Control Vector Parameterization Research Articles

Open-Loop Dynamic Optimization for Nonlinear Multi-Input Systems. Application to Recombinant Protein Production

This paper proposes a novel strategy for dynamic open-loop optimization of multivariable nonlinear systems. The methodology is based on the Fourier series and orthonormal polynomials for the control vector parameterization in a sequential direct solution approach. The advantages of this technique are that a few number of parameters is required for optimization and a smooth control profile is obtained. The proposed strategy is evaluated in the case study of recombinant protein production, that is a nonlinear system with two control actions, the substrate and inhibitor feed flow rate. The algorithms are tested through simulations and the results are compared with those published in the bibliography.

A numerical method for interval multi-objective mixed-integer optimal control problems based on quantum heuristic algorithm

This article is concerned with the numerical solution for a class of interval multi-objective mixed-integer optimal control problems (IMOMIOCPs). The IMOMIOCPs under investigation are typical NP-hard problems involve unknown-but-bounded interval parameters, multiple objectives, and mixed-integer dynamic controls. Accordingly, a new numerical method based on quantum heuristic algorithm is designed, which has the following modules: (i) Control vector parameterization and the fourth order Runge–Kutta method for model discretization, (ii) interval programming based on interval credibility for addressing interval parameters, (iii) coevolution of Quantum Annealing and Quantum Krill Herd for searching the optimal mixed-integer decisions, and (iv) the multiple populations for multi-objective technology for establishing the Pareto optimal front. The analyses on convergence and computational complexity of the proposed optimization mechanism are given. Moreover, simulation results on benchmark functions and a practical engineering IMOMIOCP verify that the proposed numerical method is more excel at achieving promising results than some classic algorithms and state-of-the-art algorithms.

Minimum motive steam consumption on full cycle optimization with cumulative fouling consideration for MED-TVC desalination system

Multi-effect distillation (MED) desalination system always faces the inevitable fouling problem, which leads to a decrease in fresh water production and restricts its long-term efficient operation. This study focuses on improving the system's operation benefits in the full cycle. Firstly, an asymptotic model for fouling accumulation is added to a MED-TVC model, which takes into account the influence of operating conditions on the fouling rate. Subsequently, the impact of fouling accumulation on the operating state of the system is analyzed in the whole cycle, indicating the decrease of operating benefits. Finally, the full cycle operation optimization method is applied for the purpose of reducing motive steam consumption as well as meeting fresh water demand using control vector parameterization strategy. The result shows a significant reduction in motive steam consumption while maintaining fresh water production capacity, which confirms that full cycle optimization has a considerable effect on dealing with fouling problem in MED desalination systems.

Dynamic Optimization of Autocatalytic Esterification in a Semi‐batch Reactor

AbstractDynamic optimization of autocatalytic esterification in a semi‐batch reactor was employed to identify the most robust optimization method. Techniques such as control vector parameterization, orthogonal collocation, and multiple shooting were selected, compared, and analyzed. Three objective functions were considered, namely, maximizing conversion, minimizing final time, and maximizing operation profit using multiple input parameters, i.e., feed rate and temperature. Results from the dynamic optimization indicated that the most effective feed flow rate and temperature trajectory were demonstrated by the orthogonal collocation. The maximum conversion, minimum final time, and maximum profit from this approach were determined.

Dynamic Process Intensification of Dimethyl Ether Reactive Distillation Based on Output Multiplicity

Dynamic process intensification technology is an emerging technology of process intensification. It adopts a nonsteady-state operation mode, which can improve the time-averaged performance of the reaction and distillation system without increasing equipment investment. This research proposes to implement dynamic process intensification in reactive distillation (RD) of methanol dehydration to dimethyl ether (DME). The multiplicity of the reactive distillation process is studied by bifurcation analysis, which uses the heat duty of reboiler as a bifurcation parameter and uses the flow rate and mole fraction of the distillate as the dependent variables (state variables). Based on the output multiplicity of the system, the dynamic intensification process for periodically switching the auxiliary products is established. Compared with the steady-state operation, the time-averaged conversion of the system under nonsteady-state operation is increased by 3.49%, and the heat duty of the reboiler is reduced by 6.00 kW. Finally, dynamic optimization (DO) of RD is studied through the control vector parametrization (CVP) method by using conversion and the heat duty of reboiler as the objective functions. The optimal cyclic steady state is obtained when the duration of the operation period with high yield is 1.1 h; correspondingly, the duration of the operation period with high purity is 0.1 h.

Engineering-oriented dynamic optimal control of a greenhouse environment using an improved genetic algorithm with engineering constraint rules

To effectively and practically solve the dynamic economic optimal control problem in greenhouse environments, an improved genetic algorithm with engineering constraint rules (R-GA) is proposed. Based on a dynamic greenhouse-crop model and control vector parameterization (CVP) method to discretize the control variables, the economic optimal control problem is transformed into a nonlinear programming (NLP) problem with finite-dimension parameters, and then R-GA is used to effectively solve the NLP problem. Three components were investigated, such as a smooth penalty function to deal with state variable path constraints, engineering constraint rules to improve the optimization performance and the algorithm feasibility, and the number of collocation points (Nc) to meet the actual control laws. The simulation results demonstrate that the proposed approach greatly improves the effectiveness and feasibility to solve the dynamic optimal control problem in greenhouse environments.

Hybrid Intelligence Assisted Sample Average Approximation Method for Chance Constrained Dynamic Optimization

Realistic industrial process is usually a dynamic process with uncertainty. Chance constraints are applicable to industrial process modeling under uncertain conditions, where constraints cannot be strictly met, or need not be fully met. Therefore, chance constrained dynamic optimization (CCDO) formulation is available to address realistic industrial process issues. Because of the dynamic and uncertainty, chance constrained dynamic optimization problems (CCDOPs) arising from practical industries are hard to cope with. In this article, a novel CCDO method is proposed to resolve this issue, where an adaptive sample average approximation method, a control vector parameterization method, and a state constraint handling strategy are integrated. Specially, a hybrid intelligent optimization algorithm is introduced to realize a global and efficient optimization performance. The proposed method is applied to CCDOPs modified by dynamic optimization standard test functions and industrial experiments to demonstrate its effectiveness. The experimental results show that the proposed method has good performance in solving CCDOPs.

Minimum attention stochastic control with homotopy optimization

A method of designing control inputs for tracking problems when models considered in the form of stochastic differential equations is proposed. Using the ideas of control vector parameterization, the control input identification problem is formulated as a parameter identification problem, which is solved using homotopy optimization. To further obtain a control with the least number of switchings, i.e., minimum attention control, a sparse recovery framework is developed. The accuracy of the proposed methods is demonstrated with the help of a linear second-order system and a non-linear quadruple tank system.

Multiplicity of solutions in model-based multiobjective optimization of wastewater treatment plants

Wastewater treatment process design involves the optimization of multiple conflicting objectives. The detection of different equivalent solutions in terms of objective values is crucial for designers in order to efficiently switch to the new optimal operation policies if changes in the process conditions or new constraints occur. In this work, the dynamic multi-objective optimization of a municipal wastewater treatment plant model is carried out. The aim is to simultaneously optimize an economic cost term and an effluent quality index. The selected process variables for the optimization are (1) an aeration factor in the aerated tank previous to the clarifier, and (2) an internal recycle flow rate. Their time profiles are approximated using the control vector parameterization technique. To solve the multi-objective problem and find the Pareto front, the NSGA-II algorithm has been used. The simulation of different realistic scenarios which impose operational constraints (e.g., maintenance operations) reveals that, indeed, multiple solutions exist at least in some areas of the Pareto front. It is observed that different control profiles can produce nearly identical results in terms of Pareto solutions. The a priori knowledge of these equivalent solutions for different scenarios provides the decision makers with alternative choices to be adapted to their organizations policies when events altering decision variables bounds or adding new constraints to the process model occur.

Dynamic Multi-Objective Optimization of Autocatalytic Esterification in Semi Batch by Using Control Vector Parameterization (CVP) and Non-Dominated Sorting Genetic Algorithm (NSGA-II)

Catalyzed Esterification of sec-butyl propionate in semi batch reactor prefers to be solved by dynamic-nonlinear programming (NLP) based optimization for determining optimal temperature and feed flowrate trajectories. In this autocatalytic esterification process, there are contrary objective functions, i.e. maximum productivity and minimum process time. Simultaneous optimization of these objectives yields in a dynamic multi-objective optimization (DMOO) problem, which is characterized by a set of multiple solutions, known as non-dominated or Pareto solutions. In this work, a control vector parameterization (CVP) and non-dominated sorting genetic algorithm (NSGA-II) approach were used to generate the Pareto solutions for two objectives: maximize conversion and minimize process time. Each point of Pareto solutions consists of different optimal temperature reactor and feed rate profiles, which lead to a variation combination of conversion and process time. These solutions give multiple alternatives in evaluating the trade-offs and selecting the most suitable operating policy.

Integrated mathematical models for drinking water reservoirs and constructed wetlands as a tool for restoration planning

In this work, we propose the integration of mechanistic models for eutrophic reservoirs and constructed wetland within a dynamic optimization framework for the optimal design of wetlands for water quality restoration purposes. A partial differential equation system results from dynamic mass balances for the main biogeochemical variables in both reservoir and wetland. The integrated model is formulated as an optimal control problem within a control vector parameterization approach. The optimization model includes wetland areas as design variables and the fractions of tributaries that are diverted through the wetlands as control variables. The objective function is the minimization of the sum of the integral of the quadratic difference between total phosphorus concentration and a desired value below eutrophication limits and the integral of the quadratic difference between total nitrogen concentration and a desired value below eutrophication limits, along a twelve-year time horizon. The case of study is the eutrophic Paso de las Piedras Reservoir which supplies drinking water to two cities in Argentina, and it has two tributaries which run through an important agricultural area in the region. Numerical results provide: a) the optimal areas for constructed wetlands next to tributaries, b) optimal temporal profiles for diverted fraction of tributaries through wetlands, and c) temporal profiles of the main biogeochemical variables within the lake and the wetlands. The integrated model, which constitutes a useful management tool for a more rational planning of water quality restoration.

Implementation of model predictive control in tracking dynamic optimal profiles of semi batch autocatalytic esterification reactor

AbstractOptimization of ester production that yields from sec‐butyl propionate in a batch operation mode need to be solved by dynamic–nonlinear programming‐based optimization because the dynamics of this autocatalytic esterification process can be described using the detailed first principle model. In order to maximize profit, the control vector parameterisation technique combined with a hybrid strategy, that is, a deterministic–stochastic nonlinear programming solver, is implemented by generating optimal temperature and feed flowrate trajectories. The final time and profit achieved is 60.0 min and RM 12.840 min−1, respectively. Model predictive control is then applied to track the optimal temperature trajectory obtained from the maximize profit study. The result reveals that the model predictive control is able to track the optimal temperature trajectory very well and consequently able to maintain the on‐spec product profit.

Full cycle dynamic optimisation maintaining the operation margin of acetylene hydrogenation fixed-bed reactor

Abstract In an acetylene hydrogenation reactor, the slowly falling catalyst activity affects ethylene plant performance in an operating cycle. To maximize the integral economic benefit in a regeneration cycle or achieve the longest regeneration cycle, the full cycle operation optimisation must be a long time-scale dynamic optimisation for which the method of control vector parameterization is adopted. Because of various uncertain disturbances, some distances must be reserved for process constraints and the desired effect of full cycle optimisation is hardly fully achieved. Thus, based on the strictly slowly-time-varying catalyst deactivation model, the operation margin consumption is estimated. Furthermore, a modified full cycle operation optimisation strategy is presented which takes the operation margin consumption as additional constraints. To obtain the maximum integral economic benefit or the longest regeneration cycle, we present a realizable full cycle optimized operation plan that considers the sufficient operation space reserved for uncertain disturbances.

Optimization of the electric efficiency of the electric steel making process

This paper reports numerical and practical results of an open-loop optimal control formulation that reduces the power consumption of the electric arc furnace (EAF) steel production process. A control vector parametrization technique is used to optimize the batch trajectory with the goal to minimize the energy losses of the process. First principles models are utilized to describe the dynamics, as well as the influence of the voltage and impedance set-points on the process. The results of the dynamic optimization provided a sequence of set-points (called a melting profile) that aligns well with intuition: the profile utilizes high power levels during the high efficiency stages of the process, and low power levels as the batch moves towards a more energy inefficient state. The benefits of the proposed optimized mode of operation are demonstrated by an experimental study case. An optimal melting profile was calculated and implemented in a fully operative ultra-high power EAF. For a series of 19 test batches, the energy consumption and the batch time of the process were reduced by 4.5% and 4.6% for one type of steel.

A Fourier-based control vector parameterization for the optimization of nonlinear dynamic processes with a finite terminal time

In this paper, a novel strategy for finding the optimal operation profiles for nonlinear dynamic processes is developed. Based on the direct sequential stochastic framework for dynamic optimization, this work proposes a technique based on Fourier series for the control vector parameterization, as an alternative to the traditional methods. This approach has the advantage of choosing a high degree of smoothness to avoid sharp changes for the input variables, which is preferred in most chemical and biological processes. On the other hand, when several arcs are present in the qualitative optimal profile, the number of parameters can be increased for a better approximation. The proposed strategy was applied to four well-studied nonlinear processes, covering batch and fed-batch reactors, and multi-input systems. The algorithm was tested through simulations. Good performances were obtained in comparison to some previous results available in the literature.

Optimal valve closure operations for pressure suppression in fluid transport pipelines

When a valve is suddenly closed in fluid transport pipelines, a pressure surge or shock is created along the pipeline due to the momentum change. This phenomenon, called hydraulic shock, can cause major damage to the pipelines. In this paper, we introduce a hyperbolic partial differential equation &#x0028 PDE &#x0029 system to describe the fluid flow in the pipeline and propose an optimal boundary control problem for pressure suppression during the valve closure. The boundary control in this system is related to the valve actuation located at the pipeline terminus through a valve closing model. To solve this optimal boundary control problem, we use the method of lines and orthogonal collocation to obtain a spatial-temporal discretization model based on the original pipeline transmission PDE system. Then, the optimal boundary control problem is reduced to a nonlinear programming &#x0028 NLP &#x0029 problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming &#x0028 SQP &#x0029. Finally, we conclude the paper with simulation results demonstrating that the full parameterization &#x0028 FP &#x0029 method eliminates pressure shock effectively and costs less computation time compared with the control vector parameterization &#x0028 CVP &#x0029 method.

Optimal grade transition of a non-isothermal continuous reactor with multi-objective dynamic optimization approach

Dynamic optimization (DO) is a useful tool for carrying out grade transitions in polymer industry. Most open literature studies on DO emphasize such grade transitions using single objective optimization. However, there are multiple criteria which must be met simultaneously for economic benefits. In this work, we solve a multi-objective DO problem for free-radical polymerization of methyl methacrylate in a non-isothermal continuous stirred tank reactor. The process objectives considered in the DO activity include minimization of off-spec, minimization of grade transition time, and minimization of the averaged feed flowrate. The manipulated variables considered for this problem are the initiator and coolant flowrates. The DO problem is solved using control vector parameterization (CVP) approach with first order interpolation. The solution of the aforementioned multi-objective DO problem is obtained in terms of a trade-off curve, pareto curve, using non-dominated sorting genetic algorithm (NSGA II). The three-dimensional pareto front is then projected to each of the three pairs of the objectives for better visualization and analysis. Furthermore, three representative pareto solution points, namely the two end points and a utopia point are further analysed for of each of the bi-objective pareto solution curves.

An investigation of initialisation strategies for dynamic temperature optimisation in beer fermentation

A wide range of optimisation methodologies exist for solving optimal control trajectory problems. With most approaches it is necessary to solve iteratively, starting from an initialising solution (often referred to as an initial guess). In this paper we investigate the performance of two different dynamic optimisation strategies for batch beer fermentation temperature control, using different initialisation profiles. A sequential method, Control Vector Parameterisation (CVP), is demonstrated as unable to produce solution profiles satisfying the industrially imposed undesirable by-product species concentration constraints. An alternative simultaneous method, Complete Parameterisation (CP), is employed in order to determine solutions satisfying these essential industrial production constraints. Blind initialisation guesses (isothermal profiles) have been shown to produce solution profiles not suitable for implementation on real fermentors; more promising candidate profile initialisations yield superior solutions. The use of a state-of-art NLP solver (IPOPT) with analytical first derivatives achieves remarkable solution robustness, eliminating initialisation and discretisation level dependence.

An adaptive wavelet method for solving mixed-integer dynamic optimization problems with discontinuous controls and application to alkali–surfactant–polymer flooding

ABSTRACTThis article presents an adaptive rationalized Haar function approximation method to solve dynamic optimization with mixed-integer and discontinuous controls. Three measures are taken to deal with the discontinuity. First, the problem is converted into a multi-stage optimization problem by non-uniform control vector parameterization. Secondly, an adaptive strategy is proposed to regulate the interval division and the order of Haar function vectors. Thirdly, a structure detection method is presented to refine the subintervals, in which adjacent arcs with the same input type are merged into one to modify redundant subintervals. During this approximation solution, the mixed-integer restriction is realized by the integer truncation strategy. Combined with the Hamiltonian function, a validation principle is shown to verify the optimality of the solution. Finally, the proposed method is applied to solve the enhanced oil recovery for alkali–surfactant–polymer flooding. The effectiveness of the method is illustrated through simulation.

Optimization for ASP flooding based on adaptive rationalized Haar function approximation

Abstract This paper presents an adaptive rationalized Haar function approximation method to obtain the optimal injection strategy for alkali-surfactant-polymer (ASP) flooding. In this process, the non-uniform control vector parameterization is introduced to convert original problem into a multistage optimization problem, in which a new normalized time variable is adopted on the combination of the subinterval length. Then the rationalized Haar function approximation method, in which an auxiliary function is introduced to dispose path constraints, is used to transform the multistage problem into a nonlinear programming. Furthermore, an adaptive strategy proposed on the basis of errors is adopted to regulate the order of Haar function vectors. Finally, the nonlinear programming for ASP flooding is solved by sequential quadratic programming. To illustrate the performance of proposed method, the experimental comparison method and control vector parameterization (CVP) method are introduced to optimize the original problem directly. By contrastive analysis of results, the accuracy and efficiency of proposed method are confirmed.