Tuning of utility function parameters to achieve smart charging of PHEVs. Semester Thesis. Author: Felix Wietor

Transcription

1 eeh power systems laboratory Tuig of utility fuctio parameters to achieve smart chargig of PHEVs Semester Thesis Author: Felix Wietor Departmet: EEH Power Systems Laboratory, ETH Zurich Expert: Prof. Dr. Göra Adersso, ETH Zurich Supervisor: Dipl. El. Ig. Matthias D. Galus, ETH Zurich Zurich, September 2010

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3 Abstract Due to risig eergy prices ad icreasig evirometal cocers, electric cars are likely to become a importat part i future trasportatio systems. These vehicles eed to charge their batteries durig certai time itervals. However, the exact timig of their chargig behavior ca be adapted to make them charge i the most suitable istats i order to shape the load curve of the power system i a useful way. With the goal of reducig the peak load (peak shavig), the possibility of dischargig at peak times should be cosidered as well. I the preset thesis, this problem was approached by tuig of the parameters of the PHEV maager approach weightig the eergy stored by the hybrid cars. The chargig/supply behavior was the simulated usig MATLAB. It was show that valley fillig ca be performed without disregardig the iterests of the owers of the respective hybrid cars. Likewise, peak shavig is also possible, but with the restrictio that battery costs eed to be eglected or compesated. iii

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5 Ackowledgmets I would like to thak my supervisor Matthias D. Galus for his guidace ad support throughout the whole semester project. Our umerous discussios ad meetigs proved very helpful to me. I would also like to express my gratitude towards Prof. Dr. Göra Adersso, head of the Power Systems Laboratory at the ETH i Zurich, for the opportuity of this iterestig ad challegig thesis. Zurich, September 2010 Felix Wietor v

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7 Cotets 1 Itroductio Motivatio for the use of plug-i hybrid electrical vehicles i smart grid applicatios The beefits of valley fillig ad peak shavig i a large power system Scope of the preset thesis Itegrated system descriptio for PHEV maagemet i multi eergy carrier systems The eergy hub as a part of the system The 4-Hub electricity ad gas etwork Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio The PHEV maager ad the cotrol structure Descriptio of the PHEV maagig algorithm for valley fillig Descriptio of the PHEV maagig algorithm for peak shavig Icorporatig battery costs ito the utility fuctio for peak shavig The exteded utility fuctio for peak shavig Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator Tuig α ad β Determiig a useful chargig behavior by takig ito accout the load curve ad the umber of PHEVs Tuig the parameters α ad β i order to achieve the desired chargig behavior Cash flow ad fiacial icetives The ifluece of the battery costs o the dischargig behavior Itegratio of the ivestigated actors i curret practice power system structures Price cosideratios from the poit of view of the PHEV aggregator 36 6 Case studies Geeral data used for the case studies Case study 1: day 1 with α = π max day vii

8 Cotets 6.3 Case study 2: day 1 for valley fillig ad peak shavig Case study 3: day 1 for valley fillig ad peak shavig, eglected battery costs Case study 4: day 2 for valley fillig ad peak shavig Case study 5: day 3,4 for valley fillig ad peak shavig Case study 6: day 2,3,4 for valley fillig ad peak shavig, eglectig the battery costs Case study 7: day 1 for valley fillig ad peak shavig ad disregardig the hub limits Case study 8: day 1 ad day 2, takig ito accout the hub limits Case study 9: day 1 for a large amout of cars with α tued for valley fillig ad peak shavig Coclusios 61 viii

9 List of Figures 1.1 The ifluece of V2G behavior o the load curve The 4-Hub electricity ad gas etwork The hierarchical structure of the cotrol system (a) θ k as fuctio of time, figures (b) ad (c) show zooms of (a) The beefit as a fuctio of the SOC The umber of cycles a battery ca perform as a fuctio of SOCk mi The margial cost ad the margial beefit as a fuctio of the SOC The reachable levels ad the areas correspodig to the charged power (red ad gree) together with the load curve The reachable level ad the hub limits The utility as a fuctio of q k for qk c 0 ad qk s The utility as a fuctio of q k for qk c < 0 ad qk s < The utility as a fuctio of q k for qk c 0 ad qk s < The utility as a fuctio of q k for qk c < 0 ad qk s > Depedecy of the battery price (red curve) o the SOC ad compariso with the value of the sold eergy (blue curve) Estimatio of the evolutio of the battery price for a price reductio of 20% (red curve) ad 30% (gree curve) every 5 years. The price curves are compared with the value of the sold eergy (blue curve) The iteractios betwee the differet actors i the system (a) PHEV load for α tued accordig to the results of chapter 4 i order to perform valley fillig; (b) price curve (gree) ad base load (blue) for the cosidered day (a) PHEV load for α tued accordig to the results of chapter 4 i order to perform valley fillig ad peak shavig while eglectig the battery costs; (b) price curve (gree) ad base load (blue) for the cosidered day Depedecy of the cost of peak shavig per kwh of supplied power o the battery price per kwh of discharged eergy Sesitivity aalysis plot displayig the obtaied profit as a fuctio of the umber of PHEVs ad π PHEV ix

10 List of Figures 6.1 The total umber of PHEVs plugged to the four hubs durig the simulatio (a) Base load ad (b) price curve for day 1 ad α = πday max (a) Base load ad (b) PHEV load for day 1 ad α = πday max Total load costructed (gree) from base load (blue) ad PHEV load (see figure 6.3) for day 1 ad α = πday max (a) The obtaied values for the average chage of the state of charge q k (T ) ad (b) the parameter α (blue) together with the price (gree) for day 1 ad α tued accordig to the results of chapter (a) Base load ad (b) PHEV load for α tued accordig to the results of chapter 4 for day Total load (gree) costructed from base load (blue) ad PHEV load (see figure 6.6) for α tued accordig to the results of chapter 4 for day (a) The obtaied values for the average chage of the state of charge q k (T ) (left) ad (b) the parameter α (blue) together with the price (gree) with α tued accordig to the results of chapter 4 for day 1 ad with eglected battery costs (a) Base load ad (b) PHEV load (with battery costs (blue), without battery costs (gree)) for day 1 with α tued accordig to the results of chapter Total load (gree) costructed from base load (blue) ad PHEV load (see 6.9) with α tued accordig to the results of chapter 4 for day 1 ad with eglected battery costs (a) Base load ad (b) price curve for day (a) The obtaied values for the average chage of the state of charge q k (T ) (left) ad (b) the parameter α (blue) together with the price (gree) for day 2 ad with α tued accordig to the results of chapter (a) Base load ad (b) PHEV load for day 2 with α tued accordig to the results of chapter Total load (gree) costructed from base load (blue) ad PHEV load (see 6.13) with α tued accordig to the results of chapter Total load (gree) costructed from base load (blue) ad PHEV load for day 3 ad with α tued accordig to the results of chapter Total load (gree) costructed from base load (blue) ad PHEV load for day 4 ad with α tued accordig to the results of chapter Total load (gree) costructed from base load (blue) ad PHEV load for day 2 ad with α tued accordig to the results of chapter 4 for eglected battery costs x

11 List of Figures 6.18 Total load (gree) costructed from base load (blue) ad PHEV load for day 3 ad with α tued accordig to the results of chapter 4 for eglected battery costs Total load (gree) costructed from base load (blue) ad PHEV load for day 4 ad with α tued accordig to the results of chapter 4 for eglected battery costs Total load (determied with pattersearch, (gree); determied with SNOPT, (red)) for day 1 ad with α tued accordig to the results of chapter 4 takig ito accout the battery costs Total load (determied with pattersearch, (gree); determied with SNOPT, (red)) for day 1 ad with α tued accordig to the results of chapter 4 for eglected battery costs Total hub limits for the 4-hub etwork Total load (gree-with hub limits; red-without hub limits) for day 1 costructed from base load (blue) ad PHEV load (see 6.13) with α tued accordig to the results of chapter 4 ad takig ito accout the hub limits Total load (gree-with hub limits; red-without hub limits) for day 2 costructed from base load (blue) ad PHEV load (see 6.13) with α tued accordig to the results of chapter 4 ad takig ito accout the hub limits Total load (gree) costructed from base load (blue) ad PHEV load for day 1 for PHEVs ad with α tued accordig to the results of chapter xi

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13 List of symbols SOC state of charge [%] SOC mi k miimal state of charge that is allowed [%] q k chage of SOC due to chargig/supplyig [%] PHEV Plug-i Hybrid Electric Vehicle C p k plug capacity of PHEV k at hub [W] Ck B battery capacity of PHEV k at hub [Wh] V2G vehicle-to-grid (smart chargig/dischargig) α/β parameters of the beefit fuctio (see chapter 3) [e/kwh] π curret spot price of electricity [e/kwh] πbatt total battery price [e/kwh] πday mi miimal spot price of the day [e/kwh] π PHEV price the PHEV customers pay for the electrical eergy [e/kwh] u k utility of PHEV k at hub (see chapter 3) [e] N PHEV umber of PHEVs at hub N PHEV umber of PHEVs for the 4-Hub etwork H set of eergy hubs PHEV set of PHEVs plugged to hub xiii

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15 1 Itroductio 1.1 Motivatio for the use of plug-i hybrid electrical vehicles i smart grid applicatios As the recet past has show, hybrid vehicles are becomig more ad more popular amog customers. Due to risig prices of the fossil fuels, classical iteral combustio egie vehicles are likely to be partly replaced by electrical or hybrid cars. As discussed later o, the price for gasolie is roughly equal to 0.43 e/kwh ad thus much higher tha the price for electrical eergy (0.14 e/kwh for private households i Germay durig 2009, see [1]). The risig umber of electrically powered cars ca be cosidered to be a challege (i [2], the expected impacts of ucotrolled chargig are aalysed) ad a opportuity for the electrical grid ad the customers (see [3]) at the same time. O the oe had, the total load is goig to be icreased, forcig the ower of the grid to either ehace ad replace his equipmet or operate the grid with a smaller stability margi. O the other had, whe properly used, PHEVs (Plug-i Hybrid Electrical Vehicles) could be used to shape the load curve. 1.2 The beefits of valley fillig ad peak shavig i a large power system Whe cosiderig a whole eergy grid over the duratio of a etire day, oe refers to the low load phases as valleys ad to the high load phases as peaks. Thus, valley fillig refers to the idea of icreasig the load durig valleys by switchig o additioal cosumers like PHEVs i order to make the productio ad electricity trasport more effective. Aalogously, the goal of peak shavig is to reduce load peaks by supplyig eergy back to the etwork or, as a last resort, by curtailig cosumers. If peak shavig is realized o a large scale, peak power plats could be temporarily or eve permaetly switched off. Peak shavig requires customers which are both capable of storig eergy ad willig to provide this service. The beefits of the combiatio of peak shavig ad valley fillig, commoly termed load levelig, are described i [4]. The load levelig (see figure 1.1) by PHEVs ca be uderstood to be oe possibility of vehicle-to-grid services, commoly abbreviated as V2G. PHEVs are particularly iterestig for load levelig, because, o the oe had, they 1

16 1 Itroductio cabepluggediorshediafastadiexpesivewayadotheother had, large fleets of PHEVs ca provide a cosiderable amout of power. For example 1 millio PHEVs with a plug power of 3.5kW each, could supply 3.5GW of power i total, thus matchig two or three fairly large geerators. I this way, eve with a market peetratio of oly a few percet (1 millio PHEVs are equivalet to a share of approximately 2 percet i Germay, as 50.2 millio cars are licesed as of 1st Jauary 2010 ([5])), PHEVs ca have a sigificat impact o the system load. peak shavig Load power supplied/ charged by the PHEVs origial load curve valley fillig Time Figure 1.1: The ifluece of V2G behavior o the load curve 1.3 Scope of the preset thesis The goal of the preset project is to explore the cocepts goverig the chargig behavior of PHEVs ad to use them to achieve the previously metioed itelliget features takig ito accout the curret state of the grid. I order to allow a simulatio of a sample of a regio or coutry, a small virtual test system used. Data about the PHEVs was geerated by simulatios at the IVT (Istitute for trasport plaig ad systems at the ETH). These PHEVs are located i differet zoes of a small fictioal area. At their parkig lot, the customers ca plug their cars i order to charge the battery ad/or to perform V2G. I additio to the data about the PHEVs, the sample was completed with real load curves (provided by EWZ, the electric utility of the city of Zurich), eergy prices (provided by EEX ) ad realistic etwork parameters icludig lie impedaces, capacities ad voltage limitatios. The V2G behavior of the PHEVs was the simulated o the metioed test system ad ivestigated usig MATLAB. 2

17 2 Itegrated system descriptio for PHEV maagemet i multi eergy carrier systems The purpose of this chapter is to preset the multi eergy carrier approach for the test etwork o which the simulatio was performed. The etwork structure preseted here is based o the etwork show i [15]. First of all, it is importat to clarify the cocept of eergy hubs as they costitute a importat part i the etwork described below. 2.1 The eergy hub as a part of the system A eergy hub is a structure disposig several iputs correspodig to differet eergy carriers (electrical, chemical, mechaical, thermal). The power from each iput eergy carrier is the used by the eergy hub to produce output eergy carriers. A eergy hub ca be described by the followig couplig matrix C: L α c αα c βα c ωα P α L β c. = αβ c ββ c ωβ P. β (2.1) L ω }{{} L } c αω c βω {{ c ωω } C P ω }{{} P The subscript letters deote the type of eergy carrier, L ad P represet the load vector ad the iput power vector of the hub, respectively. The coefficiets c αβ characterize the coversio efficiecy from eergy carrier β to carrier α. A simple type of a hub is a trasformer, trasformig electrical eergy to electrical eergy o a differet voltage level. I additio to a trasformer, a eergy hub could cosist for istace of a boiler, a CHP uit (combied heat ad power), a geerator, a furace, a battery or a fuel cell. A detailed descriptio ad explaatio of eergy hubs ca be foud i [6] ad [7]. 3

18 2 Itegrated system descriptio for PHEV maagemet i multi eergy carrier systems 2.2 The 4-Hub electricity ad gas etwork The etwork aalyzed i the preset thesis cosists of four eergy hubs, liked by a electrical grid ad a system of pipes trasportig atural gas as illustrated o figure 2.1 (figure take from [15], page 3). Figure 2.1: The 4-Hub electricity ad gas etwork Each hub supplies local customers with electrical eergy ad heat, ad cosists of a trasformer, a furace ad a CHP. The hub ca trasform electrical eergy ad gas ito electrical eergy ad heat. There are three differet ways of performig this trasformatio. First of all, usig the trasformer, electrical eergy ca be directly trasformed ito electrical eergy with a lower voltage level. Geerally, this coversio ca be doe with a high efficiecy ηee TR. Secodly, gas ca simply be bured i the furace i order to produce heat with the efficiecy ηgh F. Third, gas ca be trasformed to heat ad electrical e- 4

19 2.2 The 4-Hub electricity ad gas etwork ergy (with the respective efficiecies η CHP gh ad ηge CHP ) at the same time. This trasformatio is performed by the CHP. Whe oly heat is eeded, the CHP has a much lower efficiecy compared to the furace. But whe cosiderig the amout of heat ad electrical eergy produced by the CHP, it outperforms the furace i terms of the total coversio efficiecy. The fractio of gas itroduced ito the CHP is called the dispatch factor ν g. The share of gas bured by the furace is the equal to (1 ν g ). The specific equatio for the typeofhubusedithisthesisisgiveby: ( Le ) = L h }{{} L ( η TR ee ν g η CHP ge } 0 ν g ηgh CHP +(1 ν g ) ηgh F {{ } C ) ( ) Pe P h }{{} P (2.2) The values assumed for the coversio efficiecies durig this project are summarized i table 2.1. η TR η CHP η CHP ee 0.98 ge 0.35 gh 0.45 ηgh F 0.75 Table 2.1: Assumed coversio efficiecies for the eergy hubs Whe the etwork is strogly loaded, a trasformer at a specific hub ca be overloaded. I this case, the eergy hub ca partly compesate for a deficiecy of trasformer capacity by usig its CHP. I this way, a part of the electrical load is provided by the CHP istead of the trasformer. This ca oly be doe efficietly if the quatity of heat produced by the CHP is eeded by the customers. The etwork is cotrolled by a etwork operator (NO), which performs a AC power flow optimizatio based o the eergy spot prices ad the costraits of the eergy carrier etwork. I the implemetatio preseted i this semester thesis, the role of the etwork operator combies the tasks of the distributio system operator (DSO) ad of the eergy service provider (ESP). Thus, the etwork operator is resposible for the power flow ad the usage of coverters ad eergy carriers at the same time. The electrical load at each hub is composed of a electrical base load (regular customers i that regio) ad of the load imposed by the PHEVs. This meas that every hub has to provide a certai load for the regular customers i its regio i additio to the load imposed by the PHEVs coected to that eergy hub. Depedig o the hub, the base load correspods to a regio pricipally cosistig either of private households, offices or small ad medium-sized compaies. The heat load per hub is defied i the same way for every hub by usig a 5

20 2 Itegrated system descriptio for PHEV maagemet i multi eergy carrier systems predefied heat load vector. These heat ad electrical load curves determie the maximal available power for the PHEVs. Whilst the ifluece of the electrical load is obvious, oe should keep i mid that the heat load iflueces the usage of the CHP which i tur has a impact o the total available electrical power o the demad side. The etwork is operated usig a optimizatio based o a AC power flow ad o oliear gas flow equatios respectig the costraits imposed by the capacity of the power lies ad gas pipes as well as voltage limits, limitatios of trasformers, CHPs ad furaces. The goal of this optimizatio is to reduce the total eergy costs i the system (costs for electrical eergy ad costs for gas) whe supplyig the loads. 6

21 3 Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio Simulatios are performed usig a package of MATLAB routies as well as load curves ad spot prices provided by ENTSO-E ad EEX respectively. A two-fold optimizatio based o a local system of four eergy hubs, data about local loads ad PHEVs plugged to the specific hubs is performed. I the first step, the local load composed of the base load ad the additioal PHEV load is determied. For this purpose, a local optimizatio is performed by a so-called PHEV maager (for the cocept of the PHEV maager ad further cosideratios, see [15] ad [16]) resposible for the local area. The PHEV maager (see figure 2.1) is the cotrol structure resposible for allocatig the power to the differet PHEVs i a optimal way. Durig the preset project, the global optimum is determied usig the MATLAB algorithm pattersearch because the former algorithm SNOPT gave suboptimal results as explaied i the paragraph The choice of the optimizatio algorithm i chapter 4. I the secod step, the AC power flow ad gas flow problem metioed i chapter 2 resultig from the hub loads is solved usig the optimizatio eviromet TOMLAB with its algorithm SNOPT. I this thesis, the focus lies o the local optimizatio algorithm performed by the PHEV maagers. 3.1 The PHEV maager ad the cotrol structure A etity called the PHEV aggregator ca be evisioed to be resposible for the PHEV maagers i a certai etwork area which is cotrolled by a etwork operator. Each PHEV maager moitors the PHEVs plugged i its respective area ad operates i time slots of fiftee miutes. Thus, a PHEV is oly take ito accout at the begiig of the ext time slot after its arrival at the PHEV maager. The PHEV maager the performs a optimizatio i order to fulfill the eeds of the eergy provider ad the PHEV customers while respectig the techical limits imposed by the etwork operator. This optimizatio is based o spot prices, etwork costraits, idividual PHEV data ad the optimizatio parameters α ad β. These parameters are provided by the PHEV aggregator to all the PHEV maagers which it cotrols. The PHEV aggregator determies suitable α ad β to achieve a PHEV chargig 7

22 3 Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio behavior resultig i valley fillig ad i peak shavig. This was discussed to be potetially advatageous for the trasmissio system operator (TSO) i order to icrease the system utilizatio factor. However, the PHEV chargig behavior eeds to icorporate the local limitatios of the particular distributio system to which the fleet is coected. These are determied by the etwork operator. As already stated i chapter 2, the etwork operator uites the roles of the eergy service provider (ESP) ad the distributio system operator (DSO). The hierarchical structure discussed above is show o figure 3.1. ESP DSO Network operator hub limits PHEV load PHEV aggregator α ad β PHEV load PHEV maager 1 PHEV maager 2 PHEV maager 3 PHEV maager 4 Figure 3.1: The hierarchical structure of the cotrol system 8

23 3.2 Descriptio of the PHEV maagig algorithm for valley fillig The list cotaiig the iformatio about the PHEVs of a specific hub The PHEV maager keeps the status of the differet PHEVs i a list which cotais the followig iformatio cocerig the PHEV k at ode : The state of charge (SOC k ). This value is equivalet to the battery level ad is expressed i percet of the total battery capacity. It ca vary betwee a miimal state of charge SOC k = SOCk mi that is allowed for PHEV k at hub ad SOC k =1(correspodig to a fully charged battery). The lower boud is imposed by the fact that deep dischargig reduces the lifetime of a battery. The state of discharge (SOD k ), which is ot absolutely essetial ad is oly used for coveiece sice SOD k =1 SOC k. The time of arrival i secods The aticipated time of departure Dtime k i secods The idividual PHEV idetificatio umber TheSOD k at arrival The coectio capacity C p k i W (this is the maximal power that ca be used to charge or discharge a PHEV) Thepersoal eergy valuatio parameter θ k which idicates how urgetly a PHEV eeds to charge its batteries (see equatio (3.1) ad figure 3.2). The desired SOC at departure SOC k =SOC k +10% (cosumed SOC durig the last travel)+security margi The miimum time to recharge util the desired SOC at departure SOC k is reached (see expressio (3.2)) The battery capacity C B k i Wh 3.2 Descriptio of the PHEV maagig algorithm for valley fillig The goal of the local optimizatio is to distribute the available power i a way to maximize the total utility of the differet PHEVs. The utility u k (i e) ofphevk at ode is expressed through a utility fuctio (for more iformatio, see [15] ad [16]), which is based o the followig parameters ad variables: 9

24 3 Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio The state of charge (SOC k ) The miimal state of charge SOC mi k that is allowed. The curret price π of electricity The battery capacity C B k α ad β, which are two parameters determiig the value of the eergy that is stored i the battery of the car. Thevariableθ k which is referred to as persoal eergy valuatio ad which differs from car to car ad from time slot to time slot as well. It weights the value of the eergy stored i the battery by attributig a persoal preferece factor. The value of this factor depeds o how urgetly the eergy is eeded. At 8.00 a.m., a PHEV leavig at 8.00 p.m. will have a much lower θ k as a car leavig at a.m.. The variable θ k is computed (with ϖ(t )/4 correspodig to the chargeable eergy i oe time frame expressed i SOC, η c correspodig to the chargig efficiecy, T correspodig to the curret time slot ad Δt correspodig to the duratio of oe time slot) as follows θ k (T )=1+ρ T ( ) Dtime k SOC k SOC k (T ) Δt ϖ(t )/4 η c h. (3.1) The expressio SOC k SOC k (T ) Δt (3.2) ϖ(t )/4 η c correspods to the miimal chargig time eeded util the desired SOC SOC k ca be reached. For the preset thesis, the values of the parameters ρ ad h were set to ρ =2ad h=10. Figure 3.2 displays the chage of θ k over a time iterval of 10 time slots (correspodig to two ad a half hours) assumig that the PHEV leaves at the ed of that period. For the preset figure, it has bee assumed that o eergy is charged by the PHEV durig this period. Oe ca see that θ k icreases very strogly shortly before the departure expressig a urget eed for eergy. However, if the PHEV has largely sufficiet time left before its departure, θ k will be very close to 1 ad oly icrease slowly with time (see (b) ad (c) o figure 3.2). The trasferred amout of eergy, weighted by the battery size, expressed as the variable q k of PHEV k at hub. 10

25 3.2 Descriptio of the PHEV maagig algorithm for valley fillig 100 (a) θ k as a fuctio of time 80 θk (b) 1.15 (c) Period Figure 3.2: (a) θ k as fuctio of time, figures (b) ad (c) show zooms of (a). The state of charge SOC k (q k ) attaied after a certai amout of eergy q k, expressed i terms of the SOC, has bee charged durig oe time slot. It is defied as follows usig the state of charge SOC k already reached at the begiig of the time slot The beefit fuctio SOC k (q k )=SOC k + q k. (3.3) B k (SOC k (q k )) = α Ck B (SOC k (q k ) SOCk mi ) β Ck B (SOC k (q k ) SOC mi (3.4) k ) 2 already icorporates α, β, SOC k ad the miimal allowed state of charge SOCk mi. Figure 3.3 shows the beefit fuctio for α ad β chose accordig to the costraits set i chapter 4 (α=60 e/mwh, β=20 e/mwh, SOCk mi =0.2 ad Ck B =15kWh). The utility fuctio (oly valid for chargig behavior (q k 0)) for a sigle PHEV k at ode is defied as u k = θ k B k (SOC k (q k )) π C B k q k. (3.5) Thus, the utility is determied by two couteractig cotributios: the first oe is determied by the expressio θ k B k (SOC k (q k )) attributig a value to the eergy stored by the PHEVs. The secod oe is give through expressio π Ck B q k ad correspods to the price that has to be paid i order to icrease the respective SOC k. If the eergy q k is extracted from the eergy etwork, due to chargig losses oly η c q k (η c <1 deotes the chargig efficiecy) is stored i the battery. The 11

26 3 Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio 6 x Beefit B as a fuctio of the SOC Beefit [e] SOC Figure 3.3: The beefit as a fuctio of the SOC losses are due to coductio, coversio (power electroics) ad battery issues. This fact has to be take ito accout whe computig the beefit fuctio B. The ew beefit fuctio, modified ad takig ito accout the chargig losses, is give by B k = α Ck B (SOC k SOCk mi +η c q k ) β Ck B (SOC k SOCk mi +η c q k ) 2. (3.6) The objective of the PHEV maager at ode is to maximize the sum u of the utility fuctios of every PHEV plugged to that particular hub u = N PHEV k=1 u k (3.7) where N PHEV is the umber of coected PHEV at ode. Sice pattersearch is implemeted i order to miimize a fuctio, a cost fuctio is defied by simply multiplyig the utility fuctio by ( 1). This fuctio is the miimized by pattersearch, thus maximizig the utility fuctio. The optimizatio is subjected to supplemetary costraits cocerig the maximal trasferable eergy q max k ad N PHEV k=1 q max k q k C B k Δt ad the maximal hub power L el q k q max k (3.8) R(L el Lel,base ) (3.9) where Δt represets the duratio of a chargig iterval. The base load L el,base is imposed by all other customers tha the PHEVs demadig eergy from the 12

27 3.2 Descriptio of the PHEV maagig algorithm for valley fillig hub. The maximal hub power L el is determied by the trasformer capacity P,max Tras ad the power that ca be provided by the CHP (heat load L h,coversio efficiecy from gas to electricity η CHP ge η CHP gh ) L el = P,max Tras + Lh ηge CHP, coversio efficiecy gas to heat ηgh CHP. (3.10) 13

28 3 Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio 3.3 Descriptio of the PHEV maagig algorithm for peak shavig So far, oly the chargig behavior has bee icorporated ito the utility fuctio. I a secod phase, i order to achieve peak shavig, it is crucial to modify this fuctio slightly, takig ito accout the costs (eergy price, battery costs, losses) iduced by the peak shavig behavior of the PHEVs Icorporatig battery costs ito the utility fuctio for peak shavig I order to determie the battery costs, it is suitable to use the model preseted i [9] (p.9). The model as well as the followig parameters ad values are take from [10] (slide 18). First of all, the total cost πbatt total of a battery is calculated. With a price of 600$/kWh oe eds up with a price πbatt total =9000$ for a battery with a capacity of 15kWh. I additio, the fact, that every chargig or dischargig actio reduces the remaiig battery lifetime i a certai way, is take ito accout. Sice chargig ad dischargig is also eeded durig ormal operatio, oly the supplemetary costs due to V2G actio are cosidered. Based o this model, the umber N c of total battery cycles that ca be achieved whe dischargig to SOCk mi is give by N c = ( 1 SOC mi ) 1 a k b (3.11) with a ad b beig two empirical parameters set to a=0.65 ad b=151.5 for NiMH batteries. N c as a fuctio of the miimal state of charge SOCk mi is show o figure 3.4. As a battery cycle is equivalet to the dechargig of the N c as a fuctio of SOC mi k Nc SOC mi k Figure 3.4: The umber of cycles a battery ca perform as a fuctio of SOC mi k 14

29 3.3 Descriptio of the PHEV maagig algorithm for peak shavig battery from SOC k =1 to SOCk mi ad chargig up to SOC k =1 agai, the legth of a battery cycle is 2 (1 SOCk mi ). Sice oly the costs due to V2G actio are cosidered half the legth of a whole cycle is take ito accout. The battery cost will thus be equal to the price πbatt total of the battery multiplied with the share of a cycle with legth 1 SOCk mi that is performed due to V2G actio. I this way, the battery cost π batt ca be determied as follows π batt = 1 N c π total batt 1 1 SOCk mi The variable π batt C B k is displayed o figure The exteded utility fuctio for peak shavig 1 C B k. (3.12) I additio to the cosideratio of the battery costs, it is importat to otice that the eergy fed back by a PHEV geerally eeds to be recharged sooer or later. Assumig that the PHEV aggregator has sufficiet time to furish this amout of eergy (ad hece is able to wait for the ideal time slot whe the electrical eergy is cheap), the miimal price of electricity πday mi is subtracted from the mometary spot price i the utility fuctio for feedig. Hece, while disregardig losses durig the chargig or dischargig process, the ew utility fuctio for peak shavig looks as u k = θ k B k (SOC k (q k )) (π π mi day π batt ) C B k q k. (3.13) If the eergy q k is fed back to the eergy etwork, a larger quatity q k /η s (η s <1 deotes the dischargig efficiecy) is extracted from the battery. This fact has to be take ito accout whe computig the chage of the beefit fuctio B. The ew beefit fuctio, adapted i order to take ito accout the supplyig losses, is give by ( B k (SOC k (q k )) = α Ck B SOC k SOCk mi + q ) k η s ( β Ck B SOC k SOCk mi + q ) 2 (3.14) k. η s The miimal price πday mi πday mi = πday mi/(η s η c ). has to be adapted as follows: The divisio by η s is due to the fact that more eergy is take from the battery tha actually fed back to the grid. The divisio by η c is justified by the circumstace that the discharged eergy has to be charged back at some poit with a limited efficiecy as well. The battery costs have to be divided by η s, sice the amout of eergy take from the battery is larger tha the 15

30 3 Distributio of the scarce power resources betwee PHEVs by meas of local optimizatio oe fed back to the grid. Fially, this leads to the followig utility fuctio ( ) u k = θ k B k (SOC k (q k )) π πmi day π batt Ck B η s η c η q k. (3.15) s It should be emphasized that equatio (3.15) is oly used for PHEVs supplyig eergy to the grid (q k <0) whereas (3.5) applies for PHEVs chargig their batteries (q k 0) oly. Hece the complete utility fuctio is composed of two locally cotiuous ad differetiable fuctios defied o R 0 ad R <0 respectively ad looks as follows θ k B k (SOC k (q k )) π Ck B q k (q k [0, [) ( ) u k = θ k B k (SOC k (q k )) π πmi day π batt Ck B η s η c η q k (q k ], 0[). s (3.16) The choice of the optimizatio algorithm Sice u k is ot differetiable aroud q k = 0, SNOPT caot be used to fid the optimum of u k. This is the reaso why pattersearch is used istead. Ufortuately, pattersearch, beig a global optimizatio tool, eeds more CPU time tha the local optimizatio tool SNOPT ad ca oly deal with a lower amout of optimizatio variables. Although it is suitable for the aalyses performed i the preset thesis, it would ot be able to deal with the requiremets i a real system. A solutio could be to subdivide the day i two differet phases, depedig o whether the PHEVs are expected to charge or to feed eergy back to the grid. The the costraits for the optimizatio ca be set i a way to allow oly the expected actio (chargig or supplyig). Thus oly oe part of the utility fuctio is used per time slot. I this case, SNOPT works properly. The differece betwee this strategy ad the more geeral oe ivolvig pattersearch is ivestigated i chapter 6. 16

31 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator Whereas differet ways of fixig fairly high α for a whole day have prove useful, the success of this approach strogly depeds o the actual spot prices by EEX. I this chapter, a procedure describig how α ad β ca be chaged from oe time slot to the ext i order to give better results ad to reduce the depedecy of the outcome o the shape of the price curve is developed. I order to set α ad β i a appropriate way, it is useful to ivestigate the sigificace ad effects of α ad β first. Allowed values for α ad β I order to avoid a icosequetial chargig/dischargig behavior of the PHEVs, the beefit fuctio B requires a more detailed aalysis. It attributes a value to the amout of eergy that is stored i the battery of a specific PHEV. This value should always be positive (1st coditio). Hece, the boudary case B=0 is cosidered: B k (SOC k (q k )) = 0 α (SOC k (q k ) SOCk mi ) β (SOC k (q k ) SOCk mi ) 2 =0 (SOC k (q k ) SOCk mi )=0 or (SOC k (q k ) SOCk mi )= α β The zeros of B are foud at SOC k (q k )=SOCk mi ad SOC k (q k ) SOCk mi > 1 if ad oly if α>β. Computig the secod derivative of the beefit fuctio, oe gets (4.1) d 2 B k dsoc 2 k = 2 θ k C B k β. (4.2) As the secod derivative of the beefit fuctio is egative for β>0ad sice B is a polyomial of the secod degree i (SOC k (q k ) SOCk mi ),itappears that B has the shape of a iversed parabola. If α>βis fulfilled as well, B takes positive values betwee (SOC k (q k ) SOCk mi )=0 ad (SOC k (q k ) SOCk mi )=1, thus givig rise to a strictly positive value of the stored eergy 17

32 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator for ay allowed value of SOC mi k. Aother importat feature is to let B reach its maximum beyod SOC=1, otherwise further chargig would be puished beyod a certai SOC (2d coditio). I order to fid the maximum of B, its first derivative with respect to the SOC is computed ad set equal to zero. The obtaied equatio is the solved with respect to the SOC db k dsoc k (q k ) = θ k Ck B (α 2 β (SOC k (q k ) SOCk mi )) = 0 (SOC k (q k ) SOCk mi )= α (4.3) 2 β. If α 2 β ad β>0, the maximum of B lies beyod SOC k (q k ) SOC mi k =1 ad α>βis fulfilled as well. Thus the two previously metioed coditios are met. The ifluece of α ad β o the chargig behavior It is essetial to state that for the preset thesis, the parameter β is chose i a way to avoid chargig if the batteries are already full, that is if SOC k = 1. The critical case appears for the miimal price of the day πday mi. I this case, the icetive for the PHEVs to charge is the largest. For the followig computatios π = πday mi is thus assumed, further θ k is set to oe k, (sice, if SOC k =1, there will be o urget eed for eergy) ad the chargig ad dischargig losses are disregarded. The first derivative of the utility fuctio is the computed ad set equal to zero i order to fid its maximum du k dq k = α Ck B 2 β Ck B (SOC k + q k SOCk mi ) Ck B πday mi =0 β = α πday mi 2 (SOC k + q k SOCk mi ). (4.4) For the critical case, SOC mi k =0, SOC k =1, the goal is achieve q k =0 (o chargig). Hece, oe obtais β = α πmi day. (4.5) 2 If β is determied usig equatio (4.5), the two coditios metioed i the previous paragraph are met, sice α 2 β ad β>0. I order to clarify the 18

33 importace of α, the first derivative of the utility fuctio with respect to q k is cosidered agai for the geeral case of a chargig PHEV du k dq k = θ k α C B k 2 θ k β C B k (SOC k + q k SOC mi k ) C B k π. (4.6) Settig this expressio equal to zero (i order to fid its maximum) ad solvig the resultig equatio with respect to q k,oeobtais q k = α 2 β (SOC k SOCk mi ) π θ k. (4.7) 2 β I (4.7), substitutig α πmi day for β leads to 2 α (α πmi day q k = ) (SOC k SOCk mi ) π θ k. (4.8) α πday mi Now the derivative of (4.8) with respect to α is computed, leadig to dq k dα = ( π θ k πday mi). (4.9) (α πday mi )2 Oe ca see that icreasig α, adaptig β accordigly ad avoidig egative values for α ad β will result i a icrease of q k if π is larger tha πmi θ day. Thus, for the usual case where θ k is close to oe, if α is chose relatively high i compariso with the spot price π, apositivevalueforq k is foud. For θ k >> 1, q k decreases with risig α. The miimal value for q k that ca be reached i that case is q k =1 (SOC k SOCk mi ) for α,sice dq k dα is egative. Now, this value ca be expected to be higher tha the maximal chargeable SOC k (for reasoable values for the plug power ad SOCk mi ) durig oe time slot (see equatio (4.20)). This meas that the quality of the cotrol gets reduced if θ k is too large, sice the PHEVs will i that case charge with the maximum available power idepedetly of the value of α. This problem is tackled i two ways: first, the values for θ k are kept low by spreadig the eergy i such a way that last miute chargig is preveted; as a last resort, the hub limits are set i order to prevet the total load from exceedig the peak load. A way to demostrate the ifluece of the parameter θ is to cosider the margial beefit (MB) ad the margial cost (MC) resultig from a certai chargig behavior (as i [15] ad [16]). The margial beefit ca be derived by takig the first derivative of the beefit fuctio B with respect to SOC k (q k ) MB = db k dsoc k (q k ) = CB k θ k (α 2 β (SOC k (q k ) SOC mi k )). (4.10) 19

34 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator The margial cost ca be determied by takig the first derivative of the expressio C B k π (q k ) with respect to SOC k (q k ), givig MC=π C B k.sice both the margial beefit ad the margial cost iclude the factor C B k,itca be eglected for the followig cosideratio. Figure 4.1 displays the margial cost (black) ad the margial beefit (blue, for θ =1;red,forθ>1) asa fuctio of the SOC. As log as the margial beefit exceeds the margial cost, the PHEV will have a icetive to charge. As oe ca easily see, this icetive ca be icreased by raisig θ. α θ Margial beefit α Slope=-2 β θ Slope=-2 β Margial cost π 0 SOC-SOC mi k Figure 4.1: The margial cost ad the margial beefit as a fuctio of the SOC 20

35 4.1 Tuig α ad β 4.1 Tuig α ad β Determiig a useful chargig behavior by takig ito accout the load curve ad the umber of PHEVs I a first step the desired q k (T ), correspodig to the desired average amout of eergy charged/supplied per PHEV expressed i terms of the SOC, is determied. This q k (T ) ca of course be defied i several differet ways, depedig o the desired behavior of the PHEVs. I practice, the desired behavior depeds o the prices, limitatios ad cotrol issues of the etwork operator. The approach applied here takes ito accout the umber of plugged PHEVs at a specific time istat as well as the desired load potetially formed by the PHEVs. For a start, a particular load level amed here reachable level L reach, is computed. The reachable level is chose i a way that the area eclosed by the load curve ad the lie correspodig to the reachable level is equal to the total eergy demad E PHEV of the PHEVs forecasted for this day. This load level is idicated i figure 4.2. As metioed i chapter 6, the day ca be subdivided i several time itervals cosistig of a certai umber of cosecutive time slots. The begiig ad the ed of such a time iterval are determied by the patter of arrival ad departure of the PHEVs. Thus a certai fleet of PHEVs is expected to charge its batteries durig oe specific time iterval; the spreadig of the eeded chargig power over the differet time slots withi a time iterval ca be altered to achieve load levelig. Before each time iterval a reachable load level is the determied. This is doe i a iterative way: the miimal load is take as the startig value for the reachable level, the level is the icreased i small steps (1MW i this thesis). The iteratio eds whe the eclosed area is larger tha E PHEV η c 105%. The factor η c accouts for the limited chargig efficiecy of the PHEVs: the amout of eergy extracted from the grid has to be larger tha the eergy eeded by the PHEVs because of chargig losses. As a safety margi, the reachable level is chose for 105 percet of the expected eergy E PHEV. I order to avoid too large values for θ k resultig i a reduced cotrollability of the chargig behavior, the chargig period will be exteded to some additioal time slots. Therefore, 5 percet of the eeded eergy is spread over the remaiig time slots lyig above the reachable level. A secod reachable level L reach2 is therefore determied aalogously to the first oe: it is chose i a way that the area eclosed by the load curve ad the lie correspodig to the secod reachable level is equal to E PHEV η c 5%. For this computatio, oly the time slots lyig above the first reachable level are cosidered. L reach2 is also computed by iteratio, startig with the first reachable level L reach.the situatio is illustrated i figure 4.2. The computatio of the reachable levels 21

36 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator is doe before the respective time iterval. Load 5%E PHEV Lreach2 Lreach D 2 (T) D(T) 105%E PHEV T Time Figure 4.2: The reachable levels ad the areas correspodig to the charged power (red ad gree) together with the load curve The followig computatios are performed shortly before each time slot. For the sake of simplicity, the day is ot subdivided i differet time itervals for this explaatio but cosidered as a whole. Based o the load L(T) durig time slot T, three cases have to be distiguished: 1. L(T) L reach First the deviatio D(T) of the load L(T) at istat T from L reach durig a certai cosidered period T is defied as illustrated i figure 4.2 D(T )=L reach L(T ). (4.11) The it is ormalized to the average deviatio (D), computed for the time slots fulfillig L(T) L reach, givig rise to a factor f 1 (T ) f 1 (T )= D(T ) D. (4.12) Based o iformatios kow from experiece (the average umber of PHEVs N PHEV ad the total eergy demad throughout a day i terms of SOC E tot = E PHEV which i fact fluctuate from day to day but are C B assumed to be close to their all time average), ad the umber of time slots #ts fulfillig L(T) L reach, the average amout of q k (average over time ad PHEVs) eeded per PHEV is determied 22

37 4.1 Tuig α ad β 1.05 E tot q k =. (4.13) η c #ts N PHEV The factor 1.05 is due to the fact that 105 percet of the eergy are take ito accout for this case. The the average of q k for the curret time slot is determied by weightig q k with the factor f 1 (T ) ad the ratio give by the curret umber of PHEVs ad the average umber of PHEVs q k (T )= q k f 1 (T ) N PHEV N PHEV. (4.14) 2. L reach <L(T) L reach2 First the deviatio D 2 (T ) of the load L(T) at istat T from L reach2 durig a certai cosidered period T is defied as illustrated i figure 4.2 D 2 (T )=L reach2 L(T ). (4.15) The it is ormalized to the average deviatio (D 2 ), computed for the time slots fulfillig L reach <L(T) L reach2, givig rise to a factor f 2 (T ) f 2 (T )= D 2(T ) D 2. (4.16) Agai, based o iformatios kow from experiece (the average umber of PHEVs N PHEV ad the total eergy demad throughout a day i terms of SOC E tot = E PHEV which i fact fluctuate from day to day but C B are assumed to be close to their all time average), ad the umber of time slots #ts 2 fulfillig L reach <L(T) L reach2, the average amout of q k (average over time ad PHEVs) eeded per PHEV is determied q k = 0.05 E tot. (4.17) η c #ts 2 N PHEV The factor 0.05 is due to the fact that oly 5 percet of the eergy is take ito accout for this case. The the average of q k for the curret time slot is determied by weightig q k with the factor f 2 (T ) ad the ratio give by the curret umber of PHEVs ad the average umber of PHEVs q k (T )= q k f 2 (T ) N PHEV N PHEV. (4.18) 23

38 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator 3. L(T)> L reach2 For these very high load phases peak shavig is desired, if possible. Therefore, q k (T ) is chose i a way that the resultig load will ot be lower tha L reach2, avoidig thus that more peak shavig tha ecessary is performed. Hece q k (T ) is give by the differece betwee the actual load level ad the secod reachable level L reach2 expressed i terms of the SOC (divisio by Ck B ) ad ormalized to the average umber of PHEVs N PHEV tot plugged i the whole system. q k (T )= L reach2 L(T ) 4 C B k N PHEV tot (4.19) The factor 4 arises due to the fact that the duratio of oe time slot is oly fiftee miutes. For all these cases, the value of q k is bouded by the plug power of the PHEVs. If oe assumes a maximum plug power C p k (both for egative ad positive power flow directios) of 3.5kW ad a battery capacity Ck B of 15kWh oe eds up with Cp k 0.25 q Ck B k Cp k q Ck B k (4.20) The factor 0.25 accouts for the fact that everytimeslotlastsforaquarterof a hour; the trasferred eergy C p k 0.25 is ormalized to the battery capacity Ck B. Hece, if q k (T ) exceeds a boud for a time slot, it is replaced by the value of the specific boud. Takig ito accout the hub limits So far, the hub limits have ot bee cosidered. However, it is quite straightforward to do so. Durig the computatio of the reachable levels L reach ad L reach2, the hub limits simply have to be take ito accout: if, durig a time slot, the deviatio D(T) exceeds the sum of the hub limits (scaled to fit the umber of PHEVs used for the simulatio, sice D(T) accouts for the total deviatio of the load curve from L reach ), the cosidered area is ot delimited by the lie correspodig to L reach aymore, but by the hub limits. This is illustrated o figure 4.3. The same procedure is followed whe determiig L reach Tuig the parameters α ad β i order to achieve the desired chargig behavior After havig obtaied a reasoable value for q k (T ), the ext issue is to tue α ad β i a way to achieve a behavior close to the desired oe icludig 24

39 4.1 Tuig α ad β Load Costraits imposed by hub limits Lreach2 Lreach Time Figure 4.3: The reachable level ad the hub limits chargig i low load itervals ad feedig back i peak load itervals. Sice the PHEV maagers distribute the available power i order to maximize the total utility, they will also maximize the idividual utilities, as log as the hub limits are ot reached. Geerally, extrema of cotiuous ad derivable fuctios are determied by settig their derivative to zero ad checkig the secod derivative i order to figure out whether the extremum is a miimum or a maximum. Optimum for the utility fuctio for chargig (q k 0) The extremum value for q k ca be foud by computig the derivative of the utility fuctio (icludig efficiecies) as du k dq k = θ k α η c θ k (α π mi day ) η c (SOC k SOC mi k +q k η c ) π. (4.21) Settig it equal to zero ad solvig with respect to q k q k = α η c π η 2 c θ k α πday mi Cosiderig the secod derivative gives SOC k SOCk mi. (4.22) η c d 2 u k dq 2 k = θ k (α π mi day ) η2 c (4.23) it is obvious that the extremum give i (4.22) is the maximum of the fuctio sice the expressio (4.23) is egative, as log as β>0 α>π mi day. 25

40 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator Optimum for the utility fuctio for dischargig (q k <0) The same procedure ca be applied i order to fid the optimum for the dischargig fuctio. Cosecutively the derivative of this utility fuctio gives du k dq k =θ k α CB k η s θ k ( ) α π mi ( day Ck B η SOC k SOCk mi + q ) k s η s CB k π Ck B + πmi day η s η c + π batt C B k η s (4.24) ad settig it to zero results i q k = α η s (π η2 s πmi α πday mi Agai, the secod derivative day ηs ηc π batt η s) θ k η s (SOC k SOC mi k ). (4.25) d 2 u k dq 2 k = θ k α πmi day ηs 2 (4.26) is egative ad the extremum give i (4.25) is the maximum of the utility fuctio. Determiig α i a iterative way With the foregoig explaatios i mid, oe ca determie α i order to obtai the desired value for q k (T ). This computatio is performed before each time step T. Reasoable bouds for α were chose to be α mi = πday mi + ɛ (ɛ <<1 was chose to prevet divisio by zero, see equatios (4.22) ad (4.25)) ad ( πday mi qk max α max = π(t ) η 2 c 1 η c qk max ) + 1 SOCmi k η c (4.27) + SOCmi k η c with qk max = correspodig to the maximal chargeable SOC per PHEV ad time slot. The lower boud α mi is imposed by the fact that β shall be positive throughout the whole day. The upper boud α max is chose to allow maximal chargig for each PHEV. Equatio (4.27) is foud by takig equatio (4.22) ad settig θ k =1, SOC k =1ad qk max = represetig the boudary case. The resultig equatio is the solved with respect to α max. The followig procedure allows a fast determiatio of a suitable value for α. The iteratio is performed followig the steps below: 26

41 4.1 Tuig α ad β 1. The lower boud α mi ad the higher boud α max are set. 2. The the average of α mi ad α max is computed ad chose to be the ew α: α = α mi+α max. The iteratio stops if the differece betwee the ew 2 α ad the oe from the iteratio before is smaller tha a certai value δ α, defiig the criterio for covergece of the algorithm. For the preset thesis δ α =0.05 was chose. The last value for α is retaied. 3. q k (T ) is the calculated for α, α mi ad α max. Sice the SOCs of the differet PHEVs are kow q k ca easily be calculated takig ito accout the price curve ad usig equatios (4.22) respectively (4.25). The q k (T ) is computed by summig up all the values q k foud usig equatios (4.22) respectively (4.25), which is computatioally iexpesive. 4. Two cases are differetiated: q k (T )(α mi ) 0 ad q k (T )(α) 0 If this is the case, α mi is chose to be the ew lower boud ad α is chose as the ew upper boud. q k (T )(α) < 0 ad q k (T )(α max ) 0 If this is the case α is chose to be the ew lower boud ad α max is chose as the ew upper boud. 5. The iteratio is restarted from the first step agai. At the ed of the iteratio, the best value for α, meaig the oe that gives rise to the q k (T ) that is closest to the desired oe, is retaied. The β is determied usig equatio (4.5). 27

42 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator How to choose the correct chage q k Sice the utility fuctio cosists of two piecewise cotiuous parts, depedig o the sig of q k as recalled i (4.28), it is importat to make sure that the correct value (the oe that really optimizes the utility fuctio) is chose. θ k B k (SOC k (q k )) π Ck B q k (q k [0, [) ( ) u k = θ k B k (SOC k (q k )) π πmi day π batt Ck B η s η c η q k (q k ], 0[) s (4.28) I order to achieve this, 4 differet cases have to be cosidered. I the followig eumeratio, qk c ad qk s deote the resultig chargig ad supplyig powers obtaied from (4.22) ad from (4.25), respectively. For reasos of clarity, the utility fuctios displayed i the followig figures are deoted i the followig way u1 =θ k B k (SOC k (q k )) π C B k q k (4.29) ad ( ) u2 =θ k B k (SOC k (q k )) π πmi day π batt Ck B η s η c η q k. (4.30) s 28

43 4.1 Tuig α ad β qk c 0 ad qk s 0 I this case, the utility fuctios look as depicted o figure 4.4. Sice u2 is ot defied for positive values of q k it is obvious that the correct solutio is qk c. Utility u1 u2 (0,0) qk Figure 4.4: The utility as a fuctio of q k for q c k 0 ad q s k 0 q c k < 0 ad q s k < 0 I this case, the utility fuctios look as depicted o figure 4.5. Sice u1 is ot defied for egative values of q k the correct solutio is q s k. Utility u2 u1 (0,0) qk Figure 4.5: The utility as a fuctio of q k for q c k < 0 ad q s k < 0 29

44 4 Aalysis of the parameters α ad β of the utility fuctio used by the PHEV aggregator q c k 0 ad q s k < 0 I this case, the utility fuctios look as depicted o figure 4.6. Sice u1 ad u2 are both defied i the eighborhood of their optima, the utility fuctios have to be computed ad the solutio with the highest utility (u1 for figure 4.6) is retaied. Utility u2 u1 (0,0) qk Figure 4.6: The utility as a fuctio of q k for q c k 0 ad q s k < 0 q c k < 0 ad q s k > 0 I this case, the utility fuctios look as depicted o figure 4.7. Sice either u1 or u2 are defied i the eighborhood of their optima, the best solutio is achieved for q k =0. Utility u1 u2 (0,0) qk Figure 4.7: The utility as a fuctio of q k for q c k < 0 ad q s k > 0 30

45 5 Cash flow ad fiacial icetives 5.1 The ifluece of the battery costs o the dischargig behavior I this sectio, the battery costs will be estimated ad ivestigated. Usig equatio (3.12) agai, ad settig SOCk mi =0.2, gives π batt Ck B =3.53 $ or π batt =2.78 Ck B =3.53 e (1$ =0.79 e). If all the eergy stored i a battery of 15kWh is sold at a moderate spot price of 0.05 e/kwh, the amout of moey obtaied is equal to π sold =0.75 e. Figure 5.1 shows the gap betwee battery costs ad spot prices. Without a exteral icetive, peak shavig will oly be attractive for PHEVs which have their SOCk mi very close to oe. I such a case the battery capacity will oly be poorly used ad the rage will be extremely limited. This clearly shows that, due to the high battery costs makig peak shavig uprofitable, a further icetive has to be provided to achieve V2G behavior if battery costs are cosidered i the optimizatio. This icetive could be a remueratio paid to the PHEV aggregator by the trasmissio system operator compesatig for the battery wear costs. Whe battery costs are disregarded, however, peak shavig is possible assumig that the spot price is above a certai level, ad that α ad β are chose accordigly. Sice α ad β ca oly be chose to be withi certai bouds (as discussed ( i chapter 4), the required price must at least exceed π limit = πday mi 1 η s + 1 η s η c ). This ca be show by settig α = πday mi leadig to β =0. I this case, expressio (4.24) is a costat. Thus, peak shavig is oly achieved if this costat is egative leadig to the foregoig equatio for π limit. Evolutio of the battery costs Predictig the evolutio of the battery costs is a very difficult issue because they deped o both techological ad ecoomical factors. A educated guess ca be made based o the diagram preseted i [10] (slide 10). It predicts a icrease of the NiMH batteries sector of 100 percet withi 5 years. Likewise, for the sake of simplicity, a icrease of 100 percet i the umber of produced NiHM batteries is assumed durig the same time period. The theory of experiece curve effects (see [12]) predicts a uit price drop of percet if the 31

46 5 Cash flow ad fiacial icetives Price of the battery ad spot price as a fuctio of the SOC value of the sold eergy battery price Price [e] Miimal state of charge allowed Figure 5.1: Depedecy of the battery price (red curve) o the SOC ad compariso with the value of the sold eergy (blue curve) umber of produced uits is doubled. Figure 5.2 shows the resultig evolutio of the battery price if a 100 percet icrease of battery sales every 5 years, a costat eergy price ad SOCk mi =0.2 are assumed. Oe ca see that for a reductio of 30 percet every five years, the break eve poit will be reached before 2030, whereas it will ot be reached util 2040 for a reductio of oly 20 percet every five years. Battery costs/eergy price [e] Estimated evolutio of the battery price durig the ext 30 years value of the sold eergy battery price for a reductio of 20% every 5 years battery price for a reductio of 30% every 5 years Year Figure 5.2: Estimatio of the evolutio of the battery price for a price reductio of 20% (red curve) ad 30% (gree curve) every 5 years. The price curves are compared with the value of the sold eergy (blue curve). 32

47 5.2 Itegratio of the ivestigated actors i curret practice power system structures Reducig the battery costs Apart from the assumptio that techological advaces ad the market itself ca lower the battery costs i the future, there are several other possibilities. A potetial solutio could be the use of smaller, cheaper batteries, which could be made possible through a battery sharig system. If a large etwork of rechargig spots would be available, the batteries would ot eed to be discharged as deeply. I this way, the umber of cycles could be icreased, ad thus the battery costs could be reduced. As proposed i [11], parterships betwee maufacturers would icrease the umber of uits, thus lowerig the battery costs as displayed o figure Describig the itegratio of the ivestigated actors i curret practice power system structures I this chapter, the prices paid by the differet actors participatig i the simulated processes shall be described. Figure 5.3 (take from [17]) shows the fiacial, eergy ad iformatio flows betwee the differet actors i the system. The part of the diagram bordered i red -icludig the PHEVs, the PHEV aggregator, the DSO, the TSO ad the ISO/RTO- is i the focus of the preset thesis. Figure 5.3: The iteractios betwee the differet actors i the system PHEVs Because the PHEVs are regarded as private customers, their eergy will 33