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j.enconman.2018.08.046

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Energy Conversion and Management 174 (2018) 475–488
Contents lists available at ScienceDirect
Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman
Whole-day optimal operation of multiple combined heat and power systems
by alternating direction method of multipliers and consensus theory
T
⁎
Huynh Ngoc Trana, , Tatsuo Narikiyoa, Michihiro Kawanishia, Satoshi Kikuchib, Shozo Takabab
a
b
Control System Laboratory, Toyota Technological Institute, Nagoya, Japan
Higashifuji Technical Center, Toyota Motor Corporation, Shizuoka, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords:
Combined heat and power system
Fuel cell
Fuel cell efficiency
Distributed energy management system
Alternating direction method of multipliers
Multi-agent system
This paper proposes a distributed energy management system-based algorithm to solve the optimal schedule of
multiple combined heat and power systems in which exponential efficiencies of fuel cells are considered. To deal
with the variation in fuel cell efficiencies, the proposed algorithm utilizes the alternating direction method of
multipliers, consensus theory, and quadratic programming to solve the optimization problem with constant
efficiencies. Then, the efficiency matching is checked iteratively to verify the obtained result. The optimization
algorithm is constructed in a whole-day distributed form, such that all time slot variables can be solved simultaneously in a distributed way. With the help of this simultaneous solution method, operation of hot tanks
and combined heat and power systems can be scheduled globally. The proposed algorithm is tested successfully
with a test system having 4 combined heat and power systems, showing fast convergence in numerous simulations.
1. Introduction
Recently, combined heat and power (CHP) systems or co-generation
systems have been developed to increase the efficiency of energy systems. In CHP systems, the efficiency of energy conversion increases to
over 80% as compared to an average of 30–35% for conventional fossil
fuel fired electricity generation systems [1]. In particular, Japan’s industry has developed CHP systems based on fuel cells (FCs) as one of
the critical solutions to the high demand for clean energy [2]. The
penetration of CHP systems into energy systems requires studies on
system structure and energy management algorithm to optimally
schedule CHP operation.
Parameter sizing or structure design problems have been studied by
various approaches in many papers for single integrated CHP system or
fuel cell based systems. Ref. [3] proposes a soft-run strategy which takes
battery size into consideration for real-time and multi-objective control
algorithm design. In [4], a two-loop framework is used to solve a multiobjective optimization problem which considers both fuel economy and
system durability. In [5], genetic algorithm is utilized to perform a
multi-objective sizing optimization on a solar-hydrogen CHP system
integrated with solar-thermal collectors. Multi-objective optimization
approach is also utilized in [1] for designing integrated CHP systems for
housing complexes in which the selection of technologies, the size of
required units and operating modes of equipment are taking into
⁎
account. Ref. [6] proposes a new combined heat and power - heat pump
integration system and utilizes genetic algorithm to optimize key design
parameters, such as the prime mover capacity (PM), the outlet temperature from the heat pump and the decision value to run the PM. In
[7], a new hub planning formulation is proposed to exploit assets of
midsize/large CHPs. Optimal operation, planning, sizing and contingency operation of energy hub components are integrated and formulated as a single comprehensive mixed-integer linear programming
problem. A probabilistic approach is proposed in [8] for optimal sizing
of CHP systems in which the effect of long-term uncertainty in energy
demand is analytically investigated. As for the microgrid with penetration of CHP systems, [9] presents a power source sizing PSO-based
strategy with integrated consideration of characteristics of distributed
generations, energy storage and loads in an autonomy microgrid.
As a basic problem for the parameter sizing/structure design problem, optimal scheduling or operational strategy for CHP system and
microgrid with multiple CHP systems has been introduced in many
papers. In [10], four CHP operational strategies are tested to choose
their appropriate demand profile. Ref. [11] proposes a strategy of
sharing surplus heat and electricity produced by CHP plants in different
types of buildings, which could yield total energy cost savings of 1–9%.
Dealing with uncertainties is one of important tasks of optimal scheduling problem, therefore it is taken into account in some papers. Ref.
[12] proposes a robust optimization scheduling method to attenuate the
Corresponding author.
E-mail address: tranhuynhngoc@toyota-ti.ac.jp (H.N. Tran).
https://doi.org/10.1016/j.enconman.2018.08.046
Received 11 March 2018; Received in revised form 9 August 2018; Accepted 12 August 2018
0196-8904/ © 2018 Elsevier Ltd. All rights reserved.
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
Nomenclature
η Fp
ηBL
ηex
ηi,Fej
ηi,Fh
j
μHT
cp
CkWhMJ
G Fp
Gi, j
GiF, j
GiBL
,j
HTi, j
m
n
P jB
fuel cell efficiency of processing natural gas to hydrogen
(assumed as 0.95)
boiler efficiency (assumed as 0.99)
hot water exchange efficiency (assumed as 0.95)
electricity output efficiency of fuel cells at the ith building
at time interval j
heat recovery efficiency of fuel cells at the ith building at
time interval j
hot tank leakage ratio (assumed as 0.01)
the specific heat of water (J/kg °C)
conversion factor from kWh to MJ (3.6 MJ/kWh)
processed gas energy of fuel cells at the ith building at time
interval j (MJ)
total gas energy consumed at the ith building in time interval j (MJ)
natural gas energy consumed by fuel cells at the ith
building at time interval j (MJ)
gas energy consumed by the ith gas boiler at time interval j
(MJ)
hot tank level at the ending time of interval j at the ith
building
PiD, j
PiF, j
Pr G
Pr je
Tj
Tpdem
TpHT
Tpin
WiD, j
WiBL
,j
Wi,frj
WiHT
,j
Wito, j
number of time intervals in a whole day
number of buildings in the microgrid
power bought from main grid to microgrid via substation
at time interval j (kW)
power demand at the ith building at time interval j (kW)
power generated by fuel cells at the ith building at time
interval j (kW)
gas price (JPY/MJ)
electricity price at time interval j (JPY/kWh)
time length of time slot or time interval j (h)
temperature of hot-water demand (assumed as 40 °C)
temperature of hot water in hot tank (assumed as 70 °C)
temperature of tap water (assumed as 20 °C)
hot water demand at the ith building at time interval j (L)
hot water from gas boiler at the ith building at time interval j (L)
hot water sent from the ith building at time interval j to its
exchanged building
water discharged from hot tank at the ith building at time
interval j (L)
hot water received at the ith building at time interval j
from its exchanged building
from the demand side and large computational time. Therefore, in this
paper, we introduce a distributed energy management system (EMS)based algorithm for scheduling a microgrid with penetrated multiple
CHP systems. Our algorithm is based on the alternating direction
method of multipliers (ADMM) [25], an analytical fast-convergence
method which is suitable for large-scale optimization problems. There
have been some papers applying ADMM approach for operational optimization of energy systems. Ref. [26] utilizes ADMM to develop a
robust co-optimization scheduling model to study the coordinated optimal operation of the two energy systems. In [27], a day-ahead scheduling framework of integrated electricity and natural gas system is
proposed at a distribution level based on the fast ADMM with restart
algorithm considering demand-side response and uncertainties. As for
the distributed EMS system, there have been some proposed algorithms
[28,29] utilizing the ADMM approach and consensus theory for multiagent systems. These algorithms are proposed for application to optimal power flow [29], demand response, or real-time pricing problems
[28] in transmission power systems. Owing to the characteristics of
such problems, their corresponding algorithms are based on time-slot
sequential solving in which the whole-day schedule problem is divided
into time interval schedule problems to be solved individually. Furthermore, the objective functions of these problems have a convex
form, such that the ADMM approach can be applied easily. The microgrid with CHP systems presents some challenges for scheduling by
these proposed algorithms. First, owing to the efficiency characteristics
of the FC model, the relation between the FC input (energy of natural
gas) and output (generated electricity and heat energy) is not linear. In
this study, we assume an exponential efficiency curve for these relations
based on a certain FC efficiency curve shape. Second, with the existence
of energy storage (hot tank) in each CHP system, it is much better to
solve the whole day/whole time problem than to sequentially solve
time-interval problems. In this paper, we propose a distributed-EMSbased algorithm to overcome the two difficulties mentioned above.
First, we introduce an iteration loop to change the nonconvex problem
to a convex problem, such that the ADMM approach can be applied. The
numerical results show that the added iteration loop has a good convergence characteristic. Second, we use the ADMM approach, consensus theory, and quadratic programming to solve our problem in a
whole-day form, i.e., all time-interval problems are solved simultaneously. Furthermore, based on the case study results, we point out
disturbance of uncertainty, and derive the day-ahead scheduling decision under two strategies including electrical load tracking and thermal
load tracking. Ref. [13] presents a stochastic programming framework
for conducting optimal 24-h scheduling of CHP-based microgrids considering demand response programs and uncertainties. A revision approach [2] is proposed to deal with the error in predicted demand for
the optimal operation of fuel-cell-based residential energy systems. In
[14], a colonial competitive algorithm is used in an optimal dispatching
problem with uncertain electricity load. In terms of approach utilizing,
various approaches including mix-integer linear/nonlinear programmings and meta-heuristic optimization have been developed for the
optimization problem. Ref. [15] proposes a novel method based on
information gap decision theory to evaluate a profitable operation
strategy for combined heat and power units in a liberalized electricity
market. In [16], an optimal operation management system that integrates demand prediction into optimal scheduling by mixed-integer
linear programming and real-time control for cogeneration units is
proposed. Ref. [17] introduces a meta-heuristic optimization that could
find the global solution for a complex energy system. Ref. [18] builds a
multiobjective optimization model for integrated CHP systems and
compares its performance with the thermal-load-tracking control
strategy. Ref. [19] introduces a multiobjective mixed-integer nonlinear
programming model for the operational optimization of a large-scale
combined cooling, heat, and power system. Through multiobjective
optimization, Ref. [20] compares various system configurations that
have on-site heat and electricity generation components. In [21], a solid
oxide fuel cells based combined-cooling heating-and-power system
design and operation optimization model has been developed using the
Mixed Integer Non-linear Programming approach. A new isochronous
governor control strategy for the CHP systems is proposed in [22] to
provide zero-steady-state-error frequency regulation for a microgrid.
Ref. [23] proposes a multiparty energy management framework with
electricity and heat demand response for the CHP-Microgrid. Ref. [24]
utilizes a coalitional game approach to propose a hybrid energy sharing
framework with CHP system and photovoltaic (PV) prosumers.
Although these mentioned studies cover a wide range of approaches
for the structure design and optimal scheduling problems of CHP systems, all of their corresponding optimal algorithms/strategies are centralized, which may pose some difficulties for their application on the
multiple CHP systems, owing to such factors as restricted information
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Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
efficiencies described in [30]. Owing to a long required startup time of
FC units, we assume that all the FC units are always in the ON state.
Each FC unit has a rating of 700 W output power and a minimum output
power of 50 W. The number of FC units installed at each building can be
different, subject to the electricity and hot-water demands. This study
considers only the economic dispatch of multiple CHP system, therefore
transient and saturation features of FC are neglected.
some features of the optimal scheduling of CHP systems at the residential level. The next section describes a microgrid model with penetrated residential CHP systems. Section 3 presents the problem formulation and a preliminary discussion on the solution method. Section
4 describes our proposed whole-day distributed ADMM approach,
which is developed from a sequential distributed ADMM approach [28].
Section 5 demonstrates the whole proposed algorithm. Section 6 presents a 4-CHP test system and Section 7 shows the performance of the
proposed algorithm on test system. The obtained results show that our
proposed algorithm always converge in numerous simulations, which is
better than performance of a particle swarm optimization approach on
the same test system. In Section 8, conclusions drawn and future works
are proposed.
2.2. Combined heat power model
Fig. 2b depicts the CHP system [2] at one building and Fig. 3 describes the hot-water exchange between two CHP system of two
buildings. In this study, we assume that hot water is exchanged between
the demand sides but not between hot tanks.
Based on the gas, power, and hot-water flows in Figs. 2a and 3, the
relations between electricity, gas, and hot-water energies in the CHP
system i at time interval j are expressed as follows.
2. System modelling
In this study, we consider a microgrid model, as shown in Fig. 1,
with the following features:
2.2.1. Water demand
The hot-water demand at building i is the sum of the hot-water
volume from the mixer, boiler, and exchanged hot-water volume from/
to the building exchanging hot water with building i.
• The grid is a lossless distribution network consisting of one substa-
•
•
•
tion and n buildings. The microgrid buys electrical power from the
main grid via the substation. To maintain the electrical frequency in
the microgrid, the microgrid has to be continuously connected to the
main grid. Therefore, the bought power P jB at each time interval j is
B
required to be larger than a lower limitation of Pmin
.
Every building has its electricity and hot-water demand, which are
predicted accurately. For simplicity’s sake, we do not consider other
heating demands in this study.
A CHP system is installed at each building to partially meet the
building’s electricity demand and provide heat energy to the
building’s hot-water system. The CHP model consists of FC units,
mixers, a hot tank, and an auxiliary gas boiler, which is similar to
that in [2]. A detailed description of the model is presented in the
next subsections.
To increase the usefulness of the CHP system, we consider the hotwater exchange between buildings in the microgrid model. It is assumed that each building exchanges hot water only with its nearest
neighboring building. The optimization of the hot-water exchange
structure is out of scope of this work.
fr
BL
to
WiD, j = WiHT
, j Tp + Wi, j −Wi, j + Wi, j ,
where Tp =
ηi,Fej = f Fe (PiF, j ) = ae−be e
ηi,Fh
j
=
f Fh (PiF, j )
= ah−bh e
F
Pimax
;
−kh
be =
Fe
ηmax
−η0Fe
PiF, j
F
Pimax
,
bh =
1−e−ke
; ae = be + η0Fe ,
Fh
ηmax
−η0Fh
1−e−kh
,
ah = bh +
, and
(3)
is the hot-water volume from the
2.2.2. Hot-water exchange
The hot-water exchange volume between buildings i and r is expressed by:
ηex Wi,frj −Wrto, j = ηex Wrfr, j −Wito, j = 0,
Wi,frj
(4)
Wito, j
and
are the hot-water volume sent from and received at
where
building i, respectively, and ηex (< 1) is the hot-water exchange efficiency.
2.2.3. Fuel cell and hot tank
The natural gas energy G F , processed gas energy GpF , and FC output
power P F at one time interval are related via the processing efficiency
η Fp and FC electricity efficiency η Fe :
A FC unit processes natural gas with an constant efficiency of η Fp
and consumes the processed natural gas to provide both electricity and
heat energy with efficiencies of η Fe and η Fh , respectively, to meet their
respective demands. η Fe and η Fh are dependent on the FC operation
mode and are not often given publicly. In [2], these two efficiencies are
assumed to be proportional to the FC output power, beyond a minimum
portion. However, based on the curve shape of these efficiencies in
[2,10] and in an effort to represent the FC features accurately, we assume these two efficiencies to be exponential functions of the FC output
power, which are depicted in Fig. 2a and expressed by following formulas:
−k e
Tpdem − Tpin
WiHT
, j Tp
mixer corresponding to the amount of water WiHT
, j discharged by the hot
tank.
2.1. Fuel cell model
PiF, j
TpHT − Tpin
GiF, j =
GiFp
,j =
GiFp
,j
η Fp
,
(5)
PiF, j Tj
CkWhMJ .
ηi,Fej
(6)
The hot-water volume released from the FC to charge the hot tank is
calculated from the FC heat energy output, as shown in (7).
Fh
F
WiF, j (TpH T −Tpi n) cp = QiF, j = GiFp
, j ηi, j = Pi, j Tj CkWhMJ
ηi,Fh
j
η(Fe
i, j )
The hot-tank current level is a result of the previous-interval level,
as well as the hot-water charging and discharging, as follows:
(1)
HTi, j = (1−μHT ) HTi, j − 1 + WiF, j −WiHT
,j
η0Fh ,
(2)
where
Fe
ηmax
= 0.465,
η0Fe = 0.15,
(7)
Fh
ke = 7; ηmax
= 0.445; η0Fh = 0.25; kh
= 6.
Fig. 1. Microgrid system with multiple buildings.
Fh
Fe
The values of ηmax
and ηmax
are based on the solid oxide fuel cell
477
(8)
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
Fig. 2. (a) Assumption of FC efficiency. (b) Combined heat power system [2].
Fig. 3. Hot-water exchange between two buildings.
where HTi, j − 1 is the hot-tank level at the ending time of the previous
time interval, and HTimax is the maximum capacity (L) of the hot tank i.
2.2.4. Boiler
The gas energy required by the boiler to heat a water volume of WiBL
,j
at temperature Tpin to Tpdem is calculated by (9).
GiBL
,j
=
ηex Wi,frj −Wrto, j = ηex Wrfr, j −Wito, j = 0
(16)
fr
D
HT
to
WiBL
, j = Wi, j −Wi, j Tp + Wi, j −Wi, j ⩾ 0
(17)
fr
D
HT
to
GiBL
, j = (Wi, j −Wi, j Tp + Wi, j −Wi, j )
(Tpdem−Tpin ) cp
ηBL
WiBL
, j (Tpdem−Tpin ) cp
ηBL
(9)
0 ⩽ (1−μHT ) HTi, j − 1 + PiF, j
(1−μHT ) HTi, j − 1 + PiF, j
2.2.5. Equality and inequality constraints of the CHP system
The total gas energy of the CHP system is the sum of the gas energies
of the boiler and the FC:
F
Gi, j = GiBL
, j + Gi, j
Tj CkWhMJ ηiFh
,j
ηiFe
, j (TpHT − Tpin) cp
(18)
−WiHT
, j ⩽ HTimax ,
−WiHT
, j = HTi0 ,
i = 1…(m−1)
i=m
(19)
In (15)–(19),
(10)
PiF, j ,
WiHT
,j ,
of PiF, j
Wi,frj ,
and
Wito, j
are scheduled variables, ηi,Fej
and ηi,Fh
are functions
as shown in Fig. 2a, and WiD, j is the predicted
j
demand of the CHP system i at time interval j.
From (3), we have:
fr
D
HT
to
WiBL
, j = Wi, j −Wi, j Tp + Wi, j −Wi, j
Tj CkWhMJ ηiFh
,j
ηiFe
, j (TpHT − Tpin) cp
BL
⩽ Gimax
(11)
3. Problem formulation
From (4) and (5), we have:
GiF, j =
The scheduling algorithm aims to minimize the total cost of the
microgrid in a whole-day period:
PiF, j Tj CkWhMJ
ηi,Fej η Fp
(12)
m
Minimizing F =
Substituting (12) in (6), we have
∑
j=1
WiF, j
=
PiF, j
Tj CkWhMJ ηi,Fh
j
ηi,Fej (TpHT −Tpin ) cp
where Gi, j = (WiD, j −WiHT
, j Tp +
(13)
(Tpdem−Tpin ) cp
ηBL
(14)
As a result, the total gas energy consumed by the CHP system i at
one time interval can be expressed by:
fr
to
Gi, j = (WiD, j −WiHT
, j Tp + Wi, j −Wi, j )
(Tpdem−Tpin ) cp
ηBL
+
PiF Tj CkWhMJ
ηi,Fej η Fp
(Tp
− Tp ) cp
Wi,frj −Wito, j ) demη in
BL
(20)
+
PiF, j Tj Ck WhMJ
Fp
ηiFe
,j η
is the
energy of the gas consumed by the CHP system at the ith-building in the
jth-time interval. The electricity price Pr je is given by the time-of-use
price. The gas price Pr G is the same for all time intervals. In this study,
we consider the short-term schedule/dispatch of CHP system, therefore
the investment and maintenance costs does not affect much to the result. Hence, these costs are neglected in the objective function. We will
consider these costs in the next step of this study in which the parameter sizing problem is considered.
From (9) and (11), we have
fr
D
HT
to
GiBL
, j = (Wi, j −Wi, j Tp + Wi, j −Wi, j )
n
⎛ e B
⎞
G
⎜Pr j P j Tj + Pr ∑ Gi, j ⎟
i=1
⎝
⎠
,
(15)
3.1. Constraints
Furthermore, the CHP variables need to satisfy the constraints of
water exchange, boiler water volume, boiler limitation, and hot-tank
level, as shown in (16)–(19), respectively:
The microgrid power has to satisfy a lossless power balance between
the supply and demand sides, i.e.,
478
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
n
P jB +
∑
n
PiF, j =
i
∑
PiD, j ,
4. Whole-day distributed alternating direction method of
multipliers
j = 1…m.
(21)
i
As mentioned above, the power bought from the main grid has a
B
lower limitation Pmin
:
We first denote some vectors and matrixes to be used in the proposed algorithm. Let:
B
P jB ⩾ Pmin
,
P jF = [P1,Fj P2,Fj…PnF, j ]T ,
j = 1…m.
(22)
Wjfr = [W1,frj W2,frj…Wnfr, j ]T ,
Corresponding to the output range of the FC unit from 50 to 700 W,
the CHP system at each building constrains its output power, depending
on the number of FC units at each system.
F
F
Pimax
⩾ PiF, j ⩾ Pimin
,
j = 1…m ,
Additionally, each CHP system needs to satisfy following constraints, which are derived from (16)–(19):
(24)
fr
to
D
WiHT
, j Tp−Wi, j + Wi, j ⩽ Wi, j ,
(25)
fr
to
D
WiHT
, j Tp−Wi, j + Wi, j ⩾ Wi, j −
0 ⩽ (1−μHT ) HTi, j − 1 + PiF, j
(1−μHT ) HTi, j − 1 + PiF, j
BL
ηBL Gimax
,
(Tpdem−Tpin ) cp
Tj CkWhMJ ηiFh
,j
ηiFe
, j (TpHT − Tpin) cp
Tj CkWhMJ ηiFh
,j
ηiFe
, j (TpHT − Tpin) cp
(26)
−WiHT
, j ⩽ HTimax ,
−WiHT
, j = HTi0 ,
i = 1…(m−1)
i=m
(27)
4.1. An approach of alternating direction method of multipliers
3.2. Preliminary discussion on problem-solving
When the FC efficiencies are constant, the objective function F in
Section 3 becomes convex. Therefore, it can be transformed into the
following problem to fit with the ADMM approach in [28,25]:
Owing to the exponential characteristics of the FC efficiencies ηi,Fh
j
ηi,Fej ,
to
to
to T
W to
j = [W1, j W2, j…Wn, j ]
B T
4n + 1, j = 1…m , is the vector of unPj ≜ [P jF W jHT Wjfr W to
j Pj ] ∈ R
known variables at time interval j,
P ≜ [P1 P2…Pj…Pm ]T ∈ R(4n + 1) m is the vector of all unknown variables of the whole-day scheduling,
P ≜ [P1 P2…Pj…Pm] ∈ R(4n + 1) × m is the matrix that consists of all elements of vector P but in a matrix form. From now on, when we mention
elements of vector P , we will show them in the form of matrix P to
avoid using excessive subscript levels, and similarly for elements of
 and matrix X, as mentioned below.
vector X
With these definitions of vector P and matrix P, variables
fr
to
PiF, j , WiHT
, j , Wi, j and Wi, j are indicated by Pi, j , Pi + n, j , Pi + 2n, j and Pi + 3n, j reB
spectively. P j , which is the power purchased from the main grid at time
interval j, is indicated by P4n + 1, j .
 ≜ [X1 X2 …Xj …Xm ]T ∈ R(4n + 1) m
X
and
Let:
Xj ∈ R 4n + 1, X
≜ [X1 X2 …Xj …Xm ] ∈ R(4n + 1) × m .
 and matrix X have the same structures as vector P and
Vector X
matrix P, respectively. We utilize these additional variables to satisfy
the inequality constraints of the given problem, using an ADMM approach. Hence, the given problem can be separated into equality constraint and inequality constraint optimization problems, as shown in
the following subsection.
(23)
ηex Wi,frj −Wrto, j = ηex Wrfr, j −Wito, j = 0,
HT
HT T
W jHT = [W1,HT
j W2, j …Wn, j ]
PiF, j
and
the function of Gi, j with respect to
in (20) is nonconvex.
Therefore, the optimization problem is nonconvex, with both equality
and inequality constraints, and it could not be directly solved by convex
optimization methods. To overcome the nonconvex problem, we first
try to solve the optimization problem in which the FC efficiencies are
constant values. Second, the FC efficiency matching is used to check
whether the result is appropriate. The procedure is repeated until the
result converges. This proposal is simple, but works efficiently on the
given problem, as shown in Section 6, owing to the linearity of the
optimization problem with constant FC efficiencies. As for the first step
in which FC efficiencies are constant values, we develop a whole-day
distributed ADMM algorithm to solve the whole-day optimization
problem in a distributed manner. Our approach is developed based on
the sequential distributed ADMM algorithm presented in [28]. The sequential distributed ADMM algorithm divides the whole-day objective
function into time-interval objective functions and then solves them
sequentially. However, such an approach could not apply to our problem, in which the objective functions of all time intervals need to be
solved at the same time to optimize the energy storage (hot tank)
schedule. For instance, the hot-tank levels at a time interval, which is
solved in a sequential algorithm, may limit the optimality of those in
the next time intervals, owing to their relation in (19). In our approach,
we form the X-update problem of the ADMM algorithm into a wholeday decentralized optimization problem, such that it can be solved in a
decentralized manner, using quadratic programming. With this improvement, our approach is also valid for this class of optimization
problem with additional types of energy storage, such as batteries or
electric vehicles. In the next section, we describe the first step of our
proposed algorithm, which is the whole-day distributed ADMM algorithm, to solve the optimization problem with constant FC efficiencies.
The whole algorithm is described in Section 5.
m
Minimizing FADMM =
∑
j=1
n
⎞
⎛ e
B
G


⎜Pr j Tj P j + Pr ∑ Gi, j ⎟ + I1 (P ) + I2 (X ),
i=1
⎠
⎝
(28)
 = 0,
such that P−X
where
0 if P ∈ Π1
I1 (P ) = ⎧
⎨
⎩∞ if P ∉ Π1
(29)
Π1 = {P ∈ R(4n + 1) m :
n
P jB +
∑
n
PiF, j =
i=1
∑
PiD, j ,
j = 1…m
(30)
i=1
ηex Wi,frj −Wrto, j = ηex Wrfr, j −Wito, j = 0, (i, r ) given}
(31)
Π1 is the set of P satisfying all equality constraints of the given problem.
0 if X ∈ Π2
I2 (X ) = ⎧
if X ∉ Π2
∞
⎨
⎩
(32)
 ∈ R(4n + 1) m :
Π2 = {X
B
X4n + 1, j ⩾ Pimin
F
Pimax
⩾ X i, j ⩾
(33)
F
Pimin
Xi + n, j Tp−Xi + 2n, j + Xi + 3n, j ⩽
(34)
WiD, j
Xi + n, j Tp−Xi + 2n, j + Xi + 3n, j ⩾ WiD, j −
479
(35)
BL
ηBL Gimax
(Tpdem−Tpin ) cp
(36)
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
0 ⩽ (1−μHT ) HTi, j − 1 + Xi, j
Tj CkWhMJ η(Fh
i, j )
η(Fe
i, j ) (TpHT − Tpin ) cp
i = 1…(m−1)
HTi0 ⩽ (1−μHT ) HTi, j − 1 + Xi, j
(44) can be re-written as
−Xi + n, j ⩽ HTimax ,
k+1
 k , uk ),
= argminLρP (P, X
⎧P
⎪
n
n
⎪
gj (P ) = ∑ Pi, j + P4n + 1, j− ∑ PiD, j = 0, j = 1…m
⎨
i=1
i=1
⎪
⎪ hir , j (P ) = ηex Pi + 2n, j−Pr + 3n, j = 0, (i, r ) given.
⎩
(37)
Tj CkWhMJ η(Fh
i, j )
η(Fe
i, j ) (TpHT − Tpin ) cp
−Xi + n, j ⩽ HTi0,
i = m}
(38)
where
 satisfying all inequality constraints of the given
Π2 is the set of X
problem. Note that the second constraint in (27) involves two time
interval variables, and thus it is transformed to last two inequality
constraints in (38). With this transformation, all equality constraints of
the given problem can be separated into time interval constraints.
Based on the framework of the ADMM approach [28,30], we define
the augmented Lagrange function as follows:
m
 , η) =
Lρ (P, X
 , u) =
LρP (P, X
(46)
n
 k , uk ) s. t. g (P ) = 0,
Minimize LρP (P, X
j
ρ 
⎞
⎛ e
B
G


⎜Pr j Tj P j + Pr ∑ Gi, j ⎟ + I1 (P ) + I2 (X ) + 2 ‖P
j=1 ⎝
i=1
⎠
 ‖22 + +ηT (P−X
)
−X
(39)
k
be updated iteratively by:
k
k+1 k+1
uk + 1 = uk + P −X
k
⩽
eprimal and ‖s k‖2
⩽ edual , where
k
‖r k‖2 = ‖P −X ‖2 ,
 k −X
 k − 1)‖2 ,
‖s k‖2 = ‖ρ (X
eprimal =
edual =
 ‖2 },
n ∊abs + ∊rel max{‖P ‖2 , ‖X
k

∂gj (P )
⎧ ∂LρP (P, X , u ) = λ
, i = 1…n,
j, k + 1 ∂P
⎪ ∂P4n + 1, j
4n + 1, j
⎪
k k



∂
g
P
(
)
⎪ ∂LρP (P , X , u ) = λj, k + 1 j , i = 1…n,
∂Pi, j
∂Pi, j
⎪
⎪ ∂LρP (P, X k , uk )
∂gj (P )
⎪
, i = 1…n,
= λj, k + 1 ∂P
i + n, j
⎪ ∂Pi + n, j
(48) ⇔ ∂LρP (P, X k , uk )
∂hir , j (P )
⎨
, i = 1…n,
= λjh,irk + 1 ∂P
i + 2n, j
⎪ ∂Pi + 2n, j
⎪ ∂L (P, X k , uk )

∂hir , j (P )
⎪ ρP
, i = 1…n
= λjh,irk + 1 ∂P
i + 3n, j
⎪ ∂Pi + 3n, j
⎪ g (P ) = 0
⎪ j
⎪ hir , j (P ) = 0, (i, r ) given
⎩
(43)
when ‖r k‖2
(48)
λjh,irk + 1
where
are the Lagrange scaled multipliers (at the iteration
k + 1).
Here, we see that (48) consists of msimilar, independent problems
corresponding to m time intervals. It explains that which was mentioned in Section 3.2 about the constrains of variables P. In the next
paragraph, without loss of generality, we show the solution of (48) for
time interval j only.
1
The iteration is terminated
k
λjg, k + 1,
where u = ρ η ∈ R(4n + 1) m is a scaled dual variable or scaled Lagrange
 , and u can
multiplier. Following the procedure presented in [12], P, X
(42)

⎧ ∂LρP (P, X , u ) = λ g ∂gj (P ) + ∑ λ hir ∂hir , j (P)
j, k + 1 ∂P
j, k + 1
⎪
∂P
∂P
(i, r )
⎨
⎪ gj (P ) = 0, hir , j (P ) = 0, j = 1. .m
⎩
ρ  
⎞
⎛ e
B
G


⎜Pr j Tj P j + Pr ∑ Gi, j ⎟ + I1 (P ) + I2 (X ) + 2 ‖P −X
j=1 ⎝
i=1
⎠
Tη
η
+ u‖22 −
2ρ
(40)
 k + 1 = argminL (P k + 1, X
 , uk )
X
ρ
k
n ∊abs + ∊rel ‖ρuk‖2 .
By solving (41)–(43), the original optimization problem is separated
into an equality constraint optimization problem of P and inequality
.
constraint optimization problem of X
Note that constraints on variables P are global, equality but sepa
rated into time interval constraints, whereas constraints on variables X
are local, inequality but could not be decomposed into time interval
constraints, owing to constraints (37) and (38). Therefore, we can adopt
the average consensus protocol to solve variables P in a distributed
manner, as the sequential distributed ADMM algorithm does. In con are easily divided into local variables at each agent
trast, variables X
(building), but need to be solved in a whole-day period. In the next
subsections, detailed solution of P-update and X-update in (41) and
(42), respectively, are given.
k
(49)
Elaborating (49) give us (50)–(56)
Pr je Tj + ρ (P4n + 1, j−X4kn + 1, j + u4kn + 1, j ) = λj, k + 1,
Pr G
Tj CkWhMJ
η Fe η Fp
−Pr GTp
Pr G
+ ρ (Pi, j−Xik, j + uik, j ) = λj, k + 1,
(Tpdem−Tpin ) cp
ηBL
(Tpdem−Tpin ) cp
ηBL
(50)
i = 1…n,
+ ρ (Pi + n, j−Xik+ n, j + uik+ n, j ) = 0,
(51)
i = 1…n
+ ρ (Pi + 2n, j−Xi + 2n, j + ui + 2n, j ) = ηEx λjh,irk + 1,
(52)
i = 1…n
(53)
4.2. P-update
In this subsection, we present the solution of P in (41). From (41)
and the definition of I1 (P ) , we have:
k+1
 k , uk ),
= argminLρ (P, X
⎧ P
⎨ I1 (P ) = 0
⎩
j = 1…m
Because (47) is a convex optimization, it can be solved using the
Karush-Kuhn-Turker condition, as follows:
∑
(41)
hir , j (P ) = 0,
(47)
n
k+1
 k , uk )
P
= argminLρ (P, X
n
ρ
⎞
⎛ e
2
G
⎜Pr j Tj P1 + 4n, j + Pr ∑ Gi, j ⎟ + 2 ‖P−X + u‖2 .
i=1
⎠
⎝
k+1
is the solution of the optimization
From (45), we have that P
problem
where ρ > 0 is a scaled penalty parameter and η ∈ R(4n + 1) m is a Lagrange multiplier. Then, (39) can be rewritten as:
 , u) =
Lρ (P, X
m
∑
j=1
∑
m
(45)
−Pr G
(Tpdem−Tpin ) cp
ηBL
+ ρ (Pi + 3n, j−Xi + 3n, j + ui + 3n, j ) = −λjh,irk + 1,
i = 1…n
(54)
gj (P ) =
(44)
n
∑
i=1
480
n
Pi, j + P4n + 1, j− ∑ PiD, j = 0,
i=1
(55)
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
hir , j (P ) = ηex Pi + 2n, j−Pr + 3n, j = 0,
(i, r ) given
−1
⎧ (1 + max (card (Ni ), card (Nl ))) ,
⎪1−
∑ aik , l = i
ail =
⎨ k ∈ Ni
⎪ 0, others
⎩
(56)
From (50), (51) and (55), we have:
Pr je Tj + nPr G
Tj CkWhMJ
ηFe ηFp
n
⎞
⎛
+ ρ ⎜∑ (PiD, j −Xik, j + uik, j )−X4kn + 1, j + u4kn + 1, j ⎟
i
=
1
⎠
⎝
Ni is the set of agents that communicate to agent i, which are called
neighbor agents of agent i, card(Ni) is the number of neighbor agents of
agent i, and Tc is the consensus calculating time.
n+1
1
When Tc → ∞, as proved in [31], all x i, j → x∗, j = n + 1 ∑i = 1 Si, j . By
that consensus, λj, k + 1 can be calculated at each agent by the following
formula:
= (n + 1) λj, k + 1
⇒ λ j, k + 1 =
Pr je Tj
n+1
+
nPr GTj CkWhMJ
(n + 1) ηFe ηFp
n
+
ρ ⎛
∑ (PiD,j −Xik,j + uik,j)−X4kn+1,j
n + 1 ⎜ i=1
⎝
⎞
+ u4kn + 1, j ⎟
⎠
λ j, k + 1 =
(57)
(TpHT −Tpin ) cp
ρηBL
,
i = 1…n
⇒
+
=
(1 + ηEx ) Pr G
(Tpdem−Tpin ) cp
ηBL
k+1
(1 + ηEx ηEx )
,
i = 1…n,
(62)
 , uk )
,X
(63)

= argmin‖P
⎧X
k+1
⎨ I (X

)=0
⎩2
 + uk‖22 ,
−X
k+1
(64)
Elaborating (64) and omitting superscripts for convenience, we
obtain Eqs. (65)–(71):
(59)
 = argmin‖P−X
 + u‖22 ,
X
X4n + 1, j ⩾
F
Pimax
in (59) in a distributed way. It is easy to see that λjh,irk + 1 can be calculated
at each building i and r if there exists a communication line between
them. To calculate λj, k + 1 at each building and the substation in a distributed way, we apply an average consensus protocol, which is described in the next subsection.
(65)
B
Pimin
,
(66)
F
Pimin
(67)
Xi + n, j Tp−Xi + 2n, j + Xi + 3n, j ⩽ WiD, j
(68)
Xi + n, j Tp−Xi + 2n, j + Xi + 3n, j ⩾ wi, j
(69)
⩾ Xi, j ⩾
j
j−t
j
⎧ HTimax ⩾ αHT HTi0 + ∑t = 1 αHT (βit Xi, t −Xi + n, t )
⎨ α j HTi0 + ∑ j α j − t (β Xi, t −Xi + n, t ) ⩾ 0,
it
t = 1 HT
⎩ HT
4.3. Average consensus
j = 1…(m−1)
(70)
j
j−t
j
⎧ HTi0 ⩾ αHT HTi0 + ∑t = 1 αHT (βit Xi, t −Xi + n, t )
j
⎨ α j HTi0 + ∑ α j − t (β Xi, t −Xi + n, t ) ⩾ HTi0
it
t = 1 HT
⎩ HT
We note that in (57), the global term related to all agents (buildings
and the substation) is a nonweighted summation term
n
∑i = 1 (PiD, j −Xik, j + uik, j )−X4kn + 1, j + u4kn + 1, j . Such a summation term can be
calculated in a distributed way, using average consensus theory. In this
subsection, we apply an average consensus protocol to perform this
calculation. This consensus protocol is also applied to other summation
terms such as ‖r k‖22 , ‖s k‖22 , ‖P k‖22 , and ‖X k ‖22 in checking the stop criteria
of the whole-day distributed ADMM algorithm.
Let Si, j ≜ PiD, j −Xik, j + uik, j , i = 1…n , and Sn + 1, j ≜ −X4kn + 1, j + u4kn + 1, j .
Then, (57) can be rewritten as:
+
+ ρx∗, j.
k+1
k+1
⇒
ηBL
(1 + ηEx ηEx )
n+1
(n + 1) ηFe ηFp

= argminLρ (P
⎧X
k+1
⎨ I (X

)=0
⎩2
, i = 1…n
Obviously, Pi + n, j is calculated locally in (58) without knowledge of
other buildings. Moreover, if λj, k + 1 and λjh,irk + 1 are known,
Pi, j, Pi + n, j, Pi + 2n, j, Pi + 3n, j and P4n + 1, j can also be calculated using only local
information at each building i and the substation. Therefore, to solve
variable P in a distributed manner, we calculate λj, k + 1 in (57) and λjh,irk + 1
λ j, k + 1 =
nPr GTj CkWhMJ
 ) , we have:
From (45) and definition of I2 (X
(Tpdem − Tpin ) cp
ρ (−ηEx Xi + 2n, j + ηEx ui + 2n, j + Xr + 3n, j −ur + 3n, j )
Pr je Tj
+
4.4. X-update by quadratic programing
(1 + ηEx ηEx ) λirh, j = ρ (−ηEx Xi + 2n, j + ηEx ui + 2n, j + Xr + 3n, j −ur + 3n, j )
λjh,irk + 1
n+1
(58)
From (53), (54) and (56), we have:
+ (1 + ηEx ) Pr G
Pr je Tj
Hence, λj, k + 1 is calculated in a distributed way at each building and the
substation, by applying the average consensus protocol to an extra local
variable x i, j , as shown in (61) and (62).
From (52), we have:
Pi + n, j = Xik+ n, j −uik+ 2n, j + Pr G
l ∈ Ni
nPr GTj CkWhMJ
(n + 1) ηFe ηFp
ρ
n+1
+
ρ
n+1
i=1
Si, j
αHT = 1−μHT is the coefficient of all hot tanks,
βit =
Tt CkWhMJ ηFh (i, t )
ηFe (i, t )(TpHT − Tpin ) cp
wi, j = WiD, j − (Tp
,
BL
ηBL Gimax
dem − Tpin ) cp
.
The optimization problem (65)–(71) could not be decomposed into
time interval problems, owing to multiple-time-interval constraints (70)
and (71). However, all constraints from (66)–(71) are local constraints
at each building or the substation (each agent). Therefore, the X-update
problem can be divided into local problems, as we show next.
Denote:
(60)
n+1
∑i = 1
Si, j directly, we apply multiagent
Instead of calculating
average consensus [31,32] to calculate this term in a distributed
manner. Let
Xid ≜ [Xi,1 Xi,2 …Xi, m …Xi + 3n,1 Xi + 3n,2 …Xi + 3n, m ]T ∈ R 4m , i = 1…n ,
which consists of all variables X of the i−th building in a whole-day
period,
Xnd+ 1 = [X 4n + 1,1 X 4n + 1,1…X 4n + 1, m ]T ∈ Rm , which consists of all variables X of the substation in a whole-day period,
x i, j (0) = Si, j,
x i, j (Tc + 1) = aii x i, j (Tc ) + ∑ ail xl, j (Tc ),
(71)
where
n+1
∑
j=m
(61)
where
481
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
Pid ≜ [Pi,1 Pi,2…Pi, m…Pi + 3n,1 Pi + 3n,2…Pi + 3n, m ]T ∈ R 4m , i = 1…n ,
which
consists of all variables P of the i-th building in a whole-day period,
Pnd+ 1 = [P4n + 1,1 P4n + 1,1…P4n + 1, m ]T ∈ Rm , which consists of all variables
P of the substation in a whole-day period,
and
uid ≜ [ui,1 ui,2…ui, m…ui + 3n,1 ui + 3n,2…ui + 3n, m ]T ∈ R 4m , i = 1…n ,
und+ 1 = [u4n + 1,1 u4n + 1,1…u4n + 1, m ]T ∈ Rm .
average consensus protocol, which is similar to that described in Section 4.3. The latter is checked by first transforming it to a logic value (0
or 1) at each agent and then applying the minimum consensus, as follows:
Fh
x i = ∧ (max‖EriFe
, j (c + 1)‖ ⩽ e ηe , max‖Eri, j (c + 1)‖ ⩽ e ηh ), j = 1…m ,
x i (Tc + 1) = min(x i (Tc ), xl (Tc )),
Based on the new variable assignment, the optimization problem
(65)–(71) can be formed into whole-day period problems at each agent,
as follows:
where
EriFe
, j (c + 1) =
d
d
d 2
d
⎧ Xi = argmin‖Pi −X + ui ‖2
⎨ Ai Xid ⩽ bi , i = 1…n + 1,
⎩
Fe
ηiFe
, j (c + 1) − ηi, j (c )
EriFh
, j (c + 1) =
(72)
ηiFe
, j (c + 1)
,
Fh
ηiFh
, j (c + 1) − ηi, j (c )
ηiFh
, j (c + 1)
where
and agents l are connected to agent i.
It is easy to see that after limited times of calculation Tc , all
x i = x∗ = minx i (0) , i.e., each agent can confirm whether the stop criteria are satisfied at all agents. The stop criteria used in this study are
set as follows: ρ = 0.06 ; ∊abs = ∊rel = 10−4; e ηe = e ηh = 0.01; Tc = 50 .
I
⎡ Ai ⎤
Ai = ⎢ AiII ⎥ ∈ R(4m + 2m) × 4m , i = 1…n; An + 1 = Im ∈ Rm × m
⎥
⎢
⎢− AiII ⎥
⎦
⎣
B
bi = [biI biII biIII ]T ∈ R(4m + 2m) , i = 1…n; bn + 1 = −Pmin
1m ∈ Rm
⎡ 1
⎢− 1
I
Ai = ⎢ 0
⎢ 0
⎣
0 0
0 ⎤
0 0
0 ⎥
4m × 4m ,
Tp − 1 0 ⎥ ⊗ Im ∈ R
Tp 1 − 1⎥
⎦
6. Case study
We apply the proposed algorithm to a microgrid consisting of a
substation and 4 buildings (agents) with CHP system at each building.
Buildings 1 and 2 are residential ones and buildings 3 and 4 are complex (office and restaurant) ones. The given parameters of the tested
microgrid are as follows:
i = 1…n
0
α1 = [ αHT
0 … 0] ∈ R1 × m;
j−1
j−2
0
αj = [ αHT
… αHT
αHT
0 … 0 ] ∈ R1 × m;
6.1. Power and water demands
0
Bi1 = [ αHT
βi1 0 … 0] ∈ R1 × m;
The power and hot-water demands of four buildings are shown in
Fig. 5a and b, respectively. We estimated the power demand and hotwater demand curve shapes of the residential buildings based on the
statistical data given in [33]. The power demand curve shape of a
complex building is estimated from a report of the Japanese Agency for
Natural Resources and Energy in 2011 [34]. Here, we assume that the
power and hot-water demands are constant during each 1-h time interval. The ratio between the maximum power demand and the maximum hot-water demand is different for each building.
j−1
j−2
0
βi1 αHT
βi2 … αHT
βij 0 … 0 ⎤ ∈ R1 × m;
Bij = ⎡ αHT
⎣
⎦
AiII
⎡ B1 − α1
⎤
⋮
⎢⋮
⎥
⎢
= Bj − αj 0m × 2m ⎥ ∈ Rm × 4m
⎢
⎥
⋮
⎢⋮
⎥
⎢
⎥
⎣ Bm − αm
⎦
F
⎡ Pimax 1m ⎤
⎢ − PF 1 ⎥
imin m ⎥
∈ R 4m
biI = ⎢
⎢ {WiD, j }j ∈ {1.. m} ⎥
⎥
⎢
−
⎢
⎦
⎣ { wi, j }j ∈ {1.. m} ⎥
biII
6.2. Power limitations, hot-tank parameters, and gas-boiler capacity
Table 1 lists the power limitations, hot-tank capacities and initial
levels, and boiler capacities of the CHP systems at all buildings. Based
on the power demand at each building, the numbers of FC units installed in each CHP system are 26, 18, 23, and 16 respectively. The hottank capacities are assigned as the same level as the water demand peak
at each building, as shown in Table 1. Gas boilers with the capacities of
950 (L/h), 500 (L/h), 950 (L/h), and 500 (L/h) are installed at each
CHP system, respectively, to ensure that the water demand at each
building can be fulfilled without the FCs.
⎡ αHT HTi0 ⎤
⎡ HTimax −αHT HTi0 ⎤
2
2
⎢ αHT
⎢ HTimax −αHT
HTi0 ⎥
HTi0 ⎥
III
m
m
=⎢
⎥∈R .
⎥ ∈ R ; bi = ⎢
⋮
⋮
⎥
⎢
⎥
⎢
m
m
⎢
⎢
⎦
⎣ αHT HTi0−HTi0 ⎥
⎦
⎣ HTi0−αHT HTi0 ⎥
The optimization problem in (72) can be solved by quadratic programming, such as the interior-point method, which is embedded in
Matlab/Simulink.
6.3. Energy price
5. Algorithm
The electricity price is depicted in Fig. 6. It is a time-of-use two-level
price [2], in which the higher level is applied from the 8-th time interval
to the 22-nd time interval and the lower level is applied for the remaining time.
The proposed algorithm is depicted in Fig. 4, in which two iteration
loops are included. The first, outer iteration loop (green) is the c-loop to
deal with the variation of the FC efficiencies. The second, inner iteration loop (blue1) is exactly the whole-day distributed ADMM approach
described in Section 4. The stop criteria of both the inner and outer
loops can be checked in a distributed way. The former is checked by the
31.66 (JPY/kWh),
Pr e (t ) = ⎧
⎨
13.10
(JPY/kWh),
⎩
7 am ⩽ t ⩽ 10 pm,
othertimes
The gas price is 2.24 (JPY/MJ) [2], or 8.084 (JPY/kWh). At this gas
price, a FC unit output of 1 kWh costs Pr Gas /(η Fpη Fe ) (JPY). Therefore,
the cost range of 1 kWh from an FC unit is:
1
For interpretation of color in Fig. 4, the reader is referred to the web version
of this article.
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H.N. Tran et al.
Fig. 4. Algorithm flowchart.
Fig. 5. (a) Power demand vs. time slots. (b) Hot Water demand vs. time slots.
Table 1
Power limitations, hot-tank and gas-boiler capacity at buildings.
Buildings
Type
1
Residential
2
Residential
3
Complex
4
Complex
PDmax (kW)
FC units
25.65
26
18.2
16.65
18
12.6
21.14
23
16.1
14.4
16
11.2
F
Pmax
(kW)
F
(kW)
Pmin
WDmax (L/h)
HTmax (L)
HT0 (L)
BL
Wmax
(L/h)
18.3 (JPY) =
1.3
0.9
1.15
0.8
905.67
950
260
950
487.09
600
180
500
905.31
950
230
950
451.59
500
160
500
Fig. 6. Electricity price from main grid.
Pr Gas
Pr Gas
Pr Gas
⩽ Fp Fe ⩽ Fp Fe = 34.45 (JPY),
Fe
η η
η Fpηmax
η ηmin
Fe
Fe
= fFe (700 W) = 0.465 and ηmin
= fFe (50 W) = 0.274 . It is easy
where ηmax
to see that when operating at a high percentage of output power, a FC
could generate 1 kWh with a cost within the price range of the main
grid. However, when an FC is utilized around its minimum output
power, it costs more than the highest price of the main grid.
Fig. 7. Microgrid networks.
6.4. Power, hot-water, and information networks
Fig. 7 shows the microgrid power, hot-water, and information
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H.N. Tran et al.
utilized at maximum capacity, as shown in Fig. 11. Furthermore, hot
water volume sent from building 1 to building 3 is larger than the volume building 1 received from building 3. The gas boilers are not
needed all day.
networks. In the hot-water network, building 1 exchanges hot water
with building 3, and building 2 exchanges hot water with building 4.
The information network is utilized for the distributed calculation (in
consensus calculation). Consensus weightings ail and neighboring sets
Ni can be easily calculated from the information network structure.
Based on the convergence result of the consensus calculation presented
in [28], we choose Tc = 50 (times) to reduce the computational cost
while ensuring a good convergence in the consensus calculation.
7.1. Convergence features of the proposed method
To provide information on the convergence feature of the proposed
algorithm, we show the stop criteria parameters in Fig. 15, where
7. Test results
n
E (η Fe ) =
The schedule results for total electricity are depicted in Fig. 8a.
During the 1th –7th and 23th –24th time intervals, when the electricity
price is low, all FC operate at the minimum output power to utilize the
cheap power cost. During the 8th –22nd time intervals, when the electricity price is high, the total FC output power (Fig. 8a) shows its
fluctuation in a sawtooth shape. To maximize the FC efficiency, the
optimal schedule tends to operate the FCs at high output in as many
intervals as possible. However, owing to the limitation of hot-water
demand and the constraint of the hot-tank level at the last time interval,
the FCs cannot continuously operate at high output. Therefore, there
must be time intervals during which the FCs operate at low output.
These low outputs should be minimized to reduce the energy portion
corresponding to low FC efficiency. Therefore, the FC power fluctuates
and forms a sawtooth shape, as mentioned above. Note that in this
study FC capacities are at around 70% peak of power demand of their
respective building. If FC capacities are designed at a lower percentage,
FC utilization might be higher. However, design of optimal FC capacities is out of scope of this study. It is related to a structure, parameter
sizing problem which is our future work. Fig. 8b shows the schedule
results of the total hot-water variables. The total exchanged hot water
in the sending direction is slightly larger than that in the receiving
direction, owing to the heat losses in the exchange water pipes. These
losses also lead to a slight difference between the total hot water from
the mixers and the total hot-water demand during the 8th –23rd time
intervals.
The schedule results of buildings 1, 3, 2, and 4 are depicted in
Figs. 9, 10, 12 and 13, respectively. Hot-tank schedules of buildings 1
and 3 are shown in Fig. 11 and hot-tank schedules of buildings 2 and 4
are shown in Fig. 14. Because electricity exchange is allowed in the
microgrid, the FCs at buildings 1, 2, 3, and 4 sometimes operate at an
output higher than power demand of their respective buildings, based
on the optimal solution. Hot-water exchange occurs between buildings
1 and 3, and buildings 2 and 4. During the daytime, when the hot-water
demands of buildings 1 and 2 are low, these buildings send hot water to
buildings 3 and 4, respectively. On the contrary, at night, when the hotwater demands of buildings 1 and 2 are high, buildings 3 and 4 send hot
water to buildings 1 and 2, respectively. Buildings 1 and 3 have the
same hot-tank capacity, but building 3 requires a larger total hot-water
volume than that of building 1, therefore hot-tank of building 3 is
E (η Fh) =
m
∑∑
i=1
j=1
n
m
∑∑
i=1
ηi,Fej (c )−ηi,Fej (c−1)
ηi,Fej (c )
Fh
ηi,Fh
j (c )−ηi, j (c−1)
ηi,Fh
j (c )
j=1
Maxe (η Fe ) = Max
ηi,Fej (c )−ηi,Fej (c−1)
Maxe (η Fh) = Max
ηi,Fej (c )
Fh
ηi,Fh
j (c )−ηi, j (c−1)
ηi,Fh
j (c )
,
,
,
,
i = 1…n, j = 1…m , c = 1…
5, .
The convergence of the c-loop is depicted in Fig. 15b, and the
convergence of the whole-day distributed ADMM at the last c-loop
iteration is depicted in Fig. 15a. It can be seen that the whole-day
distributed ADMM approach converges at the last iteration of c-loop, as
two conditions ‖r k‖2 ⩽ eprimal and ‖s k‖2 ⩽ edual are both satisfied. Furthermore, the general algorithm is terminated after five iterations of the
c-loop, when the condition of matching efficiencies is satisfied.
7.2. Cost comparison
To evaluate the proposed method strictly, a Particle swarm optimization (PSO) approach is also used to search for the optimal schedule
of the test system. The approach utilized is based on the approach
presented in [35,36] and was embedded into toolbox of Matlab. The
detail algorithm and parameter setting of the approach is presented in
[37]. To make inequality constraints (25)–(27) suitable for the PSO
framework, we add hot-tank levels and hot water volume from boilers
as new decision variables into the test system. With this modification,
all inequality constraints are transformed into lower and upper bounds
of decision variables, which are available in the PSO framework, but the
number of decision variables increases from 408 (in the proposed
method) to 600. As for all equality constraints, square of their violations
are added into the objective function (20) with a weighting factor of
1000. The algorithm is terminated if the relative change in the objective
value over 20 iterations is less than a small tolerance (10−6 ).
We conduct numerous simulations with the PSO approach, and see
Fig. 8. (a) Total electricity schedule. (b) Total hot-water schedule.
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Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
Fig. 9. (a) Electricity schedule, Bldg. 1. (b) Hot-water schedule, Bldg. 1.
Fig. 10. (a) Electricity schedule, Bldg. 3. (b) Hot-water schedule, Bldg. 3.
Fig. 11. (a) Hot-tank schedule, Bldg. 1. (b) Hot-tank schedule, Bldg. 3.
Fig. 12. (a) Electricity schedule, Bldg. 2. (b) Hot-water schedule, Bldg. 2.
function is corresponding to the 65-th simulation in which the total cost
is 32,689 (JPY) and the total violation of equality constraints is 0.0313.
It turns out that the optimal cost by PSO approach is larger than that of
the proposed method, which is shown in Table 2. As the number of
variables of the test system is quite large, PSO approach might have
some difficulty to have a good convergence.
Table 2 shows the cost of bought power from the main grid, the gas
that all simulations do not converge to the same result. The convergence feature of these simulations is presented in Fig. 16. The PSO
approach is applied with various values of swarm size from 50 to 600 in
120 simulations, as shown in Fig. 17. It can be seen that the total
violation of equality constraints is quite large when the swarm size is
less than 100, and reduced when the swarm size is in the range from
100 to 600. Among results of 120 simulations, the lowest objective
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H.N. Tran et al.
Fig. 13. (a) Electricity schedule, Bldg. 4. (b) Hot-water schedule, Bldg. 4.
Fig. 14. (a) Hot-tank schedule, Bldg. 2. (b) Hot-tank schedule, Bldg. 4.
Fig. 15. (a) ADMM convergence in the terminal c-loop iteration. (b) Efficiency errors vs. c-loop iterations.
Fig. 17. PSO approach - Swarm size vs. simulations.
Table 2
Comparison of costs in cases of non-CHP and CHP scheduled by different
methods.
Fig. 16. PSO approach - Result of 120 random initial scenarios.
cost of FCs, and the gas cost of boilers, in three cases: non-CHP, CHP
scheduled by the proposed method, and CHP scheduled by the PSO
method. It is evident that the CHP schedule by the proposed method
show significant cost savings compared to the non-CHP and CHP-PSO
486
Unit (JPY)
Cost of
bought power
Cost by FC
Cost by
gas boiler
Total cost
Total cost
saved
Non-CHP
CHP-ADMM
CHP-PSO
31,773
19,097
22,078
0
8638
10,611
3230
0
0
35,003
27,735
32,689
0%
20.76%
6.61%
Energy Conversion and Management 174 (2018) 475–488
H.N. Tran et al.
Fig. 18. (a) Maximum efficiency errors vs. c-loop iterations in 250 scenarios. (b) Costs vs. 250 random initial scenarios.
feature.
To evaluate the proposed method, an PSO approach is also applied
for searching for the optimal schedule of the CHP system. However, the
PSO approach does not show a good convergence result as the proposed
method. Owing to the structure of the whole-day optimization problem,
the number of variables is quite large, and it might affect the exploration and convergence features of the PSO method.
case.
7.3. Convergence feature of random initial scenarios
To verify the robustness of the proposed algorithm for the optimization problem, the algorithm is executed 250 times by randomly as 
u, i = 1…n, j = 1…m .
signing the initial values of ηi,Fej , ηi,Fh
j , P , X , and
The convergences and costs of 250 random scenarios are shown in
Fig. 18a and b, respectively. The maximum deviations in the bought
electricity costs, FC gas costs, and total costs of these 250 initial scenarios are calculated as follows:
dCostElectricity =
dCostFC =
max(ECostRand (j ) ) − min(ECostRand (j ) )
min(ECostRand (j ) )
max(FCostRand (j ) ) − min(FCostRand (j ) )
dCostTotal =
min(FCostRand (j ) )
Supplementary data associated with this article can be found, in the
online version, at https://doi.org/10.1016/j.enconman.2018.08.046.
j = 1…250
References
= 2.91%,
max(TCostRand (j ) ) − min(TCostRand (j ) )
min(TCostRand (j ) )
= 2.47%,
Appendix A. Supplementary material
= 0.8%,
j = 1…250
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In this study, we address an optimization problem of a microgrid
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