Near-Optimal Energy Allocation for Self-Powered Wearable Systems

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1 Near-Opimal Energy Alloaion for Self-Powered Wearable Sysems Ganapai Bha, Jaehyun Park and Umi Y. Ogras Shool of Elerial Compuer and Energy Engineering, Arizona Sae Universiy, Tempe, AZ {gmbha, jpark244, Absra Wearable inerne of hings (IoT) devies are beoming popular due o heir small form faor and low os. Poenial appliaions inlude human healh and aiviy monioring by embedding sensors suh as aeleromeer, gyrosope, and hear rae sensor. However, hese devies have severely limied baery apaiy, whih requires frequen reharging. Harvesing ambien energy and opimal energy alloaion an make wearable IoT devies praial by eliminaing he harging requiremen. This paper presens a near-opimal runime energy managemen ehnique by onsidering he harvesed energy. The proposed soluion maximizes he performane of he wearable devie under minimum energy onsrains. We show ha he resuls of he proposed algorihm are, on average, wihin 3% of he opimal soluion ompued offline. I. INTRODUCTION Advanes in low power sensor, proessor and wireless ommuniaion ehnologies enable a wide range of wearable appliaions. For insane, small form faor and low os IoT devies offer a grea poenial for non-invasive healhare servies whih are no limied o any speifi ime or plae [3, 7]. Exiing, and possibly pervasive, appliaions inlude healh monioring, aiviy raking and gesure-based onrol [29]. However, small form faor and low os onsrains severely limi he baery apaiy. Therefore, harvesing ambien energy and opimal energy alloaion are ruial for he suess of wearable IoT devies. Energy limiaion is one of he major problems faed by wearable appliaions. Bulky baeries are heavy and inflexible, while small prined baeries have modes ( mwh/m 2 ) apaiy [24, 25, 4], whih requires frequen harging. Therefore, i is imperaive o exploi ambien energy soures suh as ligh, moion and hea. Reen sudies show ha phoovolai (PV) ells an provide. mw/m 2 (indoor) mw/m 2 (oudoor) power [39]. Similarly, human moion and hea an generae.73 mw/m 3 [] and.76 mw/m 2 power a T = K [22], respeively. Energy harvesing an be pariularly effeive for wearable devies, sine hey are inherenly personalized. For example, he devie an easily learn he expeed energy generaion and onsumpion paerns based on daily aiviies. Therefore, we adop energy harvesing as he primary soure. A he same ime, he inermien naure and urren soure behavior of he energy soures neessiae an energy sorage elemen, suh as a baery and super apaiane [3]. In his work, we uilize rehargeable flexible baeries as a reinforemen o provide a smooh qualiy of servie and bakup, in ase he harvesed energy falls signifianly below expeaions. The baeries we employ offer 48 mwh apaiy a a 2 35 mm 2 fooprin, have 2 mm hikness, and weigh.7 g [3]. The primary goal of his work is o provide reharge-free wearable IoT devies ha maximize he qualiy of servie (QoS). To ahieve his goal, we propose a dynami energy opimizaion framework wih a finie ime horizon. The proposed framework hannels he generaed power beween he baery and he IoT devie, while enforing minimum and arge energy onsrains o guaranee reharge-free operaion. The fundamenal omponens of he proposed framework are illusraed in Figure and desribed below. Inpus and objeive: The inpus o our opimizaion framework are he iniial baery energy and he expeed energy harvesed paern. In addiion, we also speify he minimum baery level allowed a any poin in ime and he baery energy arge a he end of he day. The minimum energy onsrain ensures ha he baery always has a reserve o perform emergeny asks. Similarly, he energy level arge ensures ha he baery will have a desired level of harge a he end of eah day. Our goal is o opimize he work performed by he IoT devie, alled he uiliy, under he baery level onsrains. We measure he uiliy using an inreasing funion of he energy alloaed o he IoT devie. This hoie apures he fa ha more energy alloaion would lead o a larger uiliy. A he same ime, i is more general han simply maximizing he alloaed energy iself, sine alloaing more energy may have a diminishing rae of reurn. Dynami opimizaion wih 24-hour horizon: The firs omponen of he proposed soluion is a finie horizon dynami opimizaion formulaion, as represened by he green paerned box in Figure. We se he finie ime horizon as 24 hours, sine he energy harvesing paern and user aiviies are repeaed on a daily basis wih poenial day-o-day variaion. The 24-hour horizon is divided ino equal inervals, e.g., one hour or one minue epohs. We derive a losedform soluion ha gives he opimal energy alloaions for eah ime inerval during he day by using Karush-Kuhn-Tuker (KKT) ondiions [23]. The opimaliy of his soluion is guaraneed if he expeed energy harvesing paern mahes wih he aual generaed energy. However, here are iner-day and iner-inerval variaions in he generaed energy due o environmenal ondiions. Therefore, we also need o perurb he energy alloaions ompued using expeed values. Perurbaion in eah inerval: The energy alloaions ompued a he beginning of eah day deviae from heir opimal values due o unerainies in he harvesed energy and load ondiions. Therefore, we also perform runime opimizaion by aking he differenes in he expeed and aual energy values ino aoun. For example, suppose ha one-day horizon is divided ino 24 one-hour inervals, and he energy harvesed during he firs hour is less han he assumed value. We ompue his differene a he end of he firs hour. Then, we refle i in he energy alloaions ompued for he res of he inervals on ha day. In his way, he deviaion from he opimal alloaions are reified a every inerval. As a resul, we oninuously adap o he hanges in he environmenal ondiions wih negligible runime overhead. Learning he daily paerns: Throughou he day, we keep rak of harvesed energy in eah inerval, and use his daa o find he expeed energy harvesing paern. Similarly, user moion paerns reveal low and high aiviy periods (e.g., sleep and exerise imes). Daily averages of his daa are fed o he proposed framework. Then, his daa is used o guide he energy alloaions, suh as alloaing

2 Eah Inerval Moion Sensor Uni CPU BLE #$"6 ",2.3) PV-Cell Energy Harvesing Uni Opimal Energy % Alloaions! "##$% (') Runime Opimizaion %! "##$% (') Wearable IoT Devie +! "%)*"# %! "%)*"# Energy & Aiviy Profiling EMG Power measuring poins 4")).,5 Baery Expeed Energy Harvesing Iniial Baery Energy Eah Day Finie Horizon (24-hr) Opimizaion / Baery Energy Targe! )",-.) (') Fig. : The proposed hardware arhieure and energy harvesing framework. The energy harvesing uni hannels he generaed urren beween he IoT devie and baery. Our proof of onep prooype uses PV-ell as he ambien energy soure, bu he proposed framework an work wih muliple energy soures. minimum energy during sleep, as desribed in Seion IV-C. We demonsrae he proposed framework using he hardware prooype presened in Seion V-A. Our prooype employs flexible PVells o harves energy from ambien ligh. The effeiveness of our opimizaion algorihm is evaluaed for differen user aiviies and energy harvesing paerns obained from an online daabase [2, 38]. The proposed runime algorihm is near opimal, sine he aual harvesed energy in a given inerval may be differen han he expeed value. Therefore, we ompare our resuls wih he maximum ahievable uiliy ompued using an orale and an offline opimizaion algorihm [2, 3]. We show ha he uiliy obained by our runime opimizaion approah is wihin 3% of he opimal uiliy, whih is no feasible sine i assumes an orale. Moreover, our resuls onverge o he opimal soluion as he differene beween he harvesed energy and is expeed value diminish. The major onribuions of his work are as follows: We presen a losed-loop soluion for finding he opimal energy onsumpion of a self-powered IoT devie when he amoun of harvesed energy is known a priori, Sine he aual harvesed and onsumed energy may differ from heir expeed values, we propose a novel runime algorihm wih onsan ime omplexiy for seing he energy onsumpion in finie horizon, We demonsrae ha our resuls are, on average, wihin 3% of he opimal soluion ompued offline for a wide range of praial senarios using a hardware prooype. We also show ha he proposed algorihm inurs negligible power onsumpion and exeuion ime penaly. The res of he paper is organized as follows: We review he relaed work in Seion II. We presen he preliminaries and he proposed algorihm in Seion III and Seion IV, respeively. Finally, we disuss he experimenal resuls in Seion V, and summarize he onlusions in Seion VI. II. RELATED WORK Wearable IoT devies have reenly araed signifian aenion due o advanes in sensing, low-power proessing, ommuniaion proool and radio ehnologies [5, 26]. In pariular, flexible hybrid eleronis ehnology offers a grea poenial for sensor-rih wearable appliaions [5, 4, 2]. Limied baery apaiy of wearable devies has led o he sudy of energy harvesing. Major omponens of an energy harvesing sysem are he energy soure, sorage, harvesing irui and harvesing-aware power managemen [3, 34]. Solar energy harvesing using PV-ells is one of he mos promising ehniques adoped by many reen sudies [, 28, 3]. Body hea and moion an also generae energy wih he help of hermoeleri [6, 35] and piezoeleri sensors [7, 32], respeively. Energy harvesing aware power managemen for wireless sensor nodes has been sudied exensively in reen years [9, 2, 4]. In pariular, he work in [2] presens a general framework for inluding energy harvesing in power managemen deisions. The auhors maximize he duy yle of a sensor node using a linear program formulaion. To avoid solving a linear program a runime, he auhors also presen a low-omplexiy heurisi o solve he linear program. Similarly, a linear quadrai raking based algorihm ha adaps he duy yle of he sensor node is presened in [4]. The auhors minimize he deviaion of he baery level from a speified arge. However, hese soluions do no onsider he appliaion requiremens when uning he duy yle of he nodes. Conurren ask sheduling and dynami volage frequeny sheduling is proposed o inrease he lifespan of energy harvesing sysems in [27]. A he beginning of eah ime inerval, heir algorihm refines he solar irradiane esimaion and adjuss he ask sheduling, bu i is unable o orre fuure energy alloaions. To ahieve long-erm reharge-free operaion, a design-ime apaiy planning and runime adjusmen mehod is presened in [6]. Their mehodology derives he baery apaiy ha an saisfy uninerruped operaion for a year. During runime, he duy raio of he devie is hanged based on he daily operaion hisory. However, his approah only reas o he harvesed energy variaions, hus leaving room for improvemen. In wearable IoT appliaions, energy an be opimized by onsidering he user aiviy and appliaion haraerisis. Our proposed approah learns he energy harvesing and user aiviy paerns. We firs alulae he opimal energy alloaion using a losed-form formula, assuming expeed harvesing paern. Then, we propose a novel runime algorihm ha boh revises he opimal alloaion dynamially and redisribues he slak from he previous inervals. III. PRELIMINARIES AND OVERVIEW We divide he one-day horizon ino T equal inervals. For example, he baery energy illusraion in Figure 2 assumes T =24, i.e., eah inerval is one hour long. The proposed approah does no pu any onsrains on he level of granulariy, provided ha he overhead of he runime energy alloaion alulaions is negligible. Energy onsrains: The baery energy a he beginning of any inerval is denoed as for apple apple T. The proposed Our implemenaion runs wih one-minue inervals wihou any signifian overhead.

3 approah an work wih muliple ambien soures suh as a PVell, hermoeleri generaor and a piezoeleri devie. In our experimens, we use a ommerial PV-ell as he ambien energy soure [8]. Suppose ha he harvesed and onsumed energies in inerval are given by E H and E, respeively. As illusraed in Figure 2, he baery energy dynamis an be expressed as: u (E ) + = + E H E, apple apple T () where is used o model he losses of he baery and power managemen iruiry inluding he PV ell and volage onverers. The effiieny is ime varying sine i is a funion of generaed urren. Regardless of he harvesed energy, he IoT devie should have enough reserves o perform an emergeny ask, suh as deeing a fall and sending an emergeny signal. Therefore, we se a minimum baery level onsrain E min. Similarly, we onsrain he energy level a he end of he day from below, suh ha here is suffiien reserve for he nex day. Hene, he onsrains on he baery energy level are given as:! ( #! " #! ' #! $%& E B T E arge and E min 8 apple apple T (2) Baery Energy (J) * +! +,! Baery energy arge Minimum baery energy T=24 Fig. 2: Illusraion of he baery level ompuaion for T = 24 hr horizon. Driver appliaions and he uiliy funion: Alhough he proposed framework does no depend on any pariular appliaion, we onsider healh monioring and aiviy raking as he driver appliaions. We monior he user aiviy using a moion sensor uni ha inegraes an aeleromeer and a gyrosope. We also employ iruiry for realime aquisiion of physiologial signals suh as eleromyography (EMG) and eleroardiogram (ECG). These signals are sampled and proessed by a miro-onroller uni (MCU). The proessing resuls are ransmied o a personal devie, suh as a smarphone, using Blueooh Low Energy (BLE) proool. The energy requiremen of he arge appliaion is deermined primarily by hree faors. The firs one is he aive power onsumpion P a(f ) as a funion of he proessing speed f during inerval. In our driver appliaions, his inludes sampling he sensors, proessing he daa in real-ime, and poenially ransmiing daa hrough BLE onneion. The oher faors are he duy raio, i.e., he perenage of ime he appliaion is aive, and he idle power onsumpion P idle. Wih hese definiions, he average appliaion power onsumpion in a given inerval an be wrien as: P = P a(f )+( )P idle (3) A given arge appliaion needs a minimum duy raio min and operaing frequeny f min o aomplish is performane requiremens. For example, i may need o guaranee a erain number of measuremens per uni ime. We use hese requiremens o ompue he minimum energy M E ha should be alloaed for eah period. Alloaing more energy an improve he QoS by delivering higher hroughpu, while less energy alloaion means lower QoS. We define M E Fig. 3: Illusraion of he appliaion uiliy funion. a uiliy funion ha expresses he qualiy of servie in erms of M E o apure his behavior. For illusraion, a linear uiliy funion E M E is ploed in Figure 3. In general, alloaing more energy has a diminishing rae of reurn, while alloaion under M E degrades qualiy a a faser rae. Hene, we employ a parameerized and generalized he uiliy funion ha apures his behavior as illusraed in Figure 3: u(e E )=ln (4) M E where he parameer is used o une he uiliy funion for a speifi user or appliaion. We noe ha he algorihm presened nex works wih any uiliy funion ha is onave and inreasing. The major parameers used in his paper are summarized in Table I. A. Problem Formulaion E IV. OPTIMAL ENERGY MANAGEMENT Our goal is o maximize he uiliy over a one-day horizon under he energy onsrains explained in Seion III. Hene, we an formulae he opimizaion problem using Equaions 4 as follows: maximize U(E,E...E T )= TX = ln E M E subje o + = + E H E apple apple T + E min apple apple T E B T E arge In his formulaion, we ompue he oal uiliy as he sum of he uiliies in eah inerval. A posiive disoun faor < apple is added o enable bias agains disan inervals. The opimal soluion o he problem given in Equaion 5 an be found offline using dynami programming [4]. However, i requires solving a se of T nonlinear equaions, whih is ompuaionally expensive for a runime algorihm. Furhermore, i relies on he knowledge of he energy ha will be harvesed in he fuure inervals (i.e., E H for apple apple T ). In wha follows, we propose a wosep soluion based on wo insighs ha enable us o overome hese hallenges. The proposed soluion leads o a near-opimal runime algorihm wih a omplexiy of O(), i.e., he omplexiy does no grow wih he ime horizon or he number of inervals. B. Opimal Closed-Form Soluion wih Relaxed Consrains The proposed soluion relies on wo key insighs: Key insigh-: We an derive a losed-form analyial soluion o his opimizaion problem, if we enaively ignore he minimum energy onsrain. Obviously, he revised soluion is no guaraneed o saisfy he minimum energy onsrain E B E min. However, we an enfore i a runime a he expense of loss in opimaliy. (5)

4 Symbol T TABLE I: Summary of he major parameers Desripion Number of onrol inervals in he finie horizon > Disouning faor for uiliy E min,e arge P,f M E Minimum and arge baery energy onsrains Power onsumpion of he IoT devie in inerval Duy raio and frequeny of he IoT devie in inerval Minimum energy required for posiive uiliy A posiive parameer o onrol he shape of he uiliy funion E H,E Harvesed and onsumed energy in inerval Baery energy a he beginning of inerval h, Deviaion from he expeed values of harvesed and onsumed energy Therefore, we find he losed-form soluion a he beginning of eah day. Then, he energy alloaions are adjused a he beginning of eah inerval, as desribed in Seion IV-C. Key insigh-2: We anno rely on he knowledge of energy ha will be harvesed or onsumed hroughou he day. However, we an learn he expeed paerns by profiling he generaed energy during eah ime inerval. This enables us o derive he opimal alloaion for eah inerval a he beginning of eah day by uilizing he expeed values. Similarly, he aual energy onsumpion may be differen han he opimal alloaion, as deailed in Seion IV-C. Therefore, we ompare he aual generaed and onsumed energies wih heir expeed values. Then, we use he differene o perurb he energy alloaions for he remaining inervals, as desribed in Seion IV-C. Sine we relax he E min onsrain, here may be ime inervals during whih he baery level drops below he minimum hreshold. Furhermore, he proposed approah an over- or under-alloae energy due o unexpeed hanges in he harvesed energy, unlike an orale-based offline opimizaion. However, hese effes do no propagae beyond one inerval, sine he proposed approah reifies over- and under-alloaions a he beginning of he nex onrol inerval. For he oninuiy of he disussion, we firs summarize he losed-form soluion wih relaxed onsrains below. Closed-form soluion: When we relax he minimum energy onsrain and assume expeed values for he harvesed energy, he opimal energy alloaion for eah inerval an be found as follows: Firs inerval : E = EB E arge + P T = EH T (6) Subsequen inervals : E + = E apple apple T The derivaion is presened in he Appendix. Noe ha he denominaor an be ompued a priori, and he oal expeed energy ha will be harvesed is available hrough profiling. Therefore, his losed-form equaion enables us ompue he energy alloaions wih onsan ime omplexiy. Nex, we explain how we employ his soluion o design a runime algorihm. C. Near-Opimal Runime Soluion This seion presens our novel algorihm ha builds on op of he losed-form soluion given by Equaion 6. The proposed algorihm perurbs he opimal alloaions found using he expeed energy values and enfores he minimum energy onsrains a runime. ) Unerainy in Expeed Energy Values: The aual energy harvesed a runime may differ from he expeed value due o faors suh as environmenal ondiions. Effiieny in soring he harvesed energy also adds o he unerainy, sine i varies wih he load. We represen he differene beween he aual energy generaion and he H. H expeed value by > ( < ) means ha aual energy harvesed during inerval is larger (smaller) han he expeed value for ha inerval. An IoT devie uses he energy alloaion arge for a given inerval o ompue he average power onsumpion allowed in ha inerval. Then, i finds he duy raio and operaing frequeny using Equaion 3 as summarized in Seion IV-C4. However, he aual energy onsumed a he end of he inerval may be differen from he arge. We subra he aual onsumpion from he alloaed energy o find he differene. Similar o he differene in he harvesed energy, > means a surplus, < means ha more energy han he alloaed arge is onsumed. Hene, he differene beween he expeed energy aumulaion and he aual values an be wrien as: = H + apple apple T (7) When is posiive, he energy surplus an be used during he remaining inervals. Oherwise, he onsumed energy is more han he alloaed arge. Therefore, he defii should be refleed in he remaining inervals. 2) Perurbaion of he Alloaed Energy Values: We need o adjus wo quaniies o aoun for he unprediable dynami variaions. Firs, he opimal soluion given in Equaion 6 needs o be orreed in ligh of he new daa available a he end of eah inerval. Seond, he over or under expendiure in he previous inerval should be disribued o fuure inervals. Correing he Fuure Alloaions: Suppose ha we adjus he opimal alloaion a he beginning of he ime inerval. The differene in expeed and aual energy aumulaed over earlier inervals {,,..., } are known a his poin. Therefore, he adjused alloaion for inerval an be found using Equaion 6 as: E = E B E arge + P T k= EH k + P k= k T Sine we are ineresed in a ompuaionally effiien reursive soluion, we an re-arrange he erms o express E in erms of E and only: E = E = E + H E B E arge + P T k= EH k + P 2 P T k k= P T k= k + k= k P T k k= Hene, Equaion 8 orres he fuure alloaions based on he mos up-o-dae energy generaion and onsumpion informaion afer eah inerval. Redisribuing he Surplus/Defii: In addiion o orreing he fuure alloaions, we need o aoun for deviaions from he revised opimal values in he pas inervals. For example, assume ha he opimal alloaion for inerval was ompued as mah, bu he harvesed energy in inerval urned ou o be signifianly lower han he expeed value. Suppose ha he opimal alloaion in inerval is orreed as 6 mah in ligh of he new measuremens. Equaion 8 orres he fuure alloaions, bu i does no laim bak 4 mah overspen in he previous inerval. In oher (8)

5 words, Equaion 8 alone does no make up for over-onsumpion, or relaim he underuilized energy alloaions in he previous inervals. Therefore, we need o disribue o he remaining inervals [, T ]. A sraighforward uniform disribuion is no suffiien, sine any adjusmen inrodued a ime affes he fuure alloaions due o he reursive rule in Equaion 8. Suppose ha we add a orreion erm o Equaion 8 as follows: E = E + P T k k= + a where a is a normalizaion oeffiien ha will ensure ha he perurbaions in he remaining inervals will add up o preisely. By grouping he erms wih, we obain: E = E + P T k= k + a (9) Sine he perurbaion erm will be muliplied wih in eah fuure inerval (due o he E erm), he sum of he perurbaions from he urren inerval hrough he las one an be wrien as: TX k= k P T k= k + a = By solving his equaion, we an find a as: 8 < T P T < < k a = k= : = T T () 3) User Aiviy and Minimum Energy Consrain: Profiling he energy onsumpion and user aiviy reveal speifi periods wih low or high aiviies. For example, i is possible o idenify sleep and exerise periods. The proposed approah enables us o easily inrodue new equaliy onsrains based on his informaion. More preisely, we se E = M E for inervals ha fall during he sleep duraion. Similarly, one an alloae a erain maximum value during expeed exerise periods. We noe ha over-alloaion does no have a signifian drawbak sine unuilized alloaions are disribued o fuure periods. However, under-alloaion may hur he uiliy if he inerval duraion is long (e.g., one hour). Therefore, low aiviy regions should be seleed onservaively. Sine hese onsrains an be inrodued as pre-alloaion, hey do no hange he formulaion. The final onsideraion is enforing he minimum energy onsrain. Equaion 9 an ause he baery energy drain below E min, sine his onsrain was relaxed o find a losed-form soluion. Therefore, we proje he remaining baery energy E+ B a runime using Equaion, and ompare i agains E min before ommiing o a soluion. If here is a violaion, we alloae he maximum energy ha saisfies E+ B = E min. Tha is, he alloaion beomes: 8 < E E + P + T k a EB + E min k= = () : E B + E H E min oherwise where + and a and are given by Equaions and, respeively. 4) Summary of he Proposed Algorihm: We onlude his seion wih a sep-by-sep desripion of he runime operaion: ) A he beginning of eah day: Compue he alloaion for he firs inerval E using Equaion 6. 2) For eah inerval apple apple T : Divide he energy alloaion E by he inerval duraion o find he arge power onsumpion P. Then, use Equaion 3 o find he duy raio. If here are muliple allowed frequeny levels f, we use he mos energy effiien f. However, any feasible ombinaion is aepable. 3) During eah inerval apple apple T : Keep rak of aual harvesed and onsumed energy. Compue a he end of he inerval by finding he differene beween he expeed and measured values. 4) Before he sar of eah inerval apple apple T : Use Equaion o find he nex alloaion E. If = T sop, oherwise inremen and go o sep 2. A. Experimenal Seup V. EXPERIMENTAL EVALUATION IoT Devie Parameers: We employ he prooype shown in Figure 4 o demonsrae he proposed algorihm under realisi senarios. I onsiss of an MPPT harger (TI BQ2554 [36]), a miroproessor (TI CC265 [37]), a moion sensor uni (InvenSense MPU-925 [9]), and EMG iruiry. We use a PV-ell from FlexSolarCells SP3-37 [8] as he energy-harvesing devie and a 2 mah Li-Po baery GMB 39 [] as he sorage elemen. We have probes o measure he power onsumpion of differen omponens, as illusraed in Figure. These measuremens are used o validae he power model given in Equaion 3 as a funion of he duy raio and frequeny. We also deermined he IoT devie parameers, suh as E min and M E, lised in Table II, based on hese measuremens. CPU TABLE II: Parameer values used during evaluaions EMG Parameer Value Parameer Value E min.75 mah E arge 8 mah P idle 2.2 mw M E.6 mah T 24 Charger (a) Anenna MPU PV Baery (b) Fig. 4: Prooype (a) fron view, (b) bak view Energy Harvesing Model: The harvesed energy is deermined by he PV-ell and he radiaion inensiy, whih is a funion of observaion ime and loaion. I-V haraerisis of SP3-37 are measured by varying he radiane from o W/m 2 wih he help of a halogen lamp. Then, his empirial daa is used o model he maximum generaed power as a funion of radiaion. This model enables us o ompue he harvesed energy, if he radiaion is known. To find he radiaion, we firs esimae he posiion of he sun a a given dae and ime using Sandia s Ephemeris model [33]. Then, we onver he posiion informaion o radiaion using Ineihen s model [8]. These hree models are used by our algorihm o predi he energy ha will be harvesed during he day. We ompare our resuls o an offline opimal algorihm implemened using he CVX pakage [3] and an orale. The orale uses he aual radiaion, whih is measured a every minue on he NREL Solar Radiaion Researh Laboraory s baseline measuremen sysem [2]. User Aiviy Model: The energy onsumpion varies as a funion of he user aiviy. To evaluae a wide range of senarios, we use

6 Energy (mah) E U (Orale) =. U =.9 E h Orale Fig. 5: Energy alloaion in January wihou learning he user paern. Energy (mah) E U (Orale) = 9.7 U = 9.6 E h Orale (mah) (mah) Fig. 7: Energy alloaion in January afer learning he user paern. Energy (mah) E U (Orale) = 32.5 U = 28. E h Orale Fig. 6: Energy alloaion in July wihou learning he user paern. Energy (mah) E U (Orale) = 27. U = 27. E h Orale Fig. 8: Energy alloaion in July afer learning he user paern (mah) (mah) differen user aiviy paerns from he Amerian Time Use Survey ondued by he US Deparmen of Labor [38]. This survey onains he ime a user spends for various aiviies. In our evaluaions, we use five aiviy aegories {sleep, work, exerise, leisure, ohers}. When he user is asleep, we alloae M E o he orresponding inerval. Oherwise, we use he proposed approah o find he opimal alloaion. B. Energy Alloaion Over Time We firs illusrae he operaion of he proposed algorihm for a speifi user and dae. Figure 5 shows he energy-harvesing profile, baery energy and opimal alloaions on January s for user-. The energy harvesing profile (blue markers) shows ha here is lile o none energy generaion unil 8 AM. During his period, he alloaed energy (red markers) is supplied by he baery, whose sored harge drops oninuously (green 4 markers). One he harvesed energy exeeds he energy alloaed wihin an inerval (around AM), he baery energy sars reovering. We observe ha our resuls mah very losely wih he resul of he offline opimizaion ha uses an orale (doed lines). We do no see a signifian differene in he alloaed energy hroughou he day, sine he baery apaiy is suffiien o absorb he variaion in he harvesed energy. However, we observe a dramaially differen behavior for July, as shown in Figure 6. The peak harvesed energy is abou 2.5 larger in July han January ( 3.5 mah versus 9 mah), and i spans a wider range. Therefore, he proposed algorihm alloaes aggressively a he beginning of he day, relying on he energy ha will be generaed laer. However, i his he minimum baery energy onsrain a 4 AM, unlike he offline opimizaion ha aouns for E min from he beginning. As soon as he baery energy drops o E min, he proposed algorihm sars alloaing subopimally only he harvesed energy o he IoT devie. This oninues unil he harvesed energy beomes suffiienly large o power he IoT devie and harge he baery (8 AM). While he alloaion during he res of he day losely follows he opimal alloaion, he IoT devie is under-powered from 3 AM o 8 AM. As a resul, he loss in uiliy wih respe o he orale is larger ompared o ha obained for January. This demonsraes he os of negleing he minimum energy onsrain a he beginning of he day. Nex, we analyze he resuls on same days by aking he user aiviy ino aoun. We idenify he periods of low aiviy, primarily he inervals aegorized as sleep, and onsrain he alloaions in hose inervals as E = M E. We add he same onsrain o he offline opimizaion for fairness. Comparing Figure 5 o Figure 7 shows ha he algorihm sars alloaing less energy a nigh. As a resul, more energy is reserved for higher aiviy inervals, whih leads o more han 3% inrease in he uiliy during hose inervals. Like before, he resuls mah very losely wih he offline opimizaion resuls. Inorporaing he user aiviy leads in even more savings in he resuls obained for July. When we aoun for user aiviy, he proposed algorihm does no over-alloae a he early hours, sine here is lile aiviy during nigh. Therefore, he baery energy does no hi o E min, and our resuls oinide wih he orale resuls, as shown in Figure 8.

7 Normalized Duy Raio Jan -Feb 4-Mar -Apr 2-May 2-Jun 7-Jul 7-Aug -Sep -O -Nov 5-De Fig. 9: Comparison of he proposed soluion o offline opimizaion for hree users over 2 monhs. C. Comparison o Offline Opimizaion Improving he duy raio is an imporan end goal. Therefore, his seion ompares he duy raio obained wih he proposed approah agains he offline opimizaion resuls, whih employ an orale. We performed he omparisons for hree differen users from he US Deparmen of Labor [38] daabase over 2 monhs. Figure 9 summarizes he normalized duy raio (our resuls divided by he offline opimizaion resuls). We observe ha he duy raio provided by our approah is, on average, wihin % of he duy raio ahieved by he orale. Moreover, he larges loss in opimaliy in he duy is less han 5%. We observe a bigger loss under wo ondiions. Firs, when he variaion beween he expeed and aual energy generaion is large, he resuls of he proposed algorihm degrade, as aniipaed. Seond, when he peak-o-peak variaion in he harvesed energy beomes omparable o he baery apaiy ( 25% of E arge), he proposed algorihm his he E min arge, as shown in Figure 6. VI. CONCLUSIONS AND LIMITATIONS Wearable IoT devies have a grea poenial o enable healh monioring, aiviy raking and gesure-based onrol appliaions. However, hey fae severe energy limiaions due o weigh and os onsrains. Therefore, harvesing energy from ambien soures, suh as ligh and body hea, and using i opimally is riial for heir suess. This paper presened a near-opimal runime algorihm for self-powered wearable IoT devies. The proposed approah is based on wo observaions ha lead o near-opimal resuls wih onsan ime omplexiy. Firs, we obain a losed-form soluion for he opimizaion problem by relaxing he minimum baery energy onsrain. Then, we use he expeed energy ha will be harvesed hroughou he day o solve he relaxed finie horizon opimizaion problem. Finally, we aoun for he deviaions from he expeed values and enfore he minimum energy onsrains a runime. We demonsrae ha our resuls are on average wihin 3% of opimal values ompued offline using an orale. The resuls degrade as he peak-o-peak variaion in he harvesed energy and deviaion from he expeed values inrease. However, he degradaion in he uiliy is small when he baery apaiy an absorb he peak-o-peak variaions. Aknowledgmen: This work was suppored by NSF CAREER award CNS REFERENCES [] C. Alippi and C. Galperi, An Adapive Sysem for Opimal Solar Energy Harvesing in Wireless Sensor Nework Nodes, IEEE Trans. Ciruis Sys. I, Reg. Papers, vol. 55, no. 6, pp , 28. [2] A. Andreas and T. Soffel, NREL Solar Radiaion Researh Laboraory (SRRL): Baseline Measuremen Sysem (BMS); Golden, Colorado (Daa); NREL Repor No. DA , 98, hp://dx.doi.org/. 5439/5222, aessed 5 Augus 27. [3] H. Banaee, M. U. Ahmed, and A. 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APPENDIX: DERIVATION OF EQUATION 6 To solve he opimizaion problem given in Equaion 5, we firs evaluae he Lagrangian of he objeive funion as: L = TX = ln E M E + + E E H TX + = µ [+ E min]+ T µ T [E B T E arge] (2) Using he Lagrangian, we an wrie he firs-order B T : : : apple ME E = apple apple T (3) + µ = apple apple T (4) T T + T µ T = i.e., T = µ T (5) In addiion o he firs-order ondiions above, he Karush-Kuhn- Tuker (KKT) ondiions are µ,, and: µ (+ E min) = apple apple T (6) µ T (E B T E arge) = (7) Sine u(e ) is onave, Equaions 3-7 give he neessary and suffiien ondiions for he opimaliy [23]. We presen a sep-bysep soluion below.. Boundary Condiion: Lagrangian mulipliers an be found using Equaion 3 as : = ME apple apple T (8) E Combining his relaion wih Equaion 5, we an onlude ha T = µ T 6=. Hene, he omplemenary slakness given by Equaion 7 implies ET B = E arge. 2. Reursion over E : We an use Equaion 4 o derive a reursion rule for and ombine i wih Equaion 8 as follows: M E E + + = + µ apple apple T = ME E + µ apple apple T (9) We plug he boundary ondiion E B T = E arge o his reursive relaion. Then, he energy alloaions in earlier inerval an be solved using he Equaion 5 and he KKT ondiion given by Equaion 6. Solving T nonlinear equaions a runime is no effiien. However, enaively ignoring he minimum energy onsrain (key insigh ), enables us o eliminae µ from Equaion 9. Tha is, we an se µ = in equaions 6 and 9. Similarly, E H is no known a priori, bu we use he expeed values (key insigh 2). The proposed algorihm presened in Seion IV-C enables us o make up for hese hoies a runime. 3. Closed-form Soluion: Afer seing µ =, apple apple T, Equaion 9 redues o: E + = E (2) We an re-arrange he baery energy dynamis in Equaion 5, and ombine wih his relaion as follows: E = E B E B + E H, E = E B E B 2 + 2E H 2,... T E = E B T E arge + T E H T Noe ha E arge in he las equaion omes from he boundary ondiion. When we summing up hese T equaions, E B ET B anel eah oher. Hene, we find E as: E = EB E arge + P T = EH T (2) Combining Equaion 2 and Equaion 2 gives he losed form soluion summarized in Equaion 6.