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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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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)