Volume 68, Issue 12 - Special Issue: Mathematical innovations fostering the energy transition – Control and optimization / Timm Faulwasser, Simone Göttlich, Karl Worthmann

We present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the network’s depot. For the former, we consider the Euler momentum and the thermal energy equation. The thermal aspects are especially challenging. Here, we develop a novel polynomial approximation that we use in the optimization model. The expansion decisions are modeled by binary variables for which we derive additional valid inequalities that greatly help to solve the highly challenging problem. Finally, we present a case study in which we identify three major aspects that strongly influence investment decisions: the estimated average power demand of potentially new consumers, the distance between the existing network and the new consumers, and thermal losses in the network.

This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p -Laplace type, where p=32p=\frac{3}{2}. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2{L^{2}}-sense. The analysis is done both for the PDE system and an ODE system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.

The energy transition is causing many stability-related challenges for power systems. Transient stability refers to the ability of a power grid’s bus angles to retain synchronism after the occurrence of a major fault. In this paper a set-based approach is presented to assess the transient stability of power systems. The approach is based on the theory of barriers , to obtain an exact description of the boundaries of admissible sets and maximal robust positively invariant sets, respectively. We decompose a power system into generator and load components, replace couplings with bounded disturbances and obtain the sets for each component separately. From this we deduce transient stability properties for the entire system. We demonstrate the results of our approach through an example of one machine connected to one load and a multi-machine system.

In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to reduce the models in order to utilize them in real-time. We reformulate the nonlinear power grid models as quadratic systems and reduce them using balanced truncation based on approximations of the reachability and observability Gramians. Finally, we present examples involving numerical simulation of reduced EN and SM models of the IEEE 57 bus and IEEE 118 bus systems.

Optimal power flow is a widely used tool in power system planning and management. Due to the complexity of the power system both in terms of number of variables, degrees of freedom and uncertainty, there is a continuous effort to find more efficient computational methods to solve optimal power flow problems. This article presents a novel method to speed-up the solution of a security constraint optimal power flow problem. An unconventional warm start based on the training of a neural network is investigated as an option to improve the computational efficiency of the optimization problem. The principle of the method and the validity of the approach is demonstrated by different analysis performed on the IEEE14 test grid and based on a linearized mathematical formulation of the problem. The results show the effectiveness of the method in reducing the number of iterations needed to converge to global optimum.

This article discusses how to use optimization-based methods to efficiently operate microgrids with a large share of renewables. We discuss how to apply a frequency-based method to tune the droop parameters in order to stabilize the grid and improve oscillation damping after disturbances. Moreover, we propose a centralized real-time feasible nonlinear model predictive control (NMPC) scheme to achieve efficient frequency and voltage control while considering economic dispatch results. Centralized NMPC for secondary control is a computationaly challenging task. We demonstrate how to reduce the computational burden using the Advanced Step Real-Time Iteration with nonuniform discretization grids. This reduces the computational burden up to 60 % compared to a standard uniform approach, while having only a minor performance loss. All methods are validated on the example of a 9-bus microgrid, which is modeled with a complex differential algebraic equation.

Accurate state-estimation is a vital prerequisite for fast feedback control methods such as Nonlinear Model Predictive Control (NMPC). For efficient process control, it is of great importance that the estimation process is carried out as fast as possible to provide the feedback mechanism with fresh information and enable fast reactions in case of any disturbances. We discuss how Multi-Level Iterations (MLI), known from NMPC, can be applied to the Moving Horizon Estimation (MHE) method for estimating the states and parameters of a system described by a Differential Algebraic Equation model. A challenging field of application for the proposed MLI-MHE method are electric microgrids. These push current control approaches to their limits due to the rising penetration of volatile renewable energy sources and the fast electrical system dynamics. We investigate the closed-loop control performance of the proposed MLI-MHE algorithm in combination with an NMPC controller for a realistic sized microgrid as a numerical example.

This thesis investigates a wireless communication network that is suitable to be deployed as a transparent intermediate wireless network within a hybrid wired-wireless network serving industrial applications with strict timing and reliability requirements. Physical layer and medium access scheme are in focus.