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Reliability-based control co-design of horizontal axis wind turbines

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Abstract

Wind energy is one of the fastest-growing energy sources due to its cleanness, sustainability, and cost-effectiveness. In the past, wind turbine design studies focused primarily on a sub-system or single-discipline design and analysis, including control, structural, aerodynamic, and electro-mechanical studies, for example. More recent studies formulated wind turbine design problems using multidisciplinary design optimization (MDO) strategies, with either static or dynamic system models, providing the potential for identifying system-level optimal designs. On the other hand, efforts have also been made to increase the reliability and robustness of wind turbines by accounting for various sources of uncertainty explicitly in the design process. In the presented study, the MDO formulation of wind turbine design problem has been extended to include both control system co-design and reliability considerations in an integrated manner. As a result, the optimal wind turbine design that has an optimal control solution and is robust to uncertainties can be obtained at an early design stage, which would benefit the controller design and maintenance design at latter phases. In this paper, the design of a horizontal axis wind turbine (HAWT) supported by a tubular tower is considered and formulated as a multi-objective control co-design problem with design parameter uncertainties and stochastic wind load. A physics-based multidisciplinary dynamic model of tubular-tower-supported pitch-controlled HAWT that captures the main design conflicts under extreme wind is provided and implemented, along with the necessary modifications to make nested control co-design comply with modern reliability-based design optimization structures, forming a new class of reliability-based co-design (RBCD) problems. In particular, we provide detailed discussions about RBCD problem formulations and implementation strategies, and with the HAWT design problem, we demonstrate the results and computational costs with integrated double-loop, single-loop, as well as decoupled methods.

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References

  • Allison JT (2013) Plant-limited co-design of an energy-efficient counterbalanced robotic manipulator. ASME J Mech Des 135(10):101003

    Article  Google Scholar 

  • Allison JT, Herber DR (2014) Multidisciplinary design optimization of dynamic engineering systems. AIAA J 52(4):691–710

    Article  Google Scholar 

  • Allison JT, Nazari S (2010) Combined plant and controller design using decomposition-based design optimization and the minimum principle. In: International design engineering technical conferences and computers and information in engineering conference, vol 44090, pp 765–774

  • Allison JT, Guo T, Han Z (2014) Co-design of an active suspension using simultaneous dynamic optimization. ASME J Mech Des 136(8):081003

    Article  Google Scholar 

  • Ashuri T, Zaaijer MB, Martins JR, Zhang J (2016) Multidisciplinary design optimization of large wind turbines—technical, economic, and design challenges. Energy Convers Manag 123:56–70

    Article  Google Scholar 

  • Azad S, Alexander-Ramos MJ (2020) A single-loop reliability-based MDSDO formulation for combined design and control optimization of stochastic dynamic systems. J Mech Des 143(2):021703

    Article  Google Scholar 

  • Box GE, Jenkins GM (1970) Time series analysis: forecasting and control Holden-day. San Francisco, p 498

  • Chiralaksanakul A, Mahadevan S (2005) First-order approximation methods in reliability-based design optimization. ASME J Mech Des 127(5):851–857

    Article  Google Scholar 

  • Cui T, Allison JT, Wang P (2020a) A comparative study of formulations and algorithms for reliability-based co-design problems. J Mech Des 142(3):031104

    Article  Google Scholar 

  • Cui T, Allison JT, Wang P (2020b) Reliability-based co-design of state-constrained stochastic dynamical systems. In: AIAA Scitech 2020 Forum, p 0413

  • Cui T, Zheng Z, Wang P (2020c) Surrogate model assisted lithium-ion battery co-design for fast charging and cycle life performances. In: ASME 2020 international design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers Digital Collection

  • Deshmukh AP, Allison JT (2016) Multidisciplinary dynamic optimization of horizontal axis wind turbine design. Struct Multidisc Optim 53(1):15–27. https://doi.org/10.1007/s00158-015-1308-y

    Article  MathSciNet  Google Scholar 

  • Deshmukh AP, Herber DR, Allison JT (2015) Bridging the gap between open-loop and closed-loop control in co-design: a framework for complete optimal plant and control architecture design. In: 2015 American control conference, pp 4916–4922. Chicago, IL, USA

  • Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. ASME J Mech Des 126(2):225–233

    Article  Google Scholar 

  • Du X, Guo J, Beeram H (2008) Sequential optimization and reliability assessment for multidisciplinary systems design. Struct Multidisc Optim 35(2):117–130. https://doi.org/10.1007/s00158-007-0121-7

    Article  MathSciNet  MATH  Google Scholar 

  • Fan X, Wang P, Hao F (2019) Reliability-based design optimization of crane bridges using kriging-based surrogate models. Struct Multidisc Optim 59(3):993–1005. https://doi.org/10.1007/s00158-018-2183-0

    Article  Google Scholar 

  • Fathy HK, Papalambros PY, Ulsoy AG, Hrovat D (2003) Nested plant/controller optimization with application to combined passive/active automotive suspensions. In: American control conference. IEEE, Denver, pp 3375–3380

  • Fathy HK, Reyer JA, Paplambros PY, Ulsoy AG (2001) On the coupling between the plant and controller optimization problems. In: Proceedings of the 2001 American control conference, vol 3, pp 1864–1869. Arlington, VA

  • Forcier LC, Joncas S (2012) Development of a structural optimization strategy for the design of next generation large thermoplastic wind turbine blades. Struct Multidisc Optim 45(6):889–906. https://doi.org/10.1007/s00158-011-0722-z

    Article  MATH  Google Scholar 

  • Grujicic M, Arakere G, Pandurangan B, Sellappan V, Vallejo A, Ozen M (2010) Multidisciplinary design optimization for glass-fiber epoxy-matrix composite 5 mw horizontal-axis wind-turbine blades. J Mater Eng Perform 19(8):1116–1127

    Article  Google Scholar 

  • He Y, Monahan AH, Jones CG, Dai A, Biner S, Caya D, Winger K (2010) Probability distributions of land surface wind speeds over North America. J Geophys Res Atmos 115(D4):1000. https://doi.org/10.1029/2008JD010708

    Article  Google Scholar 

  • Herber DR, Allison JT (2019) Nested and simultaneous solution strategies for general combined plant and control design problems. ASME J Mech Des 141(1):011402

    Article  Google Scholar 

  • Hu Z, Du X (2015) First order reliability method for time-variant problems using series expansions. Struct Multidisc Optim 51(1):1–21. https://doi.org/10.1007/s00158-014-1132-9

    Article  MathSciNet  Google Scholar 

  • Hu Z, Li H, Du X, Chandrashekhara K (2013) Simulation-based time-dependent reliability analysis for composite hydrokinetic turbine blades. Struct Multidisc Optim 47(5):765–781. https://doi.org/10.1007/s00158-012-0839-8

    Article  Google Scholar 

  • Hu W, Choi KK, Cho H (2016) Reliability-based design optimization of wind turbine blades for fatigue life under dynamic wind load uncertainty. Struct Multidisc Optim 54(4):953–970. https://doi.org/10.1007/s00158-016-1462-x

    Article  Google Scholar 

  • Jager D, Andreas A (1996) NREL National Wind Technology Center (NWTC): M2 Tower; Boulder, Colorado (Data). NREL Report DA-5500-56489. National Renewable Energy Lab. (NREL), Golden

  • Jiang X, Lu Z, Hu Y, Lei J (2021) Time-dependent performance measure approach for time-dependent failure possibility-based design optimization. Struct Multidisc Optim. https://doi.org/10.1007/s00158-020-02795-x

    Article  MathSciNet  Google Scholar 

  • Jonkman J, Butterfield S, Musial W, Scott G (2009)Definition of a 5-mw reference wind turbine for offshore system development. Tech. Rep. National Renewable Energy Lab. (NREL), Golden

  • Karush W (1939) Minima of functions of several variables with inequalities as side constraints. MSc Dissertation, Department of Mathematics, University of Chicago

  • Kooijman H, Lindenburg C, Winkelaar D, Van der Hooft E (2003) Aero-elastic modelling of the DOWEC 6 MW pre-design in PHATAS. DOWEC Dutch Offshore Wind Energy Converter 1997–2003 Public Reports

  • Kulunk E (2011) Aerodynamics of wind turbines. In: Fundamental and advanced topics in wind power. IntechOpen, London

  • Lantz EJ, Roberts JO, Nunemaker J, DeMeo E, Dykes KL, Scott GN (2019) Increasing wind turbine tower heights: Opportunities and challenges. Tech. Rep. National Renewable Energy Lab. (NREL), Golden

  • Lee G, Son H, Youn BD (2019) Sequential optimization and uncertainty propagation method for efficient optimization-based model calibration. Struct Multidisc Optim 60(4):1355–1372. https://doi.org/10.1007/s00158-019-02351-2

    Article  Google Scholar 

  • Lee U, Kang N, Lee I (2020) Shared autonomous electric vehicle design and operations under uncertainties: a reliability-based design optimization approach. Struct Multidisc Optim 61(4):1529–1545. https://doi.org/10.1007/s00158-019-02434-0

    Article  MathSciNet  Google Scholar 

  • Li H, Cho H, Sugiyama H, Choi KK, Gaul NJ (2017) Reliability-based design optimization of wind turbine drivetrain with integrated multibody gear dynamics simulation considering wind load uncertainty. Struct Multidisc Optim 56(1):183–201. https://doi.org/10.1007/s00158-017-1693-5

    Article  Google Scholar 

  • Liang J, Mourelatos Z, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. ASME J Mech Des 129(12):1215–1224

    Article  Google Scholar 

  • Liberzon D (2011) Calculus of variations and optimal control theory: a concise introduction. Princeton University Press, Princeton

    Book  Google Scholar 

  • Lindenburg C (2002) Aeroelastic modelling of the LMH64-5 blade. Technical Report No. DOWEC-02-KL-083/0. Energy Research Center of the Netherlands

  • Maki K, Sbragio R, Vlahopoulos N (2012) System design of a wind turbine using a multi-level optimization approach. Renew Energy 43:101–110

    Article  Google Scholar 

  • McWilliam MK, Barlas TK, Madsen HA, Zahle F (2018) Aero-elastic wind turbine design with active flaps for AEP maximization. Wind Energy Sci 3(1):231

    Article  Google Scholar 

  • Nguyen T, Song J, Paulino G (2010) Single-loop system reliability-based design optimization using matrix-based system reliability method: theory and applications. ASME J Mech Des 132(1):011005

    Article  Google Scholar 

  • Noh Y, Choi K, Du L (2009) Reliability-based design optimization of problems with correlated input variables using a Gaussian copula. Struct Multidisc Optim 38(1):1–16

    Article  Google Scholar 

  • NWTC (2020) NWTC Information Portal (NWTC 135-m Tower Data)

  • Park HU, Lee JW, Chung J, Behdinan K (2015) Uncertainty-based MDO for aircraft conceptual design. Aircraft Eng Aerosp Technol 87(4):345–356

    Article  Google Scholar 

  • Patterson MA, Rao AV (2016) A general-purpose MATLAB software for solving multiple-phase optimal control problems, version 2.3

  • Pavese C, Tibaldi C, Zahle F, Kim T (2017) Aeroelastic multidisciplinary design optimization of a swept wind turbine blade. Wind Energy 20(12):1941–1953

    Article  Google Scholar 

  • Sieros G, Chaviaropoulos P, Sørensen JD, Bulder BH, Jamieson P (2012) Upscaling wind turbines: theoretical and practical aspects and their impact on the cost of energy. Wind Energy 15(1):3–17

    Article  Google Scholar 

  • Sim SK, Maass P, Lind PG (2019) Wind speed modeling by nested arima processes. Energies 12(1):69

    Article  Google Scholar 

  • Tu J, Choi K, Park Y (1999) A new study on reliability- based design optimization. ASME J Mech Des 121(4):557–564

    Article  Google Scholar 

  • Vasjaliya NG, Gangadharan SN (2013) Aero-structural design optimization of composite wind turbine blade. In: Proceedings of the 10th world congress on structural and multidisciplinary optimization, vol 21, p 2014

  • Wu JS, Chiang LK (2004) Free vibrations of solid and hollow wedge beams with rectangular or circular cross-sections and carrying any number of point masses. Int J Numer Methods Eng 60(3):695–718

    Article  Google Scholar 

  • Yan H, Yan G (2009) Integrated control and mechanism design for the variable input-speed servo four-bar linkages. Mechatronics 19(2):274–285

    Article  Google Scholar 

  • Youn B, Choi K, Park Y (2003) Hybrid analysis method for reliability-based design optimization. ASME J Mech Des 125(2):221–232

    Article  Google Scholar 

  • Youn BD, Choi KK, Du L (2005) Enriched performance measure approach for reliability-based design optimization. AIAA J 43(4):874–884

    Article  Google Scholar 

  • Zhang J, Gao L, Xiao M, Lee S, Eshghi AT (2020) An active learning kriging-assisted method for reliability-based design optimization under distributional probability-box model. Struct Multidisc Optim 62(5):2341–2356. https://doi.org/10.1007/s00158-020-02604-5

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is partially supported by National Science Foundation through Faculty Early Career Development (CAREER) awards CMMI-1813111 and the Engineering Research Center for Power Optimization of Electro-Thermal Systems (POETS) with cooperative agreement EEC-1449548.

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All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

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Correspondence to Pingfeng Wang.

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The authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript.

Replication of results

This manuscript is self-contained, in that it contains all necessary theory and data to reproduce the results, including the preliminaries, i.e., the wind turbine design specifications and the theory on reliability-based co-design methodology. The design problem formulation and all parameters for the examples are provided and described in detail. Lastly, the source codes for the implementation of the wind turbine design study can be provided upon requests from interested readers.

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Responsible Editor: Jianbin Du

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Appendices

Appendices

Appendix 1: NREL 5MW wind turbine blade parameters

This appendix provides the NREL 5MW wind turbine blade parameters (Jager and Andreas 1996). Additionally, the listed airfoils are corresponding to the lift and drag coefficients extended to \([-180^\circ , 180^\circ ]\), based on the DOWEC blades (Kooijman et al. 2003; Lindenburg 2002) (Table 9).

Table 9 NREL 5MW wind turbine blade element properties

Appendix 2: HAWT deterministic co-design state and control trajectories

Appendix  2 provides state and control trajectories of the optimum design obtained from the HAWT deterministic co-design formulation.

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Appendix 3: HAWT reliability-based co-design with DLP state and control trajectories

Appendix  3 provides state and control trajectories of the optimum design obtained from the HAWT reliability-based co-design with the DLP formulation.

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Appendix 4: HAWT reliability-based co-design with SLP state and control trajectories

Appendix  4 provides state and control trajectories of the optimum design obtained from the HAWT reliability-based co-design with the SLP formulation.

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Appendix 5: HAWT reliability-based co-design with SORA state and control trajectories

Appendix  5 provides state and control trajectories of the optimum design obtained from the HAWT reliability-based co-design with the SORA formulation.

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Cui, T., Allison, J.T. & Wang, P. Reliability-based control co-design of horizontal axis wind turbines. Struct Multidisc Optim 64, 3653–3679 (2021). https://doi.org/10.1007/s00158-021-03046-3

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