Base Station Location Aware Optimization Model of the Lifetime of Wireless Sensor Networks

Mobile Netw Appl
DOI 10.1007/s11036-015-0614-3

Base Station Location -Aware Optimization Model of the Lifetime
of Wireless Sensor Networks
Nguyen Thanh Tung 1 & Huynh Thi Thanh Binh 2

# Springer Science+Business Media New York 2015

Abstract Recently, wireless sensor networks (WSNs) have
been progressively applied in various fields and areas. However, its limited energy resources is indisputably one of the
weakest point that strongly affects the network’s lifetime. A
WSN consists of a sensor node set and a base station. The
initial energy of each sensor node will be depleted continuously during data transmission to the base station either directly or through intermediate nodes, depending on the distance
between sending and receiving nodes. This paper consider
determining an optimal base station location such that the
energy consumption is kept lowest, maximizing the network’s
lifetime and propose a nonlinear programming model for this
optimizing problem. Our proposed method for solving this
problem is to combine methods mentioned in [1] respectively
named the centroid, the smallest total distances, the smallest
total squared distances and two greedy methods. Then an improved greedy method using a LP tool provided in Gusek
library is presented. Finally, all of the above methods are compared with the optimized solution over 30 randomly created
data sets. The experimental results show that a relevant location for the base station is essential.

Keywords Base station location . Wireless sensor network .
Routing . Non-linear programming

* Nguyen Thanh Tung

Huynh Thi Thanh Binh

1

International School, Vietnam National University, Hanoi, Vietnam

2

Hanoi University of Science and Technology, Hanoi, Vietnam

1 Introduction
Being invented from the purposes in the army, wireless sensor
networks (WSN) appears more and more popularly in most
areas and fields of the humanity life.
Network nodes in a WSN are sensors having capability of collecting information around their locations
then sending to the base station without physical links.
Hence, WSNs are easily deployed in dangerous or badly
situated places to provide human with requisite information. This information can be humidity, temperature,
concentration of pesticides, noise and so on; which
makes WSN applicable to many fields such as environment, heath, military, industry, agriculture, etc.
However, one disadvantage of WSNs is that sensor
nodes are operated by not frequently rechargeable and/
or limited energy resources such as batteries. These
energy resources will be depleted gradually. Then the
energy source of a sensor node runs out, this node dies,
which means it can no longer collect, exchange as well
as send information to the base station. Therefore, the
WSN will not be able to complete its mission. The
duration since the WSN began operating until the first
sensor node runs out of its energy is called the network
lifetime and is considered as one of the most important

measures to evaluate the quality of WSNs. Namely, the
longer the network lifetime is, the better the WSN is.
So, the quality of WSNs depends on speed of energy
consumption of sensor nodes. This brings out a problem
is how to use sensor nodes’ energy effectively, in other
words, to maximize the lifetime of WSNs that is considered in this paper.
There are many ways as well as methods to maximize the
lifetime of WSNs. In general, the authors approach this


Mobile Netw Appl

problem in the way to find effective routing methods in data
transmission with the given random location of sensor nodes
and the base station. However, the fact shows that the base
station location needs to be optimized, which is our approach
for the problem of maximizing the lifetime of WSN.
Specifically, we consider the model in which all sensor
nodes in the network are responsible for sending data to the
base station in every specified period. When a sensor node
sends data, its consumed energy is directly proportional to
the square of distance between it and the node which receives
data. An optimal location of the base station needs to be found
such that the network lifetime is maximized.
We modeled this problem as a nonlinear programming.
Also, five methods, the centroid, the smallest total distances,
the smallest total squared distances, the greedy and the integrated greedy method, are presented to specify the optical base
station location. Then, the nonlinear programming model is
used to evaluate our proposed methods over 30 randomly
created data sets. The experimental results show that a relevant

location for the base station is essential, which proves our
correct research way.
The rest of this paper is organized as follows: Section 2
describes the related works. Mathematical model for this problem is introduced in Section 3. Four methods for specifying
the base station location is showed in Section 4. Section 5
proposes an improved greedy method. Section 6 gives our
experiments as well as computational and comparative results.
The paper concludes with discussions and future works in
Section 7.

node i should not directly send data to node j if j ≥ i+2 because
communication over long links is not desirable. Their greedy
scheme offered an optimal placement strategies that is more
efficient than a commonly used uniform placement scheme.
In [5] proposed a network model for heterogeneous networks, a set of Ns sensors is deployed in a region in order to
monitor some physical phenomenon. The complete set of sensors that has been deployed can be referred as S = {s1……
sN}. Sensor i generates traffic at a rate of ri bps. All of the data
that is generated must eventually reach a single data sink,
labeled s0. Let qi,j be traffic on the link (i,j) during the time
T. The network scenario parameters also include the traffic
generation rate ri for each sensor. The power model in [5–9],
is used, where the amount of energy to transmit a bit can be
represented as:
The total transmission energy of a message of k bits in
sensor networks is calculated by:
E t ¼ E elec þ ε FS d 2
and the reception energy is calculated by:
E r ¼ Eelec
where Eelec represents the electronics energy, ε FS is determined by the transmitter amplifier’s efficiency and the channel conditions, d represents the distance over which data is
being communicated.

Maximize: T
Subject to:
N
X

2 Related works
Until now, the problem of maximizing the lifetime of WSNs
has received a huge interest of the researchers. According to
[2], there have two different approaches for maximizing the
network lifetime. One is the indirect approach aiming to minimize energy consumption, while the other one directly aims
to maximize network lifetime.
With the indirect approach, the authors [3] gave a method
to calculate energy consumption in WSNs depending on the
number of information packets sent or the number of nodes.
Then they proposed the optimal transmission range between
nodes to minimize total amount of consumed energy. With
this method, the total energy consumption is reduced by 15
to 38 %.
Cheng et al. formulated a constrained multivariable nonlinear programming problem to specify both the locations of the
sensor nodes and data transmission patterns [4]. The authors
proposed a greedy placement scheme in which all nodes run
out of energy at the same time. The greed of this scheme is that
each node tries to take the best advantage of its energy resource, prolonging the network lifetime. They reason that

q j;i þ ri T ¼

j¼1
N
X
À

N
X

qi; j : ∀i∈½1…N Š

ð1Þ

j¼0

X
Á
Eelec þ ε FS d 2 qi; j þ
E elec q j;i < ¼ E i : ∀i∈½1…N Šð2Þ

j¼0

qi; j >¼ 0 : ∀i; j∈½1…nŠ

N

j¼1

ð3Þ

3 Problem formulation of maximizing the lifetime
of wireless sensor networks with the base station
location
A sensor network is modeled as a complete undirected graph
G = (V, L) where V is the set of nodes including the base station

(denoted as node 0) and L be the set of links between the
nodes. The size of V is N. The link between node i and node
j shows that node i can send data to node j and vice versa. Each
node i has the initial battery energy of Ei. Let Qi be the amount
of traffic generated or sank at node i. Let dij be the distance
between node i and node j. Let T be the time until the first
sensor node runs out of energy. Let qij be the traffic on the link


Mobile Netw Appl

L(ij) during the time T. The problem of maximizing the lifetime
of the wireless sensor networks with the base station is formulated as follows [10–14]:
Maximize: T
Subject to:
N
X

q ji þ QT ¼

j¼1

N
X

qi j : ∀i∈½1…N Š

one. By using a tool to find the lifetime of WSN, we can
evaluate quality of this base station location as well as that
of these methods. Four methods are as follows:

The centroid method: defines the base station location as the
centroid of all sensor nodes. This location is calculated by (8).
N
X

ð4Þ

j¼0

h
i
X
2
2
qi j d i j 2 þ qi0 ðxi −x0 Þ þ ðyi −y0 Þ < ¼ E i : ∀i∈½1…N Š ð5Þ

x0 ¼

N

j¼1

N
X

qi0 ¼ Q0

ð6Þ

i¼1

qi j >¼ 0 : ∀i; j∈½0…N Š

ð7Þ

x0 ; y0 ; T : V ariable
In which, (xi, yi) is coordinate of node i in the twodimensional space.

4 Four methods for specifying the base station
location
To maximizing the lifetime of WSNs, the base station location
not only is close, but also balances distances with as many
sensor nodes as possible. This guarantees that sensor nodes do
not consume too much energy in transmitting data to the base
station and no sensor node depletes its energy much faster
than other nodes. The center of network seems to be in accord
with this requirement. However, there are many definitions for
the center of network, each definition gives different locations.
So this paper proposes four methods corresponding to four
different “center” definitions to specify the center of network
that is also the base station location.
These four methods are named respectively as the centroid,
the smallest total distances, the smallest total squared distances and the greedy methods. After this base station location
is determined, the model in Section 3 becomes a linear optimal

Fig. 1 Illustration of the greedy method

N
X

xi

i¼1

N −1

; y0 ¼

yi

i¼1

N −1

ð8Þ

The smallest total distances method: the base station location is a point such that the Euclidean distance summation
from it to all sensor nodes is the smallest one. This point
satisfies (9). With this definition, easily seen, the base station
location should be a point in the convex hull of all sensor
nodes. However, for the sake of simplicity, this location is
found in the smallest rectangle surrounding all sensor nodes.
M in :

N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
ðxi −x0 Þ2 þ ðyi −y0 Þ2

ð9Þ

i¼1

The smallest total squared distances method: it is similar to
the smallest total distances one, but the base station location
has to satisfy that the sum squared distances from it to all
sensor nodes is the smallest.
M in :

N
X

ðxi −x0 Þ2 þ ðyi −y0 Þ2

ð10Þ

i¼1

The greedy method: defines a sensor set includes sensor
nodes and a delegate center. If the set has only one sensor
node, its delegate center is this own sensor node. Also, we
define the distance between two sensor sets is the distance
between their two delegate centers. The main idea of this
method is that starting with one-sensor-node sets (Fig. 1a),
we merge two sets having the smallest distance (sensor node
set S1 and S2 in Fig. 1a). A new delegate center for the merged
set (the red node in Fig. 1b) is specified as follows: this center
is on the line segment connecting two old delegate centers and
splits this line into two segments with proportional by p. The
sensor sets is merged until only one set remains. The delegate

Mobile Netw Appl

center of this last set is the base station location (The green
node in Fig. 1c).
To improve the greedy method, the authors break the original algorithm into two sequential steps: finding two componential node sets and combining them into one node set with a
new delegate center (let say finding and combining steps for
short); and propose various methods for partially
optimization.To be specific, the two steps was improved with
respectively two and three changes in the algorithm, bringing
in eight different output sets. Since six of them return equallyhigh-quality results, the authors select two combinations that
have the best qualified output to present.
The first optimized greedy method: Change algorithm in
the combining step.
Step 1 The first step is kept originally, which is accomplished by looking for two sets having the smallest
distance (sensor node set S1 and S2 in Fig. 1a).
Step 2 To improve the solution, the authors propose a
change in the second step: looking for a new center
on the line segment connecting two old sets’ center
such that the average squared distance from it to two
old centers is the smallest. The average squared distance AveSD of set S having M internal nodes toward
the assuming center c is calculated as follows:


AveSD ¼ M Â ðxS −xc Þ2 þ ðyS −yc Þ2
ð11Þ

The second optimized greedy method: Change algorithm
in both steps.
Step 1 finding two componential nodes.

The authors improve the first step by searching for
two sets having the smallest average squared distance
between every pair of internal nodes of both sets. If
the set S1 has N internal nodes, and S2 has M internal
nodes, the two chosen sets are the one having:
2
XXÀ
Á2 
1
xS 1 i −xS 2 j þ yS 1 i −yS 2 j
Â
N ÂM
i¼1 j¼1
N

M in :

M

Because of that, a new concept appears—number of transmission (denote as t)—a constant value representing the average
times of data being sent through intermediate nodes/centers.
The Fig. 2 illustrates internal nodes inside node sets S1 and S2
transmit data through the sets’ center. The value of t is a
constant and is chose with the maximum value, conditioning
that the network functions well. To be specific, energy consumption for transmitting data should not over the remaining
energy of the node set.
The combination process starts with two chosen sets, the
new center is located at somewhere on the line segment
connecting two old sets’ center such that energy remained at
the new center is the greatest one. With the assumption that the

energy consumption is directly proportional to square of transmitting distance and the constant t is, the needed energy for the
set S consisting of M internal nodes to send data to the considering center c is:


Average energy consumption ¼ M  t  ðxS −xc Þ2 þ ðyS −yc Þ2

ð13Þ

5 Our proposed method for improving the current
method of finding base station location
Basically, the results derived by using [1] were nearly optimal,
which was found out by manually checking random points
that locate nearby the final optimal location. Therefore, we
propose an improving integrated method for finding the base
station location, such that it somehow combines all four published methods and inherits the advantages but limit the disadvantages of those methods.
This method applies the “divide and conquer” methodology on the original set of sensors by continuously forming
subsets of a random number of sensors, combining them and
appending their delegate point to the initial set thereafter until
only one delegate center remains. The final result is then qualified by using an intergrated LP tool included in the Gusek

ð12Þ
2. Step 2: combining two sets into one new virtual set
This improvement on the greedy method relates to the energy it takes to transmit data, which is to find the optimal
location preserving as much energy as possible. The basic idea
of this method goes with the assumption that when two original sets are combined into a new one with a delegate center,
the data transmitting flow will start from the internal nodes to
the new center before continuing to reach the base station.

Fig. 2 Two times transmitting illustration

Mobile Netw Appl
Fig. 3 Illustration of the
intergrated greedy method

library, which returns the value of maximum tval—the total
number of data transmission; the higher tval is, the better the
solution is. We called this Intergrated Greedy Method (IGM).
IGM starts with the definition of the weight of a sensor
node or set, which is defined in the original Greedy method
as the actual number of nodes that it contains. From the initial
30-node-covered set, a subset of several sensors is created
with the number of internal small-weighted nodes being limited to a specific value and needs to satisfy the condition that
there is at least one sensor node remains. In the chosen subset,
we apply one method amongst four original methods introduced in [1] with regard to specific pre-defined ratios a and b.
For the sake of high quality results, we adjusted the ratio so
that the Greedy method takes up the highest probability since
it provided the best result so far, while others have inconsiderable probability of being chosen; which, guarantees that the
final result to be at least as high as that of the Greedy method
and the minor but important changes will improve the quality
of the algorithm. The new delegate center having a new higher
weight is then appended to the subset, which means it has a
low priority to be chosen. After that, the process starts over
again repeatedly until there is only one point remains.
Pseudo code for the algorithm, in which, take_out is the
amount of sensor nodes contained in one set and ratio represent a way we classify the ratio into working variables.
Algorithm 1: Combinator
Input: A set of sensor nodes S
Output: Delegete center location (x, y)
begin

1. while size_of_subset > 1
2. take_out = rand()
3. if (take_out < threshold)
4. switch (ratio)

5. case 0–10 %: (x, y) = CentroidMethod()
6. break
7. case 10–2 %0: (x, y) = STDMethod()
8. break
9. case 20–30 %: (x, y) = STSDMethod()
10. break
11. case 30–100 %: (x, y) =
GreendyMethods()
12. break
13. end switch
14. endif
15. add (x, y) into S
16. end while
17. return (x, y)
end
The principle of this method is illustrated in the Fig. 3. In
the figures, black points represent sensor nodes that are not
either combined or considered yet, meanwhile the gray one
represent the one that has been combined at least once.
To evaluate the result derived by using this method, we
included Gusek library and based on that to calculate the
maxmimal, which represents the total number of data transmission in the WSN during its lifetime. High tval value equals
to good performance. If the result is not as good as expected,
we shall re-run the progress because the result will not be the
same next time due to many random factors taking places in

the main algorithm. Hence, the next time you repeat the entire
project, another result will be brought in and the qualification
might also be different. In addition, it is recommended that the
code should be run k times and all data are kept in a structure
file. By that way, we can easily find the peak point in those
gained from vairous earlier attempts and the avarage qualifying “tval” value for statistical purposes.


Mobile Netw Appl
The experiment parameters

Table 1
Parameter

Value

The network size
Number of sensor nodes - l
Initial energy of each node - E
Ratio p in method 4

100 m×100 m
30
1J
pffiffiffi
cx
pffiffiffi with cx, cy is the number of sensor nodes in two old sensor sets.
cy

Energy model

Eelec = d2 where d is the distance between two sensor nodes

6 Experimental results
6.1 Problem instances
In our experiments, we created 30 random instances denoted
as TPk in which k (k=1, 2,.., 30) shows ordinal number of a
instance. Each instance consists of l lines. Each line has two
numbers representing coordinate of a sensor node in the twodimensional space.
6.2 System setting
The parameters in our experiments were set as follows
Table 1:
6.3 Computational results
To prove the efficiency of our above proposal, coordinate of
the base station and the corresponding lifetime found by IGM
are presented and compared to ones found by four methods in

Table 2

[1]. Also, the optimal lifetime of each instance is presented to
evaluate quality of proposal methods.
Table 2 presents the base station location of the centroid,
the smallest total distances, the smallest total squared distances, the greedy and the integrated greedy method in BS1,
BS2, BS3, BS4 and BS5 column respectively. This table
shows that the centroid and the smallest total square distances
method gave extremely close locations over all instances. The
difference between locations found by four methods in [1] for
each instance is inconsiderable. If with each instance, we
choose the method giving the best lifetime among these four
methods, then comparing its base station location to one found

by IGM, we can see the difference between two locations is
only focus on one coordinate axis, either x-coordinate or ycoordinate. Namely, if x-coordinate of the base station locations found by the best method in four methods in [1] is near to
one found by IGM, their y-coordinate is far from each other
and vice versa. These can prove two following things: fisrtly,
all five methods have tendency of converging in one point that
is very near to optimal location. Secondly, despite having
random factor, the IGM is extremely stable.

The base station location found by five methods for 30 instances

y

BS1 (x-y)

BS2 (x-y)

BS3 (x-y)

BS4 (x-y)

TP1
TP2
TP3
TP4
TP5
TP6
TP7
TP8
TP9
TP10

TP11
TP12
TP13
TP14
TP15

55.2–39.9
39.9–55.3
55.3–42.6
42.6–56.2
56.2–42.3
42.3–56.4
56.4–42.9
42.9–56.8
56.8–41.3
41.3–58.4
58.4–42.0
42.0–58.7
58.7–39.9
39.9–59.9
59.9–38.3

54–37
36–60
55–41
40–62
57–39
39–61
58–40
39–63

59–36
36–64
62–38
36–64
61–36
36–65
62–35

55–40
40–55
55–43
43–56
56–42
42–56
56–43
43–57
57–41
41–58
58–42
42–59
59–40
40–60
60–38

53.1–50.5
40.6–54.8
52.6–53.0
42.1–55.1
52.1–53.4
41.8–55.0

53.3–54.6
44.7–53.6
54.1–53.7
45.9–54.5
54.5–53.5
46.1–54.9
57.4–43.3
46.1–48.9
58.1–41.9

BS5 (x-y)

Ins.

BS1 (x-y)

BS2 (x-y)

BS3 (x-y)

BS4 (x-y)

BS5 (x-y)

59–42
55–61
63–43
41–65
62–42
21–56

63–42
26–51
58–43
34–50
57–41
33–50
60–30

TP16
TP17
TP18
TP19
TP20
TP21
TP22
TP23
TP24
TP25
TP26
TP27
TP28
TP29
TP30

38.3–59.9
59.9–38.5
38.5–61.2
61.2–40.5
40.5–59.6
59.6–40.3

40.3–58.2
58.2–39.0
39.0–55.9
55.9–39.4
39.4–54.6
54.6–41.5
41.5–53.2
53.2–39.4
39.4–50.5

36–64
62–35
35–66
64–37
36–64
62–37
36–64
60–34
35–63
58–35
36–59
56–38
39–55
54–35
37–51

38–60
60–39
39–61
61–41

41–60
60–40
40–58
58–39
39–56
56–39
39–55
55–42
42–53
53–39
39–51

38.5–54.2
58.1–42.3
38.9–55.0
52.1–40.2
47.0–58.4
57.6–43.4
47.0–57.7
56.8–43.1
45.9–56.3
53.9–44.3
46.2–55.0
53.6–45.5
47.3–54.0
53.9–45.6
45.9–52.6

32–51
60–30

32–51
53–44
34–50
54–39
34–49
55–40
41–49
55–38
46–52
56–39
47–52
54–40
46–50


Mobile Netw Appl
Table 3

The lifetime of WSNs with the corresponding base station locations in the Table 2 and the optimal lifetime (Opt)

Ins.

BS1

BS2

BS3

BS4

TP1
TP2
TP3
TP4
TP5
TP6
TP7
TP8
TP9
TP10
TP11
TP12
TP13
TP14
TP15

870
809
867
838
878
822
931
739
941
736
1044
777
1171
762

907

811
652
845
885
839
832
926
706
910
700
1041
722
1160
739
907

867
820
866
837
869
815
925
738
948
736
1044
777

1171
762
907

890
836
865
829
877
805
906
746
907
738
971
791
1171
797
907

BS5

911
886
971
874
991
781
1006
814

1044
846
1171
826
907

Opt

Ins.

BS1

BS2

BS3

BS4

BS5

Opt

911
903
983
884
985
813
1025
836

1044
866
1171
891
907

TP16
TP17
TP18
TP19
TP20
TP21
TP22
TP23
TP24
TP25
TP26
TP27
TP28
TP29
TP30

762
907
746
893
751
1020
744
1056

695
1115
843
1040
838
988
817

739
907
726
870
695
935
682
943
587
997
741
1048
799
962
789

761
907
747
894
750
1003

745
1065
694
1116
838
1038
845
988
810

812
907
793
907
786
1130
794
1124
746
1010
893
977
862
952
861

842
907
945
907

853
1202
846
1168
808
1120
918
1068
884
988
875

952
907
946
907
879
1202
874
1178
809
1128
921
1068
884
988
884

The lifetime of 30 WSNs corresponding to 30 instances are
showed in the Table 3. These lifetime were found by using the

tool with the found base station locations in the Table 2. The
optimal liftetime of each instance is also presented in the Table 3
to easily evaluate quality of our proposal methods. The maximum lifetime of each instance among five methods is traced
with green. The optimal values are traced with blue to show that
there is at least one our method giving these optimal lifetime.
It is seen easily that IGM shows its superior to others when
having he best lifetime over all 30 instances. This method also
give the optimal value over 10 instances. On other instances,
the lifetime with the base station location found by IGM is
aproximate to the optimal, the gap between these two values is
very small, less than 2 %, especially on TP4, TP5, TP18,
TP23, so on. Comparing other methods, IGM give the better
lifetime values so far. Notably, with TP18, TP20, TP24, TP26,
the disparity in the lifetime between four remain methods and
IGM is up to from 15 to 27 %.
The centroid, the smallest total squared distances, the
greedy method gave the optimal lifetime over four instances
and the smallest total distances is over two instances.
The lifetime with the base station location of the centroid
method and the smallest total squared distances method is
about the same over all data sets, which can be explained by
the relatively same coordinate of these base station locations.
So in general, the IGM is the best method in maximizing
the network lifetime with the base station location in our proposal. The centroid is the simplest method which is suitable to
real-time or limited computing systems. And again, the difference among the network lifetimes corresponding to the base
station location gave by five methods over all instances shows

that the location for the base station should be optimized as
mentioned in the Section 1.

7 Conclusion
This paper proposed a nonlinear programming model for maximizing the lifetime of wireless sensor networks with the base
station location. We presented our intergrated greedy method
that compete with other four methods that are introduced in
[1], which shows a significant improvement on the original
greedy method, bringing in the results that is about 10%
higher than that of the other four methods. The new method
is also very close to the optimal solution. In this paper, our
proposed method was experimented on 30 random data sets.
With the found base station locations, specific lifetime of
WSNs was calculated by our model and not only offered a
high reliability about the solution but also showed that a relevant location for the base station should be essential.
Acknowledgments I would like to thank Vietnam National University,
Hanoi to sponsor in the project QG.14.57

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