Energy Budgeting for Battery-Powered Sensors with aKnown Task Schedule

Daler RakhmatovECE Department, University of VictoriaVictoria, British Columbia, CANADA

daler@uvic.ca

ABSTRACTBattery-powered wireless sensors are severely constrained bythe amount of the available energy. A method for computingthe energy budget per sensing task can be a valuable designaid for sensor network optimizations. This work presentssuch a method that computes the upper and lower boundson the task energy budget for a sensor node that must re-main operational over a specified lifetime, with a knowntask schedule. These bounds take into account nonlinearchanges in the battery voltage, capacity loss at high dis-charge rates, charge recovery, and capacity fade over time.We also propose efficient approximations replacing expen-sive calculations of the battery voltage while computing theenergy budget bounds.

Categories and Subject DescriptorsC.4 [Performance of Systems]: Performance attributes;J.6 [Computer-Aided Engineering]: Computer-aided de-sign (CAD)

General TermsDesign, Performance

KeywordsLow-power design, battery voltage modeling, energy bounds

1. INTRODUCTIONWireless sensor network systems, such as MIT's JLAMPS

[7] and Intel/Berkeley's Motes [6], have been deployed for awide spectrum of applications from environmental monitor-ing to object localization and tracking. Since sensor nodesare usually powered by batteries, efficient use of the limitedamount of the available energy is a critical concern [13]. It isimportant to know the energy consumption limit of a sensorduring its active periods, so that it can remain operationalover a required lifetime.

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission andlor a fee.ICCAD '06, November 5-9, 2006, San Jose, CACopyright 2006 ACM 1-59593-389- 1/06/0011 ..$5.00.

For example, assume that a sensor node must service a se-quence of N sensing tasks, and let tj and dj denote the starttime and duration of task j where j 1, 2 ..., N. The sen-sor must survive till the end of the last task, i.e., the requiredlifetime is (tN + dN). Let Ej denote the energy budget oftask j. For each task, the sensor energy consumption duringtask execution should not exceed the budget; otherwise, thesensor's survival is not guaranteed. If the sensor is regulatedby a DC-DC converter with power efficiency c, then

EJ = 7c dj IjS Vj, (1)

where Ij is the battery current, and Vj is the batteryvoltage. Thus, computing the sensor energy budget is re-duced to computing the battery current and voltage.

For the sake of simplicity, let Ij be constant in the timeinterval [ti t + dj] corresponding to task j and let Vj (t*)denote the voltage value at time t* E [tj, tj + dj]. Then, theenergy budget Ej is bounded as follows:

77C dj Ij min{V(t) |t c [tj, tj + dj]} = E SEj, (2)rj dX .IS -max{K(t*) Cl [t ,t3 +d3]} = Et > E.Symbols ESX and Et in Equation (2) denote, respectively,

the lower bound and the upper bound on the energybudget of task j.

1.1 Battery NonlinearitiesFigure I shows two measured voltage curves for an Itsy Li-

Ion battery [4] discharged at the constant rates of 222 mAand 814 mA. Note that the battery voltage changes non-linearly over time, even in the simplest case of a constant-current discharge. Also, the total delivered charge, or thebattery capacity, at 814 mA is less than the charge de-livered at 222 mA before reaching the 3.0-V cutoff. Thesetwo phenomena, nonlinear battery voltage changes over timeand capacity loss at high discharge rates, must be taken intoaccount during energy budget estimation. The latter effectmay especially be important for sensors drawing relativelylarge currents from the battery during wireless data com-munication.When an active interval (discharge) is followed by an idle

interval (rest), one can observe a so-called charge recoveryeffect: the battery voltage increases during that idle interval.In other words, the voltage does not decrease monotonicallyin general case. Sensors with low duty cycle and relativelyhigh currents during active periods may benefit from chargerecovery [81.

Another phenomenon to be considered is capacity fadeover time - a Li-Ion battery delivers increasingly less ca-

765

Maximram Error = 3274% Average Error = 1 13%

measuredpredicted

3.8

3222 mAO3.6

>D

t3.4-m

:814 mA

INPUT:Set of task start times St {t3 jj 1, 2,2*- NJ,Set of task durations Sd { dj 1,2, ..., NJ,Model M for computing Vj,Battery cutoff voltage V,.t,DC-DC converter efficiency qc.

OUTPUT:Set of battery currents SI {Ij j 1, 2, NSet of energy budgets SE {Ej = 1, 2, N

OBJECTIVE:maxzf Ej.

CONSTRAINT:min{Vj(t*) t* C [tj,tj +idj]} > Vc,t Vj = 1,2,...,N.

Figure 3: Energy budgeting problem.

_______________________________________________________ sensor's survival till (tN + dN), i.e., till the end of the last280 100 200 300 400 500 600 700 task N. We take the lower bound E. as a pessimistic esti-Capacity10 BatteryCurr~ ent xDischag Time,Ah mate of the energy budget Ej for task j. Note that a greater

energy budget for task implies that more work can be com-Figure 1: Battery voltage under constant load pleted during the time interval [tj, tj + dj]. It is desirable

to maximize the energy budgets for all tasks in the sensor'sschedule.

Maximum Error 2.32% Average Error 0.7604.2 Our second objective is to obtain the upper bound E+ >reaauerd4 d=_=eed Ej for comparison purposes. Ideally, the distance between

the lower and upper bounds should be small, indicating thatour estimate of the energy budget (the lower bound) is not

cuD36E_ tX _ overly pessimistic.34 This paper presents a method for computing the energy

budget bounds [E.3 Et] for each task 9. The proposedmethod uses the battery voltage estimates provided by the

3 _ wn_ analytical model reported in [9]. This model is computation-28 2 ally intensive - to improve its time and space complexity we

0 10 20 30 40 50 60Time, min also propose alternative approximations for more efficient

1500 battery voltage calculations. Thus, our contributions areE two-fold: (1) the method for energy budget estimation, and

(2) the efficient approximations for battery voltage calcula->1 500 tions.

To the best of our knowledge, this is the first work wherea 10 20 30 40 50 60 battery voltage nonlinearities are taken into account during

energy budgeting. Related research [3, 1, 5, 2, 12] consideredcharge recovery and high-rate losses, but not capacity fadingor battery voltage changes over time. A general overview ofbattery models and their applications can be found in [11].

pacity as it ages. For example, after 800 charge-dischargecycles at room temperature, a Sony 18650 battery was ableto deliver only 70% of its original capacity [10]. Capacityfade is particularly important in sensors with long lifetimesafter deployment.

Computations of the energy budget bounds using Equa-tion (2) require a model that can predict battery voltagechanges over time while taking into account high-rate losses,charge recovery, and capacity fading. Such a model has beenproposed in [9], and it will be used in this work. The modelis reasonably accurate, as indicated by Figures 1-2 showingvoltage predictions for constant-current and varying-currentexamples. The model is described in Section 3, and its pa-rameters are given in Section 7.

1.2 Objectives and ContributionsGiven the battery model and the task schedule, our first

objective is to obtain the lower bound E3 < Ej for eachtask j = 1, 2,..., N. These lower bounds must guarantee

2. ENERGY BUDGETING PROBLEMThe problem of computing the energy budget can be for-

mulated as shown in Figure 3. Given the task schedule (i.e.,St and Sd), we must first compute task currents Sw such thatthe battery voltage always remains above the cutoff value.Then, we can use Equation (2) to bound the energy budgetfor each task. By taking the lower bounds as the pessimisticenergy budget estimates SE, we ensure that all tasks will beserviced. There may be many possible combinations of bat-tery currents Si resulting in many possible energy budgetsSE The objective is to select a solution that maximizes thetotal sum of budgets over all tasks. Once the energy bud-gets of all tasks are known, a smart sensor can adjust itsconfiguration to meet those budgets on a task-by-task basis.

In this work we consider a special case assuming thatthe current ts the same for all tasks, i.e., Ij = I, VJ =

11 2, ...' N. This restriction considerably reduces the prob-lem complexity. An example of this special case is a periodic

766

Current

1,

dl d

tI t2

tj+load history(H tasks)

I1jHIHI13

d3

iI

I A

-dj

oo

Figure 4: Example of a tas

task schedule shown in Figure 5 (T00,ti,need not be the same for all tasks).

3. BATTERY MODELGiven a sequence of N battery curre

ure 4, model M computes the battery[tj,tj +dj] as follows [9]:

4. BATTERY VOLTAGE COMPUTATIONtN±+N Our battery model M is relatively complex to be used

directly. One disadvantage is the presence of the S-series(infinite functional sums). Another disadvantage is that wemust keep the history of all previous tasks in order to cal-

l| culate Ztk_ IkF,k and Z] IksF . These two sums mustN be recomputed for each j =1 2, ...1 N. For any given task j,

IN-I the sums may be recomputed several times, since a user may* check several time instances t* C [tj, tj +djd], reevaluating thedN1 dN S-series accordingly. Let X denote the time complexity of

_ , . evaluation of the S-series, and let Y denote the maximumtN-I tN lim number of checked t' per task. Then, the resulting time

complexity is O(N2XY). The space complexity is O(N),k schedule since the entire load history must be stored.

To speed up computations and reduce storage requirements,we propose alternati've approximations that avotd ustng the

and Tdlw,, however, series and the entire load history. These approximationsare, in fact, the lower and upper bounds, V- and V+ forthe battery voltage computed by model M.The bounding approximations for VJ (t') are as follows:

nts as shown in Fig-voltage at time t* e

VK (t*) V0 rIl

y+ _yP )t + ln1 +Ij Fnj+Z_1sk Fnk ]

where the F-factors are given by

Fj e--7ntj-e-7ny* + Sj

oFnke anti k~i+d5) ±Sr7a k e- anSand

Pi

p tp

t jFpj +05pjF2,,

(9)VK (t*) = V n-rIj-0k(y +7±)t* + ln n ++-1 nF+ ]ri 1F ZLi'3=lltJ

(3)

K} (t ) = Vo -lj-

[(an + 7p)t*+ Ilna +IjFF ZkjIs F-,J

(4)

Fpk = Ip(ik±dk) Pt + Spk.

The functional S-series in Equations (5)-(4) are defined inTable 1. There are nine battery parameters of interest: Vo,r, 0, a>, atp A%, ,np , and 'yp - their detailed descriptioncan be found in [9]. Briefly, a quantifies battery capacity,quantifies high-rate losses, and quantifies capacity fading.

If jnP -X oo (no high-rate losses), then each S-series eval-uates to zero:

VJ(t )Ls,pO =Vo-Irl-j (yK+-yp)t*C+ (6)

a e -7ntj +1j_j1 I* e-pntk --Yn(tk+dk

e4p _7p t J ._Y e-yp ltk~+dk ) _e-yPtko-I 'p -k 1 kh -Yp

On the other hand, if P -X 0 (no capacity fading), thenthe battery model is reduced to the following form:

VK (t*W) ln,p o = Vo-rI- (7)

q ln a,,+Ij (t +S,,,)+ZZ 115(d5+S,,5)tip- 1(t*-tj+St,)-5j1 =1 Ik (d +Sp k)

If there are no high-rate losses and no capacity fading(ideal battery), then:

Vj(t*)I1rideal = Vo -r1 a-ln t3)Zk= I lkdk

tj ) ZkE =1tikdke8(8)

(10)

These approximations use the bounds for the F-factors,which are based on the respective bounds for the S-series(see Table 1).1 For example, when computing FAk we useEquation (4) with S+;.

In Table 1 one can see that the infinite sums (functionalseries) are replaced by M-term sums where necessary. Also,at most H previous tasks are considered, instead of the en-tire load history. Hence, the corresponding time and spacecomplexities become O(NHMY) and O(H), respectively.The values of M and H are a user's choice - greatervalues will yield better approximations, but will re-quire more computational time and space.

5. BATTERY CURRENT COMPUTATIONOnce the battery voltage reaches the cutoff value V,,t,

the sensor is no longer operational. Recall that, for thesake of simplicity, we assumed that Ij I1, Vj = 1, 2, ..., N.Our objective is to maximize the value of I, which in turnmaximizes the energy budget. To ensure that the sensorsurvives all tasks, we must guarantee that VK(t') > Vce forany t* c [tj, tj + dj ], Vj = l: 2, ..., N.Computation of V(t*) using model M, given by Equa-

tion (3), is expensive. Instead we will use the lower boundVK-(t*) < V(t*) while computing the battery current. Toensure solution feasibility it is sufficient to guarantee thatVK (t*) > V02t.We perform the following three steps to find I that will

be used for computing the task energy budgets.

tDerivations of the upper and lower bounds for Sk and Spkfrom Table 1 are omitted due to the lack of space.

767

.1,"11

I

tj tjL j-H

Definitions:

|Sn 2e6-Yt z: 1 e (3. -I+yp)(t tj)3 rn 0. _2 2y2

S = m° Yn4tke+dk) e /3nm e- Ytttkp 2-+tUm -tk)

Sp - 2~5~ 1 eJp(tk+dk).e-3p2(tt k dk) -elP -tk. 0pnm2(ft tk)Spk = 2ml O3m2 +_Yp

Assumptions:

> M> 1<H<N- T* =tN+dN

Tk_ f tk+Hd+H 1 <k<j HTk l t* : j-H<k<j

Lower Bounds for Snjv and Snk:

S- 2 nt'EsM 1-,-(pumz-2-n)(t*-t5n, 2e 12m=

--n (tk tdk ),-e- n m 2 (T* -tk -dkA) _e- ntk e-On m2 (T*t tk)5nk - nZn1 Am2 i-Yn

Lower Bounds for SPJ and Spk:

S- 2 & '* EwM 1-e- (%Pm2 +JYp)(t* tj)

2S yjp(tk+dk) e_-3pm2 (T -tk-dk) -jIPtk. e-3Pm (T -t k)

pF- 2 Ym=l f3m+yS3pm2m

Upper Bounds for S7Tj and Snk:

S -2 ynF7r ImM e- 'm t3)]e L 6(3:m) zZt 1 3 2 _

gur _ 27r2e- n (tk±+dk) -e-fn (Tk-tk- ) M e-7ntk -e-¾n) 2 (T*-tk)s ~ ~~~~~~~2ZMmit3 2~t(Fnk V½tio iOn 43n(Tk tk- J) t1 m T3(0. --Y.) %/-Oe -4d h-h -dCh) 2E =l 7 -?

Upper Bounds for Sp and Spk:

S+.*-2ci I2 VF M e-(±pm2+vp)(t -fi 1St [Lip Ltm= 1 pm Ti 2

S+ 27rreYP(t k±+ dk) _T- tk) 2vM eYPtk e-3pm2(T tk)p 3Lp O-40 (Tk -tk dk) Lm2 1 LP m2 +p

Table 1: Definitions and bounds for S-series.

768

Step 1: As VN (tN + dN) > VN (tN + d), we safely let

Vcut = Vj(tN + dN) = Vo - rI- (11)4@xn±R+ 7p)(tN + dN)±lnaIF+In-' k?-1IF]

Note that t* = tN+ dN. Equation (11) has only one un-known, current I- its value can be found numerically.Since Expression (11) is monotonic in 1, we can use a fastbi'nary search to find the unknown current value.Step 2: As VN(tN) > VN (tN), it is sufficient to test

whether VN (tN) > 1cut given the current value I foundin Step 1. If the test fails, i.e., VN (tN) < VCu,, we mustdecrease I so that

V,, = VN (tN) = VO -ri-

X K(7 + ap)tN+ln ;I =& KIF$s]

(12)

In other words, unless VN (tN) > VWrt, we must performanother binary search for a smaller battery current I usingExpression (12). Note that t' = tN, and according to Equa-tions (4)-(5), FN = FPN 0 (from Table l, SnN = SpN = 0if t = tN).Step 3: In the previous two steps, we checked the con-

traint VN(t*) > Vc, for t* = tN and t* = tN + dN. Strictlyspeaking, we should also check other times instances t* C(tN,tN + dN). Based on our numerous observations, weconjecture that if the voltage is above the cutoff value atthe time boundaries of a constant-current task, then it isabove the cutoff value anywhere between those boundaries.In other words, if for some task j we have VJ (t3) > VC,,t andVj (tj + dj) > V14a, then checking whether Vj(t*) > V,et fortU G (tj, tj + dj) is not necessary. However, this conjecturehas not been proven formally. Consequently, this potentiallyredundant step is irncluded irn the algorithm. We first par-tition the time interval (tN,tN + dN) into Y subintervalsof equal length A. Then, for each y = l,2, ..., Y, we lett*= tN + yA and check whether V,N (t*) > V,,,t If the testfails, we decrease I so that VN (t*) = V1rt.

In general, considering only the last task j - N is notsufficient - recall that it is sometimes possible for recoveryeffects to mask constraint violations of the earlier tasks. Toensure solution feasibility, we repeat the above three stepsfor all tasks in reverse order, reducing the value of I asneeded. In practice, however, it is very unlikely that task Nsurvives, while one of the earlier tasks j N -1, N -2, ..., 1has failed. In other words, the original value of I found usingExpression (11) will rarely be recomputed.Note that the current I computed based on V = V,,t

(i.e., using our approximations) will be less than the optimalImal computed based on Vj VV,,, (i.e., using our modelM directly). However, since our approximations require lesscomputational time and space than M, the complexity ofcomputing I will be lower than the complexity of computing-max .

6. ENERGY BUDGET COMPUTATIONSOnce current I is known, we compute the lower bound EJ-

and the upper bound Et of the energy budget as follows:

EP, = q dj I min{jV/ (t *) |t c [tj,tj +d]}j (13)EF> q dj .I.max{KV.+ (t') I t' C [tj, tj + dj]}

A

t I t2

Tactive T idle Tactive1- -1o<

t N-1 tNTime

Figure 5: Example of a sensor load.

S. N Tactive, Tidle, Total, Active, Dutymin min h h Cycle

Si 3000 1 99 5000 50 0.01S2 3000 1 9 500 50 0.1S3 3000 0.1 0.9 50 5 0.1S4 300 0.1 9.9 50 0.5 0.01S5 300 0.1 0.9 5 0.5 0.1

Table 2: Five scenarios under consideration.

Again, instead of using our model M to compute Vj di-rectly, we have used our approximations K.- and VK+ in or-der to speed up computations. This computational speedupcomes at the expense of solution optimality, since we letE = E:P, and since K- <V<§ I < I However, theuse ofVK- instead of Vj while computing the battery currentsand energy budgets is optional, i.e., it is a user's choice totrade off solution optimality for computational complexity.

If our conjecture (see Section 5, Step 3) is true, then thelower bound EP- can be computed using a simpler equation:

EJ -= rc d} I rmin{V- (tj), Kj- (tj + dj)}. (14)

Recall that we use the lower bound EJ as the energybudget, and compare it against the upper bound Et. Ifthe difference between these two bounds is small, then theenergy budget is not overly pessimistic. Since the upperbound E+ is used only for comparison purposes, its accuracyis not critical. It can be computed using a simpler equation:

E+ = qr dj I*J max{V+ (tj), V+ (tj + dj)}. (15)

7. EVALUATION RESULTSTo illustrate the use of the proposed energy budgeting

method we consider an example of a periodic sensor loadshown in Figure 5. A sensor is activated N times: duringeach period it is active for TaCti,r minutes and idle for Tidl,minutes. We study five sensing scenarios shown in Table 2,letting the sensor lifetime range from 5 hours to 5000 hours.

For each scenario, we consider three battery variants shownin Table 3. For all three batteries, Vo = 3.76 V, r = 400 mQ,and 4) = 0.125 V. For battery Bl, we further let ca, = 15mAh. a. = 655 mAh. = 2.50 1/minm and 'p = 0.501/min. These values were used to predict measured batteryvoltages shown in Figures 1-2. Also, we let both -y. and-yp equal 0.000017 1/min. These values were obtained byfitting the cycling fade model from [9] to the experimental

769

..........w

Current

O * O

Tactive T idle Taa.-1- .1.

B. atn aI , on11 P 7n, T P:mAh| mAh 1/min 1/min 1/min 1/min

Bi JJ 15 655 2.50 0.50 0.000017 0.000017B2 15 655 1.25 0.25 0.000034 0.000034B3 15 655 00 00 0 0

N = 3000 Period = 100 mDnDuty Cycle = 0.01

B3* B1

Table 3: Three batteries under consideration.

data from [10] for a Sony battery at the room temperature(the maximum error was 3%).Battery B2 is derived from Bl by halving 3,,, and dou-

bling >r , which makes B2 more lossy than Bl. Battery B3represents an ideal case.

For our approximations we let M = H = 10. Also, weused Equations (14)-(15) while computing E- and E+.

7.1 Performance of Voltage ApproximationsThere are two contributors to the difference between the

lower and the upper bounds of the energy budget. First isthe task duration and current - a longer tasks with a highercurrent will result in a greater difference between Vj (tj) andVJ (t- + d -), which leads to a greater difference between theenergy budget bounds Et and E.. This first contributor isspecific to the task characteristics, not voltage approxima-tions. Recall that battery B3 requires no approximations,yet the energy budget bounds do not match - the greaterthe task charge (I. Tact1ve), the greater the difference be-tween Et and E For example, scenarios S4 and S5 havethe highest charge consumption per task (7.12 Coulombs),and consequently, the worst maximum difference of 0.42%among the cases involving battery B3.The second contributor is the quality of the proposed volt-

age approximations that can be seen in ten cases involvingbatteries Bi and B2. In six of those cases, the bound dif-ference is under 5%, which implies that our approximationswere somewhat accurate in comparison with modelM givenby Equation (3). In other words, if we were to use batteryvoltage model M directly, our exact results would matchapproximated solutions within 5%. The other four cases didnot perform well - in two of them (S3-B2 and S5-B2) eventhe average error exceeded 5%o. Such a poor performanceis due to the overly pessimistic lower bound.2 The use ofour approximations is not justified in those cases, unless thecomplexity of model M is an important concern.

7.2 Performance of BatteriesThe battery voltage is usually lower for later tasks, so the

energy budget is reduced accordingly. For example, in caseS3-B3 the budget of the first task is 3.27 J, while the budgetfor the last task is 2.36 J, or 28% less. Smart sensors mustbudget their energy consumption accordingly, compensatingfor changes in the battery voltage.

Cases Sl-BI and S1-B2 illustrate the effect of capacityfading over time. Compared to the 13.0-mA current for theideal battery B3, the battery current is only 9.9 mA for Bi(24% less) and 7.4 mA for B2 (43% less). The task energybudgets (lower bounds) for these three cases are shown inFigure 6. Note that scenario SI corresponds to the longestsensor lifetime of 5000 h, and capacity fading can no longer

2The output produced by model M was much closer to theupper bound than to the lower bound.

LUj 2-_

.~~~--e..........,

1

0 500 lO00 1500 2000 2500 3000

Task

Figure 6: Task Energy Budgets for Scenario Sl.

be ignored. Relatively short lifetimes of the other scenariosdo not expose fading effects.

Shorter sensing times usually lead to higher allowable dis-charge currents. For example, the sensing time of scenariosS1 and S2 is 50 h, while the sensing time of scenarios S4and S5 is only 0.5 h. This leads to a nearly 100-fold currentdifference. Due to high-rate losses, the latter scenarios needspecial consideration. Even though scenarios S4 and S5 havethe same sensing time, S4 performs better than S5 becauseof the 11-fold difference in 1tidl (9.9 min for S4 versus 0.9min for S5). Longer idle intervals allow for greater chargerecovery that mitigates high-rate losses. In other words, ca-pacity loss due to high battery currents can be lessened byenforcing very low duty cycles.Not surprisingly, battery B2 performed worse than Bi

(less lossy), i.e., the energy budgets in cases involving B2 areworse than those involving Bi. Also, note that the bounddifferences in cases S3-B2 and S5-B2 are worse than those incases S3-1Bi and S5-1Bi. Ideal battery B3 is a clear winner inTable 4, as it has no losses and requires no approximations.

7.3 General RecommendationsTime and space savings due to the lower complexity of the

approximations may be particularly important when energybudgeting is performed by a sensor itself, e.g., in responseto the change in a sensing scenario or a battery. However,the accuracy loss may exceed the acceptable error margins,which may lead to overly pessimistic energy budgets. In suchcases, which are identifiable by a large difference betweenthe lower and upper bounds, a user should work directlywith the battery voltage model M despite its complexity.The proposed methods for bounding voltages and energybudgets are likely to perform better for:* batteries with larger ~,B7,p (less high-rate losses),* batteries with smaller ya,p (less capacity fading),* scenarios with shorter Tactive,* scenarios with longer Tidl,.Last, we point out that the quality of energy budgeting

depends on the quality of the battery model. Since no modelperfectly matches reality, the computed energy budget val-ues should be used with reasonable caution.

770

Case First Task f Last Task iSE- 1i2,2 N...,JVScenario- Current Lower Bound Upper Bound Lower Bound Upper Bound Maximum AverageBattery 1, mA E1, J E+:J f EN, L E+NJ f % %SI-Bi 9.9 2.51 2.51 1.87 1.88 0.62 0.03S1-B2 7.4 1.88 1.88 1.38 1.39 0.89 0.04SI-B3 13.0 3.30 3.30 2.45 2.46 0.17 0.01S2-B2 12.7 3.22 3.22 2.33 2.36 1.45 0.04S2-B2 12.3 3.12 3.12 2.33 2.35 f 0.99_| 0.05S2-B3 13.0 3.30 3.30 2.45 2.46 0.17 0.01S3-Bi 126.9 1 3.17 3.18 ft 2.29 T 2.48 ft 8.60 1.07S3-B2 1115.3 2.88 2.90 ft 2.08 2.29 ft 14.92 7.28S3-B3 130.5 1 3.27 3.27 ft 2.36 2.36 ft 0.35 0.01S4-B I 1142.5 1 5.29 25.89 ft 20.57 20.98 ft 2.33 0.76

|S4-B2 |1105.1 24.41 25.13 ft 19.89 20.56 ft 3.82 1.24S4-B3 1186.1 1 26.63 26.74 ft 21.35 [ 21.38 ft 0.42 0.08S5-BI 990.3 22.33 22.79 ft 17.83 19.11 f 7.19 2.74S5-B2 609.1 14.32 14.58 ft 10.96 12.88 ft 17.50 8.33S5-B3 1186.1 JJ 26.63 26.74 11 21.35 21.38 ft 0.42 0.08

Table 4: Energy budget bounds for the first and last tasks and relative bound difference over all tasks.

8. CONCLUSIONBattery-powered wireless sensors are severely constrained

by the amount of the available energy. This paper describeda method for computing the lower and upper bounds for theenergy budget per task for sensors that must survive for aspecified period of time. We assumed that the sensor's taskschedule was known and each task drew the same currentfrom the battery.The proposed bounding method used an analytical model

for predicting the battery voltage given the discharge cur-rent, while taking into account battery capacity loss at highdischarge rates, charge recovery, and capacity fade over time.Since it is computationally expensive to use this model di-rectly, we also proposed alternative approximations for moreefficient voltage calculations.We considered three variants of a Li-ion battery and five

sensing scenariosl resulting in 15 different cases. The com-puted energy budget bounds were within 5% of each otherin 11 cases. Our future effort will target better voltage ap-proximations and less restrictive assumptions, e.g., allowingfor varying battery current across tasks.

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