Julian Diederichs. Optimized Time-of-Use Tariffs for Smart Charging of Plug-In Electric Vehicles. Semester Thesis

Transcription

1 eeh power sysems laboraory Julian Diederichs Opimized Time-of-Use Tariffs for Smar Charging of Plug-In Elecric Vehicles Semeser Thesis Deparmen: EEH Power Sysems Laboraory, ETH Zürich Examiner: Prof. Dr. Göran Andersson, ETH Zürich Supervisors: Dr. Luis Baringo, ETH Zürich Dipl. El. Ing. Marina González Vayá, ETH Zürich Zürich, July 2014

2 ABSTRACT 1 Absrac An anicipaed high proliferaion of plug-in hybrid elecric vehicles (PEV) challenges curren power sysem operaion. However, charging conrolschemes or smar charging can uilize he flexible PEV load o balance grid operaion and o reduce economical coss of charging. The focus is on price-based decenralized conrol schemes in which an aggregaor is responsible o se price incenives o induce a desired charging behavior. In his repor he aggregaor ses Time-of-Use (ToU) ariffs as an inpu parameer for PEV charging profile opimizaions. In order o obain opimized ToU ariffs, a bi-level opimizaion problem is solved, which on one side considers grid operaion objecives and on oher side cos minimizaion for PEV charging. Driving paerns and driver enduse consrains are aken ino accoun o deermine opimized ToU ariffs. Individual PEV cos opimizaions are aggregaed by a novel linear approach o a single opimizaion on he grid level. The problem definiion can be formulaed as a single mahemaical program wih equilibrium consrains (MPEC), which is implemened in form of a mixed ineger problem. Opimized ToU ariffs are obained by manipulaing given hourly spo aucion prices wihin symmeric bounds o obain desired valley filling PEV charging profiles. Resuls of bi-level opimizaions show ha by using opimized ToU ariffs valley filling can be increased by 40%. However, due o muliple opimal soluions of he cos opimizaion, he responses of single level aggregaed and individual cos opimizaion o he opimized ToU ariffs do no necessarily correspond o he desired charging profiles and hus yield no significan improvemens in valley filling. Therefore decenralized ToU based conrol schemes canno incenivize desired power charging scenarios. Index Terms- Smar Charging, PEV conrol schemes, Time-of-use ariffs, Bi-level opimizaion, MPEC

3 CONTENTS 2 Conens 1 Inroducion 4 2 Mehodology Driving paerns Virual sorage model Aggregaion of individual vehicles Linear approximaion of arrival and deparure energy Deparure energy Arrival energy Conrol scheme Formulaion of bi-level problem Upper level Lower level Transformaion o single level MPEC Resuls Case sudy se up Aggregaion of individual vehicles Linear approximaion of deparure energy Linear approximaion of arrival energy Comparison of aggregaion mehods Opimal seing of ToU ariffs Desired reference power profile Soluion of MPEC Discussion Accuracy of aggregaion approach Oulook Acknowledgemens 36 References 37

4 LIST OF FIGURES 3 Lis of Figures 1 Upper and lower charging power limis Upper and lower nodal energy limis for fixed iniial energy Charging responses resuling from he individual and he iniial aggregaed opimizaions Hourly ToU ariffs Comparison of linear approximaions of deparure energies vs. exac soluions Comparison of average approximaion of deparure energies vs. exac soluions Comparison of linear approximaions of arrival energies vs. exac soluions Comparison of average approximaions of arrival energies vs. exac soluions Charging responses resuling from he individual and he new aggregaed opimizaions Normalized power volume curve of July 7 h Power reference profile Charging response of lower level problem and of KKT formulaion Bi-level charging responses o opimized ToU ariffs Charging responses of lower level problem o bi-level opimized ToU ariffs Charging responses of individual cos opimizaion o bi-level opimized ToU ariffs Lis of Tables 1 Noaion able Baery specificaions Charging cos minimizaion for a PEV flee of vehicles 26 4 Charging cos minimizaion for a PEV flee of 5000 vehicles Charging cos minimizaion for a PEV flee of 1000 vehicles Bilevel charging responses Individual charging responses

5 1 INTRODUCTION 4 1 Inroducion In he conex of a urnaround in energy policies owards susainable energy sysems, a high proliferaion of plug-in elecric vehicles (PEVs) is anicipaed. A significan elecrificaion of he vehicle flee poses new challenges o power sysem operaion. Poenial consequences of unconrolled PEV charging are overloading of power sysem asses and increased peaks in demand. However, in case of charging conrol schemes or smar charging, PEVs can be employed o avoid line congesions in he grid and from an economical perspecive o reduce coss of charging [1]. A so- called aggregaor could be responsible for managing charging and purchasing elecriciy on behalf of he PEV owners, as well as for ineracion beween vehicle owners and oher energy sysem eniies, such as energy service providers, ransmission and disribuion sysem operaors [2]. PEVs have he special characerisics of being only used inermienly and hus charging wihin a flexible imeframe is possible. In order o model driver end-use consrains, he aggregaor can cluser he PEV flee in he form of a virual sorage resource wih power and energy characerisics ha depend on vehicle behavior on he grid level [3]. Advanced ways of conrol model vehicles as a sorage, which can be scheduled o eiher consume power or feed power o he grid vehicle-o-grid (V2G) [4]. As mos renewable resources are flucuaing, inegraion and uilizaion of PEVs for demand side managemen and energy sorage can become crucial for smar grid operaion. Approaches for he inegraion of elecric vehicles ino power sysems and uilizaion according o he concep of a smar grid can be divided ino direc and indirec conrol approaches. Direc conrol schemes refer o conrol approaches ha do no acively involve he vehicle owner in he conrol acion imposed on he PEV conneced o he power sysem [2], while indirec conrol schemes are fully decenralized conrol of PEVs. Direc conrol approaches allow o fully conrol PEV charging and as a resul he aggregaor can be designed o fulfill objecives such as smar charging or V2G charging. Drawbacks of direc conrol approaches are heir high requiremens for communicaion and heir low consumer accepance. In indirec conrol he auhoriy of conrol says wih he vehicle owner [5]. Mos indirec or decenralized approaches are based on exogenous price signals [6],[7], which are he inpu o individual charging coss minimizaion of each vehicle. Prices are used by individual vehicles o schedule heir charging aciviies according o heir individual emporal and spaial consrains imposed by heir driving schedule. In a decenralized conrol approach for PEV flees he exogenous inpu parameer, incenivizing a paricular behavior, is sen by an aggrega-

6 1 INTRODUCTION 5 or. The role of he aggregaor in his case would be o se incenives, e.g price incenives, o induce a paricular charging behavior. A sraighforward way of conrolling charging behavior is seing price incenives for consumers in he form of ime-of-use (ToU) ariffs. In order o opimally se up he ToU ariff, he response of he PEVs o cerain ariffs has o be considered. Simple price elasiciies do no accuraely model he response, since hey neglec he evoluion of he individual baeries sae of charge. The charging cycles of a baery are no only price sensiive, bu also consrained by fuure driving requiremens. In case a PEV owner wans o cover wo fully baery depleing rips wihin one day, he vehicle has o be charged in beween he wo rips, independen from he curren ToU ariff. In [8] boh a cenralized and decenralized price-based conrol approach are proposed. Boh, cenralized and decenralized conrol approaches yield similar resuls as mos charging is shifed o valley nigh hours wih lowes ToU ariffs, wih few excepion due o individual driving consrains. In case of sysem-wide prices a high charging simulaneiy a he lowes ToU price is obained, which are no in he ineres of grid operaors. The anicipaed increase in peak demand due o unconrolled PEV charging would require expansion of generaion and ransmission capaciy as shown in sudies for Germany [9] and he USA [10],[11]. Therefore an approch o deermine an improved ToU ariff in a decenralized conrol approach is proposed, which includes he sysem cos or sysem sress minimizaion objecive of grid operaors and he charging cos minimizaion objecive of vehicle owners. The aggregaor models he PEV flee and represens he individual PEVs opimizaions as a single aggregaed opimizaion on he grid level. This formulaion yields a bi-level problem srucure. The upper level problem represens he grid operaors objecive such as valley hour PEV charging. The lower level represens he charging cos minimizaion of he aggregaor subjec o power and energy consrains of he PEV flee. In he problem formulaion he aggregaor can manipulae given prices wihin a small range in order o obain a desired load profile. By reformulaing he lower level problem by is mahemaically equivalen Karush-Kuhn-Tucker (KKT) condiions, he bi-level problem can be ransformed ino a numerically solvable mahemaical program wih equilibrium consrains (MPEC) [12]. The resuling opimizaion yields endogenous ToU prices, since i includes a cos minimizaion of consumers and considers objecives of grid operaors when deermining opimized ToU ariffs.

7 2 METHODOLOGY 6 2 Mehodology The mehodology applied o model driving paerns, baery specificaions and grid consrains is based on [8]. Only all-elecric vehicles are considered. The mahemaical problem is formulaed as an MPEC. These ype of models are described in deail in [13]. The noaion used in he formulaion of he problem is provided in Table 1. The modelled ime horizon is one day, wih hourly ime seps. Table 1: Noaion able Indices v vehicle n node number of ime sep, for hourly ime inervals = [1, 2,.., 24] Vehicle represenaion Ev Ev,cons Ev,dep Ev,arr Pv,charged C v SOC v η v energy conen of he baery of vehicle v a ime sep energy consumed by vehicle v a ime sep deparure energy of vehicle v a ime sep arrival energy of vehicle v a ime sep charging power of vehicle v a ime sep baery capaciy of vehicle v baery sae of charge of vehicle v a ime sep charging efficiency of vehicle v Nodal represenaion En En,dep En,arr Pn,charged energy conen of nodal virual baery n a ime sep energy drop of nodal virual baery n a ime sep energy conribuion o nodal virual baery n a ime sep charging power of vehicles conneced o node n a ime sep T oun,ref reference ToU ariff a node n a ime sep T oun opimized ToU ariff a node n a ime sep η n average charging efficiency a node n χ bound of allowed ToU ariff variaions in % only 1 if vehicles arriving a node n a ime depared from γ, dep n,n dep (v) γ arr, n arr,n dep (v) node n dep a ime dep in heir previous rip only 1 if vehicles arriving a node n arr a ime arr depared from node n a ime in heir previous rip

8 2 METHODOLOGY 7 dual variables λ n,balance λ cons λ n,arr λ n,dep energy equilibrium of nodal virual sorage equaliy of consumed energy of vehicles and required energy arrival energy equilibrium based on linear approximaion deparure energy equilibrium based on linear approximaion µ min,p n lower bound on Pn,charged µ max,p n upper bound on Pn,charged µ min,e n lower bound on End µ max,e n upper bound on En µ min,e n,arr lower bound on En,arr µ max,e n,arr upper bound on En,arr µ min,e n,dep lower bound on En,dep µ max,e n,dep upper bound on En,dep binary variables ωmin,p n ωmax,p n ωmin,e n ωmax,e n ωmin,e n,arr ωmax,e n,arr ωmin,e n,dep ωmax,e n,dep lower bound on Pn,charged upper bound on Pn,charged lower bound on End upper bound on En lower bound on En,arr upper bound on En,arr lower bound on En,dep upper bound on En,dep 2.1 Driving paerns In order o derive charging schemes of elecric vehicles, realisic driving paerns based on he ranspor simulaion ool MATSim are obained [14]. MATSim is an agen-based ranspor simulaion where each agen has a se of aciviies o be performed (e.g. work, shopping). The opimizaion akes ino accoun facors such as available mehods of ransporaion and selecs he driving paerns ha maximize he agen s uiliy funcion. Driving paerns include iming, duraion and disance of rips performed by each vehicle. Informaion on parking locaions is also required o map each vehicle o a nework node. Energy consumpion of each rip is approximaed by he produc of covered disance and an assumed average PEV energy consumpion [8]. The seleced driving paerns are no price dependen. The fac ha according o he obained driving paerns vehicles are parked mos a he ime, jusifies he assumpion ha PEVs are considered a flexible nework load [2].

9 2 METHODOLOGY Virual sorage model The grid nework is simplified o he nodal level. Each parking locaion is mapped o a nework node and is defined a each ime sep by is nodal energy level according o a Virual Sorage Model [3]. From he nodal perspecive arriving PEVs increase he nodal energy level, while deparing PEVs lower he nodal energy level. The nodal virual sorage model allows o represen muliple individual PEVs in an aggregaed single way and as a resul allows o reduce he compuaional effor of he opimizaion. Equaion (1) defines he nodal energy evoluion: E n = E 1 n + P n,charged η n + E n,arr E n,dep n, (1) The nodal energy depends on he energy in he prior ime sep, energy charged in he curren ime sep, he energy conribuion of arriving vehicles and he energy drop due o deparing vehicles. I is assumed ha he iniial energy level Ev 0 of each vehicle s baery is fixed o a given value. In addiion he proposed model requires he final energy level of each vehicle a he end of he modelled ime horizon o be equal o he iniial energy. This consrain guaranees ha during he seleced ime horizon PEVs energy consumpion has o be compleely compensaed by baery charging. The resuling nodal represenaion fulfills he same condiions. The iniial oal nodal energy level is deermined by summing up all vehicle baeries conneced o he grid in he firs ime sep, which also yields a fixed iniial nodal energy level. The consumed energy can be derived from he given driving paerns by he produc of disance covered and an average PEV energy consumpion. The lower and upper border case for charging power (Pn,charged ) can be compued based on maximal charging power and known energy consumpions. The lower bound P,min n,charged is defined by a charging behavior ha only charges in case he curren (SOCv) is no sufficien o cover he nex consecuive rip. The upper bound P,max n,charged is defined by charging a each parking locaion wih maximal charging power aiming o fully charge each vehicle a each ime sep. Boh limis of Pn,charged in case of a PEV flee of vehicles in a grid nework wih one node are shown in Fig. 1. A possible assumpion for a scenario aiming a minimizing charging coss is ha vehicles will only charge he energy required for heir nex rip and consecuively arrive a heir allowed minimal energy level. However, his assumpion fails in case of muliple rips of one PEV wihin one day, since pre-charging for all fuure rips a he poin of minimal coss would resul in lower overall charging coss.

10 2 METHODOLOGY upper power bound lower power bound MW hours of day Figure 1: Upper and lower charging power limis for vehicles Assuming a fixed iniial and final energy level of he modelled ime horizon, upper and lower vehicle energy levels ((Ev,max ) and (Ev,min )) can be compued based on he following wo recursive formulas: E,max v E,min v = E 1,max v = E 1,min v + η v P,max v,charged E v,cons v, (2) + η v P,min v,charged E v,cons v, (3) The inial energy a ime sep = 0 is fixed o a given value and hus Ev 0 = Ev 0,max = Ev 0,min. The same assumpion is made for he final nodal energy Ev 24. The nodal represenaion En,max can be compued by summing a each ime sep all maximal baery energy levels Ev,max of all PEVs parked a node n. En,min is compued accordingly by adding minimal baery energy levels. The resuling nodal energy limis are shown in Fig. 2. Depending on he parameers of PEV charging, charging power and charging duraion, he nodal energy will evolve beween he known lower and upper bound. The energy conens Ev,arr and Ev,dep of he baeries of arriving and deparing vehicles depend on pas charging behavior. Based on known driving profiles Ev,arr and Ev,dep can be aggregaed o he corresponding nodal represenaion (En,arr and En,dep ). Driving profiles include informaion on deparure ime and deparure locaion and arrival ime and arrival locaion of each vehicle. By summing up he baery energy level of all vehicles which depar a one ime sep from one parking locaion n, En,dep is deermined. En,arr is compued by summing up he baery energy levels of all arriving vehicles a each ime sep a each parking locaion. By considering Ev,max or

11 2 METHODOLOGY upper energy bound lower energy bound MW hours of day Figure 2: Upper and lower nodal energy limis for fixed iniial energy for vehicles for compuing En,arr and En,dep upper and lower bounds for arrival and deparure energy can be deermined. E,min v 2.3 Aggregaion of individual vehicles An aggregaor is inroduced in order o cluser he PEV flee according o he virual sorage resource wih power and energy characerisics which depend on driver end-use consrains [3]. The applied virual sorage model is described in subsecion 2.2. By aggregaing vehicle specificaions and driving paerns on a nodal level, muliple opimizaions over individual vehicles and individual consrains can be represened by one aggregaed opimizaion. This enables o simulae charging responses of large PEV flees in reasonable imes and wih limied compuaional power. 2.4 Linear approximaion of arrival and deparure energy In [8] decenralized PEV individual cos opimizaions show lower coss han cenralized nodal opimizaions. This can be explained by he fac ha in he iniial aggregaion mehod arrival and deparure energy are approximaed as he average value of he upper and lower bounds of heir corresponding energies according o equaions (4) and (5). En,dep = E,max n,dep + E,min n,dep 2 En,arr = E,max n,arr + En,arr,min 2 n, (4) n, (5)

12 2 METHODOLOGY 11 Thereby pas charging behavior is no considered in deermining deparure and arrival energies. The upper and lower limis of arrival and deparure energy on he grid node level can be compued as described in subsecion 2.2. The resuls of he individual opimizaions and he aggregaed response wih he average approximaions of arrival and deparure energies show high deviaions in power charged (Pn,charged ) a each ime sep and hus also in he nodal energy levels (En). In order o aggregae he charging profile more accurae, arrival and deparure energy are modelled as funcions of nodal energy levels of prior ime seps insead of he average approximaion. Thereby pre-charging and pas charging behavior is aken ino consideraion. In he novel approach nodal arrival and deparure energy are approximaed by a linear funcion defined in he inerval of minimal and maximal deparure energies [E,min n,dep, E,max n,dep ] or arrival energies [En,arr,min, En,arr,max ]. Depending on pas charging behavior nodal energy levels are defined and subsequenly also he relaive energy levels of arrival and deparure energies can be deermined Deparure energy There is a logic relaion beween he energy level of each node one ime sep prior o vehicles deparing and he relaed drop in energy a ime of deparure. If he nodal energy a one ime sep is close o is maximum, all vehicles deparing in he nex ime sep will ypically also be charged close o heir maximum. Thus he coefficien αn,dep [0, 1], n, is defined as follows: E,min n α n,dep = E 1 n E 1,max n En 1,min En 1,min n, (6) represens he minimal nodal energy required o cover fuure planned rips rip, while En,max represens he upper limi of he energy levels based upon baery capaciy, maximal charging power and vehicle s energy consumpion. In case of nodal energy close o maximum αn,dep will ake a value close o 1 and as a resul he deparure energy En,dep will also be close o is maximum value and vice versa. As a resul he aggregaed deparure energy is defined by he following equaion: ( ) En,dep = E,min n,dep + α n,dep E,max n,dep E,min n,dep n, (7) Arrival energy The linear approximaion of arrival energy is also depending on he nodal energy levels one ime sep prior o deparure. In conras o he deparure energy i does no only depend on he energy level of one parking locaion,

13 2 METHODOLOGY 12 bu on he energy level of all previous parking locaions of vehicles arriving a a new parking locaion a one ime sep. Therefore α n,arr [0, 1], n, is defined as he average normalized energy level of all previous parking locaions of arriving vehicles: α n,arr = 1 v γ, dep n,n dep (v) v γ, dep n,n dep (v) E dep 1 n dep (v) E dep 1,min n dep (v) E dep 1,max n dep (v) E dep 1,min n dep (v) n, (8) Similar o he deparure energy, arrival energy will ake values close o is maximum when all arriving vehicles were fully charged a heir previous parking locaions (α n,arr = 1) or close o is minimum in case vehicles were only charged wih he minimal power required o cover heir rips a heir pas parking locaions (α n,arr = 0). Finally, arrival energy is defined he following: En,arr = En,arr,min + αn,arr ( E,max n,arr En,arr,min ) n, (9) 2.5 Conrol scheme The applied conrol scheme is indirec and hus a decenralized conrol approach, since he aggregaor manipulaes ToU ariffs o incenivize a cerain charging behavior. Thus, he aggregaor, besides aggregaing individual vehicles, ses price incenives o obain lower coss and a desired charging behavior. The desired charging behavior is defined by he grid operaor, who wans o limi PEV induced sress on he grid infrasrucure. 2.6 Formulaion of bi-level problem ToU ariffs are manipulaed by an aggregaor o incenivize a desired charging behavior. The opimal ToU ariff profile is deermined by a bi-level programming approach, which fulfills grid operaion objecives on he grid nework level and minimize charging coss of an aggregaed PEV flee. The grid nework level is referred o as he upper level problem, while he aggregaed PEV flee as he lower level problem. The lower level consrains he upper level, which requires boh problems o be joinly solved. Therefore he problem formulaion has o be ransformed ino a single level MPEC Upper level The upper level objecive funcion is defined as he minimizaion of he charging profile deviaion from a desired loading profile Pn,ref for each node a each ime sep:

14 2 METHODOLOGY 13 min P n,charged,t ou n Pn,ref P n,charged (10) n An objecive funcion defined by an absolue value can be ransformed ino a mahemaically equivalen linear opimizaion problem wihou absolue values, bu wih a new opimizaion variable Z and wo inequaliy consrains insead according o [15]: min P n,charged,z n,t ou n Zn (11) n Z n + P n,charged P n,ref n, (12) Z n P n,charged P n,ref n, (13) The upper level problem is consrained by an inequaliy consrain, enabling ToU ariffs o vary wihin cerain bounds (e.g. +/ χ%) relaive o reference prices. Thereby a rade-off beween small deviaions in reference prices and a charging behavior close o he desired loading profile according o (10) is obained Lower level (1 χ) T ou n,ref T ou n (1 + χ) T ou n,ref n, (14) The lower level problem represens he vehicle charging cos minimizaion over each PEV, enabling PEV owners o charge heir vehicles a he lowes possible price. The charging cos minimizaion over each vehicle is represened by he single aggregaed opimizaion on grid level: min P n,charged, E n, E n,arr, E n,dep T oun Pn,charged (15) n Is soluion provides he charging profile a each node and hus also he energy level evoluion a each node, including deparure and arrival energy. The dual variables, required for reformulaion of he lower level problem, are defined in Table 1. The soluion of he lower level is consrained by he following equaliy and inequaliy consrains: 1. Equaliy consrains:

15 2 METHODOLOGY 14 (a) Maximal charging power and given driving profiles, yielding En,arr and En,dep, consrain he energy profile of he nodal virual sorage model: En = En 1 + Pn,charged η n + En,arr En,dep : λ n,balance n, (16) I is assumed ha he iniial energy level of each vehicle is he same as in he end of he modelled ime horizon and each vehicle reurns o is iniial parking posiion. Thereby i is guaraneed ha vehicles charge he required power o cover heir rips wihin he modelled ime horizon. This implies also on he nodal level same iniial and final energy of each nodal virual sorage. E 0 n = E T n : λ T n,balance n (17) (b) Arrival energy and deparure energy are approximaed based upon he following inroduced formulas: En,arr = En,arr,min + αn,arr ( E,max n,arr En,arr,min ) : λ n,arr n, (18) E n,dep = E,min n,dep + α n,dep ( E,max n,dep E,min n,dep ) : λ n,dep n, (19) (c) Energy consumpion of each vehicle wihin he modelled imeframe can be compued based on he obained driving paerns. To guaranee ha each vehicle reurns o is iniial energy level, he energy charged En,charged = P n,charged of all vehicles a all imes muliplied by he given conversion efficiency η n has o exacly equal consumed energy wihin he modelled imeframe. This consrain inhibis V2G charging in order o reduce charging coss. Pn,charged η n = En,cons : λ cons (20) n n 2. Inequaliy consrains: (a) Pn,charged is consrained by minimal charging requiremens due o fuure rips and limied baery capaciies: P,min n,charged P n,charged P,max n,charged : µ min,p n, µ max,p n n, (21) (b) E n is consrained by driving paerns, charging behavior and limied baery capaciies: E,min n E n E,max n : µ min,e n, µ max,e n n, (22)

16 2 METHODOLOGY 15 (c) En,arr and En,dep are consrained by minimal and maximal obainable nodal energy levels and resuling limis of arrival and deparure energy: E,min n,arr E n,arr E,max n,arr : µ min,e n,arr, µ max,e n,arr n, (23) E,min n,dep E n,dep E,max n,dep : µ min,e n,dep, µ max,e n,dep n, (24) Transformaion o single level MPEC The lower level problem described by (15) (24) is coninuous and linear due o he fac ha T ou n is defined as an opimizaion variable of he upper level. Thus he lower level problem is convex. Convex opimizaions can be mahemaically equivalenly represened by heir Karush - Kuhn - Tucker (KKT) condiions or heir Primal Dual formulaion [13]. 1. KKT : Since he lower level problem is convex, i can be replaced by is KKT condiions. The KKT condiions can be divided ino heir opimaliy condiions and he so-called complemenariy consrains. The opimaliy condiions require he derivaive of he Lagrangian of he lower level problem in respec o each of he four opimizaion variables ( n, ) o be equal o zero: T oun + λ n,balance η n λ cons η n µ min,p n + µ max,p n = 0 n, (25) λ n,balance + λ+1 n,balance + E +1,max λ+1 n,dep E +1,min n,dep n,dep + v λ arr n arr(v) γ arr, n arr,n dep (v) v γarr, n arr,n dep (v) µ min,e n + µ max,e n = 0 n, E,max n E arr,max n arr(v) E 1,max n dep (v) E,min n E arr,min n arr(v) E 1,min n dep (v) (26) λ n,balance λ n,arr µ min,e n,arr + µ max,e n,arr = 0 n, (27) λ n,balance λ n,dep µ min,e n,dep + µ max,e n,dep = 0 n, (28) In addiion all inequaliy consrains have o be saisfied and he corresponding inequaliy dual variables µ have o be greaer han or equal o zero. Furhermore, he complemenariy condiions have o be fulfilled, which is expressed by he orhogonally requiremen of he lower level inequaliy and corresponding dual variable inequaliy:

17 2 METHODOLOGY 16 0 Pn,charged P,min n,charged µ min,p n 0 n, (29) 0 P,max n,charged P n,charged µ max,p n 0 n, (30) 0 En En,min µ min,e n 0 n, (31) 0 En,max En µ max,e n 0 n, (32) 0 En,arr En,arr,min µ min,e n,arr 0 n, (33) 0 En,arr,max En,arr µ max,e n,arr 0 n, (34) 0 En,dep E,min n,dep µ min,e n,dep 0 n, (35) 0 E,max n,dep E n,dep µ max,e n,dep 0 n, (36) The numerically criical complemenariy condiions (29) (36) can also be resaed by applying he Foruny-Ama linearizaion procedure [16], resuling in (41) (56). M imposes an upper bound for all µ and lower level inequaliy consrains. In case of bounding he lower level opimizaion variables, he upper bounds are known and M can be defined accordingly: M P n = P,max n P,min n n, (37) M E n = En,max En,min n, (38) M E n,arr = En,arr,max En,arr,min n, (39) M E n,dep = E,max n,dep E,min n,dep n, (40) For he dual variables he upper limi M was chosen o be a sufficienly large number.

18 2 METHODOLOGY 17 P n,charged P,min n,charged (1 ω min,p n ) M P n n, (41) 0 µ min,p n ω min,p n M n, (42) P,max n,charged P n,charged (1 ω max,p n ) M P n n, (43) 0 µ max,p n ω max,p n M n, (44) E n E,min n (1 ω min,e n ) M E n n, (45) 0 µ min,e n ω min,e n M n, (46) E,max n E n (1 ω max,e n ) M E n n, (47) 0 µ max,e n ω max,e n M n, (48) E n,arr E,min n,arr (1 ω min,e n,arr ) M E n,arr n, (49) 0 µ min,e n,arr ω min,e n,arr M n, (50) E,max n,arr E n,arr (1 ω max,e n,arr ) M E n,arr n, (51) 0 µ max,e n,arr ω max,e n,arr M n, (52) E n,dep E,min n,dep (1 ω min,e n,dep ) M E n,dep n, (53) 0 µ min,e n,dep ω min,e n,dep M n, (54) E,max n,dep E n,dep (1 ω max,e n,dep ) M E n,dep n, (55) 0 µ max,e n,dep ω max,e n,dep M n, (56) The new single level problem formulaion has he objecive funcion of he upper level (11): min φ Zn (57) The opimizaion variables are he opimizaion variables of he upper problem and he lower problem, including all dual and binary variables: n φ =[Zn, T oun, Pn,charged, E n, En,dep, E n,arr, λ n,balance, λ n,iniial, λ n,arr, λ n,dep, µ min,p n, µ max,p n, µ min,e n, µ max,e n, µ min,e n,arr, µ max,e n,arr, µ min,e n,dep, µ max,e n,dep, ωmin,p n, ωmax,p n, ωmin,e n, ωmax,e n, ωmin,e n,arr, ωmax,e n,arr, ωmin,e n,dep, ωmax,e n,dep ] (58)

19 2 METHODOLOGY 18 The single level consrains consis of he upper level consrains (12) (13), equaliy and inequaliy consrains of he lower level problem (16) (24) and he formulaed KKT condiions of he lower level (25) (28), (41) (56). The objecive funcions and consrains are all implemened in Yalmip for Malab and solved by CPLEX v solver [17]. 2. Primal Dual formulaion: Anoher valid reformulaion of a convex problem is based upon he principle of dualiy. In his alernaive reformulaion he lower level problem is subsiued by is primal consrains, is dual consrains and is srong dualiy equaliy according o [12]. The srong dualiy equaliy is as follows: T oun Pn,charged n = λ cons n n + n 1 n v γ, dep n,n dep (v) n + n n E n,cons µ max,p n P,max n,charged + n µ max,e n E,max n + n µ min,p n P,min n,charged µ min,e n E,min n λ n,arr(e,min n,arr (E,max n,arr E,min n,arr ) v γ, dep n,n dep (v) E dep 1,max n dep (v) µ max,e n,arr E,max n,arr + n λ n,dep (E,min n,dep En 1,max µ max,e n,dep E,dep n,arr + n E dep 1,min n dep (v) En 1,min E dep 1,min n dep (v) ) µ min,e n,arr E,min n,arr En 1,min µ min,e n,dep E,dep n,arr (E,max n,dep E,min n,dep )) (59) Boh reformulaions of he lower level are equivalen and yield in combinaion wih he upper level a MPEC. The advanage of he primal-dual over he KKT formulaion is ha i does no include any numerically expensive complemenariy condiions. In case of a very large number of complemenariy consrains, he MPEC derived from he KTT formulaion may become inracable. As a resul he primal dual formulaion is generally easier

20 2 METHODOLOGY 19 o solve. However, in his MPEC he primal dual formulaion includes he nonlinear erm n T ou n Pn,charged in is srong dualiy equaliy. Thus in his case he primal dual formulaion is no used o reformulae he lower level problem.

21 3 RESULTS 20 3 Resuls 3.1 Case sudy se up In order o simulae charging responses, PEV flee specificaions have o be defined. The PEV vehicle flee is modelled inhomogeneous in regard o baery capaciy corresponding o he specificaions of he Toyoa Prius (16 kwh) and of he Nissan Leaf (24 kwh). Limiaions and echnical specificaions of modelled baeries are provided in Table 2. Baeries require a minimal sae of charge (SOC) of 20 % and exhibi a charging efficiency of 90 %. A consan semi-fas charging power of 11 kw and ubiquious charging are assumed a each nework node. The anicipaed average PEV energy consumpion is 0.2 kwh/km and allows o compue he required energy of each vehicle o cover is rips. Table 2: Baery specificaions C v SOC v,min η v P,max v,charged 50% : 16 kwh 50% : 24 kwh kw 3.2 Aggregaion of individual vehicles The goal was o modify he aggregaion mehod for he purpose of represening he response of a PEV flee o ToU ariffs in a single opimizaion and obain soluions close o he resuls of individual PEV opimizaions wih low compuaional effor. In [8] he aggregaor models arrival and deparure energies as he mean beween he upper and lower bounds of he corresponding energy levels. This approximaion yields observable deviaions beween he individual and aggregaed response (Fig. 3) o given ToU profile (Fig. 4). Deviaions in charging profiles are clearly visible in Fig. 3 and higher coss for he aggregaed response are expeced, since he aggregaed charging profile charges less a imes of minimal prices. In order o improve he aggregaion mehod, he linear approximaion of deparure and arrival energies, described in subsecion 2.4, is inroduced. By expressing deparure and arrival energies as funcions of nodal energy levels a imes of respecive vehicles deparure, he deviaions of nodal energy levels and power charged beween he aggregaion and he individual response can be reduced. In addiion resuling coss of charging can be decreased compared o he iniial aggregaion mehod. The cos reducion can be explained by he smaller mismach of he charging profiles in case on he new aggregaion mehod, which also resuls in a smaller deviaion of

22 3 RESULTS 21 Figure 3: Charging responses of a PEV flee of vehicles in a grid wih 88 nodes resuling from he individual and he iniial aggregaed opimizaions for one given ToU ariff are shown. Figure 4: Hourly ToU ariffs on July 5h of 2014 from EPEX spor marke daa for Swizerland. charging coss Linear approximaion of deparure energy The resuls of he new linear approximaion of deparure energies are compared o he individual deparure energies in Fig. 5. Boh PEV opimizaions are simulaed for vehicles in a grid represenaion wih 88 nodes in response o ToU ariffs from July 5h o July 9h from Swiss EPEX spo aucion daa. The chosen linear approximaion yields resuls very similar o individual resuls. Deviaions are small and he linear approximaion performs significanly beer compared o he iniial average approximaion for he same inpu parameers as shown in Fig. 6.

23 3 RESULTS 22 Figure 5: Comparison of linear approximaions of deparure energies vs. exac soluions Figure 6: Comparison of average approximaion of deparure energies vs. exac soluions Linear approximaion of arrival energy The resuls of he linear approximaion of arrival energies from PEV opimizaions for vehicles in a grid represenaion wih 88 nodes in response o ToU ariffs from July 5h o July 9h from Swiss EPEX spo aucion daa, are shown in Fig. 7. Deparure energies compared o arrival energies are closer o he individual opimizaion resuls, since hey only dependen on nodal energy level a one ime sep, while arrival energies depend on all pas energy levels of arriving cars. Thereby small errors are accumulaed. In addiion here is a disconinuiy in arrival energies. Since he final energy is fixed, he denominaor of equaion (8) is equal o zero for all vehicles deparing in he firs ime sep = 1. Thus αn,arrival is no

24 3 RESULTS 23 defined in he firs sep and is se o zero. As a resul he conribuion of vehicles deparing in he firs ime sep o he arrival energy is zero. Seing αn,arrival 0 o 0.5 accordingly o SOC0 v = SOC0,max v +SOCv 0,min 2, does no yield any improvemens in he approximaion of arrival and deparure energies. Thus αn,arrival in he las ime sep is se o zero and equaion (9) is correced accordingly by reducing he number of vehicles arriving in case vehicles arriving depared in he firs ime sep. This disconinuiy combined wih he dependency on muliple energy levels explains why he linear approximaed arrival energy has more ouliers han he linear approximaed deparure energy.

25 3 RESULTS 24 Figure 7: Comparison of linear approximaions of arrival energies vs. exac soluions Figure 8: Comparison of average approximaions of arrival energies vs. exac soluions 3.3 Comparison of aggregaion mehods Smaller deviaions in he charging power profile due o he new aggregaion mehod can be observed, when comparing Fig. 3 and Fig. 9. Charging coss and errors beween individual opimizaions and aggregaed responses for hree differenly sized PEV flees are provided in Tables 3, 4 and 5. ToU ariffs are aken from EPEX spo aucion daa from July (5 h 14 h ) for Swizerland. The resuls of he new aggregaion mehod are compared o he iniial aggregaion mehod (referred o as old) and o he individual opimizaion. The derivaions of nodal energy levels and power charged beween individual opimizaions and aggregaed approaches are shown for

26 3 RESULTS 25 each flee and day. The new aggregaion ouperforms he old aggregaion for each flee size, bu performance improvemens decrease for larger flees. The obained daa proves ha on average he new aggregaion resuls in smaller deviaions from individual PEV responses and hus shows beer performance for aggregaed PEV charging opimizaions compared o he iniial formulaion in [8]. Lower coss for he new aggregaion compared o he iniial mehod are expeced, since a imes of lowes cos maximum charging power is obained. Figure 9: Charging responses of a PEV flee of vehicles in a grid wih 88 nodes resuling from he individual and he new aggregaed opimizaions are shown. In case of small PEV flees (1000 PEVs) he new aggregaion mehod can reduce absolue deviaions in nodal energy and charging profiles from he individual model by more han 60% compared o he iniial aggregaion. For larger PEV flees (10000 PEVs) he new approximaion of E n,dep and E n,arr sill ouperforms he old average approximaion in regard o absolue deviaions in power profiles compared o he resuls of individual opimizaions by more han 36%. Thus he new approximaion successfully manages o yield resuls which are beer for he purpose of flee modelling for indirec conrol approaches. The linear approximaion of E n,dep and E n,arr deerioraes in performance for very large flees due o he fac ha in case of large PEV flees a each node a each ime sep a high number of vehicles is parked. As a resul here is no srong inerdependency of nodal energy and deparure energy in case few vehicles depar from a node o which a high number of vehicles are conneced. This relaion can also be derived from he daa in Tables 3 and 5, since improvemen of power deviaions for a flee size of 1000 PEVs is 50% compared o only 30% for a flee size of PEVs. In cerain cases he new aggregaion even ouperforms he individual

27 3 RESULTS 26 opimizaions in erms of oal charging coss. The aggregaed response minimizes oal charging coss, while PEV individual opimizaions minimize charging coss for each vehicle. Toal lower charging coss in case of an aggregaed opimizaion do no guaranee ha each individual PEV is charged a lowes cos possible. As a resul cos comparisons are no relevan indicaors for performance of he new aggregaion mehod. The improved aggregaed resul in cos can be explained by he inroducion of new opimizaion variables in he aggregaed opimizaion. Explicily, En n is he new addiional opimizaion variable and can direcly influence En,dep and En,arr as shown in equaions (6-9). In large PEV flees his effec is sronger, since a small variaion in nodal energy impacs muliple nodal energy levels in fuure ime seps due o arriving vehicles. In such cases he new inerdependency can be fully uilized. The key indicaors of he precision of he aggregaion approach are deviaions from nodal energy levels and charging profiles relaive o individual PEV opimizaions resuls. Table 3: Resuls of charging cos minimizaion for a PEV flee of vehicles disribued over 88 nodes for 10 differen ToU price profiles.

28 3 RESULTS 27 Table 4: Resuls of charging cos minimizaion for a PEV flee of 5000 vehicles disribued over 88 nodes for 10 differen ToU price profiles. Table 5: Resuls of charging cos minimizaion for a PEV flee of 1000 vehicles disribued over 88 nodes for 10 differen ToU price profiles. 3.4 Opimal seing of ToU ariffs The improved aggregaion mehod is applied o formulae an MPEC o deermine opimized ToU ariffs, which on he one hand fulfill he grid operaor s objecive of following a reference load and on he oher hand minimize charging coss of PEVs Desired reference power profile The grid operaor and generaing unis favor uniform loading which reduces sress on he grid infrasrucure and a he same ime reduces ramp up and ramp down coss of generaing unis. Therefore he chosen reference power profile for PEV charging is a valley filling scenario. Since in he defined problem se V2G charging is no considered, unloading of grid infrasrucure

29 3 RESULTS 28 by depleing PEVs sored energy is no aken ino accoun. Daily power volume profiles are obained from EPEX spo aucion daa. In he chosen reference power profile valley filling is only inroduced in hours, when he aucioned power volume falls below he daily power volume mean. In Fig. 10 a daily power volume curve is shown, in which he green filled area marks he desired valley filling volume. The resuling reference load is depiced in Fig. 11. In order o reach heoreically a he opimum a minimal cos of zero on he upper level and o obain nicely inerpreable resuls, he reference load profile is normalized by he daily required energy consumpion of all vehicles o cover heir rips. Figure 10: Normalized power volume curve of July 7 h in which green filled area marks he desired valley filling volume. Figure 11: Power reference profile desired by grid operaors and generaing unis o obain uniform loading based on raded power volume on July 7 h Soluion of MPEC The single-level MPEC soluion is compared o lower level soluions and o he response of individual PEV opimizaions. The lower level aggregaed response for a given ToU ariffs is depiced in Fig. 12. The discussed unwaned peaks in load can be observed. The equiv-

30 3 RESULTS 29 alen KKT response is also ploed in Fig. 12 o prove ha he soluion of he KKT formulaion of he lower level problem exacly maches he lower level response. The desired reference profile is based on EPEX spo aucion daa of July 7 h. Load peaks a imes of lowes cos can be observed. In order o reduce he compuaional effor, simulaions are performed wih a PEV flee of PEVs and a grid represenaion wih one node. Figure 12: Responses of lower level problem and equivalen KKT reformulaed problem o real ime prices are shown. The bi-level opimizaion manipulaes hourly prices from EPEX spo aucion daa in he range of maximal +/ χ% o incenivize PEV charging according o he desired reference profile. The symmeric +/ χ% bounds were chosen o neiher favor he grid operaor and generaing unis nor he PEV owners in he opimizaion of he ToU ariffs. The bounded bi-level opimizaion yields he opimal manipulaed ToU ariffs, which enable o achieve valley filling up o 90% for cerain load and pricing scenarios as provided in Table 6. Opimizaions are performed for a PEV flee of vehicles on a grid level wih one node. Firs, resuls of bi-level opimizaions wih EPEX spo marke daa for he second week of July from 7 h 13 h are shown in Table 6.

31 3 RESULTS 30 Table 6: The Table shows charging responses of he bi-level problem by opimizing ToU ariffs based on Swiss marke daa in he second week of July. Spo marke daa is used o obain reference ToU ariffs and deermine he valley filling power profile. Two differen opimizaions ses, one wih +/ 10% allowed variaions in price and he oher wih +/ 20% are compared o he iniial soluion of he lower level problem wih non-opimized ToU ariffs. The firs se of opimizaions allows variaions in ToU ariffs by +/ 10%, yielding a weekly average valley filling of 58% compared o iniially 22%, while coss increase by 6% compared o he opimal charging coss of he lower level. The second se allows larger variaions in ToU ariffs by +/ 20%, which resul in a weekly average valley filling by 66%, bu a cos increase of 13% relae o he opimal lower level coss. The bi-level opimized ToU ariff and he resuling charging response of July 7 h is provided in Fig. 13. In Fig % of valley filling is obained, while oal charging coss rise by 5%. In he upper subplo he red line represens he original ToU ariff and he blue line shows he opimized ToU ariff. The flaening of he ToU profile around he lowes price as a resul of he opimizaion can be observed. As a resul i can be observed ha for he bi-level soluion valley filling is improved by 44% o 71% due o opimized ToU ariffs by allowed ToU variaions of +/ 10%. Depending on daily ToU profiles a differen bound on ToU variaions yields opimal resuls. The increase in charging cos is caused by a flaening of he ToU profile in order o no incenivize charging a he ime sep of lowes cos. ToU ariffs variaions of +/ 10% show a good rade-off beween successful valley filling and a low increase in coss for bi-level opimizaions. Secondly, he reponses of he individual opimizaions o he corresponding opimized ToU ariffs are shown in Table 7. Performance in valley filling is very low wih average values of 20% compared o 16% in response o non opimized ToU ariffs, independen from +/ 10% or +/ 20% allowed ToU variaions. As a resul, he opimized ToU ariffs, independen from heir variaion bounds, yield no significan improvemen in valley filling for

32 3 RESULTS 31 Figure 13: Bi-level charging responses o opimized ToU ariffs of PEVs. he applied indirec conrol scheme. Table 7: The Table shows he resuls of individual PEV opimizaions in response o he opimized ToU ariffs. The problem relaed o he bi-level opimizaion is ha he opimized ToU ariffs yield muliple opimal soluions. The ToU profile allows muliple soluion for PEV charging, which are all equal in cos of charging. Thus responses of he lower level and individual opimizaions o he opimized ToU ariffs may show charging profiles differen o he bi-level charging profile and differen o he desired power valley filling profile. The lower level response o he opimized ToU ariff obained from Fig. 13 is shown in Fig. 14 and is equivalen in charging coss o he bi-level soluion. The cos opimizaion has muliple opimal soluions, hus he response is no he same as he bi-level soluion and he desired valley filling charging profile is no

33 3 RESULTS 32 obained. Furhermore, he individual opimizaion resuls o he opimized ToU ariffs are depiced in Fig. 15. The deviaions of he charging profiles beween he lower level response and he individual response are small, proving he performance of he linear approximaion approach for PEV flee aggregaion. Boh opimizaions, he lower level and he individual, yield resuls of poor valley filling. The problem associaed wih he bad performance also lies in he naure of price incenive mechanisms. Discree price signals do always incenivize a charging peak a he lowes price and hus induce high load peaks. In case of individual cos opimizaions, vehicles will always charge a he lowes price. Therefore he appplied opimizaion approach fails o deermine opimized ToU prices ha lead o valley filling PEV charging responses for indirec charging conrol schemes. Figure 14: Charging responses of lower level problem o bi-level opimized ToU ariffs of PEVs.

34 3 RESULTS 33 Figure 15: Charging responses of individual cos opimizaion o bi-level opimized ToU ariffs of PEVs.

35 4 DISCUSSION 34 4 Discussion The opimized ToU ariffs enable o reach significan improvemens in valley filling as a resul of he bi-level opimizaion mehod. However, cos opimizaions in response o opimized ToU ariffs for he aggregaed or individual approach show no successful valley filling. The problem is ha here are muliple opimal soluions for given ToU ariffs and individual opimizaion will always charge a he lowes price available, causing load peaks a imes of lowes prices. The problem lies in he underlying characerisics of price incenive mechanisms. 4.1 Accuracy of aggregaion approach The new aggregaion approach ouperforms he iniial aggregaion and hus represens an improved basis for he bi-level opimizaion and for fuure aggregaed PEV opimizaions. The drawback of he linear approximaions concerns he discussed disconinuiy in arrival energy relaed o he fixed final energy and he decrease in performance for very large PEV flees. However, up o a PEV flee size of 1000 vehicles per grid node, he new linear approximaion significanly ouperforms he exising aggregaion approach. The approximaion of he arrival energy (9), in case of large PEV flees and high numbers of loading nodes, is numerically expensive. Especially, he formulaion of he required KKT condiions (26) relaed o arrival energy akes significan compuaional effor.

36 4 DISCUSSION Oulook In fuure simulaions nodal ToU ariffs can be inroduced o obain improved desired charging responses of vehicles. By varying ToU ariffs depending on locaion a each ime sep insead of sysem-wide ToU ariffs, he leverage of he aggregaor in indirec conrol schemes rises. The inroducion of addiional opimizaion variables due o nodal ToU ariffs can be uilized o se price incenives varying on locaion. Thereby improved desired PEV charging power responses can be obained. Advanced opimizaion approaches can also inroduce price and quaniy dependend ToU ariffs. By limiing he power quaniy provided a lower prices, demand peaks in he response can be reduced. As a resul more uniform PEV charging responses can be obained. In order o successfully implemen indirec conrol schemes, nodal or quaniy dependend ToU ariffs have o be inroduced, since hourly ToU ariffs according o he obained resuls fail o conrol PEV charging. In he fuure srong deviaions in ToU prices migh cause price dependend driving paerns. A he momen all driving paerns are modeled price independen. The proposed opimizaion approach would allow o uilize known reference prices o simulae price dependen driving paerns, since he opimized ToU ariffs will only deviae by +/ 10% from reference price.