# Optimal PMU placement for full network observability using Tabu search algorithm

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raranon,ua Unived foy obe Ofulzatiaccelerate optimization. The effectiveness and flexibility of the proposed algorithms are demonstrated by numerical results tested in IEEE 14,can be constructed. The numerical methods rely on whether theElectrical Power and Energy S* Corresponding author.E-mail address: wanghf60@hotmail.com (H.F. Wang).IEEE 57 and NE 39 bus systems.q 2005 Published by Elsevier Ltd.Keywords: Network observability analysis; Optimal PMU placement (OPP); State estimation; Tabu search (TS)1. IntroductionIn recent years, applications of phasor measurement units(PMUs) have been attracting more and more attentions inpower systems security monitoring and control. This is due tothe advantages that PMUs can offer real-time synchronizedphasor measurements (voltage, current, powers, frequency etc.)contrary to the conventional SCADA measurement devices[1,2]. The prerequisite for an efficient and accurate control isthe development of adequate meter placement scheme, whichcan realize the network full observability.As we know, observability analysis is a fundamentalcomponent of real-time state estimation, which acts as theback bone of EMS applications. The theory of networkobservability can be divided into two main classes ofalgorithms: numerical and topological methods. The topologi-cal methods are based on whether a spanning tree of full rank[39] and each of them has its own advantages and limitations.Conventionally, numerical methods involve huge matrixmanipulation, and are computationally expensive. Moreoverthe accuracy of solution is apt to suffering from thecomputation error. In theory, if all nodes in power systemhave been installed with PMUs, then the whole system state isfully observable. However, considering the cost of theequipment together with communication links, optimal PMUplacement (OPP) problem demands to reduce the number ofPMU installation and concerns about where and how manyPMUs should be implemented to a power system to achievefull-state observability at minimal cost. This problem beganbeing addressed recently and certain progress has beenachieved [1013].This paper introduces a novel topological method basedon the augment incidence matrix proposed in [13] and theTabu Search (TS) [1418]. By doing so, solution of thecombinatorial OPP problem requires less computation and isOptimal PMU placement fousing Tabu seJiangnan Peng a, Yuanzha China Power Investment Corporatib Department of Electrical Engineering, Tsinghc Department of Electronic and Electrical Engineering, UReceived 4 April 2003; received in reviseAbstractThis paper presents a fast analysis method for power system topologestimator model and uses augmented incidence matrix. In the paper, thformulated as to minimize the number of PMU installation subjecting toalgorithm, Tabu search, is proposed to solve the combinatorial optimifull network observabilitych algorithmg Sun b, H.F. Wang c,*Beijing, Peoples Republic of Chinaniversity, Beijing, Peoples Republic of Chinarsity of Bath, 39 Burnt House Road, Bath BA2 2AQ, UKrm 17 May 2005; accepted 25 May 2005servability. The method is based on the linearized power system stateptimal PMU (phasor measurement units) Placement (OPP) problem isl network observability and enough redundancy. A global optimizationon problem and a priority list based on heuristic rule is embedded toystems 28 (2006) 223231www.elsevier.com/locate/ijepesmethod using complicated matrix analysis. The paper isorganized as follows. Section 2 presents the proposedobservability analysis method based on incidence matrix forPMU applications and derives a linear phasor estimationmodel benefited from the definition of defining differentmeasurement layers. In Section 3, the OPP problem is0142-0615/$ - see front matter q 2005 Published by Elsevier Ltd.doi:10.1016/j.ijepes.2005.05.005rank. In this area, a lot of interesting work has been reported convenient than the conventional observability analysismeasurement information gain or Jacobian matrix is of full of higher robustness. The method is much faster and morerespectively. Since, the PMU measurement based on GPSd Esynchronism technology is far more accurate than traditionalanalog measurements based on SCADA, we can neglect theerror and choose directly the PMU measurement values as theestimate value. This yields the simple linear observationformulated and TS algorithm is applied to solve the combinedoptimization problem. Three IEEE test system results aregiven in Section 4 and the work is concluded in Section 5.2. Observability analysis using augment incidence matrixfor PMU applicationObservability is defined as the ability to uniquely estimatethe states of a power system using given measurements. It iswell-known that the state estimation can not work even ifmeasurements are redundant. Observability analysis is requiredto decide meter placement in order to maintain solvability ofthe observation equations in various conditions. Implemen-tation of PMU presents an opportunity for improvingobservability analysis and state estimation. Linear phasorestimation modelConsidering a power system with n nodes, the outputof the state estimation is 2nK1 dimensional state vectorXZ[V1,.,Vn, q1,.,qnK1]T containing the voltage magni-tudes and phasor angle. The measurement equation can beexpressed as [2]ZZ hxC3 (1)where measurement vector ZZ[Z1,Z2,.,Zm]T, Z2Rm;measurement function h(x), h:R2nK1/Rm; measurementerror vector 32Rm.Since PMU can directly measure phasor, in order toestablish the fast phasor estimator model, we select the optimalcandidate measurement of essential measurements as the PMUcorrelative measurement sets (CMS). CMS includes measure-ments of the voltage phasor at a bus where a PMU is placed andall the current phasors incident to the bus. CMS, denoted byZcor, can be expressed asZcorZ fZig; ZiZ _Vi; _Ii1 ;.; _IilT iZ 1;.; np (2)where np is the number of PMU placement and, il is the totalnumber of branches incident to bus i. Hence, np PMU candirectly provide 2np bus voltage and the corresponding ilcurrent phasor at all the measurement node i. Obviously, theCMS can be represented as ZcorZ[ZvZI]T, where Zv, ZI arebus voltage phasor and branch current phasor, whosedimensions are mVZnp andm1ZXnpiZ1ilJ. Peng et al. / Electrical Power an224model. For the PMU direct measured voltage VM and theadjacent non-measured voltage VC, Eq. (1) can be rewriten asfollows [2]ZVZI" #ZHxZI 0MIBYBBATMB MIBYBBATCB" #VMVC" #(3)where, MIB is ml!b measure-branch incidence matrixassociated with the current phasor measurements, YBB is b!ndiagonal matrix of branch admittances, AMB and ACB aremeasured and non-measured node-branch incidence subma-trices, respectively.Conventional observability analysis can check the satisfac-tion of the following conditionrankHZ 2nK1 (4)The above condition means that the measurement matrix isfull rank. However, it is difficult to directly assess H due to itsmassive dimension. Generally, an arbitrary PMU placement setcannot guarantee the satisfaction of Eq. (4). Moreover, Eq. (4)can not provide the quantitative index for analyzing differentPMU placement scheme. Hence, it is necessary to develop aquantitative assessment technique to guide the selection ofPMU placement sets based on the following observabilityconditions:Condition 1. If one end bus voltage and branch currentphasor of a branch is known, then the other end bus voltagephasor can be calculated via branch equation.Condition 2. For a branch with known voltage phasor at itsboth ends, its current phasor can be calculated.Condition 3. For a zero-injection node without a PMUplacement, if just only one of the incidence branch current isunknown, then the current can be calculated by KCL law.Condition 4. When all the bus voltages of an unknown zero-injection node are known, the voltage can be computed bycorresponding node voltage equation.Based on these conditions above, it is convenient to classifythe measurements, from the topological point of view, intothree layers, the measurement, pseudo-measurement, and theextension-measurement. The states variables connecteddirectly to PMUs buses (i.e. the CMS) can be defined as themeasurement layer variables. Similarly, pseudo-measurementvariables are voltages or currents those can be calculated byCMS via Ohm law (conditions 1 and 2 above) and extension-measurement variables are the voltages or currents that can beinferred by using the KC circuit laws (conditions 3 and 4above) via zero-injection node. Hence, Eq. (3) can be rewrittenaccording to measurement, pseudo-measurement andextended-measurement respectively. The relation between thePMU installation bus and its adjacent bus can be expressed asfollows:ZVZI" #ZHxZI 0YIM YIN" #VMVM" #(5)where, VNC is pseudo-measurement voltage phasor of theneighbor buses.nergy Systems 28 (2006) 223231For zero-injection bus, when all its adjacent voltage phasorsare known, it can be regarded as an extended-measurement andd ENetwork observability can be checked through the abovedifferent measurement layer variables. We can define differentvariables to express the bus voltage and branch currentaccording to their measurement layers, e.g. symbols W,V,U,for extended-measurement, pseudo-measurement andmeasurement layers respectively. Then by searching the threelayer measurements for every PMU in incidence matrix, howmany PMUs contribute to estimate the each bus voltage andline current can be estimated. Finally, the network observa-bility and measurement redundancy can be identified. Thewhole procedure can be illustrated by IEEE 14 bus system asfollows.The observability augment incidence matrix ~A of studiedsystem is defined in Figs. 1 and 2. The values of row WI, VI,UI or the column WV, VV, UV stand for times of thecorresponding branch current or bus voltage which falls intoextension-measurement, pseudo-measurement and measure-ment. It should be indicated that UV is the PMU placementset. UVZ1 means that the node is placed with PMU andUVZ0 that the node has no PMU placement. Column SV isthe node voltage observability state variable, whose valuestands for the total observable times of corresponding busvoltage. The zero-injection node is marked by initial value 0.Other columns are the incidence vector between branches mand n with the formation MmnZ 0/1mK1n/T. Row SI isbranch current observability state variable, whose valuebe calculated as followsVMAVZ" #ZI 0KYZM YZC" #K1ZZV0" #(6)where, ZZV is the known voltage phasor around the zeroinjection bus and VZ is the voltage phasor of zero injection bus.YZM and YZC represent the admittance matrix of zero injectionbus and its neighbor buses respectively. For the PMU bus andtheir neighbor bus, the state vector can be attained using Eq.(5). And the extension states can be calculated via Eq. (6). Ifthe number of measurement, pseudo and extension states for aspecified PMU is larger; the PMU has more contribution to thenetwork observability. Also we can see that the value of branchadmittance matrix YBB has no impaction in topologyobservability. Hence, we can assume it to be a unity matrix,i.e. YBBZI. Thus the measurement matrix in Eq. (4) can bewritten in a simple form as followsHZI 0MIBATMB MIBATCB" #(7)It is clear that the network observability mainly depends onthe measurements placement scheme (M) and the networktopological structure (A).2.1. Observability analysis using augment incidence matrixJ. Peng et al. / Electrical Power anstands for the total observable times of corresponding linecurrent. Other rows are the ith node incidence vector Ti,where non-zero elements mean the corresponding branchincidence to the ith node.When set a PMU on a node, the corresponding measure-ment, pseudo-measurement and extension-measurement can besearched from augment incidence matrix ~A according to thefollowing rules.(1) Measurement assigning rule. The directly measure-ments, CMS, can be found through the following search: first,set the node voltage observability UV(i)Z1 for bus i installedwith a PMU; then assign a current phasor measurement to eachbranch incident to bus i by searching the incidence vector Ti.i.e. UI(k)Z1, k2Ti, ks0.(2) Pseudo measurement assigning rule. Apply observa-bility condition 1 and search every observable current (UIZ1)branch incidence vectorMji, where there are only two non-zeroelements; Assign the non-zero element node j except themeasurement node i, a pseudo-voltage measurement, VV(j)Z1.Similarly, apply observability condition 2 and search all thepseudo-voltage nodes (VVZ1); Assign the pseudo-currentmeasurement to the unobservable branch current incident tothese nodes, VI(l)Z1, l is the line between pseudo-voltagenodes i and j; Then the system observability index can becalculated asSViZUViCVVi iZ 1; 2;.; n (8)SIjZUIjCVIj jZ 1; 2;.; b (9)(3) Extension-measurement assigning rule. For all the non-measurement (UVZ0) and observable (SVs0) zero-injectionnode Z, check whether observability condition 3 is satisfied; Ifit is true, assign the branch current as extension-currentWIZ1;When the new extension-current measurement is found, repeatapplying observability conditions 2 and 3 to find newextension-measurement, all will be assigned as extension-measurement WV, WI; Check if observability condition 4 issatisfied; assign the corresponding voltage and current asextension-measurements.The corresponding three layers of measurements of nodevoltage and their search routes are also labeled in the Figs. 1(b)and 2(b).After applying above assigning rules, the performance indexof network observability of the PMU set can be calculated asfollowsSViZUViCVViCWVi iZ 1; 2;.; n (10)SIjZUIjCVIjCWIj jZ 1; 2;.; b (11)The above search algorithm can be summarized by thefollowing flow chartThe algorithm is a searching process of the augmentincidence matrix in the case of adding a PMU in network.When removing a PMU from system, the same searchnergy Systems 28 (2006) 223231 225algorithm can be applied with the modification to subtractthe value of all found measurement to network observabilityd EJ. Peng et al. / Electrical Power an226index. In addition, the search algorithm can be extremely fastby taking advantage of the sparsity of the incidence matrix ~A.When all the np PMUs complete placement, the wholesystem node observability vector SV and branch observabilityvector SI can be computed via adding all the PMUFig. 1. Initial PMU placement results of IEEE-14 bus system. (a) Augnergy Systems 28 (2006) 223231measurement variablesSVZXmSVm SIZXmSIm mZ 1;.; np (12)The network full observability can be checked through if allthe individual part of vector SVR1. If true, the network fullment incidence matrix. (b) Node voltage measurement variables.d EJ. Peng et al. / Electrical Power anobservability is guaranteed. The following logical function isused to evaluate the checking procedureObsnp; SnpZ 15 SViR1; iZ 1; 2;.; n (13)or Obsnp; SnPZ05 at least exist one individual componentof SViZ0; ci2n (14)The number of individual value of vector SV!1 is theunobservable number of nn. In Eqs. (13) and (14), S(np) is theFig. 2. Optimal PMU placement results of IEEE-14 bus system. (a) Aunergy Systems 28 (2006) 223231 227PMU placement set and, also UV is the PMU allocationvector.The redundancy measurement index can be defined asRnp; SnpZ SVC SIKnCb (15)The maximum redundancy is the case that with the samenumber of PMUs installation, the optimal placement schemewill have the maximum redundancy. Compared with thegment incidence matrix. (b) Node voltage measurements variables.number np from observability analysis. A modified bisectingsearch is used to iteratively find the optimal PMUs number npand the updating of np employs the following equationnpZ INT0:7np_minC0:3 np_max (18)According to experience, the minimum number of np istypically around 1/4 to1/3 of the total number of system buses.Hence, a good choice for initial lower and upper limits of npsearch range is suggested to be np_minZ2, np_maxZnp_initial.For a given number of PMUs, the placement set variable S(np)or UV solution is a combinatorial optimization problem,the TS algorithm can be employed to identify the optimalplacement set.Tabu search (TS) was early proposed by Glover and hasbecome a well established combinatorial optimizationapproach which has been applied in many fields [1518].The theory of TS is well documented in [14]. In [13,15], TSis regarded superior to other heuristic approaches such as SAand GA. Unlike other heuristic techniques, the advantage ofTS algorithm is the fact that it uses a flexible memory ofIs full observable?EndYNcondition Yset the new upper limitnp_max = npSet the new lower limitnp_min=npnp_max-np_min = 1?YNFig. 3. Flow chart of OPP solution algorithm.d Etraditional observability analysis method, the methodproposed above is simple, fast and quantitative, suitable forthe applications in large power systems and convenient in theassessment of the optimal PMU placement scheme.3. Opp formulation and solution3.1. Problem formulationThe OPP problem is to determine the minimum number npand the optimal location set S(np) of PMUs to satisfy networktopology observability and preset redundancy criterion. It canbe formulated as followsJZminnpfmaxRnp; Snpg (16)s.t.Obsnp; SnpZ 1 (17)Mathematically, OPP is a large-scale combinatorial optimi-zation problem and the system observability is dependant of twofactors, the number of PMU np and the placement set S(np). So farthere has been no proposed scheme to directly obtain the optimalsolution of number np_min. Usually, it is estimated by try anderror. The search space ofnp is [0, b], and S(np) is the combinationnumber Cnpb . Obviously, the OPP is a highly nonlinear,discontinuous and multi-modal (i.e. there may exist multiplelocal optimum) problem. Moreover, its objective function isnonconvex, nonsmooth, and nodifferentiable. Hence, it isessential to employs a derivative-free optimization technique insolving OPP problem in order to keep computational cost low. Inthis paper, TS algorithm is proposed to solve the OPP problem.3.2. Set of initial PMU placementThe initial PMU setting is arranged in the following way[10](i) Place a PMU at the bus with the highest number ofincident branches in system unobservable region;(ii) Determine the measurement, pseudo and extensionmeasurements by the current PMU placement set using themethod proposed in the above section;(iii) If system is not full observable, then continue step (i)and until observed region covers the whole system.It is cleat that this procedure will lead to an initial PMU setwhich can realize whole system observable but not an optimalscheme as outlined by Eqs. (16) and (17). The initial PMUallocation of IEEE-14 bus system is shown in Fig. 1(b) and theinitial number np_initialZ5. The corresponding augmentincidence matrix result is given in Fig. 1(a). Apparently, itsatisfies the observability evaluation function Obs(np, S(np))Z1and the redundancy value is 8.3.3. Solution methodJ. Peng et al. / Electrical Power an228Fig. 3 shows the procedure of OPP solution approachproposed, which separates the searching of the minimum PMUStartIntialize PMU placement setand the intial number np_initalInitialize the upper limitnp_max = np_initaland the lower limit np_min=0.1*n Set current numbernp=[0.3np_max+0.7np_min]Execute TS to generatenew solutionIs full observable?NIs satisfied quitNYSelect current placment set Uvusing heuristic rulenergy Systems 28 (2006) 223231search history to prevent cycling and to avoid entrapment inlocal optima. It has been theoretically proved that TSPMU setting can be introduced as follows:now trialnot in the Tabu list, then put it in the Tabu list, set SnowZSbesttrialIn this paper, an improved technique has been developed tomake TS more efficient in solving OPP problem. A heuristic3.3.4. Evaluation functionThe evaluation function is devised as a penalty function asfollowsmin Jnp; SnpZ snunp; SnpCnpKl$Rnnp; Snp (19)where s is a large positive number, n (n , S(n )) is thenumber of PMUs, np. It is shown that this rule can greatlyd Energy Systems 28 (2006) 223231 229rule is suggested for solution effective moves. Whenever aPMU is removed from a node, the bus with the least branches isconsidered firstly. Whenever a PMU is added to a bus, the buswith the most branches is considered firstly. It has been provedby simulation that these two heuristic rules can speed up theTS convergence and improve solution quality. Main operationsof TS include moves, Tabu list and aspiration etc., which areexplained further as follows.3.3.1. Moves and set of candidate movesMoves are the local search process in the neighborhood ofcurrent solution Snow to generate trial solutions. The candidatemove set, C(Snow), is operated simply by just changing onePMU placement at a time, e.g. solution S(np) moves as:(01011)/(00111), (10011), etc.3.3.2. Control of Tabu listTabu list length Tl plays an important role in searching timeand solution quality. If Tl is too short, solution quality candeteriorate. However, if it is too long, it may pose massivecomputation burden. Hence, a varying-length Tabu list issuggested and used here. The length is set in the rangeTl2[Tlmin, Tlmax]. The limits are changed according to thesolutions in Tabu list and the neighboring structure of currentsolution.3.3.3. Aspiration criteriaAspiration criteria can override Tabu restrictions. Thecriteria considered here are to override the Tabu status of aand go to Step 4; if it is in the Tabu list, go to Step 3.(c) Step 3. This is to check the aspiration criterion of Sitrial. Ifit is satisfied, update Tabu restrictions and aspiration level andset SnowZSitrial.(d) Step 4. This is to check the stop criteria. If any of them issatisfied, then stop; else set kZkC1 and go back to step 2.(a) Step 1. This is to set the iteration counter kZ0 and theinitial solution S(np) to be the current solution as and the bestsolution, i.e. SbestZSnowZS0. The Tabu list is empty, HZf,with length Tl. By using incidence matrix method, the networkobservability of current placement scheme is assessed.(b) Step 2. This is to generate a set of trial solution, Strial, inthe neighborhood of the current solution based on the specifiedmoving rule, i.e. to create the candidate set C(Snow). Base onthe evaluation function value, the best trial solution Sbesttrial inC(S ) is chosen and the Tabu status of Sbest is checked; if it isalgorithm can yield global optimal solution. The basicelements and general algorithm of TS can be found in [14].TS algorithm in this particular application for the optimalJ. Peng et al. / Electrical Power anmove if this move yields a better solution than the one obtainedearlier with the same move.reduce computational time compared with the randomgeneration of the initial solutions.4. Simulation and test resultsThe proposed algorithms have been tested for the OPPproblem of IEEE 14 bus, New England 39 bus and IEEE 57 bussystems. The simulation results are given in Table 1.4.1. IEEE 14 bus systemIEEE-14 bus system only has one zero injection bus. Thereare 5 PMUs in initial placement scheme and the optimalscheme needs 3 PMUs to assure the observability conditions.The result is shown in Figs. 1 and 2.4.2. New England 39 bus systemThere are 12 zero-injection buses and the PMU placementscheme is shown in Fig. 4. Initially, the number of PMUs is 15, andthe redundancy value is 23. After optimization, the PMU numberreduces to 10 and the corresponding redundancy value is 17.Table 1Simulation resultsTest systems n np_ini Rini np_min np_min/n RminIEEE_14 14 5 8 3 0.214 2u p punobservable node number corresponding to the solution S(np),l is a relatively small positive number. This function canintegrally evaluate system observability, number of PMUs andmeasurement redundancy performances. It is clear that theminimum value of evaluation function is obtained only ifnetwork full observability (nuZ0), the minimum number ofPMUs and the maximum R(np, S(np)) are achieved.3.3.5. Stopping criterionIn our study, the search is terminated if one of the followingcriteria is satisfied: (a) the number of iterations is greater than apre-specified number before the network full observability issatisfied. This needs to increase the number of PMUs, np; (b)for a feasibility solution, the number of PMUs can not decreasefurther and at the same time the redundancy cannot improved.3.3.6. InitializationThe heuristic search rule used in TS moving operation isalso employed to initialize the solution for every updatedNE_39 39 15 23 10 0.25 17IEEE_57 57 16 37 13 0.228 144.3. IEEE 57 bus systemThere are 15 zero-injection buses and PMU placementscheme is shown in Fig. 5. Initially 16 PMUs are needed, whichis reduced to minimum 13 PMUs. The redundancy value is 37Fig. 6 shows the changes of average objective function valuesof Tuba search applied to IEEE 14 bus and NE 39 bus powersystem. From Fig. 6 it can be seen that the average objectivefunction values decrease along with the PMUs placementnumber and yet the redundancy performance also decreases.G GGGGGG GG G3039122537291726933816541827283624352221203423193310111314158 31126327PMU CurrentMeasurementNon-MeasurementG GGGGGG GG G3039122537291726933816541827283624352221203423193310111314158 31126327PMU CurrentMeasurementNon-Measurement(a) Initial PMU placement (b) Optimal PMU Placement Fig. 4. PMU allocation results in NE 39 bus system. (a) Initial PMU placement; (b) optimal PMU placement.J. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223231230and 14, respectively.G GG5171234151645GGG G30255429 5352272826 242123 22201918511078 96353433323138373614 13 1246444948475040573955414256 1143PMUCurrentMeasurementNon-measurement(a) Initial PMU placement Fig. 5. PMU allocation results in IEEE 57 bus system. (a)However, when the number of PMUs is less than theminimal np,G GG5 1234 16GGG G1730255429 5352272826 242123 22201918511078 96353433323138373614 13 12154644454948475040573955414256 1143PMU CurrentMeasurementNon-Measurement(b) Optimal PMU Placement Initial PMU placement; (b) optimal PMU placement.effectiveness of the proposed scheme is demonstrated bysimulation results in IEEE 14, NE 39 and IEEE 57 bus powersystem.5 IEEE-14 BusJAvenpJ. Peng et al. / Electrical Power and Energy Systems 28 (2006) 223231 2311 1.5 2 2.5 3 3.5 4 4.5 53.6PMU Number3.844.24.44.64.8Average Objective function Value s = 5l = 0.1the objective function value increases abruptly since theobservability condition is not satisfied at that situation.5. ConclusionsIn this paper, the OPP is solved by the Tabu searchalgorithm and the full network observability is checked byproposed fast observability analysis method. TS providessolutions with high accuracy and less computational effort.The new observability analysis method based on incidencematrix only manipulates integer numbers and can fast,conveniently and quantitatively assess the network observa-bility as far as PMU placement scheme is concerned. Thesearch algorithm for unit commitment problem. Electr Power Syst Res1999;49:718.IEEE 14 bus system9 10 11 12 13 14 1545678910 NE-39 Bus PMU NumberAverage Objective function Values = 10l = 0.1JAvenpNE 39 bus system Fig. 6. Tuba search results of average objective function values for IEEE 14 busand NE 39 bus system. IEEE 14 bus system. NE 39 bus system.AcknowledgementsThe authors acknowledge the support by the NationalNatural Science Foundation of China for Outstanding YoungInvestigators (Grant N.59825104) and National Key BasicResearch Special Fund of China (Grant N. G1998020314).References[1] Phadke AG. Synchronized phasor measurements in power systems. IEEEComput Appl Power 1993;6(2):1015.[2] Phadke AG, Thorp JS, Karimi KJ. State estimation with phasormeasurements. IEEE Trans Power Syst 1986;1(1):23341.[3] Monticelli A, Wu FF. Network observability; theory. IEEE Trans PowerAppar Syst 1985;4(5):10428.[4] Krumpholz GR, Clements KA, Davis PW. Power system observability: apractical algorithm using network topology. IEEE Trans Power ApparSyst 1980;July/August:153442 [see also 202937, November 1995].[5] Slutsker IW, Scudder JM. Network observability analysis throughmeasurement Jacobian matrix reduction. 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A new genetic-based tabuOptimal PMU placement for full network observability using Tabu search algorithmIntroductionObservability analysis using augment incidence matrix for PMU applicationObservability analysis using augment incidence matrixOpp formulation and solutionProblem formulationSet of initial PMU placementSolution methodSimulation and test resultsIEEE 14 bus systemNew England 39 bus systemIEEE 57 bus systemConclusionsAcknowledgementsReferences

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