VOL. E98-A NO. 7
JULY 2015
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IEICE TRANS. FUNDAMENTALS, VOL.E98–A, NO.7 JULY 2015

1504

PAPER

Autonomous Decentralized Mechanism for Energy Interchanges
with Accelerated Diﬀusion Based on MCMC
Yusuke SAKUMOTO†a) and Ittetsu TANIGUCHI††b) , Members

SUMMARY It is not easy to provide energy supply based on renewable energy enough to satisfy energy demand anytime and anywhere because the amount of renewable energy depends on geographical conditions and the time of day. In order to maximize the satisfaction of energy demand by renewable energy, surplus energy generated with renewable energy should be stored in batteries, and transmitted to electric loads
with high demand somewhere in the electricity system. This paper proposes a novel autonomous decentralized mechanism of energy interchanges
between distributed batteries on the basis of the diﬀusion equation and
MCMC (Markov Chain Monte Carlo) for realizing energy supply appropriately for energy demand. Experimental results show that the proposedmechanism eﬀectively works under several situations. Moreover, we discuss a method to easily estimate the behavior of the entire system by eachnode with the proposed mechanism, and the application potentiality of thisestimating method to an eﬃcient method working with non-renewable generators while minimizing the dependence of non-renewable energy, and an
incentive mechanism to prevent monopolizing energy in systems.
key words: renewable energy, microgrid, autonomous decentralized control, power routing

1. Introduction
Renewable energy (e.g., sunlight and wind) is hoped to bealternative energy of the conventional fuel (e.g., fossil fueland nuclear fuel). Since the amount of renewable energy
depends on geographical conditions and the time of day, theenergy supply based on the renewable energy cannot be provided enough to satisfy energy demand at some times andlocations. Thus, it is important challenge for renewable energy to develop an eﬀective method of the energy supply
appropriately for energy demand.
In order to prevent wasting renewable energy, surplus
energy generated with renewable energy should be stored inbatteries, and transmitted to electric loads with high demand
somewhere in the electricity system. Microgrid [1]–[3],which is an electricity system interconnecting distributed
batteries, would be a key technology to provide such theeﬀective way for surplus energy interchanges.
Some works [4]–[8] are proceeding with energy interchanges between distributed batteries for realizing the energy supply appropriately for energy demand in a microgrid.
Taniguchi et al. proposed an energy trading algorithm for
Manuscript received September 27, 2014.
Manuscript revised February 27, 2015.
†
The author is with Tokyo Metropolitan University, Hino-shi,
191-0065 Japan.
††
The author is with Ritsumeikan University, Kusatsu-shi, 5250058 Japan.
a) E-mail: sakumoto@tmu.ac.jpb) E-mail: i-tanigu@fc.ritsumei.ac.jp
DOI: 10.1587/transfun.E98.A.1504

Fig. 1 Energy interchanges with energy market and routing (left) and
autonomous decentralized mechanism (right).

energy interchanges on the basis of machine learning [5].
The algorithm supposes a virtual energy market to trade
surplus energy between distributed batteries in a microgrid,and performs energy interchanges using energy routing (seeFig. 1 (left)) based on the trades in the virtual energy market.
By using the energy trading, the algorithm balances energy
supply and demand of each node, and consequently provides the energy supply appropriately for energy demand.
However, since such the energy market is a kind of centralized management system, any energy interchanges cannot
be performed if the energy market is down. Hence, energy
interchanges should be realized by only using autonomous
decentralized manner. Therefore, we will design a methodof energy interchanges without any centralized management
system (see Fig. 1 (right)).
In a physical system, when ink spills into water, particles composing the ink are diﬀused throughout the waterover time until particle densities are eventually uniformed
without supposing any centralized entity. This phenomenon
can be described by the diﬀusion equation. If the diﬀerence
between energy supply and demand in a microgrid could
be diﬀused like the particle density, the energy supply appropriately for energy demand would be realized using autonomous decentralized manner.
This paper proposes a novel autonomous decentralizedmechanism of energy interchanges based on the diﬀusion
equation for energy supply appropriately for energy demandin microgrids. The diﬀusion equation represents the local interaction between diﬀerential volume elements in a physicalsystem, and so it can be converted into the local interaction
between adjacent nodes without necessity of any centralized
management system [9] (see Fig. 2), which achieves high robustness for disasters. We first derive an expression of theenergy interchange amount between batteries of an adjacentnode pair on the basis of the diﬀusion equation, and thenc 2015 The Institute of Electronics, Information and Communication Engineers
Copyright SAKUMOTO and TANIGUCHI: AUTONOMOUS DECENTRALIZED MECHANISM FOR ENERGY INTERCHANGES

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Table 1V
E
N

Fig. 2 The conversion from the local interaction described in the diﬀusion equation to the local interaction in a network.

speed up the derived expression by using MCMC (Markov
Chain Monte Carlo). Moreover, we visually confirm the effectiveness of the proposed mechanism, and quantitatively
investigate the speed of balancing energy supply and demand with the proposed mechanism under several situations. Finally, we discuss a method to easily estimate thebehavior of the entire system by each node with the proposed mechanism, and the application potentiality of thisestimating method to an eﬃcient method working with nonrenewable generators while minimizing the dependence of
non-renewable energy, and an incentive mechanism to prevent monopolizing energy in systems.
This paper is organized as follows: Section 2 describes
the system model, and Sect. 3 shows the proposed mechanism. Experimental results are described in Sect. 4. Section 5 discusses the extensibility of our proposed mechanism. Section 6 oncludes this paper.

Definition of symbols.

energy transmission network G
set of nodes
set of links
number of nodes

ai
bi
θi
gi (t)ci (t)qi (t)q̂i (t)Ji→ j (t)oi (t)ui (t)

node iset of adjacency nodes
battery capacity
energy demand
generated energy at time t
consumed energy at time tremaining battery amount at time t
suﬃciency level at time tenergy interchange amount to node j at time t
amount of overrun at time t
amount of underrun at time t

k
κ
ΔT J

proposed mechanism
diﬀusion coeﬃcientparameter used in Eq. (12)update interval of Ji→ j (t)2. System Model
We use a simple microgrid model to discuss fundamental
properties of the proposed mechanism. The definition ofsymbols of the system model is shown in Table 1.
The system has network G = (V, E) where V and E
are sets of nodes and links. Each node has electric equipment (i.e., a energy generator, a battery, and one or moreloads), and can transmit energy to batteries of other nodes
connected with a link in E. Let ai be the set of nodes connected to node i. We call nodes in ai “adjacency nodes”. To
discuss fundamental properties, we assume that network G
is stationary and connected.
We denote the remaining battery amount and the energy demand of node i by qi (t) and θi , respectively. We define a metric for the diﬀerence between energy supply anddemand at node i by suﬃciency level q̂i (t), which is givenbyq̂i (t) := qi (t) − θi .

(1)

Node i can know whether its adjacent nodes j needmore energy by the observation of q̂ j (t) ( j ∈ ai ). If q̂i (t) =q̂ j (t) for all node pairs (i, j), the energy supply would be
performed appropriately for the energy demand of nodes attime t.
To uniform suﬃciency levels of all nodes, each nodetransmits its energy to batteries of adjacent nodes untilq̂i (t) = q̂ j (t) for all node pairs (i, j) are fulfilled. Every ΔT J ,node i autonomously gathers suﬃciency levels q̂ j (t) from

Fig. 3The energy interchanges of node i.

adjacent nodes, and calculates energy interchange amountJi→ j (t). Node i transmits Ji→ j (t) to the battery of adjacentnodes j. Figure 3 illustrates the energy interchanges of node
i in the system model.
Let gi (t) and ci (t) be the generated energy and the consumed energy of node i. The evolution of remaining battery
amount qi (t) of node i is given by t+ΔT Jqi (t + ΔT J ) − qi (t) =gi (t) − ci (t) d t
 t
(2)

+
J j→i (t)− Ji→ j (t) .
j∈aiNote that we assume any energy loss of battery-to-battery
charging can be ignored.
3.

Autonomous Decentralized Mechanism for Energy
Interchanges

3.1

Basic Idea

We derive the expression of energy interchange amountJi→ j (t) for uniforming all suﬃciency levels on the basis of
the diﬀusion equation. The diﬀusion equation describes the
dynamics of particle density y(t, X), and is given by

IEICE TRANS. FUNDAMENTALS, VOL.E98–A, NO.7 JULY 2015

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∂y(t, X)= k y(t, X),(3)∂twhere X is a positional vector, k is the diﬀusion coeﬃcient,and  is the Laplace operator. The diﬀusion process given
by Eq. (3) continues until y(t, x) = 0, which corresponds toa uniformed state for particle density y(t, X). Hence, it ensures that particle density y(t, X) is uniformed with Eq. (3).
The diﬀusion equation represents the local interaction between diﬀerential volume elements, and so it can approximately converted to the local interaction between adjacency
nodes in a network. Hence, we can derive the expression ofenergy interchange amount Ji→ j (t) in our autonomous decentralized mechanism. Moreover, the diﬀusion equationconserves the sum of the particle densities, and so the derived expression of Ji→ j (t) can avoid any energy loss in thecalculation of energy interchange amounts.
The faster uniforming q̂i (t) is better for appropriate energy supply. However, since Eq. (3) is a linear equation if k
is constant, it is slow to converge to a uniformed state. To
speed up the uniforming q̂i (t) with Eq. (3), we derive a nonlinear diﬀusion equation on the basis of MCMC (Markov
Chain Monte Carlo), and the improved expression of Ji→ j (t)corresponding to the non-linear diﬀusion equation. Moreover, we will prove that the improved expression of Ji→ j (t)can guarantee the uniforming q̂i (t) as well as the original
expression.
3.2 Expression of Energy Interchange Amount Ji→ j (t)First, we derive the expression of energy interchangeamount Ji→ j (t) based on the diﬀusion equation. To obtain
the expression from Eq. (3), we not only replace y with q̂,but also determine the calculation way of . If the coordinate system for positional vector X can be defined,  should
be calculated with the same way in the physics. However,in network G, a coordinate system cannot be defined because each node has a diﬀerent number of its adjacent nodes.
Hence, we define the calculation way of  in network G
with considering its significance, which is the sum of diﬀerences in its surroundings. Therefore, in network G, insteadof Eq. (3), we use the following discrete diﬀusion equationof suﬃciently level q̂i (t)q̂i (t + ΔT J ) − q̂i (t)=k ΔT J
(4)q̂ j (t) − q̂i (t) ,j∈aiwhere k is the diﬀusion coeﬃcient corresponding to k in
Eq. (3). Note that k ΔT J should be small enough to stably
calculate Eq. (4). According to [10], k ΔT J should be set in
0 < k ΔT J < 1/max(|ai |) for all i. Equation (4) is also used
in a model of water supply systems [11]. One can confirm
that the diﬀusion process given by Eq. (4) also continues a
uniformed state for suﬃciency level q̂i (t).
According to Eq. (4), the expression of energy interchange amount Ji→ j (t) for uniforming q̂i (t) is given by
Ji→ j (t) = k ΔT J q̂i (t).

(5)

With Eq. (5), energy interchange amounts do not depend

on suﬃciency levels of adjacency nodes. However, theuniforming suﬃciency levels would be faster when using
Ji→ j (t) appropriately for suﬃciency level q̂ j (t) of adjacency
node j. In other words, energy interchange amount Ji→ j (t)for node pair (i, j) with a larger diﬀerence between q̂i (t) andq̂ j (t) should be larger than other energy interchange amounts
for decreasing the variance of suﬃciency levels. Hence, theproposed mechanism does not use the expression (5) derived
from Eq. (4). The proposed mechanism uses the expression
based on MCMC discussed on the following paragraphs.
To obtain the expression for fast uniforming q̂i , we consider the following non-linear diﬀusion equationq̂i (t + ΔT J ) − q̂i (t) = k fi→ j (q̂i (t), q̂ j (t))


ΔT J
q̂ j (t) − q̂i (t) ,

(6)

j∈ai

and the corresponding expression of energy interchangeamount Ji→ j (t) is given by
Ji→ j (t) = k ΔT J fi→ j (q̂i (t), q̂ j (t)) q̂i (t),(7)

where fi→ j (q̂i (t), q̂ j (t)) is a non-constant coeﬃcient that isthe diﬀerence between Eqs. (4) and (7).
On the basis of MCMC, we derive fi→ j (q̂i (t), q̂ j (t)) forfast decreasing the variance of suﬃciency levels. MCMC isa method for the probability distribution of a metric. In [12],the probability distribution of the Hamiltonian in a ferromagnetism model (Ising spin model) is controlled by using
MCMC. In this paper, we apply MCMC to control the probability distribution of variances of suﬃciency levels for fast
uniforming suﬃciency levels.
We define variance Var(Q̂(t)) of suﬃciency levels as1 q̂i (t) − E(Q̂) 2 ,|V| i=1|V|

Var(Q̂(t)) =(8)

where Q̂(t) = (q̂1 (t), q̂2 (t), ..., q̂|V| (t)), and E(Q̂(t)) is the average of suﬃciency levels, which is given by1 
q̂i (t).
|V| i=1|V|

E(Q̂(t)) =(9)

If suﬃciency levels are uniformed, Var(Q̂(t)) = 0.
We can control the probability distribution of variance
Var(Q̂(t)) by appropriately deriving the transition probability of a state transition regarding suﬃciency levels on the basis of MCMC. Here, we consider the state transition corresponding to the moving of the amount Δq̂ of the suﬃciency
level from node i to node j ( j ∈ ai ), and denote the transition
probability of the state transition by p(Δq̂|i → j). We cancontrol
distribution of variance Var(Q̂(t)) to the probability
exp −λ Var(Q̂(t)) if transition probability p(Δq̂|i → j) satisfies the following conditionexp −λVar(Q̂(t)) p(Δq̂|i → j)(10)
= p(Δq̂|i → j)exp −λVar(Q̂ (t)) ,SAKUMOTO and TANIGUCHI: AUTONOMOUS DECENTRALIZED MECHANISM FOR ENERGY INTERCHANGES

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where λ is a positive parameter (λ > 0), and Var(Q̂ (t)) is thevariance after the state transition. Probability distribution
exp[−λ x] is a monotonically decreasing function, and so it
is suitable for decreasing the variance.
According to MCMC, transition probability p(Δq̂|i →j) satisfying the condition (10) is given by
Fig. 4qi (t).

⎧⎪
1
if ΔVari→j (Q̂(t)) < 0
⎪
⎪
⎪
⎨ 
p(Δq̂|i → j) = ⎪exp−λΔVari→ j (Q̂(t))⎪⎪⎪⎩
otherwise+≈ 1−λ ΔVari→ j (Q̂(t)) ,The amounts oi (t) and ui (t) of overrun and underrun by
the energy interchange of node i are given by
⎤+
⎡

⎥⎥⎥
⎢⎢⎢

⎢
J j→i (t) − Ji→ j (t) + θi −bi⎥⎥⎥⎦ ,oi (t) = ⎢⎢⎣q̂i + ΔT j

(11)where [x]+ = max(0, x), and ΔVari→ j (Q̂(t)) is the diﬀerence
between Var(Q̂(t)) and Var(Q̂ (t)). To derive Eq. (11), we
use the first-order Taylor series approximation. Diﬀerence
ΔVari→ j (Q̂(t)) is given by

j∈ai

(14)
⎤−
⎡

⎥⎥⎥
⎢⎢⎢

⎢
ui (t) = ⎢⎣⎢q̂i + ΔT j
J j→i (t) − Ji→ j (t) + θi ⎥⎥⎥⎦ ,

ΔVari→ j (Q̂(t)) = Var(Q̂ (t)) − Var(Q̂(t))= 2(q̂ j (t) − q̂i (t) + Δq)Δq(12)≈ 2κ q̂ j (t) − q̂i (t) (q̂ j (t) − q̂i (t)),
where Δq = κ|q̂ j (t) − q̂i (t)| (κ  1). Energy interchangeamount Ji→ j (t) is also given by p(Δq̂|i → j) q̂i (t), and so wederive the following expression

(15)

j∈ai

where [x]− = min(0, x).
To avoid the overrun and underrun with oi (t) and ui (t),respectively, node i changes energy interchange amount(o)(u)
Ji→ j (t) to Ji→ j (t) + ΔJi→j (t) + ΔJi→ j (t). The diﬀerences
(o)
(u)
ΔJi→
j (t) and ΔJi→ j (t) are given by+
J j→i (t) − Ji→ j (t)oi (t)
,
+
 
ΔT J
k∈ai J j→i (t) − Ji→ j (t)
+

(t)
−
J
(t)
J
i→
j
j→i
ui (t)
(u)
.
ΔJi→
+

j (t) = Ji→ j (t) − J j→i (t) ΔT J

fi→ j (q̂i (t), q̂ j (t)) = p(Δq̂|i → j)=1−λ[ΔVari→ j (Q̂(t))]

Procedure for preventing the overrun and underrun of node energy+=1−2 λ κ q̂ j (t) − q̂i (t) (q̂ j (t) − q̂i (t))

(o)
ΔJi→
j (t) =+=1−κ q̂ j (t) − q̂i (t) (q̂ j (t) − q̂i (t))+ .
(13)
Although κ  1, κ can be set to a value larger than one
because of κ = 2 λ κ.
When using non-constant coeﬃcient fi→ j (q̂i (t), q̂ j (t))
given by Eq. (13), the uniforming q̂i becomes faster.
According to MCMC, we derive Eq. (13) for controlling the probability
 distribution of variance Var(Q̂(t)) toexp −λ Var(Q̂(t)) . As λ increases, the average of variance
Var(Q̂(t)) decreases. If λ = 0, Eq. (7) is equivalent to Eq. (5)because fi→ j (q̂i (t), q̂ j (t)) = 1. Hence, the variance when using Eq. (7) is always smaller than that when using Eq. (5).
Therefore, we can also uniform suﬃciency levels by using
Eqs. (7) and (13), and the speed for uniforming becomes
faster.

(16)

(17)

k∈ai

We illustrate the above procedures in Fig. 4.
4.

Evaluation

Through simulation experiments, we clarify the eﬀectiveness of the proposed mechanism. The goal of the proposed mechanism is to uniform suﬃciency levels of nodes
for the energy supply appropriately for the energy demandof nodes. We visually confirm that the proposed mechanism achieves the goal, and quantitatively investigate theuniforming speed of the proposed mechanism under several
situations.
4.1

Setting

3.3 Avoiding Overrun and Underrun of a Battery
In the proposed mechanism, node i transmits its remaining
battery amount only according to suﬃciency levels. Hence,
overrun (i.e., qi (t) > bi ) and underrun (i.e., qi (t) < 0) would
occur. If node i predicts overrun and underrun by its energy
transmission, it should appropriately change Ji→ j (t) to avoidthem.

We use a basic model for easily understanding fundamental properties of the proposed mechanism. In a real system,
there are several factors aﬀecting the geographical heterogeneity. However, we represent the geographical heterogeneity only by the heterogeneity of initial energies qi (0).
We assume (a) the generated energy gi (t) of node i is always
equal to the consumed energy ci (t), and (b) demand amountIEICE TRANS. FUNDAMENTALS, VOL.E98–A, NO.7 JULY 20151508

θi of node i does not vary with time. Under situations without the assumption, we will confirm the eﬀectiveness of the
proposed mechanism in the future work.
Initially, energy gi (0) of node i is given by Normal distribution N(μstq , σstq ). The variation of suﬃciency levels at
t = 0 represents the initial geographical heterogeneity. All
nodes would be suﬃcient after the uniforming q̂i (t) if wegive the initial energies according to the following condition
1 
θi .
N i=1
N

μstq >(18)

By changing the value of μstq , we can generate several situations (e.g., energy shortage and surplus situations) in simulation.
During simulation, each node transmits own energy according to the proposed mechanism (hereafter referred to as
proposal mechanism) or the mechanism with Eq. (5) (hereafter referred to as diﬀusion mechanism).
We use the Nk -th nearest neighbor network as network
G. The Nk -th nearest neighbor network is generated by
the following procedures. Initially, each node is randomly
placed on the two dimensional plane. Then, each node selects Nk nearest nodes as its adjacent nodes. Note that a node
has adjacent nodes greater than or equal to Nk . Nk -th nearest
neighbor network is a network with considering geographical dispersion of nodes and wiring cost between nodes. We
also use the two-dimensional lattice grid network for a baseline network because it is a homogeneous network where allnodes have four adjacent nodes. In a two-dimensional latticegrid network, all nodes are regularly placed in the two dimensional plane with the periodic boundary condition. Figure 5 shows examples of 5-th nearest neighbor network andFig. 5 Examples of Nk -th nearest neighbor network (Nk = 5, N = 100)and two-dimensional lattice grid network (N = 100).

Fig. 7two-dimensional lattice grid network for N = 100, respectively.
We use the parameter configuration shown in Table 2
as a default parameter configuration.
4.2

ResultFirst, we visually confirm the eﬀectiveness of the proposal
mechanism. Figure 6 illustrates the color bar corresponding
to a value for visualized results.
Figures 7(a) through (c) show the geographical distributions of node energies at t = 0, 250, and 500. In thissimulation, the average of initial energies for nodes in thecentral region (1/3 < x < 2/3 and 1/3 < y < 2/3) of thex − y plane is set to 5, and the average of initial energies
for other nodes is set to 50. According to these figures, we
confirm that the proposal mechanism can supply energies tothe nodes in the central region.
Figure 8 shows time evolution of statistics (i.e., maximum, average, and minimum) of node energies when theinitial energies shown in Fig. 7(a). In this figure, we show
the results of the proposal mechanism and the diﬀusion
mechanism for comparison. The proposal mechanism canuniform node energies faster compared with the diﬀusion
mechanism. In particular, it has high eﬀectiveness for fast
supplying the energy into nodes with severe energy shortTable 2

Parameter configuration.

networkG

number of nearest nodes
number of nodes
battery capacity of a nodeelectric demand amount of a node
diﬀusion coeﬃcientparameter in Eq. (11)interval of recalculating Ji→ javerage of initial energies
standard variance of initial energies
coeﬃcient of standard variances

Nk
N
bi
θi
k
κ
ΔT J
μstq
σstq
kσ

Fig. 6

Nk -nearest
neighbor
network
5
400
100
50
0.00011000
150
kσ μstq
0.3

Color bar corresponding to a value for Figs. 7 and 9.

Geographical distributions of node energies qi (t).

SAKUMOTO and TANIGUCHI: AUTONOMOUS DECENTRALIZED MECHANISM FOR ENERGY INTERCHANGES

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age. This result would indicate that the proposed mechanism
is more eﬀective than a method [13] based on the diﬀusion
equation.
Then, we visually confirm that the proposal mechanism can supply energies of each node according to demandamounts for each node. Figures 9(b) and 9(c) show the geographical distributions of node energies at t = 0 and t = 500
when using the node demand amounts shown in Fig. 9(a).
According to these figures, the proposal mechanism has effectiveness for uniforming the geographical heterogeneity.
Moreover, we quantitatively investigate the uniforming
speed of the proposal mechanism under several situations.
As a metric for the uniforming speed, we use time constant,which is the time required for decreasing a initial variance
Var(Q̂(0)) to e−1 Var(Q̂(0)).
Figure 10 shows the time constants of the proposal
mechanism and the diﬀusion mechanism for diﬀerent averages μstq and variances kσ μstq of initial energies. According tothis figure, the proposal mechanism has higher eﬀectivenessfor both energy shortage (i.e., small μstq ) and surplus (i.e.,large μstq ) situations. This phenomenon is caused by large
initial variances Var(Q̂(0)). Both energy shortage and surplus situations have a large variance Var(Q̂(0)) at the initiation of simulation. Hence, in these situations, each node issubjected to a strong force for decreasing the variance.
Figure 11 shows time constants for diﬀerent parameters κ and diﬀusion coeﬃcients k . According to Fig. 11,
the time constants for both mechanisms decrease as diﬀusion coeﬃcient k increases, and the diﬀerence between time
constants of both mechanisms increases as parameter κ increases because the force of decreasing the variance in the
proposal mechanism becomes strong.

Fig. 8 Time evolution of statistics (i.e., maximum, average, and minimum) of node energies qi (t).

Fig. 9Figure 12 shows time constants for diﬀerent numberof nodes, N, and networks G. Note that we show the results of 4-th nearest neighbor networks because they has almost same average number of adjacent nodes as lattice grid
networks. According to this figure, the time constants stayabout the same value when the number of nodes increases
regardless of network topology, and so the proposal mechanism also has high eﬀectiveness in networks with a large
number of nodes.
Therefore, we can confirm that the proposal mecha-

Fig. 10

Time constants for diﬀerent average of initial energies, μq .

Fig. 11

Fig. 12

Time constants for diﬀerent parameter κ .

Time constants for diﬀerent number of nodes, N.

Geographical distributions of node energies qi (t) for heterogeneous demands.

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nism has high eﬀectiveness in several situations comparedwith the diﬀusion mechanism. However, we should also
confirm the eﬀectiveness of the proposal mechanism in arealistic situation.
5. Discussion
The uniforming of suﬃciency levels by the proposed mechanism introduces the method to easily estimate the behaviorof the entire system by each node. In this section, we discuss the application potentiality of this estimating method toan eﬃcient method working with non-renewable generators
while minimizing the dependence of non-renewable energy,
and an incentive mechanism to prevent monopolizing energy in systems.

Fig. 13 An example of supplying energy from a generator using nonrenewable energy.

5.1 Supplying Energies with a Generator Using Nonrenewable Energy
There is a limitation on energy supply only using renewable energy. Hence, if energy demand is higher than energy supply, a system should receive energy supply from agenerator using non-renewable energy. However, for minimizing the dependence on non-renewable energy, the system must determine how much energy should be supplied
from a generator using non-renewable energy. Since theproposed mechanism uniforms suﬃciency levels, node i caneasily estimate the shortage energy in the entire system by
N q̂i without gathering all information from the entire system. Figure 13 illustrates an example of supplying energy
from a generator using non-renewable energy into a system
based on renewable energy.
For example, during daytime, each node generates energy with PV panels, and uniforms suﬃciency levels withthe proposed mechanism. Then, during evening, a few
nodes receives energy supply from a generator using nonrenewable energy, and each node uniforms suﬃciency levels
again for satisfying their energy demand.

Fig. 14(a) h(0) = 0, (b) h(∞) = P∗ , and (c) h(−∞) = −∞. Under
the above conditions, a cost function h(S ) is given by



(20)
h(S ) = P∗ 1 − exp −k p S ,where k p is the positive control parameter for energy priceP(S ). We illustrate cost function h(S ) in Fig. 14. If the system is a shortage of energy (S < 0), energy price P(S ) is
high. Otherwise, energy price P(S ) is low. Node i can estimate S by N q̂i , and so it can roughly know energy priceP(S ). Hence, energy price P(S ) provides an incentive forreconfiguring the demand amount of a node.
Every period (e.g., one month, one week), a management node calculates the payment for each node according
to the time series of transmission power and suﬃciency levels during the period. The time series are recorded by nodes,
and the management node gathers them for calculating thepayments of all nodes. Payment pi of node i for period[T 1 , T 2 ] is calculated bypi =5.2 Introducing an Incentive Mechanism to Prevent Monopolizing Energy
If a few nodes maliciously set its demand amount to a very
high value, the energy in a system would be monopolized
by their nodes. To prevent such monopolizing energy, we
discuss an incentive mechanism for decreasing the demandamount of a node.
We first introduce energy price P(S ) where S is the sumof suﬃciency levels of all nodes. S represents the diﬀerence
between energy supply and energy demand at a system level.
As S increases, price P(S ) should increase because the energy in the system becomes shortage. Hence, as an example,
we give energy price P(S ) by∗P(S ) = P − h(S ),(19)

where P∗ is a basic price given by the system manager,
and h(S ) is a cost function, which satisfies the conditions:

Cost function h(S ) for energy price P(S ).

T2  

J j→i (t) − Ji→ j (t) ΔT J P(S (t)).

(21)

t=T 1 j∈ai

Node i pays pi for its energy interchanges to the management system. If pi < 0, node i receives −pi from the management system.
pi = 0 is guaranteed by the amountconservation of the diﬀusion equation.
Note that the management node used in the incentive
mechanism is a centralized node, but is independent with theenergy interchange function between nodes in the proposedmechanism.
6.

Conclusion and Future WorkTo realize the energy supply based on renewable energy
appropriately for energy demand, this paper proposed anovel autonomous decentralized mechanism of energy interchanges based on the diﬀusion equation and MCMC. Since
the proposed mechanism does not rely on any centralizedSAKUMOTO and TANIGUCHI: AUTONOMOUS DECENTRALIZED MECHANISM FOR ENERGY INTERCHANGES

1511

management systems, it has high robustness for disasters.
We first derived an expression of the energy interchangeamount between batteries of an adjacent node pair on thebasis of the diﬀusion equation, and then speeded up the derived expression by using MCMC. Moreover, we visuallyconfirmed the eﬀectiveness of the proposed mechanism, andquantitatively investigated the uniforming speed of the proposed mechanism under several situations. Our findings establish that the proposed mechanism has high eﬀectivenessfor too energy shortage or surplus situations, and in networks with a large number of nodes.
To confirm the eﬀectiveness of the basic ideas for theenergy interchanges, this paper only performed experimentswith ignorable interchange losses, and the simple settingof energy generation and consumption. Hence, as future
works, we should investigate the performance of the proposed mechanism in a realistic situation with battery physical properties (e.g., the rate capacity eﬀect and the degrading its performance), energy loss of interchanges and time
variability in energy generation and consumption. Then, we
will investigate the eﬀectiveness of the method working with
non-renewable generators and the incentive mechanism explained in Sect. 5. Moreover, there are other methods foraccelerating the diﬀusion equation, so we should clarify theperformance diﬀerence between the proposed mechanismand mechanisms based on the other methods.
Acknowledgement
This work was supported by JSPS KAKENHI Grant Number 24700072.
References[1] R.H. Lasseter and P. Paigi, “Microgrid: A conceptual solution,”
Proc. PESC’04, pp.4285–4290, June 2004.
[2] S. Abu-Sharkh, R. Arnold, J. Kohler, R. Li, T. Markvart, J. Ross,
K. Steemers, P. Wilson, and R. Yao, “Can microgrids make a major contribution to UK energy supply?,” Renewable and Sustainable
Energy Reviews, vol.10, no.2, pp.78–127, April 2006.
[3] J.W. Simpson-Porco, F. Dorfler, F. Bullo, Q. Shafiee, and J.M.
Guerrero, “Stability, power sharing, & distributed secondary control in droop-controlled microgrids,” Proc. IEEE SmartGridComm
2013, pp.672–677, 2013.
[4] Y. Matsumoto and S. Yanabu, “Internet style electric energy networkarchitecture,” Proc. POWER’05, pp.332–337, Dec. 2005.
[5] T. Taniguchi and S. Yano, “Decentralized trading and demandside response in inter-intelligent renewable energy network,” Proc.
SCIS-ISIS 2012, pp.645–650, Nov. 2012.
[6] A. Delgadillo and J. Reneses, “Conjectural-variation-based equilibrium model of a single-price electricity market with a counter-trading mechanism,” IEEE Trans. Power Syst., vol.28, no.4,
pp.4181–4191, Nov. 2013.
[7] A.G. Vlachos and P.N. Biskas, “Balancing supply and demand undermixed pricing rules in multi-area electricity markets,” IEEE Trans.
Power Syst., vol.26, no.3, pp.1444–1453, Aug. 2011.
[8] T. Dai and W. Qiao, “Trading wind power in a competitive electricity
market using stochastic programing and game theory,” IEEE Trans.
Sustain. Energy, vol.4, no.3, pp.805–815, July 2013.
[9] C. Takano and M. Aida, “Diﬀusion-type autonomous decentralized
flow control for multiple flows,” IEICE Trans. Commun., vol.E90-B,

no.1, pp.21–30, Jan. 2007.
[10] R. Olfati-Saber, J.A. Fax, and R.M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol.95, no.1,
pp.215–233, Jan. 2007.
[11] R. Minciardi, M. Robba, and R. Sacile, “Decision models for sustainable groundwater planning and control,” Control Eng. Pract.,
vol.15, no.8, pp.1013–1029, 2007.
[12] M.E. Newman, G.T. Barkema, and M. Newman, “Monte carlo methods in statistical physics,” Oxford University Press., April 1999.
[13] R. Olfati-Saber and R.M. Murray, “Consensus problems in networksof agents with switching topology and time-delays,” IEEE Trans.
Automat. Contr., vol.49, no.9, pp.1520–1533, 2004.

Yusuke Sakumotoreceived M.E. and Ph.D.
degrees in the Information and Computer Sciences from Osaka University in 2008 and 2010,
respectively. He is currently an assistant professor at Graduate School of System Design, Tokyo
Metropolitan University, Japan. His research
work is in the area of autonomous decentralized
control for large-scale systems. He is a memberof IEEE, IPSJ and IEICE.

Ittetsu Taniguchi
received M.E. and Ph.D.
degrees in the Information and Computer Sciences from Osaka University in 2006 and 2009,
respectively. He is currently an lecturer at Graduate School of Science and Technology, Ritsumeikan University, Japan. His research interests are power saving and optimization of electrical systems, and design optimization of powersystems. He is a member of IEEE and IEICE.