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1110 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

Improving Service Restoration of PowerDistribution Systems Through LoadCurtailment of In-Service Customers

Michael R. Kleinberg, Student Member, IEEE, Karen Miu, Member, IEEE, and Hsiao-Dong Chiang, Fellow, IEEE

AbstractLoad curtailment programs including direct loadcontrol and demand response allow system operators to directlyand/or indirectly reduce a portion of total customer demand. Onesystem application which can be improved by implementation ofload curtailment is service restoration. As such, this work presentsa problem formulation, solution algorithm, and simulation resultsfor service restoration of power distribution systems incorporatingload curtailment of in-service customers via direct load control.It is shown that the addition of load curtailment allows for one ormore of the following: a reduction of the total number of switch

operations required, an increase to the number of customersserved, and/or an increase to the total amount of load restored.These improvements are demonstrated through simulation resultsfrom a 416-bus multiphase distribution system.

IndexTermsPower distribution control, power distribution re-liability, system restoration and load curtailment.

I. INTRODUCTION

LOAD curtailment programs including direct loadcontrol (DLC) and demand response (DR) allow system

operators to directly and/or indirectly reduce a portion of totalcustomer demand. DLC achieves curtailment through control

signals sent by system operators to turn off nonvital customerloads and/or change local controller set points [1]. Curtailmentvia DR relies on load reductions as a response to emergencyand/or real-time pricing signals [2], [3]. Advances in smart me-ters and intelligent appliances are enabling programs to playa role in all aspects of distribution system operation.

One application which can leverage via DLC is servicerestoration. By issuing DLC commands through a high-speedcommunications network following isolation of a fault, addi-tional capacity can be near instantaneously made available torestore out-of-service load. Employing in this manner pro-motes the utilitarian objective of providing a base level of ser-vice to a maximum number of customers during emergency op-

erating situations.The service restoration problem is typically formulated tomaximize the amount of out-of-service load restored followingisolation of a fault through network reconfiguration [4][6].

Manuscript received November 18, 2009; revised March 06, 2010, July 03,2010, and September 01, 2010; accepted September 07, 2010. Date of publi-cation November 18, 2010; date of current version July 22, 2011. This workwas supportedby the Office of Naval Research under grant#ONR-N0014-04-1-0404. Paper no. TPWRS-00895-2009.

M. R. Kleinberg and K. Miu are with the Department of Electrical and Com-puter Engineering, Drexel University, Philadelphia, PA 19104 USA (e-mail:[email protected]; [email protected]).

H.-D. Chiang is with the School of Electrical and Computer Engineering,Cornell University, Ithaca, NY 14853 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRS.2010.2080327

Network reconfiguration is achieved by changing the status ofnormally open tie switches and normally closed section-alizing switches among other controllable devices, suchas reclosers, existing in the network. The problem has beenextended to include network operating constraints and prioritycustomer considerations [7], [8].

For prolonged outages, the effects of cold-load pickup[9][11] and projected load changes which occur during therestoration process [12] have been captured. In [13], a frame-

work for assessing load capability is presented which allowsthe viability of control schemes to be evaluated and feasiblesequences to be determined [14]. More recently, the use ofdistributed generation to improve restoration solutions has alsobeen considered [11], [15], [16].

To determine optimal switch strategies, several solutionmethodologies have been investigated including heuristicsearch algorithms [4][10], computational intelligence methods[17], [18], fuzzy set theory [19], reliability-based methods [20],[21], and a multi-agent approach [22]. All of these schemeshowever are limited by the spare capacity in adjacent feederstying into the out-of-service areas. With widespread deploy-ment of DLC and DR, this capacity may be increased through

of in-service customers.An initial problem formulation for service restoration incor-

porating via DLC was presented in [23]. This paper expandson this formulation by providing a detailed solution algorithmand an implementation of the approach tested through extensivecase studies. The cases demonstrate improvements to restora-tion solutions achieved with the addition of and illustratethe speed at which the algorithm finds high-quality solutions.Specifically, the paper addresses:

a service restoration problem formulation for radial sys-tems incorporating via DLC and priority customers;

analytically determined indices used for ranking switchand options;

a large-scale heuristic optimization algorithm for onlinerestoration incorporating of in-service customers;

simulation results from a 416-bus test distribution systemhighlighting improvements to restoration solutions.

II. PROBLEM FORMULATION

The service restoration problem with load curtailment andpriority customers is formulated as a constrained, nondiffer-entiable, multi-objective, mixed integer nonlinear optimizationproblem. The objectives considered are to: 1) maximize pri-ority load restored, 2) maximize total load restored, 3) minimizenumber of switch operations, and 4) minimize total .

A feasible solution satisfies the equality constraints expressed

as the multiphase power flow equations and preservation of a

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KLEINBERG et al.: IMPROVING SERVICE RESTORATION OF POWER DISTRIBUTION SYSTEMS 1111

radial network structure, as well as the inequality constraintsof branch current flow, feeder loading, voltage magnitudes, andload curtailment limits.

In summary:

(1)

(2)

(3)

(4)

subject to(5)

(6)

(7)

(8)

(9)(10)

where

multiphase power flow equations;

set of all feeders;

selected network configuration;

set of possible radial configurations;

current flow entering bus , phase ;

maximum current flow or upstreamprotection setting of feeder entering bus

, whichever is least;total load current at bus ;

load to be curtailed by customers at bus, phase ;

maximum available load curtailment ofcustomers at bus , phase ;

total load curtailment current at bus ;

number of network switch operations toarrive at ;

set of all buses;

set of restorable priority customers;

set of restorable buses;set of buses with available loadcurtailment;

total apparent power entering feeder ;

maximum capacity of feeder or itssupplying transformer, whichever is least;

load curtailment scheme;

set of possible load curtailments;

vector of node voltages;

voltage at bus , phase ;

minimum operating voltage, bus ;

maximum operating voltage, bus .

The solution search space is the combinatorial product ofall radial switch configurations and all possible combina-tions. This space, while bounded [23], grows exponentially withthe number of switches and options available. As such, aheuristic solution algorithm is proposed. The search is driven byanalytically determined indices which rank candidate switchesand options.

It is noted that the proposed approach is formulated for short-ened time scales enabled through distribution automation. Thiswill allow for the work to focus on the integration of to theservice restoration problem and supporting algorithms. Subse-quently, this will also provide direct insight to incorporatingin cases of prolonged outages [9][14].

In this formulation, objectives are measured by the amount ofload restored which link directly to engineering capacity con-straints and analytically computable indices. The decision in-dices are presented in the next section. Alternative objectives,e.g., maximizing the number of customers restored, could alsobe selected; the decision indices presented and employed by thesolution algorithm will still hold. Considerations for handling

this alternative objective will be briefly discussed along with theproposed decision indices.

III. SWITCH SELECTION INDICES

In this work, four switch selection indices are utilized. Theindices allow the restoration algorithm to systematically distin-guish between network switches and options based on ana-lytically determined criteria and graph theoretical arguments.

Three of the indices are adapted from [7]. A new index,, introduced in [23] is defined to quantify capacity release

available through . In summary, the indices used are:

: the available to each ; : the spare capacity of each ; : the electrical distance of each to other buses; : the amount of load each can transfer to a .

Since the impact of is to increase the spare capacity of each, , and are reviewed here, as well as . An illus-

trative example with numerical calculations of the indices canbe found in [23].

Each feeder has a maximum amount of current it can sustainbefore a branch becomes overloaded or protection device oper-ates. For each , is defined as follows:

(11)

where is the set of upstream buses on the path from theprimary side of each to the substation. In addition to , thecritical branch at which this value is found for each candidateis stored as . If information about the expected repair time isavailable, the corresponding set of normal or emergency ratingsmay be used.

The electrical distance is given by

(12)

where is the set of all buses lying in the path between theand the bus with the voltage violation.

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The available for each , , is defined as the max-imum amount of load which may be curtailed between the crit-ical branch, , and the energized side of each candidate :

(13)

(14)

where is the set of all buses downstream of and isthe set of all buses with available . In addition, a vector of theindividual candidate values is stored for each . The partialsums of these options are denoted as . In this work, itis assumed that the values of in-service load and available cur-tailment are available to the solver, gathered from smart metermeasurements, load estimation, and/or historical data.

These indices are used to both reduce the solution searchspace and rank switches and candidates to be selected in thesolution algorithm. With an alternate objective of maximizingnumber of customers restored, an additional discrete index

would be needed with respect to the number of customersserved between sectionalizing switches. In addition, the searchalgorithm presented next would be adjusted to balance capacityindices with this customer number index.

IV. SERVICE RESTORATION ALGORITHM

The proposed algorithm solves the multi-objective optimiza-tion problem by prioritizing the objectives in the order pre-sented: (1)(4). Maximizing total outage load restored (2) isprioritized over minimizing load curtailment (4) as a means toreduce the number of sustained outages at the expense of in-creased curtailment of in-service customers. Minimization ofnetwork switch operations (3) is prioritized given that these op-

erations may cause momentary disturbances to a large numberof customers and repeated operation increases required mainte-nance and reduces lifetime.

The proposed approach is designed for online restoration inwhich solutions are calculated quickly and implemented usingprimarily automated switches and DLC commands. Outage du-rations are assumed to be on a scale which allows for load diver-sity to be maintained and energy payback effects to be minimal.Further, it is assumed that network protection devices are pro-grammed to ride through transient inrush currents which arisefrom momentary outages.

The output of the algorithm is a feasible and switch se-quence arriving at the post-fault restored network configuration.

The solution algorithm assumes: the network configuration is radial; all pre-fault network parameters are known; the fault has been isolated by a known set of breakers or

switches.If multiple out-of-service areas exist, the algorithm is appliedsequentially to each area in decreasing order of priority loadpresent followed by areas with the largest amount of total load.

Thefollowing subsections present an outline of the main stepsof the solution algorithm followed by the details of each step.

A. Main Steps

In the solution algorithm, and are selected along with

options to meet the stated objectives. The algorithm selectsoptions as needed to increase the spare capacity of each

allowing for fewer switch operations to be performed and/oradditional load to be restored in the out-of-service area. Themain steps of the algorithm are as follows.

Step 1) Build candidate tie and sectionalizing switch list.Step 2) Build candidate load curtailment list.Step 3) Select and operate one candidate and implement

to attempt full restoration of out-of-service area.

Step 4) Select and operate candidate switch pairs and imple-ment to attempt full restoration of out-of-servicearea.

Step 5) Determine which nonpriority and then priority loadsnot to restore by implementing and opening .

Step 6) Return new network configuration and load curtail-ment scheme.

The proposed algorithm accounts for both overload andvoltage violations. Overload violations refer to current orpower flows which exceed the thermal rating of a given branch.Violations are checked by running an unbalanced multiphasepower flow after each switch and set of options are op-erated. When switches are selected, they are removed from

the list of available candidates. When no violations occur, asolution is found and the algorithm exits. The details of eachstep of the restoration algorithm are reviewed next.

B. Step Details

Step 1) Build candidate switch lists.Candidate and are identified as: from energized feeders that can connect di-

rectly into the out-of-service area; located in the out-of service area.

The spare capacity, , and critical branch associated witheach are then calculated from a post-fault multiphase power

flow. In addition, the nominal outageload current is recorded as

(15)

where is the total nominal load current at bus .Step 2) Build candidate load curtailment list.

With each candidate is stored its total available, , and the partial sums of its individual

options, .Step 3) Select one .One operation is attempted and, if necessary, availableis implemented to restore the entire out-of-service area. If fullrestoration is not possible, the candidate with the largest

is chosen and the algorithm proceeds to Step 4. Specifically, aninitial is selected through the following steps:

S3.1) Create list of candidate pairs with:

.

S3.2) Sort list in increasing order of; if list is empty, go to S3.6.

S3.3) Select the next pair on list, set se-lected as ; if no additional pairs on list,go to S3.6.

S3.4) If , implement and continue;else, may be unnecessary.

S3.5) Close , run power flow, and check con-straints.

If all constraints are satisfied, setand go to Step 6.

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If an overload violation exists, openpreviouslyclosed , undo , and goto S3.3.

If a voltage violation exists, reorderlist in increasing order of and goto S3.3.

S3.6) Select with largest and set as .

S3.7) Close , run power flow, and record magni-tude of overload current of as . Set

and go to Step 4:

(16)

Step 4) Select candidate switch pairs.Switch pairs and are sequentially selected to alleviate theoverload of . For a given , candidate are those lyingin the out-of-service area and in the path from the secondaryside of the towards the substation. Switches are not pairedtogether if their operation would create a new constraint viola-

tion. Switch pairs and are selected as follows:S4.1) Create list of candidate

quad-tuples with: and

. If none exist,go to S4.6.

S4.2) Sort list in increasing order of

.

S4.3) Select next pair andset of options on the list, set switch pairas . If no more exist, go to S4.6.

S4.4) If or , implementon and/or , respectively; else,

may be unnecessary.S4.5) Operate , run power flow, and

check constraints. If all constraints are satisfied, set

and go to Step 6. If a violation exists, undo

pair and , and go to S4.3.S4.6) Create list of candidate pairs with

.S4.7) Order list in decreasing order of .S4.8) Select next pair on list and set pair

as ; if no more exist, go to Step 5.S4.9) Operate , run power flow, and

check constraints. If operation introduces no additional

constraint violations, update ,set , and go to S4.1.

If a new overload or voltage violationis introduced, go to S4.8.

Step 5) Determine which loads not to restore.If the entire out-of-service area cannot be restored throughSteps 3 and 4 without constraint violations, then nonpriorityand, if necessary, priority loads are selected to be excludedfrom restoration. This is achieved by opening in the areaserved by , referred to as , which isolate loads frombeing restored:

S5.1) Partition candidate into those which servepriority load and those which do not.

S5.2) Sort list of with no priority customersdownstream in increasing order of .

S5.3) Select next on list, set as . If no addi-tional exist, go to Step 5.8.

S5.4) Sort list of in increasing order such

that .

S5.5) Select next on list. If no more exist,operate , run power flow,update , set , and goto S5.3.

S5.6) If , implement and continue.S5.7) Operate , run power flow, and check

constraints. If no violations exist, set

and go to Step 6. Else, undoand and go to S5.5.

S5.8) Sort with priority customers downstreamin increasing order of priority load servedand go to S5.3.

Step 6) Return feasible control scheme.To implement restoration, the following and switch se-quence is performed:

S6.1) Curtail all load identified in Steps 3, 4, or 5.S6.2) Open all selected in Step 5.S6.3) Operate switch pairs identified in Step 4, first

opening and then closing of eachpair.

S6.4) Close .To demonstrate the performance of the proposed solution al-

gorithm, a set of case studies on an actual distribution system ispresented next. For each case, it will be shown that the solutionalgorithm successfully finds a Pareto optimal solution.

V. SIMULATION RESULTS

Results will be presented for service restoration with andwithout . The test cases demonstrate practical examples ofthe possible improvements to restoration solutions when ofin-service customers is considered.

A one-line diagram of a 416-bus, 1202-node, multiphase dis-tribution system used in these studies can be seen in Fig. 1. Thetotal load of the system is 37 567 kW and 17 762 kvar. A countof each system component can be found in Table I. All trans-formers and lines immediately before or after a transformer areprotected by circuit breakers. Faults occurring on these branchesare isolated by operating these devices. Faults on other branches

are isolated by operating directly upstream and downstream sec-tionalizing switches or breakers.

The service restoration program was coded in MATLAB. Allpower flow calculations were performed using an unbalancedmultiphase power flow solver [24]. It is assumed that 20% ofthe existing balanced loads are curtailable. For the followingstudies, each of these candidates has one level of load con-trol, , fixed at 30% of its total three-phasevalue [3]. For the given test case, this results in 14 curtailableloads corresponding to total nominal power values of 3793 kWand 1667 kvar of distributed throughout the network. Foradditional studies, the number and percentage of candidatesis easily adjusted in the service restoration program. Units of

power are used when presenting the simulation results to co-ordinate with nominal and/or measured values, which are then

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Fig. 1. One-line diagram of 416-bus distribution system switch locations highlighted.

TABLE INETWORK COMPONENT LIST

adjusted to a complex current value based on the correspondingpower flow solution.

For all cases, the acceptable voltage range during the emer-gency operating condition was set at 110 V126 V. Results ofthe service restoration algorithm applied to arbitrary fault loca-tions in the test system are shown for cases with and without

. For each of the cases presented, the following informationis provided:

fault locations; total outage load before and after service restoration; total number of switch operations required; specific switches selected in each step of the restoration

algorithm;

total implemented; buses not restored.

A. Service Restoration Results Without

Table II displays the service restoration algorithm results fornine arbitrary fault locations on the test system. For each of thecases presented, of in-service customers was not consideredand all loads were given equal priority. The following observa-

tions are made. Cases 14 resulted in full restoration of the out-of-service

area selecting an initial and one or more pairsas necessary.

Cases 57 and and 9 resulted in partial restoration requiring

the selection of to isolate loads from restoration. In Case 8, no portion of the out-of-service area was able to

be restored.

For each fault, an exhaustive search over all single tier andin the out-of-service area was performed. In each case, the

solution obtained by the proposed algorithm was found to bea non-inferior solution to the service restoration problem ne-

glecting . For full restoration cases, this implies no solutionexists which restores the entire out-of-service area with fewerswitch operations. For partial restoration cases, no switch com-

bination exists which restores more load in the out-of-servicearea.

B. Service Restoration Results With

Table III displays the service restoration algorithm results forthe same nine fault locations with considered. All loads

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TABLE IISERVICE RESTORATION RESULTS WITHOUT LOAD CURTAILMENT

TABLE III

SERVICE RESTORATION RESULTS WITH LOAD CURTAILMENT

were again given equal priority. The following comments aremade regarding the results.

Cases 1, 2, 5, 8, and 9 required one along with torestore the entire out-of-service area.

Cases 3 and 4 required an initial and one pairalong with to restore the entire out-of-service area.

Case 6 resulted in partial restoration selecting alongwith one to isolate loads from restoration.

Case 7 resulted in partial restoration, selecting no op-tions and one to isolate loads from restoration.

For each fault, an exhaustive search was performed over theset of all candidate switches and reduced search space of op-tions downstream from each candidate . The solution obtainedby the proposed algorithm was found to be optimal with respectto the prioritized objectives including for each case. For fullrestoration cases, this implies no solution exists, which restoresthe entire out-of-service area with fewer or the same number ofswitch operations and less . For partial restoration cases, noswitch and combination restores more out-of-service loadand any reduction in results in less outage load restored.

C. Solution Comparisons

A comparison between the restoration schemes with and

without can be seen in Table IV. This table highlights themajor improvements and tradeoffs which occur when of

TABLE IVSERVICE RESTORATION SOLUTION COMPARISONSILLUSTRATION OF RESULTS

INCLUDING ADVANTAGES AND DISADVANTAGES OF LC

in-service customers is considered. The following changes toproblem objectives were observed:

an increase to the total load restored; a reduction to the number of switch operations required; no affect to the restoration solution.

With respect to the problem objectives, the benefits of the ad-dition of are clear. In addition to the improvements of theproblem objectives, the following benefits were also observed:

an increase to the total number of customers restored; an increase to the total load served.

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TABLE VCOMPARISON OF NUMBER OF CONFIGURATIONS CHECKED FOR PROPOSED METHOD AND EXHAUSTIVE SEARCH

In Case7, the available toeach candidate isinsufficientto adequately increase spare capacity beyond that required topick up additional out-of-service load. With the exception ofCase 7, which showed no change to the restoration solution,incorporating resulted in improvements to one or more ofthe service restoration objectives. The following comments onthe results are discussed.

1) to Increase Total Load Served: When the spare ca-pacity of an adjacent feeder approaches but is less than thetotal outage load, it is possible to restore a significantly largeramount of out-of-service load through . Such is the situa-tion in Cases 8 and 9. Without , no portion of the out-of-ser-vice area could be restored. By implementing , the entireout-of-service area was restored. This resulted in an increaseto both total load served and number of customers restored.Improvements to system reliability indices, such a SAIDI andCAIDI, may then be obtained.

2) to Reduce Number of Switch Operations: For Cases16, the addition of reduced the required number of switchoperations. In Cases 5 and 6, the number of customers restoredwas also increased. Cases 14 reduced the number of switchoperations required to restore the entire out-of-service area.

When full restoration can be obtained without , the DAtool or system operator must make a decision as to the appro-priate restorative action. For systems with automated switching,the additional switch operations should be performed to avoid

the curtailment of in-service customers. However, for systemswith manual switches, an additional parameter which may influ-ence this decision is total time required to make the additionalswitch operations. If the time required prolongs the outage be-yond the reliability index reporting limits, it may be prudentto implement . Lastly, these results may also be used todrive switch placement studies to identify locations where newswitches will have the greatest impact on reliability.

3) to Increase Number of Customers Restored: In Cases5, 6, and 8, the total number of customers restored was increasedwith the addition of . As seen in Case 6, while a net de-crease to the total load served following restoration was ob-

served, the number of customers restored increased. Utilizingin this manner then allows for an improvement to reliability

indices by reducing the number of customer experiencing sus-tained interruptions.

As the simulation results have shown, improvements to theservice restoration objectives are possible when is incorpo-rated. Specifically, eight of the nine cases presented showed im-provements to one or more service restoration objectives. Theseresults demonstrate the benefits which may be obtained by uti-lizing in service restoration of distribution systems.

D. Computational Performance

The search space of the problem grows exponentially with thenumber of switches and load curtailment options. The proposedapproach is designed to quickly eliminate infeasible solutionsand rank those remaining to check operating constraints of thepotentially Pareto optimal solutions first. To check constraints, amultiphase distribution power flow must be performed. This re-quires the majority of the computation time. As such, evaluationof the computational performance of the proposed method canbe made with respect to the number of configurations checkedfor an exhaustive search over the reduced search space.

For cases in which full restoration is achievable, an exhaus-tive search must check every possible configuration which usesless load curtailment and/or fewer switch operations than the re-sults from the proposed method. When only partial restoration isachieved, all combinations of switch status and load curtailmentoptions must be attempted to determine if additional load maybe restored. The results of this analysis may be seen in Table V.As seen in the table, the proposed method can find a solution up

to 150 000 times faster than a worst-case exhaustive search.

VI. CONCLUSIONS

The system impacts of load curtailment via DLC on distribu-tion applications have been investigated. Specifically, this paperhas presented a problem formulation, solution algorithm, andsimulation results for service restoration of power distributionsystems incorporating of in-service customers. The new for-mulation, even though a more complicated mixed integer non-linear optimization problem, allows us to improve one or moreof the service restoration objectives through inclusion of inrestoration solutions. These results support the deployment of

programs as a means to improve system operation and in-

crease reliability. The inclusion of time-dependent terms intothe problem formulation will be investigated in future work.

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The proposed solution algorithm prioritizes multiple objec-tives of the service restoration problem and embeds this priori-tization into a heuristic search. The heuristic incorporates ana-lytically based decision indices to rank candidate switches and

options. The result of the solution algorithm is a sequentialswitch and procedure which arrives at a new realizable net-work configuration. In the test cases presented, the algorithm

was able to find Pareto optimal restoration solutions as con-firmed by an exhaustive search.

Results are presented which demonstrate the differences en-countered when is incorporated in the restoration solution.It has been shown that the number of network switch opera-tions may be reduced; the number of customers restored maybe increased, and/or the total amount of load served may beincreased.

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[23] M. Kleinberg, K. Miu, and H.-D. Chiang, Service restoration ofpower distribution systems incorporating load curtailment, in Proc.2009 IEEE Int. Symp. Circuits and Systems, pp. 17091712.

[24] R. D. Zimmerman, Comprehensive distribution power flow: Mod-

eling, formulation, solutionalgorithm and analysis, Ph.D.dissertation,Cornell Univ., Ithaca, NY, 1995.

Michael R. Kleinberg (S05) received the B.S. andM.S. degrees in electrical engineering from DrexelUniversity,Philadelphia, PA. He is currently pursuingthe Ph.D. degree in the Electrical and Computer En-gineering Department, Drexel University.

His research interests include analysis and opti-mization of electric power distribution systems.

Karen Miu (M98) received the B.S., M.S., andPh.D. degrees in electrical engineering from CornellUniversity, Ithaca, NY.

She is currently an Associate Professor in theElectrical and Computer Engineering Department,Drexel University, Philadelphia, PA. Her researchinterests include distribution system analysis, dis-tribution automation, and optimization techniquesapplied to power systems.

Dr. Miu is a recipientof the 2000 National ScienceFoundation Career award, the 2001 Office of Naval

Research Young Investigator Award, and the 2005 Eta Kappa Nu (HKN) Out-standing Young Electrical and Computer Engineer Award.

Hsiao-Dong Chiang (F97) received the Ph.D.degree in electrical engineering and computer sci-ences from the University of California at Berkeleyin 1986.

He is currently a Professor of electrical andcomputer engineering at Cornell University, Ithaca,NY. He holds ten U.S. patents related to powersystems and Trust-tech-based nonlinear optimizationtechniques. His research and development interestsinclude theoretical developments and practicalapplications of nonlinear system theory, computa-

tion, and application to electrical circuits, signals, systems, and medical images.

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