Lagrangian Decomposition based Multi Agent Model Predictive Control for Electric Vehicles Charging integrating Real Time Pricing

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1 1 Lagrangian Decomposiion based Muli Agen Model Predicive Conrol for Elecric Vehicles Charging inegraing Real Time Pricing Alessandro Di Giorgio, Andrea Di Maria, Francesco Liberai, Vincenzo Suraci, Francesco Delli Priscoli arxiv: v1 [cs.sy] 11 Jul 216 Absrac This paper presens a real ime disribued conrol sraegy for elecric vehicles charging covering boh drivers and grid players needs. Compuaion of he charging load curve is performed by agens working a he level of each single vehicle, wih he informaion exchanged wih grid players being resriced o he chosen load curve and energy price feedback from he marke, elaboraed according o he charging infrasrucure congesion. The disribued conrol mechanism is based on model predicive conrol mehodology and Lagrangian decomposiion of he opimizaion conrol problem a is basis. The simulaion resuls show he effeciveness of he proposed disribued approach and he muual coherence beween he compued charging load curves and he resuling energy price over he ime. Index Terms Elecric Vehicles; Uiliy funcions; Muli Agen Model Predicive Conrol; Lagrangian Decomposiion; Smar Grid. p P λ x e T c T NOMENCLATURE EV charging power Power generaion Shadow price Sorage power EV s sae of charge EV s sae of charge error Sampling ime Se of ime slos in he conrol horizon I. INTRODUCTION IN recen years Elecric Vehicles (EVs) are receiving an increasing aenion, wih heir number growing over he ime; his is due o an increasing concern abou air polluion, energy consumpion, climae change and insabiliy. A firs assessmen of EVs impac has been invesigaed in 29 (see [1]), where i is remarked ha he presence of EVs in he nework will lead o a nework characerized by a predominance of acive elemens, in addiion o he radiional passive ones. From he echnical poin of view, anoher delicae effec of The auhors are wih he Deparmen of Compuer, Conrol and Managemen Engineering, a Sapienza Universiy of Rome, Via Arioso 25, 185, Rome, Ialy, digiorgio@diag.uniroma1.i. F. Liberai is wih he SMART Engineering Soluions & Technologies (SMARTEST) Research Cener, ecampus Universiy, Via Isimbardi 1, 226, Novedrae CO, Ialy This work is parially suppored by he SAPIENZA - ATENEO 213 Planning and conrol of flexible elecriciy demand and generaion from renewable energy sources in Smar Grids projec, no. C26A13LYTB, amd he SAPIENZA - ATENEO 215 SMILE - Smar MIcrogrid of elecric Energy Projec, no. C26H15ZWK5 EVs inegraion is he harmonic disorions caused by hese new acive elemens wih such a high power demand. Beyond echnical aspecs, also he human-behaviour plays an imporan role: people ends o charge EVs during working ime a heir job locaion, hen mainly a he same hours, beween 1: AM and 6: PM [1]); his generaes high power peaks in he nework, which is expeced o have a derimenal effec on grid operaion. As a maer of fac conrolling he EVs charging power is becoming a real need, wih some echnical soluions becoming o appear in he indusrial pracice and academic lieraure. In his paper, a decenralized Model Predicive Conrol (MPC) sraegy for EVs charging is presened. An opimizaion problem is solved a each sampling ime wih he aim of esablishing he operaing sepoins for he EVs chargers and an energy sorage sysem (ESS) conribuing o he coverage of he charging load; addiionally he ne power provided by he grid is calculaed. The problem is solved in a decenralized way, resuling in a dedicaed real ime and dynamic local elecriciy marke where he power schedules raded by grid and EVs agens ieraively converge o an equilibrium guaraneeing he balance beween demand and supply. As cusomary in MPC applicaions, he firs conrol sample is acually applied for operaing he charging and sorage infrasrucure. The remainder of he paper is organized as follows. Secion II discusses he mos modern approaches o he EVs charging problem. Secion III presens he reference scenario used for formulaing he conrol problem. Secion IV provides he mahemaical formulaion and ses he basis for is soluion. Secion V presens he mahemaical soluion o he problem. Secion VI shows he resuls obained in simulaing he addressed scenario and finally he conclusions are drawn in secion VII. II. STATE OF THE ART In his secion, he curren sae of he ar for he addressed problem is presened. Firs of all i mus be said ha he problem of EV charging is ruly recen and i has been of ineres since few years. Even in his shor ime, a significan number of differen approaches have been proposed. In [2] here is a major aenion owards he baery pack of plug-in EVs. Linear and quadraic models are used; a case sudy of consrains violaion is performed, hrough baery charging schedule calculaion and execuion hrough an ordinary differenial equaion solver. I is demonsraed

2 2 ha a linear approximaion is beer han he quadraic one when processing boundary condiions violaions; i requires less compuaional effor and he violaion rae is almos he same of he quadraic case. In [3] a myopic algorihm is developed o find a minimum variance sraegy for EV baery charging. I does no know any informaion abou he fuure and i jus calculaes he acual bes move. This is achieved hrough a cenral uni, he Aggregaor, ha receives informaions from EVs: i asks heir required power absorpion level and provides a reference value (aken wihin a feasible range) o follow, based on nework congesion level. The convergence o opimaliy is reached when minimum and imum reference values are almos he same. In [4] a mehod based on nework losses minimizaion is aken ino accoun. Saring from an unconrolled case, where power flow equaions are sudied in presence of volage limis, a solver is used o find he bes siuaion (where losses in he grid are minimized) for a given peneraion level of EVs. In [5] an admission conrol scheduling algorihm is developed. The main idea is ha, when an EV arrives a a charging saion, is charging session can be acceped, rescheduled or refused. The decision is made according o a greedy profi policy, which makes a comparison beween he profi gained in acceping he charging session and nework congesion. In [6] he problem of EV charging scheduling inside a large load area is faced. Saring from a global scheduling problem (which aims o minimize coss due o EV charging), a soluion is found bu i is, in realiy, impracical o use because of he huge number of variables and he compuaion ime. Then, he problem is reduced o a local problem aking ino accoun small groups of EVs. A soluion is found and, by proper scaling, i is demonsraed o be close o he global one. In [7] he problem of EV charging is approached hrough a marke-based heory. EVs have heir own imizaion funcions in erms of power absorpion, while he marke has o minimize a congesion cos funcion. EV baery consrains and marke acual imum capaciy are aken ino accoun in order o solve he problem, and o find he opimal charging rae hrough a Lagrangian approach. In [8] a Lagrangian approach is used. By defining user uiliy funcions, which are dependen on power absorpion, and joining hem ino one single imizaion problem, he sandard formulaion applicaion of dualiy heory is achieved. Consrains are grouped ogeher and added o he primal problem in order o find a dual one. Lagrangian mulipliers are updaed following he ani gradien of he dual funcion and he opimal primal soluion is found hrough primal problem imizaion (when he dual is a is opimum). In [9] he problem of EV charging is faced in wo seps. The firs one, consiss of acquiring informaions from all nodes along a ravelling road and hen making an esimae of he congesion level of he nework; aferwards, he informaion on how many EVs can be charged is forwarded o single charging saions. In he second sep EVs acquire daa from he neares charging saion when approaching o i and hen an inner algorihm akes a decision abou sopping here or no, based on he baery level and he remaining disance o be covered by he EV. In [1] a real-ime decenralized charging algorihm is proposed. Based on a general Lagrangian approach, EVs are modeled hrough uiliy funcions which are joinly aken ino accoun in a global imizaion problem. Dualiy heory and he gradien descen algorihm are applied. When he laer one converges he primal problem converges o is opimal value and each EV is noified wih is own opimal charging rae. In [11] a global imizaion problem is defined and he alernaing direcion mehod of mulipliers is employed o reach an opimal value for he sae variables of he problem (in his case, EV charging powers). Then he mehod is joined wih sochasic predicions made for renewable power producion profiles and EVs arrival rae, in order o apply an MPC mehodology and use predicions o adjus he soluion of he algorihm. Wih respec o he approaches and differen formulaions here repored for solving he EVs charging problem, he proposed work is characerized as follows: Firs of all, he mehod ries o be as decenralized as possible. There is no need for a lo of daa o be exchanged beween he cenral uni and he agens. This is reached hrough Lagrangian relaxaion and convergence of mulipliers vecor, which is seen as he price o pay when buying power. Then, he vecors of variables mus be of small lengh, because oherwise he compuaional ime becomes higher. In order o reach his goal, only one vecor is broadcas o all agens and i is, indeed, he price vecor: i drives user choices in such a way ha hey can decide on heir own how much power o ask, wihou any knowledge of he nework (excep for price); MPC approach is used. I is effecive in order o keep he sysem, a each ime insan, on he righ evoluion owards global equilibrium and single agen opimizaion. Mos of previous case sudies, in fac, were only focused on real-ime implemenaion and no on preview abou wha could happen. A sorage elemen is inroduced in problem formulaion, in order o help he algorihm in disribuing peak requess ha oherwise would cause insabiliy and fauls in charging saion. Wih his in mind, he proposed EV charging reference scenario and sraegy is presened in he following secions. A. Acors and Sysems III. REFERENCE SCENARIO The scenario addressed in his paper is composed of hese eniies (acors and sysems). 1) Charging Saions: The Charging Saion is he main plan of he scenario. I is acually he place where all he elecric vehicles come asking for powers and i has he responsibiliy of disribuing power o hem. I is equipped wih some access poins where he EVs can plug-in heir chargers and proceed in acquiring power. I acs like a middleware beween providers and cusomers, since i has he duy of aking power from he formers and disribue i o he laer.

3 3 2) Disribuion Sysem operaor (DSO): The Disribuor Sysem Operaor is he eniy in charge of providing power o charging saion. I is equipped wih a sorage elemen ha can lend free power when needed. I is direcly conneced o he charging saion and i communicaes o his one he quaniy of power ha i has deermined o sell (a curren price). 3) Drivers: They need o have heir baery reach a desired charge level, before heir deparure ime. 4) Aggregaor: I is he eniy ha has he duy of deermining power price. I receives boh requess of EVs and DSO, hen i proceeds in deermining he bes price; i does i by ieraively asking power curves o he oher acors and adjusing price. I is usually mouned inside he charging saion (as a conrol logic) bu i can be exernal o i; in his second case i should be direcly conneced o all he oher acors. B. Use Case Here a shor descripion of eniies ineracion is given. During a cerain amoun of ime (i may be a day, a week, or an hour) an unknown number of elecric vehicles (drivers) come o a charging saion. Their arrival ime is compleely random and here will be overlapping requess (in a real case, a lo of EVs can come charging a he same ime, as discussed in inroducion). They connec asking for powers and communicae heir desired power curves, which are elaboraed based on curren predicions of power price. Charging saion asks o he Disribuor Sysem Operaor how much power i can sell a he acual prediced price and i obains he offer curve. The Aggregaor receives his daa and, by ieraively updaing price levels and collecing curves from EVs and DSO, i reaches an opimal price value o be used for all he prediced ime (hus, he opimal power curves). Afer his, he DSO provides required power and he Charging Saion disribues i o EVs. This whole process is repeaed for every ime slo of he day. IV. CONTROL PROBLEM FORMALIZATION Some key poins are se here, in order o beer specify he siuaion: Time is spli ino ime slos. The acual ime is named. The mehod is developed o work in any ime-scale. There is a predicion ime window T. I is he se of ime slos during which acual EVs will be acive. I is composed of N slos, and ranges from acual ime up o N 1 slos in he fuure. T =, + 1,, + N 1. In order o calculae T, wo mehods can be used: 1 Time window has fixed lengh N, and all he EVs mus ask o charge heir baery imum before N ime slos. 2 Time window goes from acual ime up o he imum ime (ha can be waied for charging) beween acual EVs. Since here are predicions (due o MPC), when referring o prediced quaniies (and curves) each variable is of he kind x(τ/), which means ha he variable value is previewed a ime τ wih respec o acual ime, where τ belongs o Time Window T. The number of EVs changes over ime, because hey can arrive or end heir charging process. So he number of EVs acive a ime is R. Predicion is given only for power curves. The number of EVs a ime (R ) is he same along all he acual prediced ime window T. A nex ime slo, if an EV arrives, ime window is changed and so R value. There is only one power disribuor, which is assumed s well o operae a sorage elemen. Power wihdrawn from sorage does no have a cos. To every EV a single Uiliy funcion is assigned, which is a funcion of he power exchanged wih he charging saion, namely U(p r (τ/)). This uiliy funcion has hree imporan properies: I is monoonically non-decreasing: his is due o he fac ha he level of saisfacion of each user grows up wih he level of power consumpion. I is concave: his is due o he fac ha saisfacion of he user can have sauraion. I is coninuous. Each EV ries o imize is own uiliy funcion and demands he charging saion as much power as possible. Is reques in erms of power absorpion is limied by wo imporan facors: he imum physical power capabiliy of he EV and he quaniy of power o absorb in order o reach desired sae of charge. The imizaion is done on all he emporal window ha each EV can see. U r (p r (τ/)) (1) p r(τ/) I r() τ T Subjec o he boundary condiions on EV power absorpion limis p min r < p r (τ/) < p r, r R, τ T (2) and o sae of charge error condiions e r (τ + 1/) = e r (τ/) (1 ξ)t c p r (τ/), e r ( f /) =, e r (/) = e(), τ T,r R r R r R where ξ [, 1]. Power producion, insead, ries o sell he quaniy of power ha i is more convenien. Thus is uiliy funcion ries o minimize coss of producion (ha, in curren scenario, is equal o imize profis). Is uiliy funcion depends on he quaniy of power ha i has o produce a every ime, i.e. C l (P l (τ/)). I has he following properies: I is monoonically non-decreasing: he cos of providing a cerain amoun of power should be increasing wih he level of energy capaciy. I is convex. I is coninuous. In a more explici form min C l (P l (τ/)) (3) P l (τ/) I l τ T

4 4 subjec o P min l < P l (τ/) < P l, τ T (4) The sorage elemen provides free energy o power producion when needed; i has an uiliy funcion ha ries o charge back power when i can and ries o keep he sae of charge as close as possible o a reference value. I can be formulaed as +N 1 j (x s () (i/)δ s T c x ref ) 2 (5) j= wih boundary condiion of P min s i= < (τ/) < P s, τ T (6) By summing up all he uiliy funcions of all he agens (for all he previewed ime) a Global Uiliy Funcion is obained. U r (p r (τ/)) p r (τ/) I r () r R τ T P l (τ/) I l () (τ/) I s () τ T C l (P l (τ/) (τ/)) +N 1 j= (x s () j (i/)δ s T c x ref ) 2 Wih 3 ypes of agens acing a he same ime, he only fair soluion o he problem is o give every user he imum possible degree of saisfacion wih he same prioriy of he ohers. This is acually rying o imize he Social Welfare. This mus be done while considering he physical consrains of he problem i= (7) r R p r (τ/) = P l (τ/), τ T (8) V. DECENTRALIZED SOLVING PROCEDURE In order o accomplish he ask of solving he imizaion problem, one possible formulaion is o use he global consrains wih Lagrangian Mulipliers and o inroduce hem inside he imizaion problem, as in sandard Lagrangian Theory. Local consrains are of no concerns owards he convergence of he algorihm, because hey only limi some quaniies, bu hey do no influence global mulipliers. A Lagrangian variable is needed, in order o have a unique key quaniy ha is shared among all he agens and can help regulaing price of energy. A new vecor is required and i is λ() = [λ(/), λ( + 1/),, λ( + N 1/)]. By premuliplying 8 wih λ(), he sandard formulaion of [12] is achieved. L( p r (), P l (), (), λ()) = U r (p r (τ/))+ τ T +N 1 j= (x s () r R τ T C l (P l (τ/) (τ/))+ j (i/)δ s T c x ref ) 2 + i= τ T (λ(τ/)( r R p r (τ/) P l (τ/))) (9) The Lagrangian can be spli ino #R + 1 separae problems: #R problems, for every EV, and one for disribuor. r R L( p r (), P l (), (), λ()) = τ T (U r (p r (τ/)) λ(τ/)p r (τ/))+ τ T (C l (P l (τ/) (τ/)) λ(τ/)p l (τ/)))+ +N 1 j= (x s () j (i/)δ s T c x ref ) 2 (1) i= They can be solved separaely. So he single EV problem is: { p r() r R τ T (U r (p r (τ/)) λ(τ/)p r (τ/)) subjec o consrains a,b,c,d (11) where (a),(b),(c),(d) are a p min r < p r (τ/) < p r, r R, τ T b e r (τ + 1/) = e r (τ/) (1 ξ)t c p r (τ/), τ T,r R c e r ( f /) =, r R d e r (/) = e(), r R The DSO problem is: minp r() τ T (λ(τ/)p l (τ/)) C l (P l (τ/) (τ/)) (x s () j i= (i/)δ s T c x ref ) 2 s.. Pl min P l (τ/) Pl τ T Ps min (τ/) Ps τ T (12) By solving 11 and 12, he Dual funcion D is found. where D( λ()) = p r I r () P l I l () I s () min D( λ()) (13) λ() L( p r (), P l (), (), λ()) (14) Then, in order o find he opimal value for he dual problem, a variaion of Gradien Projecion algorihm is used. The Lagrangian mulipliers are updaed wih he projeced anigradien of D. λ k+1 () = ( λ k () γ( D), ) (15)

5 5 where D = P l () r R p r () (16) Thus he final algorihm is Soluion Algorihm 1 A ime slo of he day, calculae he number of acive EVs ( R ). 2 For each of he acive EV ask is deparure ime, and se he imum value among hem as Time Horizon. This will be he limi of he opimizaion window (T ). 3 During same ime slo, execue his cycle unil exi condiion is me: a Ieraion k=, ake price of previous ime slo as saring price and se all prices along ime window T he same. b Each vehicle solves is own opimizaion problem (11), from acual ime up o is deparure ime, calculaing p r (). c The informaion of every p r () is forwarded o he charging saion and he aggregaor sums hem ogeher. Afer ha he DSO solves is own opimizaion problem (12) o calculae he power which can give away ( P l () ()). d Price vecor is updaed following he anigradien of he dual problem (Eq 15). e If he exi condiion is me, price vecor is he opimum, so exi his inner loop and go o 4. Oherwise, se k k + 1 and repea from b o e, using new prices and new powers as saring values for nex ieraion. 4 Once 3 has ended, he price value obained is he opimal price a curren ime slo, and he firs value of he vecor is aken as price updae for nex ime slo +1. Powers are exraced by solving EV and DSO problem and hey are he opimum curves o follow. 5 Se +1 (i.e. go ahead of one ime slo 1) and repea seps from 1 o end. VI. SIMULATION RESULTS Simulaions have been performed using an imac 21.5, Inel Core i5, 2,7 GHz, 12 GB RAM 1333 Mhz DDR2 compuer, running Apple OSX 1 (v.11). The conrol framework has been buil in Malab 64 bi, and he MPC problem has been solved a each ieraion by using Malab buil-in solver. The base case is a half day scenario (12 hours), wih ime spli ino 15 min ime slos, for a oal of 48 ime slos. EVs arrivals and deparures have been randomly generaed. Simulaion parameers are specified in able (I). Firs of all, in Fig. 1, an earlier simulaion has been performed wihou any sorage elemen. This is a developmen of he basis heory formalized in [13], where a similar approach has been se only for a single ime slo. The firs plo is he unconrolled power profile, which is obained when he EVs arrive a he saion and hey charge heir baery a imum power. Those peaks are he ones o be avoided. TABLE I LOAD AREA SIMULATION SCENARIO 3 Logarihmic Uiliy Funcion U r(p r()) r R # EV Variable I r() for each consumer p r() [; 22]kW Weighs w r() = 1; forevs 1 o 2 # Energy Sources 1 DSO cos funcion.6(p l () ()) (P l () ()) Iniial Prices 16 ecen /kwh Pl () 1kW Sorage cos funcion # Sorage 1 (x s( ) τ i= Ps(i/)δsTc x ref ) 2 = (x() x ref ) 2 Ps min 1kW Ps 1kW x s( ) = x ref 1kW h P Unc Σ p r λ [eurocen/kw] Fig. 1. Case Wihou Sorage The second one is he evoluion of demand curve along he day, once he conrol algorihm proposed in his paper has been applied; of course, since here is a balance equaion beween demand and offer, he demand curve exacly maches he offer curve. The hird one is he daily evoluion of price. I follows he same shape of offer curve. When a sorage elemen is inroduced (Fig. 2) demand and offer curves say almos he same of he case of no sorage(2 nd plo), bu he evoluion of prices changes (see hird plo). The evoluion of prices is heavily reduced, more or less of 1/5, wih respec o before and is shape is smooher, wih less variaions. Moreover, one can noe ha price has a differen shape wih respec o power offer curve. In Fig. 3 he reason behind he new price curve is explained: since he sorage elemen is lending free power o producion side (i.e. here is no cos of producion for ), he quaniy of power ha is produced is modified accordingly, in order o keep he Uiliy Cos Funcion of he producer a is opimal value. Since price follows producion shape, i goes alongside wih P l, which is he ne produced power. Sorage is employed o reain he imum possible reward in producing as long as i does no discharge oo much he sorage elemen.

6 6 P Unc Σ p r λ [eurocen/kw] Fig. 2. Powers and Price P l - Sorage SoC he possible role of he former and he laer s conribuion owards he smoohness of offered power curve. REFERENCES [1] C. Yang and R. McCarhy, Elecriciy grid: Impacs of plug-in elecric vehicle charging, Universiy of California, 29. [2] O. Sundsrom and C. Binding, Opimizaion mehods o plan he charging of elecric vehicle flees, IBM Research, 21. [3] Q. Li and T. Cui, On-line decenralized charging of plug-in elecric vehicles in power sysems, arxiv, Cornell Universiy Library, 211. [4] D. S. A. T. LE and M. O. VAZQUEZ, Scheduling charging of elecric vehicles for opimal disribuion sysems planning and operaion, 21s Inernaional Conference on Elecriciy Disribuion, 211. [5] Y. J. Shiyao Chen,, and L. Tong, Deadline scheduling for large scale charging of elecric vehicles wih renewable energy, School of Elecrical and Compuer Engineering Cornell Universiy, 212. [6] Y. He and B. Venkaesh, Opimal scheduling for charging and discharging of elecric vehicles, IEEE Transacions on Smar Grids, 212. [7] J. Hu and ShiYou, Coordinaed charging of elecric vehicles for congesion prevenion in he disribuion grid, IEEE Transacions on Smar Grids, 213. [8] C. R. Omid Ardakanian and S. Keshav, Disribued conrol of elecric vehicle charging, e-energy 13 Proceedings of he fourh inernaional conference on Fuure energy sysems, 213. [9] Z. Q. Azwirman Gusrialdi and M. A. Simaan, Scheduling and cooperaive conrol of elecric vehicles charging a highway service saions, he 53rd IEEE Conference on Decision and Conrol, 214. [1] J. Rivera and C. Goebel, A disribued anyime algorihm for realime ev charging congesion conrol, Conference: Proceedings of he 215 ACM Sixh Inernaional Conference on Fuure Energy Sysems, A Bangalore, India, 215. [11] D. O. Trudie Wang and H. Kamah, Dynamic conrol and opimizaion of disribued energy resources in a microgrid, Smar Grid, IEEE Transacions on (Volume:6, Issue: 6 ), 215. [12] R. Freund, Dualiy heory of consrained opimizaion, MIT courses, 24. [13] A. Q. Andrea Mercurio, Alessandro Di Giorgio, Disribued conrol approach for communiy energy managemen sysems, MED 212, 27. Fig. 3. Agregaed ne power flow and sorage behaviour. VII. CONCLUSIONS In his paper a decenralized MPC approach for EVs smar charging in a load area has been presened. The problem has been considered firs from a global perspecive, building a sandard Lagrangian formulaion wih a se of balance equaions as he unique consrains. Then a decenralizaion sep has been performed, hrough decomposiion of he global problem ino small subproblems, one for each EV and one more for he DSO, wih heir own local consrains. Then he dual problem has been inroduced. Finally, a decenralized MPC algorihm has been defined in order o conrol power curves evoluion: he mehod has been demonsraed o achieve convergence for variables of boh primal and dual problems. Simulaions have been performed for a case of an half day scenario in a small area, equipped wih a sorage elemen. They showed he effeciveness of he mehod when using he sorage and he fas convergence rae owards nework balance and users saisfacion. Some consideraions have been done regarding he shadow price variable and he sorage, assessing