Optimal Number of Clusters in Wireless Sensor ... - Semantic Scholar
|ICCCT’10|
Optimal Number of Clusters in Wireless Sensor Networks: An FCM Approach A. S. Raghuvanshi*, S Tiwari, R Tripathi and N. Kishor Motilal Nehru National Institute of Technology Allahabad - 211004, INDIA * [email protected]
Abstract— Wireless Sensor Networks (WSN) are resource constrained systems that needs efficient utilization of all resources. Clustering is well known technique for achieving high scalability and efficient resource allocation in WSN. One of the fundamental issues in cluster based networks is to determine the optimal number of clusters with the objective of minimizing the energy consumption. Considering its importance, a Fuzzy c-Means (FCM) clustering approach is proposed to determine the optimal number of clusters in WSN. The study considers the deployment of 100 nodes in 100X100 m2 area for random uniform distribution. The optimal number of clusters determined by FCM has been compared with those obtained by analytical method. Study on lifetime of wireless sensor networks is also presented with optimal clusters in network.
members. Hence the number of cluster formed should be optimal for extending the life of WSNs. To calculate the optimal number of clusters, we have used fuzzy c-Means (FCM) algorithm for partitioning the sensor nodes into clusters and thereby determining the optimal number of clusters utilizing Euclidian distance norm. Xie and Beni’s (XB) index is used for validation measure The contribution of this paper is summarized here • First, we propose a new approach using Fuzzy cMeans algorithm for determining optimal numbers of clusters in WSN
Keywords-Wireless Sensor Networks, Fuzzy clustering, optimal number of clusters, FCM,Energy efficiency.
I.
INTRODUCTION
Wireless Sensor Networks (WSN) can be defined as a self configured network formed using a large numbers of sensor nodes distributed over a geographical area with either predefined location or randomly deployed. Individual sensor nodes posses limited processing, communicating and energy recourses. These sensor nodes can sense, measure, and gather information from the vicinity, based on the local pre defined decision process, can transmit the sensed information (data) to the user or information sink [1-3]. Communication of sensed data from the sensor nodes to information sink happens to be the main cause of energy depletion in sensor nodes [4]. In order to extend the active life of WSN, it is required to process data locally and send the processed information to sink node. One of possible solution for optimal use of available energy is by using cluster based routing protocols. Hierarchical or cluster based routing is well known technique for scalability and efficient communication [5]. Clustering enables bandwidth re-use and thus can improve the system capacity [6]. Clustering enables efficient resource allocation and helps in designing better power control [7]. In the hierarchical routing the data collected by sensor nodes in the proximity of the event is routed to the sink nodes through cluster heads. These selected head nodes are responsible for routing the information from the sensor nodes to the sink node after application of proper data aggregation or fusion [6, 8]. Overall system scalability, life time and efficiency depend upon the creation of optimal number of clusters and the spatial position of cluster head. A large number of clusters will congest the area with small size clusters and very small number of clusters will exhaust the cluster head with large amount of messages transmitted from cluster
__________________________________ 978-1-4244-9034-/10/$26.00©2010 IEEE
•
Compared the results obtained, with Analytical results obtained from energy efficiency point of view.
•
Studied the impact of optimum selection on the life of Wireless Sensor Network.
Rest of the paper is organized as follows in section II we present a detailed literature survey of related work. Section III discusses the FCM approach for partitioning the sensor nodes in to clusters utilizing Euclidian distance norm and describe the Xie and Beni’s (XB) index to validate the optimal number of clusters. In Section IV we formulated the optimization problem, introduced the system model and simulation model. Finally in section V we discuss our results and findings. II.
RELATED WORK
The issue of optimizing the number of clusters in WSN has been addressed by a number of researchers with different approaches. Low Energy Adaptive clustering Hierarchy, (LEACH) [6] is the first hierarchical based routing protocol that integrates TDMA at MAC with clustering. It utilizes the randomized rotation of the cluster head (CH) to evenly distribute the energy load with in the sensor nodes. Authors have advocated the need of optimal numbers of clusters kopt for energy efficiency. The optimal number of clusters have been determined analytically and it is reported that optimality exist for 1 < kopt 1 .
Steps (i) Choose a value for N c ,
º ¼
c(l ) 2
UF
Compare
z k − ci
Where Z is the feature vectors and c 2
(4)
exponent
c(l ) 2
σ U F , C c , Z = ¦ ¦ μ ik
The minimizing criterion used to define clusters, i.e. optimal fuzzy c-partition is defined as c
Nc
¦ ª z k − ci j =1 ¬
B. Fuzzy cluster validity measures Selecting a good and effective method to find clusters in data depends on several factors, like size of the data, match of the data to the algorithm, choice of the parameters of the algorithm, etc. The number of clusters can either be selected a priori or it can be automatically determined by using cluster validity measures. Different scalar validity measures in contest of fuzzy clustering are partition coefficient (PC), classification entropy (CE), partition index (PI), separation index (SI), alternative Dunn index (ADI) and Xie and Beni’s (XB) index. .Xie and Beni’s index has been shown as better index to indicate the correct number of clusters in many practical problems [21, 22]. It aims to quantify the ratio of the total variation σ within clusters to the minimum separation sep of clusters as:
T
(1)
where,
(5)
mp
The number of clusters is determined by testing error criterion. This make the clusters evenly distributed all across the deployed field thus enhancing load balancing needed for WSN.
Z consists of vectors
the calculation of fuzzy partition matrix following constraints: μ jk ∈ [ 0, 1] , 1 ≤ j ≤ N c , 1 ≤ k ≤ N s
(l ) ¦ ª μik º¼ k =1 ¬
is reached, the process ends.
using fuzzy clustering
partitioned into c(l )
( l −1)
UF − UF
z k , k = 1,..., N s contained in its column. The vectors NC clusters, represented by prototype
ci
Ns
1
Step (iv)
such that,
approach. Consider the data matrix
=
l
sensor field with area MXM m2. Each node
NC
mp
(l ) ¦ ª μik º¼ z k k =1 ¬
Compute the new partition matrix
μik( ) =
N s ’ sensor nodes are deployed in a ni where,
optimal number of clusters
l = 1, 2,... . Compute the
Step (ii) For the iteration, cluster center
(
)
XB U F , C c , Z =
m p and ε t , a small
σ (U F , C c , Z )
( )
N s sep C c
(9)
When the partitioning is compact and well distributed across complete sensor field, the value of σ should be low, while sep should be high. Therefore, XB should have low a value when data have been appropriately clustered.
positive constant. Initialize randomly a fuzzy c-partition 0 U F satisfying Esq. (1)–(3).
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expended by the transmitting node in transmitting a L bits message to a receiver d meter away at an acceptable signal to noise ratio(SNR) is given by
Which means if the XB index for a particular tuple (Ns1, Nc1) is XB1 and that of other tuple (Ns2, Nc2) be XB2 and if XB1 < XB2 then partition corresponding to (Ns1, Nc1) is taken better than (Ns2, Nc2). The XB index has been found to be better able to indicate correct number of partitions in the data [17] for a wide range of the choice of numbers of clusters. Hence XB index is used as the criterion to get the optimized numbers of clusters. IV.
° L . E elect + L .ε fs . d 2 ETx ( L, D) = ® 4 °¯ L . E elect + L .ε mp . d (10) Where,
PROBLEM FORMULATION
if d ≤ d 0 if d > d 0
E elect is the energy dissipated per bit to run the
transmitter circuit,
ε fs d 2
or
ε mp d 4
is the energy
Overall energy efficiency is one of the most important performances metric for sensor networks. It has been established by research that energy efficiency can be enhanced using hierarchal clustering or multi clustered approach [5]. Overall energy efficiency depends upon the optimum number of clusters selected [15]. Optimal numbers of clusters depend upon the spatial distribution of nodes in the sensor field and residual energy available with each node. Initially when the sensors are just deployed and all nodes have new batteries, the optimal number of clusters will depend more on the spatial distribution of nodes. To get the optimal number of clusters at the initial state of deployment is considered in this paper.
d 2 power loss) is used; otherwise multipath 4 fading channel model ( d power loss) is used [28] . By equating above equ.(10), at d = d 0 we get
A. Network Model Consider a set of ‘N’ sensors dispersed in an M X M m2 region with following properties about the sensor networks • Sensor nodes are quasi-stationary. Stationary after deployment. This is typical for sensor network applications
are uniformly deployed and for simplicity the sink is located at the center of sensor field. As shown by [7, 19] total energy dissipated per round is equal to
•
All nodes have similar communication and processing capabilities and equal significance.
•
Nodes are deployed in Random scenarios
•
Each node is left unattended after deployment and it is impossible to recharge or replace the batteries.
•
Each node has limited transmission power level and thus limited radio coverage.
consumed depending upon the distance to the receiver. If the distance between transmitter and receiver is less than a chosen threshold distance d 0 the free space channel model (
d0 =
ε fs ε mp
. To receive a message of L bits radio
= L . E elect . Assume an area A = M X M m2 over which N s nodes expends E RX
2 2 ETot = L(2 N s Eelect + N s EDA + ε fs (kdtoBS + N s dtoCH ))
By differentiating
ETot with respect to k and equating it
to zero, optimum number of clusters can be found as
Ns M = 2π d toBS
K opt =
(11)
Average distance from cluster head to sink is given as
B. Energy Model
d toBS = ³ x 2 + y 2 A
ETX ( L, D)
Ns 2 2π 0.765
M 1 dA = 0.765 A 2
(12)
E RX (L)
if the distance of significant number of nodes is more than d 0 then following the same analysis [7] they have Eelect L
obtained:
ε Ldv Figure 1.
Radio Energy model.
kopt =
Lee et al. [25] have studied the Entire aging process of WSN in a periodic data gathering application. Most research on the lifetime of WSN primarily focus on the energy depletion of the very first node in the network, because of the reason that entire network tends to get unstable due to subsequent node failure. In order to calculate the optimal number of clusters from energy consumption point of view we have used the similar energy model and analysis as proposed in [7]. According to the radio energy model illustrated in Fig. 1, energy
Ns
∈ fs
2π
∈mp
M 2 dtoBS
(13)
where
kopt : Optimal number of cluster, N s : no. of nodes distributed randomly in a M X M region 820
|ICCCT’10|
∈ fs d 2 :
100
amplifier energy (free space).
∈mp d 4 : amplifier energy (multi-path fading).
80 Y-position (meters)
d : distance between transmitting node and base station.
d toBS : average distance between nodes and base station C. Simulation Model and parameters We have used QualNet 4.5 [23] for generating random scenario of 100 nodes with different seeds. We have generated 100 different set of spatial data for 100X 100m2 for random deployment of sensor nodes. The data thus generated is imported to MatLab 7 release 14 [24] for fuzzy clustering. Xie and Beni’s (XB) index is calculated for all simulation run and the average of 100 runs is plotted with respect to numbers of clusters. The minimum value of averaged index will indicate the optimal number of clusters. The lifetime impact of optimal selection is studied by simulating the energy model described in section IIB. The initial energy of the nodes is set to E0 = 0.5 J this value is arbitrary for the purpose of study. Position of the sink is assumed to be at the center of the sensor field. The size of message that node sends to cluster head as well as the size of the message that a cluster head sends to sink is set to 4000 bits. The radio characteristics used in our simulations are summarized in Table 1 TABLE I.
Sensor Node Base Station
90
70 60 50 40 30 20 10 0 0
20
40 60 X-position (meters)
80
100
Figure 2. Random deployment of 100 nodes
Average value of Xie Beni's index (XB)
8
RADIO CHARACTERISTICS USED IN SIMULATIONS
7 6 5 4 3 2 1 Minimum Value of XB
Operation Transmitter/Receiver Electronics
Energy Dissipated Eelect = 50nJ/bits
2
Data Aggregation
EDA = 5nJ/bit/report
3
Transmitter
∈ fs =10pJ/bit/m
Amplifier
if d ≤ d 0 4
Transmitter
amplifier
if d ≥ d 0 V.
0 2
3
4
5
6
7 8 9 10 Number of clusters
11
12
13
14
Figure 3. Xie-Beni index for 100 randomly deployed nodes
2
1
∈mp = 0.00013pJ/bit/m4
0.9 0.8 Normalised Distance X 100 m
Sl. No. 1
RESULT AND DISCUSSION
A. FCM Based Clustering Fig.2 shows a typical random distribution of 100 nodes deployed in 100X100 m2 area. Xie and Beni’s (XB) index calculated for all simulation run and averaged over 100 runs is plotted with respect to numbers of clusters as shown in the fig. 3. As discussed earlier the XB value should be least for an appropriate number of clusters in the data set. It can be seen from the plotted results in fig.3 that the optimal number of cluster is 10, where the index has least value. Fig. 4 shows the optimal clustering for the taken scenario. The center point ‘+’ of each cluster indicates the location of cluster head. Since the clustering is done using Euclidian distance norm different boundary colour indicate the varying membership of each node with the given cluster head or cluster center.
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
Normalised Distance X 100 m
Figure 4. Optimal Clusteration with FCM for Random topology
B. Sensor Network Lifetime The energy depletion analysis of a sensor network is presented for 100 nodes with 10 numbers of clusters. Fig. 821
1
|ICCCT’10| 5 shows the lifetime curve of a sensor network for optimal selection of number of clusters, drawn between numbers of alive nodes and number of rounds. Lifetime curve can be divided into two portions first the stable part (stability period) where all nodes are functional and the other unstable part (instability period) where nodes start dying till the last alive node. Although the network life time can be defined as the time interval from the deployment of network until the death of the last node, but because of instability introduced due to node failure, we have defined the lifetime for sensor network as time when first node dies. Stability region with all alive nodes
100
Instability start with first dead node
90 Number of alive nodes
550
Number of rounds for first dead node
110
affected if the number of clusters selected are less than optimal number as compare to case when number of cluster selected are more than optimal number. As understood, the number of rounds for first dead node is maximum with 10 clusters which are same as the one calculated using FCM approach.
80 70
Highest X: 10 Y: 541
500
450
60 50
400 2
4
6
40 30
8 10 Number of clusters
12
14
Figure 7. First Node Dead for different Numbers of Clusters taken
20
Next in the study, the value of
10 0 0
200
400 600 Number of rounds(time)
800
1000
Figure 5. Life time curve for 100 nodes 110 Random Optimal
100
K opt is determined for
varying node density by increasing the number of nodes from 50 to 500 in the step of 50 for the same sink position and observation area 100X100 m2. Table II represent the optimal number of clusters determined by Equ.(11) and FCM approach for different node densities in 100X100 m2 area. We can see from the result that optimal number of clusters calculated by FCM approach are in good agreement with that of calculated by analytical method.
Number of alive nodes
90 80
TABLE II.
70 60
N 50 100 150 200 250 300 350 400 450 500
50 40 30 20 10 0 0
100
200
300 400 500 600 700 Number of rounds (time)
800
900
1000
Figure 6. Comprative life time curve
We studied the life time of the entire sensor network by first selecting the optimal number as indicated by FCM with non optimal selection keeping other parameters for simulation same. The comparison is illustrated by Fig.6.shows that the first node for non optimal case dies at 409th round where as first node for optimal case dies at 541st round. Thus giving a gain of nearly 20% in terms of stability region for the network. A larger stable region is better from reliability point of view. Fig.7 shows the round location of first dead node when we vary the number of clusters from 2 to 14 and plotted the result. It can be seen from the plot that network lifetime is more adversely
kopt FOR DIFFERENT NODE
Kopt (Eq.13) 7.37 10.43 12.77 14.75 16.49 18.06 19.51 20.86 22.12 23.32
VI.
DENSITIES
Kopt (FCM) 7 10 14 16 17 18 20 21 23 25
CONCLUSIONS
This paper presented the application of FCM algorithm to obtain the optimal numbers of clusters for sensor node deployment. As an illustration, 100 nodes were randomly deployed in 100X100m2 area. The determined optimal clusters using FCM approach were compared with analytical result. Energy depletion analysis for sensor
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|ICCCT’10| network indicated that the stability region was higher for optimal selection of number of clusters. [23] REFERENCES [1]
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Optimal Number of Clusters in Wireless Sensor Networks: An FCM Approach A. S. Raghuvanshi*, S Tiwari, R Tripathi and N. Kishor Motilal Nehru National Institute of Technology Allahabad - 211004, INDIA * [email protected]
Abstract— Wireless Sensor Networks (WSN) are resource constrained systems that needs efficient utilization of all resources. Clustering is well known technique for achieving high scalability and efficient resource allocation in WSN. One of the fundamental issues in cluster based networks is to determine the optimal number of clusters with the objective of minimizing the energy consumption. Considering its importance, a Fuzzy c-Means (FCM) clustering approach is proposed to determine the optimal number of clusters in WSN. The study considers the deployment of 100 nodes in 100X100 m2 area for random uniform distribution. The optimal number of clusters determined by FCM has been compared with those obtained by analytical method. Study on lifetime of wireless sensor networks is also presented with optimal clusters in network.
members. Hence the number of cluster formed should be optimal for extending the life of WSNs. To calculate the optimal number of clusters, we have used fuzzy c-Means (FCM) algorithm for partitioning the sensor nodes into clusters and thereby determining the optimal number of clusters utilizing Euclidian distance norm. Xie and Beni’s (XB) index is used for validation measure The contribution of this paper is summarized here • First, we propose a new approach using Fuzzy cMeans algorithm for determining optimal numbers of clusters in WSN
Keywords-Wireless Sensor Networks, Fuzzy clustering, optimal number of clusters, FCM,Energy efficiency.
I.
INTRODUCTION
Wireless Sensor Networks (WSN) can be defined as a self configured network formed using a large numbers of sensor nodes distributed over a geographical area with either predefined location or randomly deployed. Individual sensor nodes posses limited processing, communicating and energy recourses. These sensor nodes can sense, measure, and gather information from the vicinity, based on the local pre defined decision process, can transmit the sensed information (data) to the user or information sink [1-3]. Communication of sensed data from the sensor nodes to information sink happens to be the main cause of energy depletion in sensor nodes [4]. In order to extend the active life of WSN, it is required to process data locally and send the processed information to sink node. One of possible solution for optimal use of available energy is by using cluster based routing protocols. Hierarchical or cluster based routing is well known technique for scalability and efficient communication [5]. Clustering enables bandwidth re-use and thus can improve the system capacity [6]. Clustering enables efficient resource allocation and helps in designing better power control [7]. In the hierarchical routing the data collected by sensor nodes in the proximity of the event is routed to the sink nodes through cluster heads. These selected head nodes are responsible for routing the information from the sensor nodes to the sink node after application of proper data aggregation or fusion [6, 8]. Overall system scalability, life time and efficiency depend upon the creation of optimal number of clusters and the spatial position of cluster head. A large number of clusters will congest the area with small size clusters and very small number of clusters will exhaust the cluster head with large amount of messages transmitted from cluster
__________________________________ 978-1-4244-9034-/10/$26.00©2010 IEEE
•
Compared the results obtained, with Analytical results obtained from energy efficiency point of view.
•
Studied the impact of optimum selection on the life of Wireless Sensor Network.
Rest of the paper is organized as follows in section II we present a detailed literature survey of related work. Section III discusses the FCM approach for partitioning the sensor nodes in to clusters utilizing Euclidian distance norm and describe the Xie and Beni’s (XB) index to validate the optimal number of clusters. In Section IV we formulated the optimization problem, introduced the system model and simulation model. Finally in section V we discuss our results and findings. II.
RELATED WORK
The issue of optimizing the number of clusters in WSN has been addressed by a number of researchers with different approaches. Low Energy Adaptive clustering Hierarchy, (LEACH) [6] is the first hierarchical based routing protocol that integrates TDMA at MAC with clustering. It utilizes the randomized rotation of the cluster head (CH) to evenly distribute the energy load with in the sensor nodes. Authors have advocated the need of optimal numbers of clusters kopt for energy efficiency. The optimal number of clusters have been determined analytically and it is reported that optimality exist for 1 < kopt 1 .
Steps (i) Choose a value for N c ,
º ¼
c(l ) 2
UF
Compare
z k − ci
Where Z is the feature vectors and c 2
(4)
exponent
c(l ) 2
σ U F , C c , Z = ¦ ¦ μ ik
The minimizing criterion used to define clusters, i.e. optimal fuzzy c-partition is defined as c
Nc
¦ ª z k − ci j =1 ¬
B. Fuzzy cluster validity measures Selecting a good and effective method to find clusters in data depends on several factors, like size of the data, match of the data to the algorithm, choice of the parameters of the algorithm, etc. The number of clusters can either be selected a priori or it can be automatically determined by using cluster validity measures. Different scalar validity measures in contest of fuzzy clustering are partition coefficient (PC), classification entropy (CE), partition index (PI), separation index (SI), alternative Dunn index (ADI) and Xie and Beni’s (XB) index. .Xie and Beni’s index has been shown as better index to indicate the correct number of clusters in many practical problems [21, 22]. It aims to quantify the ratio of the total variation σ within clusters to the minimum separation sep of clusters as:
T
(1)
where,
(5)
mp
The number of clusters is determined by testing error criterion. This make the clusters evenly distributed all across the deployed field thus enhancing load balancing needed for WSN.
Z consists of vectors
the calculation of fuzzy partition matrix following constraints: μ jk ∈ [ 0, 1] , 1 ≤ j ≤ N c , 1 ≤ k ≤ N s
(l ) ¦ ª μik º¼ k =1 ¬
is reached, the process ends.
using fuzzy clustering
partitioned into c(l )
( l −1)
UF − UF
z k , k = 1,..., N s contained in its column. The vectors NC clusters, represented by prototype
ci
Ns
1
Step (iv)
such that,
approach. Consider the data matrix
=
l
sensor field with area MXM m2. Each node
NC
mp
(l ) ¦ ª μik º¼ z k k =1 ¬
Compute the new partition matrix
μik( ) =
N s ’ sensor nodes are deployed in a ni where,
optimal number of clusters
l = 1, 2,... . Compute the
Step (ii) For the iteration, cluster center
(
)
XB U F , C c , Z =
m p and ε t , a small
σ (U F , C c , Z )
( )
N s sep C c
(9)
When the partitioning is compact and well distributed across complete sensor field, the value of σ should be low, while sep should be high. Therefore, XB should have low a value when data have been appropriately clustered.
positive constant. Initialize randomly a fuzzy c-partition 0 U F satisfying Esq. (1)–(3).
819
|ICCCT’10|
expended by the transmitting node in transmitting a L bits message to a receiver d meter away at an acceptable signal to noise ratio(SNR) is given by
Which means if the XB index for a particular tuple (Ns1, Nc1) is XB1 and that of other tuple (Ns2, Nc2) be XB2 and if XB1 < XB2 then partition corresponding to (Ns1, Nc1) is taken better than (Ns2, Nc2). The XB index has been found to be better able to indicate correct number of partitions in the data [17] for a wide range of the choice of numbers of clusters. Hence XB index is used as the criterion to get the optimized numbers of clusters. IV.
° L . E elect + L .ε fs . d 2 ETx ( L, D) = ® 4 °¯ L . E elect + L .ε mp . d (10) Where,
PROBLEM FORMULATION
if d ≤ d 0 if d > d 0
E elect is the energy dissipated per bit to run the
transmitter circuit,
ε fs d 2
or
ε mp d 4
is the energy
Overall energy efficiency is one of the most important performances metric for sensor networks. It has been established by research that energy efficiency can be enhanced using hierarchal clustering or multi clustered approach [5]. Overall energy efficiency depends upon the optimum number of clusters selected [15]. Optimal numbers of clusters depend upon the spatial distribution of nodes in the sensor field and residual energy available with each node. Initially when the sensors are just deployed and all nodes have new batteries, the optimal number of clusters will depend more on the spatial distribution of nodes. To get the optimal number of clusters at the initial state of deployment is considered in this paper.
d 2 power loss) is used; otherwise multipath 4 fading channel model ( d power loss) is used [28] . By equating above equ.(10), at d = d 0 we get
A. Network Model Consider a set of ‘N’ sensors dispersed in an M X M m2 region with following properties about the sensor networks • Sensor nodes are quasi-stationary. Stationary after deployment. This is typical for sensor network applications
are uniformly deployed and for simplicity the sink is located at the center of sensor field. As shown by [7, 19] total energy dissipated per round is equal to
•
All nodes have similar communication and processing capabilities and equal significance.
•
Nodes are deployed in Random scenarios
•
Each node is left unattended after deployment and it is impossible to recharge or replace the batteries.
•
Each node has limited transmission power level and thus limited radio coverage.
consumed depending upon the distance to the receiver. If the distance between transmitter and receiver is less than a chosen threshold distance d 0 the free space channel model (
d0 =
ε fs ε mp
. To receive a message of L bits radio
= L . E elect . Assume an area A = M X M m2 over which N s nodes expends E RX
2 2 ETot = L(2 N s Eelect + N s EDA + ε fs (kdtoBS + N s dtoCH ))
By differentiating
ETot with respect to k and equating it
to zero, optimum number of clusters can be found as
Ns M = 2π d toBS
K opt =
(11)
Average distance from cluster head to sink is given as
B. Energy Model
d toBS = ³ x 2 + y 2 A
ETX ( L, D)
Ns 2 2π 0.765
M 1 dA = 0.765 A 2
(12)
E RX (L)
if the distance of significant number of nodes is more than d 0 then following the same analysis [7] they have Eelect L
obtained:
ε Ldv Figure 1.
Radio Energy model.
kopt =
Lee et al. [25] have studied the Entire aging process of WSN in a periodic data gathering application. Most research on the lifetime of WSN primarily focus on the energy depletion of the very first node in the network, because of the reason that entire network tends to get unstable due to subsequent node failure. In order to calculate the optimal number of clusters from energy consumption point of view we have used the similar energy model and analysis as proposed in [7]. According to the radio energy model illustrated in Fig. 1, energy
Ns
∈ fs
2π
∈mp
M 2 dtoBS
(13)
where
kopt : Optimal number of cluster, N s : no. of nodes distributed randomly in a M X M region 820
|ICCCT’10|
∈ fs d 2 :
100
amplifier energy (free space).
∈mp d 4 : amplifier energy (multi-path fading).
80 Y-position (meters)
d : distance between transmitting node and base station.
d toBS : average distance between nodes and base station C. Simulation Model and parameters We have used QualNet 4.5 [23] for generating random scenario of 100 nodes with different seeds. We have generated 100 different set of spatial data for 100X 100m2 for random deployment of sensor nodes. The data thus generated is imported to MatLab 7 release 14 [24] for fuzzy clustering. Xie and Beni’s (XB) index is calculated for all simulation run and the average of 100 runs is plotted with respect to numbers of clusters. The minimum value of averaged index will indicate the optimal number of clusters. The lifetime impact of optimal selection is studied by simulating the energy model described in section IIB. The initial energy of the nodes is set to E0 = 0.5 J this value is arbitrary for the purpose of study. Position of the sink is assumed to be at the center of the sensor field. The size of message that node sends to cluster head as well as the size of the message that a cluster head sends to sink is set to 4000 bits. The radio characteristics used in our simulations are summarized in Table 1 TABLE I.
Sensor Node Base Station
90
70 60 50 40 30 20 10 0 0
20
40 60 X-position (meters)
80
100
Figure 2. Random deployment of 100 nodes
Average value of Xie Beni's index (XB)
8
RADIO CHARACTERISTICS USED IN SIMULATIONS
7 6 5 4 3 2 1 Minimum Value of XB
Operation Transmitter/Receiver Electronics
Energy Dissipated Eelect = 50nJ/bits
2
Data Aggregation
EDA = 5nJ/bit/report
3
Transmitter
∈ fs =10pJ/bit/m
Amplifier
if d ≤ d 0 4
Transmitter
amplifier
if d ≥ d 0 V.
0 2
3
4
5
6
7 8 9 10 Number of clusters
11
12
13
14
Figure 3. Xie-Beni index for 100 randomly deployed nodes
2
1
∈mp = 0.00013pJ/bit/m4
0.9 0.8 Normalised Distance X 100 m
Sl. No. 1
RESULT AND DISCUSSION
A. FCM Based Clustering Fig.2 shows a typical random distribution of 100 nodes deployed in 100X100 m2 area. Xie and Beni’s (XB) index calculated for all simulation run and averaged over 100 runs is plotted with respect to numbers of clusters as shown in the fig. 3. As discussed earlier the XB value should be least for an appropriate number of clusters in the data set. It can be seen from the plotted results in fig.3 that the optimal number of cluster is 10, where the index has least value. Fig. 4 shows the optimal clustering for the taken scenario. The center point ‘+’ of each cluster indicates the location of cluster head. Since the clustering is done using Euclidian distance norm different boundary colour indicate the varying membership of each node with the given cluster head or cluster center.
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
Normalised Distance X 100 m
Figure 4. Optimal Clusteration with FCM for Random topology
B. Sensor Network Lifetime The energy depletion analysis of a sensor network is presented for 100 nodes with 10 numbers of clusters. Fig. 821
1
|ICCCT’10| 5 shows the lifetime curve of a sensor network for optimal selection of number of clusters, drawn between numbers of alive nodes and number of rounds. Lifetime curve can be divided into two portions first the stable part (stability period) where all nodes are functional and the other unstable part (instability period) where nodes start dying till the last alive node. Although the network life time can be defined as the time interval from the deployment of network until the death of the last node, but because of instability introduced due to node failure, we have defined the lifetime for sensor network as time when first node dies. Stability region with all alive nodes
100
Instability start with first dead node
90 Number of alive nodes
550
Number of rounds for first dead node
110
affected if the number of clusters selected are less than optimal number as compare to case when number of cluster selected are more than optimal number. As understood, the number of rounds for first dead node is maximum with 10 clusters which are same as the one calculated using FCM approach.
80 70
Highest X: 10 Y: 541
500
450
60 50
400 2
4
6
40 30
8 10 Number of clusters
12
14
Figure 7. First Node Dead for different Numbers of Clusters taken
20
Next in the study, the value of
10 0 0
200
400 600 Number of rounds(time)
800
1000
Figure 5. Life time curve for 100 nodes 110 Random Optimal
100
K opt is determined for
varying node density by increasing the number of nodes from 50 to 500 in the step of 50 for the same sink position and observation area 100X100 m2. Table II represent the optimal number of clusters determined by Equ.(11) and FCM approach for different node densities in 100X100 m2 area. We can see from the result that optimal number of clusters calculated by FCM approach are in good agreement with that of calculated by analytical method.
Number of alive nodes
90 80
TABLE II.
70 60
N 50 100 150 200 250 300 350 400 450 500
50 40 30 20 10 0 0
100
200
300 400 500 600 700 Number of rounds (time)
800
900
1000
Figure 6. Comprative life time curve
We studied the life time of the entire sensor network by first selecting the optimal number as indicated by FCM with non optimal selection keeping other parameters for simulation same. The comparison is illustrated by Fig.6.shows that the first node for non optimal case dies at 409th round where as first node for optimal case dies at 541st round. Thus giving a gain of nearly 20% in terms of stability region for the network. A larger stable region is better from reliability point of view. Fig.7 shows the round location of first dead node when we vary the number of clusters from 2 to 14 and plotted the result. It can be seen from the plot that network lifetime is more adversely
kopt FOR DIFFERENT NODE
Kopt (Eq.13) 7.37 10.43 12.77 14.75 16.49 18.06 19.51 20.86 22.12 23.32
VI.
DENSITIES
Kopt (FCM) 7 10 14 16 17 18 20 21 23 25
CONCLUSIONS
This paper presented the application of FCM algorithm to obtain the optimal numbers of clusters for sensor node deployment. As an illustration, 100 nodes were randomly deployed in 100X100m2 area. The determined optimal clusters using FCM approach were compared with analytical result. Energy depletion analysis for sensor
822
|ICCCT’10| network indicated that the stability region was higher for optimal selection of number of clusters. [23] REFERENCES [1]
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