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A dynamic multi-stage design framework for staged deployment optimization of highly stochastic systems

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Abstract

The need for staged design optimization for multidisciplinary systems with strong, cross-system links and complex systems has been acknowledged in various contexts. This is prominent in fields where decisions between subsystems are dependant, as well as in cases where tactical decisions need to be made in uncertain environments. The flexibility gained by incorporating evolutionary design options has been analyzed by discretizing the time-variant uncertainties into scenarios and considering the flexible decision variables in each scenario separately. However, these problems use existing information at the decision time step. This paper presents a dynamic multi-staged design framework to solve problems that dynamically incorporate updated system information and reformulate the problem to account for the updated parameters. The importance of considering staged decisions is studied, and the benefit of the model is evaluated in cases where the stochasticity of the parameters decreases with time. The impact of considering staged deployment for highly stochastic, large-scale systems is investigated through a numerical case study as well as a case study for the IEEE 30 bus system. The case studies presented in this paper investigate multi-disciplinary design problems for large-scale complex systems as well as operational planning for highly stochastic systems. The importance of considering staged deployment for multi-disciplinary systems that have decreasing variability of their parameters with time is highlighted and demonstrated through the results of numerical and engineering case studies.

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Abbreviations

\(i\) :

Set of springs, indexed by \(i \in \{1,2,3\}\)

\(s\) :

Set of scenarios, indexed by \(s \in S\)

\(\Pi\) :

Probability of scenario occurrence

\(G\) :

Shear modulus

\(C\) :

Spring index

\(c\) :

Damping coefficients

\(m_{i}\) :

Mass for each spring subsystem \(i\)

\(Q_{i}\) :

Diagonal weighting matrix associated with the velocity and position terms in the objective

\(R_{i}\) :

Weighting parameter associated with the control term in the objective

\(w_{c_{i}}\) :

Weighting coefficient associated with the control objective function

\(w_{p_{i}}\) :

Weighting coefficient associated with the physical design objective function

\(x_{i0}\) :

Initial conditions of state variables

\(A\) :

Amplitude of oscillation

\(D_{s}\) :

Clearance constant

\(k_{i}\) :

Spring constant

\(F_{u}\) :

Maximum allowable force

\(\kappa _{1}\) :

Minimum allowable inside diameter

\(\kappa _{2}\) :

Lower bound on physical variables

\(u_{i}(t)\) :

Control variable corresponding to input applied to spring \(i\) at time t

\(x_{i}(t)\) :

Position and velocity of the \(i ^{th}\) mass at time t

\(y_{i}\) :

Local physical decision variables in \(i ^{th}\) subsystem

\(y_{s}(i,j)\) :

Shared physical decision variables in \({i }^{th}\) and \({j }^{th}\) subsystems, i.e., combined, shared physical decision variables

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Acknowledgements

This work was partially supported by the National Science Foundation (NSF) through the supplement project to the Engineering Research Center for Power Optimization of Electro-Thermal Systems (POETS) with cooperative agreement EEC-1449548, and through the Faculty Early Career Development awards CMMI-1942559 and CBET-1554018.

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Authors and Affiliations

  1. Department of Industrial and Enterprise Systems Engineering, University of Illinois Urbana-Champaign, Champaign, USA

    Bayan Hamdan, Zheng Liu & Pingfeng Wang

  2. Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, USA

    Koki Ho

  3. Grado Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, USA

    İ. Esra Büyüktahtakın

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  1. Bayan Hamdan
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  2. Zheng Liu
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  5. Pingfeng Wang
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Correspondence to Pingfeng Wang.

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Replication of Results

The full formulation for the case studies considered is provided in the paper. The scenarios were generated stochastically and changed in every run based on the method detailed in the paper. GAMS modeling software was used with BARON optimization solver used for the independent runs and the DSMD solution framework provided. The IEEE 30 Bus Case Study has been referenced to refer the reader to the full data available online. The codes are available upon request by contacting the corresponding author.

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Hamdan, B., Liu, Z., Ho, K. et al. A dynamic multi-stage design framework for staged deployment optimization of highly stochastic systems. Struct Multidisc Optim 66, 162 (2023). https://doi.org/10.1007/s00158-023-03609-6

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