Instituto Tecnológico y de Estudios Superiores de Monterrey
Campus Ciudad de México
School of Engineering and Sciences
A hybrid metaheuristic optimization approach for the synthesis of operating procedures for optimal drum-boiler startups
A dissertation presented by
Emilio Garduño Hernández
Submitted to the
School of Engineering and Sciences
in partial fulfillment of the requirements for the degree of Master
In
Engineering Science
Principal Advisor: Rafael Batres Prieto
Co-advisors: Erik Rosado Tamariz and Miguel Ángel Zúñiga García
Mexico City, June 2020
3
List of Advisors This thesis was completed under the supervision of:
Dr. Rafael Batres Prieto
Instituto Tecnológico y de Estudios Superiores de Monterrey Principal Advisor
MSc Erik Rosado Tamariz
Instituto Nacional de Electricidad y Energias Limpias Co-advisor
MSc Miguel Ángel Zúñiga García
Instituto Nacional de Electricidad y Energias Limpias Co-advisor
5 Dedication
To my mother, Ma. Isabel Hernández Torres, who taught me that love and sacrifice can achieve any goal. To my father, Mario Garduño Quiñónez, who taught me the value of discipline and hard work. For both of you, know that your work and effort had made me achieve this goal.
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Acknowledgements
This thesis for the master's degree in engineering science was completed thanks to the support in the tuition granted by the Tecnologico de Monterrey and thanks to the support for living granted by the national council of science and technology (CONACYT). I am truly grateful to both institutions for trusting in my abilities and I believe that this kind of support is the basis for the future of research in Mexico.
This research was benefited by the voices, comments and ideas of academics, whom I am greatly thankful: Dr. Rafael Batres Prieto, my main advisor who encouraged me to take on this great challenge and showed me the right path for research; My co-advisors, Erik and Miguel, who helped me lay the basis for this research; My committee members, Dr. Ponce and Dr. Noguez, for their comments and guidance on this work; My colleagues in the research group Luis and Sara who gave me their best advice.
My sincere thanks to my parents, Mario and Isabel for their eternal teaching and support in this challenge; To my sister and brother, Cristina and Mario for their words and support, from a young age until now, and to my grandparents Emilio, María, Isabel and Antolín for being the root of this family. Special recognition deserves my partner and confidant Gabriela Pérez Martínez, you were with me in the best moments of this master's degree, but also in the worst, thanks to you and your family.
Thanks to my friends, who gave me encouragement, support and a moment for relax. To all those mentioned, my most sincere thanks.
7
A hybrid metaheuristic optimization approach for the synthesis of operating procedures for optimal drum-boiler startups
by
Emilio Garduño Hernández Abstract
A steam generator serves as a power generation equipment that uses the expansive power of the steam to generate electricity. The startup process of a steam generator plays an important role in the ability of a power plant to adjust its electricity generation to changes in demand. As renewable generation plants increase, the levels of variability in electricity production increase. Fast startups become instrumental as they enable traditional power generation plants to provide the quantity of electricity missing when variable renewable energies cannot satisfy demand. A main equipment involved in the startup process of the steam generator is the drum boiler.
However, if the startup process is carried out too fast, excessive thermal stresses can occur and provoke damage to the components of the drum boiler. This thesis proposes a dynamic optimization methodology to synthesize operating valve profiles that minimize the startup time of the drum boiler while avoiding the excessive formation of thermal stresses. Since valve operations influence the time-varying behavior of the steam, dynamic simulation is needed in order to evaluate the operating procedure. This thesis proposes a dynamic optimization approach with a hybrid-metaheuristic algorithm that generates the optimal startup procedure of a drum boiler. The proposed algorithm is based on two important elements of two metaheuristic algorithms. Namely, the search zone in the cooling element from the simulated annealing algorithm and the efficient computational performance provided from the tabu search algorithm memory structures. A case study evaluates the proposed approach by comparing it against results previously published in the literature.
8 List of Figures
Figure 1. Hourly demand curve throughout 2016 of the SIN Mexico, Source
PRODESEN 2016-2030 [24]. ... 17
Figure 2. Growth in energy generation by technology 2017 vs 2018 ... 18
Figure 3. Schematic representation of the drum boiler, source Aström et al. [3] ... 19
Figure 4. Proposed approach to prove hypothesis ... 22
Figure 5. Graphical representation of exploration and exploitation in a gold treasure hunt. ... 27
Figure 6. Simulated annealing algorithm ... 28
Figure 7. Tabu search algorithm ... 29
Figure 8. Proposed methodology ... 31
Figure 9. Example of five neighbors from a current solution by changing one random element of the sequence ... 33
Figure 10. operation diagram of the Hybrid metaheuristic algorithm ... 35
Figure 11. Operation diagram of the initial solution generation algorithm ... 37
Figure 12. Schematic diagram of the drum boiler, source Aström et al. [3] ... 39
Figure 13. Simulation model graphical representation in the OMEdit Modelica software ... 42
Figure 14. Solution representation of an operating procedure for the drum boiler optimization problem ... 45
Figure 15. Mutation process from one to four mutations ... 46
9
Figure 16. probability of selecting a better or a worse solution (50 seconds worse) using different γ values, depending on the progress of a run of 1000 iterations. ... 48 Figure 17. Goal state value over time with the reference startup operating procedure.
... 50 Figure 18. Thermal stress in the drum boiler through time with the reference startup profile. ... 50 Figure 19. Comparison of the reference startup time (blue line) with the startup time obtained with the hybrid metaheuristic algorithm (green line) ... 52 Figure 20. Comparison of the Belkhir´s startup time (blue line) with the startup time obtained with the hybrid metaheuristic algorithm (green line) ... 52 Figure 21. Comparison of the Belkhir´s water level (blue line) with the startup time obtained with the hybrid metaheuristic algorithm (green line) ... 53 Figure 22. Modified tabu search algorithm ... 54 Figure 23. Modified simulated annealing algorithm ... 55
10 List of Tables
Table 1. Comparison between previous work in startup optimization and solution proposed. ... 26 Table 2. Operation translation to heat flow and valve position ... 44 Table 3. Number of mutations according to the feasibility function of the operating procedure ... 47 Table 4. Best operating procedure obtained with the hybrid metaheuristic algorithm and a randomly generated initial-solution. ... 51 Table 5. Valve operation translated from the operating procedure in table 4. ... 51 Table 6. Experiments design for experiment B ... 56 Table 7. Operating procedure for initial representative solution for startup time of the drum boiler. ... 57 Table 8.Valve operation translated from the operating procedure in table 7. ... 58 Table 9. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the hybrid algorithm, starting with a representative initial solution. ... 58 Table 10. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the tabu search algorithm, starting with a representative initial solution. ... 59 Table 11. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the simulated annealing algorithm, starting with a representative initial solution. ... 59
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Table 12. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the hybrid algorithm, starting with a random initial solution. ... 60 Table 13. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the tabu search algorithm, starting with a random initial solution. ... 61 Table 14. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the simulated annealing algorithm, starting with a random initial solution. ... 61 Table 15. Operating procedure for initial feasible solution for startup time of the drum boiler. ... 62 Table 16. Valve operation translated from the operating procedure in table 15. ... 63 Table 17. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the hybrid algorithm, starting with an initial feasible solution ... 63 Table 18. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the tabu search algorithm, starting with an initial feasible solution. ... 64 Table 19. Results obtained with the synthesis of the operating procedure for the minimize of the startup time for a drum boiler using the simulated annealing algorithm, starting with an initial feasible solution. ... 64 Table 20. Values obtained from the results starting with feasible initial solution. .. 66 Table 21. Results for t-test comparing startup time of hybrid algorithm vs simulated annealing ... 66 Table 22. Values obtained from the results starting with feasible initial solution. .. 66
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Table 23. Results for t-test comparing startup time of hybrid algorithm vs simulated annealing. ... 67 Table 24. Values obtained from the results starting with feasible initial solution. .. 67 Table 25. Results for t-test comparing computational time of hybrid algorithm vs tabu search. ... 68 Table 26. Values obtained from the results starting with feasible initial solution. .. 68 Table 27. Results for t-test comparing computational time of hybrid algorithm vs simulated annealing. ... 69 Table 28. Values obtained from the results starting with random initial solution.... 70 Table 29. Results for t-test comparing startup time of hybrid algorithm vs tabu search. ... 70 Table 30. Values obtained from the results starting with random initial solution.... 70 Table 31. Results for t-test comparing startup time of hybrid algorithm vs simulated annealing. ... 71 Table 32. Values obtained from the results starting with random initial solution.... 71 Table 33. Results for t-test comparing computational time of hybrid algorithm vs tabu search. ... 72 Table 34. Values obtained from the results of starting with random initial solution.
... 72 Table 35. Results for t-test comparing computational time of hybrid algorithm vs simulated annealing. ... 73 Table 36.Values obtained from the results starting with feasible initial solution. ... 74
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Table 37. Results for t-test comparing startup time of hybrid algorithm vs tabu search. ... 74 Table 38. Values obtained from the results starting with feasible initial solution. .. 74 Table 39. Results for t-test comparing startup time of hybrid algorithm vs simulated annealing. ... 75 Table 40. Values obtained from the results starting with feasible initial solution. .. 75 Table 41. Results for t-test comparing computational time of hybrid algorithm vs tabu search. ... 76 Table 42. Values obtained from the results of starting with feasible initial solution.
... 76 Table 43. Results for t-test comparing computational time of hybrid algorithm vs simulated annealing. ... 77 Table 44. Results obtained with the application of the T-test in experiment B. ... 77 Table 45. Conclusions obtained from the resolution of the t-test hypothesis ... 79
14 Contents
List of Advisors ...………..3
Declaration of Authorship ……….…. 4
Dedication ……… 5
Acknowledgement ……….………. 6
Abstract ……… 7
List of figures ……… 8
List of Tables ………..10
1. Introduction ... 17
1.1. Background ... 17
1.2. Problem statement ... 20
1.3. Research question ... 21
1.4. Hypothesis ... 21
1.5. Objectives ... 21
1.6. Scope ... 22
1.7. Proposed approach ... 22
1.8. Thesis structure ... 23
2. Literature review ... 24
2.1. Operating procedure synthesis ... 24
2.2. Drum boiler operational optimization ... 24
2.3. Metaheuristic optimization algorithms ... 26
15
2.3.1. Simulated annealing ... 27
2.3.2. Tabu search ... 29
3. Methodology ... 31
3.1. Simulator ... 32
3.2. Optimizer ... 32
3.2.1. Solution representation ... 32
3.3. Optimization algorithm ... 33
3.3.1. Neighbor solution ... 33
3.4. Proposed metaheuristic hybrid optimization algorithm ... 34
3.5. Generation of the initial solution ... 36
4. Case study ... 39
4.1. Drum boiler model ... 39
4.2. Drum boiler optimization problem ... 42
4.2.1. Drum boiler solution representation ... 44
4.2.2. Drum boiler feasibility function ... 45
4.2.3. Drum boiler acceptance probability function ... 47
5. Experiments and results ... 49
5.1. Experiment A ... 49
5.1.1. Benchmark solution ... 49
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5.1.2. Drum boiler optimization using the metaheuristic hybrid algorithm .... 51
5.2. Experiment B ... 54
5.2.1. Tabu search ... 54
5.2.2. Simulated Annealing ... 55
5.2.3. Statistical hypothesis-testing ... 56
5.2.4. Experiments with benchmark operating procedure used as the initial solution 57 5.2.5. Experiments with randomly generated initial solution ... 60
5.2.6. Experiment with a feasible initial solution ... 62
5.3. Result analysis ... 65
5.3.1. Experiments with a benchmark operating procedure used as the initial solution 65 5.3.2. Experiments with a randomly generated initial solution ... 69
5.3.3. Experiments with a feasible initial solution ... 73
5.3.4. Results comparison ... 77
6. Conclusions and future work ... 80
7. References ... 82
8. Appendix ... 86
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Chapter 1
1. Introduction
1.1. Background
Generation-side flexibility is the power system’s ability to adjust electricity generation rapidly as requested to match electricity demand [16]. There are dramatic changes in the power industry because of electricity market deregulation [3]. One consequence of this is that the need for rapid changes in power generation is increasing.
Figure 1 shows the enormous variability in the hourly electricity demand throughout 2016 of the National Interconnected System (SIN) of Mexico [24]
Figure 1. Hourly demand curve throughout 2016 of the SIN Mexico, Source PRODESEN 2016-2030 [24].
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In addition, according to data obtained from the Energy Information System (Sistema de Información Energética1), between 2017 and 2018 there was an increase in generation demand. This increase is most noticeable in non-conventional renewable energy. This relationship in shown in Figure 2.
Figure 2. Growth in energy generation by technology 2017 vs 2018
Keeping a balance between generation and demand is crucial for a system’s reliable operation since a mismatch can disturb power system frequency and possibly affect the reliability of system operations. This combined with the inherent intermittency of power generation based on variable renewable energy sources leads to a bigger interest in improving the response and adjustment capabilities of the electric power systems, especially in conventional power generation such as thermal power plants.
In the case of Mexico, the most important combined cycle thermal power plants are Manzanillo, which has 2,754 MW of electricity generation capacity, Tuxpan, in
1 http://sie.energia.gob.mx/
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Veracruz which produces 2,200 MW, followed by that of Tula, in Hidalgo, with 1,881 MW [25].
A thermal power plant transforms heat energy into electric power. A thermal power plant often makes use of a steam generator to take advantage of the heat obtained from its main electricity generation process. In a steam generator, water is heated, and it is turned into steam to spin a turbine, which drives an electric generator.
A steam generator is a power generation equipment that takes advantage of the expansive power of steam. To create high-temperature, high-pressure steam, fuel energy is converted to heat. Then such heat is transferred to the drum boiler. For that reason, one of the most important components of the steam generator is the drum boiler.
A typical drum boiler is shown in Figure 3. As explained by Aström, Et. Al [3] the drum boiler has a reservoir for water and steam with a water inlet and a steam outlet at the top. The drum stores the steam generated in the water tubes and acts as a phase-separator for the steam-water mixture. The difference in densities between hot and cold water along with the gravity helps in the accumulation of the "hotter"
water and saturated steam into the drum boiler.
Figure 3. Schematic representation of the drum boiler, source Aström et al. [3]
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The drum boiler has the potential to improve the competitiveness of a thermal power plant by means of the reduction of its startup time. The rate at which a boiler can be brought up to normal operating status depends on its size, and the length of time it has been shut down. In general, the larger and colder a boiler is, the longer it takes to startup.
The startup consists of operating the control valves as boiler pressure approaches its normal setpoint where the steam pressure and combustion control system can be switched over to automatic.
Although fast startups improve the competitiveness in an open power-market, if a startup is carried out too fast, excessive thermal stress can occur in the drum boiler components [23]. Therefore, the optimum synthesis of operational procedures must consider the physical constraints of the drum boiler to assure its integrity.
Operating procedure synthesis can be described as a planning problem where the sequences of actions need to be found to satisfy one or more goals. Operating process synthesis are especially useful for transient operations such as startups and shutdowns according to Batres [5].
Despite the focus on startup-time reduction, current approaches [1-5] fail short and can only obtain startup curves as the drum boiler state variables but cannot identify the corresponding control actions (operations) and their sequence.
1.2. Problem statement
This thesis focuses on how to synthesize operating procedures for the startup of a drum boiler, taking the drum boiler from an initial state to a goal state in the least amount of time, while considering the mechanical integrity of the drum boiler components but also reaching the solution in a computational efficient manner.
21 1.3. Research question
How to synthesize operating procedures that minimize the startup time of a drum boiler without compromising the structural integrity of critical components in a computationally efficient way?
1.4. Hypothesis
A methodology that couples a simulator with an optimizer using a hybrid metaheuristic algorithm based on the simulated annealing and tabu search can synthesize an operating procedure of valves for the startup of a drum boiler of a thermal power plant that takes less time to reach its goal state than a representative startup profile found in literature [6].
When using this methodology to synthesize the operating procedure that minimize the startup time of a drum boiler, the hybrid metaheuristic algorithm takes less computational time than the classical simulated annealing algorithm [17] and tabu search algorithm [14].
1.5. Objectives
To develop a methodology that synthesizes a near-optimum operating procedure that minimizes the startup time of a drum boiler without compromising the lifetime of the drum boiler and compare the performance of the proposed optimization algorithm against the classical simulated annealing algorithm and tabu search algorithm in the same drum boiler case study.
Specific objectives:
To develop and validate an optimization algorithm for the operating procedure of the startup of a drum boiler.
To compare the efficiency of the hybrid metaheuristic optimization algorithm with other existing optimization methods.
22 1.6. Scope
The operating procedure synthesis of the drum boiler will only manipulate the outlet valve and heat input flow. Two main constraints will be considered: the limit of heat flow of the drum boiler and a constraint on the thermal stress by pressure or compression.
The case study will not focus on other processes of operation of the steam generator outside the drum boiler. Furthermore, this study will not consider constraints other than the above mentioned, since it is specifically focused in the synthesis of the optimum operation procedure of the startup of the drum boiler.
1.7. Proposed approach
The research methodology is shown in Figure 4.
Figure 4. Proposed approach to prove hypothesis
23 1.8. Thesis structure
Chapter 1 presents the introduction of the research work. Here outlines the background, problem statement, research questions, hypothesis, objectives and the thesis structure.
Chapter 2 is a literature review and background about operating procedures synthesis, drum boiler operational optimization, and metaheuristic algorithms.
Chapter 3 cover the methodology proposed for the optimization of the operating procedure for systems with dynamic behavior. Focuses to explain the simulator, the optimizer and how they work together to solve the problem. It also explains the proposed optimization algorithm developed as a result for this research to find an optimal sequence for a problem with the mentioned characteristics.
Chapter 4 Explains the drum boiler case study. This covers the physical model of the drum boiler, the implementation of the simulation model and the optimization algorithm for the case study.
Chapter 5 presents the evaluation of the proposed optimization method for the case of study, showing the results using the proposed optimization algorithm together with other optimization algorithms. By means of a hypothesis test the efficiency of the proposed algorithm will be evaluated.
Chapter 6 summarizes the main conclusions of this research work and offers recommendations for future work.
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Chapter 2
2. Literature review
2.1. Operating procedure synthesis
Operating procedure synthesis (OPS) is a problem in which a set of equipment manipulations and their ordering must be generated to take the process from an initial state to a goal state. Transient operations such as startups and shutdowns are crucial for continuous and batch processes and can take advantage of OPS in terms of both safety and benefit [5].
OPS can be viewed as searching for a set of sequenced primitive operations that transform a plant system from an initial state to a pre-specified goal state through a series of intermediate states. These primitive operations must be carried out in such a way that no violations are made of any relevant process or of mechanical, safety and environmental constraints.
Most attempts to solve OPS problems have relied on simplified process behavior models [5]. The simulation-based planning approach makes use of detailed dynamic behavior models of the process, and a mathematical representation of quantitative safety constraints embedded within a rigorous dynamic optimization framework.
2.2. Drum boiler operational optimization
Much work has been done regarding the optimization of steam generation in a drum boiler from a procedural focus. Aström [3] developed a nonlinear physical model with a complexity that is suitable for dynamic optimization and OPS. The model is based on physical parameters for the plant and can be easily scaled to represent any drum power station.
Franke et al. [10] developed a nonlinear dynamic model of a drum boiler based on the Modelica language based on Aström’s physical model. Their model had three
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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 30%.
Kruger et al. [18] proposed a quadratic programming optimization approach to determine the optimal values of steam pressure and steam temperature in a startup process. Their model considered hard constraints such as control bounds and stress levels for the drum and header. They concluded that their optimization model could minimize both fuel consumption and startup time.
Belkhir et al. [6] investigated the minimization of the startup time of a drum boiler.
Their proposed startup strategy defined the initial and goal states in terms of steam mass flow rate and the pressure inside the drum. The startup process was formulated as an optimal control problem that minimized 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 optimization problem was solved by combining a framework developed on the JModelica environment and interior point optimizer algorithm (IPOPT). Their results were compared with the optimal start-up trajectories in Franke et al. [9], and the optimized profiles reached desired states in a shorter time without violating the operational and physical constraints.
Zhang et al. [28] 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 Aström and Bell model with specific parameters to simulate the dynamic analysis of the steam water 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.
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Table 1 shows a Comparison between previous work in startup optimization and solution proposed.
Table 1. Comparison between previous work in startup optimization and solution proposed.
Limitations of previous works Reference Solution proposed To minimize startup times
considering thermal stresses as the main constraint, but the optimization is directly to the state
variables regardless of how they must operate the drum boiler to achieve the goal states of the
critical state variables.
Belkhir et al. [6], Franke et al. [10] And Zhang et
al. [28]
Manage the optimization problem as a synthesis of the operating procedures of the drum boiler to
minimize the startup time considering thermal stresses as
the main constraint
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
Batres et al. [4] and Zhang et al. [28]
Use of a replaceable dynamic simulation model coupled with an
optimization algorithm for operating procedures of the dynamic system. Those elements are connected by an interface that
allows a communication between the simulation model and the
optimization algorithm.
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. They are also limited to a certain type of
optimization algorithms.
Belkhir et al. [6]
Use of a replaceable dynamic simulation model coupled with an
optimization algorithm for operating procedures of the dynamic system. Those elements are connected by an interface that
allows a communication between the simulation model and the
optimization algorithm.
2.3. Metaheuristic optimization algorithms
In computer science and mathematical optimization, a metaheuristic algorithm is a higher-level procedure or heuristic designed to find, generate, or select a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity [12]. Metaheuristic optimization
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algorithms are useful for combinatorial optimization problems. The two main characteristics of a metaheuristic algorithm are: exploration and exploitation.
Figure 5. Graphical representation of exploration and exploitation in a gold treasure hunt.
A graphical representation of exploration and exploitation is shown in Figure 5Figure 1. Exploration means generating diverse solutions, usually by randomization, to explore the most possible the search space of solutions, while exploitation means focusing on the search in a local region when a good enough solution has been found there, this is to exploit the information. Exploration keep the solutions from being trapped in local optima while exploitation ensures to use the resources in the best local search region [12].
The combination of both characteristics makes metaheuristic algorithms a suitable option to find good solution in a reasonable amount of time for large-scale nondeterministic optimization problems, or NP-hard problems. Especially when good solutions, but not necessarily the best solution, are needed within a reasonable time [12].
2.3.1. Simulated annealing
The Simulated Annealing algorithm was proposed by Kirkpatrick, et al. [17]. It is a metaheuristic optimization algorithm that uses local search methods to approximate
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global optimization in a large search space. It is often used when the search space is discrete.
The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. Heating and cooling the material affects both the temperature and the thermodynamic free energy. The simulated annealing shown in Figure 6 can be used to find an approximation of a global minimum for a function with many variables like the drum boiler problem.
Figure 6. Simulated annealing algorithm
This notion of slow cooling implemented in the simulated annealing algorithm is interpreted as a slow decrease in the probability of accepting worse solutions as the solution space is explored. Accepting worse solutions is a fundamental property of metaheuristics because it allows for a more extensive search for the global optimal solution. In general, the simulated annealing algorithms work as follows.
At each time step, the algorithm randomly selects a solution close to the current one, measures its quality, and then decides to move to it or to stay with the current solution based on the evaluation of the new solution and a probability. During the
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search, the temperature is progressively decreased from an initial positive value to zero and affects the probability function: at each step, the probability of moving to a better new solution is either kept to 1 or is changed towards a positive value; on the other hand, the probability of moving to a worse new solution is progressively changed towards zero.
2.3.2. Tabu search
The tabu search algorithm was proposed by Glover [14]. It is a metaheuristic optimization method that employs local search and a memory data structure that avoids the algorithm to repeat search in previously visited regions (Figure 7).
Local (neighborhood) search takes a potential solution to a problem and checks its immediate neighbors (that is, solutions that are similar except for very few minor details) in the hope of finding an improved solution. Traditional local search methods tend to become stuck in suboptimal regions or regions where many solutions are equally fit.
Figure 7. Tabu search algorithm
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Tabu search enhances the performance of local search by relaxing its basic rule.
First, at each step worsening moves can be accepted if no improving move is available (like when the search is stuck at a strict local minimum). In addition, prohibitions (tabu) are introduced to discourage the search from coming back to previously visited solutions.
Tabu search uses memory structures that store previously visited solutions. If a potential solution has been previously visited within a certain short-term period or if it has violated a rule, it is marked as tabu so that the algorithm does not consider that solution repeatedly, except for the case that the solution meets the aspiration criteria, which consists of a series of characteristics that allows the current solution to be selected as the new one, regardless of the tabu list. Usually, the aspiration criteria is to be better than the best solution found so far. This memory structures can be used to enhance some of the weakness in other metaheuristic algorithms like simulated annealing.
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Chapter 3
3. Methodology
In this thesis, a methodology is proposed that combines the use of a replaceable dynamic simulation model coupled with an optimization algorithm for operating procedures of the dynamic system. Those elements are connected by an interface that allows a communication between the simulation model and the optimization algorithm. This methodology is shown in Figure 8.
Figure 8. Proposed methodology
Data exchange in this methodology occurs as follows:
The user feeds the optimizer with the optimization algorithm and the simulator with the simulation model. The optimizer executes the optimization algorithm. The optimization algorithm creates an individual, which represents a solution. This individual is sent to the communication interface which creates a file that contains
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the operating procedures. Then, the communication interface translates this operating procedure to a format that can be passed to the simulator. The simulator has the function of running the plant model using the simulation model and the operating procedure. The simulator results are sent back to the optimizer through the communication interface. Then the optimizer calculates the objective function, evaluates the constraints and generate a new solution to continue the cycle of the methodology until achieving the stop criteria. Finally, the optimizer sends the final solution back to the user.
3.1. Simulator
The dynamic behavior of the drum boiler is simulated by solving an ordinary- differential-equation model. Each operating procedure generated by the optimization method is sent to the simulator in order to determine the values of the state variables along time. The simulation model was developed using the modeling and simulation environment OpenModelica.
3.2. Optimizer
The optimizer generates an operating procedure that can be considered a good solution given an objective function and a set of constraints. The optimizer can generate an initial feasible solution searching through the space of solutions (section 3.5). Then, the optimization algorithm iteratively improves the initial solution by making local changes until there is no better solution when applying such changes.
3.2.1. Solution representation
The solution is represented as a finite sequence (𝐴 , 𝐴 , 𝐴 , … , 𝐴 ) where 𝐴 = (< 𝑂 , 𝑇 >, 𝑅 ). 𝑂 is an operation, 𝑇 denotes the length in time that 𝑂 will be applied and 𝑅 denotes the number of times that the operation 𝑂 with length in time 𝑇 will be repeated.
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An operation is defined as an element of the set 𝑂 = 𝐼 , 𝑃 where 𝐼 is a unique integer number that serves as identifier of the operation, and 𝑃 is a numeric value of parameter 𝑘 assigned to remotely-operable equipment 𝑙. A control valve is an example of remotely operable equipment.
3.3. Optimization algorithm
This section represents the process of optimization of the feasible operating procedure in a general way because the methodology is designed to be able to use different optimization algorithms.
3.3.1. Neighbor solution
The proposed algorithm can be clasified as a local search algorithm. Local search algorithms move from solution to solution in the space of candidate solutions (the search space) by applying local changes by means of a neighborhood operator (NOP). These solutions are named neighbor solutions.
As mentioned in section 3.2.1 the proposed algorithm represents the solution using a sequence structure. The NOP can be defined in terms of local rearrangements, such as swapping, moving o changing one or more elements from the current solution sequence. A neighborhood operator can be applied multiple times to make significative changes in the solution to improve the diversification of solutions through the optimization process.
Figure 9. Example of five neighbors from a current solution by changing one random element of the sequence
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3.4. Proposed metaheuristic hybrid optimization algorithm
The proposed optimization algorithm is a metaheuristic hybrid algorithm shares characteristic of two well-known metaheuristic algorithms: simulated annealing by Kirkpatrick [17] and tabu search by Glover [14].
Simulated annealing is distinguished by the “cooling” element that allows the selection of new “worse” solutions at the early stages of the iterative process in order to avoid local optima. The algorithm “cools” as it converges, so the probability of selecting "worse" solutions decreases and accepting only better solutions.
Tabu search is characterized by the use of a memory structure that stores information of previously evaluated solutions in a memory data structure called tabu list. As a result, tabu search avoids visiting again solutions that have already been evaluated, improving the computational efficiency by avoiding unnecessary simulation runs.
The flow chart of the metaheuristic hybrid algorithm is shown in the Figure 10, the following is a detailed explanation of each of the steps in the metaheuristic hybrid algorithm.
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Figure 10. operation diagram of the Hybrid metaheuristic algorithm
The first step is to generate an initial solution (see Section 3.5). Subsequently, the algorithm uses this initial solution to produce a neighbor solution (see Section 3.3.1).
After selecting and evaluating the neighbor solution, that neighbor is added to the tabu list. From this point on, each new solution is tested against the tabu list.
The tabu list is a list of previously evaluated solutions, to avoid looking in the same area and avoid rework in simulation. This list works as a long-term memory for the algorithm and keeps from the first to the last solution evaluated.
If the solution appears in the tabu list, then it will be avoided until a certain number of iterations have been reached or if the solution satisfies the aspiration criteria. The aspiration criteria is a set of requirements that a solution must meet to be considered a new current solution. In the proposed algorithm, the aspiration criterium is based on the probability of acceptance of the simulated annealing algorithm.
Normally, when the neighbor solution is feasible, and the objective function of the neighbor solution is better than the objective function of the current solution the neighbor solution will be selected. If the total operation time is set as the objective function, a better neighbor solution is the one that takes less time to reach the goal state.
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The aspiration criterium is activated when the neighbor solution is feasible, and the objective function of the neighbor solution is worse than the objective function of the current solution. Then, depending on an acceptance probability function 𝑃(), that depends on the values of the objective function of the current and the neighbor solutions, and on a global time-varying parameter T the neighbor solution is accepted. A typical probability function is the Boltzman distribution (equation 1).
𝑃 = 𝑒 (( ( ) ( ( )))/ (1)
Where P is the probability of selecting a "worse" neighbor solution, 𝑇 is the temperature of the algorithm, 𝐸(𝑦) is the objective function of the neighbor solution and 𝐸(𝑥) the objective function of the current solution.
If none of the aspiration criteria are met a comeback operator is applied. A comeback operator serves to cancel the generation of the neighbor solution and keep the previous solution as the current solution [8]. Let NOP be the neighbor operator which transforms a solution 𝑥 in a neighbor solution 𝑦. The comeback operator will be the inverse NOP-1 which transforms a neighbor solution 𝑦 in the previous solution 𝑥.
If the objective function of the neighbor solution is better than the best solution found so far, the neighbor solution becomes the best solution. From this point on, a new current solution is already in place and the iterative process starts again. Throughout the search better results will be found, while new search areas are evaluated. Once the stop condition is met, the algorithm delivers the best solution obtained along the run.
3.5. Generation of the initial solution
The algorithm shown in Figure 11 is proposed for the generation of a feasible initial solution that will function as the starting point for the optimization algorithm to find the optimal solution.
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Figure 11. Operation diagram of the initial solution generation algorithm
The following is a detailed explanation of each of the step in the seed generation algorithm.
The algorithm starts by generating a random solution in a codified form explained in Section 3.2.1. This sequence will be evaluated by the simulator using the simulation model. The feasibility function 𝑓(𝑥) determines the difference between the left-hand side and the right-hand side of the constraints. If the generated solution (solution x) is feasible, 𝑓(𝑥) becomes zero. A solution is considered feasible if it reaches the goal state determined by the problem without violating any of the constraints.
If 𝑓(𝑥) > 0, a new solution (solution y) will be generated based on the current solution x. Solution y can apply multiple neighborhood operators depending on the problem and how far solution x is from being feasible. This characteristic add the exploration element to the algorithm.
Solution y will be evaluated by the simulator using the simulation model. The value of 𝑓(𝑦) can be calculated with the simulation results. Once 𝑓(𝑦) is calculated, a comparison between 𝑓(𝑥) and 𝑓(𝑦) is made. If 𝑓(𝑦) > 𝑓(𝑥) solution x remains as the current solution, and a new solution y is generated restarting the loop. If 𝑓(𝑦) <
𝑓(𝑥) solution y becomes the new current solution (new solution x), and a new
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solution y is generated to continue the loop. If 𝑓(𝑦) = 0, solution y becomes the initial solution and it goes on to the next process of the optimization method.
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Chapter 4
4. Case study
The evaluation of the proposed methodology is carried out by means of a case study on the generation of the optimum operating procedure of a drum boiler. A drum boiler is a piece of equipment that converts water to steam by applying heat energy. A fast startup of a drum boiler can satisfy the steam demand quickly. However, if startup is carried out too fast, excessive thermal stresses can occur in the drum boiler.
The problem consists of finding an operating procedure for the valves that takes the system from given pressure and steam-flowrate values to desired pressure and steam-flowrate values in the shortest time possible, while avoiding excessive thermal stresses in the metal of the wall of the drum boiler.
4.1. Drum boiler model
The dynamic simulation model of the drum boiler is based on the lumped model by Aström and Bell [3]. The simulation model consists of a water inlet, a heat supply, a water-level PI-controller and a saturated steam outlet. A schematic representation of the drum boiler is shown in Figure 12.
Figure 12. Schematic diagram of the drum boiler, source Aström et al. [3]
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Water from the condenser enters the drum through the water inlet and saturated steam is extracted. Behavior of the boiler furnace in a coal fired power plant or exhaust gases of a gas turbine are modeled by means of a heat supply system to heat and evaporate the water in the rising tubes. For simplicity, the model assumes thermodynamic equilibrium between water and steam inside the drum.
The mass balance in the drum boiler can be written as:
𝑉 (𝑙 ) + 𝑉 + (𝜌 − 𝜌 ) ( )= 𝑞 − 𝑞 (2) where 𝑉 is the water volume in the drum, 𝑉 the steam volume in the drum, 𝜌 and 𝜌 the density of water and steam, 𝑙 the water level inside the drum, 𝑝 the pressure inside the drum, 𝑞 feedwater flow y 𝑞 the steam flow rate extracted from the drum.
The energy balance in the drum boiler can be expressed as:
𝑉 ℎ + 𝜌 + 𝑉 ℎ + 𝜌 − 𝑉 + 𝑚𝐶 + (𝜌 ℎ + 𝜌 ℎ ) = 𝑄̇ + (𝑞 ℎ − 𝑞 ℎ )
(3)
where 𝑄̇ is the heat flow, 𝑉 the water-steam volume in the drum, ℎ y ℎ are the water and steam enthalpies, while 𝐶 denotes the specific heat capacity of steam.
According Franke et al. [9] thermal stresses occurs in the drum if there are spatial temperature differences. Thus, thermal stress is determined proportionately to the time derivative of the metal temperature, considering that the metal temperature is equivalent to the water saturation temperature inside the drum, as given in equation 4.
𝜎 = 𝑘 (4) where 𝜎 is the thermal stress in the thick-walled drum, 𝑘 the thermal conductivity of the wall and 𝑇 = 𝑇 (𝑝) the inner temperature in the drum.
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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 could capture the overall behavior of the drum boiler.
For the implementation of the model, the OpenModelica environment was used, which allows the development of dynamic simulators. OpenModelica has algorithms for solving differential equation systems, making it possible to observe changes in variables with respect to time, which is known as dynamic simulation. The OpenModelica model, based in the work of Rosado [23] is shown in Figure 13.
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Figure 13. Simulation model graphical representation in the OMEdit Modelica software
The integration between the OpenModelica and the optimization algorithm was carried out by implementing a two-way interface. The interface receives the operating procedure generated by the optimization algorithm, translates this procedure as a set of parameters for the simulation model, then evaluates and executes the simulation model. Conversely, the interface receives the results of the simulation, and translate them in a format suited to the optimization algorithm.
4.2. Drum boiler optimization problem
The optimization problem follows the formulation described by Belkhir et al. [6]. The optimization problem is formulated so as to take the system from an initial state to a goal state in a minimum time while satisfying both mechanical and process constraints. The initial and goal states are described in terms of inner pressure and steam mass flowrate.
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The goal is to reach a pressure of 9 MPa and a steam outlet flow of 180 kg/s by manipulating the heat inlet valve and the steam outlet valve.
The objective function is expressed as:
Min (𝐴 ∗ 𝑆) + 𝛼 𝑃 − 𝑃 + 𝛽 𝑞 − 𝑞 𝑤ℎ𝑒𝑟𝑒 𝑆 = ∑ 𝑑𝑡
(5)
subject to:
0 ≤ 𝑉𝑝𝑜𝑠 ≤ 1 (6)
0 ≤ ̇ ≤ 25 (7)
0 ≤ 𝑄̇ ≤ 500 𝑀𝑊 (8)
−10 𝑀𝑃𝑎 ≤ 𝜎𝐷 ≤ 10 𝑀𝑃𝑎 (9) Equation 5 seeks to minimize the time it takes for the drum boiler to reach the goal.
When A = 0 the problem is reduced to find a sequence of operations that is feasible but not necessarily optimal [3]. A feasible solution is a solution that reaches the goal state without violating any of the constraints.
Equation 6 limits the opening of the steam outlet valve (Vpos) to values between 0 (totally closed) and 1 (fully open). Equation 7 ensures that the heat (Q̇) per minute does not exceed 25 MW/min. Equation 7 is a nonlinear constraint because it implies that there could be different heat ramp rates during the process 25 𝑀𝑊/𝑚𝑖𝑛.
Equation 12 is a constraint of the accumulated heat limit of the drum boiler which must not exceed 500 MW and equation 13 is a constraint that avoids an excessive thermal stress in the drum boiler that must be less than 10 MPa.
The heat is supplied in MW, the water is supplied by a control system and the steam flow is controlled by a valve. The interaction between the opening of the steam outlet
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valve and the heat flow in the drum boiler generates steam at pressure (Psat) which exits at flow rate (qs). The steam can later be sent to a superheater or directly to a steam turbine [3]. For this problem the goal was set to Pgoal = 9 MPa, qgoal= 180 kg/s.
The weights were set to α = 10-4 and β = 10-4. Parameter A = 0 for obtaining the initial solution and then changed to A = 1 when used in the rest of the algorithm. The nonlinearity of equation 5 and equation 7 add complexity to the problem, which justifies the use of metaheuristic methods to find a solution.
In order to ensure that the optimization algorithm converges. The stop condition of the algorithm will be to perform 1000 iterations. Various tests with more iteration limits were performed but the algorithm always converged before 1000 iterations.
4.2.1. Drum boiler solution representation
Each operating procedure is represented according to the representation scheme explained in Section 3.2.1. Each operation is formed by combining discrete values of the heat flow and the steam flow rate. This results into the 9 operations shown in Table 2.
Table 2. Operation translation to heat flow and valve position
Operation Heat flow MW/min Vpos
1 8 0
2 8 0.6
3 8 1
4 16 0
5 16 0.6
6 16 1
7 24 0
8 24 0.6
9 24 1
The execution time per operation is set to 60, 120 or 180 seconds. The repetition parameter is set to vary between 0 to 9. The length of the sequence is fixed to 9 elements.
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Figure 14 shows an example of an operating procedure. The first element in the sequence represents operation 8 (heat flow 24 MW/min, Vpos 0.6) being executed for 60 seconds and repeated 3 times.
Figure 14. Solution representation of an operating procedure for the drum boiler optimization problem
4.2.2. Drum boiler feasibility function
The feasibility function 𝑓(𝑥) is calculated with equation 10. An extra penalty can be applied to 𝑓(𝑥) in case the total time of the generated sequence is less than 1200 seconds, a value too low to be feasible.
𝑖𝑓 𝑡 < 1200, 𝑓(𝑥) = 𝑉 + 100 𝑡 > 1200, 𝑓(𝑥) = 𝑉
𝑉 = 400 − 𝑄̇ + 𝑓 + 𝐺 (10)
𝐺 = 𝛼 𝑃 − 𝑃 + 𝛽 𝑞 − 𝑞 where Q̇ is the accumulated heat that must reach 400 MW, fs represents the number of times the thermal stress exceeded the 10 MPa limit throughout the process and G represents how far is the sequence of approaching the steam pressure and steam outflow target.
During the optimization process new solutions are generated by means of the neighborhood operator (NOP). NOP takes an existing solution and makes a mutation by randomly change one of the sequence elements (operator, time and repetition).
A neighborhood operator can be applied multiple times and corresponds to the
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penalty given to the solution for violating the constraints. Figure 15 shows an example of NOP applied four times.
Actual solution
3 4 6 3 6 6 7 4 3
60 120 180 180 60 60 60 180 120
4 2 1 5 5 7 3 7 3
One mutation
3 8 6 3 6 6 7 4 3
60 120 180 180 120 60 60 180 120
4 2 6 5 5 7 3 7 3
Two mutations
3 8 6 3 6 6 2 4 3
60 120 180 180 120 180 60 180 120
4 2 6 5 5 7 3 7 0
Three mutations
3 8 6 5 6 6 2 4 3
60 60 180 180 120 180 60 180 120
4 8 6 5 5 7 3 7 0
Four mutations
3 8 6 5 6 1 2 4 3
60 60 180 180 120 180 60 120 120
4 8 6 5 5 1 3 7 0
Figure 15. Mutation process from one to four mutations
The number of times that NOP is applied is set based on the value of 𝑓(𝑥) as shown in Table 3. The number of times NOP is applied depends on the length of the solution for the given problem. In this case study, the maximum number of mutations is 4 because it changes half the values of the previous solution in the worst scenario for 𝑓(𝑥).
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Table 3. Number of mutations according to the feasibility function of the operating procedure
𝑓(𝑥) Number of mutations
>400 4
>300 3
>50 2
<50 1
≈0 0
4.2.3. Drum boiler acceptance probability function
For the use of the hybrid metaheuristic algorithm and the simulated annealing algorithm, it is necessary to establish the acceptance probability function of selecting worse solutions. This function is represented by equation 11:
𝑃 = 𝑒 (( ( ) ( ))∗ )/ (11)
where P is the probability of selecting a worse neighbor solution, T is a parameter that gradually decreases as the algorithm proceeds2, t(y) is the final time of the neighbor solution and t(x) the final time of the actual solution because the time is the value that is sought to optimize in this problem. 𝛾 is a parameter that magnifies the difference between two solutions.
To select the value of 𝛾, the value of the Boltzman distribution was analyzed with a difference of 50 seconds between two solutions, varying the value of 𝛾 between 1 and 15. The value of 𝛾 = 10 was selected as it starts with a probability slightly larger than 50% but steadily decreasing through the iterations. Figure 16 shows the probability of selecting a worse solution (50 seconds worse) using different 𝛾 values in a run of 1000 iterations.
2 T is also known as the annealing temperature
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Figure 16. probability of selecting a better or a worse solution (50 seconds worse) using different γ values, depending on the progress of a run of 1000 iterations.
0 0.2 0.4 0.6 0.8 1 1.2
0 200 400 600 800 1000 1200
Probability
iteration
Better solution worse solution, ϒ=1 worse solution, ϒ=5 worse solution, ϒ=10 worse solution, ϒ=15
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Chapter 5
5. Experiments and results
To verify the effectiveness and efficiency on the application of the methodology for the synthesis of the operating procedure to minimize the drum boiler startup time, two experiments were carried out:
Experiment A uses a randomly generated operating-procedure as the initial solution to the metaheuristic hybrid algorithm. The results are then evaluated by a comparison , against a representative startup profile found in literature which is used as benchmark [6].
Experiment B compares the metaheuristic hybrid algorithm, the simulated annealing algorithm and the tabu search algorithm. The comparison will consider three different scenarios:
Using as an initial solution the benchmark reported in [6].
Using as an initial solution a randomly generated sequence.
Using as an initial solution a feasible solution generated by the procedure explained in Section 3.5.
5.1. Experiment A 5.1.1. Benchmark solution
The benchmark is the startup operating procedure proposed by Belkhir et al.. This operating procedure keeps the heat input valve at a constant value of 8 MW/min and the steam output valve with fully open. After executing the operating procedure with the OpenModelica simulation model, the startup completes in 3000 seconds. Figure 17 shows the behavior of function 𝑓(𝑃 , 𝑞 ) = 𝛼 𝑃 (𝑡) − 𝑃 + 𝛽 𝑞 (𝑡) − 𝑞 which measures the distance to the goal state over time.
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Figure 17. Goal state value over time with the reference startup operating procedure.
The Belkhir startup profile never exceeds the limits imposed by the thermal stress constraint, maintaining a stress value between -10 MPa and 10 Mpa, The stress profile is shown in Figure 18.
Figure 18. Thermal stress in the drum boiler through time with the reference startup profile.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0 500 1000 1500 2000 2500 3000 3500 4000
𝛼(𝑃s𝑎𝑡−𝑃𝑔𝑜𝑎𝑙)2+𝛽(𝑞𝑠−𝑞𝑔𝑜𝑎𝑙)2
Time (s)
DRUM BOILER STARTUP TIME
-2 0 2 4 6 8 10 12
0 500 1000 1500 2000 2500 3000 3500 4000
Thermal stress (MPa)
Time (s)
THERMAL STRESS OF THE DRUM BOILER
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5.1.2. Drum boiler optimization using the metaheuristic hybrid algorithm
The algorithm was initiated with a randomly generated solution and stopped after 1000 iterations. Table 4 shows the operating procedure obtained as a result of the optimization. Table 5 shows the valve operation translated from the operating procedure according to section 4.2.1. Once the operation reaches the desired goal, the heat inlet valve and the steam outlet valve take the values of 0 MW/min and totally open respectively.
Table 4. Best operating procedure obtained with the hybrid metaheuristic algorithm and a randomly generated initial-solution.
Operation 8 3 9 9 7 2 8 1 3
Time 60 120 180 120 120 60 120 120 60
Repetitions 1 7 8 3 8 5 1 9 5
Table 5. Valve operation translated from the operating procedure in table 4.
time heat inlet valve steam outlet valve accumulated time
60 24 0.6 60
840 8 1 900
660 24 1 1560
- 0 1 -
With this operation procedure the drum boiler arrives at the goal state in 1560 seconds.
The behavior of the objective function over time is shown in Figure 19
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Figure 19. Comparison of the reference startup time (blue line) with the startup time obtained with the hybrid metaheuristic algorithm (green line)
Despite the shorter startup time, the thermal stress remains within the limits imposed by the problem, so the optimum solution is considered feasible. Figure 20 shows the levels of thermal stress throughout the startup process with the generated solution and with the reference sequence, both solutions stay within the limit of ± 10 MPa.
Figure 20. Comparison of the Belkhir´s startup time (blue line) with the startup time obtained with the hybrid metaheuristic algorithm (green line)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0 500 1000 1500 2000 2500 3000 3500 4000
𝛼(𝑃s𝑎𝑡−𝑃𝑔𝑜𝑎𝑙)2+𝛽(𝑞𝑠−𝑞𝑔𝑜𝑎𝑙)2
Time (s)
DRUM BOILER STARTUP TIME
-2 0 2 4 6 8 10 12
0 500 1000 1500 2000 2500 3000 3500 4000
Thermal stress (MPa)
Time (s)
THERMAL STRESS OF THE DRUM BOILER
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The feed water flow needs to be controlled so that the water level inside the drum is kept at its set point. A PI controller is used for this purpose. Figure 21 shows the behavior of the water level throughout the drum boiler startup process.
Figure 21. Comparison of the Belkhir´s water level (blue line) with the startup time obtained with the hybrid metaheuristic algorithm (green line)
Figure 21 shows that the water level changes abruptly during the drum boiler startup process. The instability of the liquid level can be reduced with the use of a non-linear controller that can adapt to the non-linearity of the model or the use of gain scheduling approach which involves the application of different controller tuning parameters as a process transitions from one operating range to another [21].
However, the magnitude of the changes in water level is considered tolerable for this experiment.
The best result was obtained at the 654th iteration. Due to the memory strategy, the simulation of 396 previously simulated solutions was avoided. The experiment took 216 minutes on a computer, with 4.00GHz Intel Xeon W-2125 CPU and 32GB of RAM, running Windows 10 Pro.
60 61 62 63 64 65 66 67 68 69 70
0 500 1000 1500 2000 2500 3000 3500 4000
Water level
Time (s)
WATER LEVEL
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With results shown in Figure 19 and Figure 20, it can be prove that a methodology that couples a simulator with an optimizer, using a hybrid metaheuristic algorithm based on the simulated annealing and tabu search can synthesize an operating procedure of valves for the startup of a drum boiler of a thermal power plant that avoid excessive thermal stress and takes 48% less time to reach its goal state than the benchmark.
5.2. Experiment B
Experiment B compares the proposed hybrid algorithm against two well-known metaheuristic algorithms: simulated annealing and tabu search.
5.2.1. Tabu search
The classic tabu search algorithm was modified to synthesize operating procedures as shown in Figure 22. The modified tabu search algorithm works by using the tabu list as a memory structure that allows to search through a neighborhood of solutions, avoiding simulating previously evaluated solutions.
Figure 22. Modified tabu search algorithm
The algorithm works by generating a neighborhood of solutions and selecting the best solution from this neighborhood as the current new solution. Even if the best
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solution in the neighborhood is no better than the current solution. This allows the algorithm to escape local optimum zones.
The memory structure prevents the algorithm from returning to previously evaluated solutions and encourages a more extensive search. This in turn contributes to reducing computational time avoiding unnecessary simulations.
For the case study, a neighborhood of four solutions was used, as explained in the section 3.5.1.
5.2.2. Simulated Annealing
The classic simulated annealing algorithm was modified to synthesize operating procedures as shown in Figure 23. The modified simulated annealing algorithm works with a probability function that allows to select “worst” neighbor solutions at the start of the run, to increase the search space and avoid local optima.
Figure 23. Modified simulated annealing algorithm