Long-Term Maintenance Schedule Arrangement for

Power Generation Plants in Restructured Power

Systems with Integration of Pumped-Storage Units
Farhad Bahrami1, Majid Moazzami2
Aff1
Aff2
Emails@

Abstract: In a power system, the reliable performance of generating plants for the satisfaction of demand
has paramount importance. The execution of maintenance measures in a timely manner has a great impact
on forced outage rate (FOR) of generating plants and consequently has impacts on mitigation of
generation costs. In restructured power systems, regard to the independence of generation companies, the
procedure of maintenance planning is substantially changed, and it has become more complicated
compared with power systems with traditional structures. This study delves into an optimization problem,
which seeks for a long-term maintenance scheduling method for power generation plants in large-scale
power systems subject to the reduction of maintenance costs, retaining the reliability of system, and
satisfaction of prevailing constraints. By setting of maintenance costs as an objective function, a
mathematical method based on Benders decomposition technique is proposed to arrange a maintenance
schedule for generating units. With respect to the importance of electricity demand peak and the
implication of market clearing price (MCP) in the costs of generating units as well as reliability of power
network, a penalty factor is defined in order to take the lost profit of units into account in the objective
function that results in the compatibility of scheduling with the consumption patterns. In addition, by
imposing appropriate constraints and considering some factors such as forced outage rate of units,
expected energy not supplied, spinning reserve required, and integrating pumped-storage unit into the
power system, the reliability of the grid is procured. The proposed method is evaluated by use of a 24-bus
IEEE-RTS test system using GAMS software. The obtained results imply the accuracy, simplicity, high
computation speed, and great performance of the model and convey that this method can be effectively be
employed for large-scale applications.
Keywords: Maintenance scheduling, power generation plants, reliability, Benders decomposition,
GAMS;

Nomenclatures
 1- Introduction
In traditional power systems with a vertically integrated structure, the generation units are saddled with
undesired maintenance schedules imposed by system operator [1]. In other word, the arrangement of
maintenance schedules for generating units and transmission companies and the coordination between
these units are performed by the system operator in a centralized way. In restructured power systems, the
independent system operator (ISO) is responsible for the provision of security and reliability in power
systems. The generating units share their desired time-oriented maintenance schedule with ISO with the
aim of maximization of their profit and regardless of reliability and security of the entire system. ISO
must modify and manage the received offers from the generation companies (GENCOs) in a coordinated
and holistic manner. ISO assesses the offered maintenance schedules in compliance with the reliability
restrictions in order to confirm or decline the offer. Then ISO sends the rejected offers back to the
corresponding GENCOs for reconsideration of required modifications [2]. The maintenance scheduling
problem in a restructured environment is a multi-level process, in which each level deals with a multi-
objective, discrete, stochastic, and non-linear optimization problem. Some important constraints should be
considered in this optimization problem such as the permissible time of maintenance, the adequacy of the
workforce, resource availability, seasonal restrictions, power equality, transmission flow capacity, forced
outage rate (FOR) of units, and interruption duration index for active power supply. Besides, any
equipment, which affects the generation or consumption of electricity can alter reliability and
consequently has profound implications on maintenance schedules. The solution methods can be
classified into two models of mathematical and analytical solutions and metaheuristic approaches [3].
A wide range of studies has been done on the subject of maintenance scheduling for power systems and
different approaches with various constraints are proposed to solve these problems. In [4], a new criterion
based on the generators’ outage risk is defined. The occurrence of a fault in a generating unit can be
estimated by reliability-oriented methods based on reliability theory and risk assessment fundamentals. In
these concepts, the final goal is to calculate the probability of the proper functioning of a system within a
specific interval. The schedule is drawn up based on the number of generating units and their
corresponding capacity, and the grid’s structure and transmission boundaries are ignored. In [5], a
maintenance scheduling model for generation and transmission section is proposed, in which the N-1
contingency condition is incorporated. In this model, the objective is to maximize the considered items of
the utility owners and to observe N-1 restrictions. The Benders decomposition technique is used to solve
this problem, and the problem is divided into a master problem and some subproblems. In [6], the antlion
optimization algorithm (ALO) is employed as an effective tool for the preparation of a maintenance
schedule. However, the system’s constraints are restricted to the capacity of lines. In [7], a closed-loop
model for coordination between maintenance schedule and the long-term security-constrained unit
commitment is proposed in order to improve the security and optimal economic dispatch. Regard to the
complexity of the calculations, in this study, the constraints transferring techniques have been employed,
and they work in accordance with mixed-integer programming. In [8], the uncertainties corresponded
with the generation cost and load forecasting are modeled by fuzzy logic, and the maintenance scheduling
problem is solved using a dynamic non-cooperative fuzzy game. Nevertheless, the problem is solved only
by consideration of maintenance costs and the reliability indices are not taken into consideration. In [9],
the maintenance scheduling for power plants is regarded as a key part of generation expansion planning,
and the multi-objective binary gravitational search algorithm is employed to search the solution space of
the maintenance scheduling problem. There are three objectives in the objective function of this problem.
The first one incorporates the lost capacity into the model while the unit is under maintenance. The
second and third objectives include the minimization of operation and maintenance costs together with
the expected energy not supplied (EENS) reliability index. In [10], the genetic algorithm is used to
establish a maintenance schedule while the power flow model is included. In this research, the
requirements of restructured power systems are not observed. In [11], the role of microgrids in
maintenance scheduling of generating units is investigated. In this research, in order to make the load
curve smoother, a large number of microgrids are included in the model so that the 30% of the whole
demand of each bus must be met by the corresponding microgrids. In [12], a bilevel optimization method
is used to deal with maintenance schedules. At the first stage, the optimization must be conducted subject
to minimize the total cost of maintenance. Afterward, it is strived to maximize the profit of GENCOs on
the second level. In [13], an optimization model with respect to the integration of wind power potential
and forced outage rate of equipment is used for maintenance scheduling. A set of uncertainties is used to
describe the wind intermittency, and an innovative approach is proposed to control the model optimally.
In [14], the arrangement of a maintenance schedule is made while the presence of a photovoltaic unit is
evaluated. Since the problem features as a non-linear, non-smooth, and non-convex problem, the electro
search optimization algorithm (ESOA) is used while the operation and maintenance costs are only
included in the model. In [15], the maintenance scheduling for hydroelectric power plants is proposed. In
this work, the objective of maximization of profit using MILP approach is pursued. The final profit is
defined as the subtraction of the final cost of maintenance from the economic value of generated power
within the time horizon of the study added by the residual water in the dam’s reservoir. In [16], the
maintenance scheduling procedure is conducted by the employment of biography-based optimization.
However, the FOR index of units is ignored, and the proposed method is only tested on a power system
encompassing three generators.
Regard to this fact that the modeling of an optimization problem by a non-mathematical approach is
remarkably simpler, most of the conducted researches are done using non-mathematical and metaheuristic
methods. However, in these methods, the accuracy of results is less to some extent in comparison with
mathematical methods. In the studies mentioned in the literature, some of the constraints or indices are
ignored in order to simplify the solution. For instance, the FOR index of units is not considered in some
of the described works. Just the same, in some works, the EENS index is ignored too. Besides, in some
studies, the simulation is just performed with consideration of the number and capacity of units as well as
the demand peak value, and the power system structure and the transmission lines’ restrictions are not
taken into consideration. In some studies, the number of involved generating units in the maintenance
schedule is restrained and the targeted time horizon of study is cut down around some weeks in order to
simplify the model. Therefore, most of the previously conducted works are not suitable to be applied for
large-scale power systems and long-term applications.
In this study, the aim is to introduce a long-term model appropriate for power networks in large-scale.
Regard to this fact that the multi-year load forecasts and price forecasts are not accurate, it is
recommended that a yearly time horizon should be considered. Regard to the impact of a weekly demand
peak on the maintenance schedules and reliability of the grid, the new concept proposed in study deals
with the implication of the presence of pumped-hydro storage unit on the maintenance schedule along
with the reliability of the system. Another feature which is added to the model is that the lost profit is also
integrated into the maintenance schedule and maintenance costs. When a generating unit must be
switched off for maintenance, this unit does not earn revenue. Thereby, the GENCO’s profit, which may
be constituted of some generating units, will be diminished. This amount of revenue reduction is defined
as the lost profit and must be added to the maintenance cost. In addition, by definition of coefficient
proportional to the weekly demand peak, known as a penalty factor, it is tried to maintain a maintenance
schedule in alignment with the reliability of the grid.
In this study, the maintenance schedule is optimized using a mathematical approach, and the formulation
of the problem is described. Then, the Benders decomposition technique is explained. In order to evaluate
the effectiveness and performance of the proposed model, the suggested methodology is tested on an
IEEE-RTS 24-bus system. The simulation is carried out in six scenarios, in which the impacts of FOR,
EENS, spinning reserve required, and pumped storage units on the maintenance schedule are investigated.
Furthermore, the obtained results are compared with a metaheuristic method.

2- The methodology of the suggested approach

In the maintenance scheduling problem, M is an integer decision-making variable, and the amount of
generation of each unit is an integer (a positive rational) number. Thus, the long-term maintenance
scheduling of generators is counted as a mixed-integer linear programming model, which consists of
maintenance constraints and system restrictions. The main goal is to minimize the maintenance cost as
well as operation cost during the targeted time interval. This interval is usually set to be one or two years.
It is because, in longer intervals, the uncertainty of load forecasts mounts drastically. TMC stands for total
maintenance cost, and TC denotes the total cost of operation. These parameters are shown by Eqs. (1) and
(2). In Eq. (1), Mi is a binary variable so that it takes the value of 1 if the unit is in service and out of
maintenance schedule, and it takes the value of 0 if the unit is under maintenance and out of service.
(1)
(2)
The maintenance cost and operation cost are shown by the variables of MCi and OCi respectively, In Eq.
(1) a weekly coefficient of PF, which is called the penalty factor, is used. This factor can be calculated
based on the weekly demand peak (Dw), and it specifies a penalty for the unit which is switched off to
perform maintenance measures. This penalty factor is proportional to the demand for electricity in the
corresponding week. This penalty is modeled as the increase in maintenance cost.
(3)
The MCP is also important for the GENCOs. The reason is that the total maintenance cost is explained as
the summation of maintenance costs and the lost income. Equation (4) represents the total maintenance
cost.
(4)
By consideration of a yearly interval for the maintenance schedule and weekly intervals for maintenance
execution, the final maintenance cost can be achieved through Eq. (5):
(5)
The overall objective function of the schedule within various intervals as defined as the summation of
maintenance and operation costs that are expressed as Eq. (6) as below:
(6)
The optimization problem consists of two sets of constraints. The maintenance constraints are dealt with
GENCOs, and the maintenance scheduling practice will be done by utilization of these constraints subject
to minimize the costs and maximize their own profit. The other set of constraints is the network
constraints, which are imposed by ISO to assess the reliability and security criteria. If the provided
schedule plan by the GENCOs does not meet the reliability and security constraints, the schedule will be
rejected by the ISO, and it will be sent back to the GENCOs along with the required reconsideration of
the network in order to implement some modifications. In Eq. (6), the maintenance constraints indicate
the temporal restrictions of maintenance implementation of each unit that can be described by Eq. (7)
[17]:
(7)
Where Miw shows the state of maintenance if the i unit in the w week so that 0 implies that the unit is
th th

min
under maintenance and out of service whereas 1 represents the unit is in service. t i indicates the soonest
max
time for execution of maintenance of the ith unit, and similarly t i delineates the latest possible time for
 m
execution of maintenance of this unit. t i shows the duration of the maintenance process for unit i. In
max min
addition, si shows the time of implementation of maintenances. The values of t i and t i will be
determined based on the hours of operation and seasonal restrictions. The constraints of availability of
workforce and accessibility to the resources can be also included in the model. In other word, this
constraint conveys that how many units can be under maintenance simultaneously with respect to the
availability of workforce and accessibility to the resources. The maintenance measure required for each
unit have to be carried out within the specified maintenance window. This limitation is expressed by Eq.
(9):
(8)
(9)
The power system constraints are issued and controlled by ISO. Equations (10) to (14) describe these
restrictions.
(10)
(11)
(12)
(13)
(14)
In above, BL is the intersection matrix of nodes and branches. PL is the power flow vector of transmission
lines at the peak load in each week, giw stands for the generated power by the ith unit at the peak load of
the wth week, Dbw denotes the weekly peak load at each bus, rbw shows the active power interruption of bus
b at the weekly peak which shows the difference between generated and consumed active power at each
bus in each week. The summations of rb in each week delineates the EENS in that week. Eq. (10)
represents the load equality at each bus of the system. Eq. (11) exhibits the maximum and minimum
generation capability of each unit. The active power interruption at each bus has to be lower than the peak
of load at that bus. In addition, PLkw indicate the transmission flow capacity of the line k at the week w.
This parameter can be obtained according to the maximum capacity of each line (PLmax) and the number
of lines (n). In order to assure the reliability and protect the security of the system, the EENS must be
capped with a preassigned value by ISO presented by ε [17].
Mi is a binary integer variable that describes the state of being under maintenance or being in service. gi is
a continuous variable that demonstrates the amount of generation of unit i. Hence, the maintenance
scheduling problem is regarded as a mixed-integer non-linear problem. Figure 1 displays an illustrative
view of the proposed model for the maintenance scheduling problem.

Figure 1. The holistic schematic of the proposed maintenance schedule

2-1- The proposed method
In order to minimize TMC in Eq. (4), the Benders decomposition algorithm is employed. In this
algorithm, the optimization problem is divided into two parts of a master problem and one or more
subproblems. In order to solve the problem, Eq. (6) is upgraded as Eq. (15).
(15)
Eq. (16) represents the Lagrangian function of Eq. (15). In this method, the violations of inequality
constraints are penalized with respect to the Lagrangian multipliers.
(16)
Thus, the maintenance scheduling problem can be expressed by Eq. (17).
(17)
The solution derived from Eq. (17) minimizes the operation and maintenance cost. In order to ensure and
guarantee the reliability of the grid, Eq. (18) has to have a lower value than ε.
(18)
Eq. (19) represents the Lagrangian function of Eq. (18).
(19)

Hence, the maintenance scheduling problem is change as Eq. (20), in which the value of C has to be
minimized.
(20)
The master problem, which must be solved by the GENCOs, can be expressed by Eq. (21):
(21)
ISO compares the obtained results with the solution of Eq. (18). If the obtained result is not viable, an
infeasibility cut will be sent to the GENCO as in Eq. (22):
(22)
The GENCO solves the master problem again by adding the infeasibility cut as a constraint. Then the
GENCO send the solution back to ISO. This process will be repeated until the feasibility condition is
materialized. Regard to this fact that this procedure is time-consuming and sophisticated, the iterative
loop between GENCO and ISO should not be a lot. The number of these iterations are defined by ISO,
and hereby the GENCO is obliged to accept a mandatory maintenance window if the feasibility is not
reached after the dedicated amount of iterations [17].
The spinning reserve is defined as the difference between the maximum generation capacity of a unit and
the real amount of generation at a specific interval, and it is usually described as a certain percentage of
the weekly peak. If the spinning reserve is included in the operation schedule, Eq. (23) must be added to
the system’s constraints.
(23)
A pumped-hydro storage plant exploits the potential energy of water in an elevated reservoir. Once the
electrical demand level is low, water is pumped from a lower reservoir into an upper reservoir in order to
store energy and charge the storage. When the demand is mounted, the water is let to flow toward the
lower reservoir. Meanwhile, the water drives a set of hydro-turbines to generate electrical power. In
general, the most prominent application of pumped storage units is to shift the low-priced energy of off-
peak to the expensive electrical energy at peak.
A pumped storage unit has three operational modes: pumping state, idle state, and generation state.
During off-peak hours, a pumped storage unit functions as a consumer and uses electricity to store energy
in pumping mode. During mid-peak hours, the plant is in idle mode, and it is often stated in generation
mode at peak hours [18]. Peak clipping or peak shaving is the most predominant role of pumped storage
units. Hence, this unit is usually is connected to the bus, where both the demand and the MCP are high,
provided the geological condition for installation of a pumped storage unit is viable. The power equality
equation in the presence of a pumped storage unit connected to the bus b in the week w can be explained
by Eq. (24). In this equation, SP represents the hourly consumed or generated power by the pumped
storage unit.
(24)
In Fig. 2, the procedure of problem with integration of pumped storage unit into the power system is
proposed.

Figure 2. The paradigm of maintenance scheduling problem with integration of storage units
3- Simulation results
The proposed methodology for arranging a maintenance schedule for generating units is implemented on
an IEEE-RTS 24-bus standard test system. This power system contains 32 generators, 24 buses, and 38
transmission lines. In this power system, the maximum peak of the yearly duration curve is forecasted
equal with 2850 MW [18]. The time horizon of the study is targeted to be a year, and the duration of
maintenance windows are assigned to be a week. It is assumed that the concurrent maintenance of 2 unit
at the same interval is possible. The data corresponded with the weekly MCP predictions refer to [20,21].
These references address the weekly electricity price of Nordpool deregulated and restructured power
system. In order to design the maintenance schedule, six scenarios are developed as follows:

 Scenario 1: without consideration of FORi and penalty factor and with consideration of EENS
equal with 0%.
 Scenario 2: with the inclusion of FORi
 Scenario 3: with consideration of maximum EENS equal with 1%.
 Scenario 4: with the inclusion of penalty factor
 Scenario 5: assigning spinning reserve required by 10% of weekly peak.
 Scenario 6: integration of a pumped storage unit into the system
In Fig. 3, with respect to the loss of load expectation (LOLE) index, a comparison is made between two
cases regardless of maintenance scheduling and scenario 1. Due to the outage of some units for
maintenance, the LOLE index is changed from 0.58 to 1.3 (in term of day) in a year in the first scenario.

Figure 3. The LOLE index in scenario 1

In the second scenario, the FORs of units are almost considered to be equal with the reduction in
generation capacity [17]. In other word, according to Eq. (25), the maximum output of each unit in each
week corresponds with the maximum rated power, availability, and FOR of each unit.
(25)
In this scenario, due to the execution of maintenance measures, the LOLE index has risen and reached the
value of 2.08 day per year. In Fig. 4, the weekly LOLE index in the second scenario is depicted.

Figure 4. The LOLE index in scenario 2

EENS is of important indices of reliability of power systems. If this index takes a value larger than zero,
it conveys that the system may sometimes be unable to serve the loads. In the third scenario, the
maximum EENS is assigned to be equal with 1% of the weekly peak of demand. As it was expected, by
the increase of EENS the LOLE index has been substantially mounted from 2.08 to 3.5 day per year. In
Fig. 5, the weekly LOLE index for the third scenario is demonstrated.

Figure 5. The LOLE index in scenario 3

The penalty factor specifies the weekly priority of execution of maintenance measures for generating
units corresponded with the maximum peak of that week. By calculation of this factor, in accordance with
Eq. (3), a value between 1 and 2 is obtained for each week. As this value is closer to 2, it represents that
the amount of demand has been high. Thus, if a unit is forced or required to perform maintenance in that
week, its maintenance cost goes up because a penalty is charged to the unit. In Fig. 6, the weekly demand
peak and the corresponding penalty factors within 52 consecutive weeks of a year is displayed. This
factor is equal with 1 in the minimum peak of the year, and it is equal with 2 at the maximum peak of the
year. In the fourth scenario, the penalty factor is included in the scheduling presented in the last scenario.

Figure 6. The weekly demand peak and the corresponding penalty factor
By taking penalty factor into consideration, with respect to the increase of maintenance cost due to the
rise of demand peak in some weeks, the units are less prone to perform maintenance measures within
these weeks. Hence, it is predicted that reliability will be enhanced. The comparison of LOLE index in
this scenario and the previous scenario obviously implies that the value of this index has been reduced
from 3.5 to 2.02 day per year. In Fig. 7, the LOLE index of scenario 4 is demonstrated.

Figure 7. The LOLE index in scenario 4

In the fifth scenario, the spinning reserve of the grid is considered to be 10% of the weekly demand peak.
The investigation of obtained indicates that the reliability of the grid has been improved and the LOLE
index is diminished and reached the value of 1.41 day per year.
Regard to the location of generators (the corresponding bus), the type of generator connected to each bus,
and the demand of each bus, the bus 115 is chosen for the installation of a storage unit which is
considered to have the type of pumped storage. The pumped storage units resemble hydroelectric power
plants. In IEEE-RTS test system, there are hydroelectric power plants with a capacity of 50 MW. Hence,
in the sixth scenario, a pumped storage unit with a capacity of 50 MW is supposed. In Fig. 8, the weekly
peak of bus 115 in two cases of with and without the integration of pumped storage unit is presented.

Figure 8. The weekly demand peak of bus 115

In scenario 6, the results of maintenance scheduling problem with the inclusion of a pumped storage unit
imply that the reliability has been vastly improved so that the LOLE index has been reached the value of
0.81 day per year. In Fig. 9, the weekly LOLE indices for the 5th and 6th scenarios are depicted.

Figure 9. The LOLE index in scenarios 5 and 6

In Fig. 10, a comparison of LOLE index between all scenarios is made:

Figure 10. The comparison of LOLE index

In Figs 11 and 12, the maintenance cost and the operation cost of each scenario are illustrated
respectively.

Figure 11. The comparison of the maintenance cost

 Figure 12. The comparison of the operation cost
The first scenario, which is the most fundamental form of a maintenance schedule, has the lowest
maintenance cost. By consideration of FOR index, with respect to the impact of this parameter on the
maintenance schedule, the maintenance cost and operation cost is fairly enhanced. By the increase of
EENS in the 3rd scenario, the maintenance cost has had a scant increase, but the operation cost and total
cost are mitigated.
As it was expected, whereas the consideration of penalty factor leads to a remarkable increase in the
reliability of the grid, it brings about a steep rise in operation cost. The reason is that some units must
perform their maintenance measures at some specific intervals when the peak price is extremely high.
Hereby, the given penalty by ISO to these units has increased their maintenance cost.
Even though the inclusion of spinning reserve improves the reliability of the grid, it results in a slight
reduction in maintenance cost. The consideration of the impact of the pumped storage unit has provided a
visible improvement in the reliability of the system, and it has reduced the maintenance cost and total
cost. The use of pumped storage unit has not a notable impact on the maintenance cost, although it
provides a considerable reduction in operation cost due to its influence on the change of consumption
curve. Figure 12 depicts the maintenance schedule with consideration of the presence of a pumped
storage plant.

Figure 12. The maintenance schedule in the presence of a pumped storage unit
In the following part, in order to evaluate the effectiveness of the proposed approach, the obtained results
are compared with a similar work, which has employed a modified PSO algorithm to solve the
optimization problem. In [19], by applying the MPSO algorithm, the maintenance scheduling problem is
tested on a 24-bus power system while the simultaneous maintenance of 5 units is contemplated. To
assure the effectiveness, accuracy, and good performance of the proposed method using the MPSO
algorithm, the maintenance scheduling the 6 th scenario is performed with the assumption of maximum
concurrent maintenance of 5 units. In Fig. 13, the LOLE index and in Fig. 14 the maintenance schedule in
both methods are pointed out. By comparison of these two methods, it can be obviously perceived that the
proposed method in this paper provides more optimal results in shorter time for computation so that the
LOLE index in the proposed method has been taken the value of 2.18 day per year, which is better than
the results of MPSO approach with the value of 3.55 day per year.

Figure 13. The comparison of LOLE between the proposed method and MPSO approach

Figure 14. The comparison of maintenance schedule between the proposed method and MPSO approach

4- Conclusions
In this study, the long-term maintenance scheduling problem for the generating units in the restructured
power systems with the evaluation of the impacts of pumped storage plants was investigated. This
optimization problem is solved by the Benders decomposition technique. The objectives of mitigation of
maintenance and operation costs and the improvement of reliability level are assigned for the objective
function in order to use this proposed method for dealing with applications in large-scale power systems.
The proposed method is implemented on an IEEE-RTS 24-bus standard test system in order to measure
its effectiveness and accuracy. According to the obtained results, the incorporation of forced outage rate
of units, and determination of a non-zero value for energy not supplied index will result in the
deterioration of the reliability of the grid and mitigation of operation cost in contrast. By incorporation of
penalty factor into the objective function, the maintenance cost was mounted, whereas the reliability
indices were improved tremendously. In addition, the incorporation of the spinning reserve requirement in
the model has resulted in the improvement of the reliability. Ultimately, the integration of a pumped
storage facility into the targeted power system has resulted in a significant increase in the reliability levels
because it has made the demand curve smoother and it has alleviated the weekly peaks. Hereby, the
maintenance cost and operation cost were cut down. By adding a comparison between the proposed
method and a published work, which has employed modified PSO (MPSO) algorithm, it can distinctly be
perceived that the proposed method has inspired performance in terms of speed and accuracy and can be
applicable for preparing maintenance schedules in large-scale power systems.