Energy aware optimization model in chain based routing

Finally, Node 3 fuses the data of the other nodes with its own data and transmits the final fused data to the base station.. Sensor nodes that are far away from the base station will con

Energy-Aware Optimization Model in Chain-Based Routing

Nguyen Thanh Tung

Published online: 26 March 2014

# Springer Science+Business Media New York 2014

Abstract Sensor networks are deployed in numerous military

and civil applications, such as remote target detection, weather

monitoring, weather forecast, natural resource exploration and

disaster management Despite having many potential

applica-tions, wireless sensor networks still face a number of

chal-lenges due to their particular characteristics that other wireless

networks, like cellular networks or mobile ad hoc networks do

not have The most difficult challenge of the design of wireless

sensor networks is the limited energy resource of the battery of

the sensors This limited resource restricts the operational time

that wireless sensor networks can function in their

applica-tions Routing protocols play a major part in the energy

efficiency of wireless sensor networks because data

commu-nication dissipates most of the energy resource of the

net-works The above discussions imply a new family of protocols

called chain-based protocols In the protocols, all sensor nodes

sense and gather data in an energy efficient manner by

cooperating with their closest neighbors The gathering

pro-cess can be done until an elected node calculates the final data

and sends the data to the base station In our works, we have

proposed two methods to optimize the lifetime of chain-based

protocols using Integer Linear Programming (ILP)

formula-tions Also, a method to determine the bounds of the lifetime

for any energy-efficient routing protocol is presented Finally,

simulation results verify the work in this chapter Furthermore,

previous researches assume that the base station position is

randomly placed without optimization In our works, a non

convex optimization model has been developed for solving

the base station location optimization problem

Keywords Sensor Routing Chain based routing Linear programming Non convex optimization

1 Introduction Lindsey et al [5] proposed one type of chain-based protocol called PEGASIS (Power-Efficient Gathering in Sensor Information Systems), which is near optimal for gathering data in sensor networks PEGASIS forms a chain among sensor nodes so that each node will receive data from a close neighboring node and transmit data to another close neighbor Gathered data moves from a sensor node to the nearest neigh-bor, is aggregated with the neighbor’s data, and eventually reaches a determined Cluster-Head (CH) before finally being transmitted to the Base Station (BS) Figure1illustrates the ideas of the PEGASIS protocol In this round of data trans-mission, Node 3 is elected as the CH Node 5 transmits data to Node 4, and Node 4 fuses the data with its own data and transmits the fused data to Node 3 Similarly, Node 1 transmits data to Node 2, and Node 2 transmits the fused data to Node 3 Finally, Node 3 fuses the data of the other nodes with its own data and transmits the final fused data to the base station The data fusion function can be any function e.g minima, maxima and average, depending on the specific applications as discussed in [1–3] Nodes take turns equally to be the CH so that the energy spent by each node is balanced In other words, each node becomes a CH once for every n rounds of data transmission, where n is the number of sensor nodes The authors in [5] showed that building a chain to minimize the energy consumption is similar to the traveling salesman problem [6], which is known to be NP-complete They pro-posed a greedy algorithm starting from the furthest node from the base station until a near optimal chain is built as follows: 1) Add the node furthest from the base station to the chain

N T Tung ( *)

International School, Vietnam National University, 144 Xuan Thuy

Street, Cau Giay District, Ha Noi, Vietnam

e-mail: tungnt@isvnu.vn

DOI 10.1007/s11036-014-0497-8

2) This node finds a closest node from it that is not already in

the chain (Closest Euclidean distance)

3) Repeat until all nodes are added to the chain

Figure2shows the formation of a chain with five sensor

nodes Node 1 connects to Node 2, Node 2 connects to Node

3, Node 3 connects to Node 4 and Node 4 connects to Node 5

In each round, a sensor node must be selected as the

CH Each sensor node receives data from its down-stream neighbor, fuses with its own data to generate a single packet of the same length, and transmits the fused data to its upstream neighbor on the chain This process is illustrated in Fig 3 below When Node 4 is selected as the CH, Node 3 fuses data with Node 5 Node 2 fuses its data with Node 1 Node 4 fuses its data with Node 2 and Node 3 and transmits the data to the base station

2 Problem formulation

In many applications, the data reporting of all sensor nodes

is critical as in medical applications or in security applica-tions The above PEGASIS protocol tries to ensure that every node can become a CH equally This is not appro-priate for optimum system lifetime Sensor nodes that are far away from the base station will consume more energy than closer nodes to send data to the base station Also, nodes that have too little energy should not become CHs

As an equal selection of CHs will result in a reduced lifetime, a formulation to determine the CH pattern among all sensor nodes is presented below

In the next section, we have proposed two methods to optimize the lifetime using Integer Linear Programming (ILP) formulations [7, 8] The first method is applied for chain-based routing, the second method can be applied for any routing including chain-based routing

2.1 Method 1 Let us define n to be the number of sensor nodes, and xjto be the number of rounds node j becomes a CH In chain-based

N1 N2

N4 N5

N3 BS

: Cluster-head

Fig 1 A reconstructed chain

from PEGASIS method

Fig 2 Greedy algorithm to build a chain by PEGASIS method Fig 3 Data moving from all sensor nodes to the CH node

routing, only one CH is selected each round Therefore, there

are n possible choices of CHs The problem for the selection

of the CHs is formulated as follows:

Maximize :X

j¼1

n

xj

Subject to :

X

j¼1

n

cijxj≤Ei:∀i∈ 1…n½ Š

xj∈Zþ:∀j∈ 1…n½ Š

ð1Þ

where cjis the energy usage of Node i to send a unit of data in

a round, when Node j becomes CH and Eito be the initial

energy storage of Node i

The above Linear Programming problem tries to

maxi-mize the total number of rounds of transmitting data by all

sensor nodes under the battery-constraint of all sensor

nodes The energy coefficients cj of each non CH node

include the energy dissipation for the node to receive data

from its downstream neighbor and to send the fused data to

its upstream neighbor in the chain The energy coefficients

of each CH node in the formula include the energy

dissi-pation for the node to receive data from its downstream

neighbors and to send the fused data to the base station The

diagram in Fig.4shows that when Node 4 becomes a CH, c4

includes the energy dissipation to receive data from Node 1

and to send the fused data to Node 4 c4includes the energy

dissipation to receive data from Node 3 and Node 2 and to

send the fused data to the base station

2.2 Method 2

The problem of finding the optimal routing to achieve the

maximum network lifetime in a sensor network was studied as

a constrained linear program optimization in [12–15] and [16] In this work, the authors find the maximum lifetime that could be achieved by any routing cost or balancing scheme for optimizing average flows between nodes Similar to work in [12–15] and [16], we model the routing problem as below:

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 Q bps All of the data that is generated must eventually reach a single data sink, labeled s0 Let qi,jbe traffic on the link (i,j) during the time T Each node i has the initial battery energy of Ei, and the amount of energy consumed in transmitting a packet across link L(i,j)is eli,j

Maximize : T

Subject to : X

j¼1

N

qj;i þ QT ¼X

j¼0

N

qi; j:∀i∈ 1…N½ Š X

i¼1

n

qi; jeli; j<¼ Ei:∀i∈ 1…n½ Š X

i¼1

n

qi;0¼ Qn

qi; j>¼ 0 : ∀i; j∈ 1…n½ Š

ð2Þ

3 A new heuristic solution Problem formulation (1) and (2) can be solved by Linear Programming solvers These solvers are not always available and it is not easy to build these solvers inside sensors There-fore, a heuristic RE_chain algorithm is proposed In the RE_chain algorithm, the CH positions are reallocated among the sensor nodes so that the minimum residual energy of all sensor nodes is maximized The heuristic algorithm (RE_chain) is given as below:

RE_chain:

In every round of data transmission to the base station, select a sensor node as a leader for the chain in order to maximize the minimum residual energy of all sensor nodes after sending data for the round

Given:

N the number of sensor nodes indexed from 1 to N

s A current CH solution f(s) The minimum residual energy of all nodes with solution s

S Best solution so far Fig 4 Energy consumption coefficients of every sensor depends on the

position of the CH

RE_chain algorithm:

For (s from 1 to N)

δ ¼ f sð Þ−f sð Þ0

Ifδ>0 then s0=s

Result S0is the CH solution obtained from the RE_chain

algorithm

4 Simulation results

To evaluate the performance of RE_chain and compare the

performance with that of PEGASIS and LEACH protocol [1],

a number of simulators in Visual C++ were developed The

comparison between the system lifetime from Problem

for-mulation (1) and that of RE_chain is also performed In the

first set of simulations, the performance of RE_chain is

com-pared to the solution given by Formulation (1) In the

simula-tions, 100 random 100-node sensor networks are generated

Each node begins with 1 J of energy The network settings for

the simulations in this section are given below The energy

model was used in [1,3,5,9–11,16]

Network size (100m×100m)

Base station (50m,300m)

Data message size: 4000 bits

Broadcast message: 200 bits

Energy message: 20 bits

Position of sensor nodes: Uniform placed in the area

Energy model: E elec =50 ∗10 −9 J, ε fs =10 ∗10 −12 J/bit/m 2 and

ε mp =0.0013 ∗10 −12 J/bit/m 4

Figure5shows the ratio of the number of rounds of RE_chain

and the Linear Programming solution of Formulation (1) From

the simulation result, it can be said that RE_chain performs

within 1 % of the Linear Programming solution

It is also of interest to compare the performance of

RE_chain, PEGASIS, and LEACH on the network

topolo-gies On average, LEACH, PEGASIS, and RE_chain perform

602, 890, and 1,305 rounds respectively (Fig.6; Table1)

5 Determination of bounds for the lifetime

from any routing algorithm

The authors in [5] proposed a method to determine the upper

bounds of the chain-based routing system lifetime In each

round, every node must transmit its packet and some node must receive it As the total transmission energy of a message

is calculated by:

Et¼ kEelecþ εampkdn And the reception energy is calculated by:

Er¼ kEelec

where Eelecis the energy dissipation of the electronic circuitry

to encode or decode a bit, k is message size, εamp is the amplifier constant and d is the distance between the transmit-ter and the receiver On average, each non-CH node spends two times the energy for electronics and some additional energy sending data to its neighbor depending on how far the node transmits As a result, the total energy consumption

of any chain built will be at least two times the energy of the electronics multiplied by the number of sensor nodes There-fore, in the bounded solution, the authors set the energy usage

of non CH nodes to two times the energy usage for electronics

Ratio between the number of rounds of RE_chain

and RE_with ILP

0.988 0.99 0.992 0.994 0.996 0.998 1 1.002

Network topology

Fig 5 Ratio of the number of rounds between RE_chain and ILP model

Fig 6 Number of rounds over 100 random 100-node networks

The solutions of the bound Table1method are the solutions of

the ILP formulation (1) with the following coefficients:

Maximize : X

j ¼1

n

xj

Subject to :

X

j ¼1

n

cijxj≤Ei:∀i∈ 1…n½ Š

xj∈Zþ:∀j∈ 1…n½ Š

ð3Þ

If (Node i is a non CH) then

cij¼ 2kEelec:∀i; j∈ 1…n½ Š; i≠j

else

cij¼ 2kEelecþ εmpkd4to BS; i ¼ j

where dto_BSis the distance from Node i to the base station

The discussion above is true for the minimum energy

consumption of any chain but is not always true for the

maximum lifetime problem In other words, a smaller total

of energy coefficients in Formulation (3) do not always

pro-vide a better optimum We show that by an example below:

Consider two ILP problems

Example 1:

Maximize : x1þ x2

Subject to : 1x1þ 1:1x2≤6:1

1:2x1þ 1:3x2≤6:5

x1; x2∈Zþ

The sum of coefficients of constraint functions is: 1+1.1+

1.2+1.3=4.6

The optimum objective for the example is 5

Example 2:

Maximize x1þ x2

Subject to : 1x1þ 2:8x2≤6:1

0:5x1þ 0:5x2≤6:5

x1; x2∈Zþ

The sum of coefficients of constraint functions is: 1+2.8+ 0.5+0.5=4.8 The optimum objective for the example is 6 The two simple examples show that smaller sum of the coefficients of the ILP problem does not necessarily mean that the better objective solution can be obtained In other words, for the chain-based routing problem, minimizing the total energy dissipation for gathering data does not always guaran-tee an optimum solution

6 Determination of absolute upper bounds for the lifetime

of any routing problem The bounds calculated by the method in (3) are called the minimum energy bounds As discussed in the two ILP exam-ples above, the bounds cannot be proven to be the upper bounds of the system lifetime given by any chain-based routing In the section, a new method to determine the abso-lute upper bounds of the system lifetime given by any routing protocol is presented

On any routing protocol, any sensor must send data once in each round, and at least a sensor node must deliver data to the base station Let us consider Formulation (1), in which the energy coefficients are determined by:

Maximize : X

j¼1

n

xj

Subject to :

X

j¼1

n

cijxj≤Ei:∀i∈ 1…n½ Š

xj∈Zþ:∀j∈ 1…n½ Š

ð4Þ

If (Node i is a non CH) then

cij¼ kEelec:∀i; j∈ 1…n½ Š; i≠j else

cij¼ 2kEelecþ εmpkd4to BS; i ¼ j

Table 1 Results for Fig 6

90 % confidence interval

of the sample means

(876, 904) (1276, 1335) (592, 613)

where dto_BSis the distance from Node i to the base station.

Let O be the optimum solution of the ILP problem (4)

Then O is the upper bound for the lifetime that can be

achieved by any routing method

Proof Theorem 1 is stated and proved below to simplify the

process:

Theorem 1 Consider two ILP problems with the same

objec-tive function and the same variables, if the set of coefficients of

ILP problem 2 is smaller than the set of coefficients of ILP

problem 1 respectively for all of these coefficients, then the

optimal solution of Problem 2 is higher than that of Problem 1

6.1 Consider two ILP problems

Problem 1

j ¼1

n

xj

Subject to :

X

j¼1

n

cijxj≤Ei:∀i∈ 1…m½ Š

xj∈Zþ:∀j∈ 1…n½ Š

ð5Þ

Problem 2

j ¼1

n

xj

Subject to :

X

j¼1

n

c0ijxj≤Ei:∀i∈ 1…m½ Š

xj∈Zþ:∀j∈ 1…n½ Š

ð6Þ

Definition O1is the optimal solution of Problem (5) O2is the

optimal solution of Problem (6)

If c′j≤cj,∀i∈[1…m],∀j∈[1…n], then O2≥O1

Using the result from Theorem 1, the energy coefficient c′j

from any constructing algorithm:

c′ ≥c,∀i,j∈[1…n] of Formulation (4)

Therefore, any feasible solution O' obtained by any routing algorithm will satisfy:

O0≤O

As a result, O from Formulation (4) is the upper bound of any possible feasible solution (End of proof)

7 Optimization of the Base station location Previous researches in [1,3,4,10–16] assume that S the base station position is randomly placed without optimization Actually, the location needs to be optimized In order to minimize the complexities of the problem, the wireless radio energy dissipation model is not used yet A very simple energy usage model is given below

Assume that the energy to transmit a unit of data is propor-tional to the square of the distance to a destination, and there is

no energy spent at the destination

E Sð Þ ¼ d2; E Dð Þ ¼ 0; for α > 0

, where S denotes a source node, D denotes a destination node, E(S) is the energy usage of node and d is the distance from S

to D The Linear Programming model (2) becomes:

Subject to :X

j ¼1

N

qj;iþ QT ¼X

j ¼0

N

qi; j:∀i∈ 1…N½ Š ð7bÞ

X

j ¼1

N

qi; j i ; j2þqi;0hðxi−XÞ2þ yð i−YÞ2i

<¼ Ei:∀i∈ 1…N½ Š X

i ¼1

n

qi;0¼ Qn

This problem is difficult due to Constraints (7c), that are non convex Finding efficiently a solution of Problem (7) is a challenge We propose a solution approach based on DC (Difference of Convex) Programming and DCA (DC Algo-rithm) They are introduced by Pham Dinh in 1985 and have been extensively developed by Le Thi and Pham Dinh since

1994 [17–28]

In the literature, several work in non convex optimization have been developed for solving the optimization problems Several approximations have been proposed including

Proof Since c′j

i≤cj

i∀i∈[1…m],∀j∈[1…n] and O1is the op-timal solution of Problem 1, then O1is a feasible solution of

Problem 2 because O1satisfy all constraints of (5) Since is

the optimal solution of Problem 2, O2≥ O1 (End of proof)

Concave exponential approximation and logarithmic

approxi-mation of Weston, piecewise concave approxiapproxi-mation [25–27]

A common point of these approximations is that the

resulting optimization problems are all DC programs and

one can investigate DCA, an efficient method in nonconvex

programming framework for solving them

Thanks to a new result concerning with exact penalty

tech-niques in DC programming [23], we first reformulate Problem

(10) as a DC program then apply DCA to the resulting problem

Despite its local character, DCA with a good initial point quite

often converges to global solutions in practice Now, consider

the left hand side of the difficult constraints (7c) Let

fið Þ ¼z X

j ¼1

N

qi; j i; j2þqi;0hðxi−XÞ2þ yð i−YÞ2i

<¼ Ei:∀i∈ 1…N½ Š

It can be decomposed as follows:

fið Þ ¼z ρ

2

z 2− ρ

2

z 2−f zð Þ

!

fið Þ ¼ gz ið Þ−hz ið Þz

Where g zð Þ ¼ρ2jzj2 is convex and h zð Þ ¼ ρ2jzj2−f zð Þ
is

also convex with a sufficiently large numberρ Constraints

(7c) are thus rewritten as a DC constraints

By denoting K¼ z ¼ q; T; X ; Yð Þ : ∑

j¼1

N

qj;iþ QT ¼ (

∑

j ¼0

N

qi; j: ∀i∈ 1…N½ Š

Problem (7) becomes:

Or

Subject to : z; tð Þ∈K  ta; tbN

ð9bÞ

Where ta≤miniminz∈Kg(z) and tb≤maximaxz∈Kh(z) Problem (9) is written as:

Subject to : z; t; sð Þ∈K  tatbN

tiþ si−hið Þ ¼ 0 : ∀i∈ 1…Nz ½ Š ð10dÞ

Whereβ=maximax{h(z)−ti} Using penalty techniques (see [23]), Problem (10) is equiv-alent to:

Min :−T þX

i ¼1

N

riðtiþ si−hið ÞzÞ ð11aÞ

Subject to : z; t; sð Þ∈K  ta; tbN

Where riis sufficiently large

Problem (11) has the convex feasible set while whose objective function is concave, which is easily transformed to

a DC function There are many ways to decompose this function We can choose and test DCA with different DC decomposition ways (possibly use a similar way as did for f(x) in (10c)) Thus, we already transformed the origin prob-lem to a DC program All we have to do now is to use DCA

8 Conclusion This paper has focused on a new family of routing protocols for sensor networks: based routing protocols In chain-based routing, nodes form a chain connecting all nodes in the

network Data are gathered from all sensor nodes and move

along the chain toward an elected sensor The role of the

elected node is rotated between all sensor nodes to increase

the network lifetime Chain-based routing exploits the data

aggregation capability of sensor networks at maximum When

data are gathered from all sensor nodes, the data are

aggregat-ed with the data from their neighbors into a single message

The process is repeated until a single message is collected at

the elected sensor node

The previous chain-based routing (PEGASIS) selects

the CH nodes uniformly among all sensor nodes It is

demonstrated in this chapter that the selection is a bad

practice to ensure a good lifetime Depending on the

energy usage of each sensor to send data to its neighbors

and to the base station, the sensor nodes should be elected

as a leader differently The paper has then proposed a

method to optimize the selection of the CH among all

sensor nodes using Linear Programming formulations As

it is not always practical to do the Linear Programming

formulation, a simple heuristic method called RE_chain is

proposed to calculate the selection Simulations showed

that RE_chain performs very closely to the Linear

Pro-gramming formulation The performance of RE_chain

was then compared to that of LEACH, PEGASIS This

was shown that RE_chain improves the system lifetime

significantly than that of PEGASIS Also, it was observed

that RE_chain performs about 3 times better than

LEACH

Although the actual optimal lifetime for chain-based

methods is unknown, two methods were proposed to

compute the upper bounds for the lifetime of any routing

method (including chain-based routing and cluster-based

routing) The first method is called “minimum energy

bound” and the second one is called “absolute bound”

It was proved that the absolute bounds are the upper

bound for any routing method

Furthermore, previous researches assume that the base

station position is randomly placed without optimization In

our works, a non convex optimization model has been

devel-oped for solving the base station location optimization

problem

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