Maximizing Sensor Lifetime in A Rechargeable Sensor Network via Partial Energy Charging on Sensors Wenzheng Xu, Weifa Liang, Xiaohua Jia, Zichuan Xu Sichuan University, P.R. China Australian National University, Australia City University of Hong Kong, Hong Kong

2 Outline Introduction Preliminaries Algorithm for the sensor lifetime maximization problem Algorithm for the service cost minimization problem with the maximum sensor lifetime Performance Evaluation Conclusion

3 Wireless Sensor Networks Environmental and Habitat Monitoring Structural Health Monitoring Precision Agriculture Military Surveillance ……

4 Wireless Energy Transfer Energy issue: sensors will run out of their energy Wireless energy transfer is very promising for charging sensors High energy transfer efficiency –40% within two meters Deploy mobile chargers to charge sensors –high and reliable charging rate

Problem of existing studies Existing studies adopt the full-charging model –A mobile charger must charge a sensor to its full energy capacity before moving to charge the next one It takes a while to fully charge a sensor, e.g., minutes Problem: an energy-criticial sensor may run out of its energy for a long time before its charging, especially when there are a lot of to-be-charged sensors 5

Example: the full-charging model 6 Assume that there are two energy-critical sensors The 1 st sensor is charged before its energy depletion The 2 nd sensor has run out of its energy for a while when the charger has fully charged the 1 st sensor

A novel charging model: the partial-charging model 7 First, the 1 st sensor is partially charged Then, the 2 nd sensor is fully charged Finally, the 1 st sensor is also charged to its full energy capacity Both of the two sensors are charged before their energy depletions

Side effect of the partial charging model The travel distance of the mobile charger will be longer than that under the full-charging model Higher service cost 8

Contributions of this paper How to find a charging tour for the mobile charger, so that the sensor lifetime is maximized without increasing the charger’s travel distance too much, under the partial-charging model? Propose a novel algorithm for the problem Experimental results: –Avg energy expiration time per sensor is only 10% of that by the state-of-the-art –Travel distance is only 7%~18% longer than that by the state-of-the-art 9

10 Outline Introduction Preliminaries Algorithm for the sensor lifetime maximization problem Algorithm for the service cost minimization problem with the maximum sensor lifetime Performance Evaluation Conclusion

Network Model A set V of energy-critical sensors at some time t –n sensors: v 1, v 2,..., v n –Battery capacities: B 1, B 2,..., B n –Amounts of residual energy: RE 1, RE 2,..., RE n –Energy demands: B 1 -RE 1, B 2 -RE 2,..., B n -RE n –Energy consumption rates: ρ 1, ρ 2,..., ρ n –Residual lifetimes: l 1, l 2,..., l n, l i = RE i / ρ i A mobile charger located at depot r –Fixed charging rate µ to every sensor –Travel speed s 11

A Novel Partial-charging Model Energy demand of sensor v i : B i -RE i A unit amount ∆ of charging energy The amount of energy charged to sensor v i –∆, 2∆,..., k i ∆, B i -RE i, where ∆ should not be too small, so that the travel distance of the charger will not be increased too much, e.g., ∆ = B i /5 A charging tour C of the charger –(r,0)→(v’ 1, e 1 ) →(v’ 2, e 2 ) →... →(v’ n, e n’ ) →(r,0) –e j : the amount of energy charged to sensor v’ j –Each sensor v i may be charged multiple times in tour C 12

Charging time vs. travel time Travel time t travel from one sensor to the next sensor in a charging tour usually is much shorter than the charging time on a sensor –e.g., 1 minute vs 10 miminutes (for an amount ∆ of energy) The travel time t travel is considered as a small constant Divide time into equal time slots –Each time slot lasts time units 13

Normalized lifetime of a sensor Assume each sensor v i is charged c i times in a charging tour C –An amount B i -RE i of energy is required to be charged to the sensor in the c i chargings – : dead time if no charging is performed – : dead time after the c i chargings – and : live and dead durations of sensor v i from time to time Normalized sensor lifetime 14

Sum of normalized sensor lifetimes Sum of normalized lifetimes can be considered as the live probability of sensor v i at any time thus is the expected number of live sensors maintained in the network by the charging tour C 15

Problem definitions Sensor lifetime maximization problem –Find a charging tour C so that the sum of normalized lifetimes is maximized – : the maximum sum of normalized lifetimes The service cost mimimization problem with the maximum sensor lifetime –Find a charging tour C so that the length of the tour is minimized, subject to that the maximum sum of normalized lifetime is acheived 16

17 Outline Introduction Preliminaries Algorithm for the sensor lifetime maximization problem Algorithm for the service cost minimization problem with the maximum sensor lifetime Performance Evaluation Conclusion

Basic idea of the algorithm Observation: an amount ∆ of energy is charged to a sensor at every time slot A feasible solution to the problem can be considered as a matching between sensors and time slots Reduce the problem to the maximum weighted matching problem 18

Algorithm for the sensor lifetime maximization problem For each sensor v i in V, create k i virtual sensors v i,1, v i,2,..., v i,ki, where k i = V’ j : the set of the jth virtual sensors for sensors in V Iteratively create k max bipartite graphs, k max =max{k i }, –S ’: the set of time slots –weight w’ j (v i,j, s q ): the normalized lifetime of sensor v i if the charger performs the jth charging to it at time slot s q Find a maximum weighted matching M j in graph G’ j Construct a charging tour from the last matching 19

Algorithm for the sensor lifetime maximization problem Theorem: there is a heurisitc algorithm for the sensor lifetime maximization problem, which takes O(n 3 ) time. Also, the algorithm finds an optimal solution if. – indicates that the lifetime of any sensor for consuming an amount ∆ of energy is no less that the total time of charging every sensor with an amount ∆ of energy, i.e., –ρ max : the maximum energy consumption rate 20

21 Outline Introduction Preliminaries Algorithm for the sensor lifetime maximization problem Algorithm for the service cost minimization problem with the maximum sensor lifetime Performance Evaluation Conclusion

Motivation for the problem The charging tour found by the matchings in the previous section may not be cheap, since the factor of sensor locations is not taken into consideration There may be multiple charging tours with the maximum sensor lifetime How to find the shortest one? 22

Algorithm overview A virtual sensor v i,j is expired if it is matched to a time slot q after its energy expiration time l i,j +1 in matching M Kmax, i.e., q > l i,j +1 –Virtual sensor v i,j must be charged at time slot q in any charging tour with the maximum sensor lifetime Otherwise, virutal sensor v i,j is unexpired, i.e., q ≤ l i,j +1 –Sensor v i,j is still unexpired if it is charged at any time slot no later than l i,j +1 –Find a shortest charging tour subject to the constraint of the energy expiration times of unexpired virtual sensors 23

Algorithm overview Create two virtual nodes r s and r f for depot r –Have the same locations as depot r Find a minimum weighted r f -rooted treeT spanning all virtual sensors and nodes r s and r f, so that –the number of virtual sensors in the subtree rooted at each unexpired sensor v i,j is no more than its energy deadline l i,j +1 –the number for each expired sensor v i,j is equal to q Transform tree T into a path P starting from r s and ending at r f –Path P actually is a closed tour, by noting that nodes r s and r f have the same location as depot r 24

Construct tree T Partition the set V’ of virtual sensors into n’=|V’| sets –An expired sensor is contained in V’ q (q is the matched time slot) –An unexpired sensor v i,j is contained in V’ j’, j’=min{l i,j +1, n’} –V’ 1, V’ 2,..., V’ n’, some sets may contain no virtual sensor – due to the matching property –Let V’ 0 ={ r s } and V’ n’+1 ={ r f } Add nodes in V’ 0, V’ 1, V’ 2,..., V’ n’, V’ n’+1 to tree T one by one 25

Construct tree T-cont. Initially, T constains only node r s Assume that V’ 0, V’ 1, V’ 2,..., V’ j have been added Consider the next non-empty set V’ k, k>j –For each residual node v i, find a minimum subtree T i k, by expanding from node v i to a subtree contains nodes in V’ k and other nodes in a greedy way –Let subtree, where w(T i k ) is the tree weight, e i,j is the nearest edge between node v i and nodes in set V’ j –Add subtree T k to tree T 26

Transform tree T into a path P Find the path from node r s to r f in tree T Obtain a graph G by replicating the edges in tree T except the edges on the path There is a Eulerian path from r s to r f in graph G –G is a connected graph –The degrees of nodes r s and r f are odd –The degrees of the other nodes are even Find a path P from r s to r f by shortcutting repeated nodes in the Eulearian path 27

28 Outline Introduction Preliminaries Algorithm for the sensor lifetime maximization problem Algorithm for the service cost minimization problem with the maximum sensor lifetime Performance Evaluation Conclusion

29 Simulation Environment ParametersValues Network Size (Small Scale)10,000 m Sensing Field (Small Scale)100 – 600 Given Time Period T (Small Scale)200 m Network Size (Large Scale)1, 2, 4, 8, 16s Sensing Field (Large Scale)[0.4, 0.9] mJ/s Given Time Period T (Large Scale)5, 10, 30 m/s ParametersValues Sensing Field 1,000m * 1,000m Network Size n sensors Battery capacity B i B i =B= 10.8kJ Data rate [b min, b max ]= [1 kbps, 10 kbps] Monitoring Period T One year Energy charging unit ∆ B/5 ~ B Algorithm TSP The shortest tour Algorithm EDF Earliest deadline first Algorithm NETWRAP [12] Charge the sensor with the minimum weighted sum of residual lifetime and travel time Algorithm AA [13] Maximize the amount of energy charged to sensors minus the total traveling energy cost

Experimental Results--∆=B/2 30 Avg energy expiration time per sensor is only 10% of that by the state-of-the-art Travel distance is only 7%~18% longer than that by the state-of-the-art

Experimental Results--∆=B/2 Each sensor is unnecessary to be charged twice, though it is allowed to be charged twice 60% of sensors are charged only once 31

Experimental results-vary ∆ from B to B/5 Sharp decrease of avg dead duration per sensor of algorithm heuristic when decreasing ∆ from B to B/2, slow decrease from B/2 to B/5 Longer travel distance of the mobile charger for a smaller energy charging unit ∆ Alg. heuristic acheives the best trade-off when ∆=B/2 32

Conclusions Unlike existing studies that adopt the full- charging model, we were the first to propose a partial-charging model->significantly shorten sensor dead durations Formulated a problem of finding a charging tour, so as to maximize sensor lifetime while minimizing the charger’s travel distance Proposed a novel algorithm for the problem Experimental results showed that the proposed algorithm is very promising 33

34 Questions ?