Annual coordinated generation and transmission maintenancescheduling considering network constraints and energy not supplied
Ahmad Norozpour Niazi1*,†, Sajjad Khoshnoud1 and Mahdieh Goudarzi2
1Department of Electrical and Computer Engineering, Babol University of Technology, Babol, Mazandaran, Iran2Department of Electrical, Computer, and IT Engineering, Islamic Azad University of Qazvin, Qazvin, Iran
As of 1980, many countries have made improvements in forming electric power markets. The main aimwas breaking the monopoly operation pattern of tradition electric power industry and building a compet-itive power industry. Therefore, it can decrease the electric power production cost and electricity price.Besides, it can improve the power supply quality and promote the healthy development of electric powerindustry. Additional competition and increasing complexity in power generating systems as well as anecessity for high service reliability and low production costs triggered additional interests in automaticscheduling techniques for maintenance of generators, transmission, and pertinent equipment. Severaloptimization methods were applied to solve the problem, which could be sorted into three categoriescalled, heuristic methods, artificial intelligent methods, and mathematical programming methods.Heuristic methods supply the most primitive solution based on trial-and-error principles. Artificial intelli-gent methods contain expert system, simulated annealing [1, 2], fuzzy theory, neural network, evolution-ary optimization comprising evolutionary programming, evolutionary strategy, and genetic algorithm,simulated evolution, Tabu search and various combinations of artificial intelligent methods [3–7]. Finally,mathematical programming methods contain mixed integer programming (MIP), mixed integer linearprogramming, decomposition [8], branch-and-bound, dynamic programming, and various combinationsof mathematical programming methods.
*Correspondence to: Ahmad Norozpour Niazi, Department of Electrical and Computer Engineering, Babol University ofTechnology, Babol, Mazandaran, Iran.†E-mail: [email protected], [email protected]
INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMSInt. Trans. Electr. Energ. Syst. (2014)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1935
Although in the past decades, several procedures were recommended for the solution of unit andtransmission maintenance scheduling, there was no consensus on the most appropriate approach to thisproblem. Earlier, much emphasis was given to heuristic methods that could not meet the multi-objectiverequirements of the problem and could not assure a feasible solution.Most artificial intelligent techniques have the ability of referring to multi-objective requirements.
Since an inference engine must be organized according to the particular characteristics of a designedproblem, it is hard to generalize the expert system approach. The membership function in fuzzy setsis to be configured under the specific requirements of the designated power system; therefore, fuzzysets are usually used as an auxiliary tool in maintenance optimization methods. However, the liter-ature shows that, of all the possible intelligence techniques, genetic algorithms are the most suitableartificial intelligent technique for maintenance scheduling. On the other hand, there is no doubt thatmathematical programming methods supply more reliable and versatile solution to maintenancescheduling [9].
2. LITERATURE REVIEW
Generally, maintenance scheduling in a raw system may fall into two stages from time horizon per-spective, entitled, long-term and short-term scheduling [10]. Long-term maintenance scheduling(LTMS) considers the schedule of system on a horizon of 1 or 2 years in order to minimize the totalsystem operation and maintenance costs. The long-term scheduling problem tackles fuel allocation,emission, budgeting, production, and maintenance costing [11]. The solutions obtained from LTMScan then be used as guidelines and bases for addressing unit commitment and optimal power flowproblems [12–17]. The objective of short-term maintenance scheduling (STMS) is to minimize the costof operation over hourly, daily or weekly periods. Because dynamic economic dispatch is fundamentalfor real-time control of power systems, the STMS causes a commitment strategy for real-time eco-nomic dispatch to meet system requirements in an on-line operation. The dynamic economic dispatchis solved for short periods of time in which the system load conditions can be assumed constant.In deregulated power markets, independent system operator (ISO) is in charge of unit and trans-
mission maintenance scheduling as well as maintaining instantaneous balance of the system. TheISO carries out its function by controlling the dispatch of flexible power plants. Furthermore, ISO isthe sole responsible for system security and reliability. Most of researches deal with maintenancescheduling problem in both long-term and short-term scheduling regardless of system security indices.M.K.C. Marwali and S.M. Shahidehpour have implemented a long-term maintenance scheduling studyin [4,8,18]; however, impact of consumers loading on maintenance scheduling is not taken intoaccount. Furthermore, fixed maintenance periods are considered for all participant units that are neces-sarily incorrect.In some previous papers [19–22], unit maintenance scheduling with network constraints and reli-
ability indices are taken into account. This paper, to have global maintenance scheduling, proposes asecurity-constrained model for preventive long-term unit and transmission maintenance schedulingproblem in which system security and reliability indices such as transmission line safety and amountof not supplied energy are taken into account. In order to get more realistic results, a penalty factor isintroduced to study the impact of customer load curve on the proposed generation and transmissionmaintenance scheduling (GTMS) problem. Unlike some previous studies that consider a fixed periodfor all unit maintenance windows and short time period for duration of maintenance, here variousmaintenance windows as well as annual duration are considered for the system that are more realis-tic. Considering the problem that contains integer variables for unit and transmission maintenancescheduling and taking into account the proposed model that is based on optimal generation andmaintenance costs, MIP is employed to obtain the most accurate results. Among different algorithmsprovided for solving mixed integer problems, branch-and-bound algorithm is employed as the mostgeneral algorithm to solve the problem and find the optimal solution. The paper is organized as fol-low: sections 3 and 4 represent the description of system and methodology, respectively. In section 5and 6, case study and results and discussion are presented to show the accuracy of proposed model,and section 7 provides the conclusion. Finally, section 8 gives suggestions for further research.
While transmission, generation, and reliability limitations are taken into account, the proposed GTMSproblem is determining the period for which generating units and transmission lines should be off, over1- or 2-year planning horizon to lessen the total operation and maintenance cost. Leaving out the networkconstraints in maintenance scheduling may end in loss of information on scheduling problems. Whennetwork constraints are included, the problem becomes a lot more realistic and complex that could bereferred as a security-constrained maintenance scheduling. The long-term GTMS in the power marketenvironment is a large-scale optimization problem. Mathematically, it can be formulated as follow:
3.1. Objective function
The objective function of the proposed model is to minimize the total maintenance and production costsover the operational planning period. Equation (1) corresponds to a MIP problem since xit and xkt areinteger variables and git is continuous. The first and second terms of the objective function are themaintenance costs of generators and transmission lines, and the third one is the energy production cost.
Min :Xt
Xi
Cit � γt � 1� xitð Þ½ � þXt
Xk
Ckt � 1� xktð Þ½ � þXt
Xi
cit � git
( )(1)
where xit and xkt represent unit and transmission maintenance status; 0 if unit i or transmission k are off formaintenance, 1 otherwise. The Cit, Ckt, cit, γt, and git represent unit maintenance cost, transmission main-tenance cost, generation cost, weekly penalty factor, and power generation at time t, respectively.
3.1.1. Penalty factor. In order to consider the impact of the load curve demand on the maintenancescheduling problem, a penalty factor is represented as Equation (2). In fact, the penalty factor showsthe importance of loading points on the proposed GTMS based on the amount of consumptions.ISO could employ a penalty factor to patronize unit and transmission not to have maintenance in peakloads. Here, the total unit maintenance cost is the maintenance cost of the unit multiplied by the penaltyfactor. By this strategy, ISO could have more effect on system maintenance scheduling.
∀ t γt ¼ 2-Dmax-Dt
Dmax-Dmin(2)
where Dt, Dmax, and Dmin represent the vector of the demand at time t, the maximum demand at studyperiod, and the minimum demand at study period.
3.2. Maintenance constraints
In order to make the maintenance schedule feasible, certain constraints should be fulfilled. Some of ba-sic constraints which should be set up are continuousness maintenance of some units and transmissionlines, maintenance manpower, maintenance window, maintenance duration, and so on. Maintenanceconstraints in the current research could be categorized as follow:
3.2.1. Maintenance window. Equations (3)–(5) and (6)–(8) show the maintenance timetable stated interms of maintenance variables. The unit and transmission maintenance may not be scheduled beforetheir earliest period, or after the latest period allowed for maintenance (li + di, lk+ dk).
for t≤ei or t≥li þ di⇒xit ¼ 1 (3)
for Si≤t≤Si þ di⇒xit ¼ 0 (4)
for ei≤t≤li⇒xit ¼ 0 or 1 (5)
for t≤ek or t≥lk þ dk⇒xkt ¼ 1 (6)
for Sk≤t≤Sk þ dk⇒xkt ¼ 0 (7)
for ek≤t≤lk⇒xkt ¼ 0 or 1 (8)
ANNUAL GENERATION AND TRANSMISSION MAINTENANCE SCHEDULING
where Si and Sk represent the period in which maintenance of unit i and transmission k start; ei and ekdemonstrate the earliest period for the beginning of unit i and transmission k maintenance; li and lkrepresent the latest period for the beginning of unit i and transmission k maintenance.
3.2.2. Maintenance duration. The maintenance of the unit i and transmission k lasts a given number ofperiods di and dk, respectively. X
t∈T1� xitð Þ ¼ di ∀i∈I (9)
Xt∈T
1� xktð Þ ¼ dk ∀k∈K (10)
3.2.3. Maintenance period. A maximum number of maintenance is imposed in the period t; whereβit and βkt represent the maximum number of maintenance unit i and transmission k at time t,respectively. X
i∈I1� xitð Þ≤βit ∀t∈T (11)
Xk∈K
1� xktð Þ≤βit ∀t∈T (12)
3.2.4. Non-stop maintenance. The maintenance of a unit is carried out in consecutive periods; wheresvit represents the maintenance start-up variable of unit i at time t.
1� xi;t� �� 1� xi;t�1
� �≤svi;t ; ∀i∈I & ∀t∈T
for t ¼ 1 select; xi;0 ¼ 1(13)
3.2.5. Exclusion constraint. Units i and j cannot be in maintenance at the same time.
1� xi;t� �þ 1� xj;t
� �≤1 ∀t∈T (14)
3.2.6. One-time maintenance. Each unit and transmission has an outage for maintenance just oncealong the time horizon considered; where svit and svkt demonstrate maintenance start-up variable ofunit i and transmission k at time t, respectively.X
t∈Tsvit ¼ 1 ∀i∈I (15)
Xt∈T
svkt ¼ 1 ∀k∈K (16)
3.2.7. Manpower availability. If one considers that in each maintenance area, there is limited availablemanpower, the constraints will be stated as follows:X
i∈I1� xitð Þ≤Mit ∀t∈T (17)
Xk∈K
1� xktð Þ≤Mkt ∀t∈T (18)
where Mit and Mkt represent the number of manpower in area for maintenance of unit i and transmis-sion k at time t, respectively.
3.3. Network constraints
The network can be modeled as either the transportation model or a linearized power flow model.
3.3.1. Power system load balance. We apply the transportation model to exhibit system operationlimits such as load balance equation, unit capacities, and power flow limits as below:
where g, z, r, D, and f represent the vector of power generation for unit i at time t, node-branch inci-dence matrix, unserved power at weekly peak demand, vector of the demand at time t, and active lineflow, respectively.
3.3.2. Unit capacity limit. Each unit is designed to work between minimum and maximum power ca-pacity (MW). The following constraint in Equation (20) ensures that the unit is within its respectiverated minimum and maximum capacities.
∀ t gmin;i≤git≤gmax;i (20)
where gmax,i and gmin,i represent the maximum and minimum power generation for unit i.
3.3.3. Transmission flow limit. The power flows on transmission lines are constrained by line capacity.The constraint (21) represents power transmission capacity; where fmax,k demonstrates the maximumline flow capacity for transmission k.
∀ t f k;t�� ��≤f max;k (21)
3.3.4. Energy not supplied. Actually, ISO is in charge of not supplied energy in all periods of time. Itis a safety margin that usually is given as a demand proportion. Equation (22) represents not suppliedenergy constraint. This indicates that the total expected power not served of the units running at eachinterval should not be more than the specified amount of energy for that interval, where α representsthe acceptable level of expected power not served at weekly peak demand.
∀ t r≤ % α� Dt (22)
4. METHODOLOGY
Any determination problem with a purpose to be maximized or minimized in which the design vari-ables must assume non fractional or discrete values may be sorted as an integer optimization problem.An integer problem is sorted as linear if, by relaxing the integer limitation on the variables, theresulting functions are completely linear. If all the design variables are limited to integer values, theproblem is called a (pure) integer problem, otherwise a MIP [23,24].In the context of linear and mixed-integer programming problems, the function that appraises the
quality of the solution, named the objective function, should be a linear function of the design variables.A linear programming will either maximize or minimize the value of the objective function. Eventually,the determinations that must be made are subject to certain requirements and limitations of a problem.Each constraint that is a linear function needs to be either equal to, not more than, or not less than, a scalarvalue. A common condition simply states that each determination variable must be nonnegative. Actually,all LP problems can be transformed into an equivalent minimization problem with nonnegative variablesand equality constraints.Therefore, suppose that here, x1 . . . xn are our set of design variables. The LP problems are as follow:Maximize or minimize:
Here, the values ci, ∀ i = 1, . . . , n, are indicated as objective coefficients and are often connected tothe costs associated with their corresponding determinations in minimization problems or the incomegenerated from the corresponding determinations in maximization problems.
ANNUAL GENERATION AND TRANSMISSION MAINTENANCE SCHEDULING
The values b1 . . . bm are the right-hand-side values of the constraints and often depict amounts of avail-able resources (especially for ≤constraints) or requirements (especially for ≥constraints). The aij-valuesthus typically indicate how much of requirement or resource j is satisfied or consumed by decision i.In this paper, in order to find the optimal solution, branch and bound [25] is used as the most general
algorithm. Branch and bound consists of a systematic enumeration of all candidate solutions by usingupper and lower estimated bounds of the quantity being optimized. Considering the above problem,assume that the goal is to find the minimum value of a function f(x) where x ranges over some set S ofadmissible or candidate solutions. A branch-and-bound procedure requires two tools. The first one is asplitting procedure that, given a set S of candidates, returns two or more smaller sets S1, S2,…whose unioncovers S. Note that the minimum of function f(x) over S isMin (v1, v2,…), where each vi is the minimum offunction f(x)within Si. This step is called branching, since its recursive application defines a tree structurewhose nodes are the subsets of S. The second tool is a procedure that computes upper and lower boundsfor the minimum value of function f(x) within a given subset of S. This step is called bounding. The keyidea of the branch-and-bound algorithm is: if the lower bound for some tree node (set of candidates) A isgreater than the upper bound for some other node B, then Amay be safely discarded from the search. Thisstep is called pruning and is usually implemented by maintaining a global variable m that records theminimum upper bound seen among all sub regions examined so far. Any node whose lower bound isgreater thanm can be discarded. The recursion stops when the current candidate set S is reduced to a singleelement or when the upper bound for set Smatches the lower bound. Either way, any element of Swill be aminimum of the function within S.Problems of the form (23)–(27) are called linear programming since the objective function and
constraint functions are all linear. A MIP is a linear program with the added limitation that some,but not necessarily all, of the variables must be integer valued. Several studies also replace the terminteger with binary (0–1 variables) when variables are limited to take on either 0 or 1 values.A solution that fulfills all constraints is called a feasible solution. Feasible solutions that obtain the best
objective function value (according to whether one is maximizing or minimizing) are called optimal solu-tions. Sometimes, no answer exists to an MIP and the MIP itself is named infeasible. On the other hand,some feasible MIPs have no optimal solution, because it is possible to obtain limitlessly good objectivefunction values with feasible solutions. These problems are called unbounded.
5. CASE STUDY
This paper, to have global maintenance scheduling, proposes an annual security-constrained model forpreventive long term GTMS problem in which system security and reliability indices such as transmis-sion line limits, penalty factor, and energy not supplied are taken into account.In this paper, the proposed method is applied to the IEEE 24-bus reliability test system (modified
system) [26]. The system has 32 thermal units, 20 consumers, 24 buses, and 38 transmission lines(See Appendix). An annual study period is taken into account. Some unit and transmission facilities ina special area require maintenance within the study period. The maintenance area coverage is from buses1 through 10. Table I gives unit placements and capacities. Operating and maintenance characteristics ofthe units are given in Table II. Transmission line data are provided in Table III. In addition, line mainte-nance cost is assumed 0.72� 104 $/mile with 1-week maintenance duration.
Figure 1 depicts weekly peak loads as the percent of the annual peak load. As shown, the maximumpeak load are in weeks 50–52. Subsequently, weekly penalty factors are provided in Figure 2. Asindicated the highest penalty factors are applied in peak loaded weeks to avoid unit maintenanceduring peak periods, hence shifting maintenance periods towards off peak times. It is assumed thatduring a year, manpower constraint is up to three groups for generation maintenance and two groups
for transmission maintenance. Detailed system data for transmission lines, generators, and loads can beseen in the Appendix. Two scenarios are studied for maintenance scheduling problem as follow:
Scenario 1: Study on annual generation maintenance scheduling problem considering networkconstraints, penalty factor, and not supplied energy.
Scenario 2: Study on annual GTMS problem considering network constraints, penalty factor, andnot supplied energy.
6. RESULTS AND DISCUSSION
M.K.C. Marwali and S.M. Shahidehpour in [18] have implemented a long-term maintenance schedulingstudy (3months); however, impact of consumer loading on maintenance scheduling is not taken into ac-count. Furthermore, fixed maintenance periods are considered for all participant units that are necessarilyincorrect. In Case 0 [18], they considered generator maintenance scheduling. Neither network constraint,transmission maintenance, fuel limit, nor emission constraint was considered. On the other hand, in [20],authors studied generation maintenance scheduling problem (three months) considering network con-straints as well as fix timetable for maintenance scheduling unlike their capacities, and the index ofnot supplied energy as the significant factor for the system.Basically, scenario 1 shows the impacts of duration of maintenance study, network constraints, and
penalty factor on GMS. Five cases are considered for this scenario. Case 1 reports the results of [18]and case 2 reports the results of [20] which was studied in previous paper (all in a 3-month study). Inorder to consider duration of maintenance study on maintenance scheduling (MS) and different timetable
Table IV. Total operation and maintenance cost.
Case: maximum amount of notsupplied energy in each week
Total operationand
maintenancecost (106 $)
Unitmaintenancecost (106 $)
Transmissionmaintenancecost (106 $)
Operationcost (106 $)
1. Three-month GMS withoutnetwork constraints and reliabilityindices in [18].
66.52694 ……. --- …….
2. Three-month GMS with, networkconstraints, and maximum notsupplied energy (1% of the totalweekly load) in [20].
64.11698824 8.1170136 --- 55.99997464
3. Yearly GMS with networkconstraints.
230.6573 5.602 --- 225.0553
4. Yearly GMS with networkconstraints and penalty factor.
231.16901989 6.03981989 --- 225.1292
5. Yearly GMS with networkconstraints, penalty factor, andfailure on transmission lines24 and 25.
231.17291989 6.03981989 --- 225.1331
6. Three-month GTMS with networkconstraints, penalty factor, andmaximum not supplied energy (1%of the total weekly load).
67.768 8.69548 2.6064 56.46612
7. Yearly GTMS with networkconstraints, penalty factor, andmaximum not supplied energy (1%of the total weekly load).
233.8428 6.0399 2.6064 225.1965
8. GTMS with network constraints,penalty factor, and maximum notsupplied energy (5% of the totalweekly load (annual GTMS).
proportion unit’s capacity, we investigate an annual duration study for GMS in cases 3 till 5. Case 3studies a yearly GMS with network constraints. On the other hand, in case 4, the effect of the penaltyfactor on GMS is considered. Finally, in case 5, it is assumed that failure is accrued for two transmissionlines (the lines between buses 15 to 16 and 15 to 21), while penalty factor is considered as well. Table IVrepresents corresponding operation and maintenance costs in some cases. Subsequently, Tables V–VIIshow corresponding unit maintenance scheduling.As shown in Table IV (cases 1–5), duration of maintenance study, timetable of maintenance, penalty
factor, network constraints, and transmission security constraints may have an effect on maintenanceand operation costs. In this paper, in the third case, we concentrate our study on the yearly maintenancescheduling. By comparing the result of cases 2 and 3, one can conclude that having more choice toselect the time of maintenance scheduling and as the chip units are not forces to be on maintenancein 3months, the operation cost is reduced from 55.99997464million dollars to 55.2242million dollarsin summer. Next in case 4, we take into account penalty factor as well as unit maintenance scheduling.
Table V. Unit maintenance scheduling (case 3).
Table VI. Unit maintenance scheduling (case 4).
Table VII. Unit maintenance scheduling (case 5).
ANNUAL GENERATION AND TRANSMISSION MAINTENANCE SCHEDULING
Here, the operation and maintenance costs are increased due to considering load levels by means of thepenalty factor. The ISO may employ the penalty factor to patronize unit not to have maintenance inpeak loads. By this strategy, the ISO will have more effect on maintenance schedules. Tables V andVI illustrate unit maintenance scheduling. Comparing these tables, one can conclude that applyingpenalty factor results in some shifting in unit maintenance periods toward off peak, periods andbecause of them, the operation planning is changed. In fact, due to more request for energy in peakperiods (weeks 50–52), all units especially cheaper ones must be available within these periods. Thisin turn improves system reliability. However, due to penalty factor coefficients that are normally morethan unity, aggregated maintenance cost will increase. Next, in case 5, authors assumed that failure isoccurred for two transmission lines (the lines between buses 15 to 16 and 15 to 21). By occurring fail-ure for transmission lines, available units in one time period may become less attractive as compared tothose in some other time periods when availability is even more crucial. Losing lines 15–16 and 15–21affects the loading of units and the inefficient unit has to be brought on-line to supply generationdeficit. As the result demonstrated, the operation cost is increased. It should be noted that althoughthere is no regulation on maintenance periods in Tables V–VII, in order to lessen the operation costs,almost all efficient and cheap units are available in peak periods.In [20], authors considered generation maintenance scheduling (GMS) in a 3-month time horizon.
Now, in scenario 2 (case 6), we study 3-month GTMS, too. In cases 7 and 8, an annual GTMS is con-sidered. In order to investigate the impact of reliability indices on GTMS, a different level of maximumnot supplied energy is taken into account as the significant factors from system operator perspective.Here, it is assumed that during a year, total maximum not supplied energy in each week is limitedto the 1% and 5% of the total weekly load. Table IV represents corresponding unit operation and main-tenance costs. Subsequently, Tables VIII–XIII show corresponding unit and transmission maintenancescheduling during specified weeks in above mentioned cases, respectively.By comparing the results in case 6 to the results in case 2, in Table IV, as all the transmission lines
are not available for transmitting the energy in all time in case 6, the results show the change in
operating cost over the study period and indicating a shift from units that use inexpensive fuel to thosewith more expensive fuels and inefficient units. Next, in order to consider an annual GTMS, yearlyGTMS is taking into account. By comparing the result in cases 6 and 7 in this scenario and as the unitsare not forced to be maintained within 12weeks, the cost can be decreased from 67.7680milliondollars to 55.2242million dollars in summer.As it is referred in this scenario to consider the effect of the reliability indices to GTMS, we consider
a different level of maximum not supplied energy. As shown in Table IV, increasing in maximum
Table XI. Transmission maintenance scheduling (case 6).
Table X. Unit maintenance scheduling (case 8).
Table XII. Transmission maintenance scheduling (case 7).
ANNUAL GENERATION AND TRANSMISSION MAINTENANCE SCHEDULING
energy not supplied level results in decreases in operation costs and system total costs as well.Although, system total cost is reduced, however, system reliability level will be decreased.Note that in all above mentioned studies, transmission maintenance costs are the same although
there are some changes in maintenance scheduling. That is due to considering constant maintenancecost coefficients and also fixed maintenance durations; however, unit maintenance scheduling mayvary depending specified conditions.Figure 3 illustrates variation of maximum not supplied energy within all maintenance periods in
proposed GTMS. Here, as the amount of the unserved energy is directly dependent on the amountof the weekly load (i.e. weeks 19, 50, and 51), the contribution of the units to satisfying the systemrequirement are more than the other weeks. As shown, decreasing in maximum not supplied energylevel results in changes in maintenance scheduling and operational planning and will improve the
Table XIII. Transmission maintenance scheduling (case 8).
Week
Max
imum
am
ount
of
not
supp
lied
ener
gy in
eac
h w
eek
(×10
MW
)
Maximum 1% total weekly load
Maximum 5% total weekly load
Figure 3. Comparison of the amount of not supplied energy in each week.
system security; on the other hand, system aggregated costs may increase by increasing unit operationcosts and maintenance cost as well.
7. CONCLUSIONS
This paper presents a model for annual long-term GTMS in which network constraints, as well assystem security are taken into account. As demonstrated, system security and reliability constraints liketransmission safety and not supplied energy of the power system may affect maintenance schedulingand altering system maintenance and operation costs. In order to consider the effect of system loadingon the proposed LTMS problem, a heuristic penalty factor coefficient was introduced. In fact, the ISOmay employ the penalty factor to patronize unit not to have maintenance in peak loads. By thisstrategy, the ISO will have more effect on maintenance schedules. As shown, decreasing in the amountof not supplied energy of the system will improve the system security; on the other hand, systemaggregated costs may increase by increasing unit operation costs and maintenance cost as well.
8. SUGGESTIONS FOR FURTHER RESEARCH
In a universal unit maintenance-scheduling problem, we suggest to take into account air pollutionconstraints, maintenance scheduling of hydro units as well as transmission and thermal generationmaintenance scheduling problem.
9. LIST OF SYMBOLS AND ABBREVIATIONS
xit Unit maintenance status, 0 if unit is off for maintenance, 1 otherwisexkt Transmission maintenance status, 0 if transmission line is off for maintenance, 1 otherwiseSi Period in which maintenance of unit i startsSk Period in which maintenance of transmission k startsei Earliest period for the beginning of unit i maintenanceek Earliest period for the beginning of transmission k maintenanceli Latest period for the beginning of unit i maintenancelk Latest period for the beginning of transmission k maintenancesvit Maintenance start-up variable of unit i at time tsvkt Maintenance start-up variable of transmission k at time tCit Maintenance cost of unit i at time tCkt Maintenance cost of transmission k at time tcit Generation cost of unit i at time tγt Weekly penalty factordi Duration of maintenance for unit idk Duration of maintenance for transmission kgmax,i Maximum power generation for unit igmin,i Minimum power generation for unit igit Vector of power generation for unit i at time tDt Vector of the demand at time tDmax Maximum demand at study periodDmin Minimum demand at study periodz Node-branch incidence matrixα Acceptable level of expected power not served at weekly peak demandr Unserved power at weekly peak demandβit Maximum number of maintenance unit i at time tβkt Maximum number of maintenance transmission k at time tfmax,k Maximum line flow capacity for transmission k
ANNUAL GENERATION AND TRANSMISSION MAINTENANCE SCHEDULING
fk,t Active line flowN Maximum number of transmission lineMit Maximum number of maintenance crew in area for maintenance of unit i at time tMkt Maximum number of maintenance crew in area for maintenance of transmission k at time t
APPENDIX
The main criterion in select of the test system configuration was the desire to achieve a useful referencefor testing and comparison of reliability evaluation methods. In this paper, we apply the proposedmethod to the IEEE 24-bus reliability test system (modified system) [26].
IEEE 24-BUS RELIABILITY TEST SYSTEM DATA
The transmission network consists of 24 bus locations connected by 38 transmission lines (Figure 4).Impedance, length, and rating data for transmission lines are given in Table XIV. The place of gener-ating units is shown in Table XV. From these generating stations, we have decided to do maintenanceall generation units and transmission lines between buses 1 and 10. The unit operating cost data can beseen in Table XVI. Table XVII gives data on weekly peak loads in percent of the annual peak load.The annual peak load for the test system is 2850MW.
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ANNUAL GENERATION AND TRANSMISSION MAINTENANCE SCHEDULING
