PTDF-Based Automatic Restoration Path Selection

1686 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 3, AUGUST 2010

PTDF-Based Automatic Restoration Path SelectionChong Wang, Student Member, IEEE, Vijay Vittal, Fellow, IEEE, Venkat Sharma Kolluri, Senior Member, IEEE,

and Sujit Mandal, Senior Member, IEEE

Abstract—After a power system blackout, system restoration isa critical task for the dispatchers. With the stringent requirementsfor faster and more efficient system restoration, an objective trans-mission restoration path selection procedure, including the optionto check constraints, may be more responsive in handling unex-pected system changes and in providing directed guidance to thedispatchers to identify the optimal transmission path and to de-liver power to other power plants or to pick up load as needed. Thispaper presents a power transfer distribution factor (PTDF)-basedpath selection approach for large-scale power systems. Two typesof restoration performance indices are utilized considering all pos-sible restoration paths, which are then ranked according to theirexpected performance characteristics as reflected by the restora-tion performance index. PTDFs and weighting factors are used todetermine the ordered list of restoration paths, which can enablethe load to be picked up by lightly loaded lines or relieve stresson heavily loaded lines. Test results which are primarily concernedwith load restoration in the outaged system are provided based on arealistic restoration exercise for the Western region of the Entergytransmission system.

Index Terms—Power transfer distribution factor, restorationperformance indices, transmission system restoration.

I. INTRODUCTION

P OWER system restoration following a blackout is one ofthe critical tasks for system operators in the control center.

However, most utility companies and reliability regions rely onan offline restoration plan and the experience of dispatchers toselect and implement scenarios for the black start path [1], [2]and procedures to restore the system. Using a restoration plandesigned based on past experience and off line analysis may notbe the most reliable approach to come up with a black start planas it is difficult to predict changing network configurations andloading levels. To address this need, computer tools have beendeveloped and implemented in [3] and [4] for the online oper-ational environment. Since actual outages are hard to predict inthe planning stages, the restoration plan only serves as a guideto the operator. When performing system restoration, operatorsneed near real-time system information in order to make deci-sions under changing system conditions.

Manuscript received June 02, 2009; revised October 28, 2009. First publishedJanuary 26, 2010; current version published July 21, 2010. This work was sup-ported by the Power System Engineering Research Center. The views expressedherein are those of the authors and do not necessarily represent the views of En-tergy Services, Inc. or any of its affiliates. Paper no. TPWRS-00414-2009.

C. Wang and V. Vittal are with the Department of Electrical Engineering,Arizona State University, Tempe, AZ 85287 USA (e-mail: [email protected]; [email protected]).

V. S. Kolluri and S. Mandal are with Entergy Services, Inc., New Orleans,LA 70113 USA (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2009.2037820

During restoration of the bulk transmission network thefollowing constraints should be systematically verified and ap-plied to determine the sequence in which the transmission linesshould be energized while satisfying the reliability criteria.These criteria include [5]:

• real and reactive power balance;• thermal constraints on transmission lines;• sustained overvoltages during early restoration;• switching surges;• unstable phenomenon of self-excitation;• maintaining steady-state and transient stability during

restoration.In an effort to reduce the time duration and cost related to ser-

vice interruption, and to check some of these constraints, sev-eral analytical tools have been proposed, such as: expert sys-tems [6], [7] and heuristic approaches [8]. These methods in-tegrate knowledge from the operators and computational algo-rithms such as power flow and transient stability software to op-timize the restoration process and to verify that constraints arenot violated.

The restoration procedure following a power system outagespans three time periods or is a three step process: 1) sendingcranking power to non-black start generators or to the criticalloads from the black start generators, or relying on assistancefrom outside the system, 2) integration of generation and trans-mission to recreate a skeleton of the bulk power system, and 3)minimization of the unserved load [9]. This paper specificallyaddresses the constraint checking aspect of 3).

The problem of optimizing the utilization of all availablecranking power in order to maximize the generation capabilityis complex and involves a large number of combinatorialchoices [10]. A number of constraints have to be verified beforethe cranking process. In this phase, constraints mainly includevoltage and frequency transient, voltage drop and protectiverelay actions [11]. In 2) and 3) described above, thermal con-straints and stability constraints need to be checked.

This paper examines the restoration path selection for loadrestoration in blacked-out transmission systems based on ef-ficient checking of the thermal, transient stability and voltageconstraints on the transmission system after the affected areahas sufficient power supply/generation available. It should beemphasized that the algorithm determines the path selectionto restore the area without violating constraints, however, thelines identified for restoration may not to be energized simulta-neously, and each transmission switching operation should bechecked and verified carefully for safety constraints prior to en-ergizing the lines.

In June 2005, severe thunderstorms precipitated and forcedthe interruption of three critical lines in the Western region ofthe Entergy system. The loss of these lines impacted the ability

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WANG et al.: PTDF-BASED AUTOMATIC RESTORATION PATH SELECTION 1687

Fig. 1. Western region of the Entergy system.

of the electrical system to serve the Western region load becausethese lines were tie-lines into the Western region. Within sec-onds, the generation and other critical lines in the area tripped,and over several minutes the Western region separated from therest of the system [12]. The system was restored back to normalwithin four hours after the disturbance. The Western region ofthe Entergy network is shown in Fig. 1. The load consumption inthe system may be unknown and need detailed cold load pickupmodeling [13]. The load levels in this analysis at every ener-gized bus are determined from the Entergy system operationsplanning case. The proposed power transfer distribution factor(PTDF)-based restoration path selection method is applied tothe Western region storm scenario and a novel approach to re-store the Western region is discussed in the paper.

The paper is organized into four sections. Section II presentsan overview of the PTDF-based restoration path selection al-gorithm. Section III describes the steps taken to recreate thesystem conditions that existed on June 15, 2005, that led tothe storm-related outages in the Western region of the EntergySystem. Section IV presents several illustrative examples of theapplication of the proposed technique for restoring Western re-gion. Conclusions drawn from the application are presented inSection V.

II. PTDF-BASED RESTORATION PATH SELECTION

To determine the correct sequence for energizing the lines, theconcept of PTDF [14, p. 421] and weighting factors are used.This idea was originally developed in contingency analysis andevaluated for the removal of branches or the loss of generatorsat specific nodes. In this paper the novelty of the approach is toplan restoration by calculating the PTDFs for candidate lines tobe closed. Even though the concept of PTDF is well established,the use of this idea in system restoration is unique and has notbeen attempted before. In the approach developed, restoration

performance indices (RPIs) will be calculated for ranking twotypes of branch closures.

During the transmission restoration process, all the newlyadded lines can be divided into two categories:

1) radial lines—which will create a branch between an ex-isting node and a new node;

2) loop closure lines—which will complete paths betweentwo existing nodes.

The restoration time period, in which the transmission systemis restored, usually takes 3 to 4 hours [9]. In order to speed upsystem restoration in the absence of system constraint viola-tions, it is assumed that the radial lines will be candidates tobe restored first, rather than loop closure lines.

A. Radial Lines Restoration Performance Index

If a line is a radial line between buses and , the power flowon the radial line can be considered as a complex bus powerinjection into the existing system [15, pp. 199–203]. The PTDFrelating the loading in the line from bus to bus with respectto the injected complex bus power on bus , is denoted as

:

(1)

The elements in the equation above are obtained fromthe bus impedance matrix referenced to the swing bus. isthe bus voltage at bus . is the primitive impedance of theline connecting bus to bus . It is to be noted that (1) is validif voltages are near their rated value. It is important to checkand avoid voltage drops in each step described in Section III.Voltage control methods such as connecting shunt elements ortransformer tap positions adjustment may need to be applied.


If the distribution factors are arranged in a rectangular array,the power transfer distribution factor matrix can be formed as

(2)

where is the number of lines and is the number of busesin system already restored. is the power flow change on linenumber and is the power injection change on bus number.

After a new radial line with load at the end of the line is re-stored in the system, the power flow on the lines that have al-ready been restored will change. The capability of different re-stored transmission lines to sustain this power flow change willalso be different. For example, a lightly loaded line will be ableto withstand a higher power flow increase than a moderatelyloaded line when a radial line with load is restored in the system.The change in power flow with each possible line addition canbe estimated using the PTDF calculation with respect to the ex-isting system topology.

If all the restored lines are not close to their thermal limit, theexisting power flow expressed as a percentage of the thermallimit on each restored line will be used as a weighting factor

on the change in power flow (obtained from PTDFs) to eval-uate each candidate radial line path restoration. A restorationperformance index (RPI) is then evaluated for each candidateradial line considered. The RPI is the sum of the products ofthe weighting factor and power flow change in each existingtransmission line. Then, the candidate radial line with the lowestvalue of its RPI vector element is restored first. This index is re-ferred to as a Type 1 RPI:

(3)

B. Loop Closure Lines Restoration Performance Index

If a line with primitive impedance is a loop closure be-tween buses and , a new intermediate matrix shown in (4) atthe bottom of the page is formed.

To get the new bus impedance matrix , the augmentedrow and column shown in (4) is Kron reduced to obtainas shown in the following:

(5)

(6)

(7)

where and denote the th column and row of a matrix,respectively. is the primitive bus impedance matrix.

To calculate the updated PTDF relating the loading inthe line from bus to bus with respect to the injected complexbus power on bus , after adding a line from bus to bus ,substitute the new bus impedance matrix values obtained from(5) into (7). The PTDF is then given by

(8)

Then, the power transfer distribution factor matrix with the ad-dition of the line from bus to bus is obtained as

For each possible loop closure line, a specific matrixis evaluated:

(9)

This matrix captures the change in each PTDF element dueto the addition of a line. If heavily loaded lines exist in therestored system, restoration of the next radial line can result inlimit violations. In such instances, loop closure lines should befirst evaluated for restoration in order to relieve the stress on

(4)


Fig. 2. Restoration path selection algorithm flow chart.

the lines that are almost at their limit. In order to compare thecandidate loop closure lines a Type 2 RPI is defined:

(10)

is the column vector of injected complex bus powerchanges due to the closing of a candidate radial line. Then,the candidate loop closure line with the lowest Type 2 RPI isrestored first, which means that it can relieve the most stress onthe heavily loaded lines.

Fig. 2 shows the flow chart of the proposed automatic restora-tion path selection algorithm. As shown in the flow chart, thealgorithm only needs the current system state to determine thenext line to be closed, rather than power flow calculations tocheck the transmission line thermal constraints. All possibletransmission lines that can be restored are evaluated. If any lineis not available or fails to be closed, operators can consider thenext best option from the sorted restoration index list until theblackout area is fully restored.

C. Criterion and Area Determination Algorithm

During the early stages of restoration, a reasonable balanceshould be maintained between generation and load to avoid fre-quency deviations [16]. At this time, generation and load in thesystem are kept at a very low level to maintain system basic op-eration, and there might be several radial line candidates that

have RPI values that are close to each other but could result inthe restoration process bringing back to service totally differentload areas.

The load areas to be restored should be defined by systemoperators based on system configuration or load priorities. Inthe proposed approach if there are more than one load areas, thesequence in which to restore the load areas is determined basedon the NERC criterion [17] and system transient securityanalysis. The area with the largest transient stability margin andleast number of insecure contingencies will be restored first.

contingency analysis is performed on all candidateload areas to be restored using the software package TSAT( [18]). The severity of a contingency can be assessedusing the transient stability index (TSI). The TSI is calculatedas follows [18]:

(11)

is the maximum angle separation of any two generatorsin the system at the same time in the post-fault response. Ifis greater than 360 , it is defined that this contingency fails the

contingency test. Hence, and corre-spond to stable and unstable conditions, respectively. The areawith the largest stability margin and least percent insecure con-tingencies should be restored first. This ensures that the restoredload area would be least susceptible to further degradation dueto transient instabilities.

D. Line Switching Issues

When energizing lightly loaded transmission lines orunderground cables, the excessive VArs generated by theundercompensated high voltage lines can increase voltages tounacceptable high levels which are referred to as sustainedpower frequency overvoltages. If not controlled, these voltagescould cause serious reactive power imbalance resulting in gen-erator self-excitation, transformer overexcitation and harmonicdistortions.

Basically, sustained overvoltages can be controlled by ab-sorbing the reactive power generated by the lightly loaded trans-mission lines. This can be done in several ways [5]:

• having sufficient under-excitation capability on the gener-ators;

• picking up loads with low power factor;• switching on shunt reactors;• adjusting transformer taps.Due to the fact that adjustments of the control variables are

subject to the constraints imposed by plant and system oper-ating conditions, the total effect of these control variables willdetermine whether the long transmission line can be energizedsuccessfully or not.

III. CASE STUDY AND RESULTS

The proposed PTDF-based restoration path selection ap-proach is applied to the Western region of the Entergy system.The detailed illustrative examples of the various calculations inthis case are shown in Section IV.


TABLE ILINES TRIPPED IN THE SYSTEM

TABLE IISYSTEM RESTORATION TIME LOG

A. Preparation of the June 15, 2005 Storm Case

In the operations planning case representing summer peakconditions, the Western region load was approximately 1900MW. Following the storm event a total of 11 lines were trippedseparating the Western region from the rest of the Entergysystem [12]. The lines that were tripped are shown in Table I.The numbers in brackets are power flow model bus numbers.

Shortly after the disturbance event, the transmission opera-tion center (TOC) operators started to take manual actions toswitch the outaged lines back into service. The first manual ac-tion recorded in the events summary relating to the outages andthe restoration activities is the restoration of the China (97714)-Amelia (97689) line at 19:04:44, approximately 8 min after theinitial event. The restoration failed because of sustained phase Bto ground fault. The first successfully restored line was Kountze(97700)-Doucett (97694) at 19:22:28, approximately 26 minafter the initial event. The actual restoration process that wasfollowed by the system operators is detailed in Table II.

From the details provided in [12], the Lewis Creek unitswhich are the major generating units in the affected area couldnot meet the immediate load demand because they sustainedminor damage during the event. Therefore, the entire powersupply for the affected area had to be obtained from outside theaffected area. Given this premise it was imperative that criticaltie lines be restored first in order to provide an outside source

for black start. The application of the proposed path selectionalgorithm with constraint checking is detailed below.

B. Proposed System Restoration

The steps taken by the automatic restoration path selectionalgorithm is as follows:

1) Step 1: China (97714) is chosen as the power sourcebus sending cranking power to the generators inside theaffected area. China (97714) Substation is one of the fourbulk power sources into the region. China (97714)-Jacinto(97476) and China (97714)-Porter (97567) are the onlytwo 230 kV lines in the Western region. There is atotal of 1587-MVA power injection capability into theWestern region through these two lines. The transmissionline China (97714)-Amelia (97689) experienced asustained fault during the outage and is one of two 230kV paths to the China Substation. The other line China(97714)-Sabine (97716) is the first line that the TOCoperators tried to restore at 19:04:44 [12]. The first stepin the proposed approach is to provide power/voltage tothe critical generator buses inside the affected area. Thegoal in this study is to restore voltage to the Lewis Creek(97451, 97452) generating bus/station, if the station isavailable and not damaged.

2) Step 2: The Woodlands area is located in the Southwestportion of the Western region, and it has a highconcentration of residential and commercial loads. Thisarea includes Conroe (97459), Alden (97544), Goslin(97468) and some other heavily loaded buses. This areais defined as Area I, and the remaining portion of theWestern region as Area II. The transmission lines at theboundary between Area I and Area II are shown in Fig. 3and Table III.

contingency analysis is then simulated on bothareas using TSAT software as described in Section II-C.The results show that if Area I is restored first, 4 outof the 68 possible contingencies are insecure and allsecure contingencies have an average TSI value of 89.85.However, if Area II is restored first, 16 out of the possible184 contingencies are insecure and the average TSI forall secure contingencies is 87.98. Hence, based on therelative severity of the dynamic security assessment it isdetermined that load Area I should be restored beforeload Area II. Once this has been ascertained the algorithmprogresses systematically to determine the transmissionpaths to restore in order to supply the load in Area I basedon the procedure described above.

3) Step 3: The transmission lines and loads in Area Iare restored based on the RPI values of all possibletransmission lines to be restored. The assumption is thatthe same amount of load at each end of the new radiallines will be energized. The algorithm will choose theline that is closer to the generator buses or power sourcebuses. The system topology is quite similar to that of atree. When one or more transmission lines are close totheir transmission limit, the algorithm will force the treeto create some loops to relieve the stress on the branches.


Fig. 3. Loads areas boundary in Western region.

TABLE IIIBOUNDARY LINES BETWEEN AREA I AND AREA II

4) Step 4: After the transmission lines supplying the loadarea are fully restored, contingency analysis issimulated on restored buses as described in Section II-C.Several solutions to the maximum load level that can besupplied in order to guarantee that all contingencieswill be secure are determined. The resulting load pick upcurve obtained by the proposed method is then comparedto the actual load pick up curve that was obtained whenthe Western system was restored by system operatorsfollowing the storm-related outages. The plot is shownin Fig. 4.

From the plot it is observed that the new restoration pathstrategy generated by the program provides a more efficient ap-proach in terms of the numbers of operations to restore the un-served load in the system than the actual restoration operation.In addition, the transmission thermal limit and transient stabilityconstraints are also satisfied. It is to be noted that in reality, there

Fig. 4. Comparison of the load curve based on proposed method and the actualsystem operations.

may be practical limitations in energizing some of the lines dueto damages. Such factors were not considered in the analysis butcan be easily implemented by taking the specific lines out of thelist of candidate lines for potential energization.

IV. ILLUSTRATIONS OF INTERMEDIATE STEPS

A. Example I: Radial Line Ranking With Type 1 RPI

After the cranking power supply is available to the generatorbuses in the affected area, the transmission lines and loads arerestored gradually. In this case, 4 transmission lines into Area Iare evaluated using the . Assuming that 5% of the load atthe end of the lines is picked up while the transmission lines arerestored, the RPI calculation results are shown in Table IV.

Based on this ranking, the transmission line from LewisCreek (97461) to Sheawil (97466) as shown in Fig. 5 is restoredfirst. The comparison between actual power flow and the PTDF


Fig. 5. Single line diagram showing the radial line candidates and the optimal line in RPI table after generator buses are energized, Example I.

TABLE IVTYPE I RPI RESULT IN EXAMPLE I

Fig. 6. Comparison of the actual power flow and PTDF predicted power flow.

predicted power flow is shown in Fig. 6. The error observed iswithin 6%, indicating that the PTDF provides a fairly accurateestimate of system performance. The comparison of actualpower flow with that predicted by PTDF after adding the otherlines in Table IV is shown in Fig. 7. From this figure, it isobserved that the PTDF based power flow results are veryclose to those obtained by running the actual power flow. Allthe candidate radial lines are then evaluated using the RPIapproach. The RPI values indicate the restoration priority ofthe lines. It is recommended that the line with the lowest RPIvalue should be restored first. However, the final decision onrestoring the lines could be left to the operators based on theirexperience and safety considerations. During the restorationprocess with RPI value, the generation capability should bechecked and verified to maintain generator stability.

Fig. 7. Comparison of the actual power flow after adding lines in Table IV.

B. Example II: Loop Closure Line Ranking With Type II RPI

Since the power flow analysis indicates that the line97476-97543 is within its rated limit, the algorithm devel-oped to determine the sequence in which transmission linesare restored is then applied to determine the next transmissionline to be restored. PTDFs and are utilized to choose thenext line to be energized. Fig. 8 shows that when line LewisCreek (97461)-Sheawil (97466) is close to its thermal limit,the algorithm chooses line Security (97456)-Jayhawk (97542)which connects two buses that have already been energized tobe restored next. The power flow change after closing this lineis shown in Table V and Fig. 9.

These results reveal that the power flow on the heavilyloaded line 97456-97542 reduces from 140.8 MW to 102.4MW (thermal limit of this line is 206 MVA), and some of thepower flow is picked up by the line 97475-97476 (thermal limitof this line is 287 MVA). This relieves the stress on the heavilyloaded line and fully utilizes all restored lines to speed up thesystem restoration.

C. Example III: Sustained Overvoltage Checking and Control

Before energizing the transmission lines selected using theRPI approach, sustained overvoltages should be evaluated tomake sure no voltage violations occur. A generator terminalvoltage violation example in the process of restoring line 97458-97461 is depicted in Fig. 10. The generator terminal voltagesare shown in Fig. 11. Buses 97451 and 97452 are the generator


Fig. 8. Single line diagram showing the loop closure line Security (97456)-Jayhawk (97542) is energized due to line thermal limit on another line, Example II.

TABLE VTYPE II RPI RESULT IN EXAMPLE II

Fig. 9. Comparison of the power flow before and after adding loop closure line.

Fig. 10. Single line diagram showing the transmission line 97461-97458 (thedashed line) to be energized.

buses inside the area being restored, bus 97714 is the outsideblack start source, which has a large generation capability.

Fig. 11. Generator terminal voltage after restoring the line 97458-97461.

If the generator terminal voltages are reduced to 0.95 p.u.before closing the line, or a shunt reactive source is connected,the generator voltages are within the acceptable limits as shownin Figs. 12 and 13.

The results show that the sustained overvoltages can be effi-ciently controlled. It is important to note that the extent of thegenerator’s voltage reduction is usually constrained by under-excitation of generators brought about by a number of limitingfactors, including generator terminal low voltage limit, reactiveampere limit relay and minimum excitation limit relay. It maybe necessary that more than one voltage control method needsto be applied in a given system.

D. Example IV: Load Level Determination After Area I isFully Restored

After the restoration of Area I is complete, the security ofthe restored transmission lines in Area I has to be maintainedbefore restoring Area II. If the system is vulnerable to line trip-ping during the restoration process, this security check shouldbe done more frequently. After restoring Area I based on thedynamic analysis, several possible solutions to satisfy re-liability criterion are satisfied by reducing the load level on cer-tain buses as shown in Table VI. The first column in parenthesis


Fig. 12. Generator terminal voltages are reduced to 0.95 p.u. before closing theline.

Fig. 13. Generator terminal voltages with shunt reactive source connected.

TABLE VISOLUTIONS IN EXAMPLE III

shows the MW value of the load, and the percentage value rep-resents the load after reduction as a percentage of the load priorto the outage.

This analysis determines the upper bound of the load level onthese buses during the restoration process of Area I. If the actualsystem has a particular load requirement, operators can considerthe option of reducing other loads.

V. CONCLUSIONS

This paper provides a systematic method for developing anautomatic restoration path selection procedure after a blackout/

island occurs. The suggested approach uses the power transferdistribution factor algorithm and weighting factors to determinethe optimal restoration sequence for the transmission system.This path selection procedure is performed by checking systemthermal constraints, transient stability constraints and voltageconstraints. The restoration path selection algorithm is intendedto assist the system operator during restoration, by providing arestoration index. Two kinds of RPI are shown. The restorationindices are effective during the restoration of the transmissionsystem as they provide guidance to the operators on how trans-mission lines should be restored. The algorithm was tested onthe Western region of the Entergy system. The restoration se-quence for the transmission lines ensures that the thermal con-straint is satisfied during the restoration and can adapt to thechanging system conditions. The transient stability constraint isalso checked before and after each load area is restored to makesure that the system is secure and stable.

ACKNOWLEDGMENT

The authors would like to thank Mr. M. Adibi for providinginvaluable insight into the technical aspects of restoration andhelping develop a much improved paper.

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Chong Wang (S’09) received the B.E. degree in electrical engineering fromTsinghua University, Beijing, China, in 2006. He is currently pursuing the Ph.D.degree in the Electrical Engineering Department at Arizona State University,Tempe.

Vijay Vittal (S’78–F’97) received the B.E. degree in electrical engineering fromthe B.M.S. College of Engineering, Bangalore, India, in 1977, the M.Tech. de-gree from the Indian Institute of Technology, Kanpur, India, in 1979, and thePh.D. degree from Iowa State University, Ames, in 1982.

Dr. Vittal is a member of the National Academy of Engineering.

Venkat Sharma Kolluri (SM’86) received the B.S.E.E. degree from VikramUniversity, Ujjain, India, in 1973, the M.S.E.E. degree from West Virginia Uni-versity, Morgantown, in 1978, and the M.B.A. degree from the University ofDayton, Dayton, OH, in 1984.

He worked for AEP Service Corporation in Columbus, OH, from 1977through 1984 in the Bulk Transmission Planning Group. In 1984, he joinedEntergy Services Inc., New Orleans, LA, where he is currently the Manager ofTransmission Planning. His areas of interest are power system planning andoperations, stability, reactive power planning, and reliability of power systems.

Mr. Kolluri is involved in several IEEE committees and working groups andis a member of CIGRE.

Sujit Mandal (S’97–M’99–SM’08) received the B.Tech degree in electrical en-gineering from the Indian Institute of Technology (IIT), Kanpur, India, and theM.S. degree in electrical engineering from Kansas State University, Manhattan,KS, in 1997 and 1999, respectively.

He worked as a consultant at Power Technologies, Inc., Schenectady, NY,from 1999 to 2000. Presently, he is with Technical System Planning, EntergyServices, Inc., New Orleans, LA.

PTDF-Based Automatic Restoration Path Selection

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