Optimal Charging of Electric Vehicles with Renewable

Optimal Charging of Electric Vehicles with Renewable Energy in Smart Grids eingereichte DIPLOMARBEIT von B.Sc. Jose Rivera geb. am 18.12.1986 wohnhaft in: Amalienstr. 87 80799 M¨ unchen

Lehrstuhl f¨ ur STEUERUNGS- und REGELUNGSTECHNIK Technische Universität M¨ unchen Univ.-Prof. Dr.-Ing./Univ. Tokio Martin Buss Univ.-Prof. Dr.-Ing. Sandra Hirche

Betreuer: Beginn: Zwischenbericht: Abgabe:

Dr. Philipp Wolfrum (SIEMENS AG) Dipl.-Ing. Stefan Sosnowski 21.05.2012 21.08.2012 16.11.2012

Abstract This thesis focuses on the challenges presented in the management of a large Electric Vehicle (EV) population. The indirect control of EVs with an aggregator through incentive signals is considered. A decentralized EV management framework based on the Alternating Direction Method of Multipliers (ADMM) is proposed. Case studies with realistic data for various EV management concepts were carried out to show the flexibility and wide applicability of this framework. The results show that the proposed framework is very simple to implement and running on 100 CPUs, optimal management results for 1 million EVs could theoretically be obtained in less than 3.3 hours. Nevertheless, the required frequent communication of the iterative ADMM algorithm, makes this framework impracticable for on-line control. To address this issue, this thesis presents an approximate solution to the optimal EV charging problem for stochastic incentive signals. The solution of this problem is the first step required to formulate an EV management framework with infrequent communication. The presented results offer relevant practical and theoretical insights into the management of a large EV population, beyond of what is available in the current literature.

Zusammenfassung Im Rahmen der nachfolgenden Arbeit wurden die Herausforderungen bei der Steuerung großer Elektrofahrzeugflotten betrachtet. Es wurde die indirekte Steuerung von Elektrofahrzeugen durch einen Aggregator mittels Anreizsignalen betrachtet und eine dezentrale Steuerungsmethode vorgeschlagen, welcher auf Alternating Direction Method of Multipliers (ADMM) basiert. Um die Flexibilität und die breite Anwendbarkeit der vorgeschlagenen Methode aufzuzeigen, wurden Fallstudien mit empirischen Daten durchgef¨ uhrt. Die Ergebnisse zeigen, dass die Methode verhältnismäßig einfach zu implementieren ist und unter Ausnutzung von 100 CPUs innerhalb von weniger als 3,3 Stunden optimale Ergebnisse f¨ ur 1 Million Elektrofahrzeuge berechnet werden könnten. Nichts desto trotz macht die hohe Kommunikationsanforderung des iterativen ADMM-Algorithmus diese Steuerungsmethode unpraktikabel f¨ ur eine Online-Steuerung. Um dieses Problem zu lösen, wird im Rahmen der Arbeit daher auch eine Näherungslösung f¨ ur die optimale Steuerung von Aufladevorgänge mittels stochastische Anreizsignale vorgeschlagen. Die Lösung des Problems stellt einen ersten Schritt in die Steuerung von Elektrofahrzeugkosten mit geringem Kommunikationsaufwand dar. Die dargestellten Ergebnisse liefern praktisch und theoretisch relevante Erkenntnisse u ¨ ber das Management großer Elektrofahrzeugflotten und erweitern den Stand der aktuellen Fachliteratur.

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To my personal heroes, my parents: Maria Esperanza Acevedo Gutierrez Jose Adan Rivera Castillo

”We don’t know one-millionth of one percent about anything.” –Thomas A. Edison

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CONTENTS

Contents 1 Introduction 1.1 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . 1.2 Problem description and goal . . . . . . . . . . . . . . . . . . . . . . 2 The 2.1 2.2 2.3

5 5 7

Electric Vehicle Management Problem 9 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The optimal EV management problem . . . . . . . . . . . . . . . . . 13

3 Distributed Optimization for EV Management 3.1 Choosing a method for distributed optimization . . . . . . . . . . . . 3.2 Decentralized EV Management with ADMM . . . . . . . . . . . . . . 3.2.1 Deriving the ADMM solution . . . . . . . . . . . . . . . . . .

17 17 20 23

4 The EV Management Framework and Case Studies 4.1 The EV Management Framework . . . . . . . . . . . 4.2 Aggregator optimization problem . . . . . . . . . . . 4.2.1 Valley filling . . . . . . . . . . . . . . . . . . . 4.2.2 Price based . . . . . . . . . . . . . . . . . . . 4.2.3 Direct coupling . . . . . . . . . . . . . . . . . 4.3 EV optimization problem . . . . . . . . . . . . . . . . 4.3.1 Battery model . . . . . . . . . . . . . . . . . . 4.3.2 Driving profile and charging strategy . . . . . 4.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Valley filling . . . . . . . . . . . . . . . . . . . 4.4.2 Price based . . . . . . . . . . . . . . . . . . . 4.4.3 Direct coupling . . . . . . . . . . . . . . . . . 4.5 Observations and discussion . . . . . . . . . . . . . .

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5 Optimal EV Charging for Stochastic Incentive Signals 5.1 The challenge of reducing communication frequency . . . . . . . 5.2 The optimal stochastic EV charging problem . . . . . . . . . . . 5.2.1 Special case optimal solution . . . . . . . . . . . . . . . 5.3 An approximate solution to the stochastic EV charging problem

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CONTENTS

5.3.1 Review of order statistics . . . . . . . . . . 5.3.2 Special case approximate solution . . . . . 5.3.3 Approximate solution to the stochastic EV 5.4 Discussion . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . charging . . . . .

. . . . . . 61 . . . . . . 63 problem 65 . . . . . . 67

6 Summary and Outlook 69 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 List of Figures

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Bibliography

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Appendix

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A Alternating Direction Method of Multipliers (ADMM) A.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Optimality conditions and stopping criteria . . . . . . . . . A.4 Varying penalty parameter . . . . . . . . . . . . . . . . . . B Simuation Parameters

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79 79 81 82 83 85

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Chapter 1 Introduction 1.1

Motivation and background

The introduction of Electric Vehicles (EVs) for personal transportation plays a leading role in the reduction of CO2 emissions. According to a McKinsey&Company study commissioned by the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU) personal transportation accounts for 12 % of the total CO2 emissions in Germany [McK10].

CO 2 emissions from cars (g/km) caused by energy supply and vehicle operation 177 120

109 5

Average for fleets of new cars

Efficient diesel

EV using electricity mix

EV using renewable power

Figure 1.1: CO2 emissions for new combustion engine cars, combustion engine cars with efficient diesel technology, electric cars charging with the current energy mix and electric cars charging with renewable energy for the case of Germany in 2010 [PH07]. EVs can contribute more to the reduction of CO2 emissions when charging is done using renewable energy (RE), see Fig. 1.1. Thus, a coupling between renewable energy generation and EV demand is needed. There are several concepts available to implement this coupling, for example: centralized coupling, direct coupling, and price based coupling. Let us now describe some of the available coupling concepts. Centralized coupling: The Independent System Operator (ISO) takes all the information on generation and demand. A schedule is produced that takes indi-

6

CHAPTER 1. INTRODUCTION

Figure 1.2: Some concepts available for coupling renewable energy generation and EV demand. vidual parties’ goals into account. The task of the aggregator is to provide a model of the EV fleet and coordinate the individual EV to perform the schedule defined by the ISO. This concept is known in the literature as Virtual Power Plant (VPP) and is widely applied in pilot projects [JBSG10]. Direct coupling: Renewable energy (RE) generation is contracted in order to serve the EV demand. If RE generation is not able to satisfy the EV demand, the required energy may be purchased from the market. RE generation excess can also be sold in the market. However, the access to the market should be restricted. The role of an aggregator here is to match EV demand and RE generation as good as possible and reduce the amount of energy purchased [Pap11]. Price based coupling: Renewable energy and EVs are coupled through the energy market [CF09]. The aggregator bids on behalf of the EVs in the market. The price obtained from the market can be modified by the aggregator as an incentive for the EVs to support the requirements of the ISO. All of the concepts mentioned above have one thing in common, they all assume that an aggregator or EV fleet manager interacts with the rest of the system on behalf of

1.2. PROBLEM DESCRIPTION AND GOAL

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the EVs, see Fig. 1.2. Having this in mind the aggregator can be thought of as an interface between the EVs and the rest of the systems, whose specific task depends on the type of coupling. However, the aggregator always needs to balance the goals that the system has for the aggregated EV demand and the individual goals of each EV.

Besides charging EVs with renewable energy we can also use EVs as mobile storage units. This is known in the literature as Vehicle-to-Grid (V2G) and it implies that the EVs feed energy back into the grid. For this case we can also apply the previous coupling models. The only difference is that the EVs do not only charge their battery, but they are also able to discharge their battery. The ability to use the EVs as storage allows the integration of more RE generation to the gird and further reduce CO2 emissions.

As we can see, the aggregator is the key element in the coupling of RE and EVs. The design of an EV management algorithm for the aggregator is an interesting and challenging task. We have to consider that this algorithm needs to manage thousands or even millions of EVs and optimize their charging based on individual goals as well as the goals that the system has for their aggregated behavior. A centralized charging management would incur in prohibitive communication and computation requirements. Thus, a robust decentralized charging management system is required.

1.2

Problem description and goal

The goal of this thesis is to provide an algoritmic framework for a decentralized optimal management of EVs through an aggregator. We now describe and discuss the desierable characteristics that such an algorithmic framework should have. Dynamic optimization: The algorithm should be a dynamic optimization. This guarantees that we use the full potential of EVs, taking into account their dynamic behavior and the dynamic changes on the goals for their aggregated behavior. The changes in global goals can be caused by dynamic electricity prices or intermittent RE generation. Prediction or modeling of this dynamics plays a fundamental role for this kind of algorithms. Decentralized: The management of thousands or millions of EVs requires a decentralized approach. The number of variables of a dynamic optimization makes a centralized approach impracticable. Besides the computational complexity, a centralized optimization would require all the information of the EVs. This information has to be transmitted to a centralized location, optimized and then the computed schedule of each individual EV has to be transmitted back to the corresponding EV. The incurred communication costs would be simply

8

CHAPTER 1. INTRODUCTION

too high. Furthermore, we have the requirements of fairness and data privacy. A centralized management means that we have a monopolistic administrative authority. In this scenario the risk of not being fair to individual EV goals is very high and the personal data of each EV is not private. In contrast to this, a decentralized approach can offer a certain level of data privacy and fairness to each EV. Scalable: Algorithmically tractable message routing is required. This means that the messages involved in the coordination of the EVs should be simple and the routing of this messages should be easy to implement. Traversable and portable: This characteristic requires the algorithm to have a common format for communication messages and data types. The coordination protocol should be independent of the individual characteristics of the EVs and the aggregator, i.e. independent of their personal goals. From the requirements presented above we conclude that the algorithm needs to be a decentralized dynamic optimization, able to support a variety of different global and local goals with a common and simple coordination protocol. The desired indirect control structure can be seen in Fig. 1.3. Aggregator

??

???

Incentive signal ?

?? ?? ?? ??

Aggregated response

??

??

??

?? ??

??

?? ??

?? ??

Figure 1.3: Desired structure for EV management. The important elements to be defined are qustionmarked.

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Chapter 2 The Electric Vehicle Management Problem In this chapter the problem for the optimal management of Electric Vehicles (EVs) is defined. First we review some of the basic concepts and definitions in the management of EVs. Later we explore the available literature on the subject and characterize the importance of a general formulation. Based on this analysis, we then formulate a generalized optimal EV management problem.

2.1

Basic concepts

Before we can analyze the available literature, we need to get familiar with some of the basic concepts. The EV management problem can be examined from the perspective of the Independent System Operator (ISO), who is in charge of controlling the whole system. Alternatively, it can be examined from the perspective of the individual EV, who interacts with the ISO. Since we are looking for a fair EV management that takes into account the goals of the ISO and the goals of the EVs, it is important to review the basic concepts form both perspectives. If we take the perspective of the ISO, EVs are seen as an aggregate. For this case, we review the possible ISO characterization of EVs. Fixed load: A load that needs to be met with no degrees of freedom for its control. Deferrable load: A load with an energy demand that needs to be met by some time deadline. The load is considered to have no other goal of its own, besides consuming a specific amount of energy, e.g. no cost function and only constraints. The degree of freedom that the ISO can exploit is the ability to shift the load in time, in order to optimize the operation costs of the whole system. Combination of deferrable load and curtailment load: A curtailment load does not impose hard constraints as a deferrable load. Instead, it has a goal function that penalizes any shortfall of load requirements not being met, e.g. a

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CHAPTER 2. THE ELECTRIC VEHICLE MANAGEMENT PROBLEM

cost function. A combination between curtailment load and deferrable load portraits more accurately the behavior of EVs, because it allows the EV to have a cost function and constraints which need to be met by the ISO. Mobile storage: If the ISO is able to discharge the EVs, the EVs become a mobile storage. In this case the EVs are deferrable/curtailment loads which can also act as a generator. The EVs have a cost function and also constraints that need to be met, but it also tries to maximize its profit from selling energy to the system, just like any generator. Taking the the EV’s perspective there are basically three ways to interact with the grid: fast charging, smart charging and offering Vehicle to Grid (V2G) services. For all of this cases the principal goal of the EV remains the same, the EV has to charge a desired amount of energy in a given time frame. This time frame is defined between the the time of connection and disconnection. The interaction with the ISO will depend on this time frame. The types of interaction of a single EV with the grid can be seen in Fig. 2.1. Let us review the types of interaction that a single EV can have with the grid. Fast charging: The EV does not know how long it is going to remain connected. The best strategy is to ignore any benefits from charging and guarantee that the charging requirement is meet as fast as possible. This is also known as dumb charging. Smart charging : If the EV knows how long it is going to remain connected and this time is larger than the time required for charging, then the EV can use this degree of freedom to maximize its charging benefit. The charging benefits can refer, for example, to the minimization of charging price and depreciation of the battery. Vehicle to Grid (V2G) : Besides knowing the connection time the EV is allowed to discharge and feed energy back to the grid. This two degrees of freedom allows the EV to optimize its charging benefits and also its discharging benefits, e.g. the profit made from providing energy to the grid. The EV decides to charge when the benefit form charging is larger than the benefit from discharging and discharges when this is the other way around. Another relevant aspect is the type of control used to influence the EVs behavior. There are mainly two prevailing concepts: direct control and indirect control. Direct control: The ISO tells each EV directly what their charging profile has to be. This type of control is very restrictive and can be unfair to the individual goals of the EVs. Indirect control: The ISO provides an incentive signal to the EVs in order to control their behavior. This type of control is less invasive but requires a very accurate model of the behavior of the EVs to this incentive signal. The incentive signal can be a price.

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2.2. LITERATURE REVIEW

Charging benefit Disharging benefit

Time

EV benefit

Fast Charging

Charging Time

Smart Charging

Charging Time

V2G

Charging Discharging

Time

Figure 2.1: EV behavior based on its personal benefit for the case of fast charging, smart charging, and Vehicle to Grid (V2G).

2.2

Literature review

The management of EVs in the power grid is a very active field of research. A comprehensive literature research on the subject of coordination strategies for EVs is given in [LVG+ 11]. They come to the conclusion that most of the systems being proposed only consider set-point coordination for one or more coordination objectives, only considering two to six layers of control. From this point of view, different objectives seem to be in conflict with each other, as only direct costs and benefits are considered. If, however, indirect effects are included there is a trade off between objectives. The trade off comes from the vehicle owners, for instance, wanting to charge their EV at lowest cost, e.g. maximal personal benefit, and the ISO wanting to minimize its costs, e.g. operational cost, investment, etc. In [CH11] the problem of achieving controllability of electric loads is studied. They conclude that the central challenge for non disruptive load control is that there are dual, often competing, control objectives: first, to achieve desirable aggregated power consumption patterns, and second, to maintain acceptable end-use performance. They propose that these objectives can be managed by quantifying metrics of load availability (a measure of the physical capacity available for control) and willingness to participate in aggregated control activities (determined by constraints on the quality of end-use function). For the architecture of the control system they acknowledge that a hierarchical model, e.g. a control through an aggregator, may

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hold the most promise because: 1) it creates an avenue for third parties to organize loads and bid them into energy and ancillary service markets, and 2) it may allow the system operator to conceptualize load groups managed by each aggregator as individual resources, similar to individual units on the supply side. Form this conclusions we infer that the design of an EV management system through an aggregator holds the most promise. The aggregator needs to balance the conflicting goals. In order to guarantee fairness, the control of the EVs needs to be indirect. One of the most interesting studies in the real implementation of an EV management system is currently being carried out in Denmark under the project EDISON [BGJ+ 10]. They use a centralized optimization approach, i.e. Virtual Power Plant (VPP). Their goal in [SB10] is to optimize the charging cost of the EVs taking into account the renewable energy, dynamic pricing profile and predicted EV driving profiles. The constraint for the aggregated EV demand is not to charge more than the available RE. They define a linear program in order to optimize the EV fleet. In their optimization they consider a maximum number of 50 EVs, which justifies a centralized optimization. However, as the number of EVs increases in the thousands a decentralized optimization will be required. With our framework we want to provide a solution for this issue. The decentralized optimization of an EV fleet has been mostly studied in the case of the valley filling problem. The idea behind valley filling is to use the EVs as a deferrable load. Given the base demand profile, e.g. the fix load, the goal is to schedule the EVs, so that the sum of the fix demand and the demand of the EVs is as small as possible for each time step, see Fig 2.2. 1 base demand valley−filling = base demand + EVs

aggregated demand (kW)

0.9

0.8

0.7

0.6

0.5

0.4 20:00

0:00

4:00

8:00 time of day

12:00

16:00

Figure 2.2: Optimal result for the valley filling problem, where the deferrable demand of the EVs fills the valley of the base demand [GTL11]. The coordination of the EVs should be done in an indirect manner through an in-

2.3. THE OPTIMAL EV MANAGEMENT PROBLEM

13

centive signal, e.g. a price. In [MCH10] a decentralized control algorithm for a large populations of EVs to solve the valley filling problem was developed. A game theoretical framework was used to formulate a decentralized optimization of the EVs. The result, however, only appears to work for the case where all EVs have homogeneous charging requirements, e.g. they plug in at the same time and require the same amount of energy in the same time frame. This assumption was later alleviated in [GTL11], where a gradient projection algorithm was used. One of the potential issues of this method is that the goal for the aggregated EV demand has to be strictly convex. This limits the application of this method for several important problems, like the price optimization of an EV fleet, which is a linear program and not a strictly convex problem. Moreover, non of the methods above takes into account possible personal goals of the EVs. An effort to alleviate this assumption was done in [GTL12] allowing the EV to have a desired charge profile. Nevertheless, none of this methods seems to be able to deal with the conflicting objectives between the ISO goals and the personal EV goals. With our framework we want to address this problem. A general framework for the decentralized dynamic optimal management of EVs through an aggregator that can deal with the several concepts available for the coupling of EVs and RE is required. Recent work offers a framework for the decentralized dynamic energy network management [KCLB12]. They use the Alternating Direction Method of Multipliers (ADMM) to produce a message passing algorithm to coordinate the schedule of different elements of a power grid. Nevertheless, its application for the EV management problem remains yet to be studied.

2.3

The optimal EV management problem

In our literature review we concluded that a hierarchical control structure holds the most promise. An aggregator should function as an interface between the EVs and the higher layers of control. This aggregator needs to balance the competing control objectives of the ISO and the individual EVs. The aggregator represents the goals of the aggregated EV behavior. This goals may come from the EVs as an aggregate or from the ISO. There are various concepts available for the coupling of RE and EVs, and also different ideas on how the EVs should interact with the power grid. Therefore, we need a general formulation that allows the implementation of different EV management concepts as an optimization problem. We consider a discrete slotted time model where the current time slot t ∈ {1, . . . , T }. The variable t is the index for the time slot and T represents the number of time slots. The slotted time assumption portraits accurately the decisions that need to be made by the aggregator and the individual EV. Typically important information such as the energy price and the RE generation profiles are provided in 15 min intervals. The profile of the aggregated EVs is defined as the vector xa = [xa,1 , . . . , xa,T ]T .

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We consider a specific number of EVs defined by NEV . The charging profile of each EV i is defined by a vector xi = [xi,1 , . . . , xi,T ]T for i = 1, . . . , NEV . We define that for an EV i a value xi,t > 0 means that the EV is consuming energy, e.g. charging, at time slot t. Just as well for xi,t < 0 an EV i is feeding energy back to the grid, e.g. discharging. The same convention is valid for the aggregated EV profile xa . We define the optimal EV management problem as a joint optimization with a tradeoff parameter γ to represent the importance of individual EV goals compared to the importance of the aggregator goals: P EV minimize fa (xa ) + γ N i=1 fi (xi ) xa ,xi PNEV subject to xa = i=1 xi (2.1) xa ∈ Xa xi ∈ Xi ; i = 1, . . . , NEV . Variable T NEV xa xi fa (xa ) fi (xi ) Xa Xi γ

Description Number of time slots Number of available EVs Aggregated EV profile for all time slots Profile of a single EV for all time slots Cost function of the aggregator Cost function of the EV i Constrains set of the aggregator Constrains set of EV i Goals trade-off parameter

Type Scalar Scalar Vector ∈ RT Vector ∈ RT Convex function Convex function Convex set Convex set Scalar

Our formulation is equivalent to an exchange problem [Boy11a]. The aggregator and the EVs exchange a common good, i.e. energy, looking to maximize a total cost. Each of the participants has its own cost function and constrains set. The different agents are coupled by an equilibrium constraint. In our case this equilibrium constraint is the requirement that the aggregated EV profile is equal to the sum of all the EV profiles at each time step: xa =

N EV X

xi .

(2.2)

i=1

This type of formulation guarantees fairness and gives us a lot of flexibility. We can formulate an individual optimization problem for the aggregator and EVs representing their personal goals and couple this optimizations with one simple coupling constraint. The formulation of the specific optimization problem for the aggregator and the individual EV depends on the assumptions made for the EV management, e.g. the coupling concept. If the cost function for the aggregator and the EVs are convex functions and their constraint sets are convex, then the EV management optimization is a convex problem and we can obtain a global optimal solution for it [SV03]. The challenge is to

2.3. THE OPTIMAL EV MANAGEMENT PROBLEM

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design an aggregator and EV optimization problem that accurately represent the real world behavior without making the problem intractable. Therefore, we should aim for a convex formulation of the optimization problem. The EV management problem can be solved in a centralized manner. However, as discussed in section 1.2, this is impracticable, due to the high computation and communication cost. Therefor our goal is to produce a decentralized algorithm that solves this problem.

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Chapter 3 Distributed Optimization for EV Management In this chapter a decentralized method for optimal EV management is presented. First, a review on the subject of distributed optimization is made in oder to select the algorithm that better suits our goals. Later the algorithm for decentralized EV management is presented and derived.

3.1

Choosing a method for distributed optimization

There are several degrees of distribution of an optimization problem. An excellent review on the subject of distributed optimization can be found in [YJ10]. In this work we define the degree of distribution of an optimization algorithm based on the degree of information dispersion. The latter refers to the information about the whole system. In a centralized optimization there is only one computing unit and the information of the whole system needs to be collected in one location, therefor the degree of information dispersion is low. The more distributed the optimization, the more computing units we have with ever less information about the whole system, e.g. the information becomes disperse. In Fig. 3.1 we see different levels of distributed optimization. The classical approach to distributed optimization has been decomposition. Based on the specific structure of the objective function and constraints, the problem is decomposed into a number of subproblems. These subproblems can be solved independently, but typically require a coordinator to ensure that the local decisions converge to the global optimum [YJ10]. This coordinator normally acts as a gather and scatter. It gathers the results of the subproblems and then scatters a coordination signal. Given the fact that the coordinator gathers all the decision information in one location, we do not consider this approach to be distributed. Therefore, we define the approach as a decentralized optimization.

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CHAPTER 3. DISTRIBUTED OPTIMIZATION FOR EV MANAGEMENT

Centralized optimization

Distributed optimization

Decentralized optimization Separate coordinator

Decomposition

Non-cooperative optimization Eliminate coordinator

Coordinator

Low information dispersion: Information concentrated in single location.

Information dispersion High information dispersion: Information distributed in many locations.

Figure 3.1: Schematic illustration of optimization distribution levels based on information dispersion. There are several situations where the implementation of this centralized coordinator is undesirable or infeasible. For example if communication is only allowed between neighbors. For these cases we need to separate the coordinator in order to obtain a fully distributed optimization algorithm. These algorithms need to respect a predefined communication structure. If the coordinator structure is separable then the formulation of a distributed optimization is trivial. But, when the coordinator structure is not separable, distributed algorithms to scatter global information over a network are required. A widely used method for this is distributed averaging also known as the consensus algorithm. There are some applications where it is desirable to avoid coordination all together, either because it is unlikely that the nodes will actually cooperate, or as a way to eliminate complex coordination protocols and associated traffic overhead [YJ10]. For this case there are several techniques based on non-cooperative game theory. The latter allows the design of individual cost functions for each node that lead to the optimization of the original problem. For the design of our EV management framework we decide to use a decentralized optimization, that means a decomposition approach. Although the fully distributed and non-cooperative optimization offer less communication complexity, the uncertainty about the state of the whole system can be a problem. We have to remember that in the fully distributed and non-cooperative optimization none of the nodes has the aggregated decision information. For our problem this means that there is no place where we can know about the aggregated behavior of EVs as this information is spread through the nodes. In this work we consider that a decentralized optimization is an acceptable trade-off between complexity and uncertainty. Decentralized optimization bases on decomposition techniques. There are several decomposition methods. The decision to use specific method depends on the structure and characteristics of the problem, i.e. the type of coupling between the nodes and the convexity of the problem. Let us quickly review how this criteria are de-

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3.1. CHOOSING A METHOD FOR DISTRIBUTED OPTIMIZATION

fined. The type of coupling in an optimization problem for the case of two nodes can be seen in Table 3.1. Table 3.1: Type of coupling in the optimization of two nodes Coupling type Resulting problem structure

Coupling cost and coupling constraints min f (x1 , x2 ) s.t. g(x1 , x2 ) = 0

Coupling cost

min f (x1 , x2 ) s.t. g1 (x1 ) = 0 g2 (x2 ) = 0

Coupling constraints

min f1 (x1 ) + f2 (x2 ) s.t. g(x1 , x2 ) = 0

The criteria of convexity is also very important. A function f (x) is said to be convex if f (θx + (1 − θ)y) ≤ θf (x) + (1 − θy)f (y) and strictly convex if f (θx + (1 − θ)y) < θf (x) + (1 − θy)f (y) for x 6= y and 0 < θ < 1 [SV03]. Some of the methods for decentralized optimization only converge if the cost function of the optimization problem is strictly convex. This is quite a restriction, since important optimization problems, like linear programs, have convex but not strictly convex cost functions. In Table 3.2 we show a comparison of several popular algorithms for decentralized optimization and characterize their ability to deal with different coupling structures and convexity levels. As we can see the Alternating Direction Method of Multipliers (ADMM) offers a decentralized optimization algorithm that works for a wide range of problems. We take advantage of this to make our EV management framework as general as possible. A full review on the ADMM method can be found in [BPC+ 11] and also in Appendix A we review the important characteristics of this method.

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3.2

Strictly convex cost

x x

x x x x

x x x x

Non-strictly convex cost

Coupling cost

x x

Coupling constraints

[BT97] [NO09] [Boy11b] [BPC+ 11]

Coupling constraints and cost

Algorithm Gradient projection algorithm Subgradient projection algorithm Dual Decomposition ADMM

Reference

Table 3.2: Comparison of algorithms for decentralized optimization

x x

Decentralized EV Management with ADMM

In the last section we came to the conclusion that ADMM offers a decentralized method to solve a wide variety of optimization problems. We take advantage of this to formulate a distributed solution of the optimal EV management problem. First we formulate the problem in its canonical form, the exchange problem. Then we use the ADMM solution for the exchange problem in order to formulate an ADMM solution to the optimal EV management problem.

Lets remember the definition of the EV management optimization problem from section 2.3 :

P EV fa (xa ) + γ N i=1 fi (xi ) xa ,xi PNEV subject to xa = i=1 xi xa ∈ Xa xi ∈ Xi ; i = 1, . . . , NEV ,

minimize

(3.1)

21

3.2. DECENTRALIZED EV MANAGEMENT WITH ADMM

Variable T NEV xa xi fa (xa ) fi (xi ) Xa Xi γ

Description Number of time slots Number of available EVs Aggregated EV profile for all time slots Profile of a single EV for all time slots Cost function of the aggregator Cost function of the EV i Constrains set of the aggregator Constrains set of EV i Goals trade-off parameter

Type Scalar Scalar Vector ∈ RT Vector ∈ RT Convex function Convex function Convex set Convex set Scalar

This problem can be formulated as an exchange optimization problem. In order to do this we consider the EVs and the aggregator as agents and define the number of agents as: N = NEV + 1 (3.2) The aggregator is redefined as agent N, where: xN = −xa . The cost function of agent N is: fa (−xN ) fN (xN ) = ∞

(3.3)

if − xN ∈ Xa otherwise

The EVs are agents i = 1, . . . , N − 1 and their cost function is: γfi (xi ) if xi ∈ Xi fi (xi ) = ∞ otherwise

(3.4)

(3.5)

With this definition we can formulate the optimal EV management problem as the exchange problem: minimize xi

subject to Variable N xi fi (xi ) yi x

PN

i=1

PN

fi (xi )

i=1 xi = 0.

(3.6)

Description Type Number of agents Scalar Agent’s i amount bought or sold of the common good Vector ∈ RT Cost function of agent i Convex function Lagrangian variable/ price of the common good Scalar Average of all the agents Scalar

It is easy to see that the exchange optimization problem (3.6) is equal to the optimal EV management problem (3.1). The exchange problem considers N agents exchanging a common good under an equilibrium constraint. The exchange problem is a

22


canonical problem, that has many applications. In economics this formulation is used to solve marked equilibrium problems. Therefore with our formulation we basically build market between the EVs and the aggregator. The solution of the exchange problem using ADMM is presented in [Boy11a]: + xk+1 ||22 xk+1 = minxi fi (xi ) + yik xi + 2ρ ||xi − xk+1 i i yik+1 = yik + ρxk+1

(3.7)

and in scaled form using u = yi /ρ: xk+1 = minxi fi (xi ) + ρ2 ||xi − xki + xk + uk ||22 i uk+1 = uk + xk+1

(3.8)

P where x = 1/N N i=1 xi , e.g. the average of all the agents. The details on how this solution is obtained are explained in 3.2.1. With the ADMM result for the exchange problem (3.8) and taking into account the definition of the agents from formulas (3.3), (3.4) and (3.5),we get the solution to our problem. ADMM solution to the optimal EV management problem • For each EV i = 1, . . . , N − 1 : xk+1 i

=

minimize xi

γfi (xi ) + ρ2 ||xi − xki + xk + uk ||22

subject to xi ∈ Xi

(3.9)

• For the aggregator:

xk+1 N

=

minimize xN

fa (−xN ) + ρ2 ||xN − xkN + xk + uk ||22

subject to −xN ∈ Xa

(3.10)

• For the coordinator, which can also be the aggregator: P k+1 xk+1 = N1 N i=1 xi uk+1 = uk + xk+1

(3.11)

Stopping criteria. As we define in our review of the ADMM method in Appendix A.3, the stopping criteria for ADMM is given by the primal r k feasibility and dual feasibility sk . With z k = xk − 1xk we define: r k = 1xk sk = −ρN(xk − xk−1 + 1(xk−1 − xk )).

(3.12) (3.13)

23


where 1 is the identity matrix 1 ∈ RN ×N . The stopping criteria is: ||r k ||2 ≤ εpri ||sk ||2 ≤ εdual

(3.14)

where εpri > 0 and εdual > 0. The structure of using the ADMM method for optimal EV management problem can be seen in Fig. 3.2. ADMM

Figure 3.2: Structure of the optimal EV management ADMM solution.

3.2.1

Deriving the ADMM solution

For the interested reader, we now explain step by step how the ADMM solution to the exchange problem is obtained. Such a derivation is missing in the literature. Without loss of generality we assume that the xi s are scalars. In order to apply ADMM the exchange problem (3.6) is reformulated as : PN minimize i=1 fi (xi ) + g(z) xi ,z

s.t

xi = zi ;

where g(z) =

0 ∞

i = 1...,N

PN if i=1 zi = 0 otherwise

(3.15)

(3.16)

It is easy to see that the formulation (3.6) and (3.15) are equivalent. In order to solve this problem we first define the augmented Lagrangian function: Lρ (x, z, y) =

N X i=1

ρ fi (xi ) + g(z) + y T (x − z) + ||x − z||22 2

24


where ρ is the penalty parameter of the augmented term and y = [y1 , . . . , yN ]T is the vector of the Lagrangian variables. In order to solve the problem we need to minimize over the primal variables x, z and maximize over the Lagrangian variables y [Ber04b]: max min Lρ (x, z, y). y

x,z

This problem can be solved using an iterative process: xk+1 = min Lρ (x, z k , y k ) x

z

k+1

y

k+1

= min Lρ (xk+1 , z, y k ) z

= max Lρ (xk+1 , z k+1 , y). y

PN fi (xi ) + y kT (x − z k ) + 2ρ ||x − z k ||22 = If we expand xk+1 we get: xk+1 = i=1 PN ρ k k k 2 i=1 fi (xi ) + yi (xi − zi ) + 2 ||xi − zi ||2 . This problem is separable and we can write a single problem for each xk+1 . The iterative solution then can be written as: i ρ xk+1 = min fi (xi ) + yik (xi − zik ) + ||xi − zik ||22 i xi 2 ρ k+1 k+1 kT − z||22 z = min g(z) − y z + ||x z 2 y k+1 = max y T (xk+1 − z k+1 ). y

(3.17) (3.18) (3.19)

The update of z has an analytical solution and a gradient algorithm can be used to solve the y update. Let us derive this solution: • Solution to z k+1 : Taking into account the definition of g(z) in formula (3.16) we can formulate the optimization problem: PN k ρ k+1 2 minzi −y z + (x − z ) i i i i 2 Pi=1 (3.20) N s.t. z = 0 i i=1 The Lagrangian of this problem is: L(z, λ) =

N X i=1

N n o X ρ k+1 k 2 L(zi , λ) = −yi zi + (xi − zi ) + λzi 2 i=1

where λ is the Lagrangian variable. To solve this problem we need to solve maxλ minz L(z, λ): !

∇zi L(zi , λ) = 0 → zi = xk+1 + i !

∇λ L(z, λ) = 0 →

N X i=1

zi = 0, .

yik λ − ρ ρ

(3.21) (3.22)

25


With (3.21) in (3.22): N X yik λ k+1 =0 − xi + ρ ρ i=1 Solving after λ we get: λ = ρ(xk+1 +

yk ) ρ

P P k+1 k and y k+1 = 1/N N where xk+1 = 1/N N i=1 xi i=1 yi , e.g. the averages. Using this result into equation 3.21, we finally get the result: zik+1 = xk+1 − xk+1 + i

yik y k − ρ ρ

(3.23)

z k+1 = xk+1 − 1xk+1 +

yk yk −1 ρ ρ

(3.24)

or in vector form,

where 1 is the identity matrix. !

• Solution of y k+1: Done by setting ∇y Lρ (xk+1 , z k+1 , y) = 0. ∇y Lρ (xk+1 , z k+1 , y) = (xk+1 − z k+1 ). This cannot be solved for y, therefore we need to use a gradient method: y k+1 = y k + ρ(xk+1 − z k+1 ).

(3.25)

For this gradient decent method the step size ρ is exactly the same as the penalty parameter of the augmented term ρ. This step size is chosen in order to guarantee convergence. With this step size the first condition of the dual feasibility is always fulfilled [BPC+ 11]. However several extensions may include a variable step size ρk , for more details on this we refer the reader to chapter 4 in [Ber04b]. When we insert the z k+1 solution (3.23) in the y k+1 solution (3.25) the result is: y k+1 = ρ(1

yk + 1xk+1) ρ

Notice that the elements of y have the same value: y = 1yi y = yi

26


therefore, yik + xk+1 ) ρ = yik + ρxk+1

yik+1 = ρ(

Considering the latter in the z k+1 solution (3.23), the new solution becomes: zik+1 = xk+1 − xk+1 . i

(3.26)

and xk+1 from formula (3.17) is: i ρ xk+1 = min fi (xi ) + yik xi + ||xi − xki + xk ||22 . i xi 2

(3.27)

With this we arrive at the same solution for the exchange problem as in formulation (3.7): + xk+1 ||22 xk+1 = minxi fi (xi ) + yik xi + 2ρ ||xi − xk+1 i i yik+1 = y k + ρxk+1 We now proof that the scaled solution is equivalent to the original solution. Remember that u = y/ρ, then : ρ xk+1 = min fi (xi ) + ||xi − xk+1 + xk+1 + uk ||22 i i xi 2 ρ = min fi (xi ) + (x2i + 2(−xk+1 + xk+1 + uk )xi + (−xk+1 + xk+1 + uk )2 ) i i xi 2 ρ + xk+1)xi + (−xk+1 + xk+1 + uk )2 ) = min fi (xi ) + uk ρxi + (x2i + 2(−xk+1 i i xi 2 ρ k = min fi (xi ) + yi xi + ||xi − xk+1 + xk+1||22 + rest i xi 2 Notice that rest is a constant value and therefore not important to the optimization. Therefore, we get the same solution as in formulation (3.8): xk+1 = minxi fi (xi ) + ρ2 ||xi − xki + xk + uk ||22 i k+1 u = uk + xk+1

27

Chapter 4 The EV Management Framework and Case Studies In this chapter the decentralized EV management framework is presented. We first explain some of the characteristics of this framework and then present some case studies to show the flexibility and wide applicability of the proposed framework.

4.1

The EV Management Framework

In the last chapter we used the ADMM algorithm to solve the EV management problem. The presented framework is equivalent to the the general ADMM solution for decentralized EV management shown in Fig. 3.2. Aggregator

Incentive signal

For all EVs

Aggregated response

Figure 4.1: EV management framework for decentralized optimization with ADMM. This framework allows the independent definition of goals for the individual EVs and the aggregated EV profile. The proposed framework can be seen in Fig. 4.1. As we can see the aggregator defines an incentive signal to coordinate the aggregated behavior of the EVs. The incentive signal is composed of the scaled price u and the average profile of all

28

CHAPTER 4. THE EV MANAGEMENT FRAMEWORK AND CASE STUDIES

agents x. Lets rememberer that the number of agents is defined by the number of EVs and P the aggregator. The EVs response to the incentive signal is aggregated EV xEV = N1 N k=1 xi . This aggregated response is send back to the aggregator. The aggregator later adds its result for the optimization of its own variable for the aggregated EV demand xn = −xa to form the average of all agents x. With this value the aggregator then updates the scaled price u . Remember that the normal price y is defined by the scaled price as y = ρu. This process is done until convergence. The framework above fulfills the requirements we set in section 2.3: Dynamic optimization: The algorithm optimizes profiles for a defined time period. Decentralized: The optimization problem of the individual EVs and the aggregator are fully decoupled, and can be computed in parallel. Each of the agents optimizes its personal goals and is coordinated through a signal which guarantees fairness to all agents. Moreover, no agent knows about the internal structure of the other agents. This gives each agent a certain level of data privacy. Scalable: The incentive signal is the same for all EVs and the aggregator only needs the aggregated EVs response. The latter is very easy to calculate and transmit without any special knowledge about the individual EVs. This guarantees an algorithmically tractable message routing. Traversable and portable The communication messages are completely independent of the internal characteristics of the EVs and the aggregator. The optimization problems for the EVs and the aggregator can be defined separately and then following the protocol presented here the optimal result for the whole system can be obtained. In our decentralized EV management optimization a time frame of 24 hours with 15 min intervals is considered. In order for this framework to be useful we need to formulate optimization problems for the aggregator and the individual EVs that accurately portrait reality but do not pose intractable problems. We aim for a convex formulation of this problems that allow fast and global optimal results. To allow a trade off comparison between the aggregator and EVs goals, their cost function’s output should have the same unit. In our formulations the output of the cost functions should be monetary costs, i.e. with the unit [EUR]. In the next section we show possible formulations of this problems.

4.2

Aggregator optimization problem

The aggregator goals are defined by the requirements set on the aggregated EV behavior. This requirements can come from the ISO or a third party that is interested in a certain behavior of the EVs, e.g. a fleet manager. Here we define three

29

4.2. AGGREGATOR OPTIMIZATION PROBLEM

aggregators that represent some of the different concepts available for the control of an EV fleet.

4.2.1

Valley filling

In the valley filling problem the goal is to use the EVs to flatten the sum of the aggregated EVs demand and the fixed/base demand. Our formulation of the valley filling problem is similar to the one in [GTL11]. The main difference is that we multiply the cost by an empirical parameter in order to convert the squared of the aggregated demand into [EUR]. Moreover, in [GTL11] it is assumed that the EVs are deferrable loads, i.e. they do not optimize personal goals and are not allowed to discharge. In our case study we release this assumptions. The optimization for the aggregator is: minimize δ||D + xa ||22

(4.1)

xa

Variable T xa D δ

Description Number of time slots Aggregated EV profile Profile of the base demand Empirical price per demand parameter

Type Scalar Vector ∈ RT Vector ∈ RT Scalar

where the base/fixed demand D is assumed to be known and the price demand parameter δ is empirically obtained by dividing average of the scaled price and the average of the based demand: p δ= (4.2) D P P where p = 1/T Tt=1 pt and D = 1/T Tt=1 Dt . The scaled electricity price p with the unit [EUR/kW] is the original electricity price given in [EUR/MWh], scaled to [EUR/kWh] and then multiplied by the time slot duration ∆t. With formula 4.2 we obtain an empirical parameter δ that has the right unit [EUR/kW2 ]. In our framework the aggregator problem becomes: xk+1 = N

minxN δ||D − xN ||22 + 2ρ ||xN − xkN + xk + uk ||22

(4.3)

This problem can be solved analytically and its solution is: xk+1 = N

ρ (xkN ρ−2δ

− xk − uk ) −

2δ D ρ−2δ

(4.4)

30


4.2.2

Price based

In the case of a price based optimization the goal is to optimize a fleet of EVs so that the aggregated fleet behavior causes minimal costs under constraints on the maximal and minimal allowed aggregated power for a given time horizon. In [SB10] the aggregated charging is constraint by the available renewable energy (RE) generation. In this previous work the EVs are not allowed to discharge. In our case study we release this assumption and allow the EVs to discharge. Our problem for the aggregator is: minimize pT xa

(4.5)

xa

s.t. Variable T p xa xa xa

xa ≤ xa ≤ xa

Description Number of time slots Scaled electricity price profile Aggregated EV profile Aggregated maximal energy feed back Aggregated maximal consumption

Type Scalar Vector ∈ RT Vector ∈ RT Negative vector∈ RT Positive vector ∈ RT

The maximal aggregated demand profile xa , the maximal energy profile that can be fed back into the grid xa and the scaled electricity price p are assumed to be known. Note that the original electricity price given in [EUR/MWh] needs to be scaled and multiply by ∆t to get a scaled price in [EUR/kW]. The availability of renewable energy can be expressed in the maximal aggregated demand xa . This would guarantee that the EVs do not charge more energy than the available renewable energy at any point in time. In our framework the optimization problem of the aggregator is: xk+1 = N

4.2.3

minxN −pT xN + 2ρ ||xN − xkN + xk + uk ||22 s.t. −xa ≥ xN ≥ −xa

(4.6)

Direct coupling

The direct coupling of RE and EVs as explained in section 1.1 considers that RE generation has been purchased to serve the EV demand. If the RE generation is not able to satisfy the EV demand, the required energy may be purchased from the market. In our formulation RE generation excess is lost. Therefore a careful matching should be made between RE production and the EV fleet demand. The aggregator’s role herein is to match EV demand and RE generation, so that the cost of market purchased energy is minimized [Pap11]. For this case we define the following optimization problem for the aggregator:

31

4.3. EV OPTIMIZATION PROBLEM

minimize pT xp

(4.7)

xp ,xa

s.t.

Variable T p xp xa re

re + xp ≥ xa

Description Number of time slots Scaled electricity price profile Electricity purchase profile Aggregated EV profile Renewable energy profile

Type Scalar Vector Vector Vector Vector

∈ RT ∈ RT ∈ RT ∈ RT

where the renewable energy profile re and scaled energy price p are known. Note that the original electricity price given in [EUR/MWh] needs to be scaled and multiplied by ∆t to get a scaled price in [EUR/kW]. We introduce a slack variable yslack ∈ RT in order to transform the inequality constraint of our problem into an equality constraint. The problem then becomes: minimize pT xp xp ,xa ,yslack

s.t.

re + xp = xa + yslack yslack ≥ 0

(4.8)

Considering that xp = yslack − re − xN and after some mathematical reformulation the aggregator optimization problem in our framework becomes:

xk+1 = N

4.3

minxN ,yslack pT (yslack − xN ) + ρ2 ||xN − xkN + xk + uk ||22 s.t. yslack ≥ 0

(4.9)

EV optimization problem

In this thesis the personal EV i = 1, . . . , NEV optimization problem is defined as: minxi αi ||xi ||22 s.t Ri ≤ Ai xi ≤ Ri S i ≤ Bi xi ≤ S i xi ≤ xi ≤ xi

(4.10)

32


Variable T ci Tci αi xi Ai Ri Ri Bi Si Si xi xi

Description Number of time slots Number of times connected Number of time slots connected Battery depreciation parameter EV charging profile Connection matrix Minimal charging requirements Maximal charging requirements Input matrix Minimal state vector Maximal state vector Maximal charging rate Minimal charging rate

Type Scalar Scalar Scalar Scalar Vector ∈ RT Matrix ∈ Rci ×T Vector ∈ Rci Vector ∈ Rci Matrix ∈ RTci ×T Vector ∈ RTci Vector ∈ RTci Vector ∈ RT Vector ∈ RT

The first inequality constraint, Ri ≤ Ai xi ≤ Ri , sets a bound on the energy requirements of the EV for each time it is connected. This energy requirements depend on the driving profile and charging strategy of the EV. The second inequality constraint, S i ≤ Bi xi ≤ S i , comes from the state equation of the EV’s battery. It guarantees that the state of the battery is kept on a desired operation level for each time slot. Notice that the latter inequality can be ignored in the case where the EV is not allowed to discharge, since in this case the battery state constraints are included in the energy requirements. The last constraint, xi ≤ xi ≤ xi , defines a maximal and minimal charging/discharging rate. The EV optimization in our framework becomes then:

xk+1 = i

minxi γαi ||xi ||22 + ρ2 ||xi − xki + xk + uk ||22 s.t Ri ≤ Ai xi ≤ Ri S i ≤ Bi xi ≤ S i xi ≤ xi ≤ xi

(4.11)

The optimization of the EV is based on its personal goals and the incentive signal. The scaled price u can be considered as the energy price that the aggregator defines for the EVs. The mean value of all agents x can be considered as a social cost caused by the EVs not cooperating to achieve global convergence. The individual EV optimization takes this prices into account. Since this is an optimization that minimizes the charging costs of the EV, it can be characterized as smart charging. We now explain how we arrive at this formulation. The EV has three principal components that influence its decisions: its battery, its driving behavior, and its charging strategy. Since this applies for all EVs let us for the moment ignore the EV index number i.

33


4.3.1

Battery model

Battery modeling is a very active field of research. A good review on different battery models can be found in [KQ11]. Most of the battery models available are highly non linear and therefore not convex. Since we are looking for a convex formulation of the individual EV optimization problem in this work a very simple battery model is used. The most important states of a battery are the State of Charge (SOC) given in [%] and the temperature given in [K]. We ignore the dynamics of temperature and focus on the SOC. The SOC represents the amount of charge time available in the battery divided by the nominal charge capacity of the battery Capnom given in [Ah]. For this model the assumption is made that the battery has a constant voltage during operations defined as Vnom . The SOC state equation is: ˙ =η I SOC Capnom where η is an efficiency parameter and I is the current going into the battery in [A]. Another way to interpret the state of charge is through the State of Energy (SOE): ˙ =η SOE

IVnom IVnom =η Capnom Vnom Enom

where Enom is the nominal energy content of the battery. If we discretize this equation we can formulate a discrete state model of the battery: I k Vnom ∆t, Enom where k is the time slot number. Since we consider the voltage constant, we know that P = Vnom I and the state equation can be written as: SOE k+1 = SOE k + η

SOE k+1 = SOE k + η

Pk ∆t Enom

In our optimization problem we define the decision variable as xk = P k . With this definition we can write our state equation as: SOE k+1 = SOE k +

η∆t k x . Enom

Multiplying this equation with Enom we get the state equation in terms of energy: E k+1 = E k + η∆txk

(4.12)

k The bounds on the state of the battery can be defined in energy terms as Emin and k Emax . For each time step the EV has the bounds: k+1 E k+1 − E k Emin − Ek ≤ xk ≤ max η∆t η∆t

(4.13)

34


Notice that Emin and Emax change with time. This is due to the energy consumed by the battery on each trip which in our formulation can not be controlled by the decision variable. The number of occasions when the EV is connected is defined as c ∈ R. Knowing the number of time slots that the EV is connected Tc , we can write the state constraints in vector form: S ≤ Bx ≤ S

(4.14)

where B ∈ RTc ×T is the input matrix and, S ∈ RTc and S ∈ RTc represent the maximal and minimal states for each time slot. Now let us consider the EV charging requirements. Depending on the EV’s charging strategy the EV may have minimal and maximal energy requirements each time it charges. Lets suppose that for one time charging the car parks, starts charging at time slot tc and stops charging at td . Using the state equation in formula (4.12), defining E des,c as the minimal energy requirement and E des,c as the maximal energy requirement for time parked c, we can write the constraint: Pd xk ≤ E des,c E des,c ≤ E tc −1 + η∆t tk=t c E des,c − E tc −1 ≤ η∆t

Ptd

k k=tc x

≤

E des,c − E tc −1 η∆t

(4.15)

Since we know the number of times that the EV is connected, we can write this constraint in vector form: R ≤ Ax ≤ R (4.16) where A ∈ Rc×T is the connection matrix, and the energy requirements are encoded in the vectors R ∈ Rc as minimal charging requirements and R ∈ Rc as maximal charging requirements. This constraints also guarantees the state constraints of the battery in the case where the EV is not allowed to discharge. Therefore, we can ignore the state constraints defined in formula (4.14) for this case. The last constraint imposes a maximal and minimal charging rate on the battery: x≤x≤x

(4.17)

where an element of vectors xt , xt = 0 if the EV is not connected at time slot t. A definition of the parameters B, S, S, A, R, R, x and x depending on the driving profile and the charging strategy is given in the next section. Another important aspect for the battery is the cost of aging. The battery is considered one of the most expensive elements of the EV. On this account, it is in the interest of the EV to optimize its charging/discharging, so that the cost of aging is minimized. Previous works on this subject assume a linear cost of charging ,e.g.

35


αi xi . However, this assumption is not accurate since charging with a higher power leads to higher depreciation of the battery with a non-linear correlation. In this work we model this through a quadratic cost function: f = α||x||22

(4.18)

where α is an empirical parameter related to the depreciation cost of the battery. With the definition of the cost (4.18) and the constraints (4.14), (4.16) and (4.17) we obtain our EV optimization formulation as in (4.10): minx α||x||22 s.t R ≤ Ax ≤ R S ≤ Bx ≤ S x≤x≤x

4.3.2

Driving profile and charging strategy

We assume that the EV knows its driving profile and the energy required on each trip for the whole optimization period, e.g. 24 hrs. Since the same parameter definition concept applies to all EVs, let us ignore the EV index number i. The driving profile is defined by a row vector d ∈ RT , where an element of this vector dt is 0 if the EV is not connected and 1 if the EV is connected to the grid at time slot t. The number of times that the EV is connected is defined by c. With this the amount of energy required for the trips can be defined in the vector Ereq ∈ Rc . The definition of the parameters B, S, S, A, R, R, x and x depends on the driving profile and the charging strategy that the EV follows. We consider sets of EVs with three different charging strategies: home charging, greedy charging and minimal energy charging. Using the driving vector d we can define the maximal and minimal charging rate as: x = dxmax x = dxmin where xmax and xmin are the maximal charging and discharging power. Under normal operation the EV can only use a percentage of the battery’s nominal energy capacity Enom , we call this effective energy capacity Ebat . If the battery is at its lowest operational charge level, then Ebat would have to be charged to reach the highest operational charge level of the battery. Let us now explain the definition of this parameters for the different charging strategies with the help of a simple example. In our formulation we make the assumption that the battery is fully charged before the first trip. Furthermore, we make the

36


requirement that the battery must be fully charged before the first trip on the next day. The results can be traced by using the battery definitions of section 4.3.1. Home charging: In this strategy the EV charges only at home. For this strategy the EV must be able to fulfill its driving requirements for the whole day with one battery charge. Some car users exceed this range and will not be able to apply this strategy. However, most city car users drive an average of 40 km which is well within the reach of current EVs. A simple example on how to define the EV optimization parameters for this charging strategy can be seen in Fig. 4.2.

Figure 4.2: Example definition of the parameters for a home charging EV. Greedy charging: In this strategy the EV tries to fully charge its battery every time it is not driving. This assumes that the EV is able to connect to the grid every time it parks. This strategy is not likely to be use by EV owners. However, if we think of a car-sharing fleet of EVs this charging strategy becomes feasible in the real world. In car-sharing it would make sense to make one trip with the EV and then connect it so that it tries to fully charge its battery for


37

the next costumer. A simple example on how to define the EV optimization parameters for this charging strategy can be seen in Fig. 4.3. Note in the energy requirements we might have the case where the EV is connected, but the EV can not fully recharge its battery in the available time. For this cases the EV charges the most it can and backlogs the required energy for full charge for the next time it is connected.

Figure 4.3: Example definition of the parameters for a greedy charging EV.

Minimal energy charging: This is the smartest strategy, where each time the EV is connected it tries to charge the minimal amount of energy that it requires for the next trip. This strategy has a great potential of reduce the charging costs of the individual EV. However, it may be a bad strategy if the EV user’s driving profile is not predicted with high accuracy. A simple example on how to define the EV optimization parameters for this charging strategy can be seen in Fig. 4.4

38


Figure 4.4: Example definition of the parameters for a minimal energy charging EV.

4.4

Case Studies

The proposed EV management framework allows an independent formulation of the individual EV and aggregator optimization problems. This gives the framework a high level of flexibility and allows the easy application of a wide range of EV management concepts. Here we demonstrate the effectiveness and flexibility of the proposed framework in the simulation of different EV management concepts using realistic data. We consider three EV management concepts for 100 EVs: valley filling, price based and direct coupling. For each of this concepts we also include the possibility of having V2G services, i.e. allow the EVs to discharge. We simulate three charging strategies defined in 4.3.2: home, greedy and minimal energy charging. The simulation of 1 different individual EV goals importance level is also considered, e.g. γ = 0, 100 , 1. The latter trade off parameter γ balances between the energy costs incurred by the aggregator and the aging costs of the EVs. In our joint optimization formulation

39

4.4. CASE STUDIES

the total cost is: Total cost = Aggregator’s energy cost + γ

N EV X

EV’s i aging cost

i=1

The relevant simulation data of each management concept and the EVs can be found in Appendix B. For the simulations we consider the Siemens movE electric vehicle. The trip patterns come from the driving National Household Travel Survey (NHTS) data. The collected dataset is publicly available and contains trip data reported by more than 150,000 U.S. households, [Goe12]. The data was parsed in order to fit the specifications of our EV and its charging strategy. The demand and renewable energy production profiles were obtained from the Munich distribution system operator website, [SWM12]. The profiles were scaled for 100 EVs. Finally, the energy price signal was obtained from the European Energy Exchange (EEX) website, [Exc12]. The simulation environment was MATLAB. The simulations are synchronous and sequential, i.e. the agents are optimized one a time. One ADMM iteration is carried out once all agents have optimized their subproblem. For the optimizations CVXGEN was used. CVXGEN generates fast custom C code for small, QP-representable convex optimization problems [MB12]. However, the generator failed for the individual EV optimization problem, where V2G services are allowed. This is due to the fact, that when the EVs are allowed to discharge the state constraints need to be included into the problem. This considerably increases the amount of constraints to a number that CVXGEN can not handle. Therefore, to solve the latter the MATLAB native solver lsqlin was used. As we will determine in section 4.5 this has a great impact on the computational performance. In spite of this, our case studies are accurate, since the results are independent of the solver. To speed up the convergence of ADMM we use a variable penalty parameter ρk described in Appendix A.4. Now let us present the simulations results. An objective comparison between the different EV management concepts goes beyond this work. However, in section 4.5 we make a couple of observations and recommendations for each of the EV management concepts.

4.4.1

Valley filling

In the valley filling management concept the goal is to use the EVs to fill the overnight valley in the fixed demand, in order to obtain a flat aggregated demand. Notice how allowing V2G services results in an almost flat aggregated demand profile when the EVs individual goals are ignored, i.e. γ = 0 for all charging strategies.

40


EV charging home

Valley filling for EVs charging home

EV connected 600

0 Aggregated demand [kW]

550 10

20

30

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350

Total demand, γ=0, Cost=674.6303 Total demand, γ=1/100, Cost=675.2076

300

Total demand, γ=1, Cost=713.3238

250

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Figure 4.5: Valley filling results for 100 EVs charging home with and without V2G. Left the driving profiles. Right the aggregated demand of base demand and EVs. Upper right only charging and lower right for discharging (V2G). Both cases consider different levels of individual EV goals importance, γ = 0, 0.01, 1.

EV charging greedy

Valley filling for EVs charging greedy

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Figure 4.6: Valley filling results for 100 EVs charging greedy with and without V2G. Left the driving profiles. Right the aggregated demand of base demand and EVs. Upper right only charging and lower right for discharging (V2G). Both cases consider different levels of individual EV goals importance, γ = 0, 0.01, 1.

41

4.4. CASE STUDIES

EV charging minimal energy

Valley filling for EVs charging minimal energy

EV connected

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Figure 4.7: Valley filling results for 100 EVs charging minimal energy with and without V2G. Left the driving profiles. Right the aggregated demand of base demand and EVs. Upper right only charging and lower right for discharging (V2G). Both cases consider different levels of individual EV goals importance, γ = 0, 0.01, 1.

42

4.4.2


Price based

In price based optimization the goal is to minimize the charging cost of an EV fleet and maintain the demand/energy feedback within a bound. The aggregator modifies the energy price it gives to the EVs to achieve this. Notice that the price only changes if the EVs meet the bound. EV charging home

Energy price

EV connected 0.08

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Figure 4.8: Price based results for 100 EVs charging home without V2G. Left the driving profiles. Upper right new prices defined by the aggregator for the EVs. Lower right the aggregated EVs demand. Both for different levels of individual EV goals importance, γ = 0, 0.01, 1. EV charging home

Energy price

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Figure 4.9: Price based results for 100 EVs charging home with V2G. Left the driving profiles. Upper right new prices defined by the aggregator for the EVs. Lower right the aggregated EVs demand. Both for different levels of individual EV goals importance, γ = 0, 0.01, 1.

43

4.4. CASE STUDIES

EV charging greedy

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Figure 4.10: Price based results for 100 EVs charging greedy without V2G. Left the driving profiles. Upper right new prices defined by the aggregator for the EVs. Lower right the aggregated EVs demand. Both for different levels of individual EV goals importance, γ = 0, 0.01, 1.

EV charging greedy

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Figure 4.11: Price based results for 100 EVs charging greedy with V2G. Left the driving profiles. Upper right new prices defined by the aggregator for the EVs. Lower right the aggregated EVs demand. Both for different levels of individual EV goals importance, γ = 0, 0.01, 1.

44



Energy price

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Figure 4.12: Price based results for 100 EVs charging minimal energy without V2G. Left the driving profiles. Upper right new prices defined by the aggregator for the EVs. Lower right the aggregated EVs demand. Both for different levels of individual EV goals importance, γ = 0, 0.01, 1.


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Figure 4.13: Price based results for 100 EVs charging minimal energy with V2G. Left the driving profiles. Upper right new prices defined by the aggregator for the EVs. Lower right the aggregated EVs demand. Both for different levels of individual EV goals importance, γ = 0, 0.01, 1.

45

4.4. CASE STUDIES

4.4.3

Direct coupling

In direct coupling renewable energy has been purchased to service the EVs demand. The aggregator modifies the price, as an incentive for the EVs to absorb this renewable energy. This lowers the aggregator’s energy purchase costs. Notice how the new price is lowered for the periods of high renewable energy availability. EV charging home

EV connected

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Figure 4.14: Direct coupling results for 100 EVs charging home without V2G. Left the driving profiles. Upper right the energy price. Lower right the available renewable energy, the aggregated EVs demand and the aggregator purchased energy considering different levels of individual EV goals importance, γ = 0, 0.01, 1.

EV charging home

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Figure 4.15: Direct coupling results for 100 EVs charging home with V2G. Left the driving profiles. Upper right the energy price. Lower right the available renewable energy, the aggregated EVs demand and the aggregator purchased energy considering different levels of individual EV goals importance, γ = 0, 0.01, 1.

46


EV charging greedy

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Figure 4.16: Direct coupling results for 100 EVs charging greedy without V2G. Left the driving profiles. Upper right the energy price. Lower right the available renewable energy, the aggregated EVs demand and the aggregator purchased energy considering different levels of individual EV goals importance, γ = 0, 0.01, 1.

EV charging greedy

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Figure 4.17: Direct coupling results for 100 EVs charging greedy with V2G. Left the driving profiles. Upper right the energy price. Lower right the available renewable energy, the aggregated EVs demand and the aggregator purchased energy considering different levels of individual EV goals importance, γ = 0, 0.01, 1.

47

4.4. CASE STUDIES


Energy price

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Figure 4.18: Direct coupling results for 100 EVs charging minimal energy without V2G. Left the driving profiles. Upper right the energy price. Lower right the available renewable energy, the aggregated EVs demand and the aggregator purchased energy considering different levels of individual EV goals importance, γ = 0, 0.01, 1.


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Figure 4.19: Direct coupling results for 100 EVs charging minimal energy with V2G. Left the driving profiles. Upper right the energy price. Lower right the available renewable energy, the aggregated EVs demand and the aggregator purchased energy considering different levels of individual EV goals importance, γ = 0, 0.01, 1.

48


4.5

Observations and discussion

In the last section 4.4 we showed the simulation results of the case studies. We now discuss some observations that apply to all cases and explore the individual cases. Finally, we present observations on the simulations performance. We want to point out that a quantitative comparison between the different EV management concepts goes beyond this work. A true comparison would needed to take socio political elements into account. Also it is important to mention that within a case a comparison based on cost would not be accurate, since the EV goal’s importance parameter γ affects the cost function. Moreover, the datasets for the different charging strategies are different. The 100 home charging EVs require less energy than the 100 greedy and minimal energy charging EVs. Therefore, we limit ourselves to qualitative observation. Observations that apply to all cases. We notice some general patterns that apply to all cases, i.e. valley filling, price based and direct coupling: • The minimal energy charging strategy performs very good or is the best for all cases. Specially when we allow V2G services and γ = 0. This was something expected, since the minimal energy charging strategy allows more freedom in the charging requirements. Remember that for this charging strategy the charging requirements are inequality constraints and not equality constraints as in home or greedy charging. • A smaller γ leads to lower costs for the aggregator and therefore a better aggregated behavior of the EVs. This is because a smaller γ means less importance of the individual EV goals, e.g. the EVs controllability by the aggregator is higher. • Allowing V2G services leads to lower costs. This is logical, since EVs are able to service each other, reducing the use external resources. • In the case where γ = 1 allowing V2G or not allowing V2G services does not make much difference in the total cost. This basically means that when the γ = 1 the participation of the individual EV in V2G services is very small, due to the high battery depreciation costs. Now let us look at the observations and some useful recommendations that can be made by looking at the results of each individual case. Observations for valley filling: • Best result: Minimal energy charging with γ = 0 produces a completely flat aggregated demand.

4.5. OBSERVATIONS AND DISCUSSION

49

• Worst result: Greedy charging with γ = 1. No valley filling and bursty aggregated demand profile. • Recommendations: Allow V2G and keep γ small, i.e. γ < 1. Greedy charging EVs is not recommended for this management concept, except if we allow V2G and make 1 >> γ ≥ 0. Observations for price based: • Best result: Minimal energy charging EVs with V2G and γ = 0. The new price is constant. • Worst result: Greedy charging EVs without V2G. Results have a very bursty consumption profile. • Recommendations: Allow V2G and keep γ at a medium value 0 > γ > 1 in order to have a balance between reaching the constrain bounds and reducing costs. The latter compromise is recommended, since in this scenario reaching the bound for long periods of time could mean stressing power lines for long periods of time. A highly undesirable system operation state. Observations for direct coupling: • Best result: Minimal energy charging EVs with γ = 0. Absorbs better the renewable energy and more energy is purchased at a lower cost. • Worst result: Home charging EVs. A lot of the purchased renewable energy is lost. • Recommendations: Allowing V2G makes no significant difference. Use a low trade off parameter (0.01 > γ ≥ 0) to guarantee the absorption of the purchased renewable energy. Observations concerning the ADMM algorithm performance. In Tables 4.1, 4.2 and 4.3, we can see the simulation performance metrics for the valley filling, price based and direct coupling case studies. It is important to point out that in our simulations for each ADMM iteration the agents were optimized in a sequential manner. The parallelization of this processes is straight forward and would greatly improve performance. From the performance metrics we can make the following observations: • Optimization of EVs without V2G is much faster than for EVs with V2G. The difference in simulation time is of two orders of magnitude 102 . This was already expected, since the individual EVs with V2G are optimized using the standard MATLAB solver lsqlin and in the EVs with V2G the solver is a custom C code produced by CVXGEN. This large difference points out to the fact that custom efficient solvers need to be used to solve the individual EV problems.

50


• A high value of γ leads to faster solutions. A possible explanation for this is that since the EVs build the majority of the agents, then giving their goals a high priority should logically lead faster to an equilibrium, i.e. a faster convergence. • The less constrained the aggregator is, the faster the convergence. Since the aggregator constraints represents global constraints in the system, it makes sense that the less global constraints the faster the convergence. In our case studies the level of constraints is: valley filling < direct coupling < price based. This correlates to the number of ADMM iterations required to achieve convergence. As we can see the trade off parameter γ plays a fundamental role on the behavior of the EVs. On the one hand, a lower γ leads to better behavior of the aggregated EVs. On the other hand, a higher γ leads to better performance of the ADMM algorithm and also lowers aging costs for the individual EVs. The definition of this parameter will depend on the particular interests of the framework’s user. Nevertheless, we recommend a low value for the trade off parameter 0 < γ