Optimization of Generator Maintenance Scheduling with Consideration on the Equivalent Operation Hours
Optimization of Generator Maintenance Scheduling with Consideration on the Equivalent Operation Hours
Journal of Electrical Engineering and Technology. 2016. Mar, 11(2): 338-346
Copyright © 2016, The Korean Institute of Electrical Engineers This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
• Received : September 01, 2015
• Accepted : January 02, 2016
• Published : March 01, 2016 PDF e-PUB PubReader PPT Export by style
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Sangheon Han
Dept. of Electrical Engineering, Pusan National University, Korea. (hansh0518@gmail.com, {kimhyoungtae, leesw}@pusan.ac.kr)
Hyoungtae Kim
Dept. of Electrical Engineering, Pusan National University, Korea. (hansh0518@gmail.com, {kimhyoungtae, leesw}@pusan.ac.kr)
Sungwoo Lee
Dept. of Electrical Engineering, Pusan National University, Korea. (hansh0518@gmail.com, {kimhyoungtae, leesw}@pusan.ac.kr)
Wook Kim
Corresponding Author: Dept. of Electrical Engineering, Pusan National University, Korea. (kimwook@pusan.ac.kr)

Abstract
In order for the optimal solution of generators’ annual maintenance scheduling to be applicable to the actual power system it is crucial to incorporate the constraints related to the equivalent operation hours (EOHs) in the optimization model. However, most of the existing researches on the optimal maintenance scheduling are based on the assumption that the maintenances are to be performed periodically regardless of the operation hours. It is mainly because the computation time to calculate EOHs increases exponentially as the number of generators becomes larger. In this paper an efficient algorithm based on demand grouping method is proposed to calculate the approximate EOHs in an acceptable computation time. The method to calculate the approximate EOHs is incorporated into the optimization model for the maintenance scheduling with consideration on the EOHs of generators. The proposed method is successfully applied to the actual Korean power system and shows significant improvement when compared to the result of the maintenance scheduling algorithm without consideration on EOHs.
Keywords
1. Introduction
The main purpose of the generator maintenance is to maintain or improve performance and / or reliability of the generator by periodical examination and repair of the components. Proper maintenance and its optimal scheduling are very crucial to maintain stable and reliable operation of not only the generator but also the overall power system  .
The maintenance scheduling is performed either by the system operator for the entire system or by each individual generation company for its own generator(s). The objectives and technical considerations are a lot different according to the performers of the maintenance scheduling. The former put more focus on the overall system generation cost, maintenance cost or reliability [1 , 2] , whereas the latter usually tries to maximize the generation company’s profit  . In some countries, government supervises the maintenance scheduling for the stable electricity supply and the efficient operation of power system. For example, in Korea, KPX (Korea Power Exchange) has to establish an annual preventive maintenance plan every year for the entire market-participating generators based on the standards on generator maintenance scheduling published by Korean government  .
The mathematical model for the optimization of maintenance scheduling includes not only the binary decision variables to describe whether the generators are in maintenance states or not but also various nonlinear cost functions and constraints. Therefore, the maintenance scheduling problem is usually formulated as the large-scale mixed-integer nonlinear programming (MINLP).
In order to find the optimal solution of the maintenance scheduling, the existing studies are based on various MINLP optimization theories such as dynamic programming  , branch and bound  , integer programming  , Benders decomposition  , genetic algorithm  , simulated annealing  , etc. One of the significant problems of the existing researches is that they do not take into account the equivalent operating hours (EOHs) of combined cycle gas turbine (CCGT) generators in terms of system operator.
Not like the baseload generators such as coal or nuclear plants of which the maintenance schedules are decided based on the fixed period, the CCGT generators should consider the equivalent operating hours because they usually suffer frequent output fluctuation and start-ups/shutdowns and, most of all, the manufacturers only guarantee the performance of the generators only when the maintenances are performed according to the pre-specified EOHs.
To consider the EOHs of the CCGT generators, it is necessary that the mathematical model for the optimization of the annual maintenance scheduling includes the unit commitment and the economic dispatch models, which makes the overall computation time increases exponentially as the system size or the number of generators becomes larger. Most of the existing researches avert this problem by assuming that the maintenance schedules of CCGT generators are to be decided based on not the EOHs but the fixed period  . Researches on the generator maintenance scheduling with consideration on EOHs can be found in a limited amount of literature. Most of them are based on too complicated mathematical models, so they can only be applied to rather small scale power systems  .
The main contribution of this paper is to consider the EOHs of CCGT generators when finding the optimal solution of the annual maintenance scheduling. It is obvious that it is impractical and unnecessary to use the exact and detailed hourly-based unit commitment and economic dispatch models to estimate the EOHs of CCGT generators. It is necessary to simplify both models to a certain degree to reduce the computation time when the size of the system becomes larger. Lobato et al.  used demand grouping method to calculate the number of start-ups and the generator outputs, but they used such simplified mathematical model (for example, the generator outputs are modeled as the binary variables - 0 or maximum) that the calculation results of EOHs are erroneous.
In this paper an effective and practical method to estimate the EOHs of CCGT generators for the annual maintenance scheduling is proposed. The main idea is that the hourly load demands are categorized into several groups and substituted with representative value of each group. Properly categorizing the hourly load demands, the number of combinations to estimate the generation amounts and the number of start-ups/shutdowns can be enormously reduced. And consequently the EOHs can be can be easily calculated.
This paper organizes as follows. The mathematical formulation for the optimization of annual maintenance scheduling with consideration on the EOHs of CCGT generators will be explained in Section 2. Section 3 will describe the studied system, or the Korean power system with the market and system data actually used in the year of 2013. In the same section, the simulation results of the proposed method will be compared with those of the maintenance scheduling without consideration on the EOHs of CCGT generators. The conclusions will be given in the last section.
2. Mathematical Formulation of the Annual Maintenance Scheduling
- 2.1 Nomenclature
Sets and indices
ggenerator set index wmaintenance period set index dday set index Ggenerator set Wmaintenance periods set (1 year = 52 weeks) Ddays set
Parameters
fuelgfuel cost per MW (KRW/MW) supgstart-up cost per start (KRW) Resthe minimum reserve rate (%) Kstartdecision variable for start-up LimitEOHmaximum allowable EOHs (hour) CCxmaintenance duration of CCGT generator τw,dweight factor for day (business day: 5, weekend: 2) Demandw,delectricity demand (MW) Dmaxwmaximum demand (MW) Pmaxgmaximum generator output (MW) Pmingminimum generator output (MW) Mtypegtype of maintenance (period)
Variables
Fcostg,w,dtotal fuel cost (KRW) Scostg,w,dtotal start-up cost (KRW) Pg,w,dgenerator Output (MW) opg,w,doperation status (ON: 1, OFF: 0) stupg,w,dbinary variable for start-up (start-up: 1, other: 0) xg,wbinary variable for maintenance status (maintenance state: 1, other: 0) xstupg,wbinary variable for start of maintenance (start: 1, other: 0) EOHg,waccumulated EOHs (positive) (hours) dEOHg,wincrement of weekly EOHs (positive) (hours)
- 2.2 Grouping of hourly load demands
In order to consider the EOHs of CCGT generators in the optimization of annual maintenance scheduling, it is necessary to estimate the capacity factor (annual generation amount) and the number of start-ups/shutdowns of each generator. Therefore, it is necessary for the mathematical model of the problem to incorporate the economic dispatch and unit commitment models.
Although more detail models are desirable for the exact estimation of utilization and number of start-up/shutdown, it is impractical to use the detailed hourly economic dispatch and unit commitment models for annual maintenance scheduling mainly due to computation time. In this paper the method used in reference  has been adopted and slightly modified to estimate the annual generation amount and the number of start-ups/shutdowns in reasonable computation time.
Here is the summary of the procedure to estimate EOHs of CCGT generators:
1) One year is divided into 52 weeks.
2) As shown in Fig. 1 the hourly load demands of each week are classified into 6 groups (three groups for weekdays and three groups for weekend) PPT Slide
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Classification of hourly load demands of a week
3) The hourly load demands are replaced with the representative value of each demand group.
4) Mixed-integer economic dispatch algorithm is applied to each group to calculate the output of each generator.
5) By comparing the results of economic dispatches for all 312 demand groups (6 groups each week times 52 weeks), the annual generation amount and the number of start-up/shutdowns of each generator can be easily calculated.
As mentioned above, one week is divided into business days and weekend days. The hourly load demands in weekdays and weekend days are grouped into three groups, respectively: high, middle and low demands. So the hourly load demands are classified into one of the six groups (three for business days and three for weekend days), and the hourly load demands are replaced with the representative value of the group where the demand is included as shown in Fig. 1 .
Average value of the maximum and minimum demand (the middle demand) is used for the representative value of middle demand group. Similarly, average value of maximum demand and the middle demand is used for the representative value of high demand, whereas average value of minimum demand and middle demand is used for the low demand as shown in Fig. 2 . PPT Slide
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Demand clustering method
The number of hours in a day classified into high, middle and low demands are 9, 9 and 6, respectively. Hence, the number of hours for high, middle and low demands of business days are 45, 45 and 30, respectively. Similarly, 18, 18 and 12 hours are assigned to high, middle and low demand for weekend days. These values can be used as weightings to calculate the weekly generation amount of each generator.
Once the hourly load demands are grouped as described above, then the hourly demand loads are replaced with the representative value of each corresponding group. Then the economic dispatch is applied to each group.
It should be noted that the economic dispatch algorithm used here is a mixed-integer type, which means the values of generator outputs can be either zero or between minimum and maximum values. In general, economic dispatch is formulated as quadratic programming based on the assumption that the information on unit commitment is preliminarily given. But in this case, because we cannot get the unit commitment information, the economic dispatch problem should be formulated as the mixed-integer nonlinear programming (the output of generators can be either zero or in between minimum and maximum outputs). The mixed-integer economic dispatch can determine the optimal combinations of generators’ on/off status and outputs of generators to satisfy the given demand. By multiplying weightings to these values, weekly or annual generation amounts can be estimated. And also by comparing the values of generator outputs between low and middle demands, and between middle and high demands, the start-up and shutdown status of generators can approximately be determined.
Fig. 3 shows an illustrative example for the demand grouping method for the power system with only four (4) generators. By demand grouping method, three (3) times of economic dispatch calculations are enough to calculate the approximate generation amount and the number of start-ups / shutdowns as shown in the figure. PPT Slide
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An illustrative example of demand grouping method
The method described above is based on rather severe approximation and, hence, can cause an inaccurate results to estimate the EOHs of CCGT generators. However, it provides much better solution compared to the method which does not consider the EOHs at all as will be explained in the simulation section.
- 2.3 Objective function
The most frequently used type of objective function used for the maintenance scheduling problem is the total generation cost or variance of reserve rate. In this paper annual total generation cost is chosen as the objective function as follows: PPT Slide
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The objective function, Eq. (1), consists of fuel cost and start-up cost. In Eq. (2), the fuel cost, Fcost g,w,d , is calculated as the multiplication of fuel cost per MW of each generator (KRW/MW) and the generation amount (MW). The start-up cost or Scost g,w,d is defined as the multiplication of start-up cost per start and binary decision variable for start-up of operation ( stup g,w,d ) as in Eq. (3).
- 2.4 Constraints
This paper incorporates most of the general constraints commonly used in other similar studies  . However, several additional constraints related to the equivalent operation hours are added.
Demand and supply balance:
The following is the equality constraint for the demandand- supply balance condition: PPT Slide
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Operating reserve margin:
In order to secure the reliability of the power system operation, the following inequality constraint for the operating reserve rate is required: PPT Slide
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where res is the minimum value of the operating reserve rate which is normally smaller than capacity reserve rate. In this paper, the value is set to 6.0% based on the historical values.
The maximum & minimum generator outputs:
The output of generator should be either zero or in between the minimum and maximum values, which can be expressed as the following discontinuous inequality constraint: PPT Slide
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Operation status of generators during the maintenance period:
It is obvious that generator cannot operate when it is under maintenance. Mathematically it can be expressed as the following inequality constraint: PPT Slide
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where op g,w,d and x g,w are the binary variables for operation status and maintenance status, respectively. When x g,w is one, then op g,w,d can only be zero. But if x g,w is zero, then op g,w,d can either be zero or one.
Continuity condition for maintenance status
Generally, once the maintenance starts, it cannot be halted until the pre-determined type of maintenance is completed. Therefore, the following constraints must be satisfied: PPT Slide
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Maximum EOHs of CCGT generators:
As mentioned above, the maintenance schedule for CCGT generator is determined based on the EOH or its variations. For example, MHI and Alstom companies apply ‘Equivalent Operating Hour’, GE and Siemens use ‘Factored Hour’ and ‘Equivalent Based Hour/Start’, respectively  . In this paper, only ‘EOH’ is used because their concepts are all similar to each other  . EOH is calculated as 20× (The number of start-ups) + (Operating hours). Therefore it should be noted that the EOH can be higher than the number of actual hours during the period if the number of start-ups is significantly high. The following inequality constraints are used for the maintenances of CCGT generators to be scheduled based on the EOHs: PPT Slide
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The purpose of Eq. (11) is to calculate the weekly EOHs by adding the increments of EOHs, given by Eq. (15), to the accumulated EOHs of the previous week. Inequality sign in Eq. (11) along with Eq. (12) and (13) are necessary to reset the EOH to zero after the maintenance is performed. Here, M is the large number. Eq. (14) represents the condition for the minimum EOHs at which the maintenance is allowed. Here, it is set to α (80%) of the maximum allowable EOHs. Grouping of the hourly demand into ‘three’ groups can be helpful to reduce the overall computation time for optimization, but the estimation results of EOHs (or the number of start-ups) can have slight errors. Therefore, it is highly probable for the EOHs of CCGTs can be reached earlier than the optimization results due to errors of grouping and/or the demand forecasting. Empirically we set the maximum number of EOHs during the optimization at the 80% of the maximum allowable EOHs in order to eliminate this problem.
3. Case study
Simulation study is performed on the actual Korean electricity market in order to show the validity of the proposed method. Most of the data used for simulations are actual market data in the year of 2013 available from public sources
Generation Mix in Korea power system (2013) PPT Slide
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Generation Mix in Korea power system (2013)
For simplicity, it is assumed that there is only one type of CCGT generator or 7FA turbine generator with about 1,300℃ of inlet temperature which is one of the most common CCGT generators in Korea  .
Because nuclear plants should keep the stringent schedules for nuclear fuel rod exchange and safety inspection, they are excluded from the optimization of maintenance scheduling and the pre-determined maintenance schedules are applied. Hydro plants are also excluded from the optimization due to various reasons  .
The maintenance scheduling has been optimized using CPLEX solver available with GAMS (General Algebraic Modeling System)  . Two cases are compared. Firstly, the optimization on the maintenance scheduling is performed without consideration on the EOHs of CCGT generators. Instead of Eq. (11)~(16) which are the constraints related the EOHs of CCGT generators, the maintenance periods for the CCGT generators are assigned according to the pre-determined periods. Secondly, the maintenance schedules of CCGT generators are optimized with consideration on EOH constraints or Eq. (11)~(16). The results from both cases are summarized and compared in the following subsections.
- 3.1 Maintenance scheduling without consideration on EOH constraints
The optimization result without consideration on EOHs is shown in Table 2 . The total annual generation cost for this case is about 17.40 trillion KRW and its weekly operating reserve rate is shown in Fig. 4 . The average and minimum reserve rates during the planning period are 14.2% and 6.3%, respectively.
Weekly EOHs of the selected combined cycle generators (without considering EOH) [unit: hours] PPT Slide
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Note: 1) The shaded blanks denote that the CCGTs are in maintenance states. 2) The red colored numbers denote that the EOHs are higher than the maximum allowable EOHs. PPT Slide
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Operating reserve rate without considering EOH
The most significant problem of this case is that the EOHs of some CCGT generators (CC#10, CC#31 and CC#41) are higher than the maximum allowable values (LimitEOH) as shown in Table 2 and Fig. 5 . Therefore, the optimization result cannot be directly applied and needs to be manually re-adjusted. It should be noted that the maintenance schedules are unnecessarily assigned to some generators, such as CC#7, CC#20 and CC#24, of which the EOHs are significantly lower than the maximum allowable limit. During the process of the manual re-adjust of the unreasonable maintenance scheduling, the overall cost may be increased compared to the original value. PPT Slide
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EOHs of selected generators (without considering the EOH constraints)
- 3.2 Optimal maintenance scheduling with consideration on EOH constraints (proposed method)
It is obvious that the problem of the previous section can be improved if the optimization model includes the constraints related to EOHs, or Eq. (11)~(16). Table 3 shows the optimization result of the maintenance scheduling with consideration on the EOH constraints. Here the total annual generation cost is about 17.45 trillion KRW which is a little bit higher than the previous case or 17.40 trillion KRW. It should be noted that the higher generation cost of the proposed method compared to the previous case (not considering EOHs) does not infer that the proposed method is inferior to the previous method. It is natural phenomenon to have worse result when more constraints are incorporated in the optimization model.
Weekly EOHs of the selected combined cycle generators (in consideration of EOH constraints) [unit: hours] PPT Slide
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Note: 1) The shaded blanks denote that the CCGTs are in maintenance states. 2) The red colored numbers denote that the EOHs are higher than the maximum allowable EOHs.
Fig. 6 shows that the weekly operating reserve rates for the maintenance scheduling under the consideration on EOH constraints. The average and minimum operating reserve rates are 14.6% and 6.6%, respectively. PPT Slide
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Weekly operating reserve rates ‘with’ consideration on EOH constraints
Not like the previous case, the maintenance scheduling result when the EOH constraints are considered does not have the problem of the maximum EOHs violation for combined cycle generators as shown in Table 3 and Fig. 7 . PPT Slide
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EOHs of the selected CCGT generators (under the consideration on EOH constraints)
Because the CCGT generators can enter into the maintenance state if the EOH is higher than 80% of the maximum allowable EOH limit by Eq. (12), some generators may enter into the maintenances at the EOHs slightly lower than the maximum values as shown in Fig.6 . This makes the overall maintenance scheduling results more robust against the electricity demand variation.
- 3.3 Comparison of the results
As mentioned above, because the optimization of the maintenance scheduling without consideration on EOH constraints may draw unreasonable results as shown in Table 2 , it is necessary to be re-adjusted. For example, the maintenance schedules for CC#10, CC#31 and CC#41 should be adjusted to earlier time so as not to violate the maximum allowable EOH limits. The overall generation cost after the adjustment increases by small amount compared to the result of Section 3.1 as shown in Table 4 . The more significant problem is that the minimum operating reserve rate decreases to 4.9% from 6.3% which is significantly lower than the minimum reserve rate requirement (in this case we set the value to 6.0% hypothetically). At such a low operating reserve rate, a sudden equipment failure in the power system or small increase of electricity demand can cause significant stability problem in power system.
Generation cost & Reserve rates after adjustment PPT Slide
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Generation cost & Reserve rates after adjustment PPT Slide
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Comparison of reserve rates before and after adjustment
However, the proposed method does not have such problems. It does not require additional adjustment of maintenance scheduling acquired from the optimization result. Especially, the minimum operating reserve rate of the proposed method is 1.7%p higher than the result from the optimization without consideration on EOH constraints (after adjustment).
This extra minimum reserve rate of 1.7% acquired by the proposed method as shown in Table 4 is equivalent to 1,478MW of CCGT capacity (based on the peak demand in 2013). In order to build this capacity approximately 1,450 billion KRW or 116.9 billion KRW/year is required (the construction cost for CCGT is assumed to be 980,000 KRW/kW and the discount rate is 7%). This cost is much higher than the fuel cost increase of 16.7 billion KRW/year by the proposed method.
- 3.4 Reduction of the number of demand groups
In the previous section, the hourly demands are grouped into 6 groups (3 groups for weekdays and 3 groups for weekend days). It might be interesting to examine what is the impact of reducing the number of groups in order to make the overall optimization time faster. It should be noted that the optimization cannot be accomplished in a reasonable period of time if the number of groups is increased.
Table 5 and Fig. 9 show that the optimization result for case when the hourly demands are grouped into 3 groups regardless of weekdays or weekends. While the proposed method in the previous subsection needs to solve economic dispatches 6 times per week, it is necessary to solve economic dispatch only 3 times per week when the number of groups are reduced to three. It is obvious that the optimization results become less exact if the number of demand groups is reduced as shown in Table 5 .
Generation cost & Reserve rates for 3 & 6 groupings PPT Slide
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Generation cost & Reserve rates for 3 & 6 groupings PPT Slide
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Weekly operating reserve rates under 3 grouping
The minimum reserve rates for the 6 grouping (the proposed method) and the 3 grouping are 6.6% and 6.3%, respectively while there is no big difference in annual generation costs between the two methods. The reason why the 3 grouping method has lower minimum reserve rate is mainly due to the inexactness of economic dispatch results.
4. Conclusions
While the baseload generators such as nuclear or coal plants carry out maintenances periodically regardless of the actual capacity factors, CCGT generators perform maintenance based on the equivalent operation hours. Because the mathematical model for the optimization of maintenance scheduling with consideration on EOHs is formulated as the mixed-integer nonlinear programming, the overall computation to find the optimal solution increases exponentially as the system size grows. Therefore, it is common practice to set up the maintenance scheduling without consideration on EOHs even though the result may be quite erroneous.
This paper proposes a novel optimization method for the optimization of the annual maintenance scheduling for CCGTs with consideration on EOHs. An electricity demand grouping method is applied in order to estimate the annual approximate capacity factor (generation amount) and the number of start-up and shutdown of each CCGT generator in a reasonable computation time.
The proposed method has been applied to the Korean power system with the actual market data in the year of 2013. The simulations are performed on two cases: one for the case with consideration on EOHs and the other for the case without consideration on EOHs, while the effect of the EOH considerations on the maintenance scheduling is quite significant. The simulation results show that the proposed method (considering EOHs) can improve the average and minimum reserve rates by 1.7%p and 0.6%p to 6.6% and 14.6% from 4.9% and 14.0%, respectively, without significant increase of the total generation cost compared to the optimization without consideration on EOHs.
Acknowledgements
This work was supported by the Human Resources Program in Energy Technology of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (No.2015 4030200670), and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No.2015R1D1A1A 01061286).
BIO Han Sangheon received a B.S and M.S degree in Electronic and Electrical engineering from Pusan National University, Korea, in 2014 and 2016, respectively. Currently, he has been with LG chem. His research interests are power system economics, micro grid and optimization theory. Hyoungtae Kim received B.S. and M.S degrees in Electronic and Electrical engineering from Pusan National University, Korea, in 2013 and 2015, respectively. Currently, he is pursuing his Ph.D. degree in Pusan National University. His research interests are power system economics and smart grid. Sungwoo Lee received B.S. and M.S. degrees in Electronic and Electrical engineering from Pusan National University, Korea, in 2013 and 2015, respectively. Currently, he is pursuing his Ph.D. degree in Pusan National University. His research interests are power system economics and smart grid. Wook Kim received B.S, M.S. and Ph.D. degrees in Electrical Engineering from Seoul National University in 1990, 1992 and 1997, respectively. For 1997-2011, he was with LG Industrial Systems, Samsung Securities and Korea Southern Power Co. Since Sep. 2011, he has been with Electrical Engineering Department at Pusan National University. He is also a research member of the Research Institute of Computer Information and Communication and LG Electronics Smart Control Center in Korea. His research interests include power economics, electricity and carbon trading, smart grid and optimization theory.
References
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