Mathematical models for simulation and optimization of multi-carrier energy systems

A.S. Markensteijn

Research output: ThesisDissertation (TU Delft)

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Energy systems are vital in modern society, and reliable operation is crucial. Multi-carrier energy systems (MESs), which couple two or more single-carrier systems, have recently become more important, as the need for sustainable energy systems increases. Important tools for the design and operation of energy systems are steady-state simulation and optimization. Steady-state simulation, which involves solving the load flow (LF) problem, is concerned with determining the flow of energy through the system and the values of other quantities throughout the system, such as voltages and pressures, for given demands. In operational optimization, which involves solving the optimal flow (OF) problem, the distribution of generation over the various sources and the set points of controllable elements are determined such that some objective is optimized and such that the system is operated within physical limits.

LF problems and OF problems have been widely studied for single-carrier (SC) systems. However, conventional LF models for the separate single-carrier networks (SCNs) are not able to capture the full extent of the coupling. Recently, different LF models for MESs have been proposed, either using the energy hub (EH) concept, or using a case specific approach. Yet, they do not state how the graphs of the SCNs can be combined into one multi-carrier network (MCN). A good description of integrated networks of multiple energy carriers is very important. Some couplings between energy systems, while possible in practice, can lead to model problems. Although the EH concept can be applied to a general MES, it is unclear in the existing models how the EH should be represented in the graph of the MES. On the other hand, the case specific approaches are not easily applicable to general MESs. Moreover, the effect of the coupling on solvability and well-posedness of the system of nonlinear LF equations for a MES has had little attention in these models.

Operational optimization requires the detailed LF equations to be incorporated into the optimization problem. Nonlinearities of these equations cause issues with convexity and solvability of the OF problem. Hence, the formulation of the LF equations, and the way they are incorporated in the OF problem, greatly influence the solvability of the OF problem and the convergence of the optimization algorithms.

In this thesis, we address some of the existing issues and possibilities to improve on the available models. We present a graph-based framework for steady-state load flow analysis of general MESs that consist of gas, electricity, and heat. The framework is based on connecting the SCNs to heterogeneous coupling nodes, using homogeneous dummy links, to form one connected MCN. Load flow equations are associated with each network element, including the coupling nodes, which are combined with boundary conditions to form one integrated system of nonlinear equations, that needs to be solved to find the solution to the LF problem. This is the integrated approach to formulate the LF problem of a MES.

Alternatively, the model of the connected MCN can be reformulated, such that a MES is represented by a disconnected MCN that consists of the SC networks and a coupling network. This allows for a more decoupled approach to the LF problem, in which the system of nonlinear equations, now consisting of interface conditions connecting the coupling network with the SC networks and the LF equations per SC network, can be solved making use of individual solves for each SC network.

The model framework is validated using a small example MES. Using the integrated approach, we formulate the LF problem of various example MESs, of varying size, with various coupling models and topologies, and various formulations in the single-carrier parts, and solve their LF problems using the Newton-Raphson method (NR). Using these examples, we investigate the effect of coupling on the system of LF equations and discuss the problems arising due to the coupling of SC networks on the solvability of the LF problem. Based on numerical experiments, we compare the convergence behavior of NR for the various single- and multi-carrier systems. Finally, we formulate and solve the LF problem of MESs using the integrated approach and using the decoupled approach. We compare the systems of equations, and we compare the convergence of the solution methods for the two approaches.

Furthermore, in this thesis, we consider two ways to include the LF equations in the OF problem for general MESs, called formulation I and formulation II. 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, and the LF equations are included as a subsystem, which is solved to obtain the state variables for given control variables. We compare the two formulations theoretically, and we illustrate the effect of the two formulations on the solvability of the OF problem by optimizing two MESs.

This study shows that the graph-based framework can be used to formulate and solve the steady-state LF problem for general MESs that consist of gas, electricity, and heat, both with the integrated approach and with the decoupled approach. Moreover, the framework can be used with different components and models, both in the SCNs and for the coupling units. Therefore, our framework includes and extends the currently available LF models for MESs. Furthermore, the model framework provides guidelines to obtain a solvable steady-state LF problem for MESs. We find that using the decoupled approach to perform LF analysis is slower than using the integrated approach. For the LF problem of an example MES with a tree-like structure, NR is independent of the size of the network and of the coupling, and NR requires at most as many iterations as the slowest single-carrier network.

Both formulation I and formulation II result in a solvable OF problem. For the two example MESs, the optimization algorithms require significantly fewer iterations with formulation II than with formulation I.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Delft University of Technology
  • Vuik, C., Supervisor
Thesis sponsors
Award date5 Jul 2021
Print ISBNs978-94-6366-419-6
Publication statusPublished - 2021


  • Adjoint approch
  • Decoupled load flow
  • Fixed-point method
  • Gas networks
  • Heat networks
  • Integrated energy systems
  • Load flow analysis
  • Load flow problem
  • Multi-carrier energy systems
  • Multi-carrier energy networks
  • Natural gas
  • Newton-Raphson (N-R) method
  • Newton's method
  • Nonlinearly constrained optimization problem
  • Numerical analysis/modeling
  • Optimal Power Flow problem
  • Power flow analysis
  • Power grids
  • Scaling

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