Research On PDAE-constrained Dynamic Optimization Algorithms Based On Space-time Orthogonal Collocation On Finite Elements And Applications

The critical process states in energy,chemical,and biological process systems,such as concentration,temperature,pressure,and flow,distribute on both space and time domain.Dis-tributed parameter systems（DPS）presented by partial differential algebraic equations（PDAEs）can depict the state evolution with respect to space and time,providing more comprehensive and accurate state information compared to lumped parameter systems.Prediction,optimization,and control based on PDAE models contribute to deeply analyze and understand the dynamic characteristics of complex process systems,where more effective strategies for achieving safe,stable,long-term,low-emission,and efficient flexible operation can be obtained.However,PDAE-constrained dynamic optimization problems require high-quality discretization methods and dynamic optimization algorithms due to the diversity,complexity,and infinite-dimensional nature.Besides,their high computational burden is the limitation of the application in nonlinear model predictive control（NMPC）.Therefore,efficient algorithms for solving one-dimensional PDAE-constrained dynamic optimization problems are studied in this dissertation.Space-time discretization methods and discretize-then-optimize dynamic optimization algorithms are inves-tigated and then applied to NMPC for solid oxide fuel cells（SOFCs）,where a thermal-electric cooperative control strategy is designed to enhance the cell stability and durability.The main research contents include:（1）A discretization method called space–time orthogonal collocation on finite elements（ST-OCFE）is proposed for the one-dimensional PDAE dynamic model,where a fully implicit scheme with universality,high-order convergence rate,and good numerical stability is acquired.The orthogonal collocation on finite elements（OCFE）method which possesses high flexibility,high-order truncation error,and absolute stability is utilized to discretize both space and time domains.Considering different characteristics of boundary and initial value problems,distinct continuity conditions are enforced on space and time domains to make elements connected.The discrete dynamic system generated from the ST-OCFE method is derived using first-and second-order differential matrices.For convection-diffusion processes described by linear parabolic PDE,both steady-state and dynamic analytical solutions are derived.The influence of the spatial and temporal discretization parameters on the convergence order of the ST-OCFE method is quantitatively analyzed using thenorm of the absolute error.The results confirm that the proposed discretization method exhibits a high-order convergence rate.For fluid pipeline transport processes described by nonlinear hyperbolic PDEs,the influence of the spatial and temporal discretization parameters on the numerical solution of the ST-OCFE method is analyzed based on Lax-Wendroff scheme.The results demonstrate that the proposed discretization method has good numerical stability.（2）Three one-dimensional PDAE-constrained dynamic optimization algorithms,namely,ST-OCFE based single shooting approach,ST-OCFE based multiple shooting approach,and ST-OCFE based simultaneous approach,are proposed,which have widespread applicability and exhibit superior numerical accuracy and computational performance.In these algorithms,the ST-OCFE method is applied to discretize the state variables,while the OCFE method on time do-main is used to discretize the control variables.The discretized nonlinear programming problems for each PDAE-constrained dynamic optimization approach are formulated.The features and advantages of each algorithm are systematically analyzed in conjunction with their descriptions.To significantly reduce the computational cost of the exact Hessian matrix in the ST-OCFE based shooting algorithms,the first-order forward and reverse modes for calculating the first-order sen-sitivities of the discrete dynamic system,as well as the second-order reverse mode for calculating the second-order sensitivities,are derived in detail using the principles of algorithmic differen-tiation.Various optimization problems are studied to validate the effectiveness of the proposed PDAE-constrained dynamic optimization algorithms,including the optimal control problem of water hammer suppression,the optimal control problem of ethylene dichloride cracking furnace,and the optimal operation problem of a non-isothermal tubular reactor.The comparison with existing results from the literature demonstrates that these ST-OCFE based dynamic optimization algorithms can effectively handle diverse dynamic models,objective functions,control actions,and constraint while achieving superior numerical accuracy and computational performance.Furthermore,the introduction of time-scale variables enables free terminal time optimal control by transforming it into a fixed terminal time optimal control problem,thereby showcasing the benefits of variable length control horizons and variable terminal time in providing improved operational policies.（3）A rigorous mechanistic model formulated by one-dimensional PDAEs is developed for direct internal reforming solid oxide fuel cells（DIR-SOFC）to predict cell voltage and the spatio-temporal distribution of current density,temperature,and concentration.According to control-oriented assumptions of quasi-equilibrium and temperature layers incorporation,a simplified PDAE dynamic model is obtained.Steady and dynamic simulations are conducted using the proposed ST-OCFE method.Through the comparison between the one-dimensional PDAE model and the quasi one-dimensional model from literature,it is verified that the proposed discretization method can achieve high numerical accuracy with only a small amount of discretization points.Furthermore,it is demonstrated that the simplified model can accurately predict the cell voltage and capture the transient characteristics of temperature.（4）A double layer framework containing set point optimizer and nonlinear model predictive controller is designed for thermal-electric cooperative control of DIR-SOFCs,aiming to achieve high steady-state economy,fast load tracking,and efficient thermal management with safety con-straints,such as maximum temperature gradient,and physical constraints of actuators considered.In the rolling optimization of NMPC,the proposed PDAE-constrained dynamic optimization al-gorithms are employed to solve the finite horizon optimal control problem which is constrained by the simplified PDAE dynamic model.To enhance the feasibility of the optimization problem,the hard constraints of maximum temperature gradient is transformed into soft constraints by introducing slack variables.The optimal steady-state set points that maximize the DIR-SOFC power efficiency are compared under three power demand scenarios and validated through the existing research conclusions.The weights in the cost function of NMPC are determined based on the priority of the tracking objectives.The dynamic responses of the feedback closed-loop system with NMPC and the open-loop system with feed-forward control are investigated and analyzed under power demand step-up and step-down scenarios.The comparison demonstrates that the proposed NMPC strategy can realize fast,accurate,stable,and safe predictive control with acceptable real-time performance.This dissertation focuses on efficient algorithms for solving one-dimensional PDAE-constrained dynamic optimization problems.The key contributions include space–time orthog-onal collocation on finite element discretization method and ST-OCFE based single shooting,multiple shooting,and simultaneous approaches.These algorithms are employed to explore the application of NMPC in the context of thermal-electric cooperative control of fuel cells.The findings of this study provide theoretical support for optimization and control of industrial process production.