A Distributed Scheme for Power Profile
Market Clearing under High Battery
Penetration ⋆Takayuki Ishizaki, ∗ Masakazu Koike, ∗∗Nobuyuki Yamaguchi, ∗∗∗ Asami Ueda, ∗ Tu Bo, ∗Jun-ichi Imura. ∗∗Graduate School of Engineering, Tokyo Institute of Technology;
2-12-1, Ookayama, Meguro, Tokyo, 152-8552, Japan. e-mail:{ishizaki,ueda,tubo,imura}@cyb.mei.titech.ac.jp.
∗∗
Tokyo University of Marine Science and Technology; 4-5-7,
Kounan, Minato, Tokyo, 108-0075, Japan. e-mail:mkoike0@kaiyodai.ac.jp.
∗∗∗
Tokyo University of Science; 6-3-1, Niijuku, Katsushika, Tokyo
125-8585, Japan. e-mail: n-yama@ee.kagu.tus.ac.jp.
Abstract: In this paper, we formulate a problem of power profile market clearing and develop
a distributed market clearing scheme with explicit consideration of high battery penetration.
The power profile market is a multiperiod electricity market in which each aggregator aims at
making the highest profit by transacting a power profile, i.e., a time sequence of energy amounts
at several time slots, that is generated by dispatchable power generation as well as the charge
and discharge of batteries. It is theoretically shown that the clearing price profile during the time
period of interest tends to level oﬀ in the high penetration of batteries. This finding enables todevelop a distributed market clearing scheme that is implemented as a bidding strategy for the
total energy amount during the period followed by a distributed iterative algorithm for profile
imbalance minimization. Numerical simulations demonstrate the price leveling-oﬀ led by highbattery penetration and the eﬃciency of the proposed distributed scheme.
Keywords: Multiperiod electricity markets, Energy storage, Power profile balancing, Bidding
strategy, Convex analysis.
1. INTRODUCTION

energy resources including energy storage, can be a strong
stakeholder in an electricity market.

The development of a smart grid has been recognized as
one of key issues in addressing environmental and social
concerns, such as the sustainability of energy resources
and the eﬃciency of energy management [Annaswamy andAmin (2013)]. In particular, towards eﬀective integrationof dispatchable and renewable power generation, the potential of energy storage has been attracting international
attention in smart grid community. Actually, energy storage techniques can be expected as a fundamental tool
for load shifting as well as reducing the fluctuation ofrenewable energy.

With this background, we formulate an electricity market
mechanism with explicit consideration of battery penetration, which is referred to as a power profile market
mechanism. A power profile market is a multiperiod electricity market in which each aggregator aims at making
the highest profit by transacting the time sequence ofenergy amounts at several time slots. This energy amountsequence, formulated as a vector having the dimension
compatible with the number of time slots, is referred to as
a power profile. Each aggregator generates a marketable
power profile by aggregating available energy resources,
such as dispatchable and renewable power generation andthe charge and discharge of distributed batteries. As shownin Section 4.3 of [Annaswamy and Amin (2013)], such amultiperiod market is indispensable for making use of thepower shiftability of batteries and flexible loads. This isbecause their utility or cost function is not an additively
decomposable function of period-specific power consumption and generation.

The penetration of energy storage is generally supposed to
be spatially distributed due to the limitation of installation
capability. Examples of distributed energy storage include
electric vehicles, home energy storage systems, batteries inelectric devices, and so forth. Even though the impact ofthese individual materials and components on the grid may
be tiny, the aggregation of them has high potential to serve
for supply-demand balancing in power system operation.
This implies that an aggregator, a manager of available⋆ This work was supported by CREST, Japan Science and Technology Agency.

To establish a mathematically rigorous formulation of
the power profile market mechanism, we first derive aregulation cost function of marketable power profiles,

consisting of load, dispatchable power generation, andbattery charge and discharge power profiles. Then, weshow that the profile regulation cost function is necessarily
convex provided that the aggregator adopts the optimal
strategy for managing dispatchable power generators andbatteries, whose cost functions are assumed to be bothconvex. This clarification enables to formulate the power
profile market clearing problem as a convex program.
Furthermore, we develop a distributed solution scheme
to the power profile market clearing problem, which canbe implemented as an indirect communication among aggregators through an independent system operator (ISO).
The market clearing scheme is developed by theoreticallyshowing that the clearing price profile, i.e., the multiperiod
clearing price vector of the power profile market, tends
to level oﬀ in high penetration of batteries. Numericalsimulations in this paper demonstrate the price levelingoﬀ as well as the eﬃciency of the proposed distributed
scheme, which consists of a bidding strategy for the totalenergy amount during the period and a distributed iterative algorithm for profile imbalance minimization.
Finally, references related to electricity markets are discussed. As market clearing strategies, a number of bidding and dynamic pricing methods have been developedin diﬀerent settings; see [Hansen et al. (2015); He et al.
(2015); Liu et al. (2016); Shiltz et al. (2016)] and references therein. However, these existing methods are notdirectly applicable to the power profile market clearing
problem. This is due to the fact that a transacted power
profile is a high-dimensional vector and the cost functionof power profile regulation is not strictly convex becauseof the power shiftability of batteries; see Section 2.3 fordetails. Furthermore, even though the eﬃciency and significance of their methods are demonstrated numerically, thestructures and properties of market mechanisms are nottheoretically investigated. In contrast to this, by utilizing
tools from convex analysis theory, we clarify a particularimpact of high battery penetration on multiperiod market mechanisms on the basis of a simple but meaningfulmathematical formulation.
The remainder of this paper is structured as follows. In
Section 2, we first formulate the power profile market clearing problem, and then discuss the diﬃculties in addressing
it. Next, in Section 3, we develop a distributed market
clearing scheme while clarifying that the clearing price profile tends to level oﬀ in high battery penetration. Numerical simulations are provided in Section 4 and concluding
remarks are provided in Section 5. All mathematical proofs
are shown in Appendix.
Notation: We denote the set of real values by R, the set of
nonnegative real values by R+ , the image of a matrix M byim M , the all-ones vector by 1, the orthogonal projectionof a vector v onto a subspace V by projV (v), and the direct
product of sets S1 , . . . , Sn by!
S1 × · · · × Sn = i∈{1,...,n} Si .
A function F : Rn → R is said to be convex if"#
F (1 − λ)x + λx′ ≤ (1 − λ)F (x) + λF (x′ )(1)for all λ ∈ (0, 1) and for every pair of x and x′ in thedomain such that the value of F is finite. In particular,

F is said to be strictly convex if (1) holds with the strict
inequality unless x = x′ .
2. FORMULATION OF POWER PROFILE MARKETS
2.1 Aggregator Models
In this subsection, we give a model of aggregators, eachof whom transacts a power profile, i.e., the time sequenceof energy amounts at several time slots. Let A denote theindex set of aggregators and let n denote the number oftime slots during the period of interest. The power profile
equation of the αth aggregator can be described asxα = gα − lα + ηαout δαout − η1in δαin , α ∈ A(2)αwhere xα ∈ Rn denotes the resultant power profile to
the grid, gα ∈ Rn+ denotes the power generation profile
of dispatchable generators, lα ∈ Rn+ denotes the load
profile, and δαin ∈ Rn+ and δαout ∈ Rn+ denote the battery
charge and discharge power profiles. The positive constants
ηαin and ηαout denote the charge and discharge eﬃciency,
respectively, each of which takes a value in (0, 1]. Notethat the sign of xα is positive for outflow direction to thegrid.
In the following, we suppose that the load profile lα isfixed as a constant vector, whereas the dispatchable power
generation profile gα as well as the battery charge anddischarge power profiles δαin and δαout are decision variables.
To realize a desired power profile xα , each aggregator
determines gα and δα := (δαin , δαout ) as complying with theconstraints ofgα ∈ G α , δα ∈ D α ,(3)where Gα and Dα denote some connected spaces including
the origin. The left condition in (3) is given to represent
the upper and lower bounds for the dispatchable generator outputs, whereas the right is given to represent thelimitation of inverter and battery capacities.
With respect to each power profile xα , we denote thefeasible subspace of the dispatchable power generation andthe battery charge and discharge profiles as$%
Fα (xα ) := (gα , δα ) ∈ Gα × Dα : (2) is satisfied , (4)and denote the set of realizable power profiles as$%
Xα := xα ∈ Rn : Fα (xα ) ̸= ∅ .
(5)Furthermore, we denote the generation cost function of
dispatchable generators and the battery usage cost function as
Gα : G α → R+ , D α : D α → R+ .
(6)
On the basis of this formulation, we define a cost function
with respect to power profile regulation as follows.
Lemma 1. In the notation above, if the generation costfunction Gα and the battery usage cost function Dα areconvex on convex domains Gα and Dα , then the profile
regulation cost function defined by&'
Fα (xα ) :=min
Gα (gα ) + Dα (δα )(7)(gα ,δα )∈Fα (xα )is convex on the convex domain Xα .

The value of Fα (xα ) in (7) represents the minimum costto realize a power profile xα . Lemma 1 shows that the
profile regulation cost function turns out to be convex

with respect to generated power profiles, provided thatthe aggregator adopts the optimal strategy for the determination of the dispatchable power generation profile gα
and the battery charge and discharge power profile δα . In
the rest of this paper, we assume the convexity of Gα andDα , which implies that every Fα is convex.

– The clearing price profile λ∗ is determined by the ISO
without the information of the profile regulation costfunctions of the aggregators.
– Each clearing power profile x∗α is determined by anaggregator without the information of the profile regulation cost functions of the other aggregators.

2.2 Power Profile Market Clearing Problem
In this subsection, we formulate a market clearing problem
with respect to power profile transaction based on theaggregator model in Section 2.1. Let λ ∈ Rn denote theprice profile with respect to power profile transaction. Notethat this vector corresponds to the set of prices at alltime slots during the period of interest. Then, the profit
function for selling surplus power (or for buying shortage
power) during the period is defined as
Jα (xα ; λ) := λT xα − Fα (xα ),(8)Twhere λ xα represents the total income by selling thepower profile during the time period. On the basis of
this profit function, each aggregator can determine a bidfunction with respect to power profile transaction given as$%xα (λ) := xα ∈ Xα : Jα (xα ; λ) ≥ Jα (x′α ; λ), ∀x′α ∈ Xα ,(9)which corresponds to the set of xα that attain the maximum of the profit function Jα (xα ; λ) with a fixed priceprofile λ. The bid function is a set valued function, i.e., it
is mathematically referred to as a correspondencexα : Rn → Rn .
(10)
It should be noted that the graph of xα is depicted as a
one-dimensional curve for n = 1, whereas it is depicted asa higher-dimensional hyperplane for a general dimensionn.
In the rest of this paper, for simplicity of notation, we
denote the direct product of xα as!
xA (λ) := α∈A xα (λ).
Furthermore, the tuple of a symbol indexed by α ∈ A isdenoted by the corresponding symbol with the subscriptof A, for example
xA := (xα )α∈A .
The following lemma shows the existence of a clearing priceprofile, determined via the aggregation of power profile bidfunctions.
Lemma 2. Consider a set of aggregators in Section 2.1. If
at least one profile regulation cost function Fα is smooth,then there exists the unique clearing price profile, denotedas λ∗ , such that the power profile balance(∗∃x∗A ∈ xA (λ∗ ) s.t.
(11)α∈A xα = 0
is attained.
The problem of power profile market clearing is formulated
as a problem of finding the clearing price profile and thetuple of clearing power profiles, denoted as λ∗ and x∗A ,respectively, that attain the power profile balance in (11).
More formally, the power profile market clearing problem
is summarized as follows.
Problem. Consider a set of aggregators in Section 2.1.
Devise a market clearing scheme to realize the power
profile balance in (11) such that the following requirements
are satisfied.

2.3 Diﬃculties in Power Profile Market Clearing Problem
In this subsection, we overview the diﬃculties in addressing the power profile market clearing problem in Section 2.2, while reviewing several existing methods for market clearing. The problem of power profile market clearing
is equivalent to finding a solution to the convex programof((min α∈A Fα (xα ) s.t.
(12)α∈A xα = 0,
xA

whose Lagrange relaxation is given by&'(
max min α∈A Fα (xα ) − λT xα .
λ

xA

(13)

Note that the optimal Lagrange multiplier of (13) corresponds to the clearing price profile. This can be seen fromthe fact that the minimization in (13) is equivalent to
the maximization of the profit function Jα in (8); thereby
leading to the problem of finding x∗A and λ∗ such that (11)holds.
One may think that the ISO can easily find the clearing
price profile λ∗ , provided that each aggregator submits thegraph of the bid function xα in (9) to the ISO. However,
not only finding λ∗ that solves the generalized equationin (11), but also plotting the graph of the bid function
xα is not straightforward in multi-dimensional cases, i.e.,n ≥ 2. This is due to the fact that the graph of eachelement of xα is depicted as an n-dimensional hyperplane.
This implies that plotting the graph of each element of xα
requires griding the n-dimensional space of price profiles;
thereby incurring considerably large computation loads.
Thus, existing bidding strategies, such as in [Liu et al.
(2016)], are not directly applicable to the power profile
market clearing problem.
As another approach, dynamic pricing methods can befound in the literature [Hansen et al. (2015); Shiltz et al.
(2016)]. This approach is mainly based on the dual ascentalgorithm to solve the convex program in (12), or equivalently (13), namely) k+1
xα = arg min Fα (xα ) − λTα∈Ak xα ,xα(14)(
λk+1 = λk + γ k α∈A xk+1
α ,where γ k denotes a step size to update the dual variable.
Even though this algorithm can be implemented in a
distributed manner, the update of each primal variable
xkα assumes the strict convexity of the cost function Fα .
In fact, the cost function of an aggregator with battery
storage does not generally satisfy the assumption of thestrict convexity; see Section 3.1 for details.
As seen above, neither a bidding strategy nor a dynamic
pricing method is directly applicable. Thus, it is crucial todevelop a novel distributed scheme for the power profile
market clearing problem.

3. POWER PROFILE MARKET CLEARING SCHEME
3.1 Price Analysis under High Battery Penetration
In this subsection, we first analyze a property of theclearing price profile in a situation where a large amountof battery storage penetrates in aggregators. In particular,
it will be found that the clearing price profile tends to leveloﬀ in high battery penetration. Furthermore, the level-oﬀ
price can be found via a bidding strategy with respect tothe total amount of energy transaction during the period
of interest.
For the price analysis below, we introduce a notion of directional derivatives. The one-sided directional derivativeof Fα at xα with respect to v is defined as
F (xα + ϵv) − F (xα )Fα′ (xα ; v) = lim
ϵ↓0ϵ
if it exists. In particular, the one-sided directional derivative is said to be two-sided if Fα′ (xα ; −v) exists andFα′ (xα ; −v) = −Fα′ (xα ; v).
Given this notation, we define the following terminology.
Definition. Consider an aggregator in Section 2.1, and let(i, j) ∈ {1, . . . , n} × {1, . . . , n}denote a time point pair. Then, a power profile xα is said tobe (i, j)-shiftable if the directional derivative of the profile
regulation cost function Fα at xα with respect to ei − ejis two-sided and zero, where ei denotes the ith canonicalunit vector.
An aggregator with battery storage is generally endowedwith this notion of power profile shiftability. This isbecause the amounts of battery charge and discharge
power at some time slots can be shifted without changing
the total amount of charge and discharge energy during
the time period of interest. Note that Fα is not strictly
convex whenever it is (i, j)-shiftable. This implies thatthe profile regulation cost function of an aggregator withbattery storage does not satisfy the assumption of strict
convexity in general.
The following lemma shows that the power profile shiftability can lead to the leveling-oﬀ of clearing price profiles.
Lemma 3. Consider a set of aggregators in Section 2.1,
and let λ∗ denote the clearing price profile in (11). Then,
there exists at least one aggregator such that a power
profile xα ∈ xα (λ∗ ) is (i, j)-shiftable if and only if
λ∗i = λ∗j ,(15)∗
where λi denotes the ith element of λ∗ .
Lemma 3 shows that the shiftability of an optimal power
profile xα ∈ xα (λ∗ ) that maximizes the profit function
J(xα ; λ∗ ) in (8) leads to the leveling-oﬀ of the clearing
price profile λ∗ . Note that the degree of power shiftability
tends to increase as the capacities of batteries increase.
Therefore, if a suﬃciently large amount of battery storage
penetrates in aggregators, we can expect that the prices
at all time slots are close, i.e., λ∗ is close to λ∗e 1 for somescalar λ∗e . In what follows, focusing on this situation ofhigh battery penetration, we develop a distributed scheme
for profile market clearing

The following theorem shows that the value of λ∗e , called
the clearing energy price, can be determined by a bidding strategy with respect to the total amount of energy
transaction during the period of interest, where we use theenergy bid function defined byeα (λe ) := 1T xα (λe 1),(16)
whose direct product is denoted as!
eA (λe ) := α∈A eα (λe ).

Theorem 1. Consider a set of aggregators in Section 2.1. If
at least one profile regulation cost function Fα is smooth,then there exists the unique clearing energy price, denotedas λ∗e , such that the energy balance(∗∃e∗A ∈ eA (λ∗e ) s.t.
(17)α∈A eα = 0
is attained. Furthermore, assume thatλ∗ ∈ im 1(18)∗
for the clearing price profile λ in (11), or equivalently,
assume that there exists at least one aggregator such thata power profile xα ∈ xα (λ∗ ) is (i, j)-shiftable for every
time point pair (i, j). Then, λ∗ coincides with λ∗e 1.
Theorem 1 provides a bidding strategy for determining
the clearing energy price λ∗e , which can be expected
to approximate the clearing price profile as λ∗ ≃ λ∗e 1
especially in high battery penetration. It should be noted
that, in a manner similar to existing bidding strategies[Liu et al. (2016)], the ISO can easily find the value of λ∗ein (17) by aggregating the graphs of energy bid functions
submitted from individual aggregators. The graph of eαin (16) is reasonably plottable by each aggregator because
it is just a one-dimensional curve, namely
eα : R → R,
as opposed to a high-dimensional hyperplane of xα in (10).
3.2 Distributed Scheme for Imbalance Minimization
In this subsection, we develop a distributed scheme to finda tuple of power profiles that minimizes the amount of
power profile imbalance. The alternating direction methodof multipliers (ADMM) [Boyd et al. (2011)] provides a
distributed iterative scheme as follows.
Theorem 2. Consider a set of aggregators in Section 2.1.
Let λ be a price profile, and consider the iterative algorithm given as"#
xk+1= projxα (λ) xkα − ∆kα , α ∈ A(19)
αwhere the update with respect to the index α is performedin ascending order and the kth-step power profile imbalance ∆kα is defined as((∆kα := i≤α−1 xk+1
+ i≥α xki .
(20)iThen, for any initial value x0A ∈ xA (λ), it follows thatkx∞
A := lim xAk→∞is a solution to the convex program of*(*2
min 1 *xα * .
xA ∈xA (λ) 2

α∈A

(21)

In particular, if λ is given as the clearing price profile λ∗in (11), then(∞(22)α∈A xα = 0,∞
which implies that xA coincides with x∗A in (11).

Theorem 2 provides a distributed scheme to solve theimbalance minimization problem in (21). Note that the
update in (19) can be performed by individual aggregators,
provided that the power profile imbalance ∆kα in (20) is
computed and informed by the ISO. Thus, the distributed
scheme can be regarded as a communication among theaggregators through the ISO.
Combining the theoretical results in Theorems 1 and 2,
we develop a distributed scheme for power profile market
clearing as follows.
(i) Each aggregator submits the graph of the energy bid
function eα in (16) to the ISO.
(ii) The ISO determines the clearing energy price λ∗e thatattains the energy balance in (17).
(iii) Each aggregator selects an initial power profile x0α in
xα (λ∗e 1) and submits it to the ISO.
(iv) Each power profile xkα is updated according to the
distributed scheme in (19) with λ = λ∗e 1 that attains theminimization of the resultant power profile imbalance.
This power profile market clearing scheme can be expected
to work well in high battery penetration. This is owing
to the fact that the resultant power profile imbalance is
expected to be small if the approximation error between
λ∗ and λ∗e 1 is small. In particular, the power profile
market clearing problem in Section 2.2 is exactly solved if
λ∗ = λe∗ 1. The eﬃciency of this scheme will be numericallydemonstrated in Section 4 below.
4. NUMERICAL SIMULATION
4.1 Simulation Setup
We consider a power profile market that consists of threeaggregators. The period of power profile market clearing
is supposed to be six hours and the amounts of transacted power are determined at every 30 minutes, i.e., thedimension of transacted power profiles is n = 12. We
suppose that the first two aggregators manage five million
residential consumers with energy storage, whereas thethird aggregator manages a thermal generator, namelyxα = −lα + ηαout δαout − η1in δαin , α ∈ {1, 2},(23)
αand x3 = g3 . In our simulation, ηαout = ηαin = 0.95
for α ∈ {1, 2} are given for the charge and discharge
eﬃciency of batteries. The inverter and battery capacities,
corresponding to Dα in (3), are supposed to be 12 [kW]and 24 [kWh], respectively, for one residential consumer.
The load profile lα is given as a net load profile including
consumption and photovoltaic power generation during
the period of 6 [h] to 12 [h], for which we use the datameasured at houses in Ohta city, Japan.
With G3 = R12+ in (3), the fuel cost function of the thermal
generator is given as
G3 (g3 ) = 0.19 × ∥g3 ∥2 + 2.5 × 1T g3 ,(24)
which corresponds to the oil thermal generator in [Masuta
and Yokoyama (2012)]. On the other hand, the battery usage cost function Dα in (6) is supposed to be decomposedasDα (δα ) = Eα (δα ) − Sα (δα ), α ∈ {1, 2}(25)
where Eα represents a battery degradation cost function
given asEα (δα ) = 10−5 × ∥δαout ∥2 ,(26)
and Sα represents an assessment function with respect tothe stored energy at the termination time of the period ofinterest.
Let s0α denote the initial amount of stored energy, i.e., thestate of charge, which is defined as the deviation from aneutral value. Then, the amount of stored energy deviation
at the termination time can be represented assα (δα ) := s0α + 1T (δαin − δαout ).
(27)
0
In our simulation, we set sα = 0 for α ∈ {1, 2}, whose unit
is [GW0.5h]. It is reasonable to suppose that a higher levelof terminal stored energy is more preferable than a lower
level, and vice versa. To take into account this aspect,each aggregator is supposed to assess the value of terminal
stored energy as)λ̂α ηαout sα (δα ), sα (δα ) ≥ 0,Sα (δα ) :=(28)λ̂αsα (δα ) < 0,
η in sα (δα ),αwhere λ̂α ∈ R+ denotes a constant electricity price oﬀered,e.g., by an independent energy supplier having a steadycontract with the corresponding aggregator. Comparing
the energy price λe with λ̂α , each aggregator determines
own energy bid function eα in (16), i.e., the battery usage
strategy for buying and selling energy in the six-hour
period. In our simulation, we set λ̂1 = 5.0 and λ̂2 = 4.0,
whose unit is [Yen/kW0.5h]. On the basis of the above
setting, we obtain the profile regulation cost function Fα
in (7) for each aggregator.
It should be remarked that Dα in (25) is convex because
Eα in (26) is convex and Sα in (28) is concave. To show
the concavity of Sα , i.e., the convexity of −Sα , we define)λ̂α ηαout s, s ≥ 0,h(s) :=
λ̂αs < 0,
η in s,αwhich is concave for any values of ηαout , ηαin ∈ (0, 1] andλ̂α ∈ R+ . Note that sα in (27) is aﬃne andSα (δα ) = h ◦ sα (δα ).
Thus, the convexity of −Sα is proven by the fact that theinverse image of a convex function under an aﬃne mapping
is a convex function; see Appendix for details. Note thatDα in (25) is convex but it is not strictly convex.
4.2 Results
In the following, varying the levels of battery penetration,
we solve the power profile market clearing problem in
Section 2.2. In particular, we consider the three cases
where 20%, 40%, and 60% of residential consumers havethe batteries. In Fig. 1, we plot the resultant clearing priceprofile λ∗ ∈ R12 for each case, which is obtained by solving
the convex program in (13). As theoretically supported in
Lemma 3, the clearing price profile tends to level oﬀ as thebattery penetration level becomes high.
In the following, without knowing the exact value of theresultant clearing price profiles in Fig. 1, we solve thepower profile market clearing problem according to the
distributed scheme developed in Section 3.2. The graphsof the resultant energy bid functions eα in (16) are shownin Fig. 2. From the total energy bid functions given as

40%

20%

60%

5
4.5
12[h] 6[h]
12[h]
6[h]
12[h]
4 6[h]
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12Time index

Time index

Total [GW0.5h]

Agg. 3 [GW0.5h]

Agg. 2 [GW0.5h]

40%

60%

10
8
6
4
2
0

0

5

10 15 20 25 30 35 40 0

Steps

5

10 15 20 25 30 35 40 0

Steps

5

10 15 20 25 30 35 40

Steps

Fig. 3. The norms of profile imbalance in the cases ofdiﬀerent battery penetration levels.

60%
Power [GW]

Agg. 1 [GW0.5h]

20%

40%

12

Time index

Fig. 1. The clearing price profiles in the cases of diﬀerent
battery penetration levels.
40
20
0
-20
-40
-60
-80
-100
-120
40
20
0
-20
-40
-60
-80
-100
-120
180
160
140
120
100
80
60
40
20
0
-20
250
200
150
100
50
0
-50
-100
-150
-200
-250

Norm of imbalance

Price [Yen/kW0.5h]

20%
5.5

6
4
2
0
-2
-4
-6
-8
-10
-12

20%

40%

60%

Agg. 1
Agg. 2Agg. 3
Imbalance

12[h]
6[h]
12[h]
6[h]
12[h]
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12Time index

Time index

Time index

Fig. 4. The resultant power profiles in the cases of diﬀerent
battery penetration levels.
of this finding, we have developed a distributed market
clearing scheme implemented as the combination of an energy bidding strategy and a distributed iterative algorithm
for profile imbalance minimization.

4.73

2

3

4

5

4.53

6

Price [Yen/kW0.5h]

7

8 2

3

4

5

4.52

6

Price [Yen/kW0.5h]

7

8 2

3

4

5

6

7

8

Price [Yen/kW0.5h]

Fig. 2. The resultant energy bid functions in the cases ofdiﬀerent battery penetration levels.
(∗α∈A eα , we find the clearing energy price λe in (17)as 4.75, 4.53 and 4.52 [Yen/kW0.5h] for the 20%, 40%,
and 60% penetration levels, respectively. This procedurecorresponds to Steps (i) and (ii) in our distributed scheme.
Next, we implement the iterative scheme in Theorem 2that performs profile imbalance minimization. This procedure corresponds to Steps (iii) and (iv). The normsof resultant profile imbalance are plotted in Fig 3, where
the horizontal axis corresponds to the number of iteration
steps. From this figure, we see that some amount of profile
imbalance remains in the 20% penetration level, whereas
almost no amount of imbalance remains in the 40% and
60% penetration levels. This is compatible with the fact
shown in Theorem 2 that the clearing price profile in the
20% penetration level does not completely level oﬀ whereas
those in the 40% and 60% penetration levels clearly leveloﬀ; see Fig. 1. Finally, the resultant power profiles of theaggregators are plotted in Fig 4. From this figure, we seethat our distributed scheme works well especially in highbattery penetration.
5. CONCLUDING REMARKSIn this paper, we have formulated a power profile market
clearing problem and develop a distributed market clearing
scheme with explicit consideration of high battery penetration. It has been theoretically shown that the clearing
price profile during the time period of interest tends tolevel oﬀ in the high penetration of batteries. On the basis

The energy bidding strategy in this paper can be regarded
as a long-period version of the current energy market. For
example, an energy transaction is carried out at every 30
minutes in the Japanese electricity market. Our numericalsimulation suggests that, provided that the level of battery
penetration is suﬃciently high, the energy balancing at
every 30 minutes, i.e., power profile balancing, can be attained via the six-hour period energy transaction followed
by a profile imbalance minimization scheme. The power
profile market would be one of well-defined multiperiod
electricity markets that can utilize and enhance the potential of energy storage.
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APPENDIX: MATHEMATICAL PROOFS
Preliminary Facts from Convex Analysis: We first overview
a rich variety of facts shown in convex analysis theory[Boyd and Vandenberghe (2004); Rockafellar (1970)]. Let
F : Rn → R be a convex function. For A denoting a linear
mapping from Rn to Rm , the function F̂A defined byF̂ (x) := F (Ax)(29)

is convex on R . Furthermore, the function F̌A defined byF̌ (y) := inf F (x) s.t. Ax = y
(30)
n

x

is convex on Rm . The function F̂ is called the inverse imageof F under A, while F̌ is called the image of F under A;
see [Theorem 5.7, Rockafellar (1970)] for details.
The conjugate of F is defined by$ T%
F (λ) := sup
λ x − F (x) ,(31)

Proof of Lemma 2: From the definition of the conjugate
function, we notice that
sup Jα (xα ; λ) = F α (λ)
xαfor Jα in (8). Furthermore, the first derivative condition
for the supremum, namely∂Jα (xα ; λ)= 0,∂xαleads to λ ∈ ∂Fα (xα ). Because λ ∈ ∂Fα (xα ) if and only ifxα ∈ ∂F α (λ), we see that xα in (9) is identical to ∂F α .
Thus, (11) is equivalent to the generalized equation of(
0 ∈ α∈A ∂F α (λ∗ ).
(33)

Note that ∂F α is strictly monotone if F α is strictly
convex. Furthermore, F α is strictly convex if and only
if Fα is smooth. Thus, provided that at least one Fα issmooth, the sum of monotone mappings in (33) is strictly
monotone. This proves the uniqueness of λ∗ , which solves
the generalized equation in (33) with respect to a strictly
monotone mapping.
!
Proof of Lemma 3: From the fact that xα ∈ ∂F α (λ∗ ),we have λ∗ ∈ ∂Fα (xα ). As shown in [Theorem 23.4,
Rockafellar (1970)], it follows thatFα′ (xα ; v) =λ∈dom F

Furthermore, the strict convexity of F is equivalent to thesmoothness of F . In convex analysis theory, the transformation between (31) and (32) is called the LegendreFenchel transformation, and some of conjugate pairs canbe found in closed forms.
The subdiﬀerential of a convex function F is defined by∂F (x) := {λ : F (x′ ) ≥ F (x) + λT (x′ − x), ∀x′ ∈ dom F },which is a multivalued mapping to a convex set. As shownin [Corollary 31.5.2, Rockafellar (1970)], it follows that
∂F : Rn → Rn
is a monotone increasing mapping, namely"#T∂F (x) − ∂F (x′ ) (x − x′ ) ≥ 0, ∀x, x′ ∈ dom F,
if F is convex. In a similar manner, it is shown to bestrictly monotone if F is strictly convex [Theorem 2.1, Luc(1994)]. Furthermore, [Theorem 23.5, Rockafellar (1970)]
shows thatx ∈ ∂F (λ), λ ∈ ∂F (x)are equivalent. These correspond to the first derivative
conditions for the suprema in (31) and (32).
Proof of Lemma 1: We see that xα in (2) is given as anaﬃne mapping of (gα , δα ). As shown in (30), the image ofa convex function under an linear mapping, or equivalently
aﬃne mapping, is a convex function. Thus, the convexityof Fα in (7) is proven.
!

v T λ.

Thus, (i, j)-shiftability of xα ∈ ∂F α (λ∗ ) implies that
0 = Fα′ (xα ; ei − ej ) ≥ (ei − ej )T λ∗ ,0 = Fα′ (xα ; ej − ei ) ≥ (ej − ei )T λ∗ ,x∈dom F

where domF denotes the eﬀective domain of F . It is known
that F is convex and the conjugate of F coincides with F
as long as F is convex, namely$ T%
F (x) = supx λ − F (λ) .
(32)

sup
λ∈∂Fα (xα )or equivalently, (ei − ej )T λ∗ = 0. This leads to (15).

!

Proof of Theorem 1: Note that (17) is equivalent to"(#(∗
0 ∈ α∈A 1T ∂F α (λ∗e 1) = ∂α∈A F α (λe 1) ,where the sum of F α (λe 1) is strictly convex with respectto λe , provided that at least one Fα is smooth. Thus, theuniqueness of λ∗e is proven in a manner similar to the proof
of Lemma 2.
Furthermore, (18) implies that λ∗ = γ1 holds for somescalar γ. Substituting this into (33) and multiplying it by
1T from the left, we have(0 ∈ α∈A 1T ∂F α (γ1),
which implies that γ solves the generalized equation.
Because of the uniqueness of the solution, the scalar γ
is identical to λ∗e . This proves the claim.
!
Proof of Theorem 2: We follow the arguments in [Boydet al. (2011)] for deriving the distributed scheme in (19).
A proof of convergence can be found in the reference orin [Rockafellar (1976)]. Notice that the convex program in(19) is equivalent to((min 12 ∥ϵ∥2 + α∈A Ixα (λ) s.t.
α∈A xα + ϵ = 0xA ,ϵwhere Ixα (λ) denotes the indicator function of xα (λ). For
this program, we consider the augmented Lagrangian in
the scaled form as
(
(
1
1
2
2
α∈A Ixα (λ) + 2 ∥α∈A xα + ϵ + y∥2 ∥ϵ∥ +
where y denotes the scaled dual variable. Then, the iteration of the ADMM is obtained as

⎧
" "(
##
(
k+1
k+1
⎪
−
+ i>α xki + ϵk + y k⎨ xα = proj"x(i<α #xiα (λ)
kkϵk+1 = − 12α∈A xα + y#⎪
⎩ y k+1 = 1 "(kk.
α∈A xα + y2Note that ϵk and y k in the update of xkα is redundantbecause ofϵk + y k = 0.
Thus, this is reduced to (19). In particular, when λ = λ∗ ,there exists x∗A ∈ xA (λ∗ ) such that (11) holds. Thus, (22)follows.
!