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566 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013

Optimal PMU Placement Based on Probabilistic Cost/Benefit AnalysisFarrokh Aminifar, Member, IEEE, Mahmud Fotuhi-Firuzabad, Senior Member, IEEE, and

Amir Safdarian, Student Member, IEEE

Abstract—This letter proposes an analytic framework forachieving an optimal solution for phasor measurement unit(PMU) placement problem. The proposed technique is on thebasis of the cost/benefit analysis while long-term economic aspectsand existing technical issues are simultaneously accounted for.More deployment of PMU devices would result in more reliableand secure electricity services. The reduction of system risk costis hence recognized as the benefit associated with the develop-ment of wide-area measurement system (WAMS). The optimalnumber and location of PMUs are determined such that addingmore PMUs is no longer beneficial. Two systems are examined todemonstrate the performance of the proposed method.

Index Terms—Cost/benefit analysis, phasor measurement unit(PMU), risk assessment.

NOMENCLATURE

Benefit of WAMS with PMUs [$].

Development cost of WAMS with PMUs[$].

Implementation cost of PMU at bus [$].

Expected energy not supplied in bus and atperiod while PMUs are installed in thesystem [MWh].

Fixed cost of WAMS development [$].

, Index and set of system buses.

, Number of PMUs and its optimal value.

Present value factor associated with period .

Risk cost at period while the system isequipped with PMUs [$].

Discount rate.

, Index and set of periods in the study horizon[yr].

Binary parameter that is equal to 1 if PMUis installed at bus and 0 otherwise.

Value of lost load in bus and at period[$/MWh] .

Manuscript received January 26, 2012; revised April 05, 2012; accepted April25, 2012. Date of publication June 05, 2012; date of current version January 17,2013. This work was supported by the Iran National Science Foundation. Paperno. PESL-00014-2012.The authors are with the Center of Excellence in Power System Control and

Management, Electrical Engineering Department, Sharif University of Tech-nology, Tehran, Iran (e-mail: frkh_aminifar@ee.sharif.edu; fotuhi@sharif.edu;a_safdarian@ee.sharif.edu).Digital Object Identifier 10.1109/TPWRS.2012.2198312

I. INTRODUCTION

T O date, twofold classes of methods have been developedfor the optimal placement problem of phasor measure-

ment units (PMUs): 1) mathematical models seeking the min-imal number of PMUs subject to technical constraints such asthe network observability in either basic or augmented forms[1], [2], and 2) heuristic approaches based on engineering judg-ments [3]. Some recent literatures have presented few proba-bilistic studies [4]–[6]; while, no technique has been yet pro-posed to quantitatively assess the economic aspects of wide-areameasurement system (WAMS) applications. From a practicalperspective, a PMU placement scheme could not be optimalwhen the associated cost/benefit analysis is not fulfilled.The aim of this letter is to present a mathematical framework

for locating PMUs considering both cost and benefit facets. Thecalculation of implementation cost of WAMS is rather straight-forward since all the relevant costs can be summed up. On thecontrary, quantifying the benefits associated withWAMS devel-opment is seriously rigorous. Such an effort should include costterms imposed to the system when the WAMS is not employedor is used but malfunctioned. Since these situations are likelywhen the system is at risk, the probabilistic modeling and anal-ysis is crucial. Doing so, the enforcement in reliability by em-ploying the WAMS infrastructure is measured and converted toa monetary value as the WAMS development benefit. The ben-efit is then compromised with the cost value to reach to an op-timal PMU placement solution. Numerical evidences are pro-vided as well in support of practicality of the proposed algo-rithm.

II. PROPOSED METHODOLOGY

In the following, the cost and benefit associated with theWAMS development are first calculated. Then, a framework tofind the optimal placement scheme of PMUs is presented.For a given placement scheme of PMUs, the development

cost is calculated by

(1)

where are known and , is the costof PMU implementation at bus and consists of procure-ment, installation, commissioning, and periodical calibrationsof PMU and its accessories such as software, measurementdevices, panel, cables, and communication link. In (1),mainly refers to the control center hardware facilities as wellas WAMS control center software. Note that the solution underthe above investigation is subject to technical and security con-straints such as the basic network or contingency-constrainedobservability [2].

0885-8950/$31.00 © 2012 IEEE

AMINIFAR et al.: OPTIMAL PMU PLACEMENT BASED ON PROBABILISTIC COST/BENEFIT ANALYSIS 567

TABLE IRESULTS OF NINE-BUS TEST SYSTEM

The benefit of WAMS development is realized over the yearsof study horizon through the reduction of the system risk cost.That is,

(2)

where is the financial scaling factor defined as

(3)

In (2), is the system risk cost at time period when noPMU is installed. That is, the system security is just assuredthrough the conventional supervisory control and data acquisi-tion (SCADA) infrastructure. In order to evaluate and

, the expected energy not supplied (EENS) indices areevaluated via the algorithm proposed in [5]. These indices arethen converted to a monetary value by

(4)

It has to be emphasized that since multiple solutions withPMUs might exist [6], optimal solution with PMUs is the onewith the highest benefit to cost ratio (BCR). Multiple solutionsof a PMU placement problem could be readily collected by thesolution pool in CPLEX 11.0.So far, both cost and benefit associated with PMU deploy-

ment are quantified for the best placement scheme of PMUs.To determine the optimal value of , the net benefit of WAMSdevelopment, which is equal to benefit minus cost, should bemaximized. Differentiating the net benefit with respect to deci-sion variable, , and equating to zero, we have

(5)

Since is an integer number, difference operator, , is applied.Also, based on the linearity property, we have

(6)

is independent of ; consequently

(7)

Based on (7), the increase of PMUs, , is profitableto the point in which the difference of WAMS benefit,

, is greater than the difference in itscost . The next PMU would not be profitable anymore.The optimal number and locations of PMUs is accordinglyobtained.

III. CASE STUDIES

A small nine-bus system and the IEEE 57-bus test system areexamined here. These systems have been already investigatedin [2] and [4]–[6] for the PMU placement and reliability assess-ment with WAMS malfunction. All power flow and reliabilitydata of the test systems was taken from the mentioned refer-ences and the remaining data required for the analysis are

The network complete observability is considered as theunderlying security constraint. The zero-injection effect isneglected to prevent the uncertainty propagation caused bythe Kirchhoff’s current law (KCL) application. Hence, theinitial solution of nine-bus system has three PMUs at buses4, 6, and 8. The next placement scheme with four PMUsand the associated cost and benefit with their differencevalues are given in Table I. As shown in this table, increasingthe number of PMUs from three to four is profitable since

. The new PMU is in-stalled at bus 1 where the bulk generation of this system islocated and its observability and control are of extreme im-portance. However, adding further PMUs is not cost-effectiveanymore and the saturation effect in reliability improvement isobserved.The cost/benefit analysis of the PMU placement procedure

for the IEEE 57-bus network is also performed. For this system,the most economical solution is to locate 20 PMUs at buses 1,4, 6, 9, 10, 15, 20, 22, 25, 27, 29, 32, 33, 36, 39, 41, 44, 46, 49,and 53.

IV. CONCLUSIONS

An analytical framework for cost/benefit analysis of PMUplacements was proposed. As shown, the benefit of furtherimplementation of PMUs is saturated when the number ofPMUs increases more and more. Consequently, the cost asso-ciated with WAMS development restricts the number of PMUsassigned to the system. Based on such an analysis, the optimalnumber and location of PMUs were obtained.

REFERENCES

[1] T. L. Baldwin, L.Mili, M. B. Boisen, and R. Adapa, “Power system ob-servability withminimal phasormeasurement placement,” IEEE Trans.Power Syst., vol. 8, no. 2, pp. 707–715, May 1993.

[2] F. Aminifar, A. Khodaei, M. Fotuhi-Firuzabad, and M. Shahidehpour,“Contingency-constrained PMU placement in power networks,” IEEETrans. Power Syst., vol. 25, no. 1, pp. 516–523, Feb. 2010.

[3] G. B. Denegri, M. Invernizzi, and F. Milano, “A security oriented ap-proach to PMU positioning for advanced monitoring of a transmissiongrid,” in Proc. Int. Conf. Power Syst. Tech, Oct. 2002, pp. 798–803.

[4] F. Aminifar, M. Fotuhi-Firuzabad, M. Shahidehpour, and A. Khodaei,“Probabilistic multistage PMU placement in electric power systems,”IEEE Trans. Power Del., vol. 26, no. 2, pp. 841–849, Apr. 2011.

[5] F. Aminifar, M. Fotuhi-Firuzabad, M. Shahidehpour, and A. Safdarian,“Impact of WAMS malfunction on power system reliability assess-ment,” IEEE Trans. Smart Grid, to be published.

[6] F. Aminifar, M. Fotuhi-Firuzabad, M. Shahidehpour, and A. Khodaei,“Observability enhancement by optimal PMU placement consideringrandom power system outages,” Energy Syst., vol. 2, no. 1, pp. 45–65,Mar. 2011.