A Two-Stage Framework for Power Transformer Asset Maintenance Management—Part II: Validation Results

1404 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 2, MAY 2013

A Two-Stage Framework for Power TransformerAsset Maintenance Management—Part II: Validation

ResultsAmir Abiri-Jahromi, Student Member, IEEE, Masood Parvania, Student Member, IEEE,François Bouffard, Member, IEEE, and Mahmud Fotuhi-Firuzabad, Senior Member, IEEE

Abstract—A two-stage framework for transformer mainte-nance management is introduced and formulated in Part I of thistwo-part paper in the context of transmission asset managementstrategies (TAMS). The proposed model optimizes maintenanceoutage schedule over a predefined period of time by taking intoaccount the actual and expected transformer assets’ conditiondynamics in terms of failure rate and resource limitations inmidterm horizons, as well as operating constraints, economicconsiderations and N-1 reliability in the shorter term. In Part II, asmall six-bus system is first used to demonstrate how the two-stagemaintenance framework works using a step-by-step procedure.Then, IEEE-RTS is used to investigate the performance of theproposed model in more detail. In addition, the impacts of varyingthe characteristics of the proposed midterm and short-term main-tenance schedulers, such as flexibility in time horizon selection, onmaintenance scheduling results and computational efficiency areinvestigated on IEEE-RTS. The numerical studies indicate thatthe proposed framework gives appropriate results in terms of eco-nomics and technical constraints at a reasonable computationalcost.

Index Terms—Computational burden, failure rate, midterm andshort-term maintenance planning, mixed integer linear program-ming, transformers, transmission asset management.

I. INTRODUCTION

T HIS paper validates and analyzes the midterm and short-term maintenance scheduling models and formulations

developed in Part I [1] through several case studies conductedon a six-bus test system and the IEEE-RTS. In addition, thetrade-off between the computation burden and accuracy of thesolutions of the midterm and short-term maintenance sched-ulers’ is investigated using the IEEE-RTS. In these studies, themaintenance and repair tasks associated with transformers arearbitrarily divided into minor and major maintenance and repairtasks. Minor tasks stand for the tasks that would be done on-sitewith short outage durations while major tasks include major

Manuscript received February 20, 2012; revised July 03, 2012 and August 20,2012; accepted August 27, 2012. Date of publication October 16, 2012; date ofcurrent version April 18, 2013. The work of A. Abiri-Jahromi was supported inpart by a McGill Engineering Doctoral Award. Paper no. TPWRS-00168-2012.A. Abiri-Jahromi and F. Bouffard are with the Electrical Engineering

Department, McGill University, QC H3A 2A7, Canada (e-mail: [email protected]; [email protected]).M. Parvania and M. Fotuhi-Firuzabad are with the Center of Excellence in

Power System Management and Control, Electrical Engineering Department,Sharif University of Technology, Tehran, Iran (e-mail: [email protected];[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2012.2216904

transformer overhauls and have much longer outage durations.However, any other maintenance task division can be definedand used based on user preference without loss of generality.In Section II, a small six-bus test system is used to demon-

strate how the two-stage maintenance framework works using astep-by-step procedure. Accordingly, all the steps taken to reachthe final solution in midterm and short-term maintenance sched-ulers’ are explained.In Section III, the midterm maintenance scheduling model is

applied to the IEEE-RTS [2], and its capabilities are studied indepth through two studies. The studies are focused on 1) ex-amining the effects of variations in Weibull distribution param-eters on midterm maintenance schedules, 2) examining the ef-fects of resource limitations primarily labor constraint on main-tenance schedules, and 3) examining the effects of midterm timeblock duration on the computation burden and accuracy of thesolutions.In Section IV, some of the midterm maintenance schedules

obtained in Section III for IEEE-RTS is fed into short-termmaintenance scheduler and exact hourly maintenance outageschedules of transformers obtained. In this section, the char-acteristics of the short-term maintenance scheduler and howit handles requests for maintenance coming from the midtermplanning stages are examined in depth. Also, the computationburden and performance of the short-term maintenance sched-uler are investigated for different midterm time block durations.

II. ILLUSTRATIVE STUDY USING A SIX-BUS SYSTEM

An illustrative study is conducted on a six-bus system in thissection to demonstrate how the two-stage maintenance frame-work works using a step-by-step procedure. The six-bus testsystem, depicted in Fig. 1, consists of three generating units,7 transmission lines, 2 transformers and three load points. Thesystem data are given in Appendices A and B.

A. Midterm Maintenance Scheduling Model Solution Steps

The midterm maintenance horizon of the six-bus test systemis considered to be 8 weeks, which is divided into 8 weeklymidterm time blocks. The duration of the midterm time horizonand the corresponding time blocks can be selected subjectivelybased on user preference. Also, note that the duration of themidterm time blocks determines the duration of the short-termhorizon. Accordingly, since the duration of each midterm timeblock is assumed to be one week in the six-bus system, the du-ration of short-term horizon is also one week. The hourly load

0885-8950/$31.00 © 2012 IEEE

ABIRI-JAHROMI et al.: TWO-STAGE FRAMEWORK FOR POWER TRANSFORMER ASSET MAINTENANCE MANAGEMENT 1405

Fig. 1. One-line diagram of the six-bus system.

TABLE IWEIBULL DISTRIBUTION PARAMETERS OF TRANSFORMERS’TIME VARYING FAILURE RATE IN THE SIX-BUS SYSTEM

curve in the 8-week midterm horizon of the six-bus system isconsidered to be similar to the first 8 weeks load profile of theIEEE-RTS [2]. The peak load of the six-bus system is assumedto be 270 MW. The failure rates of the three generating units atthe beginning of the maintenance planning horizon are assumedto be similar and equal to 4 failures per year. In addition, thefailure rates of transmission lines at the beginning of the mainte-nance planning horizon are also assumed to be similar and equalto 0.2 failures per year. The Weibull distribution parameters as-sociated with minor and major failure rates of transformers T1and T2 are also summarized in Table I.The major solution steps performed in midterm mainte-

nance scheduler to determine the maintenance schedules oftransformers T1 and T2 in the six-bus system are given asfollows:1) Select a set of scenarios such that each scenario containsoutage of a transformer. In this study, the outage scenariosare selected based on the N-1 criterion, however, any otherapproaches can be used by user to obtain outage scenarios.Accordingly, two outage scenarios are considered in thesix-bus system, including transformer T1 outage and trans-former T2 outage.

2) Calculate the occurrence probability of each scenario se-lected in Step 1, in each hour of midterm time block [1, eq.(3)]. The results are summarized in Table II. In Table II,

denotes the th failure rate of the th transformerin midterm time block .

3) Using the NCUC simulator introduced in [1], calculate thesystem operating cost over each midterm time block for thebase case and each outage scenario selected in Step 1. TheNCUC simulator outputs for the base case and transformeroutage scenarios are summarized in Table III.

TABLE IIOCCURRENCE PROBABILITY OF SELECTED SCENARIOS

IN EACH MIDTERM TIME BLOCK

TABLE IIISIX-BUS SYSTEM OPERATING COST OVER MIDTERM TIME

BLOCKS FOR BASE CASE AND SCENARIOS ($)

4) Calculate the expected cost associated with each outagescenario in each midterm time block based on the resultsgiven in Tables II and III [1, eq. (2)].

5) Calculate the expected cost associated with transformerpreventive maintenance outage in each midterm time blockbased on the results obtained in step 4 [1, eq. (5)].

6) Calculate the explicit costs associated with the preventivemaintenance and repair of a failed transformer using thedata provided in the Appendix B[1, eqs. (6)–(8)].

7) Solve the midterm maintenance scheduling optimizationproblem, formulated by [1, eqs. (9)–(16)] using commer-cial MILP solver such as CPLEX [3].

In the six-bus test system, transformers T1 and T2 are, respec-tively, scheduled for minor and major maintenance in the 5thand 4th periods based on the data given in the Appendices A,B and the midterm maintenance model solution steps 1 to 7.The optimal midterm maintenance objective function value is$59 000.3.In the proposed midterm model, transformers’ location in

the transmission network and their aging momentums arethe decisive parameters which control the preventive mainte-nance outage schedules. Transformer location determines theexpected change in system operating cost under transformeroutage as summarized in Table III while aging momentumquantifies the expected probability of transformer failure sce-narios. In the Weibull distribution model, the transformer agingmomentum is characterized by parameters and as follows:

(1)

B. Short-Term Maintenance Scheduling Model Solution Steps

The goal here is to demonstrate how the short-term mainte-nance scheduling model, developed in Part I [1], performs. Ac-cordingly, the exact hourly major maintenance outage schedule


TABLE IVSHORT-TERM HORIZON OBJECTIVE FUNCTION VALUE

Fig. 2. Hourly scheduling of the six-bus system generating units. (a) Withoutsecurity constraint. (b) With security constraint.

of transformer T2 in the 4th midterm time block is determinedand analyzed here using the short-term maintenance schedulingmodel. The following input data are required by the short-termmaintenance model:1) transformers that should be maintained in the short-termhorizon, e.g., T2;

2) duration of the associated maintenance tasks, e.g., 100hours for major maintenance task;

3) load profile of the associated short-term horizon.First, the short-term maintenance scheduling problem of

transformer T2 is solved without considering any securityconstraint. Then, outage of transmission line L1 is consideredas a security constraint and the short-term transformer main-tenance scheduling problem is solved again with this securityconstraint and compared with the previous case to explainthe impact of security constraints on maintenance schedule,operating procedure and system cost. The resulting schedulesare summarized in Table IV. The output levels of generatingunits are also shown in Fig. 2. Table IV indicates that trans-former T2 is scheduled for major maintenance in hours 37–136in the case without security constraint. Fig. 2(a) shows thatonly units G1 and G3 are committed to satisfy the operatingconstraints. However, this case does not guarantee the securityof the system under the outage of system equipments.Next, the major maintenance outage schedule of transformer

T2 is determined considering the outage of transmission line L1

Fig. 3. Schematic representation of the midterm and short-term major mainte-nance scheduling of transformer T2.

as a security constraint. Fig. 2(b) shows that unit G2 is also com-mitted in this case in the hours that transformer T2 is scheduledfor major maintenance. In addition, generating units’ G1 and G3schedules and outputs have adjusted by the short-term mainte-nance scheduling model to satisfy the operating constraints aswell as the security constraint of transmission L1 outage. Theseadjustments in units scheduling assure that the outage of trans-mission line L1 would not endanger the system security whiletransformer T2 is onmaintenance. It can also be seen in Table IVthat the objective function value in the case where the sched-uling is robust to outage of line L1, is much higher than that ofthe case without security constraint.The schematic representation of the midterm and short-term

major maintenance scheduling of transformer T2 in the six-bussystem for the case with security constraint is also shown inFig. 3. As shown in this figure, the midterm maintenance sched-uling model has just located the midterm time block in whichmajor maintenance task should take place on transformer T2,i.e., 4th midterm time block. This is while the short-term main-tenance scheduler has determined the exact hourly outage timeof transformer T2 in the 4th midterm time block.

III. MIDTERM TRANSFORMER MAINTENANCE SCHEDULINGNUMERICAL STUDIES USING IEEE-RTS

In this section, the midterm transformer maintenance modelis applied to the IEEE-RTS to investigate the characteristics ofthemodel in depth [2]. The one-line diagram of the IEEE-RTS isdepicted in Fig. 4. This system is composed of 24 buses, 32 gen-erating units, 20 load points, 33 transmission lines and 5 trans-formers. Some of the required data for the case studies suchas generating units’ and transmission lines’ data are given in[2]. Other data such as decoupled deteriorating failure rates ofeach transformer, average duration of transformer maintenanceor failure outages, average number of working hours necessaryto perform maintenance or repair tasks, which are not given in[2], are defined subjectively and provided in the Appendix B.The annual peak demand of the system is 2850 MW and theyearly load profile of the IEEE-RTS [2] is utilized to obtain thesystem hourly load curve nodal distribution. In addition, the ca-pacities of the transformers are reduced to 60% of the valuesprovided in [2] in order to obtain a better illustration of the per-formance of the models.For the proposed model, all the problem formulations are

linear. Accordingly, the formulations were implemented in theGAMS environment and solved using CPLEX 12 [3] on a PCequipped with 3.0-GHz processor and 10 GB of RAM.A network-constrained unit commitment (NCUC) simulator

is developed to calculate the system operating costs for the basecase and five single transformer outage scenarios.


Fig. 4. One-line diagram of the IEEE-RTS.

Generally, the NCUC simulation of a large number of casesfor the whole midterm optimization horizon would be computa-tionally demanding. However, the division of themidtermmain-tenance horizon into several time blocks as we proposed in [1]provides a unique opportunity to perform the simulations onseveral processors simultaneously. This feature also provides auser flexibility to select the number and the duration of midtermpreventive maintenance periods subjectively, as discussed fur-ther later. Furthermore, the similarity of transformer locations,ratings, manufacturers and operating conditions, which is com-monplace in power systems, could further reduce the number ofrequired simulations and reduce the computation burden of theproblem.

A. Study 1

In this study, the midterm time horizon is considered to beone year divided into 13 four-week blocks. In order to illustratethe effectiveness of the proposed approach, four case studiesare conducted. In Case 1, it is assumed that transformers T1and T3 are in the wear-out stage with increasing failure rate,while the remaining transformers are in the normal life stagewith constant failure rate [4]. In Case 2, it is assumed that allthe transformers are in the wear-out stage with increasing failurerate. The Weibull distribution parameters associated with minorand major failure rates in Cases 1 and 2 are given in Tables Vand VI, respectively. In Case 3, the parameter of the minorand major failure rates of transformers T1 and T3 from Case 2

TABLE VWEIBULL DISTRIBUTION PARAMETERS OF TRANSFORMERS’

TIME VARYING FAILURE RATE IN CASE 1

TABLE VIWEIBULL DISTRIBUTION PARAMETERS OF TRANSFORMERS’

TIME VARYING FAILURE RATE IN CASE 2

TABLE VIIAVAILABLE LABOR IN EACH MIDTERM TIME BLOCK IN CASE 4

TABLE VIIIMIDTERM MAINTENANCE TASK SCHEDULES—CASE 1

TABLE IXMIDTERM MAINTENANCE TASK SCHEDULES—CASE 2

are increased by 20% to investigate the effects of the increasedaging momentum on maintenance task scheduling. This is whilethe Weibull parameters of other transformers are considered tobe similar to Case 2. In all the above cases, the labor constraintis ignored. Finally in Case 4, the labor constraint is added toCase 3 to illustrate the effect of labor resource limitations sinceCase 3 has the worst condition of aging transformers and laborrequirements. In Case 4, it is assumed that the available laborresources are limited based on Table VII. The maximum numberof maintenance task schedules in each midterm time block isalso limited to four in all cases.The midterm maintenance schedules obtained for the trans-

formers in the four cases are summarized in Tables VIII–XI. Asdiscussed earlier for the six-bus test system, transformer loca-tion in the transmission network and its agingmomentum are the


TABLE XMIDTERM MAINTENANCE TASK SCHEDULES—CASE 3

TABLE XIMIDTERM MAINTENANCE TASK SCHEDULES—CASE 4

decisive parameters which control the preventive maintenanceoutage schedules. Transformer location determines the expectedchange in system operating cost under transformer outage whileaging momentum quantifies the expected probability of trans-former failure.In Case 1, transformers T1 and T3 with high minor and major

aging momentums are scheduled for minor and major mainte-nance respectively in the 8midterm time block in which the loaddemand is fairly low, while other transformers are not main-tained. This is due to the fact that transformers T2, T4 and T5are assumed to be in the normal life stage with constant minorand major failure rates.In Case 2, transformers T1 and T2 with high are

scheduled for minor maintenance while transformers T3, T4 andT5 with high are scheduled for major maintenance. Thisindicates that the entire aging transformers are scheduled formaintenance based on their aging momentum.In Case 3, transformers T1 and T3 with very high aging

momentums are scheduled three times for maintenance whileother transformers are just scheduled once in the midterm main-tenance horizon. This Case illustrates that the proposed modelcan precisely identify the transformer maintenance require-ments based on aging momentum. Finally in Case 4, in whichthe labor resource limitation is taken into account, the numberof maintenance task schedules are reduced significantly incomparison to Case 3. In this case, all available labor resourcesare allocated to transformers T1 and T3 with the highest agingmomentum. Since the available labor resources in the midtermtime blocks four to ten is considered to be equal to zero, nomaintenance task is scheduled during these time blocks.In order to demonstrate the impact of maintenance tasks

on the failure rate of aging transformers, the minor and majorfailure rate variation of transformers T1 and T3 is depicted inFigs. 5 and 6 for Case 3. As it can be seen in these figures,the transformer failure rates reduce to their initial values in theperiods that the transformers are maintained and increased inother periods based on the Weibull distribution.The CPU times and optimal values of the midterm objective

function for the four cases are summarized in Table XII. By

Fig. 5. Variation of aging transformers T1 and T3 minor failure rate in Case 3.

Fig. 6. Variation of aging transformers T1 and T3 major failure rate in Case 3.

TABLE XIIOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME

inspecting the results, it is clear that the optimal values of theobjective function in the midterm model change significantlyfrom Case 1 to 4 due to the increased aging momentum and/orlabor resource limitation. For instance, the objective functionincreased from Case 1 to Case 2 due to the increased numberof aging transformers. The objective function increased fromCase 2 to Case 3 due to the increased aging momentum of trans-formers T1 and T3. Finally, the objective function increasedfrom Case 3 to Case 4 due to the labor resource shortage, whichresults in increased probability of transformer failure conse-quent to the delayed maintenance activities. Computation timeswere also increased from Case 1 to Case 3 due to the increasednumber of restricting constraints. However, in Case 4 the re-source limitation reduces the number of feasible maintenancetime blocks resulting in reduced CPU time.

B. Study 2

In Study 1, the yearly midterm time horizon was limited to13 time blocks. In this study, we investigate the effects of the


Fig. 7. Stair-wise failure rate using Weibull distribution.

TABLE XIIIOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPUTIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 1)

TABLE XIVOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPUTIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 2)

number of midterm time blocks on the computation burden andaccuracy of the solutions. Accordingly, the yearly midterm timehorizon is divided into 26 and 52 time blocks and the approachproposed in the companion paper is applied. The computationburden and accuracy of the results are investigated and com-pared with the solutions obtained in Study 1. It is important tonote that the maximum number of maintenance task scheduledin each block of the 26 and 52 time block models is limited totwo and one task respectively to keep the consistency with theresults obtained in Study 1. Note that in all case studies, thestair-wise values of the Weibull distributions are obtained for52 time blocks. Then, the average values of four and two subse-quent time blocks of the 52 step stair-wise Weibull distributionsare used in the 26- and 13-time block midterm studies. For clar-ification, the approach used to calculate the stair-wise value ofa typical Weibull distribution is depicted in Fig. 7.The CPU time and optimal solutions in those cases are sum-

marized in Tables XIII–XVI. As it can be seen in Tables XIVand XV, the processor encounters memory limit problems inCases 2 and 3 for the models with 26 and 52 time blocks. Thisis due to the large number of constraints for those correspondingproblems.In order to avoid the memory problems, the solutions ob-

tained for 13 time blocks problem can be used as an initialguess to define the maintenance windows for the 26 and 52 time

TABLE XVOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPUTIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 3)

TABLE XVIOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPUTIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 4)

TABLE XVIIOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPUTIME FOR DIFFERENT MIDTERM TIME BLOCKS USINGTHE MAINTENANCE WINDOW APPROACH (CASE 2)

TABLE XVIIIOPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPUTIME FOR DIFFERENT MIDTERM TIME BLOCKS USINGTHE MAINTENANCE WINDOW APPROACH (CASE 3)

block problems. Accordingly, first the 13 time blocks problemhas been solved, then the resulting maintenance schedule timeblock has been considered with its two neighboring blocks asthe maintenance window in 26 and 52 time block models. Forinstance in Case 2, transformer T1 is scheduled for minor main-tenance in the 9th time block in the 13 time block model. Giventhis information, we should only consider the 8th, 9th and 10thtime blocks as the possible maintenance window in the 26 and52 time block model for transformer T1’s minor maintenancetask. This entails as well that transformer T1’s minor main-tenance window should nominally span over the 15th to the20th block in the 26 time block model. Similarly, the trans-former T1 minor maintenance window spans over the 29th tothe 40th block in the 52 time block model. It is important tonote that the two neighboring blocks are considered in the main-tenance window to ensure the optimality and robustness of thesolutions. Restricting the maintenance windows using the 13block problem information, decreases the number of constraintsand the computation burden of the problem for the 52 and 26block models and avoids running out of memory. The requiredCPU time and the optimal solutions obtained by the mainte-nance window approach for Cases 2 and 3 are summarized inTables XVII and XVIII. Also the maintenance schedules ob-tained for 13, 26 and 52 time block periods in Cases 1 to 4 areshown in Figs. 8–11 for comparison.


Fig. 8. Midterm maintenance task schedules, 13, 26, 52 time blocks—Case 1.




Analyzing the results for different numbers of midterm timeblocks, it is recognized that the results are not matching com-pletely. Examining the results indicates that the existing discrep-ancies between the results originate from the following three ap-proximations:1) The approximation used in obtaining the stair-wiseWeibulldistribution values for different number of midterm timeblocks, i.e., Fig. 7.

2) The approximation that exists in calculating the averagevalues used in the midterm maintenance problem formu-lations are affected by the duration of the midterm timeblocks.

3) Constraining the number of maintenance tasks in each timeblock also affects the schedules obtained for different num-bers of midterm time blocks.

It can be concluded that longer midterm time blocks resultin less accurate outcomes due to the existence of higher levelsof approximation. Additionally, the longer midterm time blocksresult in lower computational cost. Thus, users should attemptto compromise between accuracy and computational cost basedon their preferences and utility practices. In turn, however, herecomputational cost is a lesser issue because of the correspondinglead time between planning and execution of the actual mainte-nance tasks.

IV. SHORT-TERM TRANSFORMER MAINTENANCE SCHEDULINGMODEL NUMERICAL STUDY USING IEEE-RTS

This section investigates the characteristics of the short-termtransformer maintenance scheduling model and its integrationwith the midterm planning process in depth. The results ob-tained by the midterm maintenance scheduler for IEEE-RTSare fed into the short-term maintenance scheduling model todetermine the exact maintenance outage time of transformersin light of the upcoming operating conditions. The short-termtransformer maintenance scheduling model is also coded in theGAMS environment and solved using CPLEX 12 [3], on thesame computer as the midterm model. To analyze the differentaspects of the short-term model, two studies are conducted onthe IEEE-RTS.

A. Study 1

In this study, the outputs obtained for Case 2 of the midtermtransformer maintenance scheduling model with 52 time blocksare utilized to analyze the characteristics of the short-termmaintenance scheduling model. As it can be seen in Fig. 9,transformers T1 and T2 are, respectively, scheduled for minormaintenance in the 34th and 28th midterm time blocks, andtransformers T3, T4 and T5 are, respectively, scheduledfor major maintenance in the 31th, 29th and 27th midtermtime blocks. The short-term transformer maintenance sched-uling model receives the short-term load forecast and thetransformer(s) scheduled for maintenance and the associatedmaintenance task duration as inputs. Then, it determines theexact outage time of transformers over duration of the associ-ated midterm time block as discussed for the six-bus system.The outputs of the short-term maintenance scheduling modelfor this study are summarized in Table XIX. As it can be seenfrom this table, the 30-hour long minor maintenance outages


TABLE XIXSHORT-TERM MAINTENANCE TASK SCHEDULES—STUDY 1

TABLE XXOBJECTIVE FUNCTION VALUES ($)—STUDY 1

Fig. 12. Average percent change in LMPs.

of transformers T1 and T2 are both scheduled in the weekendhours in which the load of the system is the lowest for the cor-responding midterm blocks. Similarly, the major maintenanceoutages of the transformers T3, T4 and T5 take place in the last100 hours of the midterm time block in which the load is lowerrelative to the beginning of the block.To obtain a better understanding of how the transformer out-

ages affect the economics of the power system operation, theobjective function values are compared in Table XX with thecases in which no maintenance is done on the transformers. Ad-ditionally, the average percent changes in the locational mar-ginal prices (LMPs) of the system buses are calculated and pre-sented in Fig. 12.As it can be seen from Table XX, the maintenance outage

of transformer T1 has slightly decreased the operating cost ofthe system in comparison with the no maintenance case. Thisphenomenon is side benefit obtained by the reduction in net-work meshing produced by the transformer disconnection. Thisis the same benefit that is exploited in the optimal transmissionswitching problem in which the outage of certain connections inthe network in some periods improves the overall economics ofthe system [5], [6]. As it can be seen from Fig. 12, the averageLMPs are slightly reduced in this case.

TABLE XXICPU TIMES—STUDY 1

The maintenance outage of transformer T2 increases the op-erating cost of the system as seen in Table XX. The associ-ated average LMPs are also increased at all buses of the system(Fig. 12). This indicates that transformer T2 is an important con-nection in the transmission network as its relatively short outageresults in increased costs of the system.As it can be seen in Table XX, the increased operation cost of

the system due to major maintenance outages of transformersT3, T4 and T5 have larger impacts in comparison to those oftransformers T1 and T2 (minor maintenance). The average per-cent changes in LMPs are also relatively larger. However, it canbe seen in Fig. 12 that the changes in average LMPs are notuniform and the maintenance outages have different impacts atdifferent nodes. For example, the average LMPs of buses B3and B24 are mostly affected following the major maintenanceoutage of transformer T3. This is due to the fact that the outageof transformer T3 results in the congestion of transformer T1which is located between buses B3 and B24.The other noteworthy outcome is the pattern of average LMP

changes as a result of the outage of transformer T5. It can be seenin Fig. 12 that the average LMPs of buses B1, B2, B4, B5, B6and B7 located at the southern part of the IEEE-RTS increasewhile those of the other buses reduce. The reason for this isthat the maintenance outage of transformer T5, located betweenbuses B10 and B12, puts the power transmission burden ontransformer T4 such that it reaches its maximum power transfercapability during most of the T5 maintenance outage hours. Theresulting transmission bottleneck limits the amount of powerthat can be transferred from northern part of the IEEE-RTS tothe southern part. Consequently, the more expensive units G4and G7, located at buses B1 and B2, are committed to meet thepower demand and satisfy N-1 security. This situation increasesthe average LMPs at the buses located in southern part, whilethe average LMPs of the other buses decreases, due to the avail-ability of the cheaper power from units located at the northernpart of the system. Considering the results provided in Table XXand Fig. 12, it can also be concluded that transformers can besorted in descending order as T4, T5, T3, T2 and T1 based ontheir impacts on system LMPs.The required CPU times to solve the short-term maintenance

scheduling model in Study 1 are presented in Table XXI. It hasto be noted that the upper bound on the duality gap is set to be0.1% in all cases. As expected, the CPU times associated withthe short-term maintenance scheduling model in which mainte-nance scheduling is considered, are much larger than those ofthe no maintenance cases. It can also be seen in Table XXI thatthe CPU times associated with the major maintenance tasks arerelatively smaller than those of minor maintenance task. This is


due to the fact that the duration of the minor maintenance tasksis relatively smaller than that of major maintenance tasks whichresults in higher computation burden to find the optimum main-tenance outage schedule.

B. Study 2

In this study, the computation burden and performance ofthe proposed short-term maintenance scheduling model fordifferent midterm time blocks duration are investigated. Ac-cordingly, the short-term major maintenance outage scheduleof transformer T3 in Case 2 for different midterm time blockdurations are determined and compared. The correspondingmidterm maintenance schedule of transformer T3 is givenin Fig. 9, respectively, for the 13, 26 and 52 midterm timeblocks. The short-term major maintenance outage schedulesof transformer T3 are illustrated in Fig. 13. In order to bettercompare the schedules in different midterm models, the majormaintenance schedules of transformer T3 are illustrated in thisfigure along with the load pattern of the period. It can be seenin Fig. 13 that although the maintenance window of the 13 and26 time block midterm models are, respectively, four times andtwo times larger than the 52 time block midterm model, theschedules almost coincide with each other.The required CPU times to solve the short-term maintenance

schedulingmodel for different midterm time block durations aregiven in Table XXII. It can be seen that the required CPU timeof 13 time blocks is approximately 21.7 times larger than thatfor the 52 time block. Additionally, the required CPU time of 26time block model is approximately 6.5 times larger than that of52 time block model. We thus see here that there is a tradeoff be-tweenmidterm and short-term computational cost. Havingmoremidterm blocks entails a harder midterm problem while it savesconsiderable time at the short-term planning stage. Given thatshort-term planning lead times are short by definition, then oneshould favor having more midterm blocks from the computa-tional point of view.To make the solutions obtained for different midterm time

block durations comparable, the short-term maintenancescheduling problem is solved again for the 4-weeks’ timeblock model shown in Fig. 13, while fixing the outage time oftransformer T3 based on the results obtained in the 26 and 52time block models. Accordingly, to make the result similar tothe solution obtained for the short-term maintenance sched-uling of transformer in the 52 time block model, transformerT3 is assumed to be on outage in hours 405 to 504 in thefour-week horizon. Similarly, to make the solution similarto the short-term maintenance scheduling problem in the 26time block model, transformer T3 is assumed to be on outageduring hours 430 to 529 during the four-week horizon. Theresults of this study are summarized in Table XXIII. As itcan be seen, the results for the 52 time blocks model are lesseconomical in comparison to the two other cases because of theshorter maintenance window. Although, the longer time blockdurations may result in more economical solutions (0.02% lessexpensive), it results in a much higher computational burden.Additionally, the operational input data to the short-termmaintenance scheduler such as load forecasts are much more

Fig. 13. Short-term major maintenance outage schedule of transformer T3 inCase 2: (a) 13 midterm time blocks, (b) 26 midterm time blocks, (c) 52 midtermtime blocks.

TABLE XXIICPU TIMES—STUDY 2

TABLE XXIIIOPERATING COSTS ($)—STUDY 2

accurate for smaller maintenance windows in comparison withthe longer ones which make the smaller maintenance windowsmore desirable in comparison to larger ones.

V. CONCLUDING REMARKS

This paper has demonstrated the characteristics of a two-stage midterm and short-term transformer maintenance sched-


TABLE XXIVGENERATING UNIT COST DATA OF THE SIX-BUS SYSTEM

TABLE XXVGENERATING UNIT OPERATING DATA OF THE SIX-BUS SYSTEM

TABLE XXVITRANSFORMER AND TRANSMISSION LINE DATA OF THE SIX-BUS SYSTEM

uling model developed in [1]. The case studies presented in thispart allow drawing the following conclusions:1) The decoupling of the maintenance scheduling probleminto midterm and short-term horizons is beneficial as it pro-vides the ability to constrain each horizon to the associatedrestrictions without adversely affecting or ignoring the in-terdependency of the horizons.

2) The decoupling of the maintenance scheduling problemalso provides a great flexibility to users to manage the com-putation burden of the problem by making a trade-off be-tween the accuracy of the results and computation burdenof the short-term and midterm maintenance horizons.

3) It is observed that the midterm maintenance model con-tains a tradeoff between the accuracy of the failure ratemodeling and computational burden. The larger midtermtime blocks involve much lower computation burden whilethe failure rate models involve higher levels of approxima-tion. Accordingly, the user needs to reach a compromisebetween the accuracy of failure rate modeling and compu-tational burden of the problem.

4) Additionally, we concluded that the shorter short-termmaintenance horizons are more desirable due to the higheraccuracy of the input data to the model and lower compu-tational burden they involve; however, the results mightnot be as economical as the ones obtained for longershort-term time horizons.

The proposed scheme can be modified and used for othertransmission assets as well. The main characteristic of thismodel is the realization of transmission asset maintenancemanagement strategy with an acceptable level of complexity,cost and computational burden.

TABLE XXVIIINPUT DATA USED IN THE SIX-BUS SYSTEM AND IEEE-RTS STUDIES

TABLE XXVIIIINPUT DATA USED IN THE SIX-BUS SYSTEM AND IEEE-RTS STUDIES

APPENDIX A

Tables XXIV–XXVI list the generating unit cost data of thesix-bus system, generating unit operating data of the six-bussystem, and transformer and transmission line data of thesix-bus system, respectively.

APPENDIX B

Table XXVII lists the input data used in the six-bus systemand IEEE-RTS studies, and Table XXVIII lists the input dataused in the six-bus system and IEEE-RTS studies.

ACKNOWLEDGMENT

The research presented in this paper benefited from the IBMAcademic Initiative through a free license for the ILOGCPLEXsolver library.

REFERENCES

[1] A. Abiri-Jahromi, M. Parvania, F. Bouffard, and M. Fotuhi-Firuzabad,“A two-stage framework for power transformer asset maintenancemanagement—Part I: Models and formulations,” IEEE Trans. PowerSyst., vol. 28, no. 2, pp. 1395–1403, May 2013.

[2] Reliability Test System Task Force, “The IEEE reliability testsystem—1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp.1010–1020, Aug. 1999.

[3] CPLEX 12 Manual, IBM Corp.[4] R. Billinton and R. N. Allan, Reliability Evaluation of Power Sys-

tems. New York: Plenum, 1996.[5] E. Fisher, R. O’Neill, and M. Ferris, “Optimal transmission

switching,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1346–1355,Aug. 2008.

[6] K. Hedman, R. O’Neill, E. Fisher, and S. Oren, “Optimal transmissionswitching—Sensitivity analysis and extensions,” IEEE Trans. PowerSyst., vol. 23, no. 3, pp. 1469–1479, Aug. 2008.


Amir Abiri-Jahromi (S’10) received the B.Sc. de-gree in electrical engineering from Shiraz University,Shiraz, Iran, in 2003 and the M.Sc. degree in energysystems engineering from Sharif University of Tech-nology, Tehran, Iran, in 2007, and he is currently pur-suing the Ph.D. degree at McGill University, Mon-treal, QC, Canada.He was a Research Assistant in the electrical

engineering department of Sharif University ofTechnology from 2008 to 2010. He was also aresearch and development engineer with UIS Com-

pany, Dubai, United Arab Emirates, from 2008 to 2010. His research interestsare asset management and automation as well as operation, reliability, andoptimization of smart electricity grids.

Masood Parvania (S’09) received the B.S. degreein electrical engineering from Iran University ofScience and Technology (IUST), Tehran, Iran, in2007, and the M.S. degree in electrical engineeringfrom Sharif University of Technology, Tehran, Iran,in 2009, where he is currently pursuing the Ph.D.degree.Since 2012, he has been a Research Associate with

the Robert W. Galvin Center For Electricity Inno-vation, Electrical and Computer Engineering depart-ment, Illinois Institute of Technology, Chicago, IL.

His research interests include power system reliability and security assessment,as well as operation and optimization of smart electricity grids.

François Bouffard (S’99–M’06) received theB.Eng. (Hon.) and the Ph.D. degrees in electricalengineering both from McGill University, Montreal,QC, Canada, in 2000 and 2006, respectively.From 2004 to 2006, he was a Faculty Lecturer

in the Department of Electrical and ComputerEngineering at McGill University. In 2006, hetook up a lectureship with the School of Electricaland Electronic Engineering at The University ofManchester, Manchester, U.K. In 2010, he re-joinedMcGill University as an Assistant Professor. His

research interests are in the fields of power system modeling, economics,reliability, control, and optimization.Dr. Bouffard is a member of the IEEE Power & Energy Society (PES). He is

an Editor of the IEEE TRANSACTIONS ON POWER SYSTEMS, and he chairs theSystem Economics Subcommittee of the IEEE PES.

Mahmud Fotuhi-Firuzabad (SM’99) received theB.Sc. degree in electrical engineering from SharifUniversity of Technology, Tehran, Iran, in 1986and the M.Sc. degree in electrical engineering fromTehran University, Tehran, Iran in 1989, and theM.Sc. and Ph.D. degrees in electrical engineeringfrom the University of Saskatchewan, Saskatoon,SK, Canada, in 1993 and 1997, respectively.Currently, he is a Professor and Head of the De-

partment of Electrical Engineering, Sharif Universityof Technology. He is also an Honorary Professor in

the Universiti Teknologi Mara (UiTM), Shah Alam, Malaysia. He is a memberof the Center of Excellence in Power System Management and Control.Dr. Fotuhi-Firuzabad serves as an Editor of the IEEE TRANSACTIONS ON

SMART GRID.

Documents

A Two-Stage Framework for Power Transformer Asset Maintenance Management—Part II: Validation Results