A Two-Stage Framework for Power Transformer Asset ... - IEEE Xplore

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

A Two-Stage Framework for Power Transformer Asset Maintenance Management—Part II: Validation Results Amir 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 maintenance management is introduced and formulated in Part I of this two-part paper in the context of transmission asset management strategies (TAMS). The proposed model optimizes maintenance outage schedule over a predefined period of time by taking into account the actual and expected transformer assets’ condition dynamics in terms of failure rate and resource limitations in midterm horizons, as well as operating constraints, economic considerations and N-1 reliability in the shorter term. In Part II, a small six-bus system is first used to demonstrate how the two-stage maintenance framework works using a step-by-step procedure. Then, IEEE-RTS is used to investigate the performance of the proposed model in more detail. In addition, the impacts of varying the characteristics of the proposed midterm and short-term maintenance schedulers, such as flexibility in time horizon selection, on maintenance scheduling results and computational efficiency are investigated on IEEE-RTS. The numerical studies indicate that the proposed framework gives appropriate results in terms of economics and technical constraints at a reasonable computational cost. Index Terms—Computational burden, failure rate, midterm and short-term maintenance planning, mixed integer linear programming, transformers, transmission asset management.

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I. INTRODUCTION

HIS paper validates and analyzes the midterm and shortterm maintenance scheduling models and formulations developed in Part I [1] through several case studies conducted on a six-bus test system and the IEEE-RTS. In addition, the trade-off between the computation burden and accuracy of the solutions of the midterm and short-term maintenance schedulers’ is investigated using the IEEE-RTS. In these studies, the maintenance and repair tasks associated with transformers are arbitrarily divided into minor and major maintenance and repair tasks. Minor tasks stand for the tasks that would be done on-site with 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 of current version April 18, 2013. The work of A. Abiri-Jahromi was supported in part 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 defined and used based on user preference without loss of generality. In Section II, a small six-bus test system is used to demonstrate how the two-stage maintenance framework works using a step-by-step procedure. Accordingly, all the steps taken to reach the final solution in midterm and short-term maintenance schedulers’ are explained. In Section III, the midterm maintenance scheduling model is applied to the IEEE-RTS [2], and its capabilities are studied in depth through two studies. The studies are focused on 1) examining the effects of variations in Weibull distribution parameters on midterm maintenance schedules, 2) examining the effects of resource limitations primarily labor constraint on maintenance schedules, and 3) examining the effects of midterm time block duration on the computation burden and accuracy of the solutions. In Section IV, some of the midterm maintenance schedules obtained in Section III for IEEE-RTS is fed into short-term maintenance scheduler and exact hourly maintenance outage schedules of transformers obtained. In this section, the characteristics of the short-term maintenance scheduler and how it handles requests for maintenance coming from the midterm planning stages are examined in depth. Also, the computation burden and performance of the short-term maintenance scheduler 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 this section to demonstrate how the two-stage maintenance framework works using a step-by-step procedure. The six-bus test system, depicted in Fig. 1, consists of three generating units, 7 transmission lines, 2 transformers and three load points. The system data are given in Appendices A and B. A. Midterm Maintenance Scheduling Model Solution Steps The midterm maintenance horizon of the six-bus test system is considered to be 8 weeks, which is divided into 8 weekly midterm time blocks. The duration of the midterm time horizon and the corresponding time blocks can be selected subjectively based on user preference. Also, note that the duration of the midterm time blocks determines the duration of the short-term horizon. Accordingly, since the duration of each midterm time block is assumed to be one week in the six-bus system, the duration of short-term horizon is also one week. The hourly load

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

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TABLE II OCCURRENCE PROBABILITY OF SELECTED SCENARIOS IN EACH MIDTERM TIME BLOCK

TABLE III SIX-BUS SYSTEM OPERATING COST OVER MIDTERM TIME BLOCKS FOR BASE CASE AND SCENARIOS ($)

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

TABLE I WEIBULL 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 is considered to be similar to the first 8 weeks load profile of the IEEE-RTS [2]. The peak load of the six-bus system is assumed to be 270 MW. The failure rates of the three generating units at the beginning of the maintenance planning horizon are assumed to be similar and equal to 4 failures per year. In addition, the failure rates of transmission lines at the beginning of the maintenance planning horizon are also assumed to be similar and equal to 0.2 failures per year. The Weibull distribution parameters associated with minor and major failure rates of transformers T1 and T2 are also summarized in Table I. The major solution steps performed in midterm maintenance scheduler to determine the maintenance schedules of transformers T1 and T2 in the six-bus system are given as follows: 1) Select a set of scenarios such that each scenario contains outage of a transformer. In this study, the outage scenarios are selected based on the N-1 criterion, however, any other approaches can be used by user to obtain outage scenarios. Accordingly, two outage scenarios are considered in the six-bus system, including transformer T1 outage and transformer T2 outage. 2) Calculate the occurrence probability of each scenario selected 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 transformer in midterm time block . 3) Using the NCUC simulator introduced in [1], calculate the system operating cost over each midterm time block for the base case and each outage scenario selected in Step 1. The NCUC simulator outputs for the base case and transformer outage scenarios are summarized in Table III.

4) Calculate the expected cost associated with each outage scenario in each midterm time block based on the results given in Tables II and III [1, eq. (2)]. 5) Calculate the expected cost associated with transformer preventive maintenance outage in each midterm time block based on the results obtained in step 4 [1, eq. (5)]. 6) Calculate the explicit costs associated with the preventive maintenance and repair of a failed transformer using the data provided in the Appendix B[1, eqs. (6)–(8)]. 7) Solve the midterm maintenance scheduling optimization problem, formulated by [1, eqs. (9)–(16)] using commercial MILP solver such as CPLEX [3]. In the six-bus test system, transformers T1 and T2 are, respectively, scheduled for minor and major maintenance in the 5th and 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 are the decisive parameters which control the preventive maintenance outage schedules. Transformer location determines the expected change in system operating cost under transformer outage as summarized in Table III while aging momentum quantifies the expected probability of transformer failure scenarios. In the Weibull distribution model, the transformer aging momentum 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 maintenance scheduling model, developed in Part I [1], performs. Accordingly, the exact hourly major maintenance outage schedule

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TABLE IV SHORT-TERM HORIZON OBJECTIVE FUNCTION VALUE

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

Fig. 2. Hourly scheduling of the six-bus system generating units. (a) Without security constraint. (b) With security constraint.

of transformer T2 in the 4th midterm time block is determined and analyzed here using the short-term maintenance scheduling model. The following input data are required by the short-term maintenance model: 1) transformers that should be maintained in the short-term horizon, e.g., T2; 2) duration of the associated maintenance tasks, e.g., 100 hours 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 security constraint. Then, outage of transmission line L1 is considered as a security constraint and the short-term transformer maintenance scheduling problem is solved again with this security constraint and compared with the previous case to explain the impact of security constraints on maintenance schedule, operating procedure and system cost. The resulting schedules are summarized in Table IV. The output levels of generating units are also shown in Fig. 2. Table IV indicates that transformer T2 is scheduled for major maintenance in hours 37–136 in the case without security constraint. Fig. 2(a) shows that only units G1 and G3 are committed to satisfy the operating constraints. However, this case does not guarantee the security of 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

as a security constraint. Fig. 2(b) shows that unit G2 is also committed in this case in the hours that transformer T2 is scheduled for major maintenance. In addition, generating units’ G1 and G3 schedules and outputs have adjusted by the short-term maintenance scheduling model to satisfy the operating constraints as well as the security constraint of transmission L1 outage. These adjustments in units scheduling assure that the outage of transmission line L1 would not endanger the system security while transformer T2 is on maintenance. It can also be seen in Table IV that the objective function value in the case where the scheduling is robust to outage of line L1, is much higher than that of the case without security constraint. The schematic representation of the midterm and short-term major maintenance scheduling of transformer T2 in the six-bus system for the case with security constraint is also shown in Fig. 3. As shown in this figure, the midterm maintenance scheduling model has just located the midterm time block in which major maintenance task should take place on transformer T2, i.e., 4th midterm time block. This is while the short-term maintenance scheduler has determined the exact hourly outage time of transformer T2 in the 4th midterm time block. III. MIDTERM TRANSFORMER MAINTENANCE SCHEDULING NUMERICAL STUDIES USING IEEE-RTS In this section, the midterm transformer maintenance model is applied to the IEEE-RTS to investigate the characteristics of the model in depth [2]. The one-line diagram of the IEEE-RTS is depicted in Fig. 4. This system is composed of 24 buses, 32 generating units, 20 load points, 33 transmission lines and 5 transformers. Some of the required data for the case studies such as generating units’ and transmission lines’ data are given in [2]. Other data such as decoupled deteriorating failure rates of each transformer, average duration of transformer maintenance or failure outages, average number of working hours necessary to 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 the yearly load profile of the IEEE-RTS [2] is utilized to obtain the system hourly load curve nodal distribution. In addition, the capacities of the transformers are reduced to 60% of the values provided in [2] in order to obtain a better illustration of the performance of the models. For the proposed model, all the problem formulations are linear. Accordingly, the formulations were implemented in the GAMS environment and solved using CPLEX 12 [3] on a PC equipped 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 base case and five single transformer outage scenarios.


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TABLE V WEIBULL DISTRIBUTION PARAMETERS OF TRANSFORMERS’ TIME VARYING FAILURE RATE IN CASE 1

TABLE VI WEIBULL DISTRIBUTION PARAMETERS OF TRANSFORMERS’ TIME VARYING FAILURE RATE IN CASE 2

TABLE VII AVAILABLE LABOR IN EACH MIDTERM TIME BLOCK IN CASE 4

TABLE VIII MIDTERM MAINTENANCE TASK SCHEDULES—CASE 1 Fig. 4. One-line diagram of the IEEE-RTS.

Generally, the NCUC simulation of a large number of cases for the whole midterm optimization horizon would be computationally demanding. However, the division of the midterm maintenance horizon into several time blocks as we proposed in [1] provides a unique opportunity to perform the simulations on several processors simultaneously. This feature also provides a user flexibility to select the number and the duration of midterm preventive maintenance periods subjectively, as discussed further later. Furthermore, the similarity of transformer locations, ratings, manufacturers and operating conditions, which is commonplace in power systems, could further reduce the number of required simulations and reduce the computation burden of the problem. A. Study 1 In this study, the midterm time horizon is considered to be one year divided into 13 four-week blocks. In order to illustrate the effectiveness of the proposed approach, four case studies are conducted. In Case 1, it is assumed that transformers T1 and T3 are in the wear-out stage with increasing failure rate, while the remaining transformers are in the normal life stage with constant failure rate [4]. In Case 2, it is assumed that all the transformers are in the wear-out stage with increasing failure rate. The Weibull distribution parameters associated with minor and major failure rates in Cases 1 and 2 are given in Tables V and VI, respectively. In Case 3, the parameter of the minor and major failure rates of transformers T1 and T3 from Case 2

TABLE IX MIDTERM MAINTENANCE TASK SCHEDULES—CASE 2

are increased by 20% to investigate the effects of the increased aging momentum on maintenance task scheduling. This is while the Weibull parameters of other transformers are considered to be similar to Case 2. In all the above cases, the labor constraint is ignored. Finally in Case 4, the labor constraint is added to Case 3 to illustrate the effect of labor resource limitations since Case 3 has the worst condition of aging transformers and labor requirements. In Case 4, it is assumed that the available labor resources are limited based on Table VII. The maximum number of maintenance task schedules in each midterm time block is also limited to four in all cases. The midterm maintenance schedules obtained for the transformers in the four cases are summarized in Tables VIII–XI. As discussed earlier for the six-bus test system, transformer location in the transmission network and its aging momentum are the

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TABLE X MIDTERM MAINTENANCE TASK SCHEDULES—CASE 3

TABLE XI MIDTERM MAINTENANCE TASK SCHEDULES—CASE 4

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

decisive parameters which control the preventive maintenance outage schedules. Transformer location determines the expected change in system operating cost under transformer outage while aging momentum quantifies the expected probability of transformer failure. In Case 1, transformers T1 and T3 with high minor and major aging momentums are scheduled for minor and major maintenance respectively in the 8 midterm time block in which the load demand is fairly low, while other transformers are not maintained. This is due to the fact that transformers T2, T4 and T5 are assumed to be in the normal life stage with constant minor and major failure rates. In Case 2, transformers T1 and T2 with high are scheduled for minor maintenance while transformers T3, T4 and T5 with high are scheduled for major maintenance. This indicates that the entire aging transformers are scheduled for maintenance based on their aging momentum. In Case 3, transformers T1 and T3 with very high aging momentums are scheduled three times for maintenance while other transformers are just scheduled once in the midterm maintenance horizon. This Case illustrates that the proposed model can precisely identify the transformer maintenance requirements based on aging momentum. Finally in Case 4, in which the labor resource limitation is taken into account, the number of maintenance task schedules are reduced significantly in comparison to Case 3. In this case, all available labor resources are allocated to transformers T1 and T3 with the highest aging momentum. Since the available labor resources in the midterm time blocks four to ten is considered to be equal to zero, no maintenance 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 major failure rate variation of transformers T1 and T3 is depicted in Figs. 5 and 6 for Case 3. As it can be seen in these figures, the transformer failure rates reduce to their initial values in the periods that the transformers are maintained and increased in other 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. 6. Variation of aging transformers T1 and T3 major failure rate in Case 3.

TABLE XII OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME

inspecting the results, it is clear that the optimal values of the objective function in the midterm model change significantly from Case 1 to 4 due to the increased aging momentum and/or labor resource limitation. For instance, the objective function increased from Case 1 to Case 2 due to the increased number of aging transformers. The objective function increased from Case 2 to Case 3 due to the increased aging momentum of transformers T1 and T3. Finally, the objective function increased from Case 3 to Case 4 due to the labor resource shortage, which results in increased probability of transformer failure consequent to the delayed maintenance activities. Computation times were also increased from Case 1 to Case 3 due to the increased number of restricting constraints. However, in Case 4 the resource limitation reduces the number of feasible maintenance time blocks resulting in reduced CPU time. B. Study 2 In Study 1, the yearly midterm time horizon was limited to 13 time blocks. In this study, we investigate the effects of the


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TABLE XV OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 3)

TABLE XVI OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 4)

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

TABLE XIII OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 1)

TABLE XVII OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME FOR DIFFERENT MIDTERM TIME BLOCKS USING THE MAINTENANCE WINDOW APPROACH (CASE 2)

TABLE XIV OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME FOR DIFFERENT MIDTERM TIME BLOCKS (CASE 2)

TABLE XVIII OPTIMAL VALUE OF THE OBJECTIVE FUNCTION AND CPU TIME FOR DIFFERENT MIDTERM TIME BLOCKS USING THE MAINTENANCE WINDOW APPROACH (CASE 3)

number of midterm time blocks on the computation burden and accuracy of the solutions. Accordingly, the yearly midterm time horizon is divided into 26 and 52 time blocks and the approach proposed in the companion paper is applied. The computation burden and accuracy of the results are investigated and compared with the solutions obtained in Study 1. It is important to note that the maximum number of maintenance task scheduled in each block of the 26 and 52 time block models is limited to two and one task respectively to keep the consistency with the results obtained in Study 1. Note that in all case studies, the stair-wise values of the Weibull distributions are obtained for 52 time blocks. Then, the average values of four and two subsequent time blocks of the 52 step stair-wise Weibull distributions are used in the 26- and 13-time block midterm studies. For clarification, the approach used to calculate the stair-wise value of a typical Weibull distribution is depicted in Fig. 7. The CPU time and optimal solutions in those cases are summarized in Tables XIII–XVI. As it can be seen in Tables XIV and XV, the processor encounters memory limit problems in Cases 2 and 3 for the models with 26 and 52 time blocks. This is due to the large number of constraints for those corresponding problems. In order to avoid the memory problems, the solutions obtained for 13 time blocks problem can be used as an initial guess to define the maintenance windows for the 26 and 52 time

block problems. Accordingly, first the 13 time blocks problem has been solved, then the resulting maintenance schedule time block has been considered with its two neighboring blocks as the maintenance window in 26 and 52 time block models. For instance in Case 2, transformer T1 is scheduled for minor maintenance in the 9th time block in the 13 time block model. Given this information, we should only consider the 8th, 9th and 10th time blocks as the possible maintenance window in the 26 and 52 time block model for transformer T1’s minor maintenance task. This entails as well that transformer T1’s minor maintenance window should nominally span over the 15th to the 20th block in the 26 time block model. Similarly, the transformer T1 minor maintenance window spans over the 29th to the 40th block in the 52 time block model. It is important to note that the two neighboring blocks are considered in the maintenance window to ensure the optimality and robustness of the solutions. Restricting the maintenance windows using the 13 block problem information, decreases the number of constraints and the computation burden of the problem for the 52 and 26 block models and avoids running out of memory. The required CPU time and the optimal solutions obtained by the maintenance window approach for Cases 2 and 3 are summarized in Tables XVII and XVIII. Also the maintenance schedules obtained for 13, 26 and 52 time block periods in Cases 1 to 4 are shown in Figs. 8–11 for comparison.

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


Analyzing the results for different numbers of midterm time blocks, it is recognized that the results are not matching completely. Examining the results indicates that the existing discrepancies between the results originate from the following three approximations: 1) The approximation used in obtaining the stair-wise Weibull distribution values for different number of midterm time blocks, i.e., Fig. 7. 2) The approximation that exists in calculating the average values used in the midterm maintenance problem formulations are affected by the duration of the midterm time blocks. 3) Constraining the number of maintenance tasks in each time block also affects the schedules obtained for different numbers of midterm time blocks. It can be concluded that longer midterm time blocks result in less accurate outcomes due to the existence of higher levels of approximation. Additionally, the longer midterm time blocks result in lower computational cost. Thus, users should attempt to compromise between accuracy and computational cost based on their preferences and utility practices. In turn, however, here computational cost is a lesser issue because of the corresponding lead time between planning and execution of the actual maintenance tasks. IV. SHORT-TERM TRANSFORMER MAINTENANCE SCHEDULING MODEL NUMERICAL STUDY USING IEEE-RTS


This section investigates the characteristics of the short-term transformer maintenance scheduling model and its integration with the midterm planning process in depth. The results obtained by the midterm maintenance scheduler for IEEE-RTS are fed into the short-term maintenance scheduling model to determine the exact maintenance outage time of transformers in light of the upcoming operating conditions. The short-term transformer maintenance scheduling model is also coded in the GAMS environment and solved using CPLEX 12 [3], on the same computer as the midterm model. To analyze the different aspects of the short-term model, two studies are conducted on the IEEE-RTS. A. Study 1



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


TABLE XIX SHORT-TERM MAINTENANCE TASK SCHEDULES—STUDY 1

TABLE XX OBJECTIVE FUNCTION VALUES ($)—STUDY 1

Fig. 12. Average percent change in LMPs.

of transformers T1 and T2 are both scheduled in the weekend hours in which the load of the system is the lowest for the corresponding midterm blocks. Similarly, the major maintenance outages of the transformers T3, T4 and T5 take place in the last 100 hours of the midterm time block in which the load is lower relative to the beginning of the block. To obtain a better understanding of how the transformer outages affect the economics of the power system operation, the objective function values are compared in Table XX with the cases in which no maintenance is done on the transformers. Additionally, the average percent changes in the locational marginal prices (LMPs) of the system buses are calculated and presented in Fig. 12. As it can be seen from Table XX, the maintenance outage of transformer T1 has slightly decreased the operating cost of the system in comparison with the no maintenance case. This phenomenon is side benefit obtained by the reduction in network meshing produced by the transformer disconnection. This is the same benefit that is exploited in the optimal transmission switching problem in which the outage of certain connections in the network in some periods improves the overall economics of the system [5], [6]. As it can be seen from Fig. 12, the average LMPs are slightly reduced in this case.

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TABLE XXI CPU TIMES—STUDY 1

The maintenance outage of transformer T2 increases the operating cost of the system as seen in Table XX. The associated average LMPs are also increased at all buses of the system (Fig. 12). This indicates that transformer T2 is an important connection in the transmission network as its relatively short outage results 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 transformers T3, T4 and T5 have larger impacts in comparison to those of transformers T1 and T2 (minor maintenance). The average percent changes in LMPs are also relatively larger. However, it can be seen in Fig. 12 that the changes in average LMPs are not uniform and the maintenance outages have different impacts at different nodes. For example, the average LMPs of buses B3 and B24 are mostly affected following the major maintenance outage of transformer T3. This is due to the fact that the outage of transformer T3 results in the congestion of transformer T1 which 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 seen in Fig. 12 that the average LMPs of buses B1, B2, B4, B5, B6 and B7 located at the southern part of the IEEE-RTS increase while those of the other buses reduce. The reason for this is that the maintenance outage of transformer T5, located between buses B10 and B12, puts the power transmission burden on transformer T4 such that it reaches its maximum power transfer capability during most of the T5 maintenance outage hours. The resulting transmission bottleneck limits the amount of power that can be transferred from northern part of the IEEE-RTS to the southern part. Consequently, the more expensive units G4 and G7, located at buses B1 and B2, are committed to meet the power demand and satisfy N-1 security. This situation increases the average LMPs at the buses located in southern part, while the average LMPs of the other buses decreases, due to the availability of the cheaper power from units located at the northern part of the system. Considering the results provided in Table XX and Fig. 12, it can also be concluded that transformers can be sorted in descending order as T4, T5, T3, T2 and T1 based on their 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 has to be noted that the upper bound on the duality gap is set to be 0.1% in all cases. As expected, the CPU times associated with the short-term maintenance scheduling model in which maintenance scheduling is considered, are much larger than those of the no maintenance cases. It can also be seen in Table XXI that the CPU times associated with the major maintenance tasks are relatively smaller than those of minor maintenance task. This is

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due to the fact that the duration of the minor maintenance tasks is relatively smaller than that of major maintenance tasks which results in higher computation burden to find the optimum maintenance outage schedule. B. Study 2 In this study, the computation burden and performance of the proposed short-term maintenance scheduling model for different midterm time blocks duration are investigated. Accordingly, the short-term major maintenance outage schedule of transformer T3 in Case 2 for different midterm time block durations are determined and compared. The corresponding midterm maintenance schedule of transformer T3 is given in Fig. 9, respectively, for the 13, 26 and 52 midterm time blocks. The short-term major maintenance outage schedules of transformer T3 are illustrated in Fig. 13. In order to better compare the schedules in different midterm models, the major maintenance schedules of transformer T3 are illustrated in this figure along with the load pattern of the period. It can be seen in Fig. 13 that although the maintenance window of the 13 and 26 time block midterm models are, respectively, four times and two times larger than the 52 time block midterm model, the schedules almost coincide with each other. The required CPU times to solve the short-term maintenance scheduling model for different midterm time block durations are given in Table XXII. It can be seen that the required CPU time of 13 time blocks is approximately 21.7 times larger than that for the 52 time block. Additionally, the required CPU time of 26 time block model is approximately 6.5 times larger than that of 52 time block model. We thus see here that there is a tradeoff between midterm and short-term computational cost. Having more midterm blocks entails a harder midterm problem while it saves considerable time at the short-term planning stage. Given that short-term planning lead times are short by definition, then one should favor having more midterm blocks from the computational point of view. To make the solutions obtained for different midterm time block durations comparable, the short-term maintenance scheduling problem is solved again for the 4-weeks’ time block model shown in Fig. 13, while fixing the outage time of transformer T3 based on the results obtained in the 26 and 52 time block models. Accordingly, to make the result similar to the solution obtained for the short-term maintenance scheduling of transformer in the 52 time block model, transformer T3 is assumed to be on outage in hours 405 to 504 in the four-week horizon. Similarly, to make the solution similar to the short-term maintenance scheduling problem in the 26 time block model, transformer T3 is assumed to be on outage during hours 430 to 529 during the four-week horizon. The results of this study are summarized in Table XXIII. As it can be seen, the results for the 52 time blocks model are less economical in comparison to the two other cases because of the shorter maintenance window. Although, the longer time block durations may result in more economical solutions (0.02% less expensive), it results in a much higher computational burden. Additionally, the operational input data to the short-term maintenance scheduler such as load forecasts are much more

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

TABLE XXII CPU TIMES—STUDY 2

TABLE XXIII OPERATING COSTS ($)—STUDY 2

accurate for smaller maintenance windows in comparison with the longer ones which make the smaller maintenance windows more desirable in comparison to larger ones. V. CONCLUDING REMARKS This paper has demonstrated the characteristics of a twostage midterm and short-term transformer maintenance sched-


TABLE XXIV GENERATING UNIT COST DATA OF THE SIX-BUS SYSTEM

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

TABLE XXV GENERATING UNIT OPERATING DATA OF THE SIX-BUS SYSTEM

TABLE XXVI TRANSFORMER AND TRANSMISSION LINE DATA OF THE SIX-BUS SYSTEM

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

uling model developed in [1]. The case studies presented in this part allow drawing the following conclusions: 1) The decoupling of the maintenance scheduling problem into midterm and short-term horizons is beneficial as it provides the ability to constrain each horizon to the associated restrictions without adversely affecting or ignoring the interdependency of the horizons. 2) The decoupling of the maintenance scheduling problem also provides a great flexibility to users to manage the computation burden of the problem by making a trade-off between the accuracy of the results and computation burden of the short-term and midterm maintenance horizons. 3) It is observed that the midterm maintenance model contains a tradeoff between the accuracy of the failure rate modeling and computational burden. The larger midterm time blocks involve much lower computation burden while the failure rate models involve higher levels of approximation. Accordingly, the user needs to reach a compromise between the accuracy of failure rate modeling and computational burden of the problem. 4) Additionally, we concluded that the shorter short-term maintenance horizons are more desirable due to the higher accuracy of the input data to the model and lower computational burden they involve; however, the results might not be as economical as the ones obtained for longer short-term time horizons. The proposed scheme can be modified and used for other transmission assets as well. The main characteristic of this model is the realization of transmission asset maintenance management strategy with an acceptable level of complexity, cost and computational burden.

APPENDIX A Tables XXIV–XXVI list the generating unit cost data of the six-bus system, generating unit operating data of the six-bus system, and transformer and transmission line data of the six-bus system, respectively. APPENDIX B Table XXVII lists the input data used in the six-bus system and IEEE-RTS studies, and Table XXVIII lists the input data used in the six-bus system and IEEE-RTS studies. ACKNOWLEDGMENT The research presented in this paper benefited from the IBM Academic Initiative through a free license for the ILOG CPLEX solver library. REFERENCES [1] A. Abiri-Jahromi, M. Parvania, F. Bouffard, and M. Fotuhi-Firuzabad, “A two-stage framework for power transformer asset maintenance management—Part I: Models and formulations,” IEEE Trans. Power Syst., vol. 28, no. 2, pp. 1395–1403, May 2013. [2] Reliability Test System Task Force, “The IEEE reliability test system—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 Systems. 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 transmission switching—Sensitivity analysis and extensions,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1469–1479, Aug. 2008.

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Amir Abiri-Jahromi (S’10) received the B.Sc. degree in electrical engineering from Shiraz University, Shiraz, Iran, in 2003 and the M.Sc. degree in energy systems engineering from Sharif University of Technology, Tehran, Iran, in 2007, and he is currently pursuing the Ph.D. degree at McGill University, Montreal, QC, Canada. He was a Research Assistant in the electrical engineering department of Sharif University of Technology from 2008 to 2010. He was also a research and development engineer with UIS Company, Dubai, United Arab Emirates, from 2008 to 2010. His research interests are asset management and automation as well as operation, reliability, and optimization of smart electricity grids.

Masood Parvania (S’09) received the B.S. degree in electrical engineering from Iran University of Science and Technology (IUST), Tehran, Iran, in 2007, and the M.S. degree in electrical engineering from 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 Innovation, Electrical and Computer Engineering department, 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 the B.Eng. (Hon.) and the Ph.D. degrees in electrical engineering 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 Computer Engineering at McGill University. In 2006, he took up a lectureship with the School of Electrical and Electronic Engineering at The University of Manchester, Manchester, U.K. In 2010, he re-joined McGill 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 the System Economics Subcommittee of the IEEE PES.

Mahmud Fotuhi-Firuzabad (SM’99) received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1986 and the M.Sc. degree in electrical engineering from Tehran University, Tehran, Iran in 1989, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Saskatchewan, Saskatoon, SK, Canada, in 1993 and 1997, respectively. Currently, he is a Professor and Head of the Department of Electrical Engineering, Sharif University of Technology. He is also an Honorary Professor in the Universiti Teknologi Mara (UiTM), Shah Alam, Malaysia. He is a member of the Center of Excellence in Power System Management and Control. Dr. Fotuhi-Firuzabad serves as an Editor of the IEEE TRANSACTIONS ON SMART GRID.