Process Scheduling Analysis and Dynamic Optimization Maintaining the Operation Margin for the Acetylene Hydrogenation Fixed-Bed Reactor

Abstract

The full-cycle operation optimization of the acetylene hydrogenation reactor should strictly adhere to the operation optimization scheme within the operation cycle, regardless of scheduling changes. However, in actual industrial processes, in order to meet temporary process scheduling requirements, the acetylene hydrogenation reactor needs to adjust its operation strategy temporarily within the remaining operation cycle based on the results of dynamic optimization for a certain period. It brings additional challenges and a research gap to the operational optimization problem. To make up for this research gap, this paper focuses on researching a type of full-cycle dynamic optimization problem where the operation optimization scheme is temporarily adjusted during the operation cycle. The methods employed for changing the operation optimization scheme include modifying the operation cycle, maximizing economic benefits, and altering the optimization goal to maximize the operation cycle. A novelty full-cycle scheduling optimization framework based on surplus margin estimate is proposed to build a platform for these methods. The paper analyzes the impact of process scheduling changes on full-cycle optimization using a dynamic optimization model that maintains the operation margin. It establishes a full-cycle scheduling optimization model and obtains the optimal scheduling strategy by a novelty method NSGBD (non-convex sensitivity-based generalized Benders decomposition). In this process, an adaptive CVP (control vector parameterization) based on a decomposition optimization algorithm is proposed, which tackles the challenge of optimizing complex acetylene hydrogenation reactor models on a large time scale. Scheduling optimization can be realized as an annualized benefit of 1.56 × 10⁶ and 1.57 × 10⁶ ¥ separately within two scheduling optimization constraints, and the computational time required is much less than previous operational optimizations.

1. Introduction

The acetylene hydrogenation reactor serves as a critical component in the ethylene industry, primarily responsible for removing trace amounts of acetylene impurities in high-concentration ethylene streams, to ensure product purity and smooth operation in subsequent production processes. Fixed-bed adiabatic reactors are widely utilized for hydrogenation in actual ethylene refining units and typically encompass three bed layers for precise control of hydrogenation depth. The principal reaction within the reactor is catalytic hydrogenation, where acetylene is converted to ethylene via hydrogenation, and during operation, catalyst activity and selectivity will gradually decline. Due to the impact of catalyst deactivation, the reactor will shut down after a period of operation to regenerate the catalyst. The long-cycle and slow-time-varying characteristics of the acetylene hydrogenation reactor enable full-cycle control and optimization to significantly prolong the reactor’s operating cycle and increase overall profitability.

The operation optimization of the acetylene hydrogenation reactor, in current literature, is based on the premise that the operation optimization strategy remains unchanged in the full cycle. However, in reality, the operation demands of the reactor may vary due to unforeseen circumstances or scheduling requirements, which leads to modifications of the operation strategy throughout the cycle.

• Research Gap: Consequently, a dynamic optimization model capable of adapting to the changing operation margin is required to obtain the optimal solution. It is a research gap for the full-cycle operation optimization of the acetylene hydrogenation reactor; adding scheduling changes means that the difficulty of system control and optimization will be substantially increased.
• Contribution: Discussion of the temporarily changed optimization strategy is the significance of the problem of the surplus operation cycle and economic benefit due to the temporary optimization strategy adjustment estimated in the long-cycle operation system, and the full-cycle dynamic optimization maintaining the operation margin of several optimization strategies in the full cycle.
• Contribution: The optimal surplus margin release characteristic is closely related to the set operation cycle for slow-time-varying systems. When the operation cycle is shortened, the surplus margin is released more quickly, resulting in an increase in the average daily economic benefit. Conversely, if the cycle is prolonged, the surplus margin is released at a slower rate, leading to a higher overall economic benefit for the entire cycle. Once the maximum cycle for optimized operation is reached, this system performance will be constrained within the boundary. It is crucial to ensure that the dynamic optimization model maintaining the operation margin with temporary changes in the operation strategy follows this rule. Based on this rule, the study contributes to developing a scheduling optimization framework based on surplus margin estimation, which yields an optimal scheduling strategy. Furthermore, the study conducts a detailed analysis and improvement of the optimization algorithm to reduce resource consumption and enhance efficiency.

2. Problem Definition and Literature Review

2.1. Problem Definition

For example, as shown in Figure 1, considering the specific requirements of industrial scheduling, the operation cycle is temporarily changed at a certain time point (point D in Figure 1) in the operation process, and the optimal released characteristic of the margin should also be changed accordingly. As shown in the dotted line in Figure 1, “curve a” and “curve c” are the curves of economic benefit and available surplus obtained by temporarily increasing and decreasing the regeneration cycle, respectively. “Curve b” is the curve of economic benefit and available surplus obtained by operating according to the original optimized operation plan without temporary change. When the operation cycle is shortened, it generates a surplus margin that is faster released (the available surplus margin of “curve a” in the figure is exhausted more rapidly). On the contrary, with the operation cycle lengthened, it generates a slowly released margin (the available surplus of “curve b” in the figure decreases more slowly), to ensure the process performance of the device. As the challenge of the temporarily changed operation cycle for the full-cycle dynamic optimization problems presented, the scheduling optimization problem is considered based on the allowance of slow-release operation optimization, the established acetylene hydrogenation operation optimization model of the full cycle of margin analysis, including: changed the operation cycle with maximized economic benefit and changed the optimization goal with maximized operation cycle. A full-cycle dynamic optimization model maintaining the operation margin including scheduling optimization is established, and the optimal scheduling and operation strategies are obtained.

Detailed description of the problem, as shown in Figure 2, includes the process scheduling, changes in the time domain D optimization strategy, the objective function changes, or switch system internal structure change, the permissible range of maximum running cycle within F, with a need to find the optimal process scheduling time ${\Gamma }_{\mathrm{c}}^{*}$ (${\Gamma }_{\mathrm{c}}^{*}\in D$) and the optimal maximum running cycle ${\Gamma }_f^{*}$(${\Gamma }_f^{*}\in F$). In the actual production process, especially in the slow-time-varying system with a full-cycle operation cycle, it occurs with a temporary scheduling situation. Considering scheduling in a certain time domain (changed the operation optimization strategy) in full cycle, it is required to ensure the optimal performance. As the fixed process scheduling with scheduling adjustment and strategy adjustment, the optimization problem is considered a scheduling optimization problem nested in the original dynamic optimization problem. Hence, the operation framework of full-cycle dynamic optimization cannot be used as a solution. To obtain the optimal scheduling policy and solve the optimal control trajectory under the optimal scheduling policy, the kind of integrated control optimization and scheduling full-cycle dynamic optimization problem is proposed. Considering solving the optimal control trajectory, full-cycle performance and process index optimization are obtained. Figure 1 is only a summary representation of the full-cycle optimization problem that is integrated control optimization and scheduling. In the actual process, multiple switches of various optimization strategies are involved, and the internal structure of the system is also changed, making the problem more complex than the general full-cycle dynamic optimization problem.

In summary, this paper aims to achieve the following three objectives: 1. Establish a scheduling optimization framework for the acetylene hydrogenation reactor to further address and respond to temporary scheduling optimization issues that arise in practical production. (Previous studies only focused on optimizing objectives throughout the entire operational cycle with unchanging strategies.) 2. Investigate and analyze the performance of scheduling optimization using the example of temporarily changing optimization strategies during the optimization process. 3. Implement full-cycle dynamic optimization maintaining the operation margin for the acetylene hydrogenation reactor, incorporating scheduling optimization.

2.2. Literature Review

Utilizing the hydrogenation reaction of acetylene in the reactor, its primary function is to convert trace amounts of acetylene into ethylene within a high-concentration ethylene flow, thereby preventing the subsequent catalyst poisoning during ethylene polymerization [1,2,3,4,5]. To meet the process requirements for the outlet acetylene content, selective catalytic hydrogenation of the trace acetylene is induced using a palladium catalyst. Thus, the catalyst’s activity becomes a crucial indicator of the plant’s operational performance and economic benefits. Within the reactor, the catalyst is influenced by temperature and reaction by-products, causing its activity to gradually decline. When the activity reaches a certain level where selectivity no longer complies with process requirements, reactor shutdown becomes necessary, and catalyst regeneration can be initiated. The acetylene hydrogenation reactor model is one of the research focuses in this field, including the gas–solid phase model by CFD Simulation [1], the hybrid model by self-adaptive iterative [2], microkinetic analysis of the acetylene hydrogenation reactor model [3], a surrogate model of the acetylene hydrogenation reactor [4], and Kinetics Insights and Active Sites Discrimination of the catalyst model [5].

The catalyst’s activity and selectivity in the reactor can be adjusted by controlling the temperature and the amount of hydrogenation at the inlet, offering significant opportunities for operational optimization. The key requirement for optimizing operations is to establish a precise model of the acetylene hydrogenation reactor. There are two main types of models for the acetylene hydrogenation reactor: the quasi-homogeneous model [6,7,8] and the heterogeneous model [9,10]. In the early research, the quasi-homogeneous model was used for calculation, and relevant results were obtained in modeling [7], control [6], and optimization [8]. The quasi-homogeneous model is relatively straightforward to implement, while the heterogeneous model [9] provides more accurate calculations but is more complex to implement [10]. In this study, an accurate catalyst deactivation model is established based on the mechanics of catalyst deactivation to account for factors such as reactor temperature and the cumulative effect of oligomers on catalyst activity [11]. Additionally, considering the challenge of detecting changes in catalyst activity over extended periods of operation, an efficient online activity estimator is designed to accurately estimate activity throughout the entire operational cycle [12]. By incorporating these models and estimators, the operation of the acetylene hydrogenation reactor can be optimized for improved performance.

Currently, the focus is on optimizing the operation of the acetylene hydrogenation reactor by taking into account the slow-time-varying characteristics that align with the changes in catalyst activity and the optimization objectives related to economic benefits, production indexes, and energy consumption. Researchers have established a mathematical optimization model to achieve full-cycle operation optimization of the reactor [11,12,13,14]. By conducting dynamic analysis, the design margin requirements for process control performance can be identified by analysis of a fluid catalytic cracking unit regenerator [15], heat exchanger network [16] and used for online optimization [17], life-cycle energy saving [18], and a multistage dynamic optimization method for long-term planning [19]. To ensure proper operation and control, it is necessary to maintain a certain margin to accommodate unforeseen factors. The extent of the design allowance depends on the level of control performance required, which is linked to control system design. The design margin consists of the process margin and control margin [20]. In practical applications, particularly in the acetylene hydrogenation reactor, a regeneration cycle can last several months, during which various factors, such as the expected impact of catalyst activity, may affect the process. During the design stage, the process margin is determined by calculating the portion of the design margin that can handle the “worst” impact of these factors. On the other hand, the control margin is part of the design margin retained to counter unpredictable influencing factors [21], which can be compensated by the control system’s operation. If the process margin does not reach the expected “worst” situation, the design margin is not fully utilized. By considering the consumed control margin and the released process margin, there is an untapped surplus margin, which provides operational flexibility within the optimization model.

To fully utilize the surplus margin, it is suggested to adopt an integrated approach that includes control and operational optimization. Currently, the “back-off” method is a well-known approach that can assess the impact of uncertain conditions on the system. This method is suitable for steady-state optimization models [22,23,24] as well as more complex nonlinear dynamic optimization models [25,26,27,28,29]. In recent years, this method is still an efficient method for simultaneous design and nonlinear model predictive control under uncertainty [30]. Similar methods can further integrate design, control, and scheduling [31], yielding enhanced control and optimization outcomes. Non-convex sensitivity analysis is the basis of decomposition optimization and solving hybrid problems in this study. The sensitivity analysis is carried out to study the behavior of the key parameters for varying decision-maker preferences, and significant strategy insights are obtained [32]. The global optimality of the cost function and decision variables are validated through classical optimization. The numerical examples confirm analytical results, and sensitivity analysis is provided for different parameters [33]. Xie et al. conducted research on an acetylene hydrogenation reactor and developed a framework for integrated optimization and control of a slow-time-varying system. They proposed a dynamic optimization method called the “margin sustained-release operation optimization method,” which effectively exploits the surplus margin [34]. In simpler terms, this method implements comprehensive dynamic optimization throughout the entire operational cycle to maintain the operational margin. It utilizes a margin estimator to estimate the surplus margin at each stage of the cycle. By optimizing the system’s performance, it maximizes the utilization of the surplus margin.

In the process of full-cycle operation optimization, if the surplus margin is consumed in advance, the operation cycle will be shortened. In the design stage, the performance of the plant is required to reach the process constraint boundary at the end of the operation cycle so that the slow-time-varying parameters reach the “worst” situation, and the surplus margin is fully released. The optimal release characteristics of the surplus margin may be correlated to the change patterns of the slow-time-varying parameters, while the release mechanism is determined by the setting of the operational cycle. In industrial production, the operation optimization scheme is changed temporarily due to production scheduling [35]. After running according to the original operation optimization strategy for a certain period, considering the temporarily changed operation strategy, the optimal release characteristics of the surplus margin and the results of full-cycle optimization could be different from the original full-cycle operation optimization that was discussed in previous literature. In this paper, based on a two-dimensional heterogeneous acetylene hydrogenation reactor model, a full-cycle dynamic optimization problem with temporary changes in the operational optimization scheme is proposed. The methods for changing the operational optimization schemes include: altering the operational cycle to maximize economic benefit and changing the optimization objective function to maximize the operational cycle. Analyzed operation optimization schemes, the influence of operation schemes on the margin release characteristics, operation cycle, and economic benefit are discussed, and the optimal scheduling strategy is solved by establishing a schedule optimization model.

3. Analysis of the Influence of Process Scheduling on Optimization Strategy

3.1. Assumptions

The results of this study are based on many assumptions, including modeling and algorithm. The relevant assumptions of the study in terms of modeling are as follows: A fixed-bed adiabatic reactor was used for acetylene removal; The device comprises three bed layers; Hydrogen enters the reactor at a constant rate and reacts adequately; The by-products in the outlet of the previous bed are cleaned and not carried into the next bed; Axial heat and diffusion are ignored in the model; No structural changes are observed in the reactor; It is assumed that the temperature and concentration of the gas–solid phase of the main body are different; The radial axial flow of green oil is not considered. The relevant assumptions of the study in terms of algorithm are as follows: After the reactor reaches the critical value of inactivation, it stops running; The delay of data transmission between systems with different time scales is not considered; Scheduling changes that change the system structure, such as adding a reactor bed, are not considered; The optimizer only considers the daily steady-state process, and the dynamic effects brought by the controller are substituted into the optimization framework in the form of control margins; The scheduling optimization problem satisfies the quasi-convex condition; In the process of NSGBD iteration, the solutions of the original problems are independent of each other; The optimization process of decomposition algorithm is carried out separately.

3.2. Full-Cycle Optimization Framework

The basis of this paper is the Acetylene Hydrogenation Reactor model, which incorporates a previous dynamic model of the reactor and a two-dimensional heterogeneous model that considers the catalyst deactivation mechanism and oligomer accumulation, as described [12]. By combining the catalyst deactivation mechanism and the two-dimensional heterogeneous model, the accuracy of the optimization is improved, as highlighted [10]. This model accurately represents the dynamic characteristics of the system affected by catalyst deactivation over a long period, fulfilling the accuracy requirements for optimizing the Acetylene Hydrogenation Reactor throughout its full cycle.

In this paper, we present an optimization model for the sustained-release operation of the Acetylene Hydrogenation Reactor, which considers a specific situation where the optimization strategy changes at a certain point in the operational cycle. Sustained-release operation optimization, proposed in [34], is an approach that estimates and gradually releases the residual margin throughout the entire operational cycle, creating more flexibility for optimization and control and improving overall optimization and control performance. Our prior work implemented this method in the Acetylene Hydrogenation Reactor, resulting in greater economic benefits compared to traditional full-cycle dynamic optimization approaches that retain all design margins for optimization. Additionally, we have established a general margin sustained-release dynamic optimization model, which has been proven suitable for slow-time-varying systems through derivation. Thus, the optimization strategy changes presented in this paper are based on the findings in Xie et al.’s literature from 2020.

In addition, due to the systems that exist in the different time scale (the slow-time-varying parameters changes are too slow compared to the rate of the control system response, unable to control and optimize under the same time scale), previous work has also established full-cycle control optimization framework slow-time-varying systems [34] that build fast and slow-time-varying systems and control optimization, respectively. In this paper, the optimization strategy for the entire cycle requires temporary changes but can be considered as an initial value different allowance dynamic optimization problem to maintain the operational margin. Therefore, the corresponding control optimization framework is similar to the previous work. Figure 3 is the control optimization frame diagram after temporarily changing the optimization strategy; ${\Gamma }_{\mathrm{c}}$ represents the point in time when the optimization policy was temporarily changed, ${\Gamma }_f^{\prime }$ represents the operation period of the reactor after the temporary change (${\Gamma }_f$ is the period before the change). The system’s surplus margin also needs to be re-estimated based on the catalyst activity ${\theta }_{\mathrm{c}}$ at the ${\Gamma }_{\mathrm{c}}$ day and state variables of the system ${\mathit{x}}_{\mathit{c}}$. The specific margin estimation model and optimization model (Models 1–4) in the figure will be described in detail. Based on the control optimization framework, the dynamic optimization that temporarily changes the operating period must re-estimate the surplus margin in the time domain after changing the operating period, so that the operating dynamic optimization maintains the operation margin. Therefore, the estimation models of the process margin, control margin, and optimal design quantity should be established.

3.3. Change Optimization Strategy

In the frame of operation optimization maintaining the operation margin, the estimation of the process margin, control margin, and surplus margin can be realized by solving the optimization model. In order to construct an objective function for dynamic optimization aimed at maximizing the economic benefits of a certain regeneration cycle over its full cycle, it is necessary to quantify the economic benefits of reactants and products. Ea, Eb, and Ec, respectively, represent the economic benefit functions of acetylene, ethylene, and hydrogen on the $\mathit{\Gamma }$ day.

$$E_{\mathrm{a}}(\Gamma )=\frac{{\mathrm{M}}_{\mathrm{a}}}{\tau }{\displaystyle \sum _{R=0}^{R_{\tau }}{\displaystyle \sum _{k=1}^3(p_{\mathrm{a}k}^{\mathrm{s}}(1,R,\Gamma )-p_{\mathrm{a}k}^{\mathrm{s}}(0,R,\Gamma )})}$$

(1)

$$E_{\mathrm{b}}(\Gamma )=\frac{{\mathrm{M}}_{\mathrm{b}}}{\tau }{\displaystyle \sum _{R=0}^{R_{\tau }}{p}_{\mathrm{b}3}^{\mathrm{s}}(1,R,\Gamma )}$$

(2)

$$E_{\mathrm{c}}(\Gamma )=\frac{{\mathrm{M}}_{\mathrm{c}}}{\tau }{\displaystyle \sum _{R=0}^{R_{\tau }}{\displaystyle \sum _{k=1}^3(p_{\mathrm{c}k}^{\mathrm{s}}(1,R,\Gamma )-p_{\mathrm{c}k}^{\mathrm{s}}(0,R,\Gamma )})}$$

(3)

The partial pressure of the gas in the reactor is represented as P. Subscripts a, b, and c are, respectively, acetylene, ethylene, and hydrogen. Subscript k is the serial number of the reactor bed, k = 1, 2, 3. The superscript s represents the catalyst phase of the heterogeneous model. R represents the length of the reactor, and $R_{\tau }$ represents the discrete section of the reactor wall. According to Equations (1)–(3), the expression of the overall economic benefit in the time domain after the operation cycle is temporarily changed is represented as Equation (4).

$$J=-{\displaystyle {\int }_{{\Gamma }_{\mathrm{c}}}^{{\Gamma }_f^{\prime }}(E_{\mathrm{b}}(\Gamma )-E_{\mathrm{a}}(\Gamma )-E_{\mathrm{c}}(\Gamma ))}\mathrm{d}\Gamma$$

(4)

To estimate the surplus margin on this period domain, the optimal design variables of the whole operation cycle d₀ should be obtained by solving Model 1. Then, considering the optimization model, the corresponding control margin and process margin are estimated.

$$\mathbf{Model} \mathbf{1} \underset{u,d_0}{\min } J$$

$$\mathrm{s}.\mathrm{t}. \mathit{f}(\overline{\mathit{x}},\mathit{u},\Gamma )=0$$

$$\mathit{h}(\mathit{x},\mathit{u},t,{\mathit{d}}_0)=0$$

$$\mathit{g}(\overline{\mathit{x}},\mathit{u},\Gamma ,{\mathit{d}}_0)\le 0$$

f(▪), h(▪), and g(▪) are, respectively, the deactivation model in the slow-time-varying system, the reactor dynamic model in the fast-time-varying system, and inequality constraints of the optimization model. The sum of the optimal design margin and process margin of the system at the ${\Gamma }_{\mathrm{c}}$ day is represented as $d_{{\mathsf{\Gamma }}_{\mathrm{c}}}$. By solving the optimization model in the time domain, the corresponding process margin estimate can be obtained when the system runs to the $\Gamma$ day, $\Gamma \in [{\Gamma }_{\mathrm{c}},{\Gamma }_f^{\prime }]$.

$$\mathbf{Model} \mathbf{2} \underset{u,d_{{\Gamma }_{\mathrm{c}}}+\Delta d}{\min }{J}_1=-E_{\mathrm{b}}(\Gamma )+E_{\mathrm{a}}(\Gamma )+E_{\mathrm{c}}(\Gamma )$$

$$\mathrm{s}.\mathrm{t}. f(\overline{\mathit{x}},\mathit{u},\Gamma )=0$$

$${\mathit{h}}_{\mathit{s}}(\mathit{x},\mathit{u},\mathit{t},{\mathit{d}}_{{\Gamma }_{\mathrm{c}}}+\Delta \mathit{d})=0$$

$$g(\overline{\mathit{x}},\mathit{u},\Gamma ,{\mathit{d}}_{{\Gamma }_{\mathrm{c}}}+\Delta \mathit{d})\le 0$$

$$\Delta d(\Gamma )\ge 0$$

For the estimation of the control margin, the steady-state equation which does not contain the control system in the fast-time-varying system is represented as $h_s$. ${\overline{\mathit{x}}}_{}$ represent state variables in the slow-time-varying system. Then, considering the control effect in the fast-time-varying system, the controller is incorporated into the optimization model to obtain the control margin $\Delta d_{\mathrm{c}}(\Gamma )$.

$$\mathbf{Model} \mathbf{3}\underset{u,d_{{\Gamma }_{\mathrm{c}}}+\Delta d+\Delta d_{\mathrm{c}}}{\min }{J}_1=-E_{\mathrm{b}}(\Gamma )+E_{\mathrm{a}}(\Gamma )+E_{\mathrm{c}}(\Gamma )$$

$$\begin{gathered} \mathrm{s}.\mathrm{t}. f(\overline{\mathit{x}},\mathit{u},\Gamma )=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathit{h}}_{\mathit{s}}(\mathit{x},\mathit{U},\mathit{t},{\mathit{d}}_{{\Gamma }_{\mathrm{c}}}+\Delta \mathit{d}+\Delta {\mathit{d}}_{\mathrm{c}})=0 \\ \mathit{g}(\overline{\mathit{x}},\mathit{u},\Gamma ,{\mathit{d}}_{{\Gamma }_{\mathrm{c}}}+\Delta \mathit{d}+\Delta {\mathit{d}}_c)\le 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{U}(t)={\mathrm{K}}_{\mathrm{P}}\left(e(t)+{\mathrm{T}}_{\mathrm{I}}^{-1}{\displaystyle {\int }_0^t\hspace{1em}e(t)\mathrm{d}t}+{\mathrm{T}}_{\mathrm{D}}\frac{\mathrm{d}e(t)}{\mathrm{d}t}\right) \end{gathered}$$

$$e(t)=\mathit{u}-\mathit{x}(t)$$

$$\Delta d_{\mathrm{c}}(\Gamma ),\Delta d(\Gamma )\ge 0$$

A margin consumption function is constructed to describe the distance between the operating point and the constraint boundary in a time domain after the operation period is temporarily changed. Based on the objective function J of Model 1, the margin consumption function is introduced to obtain the dynamic optimization model, Model 4 considering margin consumption.

$$\mathbf{Model} \mathbf{4} \min J_2=J+{\displaystyle \sum _j{\mathsf{\delta }}_j^{\mathrm{E}}{g}_j^2(\cdot )}$$

$$\mathrm{s}.\mathrm{t}. f(\overline{\mathit{x}},\mathit{u},\Gamma )=0$$

$$h(\mathit{x},\mathit{u},t)=0$$

$$g(\overline{\mathit{x}},\mathit{u},\Gamma )\le 0$$

$$0<{\delta }_j^{\mathrm{E}}{g}_j^2(\cdot )<\Delta f$$

The margin conversion factor and margin consumption function are, respectively, represented as ${\mathit{\delta }}_j^{\mathrm{E}}$ and ${\displaystyle \sum _{\mathit{j}}{\mathit{\delta }}_{\mathit{j}}^{\mathrm{E}}{\mathit{g}}_{\mathit{j}}^2(\cdot )}$. The upper limit of economic benefit obtained by operation optimization is equal to the economic benefit reduced by the influence of residual margin, and it is represented as $\Delta f$.

Then, the dynamic optimization method of the full-cycle optimization framework for the acetylene hydrogenation reactor is considered [36]. The CVP (control vector parametrization method) [37,38,39,40,41,42] is adopted to convert continuous slow-time-varying full-cycle optimization variables into several discrete parameter variables, which helps in solving the optimization model. In recent years, alternative and improved methods of CVP are still playing an important role in dynamic optimization of complex nonlinear models, such as an alternative method based on Fourier series for the control vector parameterization [43], an improved method to generate optimal evaluation for two objectives [44], a nonuniform method with time grid refinement [45], and an alternative method with control-dependent time-delayed arguments [46]. Using this method, the full-cycle economic benefit curve (Figure 4) is shown through the solutions of Model 4 (including the curve before the operation cycle remains unchanged). Parameter values and operation data of acetylene hydrogenation reactor are shown in Appendix A and Appendix B. Due to the rapid release of surplus margin within a short period, the economic benefit curve can maintain significant growth until the acetylene content of the reactor outlet exceeds the process constraint and the reaction stops, with 150 and 160 days being the maximum operating cycle. When the maximum operating cycle is 170 days, the economic benefit curve approximates the dynamic optimization trajectory without temporary adjustment of the operation cycle. Furthermore, considering the desire for a longer operation cycle, the economic benefit curve decreases rapidly when the maximum operation cycle is set to 190 days.

As shown in

Table 1. Optimization result for temporary change of optimization strategy.

${\Gamma }_f^{\prime }$/day	150	160	170	180	190
Total economic benefit of full cycle (¥)	7.40 × 105	7.88 × 105	8.09 × 105	8.47 × 105	8.00 × 105
Annualized economic benefits (¥/a)	1.53 × 106	1.55 × 106	1.50 × 106	1.49 × 106	1.33 × 106

, the maximal operation cycle of ${\Gamma }_f^{\prime }=150,160,170,190$ effects the whole period of total economic efficiency and the annual economic benefit. Considering total economic benefit (the maximum cycle of 180 day), the economic benefit obtained more than the project that temporarily changes the operating cycle, the distance between the operation of the temporary change cycle and scheduled cycle is positively related to the loss of total economic benefit. However, from the perspective of annualized economic benefits (excluding the cost of catalytic regeneration), the annualized economic benefits obtained by operation optimization maintaining the operation margin without temporary change are lower than those obtained by temporarily changing the maximum operating cycle as ${\Gamma }_f^{\prime }=150,160,170$. Moreover, ${\Gamma }_f^{\prime }=160$ is roughly considered the optimal strategy for annualized economic benefits.

After optimizing the acetylene hydrogenation reactor for a specific duration based on operation optimization, while ensuring a fixed operation cycle, the focus shifts to determining the maximum number of days the system can sustain operation. Essentially, this problem translates into optimizing the operation maintenance with the objective of maximizing the regeneration period. It is important to note that the regeneration cycle cannot be extended indefinitely due to catalyst deactivation during the reactor’s operating cycle.

4. Full-Cycle Optimization That Includes Scheduling Optimization

For the comprehensive dynamic optimization of the acetylene hydrogenation reactor, an optimization model that integrates design, control, and optimization can be established. In the second section, we investigated how adjusting the operating cycle or optimization strategy affects the operation optimization and maintenance of the operation margin throughout the full cycle. Additionally, by incorporating the scheduling strategy optimization method into the operation optimization and maintenance of the operation margin, the optimal scheduling strategy under specific conditions is presented.

The combination of scheduling optimization and operation optimization and maintenance of the operation margin can be viewed as a hybrid optimization model. This model consists of discrete optimization scheduling strategies and full-cycle dynamic optimization. The variables representing the full-cycle operation optimization and maintenance of the operation margin, as well as the full-cycle scheduling optimization decisions, are denoted as “u” and “y”, respectively. The integration of both optimization models can be represented as problem 1.

$$\mathrm{Problem} 1 \underset{u,y}{\min }\mathit{W}(u,y)$$

$$\mathrm{s}.\mathrm{t}. \mathit{G}(u,y)\le 0$$

$$u\in U_{\mathrm{n}}\subset {\mathrm{R}}^{n_u}$$

$$\mathit{y}\in Y\subset {\mathrm{R}}^{n_y}$$

Sets U_n and Y are described by lower and/or upper bound constraints. Considering complexity and nonlinearity of the acetylene hydrogenation reactor model, it is difficult to be solved by the conventional hybrid dynamic optimization method. A kind of decomposition algorithm, a nonconvex sensitivity-based generalized Benders decomposition (NSGBD), is proposed to solve the optimization problem of the operation optimization maintaining the operation margin considering process scheduling.

4.1. Decomposition Algorithm Model

NSGBD (Non-convex Sensitivity Generalized Benders Decomposition) is the generalized form of generalized Benders decomposition (GBD) algorithm [47,48,49] in separable quasi-convex cases [50]. The core idea of GBD algorithm is projection, considering integrated scheduling optimization as problem 1 in y-space. The projection problem for Problem 1 is Problem 2.

$$\mathrm{Problem} 2 \underset{\mathit{y}}{\min }v(\mathit{y})$$

(5)

$$\mathrm{s}.\mathrm{t}.\mathit{y}\in Y\cap V$$

(6)

$$\mathit{v}(\mathit{y})=\underset{\mathit{x}}{\mathit{min}}\left[W(\mathit{u},\mathit{y}) \mathrm{s}.\mathrm{t}. \mathit{G}(\mathit{u},\mathsf{y})\le \mathbf{0}, \mathit{u}\in U_{\mathrm{n}}\right]$$

(7)

$$V=\{\mathit{y}:G(\mathit{u},\mathit{y})\le \mathbf{0} \mathrm{for} \mathrm{some} \mathit{u}\in U_{\mathrm{n}}\}$$

(8)

$\mathit{v}(\mathit{y})$ is the optimal value for fixed y in Problem 1. Set V is the set of y that makes Equation (6) feasible, and $Y\cap V$ is the y-space projection of the feasible region of Problem 1. Obviously, Problem 1 and Problem 2 are equivalent. If Problem 2 is convex, U_n and Y are non-empty convex sets, and each component of $\mathit{G}(u,\mathit{y})$ and $\mathrm{W}(u,\mathit{y})$ is a convex function, then the Strong Duality Theorem holds. The maximum value is equivalent to the minimum upper bound. For a given $\mathit{\lambda }$, $\underset{\mathit{x}\in X}{\min }\left[f(\mathit{x},\mathit{y})+\mathit{\lambda }\cdot \mathit{G}(\mathit{x},\mathit{y})\right]$ is a lower estimate function of (Benders cut) $\mathit{v}(\mathit{y})$.

To obtain the strongest Benders cut about ${\mathit{y}}_a^j\in Y\cap V$, it is necessary to obtain the optimal u and corresponding to ${\mathit{y}}_a^j\in Y\cap V$ that is to solve Problem 3.

$$\mathrm{Problem} 3 ({\mathit{u}}_a^j|{\mathit{\lambda }}_a^j)=\arg \underset{\mathit{u}\in U_{\mathrm{n}}}{\min }W(\mathit{u},{\mathit{y}}_a^j)$$

(9)

$$\mathrm{s}.\mathrm{t}. \mathit{G}(\mathit{u},{\mathit{y}}_a^j)\le \mathbf{0}$$

(10)

The computational process of GBD algorithm is as follows: solving a series of master problems to obtain a sequence of nondecreasing lower bounds and for the original problems; solving a series of original problems to obtain a sequence of nonincreasing upper bounds and Benders cuts for the master problems; the algorithm terminates when the lower bounds and upper bounds satisfy the convergence criterion. In the complex industrial models of actual production, the majority of the objective functions are pseudo-convex and do not meet the conditions required by the GBD algorithm. However, pseudo-convex problems can be easily solved using solvers based on local optimality conditions. Therefore, it is possible to consider the extension of the GBD algorithm to pseudo-convex problems. The problems that NSGBD algorithm can handle can be described by separable quasi-convex conditions. For the problems that meet the separable quasi-convex conditions, the slope of effective Benders cuts can be increased by operating Benders cuts, solving the original problem, and testing the optimal conditions until the optimal solution is found.

${\mathit{\lambda }}_a^j$ is the Lagrange multiplier of $\mathit{G}(\mathit{u},{\mathit{y}}_a^j)\le \mathbf{0}$. Moreover, for a given ${\overline{\mathit{y}}}_a^j\in Y\cap V$, Problem 4 is proposed.

$$\mathrm{Problem} 4\text{:} ({\mathit{u}}_b^j,{\mathit{y}}_b^j|{\mathit{\lambda }}_b^j,{\mathit{\mu }}_a^j)=\arg \underset{\mathit{u}\in U_{\mathrm{n}},\mathit{y}\in Y}{\min }W(\mathit{u},\mathit{y})$$

(11)

$$\mathrm{s}.\mathrm{t}. \mathit{G}(\mathit{u},\mathit{y})\le \mathbf{0}$$

(12)

$$\mathit{h}(\mathit{y},{\overline{\mathit{y}}}_a^j)=\mathbf{0}$$

(13)

${\mathit{\lambda }}_b^j$ and ${\mathit{\mu }}_a^j$ are the Lagrange multipliers of $\mathit{G}(\mathit{x},\mathit{y})\le \mathbf{0}$ and $\mathit{h}(\mathit{y},{\overline{\mathit{y}}}_a^j)=\mathbf{0}$. Solve Problem 4 to obtain the optimal solution ${\mathit{u}}_a^j$ and the optimal objective function value $W({\mathit{u}}_a^j,{\mathit{y}}_a^j)$. Then, discuss and solve the problem according to the convexity of the model and Benders cut operations. Repeat this step until convergence to the optimal solution. The slow-time-varying dynamic model considering the actual process of the full-cycle operation margin optimization problem including scheduling optimization can be described as Equations (14) and (15).

$${\dot{\mathit{x}}}_d={\mathit{f}}_d(\Gamma ,{\overline{\mathit{x}}}_d(\Gamma ),{\overline{\mathit{x}}}_a(\Gamma ),\mathit{u}(\Gamma ),\mathrm{S}({\Gamma }_{\mathrm{s}},{\Gamma }_f))$$

(14)

$$\mathbf{0}={\mathit{f}}_a(\Gamma ,{\overline{\mathit{x}}}_d(\Gamma ),{\overline{\mathit{x}}}_a(\Gamma ),\mathit{u}(\Gamma ),\mathrm{S}({\Gamma }_{\mathrm{s}},{\Gamma }_f))$$

(15)

${\overline{\mathit{x}}}_d(\Gamma )\in R^{n_d}$/${\overline{\mathit{x}}}_a(\Gamma )\in R^{n_a}$ represents the differential/algebraic state variable of the slow-time-varying system; $\mathit{u}(\Gamma )$ represents the control variable of the full-cycle dynamic model; ${\mathit{f}}_d$/${\mathit{f}}_a$ denotes differential/algebraic equations; S is the scheduling policy function. Therefore, the full-cycle operation optimization maintaining the operation margin including scheduling optimization is described as Model 5.

$$\mathbf{Model} \mathbf{5} \underset{ }{\min } J_3=J(u(\Gamma ),\mathrm{S})$$

$$\mathrm{s}.\mathrm{t}. {\dot{\mathit{x}}}_d={\mathit{f}}_d(\Gamma ,{\overline{\mathit{x}}}_d(\Gamma ),{\overline{\mathit{x}}}_a(\Gamma ),\mathit{u}(\Gamma ),\mathrm{S})$$

$$\mathbf{0}={\mathit{f}}_a(\Gamma ,{\overline{\mathit{x}}}_d(\Gamma ),{\overline{\mathit{x}}}_a(\Gamma ),\mathit{u}(\Gamma ),\mathrm{S})$$

$$\mathit{h}(x_a,x_d,\mathit{u},t)=0$$

$$\mathit{g}({\overline{x}}_a,{\overline{x}}_d,\mathit{u},\Gamma ,\mathrm{S})\le 0$$

If ${\mathrm{S}}_a^j$ is feasible, it is considered to generate Benders cuts based on margin estimation, to solve Equations (16) and (17), that $J_a^j$ is the optimal value of the objective function, ${\mathit{\mu }}_a^j$ is the Lagrange multiplier of Equation (17), and $\Delta d_a$ is the surplus margin estimation.

$$({\mathbf{S}}_a^j|J_a^j)=\arg \underset{\mathrm{S}}{\min } J(\mathit{u}(\Gamma ),{\mathbf{S}}_a^{}({\Gamma }_{\mathrm{s}},{\Gamma }_f),\Delta {\mathit{d}}_a)$$

(16)

$$({\mathrm{S}}_b^j,\Delta {\mathit{d}}_a^j|{\mathit{\mu }}_a^j)=\arg \underset{{\mathrm{S}}_b^j,\Delta {\mathit{d}}_a}{\min } J(\mathit{u}(\Gamma ),\mathrm{S},\Delta {\mathit{d}}_a)$$

(17)

If ${\mathrm{S}}_a^j$ is not feasible, we consider solving the infeasible minimization problems (18) and (19) that generate feasible points and feasible domains of complex variables to support the hyperplane. ${\mathit{\mu }}_b^j$ is the Lagrange multiplier of Equation (19).

$$({\mathrm{S}}_c^j,\Delta {\mathit{d}}_a^j)= \arg \underset{\mathrm{S},\Delta {\mathit{d}}_a^{}}{\min } {‖\mathrm{S}-{\mathrm{S}}_a^j‖}_A^2$$

(18)

$$({\alpha }_d^j,{\mathrm{S}}_d^j,\Delta d_a^j|{\mathit{\mu }}_b^j)= \arg \underset{\alpha ,\mathrm{S},\Delta d_a^{}}{\min }\alpha$$

(19)

Then, solve the main problem, Equation (20), obtain the corresponding decision variables, and update ${\mathrm{S}}_a^j$. Repeat this step (solve Equations (16)–(20)) until the problem converges.

$$({\eta }_{}^j,{\mathrm{S}}_a^j|{\mathit{\mu }}_a^j,{\mathit{\mu }}_b^j)= \arg \underset{\eta ,\mathrm{S}}{\min } \eta$$

(20)

Through the processing of the decomposition algorithm, the hybrid dynamic optimization problem including scheduling optimization can be transformed into an iterative process of repeatedly solving the primary problem and the main problem, and the full-cycle optimal scheduling strategy can be solved. The detailed block diagram of the NSGBD algorithm is as follows Algorithm 1:

Algorithm 1: CVP-NSGBD

Step 1.

Select a set of appropriate internal points t0 < t1 < t2 < … < tn = tf in the time domain [t0, tf], and then parameterize the continuous control variable u(t) to ${\widehat{\mathit{u}}}_N$ (segmented constant control) by CVP method. The initial point is ${\overline{\mathit{u}}}_t^1$; let $j=1$, $j{}^{\prime }=1$, $K_{feas}=\varnothing$, $K_{infeas}=\varnothing$, $LBD=-\infty$; set hyperparameters $\gamma$, ${\epsilon }_1$, ${\epsilon }_2$, and ${\epsilon }_3$.

Step 2. Let ${\overline{\mathit{u}}}_a^j={\overline{\mathit{u}}}_{\mathit{t}}^j$, solve the original Problem 3. One of the following is necessary:

(1) If the original Problem 3 is feasible, then ${\widehat{\mathit{u}}}_{N,a}^j$ and $J_a^j$ are solved; let ${\underset{\_}{\mathit{u}}}_a^j={\overline{\mathit{u}}}_a^j$, using $({\widehat{\mathit{u}}}_{N,a}^j,{\overline{\mathit{u}}}_a^j)$ as the initial point to solve the original Problem 4, the solution is ${\mathit{\mu }}_a^j$.

① If ${\overline{\mathit{u}}}_a^j$ is the inner point, then the algorithm terminates, the optimal solution is $({\widehat{\mathit{u}}}_{N,a}^j,{\overline{\mathit{u}}}_a^j)$, otherwise let ${\mathit{\mu }}_a^l=\gamma {\mathit{\mu }}_a^l$.

② If ${\overline{\mathit{u}}}_a^j$ is the boundary point, Model 5 is solved with ${\overline{\mathit{u}}}_a^j$ as the initial point, and ${\underset{\_}{\mathit{u}}}_c^j$ is obtained. Where ${\underset{\_}{\mathit{u}}}_b^j={\overline{\mathit{u}}}_a^j$, then the algorithm terminates and the optimal solution is ${\mathit{\mu }}_a^l=\gamma {\mathit{\mu }}_a^l$.

Let $j=j+1$.

(2) If the original Problem 4 is not feasible, then solving Model 5 with respect to ${\overline{\mathit{u}}}_{\mathit{t}}^j$ yields $({\widehat{\mathit{u}}}_{N,c}^{j{}^{\prime }},{\overline{\mathit{u}}}_c^{j{}^{\prime }})$. Return to Step 2.

Step 3. To solve Model 5; ${\overline{\mathit{u}}}_{\mathit{t}}^j={\underset{\_}{\mathit{u}}}_b^j$. Return to Step 2.

The NLP problem of the NSGBD algorithm is large-scale, even though the main problem is a simple LP problem. However, the NSGBD algorithm needs to repeatedly solve the original problem and the main problem. Since solving the dynamic optimization of the original problem is the most time-consuming part, the NSGBD algorithm’s advantage is more evident in this study.

4.2. Scheduling Strategy Optimization

By analyzing the optimal operation cycle temporarily adjusted in Section 3.3, it was observed that the difference between the original optimal operation cycle and the adjusted operation cycle positively correlates with the overall economic benefits obtained throughout the full cycle. In order to accurately reflect the annual economic benefits of the acetylene hydrogenation reactor and meet the requirements of scheduling optimization, it is necessary to improve the objective function accordingly. Based on the optimization decomposition algorithm model presented in the previous section, which focuses on maintaining the operation margin and includes scheduling optimization, a specific optimization model can be derived to solve the scheduling optimization problem. This model is referred to as Model 6.

$$\mathbf{Model} \mathbf{6} J_4=(J(u(\Gamma ),{\Gamma }_c,{\Gamma }_f)-E_R)\cdot \frac{{\mathrm{n}}_{\mathrm{a}}}{{\Gamma }_f}$$

$$\mathrm{s}.\mathrm{t}. {\dot{\mathit{x}}}_d={\mathit{f}}_d(\Gamma ,{\overline{\mathit{x}}}_d(\Gamma ),{\overline{\mathit{x}}}_a(\Gamma ),\mathit{u}(\Gamma ),{\Gamma }_c,{\Gamma }_f)$$

$$\mathbf{0}={\mathit{f}}_a(\Gamma ,{\overline{\mathit{x}}}_d(\Gamma ),{\overline{\mathit{x}}}_a(\Gamma ),\mathit{u}(\Gamma ),{\Gamma }_c,{\Gamma }_f)$$

$${\Gamma }_{\mathrm{c},\min }^{}\le {\Gamma }_{\mathrm{c}}^{}\le {\Gamma }_{\mathit{c},\max }^{},{\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$$

$$\mathit{h}(x_a,x_d,\mathit{u},t)=0$$

$$\mathit{g}({\overline{x}}_a,{\overline{x}}_d,\mathit{u},\Gamma ,{\Gamma }_c,{\Gamma }_f)\le 0$$

In Model 6, E_R is the cost of cleaning and regeneration after one operation cycle of the reactor; n_a is the number of annualized days; J is the objective function for temporarily changing the operating cycle, which consists of the normal optimized part before the change and the optimized part after the change. The specific optimization model has been described in detail in Section 3.

NSGBD algorithm is used to solve Model 6. The iterative optimization process is shown in

Table 2. The influence of NSGBD algorithm iterations on the optimization results.

Number of Iterations	${\Gamma }_{\mathrm{c}}^{*}$ (day)	${\Gamma }_f^{*}$ (day)	J (¥/a)	J4 (¥/a)
1.1	120	171.19	7.38 × 105	1.34 × 106
1.2	117.70	180	8.45 × 105	1.49 × 106
2	117.65	176.74	8.09 × 105	1.48 × 106
3	90.86	181.13	8.46 × 105	1.48 × 106
4	99.36	150.33	7.42 × 105	1.54 × 106
5	108.34	158.69	7.77 × 105	1.54 × 106
6	105.81	163.69	8.02 × 105	1.55 × 106
7	105.33	163.27	8.03 × 105	1.55 × 106

. By the optimal strategy, the catalyst deactivation trajectory and margin release curve are shown in Figure 5:

Considering the need to solve the scheduling cycle initial time and termination of the two time variables, and improved iterative efficiency, the parallel strategy is employed during the first iteration, respectively, to obtain the optimal scheduling initial time ${\Gamma }_{\mathrm{c}}^{*}$ (the number of iterations 1.1), and the maximum operation cycle ${\Gamma }_f^{*}$ (the number of iterations 1.2). Then, the optimized results are substituted into the second iteration. After the 7th iteration of the algorithm, the error between lower bounds of the optimal value and the subproblem are less than 1, the iteration stops, and the optimal annualized economic benefit is 1.553 × 10⁶, the optimal scheduling policy is: ${\Gamma }_{\mathrm{c}}^{*}=105$, ${\Gamma }_f^{*}=163$. By the optimal strategy, as shown in Figure 5, increasing the hydrogenation amount of the reactor, the deactivation rate of the catalyst after the maximum operating cycle adjustment is significantly accelerated, and the consumption of surplus margin is also increased accordingly. In Section 3.3, the initial point of change scheduling is 120 day, and the actual annual economic benefit of the optimal strategy analyzed is about 1.547 × 10⁶, indicating that the scheduling optimization result is better than the analysis result in Section 3.3. The solution to the optimization model, Model 1 (with the objective function J), is essentially derived from the full-cycle optimization framework of the literature [34]. It should be noted that even though the same scheduling optimization problem is being addressed, different optimization frameworks and methods are utilized. The optimization result obtained for J is based on the full-cycle operational optimization framework used in the earlier study. On the other hand, the optimization result for J₄ is obtained using the full-cycle scheduling optimization framework proposed in this study. It is evident that the full-cycle scheduling optimization framework offers significantly higher benefits than the full-cycle operational optimization when it comes to addressing scheduling optimization problems.

The optimization results of full-cycle scheduling are related to the constraint of scheduling strategy ${\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$ and ${\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$. In addition, as the full-cycle dynamic optimization method based on CVP in this paper, the optimization results are also related to the dispersion degree of optimization variables. Hence, this paper will further analyze and discuss the influence of these two factors on the full-cycle operation optimization of the acetylene hydrogenation reactor maintaining the operation margin including scheduling optimization.

Since changes occurring in the initiation or termination of operation cycle is meaningless for the actual full-cycle operation, the time domain [40 day, 150 day] is considered dividing into 11 equidistant constraint intervals, and Model 6 is solved within the constraint interval. The optimization results are shown in

Table 3. The effects of ${\Gamma }_{\mathrm{c},\min }^{}\le {\Gamma }_{\mathrm{c}}^{}\le {\Gamma }_{\mathrm{c},\max }^{}$ on the optimization solution.

${\Gamma }_{\mathrm{c},\min }^{}\le {\Gamma }_{\mathrm{c}}^{}\le {\Gamma }_{\mathrm{c},\max }^{}$	${\Gamma }_{\mathrm{c}}^{*}$ (day)	${\Gamma }_f^{*}$ (day)	J (¥/a)	J4 (¥/a)
[40, 50]	50	173.23	8.01 × 105	1.46 × 106
[50, 60]	60	169.80	7.79 × 105	1.45 × 106
[60, 70]	70	168.73	7.73 × 105	1.44 × 106
[70, 80]	74.36	167.86	7.93 × 105	1.49 × 106
[80, 90]	90	174.77	8.07 × 105	1.46 × 106
[90, 100]	97.55	170.39	8.13 × 105	1.51 × 106
[100, 110]	105.33	163.27	8.03 × 105	1.55 × 106
[110, 120]	120	161.33	7.88 × 105	1.54 × 106
[120, 130]	120	161.33	7.88 × 105	1.54 × 106
[130, 140]	136.92	179.10	8.46 × 105	1.50 × 106
[140, 150]	140	183.74	8.62 × 105	1.49 × 106

.

Similarly, the same method can be used to study the constraint problem of the maximum operating cycle. All feasible regions of ${\Gamma }_f$ are set as [120, 200], which are divided into 8 equidistant constraint intervals, and the scheduling strategy optimization problem (Model 6) is solved. The results are shown in

Table 4. The effects of ${\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$ on the optimization solution.

	${\Gamma }_{\mathrm{c}}^{*}$ (day)	${\Gamma }_f^{*}$ (day)	J (¥/a)	J4 (¥/a)
[120, 130]	82.74	122.23	5.97 × 105	1.46 × 106
[130, 140]	90.25	140	6.68 × 105	1.46 × 106
[140, 150]	97.21	150	7.14 × 105	1.47 × 106
[150, 160]	101.24	160	7.84 × 105	1.54 × 106
[160, 170]	105.33	163.27	8.03 × 105	1.55 × 106
[170, 180]	97.55	170.39	8.13 × 105	1.51 × 106
[180, 190]	118.95	181.26	8.49 × 105	1.49 × 106
[190, 200]	127.45	190	8.01 × 105	1.33 × 106

. Similarly, compared to the optimization results in J, J₄ has a higher overall return.

Moreover, the grid division of CVP is considered as an adjustment to research and discuss the influence of the discrete degree of control variables on the optimization of the scheduling strategy.

To solve the dynamic optimization problem with scheduling optimization, the optimal scheduling strategy and the optimal economic benefits under different control variables are obtained by mesh refinement within scheduling optimization constraints ${\Gamma }_{\mathrm{c},\min }^{}\le {\Gamma }_{\mathrm{c}}^{}\le {\Gamma }_{\mathrm{c},\max }^{}$ and ${\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$.

The optimization precision of CVP can be effectively promoted by simply mesh refinement, but the dimensions of the NLP problem decision variables that are translated from the original problem can be substantially increased, which increased the complexity of the problem-solving. To balance the accuracy and computational complexity of the solution, this paper adopted a kind of adaptive CVP method to solve the problem. The basic idea is that subintervals with a large proportion of minimizing objective function are added, and subintervals with a small proportion of minimizing objective function are combined.

The algorithm consisted of two parts: solving the obtained NLP problem with a CVP solver and generating a new parameterized form adaptive based on posterior analysis. Considering posterior analysis usually involves only one simple algebraic operation on the solution, the computation amount of the adaptive fine process can be ignored compared to the large number of integral operations and large dimensional NLP problems generated by direct mesh refinement of a CVP solver. Adaptive CVP is a kind of iterative algorithm. Each iteration steps makes a posterior analysis on the solution of the previous step and then generates and solves the problem-dependent discrete problem continuously until it converges to the optimal solution. Similarly, adaptive parametrization is considered within scheduling optimization constraint ${\Gamma }_{\mathrm{c},\min }^{}\le {\Gamma }_{\mathrm{c}}^{}\le {\Gamma }_{\mathrm{c},\max }^{}$ and ${\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$ scheduling optimization constraint. respectively. The scheduling optimization results and CPU running time are shown in

Table 5. The effects of the degree of discretization of control variables in ${\Gamma }_{\mathrm{c},\min }^{}\le {\Gamma }_{\mathrm{c}}^{}\le {\Gamma }_{\mathrm{c},\max }^{}$ of scheduling change on the optimization solution.

Discretization of Control Variables	${\Gamma }_{\mathrm{c}}^{*}$ (day)	${\Gamma }_f^{*}$ (day)	J (¥/a)	J4 (¥/a)	Time (min)
10 days	105.33	163.27	8.03 × 105	1.55 × 106	983
5 days	104.74	168.82	8.05 × 105	1.55 × 106	1483
2 days	104.12	169.73	8.09 × 105	1.56 × 106	2359
1 day	103.95	171.54	8.11 × 105	1.57 × 106	3874
Self-adaption CVP	104.33	169.73	8.08 × 105	1.56 × 106	1036

and

Table 6. The effects of the degree of discretization of control variables in ${\Gamma }_{f,\min }^{}\le {\Gamma }_f^{}\le {\Gamma }_{f,\max }^{}$ of scheduling change on the optimization solution.

Discretization of Control Variables	${\Gamma }_{\mathrm{c}}^{*}$ (day)	${\Gamma }_f^{*}$ (day)	J (¥/a)	J4 (¥/a)	Time (min)
10 days	105.33	163.27	8.03 × 105	1.55 × 106	983
5 days	104.98	168.99	8.05 × 105	1.55 × 106	1389
2 days	104.66	171.48	8.08 × 105	1.56 × 106	2256
1 days	103.83	175.74	8.11 × 105	1.58 × 106	3673
Self-adaption CVP	105.10	173.58	8.10 × 105	1.57 × 106	1098

. It is noteworthy that the CPU runtime for J is not provided in the tables, considering that the benefit obtained from J is considerably lower than that of J₄. Even if the computational resource consumption is lower for J compared to J₄, it does not have any meaningful comparative significance.

The optimal economic benefit can be improved by mesh refinement in the scheduling optimization constraint interval, but the CPU running time also increased. Hence, the production scheduling decision and optimization prediction time should be kept within an acceptable short range, so it is not practical. The CVP algorithm based on adaptive grid partitioning can achieve a balance among them, more economic benefits can be obtained, and shorter CPU running time can be guaranteed.

5. Conclusions

Based on full-cycle dynamic optimization of the acetylene hydrogenation reactor, this paper discusses scheduling optimization problem that temporarily change operation cycles. Hence, a full-cycle scheduling optimization framework based on surplus margin estimate is proposed for the preliminary exploratory research. The roughly scheduling strategy ranges achieve maximum economic benefits, and the maximum annual economic benefit is obtained.

Moreover, to research the optimization of scheduling time, an iterative optimization framework based on NSGBD (nonconvex sensitive generalized Benders decomposition) is proposed, and the optimal scheduling strategy is obtained. This paper further researches the influence of each constraint interval of scheduling optimization on the scheduling optimization strategy and the influence of the dispersion degree of time grid on the optimization result. In addition, considering the contradiction that mesh refinement improves the optimization performance but reduces the operation efficiency, an adaptive time grid CVP algorithm is proposed, which can not only achieve better optimization effect, but also significantly improve the operation efficiency.

This study only considers the scheduling scenario of changing optimization objectives. In actual production, there may be multiple scheduling scenarios such as the replacement or addition of fixed beds, which still require further expansion and research. Additionally, in terms of algorithms, NSGBD has demonstrated its ability to solve large-scale mixed problems in this project, but its computational resource consumption is still considerable and requires further optimization for specific problems.