Stabilization of structure-preserving power networks with market dynamics

This paper studies the problem of maximizing the social welfare while stabilizing both the physical power network as well as the market dynamics. For the physical power grid a third-order structure-preserving model is considered involving both frequency and voltage dynamics. By applying the primal-dual gradient method to the social welfare problem, a distributed dynamic pricing algorithm in port-Hamiltonian form is obtained. After interconnection with the physical system a closed-loop port-Hamiltonian system of differential-algebraic equations is obtained, whose properties are exploited to prove local asymptotic stability of the optimal points.Comment: IFAC World Congress 2017, accepted, 6 page

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.Comment: 11 pages, submitted to TAC, accepted. arXiv admin note: text overlap with arXiv:1510.0542

Energy-based analysis and control of power networks and markets:Port-Hamiltonian modeling, optimality and game theory

This research studies the modeling, control and optimization of power networks. A unifying mathematical approach is proposed for the modeling of both the physical power network as well as market dynamics. For the physical system, several models of varying complexity describing the changes in frequency and voltages are adopted. For the electricity market, various dynamic pricing algorithms are proposed that ensure a optimal dispatch of power generation and demand (via flexible loads). Such pricing algorithms can be implemented in real-time and using only local information that is available in the network (such as the frequency). By appropriately coupling the physical dynamics with the pricing algorithms, stability of the combined physical-economical system is proven. This in particular shows how real-time dynamic pricing can be used as a control method to achieve frequency regulation and cost efficiency in the network

Coping with Algebraic Constraints in Power Networks

In the intuitive modelling of the power network, the generators and the loads are located at different subset of nodes. This corresponds to the so-called structure preserving model which is naturally expressed in terms of differential algebraic equations (DAE). The algebraic constraints in the structure preserving model are associated with the load dynamics. Motivated by the fact the presence of the algebraic constraints hinders the analysis and control of power networks, several aggregated models are reported in the literature where each bus of the grid is associated with certain load and generation. The advantage of these aggregated models is mainly due to the fact that they are described by ordinary differential equations (ODE) which facilitates the analysis of the network. However, the explicit relationship between the aggregated model and the original structure preserved model is often missing, which restricts the validity and applicability of the results. Aiming at simplified ODE description of the model together with respecting the heterogenous structure of the power network has endorsed the use of Kron reduced models; see e.g. [2]. In the Kron reduction method, the variables which are exclusive to the algebraic constraints are solved in terms of the rest of the variables. This results in a reduced graph, the (loopy) Laplaican matrix of which is the Schur complement of the (loopy) Laplacian matrix of the original graph. By construction, the Kron reduction technique restricts the class of the applicable load dynamics to linear loads. The algebraic constraints can also be solved in the case of frequency dependent loads where the active power drawn by each load consists of a constant term and a frequencydependent term [1],[3]. However, in the popular class of constant power loads, the algebraic constraints are “proper”, meaning that they are not explicitly solvable. In this talk, first we revisit the Kron reduction method for the linear case, where the Schur complement of the Laplacian matrix (which is again a Laplacian) naturally appears in the network dynamics. It turns out that the usual decomposition of the reduced Laplacian matrix leads to a state space realization which contains merely partial information of the original power network, and the frequency behavior of the loads is not visible. As a remedy for this problem, we introduce a new matrix, namely the projected pseudo incidence matrix, which yields a novel decomposition of the reduced Laplacian. Then, we derive reduced order models capturing the behavior of the original structure preserved model. Next, we turn our attention to the nonlinear case where the algebraic constraints are not readily solvable. Again by the use of the projected pseudo incidence matrix, we propose explicit reduced models expressed in terms of ordinary differential equations. We identify the loads embedded in the proposed reduced network by unveiling the conserved quantity of the system