eeh power systemslaboratory

Julian Diederichs

Optimized Time-of-Use Tariffs for Smart

Charging of Plug-In Electric Vehicles

Semester Thesis

Department:EEH – Power Systems Laboratory, ETH Zurich

Examiner:Prof. Dr. Goran Andersson, ETH Zurich

Supervisors:Dr. Luis Baringo, ETH Zurich

Dipl. El. Ing. Marina Gonzalez Vaya, ETH Zurich

Zurich, July 2014

ABSTRACT 1

Abstract

An anticipated high proliferation of plug-in hybrid electric vehicles (PEV)challenges current power system operation. However, charging control-schemes or “smart charging” can utilize the flexible PEV load to balancegrid operation and to reduce economical costs of charging. The focus is onprice-based decentralized control schemes in which an aggregator is respon-sible to set price incentives to induce a desired charging behavior. In thisreport the aggregator sets “Time-of-Use” (ToU) tariffs as an input parame-ter for PEV charging profile optimizations.

In order to obtain optimized ToU tariffs, a bi-level optimization problemis solved, which on one side considers grid operation objectives and on otherside cost minimization for PEV charging. Driving patterns and driver end-use constraints are taken into account to determine optimized ToU tariffs.Individual PEV cost optimizations are aggregated by a novel linear approachto a single optimization on the grid level. The problem definition can beformulated as a single mathematical program with equilibrium constraints(MPEC), which is implemented in form of a mixed integer problem.

Optimized ToU tariffs are obtained by manipulating given hourly spotauction prices within symmetric bounds to obtain desired valley filling PEVcharging profiles. Results of bi-level optimizations show that by using opti-mized ToU tariffs valley filling can be increased by 40%. However, due tomultiple optimal solutions of the cost optimization, the responses of singlelevel aggregated and individual cost optimization to the optimized ToU tar-iffs do not necessarily correspond to the desired charging profiles and thusyield no significant improvements in valley filling. Therefore decentralizedToU based control schemes cannot incentivize desired power charging sce-narios.

Index Terms- Smart Charging, PEV control schemes, Time-of-use tariffs,Bi-level optimization, MPEC

CONTENTS 2

Contents

1 Introduction 4

2 Methodology 62.1 Driving patterns . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Virtual storage model . . . . . . . . . . . . . . . . . . . . . . 82.3 Aggregation of individual vehicles . . . . . . . . . . . . . . . . 102.4 Linear approximation of arrival and departure energy . . . . 10

2.4.1 Departure energy . . . . . . . . . . . . . . . . . . . . . 112.4.2 Arrival energy . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Formulation of bi-level problem . . . . . . . . . . . . . . . . . 12

2.6.1 Upper level . . . . . . . . . . . . . . . . . . . . . . . . 122.6.2 Lower level . . . . . . . . . . . . . . . . . . . . . . . . 132.6.3 Transformation to single level MPEC . . . . . . . . . 15

3 Results 203.1 Case study set up . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Aggregation of individual vehicles . . . . . . . . . . . . . . . . 20

3.2.1 Linear approximation of departure energy . . . . . . . 213.2.2 Linear approximation of arrival energy . . . . . . . . . 22

3.3 Comparison of aggregation methods . . . . . . . . . . . . . . 243.4 Optimal setting of ToU tariffs . . . . . . . . . . . . . . . . . . 27

3.4.1 Desired reference power profile . . . . . . . . . . . . . 273.4.2 Solution of MPEC . . . . . . . . . . . . . . . . . . . . 28

4 Discussion 344.1 Accuracy of aggregation approach . . . . . . . . . . . . . . . 344.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Acknowledgements 36

References 37

LIST OF FIGURES 3

List of Figures

1 Upper and lower charging power limits . . . . . . . . . . . . . 92 Upper and lower nodal energy limits for fixed initial energy . 103 Charging responses resulting from the individual and the ini-

tial aggregated optimizations . . . . . . . . . . . . . . . . . . 214 Hourly ToU tariffs . . . . . . . . . . . . . . . . . . . . . . . . 215 Comparison of linear approximations of departure energies

vs. exact solutions . . . . . . . . . . . . . . . . . . . . . . . . 226 Comparison of average approximation of departure energies

vs. exact solutions . . . . . . . . . . . . . . . . . . . . . . . . 227 Comparison of linear approximations of arrival energies vs.

exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Comparison of average approximations of arrival energies vs.

exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Charging responses resulting from the individual and the new

aggregated optimizations . . . . . . . . . . . . . . . . . . . . 2510 Normalized power volume curve of July 7th . . . . . . . . . . 2811 Power reference profile . . . . . . . . . . . . . . . . . . . . . . 2812 Charging response of lower level problem and of KKT formu-

lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2913 Bi-level charging responses to optimized ToU tariffs . . . . . 3114 Charging responses of lower level problem to bi-level opti-

mized ToU tariffs . . . . . . . . . . . . . . . . . . . . . . . . 3215 Charging responses of individual cost optimization to bi-level

optimized ToU tariffs . . . . . . . . . . . . . . . . . . . . . . 33

List of Tables

1 Notation table . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Battery specifications . . . . . . . . . . . . . . . . . . . . . . 203 Charging cost minimization for a PEV fleet of 10000 vehicles 264 Charging cost minimization for a PEV fleet of 5000 vehicles . 275 Charging cost minimization for a PEV fleet of 1000 vehicles . 276 Bilevel charging responses . . . . . . . . . . . . . . . . . . . . 307 Individual charging responses . . . . . . . . . . . . . . . . . . 31

1 INTRODUCTION 4

1 Introduction

In the context of a turnaround in energy policies towards sustainable energysystems, a high proliferation of plug-in electric vehicles (PEVs) is antici-pated. A significant electrification of the vehicle fleet poses new challengesto power system operation. Potential consequences of uncontrolled PEVcharging are overloading of power system assets and increased peaks in de-mand.

However, in case of charging control schemes or “smart charging”, PEVscan be employed to avoid line congestions in the grid and from an economi-cal perspective to reduce costs of charging [1]. A so- called aggregator couldbe responsible for managing charging and purchasing electricity on behalfof the PEV owners, as well as for interaction between vehicle owners andother energy system entities, such as energy service providers, transmissionand distribution system operators [2].

PEVs have the special characteristics of being only used intermittentlyand thus charging within a flexible timeframe is possible. In order to modeldriver end-use constraints, the aggregator can cluster the PEV fleet in theform of a virtual storage resource with power and energy characteristics thatdepend on vehicle behavior on the grid level [3]. Advanced ways of controlmodel vehicles as a storage, which can be scheduled to either consume poweror feed power to the grid “vehicle-to-grid (V2G)” [4]. As most renewable re-sources are fluctuating, integration and utilization of PEVs for demand sidemanagement and energy storage can become crucial for smart grid operation.

Approaches for the integration of electric vehicles into power systems andutilization according to the concept of a “smart grid” can be divided into di-rect and indirect control approaches. Direct control schemes refer to controlapproaches that do not actively involve the vehicle owner in the control ac-tion imposed on the PEV connected to the power system [2], while indirectcontrol schemes are fully decentralized control of PEVs. Direct control ap-proaches allow to fully control PEV charging and as a result the aggregatorcan be designed to fulfill objectives such as smart charging or V2G charg-ing. Drawbacks of direct control approaches are their high requirements forcommunication and their low consumer acceptance. In indirect control theauthority of control stays with the vehicle owner [5]. Most indirect or decen-tralized approaches are based on exogenous price signals [6],[7], which arethe input to individual charging costs minimization of each vehicle. Pricesare used by individual vehicles to schedule their charging activities accordingto their individual temporal and spatial constraints imposed by their drivingschedule. In a decentralized control approach for PEV fleets the exogenousinput parameter, incentivizing a particular behavior, is sent by an aggrega-

1 INTRODUCTION 5

tor. The role of the aggregator in this case would be to set incentives, e.gprice incentives, to induce a particular charging behavior.

A straightforward way of controlling charging behavior is setting priceincentives for consumers in the form of time-of-use (ToU) tariffs. In order tooptimally set up the ToU tariff, the response of the PEVs to certain tariffshas to be considered. Simple price elasticities do not accurately model theresponse, since they neglect the evolution of the individual batteries’ stateof charge. The charging cycles of a battery are not only price sensitive, butalso constrained by future driving requirements. In case a PEV owner wantsto cover two fully battery depleting trips within one day, the vehicle has tobe charged in between the two trips, independent from the current ToU tar-iff. In [8] both a centralized and decentralized price-based control approachare proposed. Both, centralized and decentralized control approaches yieldsimilar results as most charging is shifted to valley night hours with lowestToU tariffs, with few exception due to individual driving constraints. In caseof system-wide prices a high charging simultaneity at the lowest ToU priceis obtained, which are not in the interest of grid operators. The anticipatedincrease in peak demand due to uncontrolled PEV charging would requireexpansion of generation and transmission capacity as shown in studies forGermany [9] and the USA [10],[11].

Therefore an approch to determine an improved ToU tariff in a decen-tralized control approach is proposed, which includes the system cost orsystem stress minimization objective of grid operators and the charging costminimization objective of vehicle owners. The aggregator models the PEVfleet and represents the individual PEVs optimizations as a single aggregatedoptimization on the grid level. This formulation yields a bi-level problemstructure. The upper level problem represents the grid operators’ objectivesuch as valley hour PEV charging. The lower level represents the chargingcost minimization of the aggregator subject to power and energy constraintsof the PEV fleet. In the problem formulation the aggregator can manipulategiven prices within a small range in order to obtain a desired load profile.By reformulating the lower level problem by its mathematically equivalentKarush-Kuhn-Tucker (KKT) conditions, the bi-level problem can be trans-formed into a numerically solvable mathematical program with equilibriumconstraints (MPEC) [12]. The resulting optimization yields endogenous ToUprices, since it includes a cost minimization of consumers and considers ob-jectives of grid operators when determining optimized ToU tariffs.

2 METHODOLOGY 6

2 Methodology

The methodology applied to model driving patterns, battery specificationsand grid constraints is based on [8]. Only all-electric vehicles are considered.The mathematical problem is formulated as an MPEC. These type of modelsare described in detail in [13]. The notation used in the formulation of theproblem is provided in Table 1. The modelled time horizon is one day, withhourly time steps.

Table 1: Notation table

Indices

v vehiclen nodet number of time step, for hourly time intervals t = [1, 2, .., 24]

Vehicle representation

Etv energy content of the battery of vehicle v at time step tEtv,cons energy consumed by vehicle v at time step t

Etv,dep departure energy of vehicle v at time step t

Etv,arr arrival energy of vehicle v at time step t

P tv,charged charging power of vehicle v at time step t

Cv battery capacity of vehicle vSOCv battery state of charge of vehicle v at time step tηv charging efficiency of vehicle v

Nodal representation

Etn energy content of nodal virtual battery n at time step tEtn,dep energy drop of nodal virtual battery n at time step t

Etn,arr energy contribution to nodal virtual battery n at time step t

P tn,charged charging power of vehicles connected to node n at time step t

ToU tn,ref reference ToU tariff at node n at time step t

ToU tn optimized ToU tariff at node n at time step tηn average charging efficiency at node nχ bound of allowed ToU tariff variations in %

γt,tdepn,ndep(v)

only 1 if vehicles arriving at node n at time t departed from

node ndep at time tdep in their previous trip

γtarr,tnarr,ndep(v)only 1 if vehicles arriving at node narr at time tarr

departed from node n at time t in their previous trip

2 METHODOLOGY 7

dual variables

λtn,balance energy equilibrium of nodal virtual storage

λcons equality of consumed energy of vehicles and required energyλtn,arr arrival energy equilibrium based on linear approximation

λtn,dep departure energy equilibrium based on linear approximation

µtmin,Pnlower bound on P tn,charged

µtmax,Pnupper bound on P tn,charged

µtmin,Enlower bound on Etnd

µtmax,Enupper bound on Etn

µtmin,En,arrlower bound on Etn,arr

µtmax,En,arrupper bound on Etn,arr

µtmin,En,deplower bound on Etn,dep

µtmax,En,depupper bound on Etn,dep

binary variables

ωtmin,Pnlower bound on P tn,charged

ωtmax,Pnupper bound on P tn,charged

ωtmin,Enlower bound on Etnd

ωtmax,Enupper bound on Etn

ωtmin,En,arrlower bound on Etn,arr

ωtmax,En,arrupper bound on Etn,arr

ωtmin,En,deplower bound on Etn,dep

ωtmax,En,depupper bound on Etn,dep

2.1 Driving patterns

In order to derive charging schemes of electric vehicles, realistic driving pat-terns based on the transport simulation tool MATSim are obtained [14].MATSim is an agent-based transport simulation where each agent has a setof activities to be performed (e.g. work, shopping). The optimization takesinto account factors such as available methods of transportation and selectsthe driving patterns that maximize the agent‘s utility function. Drivingpatterns include timing, duration and distance of trips performed by eachvehicle. Information on parking locations is also required to map each ve-hicle to a network node. Energy consumption of each trip is approximatedby the product of covered distance and an assumed average PEV energyconsumption [8]. The selected driving patterns are not price dependent.The fact that according to the obtained driving patterns vehicles are parkedmost at the time, justifies the assumption that PEVs are considered a flex-ible network load [2].

2 METHODOLOGY 8

2.2 Virtual storage model

The grid network is simplified to the nodal level. Each parking location ismapped to a network node and is defined at each time step by its nodalenergy level according to a “Virtual Storage Model” [3]. From the nodalperspective arriving PEVs increase the nodal energy level, while departingPEVs lower the nodal energy level. The nodal virtual storage model al-lows to represent multiple individual PEVs in an aggregated single way andas a result allows to reduce the computational effort of the optimization.Equation (1) defines the nodal energy evolution:

Etn = Et−1n + P tn,charged ηn ∆t+ Etn,arr − Etn,dep ∀n,∀t (1)

The nodal energy depends on the energy in the prior time step, energycharged in the current time step, the energy contribution of arriving vehiclesand the energy drop due to departing vehicles.

It is assumed that the initial energy level E0v of each vehicle‘s battery

is fixed to a given value. In addition the proposed model requires the fi-nal energy level of each vehicle at the end of the modelled time horizonto be equal to the initial energy. This constraint guarantees that duringthe selected time horizon PEVs’ energy consumption has to be completelycompensated by battery charging. The resulting nodal representation fulfillsthe same conditions. The initial total nodal energy level is determined bysumming up all vehicle batteries connected to the grid in the first time step,which also yields a fixed initial nodal energy level.

The consumed energy can be derived from the given driving patterns bythe product of distance covered and an average PEV energy consumption.The lower and upper border case for charging power (P tn,charged) can becomputed based on maximal charging power and known energy consump-tions. The lower bound P t,minn,charged is defined by a charging behavior that

only charges in case the current (SOCtv) is not sufficient to cover the nextconsecutive trip. The upper bound P t,maxn,charged is defined by charging at eachparking location with maximal charging power aiming to fully charge eachvehicle at each time step. Both limits of P tn,charged in case of a PEV fleet of10000 vehicles in a grid network with one node are shown in Fig. 1.

A possible assumption for a scenario aiming at minimizing charging costsis that vehicles will only charge the energy required for their next trip andconsecutively arrive at their allowed minimal energy level. However, thisassumption fails in case of multiple trips of one PEV within one day, sincepre-charging for all future trips at the point of minimal costs would resultin lower overall charging costs.

2 METHODOLOGY 9

5 10 15 200

10

20

30

40

50

60

hours of day

MW

upper power boundlower power bound

Figure 1: Upper and lower charging power limits for 10000 vehicles

Assuming a fixed initial and final energy level of the modelled time hori-zon, upper and lower vehicle energy levels ((Et,maxv ) and (Et,minv )) can becomputed based on the following two recursive formulas:

Et,maxv = Et−1,maxv + ηv P

t,maxv,charged ∆t− Etv,cons ∀v,∀t (2)

Et,minv = Et−1,minv + ηv P

t,minv,charged ∆t− Etv,cons ∀v,∀t (3)

The intial energy at time step t = 0 is fixed to a given value and thusE0v = E0,max

v = E0,minv . The same assumption is made for the final nodal

energy E24v . The nodal representation Et,maxn can be computed by summing

at each time step t all maximal battery energy levels Et,maxv of all PEVsparked at node n. Et,minn is computed accordingly by adding minimal bat-tery energy levels. The resulting nodal energy limits are shown in Fig. 2.Depending on the parameters of PEV charging, charging power and charg-ing duration, the nodal energy will evolve between the known lower andupper bound.

The energy contents Etv,arr and Etv,dep of the batteries of arriving anddeparting vehicles depend on past charging behavior. Based on known driv-ing profiles Etv,arr and Etv,dep can be aggregated to the corresponding nodal

representation (Etn,arr and Etn,dep). Driving profiles include information ondeparture time and departure location and arrival time and arrival locationof each vehicle. By summing up the battery energy level of all vehicles whichdepart at one time step t from one parking location n, Etn,dep is determined.

Etn,arr is computed by summing up the battery energy levels of all arriving

vehicles at each time step at each parking location. By considering Et,maxv or

2 METHODOLOGY 10

0 5 10 15 20

60

80

100

120

140

160

180

hours of day

MW

upper energy boundlower energy bound

Figure 2: Upper and lower nodal energy limits for fixed initial energy for10000 vehicles

Et,minv for computing Etn,arr and Etn,dep upper and lower bounds for arrivaland departure energy can be determined.

2.3 Aggregation of individual vehicles

An aggregator is introduced in order to cluster the PEV fleet according tothe virtual storage resource with power and energy characteristics whichdepend on driver end-use constraints [3]. The applied virtual storage modelis described in subsection 2.2. By aggregating vehicle specifications anddriving patterns on a nodal level, multiple optimizations over individualvehicles and individual constraints can be represented by one aggregatedoptimization. This enables to simulate charging responses of large PEVfleets in reasonable times and with limited computational power.

2.4 Linear approximation of arrival and departure energy

In [8] decentralized PEV individual cost optimizations show lower costs thancentralized nodal optimizations. This can be explained by the fact that inthe initial aggregation method arrival and departure energy are approxi-mated as the average value of the upper and lower bounds of their corre-sponding energies according to equations (4) and (5).

Etn,dep =Et,maxn,dep + Et,minn,dep

2∀n,∀t (4)

Etn,arr =Et,maxn,arr + Et,minn,arr

2∀n, ∀t (5)

2 METHODOLOGY 11

Thereby past charging behavior is not considered in determining depar-ture and arrival energies. The upper and lower limits of arrival and departureenergy on the grid node level can be computed as described in subsection2.2. The results of the individual optimizations and the aggregated responsewith the average approximations of arrival and departure energies show highdeviations in power charged (P tn,charged) at each time step and thus also in

the nodal energy levels (Etn).In order to aggregate the charging profile more accurate, arrival and depar-ture energy are modelled as functions of nodal energy levels of prior timesteps instead of the average approximation. Thereby pre-charging and pastcharging behavior is taken into consideration. In the novel approach nodalarrival and departure energy are approximated by a linear function definedin the interval of minimal and maximal departure energies [Et,minn,dep , E

t,maxn,dep ]

or arrival energies [Et,minn,arr , Et,maxn,arr ]. Depending on past charging behavior

nodal energy levels are defined and subsequently also the relative energylevels of arrival and departure energies can be determined.

2.4.1 Departure energy

There is a logic relation between the energy level of each node one timestep prior to vehicles departing and the related drop in energy at time ofdeparture. If the nodal energy at one time step is close to its maximum, allvehicles departing in the next time step will typically also be charged closeto their maximum. Thus the coefficient αtn,dep ∈ [0, 1], ∀n,∀t is defined asfollows:

αtn,dep =Et−1n − Et−1,min

n

Et−1,maxn − Et−1,min

n

∀n,∀t (6)

Et,minn represents the minimal nodal energy required to cover futureplanned trips trip, while Et,maxn represents the upper limit of the energylevels based upon battery capacity, maximal charging power and vehicle’senergy consumption. In case of nodal energy close to maximum αtn,dep will

take a value close to 1 and as a result the departure energy Etn,dep will alsobe close to its maximum value and vice versa. As a result the aggregateddeparture energy is defined by the following equation:

Etn,dep = Et,minn,dep + αtn,dep

(Et,maxn,dep − E

t,minn,dep

)∀n, ∀t (7)

2.4.2 Arrival energy

The linear approximation of arrival energy is also depending on the nodalenergy levels one time step prior to departure. In contrast to the departureenergy it does not only depend on the energy level of one parking location,

2 METHODOLOGY 12

but on the energy level of all previous parking locations of vehicles arrivingat a new parking location at one time step. Therefore αtn,arr ∈ [0, 1], ∀n,∀tis defined as the average normalized energy level of all previous parkinglocations of arriving vehicles:

αtn,arr =1∑

v γt,tdepn,ndep(v)

∑v

γt,tdepn,ndep(v)

Etdep−1

ndep(v)− Etdep−1,min

ndep(v)

Etdep−1,max

ndep(v)− Etdep−1,min

ndep(v)

∀n, ∀t (8)

Similar to the departure energy, arrival energy will take values close toits maximum when all arriving vehicles were fully charged at their previousparking locations (αtn,arr = 1) or close to its minimum in case vehicles wereonly charged with the minimal power required to cover their trips at theirpast parking locations (αtn,arr = 0). Finally, arrival energy is defined thefollowing:

Etn,arr = Et,minn,arr + αtn,arr(Et,maxn,arr − Et,minn,arr

)∀n,∀t (9)

2.5 Control scheme

The applied control scheme is indirect and thus a decentralized control ap-proach, since the aggregator manipulates ToU tariffs to incentivize a certaincharging behavior. Thus, the aggregator, besides aggregating individualvehicles, sets price incentives to obtain lower costs and a desired chargingbehavior. The desired charging behavior is defined by the grid operator,who wants to limit PEV induced stress on the grid infrastructure.

2.6 Formulation of bi-level problem

ToU tariffs are manipulated by an aggregator to incentivize a desired charg-ing behavior. The optimal ToU tariff profile is determined by a bi-levelprogramming approach, which fulfills grid operation objectives on the gridnetwork level and minimize charging costs of an aggregated PEV fleet. Thegrid network level is referred to as the upper level problem, while the aggre-gated PEV fleet as the lower level problem. The lower level constrains theupper level, which requires both problems to be jointly solved. Thereforethe problem formulation has to be transformed into a single level MPEC.

2.6.1 Upper level

The upper level objective function is defined as the minimization of thecharging profile deviation from a desired loading profile P tn,ref for each nodeat each time step:

2 METHODOLOGY 13

minP tn,charged,T oU

tn

∑t

∑n

∣∣P tn,ref − P tn,charged∣∣ (10)

An objective function defined by an absolute value can be transformedinto a mathematically equivalent linear optimization problem without ab-solute values, but with a new optimization variable Z and two inequalityconstraints instead according to [15]:

minP tn,charged,Z

tn,T oU

tn

∑t

∑n

Ztn (11)

Ztn + P tn,charged ≥ P tn,ref ∀n,∀t (12)

Ztn − P tn,charged ≥ −P tn,ref ∀n, ∀t (13)

The upper level problem is constrained by an inequality constraint, en-abling ToU tariffs to vary within certain bounds (e.g. +/− χ%) relative toreference prices. Thereby a trade-off between small deviations in referenceprices and a charging behavior close to the desired loading profile accordingto (10) is obtained.

(1− χ) ToU tn,ref ≤ ToU tn ≤ (1 + χ) ToU tn,ref ∀n,∀t (14)

2.6.2 Lower level

The lower level problem represents the vehicle charging cost minimizationover each PEV, enabling PEV owners to charge their vehicles at the lowestpossible price. The charging cost minimization over each vehicle is repre-sented by the single aggregated optimization on grid level:

minP tn,charged, E

tn, E

tn,arr, E

tn,dep

∑t

∑n

ToU tn Ptn,charged (15)

Its solution provides the charging profile at each node and thus also theenergy level evolution at each node, including departure and arrival energy. The dual variables, required for reformulation of the lower level problem,are defined in Table 1.

The solution of the lower level is constrained by the following equalityand inequality constraints:

1. Equality constraints:

2 METHODOLOGY 14

(a) Maximal charging power and given driving profiles, yielding Etn,arrand Etn,dep, constrain the energy profile of the nodal virtual stor-age model:

Etn = Et−1n +P tn,charged ηn ∆t+Etn,arr−Etn,dep : λtn,balance ∀n,∀t

(16)It is assumed that the initial energy level of each vehicle is thesame as in the end of the modelled time horizon and each vehiclereturns to its initial parking position. Thereby it is guaranteedthat vehicles charge the required power to cover their trips withinthe modelled time horizon. This implies also on the nodal levelsame initial and final energy of each nodal virtual storage.

E0n = ETn : λTn,balance ∀n (17)

(b) Arrival energy and departure energy are approximated basedupon the following introduced formulas:

Etn,arr = Et,minn,arr + αtn,arr(Et,maxn,arr − Et,minn,arr

): λtn,arr ∀n, ∀t (18)

Etn,dep = Et,minn,dep + αtn,dep

(Et,maxn,dep − E

t,minn,dep

): λtn,dep ∀n, ∀t (19)

(c) Energy consumption of each vehicle within the modelled time-frame can be computed based on the obtained driving patterns.To guarantee that each vehicle returns to its initial energy level,the energy charged Etn,charged = P tn,charged ∆t of all vehicles atall times multiplied by the given conversion efficiency ηn has toexactly equal consumed energy within the modelled timeframe.This constraint inhibits V2G charging in order to reduce chargingcosts.∑

t

∑n

P tn,charged ηn ∆t =∑t

∑n

Etn,cons : λcons (20)

2. Inequality constraints:

(a) P tn,charged is constrained by minimal charging requirements dueto future trips and limited battery capacities:

P t,minn,charged ≤ Ptn,charged ≤ P

t,maxn,charged : µtmin,Pn

, µtmax,Pn∀n, ∀t

(21)

(b) Etn is constrained by driving patterns, charging behavior and lim-ited battery capacities:

Et,minn ≤ Etn ≤ Et,maxn : µtmin,En, µtmax,En

∀n,∀t (22)

2 METHODOLOGY 15

(c) Etn,arr and Etn,dep are constrained by minimal and maximal ob-tainable nodal energy levels and resulting limits of arrival anddeparture energy:

Et,minn,arr ≤ Etn,arr ≤ Et,maxn,arr : µtmin,En,arr, µtmax,En,arr

∀n,∀t (23)

Et,minn,dep ≤ Etn,dep ≤ E

t,maxn,dep : µtmin,En,dep

, µtmax,En,dep∀n, ∀t (24)

2.6.3 Transformation to single level MPEC

The lower level problem described by (15) − (24) is continuous and lineardue to the fact that ToU tn is defined as an optimization variable of the upperlevel. Thus the lower level problem is convex. Convex optimizations can bemathematically equivalently represented by their Karush - Kuhn - Tucker(KKT) conditions or their Primal Dual formulation [13].

1. KKT : Since the lower level problem is convex, it can be replaced byits KKT conditions. The KKT conditions can be divided into theiroptimality conditions and the so-called complementarity constraints.

The optimality conditions require the derivative of the Lagrangian ofthe lower level problem in respect to each of the four optimizationvariables (∀n, ∀t) to be equal to zero:

ToU tn + λtn,balance ∆t ηn − λcons ∆t ηn − µtmin,Pn+ µtmax,Pn

= 0 ∀n, ∀t(25)

− λtn,balance + λt+1n,balance + λt+1

n,dep

Et+1,maxn,dep − Et+1,min

n,dep

Et,maxn − Et,minn

+∑v

λtarrnarr(v)

γtarr,tnarr,ndep(v)∑v γ

tarr,tnarr,ndep(v)

Etarr,maxnarr(v)− Etarr,minnarr(v)

Et−1,maxndep(v)

− Et−1,minndep(v)

− µtmin,En+ µtmax,En

= 0 ∀n, ∀t

(26)

λtn,balance − λtn,arr − µtmin,En,arr+ µtmax,En,arr

= 0 ∀n,∀t (27)

− λtn,balance − λtn,dep − µtmin,En,dep+ µtmax,En,dep

= 0 ∀n,∀t (28)

In addition all inequality constraints have to be satisfied and the cor-responding inequality dual variables µ have to be greater than or equalto zero. Furthermore, the complementarity conditions have to be ful-filled, which is expressed by the orthogonally requirement of the lowerlevel inequality and corresponding dual variable inequality:

2 METHODOLOGY 16

0 ≤ P tn,charged − Pt,minn,charged ⊥ µtmin,Pn

≥ 0 ∀n, ∀t (29)

0 ≤ P t,maxn,charged − Ptn,charged ⊥ µtmax,Pn

≥ 0 ∀n,∀t (30)

0 ≤ Etn − Et,minn ⊥ µtmin,En≥ 0 ∀n, ∀t (31)

0 ≤ Et,maxn − Etn ⊥ µtmax,En≥ 0 ∀n,∀t (32)

0 ≤ Etn,arr − Et,minn,arr ⊥ µtmin,En,arr≥ 0 ∀n, ∀t (33)

0 ≤ Et,maxn,arr − Etn,arr ⊥ µtmax,En,arr≥ 0 ∀n, ∀t (34)

0 ≤ Etn,dep − Et,minn,dep ⊥ µtmin,En,dep

≥ 0 ∀n, ∀t (35)

0 ≤ Et,maxn,dep − Etn,dep ⊥ µtmax,En,dep

≥ 0 ∀n, ∀t (36)

The numerically critical complementarity conditions (29) − (36) canalso be restated by applying the Fortuny-Amat linearization procedure[16], resulting in (41)− (56). M imposes an upper bound for all µ andlower level inequality constraints. In case of bounding the lower leveloptimization variables, the upper bounds are known and M can bedefined accordingly:

MP tn

= P t,maxn − P t,minn ∀n, ∀t (37)

MEtn

= Et,maxn − Et,minn ∀n, ∀t (38)

MEtn,arr

= Et,maxn,arr − Et,minn,arr ∀n, ∀t (39)

MEtn,dep

= Et,maxn,dep − Et,minn,dep ∀n, ∀t (40)

For the dual variables the upper limit M was chosen to be a sufficientlylarge number.

2 METHODOLOGY 17

P tn,charged − Pt,minn,charged ≤ (1− ωtmin,Pn

) MP tn∀n, ∀t (41)

0 ≤ µtmin,Pn≤ ωtmin,Pn

M ∀n, ∀t (42)

P t,maxn,charged − Ptn,charged ≤ (1− ωtmax,Pn

) MP tn∀n,∀t (43)

0 ≤ µtmax,Pn≤ ωtmax,Pn

M ∀n, ∀t (44)

Etn − Et,minn ≤ (1− ωtmin,En) MEt

n∀n, ∀t (45)

0 ≤ µtmin,En≤ ωtmin,En

M ∀n,∀t (46)

Et,maxn − Etn ≤ (1− ωtmax,En) MEt

n∀n,∀t (47)

0 ≤ µtmax,En≤ ωtmax,En

M ∀n,∀t (48)

Etn,arr − Et,minn,arr ≤ (1− ωtmin,En,arr) MEt

n,arr∀n, ∀t (49)

0 ≤ µtmin,En,arr≤ ωtmin,En,arr

M ∀n, ∀t (50)

Et,maxn,arr − Etn,arr ≤ (1− ωtmax,En,arr) MEt

n,arr∀n, ∀t (51)

0 ≤ µtmax,En,arr≤ ωtmax,En,arr

M ∀n, ∀t (52)

Etn,dep − Et,minn,dep ≤ (1− ωtmin,En,dep

) MEtn,dep

∀n, ∀t (53)

0 ≤ µtmin,En,dep≤ ωtmin,En,dep

M ∀n,∀t (54)

Et,maxn,dep − Etn,dep ≤ (1− ωtmax,En,dep

) MEtn,dep

∀n, ∀t (55)

0 ≤ µtmax,En,dep≤ ωtmax,En,dep

M ∀n,∀t (56)

The new single level problem formulation has the objective functionof the upper level (11):

minφ

∑t

∑n

Ztn (57)

The optimization variables are the optimization variables of the upperproblem and the lower problem, including all dual and binary variables:

φ =[Ztn, T oUtn, P

tn,charged, E

tn, E

tn,dep, E

tn,arr,

λtn,balance, λtn,initial, λ

tn,arr, λ

tn,dep,

µtmin,Pn, µtmax,Pn

, µtmin,En, µtmax,En

,

µtmin,En,arr, µtmax,En,arr

, µtmin,En,dep, µtmax,En,dep

,

ωtmin,Pn, ωtmax,Pn

, ωtmin,En, ωtmax,En

,

ωtmin,En,arr, ωtmax,En,arr

, ωtmin,En,dep, ωtmax,En,dep

]

(58)

2 METHODOLOGY 18

The single level constraints consist of the upper level constraints (12)−(13), equality and inequality constraints of the lower level problem(16) − (24) and the formulated KKT conditions of the lower level(25) − (28), (41) − (56). The objective functions and constraints areall implemented in Yalmip for Matlab and solved by CPLEX v. 12.4solver [17].

2. Primal Dual formulation: Another valid reformulation of a convexproblem is based upon the principle of duality. In this alternativereformulation the lower level problem is substituted by its primal con-straints, its dual constraints and its strong duality equality accordingto [12]. The strong duality equality is as follows:

∑t

∑n

ToU tn Ptn,charged

= λcons∑n

∑t

Etn,cons

−∑n

∑t

µmax,P tnP t,maxn,charged +

∑n

∑t

µmin,P tnP t,minn,charged

−∑n

∑t

µmax,EtnEt,maxn +

∑n

∑t

µmin,EtnEt,minn

+∑n

∑t

λtn,arr(Et,minn,arr − (Et,maxn,arr − Et,minn,arr )

∗ 1∑v γ

t,tdepn,ndep(v)

∑v

γt,tdepn,ndep(v)

Etdep−1,min

ndep(v)

Etdep−1,max

ndep(v)− Etdep−1,min

ndep(v)

)

−∑n

∑t

µmax,Etn,arr

Et,maxn,arr +∑n

∑t

µmin,Etn,arr

Et,minn,arr

+∑n

∑t

λtn,dep(Et,minn,dep −

Et−1,minn

Et−1,maxn − Et−1,min

n

(Et,maxn,dep − Et,minn,dep ))

−∑n

∑t

µmax,Etn,dep

Et,depn,arr +∑n

∑t

µmin,Etn,dep

Et,depn,arr

(59)

Both reformulations of the lower level are equivalent and yield in combi-nation with the upper level a MPEC. The advantage of the primal-dual overthe KKT formulation is that it does not include any numerically expensivecomplementarity conditions. In case of a very large number of complemen-tarity constraints, the MPEC derived from the KTT formulation may be-come intractable. As a result the primal dual formulation is generally easier

2 METHODOLOGY 19

to solve. However, in this MPEC the primal dual formulation includes thenonlinear term

∑t

∑n ToU

tn P

tn,charged in its strong duality equality. Thus

in this case the primal dual formulation is not used to reformulate the lowerlevel problem.

3 RESULTS 20

3 Results

3.1 Case study set up

In order to simulate charging responses, PEV fleet specifications have tobe defined. The PEV vehicle fleet is modelled inhomogeneous in regardto battery capacity corresponding to the specifications of the Toyota Prius(16 kWh) and of the Nissan Leaf (24 kWh). Limitations and technicalspecifications of modelled batteries are provided in Table 2. Batteries requirea minimal state of charge (SOC) of 20 % and exhibit a charging efficiencyof 90 %. A constant semi-fast charging power of 11 kW and ubiquitouscharging are assumed at each network node. The anticipated average PEVenergy consumption is 0.2 kWh/km and allows to compute the requiredenergy of each vehicle to cover its trips.

Table 2: Battery specifications

Cv SOCv,min ηv P t,maxv,charged

50% : 16 kWh0.2 0.9 11 kW

50% : 24 kWh

3.2 Aggregation of individual vehicles

The goal was to modify the aggregation method for the purpose of repre-senting the response of a PEV fleet to ToU tariffs in a single optimizationand obtain solutions close to the results of individual PEV optimizationswith low computational effort. In [8] the aggregator models arrival and de-parture energies as the mean between the upper and lower bounds of thecorresponding energy levels. This approximation yields observable devia-tions between the individual and aggregated response (Fig. 3) to given ToUprofile (Fig. 4). Deviations in charging profiles are clearly visible in Fig. 3and higher costs for the aggregated response are expected, since the aggre-gated charging profile charges less at times of minimal prices.

In order to improve the aggregation method, the linear approximationof departure and arrival energies, described in subsection 2.4, is introduced.By expressing departure and arrival energies as functions of nodal energylevels at times of respective vehicles’ departure, the deviations of nodalenergy levels and power charged between the aggregation and the individualresponse can be reduced. In addition resulting costs of charging can bedecreased compared to the initial aggregation method. The cost reductioncan be explained by the smaller mismatch of the charging profiles in caseon the new aggregation method, which also results in a smaller deviation of

3 RESULTS 21

Figure 3: Charging responses of a PEV fleet of 10.000 vehicles in a grid with88 nodes resulting from the individual and the initial aggregated optimiza-tions for one given ToU tariff are shown.

Figure 4: Hourly ToU tariffs on July 5th of 2014 from EPEX sport marketdata for Switzerland.

charging costs.

3.2.1 Linear approximation of departure energy

The results of the new linear approximation of departure energies are com-pared to the individual departure energies in Fig. 5. Both PEV optimiza-tions are simulated for 10.000 vehicles in a grid representation with 88 nodesin response to ToU tariffs from July 5th to July 9th from Swiss EPEX spotauction data. The chosen linear approximation yields results very similarto individual results. Deviations are small and the linear approximationperforms significantly better compared to the initial average approximationfor the same input parameters as shown in Fig. 6.

3 RESULTS 22

Figure 5: Comparison of linear approximations of departure energies vs.exact solutions

Figure 6: Comparison of average approximation of departure energies vs.exact solutions

3.2.2 Linear approximation of arrival energy

The results of the linear approximation of arrival energies from PEV op-timizations for 10000 vehicles in a grid representation with 88 nodes inresponse to ToU tariffs from July 5th to July 9th from Swiss EPEX spotauction data, are shown in Fig. 7. Departure energies compared to arrivalenergies are closer to the individual optimization results, since they onlydependent on nodal energy level at one time step, while arrival energiesdepend on all past energy levels of arriving cars. Thereby small errors areaccumulated. In addition there is a discontinuity in arrival energies. Sincethe final energy is fixed, the denominator of equation (8) is equal to zerofor all vehicles departing in the first time step t = 1. Thus αtn,arrival is not

3 RESULTS 23

defined in the first step and is set to zero. As a result the contribution ofvehicles departing in the first time step to the arrival energy is zero. Set-

ting α0n,arrival to 0.5 accordingly to SOC0

v = SOC0,maxv +SOC0,min

v2 , does not

yield any improvements in the approximation of arrival and departure en-ergies. Thus αtn,arrival in the last time step is set to zero and equation (9)is corrected accordingly by reducing the number of vehicles arriving in casevehicles arriving departed in the first time step. This discontinuity com-bined with the dependency on multiple energy levels explains why the linearapproximated arrival energy has more outliers than the linear approximateddeparture energy.

3 RESULTS 24

Figure 7: Comparison of linear approximations of arrival energies vs. exactsolutions

Figure 8: Comparison of average approximations of arrival energies vs. exactsolutions

3.3 Comparison of aggregation methods

Smaller deviations in the charging power profile due to the new aggregationmethod can be observed, when comparing Fig. 3 and Fig. 9. Chargingcosts and errors between individual optimizations and aggregated responsesfor three differently sized PEV fleets are provided in Tables 3, 4 and 5 .ToU tariffs are taken from EPEX spot auction data from July (5th − 14th)for Switzerland. The results of the new aggregation method are comparedto the initial aggregation method (referred to as old) and to the individualoptimization. The derivations of nodal energy levels and power chargedbetween individual optimizations and aggregated approaches are shown for

3 RESULTS 25

each fleet and day. The new aggregation outperforms the old aggregationfor each fleet size, but performance improvements decrease for larger fleets.The obtained data proves that on average the new aggregation results insmaller deviations from individual PEV responses and thus shows betterperformance for aggregated PEV charging optimizations compared to theinitial formulation in [8]. Lower costs for the new aggregation comparedto the initial method are expected, since at times of lowest cost maximumcharging power is obtained.

Figure 9: Charging responses of a PEV fleet of 10000 vehicles in a grid with88 nodes resulting from the individual and the new aggregated optimizationsare shown.

In case of small PEV fleets (1000 PEVs) the new aggregation methodcan reduce absolute deviations in nodal energy and charging profiles fromthe individual model by more than 60% compared to the initial aggregation.For larger PEV fleets (10000 PEVs) the new approximation of Etn,dep and

Etn,arr still outperforms the old average approximation in regard to absolutedeviations in power profiles compared to the results of individual optimiza-tions by more than 36%. Thus the new approximation successfully managesto yield results which are better for the purpose of fleet modelling for indi-rect control approaches.

The linear approximation of Etn,dep and Etn,arr deteriorates in perfor-mance for very large fleets due to the fact that in case of large PEV fleets ateach node at each time step a high number of vehicles is parked. As a resultthere is no strong interdependency of nodal energy and departure energy incase few vehicles depart from a node to which a high number of vehicles areconnected. This relation can also be derived from the data in Tables 3 and5, since improvement of power deviations for a fleet size of 1000 PEVs is50% compared to only 30% for a fleet size of 10000 PEVs.

In certain cases the new aggregation even outperforms the individual

3 RESULTS 26

optimizations in terms of total charging costs. The aggregated responseminimizes total charging costs, while PEV individual optimizations mini-mize charging costs for each vehicle. Total lower charging costs in case ofan aggregated optimization do not guarantee that each individual PEV ischarged at lowest cost possible. As a result cost comparisons are no relevantindicators for performance of the new aggregation method. The improvedaggregated result in cost can be explained by the introduction of new opti-mization variables in the aggregated optimization. Explicitly, Etn ∀n ∀t isthe new additional optimization variable and can directly influence Etn,depand Etn,arr as shown in equations (6-9). In large PEV fleets this effect isstronger, since a small variation in nodal energy impacts multiple nodal en-ergy levels in future time steps due to arriving vehicles. In such cases thenew interdependency can be fully utilized. The key indicators of the pre-cision of the aggregation approach are deviations from nodal energy levelsand charging profiles relative to individual PEV optimizations results.

Table 3: Results of charging cost minimization for a PEV fleet of 10000vehicles distributed over 88 nodes for 10 different ToU price profiles.

3 RESULTS 27

Table 4: Results of charging cost minimization for a PEV fleet of 5000vehicles distributed over 88 nodes for 10 different ToU price profiles.

Table 5: Results of charging cost minimization for a PEV fleet of 1000vehicles distributed over 88 nodes for 10 different ToU price profiles.

3.4 Optimal setting of ToU tariffs

The improved aggregation method is applied to formulate an MPEC to de-termine optimized ToU tariffs, which on the one hand fulfill the grid opera-tor’s objective of following a reference load and on the other hand minimizecharging costs of PEVs.

3.4.1 Desired reference power profile

The grid operator and generating units favor uniform loading which reducesstress on the grid infrastructure and at the same time reduces ramp up andramp down costs of generating units. Therefore the chosen reference powerprofile for PEV charging is a valley filling scenario. Since in the definedproblem set V2G charging is not considered, unloading of grid infrastructure

3 RESULTS 28

by depleting PEVs’ stored energy is not taken into account. Daily powervolume profiles are obtained from EPEX spot auction data. In the chosenreference power profile valley filling is only introduced in hours, when theauctioned power volume falls below the daily power volume mean. In Fig. 10a daily power volume curve is shown, in which the green filled area marks thedesired valley filling volume. The resulting reference load is depicted in Fig.11. In order to reach theoretically at the optimum a minimal cost of zeroon the upper level and to obtain nicely interpretable results, the referenceload profile is normalized by the daily required energy consumption of allvehicles to cover their trips.

Figure 10: Normalized power volume curve of July 7th in which green filledarea marks the desired valley filling volume.

Figure 11: Power reference profile desired by grid operators and generatingunits to obtain uniform loading based on traded power volume on July 7th.

3.4.2 Solution of MPEC

The single-level MPEC solution is compared to lower level solutions and tothe response of individual PEV optimizations.

The lower level aggregated response for a given ToU tariffs is depicted inFig. 12. The discussed unwanted peaks in load can be observed. The equiv-

3 RESULTS 29

alent KKT response is also plotted in Fig. 12 to prove that the solution ofthe KKT formulation of the lower level problem exactly matches the lowerlevel response. The desired reference profile is based on EPEX spot auctiondata of July 7th. Load peaks at times of lowest cost can be observed. Inorder to reduce the computational effort, simulations are performed with aPEV fleet of 10000 PEVs and a grid representation with one node.

Figure 12: Responses of lower level problem and equivalent KKT reformu-lated problem to real time prices are shown.

The bi-level optimization manipulates hourly prices from EPEX spotauction data in the range of maximal +/−χ% to incentivize PEV chargingaccording to the desired reference profile. The symmetric +/− χ% boundswere chosen to neither favor the grid operator and generating units nor thePEV owners in the optimization of the ToU tariffs.

The bounded bi-level optimization yields the optimal manipulated ToUtariffs, which enable to achieve valley filling up to 90% for certain load andpricing scenarios as provided in Table 6. Optimizations are performed for aPEV fleet of 10000 vehicles on a grid level with one node.

First, results of bi-level optimizations with EPEX spot market data forthe second week of July from 7th − 13th are shown in Table 6.

3 RESULTS 30

Table 6: The Table shows charging responses of the bi-level problem byoptimizing ToU tariffs based on Swiss market data in the second week ofJuly.

Spot market data is used to obtain reference ToU tariffs and determinethe valley filling power profile. Two different optimizations sets, one with+/− 10% allowed variations in price and the other with +/− 20% are com-pared to the initial solution of the lower level problem with non-optimizedToU tariffs. The first set of optimizations allows variations in ToU tariffsby +/ − 10%, yielding a weekly average valley filling of 58% compared toinitially 22%, while costs increase by 6% compared to the optimal chargingcosts of the lower level. The second set allows larger variations in ToU tariffsby +/ − 20%, which result in a weekly average valley filling by 66%, but acost increase of 13% relate to the optimal lower level costs. The bi-level opti-mized ToU tariff and the resulting charging response of July 7th is providedin Fig. 13. In Fig. 13 71% of valley filling is obtained, while total chargingcosts rise by 5%. In the upper subplot the red line represents the originalToU tariff and the blue line shows the optimized ToU tariff. The flatteningof the ToU profile around the lowest price as a result of the optimizationcan be observed. As a result it can be observed that for the bi-level solutionvalley filling is improved by 44% to 71% due to optimized ToU tariffs byallowed ToU variations of +/ − 10%. Depending on daily ToU profiles adifferent bound on ToU variations yields optimal results. The increase incharging cost is caused by a flattening of the ToU profile in order to notincentivize charging at the time step of lowest cost. ToU tariffs’ variationsof +/ − 10% show a good trade-off between successful valley filling and alow increase in costs for bi-level optimizations.

Secondly, the reponses of the individual optimizations to the correspond-ing optimized ToU tariffs are shown in Table 7. Performance in valley fillingis very low with average values of 20% compared to 16% in response to nonoptimized ToU tariffs, independent from +/ − 10% or +/ − 20% allowedToU variations. As a result, the optimized ToU tariffs, independent fromtheir variation bounds, yield no significant improvement in valley filling for

3 RESULTS 31

Figure 13: Bi-level charging responses to optimized ToU tariffs of 10000PEVs.

the applied indirect control scheme.

Table 7: The Table shows the results of individual PEV optimizations inresponse to the optimized ToU tariffs.

The problem related to the bi-level optimization is that the optimizedToU tariffs yield multiple optimal solutions. The ToU profile allows multi-ple solution for PEV charging, which are all equal in cost of charging. Thusresponses of the lower level and individual optimizations to the optimizedToU tariffs may show charging profiles different to the bi-level charging pro-file and different to the desired power valley filling profile. The lower levelresponse to the optimized ToU tariff obtained from Fig. 13 is shown in Fig.14 and is equivalent in charging costs to the bi-level solution. The cost op-timization has multiple optimal solutions, thus the response is not the sameas the bi-level solution and the desired valley filling charging profile is not

3 RESULTS 32

obtained. Furthermore, the individual optimization results to the optimizedToU tariffs are depicted in Fig. 15. The deviations of the charging pro-files between the lower level response and the individual response are small,proving the performance of the linear approximation approach for PEV fleetaggregation. Both optimizations, the lower level and the individual, yieldresults of poor valley filling. The problem associated with the bad perfor-mance also lies in the nature of price incentive mechanisms. Discrete pricesignals do always incentivize a charging peak at the lowest price and thusinduce high load peaks. In case of individual cost optimizations, vehicleswill always charge at the lowest price. Therefore the appplied optimizationapproach fails to determine optimized ToU prices that lead to valley fillingPEV charging responses for indirect charging control schemes.

Figure 14: Charging responses of lower level problem to bi-level optimizedToU tariffs of 10000 PEVs.

3 RESULTS 33

Figure 15: Charging responses of individual cost optimization to bi-leveloptimized ToU tariffs of 10000 PEVs.

4 DISCUSSION 34

4 Discussion

The optimized ToU tariffs enable to reach significant improvements in val-ley filling as a result of the bi-level optimization method. However, costoptimizations in response to optimized ToU tariffs for the aggregated or in-dividual approach show no successful valley filling.

The problem is that there are multiple optimal solutions for given ToUtariffs and individual optimization will always charge at the lowest priceavailable, causing load peaks at times of lowest prices. The problem lies inthe underlying characteristics of price incentive mechanisms.

4.1 Accuracy of aggregation approach

The new aggregation approach outperforms the initial aggregation and thusrepresents an improved basis for the bi-level optimization and for futureaggregated PEV optimizations. The drawback of the linear approximationsconcerns the discussed discontinuity in arrival energy related to the fixedfinal energy and the decrease in performance for very large PEV fleets.However, up to a PEV fleet size of 1000 vehicles per grid node, the new linearapproximation significantly outperforms the existing aggregation approach.

The approximation of the arrival energy (9), in case of large PEV fleetsand high numbers of loading nodes, is numerically expensive. Especially, theformulation of the required KKT conditions (26) related to arrival energytakes significant computational effort.

4 DISCUSSION 35

4.2 Outlook

In future simulations nodal ToU tariffs can be introduced to obtain improveddesired charging responses of vehicles. By varying ToU tariffs depending onlocation at each time step instead of system-wide ToU tariffs, the leverageof the aggregator in indirect control schemes rises. The introduction of ad-ditional optimization variables due to nodal ToU tariffs can be utilized toset price incentives varying on location. Thereby improved desired PEVcharging power responses can be obtained.

Advanced optimization approaches can also introduce price and quantitydependend ToU tariffs. By limiting the power quantity provided at lowerprices, demand peaks in the response can be reduced. As a result moreuniform PEV charging responses can be obtained.

In order to successfully implement indirect control schemes, nodal orquantity dependend ToU tariffs have to be introduced, since hourly ToUtariffs according to the obtained results fail to control PEV charging.

In the future strong deviations in ToU prices might cause price depen-dend driving patterns. At the moment all driving patterns are modeledprice independent. The proposed optimization approach would allow toutilize known reference prices to simulate price dependent driving patterns,since the optimized ToU tariffs will only deviate by +/−10% from referenceprice.

5 ACKNOWLEDGEMENTS 36

5 Acknowledgements

Having read about Marina Gonzalez Vaya research field, I asked her straightaway if it is possible to write my semester project with her and she fortu-nately agreed. I was strongly interested in her research field, but at thattime my knowledge of convex optimizations and extensive Matlab modellingwas very limited. It took me quite some time to first understand her de-tailed PEV fleet modelling and optimization formulation and secondly tocomprehend the mathematical procedures required to formulate a bi-leveloptimization problem. Dr. Luis Baringo gave me continuous support in caseof problems, especially when deriving the MPEC formulation. Unfortunatelymy results are limited in their scope, since using ToU tariffs to control PEVfleet charging seems not to work according to my results. However, at leastI was able to gain an extensive insight into the implications of anticipatedPEV proliferation and convex optimizations and in addition to significantlyimprove my Matlab programming skills. Thank you!

REFERENCES 37

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