Pricing and Electric Vehicle Charging Equilibria Trivikram Dokka Jorge BrunoSonali SenGupta Chowdhury Mohammad Sakib Anwar

2025-05-06 0 0 1.14MB 21 页 10玖币

侵权投诉

Pricing and Electric Vehicle Charging Equilibria

Trivikram Dokka Jorge Bruno†Sonali SenGupta‡

Chowdhury Mohammad Sakib Anwar§

December 13, 2022

Abstract

We study equilibria in an Electric Vehicle (EV) charging game, a cost minimization

game inherent to decentralized charging control strategy for EV power demand man-

agement. In our model, each user optimizes its total cost which is sum of direct power

cost and the indirect dissatisfaction cost. We show that taking player speciﬁc price in-

dependent dissatisfaction cost in to account, contrary to popular belief, herding only

happens at lower EV uptake. Moreover, this is true for both linear and logistic dissatis-

faction functions. We study the question of existence of price proﬁles to induce a de-

sired equilibrium. We deﬁne two types of equilibria, distributed and non-distributed

equilibria, and show that under logistic dissatisfaction, only non-distributed equilibria

are possible by feasibly setting prices. In linear case, both type of equilibria are possi-

ble but price discrimination is necessary to induce distributed equilibria. Finally, we

show that in the case of symmetric EV users, mediation cannot improve upon Nash

equilibria.

Keywords : Electric Vehicles, Pricing, Nash equilibrium, Coarse correlated

equilibrium, Mediation, Herding, Dissatisfaction cost

JEL Codes: C61, C72, D4, D11, D82

Advanced Analytics Group, Air Products Plc, United Kingdom. Email: trivikram.dokka@yahoo.co.uk

†Department of Digital Technologies, Faculty of Business and Digital Technologies, University of Winch-

ester. Email:Jorge.Bruno@winchester.ac.uk

‡Economics Section, Queens Management School, Queens University Belfast, United Kingdom. Email:

s.sengupta@qub.ac.uk

§Author for correspondences. Department of Economics, University of Winchester, United Kingdom.

Email: Sakib.Anwar@winchester.ac.uk.

arXiv:2210.15035v2 [econ.TH] 12 Dec 2022

1 MOTIVATION AND RESEARCH QUESTIONS

Electric Vehicles (EVs) are widely seen as part of a solution to economically and environ-

mentally sustainable transportation future. With more countries looking to de-carbonize

their economies at an increased pace, more incentives to EV uptake are being proposed

and implemented. However, mass scale EV uptake comes with its own challenges, pri-

mary of them being impact on existing electricity infrastructure. Several researchers in-

vestigate economic and environmental implications of residential charging of electric ve-

hicles (Clement-Nyns et al., 2009; Muratori, 2018). It is widely believed that an efﬁcient de-

mand response is essential to avoid costly infrastructure upgrades and/or blackouts. This

involves alignment of EV charging demand with supply. Such an alignment not only avoids

costly and unnecessary capacity addition but also results in shift to renewable sources. Ini-

tial ideas to achieve this alignment, naturally, were based on incentivizing people via prices

(to charge at non-peak times)(Palensky & Dietrich, 2011; Western-Power-Distribution et

al., 2015). It is argued that the same price signals received by all EV users will result in

herding behavior where, all users shift their charging to low cost periods to avoid peak

periods, resulting in new peaks (Dudek et al., 2019; Valogianni et al., 2020; Western-Power-

Distribution et al., 2015). Herding behavior formation argued in these studies relies di-

rectly on the assumption that EV users are cost minimizers. Ironically, the observations

of herding are made without any reference to the level of EV uptake. In a typical herding

scenario (with high uptake), a EV user receives less power (kWh) than is expected when

charged at full speed, due to congestion or as part of a strategy to move users to different

time of day, hence causing dissatisfaction due to less battery charge received. Without tak-

ing this dissatisfaction in to account the conclusion that charging behaviors result in herd-

ing may not be consistent with expected behavior over time. The motivation for capturing

dissatisfaction explicitly is justiﬁed because price as an instrument to control charging be-

havior is only possible when users are (or are not) willing to pay to avoid dissatisfaction. In

fact, more recently user dissatisfaction is explicitly modeled within an algorithmic charg-

ing decision-making set-up (Lin et al., 2021). Similarly, Wu et al. (2022) uses the term in-

convenience cost in the same sense and illustrate optimal mechanisms for EV charging at

public stations. Consistent with this, our ﬁrst research question is:

Question 1 When users experience indirect costs associated with dissatisfaction will herd-

ing still happen?

Grid managers and DNOs would want to use the ﬂexibility of EV charging (believed

to be ﬂexible load compared to other loads such as household power demand) to achieve

a desired consumption proﬁle which better aligns with grid management objectives (see

Valogianni et al. (2020) and references therein). Recent research suggests designing a dy-

namic and adaptive pricing schemes to achieve a desired charging proﬁle (Jacobsen &

Stewart, 2022). Our goal differs from these works in that we seek to ﬁnd if price proﬁles

exist which will lead to a desired behavior proﬁle in equilibrium, and if so under what con-

ditions, taking into account congestion aka dissatisfaction. To the best of our knowledge

we are not aware of any work that takes into account dissatisfaction cost or analyzes equi-

librium outcomes. With this motivation, we address our second research question:

Question 2 Does there exist price proﬁles that will induce a desired charging proﬁle un-

der the given player-speciﬁc price independent dissatisfaction costs?

To answer our questions we take a game-theoretic approach to model EV user’s selﬁsh

behavior and use a stylized model that captures the key aspects of EV charging behavior

as an EV charging game. Centralized versus decentralized control of charging has received

much attention in EV related literature. Decentralized setting can be seen through the

game theoretic lens, an approach only taken by relatively few researchers compared to

much more abundant empirical studies. While earlier studies, such as Tushar et al. (2012),

considered a stackelberg approach, closer to our setting, simultaneous form games were

studied in Chakraborty et al. (2017) and Chakraborty and Khargonekar (2014). However,

no studies considered dissatisfaction cost within game-theoretic setting. Our work also

complements the alternative stream of literature that takes a mechanism design approach

to pricing problem, such as Nejad et al. (2017) and Wu et al. (2022).

Price-based or otherwise, the idea of (decentralized) controlled charging relies on the

assumption that EV users will ﬁnd themselves better off when an agency such as EV aggre-

gator acts as recommender system; under the belief that such an entity has greater (tech-

nological and informational) ability to make better charging recommendations compared

to EV users deciding on their own.1The role of mediating agency is certainly not unique to

this situation, and many economic situations, whether it be resource sharing or contribut-

ing towards a public good, also have this characteristic. Theoretically, such an entity will

recommend (or implement), with user’s consent, a charging regime which may or may not

satisfy user’s complete demand but may result in lower cost. But, what if users do not ﬁnd

following agency recommendations better than their own decisions? Therefore, a conﬁr-

mation of existence of such an entity is necessary via game theoretic analysis. This leads

to our ﬁnal question.

1A number of researchers proposed optimization algorithms under decentralized scenario, see Shen et al.

(2019).

Question 3 Will co-ordination (or co-ordinated mediation) help? In other words, if an

agency (aggregator/charging manager) acts as a recommender system of how to charge, will

EV users commit to such an agency, and if they do, does it lead to a different equilibrium

than if they do not?

It is well known that in non-cooperative settings mediated communication is an efﬁ-

cient way to achieve incentive-compatible outcomes via correlation devices aka correlated

equilibrium (Aumann, 1987) and Coarse Correlated Equilibrium (CCE)(Moulin & Vial, 1978).

We adopt CCE to answer if mediated communication leads to different outcomes as against

when EV users behave on their own. CCE, in recent years has received considerable atten-

tion owing to the ﬁnding that no-regret play leads to coarse-correlated equilibria (Rough-

garden, 2015). Correlated equilibria have also been associated with evolutionary learning

(Arifovic et al., 2019).

From the structure of games point of view, the games that we study in our work could be

seen to be connected to congestion and budget games, hence a comment on connection

to this literature is in order. The extant literature on congestion games spanning areas of

economics, computer science and operations research ﬁelds, mainly focus on existence

and efﬁciency of equilibria, that too, predominantly (pure) Nash equilibria. For example,

these results commonly establish bounds on price of anarchy and stability. On the other

hand, our questions are not related to efﬁciency of equilibria, instead, our questions are

motivated by practical observations from EV ﬁeld trials. In the context of our games, a

desirable outcome may not even be the efﬁcient one as is usually deﬁned. It is conceivable

that games in our work could be modeled via congestion games frameworks, furthermore,

efﬁciency questions may also be relevant (as discussed in Chakraborty and Khargonekar

(2014)). However, this is not the main focus of our work and we leave it for future study.

The rest of the paper is organized as follows. In Section 2we present a discrete EV

charging game along with the main assumptions. In Section 3, we outline our three main

results that answer the questions stated in Section 1. In Sections 4,5, and 6we present the

details including statements and proofs of the results underlying research questions 1, 2

and 3 respectively.

2 EV CHARGING GAME:MODEL AND ASSUMPTIONS

Consider the following game we call EV charging game (EVCG). There are nplayers. A

typical Player ihas a demand ridiwhich can be fulﬁlled by choosing to charge in any of

di≤Ttime periods. That is, strategy of a player is a T-dimensional binary vector.

Given a strategy proﬁle (si,s−i), the dis-utility/cost of Player iis

c(si,s−i) = X

btgt(si,s−i) + X

ft(si,s−i), (1)

where

gt(si,s−i) = 





Ptst

k=1rkst

, if Pn

k=1rkst

k>Pt

rist

i, otherwise

ft(si,s−i) = 





hst

iPt

k=1rkst

k, if Pn

k=1rkst

k>Pt

0, otherwise

with rand Pbeing parameters of the game which are explained as follows: riis the power

rating (in kW ) of Player iwhich informs power transfer rate; Ptis the total available power

(in kW h) in time period t;st

iindicates whether Player icharges at time period tor not;

k=1st

kis the total number of users who decided to charge in time period t; and hrepre-

sents a dissatisfaction function. Furthermore, each time period is classiﬁed according to k

price slabs. In the most general case, k=T. For this reason, typically, charging choices are

modeled as discrete to allow for modeling a complete price discrimination between users,

where users may pay time of day tariffs. However, consumers rarely choose schemes which

employ real-time pricing or several price slabs during the day, this is usually explained as

fear about price volatility. Most common practice is to differentiate between peak and

non-peak times (see Jacobsen and Stewart (2022) and Western-Power-Distribution et al.

(2015)). In our model, btis the price per unit in time t, we will assume a two part price

plan as formalized in the assumption below.

Assumption 1. Two price slabs: peak and non-peak; we superscript peak and non-peak

times with D and N respectively, eg., b N

iis the non-peak price for user i .

Remark 1. Note that the scenario when gt(si,s−i)<rican be interpreted as congestion or

a deliberate delayed charging strategy as in Wu et al. (2022) to induce a desirable charging

behavior equilibrium.

Assumption 2. h(·)is a continuous and monotone function.

In our analysis we consider two functions: a linear and a logistic one. Linear dissat-

isfaction is also considered in the recent literature, the reason being linear dissatisfaction

rates are more appealing because of simplicity and associated tractability of resulting anal-

ysis. In practice, however, it is likely the dissatisfaction is different across the support. For

example, increase in dissatisfaction is probably higher when a user gets 10 kWh instead of

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

PricingandElectricVehicleChargingEquilibriaTrivikramDokkaJorgeBrunoSonaliSenGuptaChowdhuryMohammadSakibAnwar§December13,2022AbstractWestudyequilibriainanElectricVehicle(EV)charginggame,acostminimizationgameinherenttodecentralizedchargingcontrolstrategyforEVpowerdemandman-agement.Inourmodel,eachuse...

展开>> 收起<<

Pricing and Electric Vehicle Charging Equilibria Trivikram Dokka Jorge BrunoSonali SenGupta Chowdhury Mohammad Sakib Anwar.pdf

共21页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Pricing and Electric Vehicle Charging Equilibria Trivikram Dokka Jorge BrunoSonali SenGupta Chowdhury Mohammad Sakib Anwar

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: