Joint Transportation and Charging Scheduling in Public Vehicle Systems - A Game Theoretic Approach

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1 1 Join Transporaion and Charging Scheduling in Public Vehicle Sysems - A Game Theoreic Approach Ming Zhu, Xiao-Yang Liu, and Xiaodong Wang, Fellow, IEEE arxiv: v3 [cs.sy] 27 Dec 2017 Absrac Public vehicle (PV) sysems are promising ransporaion sysems for fuure smar ciies which provide dynamic ride-sharing services according o passengers requess. PVs are driverless/self-driving elecric vehicles which require frequen recharging from smar grids. For such sysems, he challenge lies in boh he efficien scheduling scheme o saisfy ransporaion demands wih service guaranee and he cos-effecive charging sraegy under he real-ime elecriciy pricing. In his paper, we sudy he join ransporaion and charging scheduling for PV sysems o balance he ransporaion and charging demands, ensuring he long-erm operaion. We adop a cake cuing game model o capure he ineracions among PV groups, he cloud and smar grids. The cloud announces sraegies o coordinae he allocaion of ransporaion and energy resources among PV groups. All he PV groups ry o maximize heir join ransporaion and charging uiliies. We propose an algorihm o obain he unique normalized Nash equilibrium poin for his problem. Simulaions are performed o confirm he effecs of our scheme under he real axi and power grid daa ses of New York Ciy. Our resuls show ha our scheme achieves almos he same ransporaion performance compared wih a heurisic scheme, namely, ransporaion wih greedy charging; however, he average energy price of he proposed scheme is 10.86% lower han he laer one. Index Terms Public vehicle sysems, ransporaion, charging, smar grids, real-ime elecriciy pricing, cake cuing game. I. INTRODUCTION The public vehicle (PV) sysems [1] [2] [3], also known as he shared inerne of vehicle sysems or inelligen ransporaion sysems, provide high-qualiy ride-sharing services in fuure smar ciies. The vehicles in PV sysems, called PVs, are ypically driverless/self-driving [4] elecric vehicles wih large capaciies jus like buses. PVs are conneced o smar grids for self-charging. A PV sysem consiss of hree main componens: a cloud, passengers/users, and PVs. The operaion flow of a PV sysem is as follows. If a passenger/user needs a rip service, he/she sends a reques o he cloud via a smar phone, including an earlies sar ime, a pickup posiion (origin) and a dropoff posiion (desinaion), ec. Then he cloud compues he ride maches beween PVs and passengers, M. Zhu is wih he Shenzhen Insiues of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China, and he Deparmen of Compuer Science and Engineering, Shanghai Jiao Tong Universiy, Shanghai, China, zhumingpassional@gmail.com, zhumingpassional@sju.edu.cn. X.-Y. Liu is wih he Elecrical Engineering Deparmen, Columbia Universiy, New York Ciy, US, and he Deparmen of Compuer Science and Engineering, Shanghai Jiao Tong Universiy, Shanghai, China. xiaoyang@ee.columbia.edu. X. Wang is wih he Deparmen of Elecrical Engineering, Columbia Universiy, New York, NY, 10027, USA ( wangx@ee.columbia.edu). Fig. 1. The V2G scenario: a PV sysem and he smar grid. and calculaes pahs for PVs, and finally schedules a suiable PV o drive him/her from he origin o he desinaion, wherein he pahs may be shared wih ohers. PVs are ypically elecric vehicles (EVs), and are conneced o smar grids for baery charging. Smar grids are envisioned as he nex-generaion power grid sysems ha can inelligenly accommodae requess by all conneced users. They are equipped wih a smar meering infrasrucure capable of sensing and measuring power consumpions from consumers wih he inegraion of advanced compuing, conrol, informaion and communicaion echnologies [5]. The smar grids will have more efficien, more economical and more reliable power generaions, disribuions, and consumpions han he convenional power grids. PVs and smar grids consiue complicaed vehicle-o-grid (V2G) ecosysems [6]. Fig. 1 shows an overview of he arge scenario including a PV sysem and a smar grid. To he bes of our knowledge, his is he firs work o consider he join ransporaion and charging scheduling problem for PV sysems. Exising works focus only on eiher he ransporaion or he charging problem for PV sysems or PV-like sysems. For example, [7] proposes pah planning sraegies wihou considering energy sorage or charging acions; whereas [8] proposes he charging sraegies from smar grids neglecing he ransporaion acions. In addiion, he charging scenario in PV sysems is differen from ha in exising works. For example, privae EVs are parked 95% of he ime [9] and he baeries are usually charged a parking los a homes or working places mos of he day. PVs have

2 2 much less parking ime han privae EVs since hey are shared by all passengers, and hey need frequen charging since hey serve more passengers wih high occupancy raes [1]. PVs roues and schedules vary wih imes and locaions for he consanly changing ransporaion demands. Therefore, how o ake advanage of he limied amoun of idle ime o charge he baeries becomes an imporan problem. We consider he join ransporaion and charging scheduling problem (JTCSP) for PV sysems. The goal is o balance he ransporaion and charging demands o guaranee he long-erm operaion of PV sysems wih less charging coss and more profis. The challenges of JTCSP are as follows: 1) The cloud should consider he ransporaion demands from passengers which vary wih imes and locaions. 2) The charging demands of PV groups should be saisfied wih less coss under he scenario of real-ime elecriciy pricing. 3) The cloud should ensure PVs o have sufficien energy o serve passengers a any ime even if he energy price is high, oherwise, some PVs may no have sufficien energy o serve passengers. Too many or oo few PVs engaged in ransporaion or charging will affec he profis and energy sorages, e.g., some PVs engaged in ransporaion may have insufficien energy, and some oher PVs engaged in charging may no provide ransporaion services o earn profis. The main conribuions of his paper include he following: 1) We adop he cake cuing game [10] o capure he ransporaion and charging paerns of PVs, and hen use a novel uiliy model o reflec he ransporaion benefis, saisfacions for charged baeries, and charging coss. Suppose each player has a differen opinion as o which par of a cake is more valuable. The aim of a cake cuing game [10] is o divide a cake among muliple players so ha everyone values his or her piece no less han any oher piece. In his paper, we make use of he cake cuing game in selecing differen ransporaion and charging vehicles for differen PV groups. 2) We analyze he exisence and uniqueness of he Nash equilibrium in his cake cuing game, and analyze he feaures of he game in he JTCSP compared wih exising works, e.g., hose ha only consider ransporaion or charging. 3) We propose an efficien soluion o JTCSP o balance he ransporaion and charging demands of PVs which achieves a unique normalized Nash equilibrium poin in he cake cuing game. 4) Simulaions are performed o demonsrae he effeciveness of he proposed soluion under he real ransporaion and power grid daa ses of New York Ciy, and hen compare he performance of our scheme wih a heurisic scheme. The remainder of he paper is organized as follows: Secion II describes he relaed work. Secion III presens he sysem model. Our problem is formulaed in Secion IV. The analysis abou he JTCSP is given in Secion V, and a soluion is proposed in Secion VI. Secion VII presens he simulaion resuls. Secion VIII concludes his paper. II. RELATED WORKS Transporaion is he mos imporan ask of PV sysems [11] [12], wherein pah planning sraegies [13] direcly deermine he qualiy of service (QoS) of passengers. Some pah planning soluions in PV sysems or PV-like sysems (e.g., axi sharing sysems) have been proposed. Zhu e al. [1] propose a pah planning sraegy wih balanced QoS (shor waiing ime and limied deour) in PV sysems. Aasoy e al. [14] propose a flexible mobiliy-on-demand sysem uilizing hree services: axi, shared-axi and mini-bus, which can balance consumer surplus and he operaor s profis. Herbawi e al. [15] propose a geneic algorihm o solve he ride-maching problem wih ime windows in ride-sharing. Fagnan e al. [16] presen a cusomer waiing ime model of shared auonomous vehicles, and find ha i can save 10 imes he number of cars needed for self-owned vehicle rips. Jung e al. [17] propose a dynamic dispaching algorihm in axi sharing sysems using he hybrid simulaed annealing aiming a minimizing he oal ravel ime and maximizing he profis. However, he above works do no consider he energy demands of vehicles. To ensure he long-erm ransporaion of EVs, charging sraegies wih low coss are required. Recenly, he charging problem of EVs becomes an imporan opic, e.g., he charging cos minimizaion problem [18] [19] and he waiing ime minimizaion problem [20] [21]. Rivera e al. [18] propose an opimizaion framework for achieving compuaional scalabiliy based on he alernaing direcions mehod of mulipliers wih wo objecives, valley filling and cos-minimal charging wih grid capaciy consrains. Franco e al. [19] presen a mixed-ineger linear programming model o minimize he oal energy coss, which corresponds o he EV charging coordinaion problem in elecrical disribuion sysems. Zhu e al. [20] propose wo cenralized algorihms o minimize he oal charging ime, i.e., EV ravelling ime, he queuing ime, and he acual charging ime a charging saions. Zhang e al. [21] use independen Markov processes o model he uncerainy of he arrival of EVs, he inermience of he renewable energy, and he variaion of grid power prices, and propose a Markov decision process framework o minimize he mean waiing ime for EVs under he long-erm consrain on he cos. Weerd e al. [22] propose a rouing policy ha minimizes EVs expeced journey ime including he waiing ime a charging saions. Game heoreic approaches are applied o ransporaion sysems o opimize raffic flows, ravel ime, ec. Groo e al. [23] propose hree approaches based on reverse Sackelberg game o reach a sysem-opimal disribuion of raffics on freeway roues, e.g., o minimize he oal rip ime and o reduce raffic emissions in urban areas. The opimizaion of he ravel ime and he raffic flows a inersecions is a key issue since i is one of he major bolenecks for urban raffic congesions. To reduce he average ravel ime delay a unconrolled inersecions, Elhenawy e al. [24] propose a chicken-game-based algorihm for conrolling auonomous vehicle movemens equipped wih cooperaive adapive cruise conrol sysems. Bui e al. [25] propose a Courno model and a Sackelberg model o opimize he raffic flows for smar raffic ligh conrol a inersecions. Farokhi e al. [26] adop an aomic congesion game model o capure he ineracions beween he car raffic and he ruck plaooning incenives. Several novel pricing mechanisms and scheduling sraegies have been proposed in on-demand mobiliy sysems. Drwal

3 3 TABLE I VARIABLES AND NOTATIONS J number of all PVs in he ciy. c baery capaciy of PVs. r charged energy in a ime slo (an hour be defaul). I se of all PVGs in ime slo. I number of all PVGs in ime slo in he ciy. m i, number of unfully charged PVs in region i in ime slo. T ime slo se. ime slo in T. x i, sraegy of PVG i in ime slo. d i, ransporaion demands of PVG i in ime slo. d oal ransporaion demands of all PVGs in ime slo. E r oal remaining energy of all PVs in ime slo. p real-ime elecriciy price in ime slo. u i, uiliy funcion of PVG i in ime slo. Ω i, feasible sraegy se of PVG i in ime slo. Ω feasible sraegy ses of all PVGs in ime slo. x sraegies of all PVGs in ime slo. u objecive funcion vecor of all PVGs in ime slo. E - consumed energy of all PVs in ime slo. E + charging demands of all PVs in ime slo. n i, number of ransporaion PVs in region i in ime slo. a i, number of all PVs in region i in ime slo. f i, number of fully (or near fully) charged PVs in region i in ime slo. φ i, number of ransporaion PVs of PVG i in ime slo. ψ i, number of charging PVs of PVG i in ime slo. e al. [27] propose pricing mechanisms o balance he demands of differen parking saions, reduce he cos of manual relocaions of vehicles, and maximize he operaor s revenue. However, his soluion only considers he ransporaion performance, and does no consider he energy demands of vehicles from smar grids. Rigas e al. [28] sudy he scenario where EVs are hired in on-demand mobiliy sysems and here exis muliple baery swap faciliy poins. Is aim is o maximize he number of passengers ha are served. They focus on he ransporaion performance wih limied ravel ranges and simplify he charging problem as swapping baeries in faciliy poins, ignoring he effecs of he real-ime pricing of smar grids, e.g., how o reduce he charging coss wih he guaranee of providing ransporaion services. Our soluion proposed in his paper considers he join ransporaion and charging sraegies of vehicles over one day and muliple days under he scenario of real-ime elecriciy pricing. III. SYSTEM MODEL We firs describe he PV model, and hen presen he smar grid model. All he imporan variables and noaions in his aricle are summarized in Table I. A. PV Model We assume all J PVs in a ciy have idenical baery capaciy c (kwh) and idenical charged energy r (kwh) in a ime slo (an hour by defaul). To explore he ime-dependen ransporaion and charging paerns, we inroduce ime slo se T = {0, 1,..., T 1}. The PVs are considered fully charged if heir remaining energy is greaer han (c r) since he charging ime is less han a ime slo, which is oo shor, and he ohers are unfully charged. Fully charged PVs can only choose o serve passengers. While unfully charged PVs have wo choices, ransporaion or charging. To ensure he ransporaion of he nex ime slo, a each ime he remaining energy of a PV should no be less han he consumed energy in a ime slo. The ciy is divided ino I regions, and in each region i, m i, denoes he number of unfully charged PVs in ime slo and hey form public vehicle group (PVG) i. All PVGs in ime slo are denoed by a se I wih he number I. We see ha I and I may vary over ime and should be updaed in each ime slo, while J does no change. PVG Transporaion Model: The sraegy of PVG i I in ime slo is denoed by x i, [0, 1], which is he raio of PVs in PVG i ha will be used for ransporaion, and he remaining porion will charge heir baeries. In each ime slo, he cloud calculaes he ransporaion demands of PVG i denoed by d i,, i.e., he number of ransporaion PVs. The oal ransporaion demands of all PVGs in ime slo is d = i d i,. PVG Charging Model: We assume ha each charging PV will charge baeries wih he idenical charged energy r in a ime slo. For PVG i, he charged energy in he ime slo [, + 1] are r m i, (1 x i, ). Discharging in peak elecriciy price periods can bring more benefis for EVs. However, we do no consider discharging in PV sysems for several reasons. 1) The mos imporan acions of PVs are ransporaion and charging since ransporaion brings profis and charging ensures he abiliy of making profis. 2) In general, he ransporaion peak ime coincides wih he elecriciy price peak ime according o he axi rip daa [29] and grid daa [30] in New York Ciy. The urgen acion a he energy consumpion peak ime [31] is charging no discharging. Thus, if we consider discharging, he ransporaion demands may no be fully saisfied. 3) Even we consider discharging, he profis hrough V2G are 90 4,000 US dollars per year per vehicle [32], which can no mach wih ha from ransporaion services. Due o he above reasons, we assume ha PVs only have wo acions: ransporaion and charging. B. Smar Grid Model Generally, smar grids serve primary consumers such as indusries, houses, and offices wih high prioriy. Afer meeing he demands of he primary consumers, smar grids wish o sell energy o secondary users, e.g., PVs. We assume ha, he cloud of PV sysems can requess a cerain amoun of energy from smar grids. In he elecriciy marke, he real-ime prices (RTPs) [33] vary wih he oal demands, which can reduce he peak-o-average load raio hrough encouraging consumers o shif heir usages o off-peak hours. Le he RTP in ime slo be p. The elecriciy pricing model is p = α 0 ( L C 0 L = i ) k0, (1) L i,, (2) where α 0 and k 0 are predefined pricing consans by he smar grids, C 0 is he capaciy of he elecriciy markes [34], L i,

4 4 is he elecriciy load of region i in ime slo, and L is he oal elecriciy load in ime slo, including ha from indusries, houses, offices and secondary users such as EVs. I has been proven ha he overall energy coss are minimal when elecriciy consumpions are balanced in each equal-size ime slo [34]. Then he charging cos of PVG i in he ime slo [, + 1] is r p m i, (1 x i, ). IV. PROBLEM FORMULATION We firs presen a novel uiliy model for PVGs, and hen formulae he JTCSP. A. PVG Uiliy Model The uiliy model for each PVG should reflec is ransporaion and charging willingness considering he rip demands of passengers, elecriciy prices, and is own energy saes. The uiliy funcion of PVG i is formulaed as u i, = (m i, x i, d i, ) 2 + α 1 m i, ln(2 x i, ) α 2 p m i, (1 x i, ), (3) where α 1 and α 2 are consans. u i, has he following erms: The erm (m i, x i, d i, ) 2 denoes he ransporaion uiliy. Clearly, when he number of ransporaion PVs of PVG i equals d i,, i obains he maximum uiliy. Noe ha if m i, x i, is larger han he criical value d i,, he uiliy decreases for several reasons such as waiing coss and parking fees, and if i is less han d i,, he uiliy also decreases since i can no saisfy he ransporaion demands. The erm α 1 m i, ln(2 x i, ) denoes he saisfacion level for charged energy wih he weigh α 1. The charged energy r in a ime slo is omied since i is included in he weigh α 1, which can also be seen in he hird erm of he uiliy model. We adop a logarihmic uiliy model o denoe he saisfacion level since i can quanify user saisfacions wih diminishing reurns [35], which is widely used in designing he uiliy for energy consumers, e.g., [36] [37]. We use (2 x i, ) insead of (1 x i, ) o ensure ha i is always posiive and he logarihmic funcion is always available. The erm α 2 p m i, (1 x i, ) denoes he charging fees wih he weigh α 2. B. Problem Saemen The ransporaion and charging sraegies of PVs are coordinaed raher han direcly conrolled for he following reasons. 1) This can reduce he cos of daa ransmission hrough 4G/5G. During he charging periods, some daa should be exchanged beween PVs and he cloud, e.g., he remaining energy of baeries, he elecriciy prices, and some informaion abou charging saions. 2) This can reduce he cos of daa sorage. PVs are driverless/self-driving EVs, which generae a large amoun of daa even wihin a shor period, e.g., one GB daa per second [38]. 3) More compuing resources can be assigned o ransporaion o provide beer services for passengers. PV sysems need more compuing resources in solving pah planning problems [3] [39] based on raffic big daa. Moreover, he cloud has o predic in real-ime he vehicle speed [40], ransporaion demands [8] in each region of smar ciies. We explore he cake cuing game [41] [10] [42] o coordinae he ransporaion and charging sraegies for all PVGs. The cake cuing game is one of he mos fundamenal games for fair division wih he aiming of dividing he cake (here, i means he ransporaion and charging resouces) fairly. The cake cuing game can encapsulae he imporan problem of allocaing heerogeneous resources among muliple players wih differen preferences. Each PVG has is own ransporaion demands, and energy saes. Therefore, we adop he cake cuing game o analyze he JTCSP in PV sysems. PVGs are noncooperaive since hey do no communicae wih each oher, bu hey may inerac wih he cloud and smar grids by he conrolled signaling hrough smar meers. The cloud calculaes he charging demands of all PVs {E + } T ahead of he day, and he ransporaion demands of each PVG {d i, } i I and he oal ransporaion demands {d } i I in each ime slo. Then PVGs selec heir bes response sraegies {x i, } i I o maximize heir uiliies. The opimizaion problem of he PVG i in ime slo is formulaed as max x i, u i,, (4) s.. r i I m i, (1 x i, ) = E +, (5) m i, x i, d, (6) i I x i, [0, 1], i I. (7) We see ha, for any PVG, he firs and second consrains are shared by all PVGs. The firs consrain indicaes ha he oal charged energy should be equal o he charging demands E + in ime slo from smar grids. The second consrain indicaes ha he oal number of ransporaion PVs should no be less han he ransporaion demands calculaed by he cloud, d. The hird consrain indicaes he bound of he PVG s sraegy x i,. Herein, he second consrain can always hold since he ransporaion demand d is deermined by he cloud using a scheduling sraegy a he beginning of each ime slo, which is described in Algorihm 2 of Secion VI-C. If he ransporaion demands are oo high o be covered for a given se of PVs, he cloud will use more PVs o serve passengers excep he charging PVs, and accordingly, he passengers may have o wai for longer ime. Le Ω i, be he feasible sraegy se of PVG i in ime slo which saisfies he hree consrains in Problem (4). Le Ω be he feasible sraegy ses of all PVGs in ime slo, i.e., Ω := Ω 1,... Ω I,. We formulae he JTCSP based on PVGs raher han individual PVs for several reasons: 1) This can reduce he operaing complexiy especially in large ciies wih a large number PVs, since he number of PVGs is much less han ha of PVs. 2) This can save a lo of compuing resources such ha he energy managemen coss can be reduced.

5 5 The JTCSP is a cake cuing game [41] [10], and in ime slo i is defined by where (I, Ω, x, u ), I denoes he players (PVGs) in ime slo in he cake cuing game; Ω := Ω 1,... Ω I, is he sraegy se of all PVGs in ime slo ; x := (x i, ) i I denoes he sraegies of all PVGs in ime slo ; u := (u i, ) i I denoes he objecive funcion vecor of all PVGs o maximize in ime slo. V. PROBLEM ANALYSIS The firs and second consrains in Problem (4) imply ha he acion of a PVG is consrained by he acions of oher PVGs, which are known as shared/coupled consrains [43]. The games wih shared consrains bring paricular complexiy o ackle. In his secion, we prove he exisence and uniqueness of equilibrium for our JTCSP. Lemma V.1. An equilibrium poin exiss in he JTCSP for PV sysems. Proof. Since all PVGs have he same shared consrains, he JTCSP is a generalized Nash equilibrium problem (GNEP). The GNEP exends he classical Nash equilibrium problem by assuming ha each player s feasible sraegy se can depend on he rival players sraegies. The uiliy funcions of PVGs are coninuous and concave and heir sraegy ses are closed and convex. Therefore, he JTCSP is a concave n-person game. According o Rosen s work (Theorem 1 in [44]), a Nash equilibrium poin exiss for every concave n-person game, Lemma V.1 is obained. From Lemma V.1, we know ha here may exis muliple Nash equilibria in he JTCSP in PV sysems. Now, consider he following opimizaion problem for each uiliy funcion: max x σ(x, w ) = max x w i, u i,, (8) i I s.. θ(x ) 0, (9) where w i, is a weigh facor, and θ(x ) = [θ 1 (x ),..., θ M (x )] T collecs M consrain ses which consiue a se M = {1,..., M}. Here, M = 3, which can be seen from he consrains form Problem (4). Denoe he Lagrange muliplier vecor for PVG i as λ i,, and λ = (λ 1,, λ 2,,..., λ I, ) T. The generalized Nash equilibrium x Ω is called a normalized Nash equilibrium (NNE) wih weighs if and only if i saisfies he following Karush-Kuhn-Tucker (KKT) [45] condiions: w i, xi, u i, + λ T i, xi, θ(x i,, x i, ) = 0, (10) λ T i, θ(x i,, x i, ) = 0, (11) λ i, 0, (12) θ(x i,, x i, ) 0, (13) where λ i, = λ w i,, i I. (14) Theorem V.1. A unique normalized Nash equilibrium (NNE) exiss for he JTCSP in PV sysems. Proof. In PV sysems, PVGs aim o maximize heir uiliies hrough buying low-cos energy from smar grids and providing ransporaion services for more profis. The objecive funcion of each player in a joinly convex GNEP is coninuously differeniable. According o Rosen s work (Theorem 2 in [44]), here exiss a unique NNE in concave n-player games if he join uiliy funcion σ(x, w ) = i I w i, u i, wih w = [w 1,,..., w I, ] is diagonally sricly concave. Nex, we prove σ(x, w ) is diagonally sricly concave. We define g(x, w ) as he pseudogradien for σ(x, w ): g(x, w ) = [w 1, x1, u 1, (x 1, ), w 2, x2, u 2, (x 2, ),..., w I, xi, u I, (x I, )] T. According o Rosen s work (Theorem 6 in [44]), a sufficien condiion ha σ(x, w ) be diagonally sricly concave for x Ω and w > 0 is ha he symmeric marix [G(x, w ) + G T (x, w )] be negaive definie, where G(x, w ) is he Jacobian wih respec o x of g(x, w ). The second derivaive on he uiliy funcion u i, in (3) wih respec o x i, is κ i, = 2 u i, x 2 i, = 2 m 2 i, α 1 m i, (x i, 2) 2. (15) Clearly, κ i, < 0, i I, T. So u i, is sricly concave. The Jacobian of g(x, w ) wih respec o x is w 1, κ 1, w 2, κ 2, 0 G(x, w ) = (16) 0 0 w I, κ I, Clearly, G(x, w ) is negaive definie, and he marix [G(x, w ) + ( G(x, w ) ) T ] = 2 G(x, w ) is also negaive definie. So σ(x, w ) is diagonally sricly concave. Hence, Theorem V.1 is proved. We see ha differen w s will yield differen NNEs. However, he NNE is unique for each fixed w. Now and henceforh, we consider he NNE wih he idenical weighs, i.e., w 1, = w 2, =... = w I, = 1, and we ge λ i, = λ, i I. (17) We know ha, all PVGs have he same Lagrange mulipliers for all consrains. Nex, we explore he relaionships beween a variaional inequaliy problem and he JTCSP. The variaional inequaliy problem VI(F, Ω) is o find a poin x Ω such ha F (x ), x x 0, x Ω, (18) where Ω is a closed and convex se, and denoes he inner produc, and F is a coninuous funcion. Theorem V.2. The variaional inequaliy problem VI(F, Ω ) wih he consrain θ(x ) 0 where F := ( xi, u i, ) i I is monoonic. The soluion o VI(F, Ω ) is he NNE wih

6 6 idenical weighs, and is also called a variaional equilibrium. Therefore, he soluion o VI(F, Ω ) is he soluion o he JTCSP in PV sysems. Proof. The KKT condiions for VI(F, Ω ) are F + λ T x θ(x ) = 0, (19) λ T θ(x ) = 0, (20) λ 0, (21) θ(x ) 0. (22) We can see ha, he KKT condiions for VI(F, Ω ) are exacly (10) (13) wih he same Lagrange mulipliers for all PVGs in he JTCSP. If w 1, = w 2, =... = w I, = 1, F is g(x, w ), and correspondingly, he Jacobian of F is G(x, w ). From Theorem V.1, we know G(x, w ) is negaive definie, so F is posiive definie. Therefore, F is sricly monoonic. So Theorem V.2 is proved. The mehod o he variaional inequaliy problem can be used o solve he GNEP [46]. VI. PROPOSED SOLUTION In his secion, we firs describe he basic idea of our proposed soluion for solving he JTCSP in PV sysems, and hen presen wo key rouines: charging scheduling and ransporaion scheduling, and finally deail he scheme o obain he unique NNE. Why is here an opimal soluion o he JTCSP in PV sysems? Essenially, boh he ransporaion and charging sraegies focus on he profis earned by providing services for passengers: ransporaion sraegies focus on he curren profis from passengers, while charging sraegies focus on he fuure profis since he charged energy will be used for ransporaion o make profis. If we only focus on he curren ransporaion, he fuure profis may no be fully obained since some PVs may no have sufficien energy. If we only focus on he curren charging, he charging coss can be minimized, while he profis from he curren ransporaion may no be maximized. A. Basic Idea We assume ha he rip requess of passengers and RTP of he nex day are known in advance. The basic idea of our soluion is as follows. 1) The cloud performs he charging scheduling one day ahead o calculae charging demands. 2) In each ime slo, he cloud performs he ransporaion scheduling according o he real-ime rip requess o calculae ransporaion demands. 3) We use a projecion mehod o solve he variaional inequaliy problem and obain he bes response sraegies of PVGs. 4) Each PVG selecs he a par of PVs o provide ransporaion services and ohers o charge baeries according o is bes sraegy. Wih respec o he ransporaion PVs, we use PCI [1] o schedule hem o serve passengers, and wih respec o he charging PVs, hey ravel o he neares charging saions o charge baeries. B. Charging Scheduling We assume ha he cloud knows he rip requess of one day ahead, herefore, he charging scheduling of PVs can be calculaed. The objecive of charging scheduling is o minimize he oal charging coss considering of energy consumpions of PVs and RTPs. Suppose ha he PVs have infinie energy and he energy consumpion is posiively correlaed wih he ravel disance. We schedule all PVs using he PCI algorihm [1] o serve passengers, and hen record he ravel disance and consumed energy of all PVs in each ime slo {E - } T. The process of PCI is as follows: Firs, all requess will be sored by heir waiing ime from he maximum o he minimum, i.e., he passengers wih longes waiing ime will have he highes scheduling rank. Second, inser each reques o he pah of each PV and hen calculae he inserion cos (oal ravel disance of PVs) and deour raio. Third, if we can find he ride-mach and pah wih he minimum inserion cos from all pahs wihin limied deour raio, schedule he PV o serve he corresponding reques, oherwise he reques will be pu o a waiing lis. Algorihm 1: Charging Scheduling One Day Ahead 1: Iniialize he remaining energy for all PVs in he ciy as infiniy; 2: Use PCI [1] o schedule PVs o serve passengers; 3: Record {E - } T ; 4: Calculae {E + } T by solving Problem (23) using a convex solver; 5: reurn {E + } T ; The opimizaion problem of charging scheduling is formulaed as min p E +, T (23) s.. E+1 r = E r E - + E +, {0,..., T 2}, (24) E r (1 + ρ) max(e -, J e min ), {1,..., T 1}, (25) E r T 1 E - T 1 + E + T 1 E r 0, (26) 0 E + (J d ) r, {0,..., T 1}, (27) 0 E r J c, {0,..., T 1}, (28) where E + and E - are he charging demands and consumed energy of all PVs in ime slo respecively, E r is he oal remaining energy of all PVs in ime slo, ρ > 0 is a consan, J is he number of all PVs in he ciy, and e min is he minimum energy of any PV o ravel o he neares he charging saion. The firs consrain is on he relaionship beween he consumed energy, he charging demands (charged energy), and he remaining energy in wo consecuive ime slos. The second consrain means ha he remaining energy in ime slo is a leas (1 + ρ) of he consumed energy used on ransporaion, which ensures he ravel of he PVs in he nex ime slo, a he same ime, he remaining energy is no less han he (1 + ρ) of he energy consumed on he ravel o charging saions. We assume ha he final remaining energy is no less han he iniial energy, which is refleced in he hird consrain. The fourh consrain implies ha he

7 7 charged energy has a upper bound if all PVs excep for he ransporaion ones decide o charge baeries. The fifh consrain poins ou ha he remaining energy should no exceed he oal energy capaciies of all PVs. Problem (23) is a sandard convex problem, and can be solved by any sandard convex solver. The procedure for charging scheduling of one day is summarized in Algorihm 1. C. Transporaion Scheduling In ime slo, he cloud calculaes ransporaion scheduling, i.e., he number of ransporaion PVs, n i,, in each region i given he rip requess using a vehicle scheduling scheme. We ge he number of unfully charged PVs in region i, i.e., he number of PVs in PVG i: m i, = a i, f i,, (29) where a i, is he number of all PVs in region i in ime slo, and f i, is he number of fully (or near fully) charged PVs in region i in ime slo. In ime slo we use PCI [1] o schedule PVs o serve passengers, and he ransporaion demands of PVG i in ime slo is d i, = max(n i, f i,, 0). (30) The oal ransporaion demands in ime slo is d = i d i,. (31) Algorihm 2: Transporaion Scheduling in Time Slo 1: Record he iniial saes of all PVs and passengers; 2: Use PCI [1] o serve he passengers in he ime slo [, + 1]; 3: Record he number of ransporaion PVs in each region {n i, } i I ; 4: Calculae d i, and d using (30) and (31) respecively; 5: Pu all PVs and passengers o iniial saes; 6: Reurn d i, and d ; Algorihm 2 shows he procedure of ransporaion scheduling in ime slo. The posiions, remaining energy, energy consuming raes, service saes of all PVs, and he service saes (pickup or dropoff) of passengers are called iniial saes, which are recorded in line 1. Line 2 uses he PCI algorihm o serve passengers considering of energy consumpions and energy limis. Line 3 records he number of ransporaion PVs of all regions. Line 4 calculaes he ransporaion demands d i, of PVG i and he oal ransporaion demands d. Line 5 means ha all PVs and passengers reurn o heir iniial saes. D. Soluion o JTCSP From Theorem V.2, we know ha F in VI(F, Ω ) is monoonic wih respec o is sraegy se. So we can use a hyperplane projecion mehod o solve i and he convergence can be guaraneed [47]. Here, we use SSPM (Solodov-Svaier projecion mehod) [48] o solve VI(F, Ω ). The projecion operaor P Ω is defined as P Ω (x ) = arg min x Ω x x. (32) We inroduce parameers γ 1 (0, 1), γ 2 (0, 1), γ 3 > 1, µ > 0, and η > 0. µ (k) is calculaed by µ (k) = min(γ 3 η (k 1), 1). (33) The projeced residual funcion is defined as ν(x (k), µ (k) ) := x (k) ( (k) P Ω x µ (k) F (x (k) ) ). (34) Le ζ (k) be he smalles nonnegaive ineger which saisfies γ ζ(k) 1 µ (k) ν(x (k) γ 2 ν(x(k) µ (k), µ (k) ) 2, (35) F (x (k), µ (k) )), ν(x (k), µ (k) ) where denoes he inner produc. η (k) and y (k) are calculaed by η (k) = γ ζ(k) 1 µ (k), (36) y (k) = x (k) The halfspace H (k) is defined as H (k) := {x (k) Algorihm 3: SSPM 1: Iniialize x (0) (0, 1), γ 3 > 1, ν(x (0) η (k) ν(x (k), µ (k) ). (37) R I F (y (k) ), x (k) y (k) 0}. (38) posiive value for a bound ɛ; 2: repea Ω, η ( 1) > 0, γ 1 (0, 1), γ 2, µ (0) ) = 1, k = 0. Se a small 3: Calculae µ (k) and ν(x (k), µ (k) ) using (33) and (34) respecively; 4: if ν(x (k), µ (k) ) < ɛ 5: break; 6: else 7: Calculae he smalles nonnegaive ineger ζ (k) saisfying (35), and hen calculae η (k) and y (k) using (36) and (37) respecively; 8: Calculae he halfsapce H (k) using (38), and hen calculae x (k+1) using (39); 9: k k + 1; 10: end if 11: unil ν(x (k), µ (k) ) < ɛ; 12: reurn x ;

8 8 Algorihm 4: Join Transporaion and Charging Scheduling (JTCS) 1: Perform charging scheduling one day ahead using Algorihm 1; 2: for T 3: Updae each PVG i, i I; 4: Perform ransporaion scheduling in ime slo using Algorihm 2; 5: The cloud announces he charging demands and ransporaion demands; 6: Calculae he bes response sraegies x = {x 1,,..., x I, } of all PVGs using SSPM (Algorihm 3) given he cloud s sraegies; 7: for PVG i I 8: φ i, m i, x i, ; 9: ψ i, m i, φ i, ; 10: end 11: end Then x (k+1) is obained by projecing x (k) ono he inersecion of is feasible se Ω and he halfspace H (k) : x (k+1) = P Ω H (k)(x(k) ). (39) The procedure of SSPM [48] is shown in Algorihm 3. Line 1 is iniializaion. Lines 2 11 consiue an ieraive process unil he projeced residual is less han a prese bound. Lines 3 and 8 imply ha only wo projecions are needed in each ieraion. Algorihm 4 shows he overall algorihm for he JTCSP. Line 1 uses Algorihm 1 o perform he charging scheduling in one day. The for loop beween lines 2 11 indicaes ha in each ime slo he SSPM mehod will be execued once. Line 3 means ha in each ime slo all PVGs will be updaed, since he unfully charged PVs in each region may change. Line 6 uses SSPM (Algorihm 3) o ge he bes response sraegies of PVGs. In lines 8 and 9, φ i, and ψ i, are he numbers of ransporaion and charging PVs respecively of PVG i in ime slo. Finally, for PVG i, he φ i, PVs wih he maximum remaining energy are seleced o provide ransporaion services and ohers o charge baeries. E. Comparison wih Exising Approaches There exis wo cakes in he JTCSP in PV sysems, a ransporaion cake and a charging cake. The proposed JTCS algorihm considers how o cu he wo cakes o allocae ransporaion and charging resources among PVGs. However, if we only consider cuing he ransporaion or charging cake, he process of cuing he oher cake will be in a disorderly sae. In he opimizaion problem (4) of PVG i in PV sysems, we see ha he firs consrain implies he condiion in cuing he charging cake, and he second consrain implies he condiion in cuing he ransporaion cake. The uiliy model in (3) reflecs he ransporaion and charging uiliies. However, if we only consider cuing he ransporaion cake, he uiliy model of PVG i is v i = (m i, x i, d i, ) 2, (40) Fig. 2. Manhaan in New York Ciy. and he corresponding opimizaion problem is max v i,, (41) x i, s.. m i, x i, d, (42) i I x i, [0, 1], i I. (43) We see ha, Problem (41) have a differen objecive funcion and differen consrains compared wih Problem (4). So if we only consider cuing ransporaion or charging cake, how o cu he oher cake will be negleced. The proposed JTCS algorihm always converge o he NNE a each imeslo in he JTCSP, since he projecion mehod SSPM in he JTCS algorihm always converges o he opimal soluion o VI(F, Ω ). In [48], i is shown ha, SSPM converges o a soluion of he variaional inequaliy problem under he only assumpion ha is funcion is coninuous and monoonic. In Theorem V.2, we know ha F is coninuous and monoonic, herefore, he soluion o VI(F, Ω ) is he NNE wih idenical weighs. VII. PERFORMANCE EVALUATION In his secion, we firs describe he simulaion seings, and hen presen he simulaion resuls based on real daa ses. We compare he JTCS algorihm wih a heurisic soluion: ransporaion wih greedy charging (TGC), where we firs use he PCI algorihm proposed in [1] o provide ransporaion services and hen schedule all he oher unfully charged PVs o charge baeries unil fully charged. A. Simulaion Seings PV Seing: We use 500 EVs, Yuong E7 [49] (China), o sudy he ransporaion and charging paerns of PVs, alhough hey are no self-driving vehicles now. The number of seas of Yuong E7 is 10 30, and here we assume ha each PV has 16 seas. Each PV has elecriciy capaciy of 45 kwh, and he maximum ravel disance wih fully charged baery is 150 km, i.e., i consumes 0.3 kwh each km. Under 220V volage, he baeries can be fully charged wihin 8 hours, i.e., he charged energy one hour is kwh. We assume ha he iniial remaining energy of all PVs follows a uniform disribuion over [32, 41] kwh. Assuming ha all PVs ravel along he shores pah beween any wo posiions (origins or desinaions of requess) in PCI algorihm wih he idenical speed 30 km/h. Road Map Daa: Fig. 2 shows he map of Manhaan wih abou 60 km 2 in New York Ciy, where black lines are roads, green poins are nodes, and red poins are charging saions,

9 9 Fig. 3. Disribuion of rip requess in Manhaan. Fig. 4. CDF of rip disance in Manhaan. Fig. 5. CDF of rip fares in Manhaan. and he blue lines divide he map ino five regions. The road map is exraced hrough he opensreemap [50], and six ypes of ways are seleced: primary, secondary, eriary, moorway, moorway link, and residenial, and ohers such as runk, unclassified, are ignored. Finally, 3,900 ways and 29,792 nodes are filered. The longiudes and laiudes of charging saions are exraced from he Google map, and hen hey are moved o he neares nodes on roads. New York Ciy Taxi Daa: We use he axi daa se (yellow records) of he New York Ciy during January 1 31, 2016 [29]. Each record conains several useful fields for our sudy, including passenger coun, pickup ime, dropoff ime, rip disance, laiudes/longiudes of origins, laiudes/longiudes of desinaions, fares, axes, ips, and oal paymen. There are 134,721 requess whose origins and desinaions boh are in Manhaan on January 5 (Tuesday), Fig. 3 shows he disribuion of hese rip requess (only in Manhaan) in each hour on January 5, The ime begins a 3:00 for hree reasons. 1) This ime is one of he mos imporan shif handover ime in axi companies in many ciies [51]. 2) The number of rip requess of his ime is almos he minimum of one day [1] [14]. 3) The base loads and energy demands a his ime are boh almos he minimum in one day [30] [52]. Fig. 4 and Fig. 5 show he CDF of rip disance and rip fares (ips and axes are no included) respecively in Mahaan on January 5, We see ha, abou 70% of rip disance is less han 2 km, and 70% of rip fares are less han 10 US dollars. To make he performance more sable, we only choose he rip requess wih ravel disance no less han 2 km, and he number of rip requess is reduced o 41,341. Smar Grid Daa: The online daa se of RTPs is shown in Fig. 6, which is provided by he Commonwealh Edison Company [53]. We see ha he RTP morning and evening peak ime is 9:00 and 17:00 respecively, and he RTP evening peak is much higher han he morning peak. The parameer seings are summarized in Table II. α 1 and α 2 should reflec he weighs of saisfacion level of charging and he charging fees, and should be esed o represen he uiliies of PVGs. As we have described in he previous secion, γ 1 (0, 1), γ 2 (0, 1), γ 3 > 1, µ > 0, η > 0, herefore, we se γ 1 = 0.4, γ 2 = 0.5, γ 3 = 1.5, µ = 1, η = 1. e min denoes he minimum energy of any PV o ravel o he neares he charging saion, and 3 kwh is generally TABLE II VALUES OF PARAMETERS IN SIMULATION SETTINGS α 1 α 2 η γ 1 γ 2 γ µ e min ρ ɛ enough for one PV in Manhaan since i can ravel a leas 10 km according o our assumpions. We se ρ = 0.2, which is generally enough for he energy in he nex ime slo. ɛ = 10 3 is a widely acceped bound in SSPM. B. Resuls We presen he performance of JTCS and TGC in erms of eigh merics: convergence of PVG sraegies, convergence of PVG uiliies, number of ransporaion PVs in each hour, consumed energy in each hour, charged energy in each hour, energy paymen in each hour, average energy price and oal paymen, and remaining energy of all PVs. Finally, he scalabiliy of JTCS and TGC is discussed. The firs meric is he convergence of PVG sraegies a he ime 3:00, which is shown by Fig. 7. We see ha, afer seven ieraions, he proposed JTCS algorihm wih five PVGs converges o he NNE. Differen PVGs have differen bes response sraegies according o heir ransporaion and charging demands. The second meric is he convergence of PVG uiliies a he ime 3:00, which is presened by Fig. 8. We see ha, all he uiliy values are negaive since wo iems in he uiliy funcion (3) are negaive and only one iem is posiive, and he uiliy values converge o he NNE wih he converges of sraegies. If he number of PVs is se o 500, JTCS and TGC almos have he same ransporaion performance in erms of he number of ransporaion PVs in each hour, and he average rip ime. In JTCS and TGC, he average rip ime of passengers is abou 14.6 minues wih he waiing ime of 6.4 minues and he ravel ime (from pickup ime o dropoff ime) of 8.2 minues. The hird meric is he number of ransporaion PVs in each hour in JTCS and TGC, which is depiced by Fig. 9. We see ha, he wo schemes almos have he same performances,

10 10 Fig. 6. Real-ime price (November 27, 2016) Fig. 7. Convergence of PVG sraegies. Fig. 8. Convergence of PVG uiliies. Fig. 9. Number of ransporaion PVs in each hour. Fig. 10. Consumed energy in each hour. Fig. 11. Charged energy in each hour. Fig. 12. Energy paymen in each hour. Fig. 13. Average energy price and oal paymen. Fig. 14. Remaining energy of all PVs. e.g., in ransporaion peak ime (e.g., 8:00), abou 400 PVs should serve passengers, i.e., a mos 100 PVs can charge baeries. Generally, he number of PVs wih sufficien energy in JTCS is smaller han ha in TGC since all PVs excep he ransporaion ones charge baeries in TGC, however, JTCS has almos he same ransporaion performance as TGC since i can coordinae he PVs in each region o saisfy he ransporaion demands of passengers, which can be seen from he firs erm in (3). The fourh meric is he consumed energy in each hour, which is shown by Fig. 10. We see ha, he energy consumpions in JTCS are similar o ha in TGC, since he cloud in JTCS can coordinae he ransporaion demands of PVs in each region of he ciy, and considers abou he energy consumpions in he nex ime slo, which is shown in he second consrain in Problem (23). The fifh meric is he charged energy in each hour, which is presened by Fig. 11. We see ha, he charged energy during 3:00-3:59, 6:00-13:59 and 16:00-22:59 in TGC is obviously higher han ha in JTCS. However, during 23:00-23:59, 1:00-2:59, and 4:00-5:59, he charged energy of JTCS is higher han ha in TGC. We conclude ha JTCS decides o charge more energy a low RTPs, however, o ensure he ransporaion performances of he nex ime slos, i decides o charge some energy even a high RTPs. The sixh meric is he energy paymen in each hour using JTCS and TGC, which is depiced by Fig. 12. We see ha, he profile of energy paymen in each hour is similar o charged

11 11 energy shown in Fig. 11. During nigh and noon ime, he energy paymen in JTCS is high, while a oher ime he energy paymen may decrease o zero. The charged energy of JTCS and TGC is 15,392 kwh and 19,315 kwh respecively. We see ha, PVs in JTCS charge less energy han ha in TGC while his does no affec he ransporaion services. The sevenh meric is he average energy price (he average energy paymen per kwh, US cens/kwh) and oal paymen (US cens) using JTCS and TGC, which are ogeher shown by Fig. 13 wih he same X-axis. We see ha, he average energy price in he JTCS algorihm is 10.86% less han TGC, and he oal energy paymen of JTCS and TGC is 355 and 505 US dollars respecively, reduced by 29.8%. From his poin of view, he average energy price in JTCS is reduced wihou reducing he ransporaion service qualiy since i considers boh he ransporaion demands and charging demands of PVs. However, TGC only considers ransporaion wih a greedy charging sraegy, herefore, he charging coss are much higher han ha in JTCS. The eighh meric is he remaining energy of all PVs in each hour, which is depiced by Fig. 14. We see ha, he remaining energy in JTCS is generally less han ha in TGC. The remaining energy in JTCS reduces quickly in ransporaion morning and evening peak ime, e.g., 8:00 and 18:00, however, he remaining energy in GTC reduces very slowly, since more PVs decide o charge baeries even he RTPs are in high levels, which can reflec he effecs of our scheme. Finally, we discuss he scalabiliy of JTCS over muliple days. In JTCS, we assume ha he final remaining energy is no less han he iniial energy in he day. To ensure he validness of JTCS over muliple days, we should guaranee he final energy of each day is no less han he consumed energy of he nex day. For example, if we expec he final remaining energy of one day be more, we can revise he hird consrain of (23) in Algorihm 1 by increasing he righ-hand bound, since he higher is he righ-hand bound, he more final remaining energy PVs will have a he end of he day, and vice versa. Our proposed approach can be easily exended o he case when he exac rip requess one day ahead are no known. In paricular, he cloud can use he pas requess of a similar day o predic he charging demands, e.g., on Monday, he cloud can use he requess of he previous Monday or he average of several Mondays. If he acual number of requess is significanly differen from he pas, he cloud can adjus he value of some parameers o adap o he new scenarios. For example, if he number of requess becomes much larger han ha of he pas similar day, he cloud can increase he charging demands E + o ensure sufficien fuure energy. VIII. CONCLUSIONS PV sysems are new ransporaion sysems for fuure smar ciies, where PVs are ypically self-driving elecric vehicles. Transporaion and charging coexis in PV sysems, and o balance ransporaion and charging demands, we use a cake cuing game o capure he feaures of he PVGs and build uiliy models. Then we analyze he exisence and uniqueness of Nash equilibrium in his game. Moreover, we propose he JTCS algorihm o achieve he unique normalized Nash equilibrium. Finally, we perform simulaions based on he axi rip daa and smar grid daa of New York o evaluae is performance. We find ha, JTCS can provide almos he same ransporaion services as TGC, however, he average energy price is reduced by 10.86% compared wih TGC. There are several fuure works on his research. As previously menioned, he proposed JTCS algorihm assumes ha he rip requess and he real-ime price are known in advance, however, in he real world, i is impossible. 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Hiskens, Decenralized charging conrol of large populaions of plug-in elecric vehicles, IEEE Transacions on Conrol Sysems Technology (TCST), vol. 21, no. 1, pp , [53] Real ime elecriciy prices. hps://hourlypricing.comed.com/live-prices/, Ming Zhu is now a posdocoral researcher and assisan researcher in Shenzhen Insiues of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China. He received he PhD degree in Compuer Science and Engineering in Shanghai Jiao Tong Universiy, Shanghai, China. A par of his work is finished in Shanghai Jiao Tong Universiy. His research ineress are in he area of big daa, arificial inelligence, inerne of hings, and wireless communicaions.

13 13 Xiao-Yang Liu received his B.Eng. degree in compuer science from Huazhong Universiy of Science and Technology, China, in He is currenly a join PhD in he Deparmen of Elecrical Engineering, Columbia Universiy, and in he Deparmen of Compuer Science and Engineer, Shanghai Jiao Tong Universiy. His research ineress include ensor heory, deep learning, nonconvex opimizaion, big daa analysis and homomorphic encrypion, cyber-securiy and wireless communicaion. Xiaodong Wang (S 98-M 98-SM 04-F 08) received he Ph.D. degree in elecrical engineering from Princeon Universiy. He is currenly a Professor of elecrical engineering wih Columbia Universiy, New York NY, USA. His research ineress fall in he general areas of compuing, signal processing, and communicaions. He has auhored exensively in hese areas. He has auhored he book eniled Wireless Communicaion Sysems: Advanced Techniques for Signal Recepion, (Prenice Hall, 2003). His curren research ineress include wireless communicaions, saisical signal processing, and genomic signal processing. He has served as an Associae Edior of he IEEE TRANSACTIONS ON COMMUNICATIONS, he IEEE TRANSACTIONS ON WIRELESS COM- MUNICATIONS, he IEEE TRANSACTIONS ON SIGNAL PROCESSING, and he IEEE TRANSACTIONS ON INFORMATION THEORY. He is an ISI Highly Cied Auhor. He received he 1999 NSF CAREER Award, he 2001 IEEE Communicaions Sociey and Informaion Theory Sociey Join Paper Award, and he 2011 IEEE Communicaion Sociey Award for Ousanding Paper on New Communicaion Topics.