1

Sharing Storage in a Smart Grid: A Coalitional
Game Approach

arXiv:1712.02909v1 [] 8 Dec 2017

Pratyush Chakraborty∗, Enrique Baeyens∗, Kameshwar Poolla, Pramod P. Khargonekar, and Pravin Varaiya

Abstract—Sharing economy is a transformative socio-economic
phenomenon built around the idea of sharing underused resources and services, e.g. transportation and housing, thereby
reducing costs and extracting value. Anticipating continued
reduction in the cost of electricity storage, we look into the
potential opportunity in electrical power system where consumers
share storage with each other. We consider two different scenarios. In the first scenario, consumers are assumed to already
have individual storage devices and they explore cooperation
to minimize the realized electricity consumption cost. In the
second scenario, a group of consumers is interested to invest in
joint storage capacity and operate it cooperatively. The resulting
system problems are modeled using cooperative game theory. In
both cases, the cooperative games are shown to have non-empty
cores and we develop efficient cost allocations in the core with
analytical expressions. Thus, sharing of storage in cooperative
manner is shown to be very effective for the electric power system.
Index Terms—Storage Sharing, Cooperative Game Theory,
Cost Allocation

I. I NTRODUCTION
A. Motivation
The sharing economy is disruptive and transformative socioeconomic trend that has already impacted transportation and
housing [1]. People rent out (rooms in) their houses and
use their cars to provide transportation services. The business
model of sharing economy leverages under utilized resources.
Like these sectors, many of the resources in electricity grid
is also under-utilized or under-exploited. There is potential
benefit in sharing the excess generation by rooftop solar
panels, sharing flexible demand, sharing unused capacity in
the storage services, etc. Motivated by the recent studies [2]
predicting a fast drop in battery storage prices, we focus on
sharing electric energy storage among consumers.
B. Literature Review
Storage prices are projected to decrease by more than 30%
by 2020. The arbitrage value and welfare effects of storage in
This research is supported by the National Science Foundation under grants
EAGER-1549945, CPS-1646612, CNS-1723856 and by the National Research
Foundation of Singapore under a grant to the Berkeley Alliance for Research
in Singapore
∗ The first two authors contribute equally
Corresponding author P. Chakraborty is with the Department of Mechanical
Engineering, University of California, Berkeley, CA, USA
E. Baeyens is with the Instituto de las Tecnologı́as Avanzadas de la
Producción, Universidad de Valladolid, Valladolid, Spain
K. Poolla and P. Varaiya are with the Department of Electrical Engineering
and Computer Science, University of California, Berkeley, CA, USA
P. P. Khargonekar is with the Department of Electrical Engineering and
Computer Science, University of California, Irvine, CA, USA

electricity markets has been explored in literature. In [3], the
value of storage arbitrage was studied in deregulated markets.
In [4], the authors studied the role of storage in wholesale
electricity markets. The economic viability of the storage
elements through price arbitrage was examined in [5]. Agentbased models to explore the tariff arbitrage opportunities for
residential storage systems were introduced in [6]. In [7], [8],
authors address the optimal control and coordination of energy
storage. All these works explore the economic value of storage
to an individual, not for shared services. Sharing of storage
among firms has been analyzed using non-cooperative game
theory in [9]. But the framework needs a spot market among
the consumers and also coordination is needed among the firms
that are originally strategic.
In this paper, we explore sharing storage in a cooperative
manner among consumers. Cooperative game theory has significant potential to model resource sharing effectively [10].
Cooperation and aggregation of renewable energy sources
bidding in a two settlement market to maximize expected
and realized profit has been analyzed using cooperative game
theory in [11]–[13]. Under a cooperative set-up, the cost
allocation to all the agents is a crucial task. A framework
for allocating cost in a fair and stable way was introduced
in [14]. Cooperative game theoretic analysis of multiple demand response aggregators in a virtual power plant and their
cost allocation has been tackled in [15]. In [16], sharing opportunities of photovoltaic systems (PV) under various billing
mechanisms were explored using cooperative game theory.
C. Contributions and Paper Organization
In this paper, we investigate the sharing of storage systems
in a time of use (TOU) price set-up using cooperative game
theory. We consider two scenarios. In the first one, a group of
consumers already own storage systems and they are willing to
operate all together to minimize their electricity consumption
cost. In a second scenario, a group of consumers wish to
invest in a shared common storage system and get benefit
for long term operation in a cooperative manner. We model
both the cases using cooperative game theory. We prove that
the resulting games developed have non-empty cores, i.e.,
cooperation is shown to be beneficial in both the cases. We also
derive closed-form and easy to compute expressions for cost
allocations in the core in both the cases. Our results suggest
that sharing of electricity storage in a cooperative manner is
an effective way to amortize storage costs and to increase
its utilization. In addition, it can be very much helpful for
consumers and at the same time to integrate renewables in the

2

Consumer 1

Consumer 2

Consumer 3

Storage

Storage

Storage

Consumer 1

Consumer 2

Consumer 3

which corresponds to γi ∈ [0, 1]. The consumers discharge
their storage during peak hours and charge them during offpeak hours.
The daily cost of storage of a consumer i ∈ N for the peak
period consumption xi depends on the capacity investment Ci
and is given by

Storage

J(xi , Ci ) = πi Ci + πh (xi − Ci )+ + πℓ min{Ci , xi },

Fig. 1. Configuration of three consumers in the two analyzed scenarios

system, because off-peak periods correspond to large presence
of renewables that can be stored for consumption during peak
periods.
The remainder of the paper is organized as follows. In
Section II, we formulate the cooperative storage problems. A
brief review of cooperative game theory is presented in Section
III. In Section IV, we state and explain our main results. A
case study illustrating our results using real data from Pecan
St. Project is presented in Section VI. Finally, we conclude
the paper in Section VII.
II. P ROBLEM F ORMULATION
A. System Model
We consider a set of consumers indexed by i ∈ N :=
{1, 2, . . . , N }. The consumers invest in storage. The consumers cooperate and share their storage with each other. We
consider two scenarios here. In the scenario I, the consumers
already have storage and they operate with storage devices
connected to each other. In the scenario II, the consumers
wish to invest in a common storage. There is a single meter
for this group of consumers. We assume that there is necessary
electrical connection between all the consumers for effective
sharing. We ignore here the capacity constraints, topology
or losses in the connecting network. The configuration of
the scenarios with three consumers are depicted in Figure 1.
Examples of the situations considered here include consumers
in an industrial park, office buildings on a campus, or homes
in a residential complex.

(4)

where πi Ci is the capital cost of acquiring Ci units of storage
capacity, πh (xi − Ci )+ is the daily cost of the electricity
purchase during peak price period, and πℓ min{xi , Ci } is the
daily cost of the electricity purchase during off-peak period to
be stored for consumption during the peak period. We ignore
the off-peak period electricity consumption of the consumer
from the expression of J as its expression is independent of
the storage capacity. The daily peak consumption of electricity
is not known in advance and we assume it to be a random
variable. Let F be the joint cumulative distribution function
(CDF) of the collection of random variables {xi : i ∈ N } that
represents the consumptions of the consumers in N . If S ⊆ N
is a subset of consumers, then xS denotes the aggregated peak
consumption of S and its CDF is FS .
The daily cost of storage of a group of P
consumers S ⊆ N
with aggregated peak consumption xS = i∈S xi and joint
storage capacity CS is
J(xS , CS ) = πS CS + πh (xS − CS )+ + πℓ min{CS , xS }
(5)
where πS is the daily capital cost of aggregated storage of the
group amortized during its life span. Note that the individual
storage costs (4) are obtained from (5) for the singleton sets
S = {i}.
The daily cost of storage given by (4) and (5) are random
variables with expected values
JS (CS ) = EJS (xS , CS ),

S ⊆ N.

(6)

In the sequel, we will distinguish between the random variables and their realized values by using bold face fonts xS for
the random variables and normal fonts xS for their realized
values.

B. Cost of Storage
Each day is divided into two periods –peak and off-peak.
There is a time-of-use pricing. The peak and off-peak period
prices are denoted by πh and πℓ respectively. The prices are
fixed and known to all the consumers.
Let πi be the daily capital cost of storage of the consumer
i ∈ N amortized over its life span. Let the arbitrage price be
defined by
πδ := πh − πℓ
(1)
and define the arbitrage constant γi as follows:
γi :=

πδ − πi
πδ

(2)

In order to have a viable arbitrage opportunity, we need
πi ≤ πδ

(3)

C. Quantifying the Benefit of Cooperation Benefit
We are interested in studying and quantifying the benefit of
cooperation in the two scenarios. In the first scenario, the consumers already have installed storage capacity {Ci : i ∈ N }
that they acquired in the past. Each of the consumers can have
a different storage technology that was acquired at a different
time compared to the other consumers. Consequently, each
consumer has a different daily capital cost πi . The consumers
aggregate their storage capacities and they operate using the
same strategy, they use the aggregated storage capacity to store
energy during off-peak periods that they will later use during
peak periods. By aggregating storage devices, the unused
capacity of some consumers is used by others producing
cost savings for the group. We analyze this scenario using
cooperative game theory and develop an efficient allocation

3

rule of the daily storage cost that is satisfactory for every
consumer.
In the second scenario, we consider a group of consumers
that join to buy storage capacity that they want to use in a
cooperative way. First, the group of consumers have to make
a decision about how much storage capacity they need to
acquire and then they have to share the expected cost among
the group participants. The decision problem is modeled as an
optimization problem where the group of consumers minimize
the expected cost of daily storage. The problem of sharing the
expected cost is modeled using cooperative game theory. We
quantify the reduction in the expected cost of storage for the
group and develop a mechanism to allocate the expected cost
among the participants that is satisfactory for all of them.
III. BACKGROUND : C OALITIONAL G AME T HEORY
C OST S HARING

FOR

Game theory deals with rational behavior of economic
agents in a mutually interactive setting [17]. Broadly speaking,
there are two major categories of games: non-cooperative
games and cooperative games. Cooperative games (or coalitional games) have been used extensively in diverse disciplines
such as social science, economics, philosophy, psychology
and communication networks [10], [18]. Here, we focus on
cooperative games for cost sharing [19].
Let N := {1, 2, . . . , N } denote a finite collection of
players. In a cooperative game for cost sharing, the players
want to minimize their joint cost and share the resulting cost
cooperatively.
Definition 1 (Coalition): A coalition is any subset S ⊆ N .
The number of players in a coalition S is denoted by its
cardinality, |S|. The set of all possible coalitions is defined
as the power set 2N of N . The grand coalition is the set of
all players, N .
Definition 2 (Game and Value): A cooperative game is
defined by a pair (N , v) where v : 2N → R is the value
function that assigns a real value to each coalition S ⊆ N .
Hence, the value of coalition S is given by v(S). For the cost
sharing game, v(S) is the total cost of the coalition.
Definition 3 (Subadditive Game): A cooperative game
(N , v) is subadditive if, for any pair of disjoint coalitions
S, T ⊂ N with S ∩T = ∅, we have v(S)+v(T ) ≥ v(S ∪T ).
Here we consider the value of the coalition v(S) is transferable among players. The central question for a subadditive
cost sharing game with transferrable value is how to fairly
distribute the coalition value among the coalition members.
Definition 4 (Cost Allocation): A cost allocation for the
coalition S ⊆ N is a vector x ∈ RN whose entry xi represents
the allocation to member i ∈ S (xi = 0, i ∈
/ S).
For any coalition S ⊆ N , let xS denote the sum
P of cost
allocations for every coalition member, i.e. xS = i∈S xi .
Definition 5 (Imputation): A cost allocation x for the grand
coalition N is said to be an imputation if it is simultaneously
efficient –i.e. v(N ) = xN , and individually rational –i.e.
v(i) ≥ xi , ∀i ∈ N . Let I denote the set of all imputations.
The fundamental solution concept for cooperative games is
the core [17].

Definition 6 (The Core): The core C for the cooperative
game (N , v) with transferable cost is defined as the set of
cost allocations such that no coalition can have cost which is
lower than the sum of the members current costs under the
given allocation.

(7)
C := x ∈ I : v(S) ≥ xS , ∀S ∈ 2N .

A classical result in cooperative game theory, known as
Bondareva-Shapley theorem, gives a necessary and sufficient
condition for a game to have nonempty core. To state this
theorem, we need the following definition.
Definition 7 (Balanced Game and Balanced Map): A cooperative game (NP
, v) for cost sharing is balanced if for any
balanced map α, S∈2N α(S)v(S) ≥ v(N ) where the map
α : 2N →
P [0, 1] is said to be balanced if for all i ∈ N ,
we have S∈2N α(S)1S (i) = 1, where 1S is the indicator
function of the set S, i.e. 1S (i) = 1 if i ∈ S and 1S (i) = 0
if i 6∈ S.
Next we state the Bondareva-Shapley theorem.
Theorem 1 (Bondareva-Shapley Theorem [10]): A coalitional game has a nonempty core if and only if it is balanced.
If a game is balanced, the nucleolus [18] is a solution that
is always in the core.
IV. M AIN R ESULTS
A. Scenario I: Realized Cost Minimization with Already Procured Storage Elements
Our first concern is to study if there is some benefit in
cooperation of the consumers by sharing the storage capacity
that they already have. To analyze this scenario we shall
formulate our problem as a coalitional game.
1) Coalitional Game and Its Properties: The players of the
cooperative game are the consumers that share their storage
and want to reduce their realized joint storage investment cost.
For any coalition S ⊆ N , the cost of the coalition is u(S)
which
is the realized cost of the joint storage investment CS =
P
C
i∈S i . Each consumer may have a different daily capital
cost of storage {πi : i ∈ N }, because they did not necessarily
their storage systems at the same time or at the same price for
KW. The realized cost
P of the joint storage for the peak period
consumption xS = i∈S xi is given by
u(S) = J(xS , CS )

(8)

where J was defined in (5). Since we are using the realized
value of the aggregated peak consumption xS , J(xS , CS ) is
not longer a random variable.
In order to show that cooperation is advantageous for the
members of the group, we have to prove that the game
is subadditive. In such a case, the joint daily investment
cost of the consumers is never greater that the sum of the
individual daily investment costs. Subadditivity of the cost
sharing coalitional game is established in Theorem 2.
Theorem 2: The cooperative game for storage investment
cost sharing (N , u) with the cost function u defined in (8) is
subadditive.

4

Proof: See appendix.
However, subadditivity is not enough to provide satisfaction
of the coalition members. We need a stabilizing allocation
mechanism of the aggregated cost. Under a stabilizing cost
sharing mechanism no member in the coalition is impelled
to break up the coalition. Such a mechanism exists if the cost
sharing coalitional game is balanced. Balancedness of the cost
sharing coalitional game is established in Theorem 3.
Theorem 3: The cooperative game for storage investment
cost sharing (N , u) with the cost function u defined in (8) is
balanced.
Proof: See the appendix.
2) Sharing of Realized Cost: Since the cost sharing cooperative game (N , u) is balanced, its core is nonempty and there
always exist cost allocations that stabilize the grand coalition.
One of this coalitions is the nucleolus while another one is the
allocation that minimizes the worst case excess [12]. However,
computing these allocations requires solving linear programs
with a number of constraints that grows exponentially with
the cardinality of the grand coalition and they can be only
applied for coalitions of moderate size. As an alternative to
these computationally intensive cost allocations, we propose
the following cost allocation.
Allocation 1: Define the cost allocation {ξi : i ∈ N } as
follows:

πi Ci + πh (xi − Ci ) + πℓ Ci , if xN ≥ CN
ξi :=
(9)
πi Ci + πℓ xi ,
if xN < CN
for all i ∈ N .
We establish in Theorem 4, this cost allocation belongs to
the core of the cost sharing cooperative game.
Theorem 4: The cost allocation {ξi : i ∈ N } defined in
Allocation 1 belongs to the core of the cost sharing cooperative
game (N , u).
Proof: See appendix.
Unlike the nucleolus or the cost allocation minimizing the
worst-case excess, Allocation 1 has an analytical expression
and can be easily obtained without any costly computation.
Thus, we have developed a strategy such that consumers
that independently invested in storage, and are subject to a
two period (peak and off-peak) TOU pricing mechanism can
reduce their costs by sharing their storage devices. Moreover,
we have proposed a cost sharing allocation rule that stabilizes
the grand coalition. This strategy can be considered a weak
cooperation because each consumer acquired its storage capacity independently of each other, but they agree to share the
joint storage capacity.
In the next section we consider a stronger cooperation
problem, where a group of consumers decide to invest jointly
in storage capacity.
B. Scenario II: Expected Cost Minimization for Joint Storage
Investment
In this scenario, we consider a group of consumers indexed
by i ∈ N , that decide to jointly invest in storage capacity.
We are interested in studying whether cooperation provides a
benefit for the coalition members for the long term.

1) Coalitional Game and Its Properties: Similar to the
previous case, only the peak consumption is relevant in the
investment decision. Let xi denote the daily peak period
consumption of consumer i ∈ N . Unlike the previous scenario, here xi is a random variable with marginal cumulative
distribution function (CDF) Fi . The daily cost of the consumer
i ∈ N depends on the storage capacity investment of the
consumer as per (4). This cost is also a random variable. If
the consumer is risk neutral, it acquires the storage capacity
Ci∗ that minimizes the expected value of the daily cost
Ci∗ = arg min Ji (Ci ),

(10)

Ji (Ci ) = EJ(xi , Ci ),

(11)

Ci ≥0

where

and πS is the daily capital cost of storage amortized over its
lifespan that in this case is the same for each of the consumers
–i.e. πi = πS for all i ∈ N , because we assume that they buy
storage devices of the same technology at the same time. This
problem has been previously solved in [9] and its solution is
given by Theorem 5.
Theorem 5 ( [9]): The storage capacity of a consumer i ∈ N
that minimizes its daily expected cost is Ci∗ , where
πδ − πS
= γS
Fi (Ci∗ ) =
πδ
and the resulting optimal cost is
Ji∗ = Ji (Ci∗ ) = πℓ E[xi ] + πS E[xi | xi ≥ Ci∗ ].

(12)

Let us consider a group of consumers S ⊆ N that decide
to join to invest in joint storage capacity.
P The joint peak
consumption of the coalition is xS =
i∈S xi with CDF
FS . We also assume that the joint CDF of all the agent’s
peak consumptions F is known or can be estimated from
historical data. By applying Theorem 5, the optimal investment
in storage capacity of the coalition S ⊆ N is CS∗ such that
FS (CS∗ ) = γS and the optimal cost is
JS∗ = JS (CS∗ ) = πℓ E[xS ] + πS E[xS | xS ≥ CS∗ ].

(13)

Consider the cost sharing cooperative game (N , v) where
the cost function v : 2N → R is defined as follows
v(S) = JS∗ = arg min JS (CS ),
CS ≥0

(14)

where JS∗ was defined in (13).
Similar to the case of consumers that already own storage
capacity and decide to join to reduce their costs, here we prove
that the cooperative game is subadditive so that the consumer
obtain a reduction of cost. This is the result in Theorem 6.
Theorem 6: The cooperative game for storage investment
cost sharing (N , v) with the cost function v defined in (14) is
subadditive.
Proof: See appendix.
We also need a cost allocation rule that is stabilizing.
Theorem 7 establishes that the game is balanced and has a
stabilizing allocation.
Theorem 7: The cooperative game for storage investment
cost sharing (N , v) with the cost function v defined in (14) is
balanced.

5

Proof: See appendix.
2) Stable Sharing of Expected Cost: Similar to the previous
scenario, we were able to develop a cost allocation rule that is
in the core. This cost allocation rule has an analytical formula
and can be efficiently computed. This allocation rule is defined
as follows.
Allocation 2: Define the cost allocation {ζi : i ∈ N } as
follows:
∗
ζi := πℓ E[xi ] + πS E[xi | xN ≥ CN
], i ∈ N .

(15)

In the next theorem, we prove that Allocation 2 provides
a sharing mechanism of the expected daily storage cost of a
coalition of agents that is in the core of the cooperative game.
Theorem 8: The cost allocation {ζi : i ∈ N } defined in
Allocation 2 belongs to the core of the cost sharing cooperative
game (N , v).
Proof: See appendix.
3) Sharing of Realized Cost: Based on the above results,
the consumers can invest on joint storage and they will make
savings for long term. But the cost allocation ζi defined by
(15) is in expectation. The realized allocation will be different
due to the randomness of the daily consumption. Here we
develop a daily cost allocation for the k-th day as
k
ρki = βi πN
,

(16)

k
where πN
is the realized cost for the grand coalition on the
k-th day and βi = PNζi ζ .
i=1 i
PN
P
N
k
As i=1 βi = 1, i=1 ρki = πN
and the cost allocation
is budget
balanced.
Also
using
strong
law of large numbers,
PK
1
k
ρ
→
ζ
as
K
→
∞
and
the
realized allocation is
i
k=1 i
K
strongly consistent with the fixed allocation ζi .

V. B ENEFIT OF C OOPERATION
A. Scenario I
The benefit of cooperation by joint operation of storage
reflected in the total reduction of cost is given by
X
X
Ji − JS = πh ( (xi − Ci )+ − (xS − CS )+ )+
i∈S

i∈S

X
πℓ (
min{Ci , xi } − min{CS , xS }),

(17)

i∈S

where the reduction for individual agent with cost allocation
(9) is

πδ (Ci − xi )+ , if xN ≥ CN
Ji − ζi :=
(18)
πδ (xi − Ci )+ , if xN < CN
B. Scenario II
The benefit of cooperation given by the reduction in the
expected cost that the coalition S obtains by jointly acquiring
and exploiting the storage is
X
Ji∗ − JS∗ =
i∈S

πS

X
i∈S

E[xi | xi ≥ Ci∗ ] − πS E[xS | xS ≥ CS∗ ],

(19)

Fig. 2. Estimated CDFs of the peak consumption of the five households and
their aggregated consumption
TABLE I
C ORRELATION COEFFICIENTS FOR THE FIVE
1
2
3
4
5

1
1.000000
0.363586
0.297733
0.292073
0.486665

2
0.363586
1.000000
0.132320
0.453056
0.157210

3
0.297733
0.132320
1.000000
0.085869
0.365212

HOUSEHOLDS

4
0.292073
0.453056
0.085868
1.000000
-0.056696

5
0.486665
0.157210
0.365212
-0.056696
1.000000

and the reduction in expected cost of each participant assuming
that the expected cost of the coalition is split using cost
allocation (15) is
Ji∗ − ζi = πS E[xi | xi ≥ Ci∗ ] − πS E[xi | xS ≥ CS∗ ]. (20)
VI. C ASE S TUDY
We develop a case study to illustrate our results. For this
case study, we used data from the Pecan St project [20]. We
consider a two-period ToU tariff with πh = 55¢/KWh, and
πℓ = 20¢/KWh. Electricity storage is currently expensive. The
amortized cost of Tesla’s Powerwall Lithium-ion battery is
around 25¢/KWh per day. But storage prize is projected to
reduce by 30% by 2020 [21]. Keeping in mind this projection,
we consider πS = 15¢/KWh.
A group of five household decide to join to acquire storage.
Using historical data of 2016, we estimate the individual CDFs
of their daily peak consumptions and the CDF of the daily
joint peak consumption. Peak consumption period in Texas
corresponds to non-holidays and non-weekends from 7h to
23h. The estimated CDFs for peak consumption are depicted
in Figure 2. From this figure, we can see that the shape of the
CDFs are quite similar for the five households. The correlation
coefficients of these five households are given in Table I.
Although the shape of the CDFs are very similar, the peak
consumptions are not completely dependent. This means that
there is room for reduction in cost by making a coalition.
The optimal investments in storage for the five households
and for the grand coalition are given in Table II. Also in
this table, we show the allocation of the expected storage
cost given by (15). The reduction in cost for the consumers
coalition is about 5%, however those with less correlation with
the other, have a larger reduction. Consumers 3 and 4 have
cost reductions higher than 7%, while consumer 1, whose
consumption is more correlated with the other, have about
2.4% of cost reduction.

6

TABLE II
O PTIMAL STORAGE CAPACITY INVESTMENTS ( IN KW H ), MINIMAL
EXPECTED STORAGE COST ( IN $) AND EXPECTED COST ALLOCATION OF
THE GRAND COALITION ( IN $)
Ci∗
Ji∗
ζi

1
22.98
899.76
882.45

2
14.09
579.79
543.10

3
12.64
600.88
550.02

4
13.21
525.51
488.20

5
29.82
1189.41
1140.35

N
95.58
3604.13
3604.13

TABLE III
A LLOCATION OF THE REALIZED COST FOR S CENARIO I FOR THE FIRST
TEN DAYS OF THE YEAR ( IN $)
Day
1
2
3
4
5
6
7
8
9
10

ξ1
492.66
464.89
541.21
675.74
761.41
646.05
654.47
583.59
640.46
604.49

ξ2
612.83
624.96
482.61
373.95
403.49
516.53
760.99
411.25
394.04
446.14

ξ3
436.88
343.61
299.84
377.64
405.52
404.89
387.80
533.00
482.85
475.46

ξ4
549.61
567.21
541.40
418.01
371.64
573.17
536.92
455.56
483.24
310.22

ξ5
904.69
947.27
820.46
734.10
799.23
812.54
797.46
831.97
787.20
791.60

R EFERENCES

Now, we assume that the five households buy storage independently and then decide to cooperate by sharing their storage
to reduce the realized cost. This corresponds to Scenario I.
For simplicity of computation and comparison with scenario
II, we consider πi = πS for all i. The realized cost is allocated
using (9). In Table III, we show the allocation of the realized
aggregated cost for the ten first days of 2016, assuming that
the households have storage capacities {Ci∗ : i ∈ N }.
Finally, in Figure 3, we depict the evolution of the average
allocation of the realized cost of storage to each household for
the 2016 year. The average allocation for D days is given by
D
1 X
ξi , i ∈ N ,
ξ¯i (D) =
D i=1

(21)

where D is the number of days. The average cost allocation is
compared to the optimal expected costs Ji∗ . Assuming stationarity of the peak consumptions random variables, the expected
allocation converge to some values ξi∞ = limD→∞ ξ̄i (D) ≤
Ji∗ for i ∈ N , as it is shown in Figure 3.
VII. C ONCLUSIONS
In this paper, we explored sharing opportunities of electricity storage elements among a group of consumers. We

Fig. 3. Average allocation of the realized storage cost

used cooperative game theory as a tool for modeling. Our
results prove that cooperation is beneficial for agents that
either already have storage capacity or want to acquire storage
capacity. In the first scenario, the different agents only need
the infrastructure to share their storage devices. In such a case
the operative scheme is really simple, because each agent only
has to storage at off-peak periods as much as possible energy
that they will consume during peak periods. At the end of the
day, the realized cost is shared among the participants. In the
second scenario, the coalition members can take an optimal
decision about how much capacity they jointly acquire by
minimizing the expected daily storage cost. We showed that
the cooperative games in both the cases are balanced. We also
developed allocation rules with analytical formulas in both
the cases. Thus, our results suggest that sharing of storage in
a cooperative way is very much useful for all the agents and
the society.

[1] H. Heinrichs, “Sharing economy: a potential new pathway to sustainability,” Gaia, vol. 22, no. 4, p. 228, 2013.
[2] N. Kittner, F. Lill, and D. M. Kammen, “Energy storage
deployment and innovation for the clean energy transition,”
Nature Energy, vol. 2, p. 17125, 07 2017. [Online]. Available:
http://dx.doi.org/10.1038/nenergy.2017.125
[3] F. Graves, T. Jenkin, and D. Murphy, “Opportunities for electricity
storage in deregulating markets,” The Electricity Journal, vol. 12, no. 8,
pp. 46–56, 1999.
[4] R. Sioshansi, P. Denholm, T. Jenkin, and J. Weiss, “Estimating the value
of electricity storage in pjm: Arbitrage and some welfare effects,” Energy
economics, vol. 31, no. 2, pp. 269–277, 2009.
[5] K. Bradbury, L. Pratson, and D. Patiño-Echeverri, “Economic viability
of energy storage systems based on price arbitrage potential in real-time
us electricity markets,” Applied Energy, vol. 114, pp. 512–519, 2014.
[6] M. Zheng, C. J. Meinrenken, and K. S. Lackner, “Agent-based model
for electricity consumption and storage to evaluate economic viability of
tariff arbitrage for residential sector demand response,” Applied Energy,
vol. 126, pp. 297–306, 2014.
[7] D. Wu, T. Yang, A. A. Stoorvogel, and J. Stoustrup, “Distributed optimal
coordination for distributed energy resources in power systems,” IEEE
Transactions on Automation Science and Engineering, vol. 14, no. 2,
pp. 414–424, 2017.
[8] P. M. van de Ven, N. Hegde, L. Massoulié, and T. Salonidis, “Optimal
control of end-user energy storage,” IEEE Transactions on Smart Grid,
vol. 4, no. 2, pp. 789–797, 2013.
[9] C. Wu, D. Kalathil, K. Poolla, and P. Varaiya, “Sharing electricity
storage,” in Decision and Control (CDC), 2016 IEEE 55th Conference
on. IEEE, 2016, pp. 813–820.
[10] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Başar, “Coalitional
game theory for communication networks,” IEEE Signal Processing
Magazine, vol. 26, no. 5, pp. 77–97, 2009.
[11] P. Chakraborty, E. Baeyens, P. P. Khargonekar, and K. Poolla, “A
cooperative game for the realized profit of an aggregation of renewable
energy producers,” in 2016 IEEE 55th Conference on Decision and
Control (CDC), Dec 2016, pp. 5805–5812.
[12] E. Baeyens, E. Y. Bitar, P. P. Khargonekar, and K. Poolla, “Coalitional
aggregation of wind power,” IEEE Transactions on Power Systems,
vol. 28, no. 4, pp. 3774–3784, 2013.
[13] P. Chakraborty, “Optimization and control of flexible demand and
renewable supply in a smart power grid,” Ph.D. dissertation, University
of Florida, 2016.
[14] P. Chakraborty, E. Baeyens, and P. P. Khargonekar, “Cost causation
based allocations of costs for market integration of renewable energy,”
IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–1, 2017.
[15] H. Nguyen and L. Le, “Bi-objective based cost allocation for cooperative
demand-side resource aggregators,” IEEE Transactions on Smart Grid,
2017.
[16] P. Chakraborty, E. Baeyens, and P. P. Khargonekar, “Analysis of solar
energy aggregation under various billing mechanisms,” arXiv preprint
arXiv:1708.05889, 2017.

7

[17] J. Von Neumann and O. Morgenstern, Theory of Games and Economic
Behavior. Princeton University Press, 1944.
[18] R. B. Myerson, Game Theory: Analysis of Conflict. Harvard University
Press, 2013.
[19] K. Jain and M. Mahdian, “Cost sharing,” Algorithmic game theory, pp.
385–410, 2007.
[20] Pecan St. Project. [Online]. Available: http://www.pecanstreet.org/
[21] How
Cheap
Can
Energy
Storage
Get?
Pretty
Darn
Cheap.
[Online].
Available:
http://rameznaam.com/2015/10/14/how-cheap-can-energy-storage-get/

A. Proof of Theorem 2
We shall prove that J defined by (4) is a subadditive
function. For any nonnegative real numbers xS , xT , CS , CT ,
we define JS = J(xS , CS ), JT = J(xT , CT ), JS∪T =
J(xS + xT , CS + CT ), then
X
JS =
πi Ci + πh (xS − CS )+ + πℓ min{CS , xS },
i∈S

JT =

πi Ci + πh (xT − CT )+ + πℓ min{CT , xT },

i∈T

JS∪T =

X

We begin by proving that the cost allocation (9) is an
imputation, i.e. ξ ∈ I. An imputation is a cost allocation
satisfying budget balance and individual rationality.
If xN ≥ CN :
X
X
ξi =
πi Ci + πh (xN − CN ) + πℓ CN = u(N ).
i∈N

i∈N

If xN < CN :

A PPENDIX

X

C. Proof of Theorem 4

X

πi Ci + πℓ xN = u(N ).

i∈N

P
Thus, i∈N ξi = u(N ) and the cost allocation {ξi : i ∈ N }
satisfies budget balance.
The individual cost is:

πi Ci + πh (xi − Ci ) + πℓ Ci xi ≥ Ci
u({i}) =
πi Ci + πℓ xi
xi < Ci
If xN ≥ CN :
ξi = πi Ci + πh (xi − Ci ) + πℓ Ci
= πi Ci + πℓ xi − πδ (Ci − xi )

πi Ci + πh (xS + xT − CS − CT )+ +

i∈S∪T

πℓ {CS + CT , xS + xT }.

= u({i}) − πδ (Ci − xi )+ .

We can distinguish four cases1 : (a) xS ≥ CS and xT ≥ CT ,
(b) xS ≥ CS , xT < CT and xS + xT ≥ CS + CT , (c)
xS ≥ CS , xT < CT and xS + xT < CS + CT , and (d)
xS < CS and xT < CT . Using simple algebra it is easy to
see that for all of these cases, JS∪T ≤ JS +JT or equivalently,
J(xS + xT , CS + CT ) ≤ J(xS , CS ) + J(xT , CT ),

(22)

and this proves subadditivity of J. Since the storage cost
function u(S) = J(xS , CS ), the cost sharing cooperative
game (N , u) is subadditive.

B. Proof of Theorem 3
We notice that the function J is positive homogeneous,
i.e, for any α ≥ 0, J(αxS , αCS ) = αJ(xS , CS ). J is also
subadditive as per Theorem 2. Thus for any arbitrary balanced
map α : 2N → [0, 1]
X
α(S)u(S)

If xN < CN :
ξi = πi Ci + πℓ xi
= u({i}) − πδ (xi − Ci )+ .
Thus, ξi ≤ v({i}) for all i ∈ N , and the cost allocation ξ is
individually rational. Since it is also budget balanced, it is an
imputation, i.e. ξ ∈ I.
Finally, to prove that the cost allocation ξ belongs
P to the
core of the cooperative game, we have to prove that i∈S ξi ≤
u(S) for any coalition S ⊆ N .
If xN ≥ CN :
X
X
ξi =
πi Ci + πh (xS − CS ) + πℓ CS
i∈S

i∈S

=

X

πi Ci + πℓ xS − πδ (CS − xS )

i∈S

= u(S) − πδ (CS − xS )+ .

S∈2N

=

X

α(S)J(xS , CS )

S∈2N

=

X

J(α(S)xS , α(S)CS ) [positive homogeneity]

S∈2N

≥ J(

X

α(S)xS ,

S∈2N

= J(

X X

X

α(S)CS ) [subadditivity]

S∈2N

α(S)1S (i)xS ,

i∈N S∈2N

= J(xN , CN ) = u(N ).

X X

α(S)1S (i)CS )

i∈N S∈2N

and this proves that the cost sharing game (N , u) is balanced.

1 Since

xS , xT , CS and, CT are arbitrary nonnegative real numbers,
any other possible case can be easily recast as one of these four cases by
interchanging S and T .

If xN < CN :
X

ξi =

i∈S

X

πS CS + πℓ xS

i∈S

= u(S) − πδ (xS − CS )+ .

P
Thus, i∈S ξi ≤ u(S) for any S ⊆ N and the cost allocation
ξ is in the core of the cooperative game (N , u).

D. Proof of Theorem 6
Let S and T two arbitrary nonempty disjoint coalitions, i.e.
S, T ⊆ N such that S ∩ T = ∅. Define
Φ(xS ) = min EJ(CS , xS ).
CS ≥0

We shall prove that Φ(xS ) is a subbadditive function.

(23)

8

From the definition of J given in (4),
J(xS , CS∗ )

+

J(xT , CT∗ )

Note that
xT , CS∗

≥ J(xS +

+

CT∗ ).

Taking expectations on both sides,

πh (xS − CS∗ )+ + πℓ min{CS∗ , xS } ≥ πh (xS − CS∗ ) + πℓ CS∗ .
and therefore,

xT , CS∗

CT∗

Φ(xS ) + Φ(xT ) ≥ EJ(xS +
+
)
≥ min EJ(xS + xT , C)

πS CS∗ + πh E[(xS − CS∗ )+ ] + πℓ E[min{CS∗ , xS }]
≥ πS CS∗ + πh E[(xS − CS∗ )] + πℓ CS∗ .

C≥0

= Φ(xS , xT ),
and this proves subadditivity of Φ.
Subadditivity of the cost sharing cooperative game (N , v)
is a consequence of the subadditivity of Φ because v(S) =
Φ(xS ) for any S ⊆ N .

E. Proof of Theorem 7
First, we prove that the function Φ defined by (23) is
positive homogeneous. Observe that if a random variable z
has CDF F , then the scaled random variable αz with α > 0
has CDF: Fα (θ) = P{αz ≤ θ} = F (θ/α). Then, for any
α ≥ 0 and γ ∈ [0, 1], γ = F (C) if and only if γ = Fα (αC).
This means that if CS is such that Φ(xS ) = EJ(xS , CS∗ ), then
Φ(αxS ) = EJ(αxS , αCS∗ ).
For any α ≥ 0, and from the definition of the daily storage
cost J (4), J(αxS , αCS∗ ) = αJ(xS , CS∗ ). Taking expectations
on both sides, Φ(αxS ) = αΦ(xS ), and this proves positive
homogeneity of Φ.
Now, balancedness of the cost sharing cooperative game
(N , v) is a consequence of the properties of function Φ
X
X
α(S)v(S) =
α(S)Φ(xS )
S∈2N

Let us define the sets A+ = {xN ∈ R+ | xN ≥ CN },
A− = R+ \A+ , and the auxiliary function ψ(xN ) as follows

πh if xN ∈ A+
ψ(xN ) =
πℓ if xN ∈ A−
Let F (xS , xN ) be the joint distribution function of the peak
consumptions (xS , xN ), then
∗
E[ψ(xN )(xN − CN
)]
Z Z
= πℓ
(xS − CS0 )dF (xS , xN )+
R+ A−
Z Z
πh
(xS − CS∗ )dF (xS , xN )
Z R+Z A+
≤ πh
(xS − CS∗ )dF (xS , xN )
ZR+ A+ ∪A−
= πh
(xS − CS∗ )dF (xS , xN )
R+

= E[(xS − CS∗ ],

and consequently,
πS CS∗ + πh E[(xS − CS∗ )] + πℓ CS∗
≥ πS CS∗ + E[πα (xS − CS∗ )] + πℓ CS∗

S∈2N

=

X

Φ(α(S)xS ) [positive homogeneity]

S∈2N

≥ Φ(

X

α(S)xS ) [subadditivity]

S∈2N

= Φ(

X X

α(S)1S (i)xS )

i∈N S∈2N

= Φ(xN ) = v(N ).
F. Proof of Theorem 8
We begin by proving that the cost allocation given by (9)
satisfies budget balance,
X
X
X
∗
ζi =
πℓ E[xi ] +
πS E[xi | xN ≥ CN
]
i∈N

i∈N

= πℓ E

i∈N

"

X

i∈N

#

xi + πS E

"

X

∗
xi | xN ≥ CN

i∈N

#

∗
= πℓ E[xN ] + πh E[xN | xN ≥ CN
]
= v(N ).

Now, we proveP
that the right hand side of the previous
expression equals i∈S ζi

πS CS∗ + E[ψ(xN )(xS − CS∗ )] + πℓ CS∗
Z Z
= πS CS∗ +
ψ(xN )(xi − Ci0 )dF (xS , xN ) + πℓ CS∗
R+ R+
Z Z
= πS CS∗ + πℓ
(xS − CS∗ )dF (xS , xN )+
R+ A− ∪A+
Z Z
(πh − πℓ )
(xS − CS∗ )dF (xS , xN ) + πℓ CS∗
R+ A+
Z Z
∗
= πS CS + πδ
(xS − CSi )dF (xS , xN ) + πℓ E[xi ]
R+ A+
Z Z
πS
∗
(xS − CS∗ )dF (xS , xN ) + πℓ E[xi ]
= πS CS +
1 − γS R+ A+
Z Z
1
xS dF (xS , xN ) + πℓ E[xi ]
= πS
P{xN ≥ CN } R+ A+
X
=
ζi
i∈S

The cost allocation is in the core if we prove that v(S) ≥
P
i∈S ζi for any coalition S ⊂ N . Please note that individual
rationality is included in the previous condition.
The storage cost for a coalition S ⊂ N is
v(S) = πℓ E[xS ] + πS E[xS | xS ≥ CS∗ ]
= πS CS∗ + πh E[(xS − CS∗ )+ ] + πℓ E[min{CS∗ , xS }].

P
Thus, i∈S ζi ≤ v(S) and the cost allocation {ζi : i ∈ N }
is an imputation in the core.

