Date of Award
8-16-2024
Document Type
Open Access Dissertation
Department
Chemical Engineering
First Advisor
Edward Gatzke
Abstract
Computational methods for optimal control are important for dealing with bottlenecks in industries. By implementing optimal operation strategies obtained from the computational methods into dynamic systems, objectives such as saving energy, reducing cost, exploiting potential, and improving efficiency can be achieved. Computational methods for optimal control have been used in a broad range of disciplines, including aeronautics and astronautics, oil and chemical engineering, power systems, clean energy, biomedical engineering, economics, and management. Due to the great application value of optimal control, its computational methods have drawn worldwide attention. Direct methods are most effective in computational methods for optimal control. Direct methods transform an optimal control problem into an approximate mathematical programming problem, then choose a mathematical programming solver to get an approximate solution to the original problem.
Two representative direct methods are: the control vector parameterization (CVP) method and the orthogonal collocation (OC) method. This work focuses exclusively on an in-depth study and comprehensive survey of the CVP method. In the context of the CVP method using variational methods, this dissertation introduces a novel approach for problem-solving under a uniform time grid where all control variables switch simultaneously at the same time points. It treats the duration of time segments as a category of decision variables and derives gradient information of the objective function and constraints with respect to the length of these time segments. Compared to the two previously published approaches under the same variational methods which takes switching time points as a category of decision variables and the other approach which employs CPET technology, the novel approach offers significant advantages. It eliminates the need for scale transformation of the time axis and avoids the complex and error-prone transfer conditions at switching time points in one of the published approaches.
This work conducts comparative studies of the three approaches and revises their sensitivity formulas. The newly written sensitivity formulas more effectively reveal an important fact. This important fact of the variational method is that the varying of decision variables after a certain time point will not affect the sensitivity information before this time point. Regarding Approach 2, which is essentially the CPET technique proposed by Dr. Teo Kok Lay, this work supplements a new time scaling transformation formula. This new formula was previously proposed and applied by Rein Luus in the IDP algorithm that he developed, it is also applicable to sensitivity methods. Both time scale transformation formulas can accurately capture transition time points, enhancing the precision and efficiency of the analysis.
In a novel approach to multiple time-grid division, where multiple control variables are allowed to switch asynchronously, this study introduces a new time scaling transformation formula. This formula uses the length of sub-time segments between two adjacent time points as the scaling factor even though the two adjacent could be from different control components. By employing this innovative time scaling transformation formula, the numerical issues of division by zero encountered in previously reported sensitivity methods for multi-time grids without time scaling transformation could be effectively circumvented. Compared to a similar sequential time scaling technique recently proposed by Teo, the method in this work is more straightforward and has been shown to require less computational time as validated by case studies.
To address the issue of parameter proliferation caused by multiple time grids, this study introduces an innovative Multi-Grid Sparse Variable Time Node (MG-SVTN) grid. This time grid is constructed using a limited number of variable time nodes, with each pair of adjacent nodes further subdivided into several equal-length time subsegments. The control values on each time subsegment are variable, while the start and end times of each subsegment are determined by the variable time nodes, and thus are not considered additional decision variables. By employing this grid subdivision method, the number of decision variables in multiple time grids is effectively reduced, thereby accelerating the decision-making process.
This study developed a model of a mixed air storage system. Due to a physical phenomenon known as choked flow, the state equation includes multiple switching subsystems, making it extremely challenging to accurately capture the timing of transitions between subsystems. In response to this challenge, a smoothed Heaviside function was employed to determine the system's state and to facilitate a seamless transition during the switching of subsystems. Moreover, the algorithm investigated in this research was applied to optimize this experimental hybrid system, thereby verifying the effectiveness of the incorporation of the proposed algorithm and the Smoothed Heaviside function.
Rights
© 2024, Hainan Wang
Recommended Citation
Wang, H.(2024). A Study of Direct Sensitivity Method Solving Optimal Control Problems with Single and Multiple Time Grids with an Application in An Experimental Hybrid System. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/7765