1

Energy Storage Sharing in Smart Grid: A Modified
Auction Based Approach

arXiv:1512.07700v1 [] 24 Dec 2015

Wayes Tushar, Member, IEEE, Bo Chai, Chau Yuen, Senior Member, IEEE, Shisheng Huang, Member, IEEE,
David B. Smith, Member, IEEE, H. Vincent Poor Fellow, IEEE, and Zaiyue Yang, Member, IEEE

Abstract—This paper studies the solution of joint energy
storage (ES) ownership sharing between multiple shared facility
controllers (SFCs) and those dwelling in a residential community.
The main objective is to enable the residential units (RUs) to
decide on the fraction of their ES capacity that they want to
share with the SFCs of the community in order to assist them
storing electricity, e.g., for fulfilling the demand of various shared
facilities. To this end, a modified auction-based mechanism is
designed that captures the interaction between the SFCs and
the RUs so as to determine the auction price and the allocation
of ES shared by the RUs that governs the proposed joint ES
ownership. The fraction of the capacity of the storage that each
RU decides to put into the market to share with the SFCs and
the auction price are determined by a noncooperative Stackelberg
game formulated between the RUs and the auctioneer. It is shown
that the proposed auction possesses the incentive compatibility
and the individual rationality properties, which are leveraged via
the unique Stackelberg equilibrium (SE) solution of the game.
Numerical experiments are provided to confirm the effectiveness
of the proposed scheme.
Index Terms—Smart grid, shared energy storage, auction
theory, Stackelberg equilibrium, strategy-proof, incentive compatibility.

I. I NTRODUCTION

E

NERGY storage (ES) devices are expected to play a
significant role in the future smart grid due to their
capabilities of giving more flexibility and balance to the grid
by providing a back-up to the renewable energy [1]–[9]. ES
can improve the electricity management in a distribution network, reduce the electricity cost through opportunistic demand
response, and improve the efficient use of energy [10]. The
distinct features of ES make it a perfect candidate to assist in
W. Tushar and C. Yuen are with Singapore University of Technology and
Design (SUTD), 8 Somapah Road, Singapore 487372 (Email: {wayes tushar,
yuenchau}@sutd.edu.sg).
B. Chai is with the State Grid Smart Grid Research Institute, Beijing,
102211, China (Email: chaibozju@gmail.com).
S. Huang is with the Ministry of Home Affairs, Singapore (Email:
shisheng@gmail.com).
D. B. Smith is with the National ICT Australia (NICTA), ACT 2601,
Australia and adjunct with the Australian National University (Email:
david.smith@nicta.com.au).
H. V. Poor is with the School of Engineering and Applied Science at
Princeton University, Princeton, NJ, USA (Email: poor@princeton.edu).
Z. Yang is with the State Key Laboratory of Industrial Control Technology
at Zhejiang University, Hangzhou, China (Email: yangzy@zju.edu.cn).
This work is supported in part by the Singapore University of Technology
and Design (SUTD) through the Energy Innovation Research Program (EIRP)
Singapore NRF2012EWT-EIRP002-045 and IDC grant IDG31500106, and in
part by the U.S. National Science Foundation under Grant ECCS-1549881. D.
B. Smith’s work is supported by NICTA, which is funded by the Australian
Government through the Department of Communications and the Australian
Research Council.

residential demand response by altering the electricity demand
due to the changes in the balance between supply and demand.
Particularly, in a residential community setting, where each
household is equipped with an ES, the use of ES devices can
significantly leverage the efficient flows of energy within the
community in terms of reducing cost, decarbonization of the
electricity grid, and enabling effective demand response (DR).
However, energy storage requires space. In particular for
large consumers like shared facility controllers (SFCs) of large
apartment buildings [11], the energy requirements are very
high, which consequently necessitates the actual installment
of very large energy storage capacity. The investment cost of
such storage can be substantial whereas due to the random
usage of the facilities (depending on the usage pattern of
different residents) some of the storage may remain unused.
Furthermore, the use of ESs for RUs is very limited for two
reasons [10]: firstly, the installation cost of ES devices is very
high and the costs are entirely borne by the users. Secondly, the
ESs are mainly used to save electricity costs for the RUs rather
than offer any support to the local energy authorities, which
further makes their use economically unattractive. Hence, there
is a need for solutions that will capture both the problems
related to space and cost constraints of storage for SFCs and
the benefit to RUs for supporting third parties.
To this end, numerous recent studies have focused on energy
management systems with ES devices as we will see in the
next section. However, most of these studies overlook the
potential benefits that local energy authorities such as SFCs
can attain by jointly sharing the ES devices belonging to the
RUs. Particularly due to recent cost reduction of small-scale
ES devices, sharing of ES devices installed in the RUs by the
SFCs has the potential to benefit both the SFCs and the RUs
of the community as we will see later. In this context, we
propose a scheme that enables joint ES ownership in smart
grid. During the sharing, each RU leases the SFCs a fraction
of its ES device to use, and charges and discharges from the
rest of its ES capacity for its own purposes. On the contrary,
each SFC exclusively uses its portion of ES devices leased
from the RUs. This work is motivated by [10], in which the
authors discussed the idea of joint ownership of ES devices
between domestic customers and local network operators, and
demonstrated the potential system-wide benefits that can be
obtained through such sharing. However, no policy has been
developed in [10] to determine how the fraction of battery
capacity, which is shared by the network operators and the
domestic users, is decided.
Note that, as an owner of an ES device, each RU can decide

IEEE Trans. Smart Grid, Dec. 2015
II. S TATE - OF -T HE A RT

whether or not to take part in the joint ownership scheme with
the SFCs and what fraction of the ES can be shared with the
SFCs. Hence, there is a need for solutions that can capture this
decision making process of the RUs by interacting with the
SFCs of the network. In this context, we propose a joint ES
ownership scheme in which by participating in storage sharing
with the SFCs, both the RUs and SFCs benefit economically.
Due to the interactive nature of the problem, we are motivated
to use auction theory to study this problem [12].
Exploiting the two-way communications aspects, auction
mechanisms can exchange information between users and electricity providers, meet users’ demands at a lower cost, and thus
contribute to the economic and environmental benefits of smart
grid1 [13]. In particular, 1) we modify the Vickrey auction
technique [14] by integrating a Stackelberg game between the
auctioneer and the RUs and show that the modified scheme
leads to a desirable joint ES ownership solution for the RUs
and the SFCs. To do this, we modify the auction price derived
from the Vickrey auction, to benefit the owner of the ES,
through the adaptation of the adopted game as well as keep
the cost savings to the SFCs at the maximum; 2) We study
the attributes of the technique, and show that the proposed
auction scheme possesses both the incentive compatibility and
the individual rationality properties leveraged by the unique
equilibrium solution of the game; 3) We propose an algorithm
for the Stackelberg game that can be executed distributedly
by the RUs and the auctioneer, and the algorithm is shown to
be guaranteed to reach the desired solution. We also discuss
how the proposed scheme can be extended to the time varying
case; and 4) Finally, we provide numerical examples to show
the effectiveness of the proposed scheme.
The importance and necessity of the proposed study with
respect to actual operation of smart grid lies in assisting
the SFCs of large apartment buildings in smart communities
to reduce space requirements and investment costs of large
energy storage units. Furthermore, by participating in storage
sharing with the SFCs, the RUs can benefit economically,
which can consequently influence them to efficiently schedule
their appliances and thus reduce the excess use of electricity.
We stress that multi-agent energy management schemes are
not new in the smart grid paradigm and have been discussed
in [11], [15] and [16]. However, the scheme discussed in the
paper differs from these existing approaches in terms of the
considered system model, chosen methodology and analysis,
and the use of the set of rules to reach the desired solution.
The remainder of the paper is organized as follows. We
provide a comprehensive literature review of the related work
in Section II followed by the considered system model in
Section III. Our proposed modified auction-based mechanism
is demonstrated in Section IV where we also discuss how the
scheme can be adopted in a time varying environment. The
numerical case studies are discussed in Section V, and finally
we draw some concluding remarks in Section VI.

In the recent years, there has been an extensive research
effort to understand the potential of ES devices for residential energy management [17]. This is mainly due to their
capabilities in reducing the intermittency of renewable energy
generation [18] as well as lowering the cost of electricity [19].
The related studies can be divided into two general categories.
The first category of studies consisting of [20], [21], which
assume that the ESs are installed within each RU premises
and are used solely by the owners in order to perform different
energy management tasks such as optimal placement, sizing
and control of charging and discharging of storage devices.
The second type of studies deal with ES devices that are
not installed within the RUs but located in a different location
such as in electric vehicles (EVs). Here, the ESs of EVs are
used to provide ancillary services for RUs [22]–[24] and local
energy providers [25]–[27]. Furthermore, another important
impact of ES devices on residential distribution grids is studied
in [28] and [29]. In particular, these studies focus on how
the use of ES devices can bring benefits for the stakeholders
in external energy markets. In [28], the authors propose a
multi-objective optimization method for siting and sizing of
ESs of a distribution grid to capture the trade-offs between
the storage stakeholders and the distribution system operators.
Furthermore, in [29], optimal storage profiles for different
stakeholders such as distribution grid operators and energy
traders are derived based on case studies with real data. Studies
of other aspects of smart grid can be found in [30]–[36].
As can be seen from the above discussion, the use of
ES devices in smart grid is not only limited to address the
intermittency of renewable generation [18] and assisting users
to take part in energy management to reduce their cost of
electricity [19], [21] but also extends to assisting the grid
(or, other similar energy entities such as an SFC) [37] and
generating revenues for stakeholders [28], [29]. However, one
similarity between most of the above mentioned literature is
that only one entity owns the ES and uses it according to its
requirements. Nonetheless, this might not always be the case if
there are large number of RUs2 in a community. In this regard,
considering the potential benefits of ES sharing, as discussed
in [10], this paper investigates the case in which the SFCs
in a smart community are allowed to share some fraction of
the ESs owned by the RUs through a third party such as an
auctioneer or a community representative.
The proposed modified auction scheme differs from the
existing techniques for multi-agent energy management such
as those in [11], [15], [16] in a number of ways. Particularly, in
contrast to these studies, the proposed auction scheme captures
the interaction between the SFCs and the RUs, whereby the
decision on the auction price is determined via a Stackelberg
game. By exploiting auction rules including the determination
rule, payment rule, and allocation rule the interaction between
the SFCs and RUs is greatly simplified. For instance, the
determination rule can easily identify the number of RUs that
are participating in the auction process, which further leverage

1 Please note that such a technique can be applied in the real distribution
network such as in electric vehicle charging stations by using the two-way
information and power flow infrastructure of smart grids [3].

2 Each RU may participate as a single entity or as a group where RUs
connected via an aggregator [38].

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IEEE Trans. Smart Grid, Dec. 2015
sicap

either due to the fact that some SFCs do not have their own
ESs [11] or that the ESs of the SFCs are not large enough
to store all the excess energy at that time. It is important to
note that the ES requirement of the SFCs can stem from any
type of intermittent generation profile that the SFCs or RUs
can adopt. For example, one can consider that the proposed
scheme is based on a hybrid generation profile comprising both
solar and wind generation. However, the proposed technique
is equally suitable for other types of intermittent generation
as well. We assume that there are N = |N | RUs, where N is
the set of all RUs in the system, that are willing to share some
parts of their ES with the SFCs of the network. The battery
cap
capacity of each RU i ∈ N is si , and each RU i wants to
put xi fraction of its ES in the market to share with the SFCs,
where

(1)
xi ≤ bi = scap
i − di .

sicap : Total capacity of ES device of RU i.
di : The amount that RU i does not sell.
bi : The maximum ES space the RU i might sell to the SFCs.

bi = sicap − di

di

Sharing price pt

x1 ≤ b1

x2 ≤ b2

Each i decides

ri

ri < pt

Yes

No

x N ≤ bN
Leaves the
et
sharing market

q1

q2

Yes

Each m decides

am

qM

Take
ke part in ES
sharing

am > pt

No

Sharing price pt

Here, bi is the maximum amount of battery space that the RU
can share with the SFCs if the cost-benefit tradeoff for the
sharing is attractive for it. di is the amount of ES that the RU
does not want to share, rather uses for its own needs, e.g., to
run the essential loads in the future if there is any electricity
disruption within the RU or if the price of electricity is very
high.

Fig. 1: The fraction of the ES capacity that an RU i is willing to share with
the SFCs of the community.

the determination of the auction price via the Stackelberg game
in the payment rule. Furthermore, on the one hand the work
here complements the existing works focusing on the potential
of ES for energy management in smart grid. On the other
hand, the proposed work has the potential to open new research
opportunities in terms of control of energy dispatch from ES,
the size of ES, and exploring other interactive techniques such
as cooperative games and bi-level optimization for ES sharing.

To this end, to offer an ES space xi , on the one hand, each
RU i decides on an reservation price ri per unit of energy.
Hereinafter, we will use ES space and energy interchangeably
to refer to the ES space that each RU might share with the
SFCs. However, if the price pt , which each RU received for
sharing its ES, is lower than ri , the RU i removes its ES
space xi from the market as the expected benefit from the
joint sharing of ES is not economically attractive for it. On
the other hand, each SFC m ∈ M, that needs to share ES
space with the RUs to store their energy, decides a reservation
bid am , which represents the maximum unit price the SFC m
is willing to pay for sharing per unit of ES with the RUs in
the smart community, to enter into the sharing market. And,
if am > pt , the SFC removes its commitment of joint ES
ownership with RUs from the market due to the same reason
as mentioned for the RU. A graphical representation of the
concept of ES sharing and their decision making process of
sharing the ES space of each RU i with the SFCs are shown
in Fig. 1. Please note that to keep the formulation simple,
we do not include any specific storage model in the scheme.
However, by suitably modeling some related parameters such
as the storage capacity scap
i , and parameters like di and bi , the
proposed scheme can be adopted for specific ES devices.

III. S YSTEM M ODEL
Let us consider a smart community that consists of a large
number of RUs. Each RU can be an individual home, a
single unit of a large apartment complex, or a large number
of units connected via an aggregator that acts as a single
entity [38]–[40]. Each RU is equipped with an ES device that
the RU can use to store electricity from the main grid or its
renewable energy sources, if there are any, or can perform
DR management according to the real-time price offered by
the grid. The ES device can be a storage device installed
within each RU premises or can be the ES used for the RU’s
electric vehicles. The entire community is considered to be
divided into a number of blocks, where each block consists
of a number of RUs and an SFC. Each SFC m ∈ M, where
M is the set of all SFCs and M = |M|, is responsible for
controlling the electrical equipment and machines such as lifts,
parking lot lights and gates, water pumps, and lights in the
corridor area of a particular block of the community, which
are shared and used by the residents of that block on regular
basis. Each SFC is assumed to have its own renewable energy
generation and is also connected to the main electricity grid
with appropriate communication protocols.
Considering the fact that the nature of energy generation
and consumption is highly sporadic [41], let us assume that
the SFCs in the community need some extra ESs to store
their electricity after meeting the demand of their respected
shared facilities at a particular time of the day. This can be

The interaction that arises from the choice of ES sharing
price between the SFCs and RUs as well as the need of the
SFCs to share the ES space to store their energy and the profits
that the RUs can reap from allowing their ESs to be shared
give rise to a market of ES sharing between the RUs and the
SFCs in the smart grid. In this market, the involved N RUs
and M SFCs will interact with each other to decide as to
how many of them will take part in sharing the ESs between
themselves, and also to agree on the ES sharing parameters
such as the trading price pt and the amount of ES space to
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IEEE Trans. Smart Grid, Dec. 2015
IV. AUCTION BASED ES OWNERSHIP
Vickrey auction is a type of sealed-bid auction scheme,
where the bidders submit their written bids to the auctioneer without knowing the bids of others participating in the
auction [14]. The highest bidder wins the auction but pays
the second highest bid price. Nevertheless, in this paper, we
modify the classical Vickrey auction [14] to model the joint
ES ownership scheme for a smart community consisting of
multiple customers (i.e., the SFCs) and multiple owners of ES
devices (i.e., the RUs). The modification is motivated by the
following factors: 1) unlike the classical Vickrey auction, the
modified scheme would enable the multiple owners and customers to decide simultaneously and independently whether to
take part in the joint ES sharing through the determination rule
of the proposed auction process, as we will see shortly; 2) the
modification of the auction provides each participating RU i
with flexibility of choosing the amount of ES space that they
may want to share with the SFCs in cases when the auction
price5 pt is lower than their expected reservation price ri ;
and 3) finally, the proposed auction scheme provides solutions
that satisfy both the incentive compatibility and individual
rationality properties, as we will see later, which are desirable
in any mechanism that adopts auction theory [41].
To this end, the proposed auction process, as shown in
Fig. 2, consists of three elements:

Fig. 2: Energy management in a smart community through auction process
consisting of multiple RUs with ES devices, an auctioneer and a number of
SFCs.

be shared. In the considered model, the RUs not only decide
on the reservation prices ri , but also on the amount of ES
space xi that they are willing to share with the SFCs. The
amount of xi is determined by the trade-off between between
the economic benefits that the RU i expects to obtain from
giving the SFCs the joint ownership of its ES device and
the associated reluctance αi of the RU for such sharing. The
reluctance to share ESs may arise from the RUs due to many
factors. For instance, sharing would enable frequent charging
and discharging of ESs that reduce the lifespan3 of an ES
device [42]. Hence, an RU i may set its αi higher so as
to increase its reluctance to participate in the ES sharing.
However, if the RU is more interested in earning revenue rather
than increasing ES life time, it can reduce its αi and thus get
more net benefits from sharing its storage. Therefore, for a
given set of bids am , ∀m and storage requirement qm , ∀m
by the SFCs, the maximum amount of ES xi that each RU
i will decide to put for sharing is strongly affected by the
trading price pt and the reluctance parameter4 αi of each
RU i ∈ N during the sharing process. In this context, we
develop an auction based joint ES ownership scheme in the
next section. We understand that the proposed scheme involves
different types of users such as auctioneers, SFCs, and RUs.
Therefore, the communication protocol used by them could
be asynchronous. However, in our study we assume that the
communication between different entities of the system are
synchronous. This is mainly due to the fact that we assume
our algorithm is executed once in a considered time slot, and
the duration of this time slot can be one hour [43]. Therefore,
synchronization is not a significant issue for the considered
case and the communication complexity is affordable. For
example, the auctioneer can wait for five minutes until it
receives all the data from SFCs and the RUs and then the
algorithm, which is proposed in Section IV-C, can be executed.

1) Owner: The RUs in set N , that own the ES devices,
and expect to earn some economic benefits, e.g., through
maximizing a utility function, by letting the SFCs to share
some fraction of their ES spaces.
2) Customer: The SFCs in set M, that are in need of ESs in
order to store some excess electricity at a particular time
of the day. The SFCs offer the RUs a price with a view
to jointly own some fraction of their ES devices.
3) Auctioneer: A third party (e.g., estate or building manager), that controls the auction process between the
owners and the customers according to some predefined
rules.
The proposed auction policies consist of A) determination
rule, B) payment rule and C) storage allocation rule. Here,
determination rule allows the auctioneer to determine the
maximum limit for the auction price pmax
and the number
t
of SFCs and RUs that will actively take part in the ES sharing
scheme once the auction process is initiated. The payment rule
enables the auctioneer to decide on the price that the customer
needs to pay to the owners for sharing their ES devices, which
allows the RUs to decide how much storage space they will
be putting into the market to share with the SFCs. Finally,
the auctioneer allocates the ES spaces for sharing for each
SFC following the allocation rule of the proposed auction.
It is important to note that although both the customers and
owners do not have any access to others private information
such as the amount of ES to be shared by an RU or the required
energy space by any SFC, the rules of auction are known to
all the participants of the joint ownership process.

3 Please note that the life time degradation due to charging and discharging
may not true for all electromechanical systems such as redox-flow system.
4 Reluctance parameter refers to the opposite of preference parameter [38].

5 Hereinafter, p will be used to refer to auction price instead of sharing or
t
trading price.

4

IEEE Trans. Smart Grid, Dec. 2015
of the SFCs and RUs in the network:

90
Cosumers participating
in auction

Owner
Customer

am < ri ; ∀m ∈ M/{1, 2, . . . , K}, ∀i ∈ N /{1, 2, . . . , J}. (4)

80

Hence, the joint ownership of ES would be a detrimental choice for the RUs and the SFCs within the set
N /{1, 2, . . . , J} and M/{1, 2, . . . , K} respectively, which
consequently remove them from the proposed auction process.
Now, one desirable property of any auction mechanism is that
no participating agents in the auction mechanism will cheat
once the payment and allocation rules are being established.
To this end, we propose that, once J and K are determined,
K − 1 SFCs and J − 1 RUs will be engaged in the joint ES
sharing process, which is a necessary condition for matching
total demand and supply while maintaining a truthful auction
scheme [44]. Nevertheless, if truthful auction is not a necessity,
SFC K and RU J can also be allowed to participate in the
joint ES ownership auction.

Price

70

60

Maximum auction price pmax
t
Vickrey price pmin
t

50

40

Owners participating
in auction

30
0

50

100

150

Storage amount

Fig. 3: Determination of the Vickrey price, the maximum auction price, and
the number of participating RUs and SFCs in the auction process.

The proposed scheme initially determines the set of SFCs
⊂ M and RUs ⊂ N that will effectively take part in the
auction mechanism once the upper bound of the auction price
pmax
is determined. Eventually, the payment and the allocation
t
rules are executed in the course of the auction plan.

B. Payment Rule
We note that the intersection of the demand and supply
curves demonstrates the highest reservation price pmax
for the
t
participating J − 1 RUs. According to the Vickrey auction
mechanism [14], the auction price for sharing the ES devices
would be the second highest reservation price, i.e., the Vickrey
price, which will be indicated as pmin
hereinafter. However, we
t
note that this second highest price might not be considerably
beneficial for all the participating RUs in the auction scheme.
In contrast, if pt is set to pt = pmax
t , the price could be
detrimental for some of the SFCs. Therefore, to make the
auction scheme attractive and beneficial to all the participating
RUs and, at the same time, to be cost effective for all the SFCs,
we strike a balance between the pmax
and pmin
t
t . To do so, we
propose a scheme for deciding on both the auction price pt
and the amount of ESs xi that RUs will put into the market for
sharing according to pt . In particular, we propose a Stackelberg
game between the auctioneer that decides on the auction price
pt to maximize the average cost savings to the SFCs as well as
satisfying their desirable needs of ESs, and the RUs, that decide on the vector of the amount of ES x = [x1 , x2 , . . . , xJ−1 ]
that they would like to put into the market for sharing such that
their benefits are maximized. Please note that the solution of
the proposed problem formulation can also be solved following
other distributed algorithms, e.g., algorithms designed via the
bi-level optimization technique [45].
Stackelberg game: Stackelberg game is a multi-level decision making process, in which the leader of the game takes
the first step to choose its strategy. The followers, on the other
hand, choose their strategy in response to the decision made by
the leader. In the proposed game, we assume the auctioneer
as the leader and the RUs as the followers. Hence, it can
be seen as a single-leader-multiple-follower Stackelberg game
(SLMFSG). We propose that the auctioneer, as a leader of
the SLMFSG Γ, will take the first step to choose a suitable
min
auction price pt from the range [pmin
t , pt ]. Meanwhile, each
RU i ∈ {1, 2, . . . , J − 1}, as a follower of the game, will
play its best strategy by choosing a suitable xi ∈ [0, bi ] in
response to the price pt offered by the auctioneer. The best

A. Determination Rule
The determination rule of the proposed scheme is executed
by the following steps (inspired from [44]):
i) The RUs of set N , i.e., the owners of the ESs, declare
their reservation price ri , ∀i in an increasing order, which
we can consider, without loss of generality, as:
r1 < r2 < . . . < rN .

(2)

The RUs submit the reservation price along with the
amount xi of ES that they are interested to share with
the SFCs to the auctioneer.
ii) The SFCs’ bidding prices, i.e., am , ∀m, are arranged in
a decreasing order, i.e.,
a1 > a2 > . . . > aM .

(3)

The SFCs submit to the auctioneer along with the quantity
qm ∀m of ES that they require.
iii) Once the auctioneer receives the ordered information from
the RUs and the SFCs, it generates the aggregated supply
(reservation price of the RUs versus the amount of ES the
RUs interested to share) and demand curves (reservation
bids am verses the quantity of ES qm needed) using (2)
and (3) respectively.
iv) The auctioneer determines the number of of participating
SFCs K and RUs J that satisfies aK ≥ rJ from
the intersection of the two curves using any standard
numerical method [44].
As soon as the SFC K ≤ M and RU J ≤ N are determined
from the intersection point, as shown in Fig. 3, an important
aspect of the auction mechanism is to determine the number
of SFCs and RUs, which will take part in the joint ownership
of ESs. We note that once the number of SFCs K and RUs
J are determined, the following relationship holds for the rest
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IEEE Trans. Smart Grid, Dec. 2015
response strategy of each RU i will stem from a utility function
Ui , which captures the benefit that an RU i can gain from
deciding on the amount of ES xi to be shared for the offered
price. Whereas the auctioneer chooses the price pt with a
view to maximize the average cost savings Z of the SFCs
in the network. Now to capture the interaction between the
auctioneer and the RUs, we formally define the SLMFSG Γ
as
Γ = {{{1, 2, . . . , J − 1}, {Auctioneer}}, {Ui }i∈{1,2,...,J−1} ,
{Xi }i∈{1,2,...,J−1} , Z, pt },
which consists of: i) the set of RUs {1, 2, . . . , J − 1} participating in the auction scheme and the auctioneer; ii) the utility
Ui that each RU i reaps from choosing a suitable strategy
xi in response to the price pt announce by the auctioneer;
iii) the strategy set Xi of each RU i ∈ {1, 2, . . . , J − 1};
iv) the average cost savings Z that incurred to each SFC
m ∈ {1, 2, . . . , K − 1} from the strategy
 by the
 chosen
max
of the
auctioneer, and v) the strategy pt ∈ pmin
t , pt
auctioneer.

In the proposed approach, each RU i iteratively responses to
the strategy pt chosen by the auctioneer independent of other
RUs in set {1, 2, . . . , J − 1}/{i}. The response of i is affected
by the offered price pt , its reluctance parameter αi and the
initial reservation price ri .
However, we note that the auctioneer does not have any
control over the decision making process of the RUs. It only
sets the auction price pt with a view to maximize the cost
savings Z, with respect to the cost with the initial bidding
price, for the SFCs. To this end, the target of auctioneer is
(5)assumed to maximize the average cost savings
! J−1
PK−1
X
(a
−
p
)
m
t
m=1
Z=
xi
(8)
K −1
i=1

by choosing an appropriate price
p to offer to each RU from
PK−1 t
min max
m=1 (am −pt )
is the average savthe range [pt , pt ]. Here,
K−1
ing in auction
price
that
the
SFCs
pay
to
the
RUs for sharing
P
the ESs and i xi is the total amount of ES that all the SFCs
share from the RUs. From Z, we note that the cost savings
will be more if pt is lower for all m ∈ {1, 2, . . . , K − 1}.
However, this is conflicted by that fact that a lower pt may
lead to the choice of lower xi ∀i ∈ {1, 2, . . . , J − 1} by the
RUs, which in turn will affect the cost to the SFCs. Hence,
to reach a desirable solution set (x∗ , p∗t ), the auctioneer and
the RUs continue to interact with each other until the game
reaches a Stackelberg equilibrium (SE).

Now, the utility function Ui , which defines the benefits that
an RU i can attain from sharing xi amount of its ES with the
SFCs, is proposed to be
Ui (xi ) = (pt − ri )xi − αi x2i , xi ≤ bi ,

(6)

where, αi is the reluctant parameter of RU i, and ri is the
reservation price set by RU i. Ui mainly consists of two parts.
The first part (pi − ri )xi is the utility in terms of its revenue
that an RU i obtains from sharing its xi portion of ES device.
The second part αi x2i , on other hand, is the negative impact
in terms of liability on the RU i stemming from sharing its
ES with the SFC. This is mainly due to the fact that once an
RU decides to share its xi amount of storage space with an
SFC, the RU can only use scap
i − xi amount of storage for its
own use. The term αi x2i captures this restriction of the RU
on the usage of its own ES. In (6), the reluctance parameter
αi is introduced as a design parameter to measure the degree
of unwillingness of an RU to take part in energy sharing. In
particular, a higher value of αi refers to the case when an RU
i is more reluctant to take part in the ES sharing, and thus, as
can be seen from (6), even with the same ES sharing attains
a lower net benefit. Thus, Ui can be seen as a net benefit
to RU i for sharing its ES. The utility function is based on
the assumption of a non-decreasing marginal utility, which
is suitable for modeling the benefits of power consumers, as
explained in [46]. In addition, the proposed utility function
also possesses the following properties: i) the utility of any
RU increases as the amount of price pt paid to it for sharing
per unit of ES increases; ii) as the reluctance parameter αi
increases, the RU i becomes more reluctant to share its ES, and
consequently the utility decreases; and iii) for a particular price
pt , the more an RU shares with the SFCs, the less interested it
becomes to share more for the joint ownership. To that end, for
a particular price pt and reluctance parameter αi , the objective
of RU i is


(7)
max (pt − ri )xi − αi x2i , xi ≤ bi .

Definition 1. Let us consider the game Γ as described in (5),
where the utility of each RU i and the average utility per SFC
are described via Ui and Z respectively. Now, Γ will reach a
SE (x∗ , p∗t ), if and only if the solution of the game satisfies
the following set of conditions:
Ui (x∗i , x∗−i , p∗t ) ≥ Ui (xi , x∗−i , p∗t ), ∀i ∈ {1, 2, . . . , J − 1},
max
∀xi ∈ Xi , p∗t ∈ [pmin
t , pt ], (9)

and
PK−1

− p∗t ) X ∗
xi ≥
K −1
i

m=1 (am

PK−1

− pt ) X ∗
xi ,
K −1
i

m=1 (am

(10)

where x−i = [x1 , x2 , . . . , xi−1 , xi+1 , . . . , xJ−1 ].

Hence, according to (9) and (10), both the RUs and the SFCs
achieve their best possible outcomes at the SE. Hence, neither
the RUs nor the auctioneer will have any incentive to change
their strategies as soon as the game Γ reaches the SE. However,
achieving an equilibrium solution in pure strategies is not
always guaranteed in non-cooperative games [38]. Therefore,
we need to investigate whether the proposed Γ possesses an
SE or not.
Theorem 1. There always exists a unique SE solution for
the proposed SLMFSG Γ between the auctioneer and the
participating RUs in set {1, 2, . . . , J − 1}.
Proof: Firstly, we note that the strategy set of the auction
max
.
eer is non-empty and continuous within the range pmin
t , pt
Hence, there will always be a non-empty strategy for the
auctioneer that will enable the RUs to offer some part of their

xi

6

IEEE Trans. Smart Grid, Dec. 2015
ES, within their limits, to the SFCs. Secondly, for any price
pt , the utility function Ui in (6) is strictly concave with respect
2
of xi , ∀i ∈ {1, 2, . . . , J − 1}, i.e., δδxU2i < 0. Hence, for any
i
 min max 
price pt ∈ pt , pt , each RU will have a unique xi , which
will be chosen from a bounded range [0, bi ] and maximize Ui .
Therefore, it is evident that as soon as the scheme will find
a unique p∗t such that the average utility Z per SFC attains a
maximum value, the SLMFSG Γ will consequently reach its
unique SE.
To this end, first we note that the amount of ES x∗i , at which
the RU i achieves its maximum utility in response to a price
pt can be obtained from (6),
pt − ri
x∗i =
.
(11)
2αi

Algorithm 1: Algorithm for SLMFSG to reach the SE
∗
1: Initialization: p∗t = pmin
t , Z = 0.
2: for Auction price pt from pmin
to pmax
do
t
t
3:
for Each RU i ∈ {1, 2, . . . , (J − 1)} do
4:
RU i adjusts its amount of ES xi to share according to

x∗i = arg
5:
6:

7:
8:

Now, replacing the value of x∗i in (8) and doing some simple
arithmetics, the auction price p∗t , which maximizes the average
cost savings to the SFCs can be found as
P
 P
 P
J−1 1
K−1
J−1
a
+ i=1 ri (K−1)
m
i=1 2αi
m=1
2αi
∗
pt =
, (12)
PJ−1 K−1
i=1

max

0≤xi ≤bi

[(pt − ri )xi − αi x2i ].

(13)

end for
The auctioneer computes the average cost savings to SFCs
! J −1
PK−1
X ∗
m=1 (am − pt )
xi .
(14)
Z=
K−1
i=1

if Z ≥ Z ∗ then
The auctioneer record the desirable price and maximum
average cost savings
p∗t = pt , Z ∗ = Z.

(15)

9:
end if
10: end for

The SE (x∗ , p∗t ) is achieved.

αi

guaranteed to reach SE of the proposed SLMFSG Γ.

where am for any m ∈ {1, 2, . . . , K − 1} and αi for any
i ∈ {1, 2, . . . , J − 1} is exclusive. Therefore, p∗t is unique for
Γ, and thus Theorem 1 is proved.

Proof: In the proposed algorithm, we note that the choice
of strategies by the RUs emanate from the choice pt of the
auctioneer, which as shown in (12) will always attain a nonempty single value p∗t at the SE due to its bounded strategy set
max
[pmin
t , pt ]. On the other hand, as the Algorithm 1 is designed,
in response to the p∗t , each RU i will choose its strategy xi
from the bounded range [0, bi ] in order to maximize its utility
function Ui . To that end, due to the bounded strategy set and
continuity of Ui with respect to xi , it is confirmed that each
RU i will always reach a fixed point x∗i for the given p∗t .
Therefore, the proposed Algorithm 1 is always guaranteed to
reach the unique SE of the SLMFSG.

C. Algorithm for payment
To attain the SE, the auctioneer, which has the information
of am , m = {1, 2, 3, . . . , K − 1}, needs to communicate with
each RU. It is considered that the auctioneer does not have
any knowledge of the private information of the RUs such
as αi , ∀i. In this regard, in order to decide on a suitable
auction price pt that will be beneficial for both the RUs and
the SFCs, the auctioneer and the RUs interact with one another.
To capture this interaction, we design an iterative algorithm,
which can be implemented by the auctioneer and the RUs in a
distributed fashion to reach the unique SE of the proposed
SLMFSG. The algorithm initiates with the auctioneer who
sets the auction price pt to pmin
and the optimal average
t
cost saving per SFC Z ∗ to 0. Now, in each iteration, after
having the information on the offered auction price by the
auctioneer, each RU i plays its best response xi ≤ bi and
submits its choice to the auctioneer. The auctioneer, on other
hand, receives the information on x = [x1 , x2 , . . . , xJ−1 ]
from all the participating RUs and determines the average
cost savings per SFC Z from its knowledge on the reservation
bids [a1 , a2 , . . . , aK−1 ] and using (8). Then, the auctioneer
compares the Z with Z ∗ . If Z > Z ∗ , the auctioneer updates
the optimal auction price to the one recently offered and
sends a new choice of price to the RUs in the next iteration.
However, if Z ≤ Z ∗ , the auctioneer keeps the same price
and offers another new price to the RUs in the next iteration.
The iteration process continues until the conditions in (9) and
(10) are satisfied, and hence the SLMFSG reaches the SE. We
show the step-by-step process of the proposed algorithm in
Algorithm 1.

D. Allocation Rule
Now, once the the amount of ES x∗i that each RU i ∈
{1, 2, . . . , J − 1} decides to put into the market for sharing in
response to the auction price p∗t is determined, the auctioneer
allocates the quantity Qi to be jointly shared by each RU i
and the SFCs according to following rule [44]:
(
PK−1
PJ−1
x∗i
if i=1 x∗i ≤ m=1 qm ,
Qi (x) =
(16)
P
P
J−1 ∗
(x∗i − ηi )+ if i=1
xi > K−1
m=1 qm ,

where (f )+P = max(0,
Pf ) and ηi is the allotment of the
J−1 ∗
excess ES i=1
xi − K−1
m=1 qm that an RU i must endure.
Essentially, the rule in (16) emphasizes that if the requirements
of the SFCs exceed the available ES space from the RUs, each
RU i will allow the SFCs to share all of the ES xi that it put
into the market. However, if the available ES exceeds the total
demand by the SFCs, then
RU P
i will have to share a
Peach
J−1
K−1
fraction of the oversupply i=1 x∗i − m=1 qm . Nonethless,
this burden, if there is any, can be distributed in different
ways among the participating RUs. For instance, the burden
can be distributed either proportionally to the amount of ES
x∗i that each RU i shared with the SFCs or proportionally to

Theorem 2. The algorithm proposed in Algorithm 1 is always
7

IEEE Trans. Smart Grid, Dec. 2015
the reservation price6 ri of each RU. Alternatively, the total
burden can also be shared equally by the RUs in the auction
scheme [44].
1) Proportional allocation: In proportional allocation [47],
a fraction of the total burden ηi is allocated to each RU i
in
to the reservation
price ri (or, x∗i ) such that
PJ−1
PK−1
P proportion
∗
i ηi =
i=1 xi −
m=1 qm , which can be implemented
as follows:
P
PK−1 
J−1 ∗
x
−
i=1 i
m=1 qm ri
P
ηi =
, i = [1, 2, . . . , J − 1]. (17)
i ri

it is clear that all the participants in the proposed auction
scheme are individually rational, which leads to the following
Corollary 1.
Corollary 1. The proposed auction technique possesses the
individual rationality property, in which the J − 1 rational
owners and K − 1 rational customers actively participate in
the mechanism to gain the higher utility.
Theorem 3. The proposed auction mechanism is incentive
compatible, i.e., truthful auction is the best strategy for any
RU i ∈ {1, 2, . . . , J − 1} and SFC m ∈ {1, 2, . . . , K − 1}.

By replacing ri with x∗i in (17), the burden allocation can be
determined in proportion to the shared ES by each RU.
2) Equal allocation: According to equal allocation [44],
each RU bears an equal burden
!
K−1
J−1
X
X
1
∗
ηi =
x −
qm , i = [1, 2, . . . , J − 1] (18)
J − 1 i=1 i m=1

Proof: To validate Theorem 3, first we note that the
choice of strategies by the RUs always guaranteed to converge
to a unique SE, i.e., x∗ = [x∗1 , x∗2 , . . . , x∗J−1 ] as proven in
Theorem 1 and Theorem 2, which confirms the stability of
their selections. Now, according to [44], once the owners of an
auction process, i.e., the RUs in this proposed case, decide on
a stable amount of commodity, i.e., x∗i ∀i ∈ {1, 2, . . . , J − 1},
to supply to or to share with the customers, the auction process
always converges to a strategy-proof auction if the allocation
of commodity is conducted according to the rules described
in (16) and (18). Therefore, neither any RU nor any SFC
will have any intention to falsify their allocation once they
adopt (16) and (18) [44] for sharing the storage space of the
RUs from their SE amount. Therefore, the auction process is
incentive compatible, and thus Theorem 3 is proved.

of the oversupply.
Here it is important to note that, although proportional
allocation allows the distribution of oversupply according to
some properties of the RUs, equal allocation is more suitable
to make the auction scheme strategy proof [44]. Strategy
proofness is important for designing auction mechanisms as
it encourages the participating players not to lie about their
private information such as reservation price [41], which
is essential for the acceptability and sustainability of such
mechanisms in energy markets. Therefore, we will use equal
allocation of (18) for the rest of the paper.

F. Adaptation to Time-Varying Case
To extend the proposed scheme to a time-varying case, we
assume that the ES sharing scheme works in a time-slotted
fashion where each time slot has a suitable time duration based
on the type of application, e.g., 1 hour [43]. It is considered
that in each time slot all the RUs and SFCs take part in the
proposed ES sharing scheme to decide on the parameters such
as the auction price and the amount of ESs that needs to be
shared. However, in a time-varying case, the amount of ES
that an RU shares at time slot t may be affected by the burden
that the RU needed to bear in the previous time slot t − 1. To
this end, first we note that once the number of participating
RUs and SFCs is decided for a particular time slot via the
determination rule, the rest of the procedures, i.e., the payment
and allocation rules are executed following the descriptions in
Section IV-B and IV-D respectively for the respective time
slot. Now, if the total number of RUs and SFCs is fixed, the
RUs and SFCs that participate in the modified auction scheme
in any time slot is determined by their respective reservation
and bidding prices for that time slot. Further, the proposed
auction process may evolve across different time slots based
on the change of the amount of ES that each participating RU
i may want to share and the change in the total amount of
ES required for the SFCs in different time slots. Now, before
discussing how the proposed modified auction scheme can be
extended to a time-varying environment7, first we define the

E. Properties of the Auction Process
We note that once the auction process is executed, there
is always a possibility that the owners of the ES might
cheat on the amount of storage that they wanted put into
the market during auction [13]. In this context, we need to
investigate whether the proposed scheme is beneficial enough,
i.e., individually rational, for the RUs such that they are not
motivated to cheat, i.e., incentive compatible, once the auction
is executed.
Now for the individual rationality property, first we note that
all the players, i.e., the RUs and the auctioneer on behalf of the
SFCs, take part in the SLMFSG to maximize their benefits in
terms of their respected utility from their choice of strategies.
The choice of the RUs is to determine vector of ES x∗ such
that each of the RU can be benefitted at its maximum. On the
other hand, the strategy of the auctioneer is to choose a price pt
to maximize the savings of the SFCs. Accordingly, once both
the RUs and the auctioneer reach such a point of the game
when neither the owners nor the customers can be benefitted
more from choosing another strategy, the SLMFSG reaches
the SE. To this end, it is already proven in Theorem 1 that the
proposed Γ in this auction process must possesses a unique
SE. Therefore, as a subsequent outcome of the Theorem 1,

7 Certain loads such as lifts and water pumps in large apartment buildings
are not easy to schedule as they are shared by different users of the buildings.
Hence, we focus on the time variation of the storage sharing process by the
RUs of the considered system.

6 Please

note that the reservation price ri indicates how much each RU
i wants to be paid forPsharing its ES with the SFCs, and thus affects the
determination of total
x∗i and the total burden.

8

IEEE Trans. Smart Grid, Dec. 2015
which households are equipped with a dedicated battery to sell
the stored electricity to the grid [48]. Nonetheless, xi,t is also
affected by the amount of burden ηi,t−1 that an RU needed to
bear due to an oversupply of ES spaces, if there was any, in
the previous time slot. To this end, the amount of ES space
that an RU i can offer to the SFCs at t can be defined as
(
xi,t−1
if i ∈
/ Jt−1
xi,t =
. (21)
max(bi,t − (xi,t−1 − ηi,t−1 ), 0) otherwise

following parameters:
t: index of time slot.
T : total number of time slot.
ri,t : the reservation price of RU i ∈ N at time slot t.
ri = [ri,1 , ri,2 , . . . , ri,T ]: is the reservation price vector for
RU i ∈ N .
xi,t : the fraction of ES space that the RU i wants to shares
with the SFCs at time slot t.
xi = [xi,1 , xi,2 , . . . , xi,T ]: the vector of ES space shared by
RU i with the SFC during the total considered times.
bi,t : maximum available ES of RU i for sharing at time slot
t.
am,t : the bidding price of each SFC m ∈ M at time slot t.
am = [am,1 , am,2 , . . . , am,T ]: is the reservation price vector
for SFC m ∈ N .
qm,t : the required ES space by each SFC m at time slot t.
pt,t : the auction price at time slot t.
Ui,t : the benefit that each RU i achieves at time slot t.
Zt : the average cost saving per SFC at time slot t.
ηi,t : the burden that is shared by each participating RU at time
slot t.
Kt : number of participating SFCs in the modified auction
scheme at time slot t.
Jt : number of participating RUs in the modified auction
scheme at time slot t.
To this end, the utility function Ui,t of each RU i and the
average cost savings Zt per SFC at time slot t can be defined
as
Ui,t (xi,t ) = (pt,t − ri,t )xi,t − αi x2i,t ,

The SFC m, on the other hand, decides on the amount of ES
qm,t that it needs to share from the RUs at t based on the
random requirement of the shared facilities at t, the available
shared ES space qm,t−1 from time slot t − 1, and the random
generation of renewable energy sources, where appropriate.
Hence,
qm,t = f (qm,t−1 , renewables, facility requirement).

Now, if we assume that the fraction of shared ES available
from previous time slot is negligible, i.e., qm,t−1 ≈ 0, the
requirement qm,t can be assumed to be random for each
time slot t considering the random nature of both renewable
generation and energy requirement of shared facilities. Note
that this assumption is particularly valid if the SFC uses all its
shared ESs from the previous time slot for meeting the demand
of the shared facilities and cannot use them in considered time
slot. Nonetheless, please note that this assumption does not
imply that the inter-temporal relationship between the auction
process across different time slots is non-existent. The auction
process in one time slot still depends on other time slots due
to the inter-temporal dependency of xi,t via (21).
To this end, for the modeled xi,t ∀i ∈ N and qm,t ∀m ∈ M,
the proposed modified auction scheme studied in Section IV
can be adopted in each time slot t = 1, 2, . . . , T with a
view to maximize (19) and (20) ∀t. It is important to note
that the reservation price vector ri of each RU i ∈ N and
the bidding price vector am of each SFC m ∈ M can be
modeled through any existing time-varying pricing schemes
such as time-of-use price [3]. Now, p∗t = [p∗t,1 , p∗t,2 , . . . , p∗t,T ]
and x∗ = [x∗1 , x∗2 , . . . , x∗N ] constitute the solutions of the
proposed modified auction scheme in a time-varying condition,
if the x∗ comprises the solution vector of all ES spaces shared
by the participating RUs in each time slot t = 1, 2, . . . , T
for the auction price vector p∗t . Further, all the auction rules
adopted in each time slot of the proposed time-varying case
will be similar to the rules discussed in Section IV. Hence, the
solution of the proposed modified auction scheme for a timevarying environment also possesses the incentive compatibility
and individual rationality properties for each time slot.

(19)

and
Zt =

PKt −1

(am,t − pt,t )
Kt − 1

m=1

! J−1
X

xi,t

(22)

(20)

i=1

respectively8.
Now, at time slot t, the determination rule of the proposed
scheme determines the number of participating RUs and SFCs
based on their reservation and bidding prices for that time slot.
The number of participation is also motivated by the available
ES space of each RU and the requirement of each SFC.
However, unlike the static case, in a time-varying environment
the offered ES space by an RU at time slot t is influenced by
its contribution to the auction process in the previous time
slot. For instance, if an RU i receives a burden ηi,t−1 in time
slot t − 1, its willingness to share ES space xi,t at time slot
t may reduce. xi,t is also affected by the maximum amount
of ES bi,t available to RU i at t. For simplicity, we assume
that bi,t and αi,t do not change over different time slots.
Therefore, an RU i can offer to share the same amount of
ES space xi,t to the SFCs at time slot t if it did not share
any amount in time slot t − 1. An analogous example of such
arrangement can be found in FIT scheme with ES device in

V. C ASE S TUDY
For numerical case studies, we consider a number of RUs
at different blocks in a smart community that are interested in
allowing the SFCs of the community to jointly share their ES
devices. We stress that when there are a large number of RU
and SFCs in the system, the reservation and bidding prices
will vary significantly from one another. Therefore, it will be
difficult to find an intersection point to determine the highest

8 Please note that in each time slot t, (19) and (20) are related with each
other in a similar manner as (7) and (8) are related for the static case. However,
unlike the static case, the execution of the auction process in each time slot t
is affected by the value of parameters such as xi,t and pt for that particular
time slot.

9

ES shared by each RU

IEEE Trans. Smart Grid, Dec. 2015
TABLE I: Change of average utility achieved by each SFC and each RU in
the network (according to Algorithm 1) due to the change of the reluctance
of each RU for sharing one kWh ES with the SFC.

RU5

300

RU3

RU2

RU4

Reluctance Parameter
!
0.001
0.01
0.1
1

200
RU1

100

0
0

5

10

RU6, RU7, RU8

15
20
Number of iteration

25

30

Average utility per RU
(Net benefit)
3450.6
2883.6 (-16.43%)
1117.3 (-67.6%)
142.37 (-95.3%)

Average utility for SFC
(Average cost savings)
7500
4578.3 (-38.9%)
1671.7 (-77.7%)
259.03 (-96%)

4

Average utility for SFC

8

x 10

6

RU to put into the market for sharing. As can be seen from
the figure, on the one hand, RU 1, RU 2, and RU 3 reach the
SE much quicker than RU 4 and RU 5. On the other hand, no
interest for sharing any ES is observed for RU 4, 5 and 6.
This is due to the fact that as the interaction between the
auctioneer and the RUs continues, the auction price pt is
updated in each iteration. In this regard, once the auction price
for any RU becomes larger than its reservation price, it put all
its reserve ES to the market with an intention to be shared by
the SFCs. Due to this reason, RU 1, RU 2, and RU 3 put their
ESs in the market much sooner, i.e., after the 2nd iteration,
than RU 4 and RU 5 with higher reservation prices, whose
interest for sharing ES reaches the SE once the auction price
is encouraging enough for them to share their ESs after the
5th and 20th iterations. Unfortunately, the utilities of RU 6, 7,
and 8 are not convenient enough to take part in the auction
process, and therefore their shared ES fractions are 0.
We note that the demonstration of the convergence of the
SLMFSG to a unique SE subsequently demonstrates the proofs
of Theorem 1, Theorem 2, Theorem 3 and Corollary 1, which
are strongly related to the SE as explained in the previous
section. Now, we would like to investigate how the reluctance
parameters of the RUs may affect their average utility from
Algorithm 1, and thus affecting their decisions to share ES.
To this end, we first determine the average utility that is
experienced by each RU and SFC for a reluctance parameter of
αi = 0.001 ∀i. Then considering the outcome as a benchmark,
we show the effect of different reluctance parameters on the
achieved average benefits of each SFC and RU in Table I. The
demonstration of this property is necessary in order to better
understand the working principle of the designed technique
for ES sharing.
According to Table I, as the reluctance of each RU increases,
it becomes more uncomfortable, i.e., lower utility, to put its
ES in the market to be jointly owned by the SFCs. As a consequence, it also affects the average utility achieved by each
SFC. As shown in Table I, the reduction in average utilities per
RU are 16.73%, 67.6% and 95.3% respectively compared to
the average utility achieved by an RU at αi = 0.001 for every
ten times reduction in the reluctance parameter. For similar
settings, the reduction of average utility for the SFCs are
38.9%, 77.7% and 96% at αi = 0.01, 0.1 and 1 respectively.
Therefore, the proposed scheme will enable the RUs to put
more storage in the auction market if the related reluctance for
this sharing is small. Note that although the current investment
cost of batteries is very high compared to their relative short
life times, it is expected that battery costs will go down in
the near future [10] and become very popular for addressing

*

Z

4
2
0
0

5

10

15
20
Number of iteration

25

30

Fig. 4: Convergence of Algorithm 1 to the SE. At SE, the average utility per
SFC reaches its maximum and the ES that each RU wants to put into the
market for share reaches a steady state level that maximize their benefits.

reservation price pmax
according to the determination rule. So,
t
in this paper, we limit ourself to around 6 − 10 RUs. However,
having 6-10 RUs can in fact cover a large community, e.g.,
through aggregation such as discussed in [38], [39]. Here,
each RU is assumed to be a group of [5, 25] households,
where each household is equipped with a battery of capacity
25 kilo-Watt hour (kWh) [49]. The reluctance parameter of
all RUs are assumed to be similar, which is taken from range
of [0, 0.1]. It is important to note that αi is considered as a
design parameter in the proposed scheme, which we used to
map the reluctance of each RU to share its ES with the SFCs.
Such reluctance of sharing can be affected by parameters like
ES capacity, the condition of the environment (if applicable)
and the RU’s own requirement. Now, considering the different
system parameters in our proposed scheme, we capture these
two extremes with 0 (not reluctant) and 0.1 (highly reluctant).
The required electricity storage for each SFC is assumed to be
within the range of [100, 500] kWh. Nevertheless, the required
ES for sharing could be different if the usage pattern by the
users changes. Since, the type of ESs (and their associated
cost) used by different RUs can vary significantly [50], the
choices of reservation price to share their ESs with the SFCs
can vary considerably as well. In this context, we consider
that the reservation price set by each RU and SFC is taken
from a range of [20, 70]. It is important to note that all chosen
parameter values are particular to this study only, and may vary
according the availability and number of RUs, requirements of
SFCs, trading policy, time of the day/year and the country.
Now, we first show the convergence of Algorithm 1 to the
SE of the SLMFG in Fig. 4. For this case study, we assume
that there are five SFCs in the smart grid community that are
taking part in an auction process with eight RUs. From Fig. 4,
first we note that the proposed SLMFG reaches the SE after 20
interations when the average cost savings per SFC reaches its
maximum. Hence, the convergence speed, which is just few
seconds, is reasonable. Nonetheless, an interesting property
can be observed when we examine the choice of ES by each
10

IEEE Trans. Smart Grid, Dec. 2015
1100

for the SFCs, it would put a higher burden on the RUs to
carry. As a consequence, the relative utility from auction is
lower. Nevertheless, if the requirement of the SFCs is higher,
the sharing brings significant benefits to the RUs as can be
seen from Fig. 5. On the other hand, for higher reluctance,
RUs tend to share a lower ES amount, which then enables
them to endure a lower burden in case of lower demands from
the SFCs. This consequently enhances their achieved utility.
Nonetheless, if the requirement is higher from the SFCs, their
utility reduces subsequently compared to the RUs with lower
reluctance parameters. Thus, from observing the effects of
different αi ’s on the average utility per RU in Fig. 5, we
understand that, if the total required ES is smaller, RUs with
higher reluctance benefit more and vice versa. This illustrates
the fact that even RUs with high unwillingness to share their
ESs can be beneficial for SFCs of the system if their required
ESs are small. However, for a higher requirement, SFCs would
benefit more from having RUs with lower reluctances as they
will be interested in sharing more to achieve higher average
utilities.
Now, we discuss the computational complexity of the proposed scheme, which is greatly reduced by the determination
rule of the modified auction scheme as this rule determines
the actual number of participating RUs and SFCs in the
auction. We also note that after determining the number of
participating SFCs and RUs, the auctioneer iteratively interacts
with each of the RUs and sets the auction price with a
view to increase the average savings for the SFC. Therefore,
the main computational complexity of the modified auction
scheme stems from the interactions between the auctioneer
and the participating RUs to decide on the auction price. In
this context, the computational complexity of the problem falls
within a category of that of a single leader multiple follower
Stackelberg game, whose computational complexity, which
can be approximated to increase linearly with the number of
followers [38], and is shown to be reasonable in numerous
studies such as in [11] and [38]. Hence, the computational
complexity is feasible for adopting the proposed scheme.
Having an insight into the properties of the proposed auction
scheme, we now demonstrate how the technique can benefit
the RUs of the smart network compared to existing ES
allocation schemes such as equal distribution (ED) [38] and
FIT schemes [48]. ED is essentially an allocation scheme
that allows the SFCs to meet their total storage requirements
by sharing the total requirement equally from each of the
participating RUs. We assume that if the shared ES amount
exceeds the total amount of reservation storage that an RU puts
into the market, the RU will share its full reservation amount.
In FIT, which is a popular scheme for energy trading between
consumers and the grid, we assume that each RU prefers to
sell the same storage amount of energy to the grid at an FIT
price rate, e.g., 22 cents/kWh [52] instead of sharing the same
fraction of storage with the SFC. To this end, the resulting
average utilities that each RU can achieve from sharing its ES
space with the SFCs by adopting the proposed, ED, and FIT
schemes are shown in Table II.
From Table II, first we note that as the amount of required
ES by the SFCs increases the average utility achieved per

α = 0.001 (more willing to share)

Average utility achieved by the RUs

1000

α = 0.01 (less willing to share)

900
800
700
600
500
400
300

η0.001 > 0

η

η0.01 > 0

η

Supply > Demand
200
100

150

200

0.001

0.01

η

>0

0.001

η

=0

0.01

=0

=0

Supply < Demand

250
300
350
400
450
500
Required battery space by the SFCs (kWh)

550

600

Fig. 5: Effect of change of required ES amount by the SFCs on the achieved
average utility per RU.

intermittency of renewables [51]. We have foreseen such a
near future when our proposed scheme will be applicable to
gain the benefit of storage sharing and thus motivate the RUs
to keep their αi ∀i small. According to the observation from
Table I, it can further be said that if the reluctance parameters
of RUs change over either different days or different time slots,
the performance of the system in terms of average utility per
RU and average cost savings per SFC will change accordingly
for the given system parameters.
Once all the participating RUs put their ES amount into
the auction market, they are distributed according to the
allocation rule described in (16) and (18). In this regard,
we investigate how the average utility of each RU is altered as the total storage amount required by the SFCs
changes from in the network. For this particular case, the
considered total ES requirement of the SFCs is assumed to
be 100, 150, 200, 250, 300, 350, 400, 450, 500, 550 and 600. In
general, as shown in Fig. 5, the average utility of each RU
initially increases with the increase required by the SFCs and
eventually becomes saturated to a stable value. This is due to
the fact that as the required amount of ES increases, the RU
can share more of its reserved ES that it put into the market
with the SFCs with the determined auction price from the
SLMFSG. Hence, its utility increases. However, each RU has
a particular fixed ES amount that it puts into the market to
share. Consequently, once the shared ES amount reaches its
maximum, even with the increase of requirement by the SFCs
the RU cannot share more, i.e., ηi = 0. Therefore, its utility
becomes stable without any further increment. Interestingly,
the proposed scheme, as can be seen in Fig. 5, favors the
RUs with higher reluctance more when the ES requirement
by the SFCs is relatively lower and favors the RUs with
lower reluctance during higher demands. This is due to the
way we have designed the proposed allocation scheme, which
is dictated by the burden in (18) and the allocation of ES
through (16). We note that, according to (11), if αi is lower,
the RU i will put a higher amount of ES in the market to
share. However, if the total required amount of ES is lower
11

IEEE Trans. Smart Grid, Dec. 2015
TABLE II: Comparison of the change of average utility per RU in the smart grid system as the required total amount of energy storage required by the SFCs
varies.

Required ES space by the SFCs
Average utility (net benefit) of RU for equal distribution (ED) scheme
Average utility (net benefit) of RU for FIT scheme
Average utility (net benefit) of RU for proposed scheme
Percentage improvement (%) compared to ED scheme
Percentage improvement (%) compared to FIT scheme

ES shared by each RU (kWh)

ES available to share (kWh)

RU also increases for all the cases. The reason for this
increment is explained in Fig. 5. Also, in all the studied
cases, the proposed scheme shows a considerable performance
improvement compared to the ED and FIT schemes. An
interesting trend of performance improvement can be observed
if we compare the performance of the proposed scheme with
the ED and FIT performances for each of the ES requirements.
In particular, the performance of the proposed scheme is higher
as the requirement of the ES increases from 200 to 350.
However, the improvement is relatively less significant as the
ES requirement switches from 400 to 450. This change in
performance can be explained as follows:
In the proposed scheme, as we have seen in Fig. 5, the
amount of ES shared by each participating RU is influenced
by their reluctance parameters. Hence, even the demand of
the SFCs could be larger, the RUs may choose not to share
more of their ES spaces once their reluctance is limited. In
this regard, the RUs in the current case study increase their
share of ES as the requirement by the SFCs increases, which
in turn produces higher revenue for the RUs. Furthermore,
once the RUs choice of ESs reach the saturation, the increase
in demand, i.e., from 200 to 350 in this case, does not affect
their share. As a consequence, their performance improvement
is not as noticeable as the previous four cases. Nonetheless, for
all the considered cases, the auction process performs superior
to the ED scheme with an average performance improvement
of 34.76%, which clearly shows the value of the proposed
methodology to adopt joint ES sharing in smart grid. The
performance improvement with respect to the FIT scheme,
which is 34.34% on average, is due to the difference between
the determined auction price and the price per unit of energy
for the FIT scheme.
Finally, we show how the decision making process of each
RU in the system is affected by its decision in the previous
time slot and the total storage requirement by the SFCs. The
total number of time slots that are considered to show this
performance analysis is four. In this context, we assume that
there are five RUs in the system with ES of 100, 200, 300, 200
and 200 kWh respectively to share with the SFCs. The total
ES requirements of the SFCs for considered four time slot
are 500, 250, 500, and 100. Please note that these numbers are
considered for this case study only and may have different
values for different scenarios. Now, in Fig. 6, we show the
available ES to each of the RUs at the begining of each time
slot and how much they are going to share if the modified
auction scheme is adopted in each time slot. For a simple
analysis, we assume that once an RU shares its total available
ES, it cannot share its ES for the remaining of the time slots.

200
536.52
537.83
629.82
17.4
17.1

250
581.85
583.16
789.82
35.74
35.43

300
624.52
626.83
944.26
51.19
50.63

350
669.85
673.16
960.09
43.32
42.61

400
715.19
717.50
960.09
34.24
33.81

450
757.85
759.16
960.09
26.68
26.46

300
RU1
RU2
RU3
RU4
RU5

200

100

0

1

2
3
Number of time slot

4

300
RU1
RU2
RU3
RU4
RU5

200

100

0

1

2
3
Number of time slot

4

Fig. 6: Demonstration of how the proposed modified auction scheme can be
extended to time varying system. The reservation ES amount varies by the
RUs varies between different time slots based on their sharing amount in the
previous time slot. The total required storage by the SFCs is chosen randomly
due to the reasons explained in Section IV-F.

The reservation prices are considered to change from one time
to the next based on a predefined time of use price scheme.
Now, as can be seen from Fig. 6, in time slot 1, RU1 and RU2
share all their available ESs with the SFC, whereby other RUs
do not share their ESs due to the reasons explained in Fig. 4.
Since, the total requirement is 500, therefore neither of RU1
and RU2 needs to carry any burden. In time slot 3, only RU3
shares its ESs of 300 to meet the requirement. As the SFC’s
requirement is lower than the supply, RU3 needs to carry a
burden of 50 kWh. Similarly, in time slot 3 and 4, all of RU3,
RU4 and RU5 take part in the energy auction scheme as they
have enough ES to share with the SFC. However, the ES to
share in time slot 4 stems from the burden of oversupply from
time slot 3. The scheme is not shown for more than time slot
4 as the available ES from all RUs is already shared by the
SFCs by the end of time slot 4. Thus, the proposed modified
auction scheme can successfully capture the time variation if
the scheme is modified as given in Section IV-F.
VI. C ONCLUSION
In this paper, we have modeled a modified auction based
joint energy storage ownership scheme between a number of
residential units (RUs) and shared facility controllers (SFCs)
in smart grid. We have designed a system and discussed the
determination, payment and allocation rule of the auction,
where the payment rule of this scheme is facilitated by a
Single-leader-multiple-follower Stackelberg game (SLMFSG)
12

IEEE Trans. Smart Grid, Dec. 2015
between the auctioneer and the RUs. The properties of the
auction scheme and the SLMFSG have been studied, and it has
been shown that the proposed auction possesses the individual
rationality and the incentive compatibility properties leveraged
by the unique Stackeberg equilibrium of the SLMFSG. We
have proposed an algorithm for the SLMFSG, which has been
shown to be guaranteed to reach the SE and that also facilitates
the auctioneer and the RUs to decide on the auction price as
well as the amount of ES to be put into the market for joint
ownership.
A compelling extension of the proposed scheme would be
to study of the feasibility of scheduling of loads such as
lifts and water machines in shared space. Another interesting
research direction would be to determine how a very large
number of SFCs or RUs with different reservation and bidding
prices can take part in such a modified auction scheme. One
potential way to look at this problem can be from a cooperative
game-theoretic point-of-view in which the SFCs and RUs may
cooperate to decide on the amount of reservation ES and
bidding price they would like to put into the market so as
to participate in the auction and benefit from sharing. Another
very important, yet interesting, extension of this work would
be to investigate how to quantify the reluctance of each RU to
participate in the ES sharing. Such quantification of reluctance
(or, convenience) will also enable the practical deployment of
many energy management schemes already described in the
literature.

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