scholarly journals Cyclic Control Optimization Algorithm for Stirling Engines

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 873
Author(s):  
Raphael Paul ◽  
Karl Heinz Hoffmann

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%.

2021 ◽  
Vol 313 ◽  
pp. 04002
Author(s):  
Michael Nicol-Seto ◽  
David Nobes

Stirling engines are a variety of heat engines which are capable of using heat from various sources including low temperature renewables. This work examines performance of a lab scale low temperature gamma type Stirling engine with a drive train modified with oval elliptical gears. The gears were added to dwell the engine piston motion to attempt to improve the thermodynamic performance of the engine by better replicating the ideal Stirling cycle. A variety of dwelling piston configurations were tested on both the displacer and power piston. It was observed that that the piston dwelling had the anticipated effect of changing the engine indicator diagrams to more closely resemble the ideal cycle, however there were no substantial improvements to maximum engine power. It was observed that dwelling the displacer piston caused substantial reductions to engine running speeds and resulted in maximum power being reduced. In the case of power piston dwelling the indicator diagram was enlarged and there were slight increases to maximum power production. Overall the added complexity of dwelled piston motion systems is not likely an advantageous method of increasing the power output of low temperature difference Stirling engines.


2020 ◽  
Vol 7 (3) ◽  
pp. 11-22
Author(s):  
VALERY ANDREEV ◽  
◽  
ALEXANDER POPOV

A reduced model has been developed to describe the time evolution of a discharge in an iron core tokamak, taking into account the nonlinear behavior of the ferromagnetic during the discharge. The calculation of the discharge scenario and program regime in the tokamak is formulated as an inverse problem - the optimal control problem. The methods for solving the problem are compared and the analysis of the correctness and stability of the control problem is carried out. A model of “quasi-optimal” control is proposed, which allows one to take into account real power sources. The discharge scenarios are calculated for the T-15 tokamak with an iron core.


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