Causal Bond Graphs of Unbalanced Multi–phase Electrical Systems
X. Roboam, G. Gandanegara
LEEI, UMR INPT-ENSEEIHT/CNRS No 5828
2 Rue Camichel BP 7122
31071 Toulouse Cedex, France
Phone: +33 5 61 58 83 29, Fax: +33 5 61 63 88 75, E-mail: Xavier.Roboam@leei.enseeiht.fr
Abstract - This paper aims at giving a systematic method
to ’’pre-process’’ a Bond Graph of any multi–phase
network in electrical engineering, whatever its balanced
or unbalanced operation and whatever its structure
(neutral connection,…) may be. Indeed, in order to obtain
a Bond Graph of an electrical multi–phase network free
of causality conflicts, some rules have to be fulfilled. The
rules expressed in this paper allow to avoid structural
singularities caused by derivative causalities. Some
examples are presented in the field of electrical
engineering :
§ a model of a synchronous machine with trapezoidal
(unbalanced) electromotive force;
§ a multi–phase electrical network with single and
three-phase loads.
Keyword : Bond graph, integral causality, electrical
networks, unbalanced operation, coupled circuit.
I. INTRODUCTION
The interest of a system modelling which respects the
physical causality rules linked with energy transfers is
now well known. For example, as presented in
[Morel et al., 2001-a][Morel et al., 2001-b] a comparison
between the FCA (Formal Causality Analysis) versus the
MNA (Modified Nodal Analysis) shows the advantages
(simulation cost reduction, solver convergence,…) of the
Causal approach from the simulation point of view.
Famous software as Spice and Saber as well as the new
standard VHDL – AMS [IEEE 1076.1] are based on the
MNA. Such methods use an implicit formulation of the
generalised Kirchoff laws. The main advantage of this
latter approach is the systematic derivation of system
equations, whatever the circuit may be or the physical
system to be analysed. Some analogue issues have been
discussed in [Cellier and Elmqvist, 1993] for an object
oriented DAE (Differential Algebraic Equations)
formulation automatically generated. Thus, even “anti
physical” systems as a voltage fed capacitance or an
inertia with an imposed speed can be simulated, and
eventually resulting problems are affected to solver
convergence. Generally, with respect to the numerical
simulation issues, avoiding derivative causality leads to
represent the system equations in an explicit way, without
algebraic loops, which helps the solver convergence.
On the other hand, representing physically the energy
transfers imposes to respect the integral causality which
allows to the devices to fit together in a global energy
system. This latter issue is essential from the system
design point of view as expressed in [Piquet et al., 2001].
Taking into account causality is one, if not the major,
advantage of the Bond Graph (BG) modelling. Causal
properties are graphically represented in BGs with the
causal stroke. From a Causal Bond Graph, several powerful
analysis tools can be used, as for system analysis
[Roboam, 2001], model reduction [Sueur and Dauphin-
Tanguy, 1991-a][Gandanegara et al., 2003] or structural
analysis [Sueur and Dauphin-Tanguy, 1991-b].
Nevertheless, many systems, as in electrical engineering,
ask questions about causality. It is sometimes not so easy
to affect integral causality whatever the circuit of physical
system may be. As we will see in this paper, it is for
example the widespread case of star connected three–
phase systems with isolated neutral and inductive
impedance. For the general case where circuit impedances
are balanced, the neutral voltage can be easily expressed
from feeding potentials, but it is not the same if the circuit
is unbalanced. The purpose is pre-processing the model to
construct a BG free of structural singularities (derivative
causality). In this framework, our paper aims at giving
simple specific rules to model multi-phase systems,
whatever its balanced or unbalanced operation and
whatever its neutral connection may be. Every presented
model can fulfil the integral causality rules. Some typical
examples are presented : the first one consists in a
synchronous machine with trapezoidal electromotive force
(these latter being naturally unbalanced). A second class
of examples deals with electrical three-phase networks
with both single and three–phase loads.
II. PROBLEM STATEMENT
In order to illustrate the causality issues in multi–phase
electrical systems, the simple but very spread example of
Fig. 1 can be considered. It is constituted of a three-phase
network connecting a voltage to a current supply, both
being with isolated neutrals (Np, Ns).
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