This paper establishes a compact and practical model for a waterwall system comprising supercritical oncethrough boilers, which can be used for automatic control or simple analysis of the entire boilerturbine system. Input and output variables of the waterwall system are defined, and balance equations are applied using a lumped parameter method. For practical purposes, the dynamic equations are developed with respect to pressure and temperature instead of density and internal energy. A comparison with results obtained using APESS, a practical thermal power plant simulator developed by Doosan Heavy Industries and Construction, is presented with respect to steady state and transient responses.
1. Introduction
In spite of environmental issues, thermal power plants generate approximately 65% of the world’s power supply. In recent years, the construction of largecapacity thermal power plants with environmental facilities has been common
[1
,
2]
.
With respect to structure, the boilers of thermal power plants are classified into two types: the drum boiler and the oncethrough boiler (OTB)
[3]
.
Fig. 1
shows a schematic of a supercritical oncethrough boilerturbine system. A supercritical oncethrough boiler comprises several heat exchangers, such as economizers, a water wall, superheaters, and reheaters. As shown in
Fig. 1
, the feedwater enters into the water wall through economizers. Then, the water is transformed into steam in the waterwall tube. The steam is superheated to generate electric power and circulated to the economizer again. Alternatively, a drumtype boiler system includes a drum, wherein saturated steam is separated from saturated fluid and provided to a superheater. The remaining saturated water reenters the waterwall tubes through downcomers
[4].
Schematic of a supercritical oncethrough boilerturbine system
Currently, oncethrough boilers are constructed more commonly than drum boilers. Compared with drum boilers, oncethrough boilers can operate at higher pressure and temperature and thus allow greater energy efficiency. Because oncethrough boilers do not have drums or largediameter downcomers, they exhibit less metal weight and smaller fluid storage capacity than drum boilers
[5]
. Therefore, although oncethrough boilers can respond rapidly to load changes, controlling them is more difficult than controlling drum boilers
[6]
.
In the system represented by
Fig. 1
, the entire surface of the lower part of the furnace wall is surrounded by waterwall tubes. When the operation conditions of the boiler exceed the critical point (22.09 MPa, 374.14℃
[7]
), the unit is called a “supercritical unit.” In the waterwall tube of a supercritical oncethrough boiler, the phase of the water changes directly from liquid to vapor without undergoing saturation. This is a significant difference between the supercritical oncethrough boiler and other subcritical boilers. The waterwall system is one of the most important components affecting the dynamics of supercritical oncethrough boilers.
Although there are wellestablished models for drum boilers, such as those proposed by Bell and Åström
[8]
, standard models for oncethrough boilers are far less common. Because of the major difference in the waterwall system between the two types of boilers, a compact and effective model of the waterwall system is currently a relevant research topic.
There are many mathematical models of a water wall for subcritical oncethrough boilers
[5
,
9

11]
; however, there are comparatively few mathematical models of a water wall for supercritical oncethrough boilers.
Dumont and Heyen developed an abridged mathematical model for the entire oncethrough boiler system
[12]
. They modified internal heat transfer coefficients and pressure drop formulations and considered the changes in the flow pattern. Li and Ren describe a waterwall system using a moving boundary
[13]
. They used enthalpy to track the moving boundary location at supercritical pressure and used mass, energy, and momentum balances to obtain the length of each section. Pan and colleagues presented a detailed waterwall model for predicting the mass flux distribution and metal temperature in the water wall of an ultrasupercritical boiler
[14]
. They treated the waterwall system as a network comprising 178 circuits, 15 pressure grids, and 7 connecting tubes; the system can be described using 195 nonlinear equations.
Recently, intelligent systems have been applied for modelling a oncethrough boiler. Chaibakhsh and colleagues developed a model for a subcritical oncethrough boiler whose parameters are adjusted on the basis of genetic algorithms
[15]
. Lee and colleagues established a model for a largescale power plant based on the neural network method
[16]
, and Liu and colleagues described a supercritical oncethrough boiler using the fuzzyneural network method
[17]
.
In the present study, we attempt to develop a compact and practical model of waterwall systems for supercritical boilers that can be used for automatic control, analysis, and modeling of entire boilerturbine systems. The objective is to develop a relatively simple waterwall model with sufficient accuracy for analysis and control rather than to describe the detailed dynamics occurring inside the waterwall tube. We use pressure and temperature as state variables; both of these are practical variables in industrial applications.
First, we establish input and output variables of waterwall systems and apply fundamental laws of physics, i.e., mass, energy, and momentum balance equations, using a lumped parameter method. Then, complicated equations and variables are approximated by adopting reasonable and applicable assumptions. To change the state variables with pressure and temperature, enthalpy and density are approximated as functions of pressure and temperature using a steam table. To verify the proposed model, a model of the waterwall system obtained using APESS, a practical thermal power plant simulator
[18]
developed by Doosan Heavy Industries and Construction, is presented and compared.
2. Basic Balance Eqs.[3,7,1922]
The fundamental principles used in developing the model are mass balance, energy balance, and momentum balance.
Table 1
shows the nomenclature used in this paper. In
Table 1
, “wall” denotes the tube wall of each heat exchanger, such as the water wall, superheater, and reheater.
Nomenclature
 2.1 Mass balance
Mass balance is represented in (1), which gives the rate of mass change for a heat exchanger system.
 2.2 Energy balance
Energy balance is represented in (2), (4), and (6) for the combustion gas, tube wall, and working fluid, respectively. The dynamics of combustion gas are represented in (2), where
Q_{gw}
is the transferred heat flow from combustion gas to the tube wall. As shown in (3),
Q_{gw}
has two terms: radiative heat transfer and convective heat transfer. The temperature change of the tube wall is represented in (4), where
Q_{wf}
is the transferred heat flow from the tube wall to the internal working fluid in (5). Therefore, the combustion energy is represented as the temperature change of the tube wall using (2) and (4). Finally, the dynamics of internal working fluid energy are represented in (6).
where,
where,
 2.3 Momentum balance
The exact momentum balance of fluid in the tube is difficult to describe theoretically because of the internal turbulent flow of fluid. However, in the momentum balance equation, the dynamic term can be neglected because the pressureflow process works faster than the mass and energy balance dynamics. In addition, the inertia term can be neglected compared with the friction term. These modifications result in the following equation, which is used in
[19]
.
Generally, heat exchangers in boiler systems, including a water wall, superheater, reheater, and economizer, can be modelled using (1)  (7). However, the major variables in these balance equations, such as pressure (
P
), temperature (
T
), density (
ρ
), enthalpy (
H
), and internal energy (
U
), are dependent variables that are functions of thermodynamic state. The thermodynamic state of water is classified into three state regions: the compressed liquid region, saturated liquidvapor region, and superheated region. The saturated liquidvapor region is called the “saturation region.”
3. Development of a WaterWall Model
 3.1 Balance equations for waterwall systems
The detailed waterwall model considers many variables
[12
,
14]
; in this paper, several major thermodynamic variables are selected on the basis of a lumped parameter method. To describe the simple waterwall system, several assumptions are required.
3.1.1 Assumptions

1. The pressure dynamics of the flue gas are negligible.

2. The flue gas exhibits ideal gas behavior.

3. The working fluid properties are uniform at any cross section.

4. The heat conduction in the axial direction is negligible.

5. The change in the thermodynamic properties of the internal working fluid is lumped.

6. The heat transfer from the flue gas to the wall is proportional to the combustion heat generated in the furnace.

7. The gaswall heat transfer dynamics are sufficiently faster than the wallfluid heat transfer dynamics.
The fundamental balance equations are modified according to the above assumptions. Four major variables — mass flow, enthalpy, pressure, and temperature at the outlet — are selected for both inputs and outputs. To consider the combustion energy, the mass flow of fuel (
W_{fl}
) is included as an input variable. The selected variables are represented in
Fig. 2
and
Table 2
.
Inputs and outputs of the waterwall model
Inputs and outputs of waterwall system
Inputs and outputs of waterwall system
Therefore, in this paper, the water wall is represented as a 5input and 4output system. The fundamental balance Eqs. (1)(7), are modified as follows.
3.1.2 Mass balance
Because the working fluid enters from the economizer, the inlet of the water wall is the outlet of the economizer. Therefore, the mass balance of the working fluid in the water wall, given by (1), is modified as follows:
3.1.3 Energy balance
Regarding the energy balance of internal working fluid, (6) can be written as follows:
where
Q_{wwwf}
is the transferred heat flow from the tube wall to the fluid, which is modified from (5) as
Regarding the dynamics of
T_{w}
, the temperature of the tube wall, given by (4), is rewritten as follows:
where
Q_{wwgw}
is the transferred heat flow from the flue gas to the tube wall. Then,
Q_{wwgw}
is
Regarding the dynamics of
T_{g}
in (12), (2) is modified as
where
Q_{c}
is included to consider the heat input by the fuel combustion. In (13), because the flue gas comes from an air preheater and leaves the furnace,
H_{aho}
is the enthalpy at the air preheater outlet and
H_{fno}
is the enthalpy at the furnace outlet. Because there is no mass flow change of the flue gas in the furnace,
W_{gi}
and
W_{go}
in (2) are unified with
W_{g}
.
Q_{c}
is given as follows:
where
K_{fl}
is the calorific value of fuel and
W_{fl}
is the fuel mass flow.
The energy balance Eqs. (8)  (14), can be directly used for the waterwall model. However, they require system variables from the other heat exchangers, such as the economizer, furnace, and air preheater, as well as additional system parameters such as heat transfer coefficients, the volumes of the furnace and wall, and the specific heat at constant volume of the wall and gas. Consequently, direct application of (8)(12) results in a complicated model, which is beyond the scope of this paper.
In this study, to make the model more compact, we assume that the heat transfer from the gas to the tube wall is proportional to the combustion heat (assumption 6). Then, (12) can be expressed as follows:
where α is the ratio of
Q_{wwgw}
to
Q_{c}
. Although α can be considered a constant, it is a function of another thermal state
[20]
. In this study, α is a function of
T_{ave}
, which is the average temperature between two outlets. That is,
where,
The three coefficients
a_{i}
can be determined using the measurement data.
Typically, heat exchange between the gas and the wall is far faster than that between the wall and the fluid (assumption 7). Therefore, the dynamics of
T_{w}
can be ignored in (11)
[3
,
20]
. Then, (11) is modified as a static equation with
Accordingly, (18) can be expressed using (14) and (15) as follows:
where,
η
represents the ratio of
Q_{wwwf}
to
W_{fl}
, which is equal to the product of
α
and
K_{fl}
.
As a result, the energy balance of the working fluid, given by (9), is simply represented using (20) as follows:
3.1.4 Momentum Balance
In the mass and energy balance equations, given in (8) and (21), the output variable
W_{wwo}
is determined using the momentum balance equation (7). Because the outlet of the water wall is the inlet of the primary superheater, the momentum balance of the working fluid at the primary superheater is given as follows:
In (22),
g
and
g_{c}
represent gravitational acceleration and the gravitational conversion factor, respectively, whose values are approximately 9.80665 [m/sec
^{2}
] and 9.80665 [kg(mass)·m/kg(weight)·sec
^{2}
], respectively. The constant 10.1772·10
^{4}
is included in the denominator to change the units from [kg(weight)/m
^{2}
] to [MPa].
Although the friction factor,
F_{ps}
, in (22) is considered a constant
[9
,
19]
,
F_{ps}
is proportional to
W_{eco}
in practice. In this study,
F_{ps}
is selected as a function of
W_{eco}
, as follows, to obtain better accuracy of the system:
The two coefficients
b_{i}
can determined using the measurement data. Finally, three balance equations for the waterwall model are given by (8), (21), and (22) using (16), (20), and (23).
 3.2 Change of state variables withPandT
The established model given by (8) and (21) explains the dynamics of density
ρ
and internal energy
U
of the working fluid. The state variables and the input and output variables are given as follows:
In industrial practice, the pressure
P
and temperature
T
of the working fluid are directly measured and importantly managed. That is, the steam table is necessary to calculate
ρ
and
U
from measured variables. Because
P
and
T
are measured outputs, they can be directly compared with measured data for a real plant. Therefore, in this study, we set pressure and temperature as state variables of the waterwall system as follows:
To change the state,
ρ_{wwo}
and
U_{wwo}
in dynamic equations (8) and (21) are set as functions of
P_{wwo}
and
T_{wwo}
in this study. Hereafter, the subscripts of
ρ, U, P, T
, and
H
are omitted for conciseness.
From the definition of enthalpy
[7]
,
the left side of (21) can be arranged as follows:
Accordingly, (8) and (21) can be written as follows:
Then, a steam table is used to represent
ρ
and
H
in (33) and (34) as functions of
P
and
T
. Because the objective system operates in the superheated region,
ρ
and
H
of the superheated vapor region of the steam table are approximated as the following simple polynomial functions of
P
and
T
:
where the coefficients are determined using the least squares method. These equations are valid only for the operation range used in the least squares method.
Using the chain rule,
and (33) and (34) are written as follows:
Next, d
p
/d
t
and d
T
/d
t
are determined using (39) and (41) with simple algebraic calculations as follows:
where,
Then, we can rearrange the final waterwall equations with notations for state, input, and output as follows:
where,
4. Simulation Results
To test the validity of the presented model, the waterwall system obtained using the APESS simulator is modeled as a target system. The presented waterwall model (46)  (55) is realized using MATLAB and a fourthorder RungeKutta algorithm is applied for the discrete simulation. Then, the steadystate and transient responses in superheated operation are compared.
For the simulation, three constants,
V_{ww}, K_{fl}
, and
L_{ps}
, are determined using the APESS simulator. The coefficients
a_{i}
for
η
and
b_{i}
for
F_{ps}
are determined using offline data from APESS in the superheated operation range. The results of interpolation using the least squares method are as follows:
Fig. 3
shows the measurements and plot of
η
, and
Fig. 4
shows the measurements and plot of
F_{ps}
. These two figures indicate that the interpolation is quite effective when considering real constants. In
Fig. 4
, the measurement value of
F_{ps}
changes from 2.8·10
^{4}
to 4.5·10
^{4}
according to the operating conditions, which explains why we do not use a constant
F_{ps}
in this study.
η as a function of T_{ave}
F_{ps} as a function of W_{eco}
The coefficients
c_{i}
for
H
and
d_{i}
for
ρ
are also determined using the least squares method with the steam table. The regions 410℃ <
T_{wwo}
< 430℃ and 25
MPa
<
P_{wwo}
< 31
MPa
in the steam table are selected to simulate the operation range of APESS. The results of the approximation are as follows:
Eqs. (46)(59) form the basis for the waterwall system in APESS. To verify the performance of the system, two types of simulations are tested: steady state responses and transient responses.
 4.1 Steadystate test
For the steadystate comparison, the APESS model is run with fixed electric power generation. Because the APESS system has internal control loops, all variables in APESS are stabilized to a steady state. Then, steadystate values of the 5 inputs and 4 outputs of the waterwall system are obtained from APESS. The same input values are applied to the presented model, and steadystate output values are compared.
Table 3
shows a comparison between the APESS system and the presented model. In the table, electric power is varied from 1000 MW to 800 MW, with which the boiler operates in the supercritical region. In
Table 3
,
W_{wwo}, P_{wwo}
, and
T_{wwo}
are directly proportional to the electric power, whereas
H_{wwo}
is inversely proportional. Percent errors of outputs are also presented, calculated as follows: Model APESS/APESS. In the table,
W_{wwo}
and
P_{wwo}
exhibit relatively small errors compared with
H_{wwo}
and
T_{wwo}
. The maximum error is 0.27% (1.14 ℃) for
T_{wwo}
with power generation of 900 MW. The average of all steady state errors is calculated to be 0.05%. Although there is no absolute criteria to determine modeling mismatch, we believe that these results are sufficient for predicting the steady state of the waterwall system.
Steadystate values of APESS and model
Steadystate values of APESS and model
 4.2 Transient response test
For the comparison of transient responses, the electric load demand of APESS is increased and decreased in steps. The load demand signal is adjusted as follows: 800 MW → 900 MW → 1000 MW → 900 MW → 800 MW. Each step is maintained for 20 minutes to attain a new steady state.
Fig. 5
shows graphs of the 5 inputs of the waterwall system obtained using APESS. The 5 inputs shown in
Fig. 5
are applied to the presented model.
Five input signals for transient responses.
Figs. 6

9
show a comparison between the APESS model and the presented model. According to these figures, the responses of the four outputs are similar to those of a firstorder system. Considering that (46) and (47) are very complicated, we find that the major dynamics of the waterwall system are quite simple.
Mass flow (W_{wwo}) of APESS and model
Enthalpy (H_{wwo}) graphs of APESS and model
Pressure (P_{wwo}) graphs of APESS and model.
Temperature (T_{wwo}) graphs of APESS and model.
According to
Figs. 6
and
8
, the responses of
W_{wwo}
and
P_{wwo}
are almost identical, as suggested by the steadystate responses. In
Fig. 7
, the initial value of enthalpy is not identical to the APESS data because the enthalpy is calculated using the pressure and temperature obtained by the approximated equation (58). Although the responses
H_{wwo}
and
T_{wwo}
exhibit different steady states, they have similar patterns with similar rising times.
5. Conclusion
We present a lumped model for the waterwall systems of supercritical oncethrough boilers. The model has two states, 5 inputs, and 4 outputs determined using a lumped parameter method. A steam table is approximated and used in the model equations to change the state
A waterwall system obtained using the APESS simulator is modeled as a target system. Comparison results consider both steady states and transient responses. In both simulations, the mass flow and pressure exhibited similar results, and enthalpy and temperature exhibited small errors
Although the presented model is quite complex, its dynamics are similar to those of a firstorder system. We believe that this model is useful for designing an automatic controller and for analysis of waterwall systems.
Acknowledgements
This research was supported by the ChungAng University Excellent Student Scholarship and by the ChungAng University Research Grant in 2013, and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: NRF20100025555) .
BIO
Geon Go received his B.S. degree in Electrical and Electronics Engineering from ChungAng University, Seoul, Korea, in 2013. He is currently an M.S Candidate in Electrical and Electronics Engineering at ChungAng University. His research interests involve the operation and modeling of fossil power plants and power system analysis.
UnChul Moon received his B.S., M.S., and Ph.D. degrees from Seoul National University, Korea, in 1991, 1993, and 1996, respectively, all in Electrical Engineering. In 2000, he joined WooSeok University, Korea, and in 2002, he joined ChungAng University, Korea, where he is currently an Associate Professor of Electrical Engineering. His current research interests are power system analysis, computational intelligence, and automation.
Liu Changliang
,
Wang Hong
2011
“An Overview of Modeling and Simulation of Thermal Power Plant”
Proc.of IEEE International Conference on Advanced Mechatronic Systems
Zhengzhou, China
86 
91
Bentarzi H.
,
Chentir R.A.
,
Ouadi A.
2011
“A New Approach Applied to Steam Turbine Controller in Thermal Power Plant”
2nd International Conference on Control, Instrumentation and Automation
Shiraz, Iran
86 
91
Robert J.
,
Tobias W.
,
Veronica O.
2012
“Dynamic Modelling of Heat Transfer Processes in a Supercritical Steam Power Plant”, M.S. thesis
Dept. Energy and Environment, Chalmers University of Technology
Göteborg, Sweden
Naghizadeh R.A.
,
Vahidi B.
,
Tavakoli M.R.B.
2011
“Estimating the Parameters of Dynamic Model of Drum Type Boilers Using Heat Balance Data as an Educational Procedure”
IEEE Trans. on Power Systems
26
(2)
775 
782
DOI : 10.1109/TPWRS.2010.2061879
Adams J.
,
Clark D.R.
,
Louis J.R.
,
Spanbauer J.P.
1965
“Mathematical Model of OnceThrough Boiler Dynamics”
IEEE Trans. on Power Systems
84
(2)
146 
156
DOI : 10.1109/TPAS.1965.4766165
Cheng Xu
,
Kephart R.W.
,
Menten C.H.
2000
“Modelbased Oncethrough Boiler Startup Water Wall Steam Temperature Control”
Proc. of IEEE International Conference on Control Applications
Anchorage, Alaska, U.S.A.
778 
783
Sonntag R.E.
,
Van Wylen G.J.
,
Borgnakke C.
2002
Fundamentals of Thermodynamics
John Wiley & Sons, Inc.
Bell D.
,
Åström K. J.
1987
Dynamic models for boilerturbinealternator units: Data logs and parameter estimation for a 160 MW unit
Lund Institute of Technology
Sweden
Ray A.
,
Bowman H.F.
1976
“A Nonlinear Dynamic Model of a OnceThrough Subcritical Steam Generator”
ASME Trans. on Dynamic Systems, Measurement, and Control
98
332 
339
DOI : 10.1115/1.3427046
Jensen J.M.
,
Tummescheit H.
2002
“Moving Boundary Models for Dynamic Simulations of TwoPhase Flows”
Proc. Of the 2nd Int. Modelica Conference
Oberpfaffenhofen, Germany
235 
244
Zheng S.
,
Luo Z.
,
Zhang X.
,
Zhou H.
2011
“Distributed parameters modeling for evaporation system in a oncethrough coalfired twinfurnace boiler”
International Journal of Thermal Sciences
50
2496 
2505
DOI : 10.1016/j.ijthermalsci.2011.07.010
Dumont M.N.
,
Heyen G.
2004
“Mathematical modelling and design of an advanced oncethrough heat recovery steam generator”
Computers & Chemical Engineering
28
651 
660
DOI : 10.1016/j.compchemeng.2004.02.034
Li YongQi
,
Ren TingJin
2009
“Moving Boundary Modeling Study on Supercritical Boiler Evaporator: By Using Enthalpy to Track Moving Boundary Location”
Power and Energy Engineering Conference
Wuhan, China
1 
4
Pan J.
,
Yang D.
,
Yu H.
,
Bi Q.C.
,
Hua H.Y.
,
Gao F.
,
Yang Z.M.
2009
“Mathematical modeling and thermalhydraulic analysis of vertical water wall in an ultra supercritical boiler”
Applied Thermal Engineering
27
2500 
2507
Chaibakhsh A.
,
Ghaffari A.
,
Moosavian A.A.
2007
“A simulated model for a oncethrough boiler by parameter adjustment based on genetic algorithms”
Simulation Modelling Practice and Theory
15
1029 
1051
DOI : 10.1016/j.simpat.2007.06.004
Lee K.Y.
,
Heo J.S.
,
Hoffman J.A.
,
Kim S.H.
,
Jung W.H.
2007
“Neural NetworkBased Modeling for A Large Scale Power Plant”
IEEE
Power Engineering Society General Meeting
1 
8
Liu X.J.
,
Kong X.B.
,
Hou G.L.
,
Wang J.H.
2013
“Modeling of a 1000MW power plant ultra supercritical boiler system using fuzzyneural network method”
Energy Conversion and Management
65
518 
527
DOI : 10.1016/j.enconman.2012.07.028
Lee K.Y.
,
Van Sickel J.H.
,
Hoffman J.A.
,
Jung W.H.
,
Kim S.H.
2010
“Controller Design for a LargeScale Ultrasupercritical OnceThrough Boiler Power Plant”
IEEE Trans. on Energy Conversion
25
(4)
1063 
1070
DOI : 10.1109/TEC.2010.2060488
Usoro Patrick Benedict
1977
“Modeling and Sumulation of a Drum BoilerTurbine Power Plant under Emergency State Control”, M.S. thesis
Dept. Mechanical Engineering, Massachusetts Institute of Technology
Cambridge, United States of America
Shinohara W.
,
Koditschek D.E.
1995
“A Simplified Model of a Supercritical Power Plant”
University of Michigan
Li H.
,
Huang X.
,
Zhang L.
2008
“A lumped parameter dynamic model of the helical coiled oncethrough steam generator with movable boundaries”
Nuclear Engineering and Design
238
1657 
1663
DOI : 10.1016/j.nucengdes.2008.01.009
Hwang J.H.
1994
“Drum Boiler Reduced Model: a Singular Perturbation Method”
20th International Conference on Electronics, Control and Instrumentation
Bologna, Italy
3
1960 
1964