A Thesis Entitled Nature Inspired Discrete Integer Cuckoo Search
A Thesis
entitled
Nature Inspired Discrete Integer Cuckoo Search Algorithm for Optimal Planned
Generator Maintenance Scheduling
by
Srinivasan Lakshminarayanan
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in
Engineering
______Dr. Devinder Kaur, Committee Chair
______Dr. Mansoor Alam, Committee Member
______Dr. Srinivasa Vemuru, Committee Member
______Dr. Patricia R. Komuniecki, Dean College of Graduate Studies
The University of Toledo
August 2015
Copyright 2015, Srinivasan Lakshminarayanan
This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of
Nature Inspired Discrete Integer Cuckoo Search Algorithm for Optimal Planned Generator Maintenance Scheduling
by
Srinivasan Lakshminarayanan
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science Degree in Engineering
The University of Toledo
August 2015
In this thesis, Discrete Integer Cuckoo Search Optimization Algorithm (DICS) is proposed for generating an Optimal Maintenance Schedule for power utility with multiple generator units and complex constraints of Man Power Availability, Load
Demand and strict Maintenance Window. The objective is to maximize the levelness of the Reserve Power over the entire planning period while satisfying the multiple constraints. This is an NP hard problem and there is no unique solution available for it.
Nature inspired Cuckoo Search algorithm has been chosen to address this problem.
Cuckoo search algorithm is a metaheuristic algorithm based on the obligate brood parasitism of cuckoo bird species, where cuckoo tries to find the best nest of other birds whose eggs resemble her own to lay her eggs to be hatched by other birds. Therefore the problem is formulated to find the best host nest. The host nest is defined according to the constraints of the power utility.
The algorithm was tested on two test systems, one with 21 generator units and the other with 9 generator units which is called IEEE RTS test system. The results obtained with the DICS on the 21 generator power utility system are compared with the work of iii previous researchers using the same test system and using the five traditional algorithms namely the Genetic Algorithm with Binary Representation (GABR), Genetic Algorithm with Integer Representation (GAIR), Discrete Particle Swarm Optimization (DPSO),
Modified Discrete Particle Swarm Optimization (MDPSO) and Hybrid Scatter Genetic
Algorithm (HSGA). The results obtained by applying DICS on the IEEE RTS test system are compared with HSGA algorithm. The results show that DICS outperformed all the other algorithms in the two test systems.
iv
Dedicated to Shree Mahaperiyava!
Acknowledgements
I offer my reverence to the Almighty for showering his abundant grace on me which made all things possible.
I would like to offer my sincere and special thanks to my advisor Dr. Devinder
Kaur for her encouragement, wise, patient and thorough guidance and support. It was a great experience to work under her.
I would like to thank Dr. Mansoor Alam for his kind and wise guidance throughout my Master's Program.
I thank Dr. Srinivasa Vemuru for his gracious support and encouragement.
I would also like to thank Dr. Devinder Kaur, Dr. Mansoor Alam, Mr. William
McCreary and EECS Department for the financial support.
I thank my parents, grandmother, sister, twin brother and friends for their love, support and encouragement in all my endeavors. I would like to specially thank my twin brother for being my constant companion and inspiring me always.
v
Table of Contents
Abstract ...... iii
Acknowledgements ...... v
Table of Contents ...... vi
List of Tables ...... viii
List of Figures ...... x
List of Abbreviations ...... xii
1 Introduction ...... 1
2 Problem Formulation and Test Systems ...... 5
2.1 Problem Formulation ...... 5
2.2 21-Generator Power Utility Test System ...... 11
2.3 IEEE RTS Test System ...... 13
3 Cuckoo Search ...... 16
4 Problem Encoding and Algorithm Steps of DICS ...... 18
4.1 Initialization of Population ...... 19
4.2 Computation of Fitness ...... 20
4.3 Discrete Lévy/Global Random Flight ...... 21
4.4 Revert Flight ...... 24
4.5 Discrete Local Walk ...... 24
4.6 Special Cases of the Local Flight...... 26 vi
5 Results and Discussion………………………… ...... 28
5.1 21 generator power utility test system……… ...... …………………28
5.1.1 Results ...... 28
5.1.2 Comparison ...... 33
5.2 IEEE RTS test system ...... 41
5.2.1 Results...... 41
5.2.2 Comparison ...... 46
6 Conclusion and Future Work ...... 49
References ...... 51
vii
List of Tables
2.1 The Weight Coefficients ...... 10
2.2 The 21-Generator Power Utility Test Dataset ...... 11
2.3 The IEEE RTS Test System Test Dataset ...... 13
2.4 Weekly Load Demand on the IEEE RTS Test System ...... 14
5.1 Generators in maintenance during each week for
the fittest host nest for 21 generator system...... 29
5.2 Power Generated, Reserve Power and Man Power Required during
weeks (1-26) for the fittest host for 21 generator system...... 30
5.3 Power Generated, Reserve Power and Man Power Required during
weeks (27-52) for the fittest host for 21 generator system ...... 31
5.4 The Maintenance Schedule arrays from the six algorithms...... 34
5.5 Pgenerated (t) during weeks (1-26) for the six algorithms for 21 generator
system……………………………………………………………………………35
5.6 Pgenerated (t) during weeks (27-52) for the six algorithms for 21 generator
system……………………………………………………………………………36
5.7 TMPRt during weeks (1-26) for the six algorithms for 21 generator
system……………………………………………………………………………38
5.8 TMPRt during weeks (27-52) for the six algorithms for 21 generator
system……………………………………………………………………………39 viii
5.9 Comparison of the result of the six algorithms for 21generator
system ...... 41
5.10 Generators in maintenance during each week for
the fittest host for IEEE RTS system ...... 42
5.10 Pgenerated (t) during each week for the fittest host nest for IEEE RTS system ...... 43
5.11 Reserve (surplus) power during each week for the
fittest host nest for IEEE RTS system...... 44
5.12 Maintenance Schedule Arrays from HSGA and DICS algorithms
for the IEEE RTS test system...... 46
5.13 Comparison of SSR for the Schedules from HSGA and DICS algorithms
for the IEEE RTS test system...... 47
ix
List of Figures
2-1 Maintenance Schedule Array ...... 6
3-1 Egg Mimicry by Cuckoo ...... 16
4-1 Solution xi in the form of integer array ...... 18
4-2 Initial host nest locations of the 'n' Cuckoos ...... 19
4-3 Addition of step-size of 0.7 to xi in Figure 4-1 resulting in real numbers in the
(new) new host-nest xi ...... 23
4-4 The real numbers are rounded to their nearest integers to make the host-nest an
integer array ...... 23
4-5 Flow Chart Summarizing the Steps in DICS ...... 27
5-1 The fittest host nest (Optimal Maintenance Schedule Array) obtained by
DICS for the 21 generator unit system ...... 28
5-2 Power Generated during each week (Pgenerated (t)) for the
fittest host for 21 generator system ...... 32
5-3 Total Man Power Required during each week (TMPRt) for the
fittest host for 21 generator system ...... 32
5-4 Reserve Power during each week for the Schedule the fittest host nest for 21
generator system…………………………………………………………………33
5-5 Pgenerated (t) during each week for the six algorithms
for 21 generator system...... 34 x
5-6 TMPRt during each week for the six algorithms
for 21 generator system...... 37
5-7 SSR computed for the six algorithms for 21 generator system ...... 40
5-8 The fittest host nest (Optimal Maintenance Schedule array) obtained by DICS for
the IEEE RTS system...... 41
5-9 Power Generated during each week (Pgenerated (t)) for the IEEE RTS test system
generated from the fittest host nest ...... 45
5-10 Reserve Power for the IEEE RTS test system generated from
the fittest host nest ...... 45
5-11 Power Generated during each week (Pgenerated (t)) for the Maintenance Schedules
obtained from DICS and HSGA algorithm for IEEE RTS test system...... 47
xi
List of Abbreviations
DICS ...... Discrete Integer Cuckoo Search DPSO ...... Discrete Particle Swarm Optimization GABR ...... Genetic Algorithm with Binary Representation GAIR ...... Genetic Algorithm with Integer Representation GMS ...... Generator Maintenance Scheduling HSGA ...... Hybrid Scatter Genetic Algorithm MDPSO ...... Modified Discrete Particle Swarm Optimization PSO ...... Particle Swarm Optimization
xii
Chapter 1
Introduction
It is essential for the generators of power system to undergo maintenance periodically during the time in which they will be turned off. The Generator maintenance scheduling (GMS) is a complex NP-hard combinatorial optimization problem with a nonlinear objective and constraint functions [1-3]. The goal is to find an optimal generator maintenance schedule of the power utility system which yields maximum leveling of Reserve Power over the entire planning period while satisfying all the constraints. The leveling of the Reserve Power across maintenance schedules enhances the reliable operation of the utility to meet the unexpected fluctuations in load demand
[1].
Early approaches to solve the GMS problem include branch and bound technique, integer programming and dynamic programming [4-7]. K. P. Dahal and J. R.
McDonald have discussed the severe drawbacks of these approaches such as inability to handle increased dimensionality and poor performance when the objective function is non-linear [3].
With the advent of Artificial Intelligence, various meta-heuristics algorithms were proposed for solving the GMS problem [1-3, 8-11]. Simulated annealing optimization 1
method with mixed integer representation was proposed by Satoh, T., and K. Nara for the thermal GMS problem [8]. However this method suffered the drawback of long computational time of about 21 hours for large systems [3]. Huang, C. J., C. E. Lin, and
C. L. Huang put forward a Fuzzy Approach for the GMS [9]. Though the fuzzy approach is flexible, it suffered from the drawback of the poor performance pertaining to traditional heuristic methods [3].
Genetic Algorithm with integer representation (GAIR) was brought forth to address the GMS problem in [1]. This approach unlike the earlier methods did not suffer the limitation of dimensionality and poor performance with non-linear objective function.
This was an effective approach to find a good solution to the complex GMS problem in short computation time though it did not guarantee to find an optimal solution [1]. A fuzzy evaluation function with the Genetic Algorithm was proposed for the GMS problem with reliability based objective and flexible Man Power Availability constraint
[9]. The fuzzy evaluation function provided a trade off in Man Power Availability to generate a better reliable maintenance schedule [9].
Swarm Intelligence based algorithms such as Particle Swarm Optimization (PSO) and Ant Colony Optimization are meta-heuristic algorithms which are inspired from the collective intelligence of living species such as flocks of birds and schools of fish.
Optimal GMS using Modified Discrete Particle Swarm Optimization (MDPSO), a variant of PSO combined with Evolutionary Strategy (ES), was proposed by Yare, Yusuf,
Ganesh K. Venayagamoorthy, and U. O. Aliyu [10]. The performance of MDPSO was superior to Genetic Algorithm with Binary Representation (GABR) and Discrete Particle
2
Swarm Optimization (DPSO). Recently, GMS using Hybrid Scatter-Genetic Algorithm
(HSGA) was proposed which generated a better result than MDPSO [11].
In this paper, the Discrete Integer Cuckoo Search (DICS) algorithm is presented to address the GMS problem. Cuckoo Search is a Swarm Intelligence based Meta- heuristic optimization algorithm inspired by the brood parasitic behavior observed in the cuckoo bird species [12]. Cuckoo Search has been shown to be superior to other algorithms such as GA and PSO in finding a global optimum of many numeric test functions [12]. Cuckoo Search has also been applied to many structural designing optimization problems such as design of wind turbines, vehicular components and welding structures [13-17]. Cuckoo Search has also been proposed for machine learning techniques like Feed Forward Neural Network Training and Document Clustering [18,
19].
Cuckoo Search has also been applied to classic combinatorial problems [20, 21].
Discrete Binary version of Cuckoo Search and quantum inspired Cuckoo Search were proposed for solving the classical 0-1 knapsack problem [20, 21].
Cuckoo Search algorithm has fewer control parameters to fine tune than other algorithms. Moreover, tweaking of control parameters for a particular problem is not needed [12]. Hence Cuckoo search is more generic and robust for many NP-hard optimization problems [12].
This thesis is organized as follows: Chapter 2 presents the mathematical formulation of the problem statement. Two standard test utility systems, one with 21 generator units and the other with 9 generator units are discussed. These two systems are used to assess the performance of the GMS algorithms [10, 11]. Chapter 3 introduces the 3
Cuckoo Search. Chapter 4 presents the mapping of Discrete Integer Cuckoo Search
(DICS) to the GMS problem. In Chapter 5, DICS is applied to the two test systems and the results are compared with the previous work. Chapter 6 presents the conclusion and future work.
4
Chapter 2
Problem Formulation and Test Systems
In this chapter, mathematical formulation of the problem statement is presented.
The two power utility test systems, one with 21 generator units and the other with 9 generator units are then discussed.
2.1 Formulation of Problem Statement
The objective of levelling the reserve power can be achieved by minimizing the
Sum of Squares of Reserve Power (SSR) over the entire maintenance period [22, 23].
The sum of the squares of the reserve power for a generating system is defined by:
2 SSR P L P M (2.1) total t i it t T i
where Ptotal is the total installed capacity which is the sum of the capacity of all the generator units as defined in (2)
Ptotal Pi (2.2) iI
5
'i' is the index to denote the generator unit i,and ‘I’ is total number of generator units, hence Pi denotes the capacity of generator unit i.
‘t’ is the index to denote the time period t, and 'T' is the total number of time periods.
Lt is the Load Demand during time period ‘t’
Mit is an element of the Maintenance Indicator Matrix 'M' of dimension [IxT].
The maintenance indicator matrix provides the maintenance status of each generator unit during each time period. Hence, Mit gives the maintenance status of generator unit i in time period 't'. Mit can be,
1 if unit 'i' is in maintenance during time period ‘t’
Mit = (2.3)
0 otherwise,
If Mit is equal to 1, unit 'i' is in maintenance during time period‘t’, otherwise it is equal to 0.
Figure 2-1 shows the Maintenance Scheduling Array ‘xi’ (i =1, 2, ..., I).
Figure 2-1: Maintenance Schedule Array
6
'xi' provides the start time (ST) of the maintenance of each unit as shown in
Figure 2-1 for a power system with total number of 'I' generator units. The maintenance schedule array should strictly abide by the maintenance constraints defined for the specific problem. The generator units have to be scheduled for maintenance for the complete duration 'duri' defined individually for each unit without interruption. Certain units have to be scheduled for maintenance in the first 26 weeks of the year and the others have to be scheduled during the second half of the year.
The maintenance indicator Matrix 'M' is generated from the maintenance schedule array using the following Pseudo-code 2-1:
Pseudo-code 2-1: Maintenance Indicator Matrix 'M ' Generation
// Initialize M with zeros for i=[1...... I] do
for t=[1.....T] do
Mit = 0
end for end for
// set Mit = 1 whenever there is maintenance for i=[1...... I] do
for t=Xi:duri-1 do
Mit = 1
end for end for
7
In addition to achieving the objective of minimizing the SSR (2.1), the constraints of Manpower availability and load constraints in each period should be satisfied. Man
Power Availability constraint is given by:
TMPRt AMPt t T (2.4
where
TMPR M MPR t T (2.5) t it it iI
TMPRt denotes the Total Man Power Required for maintenance during time period t, AMPt denotes the Available Man Power during the period t. MPRit denotes the
Man Power Required for maintenance of unit i in the time period t. Equation (4) defines that the Total Man Power Required for maintenance in any time period should always be less than or equal to the Available Man Power during that time period. The Load Demand
Constraint is defined by:
Lt Pgenerated (t) t T (2.6)
Equation (2.6) defines that the load demand (Lt) at any time period should be less than or equal to the Power Generated (Pgenerated(t)) during that time period.
Pgenerated(t) can be calculated using (2.7),
Pgenerated(t) Ptotal PDUt t T (2.7)
where
PDUt Pi Mit t T (2.8) i I
8
PDUt denotes the sum of the Power capacity of Down Units i.e. sum of the capacity of the generator units that are undergoing maintenance during the time period 't'.
Any violations in Man Power and Load Constraint should be added as a penalty to the objective function. The Man Power Violation (TMV) can be computed using (2.9) shown below,
TMV TMPRt AMPt (2.9) tT
TMV is computed only for the time periods when the Total Man Power Required for maintenance exceeds the Available Man Power i.e. when TMPRt is greater than AMP.
The Total Load Violation (TLV) is defined by,
TLV Lt Pgenerated(t) (2.10) tT
TLV is computed only for the time periods when the Load Demand exceeds the
Power Generated i.e. when Lt is greater than Pgenerated (t).
Finally the Objective Function (OF) is to minimize the weighted sum of SSR,
TLV and TMV as shown below:
OF min w0 SSR wl TLV wm TMV (2.11)
where w0, wl, and wm are the weight coefficients of SSR, TLV and TMV respectively.
w0 SSR is the weighted SSR.
wl TLV is the penalty for Violation of Man Power Availability constraint.
wm TLV is the penalty for Violation of Load Demand constraint.
9
w0, wl and wm are assigned values such that, (wl and wm) >> w0.
A very high value of wl and wm in comparison with w0 will heavily penalize the solutions which violates the man power and load constraints and will drastically reduce their fitness when compared to the solutions which abide by the constraints. Hence as the iterations in the swarm based cuckoo search algorithm advance more and more solutions emerge which do not violate the constraints.
The following values are assigned for the weight coefficients as displayed in
Table 2.1:
Table 2.1: The Weight Coefficients
w0 w1 wm
-5 10 30 10
In order to compare the performance of DICS algorithm, two standard test system dataset, one 21 generator units and the other with 9 generator units are used. These two systems are the standard test datasets which were used by previous researchers to access the performance of algorithms (GAIR, GABR, DPSO, MDPSO and HSGA) [1, 10, 11].
10
2.2 21-Generator Power Utility Test System
Table 2.2 shows the dataset details regarding the capacity, duration of maintenance and Man Power Required during each successive week of the maintenance for each unit [10].
Table 2.2: The 21-Generator Power Utility Test Dataset [1, 10, 11]
Generator Required Capacity Allowed Man Power Required during each Unit Maintenance (MW) Weeks successive week of maintenance Duration
1 555 7 10 ; 10 ; 5 ; 5 ; 5 ; 5 ; 3 2 180 2 15 ; 15 3 180 1 20 4 640 3 15 ; 15 ; 15 5 640 3 15 ; 15 ; 15 6 276 10 3 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2 ; (1-26) 7 140 4 10 ; 103 ; 5 ; 5 weeks 8 90 1 20 9 76 2 15 ; 15 10 94 4 10 ; 10 ; 10 ; 10 11 39 2 15 ; 15 12 188 2 15 ; 15 13 52 3 10 ; 10 ; 10 14 555 5 10 ; 10 ; 10 ; 5 ; 5 15 640 5 10 ; 10 ; 10 ; 10 ; 10 16 555 6 10 ; 10 ; 10 ; 5 ; 5 ; 5 17 76 3 (27-52) 10 ; 15 ; 15 18 58 1 Weeks 20 19 48 2 15 ; 15 20 137 1 15 21 469 4 10 ; 10 ; 10 ; 10 11
The planning period of the Maintenance Scheduling is 52 weeks. The
Maintenance Schedule must abide the following constraints:
1. Maintenance Window constraint:
The first 13 generator units are allowed to undergo maintenance only
during the first half of the planning period (1-26 weeks).
The remaining generator units are allowed to undergo maintenance
only in the second half of the planning period (27-52 weeks).
Each generator unit should undergo required maintenance in a single
stretch without interruption.
2. Load Demand constraint:
The power system should be able to deliver at-least 5047 MW during
any week. This amount is actually the sum of the Load demand of
4739 MW and 6.5% spinning reserve (6.5% of 4739=308 approx.).
3. Man Power Availability constraint:
The number of staff (Man Power) required during successive week of
maintenance for each unit is shown in column-4 of Table II.
The total number of staff required for maintenance during any week
should not exceed 40.
12
2.3 IEEE RTS Test System
The IEEE RTS test system [11] consists of 9 generator units with total capacity of
1360 MW. Table 2.3 shows the dataset details capacity and number of successive weeks of maintenance required for each unit of the IEEE RTS test system.
Table 2.3: The IEEE RTS Test Dataset [11]
Generator Power Required Unit Capacity Maintenance (MW) Duration 1 12 2
2 20 2
3 50 2
4 76 3
5 100 3
6 155 4
7 197 4
8 350 5
9 400 6
13
The planning period of the Maintenance Scheduling is 52 weeks. Unlike the 21 generator unit test system, here the Load Demand is not constant and it varies during each week. Table 2.4 shows the weekly load demand on the test system.
Table 2.4: Weekly Load Demand on the IEEE RTS Test System [11]
Week Lt Week Lt Number Number 1 862 27 755 2 900 28 816 3 878 29 801 4 834 30 880 5 880 31 722 6 841 32 776 7 832 33 800 8 806 34 729 9 740 35 726 10 737 36 705 11 715 37 780 12 727 38 695 13 707 39 724 14 750 40 724 15 721 41 743 16 800 42 744 17 754 43 800 18 837 44 881 19 870 45 885 20 880 46 909 21 856 47 940 22 811 48 890 23 900 49 942 24 887 50 970 25 896 51 1000 26 861 52 952
14
The Man Power Availability constraint is not defined for this test system. Hence,
Maintenance Scheduling must abide only by the following two constraints:
1. Maintenance Window constraint:
Each generator unit should undergo required maintenance duration specified
in Table 3.3 in a single stretch without interruption.
2. Load Demand constraint:
The power system should be able to atleast the deliver Load Demand during
each week mentioned in Table 2.4.
15
Chapter 3
Cuckoo Search
Cuckoo Search is a nature inspired optimization algorithm based on the cuckoo bird's parasitic breeding behavior of laying its eggs in the nests of different bird species.
The cuckoo bird deposits her eggs in the nest of a host bird whose eggs closely resemble her own eggs and uses the services of host birds to hatch her own eggs. The cuckoo search algorithm simulates the cuckoo bird's intelligent search strategy of finding the best host nest to deposit her egg. Figure 3-1 shows three different kinds of cuckoo eggs deposited in three different nests.
Figure 3-1: Egg Mimicry by Cuckoo [24]
The black arrow marks the cuckoo’s egg in each of the three nests. If the host bird discovers the egg is not her own, she abandons the nest and builds a new one. The closer 16
the resemblance of host egg with the cuckoo’s egg, lesser is the chance of discovery by the host bird and greater is the chance of survival of the cuckoo eggs. We can see from
Fig. 2 that the cuckoo eggs are almost identical with the eggs of host birds. Hence the chance of discovery by the host egg is very low.
The cuckoos make series of flights searching and evaluating one nest after the other until they find the optimal host nest to spawn their eggs. The series of flights the cuckoos make can be classified into two types namely Lévy/Global Flight and Local
Flight. The Lévy Flight is based on the Lévy Probability distribution. Many animals and insects forage using the Lévy flight for finding their prey/food location [12]. During the
Local Flight, the cuckoo examines around the host nests which the other cuckoos are using, and finds the fittest nest among them and computes the distance and direction of her Local flight based on that. The mathematical implementation of these two flights will be presented in greater detail in sections 4.3 and 4.5 of Chapter 4.
17
Chapter 4
Problem Encoding and Algorithm Steps of DICS
In this section the mapping of the DICS algorithm to the GMS problem and the algorithm steps are explained in detail. The cuckoo's host nest will represent a potential maintenance schedule array xi. Hence xi is encoded as an integer array with dimension
[1xI], where 'I' denotes the total number of generator units to be scheduled in the GMS problem. Figure 4-1shows an integer array with dimension [1x21] representing a potential maintenance schedule for a 21 unit power system.
1 7 8 22 19 9 9 14 24 13 3 25 15 39 35 28 27 44 45 51 48
Figure 4-1 Solution xi in the form of integer array
Each of the 21 integers in the array represents the starting week of maintenance of generators 1, 2, 3...., 21 respectively. The first integer ‘1’ represents that the generator 1 starts its maintenance in 1st week. Similarly, the successive 2nd to 21st integers in the array represents the starting week of maintenance of units 2 to 21. The algorithm is implemented with ‘n’ number of cuckoos and ‘n’ host nests. Each cuckoo is initially at her respective host nest ‘xi’ (i =1, 2, ..., n) as shown below in Figure 4-2. 18
Cuckoo 1 is at host-nest x1
Cuckoo 2 is at host-nest x2
Cuckoo i is at host-nest x3
Cuckoo n is at host-nest xn
Figure 4-2: Initial host nest locations of the 'n' Cuckoos
The ‘n’ cuckoos work in parallel evolving these n nests by making series of
Global and Local flights finding and evaluating new hosts nests until finally finding the fittest host-nest to lay their eggs. The fittest host-nest found will represent the optimal
GMS solution.
Various steps involved in the DICS are as follows:
4.1 Initialization of Population
Each host nest xi is initialized as follows
xij round LL j U (0,1) * UL j LL j (4.1)
th where, xij is the j dimension of the ith host nest . LLj and ULj are the lower and upper bounds for the jth integer i.e. jth generator unit.
19
LLj and ULj will define the range when each unit can be scheduled for maintenance. The lower limit will determine the earliest week the unit can be down for maintenance and the upper limit will determine the latest week the unit can start its maintenance schedule. U(0, 1) is a uniformly distributed random number in the range [0, 1]. Since the host nests are to be initialized as integer arrays, Xij is rounded as to the nearest integer shown in the initialization equation (4.1) above.
The pseudo-code for the nest Initialization is:
Pseudo-code 4-1: Nest Initialization
for i=[1...... n] do
for j=[1.....D] do
xij round LL j U (0,1) * UL j LL j
end for
end for
4.2 Computation of fitness
The cuckoos compute the fitness of their initial host-nests. The fitness of each host-nest 'xi' (Maintenance Schedule) is computed using the objective function defined in equation. (2.11) discussed before in the Chapter 2. A Maintenance schedule (host nest) with lesser SSR will be fitter than the host-nest with high SSR.
20
4.3 Discrete Lévy/Global Flight
The cuckoos then start their search process of finding even fitter host-nests. Each
new cuckoo ‘i’ makes a flight from their initial nest ‘xi’ to new host nest ‘xi ’ by making the
Discrete Lévy/Global Flight equation as shown below.
new xij round [xij L( )] (4.2)
The Pseudo-code for the Discrete Lévy/Global Flight is:
Pseudo-code 4-2: Discrete Lévy/Global Flight for i=[1...... n] do
for j=[1.....D] do
new xij round [xij s L( )]
end end for
th Where xij is the j dimension of the initial host nest xi,
α*L(β) is the length of flight or the step-size which the cuckoo ‘i’ makes to reach
new the new host nest ‘xij ’,
L(β) is called the Lévy probability distribution, named after the mathematician
Paul Lévy [26]. Various researches suggest that the foraging strategy of many animals and insects including the hunting strategy in nomadic hunter-gatherer society uses a search path resembling Lévy distribution [12, 25].
21
The Lévy distribution function L(β) is given by,
U L( ) (4.3) V
where,
‘β’ is a constant parameter whose value can be taken from the interval [1, 2].
β = 1.5 is used in this work. U and V are drawn from normal distributions defined by:
2 U ~ N(0,u ) (4.4)
2 V ~ N(0,v ) (4.5)
Where
0 is the mean of the normal distribution for both U and V
2 2 ϭu and ϭv are the standard deviation for U and V given by,
1 (1 ) sin( ) 2 u (4.6) ( 1) 1 2 * 2 2
v 1 (4.7)
where, Ґ denotes gamma function.
‘α’ is the step-size scaling factor [25]. The step-size scaling factor is determined based on the problem at hand. For continuous domain problems, the step-size scaling factor is usually kept very small (~0.01) so that solutions do not jump outside of the 22
design domain [25]. However, for discrete problems such as GMS which is a discrete combinatorial NP-hard problem, the scaling factor is kept higher than the continuous domain, because larger step size is needed to explore the big search space. Also, if the scaling factor is small, when we are rounding the levy flight, the cuckoo may not update to new host nest location. Hence, α is chosen to be 0.4.
(new) The new host nest xi for each cuckoo should also always be an integer array.
However, Levy flight can result in new host nests with real numbers. For example, the addition of the random step-sizes by levy flight to xi in Figure 4.1 will result in real numbers in the new host nest as shown in Figure 4-3.
1. 3 7.8 8.2 22.4 19.3 9.3 9.6 14.4 24.6 13.2 3.8 25.1
15.9 39.5 35.2 28.5 27.2 44.5 45.3 51.9 48.1
Figure 4-3: Addition of random lev to xi in Figure 4-1 resulting in real numbers in the
(new) new host-nest xi
Hence, the real numbers are rounded to their nearest integer to make the new host-nest an integer array as shown in Figure 4-4.
1 2 2 1 8 8 22 19 9 14 13 4 16 40 35 29 27 45 45 52 48 0 5 5
Figure 4-4: The real numbers are rounded to their nearest integers to make the host-nest
an integer array
After moving to the new host-nests, each cuckoo computes the fitness of her new host nest as discussed before in section 4.2.
23
4.4 Revert Flight
After all the cuckoo's move to their new host nests each cuckoo checks if their new host nest xi new is fitter than the previous one for laying her eggs. If it is not then the cuckoo returns backs to her previous host nest xi by using the following equation,
new xij xij (4.8)
The pseudo code of this revert flight process is:
Pseudo-code 4-3: Revert Flight
for i=[1...... n] do
new if fitness(xi ) < fitness(xi ) then
for j=[1.....D] do
new xij =xij
end for
end if
end for
4.5 Discrete Local Walk
After the discrete global walk, some of the cuckoos make another local flight to see if they can find even better host nest in the neighborhood.
To implement this flight two random neighboring host nests are picked and the local flight is mimicked by
new x round xij s H (P ) (xr j xr j ) (4.9) ij a 1 2
24
The Pseudo-code for the Local Flight is:
Pseudo-code 5-4: Discrete Local Walk
for i=[1...... n] do
for j=[1.....D] do
new round x s H( ) (x x ) xij ij Pa r1 j r2 j
end
end for where,
th xr1j and xr2j are the j dimension of the two randomly selected host nests xr1 and xr2. The cuckoo makes a decision whether to make the local flight or not using the switching parameter Pa is randomly chosen from the interval [0, 1].
H(Pa-€) is the heavy side function. € is the random number chosen from the uniform distribution (0, 1).
If Pa ≥ € , H(Pa-€) = 1, the cuckoo makes a local walk.
If Pa ≥ € , H(Pa-€) = 0, the cuckoo does not to make the local walk.
‘s’ is the random step-size for the local walk drawn from the uniform distribution in the interval [0, 1].
The cuckoos which made the local flight, again evaluate the fitness of their new host nest, to see if the fitness has improved. If there is no improvement in the fitness, they revert back to the previous nest using equation (4.8) discussed before in section 4.3.
25
This way the ‘n’ cuckoos work in parallel making series of lévy/global and local flights discovering new host nests iteratively again and again until they find their fittest host nests.
4.6 Special Cases of the Local Flight
If we group replace the factor 's *H(Pa-€)' in local flight equation. (4.9) and with W €(0, 2), then the Local Flight becomes,
new x roundxij W (xr j xr j ) ij 1 2 (4.10)
The above equation is an important update equation in Differential Evolution
[25, 27].
Furthermore if we replace the random-host nest xr1 with current global best host nest 'gbest' (fittest nest among the 'n' host nests) and xr2 by current host-nest xi in (4.10) the Local Flight becomes,
new xij round xij W (gbest j xij ) (4.11)
The equation (4.11) is nothing but the particle swarm update equation without past best factor which is called accelerated particle swarm optimization (APSO) [25].
We can hence term Local Flight as an intelligent integration of DE and PSO [25].
26
Figure 4-5 shows the flow chart summarizing the steps of DICS.
Figure 4-5: Flow Chart Summarizing the Steps in DICS 27
Chapter 5
Results and Discussion
This chapter consists of two sections: Section 5.1 and Section 5.2. In section 5.1, the results obtained by applying DICS on the 21-generator unit system is presented. The performance of DICS on this test system is then compared with the five traditional algorithms (GA, DPSO, MDPSO and HSGA) [1, 10, 11]. In section 5.2, the results obtained by applying DICS on the 9 generator unit system is presented. The performance of DICS on this test system is then compared with HSGA [11]. For both the test cases, the DICS was applied with the following parameters settings:
1. Total number of nests n = 25;
2. Lévy Flight Step-size Scaling Factor α = 0.4;
3. Maximum number of iterations (Termination Criteria): 5000.
5.1 21-generator test system
5.1.1 Results
20 7 19 1 4 9 7 11 16 12 7 18 11 46 41 28 34 27 51 52 37
Figure 5-1: The fittest host nest (Optimal Maintenance Schedule array) obtained by DICS
for the 21 generator unit system
28
Figure 5-1 shows the fittest host-nest obtained using DICS for the 21 generator test system. The host-nest represents the Optimal Maintenance Schedule Array for the test system. As mentioned before in Chapter 4, each of the 21 integers in this array represents the starting week of maintenance of units 1, 2, 3, ...., 21 respectively. Table 5.1 lists the generators in maintenance during each week for the schedule represented by the fittest host nest.
Table 5.1: Generators in maintenance during each week for the fittest host nest for 21
generator system
Week Generators in Week Generator Units Number Maintenance Number In Maintenance 1 4 27 18 2 4 28 16 3 4 29 16 4 5 30 16 5 5 31 16 6 5 32 16 7 2 7 11 33 16 8 2 7 11 34 17 9 6 7 35 17 10 6 7 36 17 11 6 8 13 37 21 12 6 10 13 38 21 13 6 10 13 39 21 14 6 10 40 21 15 6 10 41 15 16 6 9 42 15 17 6 9 43 15 18 6 12 44 15 19 3 12 45 15 20 1 46 14 21 1 47 14 22 1 48 14 23 1 49 14 24 1 50 14 25 1 51 19 26 1 52 19 20
29
Table 5.2 shows the Power Generated, Reserve Power and Man Power Required during weeks 1-26 for the fittest host nest.
Table 5.2: Power Generated, Reserve Power and Man Power Required during weeks 1-26
for the fittest host for 21 generator system
Week Number Power Generated Reserve Power Man Power Required Available
1 5048 1 15 2 5048 1 15 3 5048 1 15 4 5048 1 15
5 5048 1 15 6 5048 1 15 7 5329 282 40
8 5329 282 40 9 5272 225 8 10 5272 225 7
11 5270 223 32 12 5266 219 22 13 5266 219 22 14 5318 271 12 15 5318 271 12 16 5336 289 17 17 5336 289 17 18 5224 177 18
19 5320 273 35 20 5133 86 10 21 5133 86 10
22 5133 86 5 23 5133 86 5 24 5133 86 5 25 5133 86 5 26 5133 86 3
30
Table 5.2 shows the Power Generated, Reserve Power and Man Power Required during weeks 27-52 for the fittest host.
Table 5.2: Power Generated, Reserve Power and Man Power Required during weeks 27-
52 for the fittest host for 21 generator system
Week Number Power Generated Reserve Power Man Power Required Available
27 5630 583 20 28 5133 86 10 29 5133 86 10
30 5133 86 10 31 5133 86 5 32 5133 86 5
33 5133 86 5 34 5612 565 10 35 5612 565 15 36 5612 565 15 37 5219 172 10
38 5219 172 10 39 5219 172 10 40 5219 172 10
41 5048 1 10 42 5048 1 10 43 5048 1 10 44 5048 1 10 45 5048 1 10
46 5133 86 10 47 5133 86 10 48 5133 86 10
49 5133 86 5 50 5133 86 5 51 5640 593 15 52 5503 456 30
31
Figure 5-2 shows the plot of the Power Generated during each week (Pgenerated (t)) for the fittest host nest obtained.
Figure 5-2: Power Generated during each week (Pgenerated (t)) for the fittest host
nest for 21 generator system
It can be seen that Pgenerated (t) is always greater than the demand of 5047 MW.
Hence the best host nest solution satisfies the Load Constraint. Figure 5-3 shows the plot
of the Total Man Power Required during each week (TMPRt) for the fittest host nest
Figure 5-3: Total Man Power Required during each week (TMPRt) for the fittest host nest for 21 generator system 32
From Figure 5-3, it can be seen that TMPRt is always lesser than the AMP of 40.
Therefore the Man Power Constraint is also satisfied.
Figure 5-4 shows the Reserve Power available during each week for the fittest host nest.
Figure 5-4: Reserve Power Available during each week for the fittest host nest for 21 generator system From Figure 5-4, we can see the distribution of the Reserve Power throughout the
Planning Period.
The objective of the paper is to find an optimal Maintenance Schedule which yields a minimum SSR which in turn implies greater leveling of reserve power over the entire planning period. The SSR was computed using (2.1) described in Chapter 2. The value of SSR obtained is 2934355 (MW)2.
5.1.2 Comparison
The performance of DICS on 21-unit test system is compared with previous work which addressed the same 21 unit GMS problem using five other bio-inspired algorithms, namely, GAIR [1], GABR [10], DPSO [10], MDPSO [10] and HSGA [11].
33
Table 5.4 shows the comparative Maintenance Schedule Arrays for the 21-unit system generated from the five algorithms by the previous researchers [1, 10, 11] and the maintenance schedule obtained by the DICS is highlighted.
Table 5.4: The Maintenance Schedule arrays from the six algorithms.
Algorit hm Maintenance Schedule Array
GAIR [1] 6 24 26 13 2 16 18 1 9 5 11 16 21 27 48 33 39 42 31 47 43
GABR [10] 1 8 25 15 14 11 1 24 22 22 23 15 1 27 39 31 47 31 45 45 49
DPSO [10] 20 17 7 4 1 7 14 19 11 8 12 18 8 48 41 27 46 28 39 34 35
MDPSO [10] 17 2 1 24 14 4 8 13 5 9 12 1 5 38 44 28 35 52 34 27 49
HSGA [11] 1 23 11 8 12 17 19 11 15 15 18 25 15 47 42 27 33 52 35 41 37
DICS 20 7 19 1 4 9 7 11 16 12 7 18 11 46 41 28 34 27 51 52 37
Figure 5-5 shows the Power Generated during each week (Pgenerated (t)) for each Maintenance Schedule. From Figure 5-5, we can see that for GAIR and GABR, the
Power Generated does not meet the Load Demand of 5047 MW for some of the weeks i.e. Load Demand constraint is violated.
Figure 5-5: Pgenerated (t) during each week for the six algorithms for 21 generator system
34
Table 5.5 lists the Pgenerated (t) for each Maintenance Schedule for weeks (1-26).
Table 5.5: Pgenerated (t) during weeks (1-26) for the six algorithms for 21 generator
system
Week Pgeneratedt Number GAIR GABR DPSO MDPSO HSGA DICS
1 5598 4941 5048 5320 5133 5048 2 5048 4941 5048 5320 5133 5048 3 5048 4941 5048 5508 5133 5048 4 5048 4993 5048 5412 5133 5048 5 5594 5133 5048 5284 5133 5048 6 5039 5133 5048 5284 5133 5048 7 5039 5133 5232 5360 5133 5329 8 5039 5508 5266 5272 5048 5329 9 5057 5508 5266 5178 5048 5272 10 5057 5688 5266 5178 5048 5272 11 5094 5412 5242 5178 5418 5270 12 5094 5412 5297 5279 5048 5266 13 5048 5412 5373 5283 5048 5266 14 5048 4772 5272 5048 5048 5318 15 5048 3944 5272 5048 5466 5318 16 5224 3944 5272 5048 5466 5336 17 5224 4772 5368 5133 5266 5336 18 5272 5412 5320 5133 5279 5224 19 5272 5412 5410 5133 5233 5320 20 5272 5412 5133 5133 5272 5133 21 5220 5688 5133 5133 5272 5133 22 5360 5518 5133 5133 5272 5133 23 5360 5479 5133 5133 5232 5133 24 5232 5465 5133 5048 5232 5133 25 5232 5414 5133 5048 5224 5133 26 5508 5688 5133 5048 5224 5133
35
Table 5.6 lists the Pgenerated (t) for each Maintenance Schedule for weeks (27-52).
Table 5.6: Pgenerated (t) during weeks (27-52) for the six algorithms for 21 generator system
Week Pgenerated (t) Number GAIR GABR DPSO MDPSO HSGA DICS
27 5133 5133 5133 5551 5133 5630 28 5133 5133 5075 5133 5133 5133 29 5133 5133 5133 5133 5133 5133 30 5133 5133 5133 5133 5133 5133 31 5085 4520 5133 5133 5133 5133 32 5640 5133 5133 5133 5133 5133 33 5133 5133 5688 5133 5612 5133 34 5133 5133 5551 5640 5612 5612 35 5133 5133 5219 5564 5564 5612 36 5133 5133 5219 5612 5640 5612 37 5133 5688 5219 5612 5219 5219 38 5133 5688 5219 5133 5219 5219 39 5612 5048 5640 5133 5219 5219 40 5612 5048 5640 5133 5219 5219 41 5612 5048 5048 5133 5551 5048 42 5630 5048 5048 5133 5048 5048 43 5219 5048 5048 5688 5048 5048 44 5219 5688 5048 5048 5048 5048 45 5219 5503 5048 5048 5048 5048 46 5219 5640 5612 5048 5048 5133 47 5551 5612 5612 5048 5133 5133 48 5048 5612 5057 5048 5133 5133 49 5048 5143 5133 5219 5133 5133 50 5048 5219 5133 5219 5133 5133 51 5048 5219 5133 5219 5133 5640 52 5048 5219 5133 5161 5630 5503
36
From Table 5.7 and Table 5.8, we can see that, for GAIR, the generated power is lower than the demand of 5048 MW during the weeks 6,7and 8. For GABR, the generated power is lower than the demand of 5048 MW during the weeks 1, 2, 3, 4, 14,
15, 16, 17 and 31. For the algorithms DPSO, MPSO, HSGA and DICS there are no violations in load constraint.
Figure 5-6 shows the Total Man Power Required during each week (TMPRt) for the six algorithms for 21 generator system
Figure 5-6: TMPRt during each week for the six algorithms for 21 generator system
We can see that for GABR [10], the Man Power Constraint is violated i.e. the TMPRt exceeds the AMPt for some of the weeks.
37
Table 5.7 lists the Total Man Power Required during weeks (1-26) for each
Maintenance Schedule.
Table 5.7: TMPRt during weeks (1-26) for the six algorithms for 21 generator
system
Week TMPRt Number GAIR GABR DPSO MDPSO HSGA DICS
1 20 30 15 35 10 15 2 15 30 15 30 10 15 3 15 20 15 15 5 15 4 15 10 15 3 5 15
5 10 5 15 27 5 15 6 20 5 15 27 5 15 7 20 3 23 12 3 40 8 15 15 22 12 15 40 9 20 15 22 22 15 8
10 20 0 22 17 15 7
11 20 3 27 17 40 32 12 18 2 32 27 15 22
13 15 2 17 38 15 22 14 15 17 12 15 15 12 15 15 47 12 15 35 12 16 18 47 8 15 35 17 17 17 17 20 10 23 17 18 12 2 30 10 27 18 19 12 2 35 5 27 35
20 7 3 10 5 12 10 21 17 0 10 5 7 10
22 12 25 5 5 7 5 23 12 40 5 3 17 5 24 17 45 5 15 17 5
25 18 30 5 15 17 5 26 20 0 3 15 18 3
38
Table 5.8 lists the Total Man Power Required during weeks (27-52) for each
Maintenance Schedule.
Table 5.8: TMPRt during weeks (27-52) for the six algorithms for 21 generator system
Week TMPRt Number GAIR GABR DPSO MDPSO HSGA DICS
27 10 10 10 15 10 20
28 10 10 30 10 10 10
29 10 10 10 10 10 10
30 5 5 5 10 5 10
31 20 35 5 5 5 5
32 15 10 5 5 5 5
33 10 10 0 5 10 5
34 10 5 15 15 15 10
35 10 5 10 25 30 15
36 5 5 10 15 15 15
37 5 0 10 15 10 10
38 5 0 10 10 10 10
39 10 10 15 10 10 10
40 15 10 15 10 10 10
41 15 10 10 5 15 10
42 20 10 10 5 10 10
43 10 10 10 0 10 10
44 10 0 10 10 10 10
45 10 30 10 10 10 10
46 10 15 10 10 10 10
47 15 10 15 10 10 10
48 10 15 25 10 10 10
49 10 25 10 10 10 5
50 10 10 10 10 5 5
51 10 10 5 10 5 15
52 10 10 5 30 20 30
39
For GABR [1], (TMPRt) exceeds the (AMPt) of 40 during weeks 15, 16 and 24.
For the other algorithms GAIR, DPSO, MPSO, HSGA and DICS there are no violations of Man Power constraint.
Figure 5-7 shows the SSR computed for the six algorithms using equation (2.1) discussed before in section 2.1 of chapter 2. When compared to other algorithms, we can see that DICS gives the Maintenance Schedule with the lowest SSR value of 2934355
(MW)2.
Fig. 5-7: SSR computed for the six algorithms for 21 generator system
40
Table 5.9 summarizes the result of the six algorithms in terms of SSR, Violation of Load Constraint and Violation of Man-Power Constraint.
Table 5.9: Comparison of the result of the six algorithms for 21 generator unit system
Violation of Load Violation of Man Power S. No Algorithm SSR (MW)2 Demand Constraint Availability Constraint
Violated during weeks 1 GAIR 3425971 No Violation 6,7,8 Violated during weeks 8691137 Violated during weeks 2 GABR 1, 2, 3, 4, 14, 15, 16, 17 (Highest) 15,16, 24 31 3 DPSO 3090335 No Violation No Violation
4 MDPSO 3073911 No Violation No Violation
5 HSGA 3010895 No Violation No Violation
2934355 6 DICS No Violation No Violation (Lowest)
From Table 5.11, we can see that the Maintenance Schedule from DICS provided the lowest SSR with no-violation of the constraints. DPSO, MDPSO and HSGA also did not violate any constraint, however their SSR were higher than DICS. The worst performing algorithm was GABR with the highest SSR and violation of Load Constraint for 9 weeks and violation of Man Power Constraint for 3 weeks.
5.2 IEEE RTS test system
5.2.1 Results
41 21 7 27 41 31 14 9 35
Figure 5-8: The fittest host nest (Optimal Maintenance Schedule array) obtained by DICS
for IEEE RTS system 41
Figure 5-8 shows the fittest host-nest obtained using DICS for the 9 generator unit
IEEE RTS test system. The host nest represents the Optimal Maintenance Schedule Array for the 9-generator unit RTS IEEE test system. Each of the 9 integers in this array represents the starting week of maintenance of units 1, 2, 3, ...., 9 respectively. Table 5.10 lists the generator units in maintenance during each week for the schedule represented by the fittest host nest in Figure 5-8.
Table 5.10: Generator units in maintenance during each week for the fittest host nest
for IEEE RTS system
Generator Units In Generator Units In Week Number Week Number Maintenance Maintenance 1 - 27 4 2 - 28 4 3 - 29 4 4 - 30 - 5 - 31 6 6 - 32 6 7 3 33 6 8 3 34 6 9 8 35 9 10 8 36 9 11 8 37 9 12 8 38 9 13 8 39 9 14 7 40 9 15 7 41 1, 5 16 7 42 1, 5 17 7 43 5 18 - 44 - 19 - 45 - 20 - 46 - 21 2 47 - 22 2 48 - 23 - 49 - 24 - 50 - 25 - 51 - 26 - 52 -
42
Table 5.11 lists the Power generated during each week for the fittest host nest obtained for IEEE RTS system.
Table 5.11: Pgenerated (t) during each week for the fittest host nest obtained for IEEE RTS
system
Week Week Number Pgenerated (t) Number Pgenerated (t) 1 1360 27 1284 2 1360 28 1284 3 1360 29 1284 4 1360 30 1360 5 1360 31 1205 6 1360 32 1205 7 1310 33 1205 8 1310 34 1205 9 1010 35 960 10 1010 36 960 11 1010 37 960 12 1010 38 960 13 1010 39 960 14 1163 40 960 15 1163 41 1248 16 1163 42 1248 17 1163 43 1260 18 1360 44 1360 19 1360 45 1360 20 1360 46 1360 21 1340 47 1360 22 1340 48 1360 23 1360 49 1360 24 1360 50 1360 25 1360 51 1360 26 1360 52 1360
43
Table 5.12 lists the reserve power generated during each week for the fittest host nest obtained for IEEE RTS system.
Table 5.12: Reserve (Surplus) power during each week for the fittest host nest obtained
IEEE RTS system
Week Week Reserve Power Reserve Power Number Number 1 498 27 529 2 460 28 468 3 482 29 483 4 526 30 480 5 480 31 483 6 519 32 429 7 478 33 405 8 504 34 476 9 270 35 234 10 273 36 255 11 295 37 180 12 283 38 265 13 303 39 236 14 413 40 236 15 442 41 505 16 363 42 504 17 409 43 460 18 523 44 479 19 490 45 475 20 480 46 451 21 484 47 420 22 529 48 470 23 460 49 418 24 473 50 390 25 464 51 360 26 499 52 408
44
Figure 5-9 shows the plot of the Power Generated during each week for the fittest host nest obtained for IEEE RTS System.
Figure 5-9: Power Generated during each week (Pgenerated (t)) for the fittest host nest
obtained for IEEE RTS System.
From Figure 5-9, we can see that the Power Generated (Pgenerated (t)) is always greater than the Load demand. Hence Load Demand Constraint is satisfied.
Figure 5-10: Reserve Power during each week for the fittest host for IEEE RTS
system. 45
Figure 5-10 shows the Reserve Power available during each week for the fittest host nest. From Figure 5-10, we can see the distribution of the Reserve Power throughout the Planning Period for the IEEE RTS system.
The SSR was computed for the fittest host nest in Figure 5-8 using (2.1) described in Chapter 2. The value of SSR obtained is 2934355 (MW)2.
5.2.2 Comparison
The performance of DICS on the IEEE RTS system is compared with previous work [11] which addressed the same came study using HSGA Algorithm.
Table 5.13 shows the two Maintenance Schedule arrays for the IEEE RTS system obtained through HSGA [11] and DICS algorithms.
Table 5.13: Maintenance Schedule Arrays obtained from HSGA and DICS algorithms
for IEEE RTS test system
Algorithm Maintenance Schedule Array
HSGA[11] 21 31 7 27 31 40 14 9 34
DICS 41 21 7 27 41 31 14 9 35
Figure 5-11 shows the plot of Power Generated during each week for the both
Schedules. The Load demand during each week is also shown along.
46
Fig. 5-11: Power Generated during each week (Pgenerated (t)) for the Maintenance
Schedules obtained from DICS and HSGA algorithm for IEEE RTS test system.
From Figure 5-11, we can see that Power Generated during each week is always greater than the Load Demand for both algorithms. Hence, for both HSGA and DICS, the
Load Demand constraint is satisfied.
Finally the SSR for both schedules were calculate using equation. (2.1) defined in
Chapter 2. Table 5.14 shows the calculated SSR values for both algorithm.
Table 5.14: Comparison of SSR obtained from HSGA and DICS for the IEEE RTS
system
S.NO. Algorithm SSR(MW)2
1 HSGA 9685531
2 DICS 9683531
From Table 5.14, we can see that DICS gives the Maintenance Schedule with lowest SSR, hence maximum levelling of Reserve Power.
47
For both test cases, DICS gave the lowest SSR with no violation in constraints.
Lower SSR implies greater leveling of reserve power resulting in even distribution of the surplus power over the entire operational planning period. This results in better reliability of the power system to meet the unexpected fluctuations in load demand. Therefore, it can be concluded that DICS outperformed all the other previously existing algorithms in generating an Optimal Maintenance Schedule for the complex combinatorial Generator
Maintenance Scheduling Problem.
48
Chapter 6
Conclusion and Future Work
Nature inspired DICS Optimization algorithm based on cuckoo bird's parasitic breeding behavior was presented for the NP-hard problem of Optimal Generator
Maintenance Scheduling. Formulation of the problem statement was presented. The mapping of the DICS to the GMS problem and the algorithm steps were discussed in detail. To verify the performance, DICS was applied on two test systems, namely, a
21-generator power utility test system and 9-generator test system. The performance of
DICS on the 21 unit test system was compared with previous works which addressed the same case study using other algorithms namely GAIR, GABR, DPSO, MDPSO and
HSGA. The performance of DICS on the 9-generator unit IEEE RTS test system was compared with HSGA. In both test cases, DICS outperformed the other algorithms.
DICS provided the Maintenance Schedule for both test systems with the lowest SSR with no-violation in Constraints, thus providing an optimal reliable Maintenance Schedule.
The superior performance of DICS can be attributed to the effective combination of exploration and intensification. This is achieved by the combination of exploratory
Le'vy/Global Flight and Intensified Local Search by Local Flight. Also, there are only
49
two control parameters viz., number of nests ‘n’ and step-size scaling factor ‘α’, which simplifies the mapping of the algorithm to a given problem.
Therefore, DICS is robust and superior over other metaheuristic algorithms.
Hence as a future work, the DICS can be used for applying to NP-hard combinatorial optimization problems as well other optimization problems with large search space.
50
References
[1] Dahal, Keshav P., and J. R. McDonald. "Generator maintenance scheduling of
electric power systems using genetic algorithms with integer representation." (1997):
456-461.
[2] K. Dahal and N. Chakpitak, 'Generator maintenance scheduling in power systems
using metaheuristic-based hybrid approaches', Electric Power Systems Research,
vol. 77, no. 7, pp. 771-779, 2007.
[3] Dahal, Keshav P., and J. R. McDonald. "A review of generator maintenance
scheduling using artificial intelligence techniques." (1997).
[4] J. Dopazo and H. Merrill, 'Optimal generator maintenance scheduling using integer
programming', IEEE Transactions on Power Apparatus and Systems, vol. 94, no. 5,
pp. 1537-1545, 1975.
[5] Edwin, K. W., and F. Curtius. "New maintenance-scheduling method with
production cost minimization via integer linear programming." International Journal
of Electrical Power & Energy Systems 12.3 (1990): 165-170.
[6] Zurn, H. H., and V. H. Quintana. "Generator maintenance scheduling via successive
approximations dynamic programming." Power Apparatus and Systems, IEEE
Transactions on 94.2 (1975): 665-671.
[7] Egan, Gerard T., Tharam S. Dillon, and Karol Morsztyn. "An experimental method
of determination of optimal maintenance schedules in power systems using the
51
branch-and-bound technique." Systems, Man and Cybernetics, IEEE Transactions
on 8 (1976): 538-547.
[8] Satoh, T., and K. Nara. "Maintenance scheduling by using simulated annealing
method [for power plants]." Power Systems, IEEE Transactions on 6.2 (1991): 850-
857.
[9] Dahal, K. P., C. J. Aldridge, and J. R. McDonald. "Generator maintenance
scheduling using a genetic algorithm with a fuzzy evaluation function." Fuzzy Sets
and Systems 102.1 (1999): 21-29.
[10] Yare, Yusuf, Ganesh K. Venayagamoorthy, and U. O. Aliyu. "Optimal generator
maintenance scheduling using a modified discrete PSO." IET generation,
transmission & distribution 2.6 (2008): 834-846.
[11] Kim, Jinho, and Zong Woo Geem. "Optimal scheduling for maintenance period of
generating units using a hybrid scatter-genetic algorithm." IET Generation,
Transmission & Distribution 9.1 (2014): 22-30.
[12] Yang, Xin-She, and Suash Deb. "Cuckoo search via Lévy flights." Nature &
Biologically Inspired Computing, 2009. NaBIC 2009. World Congress on. IEEE,
2009.
[13] X. Yang and S. Deb, 'Engineering optimisation by cuckoo search', IJMMNO, vol.
1, no. 4, p. 330, 2010.
[14] İ. Durgun and A. Yildiz, 'Structural Design Optimization of Vehicle Components
Using Cuckoo Search Algorithm', MP, vol. 54, no. 3, pp. 185-188, 2012.
52
[15] A. Yildiz, 'Cuckoo search algorithm for the selection of optimal machining
parameters in milling operations', Int J Adv Manuf Technol, vol. 64, no. 1-4, pp. 55-
61, 2012.
[16] Ernst, Benedikt, et al. ‘Implementation of the cuckoo search algorithm to optimize
the design of wind turbine rotor blades’, Proceedings of the European Wind Energy
Association (EWEA) 2012 Annual Event, Copenhagen, Denmark:[sn] (2012).
[17] A. Kaveh and T. Bakhshpoori, 'Optimum design of steel frames using Cuckoo
Search algorithm with Lévy flights', Struct. Design Tall Spec. Build., vol. 22, no. 13,
pp. 1023-1036, 2011.
[18] E. Valian, S. Mohanna and S. Tavakoli, 'Improved Cuckoo Search Algorithm for
Feed forward Neural Network Training', International Journal of Artificial
Intelligence & Applications, vol. 2, no. 3, pp. 36-43, 2011.
[19] W. MohamedAly and H. Atef Kelleny, 'Adaptation of Cuckoo search for
Documents Clustering', International Journal of Computer Applications, vol. 86, no.
1, pp. 4-10, 2014.
[20] A. Gherboudj, A. Layeb and S. Chikhi, 'Solving 0-1 knapsack problems by a
discrete binary version of cuckoo search algorithm', International Journal of Bio-
Inspired Computation, vol. 4, no. 4, p. 229, 2012.
[21] A. Layeb, 'A novel quantum inspired cuckoo search for knapsack problems',
International Journal of Bio-Inspired Computation, vol. 3, no. 5, p. 297, 2011.
[22] Song, Yong-Hua, ed. Modern optimisation techniques in power systems. Vol. 20.
Springer Science & Business Media, 1999.
53
[23] Dahal, Keshav, Kay Chen Tan, and Peter I. Cowling, eds. Evolutionary
scheduling. Vol. 49. Springer Science & Business Media, 2007.
[24] J. Coyne, 'Mimicry: The nefarious cuckoo', Why Evolution Is True, 2011.
[Online]. Available: https://whyevolutionistrue.wordpress.com/2011/03/04/mimicry-
the-nefarious-cuckoo/. [Accessed: 09- Jul- 2015].
[25] Yang, Xin-She. "Cuckoo Search and Firefly Algorithm: Overview and
Analysis."Cuckoo Search and Firefly Algorithm. Springer International Publishing,
2014. 1-26.
[26] Kamaruzaman, Anis Farhan, et al. "Levy flight algorithm for optimization
problems-a literature review." Applied Mechanics and Materials. Vol. 421. 2013.
[27] Storn, Rainer, and Kenneth Price. "Differential evolution–a simple and efficient
heuristic for global optimization over continuous spaces." Journal of global
optimization 11.4 (1997): 341-359.
54