Cascade Control Applied to a Single-Component Single-Stage Vaporizer — Modeling and Simulation

: This comprehensive research addresses a gap in the literature by providing an extensive examination of a single-component single-stage vaporizer process. The research involves sophisticated analyses such as dynamic modeling, comprehensive control system design and performance evaluation. The study systematically derives a linear state-space model from complex nonlinear dynamic models, laying the basis for the development of two highly effective control systems specifically designed for vaporizer level and temperature control. The simulations of these control structures demonstrate notable properties, including fast disturbance rejection, minimal overshoot, and virtually no steady-state error, emphasizing their robustness and precision. This research focuses on the stability and response of the system and provides insight into its transient behavior during disturbances and setpoint changes. The study's broad implications extend beyond the results and provide a path for future improvements. The results indicate ways to refine the start-up stage, minimize initial overshoot during system initialization, and further improve control strategies. This work has the potential to make a difference in advancing the field of vaporization processes, providing engineers and researchers with the tools and insights needed to improve system reliability and performance in industrial applications.


Introduction
In industrial plants for the storage and handling of liquid and/or gaseous fuels, it is common to find liquid petroleum gas (LPG) fuels.Boiling systems are used to operate LPGs, which represent one of the most important chemical operations and at the same time the most difficult to operate [1,2].One of these boiling systems is the vaporization process, where the main equipment is the vaporizer, which is usually built using a pressurized cylindrical tank [1,2].
Usually, the LPG is transported in liquid form at a certain pressure and stored in tanks, the LPG must undergo a phase change from its liquid to its gaseous state, producing the vaporization operation [2].It is for this reason that these vaporizing systems are necessary.
In order to operate a vaporizer safely while maintaining the product's chemical and thermodynamic properties, it is necessary to have suitable and well-tuned control systems to ensure the correct and safe operation of the relevant process.
Finally, a review of the LNG IFV and its heat transfer characteristics is described in [9].This article provides a comprehensive overview IFV used in the LNG industry.It examines various aspects of IFVs, including innovative design structures, thermal models, and alternative refrigerants, all aimed at improving heat transfer performance, reducing costs, and improving equipment safety.The paper discusses both subcooled and supercritical LNG vaporization processes, highlighting the significant challenges and opportunities in cryogenic heat transfer.While providing insights from experimental measurements and numerical simulations, it emphasizes the need for further research, particularly in evaluating heat transfer correlations under IFV conditions and conducting cryogenic condensation experiments to optimize IFV operation.This review serves as a valuable resource for engineers and researchers working on LNG vaporization systems, providing a comprehensive understanding of the current state of IFV technology and future directions in this critical field.
The main objective of this article is to address the gap in the literature on state-space modeling and comprehensive control system design for vaporization processes.The introduction provides an overview of this critical gap and emphasizes the significant importance of vaporization systems in different industrial contexts.The system's multifaceted nature highlights the complex interactions between parameters such as liquid level, temperature, and pressure, which require precise control to ensure operational efficiency and stability.
This primer provides a comprehensive perspective on the importance of vaporization systems, with an emphasis on LPG vaporizers in an industrial context.It emphasizes the critical role of well-designed control systems in ensuring safe and efficient operation.In addition, it effectively establishes the need for a mathematical vaporizer model and introduces control design techniques essential to achieving controlled processes.
The introduction properly identifies a gap in the literature regarding the modeling and control of vaporizers.By citing various references [3][4][5][6][7][8][9], the introduction effectively establishes the need for this research and positions it as a departure from previous efforts.The references cited strategically highlight the lack of comprehensive studies that include both modeling and control, further emphasizing the relevance of the present study.
However, improving the impact of the introduction could be achieved by increasing clarity in certain sections.Explicitly stating the primary purpose or objective of the study early on would help with clarity.In addition, a concise preview of the study's specific contributions or approach to addressing identified gaps would provide a clearer roadmap for the reader.
While the references to previous work (references [3][4][5][6][7][8][9]) contextualize the research, they could be enhanced by emphasizing their direct relevance to the challenges and gaps highlighted in the introduction.
The paper is organized as follows.In sections 2 and 3, the description of the process and the modeling of the system are given, respectively.Then, in sections 4 and 5, the design of the vaporizer control system is developed and simulation results are presented, respectively.Finally, the conclusions are given in section 6.

Process Description
The focus of this study is the vaporization system shown in Figure 1.This system consists of a pressurized cylindrical tank provided with two inlet pipes and a single outlet pipe.One of the inlet pipes is connected to the bottom of the tank, while the other is connected to a heating coil.The flow through each inlet pipe is controlled by valves 1 and 2. The heater coil incorporates an inlet and an outlet, where the outlet is connected to a condensate trap labeled T. The primary purpose of this arrangement is to vaporize propane, a specific form of LPG.LPG enters the system through one inlet pipe, while the second inlet pipe introduces a thermal steam flow into the heating coil, effectively facilitating the vaporization process.
The result of this process is the production of vaporized gas, which is exhausted from the tank through the outlet pipe.At the same time, a by-product in the nature of condensed liquid escapes from the outlet of the heating coil and is collected by the condensate trap.The introduction of LPG is characterized by parameters such as flow rate (fi(t)), temperature (Ti(t)), pressure (Pi(t)), and density (ρi).Steam is introduced with its specific flow rate (ws(t)) and pressure (Ps(t)).The vaporized gas is discharged from the system at a flow rate (fv(t)).
The whole vaporization process is controlled as described in the section 4. The control system is based on the regulation of the level (h(t)), pressure (P(t)), and outlet temperature of the heating coil (condensate temperature, Ts(t)).Moreover, the vaporizer produces two distinct phases: a liquid-and vapor-phase [1].The liquid-phase is identified by properties such as volume (V(t)), pressure (P(t)), temperature (T(t)), and density (ρ).Meanwhile, the vapor-phase is characterized by its volume (Vv(t)), pressure (Pv(t)), and density (ρv).The theoretical framework assumes homogeneity within the liquid phase of the tank.Heat is added to the system at a rate (Q(t)) to achieve the desired pressure value, and consequently the vaporization process is driven at a rate (wv(t)).The assumptions also include the neglect of heat and mass losses through the tank walls and the neglect of dynamic effects from the relatively smaller vapor phase [1].Table 1 summarizes the variables shown in the process with their engineering units.Two control strategies are developed for the correct operation of the vaporizer (details in section 4).Firstly, a single feedback loop controls the vaporizer level, i.e., h(t) by adjusting the position of the valve 1 regarding the variable k1(t).Keeping a controlled level is essential to prevent potential problems.An excessive perturbation of fi(t) away from the operating point can lead to overflow and loss of vapor-phase, while a large decrease can empty the vaporizer, causing thermal shock and damage to the heating coil, both of which negatively affect the wv(t) ratio and the vaporization process [1].
The second strategy takes advantage of a cascade control structure with two feedback loops.The primary loop controls the temperature Ts(t), which indirectly influences the secondary loop, which controls the liquidphase pressure P(t).This cascade configuration increases control accuracy and robustness, as disturbances affecting the primary loop are effectively compensated by the secondary loop.The manipulation of valve 2, referred to as k2(t), allows the fine tuning of the vaporization process and the maintenance of optimal conditions even in the presence of external disturbances or variations [2,10,11].
In this context, a potential classification of the system control variables into the designated classifications is shown in Table 2. From Table 2, the measured disturbance variables give special emphasis to the role they play in the vaporizer.In this study, it is assumed that the processes related to the LPG and steam supply are known and constant and are assumed to operate in steady state.Also, in this study, the design of the control system for the vaporization process is based on well-defined control specifications and objectives aimed at achieving optimal performance, stability, and operational efficiency within the vaporizer.Both tracking objectives and constraint satisfaction were considered in the formulation of the control strategy.These aspects are listed in Table 3.To conclude the description of the process, two critical figures of merit (FoM) are introduced to quantitatively evaluate the efficacy of the proposed control structures.The first FoM focuses on achieving an overshoot rate (OS), of less than 5 % which reflects the degree to which the system response during transient operation goes beyond its steady-state value [2,10,11].The second FoM is based on achieving a remarkably low steady-state error (SE) of less than 1 %, which emphasizes the accuracy and precision in maintaining desired set points over long periods of operation [2,10,11].These FoMs provide important measures for evaluating the performance and efficiency of the control structures developed to control the vaporization process efficiently.

Process Modeling
The dynamic model of the vaporizer is established based on the fundamental dynamic equations of the liquid-phase, as detailed in [6] and [7].Assuming equal pressures between the vapor-and liquid-phase (P(t) = Pv(t)), the vaporization rate can be defined as wv(t) = ρv • fv(t), where ρv is the vapor phase density and fv(t) is the vaporized gas flow rate.
The system includes two different balances: a mass balance and two energy balances.Using the mass balance [1,12,13] around the tank a dynamic equation is formulated as follows Equation ( 1) introduces the variable r denotes the radius of the cylindrical tank, in m.By applying an energy balance [1,12,13] around the tank, the resulting expression is defined as follows Moreover, Cp and Cv denote the heat capacities at constant pressure and volume, respectively, measured in units of kg/m 3 .On the other hand, U and A denote the total heat transfer coefficient and the heat transfer area, respectively, in units of W/m 2 -°C and m 2 .
Given the importance of keeping the tank at the desired pressure (referred to as P(t)), the temperature T(t) in (2) needs to be represented as a function of the desired pressure P(t).To achieve this, expression in (3) introduces the equation of state, with the consideration that P(t) can be reasonably assumed to be relatively low [12].
In this context, the constants M and R are important, since they symbolize the number of moles of the fluid and the ideal gas constant, respectively, both expressed in J/mol-K [1,12].By substituting (3) into (2), and assuming equal heat capacities (Cp = Cv = C), the following equations provide a simplified model that characterizes the dynamics of the vaporization process.Also, from (2) and taking into account that d(h(t)•T(t))/dt = T(t)•(dh(t)/dt) + h(t)•(dT(t)/dt), the energy balance with respect to pressure P(t) can be formulated as follows where λv denotes the average heat of vaporization [1,12].In addition, by applying an energy balance around the heating coil, the resulting expression defined as follows where, β is the latent heat of condensation in kJ/kg, while CM is the heat capacity of the heating coil in kJ/°C.Similar to the previous scenario, it is necessary to express the expression in (5) as a function of P(t).By substituting (3) into (5), the modified version of the energy balance is derived as follows The model is completed with additional equations for the control valves.According to [14], LPG can be considered liquid in nature.Therefore, a valve that handles a liquid service is suitable for the service of LPG.Based on this and according to [12,15], the expression that models the dynamics of the LPG flow through the valve 1 can be defined as follows where, Cv1 is the valve coefficient determined experimentally for each style and size of valve, using water at standard conditions as the test fluid [15].Also, g represents the gravitational acceleration in m/s².On the other hand, for steam handling, and according to [12,15], the expression that characterizes valve 2 can be defined as where Cv2 is also the valve coefficient.Finally, the set of equations composed of ( 1), ( 4), and ( 6)-( 8) represents the coupled nonlinear model of the vaporizer.
In order to provide the basis for a feedback-based control system using output linear compensators, a critical step is to linearize the vaporizer model around its equilibrium points (EPs).This involves studying the steady-state behavior of the system to identify these points.By setting the time derivatives of the nonlinear model to zero and replacing the variables with their appropriate steady-state values, denoted by the capital letter and a superscript "ss," the steady-state model is achieved.This key process yields the following expression, as described in [7] and [12].
In the process of determining the EPs, the equations of the nonlinear vaporizer model are set to zero.It should be noted that the set of known and unknown variables in steady state are {Pi ss , Ps ss , Fv ss , K1 ss , K2 ss ; Ti ss } and {H ss , P ss , Ts ss } respectively.This procedure results in the establishment of the steady-state representation of the vaporizer model, defined in (9).
By replacing the numerical parameters and the steady-state variables of the vaporizer given in Table 4 and Table 5 respectively, and solving the equation system defined in (9), the EPs are calculated and derived as follows: H ss ≈ 50 %, P ss ≈ 18.2 kPa, and Ts ss ≈ 55.1 °C.In accordance with conventional process control practices, liquid levels are often expressed as percentages.Taking this into account and considering the maximum level, denoted as Hmax (see Table 4), the steady-state value of h(t), i.e., H ss is determined and expressed in percentage format [16].Moreover, it should be emphasized that the steady-state parameters and variables documented in Table 4 and in Table 5 are derived from different scenarios shown in references [1,12].These references are valuable sources that provide the essential parameters needed to understand the steadystate characteristics of the system.
For the sake of simplicity in the manipulation of the equations during the linearization process, it is beneficial to reformulate the model in the context of constant parameters.This effort leads to a new nonlinear model of the vaporizer, which is described in (10).
The constants Ks that are included in the model (10) are defined in (11).Formulating the model in this fashion allows for a simpler process of linearization and facilitates subsequent analysis aimed at the design of control systems.
Using the techniques of Taylor series expansion and perturbation described in references [10,11,17], the nonlinear model presented in (10) can be linearized.This process yields the linear state-space model, which is briefly expressed as follows In this model, the vectors representing state, input, and output variables are denoted as x, u, and y, respectively.Specifically, the state vector x is defined as x = [h(t), P(t), Ts(t)] T , the input vector u is defined as u = [fv(t), Pi(t), Pv(t), Ti(t), k1(t), k2(t)] T .For this particular scenario, the choice is to equate the output variables to the state variables, thus resulting in y = x.Symbolically, {x, y} ϵ {ℝ 3 } and u ϵ {ℝ 6 }.Regarding the assumption x = y, it should be noted that in a state-space model such as (12), the vector x and the output vector y are related, but not necessarily equivalent.In fact, the state vector x represents the internal state variables of the system, which describe the current state of the system.The output vector y, on the other hand, represents the measurable or observable quantities of interest [10,11,17].
In many cases, the vector y may be a subset of the vector x, which means that some state variables may correspond to measured outputs [10,11,17].In this study, the state variables are measured variables of the vaporizer.Therefore, as can be seen, there is a direct relationship between x and y.
In order to derive the expressions related to the matrices of the model (12), i.e., the state matrix A, the input matrix B, the output matrix C, and the direct-transmission matrix D, the dynamic equations in (10) are first redefined in terms of the g-functions, as is follows Then the vector function g is defined as g = [g1(fi(t), fv(t)), g2(fi(t), fv(t), Ti(t), Ts(t), P(t), h(t)), g3(ws(t), Ts(t), P(t))] T , where g ϵ {ℝ 3 }.Finally, taking into account g, x, u, and y, the matrices A, B, C, and D are defined in (14).From ( 14), Q ss is the vector of equilibrium points defined as Q ss = [H ss , P ss , Ts ss ], where Q ss ϵ {ℝ 3 }.Also, the functions gj, denoted by j ϵ {1, 2, 3, 4}, have been intentionally presented in (14) without explicitly including the time dependence of their component variables.This omission improves the clarity and readability of the (14).Moreover, the matrices I3x3 and 03x6 are the identity and zero matrices of dimension 3x3 and 3x6 respectively.Symbolically, {A, C} ϵ ℳ3x3 {ℝ} and {B, D} ϵ ℳ3x6 {ℝ}.
It should be noted that the Jacobians (equation ( 14)) play a key role in calculating the A, B, C, and D matrices for the linear state-space model.These matrices describe the system's dynamic behavior, input-output relationships, and interactions [10,11,17].The Jacobian matrix ∂g/∂x, evaluated at the equilibrium points grouped in Q ss , provides the matrix A, which represents the state transitions [10,11,17].Similarly, the Jacobian matrix ∂g/∂u, also at Q ss , builds matrix B, reflecting input influences on the system [10,11,17].Matrix C, which describes output relations, is derived from the Jacobian matrix ∂y/∂x evaluated at Q ss .This indicates how changes in state variables affect the output of the system.Finally, the D matrix, representing direct transmission, is derived from the Jacobian matrix ∂y/∂u, again at Q ss .It shows how changes in the inputs directly affect the outputs [10,11,17].
An important technical consideration arises when dealing with the derivation of the Jacobians in (14).This procedure requires a generalized format, since, in this case, the application of the Jacobians depends on the evaluation of the g-functions.In particular, this approach avoids the generation of long expressions that arise when Jacobians are evaluated based on the g-functions.
The section covers the dynamic modeling of a vaporization system, emphasizing equations that describe mass and energy balances.An equation of state linking pressure, temperature, and density is introduced.Control valve dynamics are discussed for both liquid and vapor services.The nonlinear model is established using gfunctions, setting the stage for linearization.The importance of Jacobian matrices is emphasized in the derivation of A, B, C, and D for the linear state-space model.Equating state and output variables is justified, simplifying the model.Overall, this section provides a thorough exploration of vaporization system dynamics.

Control System Design
As discussed in the previous sections of this article, the control system designed for the vaporizer was selected using two control structures.A single-loop structure and a cascade control-loop structure.The proposed control system is shown in Figure 2. The use of a single-loop structure and a cascade-loop control strategy in the proposed vaporizer process control system is based on a detailed analysis of the system dynamics and control goals (section 2 and section 3).The choice of a single-loop control structure for level control, as shown in Figure 2, is based on the need to maintain a constant h(t) within the vaporizer.By using valve 1 and its manipulating variable k1(t), this approach focuses the control objective on maintaining h(t) only.This emphasis allows the system to effectively deal with fluctuations in liquid flow rates and disturbances that directly affect the liquid level.This focused approach simplifies the control process, mitigates the potential for cross-variable disturbances, and establishes a robust and rapid h(t) control mechanism, thereby improving overall stability and reliability [10,11,17,18].
On the other hand, and according to Figure 2, the choice of a cascade-loop control structure for temperature (Ts(t)) and pressure (P(t)) regulation takes into account the complex relationship between these variables and their influence on vaporization efficiency.Temperature control within the vaporizer is closely related to the pressure of the liquid-phase.The cascade structure provides a good solution to address this coupling.The external temperature compensator (TC) focuses on precise temperature control, generating a reference pressure (P * (t)) that optimizes temperature control by adjusting the pressure by manipulating valve 2, i.e., k2(t).The internal pressure compensator (PC) then provides precise pressure control by fine-tuning the position of valve 2. This hierarchical approach addresses the fact that effective temperature control requires dynamic pressure adjustments, resulting in improved process efficiency and stability [10,11,17].
The combined use of these two control structures provides a balanced and comprehensive approach.The single-loop design provides direct and reliable control of the critical liquid level, ensuring consistent operation.At the same time, the cascade loop configuration handles the complex interaction between temperature and pressure to improve overall control performance and adaptability.This dual strategy uses the benefits of each approach to effectively address the process complexities of the vaporizer, resulting in enhanced stability, efficiency, and reaction time in real-world operating scenarios.
The study of the control system's process plants requires a transformation of the linear model expressed in (12) from its time domain to the Laplace domain (s-domain), which is achieved by applying the Laplace transform.As described in [12] and [11], the model initially defined in ( 12) is transformed to the s-domain, yielding the following expression: This equation, (15), incorporates a set of transfer functions (TFs) that closely represent the behavior of the system in the Laplace domain.
In (15), the identity matrix I shares equivalent dimensions with the matrix A, i.e., I3x3.Due to the complexity of the multi-variable process, equation ( 15) includes a set of 18 TFs that together build the linear model within the Laplace domain, customized to the specifics of the vaporizer process under study.The vectors Y(s) and U(s) are characterized as Y(s) = [H(s), P(s), Ts(s)] T and U(s) = [Pi(s), Ps(s), K1(s), K2(s), Fv(s), Ti(s)] T , respectively.It is worth noting that Y(s) ϵ {ℂ³}, while U(s) ϵ {ℂ⁶}.
From the information given in (15) and according to the TFs defined in Gij (s), a block diagram of the linear model in (12) can be determined in the s-domain.The s-domain model is shown in Figure 3.
After calculating the TFs and establishing the linear s-domain model of the vaporizer, the next task is to design the compensators that are essential for the operation of the system.In this context, linear feedback output compensators, specifically of the proportional-integral (PI) type, are selected for implementation.This choice is influenced by the extensive literature supporting the use of PI compensators in industrial systems.The benefits of using PI compensators are widely recognized, as evidenced by numerous articles discussing their effectiveness in improving system performance and control.
For instance, the article reported by [19], explores several applications of proportion-integral-derivative compensators and highlights their effectiveness in areas such as dc motor control, power system stability, automatic voltage regulation, controlled dc drives, time delay compensation, load frequency problems, and stability improvement.Specifically, PI compensators offer benefits such as improved stability and steady-state accuracy in load frequency control, precise time delay compensation, fast response and stability in adjustingvoltage regulators systems, accurate speed and torque control in dc motor control.However, careful tuning is critical to optimize performance and prevent transient overshoot.Overall, PI compensators play a key role in improving control processes in these applications.
The study in [20], discusses the application of PI compensators in several systems, highlighting their importance in achieving stability, improved performance, and effective control.The article discusses the use of PI compensators in systems such as robotic manipulators, industrial processes, and thermal plants.The benefits of using PI compensators include improved tracking accuracy, reduced steady-state error, and robust performance.The multivariable nature of the system is evident from the presence of 6 transfer functions per system output.This indicates that there are multiple input-output relationships that need to be considered in the control design and analysis of the vaporizer.
These compensators contribute to precise position control in robotic systems, precise control of industrial processes, and temperature control in thermal plants.However, it is important to carefully tune the compensator parameters to match the system characteristics and avoid problems such as overshooting.In summary, PI compensators play an important role in improving the performance and stability of control systems in a wide range of applications.
Reference [21] is a study that analyzes control strategies for a complex chemical process involving a reactor, flash tank, and recycle tank.The application and performance of PI control, dynamic matrix control (DMC), and generic model control (GMC) are examined through simulation.The process dynamics are represented by differential equations.The study compares these control methods for set point tracking and disturbance rejection.DMC emerges as the superior option, showing faster settling time and improved control performance compared to PI and GMC.This study emphasizes the importance of parameter tuning and control strategies for effective process control.This article also identifies some advantages of using a PI compensator.This compensator offers several advantages in industrial control applications.It is a fundamental and widely used control strategy due to its simplicity and robustness.PI control provides stability and steady-state accuracy for several processes.It is easy to implement and does not require complex calculations, making it suitable for real-time control systems.The integral action of the PI compensator helps eliminate steady-state errors and adapt to changes in set points or disturbances.PI compensators are effective for controlling processes with predictable behavior and minimal interactions.In addition, PI compensators are relatively easy to tune, making them suitable for a wide range of control tasks and industries.
From the information extracted in [19][20][21] and taking into account [10,11,17], it can be seen that choosing PI compensators over other control strategies is a reasoned decision based on a thorough evaluation of the benefits and drawbacks, taking into account the specific characteristics of the vaporizer process and the desired control objectives.A comprehensive analysis can be done as follow:  Benefits of PI compensators 1) Steady-state and transient performance: PI compensators provide accurate steady-state control and robust transient response.In the vaporizer process, maintaining precise temperature, pressure, and liquid levels is essential for efficiency.The integral action of PI compensators eliminates steady-state errors and ensures prompt disturbance rejection, thus achieving desired operating conditions effectively.2) Integral action for disturbance rejection: Vaporizer processes are often subject to disturbances such as varying liquid flow rates or external temperature changes.The integral term in PI compensators provides effective compensation for these disturbances by continuously adjusting the control effort based on the accumulated error.This integral action minimizes the effects of disturbances and increases system stability.3) Adaptability to process dynamics: PI compensators offer inherent adaptability to dynamic variations in the vaporizer process, such as changes in composition or heat transfer characteristics.Their proportional component facilitates rapid response to error changes, while the integral component allows control effort to be adjusted over time, ensuring consistent performance under varying conditions.4) Overshoot and settling time control: Achieving precise control with minimal overshoot and settling time is essential to prevent oscillation and ensure efficient use of energy.PI compensators balance fast response (proportional action) with optimal steady-state error correction (integral action) to provide stable control without excessive oscillation.5) Simplicity and ease of tuning: PI compensators have fewer adjustable parameters than complex strategies such as proportional-integral-derivative compensators or model-based approaches.This simplicity leads to easier tuning and implementation, reducing the risk of incorrect settings that could disrupt system behavior.6) Robustness and reliability: PI compensators are well established and widely used due to their robustness to parameter variations, uncertainties, and sensor noise.This reliability ensures consistent control performance over time, contributing to stable and reliable operation. Drawbacks of PI compensators: 1) Limited handling of complex dynamics: PI compensators may have difficulty handling highly nonlinear or time-varying processes that require more sophisticated control strategies.In such cases, advanced techniques such as model predictive control or adaptive control may provide better performance.2) Susceptibility to model mismatch: PI compensators rely on accurate process models for effective tuning.Model imprecision or changes in process behavior over time can lead to suboptimal control performance.3) Set point changes and large disturbances: While integral action compensates for steady-state errors, sudden large disturbances or set point changes can cause integral action to wind-up, resulting in overshoot or slow response until the accumulated error is resolved.4) Limited optimal performance for all scenarios: PI compensators are tuned for a specific range of operating conditions.Large changes in process dynamics or operating points can result in less than optimal control performance.In summary, the adoption of PI compensators for the vaporizer process balances their well-established benefits in achieving accurate and stable control with their drawbacks in handling complex dynamics and varying scenarios.The selection meets the process requirements for steady-state accuracy, robustness, and ease of operation, while recognizing the need for careful tuning and consideration of potential limitations in certain scenarios.
The PI compensator designs related to the single loop control structure and the cascade loop control structure are presented below.From Figure 2, it can be seen that the single-loop control is labeled as LC based on the control of the level h(t).On the other hand, the cascade loop is composed of a PC compensator (controls P(t)) and a TC (controls Ts(t)).

LC Control Loop Design
For the purpose of designing the LC compensator, the plant related to this control loop can be identified by considering Figure 3 and (15).The TF corresponds to the expression defined in (16).
3.161 1.0272 7.205 10 1.38 According to [17,22], using an approximate model of ( 16) of the first-or-plus-dead-time (FOPDT) type, it is possible to design PI compensators using tables already developed for this purpose.Also according to [22], the use of a FOPDT approximation for TF modeling and subsequent PI compensator design offers several important advantages.First, FOPDT models are simple and interpretable, providing a compact representation of the system dynamics.This simplicity facilitates initial analysis and design.Second, FOPDT models often provide physical insight into the dominant time constants and dead-time behavior, improving the understanding of the system.In addition, parameter estimation is more straightforward due to fewer model parameters, useful when working with limited or noisy data.FOPDT approximations provide essential frequency-domain characteristics to assist in control design and stability analysis.These models act as a starting point for PI controller tuning, ensuring effective control and stability.The computational efficiency of FOPDTs is suitable for real-time control, and their applicability to industrial processes emphasizes their relevance.In addition, FOPDT models are valuable teaching tools that bridge theory and real-world applications.
Overall, FOPDT approximations offer a balance of simplicity, accuracy, and ease of use, providing a strong base for designing effective control systems while allowing a deeper understanding of the fundamental dynamics.For the reasons given above, the model in ( 16) is approximated to an FOPTD model.In (17), the model in ( 16) is defined in its canonical form [10].
13 1 0.316 1 3.266 10 1.388 10 1 0.725 1 From ( 17), the MATLAB procest command is applied, which is a part of the MATLAB system identification toolbox [23].The MATLAB procest command is a powerful tool for identifying and modeling dynamic processes.It uses input-output data to estimate system dynamics by fitting models such as TFs or statespace models through optimization.It takes into account disturbances, noise, and time delays, enabling comprehensive system analysis and control design.First, a square-wave test signal is generated and used to stimulate (17).Next, a variable is formulated to simulate the output behavior of the model defined by (17).Once the procest command is performed, the approximate FODPT model is obtained as follows in (18).From here, the FOPTD parameters given by Kp = 3.2593×10 -13 , the pole time constant τp1 = 0.49975 s, and the dead time value τd = 0.0001 s are given.Use "getpvec", "getcov" for parameters a nd th Status: Estimated using PROCEST on time domain d ata "sys_iddata".
Fit to estimation data: 95.84% To compare model (17) with model ( 18), the compare MATLAB command is used.This command generates a plot of both models showing the error between the two models in percent.This plot is shown in Figure 4a.From Figure 4a, it can be seen that the approximate model in (18) is quite close to the original model in (17).In fact, the error between the two models when excited by a square-wave test signal is close to 4.13 % (see equation ( 18)).
Finally, Figure 4b shows the step responses of the model in (17) and the model in (18).It can also be qualitatively verified that both curves are close, thus verifying that the approximation is proper.17) and ( 18).(a) Comparison of simulated responses based on ( 17) and (18).The error between the two models is 4.13 %.The sysiddata signal is the model in (17) in input/output object format.The sysFOPDT signal is the approximate model in (18).(b) Step response for models (17) and (18).The next step is to tune the PI compensator.For this purpose, the transfer function Gc(s) of the PI compensator in the s-domain is defined as follows Here, kc and τi represent the tuning parameters of the compensator, which are the gain and the integral time (reset) of the compensator, respectively [10].These parameters are determined using the off-line Ziegler-Nichols method [24] applied to processes represented by FOPTD models.This process follows a control diagram with a unitary feedback loop, as shown in Figure 5a.According to [24] and considering (18), the specific formula for calculating the tuning parameters of the compensator is derived as follows Evaluating (20), the values found for the tuning parameters are kc = 1.533•10 16 and τi = 3.33•10 -4 s.These initial values operate as a starting point for effective configuration of the PI compensator.However, it is important to note that these parameters are subject to refinement once the PI compensator is integrated into the system simulator, for example.
This tuning process involves fine-tuning kc and τi by making incremental changes based on the system's response characteristics and desired performance criteria.Typically, manual iteration is used, where kc and τi are adjusted incrementally in accordance with observed system behavior and target objectives [12,24].

Design of the Cascade Control Loop
To design the cascade control loop, the control diagram shown in Figure 5b is drawn, including an inner loop configured by the PI compensator Gcs(s) (PC) and the plant G24(s), and an outer loop composed of the transfer function P * (s)/P(s), the gain KTs, and the PI compensator labeled Gcm(s) (TC).From Figure 5b, the plant G24(s) is derived from the solution of ( 15), which is defined as follows Using a methodology similar to that used to derive (18), the analogous process is applied to the model defined in (22) to obtain its approximate FOPDT expression defined in (23), where the FOPTD parameters given by Kp = 0.70326, the pole time constant τp1 = 83.457s, and the dead time value τd = 0.7161 s are derived.Note that the error between the models (22) and ( 23) is comparatively smaller than the error between the expressions (17) and (18).This error is quantified to be 0.32 %, highlighting the close similarity in dynamics between models (22) and (23).Finally, Figure 6 shows the step responses of models (22) and (23).From Figure 6, the similarities of their dynamics can be verified again.Using the expressions presented in (19) and (20) and in accordance with the model given in (23) and [24], the tuning parameters for the compensator Gcs(s) are derived.That is kc = 1.422 and τi = 2.3846 s.Step response for models (22) and (23).

 
Finally, based on expression (15) and considering Figure 3, a direct relationship between P(s) and KTs can be established, which can be expressed as P(s) ≈ 0.125•Ts(s).This implies that Ts(s) ≈ 8•P(s), and therefore KTs = 8.
Using the information developed so far, it is possible to reduce the diagram shown in Figure 5b by applying the block reduction technique [10].The reduced diagram of Figure 5b is shown in Figure 7.The Gcm(s) compensator is designed using frequency techniques and the concept of phase margin (PM).Phase margin is a stability measure in control theory that evaluates a system's ability to handle disturbances.It quantifies the phase shift introduced by the system when the open-loop gain exceeds unity (0 dB).A higher phase margin means better stability and noise rejection, while a lower margin can lead to oscillations or instability.It is measured in degrees as the difference between 180°and the phase angle at unity gain frequency.Systems with a phase margin greater than 45°-60°are stable and robust, ensuring reliable performance [10,11,17].
As mentioned above, one of the key parameters in this design is the determination and adjustment of frequency (fs) at which the compensator should be designed.To determine the fs, the Bode diagram of the transfer function P * (s)/Ts(s) shown in Figure 7 is taken and plotted in Figure 8.It should be noted that, the expression for G24(s) using in Figure 7 corresponds to (22).From Figure 8, it can be seen that the crossover frequency (fc) is approximately 0.22 Hz.This value of fc is reasonable considering that the process involved is typically characterized by low-frequency dynamics, such as thermal and level processes [1,12].Therefore, according to [10,11,17], a fs that is an order of magnitude smaller than fc can be chosen to avoid potential frequency disturbances in the control elements.Thus, fs = fc /10.
Based on Figure 7 and by removing the feedback path, the transfer function of the open-loop model is defined in as follows where kcm and τim are the tuning parameters of the compensator Gcm(s).It should be noted that in (24) the only unknown parameters are kcm and τim, since the other TFs were already evaluated previously.Finally, the calculation of the parameters kcm and τim are obtained by solving the system of equations defined in (25), where, ws = 2••fs.Also, pm is the angle of the PM, which in this case is set to 60°.The tuning parameters of the Gcm(s) are derived and set to kcm ≈ 0.08 and τim ≈ 0.01.

Journal of Electronics and Electrical Engineering
Thus far, the focus has been on developing compensators in the continuous time domain.Nevertheless, a significant determinative in control systems involves selecting between continuous and discrete compensators.
Continuous compensators have traditionally been more extensively used as they easily integrate with analog systems [25].Nonetheless, they are subject to limitations.Their performance is affected by the physical constraints of analog hardware as they work in the analog domain [25].The digital revolution has introduced discrete compensators as an alternative for digital control systems.Ideal for applications requiring precise realtime control and adaptability, these compensators use digital signal processing to enable rapid sampling, adaptive control, and advanced signal processing [26,27].Nevertheless, they also present certain challenges, such as potential problems with sampling, quantization, and numerical stability [26,27].The choice between continuous or discrete compensators depends on control system requirements and the underlying hardware.While continuous compensators have been established and are applicable for various functions, discrete compensators offer advantages for digital control and adaptability [26,27].
It is important to objectively evaluate the performance of compensators designed in continuous time versus those designed in discrete time, as previously noted.The discrete compensators are achieved through a method that converts their continuous time counterparts into discrete versions, as described in [28].This conversion was performed using the MATLAB-Simulink control systems toolbox.The integral terms of the PI compensators are estimated with their bilinear form, using the trapezoidal method.A sampling time of 0.1 seconds was specified for the design.The tuning parameters of each compensator were adjusted through heuristics by trial and error.Table 6 displays these tuning parameters.This section discusses the design of a control system for a vaporizer process using both single-loop and cascade control structures.The single-loop approach focuses on maintaining a steady liquid level, while the cascade structure addresses the complex relationship between temperature and pressure.The mathematical model of the control system is translated to the Laplace domain, allowing PI compensators to be designed using FOPTD approximations.Cascade loop design utilizes frequency techniques and PM considerations.The design process carefully weighs the advantages and disadvantages of PI compensators and effectively utilizes the advantages of cascade control.The result is a comprehensive control strategy that increases the stability and control of the vaporizer process.

Simulation Results
The simulation results utilized the MATLAB-Simulink platform to model and evaluate the system defined by ( 10) and ( 11), using values from Table 4 and Table 5.The outcomes displayed in Figure 9 and 10 provide valuable insights into the transient behaviors of variables related to both continuous-time compensators (hc(t), Tsc(t), and Pc(t)) and discrete-time compensators (hd(t), Tsd(t), and Pd(t)).Additionally, the analysis in Figure 11 examines k1c(t), k1d(t), k2c(t), and k2d(t) dynamics.In the subsequent analysis, the generic dynamics of the variables displayed in Figure 9-Figure 11 will be studied without specifying their origin from continuous or discrete compensators.During the initial stages of the system when both valves are fully closed, specific setpoints were established at h(t) = 50 % and Ts(t) = 80°C.Based on the simulation results in Figure 9, h(t), Ts(t), and P(t) steadily converge to their desired values, with precise values of 50 %, 47.22 kPa, and 80°C at 20 s, 100 s, and 120 s, respectively.Similarly, it can be observed from Figure 10 that both fi(t) and ws(t) attain their steady-state levels at 92.246 s and 124.665 s, with values of 45.31 m3/s and 16.59 kg/s, respectively.Furthermore, Figure 11 demonstrates that k1(t) and k2(t) also reach their steady-state positions at 126.954 s and 150 s, respectively, with values of 0.613 pu (61.3 %) and 0.421 pu (42.1 %).At 200 s, a perturbation is introduced in Ts(t), which changes it to 60°C and then stabilizes around this value for about 274 s.This perturbation has a ripple effect on fi(t) and ws(t), which both experience a decrease.This decrease can be attributed to the fact that the temperature change affects the efficiency of the evaporation process, requiring adjustments in fi(t) and ws(t).These changes are reflected in the k1(t) and k2(t) positions, where k2(t) goes through a complete cycle from full closure to full opening before returning to its predisturbance level at 562.3 s.
Then, at 400 s, another perturbation affects h(t), increasing it to 80% at 450 s.The perturbation affects variables such as Ts(t), P(t), ws(t), and fi(t).While Ts(t) is disturbed, it returns to its pre-disturbance steady state value.The P(t) experiences a similar perturbation, but stabilizes at 47.22 kPa before the perturbation.Notably, the behavior of the system during this perturbation appears more pronounced, particularly in Ts(t), indicating its effect on the energy and mass balances.The flow variable fi(t) stabilizes at a new value of 200 m 3 /s, significantly higher than its initial value of 39.25 m 3 /s.
Analyzing the data presented in Figure 9 and Figure 11, it is clear that the variables controlled by discrete compensators show superior performance in terms of the defined FoM, specifically overshoot (OS) and steadystate error (SE).The values in Table 7 confirm that hd(t), Tsd(t), and Pd(t) consistently exhibit significantly lower FoM values compared to their continuous-time counterparts (hc(t), Tsc(t), and Pc(t)).The implication is that the system's discrete compensators result in lower OS, which is a significant advantage.Furthermore, in the 200 second, the OS of variables such as hc(t) and Pc(t) improves significantly, and Tsc(t) even eliminates the OS altogether.Most importantly, hd(t), Tsd(t), and Pd(t) show remarkable reductions in SE, further demonstrating their superior performance.Throughout, minimal SE values reinforce the system's rapid stabilization around set points.
Although the system does not fully meet the specified constraints on OS and SE, where the values should not exceed 5 % and 1 %, respectively, the system begins to meet the OS requirements after the system change at 200 s, while SE remains consistently well below the defined constraint.This is true for all variables (hi(t), Tsi(t), and Pi(t), where i ϵ {c, d}).These results highlight the potential for further optimization to minimize the initial overshoot during system startup and to fine-tune system performance to meet even more stringent settling error targets.
In conclusion, the simulation results highlight the effectiveness of the designed compensators in achieving stable and responsive control in the vaporizer process, and in particular, the superiority of discrete-time compensators in terms of overshoot and settling error.

Conclusions
In this thoroughly designed and rigorously reviewed study, a comprehensive analysis of a vaporizer process and its control system is presented.The introduction provides a fundamental understanding of vaporizer systems' importance in industrial applications and the critical requirement for advanced control strategies to achieve efficiency and stability.The vaporization process description describes the complexities of the vaporization process, highlighting critical parameters such as liquid level, temperature, and pressure that require precise control for optimal performance.
The process modeling section provided the foundation for the subsequent control system design.Complex mathematical formulations (equations (10) and (11)) and data from Tables 4 and 5 played a vital role in formulating the models.The detailed outline of the vaporization process framework included continuous-time compensators (hc(t), Tsc(t), and Pc(t)) as well as discrete-time compensators (hd(t), Tsd(t), and Pd(t)).This framework laid the foundation for the central focus of this study: control system design.
The design of the control system was based on a thorough comprehension of process dynamics, leading to the development of compensators that demonstrated accuracy in maintaining setpoints as well as resilience against perturbations.The use of the MATLAB-Simulink platform for simulations provided significant insights into the transient behaviors of these variables.The control system's ability to direct the process towards desired conditions while withstanding perturbations like temperature and liquid level changes demonstrated its efficacy.
One of the key findings was the significant advantages provided by the discrete-time compensators (hd(t), Tsd(t), and Pd(t)) in terms of critical performance metrics, particularly overshoot and steady-state error.The findings in Table 6 confirm that discrete-time compensators consistently perform better than their continuoustime counterparts.This advantage arises from the reduced overshoot in discrete compensators, which improves the overall system stability.Additionally, the substantial reduction in steady-state error results in a faster convergence of the system to setpoints.
Although the system did not entirely satisfy the predefined constraints regarding overshoot and settling error, where values should not exceed 5 % and 1 %, respectively, there was a noticeable improvement in overshoot after the system changes at 200 seconds.The steady-state error consistently remained low, indicating promising opportunities for continued optimization.
The study's conclusion emphasizes the strong effectiveness of the designed compensators in achieving stability and responsiveness in the vaporization process.Additionally, it highlights the superior performance of discrete-time compensators in minimizing overshoot and settling errors.This work provides a strong foundation for future improvements and optimizations to meet more challenging control objectives and ensure process reliability in industrial applications.

Figure 2 .
Figure 2.This figure shows the control structure of the vaporizer with two loops: a level control loop with a level compensator (LC) and a cascade control structure with a temperature compensator (TC) and a pressure compensator (PC).Both loops use control valves to effectively regulate temperature, pressure and liquid level.This integration increases system efficiency and ensures optimum operation.

Figure 3 .
Figure3.The block diagram of the vaporizer process represents its overall structure.The multivariable nature of the system is evident from the presence of 6 transfer functions per system output.This indicates that there are multiple input-output relationships that need to be considered in the control design and analysis of the vaporizer.

Figure 4 .
Figure 4. Dynamics under excitation of models (17) and (18).(a) Comparison of simulated responses based on (17) and(18).The error between the two models is 4.13 %.The sysiddata signal is the model in(17) in input/output object format.The sysFOPDT signal is the approximate model in(18).(b) Step response for models(17) and(18).

Figure 5 .
Figure 5. Block diagrams of the vaporizer control loops.(a) Control diagram of the level h(t) built by a single feedback loop.(b) Cascade control loop regulating the temperature Ts(t) and the pressure P(t).The two control loops can be seen; the inner and the outer one.

Figure 7 .
Figure 7.A simplified model of the block diagram shown in Figure 5b.

Figure 8 .
Figure 8. Bode diagram of the transfer function P*(s)/Ts(s) in open-loop.In this plot, fc  0.22 Hz.        

Figure 9 .
Figure 9. Simulation results under transient operation of the process taking place in the vaporizer.Reference values h * (t) = 50 %, and Ts * (t) = 80 °C.Step change in Ts(t) and h(t) at 200 s and 400 s respectively.(a).Dynamic of hc(t) and hd(t).(b) Dynamic of Tsc(t) and Tsd(t).(c) Dynamic of Pc(t) and Pd(t)

Figure 10 .
Figure 10.Simulation results under transient operation of the process taking place in the vaporizer.Reference values h * (t) = 50 %, and Ts * (t) = 80 °C.Step change in Ts(t) and h(t) at 200 s and 400 s respectively.Dynamic of fi(t).(b) Dynamic of ws(t).

Figure 11 .
Figure 11.Simulation results under transient operation of the process taking place in the vaporizer.Reference values h * (t) = 50 %, and Ts * (t) = 80 °C.Step change in Ts(t) and h(t) at 200 s and 400 s respectively.(a) Dynamic of k1c(t) and k1d(t).(b) Dynamic of k2c(t) and k2d(t).

Table 1 .
Summary of Process Variables

Table 2 .
Summary of Variables and Functions in Control Structures for Vaporization System

Table 3 .
Detailed Control Specifications for Vaporizer Process

Table 4 .
Parameters of the Process

Table 5 .
Steady-State Variables of the process

Table 6 .
Tuning Parameters of the Vaporizer Digital Compensators

Table 7 .
Summary of FoMs, i.e., OS and SE regarding operation of the Vaporizer