Optimal flow for general multi-carrier energy systems, including load flow equations

A.S. Markensteijn, J.E. Romate, C. Vuik

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Optimization is an important tool for the operation of an energy system. Multi-carrier energy systems (MESs) have recently become more important. Load flow (LF) equations are used within optimization to determine if physical network limits are violated. Due to nonlinearities, the solvability of the OF problem and the convergence of the optimization algorithms are influenced by how the LF equations are included in the optimal flow (OF) problem. In addition, scaling greatly influences the practical solvability of OF problems. This paper considers two ways to include the LF equations within the OF problem for general MESs. In formulation I, optimization is over the combined control and state variables, with the LF equations included explicitly as equality constraints. In formulation II, optimization is over the control variables only. The state variables are solved from the LF equations in a separate subsystem, for given control variables. Hence, the LF equations are included only implicitly in formulation II. The two formulations are compared qualitatively, from a theoretical perspective and based on numerical experiments. Both formulation I and formulation II result in a solvable OF problem. Formulation I is easier to implement and more efficient in terms of CPU time. However, formulation II ensures feasibility and can be used for optimization in combination with dedicated load flow solvers. Both matrix scaling and per unit scaling can be used to solve the OF problem, but they are not equivalent.
Original languageEnglish
Article number100050
Pages (from-to)1-19
Number of pages19
JournalResults in Control and Optimization
Publication statusPublished - 2021


  • Adjoint approach
  • Gas networks
  • Heat networks
  • Multi-carrier energy networks
  • Nonlinearly constrained optimization problem
  • Optimal flow problem
  • Power grids


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