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.