1

The Sharing Economy for the Smart Gridπ

arXiv:1608.06990v2 [] 6 Sep 2016

Dileep Kalathila , Chenye Wua , Kameshwar Poollab , Pravin Varaiyaa

Abstract—The sharing economy has upset market for housing
and transportation services. Homeowners can rent out their
property when they are away on vacation, car owners can offer
ridesharing services. These sharing economy business models
are based on monetizing under-utilized infrastructure. They are
enabled by peer-to-peer platforms that match eager sellers with
willing buyers.
Are there compelling sharing economy opportunities in the
electricity sector? What products or services can be shared in
tomorrow’s Smart Grid? We begin by exploring sharing economy
opportunities in the electricity sector, and discuss regulatory and
technical obstacles to these opportunities. We then study the
specific problem of a collection of firms sharing their electricity
storage. We characterize equilibrium prices for shared storage
in a spot market. We formulate storage investment decisions of
the firms as a non-convex non-cooperative game. We show that
under a mild alignment condition, a Nash equilibrium exists, it is
unique, and it supports the social welfare. We discuss technology
platforms necessary for the physical exchange of power, and
market platforms necessary to trade electricity storage. We close
with synthetic examples to illustrate our ideas.
Keywords—Sharing economy, electricity storage, time-of-use pricing, Nash equilibrium

I. S HARING IN THE E LECTRICITY S ECTOR
The sharing economy. It is all the rage. Going on vacation?
Rent out your home for extra income! Not using your car.
Rent it out for extra income! Companies such as AirBnB,
VRBO, Lyft, and Uber are disrupting certain business sectors
[3]. Their innovative business models are based on resource
sharing that leverage underutilized infrastructure. And much
of our infrastructure is indeed underused. Some 90% of cars
are simply parked 90% of the time. Investors have rewarded
companies in this new sharing economy model. For example,
privately held Uber reached a valuation of $60 B in December
2015. more valuable faster than both Google and Facebook.

A. Sharing Opportunities
To date, sharing economy successes in the grid have been
confined to crowd-funding for capital projects [7]. What other
products or services could be shared in tomorrow’s grid? We
can imagine three possibilities. Many others surely exist.
π Supported in part by CERTS under sub-award 09-206, NSF under Grants
ECCS-0925337, 1129061, CNS-1239178.
a Electrical Engineering & Computer Science, Univ. of California, Berkeley, USA.
b Corresponding author, Electrical Engineering & Computer Science, Univ.
of California, Berkeley, USA, poolla@berkeley.edu

(a) Sharing excess generation from rooftop PV
Surplus residential PV generation is sold today through netmetering programs. Here, utilities purchase excess residential
generation at a fixed price πnm . Usually πnm is the retail
price and there is an annual cap, so households cannot be
net energy producers over the course of a year. Utilities are
mandated to offer net-metering, but do so reluctantly and view
such programs with hostility as it threatens their profitability
and business model [4]. Net-metering is not, strictly-speaking,
sharing. True resource sharing would pool excess PV generation and trade this over a spot market. Utilities could be paid
a toll for access to their distribution infrastructure.
(b) Sharing flexible demand recruited by a utility
Many consumers have flexibility in their electricity consumption patterns. Some consumers can defer charging their electric
vehicles, or modulate use of their AC systems. Utilities are
recognizing and monetizing the value of this demand flexibility. Excess recruited demand flexibility could be shared and
used at other buses where generation is expensive. Trading
this shared resource requires infrastructure to coordinate the
physical power transactions and support financial transactions.
(c) Sharing unused capacity in installed electricity storage
Firms faced with time-of-use (ToU) pricing might invest in
storage if it is sufficiently cheap. These firms could displace
some of their peak period consumption by charging their
storage during off-peak periods when electricity is cheap, and
discharging it during peak periods when it is dear. On days
when their peak period consumption happens to be low, these
firms may have unused storage capacity. This could be sold
to other firms. This sharing economy opportunity is the focus
of this paper.

B. Challenges to Electricity Sharing Business Models
A principal difficulty with sharing economy business models
for electricity is in tracing power flow point-to-point [8].
Electricity injected at various nodes and extracted at others
flows according to Kirchoff’s Laws, and we cannot assert
that a KWh of electricity was sold by party α to party β.
As a result, supporting peer-to-peer shared electricity services
requires coordination in the hardware that transfers power [2].
An alternative is to devise pooled markets which is possible
because electricity is an undifferentiated good. Regulatory and
policy obstacles may impede wider adoption of sharing [5].
The early adopters will use behind-the-meter opportunities
such as in industrial parks or campuses, where sharing can
be conducted privately without utility interference.

2

The successes of AirBnB or Uber are, to a large extent,
resulted by their peer-to-peer sharing platform. This brings
together willing sellers and eager buyers and enables to settle
on mutually beneficial transactions. In the electricity sector,
the challenges are to develop (a) software platforms that
support trading, and (b) hardware platforms that realize the
associated physical transfers of electricity. These must be
scalable, support security, and accommodate various market
designs.
C. Our Research Contributions
We study the specific problem of a collection of firms sharing
their electricity storage. The principal contributions of this
paper are:
− Spot Market for Sharing: We formulate storage sharing in
a spot market, and characterize the random clearing prices
in this market.
− Investment Decisions: We explicitly determine the optimal
investment decisions of a firm under sharing as the solution
of a two-stage optimization problem.
− Equilibrium Characterization: We formulate the investment
decisions of a collection of firms as a Storage Sharing Game.
This is a nonconvex game. Under a mild alignment condition, we show that this game admits a Nash Equilibrium. We
further show that if a Nash equilibrum exists, it is unique.
We explicitly characterize the optimal investment decisions
at this Nash equilibrium.
− Social Welfare: We show that these optimal investment
decisions form a Nash equilibrium which supports the social
welfare.
− Neutrality of Aggregator: We show that the aggregator
serves to inform firms of their optimal storage investments
while protecting their private information.
− Coalitional Stability: We prove that at this Nash equilibrium, no firm or subset of firms is better off defecting to
form their own coalition.
− Implementation: Under sequential decision making, we
propose a natural payment mechanism for new firms to join
the sharing coalition and realize this Nash Equilibrium.
D. Related Work
There is recent literature on estimating the arbitrage value and
welfare effects of storage in electricity markets. Graves et al.
study the value of storage arbitrage in deregulated markets
[11]. Sioshansi et al. explore the role of storage in wholesale
electricity markets [15]. Bradbury et al. examine the economic
viability of the storage systems through price arbitrage in [12].
Van de Ven et al. propose an optimal control framework for
end-user energy storage devices in [14]. Zheng et al. introduce
agent-based models to explore tariff arbitrage opportunities for
residential storage systems [13]. Bitar et al. characterize the
marginal value of co-located storage in firming intermittent
wind power [24]. While these previous works illuminate the
economic value of storage to an individual, to the best of our

knowledge, the analysis of shared electricity services has not
been addressed in the literature.

II. P ROBLEM F ORMULATION
For a random variable X, its expectation is written E [X], the
probability of some event A is written Prob (A), and we define
x+ = max{x, 0}.
Consider a collection of n firms indexed by k that use
electricity. An aggregator interfaces between these firms and
the utility. The aggregator itself does not consume electricity.
It purchases the collective electricity needed by the firms from
the utility, and resells this to the firms at cost. Firms can
trade electricity with each other, or purchase electricity from
the utility through the intermediary aggregator. The physical
delivery of electricity for these transactions are conducted over
a private distribution system within the aggregators purview.
Prices imposed by the utility are passed through to the firms.
The aggregator does not have the opportunity to sell excess
electricity back to the utility (no net metering). The situation
we consider is illustrated in Figure 1.

Fig. 1: Agents and interactions.

Remark 1: Examples of the situation we consider include
firms in an industrial park, office buildings on a campus,
or homes in a residential complex. The aggregator might be
the owner of the industrial park, the university campus, or
the housing complex community. There is a single point of
coupling or metered connection to the utility. Exchanges of
energy between firms, buildings, or homes are behind-themeter private transactions outside the regulatory jurisdiction of
the utility. The distribution grid serving these firms, buildings,
or homes is private. It could be communally owned or provided
by the aggregator for a fee. If this fee is a fixed connection
charge, our results are unaffected. Analysis of sharing when
the distribution system charge is proportional to use is substantially more complex and falls outside the scope of this paper.
We ignore capacity constraints in this private distribution grid,
and our results are agnostic to its topology. We also ignore line
losses in the distribution system.
2
Each day is divided into two fixed, contiguous periods –
peak and off-peak. The firms face common time-of-use (ToU)
prices. During peak hours, they face a price πh , while during
off-peak hours, they face a lower price π` . These prices are

3

In this case, 0 < γ < 1. With this assumption it is profitable
for firms to invest in storage. They charge their storage during
off-peak hours when electricity is cheap, and discharge it
during peak hours when it is dear. Note that the energy that is
held in storage is always acquired at price π` /MWh. Let Ck
be the storage investment of firm k, and let
X
Cc =
Ck
(5)
k

be the collective storage investment of all the firms.
Fig. 2: Consumption and Time-of-Use Pricing.

fixed and known. Our formulation considers the simplest situation of two-period ToU pricing. Weekday only ToU pricing
is easily handled.
The consumption of firm k during peak and off-peak hours on
day t are the random processes Xk (t) and Yk (t) respectively.
We model Xk as an independent identically distributed random
sequence. Let Fk (·) be the cumulative distribution function
of Xk (t) for any day t. Let fk (·) be the probability density
function of Xk (t) for any day t. Empirical distributions of Xk
may be estimated from historical data using standard methods
[19]. Let
X
Xc =
Xk
(1)
k

be the collective peak-period consumption of all the firms.
The cumulative distribution function and probability density
function for Xc are Fc (·) and fc (·) respectively. Consumption
and pricing are illustrated in Figure 2.
The off-peak consumption Yk is not material to our results.
This is because it is serviced at π` which is the lowest price at
which electricity is available. The use of storage cannot reduce
this expense. We therefore disregard Yk in the remainder of
this paper.

Remark 3: Electricity storage is expensive. The amortized
cost of Tesla’s Powerwall Lithium-ion battery is around
25¢/KWh per day [20] (assuming one charge-discharge cycle
per day over its 5 year lifetime). At current storage prices,
ToU pricing rarely offers arbitrage opportunities. An exception
is the three-period PG&E A6 program [21] under which the
electricity prices per KWh are 54¢ for peak hours (12:00pm
to 6:00pm), 25¢ for part-peak hours (8:30am to 12:00pm, and
6:00pm to 9:30pm), and 18¢ for off peak hours (rest of the
day). Storage prices are projected to decrease by 30% by 2020
[20]. Our results offer a framework for the analysis of sharing
in this future of cheap electricity storage prices with lucrative
sharing opportunities.
2
We make the following assumptions.
A.1 Arbitrage opportunity exists: πδ > πs .
A.2 Firms are price takers: the total storage investment is
modest and does not influence the ToU pricing offered by
the utility.
A.3 Inelastic demand: the statistics of the demand Xk , Yk for
firm k are not affected by savings from ToU arbitrage.
A.4 Statistical assumptions: fk (·) is continuously differentiable and fc (x) > 0 for x ≥ 0.
A.5 Electricity storage is ideal: it is lossless, and perfectly
efficient in charging/discharging.
A.6 Storage investments by the firms are made simultaneously.

Remark 2: We can just as easily treat the case where Xk is
not stationary. In this case, all stated results hold with Fk
being replaced by the average of the cumulative distribution
functions over the T -day lifetime of electricity storage:
1X
Fk (·) →
FXk (t) (·)
2
T t

Assumptions A.1 through A.3 are essential. Assumption A.4
is needed only to simplify our exposition. We will dispense
with A.5 and A.6 in Sections V and VI respectively.

If storage is sufficiently cheap, firms will invest in storage
to arbitrage ToU pricing. Let πs be the daily capital cost of
storage amortized over its lifespan. Define

A. Optimal Investment Decisions

arbitrage price
arbitrage constant

πδ = πh − π` > 0
πδ − πs
γ=
πδ

(2)
(3)

For storage to offer a viable arbitrage opportunity we clearly
require
πs < πδ

(4)

III. M AIN R ESULTS : N O S HARING

We first consider a single firm which elects to invest in
storage capacity C. Let X, Y be the random peak and offpeak consumption of this firm. The firm will choose to service
X first using its cheaper stored energy, and will purchase the
deficit (X − C)+ at the peak period price πh . During the
off-peak period, it will service its demand Y and recharge its
storage at the lower price π` . The firm will elect to completely
recharge the storage as we have assumed the storage is ideal,
and holding costs are zero. The daily expected cost of the firm

4

is therefore


J(C) = πs C + πh E (X − C)+ + π` E [min{C, X}] (6)
|{z} |
{z
}
{z
} |
cap ex

buy deficit

recharge storage

This firm will choose to invest in storage capacity

The optimal storage investment of the aggregate entity is Do =
−1
FX
(γ) which is

√
2γ
if γ ∈ [0, 0.5]
−1
√
D o = FX
(γ) =
2 + 2 − 2γ if γ ∈ [0.5, 1]
Plotted in Figure 4 are Cc , and Do as functions of γ. Notice
that Do < Cc when γ < 0.5 and Do > Cc if γ > 0.5.
2

C ∗ = arg min J(C)
C

Theorem 4: The optimal decision of a standalone firm under
no sharing is to purchase C o units of storage where
F (C o ) =

πδ − πs
=γ
πδ

(7)

The resulting optimal cost is
J o = J(C o ) = π` E [X] + πs E [X | X ≥ C o ]

(8)

Remark 5: The result above is illustrated in Figure 3. It is easy
to show that the optimal storage investment C o is monotone
decreasing in the amortized storage price πs , and monotone
increasing in the arbitrage price πδ .
2

Fig. 4: Example: Under- and over-investment.

Remark 7: The example above reveals that without sharing,
firms might over-invest in storage because they are going it
alone and do not have the opportunity to buy stored electricity
from other firms. This happens when γ is large. They might
also under-invest because they forgo revenue opportunities that
arise from selling their stored electricity to other firms. This
happens when γ is small.
2
IV. M AIN R ESULTS : W ITH S HARING

Fig. 3: Optimal Storage C o for a standalone firm.

Example 6: Consider two firms, indexed by k = 1, 2, whose
peak period demands are the random variables X1 , X2 respectively. Suppose X1 , X2 are independent and uniformly
distributed on [0, 1] Then, we have

 0 if x < 0
Fk (x) =
x if x ∈ [0, 1)

1 if x > 1
Fix γ ∈ [0, 1]. The optimal storage investment of both firms
is identical, and using Theorem 4, we calculate this to be
C o = Fk−1 (γ) = γ. Their combined storage investment is
Cc = 2C o = 2γ.
Now consider the entity formed by merging these firms. Sharing electricity between firms is an internal exchange within
the entity. This entity has combined peak period demand
X = X1 + X2 . Notice that

0
if x < 0



2
0.5x
if x ∈ [0, 1)
FX (x) =
2

1
−
0.5(x
−
2)
if
x ∈ [1, 2)


1
if x ≥ 2

Consider n firms. Firm k has chosen to invest in Ck units
of storage to arbitrage against the ToU pricing it faces. On
a given day, suppose the total peak-period energy demand of
firm k is Xk . The firm will choose to first service Xk using
its cheaper stored energy. This may leave a surplus of stored
energy (Ck − Xk )+ . This excess energy available to firm k
in its storage can be sold to other firms. Conversely, it may
happen that firm k faces a deficit in its demand of (Xk −
Ck )+ even after using its stored energy. This deficit could be
purchased from other firms that have a surplus, or from the
utility.

A. The Spot Market for Stored Energy
We consider a spot market for trading excess energy in the
electricity storage of the collective of firms. Let S be the total
supply of energy available from storage from the collective
after they service their own peak period demand. Let D be the
total deficit of energy that must be acquired by the collective
of firms after they service their own peak period demand. So,
X
X
S=
(Ck − Xk )+ , D =
(Xk − Ck )+
k

k

If S > D, the suppliers compete against each other and
drive the price down to their (common) acquisition cost of π` .
Note that unsold supply is simply held. Since the storage is

5

perfectly efficient and lossless (see Assumption A.4), there are
no holding costs. This is equivalent to selling unsold supply
at π` to an imaginary firm, and buying it back during the next
off-peak period at price π` . As a result, storage is completely
discharged during the peak period, and fully recharged during
the subsequent off-peak period. The entire supply S is traded
at π` if S > D.
If S < D, some electricity must be purchased from the utility
which is the supplier of last resort. Consumers compete and
drive up the price πh offered by the utility. As a result, the
excess energy S in storage is traded at πh when S < D.

Suppose firms i, i 6= k have invested in Ci units of storage.
The optimal investment of firm k under sharing is to purchase
Cko units of storage where
Cko = arg min J(Ck , C−k )
Ck

Unfortunately, the cost function Jk is, in general, non-convex.
It may have multiple local minima, and nonunique global minimizers. Determining Cko analytically can be extremely difficult.
Remarkably, in a non-cooperative game setting when all firms
seek to miminize their cost, we can explicitly characterize
optimal investment decisions (see Theorem 11).

C. The Social Cost
The social cost is the sum of the daily expected costs of all
the firms:
X
J(Ck , C−k )
Jc (C1 , · · · , Cn ) =
k

=

We can view trading excess storage as internal transactions
within the collective. From this observation, it is straightforward to verify that the social cost depends only on the
collective investment Cc and that


Jc (Cc ) = πs Cc + π` E [Yc ] + πh E (Xc − Cc )+

Fig. 5: Equilibrium Price: (a) left panel S > D, (b) right
panel S < D.

The market clearing price is therefore

π` if S > D
πeq =
πh if S < D

+ π` E [min{Cc , Xc }]
(9)

Note that the clearing price πeq is random and depends on
supply-demand conditions in each peak period. This competitive equilibrium is illustrated in Figure 5. Note that
X

S−D =
(Ck − Xk )+ − (Xk − Ck )+
=

k
X

Ck −

k

X

Xk = Cc − Xc

k

where Cc is the collective storage installed by the firms,
and Xc is their collective demand. We can then re-write the
clearing price as

π` if Cc > Xc
πeq =
(10)
πh if Cc < Xc

B. Optimal Investment Decisions under Sharing
Consider a collection on n firms. Suppose firm k has chosen
to invest in Ck units of storage. The expected daily cost for
firm k under sharing is
Jk (Ck , C−k ) = πs Ck + π` Ck + E [πeq (Xk − Ck )] (11)
| {z } | {z }
|
{z
}
cap ex

recharge

π` Cc + π` E [Yc ] + E [πeq (Xc − Cc )]

trade surplus/deficit

Note that this expected cost depends on the decisions C−k of
all the other firms. This dependence appears implicitly through
the random clearing price πeq for shared energy.

(12)

This can be regarded as the daily expected cost of the
collective firm under no sharing (see (6)). A social planner
would minimize the social cost and select Cc = C ∗ where
Fc (C ∗ ) = γ (see Theorem 4). No prescription would be made
on allocating the total investment C ∗ among the firms.

D. Non-cooperative Game Formulation
We stress that the optimal storage decision Cko of firm k
depends on the investment choices Ci , i 6= k made by all
other firms. This leads naturally to a non-cooperative gametheoretic formulation of the Storage Investment Game G. The
players are the n firms. The decision of firm k is Ck and its
cost function is Jk (Ck , C−k ). We explore pure strategy Nash
equilibria for this game.
Theorem 8: Suppose for k = 1, · · · , n,
dE [Xk | Xc = β]
≥0
dβ

(13)

Then the Storage Investment Game admits a Nash Equilibrium.
Remark 9: The alignment condition (13) needed to establish
existence of a Nash equilibrium has a natural interpretation
- the expected demand Xk of firm k increases if the total
demand Xc increases. This is not unreasonable. This is not
unreasonable. For example, if Xk and Xc are jointly Gaussian,
then (13) hold if they are positively correlated.
2

6

Remark 13: Our problem formulation above does not assume
a perfect competition model. Indeed, under perfect competition, the celebrated welfare theorems assure existence of a
unique Nash equilibrium. In our analysis, we allow firm k to
take into account the influence its investment decision Ck has
on the statistics of the clearing price πeq . This is a Cournot
model of competition [10], under which Nash equilibria do
not necessarily exist.
2
Theorem 14: The unique Nash equilibrium of Theorem 11 has
the following properties:

Fig. 6: Example: no Nash Equilibrium.

Example 10: We can construct examples with exotic demand
distributions where a Nash equilibrium does not exist. Let W
be a random variable uniformly distributed on [0, 10]. Define
the peak period consumption for two firms by
X1 = W sin2 (W ), X2 = W cos2 (W )
Notice that the collective demand is Xc = X1 +X2 = W . The
support of X1 , X2 in the plane in shown in the right panel of
Figure 6. We calculate
E [X1 | X1 + X2 = β] = β sin2 (β)
This is not a non-decreasing function of β. Thus the alignment
condition (13) is violated.
We choose πs = 0.3, πδ = 1. Then, Fc (Q) = 0.7 = β, which
implies Q = 7. If a Nash equilibrium exists, it is unique and
must be given by (see Theorem 11)
C1∗

=

E [X1 | Xc = 7] = 7 sin2 (7) = 3.02

C2∗

=

E [X2 | Xc = 7] = 7 cos2 (7) = 3.98

The cost function for firm 1 is
Z
J1 (C1 , C2∗ ) = 0.3C1 + 0.1

10

(W sin2 (W ) − C1 )dW

C1 +3.98

This function is plotted in the left panel of Figure 6. Notice that
C1∗ is a not a global minimizer, proving that a Nash equilibrium
does not exist.
2
Theorem 11: Suppose the Storage Investment Game admits a
Nash Equilibrium. Then it is unique and given by
Ck∗ = E[Xk |Xc = Q], k = 1, · · · , n

(14)

where Q is the unique solution of
Fc (Q) =

πδ − πs
=γ
πδ

(15)

The resulting optimal cost is
J ∗ = π` E [Xk ] + πs E [Xk | Xc ≥ C ∗ ]

(a) The Nash equilibrium supports the social welfare: the
collective investment Cc of the firms coincides with the
optimal investment
P of the collective firm with peak period
demand Xc = k Xk .
(b) Individual rationality: No firm is better off on its own as
a standalone firm without sharing.
(c) Coalitional stability: Assume the alignmnet condition (13)
holds. No subset of firms is better off defecting to form their
own coalition.
(d) No arbitrage: At the Nash equilibrium, we have
E [πeq ] = πs + π` .
(e) Neutrality of Aggregator: If firm k has peak-period consumption Xk = 0, it will not invest in storage, i.e. Ck∗ = 0.
Remark 15: Part (e) of this Theorem establishes that there is
no pure-storage play. A firm that does not consume electricity
in the peak period has no profit incentive to invest in storage.
The aggregator is in this position. An important consequence is
that the aggregator is in a position of neutrality with respect
to the firms. It can therefore act to supply the information
necessary for firm k to make its optimal investment choice.
This information consists of (i) the joint statistics of Xk and
X c , and (ii) the cumulative optimal investment C c of all the
firms. With this information, firm k can compute its share of
the optimal storage investment C c as in (14). As a result, the
private information Xi , i 6= k of the other firms is protected.
The neutrality of the aggregator affords it a position to operate
the market, determine the market clearing price of shared
storage, conduct audits, and settle transactions.
2
Remark 16: We can also treat storage sharing in a cooperative
game formulation. In this setting, we infer from Theorem
∗
14 that the allocation ak = −Jk (Ck∗ , C−k
) to firm k is an
imputation. It is budget balanced, individually rational, and
coalitionally stable. This imputation therefore lies in the core
of the cooperative game. There may be other imputations in
the core.
2

(16)

Remark 12: The game G is nonconvex. It is remarkable that
it admits an explicit characterization of its unique Nash
equilibrium should one exist. It commonly happens that the
∗
cost function Jk (Ck , C−k
) of firm k given optimal decisions
of other firms has multiple local minima. It is surprising that
Ck∗ is the unique global minimizer of this function.
2

V. N ON - IDEAL S TORAGE
We generalize our results to accommodate certain aspects of
non-ideal storage. Let ηi , ηo be the charging and discharging
efficiency respectively. We do not address maximum rates of
charge or discharge, or treat leakage. Incorporating leakage is

challenging because it affects the control strategy of storage
and the clearing price in the spot-market.
Theorem 17: With non-ideal storage, the optimal decision of
a firm under no sharing is to invest in C o units of storage
where
πh ηo − π` /ηi − πs
(17)
F (ηo C o ) =
πh ηo − π` /ηi

Electricity Price (cents/kWh)

7

Charging inefficiency has the effect of inflating the off-peak
price π` , and discharging inefficiency discounts the peakperiod price πh . These together reduce the arbitrage opportunity. Our results on optimal decisions under sharing also
generalize easily.

Qn+1 ≥ Qn .
This result shows that the optimal storage investment is
extensive. As firms join the collective, the optimal storage
investment must increase. As a result, the collective of firms
does not have to divest any storage investments already made.
It must merely purchase Qn+1 −Qn additional units of storage.
In addition, the optimal storage investments of firms in C
change when the new firm joins the collective. As a result,
these firms will have to rearrange their fraction of storage
ownership requiring an internal exchange of payments. If the
storage is co-located and managed at the aggregator, this
rearrangement reduces to simple financial transactions.
It is clear from the coalitional stability result of Theorem
14(b), that both the original collective C and the new firm are
better off joining forces. However, simple examples reveal that
some individual firms in C may be worse off in the expanded
collective C ∪ Fn+1 . This raises interesting issues on voting
rights. Under veto power the new firm may not be invited to
expand the coalition. Under a cost-weighted majority vote, the
new firm will always be invited to expand the coalition. These
questions require further exploration.

40

30

30

20

20

10

10
12:00 18:00 24:00

Time

(a)

1

We have thus far assumed that all firms make their storage
investment decisions simultaneously. A better model would
allow for sequential capital investment decisions. We explore
the situation when a collective of firms C has made optimal
storage investments, and a new firm Fn+1 wishes to join.

0
0:00

12:00 18:00 24:00

Time

0.8

0.6

0.6

0.4

0.4

0.2

0.2
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(c)

1

0

0.8

0.6

0.6

0.4

0.4

0.2

0.2
0

20

X (kWh)

0

40

0

10

20

30

(d)

1

0.8

0

(b)

1

0.8

0

F(X)

Theorem 18: Consider a collective of n firms. Let Qn be the
optimal combined storage investment of these firms under
sharing. Suppose a new firm joins the collective. Let Qn+1
be new combined optimal storage investment of these n + 1
firms under sharing. Then,

40

0
0:00

(b)

55
50

Fig. 7: ToU pricing: (a) real three-period pricing, (b)
simplified two-period pricing.

F(X)

VI. J OINING THE C LUB

(a)

55
50

0

20

X (kWh)

40

Fig. 8: Sample cumulative distribution functions of
peak-period demand Xk for 4 households.

during summer months. We approximate this by the twoperiod ToU tariff shown in Figure 7(b). We set πh = 54¢/kWh,
and π` = 21.5¢/kWh (average of the partial peak and off-peak
prices). We use the publicly available Pecan Street data set [23]
which offers 1-minute resolution consumption data from 1000
households. From historical data, we use standard methods to
estimate demand statistics. Figure 8 shows sample empirical
cumulative distribution functions (cdfs) Fk (·) for 4 household
k = 1, · · · , 4. It is apparent that there is considerable statistical diversity in peak-period consumption. Panel (a) shows
a representative cdf, panel (b) suggests that some households
are consistently vacant during peak hours, panel (c) shows a
constant background load, panel (d) suggests some users have
low variability in their peak-period demand.

VII. S IMULATION S TUDIES

For storage sharing to be beneficial, we require statistical
diversity in the peak period demands Xk . A histogram of the
pairwise correlation coefficients is shown in Figure 9. The
average pairwise correlation coefficient between households
is approximately 0.5, and there are many pairs of households
with negatively correlated demands.

We illuminate the analytical development of sharing storage
using synthetic simulations. Figure 7(a) shows the three-period
(peak, partial peak, and off-peak) A6 tariff offered by PG&E

We compare two cases: (a) without sharing, and (b) with
sharing. We have already seen (see Example 6) that firms
may over- or under-invest under no sharing depending on

8

Number of Instances

600
500
400
300
200
100
0
-1

-0.5

0

0.5

Pairwise correlation coefficient ρ

1

Fig. 9: Histogram of pairwise correlation coefficients.

0.75

18.2
18

without sharing
with sharing

17.8
17.6
17.4
17.2
17

average net profits ($ per day)

Average Storage Capacity (kWh)

700

0.65
0.6
0.55
0.5
0.45
0.4

0

100

200

300

400

number of participants

Fig. 10: Average optimal investment.

the statistics of their demand. For this data set, Figure 10
shows that the average storage investment is ≈5% lower under
sharing. Finally, we compute the average financial benefit
under sharing. The expected daily cost of a household that
chooses not to invest in storage is J = πh E [Xk ]. If this
household invests optimally in storage, but does not participate
in sharing, its expected daily cost is (see equation (8))
J o = π` E [Xk ] + πs E [Xk | Xk ≥ C o ]
where C o is prescribed by Theorem 4. If this household invests
optimally while participating in the spot market for storage,
its expected daily cost is (see equation (16))
J ∗ = π` E [Xk ] + πs E [Xk | Xc ≥ C ∗ ]
as shown in Theorem 11. The expected daily arbitrage revenue
from using storage without sharing is ∆ns = J − J o , and is
∆s = J − J ∗ under sharing. We plot ∆ns , ∆s averaged across
users. Figure 11 shows that although the users earn ≈55¢ per
day without sharing on average vs ≈75¢ under sharing which
represents a 50% better return that without sharing.
VIII. P HYSICAL AND M ARKET I MPLEMENTATIONS
Sharing electricity services requires coordination. For example, if the service is a delivery of power from one rooftop PV
to a remote consumer in the community, signals must be sent
to coordinate the equipment of both. Set point schedules for
inverters must be communicated in advance of the physical
exchange of energy.
Our focus has been on sharing energy services. The transactions involved are exchanges of energy during the peak
period without stipulating when this energy should be delivered. Small businesses could invest in electricity storage for
many reasons beyond time-of-use price arbitrage, including
smoothing high-frequency variations in PV production, or
protection from critical peak pricing (CPP). CPP [9] is a
common tariff where firms face a substantial surcharge based
on their monthly peak demand. Sharing surplus energy in
electricity storage does not preclude these other applications.
In our framework, electricity storage could be physically distributed among the firms, or centralized. A distributed storage

without sharing
with sharing

0.7

0

100

200

300

number of participants

400

Fig. 11: Average savings using storage
for ToU price arbitrage.

architecture requires n inverters and results in larger losses.
Centralized storage co-located, installed, and managed by the
aggregator, requires a single inverter and promises economies
of scale. Firms can lease their fair share of storage capacity
directly from the aggregator.
We have studied a spot market for trading energy in electricity
storage. Other arrangements are possible – bilateral trades,
auctions, or bulletin boards to match buyers and sellers. In
any event, realizing a sharing economy for electricity services
requires a scalable software platform to accept supply/demand
bids, clear markets, publish prices, and conduct audits.
The physical and market infrastructure necessary to support
the broader sharing economy for the smart grid is a topic that
demand deeper exploration.

R EFERENCES
[1] H. Heinrichs, “Sharing economy: a potential new pathway to sustainability,” GAIA-Ecological Perspectives for Science and Society, vol. 24,
pp: 228-231, 2013.
[2] M. He, E. M. Reutzel, X. Jiang, R. Katz, S. Sanders, D. Culler, K.
Lutz, “An Architecture for Local Energy Generation, Distribution, and
Sharing,” IEEE Energy2030, Atlanta, November 2008.
[3] “The rise of the sharing economy,” The Economist, March 2013.
[4] J. Warrick, “Utilities wage campaign against rooftop solar,” Washington
Post, March 7, 2015.
[5] S. Cannon and L. Summers, “How Uber and the Sharing Economy Can
Win Over Regulators,” Harvard Business Review, October 2014.
[6] J. Mather, E. Baeyens, K. Poolla, P. P. Varaiya, “The Real Value of Load
Flexibility - Congestion Free Dispatch,” submitted to the Automatic
Control Conference, Chicago, 2015.
[7] M. Crosby, “Will There Ever Be an Airbnb or Uber for the Electricity
Grid?,” GreenTech Media, September 2014.
[8] J. Bialek, “Tracing the flow of electricity,” IEE Proceedings on Generation, Transmission, and Distributions, August 2002.
[9] “The Answer to Peak Pricing - Energy Storage Solutions,” Virtual
Power Solutions, www.slideshare.net/emexlondon/the-answer-to-peaktime-pricing-energy-storage-solutions
[10] J. Tirole, The Theory of Industrial Organization, MIT Press, 1988.
[11] F. Graves, T. Jenkin, D. Murphy, “Opportunities for electricity storage in
deregulating markets,” The Electricity Journal, vol. 12, no. 8, pp:46-56,
1999.
[12] K. Bradbury, L. Pratson, 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.
[13] M. Zheng, C. J. Meinrenken, 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.

9

[14] P. M. van de Ven, N. Hegde, L. Massoulié, T. Salonidis, “Optimal control
of end-user energy storage,” IEEE Transactions on Smart Grid, vol. 4,
no. 2, pp:789–797, 2013.
[15] R. Sioshansi, P. Denholm, T. Jenkin, 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.
[16] “The Answer to Peak Pricing Energy Storage Solutions,” Virtual Power
Solutions, online at http://goo.gl/r1wyhA
[17] 2015CF - Crowdfunding Industry Report, Massolution, April 2015.
[18] J. M. Buchanan, “An economic theory of clubs,” Economica, vol. 32,
no. 125, pp:1–14, 1965.
[19] A. W. Van der Vaart, Asymptotic statistics, Cambridge University Press,
2000.
[20] R. Naam, “How Cheap Can Energy Storage Get?,” online at
http://goo.gl/jbqOo8
[21] PG&E Time of Use Tariffs, online at http://goo.gl/nRI28y
[22] “Rapid growth in sharing economy is fueled by IT,” Korea Joon-Ang
Daily, on line at http://goo.gl/Gllbeo
[23] Pecan Street Data, www.pecanstreet.org
[24] E. Bitar, P. Khargonekar, K. Poolla, “On the Marginal Value of Electricity Storage,” IEEE Transactions on Power Systems, in review, 2016.

The expected daily cost of firm k defined on Ck ≥ 0 is
∗
Jk (Ck | C−k
) = πs Ck + π` Ck + E [πeq (Xk − Ck )]

We have to show that Ck∗ is a global minimizer of
∗
Jk (Ck , C−k
).
Using the characterization (10) of the spot market clearing
price πeq , and noting that the statistics of Xk are not influenced
by Ck (see Assumption A.3), this cost function simplifies to:
Z∞

Z∞
(Xk − Ck ) fXk Xc (Xk , Xc )dXk dXc

πs Ck + πδ
Xk =0

Xc =Ck +α

It is easy to show using the Leibniz rule that
dJk
dCk

= − πδ · fc (Ck + α) · (E [Xk | Xc = Ck + α] − Ck )
|
{z
}
ψ(Ck )

+ πs − πδ + πδ Fc (Ck + α)
{z
}
|

A PPENDIX :P ROOFS

(18)

φ(Ck )

A. Proof of Theorem 4

We explore properties of the functions φ and ψ. We have

The daily expected cost for a single firm under no sharing is
(see equation (6))


J(C) = πs C + E πh (X − C)+ + π` Y + π` min{C, X}
Z ∞
= πs C + πh
(x − C)fX (x)dx + E [π` Y ]
C
Z C
+π` C · Prob (X ≥ C) + π`
xfX (x)dx
0

It is straightforward to verify that this is strictly convex in C.
As a result, the optimal investment C o is the unique solution
of the first-order optimality condition
Z ∞
dJ
0 =
= πs − πh
fX (x)dx + π` Prob (X ≥ C)
dC
C
= πs + (π` − πh ) (1 − F (C))
Rearranging this expression yields
F (C) =

πs − πδ
πδ
2

proving the claim.

φ(Ck )

is monotone increasing in Ck

φ(Ck∗ )

πδ Fc (Ck∗

=

πs − πδ +

(19)

+ α)

= πs − πδ + πδ Fc (Q) = 0

(20)

Here, we have made use of the assumption that fc (·) > 0.
Thus φ is monotone increasing and vanishes at Ck = Ck∗ .
Next observe that
X
β =
E [Xk | Xc = β]
k

1

X dE [Xk | Xc = β]
dE [Xk | Xc = β]
=
=⇒
≤1
dβ
dβ
k |
{z
}
≥0 by equation 13

Here we have made critical use of the technical condition (13)
needed to establish existence of a Nash equilibrium. It then
follows that
dψ(Ck )
dE [Xk | Xc = β]
=
−1≤0
dCk
dβ
∗
∗
ψ(Ck ) = E [Xk | Xc = Ck + α] − Ck∗
= E [Xk | Xc = Q] − Ck∗ = 0

B. Proof of Theorem 8
Consider firm k. Its decision is Ck . Fix the decisions of all
∗
other firms C−k
where
Ci∗ = E [Xi | Xc = Q] , Fc (Q) = γ =
Simple algebra reveals that
πs − πδ + πδ Fc (Q) = 0
Define the quantity
X
α=
Ci∗
i6=k

πδ − πs
πδ

Thus ψ is monotone non-increasing and vanishes at Ck = Ck∗ .
As a result,

≤ 0 Ck ≤ Ck∗
−πδ · fc (Ck + α) · ψ(Ck )
≥ 0 Ck ≥ Ck∗
Combining this with the properties of φ in equations (19, 20),
we get

< 0 Ck < Ck∗
dJk 
= 0 Ck = Ck∗
dCk 
> 0 Ck > Ck∗
∗
This proves that Ck∗ is the global minimizer of Jk (Ck , C−k
),
∗
establishing that C is a Nash equilibrium.
2

10

So for k = 1, · · · , n,

C. Proof of Theorem 11
Let Dk , k = 1, · · · , n be any Nash equilibrium. We show that
D = C ∗ where
πδ − πs
Ck∗ = E [Xk | Xc = Q] , Fc (Q) = γ =
πδ
Simple algebra reveals that Q is the unique solution of

P

k

Dk , and define the constants
K1

=

πs − πδ + πδ Fc (β)

=

πδ fXc (β) > 0

(a) Notice that
X
X
Ck∗ =
E [Xk | Xc = Q] = E [Xc | Xc = Q] = Q

Define the index sets

k

N = {j : Dj = 0}

Since D is a Nash equilibrium, it follows that Dk is a
global minimizer of Jk (Ck | D−k ). We write the first-order
optimality conditions for i ∈ M:
0

=

dJi (Ci | D−i )
dCi

D

= K1 − K2 · E [Xi − Di | Xc = β]

(21)

The first-order optimality conditions for j ∈ N are:
0

≤

dJj (Cj | D−j )
dCj

D

= K1 − K2 · E [Xj − Dj | Xc = β]

(22)

Summing these conditions, we get
≤ nK1 − K2 ·

0

n
X

∗
∗
J o ≥ Jk (C o , C−k
) ≥ Jk (Ck∗ , C−k
) = J∗

As a result, J o ≥ J ∗ , proving the claim.
(c) We next prove coalitional stability. Consider the Storage
Investment Game G. We form coalitions Aj ⊆ {1, · · · , n}
such that

= nK1
Thus, K1 ≥ 0. Using (21), for any i ∈ M we write
K1
≥0
K2

(23)

n
X

E [Xk − Dk | Xc = β]

k=1

=

X

E [Xi − Di | Xc = β] +

i∈M

≥

X

Ai ∪ Aj = φ, ∪k Ak = {1, · · · , n}
The game G induces a new game H with players Ai and
associated cost
X
JAi =
Jk (C1 , · · · , Cn )

Next, we have
=

Since Fc (Q) = γ, it follows that the Nash equilibrium (14)
supports the social welfare.
(b) We first show individual rationality - that no firm is better
off defecting from the grand coalition. Consider firm k on its
own. Its optimal investment decision is C o given by Theorem
4 and its optimal expected daily cost is J o . The standalone
firm (a) buys its shortfall (Xk − Ck )+ from the utility at πh ,
and (b) spills its surplus (Ck − Xk )+ . Under any sharing
arrangement, firm k benefits by (a) buying its shortfall at the
possibly lower price πeq , and (b) selling its surplus at the
possibly higher price πeq . Suppose this firm were to retain
its investment choice at C o , but participate in sharing with
∗
the grand coalition. Its new cost function is Jk (C o , C−k
).
Since any sharing arrangement reduces the cost of firm k,
we have

Since C ∗ is a Nash equilibrium, we have

E [Xk − Dk | Xc = β]

k=1

E [Xi − Di | Xc = β] =

k

∗
J o ≥ Jk (C o , C−k
)

= nK1 − K2 · E [Xc − β | Xc = β]

0

0 = K1 = πs − πδ + πδ Fc (β)

D. Proof of Theorem 14

K2

M = {i : Di > 0},

Using this in (21) yields

This implies β = Q, and it follows that Dk = Ck∗ for all k,
proving the claim.
2

πs − πδ + πδ Fc (Q) = 0
Let β =

0 = E [Xk − Dk | Xc = β] ⇐⇒ Dk = E [Xk | Xc = β]

X

E [Xj | Xc = β]

j∈N

E [Xi − Di | Xc = β]

i∈M

≥ 0

k∈Ai

Since the alignment condition (13) holds for G, we have for
any coalition Ai ,
X dE [Xk | Xc = β]
dE [XAi | Xc = β]
=
≥0
dβ
dβ
k∈Ai

Here, we have used (23) and the fact that the random variables
Xj are non-negative. As a result, we have
for i ∈ M:

E [Xi − Di | Xc = β] = 0

for j ∈ N:

E [Xj − Dj | Xc = β] = E [Xj | Xc = β] = 0

Thus, the alignment condition holds for the induced game
H. It therefore admits a unique Nash equilibrium D∗ where
X
Di∗ = E [XAi |Xc = Q] =
Ck∗
k∈Ai

11

Now individual rationality of D∗ in game H is equivalent
to coalitional stability of C ∗ in game H, proving the claim.
(d) Using the characterization (10) of πeq , we have
E [πeq ] = π` Fc (Cc ) + πh (1 − Fc (Cc )) = πh − πδ Fc (Cc )
At the Nash equilibrium we have Cc = Q, where Fc (Q) =
γ = (πδ − πs )/πδ . Then,
E [πeq ] = πh − πδ + πs = π` + πs
2

(e) Follows immediately from (14) with Xk ≡ 0.

E. Proof of Theorem 17
The firm can withdraw at most ηo C power from its fully
charged storage. Therefore, the firm must purchase its peakperiod deficit (X −ηo C)+ from the utility. During the off-peak
period, the firm will fully recharge its storage because there
are no holding costs (no leakage). The cost function for the
firm is then


π`
E [min{C, X}]
J(C) = πs C + πh E (X − ηo C)+ +
ηi η0
It is straightforward to verify that this function is strictly
convex. Writing the first-order optimality condition, it follows
that the optimal investment C o solves
0=

π`
dJ
= πs + πh ηo Pr (X ≥ ηo C) +
Pr (X ≥ ηo C)
dC
ηi
2

Rearranging terms establishes the claim.

F. Proof of Theorem 18
First note that the storage investments Qn and Qn+1 are
optimal. Define the collective peak demands
A=

n
X

Xk , B = A + Xn+1

k=1

Using Theorem 4, we have
FA (Qn ) = FB (Qn+1 ) = γ
As a result,
Prob (A ≤ Qn )

= Prob A + X n+1 ≤ Qn+1

≤ Prob A ≤ Qn+1



where the last inequality follows from Xn+1 ≥ 0 (demand is
non-negative). This forces Qn+1 ≥ Qn , proving the claim. 2

