Cyclic Control Optimization Algorithm for Stirling Engines

Abstract

The ideal Stirling cycle describes a specific way to operate an equilibrium Stirling engine. This cycle consists of two isothermal and two isochoric strokes. For non-equilibrium Stirling engines, which may feature various irreversibilities and whose dynamics is characterized by a set of coupled ordinary differential equations, a control strategy that is based on the ideal cycle will not necessarily yield the best performance—for example, it will not generally lead to maximum power. In this paper, we present a method to optimize the engine’s piston paths for different objectives; in particular, power and efficiency. Here, the focus is on an indirect iterative gradient algorithm that we use to solve the cyclic optimal control problem. The cyclic optimal control problem leads to a Hamiltonian system that features a symmetry between its state and costate subproblems. The symmetry manifests itself in the existence of mutually related attractive and repulsive limit cycles. Our algorithm exploits these limit cycles to solve the state and costate problems with periodic boundary conditions. A description of the algorithm is provided and it is explained how the control can be embedded in the system dynamics. Moreover, the optimization results obtained for an exemplary Stirling engine model are discussed. For this Stirling engine model, a comparison of the optimized piston paths against harmonic piston paths shows significant gains in both power and efficiency. At the maximum power point, the relative power gain due to the power-optimal control is ca. 28%, whereas the relative efficiency gain due to the efficiency-optimal control at the maximum efficiency point is ca. 10%.

Cooling Cycle Optimization for a Vuilleumier Refrigerator

“…It is interesting to note that this optimized volume dynamics is very similar to the "D-motion" considered in [28,29] for a free-piston Vuilleumier heat pump. Moreover, the respective displacer piston motion is also rather similar to the optimal working piston motion of the alpha-Stirling engines investigated in [24,26].…”

A Class of Reduced-Order Regenerator Models

“…Due to their low number of degrees of freedom and reduced numerical effort, the developed endoreversible regenerator models are predestined for applications such as design and control optimizations. In fact, the EEn-regenerator model has already been applied for piston path optimizations of a Stirling engine [52,53,55]. In future research work, the developed endoreversible regenerator models should be validated for a wider range of design parameters and operational conditions.…”

Performance Analysis and Four-Objective Optimization of an Irreversible Rectangular Cycle

“…P = P/P max (15) According to the definition of effective power (W ep ) in [27,28], the W ep of the RC is…”

“…After decades of development, a series of instructive and practical achievements have been obtained in finite time thermodynamics [1][2][3][4][5][6][7][8], the research objects of which include heat engines [9][10][11][12][13][14][15], refrigerators [16,17], heat pumps [18], chemical cycles [19], and quantum cycles [20,21]. The rectangular cycle (RC) is composed of four thermodynamic processes, its endothermic processes are closed to the endothermic processes of the dual cycle, and its exothermic processes are closed to the exothermic processes of the Miller cycle.…”