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 %.

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斯特林发动机的循环控制优化算法

理想的斯特林循环描述了一种操作平衡斯特林发动机的特定方法。该循环包括两个等温和两个等速冲程。对于可能具有多种不可逆性且其动力学由一组耦合的常微分方程表示的非平衡斯特林发动机，基于理想循环的控制策略不一定会产生最佳性能，例如，它不会通常会导致最大功率。在本文中，我们提出了一种针对不同目标优化发动机活塞路径的方法。特别是功率和效率。在此，重点是用于解决循环最优控制问题的间接迭代梯度算法。循环最优控制问题导致了哈密顿系统，该系统的状态和代价高昂的子问题之间具有对称性。对称性体现在相互关联的吸引和排斥极限环的存在上。我们的算法利用这些极限环来解决周期性边界条件下的状态和成本问题。提供了该算法的说明，并说明了如何将控件嵌入到系统动力学中。此外，讨论了对于示例性斯特林发动机模型获得的优化结果。对于此斯特林发动机模型，将优化的活塞路径与谐波活塞路径进行比较显示出功率和效率均得到了显着提高。在最大功率点，由于功率最佳控制而产生的相对功率增益约为。28％，而由于在最大效率点进行效率最佳控制而产生的相对效率增益约为ca。10％。