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1
Abstract—
Energy flow calculation (EFC) plays an im-
portant role in steady-state analysis of multi-energy sys-
tems (MESs). However, the independent management of
sub-energy systems (subsystems) poses a considerable
challenge to solve the high-order nonlinear energy flow
model due to the limited information exchange between
these subsystems. In this paper, a fixed-point based dis-
tributed method is proposed for EFC in an electrici-
ty-gas-heating system. Firstly, the mathematical modeling
of each subsystem with coupling units is introduced. Then,
two information exchange structures among subsystems
are presented as sequential and parallel structures. Based
on the fixed-point theorem, novel distributed sequential
and parallel methods for EFC are proposed to calculate
energy flow distribution in MESs. In our proposed method,
the EFC in subsystems is implemented by the individual
system operators, with limited information exchange be-
tween subsystems. Therefore, the information privacy of
subsystems can be preserved in this solution process.
Moreover, the convergence of the proposed method is
guaranteed, and the sufficient conditions for the conver-
gence are presented. Lastly, simulations on a MES demon-
strate the effectiveness of the proposed method and the
quantified superiority over the existing methods in com-
putation time, accuracy and reliability.
1
Index Terms—
Distributed method, energy flow calculation,
fixed-point, high-order nonlinear equation, multi-energy system.
NOMENCLATURE
1) Variables and Parameters in Electricity Systems
B
ij
, G
ij
Susceptance and conductance of line ij.
N
e
Total number of electrical buses.
N
1
e
, N
2
e
, N
3
e
Number of slack node, PQ node and PV node.
N
total
e
Total number of electrical equations.
P
g
i
, Q
g
i
The injected active and reactive power at bus i.
P
ge
The power generated by the units except for those
at electrical and heating slack nodes.
P
l
i
, Q
l
i
The active and reactive loads at bus i.
P
loss
The power loss of whole networks.
∆P
i
, ∆Q
i
Active and reactive power mismatches at bus i.
G. Zhang and F. Zhang are with the Key Laboratory of Power System Intel-
ligent Dispatch and Control, Ministry of Education, Shandong University,
Jinan, 250061, China (e-mail: fengzhang@sdu.edu.cn).
K. Meng and Z. Y. Dong are with the School of Electrical Engineering and
Telecommunications, The University of New South Wales, NSW 2052, Aus-
tralia (e-mail: kemeng@ieee.org, zydong@ieee.org).
X. Zhang is with the Energy and Power Theme, School of Water, Energy
and Environment, Cranfield University, Cranfield MK43 0AL, U.K. (e-mail:,
xin.sam.zhang@gmail.com).
P
le
The general electric load.
|V
i
|, θ
i
Voltage magnitude and angle at bus i and j.
θ
ij
Voltage angle difference between bus i and j.
2) Variables and Parameters in Natural Gas Systems
C
mn
The pipeline constant.
f
in
The gas flow pressurized by the compressor.
f
comp
mn
The gas flow consumed by the compressor mn;
f
l
m
Gas flow consumed by the gas load at node m.
f
p
mn
Gas flow through pipeline mn.
f
s
m
Gas flow extracted from gas sources at node m.
∆f
m
The mismatch of nodal gas flow at node m.
N
g
Total number of gas nodes.
p
comp
mn
The active power consumed by compressor mn;
N
total
g
Total number of gas equations.
N
1
g
, N
2
g
Number of slack node and known-injection
node.
The polytropic exponent.
γ
comp1
mn
, γ
comp2
mn
, γ
comp3
mn
Consumption coefficients of compressor
mn.
comp
mn
The compressor efficiency.
ρ
in
, ρ
o
Inlet and outlet pressures of the compressor.
ρ
m
The gas pressures at nodes m.
3) Variables and Parameters in Heating Systems
c
p
The specific heat of water.
L
ab
The length of pipeline ab.
m
ab
Mass flow from node a to b.
m
l
a
m
s
a
Mass flow of heating load and source at node a.
m
p
a
The water mass to be pressured by the pump a.
∆m
a
The mismatch of water mass at node a.
N
h
Total number of heating nodes.
N
1
h
, N
2
h
, N
3
h
The number of slack node, ϕT
s
node and ϕT
r
node.
N
l
h
, N
loop
h
, N
s
h
Number of demand nodes, loops and source nodes.
N
st
h
The number of heating sources at one node.
N
total
h
Total number of heating equations.
P
p
a
The electrical power of pump a.
pr
p
a
The water pressure at node a.
∆p
ab
The pressure losses in pipeline ab.
∆p
l
k
The pressure mismatch of the k
th
loop.
T
g
The ambient temperature.
T
s
a
, T
r
a
The supply and return temperatures
T
s,s
a
The supply temperature of heat sources at node a.
T
r,l
a
The return temperature of heating load at node a.
∆T
s
a
, ∆T
r
a
The mismatches of supply and return temperature
at node a.
U
The heat transfer coefficient per unit length.
A Fixed-point Based Distributed Method for Energy
Flow Calculation in Multi-Energy Systems
Gang Zhang, Student Member, IEEE, Feng Zhang, Member, IEEE, Ke Meng, Member, IEEE, Xin Zhang,
Member, IEEE, and Zhao Yang Dong, Fellow, IEEE
IEEE Transactions on Sustainable Energy, Volume 11, Issue 4, October 2020, pp. 2567 - 2580
DOI:10.1109/TSTE.2020.2966737
© 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for
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2
ϕ
l
a
, ϕ
s
a
Heating power of demand and source at node a.
∆ϕ
l
a
, ∆ϕ
s
a
The mismatch of heating power of source and
demand at node a.
p
a
The efficiency of the pump at node a.
ρ
w
The water density.
4) Variables and Parameters Related to Coupling Units
a
C
i
, b
C
i
, d
C
i
, L
1
i
, L
2
i
, r
1
i
, r
2
i
Coefficients of the CHP model;
f
C
i
, f
B
a
, f
G
i
The gas flows consumed by the CHP at bus i, GB
at node a and GT at bus i.
P
C
i
, P
G
i
The active power of the CHP and GT at bus i.
P
le
The general electric load
q
gas
The heat value of natural gas
T
C
i
The supplied temperature of the CHP at bus i.
γ
B
1
a
, γ
B
2
a
, γ
B
3
a
The consumption coefficients of the GB at node a.
γ
G
1
i
, γ
G
2
i
, γ
G
3
i
The consumption coefficients of GT at bus i.
ϕ
B
a
,ϕ
C
i
Heating power of GB at node a and CHP at bus i.
ϕ
min
i
, ϕ
max
i
The minimum and rated heating power of the CHP
at bus i.
η
C
i
The efficiency of CHP at bus i.
μ
The spectral radius.
5) Additional Unknown Variables
f
B
hs
, f
C
hs
Gas flow consumed by the GB and CHP at heating
slack node hs.
f
C
es
, f
G
es
,
Gas flow consumed by the CHP and GT at elec-
trical slack node es.
P
C
es
, P
G
es
Active power of the CHP and GT at electrical
slack node es.
P
C
hs
Active power of the CHP at heating slack node hs.
P
comp
The active power consumed by compressors.
P
p
The electrical power of pumps.
ϕ
C
es
Heating power of CHP at electrical slack node es.
ϕ
B
hs
, ϕ
C
hs
Heating power of the GB and CHP at heating slack
node hs.
6) Acronyms
CHP
Combined heat and power plant.
EFC
Energy flow calculation.
EH
Energy hub.
ESO
Electricity system operator.
FPDPM
Fixed-point based distributed parallel method.
FPDSM
Fixed-point based distributed sequential method.
GB
Gas boiler.
GSO
Gas system operator.
GT
Gas turbines.
HSO
Heating system operator.
IH
Information hub.
MES
Multi-energy system.
UNM
Unified Newton-Raphson method.
I. INTRODUCTION
ULTI-ENERGY systems (MESs) were initially proposed to
link independent sub-energy systems (subsystems) to-
gether as a whole energy system to improve techno-economic
and environmental performance, which is considered as an
effective solution to tackle climate change and energy crisis
[1]-[3]. The interaction and interdependency of MESs are
strengthened by the increasing penetration of cogeneration
systems, such as combined heat and power plants (CHPs) with
high energy conversion efficiency [4]-[6]. To achieve optimal
planning and operation of a MES, the coordinated analysis of
multi-energy carriers is desirable [7]-[8].
As a basic tool, energy flow calculation (EFC) plays a sig-
nificant role in steady-state analysis of MESs, such as
day-ahead dispatch [9], static security analysis [10
] and service
restoration [11]. However, high-order nonlinear EFC models
are challenging to solve due to the limited information sharing
between subsystems, which are generally managed by different
operators.
Studies have been conducted to solve EFC models in an in-
dividual electricity, gas or heating system, such as Newton‘s
method and holomorphic embedding (HE) for electrical power
flow calculation[12]-[14], Newton‘s method for gas flow cal-
culation[15], and graph theory method for heating flow calcu-
lation [16]-[17]. However, these EFC methods for individual
energy systems cannot be directly employed in subsystems of
MES, because of additional unknown variables from other
subsystems that lead to the EFC non-executable. For example,
electric load related information is well given in the traditional
electricity system for solving EFC problem. However, in MES,
the electrical power consumption of compressors is determined
by gas flow distribution, which is treated as an unknown load
variable of EFC in electrical subsystem. Hence, the imple-
mentation of EFC in electrical subsystem relies on the gas
subsystem, and the previous methods for electrical EFC are no
longer effective. Consequently, the interdependence between
electricity, gas and heating subsystems of MES should be
comprehensively studied, and an efficient method is required to
solve the EFC in MES.
Based on the interaction mechanism between subsystems,
the unified Newton-Raphson method (UNM) has been cus-
tomized for the EFC in electricity-gas systems [18], electrici-
ty-heating system [19] and electricity-gas-heating systems
[20]-[22]. In UNM, all EFC equations related to subsystems are
simultaneously solved in a central place, so that the information
of whole MES need to be shared and aggregated by a joint
operator [23]. However, this approach is normally impractical,
because electricity, gas and heating systems are generally
managed by different entities. Due to the risk aversion and
technical limitation of data management, subsystem operators
tend to preserve the information privacy rather than collabora-
tive data sharing [24]. Furthermore, without a robust and digi-
talized energy system, intensively sharing large amounts of
information in the UNM brings the increased communication
burden, and the information sharing scheme threatens the ro-
bustness of the UNM solution under the situation of possible
data loss and incomplete dataset. In addition, a large number of
variables in a MES will significantly increase the dimension of
Jacobian matrix in the UNM, which will generally lead to slow
or non- convergence. Consequently, it is necessary to develop a
distributed and decentralized method for the joint EFC in MESs
because 1) computationally, the dimension of the distributed
M
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Transactions on Sustainable Energy
3
method is significantly decreased by decomposing the EFC of
MESs into several sub-EFCs in the respective subsystems; 2)
effective solution methods for EFC in subsystems, such as HE,
can be utilized to accelerate the EFC process; and 3) the dis-
tributed method can preserve the autonomy of subsystems and
enhance robustness against data loss.
The structure of information exchange among subsystems is
vital to develop the distributed method for the EFC in MESs,
which determines information flow in the solution process.
Two different information exchange structures in MESs can be
implemented in practice, denoted as Structures (a) and (b) [23].
In Structure (a), the information flows as a loop in a sequential
way across the electricity system operator (ESO), gas system
operator (GSO) and heating system operator (HSO). In Struc-
ture (b), the information flows under a radial structure, i.e.,
ESO, GSO and HSO can simultaneously exchange certain
information through the information hub (IH) at the root bus.
However, the existing solution methods for EFC are
non-compatible with both ‗loop‘ and ‗radial‘ structures.
To enable the compatible solution that could adapt to various
information exchange structures across the subsystems, a
fixed-point based distributed method is proposed in this paper.
The contributions of this paper are summarized as follows:
1) According to Structure (a), a novel fixed-point based
distributed sequential method (FPDSM) is presented. In this
case, subsystem operators have independent control over indi-
vidual subsystems, and the overall EFC can be carried out in a
distributed sequential way based on the loop information flow.
2) According to Structure (b), a novel fixed-point based
distributed parallel method (FPDPM) is proposed. In the
FPDPM, certain information is exchanged between an IH and
subsystem operators. Specifically, the IH processes the infor-
mation from subsystems and exchanges the information to
subsystem operators. Then, subsystem operators can carry out
their EFCs in parallel.
3) The proposed method can converge to the fixed point in
finite iterations. Moreover, simulations on a MES demonstrate
that the FPDSM and FPDPM have improved performance over
existing methods in computation time, accuracy and robustness
against data loss.
This paper is organized as follows. The schematic overview
of the modeling methodology is shown in Section II. The MES
is modeled in Section III. The distributed method for the EFC is
proposed in Section IV. Simulation results are calculated in
Section V. Finally, our conclusion is drawn in Section VI.
II. OVERVIEW OF THE MODELING METHODOLOGY
In this section, the structure of the modeling methods is
summarized. As shown in Fig.1, the MES is modeled as elec-
tricity, gas and heating subsystems with coupling units in Sec-
tion III. Then, the distributed EFC method is proposed to solve
the MES model in Section IV. Specifically, the additional un-
known variables in EFC of subsystems are presented in Section
IV-A, and then the loop and radial structures of information
exchange among subsystems are designed in Section IV-B.
According to the two different structures of information ex-
change, fixed-point based EFC methods are proposed respec-
tively in Section IV-C, including FPDSM and FPDPM. Sub-
sequently, conditions and supplements of the distributed EFC
method are presented, including sufficient conditions for con-
vergence in Section IV-D, initial value estimation for unknown
variables in Section IV-E, superiority of the proposed method
over the independent EFC method and the UNM method in
Section IV-F, and discussions of model adaptability to other
coupling networks and the application scope in Section IV-G.
Lastly, the effectiveness of the proposed method is demon-
strated, and the superiority over other existing method is vali-
dated by numerical tests in Section V.
Fig. 1. Schematic overview of the modeling methodology
III. MULTI-ENERGY SYSTEM MODELING
In this section, a MES consists of electricity, gas, and heating
subsystems as well as various coupling units, such as combined
heat and power (CHPs), gas boilers (GBs) and gas turbines
(GTs), are comprehensively modeled.
A. Electricity System
The modeling of electricity system consists of active and
reactive power nodal balance equations [25]-[26], as shown in
(1) and (2), respectively. In the classic electricity model, there
are total number of N
total
e
=2·N
2
e
+N
3
e
equations corresponding to
(2·N
2
e
+ N
3
e
) unknown variables, i.e., voltage magnitudes and
angles with the number of N
2
e
and N
2
e
+ N
3
e
, respectively.
23
1
cos sin , 1,2, ,
e
N
g
l
i i i j ij ij ij ij e e
i
j
P P P V V G B i N N
(1)
2
1
sin cos , 1,2, ,
e
N
g
l
i i i j ij ij ij ij e
i
j
Q Q Q V V G B i N
(2)
B. Gas System
The modeling of natural gas system contains nodal gas flow
balance equations (3) [20], which are built for all
known-injection nodes. Consequently, there are total number of
N
total
g
=N
2
g
equations corresponding to N
2
g
pressure variables.
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Transactions on Sustainable Energy
4
2
1
+ =0, 1,2, ,
g
N
s l p
m m m mn g
n
f f f f m N
(3)
In addition, the gas flow equation for general pipelines (4),
and the power and gas consumption model of compressors (5-6)
are shown as supplementary equations to Eq. (3) [20]. It is
noted that sign(ρ
m
, ρ
n
) in (4) denotes the direction of the gas
flow in pipeline mn. For example, sign(ρ
m
, ρ
n
) =1 represents
ρ
m
>ρ
n
, and gas flows from node m to node n.
0.5
22
sign , sign ,
p
mn mn m n m n m n
fC
(4)
1/
/ 1
1
in
comp
mn o in
comp
mn
f
P
(5)
1 2 3
2
()
comp comp comp comp comp comp
mn mn mn mn mn mn
f P P
(6)
C. Heating System
The heating system model comprises of the nodal supply and
return temperature differences (7)-(8), nodal heating power
demand equation (9), nodal heating power source equation (10),
nodal water mass balance equation (11) and head loss equation
(12) [16]. It is noted that sign
1
(m
ba
) in (11) is the sign function,
where sign
1
(m
ba
) =1 when m
ba
>0, and otherwise sign
1
(m
ba
)=0;
and sign
2
(m
ab
) in (12) is a sign function with a value of +1 if m
ab
is in the k loop and its direction is same as the predefined loop
direction, 1 if opposite, and 0 if ab is not in the loop.
,
11
11
g
sign sign
exp( ) , 1,2, ,
hh
NN
s s l s s s
a a a ab ab a a ba
bb
ab
s
ba b g h
p ba
T T m m m m T m
UL
m T T T a N
cm
(7)
,
11
11
gg
sign sign
exp( ) , 1,2, ,
hh
NN
r r s l r l
a a a ba ba a a ab
bb
ab
r
ab b h
p ab
T T m m m m T m
UL
m T T T a N
cm
(8)
,
, 1,2, ,
l l l s r l l
a a a p a a h
m c T T a N
(9)
,
, 1,2, ,
s s s s s r s
a a a p a a h
m c T T a N
(10)
11
1
sign ( ) sign ( ) , 1,2, ,
h
N
sl
a a a ba ba ab ab h
b
m
m m m m m m a N
(11)
2
11
sign , 1,2, ,
hh
k
NN
loop
l
ab ab
h
ab
p m p k N
(12)
Furthermore, if the number of heating sources at node a is
N
st
h
, the number of equations derived from Eq. (10) is N
st
h
×N
s
h
,
and the total number of equations is N
total
h
=3N
h
+N
l
h
+N
st
h
×N
s
h
+
N
loop
h
. Correspondingly, the number of unknown variables is
3N
h
+N
l
h
+N
st
h
×N
s
h
+N
loop
h
, i.e., T
s
a
, T
r
a
, m
l
a
, m
s
a
, m
ab
and the heating
power of CHP or GB at the heating slack node, with the number
of N
h
, N
h
, N
l
h
, N
st
h
×N
s
h
, N
h
+N
loop
h
1 and 1, respectively.
The power consumption of pumps P
p
a
is modeled by Eq. (13).
6
/ / 10
p p p w p
a a a a
P pr m
(13)
D. Coupling Units
The coupling units contain CHPs, GTs, GBs, electric pumps
and compressors. In addition to the models of electric com-
pressors and pumps in Eqs. (5)-(6) and (13), an three line model
of CHPs is adopted in this paper, which takes into account the
changes of the power production at part load operation [27],
details are shown in (14)-(17). Moreover, models of GBs in Eq.
(18) and GTs in Eq. (19) are introduced, and these models have
been widely employed in the EFC of MESs [17], [20].
1 max max
1 2 max 1 max
1 2 min 2 max
,
,
,
C C C C C C
i i i i i i i i i
C C C C C C C
i i i i i i i i i i i i
C C C C C C
i i i i i i i i i i i
a b T d L
P a b T d w L L
a b T d w w L
(14)
1 1 max 1
C
i i i i i
w L r
(15)
2 2 max 2
C
i i i i i
w L r
(16)
gas
/
C C C C
i i i i
f P q
(17)
1 2 3
2
B B B B B B
a a a a a a
f
(18)
3
12
2
G
GG
G G G
i i i
i i i
f P P
(19)
An energy hub (EH) is adopted in this paper to manage
coupling units [28], and a typical EH model is shown in Fig. 2.
Briefly, the CHP consumes gas from gas networks and gener-
ates electrical and heating power, and GB and GT consume gas
to generate heating and electrical power, respectively.
Fig. 2. A typical EH model
IV. DISTRIBUTED EFC METHOD
A novel fixed point distributed method for EFC in MESs is
presented in this section. Firstly, additional unknown variables
incurred by interconnection of multi-energy subsystems are
presented. Then, two structures of information exchange are
proposed. Lastly, the distributed EFC methods are proposed,
and the sufficient condition for convergence is derived.
A. Additional Unknown Variables
When the EFCs of individual subsystems are interconnected
in MESs, additional unknown variables appear through the
coupling units as described in Table I, where hs and es denote
the heating slack node and electrical slack node, respectively.
Additional unknown variables may cause EFC non-executable
in a given subsystem if certain key variables are unknown. For
example, the EFC in electricity subsystem cannot be conducted
without the special electric load P
comp
, which is determined by
gas systems. This indicates that EFC in electricity system relies
on the gas flow distribution. The key variables that impact EFC
across subsystems are identified in Table I.
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Transactions on Sustainable Energy
5
TABLE I ADDITIONAL UNKNOWN VARIABLES UNDER THE MES
System types
Additional unknown variables
Coupled systems
Electricity
system
P
comp
Gas system
P
p
, P
C
hs
Heating system
Heating system
ϕ
C
es
Electricity system
Gas system
f
C
hs
, f
B
hs
Heating system
f
G
es
, f
C
es
Electricity system
B. Structures of Information Exchange
To enable the convergence of EFCs in MES, the information
exchange through subsystems plays an important role in de-
signing the distributed methods. Generally, there are two
structures of information exchange, which can be implemented
in practice. In Structure (a), as shown in Fig. 3-a, the infor-
mation related to additional unknown variables flows among
subsystem with a peer-to-peer structure, so a loop diagram can
be formed. In Structure (b), as shown in Fig. 3-b, the infor-
mation regarding additional unknown variables from the ESO,
GSO and HSO, i.e., (P
comp
, P
C
es
, P
G
es
, P
p
, ϕ
C
hs
, ϕ
B
h
), have been ag-
gregated by an information hub (IH) at the root bus. In this
radial structure, (P
comp
, P
C
es
, P
G
es
, P
p
, ϕ
C
hs
, ϕ
B
h
) are further translated
to (f
C
hs
, f
B
hs
, f
C
es
, f
G
es
) for the GSO, (P
C
hs
, P
p
, P
comp
) for the ESO and (ϕ
C
es
) for the HSO. Finally, the processed variables are distributed
to the GSO, ESO and HSO, so a radial diagram is developed.
C. The Fixed-Point Based Distributed Method
Based on the two structures of information exchange, the
corresponding FPDSM and FPDPM are proposed in this sec-
tion. As an example, the most comprehensive coupling rela-
tionship among subsystems is chosen for the proposed method.
In this coupling structure, the electrical and heating slack nodes
are both powered by the CHPs which simultaneously couple the
electrical, gas and heating systems. In general, the proposed
method can be adapted to other coupling relationships, which
will be discussed in Section IV-G.
For simplicity, the EFC models of subsystems are described
in compact form, as shown in Eqs. (20), where F
e
(‧), F
g
(‧) and
F
h
(‧) are the electrical EFC model (1)-(2), gas EFC model (3)-(6)
and heating EFC model (7)-(13), respectively; [‧]
e
, [‧]
g
, [‧]
h
are
variable sets that can be obtained by conducting EFC in the
electrical, gas and heating systems, respectively; and (‧) denotes
sets of additional unknown variable that need to be
pre-determined by other system operators.
[ ] =arg{F , , 0}
[ , ] =arg{F 0}
[ ] arg{F , 0}
C comp p C
es e e hs
p C C
hs h h es
comp C C
g g hs es
P P P P
P
P f f
(20)
In Structure (a), the subsystem operators exchange infor-
mation in a peer-to-peer way. Consequently, a novel FPDSM is
developed where the EFC in heating, gas and electrical sub-
systems are sequentially implemented. The detailed FPDSM
and information flow are shown in Algorithm 1 and Figure 4-a,
which match the loop diagram in Fig. 3-a.
Algorithm 1: The FPDSM based on Structure (a).
1: Initialization. Define tolerance ε, the indices of iterations
k=0; pre-estimate the initial value of P
C
es
, termed as P
C
es
(0)
.
2: EH at electrical slack node. Solve Eq. (21), and obtain f
C
es
(k)
and ϕ
C
es
(k)
. Pass ϕ
C
es
(k)
and f
C
es
(k)
to HSO and GSO, respectively.
( ) 1 max max
( ) ( ) 1 2 max 1 max
( ) 1 2 min 2 max
,
,
,
C C k C C C C
es es es es es es es es es
C k C C k C C C C
es es es es es es es es es es es es
C C k C C C C
es es es es es es es es es es es
a b T d L
P a b T d w L L
a b T d w w L
( ) ( ) ( )
gas
( )/( )
C k C k C k C
es es es es
f P q
(21)
3: HSO. According to ϕ
C
es
(k)
, solve heating EFC problem (22)
and obtain P
p(k)
, ϕ
C
hs
(k)
. Then, pass P
p(k)
to the ESO.
( ) ( ) ( )
[ , ] =arg{F ( ) 0}
p k C k C k
hs h h es
P
(22)
4: EH at heating slack node. Solve Eq. (23) and obtain f
C
hs
(k)
,
P
C
hs
(k)
. Then, pass f
C
hs
(k)
and P
C
hs
(k)
to the GSO and ESO.
( ) 1 max max
( ) ( ) 1 2 max 1 max
( ) 1 2 min 2 max
,
,
,
C C k C C C C
hs hs hs hs hs hs hs hs hs
C k C C k C C C C
hs hs hs hs hs hs hs hs hs hs hs hs
C C k C C C C
hs hs hs hs hs hs hs hs hs hs hs
a b T d L
P a b T d w L L
a b T d w w L
( ) ( ) ( )
gas
( )/( )
C k C k C k C
hs hs hs hs
f P q
(23)
5: GSO. According to f
C
hs
(k)
and f
C
es
(k)
, solve gas EFC problem
(24) and obtain P
comp(k)
. Then, pass P
comp(k)
to the ESO.
( ) ( ) ( )
[ ] arg F , 0
comp k C k C k
g g hs es
P f f
(24)
6: ESO. According to P
comp(k)
, P
p(k)
and P
C
hs
(k)
, solve the elec-
trical EFC problem (25) and obtain P
C
es
(k+1)
.
C( +1) ( ) ( ) ( )
[ ] =arg{F ( , , ) 0}
k comp k p k C k
es e e hs
P P P P
(25)
7: If |P
C
es
(k+1)
P
C
es
(k)
|≤ ε, the iterative algorithm converges; Else,
k=k+1, and repeat from step 2.
Fig. 4-a. The diagram of the FPDSM. Fig. 4-b. The diagram of the FPDPM
In Structure (b), the subsystem operators, i.e., the ESO, GSO
and HSO, simultaneously exchange information with an IH in a
radial structure. Consequently, a novel FPDPM is proposed
where the EFC in subsystems can be carried out in parallel
based on the information exchange from the IH. The detailed
FPDPM and information flow are shown in Algorithm 2 and
Fig. 4-b, as shown in radial diagram of Fig. 3-b.
GSO
HSO
ESO
comp
P
C
hs
,
CG
es es
ff
,
CB
hs hs
ff
,
Cp
hs
PP
GSO
HSO
ESO
IH
p
P
, , ,
C B C G
hs hs es es
f f f f
,
CG
es es
PP
,
CB
hs hs
C
es
, ,
C
comp p
hs
P P P
comp
P
Fig. 3-a. The loop diagram of infor-
mation exchange in Structure (a)
Fig. 3-b. The radial diagram of infor-
mation exchange in Structure (b)