Tjerk Stegink

A Unifying Energy-Based Approach to Stability of Power Grids With Market Dynamics

In this paper, a unifying energy-based approach is provided to the modeling and stability analysis of power systems coupled with market dynamics. We consider a standard model of the power network with a third-order model for the synchronous generators involving voltage dynamics. By applying the primal-dual gradient method to a social welfare optimization, a distributed dynamic pricing algorithm is obtained, which can be naturally formulated in port-Hamiltonian form. By interconnection with the physical model a closed-loop port-Hamiltonian system is obtained, whose properties are exploited to prove asymptotic stability to the set of optimal points. This result is extended to the case that also general nodal power constraints are included into the social welfare problem. Additionally, the case of line congestion and power transmission costs in acyclic networks is covered. Finally, a dynamic pricing algorithm is proposed that does not require knowledge about the power supply and demand.

Perspectives in modeling for control of power networks

Abstract Increasing integration of renewable energy sources in the power grid leads to a complete re-thinking of its operation. This necessitates the consideration of new modeling and analysis approaches, which can serve as natural starting points for robust and scalable control and design strategies. In this ‘vision’ paper we will highlight two topics within the broad area of power networks: the modeling and analysis of the synchronous generator, and the modeling and analysis of power networks using the swing equation as an approximate model for the generator. In both cases we will discuss a port-Hamiltonian formulation, which reflects the underlying physics of power flow and energy storage. Although the port-Hamiltonian model of the synchronous generator reveals a clear structure it still poses fundamental challenges for its non-zero steady state stability analysis. It is shown how the swing equation can be directly deduced from the power balance of the port-Hamiltonian model of the synchronous generator. Under the phasor assumption, this leads to a port-Hamiltonian model of power networks of generators, which enables a straightforward stability analysis and provides a starting point for control.

Optimal power dispatch in networks of high-dimensional models of synchronous machines

This paper investigates the problem of optimal frequency regulation of multi-machine power networks where each synchronous machine is described by a sixth order model. By analyzing the physical energy stored in the network and the generators, a port-Hamiltonian representation of the multi-machine system is obtained. Moreover, it is shown that the open-loop system is passive with respect to its steady states which allows the construction of passive controllers to control the multi-machine network. As a special case, a distributed consensus based controller is designed that regulates the frequency and minimizes a global quadratic generation cost in the presence of a constant unknown demand. In addition, the proposed controller allows freedom in choosing any desired connected undirected weighted communication graph.

Stable Interconnection of Continuous-Time Price-Bidding Mechanisms with Power Network Dynamics

We study price-based bidding mechanisms in power networks fo r real-time dispatch and frequency regulation. On the market side, we consider the interaction between the independent system operator (ISO) and a group of generators involved in a Bertrand game of competition. The generators seek to maximize their individual profit while the ISO aims to solve the economic dispatch problem and to regulate the frequency. Since the generators are strategic and do not share their cost functions, the ISO engages the generators in a continuous-time price-based bidding process. This results in a coupling between the ISO-generator dynamics and swing dynamics of the network. We analyze its stability, establishing frequency regulation and the convergence to the efficient Nash equilibrium and the optimal generation levels. Simulation illustrate our results.

A Complement on Elimination and Realization in Rational Representations

In this paper we study a number of problems in the context of rational representations of behaviors. In that context, a given proper real rational matrix can represent three behaviors. In the first place it can represent an input–output behavior. Second, it can represent the kernel behavior of the rational ‘differential operator’ associated with the rational matrix. Third, it can represent the image behavior asociated with the rational matrix. On the other hand, every proper real rational matrix admits a realization as a finite-dimensional linear state-space system. Such realization can represent three system behaviors: an input-state-output behavior, an output nulling behavior, or a driving variable behavior. In this paper we will study the relation between the three external behaviors of these state representations, and the behaviors given by the three rational representations associated with the underlying rational matrix. Preliminary results from [5] will be complemented to obtain necessary and sufficient conditions such that the respective external behaviors are equal.