{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T10:00:55Z","timestamp":1770631255937,"version":"3.49.0"},"reference-count":31,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T00:00:00Z","timestamp":1620864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German Federal Ministry of Education and Research","award":["FKZ01LY1706B"],"award-info":[{"award-number":["FKZ01LY1706B"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2014for example, it will not generally lead to maximum power. In this paper, we present a method to optimize the engine\u2019s 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%.<\/jats:p>","DOI":"10.3390\/sym13050873","type":"journal-article","created":{"date-parts":[[2021,5,14]],"date-time":"2021-05-14T03:28:36Z","timestamp":1620962916000},"page":"873","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Cyclic Control Optimization Algorithm for Stirling Engines"],"prefix":"10.3390","volume":"13","author":[{"given":"Raphael","family":"Paul","sequence":"first","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl Heinz","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,13]]},"reference":[{"key":"ref_1","unstructured":"Swedish Stirling AB. (2021, January 19). Residual Gases\u2014A Huge Market Potential for PWR BLOK. Available online: https:\/\/swedishstirling.com\/en\/marknad\/."},{"key":"ref_2","unstructured":"Microgen Engine Corporation Group (2021, January 19). Applications. Available online: https:\/\/www.microgen-engine.com\/applications\/."},{"key":"ref_3","unstructured":"Azelio (2021, January 19). Building a Renewable Future. Available online: https:\/\/www.azelio.com\/product\/."},{"key":"ref_4","unstructured":"Qnergy (2021, January 19). Remote Power. Available online: https:\/\/www.qnergy.com\/remote-power\/."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1063\/1.2916405","article-title":"Thermodynamics in Finite Time","volume":"37","author":"Andresen","year":"1984","journal-title":"Phys. Today"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1002\/anie.201001411","article-title":"Current Trends in Finite-Time Thermodynamics","volume":"50","author":"Andresen","year":"2011","journal-title":"Angew. Chem."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1103\/PhysRevA.19.1272","article-title":"Optimal Configuration of a Class of Irreversible Heat Engines. I","volume":"19","author":"Rubin","year":"1979","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1277","DOI":"10.1103\/PhysRevA.19.1277","article-title":"Optimal Configuration of a Class of Irreversible Heat Engines. II","volume":"19","author":"Rubin","year":"1979","journal-title":"Phys. Rev. A"},{"key":"ref_9","first-page":"311","article-title":"Endoreversible Thermodynamics","volume":"22","author":"Hoffmann","year":"1997","journal-title":"J. Non Equilib. Thermodyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1515\/JNETDY.2003.015","article-title":"Optimal Process Paths for Endoreversible Systems","volume":"28","author":"Hoffmann","year":"2003","journal-title":"J. Non Equilib. Thermodyn."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2125","DOI":"10.1063\/1.335977","article-title":"Optimal Paths for Thermodynamic Systems: The Ideal Diesel Cycle","volume":"58","author":"Hoffmann","year":"1985","journal-title":"J. Appl. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Stanislaw Sieniutycz, S., and de Vos, A. (2000). Optimal Piston Paths for Diesel Engines. Thermodynamics of Energy Conversion and Transport, Springer.","DOI":"10.1007\/978-1-4612-1286-7"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1016\/j.mcm.2011.05.014","article-title":"Optimizing piston velocity profile for maximum work output from a generalized radiative law Diesel engine","volume":"54","author":"Chen","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.ijthermalsci.2011.08.005","article-title":"Engine performance improved by controlling piston motion: Linear phenomenological law system Diesel cycle","volume":"51","author":"Xia","year":"2012","journal-title":"Int. J. Therm. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1063\/1.329894","article-title":"Optimal Paths for Thermodynamic Systems: The ideal Otto Cycle","volume":"53","author":"Mozurkewich","year":"1982","journal-title":"J. Appl. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1515\/JNETDY.2004.002","article-title":"Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak?","volume":"29","author":"Fischer","year":"2004","journal-title":"J. Non Equilib. Thermodyn."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1179\/1743967112Z.00000000025","article-title":"Optimal path of piston motion of irreversible Otto cycle for minimum entropy generation with radiative heat transfer law","volume":"85","author":"Ge","year":"2012","journal-title":"J. Energy Inst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2893","DOI":"10.1063\/1.336281","article-title":"Intrinsically Irreversible Light-Driven Engine","volume":"58","author":"Watowich","year":"1985","journal-title":"J. Appl. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/BF02906311","article-title":"Optimal Paths for a Bimolecular, Light-Driven Engine","volume":"104","author":"Watowich","year":"1989","journal-title":"Il Nuovo Cim. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1007\/s11426-009-0172-2","article-title":"Optimal paths for a light-driven engine with a linear phenomenological heat transfer law","volume":"53","author":"Ma","year":"2010","journal-title":"Sci. China Chem."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1515\/jnet-2016-0031","article-title":"Maximum Work of Free-Piston Stirling Engine Generators","volume":"42","author":"Kojima","year":"2017","journal-title":"J. Non Equilib. Thermodyn."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Masser, R., Khodja, A., Scheunert, M., Schwalbe, K., Fischer, A., Paul, R., and Hoffmann, K.H. (2020). Optimized Piston Motion for an Alpha-Type Stirling Engine. Entropy, 22.","DOI":"10.3390\/e22060700"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Scheunert, M., Masser, R., Khodja, A., Paul, R., Schwalbe, K., Fischer, A., and Hoffmann, K.H. (2020). Power-Optimized Sinusoidal Piston Motion and Its Performance Gain for an Alpha-Type Stirling Engine with Limited Regeneration. Energies, 13.","DOI":"10.3390\/en13174564"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"071002","DOI":"10.1115\/1.4029682","article-title":"Optimal Periodic Control of an Ideal Stirling Engine Model","volume":"137","author":"Craun","year":"2015","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_25","unstructured":"Paul, R.R. (2020). Optimal Control of Stirling Engines. [Ph.D. Thesis, Technische Universit\u00e4t Chemnitz]."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Papageorgiou, M., Leibold, M., and Buss, M. (2015). Optimierung\u2014Statische, Dynamische, Stochastische Verfahren f\u00fcr die Anwendung, Springer Vieweg.","DOI":"10.1007\/978-3-662-46936-1"},{"key":"ref_27","unstructured":"Berry, R.S., Kazakov, V.A., Sieniutycz, S., Szwast, Z., and Tsirlin, A.M. (2000). Thermodynamic Optimization of Finite-Time Processes, John Wiley & Sons."},{"key":"ref_28","unstructured":"Craun, M.J. (2015). Modeling and Control of an Actuated Stirling Engine. [Ph.D. Thesis, University of California Santa Barbara]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1021\/i260021a005","article-title":"Periodic processes: A variational approach","volume":"6","author":"Horn","year":"1967","journal-title":"Ind. Eng. Chem. Process Des. Dev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.1002\/aic.690180619","article-title":"The optimal control of a periodic adsorber: Part II. Theory","volume":"18","author":"Kowler","year":"1972","journal-title":"AIChE J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4115","DOI":"10.1016\/S0009-2509(03)00274-4","article-title":"Optimization of cyclically operated reactors and separators","volume":"58","author":"Bliek","year":"2003","journal-title":"Chem. Eng. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/873\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:00:28Z","timestamp":1760162428000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/873"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,13]]},"references-count":31,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["sym13050873"],"URL":"https:\/\/doi.org\/10.3390\/sym13050873","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,13]]}}}