energies
Article
A Framework for the Synthesis of Optimum
Operating Profiles Based on Dynamic Simulation
and a Micro Genetic Algorithm
Erik Rosado-Tamariz
1,2,
*, Miguel A. Zuniga-Garcia
1,2,
* and Alfonso Campos-Amezcua
2,
*
and Rafael Batres
1,
*
1
School of Engineering and Sciences, Tecnologico de Monterrey, Av. Eugenio Garza Sada Sur No. 2501,
Col. Tecnologico, Monterrey 64849, Mexico
2
Instituto Nacional de Electricidad y Energías Limpias (INEEL), Av. Reforma 113, Col. Palmira,
Cuernavaca CP 62490, Mexico
* Correspondence: erik.rosado@tec.mx (E.R.-T.); miguel.zugar@gmail.com (M.A.Z.-G.);
acampos@ineel.mx (A.C.-A.); rafael.batres@tec.mx (R.B.)
Received: 19 December 2019; Accepted: 7 January 2020; Published: 5 February 2020
Abstract:
This paper presents an approach to managing the thermal power plant’s flexible operation
based on the steam generation process optimization. A strategy at the process level, as a first step in
the operational optimization of the entire power plant, is proposed. The proposed approach focuses
on minimizing the drum boiler startup time, since it is considered the most critical element in the
steam generation process and in the thermal power plant’s efficient operation. An approach that
addresses the problem to find the optimal sequences of control valves that minimize the drum boiler
startup time as a dynamic optimization problem is proposed. To solve the optimization problem,
a dynamic optimization framework based on a micro genetic algorithm (mGA) coupled with a
dynamic simulation model is implemented. The dynamic simulation model is validated against data
available in the literature, and the proposed optimization algorithm is characterized by the use of
variable length chromosomes and the use of small population sizes. The results show that optimized
operating profiles minimize the drum boiler startup time by at least 35 percent and generate control
valve operating sequences that must be carried out to achieve the desired profile, while the structural
integrity constraints are fulfilled at all times.
Keywords:
thermal power plant; steam generation process; dynamic simulation; OpenModelica;
optimum operating profiles; micro genetic algorithm
1. Introduction
In order to achieve a reliable energy supply, the balance of electricity supply and demand must
be guaranteed at all times. Even a small mismatch in the balance may disturb the power system
frequency and affect its reliability and availability [
1
]. Furthermore, an optimal co-existence of both
conventional and renewable technologies is required. The operational flexibility of the electric power
system plays a key role in achieving such balance. This operational flexibility is the capability of a
power system to cope with the expected and unexpected changes in electricity demand and energy
supply [
2
]. The electric power system operational flexibility can be addressed from different fronts
such as demand response, energy storage, and flexible generation [
3
]. Demand response refers to
demand-side management programs, in which demand patterns are fitted to better match the changes
in the electricity supply [
4
]. Likewise, demand response provides consumers an opportunity to play a
role in the operation of the grid by regulating their electricity consumption. To best balance electricity
demand and energy supply through demand management, mainly demand and renewable energy
Energies 2020, 13, 677; doi:10.3390/en13030677 www.mdpi.com/journal/energies
Energies 2020, 13, 677 2 of 23
forecasting methods are used, as well as smart grid designs [
5
]. Operational flexibility can also be
achieved by means of energy storage. Energy storage relates to the accumulation of hydro, electrical,
or thermal energy to later be used. Even though hydro storage is dominant, it is mainly limited by its
geographic location and water availability, limiting its growth. Regarding the electrical and thermal
energy storage, several technologies are available on the market with different degrees of maturity,
capacity, and discharge duration. However, cost effective and commercially available large scale
energy storage technology is yet to be developed [6].
Flexible generation focuses on satisfying the residual load of the electric power system through
conventional power plants. The residual load is represented by subtracting the feed-in of renewable
energy sources from the load profile [
7
]. Flexible generation can be improved by redesigning
critical power plant components, identifying new market mechanisms, and defining new operational
strategies [8].
According to data from the International Energy Agency [
5
], in the short term to medium
term, flexible generation will be considered as the most effective solution to deliver power system
operational flexibility. Since it is a more mature approach, it is less constrained by capacity and
geographic location than demand response and energy storage. In this regard, the evaluation of
flexible generation capacity in conventional thermal power plants has been reported in previous works
such as those by Kubik et al. [
9
], Hentschel and Spliethoff [
10
], and Gonzalez-Salazar et al. [
3
], where
they concluded that the most suitable power plants to provide operational flexibility to the electric
power system are the simple cycle gas turbine, high efficiency coal fired, and combined cycle power
plants. Likewise, they identified that the development trend of these power plants is focused on
operational improvement in terms of increasing ramp rates, decreasing minimum power load, and
the development of the improvements of cyclic operational capabilities. In this context, research such
as that developed by Casella et al. [
11
], Almodarra et al. [
12
], Anisimov et al. [
13
], Rossi et al. [
14
],
Mei Ji et al. [
15
], and Liu and Karimi [
16
] addressed the flexible generation challenge through the
conventional thermal power plants operating profiles’ optimization. On the other hand, as a first step
for the analysis of the whole power plant and using a hierarchical optimization strategy, the works
of Franke et al. [
17
], El-Guindy et al. [
18
], Elshafei et al. [
19
], Belkhir et al. [
20
], and Zhang et al. [
21
]
proposed to manage the operation of the power plant using a process level approach. Since it is widely
known that the steam generation process is one of the most important in thermal power plants, they
focused their efforts on designing optimal steam generation profiles, obtaining encouraging results in
their research.
In thermal power plants, steam generation is carried out through a steam generator or a boiler.
The steam generator is a device that produces high pressure and high temperature steam for energy
generation. This process is carried out by transferring heat from flue gases from a furnace or a gas
turbine exhaust to water contained in the riser and downcomer waterwalls in order to produce steam
through a boiling process. Then, in a pressure vessel known as a drum boiler, the saturated steam is
separated from liquid water. The steam is dried inside the drum boiler and sent to the superheater to
be heated above the saturation temperature, then it is piped through the main steam lines to the steam
turbine in order to produce electrical power. Due to its important function in the steam generation
process, the drum boiler is considered the most critical element in the steam generator, since it is in
this equipment where the steam quality and steam flow rate that influence the generation of energy
in the steam turbine are regulated. The steam generator typically is composed of the drum boiler,
mud drum, a water circulation loop, feedwater system, and recirculation pumps, as well as a thermal
energy supply system. The water circulation loop is composed of the downcomer and riser waterwalls.
A drum boiler’s basic configuration is illustrated in Figure 1.
The heat supply and feedwater flow rate are the main controlled variables that determine the
behavior of the steam generation process. Therefore, their operation must be improved in order to
avoid dangerous operating scenarios, such as boiler swell/shrink phenomena and overheating of
thick-walled devices. Boiler swell/shrink phenomena are produced when a sudden change in the
Energies 2020, 13, 677 3 of 23
steam pressure causes the steam drum to go into violent oscillation, which can cause the boiler to
trip [
19
]. Likewise, boiler swell/shrink is related to an instant change of the liquid water level in the
drum boiler, induced by steam quality variations in the riser waterwall [
19
]. Overheating is related to
a fast heat supply, which induces severe temperature variations in the thick-walled devices causing
high thermal stresses. Since these stresses lead to fatigue or even material failures, it needs to be held
within given limits [
22
]. Therefore, approaches are needed to obtain operating profiles that minimize
the drum boiler startup time to satisfy the steam demand required by the power plant while reducing
thermal stresses.
Figure 1. A drum boiler’s basic configuration.
This paper is structured as follows: Section 2 contains the literature review. A description of
the proposed approach based on dynamic simulation and a micro genetic algorithm is presented in
Section 3. Next, the case study is described in Section 4. Then, in Section 5, the experiments and results
are presented. Finally, Section 6 contains the conclusions and future work of this research project.
2. Literature Review
Researchers have focused their attention on studying dynamic simulation and optimization of
steam generators in order to improve the flexible generation capabilities of thermal power plants.
Specifically, previous work focuses on the study of power plant transient behavior as a means to
propose and design optimal operational strategies. To that end, dynamic simulation models were
developed based on mass, momentum, and energy conservation laws.
Early simulation models of power plants steam generators were based on a simple nonlinear
boiler turbine unit. Some of these models were proposed by Astrom and Euckland [
23
]. Astrom
and Bell [
24
] improved the linear model developed by Astrom and Euckland and validated it with
experimental data, finding that the model was capable of capturing the major dynamical behavior of
the system. Later, Peet and Leung [
25
] proposed a dynamic simulation model to design a drum boiler
based on the requirements of a conventional thermal power plant’s operation to achieve flexible and
economic production of steam. Subsequently, Bell and Astrom [
26
] developed a nonlinear model of a
drum boiler based on the principles of their initial model, and its transient performance was validated
against real power plant data. In the control arena, Flynn and O’Malley [
27
] developed a drum boiler
Energies 2020, 13, 677 4 of 23
dynamic simulation model and used it in the study and design of a new control philosophy to meet the
operational requirements of a large fossil fuel power plant. Over the years, drum boiler models grew
in complexity and accuracy, mainly by replacing empirical coefficients with real operating parameters.
A commonly cited model is that of Astrom and Bell [
28
], which was a nonlinear dynamic model
in which the downcomer, riser, and drum dynamic behaviors were based on a global balance of
conservation laws and required few physical parameters to have a simple and robust system.
Subsequently, Wen and Ydstie [
29
], El-Guindy et al. [
18
], Elshafei et al. [
19
], Belkhir et al. [
20
], and
Zhang et al. [
21
] developed simulation models based on the theoretical model proposed by [
28
], using
advanced modeling and simulation techniques.
Regarding the operation of power plants, several works have been reported in the literature
that dealt with the optimization of steam generation from a process point of view. Franke et al. [
17
]
developed a nonlinear dynamic model of a drum boiler based on the Modelica language using
the fluid libraries [
30
]. Their model had three control inputs in terms of feedwater flow rate, heat
supply, and steam outlet. They solved a dynamic optimization problem using a sequential quadratic
programming (SQP) algorithm. Using this approach, the startup time could be reduced by about
30%. Kruger et al. [
31
] proposed a quadratic programming optimization approach to determine the
optimal values of steam pressure and steam temperature in a startup process. Their model took into
account hard constraints such as control bounds and stress levels for the drum and header. They
concluded that their optimization model was capable of minimizing both fuel consumption and startup
time. Li et al. [
32
] developed a drum boiler startup simulation program that focused on reducing the
operational time and minimizing fuel consumption during startup or shutdown scenarios. They used a
distributed parameter method to simulate the heat transfer process in the waterwalls, the superheater,
the reheater, and the economizer, while heat transfer in the drum and the downcomer was simulated
by lumped parameter analysis. Moreover, Belkhir et al. [
20
] minimize the startup time of a steam
generator. The proposed startup strategy was focused on achieving reference state variables in terms
of steam mass flow rate and the pressure inside the drum to fulfill the steam requirements in the power
train. The startup process was formulated as an optimal control problem that focused on minimizing
a quadratic objective function under physical and operational constraints. The physical constraints
were related to the structural integrity of thick-walled components due to higher thermal stresses.
The drum boiler model was developed in the commercial Modelica environment Dymola using the
fluid and thermal libraries. A framework developed on the JModelica environment and interior point
optimizer algorithm (IPOPT) was used to solve the optimization problem. Their results were compared
against a classical startup strategy, and the optimized profiles reached desired states in a shorter time
without violating the operational and physical constraints. Zhang et al. [
21
] presented a numerical
investigation on the dynamic analysis of the steam and water system of the natural circulation boiler
developed in the environment of MATLAB/Simulink. They proposed a boiler modeling based on
the Astrom–Bell model with specific parameters to simulate the dynamic analysis of the steam-water
system. Their model assumed that steam was saturated along with the whole evaporating system.
They solved the model using the ode45 algorithm, which is based on the fourth-order Runge–Kutta
and Dormand–Prince methods. The boiler startup was formulated to get a better curve of the startup in
order to save water and fuel. The input parameters were heat flow, the mass flow rate of steam, and the
mass flow rate of feedwater, which were changing with time. After that, they brought about a plan for
a cold startup, obtaining a cold start-up curve that could be used as a reference for practical production.
In general, there has been some research focused on optimizing the drum boiler startup process
with the aim to enhance the operating capabilities of thermal power plants. Likewise, advanced
techniques of simulation, control, and optimization have been used to solve the optimization problem
and thereby minimize the transition time in the operation of thermal power plants. Despite this,
previous work previous has not taken into account the identification and generation of actions required
to reach this objective and thus having an integral design of optimal operating procedures.
Energies 2020, 13, 677 5 of 23
Investigations such as those reported by [
17
,
20
,
21
] showed encouraging results; however, they
had limited applicability since their objective was to minimize startup times considering thermal stresses
as the main constraint, regardless of how they must operate the drum boiler to achieve the goal states of
the critical state variables. On the other hand, reported works with interesting proposals [
21
,
33
,
34
] limited
their applicability, since they were developments for specific problems; in many cases, the simulation
model was embedded within the optimization tool and it was not possible to scale them for more
complex problems such as a power plant. In the same way, the works in [
20
,
33
,
35
,
36
] propose
approaches using commercial tools for the coupling of a simulation optimization integral system.
The drawback is that these tools operate as black boxes, which limited the development and scaling of
the research carried out. Moreover, they are also limited to a certain type of optimization algorithms.
Although gradient based methods such as [
37
,
38
] can solve dynamic optimization problems
quickly, they have three key weaknesses: difficult implementation (depending on the problem);
intolerance to noisy objective function spaces; and location of local optimum solutions rather than a
global optimum [39].
To overcome the limitations of previous works, the present research aims to design, in an integral
way, the drum boiler startup optimal operating procedure in order to enhance the operational capacities
of thermal power plants. A dynamic optimization framework based on a micro genetic algorithm
(mGA) coupled with a dynamic simulation model is proposed. The integral design of the optimal
operating procedure involves the operating profiles that minimize the drum boiler startup times
to satisfy the steam demand required by the power plant, as well as the corresponding control
actions (operations) and their sequence to take the drum boiler from an initial state to a goal state.
A metaheuristic optimization algorithm characterized by the uses of variable length chromosomes
and the use of small population sizes is implemented. A scalable dynamic simulation model using the
modeling and simulation environment OpenModelica is developed. Likewise, an open interface based
on the C# code is developed in order to connect the dynamic simulator with the optimization module.
3. Proposed Approach
3.1. Problem Statement
The problem consisted of finding the optimal sequence of control valves that minimized the time
needed to take the drum boiler from an initial state to a goal state. This problem was formulated as a
dynamic optimization problem, involving a dynamic simulation model with variables whose values
changed in time. The optimization was constrained by thermal stress in the thick-walled drum, caused
by a spatial temperature difference.
A typical drum boiler configuration consists of a water circulation loop and a heat energy
system [
40
]. The water circulation loop is composed of a steam drum, mud drum, the downcomer
waterwalls tubes, and the riser waterwalls tubes. The steam drum is the top drum of a boiler where all
of the generated steam is collected before entering the distribution system. The steam drum has the
function of controlling the steam generator water level since the loss of water level can damage boiler
equipment, and excessively high water levels can result in wet steam, which can cause operational
upsets. The mud drum is the lower drum in a boiler. The mud drum is filled completely with water
and has the function of a settling point for solids in the boiler feedwater. Sediment accumulated in the
bottom of the mud drum is removed by water blowdown. Downcomer waterwalls are a set of pipes
leading from the top to the bottom of the drum boiler, and through them, the water is transferred from
the steam drum to the mud drum. The downcomer is the cooler water line that goes from the upper
drum to the lower drum. Riser waterwall tubes contain boiler feedwater that is heated by radiant
heat from the flue gas and boiled to produce steam that flows upward to the steam drum. The riser
is the hotter water line that goes from the mud drum to the steam drum. The heat energy system
supplies heat from the flue gases to the water flowing down the riser waterwall tubes in order to
regulate the boiling process. Gravity induces the saturated steam to rise, leading to circulation in the
Energies 2020, 13, 677 6 of 23
riser-drum-downcomer loop. Through a centrifugal pump, the feedwater is supplied to the steam
drum, and saturated steam is taken from the drum through a control valve to the superheaters and the
turbine. A 3D model of a typical drum boiler configuration being applied to a 500 MW thermal power
plant and a saturated steam mass-flow rate of 185 kg/s is shown in Figure 2.
Figure 2.
A 3D model of a typical drum boiler configuration. mGA, micro genetic optimization
algorithm.
3.2. Proposed Framework
In order to address the problem of finding the optimal control valve sequences that minimize the
drum boiler startup time, a dynamic optimization framework based on a micro genetic optimization
algorithm (mGA) coupled with a dynamic simulation model was proposed. The architecture of the
framework is shown in Figure 3.
Figure 3. Implementation of the framework.
This framework consisted mainly of four modules: a simulator developed on the OpenModelica
environment, the optimizer, a solution generator, and the evaluator, which were implemented in C#
code. Likewise, an interface based on C# code was developed in order to connect the drum boiler
simulator with the framework optimization modules.
Energies 2020, 13, 677 7 of 23
To solve the dynamic optimization problem, the optimizer executes the micro-genetic algorithm
and interacts with the solution generator, submitting requests for new solutions. The main role of the
solution generator is to create an individual, which represents an operation profile. Each individual
is composed of three chromosomes, which contain information about the control valve positions,
the valve positions times, and the number of repetitions of each valve position. For each individual
that is generated, the solution generator creates a file that contains the valves operating sequences of the
heat supply and steam flow rate at the outlet of the drum boiler. Then, the C# interface translates this
file to Modelica code and merges it with the Modelica.mo file of the simulator. The Modelica.mos file
contains the simulator script and has the function of running the simulations using the OpenModelica
OMC compiler. The simulation results file .mat through the C# interface is sent to the evaluator module
in order to calculate the objective function and evaluates the constraints. This information is then
passed to the solution generator to continue the cycle of the framework through the mGA algorithm
until achieving the stop criteria.
3.3. Simulation Model
The simulation model was based on the Astrom and Bell model [
28
]. The model assumed a global
mass balance and water co-existence in two phases: liquid and steam inside the drum, as well as a
water thermodynamic state at the phase boundary. Since the drum boiler had complex configurations
and geometries in this model, global system flows, volumes, and masses were considered. The model
ignored spatial variations in the process variables such as individual geometric features and fin and
pipes arrangements in the risers and downcomers. Moreover, this model did not consider heat losses
between the water inside the drum and the drum and pipes’ metal walls. Therefore, it was assumed
that the water and metal temperatures were in thermodynamic equilibrium within the drum. Despite
these simplifications, the resulting lumped parameter model was capable of capturing the overall
behavior of the drum boiler. The behavior of the boiler furnace in a coal fired power plant or exhaust
gases of a gas turbine was modeled by means of a heat supply system that heated and evaporated the
water in the rising tubes.
The global mass balance in the drum boiler is shown in Equation (1):
d
dt
[ρ
s
V
st
+ ρ
w
V
wt
] = q
f
− q
s
(1)
where
ρ
s
is the specific steam density,
V
st
is the total system steam volume,
ρ
w
is the specific water
density,
V
wt
is the total system water volume,
q
f
is the feedwater mass flow rate, and
q
s
is the steam
mass flow rate.
The global energy balance in the drum boiler can be written as:
d
dt
[ρ
s
h
s
V
st
+ ρ
w
h
w
V
wt
− pV
t
+ m
t
C
p
t
m
] = Q + q
f
h
f
− q
s
h
s
(2)
where
h
s
is the specific steam enthalpy,
h
w
is the specific water enthalpy,
p
is the mixture pressure,
V
t
is
the total system volume,
m
t
is the total mass of the metal tubes and the drum,
C
p
is the specific heat of
the metal, Q is the heat supplied to the tube, and h
f
is the specific feedwater enthalpy.
The total volume of the drum, downcomer, and riser (
V
t
) is determined by the total steam and
water volumes as shown below:
V
t
= V
st
+ V
wt
(3)
The global mass and energy balance for the riser section is represented by Equations (4) and (5),
respectively:
d
dt
[ρ
s
α
v
V
r
+ ρ
w
(1 − α
v
)V
r
] = q
dc
− q
r
(4)
d
dt
[ρ
s
h
s
α
v
V
r
+ ρ
w
h
w
(1 − α
v
)V
r
− pV
r
+ m
r
C
p
t
s
] = Q + q
dc
h
w
− (α
r
h
c
+ h
w
)q
r
(5)
Energies 2020, 13, 677 8 of 23
where
α
v
is the average volume fraction,
V
r
is the riser volume,
m
r
is the riser mass,
t
s
is the steam
temperature,
q
dc
is the downcomer flow rate,
α
r
is the steam quality at the riser outlet, and
h
c
=
h
s
−
h
w
is the condensation enthalpy.
The momentum balance for the downcomer-riser loop is:
(L
r
− L
dc
)
dq
dc
dt
= (ρ
w
− ρ
s
)α
v
V
r
g −
k(q
dc
)
2
2ρ
w
A
dc
(6)
where
L
r
is the riser lengths,
L
dc
is the downcomer lengths,
A
dc
is the downcomer area, and
k
is a
dimensionless friction coefficient.
The mass balance for the steam under the liquid level in the steam drum is:
d
dt
(ρ
s
V
sd
) = α
r
q
r
− q
sd
− q
cd
(7)
where
V
sd
is the volume of steam under the liquid level in the drum,
q
sd
is the steam flow rate through
the liquid surface in the drum, q
r
is the flow rate out of the risers, and q
cd
is condensation flow.
The proposed simulation model formulation assumed a thermodynamic equilibrium between
water and steam inside the steam drum. Feedwater from the condenser enters the steam drum, and
saturated steam is extracted.
The behavior of condensation flow in the drum and the steam flow rate through the liquid surface
in the drum are given by Equations (8) and (9):
q
cd
=
h
w
− h
f
h
c
q
f
+
1
h
c
ρ
s
V
sd
dh
s
dt
+ ρ
w
V
wd
dh
w
dt
+ (V
sd
− V
wd
)
dp
dt
+ m
d
C
p
dt
s
dt
(8)
q
sd
=
ρ
s
T
d
(V
sd
− (V
sd
)
0
) + α
r
q
dc
+ α
r
β(q
dc
− q
r
) (9)
where
V
wd
is the volume of water under the liquid level in the drum,
m
d
is the mass in the drum, (
V
sd
)
0
denotes the volume of steam in the drum in the hypothetical situation when there is no condensation
of steam in the drum, and
T
d
is the residence time of the steam in the drum, which is approximated by:
T
d
=
ρ
s
(V
sd
)
0
q
s
(10)
From the distribution of the steam below the drum level, the drum level can be modeled using
the equation of water in the drum:
V
wd
= V
wt
− V
dc
− (1 − α
v
)V
r
(11)
Since the drum has a complex geometry configuration, the liquid level changes can be described
by the wet surface
A
d
at the operating level. The deviation of the drum level
l
measured from its
normal operating level is:
l =
V
wd
+ V
sd
A
d
= l
w
− l
s
(12)
The term
l
w
represents level variations caused by changes in the amount of water in the drum,
and the term l
s
represents variations caused by the steam in the drum.
In summary, the state variables that describe the behavior of the system are: drum pressure
p
,
total water volume
V
wt
, steam quality at the riser outlet
α
r
, and volume of steam under the liquid
level in the drum
V
sd
. The parameters required by the model are: drum volume
V
d
, riser volume
V
r
,
downcomer volume
V
dc
, drum area
A
d
at the normal operating level, total metal mass
m
t
, total riser
mass
m
r
, friction coefficient in downcomer-riser loop
k
, residence time
T
d
of steam in the drum, and
parameter β in the empirical equation steam flow rate through the liquid surface in the drum q
sd
.
Energies 2020, 13, 677 9 of 23
3.4. Thermal Stress Modeling
The thermal stresses were determined according to the general equations for radial, tangential,
and axial thermal stresses in a thick-walled pressure vessel under a radial thermal gradient shown in
the work of Mirandola et al. [41]:
σ
r
=
αE
(1 − µ)r
2
r
2
− a
2
b
2
− a
2
Z
b
a
Trdr −
Z
r
a
Trdr
(13)
σ
t
=
αE
(1 − µ)r
2
r
2
+ a
2
b
2
− a
2
Z
b
a
Trdr −
Z
r
a
Trdr − Tr
2
(14)
σ
a
=
αE
(1 − µ)
2
b
2
− a
2
Z
b
a
Trdr − T
(15)
σ
VM
=
q
(σ
1
)
2
+ (σ
2
)
2
+ (σ
3
)
2
− σ
1
σ
2
− σ
1
σ
3
− σ
2
σ
3
(16)
where
σ
r
,
σ
t
, and
σ
a
are the radial, tangential, and axial stresses, respectively.
α
is the thermal expansion.
E
is Young’s modulus.
µ
is Poisson’s ratio.
r
is a variable radius.
a
is the inside radius.
b
is the outside
radius.
σ
VM
is the von Mises stress.
σ
1
,
σ
2
, and
σ
3
are the main stresses.
T
represents the metal
temperature, which is considered equivalent to the water saturation temperature inside the drum.
3.5. Optimization Algorithm
To solve the drum boiler startup problem, we used the micro genetic algorithm (mGA) proposed
by Batres [
34
]. Like a traditional genetic algorithm (GA), a micro genetic algorithm (mGA) solves
optimization problems with or without constraints, using small populations of individuals (solutions)
based on a natural selection process that emulates biological evolution. The operation diagram of the
mGA is shown in Figure 4. It consisted of an outer loop and an inner loop. The outer loop consisted of
creating a new random population, transferring the best individual from the inner loop and restarting
the inner loop. The amount of individuals that formed the random population was a parameter of the
algorithm. The traditional genetic algorithm was used as an inner loop, consisting of the evaluation of
the fitness of each member of the population, the selection of parent chromosomes, the generation of a
new population by means of the crossover and mutation operations, and the separation of the best-fit
individual after convergence.
Figure 4. Operation diagram of the mGA.
In this paper, each cycle in which the inner loop was restarted is called an epoch, and every cycle
of the inner loop is called a generation. The following is a detailed explanation of each of the steps in
the mGA.
Energies 2020, 13, 677 10 of 23
Generate a random population: In this step, a random group of individuals with the correspondent
format was generated. It is worth mentioning that the individuals had a specific format designed for
each problem. The format for this problem was a three chromosome configuration. This configuration
is explained in detail later. In Figure 5, an illustration of a random generated population is shown.
Figure 5. Example of a random population of four individuals.
Use crossover and mutation operators to generate a new population: The proposed mGA
implemented two genetic operators, which are controlled by the crossover probability and the mutation
probability, respectively. The crossover probability is a parameter that determines how often the
crossover will be performed. Similarly, the mutation probability determines the frequency in which a
mutation occurs. The crossover operator firstly selects two individuals (mom and dad individuals).
These individuals are selected based on the roulette-wheel scheme [
42
]. Then, it selects two random
genes in the chromosomes (which cannot be the first or the last gene). Then, the mom and dad
individuals are split by the selected genes into three pieces. Then, from the pieces of the mom and
dad individuals, two new individuals are formed (daughter and son). Son and daughter individuals
have the middle part of mom and dad, respectively. Then, the left and right part of the dad individual
becomes the left and right part of the daughter individual, and the left and right part of the mom
individual becomes the left and right part of the son individual. In Figure 6, a graphical representation
of this process is presented.
Figure 6. Graphical representation of the crossovergenetic operator in the mGA.
Energies 2020, 13, 677 11 of 23
When a mutation occurs, mGA first selects one individual. This individual is also selected by the
roulette-wheel scheme [
42
]. Then, it selects two random genes in the chromosomes (which cannot be a
gene with the repetitions = 0). Then, the selected genes inside the individuals are swapped. In Figure 7,
a graphical representation of this process is presented.
Figure 7. Graphical representation of the mutation genetic operator in the mGA.
In Figure 8, an example is given to illustrate how the population in Figure 8 was modified using
the crossover and mutation operators.
Figure 8. The new population after the crossover and mutation operators.
Evaluate the fitness of each member of the population: In this step, every member (individual)
of the population is evaluated by means of a fitness function. The fitness function is related to the
performance of the individual in terms of the objective function. In Figure 9, an illustration of the
fitness result associated with each member of the population is shown.
Figure 9. An example of the fitness associated with each member of the population.
Energies 2020, 13, 677 12 of 23
Select the best individual: If in an iteration of the inner loop, a better individual is found, that
individual is labeled as the new best individual. In Figure 10, an illustration of how the best individual
is selected is shown.
Figure 10. An example of the selection of the best individual of the population.
Create a new random population, and insert the best individual: In this step, the best individual
found in the inner loop is inserted into a new random population to be used in the outer loop.
In Figure 11, an illustration of how the best individual is inserted into the new random population
is shown.
Figure 11. An example of the insertion of the best individual into the new random population.
In the proposed mGA algorithm, each individual is represented as an ordered list of operations.
As in other genetic algorithms, each candidate solution in the population (an individual) is represented
by a data structure called the chromosome. However, this paper proposes a three chromosome
structure. The solution is represented by means of a three chromosome data structure. The first
chromosome represents the sequence of valve actions. The second chromosome represents the action
duration, and the third chromosome represents the number of times that the same action is repeated.
In Appendix A, the pseudocode of the mGA is shown.
The action duration is the execution time associated with each action for which valve positions
are kept unchanged. The repetition parameter shows the number of times that action is carried out.
Both the valve position and the action durations are discretized. An indexed list was created containing
Energies 2020, 13, 677 13 of 23
the possible combinations between valve openings and action durations. Figure 12 shows an example
of an individual represented by the proposed three chromosome scheme.
Figure 12. Example of an individual represented by three chromosomes.
4. Case Study
The proposed approach is illustrated with a case study that focused on the generation of operation
profiles and their sequences of control valve operations in a drum boiler of a thermal power plant.
A simulation model was developed using the modeling and simulation environment OpenModelica [
43
].
Figure 13 shows the drum boiler model in the OpenModelica OMEdit graphic environment.
Figure 13. Drum boiler simulator: OpenModelica.
The drum boiler model was validated by executing the reference startup sequence published by
Belkhir et al. [
20
] and comparing the pressure, temperature, thermal stress, heat supplied, steam flow,
and steam flow regulation profiles. Figure 14 shows the reference results reported by Belkhir et al. [
20
]
and those obtained with our model implementation, respectively. From the comparison of these
profiles, it could be concluded that the simulation of the implemented model with OpenModelica
was validated.
Energies 2020, 13, 677 14 of 23
Figure 14. Drum boiler simulation results. Belkhir et al. [20] (blue line); proposed approach (red line).
The optimization problem focused on the design of an optimal drum boiler startup profile with its
corresponding control valve operation sequence. The objective was to achieve a given state in terms of
steam temperature and pressure in the shortest time possible. Based on the work of Belkhir et al. [
20
]
and Franke et al. [
17
], we formulated the optimization problem in terms of the drum-boiler internal
pressure and the output steam mass-flow rate of the drum boiler. Therefore, the formulation of the
optimization problem for the startup operating procedure can be written as follows:
Energies 2020, 13, 677 15 of 23
Objective function:
min
t
f
∑
t
0
w
1
[p(t
f
) − p
goal
]
2
+ w
2
[q(t
f
) − q
goal
]
2
dt (17)
where
p
goal
is the desired internal pressure,
q
goal
is the desired steam mass-flow rate, and
w
1
and
w
2
are the weights. Based on Belkhir et al. [
20
], the desired internal pressure and the desired steam
mass-flow rate were set to 90 bar and 185 kg/s, respectively, while
w
1
and
w
2
were set to 0.01 and 0.001,
respectively. Model constraints must be fulfilled any time during the optimization horizon
t ∈ [t
0
,
t
f
]
.
The behavior of the system was controlled by regulating the heat flow at the input of the drum boiler
and the output flow rate of the steam that was extracted from the drum boiler:
0 ≤ Q ≤ 500MW (18)
0 ≤ V
pos
≤ 1 (19)
where
Q
is the heat flow and
V
pos
is the valve position (in percent) of the steam output valve. In order
to avoid sudden changes in the state variables, the heat flow rate (dQ/dt) is constrained as follows:
0
MW
min
≤
dQ
dt
≤ 25
MW
min
(20)
In the same way, a feasible solution must be able to achieve the objective by fulfilling the process
constraints. In this context, we set up inequalities constraints that determined the state variables
operational range such as temperature and pressure inside the drum boiler:
p(t
f
)
min
< p( t
f
) < p(t
f
)
max
(21)
T(t
f
)
min
< T(t
f
) < T(t
f
)
max
(22)
These inequalities set the state variables limits and the mix quality in the drum boiler, for which it
must be fulfilled that
p(t
f
)
min
≥ p
atm
,
p(t
f
)
max
=
p
nom
,
T(t
f
)
min
≥ T
eco
, and
T(t
f
)
max
=
T
nom
, where,
p
atm
is the atmospheric pressure,
T
eco
is the temperature in the economizer,
p
nom
is the full load nominal
pressure, and T
mom
is the full load nominal temperature.
Moreover, an output constraint was introduced in order to avoid large thermal stresses in the
drum boiler thick wall.
− 10
N
mm
2
≤ σ
VM
≤ 10
N
mm
2
(23)
In order to include this constraint evaluation, the objective function (Equation (17)) was modified
by means of a penalty function (PF):
PF =
(
if PenNum ≥ 1 W
1
t
f
+ 50(PenNum)
2
+ 50(PenStress)
2
+ W
2
(OF)
Otherwise W
1
t
f
+ W
2
OF
(24)
where
W
1
=
0.01,
W
2
= (
1
− W
1
)
,
OF
is the result of the objective function,
PenNum
is the amount
of times the individual violates the constraint, and
PenStress
is the accumulated stress violation.
Both
PenNum
and
PenStress
were evaluated along with the dynamic simulation from the initial state
to the final state.
According to the data structure of the micro genetic algorithm, the solution is represented by three
chromosomes. The first chromosome represents the sequence of valve actions. The second chromosome
represents the execution time per action, and the third chromosome represents the number of times
that the same action is repeated. Each element in the first chromosome is an integer that points to a
Energies 2020, 13, 677 16 of 23
combination of the valve position of the steam outlet valve and the heat flow rate. Table 1 shows the
set of actions considered for this problem.
Table 1. Combinations of the heat and steam valve of each action.
Action dQ/dt V
pos
1 8 0.0
2 8 0.6
3 8 1.0
4 16 0.0
5 16 0.6
6 16 1.0
7 24 0.0
8 24 0.6
9 24 1.0
To solve the optimization problem, the micro genetic algorithm (mGA) was implemented together
with an interface that connects to the drum boiler simulation model in OpenModelica. Both the
algorithm and the interface were implemented in C# code.
5. Experiments and Results
The experiments considered three action durations (60, 120, and 180 s) and three valve positions.
As a result, eight different actions were obtained, resulting from the combination of three valve
positions for the two valves (the case of both valves closed was not considered). The repetition
parameter was set to take integer values from 0 to 10. The mGA probabilities used in the numerical
experiments were 10% for mutation and 20% for crossover. The population of the mGA consisted of
5 individuals, and the termination criteria were set to a maximum of 40 generations and 20 epochs,
respectively. Figure 15 shows the global objective function convergence for the optimization problem,
as well as the goal states convergence in terms of the desired drum boiler internal pressure and the
desired steam mass-flow rate.
Figure 15. mGA objective function convergence.
All the experiments were carried out on a 3.4 GHz Intel Xeon E3-1245 V2 computer with 16 GB of
RAM, running Windows 10 pro.
Energies 2020, 13, 677 17 of 23
Figures 16–21 show the comparison of the results of this research with those obtained by [
17
,
20
].
In these figures, red lines correspond to the reference model, while the green and blue lines are the
results presented by [
17
,
20
], who solved the optimization problem using a nonlinear model predictive
control (NMPC) and the interior point method (IPOPT), respectively. The black lines correspond to the
optimized drum boiler startup profile according to the dynamic optimization framework proposed in
this research.
Figures 16 and 17 show the comparison of the operating profiles. Figures 18 and 19 show the
operation of the valves that control the heat supplied to the system and the steam flow that is sent
to the power train. Finally, the system structural constraint and the maximum power generated are
shown in Figures 20 and 21.
Figure 16.
Results comparison between the curves of the operating reference profile available in the
literature (red line), the optimized profiles reported by [
17
] (green line) and [
20
] (blue line), and the
proposed approach (black line) for the output flow rate of the steam that exits from the drum boiler.
NMPC, nonlinear model predictive control; IPOPT, interior point optimizer algorithm.
Figure 17.
Results comparison between the curves of the operating reference profile available in the
literature (red line), the optimized profiles reported by [
17
] (green line) and [
20
] (blue line), and the
proposed approach (black line) for the pressure in the drum boiler.
Energies 2020, 13, 677 18 of 23
Figure 18.
Results comparison between the curves of the operating reference profile available in the
literature (red line), the optimized profiles reported by [
17
] (green line) and [
20
] (blue line), and the
proposed approach (black line) for the steam regulator valve position.
Figure 19.
Results comparison between the curves of the operating reference profile available in the
literature (red line), the optimized profiles reported by [
17
] (green line) and [
20
] (blue line), and the
proposed approach (black line) for the heat flow supplied to the system.
For both state variables (Figures 16 and 17), it can be observed that the dynamic optimization
framework can achieve the desired startup goal in less time than the reference model and with a
startup time of the same order of magnitude as the optimized profiles reported by [
17
,
20
], with a
small number of iterations in the mGA optimization algorithm, as shown in Figure 15. Likewise,
the proposed framework was capable of generating the sequence of valve operations of the steam
regulation valve and the heat flow supplied to the system, which were the main controlled process
variables that determine the efficiency of the steam generation process in a thermal power plant.
As shown in Figure 18, in the reference profile, the saturated steam regulation valve that supplied
energy to the powertrain remained fully open during the entire startup process, while for the optimized
profile presented by [
17
], the steam flow regulation was carried out during the period of 1000 to 1400 s,
displaying large instabilities in the state variables. Regarding the profile optimization reported
by [
20
], the steam valve regulation was minimal, since the valve changes ranged from fully open
Energies 2020, 13, 677 19 of 23
to 98% and 99% open, with instabilities in the state variables. In contrast, the proposed dynamic
optimization framework suggests that the steam flow control valve should be operated sequentially
and gradually in order to achieve the goal state efficiently, generating stable and continuous profiles
for the state variables.
Figure 20.
Results comparison between the curves of the operating reference profile available in the
literature (red line), the optimized profiles reported by [
17
] (green line) and [
20
] (blue line), and the
proposed approach (black line) for the thick-walled von Mises stresses.
Figure 21.
Results comparison between the curves of the operating reference profile available in the
literature (red line), the optimized profiles reported by [
17
] (green line) and [
20
] (blue line), and the
proposed approach (black line) for the power generated.
In the same way, the heat supply for the reference profile was carried out in a continuous and
constant way from the beginning of the process until the system reached the goal state. For the profiles
proposed by [
17
,
20
], the heat supply was achieved by oscillating and intermittent patterns. In the case
of the proposed framework, the heat supply is continuously applied and gradually increased until the
goal state reached. The heat supply profiles of the system for all startup processes evaluated in this
paper are shown in Figure 19.
Finally, to avoid hazardous scenarios in which the proposed profiles could result in a decrease of
useful life and the structural integrity of the thick-walled components constraint must be monitored.
In this context, in the case of the optimized profiles presented by [
17
,
20
], a decrease in the useful life
Energies 2020, 13, 677 20 of 23
of the thick-walled component’s is expected since more alternating tension and compression stresses
occur in comparison to the reference profile [
20
]. The thermal stress profile generated by the proposed
dynamic optimization framework has comparable pattern, shape, and magnitude as in the reference
profile, thus useful life of the thick-walled components is similar.
6. Conclusions and Future Work
An approach to managing the thermal power plant’s flexible operation based on the steam
generation process optimization was presented. A strategy at the process level, as a first step in the
operational optimization of the entire power plant, was used. The case study focused on the drum
boiler since it was considered the most critical element in the steam generation process. This paper
proposed a dynamic optimization framework in order to find the optimum valve sequences that
minimized the startup time.
The proposed framework had four main components: optimizer, solution generator, simulator,
and evaluator. The Optimizer ran the optimization algorithm and requested solutions from the Solution
Generator. The Solution Generator proposed new solutions based on previous ones and requested the
Simulator to solve the simulation model. After each simulation, the Evaluator calculated the objective
function and evaluated the constraints.
In the proposed approach, the problem of finding the optimal sequences of control valves that
minimized the time needed to take the drum boiler from an initial state to a goal state was formulated
as a dynamic optimization problem. To solve the optimization problem, the micro genetic algorithm
(mGA) was implemented together with an interface that connected it to the drum boiler simulation
model in OpenModelica. The drum boiler simulation model was validated against the data available
in the literature. The proposed optimization algorithm was characterized by the use of variable length
chromosomes and the use of small population sizes.
The dynamic optimization framework took 18,000 s to find the optimal operation sequence,
considering the mGA optimization algorithm stop criteria and that each drum boiler dynamic
simulation was carried out in 22.5 s on average. Likewise, numeric results showed that with the
optimal operation sequence found with the proposed approach, the steam production goal was
reached in 35% less time compared to the baseline startup strategy. As future work, we will investigate
an optimization approach based on exergy or entropy. Likewise, this approach will be tested using a
gas turbine combined heat and power system coupled with mGA.
Compared to gradient based methods, genetic algorithms are easy to implement, have tolerance
of noise in the objective function and usually find a global optimum. However, genetic algorithms
have a slow convergence and need a termination criterion, which does not guarantee that a global
optimum is found every time the algorithm is executed.
In summary, the proposed dynamic optimization framework aimed at designing the operation
sequence that minimized the drum boiler startup times to satisfy the steam demand required by the
power plant, identifying the corresponding control actions and their sequence in order to design in
an integral way the optimal operating procedure, without compromising the structural integrity of
critical components. Likewise, a scalable tool was developed focused on being implemented in more
complex processes and applications, whose applications involved advanced dynamic simulation and
optimization techniques aimed at improving the designs of the operating procedures.
Author Contributions:
All the authors contributed to this research. The following are the specific contributions per
author: E.R.-T., conceptualization, formal analysis, investigation, methodology, validation, and writing, original
draft; M.A.Z.-G., formal analysis, software, investigation, and writing, original draft; A.C.-A., supervision,
validation, and writing, review and editing; R.B., methodology, supervision, writing, review and editing, project
administration, and funding acquisition. All authors read and agreed to the published version of the manuscript.
Funding:
This research was funded by the CONACYT SENER Fund for Energy Sustainability Grant
Number S0019201401.
Energies 2020, 13, 677 21 of 23
Acknowledgments:
This research is a result of the Project 266632 “Laboratorio Binacional para la Gestión Inteligente
de la Sustentabilidad Energética y la Formación Tecnológica” (“Bi-National Laboratory on Smart Sustainable Energy
Management and Technology Training”), funded by the CONACYT SENER Fund for Energy Sustainability
(Agreement: S0019201401).
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. The Micro Genetic Algorithm
The pseudocode of the micro genetic algorithm is shown as follows.
Figure A1. The pseudocode of the micro genetic algorithm.
References
1. Kirby, B. Ancillary services: Technical and commercial insights. Retrieved Oct. 2007, 4, 2012.
2.
Cochran, J.; Miller, M.; Zinaman, O.; Milligan, M.; Arent, D.; Palmintier, B.; O’Malley, M.; Mueller, S.;
Lannoye, E.; Tuohy, A.; et al. Flexibility in 21st Century Power Systems; Technical Report; National Renewable
Energy Lab. (NREL): Golden, CO, USA, 2014.
Energies 2020, 13, 677 22 of 23
3.
Gonzalez-Salazar, M.A.; Kirsten, T.; Prchlik, L. Review of the operational flexibility and emissions of gas-and
coal-fired power plants in a future with growing renewables. Renew. Sustain. Energy Rev.
2018
, 82, 1497–1513.
[CrossRef]
4.
Taibi, E.; Nikolakakis, T.; Gutierrez, L.; Fernandez, C.; Kiviluoma, J.; Rissanen, S.; Lindroos, T.J. Power System
Flexibility for the Energy Transition: Part 1, Overview for Policy Makers; IRENA: Abu Dhabi, UAE, 2018.
5.
Elzinga, D.; Baritaud, M.; Bennett, S.; Burnard, K.; Pales, A.; Philibert, C.; Cuenot, F.; D’Ambrosio, D.;
Dulac, J.; Heinen, S.; et al. Energy Technology Perspectives 2014: harnessing electricity’s Potential; International
Energy Agency (IEA): Paris, France, 2014.
6. Lott, M.; Kim, S. Technology Roadmap: Energy Storage; International Energy Agency: Paris, France, 2014.
7.
von Roon, S.; Huber, M. Modeling spot market pricing with the residual load. In Proceedings of the
Enerday-5th Conference on Energy Economics and Technology, Dresden, Germany, 16 April 2010.
8.
Glensk, B.; Rosen, C.; Schiavo, R.B.; Rabiee, S.; Madlener, R.; De Doncker, R. Economic and Technical
Evaluation of Enhancing the Flexibility of Conventional Power Plants. Available online: https://d-nb.info/
1159380031/34 (accessed on 14 December 2019).
9.
Kubik, M.; Coker, P.J.; Barlow, J.F. Increasing thermal plant flexibility in a high renewables power system.
Appl. Energy 2015, 154, 102–111. [CrossRef]
10.
Hentschel, J.; Spliethoff, H. A parametric approach for the valuation of power plant flexibility options.
Energy Rep. 2016, 2, 40–47. [CrossRef]
11.
Casella, F.; Farina, M.; Righetti, F.; Faille, D.; Tica, A.; Gueguen, H.; Scattolini, R.; Davelaar, F.; Dumur, D.
An optimization procedure of the start-up of combined cycle power plants. IFAC Proc. Vol.
2011
, 44, 7043–7048.
[CrossRef]
12.
Almodarra, S.; Alabdulkarem, A. Efficiency Optimization of Four Gas Turbine Power Plant
Configurations. In ASME 2016 Power Conference collocated with the ASME 2016 10th International
Conference on Energy Sustainability and the ASME 2016 14th International Conference on Fuel Cell
Science, Engineering and Technology; American Society of Mechanical Engineers Digital Collection, 2016.
Available online: https://asmedigitalcollection.asme.org/POWER/proceedings-abstract/POWER2016/
50213/V001T02A002/365710 (accessed on 14 December 2019).
13.
Anisimov, A.; Klub, M.; Sargsyan, K.; Eritsyan, S.K.; Petrosyan, G.; Avtandilyan, A.; Gevorgyan, A. Optimization
of Start-Up of a Fully Fired Combined-Cycle Plant with GT13E2 Gas Turbine. Power Technol. Eng.
2016
,
49, 359–364. [CrossRef]
14.
Rossi, I.; Sorce, A.; Traverso, A. Gas turbine combined cycle start-up and stress evaluation: A simplified
dynamic approach. Appl. Energy 2017, 190, 880–890. [CrossRef]
15.
Ji, D.-M.; Sun, J.-Q.; Sun, Q.; Guo, H.-C.; Ren, J.-X.; Zhu, Q.-J. Optimization of start-up scheduling and life
assessment for a steam turbine. Energy 2018, 160, 19–32.
16.
Liu, Z.; Karimi, I. Simulation and optimization of a combined cycle gas turbine power plant for part-load
operation. Chem. Eng. Res. Des. 2018, 131, 29–40. [CrossRef]
17.
Franke, R.; Rode, M.; Krüger, K. On-line optimization of drum boiler startup. In Proceedings of the 3rd
International Modelica Conference, Linköping, Sweden, 3–4 November 2003; Volume 3.
18.
El-Guindy, A.; Rünzi, S.; Michels, K. Optimizing drum-boiler water level control performance: A practical
approach. In Proceedings of the 2014 IEEE Conference on Control Applications (CCA), Juan Les Antibes,
France, 8–10 October 2014; pp. 1675–1680.
19.
Elshafei, M.; Habib, M.A.; Al-Zaharnah, I.; Nemitallah, M.A. Boilers optimal control for maximum load
change rate. J. Energy Resour. Technol. 2014, 136, 031301. [CrossRef]
20.
Belkhir, F.; Cabo, D.K.; Feigner, F.; Frey, G. Optimal startup control of a steam power plant using the
Jmodelica Platform. IFAC-PapersOnLine 2015, 48, 204–209. [CrossRef]
21.
Zhang, T.; Zhao, Z.; Li, Y.; Zhu, X. The simulation of start-up of natural circulation boiler based on the
Astrom-Bell model. In AIP Conference Proceedings; AIP Publishing: Melville, NY, USA, 2017.
22.
Taler, J.; Dzierwa, P.; Taler, D.; Harchut, P. Optimization of the boiler start-up taking into account thermal
stresses. Energy 2015, 92, 160–170. [CrossRef]
23.
Åström, K.J.; Eklund, K. A simplified non-linear model of a drum boiler-turbine unit. Int. J. Control
1972
,
16, 145–169. [CrossRef]
24.
Åström, K.J.; Bell, R.D. Simple drum-boiler models. In Power Systems: Modelling and Control Applications;
Elsevier: Amsterdam, The Netherlands, 1989; pp. 123–127.
Energies 2020, 13, 677 23 of 23
25.
Peet, W.; Leung, T. Development and Application of a Dynamic Simulation Model for a Drum Type Boiler
with Turbine Bypass System. Available online: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.
126.1142 (accessed on 14 December 2019).
26.
Bell, R.D.; Åström, K.J. A fourth order non-linear model for drum-boiler dynamics. IFAC Proc. Vol.
1996
,
29, 6873–6878. [CrossRef]
27.
Flynn, M.; O’Malley, M. A drum boiler model for long term power system dynamic simulation. IEEE Trans.
Power Syst. 1999, 14, 209–217. [CrossRef]
28. Åström, K.J.; Bell, R.D. Drum-boiler dynamics. Automatica 2000, 36, 363–378. [CrossRef]
29.
Wen, C.; Ydstie, B.E. Passivity based control of drum boiler. In Proceedings of the 2009 American Control
Conference, St. Louis, MO, USA, 10–12 June 2009; pp. 1586–1591.
30.
Elmqvist, H.; Tummescheit, H.; Otter, M. Modeling of thermo-fluid systems–Modelica. Media and Modelica.
Fluid. In Proceedings of the 3rd International Modelica Conference, Linköping, Sweden, 3–4 November 2003.
31.
Krüger, K.; Franke, R.; Rode, M. Optimization of boiler start-up using a nonlinear boiler model and hard
constraints. Energy 2004, 29, 2239–2251. [CrossRef]
32.
Li, B.; Chen, T.; Yang, D. DBSSP—A computer program for simulation of controlled circulation boiler and
natural circulation boiler start up behavior. Energy Convers. Manag. 2005, 46, 533–549. [CrossRef]
33.
Zhu, G.Q.; Yang, L.; Cheng, G. Dynamic Optimization of Ship Boiler Startup Based on Modelica and
JModelica. org. In Applied Mechanics and Materials; Trans Tech Publications: Zurich, Switzerland, 2014;
Volume 662, pp. 191–195.
34.
Batres, R. Generation of operating procedures for a mixing tank with a micro genetic algorithm.
Comput. Chem. Eng. 2013, 57, 112–121. [CrossRef]
35.
Pelusi, D.; Mascella, R.; Tallini, L.; Vazquez, L.; Diaz, D. Control of Drum Boiler dynamics via an optimized
fuzzy controller. Int. J. Simul. Syst. Sci. Technol 2016, 17, 1–7.
36.
Krishnan, P.H.; Prathyusha, S. Optimization of Main Boiler Parameters Using Soft Computing
Techniques. Available online: https://ijret.org/volumes/2014v03/i19/IJRET20140319099.pdf (accessed on
14 December 2019).
37.
Bartl, M.; Li, P.; Biegler, L.T. Improvement of state profile accuracy in nonlinear dynamic optimization with
the quasi-sequential approach. AIChE J. 2011, 57, 2185–2197, doi:10.1002/aic.12437. [CrossRef]
38.
Hong, W.; Wang, S.; Li, P.; Wozny, G.; Biegler, L.T. A quasi-sequential approach to large-scale dynamic
optimization problems. AIChE J. 2006, 52, 255–268, doi:10.1002/aic.10625. [CrossRef]
39.
Zingg, D.W.; Nemec, M.; Pulliam, T.H. A comparative evaluation of genetic and gradient-based algorithms
applied to aerodynamic optimization. Eur. J. of Comput. Mech. /Rev. Eur. Méc. Numér.
2008
, 17, 103–126.
[CrossRef]
40. Cannon, G. Process Technology Equipment; Pearson Education, Incorporated: London, UK, 2019.
41.
Mirandola, A.; Stoppato, A.; Casto, E.L. Evaluation of the effects of the operation strategy of a steam power
plant on the residual life of its devices. Energy 2010, 35, 1024–1032. [CrossRef]
42. Davis, L Handbook of Genetic Algorithms; Van Nostrand Reinhold, New York, NY, USA, 1991.
43.
Fritzson, P.; Pop, A.; Asghar, A.; Bachmann, B.; Braun, W.; Braun, R.; Buffoni, L.; Casella, F.; Castro, R.;
Danós, A.; et al. The OpenModelica Integrated Modeling, Simulation, and Optimization Environment.
In Proceedings of The American Modelica Conference 2018, Cambridge MA, USA, 9–10 October 2019; Linköping
University Electronic Press: Linköping, Sweden, 2019; pp. 206–219.
c
2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
