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.
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.
is the optimal value for fixed y in Problem 1. Set V is the set of y that makes Equation (6) feasible, and 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, Un and Y are non-empty convex sets, and each component of and is a convex function, then the Strong Duality Theorem holds. The maximum value is equivalent to the minimum upper bound. For a given , is a lower estimate function of (Benders cut) .
To obtain the strongest Benders cut about
, it is necessary to obtain the optimal
u and corresponding to
that is to solve Problem 3.
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.
is the Lagrange multiplier of
. Moreover, for a given
, Problem 4 is proposed.
and
are the Lagrange multipliers of
and
. Solve Problem 4 to obtain the optimal solution
and the optimal objective function value
. 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).
/ represents the differential/algebraic state variable of the slow-time-varying system; represents the control variable of the full-cycle dynamic model; / 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.
If
is feasible, it is considered to generate Benders cuts based on margin estimation, to solve Equations (16) and (17), that
is the optimal value of the objective function,
is the Lagrange multiplier of Equation (17), and
is the surplus margin estimation.
If
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.
is the Lagrange multiplier of Equation (19).
Then, solve the main problem, Equation (20), obtain the corresponding decision variables, and update
. Repeat this step (solve Equations (16)–(20)) until the problem converges.
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 (segmented constant control) by CVP method. The initial point is ; let , , , , ; set hyperparameters , , , and . |
Step 2. Let , solve the original Problem 3. One of the following is necessary: |
(1) If the original Problem 3 is feasible, then and are solved; let , using as the initial point to solve the original Problem 4, the solution is .
|
① If is the inner point, then the algorithm terminates, the optimal solution is , otherwise let . |
② If is the boundary point, Model 5 is solved with as the initial point, and is obtained. Where , then the algorithm terminates and the optimal solution is . |
Let . |
(2) If the original Problem 4 is not feasible, then solving Model 5 with respect to yields . Return to Step 2. |
Step 3. To solve Model 5; . 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.
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. 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
(the number of iterations 1.1), and the maximum operation cycle
(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 × 10
6, the optimal scheduling policy is:
,
. 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 × 10
6, 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 J
4 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 and . 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.
Similarly, the same method can be used to study the constraint problem of the maximum operating cycle. All feasible regions of
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. Similarly, compared to the optimization results in J, J
4 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 and .
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
and
scheduling optimization constraint. respectively. The scheduling optimization results and CPU running time are shown in
Table 5 and
Table 6. 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 J
4. Even if the computational resource consumption is lower for J compared to J
4, 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.