IEEE Circuits and Systems Magazine - Q1 2023 - 40

The interface of an a-causal model is made of ports,
not inputs and outputs. Ports naturally lend themselves
to represent physical connection points, like the pins
of a resistor. They carry variables, that can be of two
kinds: effort variables, that make sense with respect to a
reference or as the difference between two points (such
as voltage or temperature) and flow variables, that conversely
make sense through a surface (such as current
or heat rate). There is a third kind of variables named
stream, but this is better introduced later on.
When connected, ports generate connection equations.
These equations state that all their effort variables
with the same name (think of the voltages in a set
of pins soldered together) are equal, while all their flow
variables with the same name (think of the currents in
the same set and with uniform convention, e.g., positive
if entering the pin) sum to zero. Together with the constitutive
equations, the connection ones give rise to a
closed compound model.
IV. Minimal Introduction to EBM
This section is devoted to a nutshell-size and operational
introduction to EBM. We employ to this end a simple example,
with which we also start introducing the Modelica
syntax. In the example we refer to an electrical case for
simplicity, as such a system is most likely familiar for the
reader. The goal is first to introduce the core modeling
concepts required in this research. Of course, starting
from the next section, we will switch to thermal models.
Coming to the example, we want to model a simple
circuit made of step voltage generators, conductors,
capacitors and ground. Listing 1 defines the required
components from scratch, in declarative form. For completeness,
we have to say that all the components in
Listing 1 are already available as part of the Modelica
Standard Library (MSL for short), that we shall mention
again later. For a complete and self-contained explanation,
however, at line 1 we here took from the MSL just
the definition of physical quantities-voltage, current
and so on-and their SI units of measurement (UoM).
Note the presence of inheritance, typical of objectoriented
languages like Modelica: lines 8-16 define a
generic component with two pins, and then this is specialized
to be conductor, capacitor and voltage generator.
By just inheriting (in Modelica, extending) TwoPin
and adding the convenient v to i relationship, one can
obtain resistor, inductor, diode, and so forth. Note also
(line 41) the availability of conditional equations. Finally,
line 22 could obviously be written as i = G * v, and analogously
for lines 30 and 32, but we wanted to stress that
component models contain equations and not assignment
statements.
40
IEEE CIRCUITS AND SYSTEMS MAGAZINE
Listing 1. EBM introductory example-components.
The Modelica listing on the right in Fig. 1 shows how
the defined components are used to build the simple
circuit on the left, also overriding some parameter defaults.
One can of course insert more occurrences of the
same component, each with its own parameter values.
The connect statements at lines 7-10 join connectors
(pins) into three connection sets. As said, a set of n ≥ 2
connectors generates n equations per effort variable to
set all its values equal, and one per flow variable to set
the sum of its values to zero. For example, the blue connection
set (of 3 pins) generates
S.b.v = gnd.a.v,
C.b.v = gnd.a.v,
S.b.i+C.b.i+gnd.a.i = 0.
(1)
An open pin would be treated as a connection set
with n=1, resulting in the one equation i = 0 without any
inconsistency.
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