Mathematical Modeling, Simulation and Optimization for Power Engineering and Management

Future electricity markets face new challenges such as increasing variation in supply due to the dominance of renewable energy providers or variation in demand due to the presence of price sensitive customers. In this contribution, we survey the first step to modeling the current demand process for electricity in Germany. Besides standard affine-linear diffusion processes, we aim to model the intraday electricity demand via a Jacobi process that has attractive properties for our applications. Further, we demonstrate the usefulness of the new models by conducting a comprehensive data analysis.

Examples for the use of continuous stochastic processes in the modelling of energy markets are discussed. Two practical use-cases are chosen to illustrate applications. For the stochastic modelling of the economics behind energy markets, models for temperature, demand, and renewable electricity generation are discussed. The modelling of prices on energy markets themselves is debated as a second application. Possible open source data sets for an application of the models are listed.

Solar power generation at solar plants is a strongly fluctuating non-deterministic variable depending on many influencing factors. In general, it is not clear which and how certain variables influence solar power supply at feed-in points in a distribution network. Therefore, analyzing the dependence structure of measured solar power supply and other variables is very informative and can be helpful in designing probabilistic prediction models. In this paper multivariate D-vine copulas are fitted to investigate the relationship between solar power supply and certain meteorological variables in the current time period of one hour length as well as solar power supply in previous time periods. The meteorological variables considered in this analysis are global horizontal irradiation, temperature, wind speed, humidity, precipitation and pressure. By applying parametric D-vine copulas useful insight is gained into the dependence structure of solar power supply and the considered meteorological variables. The main goal lies in determining suitable explanatory variables for the design of probabilistic prediction models for solar power supply at single feed-in points and analyzing their impact on the validation of conditional level-crossing probabilities.

This paper describes the project GivEn that develops a novel multiobjective optimization process for gas turbine blades and vanes using modern “adjoint” shape optimization algorithms. Given the many start and shut-down processes of gas power plants in volatile energy grids, besides optimizing gas turbine geometries for efficiency, the durability understood as minimization of the probability of failure is a design objective of increasing importance. We also describe the underlying coupling structure of the multiphysical simulations and use modern, gradient based multiobjective optimization procedures to enhance the exploration of Pareto-optimal solutions.

This paper examines the measurement uncertainty calculation for results from network state reconstruction and gas quality tracking in gas transport and distribution networks. The Monte Carlo method is used for this purpose. It analyses how many Monte Carlo runs are needed to determine the measurement uncertainty with sufficient accuracy and reliability. The analysis shows that the number of Monte Carlo runs depends on the stochastic properties of the calculation results, particularly on the excess of probability distribution. It shows how Stein’s two-stage procedure can be adapted to determine the measurement uncertainty reliably.

We consider mathematical models and optimization techniques for a melting furnace in phosphate production. In this high temperature process radiation plays a predominant role. The main design goals are a reduction of the energy consumption as well as the product quality. In particular, we are going to focus on shape optimization techniques for an improved design of the melting furnace, which will rely on a hierarchy of models incorporating the multi-physics of the process.

A novel benchmark superstructure is defined for the production of syngas from renewable energy, \(\mathrm {H_2}\), \(\mathrm {CO_2}\) and biogas. Fixed bed reactors (FBR) for the dry reforming (DR) or the reverse water gas shift (RWGS) are used to convert the reactants into raw syngas, which is then purified in a sequence of pressure and/or temperature swing adsorption (PSA/TSA) steps. We discuss the simulation of the resulting model equations. The complexity of the dynamic process model is tackled by model order reduction and parallel-in-time integration using parareal in order to allow a faster model evaluation in particular for the multi-query context arising in a design optimization process. Future challenges regarding mixed-integer optimization and more complex model dynamics are discussed.

Triggered by the increasing number of renewable energy sources, the German electricity grid is undergoing a fundamental change from mono to bidirectional power flow. This paradigm shift confronts grid operators with new problems but also new opportunities. In this chapter we point out some of these problems arising on different layers of the grid hierarchy and sketch mathematical methods to handle them. While the transmission system operator’s main concern is stability and security of the system in case of contingencies, the distribution system operator aims to exploit inherent flexibilities. We identify possible interconnections among the layers to make the flexibility from the distribution grid available within the whole network. Our presented approaches include: the distributed control of energy storage devices on a residential level; transient stability analysis via a new set-based approach; a new clustering-based model-order reduction technique; and a modeling framework for the power flow problem on the transmission level which incorporates new grid technologies.

The computation of the optimum of a dynamical or multi-period Optimal Power Flow problem assuming an Interior Point Method (IPM) leads to linear systems of equations whose size is proportional to the number of considered time steps. In this paper we propose a new approach to reduce the amount of time steps needed. Assuming that the power grid’s dynamic is mainly determined by changes of the residual demand, we drop time steps in case it does not change substantially. Hence, the size of the linear systems can be reduced. We tested this method for the German Power Grid of the year 2023 and a synthetic 960 h profile. We were able to reduce the amount of time steps by 40% without changing the quality objective function’s value significantly.

Modern smart grids are required to transport electricity along transmission lines from the renewable energy sources to the customer’s demand in an efficient manner. It is inevitable that power is lost along these lines due to active as well as reactive power flows. However, the losses caused by reactive power flows can be reduced by optimizing the power factor. Therefore, we propose a power flow optimization problem aiming to reduce losses by controlling the power factors within the low-voltage electricity grid online. Furthermore, we show the potential of the proposed scheme in a numerical case study for two scenarios based on real-world data provided by a German distribution system operator.

For a sustainable and CO\(_2\) neutral power supply, the entire energy cycles for power, gas and heating grids have to be taken into account simultaneously. Despite rapid progress, the energy industry is insufficiently equipped for the superordinate planning, monitoring and control tasks, based on increasingly large and coupled network simulation models. The German MathEnergy project aims to overcome these shortcomings by developing selected mathematical key technologies for energy networks and respective software. This chapter gives an overview of MathEnergy by discussing selected new developments related to model order reduction for gas networks, state estimation for gas and power networks, as well as cross-sectoral modeling, simulation and ensemble analysis. Several new theoretical results as well as related software prototypes are introduced. Results for selected gas and power networks are presented, including a first version of the partDE-Hy demonstrator for analysis of power-to-gas scenarios.

In this contribution, we present a gas-power benchmark problem tailored to simulation and optimization of coupled electrical grids and gas networks in a time-varying setting. Based on realistic data sets from the IEEE database and the GasLib suite, we describe the full set up of the underlying equations and motivate the choice of parameters. The illustrative simulation results demonstrate the applicability of the proposed approach and also allow for a clear visualization of gas-power conversion. Moreover, they show that the proposed benchmark problem warrants further investigations.

The modelling of gas networks requires the development of coupling techniques at junctions. Recent work on the coupling of hyperbolic systems based on solving two half Riemann problems can be useful also for the coupling issue in gas networks. This strategy is exemplified here for the coupling of a fluid with a solid modelled by the Euler equations supplemented with a stiffened gas equation and a linear elastic model, respectively. This framework may serve as a basis for investigations of coupling conditions on nodes of a gas network.

District heating networks will play a prominent role in sector coupling. On the one hand, they can help compensating for fluctuations in renewable power generation. On the other hand, they allow to use waste heat from industrial processes instead of natural gas. However, this new role of district heating will also require new operating modes, deeper insight into the network and, consequently, more sophisticated simulation and optimization tools. Here, we deal with an optimal control problem which is dominated by time-varying delays between heat source and consumers: optimal preheating. The goal is to satisfy a periodic demand with a constant heat supply. This problem is still simple, as no devices like turbines are involved. But it contains already all challenges of simulating dynamic networks and, therefore, represents an ideal benchmark. We present a suitable mathematical model, some illustrative analytic examples, an efficient numerical scheme, and a solution to optimal preheating for a real municipal network. The model is both, accurate enough to predict pressure drop or cooling, but also simple enough to allow for fast numerical solution by a method of characteristics. Automatic differentiation is used for both, computing exact Jacobians within Newton’s method and providing an optimizer with sensitivities.