Skip to content

Commit

Permalink
Remove exaggerated emphasis with \textbf in equation/variable balance…
Browse files Browse the repository at this point in the history
… examples

As it's clear from the context that these examples are all about these concepts anyway, it didn't seem motivated to make these terms stand out more with something like \emph.
  • Loading branch information
henrikt-ma committed Jun 7, 2020
1 parent a6e2871 commit 7d24d5d
Showing 1 changed file with 20 additions and 39 deletions.
59 changes: 20 additions & 39 deletions chapters/classes.tex
Original file line number Diff line number Diff line change
Expand Up @@ -1146,25 +1146,16 @@ \section{Balanced Models}\doublelabel{balanced-models}
\end{align*}
and 2 equations corresponding to the 2 flow-variables \lstinline!p.i! and \lstinline!n.i!.

These are 5 equations in 5 unknowns (\textbf{locally}
\textbf{balanced model}). A more detailed analysis would reveal that
this is \textbf{structurally non-singular}, i.e. that the hybrid DAE
will not contain a singularity independent of actual values.

If the equation \lstinline!u = p.v - n.v! would be missing in the
Capacitor model, there would be 4 equations in 5 unknowns and the model
would be \textbf{locally} \textbf{unbalanced} and thus simulation models
in which this model is used would be usually structurally singular and
thus not solvable.

If the equation \lstinline!u = p.v - n.v! would be replaced by the
equation \lstinline!u = 0! and the equation \lstinline!C*der(u) = p.i! would
be replaced by the equation \lstinline!C*der(u) = 0!, there would be 5
equations in 5 unknowns (\textbf{locally} \textbf{balanced}), but the
equations would be \textbf{singular}, regardless of how the equations
corresponding to the flow-variables are constructed because the
information that \lstinline!u! is constant is given twice in a slightly
different form.
These are 5 equations in 5 unknowns (locally balanced model). A more detailed analysis would reveal that this is structurally non-singular, i.e. that
the hybrid DAE will not contain a singularity independent of actual values.

If the equation \lstinline!u = p.v - n.v! would be missing in the \lstinline!Capacitor! model, there would be 4 equations in 5 unknowns and the model
would be locally unbalanced and thus simulation models in which this model is used would be usually structurally singular and thus not solvable.

If the equation \lstinline!u = p.v - n.v! would be replaced by the equation \lstinline!u = 0! and the equation \lstinline!C*der(u) = p.i! would be
replaced by the equation \lstinline!C*der(u) = 0!, there would be 5 equations in 5 unknowns (locally balanced), but the equations would be singular,
regardless of how the equations corresponding to the flow-variables are constructed because the information that \lstinline!u! is constant is given twice
in a slightly different form.
\end{example}

\begin{example}
Expand Down Expand Up @@ -1200,9 +1191,7 @@ \section{Balanced Models}\doublelabel{balanced-models}
end Circuit;
\end{lstlisting}

Since \lstinline!t! is partial we cannot check whether this is a
\textbf{globally balanced model}, but we can check that \lstinline!Circuit!
is \textbf{locally balanced}.
Since \lstinline!t! is partial we cannot check whether this is a globally balanced model, but we can check that \lstinline!Circuit! is locally balanced.

Counting on model \lstinline!Circuit! results in the following balance sheet:

Expand All @@ -1219,9 +1208,7 @@ \section{Balanced Models}\doublelabel{balanced-models}
\end{align*}
and 2 equation corresponding to the flow variables \lstinline!p.i!, \lstinline!n.i!.

In total we have 8 scalar unknowns and 8 scalar equations, i.e., a
\textbf{locally} \textbf{balanced model} (and this feature holds for any
models used for the replaceable component \lstinline!t!).
In total we have 8 scalar unknowns and 8 scalar equations, i.e., a locally balanced model (and this feature holds for any models used for the replaceable component \lstinline!t!).

Some more analysis reveals that this local set of equations and
unknowns is structurally non-singular. However, this does not provide
Expand Down Expand Up @@ -1359,7 +1346,7 @@ \section{Balanced Models}\doublelabel{balanced-models}
$2 + \text{\lstinline!nXi!}$ flow variables in the \lstinline!port! connector.
\end{itemize}

Therefore, \lstinline!DynamicVolume! is a \textbf{locally balanced} model.
Therefore, \lstinline!DynamicVolume! is a locally balanced model.

Note, when the \lstinline!DynamicVolume! is used and the \lstinline!Medium!
model is redeclared to \lstinline!SimpleAir!, then a tool will try
Expand Down Expand Up @@ -1410,19 +1397,13 @@ \section{Balanced Models}\doublelabel{balanced-models}
$2 + \text{\lstinline!nXi!}$ flow variables in the \lstinline!port! connector.
\end{itemize}

Therefore, \lstinline!FixedBoundary_pTX! is a \textbf{locally
balanced} model. The predefined boundary variables \lstinline!p! and \lstinline!Xi!
are provided via equations to the input arguments \lstinline!medium.p!
and \lstinline!medium.Xi!, in addition there is an equation for \lstinline!T!
in the same way -- even though \lstinline!T! is not an input. Depending
on the flow direction, either the specific enthalpy in the port
(\lstinline!port.h!) or h is used to compute the enthalpy flow rate
\lstinline!H_flow!. \lstinline!h! is provided as binding equation to the medium. With
the equation \lstinline!medium.T = T!, the specific enthalpy \lstinline!h! of the
reservoir is indirectly computed via the medium equations. Again, this
demonstrates, that an \lstinline!input! just defines the number of equations
have to be provided, but that it not necessarily defines the
computational causality.
Therefore, \lstinline!FixedBoundary_pTX! is a locally balanced model. The predefined boundary variables \lstinline!p! and \lstinline!Xi! are
provided via equations to the input arguments \lstinline!medium.p! and \lstinline!medium.Xi!, in addition there is an equation for \lstinline!T!
in the same way --- even though \lstinline!T! is not an input. Depending on the flow direction, either the specific enthalpy in the port
(\lstinline!port.h!) or \lstinline!h! is used to compute the enthalpy flow rate \lstinline!H_flow!. \lstinline!h! is provided as binding equation
to the medium. With the equation \lstinline!medium.T = T!, the specific enthalpy \lstinline!h! of the reservoir is indirectly computed via the
medium equations. Again, this demonstrates, that an \lstinline!input! just defines the number of equations have to be provided, but that it not
necessarily defines the computational causality.
\end{example}

\section{Predefined Types and Classes}\doublelabel{predefined-types-and-classes}
Expand Down

0 comments on commit 7d24d5d

Please sign in to comment.