This document discusses using linear approximation techniques to solve a nonlinear optimization problem for renewable sensor networks with wireless energy transfer. It presents a case study where the objective is to maximize the vacation time of a wireless charging vehicle over each cycle time by determining its optimal traveling path, charging schedule at each sensor node, and multi-hop data routing. The nonlinear problem is approximated using piecewise linear functions and solved to provide a near-optimal solution within a specified performance gap. Numerical results demonstrate the approach for a 50-node and 100-node network.

1. 1
Linear Approximation

2. 2
Outline
 Review of Linear Approximation for
Nonlinear Terms
 A Case Study
 Renewable Sensor Networks with Wireless
Energy Transfer

3. Linear Approximation
 Motivation
 Nonlinear optimization is difficult
 Linear programming is easy to solve with many
efficient tools available
 Basic idea: Replacing nonlinear terms by some
linear approximation approach
 Demonstrating piecewise linear approximation in this
chapter
3

4. Separable Problem
4
 Objective function and constraint functions
are additively separable
 Constraint functions are linear
 and are constants

5. Piecewise Linear Approximation
휇 = 휆푘−1휇푘−1 + 휆푘 휇푘
휃 (휇) = 휆푘−1휃(휇푘−1) + 휆푘휃(휇푘)
 : A continuous nonlinear function
 : A linear approximation of
5
휃(·)
휃 (·) 휃 (휇)
휃(휇)
휇0 = 푎 휇1 휇휇2
푘−1 휇 휇푘 휇푚 = 푏

6. Piecewise Linear Approximation (cont’d)
 General math expression for
 : Weight for each grid point
 Only two continuous and can be positive
 All other must be zero
 휃 푢 is defined as
6

7. Piecewise Linear Approximation (cont’d)
 : A binary variable indicates whether falls within
the k-th segment
 Relationship between and
7

8. Piecewise Linear Approximation (cont’d)
 Back to the separable problem
 Replacing each nonlinear function and
by its piecewise linear approximation
 The solution to the linear approximating problem
may or may not be feasible to the original
separable problem
 Feasible: Accuracy/complexity vs. number of grid points
 Infeasible: Construct a feasible solution by local search
8

9. 9
Outline
 Review of Linear Approximation for
Nonlinear Terms
 A Case Study
 Renewable Sensor Networks with Wireless
Energy Transfer

10. Wireless Sensor Networks (WSNs)
 Sensor nodes
 Sensing multi-media (video, audio
etc.) and scalar data (temperature,
pressure, light etc. )
 Transmitting data to the base
station via multi-hop
 Battery-powered
 Network lifetime
 Limited by battery outage
 Tremendous research efforts
 Energy-harvesting
10

11. WSNs (cont’d)
 But lifetime remains a major performance
bottleneck
 Our objective
 Remove the fundamental performance bottleneck
 Motivated by the recent breakthrough in wireless
energy transfer
11

12. Wireless Energy Transfer
 Dated back to the early 1900s
by Nikola Tesla
 Radiative mode by omni-directional
antennas
 Low efficiency
 Never put into practice use
12

13. Wireless Energy Transfer (cont’d)
 Little progress over nearly a century
 In early 1990, the critical need of
wireless power transfer re-emerged
 Example: electric toothbrush
 Limited applications due to
stringent requirements
 Omni-directional antennas: close in contact
 Uni-directional antennas: accurate alignment in charging
direction, uninterrupted line of sight
13

14. Wireless Energy Transfer (cont’d)
 A breakthrough technology: Magnetic resonance
 Published in Science 2007
 Non-radiative, based on resonant coupling
 Omni-directional, mid-range (up to 2 meters)
and efficient energy transfer
A 60-W light bulb is lighted up from a distance
of 2 meters away (by Kurs et al.)
14

15. Experiment Setting
15

16. Wireless Energy Transfer (cont’d)
The emerging technology
comes to reality!
Demonstrated in 2009 TED Global
The first No-tail TV was released at
2010 International CES
Wireless power
consortium is setting
the international
standard for
interoperable wireless
charging
Revolutionize how energy is
exchanged in the near
future!
16

17. Problem Description
Objective:
1) Make sensor network
work forever
2) Maximize the
percentage of
vacation time
Mobile Wireless
Charging Vehicle (WCV)
Sensor Node
Base Station
Service
Station
17

18. Three Basic Components
 Traveling path
(including direction)
for WCV
 Charging schedule
at each node
 Multi-hop data
routing
18

19. 19
Outline of Case Study
 A Solution Procedure
 Renewable energy cycle
 The optimal traveling path for WCV
 The optimal charging schedule and multi-hop data
routing
 Numerical Results

20. Renewable Energy Cycle
i
Arrival
time
Sensor Energy Behavior
Departure
time
Vacation
Time
Two requirements for renewable
energy cycle are:
1. Starts and ends at the same
energy level over a period of cycle.
2. Energy level never falls below 퐸푚푖푛
20

21. Optimal Traveling Path
 In an optimal solution, the WCV must move
along the shortest Hamiltonian cycle that crosses
all sensor nodes and the service station
 Obtained by solving the Traveling Salesman
Problem (TSP)
 The shortest cycle may not be unique
 WCV can follow either direction
 Any shortest path can achieve the same optimal
objective
21

22. Charging Schedule and Data Routing
 Traveling path for
WCV
Charging schedule
at each node
Multi-hop data
routing
22

23. Optimization Problem
The ratio of the WCV’s vacation time to a
cycle time
Data
routing
Charging
schedule
Consumed
energy
Nonlinear optimization problem: NP-hard in general.
Routing
constraint
Energy
consumption
Time constraint
Cycle
constraints
23

24. A Near-Optimal Solution
24
OPT-R
Problem OPT
Problem OPT-R
Piecewise linear
approximation
Problem OPT-L
Now only a
nonlinear
term 휂2
푖
exists for
each 푖
An illustration of piecewise linear
approximation
(with 푚 = 4)
2 ≤
The approximation error 휁푖 − 휂푖
1
4푚2
Change-of-variable
technique
Linear
relaxed
formulation

25. Performance Gap vs. Segment Number
An upper bound is obtained
by linear relaxation OPT-L
Unknown optimal objective
value of problem OPT
UB
LB A lower bound is obtained by
feasible solution construction
Estimated
performance
gap
Performance
gap
Estimated performance gap is a decreasing function of m
(the number of segments in approximation)
25

26. Choosing Segment Number
An upper bound by
linear relaxation
Unknown optimal
objective value
A feasible solution
is constructed
UB
LB
3) Construct a provable near-optimal solution
Estimated
performance
gap < є
Target
performance
gap є
1) Given an arbitrarily small performance gap ϵ
2) Choose an appropriate 푚 (a function of ϵ)
26

27. Summary of Solution Approach
 Traveling path for WCV
 Charging schedule at each node
 Multi-hop data routing
The performance gap is no more than requirement є
27

28. Sensor Energy Behavior
- Complete Process
1st renewable
energy cycle
The first renewable
energy cycle starts.
2nd renewable
energy cycle
Initial transient
cycle
The first renewable
energy cycle ends, i.e.,
the second renewable
energy cycle starts.
0
28

29. 29
Outline of Case Study
 A Solution Procedure
 Renewable energy cycle
 The optimal traveling path for WCV
 The optimal charging schedule and multi-hop data
routing
 Numerical Results

30. A 50-node Network
ϵ = 0.01
m = 4
Cycle time = 30.73 hours
Vacation time = 26.82 hours
Percentage = 87.27%
Optimal traveling path Charging schedule
30

31. Results under Different Directions
Identical charging
schedule and flow
routing
Different arrival time
and renewable cycle
starting energy
31

32. Bottleneck Node
Provably show the
existence of at least
one “bottleneck”
node in an optimal
solution
32
Bottleneck node

33. Bottleneck Node (cont’d)
The energy behavior of the bottleneck node
33

34. Multi-hop Data Routing
34

35. A 100-node network
35
ϵ = 0.01
m = 5
Cycle time = 58.52 hours
Vacation time = 50.30 hours
Percentage = 85.95%

36. Summary
36
 Review of Linear Approximation for Nonlinear
Terms
 A case study
 Exploited recent breakthrough in wireless energy
transfer technology for a WSN
 Showed that a WSN can remain operational forever!
 Studied a practical optimization problem
 Objective: Maximizing the ratio of the WCV’s vacation
time over the cycle time
 Proved that the optimal traveling path is the shortest
Hamiltonian cycle
 Developed a provable near-optimal solution for both
charging schedule and flow routing