannel constraint.

5.1. Algorithm procedure

• The first step of the CONITPLAC algorithm is to identify the nodewith the highest degree. If there are several nodes with the samedegree, then the node with the highest degree sum is chosen. Ifthere is still a tie between any nodes at this point, then start-ing node may be chosen at random. The chosen starting node ismarked, and then placed into the pool, denoted as P.

• The rest of the algorithm is iterative. The neighbors of every nodeni ∈ P are considered separately. From these, the next node ischosen in the same was as the starting node, namely – first bydegree and then by the degree sum. This node is added to P alongwith all edges it shares with ni ∈ P. These new edges must beproperly marked to maintain the minimum cover.

• In most cases, deciding how to cover the new edges is verystraightforward. Each relevant node of the augmented graph isconsidered individually. If a neighbor of this node has less thank edges marked, where k is channel constraint (i.e., the num-ber of line currents that a PMU will observe) then this edge isadded to the cover set of the node. If this condition is not met,then the new node is marked (if it hasn’t already been), and theedge gets marked automatically by the new node. If this node isalready responsible for k edges, then this graph will be deemedunsolvable under the constraint k.

• Following this procedure, all of the nodes in the graph are addedone by one. This will result in a minimum vertex cover containingeach marked node, with the node marking the edges determinedby the algorithm. After all of the nodes have been added, one finalsweep of the graph is made to attempt to minimize the graphfurther.

• Some nodes may be under-utilized: that is, they may be marked,but covering less than k edges. For each of these under-utilized nodes, we determine which of them (whether markedor unmarked) may be connected to a marked edge, and prepare

a list. Each of these lists is then compared. If several of these listscontain the same node, it means that there is a chance that edgescan be exported to this node to decrease the number of markednodes.

58 J.E. Anderson, A. Chakrabortty / Electric Power Systems Research 106 (2014) 51– 61

(a) Expanded graph (b) Marked graph

(c) Case 1 (d) Case 2

(e) Case 2 (f) Case 2

Fig. 8. PMU placement algorithm with channel measurement constraints (Steps 1, 2, and 3).

J.E. Anderson, A. Chakrabortty / Electric Power Systems Research 106 (2014) 51– 61 59

(a) Case 3-1 (b) Case 3-2

(c) Case 3-3 (d) Case 3-4

(e) Case 3-5 (f) Case 3-6

Fig. 9. PMU placement algorithm with channel measurement constraints (Step 3).

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After this initial cover has been obtained, new nodes can be addedto the cover by following the same process. Removal of a node isas simple as re-running the final sweep for minimization, if theremoved node was not marked. If this node was marked, then theedges that it covered must be exported to its neighboring nodes,and some previously unmarked neighboring nodes may need tobe marked before the final sweep can be run.

.2. Example: IEEE 34-bus model

We illustrate the algorithm on a 34-bus power system, which iserived from the IEEE 118-bus system model, as shown in Fig. 5.nly the double-circuit lines of the 118-bus model are considered,ssuming that PMUs will be placed in the high voltage transmissionines to enhance the visibility of inter-area oscillations. Each nodes labeled with the name of the bus that it represents, as well as aumber indicating the order in which each node was added to theool. The constraint is chosen as k = 2. The steps of the algorithmre listed as follows:

First, the starting node is selected. There are two nodes with adegree of 6 in this graph, but since the degree sum of Philip Spornis 23 while that of Philo is only 17, Philip Sporn is chosen, marked,and added to the pool.Next, a node is selected for addition. Since the pool consists ofonly node 1, all neighbors of 1 are compared for their degree. Thedegree of Philo is 6, meaning that it is chosen next and added tothe pool. Note that the only edge under consideration for markingis that between the nodes of the pool, which is only that betweennodes 1 and 2. Since node 1 is marked and covering less than k = 2edges (it is presently covering no edges), then this edge is alreadycovered by the marked node 1. Since there are no more edges toconsider, node 2 is left unmarked.Now there are 9 neighbors to consider in choosing the next addi-tion to the pool. Both nodes 5 and 3 have a degree of 4, but sincethe degree sum of node 3 (18) is slightly larger than that of node5 (16), the former is chosen. The only relevant edge added hereis that between nodes 1 and 3, and since 1 is marked and stillcovering less than k = 2 edges, it is made to cover this edge aswell.Next 4 is included as it is the available node of the highest degree(5). The only new edge this adds is that between 3 and 4, andsince node 3 is unmarked node 4 will be marked and made tocover this edge.Node 5 is added next, with a degree of 4 and a degree sum of 16.Only two edges are added by this inclusion, those shared withnodes 1 and 3. Both of these require node 5 to be marked andmade to cover those edges. Node 6 is chosen as it is the lastdegree-4 node, and is left unmarked, its only added edge beingmarked by node 4. This pattern continues over the rest of thegraph.The final result of simply adding each node iteratively results ina minimum vertex cover containing 26 nodes, with 8 nodes notin the graph (nodes 2, 3, 6, 16, 25, 30, 31, and 32).The final step of the algorithm is to run the final sweep to reducethe cover even further. The under-utilized nodes in this coverare 17, 20, 21, 33, and 34. These nodes are only responsible forcovering one edge, even though they are capable of covering twoas per the constraint. The first part of our sweep is to make alist of nodes that these marked edges could be exported to, sothat the under-utilized node may be unmarked (and decreasethe number of marked nodes). These lists are then checked to

see if they contain common unmarked or under-utilized nodes.Since the lists for 33 and 17 both contain node 2, and exportingthe marks on these edges would unmark both nodes, thereforenodes 33 and 17 are unmarked and node 2 is marked in their

er Systems Research 106 (2014) 51– 61

place. This is the only opportunity that exists for simplifying thegraph, and so this is the minimum vertex cover approximationyielded by the algorithm.

• The final cover consists of 25 marked nodes, with 3 of them beingunder-utilized.

The final result can be seen in Fig. 5. Yellow coloring desig-nates that a marked node is under-utilized. The light blue nodesare unmarked, and not members of this vertex cover. The sub-optimality of this solution can be found by comparing the number ofmarked nodes with the minimum number needed to cover all edgeswith a constraint of k. In this graph there are 47 edges, meaning thatif each mark is placed perfectly 24 (the ceiling of 47/2) nodes will beneeded to monitor every edge. The algorithm has placed 25 markson various nodes of the graph. This is an acceptable solution withonly 1 extra mark (i.e., 96% optimal) being placed in the cover.

Another advantageous property is the ability of this algorithmto be run in parallel. For example, dividing a graph into two halvesand running the algorithm on each half will give a minimum ver-tex cover of the halves, which can then be recombined to yield aminimum cover of the original graph. The details of this splittingand rejoining are beyond the scope of this paper, and is left as ourfuture work.

6. Algorithm testing and sensitivity analysis

In this section we test the CONPLAC algorithm in terms of threefactors – the network size, the degree of meshness and the con-straint placed on the number of observable channels. We performa statistical analysis by considering a large set of bipartite graphs,similar to as in a Monte Carlo simulation.

6.1. Network size

The effect of running the CONPLAC algorithm on bipartite graphsof varying sizes can be seen in Fig. 6. The X and Y axes in the figuresrepresent the number of nodes on each set of the bipartite graph,while the Z axis shows the parameter under inspection. Both theexecution time of the algorithm (right figure), and the number ofmarked nodes (middle) increase monotonically with the networksize. The effect of the constraint on the number of nodes markedcan be seen by comparing the unconstrained and constrained cases.For both cases, the number of marked nodes increase linearly withthe constraint. However, if there are many nodes on one side ofthe graph and few on the other, the unconstrained case can sim-ply cover all nodes on the side with fewer nodes. This will notbe true in the constrained case as nodes with multiple edges can-not be covered. Hence, the surface plot for the constrained case ismore planar than for the unconstrained case, which has a distinctpyramid shape.

6.2. Degree of mesh

To analyze the effect of the nodal degree, random graphs with 20nodes on each side are selected, and the constraint on the numberof observable edges is fixed to a constant value of 5. Assuming equaldegree for the nodes, the degree was changed from 1 to 20, withthe final case resulting in a complete bipartite graph. The result-ing values are shown in Fig. 7. The runtime increases exponentiallywith the degree (right figure). Both the constrained and uncon-strained cases saturate at a particular number of covered nodes

(left and middle figures, respectively). The saturation point for theconstrained case (where all nodes are covered) serves as the pointof impossibility of the cover, indicating that it will not be possibleto cover all edges of the graph with the constraint applied.

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.3. Constraint of observable channels

When k is small it does not affect the runtime significantly, buts it approaches a value leading to fewer supernodes the runtimeecreases exponentially. This is natural as it implies that both algo-ithms level off beyond a critical value of k due to the unsolvabilityf the optimal problem, and settling to only a sub-optimal solution.

. Conclusions

In this paper we presented two graph-theoretic algorithms forMU placement in power system networks for the purpose of iden-ifying equivalent network models, especially when there is a hardonstraint on the number of current channels that can be mea-ured by each PMU. The problem can also be looked upon in lightf a PMU usage problem rather than simply a placement problem,ndicating which PMUs should be used for network equivalencingrom a given set of large number of PMUs measuring redundantata. Future work will include extension of the algorithm to a moreon-heuristic structure that can handle large networks with a rea-onable execution time, combining our constrined approach withhe NP-complete methods of [8], as well as the important problemf PMU reassignment in case of unit failures.

eferences

[1] A.G. Phadke, J.S. Thorp, Synchronized Phasor Measurements and Their Appli-cations, Springer, New York, NY, USA, 2008.

[2] T.L. Baldwin, L. Mili, M.B. Boisen, R. Adapa, Power system observability withminimal phasor measurement placement, IEEE Transactions on Power Systems8 (2) (1993) 707–715.

[3] K.A. Clements, Observability methods and optimal meter placement, ElectricalPower and Energy Systems 12 (2) (1990) 88–93.

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er Systems Research 106 (2014) 51– 61 61

[4] F. Aminifar, A. Khodaei, M. Fotuhi-Firuzabad, M. Shahidehpour, Contingency-constrained PMU placement in power networks, IEEE Transactions on PowerSystems 25 (1) (2010) 516–523.

[5] B. Xu, A. Abur, Observability analysis and measurement placement for sys-tems with PMUs, in: Proceedings of IEEE PES Power Systems Conference andExposition, Oct 2004, pp. 943–946.

[6] F. Lin, P.L. Chiu, A near-optimal sensor placement algorithm to achievecomplete coverage/discrimination in sensor networks, IEEE CommunicationsLetters 9 (2005).

[7] X. Wang, J. Ma, S. Wang, D. Bi, Distributed particle swarm optimization andsimulated annealing for energy-efficient coverage in wireless sensor networks,Sensors (2007).

[8] D. Gyllstrom, E. Rosensweig, J. Kurose, On the Impact of PMU Placement onObservability and Cross-Validation, Department of Computer Science, Univer-sity of Massachusetts Amherst, 2012 (Technical Report UM-CS-2012-001).

[9] R. Emami, A. Abur, Reliable placement of synchronized phasor measurementson network branches, in: IEEE PES Power Systems Conference and Exposition,2009.

10] M. Patel, S. Aivaliotis, E. Ellen, et al., NERC real-time application of synchropha-sors for improving reliability, 2010 (October, NASPI Report).

11] T. Gentile, V. Balasubramanuam, L. Beard, J.H. Chow, D. Sobajic, D.D. Tran, NASPIprocess applied to locate phasor measurement units (PMUs) within the NewYork Control Area (NYCA), in: proceedings of the IEEE PES General Meeting, SanDiego, July, 2012.

12] N.M. Manousakis, G.N. Korres, P.S. Georgilakis, Taxonomy of PMU placementmethodologies, IEEE Transactions on Power Systems 27 (May (2)) (2012).

13] A. Chakrabortty, J.H. Chow, A. Salazar, A measurement-based framework fordynamic equivalencing of power systems using wide-area phasor measure-ments, IEEE Transactions on Smart Grid 1 (2) (2011).

14] I. Dobson, M. Parashar, C. Carter, Combining phasor measurements to monitorcutset angles, in: 43rd Hawaii International Conference on System Sciences,Kauai, HI, Jan 2010.

15] J.E. Anderson, A. Chakrabortty, A minimum cover algorithm for PMU placementin power system networks under line observability constraints, in: IEEE PES

UK, 1976, pp. 203–207.17] B. Korte, J. Vygen, Combinatorial Optimization: Theory and Algorithms,

Springer, NY, 2006.