The Modelling of Organic Rankine Cycle Systems Intended for Industrial Optimization
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The Organic Rankine Cycle (ORC) is one of the most efficient heat recovery technologies for low-temperature heat resources that are used in the geothermal power plant industry and waste heat recovery systems. As such, ORC systems are proving to be a sustainable technology that can help address some of the current concerns surrounding global warming and environmental pollution from using non-renewable resources. However, optimizing large and advanced systems can be a complex task that involves various fields of many variables from the thermodynamics to the plant's topologies to the environmental regulations. Therefore, this thesis will focus on providing a systematic modelling framework that can be used to optimize ORC systems efficiently. Commonly in the literature, ORC systems are modelled using the sequential-modular (SM) approach where the unit operation modules are connected in the order of the plant's process. This forms the flowsheet of the ORC system, which is then solved using a nonlinear equation solver to converge to a feasible operating point. Generally, the SM model is optimized by manually varying the plant parameters or by using advanced optimization algorithms to maximize/minimize the objective function. This is not an efficient approach and can lead to various optimization and numerical issues, such as failure to converge to a solution and long execution times. A more efficient method, but often more difficult to construct and troubleshoot, is to model the ORC system using the equation-oriented (EO) approach where the system is expressed as a set of equations. Provided the equations are algebraic and twice differentiable, the optimization solvers can exploit the model structure, the underlying equations and the relationship between the decision variables to effectively optimize the ORC system. Generally, the equations are not algebraic and consist of thermodynamic routines or external functions that are not differentiable and incompatible with white-box solvers that can deterministically guarantee global optimality. Therefore, this thesis will propose a modelling approach and provide a set of tools to model an ORC system in order to tailor the model for derivative-based and white-box solvers. This involves deriving the set of equations that describe the ORC system and approximating the nonlinear terms that are not differentiable using regression tools. As a result, the optimization performance of the algebraic EO model can be more than 29000x faster than the SM model. The problem with any approximated model is to ensure that the model can accurately represent the original system. While the accuracy of the model can sometimes be improved by using highly nonlinear model fits, such as higher order polynomial functions, they can contribute to the nonlinearity of the model and degrade the optimization performance. Therefore, this thesis will introduce a piecewise regression approach to improve the accuracy of the approximated model and decrease the nonlinearity of the optimization problem. As a result, the performance of some solvers can be significantly improved and, in some cases, more than 6x faster than using the single fit approximations. Building on the piecewise fit approximation work, the final study that was carried out in this research focused on a mixed-integer linear programming (MILP) formulation of the ORC model. This is to address the gap in the literature where the MILP formulations of ORC models have not been extensively investigated, despite the general view that nonlinear problems are harder to solve than linear problems. This involves utilizing existing integer and linear programming techniques and piecewise linear approximations. Consequently, the size of the optimization problem increases considerably due to the auxiliary variables and constraints from the linearization procedure, which can degrade the optimization performance. This research provides an alternative approach to the SM model for modelling large and complex ORC systems that are robust and efficient for optimization. The proposed modelling framework will be implemented on three real-world geothermal power plants that vary in size and topology. In addition, this research shows that while decreasing the nonlinearity of the optimization problem can improve the performance, it is not advisable to completely linearize the model as it can have an adverse effect due to the large number of auxiliary variables and constraints that are generated.