## Optimization of Energy Systems

The EDOM group has been active in the optimization of energy systems for several years. In a number of projects we have cooperated with electrical and civil engineers, architects, economists, and other mathematicians from universities, research institutes, and industry alike. We investigate problems arising from the planning and operation of energy networks as well as from the analysis of energy markets. We model these as optimization problems, where we can tackle discrete (e.g., yes/no) decisions but also the physical and technical restrictions. In addition, we include stochastic components and multilevel structures in the models where appropriate.

If you have further questions regarding our projects; please contact Alexander Martin (alexander.martin[at]fau.de), Martin Schmidt (mar.schmidt[at]fau.de), or Lars Schewe (lars.schewe[at]math.uni-erlangen.de).

## Ongoing Projects

## CRC/Transregio 154 - Mathematical Modelling, Simulation and Optimization using the Example of Gas NetworksGas transport through pipeline systems has been an important research area in applied mathematics since several decades. In particular, the disciplines of mathematical modelling, simulation, and optimization have been applied to problems from gas transport. However, new developments related to the gas market demand further progress in these mathematical disciplines. The work on these challenges will also extend the range of the yet known mathematical methods. Recently, the necessary fundamental research for this is funded by the Deutsche Forschungsgemeinschaft by implementing the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation, and Optimization using the Example of Gas Networks" in October 2014. The research will include not only progress in each of the mentioned areas. Rather, the main goal is a tighter linkage as a key to answer theoretical as well as applied questions associated to gas transport.Spokesperson of this CRC/Transregio is Alexander Martin. Details |

## Energie Campus NürnbergThe structurally adjusted treatment of diverse forms of energy, their availability on different scales in time and space, and their feed-in and transportation through a holistically designed “energy grid” are among the major challenges for the power industry and power-related sciences. The Energie Campus Nürnberg (EnCN) is an interdisciplinary research center that combines scientific work from the areas engineering and natural sciences, computer sciences, socio-economic, architecture, and mathematics. Its goal is to put to the vision of a sustainable power society based on renewable energy into practice. Currently, the EnCN is divided into ten research projects that strongly interact with each other. The chair of EDOM is active in the project EnCN Simulation that acts as a link between the other EnCN projects Transport, Networks, Process, Building, and (in particular) Economy. Examples of encompassing goals are the optimal layout, planning, and coupling of networks or the development of models and methods to increase robustness against fluctuating availability of energy forms, quantities, market economies and consumption.Details |

## Robustification of Physical Parameters in Gas NetworksIn many real-world optimization problems, not every model parameter can be measured precisely. With the help of robust optimization, uncertain problems can be immunized against perturbations of the input parameters. We focus on the nomination validation problem under uncertain roughness values of the networks’ pipes. As the simple strict robustification is too conservative for this setting, a more flexibel two-stage approach is developed. A particular challenge is the incorporation of binary decision variables in this framework. The goal of this research project is the development of tractable robust counterparts for global optimization problems, with a focus on gas networks.Details |

## Adaptive MIP-Relaxations for MINLPsGoal of the project is the analysis and solution of large-scale MINLPs, especially from the application of instationary gas network optimization, using adaptive MIP models. We approximate the nonlinearities with piecewise-linear functions to construct MIP relaxations of the underlying MINLP. In addition, theoretical results linking the complexity of the relaxations to structural properties of the nonlinear functions and the linearization error shall be derived, whereby known statements of approximation theory are to be combined with techniques of polyhedral combinatorics. Furthermore the polyhedral structure of the resulting MIP relaxations shall be investigated.Details |

## Analysis of the German Electricity MarketThe analysis and evaluation of the German energy market design leads to challenging mathematical optimization problems. For instance, the behavior of different agents like regulated transport system operators investing in network infrastructure and profit-maximizing private firms investing in generation capacity results in multilevel optimization models. These models also exhibit mixed-integer as well as nonlinear aspects due to, e.g., network design subproblems and the modeling of energy flows.In this interdisciplinary project we study these kinds of optimization models and develop problem-specific solution approaches as well as the required theory in order to evaluate the current energy market design and to make proposals for improvement. Details |

## MIP-based Alternating Direction Methods for High-Detail Stationary Gas Transport MINLPsComputing a discrete control of active gas network elements with respect to the nonlinear physics of gas flow leads to challanging mixed-integer nonlinear and nonconvex optimization and feasibility problems. In this project we investigate the strength of mixed-integer linear programming approaches in problem-tailored alternating direction methods (ADMs) for solving large-scale real-world problems from steady-state gas transport. To this end, we develop ADM-tailored model reformulations to aim on extending the theory of classical alternating direction methods in order to implement efficient MIP-based ADMs for solving stationary mixed-integer nonlinear gas transport models that incorporate both, mixing of specific gas quality parameters and a highly detailed model for compressor machines and drives.Details |

## Decomposition Methods for Mixed-Integer Optimal ControlThe objective of this project is the development of mathematical algorithms to find an optimal control for mixed-integer problems on transport networks with the help of decomposition methods. For the sake of synergy inside of the TRR 154 the focus is on gas networks, but the methods should also be useful for water networks or other energy networks. The optimization problems are planned to be decomponed with respect to variables but also with respect to subsystems, with the result that we are getting a time-expansive MINLP with a hierarchic structure. The focus of the sub-project A05 is on the mathematical analysis of structured MINLPs in the light of hierarchic models. The methods of many classical decomposition approaches like Benders, Outer Approximation or Dantzig-Wolfe focus on a generation of cutting planes in the subproblem, which tighten the relaxed set in the masterproblem to achieve a convergence between the values of the objective functions of the masterproblem (dual bound) and the subproblem (primal bounds). In this sub-project we want the subproblem to provide disjunctions for the masterproblem as well, because such an approach enables the algorithm to find global optima for non-convex problems as well.Details |

## Energy System AnalysisAn Energy System comprises of energy production, energy transportation and storage as well as energy consumption. The aim of the energy system Analysis in the context of the energy turnaround agreed on by the German government, is to suggest an optimal capacity expansion plan for renewable energy systems and fossile power plants in order to meet the governments targets as well as ensure maximum security of energy supplies for the planning period until 2023 when all nuclear power plants will be turned off. This leads to a time expanded unit commitment problem ensuring energy demand satisfaction taking the fluctuating feed in of renewable energy systems into account combined with a capacity expansion problem compensating the turned off nuclear power plants and yielding the renewable target proportions set by the government.Details |

## Robust Power Load Balancing in Railway NetworksThe aim of this project are robust train schedules with respect to the power consumption from the power supply stations. The input is a given schedule which is slightly adapted to desynchronize simultaneous train departures. On the other hand, train departures are synchronized with the recuperation phases of other trains to make use of their braking energy. Preliminary results show that significant savings with respect to the provision of reserve power can be achieved.Details |

## Smart Grid OptimizationCurrent trend towards decentralized power generation requires management down to the level of a single household. We have to solve problems concerning both planning of a micro smart grid in question and scheduling of power generation for the immediate future. Our model includes combined heat and power generation unit together with heat and electrical power storage facilities. Additional complications for future scheduling are introduced by the use of solar and wind power.Details |

## Optimal Control of Electrical Distribution Networks with Uncertain Solar Feed-InThe objective of this project is to develop innovative techniques for modelling, simulation and optimization of electrical distribution networks with regard to uncertain solar feed-in. We want to study and optimize fluctuating energy feed and corresponding power flows. Therefore new mathematical approaches will be combined to guarantee a efficient design and stable real-time operation.Details Participants: Frauke Liers, Alexander Martin, Kevin-Martin Aigner |

## MINOA: Mixed-Integer Non-Linear Optimisation: Algorithms and ApplicationsMINOA will train a new generation of scientists in the rather young but fast growing field of mixed-integer nonlinear optimisation applications and algorithms, by enhancing research-related and transferable competences and exposure to the non-academic sector. Through self-organizing training events, the young researchers take responsibility at an early stage of their career. The settings provided by the hosting institutions empower the ESRs to become independent and creative researchers, which increases their employability. Mobility and internationality is provided through secondments within our international consortium that includes institutions from 6 European countries. Furthermore, network-wide events take place regularly.Details Participants: Frauke Liers, Martin Schmidt, Dennis Adelhütte |

## Finished Projects

## Sustainable Business Models in Energy MarketsThe liberalization of electricity markets and the increasing advancement of renewable energy sources pose important new challenges and requirements for our energy system with regard to grid expansion, energy production, transmission, distribution, and innovative storage systems. A successful transformation into a smart energy system thereby crucially depends on adequate investment incentives and the attractiveness of the business models of involved stakeholders. The purpose of the research project "Sustainable Business Models in Energy Markets: Perspectives for the Implementation of Smart Energy Systems" is to provide a comprehensive analysis of the energy system, including the specific economic incentives and business models of all relevant stakeholders, the institutional environment and the technological perspectives. The aim of the project is to develop new and urgently needed insight into the interaction between business models and regulation while taking into account the technological framework, and to allow a more informed discussion and advice regarding political and regulatory frameworks to ensure a successful transition towards a smart energy system.Details |

## Optimal Allocation of Gas Network CapacitiesDue to regulations gas network operators face the new challenge of allocating free capacity at all entry and exit points. Customers may then book within the reported capacity intervals separately at the entry and exit points. Operators have to guarantee that all expected requests within these intervals (called nominations) can be transported through the network. We develop an algorithm for solving a relaxed variant. Therein a nomination validation tool is used as black box.Details |

## Smart Grid SolarThe partners in "Smart Grid Solar" are implementing and field-testing components of a smart grid in the region of Upper Franconia. With a special focus on photovoltaics, simulation and optimisation is used to study the integration of large power production in low-voltage grids using both small private and larger proximity storages.Several storage systems are analysed and evaluated according to seasonal or short-term electrical storage potentials. Furthermore a common framework is used to derive strategies for optimal (dis)charge schedules that exploit a great variety of input data including residual loads, weather forecasts, market prices for electricity and storage properties. The research project ”Smart Grid Solar” is co-financed by the European Union through the European Regional Development Fund and by the Free State of Bavaria. Details |

## EWave - Water Supply Energy Management SystemOver the past few years, requirements for drinking water supply in Germany have become more and more demanding. While the secure supply of high-quality drinking water for the public was the priority during the past decades, rising energy costs and the energy reform being implemented by the German Federal Government now also require that energy is used efficiently. The joint research project EWave aims to develop an innovative energy management system. The objective is to devise energy-optimized operating plans for plants for water collection, treatment, and distribution within the supply system by using state-of-the-art methods, algorithms and software from discrete and continuous optimization.Details |

## Optimization of Hybrid Energy SystemsHybrid energy systems usually consist of two or more energy sources with at least one renewable source and one completely controllable source. In our case the hybrid system also comprises of energy storages, different types of energy consumers and a mini-grid connecting a small number of households. The aim of the project is the optimization of the internal and external power distribution, i. e. inside the individual households as well as between the different households, in order to minimize the energy costs while satisfying the demands. |

## Clearing Coupled Day-Ahead Electricity MarketsThe European power grid can be divided into several market areas where the price of electricity is determined in a day-ahead auction. Market participants can provide continuous hourly bid curves and combinatorial bids with associated quantities given the prices. The goal of the auction is to determine cross-border flow and market clearing prices. Whereas this can be done rather efficiently in the absence of combinatorial structure, in the case of electricity markets the determination of a market clearing price is hard. We solve a non-discriminatory market model to determine clearing prices that maximize the economic surplus of all participants. The determined prices are consistent throughout the market areas. |