Generation Maintenance Scheduling Problem in Power System

BRIEF CONTENT

Preface

1. Abstract

2. List of Symbols and Abbreviations

CHAPTER NO.

TITLE

PAGENO.

1 INTRODUCTION
1.1 EQUIPMENT MAINTENANCE
1.1.1 Maintenance scheduling
1.1.1.1 Maintenance activity
1.2 MAINTENANCE SCHEDULING METHODS
1.2.1 Existing methods
1.2.2 Objective function
1.2.2.1 Economic cost objective function
1.2.2.2 Reliability objective function
1.3 LITERATURE REVIEW
1.4 OBJECTIVES
1.5 SUMMARY

2 SOFT COMPUTING TECHNIQUE BASED LEVELIZED RESERVE BASED MAINTENANCE SCHEDULING
2.1 INTRODUCTION
2.2 PROBLEM FORMULATION
2.2.1 Levelized reserve capacity method
2.2.2 Levelized reserve rate method
2.3 MAINTENANCE SCHEDULING CONSTRAINTS
2.3.1 Time constraint
2.3.2 Maintenance crew constraint
2.3.3 Reserve constraint
2.4 PSO BASED APPROACH FOR MAiNTENANCE
SCHEDULING
2.4.1 Introduction
2.4.2 Overview of PSO
2.4.3 Development of the proposed PSO based MS
algorithm
2.5 BPSO BASED MAiNTENANCE SCHEDULING
2.6 IMPROVED BPSO BASED MAiNTENANCE
SCHEDULING
2.6.1 Execution of proposed IBPSO based MS algorithm
2.6.2 Implementation of MS using BPSO and IBPSO
2.7 RESULTS AND DISCUSSION
2.7.1 Case study 2–IEEE RTS
2.7.2 Comparison of results
2.8 SUMMARY

3 LEVELIZED RISK BASED MAINTENANCE SCHEDULING
3.1 INTRODUCTION
3.1.1 Loss of load probability
3.1.2 Loss of load expectation
3.2 LEVELIZED RISK METHOD
3.2.1 Capacity outage probability table
3.2.2 Risk characteristic coefficient
3.3 PROBLEM FORMULATION
3.3.1 Objective function
3.3.2 Encoding scheme for IBPSO based levelized risk
method
3.3.4 Implementation of IBPSO based levelized risk method
3.4 RESULTS AND DISCUSSION
3.5 SUMMARY

4 RESEARCH CONCLUSIONS
4.1. RESEARCH SUMMARY AND CONCLUSIONS
4.2 SCOPE FOR FUTURE WORK

REFERENCES

PREFACE

Power system components are regularly maintained in order to remain in a working condition during the entire period of life. The purpose of Maintenance Scheduling (MS) is to find the time table of the maintenance outage of power system units over a given period of time. The present-day need for MS is now more important than ever before, and continues to grow constantly. In order to avoid premature aging and failure of generators in a power system leading to unplanned and costly power outages, it is important to carry out preventive maintenance at regular intervals.

As power systems become larger and the demand for electricity increases continually, the difficulty in finding maintenance schedules increases in complexity, especially in systems with small reserve margins and high levels of constrictions. An automated technique allows operation personnel to utilize the computing power available on real-time control systems to revise and report schedules quickly. The use of a computer program can also eliminate, or reduce substantially, the effort required to prepare schedules and maintain records. Therefore, the MS is a significant part of the overall operations scheduling problem. Motivation of this research work includes the MS affects many short-and long term planning functions like unit commitment, fuel scheduling etc.

Traditional techniques such as linear programming based branch and bound and Bender’s decomposition are employed to solve the MS. The main problem with the exact mathematical methods is that the number of combinations of states that are searched increases exponentially with the size of problem and becomes computationally prohibitive. Hence there is a need for new optimization algorithm to solve MS problem. Hence, in this research Improved Binary PSO is used to solve the MS problem.

Improved Binary Particle Swarm Optimization based Maintenance Scheduling using Levelized Reserve and Risk method

Dr. K. Suresh, Professor,

Department of Electrical and Electronics Engineering

Sri Manakula Vinayagar Engineering College,Puducherry, India

ABSTRACT

The fundamental concern of Maintenance Scheduling (MS) is to reduce the generator failures and extend the generator’s lifespan thereby increasing the system reliability. The objective function of MS problem is to reduce the LOLP and minimizing the annual supply reserve ratio deviation for a power system which are considered as a measure of power system reliability. The typical Particle Swarm Optimization (PSO) is designed for continuous function optimization problems, and not for discrete function optimization problems. The MS problem has discrete decision variables. Hence, the Improved Binary PSO (BPSO) is utilized to solve the MS problem. Furthermore, Improved BPSO (IBPSO) based and levelized reserve rate methods are developed to solve the MS problem. In order to test the effectiveness of the IBPSO based MS method IEEE Reliability Test System (RTS) case study is considered

A comparison is made between the MS results obtained from the levelized reserve and the levelized risk methods. Consequently, the comparison reveals that the levelized risk method is obviously superior to the levelized reserve rate method which is deterministic methods neglect the influence of the generating unit’s random outages.

The IBPSO generates optimal MS solution and overcomes the limitation of the conventional methods such as extensive computational effort which increases significantly as the size of the problem increases. The proposed method yields better result by means of improved search performance and better convergence characteristics which are compared to the other optimization methods and conventional method.

Keywords Maintenance Scheduling (MS), Loss of Load Probability (LOLP), Improved BPSO (IBPSO)

LIST OF SYMBOLS AND ABBREVIATIONS

LIST OF SYMBOLS

A.Sets/Indices

T – Number of maintenance time intervals

n – Number of days in a week (n =7)

t – Maintenance intervals

k – Generating unit

N – Number of generating units

Tc – Total system capacity in MW

S – Set of generating units involved in maintenance in the period under examination

Vr – Maintenance crew

Vrt – Maximum number of generating units that the maintenance crew can work simultaneously in any stage t

Abbildung in dieser Leseprobe nicht enthalten – ith particle position of reserve values

Abbildung in dieser Leseprobe nicht enthalten – ith particle position of reserve rate values

Abbildung in dieser Leseprobe nicht enthalten – i th particle position of risk characteristic coefficient values

B.Parameters

Xtk – Abbildung in dieser Leseprobe nicht enthalten

rand( ) – uniform random number in the range [0, 1].

Abbildung in dieser Leseprobe nicht enthalten ,Abbildung in dieser Leseprobe nicht enthalten – Random binary integer numbers uniformly distributed in the range of [0,1]

C.Operators

Abbildung in dieser Leseprobe nicht enthalten – AND operator

Abbildung in dieser Leseprobe nicht enthalten – XOR operator

+ – OR operator

D.Variables

LOLPi ,LOLPj – System risks in any maintenance interval

LOLE p – Loss of load expectation in the maintenance period p

Abbildung in dieser Leseprobe nicht enthalten – Probability of loss of load on day i

Ci – Available capacity on day i

Li – Forecast peak load on day i

LOLE a – Annual LOLE in a year

Abbildung in dieser Leseprobe nicht enthalten – Objective function

Abbildung in dieser Leseprobe nicht enthalten – Net reserve rate of the r th maintenance time interval t

Abbildung in dieser Leseprobe nicht enthalten – Average reserve rate in the r th maintenance time interval t

R – Net reserve in MW

Pt – Predicted maximum load in MW during maintenance time interval t

Ck – Capacity of unit k in MW

tk – Starting stage for maintenance

Sk – Number of stages for maintenance

Sk-1 – Number of intervals for maintenance till k-1 th weeks

XA – Outage at point A

XB – Outage at point B

P (X) – Probability of outage

P (XA) – Probability of outage at point A

P (XB) – Probability of outage at point B

m – Risk characteristic coefficient

C – Capacity of the generator being added

Ce – Effective load carrying capacity of a generator in MW

p – Availability of a generator

q – Unavailability or forced outage rate of a generator

Le – Equivalent load in MW

Lm – Maximum load of the stage under study in MW

Lj – Daily maximum load under the stage under study in MW

Tp – Number of days in a maintenance stage

K – Population size

[ Xmin, Xmax ] – Particle position in the minimum and maximum range

[ Vmin, Vmax ] – Particle velocities in the minimum and maximum range

Na – Number of iterations

Abbildung in dieser Leseprobe nicht enthalten – Updated velocity of jth dimension in ith particle at iteration k+1

Abbildung in dieser Leseprobe nicht enthalten – Particle best of individual jth dimension in ith particle at iteration k

Abbildung in dieser Leseprobe nicht enthalten – Global best of individual jth dimension in ith particle at iteration k

Abbildung in dieser Leseprobe nicht enthalten – Updated current position of the jth dimension in ith particle at iteration k+1

Abbildung in dieser Leseprobe nicht enthalten – Current position of the jth dimension in ith particle at iteration k

C1,C2 – Acceleration constant

Wmax, Wmin – Maximum and minimum inertia weight

iter – Iteration number

itermax – Maximum number of iterations

Xmax, Xmin – Maximum and minimum positions of the particle

LOLPobj,i – Objective level of the LOLP values

List of abbreviations

MS – Maintenance Scheduling

LOLP – Loss of Load Probability

LOLE – Loss of Load Expectation

PSO – Particle Swarm Optimization

GA – Genetic Algorithm

EP – Evolutionary Programming

SA – Simulated Annealing

TS – Tabu Search

FOR – Forced Outage Rate

COPT – Capacity Outage Probability Table

UC – Unit commitment

ED – Economic Dispatch

RTS – Reliability Test System

BPSO – Binary Particle Swarm Optimization

IBPSO – Improved Binary Particle Swarm Optimization

CHAPTER 1

INTRODUCTION

1.1 EQUIPMENT MAINTENANCE

A failure in a generating unit results in the unit being removed from service in order to be repaired or replaced. This event is known as an outage. Such outages can compromise the ability of the system to supply the load and affect system reliability. Consequently, the generator MS for a large power system has become a complex, multi-object-constrained optimization problem. Both research and practice show that power system maintenance schedule is in fact a constrained optimization problem. The maintenance schedule that satisfies all the constraints is called a “feasible” schedule.

Preventive MS of the generating unit is an important requirement of power system planning. The MS of generating units attract great attention in power system operation planning. Modern power system is experiencing increased demand for electricity with related expansions in system size, which has resulted in a higher number of generators making MS problem more complicated [1] (1972).. The maintenance of generators is directly associated with the overall reliability of the power system. It is important to supply reliable and economical electricity to the customers. It can be accomplished by optimal schedules of system operation and planning.

The maintenance of power system equipment, especially, the maintenance of generating units, is implicitly related to power system reliability. Therefore, maintenance problem has always been investigated together with system reliability problems and is one of the main subjects in reliability engineering research [7]

(1994).In reality, the MS problem is essentially a part of the total problem of economic operation and control which involves ED and UC [9] (1996).

1.1.1 Maintenance scheduling

The preventive MS of generating units occupy a significant place in power system operation and expansion planning, at the same time it is a challenging optimization problem. This chapter deals with the introduction of preventive MS of generating units. Maintenance means taking a unit out of service at a selected time, usually for the purpose of preventive (or) corrective measures.

- Scheduled maintenance

It is maintenance carried out on power system units in a regular scheduled manner.

- Unscheduled maintenance

It is maintenance carried out on power system units due to sudden outage.

The MS problem is vital to plan the secure and reliable operation of a power system, primarily because other short and long term planning activities, such as UC, generation dispatch, import/export of power and generation expansion planning are directly affected by such decisions [9] (1996).

1.1.1.1 Maintenance activity

The main activities which have to be carried out in order to achieve the goals are as follows, preventive checkups necessitated for all equipment components include boiler, turbine, generator and auxiliary equipments. Repair or replacement of incorrect components is needed where working characteristics have fallen below acceptable limit due to wear (or) aging. Major maintenance activities include checkup and maintenance of boiler (typically once a year) and turbine (typically once every five years).

The generator like any other electrical component is subjected to different operating conditions, due to which deterioration and wearing takes place resulting in premature aging. As a result of this, the reliability of the component decreases. The stator windings are the most important components of the generator, failures related to the stator windings are the main reasons for the majority of failures in the generator.

Basically, the stator windings are copper conductors which are equally distributed in the stator core slots to engage symmetrical linkage with the flux produced by the rotor. To minimize the effect of any eddy currents these stator windings are made up of many number of copper strands which are insulated from each other. Most of the failures in stator windings are related to insulation failures, which could be due to ineffective cooling system, thermal cycling, thermal strength of insulating material, operation under abnormal conditions etc. Overheating of the stator core eventually leads to thermal aging of the generator.

Excessive eddy current flow consequently leads to breakdown of inter-laminar insulation due to overheating by eddy currents. While inspecting the rotor, the condition of the rotor winding insulation, rotor vibration and coupling with the turbine are checked. Besides the stator and rotor, the integrity of the mechanical components is important for the optimal performance of the generator. The mechanical components like stator frame, bearings, brake etc, are essential for a reliable operation of the generator as envisaged by designers.

Any malfunction can be risky for the safety of the personnel as well as lead to complete outage of the unit with the loss of revenue and extra cost of repair and replacement. One of the most important mechanical component of the generator is the bearing, which should be in good condition for mechanical integrity of the generator.

A generation unit running may operate under de-rated capacity due to removal of coils after previous winding failures, insulation break down and ineffective cooling systems.

1.2 MAINTENANCE SCHEDULING METHODS

The MS problem has been studied for several decades and various methods have been applied by the researchers to solve the MS problem. Mathematical methods are mainly based on integer programming, branch-and-bound and dynamic programming. Mathematically, MS problem is a multiple constraint, non linear and stochastic optimization problem. In mathematical programming methods, the branch–and–bound method is suitable for solving this problem, while other methods such as integer programming and dynamic programming have got limited application, because they either cannot accurately simulate power system operations or may not guarantee the global optimal solution for a complex optimization problem [13] (2000).

1.2.1 Existing methods

The conventional approaches suffer from ‘curse of dimensionality’ with the increase of a more number of system variables which depend on the dimension size of the MS problem. The computational effort and complexity will grow prohibitively with the problem size. Some of the popular methods are explained [7] (1994). Some of the various methods developed are listed below.

- Levelized reserve capacity method
- Levelized reserve rate method
- Levelized risk method

The evolutionary approaches have been applied to solve a range of optimization problems in electrical power systems with encouraging results [6] (1992). The most widely used optimization techniques are listed below.

- Genetic Algorithm (GA) [13] (1999)
- Particle Swarm Optimization (PSO) [15] (2003), [19] (2008)
- Simulated Annealing (SA)
- Tabu Search (TS)
- Evolutionary Programming (EP)

The heuristic-based techniques use a specialized method to evaluate the objective function in the time interval under examination. In order to overcome some of these limitations a number of metaheuristic and soft computing based approaches for maintenance have been used. These include GA, SA, EP, fuzzy logic and their hybrids which are used to solve the MS problem [12] (2000)

a. Mathematical programming

Mathematical programming is one of the oldest methods for obtaining the optimum maintenance schedule. Coincidentally, it is also one of the most cumbersome when the size of the system increases [10] (1998). Theoretically, a dynamic programming technique is more suitable for solutions since MS is a multi-stage policy-making process [7] (1994). However, the practical applications of the method are limited by the ‘dimension barrier’ problem. In order to greatly reduce the state space and the computation time and to make the method more practical, the successive approximation dynamic programming method can be used in MS method. There are other methods for maintenance optimization using mathematical programming. For example, the maintenance problem can be transformed into a mixed integer linear programming problem in order to reduce the computation time.

Integer programming, branch-and-bound techniques and dynamic programming are generally not suitable for the large scale generator maintenance schedule problem and their computational time grows prohibitively with problem size. It is obvious that it is difficult to use mathematical approach in large systems particularly with variable load models. However, there are two difficulties in the practical applications of this method. In mathematical approach the computation time is very high and there are certain large scale problems that can exceed the computer processing capabilities. The main problem with the exact mathematical methods is that the number of combinations of states that must be searched increases exponentially with the size of the problem and becomes computationally prohibitive [7] (1994). Levelized reserve method, Levelized risk methods are explained in Chapter 2.

1.2.2 Objective function

In MS there are normally two categories of objective function in power system maintenance. Generally, there are two objectives in the operation of an electric power system. They are minimizing the total operating costs to the utility and maximizing the system reliability. The objective functions for MS can be classified in to following two categories: Economic cost objective function and reliability objective function [7] (1994).

1.2.2.1 Economic cost objective function

The economic cost objective function includes maintenance cost and production cost. The main objective under this is simply expressed as the cost involving the maintenance of generating unit and the production cost of power. The maintenance cost is further divided into two, one is a fixed maintenance cost, which is normally constant and is not related to the generating unit often being in the operation state. The other part is the cost arising from wear due to frequent switching between the ‘ON’ state and ‘OFF’ state or a prolonged operation and the difference in the maintenance standard. The production cost is the cost of the fuel necessary to generate a certain amount of electrical energy when the maintenance schedule is already formulated.

1.

1.2

1.2.2.2 Reliability objective function

The reliability objective function is either deterministic (or) random. The deterministic reliability objective function is for maximizing the system’s net reserve (the system installed capacity minus the maximum load and the maintenance capacity during the period of examination) when unit maintenance outages are considered. The main drawback is that the deterministic reliability objective function neglects the randomness of the available generating units capacity, meaning that the reliability is not necessarily the same for a system with an identical net reserve during different time intervals (the units under maintenance are different)

The random reliability objective function, considers the generating unit’s forced outages when formulating the maintenance schedule. The random reliability objective function remedies the above mentioned defect as the influence of the power-generating unit’s random forced outages is taken into account.

1.3 LITERATURE REVIEW

Christiaanse, W.R. and Palmer, A.H. [1] (1972) described a technique and a computer program which developed as an operating and planning tool for scheduling the maintenance of generating facilities. The method included a description of the MS problem, the model on which the scheduling technique was based and the optimization technique.

. Wang, L. [2] (1977) proposed an analysis of how the LOLP was affected by uncertainties in the estimated forced outage rates of generating units and the forecast peak loads.

Contaxis, G.C. et al [4] (1989) described an interactive computer package for evaluating the risk level of a power system and for scheduling the preventive maintenance of the system’s generating units.

Lin, C.E. et al [6] (1992) presented a knowledge-based expert system for generator MS. The MS was considered as a constrained optimization problem. An operation index was suggested to determine an appropriate strategy for a decision-making process.

Chattopadhyay, D. [10] (1998) designed an improved MS methodology that employed linear programming. It marked significant improvement over the integer programming based MS algorithms both in terms of computational time and reliability of convergence.

Burke, E.K and Smith, A.J. [12] (2000) investigated the use of a memetic algorithm that employs a tabu-search operator for the thermal generator MS problem. The local search operators were found to produce good quality results.

Wang, J. and Handshin, E. [13] (2000) presented a GA based MS of power systems. This method improved the GA computation performance by adopting a code-specific and constraint-transparent integral coding method.

Baskar, S. et al [14] (2003) presented the applicability of GA to the generator MS problem with modified genetic operators such as string reversal and reciprocal exchange mutation.

. Chin Aik Koay and Srinivasan, D. [15] (2003) introduced a PSO based method for solving a multi-objective generator MS problem with many constraints. This method also introduced a concept for spawning and selection mechanism.

Rong-Ceng Leou [17] (2006) adopted the GA combined with the SA method as a solution method. This formulation led to minimum cost for operation and maintenance under a condition of sufficient spinning reserve.

Yare, Y. et al [22] (2010) introduced a multiple swarm concept for the Modified Discrete Particle Swarm Optimization algorithm (MDPSO) to solve the MS problem. This algorithm was referred to by the authors as multiple swarms-modified particle swarm optimization.

Joao Tome Saraiva et al [23] (2011) presented simulated annealing based approach to solve the generator MS problem. The objective function was formulated as a mixed integer optimization problem in which minimization of the operation cost along with the scheduling period plus a penalty on energy not supplied were considered.

Murat Yildirim . et al [24] (2016) introduced In Part 1, we propose a new mixed-integer optimization model for generation maintenance scheduling and a framework for maintenance on network operation by coordinating generator maintenance schedules with the unit commitment and dispatch decisions.

1.4 OBJECTIVES

In this research, a generating unit MS method that emphasizes the reliability is selected as a main criterion. In order to meet the goal, a number of objectives are addressed below.

· To develop a coordinated levelized reserve and levelized risk based MS formulation in the MS problem is not explored in the literature. To propose a new goal of the MS, to levelize risk and annual supply reserve ratio deviation in the maintenance time intervals.

· To implement the various intelligent techniques that include PSO, Binary Particle Swarm Optimization (BPSO) and Improved BPSO (IBPSO) for levelized reserve capacity and levelized reserve rate methods. To carry out the comparison between IBPSO based levelized reserve and levelized risk methods.

· To compare the results obtained from the proposed IBPSO based coordinated MS with the PSO, BPSO levelized reserve rate method.

1.5 SUMMARY

In this chapter, the background of the research work presented in this research is reviewed. Initially a general overview of the power systems reliability evaluation is explained. In particular, the definition of power system MS optimization adopted in the past studies and the methods previously applied to this problem are discussed. Moreover, the importance of pertinent MS problem is addressed and various MS methods are explained.

CHAPTER 2

SOFT COMPUTING TECHNIQUE BASED LEVELIZED RESERVE BASED MAINTENANCE SCHEDULING

2.1 INTRODUCTION

Evolution is certainly the unifying principle of modern biology. Evolutionary computation is the standard term that encompasses all these biologically motivated techniques. The MS is formulated by implementing the following evolutionary techniques. The best-known algorithms in the class include GA, PSO, SA, TS etc.

2.2 PROBLEM FORMULATION

MS is a sub problem of integral long-term operations planning problem and relates to the utilization and maintenance of the power and energy resources during the period from several weeks in the future to several months or years [4] (1989).

2.2.1 Levelized reserve capacity method

Levelized reserve capacity method gives the result of MS where the net reserve capacity is more or less same throughout the year. The reserve value of any particular week is given by the difference between the total capacity of the system and the load value of that particular week. The reserve capacity in the maintenance time intervals (including the maintenance) is calculated using the Equation (2.1). The total capacity of the generating unit is calculated using the Equation (2.2).

R = Tc – Pt – Abbildung in dieser Leseprobe nicht enthalten (2.1)

where

Xtk – Abbildung in dieser Leseprobe nicht enthalten

Abbildung in dieser Leseprobe nicht enthalten (2.2)

The MS constraints are given by Equations (2.6)–(2.8). The generating unit is to be scheduled consecutively (Time constraint). The maintenance completion constraint ensures that once a unit is removed from the system for maintenance, it completes the maintenance without interruption. Usually, two generating units in the same group cannot be scheduled for maintenance together at the same maintenance time interval (Maintenance crew constraint). In all the maintenance time intervals the net reserve of the power system is greater or equal to zero [7] (1994).

The algorithm for levelized reserve capacity is as follows

Step 1: Get the generator data viz the number of generators, their rated capacity

Step 2: Get the load data for 52 weeks.

Step 3: Calculate the reserve values in the maintenance interval using the Equation (2.1).

Step 4: Arrange the generating units for maintenance in a certain order, i.e. ranging from low to high, according to the generating unit’s capacities.

Step 5: Start from maintenance time interval with the highest reserve capacity and then search on both, right and left of the load curve and schedule the generator using heuristic approach until the time interval is continuously scheduled.

Step 6: Find the next larger capacity generator and repeat step 5 for all the generators to find MS. Check whether all the MS constraints are satisfied using the Equations (2.6)–(2.8). If the obtained MS is feasible, go to the next step and revise the maintenance schedule. Otherwise go to step 5.

Step 7: Terminate the program and print the optimal maintenance schedule.

2.2.2 Levelized reserve rate method

The levelized reserve rate method uses the ratio of the net reserve capacity in each sub-interval over the maximum load in the sub-interval as the objective function. The procedure is the same as in the reserve capacity method. The only difference being that the search for the optimal maintenance position starts not from the sub- interval with the maximum reserve capacity but from the sub-interval with the maximum reserve rate. The objective function of the deterministic levelized reserve rate method is as given in the following Equations (2.3)-(2.5)

Abbildung in dieser Leseprobe nicht enthaltenAbbildung in dieser Leseprobe nicht enthalten (2.3)

The objective of the deterministic levelized reserve rate method is to minimize the sum of squares of the reserves. The MS problem must ensure that a sufficient minimum net reserve exists to provide a secure supply [12] (2000).

Abbildung in dieser Leseprobe nicht enthalten (2.4)

Abbildung in dieser Leseprobe nicht enthalten = Abbildung in dieser Leseprobe nicht enthalten/Abbildung in dieser Leseprobe nicht enthalten (2.5)

The algorithm for levelized reserve rate is as follows.

Step 1: Get the generator data viz the number of generators, their rated capacity.

Step 2: Get the load data for 52 weeks.

Step 3: Calculate the reserve rate in the maintenance interval using the Equation (2.4).

Step 4: Arrange the generating units for maintenance in a certain order, i.e. ranging from low to high, according to the generating unit’s reserve rate. Calculate the minimum annual supply reserve ratio deviation using the Equation (2.3).

Step 5: Start from maintenance time interval with the highest reserve rate and then search on both, right and left of the load curve and schedule the generator using heuristic approach until time interval is continuously scheduled.

Step 6: Find the next larger capacity generator and repeat step 5 for all the generators to find MS. Check whether all the MS constraints are satisfied using Equations (2.6)–(2.8). If the obtained MS is feasible, go to the next step and revise the maintenance schedule. Otherwise go to step 5.

Step 7: Terminate the program and print the optimal maintenance schedule.

2.3 MAINTENANCE SCHEDULING CONSTRAINTS

Certain constraints should be set up according to the real conditions of the power system in order to make maintenance schedule feasible. The resource constraint relates to the maximum resources that can be provided for certain maintenance, e.g. test machinery, necessary spare parts and materials etc. As a problem is subject to a variety of constraints, the problem of MS also has its share of important constraints which are listed below.

2.3.1 Time constraint

Generators must be scheduled at certain intervals. In addition to this, one more consideration is the continuity of the maintenance activity. Maintenance once started must be completed in continuous maintenance intervals.

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten (2.6)

2.3.2 Maintenance crew constraint

Normally, two generating units cannot be scheduled for maintenance together in the same power plant and at the same time, i.e. Abbildung in dieser Leseprobe nicht enthalten. Only a few power plants with considerable maintenance resources can allow Abbildung in dieser Leseprobe nicht enthalten. However, the following constraint is also met.

Abbildung in dieser Leseprobe nicht enthalten (2.7)

2.3.3 Reserve constraint

At any maintenance interval, the total capacity of the generating units should be greater than the predicted load, i.e.

Abbildung in dieser Leseprobe nicht enthalten (2.8)

The Equations (2.6)–(2.8) represent the important maintenance constraints which are considered in this research.

2.4 PSO BASED APPROACH TO MAiNTENANCE SCHEDULING

In this section the PSO is utilized to solve the levelized reserve capacity and the reserve rate based MS. The IEEE RTS case study is presented for the PSO based MS method.

2.4.1 Introduction

In this part, the focus is on the application of PSO concept to solve a MS problem considering the reliability objective function that has various constraints. The objective functions for levelized reserve capacity and reserve rate MS and constraints are outlined in section 2.2.1, 2.2.2 and 2.3 respectively.

2.4.2 Overview of PSO

Particle Swarm Optimization is a population based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy in the year 1995 after getting inspiration from the social behaviour of bird flocking or fish schooling [8] (1995). Basically, the system is initialized with a population of random solutions and a search is performed for the optimum solution by updating the generations. Let X be the ‘particle’ co-ordinates (position) and V its speed (velocity) in a search space. Consider i as a particle in the total population (swam). Now the ith particle position can be represented as X1= (Xi1, Xi2, Xi3.XiN) in the N -dimensional space. The previous best position of the i th particle is stored and represented as pbesti= (pbesti1, pbesti2, pbestij) . All the pbest are evaluated by using a fitness function, which differs for different problems. The velocity of the i th particle is represented as Vi= (Vi1, Vi2…..Vij) . The modified velocity and position update of each particle is calculated using the following Equations (2.9)-(2.10).

Abbildung in dieser Leseprobe nicht enthalten (2.9)

Abbildung in dieser Leseprobe nicht enthalten (2.10)

A suitable selection of the W can provide a balance between global and local exploration abilities and thus requires less iteration on average to find the optimum. W can be calculated according to the following Equation (2.11).

Abbildung in dieser Leseprobe nicht enthalten (2.11)

The positions of the particles are updated using the Equation (2.10). The following procedure is used for implementing the standard PSO algorithm. For each particle, the particle parameters including both position and velocity are first initialized. Then its fitness value is calculated according to the fitness measure pre-specified. If the position is superior with respect to the best position found so far, the current value is set as the new pbest.

The particle with the best fitness value of all the particles is chosen as the gbest. Then, the particle velocity and the particle position are updated according to certain rules. Finally, the stopping criteria such as maximum iterations or minimum error are checked to see if the algorithm should halt or the above process should be repeated until the termination criteria are satisfied.

2.4.3 Development of the proposed PSO based MS algorithm

The maximum reserve capacity and reserve rate are taken as control variables in the PSO based levelized reserve capacity and reserve rate MS. The particles are generated by randomly selecting a value with uniform probability over the search space between the maximum and minimum reserve rate values in the maintenance time intervals [ xmin, xmax ]. The ith particle position of ‘ R ’ is represented as follows for the levelized reserve capacity method:Abbildung in dieser Leseprobe nicht enthalten = (Abbildung in dieser Leseprobe nicht enthalten,Abbildung in dieser Leseprobe nicht enthalten,Abbildung in dieser Leseprobe nicht enthalten, ,Abbildung in dieser Leseprobe nicht enthalten). Similarly for the levelized reserve rate method the ith particle position of ‘ Abbildung in dieser Leseprobe nicht enthalten’ is represented as follows:Abbildung in dieser Leseprobe nicht enthalten = (Abbildung in dieser Leseprobe nicht enthalten, Abbildung in dieser Leseprobe nicht enthalten,Abbildung in dieser Leseprobe nicht enthalten, , Abbildung in dieser Leseprobe nicht enthalten).

The random initial velocity of all the particles are generated over the search space [ Vmin, Vmax ] . Maximum velocity of a particular dimension is given by the following Equation.

Vk max= (Xmax-Xmin) /Na (2.12)

The evaluated fitness values are considered as pbest values for each particle, gbest value is identified and corresponds to the particle shown by for the levelized reserve capacity method Abbildung in dieser Leseprobe nicht enthalten(0) = (Abbildung in dieser Leseprobe nicht enthalten t (0), .…,Abbildung in dieser Leseprobe nicht enthalten N (0)) , where Abbildung in dieser Leseprobe nicht enthalten(0) is the initial particle position. Likewise for the levelized reserve rate method Abbildung in dieser Leseprobe nicht enthalten (0) = (Abbildung in dieser Leseprobe nicht enthalten t (0), .…,Abbildung in dieser Leseprobe nicht enthalten N (0)) , where Abbildung in dieser Leseprobe nicht enthalten i (0) is the initial particle position. The velocity is updated using the gbest and the pbest of each particle, the ith particle’s velocities in the j th dimensions are updated.

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